The HYDRUS-1D Software Package for Simulating the
One-Dimensional Movement of Water, Heat, andMultiple Solutes in Variably-Saturated Media

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Version 4.08
January 2009
DEPARTMENT OF ENVIRONMENTAL SCIENCES
UNIVERSITY OF CALIFORNIA RIVERSIDERIVERSIDE, CALIFORNIA

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The HYDRUS-1D Software Package for Simulating the
One-Dimensional Movement of Water, Heat, andMultiple Solutes in Variably-Saturated Media

Version 4.08

byJ. Šimůnek, M. Šejna, H. Saito, M. Sakai, and M. Th. van Genuchten

January 2009

DEPARTMENT OF ENVIRONMENTAL SCIENCES
UNIVERSITY OF CALIFORNIA RIVERSIDERIVERSIDE, CALIFORNIADISCLAIMER
This report documents version 4.08 of HYDRUS-1D, a software package for simulatingwater, heat and solute movement in one-dimensional variably-saturated media. The software has
been verified against a large number of test cases. However, no warranty is given that the
program is completely error-free. If you do encounter problems with the code, find errors, or
have suggestions for improvement, please contact
Jirka Šimůnek
Department of Environmental Sciences
University of California Riverside
Riverside, CA 92521
USA
Phone/Fax: (951) 827-7854
Email: Jiri.Simunek@ucr.edu

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ABSTRACTŠimůnek, J., M. Šejna, H. Saito, M. Sakai, and M. Th. van Genuchten, The HYDRUS-1D
Software Package for Simulating the Movement of Water, Heat, and Multiple Solutes in
Variably Saturated Media, Version 4.08, HYDRUS Software Series 3, Department ofEnvironmental Sciences, University of California Riverside, Riverside, California, USA, pp.
330, 2008.
This report documents version 4.0 of HYDRUS-1D, a software package for simulatingwater, heat and solute movement in one-dimensional variably saturated media. The software
consists of the HYDRUS computer program, and the HYDRUS1D interactive graphics-baseduser interface. The HYDRUS program numerically solves the Richards equation for variablysaturated water flow and advection-dispersion type equations for heat and solute transport. The
flow equation incorporates a sink term to account for water uptake by plant roots. The flow
equation may also consider dual-porosity type flow in which one fraction of the water content ismobile and another fraction immobile, or dual-permeability type flow involving two mobile
regions, one representing the matrix and one the macropores. The heat transport equationconsiders transport due to conduction and convection with flowing water. Coupled water, vapor,and energy transport can be considered as well. The solute transport equations consider
advective-dispersive transport in the liquid phase, as well as diffusion in the gaseous phase. The
transport equations also include provisions for nonlinear nonequilibrium reactions between thesolid and liquid phases, linear equilibrium reactions between the liquid and gaseous phases, zeroorder production, and two first-order degradation reactions: one which is independent of othersolutes, and one which provides the coupling between solutes involved in sequential first-orderdecay reactions. In addition, physical nonequilibrium solute transport can be accounted for byassuming a two-region, dual-porosity type formulation which partition the liquid phase into
mobile and immobile regions. Alternatively, the transport equations include provisions for
kinetic attachment/detachment of solute to the solid phase and it can be thus used to simulatetransport of viruses, colloids, or bacteria.
The HYDRUS software package also includes modules for simulating carbon dioxideand major ion solute movement. Diffusion in both liquid and gas phases and convection in theliquid phase are considered as CO2 transport mechanisms. The CO2 production model is
described. The major variables of the chemical system are Ca, Mg, Na, K, SO4, Cl, NO3, H4SiO4,v

alkalinity, and CO2. The model accounts for equilibrium chemical reactions between thesecomponents such as complexation, cation exchange and precipitation-dissolution. For the
precipitation-dissolution of calcite and dissolution of dolomite, either equilibrium or
multicomponent kinetic expressions are used which include both forward and back reactions.
Other dissolution-precipitation reactions considered include gypsum, hydromagnesite,nesquehonite, and sepiolite. Since the ionic strength of soil solutions can vary considerably withtime and space and often reach high values, both modified Debye-Hückel and Pitzer expressions
were incorporated into the model as options to calculate single ion activities.
The program may be used to analyze water and solute movement in unsaturated, partiallysaturated, or fully saturated porous media. The flow region may be composed of nonuniform
soils. Flow and transport can occur in the vertical, horizontal, or a generally inclined direction.
The water flow part of the model can deal with prescribed head and flux boundaries, boundaries
controlled by atmospheric conditions, as well as free drainage boundary conditions. The
governing flow and transport equations are solved numerically using Galerkin-type linear finite
element schemes. HYDRUS also includes a Marquardt-Levenberg type parameter optimization
algorithm for inverse estimation of soil hydraulic and/or solute transport and reaction parametersfrom measured transient or steady-state flow and/or transport data.
New features in version 4.0 of HYDRUS-1D as compared to version 3.0 include:a) Vapor flow,b) Coupled water, vapor, and energy transport,c) Dual-permeability type water flow and solute transport,d) Dual-porosity water flow and solute transport, with solute transport subjected to two-site
sorption in the mobile zone,e) Potential evapotranspiration as calculated with the Penman-Monteith combination
equation or with Hargreaves equation,f) Daily variations in the evaporation, transpiration, and precipitation rates,g) Support for the HP1 code, which was obtained by coupling HYDRUS with the
PHREEQC biogechemical code.
This report serves as both a user manual and reference document. Detailed instructions
are given for data input preparation. A graphical user interface, HYDRUS1D, for easy datapreparation and output display in the MS Windows environment is described in the second partof the manual. The software package can be freely downloaded from www.hydrus2d.com (orwww.pc-progress.cz).

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New features in version 4.07 of HYDRUS-1D as compared to version 4.0 include:a. Option to specify the nonequilibrium phase concentration initially at equilibrium with theequilibrium phase concentration,b. Option to specify initial conditions in total (instead of liquid) concentrations,c. Option to print fluxes instead of temperatures for observation nodes,
d. Linking of optimized parameters of different soil layers,e. Constant mobile water content in multiple layers (in the Mobile-Immobile Water Model)
when optimizing immobile water content,
f. HP1 – support of dual-porosity models, higher user friendliness for HP1,
g. The Per Moldrup’s tortuosity model was implemented as an alternative to the Millington
and Quirk (1960) model,h. Surface Energy Balance (i.e., the balance of latent, heat, and sensible fluxes) for baresoils can be considered,i. Daily variations of meteorological variables can be generated by the model using simple
meteorological models.
New features in version 4.08 of HYDRUS-1D as compared to version 4.07 include:a) Option to consider root solute uptake, including both passive and active uptake.

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TABLE OF CONTENTSDISCLAIMER ............................................................................................................................... iii
ABSTRACT.....................................................................................................................................v
TABLE OF CONTENTS............................................................................................................... ix
LIST OF FIGURES ..................................................................................................................... xiii
LIST OF TABLES...................................................................................................................... xvii
LIST OF ALL VARIABLES....................................................................................................... xix
GENERAL INTRODUCTION AND OVERVIEW OF HYDRUS-1D ..........................................1
PART A - The HYDRUS Code for Simulating the One-Dimensional Movement of .............7
Water, Heat and Multiple Solutes in Variably Saturated MediaJ. Šimůnek and M. Th. van Genuchten
1. INTRODUCTION......................................................................................................................92. VARIABLY SATURATED WATER FLOW.........................................................................112.1. Governing Water Flow Equations ..................................................................................112.1.1. Uniform Water Flow ...........................................................................................112.1.2. Uniform Water Flow and Vapor Transport ........................................................122.1.3. Flow in a Dual-Porosity System .........................................................................122.1.4. Flow in a Dual-Permeability System ..................................................................132.2. Root Water Uptake..........................................................................................................132.2.1. Root Water Uptake Without Compensation........................................................15
2.2.2. Root Water Uptake With Compensation.............................................................20
2.3. The Unsaturated Soil Hydraulic Properties ...................................................................222.3.1. Uniform Water Flow System...............................................................................222.3.2. Uniform Water Flow and Vapor Transport System............................................27
2.4. Scaling in the Soil Hydraulic Functions .........................................................................29
2.5. Temperature Dependence of the Soil Hydraulic Functions ...........................................30
2.6. Hysteresis in the Soil Hydraulic Properties ...................................................................312.7. Initial and Boundary Conditions ....................................................................................35
2.7.1. System-Independent Boundary Conditions ........................................................35
2.7.2. System-Dependent Boundary Conditions ..........................................................36ix

2.7.3. Penman-Monteith Combination Equation .........................................................40
2.7.4. Hargreaves Formula ..........................................................................................41
2.7.5. Surface Energy Balance Equation .....................................................................422.8. Water Mass Transfer ......................................................................................................44
3. NONEQUILIBRIUM TRANSPORT OF SOLUTES INVOLVED IN SEQUENTIAL
FIRST-ORDER DECAY REACTIONS..................................................................................47
3.1. Governing Solute Transport Equations ..........................................................................47
3.1.1. Two-Site Sorption Model (Chemical Nonequilibrium) .......................................50
3.1.2. Attachment-Detachment Model (Two Kinetic Sites Model) ...............................53
3.1.3. Dual-Porosity Model (Physical Nonequilibrium) ...............................................56
3.1.4. Dual-Porosity Model with One Kinetic Site (Physical and chemical
Nonequilibrium) .................................................................................................59
3.1.5. Dual-Permeability Model (Physical Nonequilibrium) .......................................60
3.1.6. Dual-Permeability Model with Immobile Water (Physical Nonequilibrium) .....61
3.1.7. Dual-Permeability Model with Two-Site Sorption (Physical andChemical Nonequilibrium) .................................................................................62
3.2. Initial and Boundary Conditions ....................................................................................59
3.3. Effective Dispersion Coefficient .....................................................................................64
3.4. Temperature and Water Content Dependence of Transport and Reaction
Coefficients .....................................................................................................................65
3.5. Root Solute Uptake .........................................................................................................66
3.5.1. Uncompensated Root Nutrient Uptake Model ....................................................66
3.5.2. Compensated Root Nutrient Uptake Model ........................................................68
4. HEAT TRANSPORT...............................................................................................................71
4.1. Governing Heat Transport Equation..............................................................................71
4.1.1. Heat Transport Without Vapor Transport ............................................................71
4.1.2. Heat Transport With Vapor Transport .................................................................72
4.2. Apparent Thermal Conductivity......................................................................................72
4.3. Initial and Boundary Conditions ....................................................................................73
5. CARBON DIOXIDE TRANSPORT AND PRODUCTION...................................................75
5.1. Governing CO2 Transport Equations...........................................................................75
5.2. Effective Dispersion Coefficient...................................................................................78
5.3. Initial and Boundary Conditions..................................................................................79
5.4. Production of Carbon Dioxide .....................................................................................80
6. CARBONATE CHEMISTRY .................................................................................................85
6.1. Mass and Charge Balance Equations ..........................................................................86
6.2. CO2 - H2O System.........................................................................................................87
6.3. Complexation Reactions...............................................................................................88

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6.4.
6.5.
6.6.
6.7.
6.8.
6.9.

Cation Exchange and Selectivity ..................................................................................88
Precipitation-Dissolution Reactions ............................................................................89
Kinetic Model of Calcite Precipitation-Dissolution.....................................................91
Kinetic Model of Dolomite Dissolution........................................................................92
Silica Concentration in Soil Solution ...........................................................................93
Activity Coefficients......................................................................................................93
6.9.1. Extended Debye-Hückel Expression.................................................................93
6.9.2. Pitzer Expressions.............................................................................................94
6.10. Temperature Effects .....................................................................................................95
6.11. Osmotic Coefficient and Osmotic Pressure..................................................................96
6.13. Effect of Solution Composition on Hydraulic Conductivity .........................................96
7. NUMERICAL SOLUTION OF THE VARIABLY SATURATED FLOW EQUATION....101
7.1. Space and Time Discretization .....................................................................................101
7.2. Treatment of Pressure Head Boundary Conditions......................................................103
7.3. Treatment of Flux Boundary Conditions ......................................................................103
7.4. Numerical Solution Strategy.........................................................................................105
7.4.1. Iterative Process ...............................................................................................105
7.4.2. Time Control .....................................................................................................105
7.4.3. Atmospheric Boundary Conditions and Seepage Faces ...................................106
7.4.4. Water Balance Computations ...........................................................................106
7.4.5. Computation of Nodal Fluxes ...........................................................................108
7.4.6. Water Uptake by Plant Roots............................................................................1087.4.7. Evaluation of the Soil Hydraulic Properties.....................................................109
8. NUMERICAL SOLUTION OF THE SOLUTE TRANSPORT EQUATION ......................111
8.1. Space Discretization .....................................................................................................111
8.2. Time Discretization.......................................................................................................114
8.3. Numerical Solution for Linear Nonequilibrium Solute Transport ...............................1198.4. Numerical Solution Strategy.........................................................................................120
8.4.1. Solution Process................................................................................................120
8.4.2. Upstream Weighted Formulation......................................................................122
8.4.3. Reverse Back-Step Particle Tracking................................................................122
8.4.4. Mass Balance Calculations...............................................................................123
8.4.5. Oscillatory Behavior .........................................................................................126
9. PARAMETER OPTIMIZATION...........................................................................................129
9.1. Definition of the Objective Function .............................................................................129
9.2. Marquardt-Levenberg Optimization Algorithm ............................................................130
9.3. Statistics of the Inverse Solution ...................................................................................131
10. PROBLEM DEFINITION .....................................................................................................139xi

10.1. Construction of Finite Element Mesh ...........................................................................139
10.2. Coding of Soil Types and Subregions ...........................................................................139
10.3. Coding of Boundary Conditions ...................................................................................140
10.4. Program Memory Requirements...................................................................................145
11. EXAMPLE PROBLEMS.......................................................................................................147
11.1. Example 1 - Column Infiltration Test ...........................................................................148
11.2. Example 2 - Water Flow in a Field Soil Profile Under Grass .....................................151
11.3. Example 3 - Solute Transport with Nitrification Chain................................................156
11.4. Example 4 - Solute Transport with Nonlinear Cation Adsorption ...............................160
11.5. Example 5 - Solute Transport with Nonequilibrium Adsorption ..................................164
11.6. Example 6 - Inverse Analysis of a One-Step Outflow Experiment ..............................166
11.7. Example 7 - Inverse Analysis of a Multi-Step Outflow Experiment ............................169
11.8. Example 8 - Inverse Analysis of an Evaporation Experiment ......................................171
11.9. Example 9 – Infiltration into Structured Soil ...............................................................175
11.10. Example 10 – Evaluation of Hysteretic Models ..........................................................177
11.11. Example 11 – Coupled Water, Vapor, and Heat Transport........................................179
12. INPUT DATA......................................................................................................................177
13. OUTPUT DATA..................................................................................................................227
15. REFERENCES.....................................................................................................................243
Appendix A – Penman-Montheith Variables...............................................................................263
Appendix B – Surface Energy Balance Variables ......................................................................268
Appendix C - Daily Variations of Meteorological Variables ......................................................271
PART B - Interactive Graphics-Based User Interface HYDRUS1D ..................................277
J. Šimůnek, M. Šejna and M. Th. van Genuchten
B. Brief Description of Selected Modules...................................................................................281
B.1. Module HYDRUS1D ....................................................................................................281
B.2. Module POSITION ......................................................................................................292
B.3. Module PROFILE ........................................................................................................293
B.3.1. Soil Profile Discretization.................................................................................293
B.3.2. Specification of Soil Properties within the Soil Profile ....................................295xii

LIST OF FIGURES
FigurePage

Fig. 2.1.

Conceptual physical nonequilibrium models for water flow and solutetransport. In the plots, θ is the water content, θmo and θim in (b) and (c) arewater contents of the mobile and immobile flow regions, respectively; θMand θF in (d) are water contents of the matrix and macropore (fracture)regions, respectively; θM,mo, θM,im, and θF in (e) are water contents of themobile and immobile flow regions of the matrix domain, and of themacropore (fracture) domain, respectively; c are concentrations ofcorresponding regions, with subscripts having the same meaning as forwater contents, while S is the total solute content of the liquid phase...................14

Fig. 2.2.

Schematic of the plant water stress response function, α(h), as used bya) Feddes et al. [1978] and b) van Genuchten [1987]............................................18

Fig. 2.3.

Schematic of the potential water uptake distribution function, b(x), in thesoil root zone. .........................................................................................................18

Fig. 2.4.

Ratio of actual to potential transpiration as a function of the stress index ω.........18

Fig. 2.5.

Schematics of the soil water retention (a) and hydraulic conductivity(b) functions as given by equations (2.31) and (2.32), respectively. .....................25

Fig. 2.6.

Example of composite retention (left) and hydraulic conductivity (right)
functions (θr=0.00, θs=0.50, α1=0.01 cm-1, n1=1.50, l=0.5, Ks=1 cm d-1,
w1=0.975, w2=0.025, α2=1.00 cm-1, n2=5.00). ......................................................27

Fig. 2.7.

Example of a water retention curve showing hysteresis. Shown are theboundary wetting curve, θw(h), and the boundary drying curve, θd(h). .................32

Fig. 3.1.

Conceptual chemical nonequilibrium models for reactive solute transport.
In the plots, θ is the water content, θmo and θim in (d) are water contents ofthe mobile and immobile flow regions, respectively; θM and θF in (e) arewater contents of the matrix and macropore (fracture) regions,
respectively; c are concentrations of the corresponding regions, se aresorbed concentrations in equilibrium with the liquid concentrations of thecorresponding regions, and sk are kinetically sorbed solute concentrationsof the corresponding regions..................................................................................58

xiii

Fig. 11.1.

Soil water retention and relative hydraulic conductivity functions forexample 1. The solid circles are UNSAT2 input data [Davis and
Neuman, 1983]. ....................................................................................................149

Fig. 11.2.

Instantaneous, q0, and cumulative, I0, infiltration rates simulated with
the HYDRUS (solid lines) and UNSAT2 (solid circles) computercodes (example 1). ...............................................................................................150

Fig. 11.3.

Unsaturated hydraulic properties of the first and second soil layers(example 2)...........................................................................................................152

Fig. 11.4.

Precipitation and potential transpiration rates (example 2). ................................153

Fig. 11.5.

Cumulative values for the actual transpiration and bottom leachingrates as simulated with the HYDRUS (solid line) and SWATRE (solid
circles) computer codes (example 2). ..................................................................154

Fig. 11.6.

Pressure head at the soil surface and mean pressure head of the rootzone as simulated with the HYDRUS (solid lines) and SWATRE (solid
circles) computer codes (example 2). ..................................................................155

Fig. 11.7.

Location of the groundwater table versus time as simulated with the
HYDRUS (solid line) and SWATRE (solid circles) computer codes(example 2)...........................................................................................................156

Fig. 11.8.

Analytically and numerically calculated concentration profiles for
NH4+, NO2-, and NO3- after 200 hours (example 3). ............................................158

Fig. 11.9.

Analytically and numerically calculated concentration profiles for NH4+
(top), NO2- (middle), and NO3- (bottom) after 50, 100, and 200 hours(example 3).......................................................................................................... 159

Fig. 11.10.

Mg breakthrough curves for Abist loam calculated with the MONOD,HYDRUS, and new HYDRUS codes (data points from Selim et al.,1987) (example 4). ...............................................................................................161

Fig. 11.11.

Ca breakthrough curves for Abist loam calculated with the MONOD
and HYDRUS codes (data points from Selim et al., 1987) (example 4). ............163

Fig. 11.12.

Observed and calculated effluent curves for boron movement throughGlendale clay loam (data points from van Genuchten [1981])(example 5)...........................................................................................................165

xiv

Fig. 11.13.

Measured and optimized cumulative outflow versus time for a onestep
outflow experiment (example 6). .........................................................................167

Fig. 11.14.

Observed and predicted retention characteristics, and calculated andpredicted diffusivities (example 6). .....................................................................168

Fig. 11.15.

Measured and optimized cumulative bottom flux for a multistep
outflow experiment (example 7). ........................................................................170

Fig. 11.16.

Measured and optimized pressure heads inside the soil sample for a
multistep outflow experiment (example 7). ........................................................170

Fig. 11.17.

Water retention (a) and hydraulic conductivity (b) functions determinedwith inverse parameter estimation and Wind's method (example 8). ..................172

Fig. 11.18.

Measured and fitted tensiometer readings as a function of time (a) and
depth (b) (example 8). ..........................................................................................174

Fig. 11.19.

Water content (a) and concentration (b) profiles in the fracture(mobile) domain, the matrix (immobile) domain, and both domains
combined, as well as the water (c) and solute (d) mass transfer terms
as calculated with the dual-porosity model..........................................................176

Fig. 11.20.

Main drainage curves, and imbibition and drainage scanning curves, ascalculated at elevations of 40 and 50 (left figure) and 30 and 60 cm(right figure) using the original (top) and new (bottom) hysteresis models. .......178

Fig. 11.21.

Calculated pressure heads (left) and water contents (right) atelevations of 20 (N6), 30 (N5), 40 (N4), 50 (N3), 60 (N2), and 70 (N1) cm. .....179

Fig. 11.22.

Water content (a), total flux (b), temperature (c), and soluteconcentration (d) distributions in a 10-cm long vertical soil sample
with zero water fluxes across the top and bottom boundaries, and withthe temperature increasing from top to bottom.. ..................................................180

Fig. B.1.

Main window of the HYDRUS1D module, including the project
manager. ...............................................................................................................282

Fig. B.2.

Main window of the PROFILE module, when used for soil profile
discretization. .......................................................................................................293xv

Fig. B.3.

The main window of the PROFILE module, when used forspecification of soil properties. ............................................................................295xvi

LIST OF TABLES
Page

Table
Table 6.1.

Chemical species considered in the carbonate chemistry module. ........................85

Table 8.1.

Values of the diagonal entries di , and off-diagonal entries bi and ei of
matrix [Ps] for linear finite elements....................................................................117

Table 8.2.

Values of the diagonal entries di , and off-diagonal entries bi and ei of
matrix [Ps] for linear finite elements with upstream weighting...........................118

Table 9.1.

Definition of the objective function for different water flow and solutetransport models. . ................................................................................................133

Table 10.1.

Initial settings of KodTop (KodBot), rTop (rBot), and h(n) for constantboundary conditions. ............................................................................................140

Table 10.2.

Initial settings of KodTop (KodBot), rTop (rBot), and h(n) for timevariable boundary conditions. ..............................................................................141

Table 10.3.

Definition of the variables KodTop, rTop, and h(n) when an atmosphericboundary condition is applied. .............................................................................142

Table 10.4.

Definition of the variables KodTop (KodBot), rTop (rBot), and h(n) when
variable head or flux boundary conditions are applied. .......................................142

Table 10.5.

Initial setting of KodBot, rBot, and h(n) for seepage faces..................................144

Table 10.6.

List of the array dimensions.................................................................................145

Table 11.1.

Input parameters for example 3............................................................................158

Table 11.2.

Input parameters for example 4............................................................................160

Table 11.3.

Input parameters for example 5............................................................................164

Table 11.4.

Initial estimates and optimized parameters for example 6...................................167

Table 11.5.

Initial estimates and optimized parameters for example 7...................................169

Table 11.6.

Hydraulic parameters obtained from an evaporation experiment usingparameter estimation and Wind's method (example 8). .......................................173

Table 12.1.

Block A - Basic information. ...............................................................................183

Table 12.2.

Block B - Water flow information. ......................................................................185

Table 12.3.

Block C - Time information.................................................................................191

Table 12.4.

Block D - Root growth information. ....................................................................193

Table 12.5.

Block E - Heat transport information...................................................................194xvii

Table 12.6.

Block F - Solute transport information. ...............................................................195

Table 12.7.

Block G - Root water uptake information............................................................202

Table 12.8.

Block H - Nodal information. ..............................................................................205

Table 12.9.

Block I - Atmospheric information. .....................................................................208

Table 12.10. Block J - Inverse solution information ................................................................210
Table 12.11. Block K – Carbon dioxide transport information ...............................................216
Table 12.12. Block L – Major ion chemistry information .......................................................218
Table 12.13. Block M – Meteorological information ..............................................................222
Table 13.1.

T_LEVEL.OUT - pressure heads and fluxes at boundaries and in the rootzone. .....................................................................................................................229

Table 13.2.

RUN_INF.OUT - time and iteration information. ...............................................230

Table 13.3.

SOLUTE.OUT - actual and cumulative concentration fluxes. ............................231

Table 13.4.

NOD_INF.OUT - profile information..................................................................232

Table 13.5.

BALANCE.OUT - mass balance variables..........................................................233

Table 13.6.

A_LEVEL.OUT - pressure heads and cumulative fluxes at the boundary
in the root zone.....................................................................................................235

Table 13.7.

FIT.OUT - parameter estimation related information .........................................236

Table 13.8.

CO2_INF.OUT - CO2 concentrations and CO2 fluxes at boundaries andand in the root zone. ............................................................................................237

Table 13.9.

CONC.OUT - solute concentration information. ................................................238

Table 13.10. SOLID.OUT - precipitated and adsorbed concentrations. ..................................239
Table 13.11. EQUIL.OUT - chemical information. .................................................................240
Table 13.12. METEO.OUT – meteorological information. .....................................................241
Table B.1.

Main modules of the HYDRUS-1D software package. .......................................279

Table B.2.

Menu commands in the main module HYDRUS1D............................................283

Table B.3.

Description of all menu commands in the main module HYDRUS1D. ..............285

Table B.4.

Graph options in the HYDRUS-1D interface. .....................................................288

Table B.5.

Information in the HYDRUS1D.DAT file...........................................................292

Table B.6.

Definition of terms in module PROFILE.............................................................294

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LIST OF ALL VARIABLESa

ion activity on the exchange surfaces [-]a*parameter in the exponential depth distribution function [L-1]adrgeometry factor for radial flow depending on the flow conditions [-]aiconversion factor from concentration to osmotic head [L4M-1]

arvalue of coefficient a at reference temperature TrA

aTvalue of coefficient a at temperature TAav

air content [L3L-3]A

amplitude of temperature sine wave at the soil surface [K]Aqhempirical parameter in the deep drainage function, Eq. (7.1) [LT-1]

As

correction factor [-]b

normalized root water uptake distribution [L-1]b'arbitrary root water uptake distribution [L-1]bicoefficients in the global matrix equations for water flow [T-1] and solute transport[LT-1]

b1,b2,b3

empirical parameters to calculate the soil thermal conductivity, λ0 [MLT-3K-1] (e.g.
Wm-1K-1)B

convective term in the solute transport equation [LT-1]

Bqhempirical parameter in the deep drainage function, Eq. (7.1) [L-1]csolution concentration [ML-3]c'finite element approximation of c [ML-3]

ci

initial solution concentration [ML-3]cim,msolute concentration of the immobile zone of the matrix region [ML-3]cm,msolute concentration of the mobile zone of the matrix region [ML-3]

cnvalue of the concentration at node n [ML-3]

cp

specific heat of moist air [L2T-2K-1] (i.e., 1013 J kg-1 oC-1)

cr

concentration of the sink term [ML-3]

c0

prescribed concentration boundary condition [ML-3]

Csoil water capacity [L-1]

Ca

volumetric heat capacity of the gas phase [ML-1T-2K-1] (e.g. Jm-3K-1)

xix

Cn

volumetric heat capacity of the solid phase [ML-1T-2K-1] (e.g. Jm-3K-1)

Co

volumetric heat capacity of the organic matter [ML-1T-2K-1] (e.g. Jm-3K-1)

Cp

volumetric heat capacity of the porous medium [ML-1T-2K-1] (e.g. Jm-3K-1)

CT

total solution concentration [ML-3] (mmolcl-1)

Cw

volumetric heat capacity of the liquid phase [ML-1T-2K-1] (e.g. Jm-3K-1)

Crie

local Courant number for element e [-]dthickness of stagnant boundary layer above soil surface [L]deffective ‘diffusion’ pathlength (i.e. half the aggregate width or half the fracture
spacing) [L]dc

diameter of the sand grains [L]dicoefficients in the global matrix equations for water flow [T-1] and solute transport[LT-1]

dr

relative distance between Earth and Sun [-]Deffective dispersion coefficient of the soil matrix [L2T-1]

D-, D+

effective dispersion coefficients corrected for higher-order approximation [L2T-1]

Da

diffusivity of water vapor in air [L2T-1] at temperature T

Deq

equivalent depth [L]Dgdiffusion coefficient for the gas phase [L2T-1]

Dwsolute dispersion coefficient for the liquid phase [L2T-1]

Dg

molecular diffusion coefficient of the gas phase [L2T-1]

DLDL

longitudinal dispersivity [L]*additional longitudinal dispersivity [L] to fulfill the performance index ωsDrthickness of layers over which radial drainage flow is considered [L]

Dv

thickness of layers over which vertical drainage flow is considered [L]Dvvapor diffusivity in soil [L2T-1]

Dw

molecular diffusion coefficient in free water [L2T-1]

eelement number [-]ea

saturation vapor pressure at temperature T [ML-1T-2] (e.g., kPa)(ea-ed)vapor pressure deficit [ML-1T-2] (e.g., kPa)ed

vapor pressure at dew point [ML-1T-2] (e.g., kPa)

xx

ei

coefficients in the global matrix equations for water flow [T-1] and solute transport[LT-1]eR

elements located in the root zone [-]E

maximum (potential) rate of infiltration or evaporation under the prevailing
atmospheric conditions [LT-1]

E

dispersive term in the solute transport equation [L2T-1]

Eaactivation energy of a chemical reaction or process [ML2T-2M-1] (e.g., kgm2s-2mol-1)

ET0reference crop evapotranspiration [LT-1] (e.g., mm d-1)

ffraction of exchange sites assumed to be at equilibrium with the solution concentration
[-]

fem

fraction of sorption sites in equilibrium with the mobile liquid phase (remaining sites arein contact with the mobile liquid phase) (dual-porosity model with two-site sorption in
the mobile phase) [-]ff

fraction of the exchange sites assumed to be in equilibrium with the solution phase [-]of the fracture domain (dual-permeability model) [-]fi

coefficients in the global matrix equations for water flow [LT-1] and solute transport[ML-2T-1]

fm

fraction of sorption sites in contact with the mobile region of the matrix (dualpermeability model with immobile zone in the matrix) [-]fm

fraction of the exchange sites assumed to be in equilibrium with the solution phase [-]of the matrix domain (dual-permeability model) [-]fmo

fraction of sorption sites in contact with mobile water (the remainder is in contact withimmobile water) (dual-porosity model) [-]fr

root growth coefficient [-]{f}vector in the global matrix equation for solute transport [MT-1L-2]

F

first-order decay term in the solute transport equation [T-1]{Fw}coefficient vector in the global matrix equation for water flow [LT-1]

ggas phase concentration [ML-3]ggravitational acceleration (= 9.81 m s-2) [LT-2]

gatm

gas phase concentration above the stagnant boundary layer [ML-3]

G

zero-order decay term in the solute transport equation [ML-3T-1]

Gsoil heat flux [MT-3] (e.g., MJ m-2 d-1)

Gscsolar constant [MT-3] (e.g., J m-2s-1, 1360 W m-2)

xxi

GwT

gain factor [-] (equal 7 for sand)hpressure head [L]h*scaled pressure head [L]h'finite element approximation of h [L]hA

minimum pressure head allowed at the soil surface [L]hdrwatertable height above drain level at midpoint between drains, i.e. hydraulic headneeded for calculating subsurface flow into drains [L]hf

pressure head in the macropore region of the dual-permeability model [L]hiinitial condition for the pressure head [L]hmpressure head in the matrix region of the dual-permeability model [L]hnnodal value of the pressure head [L]hrefpressure head at reference temperature Tref [L]hs

air-entry value in the Brooks and Corey soil water retention function [L]hS

maximum pressure head allowed at the soil surface [L]hT

pressure head at soil temperature T [L]h0surface boundary condition for the pressure head [L]h50

pressure head at which root water uptake is reduced by 50 % [L]hφosmotic head [L]hφ50

osmotic head at which root water uptake is reduced by 50 % [L]hΔ

pressure head at the reversal point in a hysteretic retention function [L]H

Hamaker constant (= 1e-20 J) [ML2T-2]

Hrrelative humidity [-]Jnumber of the day in the year [-]kkth chain number [-]kBoltzman constant (= 1.38048e-23 J/K) [M L2T-2K-1]

ka

first-order deposition (attachment) coefficient [T-1]kd

first-order entrainment (detachment) coefficient [T-1]kgempirical constant relating the solution and gas phase concentrations [-]ks

empirical constant relating the solution and adsorbed concentrations [L3M-1]

K

unsaturated soil hydraulic conductivity [LT-1]

xxii

K*

scaled unsaturated soil hydraulic conductivity [LT-1]

Kd

unsaturated hydraulic conductivity of the main drying branch [LT-1]

Kw

unsaturated hydraulic conductivity of the main wetting branch [LT-1]

Ka

effective hydraulic conductivity of the fracture-matrix interface [LT-1]

Kexdimensionless thermodynamic equilibrium constant [-]Kf

unsaturated soil hydraulic conductivity of the macropore (fracture) region in the dualpermeability model [LT-1]

KH

Henry's law constant [MT2M-1L-2]

KhBot

horizontal saturated hydraulic conductivity below the drain system [LT-1]

KhTophorizontal saturated hydraulic conductivity above the drain system [LT-1]

Kkmeasured value of the unsaturated soil hydraulic conductivity at θk [LT-1]

KLT

thermal hydraulic conductivity of the liquid phase [L2K-1T-1]

Km

unsaturated soil hydraulic conductivity of the matrix region in the dual-permeability
model [LT-1]

Kr

relative soil hydraulic conductivity [-]Kr'

saturated hydraulic conductivity in the layer with radial drainage flow [LT-1]

Krefhydraulic conductivity at reference temperature Tref [LT-1]

Ks

saturated hydraulic conductivity [LT-1]

Ksd

saturated hydraulic conductivity associated with the main drying branch [LT-1]

Ksw

saturated hydraulic conductivity associated with the main wetting branch [LT-1]

KT

hydraulic conductivity at soil temperature T [LT-1]

Kv

Vanselow selectivity coefficient [-]Kv'

saturated hydraulic conductivity in the layer with vertical drainage flow [LT-1]

Kvh

isothermal vapor hydraulic conductivity [LT-1]

KvT

thermal vapor hydraulic conductivity [L2K-1T-1]

K12

selectivity coefficient [-]KΔ

unsaturated hydraulic conductivity at the reversal point in a hysteretic conductivity
function [LT-1]

lpore-connectivity parameter [-]L

x-coordinate (depth of the soil profile) of the soil surface above a certain reference
plane [L]

xxiii

Ldr

drain spacing [L]Lmmaximum rooting depth [L]LRroot depth [L]L0initial value of the rooting depth at the beginning of the growth period [L]

L0

volumetric latent heat of vaporization of liquid water [ML-1T-2] (e.g., Jm-3)mparameter in the soil water retention function [-]Mtotal amount of mass in the entire flow domain [ML-2]

M

molecular weight of water [M mol-1] (=0.018015 kg mol-1)

M0

cumulative amount of solute removed from the flow region by zero-order reactions[ML-2]

M1

cumulative amount of solute removed from the flow region by first-order reactions[ML-2]

Mr

cumulative amount of solute removed from the flow region by root water uptake[ML-2]

Mt

total amount of solute in the flow region at time t [ML-2]

M teamount of solute in element e at time t [ML-2]

M0

amount of solute in the flow region at the beginning of the simulation [ML-2]

M0eamount of solute in element e at the beginning of the simulation [ML-2]nexponent in the soil water retention function [-]ndexponent in the soil water retention function; drying branch [-]nw

exponent in the soil water retention function; wetting branch [-]ns

number of solutes involved in the consecutive solute chain [-]n/Nrelative sunshine fraction [-]Ntotal number of nodes [-]NPe

Peclet number [-]NRinterception number [-]NG

gravitation number [-]NLocontribution of particle London-van der Walls attractive forces to particle removal [-]O

actual rate of inflow/outflow to/from a subregion [LT-1]

p

exponent in the water and osmotic stress response function [-]pt

period of time necessary to complete one temperature cycle (1 day) [T]

xxiv

p1

exponent in the water stress response function [-]p2

exponent in the osmotic stress response function [-]Patmospheric pressure [ML-1T-2] (e.g., kPa)

Peie

local Peclet number for element e [-][Ps]

coefficient matrix in the global matrix equation for solute transport [LT-1][Pw]coefficient matrix in the global matrix equation for water flow [T-1]qDarcian fluid flux density [LT-1]

qdraindrain discharge rate per unit surface area [LT-1]

qN

water flux boundary condition at the soil surface [LT-1]

qs

solute flux [ML-2T-1]

qs0

solute flux boundary condition at the bottom of the soil profile [LT-1]

qsN

solute flux boundary condition at the soil surface [LT-1]

qv

vapor flux density [LT-1]

q0

water flux boundary condition at the bottom of the soil profile [LT-1][Q]

coefficient matrix in the global matrix equation for solute transport [L]rgrowth rate [T-1]ricoefficients in the global matrix equation for solute transport [ML-2T-1]

R

solute retardation factor [-]Ra

extraterrestrial radiation [MT-3] (e.g., MJ m-2d-1)

Rn

net radiation at crop surface [MT-3] (e.g., MJ m-2d-1)

Rnl

net longwave radiation [MT-3] (e.g., MJ m-2d-1)

Rns

net shortwave radiation [MT-3] (e.g., MJ m-2d-1)

Ruuniversal gas constant [ML2T-2K-1M-1] (= 8.314 kg m2s-2K-1mol-1, J mol-1K-1){R}

vector in the global matrix equation for solute transport [ML-2T-1]sadsorbed solute concentration [-]seadsorbed solute concentration on type-1 sites [-]ksorbed concentration of type-2 (kinetic) sites in the fracture domain (dual-permeability
model) [-]sf
siinitial value of adsorbed solute concentration [-]smax

maximum solid phase concentration [MM-1]

xxv

sk

adsorbed solute concentration on type-2 sites [-]smk

sorbed concentration of type-2 (kinetic) sites in the matrix domain (dual-permeability
model) [-]esmosorbed concentration in equilibrium with the liquid phase concentration of the mobile
region of the dual-porosity model [MM-1]

k
smo
,e

sorbed concentration of kinetic sites in contact with the mobile region of the dualporosity model when at equilibrium [MM-1]

Ssink term in the flow equation [T-1]Se

effective saturation [-]Seim

effective fluid saturation of the immobile (matrix) regionSem

effective fluid saturations of the mobile (fracture) regionSekeffective saturation at θk [-]Sfsink term in the flow equation for the macropore (fracture) reqion of the dualpermeability model [T-1]Simsink term in the flow equation for the immobile flow region [T-1]Smsink term in the flow equation for the matrix reqion of the dual-permeability model
[T-1]

Smo

sink term in the flow equation for the mobile flow region [T-1]Spspatial distribution of the potential transpiration rate over the soil profile [T-1]STcation exchange capacity [MM-1] (mmolckg-1)[S]coefficient matrix in the global matrix equation for solute transport [LT-1]

ttime [T]t0time when simulation begins [T]tpperiod of time covering one complete cycle of the temperature sine wave [T]T

temperature [K, oC]Taverage temperature at the soil surface during period tp [K]A

Tabsolute temperature [K]Taactual transpiration rate per unit soil surface [LT-1]

Ti

initial temperature [K]Tmdaily mean air temperature [K]Tmaxmaximum daily air temperature [K]

xxvi

Tmin

minimum daily air temperature [K]Tppotential transpiration rate [LT-1]

TrA

reference absolute temperature [K] (293.15K = 20oC)

Tref

reference temperature [K]T0prescribed temperature boundary condition [K]TRtemperature range between mean daily maximum and minimum air temperatures [oC][T]coefficient matrix in the global matrix equation for solute transport [LT-1]

uwet perimeter of the drain [L]U2windspeed measured at 2 m height [LT-1] (e.g., m s-1)v

average pore-water velocity [LT-1]

V

volume of water in each subregion [L]Vnew

volume of water in each subregion at the new time level [L]

Vold

volume of water in each subregion at the previous time level [L]Vt

volume of water in the flow domain at time t [L]Vtevolume of water in element e at time t [L]V0

volume of water in the flow domain at initial time t0 [L]V0evolume of water in element e at initial time t0 [L]w

ratio of the volumes of the macropore or fracture domain and the total soil system inthe dual-permeability model [-]W

total amount of energy in the flow region [MT-2]

wiweighting factors for the two overlapping regions of the dual-porosity model [-]x

spatial coordinate [L] (positive upward)

x0

coordinate of the location where the straining process starts [L]αdimensionless water stress response function [-]αparameter in the soil water retention function [L-1]αsticking efficiency (ratio of the rate of particles that stick to a collector to the rate they
strike the collector) [-]αdvalue of α for a drying branch of the soil water retention function [L-1]

αwvalue of α for a wetting branch of the soil water retention function [L-1]αwweighing factor [-]

xxvii

αch

first-order rate constant [T-1] accounting for chemical rate processes

αch,mfirst-order rate constant for the matrix domain [T-1]αhscaling factor for the pressure head [-]αh*temperature scaling factor for the pressure head [-]αKscaling factor for the hydraulic conductivity [-]αK*temperature scaling factor for the hydraulic conductivity [-]αθscaling factor for the water content [-]αw

first-order mass transfer coefficient [L-1T-1]

βempirical constant in the adsorption isotherm [-]β

empirical factor in the straining function [-]βshape factor that depends on the geometry [-]βtthermal dispersivity [L]γsurface tension of soil water [MT-2] (e.g., Jm-2)γ

psychrometric constant [ML-1T-2K-1] (e.g., kPa oC-1)

γ0surface tension at 25oC [MT-2] (= 71.89 g s-2)

γentr

entrance resistance into the drains [T]γgzero-order rate constant for solutes in the gas phase [ML-3T-1]

γiactivity coefficient in the soil solution [L3M-1] (l mol-1)

γszero-order rate constant for solutes adsorbed onto the solid phase [T-1]γw

zero-order rate constants for solutes in the liquid phase [ML-3T-1]

γw

scaling factor in the mass transfer function (=0.4) [-]Γw

transfer rate for water from the inter- to the intra-aggregate pores [T-1]Γs1

mass transfer term for solute exchange between the mobile and immobile regions(dual-porosity model) [ML-3T-1]

Γs2

mass transfer to kinetic sorption sites in the mobile region (two-site sorption model)[ML-3T-1]

Γ s*mass transfer term for solutes between the mobile and immobile regions of the matrix
domain (dual-permeability model with immobile region in the matrix) [ML-3T-1]

δsolar declination [rad]δijKronecker delta [-]Δ

slope of the vapor pressure curve [ML-1T-2K-1] (e.g., kPa oC-1]

xxviiiΔttime increment [T]Δtmax

maximum permitted time increment [T]Δtmin

minimum permitted time increment [T]Δxsize of the elements [L]εtemporal weighing factor [-]εratio of the molecular weights of water vapor and dry air [-], i.e., 0.622

εac

absolute error in the solute mass balance [ML-2]

εaw

absolute error in the water mass balance [L]ε rcrelative error in the solute mass balance [%]ε rw

relative error in the water mass balance [%]η

empirical constant in the adsorption isotherm [L3M-1]

ηsingle-collector efficiency [-]ηeenhancement factor [-]θ

volumetric water content [L3L-3]

θ*

scaled volumetric water content [L3L-3]

θaparameter in the soil water retention function [L3L-3]

θfvolumetric water content of the macropore (fracture) region in the dual-permeability
model [L3L-3]

θF

volumetric water content (absolute) of the fracture region in the dual-permeability
model (=(1-w)θf) [L3L-3]

θimwater content in the immobile (stagnant, intra-aggregate) region [L3L-3]

θm,mwater content of the immobile (stagnant, intra -aggregate) region in the matrix (dualpermeability model) [L3L-3]

θkvolumetric water content corresponding to Kk [L3L-3]

θmparameter in the soil water retention function [L3L-3]

θmvolumetric water content of the matrix region in the dual-permeability model [L3L-3]

θmowater content in the mobile (flowing, inter-aggregate) region [L3L-3]

θmdparameter in soil water retention function; drying branch [L3L-3]

θmw

parameter in soil water retention function; wetting branch [L3L-3]

θm,mwater content of the mobile (flowing, inter-aggregate) region of the matrix (dualpermeability model) [L3L-3]

xxix

θM

volumetric water content (absolute) of the matrix region in the dual-permeability
model (=wθm) [L3L-3]

θnvolumetric solid phase fraction [L3L-3]

θovolumetric organic matter fraction [L3L-3]

θrresidual soil water content [L3L-3]

θr *scaled residual soil water content [L3L-3]

θr dresidual soil water content of the main drying branch [L3L-3]

θr w

residual soil water content of the main wetting branch [L3L-3]

θssaturated soil water content [L3L-3]

θs dsaturated soil water content of the main drying branch [L3L-3]

θs w

saturated soil water content of the main wetting branch [L3L-3]

θT

total volumetric water vapor content (being the sum of θ and θv) [L3L-3]

θv

volumetric water vapor content (expressed as an equivalent water content) [L3L-3]

θΔ

water content at the reversal point of a hysteretic retention function [L3L-3]

λapparent thermal conductivity of the soil [MLT-3K-1] (e.g. Wm-1K-1)λlatent heat of vaporization of water [L2T-2] (e.g., Jkg-1)

λ0

thermal conductivity of porous medium in the absence of water flow [MLT-3K-1] (e.g.
Wm-1K-1)λET0latent heat flux of evaporation [MT-3] (e.g., J m-2s-1)

μfluid viscosity [ML-1T-1] (= 0.00093 Pa s)

μg

first-order rate constant for solutes in the gas phase [T-1]

μrefdynamic viscosity at reference temperature Tref [MT-1L-1]

μs

first-order rate constant for solutes adsorbed onto the solid phase [T-1]

μT

dynamic viscosity at temperature T [MT-1L-1]

μw

first-order rate constant for solutes in the liquid phase [T-1]μg'

first-order rate constant for decay chain solutes in the gas phase [T-1]μs'

first-order rate constant for decay chain solutes adsorbed onto the solid phase [T-1]μw'

first-order rate constant for decay chain solutes in the liquid phase [T-1]ζiactivity coefficient for the exchange surfaces [MM-1] (kg mol-1)

ρbulk density of porous medium [ML-3]

xxx

ρf

fluid density (= 998 kg m-3) [ML-3]

ρrefdensity of soil water at reference temperature Tref [ML-3]

ρpbacterial density (= 1080 kg m-3) [ML-3]

ρT

density of soil water at temperature T [ML-3]

ρvssaturated vapor density [ML-3]

ρwdensity of liquid water [ML-3]ψdimensionless colloid retention function [-]σsurface tension [MT-2]

σrefsurface tension at reference temperature Tref [MT-2]

σTsurface tension at temperature T [MT-2]

τgtortuosity factor in the gas phase [-]τw

tortuosity factor in the liquid phase [-]φmo, φmi, φmo,k sink/source terms for the equilibrium phases of the mobile zone, the immobile
zone, and for the kinetic sorption sites [ML-3T-1], respectively

φm,m, φim,mvarious reactions in the mobile and immobile parts of the matrix [ML-3T-1]

ϕsite latitude [rad]ϕnlinear basis functions [-]ϕnu

upstream weighted basis functions [-]ωfirst-order adsorption rate constant [T-1]ω

first-order rate coefficient [T-1] in the water mass transfer equationωdpm

mass transfer coefficient between the mobile and immobile zones of the matrix region
[T-1]

ωph

first-order rate constant [T-1] accounting for physical rate processesωsperformance index for minimizing or eliminating numerical oscillations [-]ωssunset hour angle [rad]ξ

local coordinate [-]xxxi

List of additional variables in the carbon dioxide and major ion chemistry modulesaparameter in the exponential depth reduction function [L-1]aiactivity of the ith ion [-]A

Debye-Hückel constant (kg0.5mol-0.5)

Alkalkalinity (molckg-1)

B

Debye-Hückel constant (kg0.5cm-1mol-0.5)

Bij

Pitzer specific virial coefficient for double ion interaction

csurface species concentration [MM-1]c$solid phase concentration [MM-1]

ca, cw

CO2 concentrations in the gas and liquid phase, respectively [L3L-3]

cai

initial CO2 concentration in the gas phase [L3L-3]

cas

CO2 concentration in the soil gas at the soil surface [L3L-3]

catm

CO2 concentration at the top of the stagnant boundary layer [L3L-3]

ca0boundary condition for CO2 concentration in the gas phase [L3L-3]

cT

total volumetric CO2 concentration [L3L-3]CT

cation exchange capacity (molckg-1)

C0

total salt concentration (mol l-1)

Cijk

Pitzer specific virial coefficient for triple ion interaction

CEC

cation exchange capacity (mol kg-1)

Da, Dw

effective soil matrix diffusion coefficients of CO2 in the gas and liquid phase,respectively [L2T-1]

Das, Dws diffusion coefficients of CO2 in the gas and liquid phase, respectively [L2T-1]
DE

effective dispersion coefficient in the soil matrix [L2T-1]

ECelectric conductivity of the solution (dS m-1)

ESP*

exchangeable sodium percentage

ESP

adjusted exchangeable sodium percentage

fmont

weight fraction of montmorillonite in soil [-]fp , fs

reduction functions for CO2 production by plant roots and by soil microorganisms,
respectively [-]xxxii

h1, h3pressure head when CO2 production ceases [L]h2pressure head when CO2 production is optimal [L]I

ionic strength (mol kg-1)

IAPC

ion activity product for calcite [-]IAPG

ion activity product for gypsum [-]Jca, Jcw

CO2 fluxes caused by convection in the gas and liquid phase, respectively [LT-1]

Jda, Jdw

CO2 fluxes caused by diffusion in the gas and liquid phase, respectively [LT-1]

ka1

first dissociation constant of carbonic acid [-]ka2

second dissociation constant of carbonic acid [-]kx

multiplication factor

KCO2

Henry's Law constant [MT2M-1L-2]

KM, KM* Michaelis' constants for O2 and CO2 concentrations, respectively [L3L-3]
KSPC

solubility product for calcite [-]KSPGsolubility product for gypsum [-]KSPHsolubility product for hydromagnesite [-]KSPNsolubility product for nesquehonite [-]KSPSsolubility product for freshly precipitated sepiolite [-]KW

dissociation constant for water [-]K1,..K12

equilibrium constants for complexation reactions [-]K13,..K15 selectivity constants for cation exchange reactions [-]mimolality (mol kg-1)

m0

unit molality (1 mol kg-1)

M

number of species in the solution mixture [-]MCO

total amount of CO2 in the entire flow domain [L]Mlamount of solute in the liquid phase in the flow region at time t [ML-2]

Mpamount of solute in the precipitated phase in the flow region at time t [ML-2]

Ms

amount of solute in the sorbed phase in the flow region at time t [ML-2]

MS

molar weight (mol-1)

Pproduction/sink term for CO2 [L3L-3T-1]

PCO2

partial pressure of CO2 [ML-1T-2] (atm)xxxiiiPT

actual CO2 production rate [L3L-2T-1]

Pφosmotic pressure of electrolyte solution [ML-1T-2] (Pa)pH

negative logarithm of hydrogen activity [-]pIAPnegative logarithm of the ion activity product [-]q

oxygen uptake rate [L3L-3T-1]

qa, qw

soil air and soil water fluxes, respectively [LT-1]qE

CO2 effective velocity [LT-1]

qE0

prescribed CO2 effective boundary flux [LT-1]

qmax

maximum oxygen uptake rate [L3L-3T-1]

rscaling factor which represents the effect of solution composition (SAR, C0, pH) on
the hydraulic conductivity [-]r1scaling factor which represents the effect of solution composition (SAR, C0) on thehydraulic conductivity [-]r2

scaling factor which represents the effect of solution pH on the hydraulic conductivity[-]

R

universal gas constant [ML2T-2K-1M-1]

RC

calcite dissolution-precipitation rate (mmol cm-2s-1)

RD

dolomite dissolution rate (mmol cm-2s-1)

ssurface species concentration [-]S*

CO2 uptake rate associated with root water uptake [T-1]

SAR

sodium adsorption ratio (mmol0.5l-0.5)W

molecular weight of waterxswelling factor [-]γiactivity coefficient of the ith solute ion [-]γiDHmodified Debye-Hückel activity coefficient of the ith solute ion [-]γp, γsγp0, γs0

actual CO2 production rate of plant roots and soil microorganisms, respectively [L3LT ]3 -1

optimal CO2 production rate of plant roots and soil microorganisms (20oC),
respectively [L3L-2T-1]

xxxiv

GENERAL INTRODUCTION AND OVERVIEW OF HYDRUS-1D
The importance of the unsaturated zone as an integral part of the hydrological cycle haslong been recognized. The vadose zone plays an inextricable role in many aspects of hydrology,
including infiltration, soil moisture storage, evaporation, plant water uptake, groundwater
recharge, runoff and erosion. Initial studies of the unsaturated (vadose) zone focused primarily
on water supply studies, inspired in part by attempts to optimally manage the root zone of
agricultural soils for maximum crop production. Interest in the unsaturated zone has dramaticallyincreased in recent years because of growing concern that the quality of the subsurface
environment is being adversely affected by agricultural, industrial and municipal activities.
Federal, state and local action and planning agencies, as well as the public at large, are now
scrutinizing the intentional or accidental release of surface-applied and soil-incorporated
chemicals into the environment. Fertilizers and pesticides applied to agricultural lands inevitably
move below the soil root zone and may contaminate underlying groundwater reservoirs.
Chemicals migrating from municipal and industrial disposal sites also represent environmental
hazards. The same is true for radionuclides emanating from energy waste disposal facilities.
The past several decades has seen considerable progress in the conceptual understanding
and mathematical description of water flow and solute transport processes in the unsaturated
zone. A variety of analytical and numerical models are now available to predict water and/or
solute transfer processes between the soil surface and the groundwater table. The most popularmodels remain the Richards equation for variably saturated flow, and the Fickian-based
convection-dispersion equation for solute transport. Deterministic solutions of these classicalequations have been used, and likely will continue to be used in the near future, for predicting
water and solute movement in the vadose zone, and for analyzing specific laboratory or field
experiments involving unsaturated water flow and/or solute transport. Models of this type arealso helpful tools for extrapolating information from a limited number of field experiments todifferent soil, crop and climatic conditions, as well as to different tillage and water management
schemes.
Once released into the subsurface environment, industrial and agricultural chemicals are
generally subjected to a large number of simultaneous physical, chemical, and biologicalprocesses, including sorption-desorption, volatilization, photolysis, and biodegradation, as wellas their kinetics. The extent of degradation, sorption and volatilization largely determines thepersistence of a pollutant in the subsurface [Chiou, 1989]. For example, the fate of organic

1chemicals in soils is known to be strongly affected by the kinetics of biological degradation.

