Fundamental Flux Equations for Fracture-Matrix Interactions with Linear Diffusion

The conventional dual-continuum models are only applicable for late-time behavior of pressure propagation in fractured rock, while discrete-fracture-network models may explicitly deal with matrix blocks at high computational expense. To address these issues, we developed a unified-form diffusive flux equation for 1D isotropic (spheres, cylinders, slabs) and 2D/3D rectangular matrix blocks (squares, cubes, rectangles, and rectangular parallelepipeds) by partitioning the entire dimensionless-time domain (Zhou et al., 2017a, b). For each matrix block, this flux equation consists of the early-time solution up until a switch-over time after which the late-time solution is applied to create continuity from early to late time. The early-time solutions are based on three-term polynomial functions in terms of square root of dimensionless time, with the coefficients dependent on dimensionless area-to-volume ratio and aspect ratios for rectangular blocks. For the late-time solutions, one exponential term is needed for isotropic blocks, while a few additional exponential terms are needed for highly anisotropic blocks. The time-partitioning method was also used for calculating pressure/concentration/temperature distribution within a matrix block. The approximate solution contains an error-function solution for early times and an exponential solution for late times, with relative errors less than 0.003. These solutions form the kernel of multirate and multidimensional hydraulic, solute and thermal diffusion in fractured reservoirs.