Effective Forchheimer Coefficient for Layered Porous Media

Abstract

Inertial flow in porous media, governed by the Forchheimer equation, is affected by domain heterogeneity at the field scale. We propose a method to derive formulae of the effective Forchheimer coefficient with application to a perfectly stratified medium. Consider uniform flow under a constant pressure gradient Delta P/L in a layered permeability field with a given probability distribution. The local Forchheimer coefficient beta is related to the local permeability k via the relation beta =a/k^c, where a>0 being a constant and cin [0,2]. Under ergodicity, an effective value of beta is derived for flow (i) perpendicular and (ii) parallel to layers. Expressions for effective Forchheimer coefficient, beta _e, generalize previous formulations for discrete permeability variations. Closed-form beta _e expressions are derived for flow perpendicular to layers and under two limit cases, Fll 1 and Fgg 1, for flow parallel to layering, with F a Forchheimer number depending on the pressure gradient. For F of order unity, beta _e is obtained numerically: when realistic values of Delta P/L and a are adopted, beta _e approaches the results valid for the high Forchheimer approximation. Further, beta _{e} increases with heterogeneity, with values always larger than those it would take if the beta - k relationship was applied to the mean permeability; it increases (decreases) with increasing (decreasing) exponent c for flow perpendicular (parallel) to layers. beta _{e} is also moderately sensitive to the permeability distribution, and is larger for the gamma than for the lognormal distribution.