Diffusion Processes in Composite Porous Media and Their Numerical Integration by Random Walks: Generalized Stochastic Differential Equations With Discontinuous Coefficients

Abstract

Discontinuities in effective subsurface transport properties commonly
arise (l) at abrupt contacts between geologic materials (i.e., in
composite porous media) and (2) in discrete velocity fields of numerical
groundwater-flow solutions. However, standard random-walk methods for
simulating transport and the theory on which they are based (diffusion
theory and the theory of stochastic differential equations (SDEs)) only
apply when effective transport properties are sufficiently smooth.
Limitations of standard theory have precluded development of random-walk
methods (diffusion processes) that obey advection dispersion equations
in composite porous media. In this paper we (1) generalize SDEs to the
case of discontinuous coefficients (i.e., step functions) and (2)
develop random-walk methods to numerically integrate these equations.
The new random-walk methods obey advection-dispersion equations, even in
composite media. The techniques retain many of the computational
advantages of standard random-walk methods, including the ability to
efficiently simulate solute-mass distributions and arrival times while
suppressing errors such as numerical dispersion. Examples relevant to
the simulation of subsurface transport demonstrate the new theory and
methods. The results apply to problems found in many scientific
disciplines and offer a unique contribution to diffusion theory and the
theory of SDEs.

Full text

Diffusion processes in composite porous media and their
numerical integration by random walks: Generalized stochastic
differential equations with discontinuous coefficients
Eric M. LaBolle,
1
Jeremy Quastel,
2
Graham E. Fogg,
3
and Janko Gravner
4
Abstract. Discontinuities in effective subsurface transport properties commonly arise (1)
at abrupt contacts between geologic materials (i.e., in composite porous media) and (2) in
discrete velocity fields of numerical groundwater-flow solutions. However, standard
random-walk methods for simulating transport and the theory on which they are based
(diffusion theory and the theory of stochastic differential equations (SDEs)) only apply
when effective transport properties are sufficiently smooth. Limitations of standard theory
have precluded development of random-walk methods (diffusion processes) that obey
advection dispersion equations in composite porous media. In this paper we (1) generalize
SDEs to the case of discontinuous coefficients (i.e., step functions) and (2) develop
random-walk methods to numerically integrate these equations. The new random-walk
methods obey advection-dispersion equations, even in composite media. The techniques
retain many of the computational advantages of standard random-walk methods, including
the ability to efficiently simulate solute-mass distributions and arrival times while
suppressing errors such as numerical dispersion. Examples relevant to the simulation of
subsurface transport demonstrate the new theory and methods. The results apply to
problems found in many scientific disciplines and offer a unique contribution to diffusion
theory and the theory of SDEs.
1. Introduction
Facilitated by geostatistical methods, detailed characteriza-
tions of the subsurface can capture the character of intricate
heterogeneities that strongly control transport [Copty and Ru-
bin, 1995; Sheibe and Freyberg, 1995; McKenna and Poeter,
1995; Carle et al., 1998]. Adequately resolving subsurface het-
erogeneity can yield immense computational grids, commonly
with greater than 10
6
nodes [e.g., Tompson, 1993], that demand
specialized numerical techniques to solve governing transport
equations. In many cases, random-walk methods are favored
over finite difference, finite element, and method of charac-
teristic techniques for large problems of this type [e.g., see
Tompson et al., 1987; Tompson and Gelhar, 1990; LaBolle et al.,
1996] because of their ability to efficiently simulate solute-mass
distributions and arrival times while suppressing errors such as
numerical dispersion [Prickett et al., 1981; Uffink, 1985; Ahl-
strom et al., 1977; Kinzelbach, 1988; Tompson et al., 1987].
Spatial averaging [Plumb and Whitaker, 1990] of pore-scale
equations for transport by advection and molecular diffusion in
porous media gives rise to advection-dispersion equations
(ADEs) commonly used to model subsurface transport:
⭸
⭸tx,tcx,t ⫽
冘
i
⭸
⭸xi
vix,tx,tcx,t
⫹
冘
i,j
⭸
⭸xi
冋
x,tDijx,t⭸cx,t
⭸xj
册
, (1)
where tis time, c[ML
3
] is concentration, v
i
[LT
1
]is
velocity, [L
3
L
3
] is effective porosity, and D
ij
[L
2
T
1
]is
a real symmetric dispersion tensor. Standard random-walk
methods [e.g., Kinzelbach, 1988; Tompson et al., 1987] approx-
imate solutions to (1) by simulating sample (particle) paths
corresponding to diffusion processes described by stochastic
differential equations (SDEs). Here “diffusion processes” re-
fer to Markov processes with continuous sample paths as math-
ematical models of real subsurface-transport phenomena.
Therefore, in the present context, the term “diffusion process”
refers to “advection dispersion process.”
Use of (1) as a model of transport in heterogeneous porous
media poses problems in the context of random-walk simula-
tion methods: Standard methods, and the theory on which they
are based (i.e., diffusion theory and the theory of SDEs), only
apply when coefficients, porosity and dispersion tensors are
sufficiently smooth functions of space [LaBolle et al., 1996,
1998]. Discontinuities in effective transport properties, how-
ever, commonly arise (1) at abrupt contacts between geologic
materials (i.e., in composite porous media) and (2) in discrete
velocity fields of numerical groundwater-flow solutions. As a
result, standard random-walk methods (and SDEs) cannot
simulate transport in heterogeneous porous media with abrupt
contacts between geologic materials.
Both interpolation [LaBolle et al., 1996] and “reflection”
[Uffink, 1985; Ackerer, 1985; Cordes et al., 1991; Semra et al.,
1993; LaBolle et al., 1998; LaBolle and Fogg, this issue]
1
Hydrologic Sciences, University of California, Davis.
2
Department of Mathematics, University of Toronto, Toronto, On-
tario, Canada.
3
Hydrologic Sciences and Department of Geology, University of
California, Davis.
4
Department of Mathematics, University of California, Davis.
Copyright 2000 by the American Geophysical Union.
Paper number 1999WR900224.
0043-1397/00/1999WR900224$09.00
WATER RESOURCES RESEARCH, VOL. 36, NO. 3, PAGES 651–662, MARCH 2000
651

