HYDRUS
Technical Manual
Version 3

Software Package for Simulating
the Two- and Three-Dimensional Movementof Water, Heat and Multiple Solutes
in Variably-Saturated MediaJanuary 2018, PC-Progress, Prague, Czech Republic© 2018 J.šimůnek and M. Šejna. All rights reserved

The HYDRUS Software Package for Simulating
the Two- and Three-Dimensional Movementof Water, Heat, and Multiple Solutes
in Variably-Saturated Porous Media

Technical Manual
Version 3.03

J. Šimůnek1, M. Th. van Genuchten2,3, and M. Šejna4

December 2020

1

2

University of California Riverside, Riverside, CA
Department of Nuclear Engineering, Federal University of Rio de Janeiro, Brazil
3
Department of Earth Sciences, Utrecht University, Netherlands
4

PC Progress, Prague, Czech Republic

ii

DISCLAIMER
This report documents version 3.0 of HYDRUS, a software package for simulating water, heat, andsolute movement in two- and three-dimensional variably saturated porous 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 havesuggestions for improvement, please contact one of the authors at

Tel. 1-951-827-7854 (J. Šimůnek)Tel. +55-21-32-64-70-25 (M. Th. van Genuchten)
Tel. +420-222-514-225 (M. Šejna)
Fax. 1-951-787-3993
E-mail Jiri.Simunek@ucr.edu
rvangenuchten@hotmail.com
m.sejna@pc-progress.cz

iii

iv

ABSTRACTŠimůnek, J., M. Th. van Genuchten, and M. Šejna, The HYDRUS Software Package for Simulating
Two- and Three Dimensional Movement of Water, Heat, and Multiple Solutes in VariablySaturated Porous Media, Technical Manual, Version 3.0, PC Progress, Prague, Czech Republic, 274
pp., 2020.
This report documents version 3.0 of HYDRUS, a general software package for simulatingwater, heat, and solute movement in two- and three-dimensional variably saturated porous media.
The software package consists of the computation computer program and the interactive graphicsbased user interface. The HYDRUS program numerically solves the Richards equation forsaturated-unsaturated water flow and the convection-dispersion equation for heat and solutetransport. The flow equation incorporates a sink term to account for water uptake by plant roots.
The heat transport equation considers transport due to conduction and convection with flowing
water.
The solute transport equations consider convective-dispersive transport in the liquid phase,
as well as diffusion in the gaseous phase. The transport equations also include provisions fornonlinear 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. In addition, physical nonequilibrium
solute transport can be accounted for by assuming a two-region, dual-porosity type formulation,
which partitions the liquid phase into mobile and immobile regions. Attachment/detachment theory,
including the filtration theory, is included to simulate the transport of viruses, colloids, and/or
bacteria.
The program may be used to analyze water and solute movement in unsaturated, partiallysaturated, or fully saturated porous media. HYDRUS can handle flow regions delineated byirregular boundaries. The flow region itself may be composed of nonuniform soils having an
arbitrary degree of local anisotropy. Flow and transport can occur in the vertical plane, thehorizontal plane, a three-dimensional region exhibiting radial symmetry about the vertical axis, or a
fully three-dimensional domain. The water flow part of the model can deal with prescribed head andflux boundaries, boundaries controlled by atmospheric conditions, free drainage boundary
conditions, as well as a simplified representation of nodal drains using results of electric analogv

experiments. The two-dimensional part of this program also includes a Marquardt-Levenberg typeparameter 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 for twodimensional problems.
Versions 2.0 and 3.0 also include several additional modules for simulating transport andreactions of major ions (UnsatChem) [Šimůnek and Suarez, 1994; Šimůnek et al., 1996, 2012a],reactions in constructed wetlands (Wetlands) [Langergraber and Šimůnek, 2005, 2006], colloidfacilitated solute transport (C-Ride) [Šimůnek et al., 2012c], general biogeochemical reactions(HP2) [Šimůnek et al., 2012d], and flow and transport dual-permeability porous media(DualPerm) [Šimůnek et al., 2012b]. However, these modules are not described in this manual.
The governing flow and transport equations are solved numerically using Galerkin-typelinear finite element schemes. Depending upon the problem's size, the matrix equations resulting
from the discretization of the governing equations are solved using either Gaussian elimination forbanded matrices or a conjugate gradient method for symmetric matrices and the ORTHOMIN
method for asymmetric matrices.
The program is distributed by means of several different options (Levels). Levels 2D-Lightand 2D-Standard are for the programs and the graphical interface for two-dimensional problems
with either a structured mesh generator for relatively simple flow domain geometries or a CAD
program for more general domain geometries, and the MESHGEN2D mesh generator for an
unstructured finite element mesh specifically designed for variably-saturated subsurface flowtransport problems, respectively. Levels 3D-Light and 3D-Standard include the two-dimensionalversion and, additionally, the three-dimensional versions for simple hexagonal or more generallayered geometries, respectively. Finally, Level 3D-Professional supports complex general threedimensional geometries.
Features in version 2.0 of HYDRUS as compared to version 1.0 included:1) A new, more efficient algorithm for particle tracking and a time-step control to guaranteesmooth particle paths.
2) Initial conditions can be specified in the total solute mass (previously, only liquid phase
concentrations were allowed).
3) Initial equilibration of nonequilibrium solute phases with equilibrium solute phase (given ininitial conditions).
4) Gradient Boundary Conditions (users can specify other than unit (free drainage) gradients
boundary conditions).

vi5) A subsurface drip boundary condition (with a drip characteristic function reducing irrigation
flux based on the back pressure) [Lazarovitch et al., 2005] (only in 2D).
6) A surface drip boundary condition with dynamic wetting radius [Gärdenäs et al., 2005](only in 2D).
7) A seepage face boundary condition with a specified pressure head.
8) Triggered Irrigation, i.e., irrigation can be triggered by the program when the pressure headat a particular observation node drops below a specified value (only in 2D).
9) Time-variable internal pressure head or flux nodal sinks/sources (previously only constantinternal sinks/sources).
10) Fluxes across meshlines in the computational module for multiple solutes (previously only
for one solute).
11) HYDRUS calculates and reports surface runoff, evaporation, and infiltration fluxes for theatmospheric boundary.
12) Water content dependence of solute reactions parameters using Walker’s [1974] formula
was implemented.
13) An option to consider root solute uptake, including both passive and active uptake (onlyin 2D) [Šimůnek and Hopmans, 2009].
14) The Per Moldrup’s tortuosity models [Moldrup et al., 1997, 2000] were implemented as
an alternative to the Millington and Quirk [1961] model.
15) An option to use a set of Boundary Condition records multiple times.
16) Options related to the fumigant transport (e.g., removal of a tarp, temperature-dependent
tarp properties, an additional injection of fumigant).
New features in version 3.0 of HYDRUS as compared to version 2.0 included:
1) Reservoir Boundary Condition (see Section 2.7.2.10).
2) Root growth (see Section 2.2.4).
3) The Inverse module also for three-dimensional problems.
This report serves as both a technical manual and reference document. Detailed instructions
are given for data input preparation. The graphical user interface (GUI) of the Hydrus software
package is documented in a separate user manual [Šejna et al., 2018].
Other Useful Information Online:
1. Outside reviews of the HYDRUS software packages:
http://www.pc-progress.com/en/Default.aspx?h3d-reviewsvii

2. HYDRUS 2D/3D Selected References:
http://www.pc-progress.com/en/Default.aspx?h3d-references
3. HYDRUS Quick Tour and Tutorials:
4. http://www.pc-progress.com/en/Default.aspx?h3d-tutorials
5. HYDRUS Applications and Public Library of HYDRUS Projects:
http://www.pc-progress.com/en/Default.aspx?h3d-applications
6. HYDRUS Book: Soil Physics with HYDRUS: Modeling and Applications:
http://www.pc-progress.com/en/Default.aspx?h3d-book
7. HYDRUS FAQ:
http://www.pc-progress.com/en/Default.aspx?hydrus-faq
8. HYDRUS (2D/3D) Troubleshooting:
http://www.pc-progress.com/en/Default.aspx?h3d-troubleshooting

A detailed description of the history of the development of various HYDRUS versions, areview of its applications, or the description of recently implemented new options are given in:Šimůnek, J. and M. Th. van Genuchten, Modeling nonequilibrium flow and transport with
HYDRUS, Vadose Zone Journal, 7(2), 782-797, doi:10.2136/VZJ2007.0074, 2008.
Šimůnek, J., M. Th. van Genuchten, and M. Šejna, Recent developments and applications of the
HYDRUS computer software packages, Vadose Zone Journal, 15(7), pp. 25, doi:
10.2136/vzj2016.04.0033, 2016.
Šimůnek, J., M. Šejna, and M. Th. van Genuchten, New features of the version 3 of the
HYDRUS (2D/3D) computer software package, Journal of Hydrology andHydromechanics, 66(2), 133-142, doi: 10.1515/johh-2017-0050, 2018.

viii

TABLE OF CONTENTSDISCLAIMER....................................................................................................................................... 3
ABSTRACT .......................................................................................................................................... 5
TABLE OF CONTENTS ...................................................................................................................... 9
LIST OF FIGURES ............................................................................................................................. 13
LIST OF TABLES .............................................................................................................................. 17
LIST OF VARIABLES ....................................................................................................................... 19
1. INTRODUCTION ........................................................................................................................... 1
2. VARIABLY SATURATED WATER FLOW ............................................................................... 7
2.1. Governing Flow Equation ...................................................................................................... 7
2.1.1. Uniform Flow ........................................................................................................7
2.1.2. Flow in a Dual-Porosity System ...........................................................................7
2.2. Root Water Uptake ................................................................................................................. 9
2.2.1. Root Water Uptake without Compensation...........................................................92.2.2. Root Water Uptake with Compensation..............................................................132.2.3. Spatial Root Distribution Functions ...................................................................142.2.4. Root Growth ........................................................................................................15
2.3. The Unsaturated Soil Hydraulic Properties ........................................................................ 172.4. Scaling in the Soil Hydraulic Functions .............................................................................. 222.5. Temperature Dependence of the Soil Hydraulic Functions................................................. 23
2.6. Hysteresis in the Soil Hydraulic Properties ......................................................................... 24
2.7. Initial and Boundary Conditions .......................................................................................... 28
2.7.1. System-Independent Boundary Conditions ........................................................28
2.7.2. System-Dependent Boundary Conditions ..........................................................30
2.8. Water Mass Transfer ......................................................................................................40
3. NONEQUILIBRIUM TRANSPORT OF SOLUTES INVOLVED IN SEQUENTIAL
FIRST-ORDER DECAY REACTIONS....................................................................................... 43
3.1. Governing Solute Transport Equations ............................................................................... 43
3.1.1. Chemical Nonequilibrium ...................................................................................... 46
3.1.2. Physical Nonequilibrium ........................................................................................ 49
3.1.3. Attachment-Detachment Model ............................................................................ 50
3.2. Initial and Boundary Conditions .......................................................................................... 54
3.3. Effective Dispersion Coefficient ........................................................................................... 57ix

3.4. Temperature and Water Content Dependence of Transport and Reaction Coefficients..... 59
3.5. Root Solute Uptake .........................................................................................................60
3.5.1. Uncompensated Root Nutrient Uptake Model ....................................................60
3.5.2. Compensated Root Nutrient Uptake Model ........................................................62
4. HEAT TRANSPORT .................................................................................................................... 65
4.1. Governing Heat Transport Equation ................................................................................... 65
4.2. Apparent Thermal Conductivity Coefficient ........................................................................ 65
4.3. Initial and Boundary Conditions .......................................................................................... 67
5. NUMERICAL SOLUTION OF THE WATER FLOW EQUATION ......................................... 69
5.1. Space Discretization ............................................................................................................. 69
5.2. Time Discretization............................................................................................................... 74
5.3. Numerical Solution Strategy ................................................................................................ 74
5.3.1. Iterative Process ...................................................................................................... 74
5.3.2. Treatment of the Water Capacity Term................................................................... 75
5.3.3. Time Control............................................................................................................ 76
5.3.4. Treatment of Pressure Head Boundary Conditions................................................ 77
5.3.5. Flux and Gradient Boundary Conditions ............................................................... 77
5.3.6. Atmospheric Boundary Conditions and Seepage Faces ......................................... 77
5.3.7. Tile Drains as Boundary Conditions ...................................................................... 78
5.3.8. Water Balance Computations ................................................................................. 79
5.3.9. Computation of Nodal Fluxes ................................................................................. 81
5.3.10. Water Uptake by Plant Roots .................................................................................. 82
5.3.11. Evaluation of the Soil Hydraulic Properties ........................................................... 83
5.3.12. Implementation of Hydraulic Conductivity Anisotropy .......................................... 83
5.3.13. Steady-State Analysis .............................................................................................. 84
6. NUMERICAL SOLUTION OF THE SOLUTE TRANSPORT EQUATION ........................... 87
6.1. Space Discretization ............................................................................................................. 87
6.2. Time Discretization............................................................................................................... 89
6.3. Numerical Solution for Linear Nonequilibrium Solute Transport....................................... 90
6.4. Numerical Solution Strategy ................................................................................................ 92
6.4.1. Solution Process ...................................................................................................... 92
6.4.2. Upstream Weighted Formulation............................................................................ 93
6.4.3. Implementation of First-Type Boundary Conditions .............................................. 96
6.4.4. Implementation of Third-Type Boundary Conditions ............................................. 97
6.4.5. Mass Balance Calculations ..................................................................................... 98
6.4.6. Oscillatory Behavior ............................................................................................. 101
7. PARAMETER OPTIMIZATION................................................................................................ 103
7.1. Definition of the Objective Function ................................................................................. 103

x7.2. Marquardt-Levenberg Optimization Algorithm ................................................................ 104
7.3. Statistics of the Inverse Solution ........................................................................................ 105
8. PROBLEM DEFINITION........................................................................................................... 107
8.1. Construction of Finite Element Mesh................................................................................. 107
8.2. Coding of Soil Types and Subregions ................................................................................ 110
8.3. Coding of Boundary Conditions......................................................................................... 110
8.4. Program Memory Requirements ........................................................................................ 116
8.5. Matrix Equation Solvers ..................................................................................................... 118
9. EXAMPLE PROBLEMS ............................................................................................................ 121
9.1. Direct Example Problems .................................................................................................. 121
9.1.1. Example 1 - Column Infiltration Test.................................................................... 122
9.1.2. Example 2 - Water Flow in a Field Soil Profile Under Grass ............................. 125
9.1.3a. Example 3A - Two-Dimensional Unidirectional Solute Transport ...................... 131
9.1.3b. Example 3B - Three-Dimensional Unidirectional Solute Transport .................... 135
9.1.4. Example 4 - One-Dimensional Solute Transport with Nitrification Chain .......... 137
9.1.5. Example 5 - 1D Solute Transport with Nonlinear CationAdsorption ............... 141
9.1.6. Example 6 - 1D Solute Transport with Nonequilibrium Adsorption ................. 145
9.1.7. Example 7 - Water and Solute Infiltration Test .................................................... 147
9.1.8. Example 8 - Contaminant Transport from a Waste Disposal Site ..................... 157
9.2. Inverse Example Problems ................................................................................................. 167
9.2.1. Example 9 - Tension Disc Infiltrometer ................................................................ 167
9.2.2. Example 10 - Cone Penetrometer ......................................................................... 169
9.2.3. Example 11 - Multiple-Step Extraction Experiment ............................................. 171
10. INPUT DATA ............................................................................................................................ 173
11. OUTPUT DATA ........................................................................................................................ 219
12. REFERENCES ........................................................................................................................... 233

xi

xii

LIST OF FIGURES

FigurePage

Figure 2.1.

Schematic of the plant water stress response function, α(h), as used bya) Feddes et al. [1978] and b) van Genuchten [1987] ................................................ 11

Figure 2.2.

Schematic of the potential water uptake distribution function, b(x,z), in
the soil root zone ......................................................................................................... 12

Figure 2.3.

The ratio of actual to potential transpiration as a function of the stress index ω. ..13

Figure 2.4.

Schematics of the soil water retention (a) and hydraulic conductivity (b)functions as given by equations (2.32) and (2.33), respectively ................................ 20

Figure 2.5.

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). .......................................................13

Figure 2.6.

Example of a water retention curve showing hysteresis. Shown are theboundary wetting curve, θ w(h), and the boundary drying curve, θ d(h). ..................25

Figure 2.7.

Three different types of the reservoir boundary condition: a well, a furrow,and a wetland. ........................................................................................................35

Figure 6.1.

Direction definition for the upstream weighting factors αij. ....................................94

Figure 8.1.

Finite elements and subelements used to discretize the 3-D domain:1) tetrahedral, 2) hexahedral, and 3) triangular prism.. ........................................ 109

Figure 9.1.

Flow system and finite element mesh for example 1 ............................................... 122

Figure 9.2.

Retention and relative hydraulic conductivity functions for example 1.
The solid circles are UNSAT2 input data [Davis and Neuman, 1983].................... 123

Figure 9.3.

Instantaneous, q0, and cumulative, I0, infiltration rates simulated with
the HYDRUS (solid lines) and UNSAT2 (solid circles) codes for example 1 ........ 124

Figure 9.4.

Flow system and finite element mesh for example 2 ............................................... 125

Figure 9.5.

Unsaturated hydraulic properties of the first and second soil layers forxiiiexample 2 .................................................................................................................. 127
Figure 9.6.

Precipitation and potential transpiration rates for example 2 ................................... 128

Figure 9.7.

Cumulative values for the actual transpiration and bottom discharge rates
for example 2 as simulated by HYDRUS (solid line) and SWATRE(solid circles) ............................................................................................................. 129

Figure 9.8.

Pressure head at the soil surface and mean pressure head of the root zone
for example 2 as simulated by HYDRUS (solid lines) and SWATRE(solid circles) ............................................................................................................. 131

Figure 9.9.

Location of the groundwater table versus time for example 2 as simulatedby HYDRUS (solid line) and SWATRE (solid circles) computer programs .......... 131

Figure 9.10.

Advancement of the concentration front (c=0.1) for example 3a as
calculated with HYDRUS (dotted lines) and the analytical solution(solid lines)................................................................................................................ 133

Figure 9.11.

Concentration profile at the end of the simulation (t=365 days) for
example 3a as calculated with HYDRUS (dotted lines) and the analyticalsolution (solid lines).................................................................................................. 134

Figure 9.12.

Advancement of the concentration front (c=0.1) for example 3b as
calculated by HYDRUS (dotted lines) and the analytical solution (solid lines)...... 134

Figure 9.13.

Concentration profile at the end of the simulation (t=365 days) for
example 3b as calculated with HYDRUS (dotted line) and the analyticalsolution (solid lines).................................................................................................. 135

Figure 9.14.

Schematic of the transport system for example 3B. ................................................ 137

Figure 9.15.

Analytically and numerically calculated concentration profiles for
NH4+, NO2-, and NO3- after 200 hours for example 4. ............................................. 139

Figure 9.16.

Analytically and numerically calculated concentration profiles for
NH4+(top), NO2- (middle), and NO3- (bottom) after 50, 100, and 200
hours for example 4. ................................................................................................ 140

Figure 9.17.

Mg breakthrough curves for Abist loam calculated with the MONOD,HYDRUS, and HYDRUS codes (data points from Selim et al. [1987])(example 5). .............................................................................................................. 142

xivFigure 9.18.

Ca breakthrough curves for Abist loam calculated with the MONOD and
HYDRUS codes (data points from Selim et al. [1987]) (example 5). ..................... 144

Figure 9.19.

Observed and calculated effluent curves for Boron movement throughGlendale clay loam (data points from van Genuchten [1981]) (example 6)............ 146

Figure 9.20.

Flow system and finite element mesh for example 7 ............................................... 151

Figure 9.21.

Initial (top) and steady state (bottom) pressure head profiles forexample 7. ................................................................................................................. 153

Figure 9.22.

Temperature profiles after 1 (top) and 10 days (bottom) for example 7.................. 153

Figure 9.23.

Concentration profiles for the first solute after 2.5, 5, 7.5, and 10 days forexample 7. ................................................................................................................. 154

Figure 9.24.

Concentration profiles for the second solute after 2.5, 5, 7.5, and 10 daysfor example 7. ........................................................................................................... 155

Figure 9.25.

Concentration profiles for the third solute after 2.5, 5, 7.5, and 10 days forexample 7. ................................................................................................................. 156

Figure 9.26.

Geometry and boundary conditions for example 8 simulating threedimensional flow and contaminant transport in a ponded variably-saturated
aquifer. ...................................................................................................................... 158

Figure 9.27.

Finite element mesh for example 8........................................................................... 160

Figure 9.28.

Calculated (a) longitudinal (y=0) and (b) transverse (x=170 m) elevationsof the groundwater table. .......................................................................................... 161

Figure 9.29.

Computed velocity field and streamlines at t = 10 days. ......................................... 163

Figure 9.30.

Concentration contour plots for (a) c = 0.1 in a longitudinal cross-section(y = 0), and (b) c = 0.05 in a transverse cross-section (x = 170 m). ......................... 164

Figure 9.31.

Concentration distributions in a horizontal plane located at z = 20 m
for t = 10, 50, 100, and 200 days. ............................................................................. 165

Figure 9.32.

Breakthrough curves observed at observation node 1 (x = 40 m, z = 32 m),
node 2 (x = 150 m, z = 24 m), node 3 (x = 170 m, z = 18 m), and node 4xv

(x = 200 m, z = 20 m)................................................................................................ 166
Figure 9.33.

Measured and optimized cumulative infiltration curves for a tension disc
infiltrometer experiment (example 9). ...................................................................... 168

Figure 9.34.

Unsaturated hydraulic conductivities at particular pressure heads
calculated using Wooding's analytical solution and the complete function
obtained by numerical inversion (example 9). ......................................................... 168

Figure 9.35.

Schematic of the modified cone penetrometer (example 10)................................... 170

Figure 9.36.

Comparison of observed and optimized cumulative infiltration curves andtensiometer readings for the modified cone penetrometer test
(example 10). ............................................................................................................ 170

Figure 9.37.

The layout of the laboratory multistep extraction experiment (example 11)........... 172

Figure 9.38.

Comparison of measured (symbols) and optimized (lines) cumulative
extraction (a) and pressure head (b) values (example 11). ....................................... 172

xvi

LIST OF TABLES
Table

Page

Table 8.1.

Initial settings of Kode(n), Q(n), and h(n) for constant boundary
conditions .................................................................................................................. 110

Table 8.2.

Initial settings of Kode(n), Q(n), and h(n) for variable boundary conditions .......... 111

Table 8.3.

Definition of the variables Kode(n), Q(n), and h(n) when an atmosphericboundary condition is applied................................................................................... 112

Table 8.4.

Definition of the variables Kode(n), Q(n), and h(n) when variable
head or flux boundary conditions are applied .......................................................... 112

Table 8.5.

Initial setting of Kode(n), Q(n), and h(n) for seepage faces ..................................... 114

Table 8.6.

Initial setting of Kode(n), Q(n), and h(n) for drains ................................................. 114

Table 8.7.

Summary of Boundary Coding ............................................................................... 117

Table 8.8.

List of array dimensions in HYDRUS ..................................................................... 118

Table 9.1.

Input parameters for example 3 ................................................................................ 133

Table 9.2.

Input parameters for example 4 ................................................................................ 139

Table 9.3.

Input parameters for example 5 ................................................................................ 141

Table 9.4.

Input parameters for example 6 ................................................................................ 145

Table 9.5.

Hydraulic input parameters for example 7 ............................................................... 149

Table 9.6.

Heat transport input parameters for example 7 ........................................................ 149

Table 9.7.

Solute transport input parameters for example 7...................................................... 150

Table 10.1.

Block A - Basic information..................................................................................... 175

Table 10.2.

Block B - Material information ................................................................................ 178

Table 10.3.

Block C - Time information ..................................................................................... 181

Table 10.4.

Block D - Solute transport information .................................................................... 182

Table 10.5.

Block E - Heat transport information ....................................................................... 188

xvii

Table 10.6.

Block F - Root water uptake information ................................................................. 189

Table 10.7.

Block F - Root water uptake information ................................................................. 192

Table 10.8a.

Block G - Finite element mesh information for two-dimensional applications....... 194

Table 10.8b.

Block G - Finite element mesh information for three-dimensional applications..... 195

Table 10.9.

Block H - Nodal information .................................................................................... 197

Table 10.10.

Block I - Element information .................................................................................. 199

Table 10.11.

Block J - Boundary information ............................................................................... 200

Table 10.12.

Block K - Atmospheric information ......................................................................... 207

Table 10.13.

Block L - Dimension information ............................................................................ 209

Table 10.14.

Block M - Inverse solution information ................................................................... 210

Table 10.15.

Block N – Mesh-lines information ........................................................................... 217

Table 10.16.

The Mater.in input file. ............................................................................................. 218

Table 11.1.

H_MEAN.OUT - mean pressure heads.................................................................... 221

Table 11.2.

V_MEAN.OUT - mean and total water fluxes ........................................................ 222

Table 11.3.

CUM_Q.OUT - total cumulative water fluxes ......................................................... 223

Table 11.4.

RUN_INF.OUT - time and iteration information .................................................... 224

Table 11.5.

SOLUTEx.OUT - actual and cumulative concentration fluxes ............................... 225

Table 11.6.

BALANCE.OUT - mass balance variables .............................................................. 226

Table 11.7.

A_LEVEL.OUT - mean pressure heads and total cumulative fluxes ...................... 227

Table 11.8.

FIT.OUT - information related to parameter estimation. ......................................... 228

Table 11.9.

FUMIGANT.OUT - actual and cumulative fumigant. .......................................... 229

Table 11.10.

WELL.OUT - volume and fluxes for reservoir boundary condition. ................... 230

Table 11.11.

IRRIG.OUT - information about triggered irrigations. ......................................... 231

xviii

LIST OF VARIABLESa

ion activity in soil solution [-]ahalf-width of the furrow (for reservoir BC) [L]aiconversion factor from concentration to osmotic head [L4M-1]

av

air content [L3L-3]a

ion activity on the exchange surfaces [-]Aamplitude of temperature sine wave [K]Aearea of a triangular element [L2]Aqh

parameter in equation (8.1) [LT-1]

As

correction factor [-][A]

coefficient matrix in the global matrix equation for water flow, [LT-1] or [L2T-1]+bnormalized root water uptake distribution, [L-2] or [L-3]+bwidth of the water table, [L] or [L2]+b’arbitrary root water uptake distribution, [L-2] or [L-3]+

bi , ci

geometrical shape factors [L]b1 , b2 , b3

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

Bqh

parameter in equation (8.1) [L-1]{B}

vector in the global matrix equation for water flow, [L2T-1] or [L3T-1]+csolution concentration, [ML-3] or [NcL-3]

c’finite element approximation of c [ML-3]

ci

initial solution concentration [ML-3]

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

cpsolute concentration of water injected into a reservoir [ML-3]

cr

concentration of the sink term [ML-3]

c0

prescribed concentration boundary condition [ML-3]

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

Cdfactor used to adjust the hydraulic conductivity of elements in the vicinity of drains [-]Cgvolumetric heat capacity of gas phase [ML-1T-2K-1] (e.g. Jm-3K-1)

Cn

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

xix

Co

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

CT

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

Cw

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

Crie

local Courant number [-]dthickness of stagnant boundary layer [L]dc

diameter of the sand grains [L]deeffective drain diameter [L]dp

diameter of the particle (e.g., virus, bacteria) (= 0.95 µm = 0.95e-6 m) [L]Dside length of the square in the finite element mesh surrounding a drain (elements haveadjusted hydraulic conductivities) [L]Dg

ionic or molecular diffusion coefficient in the gas phase [L2T-1]

Dij

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

Dijg

diffusion coefficient tensor for the gas phase [L2T-1]

Dijw

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

DLlongitudinal dispersivity [L]DTtransverse dispersivity [L]Dw

ionic or molecular diffusion coefficient in free water [L2T-1]{D}vector in the global matrix equation for water flow, [L2T-1] or [L3T-1]+

ensubelements which contain node n [-]Emaximum (potential) rate of infiltration or evaporation under the prevailing atmosphericconditions [LT-1]

Eaactivation energy of a reaction or process [ML2T-2M-1] (m2s-2mol-1)

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

fr(t)

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

coefficient matrix in the global matrix equation for water flow, [L2] or [L3]+ggas concentration [ML-3]ggravitational acceleration (= 9.81 m s-2) [LT-2]

gatm

gas concentration above the stagnant boundary layer [ML-3]{g}vector in the global matrix equation for solute transport, [MT-1L-1] or [MT-1]+

xx

[G]

coefficient matrix in the global matrix equation for solute transport, [L2T-1] or [L3T-1]+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]hnnodal values 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]hwwater level in the reservoir [L]h0initial 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]h∆pressure head at the reversal point in a hysteretic retention function [L]H

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

empirical constant relating the solution and adsorbed concentrations; for linear sorption
units are [L3M-1], for nonlinear Freundlich sorption, they are [L3βM-β]K

unsaturated hydraulic conductivity [LT-1]

K*

scaled unsaturated hydraulic conductivity [LT-1]

KAdimensionless anisotropy tensor for the unsaturated hydraulic conductivity K [-]Kd

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

Kw

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

Kdrainadjusted hydraulic conductivity in the elements surrounding a drain [LT-1]

xxi

Kex

dimensionless thermodynamic equilibrium constant [-]KH

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

KijA

components of the dimensionless anisotropy tensor KA [-]Kkmeasured value of the unsaturated hydraulic conductivity at θk [LT-1]

Kr

relative hydraulic conductivity [-]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 [-]K12

selectivity coefficient [-]K∆

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

lpore-connectivity parameter [-]Llength of the side of an element [L]Li

local coordinate [-]Lmthe maximum rooting depth [L]Lnlength of a boundary segment [L]LRrooting depth [L]Lxwidth of the root zone [L]Lz

depth of the root zone [L]L0initial value of the rooting depth at the beginning of the growing season [L]mparameter in the soil water retention function [-]M0cumulative amount of solute removed from the flow region by zero-order reactions,[ML-1] or [M]+M1

cumulative amount of solute removed from the flow region by first-order reactions,[ML-1] or [M]+Mr

cumulative amount of solute removed from the flow region by root water uptake, [ML] or [M]+1

Mtamount of solute in the flow region at time t, [ML-1] or [M]+

xxii

Mte

amount of solute in element e at time t, [ML-1] or [M]+M0amount of solute in the flow region at the beginning of the simulation, [ML-1] or [M]+M0eamount of solute in element e at the beginning of the simulation, [ML-1] or [M]+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 [-]Ncnumber of colloids (particles)ni

components of the outward unit vector normal to boundary ΓN or ΓG [-]ns

number of solutes involved in the chain reaction [-]Ntotal number of nodes [-]Nenumber of subelements en, which contain node n [-]NGgravitation number [-]NLocontribution of particle London-van der Walls attractive forces to particle removal [-]NPe

Peclet number in the single-collector efficiency coefficient [-]NRinterception number [-]O

actual rate of inflow/outflow to/from a subregion, [L2T-1] or [L3T-1]+

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

period of time necessary to complete one temperature cycle (1 day) [T]p1

exponent in the water stress response function [-]p2

exponent in the osmotic stress response function [-]px

empirical parameter in the root distribution function [-]pyempirical parameter in the root distribution function [-]pz

empirical parameter in the root distribution function [-]Peielocal Peclet number [-]qi

components of the Darcian fluid flux density [LT-1]

Qinwater flow rate into the well from the soil profile across the well wall (or its screenedpart), [L2T-1] or [L3T-1]+

Qp

pumping rate from the reservoir (positive for removal of water, negative for adding
water), [L2T-1] or [L3T-1]+

QnA

convective solute flux at node n, [MT-1L-1] or [MT-1]+

xxiii

QnD

dispersive solute flux at node n, [MT-1L-1] or [MT-1]+

QnT

total solute flux at node n, [MT-1L-1] or [MT-1]+{Q}

vector in the global matrix equation for water flow, [L2T-1] or [L3T-1]+[Q]

coefficient matrix in the global matrix equation for solute transport, [L2] or [L3]+rroot growth rate [T-1]rwradius of the well [L]rmaxmaximum width (radius) of the wetland [L]Rsolute retardation factor [-]Ruuniversal gas constant [ML2T-2K-1M-1] (=8.314kg m2s-2K-1mol-1)sadsorbed solute concentration, [-] or [NcM-1]

seadsorbed solute concentration on type-1 sites [-]siinitial value of adsorbed solute concentration [-]skadsorbed solute concentration on type-2 sites [-]smax

maximum solid phase concentration [NcM-1]

Ssink term [T-1]

S

volume of water in the reservoir, [L2] or [L3]+Se

degree of saturation [-]Sek

degree of saturation at θk [-]Spspatial distribution of the potential transpiration rate [T-1]St

width of soil surface associated with transpiration, [L] or [L2]+ST

cation exchange capacity [MM-1] (mmolckg-1)[S]coefficient matrix in the global matrix equation for solute transport, [L2T-1] or [L3T-1]+

ttime [T]t*

local time within the time period tp [T]tpperiod of time covering one complete cycle of the temperature sine wave [T] T
temperature [K]Taactual transpiration rate per unit surface length [LT-1]

Taverage temperature at soil surface during period tp [K]TAabsolute temperature [K]Tiinitial temperature [K]

xxiv

Tp

potential transpiration rate [LT-1]

TrA

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

T0

prescribed temperature boundary condition [K]v

average pore-water velocity [LT-1]

V

volume of water in each subregion, [L2] or [L3]+

Vnew

volume of water in each subregion at the new time level, [L2] or [L3]+

Vold

volume of water in each subregion at the previous time level, [L2] or [L3]+Vt

volume of water in the flow domain at time t, [L2] or [L3]+Vtevolume of water in element e at time t, [L2] or [L3]+Vwvolume of water in the well, [L2] or [L3]+V0

volume of water in the flow domain at time zero, [L2] or [L3]+V0evolume of water in element e at time zero, [L2] or [L3]+W

total amount of energy in the flow region, [MLT-2] or [ML2T-2]+x*empirical parameter in the root distribution function [L]xispatial coordinates (i=1,2,3) [L]Xm

maximum rooting length in the x-direction [L]y*empirical parameter in the root distribution function [L]Ymmaximum rooting length in the y-direction [L]z*empirical parameter in the root distribution function [L]z0

coordinate of the location where the straining process starts [L]zmaxmaximum depth of the wetland [L]Z0characteristic impedance of a transmission line analog to drain

Z0’characteristic impedance of free space (≈376.7 ohms)

Zmmaximum rooting length in the z-direction [L]αcoefficient in the soil water retention function [L-1]αdimensionless water stress response function [-]αsticking efficiency (ratio of the rate of particles that stick to a collector to the ratethey strike the collector) [-]αslope of the furrow side (for reservoir BC) [-]αdvalue of α for a drying branch of the soil water retention function [L-1]

xxv

αw

value of α for a wetting branch of the soil water retention function [L-1]αwweighing factor [-]αhscaling factor for the pressure head [-]α h*temperature scaling factor for the pressure head [-]αKscaling factor for the hydraulic conductivity [-]α K*temperature scaling factor for the hydraulic conductivity [-]αθscaling factor for the water content [-]βempirical constant in adsorption isotherm [-]β

empirical factor in the blocking function [-]γgzero-order rate constant for solutes in the gas phase [ML-3T-1]

γiactivity coefficient in 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]

γeboundary segments connected to node nΓDpart of flow domain boundary where Dirichlet type conditions are specified

ΓGpart of flow domain boundary where gradient type conditions are specified

ΓNpart of flow domain boundary where Neumann type conditions are specified

ΓCpart of flow domain boundary where Cauchy type conditions are specifiedδij

Kronecker delta [-]∆ttime increment [T]∆tmax

maximum permitted time increment [T]∆tminminimum permitted time increment [T]εtemporal weighing factor [-]εacabsolute error in the solute mass balance, [ML-1] or [M]+εawabsolute error in the water mass balance, [L2] or [L3]+εrcrelative error in the solute mass balance [%]εrw

relative error in the water mass balance [%]ε0permittivity of free space (used in the electric analog representation of drains)ηempirical constant in adsorption isotherm [L3M-3]

ηsingle-collector efficiency [-]

xxvi

θ

volumetric water content [L3L-3]

θ*

scaled volumetric water content [L3L-3]

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

θkvolumetric water content corresponding to Kk [L3L-3]

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

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

θmw

parameter in soil water retention function; wetting branch [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]

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

θrw

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

θssaturated soil water content [L3L-3]

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

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

θ∆

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

κparameter which depends on the type of flow being analyzed, [-] or [L]+λfirst-order rate constant [T-1]

λij

apparent thermal conductivity tensor of the soil [MLT-3K-1] (e.g., Wm-1K-1)

λL

longitudinal thermal dispersivity [L]λTtransverse thermal dispersivity [L]λ0thermal conductivity of porous medium in the absence of flow [MLT-3K-1] (e.g., W m1 -1
K )µfluid viscosity (= 0.00093 Pa s) [ML-1T-1]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]

xxvii

µg’

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

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

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

ρbulk density of porous medium [ML-3]ρddimensionless ratio between the side of the square in the finite element meshsurrounding the drain, D, and the effective diameter of a drain, de [-]ρffluid 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]

σrefsurface tension at reference temperature Tref [MT-2]

σTsurface tension at temperature T [MT-2]

σ1

prescribed flux boundary condition at boundary ΓN [LT-1]

σ2

prescribed gradient boundary condition at boundary ΓG [-]τatortuosity factor in the gas phase [-]τw

tortuosity factor in the liquid phase [-]φnlinear basis functions [-]φnu

upstream weighted basis functions [-]ψprescribed pressure head boundary condition at boundary ΓD [L]ψdimensionless colloid retention function [-]ωfirst-order adsorption rate constant [T-1]ωaangle between principal direction of K1A and the x-axis of the global coordinate system[-]ωsperformance index used as a criterion to minimize or eliminate numerical oscillations[-]Ω

flow regionΩedomain occupied by element eΩRregion occupied by the root zone+

for plane (2D) and axisymmetric (or 3D) flow, respectively

xxviii1. INTRODUCTION
The importance of the unsaturated zone as an integral part of the hydrological cycle has long
been recognized. The 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 dramatically increased in recent
years because of growing concern that the quality of the subsurface environment is adversely
affected by agricultural, industrial, and municipal activities. Federal, state, and local action andplanning 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 andpesticides applied to agricultural lands inevitably move below the soil root zone and contaminate
underlying groundwater reservoirs. Chemicals migrating from municipal and industrial disposalsites similarly represent environmental hazards. The same is true for radionuclides emanating from
nuclear waste disposal facilities.
The past several decades have 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 solutetransfer processes between the soil surface and the groundwater table. The most popular models
remain the Richards equation for variably saturated flow and the Fickian-based convectiondispersion equation for solute transport. Deterministic solutions of these classical equations havebeen used, and likely will continue to be used in the near future, for (1) predicting water and solute
movement in the vadose zone, (2) analyzing specific laboratory or field experiments involving
unsaturated water flow and/or solute transport, and (3) extrapolating information from a limited
number of field experiments to different soil, crop, and climatic conditions, as well as to different
soil 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 biological processes,
including sorption-desorption, volatilization, photolysis, and biodegradation, as well as theirkinetics. The extent of degradation, sorption, and volatilization largely determines a pollutant's
persistence in the subsurface [Chiou, 1989]. For example, the fate of organic chemicals in soils isknown to be strongly affected by the kinetics of biological degradation. Alexander and Scow [1989]1

gave a review of some of the equations used to represent biodegradation kinetics. 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 base, they are commonly used only empirically by fitting theequations to observed data. Zero- and first-order kinetic equations remain the most popular fordescribing biodegradation of organic compounds, mostly because of their simplicity and the ease atwhich they can be incorporated in solute transport models. Conditions for the application of these
two equations are described by Alexander 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], and the transport of certain pesticides and their metabolites [Bromilow
and Leistra, 1980; Wagenet and Hutson, 1987].
While most pesticides were regarded as involatile in the past, volatilization is now
increasingly recognized as being an important process affecting the fate of pesticides in field soils[Glotfelty and Schomburg, 1989; Spencer, 1991]. Another process affecting pesticide fate andtransport 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 evenexclusively, in the liquid phase [Pignatello, 1989]. On the other side, radioactive decay occurs
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 more complex
situations, such as for transient water flow or the nonequilibrium solute transport with nonlinear
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 to occur inthe solid, liquid, and gaseous phases, as well as for a correct distribution of the solutes among thedifferent phases.

