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

Fracture matrix transfer functions have long been recognized as tools in modelling naturally fractured reservoirs. If a significant degree of fracturing is present, models involving single matrix blocks and matrix block distributions become relevant. However, this captures only the largest fracture sets and treats the matrix blocks as homogeneous, though possibly anisotropic. Herein, we produce the steady and transient baseline solutions for depletion for such models. Multiscale models pass below grid scale information to the larger scale system with some numerical cost. Instead, for below block scale information, we take the analytic solution to the Diffusivity Equation for transient inflow performance of wells of arbitrary trajectory, originally developed for Neumann boundary conditions, and recast it for Dirichlet boundaries with possible internal fractures of variable density, length, and orientation. As such, it represents the analytical solution for a heterogeneous matrix block surrounded by a constant pressure sink, we take to be the primary fracture system. Instead of using a constant rate internal boundary condition on a fracture surrounded by matrix, we segment the fracture and, through imposed material balance, 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 infinite conductivity subscale fractures that impact the overall drainage into the surrounding fracture system. We vary the internal fracture structure and delineate sensitivity to fracture spacing and extent of fracturing. We generate 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 extended to 3D, generalized floating fractures of arbitrary inclination, and internal complex fracture networks.