sugwfoam: an open source saturated …water.engr.psu.edu/liu/publications/sugwfoam_tr.pdfcivil and...
TRANSCRIPT
![Page 1: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/1.jpg)
Civil and Environmental Engineering Studies Technical Report No. 01-01
suGWFoam: An Open Source Saturated-Unsaturated GroundWater Flow Solver based on OpenFOAM®
Xiaofeng Liu, Ph.D., P.E.
Assistant Professor Department of Civil and Environmental Engineering
University of Texas at San Antonio One UTSA Circle, San Antonio, Texas, U.S.A. 78249
Email: [email protected]
Web: http://engineering.utsa.edu/~xiaofengliu
November 2012
![Page 2: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/2.jpg)
Disclaimer: This offering is not approved or endorsed by ESI Group, the producer of the OpenFOAM® software and owner of the OpenFOAM® trademark.
Suggested Citation: X. Liu (2012). “suGWFoam: An Open Source Saturated-Unsaturated GroundWater Flow Solver based on OpenFOAM®”, Civil and Environmental Engineering Studies Technical Report No. 01-01, University of Texas at San Antonio, San Antonio, Texas.
![Page 3: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/3.jpg)
ACKNOWLEDGEMENTS
This work was supported by the author’s startup funding from the University of Texas at San
Antonio, USA. The author also wants to acknowledge the Texas Advanced Computing Center
(TACC) at the University of Texas at Austin for providing HPC resources that have contributed
to the parallel computation results reported within this report.
![Page 4: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/4.jpg)
Abstract
In this technical report, we document a three-dimensional open source model suGWFoam, based
on the multiphysics platform OpenFOAM for variably saturated flow in porous media. The
nonlinear Richards equation is spatially discretized on unstructured meshes using finite volume
method in conjunction with the modified Picard iterative method, which inherently guarantees
conservation of mass. A switching algorithm between different forms of the Richards equation
is implemented and three choices of convergence criteria suitable for different simulation sce-
narios are provided. The linearized equation results in a symmetric matrix system which can
be solved with one of the following Krylov subspace linear solvers: preconditioned conjugate
gradient (PCG), incomplete Cholesky preconditioned conjugate gradient (ICCG), geometric ag-
glomerated algebraic multigrid (GAMG). Various preconditioners are runtime selectable. Paral-
lel computation through domain decomposition shows good speed up for large scale simulations.
Comparing to other available codes for ground water, the merits of this model reside in the ar-
chitecture of OpenFOAM. Built upon the modularized and hierarchy design of the platform, this
model has less than several hundreds of lines of code on its top level and yet has the full func-
tionalities for three-dimensional unstructured mesh, parallel computing, and numerous choices
of spatial and temporal discretization schemes. This liberates the researchers in the water re-
source community from the tedious and long process of coding. It is also extremely easy to
couple groundwater flow with surface water hydrodynamics using existing flow solvers (such as
channel flows and waves) in OpenFOAM. This model is fully functional and validations show
excellent agreement with data in literature for a wide range of applications. It can be used for
both fundamental researches and real world applications.
This report was based on a manuscript under review for possible publication on Engineering
Application of Computational Fluid Mechanics.
![Page 5: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/5.jpg)
ii
![Page 6: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/6.jpg)
Contents
1 Introduction 1
2 Model Formulations and Implementations 5
2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Finite volume discretization . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Temporal discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.3 Nonlinearity and modified Picard iterations . . . . . . . . . . . . . . . . 13
2.2.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.5 Solution of linear systems . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.6 Convergence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.7 Parallel computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Simulation Examples 22
3.1 1D infiltration/redistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 2D recharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 3D aquifer pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 3D aquifer pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
iii
![Page 7: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/7.jpg)
iv CONTENTS
3.5 Parallel computation and mass balance . . . . . . . . . . . . . . . . . . . . . . . 34
4 Summary and Conclusions 38
![Page 8: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/8.jpg)
List of Figures
2.1 Finite volume discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1 Comparison of results for one-dimensional infiltration problem . . . . . . . . . . 24
3.2 Comparison of results for two-dimensional, unsteady, water table recharge problem 25
3.3 Three-dimensional field scale pumping test described in Nwankwor et al. [1992] . 28
3.4 Schematic of the 3D pumping problem in 3DFEMWATER Yeh and Cheng [1994] 30
3.5 Steady state results for the 3D pumping problem in 3DFEMWATER . . . . . . . 31
3.6 Parallel computation and mass balance analysis . . . . . . . . . . . . . . . . . . 36
v
![Page 9: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/9.jpg)
List of Tables
3.1 Values of parameters for 1D infiltration problem and 2D water table recharge
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Domain decomposition and parallel computing analysis results for the refined
case in Nwankwor et al. [1992] . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
vi
![Page 10: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/10.jpg)
Chapter 1
Introduction
Flow and species transport in variably saturated porous media are essential to understand the
processes of many environmental and water resources problems. Flow in porous media usu-
ally involves multiphases [Adler and Brenner, 1988] and complex physical/chemical processes
[Domenico and Schwartz, 1990]. Variably saturated porous media flows can be modeled by
the classic nonlinear Richards equation [Richards, 1931] and closed by constitutive relations de-
scribing the functional dependence of moisture content and hydraulic conductivity on the pres-
sure head [Miller et al., 1998]. Due to the high nonlinearity and hysteresis in the constitutive
relations, analytical solutions to the Richards equations rarely exist except for a limited number
of simple configurations. In addition, the complications due to irregular domain geometry and
nonuniform initial/boundary conditions make numerical solutions the typical choice in practice
[Celia et al., 1990].
There exists a wealth of models, whose core capability is to solve the nonlinear Richards
equation using different numerical methods. Space limit prohibits a thorough discussion of these
models. In general, these models can be classified by the numerical schemes they use.
Finite difference method (FDM), though simple, has been successfully used in many popular
models, for example MODFLOW [McDonald and Harbaugh, 1988] and SWAP [van Dam et al.,
1
![Page 11: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/11.jpg)
2 CHAPTER 1. INTRODUCTION
2008]. Clement et al. [1994] proposed a physically based form of the general, variably saturated
flow equation which is solved using finite differences (centered in space, fully implicit in time) in
2D employing the modified Picard iteration scheme. Other FDM models include VS2D [Lappala
et al., 1987] and SU3D [Dogan and Motz, 2005a,b], among many others. Since groundwater
flow is a part of the hydrological cycle, it is often embedded in a large hydrological model.
For example, MIKE SHE is a modeling system for building and simulating surface water flow
and groundwater flow. MIKE SHE calculates the unsaturated flow using a fully implicit finite
difference solution [Refsgaard and Storm, 1995]. It employs an iterative coupling procedure
between the unsaturated zone and the saturated zone. One of the drawbacks of FDM models is
that it is not easy to model complex domain geometries without loss of accuracy.
Finite element method (FEM) and finite volume method (FVM) can use unstructured meshes
and inherently deal with arbitrary domain with ease. Examples of finite element method (FEM)
models include 3DFEMWATER [Yeh et al., 1992, Yeh and Cheng, 1994],SutraSuite [Voss and
Provost, 2010], HYDROGEOCHEM [Yeh et al., 2004], HydroGeoSphere [Therrien et al., 2010],
and FEFLOW [Diersch, 2009]. Models in the literature using finite volume method (FVM),
though not as many as FDM and FEM models, are emerging. Manzini and Ferraris [2004]
solved the 2D Richards equation on unstructured grid using a mass-conservative finite volume
method where parameters are approximated at the cell centers. McBride et al. [2006] developed
a 3D finite volume model for variably saturated flows based on the general-purpose commercial
computational fluid dynamic (CFD) code PHYSICA, which is similar to OpenFOAM. Kumar
et al. [2009] presented a second-order accurate, finite-volume based 3D model which couples
the surface and subsurface flows. Similar coupling has also been investigated in Pham-Van et al.
[2011]. Solving the governing equations using FVM automatically guarantees the conservation
of mass at the equation level, i.e., it will not introduce mass balance errors due to numerical
discretizations. Combined with special treatment of the differential equation (such as the classic
modified Picard iteration of Celia et al. [1990]), excellent mass balance is guaranteed.
