Beyond Beta Factors: A Complete Model for Darcy, Forchheimer, and Trans-Forchheimer Flow in Porous Media

Abstract The limitations of Darcy's Law to a relatively small velocity region have long been recognized. A commonly accepted approach has been to use Forchheimer's equation, and its inertial flow parameter (β), as an extension of Darcy's Law beyond the linear flow region. This "trans-Darcy" flow is especially important in hydraulic fracture conductivity calculations, where flow velocities in the proppant pack are much higher than in the surrounding reservoir. The computation and presentation of β values as functions of closure stress and permeability has become an important consideration in proppant selection and fracture treatment design. New experimental data, under very high-rate flow conditions, has shown conclusively that Forchheimer's equation, like Darcy's Law, has a limited range of applicability. At high potential gradients the flow rate cannot be predicted from Darcy or Forchheimer equations. These data also show that β is not a single-valued function of permeability, as has been expected, but is as much a function of Reynolds Number as the apparent Darcy permeability. This leads to different values of β for the same proppant, depending on the range of flow rates used for the measurement. This paper presents a single new equation that describes the relationship between rate and potential gradient for porous media flow over the entire range of Reynolds Number. The equation simplifies to both the Forchheimer and Darcy equations under their governing assumptions. The equation can be used to determine the correct theoretical β value and to demonstrate the limits of applicability of β and the Forchheimer equation. A new method for describing porous media flow using different coefficients, and the relationship of these coefficients to physical parameters, is presented. The development of the complete porous media flow model is supported by extensive laboratory data on various proppant packs.