High-Resolution Finite Volume Modeling of Wave Propagation in   Orthotropic Poroelastic Media
  Poroelasticity theory models the dynamics of porous, fluid-saturated media. It was pioneered by Maurice Biot in the 1930s through 1960s, and has applications in several fields, including geophysics and modeling of in vivo bone. A wide variety of methods have been used to model poroelasticity, including finite difference, finite element, pseudospectral, and discontinuous Galerkin methods. In this work we use a Cartesian-grid high-resolution finite volume method to numerically solve Biot's equations in the time domain for orthotropic materials, with the stiff relaxation source term in the equations incorporated using operator splitting. This class of finite volume method has several useful properties, including the ability to use wave limiters to reduce numerical artifacts in the solution, ease of incorporating material inhomogeneities, low memory overhead, and an explicit time-stepping approach. To the authors' knowledge, this is the first use of high-resolution finite volume methods to model poroelasticity. The solution code uses the CLAWPACK finite volume method software, which also includes block-structured adaptive mesh refinement in its AMRCLAW variant. We present convergence results for known analytic plane wave solutions, achieving second-order convergence rates outside of the stiff regime of the system. Our convergence rates are degraded in the stiff regime, but we still achieve similar levels of error on the finest grids examined. We also demonstrate good agreement against other numerical results from the literature. To aid in reproducibility, we provide all of the code used to generate the results of this paper, at https://bitbucket.org/grady_lemoine/poro-2d-cartesian-archive . 
Grady I. Lemoine, M. Yvonne Ou, Randall J. LeVeque
math.NA cs.NA