Unsteady-State Liquid Flow Through Porous Media Having Elliptic Boundaries

Abstract

Published in Petroleum Transactions, AIME, Volume 216, 1959, pages 460–464. Introduction A large number of boundary value problems encountered in unsteady-state heat transfer, fluid flow through porous media, neutron diffusion and mass transfer involve the solution of a linear, parabolic partial differential equation commonly referred to as the diffusivity equation, ................... (1) where U is the dependent potential variable, K is the diffusivity (hydraulic, thermal, neutron, etc.) and t is the time variable. Solutions to Eq. 1 are available in the literature for a wide variety of initial and boundary conditions. The great majority of these solutions are obtained for geometric boundaries corresponding to linear, cylindrical or spherical flow models. A typical engineering application where the solution to Eq. 1 is required is the calculation of underground water encroachment across the boundaries of oil or natural gas reservoirs In this particular area of application, the reservoir boundary is invariably approximated by circular geometry. However, the areal shape of many reservoirs can be better approximated by elliptic rather than circular boundaries. Thus the need for a general method of solving the diffusivity equation in elliptic coordinates arises in this problem as well as in other engineering applications involving elliptic boundaries. The solution to the diffusivity equation usually involves the Error Function for the linear flow model, Bessel functions for the radial flow model and trigonometric or Legendre functions for the spherical flow model. It is well known that the general solution to the diffusivity equation in elliptic coordinates involves Mathieu functions. The significance of Mathieu functions in the analytical treatment of the diffusivity equation in elliptic coordinates is discussed in the literature. However, these references do not provide analytical solutions useful in practical engineering problems. The objectives of this paper are the development of the equations describing the unsteady-state liquid flow through a porous medium with an elliptic inner boundary, the development of a numerical method of solving these equations and, finally, a comparison of the water encroachment quantities calculated from the elliptic flow equation with those calculated from the radial flow equation. While the specific problem treated in this paper relates to unsteady-state liquid flow through a porous medium, the basic equations and computational techniques developed will apply equally well to problems occurring in the other areas of engineering interest mentioned previously. The solution given here is limited to a single case in which the outer boundary encloses an area 100 times that of the inner boundary.