2

The purpose of this technical report is to document version 2.0 of the HYDRUS software
package simulating two- and three-dimensional variably-saturated water flow, heat movement, andtransport of solutes involved in sequential first-order decay reactions. The software package consistsof several computational programs [h2d_calc.exe (2D direct), h2d_clci.exe (2D inverse),
h2d_wetl.exe (2D wetland), h2d_unsc.exe (2D major ions), and h3d_calc.exe (3D direct)] and theinteractive graphics-based user interface HYDRUS. The HYDRUS program numerically solves theRichards equation for saturated-unsaturated water flow and convection-dispersion type equations
for heat and solute transport. The flow equation incorporates a sink term to account for water uptake
by plant roots. The heat transport equation considers movement by conduction as well as convection
with flowing water.
The governing convection-dispersion solute transport equations are written in a very generalform by including provisions for nonlinear nonequilibrium reactions between the solid and liquid
phases, and linear equilibrium reaction 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 zero-order production, first-order degradation independent of othersolutes, and first-order decay/production reactions that provide the required coupling between thesolutes involved in the sequential first-order chain. The transport models also account for
convection and dispersion in the liquid phase, as well as for diffusion in the gas phase, thus
permitting one to simulate solute transport simultaneously in both the liquid and gaseous phases.
HYDRUS at present considers up to fifteen solutes, which can be either coupled in a unidirectional
chain or may move independently of each other. Physical nonequilibrium solute transport can be
accounted for by assuming a two-region, dual-porosity type formulation, which partitions the liquid
phase into mobile and immobile regions. Attachment/detachment theory, including the filtration
theory, is included to simulate the transport of viruses, colloids, and/or bacteria.
Version 2.0 includes two additional modules for simulating transport and reactions of major
ions [Šimůnek and Suarez, 1994; Šimůnek et al., 1996] and reactions in constructed wetlands[Langergraber and Šimůnek, 2005, 2006]. However, these two modules are not described in thismanual.
The program may be used to analyze water and solute movement in unsaturated, partiallysaturated, or fully saturated porous media. HYDRUS can handle flow domains delineated byirregular boundaries. The flow region itself may be composed of nonuniform soils having an
arbitrary degree of local anisotropy. Flow and transport can occur in the vertical plane, thehorizontal plane, a three-dimensional region exhibiting radial symmetry about a vertical axis, or in a3

three-dimensional region. The water flow part of the model considers prescribed head and flux
boundaries, boundaries controlled by atmospheric conditions, free drainage boundary conditions, aswell as a simplified representation of nodal drains using results of electric analog experiments. Firstor third-type boundary conditions can be implemented in both the solute and heat transport parts ofthe model. In addition, HYDRUS implements a Marquardt-Levenberg type parameter estimation
scheme for inverse estimation of soil hydraulic and/or solute transport and reaction parameters frommeasured transient or steady-state flow and/or transport data for two-dimensional problems.
The governing flow and transport equations are solved numerically using Galerkin-typelinear finite element schemes. Depending upon the size of the problem, the matrix equations
resulting from the discretization of the governing equations are solved using either Gaussianelimination for banded matrices or the conjugate gradient method for symmetric matrices and the
ORTHOMIN method for asymmetric matrices [Mendoza et al., 1991]. The program is an extensionof the variably saturated flow codes HYDRUS-2D of Šimůnek et al. [1999], SWMS_3D of Šimůnek
et al. [1995], SWMS_2D of Šimůnek et al. [1992] and CHAIN_2D of Šimůnek and van Genuchten[1994], which in turn were based in part on the early numerical work of Vogel [1987] and Neumanand colleagues [Neuman, 1972, 1973, Neuman et al., 1974; Neuman, 1975; Davis and Neuman,
1983].
New features in version 2.0 of HYDRUS as compared to version 1.0 include:a) A new, more efficient algorithm for particle tracking and a time-step control to guaranteesmooth particle paths.
b) Initial conditions can be specified in the total solute mass (previously, only liquid phase
concentrations were allowed).
c) Initial equilibration of nonequilibrium solute phases with equilibrium solute phase (given ininitial conditions).
d) Gradient Boundary Conditions (users can specify other than unit (free drainage) gradients
boundary conditions).
e) A subsurface drip boundary condition (with a drip characteristic function reducing irrigation
flux based on the back pressure) [Lazarovitch et al., 2005] (only in 2D).
f) A surface drip boundary condition with dynamic wetting radius [Gärdenäs et al., 2005](only in 2D).
g) A seepage face boundary condition with a specified pressure head.
h) Triggered Irrigation, i.e., irrigation can be triggered by the program when the pressure headat a particular observation node drops below a specified value (only in 2D).

4i) Time-variable internal pressure head or flux nodal sinks/sources (previously only constantinternal sinks/sources).
j) Fluxes across meshlines in the computational module for multiple solutes (previously only
for one solute).
k) HYDRUS calculates and reports surface runoff, evaporation, and infiltration fluxes for theatmospheric boundary.
l) Water content dependence of solute reactions parameters using Walker’s [1974] formula
was implemented.
m) An option to consider root solute uptake, including both passive and active uptake (onlyin 2D) [Šimůnek and Hopmans, 2009].
n) The Per Moldrup’s tortuosity models [Moldrup et al., 1997, 2000] were implemented as
an alternative to the Millington and Quirk [1961] model.
o) An option to use a set of Boundary Condition records multiple times.
p) Options related to the fumigant transport (e.g., removal of a tarp, temperature-dependent
tarp properties, an additional injection of fumigant).
New features in version 3.0 of HYDRUS as compared to version 2.0 include:a) A new "Reservoir" boundary condition” (BC).
b) A dynamic root system.
Even with an abundance of well-documented models now being available, one major
problem often preventing their optimal use is the extensive work required for data preparation,
numerical grid design, and graphical presentation of the output results. Hence, the more widespreaduse of multi-dimensional models requires ways that make it easier to create, manipulate, and display
large data files and which facilitate interactive data management. Introducing such techniques will
free users from cumbersome manual data processing and should enhance the efficiency in which
programs are being implemented for a particular example. To avoid or simplify the preparation andmanagement of relatively complex input data files for two- and three-dimensional applications, andto graphically display the final simulation results, we developed an interactive graphics-based userfriendly interface HYDRUS for the MS Windows 95, 98, NT, ME, XP, Vistas, 7, and 10
environments.
The HYDRUS software is distributed in the following forms (Levels):1) Level 2D-Light includes the executable code HYDRUS (h2d_calc.exe and h2d_clci.exe)

5and a graphics-based user interface so as to facilitate data preparation and output display in the MS
WINDOWS 95, 98, 2000, NT, and/or XP environments. A mesh generator for relatively simple
rectangular domain geometry is made part of option A. The user interface is written in MS Visual
C++. Because HYDRUS was written in Microsoft FORTRAN, this code uses several extensions
that are not part of ANSI-standard FORTRAN, such as dynamically allocated arrays.
2) Level 2D-Standard consists of 2D-Light, but further augmented with a CAD program
MESHGEN2D for designing more general domain geometry and its discretization into an
unstructured finite element mesh for a variety of problems involving variably-saturated subsurface
flow and transport. Option B is also distributed on a CD ROM and web.
3) Level 3D-Light includes 2D-Standard above and a mesh generator for relatively simple
hexagonal three-dimensional domain geometry.
4) Level 3D-Standard includes 3D-Light, and a mesh generator that can generate
unstructured meshes for general two-dimensional geometries that can be extended into threedimensional transport domains.
5) Finally, 3D-Professional includes 3D-Standard, and additionally also supports flowtransport in complex general three-dimensional geometries.
A general overview of the graphics-based interface is described in the accompanying user
manual. In addition to the detailed description in the accompanying user manual, extensive on-line
help files are available with each module of the user interface.
Demo version of the program, the input and output files of examples discussed in this report,
plus many additional examples which illustrate the interface and the program in its full complexity,
can be downloaded from www.hydrus3d.com.

62. VARIABLY SATURATED WATER FLOW
2.1. Governing Flow Equation
2.1.1. Uniform Flow
Consider two- and/or three-dimensional isothermal uniform Darcian flow of water in avariably saturated rigid porous medium and assume that the air phase plays an insignificant role in
the liquid flow process. The governing flow equation for these conditions is given by the following
modified form of the Richards' equation:∂θ∂   A ∂h=
+ K izA   - S
 K  K ij∂t ∂xi  
∂x j


(2.1)

where θ is the volumetric water content [L3L-3], h is the pressure head [L], S is a sink term [T-1], xi(i=1,2,3) are the spatial coordinates [L], t is time [T], KijA are components of a dimensionlessanisotropy tensor KA, and K is the unsaturated hydraulic conductivity function [LT-1] given by
K ( h, x , y , z ) = K s ( x , y , z ) K r ( h, x , y , z )(2.2)

where Kr is the relative hydraulic conductivity and Ks the saturated hydraulic conductivity [LT-1].
The anisotropy tensor KijA in (2.1) is used to account for an anisotropic medium. The diagonal
entries of KijA equal one and the off-diagonal entries zero for an isotropic medium. If (2.1) is applied
to planar flow in a vertical cross-section, x1=x is the horizontal coordinate and x2=z is the verticalcoordinate, the latter taken to be positive upward. Einstein's summation convention is used in (2.1)
and throughout this report. Hence, when an index appears twice in an algebraic term, this particularterm must be summed over all possible values of the index.
2.1.2. 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 (intra-aggregate pores or the rock7

matrix) does not move at all. These models assume that the matrix, consisting of 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), θm, and immobile (stagnant, intra-aggregate), θim, regions:θ = θ m + θim(2.3)

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, intraaggregate 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 m∂   A ∂h=
+ K izA   − S m − Γ w
 K  K ij∂t∂xi  
∂x j

∂θim
=
− Sim + Γ w
∂t

(2.4)

where Sm and Sim (Sim is assumed to be zero in the current HYDRUS version) are sink terms forboth regions, and Γw is the transfer rate for water from the inter- 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 the gravitational movement of water in macropores. Although dual-porosity models
have been popularly used for solute transport studies (e.g., van Genuchten [1981]), they have notthus far been used for water flow problems.

8

2.2. Root Water Uptake
2.2.1. Root Water Uptake without Compensation
The sink term, S, in (2.1) represents the volume of water removed per unit time from a unit
volume of soil due to plant water uptake. Feddes et al. [1978] defined S as
S ( h) = α ( h) S p(2.5)

where the water stress response function α(h) is a prescribed dimensionless function (Fig. 2.1) of
the soil water pressure head (0≤α ≤1), and Sp is the potential water uptake rate [T-1]. Figure 2.1
gives a schematic plot of the stress response function as used by Feddes et al. [1978]. Notice thatwater uptake is assumed to be zero close to saturation (i.e., wetter than some arbitrary "anaerobiosis
point", h1). For h<h4 (the wilting point pressure head), water uptake is also 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 (2.5) is equal to the water uptake rate during periods of no water stress when α (h)=1. Whenroot water uptake can occur at its optimal value even at full saturation, set both h1 and h2 equal tozero.
van Genuchten [1987] expanded the formulation of Feddes by including osmotic stress asfollows
S (h, hφ ) = α (h, hφ ) S p(2.6)

where hφ is the osmotic head [L], which is assumed here to be given by a linear combination of theconcentrations, ci, of all solutes present, i.e.,hφ = ai ci(2.7)

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 stressresponse function (Fig. 2.1) and suggested that the influence of the osmotic head reduction can beeither additive or multiplicative as follows

9α (h, hφ ) =1

 h + hφ 1+  h50 α (h, hφ ) = 0.

ph + hφ > h4
(2.8)
h + hφ < h4or

α (h, hφ ) =11
p1
1 + (h/h50 ) 1 + (hφ /hφ 50 ) p 2(2.9)

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 parameter h50
in (2.8) and (2.9) 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 the stress response function,α(h,hφ), does not consider a transpiration reduction near saturation. Such a simplification seems
justified when saturation occurs for only relatively short periods of time. When the multiplicativemodel is used for salinity stress, one can use the Threshold Model of Maas [1990] as analternative to the S-Shaped Model of van Genuchten [1985].
Note that different sensitivity of the root water uptake to water and salinity stresses canbe considered using the additive formulation (2.8) as follows:=α (h, hφ )

1
1
1
=
=
p
p
p
 h
 h + a1hφ  h + a1 ( ai ci ) 
hφ 
+
1+ 
 1 +  1+ 

 h
h
h50
h
50




φ
50
50



(2.10)

where a1 is a coefficient accounting for different responses to different stresses (=h50/hφ50).
When the potential water uptake rate is equally distributed over a two-dimensional
rectangular root domain, Sp becomes
Sp=

1
Lx Lz

St T p(2.11)

where Tp is the potential transpiration rate [LT-1], Lz is the depth [L] of the root zone, Lx is thewidth [L] of the root zone, and St is the width [L] of the soil surface associated with thetranspiration process. Notice that Sp reduces to Tp/Lz when St=Lx. For three-dimensional
problems, (2.11) additionally includes the width of the y-direction, Lx [L], and St becomes the10soil surface associated with transpiration [L2] (Figure 2.2, right).
Equation (2.11) may be generalized by introducing a nonuniform distribution of thepotential water uptake rate over a root zone of arbitrary shape [Vogel, 1987]:(2.12)

S p = b ( x, y , z ) S t T p

Figure 2.1. Schematic of the plant water stress response function, α(h),as used by a) Feddes et al. [1978] and b) van Genuchten [1987].
LtTab(x,z)

S

ΩR

z

Ω

y
z

x

X

Figure 2.2. Schematic of the potential water uptake distribution function, b(x,y,z),
in the soil root zone.

11where b(x,y,z) is the normalized water uptake distribution [L-2 or L-3]. This function describes thespatial variation of the potential extraction term, Sp, over the root zone (Fig. 2.2), and is obtained
from b’(x,y,z) as followsb ( x, y , z ) =b ′( x, y, z )∫ b′( x, y, z )d Ω

(2.13)ΩR

where ΩR is the region occupied by the root zone, and b’(x,y,z) is an arbitrarily prescribed
distribution function. Normalizing the uptake distribution ensures that b(x,y,z) integrates to unity
over the flow domain, i.e.,∫ b ( x, y , z ) d Ω = 1(2.14)ΩR

From (2.12) and (2.14) it follows that Sp is related to Tp by the expression
1
S p d Ω= Tp
St Ω∫R(2.15)

The actual water uptake distribution is obtained by substituting (2.12) into (2.5):
S ( h, hφ , x, y , z ) = α ( h,hφ , x, y , z )b( x, y , z ) St Tp(2.16)

whereas the actual transpiration rate, Ta, is obtained by integrating (2.16) as follows

Ta =1
Sd Ω = Tp ∫ α (h, hφ , x, y, z )b( x, y, z )d Ω
St Ω∫RΩR

12(2.17)

2.2.2. Root Water Uptake with Compensation
The ratio of actual to potential transpiration of the root uptake without compensation isdefined as follows:Ta1=
S (h,hφ , x, y, z )d Ω = ∫ α (h,hφ , x, y, z ) b( x, y, z )d Ω =ωTp Tp Ω∫R
ΩR

(2.18)

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 root adaptability factor, which
represents a threshold value above which root water uptake reduced in stressed parts of the rootzone is fully compensated by increased uptake from other parts. However, some reduction in
potential transpiration will occur below this threshold value, although smaller than for water
uptake without compensation.
11Compensated

Tac/Tp

Non-Compensated

Ta/Tp

ωc

0
0

ω

0
1

Figure 2.3. The 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

13

Tp

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

Ta LR
=
Tp

(2.19)

=

ω

ω

ω
= 1
ω

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

T a = LRTp

ωc(2.20)

ω
<1
ωc

=

ωc

Tp

When the parameter ωc is equal to one we hence have noncompensated root water uptake, andwhen ωc is equal to zero we obtain fully compensated uptake.
2.2.3. Spatial Root Distribution Functions
Following two- and three-dimensional root distribution functions are implemented into
HYDRUS [Vrught et al., 2001, 2002]:
 p

p



z
r

z 
x  − Zm z − z + X m x − x b ( x, z ) =
1 −
1 −
e
 Z m  X m 
*

 p

*

py

(2.21)
p



x
z

x 
y 
z  − X m x − x + Ym y − y + Zm z − z b ( x, y , z ) =
1
−
1
−
1
−
e




 X m  Ym  Z m ***(2.22)

where Xm, Ym, and Zm are the maximum rooting lengths in the x-, y-, and z- directions [L],respectively; x, y, and z are distances from the origin of the tree in the x-, y-, and z- directions[L], respectively; px [-], py [-], pz [-], x* [L], y* [L], and z* [L] are empirical parameters, andb(x,z) and b(x,y,z) denote two- and three-dimensional spatial distribution of the potential root
water uptake [-]. Following Vrugt et al. [2002], we set px, py, and pz to unity for z>z*, x>x*, and14

y>y*, respectively. Vrugt et al. [2001ab] showed that the root water uptake in (2.21) is extremely
flexible and allows spatial variations of water uptake as influenced by non-uniform water
application (e.g. , drip irrigation) and root length density patterns. See Vrugt et al. [2001ab] fordifferent configurations of the normalized spatial distribution of potential root water uptake.
Alternatively, one can use the following function (Hoffman and van Genuchten, 1983):

1.667


Zm

 2.0833 L−z=b( z ) 
1 −

Zm 
 Zm 
0
z > L − 0.2 Z mz ∈ ( L − Z m ; L − 0.2 Z m )(2.23)

z < L − Zm

where L is the x-coordinate of the soil surface [L] and Zm is the rooting depth [L].
2.2.4. Root Growth
The previous Version 2 of HYDRUS(2D/3D) includes a relatively comprehensive
macroscopic root water and solute uptake model (Šimůnek and Hopmans, 2009) to account for
the effects of both water and salinity stresses on root water uptake, while additionally accounting
for possible active and passive root contaminant or nutrient uptake. Root water and solute
uptake both can furthermore be treated as being either non-compensated or compensated, whileusers can select the degree of compensation (Šimůnek and Hopmans, 2009).
HYDRUS-1D additionally allows users to externally prescribe a time-variable rooting
depth, either using the logistic growth function or a tabulated form. This feature has thus far not
been available in HYDRUS (2D/3D), which has been noted by several HYDRUS users (e.g.,
Ramos et al., 2012; Karandish and Šimůnek, 2016; among others). The spatial distribution ofroots in the root zone was assumed to remain constant during the simulations, and only theintensity of uptake (as a result of time-variable potential transpiration rates) could be specified as
input. The HYDRUS models also did not allow the spatial extent of the rooting zone to change
actively as a result of certain environmental stresses (e.g., Hartman et al., 2017).
To overcome these deficiencies, several studies either further modified the HYDRUS

15

models (or their predecessors such as CHAIN-2D or SWMS-3D), or coupled the models withexisting crop growth or root growth models. For example, Javaux et al. (2008, 2013) developed
R-SWMS, a three-dimensional root growth model that couples the model of Somma et al. (1998)(based on SWMS-3D) with the root architecture model of Doussan et al. (1998). Zhou et al.
(2012) coupled HYDRUS-1D with the WOFOST crop growth model (Boogaard et al., 1998),
and used the resulting code to simulate the growth and yield of irrigated wheat and maize (Li etal., 2012, 2014). Han et al. (2015) similarly coupled HYDRUS-1D with a simplified crop growthversion used in SWAT to simulate the contribution and impact of groundwater on cotton growth
and root zone water balance. Wang et al. (2014) and Wang et al. (2015) coupled the crop growthmodel EPIC (Williams et al., 1989) with CHAIN-2D and HYDRUS-1D to assess the effects offurrow and sprinkler irrigation, respectively, on crop growth. Finally, Hartmann et al. (2017)
implemented into both HYDRUS-1D and HYDRUS (2/3D) the root growth model developed byJones et al. (1991). Their model assumes that various environmental factors, as characterized by
growth stress factors, can influence root development under suboptimal conditions. Of thevarious software, only the models of Hartmann et al. (2017) are directly available from the
HYDRUS website as a non-standard HYDRUS module (i.e., not fully supported by the
HYDRUS GUI).
To extend the capabilities of the standard two-dimensional module of HYDRUS(2D/3D), a simple root growth model with similar capabilities as those in HYDRUS-1D was
implemented into Version 3. The rooting depth, LR, can now be either constant (the standard
approach) or variable during the simulations. For annual vegetation, a growth model is required
to simulate changes in rooting depth with time. Time-variable rooting depth values can beprovided either using a table on input, or calculated with the program assuming that the actual
rooting depth is the product of the maximum rooting depth, Lm [L], and a root growth coefficient,
fr(t) [-], given by:
LR (t ) = Lm f r (t )(2.24)For the root growth coefficient, fr(t), we use the classical Verhulst-Pearl logistic growth function

f r (t ) =L0
L0 + ( Lm - L0 ) e- rt

16(2.25)

where L0 is the initial value of the rooting depth at the beginning of the growing season [L], and r
the growth rate [T-1]. The growth rate can be calculated either from the assumption that 50% of
the rooting depth will be reached after 50% of the growing season has elapsed or from given dataof the rooting depth at a specified time. The same approach is also used for the horizontal extent
of the rooting zone, except that the maximum extent of the rooting zone in the horizontal
direction is used instead of the maximum rooting depth.

2.3. The Unsaturated Soil Hydraulic Properties
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; and Vogel andCíslerová, 1988; Kosugi, 1995, 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/ αh ≥ - 1/ α

K = K s Se2 / n+l+2(2.26)(2.27)

respectively, where Se is the effective water content,Se =θ - θrθs - θr(2.28)

in which θr and θs denote the residual and saturated water content, respectively; Ks is the saturated
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 the original study of
Brooks and Corey [1964]. The parameters α, n, and l in HYDRUS are considered to be merely

17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 theunsaturated hydraulic conductivity function in terms of soil water retention parameters. The
expressions of van Genuchten [1980] are given byθs - θr θr +
θ ( h) = [1+ |α h| n] m
 θ
s


h<0

(2.29)h≥02

K ( h ) = K s Sel [1 - (1 - Se1 / m ) m ](2.30)

m = 1 - 1/n , n > 1(2.31)

where

The above equations contain six independent parameters: θr, θs, α, n, Ks, and l. The poreconnectivity parameter l in the hydraulic conductivity function was estimated [Mualem, 1976] to beabout 0.5 as an average for many soils.
The 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), and hydraulicconductivity, K(h), functions of Vogel and Císlerová [1988] are given by (Fig. 2.4)

θm - θam θa +
θ ( h) = (1+ |α h| n)θsh < hs(2.32)h ≥ hsand K s K r ( h)(h − hk ) ( K s − K k )K (h=) Kk +
hs − hk K s18h ≤ hshk < h < hsh ≥ hs(2.33)

respectively, where
1/ 2

K S 
Kr = k  e 
K s  Sek 

 F (θ r ) − F (θ )  F (θ r ) − F (θ k )   θ − θ 1/ m a
F (θ =) 1 −    θ m − θ a  Sek =

θk − θrθs − θr2(2.34)

m(2.35)(2.36)

The above equations allow for a non-zero minimum capillary height, hs, by replacing the parameterθs in van Genuchten's retention function with a fictitious (extrapolated) parameter θm slightly larger
than θs as shown in Figure 2.4. While this change from θs to θm has little or no effect on theretention curve, the effect on the shape and value of the hydraulic conductivity function can beconsiderable, 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 retention function was
replaced by the fictitious (extrapolated) parameter θa≤θr. The approach maintains the physicalmeaning of θr and θs as measurable quantities. Equation (2.34) assumes that the predicted hydraulicconductivity function is matched to a measured value of the hydraulic conductivity, Kk=K(θk), atsome water content, θk, less than or equal to the saturated water content, i.e., θk≤θs and Kk≤Ks [Vogel
and Císlerová, 1988; Luckner et al., 1989]. Inspection of (2.32) through (2.34) shows that thehydraulic characteristics contain 9 unknown parameters: θr, θs, θa, θm, α, n, Ks, Kk, and θk. Whenθa=θr, θm=θk=θs, and Kk=Ks, the soil hydraulic functions of Vogel and Císlerová [1988] reduce tothe original expressions of van Genuchten [1980].

19Figure 2.4. Schematics of the soil water retention (a) and hydraulic conductivity (b) functions
as given by equations (2.32) and (2.33), respectively.

Version 3.0 of Hydrus allows the soil hydraulic properties to be also defined according to
Kosugi [1996], who suggested the following lognormal distribution model for Se(h):1 ln ( h / α ) 
2n 

θ − θ r  erfc 
=
Se =
2

θs − θr 
1( h < 0)(2.37)( h ≥ 0)Application of Mualem's pore-size distribution model [Mualem, 1976] now leads to thefollowing hydraulic conductivity function:

20

2





α
h
ln
/
)
(
n
1


 K S 1/ 2+
erfc 
K =  s e  22n2  Ks( h < 0)(2.38)( h ≥ 0)Note that in this text, we use symbols α [L] instead of h0 [L] 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

(2.39)

Combining this retention model with Mualem’s [1976] pore-size distribution model leads now to:K ( S e) = K sdw1Se1 + w2 Se2ielw1α 1 [1 - (1 - S 1e1/ m1 ) 1 ] + w2α 2 [1 - (1 - S 1e2/ m2 ) 2 ]mm

bw α + w α g
1

1

2

2

j

2

(2.40)

2

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 also beenused for fractured rock by Peters and Klavetter [1988], Pruess and Wang [1987], and Flint et al.
[2001], among others.

21

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]|)

Figure 2.5. 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).

2.4. Scaling in the Soil Hydraulic Functions
HYDRUS implements a scaling procedure designed to simplify the description of the spatial
variability of the unsaturated soil hydraulic properties in the flow domain. The code assumes thatthe hydraulic variability in a given area can be approximated by means of a set of linear scalingtransformations that relate the individual soil hydraulic characteristics θ(h) and K(h) to reference
characteristics θ*(h*) and K*(h*). The technique is based on the similar media concept introduced byMiller and Miller [1956] for porous media, which differ only in the scale of their internal geometry.
The concept was extended by Simmons et al. [1979] to materials that differ in morphologicalproperties, 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 ]h = αh h(2.41)*

in which, for the most general case, αθ, αh, and αK are mutually independent scaling factors for the22

water content, the pressure head, and the hydraulic conductivity, respectively. Less general scaling
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), and αK=αh-2. A
detailed discussion of the scaling relationships given by (2.41), and their application to the hydraulicdescription of heterogeneous soil profiles, is given by Vogel et al. [1991].

2.5. Temperature Dependence of the Soil Hydraulic Functions
As described above, a similar scaling technique is used in HYDRUS to express thetemperature dependence of the soil hydraulic functions. Based on the 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 dT(2.42)

where T is temperature [K] and σ is the surface tension at the air-water interface [MT-2]. From
(2.42) it follows thathT =σThref = α h* href
σ ref(2.43)

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

µ ref ρ T=
K ref (θ ) α K* K ref (θ )µ T ρ ref23

(2.44)

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 point in
the soil profile. While such a simplification may be acceptable for some flow simulations, manycases require a more realistic description involving hysteresis in the soil hydraulic properties. The
HYDRUS code incorporates hysteresis by using the empirical model introduced by Scott et al.
[1983]. This model was also employed by Kool and Parker [1987], who modified the formulationto account for air entrapment. The present version of HYDRUS further extends the model of Kool
and Parker, according to Vogel et al. [1996], by also considering hysteresis in the hydraulicconductivity function.
The adopted procedure for modeling hysteresis in the retention function requires that boththe main drying and main wetting curves be known (Fig. 2.6). These two curves are described with
(2.30) 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(2.45)We also invoke the often-assumed restrictionnd = nw(2.46)

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

24

w
θ m=
θr +

θ sw - θ r d(θ - θ )θ sd - θ r m r(2.47)

Figure 2.6. Example of a water retention curve showing hysteresis. Shown are theboundary wetting curve, θ w(h), and the boundary drying curve, θ d(h).

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 from themain 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 a reference
curve in scaling equation (2.41) (keeping α h = 1 to scale only in the water content direction), i.e.:25θ ( h=) θ r' + αθ [θ d ( h ) - θ rd ](2.48)

and forcing each scanning curve, θ(h), to pass through the point (θ∆, h∆), characterizing the latest
reversal from wetting to drying. Substituting this reversal point into (2.48), and assuming thatθr = θrd, leads toαθ =θ∆ - θrθ ( h∆ ) - θ r
d(2.49)

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 curves) θ r' + αθ [θ w ( h ) - θ r ]θ ( h=(2.50)

in which the fictitious parameter θr’ is now used (again possibly scaled outside of the main loop).
The scaling factor αθ for a particular scanning curve can be obtained by substituting the reversal
point (θ∆, h∆) and the full saturation point (θs, 0) into (2.50), and subtracting the two resulting
equations to eliminate θr’ to giveαθ =θ∆ - θsθ (h∆ ) - θ sw
w

(2.51)The parameter θr’ is subsequently determined from (2.50) 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 sd - θ ∆11dθ s θ=
, R
- d d
=
s
d
d
w
1 + R (θ s - θ ∆ )θs - θs θs - θr

26

(2.52)

An analogous hysteretic procedure can be applied to the unsaturated hydraulic conductivity
function K(h). The main branches Kd(h) and Kw(h) of the hysteresis loop are characterized by thesame set of parameters as the corresponding retention curves θd(h) and θw(h), and by the saturated
conductivities Ksd and Ksw according to Eq. (2.30). For drying scanning curves, we obtain from
(2.41)
K (h) = α K K d (h)(2.53)

From knowledge of the reversal point (h∆, K∆) we obtain

αK =

K∆
K ( h∆ )d(2.54)For a wetting scanning curve, we have now) K r' + α K K w ( h )K ( h=(2.55)

where Kr’ is a fictitious parameter. Substituting the reversal point (h∆, K∆) and the saturation point(0, Ks) into (2.40) and solving for αK yields

αK =K∆ - Ks
K ( h∆ ) - K sw
w

(2.56)

The fictitious conductivity parameter Kr’ may be obtained from (2.55) as Kr’ = Ks - αK Ksw. If themain hysteresis loop is not closed at saturation, the hydraulic conductivity at saturation for a wetting
scanning curve is evaluated using equations similar to (2.52), i.e.,
K sd - K ∆11d=
K s K=
, R
- d
s
d
d
w
1 + R ( Ks - K∆ )Ks - Ks Ks

27(2.57)

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 heador the water content within the flow domain, Ω:h( x, y, z , t ) = h0 ( x, y, z )for t = 0

θ ( x, y , z , t ) = θ 0 ( x, y , z )for t = 0(2.58)

where h0 is a prescribed function of x, y and z. The initial condition can also be set (in thegraphical user interface) to be equal to water content at field capacity, which is calculated asfollows [Twarakavi et al., 2009]:

S fc =

θ fc − θ r= n −0.60(2+ log (K ))θs −θr10s(2.59)

where θfc and Sfc are the water content and saturation at field capacity, and θr, θs, n, and Ks are
the soil hydraulic parameters for the van Genuchten [1980] model. Note that the water content at
field capacity corresponds to the hydraulic conductivity of about 0.01 cm/d [Twarakavi et al.,2009]. The initial pressure head at field capacity is calculated from the water content at field
capacity using the van Genuchten [1980] retention curve model.
2.7.1. System-Independent Boundary Conditions
HYDRUS implements three types of conditions to describe system-independent interactions

28

along the boundaries of the flow region. These conditions are specified pressure head (Dirichlettype) boundary conditions of the formh ( x, y , z , t ) = ψ ( x, y , z , t )for ( x, y, z ) ∈ Γ D(2.60)

specified flux (Neumann type) boundary conditions given by-[ K ( K ijA∂h+ K izA )]ni = σ 1 ( x, y, z , t )∂x jfor ( x, y, z ) ∈ Γ N(2.61)for ( x, y, z ) ∈ Γ g(2.62)and specified gradient boundary conditions( K ijA∂h+ K izA )]ni = σ 2 ( x, y, z , t )∂x jwhere ΓD, ΓN, and ΓG indicate Dirichlet, Neumann, and gradient type boundary segments,
respectively; ψ [L], σ1 [LT-1], and σ 2 [-] are prescribed functions of x, z, and t; and ni are thecomponents of the outward unit vector normal to boundary ΓN or ΓG. As pointed out by McCord[1991], the use of the term "Neumann type boundary condition" for the flux boundary is not very
appropriate since this term should hold for a gradient type condition (see also Section 3.2 for solutetransport). However, since the use of the Neumann condition is standard in the hydrologic literature[Neuman, 1972; Neuman et al., 1974], we shall also use this term to indicate flux boundaries
throughout this report. HYDRUS implements the gradient boundary condition only in terms of aunit vertical hydraulic gradient simulating free drainage from a relatively deep soil profile. Thissituation is often observed in field studies of water flow and drainage in the vadose zone [Sisson,
1987; McCord, 1991]. McCord [1991] states that the most pertinent application of (2.62) is its use
as a bottom outflow boundary condition for situations where the water table is situated far below thedomain of interest.

29

2.7.2. System-Dependent Boundary Conditions
2.7.2.1. Atmospheric Boundary Conditions
In addition to the system-independent boundary conditions given by (2.60), (2.61), and (2.
62), HYDRUS considers several different types of system-dependent boundary conditions thatcannot be defined a priori. One of these involves soil-air interfaces, which are exposed toatmospheric conditions. The potential fluid flux across these interfaces is controlled exclusively byexternal conditions. However, the actual flux also depends on the prevailing (transient) soil moisture
conditions. Soil surface boundary conditions may change from prescribed flux to prescribed head
type conditions (and vice-versa). In the absence of surface ponding, the numerical solution of (2.1)
is obtained by limiting the absolute value of the flux such that the following two conditions are
satisfied [Neuman et al., 1974]:| K ( K ijA∂h+ K izA )ni | ≤ E∂x j(2.63)hA ≤ h ≤ hS

(2.64)

andwhere E is the maximum potential rate of infiltration or evaporation under the current atmosphericconditions, h is the pressure head at the soil surface, and hA and hS are, respectively, minimum andmaximum pressure heads allowed under the prevailing soil conditions. The value for hA isdetermined from the equilibrium conditions between soil water and atmospheric water vapor,whereas hS is usually set equal to zero. HYDRUS assumes that any excess water on the soil surface
is immediately removed. When one of the endpoints of (2.64) 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].
2.7.2.2. Seepage Face
The second type of system-dependent boundary condition considered in HYDRUS is aseepage face through which water leaves the saturated part of the flow domain. In this case, thelength of the seepage face is not known a priori. HYDRUS assumes that the pressure head is always
uniformly equal to zero along a seepage face. Additionally, the code assumes that water leaving the30saturated zone across a seepage face is immediately removed by overland flow or some otherremoval process. In Version 2.0, users can specify the pressure head other than zero for activating
the seepage face.
2.7.2.3. Tile Drains
The third class of system-dependent boundary conditions in HYDRUS concerns tile drains.
Similarly, as for the seepage phase, HYDRUS assumes that as long as a drain is located in thesaturated zone, the pressure head along the drain will be equal to zero; the drain then acts as apressure head sink. However, the drain will behave as a nodal sink/source with zero recharge whenlocated in the unsaturated zone. More information can be found in Section 5.3.7.
2.7.2.4. Special Time-Variable Flux/Head Boundary Conditions
In addition to system-dependent boundary conditions available in version 2.x of HYDRUS-2D,
several new options are available in HYDRUS:a) While in earlier versions, all boundary conditions (i.e., fluxes or pressure heads) changed
in abrupt steps, the new version allows boundary pressure heads to change smoothly withtime. Abrupt changes in the pressure heads lead to sudden changes in fluxes, whilesmoothly changing pressure heads provide smoothly changing fluxes. An example ofsuch a boundary condition is the water level in a stream or furrow.
b) While earlier versions only allowed either time-variable pressure heads or time-variable
fluxes on a particular part of the boundary, the new version allows boundary conditions
to change from variable pressure heads to a zero flux and vice-versa. This boundary
condition can be used, for example, for a disc permeameter where the specified headchanges to a zero flux during time periods when the permeameter is re-supplied withwater. The zero flux is initiated by specifying a value larger than 999999.
c) When a time-variable pressure head boundary condition is specified along a boundary,then the specified value is assigned to the lowest nodal point of a particular boundary,while pressure heads at other nodes are adjusted based on the z-coordinate. When thisoption is selected, then nodes with calculated negative pressure heads are not associatedwith a Dirichlet boundary condition, but rather with a zero flux. A fluctuating water levelin a stream or furrow is an example of this type of boundary condition. While positive

31

pressure head values are below the water table, negative values occur above the watertable.
d) This is similar to c) except that an atmospheric boundary condition is assigned to nodes
with negative calculated pressure heads.
e) This is similar to c) except that a seepage face boundary condition is assigned to nodes
with negative calculated pressure heads.
f) When this type of system-dependent boundary condition is selected, then HYDRUS
treats the time-variable flux boundary conditions similarly as atmospheric fluxes. Thismeans that pressure heads have two limiting values, with the maximum pressure headequal to hCritS and the minimum pressure head equal to hCritA.
g) While in earlier versions of the code, the flux across the nonactive part of the seepage
face was always equal to zero, the new version can apply atmospheric boundary
conditions on a nonactive seepage face.
Boundary condition options a) through g) can be used only with the first time-variable head/flux
condition.
2.7.2.5. Snow Boundary Condition
When heat transport is simulated simultaneously with water flow and atmosphericboundary conditions, then snow accumulation on top of the soil surface can be simulated. The
code then assumes that when the air temperature is below -2oC all precipitation is in the form ofsnow. When the air temperature is above +2oC all precipitation is in the form of liquid, while alinear transition is used between the two limiting temperatures (-2,2). The code further assumes
that when the air temperature is above zero, the existing snow layer (if it exists) melts
proportionally to the air temperature.
2.7.2.6. Gradient-Type Boundary Condition
Version 1.xx of HYDRUS implements gradient type boundary condition only as the Free
Drainage boundary condition or the unit gradient boundary condition. However, in many
situations, one needs a non-unit gradient BC. For example, it is difficult to select appropriate
boundary conditions for vertical boundaries for flow in a hill slope where the side-gradient ismore or less parallel with the direction of the slope. We have implemented in Version 2.0 of
HYDRUS an option, using which users can specify a gradient other than one (unit gradient).
This option, i.e., non-unit gradient boundary condition, needs to be selected from the Special

32

Boundary Conditions tab of the Boundary Conditions Options dialog window. Gradients are
positive for flow against a particular axis, i.e., from right to left (in the x-direction) and fromback to front (in the y-direction), and should be used only (or mainly) on sides of the transportdomain. In 3D, gradient BCs can be specified only in one direction (i.e., in either x- or ydirection).
2.7.2.7. Subsurface Drip Characteristic Function
The infiltration rate of water from a subsurface cavity (dripper) is affected by manyfactors, including the pressure in the cavity, its size and geometry, and the hydraulic properties ofthe surrounding soil [Lazarovitch et al., 2005]. When a predetermined discharge of a subsurface
source (e.g., a subsurface emitter) is larger than the soil infiltration capacity, the pressure head in
the source outlet increases and becomes positive. The built-up pressure may significantly reduce
the source discharge rate. HYDRUS implements a special system-dependent boundary condition
for flow from subsurface sources that uses the drip characteristic function:
Q = Q0 ( hin - hs )c(2.65)

where hs [L] is the pressure head at the source-soil interface, often called the back pressure, Q0[L3T-1] ([L2T-1] in 2D) is the nominal discharge (Optimal Flux in the HYDRUS GUI) of thesource for the reference inlet pressure hin (usually being 10 m) and the back pressure equal tozero, Q [L3T-1] ([L2T-1] in 2D) is the source discharge for the actual back pressure hs, and c [-] isan empirical constant (Exponent in the HYDRUS GUI) that reflects the flow characteristics of theemitter. Normally, c = 0.5 corresponds to a turbulent flow emitter and c = 1 to a laminar one. Thisequation states that when hs increases, the pressure difference between the soil and the source
inlet decreases, and the discharge rate correspondingly decreases [Lazarovitch et al., 2005]. This
option is currently available only for two-dimensional and axisymmetrical geometries.
2.7.2.8. Surface Drip Irrigation – Dynamic Evaluation of the Wetted Area
The radius of the wetted area for drip irrigation for transient conditions can be calculated
in Hydrus as follows [Gärdenäs et al., 2005]. The irrigation flux Q is applied to the single
boundary node that represents the dripper with the Neumann (flux) boundary condition. If thepressure head required to accommodate the specified flux Q is larger than zero, the boundary
condition in this particular node is changed into the Dirichlet (head) boundary condition with a33

zero pressure head value, and the actual infiltration flux Qa through this node is calculated. The
excess flux (Q - Qa) is then applied to the neighboring node, again with the specified Neumannboundary condition. This procedure is iteratively repeated until the entire irrigation flux Q is
accounted for, and the radius of the wetted area is obtained. Since the infiltration flux into thedry soil is larger for early times, the wetted area continuously increases as irrigation proceeds.
This option is currently available only for two-dimensional and axisymmetrical geometries.
2.7.2.9. Irrigation Scheduling (Triggered Irrigation)
Irrigation can be triggered by a user-specified Pressure Head (input) reached in theselected Observation Node (input). The irrigation starts after a user-specified Lag Period (input)
at a user-specified Irrigation Rate or Pressure Head (input). The Duration of Irrigation is also
specified (input). This option is currently available only for two-dimensional and axisymmetricalgeometries.
2.7.2.10. Reservoir Boundary Conditions
Version 3 of the two-dimensional module of HYDRUS (2D/3D) offers a new systemdependent boundary condition, further referred to as the Reservoir Boundary Condition[Šimůnek et al., 2018]. This option allows users to consider a reservoir that is external to the
HYDRUS transport domain, while water can be added (injected) to or removed (pumped) fromthe reservoir. Flow into or out of the reservoir through its interface with the subsurface transportdomain depends on the prevailing conditions in the transport domain (e.g., the position of thegroundwater table) and external fluxes. Since mass balances of water and solute in the external
reservoir are constantly being updated (based on all incoming and outgoing fluxes), the boundary
conditions along the transport domain are dynamically adjusted depending upon the water levelin the reservoir.
Reservoir boundary conditions potentially have a large number of applications, such as
estimating dynamically the water level in wells, in furrows during irrigation, and in wetlands.
Another application concerns a relatively new approach to land development known as LowImpact Development (LID), which is a “green” approach to stormwater management that seeks
to mimic the natural hydrology of a site using decentralized microscale control measures(Coffman, 2002). Low-impact development practices range from the use of bioretention cells,
infiltration (dry) wells or trenches, stormwater wetlands, wet ponds, level spreaders, permeable
pavements, swales, green roofs, vegetated filter and buffer strips, sand filters, smaller culverts, to34

water harvesting systems (Brunetti et al., 2016, 2017). We provide later two applications where
the reservoir BC is used to evaluate fluxes into or out of wells, and to simulate a falling headexperiment
The Reservoir BC is implemented in HYDRUS (2D/3D) for three different geometries(Fig. 2.7): a well, a furrow, and a wetland. While the furrow BC is implemented only for twodimensional transport domains, the well and wetland BCs are implemented for both twodimensional planar and axisymmetric transport domains.