![Page 12: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/12.jpg)
3
The governing equation of the variably saturated porous media flow can be casted into dif-
ferent forms, namely h-based, θ-based, and mixed forms. There has been a long history of
discussion and debate on these forms. In general, the h-based form can be used for both sat-
urated and unsaturated conditions. Good mass balance can be achieved if the change of h and
consequently the change of C(h) are small during one time step. However, this form suffers
from severe mass-balance error under highly nonlinear conditions, such infiltration into very dry
heterogeneous soils [Celia et al., 1990, McBride et al., 2006]. The mass imbalance is due to the
fact that ∂θ/∂t is mathematically equivalent to C(h)∂h/∂t in the continuous space, but no so
in their discrete forms. In addition, the nonlinear functional form of the moisture content θ(h)
further increases the severity. Many authors have proposed different method to overcome these
difficulties. Kirkland et al. [1992] proposed an innovative alternative by defining a new variable
which equals the saturation in unsaturated zones and the pressure head in saturated zones and
therefore overcomes the singularity. Similar techniques of solving moisture content and pressure
head in different regions or different stages of the simulation have also been used by others, for
example Forsyth et al. [1995] and Diersch and Perrochet [1999].
For the θ-based form, despite its very obvious advantage of perfect mass conservation in both
continuous and discrete spaces, it is not suitable for fully saturated conditions because hydraulic
diffusivity is infinity in saturated regions and a head-saturation relationship does not exist. Hills
et al. [1989] gave a very detailed discussion of the advantages and disadvantages of the h-based
and θ-based forms of the Richards equation.
The mixed form retains both the moisture content θ and pressure head h in the differential
equation [Soraganvi and Kumar, 2009]. Some of the aforementioned difficulties are avoided
when using the mixed-form which possess much better properties with respect to mass conserva-
tion. Celia et al. [1990] solved the mixed form by a modified Picard iteration scheme. Within the
iterative procedure the pressure head is used as the primary variable for the solution at a new it-
eration step. This mixed Picard technique has been successfully used in many models regardless
![Page 13: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/13.jpg)
4 CHAPTER 1. INTRODUCTION
of the numerical schemes they use.
Despite the abundance of existing models for variably saturated flows, this report introduces
a new 3D variably saturated flow model, suGWFoam (saturated-unsaturated groundwater model
using OpenFOAM [OpenFOAM Foundation, 2011]). Compared with other models, the unique-
ness of this model hinges on the advances in computational continuum mechanics and modern
software development. It is based on the open-source finite-volume platform OpenFOAM. This
model builds on the hierarchy structure of OpenFOAM which makes it versatile and efficient to
write new partial differential equation solvers. It also provides a library of numerical schemes
necessary for the discretizations. Parallel computation is easily implemented with the domain
decomposition functionality of the platform. All these features are built-in automatically into
any code developed in OpenFOAM. It is therefore appealing to the porous media flow modeling
community since it is not necessary for the model developers and researchers to write the code
from scratch. They can focus on the physics instead of numerical details.
The structure of this report is as follows. The governing equation and its different forms will
be introduced first. Then the finite volume discretization of the equations and their implemen-
tation in OpenFOAM are presented in detail. Numerical examples covering multiple aspects of
the variably saturated flow will be shown to demonstrate the accuracy of the new model. Results
from large scale parallel computations up to 1024 cores will showcase the excellent speedup and
efficiency of suGWFoam. At the end, this report will conclude with a discussion and conclusion.
![Page 14: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/14.jpg)
Chapter 2
Model Formulations and Implementations
2.1 Governing equations
2.1.1 Conservation of mass
The governing equation which describes the flow in variably saturated porous media is the non-
linear Richards equation Richards [1931]. Its derivation is through the combination of the con-
servation of mass and the Darcy’s law. To solve the equation, constitutive relations between the
pressure head and moisture content, as well as the hydraulic conductivities, have to be specified.
The Richards equation can be written into different forms, namely h-based form, θ-based form,
and mixed form.
• h-based form:
[C(h) + SsSw]∂h
∂t= ∇ · [K(h) · ∇ (h+ z)] +Qs, (2.1)
where C(h) is the specific moisture capacity (L−1) which is defined as
C(h) =∂θ
∂h, (2.2)
5
![Page 15: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/15.jpg)
6 CHAPTER 2. MODEL FORMULATIONS AND IMPLEMENTATIONS
and θ is the moisture content, h is the pressure head (L), Qs is the volumetric source/sink
term (L3T−1L−3). Ss is the specific storage (L−1), Sw is the saturation ratio (= θ/η), η is
the porosity. K(h) is the hydraulic conductivity (L/T ), which is a second order tensor. In
the equation, ‘·’ is the inner product operator.
• θ-based form:
∂θ
∂t+ SsSw
∂h
∂t= ∇ · [D(θ) · ∇θ] +∇ · [K(h) · ∇z] +Qs, (2.3)
where D(θ) is the hydraulic diffusivity (L2T−1).
• Mixed form:∂θ
∂t+ SsSw
∂h
∂t= ∇ · [K(h) · ∇ (h+ z)] +Qs (2.4)
Each of the aforementioned forms of Richards equations has its strength and drawbacks as
discussed in Chapter 1. Therefore it makes sense to combine them and switch the forms de-
pending on the status of the modeled systems. This is the rationale behind the switching algo-
rithm proposed by Hao et al. [2005], where the h-based form and the mixed form are selected
dynamically during the simulation. Diersch and Perrochet [1999] proposed another switching
methodology where the primary variables to be solved, h or θ, is determined dynamically. It is
demonstrated that the primary variable switching technique is mass-conservative for an arbitrary
time step size and provides a much better convergence behavior compared to both the h-based
and mixed forms of the Richards equation. Diersch and Perrochet [1999] presented the algorithm
in the context of finite element method using Newton’s method which involves the evaluation of
Jacobian matrices. This report uses finite volume method and Picard iterations. Therefore, we
adopt the switching algorithm of Hao et al. [2005], which was also used in McBride et al. [2006].
The essence of the switching algorithm is that within a time step, the magnitude of the pressure
head change is calculated as hc = |hn+1 − hn| and compared with a specified threshold value h0.
![Page 16: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/16.jpg)
2.1. GOVERNING EQUATIONS 7
The superscripts n and n+ 1 represent the iteration numbers within a time step. If hc < h0, then
the h-based form is solved, otherwise the mixed form is solved. The specification of the thresh-
old value depends on many factors and it affects the accuracy and convergence of the simulation
Hao et al. [2005]. For all the simulations in this report, the default value of h0 was 3 cm as in
McBride et al. [2006].
2.1.2 Constitutive relations
The constitutive relations refer to the functional forms of the moisture content θ(h) and hydraulic
conductivity K(h). A library of commonly used constitutive models have been implemented in
suGWFoam using the OpenFOAM runtime selectable mechanism where the user only need to
specify the model name and relevant parameters. The runtime selection mechanism is a tem-
plate implementation of the so called ”virtual constructor” in object-orient programming, which
simplifies the top-level code and makes future expansion easy. For example, when a new consti-
tutive relation needs to be added, the only thing needs to be done is to make a copy of existing
relationship and modify accordingly. The model can automatically load the library and make it
available. Another advantage is that the constitutive relations are runtime modifiable. During the
simulation, users can switch between different constitutive relations without interrupting. For
example, this functionality is useful to get a good initial condition using less nonlinear relations
and then switch to a more physical but more nonlinear relation.
The common constitutive relations implemented in suGWFoam include the Brooks and Corey
[1964] model, the van Genuchten [1980] model, the van Genuchten and Nielsen [1985] model,
the modified van Genuchten and Nielsen’s model in Paniconi et al. [1991], etc. The details of
these relations can be found in any groundwater modeling textbook and therefore the details are
omitted. In any constitutive relation, the routines in the model should calculate the moisture
content and hydraulic conductivity given the values of h. It should also calculate the specific
moisture capacity ∂θ/∂h. For the purpose of demonstration, the van Genuchten [1980] model
![Page 17: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/17.jpg)
8 CHAPTER 2. MODEL FORMULATIONS AND IMPLEMENTATIONS
has the following formulas for moisture content θ and the relative hydraulic conductivity kr
θ(h) =
θr + θs−θr[1+(α|h|)n]m
if h < 0
θs otherwise(2.5)
kr(θ) =K
Ks
=
(θ − θrθs − θr
)0.5{
1−
[1−
(θ − θrθs − θr
)1/m]m}2
(2.6)
where α and n are the parameters controlling the shape of the soil constitutive curves, and m =
1− 1/n.
suGWFoam also has some ad-hoc models which are from the fitting of experimental data into
specific functional forms. The coefficients are usually determined by regressions. Most of these
models are provided for the purpose of validations using the cases in the literature which will be
demonstrated in Chapter 3.
2.2 Numerical solutions
2.2.1 Finite volume discretization
suGWFoam is implemented in OpenFOAM, an open source platform for solving partial differ-
ential equations using finite volume method on unstructured meshes OpenFOAM Foundation
[2011]. OpenFOAM, acronym for Open source Field Operations And Manipulations, incorpo-
rates a tensorial approach to computational continuum mechanics based on the object-oriented
programming technique Weller et al. [1998]. It is a versatile toolbox which can be easily used to
create a solver for complex physics problems as long as they can be mathematically described in
the form of differential equations with appropriate boundary and initial conditions. Various dif-
ferential operators, such as divergence, gradient, Laplacian, and time derivative, are defined for
physical variable fields defined on the solution domain. The details of the discretisation of spatial
![Page 18: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/18.jpg)
2.2. NUMERICAL SOLUTIONS 9
terms and temporal terms can be found in Jasak [1996]. In this work, we have used version 2.0.1
of OpenFOAM.
