Analytical model for rock effective elastic properties with aligned elliptical cracks embedded in transversely isotropic background

Seismic methods are often used for fracture detection and characterization, for which many analytical rock physics models have been developed to link fracture parameters to rock elastic properties. However, most of these models are limited to isotropic background and circular (penny-shaped) fractures. In this work, we derived a more general analytical model for effective elastic properties of rocks embedded with aligned elliptical cracks in transversely isotropic background. The solutions for the case with dry cracks parallel to isotropic plane of transversely isotropic background were first derived using the corresponding Eshelby's tensor. Then, the results were extended to the case with inclined and rotated cracks under the assumption of constant crack opening displacement tensor. Finally, to obtain the results for fluid-filled crack case, the Brown–Korringa equation was applied. To investigate the controlling factors on rock elastic properties, we studied one fractured tight sandstone sample with transversely isotropic background. The results show that rock elastic properties are insensitive to crack spin angle, but sensitive to crack inclination angle. Depending on crack inclination angle and fluid saturation, background anisotropy can either amplify or reduce the anisotropy of seismic wave velocity. Crack density and aspect ratio are found to be controlling crack parameters for rock elastic properties. The crack plane shape has negligible direct influence on rock elastic properties, whereas it has indirect influence by affecting values of crack density and aspect ratio. Compared with the numerical model and the previous analytical model for the special case with circular cracks, our model agrees well with them, which validates the accuracy of our model.