Figure 2.7. Three different types of the reservoir boundary condition: a well, a furrow, and a wetland. In
the figure hw is the water level in the reservoir [L], S is the volume of water in the reservoir ([L2] or [L3]for two-dimensional and axisymmetric systems, respectively), Qp is the pumping rate (positive forremoval of water, negative for adding water) ([L2T-1] or [L3T-1] for two-dimensional and axisymmetricsystems, respectively), c is the solute concentration in reservoir water [ML-3], cp is the soluteconcentration in injected water [ML-3], P and E refer to the precipitation and evaporation rates [LT-1], rw
is the radius of the well [L], a is the a half-width of the furrow [L], α is the slope of the furrow side [-],
rmax is the maximum width (radius) of the wetland [L], zmax is the maximum depth of the wetland [L], and b
represents the width of the water table ([L] or [L2] for two-dimensional and axisymmetric systems,
respectively).

a. The Well Reservoir Boundary Condition
The position of the water level in the well is obtained by solving the following massbalance equation:
dVw
dhw
=
π rw2=Qin ( t ) − Q p ( t )
dt
dt

35

(2.66)

which in HYDRUS is implemented in terms of the finite difference discretization:
hwj +1 − hwj= Qin − Q p∆t∆t
hwj +1 =hwj + 2 (Qin − Q p )
π rw

π rw2

(2.67)

where Vw is the volume of water in the well [L3], rw is the well radius [L], hw is the water level inthe well [L], Qin is the water flow rate into the well from the soil profile across the well wall (orits screened part) [L3T-1], Qp is the pumping rate [L3T-1], ∆t is the time step [T], and hwj and hwj +1
are water levels in the well at the previous and current time levels [L], respectively (Fig. 2.7a).
The fluxes Qin and Qp can have negative values, in which case they represent outflow from thewell (infiltration into the soil profile) and the water flow rate entering the well, respectively.
Equation (2.66) can be solved, providing that the initial position of the water level in thewell, hw,init, and the pumping rate are known. The parts of the boundary below and above thewater level in the well are then assigned internally to be (time-variable) pressure head (Dirichlet)
and seepage face boundary conditions, respectively. During execution, HYDRUS calculates
which part of the seepage face boundary is active (with a prescribed zero pressure head) andwhich is inactive (with a prescribed zero flux). HYDRUS also calculates and reports fluxes
separately across these two parts of the boundary along a well, and uses these fluxes in Eq.
(2.66). Note that this option can also be used for 2D geometries. The surface area πrw2 is thenreplaced with the width, rw, of the well bottom.
A similar mass balance equation as for water flow is also used for solute transport:
dVwc
dV
dc
=Vw
+ c w =Qin cin − Q p c p
dt
dt
dt
dc
Vw
= Qin cin − Q p c p − c ( Qin − Q p )dt(2.68)

where c is the solute concentration in the reservoir [ML-3], cin is solute concentration associatedwith mass transfer between the soil profile and water reservoir [ML-3] (cin is equal to c whenwater infiltrates into the soil profile and equal to the solute concentration in the soil profile when36

water exfiltrates into the reservoir), and cp is the solute concentration associated with pumping or
injection (equal to c for pumping or to the concentration of water being injected into thereservoir) [ML-3].
b. The Furrow Reservoir Boundary Condition
Similarly, as for wells, the reservoir BC for furrows within HYDRUS is based on massbalance considerations to determine the position of the water level in the furrow, hw [L]. The
volume of water, S [L2], in the furrow, and its change in time depending upon the inflow andoutflow rates, are described using the equations (Šimůnek et al., 2016b):
hw 
hw2
1
1 
= hw a +S=h(a + b)=hw a + a +22 tan α 2 tan αdS
= Qin ( t ) − Q p ( t ) + ( P − E )bdtdh
h dhw h  dh=  a + w  w = Qin ( t ) − Q p ( t ) + ( P − E )b
a w + wdt
tan α dt
tan α  dt
(2.69)or in its finite difference discretization:
hw  hwj +1 − hwj+=aQin − Q ptan α  ∆t
∆t tan α
hwj +1 =hwj + j( Qin − Qp )hw + a tan α(2.70)

where a is the half-width of the bottom of the furrow [L], b is the half-width of the water level[L], hw is the water level in the furrow [L], Qin the water infiltration rate from the furrow to thesoil profile across the furrow walls [L2T-1], Qp is the pumping rate (positive for removal of water,
negative for adding water) [L2T-1], ∆t is the time step [T], P and E are precipitation andevaporation rates [LT-1], respectively, and hwj and hwj +1 are water levels in the well at theprevious and current time levels [L], respectively (Fig. 2.7b). Similarly, as for the well BC, partsof the boundary below and above the water level in the furrow are then assigned time-variable
pressure head (Dirichlet) and seepage face boundary conditions, respectively.
A similar solute mass balance approach as for wells is also used for the furrow BC

37

(Šimůnek et al., 2016b):d ( Sc )
dt

=S

dc
dS
+c
= Q p ( t ) c p − Qin ( t ) c
dt
dt

dc
S= Q p ( t ) c p − Qin ( t ) c − c Q p ( t ) − Qin ( t ) + ( P − E )b dtdc
SQ p ( t ) ( c p − c ) − c [ ( P − E )b ]=dt

(2.71)

where cp is the solute concentration of irrigation water (fertigation) [ML-3; or dimensionless], andc is the average solute concentration of the furrow water and hence the concentration of theinfiltrating water [ML-3]. Note that we assume that precipitation and evaporation fluxes aredevoid of solutes and that there is an instantaneous and complete mixing of solutes within the
furrow. Precipitation will therefore lead to dilution of solute in the furrow water, whileevaporation will lead to increasing concentrations. The furrow BC module requires as input thesolute concentration in the irrigation water (cp) and the timing of fertigation (i.e., the beginningand end time of fertigation). The module then calculates the solute concentration in the furrow
water, c, which is subsequently used in a Cauchy (concentration flux) boundary condition
together with the local infiltration flux calculated with HYDRUS.
c. The Wetlands Reservoir Boundary Condition
Similarly, as for the well and furrow boundary conditions, users can specify a reservoir
boundary condition along the wetland boundary (Fig. 2.7c). The bottom coordinates of a wetland
are described using the following equations: r 
z = zmax  rmax 

p

1/ p

 z 
r = rmax  zmax (2.72)

where zmax is the depth of a wetland [L], rmax is the radius of a wetland [L], and p is a shapeparameter (Fig. 2.7c). The volume of water in the wetland, Vw ([L2] or [L3] for two- and threedimensional problems, respectively), and the water depth, hw [L], for two-dimensional problems

38

are calculated using:
1/ p

 z 
dV
= rdz
= rmax  dzz max rmax
rmax  p  1/ p +1
V
z1/ p dz==z
1/ p ∫
1/ p 
( zmax )( zmax )  1 + p  1 + p  ( zmax )hw = 
 p  rmax

1/ p


V


(2.73)

p1+ p(2.74)

and for axisymmetric problems using:
 z 
=
dV π=
r dz π ( rmax )  zmax 

2/ p

2

2

dz

 r
  p  2 / p +1
( rmax )=V π=
z 2 / p dz π  max1/ p  z2/ p ∫( zmax ) zmax   2 + p 22

 1  2 + p  z max
hw =   V π  p   rmax  
1/ p

2

(2.75)

p2+ p(2.76)

A similar solute mass balance equation as for the well and furrow reservoir BCs is also used forthe wetlands reservoir BC.
We note that water flow and solute transport can occur in both directions between areservoir and the transport domain depending upon the prescribed initial and boundary
conditions. For example, a reservoir can be initially empty while the groundwater table is present
in the transport domain. Water will then flow from the transport domain into the reservoir, with
HYDRUS dynamically adjusting the boundary conditions at the reservoir walls. The code willassign a time-variable pressure head boundary condition to the reservoir walls that are below thewater level in the reservoir and a seepage face boundary condition to the reservoir walls above
the water level. At the same time, water may be pumped from or added to the reservoir using
other boundary conditions. Alternatively, water may be initially present in or be added to thereservoir, while the transport domain is initially still dry. Water will then flow from the reservoir

39

into the transport domain, in which case the water level in the reservoir and corresponding
boundary conditions will be adjusted dynamically based on the fluxes into and out of thereservoir.
Additionally, the interface between the reservoir and the transport domain is also selectedby the software users. For example, in the case of a well reservoir, users can select that only thebottom or the walls of the well are permeable, or that some part of the well is lined (impermeable
to water) or in contact only with certain soil horizons. Since the initial and boundary conditions
for the reservoir and the transport domain are specified independently, many different
combinations of these various scenarios can be accommodated with the new reservoir boundary
condition module is HYDRUS.
2.8. Water Mass Transfer
The mass transfer rate, Γw, in (2.4) 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(2.77)

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 (2.77) 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 (2.77) is that the water retention properties of the matrix and thefracture domains are identical. For this reason, equation (2.77) 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 (2.77) 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 water40

contents. Coupling (2.77) 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 )(2.78)

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(2.79)

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 ) =

(2.80)The use of (2.79) 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 are41

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 (2.79) may not be identifiable. Hence, rather
than using (2.79) 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 )(2.81)

in which case Ka* can be used as a calibration parameter (this variable is an input parameter to
HYDRUS).

42

3. NONEQUILIBRIUM TRANSPORT OF SOLUTES INVOLVED IN SEQUENTIAL
FIRST-ORDER DECAY REACTIONS
3.1. Governing Transport Equation
We assume that solutes can exist in all three phases (liquid, solid, and gaseous) and that the
decay and production processes can be different in each phase. Interactions between the solid andliquid 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 thatthe solutes are transported by convection and dispersion in the liquid phase, as well as by diffusionin 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

Productsµg,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

γg,1 γw,1 γs,1

γg,2 γw,2 γs,2

Products

Products

µw,2
µs,2
µg,2

C ...

where c, s, and g represent concentrations in the liquid, solid, and gaseous phases, respectively; thesubscripts 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:43

1. Radionuclides [van Genuchten, 1985]

→

238

Pu

→

234

U

c1 s1

c2 s2

230

Th

→

226

Ra

c3 s3

c4 s4

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

g2(NH2)2CO

↑

→

→

NH4+

c1 s1

NO2-

c2 s2

→

c3

NO3-



c4

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

44→

Product

HYDRUS at present considers up to fifteen solutes, which can be either coupled in a unidirectional
chain or may move independently of each other.
The partial differential equations governing nonequilibrium chemical transport of solutes
involved in a sequential first-order decay chain during transient water flow in a variably saturated
rigid porous medium are taken as ∂  ∂qi c1
∂θ c1 ∂ρ s1 ∂av g1
∂ 
w ∂c1
g ∂g1
+
+
=
- Scr ,1  θ Dij ,1 + av Dij ,1
∂t
∂t
∂t
∂xi 
∂x j  ∂xi ∂x j  ∂ xi

(3.1)-( µ w,1 + µ w' ,1 )θ c1 - ( µ s ,1 + µ s' ,1 ) ρ s1 - ( µ g ,1 + µ g' ,1 )av g1 + γ w,1θ + γ s ,1 ρ + γ g ,1 av
 ∂  ∂qi ck∂θ ck ∂ρ sk ∂av g k
∂ 
w ∂ck
g ∂g k++= θ Dij ,k + a vDij ,k
 ∂t
∂t
∂t
∂xi 
∂x j  ∂xi ∂x j  ∂xi-( µ w,1 + 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 −1 a vg k −1 + γ w,k θ + γ s ,k ρ + γ g ,k av - Scr ,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 [ML-3],
phases, respectively; qi is the i-th component of 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’ are similar 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 [ML-3], av is the air content [L3L-3], S is thesink term in the water flow equation (2.1), cr is the concentration of the sink term [ML-3], Dijw is thedispersion coefficient tensor [L2T-1] for the liquid phase, and Dijg is the diffusion coefficient tensor[L2T-1] for the gas phase. As before, the subscripts w, s, and g correspond with the liquid, solid, andgas phases, respectively; while the subscript k represents the kth chain number, and ns is the numberof solutes involved in the chain reaction. The indicial notation used in this report assumes
summations over indices i and j (i,j = 1,2), but not over index k. The nine zero- and first-order rate
constants in (3.1) and (3.2) may be used to represent a variety of reactions or transformations,
including biodegradation, volatilization, and precipitation.

45

HYDRUS assumes nonequilibrium interaction between the solution (c) and adsorbed (s)concentrations, and equilibrium interaction between the solution (c) and gas (g) concentrations of
the solute in the soil system. The adsorption isotherm relating sk and ck is described by a generalized
nonlinear equation of the form

=
sk

β

k s ,k c k k
1 + η k c kβ kk ∈ (1, ns )β −1
∂k s ,kk s ,k ck2 βk ∂η k k s ,k ckβk ln ck ∂ β k∂sk ks ,k β k ck k ∂ck
ckβk
=+
+
2
∂t (1 + ηk ckβk ) 2 ∂t 1 + ηk ckβk ∂t (1 + ηk ckβk ) 2 ∂t
(1 + ηk ckβk ) ∂t

(3.3)

where ks,k [for linear sorption, L3M-1, for Freundlich sorpion L3βM-β], βk [-] and ηk [L3M-1] areempirical coefficients. The Freundlich, Langmuir, and linear adsorption equations are special casesof 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 alinear adsorption isotherm. Solute transport without adsorption is described with ks,k=0. While thecoefficients ks,k, βk, and ηk in equation (3.3) are assumed to be independent of concentration, they
are permitted to change as a function 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 ∈ (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 isabsolute temperature [K].

3.1.1. Chemical Nonequilibrium
The concept of two-site sorption [Selim et al., 1977; van Genuchten and Wagenet, 1989] is
implemented in HYDRUS to permit consideration of nonequilibrium adsorption-desorption
reactions. The two-site sorption concept assumes that the sorption sites can be divided into two
fractions:

46

sk =
ske + skkk ∈ (1, ns )(3.5)

Sorption, ske [MM-1], on one fraction of the sites (the type-1 sites) is assumed to be instantaneous,while sorption, skk [MM-1], on the remaining (type-2) sites is considered to be time-dependent. At
equilibrium, we have for the type-1 (equilibrium) and type-2 (kinetic) sites, respectively=
ske fskk ∈ (1, ns )(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 the sorption
rate for the type-1 equilibrium sites immediately:=skk (1- f ) skk ∈ (1, ns )

(3.7)

∂ske
∂s
=
f k
∂t
∂t

k ∈ (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
β


∂skk
k c k
ωk (1- f ) s ,k k β k - skk  - ( µ s ,k + µ s' ,k ) skk + (1- f )γ s ,k=∂t
1 +η k ckk ∈ (1, ns )(3.9)

where ωk is the first-order rate constant for the kth solute [T-1].
Substituting (3.3) through (3.9) into (3.1) and (3.2), and using the continuity equation
describing the isothermal Darcian flow of water in a variably saturated porous medium, i.e.,∂q∂θ=- i -S∂t∂xi

47(3.10)

leads to the following equation

-θ Rk

∂ck
∂c
∂c 
∂ 
- qi k + θ Dij ,k k  + Fk ck +Gk = 0∂t∂xi ∂xi ∂x j k ∈ (1, ns )(3.11)

in which Dij [L2T-1] is an effective dispersion coefficient tensor given byθ Dij ,k = θ Dijw,k + av Dijg,k k g ,kk ∈ (1, ns )(3.12)

The coefficients Fk and Gk in (3.11) are defined as

Fk (ck ) = -( µ w,k + µ w' ,k )θ - ( µ s ,k + µ s' ,k ) ρ f

ks ,k ckβk −1
βk1+η k ck

- ( µ g ,k + µ g' ,k )av k g ,k +∂k g ,k(1 - f )k s ,k ckβk −1∂θ− g k (ck )+S + k g ,k
- av
- ωk ρ
∂t∂t
1+ηk ckβk(3.13)k ∈ (1, ns )G1 (c1 ) = γ w,1θ + γ s ,1 f ρ + γ g ,1av - Scr ,1 + ω1 ρ s1k
Gk (ck ) = ( µ w' ,k −1θ + µ s' ,k −1 f ρ

k s ,k −1ckβ−k1−1 −1
1+ η k -1 ckβ−k1−1+ µ g' ,k −1av k g ,k −1 )ck −1 ++ µ s' ,k −1 ρ skk−1 + γ w,kθ + γ s ,k f ρ + γ g ,k av - Scr ,k + ωk ρ skk(3.14)k ∈ (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):

 ckβk −1 ∂ k s ,k k s ,kck2 βk −1 ∂ηk k s ,k ln ck ckβk −1 ∂β k =+
g k (ck ) ρ f 

βk
βk 2
(1 + ηk ckβk ) 2 ∂t 
 1 + ηk ck ∂t (1 + ηk ck ) ∂t

The retardation factor Rk [-] in (3.11) is given by48

k ∈ (1,ns ) (3.15)

β −1
av k g ,kρ fks ,k β k ck k1+
Rk (ck ) =+βk 2
θ (1 + ηk ck )θk ∈ (1, ns )(3.16)

3.1.2. 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 physical nonequilibriumtransport. The two-region concept assumes that the liquid phase can be partitioned into mobile(flowing), θm [L3L-3], and immobile (stagnant), θim [L3L-3], regions:θ = θ m + θim(3.17)and that solute exchange between the two liquid regions can be modeled as a first-order process, i.e.,
ks ,k β k ckβ,kim−1  ∂ck ,im
θ
+
ρ
(1f
)
 im
 = ωk (ck - ck ,im ) + γ w,kθim + (1- f ) ργ s ,k (1 + ηk ckβ,kim ) 2  ∂tks ,k ckβ,kim−1 ''- θim ( µ w,k + µ w,k ) + ρ ( µ s ,k + µ s ,k )(1- f )k ∈ (1, ns )c
ηk ckβ,kim  k ,im(3.18)

where cim [ML-3] is the concentration in the immobile region and ωk is the mass transfer coefficientfor the kth solute [T-1].
Substituting (3.17) and (3.18) into (3.1) and (3.2), the latter two equations modified formobile and immobile regions as shown by van Genuchten and Wagenet [1989], leads to equation(3.11) in which θ is replaced with θm and with the coefficients Fk and Gk redefined as follows
Fk (ck ) = -( µ w,k + µ'w, k)θ m - ( µ s ,k + µ ) ρ f
'
s ,k

49

ks ,k ckβk −1
1 +ηk ckβk

- ( µ g ,k + µ g' ,k )av k g ,k - ωk

(3.19)

G1 (c1 ) = γ w,1θ m + γ s ,1 f ρ + γ g ,1av - Scr ,1 + ω1c1.im - ρ fg1 (c1 )Gk (ck ) = [ µ
+[ µ

θ +µ

'
s , k −1

θ +µ's , k −1'
w , k −1 m'
w , k −1 imfρ

ks ,k −1ckβ−k1−1 −1
1 + ηk −1ckβ−k1−1(1- f ) ρ

+ µ g' ,k −1av k g ,k −1 ]ck −1 +−1
ks ,k −1ckβ−k1,−1im

1 + ηk −1ckβ−k1,−1im(3.20)]ck −1.im ++ γ w,kθ m + γ s ,k f ρ + γ g ,k av - Scr ,k + ωk ck .im - ρ fg k (ck )k ∈ (2 ,ns )In order to solve equation (3.11), it is necessary to know the water content θ and thevolumetric flux 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 ofparticular coefficients (i.e., γw , γs , γg, µw, µs, µg, µw’, µs’, µg’, η, ks, kg, f, β, ω) the entire system canbe 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 (3.20) simplifies into a set of equations
describing the transport of mutually independent solutes, i.e., single-ion transport as applicable to:θR ∂qi c
∂c
∂ 
w ∂c
=
+ Fc + G θ Dij∂t ∂xi ∂x j  ∂xi(3.21)

3.1.3. Attachment-Detachment Model
Virus, colloid, and bacteria transport and fate models commonly employ a modified formof the convection-dispersion equation. In this study, we define the mass balance equation forthese applications as: ∂qi c
∂s
∂s
∂s
∂θ c
∂ 
w ∂c
=
+ρ e +ρ 1 +ρ 2
- µ wθ c - µ sρ ( se + s1 + s2 ) θ Dij
∂t
∂t
∂t
∂t
∂xi 
∂x j  ∂xi(3.22)

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.

50

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):∂s=ρθ kaψ c − kd ρ s∂t(3.23)

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 (3.23) have been found to strongly depend uponwater 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
phenomenon:

ψ=smax − s
s
= 1−
smax
smax

(3.24)

in which smax is the maximum solid-phase concentration [NcM-1]. Blocking coefficients listed
below are implemented only into a two-dimensional formulation. 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)(3.25)

Johnson and Elimelech [1995] proposed the so-called random sequential adsorption
model to describe the blocking of the sorption sites:51ψ =1 − 4a + 3.308a 2 + 1.4069a 3=ψb=(1 − bs )fors < 0.8smax

fors > 0.8smax

3

2d c 2 b3(3.26)

1smax

a = 0.546

ssmax

Finally, Bradford et al. [2003] hypothesized that the influence of straining andattachment processes on colloid retention could be separated into two distinct components. Theysuggested the following depth-dependent blocking coefficient for the straining process: d + z − z0 ψ = c

dc



−β

(3.27)

where dc is the diameter of the sand grains [L], z0 is the coordinate of the location where thestraining process starts [L] (the surface of the soil profile, or the 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],
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

(3.28)

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 [-]:

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

52(3.29)

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 6γ=(3.30)(1 − θ )1/ 3

The dimensionless Peclet number in (3.29) is calculated as follows:
N Pe =

3π µ d p d c q
kT(3.31)

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, theinterception number, NR, the gravitation number, NG, and the number representing London-van
der Walls attractive forces, NLo, in (3.29) 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

(3.32)(3.33)(3.34)

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].

53

The model described above using equation (3.22) can be used in many different ways.
For example, one can assume that the soil has two sorption sites, s1 and s2, each having its 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 the removal of particles by means of attachment to the air-water interface.

3.2. Initial and Boundary Conditions
The solution of (3.11) requires knowledge of the initial concentration within the flow region,Ω, i.e.,c( x, y, z , 0) = ci ( x, y, z )s k ( x, y, z 0) = sik ( x, y, z )(3.35)cim ( x, y, z 0) = cim ,i ( x, y, z )where ci [ML-3], cim,i [ML-3] and sik [-] are prescribed functions of x and z. The initial condition forsik must be specified only when non-equilibrium adsorption is considered. The subscript k is
dropped in (3.35) and throughout the remainder of this report, thus assuming that the transportrelated equations in the theoretical development and numerical solution apply to each of the solutes
in the decay chain.
Solute transport initial conditions can be alternatively specified in terms of the total
concentration S [ML-3; the mass of solute/volume of soil] instead of the liquid concentration c[ML-3; the mass of solute/volume of water] as in (3.35). The liquid phase concentration is thencalculated for linear sorption as follows:

S = θ c + ρ s + av g = θ c + ρ k s c + av k g c = c(θ + ρ k s + av k g )c=(3.36)

S

θ + ρ k s + av k g

and for nonlinear sorption by finding a root of the following nonlinear equation:54

S = θ c + ρ s + av g = θ c +ρ ks c β+ av k g c
1 + ηc β(3.37)For the two kinetic sorption sites model (only a linear case is implemented, i.e., without
blocking) the distribution coefficient ks for (3.35) is assumed to be defined as:
ks =

θ ka
ρ kd

(3.38)

Rather than specifying concentrations directly in the nonequilibrium phases in (3.35), thenonequilibrium phase concentration can be specified to be initially at equilibrium withequilibrium phase concentration. In such a case, the immobile water content concentration will
be set equal to the mobile water content concentration for the dual-porosity models. The sorbed
concentration at the kinetic sorption sites will be set equal to:s= (1 − f )k s c(3.39)

ρ ks c β
1 + ηc β(3.40)

for the linear sorption and:s= (1 − f )for the nonlinear sorption. For the two kinetic sorption sites model, the sorbed concentrations are
set equal to:
s=

θ ka
c
ρ kd

(3.41)

where ka and kd are the attachment and detachment coefficients, respectively.
Two types of boundary conditions (Dirichlet and Cauchy type conditions) can be specified

55

along the boundary of Ω. First-type (or Dirichlet type) boundary conditions prescribe theconcentration along a boundary segment ΓD:c( x, y, z , t ) = c0 ( x, y, z , t )for ( x, y, z ) ∈ Γ D(3.42)

whereas third-type (Cauchy type) boundary conditions may be used to prescribe the concentration
flux along a boundary segment ΓC as follows:-θ Dij∂cni + qi ni c = qi ni c0∂x jfor ( x, y, z ) ∈ Γ C(3.43)

in which qi ni represents the outward fluid flux, ni is the outward unit normal vector, and c0 is theconcentration of the incoming fluid [ML-3]. In some cases, for example, when ΓC is an impermeableboundary (qini=0) or when water flow is directed out of the region, (3.43) reduces to a second-type(Neumann type) boundary condition of the form:

θ Dij

∂c
ni = 0∂x jfor ( x, y, z ) ∈ Γ N(3.44)

A different type of 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, which hason the right-hand side an additional term to account for gas diffusion through a stagnant boundary
layer of thickness d [L] on the soil surface. The additional solute flux is proportional to thedifference in gas concentrations above and below this boundary layer [Jury et al., 1983]. This
modified boundary condition has the form-θ DijD∂cni + qi ni c = qi ni c0 + g (k g c - g atm )∂x jdfor ( x, y, z ) ∈ Γ C(3.45)

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]. We note that Jury et al. [1983] assumed
56

gatm to be zero. Similarly, as for (3.43), (3.45) reduces to a second-type (Neumann type) boundary
condition when water flow is zero or directed out of the region:-θ DijDg∂c(k g c - g atm )
ni =
d
∂x j

for ( x, y, z ) ∈ Γ N(3.46)

Equations (3.45) and (3.46) can only be used when the additional gas diffusion flux is positive. Juryet al. [1983] discussed how to estimate the thickness of the boundary layer, d, and recommended
d=0.5 cm as a good average value for a bare surface.

3.3. Effective Dispersion Coefficient
The components of the dispersion tensor in the liquid phase, Dijw, are given by [Bear,1972]

θ Dijw = DT | q | δ ij + ( DL - DT )q j qi|q |+ θ Dwτ wδ ij(3.47)

where Dw is the molecular diffusion coefficient in free water [L2T-1], τw is a tortuosity factor in theliquid phase [-], |q| is the absolute value of the Darcian fluid flux density [LT-1], δij is the Kronecker
delta function (δij=1 if i=j, and δij=0 if i≠j), and DL and DT are the longitudinal and transverse
dispersivities, respectively [L]. After adding the diffusion contribution from the gas phase, theindividual components of the effective dispersion tensor in the soil matrix for three-dimensionaltransport are as follows:

57

q y2
qx2
qz2
θ Dxx = DL
+ DT
+ DT
+ θ Dwτ w + av Dg k gτ g
|q |
|q |
|q |

θ Dyy = DL

q y2
|q |

+ DT

qx2
q2
+ DT z + θ Dwτ w + av Dg k gτ g
|q |
|q |

qxy2
qz2
qz2
θ Dzz = DL
+ DT
+ DT
+ θ Dwτ w + av Dg k gτ g
|q |
|q |
|q |
qx q yθ Dxy = ( DL - DT )|q |qqθ Dxz = ( DL - DT ) x z|q |q y qzθ Dyz = ( DL - DT )|q |(3.48)

where Dg is the molecular diffusion coefficient in the gas phase [L2T-1] and τg is a tortuosity factor
in the gas phase [-]. Terms with y in (3.48) are equal to zero in a two-dimensional formulation.
The tortuosity factors for both phases are evaluated in HYDRUS as a function of the waterand gas contents using the relationship of Millington and Quirk [1961]:

τw =

θ 7/3
θ s2
av7 / 3τg =(3.49)

θ s2

Alternative relationships are available in version 4.05 of HYDRUS-1D and higher todescribe tortuosity coefficients in both phases. Moldrup et al. [2000] suggested the following
formula for calculating the tortuosity factor in the gaseous phase for sieved and repacked soils:

τg =

av1.5

θs

(3.50)

This formulation was found to provide superior predictions of several porosity-based
relationships by Werner et al. [2004]. Similarly, Moldrup et al. [1997] suggested an alternative
relationship for calculating the tortuosity coefficient in the liquid phase:

58

θ 
τ w = 0.66   θs 

8/3(3.51)The Millington-Quirk [1961] tortuosity models are expected to perform well for sands (since
they were derived assuming randomly distributed particles of equal size) while Modrup’s
tortuosity models are expected to perform better across soil types.

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-order degradation(µw’, µs’, µg’, µw, µs, and µg), and adsorption (ks, kg, β, η, ω) coefficients may be strongly
dependent upon temperature. HYDRUS assumes that this dependency can be expressed by the
Arrhenius equation [Stumm and Morgan, 1981]. After some modification, this equation can be
expressed in the general form
 E (T A - T A) 
aT = ar exp  a A Ar 
 R uT T r (3.52)

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, and Ea[ML2T-2M-1] is the activation energy of the particular reaction or process being modeled.
The water content dependence of degradation coefficients is implemented using amodified equation of Walker [1974]:  θ B µ (θ ) = µr (θ r ) min 1,     θ r  

(3.53)

where µr is the values of a particular coefficient (rate constant) at the reference water content θr,µ is the value at the actual water content θ, and B is a solute dependent parameter (usually 0.7).

59

The reference water content, θr, which may be different for different soil layers, is calculated
from the reference pressure head, hr, which is considered to be constant for a particularcompound.

3.5. Root Solute Uptake
The root solute uptake models implemented in HYDRUS-2D was developed by Šimůnek
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:r=pa ( x, t ) + aa ( x, t )a ( x, t )R=
Pa (t ) + Aa (t )a (t )(3.54)(3.55)

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), orpa ( x, t ) = s* ( x, t ) min [ c( x, t ), cmax ](3.56)

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 dissolved60

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 canbe 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 )p ( x, t )dx ∫ s ( x, t ) min [ c( x=, t ), c ] dx
∫=
*

a

LR

=

max

LR

Tp (t )

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

(3.57)

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(3.58)

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-3T1], 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:61

a p ( x, t ) = b( x, t ) Lt Ap (t )(3.59)

Using Michaelis-Menten kinetics (e.g., Jungk [1991]) provides for actual distributed values ofactive 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 )(3.60)

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, the total active uncompensated root nutrient uptake rate, Aa [ML-2T-1], is calculated
by integrating 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:=Aa (t )c ( x, t )a ( x, t )dx A (t ) ∫b( x, t )dx∫=
K + c ( x, t )
a

p

LR

LR

(3.61)

m

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. (3.57).
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 )

(3.62)

After substitution of the active total root nutrient uptake rate value from Eq. (3.61) above, this
newly defined nutrient stress index (π) is equal to:62c ( x, t )b( x, t )dx
+
K
c
x
t
(
,
)
m
LR

π (t ) = ∫(3.63)

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. (3.60), or:aac ( x, t ) =Ap (t )c ( x, t )b ( x, t )K m + c ( x, t )max [π (t ), π c ](3.64)

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 )c

LR(3.65)m

Equation (3.57) implies that a 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:
T (t )=Ap (t ) max  R p (t ) ac − Pa (t ),0 
Tp (t )
63

(3.66)

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. (3.60). 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.

644. HEAT TRANSPORT
4.1. Governing Heat Transport Equations
Neglecting the effects of water vapor diffusion, two- and three-dimensional heat transport
can be described as [Sophocleous, 1979]:

C (θ )

∂T
∂ 
∂T 
∂T
=
 λij (θ ) - Cw qi∂t ∂xi ∂x j 
∂xi(4.1)

where λij(θ) is the apparent thermal conductivity of the soil [MLT-3K-1] (e.g., Wm-1K-1), and C(θ)
and Cw are the volumetric heat capacities [ML-1T-2K-1] (e.g., Jm-3K-1) of the porous medium and theliquid phase, respectively. Volumetric heat capacity is defined as the product of the bulk density andgravimetric heat capacity. The first term on the right-hand side of (4.1) represents heat flow due to
conduction, and the second term accounts for heat being transported by flowing water. We do notconsider the transfer of latent heat by vapor movement.According to de Vries [1963] the volumetricheat capacity can be expressed as

C p (θ ) = Cnθ n + Coθ o + Cwθ + Cg av ≈ (1.92θ n + 2.51θ o + 4.18θ )106

( J m -3 o C-1 )(4.2)

where θ refers to a volumetric fraction [L3L-3], and subscripts n, o, g, w represent solid phase,organic matter, gas phase, and liquid phase, respectively.

4.2. Apparent Thermal Conductivity Coefficient
The apparent thermal conductivity, λij(θ), combines the thermal conductivity λ0(θ) of theporous medium (solid plus water) in the absence of flow, and the macrodispersivity, which isassumed to be a linear function of the velocity [de Marsily, 1986]. In analogy with the dispersion
coefficient for solute transport, the apparent thermal conductivity λij(θ) is given by [Šimůnek andSuarez, 1993b]

65

λij (θ ) = λT Cw | q | δ ij + (λL - λT )cw

q j qi|q |+ λ0 (θ )δ ij(4.3)

where |q| is the absolute value of the Darcian fluid flux density [LT-1], δij is the Kronecker delta
function as before, and λL and λT are the longitudinal and transverse thermal dispersivities [L],respectively. The individual components of the thermal conductivity tensor for three-dimensionaltransport are as follows:
x y2
xx2
xz2
+ λT Cw+ λT Cw+ λ0
λxx = λL Cw
|q |
|q |
|q |

λ yy = λL Cw

x y2
|q |

+ λT Cw

xx2x2+ λT Cw z + λ0
|q |
|q |

x y2
xz2
xz2
+ λT Cw+ λT Cw+ λ0
λzz = λL Cw
|q |
|q |
|q |
qx q yλ xy = (λL - λT )Cw
|q |
qx qz
λ xz = (λL - λT )Cw|q |q y qz
λ yz = (λL - λT )Cw
|q |

(4.4)

The volumetric heat capacity of the liquid phase is included here in the definition of the thermal
conductivity in order to have the dimensions of the thermal dispersivities in the length units [deMarsily, 1986]. The thermal conductivity, λ0(θ), accounts for the tortuosity of the porous medium
and is described with the simple equation [Chung and Horton, 1987]

λ0 (θ ) = b1 +b2θ w +b3θ w0.5
where b1, b2 and b3 are empirical parameters [MLT-3K-1] (e.g. Wm-1K-1).

66

(4.5)

4.3. Initial and Boundary Conditions
Equation (4.1) will be solved subject to the general initial condition
T ( x, y, z , 0) = Ti ( x, y, z )(4.6)

where Ti is a prescribed function of x and z.
Two types of boundary conditions (Dirichlet and Cauchy type conditions) can again be
specified along the boundary of Ω. First-type (or Dirichlet type) boundary conditions prescribe thetemperature along a boundary segment ΓD:
for ( x, y, z ) ∈ Γ D

T ( x, y, z , t ) = T0 ( x, y, z , t )(4.7)

whereas third-type (Cauchy type) boundary conditions prescribe the heat flux along a boundary
segment ΓC as follows

-λij

∂T
ni +TCw qi ni = T0Cw qi ni∂x jfor ( x, y, z ) ∈ Γ C(4.8)

in which qi ni represents the outward fluid flux, ni is the outward unit normal vector, and T0 is thetemperature of the incoming fluid. When ΓC is an impermeable boundary (qini=0) or when waterflow is directed out of the region, (4.8) reduces to a second-type (Neumann type) boundary
condition of the form:λij∂Tni = 0∂x jfor ( x, y, z ) ∈ Γ N(4.9)

The atmospheric boundary condition for soil temperature is assumed to be given by a sinefunction as follows [Kirkham and Powers, 1972]:67 2π t * 7π 
T0 = T + A sin 

 t
12 
p


(4.10)

where tp is the 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 period tp, A is the amplitude of thesine wave [K], and t* is the local time [T] within the period tp. The second term within the argumentof the sine function is included to allow the highest temperature to occur at 1 p.m.