techniques have been proposed to address the aforementioned
limitations of standard random-walk methods. By spatially in-
terpolating coefficients, one can ensure they remain sufficiently
smooth throughout the domain such that standard random-
walk (or Itoˆ-Euler integration) methods can be applied. Accu-
racy of this approach, however, suffers unless one refines the
interpolation (i.e., the region over which coefficients are
smoothed) and time step simultaneously [LaBolle et al., 1996],
which commonly leads to undesirable increases in computa-
tional effort. Furthermore, in the limit, refining the interpola-
tion ultimately gives rise to the original problem: discontinuous
coefficients.
LaBolle et al. [1998] developed necessary conditions for the
convergence of diffusion processes to ADEs in composite po-
rous media and applied the new theory to test four one-
dimensional “reflection” techniques [Uffink, 1985; Ackerer,
1985; Cordes et al., 1991; Semra et al., 1993]. The term “reflec-
tion” is derived from the usual method of reflecting particles to
maintain no-flux boundary conditions in a random walk [see
Tompson et al., 1987]. These techniques rely on either an an-
alytical solution to the specific problem or the specialized nu-
merical treatment of particle displacements to maintain mass
balance at an interface between regions with constant, but
different, diffusion coefficients. LaBolle et al. [1998] showed
that two of these reflection techniques fail to solve the speci-
fied problem, while the methods of Uffink [1985] and Semra et
al. [1993] succeed. One can show that these methods relate to
an analytical solution to the problem of one-dimensional dif-
fusion at an interface [LaBolle and Fogg, this issue]. The
method of Semra et al. [1993] has been recently extended to
three dimensions [Semra, 1994; Ackerer, 1999] for constant
coefficients within subdomains. General mathematical repre-
sentations of multidimensional diffusion processes obeying
ADEs in composite porous media (i.e., porous media charac-
terized by discontinuous coefficients) have remained undevel-
oped [LaBolle et al., 1998].
In this paper we generalize SDEs to the case of discontinu-
ous coefficients to develop (1) new mathematical representa-
tions of diffusion processes that simulate advection and dis-
persion in composite porous media and (2) random-walk
methods for numerical integration of these equations. The new
methods retain many of the computational advantages of stan-
dard methods [e.g., Kinzelbach, 1988; Tompson et al., 1987].
Examples demonstrate application of the new theory and
methods to problems of transport in porous media. However,
our results apply to problems found in numerous scientific
disciplines. Further, since the treatment of diffusion processes
with discontinuous coefficients is presently not covered in sto-
chastic theory (e.g., as described by Arnold [1992]), our results
offer a unique contribution to diffusion theory and the theory
of SDEs. Before considering the new approximations, we re-
view standard stochastic methods for simulating diffusion pro-
cesses.
2. Standard Methods for Simulating Diffusion
Processes
Standard stochastic methods for simulating subsurface
transport [e.g., Kinzelbach, 1988; Tompson, 1987] may be ap-
plied when effective transport properties vary smoothly in
space. In this case, diffusions corresponding to (1) are com-
monly represented by an Itoˆ SDE [Itoˆ and McKean, 1961]:
冕
t0
t
dXit ⫽
冕
t0
t
AiX,t dt⫹I
冕
t0
t
冘
j
BijX,t dWjt
(2a)
AiX,t⫽viX,t⫹1X,t
冘
j
⭸
⭸xj
X,tDijX,t,
(2b)
where the last integral in (2a) is referred to as a stochastic
integral, (I) denotes the Itoˆ interpretation of this integral (de-
fined below), X
i
(t)[L] is a sample path in space, B
ik
B
jk

2D
ij
, and W
j
(t)[T
1/2
] is a Brownian motion process such
that W
j
W
j
(t)W
j
(t
0
) has mean zero and covariance
t
␦
ij
. Note that B
ij
is generally not unique. The coefficients A
i
[LT
1
] and D
ij
are referred to as the drift vector and diffu-
sion tensor, respectively, and are defined as
lim
t30
1
tXit⫺Xit0 ⫽Ai⫽vi⫹1
冘
j
⭸
⭸xj
Dij(3a)
lim
t30
1
tXit⫺Xit0Xjt⫺Xjt0 ⫽
冘
k
BikBjk ⫽2Dij,
(3b)
where angle brackets denote the expectation. We will refer to
a component of the drift that involves gradient terms, such as
the second term in the right-hand side of (3a), as a “gradient
drift term.”
In general, to arrive at a unique definition for a stochastic
integral, it is necessary to specify how it is to be evaluated [see
Arnold, 1992, chapter 2]. The Itoˆ stochastic integral specified in
(2a) is defined as
I
冕
t0
t
冘
j
BijX,t dWjt
⫽mslim
n3
冘
k1
n
冘
j
BijXtk1,tk1Wjtk⫺Wjtk1, (4)
where ms-lim denotes the limit in the mean square [Gardiner,
1990]. The Itoˆ definition given in (4) evaluates B
ij
at location
X(t
k1
) rendering B
ij
statistically independent of dW
j
thus
ensuring that the integral in (4) has mean zero. One well-
known alternative to the Itoˆ interpretation of the stochastic
integral is that of Stratonovich [1963], in which Bis evaluated
at location [X(t
k1
)X(t
k
)] / 2, that is,
S
冕
t0
t
冘
j
BijX,t dWjt
⫽ms lim
n3
冘
k1
n
冘
j
Bij
冋
Xtk⫹Xtk1
2,tk1
册
䡠Wjtk⫺Wjtk1, (5)
where the (S) denotes the Stratonovich interpretation. For
convenience, herein we will adopt the following notation:
I
冘
j
BijX,tdWjt⫽
冘
j
BijX,tdWjt(6a)
LABOLLE ET AL.: DIFFUSION PROCESSES IN COMPOSITE POROUS MEDIA652

