A master equation for reactive solute transport in ... - Springer Link

Stochastic Hydrol. Hydraul. 7 (1993) 255-268

StochasticHydrology and Hydraufics 0 Springer-Verlag 1993

A master equation for reactive solute transport in porous media Z. J. Kabala and A. Hunt

Department of Soil and Environmental Sciences, University of California, Riverside, CA 92521, USA Abstract: The mean value of a density of a "cloud of points" described by a generalized Liouville equation associated with a convection dispersion equation governing adsorbing solute transport )fields a joint concentration probability density. The general technique can be applied for either linear or nonlinear adsorption; here the application is restricted to linear adsorption in one-dimensional transport. The equation generated for the joint concentration probability density is in the general form of a Fokker-Planck equation, but with a suitable coordinate transformation, it is possible to represent it as a diffusion equation with variable coefficients. Key words: Groundwater hydrology, solute transport, master equation. 1 Introduction Impressive advances in the theory of solute transport in natural porous formations have been made in the last decade [Dagan, 1989]. Deterministic models based on homogeneous idealization of aquifers no longer dominate groundwater hydrology. Small-scale heterogeneities in the flow velocities and in parameters characterizing solute sorption are now accounted for by stochastic models. Most of the research has focused on modeling the ensemble mean concentration, whereas concentration uncertainty, the concentration higher moments, and the concentration probability distribution function (pdf) received significantly less attention [Sposito et al., 1986; Dagan, 1989]. In addition, the existing stochastic models of contaminant transport address phenomena governed only by linear stochastic partial differential equations.

1.1 Stochastic models of reactive solute transport The field experiment at the Borden site [Roberts et al., 1986] highlighted a significant growth in time of the field-scale retardation factor, a new feature in the field-scale reactive solute transport. It motivated the recent development of theoretical stochastic models [Garabedian et al., 1988; Dagan, 1989; Cvetkovic and Shapiro, 1990; Kabala and Sposito, 1991], as well as the Monte Carlo studies [Robin et al., 1991; Tompson, 1993]. A number of mechanisms has been proposed to account for this phenomenon. Roberts et al. [t98@ Goltz and Roberts [t986, 1988], Brusseau et al. [1989, 1991], Ball and Roberts [1991], and Brusseau [1992]

256 proved in laboratory studies that rate-limited sorption through intraparticle diffusion is a viable mechanism, whereas Kabala and Sposito [1991] demonstrated that cross-correlation between the flow field and the field of local retardation can produce the same effect. Roberts et al. [1986], following Van Genuchten and Wierenga [1976], suggested that nonlinear equilibrium sorption is yet another possibility for explaining the growth of the fieldscale retardation factor. This, however, leads to a governing nonlinear stochastic partial differential equation. A stochastic model of solute transport that accounts for nonlinear sorption has not been developed yet. One cannot evaluate the effect of nonlinear sorption on the field-scale retardation factor without such a model.

1.2 Nonlinear stochastic differential equations Traditional treatments of nonlinear stochastic differential equations (in groundwater studies) are significantly more complicated than those of linear equations. The main reason for this is that one cannot derive directly a differential equation for the mean dependent variable when the dependent variable obeys a nonlinear stochastic differential equation, because the nonlinearity brings in the higher moments and a consistent closure is not possible. This difficulty was noted by van Kampen [1976, p. 208] and Fox [1978]. However, it is possible to formulate an equation that governs the evolution of the probability density function of the dependent variable [van Kampen, 1976; Fox, 1978; and Gardiner, 1990]. If one knows this probability density function, one can calculate the mean value, as well as any higher moment. The cited method~ which will be used here, is not restricted in its applicability to either linear, or nonlinear equations. The resulting generalized Liouville equation is linear in its dependent variable whose mean value is the de,sired probability density function of the dependent variable in the original equation. Analytical solution of the resulting equation for the probability density is possible only when the solutions of the "sure" (i.e. non-stochastic, or unperturbed) equation are known, because these unperturbed solutions appear in integrals which constitute coefficients of a Fokker-Planck equation. So, in order to be concrete, it proves useful to apply the method first to a linear equation for which analytical solutions to the non-stochastic portion are available. In later papers applications of this approach will be made to nonlinear adsorption, and to more complex nonlinear systems involving unsaturated soils. The equation developed is called "the master equation". The recent physics literature is replete with derivations and applications of master equations. For example, Breuer and Petruccione [1993] use it to describe the fluctuating hydrodynamics, Park [1992] to deal with multidimensional nonlinear field theories, Gangopadhyay and Ray [1992] to describe nonlinear dissipative systems, whereas Rudenko and Bassler [1991] to deal with diffusion processes in random media.

