Spectral Lanczos decomposition method for solving single‐phase fluid flow porous media

Abstract

AbstractA three‐dimensional well model (r − θ − z) for the simulation of single‐phase fluid flow in porous media is developed. Rather than directly solving the 3‐D parabolic PDE (partial differential equation) for fluid flow, the PDE is transformed to a linear operator problem that is defined as u = f(A)σ, where A is a real symmetric square matrix and σ is a vector. The linear operator problem is solved by using the spectral Lanczos decomposition method. This formulation gives continuous solutions in time. A 7‐point finite difference scheme is used for the spatial discretization. The model is useful for well testing problems as well as for the simulation of the wireline formation tester tool behavior in heterogeneous reservoirs. The linear operator formulation also permits us to obtain solutions in the Laplace domain, where the wellbore storage and skin can be incorporated analytically. The infinite‐conductivity (uniform pressure) wellbore condition is preserved when mixed boundary conditions, such as partial penetration, occur. The numerical solutions are compared with the analytical solutions for fully and partially penetrated wells in a homogeneous reservoir. © 1994 John Wiley & Sons, Inc.