Fractures in the subsurface serve as conduits for fluids and gas, connecting remote hydrocarbon reservoir sections to production wells. Seismic and sonic data are popular sources for information on fracture properties. The most commonly used model to extract fracture information from such data is based on the paradigm of the displacement discontinuity interface, without a direct link to relevant characteristics such as the surface roughness properties of a fracture. Indeed, fractures can be modeled as displacement discontinuity surfaces, and in this sense they resemble traction-free cracks. In literature, cracks and fractures are not always properly distinguished, perhaps because the terms are often perceived as synonyms. However, microstructural parameters that control magnitudes of the discontinuities — and thus the effective stiffnesses — are entirely different: statistics of contacts for fractures versus crack density for traction-free cracks. We explore the effective elasticity of rocks containing multiple fractures using a model of a fracture as two rough surfaces with isolated contacts. This is done in the context of the incremental, linear elastic response to small stress changes, typical in wave-propagation problems. Fractures are dry or may have diverse orientations, and contacts may or may not be Hertzian. A link exists between contact characteristics and effective stiffness of single and multiple fractures. Our work examines and accounts for the strong effect of interactions between individual contacts by means of a double sum over mutual positions as well as outlines the differences and similarities between theories for cracks and fractures.

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