All methods of seismic characterization of fractured reservoirs are based on effective media theories that relate geometrical and material properties of fractures and surrounding rock to the effective stiffnesses. In exploration seismology, the first-order theory of Hudson is the most popular. It describes the effective model caused by the presence of a single set of thin, aligned vertical fractures in otherwise isotropic rock. This model is known to be transversely isotropic with a horizontal symmetry axis (HTI). Following the theory, one can invert the effective anisotropy for the crack density and type of fluid infill of fractures, the quantities of great importance for reservoir appraisal and management.
Here I compute effective media numerically using the finite element method. I deliberately construct models that contain a single set of vertical, ellipsoidal, non-intersecting and non-interconnected fractures to check validity of the first-order Hudson’s theory and establish the limits of its applicability. Contrary to conventional wisdom that Hudson’s results are accurate up to crack density e ≈ 0.1, I show that they consistently overestimate the magnitudes of all effective anisotropic coefficients ε(V), δ(V), and γ(V). Accuracy of theoretically derived anisotropy depends on the type of fluid infill and typically deteriorates as e grows. While the theory gives | ε(V)|, |δ(V)|, |γ(V)| and close to the upper bound of the corresponding numerically obtained values for randomly distributed liquid-filled fractures, theoretical predictions of ε(V), δ(V) are not supported by numerical computations when the cracks are dry. This happens primarily because the first-order Hudson’s theory makes no attempt to account for fracture interaction which contributes to the final result much stronger for gas- than for liquid-filled cracks. I find that Mori-Tanaka’s theory is superior to Hudson’s for all examined crack densities and both types of fluid infill.
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The paper was presented at the 11th International Workshop on Seismic Anisotropy (11IWSA) held in St. John’s, Canada in 2004.
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Grechka, V. Penny-shaped fractures revisited. Stud Geophys Geod 49, 365–381 (2005). https://doi.org/10.1007/s11200-005-0015-3
- effective media
- single set
- horizontal transverse isotropy