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Approximate solutions for diffusive fracture‐matrix transfer: Application to storage of dissolved CO 2 in fractured rocks
Author(s) -
Zhou Quanlin,
Oldenburg Curtis M.,
Spangler Lee H.,
Birkholzer Jens T.
Publication year - 2017
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1002/2016wr019868
Subject(s) - dimensionless quantity , isotropy , exponential function , permeability (electromagnetism) , anisotropy , thermal diffusivity , matrix (chemical analysis) , mathematics , mathematical analysis , mechanics , materials science , physics , thermodynamics , chemistry , biochemistry , quantum mechanics , membrane , composite material
Analytical solutions with infinite exponential series are available to calculate the rate of diffusive transfer between low‐permeability blocks and high‐permeability zones in the subsurface. Truncation of these series is often employed by neglecting the early‐time regime. In this paper, we present unified‐form approximate solutions in which the early‐time and the late‐time solutions are continuous at a switchover time. The early‐time solutions are based on three‐term polynomial functions in terms of square root of dimensionless time, with the first coefficient dependent only on the dimensionless area‐to‐volume ratio. The last two coefficients are either determined analytically for isotropic blocks (e.g., spheres and slabs) or obtained by fitting the exact solutions, and they solely depend on the aspect ratios for rectangular columns and parallelepipeds. For the late‐time solutions, only the leading exponential term is needed for isotropic blocks, while a few additional exponential terms are needed for highly anisotropic rectangular blocks. The optimal switchover time is between 0.157 and 0.229, with highest relative approximation error less than 0.2%. The solutions are used to demonstrate the storage of dissolved CO 2 in fractured reservoirs with low‐permeability matrix blocks of single and multiple shapes and sizes. These approximate solutions are building blocks for development of analytical and numerical tools for hydraulic, solute, and thermal diffusion processes in low‐permeability matrix blocks.