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Exact Averaging of Stochastic Equations for Flow in Porous Media
Mark Shvidler and Kenzi Karasaki
Earth Sciences Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd, MS 90-
1116, Berkeley, CA 94720; Mshvidler@lbl.gov, email@example.com
It is well known that at present, exact averaging of the equations for flow and transport in
random porous media have been proposed for limited special fields. Moreover, approximate
averaging methods-for example, the convergence behavior and the accuracy of truncated
perturbation series-are not well studied, and in addition, calculation of high-order perturbations
is very complicated. These problems have for a long time stimulated attempts to find the answer
to the question: Are there in existence some, exact, and sufficiently general forms of averaged
equations? Here, we present an approach for finding the general exactly averaged system of
basic equations for steady flow with sources in unbounded stochastically homogeneous fields.
We do this by using (1) the existence and some general properties of Green's functions for the
appropriate stochastic problem, and (2) some information about the random field of conductivity.
This approach enables us to find the form of the averaged equations without directly solving the
stochastic equations or using the usual assumption regarding any small parameters. In the
common case of a stochastically homogeneous conductivity field we present the exactly
averaged new basic nonlocal equation with a unique kernel-vector. We show that in the case of
some type of global symmetry (isotropy, transversal isotropy, or orthotropy), we can for three-
dimensional and two-dimensional flow in the same way derive the exact averaged nonlocal
equations with a unique kernel-tensor. When global symmetry does not exist, the nonlocal
equation with a kernel-tensor involves complications and leads to an ill-posed problem.
Key words: heterogeneous porous media, random, flow, exact, averaging, nonlocal
Recently, methods for analyzing flow and transport in random media have been finding ever-
widening applications. An effective description of flow and transport in irregular porous media
entails interpreting permeability or conductivity fields as random functions of spatial
coordinates, and flow velocity as a random function of spatial coordinates and time. Such a
description also involves averaging of the stochastic system of flow and transport equations
containing these functions (conservation laws, Darcy's law, and closing relations). The averaging
problem involves finding the relationship between the nonrandom functionals of the unknown
and the given fields-means, variations, distributions, densities, etc.-or a closed set of
equations that contain these functionals. A certain interest attaches to the equations for the
averaged functionals that are the laws of conservation of mass, momentum, and energy, which
are invariant with respect to some set of conditions that uniquely determine the process (for
example, the initial and boundary conditions). This is fundamentally possible, for example, in
cases where the length scales of heterogeneity are extremely small. In physics and mathematical
literature, this phenomenon is sometimes referred to as self-averaging (Lifshitz et al., 1998) and
homogenization (Zhikov et al., 1994)
It is apparent that (in general) this is impossible, because in practical situations the process
depends on a set of parameters that are usually not small, and thus an averaged description is
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Karasaki, Kenzi; Shvidler, Mark & Karasaki, Kenzi. Exact Averaging of Stochastic Equations for Flow in Porous Media, article, March 15, 2008; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc895182/m1/1/: accessed November 19, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.