Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue7.2000:

Advances in Water Resources 23 (2000) 731±745
www.elsevier.com/locate/advwaters

A reduced degree of freedom method for simulating non-isothermal
multi-phase ¯ow in a porous medium
Peng-Hsiang Tseng *, George A. Zyvoloski
Geoanalysis Group, Los Alamos National Laboratory, EES-5, MS-C306, Los Alamos, NM 87545, USA
Received 22 May 1999; received in revised form 6 December 1999; accepted 19 January 2000

Abstract
Two solution algorithms, with guaranteed memory savings over fully implicit methods, are presented for solving coupled processes in subsurface hydrology. They are applied to problems of non-isothermal multi-phase ¯ow in porous media using both an
equivalent continuum approximation and a dual-permeability approach. One algorithm uses an approximation of the Jacobian
matrix during the Newton±Raphson iteration such that the coupled system of equations can be partitioned into a solution of some
selected primary variables plus a back substitution procedure for the solution of the other variables. The second algorithm uses a
similar procedure, except the operations are performed during the preconditioning phase of the solution of linear equations. Numerical forms were derived and the solution procedures were illustrated to reduce the problem from three unknowns per node to one
for the single continuum formulation, and from six unknowns per node to two for the dual-permeability approximation. Simulation
examples showed that the proposed method produced nearly identical results compared to the traditional fully implicit method while
enjoying large memory savings and competitive CPU times. Ó 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Multi-phase ¯ow; Non-isothermal ¯ow; Dual-permeability model; Reduced degree of freedom method

1. Introduction

Coupled processes are involved in many important
engineering and technological problems facing the world
today. These problems involve the dynamic interaction
of two or more physically distinct subsystems treated as
a single system to account for the overall physical e€ects
arising from the coupling. In the areas of ¯ow and
transport in porous media, coupled processes represent
an abundance of ongoing and new challenges of research for problems such as aquifer restoration, subsurface waste isolation, geothermal reservoir simulation,
and enhanced oil recovery. The solution to these problems typically require sophisticated three-dimensional
simulations using relevant governing equations that are
often highly coupled. The present work was motivated
by a need to understand the thermal behavior of a potential nuclear waste repository. To characterize the
entire physical system requires not only an integrated

*

Corresponding author. Tel.: +1-505-667-5518; fax: +1-505-6658244.
E-mail address: tseng@lanl.gov (P.-H. Tseng).

understanding of all the coupling processes but also a

proper treatment of the solution procedures.
Numerical solution of coupled processes can pose
varying degrees of complexity depending on the nature
of the processes and their interaction mechanisms.
Traditionally, a system of coupled processes has been
treated and solved as a single computational entity
known as the monolithic augmentation approach
[15,30]. Diculty was soon encountered due to the orders-of-magnitude increases in computational requirements that makes this approach inappropriate for most
practical uses. Even for some single-®eld single-equation
solvers the same diculty remains when a large scale
real-world scenario simulation is desired. This has motivated the development of an improved numerical
solution method that requires minimal computational
e€ort while maintaining satisfactory numerical stability
and accuracy.
In an earlier attempt, Stone and Garder [25] applied a
so-called ®eld elimination approach combined with a
subsequent explicit updating procedure to solve the
coupled one-dimensional equations of gas and oil ¯ow
in petroleum reservoir applications. Their method is
typically referred to as the implicit pressure±explicit

0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 0 6 - 3

732

P.-H. Tseng, G.A. Zyvoloski / Advances in Water Resources 23 (2000) 731±745

saturation (IMPES) approach. Although this method
avoided the simultaneous solution of coupled equations,
the resulting scheme is only conditionally stable. In
addition, the many drawbacks of the ®eld elimination
approach, as discussed in more detail by Felippa and
Park [6] and Park and Felippa [15,16], have greatly restricted its application to other general problems of
coupled systems. A type of mixed implicit and explicit
scheme was developed, on either a ®xed or an adaptive
basis, for the solutions of both coupled systems [7,26,27]
and single-®eld problems [12,13,30]. The basic idea behind this type of scheme is that a more ecient explicit
formulation can usually be applied to part of the solution domain when the system response in that region is
relatively steady. The single-®eld problem is formulated

as a combination of two separate implicit and explicit
domains in the context of numerical methods and hence
can be de®ned as a coupled problem in a broader sense
[32]. The more appealing adaptive scheme requires a
relatively sophisticated bookkeeping process, when the
number of coupled equations becomes large, to keep
track of the number of nodes to be handled in each
domain. For these methods, computer time savings and
memory savings are problem dependent. For nonlinear
problems, identifying a threshold value to achieve optimum separation of the solution domain depends
largely on experience and hence may sometimes overwhelm the advantages of this scheme.
More recently, a method known as the partitioned
solution procedure has been pursued and applied successfully to a variety of coupled ®eld problems
[5,11,14,17,20,22,23,28,33]. In this approach, the timeintegration process is carried out over each suitably
decomposed subsystem treated as an isolated entity and
recoupled subsequently by appropriate temporal extrapolation techniques. This method greatly enhanced
computational eciency and program modularity and
has become popular for many practical applications.
Unfortunately, the numerical stability and accuracy may
be a€ected by the partitioned treatment and are problem

speci®c. The success of this scheme requires considerable
investment in methodology issues to assure a stable and
accurate solution. This diculty has greatly prevented
the generalization of this approach to some geoenvironmental applications which may require a solution
of six coupled nonlinear equations. An alternative approach, ®rst introduced by Zyvoloski et al. [35] for the
applications of nonlinear geothermal problems, circumvents this diculty while preserving computer
memory savings and, to a certain degree, computational
eciency. This approach, while leaving the problem
physics unchanged, reduces the degree of freedom at
each computational node with some suitable simpli®cations during the Newton±Raphson iteration approximation. A successive over relaxation (SOR) iteration
scheme was later introduced by Zyvoloski [34] as an

e€ective recoupling procedure. This method has the
potential to be applied to a number of coupled-®eld
problems, especially when the number of state variables
is large.
The objective of this study is to apply the reduced
degree of freedom (RDOF) method to the solution of
coupled processes involving two-phase ¯ow in a nonisothermal porous medium. Both an equivalent continuum approximation and dual-permeability approach
will be considered. For the later case, the solution of six

coupled governing equations is required. Simulation
examples are presented to illustrate the performance of
the reduced degree of freedom method under varying
amounts of complexity arising in subsurface ¯ow and
energy transport. The simulations were conducted using
the Los Alamos National Laboratory code FEHM [36].
The sensitivity of model performance to selected ¯ow
parameters is discussed.

