Numerical Modeling of Poroelastic-Fluid Systems Using High-Resolution Finite Volume Methods
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Poroelasticity theory models the mechanics of porous, uid-saturated, deformable solids. It was originally developed by Maurice Biot to model geophysical problems, such as seismic waves in oil reservoirs, but has also been applied to modeling living bone and other porous media. Poroelastic media often interact with uids, such as in ocean bottom acoustics or propagation of waves from soft tissue into bone. This thesis describes the development and testing of high-resolution nite volume numerical methods, and simulation codes implementing these methods, for modeling systems of poroelastic media and uids in two and three dimensions. These methods operate on both rectilinear grids and logically rectangular mapped grids. To allow the use of these methods, Biot's equations of poroelasticity are formulated as a rst-order hyperbolic system with a source term; this source term is incorporated using operator splitting. Some modi cations are required to the classical high-resolution nite volume method. Obtaining correct solutions at interfaces between poroelastic media and uids requires a novel transverse propagation scheme and the removal of the classical second-order correction term at the interface, and in three dimensions a new wave limiting algorithm is also needed to correctly limit shear waves. The accuracy and convergence rates of the methods of this thesis are examined for a variety of analytical solutions, including simple plane waves, re ection and transmission of waves at an interface between di erent media, and scattering of acoustic waves by a poroelastic cylinder. Solutions are also computed for a variety of test problems from the computational poroelasticity literature, as well as some original test problems designed to mimic possible applications for the simulation code.
- Applied mathematics