fluid =·
vv ¼
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 ¹ v
x ·=c
x
, t
3a With the addition of the deterministic initial condition
c x
, t
o
¼ f
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
¼ 4a
dx t
dt ¼
v
[
x t
] 4b
Initial condition : c [
x t
o
, t
o
] ¼
f [
x t
o
] 4c
Eqn 4 represents an initial value problem, and integrating once leads to the solution
c [
x t
, t
] ¼
f [
x t
o
] 5a
x t
¼ x
t
o
þ Z
t t
o
dt v
[
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
] ;
Z
t t
o
dt v
[
x t
] 6
where L[xt] represents the displacement trajectory between the initial location xt
o
and the location at time
t given by xt Fig. 1. If we adopt xt as the independent variable, eqn 5 can be solved by specifying the location
xt, 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
¼ f
x ¹ L
[
x t
] 7a
x t
o
¼ x ¹ L
[
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 xt
o
at time t
o
such that it arrives at
the location x at time t is traced backwards to the initial time eqn 7b. Then, cx,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 t
o
¼ 0, and the initial location
set at x ¼ 0. Then, the displacement expressed by eqn 6 is given by Lx,t ¼ vt, and the traced-back location is given
by xt
o
¼ x ¹ vt for all time. The solution for the concen- tration specified by eqn 7a is given by the simple relation
cx,t ¼ fx ¹ vt. 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
i ¼ h f
x ¹ L
[
x t
] i
8 where h
… i represents 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 ¹ L[xt] that starts
from the point located at x at time t and traces back to the initial location at time t
o
, 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[xt] 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 xt
o
at time t
o
; the concentration at this traced back location i.e., the initial con-
dition is the solution for the location x at time t.
Connection between Lagrangian and cumulant expansion approaches 321
form of a path integral or functional integral Bogoliubov and Shirkov
4
, ch. 8; Feynman
20
; Feynman and Hibbs
21
, ch. 12. In the notation of Feynman this is
hc
x
, t
i ¼ Z
Q
f x ¹ L
[
x t
] p
L [
x t
] DL
[
x t
] 9
where Q represents the space of all possible paths between the deterministic location x at time t, and the random initial
location xt
o
at time t
o
. The quantity pL[xt] DL[xt]
represents the differential probability measure associated with a particular trajectory, subject to the normalization
condition Z
Q
p L
[
x t
] DL
[
x t
] ; 1
10 Although not explicitly indicated, the probability functional
pL[xt] 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 measure
34
, 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., Hopf
32
, so we will adopt the perspective of Feynman and Hibbs
21
that ‘‘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 xt
o
at time t
o
, 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
i ¼ Z
Q
Z
R
3
f x ¹ y
d y ¹ L
[
x t
] dy p
L [
x t
] 3 DL
[
x t
] ð
11Þ 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
i ¼ Z
R
3
dy p y; x
f x ¹ y
12
where y represents the displacement rather than the trajec- tory, as indicated in Fig. 1, and
p y; x
¼ h d
y ¹ L
[
x t
] i
13
Here the notation py;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 Simmons
49
, and a similar result has been presented by Dagan
13,15
. 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 py;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 by q
1
, q
2
, and q
3
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 by q
1
and q
2
that have coin- ciding points at times t
o
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
given by eqn 12.
322 B. D. Wood
2.2 Renormalized cumulant expansion approach