Alexander and Scow [1989] gave a review of some of the equations used to represent the kinetics
of biodegradation. These equations include zero-order, half-order, first-order, three-half-order,
mixed-order, logistic, logarithmic, Michaelis-Menton, and Monod type (with or without growth)
expressions. While most of these expressions have a theoretical basis, they are commonly used
only in an empirical fashion by fitting the equations to observed data. Zero- and first-orderkinetic equations remain the most popular for describing biodegradation of organic compounds,
mostly because of their simplicity and the ease at which they can be incorporated in solutetransport models. Conditions for the application of these two equations are described byAlexander and Scow [1989].
One special group of degradation reactions involves decay chains in which solutes are
subject to sequential (or consecutive) decay reactions. Problems of solute transport involving
sequential first-order decay reactions frequently occur in soil and groundwater systems.
Examples are the migration of various radionuclides [Lester et al., 1975; Rogers, 1978;
Gureghian, 1981; Gureghian and Jansen, 1983], the simultaneous movement of interacting
nitrogen species [Cho, 1971; Misra et al., 1974; Wagenet et al., 1976; Tillotson et al., 1980],
organic phosphate transport [Castro and Rolston, 1977], the transport of certain pesticides andtheir metabolites [Bromilow and Leistra, 1980; Wagenet and Hutson, 1987], the transport ofsequential biodegradation of chlorinated hydrocarbons [e.g., Schaerlaekens et al., 1999; Caseyand Šimůnek, 2001], and the transport of various hormones such as estrogen and testosterone[e.g., Casey et al., 2003, 2004].
While in the past most pesticides were regarded as involatile, volatilization is now
increasingly recognized as being an important process affecting the fate of pesticides in fieldsoils [Glotfelty and Schomburg, 1989; Spencer, 1991]. Another process affecting pesticide fate
and transport is the relative reactivity of solutes in the sorbed and solution phases. Several
processes such as gaseous and liquid phase molecular diffusion, and convective-dispersive
transport, act only on solutes that are not adsorbed. Degradation of organic compounds likely
occurs mainly, or even exclusively, in the liquid phase [Pignatello, 1989]. On the other side,
radioactive decay takes place equally in the solution and adsorbed phases, while other reactions
or transformations may occur only or primarily in the sorbed phase.
Several analytical solutions have been published for simplified transport systems
involving consecutive decay reactions [Cho, 1971; Wagenet et al., 1976; Harada et al., 1980;
Higashi and Pigford, 1980; van Genuchten, 1985]. Unfortunately, analytical solutions for more2complex situations, such as for transient water flow or the nonequilibrium solute transport withnonlinear reactions, are not available and/or cannot be derived, in which case numerical models
must be employed. To be useful, such numerical models must allow for different reaction rates totake place in the solid, liquid, and gaseous phases, as well as for a correct distribution of thesolutes among the different phases.
The processes of evaporation and plant transpiration also exert a major influence onwater and solute distributions in near-surface environments. These processes concentrate salts bydecreasing the amount of water in the soil, and when combined with irrigation in arid regions
can lead to highly saline conditions. Ion activities for such chemical conditions should be
calculated with expressions suitable for use in brines, rather than with the more standard
formulations for dilute solutions. The interaction of evapotranspiration, changing soil gas
composition, ion exchange and soil-water reactions may cause precipitation and dissolution of avariety of minerals. Major ions (mainly of Ca2+, Mg2+, Na+, K+, Cl-, SO42-, HCO3-, CO32-, andNO3-) may accumulate in certain parts of the soil profile in such amounts that crop yield can beseriously reduced. Hence, models used to predict the solution chemistry of major ions in theunsaturated zone should include all of these processes and variables [Šimůnek et al., 1987].
The purpose of this report is to document the HYDRUS-1D software package forsimulating one-dimensional variably saturated water flow, heat movement, and the transport ofsolutes involved in sequential first-order decay reactions. To be able to simulate the salinization
processes described in the previous paragraph, we also implemented into the HYDRUS software
the carbon dioxide transport and production, and major ion chemistry modules originallydeveloped for the UNSATCHEM program [Šimůnek et al., 1996]. HYDRUS-1D consists of the
HYDRUS computer program, and the HYDRUS1D interactive graphics-based user interface.
The HYDRUS program numerically solves the Richards equation for saturated-unsaturated
water flow and advection-dispersion type equations for heat and solute transport. The water flow
equation incorporates a sink term to account for water uptake by plant roots. The flow equation
may also consider dual-porosity type flow in which one fraction of water content is mobile andanother fraction immobile, or dual-permeability type flow involving two mobile regions, one
representing the matrix and one the macropores. The heat transport equation considers
movement by conduction as well as convection with flowing water. Coupled water, vapor, andenergy transport can be considered as well. The governing advection-dispersion solute transport
equations are written in a very general form by including provisions for nonlinear
nonequilibrium reactions between the solid and liquid phases, and linear equilibrium reaction

3

between the liquid and gaseous phases. Hence, both adsorbed and volatile solutes such as
pesticides can be considered. The solute transport equations also incorporate the effects of zeroorder production, first-order degradation independent of other solutes, and first-orderproduction/decay reactions that provide the required coupling between the solutes involved in
the sequential first-order decay chain. The transport models also account for advection anddispersion in the liquid phase, as well as for diffusion in the gas phase, thus permitting one tosimulate solute transport simultaneously in both the liquid and gaseous phases. HYDRUS-1D atpresent considers up to five solutes, which can be either coupled in a unidirectional chain or may
move independently of each other. Physical nonequilibrium solute transport can be accounted forby assuming a two-region, dual-porosity type formulation which partition the liquid phase into
separate mobile and immobile regions. Additionally, the transport equations may include
provisions for kinetic attachment/detachment of solutes to the solid phase, thus permitting
simulations of the transport of viruses, colloids, or bacteria.
The HYDRUS software package also includes modules for simulating carbon dioxideand major ion solute movement. Diffusion in both liquid and gas phases and convection in theliquid phase are considered as CO2 transport mechanisms. The CO2 production model is
described. The major variables of the chemical system are Ca, Mg, Na, K, SO4, Cl, NO3, H4SiO4,
alkalinity, and CO2. The model accounts for equilibrium chemical reactions between thesecomponents such as complexation, cation exchange and precipitation-dissolution. For the
precipitation-dissolution of calcite and dissolution of dolomite, either equilibrium or
multicomponent kinetic expressions are used which include both forward and back reactions.
Other dissolution-precipitation reactions considered include gypsum, hydromagnesite,nesquehonite, and sepiolite. Since the ionic strength of soil solutions can vary considerably withtime and space and often reach high values, both modified Debye-Hückel and Pitzer expressions
were incorporated into the model as options to calculate single ion activities.
The HYDRUS-1D code may be used to analyze water and solute movement in
unsaturated, partially saturated, or fully saturated porous media. The flow region itself may becomposed of nonuniform soils. Flow and transport can occur in the vertical, horizontal, or in a
generally inclined direction. The water flow part of the model considers prescribed head and flux
boundaries, as well as boundaries controlled by atmospheric conditions, free drainage, or flow tohorizontal drains. First and third-type boundary conditions can be implemented in both the soluteand heat transport parts of the model. In addition, HYDRUS-1D implements a MarquardtLevenberg type parameter estimation technique for inverse estimation of soil hydraulic and/or

4

solute transport and reaction parameters from measured transient or steady-state flow and/ortransport data.
The governing flow and transport equations are solved numerically using standard
Galerkin-type linear finite element schemes, or modification thereof. The program is a onedimensional version of the HYDRUS-2D and HYDRUS (2D/3D) codes simulating water, heatand solute movement in two- or three-dimensional variably saturated media [Šimůnek et al.,
1999; 2006a,b], while incorporating various features of earlier related codes such as SUMATRA[van Genuchten, 1978], WORM [van Genuchten, 1987], HYDRUS 3.0 [Kool and van Genuchten, 1991], SWMI [Vogel, 1990], SWMI_ST [Šimůnek, 1993], HYDRUS 5.0 [Vogel et al.,1996], and HYDRUS-1D, version 3.0 [Šimůnek et al., 2005]. The method of incorporating
hysteresis in the soil hydraulic properties, as well as several other features, was adopted fromHYDRUS 5.0 [Vogel et al., 1996]. Carbon dioxide transport and major ion chemistry modules
were adopted from the UNSATCHEM program [Šimůnek et al., 1996]. The text in this manualdraws heavily upon texts in the early versions of these various programs. The source code wasdeveloped and tested on a Pentium 4 PC using the Microsoft's Fortran PowerStation compiler.
Several extensions of the MS Fortran beyond the ANSI standard were used to enablecommunication with graphic based user-friendly interface. New main features of the version 3.0,
compared to version 2.0, were a) new analytical models for the soil hydraulic properties, b)compensated root water uptake, c) the dual-porosity features for water flow, d) the
attachment/detachment models for solute transport to allow simulations of the virus, colloid, andbacteria transport, and e) the carbon dioxide and f) major ion chemistry modules. Major new
features of the current version 4.0, compared to version 3.0, are consideration of a) vapor flow,b) coupled water, vapor, and energy transport, c) dual-permeability type water flow and solutetransport, d) dual-porosity water flow with solute transport considering two-site sorption in themobile zone, e) potential evapotranspiration as calculated with the Penman-Monteith
combination and Hargreaves equations, f) daily variations in evaporation, transpiration, andprecipitation, and g) support for the HP1 code [Jacques and Šimůnek, 2005; Šimůnek et al., 2006c;
Jacques et al., 2007], obtained by coupling HYDRUS with the PHREEQC biogechemical code
[Parkhurst and Appelo, 1999]. New features in version 4.07 of HYDRUS-1D as compared toversion 4.0 include: a) an option to specify the nonequilibrium phase concentration initially atequilibrium with the equilibrium phase concentration, b) an option to specify initial conditions in
total (instead of liquid) concentrations, c) an option to print fluxes instead of temperatures forobservation nodes, d) linking of optimized parameters of different soil layers, e) constant mobile

5

water content in multiple layers (in the Mobile-Immobile Water Model) when optimizing
immobile water content, f) Per Moldrup’s tortuosity model was implemented as an alternative to
the Millington and Quirk (1960) model, g) SSurface Energy Balance (i.e., the balance of latent,
heat, and sensible fluxes) for bare soils can be considered, and h) daily variations ofmeteorological variables can be generated by the model using simple meteorological models. A
new features in version 4.08 of HYDRUS-1D as compared to version 4.07 is an option toconsider root solute uptake, including both passive and active uptake.
One major problem which often prevents the widespread use of otherwise welldocumented numerical computer codes is the extensive work generally required for input datapreparation, finite element grid design, and graphical presentation of the output results. Hence,techniques are needed which make it easier to create, manipulate and display large data files, andwhich facilitate interactive data management. Such techniques will free users from cumbersome
manual data processing, and should enhance the efficiency in which programs are being
implemented for a particular problem. To avoid or simplify the preparation and management ofrelatively complex input data files for flow problems, and to graphically display the final
simulation results, we developed the HYDRUS1D interactive graphics-based user-friendly
interface for the MS Windows environment. The HYDRUS1D interface is directly connected to
the HYDRUS computational programs. The software package is distributed on a CD-ROM
containing all necessary files needed to run the interface, the input and output files of various
examples discussed in this report and several other examples. The software package can also be
freely downloaded from www.hydrus2d.com (orm www.pc-progress.cz).
A general overview of the HYDRUS1D graphics-based interface is described in Part B ofthis manual. We note that in addition to the detailed descriptions in this section, extensive online help files are also available in each module of the user interface.

6PART AThe HYDRUS Code for Simulating the One-Dimensional
Movement of Water, Heat, and Multiple Solutes
in Variably-Saturated Porous MediabyJ. Šimůnek and M. Th. van Genuchten

781. INTRODUCTION
This part of the manual gives a detailed description of the HYDRUS computer code,
which numerically solves the Richards equation for variably-saturated water flow and advectiondispersion type equations for heat and solute transport. The flow equation incorporates a sinkterm to account for water uptake by plant roots. The flow equation may also consider dual-porosity
type flow in which one fraction of the water content is mobile and another fraction immobile, or
dual-permeability type flow involving two mobile regions, one representing the matrix and one themacropores. The heat transport equation considers transport due to conduction and convection
with flowing water. Coupled water, vapor, and energy transport can be considered as well. The
solute transport equations consider advective-dispersive transport in the liquid phase, as well as
diffusion in the gaseous phase. The transport equations also include provisions for nonlinear
nonequilibrium reactions between the solid and liquid phases, linear equilibrium reactions
between the liquid and gaseous phases, zero-order production, and two first-order degradation
reactions: one which is independent of other solutes, and one which provides the coupling
between solutes involved in sequential first-order decay reactions. Physical nonequilibrium
solute transport can be accounted for by assuming a two-region, dual-porosity type formulation
which partitions the liquid phase into separate mobile and immobile regions. Additionally, thetransport equations may include provisions for kinetic attachment/detachment of solutes to thesolid phase, thus permitting simulations of the transport of viruses, colloids, or bacteria.
The program may be used to analyze water and solute movement in unsaturated, partiallysaturated, or fully saturated porous media. The flow region may be composed of nonuniform
soils. Flow and transport can occur in the vertical, horizontal, or a generally inclined direction.
The water flow part of the model can deal with prescribed head and flux boundaries, boundaries
controlled by atmospheric conditions, as well as free drainage boundary conditions. The
governing flow and transport equations are solved numerically using Galerkin-type linear finite
element schemes. HYDRUS-1D also includes a Marquardt-Levenberg type parameter
optimization algorithm for inverse estimation of soil hydraulic and/or solute transport andreaction parameters from measured transient or steady-state flow and/or transport data.
HYDRUS-1D further incorporates modules simulating carbon dioxide production andmajor ion solute movement. The CO2 transport processes include diffusion in both the liquid andgas phases and advection in the liquid phase. The CO2 production model is described in detail.
The major variables of the chemical system are Ca, Mg, Na, K, SO4, Cl, NO3, H4SiO4, alkalinity,9and CO2. The model accounts for equilibrium chemical reactions between these components
such as complexation, cation exchange and precipitation-dissolution. For the precipitationdissolution of calcite and dissolution of dolomite, either equilibrium or multicomponent kinetic
expressions are used which include both forward and back reactions. Other dissolutionprecipitation reactions considered include gypsum, hydromagnesite, nesquehonite, and sepiolite.
Since the ionic strength of soil solutions can vary considerably with time and space and often
reach high values, both modified Debye-Hückel and Pitzer expressions were incorporated into
the model as options to calculate single ion activities.

102. VARIABLY SATURATED WATER FLOW
2.1. Governing Water Flow Equations
2.1.1. Uniform Water Flow
One-dimensional uniform (equilibrium) water movement in a partially saturated rigid
porous medium (Fig. 2.1a) is described by a modified form of the Richards equation using theassumptions that the air phase plays an insignificant role in the liquid flow process and that waterflow due to thermal gradients can be neglected:∂θ∂ ⎡ ⎛ ∂h⎞⎤=K ⎜ + cos α ⎟ ⎥ - S⎢∂t ∂x ⎣ ⎝ ∂x
⎠⎦

(2.1)

where h is the water pressure head [L], θ is the volumetric water content [L3L-3], t is time [T], x
is the spatial coordinate [L] (positive upward), S is the sink term [L3L-3T-1], α is the angle
between the flow direction and the vertical axis (i.e., α = 00 for vertical flow, 900 for horizontal
flow, and 00 < α < 900 for inclined flow), and K is the unsaturated hydraulic conductivity
function [LT-1] given by

K (h, x) = K s ( x) K r (h, x)(2.2)

where Kr is the relative hydraulic conductivity [-] and Ks the saturated hydraulic conductivity[LT-1].
2.1.2. Uniform Water Flow and Vapor Transport
The Richards equation (2.1) considers only water flow in the liquid phase and ignores theeffects of the vapor phase on the overall water mass balance. While this assumption is justified
for the majority of applications, a number of problems exist in which the effect of vapor flow can
not be neglected. Vapor movement is often an important part of the total water flux when the soil
is relatively dry. Scanlon et al. [2003] showed that water fluxes in deep vadose zone profiles of11the arid and semiarid regions of the western U.S. are often dominated by thermal vapor fluxes.
Nonisothermal liquid and vapor flow in HYDRUS is described as follows (e.g., Saito et al.
[2006]):∂θT ( h ) ∂ ⎡
∂h
∂T ⎤
=
( K + Kvh ) ⎛⎜ + cos α ⎞⎟ + ( K LT + KvT ) ⎥ - S (h)
⎢
∂t
∂x ⎣
∂x ⎦
⎝ ∂x
⎠

(2.3)

where θT is the total volumetric water content [L3L-3], being the sum (θT=θ+θv) of the volumetricliquid water content, θ, and the volumetric water vapor content, θv (both expressed in terms ofequivalent water contents) [L3L-3]; T is temperature [K]; K is the isothermal hydraulicconductivity of the liquid phase [LT-1]; KLT is the thermal hydraulic conductivity of the liquid
phase [L2K-1T-1]; Kvh is the isothermal vapor hydraulic conductivity [LT-1]; and KvT is thethermal vapor hydraulic conductivity [L2K-1T-1]. Overall water flow in (2.3) is given as the sumof isothermal liquid flow, isothermal vapor flow, gravitational liquid flow, thermal liquid flow,
and thermal vapor flow. Since several terms of (2.3) are a function of temperature, this equation
should be solved simultaneously with the heat transport equation (4.5) to properly account for
temporal and spatial changes in soil temperature.
2.1.3. Flow in a Dual-Porosity System
Dual-porosity models assume that water flow is restricted to the fractures (or interaggregate pores and macropores), and that water in the matrix (the intra-aggregate pores or therock matrix) does not move at all (Fig. 2.1bc). These models assume that the matrix, consistingof immobile water pockets, can exchange, retain, and store water, but does not permit convective
flow. This conceptualization leads to two-region, dual-porosity type flow and transport models[Philip, 1968; van Genuchten and Wierenga, 1976] that partition the liquid phase into mobile(flowing, inter-aggregate), θmo, and immobile (stagnant, intra-aggregate), θim, regions:θ = θ mo + θim(2.4)

with some exchange of water and/or solutes possible between the two regions, usually calculated
by means of a first-order rate equation. We will use here the subscript m to represent fractures,inter-aggregate pores, or macropores, and the subscript im to represent the soil matrix, intra-

12

aggregate pores, or the rock matrix.
The dual-porosity formulation for water flow as used in HYDRUS-1D is based on amixed formulation, which uses Richards equation (2.1) to describe water flow in the fractures(macropores), and a simple mass balance equation to describe moisture dynamics in the matrix
as follows [Šimůnek et al., 2003]:∂θ mo ∂ ⎡⎛ ∂h⎞⎤
= ⎢ K ( h ) ⎜ + cos α ⎟ ⎥ − Smo − Γ w
∂t
∂x ⎣
⎝ ∂x
⎠⎦
∂θim
=
∂t

(2.5)

− Sim + Γ w

where Sm and Sim are sink terms for both regions, and Γw is the transfer rate for water from theinter- to the intra-aggregate pores.
An alternative dual-porosity approach, not implemented in HYDRUS-1D, was suggested
by Germann [1985] and Germann and Beven [1985], who used a kinematic wave equation todescribe gravitational movement of water in macropores. Although dual-porosity models havebeen popularly used for solute transport studies (e.g. van Genuchten [1981]), they have thus far
not been used extensively for water flow problems.
2.1.4. Flow in a Dual-Permeability System
While dual-porosity models assume that water in the matrix is stagnant, dualpermeability models (Fig. 2.1de) allow for water flow in the matrix as well. The approach of
Gerke and van Genuchten [1993a, 1996], who applied Richards equations to each of two poreregions is implemented in HYDRUS-1D. The flow equations for the macropore or fracture(subscript f) and matrix (subscript m) pore systems in this approach are given by:∂θ f (h f )∂t=⎛ ∂h f⎞⎤∂ ⎡+ cos α ⎟ ⎥ - S f (h f ) - Γ w⎢ K f (h f ) ⎜
w
∂x ⎣⎢
⎝ ∂x
⎠ ⎦⎥

∂θ m (hm ) ∂ ⎡
⎛ ∂h
⎞⎤
=
K m (hm ) ⎜ m + cos α ⎟ ⎥ - S m (hm ) + Γ w
⎢
∂t
∂x ⎣
1- w
⎝ ∂x
⎠⎦

13

(2.6)

Equilibrium Model ---------------------------------------Non-Equilibrium Models -------------------------------------a. Uniform Flowb. Mobile-Immobile Water

Water

Water

Soluteθ
S = θcFigure 2.1.

c. Dual-Porosity

Immob.

SoluteImmob.

Water

MobileSoluteMobileImmob.

θ = θ im + θ mo
S = θ im cim + θ mo cmo

Mobileθ = θ im + θ mo
S = θ im cim + θ mo cmo

d. Dual-Permeability e. Dual-Permeability with MIM

Water

Slow

Water

Fast

Slow

SoluteSlow

Fast

SoluteFastθ = θM + θFS = θ M cm + θ F c f

Im.

Slow

Fast

θ = θ M ,im + θ M ,mo + θ F
S = θ M ,im cM ,im + θ M ,mo cM ,mo + θ F c f

Conceptual physical nonequilibrium models for water flow and solute transport. In the plots, θ is the water content, θmo and θim in (b)and (c) are water contents of the mobile and immobile flow regions, respectively; θM and θF in (d) are water contents of the matrix andmacropore (fracture) regions, respectively, and θM,mo, θM,im, and θF in (e) are water contents of the mobile and immobile flow regionsof the matrix domain, and of the macropore (fracture) domain, respectively; c are concentrations of corresponding regions, with
subscripts having the same meaning as for water contents, while S is the total solute content of the liquid phase.

14respectively, where w is the ratio of the volumes of the macropore or fracture domain and thetotal soil system [-]. Note that the water contents θf and θm in (2.6) have different meanings thanthose in (2.5) where they represented water contents of the total pore space (i.e., θ = θmo + θim),while in (2.6) they refer to water contents of the two separate (fracture or matrix) pore domains
such that θ = wθf + (1-w)θm(=θF + θM, where θM and θF are absolute water contents in the matrix
and macropore (fracture) regions, respectively).
2.2. Root Water Uptake
2.2.1. Root Water Uptake Without Compensation
The sink term, S, is defined as the volume of water removed from a unit volume of soil
per unit time due to plant water uptake. Feddes et al. [1978] defined S as

S (h ) = α (h ) S p(2.7)

where the root-water uptake water stress response function α(h) is a prescribed dimensionlessfunction (Fig. 2.1) of the soil water pressure head (0 ≤ α ≤ 1), and Sp the potential water uptake

rate [T-1]. Figure 2.1 gives a schematic of the stress response function as used by Feddes et al.
[1978]. Notice that water uptake is assumed to be zero close to saturation (i.e., wetter than somearbitrary "anaerobiosis point", h1). For h<h4 (the wilting point pressure head), water uptake isalso assumed to be zero. Water uptake is considered optimal between pressure heads h2 and h3,
whereas for pressure head between h3 and h4 (or h1 and h2), water uptake decreases (or increases)
linearly with h. The variable Sp in (2.7) is equal to the water uptake rate during periods of nowater stress when α(h)=1.
Van Genuchten [1987] expanded formulation of Feddes et al. [1978] by including
osmotic stress as follows
S ( h, hφ ) = α ( h, hφ ) S p(2.8)

where hφ is the osmotic head [L], which is assumed here to be given by a linear combination of
the concentrations, ci, of all solutes present, i.e.,15hφ = ai ci(2.9)

in which ai are experimental coefficients [L4M] converting concentrations into osmotic heads.
van Genuchten [1987] proposed an alternative S-shaped function to describe the water uptake
stress response function (Fig. 2.1), and suggested that the influence of the osmotic head
reduction can be either additive or multiplicative as follows

α ( h, hφ ) =1⎛ h + hφ ⎞1+ ⎜⎟⎝ h50 ⎠

p(2.10)orα ( h, hφ ) =11
p1
1 + ( h / h50 ) 1 + ( hφ / hφ 50 ) p 2(2.11)

respectively, where p, p1, and p2 are experimental constants. The exponent p was found to beapproximately 3 when applied to salinity stress data only [van Genuchten, 1987]. The parameterh50 in (2.10) and (2.11) represents the pressure head at which the water extraction rate is reduced
by 50% during conditions of negligible osmotic stress. Similarly, hφ50 represents the osmotic
head at which the water extraction rate is reduced by 50% during conditions of negligible water
stress. Note that, in contrast to the expression of Feddes et al. [1978], this formulation of thestress response function, α(h,hφ), does not consider the transpiration reduction near saturation.
This simplification seems justified when saturated or near-saturated conditions occur for only
relatively short periods of time.
When the potential water uptake rate is equally distributed over the root zone, Sp
becomes

Sp =

1
Tp
LR

16

(2.12)

where Tp is the potential transpiration rate [LT-1] and LR the depth [L] of the root zone. Equation(2.12) may be generalized by introducing a non-uniform distribution of the potential water
uptake rate over a root zone of arbitrary shape:
S p = b( x )Tp(2.13)

Fig. 2.2. Schematic of the plant water stress response function, α(h),as used by a) Feddes et al. [1978] and b) van Genuchten [1987].

where b(x) is the normalized water uptake distribution [L-1]. This function describes the spatial
variation of the potential extraction term, Sp, over the root zone (Fig. 2.2), and is obtained bynormalizing any arbitrarily measured or prescribed root distribution function, bΝ(x), as followsb( x ) =b′( x)∫ b′( x) dx
LR17(2.14)

where LR is the region occupied by the root zone. Normalizing the uptake distribution ensuresthat b(x) integrates to unity over the flow domain, i.e.,∫ b( x)dx = 1(2.15)

LR

Fig. 2.3. Schematic of the potential water uptake distribution function, b(x),in the soil root zone.

There are many ways to express the function b(x): constant with depth, linear [Feddes et al.,1978], or the following function [Hoffman and van Genuchten, 1983]:

18

1.66667⎧⎪
LR⎪⎪ 2.0833 ⎛x0 − x ⎞b( x ) = ⎨
⎜1 −
⎟
LR ⎠
⎪ LR ⎝⎪
0⎪⎪⎩x > L − 0.2 LR
x ∈ ( L − LR ; L − 0.2 LR )(2.16)

x < L − LR

where L is the x-coordinate of the soil surface [L] and LR is the root depth [L]. HYDRUS allows
a user to prescribe virtually any shape of the water uptake distribution function, provided that
this function is constant during the simulation. When the rooting depth varies in time (asdescribed later), only the Hoffman and van Genuchten [1983] is used. Note that in the above
development, and throughout this manual, the bottom of the soil profile is located at x = 0 andthe soil surface at x = L.
From (2.13) and (2.15) it follows that Sp is related to Tp by the expression∫ S dx = Tpp(2.17)

LR

The actual water uptake distribution is obtained by substituting (2.13) into (2.7):
S ( h, hφ , x ) = α ( h, hφ , x ) b( x )Tp(2.18)

whereas the actual transpiration rate, Ta, is obtained by integrating (2.18) as follows
Ta = ∫ S ( h, hφ , x ) dx = Tp ∫ a ( h, hφ , x ) b( x ) dx
LR

(2.19)LR

The root depth, LR, can be either constant or variable during the simulation. For annual
vegetation a growth model is required to simulate the change in rooting depth with time.
HYDRUS assumes that the actual root depth is the product of the maximum rooting depth, Lm[L], and a root growth coefficient, fr(t) [-] [Šimůnek and Suarez, 1993a]:19

LR (t ) = Lm f r (t )(2.20)For the root growth coefficient, fr(t), we use the classical Verhulst-Pearl logistic growth
functionf r (t ) =L0
L0 + ( Lm - L0 ) e - rt

(2.21)

where L0 is the initial value of the rooting depth at the beginning of the growing season [L], andr the growth rate [T-1]. The growth rate is calculated either from the assumption that 50% of therooting depth will be reached after 50% of the growing season has elapsed, or from given data.

2.2.2. Root Water Uptake With Compensation
The ratio of actual to potential transpiration of the root uptake without compensation isdefined as follows:Ta
1=
Sdx = ∫ α ( h,hφ , x ) b( x )dx = ω
Tp Tp L∫R
LR

(2.22)

where ω is a dimensionless water stress index [Jarvis, 1989]. Following Jarvis [1989], we
introduce a critical value of the water stress index ωc, a so-called the root adaptability factor,
which represents a threshold value above which root water uptake reduced in stressed parts of
the root zone is fully compensated by increased uptake from other parts. However, some
reduction in potential transpiration will occur below this threshold value, although smaller thanfor water uptake without compensation.

20

1

1
Compensated

Tac/Tp

Non-Compensated

Ta/Tp

ωc

0

0

ω

0

1

Fig. 2.4. Ratio of actual to potential transpiration as a function of the stress index ω.

Thus, for the interval when ω is larger than the threshold value ωc (Fig. 2.3), one obtains

S ( h,hφ , x ) = α ( h,hφ , x ) b( x )∫α (h,hφ , x ) b( x )dx

Ta LR
=
Tp

ω

=

Tp

ω
ω
=1
ω

(2.23)

While for the interval when ω is smaller than the threshold value ωc, one has
S ( h,hφ ,x ) = α (h,hφ , x ) b( x )∫α (h,hφ , x ) b( x )dx

T a LR
=
Tp

ωc

Tp

ωc
(2.24)=

ω<1
ωcWhen the parameter ωc is equal to one we hence have noncompensated root water uptake, and21

when ωc is equal to zero we obtain fully compensated uptake.
2.3. The Unsaturated Soil Hydraulic Properties
2.3.1. Uniform Water Flow System
The unsaturated soil hydraulic properties, θ(h) and K(h), in (2.1) are in general highly
nonlinear functions of the pressure head. HYDRUS permits the use of five different analyticalmodels for the hydraulic properties [Brooks and Corey, 1964; van Genuchten, 1980; Vogel andCíslerová, 1988; Kosugi, 1996; and Durner, 1994].
The soil water retention, θ(h), and hydraulic conductivity, K(h), functions according to
Brooks and Corey [1964] are given by⎧ |α h |- n⎪
Se = ⎨⎪ 1⎩

h < -1/α(2.25)h ≥ -1/α

K = K s Se2 / n + l + 2(2.26)

respectively, where Se is effective saturation:Se =θ - θrθs - θr(2.27)

in which θr and θs denote the residual and saturated water contents, respectively; Ks is thesaturated hydraulic conductivity, α is the inverse of the air-entry value (or bubbling pressure), n
is a pore-size distribution index, and l is a pore-connectivity parameter assumed to be 2.0 in theoriginal study of Brooks and Corey [1964]. The parameters α, n and l in HYDRUS areconsidered to be empirical coefficients affecting the shape of the hydraulic functions.
HYDRUS also implements the soil-hydraulic functions of van Genuchten [1980] who
used the statistical pore-size distribution model of Mualem [1976] to obtain a predictive equation
for the unsaturated hydraulic conductivity function in terms of soil water retention parameters.

22

The expressions of van Genuchten [1980] are given byθs - θr⎧⎪ θ r + [1+ |α h|n ]m⎪⎪
θ (h) = ⎨
⎪ θ
s
⎪
⎪⎩

h<0
(2.28)h≥02

K ( h ) = K s Sel [1- (1- Se1 / m ) m ](2.29)

m = 1 - 1/n , n > 1(2.30)

where

The above equations contain five independent parameters: θr, θs, α, n, and Ks. The poreconnectivity parameter l in the hydraulic conductivity function was estimated [Mualem, 1976] tobe about 0.5 as an average for many soils.
A third set of hydraulic equations implemented in HYDRUS are those by Vogel andCíslerová [1988] who modified the equations of van Genuchten [1980] to add flexibility in thedescription of the hydraulic properties near saturation. The soil water retention, θ (h), andhydraulic conductivity, K(h), functions of Vogel and Císlerová [1988] are given by (Fig. 2.3)

θm - θa⎧m⎪ θa +(1+ |α h|n )⎪⎪θ (h) = ⎨
⎪
θs
⎪
⎪⎩
and

23

h < hs(2.31)h ≥ hs⎧K s K r (h)h ≤ hk⎪⎪⎪⎪( h - hk )( K s - K k )K (h) = ⎨ Kk +hk < h < hs
hs - hk⎪⎪⎪Ksh ≥ hs⎪⎩(2.32)

respectively, whereK ⎛ S ⎞ ⎡ F (θ r ) - F (θ ) ⎤Kr = k ⎜ e ⎟ ⎢⎥K s ⎝ Sek ⎠ ⎣ F (θ r ) - F (θ kr ) ⎦⎡ ⎛ θ - θ ⎞1 / m ⎤aF (θ ) = ⎢1- ⎜⎟ ⎥θθ⎢⎣ ⎝ m a ⎠ ⎥⎦Sek =

θk - θrθs - θr2(2.33)

m(2.34)(2.35)

The above equations allow for a non-zero minimum capillary height, hs, by replacing theparameter θs in van Genuchten's retention function by a fictitious (extrapolated) parameter θmslightly larger than θs as shown in Fig. 2.4. While this change from θs to θm has little or no effecton the retention curve, the effect on the shape and value of the hydraulic conductivity function
can be considerable, especially for fine-textured soils when n is relatively small (e.g., 1.0 < n <1.3). To increase the flexibility of the analytical expressions, the parameter θr in the retentionfunction was replaced by the fictitious (extrapolated) parameter θa≤θr. The approach maintains
the physical meaning of θr and θs as measurable quantities. Equation (2.33) assumes that thepredicted hydraulic conductivity function is matched to a measured value of the hydraulicconductivity, Kk=K(θk), at some water content, θk, less that or equal to the saturated watercontent, i.e., θk≤θs and Kk≤Ks [Vogel and Císlerová, 1988; Luckner et al., 1989]. Inspection of(2.31) through (2.34) shows that the hydraulic characteristics contain 9 unknown parameters: θr ,
θs , θa , θm , α, n, Ks , Kk , and θk . When θa=θr, θm=θk=θs and Kk=Ks, the soil hydraulic functions
of Vogel and Císlerová [1988] reduce to the original expressions of van Genuchten [1980].

24

Fig. 2.5. Schematics of the soil water retention (a) and hydraulic conductivity (b) functions
as given by equations (2.31) and (2.32), respectively.

Vogel and Císlerová [1988] model (2.31) in which θm is calculated so that the air entry
value hs [L] is equal to – 2 cm is implemented as “van Genuchten-Mualem with air-entry valueof -2 cm”. We recommend that this model be used for heavy textured soils (e.g., clays).
Version 4.0 of HYDRUS allows the soil hydraulic properties to be defined also according
to Kosugi [1996], who suggested the following lognormal distribution model for Se(h):⎧1⎪⎧ ln ( h / α ) ⎪⎫⎬2n ⎭⎪

θ − θ r ⎪ erfc ⎨
=⎨2
Se =
⎩⎪
θs − θr ⎪⎩1( h < 0)(2.36)( h ≥ 0)Application of Mualem's pore-size distribution model [Mualem, 1976] now leads to thefollowing hydraulic conductivity function:

25

2
⎧
⎧
⎡ ln ( h / α ) n ⎤ ⎫⎪l ⎪1+
erfc ⎢⎪⎪ K S⎥⎬K = ⎨ s e ⎨⎩⎪ 22n2 ⎦ ⎭⎪⎣⎪
Ks⎪⎩(h < 0)(2.37)( h ≥ 0)Note that in this manual we use the symbol α instead of h0 and n instead of σ as used in Kosugi[1996].
Durner [1994] divided the porous medium into two (or more) overlapping regions andsuggested to use for each of these regions a van Genuchten-Mualem type function [vanGenuchten, 1980] of the soil hydraulic properties. Linear superposition of the functions for eachparticular region gives then the functions for the composite multimodal pore system [Durner etal., 1999]:
S e = w1[1 + (α1h ) 1 ] 1 + w2 [1 + (α 2 h ) 2 ] 2
n

-m

n

-m

(2.38)

Combining this retention model with Mualem’s [1976] pore-size distribution model leads nowto:K ( S e) = K s(

w1Se1 + w2 Se2)(lw1α1[1 - (1 - S 1e1/m1 ) 1 ] + w2α 2 [1 - (1 - S 1e2/m2 ) 2 ]m( w1α1 + w2α 2 )

m

2

)

2

(2.39)

where wi are the weighting factors for the two overlapping regions, and αi, ni, mi (=1-1/ni), and l
are empirical parameters of the separate hydraulic functions (i=1,2).
An example of composite retention and hydraulic conductivity functions for two
overlapping porous media is shown in Figure 2.5. Note that the pressure head axes are on a log
scale, which causes the near-saturated values to be significantly enlarged. The fracture domain inthis example represents only 2.5% of the entire pore space, but accounts for almost 90% of thehydraulic conductivity close to saturation. Curves similar to those in Figure 2.5 have been usedalso for fractured rock by Peters and Klavetter [1988], Pruess and Wang [1987], and Flint et al.
[2001], among others.

26

0.6
Total

0.5

Water Content [-]Log(Conductivity [cm/days])0

MatrixFracture

0.4
0.3
0.2
0.1
0

-2
Total
Matrix-4

Fracture
-6

-8

-10
-1

0

1

2

3

4

5

-1

Log(|Pressure Head [cm]|)

0

1

2

3

4

5

Log(|Pressure Head [cm]|)

Fig. 2.6. Example of composite retention (left) and hydraulic conductivity (right) functions (θr=0.00,
θs=0.50, α1=0.01 cm-1, n1=1.50, l=0.5, Ks=1 cm d-1, w1=0.975, w2=0.025, α2=1.00 cm-1, n2=5.00).

2.3.2. Uniform Water Flow and Vapor Transport System

The thermal hydraulic conductivity function, KLT, in (2.3) may be defined as (e.g.,Noborio et al. [1996ab], Saito et al. [2006]):⎛
1 dγ ⎞K LT (T ) = K Lh ( h ) ⎜ hGwT⎟
γ 0 dT ⎠⎝(2.40)

where GwT is the gain factor (7 for sand), which quantifies the temperature dependence of thesoil water retention curve [Nimmo and Miller, 1986], γ is the surface tension of soil water [MT-2,
Jm-2], and γ0 is the surface tension at 25oC (= 71.89 g s-2). The temperature dependence of γ is asfollows (γ is in [g s-2] and T in [oC]):

γ = 75.6 − 0.1425T − 2.38 ⋅ 10 −4 T 2(2.41)The isothermal, Kvh, and thermal, KvT, vapor hydraulic conductivities are described as27

(e.g., Nassar and Horton [1989], Noborio et al. [1996b], Fayer [2000]):

K vh =
K vT =

Dv

ρ vs

Dv

ηe H r

ρw
ρw

Mg
Hr
RuT

(2.42)d ρ vsdT(2.43)

where D is the vapor diffusivity in soil [L2T-1], ρvs is the saturated vapor density [ML-3], M is themolecular weight of water [M mol-1] (=0.018015 kg mol-1), g is the gravitational acceleration[LT-2] (=9.81 m s-2), Ru is the universal gas constant [J mol-1K-1, ML2T-2mol-1K-1] (=8.314 J mol1K-1), ηe is the enhancement factor [-] [Cass et al., 1984], and Hr is the relative humidity [-]. The

vapor diffusivity, Dv, in soil is defined as:
Dv = τ g av Da(2.44)

where av is the air-filled porosity [-], τg is the tortuosity factor as defined by Millington and
Quirk (1961) and Da is the diffusivity of water vapor in air [L2T-1] at temperature T [K]:⎛ T ⎞Da = 2.12 ⋅ 10 ⎜⎟⎝ 273.15 ⎠2−5(2.45)The saturated vapor density, ρvs [ML-3] (in kg m-3), as a function of temperature may be
expressed as:
6014.79⎞⎛exp⎜ 31.3716 −− 7.92495 ⋅ 10 −3 T ⎟T⎠⎝
ρ vs = 10 −3Tand the relative humidity, Hr [Philip and de Vries, 1957]:28(2.46)

⎡ hMg ⎤H r = exp ⎢ −⎣ RT ⎥⎦(2.47)

When the liquid and vapor phases of water in soil pores are in equilibrium, the vapor
density of the soil can be expressed as the product of the saturated vapor density and the relative
humidity:ρ v = ρ vs H r(2.48)The volumetric water vapor content, θv, is given here in terms of an equivalent water content[L3L-3] as follows:θv = ρ v

θs − θθ −θ
= ρ vs H r s
ρw
ρw(2.49)The HYDRUS-1D code uses the enhancement factor, ηe, to describe increases in thethermal vapor flux as a result of liquid-island and increased temperature gradients in the airphase [Philip and de Vries, 1957]. The enhancement factor as first formulated by Cass et al.
[1984] may be expressed as [Campbell, 1985]:
⎧

4

⎫

θ
2.6 ⎞ θ ⎤ ⎪⎪ ⎡⎛ηe = 9.5 + 3 − 8.5exp ⎨ − ⎢⎜ 1 +⎟⎟ ⎥ ⎬
⎜
θs
f
⎢
c
⎪ ⎣⎝
⎠ θ s ⎥⎦ ⎪⎩(2.50)⎭

where fc is the mass fraction of clay in the soil [-].
2.4. Scaling in the Soil Hydraulic Functions
HYDRUS implements a scaling procedure designed to simplify the description of thespatial variability in the unsaturated soil hydraulic properties in the flow domain. The code29

assumes that variability in the hydraulic properties of a given soil profile can be approximated bymeans of a set of linear scaling transformations which relate the soil hydraulic characteristics
θ(h) and K(h) of the individual soil layers to reference characteristics θ*(h*) and K*(h*). The
technique is based on the similar media concept introduced by Miller and Miller [1956] forporous media which differ only in the scale of their internal geometry. The concept was extended
by Simmons et al. [1979] to materials which differ in morphological properties, but which
exhibit 'scale-similar' soil hydraulic functions. Three independent scaling factors are embodied in
HYDRUS. These three scaling parameters may be used to define a linear model of the actual
spatial variability in the soil hydraulic properties as follows [Vogel et al., 1991]:
K ( h ) = α K K * ( h* )

θ ( h ) = θ r + αθ [θ * ( h* ) - θ *r ](2.51)

h = α h h*
in which, for the most general case, αθ, αh and αK are mutually independent scaling factors forthe water content, the pressure head and the hydraulic conductivity, respectively. Less generalscaling methods arise by invoking certain relationships between αθ, αh and/or αK. For example,
the original Miller-Miller scaling procedure is obtained by assuming αθ =1 (with θr* = θr), andαK=αh-2. A detailed discussion of the scaling relationships given by (2.51), and their application
to the hydraulic description of heterogeneous soil profiles, is given by Vogel et al. [1991].
2.5. Temperature Dependence of the Soil Hydraulic Functions
A similar scaling technique as described above is used in HYDRUS to express thetemperature dependence of the soil hydraulic functions. Based on capillary theory that assumes
that the influence of temperature on the soil water pressure head can be quantitatively predictedfrom the influence of temperature on surface tension, Philip and de Vries [1957] derived thefollowing equation
dh h dσ=
dT σ dT30(2.52)

where T is temperature [K] and σ is the surface tension at the air-water interface [MT-2]. From
(2.52) it follows that

hT =

σT
href = α h* href
σ ref(2.53)

where hT and href (σT and σref) are pressure heads (surface tensions) at temperature T andreference temperature Tref , respectively; and αh* is the temperature scaling factor for the pressure
head.
Following Constantz [1982], the temperature dependence of the hydraulic conductivity
can be expressed as

KT (θ ) =μ ref ρ T
K ref (θ ) = α K* K ref (θ )μ T ρ ref(2.54)

where Kref and KT denote hydraulic conductivities at the reference temperature Tref and soil
temperature T, respectively; μref and μT (ρref and ρT) represent the dynamic viscosity [ML-1T-1](density of soil water [ML-3]) at temperatures Tref and T, respectively; and αK* is the temperaturescaling factor for the hydraulic conductivity.
2.6. Hysteresis in the Soil Hydraulic Properties
Applications of unsaturated flow models often assume unique, single-valued (nonhysteretic) functions for θ(h) and K(h) to characterize the hydraulic properties at a certain pointin the soil profile. While such a simplification may be acceptable for many flow simulations,other cases require a more realistic description involving hysteresis in the soil hydraulicproperties. The HYDRUS code incorporates hysteresis by using the empirical model introducedby Scott et al. [1983]. This model was also employed by Kool and Parker [1987], who modified
the formulation to account for air entrapment. Following Vogel et al. [1996], the present versionof HYDRUS further extends the model of Kool and Parker by considering also hysteresis in thehydraulic conductivity function.
The adopted procedure for modeling hysteresis in the retention function requires that31both the main drying and main wetting curves are known (Fig. 2.4). These two curves aredescribed with (2.31) using the parameter vectors (θrd, θsd, θmd, αd, nd) and (θrw, θsw, θmw, αw, nw),
respectively, where the subscripts d and w indicate wetting and drying, respectively. The
following restrictions are expected to hold in most practical applications:θ rd = θ rw , α d ≤ α w(2.55)We also invoke the often assumed restrictionnd = n w(2.56)

If data are lacking, one may use αw = 2αd as a reasonable first approximation [Kool and Parker,
1987; Nielsen and Luckner, 1992]. We further assume

Fig. 2.7. Example of a water retention curve showing hysteresis. Shown are theboundary wetting curve, θw(h), and the boundary drying curve, θd(h).

32θ mw = θ r +θ sw - θ r d(θ - θ )θ sd - θ r m r(2.57)

so that the parameters θs and α are the only independent parameters describing hysteresis in theretention function. According to the hysteresis model, drying scanning curves are scaled fromthe main drying curve, and wetting scanning curves from the main wetting curve. The scaling
factors for the drying scanning curves can be obtained by considering the main drying curve as areference curve in scaling equation (2.51) (keeping αh = 1 to scale only in the water content
direction), i.e.:θ ( h ) = θ r' + αθ [θ d ( h ) - θ rd ](2.58)

and forcing each scanning curve, θ(h), to pass through the point (θΔ, hΔ) characterizing the latest
reversal from wetting to drying. Substituting this reversal point into (2.57), and assuming thatθr=θrd, leads toαθ =θΔ - θrθ ( hΔ ) - θ r
d(2.59)

Note that the scaling procedure results in a fictitious value of the parameter θs’ for the drying
scanning curve (this parameter may be located outside of the main hysteresis loop). The scaling
relationship is similarly for the wetting scanning curvesθ ( h ) = θ r' + αθ [θ w ( h ) - θ r ](2.60)

in which the fictitious parameter θr’ is now used (again possibly scaled outside of the mainloop). The scaling factor αθ for a particular scanning curve can be obtained by substituting thereversal point (θΔ, hΔ) and the full saturation point (θs, 0) into (2.59), and subtracting the two
resulting equations to eliminate θr’ to give

33

αθ =

θΔ - θsθ ( hΔ ) - θ sw

(2.61)w

The parameter θr’ is subsequently determined from (2.59) as θr’ = θs - αθ(θsw - θr). If the main
hysteresis loop is not closed at saturation, the water content at saturation for a particular wetting
scanning curve is evaluated using the empirical relationship of Aziz and Settari [1979]θ s = θ sd -θ sd - θ Δ,
1 + R (θ sd - θ Δ )

R=

1
1
- d d
w
θ - θs θs - θrds

(2.62)An analogous hysteretic procedure can be applied to the unsaturated hydraulicconductivity function K(h). The main branches Kd(h) and Kw(h) of the hysteresis loop arecharacterized by the same set of parameters as the corresponding retention curves θd(h) andθw(h), and by the saturated conductivities Ksd and Ksw according to Eq. (2.28). For drying
scanning curves we obtain from (2.51)
K (h ) = α K K d (h )(2. 63)

From knowledge of the reversal point (hΔ, KΔ) we obtain

αK =

KΔ
K ( hΔ )d(2.64)For a wetting scanning curve we have now
K ( h ) = K r' + α K K w ( h )(2.65)

where Kr’ is a fictitious parameter. Substituting the reversal point (hΔ, KΔ) and the saturation
point (0, Ks) into (2.64) and solving for αK yields

34

αK =

KΔ - Ks
K ( hΔ ) - K sw
w

(2.66)

The fictitious conductivity parameter Kr’ may be obtained from (2.64) as Kr’ = Ks - αK Ksw. If themain hysteresis loop is not closed at saturation, the hydraulic conductivity at saturation for awetting scanning curve is evaluated using equations similar to (2.61), i.e.,
K s = K sd -K sd - K Δ
1
1, R= d- ddw
1 + R ( Ks - KΔ )Ks - Ks Ks(2.67)

While relatively simple to implement, the above model has been found to suffer from aso-called pumping effect, in which the hysteresis loops can move to physically unrealistic partsof the retention function. As an alternative, we also incorporated in HYDRUS the hysteresis
model of Lenhard et al. [1991] and Lenhard and Parker [1992] that eliminates pumping bykeeping track of historical reversal points. We greatly acknowledge the help of Robert Lenhard
in this effort.
2.7. Initial and Boundary Conditions
The solution of Eq. (2.1) requires knowledge of the initial distribution of the pressure
head within the flow domain:h( x, t ) = hi ( x )t = t0(2.68)

where hi [L] is a prescribed function of x, and to is the time when the simulation begins.
2.7.1. System-Independent Boundary Conditions
One of the following boundary conditions must be specified at the soil surface (x=L) or at
the bottom of the soil profile (x=0):35

h( x, t ) = h0 (t )at x = 0 or x = L⎛ ∂h⎞- K ⎜ + cos α ⎟ = q0 (t ) at x = 0 or x = L
⎝ ∂x
⎠
∂h
=0
at x = 0∂x(2.69)

where h0 [L] and q0 [LT-1] are the prescribed values of the pressure head and the soil water flux
at the boundary, respectively.
2.7.2. System-Dependent Boundary Conditions
In addition to the system-independent boundary conditions given by (2.68), we consider
two system-dependent boundary conditions, which cannot be defined a priori. One of these
involves the soil-air interface, which is exposed to atmospheric conditions. The potential fluid
flux across this interface is controlled exclusively by external conditions. However, the actual
flux depends also on the prevailing (transient) soil moisture conditions near the surface. The soil
surface boundary condition may change from a prescribed flux to a prescribed head typecondition (and vice-versa). The numerical solution of (2.1) is obtained by limiting the absolutevalue of the surface flux by the following two conditions [Neuman et al., 1974]:∂h- K |≤ E∂x

at x = L(2.70)hA ≤ h ≤ hS

at x = L(2.71)|- K
andwhere E is the maximum potential rate of infiltration or evaporation under the current
atmospheric conditions [LT-1], and hA and hS are, respectively, minimum and maximum pressure
head at the soil surface allowed under the prevailing soil conditions [L]. The value for hA isdetermined from the equilibrium conditions between soil water and atmospheric water vapor,whereas hS is usually set equal to zero; if positive, hS represents a small layer of water ponded
which can form on top of the soil surface during heavy rains before initiation of runoff. One
options in HYDRUS is to assume that any excess water on the soil surface above zero will be

36immediately removed. When one of the end points of (2.69) is reached, a prescribed headboundary condition will be used to calculate the actual surface flux. Methods of calculating E
and hA on the basis of atmospheric data have been discussed by Feddes et al. [1974]. The
minimum pressure head at the soil surface allowed under the prevailing soil conditions, hA [L],can be calculated from the air humidity, Hr [-], as follows:⎡ h Mg ⎤H r = exp ⎢ − A⎣ RT ⎥⎦
RT
hA = −ln( H r )Mg

(2.72)

where M is the molecular weight of water [M mol-1] (=0.018015 kg mol-1), g is the gravitational
acceleration [LT-2], (=9.81 m s-2), and R is the gas constant [J mol-1 K-1] (=8.314 J mol-1K-1)[ML2T-2mol-1K-1].
Variations in potential evaporation and transpiration during the day can be generated
with HYDRUS-1D using the assumptions that hourly values between 0-6 a.m. and 18-24 p.m.
represent 1% of the total daily value and that a sinusoidal shape is followed during the rest of theday [Fayer, 2000], i.e.,
Tp (t ) = 0.24Tp

t < 0.264d, t > 0.736d⎛ 2π t π ⎞Tp (t ) = 2.75Tp sin ⎜− ⎟ t ∈ (0.264d, 0.736d)⎝ 1day 2 ⎠(2.73)

where Tp is the daily value of potential transpiration (or evaporation). Similarly, variation of
precipitation can be approximated using a cosine function as follows:⎛⎛ 2π t⎞⎞−π ⎟⎟P(t ) = P ⎜ 1 + cos ⎜
⎝ Δt
⎠⎠
⎝

(2.74)

where P is the average precipitation rate of duration Δt.
Potential evaporation and transpiration fluxes can also be calculated from potential

37

evapotranspiration using Beer’s law that partitions the solar radiation component of the energy
budget via interception by the canopy [Ritchie, 1972] as follows:
Tp = ETp (1 − e − k LAI ) = ETp SCF
E p = ETp e − k LAI = ETp (1 − SCF )(2.75)

where ETp, Tp, and Ep are potential evapotranspiration, transpiration and evaporation fluxes [LT-1],
respectively, LAI is the leaf area index [-], SCF is the soil cover fraction [-], and k is a constantgoverning the radiation extinction by the canopy [-] as a function of sun angle, the distribution ofplants, and the arrangement of leaves (between 0.5-0.75).
Another option in HYDRUS is to permit water to build up on the surface. If surface
ponding is expected, a "surface reservoir" boundary condition of the type [Mls, 1982]
dh
⎛ ∂h
⎞
- K ⎜ + cos α ⎟ = q0 (t ) dt
⎝ ∂z⎠

at x = L(2.76)

may be applied. The flux q0 in this equation is the net infiltration rate, i.e., the difference
between precipitation and evaporation. Equation (2.75) shows that the height h(L,t) of thesurface water layer increases due to precipitation, and reduces because of infiltration andevaporation.
A third system-dependent type boundary condition considered in HYDRUS is a seepage
face at the bottom of the soil profile through which water can leave the saturated part of the flow
domain. This type of boundary condition assumes that a zero-flux boundary condition applies aslong as the local pressure head at the bottom of the soil profile (x = 0) is negative (or below somespecified value hSeep). However, a zero pressure head (or the specified value hSeep) will be
used as soon as the bottom of the profile becomes saturated. This type of boundary condition
often applies to finite lysimeters that are allowed to drain under gravity.
Another system-dependent boundary condition, which can be used at the bottom of thesoil profile involves flow to a horizontal subsurface tile drains. HYDRUS-1D permits two
different analytical solutions to be used to approximate tile drainage. The first solution is known
as the Hooghoudt equation [Houghoudt, 1940; van Hoorn, 1998; van Dam et al., 1997]:

38

qdrain =8K hBot Deq hdr + 4 K hTop h 2dr
2
dr

L

+

hdr

γ entr(2.77)

where qdrain is the drain discharge rate per unit surface area [LT-1], KhTop and KhBot are thehorizontal saturated hydraulic conductivities above and below the drain system [LT-1],
respectively; hdr is the watertable height above the drain at the midpoint between the drains, i.e.,the hydraulic head needed for calculating subsurface flow into the drains [L], Ldr is the drain
spacing [L], γentr is the entrance resistance into the drains [T], and Deq is the equivalent depth [L].
The equivalent depth as introduced by Hooghoudt is a function of Ldr, the depth to an impervious
layer, and the drain radius. HYDRUS-1D adopts a numerical scheme as used in the SWAT
model [van Dam et al., 1997]. When the drains are located in a homogeneous soil profile justabove an impervious layer, (2.76) simplifies as follows:
4 K h hdr2
h+ dr
qdrain =2
γ entr
Ldr(2.78)

where Kh is the horizontal saturated hydraulic conductivity [LT-1].
The second analytical solution in HYDRUS was derived by Ernst [Ernst, 1962; van
Hoorn, 1997] for a layered soil profile:qdrain =K v ' hdr 8 ∑( K D ) h hdr
hπ K r ' hdr
+
+
+ dr
2
α
Dv
Ldr
Ldr ln dr D r γ entr
u(2.79)

where Kv' and Kr' are the saturated hydraulic conductivities in the layers with vertical and radialflow [LT-1], respectively; Dv and Dr are the thicknesses of the layers in which vertical and radialflow is considered [L], respectively; ∑(KD)h is the transmissivity of the soil layers throughwhich horizontal flow is considered [L2T-1], u is the wet perimeter of the drain [L], and adr is ageometry factor [-] for radial flow whose value depends on the flow condition (see Table 12.2).
Still another system-dependent lower boundary condition may be imposed in cases where39

a functional relationship between the position of the water table and drainage from the soil
profile can be established. One possible relationship of this type is discussed in Section 10.3.
HYDRUS additionally allows snow accumulation at the atmospheric boundary when air
temperatures are entered. The code in that case assumes that all precipitation is in the form ofsnow when the air temperature is below -2oC, all precipitation is in the form of liquid when the
air temperature is above +2oC, and that a linear transition exists between these two limiting
temperatures at -2 and 2 oC [Jarvis, 1989]. The code further assumes that when the airtemperature is above zero, the existing snow layer (if it exists) melts proportionally to the airtemperature.
2.7.3. Penman-Monteith Combination Equation
Potential evapotranspiration may be calculated in HYDRUS-1D using either the FAO
recommended Penman-Monteith combination equation for evapotranspiration (ET0) [Monteith,
1981; Monteith and Unsworth, 1990; FAO, 1990] or the Hargreaves equation [Hargreaves, 1994;
Jensen et al., 1997]. With the Penman-Monteith approach, ET0 is determined using a combination
equation that combines the radiation and aerodynamic terms as follows [FAO, 1990]:

ET0 = ETrad + ETaero =ρ c p (ea - ed ) / ra ⎤1 ⎡ Δ( Rn - G )+⎢⎥λ ⎣ Δ + γ (1 + rc / ra ) Δ + γ (1 + rc / ra ) ⎦(2.80)

where ET0 is the evapotranspiration rate [mm d-1], ETrad is the radiation term [mm d-1], ETaero is the
aerodynamic term [mm d-1], λ is the latent heat of vaporization [MJ kg-1], Rn is net radiation at
surface [MJ m-2d-1], G is the soil heat flux [MJ m-2d-1], ρ is the atmospheric density [kg m-3], cp isthe specific heat of moist air [i.e., 1.013 kJ kg-1 oC-1], (ea-ed) is the vapor pressure deficit [kPa], ea isthe saturation vapor pressure at temperature T [kPa], ed is the actual vapor pressure [kPa], rc is thecrop canopy resistance [s m-1], and ra is the aerodynamic resistance [s m-1]. The slope of the vapor
pressure curve, Δ [kPa oC-1] [Tetens, 1930; Murray 1967], and the psychrometric constant, γ [kPao -1
C ] [Brunt, 1952], are defined as follows:Δ=4098 ea(T + 237.3)

402(2.81)

γ=

cp P

ελ

*10−3 = 0.00163

P

λ(2.82)

respectively, where T is the average air temperature [oC], P is the atmospheric pressure [kPa], ε isthe ratio of the molecular weights of water vapor and dry air (i.e., 0.622), and λ is the latent heat
[MJ kg-1].
When no measured radiation data are available, the net radiation can be estimated as:
Rn = Rns − Rnln⎞⎛Rns = (1 − α ) ⎜ as + bs ⎟ RaN⎠⎝
4
4
Tmax + Tmin()
Rnl = f ε ′σ2
G = 0.14(Tmonth , n - Tmonth , n −1 ) ≈ 0(2.83)(2.84)(2.85)(2.86)

where Rn is net radiation [MJ m-2d-1], Rns is net shortwave radiation [MJ m-2d-1], Rnl is net longwave
radiation [MJ m-2d-1], Ra is extraterrestrial radiation [MJ m-2d-1], α is the albedo or the canopy
reflection coefficient (i.e., 0.23), as and bs are parameters for the fraction of radiation (i.e., as =0.25, bs = 0.5), n/N is the relative sunshine fraction [-], f is the cloudiness factor [-], ε’ is the net
emissivity, σ is the Stefan-Boltzmann constant (4.90×10-9 MJ m-2K-4d-1), Tmax and Tmin are themaximum and minimum daily air temperatures [K], ed is the actual vapor pressure [kPa], and G is
the soil heat flux [MJ m-2d-1]. Further details can be found in FAO [1990] and in Appendix A.
2.7.4. Hargreaves Formula
The potential evapotranspiration can also be evaluated using the much simpler
Hargreaves formula [e.g., Hargreaves, 1994; Jensen et al., 1997]:
ETp = 0.0023Ra (Tm + 17.8 ) TR

(2.87)

where Ra is extraterrestrial radiation in the same units as ETp [e.g., mm d-1 or J m-2s-1], Tm is thedaily mean air temperature, computed as an average of the maximum and minimum air

41

temperatures [oC], TR is the temperature range between the mean daily maximum and minimum airtemperatures [oC]. Extraterrestrial radiation, Ra [J m-2s-1], can be calculated as follows

Ra =

Gsc

π

d r (ωs sin ϕ sin δ + cos ϕ cos δ sin ωs )(2.88)

where Gsc is the solar constant [J m-2s-1] (1360 W m-2), ϕ is the site latitude [rad], ωs is the sunset
hour angle [rad], dr is the relative distance between Earth and Sun [-], and δ is the solar declination[rad]. The last three variables are calculated as follows:ωs = arccos(- tan ϕ tan δ )⎛ 2π ⎞

d r = 1 + 0.033 cos ⎜J⎟⎝ 365 ⎠⎛ 2π⎞
J -1.39 ⎟δ = 0.409 sin ⎜⎝ 365⎠(2.89)(2.90)(2.91)

where J is the number of the day in the year [-].
2.7.5. Surface Energy Balance Equation
Surface precipitation, irrigation, evaporation, and heat fluxes are used as boundary
conditions for the coupled liquid water and water vapor flow and heat transport in field soils.
Surface water and heat fluxes for bare soils can be calculated from the surface energy balance
equation using available meteorological models. This section presents descriptions of eachcompontent of the surface energy balance equation, while its parameterization is presented in the
Appendix B. The energy balance equation at the bare soil surface is expressed using net radiation,
sensible heat flux, latent heat flux, and soil heat flux [e.g., van Bavel and Hillel, 1976; Saito et al.,2006].
Rn − H − λ E − G = 0(2.92)

where Rn is the net radiation [Wm-2], H is the sensible heat flux density [Wm-2], λE is the latent42

heat flux density [Wm-2], λ is the latent heat of vaporiazation [J kg-1], E is the evaporation rate
[kg m-2s-1], and G is the surface heat flux density [Wm-2]. While Rn and G are positive
downward, λE and H are positive upward.
Radiation term
Net radiation, Rn, is defined as [e.g., Campbell, 1977; Monteith and Unsworth, 1990;
Sharratt et al., 1992]:{

Rn = Rns + Rnl = (1 − α ) Rs + ⎡⎣(1 − 0.84c ) ⋅ ε a + 0.84c ⎤⎦ σ Ta4 − ε sσ Ts4}(2.93)

where Rns is the net shortwave radiation [Wm-2], Rnl is the net longwave radiation [Wm-2], α is thesurface albedo [-], Rs is the incoming shortwave solar radiation [Wm-2] εa is the atmosphericemissivity of clear sky [-], εs is the soil surface emissivity [-] representing the reflection of the
longwave radiation at the soil surface, c is the fraction of cloud cover [-], σ is Stefan-Boltzmann
constant (= 4.90×10-9 MJm-2K-4d-1), Ta is the air temperature [K], and Ts is the soil surface
temperature [K].

