A Generalized Multirate Memory Function for Modeling Flow and Transport in Fractured Reservoirs

Fractures and reservoir heterogeneity complicate subsurface flow and transport relevant to many applications, such as geological CO₂ storage, geothermal energy production, nuclear waste disposal, and groundwater remediation. To simplify fracture-induced modeling complexity, we developed a generalized flux equation for linear diffusion (Zhou et al., WRR, 2017a, b). The diffusive flux equation contains an early-time solution with a three-term polynomial in terms of square root of dimensionless time and a late-time exponential solution, the two of which are continuous at the switch-over time. The flux equation is applicable to any regular 1-D, 2-D, and 3-D matrix blocks (e.g., slabs, rectangles, and rectangular parallelepipeds) formed by one, two, and three sets of orthogonal fractures. On the basis of the flux equation, we have now developed a multirate memory function in the Laplace domain to account for hydraulic, solute, or thermal exchange between well-mixed fractures and matrix blocks of different shapes, sizes, volume fractions, and properties in a representative elementary volume of fractured media. We have plugged the memory function into global transfer functions of solute/heat transport in fractures without fracture-matrix coupling and obtained a suite of analytical solutions for fractured reservoirs with 1-D linear and radial and 2-D dipole flow. We have also performed modeling of global transport through focused flow channels to show the effect of flow channeling on early solute/thermal breakthroughs. We have compared the performance (computational efficiency and accuracy) of the developed modeling methodology to that of conventional dual-continuum methods, and discussed discrete fracture network and discrete fracture-matrix modeling approaches and applicability in various fractured reservoir systems.