The Application of the Laplace Transformation to Flow Problems in Reservoirs

Abstract For several years the authors have felt the need for a source from whichreservoir engineers could obtain fundamental theory and data on the flow offluids through permeable media in the unsteady state. The data on the unsteadystate flow are composed of solutions of the equation (Equation). Two sets of solutions of this equation are developed, namely, for "theconstant terminal pressure case" and "the constant terminal ratecase." In the constant terminal pressure case the pressure at the terminalboundary is lowered by unity at zero time, kept constant thereafter, and thecumulative amount of fluid flowing across the boundary is computed, as afunction of the time. In the constant terminal rate case a unit rate ofproduction is made to flow across the terminal boundary (from time zero onward)and the ensuing pressure drop is computed as a function of the time.Considerable effort has been made to compile complete tables from which curvescan be constructed for the constant terminal pressure and constant terminalrate cases, both for finite and infinite reservoirs. These curves can beemployed to reproduce the effect of any pressure or rate history encountered inpractice. Most of the information is obtained by the help of the Laplacetransformations, which proved to be extremely helpful for analyzing theproblems encountered in fluid flow. The application of this method simplifiesthe more tedious mathematical analyses employed in the past. With the help ofLaplace transformations some original developments were obtained (andpresented) which could not have been easily foreseen by the earliermethods. Introduction This paper represents a compilation of the work done over the past few yearson the flow of fluid in porous media. It concerns itself primarily with thetransient conditions prevailing in oil reservoirs during the time they areproduced. The study is limited to conditions where the flow of fluid obeys thediffusivity equation. Multiple-phase fluid flow has not been considered. T.P. 2732