For the successful development and operation of hydrocarbon or geothermal reservoirs, knowledge of the hydraulic transport is of crucial importance. Because fundamental physical processes of borehole fluid injections are still insufficiently understood, gathering information about transport properties of rocks under field conditions is quite difficult. However, a substantial contribution in determining the permeability evolution can be obtained by understanding the distribution of induced seismicity in space and time. We have analyzed spatio-temporal characteristics of seismicity recorded during a hydraulic fracturing treatment in the Barnett Shale. In this study, we show that the fluid-rock interaction is nonlinear. To explain corresponding spatio-temporal features of induced seismicity, we considered pore pressure diffusion based on a power-law pressure dependence of permeability. A scaling approach was used to transform clouds of hypocenters of events obtained in a hydraulically anisotropic nonlinear medium into a cloud which would be obtained in an equivalent isotropic but still nonlinear medium. For this, we used a concept of a factorized anisotropic pressure dependence of permeability and found that it is in agreement with the microseismic data under consideration. We used a numerical modeling approach to generate synthetic seismicity by solving nonlinear diffusion equations. The pore-pressure field obtained from flow rates was calibrated with the pore-pressure field computed for injection pressures. This yielded an estimate of the uniaxial storage coefficient and permitted us to compute the permeability evolution inside the fracture stimulated reservoir. Following our modeling, we generated synthetic seismicity whose spatio-temporal features are similar to the ones observed in the case study. This indicates that a nonlinear diffusion with a pressure-dependent permeability seems to provide a reasonable model of the hydraulic-fracture stimulation under consideration. A power-law pressure dependence of stimulated permeability may be a more general characteristic for shales.