Applicability of the Forchheimer Equation for Non-Darcy Flow in Porous Media

Summary The subject of non-Darcy flow in hydraulically fractured wells has generated intense debates recently. One aspect of the discussion concerns the inertia resistance factor, or the so-called beta factor, ß, in the Forchheimer equation, and whether the beta factor ß of a proppant pack should be constant over the range of flow rates of practical interests. The problem was highlighted in a recent discussion by van Batenburg and Milton-Tayler (2005) and the reply by Barree and Conway (2005) regarding paper SPE 89325 (Barree and Conway 2004) in the August 2005 JPT. This discussion in essence revolves around the applicability of the Forchheimer equation and whether the Forchheimer equation is adequate to describe the experimental results of high rate flow in proppant packs. In order to properly assess the arguments in this debate, and to get a better understanding of the state-of-the-art on non-Darcy flow in porous media in general, literature concerning the theoretical basis of the Forchheimer equation and experimental work on the identification of flow regimes is reviewed. These areas of work provide insights into the applicability of the Forchheimer equation to conventional oilfield flow tests for proppant packs. Models for flow beyond the Forchheimer regime are also suggested. Introduction The effect of non-Darcy flow as one of the most critical factors in reducing the productivity of hydraulically fractured high-rate wells has been documented extensively with examples of field cases (Barree and Conway 2004; Holditch and Morse 1976; Olson et al. 2004; Smith et al. 2004; Vincent et al. 1999). The inertia resistance factor, or the so-called beta factor, a parameter in the Forchheimer equation for quantifying the non-Darcy flow effect, is now routinely measured for proppant packs. Nevertheless, how to derive the beta factor from experimental data is still controversial. In the August 2005 issue of JPT, there was a discussion by van Batenburg and Milton-Tayler (2005) and a reply by Barree and Conway (2005) regarding paper SPE 89325 (Barree and Conway 2004) on whether the beta factor ß of a proppant pack should be constant over the range of flow rates of practical interests. The so-called non-Darcy flow in porous media occurs if the flow velocity becomes large enough so that Darcy's law (Darcy 1856) for the pressure gradient and the flow velocity, i.e.,(Eq. 1) is no longer sufficient. In Eq. 1, permeability k is an intrinsic property of porous media. To describe the nonlinear flow situation, a quadratic term was included by Dupuit (1863) and Forchheimer (1901) to generalize the flow equation, i.e.,(Eq. 2) is commonly known as the Forchheimer equation.