S
冘
j
BijX,tdWjt⫽
冘
j
BijX⫹dY,tdWjt(6b)
dYt⫽1
2dXt, (6c)
where we have written stochastic integrals as differentials, leav-
ing all integration implied. We will say that B
ij
is evaluated at
location Xand XdYin the Itoˆ and Stratonovich integrals of
(6a) and (6b), respectively. For dX
i
B
ij
(XdY,t)䡠dW
j
and assuming that B
ij
is sufficiently smooth, expanding the
Stratonovich equation (6b) in a Taylor series shows the rela-
tionship between Itoˆ and Stratonovich integrals is given as
dXi⫽S
冘
j
BijX,tdWj⫽
冘
j
BijX⫹1
2dX,tdWj
⫽
冘
j
BijX,tdWj⫹1
2
冘
j,l
⭸BijX,t
⭸xl
dXldWj
⫽
冘
j
BijX,tdWj⫹1
2
冘
j,l,k
⭸BijX,t
⭸xl
BlkX,tdWkdWj
⫽
冘
j
BijX,tdWj⫹1
2
冘
j,l
⭸BijX,t
⭸xl
BljX,tdt
⫽I
冘
j
BijX,tdWj⫹1
2
冘
j,l
⭸BijX,t
⭸xl
BljX,tdt,
(7)
where we have used dW
k
dW
j
dt
␦
kj
. Assuming constant ,
one can show that the following Stratonovich SDE obeys (1)
and is equivalent to the Itoˆ SDE (2a):
dXit⫽
冋
viX,t⫹1
2
冘
j,l
Bij
⭸
⭸xl
BljX,t
册
dt
⫹
冘
j
BijX⫹1
2dX,tdWjt, (8)
where B
ij
of the stochastic integral is evaluated at time tand
location given by the vector X1
2dX. Therefore evaluating B
ij
at various locations, which differ infinitesimally from the cur-
rent location, and adding or subtracting necessary gradient
drift terms allows one to formulate a variety of equations that
are mathematically equivalent to (2a).
Standard random-walk methods for simulating (1) are nor-
mally based on Euler integration of the Itoˆ SDE in (2a) [e.g.,
see Tompson et al., 1987]. However, equation (2a) only applies
when coefficients and D
ij
are sufficiently smooth. Therefore,
when either D
ij
or are discontinuous, these standard meth-
ods fail [LaBolle, 1996]. For example, Euler approximations to
(2a) that either evaluate gradient drift terms by finite differ-
ences [Tompson et al., 1987] or neglect gradient drift terms all
together [Prickett et al., 1981] cannot simulate (1) in composite
media. Further, results from these methods cannot be im-
proved by refining the time step of the integration scheme. In
summary, standard stochastic theory fails to provide a defini-
tion for such equations in composite media in which and D
ij
are discontinuous. We address this problem in section 3.
3. Diffusion Processes in Composite Media
The methods developed here stem from the premise that
SDEs may be generalized to consider discontinuous coeffi-
cients, yielding diffusion processes that obey (1) for both
smooth and discontinuous transport properties. By inspection
the principle objection to applying the SDE (2a) in composite
media is the presence of the gradient drift term therein, not
formally defined when and/or D
ij
are discontinuous, that is,
step functions. We will show that one can formulate equations
equivalent in meaning to (2a) but free of gradient drift terms.
Numerical integration by random walks will demonstrate that
these equations correspond to (1) in composite media, where
and D
ij
are discontinuous. These new methods preserve
many of the computational advantages of standard random-
walk techniques. In Appendix A we consider the new methods
in one dimension and show that they, indeed, correspond to (1)
in composite media. In Appendix B we show that a similar
result may be obtained by applying a stochastic calculus using
generalized functions. Next we present these methods and
demonstrate their application to subsurface transport prob-
lems beginning with the simple case of isotropic dispersion in
composite media, and then we consider the case of anisotropic
dispersion.
3.1. Isotropic Diffusions
The mathematical representation of isotropic diffusion pro-
cesses arising from advection and dispersion in composite po-
rous media is relevant to the simulation of transport in heter-
ogeneous porous media. As we will demonstrate in a
subsequent paper, the dispersion tensor can often be approx-
imated as isotropic without loss of accuracy provided the pore-
scale dispersion tensor can be approximated as isotropic with
respect to its minor axes (assumed to be orthogonal to the
velocity vector) and provided longitudinal spreading due to
explicitly modeled heterogeneities is much greater than longi-
tudinal spreading represented by the pore-scale dispersion
process. Beginning with the case of constant and isotropic
D
ij
, we develop an equation that is equivalent to (2a) and yet
free of gradient terms. We seek an equation that takes the
form of (2a) after expanding in Taylor series. By inspection we
arrive at the following result:
dXit⫽viX,tdt⫹
冘
j
BijX⫹dX,tdWjt(9a)
Bijx,t⫽
冑
␭
x,t
␦
ij, (9b)
where
␭
is the eigenvalue of 2D
ij
and B
ij
is evaluated at time
tand location given by the vector XdX. For smooth coef-
ficients, (9a) has been referred to as the “backward Itoˆ” sto-
chastic integral [Karatzas and Shreve, 1991]. Expanding (9a) in
a Taylor series shows that (9a) and (2a) are, indeed, equivalent
for constant and smooth isotropic D
ij
. For all D
ij
we have
dXi⫽viX,tdt⫹
冘
j
BijX⫹dX,tdWj⫽viX,tdt
⫹
冘
j
冋
BijX,t⫹
冘
k,l
⭸BijX,t
⭸xl
BlkX,tdWk⫹···
册
dWj
⫽viX,tdt⫹
冘
j,l
⭸BijX,t
⭸xl
BljX,tdt⫹
冘
j
BijX,tdWj,
(10a)
and when D
ij
is diagonal,
653LABOLLE ET AL.: DIFFUSION PROCESSES IN COMPOSITE POROUS MEDIA

