Abstract

We solve a problem of fluid dynamics in an arbitrarily shaped fracture containing proppant. The fracture is embedded in an absolutely rigid solid. The fluid flow is induced by normal harmonic oscillations of fracture walls that are permeable, allowing the fluid to filtrate into the surrounding formation. The amplitude of oscillations can be described as a function of a spatial coordinate along the fracture. Fluid flow inside the fracture is described by Biot's modification of Darcy's law that includes both inertia and filtration terms. Filtration inside the fracture may drastically change its acoustic characteristics. The acoustic resonance along the length of the fracture filled with proppant can be observed only if the permeability of the proppant is extremely high; otherwise filtration of the fluid in the fracture dampens the resonance. If the resonance is observed, its period differs from that in a fracture without internal filtration. The apparent acoustic velocity in the fracture is lower than sound velocity in the fluid because of coupling between the fluid and the proppant. In a fracture without proppant, fluid filtration into the surrounding rock acts to decrease resonance peaks and to increase resonance period. Resonance period changes significantly in fractures of varying shape, an effect which is important when estimating the size of a tapered fracture.

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