Numerical simulation of borehole Stoneley wave reflection by a fracture based on variable grid spacing method

Abstract

In this paper, the finite difference method is used to model the Stoneley wave reflection by a horizontal fracture in a borehole. The fracture shape is described by some finite difference grids. Therefore, the fracture aperture can be varied in the radial direction, thus extending previous researches on the assumption that the fracture aperture is constant throughout the fracture. Finite difference grids can also be used to describe a fracture which extends a finite distance in the radial direction. In addition, the finite difference algorithm can deal with the problem of inhomogeneous formation. Therefore, it allows the variation of formation elasticity in the model. Fine grids are needed to describe the small fracture aperture, and variable grid spacing is employed by finite difference method to improve computational efficiency. The Stoneley wave propagation is simulated by the variable grid spacing finite difference method in several models with variable fracture aperture, finite extension fracture and models with heterogeneous formation. The variable grid spacing finite difference method is validated through a comparison with real axis integration method and the analytical method. We get some conclusions by simulating and investigating effects of the variation of fracture aperture, the fracture of finite extension and inhomogeneity of formation on the Stoneley wave reflection. Although the fracture aperture changes along the fracture extension direction, the reflection coefficient of Stoneley wave is mainly controlled by the fracture aperture near the borehole. The Stoneley wave in the fracture is reflected back into the borehole, by the tip of finite extension fracture, which results in some notches in the reflection coefficient curve. If the Stoneley wave propagates from the formation with small elastic modulus to the formation with large elastic modulus, the reflection coefficient of Stoneley wave will be larger than that of homogeneous model with small elastic modulus. And if Stoneley wave propagates from the formation with large elastic modulus to the formation with small elastic modulus, the reflection coefficient of Stoneley wave will be smaller that of homogeneous model with large elastic modulus. These results provide some basis for the use of Stoneley wave to detect the fracture properties in formation.