Modelling Variably
Saturated Flow with
HYDRUS-2D
D. Rassam, J. Šimůnek, and M. Th. van Genuchten

Modelling Variably Saturated Flow with HYDRUS-2DFirst Edition
Copyright 2003. Exclusive rights by the publisher ‘ND Consult’, Brisbane, Australia.
All rights reserved. No part of this book may be used or reproduced in any form or by
any means without prior written permission of the publisher.
ISBN: 0-646-42309-6
Contact email: hydrus@optusnet.com.au
Contact fax : + 61 7 3376 7454

Acknowledgment
The lead author would like to acknowledge the efforts of Drs. John Knight andFreeman Cook in helping initiate the project. Special thanks to Dr. Eileen Poeter,
Director of IGWMC, for her support.

Contentsi
CONTENTS

Prefacev

Introductory Examples

ix

Example 1: Project ‘1D-Infil’
Example 2: Project ‘2D-infil’ix
xx

SECTION 1: A Journey Through HYDRUS Windows

Project Manager

1.1

1.1 Pre-Processing

1.3

1.1.1
1.1.2
1.1.3
1.1.4
1.1.5
1.1.6
1.1.7
1.1.8
1.1.9

Main Processes
Geometry Information
Time Information
Print Information
Iteration Criteria
Soil Hydraulic Model
Water Flow Parameters
Time Variable Boundary Conditions
Geometry and Finite Element Mesh Editor
1.1.9.1 Rectangular Grids
1.1.9.2 General Grids, MeshGen
1.1.10 Boundary Conditions Editor
1.1.10.1 Water Flow Boundary Conditions
1.1.10.2 Material Distribution
1.1.10.3 Root Distribution
1.1.10.4 Initial Conditions
1.1.10.5 Subregions
1.1.10.6 Scaling Factors
1.1.10.7 Observation Nodes
1.1.10.8 Nodal Recharge
1.1.10.9 Local Anisotropy

1.3
1.4
1.5
1.6
1.8
1.10
1.11
1.12
1.13
1.13
1.15
1.26
1.27
1.35
1.37
1.37
1.41
1.42
1.43
1.43
1.43

Post-Processing

1.44

1.2.1
1.2.2
1.2.3
1.2.4
1.2.5
1.2.6
1.2.7
1.2.8

1.44
1.50
1.50
1.50
1.51
1.52
1.52
1.53

1.2

Graphical Display of Results
Pressure Heads
Water Boundary Fluxes
Cumulative Water Boundary Fluxes
Soil Hydraulic Properties
Run Time Information
Mass Balance Information
Convert Output to ASCII

Modelling variably saturated flow with HYDRUS-2D

Contents

ii
SECTION 2: HYDRUS Output Files

2.1 Output Files

2.1

2.1.1 Boundary.out
2.1.2 ObsNod.out
2.1.3 h_mean.out
2.1.4 v_Mean.out
2.1.5 Cum_Q.out
2.1.6 Check.out
2.1.7 Run_Inf.out
2.1.8 Balance.out
2.1.9 Optional Files
2.1.10 Fit.out
2.1.11 A_Level.out

2.1
2.2
2.2
2.2
2.3
2.3
2.4
2.4
2.5
2.5
2.6
2.6

2.2 Examples
2.2.1 Example Project WT-1
2.2.2 Example Project Mbal-2
2.2.3 Example Project Mbal-3

2.6
2.8
2.12

SECTION 3: Root Water Uptake

3.1 Relevant Windows in HYDRUS
3.2 Root Water Uptake Simulations

3.1
3.3

SECTION 4: Example Applications

Vertical Flow Examples
4.1 Fluctuating Stream Level
4.2 Modelling Evaporation
4.3 Capillary Barrier
4.4 Modelling Hysteresis
4.5 Estimating Flow Rates Inside a Domain
4.6 Effect of Mesh Spatial Discretization and Tolerance Limits
4.7 Simulating Surface Runoff
4.8 Local Anisotropy

4.1
4.3
4.6
4.8
4.12
4.16
4.17
4.18

Modelling variably saturated flow with HYDRUS-2D

Contentsiii
Axisymmetric Flow Examples
4.9 Tension Disc Infiltrometer4.10 Dewatering of a Cylindrical Pit

4.21
4.23

Horizontal Flow Example
4.11 Steady State Horizontal Flow

4.25

Miscellaneous Examples
4.12 Print Times
4.13 Running Multiple Sequential Simulations

4.26
4.27

SECTION 5: Inverse Solution

5.1 Inverse Solution Options in HYDRUS

5.1

5.2 Inverse Modelling of Controlled Laboratory Column
5.2.1 Simulating Wetting Cycle ‘a’
5.2.2 Simulating Wetting Cycle ‘b’
5.2.3 Optimising Parameter ‘L’

5.4
5.4
5.10
5.14

5.3 Effect of Model’s Initial Parameters on Inverse Solution

5.15

5.4 Long Inverse Trial With a Multiple-Layer Soil Profile
5.4.1 Sensitivity Analysis
5.4.2 Inverse Simulations With Layered System

5.16
5.17
5.18

5.5 General Rules; Do’s and Don’ts In Inverse Modelling

5.24

SECTION 6: Trouble Shooting

6.1 Infiltration After Long Dry Period
6.2 Intense Infiltration Into a Two-Layered Soil System
6.3 Simulations With Highly Unstable Initial Conditions;
Drainage of a Saturated Coarse-Grained Soil
6.4 Infiltration Into a Thick Clay Layer After a Dry Period
6.5 Simulations With Highly Unstable Initial Conditions;
Very High Pressure Gradient
6.6 Concluding Remarks

6.1
6.4
6.8
6.10
6.15
6.20

Modelling variably saturated flow with HYDRUS-2D

Contents

iv
APPENDIXES

Appendix I

Soil Hydraulic Properties

I.1 Water Retention Parameters
I.2 Hydraulic Conductivity Function
I.3 Sensitivity Analysis
Appendix II

Concepts Related to Modelling Evaporation

II.1 Significance of Suctions at Soil Surface
II.2 The Concept of hCritA
Appendix III

III.1
III.3
III.4

Scaling Factors

IV.1 BackgroundIV.2 Stochastic Scaling Factors
Appendix V

II.1
II.3

Root Water Uptake

III.1 Root Water Uptake Reduction Models
III.2 Root Density
III.3 Root Distribution Functions
Appendix IV

I.1
I.6
I.9

IV.1
IV.3

Inverse Solution

V.1 The Inverse Method
V.2 Statistical Issues
V.2.1 Distribution of Residuals
V.2.2 Confidence Limits
V.2.3 Goodness of Fit and Errors
V.2.4 Correlation Matrix
V.2.5 Weighting Parameters
V.3 Intelligent Data Acquisition

V.1
V.2
V.2
V.4
V.5
V.6
V.7
V.9

Appendix VI

Introductory Examples on Solute Transport

VI.1

Appendix VII

Index of HYDRUS Windows

VII.1

Appendix VIII

Description of Toolbars

VIII.1REFERENCES

Ref.1

ALPHABETICAL INDEX

Index.1

Modelling variably saturated flow with HYDRUS-2D

Prefacev
PREFACEThis document is a user manual for the HYDRUS-2D software package (Šimůnek etal., 1999), which simulates water flow and solute transport in two-dimensionalvariably saturated media. The manual mostly relates to modelling water flow;
however, several introductory examples on solute transport are discussed in an
appendix. Over one hundred example projects are included on the accompanying CD.
Most of the example projects on the accompanying CD were prepared using
HYDRUS-2D version 2.007; hence you might encounter problems when using older
versions of the software.
This manual covers in details all aspects of modelling water flow that can beaccomplished with HYDRUS-2D. It includes step-by-step procedures for beginners,
as well as techniques and tips for advanced users. Many of the example applications
and tips were inspired by numerous questions and comments put forward by users
through the HYDRUS discussion group at www.pc-progress.cz.
This manual is organised into eight major parts:• Introductory Examples: Includes two basic examples designed to demonstrate thecapabilities of the software for first-time users.
• Section 1: This section takes the user on a journey through all HYDRUS
windows. It covers various aspects that include starting new projects, setting up
finite element grids, assigning boundary conditions, running the model, and viewing
the results.
• Section 2: This section explains the various output files that HYDRUS produces.
It explains the meaning of the different variables involved, and provides examples
on how to manipulate the output data.
• Section 3: This section includes example projects on root water uptake.
• Section 4: This section includes 13 example applications. Examples 1-8 are‘Vertical Plane Flow’ examples. Examples 9 and 10 are ‘Axisymmetric Vertical
Flow’ examples. Example 11 is a ‘Horizontal Plane Flow’ example. Examples 12
and 13 show extra features for advanced users.
• Section 5: This is a comprehensive section on inverse parameter estimation. It
demonstrates through numerous examples important issues such as non-uniquenessof a solution, effects of temporal and spatial distributions of the calibration data,
effects of initial values of the optimised parameters, statistical issues, and validationof results. This section ends with a list of recommendations on inverse modelling.
• Section 6: This section relates to trouble-shooting. We demonstrate through many
examples cases where HYDRUS may not perform as well as it should, present likely
reasons, and put forward measures for avoiding such problems.
• Appendixes: In Appendixes I to V we present theoretical backgrounds related tosoil hydraulic properties, modelling evaporation and root water uptake, the meaningof scaling factors, and various aspects related to inverse parameter estimation.
Appendix VI presents two basic examples on modelling solute transport. Appendix
VII includes an alphabetical index of HYDRUS windows with cross-referencing torelated pages and examples in the manual. Finally, Appendix VIII includes a list ofavailable toolbars and their meaning.
Modelling variably saturated flow with HYDRUS-2D

Prefacevi
Example projects on the CD

Follow the instructions below if you want to use the ‘HYDRUS Project Manager’ toaccess the example projects shipped on the CD that accompanies the manual:
1. Copy directory ‘HYDRUS Manual Projects’ from the CD to your local hard
drive.
2.

In ‘Windows Explorer’, right click on the
directory and select ‘Properties’; thiswindow will then appear.

3. Uncheck the ‘Read only’ box and hit OK.

4.

You will be prompted to confirm theattribute changes. Check the lower box‘Apply changes to this folder, subfolders
and files’, and hit OK. This will allow
HYDRUS to access the project files.

5. Since these projects were
copied from an externalsource, you will need to
inform HYDRUS that they
exist so that you can accessthem from ‘Project Manager’.
In this example, we are setting
a workspace for the projects
saved in the ‘Appendix’ subdirectory (where all projects related to theappendixes are stored). Start HYDRUS, ‘File/ProjectManager/Workspaces/New’, enter the information as shown here (in Browse,
you should provide the relevant directory path on your computer). Hit OK.
Note that ‘D:\..’ here refers to the path on your hard drive.

Modelling variably saturated flow with HYDRUS-2D

Prefacevii
6. You will see that the ‘Workspaces’ window will now look like this.

7.

Double click on the ‘Appendix’ Workspace (or select ‘Appendix’ and clickon ‘Projects’). The ‘Projects’ window should look like this (you should have 9
projects in this workspace).

8. Repeat this process 7 times, each time providing the relevant directory
structure. The final ‘Workspaces’ window should look like this.

Modelling variably saturated flow with HYDRUS-2D

viii

PrefaceModelling variably saturated flow with HYDRUS-2D

Introductory Examples

ix
INTRODUCTORY EXAMPLESIn this introduction, we will present two water-flow examples that will cover some of
the basic features of HYDRUS-2D. The examples are meant to demonstrate the basic
capabilities of the software package and to guide first-time users.
The examples represent a 200 cm-deep soil block having two horizons (the top 30 cm
has a lower hydraulic conductivity). The soil initially has a water content of 0.1. The
water table is located far below the freely draining bottom boundary. We model 5days of surface saturation. The simulation will provide the following information:•

Inundatedsurface

How the water content of the soil changes at

two locations, ‘a’ and ‘b’ in the profile• Advance of wetting boundary after 1 day. 50 cm• How much water infiltrated the soil profile.
• How much water drained from the soil profile.
• How much water was stored in the soil profile
during the 5-day period.

K1= 10 cm/day

a

b
40 cm

200 cmK2= 25 cm/day

100 cmFree-draining
boundary

Example 1: 1-Dimensional infiltration problem.
Example 2: 2-Dimensional infiltration problem. This example is almost identical to
the first; we convert example 1 into a 2-dimensional problem by simply changing thewidth over which infiltration is applied.

1. Project ‘1D-Infil’
We will create this project step-by-step. The
final project is found on the CD (..\HYDRUS
Manual Projects\Int-exam).
Start HYDRUS-2D
File, Project Manager ( )*Note that as you move the mouseover ‘Project Manager’, thedynamic help appears in thewindow explaining what it does.

Modelling variably saturated flow with HYDRUS-2D

Introductory Examples

xThis is the ‘Project Manager’ window.
Select the first line ‘Default’; this is
the directory where the projectwill be saved.
Click ‘Projects’
A new window will appear with a listof projects saved in this workspace; if
you are trying this for the first time,
this workspace will be empty. We will
now create a new project in this workspace.

In the ‘Projects’ window, select ‘New’Enter the details as shown here.
Select OK.

This window will appear.
This is the main HYDRUS
window, through which you
can access all available
functionalities in thesoftware. Alternatively, you
can access them also fromthe menu.
INPUT

OUTPUT

The easiest way to set a new project is to
go step-by-step through the PreProcessing section shown above.

Double-Click ‘Main Processes’Enter the Heading section and check thebox Water Flow.
Next….

Modelling variably saturated flow with HYDRUS-2D

Introductory Examples

xiEnter the information as shown here:• We are using centimetresfor length units.
• We are modelling flow in a
vertical plane.
• We are using a basic
rectangular geometry.
• We have 2 materials.
• We want 2 mass balance
regions (more relevant to next example).
Next….

Enter the information as shownhere:• We are using days for timeunits.
• The simulation length is 5
days.
• The remaining numbers relate
to the numerical time-stepping
scheme; they are explained in
Section 1.1.3.
Next….

Enter 10 for number of print times
Click ‘Select Print Times’

Select ‘Print Times’, in thiscase click ‘Default’ for uniform
spacing of print times(5/10=0.5 day).

OK….Next….

Modelling variably saturated flow with HYDRUS-2D

Introductory Examples

xii
Leave all default values in thiswindow except ‘Initial
Conditions’.
Details on the remaining
parameters are found in Section
1.1.5.
Check the radio button “In the
Water Content” as shown here.
This means that our initialconditions will be interpreted aswater contents rather thanpressure heads.

Next….

Leave the default model as shownhere.
We use the van Genuchten-Mualem
model to describe the soil hydraulicproperties.

Next….

Enter the following
water retention
parameters. Note thatthe row number here
refers to the material
number (twomaterials in our case).

Next….

Modelling variably saturated flow with HYDRUS-2D

Introductory Examples

xiiiEnter the values as shown here.
• Horizontal rectangular dimension(width) is 100 cm.
• Vertical rectangular dimension(depth) is 200 cm.
• We chose to split (discretize) thewidth into 11 columns (10 cells).
• We chose to split (discretize) the
depth into 21 columns (20 cells orelements).
Next….

Accept the default uniform
grid shown in this window.
The discretization resulted ina uniform 10-cm square grid
system.

Next….

OK….

Modelling variably saturated flow with HYDRUS-2D

Introductory Examples

xiv
Assign Water Flow Boundary Conditions( ):We will assign a Constant Pressure
boundary conditions (BC) to the surface
nodes and a Free Drainage BC to the lower
nodes.
Click the Const. Pressure button, and noticethe change in the cursor.

Left-click and drag the cursor over the top
nodes, and then left-click again to end. The
following window will apear.

Enter the value 0.43, which corresponds
to saturated water content conditions.
Although the window prompts forconstant pressure, remember that we
opted to enter initial conditions as ‘In
the Water Content’.
Click OK.
Click on Free Drainage, and repeat the same for the lower nodes. There is no value
associated with this BC.

Assign Material Distribution:
Condition, Material Distribution ( )Click Select (from sidebar), the cursor willchange.
Left-click slightly to the left of the first column
and above the fourth row and drag below
the bottom row and last column to select allnodes below row 3.
Click Set Value (sidebar buttons)
A window will appear, enter number 2. This means that all selected nodes below thethird row are assigned material number 2 (which corresponds to the row numbers in
the window ‘Water Flow Parameters’). The top 3 rows default to material number 1.

Modelling variably saturated flow with HYDRUS-2D

Introductory Examples

xv
Assign Initial Conditions:
Conditions/Initial Conditions ( )Pressure Head/Water Content
Click Select (sidebar button as done previously inMaterial Distribution).
Select all nodes from row 2 right down to thebottom boundary. Note that we exclude the firstrow because it is a constant head BC and its
value has already been prescribed.
Click Set Value (sidebar button), and thiswindow will appear. Check ‘Same value for all
nodes’, and enter the initial water content 0.1

Assign Subregions:
Condition/Subregions ( ) sets the number of mass balance
regions. Remember that we earlier decided on 2 in the “Geometry
Information’ window.
Click Select (sidebar button); select the nodes as shown here.
Click Set Value (sidebar button); enter the number 2 meaning thatthe right-hand side rectangle will be subregion 2.

Assign Observation Nodes:

Condition/Observation Nodes ()
1

2

Click Insert (sidebar buttons); cursor changes to open square.
Select the nodes as shown here.

Note: If you fail to enter the right number of materials or subregions then a warning
will be issued in this window and you will be requested to do so.
Click the save button, or File/Close, and you will be prompted to save.

Modelling variably saturated flow with HYDRUS-2D

Introductory Examples

xvi
Running the simulation
Click, Calculation, Run HYDRUS-2D ( ).
Click OK to run.
When Screen Output is checked in the ‘Print
Information’ window, the results will be
dynamically displayed as the simulation proceeds.

For our current problem, this DOS window provides the following information:

Heading in ‘Main
Processes’ window

Column 1
Column 2
Column 3
Column 4
Column 5
Column 6
Column 7
Column 8

Time
Number of iterations at each time step
Cumulative number of iterations
Constant head/saturation BC flux
Cumulative volume for constant head/saturation/flux BC. Flux acrossboundary with prescribed constant head/saturation/flux BC
Flux across seepage face (not assigned in this example)
Average pressure head at constant head/saturation BC
Average pressure head at seepage face (not assigned in this example)

At the end of the simulation, hit the Enter keyboard button to return to the HYDRUS
window (you can always hit the Enter button before the simulation ends).

Display the results in the Post-processing window• How the water content of the soil changes with time at two locations (observation
nodes) ‘1’ and ‘2’ in the profile (on the previous page).

In Post-processing part (right side of the HYDRUS main window), double-click on
Observation Points

Modelling variably saturated flow with HYDRUS-2D

Introductory Examples

xvii
In Vertical Variable list box, choose Water
Content and this figure will show.
Notice that the two observation points
produced identical results. Although thisproblem is run as a 2-dimensional domain, it isbasically a 1-dimensional problem. That is,results are the same at any point along a
horizontal line across the domain.

Click Close.

•

Advance of wetting boundary after 1 day

Double-click on Graphical Display of Results ( )On the sidebar buttons, scroll the Time-value bar to 1 (day).
Quantity/Water Content
Options/Grid Settings ( ), enter the values asshown.
Click OK.

1_D Graph/Cross Section ()

Select a node on the upper boundary (anywhere
since results do not vary horizontally), movethe mouse vertically and click on the lower
boundary. This graph showing the watercontent profile after 1 day will appear (noticethe shape of the wetting front).
Click OK.
File/Close; Return to the main Hydrus-2D
module.

Modelling variably saturated flow with HYDRUS-2D

Introductory Examples

xviii

•

How much water infiltrated into the profile?

Double-click on Cumulative Water Boundary
Fluxes.
For the Vertical Variable, select ConstantBoundary Flux.
Notice that fluxes into the transport domain are
negative.

Right-click on the chart
Select Edit Chart Data

Scroll down until C1 reads 5, which isthe final time (5 days). The volume ofwater that has infiltrated from theconstant-head boundary (soil surface) is
5190 cm2 (cm3/cm); the negative sign
indicates water entering the domain.
Click OK (leave the Cumulative Water
Boundary Fluxes dialog open).

•

How much water drained from the profile?

As Vertical Variable, select Free or Deep Drainage
Boundary Flux
This window shows that drainage from the lower
boundary is almost zero.
Notice that fluxes out of the transport domain are
positive.
Click Close.

Modelling variably saturated flow with HYDRUS-2D

Introductory Examples

xix•

How much water was stored in the profile.

HYDRUS keeps track of themass balance and provides
results at print times.
Double-click on Mass Balance
Information.
We are looking here at the
total mass balance, that is, for
the whole region regardless of
subregions.

At time 0, Volume = .217E+4 (volume of water stored in the soil)

Scroll down to the final time,
as shown here.

At time 5, Volume = .735E+4

Therefore, the change in storage=7350-2170=5180 (gain, increase in storage).
This value is very close to the cumulative infiltration (5190). The difference is theerror in the mass balance (remember that free drainage is almost zero).

Modelling variably saturated flow with HYDRUS-2D

Introductory Examplesxx
2. Project ‘2D-Infil’

This project is found on the CD (..\HYDRUS Manual Projects\Int-exam)
Project Manager ( )Select Project ‘1D-Infil’
CopyEnter the name as shown
OK
Open
This will create an identical copyof the previous project.
Double click on Boundary Condition Editor ( )Select No Flux, from the sidebar buttons
Left-click and drag over the right hand surface nodes as shown
here.
Left-click again to end.
Conditions/Initial Conditions/
Pressure Head/Water Content
Click SelectSelect the same surface nodes (shown in figure here) to change theinitial condition from 0.43 (specified here as a boundary condition for the previousrun)
Set the initial water content to 0.1
Save the changes; a warning will appear that the existing output files will be deleted,click OK.
Calculation/Run HYDRUS-2D/OK ( )View results:• How the water content of the soilchanges at two locations ‘a’ and ‘b’ in the
profileIn Post-processing, double-click on
Observation Points.
As Vertical Variable, choose Water Content.
We see here that the response is now
different for the two observation points. The
problem is not 1-dimensional any more.

Modelling variably saturated flow with HYDRUS-2D

Introductory Examplesxxi•

Advance of wetting boundary

Double-click on Graphical Display of Results( )On the sidebar buttons, slide the Time-value
scroll bar to 1 (day).
Quantity/Water Content
2-D Graph/Spectrum +Isoline ( )The figure here shows that the wetting boundary advances in a 2-dimensional manner.
•

How much water infiltrated the profile?

Repeat the same procedure as before to access the cumulative constant boundary flux;
it is equal to 3000 cm2 (much less than before).
•

How much water drained from the profile?

Free drainage is still close tozero.
File/Close - Close theGraphics module• How much water wasstored in the profile during
the 5-day period?
Click on Mass Balance
Information
At time 0
Total volume =1080+1010=2090 cm2 (sum for the two regions).
Notice that the initial volume in region 1 is slightly higher because the surface
constant head BC nodes were saturated.

Modelling variably saturated flow with HYDRUS-2D

xxii

Introductory Examples

At time =5 days
Total volume=3000+2080=5090 cm2
Notice that region 1 (below
the constant head BC)
contains more water.

The increase in storage during the 5-day period=5090-2090=3000 cm2 (equal to theconstant head BC cumulative flux).

Modelling variably saturated flow with HYDRUS-2D

1.1

Section 1: HYDRUS Windows
1. A JOURNEY THROUGH HYDRUS WINDOWS

Opening Existing HYDRUS Projects
File/Open:This window will appear. The file typethat HYDRUS looks for has the extension“.h2d”. Each project has such a file that is
located in its parent directory. In thewindow shown here, directory“MeshGen” contains three projects,
Mesh-1, 2, and 3. Notice that there arethree “*.h2d” files that have names
identical to the directories where theprojects are stored (the directory name isidentical to the project name). The *.h2d file is not project-specific, that is, it does not
relate to the contents of the project in any way. Hence, it is merely a project name file.
It is possible to copy this file, re-name, and then use it with another project.
Alternatively, and much easier, a project can be opened directly from the ProjectManager.

Project Manager

Workspaces:
Workspaces refer to directories that contain HYDRUS projects. Each workspace ischaracterized by its name, brief description and path. The window here shows thedefault workspaces that contain the project examples shipped with the software.

Modelling variably saturated flow with HYDRUS-2D1.2

Section 1: HYDRUS Windows

New: Will set a new workspace.
Enter a name, provide a
description, and browse to specify
a path.
Change: Select an existing
workspace and click “change”. A
similar window will then appear,
which allows changes to be madeto the details of the selected workspace.
Remove: Will remove a workspace from the list (will not delete existing projects).
Set as current: Select the workspace that you will work in. The projects in thisworkspace will appear whenever you open Project Manager.

Projects:
Projects refer to a particular HYDRUS run. Its input and output files are stored in one
subdirectory. A project is characterized by its name (up to 8 characters) and a brief
description. Select a workspace, then click on the Projects button on top of the ProjectManager Window.

New: Creates a new project
Copy: Copies projects. You
should provide a new namefor the copied project.
Rename: Change any of thedetails of the project, like its
name or description.
Remove: This option deletes
projects; it is irreversible
and hence gives you awarning.
Open: Select a project from the list and click “open” to work on that project.
Alternatively, projects may be opened also by double-clicking on its name or
description.
We recommend that project manipulations be carried out within the Project Manager
rather than using Windows Explorer or the File/Open command.

Modelling variably saturated flow with HYDRUS-2D

1.3

Section 1: HYDRUS Windows

Start New Project
Click the “New Project Icon” ( ), or go toFile/Project Manager/Projects/New• Enter the project name (use up to 8 characters;
longer names are possible, but then projects cannot
be manipulated using Project Manager).
•Enter a brief description.

•

Specify a directory where the project is to be
saved.

1.1 Pre-Processing• For a new project, we recommend that you go sequentially through the windows.
Start by double clicking on “Main Processes” (or ), and then click button “next” to
go to subsequent windows.
• If changes are to be made to an existing project, you can go directly to the relevant
window.

1.1.1

Main Processes• Heading: Type a heading that you want toappear in the output files.
• Simulate: Specify the scope of your problem.
You can simulate water flow, solute transport, root
water uptake, heat transport, or any combination ofthese processes. The number of options (orwindows) available under “pre-processing” will depend upon the options that you
select in this window. Deselecting any of these options in existing projects causes you
to lose all data relevant to the unchecked option.
• Inverse Solution: For normal simulations or “forward problems”, you provide theknown input parameters and HYDRUS will run once through the specified time
duration. However, under the “inverse solution” option, HYDRUS will compare thesimulation results with observed experimental data (that you provide), and then re-run
the model a number of times (the number depends on how correct the initial input
parameters were) with a new set of parameters until a close fit to the observed
experimental data is obtained. This process is alternatively referred to as “Modelcalibration”, or “Inverse parameter estimation”. Refer to Section 5 and Appendix V
for more details.
• Note that unchecking all boxes will result in a steady-state analysis of water flow
for specified boundary conditions.
Click next….

Modelling variably saturated flow with HYDRUS-2D

1.4

Section 1: HYDRUS Windows

1.1.2 Geometry Information• Length Units: Selects the length
units. You have to be consistentthroughout the project. We will beusing mostly “cm” throughout thisdocument. Note that any later change inunits will lead to conversion of theentire input data set to new units. We
do not recommend this action without
additional checking of consistency of input data.
• Geometry Type: The “Rectangular” option defines a simple structured finite
element mesh. The “General” option, which requires the MESHGEN2D interface, canhandle more complex geometries (see Figure 1.1).
.

Rectangular

GeneralFigure 1.1: Rectangular and General meshes
Section “A”

Y-axis, invisible in 2D problems
Infinite soil block

Type of Flow:• Vertical Flow models an infinite
soil block in the y-direction asdemonstrated in Figure 1.2a.
Z-axis in HYDRUS;
represents soil depth

X-axis in
HYDRUS

Sections “A” and “B” are identical

Section “B”

Figure 1.2a: 2-Dimensional vertical plane flowR

• Axisymmetric Flow considers
radially finite, 3-dimensional
geometry as shown in Figure 1.2b
where the modelled 2-D plane
represents a 3-D cylindrical shape.

R

X-Axis in
HYDRUS

Z-Axis in HYDRUS
represents depth

Figure 1.2b: 3-Dimensional axisymmetric flow
Modelling variably saturated flow with HYDRUS-2D1.5

Section 1: HYDRUS Windows• Horizontal Flow is modelled in the x-y plane that ignores the effect of gravity (seex-y plane in Figure 1.2a). Refer to example 4.11.
Soil Profile:• Number of Materials: You can introduce heterogeneity by selecting more than one
material. You’ll be prompted later to enter several sets of soil hydraulic or otherparameters, equal to the material number you enter here. In addition, you’ll have todefine where these materials are located within the domain (in the “Boundary
Conditions Editor”).
• Number of Layers (mass balance sub-regions) does not affect the solution but
provides more detailed mass balance calculations relevant to particular regions ofinterest that you will need to specify later. If you leave the default value of 1,
HYDRUS will provide at each print time the mass balance calculation only for theentire domain.
For example, in a two-dimensional drainage problem you might like to know how theblock next to the drain is draining compared to another part in the domain that is awayfrom the drain.
Under the section “Boundary conditions”, you’ll have to define later a number of subregions equal to the number of mass balance layers that you entered in this window.
If you define a different number of sub-regions, a warning will be issued and you will
be requested to enter the correct number.
Click next....
1.1.3 Time Information• Time Units: Select the time unit. You have to
be consistent throughout the project. HYDRUS
re-adjusts all time-related variables if the unit
system is changed after setting the model.
• Time Discretization: Enter initial and final
times. Initial time is usually zero, except for moreadvanced applications when re-starting existingruns (see Section 4.1).
Selection of the initial time step relates to the numerical solution, which is a selfadjusting time marching scheme. This is the initial time step that HYDRUS adopts atthe beginning of the solution and whenever boundary conditions change significantly.
As the iterative numerical solution finds it more difficult to converge, the time step is
automatically reduced. However, a limit is introduced on how small the time step isallowed to become. This limit is the minimum time step. We recommend allowing theminimum time step to be on the order of 1 s.
On the other hand, if the solution is converging fast, the time step is increased. The
maximum time step is a limit on how large the time step can become.
More details on the time marching scheme are provided in Section 1.1.5 and Chart
1.1.

Modelling variably saturated flow with HYDRUS-2D

1.6

Section 1: HYDRUS Windows• Time-Variable Boundary Condition allows a user to include atmospheric datasuch as precipitation and evaporation, plant transpiration, and time variable BCs suchas pressure heads and/or fluxes; the relevant data is input as a time-series. Once thisoption is selected, the box ‘Number of Time-Variable Boundary Records’ will be
activated and prompt the user to enter an integer ≥1.
For example, when simulating a constant evaporation rate of 2 mm/day for 100 days,
the number of time-variable boundary records is 1, which indicates that 1 constantevaporation rate is being simulated. However, when using variable daily weather data,
the number of time-variable boundary records is 100; you should later then provide atime series that comprises 100 evaporation rates, one for each day being simulated(there will be a table having 100 empty lines under “Variable Boundary Conditions”,
which should be filled with the evaporation data).
Click next….
1.1.4 Print Information
Print Information• Time Level Information: If this option ischecked, then detailed results of fluxes, pressure
heads, and other variables are printed at each time
step. This may lead in some cases to enormous
output files, especially if the simulation time islong or the solution is not converging (time steps
can become very small and HYDRUS writes to
output files at the end of each time step). In suchcases, the information provided might beredundant and may unnecessarily increase the run time. Manipulating extremely large
output data sets may become problematic, and hence it may be advantageous to turnthis option off if there is no real need for the transient data; in that case theinformation will be provided at prescribed print times only, which could be increased
to provide the necessary details.
• Screen Output: This option decides whether or not results are dynamically shownon the computer screen during a simulation. We recommend to always use this option,
especially for new projects so as to monitor their progress. It is, however,recommended to uncheck this option for “inverse solution”.
• Print Fluxes: If this box is unchecked, nodal fluxes will not be printed, that is, inthe “Graphical Display of Results”, velocities will not be available. In ‘Convert
Output to ASCII’, the 4th option ‘Velocities’ will not be available (the ‘V.OUT’ file
will be empty).
• Print times: These are prescribed times at which detailed run information isprinted to the output files, such as fluxes, pressure heads, water contents, andconcentrations. The data provided at print times could be of particular significance inmany ways:

Modelling variably saturated flow with HYDRUS-2D

1.7

Section 1: HYDRUS Windows
1. They provide a full suite of results at specified transient times. Information in
the following output files is only provided at print times: ‘h.out’, ‘th.out’, ‘v.out’,‘concx.out’, ‘sorbx.out’, ‘temp.out’, ‘boundary.out’, and ‘balance.out’.
2. They are used for animating the simulation, that is, the more print times thesmoother the animation.
3. If T-level information is turned off, you will get fluxes only at the print times.

Number of Print Times: Specify the number of print times (250 maximum in theGraphical Interface; for a higher number see Section 4.12).
Select Print Times: The default option will provide equally spaced print times, that is,
run time divided by the number of print times. You can enter times manually provided
they are listed sequentially and are less or equal to the total run time. It is useful tochoose the default option and then to manually change some of the default times. Theinterval between print times cannot be less than the minimum allowed time step.
An optimal distribution of the print times generally depends upon the type of problem.
Figure 1.3 holds for a high-conductivity soil (loamy sand, Ksat=350 cm/day) that isfreely draining for 10 days. Most of the drainage takes place during the first half dayof the simulation. To get a good animation, and to obtain useful nodal fluxes during
the early drainage period, you must have most print times during this period(uniformly distributed default print times will then be less useful). If you prefer alarge number of print times (50 or 100), it would be very tedious to enter thesemanually. It is then a good idea to prepare them first in a spreadsheet in one row(make sure the number is identical to that specified under ‘number of print times’),copy the row, then click on ‘Select Print Times’ in/ Whenever you
HYDRUS, select the first cell, and use “Ctrl_V” to paste thechange run time
print times. We include the spreadsheet “Print times.xls”,(especially
if youwhich has templates for 10 to 250 print times. You thenshorten it), don’t forget
only need to provide the total run time. The resulting print
to change print times
times vary according to a power function where the user
specifies the power (positive or negative; 0 results in uniformly-spaced times). Copy
the results and paste into HYDRUS.
Free or Deep Drainage Boundary Flux
140000120000100000
80000
60000
40000
20000
0
0

1

2

3

4

5

6

7

8

9

10

Time [days]

Figure 1.3: Example where closely spaced print times are required

Modelling variably saturated flow with HYDRUS-2D1.8

Section 1: HYDRUS Windows

1.1.5 Water Flow Iteration Criteria
Iteration Criteria, Time Step Control, and InternalInterpolation Tables are well described in thebuilt-in online HELP in HYDRUS. Part of thefollowing description about this window was
extracted from the HYDRUS Help.
Because of the nonlinear nature of the Richards
equation, an iterative process must be used ateach new time step. This iterative processcontinues until a satisfactory degree of convergence is obtained, i.e., until the change
in pressure head (or water content) at all nodes between two successive iterations
becomes less than a small value (i.e., the absolute pressure head (or water content)
tolerance).
•

Iteration Criteria

Maximum Number of Iterations: The maximum number of iterations allowed during
any time step. If the maximum number of iteration is reached without reaching asolution, the time step is divided by 3 and the computation at the current time level
restarted.
Water Content Tolerance: Absolute water content tolerance for nodes in theunsaturated part of the flow region. This parameter represents the maximum allowed
absolute change in the value of the water content between two successive iterations
during a particular time step.
Pressure Head Tolerance: Absolute pressure head tolerance for nodes in the saturated
part of the flow region [L]. This parameter represents the maximum allowed absolutechange in the value of the pressure head between two successive iterations during aparticular time step.
•Time step control

Refer to online Help for details about the optimal iteration range and the time step
multiplication factor. The information required in this window relates mainly to thenumerical time-marching scheme. We recommend not to change the default values
unless you are an advanced user. For more details on ‘Time Control’, refer to the
HYDRUS Technical Manual (p.48). Chart 1.1 demonstrates the time marching
scheme and explains the meaning of the various parameters in this window. Note thatfor Chart 1.1:
o The indicated ‘Time Step Control’ parameters are all default values.
o ΔTmin and ΔTmax are the minimum and maximum allowed time steps,
respectively (specified in the window “Time Information”), ΔTinit is theinitial time step, and Tinit the initial time.
o Time discretization associated with implementing the boundary
condition (herein called Ta): refers to times when the imposed BCs change(such as variable pressure heads or flux BCs), or times when the
introduced fluxes change (such as precipitation in atmospheric BCs).
o The time discretisation associated with print times is Tp.

Modelling variably saturated flow with HYDRUS-2D

1.9

Section 1: HYDRUS Windows

Iterative numerical solution starts
1 Time step (Time-level 1); Time=Tinit
Δt1 = ΔTinit; T1 = Tinit + Δt1
ΔTmin ≤ time between any consecutive Tp’s or Ta’sst

Iterate until I=maximum
number of iterations;
Terminate solution;
New Δt2 =Δt1/3

Solution converging?
No
Yes
Solution converges after ‘I’ iterations
Is adjustment of time step needed?

If I≥7 then new time step Δt2= Δt1×0.3

If I≤3 then new time step Δt2= Δt1×1.3

2nd Time step (Time-level 2); T2=T1+ Δt2
Time step= Δt2
Solution converges after ‘I2’ iterations
This time increment is automatically adjusted at each time level so that it coincides withtimes Tp and Ta
Is adjustment of time step needed?

If I2≥7 then new time
step= Δt2 ×0.3
Time-step can be
reduced down to
ΔTmin

If I2≤3 then new time
step= Δt2 ×1.3
Time-step can only be
increased up to ΔTmax

Continue up to Final time
Chart 1.1: Time marching scheme in HYDRUS
Modelling variably saturated flow with HYDRUS-2D1.10

Section 1: HYDRUS Windows•

Internal Interpolation Tables

At the beginning of a numerical simulation, HYDRUS generates for each soil type in
the flow domain a table of water contents, hydraulic conductivities, and specific watercapacities from the specified set of hydraulic parameters. Values of the hydraulicproperties are then computed during the iterative solution process using linear
interpolation between entries in the table. Doing so has been shown to speed up thecalculations.
Interpolation in the tables can be avoided by specifying both values equal to zero.
Then the soil hydraulic properties are always evaluated analytically directly from thehydraulic functions, i.e., without interpolation. The soil hydraulic properties are also
evaluated directly for values of the pressure head outside of the pressure head interval.
Lower limit of the tension interval: Lower limit [L] of the tension (absolute value ofthe pressure head) interval for which a table of hydraulic properties will be generated
internally for each material.
Upper limit of the tension interval: Upper limit [L] of the tension interval for which atable of hydraulic properties will be generated internally for each material.
•

Initial Conditions

Initial water flow conditions can either be described in terms of volumetric watercontents or pressure heads; they describe the state of the system prior to thesimulation. The initial conditions themselves are later set in “Boundary Conditions
Editor/Initial Conditions”. There are two options, pressure-head or water content. It is
the “Initial Condition” option here under “Iteration Criteria” that will decide whether
the initial soil conditions to be entered later are to be interpreted as water contents orpressure heads.
The ‘Iteration Criteria’ and ‘Internal Interpolation Tables’ parameters may be changed
in cases where numerical instabilities occur because of extreme conditions; see
Section 6.4.

1.1.6 Water Flow-Soil Hydraulic ModelHYDRUS allows users to select three types ofmodels to describe the soil hydraulic properties: van
Genuchten (1980), Brooks and Corey (1964) andmodified van Genuchten type equations (Vogel andCislerova, 1988).
Those models describe the water retention
parameters of the soil as well as the hydraulicconductivity function, often referred to also as theconstitutive relationships. They relate water content
and hydraulic conductivity to the pressure head. A
detailed explanation of these models is found in Appendix I. For details on hysteresis,
refer to Section 4.4.
Modelling variably saturated flow with HYDRUS-2D

1.11

Section 1: HYDRUS Windows

1.1.7 Water Flow-Soil Hydraulic Parameters
The required input parameters
vary with each model. The
meaning of the input
parameters for each of thethree models is outlined indetail in Appendix I.
More parameters will be
required if hysteresis ismodelled, see Section 4.4. If
you have directly measured
estimates, these can be entered into the “Water Flow Parameters” window.
HYDRUS provides two levels of help in providing users with soil hydraulicparameters. In cases where only the soil textural class is known (e.g., silt or clayloam), a drop down list (shown above) provides all of the required parameters. The
user must accept the fact that these only represent very approximate averages for thedifferent textural classes. The parameters were taken from Carsel and Parrish (1988).
Another possibility is to use the Neural Network Prediction option.
Neural Network Prediction (Rosetta)
HYDRUS was coupled with the Rosetta Lite DLL (Dynamically Linked Library),
which was independently developed by Marcel Schaap at the U.S. Salinity
Laboratory. Rosetta implements pedotransfer functions (PTFs) which predict van
Genuchten (1980) water retention parameters and the saturated hydraulic conductivity(Ks) in a hierarchical manner using soil textural class information, the soil textural
distribution, bulk density and one or two water retention points as input. Rosetta has
its own help features containing all relevant information and references.
Click the “Neural Network Prediction” button and this window will appear.

Modelling variably saturated flow with HYDRUS-2D1.12

Section 1: HYDRUS Windows

Depending upon the availability of soil information, you have five options; select theappropriate option under “Select Model”. The relevant cells will activate accordingly:
1. Textural class: you are prompted to only enter the textural class, more or lesslike the built-in drop-down list in HYDRUS.
2. % Sand, Silt, and Clay; enter the required percentage values (SSC).
3. As 2 above (SSC), but additionally enter the measured bulk density (BD)(g/cm3).
4. As 3 above (SSCBD), but also enter the measured water content at 33 kPa (3.3m, or one third bar).
5. As 4 above, plus the measured water content at 1,500 kPa (150 m, or 15 bar).
Once you finish, click “Predict”. If you are not happy with the result, re-enter values
and re-predict. Finally, click “Accept”, which will automatically import the predictedvalues into the “Water Flow Parameters” window.

1.1.8 Variable Boundary Conditions
In this window, the user is prompted to enter boundary conditions that vary withsimulation time. These conditions are dynamic (variable) through the simulation but
static (constant) through a defined period of time. That is, the modeller discretizes thetotal simulation time into portions with different boundary values. For example, referto the precipitation values below:
0-10 days
10-50 days
50-100 days

1 cm/day0
2 cm/dayModelling variably saturated flow with HYDRUS-2D

1.13

Section 1: HYDRUS Windows

The number of rows (3 here) in this window depends on the number of “TimeVariable Boundary Records” specified earlier in the “Time Information” window.
Note that the last “Time” must be equal to the “Final Time” specified in the “TimeInformation” window.
Time
PrecipEvap
Trans
hCritArGWL
GWL

Time for which a data record is provided (T)Precipitation or rainfall rate (L/T)
Potential evaporation rate (L/T)
Potential transpiration rate (L/T)
Absolute value of the minimum allowed suction at the soil surface (L).
Refer to Appendix II for more details.
Drainage flux (L/T) across the bottom boundary (inflow is negative),or other time-dependent prescribed flux boundary condition; set to zero
when no time-dependent flux boundary condition is specified.
Groundwater level (L), or other time-dependent prescribed headboundary condition; set equal to zero when no time-dependent headboundary condition is specified

More details are found in Section 1.1.10 under points 4, 5, and 9.

1.1.9 GEOMETRY AND FINITE ELEMENT MESH EDITOR
1.1.9.1 Rectangular Grids
When the basic shape of the domain isrectangular, this is the quickest way toprepare the finite element mesh. Actually, itis the only option in the absence of the
MeshGen2D grid generator. The use of
Rectangular grids makes it possible toconstruct an essentially rectangular domain
made up of a number of triangular elements. The domain is a simple continuous
square or rectangle with some undulating surface topography.
This option is activated when you select the “Rectangular” option of geometry type.
Please go back and do so as needed, and then click OK.
Next double click ‘Geometry and Finite
Element (FEM) Editor’ (or ); this window
will appear. Enter values as shown.
Horizontal and Vertical Dimensions: Enterthe dimensions of the domain.
Slope of the Base: refers to the slope of theentire mesh.