The finite volume discretizaion is to enforce the governing equations over each control vol-
ume VP around the center point P in the integral form (see Figure 2.1). For instance, the h-based
form can be discretized in space as
∂
∂t
∫VP
[C(h) + SsSw]hdV =
∫VP
∇ · [K(h) · ∇h] dV +
∫VP
∇ · [K(h) · ∇z] dV
+
∫VP
QsdV, (2.7)
A second order spatial discretization is achieved by assuming the variation of a variable is
linear in a control volume VP . The generalized Gauss divergence theorem is employed to convert
the volume integration into appropriate surface integrations. Both the h-based and mixed forms
only have terms involve time derivative, Laplacian (diffusion) and source/sinks. The Laplacian
term involving z is treated as an extra source term. The Laplacian term for h is discretized as
∫VP
∇ · [K(h) · ∇h] dV =∑f
S · [K(h) · ∇h]f (2.8)
where S is the face area vector constructed for each face connecting neighboring control volumes.
S is in the same direction of the outward normal vector of the face and its magnitude equals to
the face area. For each control volume P and neighboring volumeN , where P andN are labeled
at the centroids of the cells, the common face is denoted f as shown in Figure 2.1. The vector
connecting the centroid P to N is termed vector d. For a orthogonal mesh, d and S are parallel.
However, it is not the case for non-orthogonal meshes. In equation 2.8, the subscript f means
the variable is evaluated at the face f . To calculate the summation on the right hand side of the
equation, it is necessary to evaluate the face values (the hydraulic gradient ∇h in this case). For
![Page 19: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/19.jpg)
10 CHAPTER 2. MODEL FORMULATIONS AND IMPLEMENTATIONS
(a) (b)
Figure 2.1: Finite volume discretization: (a) 3D view of the neighboring polyhedrons, adaptedfrom OpenFOAM Foundation [2011], (b) non-orthogonality treatment, adapted from Jasak[1996]
a orthogonal mesh, it is straightforward to approximate the gradient of the head as
S · [K(h) · ∇h]f = [K(h)f · S] · S
|S|hP − hN|d|
. (2.9)
This is essentially the flux across the face f since right hand side is the approximation of Darcy’s
flux times the surface area. The hydraulic conductivity has to be interpolated into the surface
from the cell centers.
In the case of non-orthogonal mesh, special treatment is needed to preserve the second order
spatial accuracy. Jasak [1996] proposed to split the product S · [K(h) · ∇h]f into two parts (see
Figure 2.1)
S · [K(h) · ∇h]f = ∆ · [K(h) · ∇h]f + k · [K(h) · ∇h]f (2.10)
where the first and second terms on the right hand side are the contribution from orthogonal and
![Page 20: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/20.jpg)
2.2. NUMERICAL SOLUTIONS 11
non-orthogonal parts, respectively. The vectors satisfy the simple geometric relationship of
S = ∆ + k, (2.11)
where ∆ is in the same direction of d. The rationale of the splitting is that the orthogonal
contribution can be treated as if the mesh is orthogonal and use the formula in Equation 2.9. The
non-orthogonal contribution, also termed ”correction term” in OpenFOAM, is ignored by many
finite volume models for variably saturated porous media flows, such as the model presented in
McBride et al. [2006]. The loss of accuracy by discarding the correction term increases with the
degree of mesh non-orthogonality. To circumvent that situation, novel numerical techniques have
been used to calculate high accuracy hydraulic gradient based on sophisticated reconstruction
procedures. In their layered 3D model, Kumar et al. [2009] used a second-order accurate, pseudo-
Laplacian method to interpolate the cell center values into vertexes, whose values are used in turn
to reconstruct the gradient at edges to calculate lateral fluxes. Note that their model is layered
in horizontal. In the vertical direction, the flux is assumed to be normal to the interface between
two layers, which essentially assumes orthogonality in the vertical direction and discard the
”correction term”. For their case, if the terrain of the land does not change dramatically and the
layers are so arranged that they are almost parallel to the horizontal plane, this treatment is a good
approximation. Otherwise, corrections similar to the one proposed in this chapter are necessary.
2.2.2 Temporal discretization
The first time derivative φ∂h/∂t is integrated over a control volume as
∂
∂t
∫V
φhdV (2.12)
![Page 21: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/21.jpg)
12 CHAPTER 2. MODEL FORMULATIONS AND IMPLEMENTATIONS
where φ is a generic variable representing the coefficient before the ∂h/∂t term depending the
specific form of the governing equations. The term is discretised by finite differencing in time
using the values at different time levels, hn+1, hn, and hn−1, where the superscripts n+ 1, n, and
n − 1 denotes the values at current time step, previous time step and the time step previous to
the last. Two temporal discretization schemes have been implemented, namely, first-order Euler
implicit and second-order backward implicit.
For the first-order accurate Euler scheme, the time derivative is approximated as
∂
∂t
∫V
φhdV =(φhPV )n+1 − (φhPV )n
∆t, (2.13)
and for the second-order accurate backward scheme, the time derivative is approximated as
∂
∂t
∫V
φhdV =3 (φhPV )n+1 − 4 (φhPV )n + (φhPV )n−1
s∆t. (2.14)
For a transient problem, one also needs to treat the spatial derivatives and decides at which
time level they should be evaluated. Let’s denote the Ah as the generic spatial operator on h,
such as the Laplacian encountered in the porous media flow equation. Integrating in time gives
the form ∫ t+∆t
t
[∫V
AhdV]dt =
∫ t+∆t
t
A∗hdt, (2.15)
whereA∗ is the spatial discretization of the operatorA. OpenFOAM provides three ways to treat
the time integral,
Euler implicit:∫ t+∆t
t
A∗hdt = A∗hn+1∆t, (2.16)
Explicit:∫ t+∆t
t
A∗hdt = A∗hn∆t, (2.17)
![Page 22: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/22.jpg)
2.2. NUMERICAL SOLUTIONS 13
Crank Nicolson:∫ t+∆t
t
A∗hdt = A∗(hn+1 + hn
2
)∆t. (2.18)
Both the Euler implicit scheme and explicit scheme are first order accurate in time. Euler implicit
guarantees boundedness of the variable h and is unconditionally stable. Explicit scheme could be
unstable if the Courant number defined by the Darcy flux and time step size is greater than 1. The
Crank Nicolson scheme is second order accurate in time and unconditionally stable. However,
it does not guarantee boundedness. One should keep accuracy, stability, and boundedness of
these schemes in mind when making a choice. The time scheme is also implemented using the
run time selection mechanism in OpenFOAM and the selection is declared using the timeScheme
keyword in one of the control files. It is runtime modifiable which makes it possible to do
dynamical control of the temporal discretization during the simulation. This feature will be ideal
for large cases and long-term transient simulations where multiple time scales are involved.
2.2.3 Nonlinearity and modified Picard iterations
The nonlinearity in the constitutive relations makes direct solution of the governing equations
impractical if not impossible. Instead, iterative methods, usually fixed point Picard iterations for
nonlinear equations, are pursued. In this report, the classic method of Celia et al. [1990] is used,
which has the following formula for the mixed form (Equation 2.4)
θn+1,m+1 − θn
∆t+ SsS
nw
hn+1,m+1 − hn
∆t= ∇ ·
[K(hn) · ∇
(hn+1,m+1 + z
)]+Qn+1
s , (2.19)
where the superscripts n and m denote time level and iteration level, respectively. A truncated
Taylor series for the water content at the new time step and new iteration θn+1,m+1 with respective
![Page 23: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/23.jpg)
14 CHAPTER 2. MODEL FORMULATIONS AND IMPLEMENTATIONS
to h has the form
θn+1,m+1 ≈ θn+1,m +
(dθ
dh
)n+1,m (hn+1,m+1 − hn+1,m
). (2.20)
It is substituted into Equation 2.19 and the resulting linear equation for h is
[(dθ
dh
)n+1,m
+ SsSnw
]hn+1,m+1 − hn+1,m
∆t= ∇ ·
[K(hn) · ∇
(hn+1,m+1 + z
)]+S, (2.21)
where the new source term S needs to be updated at each iteration level and has the form
S = Qn+1s +
θn − θn+1,m
∆t+ SsS
nw
hn − hn+1,m
∆t. (2.22)
At each iteration level, Equation 2.21 represents an unsteady diffusion equation with some
source terms for h. All the coefficients and source terms are known and the solver is straight
forward to be implemented in OpenFOAM by assembling the differential operators.