Latent heat flux term
The latent heat flux density is described as the multiplication of the latent heat ofvaporization, λ, and the evaporation rate, E. Evaporation from the soil surface, in general, iscontrolled by atmospheric conditions, surface moisture, and moisture transport in the soil. A
model that accounts for all of these factors can be expressed as [Camillo and Gurney, 1986]:E=ρs − ρara + rs(2.94)

where ρs is the water vapor density at the soil surface [kg m-3] (Eq. (2.48)), ρa is the atmosphericvapor density [kg m-3], ra is the aerodynamic resistance to water vapour flow [s m-1], and rs is thesoil surface resistance to water vapor flow [s m-1].
Sensible heat flux term
The sensible heat flux is defined as [e.g., van Bavel and Hillel, 1976]:43

H = CaTs − Tarh(2.95)

where Ta is the air temperature [K], Ts is the soil surface temperature [K], Ca is the volumetric heat
capacity of air (= 1200) [Jm-3K-1], and rh is the aerodynamic resistance to heat transfer [sm-1]. Since
the aerodynamic resistance to heat flow is usually very close to the aerodynamic resistance tovapor flow [e.g., Aluwihare and Watanabe, 2003; van Bavel and Hillel, 1976], the sensible heat
flux is calculated using rh equal to ra.

2.8. Water Mass Transfer
The mass transfer rate, Γw, in (2.5) for water between the fracture and matrix regions inseveral dual-porosity studies (e.g. Phillip [1968]; Šimůnek et al. [2003]) has been assumed to be
proportional to the difference in effective saturations of the two regions using the first-order rateequation:

Γw =

∂θim
= ω ⎡⎣ Sem − Seim ⎤⎦∂t(2.96)

where θim is the matrix water content, ω is a first-order rate coefficient (T-1), and Sem and Seim are
effective fluid saturations of the mobile (fracture) and immobile (matrix) regions, respectively.
Equation (2.96) assumes that the mass transfer rate is proportional to the difference in effective
water contents, rather than pressure heads [Gerke and van Genuchten, 1993b], which should
provide a more realistic description of the exchange rate between the fracture and matrix regions.
An inherent assumption of (2.96) is that the water retention properties of the matrix and thefracture domains are identical. For this reason, equation (2.96) must be used with some caution
and probably only for dual-porosity models. The approach has nevertheless been usedsuccessfully in multiple studies (e.g., Köhne et al. [2004, 2005]).
An important advantage of (2.96) is the fact that the dual-porosity model based on thismass transfer equation requires significantly fewer parameters since one does not need to know
the retention function for the matrix region explicitly, but only its residual and saturated water44

contents. Coupling (2.96) with a dual-porosity nonequilibrium flow model leads to the usual soil
hydraulic parameters needed for the equilibrium model, two additional parameters characterizing
the matrix region (i.e. its residual, θrim, and saturated, θsim, water contents), and the first-ordermass transfer coefficient ω. By additionally assuming that the residual water content of thefracture region is equal to zero (and hence that residual water is present only in the immobile
region), one could further decrease the number of model parameters.
When the rate of exchange of water between the fracture and matrix regions is assumed
to be proportional to the difference in pressure heads between the two pore regions [Gerke andvan Genuchten, 1993a], the coupling term, Γw, becomes:

Γ w = α w (h f - hm )(2.97)

in which αw is a first-order mass transfer coefficient [L-1T-1]. Since pressure heads are now
needed for both regions, this approach requires retention curves for both pore regions. For
porous media with well-defined geometries, the first-order mass transfer coefficient, αw, can be
defined as follows [Gerke and van Genuchten, 1993b]:

αw =

β
d

2

K aγ w(2.98)

where d is an effective ‘diffusion’ pathlength (i.e. half the aggregate width or half the fracture
spacing) [L], β is a shape factor that depends on the geometry [-], and γw is a scaling factor(=0.4) obtained by matching the results of the first-order approach at the half-time level of the
cumulative infiltration curve to the numerical solution of the horizontal infiltration equation(Gerke and van Genuchten, 1993b). Gerke and van Genuchten [1996] evaluated the effective
hydraulic conductivity Ka [LT-1] of the fracture-matrix interface using a simple arithmetic
average involving both hf and hm as follows

K a ( h ) = 0.5 ⎡⎣ K a ( h f ) + K a ( hm ) ⎤⎦(2.99)The use of (2.98) implies that the medium contains geometrically well-defined rectangular orother types of macropores or fractures (e.g. Edwards et al. [1979], van Genuchten and Dalton[1986], and Gerke and van Genuchten [1996]). While geometrically based models are45

conceptually attractive, they may be too difficult to use for field applications, partly because
structured soils and rocks usually contain mixtures of aggregates and matrix blocks of various
sizes and shapes, but also because the parameters in (2.98) may not be identifiable. Hence, rather
than using (2.98) directly, one could also lump β, d, and γw into one effective hydraulicconductivity Ka* of the fracture-matrix interface to give

α w = K a* (h )(2.100)in which case Ka* can be used as a calibration parameter (this variable is an input parameter to
HYDRUS).

46

3. NONEQUILIBRIUM TRANSPORT OF SOLUTES INVOLVED IN SEQUENTIAL
FIRST-ORDER DECAY REACTIONS
3.1. Governing Solute Transport Equations
We assume that solutes can exist in all three phases (liquid, solid, and gaseous) and thatthe decay and production processes can be different in each phase. Interactions between the solidand liquid phases may be described by nonlinear nonequilibrium equations, while interactions
between the liquid and gaseous phases are assumed to be linear and instantaneous. We further
assume that the solutes are transported by convection and dispersion in the liquid phase, as well
as by diffusion in the gas phase. A general structure of the system of first-order decay reactions
for three solutes (A, B and C) is as follows [Šimůnek and van Genuchten, 1995]:

Products

Products

μg,1 μw,1 μs,1

μg,2 μw,2 μs,2A
g1 c1 s1
kg,1 ks,1

μw,1
μs,1
μg,1

B
g2 c2 s2
kg,2 ks,2

μw,2
μs,2
μg,2

γg,1 γw,1 γs,1

γg,2 γw,2 γs,2

Products

Products

C ...

where c, s, and g represent concentrations in the liquid, solid, and gaseous phases, respectively;
the subscripts s, w, and g refer to solid, liquid and gaseous phases, respectively; straight arrows
represent the different zero-order (γ) and first-order (μ, μ') rate reactions, and circular arrows (kg,ks) indicate equilibrium distribution coefficients between phases.
Typical examples of sequential first-order decay chains are:47

1. Radionuclides [van Genuchten, 1985]

238

→

Pu

234

c1 s1

→

U

c2 s2

230

Th

→

226

c3 s3

Ra

c4 s4

2. Nitrogen [Tillotson et al., 1980]
N2

g2(NH2)2CO

↑

→

→

NH4+

c1 s1

NO2-

c2 s2

→

c3

NO3c4

N2O

3. Pesticides [Wagenet and Hutson, 1987]:a) Uninterrupted chain - one reaction path:
Gas
g1

↑
Parentpesticide→

Daughterproduct 1→

Daughterproduct 2

c1 s1

c2 s2

c3 s3

Product

Product

Product↓↓→

Products↓b) Interrupted chain - two independent reaction paths:
Gas

Gas

g1

g4

↑
Parentpesticide 1↑→

Daughterproduct 1

c1 s1

c2 s2

Product

Product↓↓→

ParentProduct

Pesticide 2

c3 s3

c4 s4

↓

Product

48→

ProductOther examples of chemicals involved in sequential biodegradation chains are hormones[Casey et al., 2003, 2004], chlorinated aliphatic hydrocarbons [Schaerlaekens et al., 1999; Caseyand Šimůnek, 2001], and explosives [Dontsova et al., 2006]. HYDRUS at present considers up toten solutes (five for the dual-permeability model), which either can be coupled in aunidirectional chain or are allowed to move independently of each other.
The partial differential equations governing one-dimensional nonequilibrium chemicaltransport of solutes involved in a sequential first-order decay chain during transient water flow ina variably saturated rigid porous medium are taken as [Šimůnek and van Genuchten, 1995]:∂θ c1 ∂ρ s1 ∂av g1 ∂ ⎛ w ∂c1 ⎞ ∂ ⎛g ∂g1 ⎞ ∂qc1++
= ⎜ θ D1- ra ,1 ⎟ + ⎜ av D1
⎟∂t
∂t
∂t
∂x ⎝
∂x ⎠ ∂x ⎝∂x ⎠ ∂x(3.1)-( μ w,1 + μ )θ c1 - ( μ s ,1 + μ ) ρ s1 - ( μ g ,1 + μ )av g1 + γ w,1θ + γ s ,1 ρ + γ g ,1av
'
w ,1

'
s ,1

'
g ,1

∂θ ck ∂ρ sk ∂av g k
∂c ⎞
∂ ⎛
∂ ⎛
g ∂g k ⎞ ∂qck++
= ⎜ θ Dkw k ⎟ +⎜ a vDk
⎟∂t
∂t
∂t
∂x ⎝
∂x ⎠
∂x ⎝
∂x ⎠ ∂x-( μ w, k + μ w' ,k )θ ck - ( μ s , k + μ s' ,k ) ρ sk - ( μ g , k + μ g' ,k )av g k + μ w' , k −1θ ck −1 ++ μ s' , k −1 ρ sk −1 + μ g' ,k −1av g k −1 + γ w,kθ + γ s ,k ρ + γ g , k av - ra , k(3.2)

k ε (2, ns )where c, s, and g are solute concentrations in the liquid [ML-3], solid [MM-1], and gaseous [ML3], phases, respectively; q is the volumetric flux density [LT-1], μw, μs , and μg are first-order rate
constants for solutes in the liquid, solid, and gas phases [T-1], respectively; μw', μs', and μg' aresimilar first-order rate constants providing connections between individual chain species, γw , γs ,and γg are zero-order rate constants for the liquid [ML-3T-1], solid [T-1], and gas [ML-3T-1]
phases, respectively; ρ is the soil bulk density [M L-3], av is the air content [L3L-3], S is the sinkterm in the water flow equation (2.1), ra is the root nutrient uptake term [ML-3 T-1] that for
passive uptake is equal to the product of the sink term S in the water flow equation (2.1) and theconcentration of the sink term cr [ML-3], Dw is the dispersion coefficient [L2T-1] for the liquid
phase, and Dg is the diffusion coefficient [L2T-1] for the gas phase. As before, the subscripts w, s,
and g correspond with the liquid, solid and gas phases, respectively; while the subscript krepresents the kth chain number, and ns is the number of solutes involved in the chain reaction.
The nine zero- and first-order rate constants in (3.1) and (3.2) may be used to represent a varietyof reactions or transformations including biodegradation, volatilization, and precipitation.

49

HYDRUS assumes nonequilibrium interaction between the solution (c) and adsorbed (s)concentrations, and equilibrium interaction between the solution (c) and gas (g) concentrationsof the solute in the soil system. The adsorption isotherm relating sk and ck is described by ageneralized nonlinear equation of the formsk =k s , k ckβkk ε (1, ns )1 + ηk ckβk∂k s , kk β c βk −1 ∂ck s , k ck2 βk ∂ηk k s , k ckβk ln ck ∂β k
ckβk∂sk
= s,k k k 2 k +−+2
β k 2 ∂t∂t (1 + η c βk ) ∂t 1 + ηk ckβk ∂t(1 + η c )(1 + η c βk ) ∂t
k k

k k

(3.3)

k k

where ks,k [L3M-1], βk [-] and ηk [L3M-1] are empirical coefficients. The Freundlich, Langmuir,and linear adsorption equations are special cases of equation (3.3). When βk=1, equation (3.3)
becomes the Langmuir equation, when ηk=0, equation (3.3) becomes the Freundlich equation,
and when both βk=1 and ηk=0, equation (3.3) leads to a linear adsorption isotherm. Solutetransport without adsorption is described with ks,k=0. While the coefficients ks,k , βk , and ηk inequation (3.3) are assumed to be independent of concentration, they are permitted to change as afunction of time through their dependency on temperature. This feature will be discussed later.
The concentrations gk and ck are related by a linear expression of the formk ε (1, ns )g k = k g ,k ck(3.4)

where kg,k is an empirical constant [-] equal to (KHRuTA)-1 [Stumm and Morgan, 1981], in which
KH is Henry's Law constant [MT2M-1L-2], Ru is the universal gas constant [ML2T-2K-1 M-1] and TA
is absolute temperature [K].
3.1.1. Two-Site Sorption Model (Chemical Nonequilibrium)
The concept of two-site sorption [Selim et al., 1977; van Genuchten and Wagenet, 1989](Fig. 3.1b) is implemented in HYDRUS to permit consideration of nonequilibrium adsorptiondesorption reactions. The two-site sorption concept assumes that the sorption sites can bedivided into two fractions:50

k ε (1, ns )sk = ske + skk(3.5)

Sorption, ske [MM-1], on one fraction of the sites (the type-1 sites) is assumed to beinstantaneous, while sorption, skk [MM-1], on the remaining (type-2) sites is considered to betime-dependent. At equilibrium we have for the type-1 (equilibrium) and type-2 (kinetic) sites,
respectivelyk ε (1, ns )ske = fsk

(3.6)

where f is the fraction of exchange sites assumed to be in equilibrium with the solution phase [-].
Because type-1 sorption sites are always at equilibrium, differentiation of (3.6) gives
immediately the sorption rate for the type-1 equilibrium sites:
skk = (1- f ) skk ε (1, ns )

(3.7)

∂ske
∂s
= f k
∂t
∂t

k ε (1, ns )(3.8)

Sorption on the type-2 nonequilibrium sites is assumed to be a first-order kinetic rate process.
Following Toride et al. [1993], the mass balance equation for the type-2 sites in the presence of
production and degradation is given by⎡⎤k c βk∂skk= ωk ⎢(1- f ) s ,k k βk - skk ⎥ - ( μs ,k + μs' ,k ) skk + (1- f )γ s ,k∂t
1 + ηk ck⎣⎦

k ε (1, ns )(3.9)

where ωk is the first-order rate constant for the kth solute [T-1].
Substituting (3.3) through (3.4) into (3.1) and (3.2) leads to the following equation∂θ Rk 1 ck∂c
∂ ⎛ ∂c ⎞ ∂B c
+ θ Rk 2 k = ⎜ Ek k ⎟ - k k + Fk ck + Gk = 0∂t∂t ∂x ⎝
∂x ⎠
∂x

51

k ε (1, ns )(3.10)

in which Ek [L2T-1] and Bk [LT-1] are an effective dispersion coefficient and effective velocity
given by
Ek = θ Dkw + av Dkg k g , k

Bk = q - av Dkg∂k g , k∂xk ε (1, ns )(3.11)k ε (1, ns )(3.12)

respectively. The coefficients Fk and Gk in (3.10) are defined as

Fk ( ck ) = -( μw,k + μw' ,k )θ - ( μs ,k + μs' ,k ) ρ f

k s ,k ckβk −1- ( μ g ,k + μ g' ,k ) av k g ,k βk
1 + ηk ck

k s ,k ckβk −1-ωk ρ (1- f )1 + ηk ckβk

(3.13)

G1 (c1 ) = γ w,1θ + γ s ,1 f ρ + γ g ,1 a v - ra ,1 + ω1 ρ s1k - ρ fg1 (c1 )Gk (ck ) = ( μθ +μ'w , k −1's , k −1fρ

k s , k −1ckβ−k1−1 −1
β k −1

1 + η k −1ck −1+ μ g' ,k −1av k g ,k −1 ) ck −1 + μ s' ,k −1 ρ skk−1 + γ w, kθ + (3.14)+ γ s , k f ρ + γ g ,k av - ra ,k + ωk ρ skk - ρ fg k (ck )k ε (2 , ns )where the variable gk accounts for possible changes in the adsorption parameters caused bytemperature changes in the system as follows (see also section 3.4):

g k ( ck ) =∂k s ,kk s ,k ck2 βk ∂η k k s ,k ln ck ckβk ∂β k
ckβk+
1 + ηk ckβk ∂t (1 + ηk ckβk )2 ∂t(1 + ηk ckβk ) 2 ∂tk ε (1,ns )(3.15)

Because of numerical and programming consideration, we divided the total retardation factor Rk[-] for use in (3.10) into one part, Rk1, associated with the liquid and gaseous phases, and anotherpart, Rk2, associated with the solid phase:52

Rk 1 ( ck ) = 1 +a v k g ,kθ
β −1
ρ fk s ,k β k ck
Rk 2 ( ck ) =θ (1 + ηk ckβ )2kkk ε (1, ns )k ε (1, ns )(3.16)(3.17)

3.1.2. Attachment-Detachment Model (Two Kinetic Sites Model)
Virus, colloid, and bacteria transport and fate models commonly employ a modified formof the convection-dispersion equation (Fig. 3.1c). In this study we define the mass balance
equation for these applications as:
∂θ c
∂s
∂s
∂s
∂ ⎛
∂c ⎞ ∂qc+ ρ e + ρ 1 + ρ 2 = ⎜θ D ⎟ −- μ wθ c - μ sρ ( se + s1 + s2 )∂t∂t∂t∂t ∂x ⎝∂x ⎠ ∂x(3.18)

where c is the (colloid, virus, bacteria) concentration in the aqueous phase [NcL-3], s is the solid
phase (colloid, virus, bacteria) concentration [NcM-1], subscripts e, 1, and 2 represent equilibrium
and two kinetic sorption sites, respectively, Nc is a number of colloids (particles), and μw and μs
represent inactivation and degradation processes in the liquid and solid phases, respectively.
While sorption to equilibrium sites can be described similarly as before using (3.3), masstransfer between the aqueous and both solid kinetic phases can be described as (note that we now
dropped subscripts 1 and 2):ρ∂s= θ kaψ c − kd ρ s∂t(3.19)

where ka is the first-order deposition (attachment) coefficient [T-1], kd is the first-order
entrainment (detachment) coefficient [T-1], and ψ is a dimensionless colloid retention function []. The attachment and detachment coefficients in (3.19) have been found to strongly depend
upon water content, with attachment significantly increasing as the water content decreases.
To simulate reductions in the attachment coefficient due to filling of favorable sorption
sites, ψ is sometimes assumed to decrease with increasing colloid mass retention. A Langmuirian
dynamics [Adamczyk et al., 1994] equation has been proposed for ψ to describe this blocking

53phenomenon:sψ = max

−s

smax

= 1−

s

(3.20)

smax

in which smax is the maximum solid phase concentration [NcM-1]. Conversely, enhanced colloid
retention during porous medium ripening can theoretically be described using a functional formof ψ that increases with increasing mass of retained colloids:ψ = max (1, s smax)(3.21)

Johnson and Elimelech [1995] proposed the so-called random sequential adsorption
model to describe blocking of the sorption sites:ψ = 1 − 4a + 3.308a 2 + 1.4069a 3ψ=b=(1 − bs )fors < 0.8smax

fors > 0.8smax

3

2d 50 2b3

(3.22)

1smax

a = 0.546

ssmax

Finally, Bradford et al. [2003] hypothesized that the influence of straining andattachment processes on colloid retention can be separated into two distinct components. Theysuggested the following depth-dependent blocking coefficient for the straining process:⎛ d + x − x0 ⎞
ψ =⎜ c
⎟
dc
⎝
⎠

−β

(3.23)

where dc is the diameter of the sand grains [L], x0 is the coordinate of the location where thestraining process starts [L] (the surface of the soil profile, or interface between soil layers), and β
is an empirical factor (with an optimal value of 0.43 [Bradford et al., 2003]) [-].
The attachment coefficient is often calculated using filtration theory [Logan et al., 1995],

54

a quasi-empirical formulation in terms of the median grain diameter of the porous medium (often
termed the collector), the pore-water velocity, and collector and collision (or sticking)efficiencies accounting for colloid removal due to diffusion, interception and gravitational
sedimentation [Rajagopalan and Tien, 1976; Logan et al., 1995]:ka =3(1 − θ )
ηα v
2d c

(3.24)

where dc is the diameter of the sand grains [L], α is the sticking efficiency (ratio of the rate ofparticles that stick to a collector to the rate they strike the collector) [-], v is the pore watervelocity [LT-1], and η is the single-collector efficiency [-]:

−2 / 3
1/ 8 15/ 8
η = 4 As1/ 3 N Pe
+ As N Lo
N R + 0.00338 As N G1.2 N R−0.4(3.25)

where the first, second, and third terms represent removal by diffusion, interception, andgravitational sedimentation, respectively, and where NPe is the Peclet number [-], NR is theinterception number [-], NG is the gravitation number [-], NLo accounts for the contribution ofparticle London-van der Walls attractive forces to particle removal [-], and As is a correction
factor [-] as follows:As =2 (1 − γ 5 )2 − 3γ + 3γ 5 − 2γ 6

(3.26)

γ = (1 − θ )1/ 3

The dimensionless Peclet number in (3.25) is calculated as follows:
N Pe =

3π μ d p d c q
kT(3.27)

where μ is the fluid viscosity (= 0.00093 Pa s) [ML-1T-1], dp is the diameter of the particle (e.g.,virus, bacteria) (= 0.95 μm = 0.95e-6 m) [L], q is the Darcy’s flux [LT-1], k is the Boltzman
constant (= 1.38048e-23 J/K) [M L2T-2K-1], and T is the temperature (= 298 K) [K]. Finally, the55interception number, NR, the gravitation number, NG, and the number representing London-van
der Walls attractive forces, NLo, in (3.25) are calculated using:
NR =
NG =

dp
dc

g ( ρ p − ρ f ) d p2

18 μ q
4H
N Lo =
9π μ d p2 q

(3.28)(3.29)(3.30)

where H is the Hamaker constant (= 1e-20 J) [ML2T-2], g is the gravitational acceleration (= 9.81
m s-2) [LT-2], ρp is the bacterial density (= 1080 kg m-3) [ML-3], and ρf is the fluid density (= 998
kg m-3) [ML-3].
The model described above using equation (3.18) can be used in many different ways.
For example, one can assume that the soil has two sorption sites, s1 and s2, each having their own
attachment and detachment constants. This model has been used to describe virus transport insand dunes by Schijven and Šimůnek [2002]. Sorption sites s1 and s2 can be used to describe
straining and attachment, respectively, as was done by Bradford et al. [2002, 2003, 2004]. One
may also assume that one sorption site represents sorption to the solid phase, while the other site
represents removal of particles by means of attachment to the air-water interface.
3.1.3. Dual-Porosity Model (Physical Nonequilibrium)
The concept of two-region, dual-porosity type solute transport [van Genuchten and
Wierenga, 1976] is implemented in HYDRUS to permit consideration of physicalnonequilibrium transport. The two-region concept assumes that the liquid phase can bepartitioned into mobile (flowing), θmo [L3L-3], and immobile (stagnant), θim [L3L-3], regions suchthat:

θ = θ mo + θim(3.31)

while solute exchange between the two liquid regions is modeled as a first-order process, i.e.,56⎡k s ,k β k ckβ,kim−1 ⎤ ∂ck ,im
θ
ρ
(1f
)
+
= ωk (ck - ck ,im ) + γ w,kθim + (1- f ) ργ s ,k ⎢ im
2⎥
βk
t
∂
η
(1
c
+
)
k k ,im⎣⎦⎡k s ,k ckβ,kim−1 ⎤''- ⎢θim ( μw,k + μw,k ) + ρ ( μs ,k + μ s ,k )(1- f )⎥ ck ,im
1 + ηk ckβ,kim ⎦⎣(3.32)k ε (1, ns )where cim [ML-3] is the concentration of the immobile region and ωk is mass transfer coefficientfor the kth solute [T-1].
Substituting (3.31) and (3.32) into (3.1) and (3.2), with the latter two equations modified
for mobile and immobile regions as shown by van Genuchten and Wagenet [1989] for simplified
one-dimensional conditions, leads to equation (3.10) in which θ is replaced by θmo and with thecoefficients Fk and Gk redefined as follows
k s ,k ckβk −1- ( μ g ,k + μ g' ,k )av k g ,k - ωk
1 + ηk ckβk
G1 ( c1 ) = γ w,1θ mo + γ s ,1 f ρ + γ g ,1 a v - Scr ,1 + ω1c1,im - ρ fg1 ( c1 )Fk ( ck ) = -( μw,k + μw' ,k )θ mo - ( μ s ,k + μs' ,k ) ρ f⎡⎤
k c βk −1 −1
Gk ( ck ) = ⎢ μw' ,k −1θ mo + μ s' ,k −1 f ρ s ,k −1 k −1βk −1 + μ g' ,k −1av k g ,k −1 ⎥ ck −1 +1 + ηk −1ck −1⎣⎦⎡ 'k s ,k −1 ckβ−k1,−1im−1 ⎤'+ ⎢ μw,k −1θim + μ s ,k −1 (1- f ) ρ⎥ ck −1,im +1 + μk −1ckβ−k1,−1im ⎦⎣+ γ w,kθ mo + γ s ,k f ρ + γ g ,k av - Scr ,k + ωk ck ,im - ρ fg k (ck )(3.33)(3.34)k ε (2 , ns )In order to solve equation (3.10), it is necessary to know the water content θ and thevolumetric flux density q. Both variables are obtained from solutions of the Richards equation.
The above equations may appear to be relatively complicated. However, by selecting proper
values of particular coefficients (i.e., γw , γs , γg, μw, μs, μg, μw’, μs’, μg’, η, ks, kg, f, β, ω) theentire system can be simplified significantly. Assuming for example that μw’, μs’, μg’, η, and kg
are zero, and f and β are equal to one, the entire system of equations (3.1) through (3.17)
simplifies into a set of equations describing the transport of mutually independent solutes (i.e.,single-ion transport as applicable):∂θ Rc ∂ ⎛ w ∂c ⎞ ∂qc= ⎜θ D+ Fc +G⎟∂t∂x ⎝∂x ⎠ ∂x57(3.35)

a. One Kinetic Site Modelθb. Two-Site Modelseθc. Two Kinetic Sites Models 1kd. Dual-Porosity Model withOne Kinetic Sitee. Dual-permeability Modelwith Two-Site Sorption

Immob. Mob.

Slow Fast

θ

sk

θim

θmo

cim

cmo

smoe

sme

smok

smk

θM

θF

cm

cf

sfe

sime
c

sk

c

s 2k

c

sfk

Figure 3.1. Conceptual chemical nonequilibrium models for reactive solute transport. In the plots, θ is the water content, θmo and θim in (d) are watercontents of the mobile and immobile flow regions, respectively; θM and θF in (e) are water contents of the matrix and macropore(fracture) regions, respectively; c are concentrations of the corresponding regions, se are sorbed concentrations in equilibrium with theliquid concentrations of the corresponding regions, and sk are kinetically sorbed solute concentrations of the corresponding regions.

58

3.1.4. Dual-Porosity Model with One Kinetic Site (Physical and Chemical
Nonequilibrium)This model (Fig. 3.1d) is similar as the Dual-Porosity Model (section 3.1.3) in that theporous medium is divided into mobile and immobile domains such that θ = θmo + θim. The current
model, however, additionally divides the sorption sites in contact with the mobile zone, similarly as
the Two-Site Model (section 3.1.1), into two fractions involving instantaneous and kinetic sorption
such that the total sorbed concentration at equilibrium is given by:
e
k
+ smo
s = (1 − f mo ) sim + f mo smo = (1 − f mo ) sim + f mo ( smo,e ) == (1 − f mo ) K d cmo + f mo f em K d cmo + f mo (1 − f em ) K d cmo = K d cmo(3.36)ewhere smo
is the sorbed concentration in equilibrium with the liquid phase concentration of thekmobile region of the Dual-Porosity Model [MM-1], smo, e is the sorbed concentration of the

kinetic sites in contact with the mobile region of the Dual-Porosity Model when at equilibrium[MM-1], fmo is the fraction of sorption sites in contact with mobile water (the remainder is in contact
with immobile water), and fem is the fraction of sorption sites in equilibrium with the mobile liquid
phase (the remaining sites are in contact with the mobile liquid phase). The complete Dual-Porosity
Model with One Kinetic Site is described using the following equations [Šimůnek and van
Genuchten, 2008]:e∂θmocmo∂smo∂⎛∂c ⎞ ∂q c+ fmoρ= ⎜θmo Dmo mo ⎟ - mo mo - φmo − Γ s1 − Γ s2∂t∂t ∂x ⎝∂x ⎠∂z
∂θimcim∂s+(1 − fmo )ρ im = Γ s1 - φim
∂t
∂t
k
∂s
fmoρ mo = Γ s2 - φmo,k∂t
Γ s1 = ωph (cmo − cim )kkΓ s2 = αch ( smo,e − smo )e= femKd cmo
smo
k
smo
,e = (1 − fem )Kd cmo

59(3.37)

where ωph and αch are first-order rate constants [T-1] accounting for physical and chemical rate
processes, respectively; Γs1 is the mass transfer term for solute exchange between the mobile andimmobile regions [ML-3T-1], Γs2 represents mass transfer to the kinetic sorption sites in themobile region [ML-3T-1], and φmo, φmi and φmo,k represent sink/source terms for the equilibriumphases in the mobile zone, for the immobile zone, and for the kinetic sorption sites [ML-3T-1],
respectively. The first equation of (3.37) describes transport in the mobile phase, the second is amass balance for the immobile phase, and the third equation a mass balance for the kinetic
sorption sites in contact with the mobile zone. The fourth and fifth equations describe masstransfer rates between the mobile and immobile zones, and to the kinetic sorption sites,
respectively, while the sixth and seventh equations represent sorption onto the equilibrium andkinetic sorption sites in contact with the mobile zone, respectively. Note that (3.36) and (3.37)(and many of the transport equation later on) are written in terms of the distribution coefficient
Kd (instead of ks,k as above), although a full nonlinear Frendlich-Langmuir equation (3.3) can be
implemented.
3.1.5. Dual-Permeability Model (Physical Nonequilibrium)
Analogous to Eq. (2.6), the dual-permeability formulation for solute transport is based on
advection-dispersion type equations for transport in both the fracture and matrix regions as follows[Gerke and van Genuchten, 1993a,b] (Figs. 2.1d, 3.1d):∂θ f c f
∂t

+ρ

∂s f
∂t

=

∂c f ⎞ ∂q f c fΓ∂ ⎛-φf − s⎜θ f D f
⎟w
∂z ⎝
∂z ⎠
∂z

Γ
∂θ m cm∂s∂c ⎞ ∂q c∂ ⎛+ ρ m = ⎜ θ m Dm m ⎟ - m m - φm + s
1− w
∂t
∂t
∂z ⎝
∂z ⎠
∂z

(3.38)

Γ s = ωdp (1 − w)θ m (c f − cm ) + Γ w c *The variables in (3.38) have similar meaning as in (3.37), except that they refer now to two
overlapping domains, i.e., the matrix (subscript m) and fracture (subscript f) domains. The firstequation of (3.38) describes solute transport in the fracture domain, the second equationtransport in the matrix domain, and the third equation advective-dispersive mass transfer

60between the fracture and matrix domains. Equation (3.38) assume complete advective-dispersive
transport descriptions for both the fractures and the matrix. van Genuchten and Dalton [1986]
and Gerke and van Genuchten [1996], among others, discussed possible expressions for the firstorder solute mass transfer coefficient, ωdp [T-1]. c* is (3.38) is equal to cf for Γw>0 and cm forΓw<0.
3.1.6. Dual-Permeability Model with Immobile Water (Physical Nonequilibrium)
The Dual-Permeability Model with Immobile Water (Fig. 2.1e, 3.1e) assumes that theliquid phase of the matrix can be further partitioned into mobile (flowing), θm,m [L3L-3], andimmobile (stagnant), θim,m [L3L-3], regions as follows:θ m = θ m ,m + θim,m(3.39)

where θm is the volumetric water content of the matrix pore system [L3L-3]. The governing
advection-dispersion equation for transport in the matrix region (second equation of (3.38)) isthen replaced [e.g., Pot et al., 2005; Šimůnek and van Genuchten, 2008] to yield:∂θ f c f∂t∂θ m ,m cm ,m∂t+ρf+ ρm f m∂s f∂t∂sm ,m∂t∂θim ,m cim ,m==∂c f ⎞ ∂q f c fΓ∂ ⎛-φf − s⎜θ f D f
⎟∂z ⎝
∂z ⎠
∂z
w

∂cm ,m ⎞ ∂qm cm ,m
Γ∂ ⎛- φm , m + s − Γ s *⎜ θ m,m Dm
⎟∂z ⎝
∂z ⎠
∂z
1− w

∂ s im ,m= Γ s* - φim ,m∂t∂t
Γ s = ωdp (1 − w)θ m (c f − cm ,m ) + Γ w c *+ ρ m (1 − f m )(3.40)

Γ s* = ωdpm (c m ,m - c im ,m)where cim,m and cm,m are solute concentrations in the immobile and mobile zones of the matrix
region [ML-3], respectively; φm,m and φim,m represent various reactions in the mobile andimmobile parts of the matrix [ML-3T-1], respectively; fm is again the fraction of sorption sites incontact with the mobile region of the matrix [-], ωdpm is the mass transfer coefficient between
mobile and immobile zones of the matrix region [T-1], and Γs* is the mass transfer term for61solutes between the mobile and immobile regions of the matrix domain [ML-3T-1]. The firstequation of (3.40) now describes solute transport in the fracture domain, the second equationtransport in the mobile zone of the matrix domain, the third equation is a mass balance for theimmobile zone of the matrix domain, the fourth equation describes mass transfer between thefracture and matrix domains, while the fifth equation describes mass transfer between the mobile
and immobile zones within the matrix domain.

3.1.7. Dual-Permeability Model with Two-Site Sorption (Physical and Chemical
Nonequilibrium)
Finally, simultaneous physical and chemical nonequilibrium processes are implemented
in HYDRUS-1D by assuming applicability of the Dual-Permeability Model [Gerke and van
Genuchten, 1993a; Šimůnek et al., 2003; Šimůnek and van Genuchten, 2008] and dividing thesorption sites of both the fracture and matrix domains into equilibrium and kinetic sites (Fig.
3.3e). This model leads to the following set of equations [Pot et al., 2005]:∂θ f c f
∂t

+ ρf

∂s ef
∂t

=

∂c f ⎞ ∂q f c fΓ∂ ⎛-φf − s − Γ f⎜θ f D f
⎟w
∂z ⎝
∂z ⎠
∂z

∂θ m cm∂s e∂c ⎞ ∂q cΓ∂ ⎛+ ρ m m = ⎜ θ m Dm m ⎟ - m m - φm + s − Γ m
∂t
∂t
∂z ⎝
∂z ⎠
∂z
1− w

ρ

∂s kf

= Γ f - φ f ,k∂t∂s k
ρ m = Γ m - φm , k∂tΓ s = ωdp (1 − w)θ m (c f − cm ) + Γ w c *(3.41)

Γ f = ρ f α ch , f ⎡⎣ (1 - f f ) K df c f - s kf ⎤⎦Γ m = ρ mα ch , m ⎡⎣(1 - f m ) K dm cm - smk ⎤⎦where smk and sfk are sorbed concentrations of type-2 (kinetic) sites in the matrix and fracturedomains, respectively; fm and ff are fractions of the exchange sites assumed to be in equilibriumwith the solution phases [-] of the matrix and fracture domains, respectively; φf, φm, φf,k and φm,k62

represent reactions in the equilibrium phases of the fracture and matrix domains, and at the kinetic
sites of the fracture and matrix domains [ML-3T-1], respectively; and αch,m and αch,f are again firstorder rate constants for the matrix and fracture domains [T-1], respectively. Note that the
distribution coefficients can be different in the different regions (i.e., Kdf ≠ Kdm).
3.2. Initial and Boundary Conditions
The solution of (3.10) requires knowledge of the initial concentration within the flow
region, Ω, i.e.,c( x,0) = ci ( x )s k ( x,0) = sik ( x )(3.42)cim ( x,0) = cim , m ( x )where ci [ML-3], cim,i [ML-3] and sik [-] are prescribed functions of x. The initial condition for sik
must be specified only when nonequilibrium adsorption is considered. The subscript k is droppedin (3.42) and throughout the remainder of this report, thus assuming that the transport-related
equations in the theoretical development and the numerical solution apply to each of the solutes
in the decay chain.
Two types of boundary conditions (Dirichlet and Cauchy type conditions) can be applied
to the upper or lower boundaries. First-type (or Dirichlet type) boundary conditions prescribe theconcentration at a boundary:c( x, t ) = c0 ( x, t )at x = 0 or x = L(3.43)

whereas third-type (Cauchy type) boundary conditions may be used to prescribe theconcentration flux at the upper or lower boundary as follows:-θ D∂c+ qc = q0c0∂x

at x = 0 or x = L

63(3.44)

in which q0 represents the upward fluid flux and c0 is the concentration of the incoming fluid[ML-3]. In some cases, for example when boundary is impermeable (q0=0) or when water flow is
directed out of the region, (3.44) reduces to a second-type (Neumann type) boundary conditionof the form:

θD

∂c
=0
∂x

at x = 0 or x = L(3.45)

A different type of soil surface boundary condition is needed for volatile solutes when
they are present in both the liquid and gas phases. This situation requires a third-type boundary
condition as before, but with an additional term to account for gaseous diffusion through astagnant boundary layer of thickness d [L] on the soil surface. The additional solute flux is
proportional to the difference in gas concentrations above and below the boundary layer [Jury etal., 1983]. The modified boundary condition has the form
-θ D

D
∂c
+ qc = q0c0 + g ( k g c - g atm )∂x
d

at x = L(3.46)

where Dg is the molecular diffusion coefficient in the gas phase [L2T-1] and gatm is the gas
concentration above the stagnant boundary layer [ML-3] (Jury et al. [1983] assumed gatm to bezero). Similarly as for (3.44), (3.46) reduces to a second-type (or Neumann type) boundary
condition when water flow is zero or directed out of the region:
-θ D

∂c Dg
=
( k g c - g atm )∂x
d

at x = L(3.47)

Equations (3.46) and (3.47) can only be used when the additional gas diffusion flux is positive.
Jury et al. [1983] discussed how to estimate the thickness of the boundary layer, d; they
recommended a value of 0.5 cm for d as a good average for a bare surface.
3.3. Effective Dispersion Coefficient
The dispersion coefficient in the liquid phase, Dw, is given by [Bear, 1972]64θ D w = DL | q |+θ Dwτ w(3.48)

where Dw is the molecular diffusion coefficient in free water [L2T-1], τw is a tortuosity factor in
the liquid phase [-], |q| is the absolute value of the Darcian fluid flux density [LT-1], and DL is thelongitudinal dispersivity [L]. After adding the diffusion contribution from the gas phase, theeffective dispersion coefficient in the soil matrix for one-dimensional transport is as follows:θ D = DL | q |+θ Dwτ w + a v D g k gτ g

(3.49)

where Dg is the molecular diffusion coefficient in the gas phase [L2T-1] and τg is a tortuosity
factor in the gas phase [-].
The tortuosity factors for both phases are evaluated in HYDRUS as a function of thewater and air contents using the relationship of Millington and Quirk [1961]:

τw =
τg =

θ 7/3
θ s2
av7 / 3(3.50)

θ s2

3.4. Temperature and Water Content Dependence of Transport and Reaction Coefficients
Several of the diffusion (Dw , Dg), zero-order production (γw , γs , γg ), first-orderdegradation (μw’, μs’, μg’, μw , μs , and μg), and adsorption (ks , kg , β, η, ω) coefficients may bestrongly dependent upon temperature. HYDRUS assumes that this dependency can be expressedby the Arrhenius equation [Stumm and Morgan, 1981]. After some modification, this equation
can be expressed in the general form [Šimůnek and Suarez, 1993a]⎡ E (T A - TrA ) ⎤aT = ar exp ⎢ a⎥
A A⎣ RuT Tr ⎦(3.51)

where ar and aT are the values of the coefficient being considered at a reference absolutetemperature TrA and absolute temperature TA, respectively; Ru is the universal gas constant, andEa [ML2T-2M-1] is the activation energy of the particular reaction or process being modeled.

65

The water content dependence of degradation coefficients is implemented using amodified equation of Walker [1974]:⎡ ⎛ θ ⎞B ⎤a(θ ) = ar (θ ref ) min ⎢1, ⎜⎟ ⎥⎢ ⎜⎝ θ ref ⎟⎠ ⎥⎣⎦(3.52)

where ar is the values of the coefficient at a reference water content θref, a is the value at theactual water content θ, and B is a solute dependent parameter (usually 0.7). The reference watercontent, θref, which may be different for different soil layers, is calculated from the reference
pressure head, href, which is considered to be constant for a particular compound.

3.5. Root Solute Uptake
The root solute uptake models implemented in HYDRUS-1D was developed by Simunek
and Hopmans [2009] and only a brief description is given below.
3.5.1. Uncompensated Nutrient Uptake Model
To clearly differentiate between point and root domain nutrient uptake rate values, we define
lower case variables to represent point root nutrient uptake rates [ML-3T-1], while upper casevariables represent nutrient uptake rates [ML-2T-1] over the entire root zone, LR. Both point and root
domain nutrient uptakes are assumed to be the sum of their passive and active components, or:ra ( x, t ) = pa ( x, t ) + aa ( x, t )Ra (t ) = Pa (t ) + Aa (t )(3.53)(3.54)

where ra, pa, and aa define total actual (subscript a) passive and active root nutrient uptake rates[ML-3T-1], respectively, at any point, and Ra, Pa, and Aa denote actual total, passive and active root
nutrient uptake rates [ML-2T-1], respectively, for the root zone domain.
Passive nutrient uptake is simulated by multiplying root water uptake (compensated or
uncompensated) with the dissolved nutrient concentration, for concentration values below apriori defined maximum concentration (cmax), or66

pa ( x, t ) = s* ( x, t ) min [ c( x, t ), cmax ](3.55)

where c is the dissolved nutrient concentration [ML-3] and cmax is the maximum allowed solution
concentration [ML-3] that can be taken up by plant roots during passive root uptake. All nutrient
dissolved in water is taken up by plant roots when cmax is large (larger than the dissolved
concentration c), while no nutrient is taken up when cmax is equal to zero, with only active uptake
remaining in that case. The maximum solution concentration for passive root uptake, cmax, thus
controls the relative proportion of passive root water uptake to total uptake. Using this flexible
formulation, uptake mechanisms can vary between specific nutrients. For example, Na uptake can be
excluded by setting cmax equal to zero, passive Ca uptake can be limited by defining a finite cmax
value, or all soil solution available P or N is allowed to be taken up passively, by setting cmax to avery large value. Note that the cmax parameter is introduced as a control model parameter that
does not necessarily have a physiological meaning.
Passive actual root nutrient uptake for the whole root domain, Pa [ML-2T-1], is calculated byintegrating the local passive root nutrient uptake rate, pa, over the entire root zone:
Pa (t ) = ∫ pa ( x, t )dx = ∫ s* ( x, t ) min [ c( x, t ), cmax ] dx =
LR

=

LR

Tp (t )

max [ω (t ), ωc ] Ω∫Rα (h,hφ , x, t )b( x, t ) min [ c( x, t ), cmax ] dx

(3.56)

Defining Rp as the potential (subscript p) nutrient demand [ML-2T-1], the potential active nutrient
uptake rate, Ap [ML-2T-1], is computed from:Ap (t ) = max ⎡⎣ R p (t ) − Pa (t ),0⎤⎦(3.57)

Thus, using this formulation, we assume that active nutrient uptake will be invoked only if thepassive root nutrient uptake term does not fully satisfy the potential nutrient demand of the plant.
However, as was discussed earlier, the passive uptake can be reduced or completely turned off(cmax=0), thus allowing the potential active nutrient uptake (Ap) to be equal to the potential nutrient
demand (Rp). Once Ap is known, the point values of potential active nutrient uptake rates, ap [ML-3T-

67

1], are obtained by distributing the potential root zone active nutrient uptake rate, Ap [ML-2T-1], over
the root zone, using a predefined spatial root distribution, b(x, t), as was done for root water uptake,or:a p ( x, t ) = b( x, t ) Lt Ap (t )(3.58)

Using Michaelis-Menten kinetics (e.g., Jungk [1991]) provides for actual distributed values of active
nutrient uptake rates, aa [ML-3T-1], allowing for nutrient concentration dependency, or:aa ( x, z , t ) =c ( x, z , t )c ( x, z , t )a p ( x, z , t ) =b( x, z , t ) Lt Ap (t )K m + c ( x, z , t )K m + c ( x, z , t )(3.59)

where Km is the Michaelis-Menten constant [ML-3] [Jungk, 1991]. The Michaelis-Menten
constants for selected nutrients (e.g., N, P, and K) and plant species (e.g., corn, soybean, wheat,tomato, pepper, lettuce, and barley) can be found in the literature (e.g., Bar-Yosef [1999]).
Finally, total active uncompensated root nutrient uptake rate, Aa [ML-2T-1], is calculated byintegrating the actual active root nutrient uptake rate, aa, at each point, over the root zone LR, in
analogy with the non-compensated root water uptake term, or:c ( x, t )b( x, t )dx
K m + c ( x, t )LRAa (t ) = ∫ aa ( x, t )dx = Ap (t ) ∫LR(3.60)

3.5.2. Compensated Nutrient Uptake Model
The above nutrient uptake model includes compensation of the passive nutrient uptake, by
way of the root water uptake compensation term, sc, and root adaptability factor, ωc, in Eq. (3.56).
A similar compensation concept as used for root water uptake above, was implemented for active
nutrient uptake rate, by invoking a so-called nutrient stress index π:π (t ) =Aa (t )Ap (t )

68

(3.61)

After substitution of the active total root nutrient uptake rate value from Eq. (3.60) above, this
newly defined nutrient stress index (π) is equal to:c ( x, t )b( x, t )dx
K
c
(
x
,
t
)
+
m
LR

π (t ) = ∫(3.62)

After defining the critical value of the nutrient stress index πc, above which value active nutrient
uptake is fully compensated for by active uptake in other more-available (less stressed) soil
regions, the local compensated active root nutrient uptake rate, aac [ML-3T-1], is obtained byincluding the nutrient-stress index function in the denominator of Eq. (3.59), or:aac ( x, t ) =Ap (t )c ( x, t )b ( x, t )K m + c ( x, t )max [π (t ), π c ](3.63)

from which the total compensated active root nutrient uptake rate, Aac [ML-2T-1] in the twodimensional root domain, LR, is calculated, in analogy with the compensated root water uptake
term, as follows:
Aac (t ) = ∫ aac ( x, t )dx =LR

Ap (t )c ( x, t )b( x, t )dx
max [π (t ), π ] ∫ K + c( x, t )cLR(3.64)m

Equation (3.57) implies that reduction in root water uptake will decrease passive nutrient uptake,thereby increasing active nutrient uptake proportionally. In other words, total nutrient uptake is notaffected by soil water stress, as computed by the proportion of actual to potential root water uptake.
This is not realistic since one would expect that plant nutrient requirements will be reduced forwater-stressed plants. For that reason, the uptake model includes additional flexibility, by reducing
the potential nutrient demand Rp [ML-2T-1], in proportion to the reduction of root water uptake, as
defined by the actual to potential transpiration ratio, or:

69

⎡
⎤
T (t )Ap (t ) = max ⎢ R p (t ) ac − Pa (t ),0 ⎥T p (t )⎢⎣⎥⎦(3.65)

In summary, the presented root nutrient uptake model with compensation requires as input thepotential nutrient uptake rate (demand), Rp, the spatial root distribution function b(x,z,t) as
needed for the water uptake term, the Michaelis-Menten constant Km, the maximum nutrient
concentration that can be taken up passively by plant roots cmax, the minimum concentration cminneeded to initiate active nutrient uptake, and the critical nutrient stress index πc. The passive
nutrient uptake term can be turned off by selecting cmax equal to zero. Moreover, active nutrient
uptake can be eliminated by specifying a zero value for Rp in Eq. (3.57), or by selecting a verylarge cmin value in Eq. (3.59). It is likely that values of these parameters are nutrient- and plantspecific. Similarly as for root water uptake, it can be expected that πc for agricultural crops isrelatively high when compared to natural plants that are likely to have more ability to compensate
for soil environmental stresses. Other parameters, such as cmax will likely need to be calibrated to
specific conditions before the model can be used for predictive purposes.

70

4. HEAT TRANSPORT
4.1. Governing Heat Transport Equations
4.1.1. Heat Transport Without Vapor Transport
Neglecting the effect of water vapor diffusion on transport, one-dimensional heat transfer
can be described with a convection-dispersion equation of the form
∂C p (θ )T
∂t

=

∂ ⎡
∂T ⎤
∂qT
(
)
- Cw ST
λ
θ
C
w
∂x ⎢⎣
∂x ⎥⎦
∂x

(4.1)

or equivalently as [Sophocleous, 1979]:
C p (θ )

∂T
∂ ⎡
∂T ⎤
∂T
=
C
q
λ
(
θ
)
w
∂t ∂x ⎣⎢
∂x ⎦⎥
∂x

(4.2)

where λ(θ) is the coefficient of the apparent thermal conductivity of the soil [MLT-3K-1] andCp(θ) and Cw are the volumetric heat capacities [ML-1T-2K-1] of the porous medium and theliquid phase, respectively. The volumetric heat capacity is defined as the product of the bulk
density and gravimetric heat capacity. The first term on the right-hand side of (4.1) represents
heat flow due to conduction, the second term heat transported by flowing water, and the third
term energy uptake by plant roots associated with root water uptake. We do not consider thetransfer of latent heat by vapor movement. Equation (4.2) is derived from (4.1) by making use of
the continuity equation describing isothermal Darcian flow of water in a variably-saturated
porous medium
∂θ
∂q
=- -S
∂t
∂x

(4.3)

According to de Vries [1963] the volumetric heat capacity is given by
C p (θ ) = Cnθ n + Coθo + Cwθ + Ca av ≈ (1.92θ n + 2.51θ o + 4.18θ ) 10 671

( J m -3 oC -1 ) (4.4)

where θ refers to a volumetric fraction [L3L-3], and subscripts n, o, a, w represent solid phase,organic matter, gas phase and liquid phase, respectively.
4.1.2. Heat Transport With Vapor Transport
When the effects of water vapor diffusion can not be neglected, the heat transport must be
expanded to the form [e.g., Saito et al., 2006]:

C p (θ )

∂T
∂θ
∂ ⎛
∂T ⎞
∂T
∂q T
∂q
+ L0 v = ⎜ λ (θ )− Cv v − L0 v
⎟ - Cw q
∂t
∂t
∂x ⎝
∂x ⎠
∂x
∂x
∂x

(4.5)

where L0 is the volumetric latent heat of vaporization of liquid water [ML-1T-2] (e.g., Jm-3) and qv
is the vapor flux density [LT-1] (see also (2.3)):
∂T
⎛ ∂h
⎞
qv = − K vh ⎜ + cos α ⎟ − K vT
∂x
⎝ ∂x
⎠

(4.6)

In equation (4.5), the total heat flux density is defined as the sum of the conduction of sensible
heat as described by Fourier’s law (the first term on the right side), convection of sensible heatby liquid water (the second term) and water vapor (the third term), and convection of latent heatby vapor flow (the fourth term) [de Vries, 1958].
4.2. Apparent Thermal Conductivity
The apparent thermal conductivity λ(θ) combines the thermal conductivity λ0(θ) of theporous medium (solid plus water) in the absence of flow and the macrodispersivity, which is alinear function of the velocity [de Marsily, 1986]:λ (θ ) = λ0 (θ ) + β tCw | q |(4.7)

where βt is the thermal dispersivity [L]. The volumetric heat capacity of the liquid phase isincluded in the definition of the thermal conductivity in order to have the dimensions of thethermal dispersivity in length units. Chung and Horton [1987] described the thermal72conductivity using the equation

λ0 (θ ) = b1 + b2θ + b3θ 0.5(4.8)

where b1, b2, and b3 are empirical parameters [MLT-3K-1]. Alternatively, one can use a function
suggested by Campbell [1985]:

λ0 (θ ) = A + Bθ − ( A − D) exp[−(Cθ ) E ]A=

0.57 + 1.73θ q + 0.93θ m
1 − 0.74θ q − 0.49θ m− 2.8θ n (1 − θ n )B = 2.8θ n(4.9)

C = 1 + 2.6θ c−1 / 2
D = 0.03 + 0.7θ n2
E=4Where the subscripts n, q, c, and m refer to solid, quartz, clay, and other minerals.
4.3. Initial and Boundary Conditions
The solution of (4.1) requires knowledge of the initial temperature within the flow
region, i.e.,T ( x,0) = Ti ( x )t=0(4.10)

where Ti is a prescribed function of x.
Two types of boundary conditions (Dirichlet and Cauchy type conditions) can be
specified at the top and bottom boundaries of the soil profile. First-type (or Dirichlet type)boundary conditions prescribe the temperature:T ( x, t ) = T0 (t )at x = 0 or x = L(4.11)

whereas third-type (or Cauchy type) boundary conditions may be used to prescribe the heat flux
as follows

73

-λ

∂T
+ TCw q = T0Cw q0∂xatx = 0 or x = L(4.12)

in which T0 is either the temperature of the incoming fluid or the temperature at the boundary. In
some cases, for example for an impermeable boundary (q=0) or when water flow is directed outof the region, (4.12) reduces to a second-type (Neumann type) boundary condition of the form:
∂T
=0
∂x

at

x=0

(4.13)

Atmospheric boundary conditions for daily fluctuations in soil temperature are often
represented by a sine function as follows [Kirkham and Powers, 1972]:⎛ 2π t 7π ⎞T0 = T + A sin ⎜⎟⎝ pt 12 ⎠(4.14)

where pt is a period of time [T] necessary to complete one cycle of the sine wave (taken to be 1day), T is the average temperature at the soil surface [K] during the period pt, and A is theamplitude of the sine wave [K]. The second part of the sine term is included to force themaximum in the daily temperature to occur at 1 p.m.

