The Flow of Compressible Fluids Through Porous Media and Some Problems in Heat Conduction

Abstract

In Part I the general partial differential equations are derived for the flow of compressible liquids and of gases through porous media, on the basis of the generalized Darcy law that: for all homogeneous fluids the fluid velocity is directly proportional to the pressure gradient. The equation for compressible liquids turns out to be the Fourier heat conduction equation with the density as the dependent variable, while for gases a nonlinear parabolic equation is obtained. The steady state solutions for these equations for linear and plane radial flow are derived and discussed in Part II. The heat conduction equation, in polar coordinates with radial symmetry, for compressible liquids, is then solved in Part III for systems in which (a) the density is specified over both concentric circular boundaries of an annular region, (b) the density is given over one boundary and the normal derivative over the other, and (c) the normal derivative is given over both boundaries. For the last case a new elementary solution is introduced, not given in the standard English texts. The cases where the internal boundary reduces to an infinitesimal sink are also given explicit solutions. Applications are made to (1) production history from a well whose pressure is reduced discontinuously to its final value; (2) production history from a well whose pressure is lowered so that the fluid density at the well drops linearly with the time; (3) the pressure rise in a well after shutting in; (4) pressure decline in the ``East Texas'' oil field; (5) production decline from a single well at constant pressure in a closed reservoir; and (6) pressure decline in a closed reservoir drained at a constant rate by a central well. The Green's function, corresponding to a well displaced from the center of a closed reservoir, is derived and is applied to the problem of the interference between two wells draining the same circular reservoir. The solutions for all these problems correspond to similar heat conduction problems which have not been previously solved explicitly in the literature. In Part IV an approximation theory is given for the non-isothermal gas flow from a closed circular reservoir into a central well, and is illustrated by a numerical example showing the pressure and production decline for such a reservoir on a sudden lowering of the well pressure from an initially uniform value.