Sensitivity of time-lapse seismic data to fracture compliance in hydraulic fracturing

We study the sensitivity of seismic waves to changes in the fracture normal and tangential compliances by analysing and numerically solving the fracture sensitivity wave equation, which is derived by differentiating the elastic wave equation with respect to the fracture compliance. The sources for the sensitivity wavefield are the sensitivity moments, which are functions of fracture compliance, background elastic properties and the stress acting on the fracture surface. Based on the analysis of the fracture sensitivity wave equation, we give the condition for the weak scattering approximation to be valid for fracture scattering. We study the sensitivity of P and S waves to fracture normal and tangential compliances, respectively, by separating the seismic wavefield and the sensitivity field into P and S components. In the numerical simulations of a single fracture in a homogeneous medium, we study the effect of fracture compliances, source incident angle and background elastic properties on the sensitivity field. We also discuss the sensitivity of seismic data to the compliances of vertical and horizontal fractures, respectively, for surface and borehole acquisitions. Under the weak scattering approximation, we find that the percentage change of fracture compliance in hydraulic fracturing is equal to the percentage change of the recorded time-lapse seismic data. This could provide a means for designing and interpreting experiments that can potentially be used to monitor the opening/closing of a fracture in hydraulic fracturing through time-lapse seismic surveys.