SVFlux Verification Manual

2D / 3D Seepage Modeling Software

Verification Manual

Written by:

Robert Thode, P.Eng., B.Sc.G.E.

Edited by: Murray Fredlund, P.Eng., Ph.D.

SoilVision Systems Ltd. Saskatoon, Saskatchewan, Canada

Software License The software described in this manual is furnished under a license agreement. The software may be used or copied only in accordance with the terms of the agreement. Software Support Support for the software is furnished under the terms of a support agreement. Copyright Information contained within this Verification Manual is copyrighted and all rights are reserved by SoilVision Systems Ltd. The SVFLUX software is a proprietary product and trade secret of SoilVision Systems. The Verification Manual may be reproduced or copied in whole or in part by the software licensee for use with running the software. The Verification Manual may not be reproduced or copied in any form or by any means for the purpose of selling the copies. Disclaimer of Warranty SoilVision Systems Ltd. reserves the right to make periodic modifications of this product without obligation to notify any person of such revision. SoilVision does not guarantee, warrant, or make any representation regarding the use of, or the results of, the programs in terms of correctness, accuracy, reliability, currentness, or otherwise; the user is expected to make the final evaluation in the context of his (her) own problems. Trademarks Windows is a registered trademark of Microsoft Corporation. SoilVision is a registered trademark of SoilVision Systems Ltd. SVOFFICE is a trademark of SoilVision Systems, Ltd. CHEMFLUX is a registered trademark of SoilVision Systems Ltd. SVFLUX is a trademark of SoilVision Systems Ltd. SVHEAT is a trademark of SoilVision Systems Ltd. SVAIRFLOW is a trademark of SoilVision Systems Ltd. SVSOLID is a trademark of SoilVision Systems Ltd. SVSLOPE is a registered trademark of SoilVision Systems Ltd. ACUMESH is a trademark of SoilVision Systems Ltd. FlexPDE is a registered trademark of PDE Solutions Inc.

Copyright 2012 by

SoilVision Systems Ltd. Saskatoon, Saskatchewan, Canada

ALL RIGHTS RESERVED Printed in Canada

Last Updated: April 7, 2015

SoilVision Systems Ltd. Table of Contents 3 of 93

1 INTRODUCTION .............................................................................................................. 6

2 ONE-DIMENSIONAL SEEPAGE ..................................................................................... 7

2.1 TRANSIENT-STATE ............................................................................................................. 7 2.1.1 Haverkamp (1977)....................................................................................................... 7 2.1.2 Celia (1990) ................................................................................................................ 8 2.1.3 1D Mass Balance ........................................................................................................ 8 2.1.4 SoilCover Comparison ................................................................................................ 9 2.1.5 SoilCover Comparison #2 .......................................................................................... 12

2.1.5.1 Model Geometry ...............................................................................................................13 2.1.5.2 Material Properties ............................................................................................................13 2.1.5.3 Boundary Condition ..........................................................................................................14 2.1.5.4 Climate Data .....................................................................................................................14 2.1.5.5 Comparison.......................................................................................................................15

2.1.6 Evaporation - Wilson (1990) ...................................................................................... 17 2.1.6.1 Model geometry and boundary conditions ..........................................................................17 2.1.6.2 Material properties ............................................................................................................18 2.1.6.3 Results and discussion .......................................................................................................20

2.1.7 Evapotranspiration - Tratch (1995) ........................................................................... 21 2.1.8 Gitirana Infiltration Examples ................................................................................... 23

3 TWO-DIMENSIONAL SEEPAGE .................................................................................. 25

3.1 STEADY-STATE ........................................................................................................... 25 3.1.1 2D Cutoff .................................................................................................................. 25 3.1.2 2D Earth Fill Dam .................................................................................................... 28

3.1.2.1 Review Boundary ..............................................................................................................28 3.1.2.2 Filter Scenario ...................................................................................................................29

3.1.3 X Component of Left to Right Flow ............................................................................ 30 3.1.4 Simple Water Balance ............................................................................................... 31 3.1.5 Decreasing Pipe Size ................................................................................................. 31 3.1.6 Axisymmetric Verification .......................................................................................... 32 3.1.7 Drain-Down Verification ........................................................................................... 34 3.1.8 Roadways Subgrade Infiltration ................................................................................. 36 3.1.9 Refraction Flow Example .......................................................................................... 40 3.1.10 Axisymmetric Aquifer Pumping Well ....................................................................... 40 3.1.11 Dam Flow .............................................................................................................. 41 3.1.12 Dupuit Model ......................................................................................................... 42 3.1.13 Well Object vs Rectangle ........................................................................................ 43 3.1.14 2D Well Object with Head Boundary Condition ...................................................... 45

3.1.14.1 Purpose.............................................................................................................................46 3.1.14.2 Geometry and Boundary Conditions ..................................................................................46 3.1.14.3 Material Properties ............................................................................................................47 3.1.14.4 Results and Discussions ....................................................................................................47

3.1.14.4.1 Distributions of the Pore-Water Pressure (uw) ..............................................................47 3.1.14.4.2 Distributions of the Head (h) .......................................................................................49 3.1.14.4.3 Flux Flow ...................................................................................................................50

3.1.15 2D Well Object with Review by Pressure Boundary Condition ................................ 50


3.1.15.1 Purpose.............................................................................................................................50 3.1.15.2 Geometry and Boundary Conditions ..................................................................................50 3.1.15.3 Material Properties ............................................................................................................51 3.1.15.4 Results and Discussions ....................................................................................................52

3.1.15.4.1 Distributions of the Pore-Water Pressure (uw) ..............................................................52 3.1.15.4.2 Distributions of the Head (h) .......................................................................................53 3.1.15.4.3 Flux Flow ...................................................................................................................54

3.2 TRANSIENT STATE ........................................................................................................... 54 3.2.1 Transient Reservoir Filling ........................................................................................ 54 3.2.2 Celia Infiltration Example ......................................................................................... 58 3.2.3 Evapotranspiration - Triangular and Rectangular Root Distributions......................... 60 3.2.4 A Transient 2-D Infiltration Problem ......................................................................... 62

4 THREE-DIMENSIONAL SEEPAGE .............................................................................. 64

4.1 STEADY-STATE ................................................................................................................ 64 4.1.1 Wedge Example ......................................................................................................... 64 4.1.2 Cube Example ........................................................................................................... 65 4.1.3 Toe Example ............................................................................................................. 65 4.1.4 3D Well Object vs Cylinder ....................................................................................... 66 4.1.5 3D Well Object with Head Boundary Condition ......................................................... 68

4.1.5.1 Purpose .............................................................................................................................69 4.1.5.2 Geometry and Boundary Conditions ..................................................................................69 4.1.5.3 Material Properties ............................................................................................................70 4.1.5.4 Results and Discussions .....................................................................................................70

4.1.5.4.1 Distributions of the Pore-Water Pressure (uw)................................................................70 4.1.5.4.2 Distributions of the Head (h) .........................................................................................71

4.1.6 3D Well Object with Review by Pressure Boundary Condition ................................... 72 4.1.6.1 Purpose .............................................................................................................................72 4.1.6.2 Geometry and Boundary Conditions ..................................................................................72 4.1.6.3 Material Properties ............................................................................................................74 4.1.6.4 Results and Discussions .....................................................................................................74

4.1.6.4.1 Distributions of the Pore-Water Pressure (uw)................................................................74 4.1.6.4.2 Distributions of the Head (h) .........................................................................................75

4.1.7 Confined Aquifer 3D Ideal ......................................................................................... 76 4.1.7.1 Purpose .............................................................................................................................76 4.1.7.2 Theis System.....................................................................................................................76 4.1.7.3 Model ...............................................................................................................................77 4.1.7.4 Geometry and Properties ...................................................................................................78 4.1.7.5 Material Properties ............................................................................................................79 4.1.7.6 Results and Discussions .....................................................................................................79 4.1.7.7 Remarks............................................................................................................................82

4.2 TRANSIENT STATE ........................................................................................................... 83 4.2.1 Drain-Down Example ................................................................................................ 83 4.2.2 Cube Drain-Down Example ....................................................................................... 83 4.2.3 Cube, Wedge, and Toe Transient Example ................................................................. 84

5 SENSITIVITY OF THE FINITE ELEMENT MESH ..................................................... 85

5.1 MODEL DESCRIPTION ....................................................................................................... 85


5.2 INPUT PARAMETERS FOR SENSITIVITY ANALYSIS .............................................................. 86 5.3 SIMULATION RESULTS AND DISCUSSIONS.......................................................................... 87

6 REFERENCES ................................................................................................................. 92

SoilVision Systems Ltd. Introduction 6 of 93

1 INTRODUCTION The word “Verification”, when used in connection with computer software can be defined as “the ability of the computer code to provide a solution consistent with the physics defined by the governing partial differential equation, PDE”. There are also other factors such as initial conditions, boundary conditions, and control variables that also affect the accuracy of the code to perform as stated. “Verification” is generally achieved by solving a series of so-called “benchmark” problems. “Benchmark” problems are problems for which there is a closed-form solution or for which the solution has become “reasonably certain” as a result of long-hand calculations that have been performed. Publication of the “benchmark” solutions in research journals or textbooks also lends credibility to the solution. There are also example problems that have been solved and published in User Manual documentation associated with other comparable software packages. While these are valuables checks to perform, it must be realized that it is possible that errors can be transferred from one’s software solution to another. Consequently, care must be taken in performing the “verification” process on a particular software package. It must also be remembered there is never such a thing as complete software verification for “all” possible problems. Rather, it is an ongoing process that establishes credibility with time. SoilVision Systems takes the process of “verification” most seriously and has undertaken a wide range of steps to ensure that the SVFLUX software will perform as intended by the theory of saturated-unsaturated water seepage. The following models represent comparisons made to textbook solutions, hand calculations, and other software packages. We at SoilVision Systems Ltd. are dedicated to providing our clients with reliable and tested software. While the following list of example models is comprehensive, it does not reflect the entirety of models, which may be posed to the SVFLUX software. It is our recommendation that water balance checking be performed on all model runs prior to presentation of results. It is also our recommendation that the modeling process move from simple to complex models with simpler models being verified through the use of hand calculations or simple spreadsheet calculations.


2 ONE-DIMENSIONAL SEEPAGE The following examples compare the results of SVFLUX against published 1D solutions presented in textbooks or journal papers. One-dimensional scenarios were entered in SVFLUX through the use of a thin 1D column.

2.1 TRANSIENT-STATE Transient or time-dependent models allow the benchmarking of time-stepping aspects of the SVFLUX software.

2.1.1 Haverkamp (1977) The Haverkamp (1977) model involves infiltration into a 1D column of material. A series of infiltration experiments were performed by Haverkamp in the laboratory using a plexiglass column uniformly packed with sand to verify the numerical results. The model was originally solved using 1D finite element and 1D finite difference solution methods. Time-steps used in the analysis were varied in the original work to determine their effect on the solution. The best solution presented is with small time-steps (10 seconds) and a dense grid. Project: WaterFlow Model: Haverkamp1977 The material properties used in Haverkamp’s analysis were custom equations defined in terms of elevation head rather than soil suction. The model was initially set up in SVFLUX and then minor modifications were made to the finite element script file to duplicate the solution exactly. The script file presenting the exact comparison of results can be provided upon request. The results of the comparison may be seen in Figure 1. Previous issues with varying timesteps are rendered insignificant given the automatic time-step refinement present in the SVFLUX software. It can be seen that the mesh selected by SVFLUX automatically duplicates the best results presented by the Haverkamp solution. The solution took 26 seconds on a P-4 2.8GHz computer and used a total of 604 nodes.

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-10

0 5 10 15 20 25 30 35 40

Depth (cm)

Pres

sure

hea

d (c

m)

dt = 10 secdt = 30 secdt = 120 secDense GridSVFlux

Figure 1 Comparison between SVFLUX and Haverkamp (1977) as presented by Celia (1990)


2.1.2 Celia (1990) Celia (1990) performed comparisons of 1D solvers by varying the time-steps and the solution methods (finite difference or finite element). His results are considered classic solutions and are commonly used to benchmark the validity of 1D infiltration models. The solution presented by Celia used the h-based formulation of Richard’s equation and a Newton-Raphson iterative method. Project: WaterFlow Model: Celia1990 A replica of Celia’s model was set up using the SVFLUX software. Celia presented the material properties for the model as van Genuchten’s equation for the soil-water characteristic curve and as van Genuchten and Mualem’s equation for representing the unsaturated hydraulic conductivity curve. Since both methods are implemented in the SVFLUX software the parameters used for the material could be input directly. The results of the comparison may be seen in Figure 2. As in the previous model it can be seen that the automatic mesh generation and automatic time-step refinement allow quick convergence to the correct solution. A solution for this model was achieved in 43 minutes using an average of 897 nodes.