Modelling variably saturated flow with HYDRUS-2D1.14

Section 1: HYDRUS Windows

Number of Vertical and Horizontal Columns: refers to the number of rows andcolumns that will be used to construct the mesh; the minimum number is two. Twocolumns are needed to simulate one dimensional problems (see Test1, 2, 4, 5, and 6 in
the Workspace ‘Direct’). The maximum allowed is 10,000.
Click next….
If you accept all default options in the next window, you’ll get a grid with a slope of
45o, (slope=1, which is the tangent of 45o; see Figure 1.4 left side). However, if aslope=0 is selected, you will get the grid shown in Figure 1.4, right side.
You can view these meshes graphically in the “Boundary Conditions Editor ( )”.

45O; slope

Slope of base = 1
Slope of base = 0
Figure 1.4: Rectangular grids with different slopes
Horizontal Discretization: The
default discretization in thehorizontal (x) direction is
simply the horizontal
dimension divided into thenumber of vertical columns
minus one. You can overwritethese values by entering newvalues manually into theindividual cells.
You can also prepare thecoordinate data set in a
spreadsheet, and copy it; click
the relevant cell and thenCtrl_V to paste the data into HYDRUS.

Modelling variably saturated flow with HYDRUS-2D1.15

Section 1: HYDRUS Windows

You have the option of introducing an undulating surface
topography as shown here. This is done manually; click eachcell along “dz” and enter a vertical increment (positive or
negative) to the surface node coordinate. The resulting meshis shown in Figure 1.5.

Figure 1.5: Rectangular gridwith surface topography
Vertical Discretization: You have more control over the vertical discretization. You
can change the density of the vertical elements either from the top or from the bottom.
Density at the Top, and Density at the Bottom: The densities decide the mesh sizes;
the smaller the density the finer the mesh near that end. The only thing that matters is
the ratio between the two values. Assigning any set of equal values for both the top
and the bottom will result in the default uniform vertical discretization, that is,elements along that column are of equal height.
To obtain a finer mesh at the top end of the grid, specify
for example 0.1 (Top density/Bottom density < 1) for“Density at the Top” then click Regenerate Coordinates.
Notice that the distances between coordinates in thevertical column are now smaller near the top than near
the bottom. Again, you can view the mesh in the“Boundary Conditions Editor ( )”.

Figure 1.6: Rectangular gridwith fine density at top
1.1.9.2 General Grids; Meshgen in HYDRUS-2D
If under ‘Geometry Information/Geometry Type’,
you chose ‘General’, HYDRUS will proceed asfollows to create your mesh using the embedded
finite element mesh generator ‘MeshGen2D’. Go
first to “Geometry and FEM Generator” (or ).
The Set Viewport window (may be edited later, see
View/Viewport) will appear.
 Window Centre and Extents: You should know
the physical size of your problem, that is, the width and height of the domain so thatthe screen can accommodate and make it visible. Enter the window centre andModelling variably saturated flow with HYDRUS-2D1.16

Section 1: HYDRUS Windows

extents using the units (e.g., cm) that you specified earlier (in ‘Geometry
Information’).
 Aspect Ratio: The aspect ratio (Y/X dimensions) of your domain does notnecessarily match that of the screen; you have two options:
Either keep the extents and stretch (deform) the domain, or keep the stretching factor(default value=1 keeps original shape of grid) and modify the domain extents(HYDRUS will accommodate the larger of the two dimensions and change the otherone according to the aspect ratio (about 0.67)).
The latter option is the default and is the preferred option; it is generally better to startwith an undeformed domain to get a good feeling of the problem; you can always
change the stretching factor later.
Upgrade Grid Settings: It is also useful to have Grid Setting change automaticallyaccording to the size of the domain so leave the box checked. Alternatively, the Grid
Settings may be changed manually using Tool/Grid Settings.
There are basically three stages during
construction of the finite element mesh; they
are found under ‘View’ or at the toolbar andshould be processed sequentially:
1- Geometry ( ) lets you draw thedomain of the problem,
2- Boundary Points ( )lets you control
the density of the mesh by controllingthe density of boundary points(external or internal), and
3- FEM Mesh ( ) actually creates themesh. If you are not happy with theresults then repeat the three steps again.
We will now go through a few mesh examples
that cover the most important aspects of the
Meshgen package.

Example 1: Draining a soil blocknext to a stream; Project ‘Grid1’ (CD\HYDRUS Manual
Projects\Sect-1)
Start a new project as shown;
select a suitable directory using theBrowse button.
We will create a mesh that represents a soil block located next to a shallow stream anddrained along the other side. The block is 6 m wide and 2 m high.

Modelling variably saturated flow with HYDRUS-2D

1.17

Section 1: HYDRUS Windows

The following example demonstrates how the finite element mesh is constructed. To
do so we will skip all other options and start directly with the Mesh Editor. We will
use the default length units (cm).
Double click on Geometry FEM Mesh Editor (or).

Window Centre: 300,200
Window Extents: 650,400 (it is a good idea to make the extents slightly larger thanthe maximum dimension in the grid to ensure that the 0,0 origin point is visible). If
the origin is not visible, use the scroll bars to show it.
Leave the remaining options at default.
We will draw part of the grid graphically (using the mouse). With this option,
coordinates can be precisely located only using the “snap to grid” option, which isturned on by default. You can change the grid spacing to suit your needs.
Tools/Grid Settings (or ); change the grid spacing to 10,10. When the grid spacing
is too fine compared to the dimensions of the view window, the grid will not be
displayed, even if active in the background.
Insert Line/Graphically (or ); a special cursor (a pair of cross-hairs) will appear.
Click on the following coordinates (coordinates appear dynamically on the lower
RHS of the screen):(0,0), (600,0), (600,200), (90,200)
Right-click to end.
We have drawn a polyline consisting of three segments (still one entity)
Tools/Arc Input Mode: Select Centre, Radius, and Two
Angles (or )Insert Arc: Select Numerically (or use)

Enter the data as shown here and click OK.
We will now close the grid by drawing two lines
numerically.
Insert Line/Numerically (or use )Under Current Point
X:40, Y:150
click Add New Point.
X:0, Y:150
click Add New Point.
X:0, Y:0
click Close.
Tools/Circle Input Mode: Select Centre andRadius (or )Insert/Circle: Select numerically ( )Modelling variably saturated flow with HYDRUS-2D

1.18

Section 1: HYDRUS WindowsEnter coordinates as shown here.

View/Boundary Points ( ), Discretization of Boundary Nodes
From the sidebar select: Number of Points
The cursor will turn into a hand. Click any boundary to find the number of points onthat boundary. In cases where you have only one boundary, the number defaults to120. The default number of points in this case isNodes on outer boundary =113
Nodes on inner circle =15
Note that we included this step only for demonstration purposes.
View/FEM Mesh ( ), Finite element discretization
From the sidebar select: Make Mesh
You should see the mesh shown below.
File/Close and Return ( ) takes you back to HYDRUS,
Save changes? YES

Modelling variably saturated flow with HYDRUS-2D

1.19

Section 1: HYDRUS Windows

Referring to the MeshGen window on the previous page, the encircled numbers referto the following:
1106 Number of nodes.
2086 Number of mesh-triangles (elements).
3192 Number of edges.
Alter the density of the mesh: produce a coarse uniform mesh
Double click on Geometry and FEM Mesh Editor (or ):View/Boundary Points
Number of Point (sidebar); a warning will appear, click YES.
Click anywhere on the outer boundary and enter80.
View/FEM Mesh ( )Make Mesh (sidebar)
Figure 1.7 shows that the resulting mesh is much coarser.

Figure 1.7: Finite element mesh with fewer boundary points
File/Close and Return ( ),Save changes? YES

Alter the density of the mesh: produce a mesh of varying density
Double click on Geometry and FEM Mesh Editor (or ):View/Boundary Points ( )All default (sidebar); this will bring back the default values (outer number ofnodes=113)
We will now try to alter the density of the node distribution along the outer boundary
to get a coarse mesh on the lower LHS and a finermesh on the lower RHS.
Fixed Point/Density (sidebar)
Select the lower LHS fixed point; pink coloured point
will turn yellow.
Right click and enter the values as shown.
Modelling variably saturated flow with HYDRUS-2D1.20

Section 1: HYDRUS Windows

Note that if you want the density to be equal on both sides, leave the box “Use Left
Value for Both” checked and enter a value in “From the Left”.
The arrows in Figure 1.8 show the convention used for right and left (notice that you
are always looking at the grid from the outside).
Select the lower RHS fixed point, and enter thevalues shown here. Note that when value is <1 thisresults in a finer mesh and vice versa.

View/FEM Mesh ( )Make Mesh (sidebar)
Figure 1.8 shows that the lower boundary has an increasing number of elements
towards the RHS of the mesh.
Fixed point
Right
Left
Right

Left
Right

Left
Left

Right

Figure 1.8: Convention for changing mesh density
File/Close and Return ( ),Save Changes? YES
Example 2: Hill, 30 m wide sloping at 15%; Project ‘Grid-2’ (CD\HYDRUS
Manual Projects\Sect-1)
Start New Project:
Project Name: Grid-2
Double click on Geometry and FEM Mesh Editor (or ):Enter the following extents: Centre; 1500, 500
Extents; 3500, 1000
Insert Line/Graphically (or); click on the following coordinates:(0,0) (3000,0) (3000,500) (0,500) (0,0)

Right-click to end

Modelling variably saturated flow with HYDRUS-2D

1.21

Section 1: HYDRUS Windows

We will now introduce a slope to the square grid.
Tools/Manipulation Mode/Numerical
From the side menu: click Move
Select the upper LHS node; enter the new Y-coordinate=950 (500+3,000×15%)
Select the lower LHS node; enter the new Y-coordinate=450 (0.0+3,000×15%)

OK
OK

We have a neutron probe at a distance of 1000 cm from the LHS of the meshmeasuring water contents at four depths (100, 200, 300, and 400 cm from the soil
surface). To be able to compare simulated and measured values, we must make surethat nodes exist at these locations.
Insert Line/Numerically (or )Under Current Point
X:1000, Y:700 (at x=1000, soil surface level is 800 cm)
X:1000, Y:400

click Add New Point
click Close

View Boundary Points (or )Fixed Points/Insert (sidebar, inserts one point at a time)
Click on 1000,600 and 1000,500 (it is best to do this with the snap option on).
This should result in a line having four fixed points at the desired locations (by adding
fixed points you are making sure that nodes exist at these locations).
You can control the density of the mesh by changing the number of points on the line,
which defaulted to 15. We will slightly coarsen the mesh around the line:
Boundary/Number of Points (sidebar)
Click on the line and change to 10
We now refine the upper part of the mesh (only the area above the inserted line). To
do this we need to add a fixed point to define a density around it:
Fixed Points/Insert (sidebar)
Click on 1000, 800
Fixed Points/Density (sidebar); select fixed point(1000, 800), right-click and enter values as shown
here.
Fixed Points/Density (sidebar), select fixed point (0, 950), right-click, and enter thefollowing values:
From the left: 0.5
From the right: 1
Uncheck ‘Use Left Value for Both’.
View/FEM Mesh (or )Make Mesh (sidebar) (final result shown in Figure 1.9)

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Section 1: HYDRUS Windows

File/Close and Return ( ),Save Changes? YES

Figure 1.9: Sloping mesh with high density along a vertical axis
Example 3: Hypothetical mesh; Project ‘Grid-3’ (CD\HYDRUS Manual
Projects\Sect-1)
Start New Project:
Project Name: Grid-3
Double click on Geometry and FEM Mesh Editor (or ):Enter the following extents: Centre; 8, 0
Extents; 25, 17
Insert/Object from File
Select SplineEnter file name: Wavy Sine Mesh.txt (CD\HYDRUS
Manual Projects\Sect-1). Click Help to read more
about file formatting.
Continue: change file type to ‘all files’ since default is‘.h2d’.
You should see a boundary as shown in Figure 1.10. The coordinates were generated
in a spreadsheet using a sine function. This option is useful when you have a large
surveyed landscape (e.g., survey data for a large hill slope).

Figure 1.10: Imported spline from file

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Section 1: HYDRUS Windows

Use the scroll bar at the bottom of the screen to move the view to the right until the
point (0,0) is near the RHS of the screen.
View/Enlarge View (space to the left.

) 2-3 times to zoom in on point 0,0 and leave some working

Tools/Grid Settings/Snap off ( )(do not use the snap option since the coordinatesof the spline do not conform to the grid)
Insert/Spline/Graphically ( ) (we could draw this exactly using the numerical option,
but in this exercise we try to demonstrate most of the available features)
1st point: (0,0) (end of sine curve)
2nd point: (-0.5, -0.5) (approximately)
3rd point: (-1.5, -0.9) (approximately)
4th point: (-3.0, -1.0) (approximately)

Right-click to end.

Tools/Manipulation Mode/Numerical
Object’s Points/Move (sidebar)
Select the last point on the spline, and change the coordinates to exactly -3.0, -1.0
(with snap off, you can never get precise coordinates graphically).
View/View All ( )Tools/Grid Settings/Snap on ( )Tools/Manipulation Mode/GraphicalSelect the wavy boundary by clicking on the line between the nodes (should turnyellow when selected).
Press and hold the SHIFT key; select the other spline also.
From the Sidebar
Selection Move (cursor turns into a square)
Select the middle peak on the sine-shaped boundary; the entire drawing should now
move when you move the cursor. Click on nearby grid point (7,1) (we want the peak
to coincide with a grid point so that we can draw a horizontal line that starts from apoint directly underneath the peak).
Now we want to know the coordinates of the endof the spline.
Object’s Point/MoveIf you click on the last point of the spline, you’ll
see its coordinates as in this figure. The xcoordinate (unknown because we moved the grid)
will be highlighted; right click on this coordinate
and copy it to the clipboard. Select Cancel.

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Section 1: HYDRUS Windows

Alternatively,
View/Zoom ( )Select a small window around the end point of the spline (high magnification)
Edit/Geometry/Object’s Points/Coordinates
Copy the x-coordinate onto Clipboard
View/Zoom Previous ( )Insert/ Line/Numerically(draw Polyline)(7, -5)
Add new point(–3.85398, -5) (you can paste the x-coordinate from the clipboard) Add new point(–3.85398, -1)
Close
Tools/Manipulation Mode/Graphical (the snap feature should be on)
Select the Polyline and hold the SHIFT key to select the spline also.
Selection/Reflect: you’ll be prompted for an axis for reflection,
Click points 7,1 (peak of sine shape) and 7, -5 (end of Polyline).
The boundary should now look as in Figure 1.11.

Figure 1.11: Outer boundary of domain
We now vary the density of the grid around the middle
mound.
Tools/Circle Input Mode/Radius and Centre (orInsert/Circle/Numerically (or ))Enter the coordinates as shown here.

View/Boundary Points (or )HoleSelect the circle
Check ‘Internal Curve’ as shown here. This
will convert the circle from an opening in thetransport domain to an internal curve that will help uschange the local grid density.

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Section 1: HYDRUS Windows

View/Boundary Points (or )Number of Points: click on the circle and change the number to 25.
View/FEM Mesh (or )Make Mesh (sidebar) (final result shown in Figure 1.12)

Figure 1.12: Final mesh
File/Close and Return (orSave Changes? YES),

Several other useful features exist that have not been covered:
FILE, you can:Print the mesh ( )Print Preview before printing
Copy mesh to clipboard and paste into another software
Save the mesh details into a text file
Edit, you can:Undo/redo ( / ); is very useful to reverse the last action
Geometry split/merge; lets you join and split similar objects like polylines or splines.
View/Display Options; you can show/hide edges, nodes, elements, and their
numbers.
Tools/Check Geometry ( ); this is usually done automatically when you finalise thegeometry of the mesh and move to the boundary points. You can also check the
integrity of the grid manually using this option (e.g., the outer boundary must form
one continuous closed shape, and/or no intersecting lines can exist in the grid).
Tools/Find; you can find a particular Node/Edge/Triangle by entering its number.
Tools/Point Picking; when checked during drawing in Graphical Mode, a squareappears around the cross hairs of the cursor. This causes one to snap to an existing
point (when the square encloses the point). This snap-to-point feature can be very
useful; we recommend leaving it on.
Help/Context Sensitive ( , ); click this option and then click on any command inboth the drop-down and sidebar menus and you will get help on them.
Note that “Insert/Periodical Condition” is outside the scope of HYDRUS.
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Section 1: HYDRUS Windows

1.1.10 Boundary Conditions and Domain Properties Editor
Specifying appropriate boundary conditions (BCs) is one of the most critical tasks
when constructing a numerical model. A water flow BC is a known value of the flux,
head, or gradient along the outer boundary of the FE mesh (it means the externalboundary of the selected flow domain; it is the interface between the soil and theoutside world). Solving the governing equations for saturated/unsaturated flow, whichmeans finding the new head at each node in the FE mesh in a time-marching scheme,
requires knowledge of those BCs. Otherwise the problem becomes mathematicallyindeterminate. The BCs in a model must simulate real-life conditions and hence must
be selected with extreme care.
In HYDRUS, boundary conditions are categorised as follows:• System-dependent: or dynamic BCs, meaning that they can change during thesimulation (i.e., they depend on the solution at the end of each time step). They may
depend on saturation conditions (as in a seepage face or a drain), or on soil properties
and/or climate conditions (as at soil/atmosphere interfaces).
• System-independent: This type of BC is entirely known a priori, is implemented
by the user, and is independent of the simulation results.

System-independent water flow BCs include:•••

A known head (as in Constant Pressure and Variable Pressure)
A known flux (as in No Flux, Constant Flux, Variable Flux, and Deep Drainage)
A known gradient (as in Free Drainage)

Refer to the HYDRUS Technical Manual p. 24-26 for more details.

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Section 1: HYDRUS Windows

1.1.10.1 Water Flow Boundary Condition ( ):A Water flow BC is set as follows:• Double-click on Boundary Condition Editor (or ).
• The appropriate condition is selected from the drop-down menu
(Condition/Boundary Condition-Water flow) or from the buttons located on the left
hand side of the window (this option is quicker). The cursor turns into a finger-likepointer.
• Left-click and drag on the required nodes, and left-click to end the node selection,or right-click to abort this task.
• If a certain value is associated with the BC as in ‘Constant Head’, ‘Constant Flux’,or ‘Deep Drainage’, a window will appear where you are required to enter the
relevant data.
• Repeat the procedure in case of multiple BCs.
• If you have made a mistake, you can make changes using the LHS buttons while
the appropriate BC button is still activated. Use “Edit condition” and then “Select”,
which activates “Set value” that changes the boundary value.
We will now explain in detail the meaning of each water flow BC:
1. No Flux: this BC means that the boundary is sealed from the outside world; nowater flows into or out of the domain through this boundary; all boundary
conditions default to this option if nothing is specified.
2. Constant Head: this BC refers to a constant pressure head during the entire
simulation. Values for the pressure head (positive or negative) are entered in thewindow shown below. We recommend that the initial conditions be set first
followed by this BC (if the reverse is done, initial conditions will overwrite BCs,
since constant head BCs are stored in the same vector as the initial condition).
We illustrate a variety of cases where a constant head boundary condition isapplicable (refer to Figure 1.13):
BC-1: A tension disc infiltrometer; head = h1–h2 (negative) on the nodes representing
the soil immediately beneath the disc.
BC-2 and BC-3: A constant head from an overflowing tank or a Mariotte Bottle
assembly; head = h.
BC-4: Linearly varying hydrostatic head on nodes
representing the soil beside a stream. When
prompted for the head value, check the“Equilibrium from the lowest located nodal point”
box and enter the value for head at the lowest
point, which is equal to ‘h’.
BC-5: A constant negative head resulting from a hanging water column assembly;
head=-h.

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Section 1: HYDRUS Windows

1. Disc infiltrometer

2. Overflowing tank

3. Mariotte Bottle
Overflow

h
h2
h
h1

4. Stream

h1-h2 (negative)

BC-2

BC-1

BC-3
BC-4

h

Modelled soil blockBC-5
Porous stone and sealed
water compartmenth (negative)

5. Hanging water column

Figure 1.13: Various applications for aconstant head boundary condition
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Section 1: HYDRUS Windows

3. Constant Flux: this BC refers to a constant flux during the entire simulation.
The flux (v) has the units (Length/time, L/T), like rainfall rate (cm/day). The flux
is applied over a certain width of the outer boundary of the FE mesh (W). The
applied flux is entered after selection of the nodes.
If you need to know the total flow rate over the constant-flux boundary (Q, L2/T),
the flux is integrated (summed) over the width (W); Q (L2/T)=v (L/T) × W(L).
Refer to p. 2.9 for an example.
4. Variable Pressure: this BC refers to a variable pressure head that is input as atime series. The time series has as many records as the number of times the headchanges over the boundary (you can enter as many records as you like). For eachrecord in the time series, you enter a constant head that is effective during theperiod between the time of the previous record and the current one.
This BC may be applied also to a sloping or vertical surface where the specified
head is assigned to the lowest vertical coordinate of the boundary; the upper
points are then assigned heads that satisfy hydrostatic equilibrium (e.g., a variable
water level along a stream bank; see Section 4.1). Another example application is
on a horizontal surface as in a multi-stage, tension disc infiltrometer experiment.
We refer to one of the examples included in the HYDRUS package: Project‘Disc’, which is found in the ‘Inverse examples’ workspace. That example
considers a three-stage tension disc infiltrometer experiment.
• Check ‘Time
Information/Boundary Conditions’,and enter 3 in the box (as shown).

• Under ‘Variable Boundary Conditions’, enter the Time and GWL(applied head). Referring to the figure (hCritA is irrelevant here):i. 0 <Time ≤3400 seconds, h=-20 cm
ii. 3400 <Time≤6450 seconds, h=-10 cm
iii. 6450 <Time ≤10505 seconds, h=-3 cm

Prescribed
Time-variable
BC

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Section 1: HYDRUS Windows

5. Variable Flux: this BC refers to a variable flux that is input as a time series.
The data is input in a similar manner as the ‘Variable Head’ boundary, but insteadof ‘head’ (GWL) enter ‘flux’ (rGWL) in the time series. An example is drip
irrigation with variable intensity.
We refer to one of the examples included in the HYDRUS package: Project‘Drip’, which is found in the ‘Direct1’ workspace. In that example, the intensity isvaried 6 times during a 10-day period.
Check the ‘Time Information/Boundary
Conditions’, and enter 6 in the box (as shown).

• Under ‘Variable Boundary Conditions’, enter the Time and rGWL
(flux). Referring to the window below:
i. 0 <Time ≤ 1 day, flux = -0.15 m/day (negative is inflow)ii. 1 <Time ≤ 2 day, flux = 0iii. 2 <Time ≤ 3 day, flux = -0.25 m/day
iv. 3 <Time ≤ 4 day, flux = 0
v. 4 <Time ≤ 5 day, flux = -0.25 m/day
vi. 5 <Time ≤ 10 day, flux = 0

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Section 1: HYDRUS Windows

6. Free Drainage: this boundary conditions specifies a unit gradient along the lower
boundary (outflow, drainage) of the FE mesh. It is applicable in cases where the watertable is located far below the domain of interest. This BC assumes a unit total verticalhydraulic gradient, that is, gravity flow with no pressure head gradient. This boundary
condition should never be used along sides of the transport domain. It should be used
only at the bottom of the domain.
7. Deep Drainage: this is a variable flux BC based on an observed experimental
relationship between head and discharge. It is based on an expression proposed by
Ernst and Feddes (1979) (as reported by Hopmans and Stricker, 1989):
q = aeb h −h o(1.1)

where q is the discharge, a and b are fitting parameters (obtained from experimental
observations ), h is the groundwater level, and h0 some equilibrium level of thegroundwater table. The parameters a and b account for the effects of soil hydraulicproperties (of the deeper layers) and regional flow on changes in the groundwatertable.
The nodal flux at each time during the simulation is calculated using Equation 1.1.
This flux depends upon the pressure head, which varies during the simulation. You
are required to enter three values:o The parameters a and b (ofEquation 1.1).
o The reference position of thegroundwater table, h0, which is like adatum value.
The calculated flux is assigned to thenode at each time step. Applications of
this BC are limited because it requires field data to evaluate the fitting parameters.
8. Seepage Face: this is a dynamic outflow (or drainage) BC that changes according
to the flow conditions during the simulation. The user selects the potential seepage
nodes through which water may seep out of the domain during the simulation.
HYDRUS assumes a uniform pressure head equal to zero along the saturated (active)
part of the seepage face through which water seeps out from the saturated part of thedomain. Along the unsaturated segment of the seepage face, where water is still held
up in the capillaries (under tension), the flux across the boundary is assumed to be
zero. Since the length of the active seepage face is dynamic (i.e., it changes during thesimulation), the active seepage nodes may decrease or increase during any time step.
This forces the numerical solution to be iterative.

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Section 1: HYDRUS Windows

A seepage face occurs in many cases like:• Water freely draining through a soil surface exposed to the outer atmosphere, likethe open bottom boundary of a finite soil column, or to a tile drain.
Figure 1.14: Seepage across thebottom of an open-ended finite soil

Figure 1.15: Seepage into a tile drain
Example: Direct\Furrow• Water draining through a soil surface in contact with a filter material having alarge pore structure like gravels or coarse sands as in ditch drains.

Figure 1.16: Seepage across theface of a ditch drain• Seepage occurs at the downstream side of a dam where water seeps above thephreatic level.
Figure 1.17: Seepage
from a dam
Example: Direct1\Dike•

Water may seep through an unlined tunnel.

Figure 1.18: Seepage into anunderground tunnel
Example: Direct1\Tunnel

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Section 1: HYDRUS Windows

9. Atmospheric Boundary
Condition: this boundary conditionlets us incorporate climaticconditions like rainfall (precipitation)and evaporation, or transpiration(root uptake) by plants. The latter isonly activated if root water uptake ischecked in ‘Main Processes’ asshown.

The window shown here will appear only
if ‘Time-Variable BC’ is checked in the‘Time Information’ window.
The data are entered as a time series. The
number of records in the series depends onthe variability of the data, not the length of
the simulation. For example, you may
have a 10-day simulation where theprecipitation is recorded at an average rateof 5 cm/day throughout the 10-day period,in which case you will need only one
record as shown here.

Precipitation,
evaporation,and plant
transpiration
data

See Appendix II
for more details

However, you may also have rainfall data obtained from a 0.5 mm tipping bucket fora 12-hour duration. The raw data are shown in columns 1 and 2 of Table 1.1.
Precipitation in HYDRUS is entered as a rate (having the units chosen for thesimulation, cm/day in this example). Using the tipping-bucket data, we must calculate
a rate during each tipping period as follows (refer to Table 1.1):• Column 3: calculate the duration between consecutive tips. Convert ‘min’to ‘day’ in Column 4.
• Column 5: convert rainfall to cm.
• Column 6: calculate cumulative time (simulation time).
• Column 7: calculate rainfall rate,
Column5/Column4• Columns 6 and 7 are ready to beentered as ‘Time Variable Boundary
Conditions’ under ‘Time’ and‘Precip.’. In many cases (especiallyfor very long simulations) the weather
data are best entered from a
spreadsheet. The easiest way to enter
the data is as follows:• Enter the exact number of Time-Variable Boundary Records, 8 (Numberof Time-Variable Boundary Records) in this case.
Modelling variably saturated flow with HYDRUS-2D

1.34

Section 1: HYDRUS Windows• Copy the 8 cells from Columns
6 and 7 (shaded cells in Table 1.1).
• Open ‘Variable Boundary
Conditions’; the table should
contain 8 rows. Select the first cellin Row 1 under column ‘Time’, and
Press “Ctrl_v” on the keyboard; thedata should appear as in the
Window shown here.
Table 1.1: Rainfall dataTime
5:36:00
5:42:00
5:48:00
6:00:00
6:12:00
6:36:00
8:18:00
14:00:00
17:36:00

Rainfall(mm)

Duration(min)

Duration
(day)

1
0.5
0.5
0.5
0.5
0.5
0.5
0.5

6
6
12
12
24
102
342
216

0.004170.004170.00833
0.00833
0.016670.07083
0.237500.15000

Time (day)Rainfall (cm) Cumulative
0.1
0.05
0.05
0.05
0.05
0.05
0.05
0.05

0.00420.00830.01670.02500.04170.11250.35000.5000Rate(cm/day)
24
12
6
6
3
0.70590.21050.3333What if the specified flux is higher than what the soil can accommodate?
When an atmospheric flux boundary condition is used and the specified flux
into the soil is higher than the infiltration capacity of the soil, HYDRUS
switches the BC from a flux to a pressure head BC, with the pressure headthen becoming zero along that boundary. The amount of water infiltrating thesurface is subsequently calculated according to the hydraulic conductivity
and pressure head gradient at the soil surface (using Darcy’s law), while
excess water is instantly removed as surface runoff. Similarly, when the soil
cannot supply the specified flux out of the soil, some generally low(negative) constant pressure head (hCritA) is specified, and the flux out ofthe profile is calculated.
How many records can one enter using time-variable boundary conditions?
There is no limit on the number of time-variable BCs in the computational
module. The interface, however, is limited to 15,000 data records. If you feel
it is difficult to manually enter thousands of data or want to use more than
15,000 you can enter the data directly into the “atmosph.in” input file using a
spreadsheet. Specify the total number under ‘MaxAl’, then enter as manycolumns as needed.

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Section 1: HYDRUS Windows

10. Drain Boundary Condition: this boundary condition is only available for use
with ‘Rectangular Geometry’ (if you select ‘General’ in ‘Geometry Information’, the‘Drain’ option in the ‘Boundary Condition Editor’ will be inactive).

Select Rectangular

Boundary Conditions/Conditions/Drains/Insert ( )When you are finished inserting drains, click the right mouse button, and enterparameters in this window. This option implements a drain by means of an equivalent
nodal sink. Refer to example (Direct1\drainage) included in the HYDRUS package.

1.1.10.2 Material Distribution ( ):HYDRUS defaults to a homogeneous domain having only one material (with one setof material properties). However, you can model heterogeneous domains byintroducing more than one material. This is done as follows:

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Section 1: HYDRUS Windows

In “Geometry Information, Soil
Profile”: enter the number ofmaterials that best represents yourfield conditions (up to 100). We
selected here 3 materials.
In “Water Flow Parameters”, youwill see now a number of rows
equal to the number of materials
that you specified, 3 in this case.
You should enter the water flow
parameters for these soils.
In the “Boundary Condition Editor”, select“Condition/Material Distribution” or click
the relevant button on the toolbar ( ). Allof the domain defaults to Material 1. Selectthe nodes that you want to change. The“Edit Selection, Set Value” button on the
sidebar will then be activated; click andenter the material number. Repeat for thethird material. You will see the colour
legend in the LHS of the window. Make
sure to set the right number of materials
before closing the window; otherwise awarning will be issued and you will not beallowed to leave the Boundary Conditions Editor until you do so.
For problems having a sloping soil profile (like Project “Mesh-2”), selecting theappropriate nodes with an orthogonal window is not easy.
You may want to make the selection along the slope of the profile at any appropriate
angle (parallel to the slope of the grid).
In the “Boundary Condition Editor” select Condition/Material Distribution ( ).
In this example, you want to select all nodes along the upper boundary. You need touse the command“Edit/Select by Rhomboid”

Figure 1.19a: Node selection by rhomboid
Click on the domain to define a line whose angle will set the orientation of thewindow as shown in the Figure 1.19a (the two points defined by the two arrows).
Click and drag, and you will see that the selection window is no longer orthogonal.

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Section 1: HYDRUS Windows

1.1.10.3 Root Distribution ( ):For all information related to root water uptake, refer to Section 3 and Appendix III.

1.1.10.4 Initial Conditions ( ):These refer to the initial soil conditions prior to the simulation. They could either be
specified in terms of the water content or the pressure head, depending upon available
data such as water contents from neutron probes/TDR probes, pressure heads fromtensiometers (unsaturated zone), or pressure heads from pressure transducers(saturated zone). The values entered in the BC editor will be interpreted as watercontents or pressure heads depending upon which box you selected under “Iteration
Criteria, Initial Condition” as shown below.

Select the relevant nodes. Either assign values to individual nodes or select the entire
domain and choose one of the following four options:
1. Assign equal values to all nodes (for both water contents and pressure heads).
2. Linear distribution (for both water contents and pressure heads); specify the
top and bottom values; HYDRUS will then linearly interpolate between them.
This option is useful when imposing linearly variable gradients that are
different from hydrostatic (not equal to 1; see Section 6.5).

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Section 1: HYDRUS Windows

3. Assign hydrostatic equilibrium conditions (for pressure heads only); you haveto specify the pressure head at the lowest point, and HYDRUS will calculate
the rest. For example, if you have a 100-cm profile and you want the watertable to be at the surface, enter ‘Bottom Pressure Head Value’=100 (seebelow).

3. When we model a hill slope and want to locate the water table at the surface ofthe slope, do the following (refer to Project ‘Mesh-2’):


Select ‘Equilibrium From the Lowest Located Nodal Point’Check ‘Slope (tangent alpha)’ Enter the slope (tangent of angle); the slope
in the figure below represents a negative tangent(the sign merely reflects the orientation of theslope).
The resulting head distribution is shown below.

0

100

200

300

400

500

600

500 cmFigure 1.19b: Pressure head distribution

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Section 1: HYDRUS Windows

Another possible option is to import initial conditions (water contents or pressure
heads) from a previous simulation, provided that problem had the same FE mesh. You
may import the pressure heads for all nodes from any previous print time; HYDRUS
then accesses the output ‘h.out’ file and prompts you to enter the time level. This
option is useful in many cases, for example:• Starting a simulation from a steady-state condition (will be demonstrated in thissection).
• The boundary condition could change as in the case of a stream with a fluctuatingwater level (refer to Section 4.1 for an example).

Establishing a Steady-State Solution:
HYDRUS-2D can find the steady state flow profile using two approaches. First, one
can specify initial and boundary conditions and run the program for a long time untilpressure heads and/or water contents do not change anymore. Second, one can try tofind the steady state solution in a single step by unselecting “Water Flow” in the“Main Processes” window. We will demonstrate this option by means of thefollowing example.
Refer to example ‘Dike-ST’ on your CD (..\HYDRUS Manual Projects\Sect-1).
This example represents a dike with a hydrostatic constant pressure distribution along
the LHS (h=50 m at toe), and a seepage face on the RHS (see Figure 1.20a). Since theinitial condition shouldn’t matter that much, we have chosen equal pressure heads at
all nodes, h=0. The problem is similar to ‘Direct1\Dike’ included in the HYDRUS
examples.
Note that the steady-state solution is more suited for fully saturated conditions. It canalso be found for unsaturated conditions when the soil parameters are not highly
nonlinear and the initial conditions are close to the final steady state. In cases where
the steady-state conditions could not be established, run a transient simulation for along time until no changes are detected in the solution (can be done by monitoring
observation nodes).
To obtain a steady-state solution, un-check all options in the ‘Main Processes’
window.
Click ‘Calculation/Run HYDRUS ( )’The solution should be quick; a message will say ‘Steady state was reached after –iterations’.

Constant head

Seepage face

Figure 1.20a: Boundary conditions for Project dike

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Section 1: HYDRUS Windows

Import the Steady-State Solution
File/Project Manager ( )Select project ‘Dike-ST’
Copy
Rename as in figure; in this way weobtain an identical copy of the
project. We will import the steadystate solution as an initial state for the newly copied project ‘Dike’.

Boundary Condition Editor/Condition/Initial Condition/Import Pressure-Water
Content
You will be prompted for a HYDRUS output filefrom a previous project
Browse and select the directory: ‘Dike-ST’
Click on ‘h.out’ (since we are importing initialpressure heads)
Open

This solution has only two time levels since it is a
steady-state solution (initial and final steady state);
print times hence are irrelevant. For othersimulations (transient state) you have to select thetime level you want to import as initial state (2means the second print time). Click OK

Next you will be informed about the time at thatprint level. Select Yes.

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Section 1: HYDRUS Windows

The initial conditions for the model are shown in Figure 1.20b.

Figure 1.20b: Steady-state initial conditions
1.1.10.5 Sub-regions:
The domain may be subdivided into a number of sub-regions (maximum allowed is10). Sub-regions do not affect the solution in any way but provide further massbalance information. The possible use of additional mass balance information isdemonstrated in Section 2.2.3. For each sub-region, the amount of water in that subregion and the sum of inflow/outflow is provided at each print time. This is how you
specify sub-regions:Geometry Information/Soil
Profile/Number of Layers (Massbalances); enter the required
number of sub-regions.

Select the area of each sub-region in a way similar to that used when selecting the‘Material Distribution’.
The user often wants the additional mass balance calculations when more than one
material is present; HYDRUS can do this automatically. After finishing ‘Material
Distribution’, simply click:
Options/Sub-region = Material Distribution. Manual adjustment is often needed since
automatic assignment of subregions is not unambiguous; this is because materials are
assigned to nodes and subregions to elements.
Modelling variably saturated flow with HYDRUS-2D1.42

Section 1: HYDRUS Windows

1.1.10.6 Scaling Factors
The concept of scaling factors is briefly described in Appendix IV. The following
options are available in the “Boundary Conditions Editor”.
1. Explicitly assigning scaling factors to individual nodes
Boundary Conditions/Scaling Factors (or): enter the relevant scaling factors forhydraulic conductivity and/or pressure
head, and/or water content. Assigning thescaling factors to the grid nodes is donein a manner similar to that previously
described for “Material Distribution’.

2. Automatic generation of scaling
factors using stochastic
distribution
Options/Stochastic Distribution of
Scaling Factors:
In addition to explicitly assigning scaling factor to grid nodes, there is an option of
assigning stochastic (randomly distributed) scaling factors. This window appears
when that option is selected. Select the parameters that will be scaled (check therelevant boxes).
The following window appears. For each scaling factor, provide the standard
deviation, and the correlation lengths inthe x and z directions. More details on
the meaning of these parameters arefound in Appendix IV.2.
When done, click OK. The previous
window will re-appear; you can either
click OK or view the results by clicking
Re-calculate.

Modelling variably saturated flow with HYDRUS-2D1.43

Section 1: HYDRUS Windows

1.1.10.7 Observation Nodes ( ):You may specify up to 10 nodes in the FE mesh to obtain a time (T-level) series ofpressure heads, water contents, and solute concentrations.
Condition, Observation Nodes ( )Click Insert (sidebar buttons); cursor changes to an open square
Select the appropriate nodes.
You can access the results from ‘Observation Points’ in ‘Post-Processing’.

1.1.10.8 Nodal Recharge ( ):This option allows you to specify a sink/source node anywhere in the domain.
Condition, Nodal Recharge ( )Click Select (sidebar buttons); cursor changes to ahand; select the appropriate nodes.
Enter the rate as shown in the window here.
Note that a negative sign refers to a sink (water is being withdrawn from the domain).
When a nodal recharge is specified to a set of external boundary nodes, HYDRUS
considers those nodes to be constant flux nodes with the appropriate flux value. Refer
to example 4.10 for a sample calculation. For 2-dimensional flow problems the units
are L2/T, and for 3-dimensional (axisymmetric) flow the units are L3/T.

1.1.10.9 Local Anisotropy ():

HYDRUS allows you to introduce anisotropy in the hydraulic conductivity as well aschanges in the orientation of the local principal directions (in each element).
Condition, Local Anisotropy ( )Click Quantity (sidebar buttons); then sequentially select Angle, First Component(corresponds to x-axis), and Second Component (corresponds to z-axis).
Click Select (sidebar buttons); cursor changes to a hand; select the appropriateelements.
Click Set Value (sidebar buttons), then enter the appropriate values as shown below.
The default values shown below refer to isotropic conditions.

Refer to example 4.8. For a theoretical background on ‘Implementation of hydraulicconductivity anisotropy’, refer to Section 5.3.12 (p. 55; HYDRUS Technical Manual).

Modelling variably saturated flow with HYDRUS-2D

1.44

Section 1: HYDRUS Windows

1.2 Post-Processing
There are many ways of viewing and/or manipulating the simulation results. Post –processing is done in the RHS of the default window.
1.2.1 Graphical Display of Results
Open Project ‘Mbal-2’ (CD\HYDRUS Manual Projects\Sect-1); Double click on
Graphical Display of Results (or ).
You have the following options in the drop-down menu (see Figure 1.21):
1. Quantity: Choose the variable you want to view; water content, pressure head,velocity, temperature, or concentration. You may choose them from the sidebar or
the drop-down menu.
2. Time: Specify the print time you want to view. You may choose this option fromthe drop-down menu (Time/First (Next, Previous, Last) Time Level) or from the
sidebar menu (scroll bar above Flow Animation button or from the drop-down listbox – ‘Layer No:’).
3. Flow Animation: Having selected the variable to be displayed (e.g., head), click‘Flow Animation’ to view an animated flow simulation.
4. 2D-Graph: specify the preferred visualization method, such as isolines ( ),spectrum ( ),or a combination of isolines and spectrum ( ).
5. 1D-Graph (inactive when velocity vectors are chosen in the 2D-Graph): You canget 1-dimensional plots along boundaries ( , , ), or along specified lines
through the domain ( ).
6. Tools: You can search for a node, an element edge, or an element by entering its
number.
7. Options: You may change display options such as isoline parameters, velocity
vector parameters, and colour palettes.
All of these options are straightforward; we will demonstrate several of them here.

Screen Captures:

You may capture the active window by pressing Alt-Shift-Print Screen, thenpasting this window into another software like Word (Figure 1.21).
You may capture the entire screen by pressing Shift-Print Screen
Another option is to let HYDRUS do the screen capture for you. From 2-D
Graphs, choose velocity vector. Zoom in on the area of interest in the middle. Go
to:File/Copy to Clipboard, and choose view to obtain the velocity vector. Now you
can paste the captured view into other softwares (e.g. MS Word, MS PowerPoint).
Then repeat and choose spectrum. The results are shown in Figure 1.22.

Modelling variably saturated flow with HYDRUS-2D

1.45

Section 1: HYDRUS Windows

Figure 21: GraphicalInterphase

Another option is to let HYDRUS do the screen capture for you. From 2-D
Graphs, choose velocity vector. Zoom in on the area of interest in the middle. Go
to:File/Copy to Clipboard, and choose view to obtain the velocity vector. Now you
can paste captured view into other softwares (e.g. MS Word, MS PowerPoint).
Then repeat and choose spectrum. The results are shown in Figure 1.22.

Figure 22: Velocity vectors
for Project ‘Mbal-2’
0

10

20

30

40

50

60

70

80

1-D Plots Along an External Boundary: You can obtain 1-D plots of any variable (watercontent, etc.) over the entire boundary. Try this:Quantity/Water Content
1-D Graph/Boundary - Whole Curve ( ); thecursor turns into a hand. Click anywhere on theouter boundary of the domain.
You will obtain a plot as shown below. Note that
‘Length’ (x-axis) represents the perimeter of theexternal boundary of the domain.
You can get 1-D plots of any variable (water content, etc.) over part of the boundary.
Try this:
Modelling variably saturated flow with HYDRUS-2D

1.46

Section 1: HYDRUS Windows

1-D Graph/Boundary Selection( )Click and drag on the boundary part you want the plot for (this part will turn red), andclick again to finish.
For the plot below, we dragged over the top boundary nodes (Y=140, X=150 to 230).

Alternatively, you can obtain the same information by selecting ‘Boundary Indexes’( ). When using this option you need to know the node numbers that define thestarting and ending points of the boundary section you want to plot. To find the node
numbers, do the following:

Options/Display Options ( )To get the node numbers only on
the outer boundary, check the lasttwo boxes as shown in the figure
You will see that the boundary
section of interest is between nodes
82-92 (coordinates X=150-230).

Modelling variably saturated flow with HYDRUS-2D

1.47

Section 1: HYDRUS Windows

1-D Graph/Boundary Indexes ( )Enter the values as shown in the figure;you will obtain the same 1-D plot forthe defined portion of the boundary.

