AbstractFluid injection in porous rock occurs in a variety of hydrogeological operations including wastewater disposal, CO2 sequestration, enhanced oil recovery, and geothermal energy extraction. Influx of fluid into the host rock is usually accompanied by induced permeability enhancement due to increased pore fluid pressure and consequent dilation of the pore volume. The effect would, in turn, enhance the well injectivity index. This paper presents an analytical solution to the nonlinear problem of underground fluid injection in a disk‐shaped reservoir while accounting for the induced permeability enhancement. The host rock permeability is considered to depend on the rock total stress and pore fluid pressure. For this purpose, the rock stress is formulated as a nonlocal function of disturbances in the pore fluid pressure using the fundamental solution for a nucleus of strain in an elastic half space. Alongside, the nonlinear fluid transport equation that incorporates stress‐sensitive rock permeability is analytically solved using a perturbation technique. The good match with finite difference solution to the same problem verifies the validity and accuracy of the perturbation solution. Parametric study on a test case problem is conducted to examine the influence of different parameters on well injectivity. Findings reveal the higher vulnerability of shallower host reservoirs to stress‐induced permeability variations. The effect of rock poroelastic properties on fluid transport process is discussed.