Single-phase displacement of non-Newtonian power-law fluids in porous media

We analyze the dynamics of a moving stable interface in a semi-infinite porous domain saturated by two fluids, displacing and displaced, both non-Newtonian of shear-thinning power-law behavior, assuming pressure and velocity to be continuous at the interface. The flow law for both fluids is a modified Darcy’s law. Coupling the nonlinear flow law with the continuity equation, and considering compressibility effects, yields a set of nonlinear second-order partial differential equations. Considering two fluids with the same flow behavior index allows transformation of the latter equations via a self-similar variable; taking into account the conditions at the interface shows the existence of a compression front ahead of the moving interface. Solving the resulting set of nonlinear equations yields the positions of the moving interface and compression front, and the pressure field in closed form