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, β, in the Forchheimer
equation, and whether the beta factor β 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 β 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.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
3 July 2006
- Meeting paper published:
24 September 2006
- Revised manuscript received:
16 May 2007
- Manuscript approved:
29 May 2007
- Version of record:
20 March 2008
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