Cross-Sections along Outer Boundaries
We will demonstrate the use of crosssections along the lower boundary ofthe domain for Project ‘Mbal-3’. This option will allow you to view the water tableprofile at two different times during drainage.
1. Open Project (CD\HYDRUS Manual Projects\Sect-2\Mbal-3).
2. Double click on Graphical Display of Results
3. Quantity/Pressure Heads
4. Select Time Layer 10, or select Time Value = 0.15 day
5. 1_D Graph/Boundary Selection/Select the lower boundary of the domain
6. Export/give a file name/save (CD\HYDRUS Manual Projects\Sect2\head_T_value_10)
7. Repeat step 5 for Time Value=0.2 day
8. Import/enter the file name you
saved for time=0.02 day
9. Double click on a line and change
its style to dashed.
10. To add legends
Right-click on the plot
Legend/Visible, OK
Right-click on the plot
Edit chart data
Click on column heading (c1) andwrite T=0.02 day; repeat for column(c4) and write T=0.2 day
The resulting figure shows the watertable profiles at two times during drainage.

Modelling variably saturated flow with HYDRUS-2D1.48

Section 1: HYDRUS Windows

Cross-Sections Through the Domain
It is possible to get a cross section through the domain at any angle.
Here we demonstrate a vertical cross-sectionthat shows the vertical pressure head profile.
The cross-section shown here is at T=0.2 day;
x=200.
1-D Graph/Cross Section ( )Click a point on the upper boundary.
To ensure a perfect vertical section, use the
keyboard, use the ‘Page Down’ key for coarse
downward movement then use the arrow keys
to move the cross hairs vertically to a point
exactly on the lower boundary.
Alternatively, use the Snap to Grid option as described below.
If you are comparing cross-sections fromconsecutive runs, make sure that the crosssections are identical, that is, taken at exactly thesame position.
Under Options/Grid Settings ( ), check both
boxes “Snap to Grid” and “Grid Visible”.
Change the origin of the grid, and choose suitable
grid spacings (Width and Height) to ensure that the cross-section’s starting and ending
points coincide with a grid point so that the cross hairs will snap to it.
Velocity profiles can be generated also inside the domain, and then integrated
externally to obtain flow rates across a cross-section inside the domain; refer to
Section 4.5 for details.
Other Charting Options
By right clicking on the plot, you willobtain a long list of charting options. We
will not discuss these in detail (some are
actually disabled).
A few examples:
Edit Chart Data: get access to data values
Axis/Titles: changes titles, fonts, etc
Legend/Visible: adds a legend

Modelling variably saturated flow with HYDRUS-2D

1.49

Section 1: HYDRUS Windows

Chart Wizard gives these options:

Format series allows you to obtainstatistical parameters.

Other Features in Graphical Display of Results:
Options/Isoline Parameters: Allows you to change
isoline settings, their density and starting point. A
very small offset is sometimes useful to remove
extremely small numerical oscillations fromconcentration contours.

Options/Velocity Vector Parameters: Allows you to change
the size and scale of the velocity vectors.

Options/Colour Palette: Allows you to choose
from many colours, or black-and-white
palettes, or to custom-make your own. To
make your own palette, select “Custom” fromthe Palette name. Enter a number in “No. ofcolors” (6 here). Click on each colour in“Palette Preview” then change the colour byscrolling on R, G, and B (red, green, andblue). Move to the next colour.

Modelling variably saturated flow with HYDRUS-2D1.50

Section 1: HYDRUS Windows

1.2.2 Pressure Heads
The data are found in the output file “h_mean.out”.
You get a time series of average pressure heads acrossboundaries.
Unit System:
Pressure heads are reported in length units.

1.2.3 Water Boundary Fluxes
The data are found in the output file “v_mean.out”.
You get a time series of various boundary fluxes.

Unit System:
Atmospheric and root uptake (transpiration): L/T(length / time); the flux is reported per unit length of
the boundary; like precipitation units (e.g., mm/day).
It physically means: water volume/unit length ofboundary/unit length in 3rd dimension/unit time (mm3/mm/mm/day=mm/day). The 3rd
dimension is perpendicular to the 2-D domain that we model.
Remaining (free drainage, seepage, etc.): V/T (area / time); the flux is reported overthe entire boundary (volume/unit length in 3rd dimension/time). Note that V/T is thenotation used in the output files. V has units of L2 for two-dimensional problems andL3 for axisymmetrical three-dimensional problems.

1.2.4 Cumulative Water Boundary Fluxes
The data are found in the output file “Cum_Q.out”.
Unit system:
Cumulative flow is reported in units of area (V) for
2D problems, which means volume/unit length in the
3rd dimension (e.g., cm3/cm), and volume foraxisymmetrical 3D problems.
Cumulative boundary flows are obtained byModelling variably saturated flow with HYDRUS-2D

1.51

Section 1: HYDRUS Windows

integrating boundary fluxes (from the output file v_mean.out) to obtain the total
volume of water (Q). Refer to Section 2.2.2 for example calculation.
For atmospheric and root uptake boundary: we integrate (sum up) fluxes over thelength of the boundary and over time to obtain the cumulative flow.
For other boundaries: we integrate fluxes over time to obtain cumulative flow.

1.2.5 Soil Hydraulic Properties
The data are found in the output file “check.out”
You can view the input soil hydraulicproperties. For example, select pressure head forthe x-axis and water content for the y-axis toobtain a plot of the water retention curve.

The retention curve is best plotted as watercontent versus pressure head on a logarithmic
scale. Follow these steps:

After selecting the axes variables:
Right click on the chart
X-axisValue x-axis
SelectChange the values in “Format Axis” asshown.
Uncheck “Automatic scale” and entershown values
Type: select “Logarithmic”
Alternatively, a log scale for pressure heads and conductivities is offered from the list
boxes “Horizontal Variable” and “Vertical Variable”, respectively.

Modelling variably saturated flow with HYDRUS-2D1.52

Section 1: HYDRUS Windows

1.2.6 Run-Time Information
The data are found in the output file “Run_Inf.out”
Provides valuable run time information such as
number of iterations at any time step. This featureallows you to check whether or not the maximum
number of iterations has been exceeded at any
time or not.

1.2.7 Mass Balance Information
Provides mass balance information for the entire domain and for individual subregions at print times.
Area: the area (2D) or volume (3D) of the domain (constant for all print times)
Volume: volume of soil water
InFlow: the sum of fluxes going into and leaving the domain (or sub-region).
hMean: the average pressure head.
WatBalT (V): the total water balance error in the water volume of the domain.
WatBalR (%): the water balance error as a percentage.
As an example, the screen capture below is for a drainage problem.
 Volume is decreasing with time because water is draining from the soil.
 Inflow is negative, indicating a net flux out of domain.
 Inflow decreases with time because drainage is faster at the beginning and thenslows down.

Modelling variably saturated flow with HYDRUS-2D1.53

Section 1: HYDRUS Windows

1.2.8 Convert Binary Output to ASCII
During calculations HYDRUS
generates the binary files ‘h.out’,‘th.out’, ‘v.out’, ‘concx.out’,‘sorbx.out’, and ‘temp.out’ that
contain nodal values of pressure
heads, water contents, velocities,
solution and sorbed
concentrations, and temperaturesat selected print times. Thesebinary files can be converted into
ASCII files ‘h.txt’, ‘th.txt’, ‘v.txt’,‘concx.txt’, ‘sorbx.txt’, and‘temp.txt’ using the dialog window shown here.
The following information can be obtained in text format when needed. Check theappropriate boxes in the window; text files will be generated and saved in the Project
directory. The files are:
1. TH.out -> Th.txt: water content
2. V.out -> v.txt: Velocities (in ‘Print Information’; Print Fluxes box should bechecked)3. H.out -> h.out: Pressure heads
These three files provide nodal information at each print time. The data are reported
in ten columns. That is, the first row represents data for nodes 1-10 and so forth.
Output file ‘V’ has two data blocks at each print-time; one for the x-component and
one for the z-component of the velocity. This information is used for animation, andfor producing 1-D Graphs (along outer boundaries or internal cross-sections).
4. MeshTria.000 - > MeshTria.txt: mesh information
This file provides detailed mesh information by means of three data blocks; for
Project ‘Grid-1’ Section 1.1.9.2; Example 1 (p. 1.18):
First block: 1106 rows for nodes
Second block: 3192 rows for edges
Third block: 2086 rows for triangles

Modelling variably saturated flow with HYDRUS-2D1.54

Section 1: HYDRUS Windows

Modelling variably saturated flow with HYDRUS-2D

Section 2: Output files

2.1
2. HYDRUS OUTPUT FILES

1. h., th., & v.out
2. ObsNod.out
3. h_mean.out
4. v_Mean.out
5. Cum_Q.out
6. Check.out
7. Run_Inf.out
8. Balance.out
9. Optional files
10. Fit.out

2.1

Output Files

The graphical display that appears in the post-processing section uses informationfrom the output files in the manner shown above. These files are found in the local
directory of the project. The units associated with the various variables are discussed
in detail in Section 1.2.4. Note that ‘P-Level’ refers to print time-level, ‘T-Level’ torun-time, and ‘A-Level’ to a variable boundary condition time; refer to HYDRUS
Technical Manual (p.161) for more details.

2.1.1

Boundary.out (data at print time; P-level information)i

Data
Block*

n

x

z

Code

Q**
[V/T]

V**
[L/T]

Numbering Number ofofnode in FE Coordinates As in Table Nodal Velocity
Boundarymesh
2.1(flux)
flow
nodes

h
[L]
Head

th
[-]

Temp
[C]

Water Temperature
content

Conc[M/L3]
Concentration

Number of rows equal to number of outer boundary points (except for no-flux nodes)*The number of data blocks in each file is equal to the number of print times.
**You can access individual nodal fluxes and flow rates across an outer boundary atprint times in this output file. The velocities reported in this file are more accurate
than those used in the Graphical Module (obtained in ASCII form; see Section 2.1.9);
they are the ones used for the mass balance calculations.
This file is stored in the project folder and is not displayed/used by the interface. For
more details on the coding of boundary conditions (Table 2.1), refer to the HYDRUS
Technical Manual (p.77).
Modelling variably saturated flow with HYDRUS-2D

Section 2: Output files

2.2

Table 2.1: Boundary codes (column 5; ‘boundary.out’ output file)
1 Constant head
4 Atmospheric (capacity exceeded *)-1 Constant Flux
-4 Atmospheric (capacity not exceeded)-2 Potential Seepage (unsaturated) -5 Drain (unsaturated)
2 Actual Seepage (saturated)
5 Drain (saturated)
3 Variable head
6 Free or Deep Drainage-3 Variable Flux
* Either the evaporation capacity exceeded (when head<hCritA), or the infiltration
capacity exceeded (when head>hCritS).
2.1.2

ObsNod.out (data at run time; T-level information)

The output below shows that we have two observation points, and hence also two data
blocks, for nodes 431 and 601, which refer to the numbers of observation nodes in the
FE mesh. The information provided is pressure head (hNew), water content (theta),
and temperature (Temp). This file exists only when you include observation nodes.
Additional information on concentrations is displayed if solute transport is considered.

time hNew
0

head

2.1.3

Node(431)
theta

Temp

Water content

Temperature

hNew

Node(601)
theta

Temp

-1

0.429320

h_mean.out (data at run time; T-level information)

This file contains average pressure heads along various boundaries, such asatmospheric boundary, root zone, variable (flux or head) boundaries, constant (flux orhead) boundaries, seepage face, and drainage boundaries.
Time
[T]

2.1.4

HAtm
[L]

hRoot
[L]

hKode3[L]

hKode1
[L]

hSeep
[L]

hKode5[L]

hKode6[L]

Atmospheric

Root

Variable Constant Seepage

Drain

Free drainage

v_Mean.out (data at run time; T-level information)

This file contains average flux/flow rates across various boundaries, such as thepotential atmospheric fluxes, potential root uptake fluxes, actual atmospheric fluxes,
actual root uptake fluxes, variable (flux or head) boundary flow rates, constant (fluxor head) boundary flow rates, seepage boundary flow rates, and drainage boundaries
flow rates.
Note that the atmospheric and root data are reported as fluxes having units oflength/time, whereas the other fluxes are flow rates reported as area/time (arearepresents volume per unit length in the perpendicular direction) for 2D problems.
When calculating cumulative root and evaporation flow rates, the fluxes (in the 1st
four columns) are integrated over the width of the atmospheric boundary condition.

Modelling variably saturated flow with HYDRUS-2D

Section 2: Output files

2.3
Time
[T]

rAtm
[L/T]

rRoot
[L/T]

vAtm
[L/T]

vRoot
[L/T]

vKode3[V/T]

vKode1
[V/T]

vSeep
[V/T]

vDrain[V/T]vBottom...
[V/T]

Atmospheric boundaries

Potential
AtmosphericRain+Evap

Potential
Root
Flux

ActualAtm.
Flux

ActualRoot
Flux

Constant
Boundary
Variable
Flow rate(constant
Boundary
Flow ratehead/flux &nodal recharge)

Seepage
Flow rate

Drain
Flow rate

Lower
Boundary(free/deep
drainage)
Flow rate

Double click on “Boundary Water Fluxes/Vertical Variable”;
the list that appears (shown here) corresponds to the columns
in the “v_Mean.out” file (given above).

2.1.5

Cum_Q.out (data at run time; T-level information)

This file provides cumulative fluxes across various boundaries, that is, flow rates
integrated over the simulation time. All values have area units (V; volume/unit length;e.g. cm3/cm=cm2) for 2D problems and volume units (cm3) for axisymmetric 3D
problems.
Time CumQAP CumQRP CumQA CumQR CumQ3
[T]
[V]
[V]
[V]
[V]
[V]
Potential
Potential Root Uptake
Actual
Evaporation
Evaporation
Flux
Flux
Flux

2.1.6

ActualRoot
Uptake
Flux

CumQ1
[V]

CumQS
[V]

CumQ5
[V]

CumQ6
[V]

Cumulative Cumulative
Variable
Constant
Boundary
Boundary
Flux
Flux

Cumulative
Seepage
Flux

Cumulative
Drain Flow
Flux

Cumulative
Free/Deep
Drainage
Flux

Check.out

This file contains the data in the look-up table used by HYDRUS to interpolate soil
hydraulic properties during the simulation, in addition to some of the input parameters
contained in the input files. The information in this file may be used to plot water
retention and hydraulic conductivity curves of the soil. The range of values for h, K,
etc. depends on the limits entered in “Internal Interpolation Tables”, Section 1.1.5.
Mat

Qr
1

Table
theta

Qs

0.078
of

Alfa
0.43

HydraulichLog (h)nKsl

3.60E-02 1.56E+00 2.50E+01
0.5
Properties whichareinterpolated
C

K

Log(K)lModelling variably saturated flow with HYDRUS-2D

Section 2: Output files

2.4
2.1.7

Run_Inf.out (data at run time; T-level information)

This file provides run-time information, which includes time level, time, time step,
number of iterations, and cumulative number of time steps at every time step. The
first column is simply a counter for time steps, always an integer whose final value
depends on the minimum time step, the maximum time step, and how quickly or
slowly the solution converges. Refer to the Chart 1.1 (p 1.9) on how the time stepping
scheme proceeds. Additional information on the maximum nodal Peclet and Courant
numbers is displayed if solute transport is considered.
TLevelTime
Time-level‘current time step Time at current
number’time-level
1
1.00E-03
2
2.30E-03
3
3.99E-03

2.1.8

dt

Iter

Time step

Number of iterations

1.00E-03
1.30E-03
1.69E-03

2
2
2

ItCum
Cumulative
number of
iterations
2
4
6

Balance.out (data at print time; P-level information)

This file provides mass balance information at the print times. The sample file below
is for a simulation with two print times (at 0.5 and 1 day). Initial conditions arealways reported at the beginning of the file (Time=0).
The print time is printed at the beginning of each data set. The set comprises: Area,
Volume, InFlow, hMean, and water balance errors for the whole domain and each
subregion. Explanations of the terms are given below in the column ‘Explanation’.
Time

[T]

Total

Sub-region

number0
Area
VolumeInFlow
hMean

[V]
[V]
[V/T]
[L]

1.92E-04
8.24E-05
0.00E+00-1.00E+00

1
9.82E-05
4.21E-05
0.00E+00-1

2
9.38E-05
4.03E-05
0.00E+00-1

1
9.82E-05
4.21E-05
0.00E+00-1.2

2
9.38E-05
4.03E-05
0.00E+00
-1.2

0.5
Area
VolumeInFlow
hMean

[V]
[V]
[V/T]
[L]

1.92E-04
8.24E-05
0.00E+00-1.20E+00

WatBalT[V]

8.45E-07

WatBalR

[%]

.025

1.0
Area
VolumeInFlowhMean
WatBalT
WatBalR

[V]
[V]
[V/T]
[L]
[V]
[%]

1.92E-04
8.24E-05
0.00E+00-1.20E+00
1.64E-06.035

Explanation
For total grid and each subregion
Area
Water volume storedΣ In/out flow (out is +)Mean head

Absolute error in water massbalance of entire domain
Relative error in water massbalance of entire domain
1
9.82E-05
4.21E-05
0.00E+00-1.2

2
9.38E-05
4.03E-05
0.00E+00-1.2

Modelling variably saturated flow with HYDRUS-2D

Section 2: Output files

2.5

2.1.9 Optional ASCII files (data at run time; P-level information; for files ’h.out‘,’th.out‘, and ’v.out’)
During the calculations HYDRUS generates the binary files ‘h.out’, ‘th.out’, ‘v.out’,‘concx.out’, ‘sorbx.out’, and ‘temp.out’ that contain nodal values of pressure heads,
water contents, velocities, solution and sorbed concentrations, and temperatures atselected print times. These binary files can be converted into the ASCII files ‘h.txt’,‘th.Txt’, ‘v. txt’, ‘concx.Txt’, ‘sorbx.Txt’, and ‘temp.txt’ using the following dialog
window. These files are not available by default. Once this option is triggered, thiswindow is displayed.
The options are:
 Mesh Information: details finite element
mesh information (file name; MeshTria.txt)
 Pressure head: nodal pressure heads at theprint times (file name; H.txt).
 Water content: nodal water contents at theprint times (file name; TH.txt).
 Velocities: x- and z-components of thenodal velocities at the print times (filename; V.txt).
More details can be found in Section 1.2.8 (P 1.53).
2.1.10 Fit.out
This file relates to the inverse solution. The file lists the input soil hydraulicparameters, the optimisation results at each iteration, the final optimized (estimated)parameters as obtained with a non-linear least squares analysis (final results arenumbers shown in Bold), input inverse data (i.e., observed data that HYDRUS is
calibrating against; found under column ‘Obs’), and the simulated data (found undercolumn ‘Fitted’). The meaning of ‘Type’ and ‘position’ is explained in the HYDRUS
on-line Help.
Non-linear
least-squares analysis: final
results
95%Confidence limits
Variable
Value
S.E.Coeff. Lower
Upper
ALPHA
0.00082
0.01373
0.01738
1.56E-02
N
0.10786
1.94905
2.42969
2.19E+00
CONDS
0.00234
0.007480.01788
1.27E-02
RSQUARE forregression ofpredicted vsContributions to

the

final

Measurement Set&
Resi-

QUANTITY

Time
1
2
3

objective function

2:00 1.66E-01

FittedNoobserved =

33
66
100

Obs
0.113
0.121
0.142

Observed data (experimental
data; what you entered)Difference between
fitted and observed----------------------------Fitted
0.112
0.112
0.126

dual
0.001
0.008
0.016

Type

Position
2
2
2

1
1
1

Fitted data (what HYDRUS simulatedusing the final optimized parameters)

Modelling variably saturated flow with HYDRUS-2D

Section 2: Output files

2.6
Correlation matrix
CONDS
ALPHA N
ALPHA
1
N
-0.2156
1
CONDS
0.9887 -0.3591
1

The correlation matrix (shown above) indicates whether the parameters are correlated
or not, with 1 indicating perfect positive correlation and –1 indicating perfect negative
correlation.
S.E. Coeff. refers to the standard error.
RSQUARE refers to the goodness of fit.
Upper and lower limits refer to the sensitivity of the model to the parameter. Refer toAppendix V.2 for more details on statistical issues in inverse modelling.

2.1.11 A_Level.out (data at variable boundary time)
This file contains data at the variable boundary time levels, that is, the number oftimes (rows in this file) is equal to the time-variable boundary records entered in
Window ‘Time Information/Boundary Conditions’. This file will not be created if you
do not have time-variable boundary conditions (infiltration, evaporation, transpiration,
variable flux, or variable ground water table).
All

cumulative fluxes(CumQ)Time[T]

CumQAP[V]

CumQA CumQR
[V]
[V]

Time

Potential
cumulative
flow acrossatmosphericboundary

2.2

Examples Demonstrating the Use of Output Files

CumQRP[V]
Cumulative
potential root
uptake(transpiration)

are

Actualcumulative Cumulative
flow across Actual root
atmospheric uptake
boundary

positive
CumQ3[V]
Cumulative
flow acrossvariableboundary

out
hAtm
[L]

of

the

regionhRoot hKode3 A-level[L][L]

Averagehead in Average Number
Averagehead atroot zone headof ‘A‘
atmosphericvariable Level
boundaryboundary

2.2.1 Project ‘WT-1’ (CD\HYDRUS Manual Projects\Sect-2): This example
demonstrates a case where we must have access to the HYDRUS output files. This
problem considers a fully saturated soil profile (i.e., water table at the soil surface).
We simulate the decline of the water table as a result of evaporation (see Figure 2.1),
and want to plot the actual surface flux ratio (actual over potential flux) versus depth
to the water table. For this purpose we need to place an observation node at the baseof the soil profile, and then plot the flux (from ’v_Mean.out’ file) versus the pressure
head (from the ’ObsNod.out’ file).

Modelling variably saturated flow with HYDRUS-2D

Section 2: Output files

2.7

Watertable
Atmospheric BC
Evaporation rate = 0.5 cm/day

NOTE: The initial water table is at the soil surface; however,experience has shown that setting the initial headat the surface to 0 is often problematic for getting
convergence in the numerical solution (the entire
domain is then saturated). The problem isObservation node
avoided by setting the initial water table
depth at 1 cm.

1,000 cm

Figure 2.1: Domain and BC
Project WT-1
Open file Flux Ratio-1.xls on the CD (CD\HYDRUS Manual Projects\Sect-2).

The relevant output files are imported into
two worksheets, ‘v_Mean’ and ‘ObsNod’.

In order to obtain the flux ratio we normalise the flux with respect to the potential
evaporation rate, that is, we divide all fluxes by 0.5 cm/d. The depth to the water tableis calculated by subtracting the head from the initial head (remember 999 rather than1000).
Since the data in both files are at the same run time levels, we can plot flux ratios
calculated from the v_Mean.out file versus depth to water table calculated from’ObsNod.out’ file. The result is demonstrated in Figure 2.2, which shows the criticalwater table depth below which the soil cannot sustain the flux at its potential rate.
1
0.9
0.8

Flux ratio

0.7
0.6
0.5
Water table depth (critical depth)
at which soil can no longer
provide water as dictated bypotential evaporation (climatic
demand)

0.4
0.3
0.2
0.1
0
0

50

100

150

200

250

Depth to water table (cm)Figure 2.2: Change of flux ratio with water table depth

Modelling variably saturated flow with HYDRUS-2D

Section 2: Output files

2.8

2.2.2 Project “Mbal-2” (CD\HYDRUS Manual Projects\Sect-2): the domain of
the problem is shown in Figure 2.3. Part of the upper boundary is an atmospheric BC
and the lower boundary is a seepage face; rainfall is simulated at two different rates
during the 2-day simulation.
RainfallDay 1 = 30 cm/day
Day 2 = 60 cm/day
Atmospheric BC; 4 nodes
Code (4)

No flux BC

Loamy sand
Uniform initial potential = -100 cm

140 cmSeepage face BC; 39 nodes
Code (2)400 cmFigure 2.3: Domain and BCs for Project Mbal-2
We provide details by means of the following files:
1. Boundary.out
2. v_Mean.out
3. Cum_Q.out
4. A-Level
5. Balance.out
Those files contain the most valuable flux and mass balance data. We will
demonstrate how the data in those files are interrelated.
Open file Mbal-2.xls on the CD (CD\HYDRUS Manual Projects\Sect-2).
1. Boundary.out data at print time level (in Worksheet: Boundary):
Column 1: i =1, 2,…..44 represents the boundary nodes (other than no-flux); in thisproblem the sum of seepage and atmospheric boundary nodes is 44 (39+5).
Column 2: n represents the actual corresponding node number in the FE mesh.
Columns 3, 4: x and z represent the coordinates of the nodes. For example, if you
want to know the width of the atmospheric boundary condition, do the following:x(I=44) - x(I=40) = 217.8 - 184.1 = 33.7 cm
Column 5: code refers to the boundary condition code (as listed in Table 2.1). Notethat all seepage face nodes are coded (-2) up to Time=1.6 day, which means that they
are considered to be a potential seepage face (i.e., there is no actual seepage face yet).
At Time=1.8 days, nodes (I=19-21) are coded (2), which means they are now actual
seepage face nodes. At T=2 days, the seepage face extends to 5 nodes (I=18-22).

Modelling variably saturated flow with HYDRUS-2D

Section 2: Output files

2.9

Column 7: v refers to the nodal flux (cm/day). For atmospheric BCs, v is simply therate of rainfall, which is 30 and 60 cm/day, for Time=1, and 2, respectively.
Individual nodal fluxes are readily available only in this file (they can be obtained
also from “v.out”; Section 2.1.9; however, those are less accurate). The significanceof the data is demonstrated in Figure 2.4, which shows fluxes through individual
actual seepage points (Worksheet ‘Boundary’, Cells ‘C440, G444’ and Cells ‘C490,
G496’).
0

Seepage nodal flux

-50

-100

-150

HYDRUS ProjectMbal-2-200

2 Days
1.8 days
-250

-300
16

17

18

19

20

21

22

23

24

Node number

Figure 2.4: Individual nodal fluxes from ‘Boundary.out’ file
Column 6: Q refers to flow rate (cm2/day). This is the flux (v) integrated over thenodal spacings (Figure 2.5). When integrating, the first and the last nodal fluxes (forI=40 and 44) will be multiplied by half the spacing only.
Note that if the flux at two neighbouring nodes (of an element) is different, it iscommon practice in finite element methods to assume a linear distribution of the flux
along the element.
i = 414243

x = 209.5

201.1

192.7
Atmospheric Boundary
Flux = 30 cm/dayw
i=42

Q = 30w
30×8.4 = 252 cm2/day
Figure 2.5: Definition of v and Q

Columns 7, 8, 9, and 10: h, th, temp, and conc, refer to pressure head (cm), watercontent, temperature, and solute concentration, respectively.

Modelling variably saturated flow with HYDRUS-2D

Section 2: Output files

2.10

2. v_Mean.out data at run time level (in Worksheet: v_Mean):Column 1: Time (days) refers to run time. The smaller the time steps, the morerecords will be stored in this file. Time steps are controlled by the minimum andmaximum time steps and by how quickly the solution converges (if the solution does
not converge quickly, the time steps decrease and hence the number of records will
increase).
Columns 2 and 4: rAtm and vAtm refer to potential and actual atmospheric flux
(cm/day), respectively (equal in this problem).
Column 8: vSeep is the seepage flux across the boundary (cm2/day). Note that
seepage starts at Time=1.6988 days (row 931 in spreadsheet).
At time=1.8 days, vSeep=574 cm2/day. This value can be calculated from output file
“Boundary.out” as follows:
Go to print time=1.8 day in Worksheet Boundary,
Sum Q in column G, cells G441+G442+G443=573 cm2/day
The mass balance may be calculated from this file as follows: Calculate Qatm by multiplying the actual atmospheric flux by the width of theatmospheric BC (33.7 cm calculated in the previous section).
 Add Qatm to Column (I, vSeep) to get the total In/out flow.
 This is demonstrated in column P, Worksheet v_Mean. The result is shown in
Figure 2.6.
2500
Constant atmosphericinflow acoss the wholeboundary as result of 60
cm/day rainfall

2

Σ Flow in/out (cm /day)..

2000

1500
Constant atmosphericinflow acoss the wholeboundary as result of 30
cm/day rainfall

Drop as a result of
seepage flux out

1000

From Balance.out

500

From v_Mean.out

0
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (days)

Figure 2.6: Sum of inflow and outflow rates
In the next section, the results shown in Figure 2.6 will be compared with thoseobtained from the “Balance.out” file at the print times.
The flow rates may be integrated over time to obtain cumulative fluxes identical tothose found in output file “Cum_Q.out”. This is demonstrated in Columns M and N(Worksheet v_Mean) for cumulative atmospheric and seepage flows, respectively.
The results are very close to the corresponding values reported in the “Cum_Q.out”
output file (Columns E and I; Worksheet Cum_Q). The difference is less than 0.3%.
Modelling variably saturated flow with HYDRUS-2D

Section 2: Output files

2.11

3. Cum_Q.out data at run time level (in Worksheet: Cum_Q):
Columns 2 and 4: CumQAP and CumQA refer to potential and actual cumulative
atmospheric fluxes (cm2), respectively (equal in this problem) across the entire
atmospheric boundary.
Column 8: CumQS is the cumulative seepage flow volume (cm2) across the entire
seepage boundary.
4. A_level.out data at atmospheric time level (in Worksheet: A_level):
Columns 1: Time is atmospheric time, that is, any time where a variable atmosphericboundary condition has been specified. For our problem here this was at 1 and 2 days.
We specified (under variable boundary conditions) that:during time = 0-1 day, rainfall is 30 cm/day, andduring time = 1-2 day, rainfall is 60 cm/day.
Columns 2 and 4: CumQAP and CumQA refer to potential and actual cumulative
atmospheric flow volumes (cm2), respectively, across the whole atmosphericboundary. These numbers can also be found in Output file “Cum_Q” at times 1 and 2days, respectively. They are calculated as follows:
At time 1 CumQA = 30×33.7×1 = 1010 cm2
At time 1 CumQA = 1010 + 60×33.7×(2-1) = 3030 cm2

5. Balance.out data at Print time level (in Worksheet: Balance):
The mass balance can be calculated on a volume basis as follows:
Volume of water stored in soil prior to simulation, (Time=0) = 3990 cm2 (Cell D13)
Volume of water stored in soil at end of simulation, (Time=2) = 6820 cm2 (Cell D91)
Change in water storage (gain) = 6820 – 3990 =Cumulative infiltration (rainfall into domain) =Cumulative seepage (out of domain) =Mass balance (should be zero) =

2830 cm2
-3040
202
8

For all print times, write the ‘InFlow’ values (Cells D14, D20, etc.) and thecorresponding times in two neighbouring columns as shown in Cells(G12-H22). Insert the data set into Figure 2.6. Notice that you obtained thesame results as from the calculations in file “v_Mean.out”, but only at theprint times. To get a more continuous set of mass balance results, you canincrease the print times up to 250 (or up to 5,000 if done outside thegraphical interface; see Section 4.12).

Modelling variably saturated flow with HYDRUS-2D

Section 2: Output files

2.12

2.2.3 Project “Mbal-3” (CD\HYDRUS Manual Projects\Sect-2): the domain of
this problem is shown in Figure 2.7a. The upper 120 cm of the profile is sandy loam
and the lower 20 cm is silt. Due to the configuration of the soil layers and boundary
conditions, this system will promote lateral flow. The left-hand side of the upperboundary is an atmospheric BC, the lower boundary is a free drainage BC, and theright-hand vertical boundary is a seepage face. The soil profile is initially fully
saturated, rainfall is simulated at a rate of 10 cm/day during the 0.2-day simulation.
The number of print times was set to 100 to obtain a good resolution of nodal fluxes
reported at print times in the “Boundary.out” output file.
In this example we will go through the mass balance calculations by considering
subregions. Referring to Figure 2.7a, there are two subregions in this example (thedomain is split vertically). The atmospheric boundary condition is restricted tosubregion 1 (nodes 84-95), the free drainage BC goes through both regions (nodes 214 in subregion 1, and nodes 15-36 in subregion 2), while the seepage face is insubregion 2 (nodes 37-53).
Subregion 1

Subregion 2

96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 6867 6665646362616059585756555453
52
97
51
50
Atmospheric BC
98
49
No
flux
BC
48
Code (4)
99
47
100
46
120
cm,
Sandy
loam
45
101
44
102
Seepage face BC
43
103
42
Code (2)104
41Free drainage BC
105
40
Code (6)106
39

20 cm, Silt

107
1

2 3

4 5

6

7 8

38

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

Subregion 1

Subregion 2

Figure 2.7a: Domain and BC for Project Mbal-3
We are interested in quantifying the amount of lateral flow from subregion 1 tosubregion 2 (refer to the block diagram in Figure 2.7b). Let us study individual massbalances for each subregion at the end of the simulation (Time=0.2 day).
Infiltration

Region1Free drainage
from Region 1

Lateral flow from
Region 1 to 2

Free drainage
from Region 2

Region2

Seepage

Figure 2.7b: Flow through regions

Modelling variably saturated flow with HYDRUS-2D2.13

Section 2: Output files

Cumulative fluxes are reported across entire boundaries (e.g., the flux for the freedrainage boundary is reported across nodes 2-36). Hence, we must split this boundary
in order to calculate how much water leaves subregion 1 (nodes 2-14) and how muchleaves subregion 2 (nodes 15-36). This is achieved by manipulating the data reported
in the “Boundary.out” output file.
Open file Mbal-3.xls in the CD (CD\HYDRUS Manual Projects\Sect-2).
The ‘Boundary.out’ output file is imported into Worksheet ‘Boundary’.
•For each print time,o Write the print time (1st print time is in Cell C10) in ‘Cell N6’.
o Sum up Q’s for nodes n = 2-14 (G15-G27); this represent the flow ratefrom subregion 1 at the first Print Time. Write the result in ‘Cell O6’.

• Repeat 100 times to get data in Cells ‘N6-O105’ (writing a macro in Visual Basichelps).
• Integrate the flow rate over the print times to obtain volumes; if the print times areequal, simply multiply Q by the print time to get volume. If they are unequal, use an
average. For example, to get the volume (Vx) at print time Tx:
o Vx = Qx × (Tx-1 – Tx+1)/2
o The result is shown in Column ‘P’.
• Calculate cumulative volume as in Column ‘Q’.
• Repeat the same process by summing nodes 15-36 to get the cumulative volume
from free drainage under subregion 2. The results are shown in (Cells S5-V105).
• Add the final numbers from both subregions; this should be equal to that reported
in output file Cum_Q.out; referring to Worksheet ‘Boundary’:
Cell Q105 + V105 = 161.82 + 298.32 = 460.14 ≈ 460 (Sheet Cum_Q, Cell K142) OK.
• Mass balance for subregion 1 = Infiltration (water gained, negative sign) + Free
boundary flux (sum of nodes 2-14, water lost) + change in storage (note that waterdrained from the soil is a gain to the water balance, and thus has a negative sign;calculated as the difference in Volume at time 0.2 day and the initial Volume,
Worksheet Balance ) + Lateral flow from subregions ‘1’ to ‘2’ = 0 (only lateral flow
is unknown)• The amount of lateral flow that resulted in a zero-mass balance is added as ‘flow
in’ to subregion ‘2’; the water balance for subregion ‘2’ is calculated in the same way;
the lateral out flow from subregion ‘2’ should be equal to the cumulative seepage
reported in output file ‘Cum_Q.out, Cell I142”.
A sample calculation is shown in Worksheet ‘Volume-based mass balance’, and isalso shown in Table 2.2.

Modelling variably saturated flow with HYDRUS-2D

Section 2: Output files

2.14

Cumulative Boundary Flow from ‘Cum_Q.out’ output file found in Worksheet Cum_Q
Cumulative Free Boundary Flow Cell K-142
Cumulative Seepage Flow Cell I-142
Cumulative Actual Atmospheric Flow Cell E-142

Table 2.2: Volume-based massbalance (from spreadsheet of same
name in Workbook Mbal-3.xls

460
906
-257

Mass Balance Region 1
Atmospheric Boundary, Rain =Free Boundary flow out under Region 1 =Change in Soil-water storage (drained) =Summation

Spreadsheet
Cum_Q
Boundary
Balance

Cell
E142
Q105
O19

-257
161.8
-110

gainedlostgained-205.2Out of region 1 laterally into Region 2-205.18
298.3
-1000

Spreadsheet
gained This sheet
lost
Boundary
gained Balance

Mass Balance Region 2
Input from Region 1 =Free Boundary flow out under Region 1 =Change in Soil-water storage (drained) =Rain in-257

REGION 1-110 Drained from storage(gain to mass balance)
Free Boundary out
161.8

..

Summation

Cell
E13
V105
P19

-906.9 Out of region 2 laterally as Seepage

Lateral flow
-257
+
-110
+
161.8
=
-205.2REGION 2-1000 Drained fromstorage-205.2
+
-1000
+
298.3
=
906.9

Close to cumulative
seepage flow (906)in ‘Cum_Q.out’

Free Boundary out
298.3
Sum = 460.14; close to value in ‘Cum_Q.out’
Modelling variably saturated flow with HYDRUS-2D

Section 2: Output files

2.15

Compare the flow rates during the simulation from output files ‘v_Mean.out’ and‘Balance.out’.
• Flow rates at run times are found in output file ‘v_Mean.out’. In the case of
atmospheric BCs, fluxes are reported (cm/day). Hence, we sum them up over theentire boundary to obtain the total flow rate (cm2/day). This is achieved bymultiplying the flux by the width of the atmospheric BC, which is 130 cm in this case.
The calculations are shown in Worksheet ‘Flow rates, Columns C and D’.
• Add ‘Column D’ (Worksheet ‘Flow rates’) to ‘Columns I and K’ (Worksheet‘v_Mean’); find the result in ‘Column H’ (Worksheet ‘Flow rates’). This will result
in total inflow/outflow rates during the simulation.
• Plot the result versus time as shown in Figure 2.8.

5000

2

Σ Inflow/Outflow (cm /day)

10000

Hatched area = Change in storage volume = 1,100 cm2

0Flow sums from v_Mean.out file
Total InFlow from Balance.out file

-5000

-10000
0

0.02

0.04

0.06

0.08

0.1
0.12
Time (days)

0.14

0.16

0.18

0.2

Figure 2.8 : Total inflow/outflow from ‘Balance.out’ and ‘v_Mean.out’ files•••

The same data may be obtained from output file ‘Balance.out’.
Go to Worksheet ‘Balance’.
For each print time,o Write the print time (1st print time is found in Cell B11) in ‘Cell G3’o Write Inflow (1st Inflow is found in Cell D14) in ‘Cell I3’.

• Repeat 100 times to get data in Cells ‘G3-I103’.
• Plot inflow data versus time as shown in Figure 2.8; you will see that they are
similar to those obtained from the ‘v_Mean.out’ file.
If we integrate flow rates over time we obtain cumulative fluxes (or volumes). The
area under the curve shown in Figure 2.8 represents the change in storage during thesimulation (1100 cm2).

Modelling variably saturated flow with HYDRUS-2D2.16

Section 2: Output files

Modelling variably saturated flow with HYDRUS-2D

Section 3: Root Water Uptake

3.1

3. ROOT WATER UPTAKE

3.1 Relevant Windows in HYDRUS

When root water uptake (planttranspiration) is modelled, therelevant box in “Main Processes”
should be checked.

As a result, two new input lines
are added under “PreProcessing”. They are: “Root
Water Uptake Models” and“Root Water Uptake ModelsPressure Head Reduction” asshown here.

Other related windows where input is required are:•“Variable Boundary Conditions”, enter ‘potential transpiration’. Described in
Boundary conditions, Section 1.1.10-9; Atmospheric.

•“Boundary Condition Editor”
Either:o “Condition/Root Distribution” ( )Select nodes
Set Value
Water Uptake Distribution; enter appropriate value

Modelling variably saturated flow with HYDRUS-2D

Section 3: Root Water Uptake

3.2

o Select a set of nodes, and choose
one of the options as shown in thiswindow. When selecting ‘Same
value for all nodes’, only the ‘Top
Root Uptake Value’ is needed.
Refer to Appendix III.2 for details.

o Alternatively, you can enter appropriate parameters under“Option/Parameters for Root Distribution”; see Appendix III.3 fordetails.

Root Water Uptake Model:

The water uptake reduction model defines the mannerin which transpiration is reduced below the potential
rate when the soil is no longer capable of supplying
the amount of water demanded by the plant under theprevailing weather conditions. There are twoalternative reduction models: one by the Feddes et al.
(1978), further referred to as the Feddes model, andone by van Genuchten (1987), further referred to as
the S-shaped model. The former is more commonly used.
Feddes’ Model Parameters
The Feddes model assigns plant transpiration rates
according to the soil’s pressure head.
Feddes’ model parameters are shown in thiswindow. The meaning of those parameters isgraphically demonstrated in Appendix III.1.
P0: Value of the pressure head below which roots
start to extract water from the soil.
Popt: Value of the pressure head below which roots
extract water at the maximum possible rate
(potential transpiration).
P2H: Value of the limiting pressure head below which roots no longer extract water at
the maximum rate (assuming a potential transpiration rate of r2H).

Modelling variably saturated flow with HYDRUS-2D

Section 3: Root Water Uptake

3.3

P2L: As above, but for a potential transpiration rate of r2L.
P3: Value of the pressure head below which root water uptake ceases (usually taken at
the wilting point).
A database of suggested values for different plants is provided based on studies by
Wesseling (1991) and Taylor and Ashcroft (1972).
r2H: Potential transpiration rate (L/T) (currently set at 0.5 cm/day).
r2L: Potential transpiration rate (L/T) (currently set at 0.1 cm/day).
The above 2 input parameters permit one to make the variable P2 a function of thepotential transpiration rate (P2 presumably decreases at higher transpiration rates).
HYDRUS currently implements a linear interpolation scheme for this purpose (givenin the HYDRUS Technical Manual).

S-Shaped Model Parameters
van Genuchten (1987) proposed an S-shaped function
to describe root water uptake reduction, which isapplicable to water stress as well as salinity stress.
Two input parameters are required in this model: P3,
which is an experimental constant, and P50, thepressure head at which the actual uptake rate is half
the potential rate. More details about the S-shaped
function are found in Appendix III.1.

3.2 Root water uptake simulations
Projects Root-1 to Root-8 (CD\HYDRUS Manual Projects\Sect-3) demonstrate theeffects of various root water uptake parameters on root zone fluxes and soil waterpressure heads. The projects simulate root water uptake from a freely draining, 1-m
deep silty-loam soil profile.
The initial pressure head for all projects is hydrostatic equilibrium with the zeropressure head at the freely draining bottom boundary (initially set equal to zero).

Projects Root-1, 2 and 3: Effect of range of optimal root water uptake (difference
between P2H and P2L).
The Projects assume a uniform root distribution down to a depth of 50 cm (no roots
below 50 cm), and a potential transpiration rate of 0.2 cm/day. Note that in the case ofa uniform root distribution, the absolute value for the water uptake distribution does
not really matter; any number that you enter will yield the same result since the root
distribution values are internally integrated and normalized by HYDRUS .
Project Root-4: As above but the root distribution gradually decreases from unity at
the surface to zero at a depth of 50 cm.

