A discrete, arbitrarily oriented 3D plane-source analytical solution to the diffusivity equation for modeling reservoir fluid flow

A highly accurate and efficiently computable analytical solution to the diffusivity equation is presented for modeling fluid flow into a 3D, arbitrarily oriented plane sink within a box-shaped, anisotropic medium with Neumann boundary conditions. The plane sink represents a gathering system for a well stimulated by means of hydraulic fracturing. Our plane-source Neumann function arises from analytic double integration of the point-source solution to the diffusivity equation along two vectors, forming a parallelogram. A Neumann boundary condition is achieved by means of the method of images, resulting in triple infinite summations that are reduced with mathematical identities to a combination of closed-form expressions and infinite sums with exponential damping. Our solution forecasts time-dependent behavior of fractured wells, useful in interpreting field experiments for the characterization of fracturing efficacy, reservoir size, and matrix fluid-transport properties. We demonstrate our model with two applications. One is pressure-transient analysis with identified flow regimes from a pressure vs. time plot. The other is pseudosteady-state (PSS) pressure mapping, simulating inflow from multiple fractures along the trajectory of a single horizontal well, which is achieved with superposition theory and adjustment of flux strength of each plane source to achieve a common pressure at each well/fracture intersection.