2. Mathematical formulation
2.1. Flow and energy-transport equations
For two-phase Darcian ¯ow in a non-isothermal
porous medium under the assumptions that viscous
heating is negligible, the solid phase is immobile, and
local thermal equilibrium between ¯uid and solid phases
holds, the governing equations are taken as [37]
ÿr
‰…1 ÿ gv †Dmv rPv Š ÿ r
‰…1 ÿ gl †Dml rPl Š ÿ r


o

Dg qv r…1 ÿ gv † ‡ qm ‡ g‰…1 ÿ gv †Dmv qv
oz
oAm
ˆ 0;
…1a†
‡ …1 ÿ gl †Dml ql Š ‡
ot
ÿr
…Dev rPv † ÿ r
…Del rPl † ÿ r
… KrT † ‡ qe
‡

o
oAe
ˆ 0;
g…Dev qv ‡ Del ql † ‡
ot
oz

…1b†

ÿ

ÿr
…gv Dmv rPv † ÿ r
…gl Dml rPl † ÿ r
Dg qv rgv
‡ qg ‡
‡

o
g…gv Dmv qv ‡ gl Dml ql †
oz

oAg
ˆ 0;
ot

…1c†

where the subscripts v and l denote quantities for the
vapor phase and the liquid phase, respectively, P is the
phase pressure, T the temperature, K an e€ective thermal conductivity, g represents the acceleration due to
gravity, q the phase density, z the spatial coordinate
oriented in the direction of the gravity, t the time, r the

vector di€erential operator, q the source and sink terms
for the corresponding equations, and g is the mass
fraction of noncondensible gas in the corresponding
phases. By de®nition, gv ˆ qa =qv and gl ˆ Ha Pa , where

P.-H. Tseng, G.A. Zyvoloski / Advances in Water Resources 23 (2000) 731±745

the subscript a stands for air, Ha is Henry's law constant
for air, and Pa is the partial pressure of air. The density
of air qa is assumed to obey the ideal gas law. The diffusion coecient of a vapor-air mixture within the porous air space is designated as Dg ˆ sDag , where s is the
tortuosity factor, and Dag is the binary gaseous di€usion
coecient in free air. The value of Dag depends on the
temperature and pressure [37].
The mobilities Dm and De for the mass and energy
equations can be expressed as
Dmv

kRv qv
;
ˆ

lv

Dml

kRl ql
ˆ
;
ll

…2a†

The other two parameters, a length scale and a shape
factor, are used to quantify interactions between the two
pore systems [1,8].
The variables and parameters in Eqs. (1a)±(1c) are
de®ned relative to the partial volume of each pore system to account for the relative ¯ow and energy fractions
within the two subsystems. Exchange terms describe the
transfer of water, water vapor, air, and energy between
the two pore systems are de®ned as
Cm ˆ Cmv ‡ Cml ;

…4a†

Cg ˆ Cgv ‡ Cgl ;

…4b†

…2b†

f
Ce ˆ …1 ÿ dmv †Cmv hfv ‡ dmv Cmv hm
v ‡ …1 ÿ dml †Cml hl

where k is the intrinsic permeability, l the viscosity, h
the speci®c enthalpy, and R is the relative permeability.
The accumulation terms for Eqs. (1a)±(1c) are given
by

f
m
‡ dml Cml hm
l ‡ …1 ÿ dgv †Cgv hv ‡ dgv Cgv hv


C ÿ f
m
‡ …1 ÿ dgl †Cgl hfl ‡ dgl Cgl hm
;
l ‡ 2K T ÿT
L

Dev ˆ hv Dmv ;

Del ˆ hl Dml ;

Am ˆ /‰Sv qv …1 ÿ gv † ‡ Sl ql …1 ÿ gl †Š;

…3a†

Ae ˆ …1 ÿ /†qs us ‡ /…Sv qv uv ‡ Sl ql ul †;

…3b†

Ag ˆ /…Sv qv gv ‡ Sl ql gl †;

…3c†

733

…4c†

where


C kRv qv …1 ÿ gv † ÿ f
Pv ÿ Pvm ;
2
L
lv


C kRl ql …1 ÿ gl † ÿ f
Pl ÿ Plm ;
ˆ 2
L
ll

Cmv ˆ

where / is the porosity, S the phase saturation, u the
speci®c internal energy, and the subscript s refers to the
solid phase. Here, us ˆ cps T where cps is the speci®c heat
of the solid material.
The three equations (1a)±(1c), which describe the
conservations of mass of water, ¯uid-medium energy,
and mass of noncondensible gas, respectively, may be
used to model the ¯ow of air, water, water vapor, and
heat in a porous medium. However, in most practical
applications the structured nature of a porous medium
in ®eld soils or rock formations requires a more complicated bi-continuum approach to describe the behavior of the physical system. One general approach of this
type is referred to as the dual-porosity/dual-permeability
model or simply dual-permeability model.
2.2. Dual-permeability approach
The dual-permeability model approximates the
physical system of a structured medium with two distinct, but interacting, subsystems, one associated with a
dominant region of macropores or fractures and the
other with a passive region of matrix or porous blocks.
Equations which govern ¯ow and energy transport are
described separately for each of the two subsystems and
coupled by means of a source/sink term to account for
the exchange of mass and energy between the two interacting continua. Three parameters are generally used
to characterize the two pore systems and their interactions. The ®rst is a volume fraction (of a selected pore
system) which relates the material properties of the two
pore systems to their corresponding bulk properties [8].

Cml



C kRv qv gv ÿ f
Pv ÿ Pvm ;
L2
lv


C kRl ql gl ÿ f
Pl ÿ Plm
Cgl ˆ 2
L
ll

…5a†

Cgv ˆ

…5b†

and where the superscripts f and m denote the fracture
and matrix system, respectively; the quantity C is a
shape factor, and L represents the distance from the
center of a ®ctitious matrix block to the fracture
boundary.
Eq. (4a) speci®es that the exchange of water, Cm ,
between the two pore systems is the arithmatic sum of
two components: the transfer of water vapor and that of
liquid water. For positive Cmv and Cml , water transfer is
directed from the fracture into the matrix in both the
vapor and the liquid phases. Similarly, the exchange
term Cg is the transfer of noncondensible gas in both the
vapor phase and that dissolved in the liquid phase. The
exchange of energy consists of both a convection and a
conduction terms. The convection term is contributed
from the transfer of water and noncondensible gas in the
two phases. The direction of water or gas ¯ow between
the two pore systems is de®ned as a dimensionless coecient d as


C
:
…6†
d ˆ 0:5 1 ÿ
jCj
Notice that the subscripts mv, ml, gv, and gl for d and C
are neglected in Eq. (6) for simplicity. For d ˆ 0 (i.e.,
positive C), the exchange of mass is from fracture to

734

P.-H. Tseng, G.A. Zyvoloski / Advances in Water Resources 23 (2000) 731±745

matrix, while for d ˆ ‡1 (i.e., negative C), a matrix to
fracture ¯ow is observed. The conduction term in Eq.
(4c) is hypothesized to be proportional to the temperature di€erence between the fracture and the matrix pore
systems.
2.3. Constitutive relationships
Constitutive relationships, describing the measured
physical or material properties that constitute the system, are required to complete the mathematical formulation. Under the assumption that the hysteresis is
neglected, the water retention and hydraulic conductivity properties are assumed to be given by the expressions
[29]:
1=n
1  ÿ1=m
Sle ÿ 1
Sl < Slmax ;
Pc ˆ
a
…7a†
Pc ˆ 0 Sl P Slmax ;
1=2