Modelling variably saturated flow with HYDRUS-2D

Section 3: Root Water Uptake

3.4
Results:

Projects Root- 1 and 2: Figure 3.1 shows that the actual flux falls below the potential
rate at a later time when P2L is higher. In other words, root water uptake stays at thepotential rate longer.
0.2

Actual root flux (cm/day)

0.15

0.1

Project Root-1: Wide optimal range; P2H = -50; P2L = -4000
0.05
Project Root-2: Narrow optimal range; P2H = -500; P2L = -1000

0
0

25

50

75

100

Time (days)

Figure 3.1: Effect of P2H and P2L on calculated actual root fluxes

Projects Root- 1 and 2: Figure 3.2 shows that since actual root uptake is reduced at alater stage (high P2L), the pressure heads in Simulation Root-1 are lower than those in
Simulation Root-2.
0

Pressure head at surface (cm)

-2000

-4000

-6000

-8000

Project Root-1: Wide optimal range; P2H = -50; P2L = -4000
Project Root-2: Narrow optimal range; P2H = -500; P2L = -1000-10000
0

20

40

60

80

100

Time (days)

Figure 3.2: Effect of P2H and P2L on calculated soil water pressure heads

Modelling variably saturated flow with HYDRUS-2D

Section 3: Root Water Uptake

3.5

Project Root-3: Figure 3.3 shows that toward the end of the relatively long simulation,
the flux fluctuates due to upward migration of moisture caused by the gradient
between the dry soil within the root zone (at depth=50 cm) and the soil below it. This
process is the result of the selected precision for calculations. Uptake rates at larger
time are very small (<0.001 cm/d) and thus below the water content precision
tolerance (=0.001). Oscillations can be eliminated by decreasing the water content
tolerance to 0.0001.
1Time = 54 days

Actual root flux (cm/day)

0.1

0.01

0.001

Project 'Root3' has identical parameters to Project 'Root1'
0.0001
0

500

1000

1500

2000

Time (days)

Figure 3.3: Root fluxes with abrupt change in root distribution (Root-3)

Figure 3.4 shows that the soil within the root zone eventually reaches a pressure headequal to P3 (-8,000 cm in this example). Notice that at a depth of 52 cm (just outside
the root zone), the pressure head does not reach that value.
0
At surface
Depth = 50 cm within root zone
Depth = 52 cm below root zone

Pressure head (cm)

-2000

-4000

-6000

-8000
0

200

400

600

800

1000

1200

1400

1600

1800

2000

Time (days)

Figure 3.4: Calculated pressure heads at various depths (Root-3)

Modelling variably saturated flow with HYDRUS-2D

Section 3: Root Water Uptake

3.6

Projects Root- 1 and 4: in Project Root-4, the root distribution gradually decreases
from unity at the surface to zero at a depth of 50 cm.
Referring to Figure 3.5, notice that for a uniform root distribution at different depths,
there is a lag in the pressure head change; however, the trends are identical.
0Equal root density

Pressure head (cm)-2000

At surface-4000
At depth = 20 cm

-6000

-8000
0

10

20

30

40

50

60

70

80

90

100

Time (days)

Figure 3.5: Calculated pressure heads for uniform root distribution (Root-1)

Figure 3.6 shows that when the root densities are different at two locations in the root
zone, the pressure heads will evolve very differently in time.
0

Pressure head (cm)

-2000

-4000

At surface; root density = 1-6000

At depth = 20 cm; root density = 0.3

-8000
0

10

20

30

40

50

60

70

80

90

100

Time (days)

Figure 3.6: Calculated pressure heads for non-uniform root distribution

Modelling variably saturated flow with HYDRUS-2D

Section 3: Root Water Uptake

3.7

Projects Root-5: This simulation is identical to Root-1, except that the potential
transpiration rate is doubled to 0.4 cm/day. Figure 3.7 shows that increasing thepotential transpiration rate causes the pressure heads to drop much earlier; however,
the trends are similar in both cases.
0
Pt = 0.4 cm/day

Pressure head at surface (cm)Pt = 0.2 cm/day
-2000

-4000

-6000

-8000
0

10

20

30

40

50

60

70

80

90

100

Time (days)

Figure 3.7: Effect of potential transpiration rate (Pt) on calculated pressure heads

Figure 3.8 shows the effect of increasing the potential transpiration rate on cumulativeroot extraction.

Cumulative root extraction (cm2)

16

12

8

Pt = 0.4 cm/day
4
Pt = 0.2 cm/day

0
0

10

20

30

40

50

60

70

80

90

100

Time (days)

Figure 3.8: Effect of the potential transpiration rate on cumulative root extraction

Modelling variably saturated flow with HYDRUS-2D

Section 3: Root Water Uptake

3.8
Project Root-6: Effect of initial conditions

In this simulation, the initial pressure head of the soil is higher than P0 (i.e., nearly atsaturation). The lower boundary is sealed (not drained). Since the pressure head ishigher than P0 everywhere in the domain, it will stay that way and transpiration will
never start. This problem will be addressed in Projects Root- 7 and 8.
Project Root-7: The surface nodes (only the top 2 nodes) are set to a pressure equal to
P0. The results are shown in Figure 3.9.

Figure 3.9: Actual root uptake
for Project Root-7

Project Root-8: Introduce surface evaporation for a short duration of time. The time
variable BC records are increased to 2. The first record represents an evaporation
period of 0.1 day at an evaporation rate of 0.1 cm/day. This will decrease the surfacepressure head to a value below P0, which in turn induces the initiation oftranspiration.

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.1

4. EXAMPLE APPLICATIONS
4.1

Fluctuating Stream Level

In this section, we model a stream with a fluctuating water level. When the waterlevel in the stream is rising, a variable head BC is suitable. The specified pressure
head is assigned to the lowest point and hydrostatic equilibrium is assumed for the
points above it. For a falling stream stage, this boundary does not represent what
actually happens since a seepage face likely will develop. HYDRUS at present is notcapable of modelling such a problem in one project because of the changing nature of
the boundary representing the stream. Therefore, simulations must be carried out intwo stages as follows:
1. Project “Ris-St”: simulate the rising water level using a variable headboundary representing the stream.
2. Project “Fall-St”: simulate the water table drawdown that results from thelowered stream level. The initial conditions for this project are imported fromthe last time level in Project “Ris-St”.
All related Projects are found on the CD (..\HYDRUS Manual Projects\Sect-4).
Project “Ris-St”
The conceptual model for this problem is shown in Figure 4.1. The soil used is asandy loam. The flow domain is initially in hydrostatic equilibrium with the watertable that is located at the lower base of the domain. The stream rises in three stages
as shown in Figure 4.1.
Rising stream levels

Hydrostatic
equilibrium

Initial water table; level=0
OP

VariablepressureboundaryOP

80 cm

6-10 days

60 cm

2-6 days

20 cm

0-2 days

Figure 4.1: Conceptual model for Project “Ris-St”

Figure 4.2 shows the pressure head response at
50 cm and 150 cm away from the stream
obtained at two observation points (OP in
Figure 4.1).

Figure 4.2: Response to anincrease in the stream level

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.2

Figure 4.3 shows the pressure head distribution across the entire domain at Time=10
days. This distribution will be used as the initial condition for the next project.

Figure 4.3: Pressure head distribution; Time=10 days
Project Manager: Select Project “Ris-St”
Copy, and change details as shown here. This will
provide us with an identical copy of the project.
We will only change the initial conditions and theboundary conditions representing the stream.

Project: “Fall-St”
Boundary Condition Editor ( )Sidebar Menu: Water Flow Buttons/Seepage face: select the vertical points
representing the stream down to a depth of 40 cm.
Condition/Initial Condition/Import Pressure
Directory path……\Ris-St\h.out\OpenAccept the default value 2 (as shown here).
Accept the default value t=10 days (as shownhere).
The initial pressure head distribution should be
identical to that shown in Figure 4.3.

Figure 4.4 shows the pressure head response to thefalling-stream stage at the two observation nodes.
Note that the initial time of the simulation is day 10,
which is the final time of the previous simulation;
this is set in “Time Information/Initial Time”.

Figure 4.4: Response to drop in stream level

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.3
4.2

Modelling Evaporation: Effect of Discretization

The spacing of the elements near the soil surface can have a significant impact on the
results of evaporation simulations. The effect is related to movement of the drying
front through the soil profile and development of potentially very low (negative)
pressure heads (and gradients) near the soil surface. All related Projects are found on
the CD (..\HYDRUS Manual Projects\Sect-4).
4.2.1

Advance of Drying Boundary
2 cm
10 cm

Figure 4.5: Grid spacing near the soil surface
Project‘Evap2’

Project‘Evap1’

Coarse
mesh

Fine
mesh

In projects “Evap1” and “Evap2”, we track the movement of the drying front for twocontrasting meshes, coarse and fine, respectively (see Figure 4.5).

0
Coarse mesh
Fine mesh

Pressure head (cm)-20000-40000hCritA = 100,000 cm-60000-80000
1st element-100000
0

2

4

6

8

10

Time (days)

Figure 4.6: Surface pressure heads during evaporation as calculated using fine andcoarse meshes
Figure 4.6 shows the pressure head time series for the two meshes. Due to theshallower depth of the 1st element in the fine mesh, hCritA was reached at an earlier
time; evaporation stage 2 hence commenced earlier. hCritA is the minimum allowed
pressure head at the soil surface (set at a very low –100,000 cm in this example). For
more details about hCritA, refer to Appendix II.

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.4

Figure 4.7 shows pressure head data at deeper points in the profile. For the fine mesh
we see that the drying front has advanced to the second node (at 2 cm). However, thefront has not progressed as much for the coarse mesh because of the larger element
size (10 cm). For the fine mesh, at 10 cm depth, pressure heads are still very low since
the drying front moves through the soil profile on an element-by-element basis
depending upon the pressure gradients between them.

0

Pressure head (cm)-2000
Time when hCritA is reachedat surface element

Fine mesh 2nd element @ 2 cm

-4000

Coarse mesh 2nd element @ 10 cm
Fine mesh @ 10 cm
-6000

-8000
0

2

4

6

8

10

Time (days)

Figure 4.7: Pressure heads at two depths during evaporation as calculated using fineand coarse meshes
This phenomenon indicates that modellers should investigate the suitability of themesh when simulating evaporation. In the following section, we will demonstrate theeffect of mesh size on predicted evaporative fluxes.

4.2.2 Mesh Density and Evaporative Fluxes
We demonstrate the effect of mesh density by conducting four simulations
having rectangular grids of varying density near the soil surface. We achieve this bychanging the total number of elements as well as the node density at the soil surface(Density at Top, in Space Discretization). Mesh details for the four simulations arelisted in Table 4.1.
Figure 4.8 shows the significant effect of spatial discretization on the predictedevaporative fluxes. However, these changes tend to become less significant at somelevel of mesh density. Apparently the coarser mesh (“Cmesh”) is not an option; thismesh is far too coarse and overestimates the flux. The predictions tend to converge tosome value as the mesh is made finer. Of course, a mesh that is very fine may take toolong to run; see Table 4.1 for run times. We conclude that for this set of simulations
the best mesh is “Vfmesh” since it produces results very close to those obtained withmesh “VVFmesh” in less than one-tenth of the run time.

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.5

2

Cumulative actual evaporation (cm )

20

15

10
Cmesh
Fmesh
VFmeshVVFmesh

5

0
0

0.5

1

1.5

2

Time (days)

Figure 4.8: Effect of mesh density on evaporative fluxes
Table 4.1: Run details of rectangular 100-cm deep grid used to simulate evaporation
Run time
Project name
Number of
Density at top Depth of surface
element(cm)(Seconds)
elements
Cmesh
Fmesh
VFmeshVVFmesh

20
100
100
200

1
1
0.1
0.05

5
1
0.2
0.05

1
20
43
652

Concluding remarks:
1. Mesh density has a significant effect on predictions of evaporative fluxes
since it impacts how the drying front advances.
2. When modelling evaporation, it is worthwhile to investigate theappropriateness of the mesh beforehand. In addition, a value of hCritA thatbest suits the soil modelled must be identified (refer to Appendix II).

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.6
4.3

Capillary Barrier

The principle of a capillary barrier is having a fine-textured soil layer situated on topof a coarse-textured soil layer. The goal is to minimize water leaching into the lower
layer. Capillary barriers are widely used in mining applications and landfills where asoil cover (the fine material) retains the water and loses it to evaporation and/ortranspiration with a minimum leaching-component into the underlying waste material(such as mine waste rock or tailings). Project “Mesh9” is found on the CD(..\HYDRUS Manual Projects\Sect-4).
In Project Mesh9, we simulate a 1-dimensional freely draining soil profile thatcomprises 2 layers; an 80-cm fine layer that overlays a 20-cm coarse layer. The
optimum mesh parameters established in Project ‘Mesh8e’ (of Section 6.4) are
adopted here to simulate this practical application.
The results shown in the three figures below were obtained for two observation points
located close to the interface separating the two layers (see Project Mesh9). Figure 4.9
shows the pressure head data for the two layers. The wetting front arrives just after
0.4 day. Both pressure heads increase at comparable rates.
Time (day)
0
0

0.2

0.4

0.6

0.8

1

Pressure head (cm)

-20

-40

Sand layer
Clay layer

-60

-80

-100

Figure 4.9. Pressure head data for sand and clay layers
Figure 4.10 shows the manner in which the water content varies in the two layers.
Notice that there is a clear lag in response, with the sand keeping a much lower watercontent after the pressure head in the clay layer had already increased significantly.

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.7
0.18

0

0.16
-20
0.14
-40

0.1
-60
0.08
0.06

Pressure head (cm)

Water content

0.12

-80

0.04
-100

Water content; sand layer

0.02

Head; clay layer

0

-120
0

0.2

0.4

0.6

0.8

1

Time (day)Figure 4.10. Water content and head data for sand and clay layers, respectively
In Figure 4.11, the water content is represented in relative terms as degree ofsaturation. The simulation shows that the clay reached 98% saturation at t=0.5 d while
the sand is still at 3%, eventually reaching only about 30% saturation.
35

100
Sand layer% Saturation (sand layer)

Clay layer

99

25
98

20

15

97

10

% Saturation (Caly layer)

30

96
5

0

95
0

0.2

0.4

0.6

0.8

1

Time (day)Figure 4.11. Relative water content (% saturation) for sand and clay layers
Figure 4.12 shows that the cumulative free drainage flow, which may represent a
hazardous leachate, has dramatically dropped from 42.6 cm2 (for Project Mesh-8e ofSection 6.4) to only 3.77 cm2.

Figure 4.12. Cumulative drainage
boundary flux for a layered system

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.8
4.4

Modelling Hysteresis

Hysteresis refers to the non-unique relationshipbetween pressure head and water content in the soil
water retention curve. A drying cycle requires
more energy (a higher negative pressure head) toarrive at the same water content compared to awetting cycle. That is, it is easier to wet the soil
than to dry it to the same water content. It is
possible to incorporate hysteresis with HYDRUS
only when using the van Genuchten-Mualem soil
hydraulic model.
Hysteresis can be extended also to the conductivity
function.
One should specify where the simulation starts, i.e. whether the initial water contents
and pressure heads are on a drying or on a wetting curve.

Drying curve parameters
Qr: residual water content
Qs: saturated water content
Alpha, n, Ks, I, and Qm.
Note that Qm in this window refers
to a drying parameter. The meaningof Qm is explained in the modified
van Genuchten model (seeAppendix I).
Wetting curve parameters
QsW: saturated water content
AlphaW: parameter Alpha for wetting
KsW: saturated hydraulic conductivity
Qm for wetting is calculated by HYDRUS such that
(Qs-Qr)drying/(Qs-Qr)wetting = (Qm-Qr)drying/(Qm-Qr)wetting.
Qr and n are usually equal for drying and wetting, whereas Alpha for wetting is larger
than that for drying (Šimůnek et al., 1999). When no data are available for AlphaW, agood initial guess is AlphaW=2×Alpha.
In the following section, we will briefly investigate the effect of hysteresis bycomparing results obtained from three simulation series: (1) Using drying data and no
hysteresis, (2) Using wetting data and no hysteresis, and (3) Using wetting and drying

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.9

data with hysteresis. All related Projects are found on the CD (..\HYDRUS Manual
Projects\Sect-4).
The parameters used in the following simulations are listed in Table 4.2. The resulting
drying and wetting water retention curves are shown in Figure 4.13.
Table 4.2: Input parameters for hysteresis simulations
Qr
0.095

Qs
0.34

Alpha
0.0094

n
2.189

Ks
0.012

Qm
0.34

QsW
0.34

AlphaW
0.0155

KsW
0.012

0.4

Volumetric water content

0.3

0.2

Wetting curve
0.1

Drying curve

0
10

100

1000

Matric suction (cm)Figure 4.13: Drying and wetting water retention curves
Details of the simulations are outlined in Section 5.2.1. We model infiltration into an
initially dry soil sample; a drying cycle is then initiated when the surface flux is
terminated; another wetting cycle is subsequently simulated when the flux isreintroduced to the soil surface.
Refer to Project ‘Cyc-1’ on the CD (..\HYDRUS Manual Projects\Sect-4). Figure
4.14 shows that some numerically induced fluctuations were encountered during thesecond wetting cycle. Altering the iteration criteria solves the problem (see Table
4.3).
Table 4.3: Iteration criteria details for hysteresis simulations
Water content tolerance
Pressure head tolerance
Lower limit of tension interval
Upper limit of tension interval

Cyc-1
0.0001
0.1
0.01
200

Cyc-2
0.00001
0.01
0.00011000

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.10

0.3

Water content

0.25

0.2

Cyc-1
Cyc-2

0.15

0.1
0

400

800

1200

Time (minutes)

Figure 4.14: Comparison of simulations ‘Cyc-1’ and ‘Cyc-2’
For projects ‘Drying’ and ‘Wetting’, we use drying and wetting water retention
parameters without hysteresis, respectively. Figure 4.15 compares the resulting
pressure heads for the two cases. Pressure head predictions obtained from project‘Cyc-2’, which incorporates hysteresis, are compared with the previous two runs (seeFigure 4.15). Note how the pressure head prediction with hysteresis moves from thewetting to the drying cycle, and then again to the wetting cycle.
0

Pressure head (cm)

-200

-400

Wetting parameters-600

Drying parameters
Hysteresis

-800
0

200

400

600

800

1000

1200

Time (minutes)

Figure 4.15: Comparison of pressure head predictions with wetting, drying, andhysteresis parameters

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.11

Water content predictions are compared in Figure 4.16. When hysteresis is modelled,
the predicted water content is mostly higher during the wetting cycle and lower during
the drying cycle, as compared to the other two scenarios.
0.3

Water content

0.25

0.2

Wetting parameters
0.15

Drying parameters
Hysteresis

0.1
0

200

400

600

800

1000

1200

Time (minute)

Figure 4.16: Comparison of water content predictions with wetting, drying, andhysteresis parameters
Figure 4.17 shows that the three approaches gave different predictions for soil water
storage. The storage term is obtained from the output file ‘Balance.out’ (the water
volume stored in the soil is represented by the variable “Volume” in this output file).
Notice that consideration of hysteresis may have implications in the design of soil
covers.
3.5

Water volume stored (cm2)

3

2.5

2
Wetting parameters
Drying parameters

1.5

Hysteresis

1
0

400

800

1200

Time (minutes)

Figure 4.17: Comparison of water storage predictions with wetting, drying, andhysteresis parameters

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.12
4.5

Estimating Flow Rates Inside a Domain

In HYDRUS, fluxes at the run time levels are only reported along externalboundaries. Fluxes (or flow velocities) may be integrated (summed) along externalboundaries to obtain rates from parts of a boundary (done outside HYDRUS; see
Section 2.2.3). This information is not readily available for locations inside thedomain, but still can be calculated. Nodal flow velocities for the entire domain are
only available at print times. They can either be obtained in text format for the wholedomain (in file “v.out”; see Section 2.1.9), or for individual sections throughout thedomain within the graphical interface. In both cases, to obtain flow rates across aninternal line, the integration should be handled externally (e.g., in a spreadsheet).
In this exercise, the velocity profiles across internal line are obtained from the PostProcessor Graphical Interface. Turning the ‘Snap to Grid’ on facilitates the selectionof the end points. Numerical integration is performed in a spreadsheet using thetrapezoidal rule.
The project series (Pipe-) simulates flow through a hypothetical set-up where twocompartments are connected by a 10-cm wide conduit. Compartment ‘A’ has aconstant head boundary on its surface and compartment ‘B’ has a seepage faceboundary on its side. We wish to estimate the velocity distribution across various
sections along the conduit length, estimate the flow rate through the conduit, andcompare it with the inflow and outflow rates through the external boundaries.
The advantage with this project is that we have a priori knowledge of the flow rate;mass balance stipulates that at any time, the flux through the pipe must be equal to thefluxes across the constant head boundary inflow rate, and the seepage face boundary
outflow rate (all three fluxes are equal). We will demonstrate the effect of meshdiscretization on the results. All related Projects are found on the CD (..\HYDRUS
Manual Projects\Sect-4).

Project Pipe-1
Details of this project are shown in Figure 4.18. The default mesh that HYDRUS
produces is accepted.
Constant head
Inflowab

Seepage face
OutflowcCompartment B

Compartment A

a

b

c

Figure 4.18: Conceptual model and FE mesh for Project Pipe-1

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.13

Velocity (cm/day)
30

35

40

45

0
Section a-a2Section b-b

Distance (cm)Section c-c

4

6

8

10

Figure 4.19: Velocity profiles for Project Pipe-1
The velocity distributions along three cross-sections in the conduit are shown in
Figure 4.19 (refer to Figure 4.18 for the section locations). The velocity is uniform
along the middle section. Integrating the velocities along this cross-section resulted ina flow rate identical to that calculated from the constant head boundary and theseepage face.
However, Table 4.4 shows that the flow rates obtained from the other two crosssections have produced significant errors. Notice that the velocity profiles alongsections a-a and c-c are highly non-linear especially near the top. This problem will be
addressed in the following projects.
Table 4.4: Results from for various projects

Flow rates atinternal
x-sections
cm2/day
Error %
External
boundary flowrate cm2/day

Project
Pipe-1
Section b-b

Project Pipe1Sectionsa-a &c-cProject
Pipe-2

Project
Pipe-3

Pipe-3SS
Steady state
analysis418.978355; 352

890.27940.25940.250.0004815.2; 15.7

6.94

1.72

1.72

418.98

956.7

Projects Pipe-2
In this project, the seepage face extends through the entire RHS boundary (see Figure4.20) and the conduit is shorter. These changes will increase the nonlinearity in thevelocity profile in the conduit. We will obtain velocity profiles for cross-section a-a
only (highest non-linearity) and try to get the best possible results by manipulating themesh.

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.14

Constant heada
Seepage faceaFigure 4.20: Finite element mesh for Projects Pipe-2
Let us refine the mesh in this project along cross-section ‘a-a’ (see Figure 4.20). Thisshould result in a more refined velocity profile, and hence improve numericalintegration.
Velocity (cm/day)
70

80

90

100

110

120

130

140

150

160

170

0

Ditance (cm)12

Project Pipe-3

3Project Pipe-2

4
5
6
7
8
9

Section a-a

10

Figure 4.21: Velocity profiles at section a-a for Projects Pipe- 2 and 3
Figure 4.21 shows that the velocity profile is smoother; notice that we now have 28
points defining the profile as compared to only 10 in the previous example. Table 4.4
shows that error has dropped to 6.94%, but it is still significant.

Projects Pipe3
In this project, the mesh representing the entire area of the conduit is refined (seeFigure 4.22). Figure 4.21 shows now a very smooth velocity profile was obtained.
Table 4.4 indicates that the error has dropped to only 1.72%.

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.15

a

a
Figure 4.22: Finite element mesh for Projects Pipe-3
Figure 4.23 shows velocity contours obtained for this simulation; Notice the very highvelocities at the corners. The simulation was repeated to obtain a steady-statesolution; the results were identical (see Project Pipe-3SS; (CD\HYDRUS Manual
Projects\Sect-4)).

0

20

40

60

80

100

120

140

160

180

Figure 4.23: Velocity contours at steady state
Comments:







It is possible to obtain flow rates across internal cross-sections inside the domain.
Obtain the velocity profile and integrate outside HYDRUS (e.g., using a
spreadsheet).
You are likely to obtain better results if the velocity distribution across the section
is either constant or varies in a linear manner.
It is very important to investigate the existence of velocity nonlinearity across asection and discretize the mesh accordingly. That is, the higher the non-linearity,
the finer the mesh should be.
Note that this procedure is possible only at print times. Therefore, when deciding
on print times, one should take into account the temporal nonlinearity that may
exist in some problems (e.g. high early drainage rate in the case of a coarse soil).
For external boundaries, the velocities reported in the Graphical interface areslightly different (less accurate) than those reported in the “Boundary.out” file.
Velocities are secondary variables calculated by applying Darcy’s law to thepressure head (primary variable) field. Their precision is lower than the solutionof the flow field itself (as obtained with the Richards equation).

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.16
4.6

Effect of Mesh Spatial Discretization and Tolerance Limits

Projects ‘cmbal’ and ‘fMbal1’, and ‘fMBal2’ demonstrate the effects of meshdensity and tolerance limits on the simulation results, and subsequent impacts on themass balance. The Projects are found on the CD (..\HYDRUS Manual Projects\Sect4).
The three projects simulate a rectangular domain with an atmospheric BC at thesurface, a free drainage BC at the bottom, and a seepage face on part of the RHS
boundary. Infiltration is simulated as follows:
0-2 days 100 cm/day
2-4 days 4-6 days 25 cm/day
The first two projects ‘cmbal’ and ‘fMbal1’ differ in terms of their mesh size, with theformer having a relatively coarsely-spaced mesh. Project ‘fMbal2’ has a mesh size
similar to Project ‘fMbal1’ but with looser tolerances. Details are listed in Table 4.5.
Table 4.5: Summary of mesh, mass balance, run time, and tolerance details
Project
Element
Number ofTolerance
Run
Mass balance
namesize (cm)elements
time
WC
Pres.
V
%
(sec.)
cMBal
120
0.0001*
0.1*
19
-232
0.046
50×50
fMBal13000
0.0001*
0.1*
1947
-64
0.013
10×10
#
fMBal2
3000
0.01
0.5
1459
-20,200 4.134
10×10*Default values#High tolerance; not recommended.
Table 4.5 shows that the much finer grid produced a better mass balance. However,notice that this required a 100-fold increase in run time. Loosening the tolerances
resulted in a significant error in the mass balance. The run time decreased by 25 %.
Figure 4.24 compares cumulative free drainage fluxes for the three simulations.
3.0E+05

Cumulatuve free drainage flux ….

2.5E+05

2.0E+05

fMBal1; Fine grid

1.5E+05

cMBal; Coarse grid
1.0E+05

fMBal2; Fine grid/loose tolerance

5.0E+04

0.0E+00
0

1

2

3

4

5

6

7

Time (days)

Figure 4.24. Comparison of cumulative free drainage flux for three projects

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.17
4.7

Simulating Surface Runoff

HYDRUS does not evaluate surface runoff. When precipitation exceeds theinfiltration capacity of the soil, it instantaneously removes excess water from the soil
surface. However, it is possible to mimic surface runoff by using an imaginary surface
layer having a very high hydraulic conductivity and a saturated water content
approaching unity. The excess water that cannot infiltrate layer 2 will move laterally
and ultimately drain through a seepage face located at the end of this imaginary layer.
Note that this technique should not be used if evaporation is simulated. The Project isfound on the CD (..\HYDRUS Manual Projects\Sect-4).

Seepage face

Atmospheric BC

Imaginary soil layer

Observation node

Real soil layer
Free Drainage

Figure 4.25: Boundary conditions for Project ‘Runoff’
Project ‘Runoff’
The scenario shown in Figure 4.25 is modelled in this project. Figure 4.26 shows the
cumulative flow for the atmospheric and free drainage boundary, the difference is therunoff represented by the seepage component. The observation node shows that thewater ponds to a height of about 2.2 cm on top of the real soil layer (results not shown
here; see values for the observation node in the HYDRUS project).
10000
Infiltration
7500

Seepage (Runoff)
Free drainage

Cumulative flow (cm2)

5000
2500
0
-2500
-5000
Infiltration = 9760
Seepage (runoff) = 8360
Free drainage = 1400

-7500
-10000
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Time (days)

Figure 4.26: Cumulative flow components for project ‘Runoff’
Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.18
4.8:

Local Anisotropy

In this hypothetical example we demonstrate the use of the ‘Local Anisotropy’ featurein HYDRUS. Refer to Section 5.3.12 for the theoretical background on‘Implementation of hydraulic conductivity anisotropy’ (p. 55; HYDRUS Technical
Manual).
We use the FE mesh constructed in Section 1.1.9.2 (Example 2, Project ‘Grid-2’),
which represents a 15% sloping hill. We assume that a fault exists along the slope thatacts as a preferential path for water draining along the slope. The hydraulicconductivity along the preferential path is equal to 50Ks (Ks is the saturated hydraulicconductivity elsewhere in the domain). Details of the domain, the boundary
conditions, and the fault location are shown in Figure 4.27. We simulate precipitationfor 1 day; the initial pressure head is –1 cm for the entire domain.

Preferential
path

Atmospheric

30 m

o

15% slope; θ = 8.53

Free drainage

5m

Figure 4.27: Domain and BC for Project ‘Aniso’

Seepage face

We define a local coordinate system for the anisotropy
tensor oriented with the hill slope (Boundary Condition
Editor/Local Anisotropy/Angle; or /Side bar-Angle:
Select the entire domain and enter the angle as shownhere).
We define the first component of the hydraulicconductivity tensor K1: Boundary Condition
Editor/Local Anisotropy/First Component; or/Side bar- First Component: Select the elements
representing the fault (approximately parallel to theslope; see Figure 4.27; remember that ‘Edit/Selectby Rhomboid’ is a useful option here); enter the value for the first component K1 asshown here. The second component is left at the default value (=1).
The orientation of the local coordinate system for the anisotropy tensor is shown in
Figure 4.28. Note that z-axis is positive upwards.

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.19

We copy Project ‘Aniso’, and rename it ‘Iso’; remove the anisotropy (set K1=K2=1)
and run it in order to compare results from the two runs.
Global coordinate system(x-z Principal axes)oLocal coordinate systemoriented with slope angle

θ = -8.53

K1=50 (along thepreferential path)
K2=1 (perpendicular tothe preferential path)
Figure 4.28: Coordinate system for local anisotropy
Results:
Figure 4.29 shows the velocity vectors at T=1 day. Note the direction and themagnitude of the velocity vectors along the preferential flow path. Figure 4.30 shows
the velocity vectors for the isotropic domain (Ks is equal everywhere in the domain).

0

50

100

150

200

250

Figure 4.29: Velocity vectors at T=1 day; Project ‘Aniso’
0

5

10

15

20

25

30

35

40

45

50

Figure 4.30: Velocity vectors at T=1 day; Project ‘Iso’

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.20

Figure 4.31 shows the effect of the preferential path in lowering the pressure heads asa result of enhancing drainage.

Pressure heads (cm)

450
400

Isotropic

350

Anisotropic

300
250
200
150
100
50
0
-50
0

500

1000

1500

2000

2500

3000

3500

Length of lower boundary (cm)Figure 4.31: Pressure heads along the lower boundary for Projects ‘Aniso’ and ‘Iso’
Figures 4.32 and 4.33 show how the preferential path slightly reduced the cumulative
free drainage flux, but significantly increased the cumulative seepage face flux,
respectively.

Cumulative free drainage flux (cm 2) ..

20000

Isotropic

15000

Anisotropic

10000

5000

0
0

0.2

0.4

0.6

0.8

1

Time (day)Figure 4.32: Cumulative free drainage flux for the Projects ‘Aniso’ and ‘Iso’

Cumulative seepage flux (cm 2) ..

20000

Isotropic

15000

Anisotropic

10000

5000

0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (day)Figure 4.33: Cumulative seepage face flux for the Projects ‘Aniso’ and ‘Iso’

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.21
4.9 Tension Disc Infiltrometer

The axisymmetric option in Geometry may be used to simulate a three-dimensional
domain with symmetry around the vertical axis (z in HYDRUS). This option in
HYDRUS creates a domain that is essentially still two-dimensional; it represents avolume that results from revolving the 2-D domain around the symmetry z-axis (refer
to Section 1.1.2, Figure 1.2b). Practical applications include modelling flow for thefollowing conditions:• A single drip source.
• Finite-length cylindrical laboratory soil columns.
• Disc or ring infiltrometers.
• Wells
In this section we will demonstrate flow under a disc infiltrometer. In addition, weshow the effect of introducing a confining ring under the source. We use the capillarybarrier phenomenon to model the isolating effect of the ring.
Domain and boundary conditions:
The domain measures 50×50-cm (radius and depth, respectively). This means that weare modelling a 100×50-cm soil cylinder (diameter and depth, respectively). The disc
infiltrometer has a 20-cm diameter membrane; therefore, it is represented by a 10-cm
radius on top of the domain. All related Projects are found on the CD (..\HYDRUS
Manual Projects\Sect-4).
The lower boundary condition is free drainage. The disc infiltrometer provides waterat a constant negative head. However, the experiment is conducted in two stages:
0 - 0.1 day
pressure head = -20 cm
0.1 - 0.5 day pressure head = -5 cm
The pressure heads are entered as variable head data in the window “Variable
Boundary Condition”, column “GWL” as discussed in Section 1.1.10-3.
The 20-cm diameter isolating ring that forces flow from the disc infiltrometer to bevertical could of course be modelled using a no-flux condition. As an exercise, we
model the ring using the concept of capillary barriers. The subsurface for this purpose
is represented by two materials of contrasting texture, fine and coarse. Figure 4.34
shows that the technique is effectively inhibiting any lateral flow (see location ofobservation points in Figure 4.35). The effect of the ring is investigated for two soils,
namely, a silt and a loamy sand. Details of the domain and its boundary conditions areshown in Figure 4.35.
Alternative approach (use of Local Anisotropy)
The ring can be alternatively (and more precisely) simulated using only one material
but by significantly decreasing the coefficients of the anisotropy tensor in certain
locations. Refer to Project ‘DF-Aniso’ on the CD (..\HYDRUS Manual
Projects\Sect-4).

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.22
0

Inside ring

-20

Outside ring

Head (cm)

-40

-60

-80

-100

-120
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Time (days)

Figure 4.34: Effect of a capillary barrier in preventing lateral flow

Observation
PointsVariablehead BC

Capillary barrier:
Green line is coarse material
Red line is fine materialFreedrainage
BC
Figure 4.35: Domain and boundary conditions for axi-symmetric model

Project “D-FSilt without ring

Project “D-C”
Loamy sand without ring

Project “D-FR”
Silt with ring

Project “D-CR”
Loamy sand with ring

Figure 4.36: Advance of wetting front under disc infiltrometer
Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.23
4.10

De-watering of a Cylindrical Pit

Project ‘Dewat’ simulates de-watering of a cylindrical pit having a diameter of 10 m
and a depth of 8 m. A constant head BC is located at 25 m away from the edge of thepit. The initial water table is at a depth of 5 m; hydrostatic equilibrium is assumed.
Water is pumped at a rate of 50 m3/day for a period of 4 days. The cross-section of thepit is shown in Figure 4.37.

Figure 4.37: Sample calculation for nodal flux from recharge rate
The problem is modeled as axisymmetric vertical flow. The base of the pit isdiscretized into 10 nodes where each node is assigned a recharge rate of –5 m3/day (inBoundary Condition Editor/Condition/Nodal Recharge).
The space representing the excavated soil (the pit) is assigned a high-porosity(=0.999), high-conductivity material (=100 m/day) that mimics air (as in the case of
the imaginary surface layer of example 4.7). The water content profile shown in thefollowing page demonstrates that the technique works well; the figure shows avertical cross-section through the pit prior to pumping (at Time=0). The water content
between 500 and 800 cm (below the water table) in the excavated area is equal tounity (it represents water), but drops to zero just above the water table due to the highAlpha and n used. Project ‘Dewat’ is found on the CD (..\HYDRUS Manual
Projects\Sect-4).

Modelling variably saturated flow with HYDRUS-2D4.24

Section 4: Example Applications

Note that once the recharge is assigned to the ten nodes at the base of the pit, they
become constant-flux boundary nodes; HYDRUS assigns the appropriate value of theflux to each node. Select the ‘Constant Flux’ option in ‘Boundary Condition Editor’,left-click on node 5 twice, and the flux value will be displayed as shown below.
Figure 4.37 demonstrates how this flux is calculated.

The pressure heads at the base of the domain are shown below; they demonstrate thewater table depression as pumping progresses.

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.25
4.11

Steady-State Horizontal Flow

Project ‘Hor-Flow’ uses the horizontal-plane flow option. This option is mostly
relevant only to saturated water flow. The horizontal flow domain shown in Figure4.38 is a ‘Plan view’ where the third dimension represents soil depth. Figure 4.38
shows the constant head boundary conditions imposed at the two corners of the flow
domain. Project ‘Hor-Flow’ is found on the CD (..\HYDRUS Manual Projects\Sect4).
Steady-state head contours are shown in Figure 4.38 (Graphical Display ofResults/Options/Isolines Parameters/Increment=1).
Constant head = 40 m
200 m

44

46

42

100 m

48

Constant head = 50 m
Figure 4.38: Flow domain, boundary conditions, and steady-state head contours
A 3D surface representing the steady-state water table can be obtained as follows.
Obtain 11 1-D graphs at each 10 m of the y-coordinate from the graphical interface
and export them into text format. Next, import the data into a software package suchas ‘Surfer’ to plot a 3-dimensional surface (see Figure 4.39).

Figure 4.39: 3-D surface representing steady-state water table
Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.26

4.12

Print Times > 250

The graphical interface (Print Information/Number of Print Times) allows one to enter
up to 250 print times. However, the HYDRUS code (h2d_calc.exe) and the graphicaldisplay interface (Post-Processing/Graphical Display of Results) allow print times of
up to 5000. This is only possible in HYDRUS version 2.007 and later.
The user has to enter the print times that exceed 250 manually by accessing the“Selector.in” input file using a spreadsheet. Below is the relevant part of the input file.
Project “Trial” is found on the CD (..\HYDRUS Manual Projects\Sect-4).

Relevant part in Input file “Selector.in”
dt
0.001
tInit
0

dtMin
0.001
tMax
400

dtMax
1

DMul
1.3

DMul2
0.7

ItMin
3

0.08
0.56

0.16
0.64

0.24
0.72

0.32
0.8

0.4
0.88

0.48
0.96

ItMax
7

MPL
5000

• Under “MPL”, enter the number of print times (maximum 5000).
• Print times are entered in rows each containing 6 print times. Make sure that the
number of entered print times add up to MPL.
• Save the file; if you are use in Excelo Click the “Save” buttono The “Save as” window will appear
o Keep the default file name “Selector.in”
o Accept the option to overwrite the existing fileo Click “Yes” to keep the file in the existing format.
Open the project in HYDRUS; run HYDRUS
without accessing the “Print Information”
window.
Project “Trial”
Click on “Graphical Display of Results” and you
will see 5,000 time layers as shown here.

Modelling variably saturated flow with HYDRUS-2D

Section 4: Example Applications

4.27
4.13

Running Multiple Sequential Simulations

If you have a large number of simulations that need to be run sequentially, you can dothis from the Start-Run option in Windows, which is similar to the old DOS prompt.
The multiple runs are executed from a batch file. In this example, we want to run
Projects “Trial1” and “Trial2”. Follow the steps below:
1. Identify the directory path of the project you want to run; in this case it is:“D:\HYDRUS manual\trial1”. Create a text file (e.g. in Notepad) and give thefile a name; in this case “Path1.txt”; copy the path into the file and save.
2. Repeat this process. Copy the path “D:\HYDRUS manual\trial2” and save thefile as “Path2.txt”.
3. When Calculate/Run HYDRUS ( ) is clicked it executes the h2d_calc.exe
program. When the run finishes it requires one to press “Enter” on the
keyboard. Accordingly, we must have a text file that contains the hexadecimalcode for “Enter”. Create a text file in Notepad; write in it “1C” (code forEnter), call the file “Return.txt”, and save.
4. Create the batch file. First we should let HYDRUS know where the location of
the project is; this is done by copying the text file that contains the path andrenaming it “level_01.dir” (the file that HYDRUS looks for when not run fromthe graphical interface to identify the directory where the project exists).
Create the following file to run projects “Trial1” and “Trial2”. Call the file
“Run.bat”; notice that the extension of the file must be (.bat). Also notice thatspaces are present between the DOS commands.
copy path1.txt level_01.dir
h2d_calc <return.txt
copy path2.txt level_01.dir
h2d_calc <return.txt
5. Place the four files in the HYDRUS home directory (where the “h2d_calc.exe”
is stored; usually C:\Program Files\USSL\Hydrus2D); the files in this case are:“Run.bat”, “Path1.txt”, “Path2.txt”, and “Return.txt” (copies of these files arefound on the CD; (..\HYDRUS Manual Projects\Sect-4\4_13).
6. In Windows: Start/Run
Through “Browse”, go to
the HYDRUS home
directory and click on the“Run.bat” file; then click
OK.

Modelling variably saturated flow with HYDRUS-2D4.28

Section 4: Example Applications

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.1
5. INVERSE SOLUTION

5.1 Inverse Solution Options in HYDRUS
Main Processes
In normal simulations or “forward
problems”, the modeller provides the known
input parameters and HYDRUS runs once
through the specified time duration.
However, under the “Inverse Solution”
option, HYDRUS runs the problem using
the initial estimate for the parameters,compares the simulation results withobserved experimental data that the user
provides, and then re-runs the model a number of times with modified sets ofparameters until a close fit to the observed experimental data is obtained. The numberof iterations depends, among other things, on how close the initial input parameters
were to the final estimate. This process is usually referred to as “Model calibration”,
or “Inverse parameter estimation”. The technique has many valuable applications,which will be discussed in a later section. For more details, refer to Appendix V.
Click next…..
Inverse Solution – This window appears
only if you have checked “Inverse
solution?” in “Main Processes.
• Estimate: Select whether you want toestimate soil hydraulic parameters, solutetransport parameters, or both. The latter
will only be activated if solute transport issimulated (mostly outside the scope of
this document).
• Weighing of inversion data provides
the user with two different options for weighting the inversion data. The weighting
process manages differences in weights because of the use of different types ofcalibration data. Refer to Appendix V for more details.
• The maximum number of iterations is an upper limit for the number of runs if
convergence is not achieved. In such cases the user might change the starting values
for the optimised parameters or limit their number. If zero is entered, the simulation
will run once (as in usual forward modelling). The latter option is very useful since it
gives you an option to test the quality of the initial input parameters. HYDRUS willthen display the observed and predicted values under “Observation Points” and bymeans of other graphical outputs.
• The number of data points in the objective function refers to the number ofobserved laboratory and/or field data in the optimisation such as pressure head watercontent, and/or flux. For example, let us assume you conducted a 10-hour drainage
Modelling variably saturated flow with HYDRUS-2D5.2

Section 5: Inverse Solution

experiment in a laboratory soil column and recorded 10 water content readings using
a TDR device, one reading every hour. You want to use the inverse solution option toestimate the soil hydraulic parameters that would give the closest prediction to thoselaboratory measurements. You should enter 10 as the number of data points, andyou’ll be prompted to enter the time and the 10 water content values under “Data for
Inverse Solution”.
If hysteresis is checked under ‘Soil
Hydraulic Model’, then the following
window will appear. You can apply
constraints on the hysteresis model.
• ThetaSD and ThetaSW =θs for drying
and wetting cycles, respectively.
• ThetaM=θm the fictitious water content
in the modified van Genuchten model (MDand MW are for main drying and main wetting, respectively).
• AlphaD and AlphaW represent Alpha for drying and wetting cycles, respectively.
More details are found in Šimůnek et al. (1999).
This ‘Water Flow Parameters’
window will now appear .
Notice that there are checkboxes beneath each parameter.
Checking a box along the line
‘Fitted?’ means that theparticular parameter will be
optimised. You can enter
minimum and maximum values to make the search process more effective. If you do
not about those limits, leave zeros in the particular boxes.

The meaning of the various
inputs required in this window
is well explained in the on-line
Help. We will explain all
available options here. Refer toTable 5.1 for an explanation of
the window shown here.

Remember that data points containing time (i.e., type=0 through 4) must be enteredfirst before any other data (such as retention or conductivity data or priorinformation).

