Multiscale permeability and dispersion in randomly heterogeneous geologic media

Flow and transport in natural soils and rocks have been traditionally described by means of partial differential equations (pde’s). These equations are generally taken to represent basic physical principles (conservation and constitutive laws) that operate on some macroscopic scale (theoretical support volume) at which the geologic medium may be viewed as a continuum. The precise nature of this theoretical macroscopic support scale remains generally unclear though some derive comfort from associating it in the abstract with a “representative elementary volume” (REV). Unfortunately, the concept of an REV is equally difficult to define without ambiguity or to apply in practice. Flow and transport pde’s are local in the sense that all quantities (parameters; forcing functions including initial, boundary and source terms; dependent variables) which enter into them are defined at a single point (x, t) in space-time. Parameters such as permeability, porosity, and dispersivity are generally regarded as macroscopic medium properties that are well-defined, and can thus be determined (at least in principle) experimentally and more-or-less uniquely, at any point x in the flow domain.