68

5. NUMERICAL SOLUTION OF THE WATER FLOW EQUATIONThe Galerkin finite element method with linear basis functions is used to obtain a solution of
the flow equation (2.1) subject to the imposed initial and boundary conditions. Since the Galerkinmethod is relatively standard and has been covered in detail elsewhere [Neuman, 1975; Zienkiewicz,
1977; Pinder and Gray, 1977], only the most pertinent steps in the solution process are given here.

5.1. Space Discretization
The flow region is divided into a network of triangular (2D) or tetrahedral (3D) elements.
The corners of these elements are taken to be the nodal points. The dependent variable, the pressure
head function h(x,y,z,t), is approximated by a function h’(x,y,z,t) as followsNh’ ( x, y, z , t ) = ∑φ n( x, y, z )hn (t )

(5.1)

n=1

where φn are piecewise linear basis functions satisfying the condition φn(xm ,zm)=δnm, hn are unknown
coefficients representing the solution of (2.1) at the nodal points, and N is the total number of nodal
points.
The Galerkin method postulates that the differential operator associated with the Richards'
equation (2.1) is orthogonal to each of the N basis functions, i.e., ∂θ ∂  


A ∂h
A
+
+
K
K
K
S





 φn d Ω =0ijiz∫Ω ∂t ∂xi   ∂x j




 

Applying Green's first identity to (5.2), and replacing h by hΝ, leads to

69

(5.2)

 ∂θ

A ∂h ' ∂φnφ+KK dΩ =∑e ∫  ∂t n
ij
∂x j ∂xi 
Ωe 




∂φ
∂h '
∑e ∫ K  KijA ∂x + KizA  niφn d Γ + ∑e ∫  -KKizA ∂xn - Sφn  d Ω
j
i

Γe
Ωe 



(5.3)

where Ωe represents the domain occupied by element e, and Γe is a boundary segment of element e.
Natural flux-type (Neumann) and gradient type boundary conditions can be immediately
incorporated into the numerical scheme by specifying the line integral in equation (5.3).
After imposing additional simplifying assumptions to be discussed later, and performing
integration over the elements, the procedure leads to a system of time-dependent ordinary
differential equations with nonlinear coefficients. In matrix form, these equations are given by[F ]d {θ }+ [ A]{h} ={Q} -{B} -{D}
dt

(5.4)

where
l

e

=∑
e

=

κ
4 Ae

∂φn ∂φ m

K ∫φ
dΩ
∑ K=
∂x ∂x

Anm
=

A
ij

l

Ωe

i

j

K  K xxA bm bn + K xzA (cm bn + bm cn ) + K zzA cm cn Kfor 2D(5.5)∑ 36V [ K b b + K c c + K d d + K ( b c + c b ) +eA
xx m nAyy m n

A
zz

n

m

A
xy

m n

+ K xzA ( bn d m + d n bm ) + K yzA ( cm d n + d m cn )]Bn = ∑K l K izA ∫ φl
e

m n

e

Ωe

∂φn
κ
d Ω =∑ K ( K xzA bn + K zzA cn )
∂xi
e 2
K
= ∑ ( K xzA bn + K yzA cn + K zzA d n )e 6

70

for 3Dfor 2D(5.6)
for 3D=Fnm δ nm ∑=∫ φn d Ω δ nm ∑
e

Ωe

e

κ
3

Ae

for 2D(5.7)

V= δ nm ∑ e
4efor 3D

Qn = −∑ σ 1l ∫ φl φn d Γ = −∑ σ 1n Ln
Γe

e

κ

=
Dn

(5.8)

e

∑ S=
∫ φ φ d Ω ∑ 12 A (3S + S )for 2DV= ∑ e ( 4S + Sn )e 20

for 3D

l

e

l n

e

Ωe

n

e

(5.9)

where the overlined variables represent average values over an element e, the subscripts i and j arespace direction indices (i,j = 1,2(,3)). For a two-dimensional problem:

=
l 1,=
m 1,=
n 1, 2,..., N
2,..., N
2,..., NAe =ck b j - c j bk
2

bi = z j - zk

ci = xk - x j

b j = zk - zi

c j = xi - xk

bk = zi - z j

ck = x j - xiK=

Ki + K j + K k
3(5.10)

S=

Si + S j + S K
3For tetrahedral element vertexed by nodes 1, 2, 3, and 4 of a three-dimensional problem:71b1 = ( y4 − y2 )( z3 − z2 ) − ( y3 − y2 )( z4 − z2 )b2 = ( y4 − y3 )( z1 − z3 ) − ( y1 − y3 )( z4 − z3 )b3 = ( y2 − y4 )( z1 − z4 ) − ( y1 − y4 )( z2 − z4 )b4 = ( y2 − y1 )( z3 − z1 ) − ( y3 − y1 )( z2 − z1 )(5.11)
c1 = ( x3 − x2 )( z4 − z2 ) − ( x4 − x2 )( z3 − z2 )c2 = ( x1 − x3 )( z4 − z3 ) − ( x4 − x3 )( z1 − z3 )c3 = ( x1 − x4 )( z2 − z4 ) − ( x2 − x4 )( z1 − z4 )c4 = ( x3 − x1 )( z2 − z1 ) − ( x2 − x1 )( z3 − z1 )d1 = ( x4 − x2 )( y3 − y2 ) − ( x3 − x2 )( y4 − y2 )d 2 = ( x4 − x3 )( y1 − y3 ) − ( x1 − x3 )( y4 − y3 )d3 = ( x2 − x4 )( y1 − y4 ) − ( x1 − x4 )( y2 − y4 )d 4 = ( x2 − x1 )( y3 − y1 ) − ( x3 − x1 )( y2 − y1 )Ve = ( x4 − x1 ) b4 + ( y4 − y1 ) c4 + ( z4 − z1 ) d 4=
K

K1 + K 2 + K 3 + K 4
S1 + S 2 + S3 + S 4=
S4
4(5.12)

Equation (5.8) is valid for a flux-type boundary condition. For a gradient-type boundary condition,
the variable σ1 in (5.8) must be replaced by the product of the hydraulic conductivity K and theprescribed gradient σ2 (=1). Equations (5.5) through (5.9) hold for flow in a three-dimensional
Cartesian (x,y,z) domain, a two-dimensional Cartesian (x,z) domain, as well as for flow in an
axisymmetric (x,z) system in which x is used as the radial coordinate. For two-dimensional planeflow, we have

κ =1

λn =

Ln
2

while for axisymmetric flow (using two-dimensional formulation)

72(5.13)

κ = 2π

xi + x j + xk

λn = Lnπ

3x’ n + 2 xn
3(5.14)The subscripts i, j, and k in equations (5.10) and (5.14) represent the three corners of a triangularelement e. Ae is the area of a two-dimensional element e, Ve is the volume of a three-dimensional
element e, K and S are the average hydraulic conductivity and root water extraction values overelement e, Ln is the length of the boundary segment connected to node n, and x’n is the x-coordinateof a boundary node adjacent to node n. The symbol σn in equation (5.8) stands for the flux [LT-1]
across the boundary in the vicinity of boundary node n (positive when directed outward of thesystem). The boundary flux is assumed to be uniform over each boundary segment. The entries ofthe vector Qn are zero at all internal nodes, which do not act as sources or sinks for water.
The numerical procedure leading to (5.4) incorporates two important assumptions inaddition to those related to the Galerkin finite element approach. One assumption concerns the time
derivatives of the nodal values of the water content in (5.4). These time derivatives were weighted
according to
∂θ

dθ n
=
dt

∑ ∫ ∂t φ d Ωne Ωe∑ ∫ φn d Ω(5.15)e ΩeThis assumption implements mass-lumping, which has been shown to improve the rate ofconvergence of the iterative solution process [e.g., Neuman, 1973].
A second assumption in the numerical scheme is related to the anisotropy tensor KA, which
is taken to be constant over each element. By contrast, the water content θ, the hydraulicconductivity K, the soil water capacity C, and the root water extraction rate S, at a given point intime are assumed to vary linearly over each element, e. For example, the water content is expanded
over each element as follows:73=θ ( x, z )3∑ θ ( x , z ) φ ( x, z )n =1=
θ ( x, y , z )

n

n

n

4

∑ θ ( x , y , z ) φ ( x, y , z )
n =1

n

n

n

n

for ( x, z ) ∈ Ωefor ( x, y, z ) ∈ Ωe

(5.16)

where n stands for the corners of element e. The advantage of linear interpolation is that nonumerical integration is needed to evaluate the coefficients in (5.4).

5.2. Time Discretization
Integration of (5.4) in time is achieved by discretizing the time domain into a sequence offinite intervals and replacing the time derivatives by finite differences. An implicit (backward) finite
difference scheme is used for both saturated and unsaturated conditions:[F ]{θ } j +1 − {θ } j∆t j+ [ A] j +1 {h} j +1 ={Q} j − { B} j +1 − { D} j

(5.17)

where j+1 denotes the current time level at which the solution is being considered, j refers to theprevious time level, and ∆tj=tj+1-tj. Equation (5.17) represents the final set of algebraic equations tobe solved. Since the coefficients θ, A, B, D, and Q (Q for only gradient-type boundary conditions)are functions of h, the set of equations is generally highly nonlinear. Note that the vectors D and Q
are evaluated at the old-time level.

5.3. Numerical Solution Strategy
5.3.1. Iterative Process
Because of the nonlinear nature of (5.17), 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 (5.17), which, after incorporation of theboundary conditions, is solved using either Gaussian elimination or the conjugate gradient method74

(see Section 8.5). The Gaussian elimination process takes advantage of the banded and symmetricfeatures of the coefficient matrices in (5.17). After inversion, the coefficients in (5.17) are reevaluated using the first solution, and the new equations are again solved. The iterative processcontinues until a satisfactory degree of convergence is obtained, i.e., until at all nodes in thesaturated (or unsaturated) region, the absolute change in pressure head (or water content) between
two successive iterations becomes less than some small value determined by the imposed absolutepressure head (or water content) tolerance [Šimůnek and van Genuchten, 1994]. The first estimate(at zero iteration) of the unknown pressure heads at each time step is obtained by extrapolation fromthe pressure head values at the previous two-time levels.

5.3.2. Treatment of the Water Capacity Term
The iteration process is extremely sensitive to the method used for evaluating the watercontent term (∆θ/∆t) in equation (5.17). The present version of the HYDRUS code uses the "massconservative" method proposed by Celia et al. [1990]. Their method has been shown to provide
excellent results in terms of minimizing the mass balance error. The mass-conservative method
proceeds by separating the water content term into two parts:{θ } j +1 − {θ } j{θ } j +1 − {θ } j +1
k +1

=
[F ]
∆t j

[F ]

k

∆t j

{θ } j +1 − {θ } jk+ [F ]∆t j(5.18)

where k+1 and k denote the current and previous iteration levels, respectively; and j+1 and j thecurrent and previous time levels, respectively. Notice that the second term on the right-hand side of(5.18) is known prior to the current iteration. The first term on the right-hand side can be expressedin terms of the pressure head so that (5.18) becomes{θ } j +1 − {θ } j

=
[F ]
∆t j

{h} j +1 − {h} j +1
k +1

[ F ][C ] j +1

∆t j

k

{θ } j +1 − {θ } jk+ [F ]∆t j(5.19)

where Cnm=δnmCn, in which Cn represents the nodal value of the soil water capacity. The first term75

on the right-hand side of (5.19) should vanish at the end of the iteration process if the numericalsolution converges. This particular feature guarantees relatively small mass balance errors in thesolution.

5.3.3. Time Control
Three different time discretizations are introduced in HYDRUS: (1) time discretizations
associated with the numerical solution, (2) time discretizations associated with the implementationof boundary conditions, and (3) time discretizations which provide printed output of the simulation
results (e.g., nodal values of dependent variables, water and solute mass balance components, andother 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 time increment, Δt.
This time increment is automatically adjusted at each time level according to the following rules[Mls, 1982; Vogel, 1987]:
a. Discretization 1 must coincide with time values resulting from discretizations 2 and 3.
b. Time increments cannot become less than a preselected minimum time step, ∆tmin, nor
exceed a maximum time step, Δtmax (i.e., ∆tmin≤Δt ≤∆tmax).
c. If, during a particular time step, the number of iterations necessary to reach convergenceis ≤3, the time increment for the next time step is increased by multiplying ∆t by apredetermined constant >1 (usually between 1.1 and 1.5). If the number of iterations is≥7, Δt for the next time level is multiplied by a constant <1 (usually between 0.3 and0.9).
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 ∆t/3, and theiterative process restarted.
The selection of optimal time steps, ∆t, is also influenced by the solution scheme for solute transport(see Section 6.4.6).

76

5.3.4. Treatment of Pressure Head Boundary Conditions
Finite element equations corresponding to Dirichlet nodes where the pressure head isprescribed can, at least in principle, be eliminated from the global matrix equation. An alternativeand numerically simpler approach is to replace the Dirichlet finite element equations by dummy
expressions of the form [Neuman, 1974]δ nm hm = ψ n(5.20)

where δnm is the Kronecker delta, and ψn is the prescribed value of the pressure head at node n. The
values of hn in all other equations are set equal to ψn, and the appropriate entries containing ψn in theleft-hand side matrix are incorporated into the known vector on the right-hand side of the global
matrix equation. When done properly, this rearrangement will preserve symmetry in the matrix
equation. This procedure is only applied when Gaussian elimination is used to solve the matrix
equations. When the conjugate gradient solver is used, then the finite element equation representing
the Dirichlet node is modified as follows. The right-hand side of this equation is set equal to theprescribed pressure head multiplied by a large number (1030), and entry on the left-hand side
representing the Dirichlet node is set equal to this large number. After solving for all pressure heads,
the value of the flux Qn can be calculated explicitly and accurately from the original finite element
equation associated with node n [e.g., Lynch, 1984].

5.3.5. Flux and Gradient Boundary Conditions
The values of the fluxes Qn at nodal points along prescribed flux and gradient boundaries arecomputed according to equation (5.8). Internal nodes that act as Neumann type sources or sinks
have values of Qn equal to the imposed fluid injection or extraction rate.

5.3.6. Atmospheric Boundary Conditions and Seepage Faces
Atmospheric boundaries are simulated by applying either prescribed head or prescribed flux
boundary conditions depending upon whether equation (2.63) or (2.64) is satisfied [Neuman, 1974].

77

If (2.64) is not satisfied, 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 (2.47), the node will beassigned a flux equal to the potential value and treated again as a prescribed flux boundary.
All nodes expected to be part of a seepage face during code execution must be identified apriori. During each iteration, the saturated part of a potential seepage face is treated as a prescribedpressure head boundary with h=0, while the unsaturated part is treated as a prescribed flux boundarywith Q=0. The lengths of the two surface segments are continually adjusted [Neuman, 1974] during
the iterative process until the calculated values of Q (equation (5.8)) along the saturated part, and thecalculated values of h along the unsaturated part, are all negative, thus indicating that water isleaving the flow region through the saturated part of the surface boundary only.

5.3.7. Tile Drains as Boundary Conditions
The representation of tile drains as boundary conditions is based on studies by Vimoke et al.
[1963] and Fipps et al. [1986]. The approach uses results of electric analog experiments conductedby Vimoke and Taylor [1962], who reasoned that drains could be represented by nodal points in a
regular finite element mesh, provided adjustments are made in the hydraulic conductivity, K, ofneighboring elements. The adjustments should correspond to changes in the electric resistance ofconducting paper as follows
K drain = KCd(5.21)

where Kdrain is the adjusted conductivity [LT-1], and Cd is the correction factor [-]. Cd is determinedfrom the ratio of the effective radius, de [L], of the drain to the side length, D [L], of the square
formed by finite elements surrounding the drain node [Vimoke et al., 1962]:C=dµ0 / ε 0Z 0′≈
Z 0 138log10 ρ d + 6.48 − 2.34 A − 0.48 B − 0.12C(5.22)

where Z0’ is the characteristic impedance of free space (≈376.7 ohms), µ0 is the permeability of freespace, ε0 is the permittivity of free space, and Z0 is the characteristic impedance of a transmission

78

line analog of the drain. The coefficients in (5.22) are given by=ρd

1 + 0.405 ρ d−4
D=Ad
1 − 0.405 ρ d−4

1 + 0.163 ρ d−8
1 + 0.067 ρ d−12=B =C
1 − 0.163 ρ d−8
1 − 0.067 ρ d−12

(5.23)

where de is the effective drain diameter to be calculated from the number and size of small openings
in the drain tube [Mohammad and Skaggs, 1984], and D is the size of the square in the finite
element mesh surrounding the drain having adjusted hydraulic conductivities. The approach above
assumes that the node representing a drain must be surrounded by finite elements (either triangularor quadrilateral), which form a square whose hydraulic conductivities are adjusted according to(5.21). This method of implementing drains by means of a boundary condition gives an efficient,yet relatively accurate, prediction of the hydraulic head in the immediate vicinity of the drain, as
well as of the drain flow rate [Fipps et al., 1986]. More recent studies have shown that thecorrection factor, Cd, could be further reduced by a factor of 2 [Rogers and Fouss, 1989] or 4
[Tseng, 1994, personal communication]. These two studies compared numerical simulations of theflow of ponded water into a tile drain system with an analytical solution given by Kirkham [1949].
Pressure head contours calculated numerically with the original correction factor Cd (5.22), as wellas with the additionally reduced correction factor Cd/4, were compared with the analytical results inŠimůnek et al. [1994].

5.3.8. 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 ([L2] or [L3]), in that subregion, and the rate, O ([L2T-1] or[L3T-1]), of inflow or outflow to or from the subregion. V and O are given by79V = ∑ κ Aee

V = ∑ Veeθi + θ j + θ k

for 2D3
θi + θ j + θ k + θl

(5.24)
for 3D

4

andO=

Vnew − Vold∆t

(5.25)

respectively, where θi, θj θk, and θl are water contents evaluated at the corner nodes of element e,
and where Vnew and Vold are volumes of water in the subregion computed at the current and previous
time levels, respectively. The summation in (5.24) is taken over all elements within the subregion.
The absolute error in the mass balance is calculated as
t

t

0

0 nΓ

ε aw = Vt − V0 + St ∫ Ta dt − ∫ ∑ Qn dt(5.26)

where Vt and V0 are the volumes of water in the flow domain at time t and zero, respectively, as
calculated with (5.24). The third term on the right-hand side represents the cumulative root water
uptake amount, while the fourth term gives the cumulative flux through nodes, nΓ, located along theboundary of the flow domain or at internal source and sink nodes.
The accuracy of the numerical solution is evaluated in terms of the relative error, εrw [%], inthe water mass balance as follows:

ε =
w
r

ε aw
t
t


e
e
max  ∑ Vt − V0 , St ∫ Ta dt + ∫ ∑ Qn dt 
0
0 nΓ
 e


(5.27)

where Vte and V0e are the volumes of water in element e at times t and zero, respectively. Note thatHYDRUS does not relate the absolute error to the volume of water in the flow domain, but insteadto the maximum value of two quantities. The first quantity represents the sum of the absolute80changes in water content over all elements, whereas the second quantity is the sum of the absolutevalues of all fluxes in and out of the flow domain. This criterion is much stricter than the usual
criterion involving the total volume of water in the flow domain. This is because cumulative
boundary fluxes are often much smaller than the volume in the domain, especially at the beginningof the simulation.

5.3.9. 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 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- and z-components of the nodal fluxes for a two-dimensional problem are computed for each noden according to: γ ix hi + γ xj h j + γ kx hkKqx =− n ∑
+ K xzA 
N e en 
2 Ae γ iz hi + γ zj h j + γ kz hk

Kn
A
qz =
K
−
+

∑
zz 
N e en 
2 Ae
=γ nx K xxA bn + K xzA cn=
γ nz K xzA bn + K zzA cn

and x-, y- and z-components for a three-dimensional problem as follows:

81(5.28)

 γ ix hi + γ xj h j + γ kx hk + γ lx hlKn−+ K xzA qx =∑N e en 6Ve γ iy hi + γ jy h j + γ ky hk + γ ly hlKn−+ K yzA qy =∑N e en 6Ve γ iz hi + γ zj h j + γ kz hk + γ lz hlKn−+ K zzA qz =∑N e en 
6Ve(5.29)

γ nx = K xxA bn + K xyA cn + K xzA d n
γ ny = K xyA bn + K yyA cn + K yzA d n
γ nz = K xzA bn + K yzA cn + K zzA d nwhere Ne is the number of sub-elements en adjacent to node n. Einstein's summation convention is
not used in (5.28).

5.3.10. 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 rate is
assumed to vary linearly over each element; this leads to approximation (5.9) for the root water
extraction term Dn in the global matrix equation. The values of the actual root extraction rate Sn in(5.9) are evaluated with (2.16). In order to speed up the calculations, the extraction rates Sn arecalculated at the old-time level and are not updated during the iterative solution process at a given
time step. HYDRUS calculates the total rate of transpiration per unit soil surface length using theequation1∑κ Ae Se
St e

for 2D1
Ta = ∑Ve Se
St e

for 3DTa =(5.30)

in which the summation takes place over all elements within the root zone.

82

5.3.11. Evaluation of the Soil Hydraulic Properties
At the beginning of a numerical simulation, HYDRUS generates for each soil type in theflow domain a table of water contents, hydraulic conductivities, and specific water capacities fromthe specified set of hydraulic parameters. The values of θi, Ki, and Ci in the table are evaluated atprescribed pressure heads hi within a specified interval (ha, hb). The entries in the table are generated
such thathi +1= constanthi(5.31)

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 anargument h falls outside the prescribed interval (ha, hb), the hydraulic characteristics are evaluateddirectly 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 overthe entire range of pressure heads, except when very simple hydraulic models were used. The use of
the interpolation tables can be disabled by specifying both limit values (ha, hb) equal to zero.
When the Brooks and Correy model is used, the internal interpolation tables should be either
disabled or the absolute value of the upper limit should be set larger than 1/α (e.g., when α=0.09, ha >1/0.09=11.1 cm).
5.3.12. Implementation of Hydraulic Conductivity Anisotropy
Since the hydraulic conductivity anisotropy tensor, KA, is assumed to be symmetric, it ispossible to define at any point in the flow domain a local coordinate system for which the tensor KA
is diagonal (i.e., having zeroes everywhere except on the diagonal). The diagonal entries K1A andK2A (and K3A for 3D) of KA are referred to as the principal components of KA.
The HYDRUS code permits one to vary the orientation of the local principal directions fromelement to element. For this purpose, the local coordinate axes are subjected to a rotation such thatthey coincide with the principal directions of the tensor KA. The principal components K1A and K2A

83

(and K3A for 3D), together with the angle ωa (the cosines of angles) between the principal directionof K1A and the x-axis (axes) of the global coordinate system, are specified for each element. Each
locally determined tensor KA is transformed into the global coordinate system at the beginning of
the simulation using the following rules for a two-dimensional problem:
K xxA K1A cos 2 ωa + K 2A sin 2 ωa=
K zzA K1A sin 2 ωa + K 2A cos 2 ωa=(5.32)

K = ( K - K ) sin ωa cos ωa
A
xz

A
2

A
1

and for a three-dimensional problem:
K xxA = K1A a11a11 + K 2A a12 a12 + K 3A a13 a13
K yyA = K1A a12 a12 + K 2A a22 a22 + K 3A a23 a23
K zzA = K1A a13 a13 + K 2A a23 a23 + K 3A a33 a33
K xyA = K1A a11a12 + K 2A a12 a22 + K 3A a13 a23(5.33)

K xzA = K1A a11a13 + K 2A a12 a23 + K 3A a13 a33
K yzA = K1A a12 a13 + K 2A a22 a23 + K 3A a23 a33

where aij represents cosine of angle between the ith principal direction of the tensor KA and the jaxis of the global coordinate system.
5.3.13. Steady-State Analysis
All transient flow problems are solved by time marching until a prescribed time is reached.
The steady-state problem can be solved in the same way, i.e., by time marching until two successive
solutions differ less than some prescribed pressure head tolerance. HYDRUS implements a fasterway of obtaining the steady-state solution without having to go through a large number of time
steps. The steady-state solution for a set of imposed boundary conditions is obtained directly during
one set of iterations at the first time step by equating the time derivative term in the Richards'
equation (2.1) to zero. However, note that this approach is not always successful in finding thesteady-state solution. The rate of success depends on the complexity and nonlinearity of the
problem. The steady-state solution is relatively easy to find for fully saturated problems. For84

unsaturated flow problems, the steady-state solution is usually found only for relatively simple
geometries with less nonlinear soil hydraulic properties (e.g., loam, as opposed to sand or clay).

85

86

6. NUMERICAL SOLUTION OF THE SOLUTE TRANSPORT EQUATIONThe Galerkin finite element method is also used to solve the solute and heat transport
equations (equations (3.11) and (4.1), respectively) 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 (3.11), the numerical solution will be given here only forsolute transport. The solution procedure largely parallels the approach used in Section 5 for the flow
equation.
6.1. Space Discretization
The dependent variable, the concentration function c(x,y,z,t), is approximated by a finite
series c’ (x,y,z,t) of the formNc′ ( x, y, z , t ) = ∑ φn ( x, y, z ) cn ( t )

(6.1)

n =1

where φn are the selected linear basis functions, cn are the unknown time-dependent coefficients
which represent solutions of (5.11) at the finite element nodal points, and, as before, N is the total
number of nodal points. Application of the standard Galerkin method leads to the following set of N
equations


∂c
∂c
∂ 
∂c 
−
−
+
+
+
R
q
D
Fc
G
θ
θ

 φn d Ω =0iij
∫  ∂t ∂xi ∂xi  ∂x j 
Ω




(6.2)

Application of Green's theorem to the second derivatives in (6.2) and substitution of c by c’ resultsin the following system of time-dependent differential equations



∂c′

∂c′



i



∂c′ ∂φn ∑ ∫  −θ R ∂t − q ∂x + Fc′ + G  φ − θ D ∂x ∂x d Ω +
e Ωe



i

n

∂c′
niφn d Γ =0
+ ∑ ∫ θ Dij
∂x j
e Γe
N

87

ij

j

i



(6.3)

or in matrix form:

[Q ]

d {c}
+ [ S ]{c} + { f } =− {Q D }
dt

(6.4)

where
Qnm = δ nm ∑ ( −θ R )l ∫ φl φnφm d Ω = −∑
Ωe

e

κ Ae

e

12

( 3θ R + θ R ) δnnfor 2D

nm

(6.5)

V
=
−∑ e ( 4θ R + θ n Rn ) δ nm
e 20



∂φm

for 3D∂φn ∂φmΩ∑ ( −q ) ∫ φ φ ∂x d Ω − (θ D ) ∫ φ x ∂x d Ω + F ∫ φ φ φ d=

S=
nm

e

i l



l n

Ωe

ij l

i

l

Ωe

l

i

j

l n m

Ωe



κ
κ
κ
= ∑{− b m (3 q x + q xn) − c m (3q z + q zn) + Ae (3F + F n + F m)(1 + δ nm) −
24
24
60
e
−

κ 

b mb nθ D xx + (b mc n + c mb n)θ D xz + c mc nθ D zz  }for 2D4 Ae= ∑ {−
e

(6.6)

bm
c
d
( 4qx + qxn ) − m ( 4q y + q yn ) − m ( 4qz + qzn ) +
120
120
120

Ve
( 4 F + Fn + Fm ) (1 + δ nm ) −120
1[bm bnθ Dxx + cm cnθ Dyy + d m d nθ Dzz + ( bm cn + cm bn ) θ Dxy +
−
36Ve
+

+ ( bm d n + d m bn ) θ Dxz + ( cm d n + d m cn ) θ Dyz ]}

=
fn

κ Ae

∑ G=
∫ φ φ d Ω ∑ 12 ( 3G + G )
l

e

l n

Ωe

n

for 3Dfor 2De

V= ∑ e ( 4G + Gn )
e 20

88

(6.7)
for 3Din which the overlined variables represent average values over a given element e. The notation in
the above equations is similar as in (5.10). The boundary integral in (6.3) represents the dispersive
flux, QnD, across the boundary and will be discussed later in Section 6.3.4.
The derivation of equations (6.5) through (6.7) invoked several important assumptions inaddition to those involved in the Galerkin finite element approach [Huyakorn and Pinder, 1983; van
Genuchten, 1978]. First, the different coefficients under the integral signs (θR, qi, θDij, F, G) wereexpanded linearly over each element, similarly as for the dependent variable, i.e., in terms of theirnodal values and associated basis functions. Second, mass lumping was invoked by redefining thenodal values of the time derivative in (6.4) as weighted averages over the entire flow region:
∂c

dcn
=
dt

∑ ∫ θ R ∂t φ d Ωne Ωe∑ ∫ θ Rφ d Ω

(6.8)

n

e Ωe

6.2. Time Discretization
The Galerkin method is used only for approximating the spatial derivatives while the time
derivatives are discretized by means of finite differences. A first-order approximation of the time
derivatives leads to the following set of algebraic equations:[ Q ] j +ε{c} j +1 − {c} jt+ ε [ S ] j +1 {c} j +1 + (1 − ε ) [ S ] j {c} j + ε { f } j +1 + (1 − ε ){ f } j =0 (6.9)where j and j+1 denote the previous and current time levels, respectively; ∆t is the time increment,and ε is a time weighing factor. The incorporation of the dispersion flux, QnD, into matrix [Q] andvector {f} is discussed in Section 6.3.4. The coefficient matrix [Q]j+ε is evaluated using weighted
averages of the current and previous nodal values of θ and R. Equation (6.9) can be rewritten in the
form:

89

[G ]{c} j +1 = { g}(6.10)

where
=
[G ]

1
[Q ] j +ε + ε [ S ] j +1
∆t

1
=
{ g}
[Q ] j +ε {c} j − (1 − ε ) [ S ] j {c} j − ε { f } j +1 − (1 − ε ){ f } j∆t(6.11)

Higher-order approximations for the time derivative in the transport equation were derived
by van Genuchten [1976, 1978]. The higher-order effects may be incorporated into the transport
equation by introducing time-dependent dispersion corrections as follows
−
D=
Dij −
ij

+
ij

D= Dij +qi q j ∆t6θ 2 Rqi q j ∆t(6.12)

6θ 2 R

where the superscripts + and - indicate evaluation at the old and new time levels, respectively.

6.3. Numerical Solution for Linear Nonequilibrium Solute Transport
The same solution procedure as described in Sections 6.1 and 6.2 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 +∆t - s t= ε [ω (1- f )ks c - ω s k - µ s s k + (1- f )γ ]t +∆t +∆t+ (1- ε )[ω (1- f )ks c - ω s k - µ s s k + (1- f )γ ]t90(6.13)

The new adsorbed concentration for type-2 sorption sites follows directly from (6.13):=s t +∆t s t

2 - ∆t (ω + µ s )2 + ∆t (ω + µ s )

t

+
t +∆t

∆t (1 − f )[(ω k s c)t +∆t

+ (ω k s c) + γ t +∆t + γ t ]t

2 + ∆t (ω + µ s )t +∆t(6.14)

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

t +∆t
*

F

t +∆t
*

G

=F

t +∆t

 ρω∆t (1 - f )ω ks -
 2 + ∆t (ω + µ s ) t+∆t

 t 2 - ∆t (ω + µ s )t∆t (1- f )[(ω ks c)t + γ t +∆t + γ t ] =+ ρω  s
+
G

t +∆t
2 + ∆t (ω + µ s )t +∆t 2 + ∆t (ω + µ s ){′t +∆t }t +∆t(6.15)(6.16)

where F*t+∆t and G*t+∆t are the values of parameters F and G for linear nonequilibrium solutetransport, and Ft+∆t' and Gt+∆t' are the original values of F and G. The above procedure avoids
having to solve two simultaneous equations for linear nonequilibrium transport. Once the transport
equation with the modified F and G parameters is solved using the methods discussed earlier toyield the concentration ct+∆t, equation (6.14) is used to update the adsorbed concentration st+∆t.
For physical nonequilibrium (dual-porosity) transport, equation (3.18), simplified for linear
adsorption, is discretized using finite differences as follows
t +∆t
t
- cim
cim
A=
ε [ω (c - cim ) - Bcim + E ]t +∆t + (1- ε )[ω (c - cim ) - Bcim + E ]t
∆t

where

(6.17)A θim + (1- f ) ρ ks=
B θim µ w + (1- f ) ρ ks µ s=
E θim γ w + (1- f ) ργ s=

91

The new concentration in the immobile region follows directly from (6.13):∆t[(ω cim ) t +∆t + (ω cim ) t + E t +∆t + E t ]2 A - ∆t (ω + B ) t=cc+
2 A + ∆t (ω + B ) t +∆t
2 A + ∆t (ω + B ) t +∆t
t +∆t
im

t
im

(6.18)

Similarly as for the chemical nonequilibrium case, equation (6.18) is incorporated directly into F
and G to obtain following values:

t +∆t
*

F

=F

t +∆t



ω∆tω
-

 2 A + ∆t (ω + B ) t +∆t

 t 2 A − ∆t (ω + B)t∆t[(ω c)t + E t +∆t + E t ] {′t +∆t }+ ω t +∆t cim+
G*t +∆t =Gt +∆t
2 A + ∆t (ω + B)t +∆t 
 2 A + ∆t (ω + B)

(6.19)(6.20)The numerical solution for the attachment/detachment approach to nonequilibrium solute transport
is handled similarly.

6.4. Numerical Solution Strategy
6.4.1. 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 5.3.1). After achieving
convergence, the solution of the transport equation (6.10) is implemented. This is done by first
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 the system of linear algebraic equations given by (6.10). 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 require that the global matrix [G] and the92vector {g} be evaluated at only one-time level (the previous or current time level). All otherschemes require evaluation at both time levels. Also, all schemes except for the explicit
formulation (ε=0) lead to an asymmetric banded matrix [G]. The associated set of algebraic
equations is solved using either a standard asymmetric matrix equation solver [e.g., Neuman,
1972], or the ORTHOMIN method [Mendoza et al., 1991], depending upon the size of the final
matrix. By contrast, the explicit scheme leads to a diagonal matrix [G], which is much easier tosolve (but generally requires smaller time steps).
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 the solid and liquid phases
is considered. The iteration procedure for solute transport is very similar to that for water flow. The
coefficients in (6.10) are re-evaluated using each iteration, and the new equations are again solved
using the results of the previous iteration. The iterative process continues until a satisfactory degreeof convergence is obtained, i.e., until at all nodes, the absolute change in concentration between two
successive iterations becomes less than some small value determined by the imposed relative andabsolute concentration tolerances.

6.4.2. 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 second (flux) term of equation (6.3) is not weighted by regular linear basis
functions φn, but instead using the nonlinear functions φnu [Yeh and Tripathi, 1990]93

φ1u =
L1 − 3α 3w L2 L1 + 3α 2w L3 L1
φ2u =L2 − 3α1w L3 L2 + 3α 3w L1 L2

for 2Dφ =L3 − 3α L L + 3α L2 L3
u
3

w
2 1 3

w
1

(6.21)

φ1u =L1 − 3α12w L2 L1 + 3α14w L4 L1 + 3α13w L1 L3
φ2u =L2 − 3α 23w L3 L2 + 3α12w L1 L2 + 3α 24w L2 L4

for 3D

φ3u =L3 − 3α 34w L4 L3 + 3α 23w L2 L3 − 3α13w L1 L3uφ=
L4 − 3α14w L4 L1 + 3α 34w L3 L4 − 3α 24w L2 L4
4

Figure 6.1. Direction definition for the upstream weighting factors αijw.

where αiw is a weighting factor associated with the length of the element size opposite to node i, αijw
is a weighting factor associated with the line connecting nodes i and j (Figure 6.1), and Li are thelocal coordinates. The weighing factors are evaluated using the equation of Christie et al. [1976]:
 uL  2 D
α iw coth =− 2 D  uL

94

(6.22)

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 (6.3) shows that thefollowing additional terms must be added to the entries of the global matrix Snm in equation (6.6) inthe two-dimensional formulation:=
S1ej' S1ej -=S 2e 'j S 2e j =S3e j' S3e j -

bj
40
cj
40
bj

40
cj
40
bj

[2qx1 (α 2w - α 3w ) + qx 2 (α 2w - 2α 3w ) + qx 3 (2α 2w - α 3w )] -(6.23)[2qz1 (α - α ) + qz 2 (α - 2α ) + qz 3 (2α - α )]
w
2

w
2

w
3

w
2

w
3

[qx1 (2α 3w - α1w ) + 2qx 2 (α 3w - α1w ) + qx 3 (α 3w - 2α1w )] -(6.24)[qz1 (2α - α ) + 2qz 2 (α - α ) + qz 3 (α - 2α )]

40
cj

-

w
3

40

w
3

w
1

w
3

w
1

w
3

w
1

[qx1 (α1w - 2α 2w ) + qx 2 (2α1w - α 2w ) + 2qx 3 (α1w - α 2w )] -(6.25)[qz1 (α - 2α ) + qz 2 (2α - α ) + 2qz 3 (α - α )]
w
1

w
2

w
1

w
2

w
1

w
2

and in the three-dimensional formulation

S1ej′ =S1ej −bj
240[qx1 ( −2α12w + 2α14w + 2α13w ) + qx 2 ( −2α12w + α14w + α13w ) ++ qx 3 ( −α12w + α14w + 2α13w ) + qx 4 ( −α12w + 2α14w + α13w )] −−(6.26)cjd − q y1 (...............) + ..... − j  − qz1 (..............) + ....
240
240

S 2e′j = S 2e j −−w+ α 24w ) + qx 2 ( −2α 23w + 2α12w + 2α 24w ) +[qx1 ( −α 23w + 2α112
240+ qx 3 ( −2α 23w + α12w + α 24w ) + qx 4 ( −α 23w + α12w + 2α 24w )] −

bj

cj

d
 −q y1 (...............) + ..... − j  −qz1 (..............) + ....
240
240

95

(6.27)

S3e′j = S3e j −−S 4e′j = S 4e j −

−

bj
240

[qx1 ( −α 34w + α 23w − 2α13w ) + qx 2 ( −α 34w + 2α 23w − α13w ) ++ qx 3 ( −2α 34w + 2α 23w − 2α13w ) + qx 4 ( −2α 34w + α 23w − α13w )] −(6.28)cjd −q y1 (...............) + ..... − j  −qz1 (..............) + ....
240
240
bj

[qx1 ( −2α14w + α 34w − α 24w ) + qx 2 ( −α14w + α 34w − 2α 24w ) +240+ qx 3 ( −α14w + 2α 34w − α 24w ) + qx 4 ( −2α14w + 2α 34w − 2α 24w )] −(6.29)cjd −q y1 (...............) + ..... − j  −qz1 (..............) + ....
240
240

The weighting factors are applied only to those element sides that are inclined within 10
degrees from the flow direction.

