Summary
This paper presents a semianalytical model to investigate the effect of
Forchheimer’s non-Darcy flow on the transient pressure behavior of a vertical
well in an infinite homogeneous reservoir. The model uses the Forchheimer
number to accurately quantify the non-Darcy flow in the reservoir through
differentiating it from sandface-flow-rate-dependent skin factor, which is used
to model the inertial-factor variation around the wellbore caused by
perforation or formation damage/stimulation.
Type curves are documented for both drawdown and buildup tests for the first
time by use of the semianalytical model proposed. It is observed that when
non-Darcy flow in reservoirs and/or across completions is considered, the
dimensionless pressure-derivative curves of drawdown tests have a wider
transition region with gentler slopes, while those of buildup tests exhibit a
shorter transition region with steeper slopes, similar to the observations of
Kim and Kang (1994), Spivey et al. (2004) and Camacho-V et al. (1996). In the
radial-flow period, compared with the cases of non-Darcy flow only across
completions, the cases with non-Darcy flow in reservoirs for drawdown and
buildup tests possess dimensionless pressure derivatives moving downward more
gradually and smoothly to approach 0.5 at decreasing gradients. In general, the
pressure derivatives of drawdown tests are larger than those of buildup tests
before they converge to 0.5.
With this model, the skin factor for non-Darcy flow across the completion
and the dimensionless Forchheimer number for non-Darcy flow in the reservoir
can be estimated from a common drawdown or buildup test. Guidelines for
interpreting field test data are presented. Several typical cases from the
literature are analyzed, and better type-curve matches and more-reliable
results are obtained.
Introduction
In 1901, Forchheimer found Darcy’s law to be inadequate to describe
high-velocity gas flow in porous media. To account for the discrepancy, he
added a drop, which is proportional to the square of the velocity, to the
pressure drop predicted by Darcy’s law (Forchheimer 1901). This yielded the
Forchheimer flow equation:
[equation] . (1)
Different mechanisms have been presented to explain the second-order term in
Eq. 1. In the 1950s, Cornell and Katz (1953) attributed the non-Darcy flow to
turbulence; thus, they labeled ß as a turbulence factor. Since the
1970s, many researchers (Bear 1972; Scheidegger 1974; Barak 1987; Whitaker
1996) have agreed that Forchheimer’s non-Darcy flow does not result from
turbulence but from inertial effects. Thus, ß is called an inertial
factor.
One of the earliest and best discussions of non-Darcy flow was presented by
Muskat (1973). By use of a numerical method, Smith (1961) and Swift and Kiel
(1962) investigated the effects of non-Darcy flow on gas-well testing and
suggested that non-Darcy flow of gas leads to an additional pressure drop near
the wellbore that can be treated as a flow-rate-dependent skin factor, which is
also called a non-Darcy skin factor. Ramey (1965) integrated wellbore storage
with the non-Darcy skin factor and proposed
[equation] . (2)
where
[equation] . (3)
Ramey further concluded the non-Darcy-flow coefficient, D, should be
computed from at least two tests under different flow rates by plotting the
total effective skin factor s’ vs. q. Therefore, flow after flow
tests (Rawlins and Schellhardt 1936), isochronal tests (Jones et al. 1976;
Kelkar 2000; Cullender 1955), and modified isochronal tests (Brar and Aziz
1978) have been proposed to estimate the coefficient D.
Eq. 2, however, could cause errors in estimating kh value and well
productivity. Wattenbarger and Ramey (1968) observed that the kh value
calculated from a drawdown test could be underestimated by a factor of 36% when
non-Darcy flow was present, while the buildup test could be interpreted
accurately even with extreme non-Darcy flow. Through experimental study, Nguyen
(1986) showed the standard Darcy flow analysis when applied for non-Darcy flow
through perforations could overpredict the productivity by as much as 100%.
Instead of treating the rate-dependent skin factor as Dqsc
, Kim and Kang (1994) and Spivey et al. (2004) treated the rate-dependent skin
factor as being proportional to sandface flow rate (i.e.,
Dqsf ). Their studies on the buildup test with wellbore
storage and non-Darcy flow revealed the unique pressure-derivative feature
between wellbore storage and radial-flow regions, which made it possible to
estimate the non-Darcy coefficient, D, from a single buildup test.
Instead of using Eq. 2, Guppy et al. (1982) derived a dimensionless
non-Darcy-flow-rate factor from Forchheimer’s equation to describe non-Darcy
flow in a 1D fracture. Lee et al. (1987) used a dimensionless
turbulence-intensity number similar to the non-Darcy-flow-rate factor of Guppy
et al. (1982) to model non-Darcy flow in a radial system.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
5 March 2006
- Meeting paper published:
15 May 2006
- Revised manuscript received:
3 October 2007
- Manuscript approved:
6 October 2007
- Version of record:
25 April 2008
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