Formation permeability of hydraulically fractured reservoirs is pressure sensitive and spatially variant. Earlier analytical models describing pressure behavior of those reservoirs either account for spatial dependence or stress-dependence effect on permeability field. In this study, a novel analytical approach is presented to account for the combined effect of stress-sensitivity and spatial variability of formation permeability.
The model considers that the reservoir fluid flows linearly in the stimulated reservoir volume (SRV) and discharges to the infinitely conductive hydraulic fractures. A newly derived diffusivity equation for single porosity continuum is employed to seek solution for the constant terminal rate condition. As the hydraulic permeability of SRV is a function of space and pressure, the resulting diffusivity equation is highly nonlinear. After weakening the nonlinearity by applying Pedrosa's transformation, features of the perturbation technique and modified Bessel functions are implemented to determine the analytical solution in Laplace space. Finally, the real-time solution is obtained by inverting through Stehfest and GWR method.
The model captures the combined effect by incorporating two characteristic parameters: permeability modulus and threshold permeability. These parameters were turned off to compare and validate the obtained solutions with the corresponding conventional analytical solutions. It was found that the earlier models may underestimate the pressure behavior significantly throughout the reservoir lifetime if the temporal and spatial variability of permeability is disregarded. Moreover, it was observed that the permeability modulus has a comparatively more significant effect on pressure drawdown behavior than that of the stimulation ratio at the late times. Pressure propagation profiles for different sets of model parameters were generated. The effect of dimensionless permeability modulus and threshold permeability were analyzed to draw a contrast among constant, spatially varying and combined-effect case. Error propagation among those three conceptualized cases were studied.
The proposed analytical approach describes the dynamic behavior of SRV continuum more realistically by taking the spatial and pressure-dependent effect of permeability into account. Compared to the available analytical models, this approach can be employed to glean more accurate attributes of the hydraulically fractured horizontal well. The proposed concept could further be extended to develop solutions for the dual-porosity continuum.