1.3 Objectives and organization of this paper The goal of this paper is the generation of a partial differential equation describing the temporal evolution of the joint concentration probability density in terms of measurable stochastic properties such as the retardation, R, or its inverse. The differential equation governing this development should be of a form independent of the initial time, and soluble in principle, in terms of initial conditions describing a known initial concentration. The procedure starts with the discretization of the partial differential equation describing linear adsorption, and continues with the application of a method to generate a master equation for a finite number of stochastic variables. Since the method used applies equally to linear and to nonlinear adsorption, and since an analytical representation of the resulting equation for the joint probability density is possible in the linear case, we restrict ourselves here to the linear case. The full probability density allowing generation of arbitrary moments has not yet been

257 obtained in the case of linear adsorption, so a method to solve this problem is also useful per se. In addressing the problem of linear adsorption, the practical limitations of the method will become apparent. At the same time, it has been possible to identify a technique for putting the nonlinear equation into a form such that the stochastic properties of the concentration can be expressed in terms of the stochastic properties of the adsorption parameter. This form will also be described in the present treatment. This paper is organized as follows. First, the equation to be treated is introduced, and its significance and applicability are discussed. Then a brief review of the mathematical treatment to be applied is given. The calculations are presented together with their limitations. Then the results are discussed and interpreted. Conclusions follow. 2 The physical model The present treatment is a direct application of the method described in the review of van Kampen [1976] (this reference will be referred to as VK) to a discretized approximation of the partial differential equation,

--•=

-n(x,0 ~ 0x

(t)

where c is the solute concentration, x is the spatial coordinate, and t stands for time. Equation (t) describes one-dimensional contaminant transport. Here u(x,t) is an effective (retarded) stochastic convective solute velocity arising from stochastic adsorption, i.e n(x,t) -

v°

R(x,t)

(2)

where v0 is the convective fluid velocity (possibly a random variable), whereas the retardation factor throughout the paper is given simply in terms of the stochastic adsorption parameter, a, i.e.

a(x,O ~ 1 + a(x,t)

(3)

The stochastic properties of u(x,t) are assumed to be known. Discretization of equation (1) allows application of the nonlinear VK method. Let us note in passing that nonlinear adsorption corresponds to making R a function of the concentration c as well, R(x,t) - 1 + ac"

(4)

and can also, in principle, be treated. In this case, the stochastic properties of v0 and a(x,t) would need to be known. In order to obtain, by the chosen method, the statistical properties of the concentration from those of the stochastic velocity u for linear adsorption, or from the stochastic adsorption parameter a and the possibly random fluid velocity v0 for nonlinear adsorption, it is necessary to take spatial averages of operators in the generalized Liouville equation (discussed by VK). If the independent stochastic variable (or variables) can be decoupled from the dependent one (the concentration) then it is possible to generate