Modelling variably saturated flow with HYDRUS-2D5.3

Section 5: Inverse Solution

Table 5.1: Details corresponding to window ‘Data on Inverse Solution’ (page 5.2)xytype
Positionwt
1Time=
Observed inflow(0) code for(1) code for constant 1
1 dayvalue= -2
Cumulative fluxpressure boundary
2
Time=
Observed(0) code for(2) code for seepage 1
2 daysoutflow
Cumulative flux
face boundary
value= 3
3
Time=
Observed inflow(0) code for(3) code for variable 1
3 daysvalue=-2
Cumulative fluxpressure boundary
4
Time=
Observed inflow(0) code for(4) code for1
4 daysvalue=-5
Cumulative flux
atmosphericboundary
5
Time=
Observed(0) code for(5) code for drain
1
5 daysoutflow
Cumulative flux
boundaryvalue=0.5
6Time=
Observed(0) code for(6) code for free1
6 daysoutflow value=7 Cumulative flux
drainage boundary
7
Time=
Observed(1) code for(1) Observation
1
10 dayspressure headpressure headpoint number
value=5.5 cm
8Time=
Observed water(2) code for(1) Observation1
20 dayscontent
water content
point number
value= 0.265
9Time=
Observed flow(3) code for(1) code for constant 1
30 daysvalue= 0.01
boundary flowpressure boundary
10
Time
Observed(2) code for(0) code indicating1(400days)
average water
water content
that observed watercontent
content is for wholevalue= 0.16
domain11
Observed
Observed water(5) code for(1) code representing 1
pressure head
content
point on water
material number
value=-5 cm
value= 0.278
retention curve
12
Observed
Observed(6) code for(2) code representing 1
pressure head
hydraulicpoint onmaterial number
value=-20 cm
conductivity
hydraulicvalue= 0.0015
conductivity
function13
dummy
Observed Alpha(7) code for(1) code representing 1(10000)parameterparameter Alphamaterial number
value= 0.0023

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.4

5.2 Inverse Modelling of Controlled Laboratory Column
In this section we will explore inverse parameter estimation by modelling flow
through a controlled laboratory column. We highlight cases that demonstrate thefollowing aspects:






Non-uniqueness of the inverse solution
Temporal and spatial distribution of calibration data
Effect of initial parameters
Correlation of variables and statistical issues
Validation of results

We also provide some guidelines that you should keep in mind when using thistechnique.
In this series of simulations we use data from Rassam and Williams (1999a). First we
give a brief description of the experiment: a 13-cm tailings sample is packed into aflow cell where transient water contents and initial and final pressure heads aremeasured. The lower boundary is a constant suction (negative head) and the upperboundary is a constant flux. Two wetting cycles covering different suction ranges are
simulated in the same flow cell containing the same soil sample (wetting cycles ‘a’and ‘b’; see Figure 7 of paper).
The water retention parameters obtained from the inverse solution and the transient
data are compared against those measured experimentally under static conditions inthe same flow cell. In addition, unsaturated hydraulic conductivities predicted fromthe water retention curve are also compared to the experimental measurements. All
related projects are found on the CD (..\HYDRUS Manual Projects\Sect-5).
5.2.1 Simulating Wetting Cycle ‘a’
Refer to Project series ‘a1-a8‘ on the CD. The initial and boundary conditions are:
 Initial pressure head = -780 cm
 Lower BC is a constant head = -780 cm
 Upper BC is a constant flux = 0.00643 cm/min (=3.86 mm/hr)
 Qs=0.3418; Qr=0.1
Project ‘a1’
Calibration data = 12 water content measurements.
Optimise Alpha, n, and Ks.
Project ‘a2’
Calibration data = 7 water content measurements covering the same range as in Run‘a1’ (i.e. every other point was eliminated from the data set).
Optimise Alpha, n, and Ks.
Comments on Projects ‘a1’ and ‘a2’:
Figures 5.1, 5.2, and 5.3 show that runs ‘a1’ and ‘a2’ produced almost identicalresults. Apparently the number of calibration points did not matter as long as the
temporal distribution was similar. The optimised parameters were found to be highly
correlated.
Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.5

0.08
0.0738

0.07
0.06

Alpha

0.05
0.04
0.03
0.02234

0.02

0.022639

0.020324

0.015554
0.012480.01

0.00857

0.00874

Run a1Run a20

Run a3Run a4

Run a5

Run a6

Run a7

Run a8Figure 5.1: Comparison of parameter Alpha for Projects ‘a1-a8’

4.5
3.96

4
3.5
3
2.5

2.5

2.5

2.465

2.5
2.306

n

2.189

2

1.697

1.5
1
0.5
0

Run a1Run a2Run a3Run a4

Run a5

Run a6

Run a7

Run a8Figure 5.2: Comparison of parameter n for Projects ‘a1-a8’

0.014
0.01268

Conductivity (cm/min)

0.012

0.012279
0.011310.01
0.008831

0.008
0.0062198

0.00668

0.006
0.004
0.002
0

Run a1Run a2Run a3Run a4

Run a5

Run a6

Figure 5.3: Comparison of parameter Ks for Projects ‘a1-a6’

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.6
Project ‘a3’

Calibration data = 12 water content measurements and a final pressure head at thesurface.
Optimise = Alpha, n, and Ks.
Comments on Projects ‘a1’ and ‘a3’:
Figures 5.1, 5.2, and 5.3 show that although Projects ‘a1’ and ‘a3’ use identical watercontent calibration data, the additional calibration point (pressure head) had asignificant impact on the results.
Figure 5.4 shows that incorporating the pressure head at the end of the simulation
greatly impacted the simulated heads at time>100 minutes and produced better
agreement with the measurement.
Figure 5.5 shows that the simulated water contents for Projects ‘a1’ and ‘a3’ are very
similar; both match the observed values. However, a close look at the ‘Inverse
Solution’ results in ‘Post-processing’ (or in the Fit.out output file) shows that the R2 islower for Project ‘a1’.
Adding the additional calibration point in Project ‘a3’ (final pressure head) resulted inmuch lower correlation coefficients between the parameters compared to Project ‘a1’.
0

Head (cm)

-200

-400

Observed
Run a1Run a3

-600

-800
0

50

100

150

200

250

300

350

400

Time (min.)Figure 5.4: Comparison of simulated head for Projects ‘a1’ and ‘a3’
0.3
Exp.

0.28

a1
Volumetric water content

0.26

a3
a4

0.24

a6

0.22

Range of inverse data for runs 4&6

a6

0.2
0.18
0.16
0.14
0.12
0.1
0

50

100

150

200

250

300

350

400

Time (min)Figure 5.5: Comparison of simulated and observed water contents for Projects ‘a-1‘,‘a-3’, ‘a-4’, and ‘a-6’

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.7
Project ‘a4’

Calibration data = 7 water content measurements but now at times>200 minutes (i.e.,the calibration is biased toward the wet end of the curve).
Optimise = Alpha, n, and Ks.
Comments on Project ‘a4’:
Figures 5.1, 5.2, and 5.3 show that the process resulted in a different set ofparameters.
Figure 5.5 shows good agreement between observed and simulated water contents (inthe calibrated wet range, time>200 min).
Project ‘a5’
Calibration data and parameter optimisation = As in Project ‘a4’.
Apply a more stringent water content tolerance.
Comments on Project ‘a5’:
Figures 5.1, 5.2, and 5.3 show that using a more stringent tolerance slightly impacted
the hydraulic conductivity.
The run time was doubled.
Project ‘a6’
Calibration data and parameter optimisation = As in Project ‘a4’.
Different set of initial parameter values Alpha, n, and k.
Comments on Project ‘a6’:
Figures 5.1, 5.2, and 5.3 show that the optimised parameters are now significantly
different.
Figure 5.5 shows that good agreement was obtained between simulated and observed
water contents in the calibrated wet range.
However, the water content predictions diverged from the observed values at the
beginning of the simulation (0<time<200) for which no calibration data was used.
Project ‘a7’
Calibration data = As in Project ‘a1’.
Optimise Alpha and n (Ks excluded).
Assume that a good initial estimate for the hydraulic conductivity was provided.
Comments on Project ‘a7’:
Figures 5.1, 5.2, and 5.3 show that the optimised parameters are different from thoseobtained in Project ‘a1’.
The correlation coefficient was reduced to –0.2954.

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.8
Project ‘a8’

Calibration data and parameter optimisation = As in Project ‘a7’.
A poor initial estimate for the hydraulic conductivity was provided.
Comments on Project ‘a8’:
Figures 5.1, 5.2, and 5.3 show that the optimised parameters are now significantly
different from those obtained in Project ‘a1’.
The correlation coefficient was low at 0.2247.

Validation of Inverse Solution Results for Projects ‘a-‘
The eight previous simulations produced eight sets of results. Which one is the best?
Fortunately, we have two independent data sets against which we can validate ourresults. Firstly, a wetting water retention curve, and, secondly, a number ofunsaturated hydraulic conductivities measured over a limited suction range. Both data
sets were experimentally obtained from the same experiment, which means that they
can be given a high degree of confidence.
The optimised parameters obtained from the inverse solution are used to plot thecorresponding water retention curves. They are compared with the experimental
curve. Results from simulations ‘a2’ and ‘a5’ are not shown because they are similar
to simulations ‘a1’ and ‘a4’, respectively. Refer to Figure 5.6:
Simulation ‘a1’ (12 water content data points) resulted in an over-prediction of thewater contents (curve shifted upwards).
Simulation ‘a3’ (12 water content data points and final head) provided the best fit to
the experimental measurements.
Simulation ‘a4’ (7 water content data points at time>200) resulted in an underprediction of the water contents (curve shifted downwards).
Simulation ‘a6’ (as in simulation ‘a4’ but different initial parameters) also resulted ina poor fit to the experimental values.
Simulation ‘a7’ (as in ‘a1’ except that Ks was not optimised; a good estimate wasavailable) resulted in a better prediction than that provided by simulation ‘a1’.
Therefore, provided the estimate for Ks was good, excluding Ks from the optimisation
procedure gave a better solution with relatively low correlation coefficients.
Simulation ‘a8’ (as in simulation ‘a7’ but with a poor initial estimate for Ks) produced
poor results.
Figure 5.7 shows similar trends when unsaturated hydraulic conductivities predictedwith the water retention parameters obtained from Projects ‘a1-a8’ are compared tothe experimental data. Project “a3” still gives the best fit to the measured hydraulicconductivities.

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.9
0.35

a4

Volumetric water content ...

0.3
a8

0.25
a1

a7

Experimental

0.2

a1
a3

a3
a4
0.15

a6
a7

a6

a8
0.1
10

100

1000

Suction (cm)Figure 5.6: Validation of water retention parameters obtained from Projects ‘a1-a8’

Hydraulic conductivity (cm/min)...

1.E-2

a8

a1

1.E-4

Experimental
a3

a1
1.E-6

a3
a4
a6

1.E-8

a4

a7

a7
a8
a6

1.E-10
10

100

1000

Suction (cm)Figure 5.7: Validation of unsaturated hydraulic conductivity data predicted using
water retention parameters obtained from Projects ‘a1-a8’
Concluding Comments on Project Series ‘a-‘:Note that all simulations produced a good visual fit to the experimental transientwater content data. Therefore, a good visual fit does not ensure that the correct results
are obtained; one should closely look at the statistical information provided in the‘Fit.out’ file.
Having additional data can be very advantageous (e.g., the final pressure head in
simulation ‘a3’). This may significantly reduce the correlation coefficients.
If you have a good experimental measurement of a certain parameter (e.g., Ks in our
example), it is advantageous not include that parameter in the optimisation. This may
reduce correlation between the remaining parameters and lead to a better solution.
However, if the initial estimate is poor, this may have the opposite effect andadversely affect the solution. Hence, it is crucial to know “how much you trust theparameters”.
Initial parameters have an impact on the simulation results if the solution is nonunique. This phenomenon will be further investigated in Section 5.4

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.10
5.2.2 Simulating Wetting Cycle ‘b’

Refer to Project series ‘b1-b6‘ on the CD. The initial and boundary conditions for thisseries are:
 Initial head = -208 cm
 Lower BC is a constant head = -208 cm
 Upper BC is a constant flux = 0.008467 cm/min (=5.08 mm/hr)
 Qs=0.3418; Qr=0.1
Project ‘b1’
Calibration data = 6 water content measurements.
Optimise Alpha, n, and Ks.
Comments on Project ‘b1’:
Figure 5.8 shows that the calibration produced an excellent fit to the observed values.
The optimised parameters are plotted in Figures 5.9, 5.10, and 5.11.
Correlation coefficients are high.
0.31b5Volumetrica water content

0.29
0.27

b1

0.25
0.23
Exp
b1
b2
b3
b4
b5
b6

0.21
b5
0.19
b6
0.17
0.15
0

40

80

120

160

200

Time (min)Figure 5.8: Comparison of simulated and observed water contents for Projects ‘b-1‘
through ‘b-6’.

Project ‘b2’
Calibration data = 6 water content measurements and final pressure head.
Optimise Alpha, n, and Ks.

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.11
0.025

0.02

0.01932
0.017026

0.016953

0.016674
0.015183

Alpha

0.015

0.010248

0.01

0.005

0
Run b1Run b2Run b3Run b4

Run b5Run b6

Figure 5.9: Comparison of parameter Alpha for Projects ‘b1-b6’

3
2.4939

2.5

2.281
2.159

n

2

1.94311.91391.9257Run b2Run b3Run b4

1.5

1

0.5

0
Run b1Run b5Run b6

Figure 5.10: Comparison of parameter n for Projects ‘b1-b6’

0.016
0.014757

0.01491

0.01490.014194

0.014

Conductivity (cm/min) ..

0.012
0.01
0.008

0.00890.007688

0.006
0.004
0.002
0

Run b1Run b2Run b3Run b4

Run b5Run b6

Figure 5.11: Comparison of parameter Ks for Projects ‘b1-b6’

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.12
Project ‘b3’

Calibration data = 6 water content measurements and 2 unsaturated hydraulicconductivity measurements (weight=2 for the K measurements).
Optimise Alpha, n, and Ks.
Project ‘b4’
Calibration data = 6 water content measurements, final pressure head, and 2
unsaturated hydraulic conductivity measurements (weight=2 for K and the pressure
head measurements).
Optimise Alpha, n, and Ks.
Comments on Projects ‘b2’, ‘b3’, and ‘b4’:
Figure 5.8 shows that the calibration resulted in an excellent fit to the observed values
for the three Projects ‘b2-b4’. The optimised parameters are plotted in Figures 5.9,
5.10, and 5.11.
The optimised parameters for the three projects are in close agreement. Adding
additional calibration data types (pressure heads or hydraulic conductivities) was notparticularly advantageous in this example. Results would have improved if we
optimised more parameters.
Correlation coefficients are slightly lower for Project ‘b2’.
Correlation coefficients remained high for Project ‘b3’.
Correlation coefficients are much lower for Project ‘b4’.
Project ‘b5’
Calibration data = 6 water content measurements, final pressure head, 2 unsaturated
hydraulic conductivity data, and one point on the dry end of the water retention curve(weight=2 for K, the pressure heads, and point on water retention curve).
Optimise Alpha, n, Ks, and Qr.
Project ‘b6’
Calibration data = 6 water content measurements, final pressure head, 2 unsaturated
hydraulic conductivity data, and a point on the dry end of the water retention curve(weight=2 for K and the pressure heads; weight=5 for point on water retention curve).
Optimise Alpha, n, Ks, and Qr.

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.13
Comments on Project ‘b5’ and ‘b6’:

Figure 5.8 shows that optimising Qr caused skewing of the calibration for Project‘b5’. For calibration point no. 9 (type 5; head=-306 cm, water content =0.132), thefitted water content was 0.151 (see file ‘Fit.out” for Project ‘b5’).
For Project ‘b6’, putting a higher weight on the water retention data point (calibration
point no. 9) resulted in a better estimate for the water content (predicted=0.142;
observed=0.132). However, Figure 5.8 shows that this resulted in a worse fit to thetransient water content data.

Validation of Inverse Solution Results for Projects ‘b-‘
Simulation ‘b1’: Figure 5.12 shows that calibration against water content data alone
produced the worst fit to the observed values.
Simulations ‘b2-b4’: Figure 5.12 shows that adding an additional data type (final
pressure head or Ks values) improved the calibration results. However, including both
did not improve the results much further, that is, the extra information was redundant
but had the advantage of reducing the correlation coefficients.
Simulations ‘b2-b4’: The three simulations failed to provide a good description of thedry end of the curve (suction>100 cm). This is attributable to the fact that the pressure
heads covered in the physical model (wetting cycle b) cover the wetter end of thecurve (starting head=-208 cm). Note that we had a better overall prediction of thewater retention curve in first set of simulations (wetting cycle a; simulation ‘a3’)
because the pressure heads covered the dry end of the curve (the starting pressure
head was -780 cm).
Simulation ‘b5’: Figure 5.12 shows that including a point on the dry end of the water
retention curve (head=-306;water content=0.132) resulted in a better agreement
towards the dry end of the water retention curve.

1

Volumetric water content

0.3

1- Run b1
2- Run b2
0.2

3- Run b3

5

4- Run b4

6

5- Run b5

2, 3, & 4

6- Run b6
Experimental
0.1
1

10

100

1000

Matric suction (cm)Figure 5.12: Validation of water retention parameters obtained from Projects ‘b1-b6’

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.14

Simulation ‘b6’: Figure 5.12 shows that increasing the weight (to 5) for the point on
the dry end of the water retention curve (pressure head=-306;water content=0.132)
improved the agreement towards the dry end of the water retention curve. However,the prediction was now worse in the wet range of the curve.

Concluding Comments on Project Series ‘b-‘:
Adding an extra data type improved the model predictions, but further additions
seemed redundant.
Due to the fact that the experimental wetting cycle covered the wet end of the water
retention curve, inferior prediction of the dry end of the curve resulted; wetting cycle‘b’ provided a worse prediction than wetting cycle ‘a’.
Including a point from the water retention curve in the calibration process improved
the fit in the dry end of the curve. However, putting a high weight on this data point
resulted in a much poorer prediction of the wet end of the curve.

5.2.3 Optimising Parameter L in the Hydraulic Conductivity Function
This is the tortuosity factor in the hydraulic conductivity function. Its value was foundto be 0.5 by Mualem, but there is no general agreement about this in the literature.
Parameter L is an input parameter in HYDRUS and can hence be optimized.
The shape of the hydraulic conductivity function is controlled mainly by the retention
curve parameters ‘alpha’ and ‘n’. The saturated hydraulic conductivity merely scales
the hydraulic conductivity function; it provides us with absolute values for the
conductivity rather than only relative values provided by the analytical function.
Optimizing the tortuosity parameter permits one to change the shape of the hydraulicconductivity function independently of the retention function. Since the older versionof HYDRUS could not jump over zero, we conduct below two runs, one starting witha positive L value and one with a negative L value.
Projects “L-Par” and “L-Par-ve”:
In these Projects we use the problem modeled in series “a-“. Let us use the optimized
Alpha and n from Project “a3”, which resulted in the best fit to the water retention
data. We calibrate against six measured unsaturated hydraulic conductivity values, inaddition to the measured final pressure head (water contents are excluded). The
hydraulic conductivity and the tortuosity factors are optimized.
Table 5.2: Optimization results of Projects “L-Par” and “L-par-ve’
Parameter L
Ksat (cm/minute)
Project “a3”0.5 (not optimised)0.01268-60.00969
Project “L-Par”
2.4×10 (optimised)
Project “L-Par-ve”-0.137 (optimised)0.00894

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.15

Water content

Hydraulic conductivity (cm/min) .

0.00160.00120.0008

0.3
0.28
0.26
0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1

FittedObserved
0

100

200

300

400

500

Time (min)0.0004

Observed conductivity
Fitted conductivity; positive L range
Fitted conductivity; negative L range
0
50

70

90

110

130

Suction (cm)Figure 5.13: Measured and fitted conductivity for Projects “L-Par” and “L-Par-ve”
Table 5.2 shows that for Project “L-Parameter”, the best fit to the measured
conductivity data arose using a slightly lower saturated hydraulic conductivity and a
tortuosity parameters of 2.4×10-6. For Project “L-Par-ve”, the best fit was obtainedwith a tortuosity parameter of –0.137. Figure 5.13 shows the fit to the experimental
data; the negative tortuosity parameter produced a better match to the observed
conductivity at suction=54.87 cm. The water content data show that the integrity ofthe previous optimisation was preserved (i.e., the new conductivity data did not
disrupt the water content predictions).
5.3 Effects of Model’s Initial Hydraulic Parameters on Inverse Solution
In this section a series of simulations will demonstrate how the starting hydraulicparameters can affect the final solution and run time. The results obtained from the six
simulations are compared with those obtained from run ‘a3’. In the following
simulation, we only vary parameters Alpha and n; the other parameters are fixed attheir optimum value. Details of the six simulations are given in Table 5.3.
Table 5.3: Effect of initial parameters on inverse runs
Initial parameters
Simulation
type
Alpha
n
Sen 1
0.0015
1.5
ForwardSen 20.0015
1.5
Inverse
Sen 3
0.008
2
Forward
Sen 4
0.008
2
Inverse
Sen 5
0.02
2
Forward
Sen 6
0.02
2
InverseQuality of
results
Very bad
Very bad
Fair
Very good
Good
Very good

Run time(seconds)
1
12
1
24
1
14

1. Simulation ‘Sen-1’: We start with arbitrary estimates of Alpha and n.
Figure-5.14 shows erroneous results that do not match the observed values.
2. Simulation ‘Sen-2’: Here we run Project ‘Sen-1’ in inverse mode to produce apoor estimate for the two optimised parameters.

Modelling variably saturated flow with HYDRUS-2D5.16

Section 5: Inverse Solution

3. Simulation ‘Sen-3’: Based on concepts outlined in Appendix I.3, we ran a few
trial forward runs and came up with a refined estimate for the initial values for
Alpha and n. Figure 5.14 shows that the results now at least are within therange of observed values.
4. Simulation ‘Sen-4’: Use the initial parameters of Project ‘Sen-3’ and run ininverse mode to refine the parameters. The final solution produces results
identical to those obtained with run ‘a3’ (Sen-4 is hence not included in Figure5.14).
5. Simulation ‘Sen-5’: Further manual refinement of the parameters results in acloser fit than that obtained from run ‘a3’.
6. Simulation ‘Sen-6’: Use the initial parameters of Project ‘Sen-5’ and run the
inverse solution to refine the parameters. We obtain the same results as fromsimulation ‘Sen-4’, so why going through the additional effort? The run timeis reduced by only 10 seconds! Is this worth it? Yes, if we think in relative
terms; 40% less. For complex 2-D runs, the run time could amount to days andsuch a saving in run time may be significant.

5.4 Long Inverse Trials With a Multiple-Layer Soil Profile
In this section we demonstrate further complications introduced as a result of having a
multiple-layer soil profile. The data set was obtained from the Department of Land &Water Conservation, Center of Natural Resources, Soils & Salinity Unit, NSW,
Australia (A. Rancic; personal communications).
Simulation ‘LongRun’ shows such a scenario. A 2-m, 3-layered soil system ismodelled (layer interfaces are at 30 cm and 110 cm). The lower BC is freely drained
and the surface is open to the atmosphere. A 2-year climate data series withtranspiration is simulated (root water uptake is modelled). We have water content
calibration data (such as neutron probe data) at three depths; 10, 50, and 140cm.

5.4.1 Sensitivity Analysis:
In addition to concepts related to the sensitivity analysis in Appendix I.3, we will
highlight several points here: the effect of a nearby boundary, and the interaction ofparameters from various layers. The input parameters for all simulations are listed inTable 5.4. The original parameters belong to Project “LongRun”; the three variations
for the sensitivity analysis are marked Runs A, B, and C (they are not included on theCD).
1. The effect of a nearby boundary: Refer to Figure 5.15 (Run A). Doubling thehydraulic conductivity (from 50 to 100) of the surface layer had very little effect on
the water content predictions at 10 cm.

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.17
0.35
Sen-2

Volumetrica water content

0.3
Sen-1

a3

Sen-5

0.25

0.2

Run a3

Sen-3

Sen-1
Sen-2

0.15

Sen-3
Sen-5

0.1
0

50

100

150

200

250

300

350

400

Time (min)Figure 5.14: Effect of initial parameter values on inverse solution
0.4
0.35

Volumetric water content

0.3
0.25
0.2
0.15
0.1
Observed at 10 cm
0.05

Modelled with optimum paramerts
Modelled with higher k at layer 1

0
0

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

Time (days)

Figure 5.15: Run A (10 cm data), higher Ks for layer 1
Table 5.4: Input parameters for simulations ‘LongRun’
Layer 1; 0-30 cm
Alpha
LongRun
Run A
Run B
Run C

0.036
0.036
0.036
0.01

n

k

1.56

50
100
50
50

Layer 2; 30-110 cm

Layer 3; 110-200 cm

Alpha

Alpha

0.02

n
1.41

k
10.8

0.036

n

k

1.56

25
25
50
25

However, Figure 5.16 shows that doubling the hydraulic conductivity (from 25 to 50)
of layer 3 (Run B) had a much more significant effect on the simulation results at 160
cm. This is attributable to the fact that this layer is freely drained. Subtle changes in
the hydraulic conductivity can have a direct impact on the amount of water exiting thesystem, and hence would affect the simulation results.
Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.18

0.3

0.25

Volumetric water content

1
0.2
3
2
0.15

0.1
Observed at 160 cm
1- Modelled with optimum paramerts

0.05

2- Modelled higher k at layer 3
3- Modelled with lower Alpha at layer 1
0
0

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

Time (days)

Figure 5.16: Runs B and C (160 cm data), higher Ks for layer 3, and lower Alpha forlayer 1, respectively
2. Parameter interaction: Figures 5.16 and 5.17 show that changing Alpha for the 1st
layer had a major impact on the simulation results for the 2nd and 3rd layers (depths of
50 and 160 cm, respectively).

5.4.2 Inverse Simulations With Layered System
Simulations presented in Section 5.4.1 are often conducted to evaluate the amount ofrecharge to the groundwater table from cultivated land (represented by cumulative
free drainage). They are highly relevant for studying salinity in arid regions.
Note that simulation ‘LongRun’ has the correct water retention parameters (listed inTable 5.4). This case yields a cumulative free drainage of 7.7 cm2; this is ourreference key prediction from the inverse modelling process. Predictions from all
subsequent simulations will be compared to this value.
Our objective here is not to come up with a magical procedure that yields the correct
answer, and is applicable to all situations. The objective is to demonstrate thecomplexity of the problem, to show how different procedures may lead to different
answers, and how to best tackle such a problem.
Projects ‘Long-‘
Simulation ‘Long-10’: We calibrate the parameters of the first layer against thecalibration data obtained from the same layer. Parameters for layers 2 and 3 are set attheir original values. Table 5.5 shows that the resulting parameters are very close to
the original parameters listed in Table 5.4 (with the exception of K). This simulation
also produced the best estimate for free drainage (result was identical to that from‘LongRun’ and is hence not included in Figure 5.18). This is attributable to the factthat only three parameters for the first layer were optimised.

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.19
0.4
0.35

Volumetric water content

0.3
0.25
0.2
0.15
0.1

Observed at 50 cm
Modelled with optimum paramerts

0.05

Modelled with lower Alpha at layer 1
0
0

150

300

450

600

750

Time (days)

Figure 5.17: Run C (50 cm data), lower Alpha for layer 1
Simulation ‘Long-50’: We calibrate the parameters of the three layers against thecalibration data obtained from the second layer. Table 5.5 shows that the optimisedparameters for layer 2 are in close agreement to the original. However, the results aredifferent for the other two layers.
Simulation ‘Long-All1’: We calibrate the parameters of the three layers against thecalibration data obtained from the three layers. A different set of results is now
obtained. Note the high value of the optimised Ks for layer 3; this is a typical casewhere imposing a maximum limit becomes beneficial and can stop the model fromassigning physically unrealistically high values.
Simulation ‘Long-All2’: As above, but use different initial parameters. Yet anotherset of results is obtained.
Table 5.5 and Figure 5.18 shows that the cumulative free drainage flux is different for
the four simulations. However, results obtained from Project ‘LongAll2’ aresignificantly higher than the rest.
40

Cumulative free drainage (cm2)

35

5

30

1- LongRun
2- LongAll3

25

3- LongAll1
4- Long-50

20

5- LongAll2

Error
duringthisperiod15
10Higher confidence in these results

5

1

0
0

150

300

450

600

750

Time (days)

Figure 5.18: Cumulative free drainage flow from the four simulations
Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.20

Note that although the free drainage prediction obtained from Project ‘LongAll2’ isconsidered erroneous; the fit to the observed values was excellent (see Figures 5.195.21).
0.4
0.35

Volumetric water content

0.3
0.25
0.2
0.15
0.1
Observed; layer 1

0.05

Modelled; layer 1
0
0

100

200

300

400

500

600

700

800

Time (days)

Figure 5.19: Observed and fitted water contents at depth=10 cm; Project ‘LongAll2’
0.35

Volumetric water content

0.3
0.25
0.2
0.15
0.1
0.05

Observed; layer 2
Modelled; layer 2

0
0

100

200

300

400

500

600

700

800

Time (days)

Figure 5.20: Observed and fitted water contents at depth=50 cm; Project ‘LongAll2’

0.3

Volumetric water content

0.25

0.2

0.15

0.1

0.05

Observed; layer 3
Modelled; layer 3

0
0

100

200

300

400

500

600

700

800

Time (days)

Figure 5.21: Observed and fitted water contents at depth=140 cm; Project ‘LongAll2’Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.21

Table 5.5: details for Project series ‘Long-‘
Project nameSoil layer

Long-10Long-50

Long-All1

Long-All2

Long-All3

L-1*

L-2

L-3

L-1

L-2

L-3

L-1

L-2

L-3

L-1

L-2

L-3

L-1

L-2

L-3

Alpha

0.02

0.02 0.036

0.02

0.03

0.04

0.02

0.04

0.05

0.001

0.002

0.001

0.001

0.002

0.01

n

1.4

1.41

1.56

1.5

1.41

1.65

1.4

1.55

1.3

2

1.25

2

2

1.25

2

Ks

40

10.8

25

30

10.8

40

40

20

100

100

20

25

100

20

25

-

-

0.024 0.021 0.034 0.025 0.016 0.073 0.0207 0.0123 0.00198

Input
parameters

Alpha 0.037
Optimisedparameters0.031571 0.01649 0.044808
n

1.553

-

-

1.68

1.38

1.6

1.68

1.43

1.36

1.726

1.435

1.509
1.6132

Calibration data

Data at 10 cm
depth

Data at 50 cm depth

36.2
5.5
608
Data at depths 10,
50, and 140 cm

Cumulative freedrainage cm2**

7.28

10.5

9.61

Ks

79

-

-

22.7

10.7

74.3

1.45581.4706

28.35
2.18
189.3
Data at depths 10, 50, and140 cm50.5895.176
100
Data at depths 10, 50, and140 cm38.4

9.01*L refers to Layer**Reference value = 7.7

21

Section 5: Inverse Solution

5.22

What do the results in Figure 5.18 tell us? In real-life situations we generally do notknow what the correct answer is, but we may have a good feel for it (know its order;
for example, we may think it should be 5 not 50). The comparison in Figure 5.18
indicates a high likelihood that the result obtained from Project ‘LongAll2’ iserroneous, or at least requires further investigation. The fact that three scenarios
produced results in the vicinity of 7-10 cm2 makes us believe these figures more anddisbelieve those obtained from Project ‘LongAll2’. This is another case demonstrating
the effects of the initial parameters on the inverse solution, and the crucial role of themodeller’s personal judgement.
Figure 5.22 shows the source of the error. It occurred mainly at the beginning of thesimulation; notice the high water content predictions for times<14 (prior to the firstcalibration data). The prediction at time=14 is excellent (observed=0.3; fitted=0.312);
hence, plotting observed versus fitted points only at observation times as shown in
Figure 5.21 might mislead the modeller (data from “Fit.out” output file). This
discrepancy for times<14 days could very well be overlooked. HYDRUS provides
comparisons of fitted data with observation node data, which gives a more complete
picture.
Let us revisit the “Variable Boundary Conditions” and look at the precipitation data.
We see that the rainfall up to time =14 does not justify the high free drainage shownin Figure 5.18. A close look at the water content data shows that the water content is
0.42 at time=0. If we go back to the sensitivity analysis performed in Appendix I.3,
we see that an overestimation of the initial water content is caused by a low Alpha(you can also see this by simply substituting values in the van Genuchten model).

0.45
0.4

Observed at 140 cm

Volumetric water content

LongAll1

Error source, might looktrivial !!

0.35

LongAll 2

0.3
0.25
0.2
0.15
0.1
0.05
0
0

100

200

300

400

500

600

700

800

Time (days)

Figure 5.22: comparison of observed and predicted water contents for two simulations
Simulation ‘Long-All3’: We will attempt to solve the problem in Project ‘LongAll2’
by following two steps:
1. Increase the initial Alpha for layer 3 to 0.01. This will result in an initialwater content of 0.327 at depth =140 cm (substitute initial values in the van
Genuchten model).
2. Introduce minimum and maximum limits to the parameter values to avoid
unrealistic estimates obtained in the previous runs (very high K values for

Modelling variably saturated flow with HYDRUS-2D

Section 5: Inverse Solution

5.23

layers 2 and 3). Note that the model failed to run without these limits (themodel crashed).
0.35

Observed
Predicted; n=2

0.3

Predicted; n=1.5
Volumetric water content

0.25

0.2

0.15

0.1

0.05

Individual forward run time = 20 seconds

0
0

100

200

300

400

500

600

700

800

Time (days)

Figure 5.23: Comparison of results from two forward simulations

3. We can investigate the effect of increasing Alpha to 0.01 by running aforward simulation (under ‘Inverse Solution’, set maximum number of
iterations to zero). The result is shown in Figure 5.23 (the forward run took
about 12 seconds). Notice that the initial water content is 0.32. The entire
curve seems to be shifted downward. We demonstrated in Appendix I.3 thatthe curve could be translated upwards by reducing the parameter n. Use n=1.5
and re-run. Figure 5.23 shows that the prediction is now very close to theobserved values. We will keep the initial n=2 and run simulation ‘Long-All3’
in inverse mode.
Figure 5.18 shows that the cumulative free drainage predictions are reasonable and inagreement with the other runs. The final “optimum” parameters are listed in Table
5.5.
Concluding Comments
Users should be wary of the complications associated with inverse simulations. Close
agreement between predicted and observed data does not necessarily mean that thecorrect answer is obtained.
One should investigate different scenarios, especially for relatively complex problems
involving many unknown parameters. Such scenarios include different initial
parameters, calibrating various soil layers against the different calibration data, andperhaps excluding results that deviate significantly from other scenarios.
For a multi-layered soil system, a parameter belonging to one layer may affect thepredictions for another layer.
Setting minimum and maximum limits for the parameter limits avoids the model fromassigning physically unrealistic values, which is advantageous in two ways; firstly, it
limits the search domain, thus reducing both run time and the likelihood of obtaining
erroneous results, and secondly, reduces the risk of crashing as a result of numericalinstabilities that are likely to occur when extremely low or high parameter values are

Modelling variably saturated flow with HYDRUS-2D

5.24

Section 5: Inverse Solution

assigned. It is always a good idea to specify the minimum for the n parameter(n>1.05) when using the van Genuchten’s model.
The model may be more sensitive to parameters belonging to the soil located next toan outer boundary such as a free drainage BC or a seepage face.

5.5 General Rules; Dos and Don’ts in Inverse Modelling
1. Having the correct conceptual model is always a crucial element of goodmodelling (e.g., correct BCs). When conducting an inverse analysis,
sometimes a completely insensible result may indicate inadequate
conceptualisation of the problem.
2. Even if the model fitted the experimental data nicely, that does not necessarilymean that the correct result has been achieved. Some degree of scepticism isalways useful.
3. Limit the number of optimised parameters as much as possible, especially ifthere is a high correlation between the optimised parameters. The number of
optimised parameters should always be equal or less than the number ofcalibration points. This is like solving a set of linear equations (eachcalibration point represents an equation): the number of unknowns (optimisedparameters) that you can estimate must be equal to the number of equations.
For example, if you have four calibration points and five unknown parameters,
there will be an infinite number of solutions for such a problem.
4. It is highly recommended to have a second type of calibration data; even one
data point could make a tremendous difference. For example, if the calibration
data set comprises transient pressure heads, a final water content measurement
would be extremely useful.
5. If no good initial estimates for the model parameters are available, try runningthe model first a few times in forward mode to get a feel for the final results.
This should produce a good set of initial parameters in addition to being able
to constrain minimum and maximum values. The procedure will be timesaving in the end, especially when running long simulations.
6. Even if a good set of starting parameters was selected, try other values tomake sure that the model converges to the same solution.
7. Define the final objective (the key prediction) of the modelling process. For
example, let’s assume that the goal is to estimate the cumulative flow across aboundary, like seepage from a dam. Identify first which parameter the flow ismost sensitive to. For a multi-layered soil system, common sense suggests thatparameters belonging to the soil next to the seepage boundary is the most
crucial; hence, there should be a good calibration data set for that area.
Another example involves high-intensity rainfall areas where the saturated
conductivity of the surface soil will determine how much water will infiltrate
into the soil; this parameter hence is likely the most important.
8. Make sure that your calibration data are evenly distributed spatially andtemporally to eliminate any bias in the results. For example, if you are trying
to evaluate the water retention curve of a soil, your calibration data should
cover water contents from saturation down to points where the retention curvechanges its slope (where it starts curving toward the residual value). High data

Modelling variably saturated flow with HYDRUS-2D

5.25

Section 5: Inverse Solution

density (at some place or during some time period) results in skewing thecalibration.
9. Highly correlated parameters often lead to non-unique solutions. That is, morethan one set of parameters will result in a similar fit to the calibration data. In
these cases, either eliminate one of the correlated parameters from theoptimisation procedure (if you have a reasonably good independent estimate
for that parameter), or introduce new types of calibration data. For example, if
your calibration data consist of pressure heads in the saturated zone and the Ks
values of a draining two-layered soil system are highly correlated (model
responds to K1/K2 ratio), you probably should try to measure the flow rate.
The more complex a model, the more likely non-uniqueness will become aproblem.
10. Weighing factors are useful, but do not use them unless you are very confidentof the data point to which you are assigning a high weight. For example, see
the ‘Crust’ example included in the HYDRUS package; a single measurementon the retention curve was given a weight of 10. You can also assign higher
weights to data obtained from areas where predictions are important (e.g. data
next to a boundary where flow needs to be predicted).
11. Since inverse runs are generally time-consuming, it is always worthwhile tofirst invest some time in devising a mesh that runs as fast and accurately as
possible. That is, use the coarsest possible mesh and loosen tolerances (whilestill getting a reasonable mass balance). Once you come up with a good
preliminary result, tighten the tolerances to obtain the final answer.
12. Even if you obtain a nice fit between observed and modelled data, always
closely investigate “Inverse Solution” and “Mass Balance” under Postprocessing. Look at the statistical information provided and make sure that theresults are physically realistic.
13. Parameters with very wide confidence limits may be eliminated from theoptimisation process since the model is not sensitive to them.
14. Validation is always another way of confirming that you obtained the correct
results. You must validate against data that were not used in the calibration
process.
15. And finally: remember that the inverse method is not the black box
approach that always gives the right answer; some subjective judgementof the modeller may well be critical in many problems.
Final remark: We refer readers to Section 1.6, 3.4.2, and 3.6 of the “Method of Soil
Analyses” book of the Soil Science Society of America for a detailed discussion of
the theory behind parameter estimation technique, and various applications for
estimating soil hydraulic and solute transport parameters (Šimůnek and Hopmans,
2002; Hopmans et al., 2002; and Šimůnek et al., 2002).

Modelling variably saturated flow with HYDRUS-2D

5.26

Section 5: Inverse Solution

Modelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.1

6. TROUBLE SHOOTING
6.1

Infiltration After Long Dry Period

Project Mesh-1:
This project simulates infiltration into a silty soil after a long evaporation period.
Project ‘Mesh1’ is run using the default time stepping parameters. The results areshown below. Projects ‘Mesh 1 & 2’ are found on the CD (..\HYDRUS Manual
Projects\Sect-6).
Checking the actual atmospheric flux reveals
an error at time=50 days; notice themagnitude of the flux (1010 cm is equal to500,000 kilometres!!).

Checking the massbalance information
confirms the presenceof serious errors
between times 50 and
51 daysModelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.2

The sudden increase in the number of
iterations at around time = 50 days is also
suspicious and should be investigated, eventhough it may not necessarily indicate an error.

This figure suggests that the solution did notconverge at time=50 days (number of
iterations exceeded the maximum limit of 20).

Spreadsheet ‘Mesh1.xls’ shows the output file ‘h-mean.out’; notice the errors attime=50 day (shown below). If you try to access the pressure head data in HYDRUS,
you will get an error message; this is due to the errors shown below in ‘h-mean.out’
50.002 -988063
50.003 59811490
50.0040***********
050.0050***********
050.0060***********
050.0070***********
0

0
0
0
0
0
0

0
0
0
0
0
0

0
0
0
0
0
0

0
0
0
0
0
0

0
0
-138.3-138.3-138.3-138.3-138.3-138.3Modelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.3
Project Mesh-2:

The problem encountered in Project ‘Mesh1’ is fixed by lowering the minimum time
step to 1E-5 day. This will allow the model to lower the time step to a small enoughvalue that will accommodate the numerical instabilities encountered when
precipitation is suddenly introduced at time=50 days.

Actual atmospheric flux is now correct.

There is now no divergence in thesolution at time=50 days.

The mass balance results are now acceptable.

Conclusion: Always allow small time steps by specifying a “Minimum Time Step” onthe order of seconds.

Modelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.4
6.2

High Infiltration Rate Into a Two-Layer Soil System

Project Mesh-3:
This project simulates infiltration into a two-layered coarse-above-fine soil system.
The soil profile is initially relatively dry (uniform pressure head=-100 cm). The
profile further dried out due to evaporation for 1 day. An intense rainstorm thenoccurred at a rate of 300 cm/day. Project ‘Mesh3’ is run using the default time
stepping parameters. The results are shown below. Projects ‘Mesh 3 & 4’ are foundon the CD (..\HYDRUS Manual Projects\Sect-6).

Checking the actual atmospheric flux
does not reveal a problem. The
magnitude of the flux is equal to thepotential rate (0.2×300×10=600),
which is time×intensity×width.
This value could be believable if you
overlook the fact that the introduced
flux exceeds the saturated conductivityof the surface layer.

The rapid increase in the cumulative
number of iterations at time=1 day is
expected but should be investigated.

Modelling variably saturated flow with HYDRUS-2D6.5

Section 6: Trouble Shooting

The atmospheric boundary pressure head drops to hCritA (-10,000) due toevaporation, which is correct. However, the pressure head does not increase due torainfall at time=1. Why??

A graphical display of the results reveals that the soil surface at the end of thesimulation is completely dry; No rainfall was apparently introduced.

Modelling variably saturated flow with HYDRUS-2D6.6

Section 6: Trouble Shooting

Checking the mass balance at time >1 day reveals extremely large errors.

Project Mesh-4:
The problem encountered in example Mesh3 is solved here using a simple trick. The
variable boundary condition is discretized into 3 stages. A very short (0.001 day, lessthan 1.5 minute) imaginary period of moderate rainfall (5 cm/day) is introduced
before the intensive-rainfall period. The problem is solved.

Mass balance isrestored.

The pressure head at the soil surface increases to zero when rainfall starts.

Modelling variably saturated flow with HYDRUS-2D6.7

Section 6: Trouble Shooting

The cumulative flux is now much lower than the potential, which is believable since
the rainfall intensity is much higher than the saturated hydraulic conductivity of thesoil.

Remark: This problem could have been solved also by allowing smaller time steps,i.e., a “Minimum Time Step”=1e-6 d.

Modelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.8

6.3
Simulations With Potentially Unstable Initial Conditions; Drainage of a
Saturated Coarse-Grained Soil

Project Mesh-5
This example demonstrates a case where the initial conditions are potentially unstable,
which could lead to divergence problems. The instability is related to the presence ofa fully saturated coarse-textured soil profile with a free-draining base and a seepage
face along the side. We proceed with setting up the problem and accepting all default
values that HYDRUS provides (initial time step=0.1 day; minimum time step=0.001
day; maximum number of iterations=20). Projects ‘Mesh 5, 6 & 7’ are found on the
CD (..\HYDRUS Manual Projects\Sect-6).
The results areshown here. During
the first time steps
the maximum
number of iterations
is exceeded.

