Numerical solution of the nonlinear diffusivity equation in heterogeneous reservoirs with wellbore phase redistribution

We consider the application of the Finite Element Method (FEM) for numerical pressure transient analysis under conditions where no reliable analytical solution is available. Pressure transient analysis is normally based on various analytical solutions of the linear one-dimensional diffusion equation under restrictive assumptions about the formation and its boundaries. For example, the formation is either assumed isotropic or a restrictive a priori assumption is made about its heterogeneity. The wellbore storage effect is also often considered without regard to the possibility of phase redistribution. In many practical situations such restrictions are not justified and analytical solutions do not exist. Here we present a numerical solution of the nonlinear diffusion equation based on the FEM that can be used without any restrictive a priori assumptions. Through the use of the weak formulation of the FEM, solution can be obtained for a heterogeneous medium with discontinuous or nonlinear properties. The weak formulation also enables the handling of time dependent boundary conditions and hence problems involving wellbore storage with significant phase redistribution. The speed and accuracy of the numerical technique is first confirmed by comparison with simple test cases that admit an analytical solution. The practical utility of the proposed method is then demonstrated for a number of test cases that involve discontinuous and nonlinear formation properties and/or wellbore storage with phase redistribution.