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.
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:
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
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.