A connection between the Lagrangian
stochastic–convective and cumulant expansion
approaches for describing solute transport in
heterogeneous porous media
Brian D. Wood
1Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, USA
(Received 6 January 1998; revised 2 June 1998; accepted 8 June 1998)
The equation describing the ensemble-average solute concentration in a heterogeneous porous media can be developed from the Lagrangian (stochastic–convective) approach and from a method that uses a renormalized cumulant expansion. These two approaches are compared for the case of steady flow, and it is shown that they are related. The cumulant expansion approach can be interpreted as a series expansion of the convolution path integral that defines the ensemble-average concentration in the Lagrangian approach. The two methods can be used independently to develop the classical form for the convection–dispersion equation, and are shown to lead to identical transport equations under certain simplifying assumptions. In the develop-ment of such transport equations, the cumulant expansion does not require a priori the assumption of any particular distribution for the Lagrangian displacements or velocity field, and does not require one to approximate trajectories with their ensemble-average. In order to obtain a second-order equation, the cumulant expansion method does require truncation of a series, but this truncation is done rationally by the development of a constraint in terms of parameters of the transport field. This constraint is less demanding than requiring that the distribution for the Lagrangian displacements be strictly Gaussian, and it indicates under what velocity field conditions a second-order transport equation is a reasonable approximation.q1998 Elsevier Science Limited. All rights reserved
Keywords: stochastic groundwater transport, cumulant expansion, stochastic-convective dispersion.
1 INTRODUCTION
Stochastic methods have become indispensable to the devel-opment of solute transport equations in heterogeneous por-ous media. This is due largely to the realization that the variations in the velocity field in the subsurface systems dominate the spreading of solutes as they are transported, and by the fact that the important scales of heterogeneity in the conductivity field (and, hence, the velocity field) can not be sufficiently resolved by measurement to provide a fully deterministic representation. A variety of stochastic
methods have been developed to describe the spreading of solutes from both the Lagrangian and Eulerian perspective. Much of the fundamental work in this area has been reviewed by Dagan13,14, Gelhar27, Cushman11. In particular, the work of Dagan13,14 provides a comprehensive review that describes the development of stochastic transport equations from both of the Eulerian and Lagrangian perspectives.
The Lagrangian approach to stochastic transport was developed in application to turbulent diffusion by Taylor52, and introduced for nonreactive transport in porous media by Dagan and Bresler17 and Simmons49. This approach has been widely applied to subsurface hydrology, and examples of such applications can be found in the work of Dagan15, Neuman and Zhang44, Cushman and Ginn10, Rajaram and
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Gelhar48, Cvetkovic and Dagan12, Zhang59,60and others. In the hydrology literature there have also been many efforts formulated from the so-called Eulerian perspective. One subset of this literature involves expansions that do not rely on regular perturbations and the infinite hierarchy of equations that results. In these works, the equation for the ensemble-average concentration is developed in terms of a perturbation series expansion; the resulting equations are not always strictly Eulerian because Lagrangian quantities are sometimes produced. Examples of these approaches can be found in Winter et al.57, Kabala and Sposito33, Neuman43, Zhang59,60, Kavvas and Karakas36, and Cushman and Hu8. In particular, we are interested in the
approach of Kabala and Sposito33 and more recently
Kavvas and Karakas37; in both works a cumulant expansion has been adopted for development of an ensemble-average expression for the solute concentration. The cumulant expansion described in this paper is renormalized by the ensemble-average velocity field, and this will be shown to have some advantageous properties.
Direct comparisons among the results for various stochastic methods are often difficult because of inherent differences the techniques that are used. When such com-parisons are available, they can be helpful in understanding and interpreting the results that are obtained. The purpose of this paper is to compare the Lagrangian stochastic–convec-tive and renormalized cumulant expansion methods. The Lagrangian stochastic–convective solution for the ensem-ble-average concentration is expressed as a convolution of the Lagrangian displacement probability density function (pdf) and the initial condition function. The pdf that arises must be determined by measurement, or by making assump-tions about the statistics of the Lagrangian displacements. If, under certain simplifying assumptions, the pdf is assumed to be Gaussian, then the conventional convection –dispersion equation can be recovered. We show that the cumulant expansion approach can be considered to be a series expansion of the convolution integral that arises from the Lagrangian stochastic–convective approach. The utility of the cumulant expansion is that it represents the solution in terms of the cumulants of the velocity field directly, without the assumption of a particular form for the pdf. Addition-ally, a constraint can be developed to indicate under what conditions the expansion can be rationally truncated at sec-ond order to yield a closed-form solute transport equation. When this constraint is met, the cumulant expansion also leads to the conventional convection–dispersion equation. This constraint is less restrictive than the requirement that the velocity field be strictly Gaussian, and has the additional advan-tage of indicating under what conditions the velocity field is close enough to Gaussian such that a second-order truncation reasonably approximates the governing transport equation.
