The flow of real gases though porous media

Al-Hussainy, Ramey y Crawford introdujeron en 1966 el concepto de la función pseudo presin m(p) para gases la resolución de esta ecuación removio la suposición de que los gradientes de presión tenían que ser peque;os para obtener una ecuación de flujo en yacimientos de gas, definieron condiciones de aplicabilidad e incluso extendieron el análisis de pruebas de presión a liquidos a gasesutilizando la función m(P)

PAPER
The Flow of Real Gases Through Porous Media
R. AL-HUSSAINY I
JUNIOR MEMBER AlME
MEMBERS AlME I

ABSTRACT

The efject of variations of pressure-dependent viscosity and gas law deviation factor on the flow of real gases t h r o ~ ~ gpho rous media has been considered. A rigorous gas Pow9 equation was developed which is a secorrd order, non-linear partialdiflerential equation with variahle coeficients. This equation was reduced by a change of variable to a form similar to the diflusivity equation, but with potential-dependent di.fusivity. The change of variable can be used as a new psezdo-pressure for gus flow which replaces pressure or prc>ssur.e-squared (4s currently rrpplied to gas flow.

Substitution of the real gas pseudo-pressure has a nrtmberof important consequences. First, second degree pressure gradient terms which have cotnmonly 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 rerious errors in estimated pressure distributions for tight formations. Second, flow equations in terrns of the real ,sas pseudo-pressuredo 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 nnmerically in t e r m of pseudo-reduced pressures and temperatures from existing physical property correlations to provide generally useful information. The real gas pseudo-pressure wasdetermined 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 di.#usivity equation solutions. Applications of the real gas pseudo-pressnre to radial pow systems under transient, steady-state or approximatepseudo-steady-state injection or production have heen considered.
Superposition of the linearized real gas pow solutions to generate variable rate performance was investigated and found satisfactory. This provides justificationfor pressure build-up testing. It is believed that the mncept of the real gas pseudo-pressure will lead to improved interpretation of results of current gas well testing procedures, bothsteady 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 gasflow. 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. Suchsolutions, though widely used and easy to apply to engineering problems, are of limited value bemuse of idealized assu~mptions 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 a/..' MacRobertsl and Janicek and...