Using a transient radial gas flow model in which slippage, visco-inertial effects, real gas behaviour and rock compressibility are permitted to operate simultaneously, solutions have been obtained for two types of boundary conditions, i.e. the constant terminal rate case with sealed external boundary, and the constant terminal pressure case with constant pressure at the external boundary.
Results presented in this paper emphasize observed deviations from Darcy flow in predictions of pressure and mass flux distributions for low and high permeability rocks. For this purpose different permeability, slip coefficient, inertial parameter and rock compressibility values have been considered. In addition, the predictions take into account changes in gas properties with pressure.
Solutions for these two types of boundary conditions are shown at much larger values of the dimensionless time than those used in the other two boundary conditions which were presented previously in the first part of this study. Once again, the predictions show that each of the rock parameters has a strong influence on gas reservoir performance.
In the first part of this study a mathematical model was presented for solving the non-linear partial differential equations describing transient radial gas flow through porous media. The development of the model and the assumptions involved are completely described in the literature.1 The model has already been used to predict reservoir performance for two sets of boundary conditions.2 Specifically, solutions were obtained for the constant terminal rate case with sealed external boundary (Case I), and for the constant terminal pressure case with constant pressure at the external boundary (Case IV). Table 1 summarizes the four most common boundary conditions for transient radial gas flow.
In this second part, a detailed study of of the remaining Cases II and III has been undertaken to examine deviation from Darcy flow for these conditions and to investigate the behavior at larger values of the dimensionless time. As in the first part, the wellbore radius was 0.5 feet, the external radius was 500 feet, and the initial pressure was 2000 psia and the flowing fluid was nitrogen.
The set of rock properties considered was as follows:
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Solutions for the low permeability type reservoir are summarized in Figures 1 through 5. The dimensionless mass flux applied to the inner boundary was 0.4 × 10–4. Figure 1 presents the dimensionless pressure-squared profile showing the effects of the individual rock parameters at a dimensionless time E = 0.21 × 10 4. The comparison of the four curves shown in Figure 1 clearly points out deviations from Darcy flow when slippage, rock compressibility or inertial effect is considered. Larger pressure drawdowns are observed whenever slippage or inertial effects are considered. This drawdown is larger when the latter effect is included in the mathematical model. If only rock compressibility is considered, predictions show lower pressure drawdowns as should be expected.