dXi⫽viX,tdt⫹
冘
j,l
⭸BijX,t
⭸xl
BljX,tdt
⫹
冘
j
BijX,tdWj
⫽viX,tdt⫹
冘
j
⭸DijX,t
⭸xj
dt
⫹
冘
j
BijX,tdWj. (10b)
Formally, this Taylor series expansion is not allowed for
discontinuous D
ij
; however, in Appendix B we show that one
can expand (9a) for discontinuous D
ij
using generalized func-
tions. In the following sections we introduce a method for
numerically integrating (9a) and apply this method to clearly
demonstrate convergence to (1) in composite media.
3.1.1. Numerical integration. Equation (9a) may be inte-
grated over a time step tby taking two particle displace-
ments, Y
l
and X
i
,as
Xi⫽vit⫹
冘
j
BijXl⫹Yl,tWj(11a)
Yl⫽
冘
k
BlkX,tWk. (11b)
Note that advective transport is not included in (11b); here
velocity only contributes to higher-order terms that can be
neglected in the limit. To implement this algorithm, (11b) is
first evaluated to determine the particle displacement Y
l
.
The result is used in (11a) to determine X
l
. This simulation
method is illustrated in Figure 1. Boundary conditions must be
implemented in the application of both (11a) and (11b). If a
particle exits an absorbing or reflecting boundary in the appli-
cation of (11a) or (11b), it is either removed from the simula-
tion or reflected in the usual way, respectively [see Tompson et
al., 1987].
In Appendix A we show that distributions approximated by
(11a) satisfy (1). In Appendix B we show that a similar result
can be obtained through the use of a stochastic calculus using
generalized functions. To our knowledge this result has not
been described elsewhere in the literature. We do not, how-
ever, present a formal proof of convergence of these approx-
imations. Such a lengthy proof usually demonstrates conver-
gence only; it does not guarantee a robust approximation. To
this end we will demonstrate convergence and applicability of
these approximations through numerical examples. In all ex-
amples, Brownian motions will be simulated by uniformly dis-
tributed random variables with mean 0 and variance t.
3.1.2. Diffusion in one-dimensional composite media. In
this example we consider one-dimensional diffusion in an un-
bounded domain with governing equation (1), constant ,v
0, and initial and boundary conditions
cix,0⫽
␦
x⫺x0,x0僆i(12a)
c1,t⫽c2,t⫽0 (12b)
c1x,t⫽c2x,t,x⫽0 (12c)
lim
x30
D1
⭸c1x,t
⭸x⫽lim
x30
D2
⭸c2x,t
⭸x(12d)
Dx⫽
再
D1,x⬍0
D2,x⬎0(12e)
where i1, 2 denotes quantities within subdomains 
1
and

2
to the left and right, respectively, of an interface located at
x0.
First, we apply (11a) performing three simulations corre-
sponding to t100.0, 10.0, and 1.0 to demonstrate con-
vergence with decreasing time step of simulated moments to
analytical moments of the distribution. In each simulation we
use D
1
10.0 and D
2
1.0 and begin with 1000 particles
located at x
0
1.0. Figure 2 plots the mean and standard
deviation computed from an analytical solution [see Carslaw
and Jaeger, 1959] against values computed from particle dis-
placements, given as
X
៮
t⫽Xt ⫽1
Np
冘
p1
Np
Xpt(13a)
t⫽Xt⫺X
៮
t21/ 2
⫽
再
1
Np
冘
p1
Np
Xpt⫺X
៮
t2
冎
1/ 2
(13b)
respectively, where N
p
is the total number of particles and
X
p
(t) is the location of the pth particle at time t.
Second, using (11a), we perform three simulations corre-
sponding to D
2
2.5, 0.5, and 0.05. In each simulation we
use D
1
5.0, t0.01, x
0
5.5 and 10
5
particles.
Figure 3 compares simulated density at time t6, computed
by summing particle masses within unit lengths along the x
axis, with analytical distributions [see Carslaw and Jaeger, 1959]
to (1) with initial and boundary conditions (12a)–(12e).
Numerical simulation results presented here compare well
with analytical solutions and demonstrate convergence of (11a)
and (9a) to (1) in one dimension with discontinuous coeffi-
cients. As we will show, extension to (multidimensional) iso-
tropic diffusions for hyperplane interfaces follows directly from
Figure 1. The two-step process of random-walk simulation
of advection and isotropic dispersion in composite porous me-
dia for the algorithm given in equations (11a) and (11b).
LABOLLE ET AL.: DIFFUSION PROCESSES IN COMPOSITE POROUS MEDIA654

this result by noting that diffusion processes in the different
coordinate directions are independent in this case.
3.1.3. Effective diffusivity of composite media. In this ex-
ample, we estimate effective diffusivity of composite media.
The specific geometry considered here is that of circular cyl-
inders packed in regular square arrays within a matrix of con-
trasting material as shown in Figure 4. Dispersion tensors of
both materials are constant and isotropic. Transport is de-
scribed by (1) with constant and
␯
0. This classic problem
has received much attention [e.g., see Keller, 1963; Sangani and
Acrivos, 1983; Quintard and Whitaker, 1993], and accurate ex-
perimental and analytical values of effective diffusivity are
available [Perrins et al., 1979].
We estimate the effective diffusivity of the periodic array by
specifying reflecting boundaries at lines ABC and DEF and
absorbing boundaries at lines AF and CD as shown in Figure
4. Simulation proceeds by releasing particles at time t0 and
random locations with uniform distribution along the line BE
that bisects the system. Particle paths are simulated using (11a)
until particles exit the system by crossing absorbing boundaries
at lines AF or CD. Effective diffusivity is given by the relation-
ship
D
៮
⫽LAB
2
冉
2
Np
冘
p1
Np
␶
p
冊
1
, (14)
where
␶
p
is the elapsed time from release until particle pexits
the system and L
AB
is the length of line AB.
The matrix diffusivity is arbitrarily chosen as 1.0. Simulations
are performed for a range of cylinder volume fractions, con-
trolled by varying cylinder diameter, and diffusivities. In each
simulation, 1000 particles are released; time step is dynamically
controlled to ensure particles cannot bypass subdomains in any
single step of the algorithm given by (11a). Figure 5 compares
simulated effective diffusivities with values reported by Perrins
et al. [1979] for a range of cylinder volume fractions and dif-
fusivities. Simulation results compare well with reported values
Figure 2. Simulation results for (a) X
៮(t) and (b) ¥(t) are
compared with analytical moments for t100, 10, and 1.
Figure 3. Simulation results for density are compared with
analytical solutions for D
1
5.0 and D
2
2.5, 0.5, and0.05.
Figure 4. Square array of cylinders with known diffusivity D
2
embedded in a matrix with contrasting diffusivity D
1
.
Figure 5. Simulated effective diffusivities are compared with
values reported by Perrins et al. [1979] for cylinder volume
fractions of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, and 0.75, matrix
diffusivity D
1
1, and cylinder diffusivities, D
2
2 (circles),
5 (diamonds), 10 (triangles), 20 (squares), and 50 (plus signs).
655LABOLLE ET AL.: DIFFUSION PROCESSES IN COMPOSITE POROUS MEDIA

