×

zbMATH — the first resource for mathematics

Three-dimensional mapped-grid finite volume modeling of poroelastic-fluid wave propagation. (English) Zbl 1456.65084
Summary: This paper extends the author’s previous two-dimensional work with Ou and LeVeque to high-resolution finite volume modeling of systems of fluids and poroelastic media in three dimensions, using logically rectangular mapped grids. A method is described for calculating consistent cell face areas and normal vectors for a finite volume method on a general nonrectilinear hexahedral grid. A novel limiting algorithm is also developed to cope with difficulties encountered in implementing high-resolution finite volume methods for anisotropic media on nonrectilinear grids; the new limiting approach is compatible with any limiter function and typically reduces solution error even in situations where it is not necessary for correct functioning of the numerical method. Dimensional splitting is used to reduce the computational cost of the solution. The code implementing the three-dimensional algorithms is verified against known plane wave solutions, with particular attention to the performance of the new limiter algorithm in comparison to the classical one. An acoustic wave in brine striking an uneven bed of orthotropic layered sandstone is also simulated in order to demonstrate the capabilities of the simulation code.
Reviewer: Reviewer (Berlin)

MSC:
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
74S10 Finite volume methods applied to problems in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74J10 Bulk waves in solid mechanics
74L05 Geophysical solid mechanics
74L15 Biomechanical solid mechanics
86-08 Computational methods for problems pertaining to geophysics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] D. F. Aldridge, N. P. Symons, and L. C. Bartel, Poroelastic wave propagation with a velocity-stress-pressure algorithm, in Proceedings of Poromechanics III, Norman, OK, 2005, pp. 253–258.
[2] A. Alghamdi, A. Ahmadia, D. I. Ketcheson, M. G. Knepley, K. T. Mandli, and L. Dalcin, Petclaw: A scalable parallel nonlinear wave propagation solver for python, in Proceedings of the 19th High Performance Computing Symposia, Society for Computer Simulation International, San Diego, CA, 2011, pp. 96–103.
[3] K. Attenborough, D. L. Berry, and Y. Chen, Acoustic scattering by near-surface inhomogeneities in porous media, Tech. report, Defense Technical Information Center OAI-PMH Repository, http://stinet.dtic.mil/oai/oai, 1998.
[4] D. S. Bale, R. J. LeVeque, S. Mitran, and J. A. Rossmanith, A wave propagation method for conservation laws and balance laws with spatially varying flux functions, SIAM J. Sci. Comput., 24 (2002), pp. 955–978. · Zbl 1034.65068
[5] M. A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range, J. Acoustical Soc. Amer., 28 (1956), pp. 168–178.
[6] M. A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range, J. Acoustical Soc. Amer., 28 (1956), pp. 179–191.
[7] M. A. Biot, Mechanics of deformation and acoustic propagation in porous media, J. Appl. Phys., 33 (1962), pp. 1482–1498. · Zbl 0104.21401
[8] J. L. Buchanan and R. P. Gilbert, Determination of the parameters of cancellous bone using high frequency acoustic measurements, Math. Comput. Model., 45 (2007), pp. 281–308. · Zbl 1170.74026
[9] J. L. Buchanan and R. P. Gilbert, Determination of the parameters of cancellous bone using high frequency acoustic measurements II: Inverse problems, J. Comput. Acoustics, 15 (2007), pp. 199–220. · Zbl 1198.74050
[10] J. L. Buchanan, R. P. Gilbert, and K. Khashanah, Determination of the parameters of cancellous bone using low frequency acoustic measurements, J. Comput. Acoustics, 12 (2004), pp. 99–126. · Zbl 1235.76142
[11] J. L. Buchanan, R. P. Gilbert, A. Wirgin, and Y. Xu, Marine Acoustics: Direct and Inverse Problems, SIAM, Philadelphia, 2004. · Zbl 1055.35134
[12] J. M. Carcione, Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, and Porous Media, Elsevier, Oxford, UK, 2001.
[13] G.-Q. Chen, C. D. Levermore, and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), pp. 787–830. · Zbl 0806.35112
[14] G. Chiavassa and B. Lombard, Wave propagation across acoustic/Biot’s media: A finite-difference method, Commun. Comput. Phys., 13 (2013), pp. 985–1012. · Zbl 1373.76042