The solution diverges and the programis terminated.

Project Mesh-6
Trying to solve the problem of Mesh-5 by increasing the maximum number of
iterations does not work.
Reducing the initial time step to 0.001 day does not work either.
Try doing both. Increase the maximum number of iterations to 50 and reduce theinitial time step to 0.001 day.

Modelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.9

The solution converges and the problem is solved. Note that the number of iterations
during the first time step is 31, higher than the default value of 20.

Project Mesh-7
Alternatively, solve the problem of Project ‘Mesh-5’ by reducing the initial time step
to 0.0001 day (better option).

Changing the initial time step to0.0001 day should solve this problem.
Make sure that the minimum time stepis simultaneously reduced to a value ≤to the initial time step.
The maximum number of iterations
should be reset to 20 (to see if
reducing the initial time step would
solve the divergence problem).

The problem is solved!
Conclusion: Allowing smaller time steps is almost always a good idea!

Modelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.10
6.4

Infiltration Into a Thick Clay Soil Layer After a Dry Period

Project Mesh-8:
This project simulates a drying and a wetting cycle in a 100-cm thick fine-textured(clay) soil block. The soil is at a uniform initial pressure head of –100 cm.
Evaporation is simulated for a period of 0.2 days followed by infiltration up to 1 day.
The precipitation rate is equal to the infiltration capacity of the soil (equal to thesaturated hydraulic conductivity of 4.8 cm/day). Boundary conditions are atmosphericat the surface and free drainage at the base.
Note that for the first two cases below (1 and 2) we used the optimum time
discretisation and iteration criteria (of run Mesh8e). This to be able to isolate theeffects of each problem, that is, the high precipitation rate (1), and the soil hydraulicmodel (2). All related projects are found on the CD (..\HYDRUS Manual
Projects\Sect-6).
Project Mesh8: use this project to try (1 and 2)
Problems that are likely to be encountered in such a simulation:
1. Precipitation ≥ saturated hydraulic conductivity (4.8 cm/day)
Make sure you check the‘with air entry value of–2 cm’ box in the soil
hydraulic model. Undervariable BC, set the
precipitation rate to 4.8
cm/day, and run.
Notice how thesimulation does notprogress beyond time >0.467 day.

Solution: decrease the precipitation slighlty to 4.7 cm/day and the simulation will run
in only 48 seconds with a total number of iterations of 16,239.

Modelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.11

2. Soil hydraulic model used
The preferred model for very fine-textured (clayey) soil is the van Genuchten-Mualem
model with an air entry value of –2 cm. This model improves numerical stability for asoil that has a highly nonlinear hydraulic conductivity function close to saturation.
Reset the precipitation rate to 4.7 cm/day in Project Mesh8, and uncheck the ‘with air
entry value of –2 cm’ box in soil hydraulic model. Run the model.
The screen capture below shows the progress of the simulation when the normal van
Genuchten-Mualem model is used. Notice the number of iterations and the time. The
simulation almost does not progress beyond time>0.246 day.

Solution: Select the soil hydraulic model as shown below

Modelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.12

3. Time discretization and iteration criteria
For Projects Mesh-8a to Mesh-8e: use the van Genuchten model with a –2 cm air
entry value, and a surface flux=4.7 cm/day.
• Project Mesh-8a: using default time step and iteration criteria (runtime = 11 sec.)
The mass Balance results are very poor
WatBalT [V]
WatBalR [%]0.53537E+09
100.000

The cumulative actual atmospheric flux is incorrect; HOWEVER, the free drainage
flux is correct!!!

Modelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.13• Project Mesh-8b: default time step and iteration
criteria, but we reduced the number of print times to
1.
The mass balance improved, and the actual
atmospheric flux is correct!!
WatBalT [V]
WatBalR [%]-.13266E+01
1.100

This suggests that in the previous run, print times must have interfered with the time
stepping scheme. This is not a solution but an indication that the time stepping
parameters were not correct.
•

Project Mesh-8c: change the minimum time step and the water content tolerance• We decrease the initial and minimum time steps to as low as 1E-15 but it
does not solve the problem.
• Try initial and minimum time step = 1E-10 day and tighten the watercontent tolerance to 1E-7. Run time is 64 sec. And the results are correct. The
run time information reveals no divergence problems.
Mass balance:
WatBalT [V]
WatBalR [%].45776E-04.000

Total number of iterations = 22,161

Modelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.14•

Project Mesh-8d: Modify the lower tension intervals

Lower limit of the tension interval = 0.001
WatBalT [V].75912E-03
WatBalR [%].001

Number of iterations is well below the maximum.
Total number of iterations = 20,097•

Project Mesh-8e: Modify the upper tension intervals

Upper limit of the tension interval = 200
Mass balance:
WatBalT [V].12970E-02
WatBalR [%].001
Run time = 51 sec.
Total number of iterations = 16,239.
The cumulative free drainage flow isshown here; compare the result with thatobtained from project ‘Mesh8a’ and‘Mesh8c’.

Conclusions: Specified infiltration rates close to the Ks value of heavy textured soils
may cause numerical instabilities. We showed above several ways of overcoming
these instabilities.
Modelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.15

6.5
Simulations With Highly Unstable Initial Conditions; Very High-Pressure
Gradient
This series of projects (“Grad-”) demonstrates the difficulties that are likely to be
encountered when highly unstable initial conditions are assigned to the domain. The
runs simulate infiltration into a 10-cm soil column with extremely high initial pressure
gradients of up to 250. The pressure head is assumed to be linearly distributed withdepth and the pressure head is set to zero at the free drainage boundary for all
simulations. All related projects are found on the CD (..\HYDRUS Manual
Projects\Sect-6).
Projects ‘Grad-1’ and ‘Grad-1a’

For simulation ‘Grad1’, the elements were evenly spaced at 1 cm. The pressure headat the surface was set at –1200 cm. Table 6.1 indicates that this grid produced arelatively small mass balance error. Figure 6.1 shows that the number of iterations has
exceeded the maximum limit (solution did not converge) during the first 16 time
steps.
In Project ‘Grad1a’, we try to reduce the initial time step but this produces extremely
poor results! This suggests that the grid should be refined.
Table 6.1 Summary of the results obtained from the seven runs.
Project

Simulation
time
(day)

Run
time
(sec)

Surfacepotential
(cm)

Mesh
size*

Initial andminimum time
step (day)Massbalance**11
0.13
default#
1
1.729
11
1340
Grad 1a
1
1
-1200
1E-6
1
99.43
51
0.00466
Grad 2
1
2-1200
default
1
0.062
101
0.00028
Grad 31
6-1500
default
0.1
0.004
101
1.1E6
Grad 41
5-2500
0.0001
0.1
52.82
101
0.0108@
Grad 50.1
81-2500
0.00001
0.1
1.798
201
0.0000348
Grad 6
0.1
144
-2500
0.00001
0.1
0.006* Number of elements and density at top of mesh, respectively** Mass balance volume and % error, respectively#
Default values are: Initial time step=0.1; minimum time step=0.001@
Reducing the time step to 1E-12 did not improve the results
Grad1

1

2

-1200

Notes onmass
balance

fair
poor
good
good
poor
fair
good

Modelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.16

25

Grad 1
Grad 2

Number of iterations .

20

15
Maximum numberof iterations = 20
10

5

0
0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Time (day)Figure 6.1. Run-time information for projects ‘Grad1’ and ‘Grad2’Projects ‘Grad-2’

For simulation ‘Grad2’, the grid was refined. The elements were evenly spaced at
0.2 cm (see Table 6.1). The mass balance is now improved. Figure 6.1 shows that the
maximum number of iterations was exceeded once only. Figure 6.2 shows that the
cumulative free drainage flow was different for the two simulations. The trend
obtained for Project ‘Grad 1’ shows an anomaly at the beginning of the simulation.

3

Grad 1
Grad 2

Cumulative free drainage flow .

2.5

2

1.5

1

0.5

0
0

0.2

0.4

0.6

0.8

1

1.2

Time (day)Figure 6.2. Cumulative free-drainage flow for projects ‘Grad1’ and ‘Grad2’

Modelling variably saturated flow with HYDRUS-2D

6.17


Section 6: Trouble Shooting

Projects ‘Grad-3’

For simulation ‘Grad3’, the surface pressure head was decreased to –1500 cm. In
order to get a stable run with a good mass balance, the grid had to be further refined.
The total number of elements was increased to 101 and the density at the top was
reduced to 0.1; this resulted in a surface element of 0.02 cm. Results were acceptable(see Table 6.1).
Projects ‘Grad-4’

For simulation ‘Grad4’, the surface pressure head was decreased to –2500 cm. We
used the same grid spacing as that for Project ‘Grad3’. In order to get the HYDRUS torun, the initial time step had to be reduced to 0.0001 day. The simulation proceeded asshown below. Notice that the number of iterations always exceeded the maximum
limit of 20; also notice that ‘hAtm’ and ‘hDrain’ fluctuate in a chaotic way.
Nevertheless, the run proceeds and results are displayed in a normal manner (the run
does not terminate automatically as in Project ‘Grad 1’). We will examine theerroneous results. If you encounter such a run, there is no point in continuing it; better
terminate it by pressing Ctrl-C.

If you did not look at the screen information during the run, you will likely only checkthe final displayed results. If you have lots of confidence in the model (and yourself!!) and don’t check everything, you may overlook errors.
Suppose that you are only interested in free drainage flow. You check atmosphericflow, but instead of checking the actual atmospheric flow, you only check thepotential flow. The latter looks correct as shown (next page); in fact the potential
atmospheric flow is always correct since it represents what could potentially go
through the boundary, not what actually entered the boundary.

Modelling variably saturated flow with HYDRUS-2D6.18

Section 6: Trouble Shooting

The free drainage flux also looks reasonable if you do not check its magnitude. A
quick assessment of the reported value suggests that it must be wrong since it is 5
times the potential atmospheric flux (initial soil profile was dry so it can’t be higher
than the potential atmospheric flux). Close examination of the actual atmospheric flux
shows the source of the error.

Table 6.1 shows that the mass balance is very poor. Notice that the run time was very
short (even shorter than Project ‘Grad3’). When the initial conditions are unstable andthe initial time step is very small, one should expect a much longer run time. Hence, if
the simulation is unexpectedly short, that should be a warning.

Projects ‘Grad-5’

In simulation ‘Grad5’, the initial time step is further reduced to 0.00001 day. Table
6.1 indicates that mass balance results improved, but the errors are still significant.
Further reduction in the initial time step does not improve the results. Notice thesignificant increase in run time (also notice that the simulation time is only 0.1 day,Modelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.19

which is 1/10 of the previous runs). The screen capture at the beginning of the run
indicates divergence of the solution during the first two time steps, which explains
why the mass balance results were not good.

Projects ‘Grad-6’
For simulation ‘Grad6’, the total number of elements was increased to 201; thisresulted in a surface element of only 0.01 cm. Table 6.1 indicates that the massbalance results are now satisfactory. The run time has increased significantly. Figure
6.3 shows that the cumulative free drainage flux is different for the two simulations.
0.0025
Grad6
Grad5
Cumulative free drainage flow

0.002

0.00150.001

0.0005

0
0

0.02

0.04

0.06

0.08

0.1

Time (day)Figure 6.3. Free drainage flow for runs ‘Grad5’ and ‘Grad6’
Hence, one may expect that the presence of very steep pressure gradients will cause
more difficulties in getting the model to run smoothly. In addition, the run time should
also increase dramatically.

Modelling variably saturated flow with HYDRUS-2D

Section 6: Trouble Shooting

6.20
6.6

Concluding Remarks

1. At the conclusion of a simulation, always inspect mass balance and run time
information; also make sure that reported fluxes are realistic. Whenever the
maximum number of iterations is used, closely inspect the results for possible
inconsistencies.
2. For each condition, try to get an optimum combination of mesh size and time
stepping scheme that runs as fast as possible, while still producing good massbalance results.
3. Avoid initially fully saturated profiles; they may result in a diverging numericalsolution (refer to Section 2.2.1); setting a few surface nodes at a very low negative
pressure heads usually helps avoiding the problem.
4. Avoid introducing fluxes that are equal or higher than Ks into the domain.
5. Use the van Genuchten model with AEV=-2 cm for fine-textured clayey soils.
6. Some inverse simulations may stall; providing realistic minimum and maximum
limits for the optimized parameters should overcome this problem.
7. Complex simulations that run unexpectedly fast indicate a problem.
8. It is always a good idea to monitor the advance of the simulation; screens similar
to those shown in pages 6.12 and 6.17 indicate errors. Such simulation may well
be terminated using ‘Ctrl_C’.
9. Extremely slow simulations usually indicate a flawed set up; check minimum andinitial time steps, the density of the grid, and the suitability of the hydraulic model.
10. It is always a good idea to allow small initial and minimum time steps (in theorder of seconds).
11. Introducing intense precipitation into an extremely dry soil profile may beproblematic (refer to Section 6.2).
12. Leave the default ‘Time Step Control’ unchanged; in most cases divergenceproblems are solved by changing the mesh density, the time discretization, and theiteration criteria controls.
13. Using unrealistic initial pressure head profiles is problematic; either use stable
initial conditions such as hydrostatic equilibrium or import initial conditions fromprevious simulations when available.

Modelling variably saturated flow with HYDRUS-2D

References

Ref-1
REFERENCES

Brooks, R.H. and Corey, A.T. (1966). Properties of porous media affecting fluid flow.
Journal of the Irrigation and Drainage Division ASCE, 92(IR2): 61-88.
Brooks, R.H., and Corey, A.T. (1964). Hydraulic properties of porous media.
Hydrology paper No. 3, Civil Engineering Department, Colorado State
University, Fort Collins, Colorado.
Campbell, G.S. (1985). Soil physics with BASIC, Elsivier, New York.
Carsel, R. F., and Parrish, R. S. (1988). Developing joint probability distributions ofsoil water retention characteristics, Water Resources Research, 24: 755-769.
Collis-George, N. (1955). Hysteresis in moisture content-suction relationships insoils. Proceedings of the National Academy of Science (India), 24a: 80-85.
Croney, D. (1952). The movement and distribution of water in soils. Geotechnique,3(1): 1-16.
Croney, D., and Coleman, J.D. (1961). Pore pressure and suction in soils. Proceedingsof the Conference on Pore Pressure and Suction in Soils. Butterworths,
London: 31-37.
Edlefson NE, and Anderson, A.B.C. (1943). Thermodynamics of Soil Moisture.
Hilgardia, 15(2): 31-298.
Everett, EG. (1993). Vadose zone monitoring. In Geotechnical Practice for Waste
Management, Ed. D.E. Daniel, Chapman and Hall, London: 651-675.
Feddes, R.A., Kowalik, P.J., and Zarandny, H. (1978). Simulation of field water use
and crop yield, John Wiley & Sons, New York, NY.
Gray (1970). Handbook on the principles of hydrology. Canadian National Committee
for the International Hydrological Decade, National Research Council of
Canada, Ottawa.
Hopmans, J.W. (1987). A comparison of various methods to scale soil hydraulicproperties. Journal of Hydrology, 93: 241-256.
Hopmans, J.W., and Stricker, J.N. (1989) Stochastic analysis of soil water regime in awatershed, Journal of Hydrology, 105, 57-84.
Hopmans, J. W., Šimůnek, J., Romano, N., and Durner, W. (2002). Inverse Modeling ofTransient Water Flow, In: Methods of Soil Analysis, Part 1, Physical Methods,
Chapter 3.6.2, Eds. J. H. Dane and G. C. Topp, Third edition, SSSA, Madison,
WI, 963-1008.
Koorevaar, P., Menelik, G., and Dirksen, C. (1983). Elements of soil physics, Elsevier
Science Publications, Amsterdam, The Netherlands.
Kutilek, M, and Nielsen, D. (1994). Soil Hydrology. Cremlingen-Destedt, Germany.
Miller, E.E., and Miller, R.D. (1956). Physical theory for capillary flow phenomena,
Journal of Applied Physics, 27: 324-332.

Modelling variably saturated flow with HYDRUS-2D

Ref-2

References

Mualem, Y. (1976). A new model for predicting the hydraulic conductivity ofunsaturated porous media. Water Resources Engineering, 12: 513-522.
Peck, A.J., Luxmoore, R.J., and Stolzy, J.L. (1977). Effects of spatial variability ofsoil hydraulic properties in water budget modelling. Water Resources
Research, 13(2): 348-354.
Rancic, A. (2002). Personal communications. Department of Land and Water
Conservation, NSW, Australia.
Rassam, D.W., and Williams, D.J. (1999a). Unsaturated hydraulic conductivity ofmine tailings under wetting and drying conditions. Geotechnical Testing
Journal ASTM, 22: 138-146.
Rassam, D.W., and Williams, D.J. (1999b). A numerical study of steady-stateevaporative conditions applied to mine tailings. Canadian Geotechnical
Journal, 36: 640-650.
Rawls, W. J., Brakensiek, D. L., and Saxton, K. E., 1982, Estimating soil water
properties, Transactions, ASAE, 25(5), 1316-1320 and 1328.
Richards, B.G. (1965). Determination of the unsaturated permeability and diffusivity
functions from pressure plate outflow data with non-negligible membrane
impedance. In G.D. Aitchison (Ed.), Moisture equilibria and moisture changes
in soils beneath covered areas, Butterworths, Australia: 47-54.
Šimůnek, J., Kodesova, R., Gribb, M., and van Genuchten, M. Th. (1999). Estimating
hysteresis in the soil water retention function from cone permeameter
experiments. Water Resources Research, 35(5): 1329-1345.
Šimůnek, J., Sejna, M., and van Genuchten, M. Th. (1999). The HYDRUS-2D
software package for simulating two-dimensional movement of water, heat,and multiple solutes in variably saturated media. Version 2.0, IGWMC - TPS 53, International Ground Water Modeling Center, Colorado School of Mines,
Golden, Colorado.
Šimůnek, J., and Hopmans, J. W. (2002). Parameter Optimization and Nonlinear Fitting,
In: Methods of Soil Analysis, Part 1, Physical Methods, Chapter 1.7, Eds. J. H.
Dane and G. C. Topp, Third edition, SSSA, Madison, WI, 139-157.
Šimůnek, J., Jacques, D., Hopmans, J. W., Inoue, M., Flury, M., and van Genuchten, M.
Th. (2002). Solute Transport During Variably-Saturated Flow - Inverse
Methods, In: Methods of Soil Analysis, Part 1, Physical Methods, Chapter 6.6,
Eds. J. H. Dane and G. C. Topp, Third edition, SSSA, Madison, WI, 1435-1449.
Taylor, S. A., and Ashcroft, G. M. (1972). Physical Edaphology. Freeman and Co.,San Francisco, California, p. 434-435.
Tinjum, J.M., Benson, C.H., and Blotz, L.R (1997). “Soil-water characteristic curves
for compacted clays.” Journal of Geotechnical and Geoenvironmental
Engineering, ASCE, 123(11): 1060-1069.
van Genuchten, M. Th. (1987). A numerical model for water and solute movement
in and below the root zone. Research Report N. 121, U.S. Salinity Laboratory,
USDA, ARS, Riverside, California.

Modelling variably saturated flow with HYDRUS-2D

Ref-3

References

van Genuchten, M.Th. (1980). A closed-form equation for predicting the hydraulicconductivity of unsaturated soils. Soil Science Society of America Journal, 44:892-898.
Vogel, T., and Cislerova, M. (1988). On the reliability of unsaturated hydraulicconductivity calculated from the moisture retention curve, Transport in PorousMedia, 3: 1-15.
Vogel, T., Cislerova, M., and Hopmans, J.W. (1991). Porous media with linearlyvariable hydraulic properties. Water Resources Research, 27(10): 2735-2741.
Vrught, J.A., Hopmans, J.W., and Simunek, J. (2001). Calibration of a twodimensional root water uptake model. Soil Science Society of America
Journal, 65: 1027-1037.
Wesseling, J. G. (1991). Meerjarige simulaties van grondwateronttrekking voor
verschillende bodemprofielen, grondwatertrappen en gewassen met het model
SWTRE. Report 152, Winand Staring Centre, Wageningen.
Wilson, G.W., Fredlund, D.G., and Barbour, S.L. (1997). The effect of soil suction on
evaporative fluxes from soil surfaces. Canadian Geotechnical Journal, 34:
145-155.

Modelling variably saturated flow with HYDRUS-2D

Ref-4

References

Modelling variably saturated flow with HYDRUS-2D

Appendix I

I.1
APPENDIX ISOIL HYDRAULIC PROPERTIES
I.1

Water Retention Properties

The fundamental relationship between soil moisture content (usually calculated on a
volumetric basis), and pressure head (or matric suction) is referred to as the soil-water
retention curve (SWRC), or the soil water-characteristic curve. Matric suction is thedifference between pore air pressure and pore water pressure. If a change in moisture
content occurs at any point in a mass of soil initially in a state of moistureequilibrium, the soil matric suction (or the negative pore water pressure) at that pointchanges (assuming atmospheric conditions, pore air pressure = 0). Following thischange, there will be a differential in the energy level between the point where thechange occurred and the surrounding soil, which will cause moisture to flow in thesoil until a new condition of suction equilibrium is established (Croney, 1952).
Knowledge of the water retention curve is essential when investigating how soil
moisture reacts following changes in the pressure head resulting from events such as
infiltration, evaporation and drainage. Due to the hysteretic behaviour of soils uponwetting and drying, different SWRC curves are obtained depending upon whether thesoil is undergoing a drying cycle, a wetting cycle, or a re-drying cycle (Collis-George,
1955).
I.1.1

Shape of Soil-Water Retention Curve

The general shape of the SWRC under wetting and drying conditions is shown in
Figure I.1 (semi-log scale). The following parameters are identified:••

The saturated volumetric water content θs, which theoretically corresponds to thesoil’s porosity, but practically is often about 10-25% less because of dissolved
and/or entrapped air.
The air-entry value, AEV, is the critical suction value at which the largest pores inthe soil matrix begin to lose water.
The residual water content θr, is the water content beyond which further increasein the soil’s suction results in only marginal changes in water content.
0.5θsVolumetric water content•

AEV

Drying curve

0.4

Wetting curve

0.3

0.2

0.1

θr
0
0.1

1

10

100

1000

10000

1000001000000

Matric suction (kPa)

Modelling variably saturated flow with HYDRUS-2D

Appendix I

I.2

Figure I.1: Typical SWRC
The total suction corresponding to zero water content is essentially the same for all
soils. This value is about 106 kPa (107 cm) and has been supported experimentally by
Croney and Coleman (1961), Koorevaar et al. (1983), and by thermodynamic
considerations (Richards, 1965).

Definition of the Residual Water Content
The residual water content and the corresponding suction, termed residual suction,
have been defined in a number of ways in the literature; however, the accepted use of
these terms is not always agreed upon. Tinjum et al. (1997) defined it as the watercontent corresponding to the asymptote of the SWRC at low degrees of saturation.
van Genuchten (1980) defined it as the water content for which its rate of change withrespect to suction becomes zero, or when liquid flow of water ceases (excluding theregion near saturation); he suggested that often it is sufficient to define it as the watercontent at a suction equal to 1500 kPa.
Classically, moisture in the unsaturated zone is separated into three components:
gravity, pellicular, and hygroscopic water (Everett 1993). Gravity water is the
moisture in a soil that can be drained by gravitational forces; pellicular water is the
moisture in a soil that cannot be drained by gravity forces, but can be lost by
evaporation; and hygroscopic water is moisture that will never be lost through theabove natural forces. The residual water content is the water content at the limit ofliquid water extraction. De-saturation beyond residual conditions occurs primarily asa result of vapor movement up to the point where the soil water content is inequilibrium with the vapor pressure of its surrounding. Brooks and Corey (1966)
pointed out that it is difficult to determine residual saturation for clayey materials;
they related residual saturation to the clay content of the soil. The well-defined,
residual state for sands, silts, and their mixtures can reasonably well be predicted fromthe SWRC as shown in Figure I.1.
1.1.2

Models Describing SWRC

A mathematical representation of the SWRC is essential when using numericalmodels for simulating fluid flow and mass transfer in the unsaturated zone. HYDRUS
allows users to select one of three types of models to describe the soil hydraulicproperties: van Genuchten (1980), modified van Genuchten (Vogel and Cislerova,
1988), and Brooks and Corey (1964). Those three models are described briefly in thefollowing section.

a.

van Genuchten Model

Van Genuchten (1980) proposed a mathematical representation of the SWRC, which
depicts an S-shaped curve when plotted on a semi-log scale. It is given by:(

Se = 1 + αψ

)

n −m

(I.1)

Modelling variably saturated flow with HYDRUS-2D

Appendix I

I.3

where α, m, and n are fitting parameters (usually m=1-1/n), and Se is the normalised
volumetric water content (also called effective saturation) given by:Se =θ − θrθs − θ r(I.2)

where θ is the volumetric water content at any pressure head, and θs and θr are thesaturated and residual water contents, respectively.
Each of the three fitting parameters (α, m, and n) control part of the S-shape curve.
Figure I.2 shows the effect of α (Alpha), which is closely related to the reciprocal of
the air-entry value (AEV) of the soil. Coarse-grained soils have a low AEV and high
α whereas fine-textured soils have a lower α.
0.35Volumetric water content

0.3

0.25

0.2

0.15

Alpha = 0.015
Alpha = 0.05

0.1

Alpha = 0.1

0.05
1

10

100

1000

Suction (cm)Figure I.2: Effect of α in van Genuchten model
Figure I.3 shows the effect of n, which controls the slope of the curve and reflects theparticle size distribution. The plots indicate that a unit increase in suction at or near
the steepest part of the curve causes more water to be extracted from coarse-textured
soils (high n) than from fine-textured soils (low n), but not at other places of thecurve.
0.35

Volumetric water content

0.3
n=1.5
0.25

n=3
n=5

0.2

0.15

0.1

0.05
10

100

1000

Suction (cm)Figure I.3: Effect of n in van Genuchten model

Modelling variably saturated flow with HYDRUS-2D

Appendix I

I.4

The parameters n and m are usually correlated as m=1-1/n, which results in a
relatively simple equation for the hydraulic conductivity when combined with atheoretical model for K (see section I.2). Figure I.4 shows that when the twoparameters are correlated, the curvatures (A and B) of the S-shape curve becomecorrelated. When individual values are assigned to n and m the curvatures become
independent.
0.35
m = 1-1/n = 0.66

A

Volumetric water content

0.3

m = 0.3

Curvatures A &B are dissimilar

Alpha = 0.015
n=3

0.25

0.2

0.15

Curvatures A & B are similar(correlated; m=1-1/n)

B

0.1

0.05
10

100

1000

10000

Matric suction (cm)Figure I.4: Relation of m and n in van Genuchten modelb.

van Genuchten Model with AEV of -2 cm

This option in the van Genuchten model implements a small non-zero air-entry value(AEV) of -2 cm. Figure I.5 shows that the effect is very marginal on the SWRC. In
fact one can only notice the difference when that part near saturation is significantly
enlarged. Figure I.5 shows that the water content remains at saturation for suctions
below 2 cm.
0.45
Modified VG; AEV=-2 cm
VG

0.43

0.25

Volumetric water conten

Volumetric water content

0.35

0.15

0.42
Modified VG
VG

0.41
1

Suction (cm)

10

0.05
1

10

100

1000

Suction (cm)Figure I.5: van Genuchten model with AEV of –2 cm

Modelling variably saturated flow with HYDRUS-2D

Appendix I

I.5

We will show later that implementing this low non-zero AEV has a significant effecton the hydraulic conductivity function. This option is particularly relevant to veryfine-textured (clayey) soils (see example in section 6.4).

c.

Modified van Genuchten Model (Vogel and Cislerova)

Vogel and Cislerova (1988) modified the van Genuchten model by incorporating a
non-zero AEV into the model. The modification is implemented by introducing a
fictitious water content θm that is higher than θs and replaces θs in the van Genuchten
model. This fictitious water content is used only when h< AEV. Above AEV thewater content is equal to θs.
θ m − θ a ⎫ h < AEV
⎧
θ +
⎪
−m ⎪
⎪⎪ a
θ=⎨
1 + αh n⎬⎪⎪ h ≥ AEV
⎩⎪θs
⎭⎪

(

)

(I.3)

Figure I.6 shows how the model is implemented. The data points at suctions<17 cm
are only a hypothetical continuation to the S-shape curve. That is, when the head is
below this value, the water content is fixed at θs.
0.5
Theta_m = 0.48
Theta_s = 0.43

Volumetric water content

0.4

0.3

Vogel & Cislerova
0.2
van Genuchten

0.1

AEV (hs) =17 cm

0
1

10

100

1000

Suction (cm)Figure I.6: Modified van Genuchten modeld.

Brooks and Corey Model

Brooks and Corey (1964, 1966) concluded from comparisons of a large number ofexperimental data that the SWRC could be described by the following formula:⎛ AEV ⎞Se = ⎜
⎟
⎝ h ⎠

n

(I.4)

where AEV is the air entry value (corresponds approximately to 1/α in HYDRUS),
and n is a soil characteristic parameter, called the pore-size-distribution index.
Experimental data of suction and water content are often plotted as log ⎪h⎪ versus log

Modelling variably saturated flow with HYDRUS-2D

Appendix I

I.6

θ. Linear regression can then be performed on the straight-line relationship. The slope
and intercept of the best-fit line correspond to 1/n and AEV, respectively. Figure I.7
shows a comparison of this model with the van Genuchten model using theparameters listed in Table I.1.

Table I.1: Parameters for two SWRC for Figure I.7.
Theta_r
Theta_s
Van Genuchten
0.078
0.43
Brooks & Corey
0.027
0.434

Alpha
0.036
0.0897

n
1.56
0.293

0.45

Volumetric water content

0.35

0.25

Brooks and Corey

0.15

van Genuchten

0.05
1

10

100

1000

Suction (cm)Figure I.7: Comparison of Brooks & Corey and van Genuchten models
I.2

Hydraulic Conductivity Functions

a.

van Genuchten-Mualem Model

Mualem (1976) developed the following equation for predicting the relative hydraulicconductivity Kr from knowledge of the SWRC:⎛Θ 1
kr = Se ⎜⎜ ∫dx
⎝ 0 h ( x)⎞1dx∫0 h ( x ) ⎟⎟
⎠
1

2

(I.5)

where h is the pressure head, given as a function of the dimensionless water content
Se defined by Equation (I.2).
For the special case where m = 1-1/n (fitting parameters of Equation (I.1)), a closedform solution was obtained by van Genuchten (1980) as follows:(k (h ) = K sSle ⎡1 − 1 − S1e m
⎢⎣

) ⎤⎥⎦
m

2

(I.6)

Modelling variably saturated flow with HYDRUS-2D

Appendix I

I.7

where l is a pore connectivity parameter established to be 0.5 by Mualem (1976).
Alternatively, this equation may be written in terms of the effective water content (seeonline HYDRUS Help). When the –2 cm AEV option is used, Equation I.6 becomesapplicable only for heads<- 2 cm. Figure I.8 shows the impact of using this option.
Note that hydraulic conductivity functions are usually plotted to log-log scale, but inthis case we chose to plot it on a semi-log scale to emphasize the dramatic differences
near saturation.

25

Hydraulic conductivity (cm/day)van Genuchten
van Genuchten with AEV=-2 cm

20

Vogel & Cislerova

15

10

5
LOAM
0
1

10

100

1000

Suction (cm)Figure I.8: Effect of non-zero AEV on hydraulic conductivity functionb.

Modified van Genuchten Model (Vogel and Cislerova)

When using this model, the hydraulic conductivity function is discretized into threezones as shown in Figure I.9:
h >= hs (AEV)

K(h) = Ks

hk < h < hs

K(h) = linear value between Ks and Kk

h <= hk

K(h) = non-linear value (Mualem model)

where hk is the head corresponding to a user-defined water content θk (Qk in
HYDRUS); θk is water content below saturation where a measurement of thehydraulic conductivity Kk is available. The mathematical formulation for this model isfound in the online Help and will not be repeated here. A comparison between the
results of this model and the original van Genuchten model is shown in Figure I.10.
Figure I.8 shows the significant differences near saturation. The modification wasmade to enable relatively large increases in K near saturation that are often observedin structured (macroporous) soils.

Modelling variably saturated flow with HYDRUS-2D

Appendix I

I.8
25

0.5

Conductivity versus head

20

0.4

kk = 15
15

Qk = 0.38

10
0.3

Volumetric water conten

Hydraulic conductivity (cm/day

Water content versus head5hs ; AEV

hk

0

0.2
0

25

50

75

100

125

150

Head (-cm)

Figure I.9: Vogel and Cislerova hydraulic conductivity function

Hydraulic conductivity (cm/day)

1.0E+01

1.0E-01

1.0E-03

van Genuchten
Vogel & Cislerova

1.0E-05

1.0E-07

1.0E-09
1

10

100

1000

10000

Suction (cm)Figure I.10: van Genuchten and Vogel & Cislerova hydraulic conductivity functions

c.

Brooks and Corey Model

For heads <AEV, the hydraulic conductivity is defined as follows:⎛ AEV ⎞k (h ) = k s ⎜
⎟
⎝ h ⎠

2+3 n

(I.7)

A comparison of the results obtained with this model, and those obtained with the van
Genuchten model is shown in Figure I.11. The hydraulic parameters listed in Table I.1
were used for this purpose.

Modelling variably saturated flow with HYDRUS-2D

Appendix I

I.9

1.0E+03
van Genuchten
Hydraulic conductivity (cm/day)

1.0E+01

Brooks & Corey

1.0E-01

1.0E-03

1.0E-05

1.0E-07

1.0E-09
1

10

100

1000

10000

Suction (cm)Figure I.11: van Genuchten and Brooks & Corey hydraulic conductivity functions
I.3 Sensitivity Analysis; Understanding How the Hydraulic Parameters of a Soil
Can Affect Simulation Results
It is crucial for users to understand how each of the many input parameters impacts
the simulation results. If the model is more sensitive to a particular parameter, one
should invest more resources and effort into getting a high-confidence estimate for it.
This is also crucial when conducting inverse simulations where the task is estimating
a set of parameters for the model. Parameters that the model is insensitive to may beomitted from the optimisation list, thus reducing correlations between the otheroptimised parameters. In addition, modellers may not have good prior knowledge ofthe soil hydraulic parameters but they still need to come up with realistic initialstarting parameters before running the inverse mode.
In this section we will investigate the effect of each of the five hydraulic parameters
Alpha, n, Ks, Qs, and Qr. This sensitivity analysis is conducted using Project “a3”,section 5.2.1.
This exercise enhances our physical understanding of the effect of various hydraulicparameters. In addition, it helps when we manually choose initial parameters forinverse runs, which may have a great impact on the optimisation result (Section 5.3).
That is, if we know the effects of each of the parameters, we can manually change
them and run trial forward runs until a reasonable agreement is obtained with theobserved data, and then let the inverse solution do the final refinement (see Section
5.3). This is advantageous in two aspects: firstly, it excludes unrealistic results thatarise when starting with physically unrealistic initial parameters, and, secondly, it
reduces the run time (very advantageous for relatively long, complex simulations).
I.3.1 Parameter Alpha. Increasing this parameter leads to (see Figure I.12):


Lower initial water contents, since in this case the soils initial condition isrepresented by the pressure head.

Modelling variably saturated flow with HYDRUS-2D

Appendix I

I.10



Later arrival of the wetting boundary.
Higher final water contents.
0.3
Run a3; Alpha=0.0155
High Alpha=0.02
Low Alpha=0.01
Volumetric water content

0.25

A higer Alpha results in:

0.2

Extents of wetting
is extended

0.15
Lower initialwater content

Later arrival of wettingboundary

0.1
0

50

100

150

200

250

300

350

400

Time (min)Figure I.12: Effect of parameter Alpha
I.3.2 Parameter n. Increasing this parameter leads to (see Figure I.13):


Downward translation of the entire curve.
0.3
Run a3; n=2.189
High n=2.4
Low n=2.0
Volumetric water content

0.25
Downward translationof whole curve

A higer n results in:

0.2

0.15

0.1
0

50

100

150

200

250

300

350

400

Time (min)Figure I.13: Effect of parameter n
I.3.3 Parameter Ks. Increasing this parameter leads to (see Figure I.14):


Earlier arrival of the wetting boundary.
Lower final water contents.

Modelling variably saturated flow with HYDRUS-2D

Appendix I

I.11
0.3
Run a3; k=0.012
High k=0.016
Low k=0.008
Volumetric water content

0.25

Curve is stretchedhorizontally

A higer k results in:

0.2

0.15
Earlier arrival ofwetting boundary

0.1
0

50

100

150

200

250

300

350

400

Time (min)Figure I.14: Effect of parameter k

I.3.4 Parameter θs. Increasing this parameter leads to (see Figure I.15):


Slightly later arrival of wetting front
Changes in the shape of the curve in a manner opposite to thatwhen Ks is increased.

0.3
Run a3; Qs=0.34
High Qs=0.35
Low Qs=0.33
Volumetric water content

0.25

Curve is squeezed horizontally(opposite to effect of k)

A higer Qs results in:

0.2

0.15
Slight later arrival ofwetting boundary

0.1
0

50

100

150

200

250

300

350

400

Time (min)Figure I.15: Effect of parameter Qs

Modelling variably saturated flow with HYDRUS-2D

Appendix I

I.12

I.3.5 Parameter θr. Increasing this parameter leads to (see Figure I.16):


Higher initial water contents.
Earlier arrival of the wetting front.

0.3
Run a3; Qr=0.1
High Qr=0.11
Low Qr=0.09
Volumetric water content

0.25

Upward translation ofwhole curve

A higer Qr results in:

0.2

0.15

More significant effects
on dry end of curve
0.1
0

50

100

150

200

250

300

350

400

Time (min)Figure I.16: Effect of parameter Qr

Modelling variably saturated flow with HYDRUS-2D

Appendix II

II.1
APPENDIX IICONCEPTS RELATED TO MODELLING EVAPORATION
Evaporation from a soil surface is an important component of the water budget.
Evaporation rates are influenced by external weather conditions as well as soil
parameters. Weather conditions usually dictate what is known as “potential
evaporation”, which is the maximum rate of evaporation from a pure water surface.
Figure II.1 shows the shape of the drying curve during evaporation, which comprises
three stages. During the first stage, the evaporation rate is relatively constant, and thesoil can fully deliver water to keep the potential rate, yet remains at or near saturation(actual evaporation = potential evaporation). The second stage commences when thesoil surface starts to de-saturate and can no longer supply water at the rate dictated byatmospheric demand (actual evaporation < potential evaporation). At this stage, thesoil hydraulic and water retention parameters start to play an important role in theevaporation process. As drying proceeds, the evaporation rate continues to decline as
more water is extracted from the soil and the contribution of the liquid-water phase toflow starts to diminish. During the third stage of evaporation, the water phasebecomes discontinuous and water flow occurs predominantly through vapour
diffusion.
1
Drying soil column
No water table

Relative evaporation

Actual evaporation
Relative evaporation =Potential evaporation

Stage 1

Stage 2
Stage 3
0
0

0.2

0.4

0.6

0.8

1

1.2

Time

Figure II.1: Drying curve during evaporation

II.1

Significance of Suctions at the Soil Surface

HYDRUS does not model vapour flow, but uses a mathematical scheme that mimics
the stages of evaporation. In this section we briefly review the significance of soil
suctions at the evaporating soil surface, and how they relate to vapour flow andrelative humidity.
Based on a mass transfer equation proposed by Dalton in 1802 (reported by Gray,(1970)), actual evaporation Ea from a soil surface may be estimated using thefollowing formula (Campbell, 1985), which estimates the actual evaporative flux froma soil surface in response to a gradient in the relative humidity between the soil andthe atmosphere:

Modelling variably saturated flow with HYDRUS-2D

Appendix II

II.2

Ea =

Hs − Ha
Ep
1− Ha(II.1)

where Ep is the potential or pan evaporation rate, Hs is the humidity at the soil surface
calculated using Eq. (II.2), and Ha is the atmospheric humidity. Relative humidity isrelated to soil’s total suction as follows (Edlefson and Anderson, 1943):⎛ M ψ⎞
H r = exp⎜ w ⎟⎝ ΘRρ w ⎠(II.2)

where Mw is the mass of a mole of water (0.018 kg/mol), ψ is the soil-water potential
in Pa (Pa is Newton/m2; Newton=J/m; 1 kPa=1000 pa= 10 cm water pressure head), R
is the gas constant (8.3143 J/mole K), Θ is the absolute temperature (K; = oC+273),
and ρw is the density of water (998 kg/m3 at 20oC). Assuming a mean temperature of
20 oC, a total suction of 300 m corresponds to a 97.8% relative humidity. Equation
II.2 is plotted in Figure II.2.
1

Relative humidity

0.8

0.6

0.4

0.2

0
0

10000

20000

30000

40000

Total suction (-m)

Figure II.2: Relation of total suction to relative humidity

Figure II.3 shows a numerical simulation of evaporation with vapour flow. During the
1st stage of evaporation, flow is mainly in the liquid phase. Stage 2 starts when thesurface dries and can no longer supply moisture at the potential rate (Time>32 hrs);
the vapour component then rapidly increases. Notice how the suctions significantly
increase during this stage (Figure II.4; Time>32 hrs). Rassam and Williams (1999b)
have shown that the maximum rate of change of the surface relative humidity occurs
at a Suction of 300 m (3,000 kPa).

Modelling variably saturated flow with HYDRUS-2D

Appendix II

II.3

30

0.8

Liquid flux (mm/day)

Vapour flux

0.6

20

15

0.4

10

Vapour flux (mm/day)

Liquid flux

25

0.2
Second stage of evaporation

First stage of evaporation

5

0
10

Time (hrs)0100

Figure II.3: Liquid- and vapour-phase fluxes during evaporation
1.0E+04

Matric suction (m)1.0E+03
Maximum slope
1.0E+02
Second stage of evaporation

First stage of evaporation

1.0E+01

1.0E+00
10

100

Time (hrs)

Figure II.4: Development of surface suctions during evaporation

Wilson et al. (1997) experimentally showed that regardless of soil texture, the actual
evaporation rate from a soil surface starts to drop below the potential rate when thesurface suction reaches 300 m (3,000 kPa, a relative humidity of about 98%).

