Rock-physics modeling of elastic properties of multiscale fractured rocks

Fractures, often existing across various scales, control the mechanical and fluid flow properties of upper crustal rocks. Inferring fracture properties at different scales from multiband geophysical measurements is essential for many fields of earth and energy sciences. Under the framework of multiscale homogenization, we develop a rock-physics model to characterize the elastic and anisotropic properties of multiscale fractured rocks following a certain statistical law. The isotropic differential effective medium theory is used to model the elasticity of randomly oriented microcracks, and the linear slip theory is used to model the elasticity of oriented or randomly oriented macroscale fractures. For a multiscale fractured rock covered by six orders of fracture length (10-4 m to 102 m), it is found that velocity exhibits a decreasing trend with the increment of scale. Nevertheless, the different statistical distribution of multiscale fractures significantly affects the velocity and anisotropy variation pattern with scale. The velocity for the fractal distribution decreases significantly at the seismic scale, whereas, for the log-Gaussian distribution, the dramatic change in velocity occurs more at the ultrasonic and logging scales, depending on the length of the dominant fractures. We apply our methodology to interpret a set of multiscale geophysical data of P-wave velocity in a fractured carbonate formation and estimate that the multiscale fracture is possibly distributed in a log-Gaussian manner. Our elastic and anisotropic modeling strategy has the potential to predict the distribution pattern of fractures, especially by reconciling the multifrequency geophysical measurements (e.g., ultrasonic, logging, and seismic data).