Averaging of Stochastic Equations for Flow and Transport in Porous Media
It is well known that at present exact averaging of the equations of flow and transport in random porous media have been realized for only a small number of special fields. Moreover, the approximate averaging methods are not yet fully understood. For example, the convergence behavior and the accuracy of truncated perturbation series are not well known; and in addition, the calculation of the high-order perturbations is very complicated. These problems for a long time have stimulated attempts to find the answer for the question: Are there in existence some exact general and sufficiently universal forms of averaged equations? If the answer is positive, there arises the problem of the construction of these equations and analyzing them. There are many publications on different applications of this problem to various fields, including: Hydrodynamics, flow and transport in porous media, theory of elasticity, acoustic and electromagnetic waves in random fields, etc. Here, we present a method of finding some general form of exactly averaged equations for flow and transport in random fields by using (1) some general properties of the Green s functions for appropriate stochastic problems, and (2) some basic information about the random fields of the conductivity, porosity and flow velocity. We present general forms of exactly averaged non-local equations for the following cases: (1) steady-state flow with sources in porous media with random conductivity, (2) transient flow with sources in compressible media with random conductivity and porosity; and (3) Nonreactive solute transport in random porous media. We discuss the problem of uniqueness and the properties of the non-local averaged equations for cases with some type of symmetry (isotropic, transversal isotropic and orthotropic), and we analyze the structure of the nonlocal equations in the general case of stochastically homogeneous fields.