Discontinuous Galerkin Method For The Solution Of The Wave Equations In Two-Phase And Viscoelastic Media

Seismic explorationis an important area ofGeophysicsscientific research,in which,the propagation of seismic wave in the complex media is a basic work of this study.The actual oil and gas reservoirs shows crack,fracture,and pore,it usually comes as some multiphase media.Porous media model considers the each aspect of its characteristics,so it can describe the properties of stratum structure better.Based on several assumptions,Biot established the wave theory in two-phase elastic medium at a macro level.It can describe the elastic wave propagation in two-phase media well.The discontinuous Galerkin method has much suitability in some complex geological model,so we try to use the discontinuous Galerkin method to solve this problem.In order to use the discontinuous Galerkin method based on numerical flux,we need to transformationelastic wave equation into some form,which meets the form of hyperbolic conservation law.We introduce some new variables in the original equation and integrate this new form to reduce the derivative degree of the original function.In line with thehyperbolic conservation law,this problem can be solved by the discontinuous Galerkin method.Using the discontinuous Galerkin method to solve this problem,we need to discrete the form in line with hyperbolic conservation in time and space.Using an apposite numerical fluxalong with the basis functions and numerical integration formulas to transform the original function into an ODE problem.Then,using an implicit diagonal method(WRKDG)to discrete time and solve the ODE function.At last,use the DG method to simulate the wave equationsin the two-phase media based on the Biot theory and the viscoelastic media.In the result of numerical simulation,we can hardly see anydispersion along with the waves.It can be concluded that the DG method can solve the wave equations in the two-phase and viscoelastic media effectively.