74

5. CARBON DIOXIDE TRANSPORT AND PRODUCTION
The carbon dioxide transport and production module, as well as the major ion chemistry
module described in the next section, were adopted from the UNSATCHEM software package[Šimůnek et al., 1996], and hence their description below closely mirrors those in the
UNSATCHEM manual. More detailed analyses of the CO2 transport and production module, as
well as a review of related literature, are presented by Šimůnek and Suarez [1993], Suarez andŠimůnek [1993], and Šimůnek et al. [1996].
5.1. Governing CO2 Transport Equations
We assume that the CO2 transport in the unsaturated zone can occur in both the liquidand gas phases. We furthermore assume that the CO2 concentration in the soil is governed by
two transport processes [Patwardhan et al., 1988]: convective and diffusive transport in both gas
and aqueous phases, and by CO2 production and/or removal. One-dimensional CO2 transport
hence can be described with the following mass balance equation:∂cT∂= - ( J da + J dw + J ca + J cw ) - Scw + P
∂t
∂x

(5.1)

where Jda describes the CO2 flux caused by diffusion in the gas phase [LT-1], Jdw is the CO2 flux
caused by diffusion and dispersion in the dissolved phase [LT-1], Jca is the CO2 flux caused byconvection in the gas phase [LT-1], and Jcw is the CO2 flux caused by convection in the dissolved
phase [LT-1]. The term cT is the total volumetric concentration of CO2 [L3L-3] and P is the CO2
production/sink term [L3L-3T-1]. The term Scw represents dissolved CO2 removed from the soil
by root water uptake. This uptake term assumes that when plants absorb water, dissolved CO2 isalso removed from the soil-water system. The individual flux densities in (5.1) are defined as[Patwardhan et al., 1988]

75

∂ca
∂x
∂c
J dw = -θ w Dw w
∂x
J ca = -qa ca

J da = -θ a Da

(5.2)

J cw = -qwcw
where cw and ca are the volumetric concentrations of CO2 in the dissolved and gas phases [L3L-3],
respectively, Da is the effective soil matrix diffusion coefficient of CO2 in the gas phase [L2T-1],
Dw is the effective soil matrix dispersion coefficient of CO2 in the dissolved phase [L2T-1], qa is
the soil air flux [LT-1], qw is the soil water flux [LT-1], θw is the soil water content [L3L-3] and θais the volumetric air content [L3L-3].
The total CO2 concentration, cT [L3L-3], is defined as the sum of CO2 in the gas anddissolved phases as follows

cT = caθ a + cwθ w(5.3)

Substituting (5.2) and (5.5) into (5.1) leads to∂ ( caθ a + cwθ w ) ∂
∂c
∂
∂c
∂
∂
= θ a Da a + θ w Dw w - qa ca - qw cw - Scw + P
∂t
∂x
∂x ∂x
∂x ∂x
∂x

(5.4)

The total aqueous phase CO2, cw, defined as the sum of CO2(aq) and H2CO3, is related to the CO2
concentration in the gas phase by [Stumm and Morgan, 1981]

cw = K CO 2RTca

76

(5.5)

where KCO2 is Henry's Law constant [MT2M-1L-2], R is the universal gas constant (8.314 kg m2 s2 -1
K mol-1) [ML2T-2K-1M-1] and T is the absolute temperature [K]. The value of KCO2 as a functionof temperature was taken from Harned and Davis [1943]. Any interaction of dissolved CO2 withthe solid phase is neglected. The quantity of CO2 added or removed by mineraldissolution/precipitation reactions is relatively small compared to the production and flux values
in root-zone environment. The assumption is generally not suitable for saturated conditions or atlarge depths.
Substituting equation (5.5) into (5.5) gives∂R f ca
∂t

=

∂
∂c ∂
DE a - qE ca - S *ca + P
∂x
∂x ∂x

(5.6)

where Rf is the CO2 retardation factor [-], DE is the effective dispersion coefficient for CO2 in thesoil matrix [L2T-1], qE is the effective velocity of CO2 [LT-1], and S* is the CO2 uptake rate [T-1]
associated with root water uptake. These parameters are defined as

R f = θ a + K CO 2RT θ w
DE = θ a Da + K CO 2RT θ w Dw
qE = qa + K CO 2RTqw(5.7)

θa = p - θw
S * = SKCO2 RT

Equation (5.6) is a nonlinear partial differential equation in which all parameters, except
for ca and qa, are either known or can be obtained from solutions of the variably saturated flow
equation. The nonlinearity of (5.6) is caused by the term P, which depends upon the CO2
concentration, ca. Since HYDRUS does not consider coupled water and air movement, the fluxof air, qa, is unknown, and thus must be approximated somehow using additional assumptions.
One possibility is to assume that advective transport of CO2 in response to the total pressure
gradient is negligible compared to CO2 diffusion, and therefore to assume a stagnant gas phase inwhich only diffusive transport occurs (qa=0). Another possibility is to assume that because of themuch lower viscosity of air in comparison to water, a relatively small pressure gradient will leadto significant gas flow. This assumption seems more realistic in that only rarely will the gas

77phase not be at atmospheric pressure throughout the unsaturated zone. Therefore, under mostconditions, the compressibility of the air can be neglected. With the additional assumptions thatthe air flux is zero at the lower soil boundary and that changes in the total volume of water in thesoil profile caused by water flow must be immediately matched by corresponding changes in thegas volume, we obtain then the following equation [Šimůnek and Suarez, 1993]:xqa ( z ) = qa (0) - qa ( x ) + ∫ S ( x )dx

(5.8)

L- LR

This approach seems reasonable since any water leaving the soil system due to evaporation androot water uptake is matched by an equivalent amount of air entering through the soil surface.
Similarly, any water entering the soil during precipitation and irrigation events will lead to asimilar amount of soil air leaving the soil profile. Only when the soil becomes locally saturated(typically at the soil surface) will air not be able do escape (leading to compressed air below thewetting front).
5.2. Effective Dispersion Coefficient
We define the dispersion coefficients, Dw, and the diffusion coefficient, Da, as

D w = D wsτ w + λ w |qw

θw|= D ws

D a = D asτ a = D as

θ w7 / 3 +p2

θ a7 / 3
p

2

λw|

qw

θw

|

(5.9)

(5.10)

where Das and Dws are the diffusion coefficients [L2T-1] of CO2 in the gas and dissolved phases,
respectively, τa and τw are the tortuosity factors [LL-1] in both phases, respectively, p is porosity[L3L-3] assumed to be equal to the saturated water content, θs, and λw is the dispersivity in thewater phase [L].

78The tortuosity factors τa and τw include not only the tortuosity of the flow paths but also
the amount of air and liquid space available for diffusion, respectively. The tortuosity factors inboth phases are defined in a manner similar to that used by Millington and Quirk [1961]. The
first term of (5.9) represents the diffusion component and the second term the hydrodynamic
dispersion component of the dispersion coefficient. We did not consider mechanical dispersion
in the gas phase since diffusion is the dominant process of CO2 transport in this phase except forvery high air velocities. The diffusion coefficients Das and Dws, as functions of temperature, weretaken from Glinski and Stepniewski [1985].
5.3. Initial and Boundary Conditions
The initial condition for the CO2 concentration in the gas phase is given byca ( x, t ) = cai ( x )t=0(5.11)

where cai(x) is a prescribed function of z [L3L-3].
First-type or third-type boundary conditions may be specified at the surface (or at thebottom) of the soil profile of the form

or
- DE

ca ( x, t ) = ca 0 (t )at x = 0 or x = L(5.12)∂ca+ qE ca = qE 0 ca 0∂x

at x = 0 or x = L(5.13)

respectively, where qE0 is the prescribed effective total CO2 flux [LT-1] and ca0 is concentration[L3L-3] associated with this flux or prescribed at the boundary. This concentration represents theequilibrium concentration of CO2 in the atmosphere (0.035%). The first-type boundary condition(5.12) allows the maximum CO2 flux into the atmosphere. It is difficult to apply the third-typeboundary condition (5.13), since the parameter qE includes both the soil air and soil water fluxes,
which are not known a priori and are obtained from solution of the Richards equation. Anotheroption is to neglect the convective fluxes and to assume that a stagnant boundary layer ofthickness d [L] exists at the soil surface through which the transport of a gas occurs by vapor
diffusion only [Jury et al., 1983,1990; Sleep and Sykes, 1989]. This leads to the following

79equation

- DE

∂ca
D
+ qE ca = as ( cas - catm )∂x
d

at x = L(5.14)

where cas is the concentration in the soil gas at the soil surface [L3L-3] and catm the concentration
at the top of the stagnant boundary layer [L3L-3]. Jury et al. [1983] referred to Das/d as theboundary transfer coefficient [LT-1] and discussed various ways of estimating this coefficient.
At the bottom of the soil profile either a continuous concentration profile may be
assumed, i.e.,∂ca( x, t ) = 0∂x

at x = 0(5.15)or the third-type boundary condition (5.13) may be used, in which case the convective fluxes qa0
and qa are equal to zero as discussed earlier. Boundary condition (5.15) implies that thedispersive flux is equal to zero and that the flux through the lower boundary is only due toconvection. A discussion of the applicability of different types of boundary conditions is given
by Baehr [1987] and Patwardhan et al. [1988], among others.
5.4. Production of Carbon Dioxide
We assume that the individual CO2 production processes are additive (5.16) and that it ispossible to superpose individual processes which reduce production from the optimal value(5.17) [Šimůnek and Suarez, 1993]. The production of CO2 is then considered as the sum of the
production by soil microorganisms, γs [L3L-3T-1], and that by plant roots, γp [L3L-3T-1], asfollows:

S = γ s +γ p

γ s = γ s 0∏ f sii

(5.16)

γ p = γ p 0∏ f pi
i80

(5.17)

∏ f = f ( x ) f (T ) f (c ) f (h) f (hφ ) f (t )ia(5.18)iwhere the subscript s refers to soil microorganisms and the subscript p to plant roots, f(x) is the
reduction coefficient as a function of depth [L-1], f(T) is the reduction coefficient as a function oftemperature [-], f(ca) depends similarly on the CO2 concentration [-], f(h) on the pressure head(or the soil water content) [-], f(hφ) on the osmotic head [-], and f(t) on time [-]. The parameters
γs0 and γp0 represent the optimal CO2 production rates by the soil microorganisms or plant roots,
respectively, for the entire soil profile at 20oC under optimal water, solute and CO2 concentration
conditions [L3L-2T-1] [Šimůnek and Suarez, 1993]. Definitions of the various reduction
coefficients are given by Šimůnek and Suarez [1993].
CO2 production generally decreases rapidly with depth because of less root mass andreadily decomposable organic matter. Glinski and Stepniewski [1985] stated that over 90% of
soil respiration activity is concentrated in the humus horizon of the soil. Many expressions are
possible to relate the dependence of the production term fs(z) on soil depth. One example is anexpression similar to the normalized distribution function β(z) given by van Genuchten [1987]
for root water uptake. Another possibility is to use again an exponential distribution with depth[Raats, 1974]:

f s( x ) = ae -a ( L- x )(5.19)

where a is an empirical constant [L-1]. We assume that at any time t the dependence of CO2
production by plant roots corresponds to the distribution function β(x) used for water uptake byplant roots (see Section 2.2).
The respiration rate of soil microorganisms has been found to decrease at relatively low
as well as at high water contents. Poor accessibility of soil water causes a reduction in CO2
production in relatively dry soils (low pressure heads) [Ekpete and Cornfield, 1965; Wilson and
Griffin, 1975]. The observed reduction of the respiration rate near saturation is explained by theunavailability of oxygen because of the high water content and, therefore, its low diffusion rate
through the soil. Because of this and consistent with the experimental data of Williams et al.
[1972] and Rixon [1968], the CO2 reduction coefficient fs(h) as a function of the soil watercontent for soil microorganisms is expressed as81

f s (h ) =f s (h) = 1

h ε ( h2 ,+∞)

log | h | - log | h3 |log | h2 | - log | h3 |h ε ( h3 , h2 )f s (h ) = 0h ε ( -∞ , h3 )(5.20)

where h2 is the pressure head when CO2 production is optimal [L] and h3 is the pressure head
when production ceases [L]. Note that no reduction in fs(h) occurs close to saturation for pressure
heads above h2 [L]. Rather than treating the oxygen stress with a pressure head relation, it seems
preferable to consider a separate response function f(ca). The dependence of the reduction term
fp(h) on the pressure head is represented by expressions similar to the reduction function αs(h)given by (2.7).
The influence of temperature on chemical processes is described by the Arrhenius
equation [Stumm and Morgan, 1981]. This equation together with the Van 't Hoff equation has
been used successfully by many authors to represent the influence of temperature on soil androot CO2 production. Assuming that f(T)=1 at temperature T20=293.15 K (20oC), the temperature
reduction coefficient can be expressed as⎡ E (T - T20 ) ⎤f (T ) = exp ⎢⎥⎣ RTT20 ⎦(5.21)

where T is absolute temperature [K] and E the activation energy of the reaction [ML2T-2M-1].
The use of the term "reduction coefficient" for f(T) may seem inappropriate since this coefficientis greater than 1 for temperatures above 20oC. We use the term to characterize the change inproduction with temperature, with values greater than 1 above 20 oC and less than 1 below 20
oC.
The dependence of CO2 production on its own concentration (actually O2 deficiency) canbe expressed with the Michaelis-Menton equation [Glinski and Stepniewski, 1985]

82

q=

q max
K
1+ M
cO 2

(5.22)

where KM is the Michaelis constant [L3L-3], i.e., the oxygen concentration, cO2, at which oxygen
uptake is equal to 1/2 qmax, and where q is the oxygen uptake rate and qmax is the maximum
oxygen uptake rate [L3L-3T-1]. Assuming that the respiratory quotient is equal to unity, then the
Michaelis constant for the CO2 concentration is given by KM*= 0.21-KM, while ca = 0.21 - cO2, in
which case the reduction coefficient is given byf (c a ) =0.21 - ca
cO 2=*
cO 2 + K M 0.42 - ca - K m(5.23)

Disadvantage of this expression is that if ca=0 the value for f(ca) is not equal to one. The values
for the optimal production rates γp0 and γs0 must therefore be adjusted accordingly.
The coefficient f(t) introduces a time dependence in the production term. This coefficient
should describe the diurnal and seasonal dynamics of soil and plant respiration. We assume thatthe diurnal dynamics for both soil and plant respiration is sufficiently reflected by thetemperature dependent coefficient f(T) and that the seasonal dynamics of soil production of CO2
is sufficiently described by other reduction coefficients. We hence use this coefficient only forchanges in CO2 production caused by the different growth stage of plants. The coefficient f(t)can be described using the growth degree day, GDD, concept.
Finally, the actual CO2 production rate, PT [L3L-2T-1], is obtained by integrating the CO2
production rate throughout the entire soil profile as follows
L

PT = γ s 0 f s (t ) ∫ f s ( x ) f s ( h ) f s (T ) f s ( ca ) f s ( hφ )dx +o(5.24)Lr+ γ p 0 f p (t ) ∫ f p ( x ) f p ( h ) f p (T ) f p ( ca ) f p ( hφ )dx
083

5.5. Parameter Selection for the CO2 Production Model
Comprehensive reviews of the selection of the values for optimal CO2 production, as wellas coefficients for particular reduction functions, were given by Suarez and Šimůnek [1993] andŠimůnek et al. [1996]. Values of the different reduction coefficients as suggested in thosereviews were used as default values in the graphics-based user interface (see Part B).

84

6. CARBONATE CHEMISTRY
The carbonate chemistry module, as well as the CO2 transport and production module as
described in the previous section, were adopted from the UNSATCHEM software package[Šimůnek et al., 1996], and thus their description closely mirrors material in the UNSATCHEM
manual. Additional details can be found in the original manual [Šimůnek et al., 1996].
When using the ion-association model (and Debye-Hückel activity coefficient
calculations) we assume that the chemical system for predicting major ion solute chemistry of
the unsaturated zone includes 37 chemical species. These species are divided into six groups aslisted in Table 6.1. They include 7 primary dissolved species (calcium, magnesium, sodium,
potassium, sulfate, chloride, and nitrate), 10 complex aqueous species, 6 possible solid phases(calcite, gypsum, nesquehonite, hydromagnesite, sepiolite and dolomite), 4 surface species, 7
species which form the CO2-H2O system, and 3 silica species. The species from the last twogroups could have been included also in other groups (e.g., CO32-, H2SiO42-, and H+ could be
included in the first group). Their consideration into separate groups is mainly due to theirdifferent treatment compared to the other species. For example, complex species of these groups
are considered also at high ionic strength when the Pitzer equations are used to calculate activity
coefficients, while the species of the second group are in that case dropped from the system, as
Table 6.1. Chemical species considered in the carbonate chemistry module.
1

Aqueous components

7Ca2+, Mg2+, Na+, K+, SO42-, Cl-, NO3-

2

Complexed species

10

CaCO3o, CaHCO3+, CaSO4o, MgCO3o, MgHCO3+,
MgSO4o, NaCO3-, NaHCO3o, NaSO4-, KSO4-

3

Precipitated species

6

CaCO3, CaSO4⋅ 2H2O, MgCO3⋅ 3H2O,
Mg5(CO3)4(OH)2⋅ 4H2O, Mg2Si3O7.5(OH) ⋅ 3H2O,
CaMg(CO3)2

4

Sorbed species

4

Ca,M g,N a,K

5

CO2-H2O species

7

PCO2, H2CO3*, CO32-, HCO3-, H+, OH-, H2O

6

Silica species

3

H4SiO4, H3SiO4-, H2SiO42-

85

discussed later. One of the solid phases (dolomite) is not included in the equilibrium systemsince its dissolution is always treated kinetically. Also, exclusion of calcite from the equilibrium
system is optional since its precipitation-dissolution can be treated as a kinetic process. As aresult, either 36 or 35 independent equations are needed to solve this system. In the following
sections we present this set of equations, while the method of solution is discussed in Šimůnek et
al. [1996].
6.1. Mass and Charge Balance Equations
Seven mass balance equations for the primary species in the 1st group and one for thesilica species in the 6th group of Table 6.1 are defined:2+oo+
Ca T = [Ca ] + [CaSO 4] + [CaCO 3 ] + [CaHCO 3 ]2+oo+
Mg T = [Mg ] + [MgSO 4] + [MgCO 3 ] + [MgHCO 3 ]
+

-

-

o

Na T = [ Na ] + [ NaSO 4] + [ NaCO 3] + [ NaHCO3 ]+K T = [K ] + [KSO 4]
2-

o

o

-

-

SO 4 T = [SO 4 ] + [CaSO 4] + [MgSO 4] + [ NaSO 4] + [KSO 4]
Cl T = [Cl ](6.1)

NO 3 T = [ NO 3]
2SiO 4 T = [H 4SiO 4] + [H 3SiO 4] + [H 2SiO 4 ]in which variables with subscript T represent the total analytical concentration in solution of thatparticular species, and where brackets refer to molalities (mol kg-1). Two mass balance equations
for the total analytical concentration of carbonate and bicarbonate are defined as follows
2-

o

o

-

CO 3 T = [CO 3 ] + [CaCO3 ] + [MgCO 3 ] + [ NaCO 3]
-

+

+

o

HCO 3 T = [HCO 3] + [CaHCO3 ] + [MgHCO 3 ] + [ NaHCO 3 ](6.2)

which are used to calculate inorganic alkalinity, Alk (molckg-1):
Alk = 2CO3T + H CO3T + [OH -] - [ H + ]

86

(6.3)

Most chemical and multicomponent transport models consider the total inorganic carbon to be aconservative property [e.g., Westal et al., 1986; Liu and Narasimhan, 1989; Yeh and Tripathi,
1991]. However, this approach can be used only for closed systems. In a soil profile withfluctuating CO2 concentrations, the approach is inappropriate and use of alkalinity as aconservative property is preferable.
In addition to the mass balance equations, the overall charge balance equation for thesolution is given as
2 [Ca 2+ ] + 2 [ Mg 2+ ] + [ Na + ] + [ K + ] + [CaHCO 3+ ] + [ MgHCO 3+ ] + [ H + ] - 2 [CO 32-]-[ HCO -3] - 2 [SO 2-4 ] - [Cl -] - [ NO -3] - [OH -] - [ NaCO -3] - [ NaSO -4 ] - [ KSO -4 ]= 0

6.4)

6.2. CO2 - H2O System
The activities of the species present in solution at equilibrium are related by the massaction equations. The dissociation of water is written as follows

H2 O(H +) (OH -)=KW(H 2 O )H + OH+(6.5)

where KW is the dissociation constant for water [-], while the parentheses denote ion activities.
Methods for calculating ion activities will be discussed later.
The solubility of CO2(g) in water is described with Henry's Law:

H 2CO

*
3

CO 2 (g) + H 2 O

K CO 2 =(H 2CO*3)
P CO 2 (H 2 O)

(6.6)

where the activity of CO2(g) is expressed in terms of the partial pressure PCO2 (atm), KCO2 isHenry's law constant and H2CO3* represents both aqueous CO2 and H2CO3.
Protolysis reactions of dissolved CO2 are written as

87

*

+
H + HCO 3-2+
H + CO 3

H 2CO 3
HCO 3

(H +) (HCO 3- )=K a1
(H 2CO*3)(CO 32- ) (H +)K a2 =(HCO 3- )

(6.7)
(6.8)

where Ka1 and Ka2 are the first and the second dissociation constants of carbonic acid [-],
respectively.
6.3. Complexation Reactions
Each complexation reaction for the species in the second group of Table 6.1 and for thesilica species can be represented by a mass action law:(Ca 2+) (SO 42- )(CaSO o4)

K2 =(Ca 2+) (CO 32- )(CaCO 3o)K3 =(Ca 2+) ( HCO 3- )(CaHCO 3+)(6.9)(Mg 2+) (SO 2-4 )(MgSO o4)K5 =( Mg 2+) (CO 32-)
(MgCO 3o )K6 =(Mg 2+) (HCO 3- )(MgHCO 3+)(6.10)( Na +) (SO 2-4 )( NaSO -4)

K8 =

K9 =

( Na + ) (HCO 3- )( NaHCO 3o )(6.11)

K1=
K4=

K7=

( Na +) (CO 32-)
( NaCO 3- )(K +) (SO 24)
K 10 =(KSO 4)(H +) (H 3SiO -4)
K 11 =(H 4SiO 4)(6.12)
2(H +) (H 2SiO 24)
K 12 =(H 4SiO 4)(6.13)

where Ki are the equilibrium constants of the ith complexed species [-].
6.4. Cation Exchange and Selectivity
Partitioning between the solid and solution phases is described with the Gapon equation[White and Zelazny, 1986]

88

y+

x + 1/ xc i (c j )K ij = x + y + 1/ yc j (c i )(6.14)

where y and x are the valences of species i and j, respectively, and Kij is the Gapon selectivity
coefficient [-]. The adsorbed concentration is expressed in (molckg-1 soil). It is assumed that the
cation exchange capacity cT (molckg-1 soil) is constant and independent of pH.
cT = Σ c i(6.15)

When four cations ( Ca , Mg , Na and K ) are involved in the exchange reactions, thefollowing system of equations results:
2+

2+
+
+
cT =Ca + Mg + Na + K(6.16)

2+(Ca 2+)1/ 2
K 13 = 2+
2+ 1/ 2Ca (Mg )Mg

2++( Na )
Ca+
2+ 1/ 2
Na (Ca )
2+
(K +)
Ca
=
K 15
+
2+ 1/ 2
K (Ca )K 14 =(6.17)

6.5. Precipitation-Dissolution Reactions
The carbonate chemistry module considers four solid phases which, if specified orapproached from oversaturation, must be in equilibrium with the solution: gypsum,nesquehonite, hydromagnesite and sepiolite. Precipitation-dissolution of calcite can be optionallytreated assuming either equilibrium or by means of rate equations. In the latter case the equation
corresponding to calcite equilibrium presented in this section is omitted from the equilibrium
system and the rate of calcite precipitation-dissolution is calculated from the rate equation as
described later. Dissolution of dolomite, which will also be discussed later, is always considered
as a kinetic process and never included in an equilibrium system since ordered dolomite almost
never precipitates in soils. We refer to Suarez and Šimůnek [1997] for a detailed discussion on

89

how to select and consider these solids. The precipitation or dissolution of gypsum, calcite (ifconsidered in the equilibrium system), nesquehonite, hydromagnesite and sepiolite in thepresence of CO2 are described by
CaSO 4 • 2 H 2O

Ca2+

+ SO 24 + 2 H 2O

(6.18)+ 2HCO3−(6.19)+ 2HCO3− + 2H 2O(6.20)

CaCO3 + CO 2 (g) + H 2O
MgCO3 • 3H 2O + CO 2 (g)

Ca
Ca

2+

2+

5 Mg 2+ + 10 HCO3−

Mg 5(CO3) 4(OH ) 2 • 4H 2O + 6 CO2 (g)Mg 2Si 3O7.5(OH) • 3H 2O + 4.5 H 2O + 4 CO 2 (g)(6.21)

2 Mg 2+ + 3H 4SiO 4 + 4 HCO3− (6.22)

with the corresponding solubility products KSP [-] given by2
2+
2G
K SP = (Ca ) (SO 4 ) (H 2O)(6.23)

2−
2+
C
K SP = (Ca ) (CO3 )(6.24)

2+
3
2−
N
K SP = ( Mg ) (CO3 )(H 2O)(6.25)

2−H
K SP = ( Mg ) (CO3 ) (OH ) (H 2O)
2+ 5

K

S
SP

4

2

( Mg 2+ ) 2 ( H 4SiO 4 ) 3 (OH -) 4=
(H 2O) 4.5

4(6.26)(6.27)

where the indexes G, C, N, H, and S refer to gypsum, calcite, nesquehonite, hydromagnesite andsepiolite, respectively.
Substituting (6.5) through (6.8) into (6.27) through (6.27) gives the solubility products
for the carbonate solids expressed in terms of bicarbonate, which is almost always the majorcarbonate ion for conditions (6<pH<10.5) in which the carbonate chemistry module is assumed
to be applicable:90

(H O)(Ca 2+ ) ( HCO3- ) 2 = K CSP K CO 2 K a1 P CO 2 2
K a2
P CO 2N K CO 2 K a1
( Mg 2+ ) ( HCO-3) 2 = K SP2
K a 2 (H 2O)
6

6

- 10
3

(6.29)

6CO 2(Mg ) (HCO ) = K K CO 2 K4 a1 P2
K a2 K w
4.5
4
4
4
2+ 2
- 4
S K CO 2 K a1 P CO 2 (H 2 O)
( Mg ) ( HCO3) = K SP
34
K w ( H 4SiO 4 )2+ 5(6.28)

H
SP(6.30)(6.31)

Expressing the solubility products in this way significantly decreases the number of numericaliterations necessary to reach equilibrium as compared to when equations (6.27) through (6.27)
are used.

6.6. Kinetic Model for Calcite Precipitation-Dissolution
The reaction rates of calcite precipitation-dissolution in the absence of inhibitors such as"foreign ions" and dissolved organic matter, RC (mmol cm-2s-1), were calculated with the rateequation of Plummer et al. [1978]

C

*

+

R = k1 ( H ) + k2 ( H 2CO3) + k3 (H 2O) - k4

K a 2 ( 2+ ) (
HCO3)
C Ca
K SP

(6.32)

where
k4 = k1 +

1
(H

+
S

⎡⎣ k2 ( H 2CO*3) + k3 (H 2O) ⎤⎦)(6.33)

and where k1, k2, and k3 are temperature-dependent first-order rate constants representing theforward reactions (mmol cm-2s-1), and k4 is a function dependent on both temperature and CO2
concentration representing the back reactions (mmol cm-2s-1). The precipitation-dissolution rate
RC is expressed in mmol of calcite per cm2 of surface area per second. The term (HS+) is the H+
activity at the calcite surface. Its value is assumed to be (H+) at calcite saturation where activities

91

of H2CO3* and H2O at the calcite surface are equal to their bulk fluid values [Plummer et al.,1978]. The temperature dependency of the constants k1, k2, k3 is expressed aslog k = a1 +a2T

(6.34)

where values of the empirical constants a1 and a2 are given by Plummer et al. [1978]. For
conditions where pH>8 and pCO2<1000 Pa, an alternative expression for the precipitation rate isused, which is considered more accurate for those conditions [Inskeep and Bloom, 1985]
2+
2C
C
R = - 11.82 [(Ca ) (CO 3 ) - K SP ](6.35)

with an apparent Arrhenius activation energy of 48.1 kJ mol-1 for the precipitation rate constant[Inskeep and Bloom, 1985].
The precipitation or dissolution rate of calcite is reduced by the presence of various
inhibitors. Suarez and Šimůnek [1997] developed the following function for the reduction of the
precipitation-dissolution rates due to surface poisoning by dissolved organic carbon, based on
experimental data from Inskeep and Bloom [1986]
r = exp(- b1x - b 2 x 2 - b 3 x 0.5)

(6.36)

where r is the reduction constant [-], x is dissolved organic carbon (μmol l-1) and b1, b2, and b3
are regression coefficients (0.005104, 0.000426, 0.069111, respectively).
6.7. Kinetic Model of Dolomite Dissolution
The reaction rates of dolomite dissolution, RD (mmol cm-2s-1), were calculated with therate equation of Busenberg and Plummer [1982]
0.5
* 0.5
+ 0.5
D
R = k1 ( H ) + k2 ( H 2CO3) + k3 (H 2O) - k4 ( HCO3 )92(6.37)

where the temperature dependent first-order rate constants k1, k2, k3 (mmol cm-2s-1), representing
the forward reactions, and k4 (mmol cm-2s-1), representing the back reaction, are given by (6.34)
with empirical constants a1 and a2 given by Busenberg and Plummer [1982]. The dissolution rate
RD is again expressed in mmol of dolomite per cm2 of surface area per second. These rate
constants are used for ion activity products IAPD<10-19. For values below 10-19 the rate isextremely small and assumed to be zero [Busenberg and Plummer, 1982] in the absence ofadditional data.
6.8. Silica Concentration in Soil Solution
Relatively little information exists about Si concentrations in soil water. Use of
equilibrium calculations of silica solubility from the stable mineral (quartz) results in the
unrealistic prediction that solution concentrations are independent of pH up to pH 8, above
which the solubility will increase due to the dissociation of silicic acid. Si concentrations in soils
are usually not fixed by quartz solubility but rather by dissolution (and possibly precipitation) ofaluminosilicates including poorly crystallized phases and Si adsorption-desorption onto oxides
and aluminosilicates. As a result of these reactions Si concentrations in soil solutions follow a U
shaped curve with pH, similar to Al oxide solubility, with a Si minimum around pH 7.5 [Suarez,1977]. Suarez [1977] developed a simple relation between silica content in the soil solution andthe soil pH:
SiO 4T = d 1 + d 2 p H + d 3 p H

2(6.38)

in which the empirical constants d1, d2, and d3 are equal to 6340, 1430, and 81.9, respectively,
and where SiO4T is the sum of all silica species expressed in mol l-1. We utilize this expression
and the dissociation expressions for K11 and K12 (eq. (6.12)) only to obtain estimates of H4SiO4from total SiO4. As a result sepiolite reactions are not expressed in terms of H3SiO4- andH2SiO42-, which are not included in the charge balance expressions. Only the species H4SiO4 isused in the carbonate chemistry module.
6.9. Activity Coefficients
6.9.1. Extended Debye-Hückel Expression

93

Activity coefficient in the dilute to moderately saline solution range are calculated using
the extended version of the Debye-Hückel equation [Truesdell and Jones, 1974]:

ln γ = -Az 2 I+ bI
1+ Ba I

(6.39)

where A (kg0.5mol-0.5) and B (kg0.5cm-1mol-0.5) are constants depending only upon the dielectric
constant, density, and temperature, z is the ionic charge in protonic units, a (cm) and b (kg mol-1)
are two adjustable parameters, and I is the ionic strength (mol kg-1):
M

I = 0.5 ∑ z i2 c i(6.40)

i=1

where M is the number of species in the solution mixture. The adjustable parameters a and b forindividual species are given by Truesdell and Jones [1974]. The activities of neutral species arecalculated asln γ = a′ I

(6.41)

where a' is an empirical parameter.
If the extended Debye-Hückel theory is used to calculate activity coefficients, the activityof water is then calculated in the same way as in the WATEQ program [Truesdell and Jones,
1974] using the approximate relationM(H 2 O) = 1 - 0.017 ∑m i

(6.42)

i =1

6.9.2. Pitzer Expressions
At high ionic strength activity coefficients are no longer universal functions of ionic94strength, but depend on the relative concentration of the various ions present in solution [Felmyand Weare, 1986]. The activity coefficients can then be expressed in terms of a virial-type
expansion of the form [Pitzer, 1979]ln γ i = ln γ iDH + ∑ B ij ( I )m j + ∑∑C ijk m j m k + ...
jj(6.43)kwhere γiDH is a modified Debye-Hückel activity coefficient which is a universal function of ionicstrength, and Bij and Cijk are specific coefficients for each interaction. The subroutines forcalculation of the Pitzer activity coefficients were adopted from the GMIN code [Felmy, 1990].
This model is considered accurate also for solutions with very high ionic strength (up to 20 molkg-1), and can be used down to infinite dilution.
If the Pitzer theory is used, then the activity of water is obtained from the expression[Felmy and Weare, 1986]ln (H 2 O) = -W ⎛M ⎞∑mi φ1000 ⎜⎝ i =1 ⎟⎠(6.44)

where W is the molecular weight of water and φ the osmotic coefficient (defined in Section
6.11). We refer to Felmy and Weare [1986] for a more detailed discussion.
6.10. Temperature Effects
Most thermodynamic equilibrium constants depend on both the temperature and pressureof the system. The temperature dependence of the thermodynamic equilibrium constants is often
expressed as a power function of the absolute temperature:log K = a1 +aa2
+ a3T + a4 log T + 52
TT

(6.45)

where T is absolute temperature [K], and a1 through a5 are empirical constants. The pressure
dependence can be neglected for most soils. The empirical constants for the temperature95

dependent thermodynamic constants used in the calculations are listed in Šimůnek et al. [1996].
The temperature dependence of the equilibrium constants for which the constants of equation(6.45) do not exist is expressed with the enthalpy of reaction and the Van 't Hoff expression[Truesdell and Jones, 1974].
6.11. Osmotic Coefficient and Osmotic Pressure Head
We use the semiempirical equation of Pitzer [1973] and co-workers to calculate theosmotic coefficient φ. The osmotic pressure of electrolyte solutions, Pφ (Pa), is related to theosmotic coefficient φ and molality as follows [Stokes, 1979]

Pφ = RTM s ν mφ0
Vs m

(6.46)

where Vs is the partial molar volume of the solvent (cm3mol-1), m0 is unit molality (1 mol kg-1),
and Ms is molar weight (mol-1). The osmotic pressure head, hφ [L], is related to the osmoticpressure by

hφ =

Pφ
ρg

(6.47)

where ρ is the density of water [ML-3] and g is the gravitational constant [L2T-1].
6.12. Effect of Solution Composition on Hydraulic Conductivity
The accumulation of monovalent cations, such as sodium and potassium, often leads toclay dispersion, swelling, flocculation and overall poor soil physico-mechanical properties.
These processes have an adverse effect on the soil hydraulic properties including hydraulicconductivity, infiltration rates and soil water retention as a result of swelling and clay dispersion.
These negative effects are usually explained based on the diffuse double layer theory. A
consequence of the more diffuse double layer in the presence of monovalent ions as compared todivalent ions is the greater repulsion force or swelling pressure between neighboring clay96

platelets. These negative effects become more pronounced with decreasing salt concentrationand valence of the adsorbed ions [Shainberg and Levy, 1992]. In addition, Suarez et al. [1984]
determined that elevated levels of pH also had an adverse effect on the saturated hydraulicconductivity.
The effect of solution chemistry on the hydraulic conductivity in the major ion chemistry
module is calculated as follows
K ( h, pH , SAR, C0 ) = r ( pH , SAR, C0 ) K ( h )(6.48)

where SAR is the sodium adsorption ratio, C0 is the total salt concentration of the ambient
solution in mmolcl-1, and r is a scaling factor which represents the effect of the solution
composition on the final hydraulic conductivity [-], and which is related to pH, SAR and salinity.
The hydraulic conductivity without the scaling factor r can be assumed to be the optimal value
under favorable chemical conditions in terms of optimal pH, SAR and salinity. Although soil
specific, the effects of solution chemistry are too important to ignore. We include reduction
functions calculated for some illitic soils of California based on the experimental work of
McNeal [1968] and Suarez et al. [1981]. The overall scaling factor r in equation (6.48) for thispurpose is divided into two partsr ( pH , SAR, C0 ) = r1 ( SAR, C0 ) r2 ( pH )(6.49)

where the first part, r1 [-], reflects the effect of the exchangeable sodium percentage and dilutionof the solution on the hydraulic conductivity, while the second part, r2 [-], represents the effect of
the soil solution pH. The first term is based on a simple clay-swelling model, which treatsmixed-ions clays as simple mixture of homoionic sodium and calcium clay. Clay swelling is thenrelated to a decrease in soil hydraulic conductivity [McNeal, 1974]. The r1 term was defined by
McNeal [1968] ascx n
r1 = 1 1 + cx n

(6.50)

where c and n are empirical parameters, and x is a swelling factor. The interlayer swelling of soil

97montmorillonite, x, is defined in the following way
x = f mont 3.6 •10−4 ESP*d *(6.51)

where fmont is the weight fraction of montmorillonite in the soil, d* is the adjusted interlayer
spacing [L] and ESP* is the adjusted exchangeable sodium percentage. For most soils one canuse the assumption that fmont = 0.1 [McNeal, 1968]. The adjusted exchangeable sodium
percentage is calculated as
ESP* = max[0, ESP-(1.24 + 11.63 log C0 )](6.52)

where C0 is total salt concentration of the ambient solution in mmolcl-1, and ESP is defined as

ESP = 100

Na
CEC

(6.53)

where CEC is the soil cation exchange capacity (mmolckg-1) and Na the exchangeable sodium
concentration (mmolckg-1). The adjusted interlayer spacing, d*, is given by
*

d =0
*
-1/ 2
d = 356.4 C 0 + 1.2

for C0 > 300 mmol c liter -1
for C0 < 300 mmol c liter -1(6.54)

McNeal [1968] reported that the values of the empirical factor n in equation (6.50) depends
primarily on the soil ESP and that as a first approximation n values may be estimated usingn =1

for ESP < 25

n=2

for 25 ≤ ESP ≤ 50

n=3

for ESP > 50

(6.55)

Only the values of the empirical factor c vary from soil to soil. HYDRUS uses values reported

98

by McNeal [1968]:
c = 35c = 932

for ESP < 25
for 25 ≤ ESP ≤ 50

c = 25 000

for ESP > 50

(6.56)The reduction factor, r2, for the effect of pH on hydraulic conductivity was calculated
from experimental data of Suarez et al. [1984] after first correcting for the adverse effects of low
salinity and high exchangeable sodium using the r1 values. The following equation was used
r2 =1
r 2 = 3.46 - 0.36 p H

for p H< 6.83

r 2 = 0.1

for p H > 9.3

for 6.83 ≤ p H≤ 9.3(6.57)

Note, that although the models for reductions in the hydraulic conductivity due tochanges in the solution composition were derived from data on the saturated hydraulicconductivity, the same reduction factors were used for the entire range of the pressure heads. The
assumption that the r values for saturated conditions can be applied to the entire range ofpressure heads has not been closely examined thus far in the literature.

99

100

7. NUMERICAL SOLUTION OF THE VARIABLY SATURATED FLOW EQUATION
7.1. Space and Time Discretization
The soil profile is first discretized into N-1 adjoining elements, with the ends of theelements located at the nodal points, and N being the number of nodes. The same spatial
discretization is used for water flow, solute transport and heat movement. HYDRUS assumes
that the vertical coordinate x is directed positive upward.
A mass-lumped linear finite elements scheme was used for discretization of the mixed
form of the Richards' equation (2.1) (the numerical solution for equation (2.3) is in principle
similar to the solution to equation (2.1)). Since the mass-lumped scheme results in an equivalentand somewhat standard finite difference scheme [e.g., Vogel et al., 1996], we omit the detailed
finite element development and give immediately the invoked final finite difference scheme:

θij +1, k +1

Δt-θ i j=j +1, k +1
1 ⎛ j +1,k h ij++11, k +1 - h ij +1, k +1
- h ij-1+1,k +1 ⎞j +1, k h i
K
K
i
+
1/
2
i
-1/
2
⎜
⎟+
Δx ⎝
Δxi
Δ x i -1⎠

+K

j +1, ki +1/ 2

+1, k
- K ij-1/
2
cos α - S ij
Δx

(7.1)

where
Δt = t j +1 - t j
x i +1 - x i -1
Δ x i = x i +1 - x i
Δ x i -1 = x i - x i -12
j +1, kj +1, kj +1, kj +1, kK i +1 + K i
K i + K i -1
j +1, kj +1, k==K i +1/ 2
K i -1/ 2
2
2
Δx =

(7.2)

in which subscripts i-1, i, and i+1 indicate the position in the finite difference mesh; superscripts
k and k+1 denote the previous and current iteration levels, respectively; and superscripts j andj+1 represent the previous and current time levels, respectively. Equation (7.1) is based on a
fully implicit discretization of the time derivative, and will be solved with a Picard iterative
solution scheme. Notice also that the sink term, S, is evaluated at the previous time level. The
mass-conservative method proposed by Celia et al. [1990], in which θj+1,k+1 is expanded in atruncated Taylor series with respect to h about the expansion point hj+1,k, is used in the timedifference scheme of (7.1):

101θ ij +1,k +1 - θ ij =ΔtC

j +1, ki

j +1, k +1

hi

- h ij +1,k θ ij +1, k - θ ij
+
Δt
Δt

(7.3)

where Ci represents the nodal value of the soil water capacity [L-1]:

C

j +1, kidθ=dh

j +1, k(7.4)

This method has been shown to provide excellent results in terms of minimizing the massbalance error. Notice that the second term on the right hand size of (7.3) is known prior to thecurrent iteration. The first term on the right hand side of (7.3) should vanish at the end of theiteration process if the numerical solution converges. The derivation leads to the following
matrix equation with matrix [Pw] and vectors {h} and {Fw}
[ P w] j +1,k{h} j +1, k +1 = {F w}(7.5)

The symmetrical tridiagonal matrix [Pw] in (7.5) has the form

[P w ] =0d 1 e1 00
e1 d 2 e 2 00 e 2 d 3 e3 0
0
.
.
.
.
.
.
0
0 e N -3 d N -2 e N -2 000 e N -2 d N -1 e N -1
0
0 e N -1 d N(7.6)

where the diagonal entries di and above-diagonal entries ei of the matrix [Pw], and the entries fi ofvector {Fw}, are given by

102

di =

Δx j +1,k K ij++11,k + K ij +1,k K ij +1,k + K ij-1+1,k
+
Ci +
Δt
2Δ x i2Δ x i -1+ K ij++11,k2Δ x i(7.7)

j +1, kKi
ei = -fi=

Δx j +1,k j +1,k Δx j +1, k j K ij++11,k - K ij-1+1, k(θ i - θ i ) +cos α - S ijΔx
C i hi Δt
Δt
2

(7.8)

(7.9)

The tridiagonal matrix [Pw] is symmetric and therefore the below-diagonal entries are equal tothe above-diagonal entries. The entries d1, e1, f1, and eN-1, dN, fN are dependent upon theprescribed boundary conditions.
7.2. Treatment of Pressure Head Boundary Conditions
If a first-type (Dirichlet) boundary condition is specified at the top or bottom of the soil
profile, then the terms d1 or dN are equal to unity, e1 or eN-1 reduce to zero, and f1 or fN equal tothe prescribed pressure head, h0. Some additional rearrangement of matrix [Pw] is also necessary
to preserve its symmetry. The appropriate entries in the second or (N-1)st equations containing
the prescribe boundary pressure head h0 in the left-hand side matrix must then be incorporated
into the known vector on the right-hand side of the global matrix equation. When done properly,
this rearrangement will restore symmetry in [Pw].
7.3. Treatment of Flux Boundary Conditions
If a third-type (Neumann) boundary condition at the bottom of the profile is specified,then the individual entries are obtained by discretization of Darcy's law, i.e.,q = -K∂h-K∂xsuch that d1 and f1 in [Pw] attain the values

103(7.10)

f1=+ K 2j +1,k
2 Δ x1(7.11)+ K 2j +1, k
j +1
+ q0
2

(7.12)

j +1, kK1

d1 =j +1, kK1

where q0 is the prescribed bottom boundary flux [LT-1] and where e1 is described by (7.8). A
similar discretization of Darcy's law is possible to incorporate flux boundary condition at the topof the soil profile. This approach, however, can quickly lead to relatively unstable solutions
when the boundary fluxes at the soil surface vary strongly with time (erratic irrigation or rainfall
rates). A more stable and mass-conservative solution results when the mass balance equation
instead of Darcy's law is discretized.
∂θ
∂q
=- -S
∂t
∂x

(7.13)

2(q Nj +1 - q Nj +-1/1,k2) j-SN=Δ x N -1(7.14)

Discretization of (7.12) gives
j +1, k +1

θNΔt

- θ NjExpanding the time derivative on the left hand side of (7.14) as in (7.3), and using the discretized
form of Darcy's law for qN-1/2 leads to
Δ x N -1 j +1, k K Nj +1, k + K Nj +-11,k
dN =
CN +
2Δt
2Δ x N -1
fN=(7.15)

j +1, kΔ x N -1 j +1, k j +1, k Δx
+ K Nj +-11,k
Δ x N -1 j
j +1
(7.16)(θ Nj +1,k - θ Nj ) - K N
cos α C N hN S N - qN
2Δt
2Δt
2
2

where qN is the prescribed soil surface boundary flux. Implementation of a third-type boundary
condition always preserves symmetry of the matrix [Pw].

1047.4. Numerical Solution Strategy
7.4.1. Iterative Process
Because of the nonlinear nature of (7.5), an iterative process must be used to obtainsolutions of the global matrix equation at each new time step. For each iteration a system of
linearized algebraic equations is first derived from (7.5), which, after incorporation of theboundary conditions, is solved using Gaussian elimination. The Gaussian elimination process
takes advantage of the tridiagonal and symmetric features of the coefficient matrix in (7.5). After
solving (7.5) the first time, the coefficients in (7.5) are re-evaluated using this first solution, andthe new equations are again solved. The iterative process continues until a satisfactory degree of
convergence is obtained, i.e., until at all nodes in the saturated (or unsaturated) region theabsolute change in pressure head (or water content) between two successive iterations becomesless than some small value determined by the imposed absolute pressure head (or water content)
tolerance. The first estimate (at zero iteration) of the unknown pressure heads at each time step isobtained by extrapolation from the pressure head values at the previous two time levels.
7.4.2. Time Control
Three different time discretizations are introduced in HYDRUS: (1) time discretizations
associated with the numerical solution, (2) time discretizations associated with theimplementation of boundary conditions, and (3) time discretizations which provide printed
output of the simulation results (e.g., nodal values of dependent variables, water, solute massbalance components, and other information about the flow regime).
Discretizations 2 and 3 are mutually independent; they generally involve variable time
steps as described in the input data file. Discretization 1 starts with a prescribed initial timeincrement, Δt. This time increment is automatically adjusted at each time level according to thefollowing rules [Mls, 1982; Šimůnek et al., 1992]:
a. Discretization 1 must coincide with time values resulting from time discretizations 2
and 3.
b. Time increments cannot become less than a preselected minimum time step, Δtmin, nor
exceed a maximum time step, Δtmax (i.e., Δtmin ≤ Δt ≤ Δtmax).

105

c. If, during a particular time step, the number of iterations necessary to reachconvergence is ≤3, the time increment for the next time step is increased bymultiplying Δt by a predetermined constant >1 (usually between 1.1 and 1.5). If thenumber of iterations is ≥7, Δt for the next time level is multiplied by a constant <1(usually between 0.3 and 0.9).
d. If, during a particular time step, the number of iterations at any time level becomes
greater than a prescribed maximum (usually between 10 and 50), the iterative process
for that time level is terminated. The time step is subsequently reset to Δt/3, and theiterative process restarted.
7.4.3. Atmospheric Boundary Conditions and Seepage Faces
Atmospheric boundaries are simulated by applying either prescribed head or prescribed
flux boundary conditions depending upon whether equation (2.69) or (2.70) is satisfied[Neuman, 1974]. If (2.70) is not satisfied, boundary node n becomes a prescribed head boundary.
If, at any point in time during the computations, the calculated flux exceeds the specified
potential flux in (2.69), the node will be assigned a flux equal to the potential value and treatedagain as a prescribed flux boundary.
If a seepage face is considered as the lower boundary condition and if during each
iteration the lower part of the soil profile is saturated then the last node is treated as a prescribedpressure head boundary with h=0. However, if this node is unsaturated then a prescribed flux
boundary with q=0 is imposed at the lower boundary. Alternatively, a certain non-zero value ofhSeep can also be specified as the limiting pressure head.
7.4.4. Water Balance Computations
The HYDRUS code performs water balance computations at prescribed times for several
preselected subregions of the flow domain. The water balance information for each subregion
consists of the actual volume of water, V, in that subregion, and the rate, O [LT-1], of inflow or
outflow to or from the subregion. These variables V and O are evaluated in HYDRUS by meansof106V = ∑Δxi

θ i + θ i+1e(7.17)2

andO=

V new - V old
Δt(7.18)

respectively, where θi and θi+1 are water contents evaluated at the corner nodes of element e, Δxi
is the size of the element, and Vnew and Vold are volumes of water in the subregion computed at
the current and previous time levels, respectively. The summation in (7.17) is taken over all
elements within the subregion. Similar calculations are carried out for the mobile and immobile
regions of the dual-porosity model and for the matrix and fracture regions of the dualpermeability model.
The absolute error in the mass balance of the flow domain is calculated as
t

t

0

0

ε = V t - V 0 + ∫T a dt - ∫ (q 0 - q N ) dt
w
a

(7.19)

where Vt and V0 are the volumes of water in the flow domain, Eq. (7.17), evaluated at times t andzero, respectively. The third term on the right-hand side of (7.19) represents the cumulative root
water uptake amount, while the fourth term gives the net cumulative flux through bothboundaries.
The accuracy of the numerical solution is evaluated by the relative error, εrw [%], in thewater mass balance as follows:

ε rw =| ε aw |⎛max ⎜ ∑ | V te - V 0e | ,⎝ e⎞∫0 T a dt + ∫0 ( | q N | + | q 0 | ) dt ⎟⎠
t

t

100

(7.20)

where Vte and V0e are the volumes of water in element e at times t and zero, respectively. Notethat HYDRUS does not relate the absolute error to the volume of water in the flow domain, butinstead to the maximum value of two quantities. The first quantity represents the sum of the107absolute changes in water content over all elements, whereas the second quantity is the sum of
the absolute values of all fluxes in and out of the flow domain.

7.4.5. Computation of Nodal Fluxes
Components of the Darcian flux are computed at each time level during the simulation
only when the water flow and solute (or heat) transport equations are solved simultaneously.
When the flow equation is being solved alone, the flux components are calculated only at
selected print times. The x-components of the nodal fluxes are computed for each node n
according to⎛ h j +1 - h1j +1
⎞
j +1
+ 1⎟
q1 = - K 1j++1/1 2 ⎜ 2⎝ Δ xi⎠⎛ j +1 - j +1 ⎞⎛ j +1 - j +1⎞
- K ij++1/1 2 ⎜ h i +1 h i + 1⎟ Δ x i -1 - K ij-1/+12 ⎜ h i h i -1 + 1⎟ Δ x i
j +1⎝ Δ xi⎠⎝ Δ x i -1⎠
qi =Δ x i -1 + Δ x i⎞⎛ h j +1 - h Nj +-11 ⎞ Δ x N -1 ⎛ θ Nj +1 - θ Nj
j +1
+ S Nj ⎟+ 1⎟ q N = - K Nj +-1/1 2 ⎜ N⎜
2 ⎝ Δt⎠⎝ Δ x N -1⎠

(7.21)

7.4.6. Water Uptake by Plant RootsHYDRUS considers the root zone to consist of all nodes, n, for which the potential root
water uptake distribution, b (see Section 2.2), is greater than zero. The root water extraction rateis assumed to vary linearly over each element. The values of actual root extraction rate Si in (7.1)
are evaluated with (2.8). HYDRUS calculates the total rate of transpiration using the equation
T a = ∑Δ x ie

S i + S i +1
2

(7.22)

in which the summation takes place over all elements within the root zone, and where Si and Si+1
are the root water uptake rates evaluated at the corner nodes of element e.

108

7.4.7. Evaluation of the Soil Hydraulic Properties
At the beginning of a simulation, HYDRUS generates for each soil type in the flow
domain a table of water contents, hydraulic conductivities, and specific water capacities from thespecified set of hydraulic parameters [Vogel, 1987]. The values of θi, Ki and Ci in the table are
evaluated at prescribed pressure heads hi within a specified interval (ha, hb). The entries in thetable are generated such thath i +1= constanthi(7.23)

which means that the spacing between two consecutive pressure head values increases in alogarithmic fashion. Values for the hydraulic properties, θ(h), K(h) and C(h), are computed
during the iterative solution process using linear interpolation between the entries in the table. If
an argument h falls outside the prescribed interval (ha, hb), the hydraulic characteristics areevaluated directly from the hydraulic functions, i.e., without interpolation. The above
interpolation technique was found to be much faster computationally than direct evaluation of
the hydraulic functions over the entire range of pressure heads, except when very simple
hydraulic models are used.

109

110

8. NUMERICAL SOLUTION OF THE SOLUTE TRANSPORT EQUATIONThe Galerkin finite element method was used to solve the solute and heat transport
equations subject to appropriate initial and boundary conditions. Since the heat transport
equation (4.1) has the same mathematical form as the (linearized) solute transport equation(3.10), the numerical solution will be given here only for solute transport.
8.1. Space Discretization
The finite element method assumes that the dependent variable, the concentration
function c(x,t), can be approximated by a finite series c’(x,t) of the formNc’( x, t ) = ∑φ m( x) c m(t )

(8.1)

m =1

where φm are the selected linear basis functions that fulfill the condition φm(xn)=δnm, δnm is
Kronecker delta (δnm=1 for m=n, and δnm=0 for m≠n), cm are the unknown time-dependent
coefficients which represent solutions of (3.10) at the finite element nodal points, and N is thetotal number of nodal points. Linear basis functions have the following form:

φ 1 = 1- ξ
φ2 =ξ

(8.2)

where ξ is the distance in the local coordinate system [-]. In the global coordinate system ξ isdefined asξ=x - x1Δxx1 ≤ x ≤ x2

(8.3)

where Δx (=x2-x1) is the size of a finite element [L], i.e., the distance between two neighboring
nodal points. The approximate solution c’(x,t) converges to the correct solution c(x,t) as thenumber of basis functions N increases.

111

Application of the Galerkin method which postulates that the differential operator
associated with the transport equation is orthogonal to each of the N basis functions, we obtainthe following system of N time-dependent differential equations with N unknown values cn(t).
⎡ ∂θ R1c
⎤
∂
∂c
2 ∂c
∫0 ⎢⎣- ∂t - θ R ∂t + ∂x ( E ∂x - Bc) + Fc + G ⎥⎦ φ n dx = 0
L

(8.4)

where, for notational convenience we have dropped the index k referring to the kth decay chain
number. Integrating by parts the terms containing spatial derivatives leads to the following
equation⎡ ∂θ R1c⎤2 ∂c∫0 ⎢⎣- ∂t - θ R ∂t + Fc + G ⎥⎦ φ n dx L

⎛ ∂c
⎞ ∂φ
− ∫ ⎜ E - Bc ⎟ n dx - q sL φ n ( L) + q s 0 φ n (0) = 0
∂x
⎠ ∂x
0⎝
L

(8.5)

where qs0 and qsL are solute fluxes across the lower and upper boundaries, respectively. By
substituting (8.1) for c(x,t) we obtain⎡ ∂θ R1cm⎤∂c∑e ∫ ⎢- ∂t φm - θ R 2 ∂tm φm - Fcmφm + G ⎥ φ n dx ⎦
0 ⎣
Le

∂φ
⎛
⎞ ∂φ
− ∑ ∫ ⎜ Ecm m - Bcmφm ⎟ n dx - q s Lφn ( L) + q s 0φn (0) = 0
∂x
⎠ ∂x
e 0 ⎝
Le

(8.6)

Equation (8.6) can be rewritten in matrix form asd ([Q1]{c})d{c}+ [Q 2]+ [ S ]{c} = { f }dtdtwhere the vector {c} contains the unknown values of the nodal concentrations, and where

112

(8.7)

Q

1
nm

Le

= ∫θ R1φ mφ ndx

(8.8)

0

Le

Q nm = ∫θ R 2 φ mφ n dx

(8.9)

dφ m dφ n
dφ- B n φ m - F φ mφ n ]dx
dx dx
dx

(8.10)

20

Le

S nm = ∫ [ E0Le

f n = ∫Gφ ndx - q sL φ n( L) + q s 0φ n (0)(8.11)

0q s = -θ D∂c’
+ q c’∂x(8.12)In addition to the basic assumptions involving the Galerkin method, several additional
assumptions are now made (van Genuchten [1978]). First, within each element and at a given
time, the different coefficients or groups of coefficients in equations (8.12) through (8.12) (i.e.,
θR, θD, q, F, and G) are assumed to change linearly according to the expressions:2θ R(t , x) = ∑θ R(t , x m) φ m( x)m =12

E (t , x) = ∑E (t , x m) φ m( x)m =12

B(t , x) = ∑B(t , x m) φ m( x)(8.13)m =1
2

F (t , x) = ∑F (t , x m) φ m( x)m =12

G (t , x) = ∑G (t , x m) φ m( x)m =1Because of (8.13) it is now not necessary to use numerical integration for evaluating the113coefficients from equation (8.7). Second, mass lumping will be invoked by redefining the nodal
values of the time derivative in (8.4) as weighted averages over the entire flow region:∂c′

L∫θ R ∂t φ dxn

dc n 0= L
dt(8.14)∫θ Rφ dxn0

The above expansions lead to the following element matrices associated with global matrix
equation (8.7). Note that [S] = [S1] + [S2] + [S3].
2

Q nm = ∫θ R φ mφ n dx =
e

1

2

S

e
nm 1

= ∫E
1

Δx 3 θ 1 R1 + θ 2 R 2
12 θ 1 R1 + θ 2 R 2

θ 1 R1 + θ 2 R 2
θ 2 R 2 + 3θ 2 R 2(8.15)

dφ m dφ n
1 E1 + E 2 - E1 - E 2
dx =
dx dx
2Δx - E 1 - E 2 E 2 + E 2(8.16)dφ n
1 2 B1 + B 2
φ mdx =
6 -2 B1 - B 2dx

B1 + 2 B 2- B1 - 2 B 2(8.17)

Δx 3F 1 + F 2
12 F 1 + F 2

F1 + F 2
F 1 + 3F 2

(8.18)

2

S nm 2 = ∫Be

1

2

S nm 3 = ∫F φ mφ ndx =e

12

f n = ∫G φ n dx =e1

Δx 2 G1 + G 2
6
G1 + 2 G 2(8.19)

8.2. Time Discretization
The Galerkin method is used only for approximating the spatial derivatives while thetime derivatives are discretized by means of finite differences as follows

114

[Q1] j +1{c} j +1 -[Q1] j{c} j{c} j +1 -{c} j+ [Q 2] j + ε+ΔtΔtε [ S ] j +1{c} j +1 + (1- ε )[ S ] j{c} j = ε { f } j +1 + (1- ε ){ f } j(8.20)

where j and j+1 indicate previous and actual time level and Δt is time step, and where ε is atemporal weighting coefficient. Different finite difference schemes results depending upon thevalue of ε (=0: explicit scheme, =0.5: Crank-Nicholson scheme, =1.: fully implicit scheme).
Equation (8.20) can be rewritten as:[ P s ]{c} j +1 = [T ]{c} j + {R }(8.21)

where

(

)

1
[Q1] j + 1 + [Q 2] j +ε + ε [ S ] j +1
Δt1
[T ] =[Q1] j + [Q 2] j +ε - (1- ε ) [ S ] j
Δt{R} = ε { f } j +1 + (1- ε ){ f } j[ P s] =()(8.22)

Notice that we separated the retardation factor R into two parts, R1 and R2, leading to two
matrices, [Q1] and [Q2], which are evaluated at different time levels. This approach was found tolead to much faster convergence when nonlinear adsorption isotherm is considered. Matrix [Q1]is evaluated at the previous and current time levels, while matrix [Q2] is evaluated using
weighted averages of the current and previous nodal values of θ and R.
Higher-order approximations for the time derivative in the transport equation were
derived by van Genuchten [1976, 1978]. The higher-order approximations may be incorporated
into the transport equation by introducing time-dependent dispersion corrections as follows

D = D-2q Δt2
6θ ( R1 + R 2)2

q Δt
D = D+ 2 1
6θ ( R + R 2)+

115

(8.23)

where the superscripts + and - indicate evaluation at the old and the new time levels,
respectively.
Evaluation of all integrals eventually leads to the following tridiagonal global matrices[Ps] and [T]
d 1 e1 0
b2 d 2 e2 0
0 b3 d 3 e3 0
.

