Vertically fractured transversely isotropic media: dimensionality and deconstruction |
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| Authors: | Michael A Schoenberg |
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| Institution: | Earth Sciences Division, Lawrence Berkeley National Lab, 1 Cyclotron Road, Berkeley, CA 74720, USA |
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| Abstract: | A vertically fractured transversely isotropic (VFTI) elastic medium is one in which any number of sets of vertical aligned fractures (each set has its normal lying in the horizontal x1, x2‐plane) pervade the medium and the sets of aligned fractures are the only features of the medium disturbing the axi‐symmetry about the x3‐axis implying that in the absence of fractures, the background medium is transversely isotropic (TI). Under the assumptions of long wavelength equivalent medium theory, the compliance matrix of a fractured medium is the sum of the background medium's compliance matrix and a fracture compliance matrix. For sets of parallel rotationally symmetric fractures (on average), the fracture compliance matrix is dependent on 3 parameters ? its normal and tangential compliance and its strike direction. When one fracture set is present, the medium is orthorhombic and the analysis is straightforward. When two (non‐orthogonal) or more sets are present, the overall medium is in general elastically monoclinic; its compliance tensor components are subject to two equalities yielding an 11 parameter monoclinic medium. Constructing a monoclinic VFTI medium with n embedded vertical fracture sets, requires 5 TI parameters plus 3×n fracture set parameters. A deconstruction of such an 11 parameter monoclinic medium involves using its compliance tensor to find a background transversely isotropic medium and several sets of vertical fractures which, in the long wavelength limit, will behave exactly as the original 11 parameter monoclinic medium. A minimal deconstruction, would be to determine, from the 11 independent components, the transversely isotropic background (5 parameters) and two fracture sets (specified by 2 × 3 = 6 parameters). Two of the background TI medium's compliance matrix components are known immediately by inspection, leaving nine monoclinic components to be used in the minimal deconstruction of the VFTI medium. The use of the properties of a TI medium, which are linear relations on its compliance components, allows the deconstruction to be reduced to solving a pair of non‐linear equations on the orientations of two fracture sets. A single root yielding a physically meaningful minimum deconstruction yields a unique minimal representation of the monoclinic medium as a VFTI medium. When no such root exists, deconstruction requires an additional fracture set and uniqueness is lost. The boundary between those monoclinic media that have a unique minimal representation and those that do not is yet to be determined. |
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