Fractional diffusion in rocks produced by horizontal wells with multiple, transverse hydraulic fractures of finite conductivity

Assuming diffusion in the rocks to be anomalous, a flux law that is nonlocal in space and time is used to develop a mathematical model for fractured rocks drained by a horizontal well produced through multiple transverse hydraulic fractures. As a result the governing differential equation is fractional in character. The conductivity of the fractures is assumed to be finite and their properties (width, length, permeability, etc.) may be variable. Expressions for the well response that produces at a constant rate or at a constant pressure are derived in terms of the Laplace transformation. Approximate analytical solutions are derived and the analytical development provides perspectives on short and long-time well behaviors. In addition to outlining characteristic features of the model, the analytical solutions are useful in verifying numerical computations. The computational results obtained by the Stehfest algorithm establish the robustness and viability of the mathematical model. Comparisons with classical diffusion are noted.