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 ΔP/L\documentclass[12pt]{minimal}
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\begin{document}$$\Delta P/L$$\end{document} in a layered permeability field with a given probability distribution. The local Forchheimer coefficient β\documentclass[12pt]{minimal}
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\begin{document}$$\beta$$\end{document} is related to the local permeability k via the relation β=a/kc\documentclass[12pt]{minimal}
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\begin{document}$$\beta =a/k^c$$\end{document}, where a>0\documentclass[12pt]{minimal}
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\begin{document}$$a>0$$\end{document} being a constant and c∈[0,2]\documentclass[12pt]{minimal}
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\begin{document}$$c\in [0,2]$$\end{document}. Under ergodicity, an effective value of β\documentclass[12pt]{minimal}
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\begin{document}$$\beta$$\end{document} is derived for flow (i) perpendicular and (ii) parallel to layers. Expressions for effective Forchheimer coefficient, βe\documentclass[12pt]{minimal}
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\begin{document}$$\beta _e$$\end{document}, generalize previous formulations for discrete permeability variations. Closed-form βe\documentclass[12pt]{minimal}
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\begin{document}$$\beta _e$$\end{document} expressions are derived for flow perpendicular to layers and under two limit cases, F≪1\documentclass[12pt]{minimal}
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\begin{document}$$F\ll 1$$\end{document} and F≫1\documentclass[12pt]{minimal}
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\begin{document}$$F\gg 1$$\end{document}, for flow parallel to layering, with F a Forchheimer number depending on the pressure gradient. For F of order unity, βe\documentclass[12pt]{minimal}
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\begin{document}$$\beta _e$$\end{document} is obtained numerically: when realistic values of ΔP/L\documentclass[12pt]{minimal}
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\begin{document}$$\Delta P/L$$\end{document} and a are adopted, βe\documentclass[12pt]{minimal}
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\begin{document}$$\beta _e$$\end{document} approaches the results valid for the high Forchheimer approximation. Further, βe\documentclass[12pt]{minimal}
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\begin{document}$$\beta _{e}$$\end{document} increases with heterogeneity, with values always larger than those it would take if the β-k\documentclass[12pt]{minimal}
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\begin{document}$$\beta - k$$\end{document} relationship was applied to the mean permeability; it increases (decreases) with increasing (decreasing) exponent c for flow perpendicular (parallel) to layers. βe\documentclass[12pt]{minimal}
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\begin{document}$$\beta _{e}$$\end{document} is also moderately sensitive to the permeability distribution, and is larger for the gamma than for the lognormal distribution.
