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EVALUATION OF TRANSFORM-BASED NUMERICAL MEI'HODS APPLEXI TO RICHARDS' EQUATION

Robert G. Baca Center for Nuclear Waste Regulatory Analyses Southwest Research Institute, San Antonio, TX

Jacob N. Chung and David A. Stock Washington State University, Pullman, WA

ABSTRACT

Transform-based numerical approaches have been emerging as very promising computational

approaches for solving the nonlinear infiltration model described by Richards' equation. These novel

methods, which are easy to implement with finite differences and finite elements, have been shown to

yield improved robustness and computational speed. A numerical study is conducted to examine the

relative advantages of three particular transforms: (i) natural log, (ii) inverse hyperbolic sine, and

(iii) algebraic. These three transform-based methods are evaluated and compared using various test

problems involving flow into very dry and heterogeneous porous media. The performance of the

numerical techniques is quantified in terms of computer execution time and global mass balance. The

sensitivity of these two performance measures to boundary conditions, initial conditions, and soil

heterogeneity is investigated. Numerical solutions obtained using the transform-based approaches in

conjunction with a finite element technique are compared against the untransformed modified Picard and

Newton-Galerkin finite element methods. The results of this comparative study show that all three

transform-based numerical algorithms and the untramformed Newton-Galerkin method are

computationally faster than the untransformed modified Picard method. In addition, these methods achieve

global mass balance equivalent to that of the modified Picard method. However, the principal finding of

this study is that the Newton-Galerkin method, when applied directly to the mixed formulation of

Richards' equation, is superior to all three transform-based numerical methods.

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INTRODUCTION

Regulatory evaluations aimed at assessing the risks of waste disposal sites are making increasing

use of detailed numerical models of flow and transport phenomena. Models of infiltration and deep

percolation, in particular, are playing a central role in studies of waste disposal safety. In the case of low-

and high-level nuclear waste disposal, for example, numerical models are currently being applied to

forecast infiltration rates that may occur as a result of climate changes (typically spanning thousands of

years). In steady-state analyses, Monte Carlo simulations of flow are generally performed to develop

probabilistic estimates of water fluxes (Childs and Long, 1992) and travel times (Bagtzoglou and Baca,

1994). Such models require fast, accurate, and robust numerical algorithms in order for these specialized

applications to become routine. This use of detailed simulation models has motivated research to pursue

improved numerical techniques to solve the governing flow equation in a more efficient manner.

Wile a variety of finite difference (Hills et al. 1989; Celia et al., 1990) and finite element

techniques (Huyakorn and Pinder, 1983; Huyakorn et al., 1984) are currently available for solving the

nonlinear Richards’ flow equation, they often work well for only a limited class of flow conditions and

simple hydrostratigraphic models. Moreover, existing techniques rarely exhibit the robustness and

convergence rate desired for efficient three-dimensional, high-resolution simulations of flow in

heterogeneous media (Ababou and Gelhar, 1988). Three particular types of simulation problems that pose

the greatest computational difficulty are those involving: (i) surface ponding on and high infiltration rates

into very dry soils, (ii) flow involving the formation and dissipation of perched water zones, and (iii) flow

through soil layers with large permeability contrasts. Test cases encompassing these types of hydraulic

conditions and flow characteristics are considered in this paper.

Although transform methods have historically been used in developing analytical solutions

(Kirkham and Powers, 1972) to Richards’ equation, only recently have researchers noted that numerical

approximations to the various forms of Richards’ equation (i.e., mixed, head-based, and moisture-based)

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can be improved by introducing a special transform. Various transform-based numerical approaches have

been investigated by Baca et al. (1978), Ross (1990)’ Kirkland et al. (1992), and Pan and Wierenga

(1995). These distinct approaches can be unified through the use of a ‘partitioned transform” formulation.

In this study, a general numerical solution algorithm is developed by discretizing the transformed

governing equation using the NewtonGalerkin finite element method. The computational performance

of these transform-based numerical algorithms is compared with that of the very competitive modified

Picard finite element method (Celia et al., 1990) and the untransformed Newton-Galerkin method.