.

.

.

.

.

[Ps ] =
0
0
0

0
0
0

(8.24)

0 b n-2 d n -2 e n -2 0
0 b n-1 d n-1 e n-1
0 bn d nwith the individual entries of [Ps] given in Table 8.1. In this table Δxi-1=xi-xi-1, Δxi=xi+1-xi,

Δx=(xi+1-xi-1)/2, Δt=tj+1-tj, i is the nodal index (increasing in the direction of the x-coordinate,i=1,2,....,n), and j is the time index. Individual entries of the vector {R} have the following form
j +1
jr 1 = Δ x i -1(2s1 + s 2) + ε q s (0, t ) + (1- ε )q s (0, t )r i = Δ x i -1 ( s i -1 + 2 s i ) + Δ x i (2 s i + s i +1)

r n = Δ x i ( s n-1 + 2 s n) - ε q s ( L, t ) - (1- ε ) q s ( L, t )1
j +1
js i = [ε G i + (1- ε )G i ]
6
j +1

j

(8.25)

From equation (8.22) it follows that matrixes [Ps] and [T] are identical if the variables D-,
F, q and ε in [Ps] are replaced by -D+, -F, -q and (1- ε) to yield [T].

116

Table 8.1. Values of the diagonal entries di, and off-diagonal entries bi and ei of matrix [Ps] for linear finite elements.

Δ x1
ε
ε
ε Δ x1(3θ 1R1 + θ 2 R 2) +( E 1 + E 2) + (2 B1 + B 2) +(3F 1 + F 2)
12Δt
2 Δ x16
12(8.26)

Δ x i -1εεεΔ x i -1(θ i -1R i -1 + θ i R i ) ( E i -1 + E i ) - (2 B i -1 + B i ) +( F i -1 + F i )12Δt
2Δ x i -1
612(8.27)

Δ x i -1Δ xiεε(θ i -1R i -1 + 3θ i R i ) +(3θ i R i + θ i +1R i +1) +( E i -1 + E i ) +( E i + E i +1) +
12Δt
12Δt
2Δ x i -1
2Δ x iεεΔ x i -1
εΔ x i( F i -1 + 3F i ) +(3F i + F i +1)(i = 2,..., n -1)+ ( B i +1 - B i -1) +61212(8.28)Δ xi
εεεΔ x i(θ i R i + θ i +1R i +1) ( E i + E i +1) + (2 B i +1 + B i ) +( F i + F i +1)
12Δt
2Δ x i612(8.29)

Δ x N -1εεεΔ x N -1(θ N -1R N -1 + 3θ N R N ) +( E N -1 + E N ) - ( B N -1 + 2 B N ) +( F N -1 + 3F N )12Δt
2Δ x N -1
612(8.30)

d1 =

bi =

di =

ei =

dN =

117

Table 8.2. Values of the diagonal entries di, and off-diagonal entries bi and ei of matrix [Ps] for linear finite elements with upstream
weighting.

Δ x1
ε
ε
εΔ x1
(3θ 1R1 + θ 2 R 2) +( E 1 + E 2) + [(2 + 3α + ) B1 + B 2] +(3F 1 + F 2)
12Δt
2 Δ x16
12(8.31)

Δ x i -1εεεΔ x i -1(θ i -1R i -1 + θ i R i ) ( E i -1 + E i ) - [(2 + 3α -) B i -1 + B i ] +( F i -1 + F i )12Δt
2Δ x i -1
612(8.32)

Δ x i -1Δ xiεε(θ i -1R i -1 + 3θ i R i ) +(3θ i R i + θ i +1R i +1) +( E i -1 + E i ) +( E i + E i +1) +
12Δt
12Δt
2Δ x i -1
2Δ x iεεΔ x i -1
εΔ x i( F i -1 + 3F i ) +(3F i + F i +1) (i = 2,..., n -1)+ [ B i +1 + 3B i (α + + α -) - B i -1] +6
12
12(8.33)Δ xi
εεεΔ x i(θ i R i + θ i +1R i +1) ( E i + E i +1) + [(2 - 3α + ) B i +1 + B i ] +( F i + F i +1)
12Δt
2Δ x i612(8.34)

Δ x N -1εεεΔ x N -1(θ N -1R N -1 + 3θ N R N ) +( E N -1 + E N ) - [ B N -1 + (2 - α -) B N ] +( F N -1 + 3F N )12Δt
2Δ x N -1
612(8.35)

d1 =

bi =

di =

ei =

dN =

118

8.3. Numerical Solution for Linear Nonequilibrium Solute Transport
The same solution procedure as described in Sections 8.1 and 8.2 is used here for either
linear equilibrium or nonlinear (both equilibrium and nonequilibrium) solute transport. However,linear nonequilibrium transport is implemented somewhat differently. First, equation (3.9),
simplified for linear adsorption, is discretized using finite differences as follows

s

t+Δt

- st
= ε [ω (1- f )k s c - ω s k - μs s k + (1- f )γ ]t +Δt +Δt+ (1- ε )[ω (1- f )k s c - ω s k - μs s k + (1- f )γ ]t(8.36)The new adsorbed concentration for type-2 sorption sites follows directly from (8.35):

s

t +Δt

2 - Δt (ω + μs ) tΔt (1- f )[(ωk s c ) t +Δt + (ωk s c ) t + γ t +Δt + γ t ]=s+
2 + Δt (ω + μs ) t +Δt
2 + Δt (ω + μs ) t +Δtt(8.37)

This term is incorporated directly into F and G so that they have the following values:

F

G

t +Δt
*

=G

{′t +Δt }

+ ρω

t +Δt
*

=F

t +Δt

⎧ ρωΔt (1- f )ωk s ⎫-⎨⎬⎩ 2 + Δt (ω + μs ) ⎭t +Δt

2 - Δt (ω + μs ) tΔt (1- f )[(ωk s c ) t + γ t +Δt + γ t ]{s}+
2 + Δt (ω + μs ) t +Δt
2 + Δt (ω + μs ) t +Δt

t +Δt

t

(8.38)(8.39)

where F*t+Δt and G*t+Δt are the values of parameters F and G for linear nonequilibrium solutetransport, and Ft+Δt' and Gt+Δt' are the original values of F and G. The above procedure avoids
having to solve two simultaneous equations for linear nonequilibrium transport. Once thetransport equation with the modified F and G parameters is solved using the methods discussed
earlier to yield the concentration ct+Δt, equation (8.37) is used to update the adsorbed
concentration st+Δt.
For physical nonequilibrium (dual-porosity) transport, equation (3.32), simplified forlinear adsorption, is discretized using finite differences as follows

119

A

t + Δtt
c im - c im
= ε [ω ( c - cim ) - Bcim + E ]t + Δt + (1- ε )[ω ( c - cim ) - Bcim + E ]tΔt
where
A = θim + (1- f ) ρ k s(8.40)

B = θim μw + (1- f ) ρ k s μs
E = θimγ w + (1- f ) ργ s
The new concentration in the immobile region follows directly from (8.39):ct + Δtim

2 A - Δt (ω + B )tΔt[(ω cim )t + Δt + (ωcim )t + E t + Δt + E t ]=c+
2 A + Δt (ω + B )t + Δt
2 A + Δt (ω + B )t + Δtt
im

(8.41)

Similarly as for the chemical nonequilibrium case, equation (8.40) is incorporated directly into F
and G to obtain following values:

F

G

t +Δt
*

t +Δt
*

=F

t
= G{′t +Δt } + ω t +Δt{c im

t +Δt

⎧
⎫
ωΔtω
-⎨
⎬
⎩ 2 A + Δt (ω + B) ⎭t +Δt

2 A - Δt (ω + B) t
Δt[(ω c) t + E t +Δt + E t ]+}2 A + Δt (ω + B) t +Δt
2 A + Δt (ω + B)t +Δt(8.42)(8.43)

Numerical approaches very similar to those described above for the chemical nonequilibrium
model were used also for the attachment/detachment model for colloid transport.
8.4. Numerical Solution Strategy
8.4.1. Solution Process
The solution process at each time step proceeds as follows. First, an iterative procedure isused to obtain the solution of the Richards' equation (2.1) (see Section 7.4.1). After achieving
convergence, the solution of the transport equation (8.7) is implemented. This is done by first120

determining the nodal values of the fluid flux from nodal values of the pressure head by applying
Darcy's law. Nodal values of the water content and the fluid flux at the previous time level are
already known from the solution at the previous time step. Values for the water content and thefluid flux are subsequently used as input to the transport equations (first for heat transport andthen for solute transport), leading to a system of linear algebraic equations given by (8.7). The
structure of the final set of equations depends upon the value of the temporal weighing factor, ε.
The explicit (ε =0) and fully implicit (ε =1) schemes for the transport equation require that the
global matrices [Ps] and [T] and the vector {R} be evaluated at only one time level (the previous
or current time level). All other schemes require evaluation at both time levels. Also, all schemes
except for the explicit formulation (ε =0) lead to an asymmetric banded matrix [Ps].
Since the heat transport equation is linear, there is no need for an iterative solution
process for heat flow. The same is true for the transport of solutes undergoing only linear
sorption reactions. On the other hand, iteration is needed when a nonlinear reaction between thesolid and liquid phase is considered. The iterative procedure for solute transport is very similar
to that for water flow. The nonlinear coefficients in (8.7) are then re-evaluated at each iteration,and the new equations solved using results of the previous iteration. The iterative processcontinues until a satisfactory degree of convergence is obtained, i.e., until at all nodes theabsolute change in concentration between two successive iterations becomes less than somesmall value determined by the imposed relative and absolute concentration tolerances.
The solution process for the carbon dioxide and major ion chemistry modules proceeds ina very similar manner. The nodal values of water content, velocity and temperature, obtained
from solutions of the water flow and heat transport equations, are used to evaluate thecoefficients of the discretized CO2 transport equation (5.6). The CO2 transport equation (5.6) is
not linear because of the dependency of the production term P on the concentration of CO2. To
avoid the need to iterate, we evaluated this term using CO2 concentrations from the previous
time step. Finally, multicomponent solute transport was solved based on knowledge of the watercontents, flow velocities, temperatures and CO2 concentrations from the previous solution. The
solution of the multicomponent chemical system and its coupling with solute transport is givenin Šimůnek et al. [1996]. Water flow was considered to be invariant with respect to temperature,
CO2 and solute transport, while heat transport was similarly considered to be invariant withrespect to both CO2 and solute transport. Finally CO2 transport was assumed to be independentof multicomponent solute transport. These assumptions make it possible to solve the various
processes sequentionally, rather than needing to solve all equations simultaneously.

1218.4.2. Upstream Weighted Formulation
Upstream weighing is provided as an option in the HYDRUS to minimize some of the
problems with numerical oscillations when relatively steep concentration fronts are being
simulated. For this purpose the fourth (flux) term of equation (8.4) is not weighted using regularlinear basis functions φn, but instead with the nonlinear functions φnu

φ 1u = φ 1 - 3α w φ 1φ 2
φ u2 = φ 2 + 3α w φ 1φ 2(8.44)

where αiw is a weighing factor associated with the length of the element size. The weighing
factors are evaluated using the equation of Christie et al. [1976]:

α w = coth(uL 2 D)2 D uL(8.45)

where u, D and L are the flow velocity, dispersion coefficient and length associated with side i.
The weighing functions φu ensure that relatively more weight is placed on the flow velocities ofnodes located at the upstream side of an element. Evaluating the integrals in (8.19) shows thatthe following terms must replace the entries of the global matrix Snm:2

S nm 2 = ∫B
e

1

w
w
dϕ n
1 B1 (2 + 3α ) + B2 B1 + B2 (2 - 3α )φ mdx =dx
6 - B1 (2 + 3α w ) - B2 - B1 - B2 (2 - 3α w )(8.46)

The coefficients of matrix [Ps] (8.24) for upstream weighting formulation are given in Table 8.2.
8.4.3. Reverse Back-Step Particle Tracking
The reverse back-step particle tracking method is another approach to stabilize numericalsolutions of the convective-dispersive equation. A two-step procedure was followed for themixed Lagrangian-Eulerian approach [Molz, 1981]. First, convective transport is consideredusing a Lagrangian approach in which the Lagrangian concentrations are estimated from particle
122

trajectories. Subsequently, all other processes including sinks and sources are modeled using thestandard Eulerian approach involving the finite element method, thus leading to the final
concentrations.
The single-step reverse particle tracking method [Molz, 1981] allows the initial positionof particles arriving at the end of a time step at fixed nodal points to be calculated at each time
step usingt k+1

x n ′ = x n - ∫ v* dt(8.47)

tk

where v* represents the pore-water velocity (v/R) which accounts for all retardation processes.
This equation states that a particle leaving location xn' at time tk will reach the grid point location
zn exactly at time tk+1. The concentration at location xn' at time tk is then used in the discretizedtransport equation. The particle tracking method is used only for the transport of major ions.
8.4.4. Mass Balance Calculations
The total amount of mass in the entire flow domain, or in a preselected subregion, isgiven byM = ∑∫ (θ c + av g + ρ s )dx = ∑∫[(θ + av k g + ρ f
e e

e e

β -1

k sc
)c + ρ s k ] dx
β
1 + ηc(8.48)

where summation is taken over all elements within the specified region. The equations in thissection pertain only to the equilibrium or chemical nonequilibrium models; the mass balance
equations for physical nonequilibrium transport are very similar as those for chemical
nonequilibrium. Similar calculations are carried out for the mobile and immobile regions of thedual-porosity model and for the matrix and fracture regions of the dual-permeability model.
The cumulative amounts M0 and M1 of solute removed from the flow region by zero- andfirst-order reactions, respectively, are calculated as follows

123

t

β -1

k sc
+ ( μ g + μ g' )av k g ]c +M = ∫ ∑∫{[( μ w + μ )θ + ( μ s + μ ) ρ f
β
1 + ηc
0 e e
'
w

1

'
s

(8.49)+ ( μ s + μs' ) ρ s k }dxdtt

M = -∫ ∑ ∫ (γ w,1θ + γ s ,1ρ + γ g ,1 a v ) dx dt
0
1

0 e e

t

'
'
M = -∫ ∑ ∫[( μ w,k −1 θ + μ s ,k −1 ρ f
0
k

0 e e

β

-1

k s , k -1 c k -1k -1+ μ g' ,k −1 a v k g , k -1)c k -1 +β k -1
1 + η k -1 c k -1

(8.50)+ μ s' ,k −1 ρ s kk -1 + γ w, k θ + γ s , k ρ + γ g ,k a v

whereas the cumulative amount, Mr, of solute taken up by plant roots is given byt

M r = ∫ ∑ ∫ S c r dx dt(8.51)

0 eR ewhere eR represents the elements making up the root zone.
When the major ion chemistry module is considered, the total amount of mass in theentire flow domain, or in a preselected subregion, in solvent (Ml), mineral phase (Mp), andsurface species (Ms), is given by
M l = ∑∫ θ cdx = ∑Δxi

θ i c i + θ i +1c i +1

M p = ∑∫ ρ cdx = ∑Δxi

ρ i ci + ρ i +1c i +1

ˆ = ∑Δxi
M s = ∑ ∫ ρ cdx

ρ i cˆ i + ρ i +1cˆ i +1

e e

e e

e

e

e

e

e

2
2

(8.52)

2

where θi , θi+1 , ρi , ρi+1, ci , ci+1 , ci , ci +1 , cˆ i , and cˆ i +1 represent, respectively, water contents, bulk
densities and aqueous, mineral phase and surface concentrations evaluated at the corner nodes of
element e. The summation is taken over all elements within the specified region. The total
amount of solute in the entire flow domain, MT [ML-1], is then calculated as124

M T = M l+M p+M s(8.53)

Finally, when all boundary material fluxes, decay reactions, and root uptake mass fluxes
have been computed, the following mass balance should hold for the flow domain as a whole:
t

t

0

0

M t - M 0 = + ∫ q s 0 dt - ∫ q sL dt - M 0 - M 1 - M r(8.54)

where Mt and M0 are the amounts of solute in the flow region at times t and zero, respectively, as
calculated with (8.48). The difference between the left- and right-hand sides of (8.54) represents
the absolute error, εac, in the solute mass balance. Similarly as for water flow, the accuracy of thenumerical solution for solute transport is evaluated by using the relative error, εrc [%], in thesolute mass balance as follows

ε cr =

100 | ε ca |t⎛⎞max ⎜ ∑ | M te - M 0e |, | M 0 | + | M 1 | + | M r | + ∫ ( | q s 0 | + | q sL |) dt ⎟0
⎝ e⎠(8.55)

where M0e and Mte are the amounts of solute in element e at times 0 and t, respectively. Noteagain that HYDRUS does not relate the absolute error to the total amount of mass in the flow
region. Instead, the program uses as a reference the maximum value of (1) the absolute change inelement concentrations as summed over all elements, and (2) the sum of the absolute values of
all cumulative solute fluxes across the flow boundaries including those resulting from sources
and sinks in the flow domain.
The total amount of heat energy in the entire flow domain, or in a preselected subregion,
is given by
W = ∑ ∫ (C nθ n + C oθ o + C wθ + C g a v)T Adx(8.56)e e

where TA is the absolute temperature [K]. The summation is again taken over all elements within125the specified region.
Total amount of carbon dioxide, MCO [L], in the flow domain or in a preselected
subregion, is given by
M CO = ∑∫ (θ wc w + θ a c a )dx = ∑Δxie eec a i (θ a i + K CO2 iRT iθ w i ) + c a i +1(θ a i +1 + K CO2 i +1RT i +1θ w i +1)
2(8.57)

The absolute error in the carbon dioxide mass balance εaCO at time t is given by
t

ε

CO
a

=M

t
CO

-M

0
CO

- ∫ (q CO- q CO+ P - S c w) dt
0L(8.58)

0where MCO0 and MCOt are the amounts of carbon dioxide in the flow region at times zero and t
respectively, while the integral represents the amount of carbon dioxide added/removed from theflow region by boundary fluxes, CO2 production and CO2 root uptake.
8.4.5. Oscillatory Behavior
Numerical solutions of the transport equation often exhibit oscillatory behavior and/or
excessive numerical dispersion near relatively sharp concentration fronts. These problems can beespecially serious for convection-dominated transport characterized by small dispersivities. One
way to partially circumvent numerical oscillations is to use upstream weighing as discussed in
Section 8.4.2. Undesired oscillations can often be prevented also by selecting an appropriate
combination of space and time discretizations. Two dimensionless numbers may be used tocharacterize the space and time discretizations. One of these is the grid Peclet number, Pee,
which defines the predominant type of the solute transport (notably the ratio of the convective
and dispersive transport terms) in relation to coarseness of the finite element grid:

e

Pe =

qΔ x
θD

(8.59)

where Δx is the characteristic length of a finite element. The Peclet number increases when theconvective part of the transport equation dominates the dispersive part, i.e., when a relatively
steep concentration front is present. To achieve acceptable numerical results, the spatial
discretization must be kept relatively fine to maintain a low Peclet number. Numerical126

oscillations can be virtually eliminated when the local Peclet numbers do not exceed about 5.
However, acceptably small oscillations may be obtained with local Peclet numbers as high as 10[Huyakorn and Pinder, 1983]. Undesired oscillations for higher Peclet numbers can beeffectively eliminated by using upstream weighing (see Section 8.4.2).
A second dimensionless number which characterizes the relative extent of numericaloscillations is the Courant number, Cre. The Courant number is associated with the time
discretization as follows

e

Cr =

q Δt
θ RΔ x(8.60)

Three stabilizing options are used in HYDRUS to avoid oscillations in the numericalsolution of the solute transport equation [Šimůnek and van Genuchten, 1994]. One option is
upstream weighing (see Section 8.4.2), which effectively eliminates undesired oscillations at
relatively high Peclet numbers. A second option for minimizing or eliminating numericaloscillations uses the criterion developed by Perrochet and Berod [1993]

Pe • Cr ≤ ω s (= 2)(8.61)

where ωs is the performance index [-]. This criterion indicates that convection-dominatedtransport problems having large Pe numbers can be safely simulated provided Cr is reduced
according to (8. 60) [Perrochet and Berod, 1993]. When small oscillations in the solution can be
tolerated, ωs can be increased to about 5 or 10.
A third stabilization option implemented in HYDRUS also utilizes criterion (8.60).
However, instead of decreasing Cr to satisfy equation (8.56), this option introduces artificial
dispersion to decrease the Peclet number. The amount of additional longitudinal dispersion, DL[L], is given by [Perrochet and Berod, 1993]

DL - =|q|Δ tθ τ- DL - Dw
Rθ ωs
|q|

(8.62)The maximum permitted time step is calculated for all three options, as well as with theadditional requirement that the Courant number must remain less than or equal to 1. The time127

step calculated in this way is subsequently used as one of the time discretization rules (rule No.
B) discussed in section 7.4.2.

128

9. PARAMETER OPTIMIZATION
Parameter optimization is an indirect approach for the estimation of soil hydraulic and/or
solute transport parameters from transient flow and/or transport data. Inverse methods are
typically based upon the minimization of a suitable objective function, which expresses thediscrepancy between the observed values and the predicted system response. Soil hydraulicproperties for this purpose are assumed to be described by an analytical model with unknown
parameter values (see Section 2.3). The system response is represented by a numerical solutionof the flow equation, augmented with the parameterized hydraulic functions, selected transportparameters, and suitable initial and boundary conditions. Initial estimates of the optimized
system parameters are then iteratively improved during the minimization process until a desireddegree of precision is obtained. This methodology was originally applied to one-step and multistep column outflow data generated in the laboratory [see for example Kool et al., 1985; vanDam et al., 1994], and laboratory or field transport data during steady-state water flow [vanGenuchten, 1981; Toride et al., 1995]. HYDRUS now implements parameter optimization also
for the estimation of the solute transport and reaction parameters from transient water flow andsolute transport experiments.
9.1. Definition of the Objective Function
The objective function Φ to be minimized during the parameter estimation process may
be defined as [Šimůnek et al., 1998]:
mq

nq j

Φ (b, q, p ) = ∑v j ∑ wi , j[q j ( x, t i ) - q j ( x, t i, b)]2 +
j =1

i =1

mp

np j

*

∑v ∑w [ p (θ ) - p (θ , b)] +
j

j =1

i =1

*
j

i, j

2

i

nb

j

∑vˆ [b -b ]
j

*
j

i

(9.1)

2

j

j =1

where the first term on the right-hand side represents deviations between measured andcalculated space-time variables, such as pressure heads, water contents, and/or concentrations at
different locations and/or times in the flow domain, or actual or cumulative fluxes versus time
across a certain boundary. Table 9.1 at the end of this chapter lists various options for defining

129the first term of Eq. (9.1) for the different equilibrium and nonequilibrium water flow and solutetransport models. In the first term, mq is the number of different sets of measurements, nqj is thenumber of measurements in a particular measurement set, qj*(x,ti) represents specificmeasurements at time ti for the jth measurement set at location x(r,z), qj(x,ti ,b) are thecorresponding model predictions for the vector of optimized parameters b (e.g., θr , θs , α, n, Ks,
Dl, kg,k, ...), and vj and wi,j are weights associated with a particular measurement set or point,respectively. The second term of (9.1) represents differences between independently measured
and predicted soil hydraulic properties (e.g., retention, θ(h) and/or hydraulic conductivity, K(θ)
or K(h) data), while the terms mp, npj, pj*(θi), pj(θi ,b), v j and w i , j have similar meanings as forthe first term but now for the soil hydraulic properties. The last term of (9.1) represents a penalty
function for deviations between prior knowledge of the soil hydraulic parameters, bj*, and their
final estimates, bj, with nb being the number of parameters with prior knowledge and vˆ jrepresenting pre-assigned weights. Estimates, which make use of prior information (such as
those used in the third term of (9.1)) are known as Bayesian estimates. We note that thecovariance (weighting) matrices, which provide information about the measurement accuracy, as
well as any possible correlation between measurement errors and/or parameters, are assumed tobe diagonal in this study. The weighting coefficients vj , which minimize differences inweighting between different data types because of different absolute values and numbers of data
involved, are given by [Clausnitzer and Hopmans, 1995]:vj=

1n jσ j2(9.2)

which causes the objective function to become the average weighted squared deviation
normalized by the measurement variances σj2.
9.2. Marquardt-Levenberg Optimization Algorithm
Minimization of the objective function Φ is accomplished by using the LevenbergMarquardt nonlinear minimization method (a weighted least-squares approach based on
Marquardt's maximum neighborhood method) [Marquardt, 1963]. This method combines the
Newton and steepest descend methods, and generates confidence intervals for the optimized
parameters. The method was found to be very effective and has become a standard in nonlinear
least-squares fitting among soil scientists and hydrologists [van Genuchten, 1981; Kool et al.,130

1985, 1987].
9.3. Statistics of the Inverse Solution
As part of the inverse solution, HYDRUS produces a correlation matrix, which specifies
degree of correlation between the fitted coefficients. The correlation matrix quantifies changes inmodel predictions caused by small changes in the final estimate of a particular parameter,
relative to similar changes as a result of changes in the other parameters. The correlation matrix
reflects the nonorthogonality between two parameter values. A value of ±1 suggests a perfectlinear correlation whereas 0 indicates no correlation at all. The correlation matrix may be used to
select which parameters, if any, are best kept constant in the parameter estimation process
because of high correlation.
An important measure of the goodness of fit is the r2 value for regression of the observed,y^ , versus fitted, y (b), values:ii2∑ yˆ i ∑ y i ⎤⎡⎢ ∑ wi yˆ i y i - ∑
⎥
wi ⎦
⎣
2
r =
⎡
ˆ i )2 ⎤ ⎡(∑ y i ) 2 ⎤2 (∑ y2ˆ∑∑−⎢ wi y i⎥ ⎢ yi⎥∑ wi ⎦ ⎣∑ wi ⎦⎣(9.3)

The r2 value is a measure of the relative magnitude of the total sum of squares associated withthe fitted equation; a value of 1 indicates a perfect correlation between the fitted and observed
values.
HYDRUS provides additional statistical information about the fitted parameters such as
the mean, standard error, T-value, and the lower and upper confidence limits (given in output file
FIT.OUT). The standard error, s(bj), is estimated from knowledge of the objective function, thenumber of observations, the number of unknown parameters to be fitted, and an inverse matrix[Daniel and Wood, 1971]. The T-value is obtained from the mean and standard error using theequationT=bjs (b j )(9.4)

The values for T and s(bj) provide absolute and relative measures of the deviations around themean. HYDRUS also specifies the upper and lower bounds of the 95% confidence level around131each fitted parameter bj. It is desirable that the real value of the target parameter always belocated in a narrow interval around the estimated mean as obtained with the optimization
program. Large confidence limits indicate that the results are not very sensitive to the value of aparticular parameter.
Finally, because of possible problems related to convergence and parameter uniqueness,
we recommend to routinely rerun the program with different initial parameter estimates to verify
that the program indeed converges to the same global minimum in the objective function. This isespecially important for field data sets, which often exhibit considerable scatter in themeasurements, or may cover only a narrow range of soil water contents, pressure heads, and/or
concentrations. Whereas HYDRUS will not accept initial estimates that are out of range, it isultimately the user's responsibility to select meaningful initial estimates.
Comprehensive reviews of issues related to inverse parameter estimation were recently
given by Hopmans and Šimůnek [1999], Šimůnek and Hopmans [2002], Hopmans et al. [2002],
and Šimůnek et al. [2002].

132

Table 9.1. Definition of the objective function for different water flow and solute transport models.
Code(.iConcType)0

1

2

Location

X

Definition of Y

1

Time

Cumulative surface flux

2

Time

Cumulative bottom flux

3
4
5

Time
Time
Time

Cumulative surface matrix flux
Cumulative surface fracture flux
Cumulative bottom matrix flux

6

Time

Cumulative bottom fracture flux

iObs

Time

Pressure head at observation node
iObs (W2: mobile zone; W3:
matrix)

-iObs

Time

iObs

Time

-iObs

Time

nObs+iObs

Time

0

Time

Fracturepressureheadobservation node iObs
Water content at observation node W1: θ(iObs)
iObs
W2: θ(iObs)=θmo(iObs)+θim(iObs)
W3: θ(iObs)=wθf(iObs)+(1-w)θm(iObs)
Fracturewater
content
at W3: θF(iObs)=wθf(iObs)
observation node iObs
Matrixwater
content
at W3: θM(iObs)=(1-w)θm(iObs)
observation node iObs
Top
Volume of water in the soil
1
W1: W =
profile∫ θ dzL BottomY

W1-W2: cum(qtop)
W3: cum(qtop)=cum[qm(1-w)+qf w]
W1-W2: cum(qbottom)
W3: cum(qbottom)=cum[qm(1-w)+qf w]
W3: cum(qM,top)=cum[qm(1-w)]
W3: cum(qF,top)=cum[qf w]
W3: cum(qM,bottom)=cum[qm(1-w)]
W3: cum(qF,bottom)=cum[qf w]
W1: h(iObs)
W2: hmo(iObs)
W3: hm(iObs)
at W3: hf(iObs)

Top

W2: W =133

1
∫ ( θm + θim ) dz
L BottomTop

W3: W =
-iLay

Time

Volume of water in the iLay
subregion1∫ ⎡⎣ wθ f + (1 − w)θm ⎤⎦ dz
L Bottomzi ,Top

1
W1: Wi =∫ θ dzLi zi , Bottom
zi ,Top

1
W2: Wi =∫ (θ m + θim )dz
Li zi , Bottom-(nObs+iLay)

Time

Volume of water in the iLay
subregion

1
W3: Wi =∫ ⎡⎣ wθ f + (1 − w)θm ⎤⎦ dz
Li zi , Bottom

1

Time

Surface Flux

2

Time

Bottom Flux

3
4
5

Time
Time
Time

Surface Matrix Flux
Surface Fracture Flux
Bottom Matrix Flux

W1-W2: qtop
W3: qtop=qm(1-w)+qf w
W1-W2: qbottom
W3: qbottom=qm(1-w)+qf w
W3: qM,top=qm(1-w)
W3: qF,top= qf w

6

Time

Bottom Fracture Flux

4.0

0

Time

4.0

iObs

Time

4.0

-iObs

Time

4.1

0

Time

4.1

iObs

Time

Solute mass in the transportdomainLiquid resident concentration at
observation node iObs (W2:
mobile zone; W3: matrix)
Liquid resident concentration of
the second solute at observation
node iObs
Logarithm of the solute mass inthe transport domain
Logarithm of the liquid resident

3134

zi ,Top

W3: qM,bottom= qm(1-w)
W3: qF,bottom= qf w
S1-S4: c(iObs)
S5-S6: cmo(iObs)
S7-S9: cm(iObs)
S1-S4: c2(iObs)
S5-S6: c2,mo(iObs)

S1-S4: log[c(iObs)]

iObs

Time

4.1

-iObs

Time

4.2

iObs

Time

4.3

iObs

Time

concentration at observation nodeiObs
Liquid resident fracture
concentration at observation nodeiObs
Logarithm of the liquid resident
concentration of the second soluteat observation node iObs
S1-S6: Flux concentration at theobservation node iObs
S7-S9: Bottom flux concentration

Total solute mass at observation
node iObs

S5-S6: log[cmo(iObs)]
S7-S9: cf(iObs)
S1-S4: log[c2(iObs)]
S5-S6: log[c2,mo(iObs)]
Dθ ∂c
q ∂z
D θ ∂c
S5-S6: cf,mo(iObs)= cmo − mo mo mo∂z
qmo
wc f q f + (1 − w)cm qm
S7-S9:
wq f + (1 − w) qm
S1-S4: cf(iObs)= c −S1: c (θ + ρ K d )S2: cθ + ρ s k
S3: c (θ + f e ρ K d ) + ρ s k
S4: cθ + ρ s1k + ρ s2k
S5: cmo (θ mo + f mo ρ K d ) + cim ⎡⎣θim + (1 − f mo ) ρ K d ⎤⎦S6:kcmo (θ mo + f mo f em ρ K d ) + f mo ρ smo++ cim ⎡⎣θim + (1 − f mo ) ρ K d ⎤⎦S7: w ⎡⎣ c f (θ f + ρ K df ) ⎤⎦ + (1 − w ) ⎡⎣ cm (θ m + ρ K dm ) ⎤⎦
S8:
S9:

135

w ⎡⎣c f (θ f + f f ρ K df ) + ρ s kf ⎤⎦ +(1 − w ) ⎡⎣cm (θ m + f m ρ K dm ) + ρ smk ⎤⎦w ⎡⎣c f (θ f + ρ K df ) ⎤⎦ + (1 − w )⎡ cm, mo (θ m ,mo + f mo ρ K dm ) + cm ,im (θ m,im + (1 − f mo ) ρ K dm ) ⎤
⎣
⎦
4.4

iObs

Time

Liquid resident concentration at
observation node iObs for thedual-porosity model

5

iMat

h

6

iMat

h

Water content, θ, at pressure
head, h, for soil material iMat
Hydraulic conductivity, K, atpressure head, h, for soil material
iMat

7
8
9
10

iMat
iMat
iMat
iMat

11

iMat

12

PLevel

x

h(x)

13

PLevel

x

θ(x)

14.0

PLevel

x

c(x)

α

n

θr
θs
Κs

136

Only for linear and isothermal transportc θ +c θ
S5-S6: c(iObs ) = mo mo im imθ mo + θim
wc f θ f + (1 − w)cmθ m
S7-S9: c(iObs ) =wθ f + (1 − w)θ m

θ(h)
Κ(h)
Retention curve parameter α for soil material iMat
Retention curve parameter n for soil material iMat
Residual water content θr for soil material iMat
W1: Saturated water content θs for soil material iMat
W2: θs=θsm+θsim
W3: θs=wθsf+(1-w)θsm
W1-W2: Saturated hydraulic conductivity Κs for soilmaterial iMat
W3: Κs=wΚsf+(1-w)Κsm
W1-W2: Pressure head h (hmo) at depth x at print time
PLevelW3: Fracture pressure head hf
W1: Water content θ at depth x at print time PLevel
W2: θ=θmo+θim
W3: θ=wθf+(1-w)θm
S1-S6: Resident liquid concentration c at depth x at
print time PLevel14.1

PLevelxlog[c(x)]14.2

PLevel

x

c(x)

14.3

PLevelx

14.4

PLevelx

15

PLevelx

Total solute mass at depth x at
print time PLevel
Liquid resident concentration at
depth x at print time PLevel
Total sorbed concentration at
depth x at print time PLevel

S1: Uniform transport model
S2: One kinetic site model
S3: Two-site model
S4: Two kinetic sites model
S5: Dual-porosity model
S6: Dual-porosity model with one kinetic site
S7: Dual-permeability model
S8: Dual-permeability model with two-site model
S9: Dual-permeability model with immobile region in the matrix
W1: Uniform water flow model
W2: Dual-porosity model
W3: Dual-permeability model

137

S7-S9: Resident liquid concentration cf in the fracturedomain at depth x at print time PLevel
S1-S6: Logarithm of the resident liquid concentration cat depth x at the print time PLevel
S7-S9: Resident liquid concentration c in the matrix at
depth x at print time PLevel
S1-S6: Flux concentration c at depth x at print time
PLevel
S7-S9: not implemented
See 4.3 above for definitions
See 4.4 above for definitions
s1+s2 (usually used for attached and strained colloids)

138

10. PROBLEM DEFINITION
10.1. Construction of Finite Element Mesh
The finite element mesh is constructed by dividing the soil profile into linear elements
whose sizes are defined by the x-coordinates of the nodes that form the element corners.
Neighboring elements should have approximately the same size. The ratio of the sizes of twoneighboring elements is not recommended to exceed about 1.5. The nodes are numberedsequentially from 1 to NumNP (total number of nodes) from the bottom of the soil profile to thesoil surface.
The element dimensions must be adjusted to a particular problem. They should be maderelatively small at locations where large hydraulic gradients are expected. Such a region isusually located close to the soil surface where highly variable meteorological factors can cause
rapid changes in the soil water content and corresponding pressure heads. Therefore, it is usually
recommended to use relatively small elements near the soil surface, and gradually larger sizes
with depth. The element dimensions are also dependent on soil hydraulic properties. Coarse
textured soils generally require a finer discretization than fine-textured soils (loams, clays). No
special restrictions are necessary to facilitate the soil root zone.
10.2. Coding of Soil Types and Subregions
Soil Types - An integer code beginning with 1 and ending with NMat (the total number ofsoil materials) is assigned to each soil type in the flow region. The appropriate material code issubsequently assigned to each nodal point n of the finite element mesh.
Interior material interfaces do not coincide with element boundaries. When different
material numbers are assigned to the nodes of a certain element, the finite element algorithm willassume that the material properties will change linearly over the element. This procedure will
somewhat smooth soil interfaces. A set of soil hydraulic parameters, and solute and heattransport characteristics must be specified for each soil material.
Subregions - Water and solute mass balances are computed separately for each specified
subregion. The subregions may or may not coincide with the material regions. Subregions arecharacterized by an integer code, which runs from 1 to NLay (the total number of subregions). A
subregion code is assigned to each element in the flow domain.

139

10.3. Coding of Boundary Conditions
Boundary codes KodTop and KodBot must be assigned to surface and bottom boundary
nodes, respectively. If a boundary node is to have a prescribed pressure head during a time step(a Dirichlet boundary condition), KodTop and KodBot must be set positive during that time step.
If the volumetric flux of water entering or leaving the system is prescribed during a certain time
step (a Neumann boundary condition), KodTop and KodBot must be negative or zero.
Constant Boundary Conditions - The value of a constant boundary condition for aparticular boundary node, n, is given by the initial value of the pressure head, h(n), in case of
Dirichlet boundary conditions, or by the initial value of the recharge/discharge flux, rTop orrBot, in case of Neumann boundary conditions. Table 10.1 summarizes the use of the variables
KodTop (KodBot), rTop (rBot), and h(n) for various types of nodes.

Table 10.1. Initial settings of KodTop (KodBot), rTop (rBot), and h(n) for constantboundary conditions.
Node Type

KodTop (KodBot) rTop (rBot)h(n)Specified Head Boundary

1

0.0

PrescribedSpecified Flux Boundary-1

Prescribed

Initial Value

Variable Boundary Conditions - Four types of variable boundary conditions can be
imposed:1. Atmospheric boundary conditions for which TopInf=AtmInf=.true.,
2. Variable pressure head boundary conditions for which TopInf=.true. and KodTop=+3,
or BotInf=.true. and KodBot=+3, or
3. Variable flux boundary conditions for which TopInf=.true. and KodTop=-3, or
BotInf=.true. and KodBot=-3.
4. Variable pressure head/flux boundary conditions for which TopInf=.true. and
KodTop=-3 or +3.
Initial settings of the variables KodTop (KodBot), rTop (rBot), and h(n) for the time-dependent
boundary conditions are given in Table 10.2.
140

Table 10.2. Initial settings of KodTop (KodBot), rTop (rBot), and h(n) for time-variable boundary
conditions.
Node Type

KodTop (KodBot) rTop (rBot)h(n)Atmospheric Boundary

-4

0.0

Initial Value

Variable Head Boundary

+3

0.0

Initial Value

Variable Flux Boundary

-3

0.0

Initial Value

Atmospheric boundary conditions are implemented when TopInf=AtmInf=.true., inwhich case time-dependent input data for the precipitation, Prec, and evaporation, rSoil, rates
must be specified in the input file ATMOSPH.IN. The potential fluid flux across the soil surface
is determined by rAtm=rSoil-Prec. The actual surface flux is calculated internally by the
program. Two limiting values of surface pressure head must also be provided: hCritS which
specifies the maximum allowed pressure head at the soil surface (usually 0.0), and hCritA which
specifies the minimum allowed surface pressure head (defined from equilibrium conditions
between soil water and atmospheric vapor). The program automatically switches the value of
KodTop from -4 to +4 if one of these two limiting points is reached. Table 10.3 summarizes theuse of the variables rAtm, hCritS and hCritA during program execution.
Variable head or flux boundary conditions on the soil surface (bottom of the soil profile)
are implemented when KodTop (KodBot)=+3 or -3 and TopInf (BotInf)=.true., respectively. In
that case, the input file ATMOSPH.IN must contain the prescribed time-dependent values of thepressure head, hT (hB), or the flux, rT (rB), imposed on the boundary. The values of hT (hB) orrT (rB) are assigned to particular nodes at specified times according to rules given in Table 10.4.

141

Table 10.3. Definition of the variables KodTop, rTop, and h(n) whenan atmospheric boundary condition is applied.
KodTop

rTop

h(n)

Event

-4

rAtm

UnknownrAtm=rSoil-Prec+4

Unknown

hCritA

Evaporation capacity
is exceeded+4

Unknown

hCritS

Infiltration capacity
is exceeded

Table 10.4. Definition of the variables KodTop (KodBot), rTop (rBot), and h(n)when variable head or flux boundary conditions are applied.
Node Type

KodTop (KodBot) rTop (rBot)h(n)Variable Head Boundary+-3

UnknownhT (hB)

Variable Flux Boundary-3

rT (rB)

Unknown

Water Uptake by Plant Roots - The program calculates the rate at which plants extract
water from the root zone by evaluating equation (2.8). Values of the potential transpiration rate,rRoot, must be specified at preselected times in the input file ATMOSPH.IN. These timedependent values must be provided by the user and can be calculated in various ways, such as
from the temperature and crop coefficients. Actual transpiration rates are calculated internally by
the program as discussed in Section 2.2. The root water uptake parameters are taken from an
input file, SELECTOR.IN. Values of the function Beta(n), which describes the potential water
uptake distribution over the root zone, must be specified for each node in the flow domain. If theroot growth model is considered, then the exponential function for the spatial distribution of thepotential root water uptake is used (equation (2.16)). All parts of the flow region whereBeta(n)>0 are treated as the soil root zone.
Root Growth Model - The program calculates the time variable rooting depth if thelogical variable lRoot in input file SELECTOR.IN is equal to .true.. The classical Verhulst-Pearllogistic function (2.21) (see Section 2.2) is used to model the rooting depth. The exponential142(2.16) spatial distribution function for the root water uptake function is always used along withthe time-variable rooting depth option. The root growth factor, r, can be calculated either fromthe known value of root depth (xRMed) at a specified time (tRMed), or from the assumption that
50% of the rooting depth is reached after 50% of the growing season.
Deep Drainage from the Soil Profile - Vertical drainage, q(h), across the lower boundaryof the soil profile is sometimes approximated by a flux which depends on the position of thegroundwater level [e.g., Hopmans and Stricker, 1989]. If available, such a relationship can be
implemented in the form of a variable flux boundary condition; the code in that case internallysets the variable KodBot equal to -7. This boundary condition will be implemented in HYDRUS
if the logical variable qGWLF in the input file SELECTOR.IN is set equal to .true.. The
discharge rate q(n) assigned to bottom node n is determined by the program as q(n)=q(h), where
h is the local value of the pressure head, and q(h) is given by
q (h) = - Aqh exp( B qh | h - GWL0 L |)(10.1)

where Aqh and Bqh are empirical parameters which must be specified in input file
SELECTOR.IN, together with GWL0L which represents the reference position of thegroundwater level (sometimes set equal to the x-coordinate of the soil surface).
Free Drainage - Unit vertical hydraulic gradient boundary conditions can be
implemented in the form of a variable flux boundary condition. The program in that case willinternally set the variable KodBot equal to -5. This boundary condition is implemented in
HYDRUS by setting the logical variable FreeD in the input file SELECTOR.IN equal to .true..
The discharge rate q(n) assigned to bottom node n is determined by the program as q(n)=-K(h),where h is the local value of the pressure head, and K(h) is the hydraulic conductivity
corresponding to this pressure head.
Seepage Faces - The initial settings of the variables KodBot, rBot and h(n) for node on a
seepage face are summarized in Table 10.5. This boundary condition is implemented in
HYDRUS by setting the logical variable SeepF in the input file SELECTOR.IN equal to .true..

143

Table 10.5. Initial settings of KodBot, rBot, and h(n) for seepage faces.
Node Type

KodBotrBoth(n)Seepage Face
(initially saturated)

+2

0.0

0.0 (hSeep)

Seepage Face
(initially unsaturated)

-2

0.0

Initial Value

Flow to Horizontal Drains - This boundary condition is implemented when the logicalvariable lDrain in the input file SELECTOR.IN is equal to .true.. Five conceptual models can beused to describe the tile-drained soil profile:a) homogeneous soil profile; drain is located immediately above the impervious layer,b) homogeneous soil profile; drain is located some distance above the impervious layer,c) layered soil profile (two layers); drain is located at interface between soil layers,d) layered soil profile (two layers); drain is located in the bottom layer,e) layered soil profile (two layers); drain is located in the top layer.
The first three cases are solved with the Hooghoudt equation (2.51), and the last two cases with
the Ernst equation (2.53).
Heat Transport Boundary Conditions - The type of applied boundary condition is
specified by the input variables kTopT and kBotT for the upper and lower boundaries,
respectively. Positive values for these variables means that a first-type boundary condition isused. When kTopT or kBotT is negative, then a third-type boundary condition is applied. On theother hand, when kBotT is equal to zero, a Neumann boundary condition with zero gradient is
implemented. All initial and boundary conditions must be specified in oC.
Solute Transport Boundary Conditions - The type of applied boundary condition is
specified by the input variables kTopCh and kBotCh for the upper and lower solute transport
boundaries, respectively. Similarly as for heat transport, positive values for these variables
means that a first-type boundary condition will be assumed. When kTopCh or kBotCh is
negative, then a third-type boundary condition is applied. When kBotCh is equal to zero, a
Neumann boundary condition with zero gradient is used.

144

10.4. Program Memory Requirements
One single parameter statement is used at the beginning of the code to define the problem
dimensions. All major arrays in the program are adjusted automatically according to these
dimensions. This feature makes it possible to change the dimensions of the problem to be
simulated without having to recompile all program subroutines. Different problems can beinvestigated by changing the dimensions in the parameter statement at the beginning of the main
program, and subsequently linking all previously compiled subroutines with the main program
when creating an executable file. Table 10.6 lists the array dimensions, which must be defined in
the parameter statement.

Table 10.6. List of the array dimensions.
Dimension

Current setting

Description

NumNPD

1001

Maximum number of nodes in finite element mesh

NMatD

20

Maximum number of materials

NTabD

100

Maximum number of items in the table of hydraulicproperties generated by the program for each soil material

NObsD

10

Maximum number of observation nodes

145

146

11. EXAMPLE PROBLEMS
Eight example problems demonstrating direct problems and three examples for inverse
problems are presented in this section. The first two examples are identical to those described in
the SWMS_2D manual [Šimůnek et al., 1992], while three other examples are the same as those
discussed in the CHAIN_2D manual [Šimůnek et al., 1994]. The three CHAIN_2D examples
were included mainly for mathematical verification purposes, and for demonstrating newfeatures of version 2.0 of HYDRUS, i.e., non-equilibrium and nonlinear adsorption, andsequential first-order decay reactions. Examples 9 through 11 demonstrate new features
implemented into versions 3.0 and 4.0 of HYDRUS.
Examples 1 and 2 provide comparisons of the water flow part of the HYDRUS code withresults from both the UNSAT2 code of Neuman [1972] and the SWATRE code of Belmans et al.
[1983]. The results obtained with HYDRUS for these two examples were identical to thoseobtained with SWMS_2D. Example 3 serves to verify the accuracy of HYDRUS by comparing
numerical results for a problem with three solutes involved in a sequential first-order decay
reaction against results obtained with an analytical solution during one-dimensional steady-statewater flow [van Genuchten, 1985]. Example 4 considers one-dimensional transport of a solute
undergoing nonlinear cation adsorption. Numerical results are compared with experimental dataand previous numerical solutions obtained with the MONOC code of Selim et al. [1987] and theprevious version of HYDRUS (version 5.0) of Vogel et al. [1996]. Example 5 serves to test theperformance of HYDRUS for nonequilibrium adsorption by comparing numerical results against
experimental data and previous numerical predictions during one-dimensional steady-state waterflow [van Genuchten, 1981]. Examples 6 through 8 demonstrate the inverse analyses of one- andmulti-step outflow experiments, as well as of an evaporation experiment, for the purpose of
estimating the unsaturated soil hydraulic parameters. Example 9 shows an application of thedual-porosity model, while example 10 demonstrates the new hysteresis model. Finally, example
11 demonstrates capabilities of HYDRUS-1D to simulate coupled water, vapor and heattransport.
Four additional examples are distributed with the HYDRUS-1D installation package
demonstrating its use for simulating carbon dioxide transport and major ion chemistry. The firstexample simulates infiltration into a relatively dry soil column. This example shows predicteddifferences between the equilibrium and kinetic precipitation-dissolution model. The second
example involving an irrigation problem with root water uptake demonstrates the importance ofconsidering a proper model for calcite precipitation-dissolution. The third example simulating anirrigation problem with a periodic upper boundary condition demonstrates the important effects147

of CO2 concentration/transport on solution chemistry. The last example demonstrates thecapabilities of HYDRUS for predicting carbon dioxide production and transport. The various
major ion chemistry examples are not further described here since they are discussed in detail inthe original UNSATCHEM manual [Šimůnek et al., 1997].
A comprehensive list of publications showing a large number of HYDURS-1D applications
can be found at www.pc-progress.cz/Pg_Hydrus1D_References.htm (for HYDRUS-1D andrelated software). Additional examples demonstrating HYDRUS capabilities to simulate
nonequilibrium flow and transport can be downloaded from the HYDRUS web site (www.pcprogress.cz).
11.1. Example 1 - Column Infiltration Test
This example simulates a one-dimensional laboratory infiltration experiment initiallydiscussed by Skaggs et al. [1970], and later used by Davis and Neuman [1983] as a test problem
for the UNSAT2 code. Hence, the example provides a means of comparing results obtained with
the HYDRUS and UNSAT2 codes.
The soil water retention and relative hydraulic conductivity functions of the sandy soil
are presented in Figure 11.1. The sand was assumed to be at an initial pressure head of -150 cm.
The soil was assumed to be homogenous and isotropic with a saturated hydraulic conductivity of
0.0433 cm/min. The column was subjected to ponded infiltration (a Dirichlet boundary
condition) at the soil surface, resulting in one-dimensional vertical water flow. The open bottom
boundary of the soil column was simulated by implementing a no-flow boundary condition
during unsaturated flow (h<0), and a seepage face with h=0 when the bottom of the column
becomes saturated (this last condition was not reached during the simulation).
The simulation was carried out for 90 min, which corresponds to the total time durationof the experiment. Figure 11.2 shows the calculated instantaneous (q0) and cumulative (I0)
infiltration rates simulated with HYDRUS. The calculated results agree closely with thoseobtained by Davis and Neuman [1983] using their UNSAT2 code.

148

Fig. 11.1. Soil water retention and relative hydraulic conductivity functions for example 1. The solid
circles are UNSAT2 input data [Davis and Neuman, 1983].

149

Fig. 11.2. Instantaneous, q0, and cumulative, I0, infiltration rates simulated with the HYDRUS (solid lines)
and UNSAT2 (solid circles) computer codes (example 1).

150

11.2. Example 2 - Water Flow in a Field Soil Profile Under Grass
This example considers one-dimensional water flow in a field profile of the Hupselse
Beek watershed in the Netherlands. Atmospheric data and observed ground water levels
provided the required boundary conditions for the numerical model. Calculations were
performed for the period of April 1 to September 30 of the relatively dry year 1982. Simulation
results obtained with HYDRUS will be compared with those generated with the SWATRE
computer program [Feddes et al., 1978, Belmans et al., 1983].
The soil profile consisted of two layers: a 40-cm thick A-horizon, and a B/C-horizon
which extended to a depth of about 300 cm. The depth of the root zone was 30 cm. The mean
scaled hydraulic functions of the two soil layers in the Hupselse Beek area [Císlerová, 1987;
Hopmans and Stricker, 1989] are presented in Figure 11.3.
The soil surface boundary condition involved actual precipitation and potential
transpiration rates for a grass cover. The surface fluxes were incorporated by using average dailyrates distributed uniformly over each day. The bottom boundary condition consisted of aprescribed drainage flux - groundwater level relationship, q(h), as given by equation (10.1). The
groundwater level was initially set at 55 cm below the soil surface. The initial moisture profile
was taken to be in equilibrium with the initial ground water level.
Figure 11.4 presents input values of the precipitation and potential transpiration rates.
Calculated cumulative transpiration and cumulative drainage amounts as obtained with the
HYDRUS and SWATRE codes are shown in Figure 11.5. The pressure head at the soil surface
and the arithmetic mean pressure head of the root zone during the simulated season are presented
in Figure 11.6. Finally, Figure 11.7 shows variations in the calculated groundwater level with
time.

151

Fig. 11.3. Unsaturated hydraulic properties of the first and second soil layers (example 2).

152

Fig. 11.4. Precipitation and potential transpiration rates (example 2).

153

Fig. 11.5. Cumulative values for the actual transpiration and bottom leaching rates as simulated with the
HYDRUS (solid line) and SWATRE (solid circles) computer codes (example 2).

154

Fig. 11.6. Pressure head at the soil surface and mean pressure head of the root zone as simulated with the
HYDRUS (solid lines) and SWATRE (solid circles) computer codes (example 2).

155

Fig. 11.7. Location of the groundwater table versus time as simulated with the HYDRUS (solid line) and
SWATRE (solid circles) computer codes (example 2).

11.3. Example 3 - Solute Transport with Nitrification Chain
This example was used to verify in part the mathematical accuracy of the solute transportpart of HYDRUS. Numerical results will be compared with results generated with an analyticalsolution published by van Genuchten [1985] for one-dimensional convective-dispersive transportof solutes involved in sequential first-order decay reactions. The analytical solution holds forsolute transport in a homogeneous, isotropic porous medium during steady-state unidirectional
groundwater flow. Solute transport equations (3.1) and (3.2) for this situation reduce to2∂ c1∂ c 1 ∂ c1
D-v
- μ 1 R 1 c1
=
R1
∂t
∂ x2
∂x

156

(11.1)

Ri

2
∂ci
∂c
∂c
= D 2i - v i + μ i -1R i -1c i -1 - μ i R i c i
∂t
∂x
∂x

i = 2, 3(11.2)

where μ is a first-order degradation constant, D is the dispersion coefficient, v is the averagepore water velocity (q/θ) in the flow direction, x is the spatial coordinate in the direction of flow,
and where it is assumed that 3 solutes participate in the decay chain. The specific example usedhere applies to the three-species nitrification chain
NH 4 → NO 2 → NO 3+--

(11.3)and is the same as described by van Genuchten [1985], and earlier by Cho [1971]. The boundary
conditions may be written as:⎛ ∂ c1⎞+ v c1 ⎟ = v c 0,1(0, t )
⎜ -D
∂x
⎝
⎠
⎛ ∂ci
⎞
i = 2, 3+ v ci ⎟ = 0
⎜ -D
∂x
⎝
⎠
∂ ci
=0
i = 1, 2, 3
lim
x →∞ ∂x(11.4)The experiment involves the application of a NH4+ solution to an initially solute-free
medium (ci = 0). The input transport parameters for the simulation are listed in Table 11.2.
Figure 11.8 shows concentration profiles for all three solutes at time 200 hours,
calculated both numerically with HYDRUS and analytically with the CHAIN code of van
Genuchten [1985]. Figure 11.9 shows the concentration profiles at three different times (50, 100,
and 200 hours) for NH4+, NO2-, and NO3-, respectively. The numerical results in each caseduplicated the analytical results.

157

Table 11.1. Input parameters for example 3.
ParameterValuev [cm/hour]
D [cm2/hour]
μ1 [hour-1]
μ 2 [hour-1]
μ 3 [hour-1]
R1 [-]
R2 [-]
R3 [-]
ci [-]

1.0
0.18
0.005
0.1
0.0
2.0
1.0
1.0
0.0

c0,1 [-]

1.0

Fig. 11.8. Analytically and numerically calculated concentration profiles for NH4+, NO2-, and NO3after 200 hours (example 3).

158

Fig. 11.9. Analytically and numerically calculated concentration profiles for NH4+ (top), NO2(middle), NO3- (bottom) after 50, 100, and 200 hours (example 3).