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0

0 20 40 60 80 100

Depth (cm)

Pres

sure

hea

d (c

m) Dense Grid

dt = 20 secdt = 2.4 mindt = 12 mindt = 60 minSVFlux

Figure 2 Comparison between SVFLUX and results presented by Celia (1990)

2.1.3 1D Mass Balance An infiltration model was created which verifies the mass-balance of a simulated rainfall for a period of 1 day. A single storm event is input into SVFLUX and a 24-hour period is run. The reported total flow into the 1D column should be equal to the amount of rainfall if calculations are correct. Zero flux boundaries on three sides of the model disallow any flux in or out of the model with the exception of the top boundary. Project: Columns Model: Day1 Precipitation: 0.1 m3/day/m2

Material: Grey Till ksat: 0.9 m/day


0

0.05

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0.35

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1e-3 1e-2 1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6

Soil suction (kPa)

Laboratory Interpolated Fredund and Xing Fit Laboratory Data

Figure 3 Soil-water characteristic curve for 1D mass-balance model

The results of the 1D model may be summarized as follows: Application: 0.1 m3/day/m2 x 0.1m = 0.01 m3/day Intensity: start 9:00 am end 17:00 (5:00pm) Reported flux in: 0.010043 m3/day Error: 0.43%

2.1.4 SoilCover Comparison The question of how the results of SVFLUX compare to the traditionally accepted SoilCover program has surfaced in the past while. The purpose of this set of benchmark models is to explain the similarities and differences between the two software packages. The primary theoretical difference is that the current version of SVFLUX does not couple in heat flow. How significant is thermal coupling in standard models? Two examples are set up and the results of the two programs are compared in order to attempt to determine computational differences. A cover scenario was chosen for the comparison and the results are presented in the following paragraphs. In this example a 1m cover is placed over a 3m tailings material. The bottom boundary is forced to a constant suction of –10 kPa. The initial conditions are considered hydrostatic through the suction of –10 kPa at elevation zero. The material properties for the cover material are as follows.


0.000

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0 0 1 10 100 1000 10000 100000 1000000

Matric Suction (kPa)

Wat

erC

onte

nt(d

ec.)

1.00E-10

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1.00E-02

1.00E-01

Slop

eFu

nctio

n(1

/kPa

)

User Points

Curve Fit

Slope Function

Figure 4 Soil-water characteristic curve for Cover material

1.00E-14

1.00E-13

1.00E-12

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Rela

tive

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eabi

lity

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Volu

met

ricW

ater

Con

tent

Figure 5 Hydraulic conductivity for the Cover material (ksat=5e-2 cm/s)

Likewise the material properties for the tailings material are as follows.

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onte

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ec.)

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Slop

eFu

nctio

n(1

/kPa

)

User PointsCurve FitSlope Function

Figure 6 Soil-water characteristic curve for the Tailings material


1.00E-14

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1.00E-09

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1.00E+000 1 10 100 1000 10000 100000 1000000


Rel

ativ

ePe

rmea

bilit

y

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met

ricW

ater

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tent

Figure 7 Hydraulic conductivity for the Tailings material (ksat=5.7e-5 cm/s)

The model is run for a total of 184 days with relative humidity set at 60%. A total of 87 nodes are used for the SoilCover analysis. Ground temperatures are calculated within SoilCover by looking up latitudes. Evapotranspiration is included in the SoilCover analysis, but does not significantly affect the end result. Vegetative and freeze/thaw options were turned off in the SoilCover analysis. The cumulative results of the SoilCover analysis may be seen in the following figure.

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0 20 40 60 80 100 120 140 160 180 200Day

Flux

(mm

)

PE AE PT AT ET Precip Run Off Infil.

Figure 8 SoilCover results of numerical model with cover


-0.6-0.5-0.4-0.3-0.2-0.1

00.10.20.30.40.5

0 50 100 150 200

Time (days)

Flux

(m)

PE AE Precipitation Net Flux

Figure 9 SVFLUX results of numerical model with cover

Items to note regarding the comparative analysis are as follows:

AE separates quickly from PE at around day 33 in the SoilCover analysis. This is

impossible because there has not been enough potential evaporation at this time to drive

suctions up past 3000 kPa. AE and PE only separate past a suction of 3000 kPa (Wilson,

1997).

Generally the results are the same and indicate good agreement between SoilCover and

SVFLUX.

The source of the difference in the initial split was investigated. It was found that AE and PE split on approximately day 30 in SoilCover while not splitting until day 38-40 in SVFLUX. This variation results in the difference in the predicted AE between the two packages. The suction profiles at day 46 were plotted for each software package and it was found that the cause of the difference is due to a single node going to a very high suction (18,000 kPa) in SoilCover. This is an error caused by a lack of nodal resolution near the upper boundary.

2.1.5 SoilCover Comparison #2 A comparison between SVFLUX and SoillCover is presented in this section. SVFLUX version 5.80 and SoilCover version 4.01 will be used to run the same 1-D problem. The problem is simulated in a period of one month (i.e., 31 days). Comparisons on soil suction distributions and actual evaporation calculated using the two programs are reported in this verification manual. Project: Theoretical_Verification (SoilCover)

CompareSoilCover (SVFLUX) Model: GH-1-95R (for SoilCover) and

GH95AVE under


2.1.5.1 Model Geometry The material column is 2 m in height as shown in Figure 10. The elevation of the bottom of the problem is set at 0.00 m.

Figure 10 Schematic illustration of the 1-D material column that used in the comparison

2.1.5.2 Material Properties The material used in this comparison was gravel. The soil-water characteristic curve of the gravel is presented in Figure 11. It can be seen from Figure 11 that the air entry value of the material is approximately equal to 0.1 kPa. Plot of the unsaturated hydraulic conductivity is shown in Figure 12.

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1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

Soil suction (kPa)

Volu

met

ricw

ater

cont

ent

Figure 11 Plot of the soil-water characteristic curve for the gravel material


1.0E-17

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Soil suction (kPa

Hyd

raul

icco

nduc

tivity

(m/d

a

Figure 12 Plot of the unsaturated hydraulic conductivity for the gravel material

2.1.5.3 Boundary Condition The top of the material column is subjected to both precipitation and evaporation. More details on the precipitation and evaporation can be seen in the Climate Data section. The bottom of the material column is subjected to a constant head boundary condition (i.e., pore-water pressure equal to -5 kPa). It is noted that SoilCover requires input temperature at the bottom of the problem. A temperature of 5 degree was set for the bottom of the problem.

2.1.5.4 Climate Data Climate data used in this comparison is the data measured by weather station. Plots of the temperature data are presented in Figure 13. Plots of the relative humidity are presented in Figure 14. Plots of precipitation and potential evaporation are shown in Figure 15.

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Tem

pera

ture

(C)

MaxMinMean

Figure 13 Plot of the max, min and mean air temperatures


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ativ

ehu

mid

ity(R

H%

)

MaxMinMean

Figure 14 Plot of the max, min and mean relative humidity values

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Time (days)

Volu

me

ofw

ater

(mm

/day

)

Precipitation

Potential Evaporation

Figure 15 Plot of precipitation and potential evaporation

2.1.5.5 Comparison The 1-D problem was modeled in SoilCover with 98 nodes along the material column and 1055 nodes for SVFLUX. Calculated results for pore-water pressures along the material column using SoilCover and SVFLUX are presented in Figure 16. Calculated soil suction profiles for the material column at the end of day 31 using SVFLUX and SoilCover are presented in Figure 17. Calculated actual evaporations for the simulation period using the two programs are presented in Figure 18. The calculation results show that the two programs essentially agreed with another. There was a slight difference between the calculation results which is likely due to the following reasons:

Number of nodes,

A small difference in the best-fitted soil-water characteristic curve and the best-fitted

unsaturated hydraulic conductivity function (i.e., due to the requirements in each program

are slightly different),


Interpolation technique used in the two programs (i.e., for unsaturated hydraulic

conductivity on logarithmic scale) and

The temperature boundary condition at the bottom of the problem in SoilCover program.

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-1

0

0 5 10 15 20 25 30

Time (days)

Pore

-wat

erpr

essu

re(k

Pa)

SVFlux 1.902 SVFlux 1.799 SVFlux 1.701 SVFlux 1.599SVFlux 1.408 SVFlux 1.04 SoilCover 1.902 SoilCover 1.799SoilCover 1.701 SoilCover 1.599 SoilCover 1.408 SoilCover 1.04

Figure 16 Comparison between pore-water pressures calculated using SVFLUX and SoilCover at different elevations along the material column (i.e., 1.902 m, 1.799 m, 1.701 m, 1.599 m, 1.408 m and 1.04

m)

0

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2

-6-5-4-3-2-10

Pore-water pressure (kPa)

Elev

atio

nof

the

soil

colu

mn

(m)

SVFluxSoilCover

Figure 17 Comparison between pore-water pressure profiles calculated using SVFLUX and SoilCover at

the end of day 31


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Time (days)

Accu

mul

ativ

eof

actu

alev

apor

atio

n(m

m)

SVFlux

SoilCover

Figure 18 Comparison between cumulative actual evaporations calculated using SVFLUX and SoilCover

for 31 days of simulation.

2.1.6 Evaporation - Wilson (1990) Project: USMEP Model: LimitingFunction1997_SVFlux, WilsonPenman1994_SVFlux, EmpiricalAE_SVFlux The classic solution to the coupling of material-atmosphere equations is presented by Wilson (1990). In the PhD thesis a column of sand was subjected to drying in a laboratory environment in which the temperature and relative humidity were controlled. Measurements of actual evaporation and the distributions of temperature along the column depth were obtained, providing several measures that can be used for the verification of the numerical model.

2.1.6.1 Model geometry and boundary conditions A Modified Penman approach to the calculation of actual evaporation was presented in the thesis (hereafter termed the Wilson-Penman method). Wilson (1990) coded a 1D finite element package termed “Flux” in order to compare the physical results to a numerical solution. The geometry and configuration of the column may be seen in the following figure.

Figure 19 Numerical simulation of the drying column test (Wilson, 1990)

SoilVision Systems Ltd. Introduction 18 of 93 An initial comparison to the results obtained by Wilson (1990) was performed by Gitirana (2004) using the FlexPDE solver used by SVFLUX. The FlexPDE formulation presented by Gitirana included full coupling of the temperature partial differential equations. The results of this work are presented in Figure 20.

Figure 20 Results of Gitirana (2004) as compared to Wilson (1990)

Three approaches are available to calculate the actual evaporation: Wilson-Penman AE (Wilson, 1994), Limiting-Function AE (Wilson, Fredlund, and Barbor, 1997), and Empirical AE (Wilson, Fredlund, and Barbor, 1997). Each approach can be simulated with fully coupled water flow and heat using SVFlux and SVHeat. However this benchmark only presents uncoupled evaporative simulations using Svflux package. Please see the SVHeat Verification Manual for the results of the fully coupling simulations. NOTE: 1. The model is required to set the “Apply Surface Suction Correction” option in the

Suction tab of SVFlux model settings dialog, and 2. The correction factor is set to be –1.8.

2.1.6.2 Material properties The material properties in Wilson’s thesis are presented as follows. The ksat value used in the numerical modeling is presented as 3e-5 m/s. The unsaturated hydraulic conductivity and gravimetric water content values calculated using the Brooks and Corey estimation method are presented in Table 6.2 (p. 252). In the “FLUX” code developed by Wilson the Brooks and Corey method of representing the SWCC and unsaturated hydraulic conductivity function. General hydraulic properties of the Beaver Creek sand are presented in Table 4.1 (p. 115). In this benchmark the soil water characteristic curve (SWCC) is approximated with Fredlund and Xing (1994) approach based on the Wilson’s measured data. The parameters for SWCC and hydraulic conductivity are presented inTable 1, Figure 21 and Figure 22.