The paper is organized as follows. In Section 2 the problem of pure convection in a random velocity field is set up, and solutions are obtained by the Lagrangian stochastic–convective method and by the cumulant expan-sion approach. In this section it is shown that the expectation
over Lagrangian trajectories leads to a convolution path integral formulation for the Lagrangian stochastic–convec-tive method, and that the renormalized cumulant expansion can be interpreted as a series expansion of this integral. In Section 3 the conventional convection–dispersion equation is developed starting from the results for each of the meth-ods that were developed in Section 2. Here, it is shown that under certain limiting assumptions the two approaches lead to results that are similar in form, and the approximations required for each of the approaches are examined and com-pared. In Section 4 these results are summarized. Although the importance of local dispersion under certain conditions has been well established9,16,35, in the interest of simplicity of notation and clarity in the developments, local dispersion is neglected. Neglecting local dispersion is not at all required for use of the renormalized cumulant expansion method (cf. Fox22, Kabala and Sposito33, Kavvas and Karakas36); it is neglected here only because it would com-plicate the comparison of these two methods without adding much to the meaning. Interested readers can find developments that include the effects of local dispersion from both the Lagrangian stochastic–convective perspective (e.g., Dagan15) and the cumulant expansion perspective22,36. A brief discus-sion of how includiscus-sion of the local disperdiscus-sion tensor alters the results obtained in this work is provided in Section 4.
2 COMPARISON OF THE METHODS
Although the stochastic–convective approach can be cast in terms of an Eulerian travel time perspective28,50we will be adopting a Lagrangian approach here. This approach has been described by Simmons49 and Dagan15, and we re-derive it from a slightly different perspective. To begin with, we consider the problem of solute transport in a steady flow field. We do not assume that the porosity or water content are necessarily constant; therefore, the velo-city field is not in general divergence-free (i.e.,=·v(x)Þ0).
We are interested mainly in the macroscopic spreading due to the velocity field variations, and as discussed above we will neglect local-scale dispersion for this example (cf. Cvetkovic and Dagan12, Simmons49, Simmons50). It is assumed that the solute initial distribution is known and that there are no solute fluxes on the boundaries so that the system may be treated as an initial value problem (cf. Dagan15). The conservation equations that apply for describing the solute transport and the steady flow of fluid at the Darcy scale can be determined by an appropriate upscaling technique1,6,7,31,42,47. These conservation equa-tions apply to either saturated (where vrepresents the por-osity) or unsaturated (wherevrepresents the water content) conditions, and are listed here as
solute
](vc)
fluid
=·(vv)¼0 (2)
In the development of eqn (1) and eqn (2) it has been assumed that the fluid density is constant. The porosity (or water content), velocity, and concentration are all inter-preted here in a macroscopic sense that is defined in the particular upscaling technique that is used to develop the Darcy-scale conservation equations. Eqn (1) and eqn (2) can be combined to give a single transport equation in which the porosity (or water content) has been elimi-nated
]c(x,t)
]t ¼ ¹v(x)·=c(x,t) (3a) With the addition of the deterministic initial condition
c(x,to)¼f(x) (3b)
the problem is fully specified.
2.1 Lagrangian stochastic–convective approach
Eqn (3) is easily cast in terms of a dynamical system where x is now considered to be an explicit function of time, and this system takes the form
dc[x(t),t]
dt ¼0 (4a)
dx(t)
dt ¼v[x(t)] (4b)
Initial condition : c[x(to),to]¼f[x(to)] (4c)
Eqn (4) represents an initial value problem, and integrating once leads to the solution
c[x(t),t]¼f[x(to)] (5a)
x(t)¼x(to)þ
Zt
to
dtv[x(t)] (5b)
Eqn (5a) is the mathematical statement that the solute concentration does not change in time if one adopts the coordinate system defined by the trajectories, and eqn (5b) defines the trajectories of the velocity field. In the developments that follow, we will make use of the defini-tion
L[x(t)];
Zt
to
dtv[x(t)] (6)
where L[x(t)] represents the displacement trajectory between the initial location x(to) and the location at time t given by x(t) (Fig. 1). If we adopt x(t) as the independent variable, eqn (5) can be solved by specifying the location x(t), that is fixed at the time and space coordinates for which we want a solution. These coordinates can be speci-fied as an ordered pair (x,t), allowing us to express eqn (5) more compactly as
c(x,t)¼f(x¹L[x(t)]) (7a)
x(to)¼x¹L[x(t)] (7b)
Some interpretation may be helpful at this point. Eqn (7) can be used to determine the solution of the initial value problem posed by eqn (5) as follows. First, we fix a point in space and time (x,t) for which we want a solution. The velocity field trajectory (the path at particle would follow from its initial location x(to) at time tosuch that it arrives at the location x at time t) is traced backwards to the initial time (eqn (7b)). Then, c(x,t) is given by the initial concen-tration at this traced back location, as indicated by eqn (7a) (Fig. 1). As a simple example, consider transport in one-dimension with a constant velocity given by v. Let the initial time be set such that to¼0, and the initial location set at x¼0. Then, the displacement expressed by eqn (6) is given by L(x,t)¼vt, and the traced-back location is given by x(to)¼x¹vt for all time. The solution for the concen-tration specified by eqn (7a) is given by the simple relation
c(x,t)¼f(x¹vt). Keeping this example in mind will help
in interpreting the developments that follow.