as expected because of the previous success of (11a) in one
dimension.
3.1.4. Discussion. Our results show that diffusions de-
scribed by (9a) obey (1) in composite media with discontinui-
ties in effective transport properties. Furthermore, the numer-
ical integration method (11a) “accurately” solves this equation.
Convergence and analysis of errors associated with the nu-
merical simulation of SDEs is a developing field [e.g., see
Kloeden and Platen, 1992]. Verification of convergence and
quantification of accuracy may generally be addressed by com-
paring numerical results with known analytical solutions, as we
have done here, or by using a bench mark against established
numerical methods as in sections 3.2.1 and 3.2.2. Since we
cannot consider all of the many problems to which (11a), and
the other approximations that follow, may be applied, we sug-
gest verification by a similar procedure to assess accuracy and
convergence when the methods are used under circumstances
other than those considered herein.
The examples presented above were for isotropic diffusions
with constant and v0. Next we present a more general
multidimensional approximation for anisotropic diffusions. Ex-
amples that follow will consider advection and dispersion in
composite porous media.
3.2. Anisotropic Diffusions
As with the isotropic case, we find ourselves faced with the
following problem: eliminating the gradient drift term in (2a)
in the anisotropic case. Note that this problem is not trivial.
Nevertheless, with considerable effort we arrived at the follow-
ing inspired result:
dXit⫽viX,tdt⫹1
2
冘
j,k
1/ 2X,tB
ˆijk
䡠
冋
Xl⫹1/ 2X,t
冘
m,n
B
ˆlmnX,tdWm,t
册
dWj
⫹1
2
冘
j,k,m,n
ZimnX,t1/ 2X,t
䡠B
ˆmjkXl⫹
␭
n
1/ 2X,tdWl,tdWj(15a)
B
ˆijkx,t⫽1/ 2x,t
␭
k
1/ 2x,tZijkx,t(15b)
Zijkx,t⫽ekx,tiekx,tj, (15c)
where e
k
is the normalized eigenvector corresponding to the
kth eigenvalue
␭
k
of 2D
ij
, and, for example, in the first term
on the right-hand side of (15a), B
ˆ
ijk
is evaluated at time tand
location given by the vector whose lth component is X
l


1/2
(X,t)¥
m,n
B
ˆ
lmn
(X,t)dW
m
. For isotropic D
ij
,
␭

␭
k
,@k, and using relationships given in Appendix C, (15a) and
(15b) simplify considerably:
dXit⫽viX,tdt⫹1/ 2X,t
䡠
冘
j
BijX⫹1/ 2X,tdX,tdWjt(16a)
Bijx,t⫽
冑
x,t
␭
x,t
␦
ij. (16b)
Further, for constant , (16a) and (16b) reduce to (9a) and
(9b). Therefore (9a) can be viewed as a special case of (15a).
Integration of (15a) may proceed according to a discrete-
time random walk that includes the approximations
Xi⫽viX,tt⫹1
21/ 2X,t
冘
j,k
B
ˆijkXl⫹Y
ˆl,tWj
⫹1
21/ 2X,t
冘
j,k,m,n
ZimnX,tB
ˆmjkXl⫹Unl,tWj
(17a)
Y
ˆl⫽1/ 2X,t
冘
m,n
B
ˆlmnX,tWm(17b)
Unl ⫽
␭
n
1/ 2X,tWl. (17c)
First, equations (17b) and (17c) are evaluated to determine
Y
ˆ
l
and U
nl
,@n. These results are used in (17a) to deter-
mine X. As with the (11a) and (11b), boundary conditions
must be implemented in the application of each step of algo-
rithm (17a)–(17c).
When coefficients are sufficiently smooth, (15a) is a SDE. In
Appendix C we expand this SDE in Taylor series to show that
it is, indeed, equivalent to (2a) in this case. Standard stochastic
theory [e.g., see Arnold, 1992] shows that diffusions described
by (2a) and therefore (15a) obey (1) when coefficients are
smooth. Next we demonstrate that (17a) converges to (1) in
composite media and is applicable to the simulation of subsur-
face transport.
3.2.1. Anisotropic diffusion in composite media. In this
example we simulate anisotropic diffusion in composite media.
The system illustrated in Figure 6 is a composite of two media
with contrasting diffusion (dispersion) tensors whose principle
axes, designated by
␭
1
, are oriented at an angle
␾
with the x
axis; eigenvectors corresponding to
␭
1
and
␭
2
are given as e
1

(sin
␾
, cos
␾
) and e
2
(sin
␾
, cos
␾
), respectively. Two
problems are considered. In the first problem we test the ability
of approximation (17a) to maintain the invariant distribution,
that is, a uniform number density, for the system in Figure 6
given periodic boundaries in the xdirection and reflective,
no-flux boundaries at y0 and y2. In each of the four
simulations a total of 5000 particles are initially distributed
uniformly over the domain, and the system is evolved over time
using the parameters specified in Table 1. Results for first
moments in the xand ydirections plotted in Figures 7a–7d
indicate that approximation (17a) can successfully maintain
the invariant distribution for these problems.
Figure 6. Composite media with anisotropic diffusion ten-
sors.
LABOLLE ET AL.: DIFFUSION PROCESSES IN COMPOSITE POROUS MEDIA656