Table 1 Material properties used in the simulation of Wilson’s evaporation benchmark

Material name Material properties Method and parameters

Value unit

Beaver Creek Sand SWCC Sat vwc 0.405 m3/m3 Fredlund and Xing af 4.046 kPa nf 1.692 mf 1.181 hr 12.415 kPa Hydraulic

conductivity Saturated k 2.592 m/day

Modified Campbell Estimation

k min 1E-7 m/day Mcampbell p 15

Figure 21 Soil water characteristic curve of Beaver Creek Sand used in Wilson’s evaporation benchmark


Figure 22 Hydraulic conductivity of unsaturated Beaver Creek Sand used in Wilson evaporation

benchmark

2.1.6.3 Results and discussion Exactly replicating this benchmark is technically challenging because i) the original code made use of an “L” parameter in the Brooks and Corey estimation in order to adjust the prediction. The use of such an “L” parameter is not currently implemented in SVFlux. Therefore the Fredlund & Xing SWCC fitting curve and the Modified Campbell hydraulic conductivity fitting curve were used and adjusted to fit the data originally published by Wilson in Table 6.2. The experimental results obtained by Wilson (1990) were then again compared to SVFLUX. The results are shown in Figure 23. It can be seen from the results that a reasonable comparison is obtained. It was found the correction number is related to material properties such as the value of k min (seeTable 1). It was also found in the course of the comparison that i) the separation point between the AE and PE as well as ii) the calculated AE later on in the calculation is highly sensitive to slight variations in the representation of the SWCC and the unsaturated hydraulic conductivity curve. To improve the modeling stability, an empirical correction number in SVFlux is utilized to account for the steep suction gradient at the soil surface (see SVFlux Theory Manual for details). The correction number in this benchmark is determined by trial and error.


0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

0 5 10 15 20 25 30 35 40

Time (day)

Evap

orat

ion

(mm

/day

)

Measured Potential Evaporation

Measured AE, Column A (Wilson 1992)

Measured AE, Column B (Wilson 1992)

Simulation of Wilson PhD thesis (1990)

SVFlux simulation, Limiting-Function (1997) Surface suction correction factor = 1.8

SVFlux simulation, Wilson-Penman (1994) Surface suction correction factor = 1.8

SVFlux simulation, Empirical AE (1997) Surface suction correction factor = 1.8

Figure 23 Comparison of evaporation simulated using SVFlux with laboratory data and numerical result by Wilson (1990)

2.1.7 Evapotranspiration - Tratch (1995) Project: Evapotranspiration Model: TratchThesis1D_Final The evapotranspiration simulations performed by Tratch (1995) examine the effects of a vegetation cover on a column of material. The plant cover was allowed to develop over an entire growing season and the measured evapotranspiration fluxes were used to calibrate a 1D finite element computer model, SoilCover (Mend, 1993). The evapotranspiration features of SVFLUX were used to duplicate the experimental and numerical results. A vertical 1D model was set up with a 0.6m depth. An error limit of 0.0001 and 437 nodes were used in the SVFLUX solver to achieve the desired accuracy. The model was run for an 86 day time period allowing SVFLUX to automatically adjust the time-steps as required. The base of the model consists of a flow boundary condition using the data from the Tratch thesis in table B.5. During the experiment the base of the column was held at a constant head, therefore

SoilVision Systems Ltd. Introduction 22 of 93 the basal flow rates represent the water that entered the model from a reservoir. An initial head = 0 kPa was entered into SVFLUX, which means that the column is fully saturated at the start of the analysis. An evapotranspiration boundary condition was applied to the top of the model. This caused an evaporative flux to be applied to the top node and additionally a transpiration sink to be applied below the surface. The evaporative flux data was entered as potential evaporation as presented by Tratch, then SVFLUX computes the actual evaporation after Wilson (1997). A constant temperature of 20oC and a constant relative humidity of 85% were used in SVFLUX instead of the exhaustive diurnal datasets used by Tratch. The transpiration sink is applied below the surface to a depth corresponding to the root zone. A triangular root zone distribution was used. The root depth was held at 0 for 2 days and then increased linearly as the growing season progressed to 0.6m at day 86. The transpiration sink is also a function of the vegetative parameters of the plant cover. The leaf-area index (LAI) versus time data (Figure 24) modifies the potential evaporation to give the potential evapotranspiration. The plant limiting function (PLF) is determined from moisture limiting point of 100 kPa and a plant wilting point of 200 kPa. As the suction increases in the model, the PLF decreases from 1 at the limiting point to 0 at the wilting point. The calculated transpiration sink is a function of the potential evapotranspiration, PLF, and active root zone.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 10 20 30 40 50 60 70 80 90

Time (days)

LAI

Figure 24 Leaf-Area Index for Evapotranspiration Verification

The column was filled with an uniform silt. Tratch estimated the soil-water characteristic curve (SWCC) for the silt. The SWCC data was fit with the Fredlund and Xing fit in SVFLUX. The resulting parameters are a saturated volumetric water content of 0.371, an air-entry value of 25, an n parameter of 3, m parameter of 0.58, and hr of 137 kPa. A saturated hydraulic conductivity, ksat, of 0.001296 m/day was used for the silt and the unsaturated hydraulic conductivity is estimated from the SWCC data using the Modified Campbell method with a p = 8 and a minimum hydraulic conductivity of 1E-9 m/day. In Figure 25, it shows the results of the SVFLUX analysis compared to the measured and computed values presented by Tratch. The model was solved in 45 minutes on a Pentium 4 - 2.8 GHz computer. The SVFLUX values compare well to the Tratch analyses. The difference between the evaporation curves is likely due to choice of exact material properties and curve fitting parameters. The model is particularly sensitive to hydraulic conductivity parameters. The lower value of the transpiration sink calculated by SVFLUX can be attributed to the same effects. The divergence of the SVFLUX transpiration from the measured values after day 70 is likely due to the upper limit on the active root zone not considered for the sake of simplicity. Tratch documents a trial and error method used to calibrate the active root zone in SoilCover to match the measured data.


Figure 25 Comparisons between SVFLUX and results presented by Tratch (1995)

2.1.8 Gitirana Infiltration Examples Project: Columns Models: InfiltrationWithRO_Gitirana2005_prec1ksat;

InfiltrationWithRO_Gitirana2005_prec1p5ksat; InfiltrationWithRO_Gitirana2005_prec2ksat; InfiltrationWithRO_Gitirana2005_prec4ksat; InfiltrationWithRO_Gitirana2005_prec10ksat

Gitirana (2005) presented a series of numerical models designed to test the ability of seepage software to handle cases of varying infiltration. The specific initiative involved determining the reasonableness of runoff calculations given increasing application of top boundary flux. For the series of models created each one had a varying application intensity scaled to the saturated hydraulic conductivity of the model. The models were all unsaturated and homogeneous models but each displayed the appropriate decay in actual infiltration, which would occur as the models became saturated and the amount of runoff therefore increased. Application rates of 1x, 1.5x, 2x, 4x, and 10x saturated hydraulic conductivity were set up. The benchmarks also demonstrate the robustness of the numerical model in handling cases of increasingly high precipitation events. As the intensity of the precipitation increases it becomes increasingly difficult to handle the increase numerically. SVFlux admirably handles intensity applications up to 10x ksat. It is therefore recommended to evaluate soil cover models in light of how the intensities compare to the ksat of the top material in the numerical model. In the series of models created a constant flux is applied to the top of the model. The model eventually saturates and runoff begins to occur. This is demonstrated in the following figures which match well with the original results presented by Gitirana (2005).


0

100

200

300

400

500

600

700

800

0 1 2 3 4 5 6 7

Infil

trat

ion

rate

, mm

/day

Time, days

Precipitation rate, P = 10 ksat

P = 4 ksat

P = 2 ksat

P = 1.5 ksat

P = 1 ksat

Figure 26 Graph of infiltration rate versus time

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 1 2 3 4 5 6 7

Infil

tratio

n ra

te /

Prec

ipita

tion

rate

Time, days

Precipitation rate, P = 1 ksat

P = 4 ksat

P = 2 ksat

P = 1.5 ksat

P = 10 ksat

Figure 27 Graph of ratio of infiltration rate / precipitation rate versus time

It can be seen from the preceding figures that the SVFLUX software performs exceptionally well in solving under extreme conditions.


3 TWO-DIMENSIONAL SEEPAGE An assorted allotment of models are used to verify the validity of the solutions provided by the SVFLUX software. Comparisons are made either to textbook solutions, journal-published solutions, or other software packages.

3.1 STEADY-STATE The first steady-state model used to compare the two software packages involves flow beneath a concrete gravity dam. The second model involves flow through an earth fill dam. Each scenario begins with a brief description of the model followed by a comparison of the results from Seep/W and SVFLUX.

3.1.1 2D Cutoff Project: EarthDams Model: Cutoff

Figure 28 Mesh from SVFLUX solver (Pentland, 2000)

Figure 29 Mesh from Seep/W (Pentland, 2000)

SoilVision Systems Ltd. Introduction 26 of 93 On the left hand side of the model a reservoir is simulated by applying a head of 60m while on the right side the water table is placed at the ground surface by setting a head of 40m. All other boundaries are set to zero flow. The Figure 28 provides a view of the mesh automatically created by the SVFLUX solver. 1 With manual preparation of data for a large and complex model, the processes of subdivision and generation of error-free input may be much more costly and time-consuming than the computer execution of the model according to Desai and Abel (1972). Automatic mesh generation not only saves time in model creation but can also show where the model gradients are high. In seepage analysis a rapidly changing head can result in high gradients and these are of utmost importance when analyzing a concrete gravity dam. Seep/W (version 5.0 and earlier) does not provide fully automatic mesh generation and requires the user to draw and refine their own mesh. The user may encounter two models if they do not correctly identify areas of high gradients and refine the mesh accordingly. The first model will involve a lack of mesh resolution in areas of high gradients. Lack of proper mesh refinement decreases the chances of convergence and overall solution accuracy. The second model may involve refining the mesh in areas where gradients are minimal. Unnecessary refinement can result in more nodes than necessary and the model takes longer than necessary to solve. The SVFLUX solver ensures that there is a proper number of nodes at the beginning of the model. The SVFLUX solver also goes one step further by offering automatic mesh refinement while the model is being analyzed. This ensures that at any time during the model solution the users can be assured that the requirements of solution accuracy are being met. There is also a greater assurance that there will be a proper convergence of your seepage model.

Figure 30 Comparison of computed head contours (Seep/W results in black, SVFLUX solver in color) (Pentland, 2000)

The comparison of computed head shows that there is good agreement between the SVFLUX solver and Seep/W. However, attention should be given to two details in Figure 30. The first is the agreement in computed head near the downstream toe of the dam. From the model description it can be seen that the mesh drawn in SVFLUX near the downstream toe has greater resolution than the mesh provided by Seep/W. However, the heads computed by both software packages are essentially the same. It can be concluded that Seep/W used more nodes than required to provide the necessary accuracy and solution efficiency in this area. A second detail is the increasing difference in computed head closer to the cutoff.

SoilVision Systems Ltd. Introduction 27 of 93 From Figure 28 and Figure 29 in the model description it can be seen that the SVFLUX solver has provided a much denser mesh than Seep/W in this area. The accuracy of a finite element solution can be improved by one of two methods: either by refining the mesh, or by selecting higher order displacement models (Desai and Abel, 1972). It appears that the difference in the solved heads are the result of the denser mesh provided by the SVFLUX solver. While the differences are small in this model, the differences may become more apparent in a more complex model.

Figure 31 Computed vectors for SVFLUX solver (Pentland, 2000)

Figure 32 Computed vectors for Seep/W (Pentland, 2000)

The comparison of computed gradients shows general agreement between the two software packages. Hydraulic gradient, according to Darcy’s law is the change in head over a change in length, (Freeze and Cherry, 1979). It may be hard to detect in Figure 31 and Figure 32, but there appears to be differences in the computed gradients for the same reason as stated for the comparison of the computed heads.


