Gravity Drainage Theory

Abstract

Abstract This paper presents a theory for estimating the rate of gravity drainage ofa liquid out of a sand column. Account is taken of the variation inpermeability to the liquid as the saturation in the upper part of the sandbecomes less than 100 pct. The theory is confirmed by previously published experimental data. Introduction Petroleum engineers have expressed the need for a theory of gravitydrainage. Brunner, in particular, has pointed out that some type ofmathematical theory is necessary to begin the application of laboratory data tofield problems. Muskat and his associates have recently made contributions to the theory ofgas-drive behavior and have indicated an intention to apply their methods towaterdrive systems. No theory of gravity drainage rates has been developed, however, and it seems desirable to formulate one at this time. Differential Equations of Capillary Flow The flow of liquids in partially saturated porous media has been studied bymany investigators. Richards presented derivations of fundamental differentialequations governing two-phase capillary flow; and used simplified forms ofthose equations in solving a steady-state problem. Muskat and Meres presentedand used equations different from those of Richards. Their equations did notexplicitly involve capillary pressure gradients; but included, on the otherhand, terms expressing the effects of the evolution of gas from the liquidphase during flow. Leverett stated in 1940 that "previous work on the flow of fluidmixtures in porous solids [had] failed adequately to account for all of thethree influences that cause motion of the fluids: capillarity, gravity, andimpressed external pressure differentials." Leverett's basic equations, however, were specialized forms of the general equations of Richards, which hadactually taken account of the three influences mentioned by Leverett, but hadnot been used in a problem involving all three. The fact is that our knowledge of capillary flow and our ability to expressthis knowledge in differential equations exceed our ability to solve theequations except in a few cases. General differential equations have usuallybeen of little more than formal value. In solving practical problems, it hasbeen necessary to develop specific equations, preserving terms that involvedthe factors important in those problems, and purposefully neglecting otherterms that were not of predominating influence. This is the method followedhere. It is believed that the solution of the particular problem and the schemeof the solution itself are new. T.P. 2464