II.2

Concept of hCritAHYDRUS implements a scheme whereby the actual evaporative flux remains equal tothe potential flux up to some user-defined suction called “hCritA”. This value isspecified in the “Variable Boundary Condition” as a positive value that represents
suction (negative head) using the unit that was adopted in the simulation (m, cm, ormm). This value may vary with soil type and could have a significant effect on thesimulation results.
Project “Evap” simulates evaporation from a 3-m deep soil profile with a constanthead boundary at the bottom. This project is identical to projects ‘Evap1’ and ‘Evap2’(CD..\HYDRUS Manual Projects\Sect-4). Various combinations of soil types andhCritA values were used; results are shown in figures II.5, II.6, and II.7. The time atwhich the evaporative flux falls below potential is hereby termed Tcrit. In this analysis

Modelling variably saturated flow with HYDRUS-2D

Appendix II

II.4

we assume that the suitable hCritA is that value beyond which no significant change
in Tcrit occurs.
1

LOAMY
SAND

hCritA = 100 m
hCritA = 500 m

0.8
Evaporative flux ratio

hCritA = 1,000 m
0.6

0.4

0.2

hCritA
0
1

10

100

Time (days)

Figure II.5: Effect of hCritA on evaporative flux for loamy sandFigure II.5 shows that the effect of hCritA is marginal for a loamy sand (a fairly
coarse-texture soil). The trends for values of 100 m and 500 m are almost identical.
Figure II.6 shows that for a silty soil, a higher value is warranted. Figure II.7 indicatesthat a clayey (fine-textured) soil is the most sensitive to hCritA. Table II.1 shows therecommended values for several soil classes.
Notice the effect of soil type on Tcrit, where silt continues to provide flux at potential
for periods much longer than the two other soil types.
Table II.1: hCritA for various soil classes
Soil class
hCritA (m)
Sandy
500
Silty
1,000
Clayey3,000

1hCritA = 100 m
hCritA = 500 m

0.8
Evaporative flux ratio

hCritA = 1,000 m

0.6

0.4

0.2

SILT
0
1

10
Time (days)

100

Figure II.6: Effect of hCritA on the evaporative flux for a typical silt

Modelling variably saturated flow with HYDRUS-2D

Appendix II

II.5
1hCritA=100 m
hCritA=1,000 m

0.8
Evaporative flux ratio

hCritA=3,000 m

0.6

0.4

0.2

CLAY
0
1

10

100

Time (days)

Figure II.7: Effect of hCritA on the evaporative flux for a typical clay

Modelling variably saturated flow with HYDRUS-2DII.6

Appendix II

Modelling variably saturated flow with HYDRUS-2D

Appendix III

III.1
APPENDIX III
ROOT WATER UPTAKE

III.1 Root Water Uptake Reduction Models
The water uptake reduction models define the manner in which transpiration is
reduced below potential when the soil is no longer capable of supplying the amount ofwater required by the plant under the prevailing climatic conditions. There are twoalternative models in HYDRUS: the Feddes model (Feddes et al., 1978) and an Sshaped model (van Genuchten, 1987). The former is more commonly used.
Feddes Model Parameters
The Feddes model at any point in the root zone assigns plant root water uptake rates
according to the local soil water pressure head. The meaning of the Feddes’
parameters is demonstrated in Figure III.1.
P0: Value of the pressure head below which roots start to extract water from the soil.
Popt: Value of the pressure head below which roots extract water at the maximum
possible rate (potential transpiration).
P2H: Value of the limiting pressure head below which roots no longer extract water at
the maximum rate (assuming a potential transpiration rate of r2H).
P2L: As above, but for a potential transpiration rate of r2L.
P3: Value of the pressure head below which root water uptake ceases (usually taken at
the wilting point).
A database of suggested values for different plants is provided based on studies by
Wesseling (1991) and Taylor and Ashcroft (1972).

1

Optimaluptake0.4

Water logging .

a(h)

0.6

Wilting .

0.8

0.2

0
-16

P3

P2 L

P2 H

POpt P000

Pressure head

Figure III.1: The Feddes root water uptake reduction model

Modelling variably saturated flow with HYDRUS-2D

Appendix III

III.2

r2H: Potential transpiration rate (L/T) (currently set at 0.5 cm/day).
r2L: Potential transpiration rate (L/T) (currently set at 0.1 cm/day).
The above input parameters permit one to make the variable P2 a function of thepotential transpiration rate (P2 presumably decreases at higher transpiration rates).
HYDRUS currently implements a linear interpolation scheme (defined in the
HYDRUS Technical Manual).

S-Shaped Model Parameters
van Genuchten (1987) proposed an S-shaped function to describe root water uptake
reduction, which is applicable to water stress as well as salinity stress:a (h ) =

1
⎛ P ⎞
1+ ⎜
⎟
⎝ P50 ⎠

(III.1)

P3

where P3 is an experimental constant and P50 is the pressure head at which local
uptake rate is half the potential rate. Figure III.2 below shows the S-shaped function
for a case where P50=-800 cm and P3=3.

1
P50 = -800 cm; p3 = 3

0.8

a(h)

0.6
p50 = 50% reduction in root uptake rate

0.4

Head at which 50%
reduction occurs = -800 cm

0.2

0
-2000

-1500

-1000

-500

0

Pressure head (cm)Figure III.2: S-shaped root water uptake reduction model

Modelling variably saturated flow with HYDRUS-2D

Appendix III

III.3
III.2 Root Density

Relative root densities are assigned to each node in the root zone. Values are either
manually assigned to individual nodes or HYDRUS assigns it using a root distributionfunction. Figure III.3 shows an example of a root distribution along 12 equally spaced
nodes representing a 1.2-m deep root zone.
Root density
0

1

2

3

4

5

6

7

8

9

10

0

0.2

Depth (m)0.4

is theshaded
small area
shadedareadotted(inclinedlines)
AnAisn small(inclinedlines)dottedisassociated
with
root
water
uptakeassignedtoananalogous to root water uptake assigned to an
node
individual
individual
node

An

At

0.6

0.8
AtA
ist large(horizontal
solid lines) issolid
analogous
total root water
is theshaded
largearea
shadedarea (horizontal
lines)toassociatedwithuptake
total
1

maximum root water uptake

1.2

Figure III.3: Root distribution and water uptake
Referring to Figure III.3, root density (x-axis) is the number assigned to each node(values provided in “Water Uptake Distribution”); when plotted versus depth, thisresults in a root distribution. Integrating this shape gives the total area At, which isinternally normalized by HYDRUS to be equal to one. The area is subsequently
discretized into sub-areas (An) associated with each individual node. The root water
uptake assigned to each node is:Nodal water uptake = total transpiration × An / At

Modelling variably saturated flow with HYDRUS-2D

Appendix III

III.4
III.3 Root Distribution Functions

Another way of defining a root distribution is by assigning root distribution
parameters that belong to the following root distribution function built into HYDRUS(Vrught et al., 2001):
1p

*
z
1⎡
z ⎤ − λ zm z −zβ(z ) = ⎢1 − ⎥ e
λ ⎣ zm ⎦(III.2)( ) λ1 ⎡⎢1 − zz ⎤⎥β z* =⎣

m(III.3)⎦where β(z) denotes the dimensionless spatial root distribution as a function of depth z;
zm is the maximum rooting depth; λ, Pz (parameter A in HYDRUS), and z* (depth of
maximum potential uptake ) are fitting parameters. The value of Pz is set to unity forz>z*. Figures III.4 and III.5 demonstrate the effects of varying z* and Pz for the casewhere λ=1 and zm=1. When λ=1, the root density at depth z*=1-z/zm (see Figures III.4
and III.5). For z>z*, the root distribution decays exponentially. The parameter Pz
controls the root density at depths z<z*.

Root density
0

0.2

0.4

0.6

0.8

1

0
z*
0.2

1-z*/zm
Depth

0.4
Pz=0
Pz=2

0.6

Pz=10
0.8
zm = 1z* = 0.2
λ=1
1

Figure III.4. Effect of Pz on root distribution for λ=1, zm=1, and z*=0.2

Modelling variably saturated flow with HYDRUS-2D

Appendix III

III.5

Root density
0

0.2

0.4

0.6

0.8

1

0

0.2
z*

Depth

0.4
Pz=10
0.6

Pz=2
1-z*/zm
Pz=00.8
zm = 1z* = 0.6
λ=1
1Figure III.5. Effect of Pz on root distribution for λ=1, zm=1, and z*=0.6
Figure III.6 shows the effect of the parameter λ, which scales the root density at depthz* (see Equation III.3). The effect of parameter Pz remains restricted to depths z<z* as
was demonstrated in Figures III.4 and III.5.
Root density
0

0.5

1

1.5

2

2.5

3

3.5

4

0
Lambda=0.1
Lambda=0.5

0.2z*Lambda=1

Depth

0.4

0.6

0.8
zm = 1
Pz = 2z* = 0.6
1

Figure III.6. Effect of λ on root distribution for zm=1 and z*=0.6

2-Dimensional Root Distribution
A radial root distribution may be implemented using a model similar to that depicted
by Equation II.1 (Vrught et al., 2001):r *⎡r ⎤ − rm r − rβ(r ) = ⎢1 − ⎥ e⎣ rm ⎦

p(III.4)Modelling variably saturated flow with HYDRUS-2D

III.6

Appendix III

where δ, rm and Pr have the same meaning in the radial (r) direction as λ, zm and Pz in
Equation II.1.
The 2-dimensional root distribution β(r,z) is now simply the product of Equations
III.2 and III.4.
When the check box “Horizontal Distribution” is selected as shown below, the user is
prompted to enter the required input parameters for a horizontal (radial) root
distribution. The figures below show the result of varying the parameters A and Delta(or Pz and δ).

Modelling variably saturated flow with HYDRUS-2D

Appendix IV

IV.1
APPENDIX IVSCALING FACTORS
IV.1 Background

HYDRUS-2D implements a scaling procedure designed to simplify the description of
spatial variability of the unsaturated soil hydraulic properties in the flow domain. The
code assumes that the hydraulic variability in a given area can be approximated bymeans of a set of linear scaling transformations 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 by Miller and Miller[1956].
Miller and Miller (1956) stated that it is possible to obtain detailed similitude ofinterface shapes and microscopic flow patterns between two media whose solid
geometries differ only by a constant magnifying factor. Two such media are called‘similar’ media, by analogy to the familiar term ‘similar’ triangles. A pair of similar
media in similar states is illustrated in Figure IV.1. When each of these geometries isreduced, i.e., expressed in terms of a characteristic length, λ, the resulting reduced
geometries are identical. The two similar media shown in Figure IV-1 differ only inthe scale of their internal microscopic geometries; they have equal porosities.

λm

λr

Soil at location ‘r’
Reference soil

Figure IV.1: Illustration of two similar media
The similar media concept allows results, either experimental or computed, of soil
water behaviour in one soil to be used to describe the behaviour in another by
employing reduced variables defined in terms of the macroscopic characteristic
length. The purpose of scaling is to simplify the description of statistical variations insoil hydraulic properties. By using this simplification, the pattern of spatial variation
is described by means of a set of scale factors αr relating the soil hydraulic properties
at each location r to a representative mean or reference soil (Hopmans, 1987). The
scaling factor is defined as:

αr =

λr
λm

(IV.1)

Modelling variably saturated flow with HYDRUS-2D

Appendix IV

IV.2

where r= 1, 2, ….. , R denotes location. The soil water retention and hydraulicconductivity functions at given water contents at any location r are related to a meanor reference hm and Km such that for each pressure head:hm
αr(IV.2)

K r = K m α 2r

(IV.3)

hr =and each hydraulic conductivity:

Since different soils generally do not have identical porosity values, h and K are
written in terms of degree of saturation S.
Hopmans (1987) reviewed in detail various methods of scaling. This is ademonstration of his Method II, which is derived from Peck et al. (1977). To evaluate
the scaling factor at any location, we need to calculate the mean values for both thepressure head and conductivity. These values are given by:⎧⎪ R ⎡ 1 ⎤ ⎫⎪h m (Si ) = R ⎨ ⎢⎥⎬⎪⎩ r =1 ⎣ h r (S i ) ⎦ ⎪⎭∑−1

(IV.4)andR⎤
1 ⎡K m (Si ) = 2 ⎢K r (S i ) ⎥R ⎣⎢ r =1⎦⎥∑

2(IV.5)

Equations (IV.4 and IV.5) evaluate the average value (pressure head or conductivity)
over all locations r, for any saturation level (Si). The scaling factors are thencalculated as follows:
α hr =h m (Si )1
SL i =1 h r (Si )(IV.6)

α kr =K r (Si )1
SL i=1 K m (Si )(IV.7)

SL

∑

SL

∑

where SL denotes the different pressure steps at location r; the scaling factors areaverages over all levels of saturation (SLs).
Vogel et al. (1991) demonstrated several cases where the concept of scaling may beutilised in numerical modelling applications. We summarise here one of their
exercises:
1. 20 undisturbed soil cores were obtained from the field.
2. Cumulative outflow curves were obtained during pressure desorption.
3. 20 inverse modelling runs were carried out for each of the outflow curves toobtain the soil hydraulic functions of each soil.

Modelling variably saturated flow with HYDRUS-2D

Appendix IV

IV.3
Alternatively:

1. Scale the outflow curves and obtain a set of scaling parameters for eachsample.
2. Conduct 1 inverse modelling run for the reference outflow curve to obtain thesoil hydraulic functions of the reference soil.
3. Use the scaling factors to obtain the soil hydraulic functions of each soil.
Results using both techniques compared relatively well. The latter approach is a timesaver since inverse numerical runs are iterative and can be very time-consuming.

IV.2 Stochastic Scaling Factors in HYDRUS
HYDRUS has an option for assigning stochastic scaling factors. This option can beused to:• Implement the above-described scaling concepts in a random manner.
• Assign random hydraulic conductivities to the domain. The conductivity ofeach node will then be multiplied by the scaling factor, which randomly varies
from node to node, thus resulting in a grid with randomly variableconductivities.
The stochastic distribution option requires three input parameters for each scaling
factor:1. Standard deviation: Indicates the extent of variation of the scaling factors; thehigher this value the wider the range over which the scaling factors vary(larger difference between minimum and maximum values).
2. Correlation length in x-direction: A number that indicates the manner in which
the scaling factor changes in the x-direction.
3. Correlation length in z-direction: As above for the z-direction.
The meaning of correlation length:
The measured values of any variable are correlated up to a certain distance. Thismeans that values close to a given point have a large probability to have similarvalues. The correlation length indicates the distance to where those values are fairly
similar, or correlated. The correlation length can be different in different directions(x-horizontal and z-vertical). For example, one would expect that the correlation
length in the vertical direction to be relatively small because of layering, while it
could be much larger in the horizontal direction. The correlation length can bedifferent for different properties.
The correlation length may simply be defined as “the distance over which asignificant correlation exists (Kutilek and Nielsen, 1994).
The correlation length L in a 1-dimensional transect (x or z) is commonly defined as:

Modelling variably saturated flow with HYDRUS-2D

Appendix IV

IV.4

r = ro e−xL(IV.8)

where ro represents the base correlation coefficient taken equal to 1, which means thatat the measured point itself (x=0), the correlation coefficient is equal to 1 (scaling
factor multiplier=1). At a distance equal to the “Correlation Length, L”, r is equal to e1(0.3678). Figure IV.2 shows the manner in which L affects the correlation
coefficient (and hence the scaling factors). Note that the dotted line intersects each of
the three curves at a distance equal to L.
1
L=2.0

Values change slowly awayfrom the reference point

0.8

L=1.0

Correlation coefficient

L=0.25
0.6

0.4

-1

e

Values change rapidly awayfrom the reference point

0.2

0
0

0.5

1

1.5

2

Distance from a reference point where a parameter is measured

Figure IV.2: The concept of Correlation Length
Effect of standard deviation on distribution of scaling factors:
The standard deviation affects the extent of variation in the scaling factors, where
higher deviations lead to more variation in the scaling factors. The figures below
demonstrate the impact of a 10-fold increase in the standard deviation. Note that when
the standard deviation is increased, the range of scaling factors follows course (seerange of colour palette bars in Figure IV.3; 0-4 on the right as compared to 0.6-1.3 on
the left).

Modelling variably saturated flow with HYDRUS-2D

Appendix IV

IV.5
Effect of correlation length on distribution of scaling factors:

The correlation length (L) is a measure of how similar the scaling factors are in aparticular direction. A high Lx means that the scaling factor maintains its value for along horizontal distance away from the reference point, that is, it does not change
rapidly. Conversely, a low Lx means that it changes at a small horizontal distance.
Figure IV.4 shows the effect of different L values in the x- and z-directions. When
Lx/Lz=0.1, the contours are vertically stretched, and when Lx/Lz=10, the contours arehorizontally stretched.

Standard
deviation=1.0

Standard
deviation=0.1

0.7

0.8

0.9

1.0

1.1

1.2

1.3

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Figure IV.3: Effect of standard deviation on
distribution of scaling factors

Lx=1; Lz=10

Lx=10; Lz=1

Figure IV.4: Effect of correlation length on
distribution of scaling factors

Modelling variably saturated flow with HYDRUS-2DIV.6

Appendix IV

Modelling variably saturated flow with HYDRUS-2D

Appendix V

V.1
APPENDIX V
INVERSE SOLUTION

This process is usually referred to as “Model calibration”, or “Inverse parameter
estimation”. The technique is used in many applications, some of which are listed in
the Table V.1.
Table V.1: Example applications for inverse solution
Experiment/application
Calibration data
Goal (Key Prediction)
Field drainage
Cumulative flow, pressure Field-scale Ksat
heads
Disc infiltrometer
Cumulative inflow, pressure Water retention andheads, water contents
hydraulic conductivity
parameters
Cultivated land
Neutron probe data (water Leakage (recharge to watercontents)
table)
Field tracer experiment
Solute concentration in
Ksat, flow rates
injection/capture wells
It is crucial that a correct conceptualisation of the problem be established first(involving boundary conditions and initial conditions); otherwise the inverse solution
will not yield a result or may yield physically unrealistic parameters. It is themodeller’s responsibility to distinguish between “realistic” and “unrealistic”
parameters.

V.1

Inverse Method

The ultimate aim of model calibration is to adjust input parameters such that themodel predictions fit observations in an optimal manner. The so-called “best fit” isassessed objectively by using a quantitative measure of the quality of the fit bymeasuring the errors involved. Minimising the sum of the squares of the residuals is
the most common procedure and is adopted in HYDRUS:
Sr =∑ (q p,i − q o,i )2 = ∑ R i 2
n

n

i =1

i =1

(V.1)

where n is the number of calibration data points (e.g., measured water contents), qp,i isthe ith predicted value, and qo,i is the ith observed value. The difference between
predicted and observed values in Equation V.1 is called the residual Ri (see FigureV.1). Each point in a calibration data set can be assigned a weighting factor
depending upon the degree of confidence or the significance associated withindividual calibration points (provided by the user; wij). The weighting factors giveindividual points higher or lower significance (or weight) relative to the total sum Sr.
The residuals are then called “weighted residuals”. HYDRUS implements anotherweighting factor vj for each data set to allow for differences in absolute values ofvarious data sets, such as pressure heads and water contents. Heads are measured in
cm and hence might have higher values when compared to volumetric water content.

Modelling variably saturated flow with HYDRUS-2D

Appendix V

V.2

Pressure head contributions to Sr are generally hence much higher than watercontents, which may lead to a biased calibration.
4

Data (e.g. water content)

Observed
Predicted

R3

R5

3
R2
R6

2
R4

R7

1
Residual (R1)

0
0

1

2

3

4

5

6

7

8

Time

Figure V.1. Meaning of residuals

Another contribution to bias is the use of different unit systems; for example, thehydraulic conductivity could have a very low value when expressed in m/s but bethousands of times higher when expressed in cm/day. HYDRUS overcomes thisproblem by introducing a weighting factor associated with each data type. The term
representing the sum of the weighted residuals from various data types hence
becomes:m

S wr =n∑ ∑w R
vj

j=1

i, j

2
i

(V.2)

i =1

where m is the number of data types used, vj is the weighting factor associated with adata type provided by HYDRUS, and wij is the weighting factor associated withindividual data points provided by the user. The definition of the weighting factors v j
and wi, j is found in the HYDRUS Technical Manual (p.72).
Differences between measured and predicted soil hydraulic parameters (such asretention parameters and the saturated hydraulic conductivity), and having known
hydraulic parameters (also referred to as prior information) form another two terms
that look like Equation V.2. The sum of the three terms makes what is known as the
Objective Function (see HYDRUS Technical Manual, p.71, Eq. 7.1). HYDRUS uses
the Marqquardt-Levenberg optimisation algorithm to minimise the objective function,that is, to come up with parameters that produce the minimum difference between theobserved and predicted values.

V.2 Statistical Issues
V.2.1 Distribution of Residuals
Figure V.2 shows an ideal case where the distribution of the weighted residuals is
random and centred around a zero value.

Modelling variably saturated flow with HYDRUS-2D

Appendix V

V.3

Weighted residuals

0.5

0

-0.5
0

5

10

15

20

25

30

Data

Figure V.2. Random distribution of weighted residuals

By contrast, Figure V.3 shows a case where the weighting process is inadequate. The
data between 0 and 16 are under-weighted; they may represent concentration datathat have low values, whereas data in the range 20-30 may represent much higher
pressure head values.

Weighted residuals

0.5

0

-0.5
0

5

10

15

20

25

30

Data

Figure V.3. Distribution of weighted residuals suggesting inadequate weighting

Figure V.4 shows another case with a systematic bias in the distribution. This casemay indicate an erroneous conceptualisation of the problem (e.g. an incorrectboundary condition).

Weighted residuals

0.5

0

-0.5
0

5

10

15

20

25

30

Data

Modelling variably saturated flow with HYDRUS-2D

Appendix V

V.4

Figure V.4. Distribution of weighted residuals with systematic bias

V.2.2 Confidence Limits
The lower and upper bounds of the confidence limits indicate the sensitivity level of amodel to any particular parameter measured by the response of the objective function.
Figure V.5 shows a case where the model is very sensitive to a parameter in that asmall change in the parameter values causes a dramatic change in the objectivefunction.
10High sensitivity to parameter

Objective function

8

Confidence limits:
Parameter range that produces
acceptable calibration

64

95% Confidence limits:
Acceptable departure from theminimum objective function

2
lower
bound

upper
bound

0
0

1

2

3

4

5

6

7

8

9

10

Parameter value

Figure V.5: Model with high sensitivity to parameter change

On the other hand, Figure V.6 shows a model with very low sensitivity to a particularparameter. Significant changes in the parameter value now cause only a very smallchange in the objective function. In such cases one may consider eliminating theparameter from the optimisation scheme.

Modelling variably saturated flow with HYDRUS-2D

Appendix V

V.5

10

8

Objective function

Low sensitivity to parameter
6
Minimum of objectivefunction not well defined42
Acceptable departurefrom minimum

Wider parameter range that
produces acceptable calibration

0
0

1

2

3

4

5

6

7

8

9

10

Parameter value

Figure V.6: Model with low sensitivity to parameter change

V.2.3 Goodness of Fit and Errors
The goodness of fit is measured using the r2 values. The equation for r2 is found in the
HYDRUS Technical Manual (p.73, Eq. 7.3). The r2 indicates that the best fit isobtained when its value approaches or is equal to 1. However, an r2 close to 1 does
not always guarantee a good fit. Users should always investigate this problem. It isbest to always plot the observed data along with the model predictions for a visualinspection. Figure V.7 demonstrates a case where two regressions have r2 values ofunity, but one of them has the wrong slope; note that only the 1:1 (45o) line represents
a perfect fit. This line is referred to as the “Line of Perfect Fit”.
10
Perfect calibration
82

Predicted

r =1 for both series
6

1
1

4
Bad calibration
wrong slope
2

0
0

2

4

6

8

10

Observed

Figure V.7: Goodness of fit

The standard error of the estimated parameter quantifies the spread of the model
predictions around the observed values, and is defined as follows:

Modelling variably saturated flow with HYDRUS-2D

Appendix V

V.6
S (b j ) =Srn − np(V.3)

where np is the number of unknown parameters to be fitted. The T-value provides anabsolute measure for the deviation:
T=

bj

( )

(V.4)

S bj

where bj is the final estimate of the parameter.

V.2.4 Correlation Matrix
HYDRUS produces a correlation matrix, which specifies the degree of correlation
between the fitted parameters. The correlation matrix quantifies changes in model
predictions caused by small changes in the final estimate of a particular parameter,
relative to similar changes as a result of changes in the other parameter (HYDRUS
Technical Manual). The matrix as reported in the “Fit.out” output files, is shown in
Table V.2:

Table V.2: Correlation matrix
Cond
Alpha
n
Cond

Alpha
1
-0.2160.98

n1
1

The correlation matrix indicates whether the parameters are correlated or not, where 1
indicates perfect positive correlation and –1 indicates perfect negative correlation.
The matrix shown above indicates that there is a significant correlation only between
Alpha and the saturated conductivity. The matrix diagonal has always values of unity
because it refers to correlation with the parameter itself. Figure V.8 shows anexample of two highly correlated parameters; notice that a straight line was fitted tothe data with an R2 of 0.952.

Modelling variably saturated flow with HYDRUS-2D

Appendix V

V.7
2.5
2

r = 0.952

High positive correlation

Parameter 2

2

1.5

1

0.5

0
0

1

2

3

4

5

6

7

8

9

10

Parameter 1

Figure V.8: Parameters correlation

Correlated parameters are often closely associated with non-unique solutions. Morethan one set of parameters may then produce a good fit to the observed trends. Thisphenomenon results in a high degree of predictive uncertainty. In cases where a highdegree of correlation does exist, the number of optimised parameters must be
decreased, or new types of calibration data must be added. For example, if you are
only monitoring pressure heads, try to also include flow rates. We emphasize that
increasing the number of calibration data in the set that produced the highly correlatedparameters by itself may not solve the problem.

V.2.5 Weighting Parameters
The number of the calibration data, and their absolute values, can have a profound
impact on the objective function. For example, if you use time units as hours, theabsolute value of the hydraulic conductivity will be 60 times higher compared towhen a time unit of minutes is used. The absolute difference between the observed
and predicted values hence will be higher also, and thus will have a greater impact on
the objective function (the sum of all differences). This means that you will get
different results depending upon what unit is used, which seems illogical. HYDRUS
uses two approaches to manage this problem: weighting by mean ratio and weighting
by standard deviation. For more details, refer to the HYDRUS Technical Manual P.
72.

Modelling variably saturated flow with HYDRUS-2D

Appendix V

V.8

We demonstrate how weighting factors avoid the problem introduced by changing theunits. Let us copy project ‘Inver-b5’ and name this project ‘Weight’ (CD..\HYDRUS
Manual Projects\Appendix). We change the time units to hours and re-run the
project.

The following information is copied from the output file “Fit.out”.
Refer to Table V.3 (the values under column ‘Weight’). Notice the high weighting
factors for data points 8 and 9. These are hydraulic conductivity measurements in
cm/minute (time units is minutes).
Table V.3: Inverse data from Project ‘Inver-b5’
Obs
Time
QUANTITY
Type
1
33
0.179
2
2
66
0.211
2
3
100
0.246
2
4
133
0.271
2
5
166
0.282
2
6
200
0.286
2
7
200
-46
1
8
-92
3.35E-04
69-52
1.35E-03
6
10
-306
0.132
5

Position
1
1
1
1
1
1
1
1
1
1

Weight
7.020282
7.020282
7.020282
7.020282
7.020282
7.0202822
2786.627
2786.627
14.04056

0.93352
3.761946

The data in Table V.4 are from the same project but with time unit system changed tohours (absolute value of conductivity increased 60 folds). Notice that the weighting
factors for data points 8 and 9 have decreased by exactly 60 fold. The numbers in bold
are the products of the measured value and the weighting factor; notice that they are
equal to the corresponding numbers in Table V.3. These are the numbers that will beincorporated into the objective function. All other weighting factors have remained
the same since they are independent of time (the unit we changed).
Table V.4: Inverse data from Project ‘Inver-b5’
Obs
Time
QUANTITY
Type
1
0.55
0.179
2
2
1.1
0.211
2
3
1.667
0.246
2
4
2.217
0.271
2
5
2.767
0.282
2
6
3.34
0.286
2
7
3.34
-46
1
8
-92
0.02
6
9
-52
0.081
6
10
-306
0.132
5
V.3
Intelligent Data Acquisition

Position
1
1
1
1
1
1
1
1
1
1

Weight
7.020282
7.020282
7.020282
7.020282
7.020282
7.0202822
46.44379
46.44379
14.04056

0.928876
3.761947

Modelling variably saturated flow with HYDRUS-2D

Appendix V

V.9

It is extremely important that the calibration data be as representative as possible ofany spatial and temporal variations in a model proposed for calibration. Calibration
data are either obtained from laboratory or field experiments. A soundconceptualisation and physical understanding of the system should precede theinstrumentation stage. Preliminary predictive modelling enables a modeller to identify
what type of data are needed (pressure heads, water contents, etc.), the location where
they are mostly needed (spatial distribution), and the frequency at which they shouldbe acquired (temporal distribution). Unless this process is carefully undertaken, one
might either end up with redundant data, or have data gaps in space and/or time that
could result in a biased calibration. The following simulation will shed some light on
how to obtain a useful calibration data set when simulating a drainage problem.
Project ‘Dr-evap’ (CD..\HYDRUS Manual Projects\Appendix) involves a finetextured soil block 10-m wide and 1-m deep that drains into a ditch in two
consecutive stages in the absence and presence of evaporation, respectively. The
pressure head in the saturated zone is measured, and the negative pressure head in theunsaturated zone is monitored at a depth of 10 cm.
The predictive modelling process reveals the following:
1. Figure V.9 shows that during drainage alone (day 1), pressure head changes
are less significant away from the drain. Hence, observation wells should beplaced with a higher density close to the drain.
2. Figure V.9 shows that 1 m from the drain an almost instantaneous decrease ofabout 15 cm occurs in the water table level. This suggests having a higher
sampling frequency at this location during this time period.
3. Figure V.9 shows that in the presence of evaporation (rate=1 cm/day), thewater table behaves significantly differently (water table will decline awayfrom the drain at almost the same rate as close to it). This means that thedominant processes are different in high-evaporation, arid areas.

100
9.8 m from drain
5.2 m from drain

Head in saturated zone (cm)90

1 m from drain
80
Drainge only
70

60

50
Drainge & evaporation
40
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (days)

Figure V.9: Pressure head data from the saturated zone at various distances from thedrain

Modelling variably saturated flow with HYDRUS-2D

Appendix V

V.10

4. Figure V.10 shows that during drainage alone, monitoring pressure heads(with tensiometers) in the unsaturated zone alone does not portray the full
picture. However, when evaporation is modelled, the data are more valuable.

Head in unsaturated zone (cm)0-200
Drainge only
-400
9.8 m from drain
5.2 m from drain-600

1 m from drain
Drainge & evaporation

-800
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (days)

Figure V.10: Head data from the unsaturated zone at various distances from the drain

5. Figure V.11 shows that the cumulative seepage flux neither reflects thesubtleties in the groundwater dynamics, nor the pressure heads in theunsaturated zone. Hence, we do not expect to obtain a good calibration fromdrainage data alone. However, those data may still be valuable when used
together with pressure head measurements.

Figure V.11: Cumulative
seepage outflow

Modelling variably saturated flow with HYDRUS-2D

Appendix IV

IV.1
APPENDIX IVSCALING FACTORS
IV.1 Background

HYDRUS-2D implements a scaling procedure designed to simplify the description of
spatial variability of the unsaturated soil hydraulic properties in the flow domain. The
code assumes that the hydraulic variability in a given area can be approximated bymeans of a set of linear scaling transformations 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 by Miller and Miller[1956].
Miller and Miller (1956) stated that it is possible to obtain detailed similitude ofinterface shapes and microscopic flow patterns between two media whose solid
geometries differ only by a constant magnifying factor. Two such media are called‘similar’ media, by analogy to the familiar term ‘similar’ triangles. A pair of similar
media in similar states is illustrated in Figure IV.1. When each of these geometries isreduced, i.e., expressed in terms of a characteristic length, λ, the resulting reduced
geometries are identical. The two similar media shown in Figure IV-1 differ only inthe scale of their internal microscopic geometries; they have equal porosities.

λm

λr

Soil at location ‘r’
Reference soil

Figure IV.1: Illustration of two similar media
The similar media concept allows results, either experimental or computed, of soil
water behaviour in one soil to be used to describe the behaviour in another by
employing reduced variables defined in terms of the macroscopic characteristic
length. The purpose of scaling is to simplify the description of statistical variations insoil hydraulic properties. By using this simplification, the pattern of spatial variation
is described by means of a set of scale factors αr relating the soil hydraulic properties
at each location r to a representative mean or reference soil (Hopmans, 1987). The
scaling factor is defined as:

αr =

λr
λm

(IV.1)

Modelling variably saturated flow with HYDRUS-2D

Appendix IV

IV.2

where r= 1, 2, ….. , R denotes location. The soil water retention and hydraulicconductivity functions at given water contents at any location r are related to a meanor reference hm and Km such that for each pressure head:hm
αr(IV.2)

K r = K m α 2r

(IV.3)

hr =and each hydraulic conductivity:

Since different soils generally do not have identical porosity values, h and K are
written in terms of degree of saturation S.
Hopmans (1987) reviewed in detail various methods of scaling. This is ademonstration of his Method II, which is derived from Peck et al. (1977). To evaluate
the scaling factor at any location, we need to calculate the mean values for both thepressure head and conductivity. These values are given by:⎧⎪ R ⎡ 1 ⎤ ⎫⎪h m (Si ) = R ⎨ ⎢⎥⎬⎪⎩ r =1 ⎣ h r (S i ) ⎦ ⎪⎭∑−1

(IV.4)andR⎤
1 ⎡K m (Si ) = 2 ⎢K r (S i ) ⎥R ⎣⎢ r =1⎦⎥∑

2(IV.5)

Equations (IV.4 and IV.5) evaluate the average value (pressure head or conductivity)
over all locations r, for any saturation level (Si). The scaling factors are thencalculated as follows:
α hr =h m (Si )1
SL i =1 h r (Si )(IV.6)

α kr =K r (Si )1
SL i=1 K m (Si )(IV.7)

SL

∑

SL

∑

where SL denotes the different pressure steps at location r; the scaling factors areaverages over all levels of saturation (SLs).
Vogel et al. (1991) demonstrated several cases where the concept of scaling may beutilised in numerical modelling applications. We summarise here one of their
exercises:
1. 20 undisturbed soil cores were obtained from the field.
2. Cumulative outflow curves were obtained during pressure desorption.
3. 20 inverse modelling runs were carried out for each of the outflow curves toobtain the soil hydraulic functions of each soil.

Modelling variably saturated flow with HYDRUS-2D

Appendix IV

IV.3
Alternatively:

1. Scale the outflow curves and obtain a set of scaling parameters for eachsample.
2. Conduct 1 inverse modelling run for the reference outflow curve to obtain thesoil hydraulic functions of the reference soil.
3. Use the scaling factors to obtain the soil hydraulic functions of each soil.
Results using both techniques compared relatively well. The latter approach is a timesaver since inverse numerical runs are iterative and can be very time-consuming.

IV.2 Stochastic Scaling Factors in HYDRUS
HYDRUS has an option for assigning stochastic scaling factors. This option can beused to:• Implement the above-described scaling concepts in a random manner.
• Assign random hydraulic conductivities to the domain. The conductivity ofeach node will then be multiplied by the scaling factor, which randomly varies
from node to node, thus resulting in a grid with randomly variableconductivities.
The stochastic distribution option requires three input parameters for each scaling
factor:1. Standard deviation: Indicates the extent of variation of the scaling factors; thehigher this value the wider the range over which the scaling factors vary(larger difference between minimum and maximum values).
2. Correlation length in x-direction: A number that indicates the manner in which
the scaling factor changes in the x-direction.
3. Correlation length in z-direction: As above for the z-direction.
The meaning of correlation length:
The measured values of any variable are correlated up to a certain distance. Thismeans that values close to a given point have a large probability to have similarvalues. The correlation length indicates the distance to where those values are fairly
similar, or correlated. The correlation length can be different in different directions(x-horizontal and z-vertical). For example, one would expect that the correlation
length in the vertical direction to be relatively small because of layering, while it
could be much larger in the horizontal direction. The correlation length can bedifferent for different properties.
The correlation length may simply be defined as “the distance over which asignificant correlation exists (Kutilek and Nielsen, 1994).
The correlation length L in a 1-dimensional transect (x or z) is commonly defined as:

Modelling variably saturated flow with HYDRUS-2D

Appendix IV

IV.4

r = ro e−xL(IV.8)

where ro represents the base correlation coefficient taken equal to 1, which means thatat the measured point itself (x=0), the correlation coefficient is equal to 1 (scaling
factor multiplier=1). At a distance equal to the “Correlation Length, L”, r is equal to e1(0.3678). Figure IV.2 shows the manner in which L affects the correlation
coefficient (and hence the scaling factors). Note that the dotted line intersects each of
the three curves at a distance equal to L.
1
L=2.0

Values change slowly awayfrom the reference point

0.8

L=1.0

Correlation coefficient

L=0.25
0.6

0.4

-1

e

Values change rapidly awayfrom the reference point

0.2

0
0

0.5

1

1.5

2

Distance from a reference point where a parameter is measured

Figure IV.2: The concept of Correlation Length
Effect of standard deviation on distribution of scaling factors:
The standard deviation affects the extent of variation in the scaling factors, where
higher deviations lead to more variation in the scaling factors. The figures below
demonstrate the impact of a 10-fold increase in the standard deviation. Note that when
the standard deviation is increased, the range of scaling factors follows course (seerange of colour palette bars in Figure IV.3; 0-4 on the right as compared to 0.6-1.3 on
the left).

Modelling variably saturated flow with HYDRUS-2D

Appendix IV

IV.5
Effect of correlation length on distribution of scaling factors:

The correlation length (L) is a measure of how similar the scaling factors are in aparticular direction. A high Lx means that the scaling factor maintains its value for along horizontal distance away from the reference point, that is, it does not change
rapidly. Conversely, a low Lx means that it changes at a small horizontal distance.
Figure IV.4 shows the effect of different L values in the x- and z-directions. When
Lx/Lz=0.1, the contours are vertically stretched, and when Lx/Lz=10, the contours arehorizontally stretched.

Standard
deviation=1.0

Standard
deviation=0.1

0.7

0.8

0.9

1.0

1.1

1.2

1.3

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Figure IV.3: Effect of standard deviation on
distribution of scaling factors

Lx=1; Lz=10

Lx=10; Lz=1

Figure IV.4: Effect of correlation length on
distribution of scaling factors

Modelling variably saturated flow with HYDRUS-2DIV.6

Appendix IV

Modelling variably saturated flow with HYDRUS-2D

Appendix VII: HYDRUS Windows Index

VII.1

APPENDIX VII
ALPHABETICAL INDEX OF HYDRUS WINDOWS------------------------------------------------------------------------------------------------------To display click: Geometry and FEM
Mesh Editor (MeshGen-2D)/Insert Arc
Numerically ( )Function: Inserts an arc numerically.
Required Input: Coordinates of threepoints on the arc (two end points andone intermediate).
Related pages: N/A*Display is Conditional: Select ‘Tools/Arc Input Mode/Three Points’ ( )------------------------------------------------------------------------------------------------------To display click: Geometry and FEM
Mesh Editor (MeshGen-2D)/Insert Arc
Numerically ( )Function: Inserts an arc numerically.
Required Input: Coordinates of centreof arc, radius of arc, starting andending angles; check box fororientation.
Related pages: 1.17
Related examples: Project ‘Grid-1’,
page 1.16*Display is Conditional: Select ‘Tools/Arc Input Mode /Centre, Radius, and Two
Angles’ ( )-------------------------------------------------------------------------------------------------------

Modelling variably saturated flow with HYDRUS-2D

Appendix VII: HYDRUS Windows Index

VII.2
To display click: Boundary
Condition Editor (Boundary)(in Pre-Processing) ( )Function: Sets boundary
conditions, material
distribution, root distribution,
nodal recharge, scaling factors,
initial conditions, anisotropy,
subregions, and observations
nodes.
Required Input: Input is
described in individual windows.

Related pages: 1.26 to 1.43 (General); p. xiii; Section 3 and Appendix III (Root
distribution); Appendix IV (Scaling factors).
------------------------------------------------------------------------------------------------------To display click: Pressure Heads (in PostProcessing)
Function: Provides graphical display of averagepressure heads across various boundaries.
Required Input: Choice of vertical variable, which
represents the relevant boundary type.
Related pages: 1.50; 2.1, 2.2.

------------------------------------------------------------------------------------------------------To display click: Water Boundary Fluxes (in PostProcessing)
Function: Provides graphical display of fluxes acrossvarious boundaries.
Required Input: Choice of vertical variable, which
represents the relevant boundary type.
Related pages: 1.50; 2.1,2.2.
Related examples: Project ‘Mbal-2’, page 2.8; Project‘Mbal-3’, page 2.12. These examples demonstrate theuse of the raw flux data from the relevant output files.

Modelling variably saturated flow with HYDRUS-2D

Appendix VII: HYDRUS Windows Index

VII.3
To display click: Graphical Display ofResults/Right-click on a chart/Wizard
Function: Provides a variety of charting
options.
Required Input: Gallery/Select a chart type;
Style/Select style; Layout/change title andfootnotes; Axes/changes axes titles.
Related pages: 1.48------------------------------------------------------------------------------------------------------To display click: Geometry and FEM
Mesh Editor (MeshGen-2D)/Insert Circle
Numerically ( )Function: Inserts a circle.
Required Input: Coordinates of threepoints on the circle.
Related pages: N/A*Display is Conditional: Select ‘Tools/Circle Input Mode /Three Points’ ( )------------------------------------------------------------------------------------------------------To display click: Geometry and FEM Mesh Editor(MeshGen-2D)/Insert Circle Numerically ( )Function: Inserts a circle.
Required Input: Coordinates of centre and value forradius.
Related pages: 1.24
Related examples: Project ‘Grid-3’, page 1.22
*Display is Conditional: Select ‘Tools/Circle Input Mode /Centre and Radius’ ( )-------------------------------------------------------------------------------------------------------

Modelling variably saturated flow with HYDRUS-2D

Appendix VII: HYDRUS Windows Index

VII.4

To display click: Boundary Condition Editor(Boundary)/Options/Colour Palette
Function: Provides options for colour displays;default is 32 colours.
Required Input: Select from drop-down list.
------------------------------------------------------------------------------------------------------To display click: Graphical Display of Results(Graphics)/Options/Color Palette
Function: Provides colour options to display results.
Required Input: Palette name; it is possible to createcustom palettes.
Related pages: 1.49------------------------------------------------------------------------------------------------------To display click: Boundary Condition
Editor (Boundary)/Condition/Boundary
Condition Water Flow/Constant Flux;
or LHS Water Flow Buttons/Const.
Flux
Function: Assigns a constant flux BC
for a set of selected boundary nodes.
Required Input: Value of flux; has units of L/T (e.g., cm/day).
Related pages: 1.29
Related examples: Inverse Simulations Series ‘a’, page 5.4*Display is Conditional: Select the nodes and left-click when you finish-------------------------------------------------------------------------------------------------------

Modelling variably saturated flow with HYDRUS-2D

Appendix VII: HYDRUS Windows Index

VII.5
To display click: Boundary Condition
Editor (Boundary)/Condition/
Boundary Condition Water/
Flow/Constant Head; or LHS Water
Flow Buttons/Const. Pressure
Function: Assigns a constant pressure
head BC for a set of selected boundary
nodes.

Required Input: Value of pressure head (positive or negative); has units of L(e.g., cm). When ‘Equilibrium from the lowest located nodal point’ is checked, thespecified value is assigned to the lowest selected point and hydrostatic equilibrium isassumed above that point.
Related pages: 1.27
Related examples: Projects ‘Pipe-’, page 4.12
*Display is Conditional: Select the nodes and left-click when you finish------------------------------------------------------------------------------------------------------To display click: Soil Hydraulic
Model/Next (in Pre-Processing)
Function: Provides constraints on
hysteretic parameters when using theinverse solution.
Required Input: Check appropriate boxes.
Related pages: 5.2*Display is Conditional: Select ‘Inverse Solution’ in ‘Main Processes’, and,
‘Hysteresis’ in ‘Soil Hydraulic Model’.
------------------------------------------------------------------------------------------------------To display click: Convert Output to
ASCII (in Post-Processing)
Function: Converts selected binary output
files into ASCII form.
Required Input: Check the appropriate
box.
Related pages: 1.53; 2.5-------------------------------------------------------------------------------------------------------

Modelling variably saturated flow with HYDRUS-2D

VII.6

Appendix VII: HYDRUS Windows Index

To display click: File/ProjectManager/Projects/Copy
Function: Provides copies of
HYDRUS projects.
Required Input: A new file name anddescription. The copy of the project will be saved in the same directory (workspace)by default.
Related pages: 1.2
Related examples: Project ‘Dike’, page 1.40; Project ‘Ris-st’, page 4.2------------------------------------------------------------------------------------------------------To display click: Cumulative Water Boundary
Fluxes (in Post-Processing)
Function: Provides graphical display of cumulative
flow volumes across various boundaries.
Required Input: Choice of vertical variable, which
represents the relevant boundary type.
Related pages: 1.49; 2.1,2.2.
Related examples: Project ‘Mbal-2’, page 2.8;
Project ‘Mbal-3’, page 2.12. These examples
demonstrate the use of the raw flux data from the relevant output files.
------------------------------------------------------------------------------------------------------To display click: Geometry and FEM Mesh Editor(MeshGen-2D)/View/Boundary Points/Hole (Sidebuttons)
Function: Decides whether an internal continuous
boundary is a hole (nodes on this boundary representinterface with outside world) or just an internal curve for mesh refinement purposes.
Related pages: 1.24
*Display is Conditional: Select continuous internal boundary.
-------------------------------------------------------------------------------------------------------

Modelling variably saturated flow with HYDRUS-2D

Appendix VII: HYDRUS Windows Index

VII.7
To display click: Inverse Solution Data
(in Pre-Processing)
Function: Provides an entry table forthe calibration data.
Required Input: Calibration data, data
type, temporal and spatial data details,
and weight for individual data points.
Related pages: 5.2, 5.3.
Related examples: All projects in Section 5.