Rl ˆ Sle
Rl ˆ 1

h

m i2
1=m
1 ÿ 1 ÿ Sle

Sl < Slmax ;

Sl P Slmax ;

…7b†

R v ˆ 1 ÿ Rl ;
where Sle ˆ …Sl ÿ Slr †=…Slmax ÿ Slr † is the e€ective liquid
saturation, Slmax and Slr denote the maximum and residual liquid saturation, respectively; Pc is the capillary
pressure, a, m, and n are shape parameters, and
m ˆ 1 ÿ 1=n. For 0 6 Sl 6 Slr , an extrapolation is needed
to approximate the capillary pressures in this range.
The pressure- and temperature-dependent water
properties such as the density q…P ; T †, enthalpy h…P ; T †,
and viscosity l…P ; T † of the two phases (i.e., liquid water
and steam) are approximated using rational function
representations [37] obtained by ®tting steam table data.
Complete polynomials of degree three of the form
a0 ‡ a1 P ‡ a2 P 2 ‡ a3 P 3 ‡ a4 T ‡ a5 T 2 ‡ a6 T 3
‡ a7 PT ‡ a8 P 2 T ‡ a9 PT 2

…8†

are used in both the numerator and the denominator.
The ®tting coecients ai ; …i ˆ 0


9† for the three
physical properties of water (i.e., q, h, and l) at each of
the two phases (l and v) are given in Zyvoloski et al. [37].
The pressure-temperature relationships, Pv …T † and
T …Pv †, are approximated using rational functions of
degree eight [37].

3. The reduced degree of freedom method
The coupled equations (1a)±(1c) were ®rst semi-discretized in space using standard ®nite element or control
volume procedures followed by a generalized one-step
time-integration scheme [37]. This resulted in a set of
nonlinear algebraic equations written in the form

f i …x1 ; x2 ; . . . ; xn † ˆ 0;

i ˆ 1; 2; . . . ; n;

…9†

where each xi ; i ˆ 1; 2; . . . ; n, denotes a state vector with
components x1 through xN , n is the number of state
variables per node, and N is the total number of nodes
obtained from numerical discretization. Similarly, each
f i ; i ˆ 1; 2; . . . ; n is a vector with components f1 ±fN . For
equivalent continuum model the number of n is 3 and
for dual-permeability model n becomes 6. With N nodal
points in the ®nite element discretization, there are
n  N equations and n  N unknowns to solve.
The resulting computational demand (namely,
storage and time) due to this increase in matrix size is
obvious. To solve the system of nonlinear equation (9),
it is necessary to employ an iterative approach. For
some highly nonlinear problems, it is desirable to use the
second-order convergent Newton±Raphson method instead of other slower convergent methods such as the
Picard iteration [9]. However, this advantage of fast
convergence may sometimes be outweighed by the
amount of computational operations required to perform each Newton±Raphson iteration. An improved
algorithm is introduced as follows to solve Eq. (9) of the
coupled processes.

3.1. Equivalent continuum formulation
For most coupled processes, one or more of the
processes usually changes slower than the other process.
The basic idea behind the reduced degree of freedom
method is to pre-factor one or several of those less active
degrees of freedom into one or more primary degrees of
freedom during the Newton±Raphson approximation.
As such, the method has the ¯exibility to reduce a
coupled system of n degrees of freedom per node to a
number less than n depending on problem characteristics.
It is important to note that the state variables which
dominate a coupled physical process may change with
time for certain applications. To enhance the performance of the RDOF method, additional procedures
are required to ®rst properly identify the active variables
at each time step and to swap the active and the passive
variables during the RDOF operations if necessary. The
swapping operations can easily be done using a pointer
array within the framework of a block matrix structure
without in¯ucing the fundamental concept of the RDOF
method. The selection of an optimized criterion to best
de®ne and identify the active variables, on the other
hand, remains largely a rule of thumb. In the following,
the method is illustrated to reduce the problem from
three unknowns per node to one unknown per node for
the single continuum formulation. For the dual-permeability approach, the same procedure is applied to reduce the system from six unknowns per node to two

P.-H. Tseng, G.A. Zyvoloski / Advances in Water Resources 23 (2000) 731±745

unknowns per node, i.e., one unknown for each of the
two pore systems.
The algebraic equation for the Newton±Raphson iteration can be written as
J…xk †…xk‡1 ÿ xk † ˆ ÿf …xk †;

…10†

where J is the Jacobian matrix, and the subscript k is the
iteration counter. For the three coupled equations (1a)±
(1c), the resulting Newton±Raphson iteration can be
written in block form as
Jij dxj ˆ ÿf i ;

i; j ˆ 1; 2; 3;

…11†

where dx ˆ xk‡1 ÿ xk is the correction vector,
Jij ˆ of i =oxj , and i and j represent standard doublesubscript notation for matrices. The iteration counter k
is neglected for simplicity.
The reduced degree of freedom method is carried out
by an approximation of the Jacobian matrix. In the
following derivation we assume that the variable x1 is
the active variable and x2 and x3 are the passive variables. Thus in Eq. (11), only the diagonal parts of the
arrays Ji2 and Ji3 , i ˆ 1; 2; 3, are retained. For the rest of
the submatrices, the derivatives with respect to the active
variable x1 , are left in their original form. The solution
of Eq. (11) is now reduced to a system of three decoupled equations [35]:
Bdx1 ˆ ÿb;

…12a†

A1 dx2 ˆ ÿA2 dx1 ÿ a1 ;

…12b†

A1 dx3 ˆ ÿA3 dx1 ÿ a2 ;

…12c†

where
A1 ˆ J33 J22 ÿ J23 J32 ;

…13a†

A2 ˆ J33 J21 ÿ J23 J31 ;

…13b†

A3 ˆ J22 J31 ÿ J32 J21 ;

…13c†

ÿ1
B ˆ J11 ÿ J12 Aÿ1
1 A2 ÿ J13 A1 A3 ;

…13d†

a1 ˆ J33 f 2 ÿ J23 f 3 ;

…13e†

a2 ˆ J22 f 3 ÿ J32 f 2 ;

…13f†

ÿ1
b ˆ f 1 ÿ J12 Aÿ1
1 a1 ÿ J13 A1 a2 :

