Effective Forchheimer Coefficient for Layered Porous Media

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 c is an element of [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, 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, 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.