In the second problem we compare predictions for evolution
of concentration from approximation (17a) with finite differ-
ence numerical solutions to (1) for the system in Figure 6 with
absorbing (zero concentration) boundaries on x0, x2,
y0, and y2 and initial distribution c(x,0)c
0
in the
region 0.99 x1.00 and 0.99 y1.00 and c(x,0)
0 outside this region. The finite difference algorithm is imple-
mented using an explicit updating scheme. Choosing x
y0.005 and t0.0005 satisfies criteria for numerical
stability [Peaceman, 1977]. Particle simulations are imple-
mented using 10
6
particles. Coarse particle and finite differ-
ence solution contours of c(x,t)/c
0
at t5.0 plotted in
Figures 8a–8d for parameters given in Table 1 compare well.
The results herein clearly demonstrate that diffusions de-
scribed by (15a) and approximation (17a) obey (1) for the cases
considered here.
3.2.2. Advective-dispersive transport in composite media.
In this example we simulate advective-dispersive transport ac-
cording to (1) for the system illustrated in Figure 6 with ab-
sorbing (zero concentration) boundaries on x0, x2, y
0, and y2. The functional form of the hydrodynamic-
dispersion tensor used here is given as
Dij ⫽兩v兩
␣
T⫹D*
␦
ij ⫹
␣
T⫺
␣
Lvivj/兩v兩, (18)
where
␣
T
and
␣
L
(L) are transverse and longitudinal disper-
sivities, respectively, and D* is effective molecular diffusivity
(L
2
T
1
). Velocity direction is oriented at an angle
␾
with the
xaxis as illustrated in Figure 6. We compare predictions from
approximation (17a) with finite difference numerical solutions
to (1). Again, the finite difference algorithm uses an explicit
updating scheme. Choosing xy0.005 and t
0.0005 satisfies the stability criteria referenced in section 3.2.1
and ensures a grid Peclet number Pe x/
␣
L
less than one.
Particle simulations use 10
6
particles. As in the previous sim-
ulations, the initial distribution in each simulation is c(x,0)
c
0
specified within a square region of xy0.01, as
shown in Figures 9a–9c, and c(x,0)0 outside of this region.
Coarse particle and finite difference solution contours of
c(x,t)/c
0
at t2.5, 5.0, and 5.0 plotted in Figures 9a, 9b,
and 9c, respectively, for parameters given in Table 2 compare
well. These and the previous results show that diffusions de-
Table 1. Parameters Corresponding to Simulation Results
Presented in Figures 7a–7d and 8a–8d
Figures
Region 1 Region 2
␭
1
␭
2
␾
,
deg
␭
1
␭
2
␾
,
deg
7a and 8a 2 10
2
210
3
90 2 10
2
210
3
0
7b and 8b 2 10
2
210
3
45 2 10
2
210
3
45
7c and 8c 2 10
2
210
3
30 2 10
3
210
4
30
7d and 8d 2 10
2
210
3
70 2 10
3
210
4
0
Figure 7. Simulated first moments in the xand ydirections as a function of time for the system illustrated
in Figure 6 with periodic boundaries in the xdirection and reflective, no-flux boundaries at y0 and y
2, and initially uniform number density. (a–d) Parameters given in Table 1.
657LABOLLE ET AL.: DIFFUSION PROCESSES IN COMPOSITE POROUS MEDIA

scribed by (15a) correspond with (1) and that approximation
(17a) simulates (1) in composite media with surprising accuracy.
4. Discussion and Conclusions
Standard diffusion theory only applies when effective trans-
port properties are sufficiently smooth, yet discontinuities in
transport properties arise naturally in porous media at abrupt
contacts between geologic materials with contrasting transport
properties. Limitations of standard diffusion theory have pre-
cluded development of diffusion processes that obey spatially
averaged transport equations in composite media. In this pa-
per we have (1) generalized SDEs to the case of discontinuous
coefficients and (2) developed random-walk methods for nu-
merically integrating these equations. The new results apply to
problems found in many scientific disciplines and offer a
unique contribution to diffusion theory and the theory of
SDEs. Examples demonstrated convergence of the new meth-
ods to ADEs in composite media and applications to subsur-
face-transport problems including (1) one-dimensional diffu-
sion in composite porous media with constant coefficients in
subdomains, (2) isotropic two-dimensional diffusion in a sys-
tem of circular cylinders packed in regular square arrays within
a matrix of contrasting material, (3) two-dimensional anisotro-
pic diffusion in a composite system with contrasting diffusion
tensors, and (4) transport by advection and dispersion in com-
posite porous media.
Standard theory shows that diffusions described by the new
generalized SDEs obey ADEs when coefficients are sufficiently
smooth. Further, we have demonstrated that these diffusions
obey ADEs in composite media. Thus, in cases where coefficients
are smooth, we conclude that the new methods may often be
more robust approximations than standard Itoˆ-Euler techniques.
The new simulation techniques possess the computational
advantages of standard random-walk methods, including the
Figure 8. (a–d) Contours of c(x,t)/c
0
from simulations of anisotropic diffusion in the composite media
shown in Figure 6 with parameters given Table 1. Figure 8a shows contour intervals used in Figures 8a–8d.
LABOLLE ET AL.: DIFFUSION PROCESSES IN COMPOSITE POROUS MEDIA658