Figure 33 Comparison of computed pressure contours (Seep/W results in black, SVFLUX solver in color) (Pentland, 2000)

Pressure is calculated as u = * (h-y), where (Kg/m3) is the unit weight of water, h is hydraulic head (m) and y (m) is the elevation in a two dimensional analysis. Because the calculation of pressure depends on the variable, head, it can be expected that there will be slight differences in the calculated pressures for the same reasons as there are differences in the comparison of head.

3.1.2 2D Earth Fill Dam The second steady-state example used to compare SVFLUX and Seep/W is an earth fill dam. The earth fill dam is analyzed on two scenarios. The first scenario does not include a filter material near the toe of the dam and involves the use of a review boundary condition on the downstream face of the dam to determine the length of the seepage face. The second scenario involves the use of a filter material to ensure that water does not exit the dam on the downstream face.

3.1.2.1 Review Boundary Project: EarthDams Model: Earth_Dam The first scenario involves the use of a review by pressure calculation of the location of the downstream exit point. The comparison results may be seen in the following figures.



Figure 35 Computed vectors (SVFLUX solver) (Pentland, 2000)

Figure 36 Computed vectors (Seep/W) (Pentland, 2000)

3.1.2.2 Filter Scenario Project: EarthDams Model: Earth_Fill_Dam The second scenario implements a filter underneath the downstream side the earth dam. Gradients then converge on the edge of the filter. The following figures illustrate the result comparison.


Figure 38 Computed vectors (SVFLUX solver) (Pentland, 2000)


Figure 39 Comparison of computed pressure contours (Seep/W results in black, SVFLUX solver in color) (Pentland, 2000)

3.1.3 X Component of Left to Right Flow Project: WaterFlow Model: FS_Q1_LeftRight The following model verifies the correct calculation of the x component (i.e., horizontal flow) of left-right flow. This steady-state model is verified using hand calculations.

4m

10m

X

Y

Head = 4m

Head = 3m

Flux 1 Flux 5

Figure 40 Setup of left-to-right flow

Flow calculations by hand are as follows. Q = kiA Q = 1E-4 m/s x (4m-3m)/(10m-0m) x 4m x 1m Q = 4E-5 m3/s Resultant x flux calculated by SVFLUX across sections Flux 1 and Flux 5 is 4E-5 m3/s (Error = 0.00%)


3.1.4 Simple Water Balance Project: WaterFlow Model: NatBCTest02 The following model illustrates the verification of flow across a flux section with a slightly irregular shaped model. Flow through Flux Section 1 must equal flow out of Flux Section 2 as well as the flux applied to the boundary at Flux Section 1.

Flux = 0.01 m / s / m3

2.5

5.0

10.0

2.5

2.5

Head = 2.5m

Figure 41 Confirmation of flow in and out across flux sections

The total flux applied at Flux Section 1 is equal to 0.01m3/s/ m2 x 2.5m x 1m = 0.025 m3/s. The results of the flux section comparison are as follows: Flux_1: -0.02451 m3/s Error: 2.0% Flux_2: 0.024821 m3/s Error: 0.7%

3.1.5 Decreasing Pipe Size Project: WaterFlow Model: Wedge This example confirms that mass is not lost through a decreasing pipe size. The pipe is 60m high on the left side and 10m high on the right side. If a mass of water is not lost, then Flux_1 and Flux_2 should be equal.


Y

X

Head = 10m

10.0

60.0

Head = 60m

Flux 2

Flux 1

80.0

Figure 42 Decreasing pipe model setup

Results of the analysis are as follows: Flux_2: 0.079239 m3/day Flux_1: 0.079247 m3/day

3.1.6 Axisymmetric Verification Project: WaterFlow Model: GradChange This example illustrates the design of a simple box taking the form of an Axisymmetric model. A head boundary condition of 11m is placed on the right-hand side of the model and a head boundary condition of 10m is placed on the left side of the model. The box dimensions are 10m x 10m. If the axisymmetric portion of the analysis is being properly considered then two aspects can be observed, namely:

1. The gradient in the x direction will increase from right to left, and

2. The flux sections will display the same value.


Figure 43 Axisymmetric analysis check

The final results from the flux section computations are as follows: Flux_1: 0.026200 m3/s Flux_2: 0.026189 m3/s Flux_3: 0.025805 m3/s The resultant gradient distribution is shown in the following figure. The contours show a constant increase in flow velocity from right to left.

Figure 44 Gradient change in a Axisymmetric model


3.1.7 Drain-Down Verification “Drain down” of water in an unsaturated model can be particularly prone to errors in water balance calculations. SVFLUX uses a mass-conservative formulation of the unsaturated flow equation (Celia, Bouloutas, and Zarba, 1990) intended to minimize the error associated with water balance calculations. For verification of this model a square box is described with dimensions of 1m x 1m. Initial conditions are saturated and water is allowed to drain out of the lower right of the model by slowly decreasing the head over the lower 0.1m section of wall. The final head boundary condition on the lower right section is 0.1m. It should be noted that the drainage boundary condition applied to the lower right should be applied slowly. If the boundary condition head equal to 0.1m is applied instantaneously at time equal to zero, numerical instability can result. A series of five scenarios with five different material properties were defined and run. The resultant water-balance error and material properties are presented in the following tables and figures.

Table 2 Scenarios and resultant model errors Model Title Start (m3) End (m3) Error BoxDrainT1 0.350 0.0707 3.7% BoxDrainT2 0.242 0.241888 2.3% BoxDrainT3 0.320 0.313 6.2% BoxDrainT5 0.301 0.201 4.4% BoxDrainT6 0.315 0.2397 1.0%

Table 3 Material properties for case BoxDrainT1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1e-3 1e-2 1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6

Soil suction (kPa)


1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-3 1e-2 1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6

Soil suction (kPa)

Modified Campbell



0

0.05

0.1

0.15

0.2

0.25

1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8

Soil suction (psf)


1e-10

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

1e+1

1e+2

1e+3

1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8

Soil suction (psf)

Modified Campbell


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8

Soil suction (psf)


1e-10

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

1e+1

1e+2

1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8

Soil suction (psf)

Modified Campbell


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8

Soil suction (psf)


1e-10

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

1e+1

1e+2

1e+3

1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8

Soil suction (psf)

Modified Campbell


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8

Soil suction (psf)


1e-10

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

1e+1

1e+2

1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8

Soil suction (psf)

Modified Campbell

SoilVision Systems Ltd. Introduction 36 of 93 3.1.8 Roadways Subgrade Infiltration Project: Roadways Model: TRB01, TRB02, TRB03 The following scenarios are comparisons to Seep/W solutions previously published at the Canadian Geotechnical Conference in Toronto (Barbour, Fredlund, Gan, and Wilson, 1991). A typical cross-section of a roadway was created and various infiltration rates were applied to the shoulders of the highway. The resultant plots of pore-water pressure were presented. The results obtained when using SVFLUX show reasonable agreement to the previously calculated pore-water pressures. Slight difference between the computed water pressures beneath the highway can be attributed to the increased mesh density generated by SVFLUX. The results also indicate agreement between the calculation of runoff computed by SVFLUX and Seep/W. Case 1 – Steady-State Infiltration of 0.17 mm/day

Figure 45 Seep/W results as presented in Figure 6 of Barbour et al., 1991

Figure 46 Pore-water pressures as computed by SVFLUX

SoilVision Systems Ltd. Introduction 37 of 93 Case 2 – Steady-State Infiltration of 1.7 mm/day



Case 3 – Steady-State Infiltration of 17 mm/day – Runoff Included




Case 4 – Transient Infiltration of –2.0 mm/day – Day 1

Figure 51 Seep/W results as presented in Figure 10a of Barbour et al., 1991


SoilVision Systems Ltd. Introduction 39 of 93 Case 5 – Transient Infiltration of –2.0 mm/day – Day 10

Figure 53 Seep/W results as presented in Figure 10c of Barbour et al., 1991



3.1.9 Refraction Flow Example Project: WaterFlow Model: Crespo This model provides verification of the “refraction law” when water passes from one material to another. The solution can be verified using either the flow lines or equipotentials since these lines are perpendicular in the steady-state solutions when k is isotropic. A square domain of 10m x 10m is set up. The outer perimeter is impervious with the exception of the specified head boundary conditions. The layer thicknesses are 4, 3, and 3m. The model was documented by Crespo (1993).

Figure 55 Verification of the refraction law using SVFLUX

3.1.10 Axisymmetric Aquifer Pumping Well This example model describes a pumping well that intersects a confined aquifer which is horizontal with a thickness, b. The aquifer is recharged by a constant-head lake at a distance R from the well center (Figure 56 of Todd, 1980). The model was solved analytically almost 130 years ago. The model is also presented by Chapuis, 2001.


0

5

10

15

20

25

30

35

0.1 1 10 100

Radial distance r (m)

Hyd

raul

iche

adh

(m)

Todd, 1980

SVFlux

Figure 56 Comparison between SVFLUX and results calculated by Todd (1980)

3.1.11 Dam Flow Two models are presented to show verification of flow through a dam cross-section. The solution was published by Bowles (1984). Results are presented in the following figure. Project: EarthDams Model: Bowles95a

Figure 57 Steady-state conditions in a homogeneous dam for comparison of numerical results with

approximate results of Bowles (1984)

Bowles estimated the flow rate Q by two approximate methods that yielded 1.10e-3m3 /(min*m) and 1.28e-3m3 /(min*m). SVFLUX yielded a flow rate of 1.41e-3m3 /(min*m). The emergence of the water table on the downstream slope is at an elevation of approximately 10.0m whereas Bowles estimated the exit point at approximately 6.5m.


3.1.12 Dupuit Model The following model illustrates calibration of SVFLUX to the solution originally presented by Dupuit in 1863. The model examines an unconfined aquifer with an effective infiltration W of 0.4 m3/(m2●year). The aquifer is 35 m thick and 3000 m long. The approximate solution to the model may be found in many textbooks as:

22

21

21

22 xx

KWhh [ 1 ]

A transformation changing the long dimension from 3000 m to 300 m was applied to both the Seep/W solution and the SVFLUX solution. The reduction allows a significant reduction in the number of elements, which are needed in the solution. The results are presented in the following figures. Project: WaterFlow Model: Dupuit

Figure 58 Steady-state seepage as calculated by Seep/W for an unconfined aquifer (ksat=1.0 x 10-4 m/s)

recharged by an effective infiltration W (Chapuis, 2001): (a) finite element mesh of 210 elements, (b) finite element mesh of 3660 elements

Figure 59 Steady-state seepage calculated by SVFLUX for an unconfined aquifer recharged by an

effective infiltration W


Figure 60 Finite element mesh refined according to areas of critical gradients in the unsaturated zone. A

total of 1589 nodes were used to achieve the above solution.

3.1.13 Well Object vs Rectangle The purpose of the models in this section is to verify that the Well object can successfully represent piezometric wells. A model with a classic rectangle representing a well is compared to a model with a Well object. The results of the model with rectangle are then compared to the model with the Well object. The Well object uses a sink mechanism to simulate pumping in piezometric wells (see the SVFlux Theory Manual for a further description). Project: WellPumping Model: Well_Object_2D and Narrow_Rectangle_2D

a) Well_Object_2D b) Narrow_Rectangle_2D

Figure 61 Model Well_Object_2D and model Narrow_Rectangle_2D In the model Well_Object_2D (thereafter referred to as the “Well Model”), there is a Well object in the center of the model domain. The vertical screen length of the well is 20 m which starts at elevation 5 m and proceeds up to an elevation of 25 m. The well is enclosed by flux sections Flux1, Flux2, Flux3 and Flux4 as shown in Figure 61. It is therefore possible to investigate the total flow into the well through the values reported by the flux sections. The model Narrow_Rectangle_2D (thereafter referred to as the “Rectangular Model”) is the same as the model Well_Object_2D but makes use of a narrow rectangle to replace the well object. The vertical length of the rectangle is 20 m.

