Seismic reflectivity of hydraulic fractures approximated by thin fluid layers

Abstract

We have analyzed the angle-dependent reflectivity of microseismic wavefields at a hydraulic fracture, which we modeled as an ideal thin fluid layer embedded in an elastic, isotropic solid rock. We derived full analytical solutions for the reflections of an incident P-wave, the P-P and P-S reflection coefficients, as well as for an incident S-wave, and the S-S and S-P reflection coefficients. The rather complex analytical solutions were then approximated and we found that these zero-thickness limit approximations are in good agreement with the linear slip model, representing a fracture at slip contact. We compared the analytical solutions for the P-P reflections with synthetic data that were derived using finite-difference modeling and found that the modeling confirmed our theoretical results. For typical parameters of microseismic monitoring by hydraulic fracturing, e.g., a layer thickness of [Formula: see text] and frequencies of [Formula: see text], the reflection coefficients depend on the Poisson’s ratio. Furthermore, the reflection coefficients of an incident S-wave are remarkably high. Theoretical results suggested that it is feasible to image hydraulic fractures using microseismic events as a source and to solve the inverse problem, that is, to interpret reflection coefficients extracted from microseismic data in terms of reservoir properties.

Extension of the classical linear slip model for fluid‐saturated fractures: Accounting for fluid pressure diffusion effects

“…The stiffness elements C i j can be complex‐valued and frequency‐dependent. Furthermore, the dynamic equations governing the wave propagation across a layer that is very thin compared to the prevailing wavelengths reduce to the quasi‐static equations [ Oelke et al , ] which, in turn, implies that inertial effects in the fracture are neglected. In this work, we consider fast P wave incidence.…”

“…The stiffness elements C ij can be complex-valued and frequency-dependent. Furthermore, the dynamic equations governing the wave propagation across a layer that is very thin compared to the prevailing wavelengths reduce to the quasi-static equations [Oelke et al, 2013] ∇ ⋅ = 0,…”

“…Worthington and Lubbe [] provide a summary of real‐valued normal and shear fracture compliances as functions of the fracture size based on seismic and laboratory experiments. Oelke et al [] model individual fractures as thin fluid layers embedded in an elastic background and derive the corresponding elastic compliance to be used in a LSM. Minato and Ghose [] proposed a methodology based on the LSM that allows to estimate real‐valued heterogeneous fracture compliance distributions by solving the inverse elastic scattering problem.…”

“…Seismic waves are widely employed for this purpose due to the fact that they tend to show significant attenuation and strong anisotropy in the presence of fractures [e.g., Gurevich et al , ; Müller et al , ; Rubino et al , ]. Despite the very large differences in scale typically observed between fracture apertures and prevailing seismic wavelengths, imaging of individual fractures is often possible [e.g., Pyrak‐Nolte et al , ; Oelke et al , ; Minato and Ghose , ]. To date, the seismic response of individual fractures is mostly assessed in the framework of the so‐called linear slip theory, where a fracture is modeled as a nonwelded interface and its effects on seismic wave propagation are represented by a discontinuity in displacement, while the stresses are assumed to be continuous across the interface.…”

Seismic wave modeling of fluid-saturated fractured porous rock: Including fluid pressure diffusion effects of discrete distributed large-scale fractures

“…Various local numerical schemes have been developed for discrete distributed large-scale fractures. The most widely used scheme is local effective-medium schemes (Coates and Schoenberg, 1995;Igel et al, 1997;Vlastos et al, 2003;Oelke, et al, 2013) that determine and incorporate the behavior of fracture-induced media within each spatial discretized cell. The advantage of using the local effective medium is that it requires no special treatment of the displacement discontinuity conditions on the fractures.…”