159

11.4. Example 4 - Solute Transport with Nonlinear Cation Adsorption
The experiment discussed in this example was conducted by Selim et al. [1987], and usedlater for previous versions (version 3.1 and 5.0) of HYDRUS [Kool and van Genuchten, 1991;
Vogel et al., 1996]. The soil in this experiment was Abist loam. A 10.75-cm long soil column
was first saturated with a 10 mmolcL-1 CaCl2 solution. The experiment consisted of applying a
14.26 pore volume pulse (t = 358.05 hours) of a 10 mmolcL-1 MgCl2 solution, followed by theoriginal CaCl2 solution. The adsorption isotherm was determined with the help of batch
experiments [Selim et al., 1987], and fitted with the Freundlich equation (3.3) [Kool and van
Genuchten, 1991]. The Freundlich isotherm parameters, as well as other transport parameters forthis problem, are listed in Table 11.2. First- and second-type boundary conditions were applied
to the top and bottom of the soil column, respectively.
The observed Mg breakthrough curve is shown in Figure 11.10, together with simulated
breakthrough curves obtained with HYDRUS, the MONOC code of Selim et al. [1987] and theprevious versions of HYDRUS. The results indicate a reasonable prediction of the measured
breakthrough curve using HYDRUS, and close correspondence between the simulated results
obtained with the HYDRUS and MONOC models. The HYDRUS results became identical tothose generated with previous versions of HYDRUS when a third-type boundary condition was
invoked at the top of the soil column.

Table 11.2. Input parameters for example 4.
ParameterValueq [cm/hour]
D [cm2/hour]ρ [g/cm3]θ [-]c0 [mmolc/L]
ks [cm3/g]
β [-]

0.271
1.167
0.884
0.633
10.0
1.687
1.615

160

Fig. 11.10. Mg breakthrough curves for Abist loam calculated with the MONOD, HYDRUS, and new
HYDRUS codes (data points from Selim et al., 1978).

The Langmuir adsorption isotherm can also be used to model the exchange of
homovalent ions. Parameters in the Langmuir adsorption isotherm for homovalent ion exchange
may be derived as follows. Ion exchange for two ions with valences n and m can be expressed ina generalized form as [Sposito, 1981]m⎛ a1 ⎞ ⎛ a 2 ⎞K ex = ⎜ ⎟ ⎜ ⎟⎝ a1 ⎠ ⎝ a 2 ⎠n(11.5)

where Kex is the dimensionless thermodynamic equilibrium constant, and a and a denote the ion
activities in the soil solution and on the exchange surfaces [-], respectively:161

ai = γ i ci

i = 1, 2(11.6)

ai = ξ i si

i = 1, 2

where ci [ML-3] (mmol/l) and si [MM-1] (mmol/kg) are solution and exchangeable
concentrations, respectively, and γi and ξi are activity coefficients in the soil solution [L3M-1](l/mmol) and on the exchange surfaces [MM-1] (kg/mmol), respectively. Substituting (11.6) into
(11.5) givesm

K 12 = K v

n

m

m
n
γ1
ξ2 γ1
s1 c 2==K ex m nnnmγ2ξ 1 γ 2 s 2 c1

(11.7)

where Kv denotes the Vanselow selectivity coefficient [-], while K12 will be simply referred to as
the selectivity coefficient [-]. Assuming that both the total solution concentration, CT [ML-3](mmolc/l), and the cation exchange capacity, ST [MM-1] (mmolc/kg), are time invariant, i.e.,n c1 + m c 2 = C T
n s1 + m s 2 = S T(11.8)

the Langmuir parameters ks and η in (3.3) for the incoming solute become
K 12 S TCT
ϑ ( K 12 -1)

ks =

η=

(11.9)CT

whereas for the solute initially in the soil column:
ST
K 12 C T
ϑ (1- K 12)
η=K 12 C T
ks =162(11.10)

The parameter ϑ in (11.9) and (11.10) equals 1 for monovalent ions, and 2 for divalent ions.
The selectivity coefficient K12 for example 5 was measured by Selim et al. [1987](K12=0.51). From the total solution concentration (CT=10 mmolc/l) and the known cationexchange capacity (ST=62 mmolc/kg), it follows that the parameters in the Langmuir adsorption
isotherm for the incoming solute (Mg) are ks=3.126 and η=-0.098, while those for the soluteinitially in the soil profile (Ca) the parameters are ks=12.157 and η=0.192. The observed Ca
breakthrough curve is shown in Figure 11.11, together with the simulated breakthrough curves
obtained with the HYDRUS2 and MONOC codes [Selim et al., 1987]. Note the close agreement
between the numerical results and the experimental data.

Fig. 11.11. Ca breakthrough curves for Abist loam calculated with the MONOD and HYDRUS codes(data points from Selim et al., 1978) (example 4).

163

11.5. Example 5 - Solute Transport with Nonequilibrium Adsorption
This example considers the movement of a boron (H3BO4) pulse through Glendale clay
loam [van Genuchten, 1981]. The numerical simulation uses solute transport parameters thatwere fitted to the breakthrough curve with the CFITIM parameter estimation model [vanGenuchten, 1981] assuming a two-site chemical nonequilibrium sorption model analogous to theformulation discussed in Section 3, but for steady-state water flow. Input parameters for example
5 are listed in Table 11.3. Figure 11.12 compares HYDRUS numerical results with theexperimental data, and with a numerical simulation assuming physical nonequilibrium andnonlinear adsorption [van Genuchten, 1981].

Table 11.3. Input parameters for example 5.

ParameterValueq [cm/day]
D [cm2/day]θ [-]ρ [g/cm3]
c0 [mmolc/L]
ks [cm3/g]
β [-]
η [-]
f [-]
ω [1/day]tp [day]

17.12
49.0
0.445
1.222
20.0
1.14
1.0
0.0
0.47
0.320
6.494

164

Fig. 11.12. Observed and calculated effluent curves for boron movement throughGlendale clay (data points from van Genuchten [1981]) (example 5).

16511.6. Example 6 - Inverse Analysis of a One-Step Outflow Experiment
The first example involving parameter estimation deals with a case first discussed by
Kool et al. [1985]. An undisturbed core sample, having a length of 3.95 cm long and a diameterof 5.4 cm, was equilibrated at zero tension in a Tempe pressure cell. The retention curve was
first measured for pressure heads up to -10 m. Water contents at pressure head of -30 and -150 m
were measured on disturbed samples. After resaturating, the pneumatic pressure of 10 m was
imposed at the top of the sample, and the cumulative outflow was recorded with time (Figure11.13). The position of the measuring burette was adjusted manually every time a reading was
made to maintain a constant head lower boundary condition. At the end of the experiment, thesoil was resaturated and the saturated hydraulic conductivity of the soil and porous plate were
measured with a falling head method.
Three hydraulic parameters (α, n, and θr) were estimated by numerical inversion of theobserved cumulative outflow data and the measured water content at the pressure head of -150
m. Since water exits the soil column across a ceramic plate, the flow problem involves a twolayered system. To be able to simulate flow through the ceramic bottom plate without having tomodify the code, the HYDRUS model must be able to simulate flow through materials with very
high air entry values such that the ceramic plate remains saturated at all times during the outflow
experiment. A relatively high air entry value of the plate was obtained by specifying parameter α
to be 10-20 (1/cm). The soil profile was discretized into 50 nodes with five nodes representing theceramic plate. Only a few nodes were used for the ceramic plate since the plate remains saturated
during the entire experiment, thus causing the flow process in the plate to be linear.
Figure 11.13 shows the measured cumulative outflow curve versus time, as well as thebest fit obtained with HYDRUS-1D. Initial and final parameter estimates are listed in Table
11.4. Notice the very good fit of the measured data in Fig. 11.13, with r2 being 0.9987. Figure
11.14 shows a comparison of the predicted and measured retention curves, as well as acomparison of the diffusivity curve obtained by parameter estimation and D(θ) values calculated
independently using the method of Passioura [1976]. Again, notice the very good agreement
between predicted and measured values.

166

Table 11.4. Initial estimates of and optimized parameters for example 6.

Parameter

Initial Estimate

Final Valueθr [cm3/cm3]
θs [cm3/cm3]
α [1/cm]

0.15
0.388+
0.025
1.5
5.4+
0.003+
0.5+

0.166

n [-]
Ks [cm/h]Ks,cer [cm/h]
l [-]+0.0363
1.42

Not optimized

Fig. 11.13. Measured and optimized cumulative outflow versus time for a onestep outflow
experiment (example 6).

167

Fig. 11.14. Observed and predicted retention characteristics and calculated and predicteddiffusivities (example 6).

16811.7. Example 7 - Inverse Analysis of a Multistep Outflow Experiment
Although initial applications of the inverse approach to onestep outflow data appeared
promising, later studies revealed possible problems in terms of non-uniqueness of the optimized
parameters [e.g., van Dam et al., 1992, 1994]. To circumvent uniqueness problems, van Dam etal. [1994] conducted and analyzed outflow experiments in which the pneumatic pressure wasincreased in several smaller steps. Eching et al. [1994] similarly estimated soil hydraulicparameters from simultaneous measurements of transient cumulative outflow and the soil waterpressure head inside of the soil sample during a multistep outflow experiment.
In this test example we analyze a multistep outflow experiment with simultaneous
measurement of the pressure head inside the soil sample [Hopmans, personal communication].
The experimental setup consisted of a 6-cm long soil column in a Tempe pressure cell modified
to accommodate a microtensiometer-transducer system. A tensiometer was installed, with thecup centered 3 cm below the soil surface. The soil sample was saturated from the bottom andsubsequently equilibrated to an initial soil water pressure head of -25 cm at the soil surface.
Pressures of 100, 200, 400, and 700 cm were applied subsequently in consecutive steps at 0,12.41, 48.12, and 105.92 hours, respectively.
Figure 11.15 compares the measured and optimized cumulative outflow curves for thesoil sample, while Fig. 11.16 compares measured and optimized pressure heads. Excellent
agreement was obtained for both variables. The final fit had an r2 of 0.9995. Table 11.5. lists
initial estimates and final values of the six optimized parameters.

Table 11.5. Initial estimates and optimized parameters for example 7.

Parameter

Initial Estimate

Final Valueθr [cm3/cm3]
θs [cm3/cm3]
α [1/cm]

0.078
0.43
0.036
1.56
1.04
0.5

0.197
0.438
0.0101
1.434
0.521
3.80

n [-]
Ks [cm/h]
l [-]

169

Fig. 11.15. Measured and optimized cumulative bottom flux for a multistep outflow
experiment (example 7).

Fig. 11.16. Measured and optimized pressure heads inside the soil sample for a
multistep outflow experiment (example 7).

170

11.8. Example 8 - Inverse Analysis of an Evaporation ExperimentThis example demonstrated application of the parameter estimation method to an
evaporation experiment. Details about the experiment are given in Šimůnek et al. [1998]. An
undisturbed soil core sample with a height of 10 cm and inside diameter of 10 cm was placed ona ceramic plate and saturated with deionized water. The soil had a bulk density of 1.59 g cm-3,and sand, silt and clay fractions of 7.4, 79.3 and 13.3%, respectively. Five tensiometers withcups having a length of 6 cm and outside diameter of 0.6 cm were inserted horizontally throughdrill holes into the soil cores at locations 1, 3, 5, 7, and 9 cm below the sample surface.
Following previous disc infiltrometer experiments on the same soil column, the sample was
moved onto impermeable plates for the evaporation experiment. An initial pressure head of -15.4
cm was measured in the middle of the soil sample.
Evaporation was subsequently allowed to start. After each pressure reading the soil
samples with the tensiometers were weighted to determine the evaporative water loss as afunction of time. The evaporation rate at the beginning of the experiments was artificially
increased to approximately 1.2 cm d-1 by using of a fan to blow air away from the soil surface atroom temperature [Wendroth et al., 1993]. Once the gradient between the tensiometers reached avalue between 1.5 and 2.5 m/m, the top of the soil sample was covered to prevent further
evaporation. After re-establishing hydraulic equilibrium in the samples, evaporation was allowed
to continue, without the fan, at a rate of approximately 0.2 cmd-1. Measurements were taken
every 30 min during the initial high-evaporation rate period, and every 4 hours during the second
stage when the rate was relatively low. The evaporation experiment was terminated when theupper tensiometer recorded a pressure head value of -650 cm. Water losses between consecutive
measurements were used to calculate the average evaporation rate for a given time interval; thisinformation was subsequently used as the upper boundary condition in the numericalsimulations.
The laboratory experiment was first analyzed using the modified Wind method as
described by Wendroth et al. [1993]. The soil hydraulic parameters were obtained bysimultaneous fitting of the resulting θ(h) and K(h) data using the RETC code [van Genuchten etal., 1991]. The final parameters are given in Table 11.6. Data obtained with Wind's method andthe subsequently fitted hydraulic functions are shown in Fig. 11.17.

171

0.35

Water Content [-]

a)
0.3

0.25

All tensiometers
Tensiometer 1
Tensiometer 2
Tensiometer 3
Tensiometer 4
Tensiometer 5
Fitted W ind's Points
W ind Method

0.2

0.15
0.5

1.0

1.5

2.0

2.5

3.0

log h [cm]2b)log K [cm/d]

1
0
All Tensiometers
Tensiometer 1
Tensiometer 2
Tensiometer 3
Tensiometer 4
Tensiometer 5
Fitted W ind's Points
W ind Method
Tension Disc Infiltration

-1
-2
-3
-4
0.0

0.5

1.0

1.5

2.0

2.5

3.0

log h [cm]Fig. 11.17. Water retention (a) and hydraulic conductivity (b) functions determined with inverse parameter
estimation and Wind's method (example 8).

172

The soil hydraulic parameters were also estimated from the evaporation experiment usingparameter inversion. We used for this purpose the tensiometer readings as a function of time andthe total water volume at the end of the experiment. The resulting optimized soil hydraulicparameters are listed in Table 11.6, together with the r2 of the regression between predicted andmeasured values. Soil hydraulic characteristics obtained by numerical inversion and using
Wind's method are compared in Figure 11.17.
Notice a very good correspondence between retention curves obtained by parameter
optimization and the θ(h) data points determined with Wind's method or their analytical fit (Fig.
11.17a). The soil-water retention parameters obtained with Wind's method and by parameter
optimization are almost identical and undistinguishable from each other (Fig. 11.17a). The
estimated unsaturated hydraulic conductivity functions obtained with the two methods (Fig.
11.17b) are also almost identical. Figure 11.17 also shows results of inversions when theobjective function was alternatively defined in terms of readings obtained with only one
tensiometer. Results apparently remain relatively close to those obtained when readings from all
tensiometers were used simultaneously.
Measured and fitted tensiometer readings are shown in Figure 11.18. The largest
deviations were about 5 and 20 cm for the first and second evaporation rates, respectively, withmost deviations being much lower.

Table 11.6. Hydraulic parameters obtained from an evaporation experiment usingparameter estimation and Wind's method (example 8).
θr

θs

α

n

Ks

(-)

(-)

(cm-1)(-)(cm d-1)

Parameter Estimation 0.0055

0.321

0.02741.22

93.1

0.9987Wind's Method

0.321

0.0249

1.23

73.3

0.992

Method of Analysis

0.0045

173

R2

0

a)
Pressure Head [cm]

-100
-200
-300

Measured 1
Fitted 1
Measured 2
Fitted 2
Measured 3
Fitted 3
Measured 4
Fitted 4
Measured 5
Fitted 5

-400
-500
-600
-700
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time [d]-900t = 0.043 d
t = 0.334 d
t = 1.674 d
t = 2.667 d
t = 3.507 d
t = 4.167 d
Measured-800

Pressure Head [cm]

-700
-600

b)

-500
-400
-300
-200
-100
0
-10

-8

-6

-4

-2

0

z [cm]
Fig. 11.18. Measured and fitted tensiometer readings as a function of time (a) and
depth (b) (example 8).

17411.9. Example 9 – Infiltration into Structured Soil
This example shows an application of the dual-porosity model described in Section 2.1.2.
Figure 11.19 shows computed water and solute distributions during infiltration obtained with themodel [Šimůnek et al., 2003]. The soil hydraulic parameters of the macropore (mobile) domain
were taken as follows: θr=0.0, θs=0.200, α1=0.041 cm-1, n=1.964, l=0.5, Ks=0.000722 cm s-1,
while the (immobile) matrix domain was assumed to have a saturated water content, θsm, of 0.15.
Initial conditions were set equal to a pressure head of –150 cm. We assumed that water masstransfer was described with Eq. (2.96), in which the mass transfer constant ω was set at 0.00001
s-1. For simplicity we considered only convective solute mass transfer between the two poreregions (i.e. no diffusive transfer), with the dispersivity again fixed at 2 cm. While for ponded
surface conditions water in the fracture domain quickly reached full saturation (Fig. 11.19a), thewater content of the matrix increased only gradually with time. Consequently, the total watercontent, defined as the sum of the water contents of both the fracture and matrix domains, also
increased only gradually. The total water content would be the quantity measured with most fieldwater content measurement devices, such as a TDR or neutron probe. Pressure headmeasurements using tensiometers are, on the other hand, often dominated by the wetter fracturedomain that reaches equilibrium relatively quickly. The dual-porosity model can therefore
explain often observed nonequilibrium between pressure heads and water contents (e.g. Šimůnek
et al. [1999, 2001], among others). Similar nonequilibrium profiles as for the water content werealso obtained for the solute concentration (Fig. 11.19b).

175

Fracture Domain

Matrix Domain

Both Domains

c)

0

0

10

10

10

10

20

20

20

20

30

t=0

40

50

30
t=0

40

t = 1800 s

t = 1800 s

t = 3600 s

t = 3600 s
50

t = 5400 s

0.1

0.15

0.2

60
0.02

0.25

Mobile Water Content [-]

0.04

0.05

0.06

0.07

t = 1800 s

t = 3600 s

50

t = 3600 s

50

t = 5400 s

t = 5400 s

t = 7200 s

t = 7200 s
60

0

0.08

0.05

0.1

0.15

0.2

0.25

0.3

0

0.000002

Total Water Content [-]Immobile Water Content [-]0.000004

0.000006

0.000008

0.00001

Mass Transfer [1/s]

b)

d)
0

0

10

10

10

10

20

20

20

20

30
t=0

40

30
t=0

40

t = 1800 s
t = 3600 s
50

50

t = 5400 s
t = 7200 s

30
t=0

40

t = 1800 s

t = 1800 s

t = 3600 s

t = 3600 s
50

t = 5400 s

Depth [cm]0Depth [cm]0Depth [cm]Depth [cm]t=0

t = 1800 s

60

0.03

30

40

t=0

t = 7200 s

60
0.05

30

40

t = 5400 s

t = 7200 s0Depth [cm]0Depth [cm]0Depth [cm]Depth [cm]

a)

30

40

t=0
t = 1800 s
t = 3600 s

50

t = 5400 s

t = 5400 s

t = 7200 s

t = 7200 s

t = 7200 s
60

60

60
0

0.2

0.4

0.6

Relative Concentration [-]

0.8

1

0

0.2

0.4

0.6

Relative Concentration [-]

0.8

1

60
0

0.2

0.4

0.6

Relative Concentration [-]

0.8

1

0

0.000002

0.000004

0.000006

0.000008

0.00001

Solute Mass Transfer [1/cm3/s]

Fig. 11.19. Water content (a) and concentration (b) profiles in the fracture (mobile) domain, the matrix (immobile) domain, and both domains
combined, as well as the water (c) and solute (d) mass transfer terms as calculated with the dual-porosity model [Šimůnek et al., 2003].

176

11.10. Example 10 – Evaluation of Hysteretic Models
The new hysteresis model is demonstrated on a data set reported by Lenhard et al.
[1991]. In this experiment, the sand (97.5% sand, 0.8% silt, and 1.7% clay) was packed in acolumn with a cross-sectional area of approximately 39 cm2 and 72 cm height. Porous ceramictensiometers connected to pressure transducers were installed 70, 60, 50, 40, 30, and 20 cm
above the soil bottom. Water contents were measured using a gamma radiation attenuation
system at the same locations. The water table was raised initially to the soil surface by wetting
the dry soil column. The water table at the beginning of the experiment (i.e., at t = 0 hours) was
lowered 5 cm. Every 10 min thereafter the water table was lowered an additional 5 cm until
reaching an elevation of 7 cm at t = 2 hours, where it remained stationary for an hour. At t = 3
hours the water table was raised 5 cm every 10 min to a final elevation of 42 cm where it
remained for an hour until t = 5 hours. The first lowering and raising of the water table produced
main drainage and imbibition scanning paths, respectively. A drying scanning path was
subsequently generated by lowering the water table at the prescribed rate from 42 cm at t = 5hours to 17 cm where again the water table remained stationary for an hour until t = 6.67 hours.
The final saturation path, an imbibition scanning path, was produced by raising the water table
elevation from 17 cm back to the soil surface at 72 cm, which was reached at t = 8.33 hours. In
the last path, as the water table was raised past 42 cm, all internal scanning loops should havebeen closed. The soil hydraulic parameters were also reported by Lenhard et al. [1991](θr=0.0612, θsd=0.36, αd=0.042 cm-1, n=5.25, Ks=119 cm h-1, l=0.5, θs=0.36, θsw=0.27, αw=0.084
cm-1). Since the collected data are described in detail by Lenhard et al. [1991], we show here
only the simulation results.
Figure 11.20 compares the original and newly-implemented hysteresis models byshowing the main drainage curve, and the imbibition (wetting) and drainage (drying) scanning
curves measured at elevations of 30, 40, 50, and 60 cm. The original model clearly shows thepumping effect after the process reversal. Figure 11.21 shows pressure heads and water contents
at elevations of 20, 30, 40, 50, 60, and 70 cm as calculated with the new model. Calculated
results are essentially identical to those presented by Lenhard et al. [1991].

177

-40

-50

50 cm
40 cm

Water Content [-]Water Content [-]

60 cm
30 cm

-40

-30

-20

Pumping Effect
-30

-20

-10
-10

Pumping Effect
0

0

0

0.1

0.2

0.3

0.4

0

Pressure head [cm]

0.2

0.3

0.4

Pressure head [cm]

-40

-50

50 cm
40 cm

60 cm
30 cm

-40

Water Content [-]-30

Water Content [-]

0.1

-20

Scanning loops
-30

-20

-10
-10

0

0

0

0.1

0.2

0.3

0.4

0

Pressure head [cm]

0.1

0.2

0.3

0.4

Pressure head [cm]Fig. 11.20. Main drainage curves, and imbibition and drainage scanning curves, as calculated atelevations of 40 and 50 (left figure) and 30 and 60 cm (right figure) using the original (top) and new(bottom) hysteresis models.

178

60

0.40

40

0.35
0.30

20

0.25
0
0.20
-20

0.15

-40

0.10

-60

0.05
0

1

2

3

4

5

6

7

8

9

10

0

1

2

Time [hours]

3

4

5

6

7

8

9

10

Time [hours]

Fig. 11.21. Calculated pressure heads (left) and water contents (right) at elevations of 20 (N6), 30 (N5), 40 (N4), 50(N3), 60 (N2), and 70 (N1) cm.

11.11. Example 11 – Coupled Water, Vapor, and Heat Transport
Example 11 demonstrates capabilities of HYDRUS-1D to simulate coupled water, vaporand heat transport [Šimůnek et al., 2007]. Figure 11.21 shows calculated water content, total flux,
temperature and concentration profiles for a 10-cm long soil sample with zero water fluxes atboth the top and bottom boundaries, and with a specified temperature gradient along the sample.
Increasing temperatures (Fig. 11.21c) from the top to the bottom of the sample cause vapor flow
(Fig. 11.21b) from the warmer bottom end of the sample toward the colder end. Water
evaporates at the warmer end, flows upward as vapor and condensates at the colder end. Water
contents correspondingly decrease at the warmer end, and increase at the colder bottom (Fig.
11.21a). As a consequence of changing water contents, a pressure head gradient develops in thesample, leading to water flow in a direction opposite to vapor flow. A steady-state is eventuallyreached when upward vapor flow fully balances downward liquid flow (Fig. 11.21b). Since
water evaporates at the bottom of the sample and condensates at the top, solute becomes moreconcentrated near the bottom and more diluted near the top (Fig. 11.21d). Also, the concentration
profile should eventually reach steady-state, although at a much later time, when the downward
advective solute flux balances the upward diffusive flux.

179

10

10
T=0
t=0.25 d
t=1
t=5
t=25

Depth [cm]

8

6

T=0
t=0.25 d
t=1
t=5
t=25

8

6

4

4

2

2

a

b

0

0
0

0.05

0.1

0.15

0.2

0

0.02

Water Content [-]
10

0.04

0.06

Total Flux [cm/d]
10

c

d

T=0

8

t=0.25 d8

t=1

Depth [cm]

t=5
6

6

4

4

T=0
t=0.25 d
t=1
t=5
t=25

2

t=25

2

0

0
0

10

20

30

Temperature [C]

0

2

4

6

8

10

Concentration [-]Fig. 11.22. Water content (a), total flux (b), temperature (c), and solute concentration (d) distributions in a

10-cm long vertical soil sample with zero water fluxes across the top and bottom boundaries, and withtemperature increasing from top to bottom.

18012. INPUT DATA
The input data for HYDRUS are given in five separate input files. These input files
consist of one or more input blocks identified by the letters from A through M. The input files
and blocks must be arranged as follows:
SELECTOR.IN
A. Basic Information
B. Water Flow Information
C. Time Information
D. Root Growth Information
E. Heat Transport Information
K. Carbon Dioxide Transport Information
F. Solute Transport Information
L. Major Ion Chemistry Information
G. Root Water Uptake Information
PROFILE.DAT
H. Nodal Information
ATMOSPH.IN
I. Atmospheric Information
FIT.INJ.

Inverse Solution Information

METEO.IN
M. Meteorological Information

All input files must be placed into one subdirectory. Output files are printed into the same
subdirectory. Another file, HYDRUS1D.DAT, which is not read by the executable code, enables
communication between particular modules of the user-interface and will be described in part Bof this manual. The input files can be created manually or with the graphics-based user-friendly
interface HYDRUS1D also described in part B.
Tables 12.1 through 12.13 describe the data required for each input block. All data areread in using list-directed formatting (free format). Comment lines are provided at the beginningof, and within, each input block to facilitate, among other things, proper identification of thefunction of the block and the input variables. The comment lines are ignored during program
execution; hence, they may be left blank but should not be omitted. The program assumes that all
input data are specified in a consistent set of units for mass M, length L, and time T. The values

181

of temperature should be specified in degrees Celsius. Several variables in the METEO.IN file
must be specified in the requested units.
Most of the information in Tables 12.1 through 12.13 should be self-explanatory. Table
12.8 (Block H) is used to define, among other things, the nodal coordinates and initial conditions
for the pressure head, temperature and solute concentrations. One short-cut may be used whengenerating the nodal coordinates. The short-cut is possible when two nodes (e.g., N1 and N2), notadjacent to each other, are located such that N2 is greater than N1+1. The program willautomatically generate nodes between N1 and N2, provided all of the following conditions aremet simultaneously: (1) all nodes between nodes N1 and N2 are spaced at equal intervals, (2)values of the input variables hNew(n), Beta(n), Axz(n), Bxz(n), Dxz(n), Temp(n), Conc(1,n)
through Conc(NS,n), and Sorb(1,n) through Sorb(NS,n) vary linearly between nodes N1 and N2,and (3) values of LayNum(n) and MatNum(n) are the same for all n = N1, N1+1,..., N2-1 (seeTable 12.8).
The major ion chemistry module requires four additional input files containing input data
for the Pitzer equations. These four input files are provided together with the program and should
not be changed by the user. The four input files, which were adopted from Felmy [1990], must beplaced in the same directory as the executable program.
COMP.DAT

contains the species ID numbers, species names, and species charge.

BINARYP.DAT

contains the ID number of each species in each binary interaction
considered (e.g., CaHCO3+) and the Pitzer ion interaction parameters β(0),β(1), β(2), and Cφ for binary systems.

TERNARYP.DAT

contains the Pitzer ion-interaction parameters for the common ion ternary
systems, θ, and ψ. The first two columns include the cation-cation oranion-anion ID numbers associated with the ion-interaction parameter, θ,in column three. Subsequent columns include the anion or cation ID
number and the triple ion-interaction parameter, ψ, associated with that
triple ion interaction.

LAMBDA.DAT

contains the Pitzer ion-interaction parameters for the neutral species, λand δ. The first column of this file contains the ID number for the neutral
species, and the second column the ID number for the cation or anion
involved in the neutral-cation or neutral-anion interaction parameterized
by the Pitzer λ parameter included in the third column. Subsequentcolumns are for higher-order neutral interactions.

182

Table 12.1. Block A - Basic Information.
RecordType

Variable

Description

0

Char

iVer

HYDRUS-1D version

1,2

-

-

Comment lines.

3

Char

Hed

Heading.

4--

Comment line.

5

Char

LUnit

Length unit (e.g., 'cm').

6

Char

TUnit

Time unit (e.g., 'min').

7

Char

MUnit

Mass unit for concentration (e.g., 'g', 'mol', '-').

8

-

-

Comment line.

9

LogicallWat

Set this logical variable equal to .true. when transient water flow is considered.
Set this logical variable equal to .false. when initial condition is to be kept
constant during the simulation.

9

LogicallChem

Set this logical variable equal to .true. if solute transport is to be considered.

9

LogicallTemp

Set this logical variable equal to .true. if heat transport is to be considered.

9

LogicallSink

Set this logical variable equal to .true. if water extraction from the root zone
occurs.

9

LogicallRoot

Set this logical variable equal to .true. if root growth is to be considered.

9

LogicallShort.true. if information is to be printed only at preselected times, but not at eachtime step (T-level information, see Section 10),.false. if information is to be printed at each time step.

9

LogicallWDep.true. if hydraulic properties are to be considered as temperature dependent.
.false. otherwise (see Section 2.5).

9LogicallScreen.true. if information is to be printed on the screen during code execution.

9

LogicalAtmInf.true. if variable boundary conditions are supplied via the input file
ATMOSPH.IN,.false. if the file ATMOSPH.IN is not provided (i.e., in case of time independent
boundary conditions).

9

LogicallEquil*.true. if equilibrium or no adsorption is considered in the solute transport
equation.
.false. if nonequilibrium adsorption is considered for at least one solute species.

9

LogicallInverse+.true. if inverse problem is to be solved.
.false. if direct problem is to be solved.

10

-

-

Comment line.

11

LogicallSnow

Set this logical variable equal to .true. if snow accumulation of the soil surface
is to be considered (heat transport needs to be considered as well).

183

Table 12.1. (continued)

RecordType

Variable

Description

11

LogicallHP1

Set this logical variable equal to .true. when the HP1 module (obtained bycoupling HYDRUS-1D and PHREEQC models) is to be run.

11

LogicallMeteo.true. if meteorological information is supplied via the input file METEO.IN.
.false. if the file METEO.IN is not provided.

11

LogicallVapor.true. if vapor transport is to be considered (see section 2.1.1),.false. if vapor transport is not to be considered.

11

LogicallActRSU.true. if active root solute uptake is to be considered (see section 3.5),.false. if active root solute uptake is not to be considered.

11

LogicallFlux.true. if fluxes are to be printed for observation nodes instead of temperatures,.false. if fluxes are not to be printed for observation nodes.

11

LogicallDummyDummy variable (six times) (prepared for further expansions of the HYDRUS
program). Set equal to .false..

12

-

-

Comment line.

13

Integer

NMat

Number of soil materials. Materials are identified by the material number,
MatNum, specified in Block H.

13

Integer

NLay

Number of subregions for which separate water balances are being computed.
Subregions are identified by the subregion number, LayNum, specified in Block
H.

13

Real

CosAlfa

Cosine of the angle between the flow direction and the vertical axis (i.e., cos α= 1 for vertical flow, cos α = 0 for horizontal flow, and 0 < cos α < 1 forinclined flow.

*

Parameter lEquil is replaced with parameter lCO2 when the major ion chemistry module is used, indicating
whether or not the carbon dioxide transport is to be considered.
+

Parameter lInverse is replaced with parameter lKRed when the major ion chemistry module is used, indicating thata reduction in the hydraulic conductivity due to solution composition is to be considered.

184

Table 12.2. Block B - Water Flow Information.
RecordType

Variable

Description

1,2

-

-

Comment lines.

3

Integer

MaxIt

Maximum number of iterations allowed during any time step (usually 20).

3

Real

TolTh

Absolute water content tolerance for nodes in the unsaturated part of the flow
region [-] (its recommended value is 0.0001). TolTh represents the maximum
desired absolute change in the value of the water content, θ, between two
successive iterations during a particular time step.

3

Real

TolH

Absolute pressure head tolerance for nodes in the saturated part of the flow
region [L] (its recommended value is 0.1 cm). TolH represents the maximum
desired absolute change in the value of the pressure head, h, between two
successive iterations during a particular time step.

4

-

-

Comment line.

5

LogicalTopInf.true. if time dependent boundary condition is to be imposed at the top of the
profile; data are supplied via input file ATMOSPH.IN.
.false. in the case of time independent surface boundary conditions.

5

Logical

WLayer

Set this variable equal to .true. if water can accumulate at the surface with zerosurface runoff.

5

Integer

KodTop

Code specifying type of boundary condition (BC) for water flow at the surface.
Code number is positive for Dirichlet BC and negative for Neumann BC. In thecase of 'Atmospheric BC' set KodTop=-1. Set KodTop=0 when a prescribed BC
can change from Dirichlet BC to Neumann BC and vice versa.

5

LogicallInitW

Set this variable equal to .true. if the initial condition is given in terms of thewater content. Set this variable equal to .false. if the initial condition is given interms of the pressure head

6

-

-

Comment line.

7

LogicalBotInf.true. if time dependent boundary condition is to be imposed at the bottom of
the profile; control data are supplied via input file ATMOSPH.IN.
.false. in the case of time independent bottom boundary conditions.

7

LogicalqGWLF

Set this variable equal to .true. if the discharge-groundwater level relationshipq(GWL) is applied as bottom boundary condition.

7

Logical

FreeD.true. if free drainage is to be considered as bottom boundary condition.

7

Logical

SeepF.true. if seepage face is to be considered as the bottom boundary condition.

7

Integer

KodBotCode specifying type of boundary condition for water flow at the bottom of the
profile. Code number is positive for a Dirichlet BC and negative for a Neumann
BC. In case of a seepage face or free drainage BC set KodBot=-1.

7LogicalqDrain.true. if flow to horizontal drains is considered as bottom boundary condition.

7

Real

hSeep

Pressure head (i.e., 0) that initiates flow over the seepage face bottom boundary.

185

Table 12.2. (continued)
RecordType

Variable

Description

8a

-

-

Comment line.

9a

Real

rTop

Prescribed top flux [LT-1] (in case of a Dirichlet BC set this variable equal tozero).

9a

Real

rBot

Prescribed bottom flux [LT-1] (in case of a Dirichlet BC set this variable equal tozero).

9a

Real

rRoot

Prescribed potential transpiration rate [LT-1] (if no transpiration occurs or if
transpiration is variable in time set this variable equal to zero).
Records 8a and 9a are provided only when lower or upper boundary conditions
are independent of time and at least one of them is a Neumann BC.

8b

-

-

Comment line.

9b

Real

GWL0L

Reference position of the groundwater table (e.g., the x-coordinate of the soil
surface).

9b

Real

Aqh

Value of the parameter Aqh [LT-1] in the q(GWL)-relationship, equation (10.1);
set to zero if qGWLF=.false.

9b

Real

Bqh

Value of the parameter Bqh [L-1] in the q(GWL)-relationship, equation (10.1); set
to zero if qGWLF =.false.
Records 8b and 9b are provided only when the logical variable qGWLF=.true..

8c

-

-

Comment line.

9c

Integer

iPosDrCode for position of the drain.
= 1: Homogeneous profile; drain on top of impervious layer.
= 2: Homogeneous profile; drain above impervious layer.
= 3: Layered profile; drain at interface between both soil layers.
= 4: Layered profile; drain in bottom layer.
= 5: Layered profile; drain in top layer.

10c

-

-

Comment line.

11c
11c
11c

Real
Real
Real

zBotDr
rSpacing
EntresCoordinate of the bottom of the drain system [L].
Drain spacing, Ldr [L].
Entrance resistance, γentr [T].

12c

-

-

Comment line.
The following value is specified when iPosDr = 1.

13c

Real

KhTop

Horizontal saturated hydraulic conductivity above the drain, KhTop [LT-1].
The following three values are specified when iPosDr = 2.

13c

Real

BaseGW

Coordinate of the impervious layer [L].

13c

Real

KhTop

Horizontal saturated hydraulic conductivity above the drain, KhTop [LT-1].

186

Table 12.2. (continued)
RecordType

Variable

Description

13c

Real

WetPerWet perimeter of the drain, u [L].
The following four values are specified when iPosDr = 3.

13c
13c
13c
13c

Real
Real
Real
Real

BaseGW
KhTop
KhBot
WetPerCoordinate of the impervious layer [L].
Horizontal saturated hydraulic conductivity above the drain, KhTop [LT-1].
Horizontal saturated hydraulic conductivity below the drain, KhBot [LT-1].
The wet perimeter, u, of the drain [L].
The following six values are specified when iPosDr = 4.

13c
13c
13c
13c
13c
13c

Real
Real
Real
Real
Real
Real

BaseGW
KvTop
KvBot
KhBot
WetPerzInTF

Coordinate of the impervious layer [L].
Vertical saturated hydraulic conductivity above the drain, KvTop [LT-1].
Vertical saturated hydraulic conductivity below the drain, KvBot [LT-1].
Horizontal saturated hydraulic conductivity below the drain, KhBot [LT-1].
The wet perimeter, u, of the drain [L].
Coordinate of the transition between the upper and lower soil layer [L].
The following seven values are specified when iPosDr = 5.

13c
13c
13c
13c
13c
13c
13c

10
11

Real
Real
Real
Real
Real
Real
Real

BaseGW
KhTop
KvTop
KhBot
WetPerzInTF
GeoFacCoordinate of the impervious layer [L].
Horizontal saturated hydraulic conductivity above the drain, KhTop [LT-1].
Vertical saturated hydraulic conductivity above the drain, KvTop [LT-1].
Horizontal saturated hydraulic conductivity below the drain, KhBot [LT-1].
The wet perimeter, u, of the drain [L].
Coordinate of the transition between the upper and lower soil layers [L].
Geometry factor, adr [-], as obtained by the relaxation method [Ernst, 1962] (seeTable below).

--------------------------------------------------------------------------------------------------------------------------------Dbot/Dtop
Khbot/Khtop---------------------------------------------------------------------------------------------------1 2
4
8
16
32
--------------------------------------------------------------------------------------------------------------------------------1
2.0
3.0
5.0
9.0
15.0
30.0
2
2.4
3.2
4.6
6.2
8.0
10.0
3
2.6
3.3
4.5
5.5
6.8
8.0
5
2.8
3.5
4.4
4.8
5.6
6.2
10
3.2
3.6
4.2
4.5
4.8
5.0
20
3.6
3.7
4.0
4.2
4.4
4.6
50
3.8
4.0
4.0
4.0
4.2
4.6
--------------------------------------------------------------------------------------------------------------------------------Comment line.
Real

ha

Absolute value of the upper limit [L] of the pressure head interval below whicha table of hydraulic properties will be generated internally for each material (hamust be greater than 0.0; e.g. 0.001 cm) (see Section 5.4.7).

187

Table 12.2. (continued)
RecordType

Variable

Description

11

Real

hb

Absolute value of the lower limit [L] of the pressure head interval for which atable of hydraulic properties will be generated internally for each material (e.g.
1000 m). One may assign to hb the highest (absolute) expected pressure head to
be expected during a simulation. If the absolute value of the pressure head
during program execution lies outside of the interval [ha ,hb], then appropriate
values for the hydraulic properties are computed directly from the hydraulicfunctions (i.e., without interpolation in the table).

12

-

-

Comment line.

13

IntegeriModelSoil hydraulic properties model:= 0; van Genuchten's [1980] model with six parameters.
= 1; modified van Genuchten's model with ten parameters [Vogel and Císlerová,
1988].
= 2; Brooks and Corey's [1964] model with six parameters.
= 3; van Genuchten's [1980] model with air-entry value of -2 cm and with six
parameters.
= 4; Kosugi’s [1996] model with six parameters.
= 5; dual porosity model of Durner [1994] with nine parameters.
= 6; dual-porosity system with transfer proportional to the effective saturation (9parameters) (see Sections 2.1.3. and 2.8.).
= 7; dual-porosity system with transfer proportional to the pressure head (11parameters) (see Sections 2.1.3. and 2.8.).
= 9; dual-permeability system with transfer proportional to the pressure head (17parameters) (see Sections 2.1.4. and 2.8.).
iModel>3 options are not available with the major ion chemistry module.

13

IntegeriHyst

Hysteresis in the soil hydraulic properties:= 0; No hysteresis= 1; Hysteresis in the retention curve only
= 2; Hysteresis in both the retention and hydraulic conductivity functions= 3; Hysteresis using Robert Lenhard’s model [Lenhard et al., 1991; Lenhard
and Parker, 1992]. (Not available with major ion chemistry module.)

14

-

-

Comment line.

15

IntegeriKappa= -1 if the initial condition is to be calculated from the main drying branch.
= 1 if the initial condition is to be calculated from the main wetting branch.
Records 14 and 15 are provided only when iHyst > 0.

16

-

-

Comment line.

17
17
17

Real
Real
Real

Par(1,M)
Par(2,M)
Par(3,M)

Parameter θr for material M [-].
Parameter θs for material M [-].
Parameter α for material M [L-1].

188

Table 12.2. (continued)
RecordType

Variable

Description

17
17
17

Real
Real
Real

Par(4,M)
Par(5,M)
Par(6,M)

Parameter n for material M [-].
Parameter Ks for material M [LT-1].
Parameter l for material M [-].
The previous six parameters are specified for the matrix region when iModel=9.

17
17
17
17

Real
Real
Real
Real

Par(7,M)
Par(8,M)
Par(9,M)
Par(10,M)

The following four parameters are specified only when iModel=1.
Parameter θm for material M [-].
Parameter θa for material M [-].
Parameter θk for material M [-].
Parameter Kk for material M [LT-1].

17
17
17
17

Real
Real
Real
Real

Par(7,M)
Par(8,M)
Par(9,M)
Par(10,M)

The following four parameters are specified only when iModel=0 and iHyst>1.
Parameter θm for material M [-].
Parameter θsw for material M [-].
Parameter αw for material M [L-1].
Parameter Ksw for material M [LT-1].

17

Real

Par(7,M)

17
17

Real
Real

Par(8,M)
Par(9,M)

The following three parameters are specified only when iModel=5 [Durner,
1994].
Parameter w for material M [-]. The weighting factor for the sub-curve for thesecond overlapping subregion.
Parameter α for material M [L-1] for the second overlapping subregion.
Parameter n for material M [-] for the second overlapping subregion.

Par(7,M)
Par(8,M)
Par(9,M)

The following four parameters are specified only when iModel=6 (dual-porosity
system with transfer proportional to the water content gradient).
Parameter θrim for the immobile region of material M [-].
Parameter θsim for the immobile region of material M [-].
Parameter ω (mass transfer coefficient in (2.96)) for material M [-].

Par(7,M)
Par(8,M)
Par(9,M)
Par(10,M)
Par(11,M)

The following four parameters are specified only when iModel=7 (dual-porosity
system with transfer proportional to the pressure head gradient).
Parameter θrim for the immobile region of material M [-].
Parameter θsim for the immobile region of material M [-].
Parameter αim for the immobile region of material M [-].
Parameter nim for the immobile region of material M [-].
Parameter Ka (mass transfer coefficient in (2.100)) for material M [-].

17
17
17

17
17
17
17
17

Real
Real
Real

Real
Real
Real
Real
Real

Record 17 information is provided for each material M (from 1 to NMat).
If lWDep=.true. (Block A) then the soil hydraulic parameters Par(i,M) must be
specified at reference temperature Tref=20oC.
18

-

-

Comment line.

19
19
19

Real
Real
Real

ParF(1,M)
ParF(2,M)
ParF(3,M)

Parameter θr for material M [-].
Parameter θs for material M [-].
Parameter α for material M [L-1].

189

Table 12.2. (continued)
RecordType

Variable

Description

19
19
19

Real
Real
Real

ParF(4,M)
ParF(5,M)
ParF(6,M)

Parameter n for material M [-].
Parameter Ks for material M [LT-1].
Parameter l for material M [-].
The previous six parameters are specified for the fracture region when iModel=9(dual-permeability model).

19

Real

19

Real

19
19

Real
Real

19

Real

ParAF(1,M) Parameter w for material M [-]. w is the ratio of the volumes of the macropore or
fracture domain and the total soil system (for iModel=9, i.e., the dualpermeability model).
ParAF(2,M) Parameter β (a shape factor that depends on the geometry) for material M [-].
See eq. (2.98).
ParAF(3,M) Parameter γ (a scaling factor) for material M [-]. See eq. (2.98).
ParAF(4,M) Parameter d (an effective ‘diffusion’ pathlength) for material M [L]. See eq.
(2.98).
ParAF(5,M) Parameter Ka (the effective hydraulic conductivity Ka [LT-1] of the fracturematrix interface) for material M [LT-1]. See eq. (2.98).
Record 19 information is provided for each material M (from 1 to NMat).
Records 18 and 19 are provided only when iModel = 9, i.e., for the dualpermeability model.

190

Table 12.3. Block C - Time information.
RecordType

Variable

Description

1,2

-

-

Comment lines.

3

Real

dt

Initial time increment, Δt [T]. Initial time step should be estimated in
dependence on the problem being solved. For problems with high-pressure
gradients (e.g. infiltration into an initially dry soil), Δt should be relatively
small.

3

Real

dtMin

Minimum permitted time increment, Δtmin [T].

3

Real

dtMax

Maximum permitted time increment, Δtmax [T].

3

Real

dMul

If the number of required iterations at a particular time step is less than orequal to ItMin, then Δt for the next time step is multiplied by a dimensionless
number dMul ≥ 1.0 (its value is recommended not to exceed 1.3).

3

Real

dMul2

If the number of required iterations at a particular time step is greater than orequal to ItMax, then Δt for the next time step is multiplied by dMul2 ≤ 1.0(e.g. 0.33).

3

IntegerItMin

If the number of required iterations at a particular time step is less than orequal to ItMin, then Δt for the next time step is multiplied by a dimensionless
number dMul ≥ 1.0 (its value is recommended not to exceed 1.3).

3

Integer

ItMax

If the number of required iterations at a particular time step is greater than orequal to ItMax, then Δt for the next time step is multiplied by dMul2 ≤ 1.0(e.g. 0.33).

3

Integer

MPL

Number of specified print-times at which detailed information about thepressure head, water content, flux, temperature, concentrations, and the waterand solute balances will be printed.

4

-

-

Comment line.

5

Real

tInit

Initial time of the simulation [T].

5

Real

tMax

Final time of the simulation [T].

6

-

-

Comment line.

7

LogicallPrint

Set this logical variable equal to .true. if information about the pressure heads,
water contents, temperatures, and concentrations in observation nodes, and thewater and solute fluxes is to be printed at a constant time interval
tPrintInterval.

7

IntegernPrintSteps

Information to the screen and output files is not printed at each time step, but
after each nPrintSteps.

7

Real

tPrintInterval A constant time interval after which information about the pressure heads,
water contents, temperatures, and concentrations in observation nodes, and thewater and solute fluxes is to be printed.

191

Table 12.3. (continued)
RecordType

Variable

Description

7

LogicallEnter

Set this logical variable equal to .true. if the Enter key is to be pressed at theend of simulation.

8

-

-

Comment line.

9

Real

TPrint(1)First specified print-time [T].

9
.
.
9

Real
.
.
Real

TPrint(2)Second specified print-time [T].
.
.
.
.
TPrint(MPL) Last specified print-time [T]. (Maximum six values on one line.)

192

Table 12.4. Block D - Root Growth Information.+
RecordType

SymbolDescription

1,2

-

-

Comment lines.

3

IntegeriRootIn= 0; the root depth is specified together with other time-variable boundary
condition, such as atmospheric fluxes.
= 1; the root depth is given in a table= 2; the root depth is calculated using the growth function.
The following lines 4-7 are given only when iRootIn =1, i.e., using a table of
RootDepth values.

4

-

-

Comment line

5

Integer nGrowth

Number of data points in the root depth table.

6

-

-

Comment line

7
7

Real
Real

tGrowthRootDepthDay
Rooting depth [L]Line 7 is given nGrowth times.
The following lines 8 and 9 are given only when iRootIn =2.

8

-

-

Comment lines.

IntegeriRFak

Method to calculate the root growth factor, r.

9= 0; the root growth factor is calculated from given data [xRMed, tRMed].
= 1; the root growth factor is calculated based on the assumption that 50% of
the rooting depth, (xRMax+xRMin)/2., is reached at the midpoint of thegrowing season, (tRMin+tRHarv)/2.

9

Real

tRMin

Initial time of the root growth period [T].

9

Real

tRMed

Time of known rooting depth (set equal to zero if iRFak=1) [T].

9

Real

tRHarvTime at the end of the root water uptake period [T].

9

Real

xRMin

Initial value of the rooting depth at the beginning of the growth period(recommended value = 1 cm) [L].

9

Real

xRMed

Value of known rooting depth (set equal to zero if iRFak=1) [L].

9

Real

xRMax

Maximum rooting depth, which may be reached at infinite time [L].

9

Real

tRPeriod

Time period at which the growth function repeat itself.

+

Block D is not needed if the logical variable lRoot (Block A) is set equal to .false.

193

Table 12.5. Block E - Heat transport information.+
RecordType

SymbolDescription

1,2

-

-

Comment lines.

3
3
3
3

Real
Real
Real
Real

TPar(1,M)
TPar(2,M)
TPar(3,M)
TPar(4,M)

3

Real

TPar(5,M)

3

Real

TPar(6,M)

3

Real

TPar(7,M)

3

Real

TPar(8,M)

3

Real

TPar(9,M)

Volumetric solid phase fraction of material M, θn [-].
Volumetric organic matter fraction of material M, θo [-].
Longitudinal thermal dispersivity of material M, λL [L].
Coefficient b1 in the thermal conductivity function [MLT-3K-1] (e.g.Wm-1K-1)(see equation (4.8)).
Coefficient b2 in the thermal conductivity function [MLT-3K-1] (e.g.Wm-1K-1)(see equation (4.8)).
Coefficient b3 in the thermal conductivity function [MLT-3K-1] (e.g.Wm-1K-1)(see equation (4.8)).
Volumetric heat capacity of solid phase of material M, Cn [ML-1T-2K-1] (e.g.
Jm-3K-1).
Volumetric heat capacity of organic matter of material M, Co [ML-1T-2K-1](e.g. Jm-3K-1).
Volumetric heat capacity of liquid phase of material M, Cw [ML-1T-2K-1] (e.g.
Jm-3K-1).
Record 3 is required for each soil material M (from 1 to NMat).

4

-

-

Comment line.

5

Real

Ampl

Temperature amplitude at the soil surface [K].

5

Real

tPeriod

Time interval for completion of one temperature cycle (usually 1 day) [T].

5

IntegeriCampbell

Set equal to 1 if Campbell [1985] formula is to be used to calculate thethermal conductivity. Set equal to 0, when Chung and Horton [1987] formula
is to be used.

5

Real

SnowMF

Amount of snow that will melt during one day for each oC (e.g., 0.43 cm).

5

LogicallDummyDummy variable (five times) (reserved for further expansions of HYDRUS).
Set equal to .false..

6

-

-

Comment line.

7

IntegerkTopT

Code which specifies the type of upper boundary condition=1: Dirichlet boundary condition,=-1: Cauchy boundary condition.

7

Real

tTop

Temperature of the upper boundary, or temperature of the incoming fluid [oC].

7

IntegerkBotT

Code which specifies the type of lower boundary condition=1: Dirichlet boundary condition,=0: continuous temperature profile, zero gradient,=-1: Cauchy boundary condition.

7

Real

tBot

Temperature of lower boundary, or temperature of the incoming fluid [oC].

+

Block E is not needed if logical variable lTemp (Block A) is set equal to .false.

194

Table 12.6. Block F - Solute transport information.+
RecordType

Variable

Description

1,2

-

-

Comment lines.

3

Real

Epsi

Temporal weighing coefficient.
=0.0 for an explicit scheme.
=0.5 for a Crank-Nicholson implicit scheme.
=1.0 for a fully implicit scheme.

3

LogicallUpW.true. if upstream weighing formulation is to be used.
.false. if the original Galerkin formulation is to be used.

3

LogicallArtD.true. if artificial dispersion is to be added in order to fulfill the stability
criterion PeCr (see Section 8.4.4).
.false. otherwise.

3

LogicallTDep.true. if at least one transport or reaction coefficient (ChPar) is temperaturedependent.
.false. otherwise.
If lTDep=.true., then all values of ChPar(i,M) should be specified at areference temperature Tr=20oC.

3

Real

cTolA

Absolute concentration tolerance [ML-3], the value is dependent on the units
used (set equal to zero if nonlinear adsorption is not considered).

3

Real

cTolR

Relative concentration tolerance [-] (set equal to zero if nonlinear adsorption
is not considered).

3

Integer

MaxItC

Maximum number of iterations allowed during any time step for solutetransport - usually 20 (set equal to zero if nonlinear adsorption is notconsidered).

3

Real

PeCr

Stability criteria (see Section 8.4.4). Set equal to zero when lUpW is equal
to .true..

3

IntegerNS

Number of solutes.

3

LogicallTort.true. if the tortuosity factor [Millington and Quirk, 1961] is to be used.
.false. if the tortuosity factor is assumed to be equal to one.

3

IntegeriBacter

Set equal to 1 if attachment/detachment approach is to be used to calculate
nonequilibrium transport of viruses, colloids, or bacteria. Set equal to 0 if
original formulations, i.e., physical nonequilibrium or two-site sorption is to
be used to describe nonequilibrium solute transport.

3

LogicallFiltr

Set this logical variable equal to .true. if the attachment coefficient is to beevaluated using the filtration theory (eq. (3.24)).

3

IntegernChParNumber of solute transport parameters specific for each solute.

4

-

-

Comment line.

195

Table 12.6. (continued)
RecordType

Variable

Description

5

Integer

iNonEqual

Code describing type of nonequilibrium considered for solute transport:= 0: equilibrium solute transport= 1: one-site sorption model (chemical nonequilibrium)= 2: two-site sorption model (chemical nonequilibrium)= 3: two kinetic sorption sites model (attachment/detachment; chemical
nonequilibrium). This model is often used for particle (viruses, colloids,
bacteria) transport.
= 4: two kinetic sorption sites model (attachment/detachment) (chemical
nonequilibrium). Attachment coefficients are calculated using filtration
theory.
= 5: dual-porosity model (mobile-immobile regions; physicalnonequilibrium).
= 6: dual-porosity model (mobile-immobile regions) with two-site sorption
in the mobile zone (physical and chemical nonequilibrium).
= 7: dual-permeability model (physical nonequilibrium).
= 8: dual-permeability model with either an immobile region in the matrix
domain (physical nonequilibrium) or with two-site sorption in both domains(physical and chemical nonequilibrium).

5

LogicallWatDep.true. if at least one degradation coefficient (ChPar) is water content
dependent.
.false. otherwise.
If lWatDep=.true., then values of all degradation coefficients should bespecified at a reference water content.

5

LogicallDualNEq.true. if the dual-porosity model (mobile-immobile regions) with two-site
sorption in the mobile zone (physical and chemical nonequilibrium) is to beconsidered (iNonEqual =6).

5LogicallDummyDummy variable (eight times) (reserved for further expansions of
HYDRUS). Set equal to .false..

6

-

-

Comment line.

7
7
7

Real
Real
Real

ChPar(1,M)
ChPar(2,M)
ChPar(3,M)

7

Real

ChPar(4,M)

Bulk density of material M, ρ [ML-3].
Longitudinal dispersivity for material type M, DL [L].
Dimensionless fraction of the adsorption sites classified as type-1, i.e., sites
with instantaneous sorption, when the chemical nonequilibrium option isconsidered. Set equal to 1 if equilibrium transport is to be considered.
Dimensionless fraction of the adsorption sites in contact with mobile water
when the physical nonequilibrium option is considered. Set equal to 1 if
all sorption sites are in contact with the mobile water.
Immobile water content. Set equal to 0 when the physical nonequilibrium
option is not considered.

196

Table 12.6. (continued)
RecordType

Variable

7
7

Real
Real

ChPar(2,M)
ChPar(3,M)

7

Real

ChPar(4,M)

Description
The parameters ChPar(i,M) below are to be reinterpreted when the variable
iNonEqual =7 or 8 (for the dual-permeability model).
Longitudinal dispersivity in the matrix domain for material type M, DL [L].
Dimensionless fraction of adsorption sites classified as type-1 (i.e., sites
with instantaneous sorption) in the matrix domain when the chemical
nonequilibrium option is considered (iNonEqual=2). Set equal to 1 if
equilibrium transport is to be considered (iNonEqual=0). Set equal to 0
when one-site kinetic sorption is to be considered (iNonEqual=1).
Dimensionless fraction of adsorption sites in contact with mobile water inthe matrix domain when the physical nonequilibrium option is considered(iNonEqual=5 or 6). Set equal to 1 if all sorption sites are in contact with themobile water.
Immobile water content of the matrix domain. Set equal to 0 when thephysical nonequilibrium option is not considered or if iModel=6 or 7.
Record 7 information is provided for each material M (from 1 to NMat).

8

-

-

Comment line.

9
9

Real
Real

ChPar(5,M)
ChPar(6,M)

Ionic or molecular diffusion coefficient in free water, Dw [L2T-1].
Ionic or molecular diffusion coefficient in gas phase, Dg [L2T-1].
The parameter ChPar(6,M) is to be reinterpreted when the variable
iNonEqual =7 or 8:
Mass transfer coefficient, ωdp [T-1], for solute mass transfer between thefracture and matrix domains. See eq. (3.38).

9

Real

ChPar(6,M)

10

-

-

Comment line.