6.4.3. Implementation of First-Type Boundary Conditions
Individual equations in the global matrix equation which correspond to nodes at which theconcentration is prescribed are replaced by new equations:δ nm cm = cn 0(6.30)

where cn0 is the prescribed value of the concentration at node n. This is only done when Gaussianelimination is used to solve the matrix equation. A similar procedure as for water flow (described in
Section 5.3.4) is applied when the ORTHOMIN method is used. Because of the asymmetry of the
global matrix [G], no additional manipulations are needed in the resulting system of equations, as
was the case for the water flow solution.
The total material flux, QnT, through a boundary at node n consists of the dispersive flux,
QnD, and the convective flux, QnA:T
Q=
QnD + QnAn96

(6.31)

The dispersive boundary nodal flux is not known explicitly but must be calculated from equation(6.4). Hence, the dispersion flux, QnD, for node n can be calculated as

j +1
j cmj − ε f n j +1 − (1 − ε ) f n j − Qnnj +ε− ε S nm
+ (1 − ε ) S nm
QnD =cnj +1 − cnj∆t(6.32)

The convective flux is evaluated as
QnA = Qn cn(6.33)

where the fluid flux Qn is known from the solution of the water flow equation.

6.4.4. Implementation of Third-Type Boundary Conditions
Equation (3.43) is rewritten as follows
∂c ′
∂x j

ni qi ni ( c − c0 )
θ Dij =

(6.34)

When substituted into the last term of (6.3), the boundary integral becomes∂c ′dΓ Q c − Q c∑ ∫ θ D ∂x n φ =
ij

e Γe
N

i n

n n

n n0

(6.35)

j

The first term on the right-hand side of (6.35) represents the convective flux. This term isincorporated into the coefficient matrix [S] of (6.4). The last term of (6.35) represents the totalmaterial flux, which is added to the known vector {f}.
At nodes where the free outflow of water and its dissolved solutes occurs, the exit
concentration c0 is equal to the local (nodal) concentration cn. In this case, the dispersive flux

97

becomes zero, and the total material flux through the boundary is evaluated as
QnT = Qn cn(6.36)The Cauchy boundary condition for volatile solutes is treated in a similar way. Equation(3.45) is rewritten as follows
∂c ′
∂x j

θ Dij =
ni qi ni ( c − c0 ) -Dg
d(k g c - g atm )(6.37)

Again, when substituted into the last term of (6.3), the boundary integral becomes∂c ′dΓ Q c − Q c ∑ ∫ θ D ∂x n φ =
ij

e Γe
N

i n

n n0

n n

j

Dg ∆L
d

(k g cn - g atm )(6.38)

where ∆L is the length of the boundary associated with node n. The last term of (6.38) representing
the gas diffusion flux through the stagnant boundary layer at the soil surface is directly added to thevector {f} in equation (6.9). In contrast, the term containing kg and unknown concentration cn isincorporated into the coefficient matrix [S]. The other terms on the right-hand side of (6.38) aretreated in the same way as described above for equation (6.35).

6.4.5. Mass Balance Calculations
The total amount of mass in the entire flow domain, or in a preselected subregion, is given
by

M
=

s)d Ω
∑ ∫ (θ c + av g + ρ=
e Ωe



ks c β −1 
k
a
k
f
c
s
θ
+
+
ρ
+
ρ
d Ω

∑e ∫  v g
β
1
+
η
c


Ω e 

98(6.39)

The summation is taken over all elements within the specified region. The equations in this section
pertain only to the equilibrium or chemical nonequilibrium models; the mass balance equations forphysical nonequilibrium transport are very similar to those for chemical nonequilibrium.
The cumulative amounts M0 and M1 of solute removed from the flow region by zero- andfirst-order reactions, respectively, are calculated as followst=
M 10 -∫ ∑ ∫ (γ w,1θ + γ s ,1 ρ + γ g ,1 av )d Ωdt
0 e Ωe '
ks ,k −1ckβ−k1−1 −1
k
'
'
+
µ
M1 =
a
k
c +
-∫ ∑ ∫ [ µ w,k −1θ + µ s ,k −1 ρ f
g , k −1 v g , k −1  k −1
1 + ηk −1ckβ−k1−1
0 e Ωe + µ s' ,k −1 ρ skk−1 + γ w,k θ + γ s ,k ρ + γ g ,k av ]d Ωdttks c β −1
'
'
{
(
)
(
)
f
µ
µ
θ
µ
µ
ρ
+
+
+
+ ( µ g + µ g' )av k g  c +ssβ∫0 ∑e ∫  w w
1 + ηcΩe(6.40)

t1=
M(6.41)+( µ s + µ ) ρ s }d Ωdt'skwhereas the cumulative amount, Mr, of solute taken up by plant roots is given byt=Mr∫ ∑ ∫ Sc d Ωdt(6.42)s

0 eR Ωewhere eR represents the elements making up the root zone.
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

M t −=M0∫ ∑ Q dt + M + M − M
T
n

0

1

r

(6.43)

0 nΓ

where Mt and M0 are the amounts of solute in the flow region at times t and zero, respectively, as
calculated with (6.39), and nΓ represents nodes located along the boundary of the flow domain or atinternal sinks and/or sources. The difference between the left- and right-hand sides of (6.43)

99

represents the absolute error, εac, in the solute mass balance. Similarly, as for water flow, theaccuracy of the numerical solution for solute transport is evaluated by using the relative error, εrc[%], in the solute mass balance as follows

ε =
c
r

100 ε ac
t


0
1
e
e
max  ∑ M t − M 0 , M + M + M r + ∫ ∑ QnT dt 
0 nΓ
 e


(6.44)

where M0e and Mte are the amounts of solute in element e at times 0 and t, respectively. Note againthat 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 in element
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 andsinks in the flow domain.
The total amount of heat energy in the entire flow domain, or in a preselected subregion, isgiven by=W∑ ∫ (c θ + c θ + C θ + C a )T d Ω
A

n n

o o

w

g

v

(6.45)e Ωewhere TA is the absolute temperature [K]. The summation is taken over all elements within thespecified region.

100

6.4.6. 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 6.4.2. 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, Peie, which
defines the predominant type of the solute transport (notably the ratio of the convective anddispersive transport terms) in relation to the coarseness of the finite element grid:Peie =qi ∆xi
θ Dii(6.46)

where Δxi 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 discretizationmust be kept relatively fine to maintain a low Peclet number. Numerical oscillation can be virtuallyeliminated 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 oscillation for higher Peclet numbers can be effectively eliminated by using upstream
weighing (see Section 6.4.2).
A second dimensionless number that characterizes the relative extent of numericaloscillations is the Courant number, Crie. The Courant number is associated with the time
discretization as follows

Cri e =qi ∆tθ R∆xi(6.47)

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 is101upstream weighing (see Section 6.4.2), 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(6.48)

where ωs is the performance index [-]. This criterion indicates that convection-dominated transport
problems having large Pe numbers can be safely simulated provided Cr is reduced according to(6.48) [Perrochet and Berod, 1993]. When small oscillations in the solution can be tolerated, ωs canbe increased to about 5 or 10.
A third stabilization option implemented in HYDRUS also utilizes criterion (6.48).
However, instead of decreasing Cr to satisfy equation (6.48), this option introduces artificial
dispersion to decrease the Peclet number. The amount of additional longitudinal dispersion, D L [L],is given by [Perrochet and Berod, 1993]
D=
L

q ∆t

θ Rω s− DL −

θ Ddτ
q(6.49)The maximum permitted time step is calculated for all three options, as well as with the additional
requirement that the Courant number must remain less than or equal to 1. The time step calculated
in this way is subsequently used as one of the time discretization rules (rule No. B) discussed in
section 5.3.3.

102

7. PARAMETER OPTIMIZATION
Parameter optimization is an indirect approach for estimating the unsaturated soil hydraulicand/or solute transport parameters from the 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 solution of
the flow equation, augmented with the parameterized hydraulic functions, selected transportparameters, and suitable initial and boundary conditions. Initial estimates of the optimized systemparameters are then iteratively improved during the minimization process until a desired degree ofprecision is obtained. This methodology was originally applied to one-step and multi-step column
outflow data generated in the laboratory [see, for example, Kool et al., 1985; van Dam et al., 1994],
and laboratory or field transport data during steady-state water flow [van Genuchten, 1981; Toride
et al., 1995]. HYDRUS implements parameter optimization also for estimating the solute transportand reaction parameters from transient water flow and/or solute transport experiments.

7.1. 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]:=
Φ (b, q, p)

mq

nq j

∑v j ∑wi, j [ g *j ( x, ti ) - g j ( x, ti , b)]2 +=j 1 =i 1
mp

np j+ ∑v j ∑wij [ p*j (θi ) - p j (θi , b)] 2 +(7.1)=j 1 =i 1nb+ ∑vˆ j[b*j - b j ] 2
j =1

where the first term on the right-hand side represents deviations between the measured andcalculated space-time variables (e.g., observed pressure heads, water contents, and/or concentrations
at different locations and/or time in the flow domain, or the actual or cumulative flux versus time103

across a boundary of a specified type). In this term, mq is the number of different sets ofmeasurements, nqj is the number of measurements in a particular measurement set, qj*(x,ti)
represents specific measurements at time ti for the jth measurement set at location x(r,z), qj(x,ti ,b)
are the corresponding 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 (7.1) represents differences between independently measured andpredicted 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 j and wi , j have similar meanings as for the firstterm but now for the soil hydraulic properties. The last term of (7.1) 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 j representing preassigned weights. Estimates, which make use of prior information (such as those used in the third
term of (7.1)) are known as Bayesian estimates. We note that the covariance (weighting) matrices,which provide information about the measurement accuracy, as well as any possible correlation
between measurement errors and/or parameters, are assumed to be diagonal in this study. The
weighting coefficients vj, which minimize differences in weighting between different data types
because of different absolute values and numbers of data involved, are given by [Clausnitzer andHopmans, 1995]:

νj =1
n j σ 2j

(7.2)

which causes the objective function to become the average weighted squared deviation normalized
by the measurement variances σj2.

7.2. 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 andsteepest descend methods and generates confidence intervals for the optimized parameters. The

104

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., 1985, 1987].

7.3. Statistics of the Inverse Solution
As part of the inverse solution, HYDRUS produces a correlation matrix that specifies thedegree 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 tosimilar changes as a result of changes in the other parameters. The correlation matrix reflects thenonorthogonality between two parameter values. A value of ±1 suggests a perfect linear correlation,
whereas 0 indicates no correlation at all. The correlation matrix may be used to select whichparameters, if any, are best kept constant in the parameter estimation process because of highcorrelation.
An important measure of the goodness of fit is the r2 value for regression of the observed, y^ ,iversus fitted, yi(b), values:2∑ yˆ i ∑ yi  ∑ wi yˆ i yi ∑ wi 2r =2ˆ i) 2  
2 (∑ y2 ( ∑ yi ) 
ˆ
w
∑
∑

  yi
i yi∑ wi  ∑ wi 
(7.3)

The r2 value is a measure of the relative magnitude of the total sum of squares associated with thefitted 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 themean, 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 theequation

105

T=

bj
s (b j )(7.4)

The values for T and s(bj) provide absolute and relative measures of the deviations around the mean.
HYDRUS also specifies the upper and lower bounds of the 95% confidence level around each fitted
parameter bj. It is desirable that the real value of the target parameter always be located in a narrowinterval 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 a particular parameter.
Finally, because of possible problems related to convergence and parameter uniqueness, werecommend 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 is especially
important for field data sets, which often exhibit considerable scatter in the measurements 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 is ultimately the user's
responsibility to select meaningful initial estimates. Parameter optimization is implemented only for
two-dimensional problems.

1068. PROBLEM DEFINITION
8.1. Construction of Finite Element Mesh
The finite element mesh is constructed by dividing the flow region for two-dimensional
problems into quadrilateral and/or triangular elements or for three-dimensional problems into
tetrahedral, hexahedral and/or triangular prismatic elements (Fig. 8.1) whose shapes are defined by
the coordinates of the nodes that form the element corners. The program automatically subdivides
the quadrilaterals into triangles (or hexahedrals and triangular prisms into tetrahedrals), which arethen treated as subelements. Two different ways are possible to subdivide the hexahedrals into
tetrahedrals, whereas six different possibilities exist for subdividing the triangular prisms into
tetrahedrals (see Fig. 8.1).
If two neighboring hexahedral elements are subdivided in the same way (e.g., options 2a or
2b in Fig. 8.1), the newly formed edges on a common surface will cross each other, a feature which
is not allowed. Two neighboring hexahedral elements should, therefore, always use both options 2a
and 2b as shown in Figure 8.1, so that the newly formed edges on the common surface will
coincide. Therefore, it is necessary to give not only the corner nodes, which define an element, butalso the code, which specifies how a particular element is to be subdivided into subelements (this isdone automatically if the graphical user interface is used). It is necessary to always realize how theneighboring elements are going to be subdivided, and to input also the proper code specifying thesubdivision. Having high flexibility in terms of possible subdivisions into subelements is important,especially for unstructured finite element meshes using triangular prisms.
Transverse lines [Neuman, 1974] formed by element boundaries must transect the meshalong the general direction of its shortest dimension. These transverse lines should be continuousand non-intersecting, but need not be straight. The nodes are numbered sequentially from 1 to
NumNP (total number of nodes) by proceeding along each transverse line in the same direction.
Elements are numbered in a similar manner. The maximum number of nodes on any transverse line,
IJ, is used to determine the effective size of the finite element matrix (i.e., its bandwidth). To
minimize memory and time requirements, IJ should be kept as small as possible. The above rules
for defining the finite element mesh only apply when Gaussian elimination is used to solve thematrix equations. Iterative methods (such as the conjugate gradient and ORTHOMIN methods) arenot so restrictive since only non-zero entries in the coefficient matrix are stored in memory, andsince the computational efficiency is less dependent upon the bandwidth of the matrix as compared107

to direct equation solvers.
The finite element dimensions must be adjusted to a particular problem. They should bemade relatively small in directions where large hydraulic gradients are expected. A region withsharp gradients are usually located in the vicinity of the internal sources or sinks, or close to the soil
surface where highly variable meteorological factors can cause fast changes in pressure head.
Hence, we recommend normally using relatively small elements at and near the soil surface. The
size of elements can gradually increase with depth to reflect the generally much slower changes inpressure heads at deeper depths. The element dimensions should also depend upon the soil
hydraulic properties. For example, coarse-textured soils having relatively high n-values and small
α-values (see Eqs. (2.30) and (2.32)) generally require a finer discretization than fine-textured soils.
We also recommend using elements having approximately equal sizes to decrease numerical errors.
For axisymmetric three-dimensional flow systems, the vertical axis must coincide with, or be to theleft of, the left boundary of the mesh. No special restrictions are necessary to facilitate the soil rootzone.
8.2. 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 thecorner nodes of a certain element, the material properties of this element will be averaged
automatically by the finite element algorithm. This procedure will somewhat smooth soil interfaces.
A set of soil hydraulic parameters and solute transport characteristics must be specified foreach soil material. Also, the user must define for each element the principal components of the
conductivity anisotropy tensor, as well as the angle between the local and global coordinate
systems.
As explained in Section 2.3, one additional way of changing the unsaturated soil hydraulicproperties in the flow domain is to introduce scaling factors associated with the water content, thepressure head, and the hydraulic conductivity. The scaling factors are assigned to each nodal point n
in the flow region.
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 are108characterized 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.

Figure 8.1. Finite elements and subelements used to discretize the 3-D domain: 1) tetrahedral, 2)hexahedral, 3) triangular prism.

109

8.3. Coding of Boundary Conditions
Flow boundary conditions were programmed in a fairly similar way as done in the UNSAT1and UNSAT2 models of Neuman [1972] and Neuman et al. [1974], and in the SWMS_2D [Šimůnek
et al., 1992] and CHAIN_2D [Šimůnek and van Genuchten, 1994] codes. A boundary code,
Kode(n), must be assigned to each node, n. If node n is to have a prescribed pressure head during atime step (Dirichlet boundary condition), Kode(n) must be set positive during that time step. If thevolumetric flux of water entering or leaving the system at node n is prescribed during a time step
(Neumann boundary condition), Kode(n) must be negative or zero.
Constant Boundary Conditions - The values of constant boundary conditions for a particularnode, n, are given by the initial values of the pressure head, h(n), in case of Dirichlet boundary
conditions, or by the initial values of the recharge/discharge flux, Q(n), in case of Neumannboundary conditions. Table 8.1 summarizes the use of the variables Kode(n), Q(n), and h(n) for
various types of nodes.

Table 8.1. Initial settings of Kode(n), Q(n), and h(n) for constant boundary conditions.

Node Type

Kode(n)

Q(n)

h(n)

Internal; not sink/source
Internal; sink/source

0
1

0.0
0.0

Initial Value
Prescribed(Dirichlet condition)
Internal; sink/source-1

Prescribed

Initial Value

Impermeable Boundary00.0

Initial Value

Specified Head Boundary

+

1

0.0

PrescribedSpecified Flux Boundary-1*

Prescribed

Initial Value(Neumann condition)+*

5 may also be used
-5 may also be used110

Variable Boundary Conditions - Three types of variable boundary conditions can be
imposed:1. Atmospheric boundary conditions for which Kode(n)=±4,
2. Variable pressure head boundary conditions for which Kode(n)=+3, +7, +8, +9, and3. Variable flux boundary conditions for which Kode(n)=-3, -7, -8, and -9.
The same set of Boundary Condition records can be repeated multiple times.
These conditions can be specified along any part of the boundary. Initial settings of the variables
Kode(n), Q(n), and h(n) for the time-dependent boundary conditions are given in Table 8.2.

Table 8.2. Initial settings of Kode(n), Q(n), and h(n) for variable boundary conditions.

Node Type

Kode(n)

Q(n)

h(n)

Atmospheric Boundary

-4

0.0

Initial Value

Variable Head Boundary+3, +7, +8, +90.0

Initial Value

Variable Flux Boundary-3, -7, -8, -90.0

Initial Value

Atmospheric boundary conditions are implemented when Kode(n)= ±4, in which case timedependent 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 allowedsurface pressure head (defined from equilibrium conditions between soil water and atmosphericvapor). The program automatically switches the value of Kode(n) from -4 to +4 if one of these twolimiting points is reached. Table 8.3 summarizes the use of the variables rAtm, hCritS, and hCritA
during program execution. Width(n) in the table denotes the length of the boundary segment
associated with node n.

111

Table 8.3. Definition of the variables Kode(n), Q(n) and h(n) whenan atmospheric boundary condition is applied.

Kode(n)

Q(n)

h(n)

Event

-4

-Width(n)*rAtm

UnknownrAtm=rSoil-Prec+4

Unknown

hCritA

Evaporation capacity
is exceeded+4

Unknown

hCritS

Infiltration capacity
is exceeded

Variable head and flux boundary conditions along a certain part of the boundary are implemented
when Kode(n)=+3, +7, +8, +9, and -3, -7, -8, and -9, respectively. In that case, the input file
ATMOSPH.IN must contain the prescribed time-dependent values of the pressure head, hti, or theflux, rti, imposed along the boundary. The values of hti or rti are assigned to particular nodes at
specified times according to rules given in Table 8.4. Variable flux boundary conditions with
Kode(n)=+3 are treated similarly as atmospheric boundary conditions, i.e., with limited pressure
head values of hCritA and hCritS.
Table 8.4. Definition of the variables Kode(n), Q(n) and h(n)when the variable head or flux boundary conditions are applied.

Node Type

Kode(n)

Q(n)

h(n)

Variable Head Boundary+3, +7, +8, +9

Unknownhti

Variable Flux Boundary-3, -7, -8, -9-Width(n)*rti

Unknown

Water Uptake by Plant Roots - The program calculates the rate at which plants extract waterfrom the soil root zone by evaluating the term D (equation (5.9)) in the finite element formulation.
The code requires that Kode(n) be set equal to 0 or negative for all nodes in the root zone. Values of112the potential transpiration rate, rRoot, must be specified at preselected times in the input file
ATMOSPH.IN. Actual transpiration rates are calculated internally by the program, as discussed in
Section 2.2. The root uptake parameters are taken from an input file SELECTOR.IN. Values of thefunction Beta(n), which describes the potential water uptake distribution over the root zone(equation (2.9)), must be specified for each node in the flow domain (see the description of input
Block I in Table 10.8 of Section 10). All parts of the flow region where Beta(n)>0 are treated as thesoil root zone.
Deep Drainage from the Soil Profile - Vertical drainage, q(h), across the lower boundary of
the soil profile is sometimes approximated by a flux which depends on the position of groundwaterlevel (e.g., Hopmans and Stricker, 1989). If available, such a relationship can be implemented in theform of a variable flux boundary condition for which Kode(n)=-6. This boundary condition is
implemented in HYDRUS by setting the logical variable qGWLF in the input file BOUNDARY.IN
equal to ".true.". The discharge rate Q(n) assigned to node n is determined by the program as
Q(n)=-Width(n)*q(h) where h is the local value of the pressure head, and q(h) is given by
q (h) =− Aqh exp ( Bqh h − GWL0 L )(8.1)

where Aqh and Bqh are empirical parameters that must be specified in the input file BOUNDARY.IN,
together with GWL0L, which represents the reference position of the groundwater level (usually set
equal to the z-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 for which Kode(n)=-6. This boundary condition is
implemented in HYDRUS by setting the logical variable FreeD in the input file BOUNDARY.IN
equal to ".true.". The program determines the discharge rate Q(n) assigned to node n as
Q(n)=-Width(n)*K(h), where h is the local value of the pressure head, and K(h) is the hydraulicconductivity corresponding to this pressure head. In Version 2.0, users can specify other than unit
gradient boundary conditions.
Seepage Faces - The initial settings of the variables Kode(n), Q(n), and h(n) for nodes along
a seepage face are summarized in Table 8.5. All potential seepage faces must be identified before

113

starting the numerical simulation. This is done by providing a list of nodes along each potential
seepage face (see input Block J as defined in Table 10.10 of Section 10). In Version 2.0, users can
specify the pressure head for the seepage face boundary conditions.

Table 8.5. Initial setting of Kode(n), Q(n), and h(n) for seepage faces.

Node Type

Kode(n)

Q(n)

h(n)

Seepage Face

+2

0.0

0.0 (or specified)(initially saturated)
Seepage Face

-2

0.0

Initial Value(initially unsaturated)

Drains - Table 8.6 summarizes the initial settings of the variables Kode(n), Q(n), and h(n)for nodes representing drains. All drains must be identified before starting the numerical simulation.
This is done by providing a list of nodes representing drains, together with a list of elements aroundeach drain whose hydraulic conductivities are to be adjusted according to the discussion in Section
5.3.7 (see also input Block J as defined in Table 10.10 of Section 10).

Table 8.6. Initial setting of Kode(n), Q(n), and h(n) for drains.

Node Type

Kode(n)

Q(n)

h(n)

Drain

+5

0.0

0.0

(initially saturated)
Drain

-5

0.0

Initial Value(initially unsaturated)

Surface Drip Irrigation – Dynamic Evaluation of the Wetted Area - User needs to specify“Time-Variable Flux 1” boundary condition at the surface boundary. The length of this boundaryneeds to be sufficient to accommodate the entire wetting zone. The drip discharge flux, Q, needs114to be entered in the “Var.Fl-1” column of the “Time Variable Boundary Conditions” dialog
window. Using Boundary Line Chart, one can see how the wetting zone expands in time.
Subsurface Drip Characteristic Function - User needs to specify “Time-Variable Flux 1”
boundary condition along the dripper boundary and hin in the “Var.H-1” column of the “TimeVariable Boundary Conditions” dialog window. A positive pressure indicates an irrigation
period, and a negative pressure indicates a non-irrigation period.
Solute and Heat Transport Boundary Conditions - The original version 1.1. of SWMS_2D[Šimůnek et al., 1992] assumed a strict relationship between the boundary conditions for water flow
and solute transport. A first-type boundary condition for water flow forced the boundary condition
for solute transport also to be of the first type. Similarly, a second-type boundary condition for waterflow induced a second- or third-type boundary condition for solute transport depending upon the
direction of the water flux. These strict relationships between the boundary conditions for waterflow and solute transport have been abandoned in later versions. The selection of the type ofboundary condition for the solute transport is now much more independent of the boundary
condition implemented for water flow. The type of boundary condition to be invoked for solute orheat transport is specified by the input variable KodCB or KodTB, respectively. A positive sign ofthis variable means that a first-type boundary condition will be used. When KodCB (KodTB) is
negative, HYDRUS selects a third-type boundary condition when the calculated water flux is
directed into the region, or a second-type boundary condition when the water flux is zero or directed
out of the region. One exception to these rules occurs for atmospheric boundary conditions whenKode(n)=±4 and Q(n)<0. HYDRUS assumes that solutes can leave the flow region acrossatmospheric boundaries only by gas diffusion. The solute flux in this situation becomes zero, i.e.,c0=0 in equation (6.34). Cauchy and Neumann boundary conditions are automatically applied tointernal sinks/sources depending upon the direction of water flow. The dependence or independenceof the solute and heat boundary conditions on the time or the system is still defined through thevariable Kode(n) as discussed above.
Although HYDRUS can implement first-type boundary conditions, we recommend users to
invoke third-type conditions where possible. This is because third-type conditions, in general, arephysically more realistic and preserve solute mass in the simulated system (e.g., van Genuchten andParker [1984]; Leij et al. [1991]).
For the user's convenience, Table 8.7 summarizes possible values of the different boundary
codes and their association with specific water flow and solute transport boundary conditions.

115

8.4. 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 be investigated bychanging the dimensions in the parameter statement at the beginning of the main program andsubsequently linking all previously compiled subroutines with the main program when creating an
executable file. Table 8.8 lists the array dimensions, which must be defined in the parameter
statement.

116

Table 8.7. Summary of Boundary Coding.
Boundary TypeTimeindependent

Timedependent

Water Flow

Solute Transport

Kode

Input

KodCB

Input

Impermeable

0

initial h(n), Q(n)=0

NA

NA

Constant head1prescribed h(n), Q(n)=0≠0cBound(i)Constant flux

-1

initial h(n), prescribed Q(n)(positive for inflow)≠0 (inflow)cBound(i)Source/sink(Dirichlet)

1prescribed h(n), Q(n)=0≠0 (Source)cBound(6)Source/sink(Neumann)-1initial h(n), prescribed Q(n)(positive for inflow)≠0 (Source)cBound(6)Seepage Face(saturated)

2h(n)=0, Q(n)=00-

Seepage Face(unsaturated)-2

initial h(n), Q(n)=00-

Drains (saturated)

5h(n)=0, Q(n)=00-

Drains (unsaturated)-5

initial h(n), Q(n)=00-Changing head

3, 7,
8, 9

h(t), initial h(n), Q(n)=0

≠0

ch(t)

Changing flux

-3, -7,-8, -9q(t), initial h(n), Q(n)=0≠0 (inflow)

cr(t)(negative q(t) for inflow)

Atmosphericboundary-4Prec, rSoil,hA,hS, initial h(n),Q(n)=00cPrec(t)Root zone0rRoot, initial h(n), Q(n)=0

NA

cBound(5)Deep drainage-6

Aqh, Bqh, GWL0L, initial h(n),Q(n)=00-Free drainage-6

initial h(n), Q(n)=00-i = 1, 2, ..., 9117

Table 8.8. List of array dimensions in HYDRUS.

Dimension

Description

NumNPD

Maximum number of nodes in finite element mesh

NumElD

Maximum number of elements in finite element mesh

MBandD

Maximum dimension of the bandwidth of matrix A when
Gaussian elimination is used. The maximum number of nodes
adjacent to a particular node, including itself, when iterative
matrix solvers are used.

NumBPD

Maximum number of boundary nodes for which Kode(n)≠0NSeepD

Maximum number of seepage faces

NumSPD

Maximum number of nodes along a seepage face

NDrD

Maximum number of drains

NElDrD

Maximum number of elements surrounding a drain

NMatD

Maximum number of materials

NTabD

Maximum number of items in the table of hydraulic properties
generated by the program for each soil material

NumKD

Maximum number of available code number values (equals 6 inpresent version)

NSD

Maximum number of solutes (equals 6 in present version)

NObsD

Maximum number of observation nodes for which the pressure
head, the water content, temperature, and concentration are
printed at each time level

MNorthMaximum number of orthogonalizations performed when
iterative solvers are used

8.5. Matrix Equation Solvers
Discretization of the governing partial differential equations for water flow (2.1), solutetransport (3.11) and heat movement (4.1) leads to the system of linear equations

118

[ A]{x} = {b}(8.2)

in which matrix [A] is symmetric for water flow and asymmetric for solute and heat transport.
The original version of SWMS_2D [Šimůnek et al., 1992] uses Gaussian elimination tosolve both systems of linear algebraic equations. The invoked solvers took advantage of the banded
nature of the coefficient matrices and, in the case of water flow, of the symmetric properties of thematrix. Such direct solution methods have several disadvantages as compared to iterative methods.
Direct methods require a fixed number of operations (depending upon the size of the matrix), which
increases approximately by the square of the number of nodes [Mendoza et al., 1991]. Iterative
methods, on the other hand, require a variable number of repeated steps, which increases at a muchsmaller rate (about 1.5) with the size of a problem [Mendoza et al., 1991]. A similar reduction also
holds for the memory requirement since iterative methods do not require one to store non-zero
matrix elements. Memory requirements, therefore, increase at a much lower rate with the size of the
problem when iterative solvers are used [Mendoza et al., 1991]. Round-off errors also represent lessof a problem for iterative methods as compared to direct methods. This is because round-off errors
in iterative methods are self-correcting [Letniowski, 1989]. Finally, for time-dependent problems, areasonable approximation of the solution (i.e., the solution at the previous time step) exists for
iterative methods, but not for direct methods [Letniowski, 1989]. In general, direct methods are
more appropriate for relatively small problems, while iterative methods are more suitable for larger
problems.
Many iterative methods have been used in the past for handling large sparse matrix
equations. These methods include Jacobi, Gauss-Seidel, alternating direction implicit (ADI), block
successive over-relaxation (BSSOR), successive line over-relaxation (SLOR), and strongly implicit
procedures (SIP), among others [Letniowski, 1989]. More powerful preconditioned accelerated
iterative methods, such as the preconditioned conjugate gradient method (PCG) [Behie andVinsome, 1982], were introduced more recently. Sudicky and Huyakorn [1991] gave threeadvantages of the PCG procedure as compared to other iterative methods: PCG can be readily
modified for finite element methods with irregular grids, the method does not require iterative
parameters, and PCG usually outperforms its iterative counterparts for situations involving
relatively stiff matrix conditions.
HYDRUS implements both direct and iterative methods for solving the system of linear
algebraic equations given by (8.2). Depending upon the size of matrix [A], we use either direct

119Gaussian elimination or the preconditioned conjugate gradient method [Mendoza et al., 1991] forwater flow and the ORTHOMIN (preconditioned conjugate gradient squared) procedure [Mendoza
et al., 1991] for solute transport. Gaussian elimination is used if either the bandwidth of matrix [A]is smaller than 20, or the total number of nodes is smaller than 500. The iterative methods used in
HYDRUS were adopted from the ORTHOFEM software package of Mendoza et al. [1991].
The preconditioned conjugate gradient and ORTHOMIN methods consist of two essential
parts: initial preconditioning and the iterative solution with either conjugate gradient or
ORTHOMIN acceleration [Mendoza et al., 1991]. Incomplete lower-upper (ILU) preconditioning isused in ORTHOFEM when matrix [A] is factorized into lower and upper triangular matrices bypartial Gaussian elimination. The preconditioned matrix is subsequently repeatedly inverted using
updated solution estimates to provide a new approximation of the solution. The orthogonalizationminimization acceleration technique is used to update the solution estimate. This technique ensuresthat the search direction for each new solution is orthogonal to the previous approximate solution
and that either the norm of the residuals (for conjugate gradient acceleration [Meijerink and van der
Vorst, 1981]) or the sum of the squares of the residuals (for ORTHOMIN [Behie and Vinsome,1982]) is minimized. More details about the two methods is given in the user's guide of
ORTHOFEM [Mendoza et al., 1991] or in Letniowski [1989]. Letniowski [1989] also gives acomprehensive review of accelerated iterative methods, as well as of the preconditional techniques.

120

9. EXAMPLE PROBLEMS
9.1. Direct Example Problems
Eight example problems are presented in this section. The first three examples are identicalto those provided previously with the first version of SWMS_2D [Šimůnek et al., 1992]. The otherfour examples in this section are identical to those provided with CHAIN_2D [Šimůnek and van
Genuchten, 1994], and are included mainly for mathematical verification purposes, and fordemonstrating new features of HYDRUS, i.e., non-equilibrium and nonlinear adsorption, sequential
first-order decay reactions, solute diffusion in the gas phase, and/or heat transport. These examples
have also been used with versions 1 and 2 of HYDRUS-2D [Šimůnek et al., 1996, 1999]. Finally,
example 8 is a fully three-dimensional problem. Examples 1 through 6 are solved using both twoand three-dimensional solvers.
Examples 1 and 2 provide comparisons of the water flow part of HYDRUS code withresults from both the UNSAT2 code of Neuman [1974] and the SWATRE code of Belmans et al.
[1983]. Examples 3a and 3b serve to verify the accuracy of the solute transport part of HYDRUS bycomparing numerical results against those obtained with two- and three-dimensional analyticalsolutions during steady-state groundwater flow. The results obtained with the HYDRUS codes (2D
and 3D) for these three examples are identical to the results obtained with SWMS_2D. Example 4
serves to verify the accuracy of HYDRUS by comparing numerical results for a problem with threesolutes involved in a sequential first-order decay chain against results obtained with an analyticalsolution during one-dimensional steady-state water flow [van Genuchten, 1985]. Example 5
considers one-dimensional transport of a solute undergoing nonlinear cation adsorption. Numericalresults are compared with experimental data and previous numerical solutions obtained with the
MONOC code of Selim et al. [1987] and the HYDRUS code of Kool and van Genuchten [1991].
Example 6 serves to verify the accuracy of HYDRUS in describing nonequilibrium adsorption bycomparing numerical results against experimental data, and previous numerical predictions during
one-dimensional steady-state water flow [van Genuchten, 1981]. Example 7 demonstrates
numerical results for a field infiltration experiment involving a two-layered axisymmetric threedimensional flow domain. The infiltrating water was assumed to have a higher temperature than thesoil and to contain an organic compound (parent pesticide), which in the soil profile degraded into
two sequential daughter products. Example 8 shows numerical results for contaminant transport inan unconfined aquifer subjected to well pumping.

121

9.1.1. Example 1 - Column Infiltration Test
This example simulates a one-dimensional laboratory infiltration experiment discussed by
Skaggs et al. [1970]. The example was used later 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.
Figure 9.1 gives graphical representations of the soil column and the finite element meshused for the numerical simulations with two- and three-dimensional solvers. The soil water retention
and relative hydraulic conductivity functions of the sandy soil are presented in Figure 9.2. The sandwas assumed to be at an initial pressure head of -150 cm. The soil hydraulic properties wereassumed 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 aseepage face with h=0 when the bottom boundary becomes saturated (this last condition was notreached during the simulation). The impervious sides of the column were simulated by imposing
no-flow boundary conditions.

Figure 9.1. Flow system and finite element mesh for example 1 (2D left, 3D right).

122

The simulation was carried out for 90 min, which corresponds to the total time duration of
the experiment. Figure 9.3 shows the calculated instantaneous (q0) and cumulative (I0) infiltration
rates simulated with HYDRUS. The calculated results agree closely with those obtained by Davisand Neuman [1983] using their UNSAT2 code.

Figure 9.2. Retention and relative hydraulic conductivity functions for example 1. The solid circles are
UNSAT2 input data [Davis and Neuman, 1983].

123

Figure 9.3. Instantaneous, q0, and cumulative, I0, infiltration rates simulated with the HYDRUS(solid lines) and UNSAT2 (solid circles) codes for example 1.

1249.1.2. 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 groundwater levels provided therequired boundary conditions for the numerical model. Calculations were performed for the periodof 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 etal., 1978, Belmans et al., 1983].
The soil profile (Fig. 9.4) 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 9.5.

Figure 9.4. Flow system and finite element mesh for example 2 (2D left, 3D right).

125

The soil surface boundary conditions 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 a prescribeddrainage flux - groundwater level relationship, q(h), as given by equation(8.1). The groundwaterlevel was initially set at 55 cm below the soil surface. The initial moisture profile was taken to be in
equilibrium with the initial groundwater level.
Figure 9.6 presents the 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 9.7. The pressure head at the soil surface andthe arithmetic mean pressure head of the root zone during the simulated season are presented in
Figure 9.8. Finally, Figure 9.9 shows variations in the calculated groundwater level with time.

126

Figure 9.5. Unsaturated hydraulic properties of the first and second soil layers for example 2.

127

Figure 9.6. Precipitation and potential transpiration rates for example 2.

128

Figure 9.7. Cumulative values for the actual transpiration and bottom discharge rates for example 2 assimulated with HYDRUS (solid line) and SWATRE (solid circles).

129Figure 9.8. Pressure head at the soil surface and mean pressure head of the root zone for example 2 assimulated with HYDRUS (solid lines) and SWATRE (solid circles).

130Figure 9.9. Location of the groundwater table versus time for example 2 as simulated with the HYDRUS(solid line) and SWATRE (solid circles) computer programs.

9.1.3a. Example 3A - Two-Dimensional Solute TransportThis example was used to verify the mathematical accuracy of the solute transport part of
HYDRUS. Cleary and Ungs [1978] published several analytical solutions for two-dimensional
dispersion problems. One of these solutions holds for solute transport in a homogeneous, isotropic
porous medium during steady-state unidirectional groundwater flow. The solute transport equation(3.11) for this situation reduces to∂2c∂ 2 c ∂c∂c
DT 2 + DL 2 - v - λ Rc = R
∂z
∂t
∂x
∂z

131

(9.1)

where λ is a first-order degradation constant, DL and DT are the longitudinal and transverse
dispersion coefficients, respectively; v is the average pore water velocity (qz/θ) in the flow direction,and z and x are the spatial coordinates parallel and perpendicular to the direction of flow. The
initially solute-free medium is subjected to a solute source, c0, of unit concentration. The source
covers a length 2a along the inlet boundary at z=0, and is located symmetrically about thecoordinate x=0. The transport region of interest is the half-plane (z≥0; -∞≤x≤∞). The boundary
conditions may be written as:
0, t ) c0c( x,=-a ≤ x ≤ ac( x, 0, t ) = 0

other values of x

(9.2)
∂c
=0
z →∞ ∂zlim∂c=0
x →±∞ ∂xlim

The analytical solution of the above transport problem is [Javandel et al., 1984, Leij and Bradford,1994]

c ( x, z , t )  vz  t / Rv2 z 2  −3 / 2
exp
exp
R
λ
+
τ
 
τ

∫

1/ 2
4
4
D
D
τ
4 (π DL )LL 2 DL  0




c0 z


 a-x  a+x  
. erf 
erf
dτ
+


1/ 2
1/ 2  
 2( DT τ )  2( DT τ )  


(9.3)

The input transport parameters for the two simulations are listed in Table 9.1. The width of thesource was assumed to be 100 m. Because of symmetry, calculations were carried out only for thequarter plane where x≥0 and z≥0.
Figure 9.10 shows the calculated concentration front (taken at a concentration of 0.1) atselected times for the first set of transport parameters in Table 9.1. Note the close agreement
between the analytical and numerical results. Excellent agreement is also obtained for the calculated
concentration distributions after 365 days at the end of the simulation (Fig. 9.11). Figures 9.12 and9.13 show similar results for the second set of transport parameters listed in Table 9.1.

132

Table 9.1. Input parameters for example 3A.

Parameter

Example 3a

Example 3bv [m/day]DT [m2/day]DL [m2/day]
λ [day-1]
R [-]

0.1
1.0
1.0
0.0
1.0

1.0
0.5
1.0
0.01
3.0

c0 [-]

1.0

1.0

Figure 9.10. Advancement of the concentration front (c=0.1) for example 3a, as calculated with HYDRUS(dotted lines) and the analytical solution (solid lines).