SoilVision Systems Ltd. Introduction 44 of 93 R is used to represent the pumping rate applied to the rectangle in the Rectangular Model. W is used to represent the pumping rate applied to the well in the Well Model. It should be noted that the pumping rate applied in the rectangular model is specified as a flow, q, per unit length of the sidewall (R). The applied pumping rate of the well (W) is specified as a total flow, Q, which is spread over the entire vertical height of the well screen. The applied total pumping rate, Q of the rectangle may be calculated as 2 * L * R in the Rectangular Model, where L is the side length of the rectangle. This applied total pumping rate is referred to as Applied Pumping Rates. In order to compare these two models, the rate W, is specified in the Well Model so that the Applied Pumping Rates are kept the same in the two models. The relationship between W and R is defined as W = 2 * L * R, where L is the side length of the rectangle in the Rectangular Model or the screen length in the Well Model since these two lengths are identical (20 m). The two models were run on a group of different R and W input values as shown in Table 7. It is important to realize that there are three numbers which are of significance when measuring the performance of such a system. The three numbers are as follows:

1. Applied pumping rate: This number is the applied pumping rate as measured in terms

of m3 / T or m3 / T / L.

2. Reported pumping rate: This is the rate of pumping as reported by the software. This is

reported as a single number for well objects. For rectangular objects, this number is taken

as the reported boundary flux on the well region.

3. Flux section box: This number is the reported flow across flux sections surrounding the

well. In the Rectangular model these flux sections should accurately match the applied

pumping rates. It is the assumption when the measuring flow on the Well Model that the

zone of influence of the sink function does not extend outside of the flux sections.

Table 8 Fluxes on the Well Model and the Rectangular Model

The Rectangular Model The Well Model

R (m3/day

/m2)

Reported Flux

(m3 /day) Error (%)

W (m3/day)

Reported Flux

(m3/ day)

Error (%)

Applied Pumping

Rate (m3/day) -0.1 3.98 0.50% -4 4.07 1.73% -4 -0.2 7.96 0.50% -8 8.14 1.73% -8 -0.3 11.94 0.50% -12 12.21 1.73% -12 -0.4 15.92 0.50% -16 16.27 1.67% -16 -0.5 19.9 0.50% -20 20.34 1.69% -20 -0.6 23.88 0.50% -24 24.42 1.65% -24

The column on the right-hand side in above table is the Applied Pumping Rate of the both models. The first three columns represent data from the Rectangular Model, the reported total flux across the rectangle measured from the four flux sections and the error between the actual flux and the Applied Pumping Rate. The next three columns represent data from the Well Model, the actual total flux across the well measured from the four flux sections and the error between the actual flux and the Applied Pumping Rate. The equation by which Errors on the column 3 (or 6) is calculated by the equation below:

2 * ||Column2(or 5)| – |Column7|| / (|Column2(or 5)| + | Column7|)

SoilVision Systems Ltd. Introduction 45 of 93 Since all of the errors are small, it illustrates both the well object and the narrow rectangle approaches can successfully represent the piezometric wells. The distributions of Pore Water Pressure (PWP) and Head (H) in both the Well Model and the Rectangular Model are the same for each row in the above table. The Figure 62 and Figure 63 represent the contour graphs of the Pore Water Pressure and Head of the Well Model and the Rectangular Model respectively for the case of the sixth row of the table, i.e., the R is specified as –0.6 (m3/day/m2) in the Rectangular Model and the W is specified as –24 (m3/day) in the Well Model.

a) Pore Water Pressure b) Head

Figure 62 Contours of Pore Water Pressure and Head in the Well Model when R = -24 (m3 / day)

a) Pore Water Pressure b) Head Figure 63 Contours of Pore Water Pressure and Head in the Rectangular Model when P = -0.6 (m3 / day

/ m2)

3.1.14 2D Well Object with Head Boundary Condition Project: WellPumping Model: Well_Object_2D_head and Narrow_Rectangle_2D_head These models are designed to verify the head boundary conditions as applied to a 2D well object within SVFlux.

SoilVision Systems Ltd. Introduction 46 of 93 3.1.14.1 Purpose Two models are created in this verification in order to test the implementation of a head boundary condition on a well object. A second model makes use of model geometry to verify the approach. The goal of this verification is to verify that the same results can be obtained by both approaches.

3.1.14.2 Geometry and Boundary Conditions The model Well_Object_2D_head geometry is shown in Figure 64 and the model Narrow_Rectangle_2D_head geometry is shown in Figure 65.

Figure 64 Model Well_Object_2D_head

Figure 65 Model Narrow_Rectangle_2D_head

These two models are all SVFlux Steady State 2D models.

SoilVision Systems Ltd. Introduction 47 of 93 The boundary conditions of the well object in model Well_Object_2D_head are as follows: Boundary Type: Head 20 m Influence Distance: 0.05 m Line Mesh Spacing: 0.32 m Growth Factor: 2 The boundary conditions of the narrow rectangle in model Narrow_Rectangle_2D_head are as follows:

Top and Bottom: No BC Two sides: Head 20 m

The other boundary conditions are the same for the two models. A head of 30 m boundary is defined for the two sides of the big rectangle and No BC is specified for the top and bottom of the big rectangle. 6 stages with values 10, 100, 1000, 10000, 100000, 1000000 are utilized for both models.

3.1.14.3 Material Properties The both models use the same material properties. Saturated hydraulic conductivity: 1 m/day Porosity: 0.3

Fredlund and Xing Fit parameters: af: 37 kPa nf: 1 mf: 2 hr: 346 kPa The rest of the parameters are taken as defaults of the software.

3.1.14.4 Results and Discussions

3.1.14.4.1 Distributions of the Pore-Water Pressure (uw) The distributions of the Pore-Water Pressure (uw) of model Well_Obect_2D_head are shown in Figure 66 and the distributions of the Pore-Water Pressure (uw) of the model Narrow_Rectangle_2D_head are shown in Figure 67.


Figure 66 Distributions of the Pore-Water Pressure (uw) of the Model Well_Object_2D_head at the last

stage (stage 6)

Figure 67 Distributions of the Pore-Water Pressure (uw) of the Model Narrow_Rectangle_2D_head at

the last stage (stage 6) The results of the models show the two models have very similar pore-water pressure (uw) distributions.

SoilVision Systems Ltd. Introduction 49 of 93 3.1.14.4.2 Distributions of the Head (h) The distributions of the Head (h) of model Well_Obect_2D_head are shown in Figure 68 and the distributions of the Head (h) of the model Narrow_Rectangle_2D_head are shown in Figure 69.

Figure 68 Distributions of the Head (h) of the Model Well_Object_2D_head at the last stage (stage 6)

Figure 69 Distributions of the Head (h) of the Model Narrow_Rectangle_2D_head

at the last stage (stage 6) The results of the models show the two models have very similar head (h) distributions.


3.1.14.4.3 Flux Flow The total flux flow to the well can be obtained through calculating the flows crossing each flux section. The total flux flow to the well is 27.67 (m3/day) in the mode Well_Object_2D_head. The total flux flow to the well is 27.75 (m3/day) in the mode Narrow_Rectangle_2D_head. There is very small different between these two flux flows (About 0.29 % different). The flux flow of the well object is significantly dependent on the parameter Influence Distance and needs to be further investigated.

3.1.15 2D Well Object with Review by Pressure Boundary Condition Project: WellPumping Model: Well_Object_2D_review and Narrow_Rectangle_2D_review These models are designed to verify the review by pressure boundary conditions as applied to a 2D well object within SVFlux.

3.1.15.1 Purpose Two models are created in this verification in order to test the implementation of a review by pressure boundary condition on a well object. A second model makes use of model geometry to verify the approach. The goal of this verification is to verify that the same results can be obtained by both approaches.

3.1.15.2 Geometry and Boundary Conditions The model Well_Object_2D_review geometries are shown in Figure 70 and the model Narrow_Rectangle_2D_review geometries are shown in Figure 71.

Figure 70 Model Well_Object_2D_review


Figure 71 Model Narrow_Rectangle_2D_review

These two models are all SVFlux Steady State 2D models. The boundary conditions of the well object in model Well_Object_2D_review are as follows: Boundary Type: Review by Pressure Influence Distance: 0.04 m Line Mesh Spacing: 0.02 m Growth Factor: 2 The boundary conditions of the narrow rectangle in model Narrow_Rectangle_2D_review are as follows:

Top and Bottom: No BC Two sides: Review by Pressure

The other boundary conditions are the same for the two models. A head of 30 m boundary is defined for the two sides of the big rectangle and No BC is specified for the top and bottom of the big rectangle. 6 stages with values 10, 100, 1000, 10000, 100000, 1000000 are utilized for both models.

3.1.15.3 Material Properties The both models use the same material properties. Saturated hydraulic conductivity: 1 m/day Porosity: 0.3

Fredlund and Xing Fit parameters: af: 37 kPa nf: 1 mf: 2 hr: 346 kPa The rest of the parameters are taken as defaults of the software.

SoilVision Systems Ltd. Introduction 52 of 93 3.1.15.4 Results and Discussions

3.1.15.4.1 Distributions of the Pore-Water Pressure (uw) The distributions of the Pore-Water Pressure (uw) of model Well_Obect_2D_review are shown in Figure 72 and the distributions of the Pore-Water Pressure (uw) of the model Narrow_Rectangle_2D_review are shown in Figure 73.

Figure 72 Distributions of the Pore-Water Pressure (uw) of the Model Well_Object_2D_review at the

last stage (stage 6)

Figure 73 Distributions of the Pore-Water Pressure (uw) of the Model Narrow_Rectangle_2D_review at


SoilVision Systems Ltd. Introduction 53 of 93 3.1.15.4.2 Distributions of the Head (h) The distributions of the Head (h) of model Well_Obect_2D_review are shown in Figure 74 and the distributions of the Head (h) of the model Narrow_Rectangle_2D_review are shown in Figure 75.

Figure 74 Distributions of the Head (h) of the Model Well_Object_2D_review at the last stage (stage 6)

Figure 75 Distributions of the Head (h) of the Model Narrow_Rectangle_2D_review at the last stage

(stage 6) The results of the models show the two models have very similar head (h) distributions.

SoilVision Systems Ltd. Introduction 54 of 93 3.1.15.4.3 Flux Flow The total flux flow to the well can be obtained through calculating the flows crossing each flux section. The total flux flow to the well is 44.91 (m3/day) in the mode Well_Object_2D_review. The total flux flow to the well is 44.23 (m3/day) in the mode Narrow_Rectangle_2D_review. There is very small different between these two flux flows (About 1.53 % different). The flux flow of the well object is significantly dependent on the parameter Influence Distance and needs to be investigated further.

3.2 TRANSIENT STATE A number of transient models were used to verify the SVFLUX software. The following models demonstrate the successful ability of the SVFLUX software to provide accurate transient solutions.

3.2.1 Transient Reservoir Filling The model involves the filling of a reservoir. This section begins with a brief description of the model followed by a comparison of the results obtained from both the SVFLUX and Seep/w software packages.

H=0

Q=0

Q=0

Qo=0, H=10

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56-202468

101214

Figure 76 Reservoir filling description (Pentland, 2000)

The earth fill dam considered is 28m high 52m in length and incorporates a filter on the downstream toe of the dam. The initial conditions of head were obtained by first solving a steady-state run of the model with the head on the upstream face of the dam set to 4m and a head of 0m on the lower portion of the filter. All other boundaries were set to zero flow. The results from the steady-state analysis were then imported as the initial conditions for the transient analysis. While the material properties remain the same in the transient flow model, the boundary conditions change slightly. A head of 10 m is set on the upstream face of the dam to simulate a full reservoir condition. The model is run for 16,383 hours. Below, the results from times 15, 255, 1023, 4095, and 16383 hours are provided.


1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

-200-150-100-50050100

Pore water pressure (kPa)

Hydr

aulic

cond

uctiv

ity(m

/s)

Dam material

Toe Drain Material

Figure 77 Material Properties (Pentland, 2000)

Results

Figure 78 Comparison of computed head contours at time 15 hours (Seep/W results in black, SVFLUX

solver in color) (Pentland, 2000)

Figure 79 Comparison of computed pore-water pressure contours at time 15 hours (Seep/W results in

black, SVFLUX solver in color) (Pentland, 2000)


Figure 80 Comparison of computed pore-water head contours at time 255 hours (Seep/W results in

black, SVFLUX solver in color) Pentland (2000)

Figure 81 Comparison of computed pore-water pressure contours at time 255 hours (Seep/W results in black, SVFLUX solver in color) (Pentland, 2000)

Figure 82 Comparison of computed head contours at time 1023 hours (Seep/W results in black, SVFLUX solver in color) (Pentland, 2000)


Figure 83 Comparison of computed pore-water pressure contours at time 1023 hours (Seep/W results in

black, SVFLUX solver in color) (Pentland, 2000)

Figure 84 Comparison of computed head contours at time 4095 hours (Seep/W results in black, SVFLUX

solver in color) (Pentland, 2000)

Figure 85 Comparison of computed pressure contours at time 4095 hours (Seep/W results in black, SVFLUX solver in color) (Pentland, 2000)


Figure 86 Comparison of computed head contours at time 16383 hours (Seep/W results in black, SVFLUX solver in color) (Pentland, 2000)

Figure 87 Comparison of computed pressure contours at time 16383 hours (Seep/W results in black, SVFLUX solver in color) (Pentland, 2000)

It can be seen from the above figures that there is good agreement between the results from the packages. Some differences appear, likely due to differences in temporal and spatial discretization between the two programs (Pentland, 2000).