258 decoupled from the dependent one (the concentration) then it is possible to generate expressions for the coefficients of the resulting Fokker-Planck equation without a priori knowledge of its solution. This is a necessary step for the method to be useful. For the nonlinear adsorption the appearance of the product acn in the denominator prevents direct application of the VK method to the nonlinear equation in its present form. If one, however, makes the substitution ac" --- (n, the resulting equation

a ~ _ ¢ aa + V o ~ aa Ot na & na[l+ff"] 8x

v0 a¢ 1+~" 8x

(5)

can be discretized and treated analogously to the development here. The treatment of this equation, however, is postponed to a later paper• 3 Review of stochastic differential equations The following notation and equations (6) (8) and (11) are taken directly from the VK review. The general form of stochastic differential equation treatable in the VK method can be written d~ = ~(~,~) dt

(6)

Note that the time derivative is first order, which allows an elementary formal solution. is in general a nonlinear function of the vector 7, and the stochastic parameter ~. Of course the function F can also be a linear function of the 7, in which case it must be possible to represent F as = _~ ~

(7)

Since we deal with a linear partial differential equation we are obviously interested in the linear form of F given in equation (7). Now, for any particular realization of ~S it is possible to solve equation (6); what is desired, of course is a means to find the statistical properties of the ~ given the statistical properties of ~. The method of solution involves an equation for the density p of points in ~-space which represents the individual realizations of the time evolution of the ~ in an ensemble of individual systems (with the same initial conditions). Each point in this cloud of points moves with velocity d~/dt determined by the form of equation (7) corresponding to an individual realization of ~. A sum over the product of the individual velocities and the density of points with each particular veloci_t~ value gives a current density, j. Application of the equation of continuity (ap/Ot = -v.j) to j yields a generalized Liouvilte equation

8p(~,t)_

~ a F~(~,t,~)p(~,t)----~-F(Z,t,~)p(~,t) v=l ~zv

(8)

The components of the vector ~ are denoted by the subscript v; the sum is taken over these components. Formal solution of equation (8) requires the enumeration of initial conditions and boundary conditions• If the initial values of the zffs are known as a function of position, then the initial value of p(E,t) can be expressed as

259

p(~,O) = II~'o, 6(z~-~,)

(9)

Here the Yv define the initial values of the zv's. It will be seen that initial conditions of the same form as in equation (9) will be used for the mean value of p~,t). In cases where physical conditions limit the values of the zv's, or the values of dzv/dt these boundary conditions must be specified as well. It is the relationship of p to the joint probability density of 7 which makes equation (8) of special significance. In particular, a temma, due to van Kampen [1976], states that the mean value p of p gives the probability of measuring a particular set, ~ of z values, i.e. (p(~,t))

=

p(g,t)

(t0)

This joint probability is of course defined in the usual sense. The equation (equation (19.6) of VK) for p = (p) (the mean value of P) valid to second order in the so-called Kubo number is obtained from equation (8) by the cumulant expansion. The Kubo number is defined as a product of the correlation time of the stochastic process ~7.F(g,t,~ ) and its magnitude, a. What is implied is that the random variations are relatively rapid and small compared with the average time development of 7. The following equation, (11), is the starting point for the treatment of the stochastic equation introduced in the previous section.

Op(~.,t)ot - ~7. -~o(ZD + ~2 fo dx - - ~

d(g-~))

(11)

In this equation, F0 is the "sure" part of ~ Fo + F1

(12)

F 1 the stochastic variation about the mean, and e.g. d~/dt represents the "sure" part of the time development of the coordinates ~ (arising from F0). p(~) gives the probability density of measuring a particular set of z values. The subscript -x on the del operator and the identical superscript on the 7 imply a shift in time from t to t - x resulting from the sure component of F. These shifts can be represented by operators which are functions of F and ~. The results will be demonstrated explicitly in the application, because the solution is elementary for the linear equation studied. In general, however, what is involved, is the first order variation of any particular zu arising from the controlled changeof a z# at a time earlier, and determined through the effects of the sure component of F. The relevant statistical properties of the combination of operators and variables with = P(-g) are found from the statistical properties (the average value, and the spatio-temporal correlation function) of the stochastic parameter, ~. 4 Derivation of the master equation The application of the methods of VK to the diffusion equation, (1), requires several basic steps. The substitution E for ~ is obvious. The stochastic convective contaminant velocity, appears as a muttiplicative factor in the original differential equation, (1). The discretization will put equation (1) in the form of equation (6) under the condition (7). It is necessary to define the initial and boundary conditions for the contaminant concentration. The average, (u(x,t)), and the spatio-temporal correlation function, (u(x,t)u(x+~',t+T)), of the convective contaminant velocity must be given.

