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Abstract

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 β in the Forchheimer equation and whether the beta factor β for a proppant pack should be constant over the range of flow rates of practical interests. The problem was highlighted in a recent discussion by Batenburg and Milton-Tayler¹ and the reply by Barree and Conway² regarding paper SPE 89325³ in the JPT in August 2005.

To properly assess all the arguments 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^3-7. The inertia resistance factor or the so-called beta factor b, 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 b from experimental data is still controversy. In a recent issue of the JPT in August 2005, there was a discussion by Batenburg and Milton-Tayler¹ and the reply by Barree and Conway² regarding paper SPE 89325³ on whether the beta factor β for 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 law⁸ for the pressure gradient and the flow velocity, i.e.,

                                                                       (1)

is no longer valid. In Eq. 1, permeability k is an intrinsic property of porous media. To describe the nonlinear flow situation, a quadratic term was included by Dupuit⁹ and Forchheimer¹⁰ to generalize the flow equation, i.e.,   

                                                                       (2)

Eq. 2 is commonly known as the Forchheimer equation. In the discussion of Batenburg and Milton-Tayler¹ and Barree and Conway,² it was presumed that non-Darcy flow in their experiments can be described by the Forchheimer equation.

According to the convention of the oil and gas industry, the beta factor β is generally deduced experimentally from the slope of the plot of the inverse of the apparent permeability 1/k_app vs. a dimensional pseudo Reynolds number ρV/μ (also called the Forchheimer graph). The apparent permeability k_app is defined as

                               ,                                       (3)       

after rewriting the Forchheimer equation. Based on the linear correlations obtained between 1/k_app and ρV/μ (see Fig. 1), Batenburg and Milton-Tayler¹ concluded that the beta factor β is constant for the range of flow rates of practical interests. It was recognized that the correlation, however, does not reduce to the inverse of Darcy permeability 1/k, when extrapolated to zero velocity. Barree and Conway,² on the other hand, obtained a nonlinear concave down curve shape for the variation of 1/k_app vs. ρV/μ (see Fig. 2) and concluded, therefore, that the beta factor β is not constant over the range of investigation. It was argued that the fact that a linear correlation does not reduce to 1/k at zero velocity indicates that the correlation is insufficient.