TRANSFORM-BASED APPROACHES

Transfomd Flow Equation

The mixed form of Richards’ equation for transient flow in variably-saturated porous media is

a well-known nonlinear parabolic equation that is widely used in the hydrological sciences (Jury et al.,

1993). A modified form of this flow equation is obtained by making a simple change of variable in the

spatial term. For onedimensional (1D) flow, the modified or ‘transformed” governing equation is

expressed as:

where e(*) is the volumetric water content (cm’km’), IC(*) is Ka*/aX, K ( $ ) is the hydraulic

conductivity (cds), z is the depth (cm) taken positive downward, and Jr is the pressure head (cm). In

this formulation, fluid storage associated with compressibility of both the porous medium and fluid is

neglected. The new dependent variable x is defined as a generalized partitioned transform, namely

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where f (0 ,Q) is an analytic function yet to be specified and Qo is an arbitrary match point. The

fundamental idea of this approach is to select an analytic function that transforms JI into a 'compressed"

state variable x . This compression results in a smooth and more gradually varying solution space which,

in theory, allows more accurate calculation of the spatial gradients of x . In order for the partitioned

transform to be appropriate, however, x must be C' continuous at the match point Jro. Thus, the

conditions constraining the choice of f (0 , Q) are twofold:

where 0, = e( 'Ito). The above two conditions are necessary for determining the two coefficients, a1 and

a*. With regard to selecting the proper match point pressure, Qo is typically chosen close to the bubbling

pressure, Jr,, for transforms that depend on the soil hydraulic properties (e.g., mixed transform);

otherwise; it is set to zero. Three transforms that can be written in partitioned form and conform to Eqs.

(3) and (4) are: (i) natural log, (ii) inverse hyperbolic sine, and (iii) algebraic.

The log-transform was originally developed and applied by Baca et al. (1978). These authors

implemented the log-transform in the standard 'head-based" formulation of Richards' equation, but in

a manner that limited its application to purely unsaturated conditions. In the partitioned form described

here, however, the transform is generalized and applies to both saturated and unsaturated regimes. The

log-transform is expressed as:

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where S is a normalizing constant. Applying Eqs. (3) and (4), it follows that

where a1 = -1 and a2 = 0. In this case, the match point pressure is taken to be q0 = 0.

Building on the log-transform idea, Ross (1990) formulated the inverse hyperbolic sine

transform, which he used in conjunction with the "mixed" formulation of Richards' equation given in

Eq. (1). A simple modification of Ross' partitioned formulation is adopted here, namely:

which has the associated derivative relation

where at = 1, az = 0 , and Jr, = 0. Ross noted that incorporation of the inverse hyperbolic sine in the

mixed formulation yielded a numerical algorithm that better accommodated very dry initial conditions,

but possibly less efficient than a method using the Kirchhoff transform (Ross and Bristow, 1990).

Recently, Pan and Wierenga (1995) introduced a new transform that they applied to the

semidiscrete form of Richards' equation currently referred to in the technical literature as the "modified

Picard" formulation (Celia et al., 1990). This transform, which we designate here as the algebraic

transform, is given by

5

- { - */(I l4 +

with the associated derivative

(9)

where a1 = empirical constant (e.g., -0.04 cm-') and Jlo = 0 . Pan and Wierenga (1995) present results

of a series of numerical test cases in which they demonstrate the superiority of their transform over the

standard modified Picard finite difference algorithm.

Modijied Picard Formulation

In a manner parallel to the aforementioned transform-based approaches, the so-called modified

Picard method produces a 'modified" form of Richards' equation. In the modified Picard method,

however, the governing equation is converted directly into a semidiscrete 6 -based formulation, where 6

is the incremental change in the pressure head and given by 6 = qf"*l*k*l - Vlek. The indices n andk

denote the time plane and iteration, respectively. This formulation, as described by Celia et al. (1990),

is obtained by expanding 8 in the iteration space using a Taylor series. This semidiscrete formulation

possesses two desirable properties: (i) the algorithm exhibits excellent global mass balance, and (ii) the

convergence rate is relatively insensitive to very dry soil conditions. This modified formulation is

expressed as:

where A@ = Om*l*k - 8".

