Poroelastic Solution for the Nonlinear Productivity Index of Wells in Stress-Sensitive Reservoir Rocks

Reservoir depletion may induce substantial changes in the stress state of the subsurface rock. The interaction between the pore fluid pressure and rock stress alters the reservoir rock porosity and permeability which, in turn, can reversely affect the productivity index (PI) of producing wells. A nonlinear analytical solution is developed for the drawdown-dependent PI of reservoirs under a steady-state flow regime. Biot's theory of poroelasticity is used to derive the depletion-induced changes in the reservoir rock porosity and permeability. The well-known Mindlin's solution for a nucleus of strain in a semi-infinite elastic medium is adopted as Green's function and integrated over the depleted volume of a disk-shaped reservoir to obtain the 3D distribution of rock stress and volumetric strain. The fluid transport equation is nonlinearly related to the solid mechanics side of the problem via the stress-dependent permeability coefficients. A perturbation technique is used to mathematically treat the described nonlinearity and analytically solve the equations of pore fluid flow and rock stress under steady-state flow regime. A good match is captured between the obtained analytical perturbation solution and the numerical finite difference solution of the same problem. Results confirm the expected strong dependence of the Well PI on the drawdown magnitude. The poroelastic constitutive parameters of the reservoir rock determine the extent of such dependency. The rock initial porosity has the strongest influence on the Well PI, followed by the reservoir initial permeability and solid grain modulus, while the reservoir depth to radius ratio and the Poisson's ratio are found to be the least sensitive parameters.