Transient flow in a linear reservoir for space–time fractional diffusion

• A fractional space–time transient equation for a complex porous media is derived.
• Solution derived in terms of Laplace transformation and Mittag–Leffler functions.
• Solution is suitable to evaluate early-time trends of fractured wells.

A one-dimensional, fractional-order, transient diffusion equation is constructed to model diffusion in complex geological media. Such a conceptual model permits for the incorporation of a wide range of velocities as fluid particles in high and low permeability paths perform complex motions. The transient diffusion equation is non-local in character with both spatial and temporal fractional derivatives. The pressure distribution is derived in terms of the Laplace transformation and the Mittag–Leffler function. Results are used to deduce expectations in the early-time response of a fractured well producing complex reservoirs such as unconventional shales. The flux law considered here allows for declines in rate that are faster or slower than models based on classical diffusion. A brief survey of the Mittag–Leffler function and its computation is provided. We apply the results derived to obtain solutions for the ‘trilinear’ model that is often used to evaluate horizontal well performance.

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