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This formulation, when further discretized using a finite element (or finite difference) method,

does not, in the strictest sense, produce a Picard-type iteration algorithm. It produces, in fact, a

"modified-Newton" iteration, which is evident from a direct comparison of the elemental matrices (see

Appendix) with those generated by the Newton-Galerkin method. The fact that it is a modified-Newton

explains, to a large degree, the superior convergence properties of this semidiscrete 8 -based formulation

over the classical Picard iteration method (Huyakorn and Pinder, 1983; Paniconi and Putti, 1994).

FINITEELEMENTALGORITHM

Newton-Galerkin Method

To formulate a numerical solution method for the transformed Richards' equation, an iterative

algorithm is typically used to accommodate its strong nonlinearity. An iterative algorithm can be directly

embedded in a finite element approximation by using the Newton-Galerkin method (Baca et al., 1995).

This method requires that the product of a set of weighting functions, o,(z) , and the first order Taylor

series expansion of the residual, , be orthogonal. Specifically, the Newton-Galerkin method requires

where Q denotes the spatial domain 0 s z s L, N is the number of basis functions, and 8 is obtained

directly from the transformed Richards' equation, namely

In this expression, the spatial variation of 8, x, IC, and K is approximated using a set of linear basis

functions; these functions are chosen to be identical to the set of weighting functions a, (2). It is

important to note that the numerical solution is in terms of Jr rather than 8 or x , both of which are

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often discontinuous at layer interfaces where the soil hydraulic properties change. Substituting Eq. (13)

into Eq. (12) and rearranging, the following system of algebraic equations is produced for an arbitrary

element e :

[J.] {A*) = -{Re}

where the right-hand side vector is:

and the Jacobian matrix is given by:

where ii = { o } ~ { K ) , the Kronecker deltas for the boundary flux 4& are defined as €iel = 1 for the first

element and is zero elsewhere, while 6, = 1 for the last element only. Inserting the standard

expressions for the 1D linear basis functions, { = [ 1 - €/Le, €/Le], the individual terms of Eqs. (15)

and (16) can be integrated exactly using (Pepper and Heinrich, 1992):

1 a! b!

(a + b + l)! E E dl = Lg I ; : 0

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where e is the local element coordinate, (4) are integers, and Le is the element length. Mass lumping

is applied to the time derivative terms in the previous equations to stabilize the matrices associated with

the 8-terms. The fully integrated forms of {Re} and [J,] can be found in the Appendix.

Iteration Strategy

After forming and solving the global form of Eq. (la), the nodal values of pressure head@,

are computed using the Newton iteration formula:

{JrP+' = {JrT + {WP (18)

Convergence of the Newton iteration process is based on the infinity or max norm of the residual error

vector and the maximum relative change. Specifically, the iterative solution must satisfy:

ldk+')I,, s E, and max lAJr,/Jr,l s ed (19)

where I 1, is the infinity n o m and dk+*) is the residual computed as JAh@+" + R. Values of the

control tolerances are chosen from the ranges lo-' s E, s and s s

Ti-Srepping Strategy

Even though Eqs. (15) and (16) use a fully implicit time approximation, the time step sizeAt

must be constrained because it has a strong impact on the solution accuracy, global mass balance, and

number of Newton iterations. To advance the numerical solution through time in an efficient manner, an

automatic time-stepping procedure was used. The procedure consisted of increasing the time step in

relation to the number of iterations required in the previous time plane. Specifically, the time step was

adjusted in accordance with the number of convergent iterations:

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At"+' = f (k )A t"

w

where the scale factor f (k) is a function of the total number of iterations (of the prior step) and is chosen

so that a target number of iterations is generally maintained. In addition, the time step calculated from

Eq. (20) was limited to stay withinminimum and maximum time step sizes, Le., At- s Ata+* s At,.

The maximum step size was computed at each time plane from:

where ASe is the desired maximum change in saturation (typically set to 0.05) and AS- is the

maximum change in saturation calculated in the previous time step. The minimum time step size was

specified by user input. If the solution failed to converge within the specified maximum iterations, a

smaller time step equal to half the previous value was chosen and the solution repeated.

