Unsteady Flow of Gas through Porous Media

Abstract

Since the equation of continuity governing transient flow of gases throughporous media cannot be integrated mathematically into a simple usableexpression free from series terms, empirical and approximate equations havebeen advanced relating cumulative production and future production rates fromgas reservoirs with time. In order to check the accuracy of these empiricalequations and to obtain an insight into the initial conditions pertinent to thesolution of the fundamental equation of continuity, experimental work has beenconducted on a vertical 2-in. tube, 91.6 ft. high, packed with unconsolidatedWilcox sand. The results presented in graphical form show the accuracy of theapproximate relations. Data from unsteady-flow runs on the 2-in. experimental tube are presentedgraphically, and are applied to obtain an equation for this specific case, andare offered as an aid in future investigations of this basic equation. The close agreement between equations calculated for steady-state flow andthe experiments indicates that the decline of actual gas reservoirs may betreated by an application of these equations without serious error. Introduction The steady-state flow of gas through a porous material-that is, flow whereinthe boundary conditions do not vary with time-is subject to analyticalsolution. Muskat has shown that the general differential equation is simply the La Place equation in P2. Direct solutionsmay be at once obtained. The unsteady state, on the other hand, is not analytically reducible. Sincenearly all phenomena of practical interest in gas production involve flow inthe unsteady rather than in the steady state, it has been necessary to deriveapproximate solutions to the general equation. The validity of suchapproximations has not hitherto been adequately tested. In the experiments to be described a simple, closed, linear system issubstituted for the real complex gas field. The general results, however, areindependent of the geometry of the system. The problem studied is the rate ofproduction, and the accompanying variations in pressure, when one end of alinear porous system is opened to the atmosphere and permitted to produce toexhaustion. An approximate solution to the general differential equation, basedupon the assumption of an "infinite series of steady states," is thenapplied to the results to determine the validity of this method of dealing withactual reservoirs. T.P. 1398