3.2.2 Celia Infiltration Example Celia (1990) presented an infiltration example comparing finite difference and finite element solutions. The example represents an approximate description of a field site in New Mexico. The model involved unsaturated infiltration into a column of 100cm in depth. The paper by Celia outlines the solution offered by both finite difference and finite element methods. The time-steps are varied to illustrate the possible variation in solution profiles. The resulting profiles presented by Celia are shown in Figure 88. Project: WaterFlow Model: Celia1990


Figure 88 (a) Finite difference and (b) finite element solutions using h-based equations with data of (13). Finite difference solution using Dt=2.4 min did not converge in nonlinear iteration (Celia (1990))

The model was duplicated in the SVFLUX software package. Material properties presented in the paper were converted from a functional to a digital representation. The results of SVFLUX as compared to the finite element results presented by Celia are shown in Figure 89. Preliminary sensitivity analysis indicates that differences between the solutions can be attributed to differences in the representation of material properties. SVFLUX results indicate correct solution of the infiltration model. The results also validate the automatic time-step selection used by SVFLUX in solving transient models. Numerical oscillations commonly encountered by the selection of large time-steps in finite element solvers can be minimized using SVFLUX.

-1200

-1000

-800

-600

-400

-200

0

0 20 40 60 80 100 120

Depth (cm)

Pres

sure

hea

d (c

m)

Dense griddt = 20 secdt = 2.4 mindt = 12 mindt = 60 minSVFlux

Figure 89 Difference between finite element solutions presented by Celia

and the solution obtained using SVFLUX

SoilVision Systems Ltd. Introduction 60 of 93 3.2.3 Evapotranspiration - Triangular and Rectangular Root Distributions SVFLUX supports 2 methods for specifying the distribution of roots in a model in which evapotranspiration is applied. The triangular distribution method assumes the material area occupied by roots is 0 at the maximum root depth and increases linearly to the ground surface or top of the active root zone. The rectangular distribution method assumes a constant area occupied by the roots within the root zone. Figure 90 and Figure 91 illustrate the triangular and rectangular root distributions respectively. The hypothesis for this test is that both distributions should pull the same amount of water from the model. Project: VerifySVFLUX Model: ED_Initial, T2DexampleET, T2DExampleETRect

Figure 90 Triangular Potential Root Uptake

Figure 91 Rectangular Potential Root Uptake The triangular and rectangular root distribution method both result in the same potential root uptake. The same volume of water will be pulled from the material independent of which method is used. The models T2DExampleET and T2DExampleETRect have been created to verify the above condition. The model consists of a single rectangular material region in 2D, 6m wide and 3m deep. An evapotranspiration boundary has been applied to the right half of the top boundary. A head boundary condition exists at the base of the model.

SoilVision Systems Ltd. Introduction 61 of 93 Model T2DExampleET using a triangular root distribution calculates a total transpiration flux of –0.684442 m3/day while model T2DExampleETRect using a rectangular root distribution calculates a total transpiration flux of –0.688750 m3/day. This gives a percent difference of only 0.63%. Even though the total water pulled form the model is the same the following contour plots show how the water is being pulled out at different rates depending on depth for the triangular distribution while being pulled out at the same rate for the rectangular distribution.

Figure 92 Triangular Root Distribution – Transpiration Sink Contour

Figure 93 Rectangular Root Distribution – Transpiration Sink Contour


3.2.4 A Transient 2-D Infiltration Problem Tsai et al. (1993) developed a finite-analytic (FA) method to solve problems associated with water flow in unsaturated soils. They published the solutions obtained from their method for a transient 2-D flow of water under a strip source infiltration of constant flux for a uniform clay loam soil. They also compared the solutions obtained from their method with those obtained from analytic and finite element solutions. The below figures compare the results for two different times; 36 min and 72 min. This verification is for Case 1 as present by Tsai et al. (1993). The paper also presents Case 2 (same a Case 1, but with a coarse sand) and Case 3 (similar to Case 1 and Case 2 with alternating layers of silty loam and coarse sand) Project: WaterFlow Model: Tsai1993_Case1, Tsai1993_Case2, Tsai1993_Case3

Figure 94 Contour of pressure head for simulation of infiltration with constant flux strip source for Clay

Loam: Time = 36 min (Tsai et al. 1993)


Loam: Time = 72 min (Tsai et al. 1993)

SoilVision Systems Ltd. Introduction 63 of 93 The problem is duplicated in the SVFLUX software. The pressure head contours obtained from SVFLUX are compared with those obtained from analytic solutions and finite element solutions presented by Tsai et al (1993) (Figure 96 and Figure 97). The results show that the SVFLUX solutions are comparable to the analytic solutions for all pressure head values except for the lowest pressure head value. The discrepancy between the results obtained from SVFLUX for the pressure head value of -200 cm may be attributed to the SWCC. The sensitivity analysis indicated that the solutions are sensitive to the representation of the SWCC.

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70x (cm)

y(c

m)

SVFlux

Analytical Solution from PaperH = -200 cm

H = - 160 cm

H = - 120 cm

H = -80

Finite Element Solutions Fromthe Paper


Loam resulted from SVFLUX, Analytical, and Finite Element solutions : Time = 36 min

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70x (cm)

y(c

m)

SVFlux

Analytical Solutions from PaperH = -200 cm

H = - 160 cm

H = - 120 cm

H = -80

Finite Element Solutions FromPaper


Loam resulted from SVFLUX, Analytical, and Finite Element solutions: Time = 72 min

SoilVision Systems Ltd. Three-Dimensional Seepage 64 of 93

4 THREE-DIMENSIONAL SEEPAGE Three-dimensional seepage models are presented in this chapter to provide a forum to compare the results of the SVFLUX solver to the results of other seepage software and other documented examples. Both steady-state and transient models are considered.

4.1 STEADY-STATE The following models are presented as steady-state verification when time is assumed to be infinite.

4.1.1 Wedge Example The following model illustrates the use of a wedge to perform calculations for a three-dimensional analysis. Flux sections are placed at various points in the model to compute water flow volumes. The hydraulic conductivity was set at 0.1 m/s. The error limit of the software had to be decreased to 0.00001 m/s in order to increase the solution accuracy. Project: WaterFlow Model: Wedge

Flux 1

Flux 2

Flux 3

Flux 4

508348

50

70

X

Y

Z

Figure 98 Wedge model geometry

Flux 1: 48.260 m3/s Error: 0.00% Flux 2: 48.230 m3/s Error: 0.06% Flux 3: 48.228 m3/s Error: 0.07% Flux 4: 47.705 m3/s Error: 1.15%


4.1.2 Cube Example The following example defines a cube of unit dimensions. Three vertical flux sections are then placed through the model. The hydraulic conductivity is input as 0.1 m/s. All flux sections should yield the same results if the model is being solved properly. Project: VerifySVFLUX Model:

1.0

0.2

1.0

0.8

1.0

Head = 0.2

Head = 0.8

Flux 3Flux 2Flux 1

Z

Y

Figure 99 2-D cross-section of 3-D cube

Flux 1: 0.6 m3/s Error: 0.00% Flux 2: 0.6 m3/s Error: 0.00% Flux 3: 0.6 m3/s Error: 0.00%

4.1.3 Toe Example The following model tests flow through a model containing a toe section. A toe section may be added to irregular models to increase the accuracy of the flux calculations.

Flux = 0.1m / s / m3 2

Flux 2

Flux 3

Flux 4

Head = 0.2m

Y

0.2

0.11.01.0

1.0

1.0Z

X

Figure 100 Geometry of cube and wedge model

SoilVision Systems Ltd. Three-Dimensional Seepage 66 of 93 Flux input: 0.1 m3/s Flux 2: 0.1000 m3/s Error: 0.00% Flux 3: 0.09856 m3/s Error: 1.44% Flux 4: 0.1002 m3/s Error: 0.20%

4.1.4 3D Well Object vs Cylinder The purpose of the models in this section is to verify that the well object can successfully represent piezometric wells in a 3D scenario. A model with a cylinder representing a well is first created in order to prove the new Well object approach. The results of the cylinder are then compared to the new Well object approach. The new well object approach requires far fewer nodes in order to calculate resulting fluxes so there are significant benefits with the proposed new approach. The Well object uses a sink mechanism to simulate pumping in piezometric wells (see the SVFlux Theory Manual for a further description). Project: WellPumping Model: Well_Object_3D and Slender_Cylinder_3D

Figure 101 Well_Object_3D model

Figure 102 Slender_Cylinder_3D model

In model Well_Object_3D (thereafter referred to as the “Well Model”), there is a Well object in the center of the model domain. The vertical screen length of the well is 20 m measured from the bottom of the well. The well object represents a well with zero radius. The well is enclosed by flux sections Flux1, Flux2, Flux3 and Flux4 as shown in Figure 101. So it is possible to investigate the

SoilVision Systems Ltd. Three-Dimensional Seepage 67 of 93 total flow into the well when a pumping rate is applied. The model Slender_Cylinder_3D (thereafter referred to as the “Cylinder Model”) is the same as the model Well_Object_3D but makes use of a slender cylinder to replace the well object. The length L of the cylinder is 20 m and the radius R of the cylinder is 0.3 m. R is used to represent the pumping rate applied to the cylinder in the Cylinder Model. W is used to represent the pumping rate of the well in the Well Model. Values of the R and the W are input by user. It should be noted that the pumping rate applied in the Cylinder model is specified as a flow, q, per unit area of the sidewall (R). The applied pumping rate of the well (W) is specified as a total flow, Q, which is spread over the entire vertical height of the well screen. The applied total pumping rate, Q of the cylinder may be calculated as 2 * * r * L * R in the Cylinder Model, where L is the length of the cylinder and r is the radius of the cylinder. These applied total pumping rates are referred to as Applied Pumping Rates. In order to compare these two models, the rate W, is specified in the Well Model so that the Applied Pumping Rates are kept the same in the two models. The relationship between of W and R is defined as W = 2 * * r * L * R, where L is the side length of the rectangle in the Cylinder Model or the screen length in the Well Model since these two lengths are identical (20 m). The r is the radius of the cylinder (0.3 m) The two models were run on a group of different R and W input values as shown in Table 8. It is important to realize that there are three numbers which are of significance when measuring the performance of such a system. The three numbers are as follows:

1. Applied pumping rate: This number is the applied pumping rate as measured in terms of m3 / T or m3 / T / A.

2. Reported pumping rate: This is the rate of pumping as reported by the software. This is reported as a single number for well objects. For cylinder objects, this number is taken as the reported boundary flux on the well region.

3. Flux section box: This number is the reported flow across flux sections surrounding the well. In the Cylinder model these flux sections should accurately match the applied pumping rates. It is the assumption when the measuring flow on the Well Model that the zone of influence of the sink function does not extend outside of the flux sections.

