Analytical Solution of the Nonlinear Hydraulic Diffusivity Equation with Depletion-Dependent Permeability in Oil Wells Near Sealing Faults

Identifying no-flow boundary areas is essential for a successful exploratory and production campaign during the development of an oil field. This work develops a new unsteady two-dimensional (2-D) integro-differential solution for permeability loss monitoring in a well near a sealing fault. The model presented in this study allows solving the nonlinear hydraulic diffusivity equation (NHDE) with an oil source term. Based on the well-known image method, the pressure field for the constant permeability solution is given by the sum of two exponential integral functions Ei ( t D ) . Although, this solution does not consider the nonlinear effect caused by the permeability loss. Therewith, a new deviation factor is derived and coupled to a first-order series expansion to approach this phenomenon. The proposed solution was calibrated by a finite-difference-based simulator named IMEX ® broadly used in reservoir engineering works, and the results were accurate. The results clearly present the instantaneous permeability decay effect by a deviation when compared to the linear solution in a semi-log plot. The main advantages of the proposed solution are the accuracy, availability of a wide table of the GFs, ease of implementation, and computational cost savings. It constitutes a useful and attractive mathematical tool to calibrate new models and support well-reservoir performance management.