To find the expression that governs the ensemble-average concentration, we take the expectation of eqn (7a) to yield hc(x,t)i¼hf(x¹L[x(t)])i (8)
whereh…irepresents ensemble expectation. Eqn (8) can be interpreted as the average of a Lagrangian quantity, where the average is conducted over the members of the ensemble of velocity fields. Each member of the ensemble represents a different realization for the trajectory ¹L[x(t)] that starts from the point located at x at time t and traces back to the initial location at time to, as indicated in Fig. 2; in the material that follows we will refer to this quantity as the ‘backward’ trajectory or path. The ensemble-average given in eqn (8) represents an average over all possible backward paths to the initial time. Eqn (8) can be re-expressed in the
Fig. 1. The quantity L[x(t)] represents displacement along the trajectory defined by the velocity field. The solution to eqn (4) is found by fixing the point x at time t and tracing backwards along the trajectory to the corresponding point x(to) at time to; the concentration at this traced back location (i.e., the initial
form of a path integral or functional integral (Bogoliubov and Shirkov4, ch. 8; Feynman20; Feynman and Hibbs21, ch. 12). In the notation of Feynman this is
hc(x,t)i¼ Z
Qf(x¹L[x(t)])p(L[x(t)])DL[x(t)] (9) whereQrepresents the space of all possible paths between the deterministic location x at time t, and the random initial location x(to) at time to. The quantity p(L[x(t)]) DL[x(t)] represents the (differential) probability measure associated with a particular trajectory, subject to the normalization condition
Z
Qp(L[x(t)])DL[x(t)];1 (10) Although not explicitly indicated, the probability functional p(L[x(t)]) may in general also depend upon the location x (fixed at time t) from which the trajectory is traced back-ward. The main difficulty in the use of such functional integrals is that ultimately one needs to be able to describe in what sense the backward trajectories are measurable so that the probability measure can be defined. If the particle trajectories are composed of independent Gaussian displa-cements, Feynman’s formulation leads immediately to the Wiener measure34, and for many analyses this approxima-tion will be appropriate. Technical details aside, the form-alisms of integration in functional spaces has been used in hydromechanics for some time (e.g., Hopf32), so we will adopt the perspective of Feynman and Hibbs21that ‘‘if the problem is not Gaussian, it can at least be formulated and studied using path integrals.’’
The path integral in eqn (9) is defined in terms of the trajectories, and according to eqn (5) the concentration along these trajectories is constant. This implies that between any two points fixed at x at time t and x(to) at time to, it is only the total displacement (rather than the particular path) that influences the integral in eqn (9)
(Fig. 3). This path independence allows a particular simpli-fication of eqn (9) by rewriting it in terms of an integration over a delta function
hc(x,t)i¼ Z
Q Z
R3f(x¹y)d(y¹L[x(t)])dy p(L[x(t)])
3DL[x(t)] ð11Þ
Eqn (9) and eqn (11) are equivalent; if eqn (11) is inte-grated first with respect to y, then eqn (9) is immediately recovered. However, upon interchanging the order of inte-gration, one finds that the ensemble-average concentration can be defined in terms of a new probability function that involves only the displacements y
hc(x,t)i¼ Z
R3dy p(y; x)f(x¹y) (12)
where y represents the displacement rather than the trajec-tory, as indicated in Fig. 1, and
p(y; x)¼hd(y¹L[x(t)])i (13) Here the notation p(y;x) is used to indicated that the prob-ability density function may depend upon the location x in the field from which the trajectory is traced back. Eqn (12) is the Lagrangian stochastic–convective form of Simmons49, and a similar result has been presented by Dagan13,15. Note that the simplification offered by eqn (12) over the path integral formulation given by eqn (9) is only apparent in that the particle trajectories are still required for the determination of p(y;x) as indicated by eqn (13). Also note that if the concentration were dependent upon the particular trajectory that connects two points in space (such as would be the case if spatially heterogeneous or nonlinear reactions were present), then a simplification of the form of eqn (12) would not be possible.
Fig. 2. Three realizations of trajectories (indicated byq1,q2, and
q3) that are located at point x at time t. For each such realization, the starting location (and hence concentration) is a random vari-able. The ensemble average is defined path-wise, so the resulting expectation can be interpreted in the most general case as a path
integral.
Fig. 3. An example of two trajectory realizations (where two separate realizations are indicated byq1andq2) that have coin-ciding points at times to and t. Because in the nonreactive case concentration is conserved along trajectories, the solution given by eqn (9) does not depend upon the particular trajectory taken between these two points; rather, it is only the total displacement given by the vector y that matters. This trajectory invariance allows the transformation of the trajectory-dependent form given by eqn (9) to a form that depends only upon the displacement, as
2.2 Renormalized cumulant expansion approach
The cumulant expansion method, pioneered by Kubo37, has been used in the study of multiplicative stochastic processes by physics community for some time23–26,53–55,58. This approach has been applied to subsurface hydrology by Sposito and Barry51, Kabala and Sposito33, and Kavvas and Karakas36. Here our objective is to use the cumulant expansion in combination with the definition of an operator exponential to derive a differential equation for transport directly from the ensemble-average solution to the transport problem given by eqn (8).
We begin by noting that in eqn (8) the coordinates of the initial condition function f(x) are translated backwards along trajectories of the velocity field by the amount ¹ L[x(t)]. Eqn (8) can be rewritten in terms of an operator exponential whose effect is to translate the function it oper-ates on along trajectories of the velocity field, and takes the form56
The series that appears in the second line of eqn (14) is a
Lie series29,45, and the differential operators involved
oper-ate on all terms to the right of them. The definition provided by eqn (14) technically requires that a Taylor series for f(x) exists (i.e., f(x) must be analytic). However, because we can find analytic functions that approximate most of the discontinuous initial conditions that would be of interest (e.g., a step pulse) as closely as we would like, this presents no real limitation. Eqn (14) can be proved in a straightfor-ward manner by integrating both sides of eqn (3) to deter-mine an implicit solution, and then iterating this solution. Note that in eqn (14) it is the Eulerian velocity that appears in the exponential; the Lie series automatically generates the trajectories from the Eulerian velocity field and its derivatives. One can think of the exponential in eqn (14) as the product of many infinitesimal displacement opera-tors; these displacements generate the Lagrangian particle path in terms of only the Eulerian velocity field. An illus-trative example showing how such a trajectory is generated for finite time intervals is provided in Appendix A.
