Effects of fracture boundary conditions on matrix-fracture transfer shape factor

The matrix-fracture transfer shape factor is one of the important parameters in modeling naturally fractured reservoirs. Four decades after Warren and Root (1963, SPEJ, 245-255.) introduced the double porosity concept and suggested a relation for it, this parameter is still not completely understood. Even for a single-phase flow problem, investigators report different shape factors. This study shows that for a single-phase flow in a particular matrix block, the shape factor that Warren and Root defined is not unique and depends on the pressure in the fracture and how it changes with time. We use the Laplace domain analytical solutions of the diffusivity equation for different geometries and different boundary conditions to show that the shape factor depends on the fracture pressure change with time. In particular, by imposing a constant fracture pressure as it is typically done, one obtains the shape factor that Lim and Aziz (1995, J. Petrolean Sci. Eng. 13, 169.) calculated. However, other shape factors, similar to those reported in other studies are obtained, when other boundary conditions are chosen. Although, the time variability of the boundary conditions can be accounted for by the Duhamel's theorem, in practice using large time-steps in numerical simulations can potentially introduce large errors in simulation results. However, numerical simulation models make use of a stepwise approximation of this theorem. It is shown in this paper that this approximation could lead to large errors in matrix-fracture transfer rate if large time-steps are chosen.