Summary
An analytical method for calculating an effective non-Darcy flow coefficient for a heterogeneous formation is presented. The method presented here can be used to calculate an effective non-Darcy flow coefficient for heterogeneous gridblocks in reservoir simulators. Based on this method, it is shown that the non-Darcy flow coefficient of a heterogeneous formation is larger than the non-Darcy flow coefficient of an equivalent homogenous formation. Non-Darcy flow coefficients calculated from gas well data show that non-Darcy flow coefficients obtained from well tests are significantly larger than those predicted from experimental correlations. Permeability heterogeneity is a very likely reason for the differences in non-Darcy flow coefficients often seen between laboratory and field data.
Introduction
In this paper, we present an analytical method for calculating an effective non-Darcy flow coefficient for a heterogeneous reservoir. The effect of heterogeneity on the non-Darcy flow coefficient is also shown using numerical simulations. Non-Darcy flow coefficients calculated from the analysis of welltest data from a gas condensate field are compared with experimental correlations. Such a comparison allows us to more accurately assess the importance of non-Darcy flow in gas condensate reservoirs.
Literature Review
As early as 1901, Reynolds observed, in his classical experiments of injecting dye into water flowing through glass tubes, that after some high flowrate, flow rate was no longer proportional to the pressure drop. Forchheimer1 also observed this phenomena and proposed the following quadratic equation to express the relationship between pressure drop and velocity in a porous medium:
d P d r = μ k u + β ρ u 2 . ( 1 )
This equation has come to be known asForchheimer's equation. At low Reynolds number (creeping flow conditions), the above equation reduces to Darcy's law.
Tek2 developed a generalized Darcy equation in dimensionless form which predicts the pressure drop with good agreement over all ranges of Reynolds numbers. Katz et al.3 attributed the phenomenon of non-Darcy flow to turbulence. Tek et al.4 proposed the following correlation for?:
β = 5.5 × 10 9 k 5 / 4 ϕ 3 / 4 . ( 2 )
Gewers and Nichol5 conducted experiments on microvugular carbonate cores to measure the non-Darcy flow coefficient. They also studied the effect of the presence of a second static fluid phase and the effect on plugging due to fines migration. They found that ? decreases and then increases with liquid saturation. Wong6 studied the effect of a mobile liquid saturation on ?. He used distilled water as the liquid phase and water saturated nitrogen as the gas phase on the same cores used by Gewers and Nichol. He plotted ? vs liquid saturation and found that there is an eight-fold increase in ? when the liquid saturation increases from 40% to 70%. He concluded that ? can be approximately calculated from the dry core experiments by using the effective gas permeability.
Geertsma7,8 introduced an empirical relationship between ?,k and ? based on a combination of experimental data and dimensional analysis. He noted that the observed departure from Darcy's law was due to the convective acceleration and deceleration of the fluid particles. He also defined a new Reynolds number as ?k??/?, and suggested the following correlation for ? with a constant C (k is in ft 2, ?is in 1/ft).
β = C k 0.5 ϕ 5.5 . ( 3 )
For the case of gas flowing through a core with a static liquid phase, he suggested the following correlation:
β = C ( k k r g ) 0.5 [ ϕ ( 1 − S w ) ] 5.5 . ( 4 )
Phipps and Khalil9 proposed a method for determining the exponent in a Forchheimer-type equation. Firoozabadi and Katz10 presented are view of the theory of high velocity gas flow through porous media. Evanset al.11 reviewed the various correlations. They conducted an experimental study of the effect of the immobile liquid saturation and suggested a correlation based on dimensional analysis. Nguyen12performed an experimental study of non-Darcy flow through perforations on a synthetic core using air. These experiments showed that non-Darcy flow exists in the convergence zone and the perforation tunnel. Results of this study showed that Darcy flow equations can over predict well productivity by as much as 100%.
Jones13 conducted experiments on 355 sandstone and 29 limestone cores. These tests were done for various core types: vuggy limestones, crystalline limestones, and fine grained sandstones. He presented the following correlation:
β = 6.15 × 10 10 k − 1.55 . ( 5 )
He also points out that the group ?k? which is the characteristic length used for defining a Reynolds number for porous media, should be proportional to the characteristic length k/ϕ. He developed an approximate multilayer flow model that demonstrates that the departure from the above relation is due to permeability variations. Jones suggested that heterogeneity may be the reason why all correlations involving ? exhibit so much scatter.
Teach Second Law of Thermodynamics via Analysis of Flow through Packed Beds and Consolidated Porous Media