An accurate interporosity flow equation incorporating a time-dependent shape factor is derived and verified for improved dual-porosity modeling of waterflooding in naturally fractured reservoirs. This equation expresses the interporosity exchange rate in terms of the oil phase pressure gradient in the matrix, fracture surface area, fluid effective permeability at the matrix/fracture interface, fluid viscosity and a variable matrix-block shape factor. This approach can accommodate the flow directed from matrix to fractures while representing the permeability of interconnected fractures as a tensor. The model equations are expressed in dimensionless form for convenient integration into present numerical simulators for accurate simulation of waterflooding in naturally fractured reservoirs. These features distinguish this model from the conventional sugar-cube modeling of naturally fractured reservoirs.
Fine-grid numerical simulation of a matrix block is performed to verify the flow equation using the time-dependent shape factor. Numerical experiments with various size matrix blocks indicate the shape factor varies with time and converges to the steady-state shape factor value reported in previous studies for single-phase flow. The shape factor for single-phase flow converges to a steady-state value at a speed proportional to the reciprocal of total compressibility, while the shape factor for two-phase flow converges at a speed proportional to the slope obtained from the capillary pressure curve evaluated at the average water saturation present at the matrix/fracture interface. Therefore, the single-phase shape factor converges much more rapidly to its steady-state value than the two-phase shape factor.
This study demonstrates that neglecting the time-dependency of the shape factor introduces significant errors in fractured reservoir simulation, and time-dependent shape factors are needed because of the variation of the water saturation front location in the matrix block, which moves from the fracture face to the block-center as the water flows from the fracture into the matrix.