Finite Volume Modeling of Poroelastic-Fluid Wave Propagation with Mapped Grids

Abstract

Abstract. In this work we develop a high-resolution mapped-grid finite volume method code to model wave propagation in two dimensions in systems of multiple orthotropic poroelastic media and/or fluids, with curved interfaces between different media. We use a unified formulation to simplify modeling of the various interface conditions -open pores, imperfect hydraulic contact, or sealed pores -that may exist between such media. Our numerical code is based on the clawpack framework, but in order to obtain correct results at a material interface we use a modified transverse Riemann solution scheme, and at such interfaces are forced to drop the second-order correction term typical of high-resolution finite volume methods. We verify our code against analytical solutions for reflection and transmission of waves at a material interface, and for scattering of an acoustic wave train around an isotropic poroelastic cylinder. For reflection and transmission at a flat interface, we achieve second-order convergence in the 1-norm, and first-order in the max-norm; for the cylindrical scatterer, the highly distorted grid mapping degrades performance but we still achieve convergence at a reduced rate. We also simulate an acoustic pulse striking a simplified model of a human femur bone, as an example of the capabilities of the code. To aid in reproducibility, at the web site http://dx.doi.org/10.6084/m9.figshare.701483 we provide all of the code used to generate the results here.Key words. poroelastic, wave propagation, finite-volume, high-resolution, operator splitting, mapped grid, transverse solve, interface condition, cylindrical scatterer AMS subject classifications. 65M08, 74S10, 74F10, 74J10, 74L05, 74L15, 86-08 1. Introduction. Poroelasticity theory was developed by Maurice A. Biot to model the mechanics of a fluid-saturated porous medium. It models the medium in a homogenized fashion, with solid portion treated with linear elasticity, and the fluid with linearized compressible fluid dynamics combined with Darcy's law to relate its pressure gradient to its flow rate. Biot's work is summarized in his 1956 and 1962 papers [3, 4, 5], and Carcione also provides an excellent discussion of poroelasticity in chapter 7 of his book [9]. While it was originally developed to model fluid-saturated rock and soil, Biot theory has also found applications in modeling of in vivo bone [13,14,20] and underwater acoustics with a porous sea floor [7,21,22].Biot theory predicts three different families of propagating waves within a poroelastic medium. In order of decreasing speed, these are: fast P waves, where the fluid and solid parts of the medium move roughly parallel to the propagation direction -exactly parallel for an isotropic medium -and are typically in phase with each other; S waves, where the motion of the medium is transverse to the propagation direction; and slow P waves, where the motion is again roughly parallel to the wavevector but the fluid and solid typically move 180 degrees out of phase, so that the fluid is leaving a regi...

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