2.2.4 Boundary conditions
Unique solutions to the nonlinear Richards equations require the specification of proper bound-
ary conditions. From the point of view of mathematics, there are three categories of boundary
conditions for partial differential equations, namely Dirichlet, Neumann, and Cauchy boundary
conditions. All three categories are physically possible for variably saturated flows. In suG-
WFoam, four types of basic boundary conditions are implemented which include fixed head h,
fixed total head h + z, specified flux q, and fixed value at specified internal locations. All these
boundary conditions can also be functions of time, i.e., they can be used to model the transient
effects. There are also other types of boundary conditions which in essence can be transformed
into one of the basic boundary conditions or their combinations. Examples of these other possi-
![Page 24: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/24.jpg)
2.2. NUMERICAL SOLUTIONS 15
ble boundary conditions include evaporation boundary and vegetation root uptake. They can be
implemented with ease once the physical processes are modeled by proper formulas.
• Specified head h or total head h + z: Pressure head h or the total head h + z can be
prescribed on a boundary face as
h = hD (x, y, z, t) (2.23)
and
h+ z = hD (x, y, z, t) (2.24)
where hD is a head distribution function, x,y,z are the coordinates of face center on the
boundary. Since the right hand sides can be functions of z, both boundary conditions can
be generalized into a single boundary condition as
h = hG (x, y, z, t) (2.25)
where hG is a function which can be customized. This boundary condition is typically
used to define the perimeters of water bodies, water table locations, and leaking surface
with specified water levels Yeh et al. [1992]. In the finite volume method framework, when
the hydraulic gradient on boundary faces (∇h)f are required for the Laplacian term in the
Richards equation, it is calculated using the specified boundary face value and the cell
center value
Sf · (∇h)f = |Sf |hG − hP|d|
. (2.26)
• Specified flux q: The specified flux can be written as
− Sf|Sf |· [Ks · kr (∇h+∇z)] = qc(x, y, z, t) (2.27)
![Page 25: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/25.jpg)
16 CHAPTER 2. MODEL FORMULATIONS AND IMPLEMENTATIONS
It can be simplified to the form as the surface normal gradient for pressure head
Sf|Sf |· ∇h = −qc(x, y, z, t)
ks,nkr− Sf|Sf |· [Ks · kr (∇z)] (2.28)
where ks,n is defined as the saturated hydraulic conductivity along the surface normal di-
rection which can be calculated by the inner product of the hydraulic conductivity tensor
Ks with the boundary surface normal vector and picking the component along the normal
direction. It is noted that the specified flux boundary condition is intrinsically nonlinear
for variably saturated flows due to the fact that the relative permeability kr is a nonlinear
function of the pressure head h. When the specified flux boundary is unsaturated, numeri-
cal difficulties have been observed. Another contributing factor is the explicit treatment of
the nonlinear boundary conditions. During each modified Picard iteration, the boundary
values are updated using the previous iteration pressure head which makes the boundary
condition lags one iteration behind the rest of the solution field.
Equation 2.28 specifies a surface normal gradient for h on the boundary. In the finite
volume method discretization, it can be directly substituted in the calculation of hydraulic
gradient for the Laplaican term in the Richards equation. When the value of pressure head
is needed on the boundary, it can be evaluated by
hf = hP + |d| Sf|Sf |· ∇h. (2.29)
• Fixed value at specified internal locations: In some applications, the pressure head at some
internal locations needs to be fixed and properly treated in the numerical code. It is not
strictly a boundary condition but rather an approximation for the condition in the well bore
holes. To simplify the modeling domain, some models do not really include the geometry
of the well because the scale of the well is usually much smaller than the whole domain.
![Page 26: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/26.jpg)
2.2. NUMERICAL SOLUTIONS 17
The boundary condition at the well bore hole is interesting and sometime complicates the
model. For partially penetrating wells into confined aquifers and for variably saturated
flow conditions, the head along the well bore hole is not uniform and not known a prior.
This head needs to be determined as part of the solution. For example, Cooley [1971]
specified hydraulic head at the well screen which corresponds to the water level in the
well. An iterative scheme was used to determine the hydraulic head by matching the
flux with the prescribed pumping rate and hydraulic properties of the aquifer. Another
example of treatment is given in Therrien and Sudicky [2000]. Based on the method for
saturated flows in Sudicky et al. [1995], they extended a noniterative method to the three-
dimensional variably saturated flows. The noniterative technique directly implements the
well bore boundary by superposing the one-dimensional elements representing the well
onto the three-dimensional porous medium elements. Nevertheless, specifying the head at
internal cells corresponding to the well as implemented in this chapter is the simplest way
to model the well. It may cause certain degree of error near the well region. However, it
may capture the hydraulics far away from the well and reduce the model complexity.
The numerical treatment of this boundary condition in suGWFoam is straightforward. Un-
like other boundary conditions, it is not considered when the linear system for the whole
domain is assembled (see Section 2.2.5). In the resulting linear equations system, for the
cell with specified pressure head, the off-diagonal coefficients are all set to zero and the
right hand side is set as the product of the diagonal coefficient and the desired pressure
head. In the equation for each neighboring cell, the corresponding off-diagonal coefficient
is set to zero and the contribution from current cell is moved to the right hand side. This
procedure is repeated for every cell with specified pressure head.
![Page 27: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/27.jpg)
18 CHAPTER 2. MODEL FORMULATIONS AND IMPLEMENTATIONS
2.2.5 Solution of linear systems
For each control volume, descritizing the governing equations using finite volume method pro-
duces one algebraic equation
aPhp +∑N
aNhN = rP (2.30)
where aP , aN and rP are coefficients from the spatial, temporal discretizations in addition to
the contributions from the boundary conditions. The value hP depends on the values of h in
its neighboring cells and assembling all equations creates a system of linear algebraic equations
which has the general form
Ah = r, (2.31)
where A is a sparse symmetric matrix, h is the unknown vector for pressure head at each cell
center, r is the right hand side. The diagonal entries of A are from the coefficients aP and the
off diagonal entries are from the coefficients aN . The number of off-diagonal entries equal to
the number of cells share faces with the current cell. The linear system can be solved by Krylov
subspace iterative solvers with proper choice of pre-conditioners. OpenFOAM has a variety of
choices of Krylov subspace solvers builtin and the upper level code does not need to know the
details of these solvers. The selection can be easily made at runtime by the specification in the
control file. Due to the symmetry of matrix A, three choices are available, namely preconditioned
conjugate gradient (PCG), incomplete Cholesky preconditioned conjugate gradient (ICCG), ge-
ometric agglomerated algebraic multigrid (GAMG). To make the iterative solver converge faster,
preconditioners can be applied to the original system. The choices of the preconditioners for a
symmetric system include diagonal-based incomplete Cholesky (DIC), multigrid preconditioner,
and a faster version of DIC (FDIC). The details of the preconditioners and the Krylov subspace
solvers can be found in the user guide of OpenFOAM. In all the simulations presented in this
report, the PCG solver with DIC preconditioner was used. A relative convergence tolerance of
10−6 was used to terminate the iterations.
![Page 28: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/28.jpg)
2.2. NUMERICAL SOLUTIONS 19
2.2.6 Convergence criteria
For soils with highly nonlinear hydraulic properties and simulations with very dry initial con-
ditions, the computational effort is always the major concern. For the modified Picard iteration
method, the computational time is directly linked to the convergence criteria. It is noted that this
convergence criteria is different from that for the linear system solution outlined in the previ-
ous section. Huang et al. [1996] evaluated three different convergence criteria, namely standard,
mixed, and moisture content criterion. In the standard criterion, the modified Picard iteration
continues until the difference in pressure head between two successive iteration levels becomes
less than a preset tolerance ha, i.e.,
∣∣hn+1,m+1 − hn+1,m∣∣ ≤ ha (2.32)
This criterion is the most commonly used one in solving the nonlinear Richards equation. How-
ever, the value of ha varies widely and is somehow arbitrary. Though extremely small value of
ha can produce relatively accurate solution, the computational cost becomes excessive. To im-
prove the performance, several researchers have suggested a modification which involves both
the absolute error ha and the relative error δr
∣∣hn+1,m+1 − hn+1,m∣∣ ≤ δr
∣∣hn+1,m+1∣∣+ ha. (2.33)
This formula is termed as the mixed convergence criterion and the value of δr generally varies
from 0.001 to 0.01 Huang et al. [1996]. The introduction of the relative term, in competition with
the absolute term on the right hand side, could relax the convergence criterion when δr |hn+1,m+1|
becomes much larger than ha. While it could accelerate the computation, the accuracy of the
generated results may be unacceptable, especially for cases where sharp moisture content front
develops. Based on the same reason behind the modified Picard iteration method in Celia et al.
