Analytical modeling of linear flow with variable permeability distribution in tight and shale reservoirs

Horizontal wells with multiple fractures, producing from tight and shale reservoirs, may exhibit linear flow for long periods of time. Majority of the available analytical solutions that model this flow behaviour assume uniform permeability, which most likely is an over-simplification of the unconventional reservoirs. Non-homogeneous shear or tensile failure away from the main induced (primary) hydraulic fractures can lead to a non-uniform permeability distribution that depends on the distance away from the hydraulic fractures. In this work, linear flow in a reservoir with non-uniform permeability adjacent to the primary hydraulic fracture is modeled rigorously using perturbation theory. The diffusivity equation is solved for the pressure response of a fractured well located in a reservoir of infinite extent with permeability as an arbitrary function of position. For constant terminal rate (CTR) or constant terminal pressure (CTP) conditions, the linear flow parameter (LFP = x(f) root k) is calculated from the slope of the square-root-of-time plot (plot of rate-normalized pressure vs. square root of time). It is demonstrated herein that the calculated LFP corresponds to a weighted average of permeabilities (and fracture half-lengths); different parts of the reservoir contribute differently to the LFP at different production times. The LFP is influenced most strongly by permeabilities at a distance y = 0.056 root(kt)/phi mu c(t). The derived weighting functions during CTR and CTP production can be applied in inverse mode for determining LFP distribution near the hydraulic fractures. This is particularly useful in evaluating the effectiveness of hydraulic fracturing operations and assessing the performance of different fracturing techniques in unconventional reservoirs. In addition, this work gives significant insight into the concept of distance of investigation (DOI) in tight and shale reservoirs, and the differences when producing under CTR and CTP conditions.