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

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

mailto:[email protected]

http://engineering.utsa.edu/~xiaofengliu

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.

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.

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.

ii

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

iv CONTENTS

3.5 Parallel computation and mass balance . . . . . . . . . . . . . . . . . . . . . . . 34

4 Summary and Conclusions 38

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

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

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

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.

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

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.

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

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.

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


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−


)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

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


(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


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)


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)


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


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-


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)


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.


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.


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.


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.


[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


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.

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

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.

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

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


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.


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.


(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


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 =


)2

(3.4)


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.


(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)



(c)



(d)



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

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).


(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)


3.5. PARALLEL COMPUTATION AND MASS BALANCE 37

(c)


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

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.

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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,

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