[15] S. C. Cowin, Bone poroelasticity, J. Biomech., 32 (1999), pp. 217–238.
[16] S. C. Cowin and L. Cardoso, Fabric dependence of bone ultrasound, Acta Bioengrg. Biomech., 12 (2010), pp. 3–23.
[17] N. Dai, A. Vafidis, and E. Kanasewich, Wave propagation in heterogeneous porous media: A velocity-stress, finite-difference method, Geophysics, 60 (1995), pp. 327–340.
[18] J. de la Puente, M. Dumbser, M. Käser, and H. Igel, Discontinuous Galerkin methods for wave propagation in poroelastic media, Geophysics, 73 (2008), pp. T77–T97.
[19] G. Degrande and G. De Roeck, FFT-based spectral analysis methodology for one-dimensional wave propagation in poroelastic media, Transp. Porous Media, 9 (1992), pp. 85–97.
[20] E. Detournay and A. H-D. Cheng, Poroelastic response of a borehole in a non-hydrostatic stress field, Internat. J. Rock Mech. Mining Sci. Geomech. Abstr., 25 (1988), pp. 171–182.
[21] M. Dumbser, M. Käser, and J. de la Puente, Arbitrary high-order finite volume schemes for seismic wave propagation on unstructured meshes in 2D and 3D, Geophys. J. Internat., 171 (2007), pp. 665–694.
[22] S. K. Garg, A. H. Nayfeh, and A. J. Good, Compressional waves in fluid-saturated elastic porous media, J. Appl. Phys., 45 (1974), pp. 1968–1974.
[23] R. P. Gilbert, P. Guyenne, and M. Y. Ou, A quantitative ultrasound model of the bone with blood as the interstitial fluid, Math. Comput. Model., 55 (2012), pp. 2029–2039. · Zbl 1255.74046
[24] R. P. Gilbert and Z. Lin, Acoustic field in a shallow, stratified ocean with a poro-elastic seabed, Z. Angew. Math. Mech., 77 (1997), pp. 677–688. · Zbl 0889.76079
[25] R. P. Gilbert and M. Y. Ou, Acoustic wave propagation in a composite of two different poroelastic materials with a very rough periodic interface: A homogenization approach, Internat. J. Multiscale Comput. Engrg., 1 (2003), pp. 431–440.
[26] M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), pp. 209–286. · Zbl 1242.65109
[27] F. Kemm, A comparative study of TVD-limiters—well-known limiters and an introduction of new ones, Internat. J. Numer. Methods Fluids, 67 (2011), pp. 404–440. · Zbl 1316.76058
[28] D. I. Ketcheson, K. T. Mandli, A. J. Ahmadia, A. Alghamdi, M. Quezada de Luna, M. Parsani, M. G. Knepley, and M. Emmett, PyClaw: Accessible, extensible, scalable tools for wave propagation problems, SIAM J. Sci. Comput., 34 (2012), pp. C210–C231. · Zbl 1253.65220
[29] D. I. Ketcheson, M. Parsani, and R. J. LeVeque, High-order wave propagation algorithms for hyperbolic systems, SIAM J. Sci. Comput., 35 (2013), pp. A351–A377. · Zbl 1264.65151
[30] J. O. Langseth and R. J. LeVeque, A wave propagation method for three-dimensional hyperbolic conservation laws, J. Comput. Phys., 165 (2000), pp. 126–166. · Zbl 0967.65095
[31] G. I. Lemoine, Numerical Modeling of Poroelastic-Fluid Systems Using High-Resolution Finite Volume Methods, Ph.D. thesis, University of Washington, Seattle, WA, 2013.
[32] G. I. Lemoine and M. Y. Ou, Finite Volume Modeling of Poroelastic-Fluid Wave Propagation with Mapped Grids, preprint, , 2013. · Zbl 1299.76160
[33] G. I. Lemoine, M. Y. Ou, and R. J. LeVeque, High-resolution finite volume modeling of wave propagation in orthotropic poroelastic media, SIAM J. Sci. Comput., 35 (2013), pp. B176–B206. · Zbl 1342.74173
[34] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, New York, 2002. · Zbl 1010.65040
[35] R. J. LeVeque and H. C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. Comput. Phys., 86 (1990), pp. 187–210. · Zbl 0682.76053
[36] J.-F. Lu and A. Hanyga, Wave field simulation for heterogeneous porous media with singular memory drag force, J. Comput. Phys., 208 (2005), pp. 651–674. · Zbl 1329.76338
[37] B. G. Mikhailenko, Numerical experiment in seismic investigations, J. Geophys., 58 (1985), pp. 101–124.
[38] C. Morency and J. Tromp, Spectral-element simulations of wave propagation in porous media, Geophys. J. Internat., 179 (2008), pp. 1148–1168.
[39] A. Naumovich, On finite volume discretization of the three-dimensional Biot poroelasticity system in multilayer domains, Comput. Methods Appl. Math., 6 (2006), pp. 306–325. · Zbl 1100.74060
[40] J. E. Santos and E. J. Oren͂a, Elastic wave propagation in fluid-saturate porous media, part II: The Galerkin procedures, Math. Model. Numer. Anal., 20 (1986), pp. 129–139.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.