133

Figure 9.11. Concentration profile at the end of the simulation (t=365 days) for example 3a
calculated with HYDRUS (dotted lines) and the analytical solution (solid lines).

Figure 9.12. Advancement of the concentration front (c=0.1) for example 3b, as calculated with HYDRUS

134

(dotted lines) and the analytical solution (solid lines).

Figure 9.13. Concentration profile at the end of the simulation (t=365 days) for example 3b as
calculated with HYDRUS (dotted lines) and the analytical solution (solid lines).

9.1.3b. Example 3B - Three-Dimensional Solute TransportThis example was used to verify the mathematical accuracy of the solute transport part of
SWMS_3D. Leij et al. [1991] published several analytical solutions for three-dimensional
dispersion problems. One of these solutions holds for solute transport in a homogeneous,isotropic porous medium during steady-state unidirectional groundwater flow (Figure 9.14). The
solute transport equation (3.1) for this situation reduces to

∂c
∂ 2c
∂ 2c
∂ 2c
∂c
R = DT 2 + DT 2 + DL 2 − v − µ c + λ
∂t
∂x
∂y
∂z
∂z

(9.4)

where λ and μ are zero- and first-order degradation constants, respectively; DL and DT are thelongitudinal and transverse dispersion coefficients, respectively; v (= qz /θ) is the average pore
135

water velocity in the flow direction, and z is the spatial coordinate parallel to the direction offlow, while x and y are the spatial coordinates perpendicular to the flow direction. The initiallysolute-free medium is subjected to a solute source, c0, of unit concentration. The rectangularsurface source has dimensions 2a and 2b along the inlet boundary at z=0, and is locatedsymmetrically about the coordinates x=0 and y=0 (Figure 9.14). The transport region of interestis the half-space (z≥0; -∞≤x≤∞, -∞≤y≤∞). The boundary conditions may be written as:c ( x, y=, 0, t ) c0

− a ≤ x ≤ a, − b ≤ y ≤ bc ( x, y, 0, t ) = 0

other values of x, y∂c=0z →∞ ∂z∂c=0lim
z →±∞ ∂x∂c
=0lim
z →±∞ ∂y

(9.5)

lim

The analytical solution of the above transport problem is [Leij and Bradford, 1994]
1/ 2t Rz 2 c=c ( x, y , z , t ) 0 ∫ 
4 P(t )  4π DLτ 3 

 µτ ( Rz −ντ )2 
exp  −−

4 RDLτ 
 R






x−a
x+a
erfc 

−
erfc


1/ 2
1/ 2 

 2 ( DTτ / R )  2 ( DTτ / R )  





y −b
y+b
erfc  dτ +
−
erfc


1/ 2
1/ 2 

 2 ( DTτ / R )  2 ( DTτ / R )   ντ − Rz 
 Rz +ντ  
 vz 
λ t
 µτ  

 dτ
−
exp
erfc
+
exp
−
erfc



 
1/ 2
1/ 2 
 R 
2 R ∫0 2 ( RDLτ )  2 ( RDLτ )   DL 

(9.6)

where P(t) = 0 if t<t0 and P(t) = t-t0 if t>t0, and where t0 is the duration of solute pulse. The input
transport parameters for the two simulations are listed in Table 9.1. The width of the source wasassumed to be 100 m in both the x and y directions. Because of symmetry, calculations were136

carried out only for part of the transport domain where x≥0, y≥0 and z≥0.

Figure 9.14. Schematic of the transport system for example 3B.

We will not show here the comparison of analytical and numerical results for these threedimensional problems, since the obtained figures are fairly similar to Figures 9.11 through 9.14.
The agreement between the analytical and numerical results is also in the same order as thoseshown in Figures 9.11 through 9.14.
9.1.4. Example 4 - One-Dimensional 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 transport ofsolutes involved in sequential first-order decay reactions. The analytical solution holds for solutetransport in a homogeneous, isotropic porous medium during steady-state unidirectional
groundwater flow. Solute transport equations (3.1) and (3.2) for this situation reduce to

137

∂c1
∂ 2 c1∂c= D 2 - v 1 - µ1 R1c1
R1
∂t
∂x
∂x

∂ci
∂ 2 ci∂c
Ri= D 2 - v i + µi −1 Ri −1ci −1 - µi Ri ci
∂t
∂x
∂x

(9.7)

i = 2, 3(9.8)

where µ is a first-order degradation constant, D is the dispersion coefficient, v is the average porewater velocity (qx/θ) in the flow direction, x is the spatial coordinate in the direction of flow, andwhere it is assumed that three solutes participate in the decay chain. The specific example used here
applies to the three-species nitrification chain
NH +4 → NO −2 → NO3−

(9.9)

and is the same as described by van Genuchten [1985], and earlier by Cho [1971]. The boundary
conditions may be written as: ∂c1

v c0,1 (0, t ) - D ∂x + vc1  =


 ∂ci

0
i 2, 3==i 
 - D ∂x + vc


(9.10)∂ci0=
i 1, 2, 3=
lim
x →∞ ∂x

The experiment involves the application of an NH4+ solution to an initially solute-free
medium (ci = 0). The input transport parameters for the simulation are listed in Table 9.2.
Figure 9.15 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 9.16 shows the concentration profiles at three different times (50, 100, and 200 hours) for
NH4+, NO2-, and NO3-, respectively. The numerical results in each case duplicated the analyticalresults.

138

Table 9.2. Input parameters for example 4.

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

Figure 9.15. Analytically and numerically calculated concentration profiles for NH4+, NO2-, andNO3- after 200 hours for example 4.

139

Figure 9.16. Analytically and numerically calculated concentration profiles for NH4+ (top), NO2(middle), NO3- (bottom) after 50, 100, and 200 hours for example 4.

1409.1.5. Example 5 - One-Dimensional Solute Transport with Nonlinear Cation Adsorption
The experiment discussed in this example was conducted by Selim et al. [1987] and usedlater to validate the HYDRUS code [Kool and van Genuchten, 1991]. 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 10
mmolcL-1 MgCl2 solution, followed by the original CaCl2 solution. The adsorption isotherm wasdetermined 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 asother transport parameters for this problem, are listed in Table 9.3. First- and second-type boundary
conditions were applied at the top and bottom of the soil column, respectively.
The observed Mg breakthrough curve is shown in Figure 9.17, together with simulated
breakthrough curves obtained with HYDRUS, the MONOC code of Selim et al. [1987], and version
3.1 of the HYDRUS code of Kool and van Genuchten [1991]. The results indicate a reasonableprediction 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 to those generated with HYDRUS-3.1 when a third-type boundary condition was
invoked at the top of the soil column.

Table 9.3. Input parameters for example 5.

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

141

Figure 9.17. Mg breakthrough curves for Abist loam calculated with the MONOD, HYDRUS, andHYDRUS codes (data points from Selim et al. [1978]).

The Langmuir adsorption isotherm can also be used to model the exchange of homovalentions. 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 in a generalized form
as [Sposito, 1981]ma  a 
K ex =  1   2 
 a1   a2 n(9.11)

where Kex is the dimensionless thermodynamic equilibrium constant, and a and a denote the ion

142activities in the soil solution and on the exchange surfaces [-], respectively:=ai γ=i 1, 2i ci(9.12)=
i 1, 2a i ξ=
i si

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 theexchange surfaces [MM-1] (kg/mmol), respectively. Substituting (9.12) into (9.11) givesγmγ2

ξn γ mξ1 γ 2

1
2
1
K12 K=K ex ==vmn
2

s1m c2ns2n c1m(9.13)

where Kv denotes the Vanselow selectivity coefficient [-], while K12 will be simply referred to as theselectivity 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.,
nc1 + mc2 =CT
ns1 + ms2 =ST(9.14)

the Langmuir parameters ks and η in (3.3) for the incoming solute become
ks =

η=

K12 STCT

ϑ ( K12 -1)
CT

whereas for the solute initially in the soil column:

143(9.15)

ks =

η=

ST
K12 CTϑ (1- K12 )(9.16)

K12 CT

The parameter υ in (9.15) and (9.16) 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 cation exchange
capacity (ST=62 mmolc/kg), it follows that the parameters in the Langmuir adsorption isotherm forthe incoming solute (Mg) are ks=3.126 and η=-0.098, while those for the solute initially in the soil
profile (Ca) the parameters are ks=12.157 and η=0.192. The observed Ca breakthrough curve isshown in Figure 9.18, together with the simulated breakthrough curves obtained with the HYDRUS
and MONOC codes [Selim et al., 1987]. Note the close agreement between the numerical results
and the experimental data.

Figure 9.18. Ca breakthrough curves for Abist loam calculated with the MONOD and HYDRUS codes(data points from Selim et al. [1978]) (example 5).

144

9.1.6. Example 6 - One-Dimensional 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 that werefitted to the breakthrough curve with the CFITIM parameter estimation model [van Genuchten,
1981], assuming a two-site chemical nonequilibrium sorption model analogous to the formulation
discussed in Section 3, but for steady-state water flow. Input parameters for example 6 are listed inTable 9.4. Figure 9.19 compares HYDRUS numerical results with the experimental data and with anumerical simulation assuming physical non-equilibrium and nonlinear adsorption [van Genuchten,
1981].

Table 9.4. Input parameters for example 6.

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
5.06

145

Figure 9.19. Observed and calculated effluent curves for Boron movement throughGlendale clay (data points from van Genuchten [1981]) (example 6).

1469.1.7. Example 7 - Water and Solute Infiltration Test
This example corresponds to example 4 in the SWMS_2D and HYDRUS-2D manuals[Šimůnek et al., 1992, 1999]. The example concerns the movement of water and a dissolved solutefrom a single-ring infiltrometer into the soil profile consisting of two layers: a 40-cm thickA-horizon, and an underlying B/C-horizon. The hydraulic functions of the two soil layers are thesame as those used in Example 2. The axisymmetric flow system and associated finite element
mesh for the ponded infiltration experiment are shown in Figure 9.19. The soil profile had an initial
temperature of 20oC, whereas the infiltrated water had a temperature of 30oC and contained anorganic (parent) compound (the pesticide aldicarb), which is known to degrade by oxidation into
two sequential daughter products (sulfone and sulfoxide) [Ou et al., 1988]. Each of the three solutes
also undergoes hydrolytical first-order decay that leads to products that are not simulated ormonitored during their subsequent transport (i.e., oxime, sulfone oxime, and sulfoxide oxime) [Ouet al., 1988]. All major solutes adsorb onto the solid phase, and volatilization is considered for the
first and the third solutes. The reaction pathway is schematically given by:
Product

Product

Product

µl,1

µl,2

µl,3

ParentDaughter

Daughter↑

Pesticide

↑

→

µ'l,1

↑

→

µ'l,2

Product 1

Product 2

g1 c1 s1c2 s2

g3 c3 s3

kg,1 ks,1

ks,2

kg,3 ks,3

→

µ'l,3

Product

The example is used here to illustrate variably-saturated water flow, heat movement, and solutetransport in a layered and radially symmetric three-dimensional soil profile.
Calculations were carried out over a period of 10 days. The pressure head profile obtained inexample problem 2 at the beginning of June 1982 was taken as the initial condition for the waterflow equation, similarly as in example 4 of Šimůnek et al. [1992]. The soil profile was assumed to
be initially free of any solutes. All sides of the flow region were considered to be impervious, except
for a small portion around the origin at the surface (the ponded surface inside the ring infiltrometer)where constant pressure head and concentration flux boundary conditions were imposed, as well as
the lower right corner where the groundwater level was kept constant. The concentration of theinfiltrating water was free of any solute except for the first five days of the simulation during which147

time the infiltrating water contained the parent solute of unit concentration. Boundary condition(3.46) was used for the non-ponded part of the soil surface, thereby assuming the existence of astagnant boundary layer of thickness d at the soil surface through which volatile solutes moved tothe atmosphere by gas diffusion only. Water and solute extraction by plant roots was not considered.
Tables 9.5, 9.6, and 9.7 list the unsaturated soil hydraulic, heat, and solute transportparameters, respectively. The solute and heat transport parameters were assumed to be the same forthe two soil layers. The heat transport parameters b1, b2, b3, Cn, Co, and Cw in Table 9.6 were takenfrom Chung and Horton [1987]. The solute parameters ks, kg, µw, µw’, and Dg in Table 9.7 weretaken from Wagenet and Hutson [1987].
Figure 9.21 presents the initial and steady-state pressure head profiles. The steady-state
profile for water flow was reached after approximately 2 or 3 days. Temperature profiles at times 1
and 10 days are shown in Figure 9.22. The heat front moved relatively slowly into the soil profile incomparison with the solute front (shown later) because of the high volumetric heat capacity of therelatively wet soil. Figures 9.23, 9.24, and 9.25 show concentration profiles for all three solutes attimes 2.5, 5, 7.5, and 10 days. After its application in the infiltrating water during the first five daysof the simulation (Fig. 9.23a, b), the first (parent) solute is transported further by water flow andgaseous diffusion, as well as is being degraded by two first-order decay reactions, such that the soil
profile is practically free of this solute after ten days (Fig. 9.23c, d). The second solute exists
exclusively because of the first-order degradation of the first solute. Hence, the second soluteinitially corresponds mainly with the first solute, but subsequently moves faster through the soil
profile because of less sorption. Note that the highest concentrations of the second solute werereached after complete application of the first solute (Fig. 9.24c, d). Similar features as for thesecond solute also apply to the third solute. In particular, notice that the soil profile is almost free of
this solute after 2.5 days (Fig. 9.25a), and that the highest concentrations of the third solute (while
being much smaller than those for the first two solutes) were reached at the end of the simulation(Fig. 9.25d).

148

Table 9.5. Hydraulic input parameters for example 7.

Parameter

1st layer

2nd layerθs=θm=θkθr=θa
Ks=Kk [m/day]
α [1/m]

0.399
0.000
0.298
1.74

0.339
0.000
0.454
1.39

n [-]

1.38

1.60

Table 9.6. Heat transport input parameters for example 7.

ParameterValueθn [-]
θo [-]
λL [m]
λT [m]

0.600+ (0.660*)
0.001
0.005
0.001
0.243
0.393
1.534
1.92*106
2.51*106
4.18*10620

b1 [Wm-1K-1]
b2 [Wm-1K-1]
b3 [Wm-1K-1]
Cn [Jm-3K-1]
Co [Jm-3K-1]
Cw [Jm-3K-1]
Ti [oC]T0 [oC]+*30

for the first layer
for the second layer

149

Table 9.7. Solute transport input parameters for example 7.

ParameterValueρ [kg/m3]

1300
0.00374
0.432
0.005
0.001
0.00010.00005
0.0002
1.33*10-70.0
1.33*10-3
0.2
0.01
0.005
0.36
0.024
0.00241.0
0.0
0.0

Dw [m2/day]
Dg [m2/day]DL [m]DT [m]
ks,1 [m3/kg]
ks,2 [m3/kg]
ks,3 [m3/kg]
kg,1 [-]kg,2 [-]kg,3 [-]µw,1 [1/day]µw,2 [1/day]µw,3 [1/day]µ’w,1 [1/day]µ’w,2 [1/day]µ’w,3 [1/day]
c0,1 [-]
c0,2 [-]
c0,3 [-]
d [m]

0.005

150

Figure 9.20. Flow system and finite element mesh for example 7.

151

Figure 9.21. Initial (top) and steady state (bottom) pressure head profiles for example 7.

152

Figure 9.22. Temperature profiles after 1 (top) and 10 days (bottom) for example 7.

153

Figure 9.23. Concentration profiles for the first solute after 2.5, 5, 7.5, and 10 days forexample 7.

154

Figure 9.24. Concentration profiles for the second solute after 2.5, 5, 7.5, and 10 days forexample 7.

155

Figure 9.25. Concentration profiles for the third solute after 2.5, 5, 7.5, and 10 days forexample 7.

156

9.1.8. Example 8 - Contaminant Transport From a Waste Disposal Site
This test problem concerns contaminant transport from a waste disposal site (or possibly
a landfill) into an unconfined aquifer containing a pumping well downgradient of the disposal
site, as shown in Figure 9.26. Water was assumed to infiltrate from the disposal site into theunsaturated zone under zero-head ponded conditions. The concentration of the contaminantleaving the disposal site was taken to be 1.0 during the first 50 days and zero afterward. The
waste disposal site itself had lateral dimensions of 10 x 40 m2. Initially, the water table decreased
from a height of 28 m above the base of the aquifer at the left-hand side (Figure 9.26) to 26 m on
the right-hand side of the flow domain. The initial pressure head in the unsaturated zone wasassumed to be at equilibrium with the initial water table, i.e., no vertical flow occurred. The
transport experiment started when the water table in the fully penetrated well at x = 170 m (y = 0)
was suddenly lowered to a height of 18 m above the bottom of the unconfined aquifer. We
assumed that at that same time (t = 0) infiltration started to occur from the disposal site.
Prescribed hydraulic head conditions h + z = 28 m and h + z = 26 m were imposed along the lefthand (x = 0) and right-hand (x = 260 m) side boundaries (-50≤y≤50 m). A prescribed hydraulichead condition of h + z = 18 m was used to represent the well along a vertical below the watertable (z≤18) at x = 170 m and y = 0 m, while a seepage face was defined at that location along thevertical above the water table (z > 18). No-flow conditions were assumed along all otherboundaries, including the soil interface. Hydraulic and transport parameters used in the analysis
are listed in Table 9.8. We selected the hydraulic retention parameters for a coarse-textured soil
with a relatively high saturated hydraulic conductivity, Ks, in order to test the HYDRUS code for
a comparatively difficult numerical problem.
Because of symmetry about the y axis, only half of the flow region was simulated. The
solution domain defined by 0≤x≤260, 0≤y≤50, and 0≤z≤38 m was discretized into a rectangulargrid comprised of 10560 elements and 12144 nodes (Figure 9.27). Nodal spacings were maderelatively small in regions near the disposal site and near the pumping well, where the highest
head gradients and flow velocities were expected. The variably saturated flow problem wassolved using HYDRUS, assuming an iteration head tolerance of 0.01 m and a water content
tolerance of 0.0001.

157

Figure 9.26. Geometry and boundary conditions for example 8 simulating three-dimensional flow andcontaminant transport in a pumped variably-saturated aquifer.

158

Table 9.8. Input parameters for example 8.

Hydraulic Parameters

Transport Parametersθs=θm=θk0.450

ρ [kg/m3]

1400

θr=θa

0.05

Dd [m2/day]

0.01

Ks=Kk [m/day]

5.0

DL [m]

1.0

α [1/m]

4.1

DT [m]

0.25

n [-]

2.0

k [m3/kg]

0.0

μw [1/day]0.0

μs [1/day]

0.0

γw [1/day]0.0

γs [1/day]

0.0

c0

1.0

Computed water table elevations are plotted in Figure 9.28a and 9.28b along longitudinal(y=0) and transverse (x=170 m) planes through the pumping well, respectively. The results show
a relatively strong direct interaction between the infiltrating water and the saturated zone after
only a short period of time; water flow reached an approximately steady state about 1.5 days
after the experiment started. The velocity field and streamlines in a longitudinal section throughthe pumping well are presented in Figure 9.29. Note that the length of the seepage face along thewell was determined to be approximately 5 meters. The calculated well discharge rate for the
fixed water table (z = 18 m) was calculated to be 39.6 m3/day. A concentration contour plot (c =0.1) is presented in Figure 9.30. This figure shows that contaminant transport was strongly
affected by well pumping. Note that although the contaminant source was located 10 m above
the initial groundwater table, and 150 m upgradient of the pumping well, the solute reached thepumping well after only 200 days of pumping. Figure 9.31 gives a two-dimensional view of
calculated concentration distributions at several times in a horizontal plane (z = 20 m).

159

Finally, Figure 9.32 presents solute breakthrough curves observed at observation node 1(x = 40 m, z = 32 m), node 2 (x = 150 m, z = 24 m), node 3 (x = 170 m, z = 18 m), and node 4 (x= 200 m, z = 20 m). These observation nodes are all on a vertical cross-section (y = 0) as shownin Figure 9.26b. Notice that the breakthrough curves differ considerably in shape and especiallypeak concentrations. Although the breakthrough curve at observation node 1 immediately below
the disposal site was very steep, no numerical oscillations were observed here. This shows thatHYDRUS is able to solve the present solute transport problem involving sharp concentration
distributions without generating non-physical oscillations. However, the efficiency of thenumerical simulation for this example was limited by the need for relatively small time steps soas to satisfy the grid Courant criterion (Section 6.3.6). Although water flow had reachedapproximately steady-state within less than 2 days, the time step for the solute transport problem
was only 0.073 day because of relatively large flow velocities near the well.

Figure 9.27. Finite element mesh for example 8.

160

igure 9.28. Calculated (a) longitudinal (y=0) and (b) transverse (x=170 m) elevations of thegroundwater table.

161

Figure 9.29. Computed velocity field and streamlines at t = 10 days.

163

Figure 9.30. Concentration countour plots for (a) c=0.1 in a longitudinal cross-section (y=0), and (b)c=0.05 in a transeverse cross-section (x=170 m).

164

Figure 9.31. Concentration distributions in a horizontal plane located at z = 20 m
for t = 10, 50, 100, and 200 days.

165

Figure 9.32. Breakthrough curves at observation node 1 (x = 40 m, z= 32 m), node 2 (x = 150 m, z = 24m), node 3 (x = 170 m, z = 18 m), and node 4 (x = 200 m, z = 20 m).

166

9.2. Inverse Example Problems
In this section, we discuss three methods recently proposed for estimating the soil hydraulicproperties by numerical inversion of the Richards' equation. One method currently being developed
involves the use of tension disc permeameter data [Šimůnek and van Genuchten, 1996, 1997](example 9), while a second method uses data collected with a modified cone penetrometer [Gribb,
1996; Gribb et al., 1998] (example 10). The third method involves the use of a multiple extraction
device [Inoue et al., 1998] (example 11). All these inverse problems are solved using a 2D solver.
9.2.1. Example 9 - Tension Disc InfiltrometerŠimůnek and van Genuchten [1997] suggested the use of multiple tension infiltration
experiments in combination with knowledge of the initial and final water contents for estimating
soil hydraulic properties. An evaluation of the numerical stability and parameter uniqueness using
numerically generated data with superimposed stochastic and deterministic errors showed that a
combination of the multiple cumulative tension infiltration data, a measured final water content, andan initial condition expressed in terms of the water content, provided the most promising parameter
estimation approach for practical applications [Šimůnek and van Genuchten, 1997].
The experiment was used to estimate the soil hydraulic characteristics of a two-layered soil
system involving a crusted soil in a Sahel region [Šimůnek et al., 1998]. Here we will report only
results for the sandy subsoil. Data were obtained with a tension disc diameter of 25 cm and with
supply tensions of 11.5, 9, 6, 3, 1, and 0.1 cm. Figure 9.33 shows measured and optimized
cumulative infiltration curves and their differences. The small breaks in the cumulative infiltration
curve (Fig. 9.33) were caused by brief removal of the infiltrometer from the soil surface to resupply
it with water and to adjust the tension for a new time interval. Very close agreement between themeasured and optimized cumulative infiltration curves was obtained; the largest deviations weregenerally less than 60 ml, which constituted only about 0.5% of the total infiltration volume. Figure9.34 shows a comparison of the parameter estimation results against results obtained using
Wooding's analysis. Both methods give almost identical unsaturated hydraulic conductivities forpressure heads between -2 and -10.25 cm. However, the hydraulic conductivity at the highest
pressure head interval was overestimated by a factor of two using Wooding's analysis. Šimůnek et
al. [1998] further compared the numerical inversion results with hydraulic properties estimated fromavailable soil textural information using a neural-network-based pedotransfer function approach.

167

Relatively good agreement between the inverse and neural network predictions was obtained.

Figure 9.33. Measured and optimized cumulative infiltration curves for a tension disc infiltrometer
experiment (example 9).
0.005
Numerical Optimization

0.004
K [cm/s]Wooding's Analysis
0.003
0.002
0.001
0.000
-1

-0.5

0

0.5

1

1.5

2

2.5

3

log(|h| [cm])Figure 9.34. Unsaturated hydraulic conductivities at particular pressure heads, calculated using
Wooding's analytical solution and the complete function obtained by numerical inversion (example
9).

168

9.2.2. Example 10 - Cone Penetrometer
While tension infiltrometer experiments provide relatively quick estimates of the hydraulicproperties, they can be used only at the soil surface. By comparison, a new cone penetrometer
method currently under development [Gribb, 1996; Gribb et al., 1998; Kodešová et al., 1998] canbe used at depth. Cone penetrometers were originally used to obtain soil strength characteristics bymeasuring the tip resistance and sleeve friction during penetration at a constant rate. To obtain thehydraulic properties, a modified cone penetrometer, instrumented with a porous filter close to thepenetrometer tip and two tensiometer rings 5 and 9 cm above the filter, is used (Fig. 9.35). The
device is pushed into soil to the desired depth, and a constant head is applied to the 5-cm filter. The
volume of water imbibed into the soil is monitored, as are tensiometer ring readings registering theadvancement of the wetting front for a short period of time (300-500 s).
Gribb [1996] gave a detailed numerical analysis of this experiment, including a study of theidentifiability of the soil hydraulic parameters. She showed that the inverse solution was leastsensitive to n and θs, and most sensitive to Ks and α. The method was recently used to estimate thehydraulic parameters of sandy soil in a laboratory aquifer system measuring 5 x 5 x 3 m [Gribb etal., 1998; Kodešová et al., 1998; Šimůnek et al., 1999].
Figure 9.36 shows observed flow data, as well as results of the numerical inversion.
Excellent agreement between measured and optimized values was obtained for the inverse solution
with four optimized parameters. Gribb et al. [1998] and Kodešová et al. [1998] presented retention
curves obtained with selected laboratory methods and the parameter estimation technique. The
optimized curves were close to the wetting curves determined in the laboratory. Gribb et al. [1998]
showed that the estimated saturated hydraulic conductivities were similar to those obtained withother test methods, such as the Guelph permeameter, slug tests, and laboratory constant head tests.
Finally, Šimůnek et al. [1999] further used the redistribution part of the experiment to also estimate
hysteresis in the soil water retention curve.

169

Water source(tank)

Bleed and fill port
for tensiometers
Laptopcomputer
Signal conditioner
and A/D converted

Internally threaded to allow
addition of extension tubes

Tensiometer rings

ScreenFigure 9.35. Schematic of the modified cone penetrometer (example 10).

Figure 9.36. Comparison of observed and optimized cumulative infiltration curves and tensiometerreadings for the modified cone penetrometer test (example 10).

170

9.2.3. Example 11 - Multiple-Step Extraction Experiment
The tension infiltrometer and cone penetrometer methods provide information about thewetting branches of the soil hydraulic properties. By comparison, a multiple-step extraction device
can be used to obtain draining branches. The device consists of a ceramic soil solution sampler (Fig.
9.37), which is inserted into an initially wet soil profile and subjected to a series of vacuum
extraction pressures. The cumulative amount of soil solution extracted during an experiment, as wellas pressure heads at various locations near the extraction device, are monitored during theexperiment and subsequently used in an objective function for the nonlinear minimization problem.
Inoue et al. [1998] first evaluated the feasibility of the vacuum extraction technique using
numerically generated data and concluded that the method is well suited for loamy-textured soils,
but not necessarily for sandy soils. They tested the method in the laboratory and in the field. Here
we briefly discuss one laboratory application.
The experiment [Inoue et al., 1998] was carried out on a Columbia fine sandy loam. The
center of the ceramic ring (with a radius of 3 cm, and a length of 3 cm) was located 6.3 cm below
the soil surface. The tensiometers were installed at the following positions: T1(r,z)=(4,-6.3 cm),
T2=(6,-6.3), T3=(6,-24). Five vacuum extraction steps were applied: hex=-35 cm for 25 h, hex=-65 cm
for 25 < t < 73 h, hex=-125 cm for 73 < t < 217 h, hex=-240 cm for 217 < t < 339 h, and finally hex=480 cm for 339 < t < 605 h. Experimental data, as well as the final results of the numericalinversion, are presented in Figure 9.38. The saturated hydraulic conductivity of the ceramic ring was
optimized simultaneously with van Genuchten's hydraulic parameters. The agreement between
measured and calculated values in Figure 9.38 was relatively good. Inoue et al. [1998] compared the
optimized soil hydraulic functions with independently measured data.

171

Figure 9.37. The layout of the laboratory multistep extraction experiment (example 11).

Figure 9.38. Comparison of measured (symbols) and optimized (lines) cumulative extraction (a) andpressure head (b) values (example 11).

172

10. INPUT DATA
The input data for HYDRUS are given in seven separate input files. These input files consistof one or more input blocks identified by the letters from A through O. The input files and blocks
must be arranged as follows:
SELECTOR.IN
A. Basic Information
B. Material Information
C. Time Information
D. Solute Transport Information
E. Heat Transport Information
F. Root Water Uptake Information
O. Root Growth
MESHTRIA.TXT
G. Finite Element Mesh Information
DOMAIN.DAT
H. Nodal Information
I. Element Information
BOUNDARY.IN
J. Boundary Information
ATMOSPH.IN
K. Atmospheric Information
DIMENSIO.IN
L. Dimension Information
FIT.IN
M. Inverse Solution Information
CROSSSECT.IN
N. Mesh Lines Information

The various input blocks are described in detail below. Tables 10.1 through 10.16 describe
the data required for each input block. All data are read using list-directed formatting (free format).
Comment lines are provided at the beginning of and within each input block to facilitate, among
other things, proper identification of the function of the block and the input variables. The comment
lines are ignored during program execution; hence, they may be left blank but should not be173omitted. All input files must be placed into one subdirectory. Output files are printed into the same
subdirectory. An additional file Level_01.dir, which specifies the path to the input and output filesubdirectory, must be given in the same directory as the executable HYDRUS codes if numericalsolvers are to be run outside of the GUI. The program assumes that all input data are specified in aconsistent set of units for mass M, length L, and time T. The values of temperature should bespecified in degrees Celsius.
Most of the information in Tables 10.1 through 10.16 should be self-explanatory. Several
input files can be alternatively entered using binary files, e.g., the DOMAIN.DAT file can be
replaced with the binary DOMAIN.IN file, and the MESHTRIA.TXT file can be replaced with the
MESHTRIA.000 (or MESHGEN2.PMG) binary file. Binary files are utilized by default when the
HYDRUS user interface is used.

174

Table 10.1. Block A - Basic information.
RecordType

Variable

Description

1

Char

cVersionSet text equal to "Pcp_File_Version=".

1

IntegeriVersion

Version of the Selector.in file; set equal to 3 for current version.

2,3

-

-

Comment lines.

4

Char

Hed

Heading.

5--

Comment line.

6

Char

LUnit

Length unit (e.g., 'cm').

7

Char

TUnit

Time unit (e.g., 'min').

8

Char

MUnit

Mass unit for concentration (e.g., 'g', 'mol', '-').

9

-

-

Comment line.

10

Integer

Kat

Type of flow system to be analyzed:
0 for a horizontal (areal) system
1 for axisymmetric flow
2 for vertical flow in a cross-section
Records 9 and 10 are provided only for two-dimensional problems.

11

-

-

Comment line.

12

Integer

MaxIt

Maximum number of iterations allowed during any time step (usually 10).

12

Real

TolTh

Absolute water content tolerance for nodes in the unsaturated part of the flow
region [-] (its recommended value is 0.001). TolTh represents the maximum
desired absolute change in the value of the water content, θ, between two
successive iterations during a particular time step.

12

Real

TolH

Absolute pressure head tolerance for nodes in the saturated part of the flow region[L] (its recommended value is 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.

12

LogicallInitW

Set this logical variable equal to .true. when the initial condition is specified interms of the pressure head.
Set this logical variable equal to .false. when the initial condition is specified interms of the water content.

13

-

-

Comment line.

14

LogicallWat

Set this logical variable equal to .true. when transient water flow is considered.
Set this logical variable equal to .false. when steady-state water flow is to be
calculated.

14

LogicallChem

Set this logical variable equal to .true. if solute transport is to be considered.

175

Table 10.1. (continued)

RecordType

Variable

Description

14

LogicalSinkF

Set this variable equal to .true. if water extraction by roots from the root zone is
imposed.

14

LogicalShortF.true. if the information is to be printed only at preselected times, but not at eachtime step (T-level information, see Section 11.1),.false. if the information is to be printed at each time step or at each n-time steps.

14

LogicallPrintD

Set this logical variable equal to .true. if the information about the pressure
heads, water contents, temperatures, and concentrations in observation nodes,
and the water and solute fluxes is to be printed at a constant time interval
tPrintInterval.

14

LogicallScreenSet this logical variable equal to .true. if the information is to be printed on thescreen during code execution.

14

LogicalAtmInf.true. if atmospheric 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).

14

LogicallTemp

Set this logical variable equal to .true. if heat transport is to be considered.

14

LogicallWDep.true. if hydraulic properties are to be considered as temperature-dependent.
.false. otherwise (see Section 2.5).

14

LogicallEquil.true. if equilibrium solute transport is considered.
.false. if nonequilibrium solute transport is considered for at least one solute
species.

14

LogicallExter

Set this logical variable equal to .true. if an external mesh generator is to be used
to generate the finite element mesh (i.e., when using MESHGEN-2D).
Set this logical variable equal to .false. if an internal mesh generator for simple
quadrilateral domains is to be used to generate the finite element mesh.

14

LogicallInv

Set this logical variable equal to .true. if the soil hydraulic or solute transportparameters are to be estimated from measured data (available only for 2D
applications).
Set this logical variable equal to .false. if only the direct solution for a particular
problem is to be carried out.

15

-

-

Comment line.

16

LogicallUnsatch

Set this logical variable equal to .true. if the major ion chemistry module(Unsatchem) is to be used to simulate solute transport (only in 2D).

16

LogicallCFTr

Set this logical variable equal to .true. if the colloid-facilitated solute transportmodule (C-Ride) is to be used to simulate colloid transport and colloidfacilitated solute transport (only in 2D).

176

Table 10.1. (continued)

RecordType

Variable

Description

16

LogicallHP2

Set this logical variable equal to .true. if the general biogeochemical solutetransport module (HP1, resulting from coupling HYDRUS with PHREEQC) is
to be used to simulate solute transport (only in 2D).

16

LogicallActRSU

Set this logical variable equal to .true. if the Active Solute Transport model is to
be used (only in 2D) [Šimůnek and Hopmans, 2009].

16

LogicallRootGrowth Set this logical variable equal to .true. if the root growth is to be simulated(available only in 2D).

16

LogicallDummyLogical dummy variable (repeat 7 times).

17

-

-

Comment line.

17

IntegernPrintSteps

Information to the screen and output files is not printed at each time step, but
after each nPrintSteps.17Real
tPrintInterval
A constant time interval,
after which the information about the pressure heads, water contents,
temperatures, and concentrations in observation nodes, and the water andsolute fluxes is to be printed.

17

LogicallEnter

Set this logical variable equal to .true. if the Enter key is to be pressed at theend of the simulation.

177

Table 10.2. Block B - Material information.
RecordType

Variable

Description

1,2

-

-

Comment lines.

3

Integer

NMat

Number of soil materials. Materials are identified by the material number,
MatNum, specified in Block H.

3

Integer

NLay

Number of subregions for which separate water balances are being computed.
Subregions are identified by the subregion number, LayNum, specified in Block I.

3

Real

ha

Absolute value of the upper limit [L] of the pressure head interval below which atable of hydraulic properties will be generated internally for each material (ha must
be greater than 0.0; e.g., 0.001 cm) (see Section 5.3.11). When the Brooks andCorrey model is used, the internal interpolation tables should be either disabled,or the absolute value of the upper limit should be set larger than 1/α (e.g., when
α =0.09, ha >1/0.09=11.1 cm).

3

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 forthe hydraulic properties are computed directly from the hydraulic functions (i.e.,
without interpolation in the table). Zero values for both ha and hb should bespecified when interpolation tables are not to be used.

3

Integer

NAniz

Number of anisotropy tensors used in the transport domain (minimum number is1). This value needs to be provided only for three-dimensional applications.

4

-

-

Comment line.

5

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 an air-entry value of -2 cm and with six
parameters.
= 4; Kosugi’s [1996] model with six parameters.
= 5; a dual-porosity model of Durner [1994] with nine parameters.
= 6; a dual-porosity system with transfer proportional to the effective saturation(9 parameters) (see Sections 2.1.2 and 2.8).
= 7; a dual-porosity system with transfer proportional to the pressure head (11parameters) (see Sections 2.1.2 and 2.8).
= 9; look-up tables of soil hydraulic properties. See Table 10.15 for theformatting of the Mater.in file that is required when this option is selected.

5IntegeriHyst

Hysteresis in the soil hydraulic properties:= 0; No hysteresis= 1; Hysteresis in the retention curve only178

Table 10.2. (continued)

RecordType

Variable

Description= 2; Hysteresis in both the retention and hydraulic conductivity functions= 3; Hysteresis using Bob Lenhard’s model [Lenhard et al., 1991; Lenhard andParker, 1992].

6

-

-

Comment line.

7

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 6 and 7 are provided only when iHyst > 0.

8

-

-

Comment line.

9
9
9
9
9
9

Real
Real
Real
Real
Real
Real

Par(1,M)
Par(2,M)
Par(3,M)
Par(4,M)
Par(5,M)
Par(6,M)

Parameter θr for material M [-].
Parameter θs for material M [-].
Parameter α for material M [L-1].
Parameter n for material M [-].
Parameter Ks for material M [LT-1].
Parameter l for material M [-].

9
9
9
9

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].

9
9
9
9

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].

9

Real

Par(7,M)

9
9

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 (2.77)) for material M [-].

Par(7,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 [-].

9
9
9

9

Real
Real
Real

Real

179

Table 10.2. (continued)

RecordType

Variable

Description

9
9
9
9

Real
Real
Real
Real

Par(8,M)
Par(9,M)
Par(10,M)
Par(11,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 (2.81)) for material M [-].
Record 9 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.

10

-

-

Comment line.

11

Real

c11(i)

11
11
11
11
11
11
11
11

Real
Real
Real
Real
Real
Real
Real
Real

c22(i)
c33(i)a11(i)
a22(i)a33(i)a12(i)a13(i)a23(i)First principal component, K1A, of the first dimensionless tensor KA which
describes the local anisotropy of the hydraulic conductivity.
Second principal component, K2A.
Third principal component, K3A.
Cosines of the angle between K1A and the x-coordinate axis.
Cosines of the angle between K2A and the y-coordinate axis.
Cosines of the angle between K3A and the z-coordinate axis.
Cosines of the angle between K1A and the y-coordinate axis.
Cosines of the angle between K1A and the z-coordinate axis.
Cosines of the angle between K2A and the z-coordinate axis.
Record 11 information is provided for each anisotropy tensor (i from 1 to NAniz).
Records 10 and 11 are provided only for three-dimensional problems.

180

Table 10.3. Block C - Time information.
RecordType

Variable

Description

1,2

-

-

Comment lines.

3

Real

dt

Initial time increment, ∆t [T]. The initial time step should be estimated based on
the problem being solved. For problems with high-pressure gradients (e.g.,
infiltration into an initially dry soil), ∆t should be relatively small.

3

Real

dtMin

Minimum permitted time increment, ∆tmin [T].

3

Real

dtMax

Maximum permitted time increment, ∆tmax [T].

3

Real

dMul

If the number of required iterations at a particular time step is less than or equal
to ItMin, then ∆t 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 ∆t 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 or equal
to ItMin, then ∆t 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 ∆t 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 the pressure
head, water content, flux, temperature, concentrations, and the water and solutebalances 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
7
.
.
7

Real
Real
.
.
Real

TPrint(1)First specified print-time [T].
TPrint(2)Second specified print-time [T].
.
.
.
.
TPrint(MPL) Last specified print-time [T]. (Maximum six values on one line.)