260 (u(x,t))

(13)

= uo

(u(x,t)u(x+r,t+,))

= (u 2) * r ( r a )

(14)

The stochastic convective velocity, u, has been assumed stationary. The same is evidently implied for the retardation. The mathematical problem to be solved is to find the statistical properties of c governed by equations (1), (13), and (14), and subject as well as to the following boundary conditions cO,t) = 0

(15)

c(1,t) = 0

(16)

c(x,O) = fix)

(17)

These boundary conditions describe a contaminant whose initial concentration is known as a function f(x) of the spatial coordinate, x, and which is considered to be confined to a finite (scaled) interval. Although it is the ultimate intent to develop a means to calculate the statistical properties for a continuous random variable whose time evolution is given in terms of a spatial partial derivative such as in equation (1) at this time no established method for such a calculation exists. However, such an equation can be considered as a continuum limit of a (discretized) linear first order differential equation for a vector concentration defined on a discrete subspace of the continuum. The method of VK described above is suitable to treat the discretized version. Such a discrete subspace is all that is accessible to subsequent measurements anyway; the function defining the initial condition on c can only be determined at a finite number of points. Thus the advantages of extending the treatment to the continuum equation are more conceptual, than practical. A related issue which we can not address at this time is that we do not describe the time evolution exactly in the equation, or we can not determine the initial conditions precisely. In either case, a lack of information is implied. But we will assume that it is not necessary to revise our statistics as time progresses in accord with Bayesian statistics. In practical terms this means that subsequent measurements of concentration may not be consistent with the governing differential equation. This may be most easily seen in the special case that no adsorption is considered, i.e. ff = 0. In such a case a conservation of total contaminant mass (expressed as an integral over all x of the concentration c) is guaranteed for physical reasons. Yet there is no corresponding guarantee that the sum of all the measured c-values be invariant. For the particular case studied, the analysis is not so simple, but, nevertheless, the more subsequent measurements are made, the more information about the system is gained, and the better can the initial statistical sampling be evaluated, at least in principle. Such an analysis is far beyond the scope of the present work. The procedure begins with the discretization. The discretization is elementary. Let x k = kAx, uk(t ) = U(Xk,t ), and Ck(t) = C(Xk,t). The number of discrete points is n. Then it is possible to write for the time-evolution of the concentration at each point k (using the second-order accurate central finite difference [Smith, 1985]):

261 dck - _uk(t) [Ck+l(t) - Ckq(t)] dt 2Ax

(18)

We note that a more general discretization scheme in terms of a general discretization operator [Ginn and Cushman, 1992] is also possible here, but we find the specific form of the central finite difference useful for its simplicity. Equations (15) and (16) imply that d% _ 0 - den dt dt

(19)

In terms of the discretized concentration, equation (1) now reads

(20)

at with

~=

1

2Ax

'0

0

0

0..

0

0

u2

0..

..0

-%

0

ui..

..0

0

-un_ I

0

,.0

0

0

0

(21)

Now that equation (20) is in the form of equation (6) with F ~ ) = -B E, we can immediately write the associated stochastic Liouville equation: Op(~,O at

_ a (fi ~p(~,O)

(22)

An alternative form using a generalized divergence operator (acting in c-space) is, ap(E,O _ _ 9 o . ~a~ at -

p(e,t)

(23)

262 An explicit representation of the sum is ao(ev %,

% cn) ~ ........ = bi~ekP(Cp c2..... ei..... %) (24) St i=i k=l Using the property that the chosen matrix representation of the gradient has only nondiagonal elements ao(e,0

[biCi+l - bici-1]

