Scaling theory of incompressible two phase flow in porous media with tensor permeabilities

A recent paper from this group extended the 1955 paper by Rapoport to anisotropic systems, and gave a full analysis of this seminal work [1, 2]. Rapoport’s approach is of importance since it leads to rigorous and exact (not approximate) scaling groups and expressions for the force balances in two phase incompressible flow. This current paper describes how to extend the Rapoport scaling “shape groups” to include the anisotropic case with tensor permeabilities. When this is carried out, it is then demonstrated that, with this modification, all of the other scaling groups are the same, and exact scaling rules still apply. This is demonstrated numerically using quite complex examples where viscous fingering is occurring in the system. Rescaled cases are presented with only viscous forces, with viscous and gravity forces and with all three forces present, i.e. viscous, gravity and capillary. In addition to extending scaling theory to tensor permeabilities, we also present some novel observations on the effects of tensor permeabilities on the fingering process itself, a matter which we believe has received very little attention in the literature.