Finite Wellbore Radius Solution for Gas Wells by Green's Functions

Abstract

Abstract Even when written in terms of a pseudo-pressure function, the diffusivity equation for flow of gases through porous media is, rigorously speaking, non-linear because the viscosity-compressibility product is pseudo-pressure dependent. However, several techniques and analysis procedures neglect such non-linearity. A new methodology for constructing solutions for gas reservoirs through the Green's Function (GF) technique was recently proposed in the literature. Such methodology handles the viscosity-compressibility product variation rigorously and it has been applied to solve several gas well tests problems successfully. In those problems the wellbore is always represented by a line source. This work extends a little further the theory by considering a finite wellbore radius boundary condition for a single vertical well producing at constant rate from an isotropic homogeneous and infinite gas reservoir. The presented solution does not consider non-Darcy flow effects and wellbore storage and skin as well. Results from our finite wellbore radius (f.w.r) solution are compared to a commercial finite difference reservoir simulator that shows a very close agreement. We also compared the f.w.r solution to the correspondent line source solution to study the difference between the two solutions. As expected, the pseudo-pressure solutions do not match at early times but they do agree at long times, which is exactly how f.w.r and line-source well solutions for slightly compressible fluids behave. It seems that, even for gas well problems, the wellbore can be satisfactorily represented by a line source without significant loss of generality. The line source assumption greatly simplifies the math and the computational effort. This aspect is especially attractive for complex non-linear gas well problems that remains to be solve by the GF approach.