Fracture compliance estimation using borehole tube waves

Abstract

We tested two models, one for tube-wave generation and the other for tube-wave attenuation at a fracture intersecting a borehole that can be used to estimate fracture compliance, fracture aperture, and lateral extent. In the tube-wave generation model, we consider tube-wave excitation in the borehole when a P-wave is incident on the fracture. The amplitude ratio of the pressure due to the tube wave to that of the incident P-wave is a function of fracture compliance, aperture, and length. Similarly, the attenuation of a tube wave in the borehole as it crosses a fracture intersecting the borehole is also a function of fracture properties. Numerically solving the dispersion relation in the fracture, we study tube-wave generation and the attenuation coefficient as a function of frequency. We observed that measuring amplitude ratios or attenuation near a transition frequency can help constrain the fracture properties. The transition frequency corresponds to the regime in which the viscous skin depth in the fracture is comparable to its aperture. Measurements in the high-frequency limit can place a lower bound on fracture compliance and lateral extent. We evaluated the applicability of the tube-wave generation model to a previously published VSP data set and found that compliance values of the order 10 −10-10 −9 m∕Pa are likely in the field. These observations support scaling of fracture compliance with fracture size.

Tube-wave Generation Due to Permeable Layers in a VSP Experiment - A New Model Elucidating the Effect of Dip Angles

“…Similar to the foregoing studies (e.g., Li et al, 1994;Bakku et al, 2013), the equation of continuity for the fluid flow due to the deformation of the layer is considered. This results in a partial differential equation for the fluid pressure perturbation in the layer as,…”

“…Here we linearized the equation by neglecting the higher-order terms (Bakku et al, 2013). Equation (2) is valid for dipping structures as well as for horizontal structures.…”

“…Similar to the open fracture model (Hardin et al, 1987;Bakku et al, 2013), we assume that the dynamic layer thickness L(t) is determined from the layer compliance Z L (m/Pa) and the effective stress in the layer as…”

“…Ionov (2007) removed the small-amplitude assumption of the Beydoun's approach by considering a realistic boundary condition that the fluid pressure at the fracture-borehole intersection is equivalent to the tube-wave generation amplitude (Ionov's boundary condition). Bakku et al (2013) further extended the Ionov's approach by considering the dynamic flow condition so that the theory is available for a wide range of frequencies.…”

“…Conventionally, the models of tube-wave generation have considered the fluid flow due to the deformation of (1) a parallel-wall open fracture (Beydoun et al, 1985;Hardin et al, 1987;Ionov, 2007;Bakku et al, 2013) or (2) a permeable layer (Li et al, 1994). The former is found to be useful to estimate the permeability corresponding to µm-to-mm scale fractures (e.g., Hardin et al, 1987;Hornby et al, 1989), whereas the latter is useful for larger-scale (cm-to-m) geological faults (e.g., Li et al, 1994;Kiguchi et al, 2001).…”