On Productivity Index in Pseudo-steady and Boundary- dominated Flow Regimes

We analyze non-stationary, but stabilized flow towards a well in a reservoir with impermeable external boundaries (volumetric reservoir.) Two operating conditions: fixed production rate and given wellbore pressure are considered. Determination of conditions under which the Productivity Index (PI) is time invariant is the focus of our study. For the fixed production rate case the flow regime providing constant PI is called pseudo-steady state. In case of constant wellbore pressure the flow regime characterized by time invariant PI is called boundary-dominated state. It is shown that for any reservoir-well geometry there exists a one-parameter family of initial pressure distributions for which the PI will be the same at any time, if the production rate is kept constant. This pseudo-steady state PI is independent of the actual production rate. Similarly, there exists a one-parameter family of initial pressure distributions assuring boundarydominated state if the wellbore pressure is kept constant. The boundary-dominated PI does not depend on the actual wellbore pressure. The two families of initial pressure distributions are different and the two values of the PI also differ. For the well located in the center of a closed circular reservoir we derive exact mathematical expressions for both PI-s and compare the numerical values.