181

Table 10.4. Block D - 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

3

LogicallArtD.true. if the upstream weighing formulation is to be used.
.false. if the original Galerkin formulation is to be used.
.true. if artificial dispersion is to be added to fulfill the stability criterion PeCr
(see Section 6.4.6).
.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 10 (set equal to zero if nonlinear adsorption is notconsidered).

3

Real

PeCr

Stability criteria (see Section 6.4.6). Set equal to zero when lUpW is equal to.true..

3

IntegerNS

Number of solutes in a chain reaction.

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

IntegerlBacter

Set equal to .true. if the attachment/detachment approach is to be used to
calculate nonequilibrium transport of viruses, colloids, or bacteria. Set equal
to .false. 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. (3.28)).

3
4

Integer-nChPar-Number of solute transport parameters specific for each solute.
Comment line.

5

LogicallMoistDep.true. if at least one reaction coefficient (ChPar) is water content dependent.
.false. otherwise.

182

Table 10.4. (continued)

RecordType

Variable

Description

5

LogicallMassIni.true. if the initial condition is given in the total mass of solute per unit volume
of soil [ML-3] rather than in liquid concentrations.
.false. if initial condition is given in liquid concentrations [ML-3].

5

LogicallEqInit.true. if initial concentrations in nonequilibrium phases (e.g., concentrations in
the immobile water or kinetically sorbed concentrations) are to be in
equilibrium with the liquid phase concentrations.
.false. if the initial conditions in non-equilibrium phases are specified
manually.

5

LogicallTortM.true. if Per Moldrup’s tortuosity models [Moldrup et al., 1997, 2000] are to
be used to calculate the tortuosity factor.
.false. if the Millington and Quirk [1961] model is to be used to calculate thetortuosity factor.

5

LogicallFumig.true. when additional options related to fumigant transport (e.g., tarp removal,temperature-dependent tarp properties, injection of fumigants) are to be used.
Records 4 and 5 information is provided only for version 2.0 and higher.

6

-

-

Comment line.

7
7
7
7

Real
Real
Real
Real

ChPar(1,M)
ChPar(2,M)
ChPar(3,M)
ChPar(4,M)

7

Real

ChPar(5,M)

Bulk density of material M, ρ [ML-3].
Longitudinal dispersivity for material type M, DL [L].
Transverse dispersivity for material type M, DT [L].
Dimensionless fraction of the sorption sites classified as type-1, i.e., sites withinstantaneous sorption, when the chemical nonequilibrium option isconsidered. Set equal to 1 if equilibrium transport is to be considered. The
dimensionless fraction of the sorption 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.
Immobile water content. Set equal to 0 when the physical nonequilibrium
option is not considered.
Record 7 information is provided for each material M (from 1 to NMat).

8

-

-

Comment line.

9
9

Real
Real

ChPar(6,M)
ChPar(7,M)

Ionic or molecular diffusion coefficient in free water, Dw [L2T-1].
Ionic or molecular diffusion coefficient in gas phase, Dg [L2T-1].

10

-

-

Comment line.

11

Real

ChPar(8,M)

11

Real

ChPar(9,M)

11

Real

ChPar(10,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.

183

Table 10.4. (continued)

RecordType

Variable

Description

11

Real

ChPar(11,M)

11
11
11
11

Real
Real
Real
Real

ChPar(12,M)
ChPar(13,M)
ChPar(14,M)
ChPar(15,M)

11
11
11
11
11
11

Real
Real
Real
Real
Real
Real

ChPar(16,M)
ChPar(17,M)
ChPar(18,M)
ChPar(19,M)
ChPar(20,M)
ChPar(21,M)

Equilibrium distribution constant between liquid and gas phases, kg, material
type M [-].
First-order rate constant for dissolved phase, µw, material type M [T-1].
First-order rate constant for solid phase, µs, material type M [T-1].
First-order rate constant for gas phase, µg, material type M [T-1].
Rate constant, µw’, representing a first-order decay for the first solute and zeroorder 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 dissolved phase, γw, material type M [ML-3T-1].
Zero-order rate constant for solid phase, γs, material type M [T-1].
Zero-order rate constant for gas phase, γg, material type M [ML-3T-1].
First-order rate coefficient for nonequilibrium adsorption, or the mass transfer
coefficient for solute exchange between mobile and immobile liquid regions,ω, material type M [T-1].
Parameters ChPar(14,M) through ChPar(21,M) are reinterpreted when thevariable iBacter=1.

11Integer

ChPar(14,M)

11
11

Integer
Real

ChPar(15,M)
ChPar(16,M)

11

Real

ChPar(17,M)

11

Real

ChPar(18,M)

11
11

Real
Real

ChPar(19,M)
ChPar(20,M)

11

Real

ChPar(21,M)

11
11
11

Integer
Integer
Real

ChPar(14,M)
ChPar(15,M)
ChPar(16,M)

Type of blocking, iPsi, applied in (3.23) on the second sorption sites.
= 0: No blocking.
= 1: Langmuirian dynamics, (3.24).
= 2: Ripening, (3.25).
= 3: random sequential adsorption model, (3.26).
= 4: depth-dependent blocking coefficient, (3.27).
ditto, for the first sorption sites.
Parameter in the blocking function for the second sorption sites (smax for(3.24), (3.25) and (3.26), β in (3.27)).
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.
Parameters ChPar(14,M) through ChPar(21,M) are further reinterpreted
when the variable iBacter=1 and lFilter=.true..
Diameter of the sand grains, dc [L].
Diameter of the particle, dp (e.g., virus, bacteria) (=0.95µm=0.95e-6 m) [L].
Parameter smax in the blocking function for the second sorption sites (3.24).

184

Table 10.4. (continued)

RecordType

Variable

Description

11
11

Real
Real

ChPar(17,M)
ChPar(18,M)

11
11
11

Real
Real
Real

ChPar(19,M)
ChPar(20,M)
ChPar(21,M)

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 (3.24).
Sticking efficiency, α [-], for the first sorption sites.
The first-order entrainment (detachment) coefficient, kd [T-1], for the firstsorption sites.
Record 11 information is provided for each material M (from 1 to NMat).
Record 7 through 11 information are provided for each solute (from 1 to NS).

12,13

-

-

Comment lines.

14

Real

TDep(6)

14

Real

TDep(7)

Activation energy for parameter ChPar(6,M) [ML2T-2M-1] (See Section 3.4).
This parameter should be specified in J mol-1. Set equal to 0 if ChPar(6,M) istemperature independent.
Same for parameter ChPar(7,M) [ML2T-2M-1].

15

-

-

Comment line.

16
.
.
16

Real
.
.
Real

TDep(8).
.
TDep(21)Same for parameter ChPar(8,M) [ML2T-2M-1].
.
.
Same for parameter ChPar(21,M) [ML2T-2M-1].
Records 13 through 16 information are provided only when the logicalvariable lTDep of record 3 is set equal to .true..

15

-

-

Comment line.

16

Real

cBound(1,1)

Value of the concentration for the first time-independent BC [ML-3]. Set equal
to zero if no KodCB(n)=±1 is specified.

16

Real

cBound(1,2)

.
.
16

.
.
Real

.
.
cBound(1,5)

Value of the concentration for the second time-independent BC [ML-3]. Set
equal to zero if no KodCB(n)= ±2 is specified.
.
.
Value of the concentration for the fifth time-independent BC [ML-3]. If water
uptake is specified, then cBound(1,5) is automatically used for theconcentration of water removed from the flow region by root water uptake[ML-3]. Set equal to zero if no KodCB(n)=±5 and no root solute uptake is
specified.

16

Real

cBound(1,6)

Value of the concentration for the sixth time-independent BC [ML-3]. If
internal sources are specified, then cBound(1,6) is automatically used for theconcentration of water injected into the flow region through internal sources[ML-3]. Set equal to zero if no KodCB(n)=±6 and no internal sources are
specified.

185

Table 10.4. (continued)

RecordType

Variable

Description

16

Real

cBound(1,7)

Concentration of the incoming fluid in equation (3.45) [ML-3]. Set equal tozero if no KodCB(n)=-7 is specified.

16

Real

cBound(1,8)

Concentration above the stagnant boundary layer, gatm (see equation (3.45))[ML-3]. Set equal to zero if no KodCB(n)=-7 is specified.

16

Real

cBound(1,9)

Thickness of the stagnant boundary layer, d [L] (see equation (3.45)). Set
equal to zero if no KodCB(n)=-7 is specified.
Record 16 information is provided for each solute (from 1 to NS).

17

-

-

Comment line.

18

Real

tPulseTime duration of the concentration pulse [T]. Concentrations along all
boundaries, for which no time-variable boundary conditions are specified,
are set equal to zero for times larger this time. (when lFumigant=.true., thevariable is interpreted as a time of removal of the surface tarp).

19

-

-

Comment line.

20

LogicallAddFum.true. if additional application (injection) of fumigant is to be considered.
.false. otherwise.
Records 19 and 20 information is provided only when lFumig=.true.

21

-

-

Comment line.

22

Real

tAddFum

Time of fumigant application/injection.

22

Real

FumMass

Mass of fumigant application/injection ([M/L-1] in 2D and [M] in 3D).

23

-

-

Comment line.

24

Real

xAddFL

Minimum x-coordinate of fumigant application/injection.

24

Real

xAddFR

Miximum x-coordinate of fumigant application/injection.

24

Real

zAddFB

Minimum z-coordinate of fumigant application/injection.

24

Real

zAddFT

Miximum z-coordinate of fumigant application/injection.

24

Real

yAddFF

Minimum y-coordinate of fumigant application/injection (only for 3D).

24

Real

yAddFB

Miximum y-coordinate of fumigant application/injection (only for 3D).

25,26

-

-

Comment lines.

27

IntegernPar

Number of parameters for which water content dependence is specified (bydefault, this is set to 9).

28

-

-

Comment line.

29

Real

WDep(1,1)

The exponent, B, of Walker’s equation for the first reaction coefficient [-]. Set
equal to 0 if this coefficient is water content independent.

186

Table 10.4. (continued)

RecordType

Variable

Description

.
.
29

.
.
Real

.
.
WDep(1,9).
.
The exponent, B, of Walker’s equation for the ninth reaction coefficient [-].
Set equal to 0 if this coefficient is water content independent.

30

Real

WDep(2,1)

.
.
30

.
.
Real

.
.
WDep(2,9)

The reference pressure head, hr (to be converted into the reference watercontent, θr, of Walker’s equation) for the first reaction coefficient [L]. Set
equal to 0 if this coefficient is water content independent.
.
.
The reference pressure head, hr (to be converted into the reference watercontent, θr, of Walker’s equation) for the first reaction coefficient [L]. Set
equal to 0 if this coefficient is water content independent.
Records 28 through 30 information are provided for each solute.
Records 25 through 30 information are provided only when the logicalvariable lMoistDep of record 5 is set equal to .true..

+

Block D is not needed when the logical variable lChem in Block A is set equal to .false. .

187

Table 10.5. Block E - Heat transport information.+

RecordType

SymbolDescription

1,2

-

-

Comment lines.

3
3
3
3
3

Real
Real
Real
Real
Real

TPar(1,M)
TPar(2,M)
TPar(3,M)
TPar(4,M)
TPar(5,M)

3

Real

TPar(6,M)

3

Real

TPar(7,M)

3

Real

TPar(8,M)

3

Real

TPar(9,M)

3

Real

TPar(10,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].
Transverse thermal dispersivity of material M, λT [L].
Coefficient b1 in the thermal conductivity function [MLT-3K-1] (e.g.Wm-1K-1)(see equation (4.5)).
Coefficient b2 in the thermal conductivity function [MLT-3K-1] (e.g.Wm-1K-1)(see equation (4.5)).
Coefficient b3 in the thermal conductivity function [MLT-3K-1] (e.g.Wm-1K-1)(see equation (4.5)).
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

TBound(1)Value of the first time-independent thermal boundary condition [0C]. Set
equal to zero if no KodTB(n)=±1 is specified.

5

Integer

TBound(2)

.
.
5

.
.
Integer.
.
TBound(6)Value of the second time-independent thermal boundary condition [0C]. Set
equal to zero if no KodTB(n)=±2 is specified.
.
.
Value of the sixth time-independent thermal boundary condition [0C]. Set
equal to zero if no KodTB(n)=±6 is specified. If internal sources are specified,then TBound(6) is automatically used for the temperature of water injected into
the flow region through an internal source.

6

-

-

Comment line.

7

Real

Ampl

Temperature amplitude at the soil surface [K] prescribed for nodes where
Kode(n)=±4. Set equal to zero when no Kode(n)=±4 is specified. Note that
this value should be zero when detailed temperature values (e.g., during theday) are provided.

7

Real

tPeriod

Time interval for the completion of one temperature cycle (usually 1 day) [T].

+Block E need not be supplied if logical variables lTemp (Block A) is set equal to .false.

188

Table 10.6. Block F - Root water uptake information.+

RecordType

Variable

Description

1,2

-

-

Comment lines.

3

Integer iMoSink

Type of root water uptake stress response function.
= 0; Feddes et al. [1978]= 1; S-shaped, van Genuchten [1987]

3

Real

OmegaC

Critical root water uptake index. Set equal to zero for a noncompensated root
water uptake and smaller than zero for compensated root water uptake [-].

4

-

-

Comment line.

5a

Real

P0

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
5a

Real
Real

P2L
P3

As above, but for a potential transpiration rate of r2L.
Value of the pressure head, h4, below which root water uptake ceases (usuallyequal to the wilting point).

5a
5a

Real
Real

r2H
r2L

Potential transpiration rate [LT-1] (currently set at 0.5 cm/day).
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 thepotential transpiration rate, Tp (h3 presumably decreases at higher transpiration
rates). HYDRUS currently implements the same linear interpolation scheme asused in several versions of the SWATRE code (e.g., Wesseling and Brandyk[1985]) and in the SWMS_2D [Šimůnek et al., 1992]. The scheme is based on
the following interpolation:

The following records (records 5a, 6a, 7a) are given only if iMoSink=0.
Value of the pressure head, h1 (Fig. 2.1), below which roots start to extract
water from the soil. Set equal to zero when root water uptake is allowed at fullsaturation.

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

for T a ≥ r2H

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). Set equal to zero when root water
uptake is allowed at full saturation.

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.

189

Table 10.6. (continued)

RecordType

Variable

Description

5b

Real

P50

5b

Real

P3

5b

Real

PW

Value of the pressure head, h50 (Fig. 2.1), at which the root water uptake isreduced by 50%.
Exponent p in the S-shaped root water uptake stress response function. The
recommended value is 3.
Wilting point, i.e., the pressure head below which the root water uptake ceases.

8

-

-

The following records are given only if lChem=.true..
Comment line.

9

Logical lSolRed=.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 head stress.

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 (2.8) or (2.9),
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.
The 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 of Maas
[1990], i.e., lMsSink=.false..

=α ( hφ ) 1

hφ ≤ hφ < 0M=
α ( hφ ) max[0 ,1. - 0.01( hφ - hφ ) s φ ]
M

13a

Real

c50

hφ < hφM

Value of the minimum osmotic head (the salinity threshold) hφM, above whichroot water uptake is not reduced. This value is specified only whenlSolAdd=.false..

190

Table 10.6. (continued)

RecordType

Variable

Description

13a

Real

P3c

Slope, sφ, of the curve determining fractional root water uptake decline per unit
increase in salinity below the threshold. This value is specified only whenlSolAdd=.false..

13

Real

aOsm(1)

Osmotic coefficient, a1, for the first solute [L4M-1].

13

13

Real
.
.
Real

aOsm(2).
.
aOsm(NSD)

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 non-compensated root
solute uptake and smaller than one for compensated root solute 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 F is not needed when the logical variable SinkF (Block A) is set equal to .false..

191

Table 10.7. Block O - Root growth information.+

RecordType

Variable

Description

1,2

-

-

Comment lines.

3

IntegeriRShapeType of the root distribution function to be used
=0: The Vrugt function is used.
=1: The van Genuchten-Hoffman function is used.
=2: Constant root distribution.

3

IntegeriRootGrowth

Consider or not the root growth.
=0: No root growth is considered.
=2: The rooting depth is time-variable, described using a table.
=2: The rooting depth (and horizontal extent) is time variable, described using
Verhulst-Pearl logistic growth function.

4

-

-

Comment lines.

5

LogicallHoriz5

IntegernPlantThe Vrugt function is used also for horizontal distribution, or a limited
horizontal extent of the root zone for iRShape =1 or 2.
Number of plants
The following four lines (6-9) are given only when iRShape =0.

6

-

-

Comment line

7
7
7

Real
Real
Real

rZm
rZ0
rA

The maximum rooting lengths in the z-direction [L]The depth of Maximum Intensity [L]The empirical factor pz.
The following two lines are given only when lHoriz = .true.

8

-

-

Comment line

9
9
9
9

Real
Real
Real
Real

rRm
The maximum rooting lengths in the x-direction [L]rR0
The Radius of Maximum Intensity [L]rB
The empirical factor px .
rCenter(nPlant)
x-coordinate of the center in the horizontal direction (for nPlants).
The following two lines (6-7) are given only when iRShape =1 or 2.

6

-

-

Comment line

7

Real

rZm

The maximum rooting lengths in the z-direction [L]The following values on line 7 are given only when lHoriz =.true..

7
7

Real
Real

rRm
The maximum rooting lengths in the x-direction [L]rCenter(nPlant)
x-coordinate of the center in the horizontal direction (for nPlants).
The following four lines (10-13) are only given when iRootGrowth =1 (Rootgrowth given in a Table).

10

-

-

Comment line

192

Table 10.7. (continued)

RecordType

Variable

Description

11

IntegernGrowthNumber of records with the rooting depth data.

12

-

-

Comment line
The following line is repeated nGrowth times.

13
13
13

Real
Real
Real

Time
RootDepth
RootDepth

Time of a rooting depth data [t].
Rooting depth data (z-direction) [t].
Rooting radius data (x-direction) [t].
The following six lines (10-15) are given only when iRootGrowth =1.

10

-

-

Comment line

11

Integer

iRFak

11
11
11
11

Real
Real
Real
Real

tRMin
tRMed
tRHarvtRPeriod

Method to calculate the root growth factor, r.
= 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.
Initial time of the root growth period [T].
Time of known rooting depth (set equal to zero if iRFak=1) [T].
Time at the end of the root water uptake period [T].
Time period at which the growth function repeats itself.

12

-

-

Comment line

13

Real

rZMin

13
13

Real
Real

rZMed
rZMax

Initial value of the rooting depth at the beginning of the growth period(recommended value = 1 cm) [L].
Value of known rooting depth (set equal to zero if iRFak=1) [L].
Maximum rooting depth, which may be reached at infinite time [L].
The following two lines (14-15) are given only when lHoriz = .true.

14

-

-

Comment line

15

Real

rRMin

15

Real

rRMax

Initial value of the rooting radius at the beginning of the growth period(recommended value = 1 cm) [L].
Maximum rooting radius, which may be reached at infinite time [L].

+Block O is not needed when the logical variable lRootGrowth (Block A) is set equal to .false..

193

Table 10.8a. Block G - Finite element mesh information for two-dimensional applications.
RecordType

Variable

Description

1

Char

cVersionSet text equal to "Pcp_File_Version=".

1

IntegeriVersion

Version of the Meshtria.txt file; set equal to 1 for current version.

2

IntegeriDummyDummy variable. Set equal to 1.

2

Integer

NumNP

Number of nodal points.

2

IntegernEdgesNumber of mesh edges.

2

Integer

NumEl

Number of finite elements (only triangles are allowed).

3

IntegernNodal number.

3

Real

x(n)

x-coordinate of node n [L] (always a horizontal coordinate).

3

Real

z(n)

z-coordinate of node n [L]. z is the vertical coordinate for problems involving
vertical planar or axisymmetric flow. For axisymmetric flow, z coincides withthe vertical axis of symmetry.
Record 3 information is required for each node n. The numbering of the nodes isarbitrary.

4

-

-

Comment line.

555Integer
Integer
IntegereiEpb(e)iEpe(e)Edge number. Not read by the calculation module.
Number of the beginning node of edge e. Not read by the calculation module.
Number of the ending node of edge e. Not read by the calculation module.
Record 5 information is required for each edge e. Record 5 information is notread by the calculation module.

6

-

-

Comment line.

7

Integert

Triangular element number.

7

Integer

KX(t,1)

Global nodal number of the first triangle's vertex.

7

Integer

KX(t,2)Global nodal number of the second triangle's vertex.

7

Integer

KX(t,3)Global nodal number of the third triangle's vertex.
Record 7 information is required for each triangle t.

194

Table 10.8b. Block G - Finite element mesh information for three-dimensional applications.
RecordType

Variable

Description

1

Char

cVersionSet text equal to "Pcp_File_Version=".

1

IntegeriVersion

Version of the Meshtria.txt file; set equal to 1 for current version.

2

-

-

Comment line.

3

-

-

Comment line.

4

IntegerlGenMeshSet equal to 0 if simple structured finite element mesh for hexahedral transportdomains is used.
Set equal to 1 if unstructured finite element mesh is used.

5

-

-

Comment line.

6

Integer

NumNP

Number of nodal points.

6

Integer

NumEl

Number of finite elements (only tetrahedrals are allowed).

6

Integer

IJ

Number of nodes on the base of the hexahedral domain. Set equal to zero if a
simple hexahedral domain is not used.

6

IntegernNx

Number of nodes in the x-direction of the hexahedral domain. Set equal to zero
if a simple hexahedral domain is not used.

6

IntegernNy

Number of nodes in the y-direction of the hexahedral domain. Set equal to zero
if a simple hexahedral domain is not used.

6

IntegernNz

Number of nodes in the z-direction of the hexahedral domain. Set equal to zero if
a simple hexahedral domain is not used.

7

-

-

Comment line.

8

IntegernNodal number.

8

Real

x(n)

x-coordinate of node n [L].

8

Real

y(n)

y-coordinate of node n [L].

8

Real

z(n)

z-coordinate of node n [L]. z is the vertical coordinate.
Record 3 information is required for each node n.

9

-

-

Comment line.

10

-

-

Comment line.

11

Integert

Tetrahedral element number.

11
11
11
11
11
11

Integer
Integer
Integer
Integer
Integer
Integer

KX(e,1)KX(e,2)KX(e,3)KX(e,4)KX(e,5)KX(e,6)

Global nodal number of the first corner node i.
Global nodal number of the second corner node j.
Global nodal number of the third corner node k.
Global nodal number of the forth corner node l.
Global nodal number of the fifth corner node m.
Global nodal number of the sixth corner node n.

195

Table 10.8b. (continued)

RecordType

Variable

Description

11
11

Integer
Integer

KX(e,7)KX(e,8)11

Integer

KX(e,9)Global nodal number of the seventh corner node o.
Global nodal number of the eighth corner node p. Indices i, j, k, l, m, n, o, and p,
refer to the corner nodes of an element e taken in a certain orientation as
described in Section 6.1. KX(e,5) for tetrahedral and KX(e,7) for triangularprismatic elements must be equal to zero.
Code specifying the subdivision of hexahedral and triangular prismatic elements
into tetrahedrals (See Chapter 6.1 and Figure 6.1).
Record 11 information is required for each triangle t.

196

Table 10.9. Block H - Nodal information.
RecordType

Variable

Description

1

Char

cVersionSet text equal to "Pcp_File_Version=".

1

IntegeriVersion

Version of the Domain.dat file; set equal to 2 for current version.

2,3

-

-

Comment lines.

4

IntegerNS

Number of solutes in a chain reaction.

4

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..

5,6
7IntegernComment line.
Nodal number.

7

IntegerKode(n)Code specifying the type of boundary condition applied to a particular node.
Permissible values are 0, ±1,±2, ±3, ±4,..., ±NumKD (see Section 8.3).

7

Real

hNew(n)Initial value of the pressure head at node n [L] when variable lInitW in Block A
is equal to .true.. The initial value of the water content at node n [L] when
variable lInitW in Block A is equal to .false.. If lWat=.false. in Block A, thenhNew(n) represents the initial guess of the pressure head (water content) for
steady state conditions.

7

Real

Q(n)

Prescribed recharge/discharge rate at node n; [L2T-1] for planar flow, [L3T-1] foraxisymmetric flow. Q(n) is negative when directed out of the system. When novalue for Q(n) is needed, set Q(n) equal to zero.

7

Integer

MatNum(n)Index for material whose hydraulic and transport properties are assigned to node
n.

7

Real

Beta(n)Value of the water uptake distribution, b(x,y,z) [L-2], in the soil root zone at noden. Set Beta(n) equal to zero if node n lies outside the root zone.

7

Real

Axz(n)

Nodal value of the dimensionless scaling factor αh [-] associated with thepressure head.

7

Real

Bxz(n)

Nodal value of the dimensionless scaling factor αK [-] associated with thesaturated hydraulic conductivity.

7

Real

Dxz(n)

Nodal value of the dimensionless scaling factor αθ [-] associated with the watercontent.

7Real

Temp(n)Initial value of temperature at node n [oC] (if lTemp=.false. then set equal to 0 or
any initial value, which is to be used with temperature dependent water flow andsolute transport).

7

Real

Conc(1,n)Initial value of the concentration of the first solute at node n [ML-3] (do not haveto be specified if lChem=.false.).

197

Table 10.9. (continued)

RecordType

Variable

Description

7

Real

Conc(2,n)

.
.
7

.
.
Real

.
.
Conc(i,n)Initial value of the concentration of the second solute at node n [ML-3] (must not
be specified if lChem=.true. and NS < 2).
.
.
Initial value of the concentration of the last solute at node n [ML-3] (must not be
specified if lChem=.true. and NS < i)..

7

Real

Sorb(1,n)Initial value of the adsorbed concentration on type-2 sites or initial concentrationin the immobile zone of the first solute at node n [ML-3]. This variable does nothave to be specified if lChem=.false.. or lEquil=.true..

7

Real

Sorb(2,n)

.
.
7

.
.
Real

.
.
Sorb(NS,n)Initial value of the adsorbed concentration on type-2 sites or initial concentrationin the immobile zone of the second solute at node n [ML-3]. This variable does
not have to be specified if lChem=.false. or lEquil=.true. or NS < 2.
.
.
Initial value of the adsorbed concentration on type-2 sites or initial concentrationin the immobile zone of the NSth solute at node n [ML-3]. This variable does nothave to be specified if lChem=.false. or lEquil=.true..

198

Table 10.10. Block I - Element information.
RecordType

Variable

Description

1,2

-

-

Comment lines.

3

Integer e

Element number.
The following 3 numbers (Record 3a) are only for two-dimensional applications.

3a

Real

Angle(e)Angle in degrees between K1A and the x-coordinate axis assigned to each element
e.

3a

Real

ConA1(e)First principal component, K1A, of the dimensionless tensor KA which describes the
local anisotropy of the hydraulic conductivity assigned to element e.

3a

Real

ConA2(e)Second principal component, K2A.
The following number (Record 3b) is only for two-dimensional applications.

3b

Real

iAniz(e)3Integer LayNum(e)Index of anisotropy tensor (Record 11 of Block B) assigned to each element e.
Subregion number assigned to element e.
In general, record 3 information is required for each element e, starting with e=1
and continuing sequentially until e=NumEl.

199

Table 10.11. Block J - Boundary information.
RecordType

Variable

Description

1

Char

cVersionSet text equal to "Pcp_File_Version=".

1

Integer iVersion

Version of the Boundary.in file; set equal to 3 for current version.

2,3-

Comment lines.

4

Integer NumBP

Number of boundary nodes for which Kode(n) is not equal to 0.

4

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.

4

Logical SeepF.true. if one or more seepage faces is to be considered.

4Logical FreeD

Set this variable equal to .true. if a unit vertical hydraulic gradient boundary
condition (free drainage) is used at the bottom boundary. Otherwise set equal to.false. .

4Logical DrainF

Set this logical variable equal to .true. if a drain is to be simulated by means ofa boundary condition. Otherwise set equal to .false.. Section 5.3.7 explains how
tile drains can be represented as boundary conditions in a regular finite element
mesh.

4

Logical qGWLF

Set this variable equal to .true. if the discharge-groundwater level relationshipq(GWL) given by equation (8.1) is used as the bottom boundary condition;
GWL=h-GWL0L, where h is the pressure head at the boundary.

5-

Comment line.

6

Logical lInterp

Logical variable indicating that time-variable boundary pressure heads should
be interpolated with time.

6

Logical lVarBC

Logical variable indicating that the time-variable boundary pressure headboundary condition should be changed to zero flux when the specified number
is larger than 999999.

6

Logical lFluxHead

Logical variable indicating that the zero flux boundary condition should beapplied on the part of the boundary where the time-variable boundary pressure
head is negative.

6

Logical lAtmHead

Logical variable indicating that the atmospheric boundary condition should beapplied on the part of the boundary where the time-variable boundary pressure
head is negative.

6

Logical lSeepHead

Logical variable indicating that the seepage face boundary condition should beapplied on the part of the boundary where the time-variable boundary pressure
head is negative.

6

Logical lAtmGWL

Logical variable indicating that the time-variable flux boundary condition
should be treated similarly as the atmospheric boundary conditions, i.e., withlimiting pressure head values hCritA and hCritS.

-

-

200

Table 10.11. (continued)

RecordType

Variable

Description

6

Logical lAtmSeep

Logical variable indicating that the atmospheric boundary condition should beapplied on the part of the boundary where the seepage face is not active.

6

Logical lSnow

Logical variable indicating that the snow accumulates at the atmosphericboundary condition whenever temperatures are below zero.
The following two lines are given only for Version 2.0 of HYDRUS.

7a

-

-

Comment line.

8a

Logical lGrad

Logical variable indicating whether a gradient boundary condition with agradient different than one (free drainage) is to be specified.

8a

Logical lDrip

Logical variable indicating whether a special boundary condition that considers
the effect of the back pressure on the subsurface drip irrigation is to be applied.

8a

Logical lSDrip

Logical variable indicating whether a special boundary condition with dynamic
ponding for surface drip irrigation is to be considered.

8aLogical lSeep

Logical variable indicating whether a seepage face boundary condition with a
specified pressure head (other than zero) is to be applied.

8a

Logical lIrrig

Logical variable indicating whether irrigation is to be triggered by the pressure
head at a specified observation nodes falling below a specified value.

8a

Logical lWell

Logical variable indicating whether irrigation the Reservoir Boundary
Condition (see section 2.7.2.10) is to be used.

8a

Logical lDummy

Dummy logical variable (2 times).
The following two lines are given only when lGrad=.true..

9a

-

-

Comment line.

10a

Real

xGrad

A gradient to be applied at the boundary with the gradient boundary condition.
The following two lines are given only when lDrip=.true..

9b

-

-

Comment line.

10b

Real

QDrip

The nominal discharge (Optimal Flux in the HYDRUS GUI) of the source forthe reference inlet pressure hin (usually being 10 m) and the back pressure
equal to zero ([L3T-1] ([L2T-1] in 2D)).

10b

Real

ExpDrip

An empirical constant (Exponent in the HYDRUS GUI) that reflects the flow
characteristics of the emitter. Normally, c = 0.5 corresponds to a turbulent flow
emitter and c = 1 to a laminar one.
The following two lines are given only when lSDrip=.true..

9c

-

-

Comment line.

201

Table 10.11. (continued)

RecordType

Variable

10cInteger iDripCenter

10cInteger iDripNode

Description
A node number representing the dripper with the Neumann (flux) boundary
condition, to which the irrigation flux Q is applied.
A code indicating in which direction the surface ponding spreads.
=-1: from left to right,= 0: from center in both directions=+1: from right to left
The following two lines are given only when lSeep=.true..

9d

-

-

Comment line.

10d

Real

hSeep

The pressure head for the boundary with the specified seepage face boundary
condition.
The following three lines are given only when lIrrig=.true..

9e,10e--

11e

Idteger iIrrig

Observation node number that triggers irrigation.

11e

Real

The pressure head in the selected observation node that triggers irrigation.

11e

Idteger kIrrig

Boundary code where irrigation will be applied (-4 = atmospheric (sprinkler),-3 or +3 = variable flux or pressure head (dripper)).

11e

Real

rIrrig

The irrigation rate applied on the boundary in [LT-1] if BC code = -4 or -3 orthe irrigation pressure head applied on the boundary in [L] if BC code = +3.

11e

Real

tIrrig(3)Duration of irrigation [T], if equal to zero, then duration is calculated fromcumulative transpiration from previous rain or irrigation.

11e

Real

tIrrig(4)Lag period for irrigation to start after the pressure head in a selected
observation node is reached.

hIrrigComment lines.

The following two lines are given only when lWell=.true..
9f, 10f--

Comment lines.

11

Integer iWell

A shape of a reservoir: =1: well, =1: furrow, =2: wetland

12

-

-

Comment line

13
13
13

Real
Real
Real

zWBot
zWInitRadiusz-coordinate of the bottom of a reservoir (well, furrow, wetland) (zwBot, [L])z-coordinate of the initial position of the water level in the reservoir (zwInit, [L])a well radius when iWell=1 (rw, [L]), a half-width of the bottom of the furrow
when iWell=2 (aBot, [L]), or the maximum width (radius) of the wetland wheniWell=3 (rmax, [L]).

13

Real

WellPump

Pumping rate (Qp, [L2/T] or [L3/T] for 2D and axisymmetrical problems,
respectively), when constant in time. When pumping/injection from/to areservoir varies with time, it is given as the first “Time-Variable Flux” value.

202

Table 10.11. (continued)

RecordType

Variable

Description

13

Real

cWell

The initial solute concentration in a reservoir (cw, [M/L3]). Note: Solute added
to a reservoir is given as cBound(6) for a constant water injection or cValue2
for time-variable water injection.
The following two values are provided when iWell=2 (furrow)

13

Real

TanAlfa

Tangent of the slope of the furrow wall (tan α [-])

13

Real

HwMax

z-coordinate of the maximum water level that may be reached in the furrow(hw, [L]) before inflow is turned off
The following two values are provided when iWell=3 (wetland)

13

Real

zMax

The maximum depth of the wetland (zmax, [L])

13

Real

pp

The shape parameter (exponent) (p [-])

7

-

-

Comment line.

8

Integer KXB(1)Global node number of the first of a set of sequentially numbered boundary
nodes for which Kode(n) is not equal to zero.

8.
8

Integer KXB(2).
.
Integer KXB(NumBP)

As above for the second boundary node.
.
As above for the last boundary node.

9

-

-

Comment line.

10

Real

Width(1)

Width [L] (for 2D applications) or surface area (for 3D applications) [L2] of theboundary associated with boundary node KXB(1). Width(n) includes half theboundary length for 2D applications (or one quarter of the boundary surface
area for 3D applications) of each element connected to node KXB(n) along theboundary. The type of boundary condition assigned to KXB(n) is determined by
the value of Kode(n). In case of axisymmetric flow, Width(n) represents the areaof the boundary strip [L2] associated with node KXB(n), and along a horizontal
boundary should be calculated as

Width( j )=π

3[( x j -1 + 2 x j )( x j - x j -1) + ( x j +1 + 2 x j )( x j +1 - x j )]If a unit vertical hydraulic gradient or a deep drainage boundary condition is
specified at node n, then Width(n) represents only the horizontal component of
the boundary.
10
.
.
10

Real
.
.
Real

Width(2)As above for node KXB(2).
.
.
.
.
Width(NumBP) As above for node KXB(NumBP).

11

-

-

Comment line.

203

Table 10.11. (continued)

RecordType

Variable

Description

12

Real

rLen

Width (for 2D) or surface area (for 3D) of soil surface associated withtranspiration [L]; represents surface area [L2] in case of axisymmetrical flow.
Set rLen equal to zero for problems without transpiration.

13

-

-

Comment line.

14

Real

GWL0L

Reference position of groundwater table (usually the z-coordinate of the soil
surface).

14

Real

Aqh

Value of the parameter Aqh [LT-1] in the q(GWL)-relationship (equation (8.1));
set to zero if qGWLF=.false.

14

Real

Bqh

Value of the parameter Bqh [L-1] in the q(GWL)-relationship (equation (8.1)); set
to zero if qGWLF =.false.
Records 13 and 14 are provided only when the logical variable qGWLF =.true.

15

-

-

16

Integer Node(1)16Integer Node(2)

.
.
16

.
.
.
.
Integer Node(NObs)

Comment line.
Global node number of the first observation node for which values of thepressure head, the water content, temperature (for lTemp=.true.), and thesolution and sorbed concentrations (for lChem=.true.) are printed at each timelevel. It does not have to be specified if NObs = 0.
Same as above for the second observation node. It does not have to be specified
if NObs < 2.
.
.
Same as above for the last observation node.
Records 15 and 16 are provided only when the variable NObs > 0.

17

-

-

Comment line.

18

Integer NPart

Number of fictional particles, for which their trajectory is to be calculated.

19
19
.
19

Integer NodeP(1)Integer NodeP(2).
.
Integer NodeP(NObs)

The node number of the first fictional particle.
The second fictional particle.
.
The last fictional particle.
Records 17 through 19 are provided only for two-dimensional applications andwhen particle option is enabled.

20,21

-

-

22

Integer NSeep

Number of seepage faces expected to develop.

23-

Comment line.

24
24Integer NSP(1)Integer NSP(2)-Comment lines.

Number of nodes on the first seepage face.
Number of nodes on the second seepage face.

204

Table 10.11. (continued)

RecordType

Variable

.
.
24

.
.
.
.
Integer NSP(NSeep).
.
Number of nodes on the last seepage face.

25-

Comment line.

26
26
.
.
26

Integer NP(1,1)
Integer NP(1,2).
.
.
.
Integer NP(1,NSP(1))-

Description

Sequential global number of the first node on the first seepage face.
Sequential global number of the second node on the first seepage face.
.
.
Sequential global number of the last node on the first seepage face.
Record 26 is provided for each seepage face.
Records 20 through 26 are not provided if the logical variable SeepF is set equal
to .false..

27,28

-

-

29

Integer NDr

Number of drains. See Section 5.3.7 for a discussion on how tile drains can be
represented as boundary conditions in a regular finite element mesh.

29

Real

DrCorrAdditional reduction in the correction factor Cd (See Section 5.3.7).

30

-

-

Comment line.

31
31
.
.
31

Integer ND(1)Integer ND(2).
.
.
.
Integer ND(NDr)

Global number of the first drain.
Global number of the second drain.
.
.
Global number of the last drain.

32-

Comment line.

33
33
.
.
33

Integer NElD(1)Integer NElD(2).
.
.
.
Integer NElD(NDr)

Number of elements surrounding the first drain.
Number of elements surrounding the second drain.
.
.
Number of elements surrounding the last drain.

34

-

-

Comment line.

35
35

Real
Real

EfDim(1,1)
EfDim(2,1)

Effective diameter of the first drain, de [L] (see Section 5.3.7).
Dimension of the square in finite element mesh representing the first drain, D[L] (see Section 5.3.7).

-

Comment lines.

Record 35 information is provided for each drain.
36

-

-

37
37

Integer KElDr(1,1)
Integer KElDr(1,2)

Comment line.
Global number of the first element surrounding the first drain.
Global number of the second element surrounding the first drain.

205

Table 10.11. (continued)

RecordType

Variable

Description

.
.
37

.
.
.
.
Integer KElDr(1,NElD(1)).
.
Global number of the last element surrounding the first drain.

Record 37 information is provided for each drain.
Records 27 through 37 are provided only if the logical variable DrainF is set
equal to .true..
38,39

-

-

Comment lines.