…13g†

The expressions shown in Eqs. (12a)±(13g) were derived
using the simpli®ed Jacobian matrix. Formally, the decoupled equations can be made exact without any simplifying assumptions, that is, even if none of the
submatrices are diagonal. However, with the indicated
diagonal submatrices, the matrix multiplications and
inversions are trivial operations. The matrices A2 , A3 ,
and B are sparse matrices while matrix A1 is a diagonal
matrix. After the solution for the primary variable dx1
using Eq. (12a), the variables dx2 and dx3 can be evaluated directly by Eqs. (12b) and (12c), respectively. Note

735

that the matrix B has the same sparsity structure as
Ji1 ; i ˆ 1; 2; 3, so Eq. (12a) can be solved without extra
storage. Bullivant and Zyvoloski [3] showed that the
decomposition phase (Eqs. (12a)±(13g)) was equivalent
to normalizing the algebraic equations (Eq. (9)) in block
form.
Each iteration of the Newton±Raphson method requires ®rst an evaluation of f…x† and J…x†, each component of which is a scalar function. Since f has 3N
components and J has 9N 2 elements, there are a total of
9N 2 ‡ 3N scalar functions to be evaluated. For the reduced degree of freedom method, due to the diagonal
structure assumed in six of the submatrices in J, the
number of scalar functions to be evaluated is reduced to
3N 2 ‡ 9N -about one-third that of the original Newton±
Raphson iteration when N is large.
The solution of the sparse nonsymmetric system (11)
or (12a) can be performed by a standard equation solver
such as the preconditioned conjugate gradient (PCG)
type method. A general step of the PCG method
requires at least one or more matrix±vector multiplications, various numbers of scalar products and vector±
scalar multiplications depending on the di€erent
versions or variants used [4,21], plus a factorization of
the conditioning matrix. Assume that the number of
nonzero matrix entries is directly proportional to the
order of the matrix. Matrix±vector multiplication
requires 3c  3N operations for the original system of
Eq. (11) where c is the mean number of nonzero entries
per row of each submatrix Jij . The scalar product and
the vector±scalar multiplication each requires 3N
operations. For the reduced system of Eqs. (12a)±(12c),
each submatrix has an order of N . The matrix±vector
multiplication requires cN operations for nondiagonal
matrices and N for diagonal ones. The scalar product
and the vector±scalar multiplication each requires N
operations.
In the case of the ecient GMRES algorithm [19], for
example, the average number of operations per step is
…m ‡ 2†…3N † ‡ 9cN for Eq. (11) where m is the total
number of steps. The reduced system requires
…m ‡ 2†N ‡ cN operations to solve Eq. (12a) of order N .
The back substitutions and the calculations to carry out
Eqs. (13a)±(13g) require a total of 10N ‡ 6cN operations. For the LU factorization of the conditioning
matrix, the original system requires approximately
2
…3c† …3N † operations. The reduced system, on the
other hand, requires only c2 N operations. One step of
the reduced system thus needs ‰m…m ‡ 2† ‡ 10ŠN ‡
…m ‡ 6†cN ‡ c2 N operations. It can be shown that the
reduced system is always ecient than the original system. In fact, the analysis suggests that it would be more
desirable to use the reduced degree of freedom method
when the system becomes large.
The computational eciency depends also upon the
number of Newton±Raphson iterations and the average

736

P.-H. Tseng, G.A. Zyvoloski / Advances in Water Resources 23 (2000) 731±745

time step sizes required to achieve satisfactory accuracy.
Since the Jacobian matrix is altered for the reduced
system, in general the solution requires more iterations.
Experience has shown that eciency can be greatly
improved when the decoupled system is combined with a
recoupling phase of SOR cycles [3,34]





1
1 ÿ1 1
DÿL
D
D ÿ U dx ˆ ÿf
x
x
x

…14†

in which the matrix J of the equation system (11) was
decomposed into J ˆ L ‡ D ‡ U, and the parameter x
is a relaxation factor. Note again that no additional
storage is required for the operations of Eq. (14).
A comprehensive comparative study [3,34] showed
that although the reduced degree of freedom method
required generally more iterations than solving the
original system, the increased number of iteration was
only moderate. Hence the overall CPU time used was
still far less. Previous studies revealed that in a number
of cases evaluated the reduced system allowed a larger
average time step size and produced better balance errors [34]. In some cases tested, the reduced system requires even less iterations [3]. The eciency is thus
greatly enhanced for those cases.
In addition to computational eciency, the reduced
degree of freedom method has the advantage of much
less storage demand, which makes this method suitable
for performing large real-world simulations on presentday personal computers and workstations. A comparison of the storage requirements is illustrated in Table 1
for selected algorithms and equation solvers. The fully
implicit entry in Table 1 denotes the simultaneous
solution of the coupled set of equations using fully implicit method. Both the RDOF-NR (Newton±Raphson)
and the RDOF-solver (linear equation solver) manipulate the Jacobian matrix to obtain a reduced system as
described above. The only di€erence between these two

algorithms is in di€erent stages to perform the reduced
system manipulation for the GMRES acceleration procedure. The RDOF-NR is implemented using a reduced
degree of freedom GMRES and a reduced degree of
freedom preconditioning step. The RDOF-solver, on the
other hand, uses the full degree-of-freedom GMRES but
implements a reduced degree of freedom preconditioning step. The preconditioner of the RDOF-solver is
formed using the simpli®ed Jacobian Jij ; i; j ˆ 1; 2; 3,
with only the diagonal parts of the arrays Ji2 and Ji3 are
retained. Because the RDOF-NR algorithm uses a
matched set of preconditioning and GMRES acceleration, the method can easily use existing linear equation
solver software packages. The RDOF-solver requires
some modi®cation of existing software packages. To the
authors knowledge, this paper represents the ®rst time
the RDOF-solver algorithm has been described and
studied. The storage requirements for the ILU factorization and the GMRES (or other acceleration methods)
represent the two largest sources of memory usage in the
computer code. Both RDOF methods save a factor of 9
in ILU storage and the RDOF-NR method saves a
factor of 3 in the GMRES storage.

3.2. Dual-permeability formulation
For dual-permeability approach, Eqs. (1a)±(1c) is
written for both the fracture and the matrix domains.
The resulting equation for the Newton±Raphson iteration has a similar form as Eq. (11) except that the
number of degrees of freedom per node has increased
from three to six. Using the same physical system as in
the single continuum formulation, Eq. (11) can be rewritten for the two pore systems as
"
#
 f 

Jffij Jfm
dxfj
f
ij
…15†
ˆ ÿ mi ; i; j ˆ 1; 2; 3:
m
mm
dx
f
Jmf
J
j
i
ij
ij

Table 1
Storage requirements for the three algorithms used
Fully implicit

RDOF-solver

RDOF-NR

Equivalent continuum
ILU(1)a
ILU(2)c
GMRES

9 ´ NC ´ Nb
9 ´ 1.4 ´ NC ´ Nd
3 ´ (NR + 1) ´ Ne

NC ´ N
1.4 ´ NC ´ N
3 ´ (NR + 1) ´ N

NC ´ N
1.4 ´ NC ´ N
(NR + 1) ´ N

Dual-permeability
ILU(1)
ILU(2)
GMRES

36 ´ NC ´ Nb
36 ´ 1.4 ´ NC ´ Nd
6 ´ (NR + 1) ´ Ne

4 ´ NC ´ N
4 ´ 1.4 ´ NC ´ N
6 ´ (NR + 1) ´ N

4 ´ NC ´ N
4 ´ 1.4 ´ NC ´ N
2 ´ (NR + 1) ´ N

a
ILU(1) refers to an incomplete LU factorization and (1) refers to a matrix where ®ll-in is allowed on only those terms that exist in the original
unfactored matrix.
b
N: Number of nodes in numerical grid; NC: average number of nodal connections.
c
ILU(2) refers to a matrix where ®ll-in is allowed to produce additional terms only when the operations involve the original terms of the matrix.
d
1.4: Fill-in factor for higher order incomplete factorizations (problem dependent).
e
NR: Number of retained previous solutions.