11

Real

ChPar(7,M)

11

Real

ChPar(8,M)

11

Real

ChPar(9,M)

11

Real

ChPar(10,M)

11
11
11
11

Real
Real
Real
Real

ChPar(11,M)
ChPar(12,M)
ChPar(13,M)
ChPar(14,M)

11
11
11

Real
Real
Real

ChPar(15,M)
ChPar(16,M)
ChPar(17,M)

Adsorption isotherm coefficient, ks, for material type M [L3M-1]. Set equal tozero if no adsorption is to be considered.
Adsorption isotherm coefficient, η, for material type M [L3M-1]. Set equal tozero if Langmuir adsorption isotherm is not to be considered.
Adsorption isotherm coefficient, β, for material type M [-]. Set equal to one
if Freundlich adsorption isotherm is not to be considered.
Equilibrium distribution constant between liquid and gas phases, kg, material
type M [-].
First-order rate constant for the dissolved phase, μw, material type M [T-1].
First-order rate constant for the solid phase, μs, material type M [T-1].
First-order rate constant for the gas phase, μg, material type M [T-1].
Rate constant, μw‘, representing a first-order decay for the first solute andzero-order production for the second solute in the dissolved phase, material
type M [T-1].
Same as above for the solid phase, μs‘, material type M [T-1].
Same as above for the gas phase, μg‘, material type M [T-1].
Zero-order rate constant for the dissolved phase, γw, material type M[ML-3T-1].

197

Table 12.6. (continued)
RecordType

Variable

Description

11
11
11

Real
Real
Real

ChPar(18,M)
ChPar(19,M)
ChPar(20,M)

Zero-order rate constant for the solid phase, γs, of material type M [T-1].
Zero-order rate constant for the gas phase, γg, of material type M [ML-3T-1].
First-order rate transfer coefficient for nonequilibrium adsorption, or themass transfer coefficient for solute exchange between mobile and immobile
liquid regions, ω, material type M [T-1].

11
11

Real
Integer

ChPar(6,M)
ChPar(13,M)

11
11

Integer
Real

ChPar(14,M)
ChPar(15,M)

11

Real

ChPar(16,M)

11

Real

ChPar(17,M)

11
11

Real
Real

ChPar(18,M)
ChPar(19,M)

11

Real

ChPar(20,M)

11
11

Integer
Integer

ChPar(13,M)
ChPar(14,M)

11
11
11

Real
Real
Real

ChPar(15,M)
ChPar(16,M)
ChPar(17,M)

11
11
11

Real
Real
Real

ChPar(18,M)
ChPar(19,M)
ChPar(20,M)

11

Real

ChPar(10,M)

The parameters ChPar(6,M) and ChPar(13,M) through ChPar(20,M) are to
be reinterpreted when the variable iBacter=1 (iNonEqual =3):
Diameter of the sand grains, dc [L].
Type of blocking, iPsi, used in (3.19) for the second sorption sites.
= 0: No blocking.
= 1: Langmuirian dynamics, (3.20).
= 2: ripening, (3.21).
= 3: random sequential adsorption, (3.22).
= 4: depth dependent blocking coefficient, (3.23).
Same for the first sorption sites.
Parameter in the blocking function for the second sorption sites (smax for(3.20), (3.21) and (3.22), β in (3.23)).
The first-order deposition (attachment) coefficient, ka [T-1], for the second
sorption sites.
The first-order entrainment (detachment) coefficient, kd [T-1], for the second
sorption sites.
Parameter in the blocking function for the first sorption sites.
The first-order deposition (attachment) coefficient, ka [T-1], for the firstsorption sites.
The first-order entrainment (detachment) coefficient, kd [T-1], for the firstsorption sites.
The parameters ChPar(13,M) through ChPar(20,M) are to be reinterpreted
when the variable iBacter=1 and lFilter=.true. (iNonEqual =4):
Diameter of the sand grains, dc [L].
Diameter of the particle, dp (e.g., virus, bacteria) (= 0.95 μm or 0.95e-6 m)[L].
Parameter smax in the blocking function for the second sorption sites (3.20).
Sticking efficiency, α [-], for the second sorption sites.
The first-order entrainment (detachment) coefficient, kd [T-1], for the second
sorption sites.
Parameter smax in the blocking function for the first sorption sites (3.20).
Sticking efficiency, α [-], for the first sorption sites.
The first-order entrainment (detachment) coefficient, kd [T-1], for the firstsorption sites.
The parameters ChPar(i,M) must be reinterpreted when the variable
iNonEqual =6:Must be set equal to zero.

198

Table 12.6. (continued)
RecordType

Variable

Description

11

Real

ChPar(13,M)

11

Real

ChPar(16,M)

11

Real

ChPar(20,M)

Dimensionless fraction of adsorption sites classified as type-1 (i.e., sites
with instantaneous sorption in contact with the mobile domain). Remaining
sites are classified as type-2 sites (i.e., with kinetic sorption). See eq. (3.37).
First-order rate transfer coefficient for nonequilibrium adsorption to type-2
sorption sites in the mobile region, αch, material type M [T-1]. See eq.
(3.37).
Mass transfer coefficient for solute exchange between mobile and immobile
liquid regions, ωph, for material type M [T-1]. See eq. (3.37).

11

Real

ChPar(7,M)

11

Real

ChPar(10,M)

11

Real

ChPar(13,M)

11

Real

ChPar(16,M)

11

Real

ChPar(19,M)

11

Real

ChPar(20,M)

The parameters ChPar(i,M) must be reinterpreted when the variable
iNonEqual =7 or 8 (for the dual-permeability model):Adsorption isotherm coefficient, Kdm, for the matrix domain, for material
type M [L3M-1]. Set equal to zero if no adsorption is to be considered.
First-order rate transfer coefficient for nonequilibrium adsorption in thematrix domain, αch,m, material type M [T-1]. See eq. (3.41).
Longitudinal dispersivity in the fracture domain for material type M, DL[L].
Dimensionless fraction of adsorption sites classified as type-1 (i.e., sites
with instantaneous sorption) in the fracture domain when the chemical
nonequilibrium option is considered.
Adsorption isotherm coefficient, Kdf, for the fracture domain, for material
type M [L3M-1]. Set equal to zero if no adsorption is to be considered.
First-order rate transfer coefficient for nonequilibrium adsorption in thefracture domain, αch,f, for material type M [T-1]. See eq. (3.41).
Record 11 information is provided for each material M (from 1 to NMat).
Record 7 through 11 information is provided for each solute (from 1 to NS).

12,13

-

-

Comment lines.

14

Real

TDep(5)

14

Real

TDep(6)

Activation energy for parameter ChPar(5,M) [ML2T-2M-1] (see Section
3.4). This parameter should be specified in J mol-1. Set equal to 0 if
ChPar(5,M) is temperature independent.
Same for parameter ChPar(6,M) [ML2T-2M-1].

15

-

-

Comment line.

16
.
.
16

Real
.
.
Real

TDep(7).
.
TDep(20)Same for parameter ChPar(7,M) [ML2T-2M-1].
.
.
Same for parameter ChPar(20,M) [ML2T-2M-1].
Record 12 through 16 information is provided only when the logicalvariable lTDep of record 3 is set equal to .true..

17,18

-

-

Comment lines.

19

IntegernParWD

Number of parameters whose values are water content dependent.

199

Table 12.6. (continued)
RecordType

Variable

Description

20

-

-

Comment line.

21

Real

WDep(1)

.
.
21

.
.
Real

First parameter for the water content dependence of degradation coefficients(see Section 3.4). Set equal to 0 if ChPar(11,M) is water content
independent.
.
.
.
.
WDep(nParWD) Same for parameter ChPar(19,M).

22

Real

WDep(1)

.
.
22

.
.
Real

Second parameter for the water content dependence of degradatioincoefficients (see Section 3.4). Set equal to 0 if ChPar(11,M) is water content
independent.
.
.
.
.
WDep(nParWD) Same for parameter ChPar(19,M).
Record 17 through 22 information is provided only when the logicalvariable lWatDep of record 5 is set equal to .true..

23

-

-

Comment line.

24

Integer

kTopChCode which specifies the type of upper boundary condition=1: Dirichlet boundary condition,=-1: Cauchy boundary condition.
=-2: a special type of boundary condition for volatile solutes as described byequation (3.46).

24

Real

cTop(1)

Concentration of the upper boundary, or concentration of the incoming
fluid, for the first solute [ML-3].

24

Real

cTop(2)

.
24

.
Real

.
cTop(NS)

Concentration of the upper boundary, or concentration of the incoming
fluid, for the second solute [ML-3] (not specified if NS < 2).
.
Concentration of the upper boundary, or concentration of the incoming
fluid, for the NSth solute [ML-3].

24

IntegerkBotChCode which specifies the type of lower boundary condition=1: Dirichlet boundary condition,=0: continuous concentration profile,=-1: Cauchy boundary condition.

24

Real

cBot(1)

Concentration of lower boundary, or concentration of the incoming fluid, forthe first solute [ML-3].

24

Real

cBot(2)

.
24

.
Real

.
cBot(NS)

Concentration of lower boundary, or concentration of the incoming fluid, forthe second solute [ML-3] (not specified if NS < 2).
.
Concentration of lower boundary, or concentration of the incoming fluid, for
the NSth solute [ML-3].

200

Table 12.6. (continued)
RecordType

Variable

Description

25

-

-

Comment line.

26

Real

dSurf

Thickness of the stagnant boundary layer, d [L] (see equation (3.46)).

26

Real

cAtm

Concentration above the stagnant boundary layer, gatm [ML-3] (see equation(3.46)).
Records 17 and 18 information is provided only when kTopCh=-2.

27

-

-

Comment line.

28

Real

tPulseTime duration of the concentration pulse [T].

+Block F is not needed when the logical variable lChem in Block A is set equal to .false. .

201

Table 12.7. Block G - Root water uptake information. +RecordType

Variable

Description

1,2

-

-

Comment lines.

3

Integer iMoSink

3

Real

.
.
3

.
.
Real

3

Real

OmegaC

Critical root water uptake index. Set equal to one for a noncompensated root
water uptake and smaller than one for compensated root water uptake [-].

4

-

-

Comment line.

Type of root water uptake stress response function.
= 0; Feddes et al. [1978]= 1; S-shaped, van Genuchten [1987]

cRootMax(1)Maximum allowed concentration in the root solute uptake term for the firstsolute [ML-3]. When the nodal concentration is lower than cRootMax, allsolute is taken up. When the nodal concentration is higher than cRootMax,
additional solute stays behind.
.
.
.
.
cRootMax(NS) Maximum allowed concentration in the root solute uptake term for the last
solute [ML-3].

The following records (records 5a, 6a, 7a) are given only if iMoSink=0.
5a

Real

P0

Value of the pressure head, h1 (Fig. 2.1), below which roots start to extract
water from the soil.

5a

Real

P2H

Value of the limiting pressure head, h3, below which the roots cannot extract
water at the maximum rate (assuming a potential transpiration rate of r2H).

5a

Real

P2L

As above, but for a potential transpiration rate of r2L.

5a

Real

P3

Value of the pressure head, h4, below which root water uptake ceases (usuallyequal to the wilting point).

5a

Real

r2H

Potential transpiration rate [LT-1] (currently set at 0.5 cm/day).

5a

Real

r2L

Potential transpiration rate [LT-1] (currently set at 0.1 cm/day).
The above input parameters permit one to make the variable h3 a function of
the potential transpiration rate, Tp (h3 presumably decreases at higher
transpiration rates). HYDRUS currently implements the same linear
interpolation scheme as used in several versions of the SWATRE code (e.g.,
Wesseling and Brandyk [1985]) and in the SWMS_2D [Šimůnek et al., 1992]
and HYDRUS 5.0 [Vogel et al., 1996] codes. The scheme is based on thefollowing interpolation:h 3 = P2H +P2L - P2H

( r2H - T p ) for r2L < T p < r2H
r2H - r2L
h 3 = P2L for T a ≤ r2L
h 3 = P2H

202

for T a ≥ r2H

Table 12.7. (continued)
RecordType

Variable

Description

6a

-

-

Comment line.

7a

Real

POptm(1)

Value of the pressure head, h2, below which roots start to extract water at the
maximum possible rate (material number 1).

7a
.
.
7a

Real
.
.
Real

POptm(2)As above (material number 2).
.
.
.
.
POptm(NMat) As above (for material number NMat).
The following record (record 5b) is given only if iMoSink=1.

5b

Real

P50

Value of the pressure head, h50 (Fig. 2.1), at which the root water uptake isreduced by 50%.

5b

Real

P3

Exponent, p, in the S-shaped root water uptake stress response function.
Recommended value is 3.
The following records are given only if lChem=.true. .

8

-

-

9

Logical lSolRed

Comment line.
=.true. : root water uptake is reduced due to salinity.
=.false.: otherwise.
The following records are given only if lSolRed=.true. .

10

-

-

11

Logical lSolAdd=.true. if the effect of salinity stress is additive to the pressure head stress.
=.false. if the effect of salinity stress is multiplicative to the pressure headstress.

12-

Comment line.

-

Comment line.

The following two values are specified when the root water uptake salinity
stress response function is described with the S-shaped function (2.10) or(2.11), i.e., lMsSink=.true..
13a

Real

c50

Value of the osmotic head hφ50, at which the root water uptake is reduced by
50%. This value is specified only when lSolAdd=.false..

13a

Real

P3c

Exponent, p, in the S-shaped root water uptake salinity stress response
function. Recommended value is 3. This value is specified only whenlSolAdd=.false..
The following two values are specified when the root water uptake salinity
stress response function is described with the threshold-slope function ofMaas [1990], i.e., lMsSink=.false..

203

Table 12.7. (continued)

RecordType

Variable

Descriptionα ( hφ ) = 1

M

hφ ≤ hφ < 0

α ( hφ ) = max[0 ,1. - 0.01( hφ - hφ ) s φ ]M

hφ < hφ

M

13a

Real

c50

Value of the minimum osmotic head (the salinity threshold) hφM, above whichroot water uptake is not reduced. This value is specified only whenlSolAdd=.false..

13a

Real

P3c

Slope, sφ, of the curve determining fractional root water uptake decline perunit increase in salinity below the threshold. This value is specified only whenlSolAdd=.false..

13
13
13

Real
Real
.
.
Real

aOsm(1)aOsm(2).
.
aOsm(NSD)

Osmotic coefficient, a1, for the first solute [L4M-1].
Osmotic coefficient, a2, for the second solute [L4M-1].
.
.
Osmotic coefficient, an, for the last solute [L4M-1].

13

Logical lMsSink=.true. : S-shaped root water uptake salinity stress response function.
=.false.: threshold function according Maas [1990].
The following records are given only if lActRSU=.true.

14

-

-

Comment line.

15

Real

OmegaS

Critical root solute uptake index. Set equal to one for a noncompensated root
solute uptake and smaller than one for compensated root water uptake [-].

15

Real

rKM

Michaelis-Menton constant for active root solute uptake.

15

Real

SPot

Potential plant nutrient demand [ML-2T-1].

15

Logical lOmegaW+

Logical variable indicating whether the potential plant nutrient demand is tobe reduced proportionally to the root water uptake reduction.

Block G is not needed when the logical variable SinkF (Block A) is set equal to .false. .

204

Table 12.8. Block H - Nodal information.
RecordType

Variable

Description

0

Char

iVer

HYDRUS-1D version

1

Integer

NFix

Number of fixed nodes.

2

Integeri

Fixed node.

2

Real

xFix(i)x-coordinate of the fixed node i.

2

Real

wTop(i)

Nodal density above fixed node i.

2

Real

wBot(i)

Nodal density below fixed node i.
Record 2 must be specified for each fixed node.
Records 1 and 2 have relevant information only for the module PROFILE of
the user interface. When the code is used without the user interface, then only
two fixed points (top and bottom of the soil profile) with unit nodal density
have to be specified.

3

Integer

NumNP

Number of nodal points.

3

IntegerNS

Number of solutes (set equal to zero if lChem is equal to .false.).

3

Integer

iTempThis variable is read only if the user interface is used.
= 1; initial condition for the temperature is specified (must be equal to 1 whenlTemp or lChem is equal to .true.).
= 0; initial condition for the temperature is not specified.

3

IntegeriEquilThis variable is read only if the user interface is used.
= 1; Equilibrium solute transport is considered.
= 0; Nonequilibrium solute transport is considered.
Set equal to 1 if lChem is equal to .false..

4

IntegernNodal number.

4

Real

x(n)

x-coordinate of node n [L].

4

Real

hNew(n)Initial value of the pressure head at node n [L]. If lWat=.false. in Block A,
then hNew(n) represents the pressure head which will be kept constant during
simulation.

4

Integer

MatNum(n)Index for material whose hydraulic and transport properties are assigned tonode n.

4Integer

LayNum(n)Subregion number assigned to node n.

4

Real

Beta(n)Value of the water uptake distribution, b(x) [L-1], in the soil root zone at noden. Set Beta(n) equal to zero if node n lies outside the root zone.
Following three numbers, i.e., Ah(n), Ak(n), and Ath(n), are given only whenneither carbon dioxide transport nor major ion chemistry is considered.

205

Table 12.8. (continued)

RecordType

Variable

Description

4

Real

Ah(n)

Nodal value of the dimensionless scaling factor αh [-] associated with thepressure head.

4

Real

Ak(n)

Nodal value of the dimensionless scaling factor αK [-] associated with thesaturated hydraulic conductivity.

4

Real

Ath(n)

Nodal value of the dimensionless scaling factor αθ [-] associated with thewater content.
The following number, i.e., CO2(n), is given only when either carbon dioxidetransport or major ion chemistry is considered.

4

Real

CO2(n)

Initial value of the carbon dioxide concentration at node n [L3L-3].

4

Real

Temp(n)Initial value of the temperature at node n [oC] (do not specify if both lTemp or
lChem are equal to .false.; if lTemp=.false. and lChem=.true. then set equal to
0 or any other initial value to be used later for temperature dependent waterflow and solute transport).

4

Real

Conc(1,n)4

Real

Conc(2,n)

.
.
4

.
.
Real

.
.
Conc(i,n)4

Real

Sorb(1,n)

4

Real

Sorb(2,n)

.
4

.
Real

.
Sorb(i,n)

Following dissolved and sorbed concentrations, i.e., Conc(i,n) and Sorb(i,n),are given only when neither carbon dioxide transport nor major ion chemistry
is considered.
Initial value of the concentration of the first solute at node n [ML-3] (omit iflChem=.false.).
Initial value of the concentration of the second solute at node n [ML-3] (omit iflChem=.true. and NS < 2).
.
.
Initial value of the concentration of the last solute at node n [ML-3] (omit iflChem=.true. and NS < i).
Initial value of the adsorbed concentration on type-2 sites of the first solute atnode n [MM-1]. Omit this variable if lChem=.false. or lEquil=.true. .
Initial value of the adsorbed concentration on type-2 sites of the second soluteat node n [MM-1]. Omit this variable if lChem=.false. or lEquil=.true. or NS <2.
.
Initial value of the adsorbed concentration on type-2 sites of the NSth solute atnode n [MM-1]. This variable does not have to be specified if lChem=.false. or
lEquil=.true. and NS < i).
Following three numbers, i.e., nC(n), nX(n), and nS(n), are given only whenmajor ion chemistry is considered.

4

IntegernC(n)4IntegernX(n)Code which specifies which solution concentration combination (see Block H)
is to be used as an initial condition at node n [-] (omit if lChem=.false.).
Code which specifies which surface species combination (see Block H) is to
be used as an initial condition at node n [-] (omit if lChem=.false.).

206

Table 12.8. (continued)

RecordType

Variable

Description

4

IntegernS(n)Code which specifies which mineral phase combination (see Block H) is to be
used as an initial condition at node n [-] (omit if lChem=.false.).
In general, record 4 information is required for each node n, starting with n=1
and continuing sequentially until n=NumNP. Record 4 information for certain
nodes may be skipped if several conditions are satisfied (see beginning of thissection).

5

Integer

NObs

Number of observation nodes for which values of the pressure head, the watercontent, temperature (for lTemp=.true.), and the solution and sorbed
concentrations (for lChem=.true.) are printed at each time level.

6

IntegeriObs(1)

Nodal number of the first observation node.

6

IntegeriObs(2)

Nodal number of the second observation node.

.

.

.

.

.

.

.

.

6

IntegeriObs(NObs)

Nodal number of the last observation node.

207

Table 12.9. Block I - Atmospheric information.+
RecordType

Variable

Description

0

Char

iVer

HYDRUS-1D version

1,2

-

-

Comment lines.

3

Integer MaxAl

Number of atmospheric data records.

4-

Comment line.

5

Logical lDailyVar.true. if HYDRUS-1D is to generate daily variations in evaporation andtranspiration (see section 2.7.2.).
.false. otherwise.

5

Logical lSinusVar.true. if HYDRUS-1D is to generate sinusoidal variations in precipitation (seesection 2.7.2.).
.false. otherwise.

5

LogicallLai

Logical variable indicating that potential evapotranspiration is to be divided into
potential evaporation and potential transpiration using eq. (2.75).

5LogicallDummyDummy variable (eight times) (reserved for further expansions of HYDRUS).
Set equal to .false..

-

The following two lines are given only when lLai =.true.
6a

-

-

Comment line.

7a

Real

rExtinct

A constant for the radiation extinction by the canopy (rExtinct=0.463) [-].

6b

-

-

Comment line.

7b

Real

hCritS

Maximum allowed pressure head at the soil surface [L].

8

-

-

Comment line.

9

Real

tAtm(i)Time for which the i-th data record is provided [T].

9

Real

Prec(i)

Precipitation rate [LT-1] (in absolute value).

9

Real

rSoil(i)Potential evaporation rate [LT-1] (in absolute value). rSoil(i) is interpreted as
KodTop when a time variable Dirichlet or Neumann boundary condition is
specified.

9

Real

rRoot(i)Potential transpiration rate [LT-1] (in absolute value).

9

Real

hCritA(i)Absolute value of the minimum allowed pressure head at the soil surface [L].

9

Real

rB(i)

Bottom flux [LT-1] (set equal to 0 if KodBot is positive, or if one of the logicalvariables qGWLF, FreeD or SeepF is .true.).

9

Real

hB(i)

Groundwater level [L], or any other prescribed pressure head boundary
condition as indicated by a positive value of KodBot (set equal to 0 if KodBot isnegative, or if one of the logical variables qGWLF, FreeD or SeepF is .true.).

9

Real

hT(i)

Prescribed pressure head [L] at the surface (set equal to 0 if KodBot is negative).

208

Table 12.9. (continued)
RecordType

Variable

Description

9

Real

tTop(i)Soil surface temperature [oC] (omit if both lTemp and lChem are equal to.false.).

9

RealtBot(i)Soil temperature at the bottom of the soil profile [oC] (omit if both lTemp andlChem are equal to .false., set equal to zero if kBotT=0).

9

Real

Ampl(i)

Temperature amplitude at the soil surface [K] (omit if both lTemp and lChem
are equal to .false.).
The following parameters (i.e., cTop(i,j) and cBot(i,j)) are to be specified only
when neither carbon dioxide transport nor major ion chemistry is considered.

9

Real

cTop(i,1)9Real

cTop(i,2)

.
9

.
Real

.
cTop(i,NS)

9

Real

cBot(i,1)

9

Real

cBot(i,2)

9

Real

cBot(i,NS)

Soil surface concentration [ML-3] for the first solute (not needed if lChem isequal to .false.).
Soil surface concentration [ML-3] for the second solute (not needed if lChem isequal to .false. or NS < 2).
.
Soil surface concentration [ML-3] for the NSth solute (not needed if lChem isequal to .false.).
Concentration at the bottom of the soil profile [ML-3] for the first solute (notneeded if lChem is equal to .false., set equal to zero if cBotSolute=0).
Concentration at the bottom of the soil profile [ML-3] for the second solute (notneeded if lChem is equal to .false., set equal to zero if cBotSolute=0 or NS < 2).
Concentration at the bottom of the soil profile [ML-3] for the NSth solute (notneeded if lChem is equal to .false., set equal to zero if cBotSolute=0).
Following two number, i.e., kTopCh(i) and kBotCh(i), are given only whenmajor ion chemistry is considered.

9

Real

kTopCh(i)Code which refers to the field ConcTab for the value of the solute transport
upper boundary condition. Sign of kTopCh(i) indicates whether a Dirichlet(positive) or Neumann (negative) boundary condition is to be applied at the soil
surface. ConcTab(abs(kTopCh(i)),j) is the boundary condition for the soil
surface for species j. Permissible values are ±1,±2,±3,...,±nSolConc.

9

Real

kBotCh(i)Code which refers to the field ConcTab for the value of the solute transport
lower boundary condition. Sign of kBotCh(i) indicates whether a Dirichlet(positive) or Neumann (negative) boundary condition is to be applied at thebottom of the soil profile. ConcTab(abs(kBotCh(i)),j) is the boundary condition
for the bottom of the soil profile for species j. Permissible values are±1,±2,±3,...,±nSolConc.

9

Real

RootDepth

Rooting depth [L] (given only when iRootIn =0).
The total number of atmospheric data records is MaxAl (i=1,2, ..,MaxAl).

+

Block I is not needed if the logical variable AtmInf (Block A) is set equal to .false. .

209

Table. 12.10. Block J - Inverse solution information.+
RecordType

Variable

Description

0

Char

iVer

HYDRUS-1D version

1

Integer NCase

Number of cases being considered (only for the first data set).

2
3

Char
Char

Title1
Title2

Descriptive title for simulation.
Descriptive title for simulation.

4

-

-

Comment line.

5

Integer NOBB

Number of observed data.

5

Integer MIT

Maximum number of iterations for the inverse problem.

5

Integer iWeight

Type of weighting used for the data set.
= 0; no internal weighting.
= 1; weighting by mean ratio.
= 2; weighting by standard deviation.

6-

Comment line.

7

Logical lWatF

Set this logical variable equal to .true. when the soil hydraulic parameters are to
be optimized.

7

Logical lChemF

7

Integer NMat

Set this logical variable equal to .true. when the solute transport parameters areto be optimized.
Number of soil materials. Materials are identified by the material number,
MatNum, specified in Block H.

7

Logical lTempF

Set this logical variable equal to .true. when the heat transport parameters are to
be optimized.

8-

Comment line.

9

Integer iModel

Soil hydraulic properties model:= 0; van Genuchten's [1980] model containing six parameters.
= 1; modified van Genuchten's model containing ten parameters, Vogel andCíslerová [1988].
= 2; Brooks and Corey's [1964] model containing six parameters.

9

Integer iHyst

Hysteresis in the soil hydraulic properties:= 0; no hysteresis= 1; hysteresis in the retention curve only
= 2; hysteresis in both the retention and hydraulic conductivity functions

10-

Comment line.

11

Integer iQSame

11

Logical lAw2Ad

-

-

-

Parameter constraints= 0: θsd > θsw= 1: θsd = θmd , θsw = θmw= 2: θsd = θsw = θm.true. if parameter constraint αw = 2 αd is to be considered.
.false. if no constraint on αw and αd is imposed.

210

Table. 12.10. (continued).

RecordType

Variable

Description

11

Logical lKSame.true. if parameter constraint Ksw = Ksd is to be considered.
.false. if no constraint on Ksw and Ksd is imposed.

11

Integer iKappa= -1 if the initial condition is to be calculated from the main drying branch.
= 1 if the initial condition is to be calculated from the main wetting branch.
Records 10 and 11 are provided only when iHyst > 0.
Records 8 through 11 are specified only when the logical variable lWatF isequal to .true.

12

-

-

13

Integer NS

Number of solutes (must be equal to 1).

13

Real

Type of concentration that is used in the objective function [-].
= 0: resident concentration (concentration in the mobile region)= 1: log resident concentration= 2: flux concentration= 3: total resident concentration (includes sorbed phase)= 4: resident concentration (includes mobile and immobile regions)

iConcType

Comment line.

Records 12 and 13 are specified only when the logical variable lChemF is equal
to .true.
14--

Comment line.

15
15
15
15
15
15

Real
Real
Real
Real
Real
Real

Par(1,M)
Par(2,M)
Par(3,M)
Par(4,M)
Par(5,M)
Par(6,M)

Initial estimate of parameter θr for material M [-].
Initial estimate of parameter θs for material M [-].
Initial estimate of parameter α for material M [L-1].
Initial estimate of parameter n for material M [-].
Initial estimate of parameter Ks for material M [LT-1].
Initial estimate of parameter l for material M [-].

15
15
15
15

Real
Real
Real
Real

Par(7,M)
Par(8,M)
Par(9,M)
Par(10,M)

The following four parameters are specified only when iModel=1.
Initial estimate of parameter θm for material M [-].
Initial estimate of parameter θa for material M [-].
Initial estimate of parameter θk for material M [-].
Initial estimate of parameter Kk for material M [LT-1].

15
15
15
15

Real
Real
Real
Real

Par(7,M)
Par(8,M)
Par(9,M)
Par(10,M)

The following four parameters are specified only when iModel=0 and iHyst>1.
Initial estimate of parameter θm for material M [-].
Initial estimate of parameter θsw for material M [-].
Initial estimate of parameter α w for material M [L-1].
Initial estimate of parameter Ksw for material M [LT-1].

16

Integer Index(1,M)

16

Integer Index(2,M)

Parameter estimation index for parameter θr for material M [-].
= 0; Coefficient is known and kept constant during optimization.
= 1; Coefficient is unknown and estimated by curve fitting the data.
Parameter estimation index for parameter θs for material M [-].

211

Table. 12.10. (continued).
RecordType

Variable

Description

16
16
16
16

Integer
Integer
Integer
IntegerIndex(3,M)
Index(4,M)
Index(5,M)
Index(6,M)

Parameter estimation index for parameter α for material M [-].
Parameter estimation index for parameter n for material M [-].
Parameter estimation index for parameter Ks for material M [-].
Parameter estimation index for parameter l for material M [-].

Index(7,M)
Index(8,M)
Index(9,M)
Index(10,M)

The following four parameter estimation indices are specified only wheniModel=1.
Parameter estimation index for parameter θm for material M [-].
Parameter estimation index for parameter θa for material M [-].
Parameter estimation index for parameter θk for material M [-].
Parameter estimation index for parameter Kk for material M [-].
The following four parameter estimation indices are specified only wheniModel=0 and iHyst>1.
Parameter estimation index for parameter θm for material M [-].
Parameter estimation index for parameter θsw for material M [-].
Parameter estimation index for parameter αww for material M [-].
Parameter estimation index for parameter Ksw for material M [-].

16
16
16
16

Integer
Integer
Integer
Integer

16
16
16
16

Integer
Integer
Integer
IntegerIndex(7,M)
Index(8,M)
Index(9,M)
Index(10,M)

17

Real

BMn(1,M)

17
17
17
17
17

Real
Real
Real
Real
Real

BMn(2,M)BMn(3,M)BMn(4,M)BMn(5,M)BMn(6,M)

Minimum constraint for parameter θr for material M [-] (dummy value if
Index(1,M)=0).
Minimum constraint for parameter θs for material M [-].
Minimum constraint for parameter α for material M [L-1].
Minimum constraint for parameter n for material M [-].
Minimum constraint for parameter Ks for material M [LT-1].
Minimum constraint for parameter l for material M [-].

BMn(7,M)BMn(8,M)BMn(9,M)BMn(10,M)

The following four minimum parameter constraints are specified only wheniModel=1.
Minimum constraint for parameter θm for material M [-].
Minimum constraint for parameter θa for material M [-].
Minimum constraint for parameter θk for material M [-].
Minimum constraint for parameter Kk for material M [LT-1].
The following four minimum parameter constraints are specified only wheniModel=0 and iHyst>1.
Minimum constraint for parameter θm for material M [-].
Minimum constraint for parameter θsw for material M [-].
Minimum constraint for parameter αw for material M [L-1].
Minimum constraint for parameter Ksw for material M [LT-1].

17
17
17
17

Real
Real
Real
Real

17
17
17
17

Real
Real
Real
Real

BMn(7,M)BMn(8,M)BMn(9,M)BMn(10,M)

18

Real

BMx(1,M)

18
18
18
18

Real
Real
Real
Real

BMx(2,M)BMx(3,M)BMx(4,M)BMx(5,M)

Maximum constraint for parameter θr for material M [-] (dummy value if
Index(1,M)=0).
Maximum constraint for parameter θs for material M [-].
Maximum constraint for parameter α for material M [L-1].
Maximum constraint for parameter n for material M [-].
Maximum constraint for parameter Ks for material M [LT-1].

212

Table. 12.10. (continued).
RecordType

Variable

Description

18

Real

BMx(6,M)

Maximum constraint for parameter l for material M [-].

BMx(7,M)BMx(8,M)BMx(9,M)
BMx(10,M)

The following four maximum parameter constraints are specified only wheniModel=1.
Maximum constraint for parameter θm for material M [-].
Maximum constraint for parameter θa for material M [-].
Maximum constraint for parameter θk for material M [-].
Maximum constraint for parameter Kk for material M [LT-1].

BMx(7,M)BMx(8,M)BMx(9,M)
BMx(10,M)

The following four maximum parameter constraints are specified only wheniModel=0 and iHyst>1.
Maximum constraint for parameter θm for material M [-].
Maximum constraint for parameter θsw for material M [-].
Maximum constraint for parameter αw for material M [L-1].
Maximum constraint for parameter Ksw for material M [LT-1].

18
18
18
18

18
18
18
18

Real
Real
Real
Real

Real
Real
Real
Real

Records 14 through 18 provide information for each material M (from 1 toNMat).
If lWDep=.true. (Block A) then the soil hydraulic parameters Par(i,M) must be
specified at reference temperature Tref=20oC.
19

-

-

Comment line.

20
20
20

Real
Real
Real

ChPar(1,M)
ChPar(2,M)
ChPar(3,M)

Initial estimate of bulk density of material M, ρ [ML-3].
Initial estimate of longitudinal dispersivity for material type M, DL [L].
Initial estimate of dimensionless fraction of the adsorption sites classified as
type-1, i.e., sites with instantaneous sorption when the chemical
nonequilibrium option is considered. Set equal to 1 if equilibrium transport isto be considered.
Dimensionless fraction of the adsorption sites in contact with mobile water
when the physical nonequilibrium option is considered. Set equal to 1 if all
sorption sites are in contact with mobile water.

20

Real

ChPar(4,M)

20

Real

ChPar(5,M)

20

Real

ChPar(6,M)

20

Real

ChPar(7,M)

20

Real

ChPar(8,M)

20

Real

ChPar(9,M)

Initial estimate of the immobile water content. Set equal to 0 when the physicalnonequilibrium option is not considered.
Initial estimate of the ionic or molecular diffusion coefficient in free water, Dw[L2 T-1].
Initial estimate of the ionic or molecular diffusion coefficient in the gas phase,
Dg [L2 T-1].
Initial estimate of the adsorption isotherm coefficient, ks, for material type M [L3
M-1]. Set equal to zero if no adsorption is to be considered.
Initial estimate of the adsorption isotherm coefficient, η, for material type M[L3M-1]. Set equal to zero if a Langmuir adsorption isotherm is not to beconsidered.
Initial estimate of the adsorption isotherm coefficient, β, for material type M [-].
Set equal to one if a Freundlich adsorption isotherm is not to be considered.

213

Table. 12.10. (continued).
RecordType

Variable

Description

20

Real

20

Real

20

Real

20

Real

20

Real

20
20
20

Real
Real
Real

20

Real

20

Real

20

Real

ChPar(10,M) Initial estimate of the equilibrium distribution constant between the liquid andgas phases, kg, material type M [-].
ChPar(11,M) Initial estimate of the first-order rate constant for the dissolved phase, μw,
material type M [T-1].
ChPar(12,M) Initial estimate of the first-order rate constant for the solid phase, μs, material
type M [T-1].
ChPar(13,M) Initial estimate of the first-order rate constant for the gas phase, μg, material type
M [T-1].
ChPar(14,M) Initial estimate of the rate constant, μw’, representing first-order decay for the
first solute and zero-order production for the second solute in the dissolved
phase, material type M [T-1].
ChPar(15,M) Initial estimate of the rate constant for the solid phase, μs’, material type M [T-1].
ChPar(16,M) Initial estimate of the rate constant for the gas phase, μg’, material type M [T-1].
ChPar(17,M) Initial estimate of the zero-order rate constant for the dissolved phase, γw,
material type M [ML-3T-1].
ChPar(18,M) Initial estimate of the zero-order rate constant for the solid phase, γs, material
type M [T-1].
ChPar(19,M) Initial estimate of the zero-order rate constant for the gas phase, γg, material type
M [ML-3T-1].
ChPar(20,M) Initial estimate of the first-order mass transfer coefficient for nonequilibrium
adsorption, ω, material type M [T-1].

21

Integer Index(1,M)

.
21

.
.
Integer Index(20,M)

22
.
.
22
23

Real
.
.
Real
Real

BMn(1,M).
.
BMn(20,M)
BMx(1,M)

.
.
23

.
.
Real

.
.
BMx(20,M)

Parameter estimation index for parameter ChPar(1,M).
= 0; Coefficient is known and kept constant during optimization.
= 1; Coefficient is unknown and estimated by curve fitting the data.
.
Parameter estimation index for parameter ChPar(20,M).
Minimum constraint for parameter ChPar(1,M) (dummy value if Index(1,M)=0).
.
.
Minimum constraint for parameter ChPar(20,M).
Maximum constraint for parameter ChPar(1,M) (dummy value if
Index(1,M)=0).
.
.
Maximum constraint for parameter ChPar(20,M).
Records 19 through 23 provide information for each material M (from 1 toNMat).

24--

Comment line.

25

Real

HO(i)

Observation data.
Time t for iType(i)=0,1,2,3,4;
Pressure head h for iType(i)=5,6;
Dummy for iType(i)=7,8,9,10,11;214

Table. 12.10. (continued).
RecordType

Variable

Description

25

Real

FO(i)

Observation data.
When iType(i)=2 and iPos(i)=0, then FO(i) represents the average water content
in the entire flow domain.
When iType(i)=4 and iPos(i)=0, then FO(i) represents the total solute amount in
the entire flow domain.

25

Integer iType(i)Type of observed data:= 0: cumulative boundary water flux
= 1: h(x,t) measurement= 2: θ(x,t) measurement= 3: boundary flux
= 4: Conc(x,t) measurement= 5: h(θ) measurement= 6: K(h) measurement= 7: prior knowledge of parameter α= 8: prior knowledge of parameter n= 9: prior knowledge of parameter θ r= 10: prior knowledge of parameter θ s= 11: prior knowledge of parameter Ks=12: Pressure head at location HO(i) at print time iPos(i).
=13: Water content at location HO(i) at print time iPos(i).
=14: Concentration at location HO(i) at print time iPos(i).
=15: Kinetically sorbed concentration at location HO(i) at print time iPos(i).

25Integer iPos(i)Position of the observation node for iType(i)=1,2,4; allowed values are 1,
2,...NObs.
When iType(i)=0 or 3, then iPos(i) is equal to 1 for the upper boundary and 2
for the lower boundary.
When iType(i)=5,6,7,8,9,10, or 11, then iPos(i) represents the material number
M; allowed values are 1, 2,...NMat.
When iType(i)=2 then iPos(i)=0 represents the average water content in theentire transport domain.
When iType(i)=2 then iPos(i)=-iLay represents the average water content in thesubregion iLay.

25

Real

Weight associated with a particular data point.

+

Weight(i)Block J is not needed if only the direct solution is calculated.

215

Table. 12.11. Block K – Carbon dioxide transport information.+
RecordType

Variable

Description

1..2

-

-

Comment lines.

3

LogicallStagnSet this variable equal to .true. if the gas phase is to be considered stagnant, i.e.,there is no gas convection. Otherwise the simplified gas convection expression
is considered (see Section 5.1).

4

-

-

Comment line.

5

Integer

kTopCO

Code specifying type of boundary condition (BC) for the CO2 transport at thesoil surface. Code number is positive for Dirichlet BC and negative for stagnant
boundary layer at the soil surface.

5

Real

CO2TopValue of the time independent BC at the surface [L3L-3]. For kTopCO<0
CO2Top represents the thickness of the stagnant boundary layer [L].

5

IntegerkBotCO

Code specifying type of boundary condition at the bottom of the profile. Code
number is positive for Dirichlet and negative for Cauchy BC. In the case of'Free drainage' set kBotCO=0.

5

Real

CO2BotValue of the time independent BC at the bottom of the soil profile [L3L-3]. In
case of 'Free drainage' set CO2Bot=0.

6

-

-

Comment line.

7
7
7

Real
Real
Real

Par(11,M)
Par(12,M)
Par(13,M)

Molecular diffusion coefficient of CO2 in air at 20oC, Da [L2T-1].
Molecular diffusion coefficient of CO2 in water at 20oC, Dw [L2T-1].
Longitudinal dispersivity of CO2 of material M, DL [L].
The same record as above must be provided for each material M (from 1 toNMat).

8--

Comment line.

9

Real

GamR0

Optimal CO2 production by plant roots for the entire soil profile at 20oC under
optimal water, solute, and CO2 concentration conditions, γr0 [L3L-2T-1].

9

Real

GamS0

Optimal CO2 production by soil microorganisms for the whole soil profile at
20oC under optimal water, solute, and CO2 concentration conditions, γs0 [L3 L-2
T-1].

9

Real

PDDMax

Cumulative value of temperature when CO2 production reaches its maximum
value. Set equal to zero if the degree day concept is not used to calculate thetime reduction coefficient for plant CO2 production. In that case the time
reduction coefficient is equal to one during the whole season.

9

IntegerkProd

Code specifying the type of spatial distribution function for CO2 production bysoil microorganisms.
=0: Exponential function.
=1: van Genuchten's distribution function.

10

-

-

Comment line.

216

Table 12.11. (continued)
RecordType

SymbolDescription

11

Real

rAlfa

Coefficient in the exponential function (only if kProd=0) [L-1].

11

Real

xR

Maximum depth of CO2 production in the soil profile (only if kProd=1) [L].

12

-

-

Comment line.

13

Real

B2

Activation energy for CO2 production by plant roots, E2 [ML2T-2M-1], divided
by universal gas constant, R [ML2T-2K-1M-1]; B2=E2/R [K].

13

Real

B1

Activation energy for CO2 production by soil microorganisms, E1 [ML2T-2M-1],
divided by universal gas constant, R [ML2T-2K-1M-1]; B1=E1/R [K].

13

Real

cM2

Michaelis' constant for CO2 production by plant roots [L3L-3]. Equal to the CO2
concentration at which CO2 production is reduced by half from the optimalvalue γr0.

13

Real

cM1

Michaelis' constant for CO2 production by soil microorganisms [L3L-3]. Equal to
the CO2 concentration at which CO2 production is reduced by half from theoptimal value γs0.

13

Real

hB1

Value of the pressure head at which CO2 production by soil microorganisms is
at the optimal level [L].

13

Real

hB2

Value of the pressure head below which CO2 production by soil microorganisms
ceases [L].

13

Real

P0c

The coefficient b in the CO2 soil production reduction function due to salinity
stress [-]. The recommended value is 3.

13

Real

P50c

The value hφ50 in the CO2 soil production reduction function due to salinity
stress [L]. There is a 50% reduction in the CO2 production by soil
microorganisms at this osmotic head.

†

Block K is not needed if the logical variable lCO2 (Block A) is set equal to .false..

217

Table. 12.12. Block L – Major ion chemistry information.+
RecordType

Variable

Description

1,2

-

-

Comment lines.

3

Real

Epsi

Temporal weighing coefficient.
=0.0 for an explicit scheme,=0.5 for a Crank-Nicholson implicit scheme.
=1.0 for a fully implicit scheme.

3

LogicallUpW.true. if upstream weighing formulation is to be used (see Section 8.3.2).
.false. if the original Galerkin formulation is to be used.

3

LogicallArtD.true. if artificial dispersion is to be added in order to fulfill the PeCr stability
criterion (see Section 8.4.5). .false. otherwise.

3

LogicallLagr.true. if the Eulerian-Lagrangian single-step reverse particle tracking technique
is to be used to solve the solute transport equation. This method is useful for
convection dominated problems (see Section 8.4.3).
.false. if the Eulerian finite differences approach is to be used to solve the solutetransport equation.

3

Real

PeCr

Stability criteria (see Section 8.4.5). Set equal to zero when lUpW is equal to.true..

3

LogicallTort.true. if tortuosity factor [Millington and Quirk, 1961] is to be used.
.false. if tortuosity factor is assumed to be equal to one.

4

-

-

Comment line.

5

LogicallRate

Set this logical variable equal to .true. if kinetic precipitation-dissolution of
calcite and kinetic dissolution of dolomite is to be considered.
Set this logical variable equal to .false. if only equilibrium reactions are to beconsidered.

5

LogicallSilicSet this logical variable equal to .true. if the silica content of the soil solution is
to be calculated based on the solution pH.
Set this logical variable equal to .false. if the silica content of the soil solution is
not considered.

5

Real

UCrit

Value of ionic strength below which the extended Debye-Hückel equation isused to calculate ion activity coefficients. Pitzer's virial-type equations are used
above this value. It is suggested that either one or the other be used.

5

Integer

MaxCh

Maximum number of iterations allowed during any time step between the solutetransport and chemical modules. When the maximum number of iterations isreached for MaxCh>5 then the time step is divided by three and the particulartime level is restarted again. When the maximum number of iterations is reached
for MaxCh≤5 then the code proceeds to the new time level. Recommended
value (from experience) is 5. Set equal to one if no iteration is required.

5

Real

xConv

Length conversion factor. Multiplication factor to convert the length unit LUnit
into meters.

218

Table 12.12. (continued)
RecordType

Variable

Description

5

Real

tConv

Time conversion factor. Multiplication factor to convert the time unit TUnit
into seconds.

6

-

-

Comment line.

7
7
7
7
7

Real
Real
Real
Real
Real

7

Real

7

Real

7
7
7

Real
Real
Real

Bulk density of material M [ML-3].
Molecular diffusion coefficient in free water, Dw [L2T-1].
Longitudinal dispersivity for material type M, DL [L].
Cation exchange capacity for material type M, cT (mmolc kg-1 of soil).
Calcite surface area AC (m2l-1 of soil matrix). Set this variable equal to zero
when lRate=.false..
ChPar(6,M) Dolomite surface area AD (m2l-1 of soil matrix). Set this variable equal to zero
when lRate=.false. or dolomite is not present in the soil profile.
ChPar(7,M) Dissolved organic carbon (μmol l-1). This variable is used to calculate thereduction in the precipitation-dissolution rates of calcite.
ChPar(8,M) Gapon' selectivity constant between calcium and magnesium, K13 [-].
ChPar(9,M) Gapon' selectivity constant between calcium and sodium, K14 [-].
ChPar(10,M) Gapon' selectivity constant between calcium and potassium, K15 [-].
ChPar(1,M)
ChPar(2,M)
ChPar(3,M)
ChPar(4,M)
ChPar(5,M)

Record 7 information is provided for each material M (from 1 to NMat).
8--

Comment line.

9

Integer

kTopChCode which specifies the type of upper boundary condition=+1: Dirichlet boundary condition,=-1: Cauchy boundary condition.

9

IntegernTop

Code which in case of a time-independent upper boundary condition(TopInf=.false. - see Block A), refers to the field ConcTab for the value of thesolute transport boundary condition. ConcTab(nTop,i) is the boundary
condition for the soil surface for species i. Permissible values are
1,2,3,...,nSolConc.

9

IntegerkBotChCode which specifies the type of lower boundary condition= 1: Dirichlet boundary condition,= 0: Continuous boundary condition, free drainage,=-1: Cauchy boundary condition.

9

IntegernBot

Code which in case of a time-independent upper boundary condition(BotInf=.false. - see Block A), refers to the field ConcTab for the value of thesolute transport boundary condition. ConcTab(nBot,i) is the boundary
condition at the soil surface for species i. Permissible values are
1,2,3,...,nSolConc.

10

-

-

Comment line.

11

IntegernSolConc

Number of different solutions used in a particular application.

219

Table 12.12. (continued)
RecordType

Variable

Description

11

IntegernAdsConc

Number of surface species combinations used in a particular application.

11

IntegernPrecConc

Number of mineral phase combinations used in a particular application.

12

-

-

Comment line.

13

Real

ConcTab(1,1)

13

Real

ConcTab(1,2)

13

Real

ConcTab(1,3)

13

Real

ConcTab(1,4)

13

Real

ConcTab(1,5)

13

Real

ConcTab(1,6)

13

Real

ConcTab(1,7)

13

Real

ConcTab(1,8)

Analytical concentration of calcium for the first solution concentration
combination, CaT (mmolcl-1 of solution).
Analytical concentration of magnesium for the first solution concentration
combination, MgT (mmolcl-1 of solution).
Analytical concentration of sodium for the first solution concentration
combination, NaT (mmolcl-1 of solution).
Analytical concentration of potassium for the first solution concentration
combination, KT (mmolcl-1 of solution).
Analytical concentration of alkalinity for the first solution concentration
combination, Alk (mmolcl-1 of solution).
Analytical concentration of sulfate for the first solution concentration
combination, SO4T (mmolcl-1 of solution).
Analytical concentration of chloride for the first solution concentration
combination, ClT (mmolcl-1 of solution).
Analytical concentration of hypothetical tracer for the first solution
concentration combination [-].
In general, one record as described above is required for each solution
concentration combination, starting with the first solution concentration
combination and continuing in sequence up to the nSolConcth combination.

14

-

-

Comment line.

15

Real

XConcTab(1,1) Adsorbed (surface species) calcium concentration for the first surface
species combination, a (mmolckg-1 of soil matrix).

15

Real

XConcTab(1,2) Adsorbed magnesium concentration for
combination, g (mmolckg-1 of soil matrix).

15

Real

XConcTab(1,3) Adsorbed sodium concentration for the first surface species combination, a(mmolckg-1 of soil matrix).

15

Real

XConcTab(1,4) Adsorbed potassium concentration for the surface species combination,(mmolckg-1 of soil matrix).

the

first

surfacespecies

In general, one record as described above is required for each surface
species combination, starting with the first surface species combination andcontinuing in sequence up to the nAdsConcth combination.
16

-

-

Comment line.

17

Real

SConcTab(1,1) Solid phase calcite concentration for the first mineral phase combination
expressed in mmolc of Ca per kg of soil matrix, CaCO3 (divide by 2*10-3 toobtain moles of calcite per kg of soil matrix).

220

Table 12.12. (continued)
RecordType

Variable

Description

17

Real

SConcTab(1,2) Solid phase gypsum concentration for the first mineral phase combination
expressed in mmolc of Ca per kg of soil matrix, CaSO4 (divide by 2*103 toobtain moles of gypsum per kg of soil matrix).

17

Real

SConcTab(1,3) Solid phase dolomite concentration for the first mineral phase combination
expressed in mmolc of Ca per kg of soil matrix, CaMg(CO3)2 (divide by
2*103 to obtain moles of dolomite per kg of soil matrix).

17

Real

SConcTab(1,4) Solid phase hydromagnesite concentration for the first mineral phasecombination expressed in mmolc of Mg per kg of soil matrix,
Mg5(CO3)4(OH)2· 4H2O (divide by 104 to obtain moles of hydromagnesite
per kg of soil matrix).

17

Real

SConcTab(1,5) Solid phase nesquehonite concentration for the first mineral phasecombination expressed in mmolc of Mg per kg of soil matrix, MgCO3·3H2O(divide by 2*103 to obtain moles of nesquehonite per kg of soil matrix).

17

Real

SConcTab(1,6) Solid phase sepiolite concentration for the first mineral phase combination
expressed in mmolc of Mg per kg of soil matrix, Mg2Si3O7.5(OH)·3H2O(divide by 4*103 to obtain moles of nesquehonite per kg of soil matrix).
In general, one record as described above is required for each mineral phasecombination, starting with the first mineral phase combination andcontinuing in sequence up to the nPrecConcth combination.

†

Block L is not needed when the logical variable lChem in Block A is set equal to .false. .

221

Table 12.13. Block M – Meteorological information. †RecordType

Variable

Description

1

char

cVersionSet text equal to “Pcp_File_Version=”iVer

HYDRUS-1D version-

Comment line

1
2,3

-

4

Integer MaxAlMeteo

Number of meteorological records

4Integer iRadiation

Index specifying how the radiation term is entered or calculated:=0: potential value calculated based on latitude, altitude and time of year
=1: solar radiation is specified=2: net radiation is specified

4

Logical lHargr.false. if potential evapotranspiration is to be calculated using the PenmanMonteith combination equation (2.80).
.true. if potential evapotranspiration is to be calculated using the Hargreaves
equation (2.87).

5-

Comment line

6

Logical lMetoDailyVar.true. if HYDRUS-1D is to generate daily variations in meteorologicalvariable (see Appendix C).
.false. otherwise.

6

Logical lDummy

Dummy variable (ten times) (reserved for further expansions of HYDRUS).
Set equal to .false..

-

The following records 7 through 14 are specified only when iRadiation <2.
7--Comment line

8
8

Real
Real

Latitude
AltitudeLatitude (deg)Altitude (m)

9

-

-

Comment line

10
10

Real
Real

ShortWaveRadA
ShortWaveRadB

Angstrom value as in (2.84), recommended value is 0.25
Angstrom value bs in (2.84), recommended value is 0.5

11

-

-

Comment line

12

Real

LongWaveRadA

12

Real

LongWaveRadB

Value of a1 in (A16) for calculating the effect of the cloudiness factor on longwave radiation, recommended value is 0.9
Value of b1 in (A16) for calculating the effect of the cloudiness factor on longwave radiation, recommended value is 0.1

13

-

-

14

Real

14

Real

LongWaveRadA1 Value of al in (A.14) for calculating the effect of emissivity on long wave
radiation, recommended value is 0.34
LongWaveRadB1 Value of bl in (A.14) for calculating the effect of emissivity on long wave
radiation, recommended value is –0.139

Comment line

222

Table 12.13. (continued)

RecordType

Variable

Description

15

-

-

Comment line

16
16

Real
Real

WindHeight
TempHeight

Height of wind speed measurements (usually 200 cm)
Height of temperature and humidity measurements (usually 200 cm)

17

-

-

Comment line

18

Integer iCrop=0: No crop (no values for Crop height, LAI, and RootDepth are specified)=1: Crop height, Albedo, LAI, and RootDepth are constant with time
=2: A table of Crop height, Albedo, LAI, and RootDepth versus time isprovided. Values are interpolated linearly with time between entered values.
=3: Crop height, Albedo, LAI, and RootDepth are given daily

18

Integer iSunSh=0: Sunshine hours are specified=1: Cloudiness is specified=2: The transmission coefficient is specified

18

Integer iRelHum=0: Relative humidity is specified=1: Vapor pressure is specified
The following two lines (19 and 20) are specified only when iCrop = 0:

19

-

-

Comment line

20

Real

AlbedoAlbedo
The following four lines (21 through 24) are specified only when iCrop > 0:

21

-

-

22

Integer iLai

Comment line
Defines how the leaf area index is calculated or specified:=1: calculated from crop height using equation for grass (i.e.,
LAI=0.24*CropHeight),=2: calculated from crop height using equation for alfalfa(LAI=1.5*log(CropHeight)+5.5),=3: calculated from surface fraction, SCF, using

SCF = 1 − exp ( − ai ⋅ LAI ) or LAI = −1
ln (1 − SCF ) , where ai= rExtinct=0.463
ai

22

Real

rExtinct

constant for the radiation extinction by the canopy [-]

23

-

-

Comment line

24

Integer iIntercept=0: no interception=1: interception according to223

Table 12.13. (continued)
RecordType

Variable

Description

⎛
⎞
⎜
⎟
1
I = a ⋅ LAI ⎜1 −bP ⎟⎜⎜ 1 +⎟⎟a ⋅ LAI ⎠⎝where P is precipitation, I is interception, and a and b(SCF) are empiricalconstants.
Lines 25 and 26 below are needed only when iCrop=1, i.e., when
CropHeight, Albedo, LAI, and RootDepth are constant with time.
25

-

-

Comment line

26
26
26
26

Real
Real
Real
Real

CropHeight
AlbedoLAI
RootDepth

Crop height [L]
Albedo [-]Leaf area index for iLai=1 or 2; soil cover fraction for iLai=3.
Rooting depth [L]Lines 25-28 below are needed only when iCrop=2, i.e., when using a tableof CropHeight, Albedo, LAI, and RootDepth values.

25

-

-

Comment line

26

Integer nGrowth

Number of data points in the crop growth table.

27

-

-

Comment line

28
28
28
28
28

Real
Real
Real
Real
Real

tGrowthCropHeight
AlbedoLAI
RootDepth

Day
Crop height [L]
Albedo [-]Leaf area index for iLai=1 or 2; soil cover fraction for iLai=3.
Rooting depth [L]Line 28 is given nGrowth times.
Lines 29 and 30 below are needed only when iIntercept=1 and iCrop>0.

29

-

-

Comment line

30

Real

aInterc

Constant a in the interception model (=0.25).
The previous lines 21-30 are not needed if iCrop=0.

31-33

-

-

Comment lines

34
34
34
34
34
34

Real
Real
Real
Real
Real
Real

tMeteo
Rad
TMax
TMin
RHMeanWind

Day
Net radiation flux at the surface [MJ/m2]
Maximum temperature [oC]
Minimum temperature [oC]
Relative humidity [%]Average daily wind speed at height WindHeight [km/d]

224

Table 12.13. (continued)
RecordType

Variable

Description

34

Real

SunHours

Bright sunshine hours per day [hr], or cloudiness, or the transmission
coefficient (based on iSunSh)
The next four entries are needed only when iCrop=3.

34
34
34
34
†

Real
Real
Real
Real

CropHeight
AlbedoLAI
RootDepth

Crop height [L]
Albedo [-]Leaf area index for iLai=1 or 2; soil cover fraction for iLai=3.
Rooting depth [L]Block M is not needed when the logical variable lMeteo in Block A is set equal to .false. .

