Averaged Description of Flow (Steady and Transient) and Nonreactive Solute Transport in Random Porous Media

We analyze an approach for finding the general forms of exactly averaged equations of flow and transport in porous media. At the core of this approach is the existence of appropriate random Green's functions. The existence of random Green's functions for physical linear processes is equivalent to the existence of some linear random operators for appropriate stochastic equations. If we restricted ourselves to this assumption only, we can study the processes in any dimensional bounded or unbounded fields and in addition, cases in which the random fields of conductivity and porosity are stochastically non-homogeneous, non-globally symmetrical, etc. It is clear that examining more general cases involves significant difficulty and constricts the analysis of structural types for the processes being studied. Nevertheless, we show that we obtain the essential information regarding averaged non-local equations for steady and transient flow, as well as for solute transport. We describe the general form for the exactly averaged system of basic equations of steady flow, non-steady transient flow and solute transport in arbitrary random bounded or unbounded domains of any dimension with sources and sinks. We examine the validity of the averaged descriptions and the generalized law for some nonlocal models. The approach does not require assuming the existence of any small parameters, for example, small scales of heterogeneity or small perturbation of conductivity field.