Combining eqn (8) and eqn (14), one can now express the ensemble-average concentration as the expectation of an operator acting on the initial condition
hc(x,t)i¼ exp ¹
Here f(x) has been removed from the expectation operator because it is assumed to be a deterministic function.
It is clear from eqn (14) how eqn (15) could be expanded in terms of moments by simply applying the expectation operator to each of the terms in the series. However, such
an expansion would include terms proportional to powers of t and leads to time secularities; that is, any finite number of terms of the expansion will provide a good approximation to the series for only sufficiently small times. To rectify this problem, we consider a renormalized cumulant expansion that expands eqn (15) in terms of powers of the Lagrangian correlation time of the velocity field perturbations53–55. The advantage of such an expansion is that if the correlation time is small enough, the series is well approximated by the first few terms for all time.
As a first step towards developing a renormalized cumu-lant expansion, we decompose the velocity field as the sum of a deterministic component and a random component
v(x)¼vo(x)þ˜v(x) (16)
where vo(x) is a deterministic velocity field, and ˜v(x) repre-sents a perturbation from the deterministic field. Although vo(x) can be taken as any deterministic function that proves to be useful, it is often convenient to take it to be the ensemble-average velocity. In this case, eqn (16) becomes
v(x)¼hv(x)iþ˜v(x) (17)
eqn (15) can now be written in terms of this decomposition
hc(x,t)i¼ exp ¹ The operator exponential of the sum can be expressed as the product of exponentials. Because in eqn (18) the opera-tors do not in general commute, the exponential is some-what more complicated than if only scalar functions were involved. However, there is a theorem that allows one to express the exponential of a sum of operators as the product of time-ordered exponentials19,40,41. In Appendix B we provide a simple development that proves this theorem for continuous velocity fields, and the following identity results (cf. Feynman19, Masani40,41, van Kampen55)
exp[¹(t¹to)hv(x)i·=¹(t¹to)˜v(x)·=]
Here, T← indicates that the exponential is
time-ordered19,24,37,53. In this case, the ordering symbol indicates
To simplify the notation, in the subsequent developments we set
U(þo)(to,t)¼exp[(t¹to)hv(x)i·=] (20a)
U(¹o)(t,to)¼exp[¹(t¹to)hv(x)i·=] (20b)
where the explicit dependence on space has been sup-pressed. These operators have the effect of shifting the functions or operators that they act upon along the trajec-tories of the ensemble-average velocity (either forward or backward depending on which of the two operators acts) and are often referred to as propagators2,56. It is important to note that although ˜v(x) is the Eulerian velocity field perturbation, the action of the propagator U(o)þ is to shift ˜v(x) forward along the trajectory formed by the ensem-ble-average velocity field, and this will ultimately render it Lagrangian.
With the identity provided by eqn (19) and the definitions provided by eqn (20a) and eqn (20b), the solution given in eqn (18) can be rewritten in the form
hc(x,t)i¼U(¹o)(t,to)
A similar expression has been derived by Kubo and
Hashitsume39 for a Brownian motion model of atomic
spins. Eqn (21) has essentially the same form as eqn (15); the difference is that eqn (21) is renormalized in the sense that the ensemble mean velocity field action has been removed from the random velocity field. The only random quantities left in eqn (21) are the velocity perturba-tions from the ensemble-mean velocity field. Much like eqn (15), an expansion of the exponential followed by the appli-cation of the averaging operator at this point will lead to an expansion that has secular terms. This problem can be rec-tified by considering an expansion in terms of time-ordered cumulants. A cumulant expansion can be considered a rear-rangement of the terms in a moment expansion that improves the series convergence. The expansion has been studied extensively in the physics literature, and we can
define the time-ordered cumulant expansion24,37,53 by the
following expression
whereh…icdefines the cumulant operator. The relationship between the cumulants and the moments can be found by expanding the right-hand sides of eqn (21) and eqn (22) and equating terms of the same power. The general relationship
between time-ordered cumulants and moments for all orders have been presented by van Kampen53 and Fox24; here we list only the first two such relations. First, let
B(to,t)¼U(þo)(to,tÞh˜v(x)ic·=U(¹o)(t,to) (23) where the dependence on space has been suppressed. Then the first two cumulants can be expressed in terms of moments by
hB(to,t)ic¼hB(to,t)i (24a)
hB(to,t)B(to,h)ic¼hB(to,t)B(to,h)i
¹←ThB(to,t)ihB(to,h)i ð25bÞ
Noting that for our choice of decomposition given in eqn (17), we must have thath˜v(x)i;0, so that eqn (22) can be
Eqn (12) and eqn (26) were both developed directly from eqn (8), and must therefore represent the same solution. However, in eqn (26) we obtain an expansion in terms of the velocity perturbations rather than a convolution over the displacement pdf. Note that to this point no assumptions regarding the statistics of the underlying velocity field have been imposed; therefore, this expression applies equally well to spatially stationary and nonstationary velocity fields.
3 DEVELOPMENT OF THE CONVENTIONAL CONVECTION–DISPERSION EQUATION
the assumptions that are made in that development. The first such assumption is that the porosity (v) of eqn (2) is treated as a constant, and this implies that=·v(x);0.