Gmpwer codes

The previously described numerical algorithms were incorporated into a set of FORTRAN 77

computer codes. The codes used common routines to ensure consistency and permit direct

intercomparisons. All calculations were performed in double precision. These codes were verified by

making intercomparisons with the BREATH code (Stothoff, 1995). Applications of the computer codes

to the simulation problems described below were performed on an IBM PS/2 Model 95 (66 MHZ)

computer. Numerical simulations for all test problems were performed using the following convergence

control tolerances:

DESCRIPTION OF FLOW SIMULATION PROBLEMS

5 em and 10-~ s Q.

To study the computational performance of the various transform-based numerical algorithms,

a set of inliltration test problems was developed and solved. The aim of the test problems was to examine

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the performance of each algorithm as a function of surface fluid flux, initial pressure head, and degree

of perched water conditions. These test conditions were woven into three hydrostratigraphic conceptual

models chosen to approximate multilayer systems of actual sites located at: (i) Yucca Mountain, Nevada;

(ii) Hanford Reservation, Washington; and (iii) Idaho National Engineering Laboratory (INEL), Idaho.

Case I - Yucca Mounrain Site

As part of an ongoing evaluation of Yucca Mountain for a proposed nuclear waste repository

location, the U.S. Geological Survey has been studying the patterns of shallow infiltration in the Pagany

Wash area of the Nevada Test Site. Boreholes in the alluvium channel of Pagany Wash have been used

to monitor the moisture content profiles and to develop a 1D infiltration model (Hevesi and Flint, 1993).

A hydrostratigraphic model similar to that of Hevesi et al. (1994) was adopted for this test case. Soil

hydraulic properties assigned to each layer are those reported for borehole UZN #7. Hevesi et al. (1994)

utilized the soil property model of Brooks and Corey (1966) to represent the hydraulic properties for the

soil layers at this site. Parameters for this property model, which were taken from Hevesi et al. (1994),

and the layer thicknesses are summarized in Table 1. The hydrostratigraphy was represented by a

computational mesh of 80 elements of varying size. The hydraulic conductivity of each element was

computed in the algorithm using the standard arithmetic mean value, which results from the assumption

that hydraulic conductivity varies linearly along the finite element.

Case 2 - Xanjbrd Reservation Site

A number of studies have been conducted to model the subsurface movement of a radioactive

plume that occurred at the Hanford Reservation as a result of the T-106 tank leak. This tank leak

occurred in the early 1970s at the 200 east area of the Hanford Reservation. Smoot and Sagar (1990)

modeled the migration of the liquid plume through the Hanford vadose zone to evaluate the likelihood

of groundwater contaminant. These authors modeled the stratified, glaciofluvid sediments at the site as

a five-layer system spanning 62 m. Estimates of hydraulic properties for each layer were developed by

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Smoot and Sagar (1990) by using data from various locations at the Hanford Reservation. The parameters

for the van Genuchten (1980) soil property model are summarized in Table 2, which also gives the layer

thicknesses. For this test case, the domain was discretized into a computational mesh of 63 elements of

variable thicknesses. In this case, the hydraulic conductivity of each element was computed in the

numerical algorithm using the geometric mean value.

Case 3 - Idaho Nm'onal Engineering Laboratory Site

As part of environmental studies of the Radioactive Waste Management Complex (RWMC),

researchers at INEL have modeled subsurface flow from drainage ditches to assess the potential for

accelerating contaminant movement beneath the RWMC. Details of the simulation are described in Martin

and Smith (1993). The hydrostratigraphic setting of this site is characterized by layers of vesicular and

massive basalt with a laterally continuous sedimentary interbed occurring at depth. Hydraulic properties

for these layers are based on laboratory measurements and are represented using van Genuchten

parameters. The parameters and thicknesses for each of these four layers are summarized in Table 3. A

computational mesh of 70 elements was used to represent a 35-m region of the RWMC vadose zone. In

this case, the hydraulic conductivity of each element was computed in the algorithm as the arithmetic

mean.

