Approximate Analytical Solutions for the Pressure Response at a Water-Injection Well

Abstract

Summary In this paper, we construct approximate analytical solutions for the injection wellbore pressure at vertical and horizontal water injection wells using the Thompson-Reynolds steady-state theory. The solutions are based on adding to the single-phase solution, a two-phase term which represents the existence of the two-phase zone and the movement of the water front. We first present the solutions for an isotropic reservoir and then show that we can obtain the solution to an anisotropic problem by introducing a coordinate transformation to convert an anisotropic system to an equivalent isotropic system. The analytical solutions provide insight into the behavior of injectivity tests at horizontal and vertical wells. For example, for a restricted-entry case, it is shown that the pressure derivative may be negative throughout an injection test even when the duration of the test exceeds ten or more days. We also show that for a well near a fault, the ratio of slopes reflected by derivative data will not in general be equal to two.