3.1 Lagrangian stochastic–convective development
Some of the techniques used above in the development of the renormalized cumulant expansion can also be employed to derive the transport equation from the conventional Lagrangian perspective. We begin with eqn (8), and express the translation of the initial condition in terms of a Taylor series in the Lagrangian displacements L[x(t)]. The result is
hc(x,t)i¼f(x)¹hL[x(t)]i·=f(x)þ 1
2!hL[x(t)L[x(t)]i
:==f(x)þ… ð27Þ
where LL:==is Gibbs notation for the contraction
LiLj
Eqn (27) can be expressed in a notation similar to that used in eqn (15)2
hc(x,t)i¼hexp(¹L[x(t)]·=)if(x) (28)
However, there is a crucial difference between the previous representation given in eqn (15) and the representation given in eqn (28). In eqn (15) it is the Eulerian velocity that appears in the exponential; the trajectories are gener-ated automatically by the group property of the convection operator. In that formulation, the exponential is expanded in a Lie series where the operator acts iteratively upon itself. In eqn (28) the motion along the trajectory is accounted for directly by the use of a Lagrangian velocity function. Therefore, one can expand in a conventional Taylor series using the (presumably known) displacements as done in eqn (27).
Like eqn (15), eqn (28) can be expanded in terms of cumulants. The cumulants are defined by
hc(x,t)i¼exp ¹hL[x(t)]ic·=þ
Because there is no time-ordering and the displacements freely commute with themselves, the cumulants defined in eqn (29) are related to the moments in a simpler manner than for the time-ordered case24. The first two cumulants are listed here as
hL[x(t)]ic¼hL[x(t)]i (30a)
hL[x(t)]L[x(t)]ic¼hL[x(t)]L[x(t)]i¹hL[x(t)]ihL[x(t)]i
(30b) Eqn (29) is an expression involving derivatives of infinite order; similar such results have been obtained previously by Simmons49,50 and Zhang60. Dagan15 states that if
dis-placements specified by L[x(t)] are Gaussian, then the expansion given in eqn (29) truncates identically at second order. In fact, if L[x(t)] is a Gaussian random vari-able the series in eqn (29) does truncate at second order, and we avoid the difficulties associated with discarding potentially secular terms. However, it is difficult to deter-mine under what conditions L[x(t)] can be exactly Gaus-sian. Weinstock56 has suggested that the Lagrangian displacements can never be exactly Gaussian for finite times. This does not preclude the distribution from being near Gaussian in some sense (as it must approach Gaussian for large times as dictated by the central limit theorem), but in the usual approach to this problem there are no guide-lines to indicate the conditions such that the Gaussian approximation is valid. With these limitations in mind, we will keep with the assumption made by Dagan15 and require that L[x(t)] is Gaussian and spatially stationary. This simplifies eqn (29) to
]hc(x,t)i
Substituting the definition of the displacements given by eqn (6) into eqn (31), and using the definition of the second cumulant given by eqn (30) and the velocity decom-position given by eqn (17), one obtains after some algebra
]hc(x,t)i
where vo;]hLi/]t and represents the (constant) ensemble-average velocity. Eqn (32) is not useful in its present form because it still contains the unknown trajectory realizations x(t). At this point, we make a final assumption that the displacements L[x(t)] defined in eqn (6) can be approxi-mated by replacing the unknown trajectory x(t) by the ensemble-average trajectory hx(t)i,
hx(t)i¼x(to)þ
Zt
to
dthv[x(t)]i¼x(to)þ(t¹to)vo (33)
As pointed out by Dagan15, this approximation is similar to the first-order terms of an iterative scheme for homogenous turbulence suggested by Phythian46. However, Neuman and Zhang44have criticized the use of this lowest-order approx-imation, and no restrictions that define the conditions such that this lowest-order approximation is valid appear to be available. Again, to be consistent with previous studies, we adopt this assumption with full realization of its limitations. Eqn (32) can then be expressed in the form
and this result is identical to the result of Dagan13,15. This takes the form of the conventional convection –dispersion equation by setting To be absolutely clear about the assumptions involved in this development, we recall that they are (1) the porosity can be treated as a constant, (2) the Eulerian velocity field is steady, (3) the displacements L[x(t)] are Gaussian, (4) the Lagrangian velocity field is spatially stationary, and (5) we can replace x(t) by its lowest-order approximationhx(t)i given by eqn (33).
3.2 Renormalized cumulant expansion development
Our goal is now to develop an expression similar to eqn (36) by the cumulant expansion approach rather than the Lagrangian stochastic–convective one. We will begin with eqn (26) as the starting point, and again adopt the assumptions that the porosity is constant and the velocity field is spatially stationary. Under these conditions, the meaning of the operators involved in eqn (26) becomes more transparent. When spatial stationarity holds, the opera-tors U(o)þ and U(o)¹ are independent of location, and they freely
commute with the gradient operator. Recall that the action of the operator U(o)þ is to shift the spatial coordinate of the
functions on which it operates forward along the trajectory of the ensemble-average velocity field29,45. It is easy to show that under the conditions of spatial stationarity, we have the relationships
U(þo)(to,t)˜v(x)·=U(¹o)(t,to)¼˜v(U( o)
þ(to,t)x)·=
¼˜v[xþ(t¹to)vo]·= ð37Þ
where here vois the constant ensemble-average velocity. Using the shifting property of the propagators, we can work out the effects of these operators in the exponential on the right-hand side of eqn (26). Although eqn (26) is somewhat complicated, by starting at the right-hand side and applying each of the propagators inside the brackets to everything to the right of them, we quickly arrive at the simplified expression
Note that even though the velocity that appears in eqn (26) is the Eulerian field, the propagators induce a Lagrangian
representation by transforming the coordinate system to the Lagrangian one.
