Abstract
Summary We construct a representative single matrix block model surrounded by fully penetrating vertical fractures that analytically captures both fracture and block depletion with fracture-matrix mass transfer. This is done with and without internal fracture structure in the matrix block. We solve the 1D Green’s function for a fracture system enveloping a matrix block in terms of the time evolution of average fracture pressure. We, likewise, solve the Green’s function for the average pressure in a matrix block surrounded by a constant pressure boundary. Average pressure is then easily transferred by material balance into cumulative production or instantaneous flowrate. Primary variables in assembling the interacting systems model are the volume ratio, Vf/Vm, permeability ratio, kf/kx, and geometry, (a/b)(ky/kx), with the last term accounting for both block shape and permeability anisotropy. The single block model is extended to include the influence on depletion rate for matrix blocks with internal, sub-scale fractures by solving for the Green’s function average pressure with a net-zero flux discrete fracture model. Internal fractures are infinite conductivity features that allow for increased mass transport to the matrix boundary and more rapid depletion. The single matrix block model is migrated to one for heterogeneous systems using superposition and matrix block distributions. We illustrate the signatures in pressure and Bourdet derivative for homogeneous and heterogeneous models. The influence of subscale fractures can be fully captured in influence of kx, ky/kx, or effective block size and shape, provided fracture characteristics are upscaled using this or other approaches.