The dynamics of viscous compressible fluid in a fracture

We solve a problem of fluid dynamics in an arbitrarSignificant progress has been made in the quantitative ily shaped fracture containing proppant. The fracture modeling of the dynamic interaction of a fracture with is embedded in an absolutely rigid solid. The fluid flow surrounding elastic rock. Using the finite-difference method, is induced by normal harmonic oscillations of fracture Chouet (1986) described the transient response of a fluidwalls that are permeable, allowing the fluid to filtrate driven crack to an impulsive pressure in the fluid. This into the surrounding formation. The amplitude of impulse triggers resonance with periods proportional to the oscillations can be described as a function of a spatial crack stiffness and to the width and length of the crack. The coordinate along the fracture. Fluid flow inside the crack wave sustaining the resonance is analogous to a tube fracture is described by Biot’s modification of Darcy’s wave propagating in a fluid-filled borehole. Its wave speed is law that includes both inertia and filtration terms. always lower than the acoustic velocity of the fluid. This Filtration inside the fracture may drastically change its theoretical model has been used successfully to explain the acoustic characteristics. The acoustic resonance along very long period of a volcanic tremor (Chouet, 1988). the length of the fracture filled with proppant can be Ferrazzini and Aki (1987) analytically studied normal observed only if the permeability of the proppant is modes trapped in a liquid layer sandwiched between two extremely high; otherwise filtration of the fluid in the elastic half-spaces. They also found a slow wave similar to a fracture dampens the resonance. If the resonance is tube wave. The phase velocity and amplitude of this slow observed, its period differs from that in a fracture wave are in close correlation with those obtained by numerwithout internal filtration. The apparent acoustic veical models. Using a three-dimensional (3-D) fluid-filled locity in the fracture is lower than sound velocity in crack model, Ferrazzini et al. (1990) were able to quantitathe fluid because of coupling between the fluid and the tively estimate the size of a fracture during hydrofracture proppant. In a fracture without proppant, fluid filtraexperiments at Fenton Hill, New Mexico. tion into the surrounding rock acts to decrease resoAn experimental proof to the existence of slow fracture nance peaks and to increase resonance period. Resowaves as predicted in Chouet (1986) and Ferrazzini and Aki nance period changes significantly in fractures of (1987) has been given in Tang and Cheng (1988). varying shape, an effect which is important when Hornby et al. (1989) used the low-frequency reflected estimating the size of a tapered fracture. Stoneley-wave mode to locate permeable fractures intersecting a borehole and to estimate their effective apertures. The fracture is idealized as a plane-parallel fluid slab of infinite extent. The authors show that within the considered range of INTRODUCTION parameters, the results do not change significantly when the condition of finite extent is included. The acoustic properties of both microfractured rocks and Mahrer and Mauk (1987) modeled a two-dimensional (2-D) rocks with macrofractures are strongly influenced by the hydraulic fracture surrounded by a low-velocity zone to relative motion between solid and fluid. The dependence of investigate hydraulic fracture treatment-induced microseisvelocity and attenuation of acoustic waves on the geometry micity. They found that a fixed-aperture fracture acted as a of microfractures and on fluid properties can, in principle, leaky waveguide confining seismic energy and altering propserve to evaluate material structure and transport characteragation characteristics. Another important conclusion of this istics both in situ and in the laboratory. The investigation of work was that the presence of the fracture accentuated the