- E

St

9(~,0

(25)

i=1

Corresponding to the decomposition of u into its sure and stochastic components = go + fi

(26)

is the following decomposition of the matrix g = go + t3

(27)

Both sure and stochastic components or B are represented in the identical form as in equation (21), but with the correspondingsure and stochastic components of u, respectively. The sure and stochastic components of F are given by the products with E of B0 and B, respectively. It turns out to be much easier to simplify the solution for p by first writing the equation for the time evolution of the continuous variable c in a stationary coordinate system, i.e. in which the average velocity is zero. Both means of solution will be given here. The validity of the solution is always restricted by the boundary conditions, (I5) and (16) to times for which the contaminant does not reach the boundaries of the system, but choosing a moving coordinate system means that this condition must be verified explicitly. The appropriate coordinate transformation is elementary: x -- x' - Uot'

(28)

t : t'

(29)

Applying these elementary transformations to de _ de at' -[no + al ~,--w

(30)

leads to de &'

de

de 0x 0x 0x'

a' - [%1

-I-

de

- -

de + de

-~

-

& -

(31)

St at'

(32)

263 . &

8c

(33)

ax

In the new coordinate system (u) = 0. Thus, in order to define the correlation function, equation (14), in such a way as to be independent of the coordinate system chosen, we have had to subtract the mean-square of u. Equation (25) can be solved formally for any realization of the stochastic velocity ~ in terms of an initial p('d,t). The VK method yields the mean value of p(E,t) provided the stochastic velocity is uncorretated with the initial value of p(-6,t). The result for p yields the joint probability distribution for the measurements of the individual c's and is in the form of an expansion to second order in Kubo number, a measure of the strength of the stochasticity. The precise definition involves a product of the correlation time, % and the typical magnitude A(~'.~ = ~/(~.~2) _ (~.~)2 of the variations in the stochastic parameter in equation (8). Using equation (6) equation (8), equation (14) and equation (11) yields

[112

(34)

In general (VK) it is possible to replace the upper limit t on the integralwith ~, as long as one is interested in the time evolution subsequent to %. The sure part, F 0 of the matrix 1~ = F0 + F1 (in the notation of VK) has been made to equal zero through the coordinate transformation (28) and (29). As a consequence, the Jacobian determinant, det[d~-'/dz-] = det[exp(TV-i~0)] arising from the sure component of the time evolution of the coordinates % is obviously one and can be omitted. Additionally, since (B E)k is independent of Ck, (because the diagonal elements of B are identically zero) the derivative 0/0% treats it as a constant. Note that even when B0 is non-zero, its structure (all zeroes on the diagonal) will guarantee that its trace is zero, leading to a Jacobian determinant of t (because the product of the diagonal elements of the exponential of B in its diagonal representation is 1). The general result, even in its simpler form, equation (34), is prohibitively difficult to treat and not very illuminating. Therefore, we consider some limiting cases, which are somewhat more transparent. Suppose first that the correlation length, 1, of u is much smaller than the sampling separation, Ax. Then,

(35)

r[(i-j)Ax,~] = r(o,~)8~j to a very good approximation. Consequently, equation (34) becomes

(36)

a

~o~

t 2ax

j a:?

or

(37)

264

(3s)

[ 1 I fo d~r(O,~) If a very coarse sampling grid is employed with, say, n = 4, or n = 5, then

(n = 4)

(39)

(n = 5)

(40)

which are in the form of elementary diffusion equations with variable diffusion coefficients. The effective diffusion coefficients can be written as

Di= f~ d~r(0,~)[ci+l-ci-l] 2 [ ~Ax I

(41)

The solution of equations (39) or (40) requires the enumeration of the relevant boundary and initial conditions on p. These conditions are: p(~,0) = II~ 5(ci-~i)

(42)

with 5(ci-~i) a Dirac delta function, and ~i the initial value of % and the following which derives from the condition that the probability flux to (and from) negative concentrations must vanish,