Eqn (38) has some advantages over the expansion in moments (that can be generated immediately from eqn (15)), or the Lagrangian expansion given in eqn (29). Due to the so-called cluster property of cumulants24,37, a cumulant becomes negligible (relative to its maximum value) as soon as any two terms in a particular cumulant are separated by more than the Lagrangian correlation time tc. This is useful for making order-of-magnitude estimates for the terms that appear in eqn (38), and van Kampen54,55 uses this property to develop the estimate ttnc¹1(˜v(x)·=)nfor
the nth-order term in the cumulant expansion. We can use these order-of-magnitude estimates to develop a constraint that indicates under what conditions a second-order trunca-tion of eqn (38) is appropriate. For example, if we require that the third-order and higher terms in eqn (38) be negli-gible relative to the second-order terms, we can impose the restriction54,55
and this leads to a constraint of the form
O[tc(˜v(x)·=)]pI (40)
where I is the identity operator. Realizing that in eqn (38) all operations are ultimately on the initial condition f(x), one can further refine this constraint to be
O tcv Lf
p1 (41)
where v is a characteristic measure of the velocity pertur-bations and Lfrepresents a characteristic length for changes in the initial condition field. Here we have estimated the gradient of f(x) as O(Df/Lf), and assumed that at most Df,O(f) in the development of the inequality (eqn (41)).
The dimensionless group on the left-hand side of the inequality (eqn (41)) is conventionally known as the Kubo number38,39. Similar such constraints have been developed for stochastic matrix processes by Fox25, and in the analysis of turbulent dispersion by Weinstock56. Under the condition that the constraint given by eqn (41) is met, then only the first two cumulants in eqn (38) are non-negligible, and the solution may be approximated to second order by the expression
hc(x,t)i¼U(¹o)(t,to)
are provided in Appendix C. The results derived in this appendix lead to a transport equation of the form
]hc(x,t)i
]t ¼ ¹vo·=hc(x,t)iþ Zt
to
dh
3h˜v(x)˜v(x¹[t¹h]vo)i:==hc(x,t)i ð43Þ
To be consistent with the work of Dagan15 we have
assumed that the Lagrangian velocity covariance function is spatially stationary. This assumption is equivalent to neglecting an additional term that represents the effects of solute plume shearing36. When it is not neglected, this additional term modifies the ensemble-average velocity field; it is described in more detail in Appendix C.
Some comments about the differences in the assumptions required for developing eqn (34) and eqn (43) can be made here. Although the basic assumptions of steady flow, con-stant porosity (or water content), and spatially stationary velocity field are adopted for both expressions, in eqn (43) the requirement of Gaussian displacements has been replaced by the constraint of eqn (41). This has the advan-tage of providing an explicit statement as to what conditions must be met such that a solution of the form of eqn (43) may be used. This can be important in application. Even if the Lagrangian velocity field is not truly Gaussian, the con-straint provides a notion of how close to Gaussian the dis-tribution needs to be for a second-order equation of the form of eqn (43) to suffice. Additionally, the assumption about the replacement of the particle trajectories by the ensemble mean is not required in the renormalized cumulant expan-sion approach. As mentioned previously, this assumptions has been criticized by Neuman and Zhang44, so eliminating it is a desirable result.
It is clear that eqn (43) is not quite identical to the Lagrangian result given by eqn (34). One can see immedi-ately that the two results become identical if we adopt the assumption that, rather than fixing x at any location for which we desire a solution, it is fixed only at the position dictated by the ensemble-average trajectory (i.e., by requir-ing that x¼xðtoÞ þ(t¹to)vo). However, this amounts only to a (somewhat restrictive) change of variables that recasts the problem from a backward to a forward tracking of the trajectories. Eqn (43) can be put in the classical form with-out further modification by setting
D(t)¼ Zt
to
dhh˜v(x)˜v(x¹[t¹h]vo)i (44)
Note that the expression for the dispersion tensor given in eqn (44) is no more complicated that the expression that is conventionally used in Lagrangian analyses give by eqn (35). In eqn (35), the covariance function is defined by tracing forward in time from the initial condition along the trajectories of the ensemble-average velocity field. In eqn (44), the covariance function is defined by tracing backwards in time along the velocity field trajectories from a fixed location x at time t. This means that to calcu-late the effective dispersion tensor given in eqn (44) one must know the Lagrangian covariance function for a point
fixed at location x at time t and traced backward to the initial time. However, as opposed to the problem that arises in eqn (32) (where the quantity x(t) arises and has an unknown value), in eqn (44) the location x is known, so this calculation can be carried out explicitly.
4 SUMMARY AND CONCLUSIONS
The stochastic–convective and cumulant expansion meth-ods are clearly connected, as can be seen from the develop-ments above. For the conditions described here, the cumulant expansion can be interpreted as an expansion of the convolution integral that appears in eqn (9). By the developments of Section 3, it is clear that under certain limiting conditions one can obtain identical solute transport equations directly from the stochastic–convective equation for the ensemble-average concentration, or from the equiva-lent cumulant expansion.
One might ask the question as to why the cumulant expansion is needed at all. If the desired end-result of our efforts is to derive an ensemble-average transport equation as done in Section 3, then the renormalized cumulant expan-sion provides some advantages.