![Page 29: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/29.jpg)
20 CHAPTER 2. MODEL FORMULATIONS AND IMPLEMENTATIONS
[1990], Huang et al. [1996] proposed the moisture content convergence criterion for the mix-form
based Richards equation solvers
∣∣θn+1,m+1 − θn+1,m∣∣ ≤ δθ. (2.34)
They reasoned that this moisture content convergence criterion makes more sense from both the
mathematical and the physical point of view. The comparison of the three convergence criteria
in terms of computational time and iteration numbers in Huang et al. [1996] suggests that the
last criterion produces a very efficient and accurate method for simulating variably saturated
flows. However, this criterion is more suitable for the mix-form Richards equation. For other
forms, the standard and the mixed criteria should be used with proper values for convergence.
These three different convergence criteria have been implemented using OpenFOAM’s runtime
selection mechanism. One only need to specify the choice of the criterion and the associated
parameters. In our simulations, the moisture content criterion was used and δθ was set as 10−5.
2.2.7 Parallel computation
The mechanism for parallel computation in OpenFOAM is fully embedded inside its lower level
code structure and it is completely automatic. suGWFoam utilizes this automatic parallelization
and builds its high level Richards equation solver without the details of how the parallel compu-
tation is implemented. suGWFoam can be run in both serial and parallel modes without the need
to recompile the code. The method used in OpenFOAM to realize parallel computing is known
as domain decomposition and it employs message-passing interface (MPI) protocol. The geom-
etry and the associated fields (pressure head h, hydraulic conductivity K, etc.) are decomposed
into pieces and allocated to separate processors (distributed memory) for solution OpenFOAM
Foundation [2011]. Special preprocessing routines are available to decompose the domain and
prepare for the simulation on the number of specified processors. In a parallel simulation, each
![Page 30: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/30.jpg)
2.2. NUMERICAL SOLUTIONS 21
processor treats its allocation as if it is a separate case except the interfaces shared by other
processors need to coordinate the communication and the interchange of information.
![Page 31: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/31.jpg)
Chapter 3
Simulation Examples
3.1 One-dimensional infiltration and redistribution problem
Paniconi et al. [1991] simulated one-dimensional infiltration and redistribution into a 10 m soil
column initially at hydrostatic condition to compare the performance of six time discretization
strategies they proposed. There is no exact solution to this nonlinear problem. Instead, they used
a fine grid, highly accurate Newton iterative solution as the base case ”exact” solution. They
tested two problems (Test Cases 2 and 3) with different boundary conditions (infiltration and
evaporation corresponding to Case 2 and 3, respectively). We used Case 2 as the first validation
problem for suGWFoam. This test problem has also been used to validate other models, including
the SU3D model presented in Dogan and Motz [2005b]. The boundary condition at the top
surface is a time-varying specified Darcy flux q(t) which increases linearly with time
q(t) =t
64m/h, (3.1)
while on the bottom of the soil column, a fixed head boundary condition h = 0 m is used. At
time t = 0 h, the initial condition is set as hydrostatic, i.e., h + z = 0 m. The constitutive
22
![Page 32: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/32.jpg)
3.1. 1D INFILTRATION/REDISTRIBUTION 23
Table 3.1: Values of parameters for 1D infiltration problem and 2D water table recharge problem
Parameter Symbol Value for 1D infiltra-tion problem (Case 2 ofPaniconi et al. [1991])
Value for 2D transientwater table rechargeproblem in Vauclinet al. [1979]
Unit
Residual moisture con-tent
θr 0.08 0.01
Saturated moisture con-tent
θs 0.45 0.30
Air entry pressure head hs -0.3 0.303 mvan Genuchten modelparameter
n 3.0 4.1
van Genuchten modelparameter
h0 -0.19105548 0 m
Specific storage Ss 0.001 0 m−1
Saturated hydraulicconductivity
Ks 5 0.35 m/h
relationship for the soil is described by the modified version of the van Genuchten and Nielsen
formula in Paniconi et al. [1991] which allows a nonzero value of specific moisture capacity in the
saturated zone. The parameters for the simulation, such as the saturated hydraulic conductivity
Ks, saturated and residual moisture content, are the same as in Paniconi et al. [1991] and they are
listed in Table 3.1. The one-dimensional soil column is represented by a uniform grid with 100
3D hexahedral elements stacking in the z direction. The special boundary conditions of empty in
OpenFOAM are used for the faces on the vertical sides of these hexahedra, which makes them
have no contribution to the finite volume integration and therefore renders the simulation a 1D
case. The simulation is run for 32 hours and the profiles for the pressure head at 1, 2, 4, 10, and
32 h after the infiltration are plotted in Figure 3.1. Data from Paniconi et al. [1991] and Dogan
and Motz [2005b] are also plotted. Perfect agreement has been observed for all pressure head
profiles.
![Page 33: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/33.jpg)
24 CHAPTER 3. SIMULATION EXAMPLES
Figure 3.1: Comparison of results for one-dimensional infiltration problem with Paniconi et al.[1991] and Dogan and Motz [2005b]
3.2 Two-dimensional variably saturated water table recharge
A 2D case was simulated where variably saturated water table was recharged by a fixed flux
from the top. This unsteady case was described in Vauclin et al. [1979], and was also used as a
benchmark case in Clement et al. [1994], Dogan and Motz [2005b], and Soraganvi and Kumar
[2009], among others. Infiltration and subsurface flow is due to a flux of water over a strip of 1m
on the surface of a 6mwide and 2m deep soil slab. The constant infiltration flux q = 0.148 m/h
was applied on the corresponding boundary. Due to the symmetry of the problem, only half of
the domain was simulated with the middle plane designated as symmetric boundary. The bottom
and the top (excluding the infiltration strip) were set as impervious zero-flux boundaries. The
total water head (water level) on the right side was set at a constant height of 0.65 m. Above
the fixed water table on the right side, the boundary was also set as zero flux boundary since this
problem has a steady-state solution without developing a seepage face on it [Vauclin et al., 1979,
Clement et al., 1994]. Initially, the whole domain has a hydrostatic pressure distribution with the
water table at 0.65 m.
From the experimental data presented in Vauclin et al. [1979], Clement et al. [1994] fitted
the [van Genuchten, 1980] model and estimated the relevant parameters, which are listed in
![Page 34: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/34.jpg)
3.3. 3D AQUIFER PUMPING 25
Figure 3.2: Comparison of results for two-dimensional, unsteady, water table recharge problemwith the experimental data from Vauclin et al. [1979]
Table 3.1 together with the reported values for hydraulic conductivity, saturated porosity, residual
porosity, etc. It is noted that Clement et al. [1994] also neglected the specific storage because the
compressibility effect is order of magnitude less important. In this research, the specific storage
was also set to zero.
Simulations was run for 8 hours and the transient positions of the water table at t = 2, 3,
4, and 8 hours are plotted in Figure 3.2 in comparison with the experimental data in Vauclin
et al. [1979]. Good agreement between the model prediction and the experiment demonstrates
the capability and accuracy of the proposed model.
3.3 Three-dimensional field-scale unconfined-aquifer pump-
ing test
To test the accuracy and applicability of suGWFoam on problems of field scale, the pumping test
in an unconfined aquifer described in Nwankwor et al. [1992] was used for validation purpose.
Through field experiments, they found it was inadequate to use two previously proposed models
![Page 35: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/35.jpg)
26 CHAPTER 3. SIMULATION EXAMPLES
to explain the observed delayed drawdown. They found that the drainage processes above the
water table (capillary fringe) have a direct and significant impact on the pumping response of
unconfined aquifers. This field test case has also been used by many others to calibrate and
validate their models. For example, Akindunni and Gillham [1992] numerically investigated
the delayed drainage and examine the capillary fringe hypothesis proposed in Nwankwor et al.
[1992]. They used a two-dimensional code which solved the variably saturated flow in cylindrical
coordinates. This case has also been used to validate the finite difference model in Dogan and
Motz [2005b] where the pumping well was modeled as a line source.
According to the description of the field site and the experimental procedures in Nwankwor
et al. [1992], the aquifer had a depth of 9 m bounded by an aquitard (thick deposit of clayey
silt) below. At the beginning of the pumping test, the water table was located at 2.3 m below
the ground surface. The pumping well has a inner diameter of 0.15 m. A well screen of 4 m
depth was used at the bottom of the aquifer. The pumping rate was recorded at 60 l/min and
the test lasted for 24 hours. During the test, piezometers installed at different radial distances
and depths recorded the transient response of the aquifer. In our simulation, only a quart of the
domain was simulated due to symmetry. The quarter has a radius of 100 m in horizontal and 9 m
in depth. The well and the screen were modeled in the domain using the specified flux boundary
condition at the well screen. All other faces were modeled as impervious no-flow boundary. In
horizontal, the mesh was clustered toward the well in the center to better capture the flow field
where hydraulic gradient was largest. In the vertical direction, a dense uniform grid with 81
cells was used. The mesh has a total number of 40,500 hexahedral cells with 46,002 points and
126,860 faces. The details of the 3D mesh can be found in Figure 3.3(a). Since the scales of
the well and the domain differ dramatically, in conjunction with the cylindrical geometry of the
domain, the quality of the mesh (i.e., orthogonality, aspect ratio, and skewness) was not ideal.
