Natural Gas Technology - The Flow of Real Gases Through Porous Media

The effect of variations of pressure-dependent viscosity and gas law deviation factor on the flow of real gases through porous media has been considered. A rigorous gas flow equation was developed which is a second order, non-linear partial differential equation with variable coeficients. This equation was reduced by a change of variable to a form similar to the diffusivity equation, but with potential-dependent diffusivity. The change of variable can be used as a new pseudo-pressure for gas flow which replaces pressure or pressure-squared as currently applied to gas flow. Substitution of the real gas pseudo-pressure has a nrtmber of important consequences. First, second degree pressure gradient terms which have commonly been neglected under the assumption that the pressure gradient is small everywhere in the flow system, are rigorously handled. Omission of second degree terms leads to verious errors in estimated pressure distributions for tight formations. Second, flow equations in terms of the real gas pseudo-pressure do not contain viscosity or gas law deviation factors, and thus avoid the need for selection of an average pressure to evaluate physical properties. Third, the real gas pseudo-pressure can be determined numerically in term of pseudo-reduced pressures and temperatures from existing physical property correlations to provide generally useful information. The real gas pseudo-pressure was determined by numerical integration and is presented in both tabular and graphical form in this paper. Finally, production of real gas can be correlated in terms of the real gas pseudo-pressure and shown to be similar to liquid flow as described by diffusivity equation solutions. Applications of the real gas pseudo-pressnre to radial flow systems under transient, steady-state or approximate pseudo-steady-state injection or production have been considered. Superposition of the linearized real gas flow solutions to generate variable rate performance was investigated and found satisfactory. This provides justification for pressure build-up testing. It is believed that the concept of the real gas pseudo-pressure will lead to improved interpretation of results of current gas well testing procedures, both steady and unsteady-state in nature, and improved forecasting of gas production. INTRODUCTION In recent years a considerable effort has been directed to the theory of isothermal flow of gases through porous media. The present state of knowledge is far from being fully developed. The difficulty lies in the non-linearity of partial differential equations which describe both real and ideal gas flow. Solutions which are available are approximate analytical solutions, graphical solutions, analogue solutions and numerical solutions. The earliest attempt to solve this problem involved the method of successions of steady states proposed by Muskat.' Approximate analytical solutions' were obtained by linearizing the flow equation for ideal gas to yield a diffusivity-type equation. Such solutions, though widely used and easy to apply to engineering problems, are of limited value bemuse of idealized assumptions and restrictions imposed upon the flow equation. The validity of linearized equations and the conditions under which their solutions apply have not been fully discussed in the literature. Approximate solutions are those of Heatherington et al.. MacRobertsl and Janicek and Katz.' A graphical solution of the linearized equation was given by Cornell and Katz. Also, by using the mean value of the time derivative in the flow equation, Rowan and Clegg' gave several simple approximate solutions. All the solutions were obtained assuming small pressure gradients and constant gas properties. Variation of gas properties with pressure has been neglected because of analytic difficulties. even in approximate analytic solutions. Green and Wilts8 used an electrical network for simulating one-dimensional flow of an ideal gas. Numerical methods using finite difference equations and digital computing techniques have been used extensively for solving both ideal and real gas equations. Aronofsky and Jenkins"I " and Bruce et al.11 gave numerical solutions for linear and radial gas flow. Douglas et al." gave a solution for a square drainage area. Aronofsky 13 included the effect of slippage on ideal gas flow. The most important contribution to the theory of flow of ideal gases through porous media was the conclusion reached by Aronofsky and Jenkins" that solutions for the liquid flow case'" could be used to generate approximate solutions for constant rate production of ideal gases. An equation describing the flow of real gases has been solved for special cases by a number of investigators using numerical methods. Aronofsky and Ferris 10 onsidered linear flow, while Aronofsky and Porter 17 considered radial gas flow. Gas properties were permitted to vary as linear functions of pressure. Recently, CarteP 18 proposed an empirical correlation by which gas well behavior can be estimated from solutions of the diffusivity equation using instantaneour values of pressure-dependent gas