ability to efficiently simulate solute-mass distributions and ar-
rival times while suppressing errors such as numerical disper-
sion, common to finite difference methods when Peclet num-
bers are large. As such, the new methods are appropriate for
problems characterized by immense computational grids, such
as those now commonly produced through the use of geostatis-
tical simulation techniques for subsurface characterization. Fi-
nally, unlike alternatives relying on Itoˆ-Euler integration and
spatial interpolation to ensure smooth coefficients, the new
methods will converge exactly in the limit, without the need to
simultaneously refine the interpolation scheme and time step.
Appendix A: Necessary Conditions for Weak
Convergence
LaBolle et al. [1998] use a graphical technique to show that
approximations with symmetric transition-probability density
satisfy necessary conditions for weak convergence developed
therein. Since transition-probability densities specified by
(11a) and similar approximations are asymmetric, however, the
simple graphical techniques used by LaBolle et al. [1998] are
not applicable here. Instead, here we will show mathematically
that (11a) approximates the diffusion equation of interest.
Functions p(x,t) satisfy the one-dimensional diffusion
equation
冕
fx,t⭸px,t
⭸tdx⫽
冕
Dx,tfx,t
⫹Dx,tfx,tpx,sdx(A1)
for all smooth and bounded test functions fand are weak solu-
tions of (1) with v0[LaBolle et al., 1998], where the fand f
denote the first and second partial derivatives of fwith respect
to x. For the problem at hand it is convenient to consider the
case in which D(x,t) has constant values within subdomains
and a single jump of size 兩D
2
D
1
兩at x
0
,DD
1
for x
x
0
, and v0. Here D0 except at x
0
, where it is given as
D⫽D2⫺D1
␦
x⫺x0. (A2)
Substituting this result into (A1) yields
冕
f⭸p
⭸tdx⫽
冕
Df⫹Df pdx
⫽
冕
D2⫺D1
␦
x⫺x0f⫹Df pdx
⫽
冕
Df pdx⫹D2⫺D1fp兩x0. (A3)
We will show that the density p
␧
(x,t) generated by (11a)
satisfies (A3).
To facilitate computations, we will consider the following
variation of approximation (11a):
Xt⫽1
2BX⫹BXZ⫹2BX⫺BX⫺BXZZ,
(A4)
where Zhas distribution (Z) with mean zero and variance
t␧
2
. We could work directly with (11a), but use of (A4)
simplifies the math that follows. The density p
␧
generated by
the Markov-chain approximation (A4) satisfies the equations
[LaBolle et al., 1998]
Figure 9. (a–c) Contours of c(x,t)/c
0
from simulations of
advective-dispersive transport in the composite media shown
in Figure 6 with parameters given Table 2. Figure 9a shows
contour intervals used in Figures 9a–9c.
659LABOLLE ET AL.: DIFFUSION PROCESSES IN COMPOSITE POROUS MEDIA

冕
0
t
冕
fx,s⭸p␧x,s
⭸tdxds
⫽
冕
0
t
冕
L␧fx,sp␧x,sdxds⫹o␧2(A5a)
L␧fx,t⫽1
␧2
冕
fz,t⫺fx,tp␧z,t⫹␧
2兩x,tdz,
(A5b)
where L
␧
is commonly referred to as the generator (or infin-
itesimal operator) of the Markov chain [Arnold, 1992]. Ex-
panding fin a Taylor series yields
␧2
冕
0
t
冕
fx,s⭸p␧x,s
⭸tdxds
⫽
冕
0
t
冕冕
fz,s⫺fx,s
䡠p␧z,s⫹␧
2兩x,sp␧x,sdxdzds⫹o␧2
⫽
冕
0
t
冕冕
z⫺xfx,s⫺1
2z⫺x2fx,s
䡠p␧z,s⫹␧
2兩x,sp␧x,sdxdzds⫹o␧2. (A6)
Noting that ( zx) is a realization of X(t) , we have,
retaining terms to order ␧
2
,
冕
z⫺xpz,t⫹␧
2兩x,tdz
⫽1
2
冕
Bx⫹BxZ⫹2Bx
⫺Bx⫺BxZZZdZ
⫽1
2
冕
Bx⫹BxZ⫺Bx⫺BxZZZdZ
(A7a)
1
2
冕
z⫺x2pz,t⫹␧
2兩x,tdz
⫽1
8
冕
Bx⫹BxZ⫹2Bx⫺Bx⫺BxZ2Z2ZdZ
⫽␧
2D
៮
x⫹o␧2, (A7b)
where D
៮
(D
1
D
2
)/2 at x
0
and D
៮
B
2
/ 2 elsewhere.
Substituting these relationships into (A6), we have
␧2
冕
0
t
冕
f⭸p␧
⭸sdxds
⫽
冕
0
t
冕冕
z⫺xf⫺1
2z⫺x2fp␧z,s⫹␧
2兩x,sp␧dxdzds
⫽1
2
冕
0
t
冕冕
Bx⫹BZ⫺Bx⫺BZZfp␧ZdxdZds
⫹␧
2
冕
0
t
冕
D
៮
fp␧dxds⫹o␧2, (A8)
where we have dropped explicit reference to ( x,t). On the
interval [ x
0
B
1
Z,x
0
B
2
Z], Bon either side of x
0
will
take on the value of Bfrom the remaining side. For all such
cases, in the limit,
lim
␧30
1
2␧2
冕冕
x0B1Z
x0B2Z
Bx⫹BxZ
⫺Bx⫺BxZZfp␧ZdxdZ
⫽lim
␧30
1
2␧2
冕冕
x0B1Z
x0B2Z
B2⫺B1Zfp␧ZdxdZ
⫽1
2B2⫹B1B2⫺B1fp␧兩x0⫽1
2B2
2⫺B1
2fp␧兩x0
⫽D2⫺D1fp␧兩x0. (A9)
Outside of this interval we have
lim
␧30
1
2␧2
冕冕
x僆兾x0B1Z,x0B2Z
Bx⫹BZ⫺Bx⫺BZ
䡠Zfp␧ZdxdZ⫽0 (A10)
because Dis constant in this case. From (A9) and (A10) we
have, in the limit,
lim
␧30
1
2␧2
冕冕
Bx⫹BZ⫺Bx⫺BZZfp␧ZdZdx
⫽D2⫺D1fp␧兩x0. (A11)
Substituting this result into (A8) yields, in the limit,
Table 2. Parameters Corresponding to Simulation Results Presented in Figures 9a–9c
Figure 
1

1
兩v兩
␣
L
␣
T
␾
, deg D*
Region 1
9a 1/3 
1
(10
1
)10
1
10
2
45 10
5
9b 1/3 
1
(10
1
)10
1
10
2
30 10
5
9c 1/3 
1
(10
1
)10
1
10
2
60 10
5
Region 2
9a 1/3 
1
(10
1
)10
1
10
2
45 10
5
9b 1/2 
1
(10
1
)
2
(10
2
)/
1
1.5 10
3
30 10
5
9c 1/2 
1
(5 10
2
)
2
(2 10
2
)/
1
3.0 10
3
010
5
LABOLLE ET AL.: DIFFUSION PROCESSES IN COMPOSITE POROUS MEDIA660

冕
0
t
冕
f⭸p␧
⭸tdxds
冕
0
t
冕
D
៮
fp␧dxds
冕
0
t
D2D1fp␧兩x0ds
⫽
冕
0
t
冕
Df p␧dxds⫹
冕
0
t
D2⫺D1fp␧兩x0ds, (A12)
where the last equality is justified since singular values of D
៮