(1) The renormalized cumulant expansion is not limited to certain distributions for the trajectory displacements or Lagrangian velocities. The development in terms of a renor-malized cumulant expansion is applicable regardless of the distributions involved as long as the expansion is conver-gent. More useful is the case when a correlation time constraint for the perturbations (given by the inequality eqn (41)) is valid, indicating that the cumulant expansion can be truncated at second order. Note that this is less restrictive than assuming that the pdf of the Lagrangian velocities is strictly Gaussian; this assumption indicates only that the pdf can be well approximated by the first two cumulants of the velocity field provided certain physical constraints are met. Therefore, the assumption of a Gaussian velocity field is replaced by a restriction that suggests under what conditions a second-order solute transport equation will be a reasonable approximation. Higher-order equations for the ensemble-average concentration can be developed by retaining more terms in the truncation of the cumulant expansion, although the utility of such higher-order expan-sions needs to be investigated further.
(3) Although not discussed here, the renormalized cumu-lant expansion allows a development of the ensemble-aver-age transport equations from Darcy-scale equations that do not neglect local dispersion (cf. Kavvas and Karakas36). Dagan13,15 also presents a Lagrangian development that allows for local dispersion (treated as a Brownian motion component of the velocity field), and develops a convec-tion–dispersion equation that includes this local dispersion term. Under the same assumptions that are applied to Dagan’s analysis, it is possible to show that a convection– dispersion equation of identical form can be garnered from a cumulant expansion approach36. In this case, the dispersion tensor given by eqn (44) is modified only by the addition of a local diffusion tensor.
(4) For reactive flows that are spatially inhomogeneous, the stochastic–convective approach leads to a solution that involves potentially complicated path integrations, as dis-cussed in Section 2. Because the renormalized cumulant expansion approach is a series expansion of these integrals, it results in solutions that are tractable. A publication that details how the cumulant expansion is carried out for reac-tive transport is presently in process.
Finally, we point out that there are very few restrictions that have been imposed on the development of the transport eqn (26) by the cumulant expansion method. Consequently, eqn (26) should apply equally well to spatially nonstationary velocity fields as is does to the case of the stationary velocity field examined in Section 3. Additionally, in nonstationary fields a shear effect may arise that modifies the ensemble-average velocity field, and this effect is captured by the cumulant expansion approach (as described in Appendix C). Although the cumulant expansion method has been applied to more complex nonstationary and non-steady flow fields33,36, it still represents a challenging area for con-tinued research.
NOMENCLATURE
c(x,t) solute concentration (mol/m3)
D(t) effective dispersion tensor as defined by eqn
(35) or eqn (44) (m2/s)
f(x) initial solute concentration distribution function (mol/m3)
Lf characteristic length-scale associated with the
concentration field (m)
L[x(t)] displacement trajectory, defined by eqn (6) (m) p(L[x(t)]) probability functional for trajectories, as defined
in eqn (9)
p(y;x) probability function for displacements, defined
by eqn (13)
←
T time ordering symbol, as defined in eqn (19)
Uþ
(o)
(to,t) forward propagator or translation operator, defined by eqn (20a)
U(o)¹(t,to) backward propagator or translation operator, defined by eqn (20b)
v(x) Darcy-scale fluid pore velocity (m/s)
hv(x)i ensemble-average fluid pore velocity (m/s)
˜v(x) perturbation to the fluid pore velocity, defined in eqn (17) (m/s)
vo constant ensemble-average fluid pore velocity
(m/s)
v characteristic measure of fluid pore velocity
perturbations, defined in the inequality
(eqn (41)) (m/s)
x(t) trajectory of the steady velocity field, defined by eqn (5) (m)
y net displacement vector, defined as in Fig. 1
v(x) porosity (saturated systems) or water content
(unsaturated systems)
tc Lagrangian correlation time for the velocity
field (s)
ACKNOWLEDGEMENTS
This research was supported in part by Pacific Northwest National Laboratory (Laboratory Directed Research and Development). Pacific Northwest National Laboratory is operated for DOE by Battelle Memorial Institute under con-tract DE-AC06-76RLO 1830. I gratefully acknowledge M.L. Kavvas (University of California, Davis) for introdu-cing me to cumulant expansion techniques and providing extensive discussions on the topic. A. Fannjiang, T.R. Ginn, and J. Gravner (University of California, Davis) provided helpful feedback and discussion on many miscellaneous topics. W. Kollmann (University of California, Davis) pro-vided a number of useful references on the topic of product integration. I am particularly indebted to R.F. Fox (Georgia Institute of Technology) for many helpful discussions on the applications of time-ordered cumulants, and for help in establishing the correspondence of the Lagrangian and
cumulant expansion approaches. Three anonymous
reviewers provided constructive comments that improved the paper’s presentation.
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APPENDIX A TRAJECTORIES GENERATED FROM THE OPERATOR EXPONENTIAL
In the mathematics and physics literature, integrals of the form of eqn (14) have been studied for some time (starting with the work of Volterra), and are referred to as product integrals3,18,30. The example that follows illustrates how the Lagrangian trajectories are generated by the exponential of the Eulerian velocity field, and is meant to be motivational rather than mathematically rigorous. We begin by discretiz-ing the integral in eqn (14) into a sum over N small time intervals Dti. Then, the familiar property of exponentials allows us to represent this sum as a product of the form
exp Zt
to
dtv(x)·=
¼lim
Dt→0 N→`
YN
i¹1
exp[¹Dtiv(x)·=]
(A1)
Now, for purely illustrative purposes, consider the case where only two such discrete displacements occur, as illu-strated in Fig. 4. Then, taking the time intervals to be equal and using the properties of such operator exponentials29,45
(A2) so that the action of these two exponentials is in fact to
trace backwards to the initial condition. This representation becomes exact asDt approaches zero. Note that the trans-lating action of the two operator exponentials is easy to prove by a simple Taylor series expansion. The important point is that it is only the Eulerian velocity field that appears in the exponential in eqn (A2), and that the trajec-tories of the velocity field are generated automatically by this exponential form.
