The current state-of-the-art modeling approaches for contaminant/heat transport in fractured rock include (1) discrete fracture-network (DFN) and discrete fracture-matrix (DFM) models with the fracture network and matrix blocks randomly generated, (2) numerical models based on conventional dual-continuum models, and (3) analytical models with simplified parallel fractures and slab-like matrix blocks. These models differ in the complexity of the fracture network and matrix blocks, modeling accuracy, and computational efficiency, making it difficult to compare their results through benchmark problems.
We developed several benchmark problems of contaminant/heat transport under different flow (e.g., linear, radial, and dipole) fields and provided corresponding analytical solutions to global advection-dispersion coupled with multirate diffusion in the rock matrix. The benchmark problems consist of (1) a fracture network of one, two, and three sets of orthogonal fractures with varying fracture spacing and (2) matrix blocks of various shapes (slabs, squares, rectangles, cubes, and rectangular parallelepipeds) and sizes bounded by the fracture network. The matrix blocks can be isotropic with the same fracture spacing in each direction and anisotropic with varying aspect ratios.
The multirate diffusion caused by different shapes and sizes of matrix blocks was accounted for by using a unified-form diffusive flux equation for 1D isotropic (spheres, cylinders, slabs) and 2D/3D rectangular matrix blocks (squares, cubes, rectangles, and rectangular parallelepipeds) in 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. These solutions form the kernel of multirate and multidimensional hydraulic, solute, and thermal diffusion in fractured reservoirs.
The transient flux equation for multirate diffusion was transformed to develop the analytical solutions to the benchmark problems in the Laplace domain with typical functions (e.g., Airy functions) for the global advection-dispersion equation. The benchmark solutions can bridge the gaps between the three modeling approaches with reasonable complexity of fracture network and matrix blocks, as well as high modeling accuracy and efficiency. They are very useful for benchmarking the DFN/DFM modeling, whose accuracy depends on how to capture the fracture-matrix diffusive transfer and diffusion within each matrix block.
Zhou, Q., C.M. Oldenburg, J. Rutqvist, J.T. Birkholzer, 2017. Revisiting the fundamental analytical solutions of heat and mass transfer: The kernel of multirate and multidimensional diffusion, Water Resources Research 53, doi: 10.1002/2017WR021040.
Zhou, Q., C.M. Oldenburg, L.H. Spangler, and J.T. Birkholzer, 2017. Approximate solutions for diffusive fracture-matrix transfer: Application to storage of dissolved CO2 in fractured rocks, Water Resources Research, 53(2), 1746–1762, doi:10.1002/2016WR019868.
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