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

Abstract

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 (ß), 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 ß 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 ß 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 ß 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 ß value and to demonstrate the limits of applicability of ß 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. Applicability of Darcy's Law In 1856 Henry Philibert Gaspard Darcy published the results of a series of experiments on water flow through a sand-packed column at various pressure differentials.1 His apparatus consisted of a 0.35 m diameter sand pack with 38% porosity. The height of the sand column varied from 0.58 meters to 1.7 meters and the imposed water head varied from 1 to 14 meters. From the observations of flow rate resulting from these various experiments he concluded that flow rate varied in proportion to the imposed head and inversely to the height of the sand pack. The flow rate could be related to the imposed head through a linear proportionality involving a constant, k, where "k is a coefficient dependent on the permeability of the (sand) layer." This linear proportionality is expressed in Darcy's Law which is given by Equation 1. In the equation the imposed potential gradient is ?P/?L, the fluid viscosity is µ, and the superficial velocity is v. As defined, a permeability of one darcy results in a flow of one cm/sec for a fluid of one centipoise under a gradient of one atmosphere per centimeter.Equation 1 The calculated apparent permeability values from Darcy's five experimental data sets are shown in Figure 1 as a function of the quantity ?v/µ, defined here as the pseudo-Reynolds number (Rp). The parameter Rp is not a true Reynolds Number because it is not dimensionless, but is missing a characteristic length giving it dimensions of 1/L. The plot shows that even Darcy's original observations do not show an absolutely constant permeability, but an apparent decrease in flow capacity with increasing velocity. Darcy mentions that there were experimental difficulties at the higher flow rates in his tests because his water source was a hospital. The maximum flow rate he could achieve was unstable because of other people in the hospital occasionally opening and closing other water taps.