*Display is Conditional: Check the box ‘Inverse Solution’ in ‘Main Processes’------------------------------------------------------------------------------------------------------To display click: Right click on any figure in PostProcessing/Edit chart data
Function: Displays the data set for the displayed
figure, provides an option to edit the data andchange the legend (unless fixed internally).
Required Input: Click on any cell and change its
value; click on the column headings and change thelegend.
Related pages: 1.46-1.47
Related examples: Project ‘1D-Infil’ page xvii.
------------------------------------------------------------------------------------------------------To display click: Boundary Condition
Editor (Boundary)/Condition/Boundary
Condition Water Flow/Deep Drainage;or LHS Water Flow Buttons/Deep
Drainage
Function: Assigns deep-drainage BC
parameters.
Required Input: ‘a’ and ‘b’ parameters
and reference groundwater level position.
Related pages: 1.31
*Display is Conditional: Select the nodes and left-click when you finish.
-------------------------------------------------------------------------------------------------------

Modelling variably saturated flow with HYDRUS-2D

Appendix VII: HYDRUS Windows Index

VII.8

To display click: Geometry and FEM Mesh Editor(MeshGen-2D)/View/Boundary Points/Density (sidebuttons)
Function: Assigns mesh densities on either side of
fixed points.
Required Input: Density on right and left hand sides of a fixed point.
Related pages: 1.19-1.21
Related example: Projects ‘Grid-1 &2’ page 1.20-1.21*Display is Conditional: Select a fixed point on a boundary in the mesh.
------------------------------------------------------------------------------------------------------To display click: Boundary Condition
Editor (Boundary)/Options/Display
Options ( ), or
Graphical Display of Results(Graphics)/Options/Display Options( )Function: Provides mesh display
options.
Required Input: Check the relevant box.
Related pages: 1.46------------------------------------------------------------------------------------------------------To display click: Boundary Condition
Editor (Boundary) /Condition/Drains/Insert/Left click to select node/rightclick to end
Function: Assigns drain BC parameters.
Required Input: Effective diameter of thedrain and the reduction factor.
Related pages: 1.35
*Display is Conditional: Select `Rectangular’ in ‘Geometry Information’.
-------------------------------------------------------------------------------------------------------

Modelling variably saturated flow with HYDRUS-2D

Appendix VII: HYDRUS Windows Index

VII.9

To display click: Geometry and FEM Mesh
Editor (MeshGen-2D)/Edit/Geometry/Change
Object Type
Function: Changes type of the geometric object.
Required Input: Select a new object type.
Display is Conditional: Select object to bechanged.
------------------------------------------------------------------------------------------------------To display click: Geometry and FEM
Mesh Editor (MeshGen-2D)/Edit/
Geometry/Object’s Points/Coordinates
Function: Changes polyline points
coordinates, adds new or deletes
existing points of the polyline
numerically.
Required Input: New x and ycoordinates of a point.
*Display is Conditional: Select Polyline to be modified.
------------------------------------------------------------------------------------------------------To display click: Geometry and FEM
Mesh Editor (MeshGen2D)/Edit/Geometry/Object’s
Points/Coordinates
Function: Changes spline point
coordinates, adds new and delete
existing points of the spline
numerically.
Required Input: New x and y coordinates of the point, Weight, and Standard
Deviation.
*Display is Conditional: Select Spline to be modified.
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VII.10To display click: Geometry and FEM Mesh
Editor (MeshGen-2D)/Edit/Geometry/Object’s
Point/Move
Function: Moves a point numerically.
Required Input: New x and y coordinates of thepoint.
*Display is Conditional: Select ‘Tools/Manipulation Mode/Numerical’.
------------------------------------------------------------------------------------------------------To display click: Geometry and FEM Mesh
Editor (MeshGen-2D)/Tools/Find, orBoundary Condition Editor(Boundary)/Tools/Find, or Graphical Displayof Results (Graphics)/Tools/Find
Function: Locates a triangle, a node, or anedge in the finite element mesh.
Required Input: Number of a triangle, a node, or an edge of the finite element mesh.

------------------------------------------------------------------------------------------------------To display click: Right click on achart/Axis/Axis/Select or double click on an axis ina chartFunction: Formats chart axes.
Required Input: Options/Line type and colour;
Grid/Line type, width, and colour; Scale/scale
extents and type; Ticks/Location and length.
Related pages: 1.51------------------------------------------------------------------------------------------------------To display click: Right click on a chart/General or
double click on a chart
Function: Formats chart.
Required Input: Hide/show Title, Legend, Footnote,and 2nd Y-axis; changes fill and frame options.

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VII.11

To display click: Right click on a chart/Footnote
Function: Formats Footnotes.
Required Input: Similar to ‘Format Legend’.

-----------------------------------------------------------------------------------------------------To display click: Right click on a chart/Legend
Function: Formats Legends.
Required Input: Backdrop/set fill and frame
options; Font/set font types and size; Location/set
the legend location inside the plot and its visibility.

------------------------------------------------------------------------------------------------------To display click: Right click on a chart/Plot
Function: Formats plots.
Required Input: Type/Chart type and series;
Location/default of specify location; Backdrop sets
fills and frames.
------------------------------------------------------------------------------------------------------To display click: Right click on achart/Series/Series/Select or double click on a data
series in a chart
Function: Formats chart series.
Required Input: Options/Hides or excludes aseries/Smooths using two splines options;
Line/changes line style, width, and colour; Markers/set style, colour, size, and width;
Statistics/plots minimum, maximum, mean, Standard deviation, and regression.
Related pages: 1.49-------------------------------------------------------------------------------------------------------

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VII.12

To display click: Right click on a chart/Title
Function: Formats chart Title.
Required Input: Similar option to ‘Format Legend’------------------------------------------------------------------------------------------------------To display click: Geometry
Information (in Pre-Processing)
Function: Selects the length units to
be used throughout the project, Typeof flow, geometry type (General uses
MeshGen-2D package), number ofsoil types, and number of layers(subregions).
Required Input: Check appropriate boxes, and provide number of materials and layers(subregions).
Related pages: 1.4
Related example: Project ‘1D-infil’ page x.
------------------------------------------------------------------------------------------------------To display click: Graphical Display of Results(Graphics)/1D-Graph/Cross Section
Function: Provides a cross section along aboundary or through the domain; displays watercontent, pressure head, or velocity profiles.
Required Input: Select one of the threequantities you want to display. Click one of thethree options to define the boundary.
Related pages: 1.45 to 1.48-------------------------------------------------------------------------------------------------------

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VII.13
To display click: Graphical Display ofResults (Graphics) ( )Function: Displays selected variables ina transport domain using spectrum
maps, contour maps, velocity vectors,
and animations.

Required Input: Select the quantity(water content, velocity, pressure head,
concentrations, or temperatures), choose
one of the display options, and specify a section.
Related pages: 1.44------------------------------------------------------------------------------------------------------To display click: Boundary Condition
Editor (Boundary)/Options/Grid
Settings, or Graphical Display ofResults (Graphics)/Options/Grid
Settings ( ), or Geometry and FEM
Mesh Editor (MeshGen-2D)/Tools/Grid
Settings ( )Function: Turns snap to grid and grid
visibility on/off; sets the origin and spacing of the grid.
Required Input: To activate snap and grid visibility, check the appropriate boxes.
Enter the x- and y-coordinates of the grid origin, and enter grid spacing.
Related pages: 1.48
Related example: Project ‘1D-infil’ page xvi.
------------------------------------------------------------------------------------------------------To display click: Boundary Condition
Editor (Boundary)/Condition/Scaling
Factor/Hydraulic Conductivity
Function: Assigns a scaling factor forthe hydraulic conductivity.
Required Input: Value for scaling factor.
Related pages: 1.42; Appendix IV-------------------------------------------------------------------------------------------------------

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VII.14

To display click: Boundary Condition Editor(Boundary)/Condition/Initial Condition/Import
Pressure-Water Content
Function: Confirmation for importing initialconditions at a certain Time Level from aprevious HYDRUS simulation.
Required Input: Confirms the Time Level at which the initial condition is imported.
Related pages: 1.40; 4.2
Related example: Project ‘Dike-St’ page 1.40; Project ‘Fall-St’ page 4.2*Display is Conditional: Select an ‘h.out’ file from a previous HYDRUS simulations
and specify a Time Level.
------------------------------------------------------------------------------------------------------To display click: ProjectManager/Projects/Delete
Function: Permanently deletes a projectfrom the current directory.
Related pages: 1.2------------------------------------------------------------------------------------------------------To display click: Boundary Condition Editor(Boundary)/ Condition/Initial Condition/Import
Pressure-Water Content
Function: Imports initial condition at a certain
Time Level from a previous HYDRUS simulation.
Required Input: Time Level at which the initial condition is imported.
Related pages: 1.40; 4.2
Related example: Project ‘Dike-St’ page 1.40; Project ‘Fall-St’ page 4.2*Display is Conditional: Select an ‘h.out’ file from a previous HYDRUS simulations.
-------------------------------------------------------------------------------------------------------

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VII.15
To display click: Inverse Solution (InPre-Processing)
Function: Specifies the parameter types
to be optimised and the weighting
scheme to be implemented; assigns alimit to the number of iterations and thenumber of calibration data points.
Required Input: Check the appropriate
boxes and enter values as demonstrated
in the figure here.
Related pages: 5.1
Related examples: All projects in Section 5.

*Display is Conditional: Check the box ‘Inverse Solution’ in ‘Main Processes’------------------------------------------------------------------------------------------------------To display click: Graphical Display ofResults (Graphics)/Options/Isolines
Parameters
Function: Sets isolines parameters.
Required Input: Provide increment
value (difference between successive
isolines) and offset value (e.g., usefulwhen excluding zero isolines).
Related pages: 1.49------------------------------------------------------------------------------------------------------To display click: Water Flow Iteration Criteria(in Pre-Processing)
Function: Sets iteration criteria, time step
control, and details of interpolation tables forsoil hydraulic parameters; specifies type for
initial conditions.
Required Input: Values for all boxes shown infigure here; in most cases these default values need not be changed, especially the‘Time Step Control’. Check the box for ‘Initial Condition’.
Related pages: 1.8
Related examples: Project series ‘Grad-‘, page 6.15 to 6.19.
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VII.16

To display click: Boundary Condition Editor(Boundary)/Condition/Local Anisotropy ( )/Angle
Function: Defines local principal directions along which
hydraulic conductivity anisotropy may be defined. The
angle is between the principal direction of K1 and the x-axis of the global coordinate
systemRequired Input: Angle in degrees, positive counter clockwise.
Related pages: I.43
Related example: Project ‘Aniso’, Example 4.8------------------------------------------------------------------------------------------------------To display click: Boundary Condition Editor(Boundary)/Condition/Local Anisotropy ( )/First& Second Components
Function: Defines K1 and K2; they are the principal
components of the anisotropy tensor; by default,
they coincide with the global coordinate x and z(when angle of local isotropy=0)
Required Input: values K1 and K2.
Related pages: I.43
Related example: Project ‘Aniso’, Example 4.8; Project ‘DF-Aniso’ p. 4.21.
------------------------------------------------------------------------------------------------------To display click: Main Processes (inPre-Processing) ( )Function: Determines the type ofsimulation to be conducted. The numberof available windows in Pre-Processing
and Post-Processing depends on what
options are selected here; for example,
windows related to root water uptake
will not be shown if the option is not checked here.
Required Input: Check the box(es) relevant to the simulation. Note that leaving allboxes unchecked prompts HYDRUS to run a steady-state analysis for water flow
Related pages: 1.3-------------------------------------------------------------------------------------------------------

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VII.17
To display click: Maas Balance
Information (in Post-Processing)
Function: displays mass balance
information at print times for the entire
domain and for individual sub-regions(if more than one layer -or subregionhas been specified).
Related pages: 1.52

Related examples: Project ‘1D-Infil’ page xviii, Project ‘2D-Infil’ page xx.
------------------------------------------------------------------------------------------------------To display click: Inverse Solution (inPost-Processing)
Function: Displays inverse solution
results.
Related pages: Section 5*Display is Conditional: Check thebox ‘Inverse Solution’ in ‘Main
Processes’------------------------------------------------------------------------------------------------------To display click: Boundary Condition Editor(Boundary)/Condition/Material Distribution ( )Function: Assigns a material number to a selectedgroup of nodes; only relevant when more than one
material is modelled (default value is one).
Required Input: A material number from 1-100 should be specified.
Related pages: 1.35
Related examples: Project ‘1D-Infil’ page xiii.
*Display is Conditional: Select nodes then ‘Set Value’.
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VII.18
To display click: Geometry and FEM
Mesh Editor (MeshGen-2D) ( )Function: Defines problem domain anddiscretises it into a finite element mesh.
Required Input: Domain dimensions,
number of nodes on the domainboundaries (default=120), and density ofmesh.
Related pages: 1.15

Related examples: Project series ‘Grid-’ pages 1.16 to 1.25
*Display is Conditional: Purchased MeshGen package with HYDRUS-2D; Select‘General’ in ‘Geometry Information/Geometry Type’.
------------------------------------------------------------------------------------------------------To display click: File/Project Manager/
Workspace /New; File/Project Manager/
Projects /New
Function: Sets a new working directory(Workspace), or creates new HYDRUS project.
Required Input: Workspace/Project name,description of Workspace/Project, and directory
path.
Related pages: 1.1; 1.3------------------------------------------------------------------------------------------------------To display click: Geometry and FEM Mesh
Editor (MeshGen-2D)/View/Boundary
Points/Number of Points (side buttons)
Function: Sets the number of points on an
external or internal boundary in the finite
element mesh.
Required Input: The total number of points on the selected boundary.
Related pages: 1.19
Related example: Projects ‘Grid-1, 2, &3’ pages 1.16-1.25*Display is Conditional: A boundary in the finite element mesh should be selected.
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VII.19To display click: Observation Points (in PostProcessing)
Function: Displays a time-series of pressure
heads/water contents/concentrations/temperatures at a specified location definedby ‘Observation Points’ in the grid. A
maximum of 10 observation points is allowed.
Required Input: Select ‘Vertical Variable’ as
shown here.
Related pages: xiv-xv
Related example: Project ‘Ris-St’ page 4.1
and many others.
*Display is Conditional: Specified ‘Observation Points’ in ‘Boundary Condition
Editor/Condition/Observation Node’.
------------------------------------------------------------------------------------------------------To display click: File/Open (Drop down
menu)
Function: Opens existing HYDRUS
projects.
Required Input: Select a file with an .h2d
extension and click Open.
Related pages: 1.1------------------------------------------------------------------------------------------------------To display click: Boundary Condition
Editor (Boundary)/Condition/Scaling
Factor/Pressure Head ( )Function: Assigns a scaling factor for thePressure Head.
Required Input: Value for scaling factor.
Related pages: 1.42; Appendix IV-------------------------------------------------------------------------------------------------------

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VII.20To display click: Print Information (in PreProcessing)
Function: Determines delivery of output
information to output files and screen.
Required Input: Check appropriate box for
print options and provide number of print
times then select print times (the window
below appears).
Related pages: Page 1.6 and 1.7

Related example: Project ‘Trial’ page 4.26------------------------------------------------------------------------------------------------------To display click: File/ProjectManager/Projects ( )Function: Sets new HYDRUS projects,
Copies / Renames/ Deletes /Opens existing
HYDRUS projects.
Required Input: Click one of the options;New, Copy, Rename, Delete, or Open.
Related pages: 1.2------------------------------------------------------------------------------------------------------To display click: Displays automaticallyafter abnormal program termination.
Function: Warns the user of abnormal
program termination resulting from a
problematic numerical solution.
Related pages: 6.8-------------------------------------------------------------------------------------------------------

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VII.21
To display click: File/ProjectManager/Workspace ( )Function: Sets a new working directoryfor HYDRUS projects, changes orremoves an existing working directory for
HYDRUS projects, and sets a selected
workspace as default.
Required Input: Click one of the options;New, Change, Remove, or Set as Default.

Related pages: 1.1------------------------------------------------------------------------------------------------------To display click: Graphical Display of Results(Graphics)/1-D Graph/Boundary Indexes ( )Function: Specifies a range of boundary nodes
over which a 1-D Graph is displayed.
Required Input: Starting and ending node
numbers representing the boundary section forwhich the results will be displayed.
Related page: 1.47------------------------------------------------------------------------------------------------------To display click: Geometry and FEM Mesh
Editor (MeshGen-2D)/Insert/Object from File( )Function: Imports coordinates of a set of points
representing a spline or a polyline.
Required Input: Check the appropriate box andhave a text file in the suitable format.
Related example: Project ‘Grid-3’ page 1.22-------------------------------------------------------------------------------------------------------

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VII.22

To display click: Boundary Condition Editor(Graphics)/Condition/Nodal Recharge ( )Function: Assigns a recharge or discharge rate
for a node anywhere in the domain that acts asa source or sink, respectively.
Required Input: Provide value for recharge (+) or discharge (-) rate.
Related pages: 1.43
Related example: Project ‘Dewat’, Example 4.10 (p. 4.23).
------------------------------------------------------------------------------------------------------To display click: Geometry and FEM
Mesh Editor ( ) (in Pre-Processing)
Function: Defines problem domain anddiscretises it into a rectangular finite
element mesh.
Required Input: Domain dimensions,slope of the bottom of the domain,
number of vertical and horizontal
columns for discretizing the domain.
Related pages: 1.13 to 1.15.
Related examples: Project series ‘1D-Infil’ page xii.
*Display is Conditional: Select ‘Rectangular’ in ‘Geometry Information/Geometry
Type’.
------------------------------------------------------------------------------------------------------To display click: Boundary Condition Editor(Boundary)/Condition/Root Distribution ( )Function: Assigns root density to a set of selected
nodes in the domain.
Required Input: Either check ‘Same value for all
nodes’ and provide one density value, or, check‘Linear distribution with depth’ and provide two density values.
Related pages: 3.2.
Related examples: Project ‘Root-4’ page 3.6.
*Display is Conditional: Check ‘Root Water Uptake’ in ‘Main ‘Processes’, and,
Select nodes then ‘Set Value’.

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VII.23

To display click: Boundary Condition Editor(Boundary)/Options/ Parameters for RootDistribution
Function: Assigns 1-D or 2-D root distribution
parameters.
Required Input: Parameters as shown in figure
here.
Related pages: III.4-III.6*Display is Conditional: Check ‘Root Water
Uptake’ in ‘Main ‘Processes’.
------------------------------------------------------------------------------------------------------To display click: Root Water Uptake Models (in Pre-Processing)
Function: Selects the water uptake reduction model.
Required Input: Check appropriate box
Related pages: Page 3.2, and Appendix III.
Related examples: See Section 3.
*Display is Conditional: Check ‘Root Water Uptake’ in ‘Main ‘Processes’-----------------------------------------------------------------------------------------------------To display click: Root Water Uptake Models - Pressure Head Reduction (in PreProcessing)
Function: Sets pressure heads that control root
water uptake for Feddes’ model.
Required Input: Values shown in figure(database provides suggested values).
Related pages: Page 3.2, and Appendix III.
Related examples: See Section 3.
*Display is Conditional: Check ‘Root Water
Uptake’ in ‘Main ‘Processes’, and checked‘Feddes’ in ‘Root Water Uptake Model/Root Water Uptake Model’-------------------------------------------------------------------------------------------------------

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VII.24

To display click: Root Water Uptake Models Pressure Head Reduction (in Pre-Processing)
Function: Sets reduction parameters for an SShaped model.
Required Input: P50 and P3.
Related pages: Page 3.3, and Appendix III.
*Display is Conditional: Check ‘Root Water
Uptake’ in ‘Main ‘Processes’, and checked ‘SShape’ in ‘Root Water Uptake Model/Root Water Uptake Model’------------------------------------------------------------------------------------------------------To display click: Water Flow – Soil HydraulicParameters (in Pre-Processing)/Neural
Network Prediction
Function: Provides predictions for soil
hydraulic parameters from pedotransfer
functions.
Required Input: The input depends on theselected model; soil textural class, or sand/silt/clay contents with or without bulk
density, and the latter with 1 or 2 points on the water retention curve.
Related pages: Page 1.11------------------------------------------------------------------------------------------------------To display click: Run Time Information (in PostProcessing)
Function: Provides graphical display of information
related to the numerical solution such as number of
iterations.
Required Input: Choice of vertical variable: time step,
number of iterations, or cumulative number of
iterations; choice of horizontal variable: time or timelevel.
Related pages: 1.52
Related examples: Project ‘Mesh-1 & 2’, page 6.1.
-------------------------------------------------------------------------------------------------------

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VII.25

To display click: Geometry and FEM Mesh Editor(MeshGen-2D)/View/Viewport ( ), or Boundary
Condition Editor (Boundary)/View/ Viewport, orGraphical Display of Results (Graphics)/View/
Viewport

Function: Set the screen limits and centre; also sets
aspect ratio options
Required Input: Coordinates for window centre andextents, check appropriate option for stretching.
Related pages: 1.15------------------------------------------------------------------------------------------------------To display click: Geometry and FEM
Mesh Editor (MeshGen2D)/Insert/Single Point/Numerically( )Function: Adds a point to boundary.
Required Input: x and y coordinates ofa point.

------------------------------------------------------------------------------------------------------To display click: Soil Hydraulic Model (in PreProcessing)
Function: Sets one of three soil hydraulic models;
hysteresis option available only with van
Genuchten-Mualem model.
Required Input: Check appropriate box for soil
hydraulic model.
Related pages: 1.10; Appendix I.
Related examples: All Project examples; hysteresis example page 4.8.
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VII.26

To display click: Soil Hydraulic Properties (inPost-Processing)
Function: Displays soil water retention andhydraulic conductivity function.
Required Input: Select required ‘Vertical andHorizontal Variable’.
Related pages: 1.51------------------------------------------------------------------------------------------------------To display click: Geometry and FEM Mesh
Editor (in Pre-Processing)
Function: Changes default meshdiscretization.
Required Input: May manually change
coordinates; enter density values to alter meshdensity at top or bottom.
Related pages: 1.14
*Display is Conditional: Select ‘Rectangular’ in ‘Geometry Information/Geometry
Type’.
------------------------------------------------------------------------------------------------------To display click: Boundary Condition
Editor (Boundary)/Options/Stochastic
Distribution of S.F.
Function: Assigns stochastic distribution toscaling factors.
Required Input: Check appropriate boxes
and click ‘Parameters’, the second window
here will appear; enter values for standard
deviation, and correlation length in x and z
directions.
Related pages: 1.42; Appendix IV

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VII.27

To display click: Boundary Condition Editor(Boundary)/Condition/Subregions ( )Function: Assigns a subregion number to a selectedgroup of elements; only relevant when more than
one subregion is modelled (default value is one).
Required Input: A subregion number from 2-10 should be specified.
Related pages: 1.41
Related examples: Project ‘1D-Infil’ page xiv*Display is Conditional: Select nodes then ‘Set Value’.
------------------------------------------------------------------------------------------------------To display click: Time Information (in PreProcessing)
Function: Defines time units, time
discretization, and number of time-variableboundary conditions (if any).
Required Input: Check appropriate box for timeunits, and enter time discretization details; if time-variable boundary conditions arechecked then provide a number.
Related pages: 1.5
Related examples: Project in Section 6 show the significance of ‘Time Discretization’------------------------------------------------------------------------------------------------------To display click: Variable Boundary Conditions(in Pre-Processing)
Function: Defines time-variable BC such as
precipitation (rainfall), evaporation, transpiration,and time-variable pressure head or flux.
Required Input: Time series for relevant boundary
condition.
Related pages: 1.12; 1.29-1.30; 1.33-1.34
Related examples: Project ‘Ris-St’ page 4.1; Project ‘Evap-1’ page 4.3; and Project‘Tension Disc Infiltrometer’ page 4.21
*Display is Conditional: Check ‘Variable Boundary Condition’ in ‘Time
Information’.

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VII.28

To display click: Graphical Display of Results(Graphics)/Options/Velocity Vectors Parameters
Function: Scales arrows representing velocity
vectors.
Required Input: As shown in figure here.
*Display is Conditional: Check ‘2DGraphs/Velocity Vectors’.
------------------------------------------------------------------------------------------------------To display click: Boundary Condition
Editor (Boundary)/View/Stretching, orGraphical Display of Results(Graphics)/View/Stretching, orGeometry and FEM Mesh Editor(MeshGen-2D)/View/Stretching
Function: Stretches the domain. The
option is useful when the aspect ratios of the domain and the screen are significantly
different.
Required Input: Stretching factors in x- or y-directions; note that when the factor is <1
the mesh is stretched in that direction.
------------------------------------------------------------------------------------------------------To display click: Boundary Condition
Editor (Boundary)/Condition/Scaling
Factor/Water Content
Function: Assigns a scaling factor for thewater content.
Required Input: Value for scaling factor.
Related pages: 1.42; Appendix IV-------------------------------------------------------------------------------------------------------

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VII.29To display click: Boundary Condition
Editor (Boundary)/Condition/Initial
Condition/Pressure Head-Water Content( )Function: Assigns initial pressure heads orwater contents.
Required Input: One value if ‘Same for all
nodes’ is checked; Top and Bottom values
if one of the other options is checked;
Specify slope of the initial condition.

Related pages: 1.37-1.38------------------------------------------------------------------------------------------------------To display click: Water Flow - Soil
Hydraulic Parameters (in PreProcessing)
Function: Defines the soil water
retention parameters, the saturated
hydraulic conductivity, and L(turtuosity parameter).
Required Input: Water retention parameters and measured saturated hydraulicconductivity; required values vary according to the selected hydraulic model.
Related pages: 1.11; Appendix I.
Related examples: All example Projects-------------------------------------------------------------------------------------------------------

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VII.30

Appendix VII: HYDRUS Windows Index

To display click: Water Flow Soil Hydraulic Parameters (inPre-Processing)
Function: Defines soil water
retention parameters, thesaturated hydraulic conductivity,and L (turtuosity parameter); if ‘Fitted’ is checked, those values are used as initialvalues for the inverse solution; sets upper and lower limits for the optimisedparameters. Determines which parameters are optimised.
Required Input: Water retention parameters, the saturated hydraulic conductivity, andthe tortuosity parameter; minimum and maximum limits for the optimised parameters;
along the line ‘Fitted’, check the parameters to be optimised.
Related pages: Page 5.2.
Related examples: All examples in Section 5.
*Display is Conditional: Check the box ‘Inverse Solution’ in ‘Main Processes’.
------------------------------------------------------------------------------------------------------

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Appendix VIII: Toolbars

VIII.1APPENDIX VIIITOOLBARS

Main Program Toolbars:
New Project (File Menu)
Open a Project (File Menu)
Project Manager (File Menu)
Save Project (File Menu)
Main Processes (Pre-Processing Menu)
Geometry and Finite Element Mesh Editor (Pre-Processing Menu)
Boundary and Initial Conditions (Pre-Processing Menu)
Run HYDRUS (Calculation Menu)
Graphical Display of Results (Post-Processing Menu)
Cascade Window (Window Menu)
Tile Windows Horizontally (Window Menu)
Tile Windows Vertically (Window Menu)
HYDRUS Help Topics (Help Menu)
Context Sensitive Help (Help Menu)
License Details (Help Menu)

MeshGen2D Toolbars 1:
Close MeshGen2D (File Menu)
Save Mesh (File Menu)
Print (File Menu)
Delete Mesh (Edit Menu)
Edit Geometry (Edit Menu)
Edit Boundary Points (Edit Menu)
Edit FEM Mesh (Edit Menu)
Undo (Edit Menu)
Redo (Edit Menu)
Consistency Check (Tools Menu)
Grid Settings (Tools Menu)
Enlarge View (View Menu)
Reduce View (View Menu)
Zoom View (View Menu)
Previous View (View Menu)
View All (View Menu)
Redraw (View Menu)
Set Viewport (View Menu)
Context Sensitive Help (Help Menu)
MeshGen2D Help Topics (Help Menu)

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Appendix VIII: Toolbars

VIII.2
MeshGen2D Toolbars 2:

Graphical Manipulation Mode (Tools Menu)
Numerical Manipulation Mode (Tools Menu)
Read Object from File (Insert Menu)
New Points Graphically (Insert Menu)
New Line Graphically (Insert Menu)
New Spline Graphically (Insert Menu)
New Arc Graphically (Insert Menu)
New Circle Graphically (Insert Menu)
New Points Numerically (Insert Menu)
New Line Numerically (Insert Menu)
New Spline Numerically (Insert Menu)
New Arc Numerically (Insert Menu)
New Circle Numerically (Insert Menu)
Arc by Three Points (Insert Menu)
Arc by Center, Radius, and Two Angles (Insert Menu)
Circle by Center and Radius (Insert Menu)
Circle by Three Points (Insert Menu)
Select Duplicated Points (Insert Menu)
Pick Existing Points (Tools Menu)
Edit Objects Coordinates (Tools Menu)

Boundary Conditions Editor Toolbars:

Close BC Editor (File Menu)
Print (File Menu)
Save (File Menu)
Display Options (Options Menu)
Water Flow BC (Condition Menu)
Solute Transport BC (Condition Menu)
Heat Transport BC (Condition Menu)
Material Distribution (Condition Menu)
Root Uptake Distribution (Condition Menu)
Nodal Recharge (Condition Menu)
Scaling Factors (Condition Menu)
Initial Conditions (Condition Menu)
Anisotropy (Condition Menu)
Subregions (Condition Menu)
Observation Nodes (Condition Menu)
Nodal Drains (Condition Menu)
Flowing Particle (Condition Menu, currently inactive)
Zoom In (View Menu)
Zoom Out (View Menu)
Zoom View (View Menu)
Previous View (View Menu)
View All (View Menu)

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Appendix VIII: Toolbars

VIII.3
Redraw (View Menu)
HYDRUS Help Topics (Help Menu)
Context Sensitive Help (Help Menu)
License Details (Help Menu)

Graphical Display of Results Toolbars:

Print (File Menu)
Display FE Mesh (2-D Graph Menu)
Velocity Vectors (2-D Graph Menu)
Isolines (2-D Graph Menu)
Spectral Map (2-D Graph Menu)
Spectral Map and Isolines (2-D Graph Menu)
1-D Graph along Cross-Section (1-D Graph Menu/Cross Section)
1-D Graph along Whole Boundary curve (1-D Graph Menu/Boundary Whole
Curve)
1-D Graph along a Boundary Between Two Given Points (1-D Graph
Menu/Boundary Indexes)
1-D Graph along a Boundary Between Two Graphically Selected Points (1-D
Graph Menu/Boundary Indexes)
Grid Settings (Options Menu)
Display Options (2-D Graph Menu)
Zoom In (View Menu)
Zoom Out (View Menu)
Zoom View (View Menu)
Previous View (View Menu)
View All (View Menu)
Redraw (View Menu)
HYDRUS Help Topics (Help Menu)
Context Sensitive Help (Help Menu)
License Details (Help Menu)

Modelling variably saturated flow with HYDRUS-2D

VIII.4

Appendix VIII: Toolbars

Modelling variably saturated flow with HYDRUS-2D

Alphabetical index

Index-1
ALPHABETICAL INDEX

Page for
ExamplesDescription
1D-Graph
1D-Graph/boundary indexes
1D-Graph/boundary selection
1D-Graph/Boundary whole curve
1D-Graph/Cross section
Abnormal HYDRUS termination
Actual root water uptake
Air entry value (AEV)
Alpha-water flow parameter

6.8
3.4/3.8

AlphaW-wetting Alpha parameter
Anisotropy
Arc
Aspect ratio
Axisymmetric flow
Boundary conditions (see water flow Boundary conditions)
Brooks and Corey model
Calibration data (data points in objective function)
Calibration data/choice of data
Capillary barrier
Change workspace
Check Geometry
CircleConcentration
Confidence limit
Consistency check
Constraints on hysteresis model (inverse mode solution)
Convert to ASCII
Copy projectCorrelation length/ Stochastic parameters
Correlation matrix/Inverse solution
Cumulative water boundary fluxes/Graphical display ofresults
Curve type
Data for inverse solution
Density at the bottom
Density at the top
Density of boundary points/MeshGen
Disc infiltrometer
Dispersivity
Drain BC
Drying water flow parameters/hysteresis
Evaporation
Evaporation - advance of drying boundary
Evaporation - effect of hCritA
Evaporation - effect of mesh density
Evaporation - effect of relative humidity
Evaporation - effect of suction

4.8
4.18
1.17

Page inAppen.
VII

Page invarious
Sections
1.45-1.48
1.47
1.46
1.45
1.48

VII.203.2
I.3/I.5/I.10
1.11/I.3/
I.10
VII.16
VII.1

1.43
1.16
1.4
1.26
1.10/I.5/
I.8
5.1-5.3
V.8 & 94.21-4.24

V.8
4.6/4.21

1.2
1.25
1.17/1.24
VI.2/VI.6

VII.3
VII.5
V.4
1.25
5.2
1.53/2.5
1.2
IV.5
V.6
1.50

1.40/4.2

1.24

VII.6
5.2/5.3
1.15
1.15

1.19-1.21
4.21
VI.2

VII.8
1.27/1.28
VII.8

4.8
4.3

1.35
1.13
Appen.-II

4.3/4.4
II.3
4.4
II.2
II.1

Modelling variably saturated flow with HYDRUS-2D

Alphabetical index

Index-2

Page for
ExamplesDescription
Evaporation - shape of drying curve
Evaporation - vapour flow
Feddes root water uptake reduction model
Final time
General geometry
Geometry Information window
Geometry Type/Geometry Information
Goodness of fit
Graph-1D
Graphical display of results window
Grid settings window
Guidelines on inverse modelling
GWL
hCritA-Evaporation parameter
Heading/main processes
Heterogeneous profile
hk parameter for Vogel & Cislerova model
Horizontal flow
HYDRUS (Multiple simulations)
Hysteresis
Import initial condition from previous HYDRUS
simulation
In the pressure head/Iteration Criteria
In the water content /Iteration Criteria
Initial condition/Iteration Criteria
Initial conditions
Initial parameters for inverse solution
Initial time
Initial time step
Insert Arc/MeshGen
Insert circle numerically/MeshGen
Insert fixed point/MeshGen
Insert line numerically/MeshGen
Insert object from file/MeshGen
Insert periodical condition
Insert Spline/MeshGen
Internal interpolation tables/ Iteration Criteria
Inverse solution window
Inverse solution/in Main processes
Isolines parameters
Iteration Criteria window
Key predictions in inverse simulations
Kk parameter for Vogel & Cislerova modelKs- water flow parameter (saturated hydraulicconductivity)
KsW-wetting saturated hydraulic conductivity
L-water flow parameter (Totuosity parameter)
Layers (or subregions)
Length units/ Geometry Information
Line
Lower-limit of the tension interval
Main processes window
Mass balance information/Graphical display of results
Material distribution
Maximum number of iterations

Page inAppen.
VII

1.15
VII.12

VII.13
VII.13
4.1/4.21
II.3
xvi
x/xiv
4.25
4.27
4.8
1.40/4.2/VI.7

xii

5.15/5.19
4.2
6.9
1.17
1.17/1.24
1.21
1.17
1.22

Page invarious
Sections
II.1
II.2
3.2/III.1
1.5
1.4
1.4
1.4
V.5
1.45-1.48
1.44
1.48
5.24/5.25
1.13/1.29
1.13
1.3
1.5/1.35
I.7
1.5
1.10

VII.15
VII.151.8/1.10
1.8/1.10
1.8/1.10
1.37/1.40
5.1
1.5
1.5

VII.1

VII.21
1.25

1.23
6.14
VII.15
VII.15
VII.151.10
5.1
1.3/5.1
1.49
1.8

5.18
I.7
1.11/I.10
4.9
1.11-5.14
1.41
1.4

xv
1.17
4.9/6.14
VII.16
6.6/xix
xiv
6.8

1.10
1.3
1.52
1.35/1.36
1.8/1.9

Modelling variably saturated flow with HYDRUS-2D

Alphabetical index

Index-3

Page for
ExamplesDescription
Maximum number of iterations/inverse solution
Maximum time step
Merge object/MeshGen
Mesh density
MeshGen
Minimum time step
Modified van Genuchten model (Vogel & Cislerova)
m-parameter (van Genuchten model)
Multiple HYDRUS simulations
Neural Network Prediction (Rosetta)
New project
New workspace
Nodal drain
Nodal recharge
Non-uniqueness of inverse solution
Normalised water content
Number of boundary points/MeshGen
Number of data points in objective function
Number of layers
Number of materials
Number of print times
n-water flow parameter
Objective function/ Inverse solution
Observation nodes
Open project window
Output fileso A_level.out
o Balance.out
o Boundary.out
o Check.out
o Cum_Q.out
o Fit.out
o H.out (Optionally converted to an ASCII fileh.txt)
o h_mean.out
o MeshTria.000 (Optionally convert to an ASCII
file MeshTria.txt)
o ObsNod.out
o Run_Inf.out
o Th.out (Optionally converted to an ASCII fileTh.txt)o V.out (Optionally converted to an ASCII filev.txt)
o V_mean.out
Parameter interaction in inverse solution
Pick existing points
Pointer to vector of solute BC
Precipitation
Pressure head tolerance
Pressure heads/Graphical display of results
Print fluxes
Print information window
Print Options/Print Information
Program error/abnormal HYDRUS termination

Page inAppen.
VII

4.4/4.16
1.15-1.25
6.3

Page invarious
Sections
5.1
1.5/1.9
1.25
1.15/1.20
1.15
1.5/1.9
1.10/I.5/I.7
I.2/I.4

4.27
1.11
1.2
1.2
1.35
1.43
5.4/V.6
I.3

4.23
Sec. 5.2.1
1.19

VII.18

x/xv
x/xiv
4.26

xv
VII.192.11
2.11
2.8/2.12

5.1
1.4/1.41
1.5/1.35
1.7
1.11/I.3/
I.10
V.1
1.43
1.1
2.1
2.4
2.1
2.3
2.3
2.5
2.5

2.11

2.2
2.5
2.2
2.4
2.5

2.7

2.5
2.2

2.7/2.10/2.15
5.18

1.25
VI.3
4.9/4.16

VII.20
6.8

1.13
1.8
1.50
1.6
1.6
1.6

VII.20Modelling variably saturated flow with HYDRUS-2D

Alphabetical index

Index-4

Page for
ExamplesDescription
Project manager window
Pulse duration
Qm-(Theta_m) parameter Vogel & Cislerova model(Modified van Genuchten)
Qr-water flow parameter (residual water content)
Qs- water flow parameter (saturated water content)
QsW-wetting saturated water content
Rectangular Geometry
Relative humidity
Remove workspace
Rename project
Residual water content (see Qr; Theta_r)
RGWL
Root density
Root distribution parameters
Root distribution window
Root water uptake model window
Root water uptake parameters
Rosetta window
Run time information/Graphical display of results
Runoff (see Surface flux higher than Ks)
Saturated water content (Qs; Theta_s)
Scaling factors
Screen capture
Screen output
Select by rhomboid
Select print times
Sensitivity analysis for water retention parameters
Set viewport window
Set workspace as current
Similar media
Simulate/main processes
Soil hydraulic model window
Soil hydraulic properties /Graphical display of results
Soil Profile/Geometry Information
Solute transport BC
Solute transport parameters
SplineSplit object/MeshGen
S-shaped parameters- Root water uptake
Standard deviation/ Stochastic parameters
Steady state analysis
Stochastic distribution of scaling factor window
Stochastic parameters window
Subregions (or layers)
Surface flux higher than Ks
Tension disc infiltrometer
Theta_m (Qm) parameter Vogel & Cislerova model(Modified van Genuchten)
Theta_r water flow parameter (residual water content Qr)
Theta_s water flow parameter (saturated water content Qs)
Time discretization
Time information window
Time Step Control/Iteration Criteria

Page inAppen.
VII
VII.20Page invarious
Sections
1.1

VI.2
I.5
1.11/I.2&3
1.11/I.3
4.8
1.13

3.6

VII.22
VII.23
VII.23
VII.24

6.2-6.4
4.17

1.17/1.20

VII.25

VII.25

1.4
II.2
1.2
1.2
1.11/I.2&3
1.13/1.30
III.3
III.4 to 6
3.2
3.2
3.2/III.1
1.11
1.52
1.34
1.11/I.3
1.42/IV.1
1.45
1.6
1.36
1.7
I.9 to I.12
1.15
1.2
IV.1
1.3
1.10
1.51
1.4

VI.2
VI.2
1.22 & 23

1.39/1.40/4.15
VII.26
VII.26
x/xv
4.17
4.21

VII.27

1.25
3.3/ III.2
IV.4
1.3/1.39
1.42/IV.3
1.42/IV.4
1.4/1.41
1.34
1.27/1.28
I.5
1.11/I.2/I.3
1.11/I.3
1.5/1.9
1.5
1.8/1.9

Modelling variably saturated flow with HYDRUS-2D

Alphabetical index

Index-5

Page for
ExamplesDescription
Time units
Time-variable BC, number of records allowed
Time-variable boundary condition checkbox/Time
information
Time-variable boundary condition/window

Tipping bucket rainfall data
T-level information
Tools/findTools/point picking
Tortuosity parameterTranspiration
Trouble shooting for HYDRUS-2Do Effect of minimum time stepo Effect of abrupt intense infiltrationo Unstable initial conditions/ Boundary +saturation effects
o Infiltration into clay/ Model + flux/Ks effects
o Infiltration into clay/ Time discretization +Iteration criteria effects
o Effect of high initial press. head gradientType of flow/ Geometry Information
Upper-limit of the tension interval
Validation of inverse results
van Genuchten modelvan Genuchten model with Air-entry value of –2 cm
Vapour flux during evaporation
Velocity vectors
Vertical flow
Vogel & Cislerova model (Modified van Genuchten)
Water boundary fluxes/Graphical display of results
Water content tolerance
Water flow Boundary conditionso No Fluxo Constant heado Constant flux
o Variable pressureo Variable flux
o Free drainageo Deep drainageo Seepage faceo Atmospherico DrainWater flow parameters inverse solution window
Water flow parameters window
Water flow-initial conditions
Water retention curve
Weighted residual
Weighting of inversion data/Inverse solution
Weighting parameters
Wetting water flow parameters/hysteresis
Workspace

Page inAppen.
VII

Page invarious
Sections
1.5
1.34
1.5

VII.27

1.12/1.29/
1.30/1.33/
1.34/
1.33/1.34
1.6/2.1
1.25
1.25
1.11-5.14
1.13/Sec-3
Section 6

VII.10Section 3
6.1-6.3
6.4-6.7
6.8-6.9
6.10-6.11
6.12-6.14
6.15-6.19
4.21/4.25
4.9/6.14
5.8/5.9/5.13

1.4
1.10
5.8
1.10/I.2/I.6
1.10/I.4/I.7
II.3

4.19
4.1 to 4.20

1.4
I.5/I.7
1.50
1.8

3.5/4.9/4.16/6.
13

4.12
5.4
4.1/4.21

VII.5
VII.4

4.6
VII.7
4.12/4.18
5.17/4.18
VII.30VII.29VII.29

V.7/V.8

1.27
1.27
1.27/1.28
1.29
1.29
1.30
1.31
1.31
1.31/1.32
1.33
1.35
5.2
1.11
1.38
I.1 to I.6
V.2
5.1
V.1/V.2/
V.7

4.8
1.1

Modelling variably saturated flow with HYDRUS-2D

Index-6

Alphabetical index

Modelling variably saturated flow with HYDRUS-2D