Study On The Effect Of Fractures And Porous Formation On Acoustic Waves In Borehole Based On Finite Difference Method

Acoustic logging is widely used in petroleum exploration,coal field geology,engineering,hydrology and environment.Acoustic logging data can be used to determine the location of reservoirs,estimate oil and gas reserves,and evaluate fractures.Oil and gas are usually stored in the pores,fractures and caves of the formation.Fractures are not only the channels for fluid storage and migration,but also control the distribution of oil and gas in the reservoir.Therefore,the evaluation of fracture and porous formation is of great significance in oil and gas exploration and development.The presence of fractures and porous formations can affect the propagation of acoustic waves in the borehole,and the propagation of acoustic waves can also be used to predict the information of fractures and pore formations.However,it is difficult to use an analytical method to calculate the propagation of acoustic waves due to the complex borehole acoustic wave mode,the irregular shape of fractures,the heterogeneity of formation and the effect of the elasticity of formation frame.The Finite-difference method can deal with these complex conditions,and solve the problem of acoustic wave propagation simulation in the borehole.Therefore,this paper adopts the finite-difference method to simulate monopole acoustic logging,and studies the effect of fracture and porous formation on acoustic propagation in the borehole.In the study of fractures,to consider only the effect of fractures,we assume that the formation is completely elastic.Because the fracture aperture is very small(tens of microns ~ 1 mm),several small finite difference grids are needed to describe the fracture aperture.To avoid oversampling in the non-fractured zone and save computing time,a variable grids finite difference method was used to simulate the propagation of acoustic waves in the formation containing horizontal fractures.Compared with the waveforms of the real axis integration and the results of analytical methods,the variable grids finite difference method is verified.And the effects of fractures on the propagation of the compressional wave,the shear wave and the Stonley wave are studied.In view of the fact that there are few studies on the effect of fractures on the compressional wave propagation in the borehole,and there are no reports on the specific propagation of sliding compressional wave when it encounters fractures,this paper studies the effect of the existence of fractures on the waveform of the compressional-head wave.However,many studies have shown that the amplitude of shear wave decreases obviously after it cross the horizontal fracture in the borehole.In this paper,several variables of the model are considered in simulating the shear wave attenuation caused by fractures: fracture aperture,the distance from the receiver to the fracture,the borehole radius,and the radial extension of the fracture.Using the simulated waveform,the relationship between the attenuation coefficient of the shear-head wave and these variables is studied.Stonley waves in a borehole attenuate and reflect when they encounter a fracture.In this paper,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.Therefore,the Stoneley wave propagation is simulated by the variable grids spacing finite difference method in several models with variable fracture aperture,finite extension fracture and models with heterogeneous formation.The simulation results of finite difference method show that:The compressional-head wave recorded by the receiver close to the fracture is affected by the scattering wave caused by the narrow fracture,resulting in a change in amplitude and a significant shift of the wave peak to the right.There is a good linear relationship between the attenuation coefficient of the shear-head wave and the fracture aperture.In different elastic formation,the intercept of the linear equation is greater than 0.71,which indicates that the fracture will cause strong attenuation of the shear wave.The slope of the linear equation is very small,which indicates that the attenuation coefficient of the f shear-head wave is not sensitive to the change of fracture aperture.The attenuation coefficient of the shear-head wave corresponding to the receiver near the fracture is small due to the effect of mode conversion.As the distance between the receiver and the fracture increases,the attenuation coefficient of shear-head wave increases gradually.Also due to the effect of mode conversion,the attenuation coefficient of the shear wave head wave decreases with the increase of the hole radius when the receiver is close to the fracture.The attenuation coefficient of the shear-head wave is very small when the fracture extends very close.In addition,when the fracture extension distance is not very far(less than 0.6 m),the attenuation coefficient of the shear-head wave increases with the increase of the extension distance.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.When the Stoneley wave propagates from the formation with small elastic modulus to the formation with large elastic modulus,the reflection coefficient of Stoneley wave is larger than that of the former.And when the Stoneley wave propagates from the formation with large elastic modulus to the formation with small elastic modulus,the reflection coefficient of Stoneley wave is smaller than that of the former.As for the study of Stonley wave reflection from the porous formation in the borehole,previous studies used simplified Biot-Rosenbaum theory to calculate the Stonley wave reflection coefficient of the porous formation.Such simplified theory ignores the effect of formation frame elasticity,thus can't obtain accurately the Stoneley wave reflection for the porous formation with small stiffness.In this study,to take the effect of formation frame elasticity into account,the Biot's theory in the low-frequency limit is used to simulate the Stoneley wave reflection by employing the velocity-stress finite-difference time-domain(FDTD)method.The FDTD method is verified by a comparison with the real axis integration(RAI)method with respect to the Stoneley wave propagation in a borehole surrounded by a homogeneous porous formation.In addition,as permeability of the formation varies in the axial direction,and the viscosity of the pore fluid also changes in a radial direction due to mud invasion,it is difficult to use the simplified theory to calculate the reflection coefficient of the Stoneley wave in such heterogeneous cases.Therefore,this study first investigates the effect of the formation permeability heterogeneity on Stoneley wave reflection coefficient by this FDTD method.The reflection coefficient obtained by the FDTD method is smaller than that using the simplified theory,which shows that elasticity of the formation frame affects Stoneley wave reflection: the effect of elasticity on the reflection coefficient is greater when the formation frame is less rigid or when the porosity and permeability of the formation are lower.According to the simulation results of the FDTD method,a modified simplified theory which can improve the calculation accuracy of Stoneley wave reflection coefficient is proposed.Furthermore,the effects of the permeability heterogeneity on the Stoneley wave reflection are investigated: the reflection coefficient peaks change when permeability alters in an axial direction,and the peak interval increases.For the mud invasion model,the reflection coefficient is almost identical to that of the homogeneous model,which has the same permeability as the borehole wall of the mud invasion model.