Cells near the center of pumping well are distorted. However, the simulation completed without
any problem thanks to the correction schemes, which demonstrate the robustness of the code.
![Page 36: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/36.jpg)
3.3. 3D AQUIFER PUMPING 27
The soil parameters were the same as in Akindunni and Gillham [1992] and Dogan and Motz
[2005b]. The relationship between pressure head h and moisture content θ was described by the
van Genuchten [1980] formula. The parameters α and n were determined by fitting the experi-
mental data and have values of 6.095 and 1.9 1/m, respectively. The hydraulic conductivity K
was determined as a power function of the moisture content by fitting experimental data and has
the form
Kx(θ) = Ky(θ) = aθb (3.2)
where the values for a and b are 25.92m/day and 4.72, respectively. Using the concept of equiv-
alent hydraulic conductivity of soils in series, the vertical hydraulic conductivity, and therefore
the anisotropy, was determined to be
Kz(θ) = 0.64Kx(θ) (3.3)
The saturated moisture content θs and residual moisture content θr were 0.37 and 0.07, which
were derived from the water content profiles in Nwankwor et al. [1992]. The specific storage Ss
was set as 3.25× 10−4 1/m and has the same values as in Akindunni and Gillham [1992].
As in the experiment, pressure head at two radial locations (5 m and 15 m, respectively) at
depth of 7 m was recorded during the simulation. The pressure head was then used to derive the
time history of the drawdowns at these locations. The numerical results were plotted in Figure
3.3(b) in comparison with measurement data which indicates excellent agreement and shows the
capability of suGWFoam to capture the delayed yield effect.
![Page 37: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/37.jpg)
28 CHAPTER 3. SIMULATION EXAMPLES
(a)
Figure 3.3: Three-dimensional field scale pumping test described in Nwankwor et al. [1992]: (a)Details of the mesh for the simulation (only a quart of the domain was simulated; z direction hasbeen stretched 4 times of real size) (b) Comparison of results with the field measurement data
(b)
Figure 3.3: Three-dimensional field scale pumping test described in Nwankwor et al. [1992]: (a)Details of the mesh for the simulation (only a quart of the domain was simulated; z direction hasbeen stretched 4 times of real size) (b) Comparison of results with the field measurement data
![Page 38: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/38.jpg)
3.4. 3D AQUIFER PUMPING 29
3.4 Three-dimensional pumping problem in homogeneous or-
thotropic domain
In this test case, a three-dimensional variably saturated pumping problem introduced in Yeh and
Cheng [1994] was simulated and the results are compared with those from their 3DFEMWATER
model as well as those by the integrated hydrologic model in Kumar et al. [2009] and the finite
difference model in Dogan and Motz [2005b]. The test was conducted in an artificial domain
with dimensions of 800 m by 1000 m in horizontal and 72 m in vertical (see Figure 3.4). In
the horizontal plane, the pumping well was located at (x,y)=(540,400) m. The model domain
was bounded on the left and right by hydraulically connected rivers which had a constant head
of 60 m. The front, back and bottom of the domain were bounded by impervious rocks. The
top of the domain was the interface between air and soil which was also modeled as impervious
boundaries.
At the beginning of the simulation, the whole domain was in the status of hydraulic equi-
librium with a water table of 60 m above the bottom. The water level at the pumping well was
lowered to 30 m and kept at that level during the simulation. The case was run until a steady
state has been reached. The original case in Yeh and Cheng [1994] was for anisotropic soil and it
has saturated hydraulic conductivities Kx=5 m/d, Ky=0.5 m/d, and Kz=2 m/d. It is designated
as the ”AnIsoXY” case in this report. The second case for this problem was proposed in Kumar
et al. [2009] where the hydraulic conductivity in y direction was increased to the same value as in
the x direction (Kx=Ky=5 m/d). This second numerical experiment, designated as the ”IsoXY”
case in this report, was used to explore the effect of anisotropy on the drawdowns.
The soil constitutive relation was described using a variant of the van Genuchten [1980]
model, which has the form
θ = θr +θs − θr
1 + (α |ha − h|)β, Kr =
(θ − θrθs − θr
)2
(3.4)
![Page 39: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/39.jpg)
30 CHAPTER 3. SIMULATION EXAMPLES
XY
Z
400 m
400 m540 m
460 m
30 m
60 m
72 m
(a)
Figure 3.4: Schematic of the 3D pumping problem in 3DFEMWATER Yeh and Cheng [1994]
where α, β, and ha are parameters which have the values of 2.0 1/m, 0.5 and 0 m, respec-
tively. The saturated moisture content θs and the residual moisture content θr were 0.25 and
0.0125, respectively. The whole domain was discretized into 324,000 cells with 338,428 points
and 986,220 faces. To better represent the hydraulic gradient, cells were clustered around the
pumping well and along the water table. Unlike the previous test case, the pumping well was
not modeled geometrically in the computational domain. Instead, the well was represented by
fixing the head at the corresponding cells. This approach was originally used in Yeh and Cheng
[1994] and was followed by other. For the purpose of comparison, it was also used in this report.
However, the finite volume approach based on unstructured mesh can handle any irregular shape
of domain including the wells and it is probably the most accurate way of modeling the pumping
process. The results for both the ”AnIsoXY” case and the ”IsoXY” case are plotted in Figure
3.5 in comparison with the results from 3DFEMWATER, Dogan and Motz [2005b], and Kumar
et al. [2009]. Good agreement has been observed in the steady state drawdown curves for both
cases.
![Page 40: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/40.jpg)
3.4. 3D AQUIFER PUMPING 31
(a)
Figure 3.5: Steady state results for the 3D pumping problem in 3DFEMWATER and comparisonwith the results in literature: (a) Water table profile for anisotropic hydraulic conductivity in xand y directions (b) Water table profile for isotropic hydraulic conductivity in x and y directions(c) 3D view of the water table profile for anisotropic hydraulic conductivity in x and y directions(d) 3D view of the water table profile for isotropic hydraulic conductivity in x and y directions
(b)
Figure 3.5: Steady state results for the 3D pumping problem in 3DFEMWATER and comparisonwith the results in literature: (a) Water table profile for anisotropic hydraulic conductivity in xand y directions (b) Water table profile for isotropic hydraulic conductivity in x and y directions(c) 3D view of the water table profile for anisotropic hydraulic conductivity in x and y directions(d) 3D view of the water table profile for isotropic hydraulic conductivity in x and y directions
![Page 41: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/41.jpg)
32 CHAPTER 3. SIMULATION EXAMPLES
(c)
Figure 3.5: Steady state results for the 3D pumping problem in 3DFEMWATER and comparisonwith the results in literature: (a) Water table profile for anisotropic hydraulic conductivity in xand y directions (b) Water table profile for isotropic hydraulic conductivity in x and y directions(c) 3D view of the water table profile for anisotropic hydraulic conductivity in x and y directions(d) 3D view of the water table profile for isotropic hydraulic conductivity in x and y directions
![Page 42: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/42.jpg)
3.4. 3D AQUIFER PUMPING 33
(d)
Figure 3.5: Steady state results for the 3D pumping problem in 3DFEMWATER and comparisonwith the results in literature: (a) Water table profile for anisotropic hydraulic conductivity in xand y directions (b) Water table profile for isotropic hydraulic conductivity in x and y directions(c) 3D view of the water table profile for anisotropic hydraulic conductivity in x and y directions(d) 3D view of the water table profile for isotropic hydraulic conductivity in x and y directions
![Page 43: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/43.jpg)
34 CHAPTER 3. SIMULATION EXAMPLES
3.5 Parallel computation and mass balance analysis
The efficiency of the parallel computation depends on an array of variables, including mesh size,
number of faces shared between processors, processor speed, memory, communication band-
width, etc. To test the parallel computation performance of suGWFoam a benchmark simulation
was run on the Ranger Linux cluster at the Texas Advanced Computing Center (TACC). Each
computing node of the system contains four AMD Opteron Quad-Core 64-bit processors (16
cores in all) and has 32 GB of memory. The interconnect and communications between comput-
ing nodes are through high speed InfiniBand switches which greatly improve the computational
performance. The field scale test case of Nwankwor et al. [1992] presented in Section 3.3 was
re-run with a refined mesh. The refined case had a total of about 2 million cells and was decom-
posed using the METIS algorithm which provides high quality partitions Karypis and Kumar
[1999]. Figure 3.6(a) shows the example decomposition of the domain into 16 partitions (each
color represents a different processor). The simulations were run using 2, 4, 8, 16, 32, 64, 128,
256, 512, and 1024 processors. The computing time for each run was recorded as TP , where
subscript P denotes the number of processors. A serial run (with only one processor) was also
done and the computing time was denoted as T1. The speedup of the parallel computation is
defined as SP = T1/TP and it was computed for each run. For the ideal scenario, doubling the
number of processors will reduce the computing time by half. In addition, another performance
metric, parallel efficiency EP = T1/ (pTP ) was also calculated. EP has a value between 0 (low
efficiency) and 1 (high efficiency), measuring how well the processors are used in solving the
problem compared to the overhead spent on communications (roughly proportional to the num-
ber of faces shared between processors). All the results for the parallel computations are listed
in Table 3.2. For this particular case, the speedup was excellent up to 256 cores which has the
maximum speedup of 118.5 and 46% of processor efficiency. Further increasing the number of
cores decreased the performance which is as expected. This decreasing can be explained by the
![Page 44: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/44.jpg)
3.5. PARALLEL COMPUTATION AND MASS BALANCE 35
Table 3.2: Domain decomposition and parallel computing analysis results for the refined case inNwankwor et al. [1992]
Numberof cores
Computingtime (s)
Speedup Efficiency Shared facesbetween cores
Cells per core
1 434,000 1 1 - 2,024,9992 224,004 1.94 0.97 10,329 1,012,5004 109,325 3.97 0.99 26,580 506,2508 75,310 5.76 0.72 52,315 253,12516 50,454 8.60 0.54 82,542 126,56232 26,940 16.11 0.50 125,872 63,28164 12,450 34.86 0.54 177,903 31,641128 5,606 77.42 0.60 237,613 15,820256 3,662 118.51 0.46 318,564 7,910512 6,245 69.50 0.14 412,686 3,9551024 8,327 52.12 0.05 530,756 1,978
small number of cells allocated to each core and large number of faces shared by them which
makes the communication overhead dominates the simulation. Nevertheless, from the example
problem with a mesh size of about 2 million, we can conclude that suGWFoam has excellent par-
allel performance and dramatically reduces the computing time. This feature is very important
for large scale and long term ground water simulations as well as rapid assessment of accidental
contaminant spill.