0e i

Ice_0 = 0

(43)

Even equations (39) and (40) are not elementary. The difficulty in solution arises from the variable coefficients (i.e. the c-dependence). It should be noted that on the hyperline defined by c i = el+ , etc. 0p/0t = 0, and no time evolution results. We point out that the derivation is considerably more complex if the coordinate transformations, equations (28),and (29) are not employed. Such a coordinate transformation can not be used, for example, when the average fluid velocity is a function of position. Thus it is useful to consider the more general formulation of the V K method which is appropriate when F0 is not zero. In this case, the result, equation (34) can be written

(44) a

,:,

'

N

0q--;

The main difference between this equation and the version in the moving coordinate system, is that the time development of the c's now includes a non-zero sure component and 0Ck/0CTr is no longer the Kronecker delta, 5kj. The Jacobian determinants are still exactly 1, J as mentioned. The derivative, 0cff0ci r is found by writing the solution of equation (20) as

265 (45)

e(O = exp(-§¢)e(t-O

Then it can be seen immediately that,

3ek(t) - c~k 8¢j(t-~) 0cj-"

[exp(-B~:)]~

(46)

Using this result, equation (46), in equation (44) yields

at

i=' j=l k=l

f°dT

I (t)cll ~ii

Op

(47)

Considering once again the case with four nodes (two internal) results in

a

\ 2ax %(o[ t 2 A x } ~33 +

2Ax

~

[ tZaxj~ ~x

OC--3+

~2AxJ (48)

+ fi3(t)c2(t) ~

2Ax

t 2 A x ) G0c -

k 2ax) -~3 + k 2ax)

1,2Ax)

11/

This equation (48), may be simplified (as was equation (34)) by assuming that the separation of the nodes is greater than the correlation length of u, whence

(49)

+ o2 ~ ; t

t~X-;) K + t 2 a x

266 It is interesting that this more general version of the equation is in the form of a FokkerPlanck equation, i.e. both first and second derivatives with respect to the c's appear. The fact that all Fokker-Planck equations are linear has been noted in VK to free the term linear to describe a particular class of such equations which may be unambiguously transformed into Langevin equations. However, the c-dependence generated in all the equations developed here makes them "nonlinear" in the classification scheme of VK, precluding an unambiguous transformation to a Langevin equation. If the average value of u is now assumed to be zero (corresponding to the choice of the moving coordinate system) one derives

0p(~.,t) _ &

I¢:

]

d~ I'(0,1:) %% (2Ax)2

-S'S + 0c2

(50)

(2Ax)2 0%

If one considers that the sure component of t] is zero when u 0 = 0 then one sees that E = E-r = E0, and equation (50) and equation (35) are identical. Of course, if v0 = 0 then equation (1) does not allow convective diffusion at all, but equation (50) does not contradict this fact; P(0,x) = 0 because u ~, v0. The methods of this paper may be extended to systems in which the flow is allowed to be two (or three) dimensional. The notation is made considerably more complex, but in the simplest form (equation (36)) generalization to higher dimensions is elementary; the sum over the c's merely extends to two dimensions. In particular, two indices are required, and for each cij two terms are present, one with the difference between neighboring concentrations in the x-direction, and one with a corresponding difference in the y-direction. It should be noted that the extension to higher dimensions also eliminates the condition on the uniformity of the fluid velocity. Thus, in the general case, the fluid velocity becomes a stochastic variable, and the generalization of equation (1) to higher dimensions involves a stochastic velocity even in the case of no adsorption. If a stochastic adsorption is relevant, then both stochasticities will be relevant. Although a differential equation for the time evolution of the probability density of the concentrations at different points has been derived, the resulting equation is rather complex, even for the simplest cases. Monte Carlo simulations of the time development of the probability density function via the stochastic Liouville equation are generally possible, however, and are a subject of our current research. For more complex situations, (e.g. higher dimensions, shorter sample net spacing, spatial dependence of the average velocity, nonlinearities, etc.) the complexity of the derived equations increases rapidly, making the derivations and results rather difficult to interpret, and thus reducing the utility of the method for casual groundwater studies. 5 Conclusions We arrive at the following conclusions: 1. We have applied the method of van Kampen for generating a master equation for the probability density of a stochastic variable to a problem in which the stochastic properties of the concentration are described by a spatially discretized linear contaminant transport equation. 2. In one dimension a representation of the equation describing the time evolution of the joint probability density for the discretized concentrations is in general of the form of a Fokker-Planck equation; translation to a coordinate system in which the average velocity is zero yields an equation which is in the form of a diffusion equation with variable coefficients.

