Transient pressure behavior for a reservoir with continuous permeability distribution in the invaded zone

A mathematical model is presented for a reservoir with a continuous permeability distribution in the invaded zone. An analytical solution of the system was obtained in Laplace space. Solutions in real time space for the pressure and pressure derivative are obtained by numerical inversion of the Laplace transform using Stehfest algorithm. Solutions were obtained for infinite reservoirs and for finite reservoirs with closed and constant pressure outer boundaries. The permeability in the invaded zone varies from a skin permeability K_s at the wellbore to the original permeability K_e at the end of the invaded zone R_s. The effects of the permeability ratio, the depth of invasion, the reservoir size, and the wellbore storage were investigated. Results of this study show that for this permeability distribution, the permeability ratio has a profound effect on the pressure behavior while the depth of invasion plays an insignificant role. It also showed that the use of the average permeability of the invaded zone in the steady state skin equation gives almost identical results of the additional pressure drop due to formation damage at large times as those obtained from the exact solution. An approximate solution can be used for relatively high invasion depths (R_SD > 20) which shows that the additional pressure drop is independent of the invasion depth Copyright 2009. Society of Petroleum Engineers.