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 Δ P/ L in a layered permeability field with a given probability distribution. The local Forchheimer coefficient β is related to the local permeability k via the relation β= a/ kc, where a> 0 being a constant and c∈ [0 , 2]. Under ergodicity, an effective value of β is derived for flow (i) perpendicular and (ii) parallel to layers. Expressions for effective Forchheimer coefficient, βe, generalize previous formulations for discrete permeability variations. Closed-form βe expressions are derived for flow perpendicular to layers and under two limit cases, F≪ 1 and F≫ 1 , for flow parallel to layering, with F a Forchheimer number depending on the pressure gradient. For F of order unity, βe is obtained numerically: when realistic values of Δ P/ L and a are adopted, βe approaches the results valid for the high Forchheimer approximation. Further, βe increases with heterogeneity, with values always larger than those it would take if the β- k relationship was applied to the mean permeability; it increases (decreases) with increasing (decreasing) exponent c for flow perpendicular (parallel) to layers. βe is also moderately sensitive to the permeability distribution, and is larger for the gamma than for the lognormal distribution.