Effective Forchheimer Coefficient for Layered Porous Media
Alessandro Lenci, Farhad Zeighami, Vittorio Di Federico
Abstract
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 $$\Delta P/L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>P</mml:mi> <mml:mo>/</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> in a layered permeability field with a given probability distribution. The local Forchheimer coefficient $$\beta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> is related to the local permeability k via the relation $$\beta =a/k^c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>k</mml:mi> <mml:mi>c</mml:mi> </mml:msup> </mml:mrow> </mml:math> , where $$a>0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> being a constant and $$c\in [0,2]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> . Under ergodicity, an effective value of $$\beta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> is derived for flow (i) perpendicular and (ii) parallel to layers. Expressions for effective Forchheimer coefficient, $$\beta _e$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>e</mml:mi> </mml:msub> </mml:math> , generalize previous formulations for discrete permeability variations. Closed-form $$\beta _e$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>e</mml:mi> </mml:msub> </mml:math> expressions are derived for flow perpendicular to layers and under two limit cases, $$F\ll 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and $$F\gg 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>≫</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , for flow parallel to layering, with F a Forchheimer number depending on the pressure gradient. For F of order unity, $$\beta _e$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>e</mml:mi> </mml:msub> </mml:math> is obtained numerically: when realistic values of $$\Delta P/L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>P</mml:mi> <mml:mo>/</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> and a are adopted, $$\beta _e$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>e</mml:mi> </mml:msub> </mml:math> approaches the results valid for the high Forchheimer approximation. Further, $$\beta _{e}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>e</mml:mi> </mml:msub> </mml:math> increases with heterogeneity, with values always larger than those it would take if the $$\beta - k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>-</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> relationship was applied to the mean permeability; it increases (decreases) with increasing (decreasing) exponent c for flow perpendicular (parallel) to layers. $$\beta _{e}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>e</mml:mi> </mml:msub> </mml:math> is also moderately sensitive to the permeability distribution, and is larger for the gamma than for the lognormal distribution.