High-Resolution Finite Volume Modeling of Wave Propagation in Orthotropic Poroelastic Media

Abstract

Poroelasticity theory models the dynamics of porous, fluid-saturated media. It was pioneered by Maurice Biot in the 1930s through the 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.

A time-domain numerical modeling of two-dimensional wave propagation in porous media with frequency-dependent dynamic permeability

“…The anisotropy of some porous media is sometimes a main feature, for instance, in biomechanics to model trabecular and cortical bones. 27,28 The anisotropy of a medium changes only the 4 Â 4 poroelastic tensor C in Eq. (19).…”

A discontinuous Galerkin method for poroelastic wave propagation: The two-dimensional case

“…A broad ranging review of computational poroelasticity is given 30 in [12]. We refer also to the recent papers [13,19,20] who work in a finite volume setting. DG methods have been implemented previously for poroelastic wave propagation; we mention [14,15] who worked in the time domain, while the recent paper [16] considers frequency domain solutions.…”

“…This 150 is considered by Carcione and Quiroga-Goode in [47] who used an operator splitting approach to avoid this issue and treated the viscous dissipation term analytically. In a more recent paper Lemoine et al [13] work in a finite volume setting and again implement an operator splitting on the dissipative part. We refer to Section 3 of their paper for a detailed discussion.…”

“…In this paper we deal with the isotropic case, while [13] deals with the more general case of orthotropic media.…”

“…The convergence test results indicate that the solver satisfies the theoretical convergence properties (convergence rate = polynomial order + 1) for typical parameter ranges one is likely to encounter in groundwater tomography. However, we note that 390 when the wave is highly dissipative and slow in Biot's low-frequency regime (corresponding to very low permeabilities), it will be necessary to implement an operator splitting technique or IMEX scheme to deal with the diffusive part, which in certain circumstances can be quasi-static [43,47,13]. In any event, the slow P-wave can impose a significant restraint on the grid resolution or basis order, and hence the time step, in the poroelastic part.…”