225

226

13. OUTPUT DATA
The program output consists of 9+(ns-1) output files (when major ion chemistry or dualpermeability are not considered), where ns is the number of solutes considered in the first-orderdecay chain. When major ion chemistry is considered the program output consists of 13 output
files. The output is organized into 3 groups:
T-level information
T_LEVEL.OUT
RUN_INF.OUT
SOLUTE.OUT
OBS_NODE.OUT
CO2_INF.OUT*
SOLUTEF.OUT+
SOLUTEM.OUT+
OBS_NODF.OUT+
P-level information
NOD_INF.OUT
BALANCE.OUT
CONC.OUT*SOLID.OUT*
EQUIL.OUT*CHEMBAL.OUT*
A-level information
A_LEVEL.OUT
METEO.OUT*

Major ion chemistry module output files+

Dual-permeability module output files

In addition, some of the input data are printed to files I_CHECK.OUT and
PROFILE.OUT. A separate output file SOLUTE.OUT is created for each solute. Results of theinverse solution are directed into an output file FIT.OUT. All output files are directed to thesame directory as the input files, which must be created by the user prior to program execution(the directory is created automatically if the user interface is used). The various output files aredescribed in detail in this section.
File I_CHECK.OUT contains a complete description of the space discretization, thehydraulic characteristic, and the transport properties of each soil material.

227

T-level information - This group of output files contains information, which is printed at
the end of each time step. Printing can be suppressed by setting the logical variable ShortF in
input Block A equal to .true.; the information is then printed only at selected print times. Output
files printed at the T-level are described in Tables 13.1 through 13.3. Output file OBS_NODE.
OUT gives transient values of the pressure head, water content, temperature, and solution andsorbed concentrations, as obtained during the simulation at specified observation nodes.
P-level information - P-level information is printed only at prescribed print times. The
following output files are printed at the P-level:
NOD_INF.OUT

Nodal values of the pressure head, the water content, the solution andsorbed concentrations, and temperature, etc. (Table 13.4).

BALANCE.OUT

This file gives the total amount of water, heat and solute inside eachspecified subregion, the inflow/outflow rates to/from each subregion,
together with the mean pressure head (hMean), mean temperature(TMean) and the mean concentration (cMean) of each subregion (seeTable 13.5). Absolute and relative errors in the water and solute massbalances are also printed to this file.

CONC.OUT

Nodal values of the aqueous concentrations for calcium, magnesium,
sodium, potassium, alkalinity, sulfate, chloride, and hypothetical tracer(see Table 13.9).

SOLID.OUT

Nodal values of the mineral phase concentrations for calcite, gypsum,
dolomite, hydromagnesite, nesquehonite, and sepiolite and nodal valuesof the adsorbed concentrations for calcium, magnesium, sodium, andpotassium (see Table 13.10).

EQUIL.OUT

This file contains the chemical information such as activities of calcium,
bicarbonate and water, alkalinity, pH, SAR, electric conductivity of thesolution, ionic strength, osmotic coefficient, osmotic pressure head, andion activity products for calcite, gypsum and dolomite (see Table 13.11).

CHEMBAL.OUT

This file contains the information about the total amount of particularspecies (e.g. Ca, Mg, SO4,..) in solution, mineral phase and surface
species form in the entire flow region, as well as the cumulative
boundary fluxes and absolute mass error in particular species.

A-level information - A-level information is printed each time a time-dependent boundary
condition is specified. The information is directed to output file A_LEVEL.OUT (Table 13.6).
Meteorological information is directed into the METEO.OUT output file (Table 13.14).

228

Table 13.1. T_LEVEL.OUT - pressure heads and fluxes on the boundaries and in the root zone.
Time

Time, t, at current time-level [T].

rTop

Potential surface flux [LT-1] (infiltration/evaporation: -/+).

rRoot

Potential transpiration rate [LT-1].

vTop

Actual surface flux [LT-1] (infiltration/evaporation: -/+).

vRoot

Actual transpiration rate [LT-1].

vBot

Actual flux across the bottom of the soil profile [LT-1] (inflow/outflow: +/-).

sum(rTop)

Cumulative value of the potential surface flux [L] (infiltration/evaporation: -/+).

sum(rRoot) Cumulative value of the potential transpiration rate [L].
sum(vTop)

Cumulative value of the actual surface flux [L] (infiltration/evaporation: -/+).

sum(vRoot) Cumulative value of the actual transpiration rate [L].
sum(vBot)

Cumulative value of the actual flux across the bottom of the soil profile [L] (inflow/outflow: +/-).

hTop

Pressure head at the soil surface [L].

hRootMean value of the pressure head over the region for which Beta(n)>0 (i.e., within the root zone) [L].

hBot

Pressure head at the bottom of the soil profile [L].

RunOff

Surface runoff [LT-1].

sum(RunOff) Cumulative surface runoff [L]VolumeVolume of water in the entire flow domain [L].

sum(Infil)

Cumulative infiltration [L]sum(Evap)

Cumulative evaporation [L]TLevelTime-level (current time-step number) [-].

sum(WTrans) Cumulative mass transfer of water between mobile and immobile regions for dual porosity model[L]
SnowLayer Thickness of the snow layer [L]229Table 13.2. RUN_INF.OUT - time and iteration information.
TLevelTime-level (current time-step number) [-].

Time

Time, t, at current time-level [T].

dtTime step, Δt [T].

IterW

Number of iterations necessary for solution of the water flow equation [-].

IterC

Number of iterations necessary for solution of the solute transport equation [-].

ItCum

Cumulative number of iterations [-].

KodT

Code for the boundary condition at the soil surface.

KodB

Code for the boundary condition at the bottom of the soil profile.

Converg

Information whether or not the numerical convergence was achieved at the current time-level.

PecletMaximum local Peclet number [-].

Courant

Maximum local Courant number [-].

230

Table 13.3. SOLUTE.OUT - actual and cumulative concentration fluxes.+
TimeTime, t, at current time-level [T].

cvTop

Actual solute flux across the soil surface [ML-2T-1] (inflow/outflow: +/-).

cvBot

Actual solute flux across the bottom of the soil profile [ML-2T-1] (inflow/outflow: +/-).

sum(cvTop)

Cumulative solute flux across the soil surface [ML-2] (inflow/outflow: +/-).

sum(cvBot)

Cumulative solute flux across the bottom of the soil profile [ML-2] (inflow/outflow: +/-).

sum(cvCh0)

Cumulative amount of solute removed from the flow region by zero-order reactions (positive whenremoved from the system) [ML-2].

sum(cvCh1)

Cumulative amount of solute removed from the flow region by first-order reactions [ML-2].

cTop

Solute concentration at the soil surface [ML-3].

cRoot

Mean solute concentration of the root zone [ML-3].

cBot

Solute concentration at the bottom of the soil profile [ML-3].

cvRootActual root solute uptake in the root zone [ML-2T-1].

sum(cvRoot) Cumulative amount of solute removed from the flow region by root water uptake S [ML-2].
sum(cvNEql) Cumulative mass transfer to either kinetic adsorption sites (type-2 adsorption sites), or to theimmobile liquid region [ML-2] (inflow/outflow: +/-).
TLevelTime-level (current time-step number) [-].

vFracS*

Mass transfer between the matrix and fracture domains of the dual-permeability model [ML-2T-1].

vFracS*

Cumulative mass transfer between the matrix and fracture domains of the dual-permeability model[ML-2].

+

Similar output files are created for each solute from 1 to NS.
Similar output files are created separately for the matrix (SoluteM) and fracture (SoluteF) domains of the dualpermeability model.
*
Provided only for the dual-permeability model.

231

Table 13.4. NOD_INF.OUT - profile information.
Node

Number of nodal point n.

Depth

x-coordinate of node n.

Head

Nodal value of the pressure head [L].

Moisture

Nodal value of the water content [-].

K

Nodal value of the hydraulic conductivity [LT-1].

C

Nodal value of the hydraulic capacity [L-1].

Flux

Nodal value of the Darcian velocity [LT-1].

Sink

Nodal value of the root water uptake [T-1].

Ks/KsTop

Ratio between the local hydraulic conductivity and the saturated
hydraulic conductivity at the soil surface [-].

v/KsTop

Ratio between the local velocity and the saturated hydraulicconductivity at the soil surface [-].

Temp

Nodal value of the temperature [K].

Conc(1,..,NS)

Nodal value of the concentration [ML-3]. Only given whenlChem=.true. .

Sorb(1,..,NS)

Nodal value of the sorbed concentration [MM-1] or concentrationin the immobile regions [ML-3]. Only given when lChem=.true.
and lEquil=.false. .

The following information is printed when dual-porosity models are used:
WTransWater mass transfer between mobile and immobile regions [T-1].

Im.Moist.

Water content in the immobile region [-].

STransSolute mass transfer between mobile and immobile regions [T-1].
Only given when lChem=.true.

The following information is printed when dual-permeability models are used:
HeadF

Nodal value of the pressure head of the fracture region [L].

MoistureF

Nodal value of the water content of the fracture region [-].

FluxF

Nodal value of the Darcian velocity of the fracture region [LT-1].

TransfWater mass transfer between the matrix and fracture regions[T-1].

TranS

Solute mass transfer between the matrix and fracture regions[T-1]. Only given when lChem=.true.

ConcF

Nodal value of the concentration of the fracture region [ML-3].
Only given when lChem=.true. .

232

Table 13.5. BALANCE.OUT - mass balance variables.
Area

Length of the entire flow domain or a specified subregion [L].

W-Volume

Volume of water in the entire flow domain or in a specified subregion [L].

InFlow

Inflow/outflow to/from the entire flow domain or a specified subregion [LT-1].

hMean

Mean pressure head in the entire flow domain or a specified subregion [L].

TVol

Amount of heat in the entire flow domain or a specified subregion [MT-2].

TMean

Mean temperature in the entire flow domain or a specified subregion [K].

COVol

Volume of CO2 in the entire flow domain or in a specified subregion [L3L-2].

COMeanMean CO2 concentration in the entire flow domain or in a specified subregion [L3L-3].

ConcVol

Amount of solute in the entire flow domain or a specified subregion [ML-2] excluding ConcVolIm.
This variable is given for all solutes from 1 to NS.

ConcVolIm Amount of solute in the entire flow domain, or in a specified subregion, either adsorbed at type-2(kinetic) adsorption sites or in the immobile liquid region [ML-2]. This information is given for all
solutes from 1 to NS.
cMean

Mean concentration in the entire flow domain or a specified subregion [ML-3]. This information is
given for all solutes from 1 to NS.

Top Flux

Actual surface flux [LT-1] (infiltration/evaporation: -/+).

Bot Flux

Actual flux across the bottom of the soil profile [LT-1] (inflow/outflow: +/-).

WatBalT

Absolute error in the water mass balance of the entire flow domain [L].

WatBalR

Relative error in the water mass balance of the entire flow domain [%].

CncBalT

Absolute error in the solute mass balance of the entire flow domain [%]. This information is given
for all solutes from 1 to NS.

CncBalR

Relative error in the solute mass balance of the entire flow domain [%]. This information is given for
all solutes from 1 to NS.

The following information is printed when carbon dioxide transport is considered:
CO2BalT

Absolute error in the CO2 mass balance for the entire flow domain [L].

CncBalT

Absolute error in the solute mass balance of the entire flow domain [ML-2]. This information is given
for all solutes from 1 to NS.

The following information is printed when dual-porosity models are used.
W-VolumeI Volume of water in the immobile domain of the entire flow domain or a specified subregion [L].
cMeanIm

Mean concentration in the immobile domain of the entire flow domain or a specified subregion[ML-3]. This variable is given for all solutes from 1 to NS.

233

Table 13.5. (continued)
The following information for the fracture domain is printed when dual-permeability models are used:
W-VolumeF Volume of water in the entire flow domain or in a specified subregion [L].
InFlowF

Inflow/outflow to/from the entire flow domain or a specified subregion [LT-1].

ConcVolF

Amount of solute in the entire flow domain or a specified subregion [ML-2] excluding ConcVolIm.
This variable is given for all solutes from 1 to NS.

ConcVolIF Amount of solute in the entire flow domain, or in a specified subregion, either adsorbed at type-2(kinetic) adsorption sites or in the immobile liquid region [ML-2]. This information is given for all
solutes from 1 to NS.
cMeanF

Mean concentration in the entire flow domain or a specified subregion [ML-3]. This information is
given for all solutes from 1 to NS.

WatBalTF

Absolute error in the water mass balance of the entire flow domain [L].

WatBalRF

Relative error in the water mass balance of the entire flow domain [%].

CncBalTF

Absolute error in the solute mass balance of the entire flow domain [%]. This information is given
for all solutes from 1 to NS.

CncBalRF

Relative error in the solute mass balance of the entire flow domain [%]. This information is given for
all solutes from 1 to NS.

234

Table 13.6. A_LEVEL.OUT - pressure heads and cumulative fluxes on the boundary andin the root zone.

Time

Time, t, at current time-level [T].

sum(rTop)

Cumulative potential surface flux [L] (infiltration/evaporation: -/+).

sum(rRoot)

Cumulative potential transpiration [L].

sum(vTop)

Cumulative value of the actual surface flux [L] (infiltration/evaporation: -/+).

sum(vRoot)

Cumulative value of the actual transpiration [L].

sum(vBot)

Cumulative value of the bottom boundary flux [L] (inflow/outflow: +/-).

hTop

Pressure head at the soil surface [L].

hRootMean value of the pressure head in the soil root zone for which Beta(n)>0 [L].

hBot

Pressure head at the bottom of the soil profile [L].

ALevelA-level number (current variable boundary condition number) [-].

235

Table 13.7. FIT.OUT - parameter estimation related information.

SSQ

Value of the objective function Φ being minimized during the parameter optimization process.

S.E.CoeffStandard error.

RSQUAREr2 value for regression of observed versus fitted values.

Quantity

Measured data, e.g., the pressure head, water content, cumulative flux.

Type

Type of measured data (see Table 12.10).

Position

Position of the measurement (see Table 12.10).

WeightWeight associated with a particular data point.

Residual

Residual between measured and fitted quantity.

236

Table 13.8. CO2_INF.OUT - CO2 concentrations and CO2 fluxes on the boundaries
and in the root zone. +

+

CvTop

Actual CO2 flux at the soil surface [L3L-2T-1] (inflow/outflow: -/+).

CvBot

Actual CO2 flux at the bottom of the soil profile [L3L-2T-1] (inflow/outflow: +/-).

sum(CvTop)

Cumulative CO2 flux at the soil surface [L3L-2].

sum(CvBot)

Cumulative CO2 flux at the bottom of the soil profile [L3L-2].

cTop

CO2 concentration at the soil surface [L3L-3].

cRoot

Mean CO2 concentration in the root zone [L3L-3].

cBot

CO2 concentration at the bottom of the soil profile [L3L-3].

vProd

CO2 production by soil microorganisms and plant roots in the soil profile [L3L-2T-1].

sum(vProd)

Cumulative CO2 production by soil microorganisms and plant roots in the soil profile [L3L-2].

sum(Sink)

Cumulative CO2 root uptake in the soil profile [L3L-2].

This output file is created only when carbon dioxide transport is calculated.

237

Table 13.9. CONC.OUT - solute concentration information.+

+

Node

Number of nodal point n.

Depth

x-coordinate of node n.

Ca

Analytical concentration of calcium (mmolcl-1) at node n.

Mg

Analytical concentration of magnesium (mmolcl-1) at node n.

Na

Analytical concentration of sodium (mmolcl-1) at node n.

K

Analytical concentration of potassium (mmolcl-1) at node n.

HCO3

Analytical concentration of alkalinity (mmolcl-1) at node n.

SO4

Analytical concentration of sulfate (mmolcl-1) at node n.

Cl

Analytical concentration of chloride (mmolcl-1) at node n.

TracerAnalytical concentration of hypothetical tracer [-] at node n.

This output file is created only when carbonate chemistry is considered.

238

Table 13.10. SOLID.OUT - precipitated and adsorbed concentrations.+

+

Node

Number of nodal point n.

Depth

x-coordinate of node n.

CalciteMineral phase Ca concentration present as calcite at node n (mmolckg-1).

GypsumMineral phase Ca concentration present as gypsum at node n (mmolckg-1).

Dolomite

Mineral phase Ca concentration present as dolomite at node n (mmolckg-1).

Nesqeh.

Mineral phase Mg concentration present as nesquehonite at node n (mmolckg-1).

HydroMg.

Mineral phase Mg concentration present as hydromagnesite at node n (mmolckg-1).

Sepiol.

Mineral phase Mg concentration present as sepiolite at node n (mmolckg-1).

XCa

Surface species concentration of calcium at node n (mmolckg-1).

XMg

Surface species concentration of magnesium at node n (mmolckg-1).

XNa

Surface species concentration of sodium at node n (mmolckg-1).

XK

Surface species concentration of potassium at node n (mmolckg-1).

This output file is created only when carbonate chemistry is considered.

239

Table 13.11. EQUIL.OUT - chemical information.+

+

Node

Number of nodal point n.

Depth

x-coordinate of node n.

aCa

Activity of Ca2+ [-].

aHCO3

Activity of HCO3- [-].

aH2O

Activity of water [-].

Alk

Alkalinity (mmolckg-1).

pH

Negative logarithm of hydrogen activity, -log(H), [-].

SAR

Sodium adsorption ratio, defined as [Na/(Ca+Mg)0.5] (mmol0.5l-0.5).

ECElectric conductivity of the soil solution (dSm-1).

U

Ionic strength (mol kg-1).

pIAP(c)Negative logarithm of the ion activity product for calcite, -log[(Ca2+)(CO32-)], [-].

pIAP(g)Negative logarithm of the ion activity product for gypsum, -log[(Ca2+)(SO42-)(H2O)2], [-].

pIAP(d)Negative logarithm of the ion activity product for dolomite, -log[(Ca2+)(Mg2+)(CO32-)2], [-].

phi

Osmotic coefficient [-].

hphi

Osmotic pressure head [L].

This output file is created only when carbonate chemistry is considered.

240

Table 13.12. METEO.OUT - meteorological information.+

+

Time

Time, t, at current time-level [T].

ET

Potential evapotranspiration (calculated using either the Penman-Monteith combination equation or
the Hargreaves equation) [mm/d].

Evap

Potential evaporation (evaluated from ET using LAI or SCF) [mm/d].

TranspPotential transpiration (evaluated from ET using LAI or SCF) [mm/d].

Rns

Shortwave net radiation (calculated using the Penman-Monteith combination equation) [MJ/m2/d].

Rnl

Longwave net radiation (calculated using the Penman-Monteith combination equation) [MJ/m2/d].

RadTerm

Radiation term in the Penman-Monteith combination equation [mm/d].

AeroTerm

Aerodynamic term in the Penman-Monteith combination equation [mm/d].

Prec

Precipitation [mm/d].

IntercInterception [mm/d].

ExInterc

Excess interception, i.e., potential transpiration minus interception [mm/d].

This output file is created only when meteorological information is considered.

241

242

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262

Appendix A – Penman-Monteith Variables

Variables in the Penman-Monteith combination equation (2.80) can be calculated using thefollowing equations [FAO, 1990].
1. Latent Heat of Vaporization

λ = 2.501 − (2.361 × 10−3 )T(A.1)

where λ is the latent heat of vaporization [MJ kg-1] and T is air temperature [oC] [Harrison, 1963].
2. Atmospheric Pressure⎛ T - α ( z - xo ) ⎞P = P0 ⎜ k 0
⎟
Tk 0
⎝
⎠

g

αR

⎛ 293 - 0.0065z ⎞= 101.3⎜
⎟
293
⎝
⎠

5.26

(A.2)

where P is the atmospheric pressure at elevation z [kPa], P0 is the atmospheric pressure at sea level(=101.3 kPa), z is the elevation [m], z0 is the elevation at the reference level (= 0 m), g is thegravitational acceleration (= 9.81 ms-2), R is the specific gas constant (= 287 J kg-1K-1), Tk0 is thereference temperature at z0 (= 20 oC), and α is the constant lapse rate saturated air (= 0.0065 K m-1)
[Burman et al., 1987].
3. Atmospheric Densityρ=1000 PP= 3.486
Tkv R
Tkv

where ρ is the atmospheric density [kg m-3] and Tkv is the virtual temperature [K].

263

(A.3)

-1

e ⎞
⎛
Tkv = Tk ⎜ 1- 0.378 d ⎟ ≈ 1.01 (T + 273)
P⎠
⎝

(A.4)

where Tk is the absolute temperature [K], and ed is the vapor pressure at dew point [kPa].
4. Saturation Vapor Pressure⎛ 17.27 T ⎞ea = 0.611exp ⎜⎟⎝ T + 237.3 ⎠(A.5)

where ea is the saturation vapor pressure [kPa] [Tetens, 1930].
5. Crop Canopy Resistance
rc =

Rl
200
=
0.5 LAI LAI(A.6)

where rc is the crop canopy resistance [s m-1], Rl is the average daily (24 hours) stomata resistanceof a single leaf (≅ 100 s m-1), and LAI is the leaf area index [-] [Allen et al., 1989]. LAI can becalculated using the following approaches:(a) For clipped grass (crop height hc = 0.05 - 0.15 m):
LAI = 24 hc(A.7)(b) For alfalfa and other field crops (hc = 0.10 - 0.50 m):
LAI = 5.5 + 1.5 ln ( hc )(c) From the surface cover fraction (SCF):

264

(A.8)

LAI = −1
ln (1 − SCF )
ai

(A.9)

where ai is a constant for the radiation extinction by the canopy (i.e., 0.463) [-].
6. Aerodynamic Resistance⎛ z - d ⎞ ⎛ zh - d ⎞ln ⎜ m. ln
zom ⎟⎠ ⎜⎝ zoh ⎟⎠
⎝
ra =
k2 Uz

(A.10)

where ra is the aerodynamic resistance [s m-1], zm is the height of windspeed measurements [m], zh
is the height of the temperature and humidity measurements [m], k is the von Karman constant (=0.41), Uz is the measured windspeed [m s-1] [Allen et al., 1989], and d is the zero plane
displacement of the wind profile [m] [Monteith, 1981]:

d=

2
hc
3

(A.11)

and where zom is the roughness parameter for momentum [m] [Brutsaert, 1975]:
zom = 0.123 hc(A.12)and zoh is the roughness parameter for heat and water vapor [m]:zoh = 0.1zom = 0.0123hc

265(A.13)

7. Net Emissivity(

ε ′ = (ε a − ε vs ) ≈ al + bl ed)(A.14)

where ε' is the net emissivity [-], εa is the effective emissivity of the atmosphere [-], εvs is theemissivity by vegetation and soil [-], ed is the actual vapor pressure [kPa], and al and bl arecorrelation coefficients (i.e., al = 0.34, bl = -0.14) [Brunt, 1932; Jensen et al., 1990].
8. Cloudiness Factor
The cloudiness factor f [-] can be calculated in three different ways:(a) From measured solar radiation data:f=⎞Rnl ⎛ Rs
= ⎜ ac+ bc ⎟
Rnl 0 ⎝ Rs 0⎠

(A.15)

where Rnl is the net longwave radiation [MJ m-2d-1], Rnl0 is the net longwave radiation for clear
skies [MJ m-2d-1], Rs is the daily averaged measured shortwave solar radiation [MJ m-2d-1], Rs0 is
the daily averaged shortwave solar radiation for clear skies, which can be calculated by substituting
α = 0 and n/N = 1 in Eq. (2.84) [MJ m-2d-1], and ac and bc are calibration values (i.e., ac = 1.35, bc =-0.35) [Wright and Jensen, 1972; Jensenet al., 1990]. When short interval (other than daily) solar
radiation data (e.g., hourly values or Rsh) are used, Rs0 in Eq. (A.15) should be given forcorresponding time (e.g., hourly or Rs0h) as follows:
Rs 0 h = max ( Rs 0 sin e Sum , 0 )(A.16)

where sine and Sum are defined by (C.6) and (C.7), respectively. The ratio Rsh/Rs0h in Eq. (A.15)
during nighttime when Rs0h is equal to 0, can be assumed to be 0.4-0.6 for humid climates and 0.70.8 for arid climate. Hence the average value of 0.6 is used for nighttime.

266

(b) From measured sunshine hour data:

f =

Rnl
n
= a1 + b1
Rnl 0
N(A.17)

where n is the measured sunshine hours per day [hr], N is the maximum day light hours [hr]described as 24ωs /π (ωs is shown in Eq. (2.89)), and a1 and b1 are parameters for cloudiness effect(i.e., a1 = 0.9, b1 = 0.1).

(c) From the atmospheric transmission coefficient :The atmospheric transmission coefficient for solar radiation, Tt [-], is defined as the ratio of themeasured incoming solar radiation, Rsm, and the daily extraterrestrial radiation, Ra, [Campbell,
1985]:
Tt =

Rsm
Ra

(A.18)The ratio of the sunshine hours and the maximum day light hours, n/N, can be calculated asfollows:n= 1 − c = 1 − ( 0 ; 2.33 − 3.33Tt ;1)
N(A.19)

where c is the fractional cloud cover [-]. Then the cloudiness factor, f, is calculated using Eq.
(A.17).

267

APPENDIX B – Surface Energy Balance Variables

Variables in the surface energy balance equation can be calculated using the following
equations.
1. Surface Albedo
do

α = 0.25
α = 0.35 − θ 0
α = 0.10

θ 0 < 0.1
0.1 ≤ θ 0 < 0.25
θ 0 ≥ 0.25

(B.1)

where α is the surface albedo [-] and θ0 is the water content [L3L-3] at the ground surface [vanBavel and Hillel, 1976].

2. Soil Surface Emissivity

ε s = min ( 0.90 + 0.18θ 0 ; 1.0 )(B.2)

where εs is the soil surface emissivity [-], and θ0 is the water content [L3L-3] at the ground surface[van Bavel and Hillel, 1976].
3. Atmospheric Emissivity of Clear Sky

17

⎛e ⎞
ε a = 1.24 ⋅ ⎜ d ⎟⎝ Ta ⎠(B.3)

where εa is the atmospheric emissivity of clear sky [-], ed is the actual vapor pressure [kPa], Ta is
the air temperature [K] [Brutsaert, 1975].

268

4. Soil Surface Resistance to Water Vapour Flow⎧⎪r0 exp ( 35.63 ( 0.15 − θ 0 ) ) θ 0 ≤ 0.15
rs = ⎨
θ 0 > 0.15
r0
⎪⎩

(B.4)

where rs is the soil surface resistance to water vapour flow [s m-1] and r0 is the surface resistance [sm-1], which should theoretically be equal to the resistance to molecular diffusion across the water
surface (=10) [van de Griend and Owe, 1994].
5. Aerodynamic Resistance to Water Vapor Flowra =⎤ ⎡ ⎛ zh - d + zoh ⎞⎤1 ⎡ ⎛ zm - d + zom ⎞ln ⎜⎟ + ψ m ⎥ . ⎢ln ⎜⎟ +ψ h ⎥2 ⎢Uzk ⎣ ⎝
zom
zoh
⎠
⎠
⎦ ⎣ ⎝
⎦

(B.5)

where ra is the aerodynamic resistance to water vapor flow [s m-1], zm is the height of windspeed
measurements [m], zh is the height of temperature measurements [m], k is von Karman constant (=0.41), Uz is the measured windspeed [m s-1], d is the zero plane displacement [m], zom is the surface
roughness length for momentum flux [m], zoh is the surface roughness length for heat flux [m], ψmis the atmospheric stability correction factor for momentum flux [-], and ψh is the atmosphericstability correction factor for heat flux [-] [Campbell, 1985]. For bare soils, the zero plane
displacement d is equal to zero [van de Griend and Owe, 1994], while typical surface roughness
values of 0.001 m are used for both zom and zoh [Oke, 1978].
The aerodynamic resistance depends on the so-called stability condition of theatmosphere that can be assessed using Monin-Obukhov’s stability parameter (or the MO length).
The MO length is calculated as follows [e.g., Camillo and Gurney, 1986; Aluwihare and
Watanabe, 2003]:

269

MO = −

CaTaU *3
kgH

(B.6)

where Ca is the volumetric heat capacity of air (= 1200) [Jm-3K-1], Ta is the atmospheretemperature [K] at zh, g is the gravitational acceleration (= 9.81 m s-2), H is the sensible heat flux
at the soil surface (Eq. (2.95)), Ta is the atmosphere temperature [K] at zref, and U* is thefrictional velocity defined based upon the logarithmic wind profile law [e.g., Camillo andGurney, 1986; van de Griend and Owe, 1994; Aluwihare and Watanabe, 2003]:⎡ ⎛ z −d ⎞⎤U * = uk ⎢ln ⎜ ref⎟ +ψ m ⎥⎣ ⎝ zom ⎠
⎦

−1

(B.7)

The evaluation of ψm and ψh are then determined using the atmospheric stability parameter ζ
defined as [Brutsaert, 1982]:

ζ =zm − dMO(B.8)(a) For a neutral atmosphere (|Ta - Ts| ≤ 0.01 K):

ψ h =ψ m = 0(B.9)(b) For an unstable atmosphere (Ta < Ts or MO < 0):⎛ 1 + 1 − 16ζ ⎞
⎟⎟
2
⎝
⎠

ψ h = −2 ln ⎜⎜(B.10)⎛ 1 + 1 − 16ζ ⎞⎛ 1 + (1 − 16ζ )0.25 ⎞π
0.25−⎟⎟ + 2 arctan (1 − 16ζ )⎟ − ln ⎜⎜
2
2
2
⎝
⎠
⎝
⎠

{

ψ m = −2 ln ⎜270}

(B.11)

(c) For a stable atmosphere (Ta > Ts or MO > 0):⎧5 ⋅ ζ⎩5ψ h =ψ m = ⎨0 < ζ <1
ζ >1

(B.12)

6. Fraction of Cloud Cover
The fraction of cloud cover, c, can be calculated using four different ways depending on
available data.
a) Eq. (A.19) is used when the sunshine hour per day, n, or the atmospheric transmission
coefficient, Tt, are available.
b) When the cloudiness factor, f, is available, n/N has to be first evaluated using (A.17) before
the fraction of cloud cover is calculated.
c) The parameter c can also be derived using the daily measured solar radiation, Rsm, by firstevaluating the asmospheric transmission coeffcient Tt using (A.18).
d) Finally, when short interval (e.g., hourly) solar radiation, Rsh, is used, Ra in (A.18) has to beevaluated for corresponding times as follows:
Rah = max ( Gsc sin e, 0 )(B.13)

where Gsc is the solar constant [J m-2s-1] (i.e., 1360 W m-2) and sine is defined using (C.6). During
nighttime, when Rah is equal to 0, c is assumed to be 0.6 (calculated assuming Rsh/Rs0h = 0.6 in (A.
15), and using (A.17) and (A.19)).

271

APPENDIX C – Daily Variations of Meteorological Variables

Solving the Penman-Monteith equation (2.80) or the energy balance equation (2.92) at atime interval of interest requires values of meteorological variables at the same or similar timeintervals. Weather stations or field measurements, however, do not always provide standard
meteorological data at time intervals of interest. Relatively simple approaches generating
continuous values of various meteorological variables (i.e., air temperature, relative humidity,
and solar radiation) from available daily average information have been implemented in
HYDRUS-1D [Saito et al., 2006].

1. Air Temperature
When the daily maximum and minimum air temperatures are provided, diurnal
continuous variations in the air temperature, Tair, are obtained using a trigonometric cosinus
function with a period of 24 hours as follows [Kirkham and Powers, 1972]:

Tair =

Tmax + Tmin Tmax − Tmin
⎡ ⎛ t − 13 ⎞ ⎤+⋅ cos ⎢ 2π ⎜⎟⎥22
⎣ ⎝ 24 ⎠ ⎦(C.1)

where Tmax and Tmin are the daily maximum and minimum temperatures [oC], respectively, and t
is the local time within the day [h]. The argument of the cosine function shows that the highest
temperature is assumed to occur at 1 p.m. and the lowest at 1 a.m.

2. Relative Humidity
Since daily temperature variations typically show a cyclic behavior throughout the day, it
is reasonable to assume that the relative humidity, Hr, also shows such a cyclic pattern. Similarly
as for the air temperature, a trigonometric cosinus function can be used to calculate diurnal
continuous changes in the relative humidity from daily information [e.g., Gregory et al., 1994].
Assuming that the atmospheric actual vapor pressure is constant during the day, the daily

272

maximum, Hrmax, and minimum, Hrmin, relative humidities can be calculated from the average
daily relative humidity, H r [%], and the daily maximum and minimum temperatures as follows:

H r max =eaes min⋅100,

H r min =ea
es max

⋅100

(C.2)

where esmax and esmin are the maximum and minimum saturation vapor pressures [kPa] calculatedusing Eq. (A.5) from Tmax and Tmin, respectively, and ea is the average daily actual vapor pressure[kPa]:ea =H r es min es max⋅
50 es min + es max(C.3)

Once the Hrmax and Hrmin are available, Hr, can be derived as follows:Hr =H r max + H r min H r max − H r min⎡ ⎛ t − 1 ⎞⎤+⋅ cos ⎢ 2π ⎜⎟⎥22
⎣ ⎝ 24 ⎠ ⎦(C.4)

Contrary to the air temperature, the highest relative humidity occurs at 1 a.m. and the lowest at 1
p.m.

3. Solar Radiation
A value of the incoming shortwave solar radiation, St [MJ m-2d-1], at any given time andlocation can be calculated based on Campbell [1985];

St ( t ) = max ( Rs sin e Sum , 0 )(C.5)

where Rs is the daily incoming solar radiation [MJ m-2d-1], and e is the solar elevation angle [rad]given by:273

sin e = sin ϕ sin δ + cos ϕ cos δ cos

2π( t − 12 )
24

(C.6)

where ϕ is the site latitude [rad], δ is the solar declination [rad] (2.91). Sum is defined as follows
to adjust the value of the daily incoming solar radiation:24
2π⎛
Sum = ∑ max ⎜ sin ϕ sin δ + cos ϕ cos δ cos( k − 12 ) , 0 ⎞⎟
24
⎝
⎠
k

274

(C.7)

Appendix References

Allen, R. G., M. E. Jensen, J. L. Wright and R. D. Burman, Operational estimates of
evapotranspiration, Agron. J., 81, 650-662, 1989.
Aluwihare, S., and K. Watanabe, Measurement of evaporation on bare soil and estimating
surface resistance, J. Environ. Eng., 129, 1157-1168, 2003.
Brunt, D., Notes on radiation in the atmosphere, Quart. J. Roy. Meteorol. Soc, 58, 389-418, 1932.
Brutsaert, W., The roughness length for water vapor, sensible heat and other scalars, J. Atm. Sci.,
32, 2028-2031, 1975.
Burman R. D., M. E. Jensen, and R. G. Allen, Thermodynamic factors in evapotranspiration, In:
Proc. Irrig. and Drain. Spec. Conf., James L.G. and M. J. English (eds). ASCE, Portland,Ore., July. pp. 28-30, 1987.
Camillo, P. J., and R. J. Gurney, A resistance parameter for bare-soil evaporation models, Soil
Sci., 141, 742-744, 1986.
Campbell, G. S., An Introduction to Environmental Biophysics, Springer-Verlag, New York,1977.
Campbell, G. S., Soil Physics with Basic, Transport Models for Soil-Plant Systems, Elsevier,
New York, 1985.
Harrison, L. P., Fundamental concepts and definitions relating to humidity, In: Humidity andMoisture, Vol. 3, Wexler A. (ed), Reinhold Publishing Company, New York, 1963.
Jensen, M. E., R. D. Burman, and R. G. Allen, Evapotranspiration and irrigation waterrequirements. ASCE Manual No. 70, 1990.
Kirkham, D., and W. L. Powers, Advanced soil physics. John Wiley & Sons, New York, 1972.
Monteith, J. L., Evaporation and surface temperature, Quarterly J. Royal Meteo. Soc., 107, 1-27,
1981.
Sharratt, B. S., M. J. Schwarzer, G. S. Campbell, and R. I. Papendick, Radiation balance ofridge-tillage with modeling strategies for slope and aspect in the subarctic, Soil Sci. Soc.
Am. J., 56, 1379-1384, 1992.

275

Tetens, O., Uber einige meteorologische Begriffe, Z. Geophys., 6, 297-309, 1930.
van Bavel, C. H. M., and D. I. Hillel, Calculating potential and actual evaporation from a baresoil surface by simulation of concurrent flow of water and heat, Agri. Meteo., 17, 453-476,
1976.
van de Griend, A. A., and M. Owe, Bare soil surface resistance to evaporation by vapor diffusion
under semiarid conditions, Water Resour. Res., 30, 181-188, 1994.
Wright, J. L., and M. J. Jensen, Peak water requirements of crops in Southern Idaho, J. Irrig.
Drain. Div., ASCE, 96(IR1), 193-201, 1972.

276

PART B

Interactive Graphics-Based User Interface

HYDRUS1D

Version 4.0

by

J. Šimůnek, M. Šejna, and M. Th. van Genuchten

277

278

Even with well-documented numerical computer models available, one major problem
often preventing the use of such codes is the extensive work required for data preparation, finite
element grid design, and graphical presentation of the output results. Hence, a more widespreaduse of numerical models requires techniques which make it easier to create, manipulate anddisplay large data files, and which facilitate interactive data management. Introducing suchtechniques will free users from cumbersome manual data processing, and should enhance theefficiency in which programs are being implemented for a particular example. To avoid or
simplify the preparation and management of relatively complex input data files and tographically display final simulation results, we developed an interactive graphics-based userfriendly interface HYDRUS1D for the MS Windows environment. The graphics interface isconnected directly to the HYDRUS FORTRAN code.
In addition to information given in this chapter, extensive context-sensitive on-line help
is made part of every module of the interface. By pushing the F1 button, or clicking on the Help
button while working in any window, the user obtains information about the window content. In
addition, context-sensitive help is available in every module using the "SHIFT+F1" help button.
In this mode, the mouse cursor changes to a help cursor (a combination arrow + question mark),
and the user proceeds to click on the object for which he needs help (i.e, a menu item, toolbar
button, or other features). At that point, a help file will be displayed giving information about theitem on which the user clicked. Except for the HYDRUS FORTRAN application itself, allmodules are written in C++.

279

280

Table B.1. Main modules of the HYDRUS-1D software package.
HYDRUS1D

main program unit, input parameters, output
graphics

POSITION

project manager (PCP_BASE.DLL)PROFILEtransport domain geometry, finite element meshgenerator, boundary and initial conditions,material distribution

H1D_calc

FORTRAN application for the direct solution

H1D_clci

FORTRAN application for the inverse solution

H1D_dual

FORTRAN application for the direct solution(dual-permeability model)

H1D_dlin

FORTRAN application for the inverse solution(dual-permeability model)

H1D_unsc

FORTRAN application for the major ion
chemistry model

HP1

C application for running the coupled HP1biogeochemical modelHydrus.dll

HYDRUS part of the HP1 model

Phreeqc.dll

PHREEQC part of the HP1 model

B. Brief Description of Selected Modules
B.1. Module HYDRUS1D
HYDRUS1D (Fig. B.1) is the main program unit defining the overall computational
environment of the system. This module controls execution of the program and determines
which other optional modules are necessary for a particular application. The module contains a
project manager and both the pre-processing and post-processing units. The pre-processing unit
includes specification of all necessary parameters to successfully run the HYDRUS FORTRAN

codes, a small catalog of soil hydraulic properties, and a plant salt tolerance database. Table B.2
lists all commands accessible through the menu, whereas Table B.3 gives a brief discussion of
the action taken with the particular commands. More detailed descriptions are available through281

the on-line help. The post-processing unit consists of simple x-y graphics for graphicalpresentation of soil hydraulic properties, as well as such output as transient values of a particularvariable at selected observation points in the domain, and actual or cumulative water and solute
fluxes across boundaries. Table B.4 gives an overview of the different graph options madeavailable through the interface. The HYDRUS1D and PROFILE modules mutuallycommunicate through the file HYDRUS1D.DAT, a description of which is given in Table B.5.
The work for a new project should begin by opening the Project Manager (see SectionB.2), and giving a name and brief description to this new project. Then select the Main Processes
command from the Main Information Menu. From this point on, the program will navigate theuser through the entire process of entering input files. The user may either select particularcommands from a menu, or allow the interface to lead him through the process of entering input
data by selecting the Next buttons. Alternatively, clicking the Previous button will return theuser to the previous window.

282

Fig. B.1. The main window of the HYDRUS1D module, including the project manager.

283

Table B.2. Menu commands in the main module HYDRUS1D.
Group Menu

Submenu

Sub-Submenu

A

Project

New
Open
Project Manager
Save
Save As
Exit

B

View

ToolbarStatus Bar
List Boxes for Inverse DataC

Pre-processing

Main Processes
Inverse Solution
Geometry Information
Time Information
Print Information
Water Flow

Solute Transport

Heat Transport
Carbon Dioxide Transport
Root Water Uptake
Root Growth Parameters
Meteorological Parameters
Variable Boundary Conditions
Meteorological Boundary Conditions
Data for Inverse Solution
Profile Information
Profile SummaryD

Calculation

Execute HYDRUS

284

Iteration Criteria
Hydraulic Properties Model
Soil Hydraulic Properties
Boundary Conditions
Constant Boundary Fluxes
Deep Drainage BC
General Solute Transport Information
HP1 Components
Solute Transport Parameters
Solute Reaction Parameters
Temperature Dependence
Water Content Dependence
Boundary Conditions
Heat Transport Parameters
Boundary Conditions
CO2 Transport Parameters
Soil CO2 Production Parameters
Root CO2 Production Parameters
Root Water Uptake Models
Pressure Head Reduction
Osmotic Head Reduction

Table B.2. (continued).
Group Menu

Submenu

E

Observation Points
Profile Informations

Results

Sub-Submenu
Basic Information
Solution Concentrations
Solid Concentrations
Chemical Information
Water Flow
Solute Transport
CO2 Transport

Boundary Informations
Soil Hydraulic Properties
Run Time Information
Mass Balance Information
Inverse Solution
Meteorological Information
Solute Mass Balance Information
F

Options

Program Options

G

Window

CascadeTile Horizontally
Tile Vertically
Arrange Icons

H

Help

Index
Help On
About HYDRUS-1D

Table B.3. Description of all menu commands in the main module HYDRUS1D.
Group Command

Brief description of the commandA

New

Creates a new project.

Open

Open an existing project (represented by project_name.h1d file)

Project Manager

Calls the project manager to manage data of existing projects;
helps to locate, open, copy, delete or rename the desired projects
and their data.

Save

Saves the input data of an actual project specified in the main
program module if the data were either newly created or changed
during an application run. This command deletes at the same time
all existing output files of a selected project since the output data
are no longer consistent with the changed input data. A warning isissued to the user before doing so.

Save asSaves data of a particular project under a new project name.

285

Table B.3. (continued).
Group Command

B

C

Brief description of the command

Exit

Closes a project and leaves the program. This command informs
the user before exiting the application whether or not the input
data of an actual project were changed during the application run.
If changes did occur, the user is given a possibility to save data
before exiting the application.

ToolbarShows or hides the toolbar.

Status Bar

Shows or hides the status bar.

List Boxes for Inverse Data

Shows a text information in the inverse data list.

Main Processes

Selects the title which is printed into output files, and specifies theprocesses to be simulated, i.e., water flow, multiple solutetransport, heat transport, root growth, and/or root water uptake.

Inverse Solution

Selects type of weighting of measured data, whether soil hydraulicparameters, solute transport parameters, or both are to be fitted.

Main Geometry Information

Selects the length unit, specifies the depth and inclination of thesoil profile to be analyzed, and determines the number of materials
to be used.

Main Time Information

Selects time units, and gives the time discretization information.

Print Information

Specifies print options.

Iteration Criteria

Specifies iteration criteria for the solution precision, andparameters for the time step control.
Selects the type of model used for the soil hydraulic properties,
and decides whether the hysteresis is to be considered.
Specifies parameters in the soil hydraulic model.
Specifies the types of upper and lower boundary conditions.
Specifies constant boundary fluxes and constant root water uptake
when no time-variable boundary conditions are given.
Specifies parameters for the deep drainage function.

Hydraulic Properties Model
Soil Hydraulic Properties
Boundary Conditions
Constant Boundary Fluxes
Deep Drainage BC
General Solute Transport Information
HP1 Components
Solute Transport Parameters
Solute Reaction Parameters
Temperature Dependence
Water Content Dependence
ST Boundary Conditions

Selects the time and spatial weighting schemes for numericalsolution of the solute transport equation; specifies the number ofsolutes to be considered.
Selects the main components of the biogeochemical system andpathways to the geochemical database.
Specifies solute transport parameters.
Specifies solute reaction parameters.
Specifies parameters, which define the temperature dependence ofreaction and transport parameters.
Specifies parameters, which define the water content dependenceof reaction and transport parameters.
Specifies the upper and lower boundary conditions for solutetransport.

286

Table B.3. (continued).
Group Command
Chemical Parameters
Solution Compositions
CO2 Transport Parameters

Brief description of the command
Selects
kinetic
orequilibrium
model
for
calciteprecipitation/dissolution, specifies critical ionic strength andmaximum number of iterations.
Specifies different solution, adsorbed and mineral phasecompositions.

Soil CO2 Production Parameters
Root CO2 Production Parameters

Specifies carbon dioxide transport parameters including boundary
conditions.
Specifies soil (microbial) CO2 production parameters.
Specifies root CO2 production parameters.

Heat Transport Parameters

Specifies heat transport parameters.

Heat Transport Boundary Conditions

Specifies the upper and lower boundary conditions for heat flow.

Root Water Uptake Models

Selects the root water uptake stress response models for bothsalinity and water stress.
Specifies parameters in the root water uptake water stress responsemodel.
Specifies parameters in the root water uptake salinity stressresponse model.
Specifies parameters in the Verhulst-Pearl logistic growth functionused to describe root growth during the growing season.

Pressure Head Reduction
Osmotic Head Reduction
Root GrowthMeteorological Parameters

Specifies meteorological parameters required by the PenmanMonteit combination or Hargreaves equations.

Variable Boundary Condition

Specifies time-dependent boundary conditions for all transport
processes.

Meteorological Boundary Conditions

Specifies meteorological variables (radiation, temperatures, windspeed, relative humidity) required by the Penman-Monteit
combination or Hargreaves equations.

Data for Inverse Solution

Specifies data for the inverse solution, their type, location andassociated weight.

Profile Information

Calls external module PROFILE, for users to discretize the soil
profile and to specify the vertical distribution of relevant
parameters.

Profile Summary

Summarizes in tabular form the spatial discretization and spatial
distribution of soil properties, initial conditions, and other variable.
This command allows the user to summarize and modify theparameters setup in the external module PROFILE.

D

Execute HYDRUS

Executes a HYDRUS FORTRAN application.

E

Observation Points

Graphical presentation of changes in water content, pressure head,temperature, and/or solute and sorbed concentration at specified
observation nodes.

287

Table B.3. (continued).
Group Command
Basic Informations
Solution Concentrations
Solid Concentrations
Chemical Information

Brief description of the command
Graphical presentation of pressure head, water content, flow
velocity, root water uptake, temperature, and concentration
profiles at different times.
Graphical presentation of major ion concentrations: Ca, Mg, Na,
K, alkalinity, SO4, Cl, and tracer.
Graphical presentation of surface species and mineral phase
concentrations:Ca, Mg, Na, K, calcite, gypsum, dolomite,nesquehonite, hydromagnesite, sepiolite.
Graphical presentation of major chemical information (pH, SAR,
(H2O0), (Ca2+), (HCO3-), pIAPC, pIAPD, pIAPG, EC, ..

Boundary Informations

Graphical presentation of actual and cumulative boundary waterand solute fluxes, and surface, root zone, and bottom pressure
heads and concentrations.

Carbon Dioxide Transport

Graphical presentation of actual and cumulative boundary CO2
fluxes, and surface, root zone, and bottom boundary CO2
concentrations.

Soil Hydraulic Properties

Graphical presentation of the soil hydraulic properties.

Run Time Information

Graphical presentation of information about the number of
iterations, time step, and Peclet and Courant numbers.

Mass Balance Information

Displays mass balance information and mean profile properties

Inverse Solution

Displays information about the inverse solution.

Meteorological Information

Displays components of the surface energy balance (ET, E, T,
aerodynamic and radiation terms of the Penman-Monteith
combination equation).

Solute Mass Balance Information

Displays mass balance information for particular major ions.

F

Program Options

Displays pathways for HYDRUS settings and configuration files,
as well as for HYDRUS projects.

H

Index

Offers an index of topics for which help is available.

Using Help

Provides general instructions on using help.

About HYDRUS-1D

Displays the version and authors of the HYDRUS-1D application.

288

Table B.4. Graph options in the HYDRUS-1D interface.
Command

Horizontal Axis

Vertical Axis

Observation Points

Time

Pressure Head
Water Content
Temperature
Concentration – x+
Sorbed Concentration - x+

Pressure Head
Water Content
Hydraulic Conductivity
Soil Water Capacity
Water Flux
Root Water Uptake
Temperature
Immobile Water Content*
Water Mass Transfer*
Solute Mass Transfer*
Concentration - x+
Sorbed Concentration - x+

DepthProfile Information- Basic Information- Solution Concentrations Calcium
Magnesium
SodiumPotassium
Alkalinity
SulfateChloride
Tracer

Depth

Solid Concentrations

CalciteGypsumDolomite
Nesquehonite
Hydromagnesite
Sepiolite
Adsorbed Calcium
Adsorbed Magnesium
Adsorbed Sodium
Adsorbed Potassium

Depth

Chemical Information

Calcium Activity
Bicarbonate Activity
Water Activity
Alkalinity
pH
SAR
Electric Conductivity

Depth

289

Table B.4. (continued).
Command

Horizontal Axis

Vertical Axis

Ionic Strength
pIAP Calcite
pIAP Gypsum
pIAP Dolomite
Osmotic Coefficient
Osmotic Pressure Head
T_Level Information- Water Flow

- Solute Transport+

Time

Potential Surface Flux
Potential Root Water Uptake Rate
Actual Surface Flux
Actual Root Water Uptake Rate
Bottom Flux
Cumulative Potential Surface Flux
Cumulative Potential Root Water Uptake
Cumulative Actual Surface Flux
Cumulative Actual Root Water Uptake
Cumulative Bottom Flux
Surface Pressure Head
Average Root Zone Pressure Head
Bottom Pressure Head
All Fluxes
All Cumulative Fluxes
All Pressure Heads
Surface Run-Off
Cumulative Surface Run-Off
Soil Water Storage
Cumulative Infiltration
Cumulative Evaporation
Cumulative Nonequilibrium Water Transfer
Soil Layer

Time

Surface Solute Flux
Bottom Solute Flux
Cumulative Surface Solute Flux
Cumulative Bottom Solute Flux
Cumulative Zero-Order Reactions
Cumulative First-Order Reactions
Surface Concentration
Average Root Zone Concentration
Bottom Concentration
Root Solute Uptake
Cumulative Root Solute Uptake
Cumulative Nonequilibrium Mass Transfer
All Solute Fluxes
All Cumulative Solute Fluxes

290Table B.4. (continued).
Command

Horizontal Axis

Vertical Axis

All Concentrations- CO2 Transport

+
*

Time

Surface CO2 Flux
Bottom CO2 Flux
Cumulative Surface CO2 Flux
Cumulative Bottom CO2 Flux
Surface CO2 Concentration
Average Root Zone CO2 Concentration
Bottom CO2 Concentration
CO2 Production
Cumulative CO2 Production
Cumulative CO2 Root Uptake

Soil Hydraulic Properties

Pressure Head
Log Pressure Head
Water Content

Water Content
Soil Water Capacity
Hydraulic Conductivity
Log Hydraulic Conductivity
Effective Water Content

Run-Time Information

Time Level
Time

Time Step
Number of Iterations
Cumulative Number of Iterations
Peclet Number
Courant Number
Number of Solute Iterations

Meteorological Information

Time

Potential Evapotranspiration
Potential Evaporation
Potential Transpiration
Net Short Wave Radiation
Net Long Wave Radiation
Radiation Term
Aerodynamic Term
Precipitation
Interception

This value is given for each solute
This value is given when dual porosity model is used
These variables are displayed for both matrix and fracture regions when the dual-permeability model is used291

B.2. Module POSITION
A project manager, POSITION (PCP_BASE.DLL) (Fig. B.1), is used to manage data ofexisting projects, and to help locating, opening, copying, deleting and/or renaming desired
projects or their input or output data. A "project" represents any particular problem to be solved
by HYDRUS-1D. The project name (8 letters), as well as a brief description of the project helps
to locate a particular problem. Input and output data for selected projects are grouped into
Workspaces (represented by subdirectories) which can be located anywhere on accessible harddiscs of a particular PC or network.
Table B.5. Information in the HYDRUS1D.DAT file.
Group Variable

Type

Description

Main

WaterFlowInt

SoluteTransport

Int

HeatTransport

Int

EquilibriumAdsorption

Int

RootWaterUptake

Int

RootGrowth

Int

MaterialNumbers
SubregionNumbers
SpaceUnit
TimeUnit
PrintTimes
NumberOfSolutes
CO2Transport

Int
Int
StringStringInt
Int
Int

SolutionConc
AdsorbedConc
PrecipConc
Unsatchem

Int
Int
Int
Int

Variable, which specifies whether or not transient water flow is to be
calculated.
Variable, which specifies whether or not solute transport is to be
calculated.
Variable, which specifies whether or not heat transport is to be
calculated.
Variable, which specifies whether or not adsorption is considered as
equilibrium process.
Variable, which specifies whether or not root water uptake is to be
calculated.
Variable, which specifies whether or not root water growth is to be
calculated.
Number of materials considered.
Number of subregions considered for mass balance calculation.
Space units.
Time units.
Number of print-times.
Number of solutes considered in the application.
Variable which specifies whether or not carbon dioxide transport is tobe calculated.
Number of solution combinations considered in the application.
Number of surface species combinations considered in the application.
Number of mineral phase combinations considered in the application.
Variable, which specifies whether or not major ion solute transport isto be calculated.

Profile NumberOfNodes
ProfileDepth
ObservationNodes
GridVisible

Int
Float
Int
Int

Number of nodes used to discretize the soil profile.
Depth of the soil profile.
Number of observation nodes.
Variable, which specifies whether or not the grid is to be visible.

292

Table B.4. (continued).
Group Variable

Type

Description

SnapToGrid

Int

ProfileWidth
LeftMargin
GridOrgX
GridOrgY
GridDX
GridDY

Int
Int
Real
Real
Real
Real

Variable, which specifies whether or not the mouse should move insteps defined by the grid.
Number of pixels for graphical display of the soil profile.
Number of pixels for graphical display of the nodal discretization.
X-coordinate of the grid origin.
Y-coordinate of the grid origin.
Step in the x direction between grid nodes.
Step in the y direction between grid nodes.

B.3. Module PROFILE
B.3.1. Soil Profile Discretization
The module PROFILE (Fig. B.2) is used, among other things, to discretize a onedimensional soil profile into discrete nodes. Nodes are generated by dividing the soil profile into
small elements. If no previous nodes exist, the program automatically generates a default
equidistant point distribution. The location of nodes can be edited by the user to optimize thethickness of the different elements. There are two ways of specifying appropriate distributions of
the nodes, i.e., by (1) editing the number of points, and (2) specifying fixed points and nodal
densities. The nodal density determines the relative length of the elements, and can be specified
only at fixed points. Fixed points can be inserted or deleted anywhere in the soil profile. The user
can edit the nodal density at a fixed point in order to locally refine the nodal distribution aroundthis point. Careful placement of the nodes is important since the nodal distribution determines ina very substantial manner the ultimate quality and speed of the calculations.

293

Fig. B.2. The main window of the PROFILE module; when used for soil profile discretization.
Table B.6. Definition of terms used in module PROFILE.

Nodal Point

Nodal points are nodes which discretize the soil profile and which are marked by green
crosses. These nodes are ordered from the top (node number 1) to the bottom (nodeNumNP).

Elements

Elements are layers discretizing the soil profile. They connect the generated nodal
points.

Fixed Points

Fixed points are points in the soil profile marked by purple stars. These points may be
used to adjust the local discretization density of nodal points. By default, fixed points
are placed at the top and bottom of the soil profile, but they can be inserted or deleted
also at any other point in the soil profile.

294

Nodal Density

The nodal density is a real number in the range <0.01, 100.> specifying the local
density of nodal points. The density can be specified only at fixed points. The program
distinguishes between top and bottom density. The top (bottom) density at a fixed
point specifies the relative thickness of the elements above (or below) this point. If the
top and bottom densities are equal then the nodal density is continuous throughout the
profile, i.e., both elements have the same thickness. If the top and bottom density
values are different then the element thicknesses will be different as well. For example:
if DT = 3., DB = 2. then LT/LB = 1.5, where DT and DB are the top and bottom
densities at a fixed point, respectively, and LT and LB are the thicknesses of elements
above and below that fixed point, respectively.

B.3.2. Specification of Soil Properties within the Soil Profile
The PROFILE module (Fig. B.3.) helps a user to define also the spatial distribution ofparameters characterizing the flow domain (e.g., spatial distribution of soil materials, hydraulicscaling factors, root-water uptake parameters) and/or observation nodes. All parameters in thismodule are specified in a graphical environment with the help of a mouse.
Specification of parameters characterizing the flow domain (initial conditions, material
distribution) is relatively straightforward. The user must first select that part of the transportdomain to which he/she wants to assign a particular value of the selected variable. It is possible
to select the entire transport domain, part of it, or only individual nodes. A particular part of thetransport domain can be selected as follows: the user must first click the Edit Condition button,
and then move the mouse to a selected position. The beginning and end of the selection
operation is framed by clicking the left button. The selected area is the vertical defined by thetwo mouse positions when the left button was clicked. When the selection is completed, thewindow Condition Specification pops up and the user must specify the value of a particularvariable. That value will then be assigned to the selected area. When specifying the initialcondition, the user has the possibility of assigning either a constant value to a selected domain,or specifying different values to the top and bottom of the selected region, in which case theprogram will linearly interpolate the variable within the selected region. Variables are always
assigned to nodal points, not to elements.

295

Fig. B.3. The main window of the PROFILE module; when used for specification of soil
properties296