The Flow of Real Gases Through Porous Media (1966)

Abstract The effect of variations of pressure-dependent viscosity and gas lawdeviation factor on the flow of real gasses through porous media has beenconsidered. A rigorous gas flow equation was developed which is a second order, non-linear partial differential equation with variation coefficients. Thisequation was reduced by a change of variable to a form similar to thediffusivity equation, but with potential-dependent diffusivity. The change ofvariable can be used as a new pseudo-pressure for gas flow which replacespressure or pressure-squared as currently applied to gas flow. Substitution of the real gas pseudo-pressure has a number of importantconsequences. First, second degree pressure gradient terms which have commonlybeen neglected under the assumption that the pressure gradient is smalleverywhere in the flow system, are rigorously handled. Omission of seconddegree terms leads to serious errors in estimated pressure distributions fortight formations. Second, flow equations in terms of the real gaspseudo-pressure do not contain viscosity or gas law deviation factors, and thusavoid the need for selection of an average pressure to evaluate physicalproperties. Third, the real gas pseudo-pressure can be determined numericallyin terms of pseudo-reduced pressures and temperatures from existing physicalproperty correlations to provide generally useful information. The real gaspseudo-pressure was determined by numerical integration and is presented inboth tabular and graphical form in this paper. Finally, production of real gascan be correlated in terms of the real gas pseudo-pressure and shown to besimilar to liquid flow as described by diffusivity equation solutions. Application of the real gas pseudo-pressure to radial flow systems undertransient, steady-state or approximate pseudo-steady-state injection orproduction have been considered. Superposition of the linearized real gas flowsolutions to generate variable rate performance was investigated and foundsatisfactory. This provides justification for pressure build-up testing. It isbelieved that the concept of the real gas pseudo-pressure will lead to improvedinterpretation of results of current gas well testing procedures, both steadyand unsteady-state in nature, and improved forecasting of gas production. Introduction In recent years a considerable effort has been directed to the theory ofisothermal flow of gases through porous media. The present state of knowledgeis far from being fully developed. The difficulty lies in the non-linearity ofpartial differential equations which describe both real and ideal gas flow. Solutions which are available are approximate analytical solutions, graphicalsolutions, analogue solutions and numerical solutions. The earliest attempt to solve this problem involved the method ofsuccessions of steady states proposed by Muskat.1 Approximateanalytical solutions2 were obtained by linearizing the flow equationfor ideal gas to yield a diffusivity-type equation. Such solutions, thoughwidely used and easy to apply to engineering problems, are of limited valuebecause of idealized assumptions and restrictions imposed upon the flowequation. The validity of linearized equations and the conditions under whichtheir solutions apply have not been fully discussed in the literature. Approximate solutions are those of Heatherington et al.,2MacRoberts 4 and Janicek and Katz.5 A graphical solutionof the linearized equation was given by Cornell and Katz.6 Also, byusing the mean value of the time derivative in the flow equation, Rowan andClegg 7 gave several simple approximate solutions. All the solutionswere obtained assuming small pressure gradients and constant gas properties. Variation of gas properties with pressure has been neglected because ofanalytic difficulties, even in approximate analytic solutions. Green and Wilts8 used an electrical network for simulatingone-dimensional flow of an ideal gas. Numerical methods using finite differenceequations and digital computing techniques have been used extensively forsolving both ideal and real gas equations. Aronofsky and Jenkins9,10and Bruce et al.11 gave numerical solutions for linear andradial gas flow. Douglas et al.12 gave a solution for asquare drainage area. Aronofsky13 included the effect of slippage onideal gas flow. The most important contribution to the theory of flow of idealgases through porous media was the conclusion reached by Aronofsky andJenkins14 that solutions for the liquid flow case15 couldbe used to generate approximate solutions for constant rate production of idealgases.