Exact solutions for flow through porous media with the Klinkenberg effect

The effect of gas compressibility during flow through a porous medium modifies the governing equation for pressure, and for low pressures, this leads to the so-called Klinkenberg effect. Darcy's law remains unchanged, and accounting for the equation of state leads to a parabolic governing equation for pressure. The increased flow rate at low pressure modifies the nonlinearity of this governing equation. We present two broad approaches for constructing exact solutions to the steady state problem. First, geometric reduction is employed to construct exact solutions in Cartesian and polar coordinates. Next, the governing equation is interpreted so that a stream function exists for the flow, and this is used to demonstrate that solutions can only be found when the flow is irrotational. A second, rather broad, class of exact solutions is thus constructed from potential flows and their generalization for variable permeability cases. The latter leads to a non-constant coefficient problem, and we provide both an algorithm illustrating how to use an existing linear numerical solver to solve the nonlinear problem and an explicit exact solution for an annular domain.