Stochastic Modeling in Composite Multiscale Porous Media

Most natural porous media are highly heterogeneous with hydraulic parameters, especially hydraulic conductivity, varying on multiple spatial scales. Darcy's Law has been used successfully to predict groundwater flows across a very wide range of experimental scales; however, different scales generally require different parameterizations, even within the same site. This may be due to the increasing levels of material heterogeneity that are sampled as the volumes used to average parameters expand. The problem of deriving average parameterizations is compounded by uncertain knowledge of the detailed structure of porous media at all scales. Parameters are often modeled as random fields due to sparse sampling. In realistic applications, conductivity random fields are highly non-Gaussian with complex correlation structures, reflecting geomorphological structure of aquifers. This limits applicability of stochastic theories that require log conductivity be a stationary, log-normally distributed field with small variance. Two-scaled models of conductivity fields can account for the non-stationarity of real conductivity data, but also maintain much of the computational simplicity of stationary models. In a two-scaled model, a porous medium is composed of disjoint, statistically homogeneous facies, each of which consists of a single type of material. At the larger scale, a porous medium is an arrangement of heterogeneous blocks whose extent and location are defined by uncertain boundaries. We will discuss applications of two-scale (and multi-scale) models to obtain effective conductivities and to propagate wetting fronts through highly heterogeneous porous media.