A Different Perspective on the Forchheimer and Ergun Equations

The need to better estimate the performance of hydraulically fractured wells in shale-gas formations has focused attention on the description of the non-Darcy flow in the region of the hydraulic fracture near the borehole where the converging flow may be increasingly dominated by the inertia effect. The analogy between high-rate flow in proppant-filled hydraulic fractures—best described by the Forchheimer equation—and the flow of reactants in fixed beds or granular beds in catalysis—widely described by the Ergun equation—may offer insights on how to estimate the value of the coefficient $β$ across the complete range of velocities (or, equivalently, Reynolds numbers). In this paper, we show that, although the value of $β$ varies over a wide range depending on the porosity (ϕ) and permeability (K) of the porous medium, the value of the product βK is confined in the range of 10⁻⁵ to 10⁻⁴ m, and a correlation is obtained with narrower uncertainty compared with published ones.