Definition of Care Studies

For each of the three sites described above, a series of simulations was performed using various

initial and boundary conditions chosen to produce variably saturated flow conditions, large pressure

gradients, and the formation and dissipation of perched water conditions. Four distinct combinations of

initial and boundary conditions were considered for each site. The initial pressure heads were chosen to

be representative of arid site conditions. The surface boundary conditions considered both fixed pressure

head values typical of ponded conditions and specified fluxes that are large relative to the permeability

of selected soil layers. The bottom boundary condition was set to: (i) a saturated condition to represent

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the presence of a water table boundary or (ii) a ftee drainage condition. Specific conditions used in these

cases are summarized in Table 4.

RESULTS AND DISCUSSION

Pressure head profiles computed for the various test cases using the transform-based and

untransformed numerical methods are compared in Figures 1,2, and 3. It is evident from these graphical

comparisons that, in almost all 12 test cases, the head profiles computed using the natural log, inverse

hyperbolic sine, modified Picard, and NewtonGalerkin are nearly identical. For the Hanford Site test

cases, the modified Picard method simulations were incomplete, while the algebraic transform method

runs were nonconvergent. In those cases in which the algebraic transform numerical method was

convergent, the empirical constant required a value of a1 --0.0()1, in contrast to the value of

a1 = -0.04 suggested by Pan and Wierenga (1995). The Central Processing Unit (CPU) times and mass

balance emrs (computed as the absolute value of 1 minus the mass balance ratio) for each test case are

summarized in Tables 5 and 6, respectively.

For the Yucca Mountain flow simulations, the results of the four test runs show that all the

transform-based and untransformed numerical methods were successful in handling the dry initial

conditions, surface ponding, and perched water zones. In addition, all numerical solutions showed

excellent agreement; the algebraic transform solution showed slight variations from the other solutions

in the vicinity of the steep fronts. The untransformed Newton-Galerkin required the least CPU time and,

as illustrated in Table 5, was 10 to 30 percent faster than the natural log and inverse hyperbolic sine.

Both the natural log and inverse hyperbolic sine transforms were 30 to 50 percent faster than the algebraic

transform. In contrast, the modified Picard method exhibited the longest CPU times, which were 4 to 9

times those of the Newton-Galerkin method.

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In the flow simulations for the Hanford Reservation, only the Newton-Galerkin and natural log

transform methods successfully solved the four test cases. Both methods easily accommodated the dry

initial conditions and formation of large perched water zones. The inverse hyperbolic sine and algebraic

transforms method exhibited nonconvergent solutions (designated as NC in Tables 5 and 6) for selected

cases; these results suggest some sensitivity of these methods to dry initial conditions. The CPU times

for the Newton-Galerkin, natural log, inverse hyperbolic sine, and algebraic transform methods were

relatively comparable. In all four cases, the modified Picard solutions were incomplete (designated by

IC in Tables 5 and 6). The computer runs with modified Picard code required extremely small time steps

(e.g., A f < s) in the latter part of the simulation and were subsequently terminated. Complete

solutions with the modified Picard method were projected to require several hours of CPU time.

Application of the numerical methods to INEL site test cases showed very contrasting results.

The modified Picard, Newton-Galerkin, natural log, and inverse hyperbolic sine transforms were

successful in simulating the predominantly gravitydriven flow and formation of a large perched water

zone. The numerical solutions produced by the algebraic transform method were nonconvergent for all

four test cases. Consistent with the previous test cases, the Newton-Galerkin method was faster than the

transform-based and modified Picard methods. For this series of test cases, the natural log transform

method and the inverse hyperbolic sine were comparable. The modified Picard algorithm required the

largest CPU times and were typically 1.7 to 4.5 times those for the Newton-Galerkin method.

As can be noted from the results in Table 6, all the transform-based and untransformed

numerical methods produced very small mass balance errors (reported without sign). The small mass

balance errors observed in these calculations are primarily due to the fact that all methods utilized the

mixed formulation of Richards' equation.

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SUMMARY AND CONCLUSIONS

A comparative study was performed to evaluate computational advantages of transform-based

numerical approaches for solving Richards’ equation. These approaches u t i l i the natural log, inverse

hyperbolic sine, and algebraic transforms to modify the governing equation. In turn, the transformed

Richards’ equation is discretized using a Newn-Galerkin finite element mwhod. These numerical

methods were also compared against the untransformed modified Picard and Newton-Galerkin methods.

