Dimensionless Solutions for the Time-Dependent and Rate-Dependent Productivity Index of Wells in Deformable Reservoirs

Reservoir depletion is known to reduce the porosity and permeability of stress-sensitive reservoir rocks. The effect may substantially hinder the productivity index (PI) of producing wells. This study presents analytical solutions for the time-dependent and steady-state well PIs, respectively, of a bounded, disk-shaped, elastic reservoir with no-flow and constant-pressure conditions at the outer boundary. A combination of Green's functions, the Laplace transform method, and the perturbation technique is used to solve the governing nonlinear partial differential equations of the considered coupled problems of flow and geomechanics. Dimensional analyses based on the Buckingham Π theorem are conducted to identify the dimensionless parameters groups of each problem and to express the resulting analytical solutions in the dimensionless form. In addition, necessary corrections to an existing error in the reported Green's functions for the induced strain field of a ring-shaped pressure source within an elastic half-space (Segall 1992) are made. The corrected Green's functions are used to obtain the strain induced by the pore fluid pressure distribution within a depleting disked-shaped reservoir. Consequently, a corrected permeability variation model compared to our previously published, time-independent solution for rate-dependent PI (Zhang and Mehrabian 2021a) is presented. Finally, a mechanistically rigorous formulation of the permeability modulus parameter that commonly appears in the pertinent literature is suggested. In addition to the in-house developed finite-difference solutions, the presented analytical solutions are verified against results from the finite-element simulation of the same problems using COMSOL® Multiphysics (2018). The obtained rate-dependent PI of the reservoir is controlled by four dimensionless parameters, namely, the dimensionless rock bulk modulus, the Biot-Willis effective stress coefficient, Poisson's ratio, and rock initial porosity. The pore fluid pressure solution is shown to asymptotically approach the corresponding flow-only solution for large values of the dimensionless rock bulk modulus. Parametric analysis of the solution suggests that the well productivity loss has a reverse relationship with the dimensionless bulk modulus and initial porosity of the rock, whereas a direct relationship is identified with Biot-Willis effective stress coefficient and Poisson's ratio. Compared to the reservoir with a constant-pressure outer boundary, the PI of a reservoir with a no-flow condition at the outer boundary is shown to be more significantly hindered by the stress sensitivity of the reservoir rock.