To check whether the model conserves mass, the mass balance error (defined as the total
flux into the domain through boundaries minus the total water change inside the domain) was
recorded at each time step. The cumulative error was subsequently calculated. Both errors are
plotted in Figure 3.6(c). At the beginning of the simulation, a jump in the error was observed.
This was due to the adjustment of the system and nonlinearity. After this period, the system
stabilized. Overall, the mass balance error (both at each time step and accumulative) were small
(in the order of 10−9 m3).
![Page 45: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/45.jpg)
36 CHAPTER 3. SIMULATION EXAMPLES
(a)
Figure 3.6: Parallel computation analysis: (a) Example decomposition of the domain into 16partitions, (b) Speedup and efficiency of the parallel computation using up to 1024 processors,(c) Mass balance error plots (both at each time step and cumulative)
(b)
Figure 3.6: Parallel computation analysis: (a) Example decomposition of the domain into 16partitions, (b) Speedup and efficiency of the parallel computation using up to 1024 processors,(c) Mass balance error plots (both at each time step and cumulative)
![Page 46: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/46.jpg)
3.5. PARALLEL COMPUTATION AND MASS BALANCE 37
(c)
Figure 3.6: Parallel computation analysis: (a) Example decomposition of the domain into 16partitions, (b) Speedup and efficiency of the parallel computation using up to 1024 processors,(c) Mass balance error plots (both at each time step and cumulative)
![Page 47: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/47.jpg)
Chapter 4
Summary and Conclusions
The variably saturated porous media flow model introduced in this report is a good example
of applying new computational software development techniques to a traditional field. Built
upon the open source platform of OpenFOAM, it solves different forms of the 3D nonlinear
Richards equation on unstructured meshes using finite volume method. Because of the design of
the platform, the model is versatile in almost all aspects of numerical schemes. The numerical
discretizations are selectable when the differential operators (such as Laplacian, gradient, and
divergence) are approximated using finite volume method in OpenFOAM. Non-orthogonal cor-
rections are considered to improve accuracy and boundedness. The temporal derivative term for
transient problem can be discretized using first order Euler implicit or second order backward
implicit schemes. The spatial derivative can be evaluated at different time levels using Euler
implicit, explicit, or Crank Nicolson schemes which can be selected at run time. Various con-
stitutive relations, including those most common ones, have been implemented in the model. To
leverage the merits of both h-based and θ-based forms of the governing equation, a switching
algorithm has been used. It has been demonstrated by a range of numerical examples that the
results of the new model agree well with those in the literature. In addition, the new model con-
serves mass to high accuracy which is as expected by virtue of conservative finite volume method
38
![Page 48: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/48.jpg)
39
and the modified Picard iterations. For large problems, the model has shown excellent parallel
computing performance. A maximum speedup of 118 has been observed in the parallel compu-
tation exercise using up to 1024 cores. Future development includes coupling of surface water
hydrodynamics with the groundwater flows using exiting flow solvers in OpenFOAM. This is in
fact a strength of using this platform since it is very mature in hydrodynamics modeling. In con-
clusion, suGWFoam is an innovative and useful contribution to the porous media flow modeling
community.
In the future, we will explore the multiphysics flexibility of OpenFOAM platform and couple
suGWFoam with other solvers, such as those for channel flows. In doing so (and with great ease),
it is very easy to do for example surface-water and groundwater interactions. It is also possible
to couple suGWFoam with chemical reaction solvers.
![Page 49: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/49.jpg)
Bibliography
P M Adler and H Brenner. Multiphase flow in porous media. Annu. Rev. Fluid Mech., 20(1):
35–59, January 1988. ISSN 0066-4189.
Festus F. Akindunni and R. W. Gillham. Unsaturated and saturated flow in response to pumping
of an unconfined aquifer: Numerical investigation of delayed drainage. Ground Water, 30(6):
873–884, 1992. ISSN 1745-6584. doi: 10.1111/j.1745-6584.1992.tb01570.x.
R.H. Brooks and A.T. Corey. Hydraulic properties of porous media. Hydrology paper no. 3, civil
engineering, Colorado State University, 1964.
Michael A. Celia, Efthimios T. Bouloutas, and Rebecca L. Zarba. A general mass-conservative
numerical solution for the unsaturated flow equation. Water Resour. Res., 26(7):1483–1496,
1990. ISSN 0043-1397.
T.P. Clement, William R. Wise, and Fred J. Molz. A physically based, two-dimensional, finite-
difference algorithm for modeling variably saturated flow. Journal of Hydrology, 161(1-4):71
– 90, 1994. ISSN 0022-1694. doi: DOI:10.1016/0022-1694(94)90121-X.
R.L. Cooley. A finite difference method for unsteady flow in variably saturated porous media:
Application to a single pumping well. Water Resour. Res., 7(6):1607–1625, 1971. doi: 10.
1029/WR007i006p0160.
H. J. G. Diersch and P. Perrochet. On the primary variable switching technique for simulating
40
![Page 50: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/50.jpg)
BIBLIOGRAPHY 41
unsaturated-saturated flows. Advances in Water Resources, 23(3):271 – 301, 1999. ISSN
0309-1708. doi: DOI:10.1016/S0309-1708(98)00057-8.
H.J.G Diersch. FEFLOW: Finite element subsurface flow & transport simulation system ref-
erence manual. DHI-WASY GmbH, Waltersdorfer Strasse 105, D-12526, Berlin, Germany,
2009.
Ahmet Dogan and Louis H. Motz. Saturated-unsaturated 3d groundwater model. i: Develop-
ment. Journal of Hydrologic Engineering, 10(6):492–504, 2005a. doi: 10.1061/(ASCE)
1084-0699(2005)10:6(492).
Ahmet Dogan and Louis H. Motz. Saturated-unsaturated 3d groundwater model. ii: Verification
and application. Journal of Hydrologic Engineering, 10(6):505–515, 2005b. doi: 10.1061/
(ASCE)1084-0699(2005)10:6(505).
P.A. Domenico and F.W. Schwartz. Physical and chemical hydrogeology. John Wiley & Sons,
Inc., 2nd edition, 1990.
