Effective Forchheimer Coefficient for Layered Porous Media

Abstract

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