To provide a broad basis of comparison, the transhmed-based and untxansformed finite element methods

were applied to a total of 12 test cases. These test cases considered three layered vadose zones that are

representative of actual arid sites. Various combinations of initial and boundary conditions were

considered in flow simulations for each site. The performance of the computational methods was

quantified in terms of CPU time and global mass balance error.

The results of this numerical study show that all three transform-based methods and the

Untransformed Newton-Galerkin method were computationally faster than the popular modified Picard

method. In fact, these methods were typically two to ten times faster than the modified Picard finite

element method. However, the untransformed Newton-Galerkm method is, in general, superior to all

three transform-based approaches in terms of robustness and computational speed. The Newton-Galerkin

method and the transform-based methods demonstrated comparable global mass balance to that of the

standard modified Picard method. The superiority of the Newton-GalerLin method makes it suitable for

application to multidimensional and multiphase problems.

ACKNOWLEDGMENTS

The authors wish to thank Drs. S. Stothoff, G. Wittmeyer, and B. Sagar for their reviews and comments

on the original manuscript. The authors also thank Dr. J.D. Randall of the Nuclear Regulatory

Commission (NRC) for his comments and suggestions. The research presented in this paper was funded

by the NRC under Contract No. NRC-02-93-005 and was performed at the Center for Nuclear Waste

15

Regulatory Analyses (CNWRA). The research was performed on behalf of tho NRC Office of Nuclear

Regulatory Research, Division of Regulatory Applications. The paper is an independent product of the

CNWRA and does not necessarily reflect the views or regulatory position of the NRC. The FORTRAN

codes used in the study are not under the CNWRA software configuration management procedure.

REFERENCES

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heterogeneous media, Proceedings of compurcrtional Methodr in Water Resources, Elsevier and

Computational Mechanics Publications, London, UK, 1: 173-178, 1988.

Baca, R.G., 1.P. King, and W.R. Norton, Finite element models for simultaneous heat and moisture

transport in unsaturated soils, Proceedings of the Second Imem*onal Conference on Finite

Elements in Warer Resources, Pentech Press, London, UK, 1.19-1.35, 1978.

Baca, R.G., J.N. Chung, and D.J. MuIIa, Mixed transform finite element method for solving the

equation for variably saturated flow, submitted to Inter. J. Numer. Method in Fluidr, 1995.

Bagtzoglou, A.B., and R.G. Baca, Probabilistic calculations of groundwater travel time in heterogeneous

three-dimensional porous media, Mar. Res. SOC. Symp. Proc., 333: 849-854, 1994.

Brooks, R.H., and A.T. Corey, Properties of porous media affecting fluid flow, J. Im'gazbn and

Drainage Div., Proc. ASCE, SER, IR2: 61-88, 1966.

Celia, M.A., E.T. Bodoutas, and R.L. Zarba, A general mass-conservative numerical solution for the

unsaturated flow equation, Water Resow. Res., 26(7): 1,483-1,496, 1990.

Wilds, S.W., and A. Long, Model and calculations for net infiltration, Proceeding of the lhird Annual

High-hel Radioactive Wmte Management Conference, American Nuclear Society,

2: 1,633-1,642, 1992.

16

Hevesi, J.A., and A.L. Flint, The influence of seasonal climatic variability on shallow infiltration at

Yucca Mountain, Proceedings of the Fourth Annual High-Level Radioaaiw Wmte Management

conference, American Nuclear Society, 1: 122-131, 1993.

Hevesi, J.A., A.L. Flint, and L.E. Flint, Verification of a onedimensional model for predicting shallow

infiltration at Yucca Mountain, Proceedings of the Fiiph Annual High-Lewl Radioactive Waste

Management Conference, American Nuclear Society, 2: 2,323-2,332, 1994.

Hills, R.G., I. Porro, D.B. Hudson, and P.J. Wierenga, Modeling onedimensional infiltration into very

dry soils, 1. Model development and evaluation, Wuter Resow. Res., 25(6): 1,259-1,269,

1989.

Huyakorn, P.S., and G. Pinder, COmpUtarionalMethods in Subsurface Flow, Academic Press, Inc., New

York, NY, 1983.