APPENDIX B RENORMALIZED CUMULANT EXPANSION FORM
We begin with the equations of transport given by eqn (3) and eqn (3)
]c(x,t)
]t ¼ ¹v(x)=·c(x,t) (B1a)
Initial condition : c(x,to)¼f(x) (B1b)
Making the decomposition
v(x);hv(x)iþ˜v(x) (B2)
and using the definition of the operator exponential, eqns (B1) can be integrated directly to yield
c(x,t)¼exp[¹(t¹to)hv(x)i·=¹(t¹to)˜v(x)·=]f(x)
(B3)
Now, if the velocity field that appears in Eq. (B1a) were spatially stationary, one could adopt a new coordinate sys-tem that moves with the velocity of the ensemble-average flow field. One can still adopt a transformation of variables that accomplishes much the same thing for more general flow fields. This is accomplished by passage to the so-called interaction representation19,24 which is defined by
c(x,t)¼exp[¹(t¹to)hv(x)i·=]u(x,t) (B4)
Fig. 4. The discrete approximation to a backward trajectory for two time steps. For each small time intervalDt, the Lagrangian velocity is approximated by a constant vector (whose value is given below the vector). The location along the trajectory for the previous time step can be approximated with this constant velocity (these values are given near the filled circles). Making a succession of such steps leads to an approximation for the entire trajectory, and this approximation approaches the exact trajectory asDt approaches zero. The trajectory represented in this manner can be expressed entirely in terms of the Eulerian velocity field v(x), as shown for the two-step approximation to
Taking time derivatives of both sides of eqn (B4) yields the
Substituting this result into the left-hand side of Eq. (B1a) along with the decomposition given by eqn (B2) yields after some algebra
]u(x,t)
]t ¼ ¹exp[(t¹to)hv(x)i·=]˜v(x)·=
3exp[¹(t¹to)hv(x)i·=]u(x,t) ðB6Þ
There is one important complication that arises in defining the transformed equation given by (B6). The coefficient of u(x,t) on the right-hand side of eqn (B6) is now a function of time, and is in general it represents a non-commuting operator36. Eqn (B6) can be integrated by the use of a
time-ordered exponential19,24,37,53, which can be thought of as
the operator version of an integrating factor. Integrating and returning to the original variable c (by inverting eqn (B4)) results in
where T← indicates that the exponential is
time-ordered 19,24,37,53. The ordering symbol indicates that operators with the earliest time index (i.e.,tclosest to to) occur to the far right in positional notation, whereas the operators with the latest time index are to the far left. Comparing the solutions given by eqn (B3) and eqn (B7), one finds a (somewhat simplified) version of the disentan-gling theorem proposed by Feynman19and Masani40,41.
exp[¹(t¹to)hv(x)i·=¹(t¹to)˜v(x)·=]
This is eqn (19) in the main body of the paper.
APPENDIX C DEVELOPMENT OF THE DIFFERENTIAL EQUATION
By design we are interested in obtaining the differential
equation for transport of the ensemble-average concentration. However, eqn (42) (repeated here for convenience)
hc(x,t)i¼U(¹o)(t,to)
is not in differential form. To obtain the differential equa-tion, we begin by noting the time derivative of U(o)¹ acting
on a function g(x,t) is given by a straightforward applica-tion of the chain rule
]
Taking the time derivative of both sides of eqn (C1), and use of the chain rule given in eqn (C2) leads to
]hc(x,t)i
Comparison of the first term on the right-hand side of eqn (C3) with eqn (C1) indicates that this term is simply
¹ hv(x)i·=hc(x,t)i. For the second term, we insert the identity
U(þo)(to,t)U(¹o)(t,to)¼I (C4)
concentration. Incorporating the results of these two sim-plifications yields the expression
]hc(x,t)i
]t ¼ ¹hv(x)i·=hc(x,t)i
þU(¹o)(t,to)
Zt
to
dhh˜v[xþ(t¹to)vo]
·=˜v[xþ(h¹to)vo]ic·=
U(þo)(to,t)hc(x,t)i
ðC5Þ
Finally, applying the propagators U(o)þ and U(o)¹, we get the
local equation
]hc(x,t)i
]t ¼ ¹hv(x)i·=hc(x,t)i
þ Zt
to
dhh˜v(x)·=˜v[x¹(t¹h)vo]ic·=
hc(x,t)i ðC6Þ
This result is identical to the result of Kavvas and Karakas36 for the case of a spatially stationary velocity field and a negligible the local dispersion coefficient. Note that
because the gradient operators act on all terms to the right, the term in brackets on the right-hand side of eqn (C6), by the chain rule will generate two
terms. The first of these terms is of the form
h˜v(x)·(=˜v[x¹(t¹h)vo])ic·= hc(x,t)i and represents a shear effect that modifies the ensemble-average velocity field. Such terms are not unique to this study (cf. Wein-stock56, Appendix C; Fox22, eqn (42)), and for the case of incompressible flow with a spatially stationary Lagrangian covariance function this terms is identically zero22,60. To be consistent with the work of Dagan15, we will assume that the Lagrangian covariance function is spatially stationary. The second term that arises from the application of the chain rule is of the form of the conventional Green-Kubo macrodispersion tensor5, and leads to the definition of the classical convection–dispersion equation
]hc(x,t)i
]t ¼ ¹vo·=hc(x,t)iþ
Zt
to
dhh˜v(x)˜v(x¹[t¹h]vo)i
:==hc(x,t)i ðC7Þ
