Time-dependent matrix-fracture shape factors for partially and completely immersed fractures

The modeling of multiphase flow in fractured porous media relies on the accurate description of fracture-to-matrix transfer of water. Calculation of this rate, within dual-continuum models, depends on matrix-fracture transfer functions incorporating the so-called shape factor. Typically, matrix-to-fracture transfer functions are obtained by assuming all fractures to be instantaneously immersed in water (instantly-filled), with a uniform fracture pressure distribution under pseudo-steady state conditions. The result is constant, time-independent, shape factors. Clearly, this is not necessarily true. Partially immersed fractures and other unsteady-state conditions do not lead to constant shape factors. A new time-dependent matrix-fracture transfer shape factor formulation and transfer functions for both filling- and instantly-filled fracture transfer are derived based on dimensional analysis of experimental data. The dimensional analysis of full-physics data avoids simplifications that may lead to expressions that do not represent accurately matrix-fracture transfer. The new shape factor carries information about the transient behavior of the water saturation, Sw, and so it leads to more accurate description of the matrix-fracture transfer. Good agreement was found between experimental data, an analytical model, and a proposed modified dual-porosity formulation with the new time-dependent shape factor and transfer function.