737

P.-H. Tseng, G.A. Zyvoloski / Advances in Water Resources 23 (2000) 731±745

Similar to the double-subscript notation, the doublesuperscript of J was used to identify the submatrices
that form the Jacobian matrix. For instance,
f
ff
mm
m
are sparce
Jfm
ij ˆ of i =oxj . The submatrices J and J
fm
mf
matrices while J and J are diagonal matrices which
contain the quantities relevant to the exchange terms
between the two pore systems.
In the derivations shown below, we assume one
dominant variable in each of the two pore systems. Eq.
(15) can be rearranged such that the dominant variable
x1 in both the fracture and the matrix domains appears
®rst
38
2 ff
9
dxf1 >
Jff12 Jff13 Jfm
Jfm
J11 Jfm
>
11
12
13
>
7>
6 mf
>
> m>
dx1 >
Jmf
Jmf
Jmm
Jmm
>
6 J11 Jmm
11
12
13
12
13 7>
>
>
>
7>
6 ff
<
fm
ff
ff
fm
fm
6 J21 J21 J22 J23 J22 J23 7 dxf2 =
7
6
fm
ff
ff
fm
fm 7
6 Jff
dxf >
>
6 31 J31 J32 J33 J32 J33 7>
>
> 3>
>
6 mf
mm
mf
mf
mm
mm 7>
>
>
>
dxm
4 J21 J21 J22 J23 J22 J23 5>
>
2
>
;
:
mf
mm
mf
mf
mm
mm
m
dx3
J31 J31 J32 J33 J32 J33
8 f 9
>
f >
>
>
> m1 >
>
>
>
f1 >
>
>
>
>
>
=
< f >
f2
:
…16†
ˆÿ
>
f f3 >
>
>
>
>
>
> m>
>
>
f >
>
>
> 2m >
;
:
f3

Following the same procedures as described above for
the single continuum approximation, the derivatives of
the mobilities are neglected. The submatrices Jffij and
are diagonal for j ˆ 2; 3. Note that the submatrices
Jmm
ij
Jfm
and
Jmf
ij
ij have zero entries for i 6ˆ j. To simplify the
notation, we neglect the double-superscript and rewrite
the Jacobian matrix in Eq. (16) as
0

J11
B J41
B
B
J
JˆB
B 21
B ..
@ .
J61

J14
J44
J24
..
.
J64

J12
J42
J22
..
.
J62

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

1
J16
J46 C
C
C
J26 C:
C
.. C
. A

ÿ1
Q12 ˆ J14 ‡ J12 Eÿ1
1 E6 ‡ J13 F1 F6 ;
ÿ1
Q21 ˆ J41 ‡ J45 Gÿ1
1 G5 ‡ J46 H1 H5 ;
ÿ1
Q22 ˆ J44 ÿ J45 Gÿ1
1 G6 ÿ J46 H1 H6 ;


ÿ
m
ÿ f f2 ‡ E2 f f3 ‡ E3 f m
q1 ˆ ÿf f1 ÿ J12 Eÿ1
2 ÿ E4 f 3
1
ÿ f


m
F2 f 2 ÿ f f3 ÿ F3 f m
ÿ J13 Fÿ1
2 ‡ F4 f 3 ;
1
ÿ


ÿ1
m
G2 f f2 ÿ G3 f f3 ÿ f m
q2 ˆ ÿf m
1 ÿ J45 G1
2 ‡ G4 f 3


ÿ
m
ÿ H2 f f2 ‡ H3 f f3 ‡ H4 f m
ÿ J46 Hÿ1
1
2 ÿ f3

m
f
m
E1 dxf2 ˆ ÿf f2 ‡ E2 f f3 ‡ E3 f m
2 ÿ E4 f 3 ÿ E5 x1 ‡ E6 x1 ;
m
f
m
F1 dxf3 ˆ F2 f f2 ÿ f f3 ÿ F3 f m
2 ‡ F4 f 3 ÿ F5 x1 ‡ F6 x1 ;
f
f
m
m
f
m
G1 dxm
2 ˆ G2 f 2 ÿ G3 f 3 ÿ f 2 ‡ G4 f 3 ‡ G5 x1 ÿ G6 x1 ;
f
f
m
m
f
m
H1 dxm
3 ˆ ÿH2 f 2 ‡ H3 f 3 ‡ H4 f 2 ÿ f 3 ‡ H5 x1 ÿ H6 x1 ;

…20†
where
E1 ˆ R3 ÿ R2 Rÿ1
1 R4 ;

ÿ1
ÿ1
E2 ˆ J23 Jÿ1
33 ‡ R2 R1 J63 J33 ;

ÿ1
ÿ1
E3 ˆ J25 Jÿ1
55 ‡ R2 R1 J65 J55 ;

E5 ˆ R5 ÿ R2 Rÿ1
1 R6 ;

E4 ˆ R2 Rÿ1
1 ;

E6 ˆ R8 ÿ R2 Rÿ1
1 R7 ;
…21a†

F1 ˆ S3 ÿ S2 Sÿ1
1 S4 ;
F3 ˆ S2 Sÿ1
1 ;

ÿ1
ÿ1
F2 ˆ J32 Jÿ1
22 ‡ S2 S1 J52 J22 ;

ÿ1
ÿ1
F4 ˆ J36 Jÿ1
66 ‡ S2 S1 J56 J66 ;

F5 ˆ S5 ÿ S2 Sÿ1
1 S6 ;

F6 ˆ S8 ÿ S2 Sÿ1
1 S7 ;
…21b†

G1 ˆ S1 ÿ S4 Sÿ1
3 S2 ;

ÿ1
ÿ1
G2 ˆ J52 Jÿ1
22 ‡ S4 S3 J32 J22 ;

ÿ1
ÿ1
G4 ˆ J56 Jÿ1
66 ‡ S4 S3 J36 J66 ;

G5 ˆ S6 ‡ S4 Sÿ1
3 S5 ;