0 have no effect on the diffusion process. Equation (A12) has
the same form as (A3); therefore the density generated by
(A4), and version (11a) of this approximation, satisfy (A3)
associated with this equation. Similar results can be developed
for the algorithm in (17a). In Appendix B we show how this
result implies a stochastic calculus with generalized functions.
Appendix B: Stochastic Calculus
With Generalized Functions
The forgoing result suggests that one can apply a stochastic
calculus with generalized functions to obtain the desired result.
For example, in one dimension, expanding (9a) yields
dX⫽BX⫹BdW,tdW
⫽BX,tdW⫹BX,t⭸BX,t
⭸xdWdW
⫽BX,tdW⫹⭸DX,t
⭸xdt
⫽BX,tdW⫹⭸DX,t
⭸x
冏
Xx0
dt
⫹D2X,t⫺D1X,t
␦
X⫺x0dt, (B1)
which from standard theory corresponds to the diffusion equa-
tion
⭸p
⭸t⫽D⭸2p
⭸x2⫹
冋
⭸D
⭸x
冏
xx0
⫹D2⫺D1
␦
x⫺x0
册
⭸p
⭸x. (B2)
Equation (B2) is equivalent to (A3) for the problem consid-
ered therein.
Appendix C: Taylor Series Expansion
of Equation (15a)
When coefficients are smooth, (15a) is a SDE. Here we
expand this equation to show that it is, indeed, equivalent to
the SDE given in (2a). Expanding the second term in the
right-hand side of (15a) in Taylor series for sufficiently smooth
D
ij
and yields:
1/ 2X,t
冘
j,k
B
ˆijk
冋
Xl⫹1/ 2X,t
冘
m,n
B
ˆlmnX,tdWm,t
册
dWj
⫽1/ 2X,t
冘
j
冋
冘
k
B
ˆijkX,t⫹1/ 2X,t
䡠
冘
k,l,m,n
⭸B
ˆijkX,t
⭸xl
B
ˆlmnX,tdWm⫹···
册
dWj
⫽1/ 2
冘
j,k
B
ˆijk dWj⫹1
冘
j,k,l,n
⭸B
ˆijk
⭸xl
B
ˆljn dt⫹odt
⫽
冘
j,k
␭
k
1/ 2Zijk dWj⫹1/ 2
䡠
冘
j,k,l,n
⭸
⭸xl
1/ 2
␭
k
1/ 2Zijk
␭
n
1/ 2Zljn dt⫹odt
⫽
冘
j,k
␭
k
1/ 2Zijk dWj⫹1/ 2
䡠
冘
j,k,m,n
⭸
⭸xj
1/ 2
␭
n
1/ 2Zimn
␭
k
1/ 2Zmjk dt⫹odt, (C1)
where we have omitted explicit reference to evaluation of
terms at (X,t) and used the following relationships
Zijk ⫽Zjik (C2a)
冘
k
Zijk ⫽
␦
ij (C2b)
冘
m
ZimnZmjk ⫽Zijk
␦
nk, (C2c)
where
␦
nk
1 for nkand 0 otherwise. Expanding the third
term on the right-hand side of (15a) in Taylor series yields
1/ 2X,t
冘
j,k,m,n
ZimnX,tB
ˆmjkXl⫹
␭
n
1/ 2X,tdWl,tdWj
⫽1/ 2X,t
冘
j,k,m,n
ZimnX,t
䡠
冋
B
ˆmjkX,t⫹⭸B
ˆmjkX,t
⭸xl
␭
n
1/ 2X,tdWl⫹···
册
dWj
⫽1/ 2
冘
j,k,m,n
Zimn
冉
B
ˆmjk dWj⫹⭸B
ˆmjk
⭸xj
␭
n
1/ 2 dt
冊
⫹odt
⫽
冘
j,k
␭
k
1/ 2Zijk dWj⫹1/ 2
䡠
冘
j,k,m,n
⭸
⭸xj
1/ 2
␭
k
1/ 2Zmjk
␭
n
1/ 2Zimn dt⫹odt, (C3)
where again we have used (C2a)–(C2c) and omitted explicit
reference to evaluation of terms at (X,t). Expressing (15a) in
terms of (C1) and (C3), we have
dXi⫽vidt⫹
冘
j,k
␭
k
1/ 2Zijk dWj
⫹1
21/ 2
冘
j,k,m,n
␭
k
1/ 2Zmjk
⭸
⭸xj
1/ 2
␭
n
1/ 2Zimndt
⫹1
21/ 2
冘
j,k,m,n
⭸
⭸xj
1/ 2
␭
k
1/ 2Zmjk
␭
n
1/ 2Zimn dtodt, (C4)
which one can show is equivalent to (2a) using the following
relationships:
冘
k
␭
kZmjk ⫽
冘
k,m,n
␭
n
1/ 2Zimn
␭
k
1/ 2Zmjk ⫽2Dij (C5a)
冘
k
␭
k
1/ 2Zijk ⫽
冘
k,m,n
␭
k
1/ 2ZimnZmjk ⫽2D1/ 2ij (C5b)
661LABOLLE ET AL.: DIFFUSION PROCESSES IN COMPOSITE POROUS MEDIA

1
21/ 2
冋
冘
j,k,m,n
⭸
⭸xj
1/ 2
␭
n
1/ 2Zimn
␭
k
1/ 2Zmjk
⫹
冘
j,k,m,n
␭
n
1/ 2Zimn
⭸
⭸xj
1/ 2
␭
k
1/ 2Zmjk
册
⫽1⭸
⭸xj
Dij
(C5c)
Acknowledgments. This work was supported by grants from the
U.S. EPA (R819658) Center for Ecological Health Research at U.C.
Davis, NIEHS Superfund Grant (ES-04699), University of California
Toxic Substances Teaching and Research Program, and Lawrence
Livermore National Laboratory. Although the information in this doc-
ument has been funded in part by the United States Environmental
Protection Agency, it may not necessarily reflect the views of the
Agency, and no official endorsement should be inferred.
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(Received November 5, 1998; revised July 15, 1999;
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LABOLLE ET AL.: DIFFUSION PROCESSES IN COMPOSITE POROUS MEDIA662