Analytical method of evaluating productivity index for constant production rate or constant wellbore pressure

The existing methods for evaluating the well productivity index are based on solution of transient problems. One approach is to consider the single well problem in infinite domain and subsequently apply the method of images. This puts restrictions on the geometry of the well and of the drainage volume. Another approach is to solve the transient problem in the bounded domain for late times. While in this case restrictions on the well geometry are less severe, the shape of the drainage volume is still limited to the simplest ones. In addition, for the constant rate case highly accurate wellbore pressures, for the constant pressure case highly accurate wellbore rates are required and that puts an extreme computational burden on the semi-analytical or numerical methods involved. Even with the most powerful methods and hardware available, the calculation of the productivity index of directionally drilled and partially penetrating wells, especially in more complex drainage volumes is a formidable task. An additional problem is that in general, the productivity index for a well produced under constant pressure condition is different, although very close, from the productivity index of a well produced with constant rate. In this work we present a new technique to evaluate single well productivity indices both for constant pressure and for constant rate conditions. The approach is based on the solution of two steady state boundary value problems with constant pressure prescribed on the wellbore. The two productivity indices, (for constant rate and constant wellbore pressure, respectively) are then computed as integral characteristics of the solutions of the corresponding time dependent boundary value problem. The two productivity indices are computed independently. The method can be applied to any geometry of the reservoir (both 2-D and 3-D, regular or irregular), and any direction and length of penetration of the well. Designer wells (with a freely prescribed path) can be also considered.