Table 9 Fluxes on the Well Model and the Cylinder Mode

The Cylinder Model The Well Model

P (m3/day

/m2)

Reported Flux

(m3/ day) Error (%)

Run Time (mm:ss)

R (m3/ day)

Reported Flux

(m3/day) Error (%)

Run Time (mm:ss)

Applied Pumping

Rate (m3/day)

0.1 3.72 1.33% 11:04 3.77 3.71 1.60% 0:17 3.77 0.2 7.44 1.33% 11:04 7.54 7.46 1.06% 0:16 7.54 0.3 11.16 1.33% 11:04 11.31 11.2 1.06% 0:17 11.31 0.4 14.88 1.33% 11:03 15.08 14.9 1.06% 0:17 15.08 0.5 18.6 1.33% 11:03 18.85 18.6 1.12% 0:17 18.85 0.6 22.33 1.29% 11:02 22.62 22.4 1.06% 0:17 22.62

The last column in above table is the Applied Pumping Rate of the both models. The first four columns are the data from the Cylinder Model, P, the actual total flux cross the cylinder measured from the four flux sections, the error between the actual flux and the applied pumping rate and run time. The next four columns are the data from the Well Model, the actual total flux across the well measured from the four flux sections, the error between the actual flux and the applied pumping rate and the run time. The equation by which Errors on the column 3 (or 7) is calculated by the equation below:


2 * ||Column2(or 6)| – |Column9|| / (|Column2(or 6)| + | Column9|) Since all of the errors are small, it illustrates both well object and slender cylinder approaches can successfully represent the piezometric wells. It also shows that the well model is much faster than the cylinder model. It is about 41 times faster in the current examples. The distributions of Pore Water Pressure (PWP) and Head (H) of both the Well Model and the Cylinder Model are the same for each row in above table. The Figure 103 and Figure 104 are the contour graphs of the Pore Water Pressure and Head of the Well Model and the Cylinder Model respectively for the case of the sixth row of the table, for example, the P is specified as –0.6 (m3/day/m2) in the cylinder model and the Rate is specified as –22.62 (m3/day) in the Well Model.


Figure 103 Contours of Pore Water Pressure and Head in the well model when R = -22.62 (m3 / day)


Figure 104 Contours of Pore Water Pressure and Head in the cylinder model when R = 0.62 (m3 / day / m2)

4.1.5 3D Well Object with Head Boundary Condition Project: WellPumping Model: Well_Obect_3D_head and Slender_Cylinder_3D_head These models are designed to verify the head boundary conditions as applied to a 3D well object within SVFlux.

SoilVision Systems Ltd. Three-Dimensional Seepage 69 of 93 4.1.5.1 Purpose Two models are created in this verification in order to test the implementation of a head boundary condition on a well object. A second model makes use of model geometry to verify the approach. The goal of this verification is to verify that the same results can be obtained by both approaches.

4.1.5.2 Geometry and Boundary Conditions The model Well_Obect_3D_head geometries are shown in Figure 105 and the model Slender_Cylinder_3D_head geometries are shown in Figure 106.


Figure 106 Model Slender_Cylinder_3D_head

These two models are all SVFlux Steady State 3D models. The boundary conditions of the well object in model Well_Object_3D_head are as follows: Boundary Type: Head 16 m Influence Distance: 0.23 m Line Mesh Spacing: 0.02 m Growth Factor: 3

SoilVision Systems Ltd. Three-Dimensional Seepage 70 of 93 The boundary conditions of the slender cylinder in model Slender_Cylinder_3D_head are as follows:

Top and Bottom: No BC Side: Head 16 m

The other boundary conditions are the same for the two models. A head of 20 m boundary is defined for the sides of the big cube and No BC is specified for the top and bottom of the big cube. 6 stages with values 10, 100, 1000, 10000, 100000, 1000000 are utilized for both models.

4.1.5.3 Material Properties The both models use the same material properties. Saturated hydraulic conductivity: 1 m/day Porosity: 0.3 Fredlund and Xing Fit parameters: af: 37 kPa nf: 1 mf: 2 hr: 346 kPa The rest of the parameters are taken as by default of the software.


4.1.5.4.1 Distributions of the Pore-Water Pressure (uw) The distributions of the Pore-Water Pressure (uw) of model Well_Obect_3D_head are shown in Figure 107 and the distributions of the Pore-Water Pressure (uw) of the model Slender_Cylinder_3D_head are shown in Figure 108.

Figure 107 Distributions of the Pore-Water Pressure (uw) of the Model Well_Object_3D_head at the last

stage (stage 6)


Figure 108 Distributions of the Pore-Water Pressure (uw) of the Model Slinder_Cylinder_3D_head at the


The results of the models show the two models have very similar pore-water pressure (uw) distributions.

4.1.5.4.2 Distributions of the Head (h) The distributions of the Head (h) of model Well_Obect_3D_head are shown in Figure 109 and the distributions of the Head (h) of the model Slender_Cylinder_3D_head are shown in Figure 110.

Figure 109 Distributions of the Head (h) of the Model Well_Object_3D_head at the last stage (stage 6)


Figure 110 Distributions of the Head (h) of the Model Slinder_Cylinder_3D_head at the last stage (stage

6)

The results of the models show the two models have very similar head (h) distributions.

4.1.6 3D Well Object with Review by Pressure Boundary Condition Project: WellPumping Model: Well_Obect_3D_review and Slender_Cylinder_3D_review These models are designed to verify the review by pressure boundary conditions as applied to a 3D well object within SVFlux.

4.1.6.1 Purpose Two models are created in this verification in order to test the implementation of a review by pressure boundary condition on a well object. A second model makes use of model geometry to verify the approach. The goal of this verification is to verify that the same results can be obtained by both approaches.

4.1.6.2 Geometry and Boundary Conditions The model Well_Obect_3D_review geometries are shown in Figure 111 and the model Slender_Cylinder_3D_review geometries are shown in Figure 112.



Figure 112 Model Slender_Cylinder_3D_head

These two models are all SVFlux Steady State 3D models. The boundary conditions of the well object in model Well_Object_3D_review are as follows:

Boundary Type: Review by Pressure Influence Distance: 0.23 m Line Mesh Spacing: 0.02 m Growth Factor: 3 The boundary conditions of the slender cylinder in model Slender_Cylinder_3D_review are as follows:

Top and Bottom: No BC Side: Review by Pressure

SoilVision Systems Ltd. Three-Dimensional Seepage 74 of 93 The other boundary conditions are the same for the two models. They are specified as a head 20 m for the sides of the big cube and No BC for the top and bottom of the big cube. 6 stages with values 10, 100, 1000, 10000, 100000, 1000000 are utilized for both models.

4.1.6.3 Material Properties The both models use the same material properties. Saturated hydraulic conductivity: 1 m/day Porosity: 0.3 Fredlund and Xing Fit parameters: af: 37 kPa nf: 1 mf: 2 hr: 346 kPa The rest of the parameters are taken as by default of the software.


4.1.6.4.1 Distributions of the Pore-Water Pressure (uw) The distributions of the Pore-Water Pressure (uw) of model Well_Obect_3D_review are shown in Figure 113 and the distributions of the Pore-Water Pressure (uw) of the model Slender_Cylinder_3D_review are shown in Figure 114.

Figure 113 Distributions of the Pore-Water Pressure (uw) of the Model Well_Object_3D_review at the



Figure 114 Distributions of the Pore-Water Pressure (uw) of the Model Slinder_Cylinder_3D_review at


4.1.6.4.2 Distributions of the Head (h) The distributions of the Head (h) of model Well_Obect_3D_review are shown in Figure 115 and the distributions of the Head (h) of the model Slender_Cylinder_3D_review are shown in Figure 116.

Figure 115 Distributions of the Head (h) of the Model Well_Object_3D_review at the last stage (stage 6)


Figure 116 Distributions of the Head (h) of the Model Slinder_Cylinder_3D_review at the last stage

(stage 6) The results of the models show the two models have very similar head (h) distributions.

4.1.7 Confined Aquifer 3D Ideal Project: WellPumping Model: ConfinedAquifer3DIdeal These models are designed to verify the review by pressure boundary conditions as applied to a 3D well object within SVFlux.

4.1.7.1 Purpose A three dimensional SVFlux model of an ideal confined aquifer was constructed to verify the implementation of wells against the classical Theis system. The Theis system is one-dimensional, of infinite extent, and includes the well as a boundary condition. The SVFlux 3D model contained a well at the center, and a large domain to minimize the effect of the boundary on the drawdown near the well. SVFlux models wells as a sink term in the differential equations. This eliminates the singularity in the Theis model boundary condition, but spreads the local effect of the well over an area that may be larger than the physical well. The extent of the spread is user adjustable.

4.1.7.2 Theis System Given a confined aquifer with transmissivity T, and storativity S, the Theis solution (Freeze and Cherry, 1979) for drawdown due to a pumping well is

0 ( , ) ,

4

v

uh h r et Q dv

T v

where h0 is the initial head, Q is the well pumping rate, and


2

4u r S

Tt

The Theis solution models radial flow about the well located at 0,r and assumes no vertical movement of fluid. The initial condition for this system is 0( ,0)h r h ,

and the boundary conditions are

0

0

lim ( , ) ,

lim .2

r

r

h r t

Qr

h

hr T

The Theis model is of infinite extent, and the pumping sink is a singularity.

4.1.7.3 Model Conceptual Model To simulate the boundary conditions of the Theis model, the SVFlux model is a large cylinder with a constant head of h0 at the cylinder wall, zero flux at the top and bottom, and the well at the center. See the SVFlux User’s Manual for information on setting up a well. The conceptual model is shown in Figure 1177. It is a one layer cylindrical model with radius 20,000 m and depth 50 m. The well is in the center and has screened length 50 m.

Figure 11717 Conceptual Model for Theis Solution.

Numerical Model The SVFlux model consists of two regions: ModelExtents and InnerRegion. InnerRegion was included to improve the resolution of the finite element mesh near the well.

SoilVision Systems Ltd. Three-Dimensional Seepage 78 of 93 The SVFlux model has the following properties: General System 3D Type Transient Units Metric Time Units Day World Coordinate System X: Minimum -20,000 m X: Maximum 20,000 m Y:Minimum -20,000 m Y: Maximum 20,000 m Z:Minimum 0 m Z: Maximum 50 m Time (day) Start Time 0 Initial Increment 1e-5 (approx. 1 s) Maximum Increment 1 End Time 10

4.1.7.4 Geometry and Properties The well parameters are: Line Mesh Spacing 1 m Mesh Growth Coefficient 1 Influence Distance 2 m Rate -8640 m3/day The model geometry is: Surfaces Bottom 0 m Top 50 m Region : ModelExtents Circle center (0 m, 0 m) Circle radius 20,000 m Region : InnerRegion Circle center (0 m, 0 m) Circle radius 100 m

1Aquifer compressibility. Water is treated as incompressible for this model. The finite element solver and mesh parameters are all left at their defaults.

SoilVision Systems Ltd. Three-Dimensional Seepage 79 of 93 4.1.7.5 Material Properties Material Properties (Aquifer—Sand): Hydraulic Conductivity 8.64 m/day (10-4 m/s) Transmissivity 432 m2/day (5x10-3 m2/s) Porosity 0.4 Compressibility1 10-7 kPa-1 Storativity 4.9x10-5

4.1.7.6 Results and Discussions Figure shows the drawdown as a function of distance from the well for the Theis analytical solution and the Model solution. The modelled solution is sampled at a height of z = 25 m (half depth). Figure compares the modelled drawdown with the Theis drawdown. Figure 120 is the same as Figure , but zoomed in to show detail closer to the well. The Theis solution, contains a singularity at 0u (corresponding to 0r ). The numerical computation of the integral diverges to infinity close to the singularity, hence the need to start the graphs at some distance away from the well. The minor divergence of the numerical model from the Theis solution is due to SVFlux treating the well as a sink, rather than as a boundary condition. Figure shows the mesh. The inner region causes the finer mesh at the center of the model. There were 21,933 nodes that generated 14,628 cells. The cell sides on the outer edge of the model were about 2000 m long. At the center, the cell horizontal extents range from 1 m at the center to 25 m at the boundary of the inner region. Figure shows the 2-D cross section at the well, zoomed in so that the extent of the drawdown is clearer.

Figure 118 Drawdown after 10 days. RMSD = 0.5 m.


Figure 119 Comparison of Theis and Model drawdown.


Figure 120 Drawdown after 10 days out to 100 m.

Figure 121 Mesh.


Figure 122 Drawdown after 10 days (vertical scale is zoomed 50x).

4.1.7.7 Remarks The Theis solution models drawdown for a confined aquifer of infinite extent. To verify it with SVFlux requires a model with a 20 km radius to minimize the effect of the boundary condition on drawdown near the well. At this scale, the effect of a well on the water table in the immediate vicinity of the well is of less importance than the seepage over the extent of the region. If greater detail is required close to the well, then a model that focusses on the well, rather than the region, should be created. The effect of the influence distance is only seen close to the well screen. Figure 123 shows the result of varying the influence distance.

