The Numerical Simulation Of Theoretical Acoustic Wave Trains Of Fractured Formation

Acoustic logging is a very important method in the petroleum geophysical exploration.After several decades of development and application,the theory of acoustic logging is becoming more and more complete,and the technology is mature,and the acoustic logging instrument has also developed significantly.The early acoustic logging can only obtain the velocity and amplitude of sliding p-wave,to correct the formation interface and the calculate the porosity.Up to now,it has been possible to obtain the full-wave related to the formation characters,which can be used for ground stress analysis,fracture identification and so on.Fractural reservoirs are one of the hot spots in the field of petroleum exploration in recently.Fractures are favorable for the migration and reservoirs of hydrocarbon.Accurate identification of fractures is the basis of evaluating fractural reservoirs,which requires us to fully understand propagation rules of the sound wave in the fractural formation,and to simulate the sound field of the fractural formation.The theory of fluid-saturated porous media proposed by Biot is the basis of the sound field in the borehole,which decomposes the motion process into two parts of solid phase and flow phase,which can effectively simulate the sound field in the porous medium.However,for the solution of the sound field in the complex formation,it is difficult to obtain the exact analytical solution,and it is usually solved by numerical method.The finite difference method is widely used in mathematics,mechanics,geophysics and other disciplines.And it can be applied to simulate the sound field characteristics of various complex formation,and obtain the instantaneous sound field of the formation.In order to improve the accuracy and stability of this method,some scholars have proposed the staggered-grid finite difference method based on the first order velocity-stress equation.In this paper,the staggered-grid finite difference method is used to simulate the sound wave field of fractural formation.Firstly,we establish the borehole model based on the theory of fluid-saturated porous media proposed by Biot.And the velocity and stress are decomposed into two parts: the flow phase and the solid phase.The first order velocity-stress equation is used to represent the wave equation of the acoustic wave in the model medium.To meet the accuracy requirement,the central difference scheme in domain of time and the fourth-order difference scheme in the spatial domain are used to establish the finite difference form of the wave equation.And we also discuss the dispersion and stability of the difference equation.To ensure the accuracy,the appropriate space step and time step are selected to eliminate numerical dispersion and guarantee the stability.In the process of solving the wave field,the actual space should be infinite,it is necessary to control the infinite problem in a limited range in the calculation,that requires the addition of the appropriate boundary conditions in the computational domain,The boundary does not reflect the energy of the wave field and interfering with the results of the wave field simulation.Here,we adopt the perfectly matched layer(PML)absorption boundary condition.We review the principle of PML absorption boundary,and the appropriate attenuation factor is selected to ensure the boundary can effectively absorb the radiation from wave field inside.Finally,we apply the staggered-grid difference method on the simulation of the wave field characteristics of a single horizontal crack,and study the propagation rules of the sound wave in the two-phase porous media in different conditions,obtaining the wave field characters of different fractural formation.