Analytical Modeling of Linear Flow in Pressure-Sensitive Formations

Abstract

Summary The mathematical models commonly used to describe fluid flow through porous media are dependent on various simplifying assumptions, one of them being that the permeability is independent of pore pressure. However, during fluid withdrawal, reservoir permeability may be reduced because of deformation of the porous rock. The pressure dependence of permeability is more pronounced in tight formations. When dealing with pressure-sensitive formations, the assumption of a constant permeability is inappropriate. In this study, analytical models are developed to model production from hydraulically fractured tight formations with pressure-dependent permeability, for both constant-pressure- and constant-rate production scenarios. By deriving an explicit relationship between time and pseudotime, it is shown that the analytical liquid solutions can be directly applied to pressure-dependent permeability reservoirs. This paper develops the appropriate transformation, and discusses its application by comparing the numerical solution of the nonlinear problem with the analytical solution proposed here. The close agreement between these solutions demonstrates the accuracy of the proposed methodology in forecasting the behavior of pressure-sensitive formations. We used the models developed in this work to address the following question: Is it possible that the permeability is decreased so much that a reduction in rate results when drawdown is increased? We show that the answer to the question is no. Depending on the strength of the nonlinearity, there could be a point beyond which the rate will not improve measurably as the flowing pressure is lowered. However, for a particular reservoir with a constant-permeability modulus, it is not possible to reduce the production rate by increasing the drawdown. This is contrary to previous publications that suggest that in a reservoir with pressure-dependent permeability, there is an optimum drawdown for maximum production. In a companion paper, we explore conditions where such an optimum drawdown could exist.