Transient Fluid Flow in Porous Media: Inertia-Dominated to Viscous-Dominated Transition

A one-dimensional isothermal flow is induced by a step change in the pressure at the boundary of a semi-infinite medium. The early flow is inertia-dominated, in accordance with Ergun’s equation, and is self-similar in the variable x/$t3$⁠. The late flow is viscous-dominated, in accordance with Darcy’s law, and is self-similar in the variable x/$t$⁠. Comprehensive numerical results are presented for both of these asymptotic regimes and also for the intermediate transition period which is governed by Forchheimer’s equation. The only explicit parameter is the pressure ratio, N, which is varied from N → ∞ (strong gas-compression), through N → 1 (constant compressibility liquid), to N → 0 (strong gas-rarefaction). The solution procedure is based on a generalized separation-of-variables approach which should also be useful in other problems which possess self-similar asymptotic solutions both at early times and at late times.