Abstract
This paper presents a numerical algorithm to simulate low-frequency loading of fluid-filled poroelastic materials and estimate the effective frequency-dependent strain-stress relations for such media. The algorithm solves Biot equation in quasi-static state in the frequency domain. Thus, a large-scale system of linear algebraic equations have to be solved for each temporal frequency. We use the direct solver, based on the LU decomposition to resolve the system of the linear equations. According to the presented numerical examples suggested algorithm allows reconstructing the stiffness tensor within a wide range of frequencies for the realistic large-scale samples within several minutes. Thus, the estimation of the frequency-dependent stiffness tensors can be done in a routine manner and statistical data may be accumulated.
The research was supported by the Russian Science foundation grant no. 19-77-20004.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Bazaikin, Y., et al.: Effect of CT image size and resolution on the accuracy of rock property estimates. J. Geophys. Res. Solid Earth 122(5), 3635–3647 (2017)
Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. ii. higher frequency range. J. Acoust. Soc. Am. 28, 179–191 (1956)
Biot, M.A.: Theory of propagation of elastic waves in fluid-saturated porous solid. i. low-frequency range. J. Acoust. Soc. Am. 28, 168–178 (1956)
Carcione, J.M., Cavallini, F.: A rheological model for anelastic anisotropic media with applications to seismic wave propagation. Geophys. J. Int. 119, 338–348 (1994)
Christensen, R.M.: Theory of Viscoelasticity, an Introduction. Academic Press, New York (1971)
Germán Rubino, J., Guarracino, L., Müller, T.M., Holliger, K.: Do seismic waves sense fracture connectivity? Geophys. Res. Lett. 40(4), 692–696 (2013)
Guo, J., Rubino, J.G., Glubokovskikh, S., Gurevich, B.: Effects of fracture intersections on seismic dispersion: theoretical predictions versus numerical simulations. Geophys. Prospect. 65(5), 1264–1276 (2017)
Huang, F., et al.: The first post-injection seismic monitor survey at the Ketzin pilot CO2 storage site: results from time-lapse analysis. Geophys. Prospect. 66(1), 62–84 (2018)
Hunziker, J., et al.: Seismic attenuation and stiffness modulus dispersion in porous rocks containing stochastic fracture networks. J. Geophys. Res. Solid Earth 123(1), 125–143 (2018)
Kong, L., Gurevich, B., Zhang, Y., Wang, Y.: Effect of fracture fill on frequency-dependent anisotropy of fractured porous rocks. Geophys. Prospect. 65(6), 1649–1661 (2017)
Lisitsa, V., Podgornova, O., Tcheverda, V.: On the interface error analysis for finite difference wave simulation. Comput. Geosci. 14(4), 769–778 (2010)
Marty, N.C.M., Hamm, V., Castillo, C., Thiéry, D., Kervévan, C.: Modelling water-rock interactions due to long-term cooled-brine reinjection in the Dogger carbonate aquifer (Paris basin) based on in-situ geothermal well data. Geothermics 88, 101899 (2020)
Masson, Y.J., Pride, S.R., Nihei, K.T.: Finite difference modeling of Biot’s poroelastic equations at seismic frequencies. J. Geophys. Res. Solid Earth 111(B10), 305 (2006)
Menke, H.P., Reynolds, C.A., Andrew, M.G., Pereira Nunes, J.P., Bijeljic, B., Blunt, M.J.: 4D multi-scale imaging of reactive flow in carbonates: assessing the impact of heterogeneity on dissolution regimes using streamlines at multiple length scales. Chem. Geol. 481, 27–37 (2018)
Moczo, P., Kristek, J., Vavrycuk, V., Archuleta, R.J., Halada, L.: 3D heterogeneous staggered-grid finite-difference modeling of seismic motion with volume harmonic and arithmetic averaging of elastic moduli and densities. Bull. Seismol. Soc. Am. 92(8), 3042–3066 (2002)
Muller, T.M., Gurevich, B., Lebedev, M.: Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks – a review. Geophysics 75(5), 75A147-75A164 (2010)
Novikov, M.A., Lisitsa, V.V., Bazaikin, Y.V.: Wave propagation in fractured-porous media with different percolation length of fracture systems. Lobachevskii J. Math. 41(8), 1533–1544 (2020)
Ovaysi, S., Wheeler, M., Balhoff, M.: Quantifying the representative size in porous media. Transp. Porous Media 104(2), 349–362 (2014)
Rubino, J.G., Muller, T.M., Guarracino, L., Milani, M., Holliger, K.: Seismoacoustic signatures of fracture connectivity. J. Geophys. Res. Solid Earth 119(3), 2252–2271 (2014)
Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM (2003)
Salaun, N., et al.: High-resolution 3D seismic imaging and refined velocity model building improve the image of a deep geothermal reservoir in the Upper Rhine Graben. Lead. Edge 39(12), 857–863 (2020)
Samarskii, A.A.: The Theory of Difference Schemes, Pure and Applied Mathematics, vol. 240. CRC Press (2001)
Vavrycuk, V.: Velocity, attenuation, and quality factor in anisotropic viscoelastic media: a perturbation approach. Geophysics 73(5), D63–D73 (2008)
Vishnevsky, D., Lisitsa, V., Tcheverda, V., Reshetova, G.: Numerical study of the interface errors of finite-difference simulations of seismic waves. Geophysics 79(4), T219–T232 (2014)
Xu, C., Dowd, P.A., Mardia, K.V., Fowell, R.J.: A connectivity index for discrete fracture networks. Math. Geol. 38(5), 611–634 (2006)
Zhu, Y., Tsvankin, I.: Plane-wave propagation in attenuative transversely isotropic media. Geophysics 71(2), T17–T30 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Solovyev, S., Novikov, M., Kopylova, A., Lisitsa, V. (2021). Numerical Solution of Biot Equations in Quasi-static State. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2021. ICCSA 2021. Lecture Notes in Computer Science(), vol 12949. Springer, Cham. https://doi.org/10.1007/978-3-030-86653-2_38
Download citation
DOI: https://doi.org/10.1007/978-3-030-86653-2_38
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-86652-5
Online ISBN: 978-3-030-86653-2
eBook Packages: Computer ScienceComputer Science (R0)