G6 ˆ S7 ‡ S4 Sÿ1
3 S8 ;
…21c†

…17†

J66

…19†

and for the passive variables:

G3 ˆ S4 Sÿ1
3 ;

The coupled system of 6N equations can be reduced to a
system of 2N equations for the two dominant state
variables dxf1 and dxm
1 speci®ed at the N ®nite element
m
nodes. The passive variables dxf2 , dxf3 , dxm
2 , and dx3 are
evaluated by back substitution after the dominant
variables are obtained. The resulting equations, obtained by substitution and elimination, can be expressed
as follows for the two dominant variables:

 f   
q1
Q11 Q12
dx1
;
…18†
ˆ
Q21 Q22
q
dxm
2
1
where

ÿ1
Q11 ˆ J11 ÿ J12 Eÿ1
1 E5 ÿ J13 F1 F5 ;

H1 ˆ R1 ÿ R4 Rÿ1
3 R2 ;

H2 ˆ R4 Rÿ1
3 ;

ÿ1
ÿ1
H3 ˆ J63 Jÿ1
33 ‡ R4 R3 J23 J33 ;
ÿ1
ÿ1
H4 ˆ J65 Jÿ1
55 ‡ R4 R3 J25 J55 ;

H5 ˆ R6 ÿ R4 Rÿ1
3 R5 ;

…21d†

H6 ˆ R7 ‡ R4 Rÿ1
3 R8

in these equations the following de®nitions are used:
ÿ1
R1 ˆ J66 ÿ J63 Jÿ1
33 J36 ÿ J65 J55 J56 ;
ÿ1
R2 ˆ J23 Jÿ1
33 J36 ‡ J25 J55 J56 ;
ÿ1
R3 ˆ J22 ÿ J23 Jÿ1
33 J32 ÿ J25 J55 J52 ;
ÿ1
R4 ˆ J63 Jÿ1
33 J32 ‡ J65 J55 J52 ;

R5 ˆ J21 ÿ J23 Jÿ1
33 J31 ;

R6 ˆ J63 Jÿ1
33 J31 ;

R7 ˆ J64 ÿ J65 Jÿ1
55 J54 ;

R8 ˆ J25 Jÿ1
55 J54 ;

…22a†

738

P.-H. Tseng, G.A. Zyvoloski / Advances in Water Resources 23 (2000) 731±745

ÿ1
S1 ˆ J55 ÿ J52 Jÿ1
22 J25 ÿ J56 J66 J65 ;

S2 ˆ

J32 Jÿ1
22 J25

‡

J36 Jÿ1
66 J65 ;

ÿ1
S3 ˆ J33 ÿ J32 Jÿ1
22 J23 ÿ J36 J66 J63 ;

…22b†

ÿ1
S4 ˆ J52 Jÿ1
22 J23 ‡ J56 J66 J63 ;

S5 ˆ J31 ÿ J32 Jÿ1
22 J21 ;

S6 ˆ J52 Jÿ1
22 J21 ;

S7 ˆ J54 ÿ J56 Jÿ1
66 J64 ;

S8 ˆ J36 Jÿ1
66 J64 :

It can be seen that all the inverse matrices in Eqs. (21a)±
(22b) correspond to diagonal matrices. The resulting
matrices Ri and Si are diagonal when i ˆ 1; . . . ; 4, and
sparse when i ˆ 5; . . . ; 8. While it may seem that the
operations given by Eqs. (18)±(22b) are very complex, it
should be noted that the operations are equivalent to
block-wise normalization. See [3] for details.
To carry out the calculations for matrices Ri and Si
and hence the matrices Ei , Fi , Gi and Hi require a total
of 40N ‡ 16cN operations. The matrices Qij and the
vectors qi require 20N ‡ 8cN operations. To solve
Eq. (18) by GMRES [19] and the back substitution of
Eq. (20), …m ‡ 2†…2N † ‡ 4cN and 12…N ‡ cN † operations, respectively, are needed. Finally, the LU factorization of the conditioning matrix requires 8c2 N
operations. The total number of operations required for
the reduced system is thus ‰m…2m ‡ 4† ‡ 72ŠN ‡…4m‡
36†cN ‡ 8c2 N . Although one step of the GMRES for the
original system requires …6m ‡ 12†N ‡ 36cN operations,
which is seemingly not too bad as compared to the reduced system, the LU factorization part of the original
system needs a lot more operations (…6c†2 …6N †) than that
required by the reduced system. The storage requirements to reduce a system from six unknowns per node to
two are given in Table 1.

4. Simulation examples
Simulation examples are presented using both the
equivalent continuum approximation and the dual-per-

meability approach. An idealized two-dimensional vertical pro®le (Fig. 1) of dimensions 560 m deep and 2000
m wide was designed to represent a simpli®cation of the
proposed high level nuclear waste repository site at
Yucca Mountain. This example was chosen because of
its use in repository design and because of its numerical
diculty. The solution experiences boiling, dryout, and
re-wetting of liquid water in both the fractures and rock
matrix. To compare the performance of various algorithms tested, simple uniform mesh systems of 3030
and 60  60 were used for this purpose. The discretization is indicated in Fig. 1 using the long (for 30  30
grid) and the short (for 60  60 one) tickmarks along the
x and y axes. An idealized stratigraphic structure that
represents the geologic settings at Yucca Mountain was
used. Due to the relatively coarse grid used in the simulations, a single layer in the numerical grid system may
contain several geologic layers at the Yucca Mountain
®eld site. In this case, a representative hydraulic property was assigned to a certain numerical layer using the
most relevant parameter set chosen from Table 2 [31].
The simulations for the coarse grid system …30  30†
were de®ned by 14 di€erent hydraulic property sets
while for the ®ne grid system …60  60† a total of 16
parameter sets were used as shown on the right panel of
Fig. 1. The properties assumed to be constant in the
simulations are: rock bulk density qs ˆ 2000 kg/m3 ,
speci®c heat of rock cps ˆ 1  103 J/kg K, and the
thermal conductivity of rock K ˆ 1.0 W/mK. The porosity in the fracture domain was de®ned as one and the
residual liquid saturation Slr was assumed to be 0.01.
The boundary conditions for water ¯ow were assumed to be a constant in®ltration rate of 1 mm/yr at the
surface and a water table condition at the bottom
boundary. For noncondensible gas ¯ow, a speci®ed air
pressure of 0.0985 MPa was assumed at the surface and
a no ¯ux condition at the lower boundary. For heat
transfer, both the surface and the bottom boundaries
were assumed to be a ¯ux-type boundary condition. The
magnitude of the heat ¯ux is assumed to be proportional

46

tcw1
ptn2
ptn4
ptn5
tsw2
tsw3
tsw4
tsw5
tsw6
tsw7
ch1v
ch3z
ch4z
pp3v

318

311
465

▼
Heat source

Observation points

tcw1
ptn1
ptn2
ptn4
ptn5
tsw2
tsw3
tsw4
tsw5
tsw6
tsw7
ch1v
ch1z
ch3z
ch4z
pp3v

Coarse

Fine

Grid

Grid

Fig. 1. Schematic of the two-dimensional cross section showing geologic layers and the location of a line source of heat assumed in the simulations.

