A Semianalytical Approach to Pseudopressure Calculations
- M.A. Klins (Pennsylvania State U.) | M.B. Biterge (Pennsylvania State U.)
- Document ID
- Society of Petroleum Engineers
- Journal of Petroleum Technology
- Publication Date
- April 1987
- Document Type
- Journal Paper
- 468 - 472
- 1987. Society of Petroleum Engineers
- 5.8.3 Coal Seam Gas, 4.6 Natural Gas, 5.6.4 Drillstem/Well Testing, 5.3.1 Flow in Porous Media
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Summary. Real-gas pseudopressures allow rigorous analytical solutions to the nonlinear mass/momentum equations for gas flow in porous media. These solutions, however, are in terms of pseudopressure rather than reservoir pressure. To convert pseudopressures to their complementary reservoir pressures, one of three techniques is traditionally applied: numerical integration, table look-up, or curve-fit analysis. All three interject some numerical error into the m(p) calculations. This paper introduces a new, "exact" procedure for making the pseudopressure/pressure conversion. It is applicable to a wide range of reservoir properties including, sour gases, temperatures up to 460 degrees F [238 degrees C], and pressures to 10,000 psia [70 MPa]. Sample calculations are shown and comparison with a number of other pseudopressure estimates is made.
Semianalytical Approach to Pseudopressures
Al-Hussainy et al. introduced the notion of pseudopressure, m(p), to analyze real-gas flow phenomena in porous media. Since that time, at least three approaches have been presented that convert reservoir pressures to their complementary pseudopressures: numerical integration, table look-up and interpolation, and curve-fit approximation.
Of these three, the most commonly applied approach is numerical integration by use of known gas correlations for viscosity and supercompressibility. Its major advantage is that it allows the user to select the and z correlations of his/her choice before numerical integration. The drawbacks are that it requires a large number of and z calculations to create a table of p/ z vs. pressure, that a numerical integration procedure and accompanying and z programs must be developed, and that the magnitude of error inherent in numerical integration is unknown and depends on the number of data points used in the integration.
The semianalytical approach, then, involves the actual solution of the m(p) integral defined in Eq. 1. If the viscosity and compressibility terms in Eq. 1 are replaced with selected correlations. the real-gas pseudopressure equation can be solved analytically and an exact solution (for the correlations chosen) of m(p) obtained.
Model Development. Real gas pseudopressure with a zero base pressure of integration is defined as
Substitutions for p/z, dp, and p with Eqs. 5, 9, and 10 can be made with a change in the variable of integration from p to pr introduced.
By using the imperfect-gas law, we can calculate gas densities:
where Dranchuk et al. assume that the critical gas con pressibility factor is 0.27 and
Then it follows that
With the Dranchuk et al. correlation for z factors,
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