Numerical Methods And Applications In Complex Elastic And Porous Media

Numerical modeling is an important branch of computational mathematics,which plays an indispensable role for science discovery and engineering/industry applications.The convergency,accuracy,and efficiency of numerical solutions are three key criteria.In this paper,elastic wave problems are solved by both finite difference method(FDM)and spectral element method(SEM).We apply these two algorithms to accurately model ex-tremely large-scale wave propagation in complex inhomogeneous(porous-)elastic media,on modern computing clusters or even a petascale supercomputer.First,we introduce viscoelastic wave equations in cylindrical coordinates and Cartesian coordinate,which are solved using the FDM.The viscoelatic wave equation in cylindrical coordinates can be applied to logging while drilling.When modelling the propagation of 3-D viscoelastic waves in cylindrical coordinates,a mathematical singularity appears,due to the presence of 1/r terms in the viscoelastic wave equations.In this work,we propose a simple but effective method to resolve this numerical singularity problem.Since the grids are symmetric around the pole axis,the finite difference formulation at the singular-ity point,in a cylindrical coordinate,can be then reverted back to Cartesian coordinates,where no singularity is involved.Specifically,by rotating the Cartesian coordinate sys-tem around the z-axis in cylindrical coordinates,the numerical singularity problems in both 2-D and 3-D cylindrical coordinates can be removed.The accuracy and stability of numerical solutions at the singularity point are determined by the Cartesian FDM.Fur-thermore,we introduced MPI technique to parallelize the FDM for 3D viscoelastic wave equation.As a result,it can be applied in a large-scale seismic wave model with a realistic geological model.Second,we propose a complete attenuation model for poroelastic media in full frequen-cy regime.Various mechanisms can cause the dissipation and the dispersion of poroelastic waves,such as the interaction between solid grains and pore fluid,the viscous fluid,and the viscoelastic properties of the solid frame.The poroelastic attenuation model consist of two parts:viscodynamic attenuation and viscoelastic attenuation corresponding to dis-sipation from the porous skeleton.For viscoelastic model,we propose four quality factors QKb,QKs,Qkf,and Qu.Physically speaking,they correspond to the dry bulk modulus,solid bulk modulus,fluid bulk modulus,and shear modulus,respectively.A mapping for-mulation is derived to convert the physically meaningful quality factors to those suitable for numerical implementation.For the viscodynamic operator,we implement the viscous pore flow over the full frequency spectrum,and a FDTD discrete form is proposed in high-frequency regimes for the first time.Third,we proposed a novel perfectly matched layer(PML)technique for wave equation in the 2nd order partial differential equation(PDE)form.When modeling time domain elastic wave propagation in an unbound space,the standard PML is straightforward for the first-order PDEs;whereas,the PML requires tremendous re-constructions of the gov-erning equations in the second-order PDE form.Therefore,it is imperative to explore a simple implementation of PML for the second-order system.In this work,we first system-atically extend the first-order Nearly PML(NPML)technique into second-order systems,implemented by the SEM and FDM algorithms.It merits the following advantages:the simplicity in implementation,by keeping the second-order PDE-based governing equa-tions exactly the same;the efficiency in computation,by introducing a set of auxiliary ordinary differential equations(ODEs).Mathematically,this NPML technique effective-ly hybridizes the second-order PDEs and frst-order ODEs,and locally attenuates outgoing waves,thus efficiently avoiding either spatial or temporal global convolutions.