739

P.-H. Tseng, G.A. Zyvoloski / Advances in Water Resources 23 (2000) 731±745
Table 2
Hydraulic parameter sets used to de®ne the rock properties of various geologic layers
Layer
index
tcw1
tcw2
tcw3
Ptn1
ptn2
ptn3
ptn4
ptn5
tsw1
tsw2
tsw3
tsw4
tsw5
tsw6
tsw7
ch1v
ch1z
ch2v
ch3z
ch4v
ch4z
pp3v
pp2v
bf3v
bf2z
a
b

Matrix domain

Fracture domain

Slr

a (1/m)

n

k (m2 )

/

a (1/m)

n

k (m2 )a

Vfr b

Exchange
term
L (m)

0.13
0.13
0.33
0.10
0.14
0.17
0.10
0.10
0.11
0.04
0.06
0.18
0.08
0.32
0.50
0.04
0.36
0.06
0.20
0.04
0.33
0.07
0.18
0.07
0.18

0.012
0.013
0.002
0.397
0.107
0.440
0.450
1.182
0.104
0.399
0.062
0.008
0.024
0.006
0.001
0.597
0.002
0.746
0.035
0.597
0.001
0.172
0.017
0.172
0.017

1.302
1.309
1.776
1.304
2.075
1.377
1.529
1.383
1.364
1.464
1.353
1.524
1.267
1.715
1.585
1.233
1.567
1.289
1.292
1.233
1.908
1.453
1.462
1.453
1.462

5.40e ) 18
5.40e ) 18
5.00e ) 17
1.60e ) 14
3.76e ) 15
1.43e ) 13
1.12e ) 13
2.10e ) 13
6.72e ) 15
1.33e ) 15
4.99e ) 17
1.46e ) 17
1.16e ) 16
1.30e ) 16
6.49e ) 17
1.73e ) 12
1.47e ) 17
2.38e ) 13
2.21e ) 17
1.73e ) 12
1.32e ) 17
3.20e ) 15
5.88e ) 17
3.20e ) 15
5.88e ) 17

0.066
0.066
0.140
0.369
0.234
0.353
0.469
0.469
0.042
0.146
0.135
0.089
0.115
0.071
0.020
0.265
0.193
0.321
0.240
0.265
0.169
0.274
0.197
0.274
0.197

22.79
22.79
8.77
10.58
0.72
33.17
8.78
10.67
1.48
1.09
0.57
0.78
1.21
1.27
1.14
11.35
1.29
11.35
10.96
11.35
10.96
10.96
13.65
10.96
13.65

1.969
1.969
1.969
1.984
1.957
1.927
1.976
1.969
1.969
1.969
1.969
1.969
1.969
1.969
1.969
1.969
1.934
1.969
1.969
1.969
1.969
1.969
1.969
1.969
1.969

2.58e ) 08
2.01e ) 08
3.40e ) 08
6.38e ) 09
8.24e ) 09
1.97e ) 08
1.69e ) 09
2.63e ) 09
1.35e ) 08
5.49e ) 09
8.49e ) 09
3.44e ) 09
2.77e ) 09
3.01e ) 09
2.44e ) 09
2.44e ) 09
3.34e ) 09
4.03e ) 09
2.28e ) 09
2.44e ) 09
2.28e ) 09
9.92e ) 09
2.28e ) 09
9.92e ) 09
2.28e ) 09

2.33e ) 04
2.99e ) 04
7.05e ) 05
4.84e ) 05
4.83e ) 05
1.30e ) 04
6.94e ) 05
8.32e ) 05
8.92e ) 05
1.29e ) 04
1.05e ) 04
1.24e ) 04
3.29e ) 04
3.99e ) 04
4.92e ) 04
7.14e ) 05
1.10e ) 05
7.14e ) 05
1.10e ) 05
7.14e ) 05
1.10e ) 05
7.14e ) 05
1.10e ) 05
7.14e ) 05
1.10e ) 05

0.98
0.55
0.79
1.15
3.45
3.45
1.59
1.54
0.91
0.99
1.45
0.53
0.55
0.48
0.35
2.38
14.93
2.38
14.93
2.38
14.93
2.38
14.93
2.38
14.93

The permeability of the individual fractures. The bulk permeability of the fracture continua is k ´ Vfr .
Volume fraction of the fracture domain.

to the temperature di€erences between the implicit local
boundary and a speci®ed 15°C ambient condition. In
addition, the lateral boundaries of the solution domain
were de®ned as no ¯ow boundaries for air, water, water
vapor, and heat. The active variable in this case was
selected to be the total pressure. The two passive variables were the temperature plus either the phase pressure
or the saturation depending on whether there is only one
phase exists in the system (e.g., due to boiling and
dryout).
The simulations were started by ®rst conducting a
preliminary run without a heat source present to represent repository using appropriate initial conditions
subjected to the boundary conditions as described
above. After a sucient period of time, the system
eventually reached a steady-state condition. A subsequent simulation was then carried out by adding a
transient heat source within this initially steady-state
system. The boundary conditions of the system remained unchanged. The heat source, located 200 m below the surface and extended laterally for 1000 m as
shown in Fig. 1, had an initial heat ¯ux of 0.02107 MJ/s
into the system. This value declined to 30% of its initial
heat load after 100 yr and 7% after 1000 yr, typical of
that representing a repository heat generated by decay
of high-level nuclear waste [18].

Four observation points corresponding to the node
numbers of the coarse grid system were chosen, as
shown in Fig. 1, for the purpose of comparing the
simulation results. Two RDOF methods that di€er only
in the stages to implement the RDOF manipulation as
discussed in the previous section were implemented. The
results were compared to the traditional simultaneous
solution method using a fully implicit algorithm. Figs. 2
and 3 show the equivalent continuum modeling results
of temperature and ¯uid saturation, respectively, as a
function of time for the coarse grid system. All three
algorithms generated essentially the same results as
shown in the ®gures. To enhance comparison, a window
which shows the enlarged local responses was given in
each ®gure. Only negligibly small di€erences in temperature can be found between the results generated by
the RDOF-NR and the two other algorithms. The total
mass balance errors, expressed as a relative error form,
ranged from 5  10ÿ7 to 3  10ÿ10 while the total energy
balance (relative) errors were in the range of 9  10ÿ8 to
3  10ÿ11 for the various algorithms tested. The small
conservation errors indicate that all the three algorithms
performed equally well in modeling the test problem.
For the dual-permeability model, the results obtained
from the three algorithms were also found to be indistinguishable as shown in Figs. 4 and 5. Fig. 4 presents

740

P.-H. Tseng, G.A. Zyvoloski / Advances in Water Resources 23 (2000) 731±745
125

Node 311

100

Node 318

Temperature ( oC)

Node 465
93.7

75
93.6
93.5
93.4

50

450

500

550

Node 46

25
Fully Implicit
RDOF-Solver
RDOF-NR

0
0

200

400

600

800

1000

Time (years)

Fig. 2. Evolution of temperature versus time at four selected observation points calculated by the equivalent continuum approximation using a
coarse grid system.

