Abstract
Summary Space–time fractional diffusion in a linear, bounded region is considered. An analytical expression for the pressure distribution in the bounded region is derived in terms of the Mittag-Leffler function and the Laplace transformation. Comparisons with numerical solutions indicate excellent agreement. A convenient expression that incorporates a combination of Dirichlet and Neumann boundary conditions, particularly suitable for rapid computations, has been elusive until now. Responses that may be expected in nanoporous reservoirs that are usually produced through horizontal wells containing multiple hydraulic fractures are deduced. It is shown that the existence of obstacles, discontinuities, and other complex flow paths in fractured reservoirs express themselves in the form of power-law declines, whereas long-range interactions reflective of rapid communication, interestingly, express themselves as exponential declines. An understanding of the simultaneous influences of these two effects is the purpose of this study. The application to the “trilinear” model often used to evaluate well performance in shales is demonstrated.
Published Version
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