The Generalized Dispersion Tensor Revisited: Theory and Calculation for Homogeneous and Heterogenous Porous Media

Over the last two decades a substantial number of researchers have studied the causes of anomalous dispersion. In the theory of porous media, it is well known that the concept of homogeneity is scale dependent; even a medium consisting of relatively uniform shaped and sized particles is not appropriately modeled by equations derived from the standardly proposed laws for homogeneous media (i.e. advection diffusion equation) until a sufficient portion of the media has been sampled. Cushman and Moroni developed an equation governing the displacement of conserved particles under very general conditions using what has been termed the generalized hydrodynamic approach, which is based on molecular hydrodynamics. The convolution flux presented in that paper is often overlooked in the literature, perhaps due to its generality. It is a theory providing a scale-dependent dispersion tensor, and in its most general form, is non-local in both space and time. Classical theories are at a fixed scale where required assumptions must be met. Due to the freedom from assumptions of scale, those result can be seen as a universal equation for dispersive processes.