1.0

0.930

Node 465

0.928

0.8

Fluid Saturation (%)

0.926
0.924
0.922

0.6

480

500

520

Node 318

Node 46

0.4

0.2
Fully Implicit
RDOF-Solver
RDOF-NR

Node 311

0.0
0

200

400

600

800

1000

Time (years)

Fig. 3. Fluid saturation as a function of time at four selected observation points calculated by the equivalent continuum approximation using a
coarse grid system.

the results representing the temperature history in the
matrix domain. Although the di€erences in ¯uid saturation between the fracture and the matrix domains are
large as expected, the di€erences in temperatures are
small. The maximum di€erence in temperature was
found to be 1:1°C in the vicinity of the heat source (with
higher temperature in the matrix domain) during the
early stages of heating. This di€erence drops to 0:4°C
after 9 yr and to 0:17°C after 100 yr. Fig. 4 gave only the
results representing the temperature history in the ma-

trix domain. The mass balance errors were found to be
in the range of 6:0  10ÿ6 to 1:2  10ÿ7 and the energy
balance errors were ranged from 4:1  10ÿ7 to
1:1  10ÿ8 .
All the above discussions apply to the results obtained using a coarse grid system. Further simulations
were conducted using a ®ner grid to test the performance of the RDOF method. The results obtained
with the three algorithms were found to be essentially
identical and indistinguishable when plotted in the

741

P.-H. Tseng, G.A. Zyvoloski / Advances in Water Resources 23 (2000) 731±745
150

125

Temperature ( oC)

Node 311

100
Node 318
100

75

99

Node 465

98
97
96

50

450

500

550

Node 46

25

Fully Implicit
RDOF-Solver
RDOF-NR

0
0

200

400

600

800

1000

Time (years)

Fig. 4. Evolution of temperature in the matrix versus time at four selected observation points calculated by the dual-permeability approach using a
coarse grid system.

Fracture Domain

Matrix Domain

1.0

0.20

0.135

Node 465
0.95

0.8

0.125

Node 465

Fluid Saturation (%)

0.15
0.115
0.94

230

250

270

0.6

Node 46
0.93

0.10

Node 318
230

250

270

0.4
Node 46

0.05
0.2
Fully Implicit
RDOF-Solver
RDOF-NR

Node 311

Node 318
Node 311

0.00

0.0
0

250

500

750

1000

Time (years)

0

250

500
Time (years)

750

1000

Fig. 5. Fluid saturation as a function of time at four selected observation points calculated by the dual-permeability approach using a coarse grid
system.

®gures (not shown). The total mass and energy balance
errors were also found to be in the same ranges as those
discussed above.
However, there are some subtle di€erences in the
simulation results obtained using two di€erent grid sizesas depicted in Fig. 6. The ®ner grid predicted a hottertemperature at the heat source and the region
becamesuperheated at an earlier time. On the other
hand, near the middle of the domain surface the
temperatures obtained from the ®ner grid were cooler.The primary reason responsible for such di€erences
isthe averaging of properties and variables during
thespatial and temporal discretization implemented in-

any numerical algorithms. The fact that there were
twomore hydraulic parameter sets used in de®ning the®ne grid layer properties has also in¯uenced the
solution. The results, nevertheless, indicate the dif®cultyand importance in capturing the essence of the
physical system during a simulation practice. Notice
also the di€erences between the equivalent continuum
and the dual-permeability simulation results shown in
the ®gures. Detailed discussions of the conceptual
models and the corresponding system behavior will be
presented in a separate study. The present study focuses
only on the relative performance of the solution algorithm.

742

P.-H. Tseng, G.A. Zyvoloski / Advances in Water Resources 23 (2000) 731±745
Dual-Permeability

Temperature ( oC)

Equivalent Continuum

150

150

125

125

100

100

Middle of the heat source

Middle of the heat source

75

75
Coarse grid (30x30)
Fine grid (60x60)

50

50

25

25

Middle of the domain surface

Middle of the domain surface

0

0
0

200

400

600

800

0

1000

200

400

600

800

1000

Time (years)

Time (years)

Fig. 6. Comparison of simulation results at selected observation points.

Table 3
CPU times and the relevant computational work required for the simulations
Equivalent continuum

Dual-permeability

Fully implicit

RDOF-solver

RDOF-NR

Fully implicit

RDOF-solver

RDOF-NR

Coarse grid
CPU seconds
Number of timesteps
Total N±R iterations
Total solver iterations

568.51
528
2489
20392

627.67
528
2500
25363

557.52
531
2806
24360

2581.76
530
2856
19490

2293.14
550
2938
25631

2497.67
827
4839
34985

Fine grid
CPU seconds
Number of timesteps
Total N±R iterations
Total solver iterations

9313.62
1471
8042
68975

9379.03
1408
7825
86434

11716.63
2046
12809
179218

30979.74
1021
6538
73227

37635.09
1059
6754
185052

41031.52
2273
16399
161952

5. Comparison of numerical eciency
Table 3 summarizes the CPU times for each method
applied to the repository heating problem described
earlier in this paper. In the example problems, pressure
was chosen as the active variable and temperature and
saturation were the passive variables. For the dualpermeability simulations, the equations were rearranged so the active variables were the pressure in the
fracture and the pressure in the matrix. The other
variables were taken as passive. Simulations are divided into four subproblems in increasing numerical
diculty: the equivalent continuum formulation on the
coarse grid, the dual-permeability formulation on the
coarse grid,the equivalent continuum formulation on
the ®ne grid, and the dual-permeability formulation on
the ®ne grid. The simulations show that all the methods have comparable CPU times for all the simulations
and thus are suciently robust to solve this dicult
problem. Except for the smallest problem, the RDOFNR method was the slowest of the methods and ex-

hibits the most Newton±Raphson iterations. This is to
be expected as the RDOF-NR method uses an approximate Jacobian matrix. The RDOF-solver uses the
same Jacobian matrix as the fully implicit method; its
approximation is done in the preconditioner stage of
the linear equation solver. Because of this, The number
of Newton±Raphson iterations for the RDOF-solver
and the fully implicit method are very close while the
iterations for the linear solver increases for the RDOFsolver method.
While the three solution techniques produced nearly
identical solutions, numerical errors in both spatial and
temporal discretization was high. Some indication of
this is evident from the fact that the maximum temperature di€erences between the coarse and ®ne grid was
over 20°C. It is likely that if models such as these were
used for detailed geochemica