Huyakorn, P.S., S.D. Thomas, and B.M. Thompson, Techniques for making finite elements competitive

in modeling flow in variably saturated porous media, Water Resow. Res., 20(8): 1,099-1,114,

1984.

Jury, W.A., W.R. Gardner, and W.H. Gardner, Soil Physics, Fifth Edition, John Wdey & Sons, New

York, NY, 1993.

Kirkham, D. and W.L. Powers, Advanced Soil Physics, Wiley-Interscience, New York, NY, 1972.

Kirkland, M.R., R.G. Hills, and P.J. Wierenga, Algorithms for solving Richards’ equation for variably

saturated soils, Wafer Resow. Res., 28(8): 2,049-2,058, 1992.

Martin, P., and C.S. Smith, Benchmark and Partial Validation Testing of the W H Computer code,

Version 3.0, EGG-ENV-10975, EG&G Idaho, Inc., Idaho Falls, ID, 1993.

Pan, L., and P.J. Wierenga, A transformed pressure head-based approach to solve Richards’ equation

for variably saturated soils, Water Resow. Res., 31(4): 925-931, 1995.

17

I

Paniconi, C., and M. Putti, A comparison of P iwd and Newton iteration in the numerical solution of

multidimensional variably saturated flow problems, Waer Resour. Res., 30(12): 3,357-3,374,

1994.

Pepper, D.W., and J.C. Heinrich, l7fe Finite Elemenr Method, Taylor and Francis Publishing, Bristol,

PA, 1992.

Ross, P.J., Efficient numerical methods for infiltration using Richards' equation, Waer Resour. Res.,

26(4): 279-290, 1990.

Ross, P.J., and K.L. Bristow, Simulating water movement in layered and gradational soils using the

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Smoot, J.L., and B. Saga, Three-Dimensional Conraminant Plume Llynamics in the VIldose Zone:

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APPENDIX - ELEMENTAL MATRICES

The elemental matrices for the transform-based Newton-Galerkin method are derived by

integrating Eqs. (15) and (16) using the factorial formula given in Eq. (17). The general expressions for

the right hand side vector and the Jacobian matrix are:

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with Axk = (x2 - If prescribed, the flux terms 4: and

(23)

4: contribute to {R) at the top and

bottom boundaries. The quantities and

for the element.

are computed as the arithmetic or geometric mean values

Similarly, for the untransformed Newton-Galerkin method, the above elemental matrices

simplify to the following expressions:

- I I

1 m a s , a m , j The elemental matrices for the untransformed modified Picard method are derived using the

standard Bubnov-Galerkin procedure and the integration formula given in Eq. (17). The resulting

elemental matrices are:

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A direct comparison of the last two sets of elemental matrices illustrates that the so-called modified Picard

method is a subset of the more general Newton-Galerkin method and is, therefore, a modified-Newton

method.

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Table 1. Brooks-Corey parameters for Yucca Mountain test case

Table 2. van Genuchten parameters for Hanford test case

Table 3. van Genuchten parameters for Idaho Nationai Engineering Laboratory test case

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Table 4. Initial and boundary conditions and simulation times for three case studies: Yucca Mountain (Case l), Hanford Reservation (Case 2), and Idaho National Engineering Laboratory site (Case 3)

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Table 5. Comparison of Central Processing Unit times for un-.nnsformed and -amformed numerical algorithms

Case

1.1

1.2

1.3

1.4

2.1

2.2

2.3

2.4

3.1

3.2

3.3

3.4

m

Modified I Newton- Picard Galerkin

272 31

370 62

49

240 255 I 59

IC 98

IC 98

IC 94

IC 107

100 100

104 NC 107 109

112 119

NC NC 115

115

252 208

251 251

154 152

177 174

NC NC NC NC

IC means incomplete run; NC means non-convergent

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Table 6. Comparison of global mass balance errors for untransformed and transformed numerical algorithms

