Multiscale matrix-fracture transfer functions for naturally fractured reservoirs using an analytical discrete fracture model

Fracture matrix transfer functions have long been recognized as tools in modeling naturally fractured reservoirs. If a significant degree of fracturing is present, models involving isolated matrix blocks and matrix block distributions become relevant. However, this methodology captures only the largest fracture sets and treats the matrix blocks as homogeneous, though possibly anisotropic. Herein, we produce the semi-analytic transient baseline solution for depletion for such models. More realistic multi-scale numerical models try to capture below grid scale information and pass it to the larger scale system at some numerical cost. Instead, for below block scale information, we take the semi-analytic solution to the Diffusivity Equation of Hazlett and Babu (2014, 2018) for transient inflow performance of wells of arbitrary trajectory, originally developed for Neumann boundary conditions, and recast it for Dirichlet boundaries. As such, it represents the analytical solution for a matrix block with an arbitrarily complex gathering system surrounded by a constant pressure sink, we take to be the primary fracture system. Instead of using a constant rate internal boundary condition for the gathering system, we segment the well or fracture and force the internal complex fracture feature to be a constant pressure element with net zero flux. In doing so, we create a representative matrix block with any degree of infinite conductivity subscale fractures that impact the overall drainage into the surrounding fracture system. We quantify drainage from each face, capturing the anisotropic effect of internal fractures. We vary the internal fracture structure and delineate sensitivity to fracture spacing and extent of fracturing. This approach also generates the complete transient solution, enabling new well test interpretation for such systems in characterization of block size distributions or extent of below block-scale fracturing. The initial model for fully-penetrating fractures can be further generalized with the 2D distributed source model of Bao et al. (2017) for partially penetrating fractures of arbitrary inclination, as represented by floating, intersecting parallelograms embedded in the matrix block with either infinite or finite conductivity.