A Generalized Dynamic Transfer Function for Ultra-Tight Dual-Porosity Systems

The Warren and Root (1963) transfer function laid the foundation for describing the mass transfer between the matrix and fracture blocks in dual-porosity (DP) reservoir simulation. However, the pseudo steady-state (PSS) assumption imbedded in the approach of Warren and Root is no longer applicable when the duration of the transient state is prominent (tight oil or shale gas reservoirs). Lim and Aziz (1995) derived new shape factors in a framework that avoids the PSS assumption. However, similar to the formulation of Warren and Root, the approximation of Lim and Aziz fails to capture the pressure gradients inside matrix blocks for tight rocks with substantial characteristic times for mass transfer. In this paper, we introduce a generalized dynamic transfer function that can accurately predict the pressure response of ultra-tight DP formations. Based on the Vermeulen (1953) approximate solution, we first derive the new transfer function to model fluid flow for one, two, and three sets of perpendicular fractures where the matrix blocks are approximated by planar, cylindrical, and spherical geometries, respectively. Then, we apply it for rocks with anisotropic permeability. We extend our transfer function to represent more realistic geology by considering irregular-shaped matrix blocks. The proposed transfer function accounts for physical mechanisms at play in the reservoir and is applicable to describe different diffusion-type processes. Development and testing of the dynamic transfer function were done in the open-source environment of the MATLAB Reservoir Simulation Toolbox (MRST). The implementation was validated using single-block DP calculations and fine grid single-porosity (SP) models. We report results from several examples covering a broad range of reservoir parameters. For comparison purposes, we also report the simulation results from a traditional transfer function, incorporating Lim and Aziz shape factors, in MRST and commercial simulators (CMG and ECLIPSE). We demonstrate that our proposed dynamic transfer function accurately predicts the pressure response across early and late times, while the traditional transfer function with Lim and Aziz shape factors can depart substantially from the true solution for ultra-tight DP reservoirs.