Figure 123 Drawdown near the well for various values of influence distance.


4.2 TRANSIENT STATE Transient state or time dependent models are presented in the following sections.

4.2.1 Drain-Down Example The drain-down example illustrates the use of the software to calculate drainage rates out of a model. The model is initially saturated and then allowed to drain down to a residual saturation level. The water leaving the model past the “Flux 4” flux section is compared to the integrated volume of water in the model at the start and at the end of a specified time. The difference in integrated volumes should equal the flux reported. It is assumed for this model that the volume integrals provide an accurate account of the total amount of water in the model at any given time. The geometry and the location of flux sections are the same as presented in Figure 100.

Table 10 - Process of accuracy improvement

Trial # Comments Initial Volume of Water (m3)

Final Volume of Water (m3)

Difference (m3)

Flux 4 (m3)

Error

1 First try - error limit = 0.1

0.3913 0.3682 0.0231 0.019165 -17%

2 Increase error limit to 0.01

0.3913 0.3682 0.0231 0.01938 -16%


0.3913 0.36831 0.02299 0.019912 -13%


0.3913 0.36899 0.02231 0.021313 -4%

It can be noted that the accuracy of the water balance calculations is related to the error limit specified. It should also be noted that the flux estimated as crossing the flux section was generally calculated as being less than the actual flow indicated by the volume integrals. Convergence of the model was also greatly improved by applying the outlet head boundary condition as a function over time.

4.2.2 Cube Drain-Down Example A cube of unit dimensions under initially saturated conditions was created for this example model. A head boundary condition of 0.2 was placed along a strip on the right hand side of the y-z plane of 0.2 m height. Water was then allowed to drain out of the model and the difference between the water volume integral and the values reported by a flux section along the opening was compared.


Head=0.2

Flux 1

Z

Y

1.0

0.2

1.0

1.0

Figure 124 Geometry of the cube-drainage model

4.2.3 Cube, Wedge, and Toe Transient Example A flux of 0.001 m3/s/m2 was applied to the top of the geometry shown below. If flow is properly calculated, then the results of flux sections 2, 3, and 4 should all display 0.00036 m3/s.

Flux = 0.001m / s / m3 2

Flux 2

Flux 3

Flux 4

Head = 0.2m

Y

0.2

0.11.01.0

1.00.6

0.6

1.0Z

X Figure 125 Application of flux in a 3-D transient model

Flux 2: 3.560e-4 m3/s Error: 1.11% Flux 3: 3.407e-4 m3/s Error: 5.36% Flux 4: 3.426e-4 m3/s Error: 4.83%

SoilVision Systems Ltd. Sensitivity of the Finite Element Mesh 85 of 93

5 SENSITIVITY OF THE FINITE ELEMENT MESH

Finite element mesh is very important in a numerical simulation. A finer finite element mesh commonly gives better calculation results. It has always been a question is: what is the resolution of the finite element mesh that provides reasonably accurate results? This section does not answer the question specifically, but will provide some senses of the relation between: i) finite element mesh; ii) hydraulic conductivity of the material; and iii) simulation results. A simple model was made up to study these relationships. This example helps SVFLUX users be aware of this issue. In order to run this model, the “auto mesh refinement” function in SVFLUX is turned off. Project: VerifySVFLUX Model: Madeup1&10&100&1000

5.1 MODEL DESCRIPTION The model is made up by a rectangular Sandy Loam, which has a width of 200 m and a height of 100.

Figure 126 Geometry of the model

The sandy loam has saturated volumetric water content of 0.3, Specific gravity of 2.65. The boundary conditions of the model are shown in Figure . The soil-water characteristic curve of the Sandy is plotted in Figure 118. Climate data is described for 4 running days in Table 11. The initial condition for this model is the water pressure in entire material is equal to zero (m).


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1e-3 1e-2 1e-1 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6

Soil suction (kPa)

Figure 1187 Soil-water characteristic curve of the Sandy Loam

Table 11 Description of the climate data

Day Precipitation (m/day)

Potential Evaporation

(m/day)

Temperature (0C) Relative humidity (%)

0 0 0 20 70 1 0.004 0.001 20 70 2 0 0.001 20 70 3 0 0.001 20 70

5.2 INPUT PARAMETERS FOR SENSITIVITY ANALYSIS

In this sensitivity study, 4 different cases will be implemented (Table 12). For the first three cases the saturated hydraulic conductivity of the material will be varied while the finite element mesh is kept the same. The last case use the lowest saturated permeability and has finest finite element mesh (i.e., increase number of nodes along the material surface). The Modified Campbell (Fredlund, 1997) is used for the prediction of the unsaturated hydraulic conductivity function. The “p-parameter” is slightly changed between the first three cases to make the unsaturated hydraulic conductivity functions for the three cases parallels to each other (Figure 119).

Table 12 Input parameters for the four cases

Case Saturated hydraulic conductivity (m/day)

p - parameter for the Modified Campbell model

Number of nodes in the finite element mesh

1 10 12.5 593 2 100 12 593 3 1000 11 593 4 10 12.5 2152


5 10 12.5 4292

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

0.01 0.1 1 10 100 1000 10000 100000 1000000Soil suction (kPa)

Hyd

raul

ic c

ondu

ctiv

ity (m

/day

) .

Ksat = 10 m/dayKsat = 100 m/dayKsat= 1000 m/day

Figure 1198 Hydraulic conductivity functions corresponding to three saturated hydraulic conductivity

5.3 SIMULATION RESULTS AND DISCUSSIONS This section presents the simulation results for the four cases. A brief discussion on the results and the effects of the saturated hydraulic conductivity and the finite element mesh to the simulation results are presented. The finite element mesh for the first three cases is shown in Figure 12029. Plots of the simulation results for first three cases are shown in Figure 0 to Figure . The finite element meshes for cases 4 and 5 (i.e., with 2152 nodes and 4292) are shown in Figure and Figure , respectively. The simulation results for cases 4 and 5 are shown in Figure and Figure 121, respectively. Table 13 shows the summary of the simulation results for the 5 cases.

Figure 1209 Finite element mesh for the first three cases (i.e., cases 1 to 3)


Precipitation

Net surface flux

Actual and potential evaporation

Time (days)

Wate

r (m

3 /day

)

Precipitation

Net surface flux


Time (days)

Wate

r (m

3 )

Figure 130 Plot of the precipitation, evaporation and net surface flux for the case saturated hydraulic conductivity of 10 m/day

Precipitation

Net surface flux


Time (days)

Wat

er (m

3 /day

)

Precipitation

Net surface flux


Time (days)

Wate

r (m

3 )

Figure 131 Plot of the precipitation, evaporation and net surface flux for the case saturated hydraulic

conductivity of 100 m/day


Precipitation

Net surface flux


Time (days)

Wat

er (m

3 )

Precipitation

Net surface flux


Time (days)

Wat

er (m

3 /day

)


conductivity of 1000 m/day

Figure 133 Finite element mesh for case 4 (2152 nodes)


Precipitation

Net surface flux


Time (days)

Wate

r (m

3 /day

)

Precipitation

Net surface flux


Time (days)

Wat

er (m

3 )


conductivity of 10 m/day using a fine finite element mesh (2152 nodes)

Figure 135 Finite element mesh for case 4 (4292 nodes)


Precipitation

Net surface flux


Time (days)

Wat

er (m

3 )

Precipitation

Net surface flux


Time (days)

Wat

er (m

3 /day

)


conductivity of 10 m/day using a fine finite element mesh (4292 nodes)

Table 13 - Calculation of errors for 4 cases. Case Saturated

hydraulic conductivity

(m/day)

Number of nodes

Average node distance near

material surface

Total precipitation

(m3)

Total actual evaporation

(m3)

Net surface flux (m3)

Error (%)

1 10 593 13.3 0.8 0.49 0.20 13.75 2 100 593 13.3 0.8 0.49 0.34 3.75 3 1000 593 13.3 0.8 0.49 0.32 1.25 4 10 2152 2 0.8 0.49 0.35 5 5 10 4292 1 0.8 0.49 0.32 1.25

As can be seen from Table 13, with the same finite element mesh, the higher saturated hydraulic conductivity of the material the better simulation results. Table 13 also shows that the finer finite element mesh, the better simulation results. Case 1 appears to have a low accuracy due to distances between nodes are too high. Cases 2 and 4 have reasonable good results. It shows that there is a relationship between the accuracy of the model and the ratio (hydraulic conductivity/distance between nodes near (or at) the material surface). It shows that this ratio of equal to 10 should provide sufficient good simulation results.

SoilVision Systems Ltd. References 92 of 93

6 REFERENCES Barbour, S.L., Fredlund, D.G., Gan, J. K-M., and G.W. Wilson, (1991). Prediction of Moisture

Movement in Highway Subgrade Soils, 45th Canadian Geotechnical Conference, Toronto, ON., Canada

Bowles, J.E., (1984). Physical and geotechnical properties of soils. 2nd ed. McGraw-Hill, New York. Celia, M.A. and E.T. Bouloutas, (1990). A General Mass-Conservative Numerical Solution for the

Unsaturated Flow Equation. Water Resources Research, Vol. 26, No. 7, pp. 1483-1496, July.

Chapuis, Robert P., Chenaf, D., Bussiere, B., Aubertin, M., R. Crespo, (2001). A user’s approach to

assess numerical codes for saturated and unsaturated seepage conditions. Canadian Geotechnical Journal, 38: 1113-1126.

Crespo, R., (1993). Modelisation par elements finis des ecoulements a travers les ouvrages de

retenue et de confinement des residus miniers. M.Sc.A. thesis, Ecole Polytechnique de Montreal, Montreal.

Dupuit, J., (1863). Etudes theoriques et pratiques sur le mouvement des eaux dans les canaux

decouverts et a travers les terrains permeables. 2nd ed. Dunod, Paris. Gitirana, G., (2004). Weather-Related Geo-Hazard Assessment Model for Railway Embankment

Stability, PhD Thesis, University of Saskatchewan, Saskatoon, SK, Canada. Gitirana, G.G., Fredlund, M.D., Fredlund, D.G., (2005). INFILTRATION-RUNOFF BOUNDARY

CONDITIONS IN SEEPAGE ANALYSIS, Canadian Geotechnical Conference, September 19-21, Saskatoon, Canada

Freeze, R. and Cherry, J. 1979. Groundwater. Prentice-Hall, pp314 – 319. Haverkamp, R., Vauclin, M., Touma, J., Wierenga, P.J., and G.Vachaud, (1977). A Comparison of

Numerical Simulation Models for One-Dimensional Infiltration, Soil Science Society of America Journal, Vol. 41, No. 2.

MEND 1993. SoilCover user’s manual for evaporative flux model. University of Saskatchewan,

Saskatoon, SK, Canada. Pentland, J.S., (2000). Use of a General Partial Differential Equation Solver for Solution of Heat and

Mass Transfer Problems in Soils, University of Saskatchewan, Saskatoon, SK, Canada. FlexPDE6.x Reference Manual 2007. PDE Solutions Inc. Spokane Valley, WA 99206. Freeze, R. A. and J. Cherry, (1979). Groundwater. Prentice–Hall, Inc., Englewood Cliffs, New Jersey Todd, D.K. (1980). Groundwater hydrology, 2nd ed. John Wiley & Sons, New York. Tratch, D.J., (1995). A Geotechnical Engineering Approach to Plant Transpiration and Root Water

Uptake, University of Saskatchewan, Saskatoon, SK, Canada. Tratch, D.J., Wilson, G.W., and D.G. Fredlund, (1995). An introduction to analytical modeling of

plant transpiration for geotechnical engineers, Proceedings, 48th Canadian Geotechnical Conference, Vancouver, BC, Canada, Vol. 2, pp. 771-780.

SoilVision Systems Ltd. References 93 of 93 Wilson, W., (1990). Soil Evaporative Fluxes for Geotechnical Engineering Problems. PhD Thesis,

Department of Civil Engineering, University of Saskatchewan, Saskatoon, SK, Canada.

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SVFlux Verification Manual