40

Integer KodCB(1)

40
.
.
40

Integer KodCB(2)Same as above for the second boundary node.
.
.
.
.
.
.
Integer KodCB(NumBP) Same as above for the last boundary node.

Code specifying the type of boundary condition for solute transport applied to aparticular node. Positive and negative signs indicate that first-, or second- or
third- (depending upon the calculated water flux Q), type boundary conditions
are implemented, respectively. In case of time-independent boundary conditions(Kode(1)= ±1, ±2, ±5, or ±6; See Block H), KodCB(1) also refers to the fieldcBound for the value of the solute transport BC. cBound(i,abs(KodCB(1))) is thevalue of the boundary condition for node KXB(1) (the first of a set ofsequentially numbered boundary nodes for which Kode(N) is not equal to zero).
Permissible values are ±1, ±2,..., ±5, ±6,-7. When KodCB(1) equals -7, a special
type of boundary condition for volatile solutes described by equations (3.45)
and (3.46) is applied.

Records 38 through 40 are provided only if the logical variable lChem (BlockA) is set equal to .true..
41,42

-

-

Comment line.

43

Integer KodTB(1)

43
.
43

Integer KodTB(2)Same as above for the second boundary node.
.
.
.
Integer KodTB(NumBP) Same as above for the last boundary node.

Code specifying the type of boundary condition for heat transport applied to aparticular node. Positive and negative signs mean that first-, or second- or third(depending upon the calculated water flux Q) type boundary conditions will be
implemented, respectively. In case of time-independent boundary conditions(Kode(1)= ±1, ±2, ±5, or ±6; See Block H), KodTB(1) refers to the vector
TBound for the value of the heat transport BC. TBound(abs(KodTB(1))) is thevalue of the boundary condition for node KXB(1) (the first of a set ofsequentially numbered boundary nodes for which Kode(N) is not equal to zero).
Permissible values are ±1,..., ±6.

Records 41 through 43 are provided only if the logical variable lTemp (BlockA) is set equal to .true..

206

Table 10.12. Block K - Atmospheric information.+
RecordType

Variable

Description

1

Char

cVersionSet text equal to "Pcp_File_Version=".

1

Integer iVersion

Version of the Atmosph.in file; set equal to 3 for current version.

2,3-

Comment lines.

4

Integer MaxAl

Number of atmospheric data records.

4

Integer nCycles

Number of times the set of atmospheric data records is to be repeated.

5

-

-

Comment line.

6

Real

hCritS

Maximum allowed pressure head at the soil surface [L].

7

-

-

Comment line.

8

Real

tAtm(i)Time for which the i-th data record is provided [T]. Boundary condition values
are specified for the time interval preceding time given at the same line. Thus,
the BC values specified on the first row are for the time interval between theinitial time and time specified on the same line.

8

Real

Prec(i)

Precipitation [LT-1] (in absolute value).

8

Real

rSoil(i)Potential evaporation rate [LT-1] (in absolute value).

8

Real

rRoot(i)Potential transpiration rate [LT-1] (in absolute value).

8

Real

hCritA(i)Absolute value of the minimum allowed pressure head at the soil surface [L].

8

Real

rGWL(1)

Drainage flux [LT-1] across the bottom boundary, or other time-dependentprescribed flux boundary condition (positive when the water leaves the flow
region), for nodes where Kode(n)=-3; set to zero when no Kode(n)=-3 boundary
condition is specified.

8

Real

GWL(1)

Groundwater level [L] (usually negative), or other time-dependent prescribed headboundary condition, for nodes where Kode(n)=+3; set equal to zero when no
Kode(n)=+3 is specified. The prescribed value of the pressure head is
h=GWL+GWL0L.

8

Real

rGWL(2)

Drainage flux [LT-1] for nodes where Kode(n)=-7; set to zero when no Kode(n)=-7
boundary condition is specified.

8

Real

GWL(2)

Groundwater level [L] for nodes where Kode(n)=+7; set equal to zero when no
Kode(n)=+7 is specified.

8
8
8

Real
Real
Real

rGWL(3)
GWL(3)rGWL(4)8Real

GWL(4)

ditto for Kode(n)=-8.
ditto for Kode(n)=+8.
ditto for Kode(n)=-9. This value is used for internal time-variable nodal flux sinks
or sources (if they exist) (in such case, the units are [L2T-1] or [L3T-1] for 2D or 3D
applications, respectively).
ditto for Kode(n)=+9. This value is used for internal time-variable nodal pressure
head sinks or sources (if they exist).

-207

Table 10.12. (continued)
RecordType

Variable

Description

8

Real

Temp1(i)First time-dependent temperature [K] prescribed for nodes where Kode(n)= ±3,±4, ±7, ±8, or ±9 (must not be specified if lTemp=.false. and lChem=.true.; set
equal to zero when no Kode(n)= ±3, ±4, ±7, ±8, or ±9 or when the flux is directed
out of the flow domain).

8

Real

Temp2(i)Second time-dependent temperature [K] prescribed for nodes where Kode(n)= ±3,±4, ±7, ±8, or ±9 (must not be specified if lTemp=.false. and lChem=.true.; set
equal to zero when no Kode(n)= ±3, ±4, ±7, ±8, or ±9 or when the flux is directed
out of the flow domain).

8

Real

cPrec(1,i)

First time-dependent solute concentration [ML-3] prescribed for nodes where
Kode(n)= ±3, ±4, ±7, ±8, or ±9 (does not need to be specified if lChem=.false.; set
equal to zero when no Kode(n)= ±3, ±4, ±7, ±8, or ±9, or when the flux is directed
out of the flow domain). When the atmospheric water flow boundary condition is
specified, this concentration value should be reserved for the atmosphericboundary condition. The code then internally adjusts cPrec as follows:
cPrec=Prec/(Prec-Evap)*cPrec, to account for the effect of evaporation.

8

Real

crt (1,i)Second time-dependent solute concentration [ML-3] prescribed for nodes where
Kode(n)= ±3, ±4, ±7, ±8, or ±9 (does not need to be specified if lChem=.false.; set
equal to zero when no Kode(n)= ±3, ±4, ±7, ±8, or ±9, or when the flux is directed
out of the flow domain).

8

Real

cht (1,i)

Third time-dependent solute concentration [ML-3] prescribed for nodes where
Kode(n)= ±3, ±4, ±7, ±8, or ±9 (does not need to be specified if lChem=.false.; set
equal to zero when no Kode(n)= ±3, ±4, ±7, ±8, or ±9, or when the flux is directed
out of the flow domain).

8

Real

cPrec(2,i)

Same as cPrec(1,i) for the second solute (does not need to be specified if
lChem=.false. or NS < 2).

8

Real

crt(2,i)Same as crt(1,i) for the second solute (does not need to be specified if
lChem=.false. or NS < 2).

8

Real

cht(2,i)Same as cht(1,i) for the second solute (does not need to be specified if
lChem=.false. or NS < 2).
The last three entries are entered for each solute from 1 to NS.
The total number of atmospheric data records is MaxAl (i=1,2, ..,MaxAl).

+

Block K is not read in if the logical variable AtmInf (Block A) is set equal to .false..

208

Table 10.13. Block L - Dimension Information.
RecordType

Variable

Description

1

Char

cVersionSet text equal to "Pcp_File_Version=".

1

Integer iVersion

Version of the Dimension.in file; set equal to 2 for current version.

2-

Comment lines.

3

Integer NumNPD

Maximum number of nodes in finite element mesh

3

Integer NumElD

Maximum number of elements in finite element mesh

3

Integer NumBPD

Maximum number of boundary nodes for which Kode(n)≠03Integer MBandD

Maximum dimension of the bandwidth of matrix A

3Integer NSeepD

Maximum number of seepage faces

3

Integer NumSPD

Maximum number of nodes along a seepage face

3

Integer NDrD

Maximum number of drains

3

Integer NElDrD

Maximum number of elements surrounding one drain

3Integer NMatD

Maximum number of materials

3

Integer NObsD

Maximum number of observation nodes (maximum is 10)

3Integer NSD

Maximum number of solutes in a chain reaction.

3

Integer NAnisD

Maximum number of anisotropy tensors for 3d applications. Set equal to 1 for 2D
applications.

-209

Table 10.14. Block M - Inverse solution information.+

RecordType

Variable

Description

1

Char

cVersionSet text equal to "Pcp_File_Version=".

1

Integer iVersion

Version of the Fit.in file; set equal to 3 for current version.

2

Integer NCase

Number of cases being considered (only for the first data set).

3

Char

Title1

Descriptive title for simulation.

4

Char

Title2

Descriptive title for simulation.

5

-

-

Comment line.

6

Integer NOBB

Number of observed data.

6

Integer MIT

Maximum number of iterations for the inverse problem.

6

Integer iWeight

Type of weighting used for the data set.
= 0; no internal weighting.
= 1; weighting by the mean ratio.
= 2; weighting by the standard deviation.

7
8

Logical lWatF

Comment line.
Set this logical variable equal to .true. when the soil hydraulic parameters are to
be optimized.

8

Logical lChemF

Set this logical variable equal to .true. when the solute transport parameters are to
be optimized.

8

Integer NMat

Number of soil materials. Materials are identified by the material number,
MatNum, specified in Block H.

8

Logical lTempF

Set this logical variable equal to .true. when the heat transport parameters are to
be optimized.

8

Integer iConcType

Type of concentration data that are used in the objective function.
= 0: resident concentrations in the liquid phase= 1: log resident concentrations in the liquid phase: area averaged resident concentrations in the liquid phase when iPos=0= 2: outflow concentration= 3: concentration flux
= 4: cumulative concentration flux
= 5: total concentration (both liquid and sorbed phases, both equilibrium andnonequilibrium phases)= 6: temperature

9-

Comment line.

10

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].

210

Table. 10.14. (continued).

RecordType

Variable

Description= 2; Brooks and Corey's [1964] model containing six parameters.
= 3; van Genuchten's [1980] model with an air-entry value of -2 cm and with six
parameters.
= 4; Kosugi’s [1996] model with six parameters.
= 5; the dual-porosity model of Durner [1994] with nine parameters.
= 6; the dual-porosity system with transfer proportional to the effective
saturation (9 parameters) (see Sections 2.1.2 and 2.8).
= 7; the dual-porosity system with transfer proportional to the pressure head (11parameters) (see Sections 2.1.2 and 2.8).

10Integer 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= 3; Hysteresis using Bob Lenhard’s model [Lenhard et al., 1991; Lenhard andParker, 1992].

10

Logical lAniz=.true. is the coefficient of anisotropy is to be optimized.

11-

Comment line.

12

Integer iQSame

Parameter constraints= 0: θsd > θsw= 1: θsd = θmd , θsw = θmw= 2: θsd = θsw = θm

12

Logical lAw2Ad.true. if parameter constraint αw = 2 αd is to be considered.
.false. if no constraint on αw and αd is imposed.

12

Logical lKSame.true. if parameter constraint Ksw = Ksd is to be considered.
.false. if no constraint on Ksw and Ksd is imposed.

12Integer 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 11 and 12 are provided only when iHyst > 0.
Records 9 through 12 are specified only when the logical variable lWatF is equal
to .true..
13

-

-

14

Integer NS

Number of solutes (must be equal to 1).
Records 13 and 14 are specified only when the logical variable lChemF is equal to.true..

15
16
16
16

Real
Real
Real

Comment line.
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].

Par(1,M)
Par(2,M)
Par(3,M)

Comment line.

211

Table. 10.14. (continued).

RecordType

Variable

Description

16
16
16

Real
Real
Real

Par(4,M)
Par(5,M)
Par(6,M)

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 [-].

16
16
16
16

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].

16
16
16
16

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 αw for material M [L-1].
Initial estimate of parameter Ksw for material M [LT-1].

17

Integer Index(1,M)

17
17
17
17
17

Integer
Integer
Integer
Integer
Integer

17
17
17
17

Integer
Integer
Integer
IntegerIndex(2,M)
Index(3,M)
Index(4,M)
Index(5,M)
Index(6,M)

Parameter estimation index for parameter θr, 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, material M [-].
Parameter estimation index for parameter α, material M [-].
Parameter estimation index for parameter n, material M [-].
Parameter estimation index for parameter Ks, material M [-].
Parameter estimation index for parameter l, 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, material M [-].
Parameter estimation index for parameter θa, material M [-].
Parameter estimation index for parameter θk, material M [-].
Parameter estimation index for parameter Kk, material M [-].
The following four parameter estimation indices are specified only wheniModel=0 and iHyst>1.
Parameter estimation index for parameter θm, material M [-].
Parameter estimation index for parameter θsw, material M [-].
Parameter estimation index for parameter αw, material M [-].
Parameter estimation index for parameter Ksw, material M [-].

17
17
17
17

Integer
Integer
Integer
IntegerIndex(7,M)
Index(8,M)
Index(9,M)
Index(10,M)

18

Real

BMn(1,M)

18
18
18
18
18

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, material M [-].
Minimum constraint for parameter α, material M [L-1].
Minimum constraint for parameter n, material M [-].
Minimum constraint for parameter Ks, material M [LT-1].
Minimum constraint for parameter l, material M [-].

212

Table. 10.14. (continued).

Record

18
18
18
18

Type

Real
Real
Real
Real

Variable

Description

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, material M [-].
Minimum constraint for parameter θa, material M [-].
Minimum constraint for parameter θk, material M [-].
Minimum constraint for parameter Kk, material M [LT-1].
The following four minimum parameter constraints are specified only wheniModel=0 and iHyst>1.
Minimum constraint for parameter θm, material M [-].
Minimum constraint for parameter θsw, material M [-].
Minimum constraint for parameter αw, material M [L-1].
Minimum constraint for parameter Ksw, material M [LT-1].

18
18
18
18

Real
Real
Real
Real

BMn(7,M)BMn(8,M)BMn(9,M)BMn(10,M)

19

Real

BMx(1,M)

19
19
19
19
19

Real
Real
Real
Real
Real

BMx(2,M)BMx(3,M)BMx(4,M)BMx(5,M)BMx(6,M)

Maximum constraint for parameter θr, material M [-] (dummy value if
Index(1,M)=0).
Maximum constraint for parameter θs, material M [-].
Maximum constraint for parameter α, material M [L-1].
Maximum constraint for parameter n, material M [-].
Maximum constraint for parameter Ks, material M [LT-1].
Maximum constraint for parameter l, 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, material M [-].
Maximum constraint for parameter θa, material M [-].
Maximum constraint for parameter θk, material M [-].
Maximum constraint for parameter Kk, 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, material M [-].
Maximum constraint for parameter θsw, material M [-].
Maximum constraint for parameter αw, material M [L-1].
Maximum constraint for parameter Ksw, material M [LT-1].

19
19
19
19

19
19
19
19

Real
Real
Real
Real

Real
Real
Real
Real

Records 15 through 19 are specified only when logical variable lWatF is equal to.true. and then must be 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.
20
21
21
21

Real
Real
Real

ChPar(1,M)
ChPar(2,M)
ChPar(3,M)

Comment line.
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 transverse dispersivity for material type M, DL [L].

213

Table. 10.14. (continued).

RecordType

Variable

21

Real

ChPar(4,M)

21

Real

21

Real

21

Real

21

Real

21

Real

21

Real

21

Real

21

Real

21

Real

21

Real

21

Real

21
21
21

Real
Real
Real

21

Real

21

Real

21

Real

22

Integer Index(1,M)

Description

Initial estimate of the dimensionless fraction of the sorption 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 is to be considered.
The dimensionless fraction of the sorption 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.
ChPar(5,M) Initial estimate of the immobile water content. Set equal to 0 when the physicalnonequilibrium option is not considered.
ChPar(6,M) Initial estimate of the ionic or molecular diffusion coefficient in free water, Dw [L2
T-1].
ChPar(7,M) Initial estimate of the ionic or molecular diffusion coefficient in the gas phase, Dg[L2 T-1].
ChPar(8,M) 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.
ChPar(9,M) 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.
ChPar(10,M) 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.
ChPar(11,M) Initial estimate of the equilibrium distribution constant between the liquid and gas
phases, kg, material type M [-].
ChPar(12,M) Initial estimate of the first-order rate constant for the dissolved phase, µw, material
type M [T-1].
ChPar(13,M) Initial estimate of the first-order rate constant for the solid phase, µs, material type
M [T-1].
ChPar(14,M) Initial estimate of the first-order rate constant for the gas phase, µg, material type
M [T-1].
ChPar(15,M) Initial estimate of the rate constant, µw’, representing first-order decay for the firstsolute and zero-order production for the second solute in the dissolved phase,material type M [T-1].
ChPar(16,M) Initial estimate of the rate constant for the solid phase, µs’, material type M [T-1].
ChPar(17,M) Initial estimate of the rate constant for the gas phase, µg’, material type M [T-1].
ChPar(18,M) Initial estimate of the zero-order rate constant for the dissolved phase, γw, material
type M [ML-3T-1].
ChPar(19,M) Initial estimate of the zero-order rate constant for the solid phase, γs, material type
M [T-1].
ChPar(20,M) Initial estimate of the zero-order rate constant for the gas phase, γg, material type
M [ML-3T-1].
ChPar(21,M) Initial estimate of the first-order mass transfer coefficient for nonequilibrium
adsorption, ω, material type M [T-1].
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.

214

Table. 10.14. (continued).
RecordType

Variable

Description

22

Integer Index(21,M)

Parameter estimation index for parameter ChPar(21,M).

23
.
.
23

Real
.
.
Real

BMn(1,M).
.
BMn(21,M)

Minimum constraint for parameter ChPar(1,M) (dummy value if Index(1,M)=0).
.
.
Minimum constraint for parameter ChPar(21,M).

24
.
.
24

Real
.
.
Real

BMx(1,M).
.
BMx(21,M)

Maximum constraint for parameter ChPar(1,M) (dummy value if Index(1,M)=0).
.
.
Maximum constraint for parameter ChPar(21,M).
Records 20 through 24 are specified only when the logical variable lChemF isequal to .true., in which case they must be provided for each material M (from 1
to NMat).
Records 20 through 24 are specified similarly for heat transport parameters when
the logical variable lTempF is equal to .true.. In such a case, they must be
provided for each material M (from 1 to NMat).

25

-

-

Comment line.

26

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;

26

Real

FO(i)

Observation data (see description with variable iType).
When iType(i)=2 and iPos(i)=0, then FO(i) represents the total volume of water inthe entire flow domain.
When iType(i)=4 and iPos(i)=0, then FO(i) represents the average concentrationof the entire flow domain.
Type of concentration data depends on iConcType.

26

Integer iType(i)Type of observed data:= 0: cumulative boundary water flux versus time
= 1: h(x,t) measurement= 2: θ(x,t) measurement= 3: boundary flux versus time
= 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 θr= 10: prior knowledge of parameter θs= 11: prior knowledge of parameter Ks

215

Table. 10.14. (continued).

RecordType

26

Integer iPos(i)Position of the observation node for iType(i)=1,2,4 (and iConcType=0, 1, 6);
allowed values are 1, 2,...NObs.
When iType(i)=0 or 3 (and 4 when iConcType=2, 3, 4); then iPos(i) is equal toKode(n) representing a particular type of boundary condition, e.g., 1 for a constantb.c., 2 for a seepage face, 3 for a varible b.c., 4 for an atmospheric b.c., 5 for ainternal drains, and 6 for deep or free drainage.
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.

26

Real

Weight associated with a particular data point.

+

Variable

Weight(i)Description

Block M is not needed if only the direct solution is calculated or for three-dimensional applications.

216

Table 10.155. Block N – Mesh-lines information.+

RecordType

Variable

Description

1

-

-

Comment line.

2

Integer nLines

Number of mesh-lines.

3-

Comment line.

4

Integer nNodes(1)Number of nodes defining the first mesh-line.

.

.

.

.
4

.
.
Integer nNodes(nLines).
Number of nodes defining the last mesh-line.

5-

Comment line.

6

Integer nElements(1)Number of elements defining the first mesh-line.

.

.

.

.
6

.
.
Integer nElements (nLines).
Number of elements defining the last mesh-line.

7-

Comment line.

8

Integer nn(j,i),j=1,nNodes(i))

Nodal numbers for the ith mesh-line. Information for each mesh-line isentered on a separate line.

9-

Comment line.

10

Integer ne(j,i),j=1,nElements(i))

+

-

.

-

.

-

-

Elemental numbers for the ith mesh-line. Information for each meshline is entered on a separate line.

Block N is not needed if no meshlines are defined.

217

Table 10.16. The Mater.in input file.+

RecordType

Variable

Description

1

-

-

Comment line.

2

Integer iCap=0; hydraulic capacity is calculated as a slope of the retention curve=1: hydraulic capacity is provided on line 6.

3-

Comment line.

4

Integer NTabThe number of data points for soil hydraulic properties.

5

-

-

Comment line.

6
6
6
6

Real
Real
Real
Real

Theta(i)
h(i)
K(i)
C (i)

Volumetric water content [L3L-3].
Pressure head [L].
Hydraulic conductivity [LT-1].
Hydraulic capacity [L-1]. This value is provided when iCap=1.

-

Information on line 6 is provided NTab times.
Information on lines 3-6 is provided for each material.

+

This file is only needed when iModel=8 in Block B of the Selector.in file (see Table 10.2).

Example of the Mater.in input file:21811. OUTPUT DATA
The program output consists of 15+3ns output files organized into 3 groups:
T-level information
H_MEAN.OUT
V_MEAN.OUT
CUM_Q.OUT
RUN_INF.OUT
SOLUTEx.OUT
OBSNOD.OUT
WELL.OUT

P-level information
H.OUT
TH.OUT
CONCx.OUTSORBCx.OUT
TEMP.OUT
V.OUT
BOUNDARY.OUT
BALANCE.OUT
A-level information
A_LEVEL.OUT
In addition, some of the input data are printed to filea CHECK.OUT and FIT.OUT.
Separate output files SOLUTEx.OUT, CONCx.OUT and SORBx.OUT are created for each solute.
The various output files are described in detail below. The file CHECK.OUT summarizes the mostimportant input variables, including the hydraulic and transport properties of each soil material.
Results of the inverse solution are written to an output file FIT.OUT (Table 11.8). All output files
are directed to the same directory as the input files and must be created by the user prior to program
execution (the directory is created automatically if the user interface is used).
T-level information - This group of output files contains information printed at the end ofeach time step (or after n type steps). 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 11.1 through 11.5. Output file
OBSNOD.OUT brings the information about the time change of the pressure head, water content,
temperature, and solution and sorbed concentrations in specified observation nodes.

219

P-level information - P-level information is printed only at prescribed print times. The
following output files are printed at the P-level:
H.OUT

Nodal values of the pressure head

TH.OUT

Nodal values of the water content

CONCx.OUT

Nodal values of the solution concentrations

SORBx.OUT

Nodal values of the nonequlibrium concentrations

TEMP.OUT

Nodal values of the temperature

V.OUT

Nodal values of the x- and y-components of the Darcian flux vector

BOUNDARY.OUT

This file contains information about each boundary node, n, for which
Kode(n) ≠ 0, including the discharge/recharge rate (positive into thedomain), Q(n), the boundary flux (positive out of the domain), q(n), thepressure head h(n), the water content θ(n), the temperature Temp(n), andthe concentration Conc(ns,n). The solute discharge/recharge rate, Qc(ns,n),
is printed by the 2D code, while the cumulative nodal discharge/recharge,
CumQ(n), is printed by the 3D code. The length (area) associated witheach boundary type is printed at the beginning of the Boundary.out file.

BALANCE.OUT

This file gives the total amount of water, heat and solute inside eachspecified subregion, the inflow/outflow rates to/from that subregion,
together with the mean pressure head (hMean), mean temperature(TMean), and the mean concentration (cMean) over each subregion (seeTable 11.6). Absolute and relative errors in the water and solute massbalances are also printed in this file.

The output files H.OUT, TH.OUT, CONCx.OUT, SORBx.OUT, TEMP.OUT, and V.OUT
provide a binary output of the specific variables. The user interface can convert these binary files
into the ASCII files H.TXT, TH.TXT, CONCx.TXT, SORBx.TXT, TEMP.TXT, and V.TXT.
A-level information - A-level information is printed each time a time-dependent boundary
condition is specified. The information is directed to the output file A_LEVEL.OUT (Table 11.7).
A new output file (FUMIGANT.OUT) is created by HYDRUS (2D/3D), which contains
actual, cumulative, and average fumigant fluxes through the tarped and untarped surface
boundaries, as well as the total flux through both tarped and untarped boundaries (whenFumigant option is invoked) (Table 11.9). An output file WELL.OUT is created when thereservoir boundary condition is used containing information about fluxes into and from areservoir (Table 11.10). An output file IRRIG.OUT is created when the triggered irrigation
option is used containing information about all triggered irrigation events (Table 11.11).

220

Table 11.1. H_MEAN.OUT - mean pressure heads.

hAtm

Mean value of the pressure head calculated over a set of nodes for which Kode(n)=±4 (i.e., along part ofa boundary controlled by atmospheric conditions) [L].

hRootMean value of the pressure head over a region for which Beta(n)>0 (i.e., within the root zone) [L].

hKode3

Mean value of the pressure head calculated over a set of nodes for which Kode(n)= ±3 (i.e., along partof a boundary where the groundwater level, the bottom flux, or other time-dependent pressure head
and/or flux is imposed) [L].

hKode1

Mean value of the pressure head calculated over a set of nodes for which Kode(n)= ±1 (i.e., along partof a boundary where time-independent pressure heads and/or fluxes are imposed) [L].

hSeep

Mean value of the pressure head calculated over a set of nodes for which Kode(n)= ±2 (i.e., along
seepage faces) [L].

hKode5.
.
.
hKodeN

Mean value of the pressure head calculated over a set of nodes for which Kode(n)= ±5 [L].
.
.
.
Mean value of the pressure head calculated over a set of nodes for which Kode(n)= ±NumKD [L].

221

Table 11.2. V_MEAN.OUT - mean and total water fluxes.*

rAtm

Potential surface flux per unit atmospheric boundary (Kode(n)=±4) [LT-1].

rRoot

Potential transpiration rate, Tp [LT-1].

vAtm

Mean value of actual surface flux per unit atmospheric boundary (Kode(n)=±4) [LT-1].

vRoot

Actual transpiration rate, Ta [LT-1].

vKode3

Total value of the bottom or other flux across part of a boundary where the groundwater level, thebottom flux, or other time-dependent pressure head and/or flux is imposed (Kode(n)=±3), [L2T-1] or[L3T-1]+. Note that fluxes across boundaries with Kode(n)=+3 and -3 (both are time-variableboundaries) are added together.

vKode1

Total value of the boundary flux across part of a boundary where time-independent pressure heads
and/or fluxes are imposed, including internal sinks/sources (Kode(n)=±1), [L2T-1] or [L3T-1]+. Note thatfluxes across boundaries with Kode(n)=+1 and -1 (both are constant boundaries) are added together.

vSeep

Total value of the boundary flux across a potential seepage face (Kode(n)=±2), [L2T-1] or [L3T-1]+.

vKode5.
.
.
vKodeN

Total value of the flux across a boundary containing nodes for which Kode(n)=±5, [L2T-1] or [L3T-1]+.
.
.
.
Total value of the flux across a boundary containing nodes for which Kode(n)=±NumKD, [L2T-1] or[L3T-1]+.

RunoffAverage surface runoff per unit atmospheric boundary (Kode(n)=±4) [LT-1].

EvaporAverage evaporation flux per unit atmospheric boundary (Kode(n)=±4) [LT-1].

Infiltr

Average infiltration flux per unit atmospheric boundary (Kode(n)=±4) [LT-1].

SnowLayer Surface snow layer [L].

+*For plane (2D) and axisymmetric (or 3D) flow, respectively
Boundary fluxes are positive when water is removed from the system.

222

Table 11.3. CUM_Q.OUT - total cumulative water fluxes.*
CumQAP

Cumulative total potential surface flux across the atmospheric boundary (Kode(n)=±4), [L2] or [L3]+.

CumQRP

Cumulative total potential transpiration rate, [L2] or [L3]+.

CumQA

Cumulative total actual surface flux across the atmospheric boundary (Kode(n)=±4), [L2] or [L3]+.

CumQR

Cumulative total actual transpiration rate, [L2] or [L3]+.

CumQ3

Cumulative total value of the bottom or other boundary flux across part of a boundary where thegroundwater level, the bottom flux, or other time-dependent pressure head and/or flux is imposed(Kode(n)=±3), [L2] or [L3]+. Note that fluxes across boundaries with Kode(n)=+3 and -3 (both are timevariable boundaries) are added together.

CumQ1

Cumulative total value of the flux across part of a boundary along which time-independent pressure
heads and/or fluxes are imposed, including internal sinks/sources (Kode(n)=±1), [L2] or [L3]+. Note thatfluxes across boundaries with Kode(n)=+1 and -1 (both are constant boundaries) are added together.

CumQS

Cumulative total value of the flux across a potential seepage face (Kode(n)=±2), [L2] or [L3]+.

CumQ5

Cumulative total value of the flux across a boundary containing nodes for which Kode(n)=±5, [L2] or[L3]+.
.
.
.
Cumulative total value of the flux across a boundary containing nodes for which Kode(n)=±NumKD,[L2] or [L3]+.

.
.
.
CumQN

+
*

cRunoff

Cumulative surface runoff across the atmospheric boundary (Kode(n)=±4), [L2] or [L3]+.

cEvaporCumulative evaporation flux across the atmospheric boundary (Kode(n)=±4), [L2] or [L3]+.

cInfiltr

Cumulative infiltration flux across the atmospheric boundary (Kode(n)=±4), [L2] or [L3]+.

For plane (2D) and axisymmetric (or 3D) flow, respectively
Boundary fluxes are positive when water is removed from the system.

223

Table 11.4. RUN_INF.OUT - time and iteration information.

TLevelTime-level (current time-step number) [-].

Time

Time, t, at current time-level [T].

dtTime step, ∆t [T].

IterW

Number of iterations for water flow [-].

IterC

Number of iterations for solute transport [-].

ItCum

Cumulative number of iterations [-].

PecletMaximum local Peclet number [-].

Courant

Maximum local Courant number [-].

224

Table 11.5. SOLUTEx.OUT - actual and cumulative concentration fluxes.*

CumCh0

Cumulative amount of solute removed from the flow region by zero-order reactions (positive whenremoved from the system), [ML-1] or [M]+.

CumCh1

Cumulative amount of solute removed from the flow region by first-order reactions, [ML-1] or [M]+.

CumChR

Cumulative amount of solute removed from the flow region by root water uptake S, [ML-1] or [M]+.

CumChN

Cumulative amount of solute being transferred to either kinetic adsorption sites (type-2 sorption sites),or to the immobile liquid region, [ML-1] or [M]+.

ChemS1

Cumulative solute flux across part of a boundary along which time-independent pressure heads and/or
fluxes are imposed, including internal sink/sources (Kode(n)= ±1), [ML-1] or [M]+.

ChemS2

Cumulative solute flux across a potential seepage faces (Kode(n)= ±2), [ML-1] or [M]+.

ChemS3Cumulative solute flux across part of a boundary along which the groundwater level, the bottom flux, orother time-dependent pressure head and/or flux is imposed (Kode(n)= ±3), [ML-1] or [M]+.

ChemS4

Cumulative total solute flux across the atmospheric boundary (Kode(n)= ±4), [ML-1] or [M]+.

ChemS5

Cumulative total solute flux across an internal or external boundary containing nodes for which
Kode(n)= ±5, [ML-1] or [M]+.
.
.
Cumulative total solute flux across an internal or external boundary containing nodes for which
Kode(n)= ±NumKD, [ML-1] or [M]+.

.
.
ChemSN+*

qc1

Total solute flux across part of a boundary along which time-independent pressure heads and/or fluxes
are imposed (Kode(n)= ±1), [ML-1T-1] or [MT-1]+.

qc2

Total solute flux across a potential seepage face (Kode(n)= ±2), [ML-1T-1] or [MT-1]+.

qc3

Total solute flux calculated across a boundary containing nodes for which Kode(n)= ±3 (i.e., along partof a boundary where the groundwater level, the bottom flux, or other time-dependent pressure head
and/or flux is specified), [ML-1T-1] or [MT-1]+.

qc4

Total solute flux across the atmospheric boundary (Kode(n)= ±4), [ML-1T-1] or [MT-1]+.

.
.
qcN

.
.
Total solute flux across an internal or external boundary containing nodes for which Kode(n)=±NumKD, [ML-1T-1] or [MT-1]+.

cMean1

Mean concentration across part of a boundary along which time-independent pressure heads and/or
fluxes are imposed (Kode(n)= ±1) [ML-3].

.
cMean N.
Mean concentration across an internal or external boundary containing nodes for which Kode(n)=±NumKD, [ML-3].

For plane (2D) and axisymmetric (or 3D) flow, respectively
The same output file is created for each solute from 1 to NS. Values of the solute flux and the cumulative solute flux
are positive when the solute is removed from the system.

225

Table 11.6. BALANCE.OUT - mass balance variables.

Area (Volume) Area of the entire flow domain or a specified subregion, [L2] or [L3]+.

+

VolumeW

Volume of water in the entire flow domain or in a specified subregion, [L2] or [L3]+.

InFlow

Inflow/Outflow to/from the entire flow domain or to/from a specified subregion (see eq. (5.25)),
positive for an increase in moisture [L2T-1] or [L3T-1]+. Since these calculations are carried out over one
time step, the "Inflow/Outflow" values represent an instantaneous flux.

hMean

Mean pressure head in the entire flow domain or in a specified subregion [L].

HeatVol

Amount of heat in the entire flow domain or in a specified subregion, [MLT-2] or [ML2T-2]+.

TMean

Mean temperature in the entire flow domain or in a specified subregion [K].

ConcVol

Amount of solute in the entire flow domain or in a specified subregion (in equilibrium phases, including
sorbed and gaseous, if applicable). This variable is given for all solutes from 1 to NS, [ML-1] or [M]+.

cMean

Mean concentration in the entire flow domain or in a specified subregion. This variable is given for all
solutes from 1 to NS [ML-3].

ConcVlIm

Amount of solute in the entire flow domain or in a specified subregion in the non-equilibrium phase(solute in the immobile water or sorbed to sorption sites in contact with immobile water). This variable
is given for all solutes from 1 to NS, [ML-1] or [M]+.

SorbVlIm

Amount of kinetically sorbed solute in the entire flow domain or in a specified subregion. This variable
is given for all solutes from 1 to NS (for 3D applications), [L2] or [L3]+.

cMeanIm

Mean concentration in the entire flow domain or in a specified subregion in the non-equilibrium phase(solute in the immobile water), [ML-3]. This variable is given for all solutes from 1 to NS.

sMeanIm

Mean concentration in the entire flow domain or in a specified subregion in the non-equilibrium phase(kinetically sorbed solute), [MM-1]. This variable is given for all solutes from 1 to NS.

WatBalT

Absolute error in the water mass balance of the entire flow domain, [L2] or [L3]+.

WatBalR

Relative error in the water mass balance of the entire flow domain [%].

Yield

The ratio of actual and potential transpiration [%].

CncBalT

Absolute error in the solute mass balance of the entire flow domain. This variable is given for all
solutes from 1 to NS, [ML-1] or [M]+.

CncBalR

Relative error in the solute mass balance of the entire flow domain. This variable is given for all solutes
from 1 to NS [%].

For plane (2D) and axisymmetric (or 3D) flow, respectively

226

Table 11.7. A_LEVEL.OUT - mean pressure heads and total cumulative fluxes.*+*

CumQAP

Cumulative total potential flux across the atmospheric boundary (Kode(n)=±4), [L2] or [L3]+.

CumQRP

Cumulative total potential transpiration rate, [L2] or [L3]+.

CumQA

Cumulative total actual flux across the atmospheric boundary (Kode(n)=±4), [L2] or [L3]+.

CumQR

Cumulative total actual transpiration rate, [L2] or [L3]+.

CumQ3

Cumulative total bottom or other flux across a boundary along which the groundwater level, the bottom
flux, or other time-dependent pressure head and/or flux is imposed (Kode(n)=±3), [L2] or [L3]+.

hAtm

Mean value of the pressure head calculated over a set of nodes for which Kode(n)=±4 [L].

hRootMean value of the pressure head over a region for which Beta(n)>0 (i.e., the root zone) [L].

hKode3

Mean value of the pressure head over a set of nodes for which Kode(n)=±3 [L].

For plane (2D) and axisymmetric (or 3D) flow, respectively
Boundary fluxes are positive when water is removed from the system.

227

Table 11.8. FIT.OUT - information related to parameter estimation.

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 10.13).

Position

Position of the measurement (see Table 10.13).

WeightWeight associated with a particular data point.

Residual

Residual between measured and fitted quantities.

228

Table 11.9. FUMIGANT.OUT - actual and cumulative fumigant.*

ActualFluxT

Actual fumigant flux through the tarped surface, [ML-1T-1] or [MT-1]+.

Cum.FluxT

Cumulative fumigant flux through the tarped surface, [ML-1] or [M]+.

AverageFluxT Average fumigant flux (from the previous print time) through the tarped surface, [ML-1T-1] or[MT-1]+.
ActualFluxU

Actual fumigant flux through the untarped surface, [ML-1T-1] or [MT-1]+.

Cum.FluxU

Cumulative fumigant flux through the untarped surface, [ML-1] or [M]+.

AverageFluxU Average fumigant flux (from the previous print time) through the untarped surface, [ML-1T-1] or[MT-1]+.
ActualFlux

Actual total fumigant flux through the soil surface (both tarped and untarped), [ML-1T-1] or[MT].

1 +

+

*

Cum.Flux

Cumulative total fumigant flux through the soil surface (both tarped and untarped), [ML-1] or[M]+.

AverageFlux

Average total fumigant flux (from the previous print time) through the soil surface (both tarped
and untarped), [ML-1T-1] or [MT-1]+.

For plane (2D) and axisymmetric (or 3D) flow, respectively
Boundary fluxes are positive when directed out of the system/domain.

229

Table 11.10. WELL.OUT - volume and fluxes for reservoir boundary condition.*

+

*

Time

Time, t, at current time-level [T].

VolumeVolume of water in the reservoir (Vw, [L2] or [L3]+).

WatLevelz-coordinate of the water level in the well (hw, [L]).

VolSolMass of solute in a reservoir (Sw, [ML-1] or [M] +).

cWell

Solute concentration in a reservoir (c, [ML-3]).

Pumping

Cumulative pumping/injection volume ([L2] or [L3]+) (positive for pumping, negative for injection).

Infiltration

Cumulative infiltration ([L2] or [L3]+ ) (positive into the reservoir).

Seepage

Cumulative flux across the seepage face ([L2] or [L3]+) (positive into the reservoir).

AtmFlux

Cumulative atmospheric flux into a reservoir ([L2] or [L3]+).

For plane (2D) and axisymmetric (or 3D) flow, respectively
This file is written only when the reservoir boundary condition is used.

230

Table 11.11. IRRIG.OUT - information about triggered irrigations.*

Irrigation Scheduling
iIrrigObservation node triggering irrigation

hIrrigPressure head triggering irrigation [L]kIrrigType of irrigation (Sprinkler, Dripper Flux, or Dripper Pressure Head Irrigation)rIrrigIrrigation rate [L/T] or pressure head [L]tIrrig(3)Duration of irrigation [T]tIrrig(4)Offset of irrigation [T]Individual Irrigation Events*nIrreg

The number (count) if irrigation eventst

Time when irrigation is triggered [T].

tIrrig(1)Time when triggered irrigation begins [T].

tIrrig(2)Time when triggered irrigation ends [T].

VIrrigIrrigation ammount ([LT-1] or [L2T-1] +).

This file is written only when the triggered irrigation option is used. It contains information about theirrigation scheduling scheme and all triggered irrigation events.

231

232

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