267 3.

In higher dimensions the general form of the equation for the probability densities is not changed, but the complexity is increased greatly. 4. The stochastic convective solute velocity in one dimension involves a stochastic adsorption and possibly a random fluid convective velocity; in higher dimensions both a stochastic fluid velocity and a stochastic adsorption must, in general, be considered. 5. We have expressed the nonlinear stochastic partial differential equation that governs transport of the nonlinearly adsorbing solute in a multiplicative form, (5), amenable to application of the same treatment that we used in this paper. Analysis of the resulting master equation (to be derived in a future paper) should allow to test the hypothesis of Van Genuchten and Wierenga [1976] and Roberts et al. [1986] that nonlinear equilibrium sorption may explain the growth of the field-scale retardation factor. The complexity (and dimensionality) of our final results is overwhelming. We hope, however, that the derived equations have analytic or asymptotic solutions for some simplified cases. Our optimism is motivated by the existence of some analytic solutions for the diffusion equation with the spatially dependent diffusion [ex. Yates, 1990, 1992; Chrysikopoulos, 1991]. Although a simpler treatment of the linear problem addressed in this paper is possible, i.e. the derivation of the evolution equation for the mean concentration (instead of the evolution equation for the concentration probability density function), it will not yield any information about higher concentration moments, as opposed to the treatment used here. Moreover, this simpler approach can not be applied to nonlinear stochastic differential equations, as noted by van Kampen [1976] and Fox [1978]. We would like to close by pointing out that significantly higher complexity of master equations associated with the nonlinear stochastic partial differential equations shall be expected. Acknowledgment We acknowledge an anonymous reviewer for a helpful in-depth review. This research was partially supported by a grant from the Kearney Foundation of Soil Science. It was performed while Z.J. KabaIa and A. Hunt were an assistant professor and a postdoctoral researcher, respectively, in the Department of Soil and Environmental Sciences at the University of California, Riverside. References Ball, W.P.; Roberts, P.V. 1991: Long-term sorption of halogenated organic chemicals by aquifer material. 2. Intraparticle diffusion. Env. Sci. Technol. 25(7), 1237-1249 Breuer, H.P.; Petruccione, E 1993: A Master Equation of description of fluctuation hydrodynamics Physica A 192(4), 569-588 Brusseau, M.L. 1992: Factors influencing the transport and fate of contaminants in the subsurface. J. Hazard. Materials 32(2-3), 137-143 Brusseau, M.L.; Larsen, T.; Christensen, Th. 1991: Rate-limited sorption and nonequilibrium transport of organic chemicals in low organic carbon aquifer materials. Water Resour. Res. 27(6), 1137-1148 Brusseau, M.L.; Jessup, R.E.; Rao, P.S.C. 1989: Modeling the transport of solutes influenced by multiprocess nonequilibrium. Water Resour. Res. 25(9), 1971-1988 Chrysikopoulos, C.V. 1991: An analytical solution for one-dimensional transport in heterogeneous porous media - comment. Water Resour. Res. 27(8), 2163-2163 Cvetkovic, V.D.; Shapiro, A. M. 1990: Mass arrival of sorptive solute in heterogeneous porous media. Water Resour. Res. 26(9), 2057-2067 Dagan, G. 1989. Flow in transport in porous media, Springer-Verlag, New York

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