1.1 1.6 x lo-" 1.2 1.7X

1.3 3 . 2 ~

1.4 3.1 X

2.1 IC 2.2 IC 2.3 IC 2.4 IC

3.1 4.6 X IO-'*

3.2 2.5x lo"* 3.3 9 . 9 ~ 10-7

3.4 1.4X 1 0 - ~

Global Mass Balance Error I I

Inveme Newton- 1 1 Hyperbolic Galerkin Natural Log ! h e

2.5 x 10"O 2.6 x 10"O 2.8 x 1 . 5 ~ 2.1 x 2.2x 10"O

9.3 x 9.4 x 9.3 x 10-9

8.9 x 8.7 x 9.2 x lo-'' 2.8 x 2.8 x 2.5 x 1 .O x 1.6 x lo-'' NC 3 . 6 ~ 4.8 x 10"O 4.9X

2.5 x 10"O 2.2 x 1.6 x

3 . 5 ~ lo-" 8 . 3 ~ lo-" 6.7 x lo-" 3 . 4 ~ 6 . 4 ~ 5.4x lo-" 9.9 x 9.9 x 9.9 x 10-7

1.3 x 10-7 1.3 x 10-7 1.3~10"

1.9x 10-10

2.2 x 10-10

9.8 x 10-9

1.1 x 10-~

NC NC

4.9 X

1.6 x

NC NC NC NC

IC means incomplete run; NC means non-convergent

24

Figure 1 a. Pressure head profiles for Yucca Mountain

100

-1 00

-300

-500 A

v 1 -700 -r Q

-900 Q, 0

-1 100

-1 300

-1 500

............................................................................ I I"""'

Newton -Galerkin

Natural Log --- Modified Picard

----- Hyperbolic Sine .-.-.-.- Algebraic

- - - - - - -

.........

.........

......

......

_._._._.-.-.-.-.- -.- ................................................................................... 13

1 . 1 ee ' .2

-30000 -20000 - 1 0000 0

Pressure head (cm)

Figure 1 b. Pressure h e a d profiles for Yucca Mountain

100

-1 00

-300

-500

-700

-900

-1 100

- 1 300

...........................................................................

............ N e w ton-Galerkin Modified Picard

----. Hyperbolic Sine .-.-.-.- Algebraic

Natural Log ............

............................................................................

.....

......

......

...... ._._._._._._._.-.-.-._._._.

Case 1 . 4 a--- -1 500

-30000 -20000 -1 0000 0 Pressure head (CM)

Q)

I/) U

Y

.- L 0

+

U

N

? 3 0, LL .-

..

..

-.

C

00

0 0

I ?

0 0

0

cu I

-:

0 0

0

m I

0 0

0

d- I

0

0 0

0

0 I

0

0 0

0

c\1 I 0 0

0

0

rs)

7

0 0

01

00

0

00

0

Ln

a

I\ I

I I

Dep

th (

cm)

I I

I I

I I

I \I

mV

1

-P

UN

"

00

00

00

0

00

00

00

0

IO

0

0

0

0

0

0 0

(A

0

0

0

0 I N

0

0

0

0 I 0

0

0

0

A

0

TI -7

(D

in

in c -l

(D

3

rD 0 n

0 =. CD

v) -

Gep

th (

cm

)

I 0

0 0

d

-L I 4

0

0

0 I crt

0 0 0

d

0

0

0

I I

I I

I I UI

0

A

N

0

0

N

G.r 0

VI

0

0

0

0

0

cn

0

0

0

0

0

0

0

1: ..

.*

-.

+

-.

..

-

.

..

.

.

..

..

.

.

..

.

.

.-

0

0)

-a

crl

N

n

! ..

-h

7 0 -

z

m r

cn

CD -. rc

Figure 3b. Pressure profiles for INEL Site

1 ~ 1 ' 1 ' ~ ' ~ ' 1 I n

0

- 500

n -1000 € 0 --/

-c - 1 500

a. a,

-2000

-w

-2500

- 3000

l ~ a l a l a l 1 1 ' 1 1 '

.....................................................................................

____ ----I \ .. PI e w t o n - G a 1 e r k i n

Modified Pic ard Nati i ial Log liyperbolic Sine

--- - _ _ - - - - ----.

...............................

...............................

............................... Le--

Case 3.4 Case 3.3 a-- 3- I... .................................................................................