P. A. Forsyth, Y. S. Wu, and K. Pruess. Robust numerical methods for saturated-unsaturated flow
with dry initial conditions in heterogeneous media. Advances in Water Resources, 18(1):25 –
38, 1995. ISSN 0309-1708. doi: DOI:10.1016/0309-1708(95)00020-J.
Xinmei Hao, Renduo Zhang, and Alexandra Kravchenko. A mass-conservative switching
method for simulating saturated-unsaturated flow. Journal of Hydrology, 311(1-4):254 – 265,
2005. ISSN 0022-1694. doi: DOI:10.1016/j.jhydrol.2005.01.019.
R.G. Hills, I. Porro, D.B. Hudson, and P.J. Wierenga. Modeling one-dimensional infiltration into
very dry soils: 1. model development and evaluation. Water Resour. Res., 25(6):1259–1269,
1989.
![Page 51: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/51.jpg)
42 BIBLIOGRAPHY
K. Huang, B. P. Mohanty, and M. Th. van Genuchten. A new convergence criterion for the
modified picard iteration method to solve the variably saturated flow equation. Journal of
Hydrology, 178(1-4):69 – 91, 1996. ISSN 0022-1694. doi: DOI:10.1016/0022-1694(95)
02799-8.
H. Jasak. Error analysis and estimation for the finite volume method with applications to fluid
flows. PhD thesis, Department of mechanical engineering, Imperial College of Science, Tech-
nology and Medicine, 1996.
George Karypis and Vipin Kumar. A fast and high quality multilevel scheme for partitioning
irregular graphs. SIAM Journal on Scientific Computing, 20(1):359–392, 1999.
M. R. Kirkland, R. G. Hills, and P. J. Wierenga. Algorithms for solving richards’ equation for
variably saturated soils. Water Resour. Res., 28(8):2049–2058, 1992. ISSN 0043-1397.
Mukesh Kumar, Christopher J. Duffy, and Karen M. Salvage. A second-order accurate, finite
volume-based, integrated hydrologic modeling (FIHM) framework for simulation of surface
and subsurface flow. Vadose Zone Journal, 8(4):873–890, 2009. doi: 10.2136/vzj2009.0014.
E.G. Lappala, R.W. Healy, and E.P. Weeks. Documentation of computer program vs2d to solve
the equations of fluid flow in variably saturated porous media. Technical Report Water Re-
sources Investigations Rep. 83-4099, U.S. Geological Survey, Washington, D.C., 1987.
Gianmarco Manzini and Stefano Ferraris. Mass-conservative finite volume methods on 2-d un-
structured grids for the richards’ equation. Advances in Water Resources, 27(12):1199 – 1215,
2004. ISSN 0309-1708. doi: DOI:10.1016/j.advwatres.2004.08.008.
D. McBride, M. Cross, N. Croft, C. Bennett, and J. Gebhardt. Computational modelling of
variably saturated flow in porous media with complex three-dimensional geometries. Interna-
tional Journal for Numerical Methods in Fluids, 50(9):1085–1117, 2006. ISSN 1097-0363.
doi: 10.1002/fld.1087.
![Page 52: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/52.jpg)
BIBLIOGRAPHY 43
M.G. McDonald and A.W. Harbaugh. A modular three-dimensional finite-difference ground-
water flow model. Technical report, U.S. Geological Survey Techniques of Water-Resouces
Investigations, Book 6, Chap. A1, Washington, DC, 1988.
Cass T. Miller, Glenn A. Williams, C. T. Kelley, and Michael D. Tocci. Robust solution of
richards’ equation for nonuniform porous media. Water Resour. Res., 34(10):2599–2610,
1998. ISSN 0043-1397.
G. I. Nwankwor, R. W. Gillham, G. van der Kamp, and F. F. Akindunni. Unsaturated and
saturated flow in response to pumping of an unconfined aquifer: Field evidence of delayed
drainageunsaturated and saturated flow in response to pumping of an unconfined aquifer.
Ground Water, 30(5):690–700, 1992. ISSN 1745-6584. doi: 10.1111/j.1745-6584.1992.
tb01555.x.
OpenFOAM Foundation. OpenFOAM: The open source CFD toolbox, programmer’s guide,
Version 1.6. Reading, UK, 2011.
Claudio Paniconi, Alvaro A. Aldama, and Eric F. Wood. Numerical evaluation of iterative and
noniterative methods for the solution of the nonlinear richards equation. Water Resour. Res.,
27(6):1147–1163, 1991. ISSN 0043-1397.
Song Pham-Van, Reinhard Hinkelmann, Marko Nehrig, and Isaac Martinez. A comparison of
model concepts and experiments for seepage processes through a dike with a fault zone. En-
gineering Applications of Computational Fluid Mechanics, 5(1):149–158, 2011.
J.C. Refsgaard and B. Storm. MIKE SHE, in Computer Models of Watershed Hydrology, V.P.
Singh, Ed., pages 809–846. Water Resources Publications, Colorado, USA, 1995.
L.A. Richards. Capillary conduction of liquids through porous mediums. Physics, 1(5):318–333,
1931. doi: 10.1063/1.1745010.
![Page 53: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/53.jpg)
44 BIBLIOGRAPHY
V.S. Soraganvi and M.S. Kumar. Modeling of flow and advection dominant solute transport in
variably saturated porous media. Journal of Hydrologic Engineering, 14(1):1–14, 2009. doi:
10.1061/(ASCE)1084-0699(2009)14:1(1).
E. A. Sudicky, A. J. A. Unger, and S. Lacombe. A noniterative technique for the direct imple-
mentation of well bore boundary conditions in three-dimensional heterogeneous formations.
Water Resour. Res., 31(2):411–415, 1995. ISSN 0043-1397.
R. Therrien and E. A. Sudicky. Well bore boundary conditions for variably saturated flow
modeling. Advances in Water Resources, 24(2):195 – 201, 2000. ISSN 0309-1708. doi:
DOI:10.1016/S0309-1708(00)00028-2.
R. Therrien, R.G. McLaren, E.A. Sudicky, and S.M. Panday. HydroGeoSphere: A three-
dimensional numerical model describing fully-integrated subsurface and surface flow and
solute transport (Draft). Groundwater Simulation Group, Waterloo, Ontario, Canada, 2010.
J. C. van Dam, P. Groenendijk, R.F.A. Hendriks, and J.G. Kroes. Advances of modeling water
flow in variably saturated soils with swap. Vadose Zone Journal, 7(2):640–653, 2008.
M. Th. van Genuchten. A closed-form equation for predicting the hydraulic conductivity of
unsaturated soils. Soil Science Society of America Journal, 44(5):892–898, 1980.
M. Th. van Genuchten and D.R. Nielsen. On describing and predicting the hydraulic properties
of unsaturated soils. Annales Geophysicae, 3(3):615–628, 1985.
M. Vauclin, D. Khanji, and G. Vachaud. Experimental and numerical study of a transient, two-
dimensional unsaturated-saturated water table recharge problem. Water Resour. Res., 15(5):
1089–1101, 1979. ISSN 0043-1397.
C.I. Voss and A.M. Provost. SUTRA: A model for saturated-unsaturated variable-density
![Page 54: suGWFoam: An Open Source Saturated …water.engr.psu.edu/liu/publications/suGWFoam_TR.pdfCivil and Environmental Engineering Studies . Technical Report No. 01- 01 . suGWFoam: An Open](https://reader031.vdocuments.us/reader031/viewer/2022030419/5aa612277f8b9a517d8e131d/html5/thumbnails/54.jpg)
BIBLIOGRAPHY 45
ground-water flow with solute or energy transport. Water-Resources Investigations Report 02-
4231. U.S. Geological Survey, Reston, Virginia, 2010.
H.G. Weller, G. Tabor, H. Jasak, and C. Fureby. A tensorial approach to computational contin-
uum mechanics using object-oriented techniques. Computers in Physics, 12:620–631, Novem-
ber 1998.
G.T. Yeh and J.R. Cheng. 3DFEMWATER user manual: A three-dimensional finite-element
model of water flow through saturated-unsaturated media: Version 2.0. Pennsylvania State
University, University Park, Pa., 1994.
G.T. Yeh, S. Sharp-Hansen, B. Leuter, R. Stroble, and J. Scarborough. 3dfemwater/3dfemwaste:
Numerical codes for delineating wellhead protection areas in agricultural regions based on the
assimilative capacity criterion. Technical Report EPA/600/R-92/223, Environmental Protec-
tion Agency, Athens, Ga., 1992.
G.T. Yeh, J. Sun, P.M. Jardine, W.D. Burgos, Y. Fang, M.H. Li, and M.D. Siegel. HYDRO-
GEOCHEM 5.0: A Three-Dimensional Model of Coupled Fluid Flow, Thermal Transport,
and HYDROGEOCHEMical Transport through Variably Saturated Conditions: Version 5.0.
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6302, May 2004.