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Viscoelastic substitute models for seismic attenuation caused by squirt flow and fracture leak off


We have investigated viscoelastic substitute models for seismic attenuation caused by fluid pressure diffusion in fluid-saturated porous media. Fluid pressure diffusion may locally occur associated with fracture leak off and/or squirt flow. We use a homogenization scheme with numerical model reduction (NMR), recently established in the literature, and we derive the corresponding viscoelastic material properties that are apparent at a larger scale (i.e., the observer scale). Moreover, we find that the rheology of the resulting viscoelastic model is of the Maxwell-Zener type. Based on a series of numerical experiments, we find that this method is able to accurately and efficiently predict the overall attenuation and stiffness moduli dispersion for a range of scenarios without resolving the substructure problem explicitly. Computational homogenization, together with NMR, can be useful to simulate seismic wave propagation using a viscoelastic substitute model that accurately reproduces the energy dissipation and dispersion of a heterogeneous medium in which squirt flow and/or fracture leak-off occurs.


  • Adelinet, M., J. Fortin, Y. Guéguen, A. Schubnel, and L. Geoffroy, 2010, Frequency and fluid effects on elastic properties of basalt: Experimental investigations: Geophysical Research Letters, 37, L02303, doi: 10.1029/2009GL041660.CrossrefWeb of ScienceGoogle Scholar
  • Biot, M., 1941, General theory of three-dimensional consolidation: Journal of Applied Physics, 12, 155–164, doi: 10.1063/1.1712886.CrossrefGoogle Scholar
  • Biot, M., 1962, Mechanics of deformation and acoustic propagation in porous media: Journal of Applied Physics, 33, 1482–1498, doi: 10.1063/1.1728759.CrossrefWeb of ScienceGoogle Scholar
  • Brajanovski, M., B. Gurevich, and M. Schoenberg, 2005, A model for P-wave attenuation and dispersion in a porous medium permeated by aligned fractures: Geophysical Journal International, 163, 372–384, doi: 10.1111/j.1365-246X.2005.02722.x.CrossrefWeb of ScienceGoogle Scholar
  • Carcione, J., C. Morency, and J. E. Santos, 2010, Computational poroelasticity — A review: Geophysics, 75, no. 5, 75A229–75A243, doi: 10.1190/1.3474602.AbstractWeb of ScienceGoogle Scholar
  • Gurevich, B., M. Brajanovski, R. Galvin, T. Muller, and J. Toms-Stewart, 2009, P-wave dispersion and attenuation in fractured and porous reservoirs — Poroelasticity approach: Geophysical Prospecting, 57, 225–237, doi: 10.1111/j.1365-2478.2009.00785.x.CrossrefWeb of ScienceGoogle Scholar
  • Gurevich, B., D. Makarynska, O. de Paula, and M. Pervukhina, 2010, A simple model for squirt-flow dispersion and attenuation in fluid-saturated granular rocks: Geophysics, 75, no. 6, N109–N120, doi: 10.1190/1.3509782.AbstractWeb of ScienceGoogle Scholar
  • Jänicke, R., F. Larsson, H. Steeb, and K. Runesson, 2016, Numerical identification of a viscoelastic substitute model for heterogeneous poroelastic media by a reduced order homogenization approach: Computer Methods in Applied Mechanics and Engineering, 298, 108–120, doi: 10.1016/j.cma.2015.09.024.CrossrefWeb of ScienceGoogle Scholar
  • Jänicke, R., B. Quintal, F. Larsson, and K. Runesson, 2019, Identification of viscoelastic properties from numerical model reduction of pressure diffusion in fluid-saturated porous rock with fractures: Computational Mechanics, 63, 49–67, doi: 10.1007/s00466-018-1584-7.CrossrefWeb of ScienceGoogle Scholar
  • Jänicke, R., B. Quintal, and H. Steeb, 2015, Numerical homogenization of mesoscopic loss in poroelastic media: European Journal of Mechanics — A/Solids, 49, 382–395, doi: 10.1016/j.euromechsol.2014.08.011.CrossrefWeb of ScienceGoogle Scholar
  • Müller, T., B. Gurevich, and M. Lebedev, 2010, Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks — A review: Geophysics, 75, no. 5, A147–A164, doi: 10.1190/1.3463417.AbstractWeb of ScienceGoogle Scholar
  • Murphy, W., K. Winkler, and R. Kleinberg, 1986, Acoustic relaxation in sedimentary rocks: Dependence on grain contacts and fluid saturation: Geophysics, 51, 757–766, doi: 10.1190/1.1442128.AbstractWeb of ScienceGoogle Scholar
  • O’Connell, R., and B. Budiansky, 1977, Viscoelastic properties of fluid-saturated cracked solids: Journal of Geophysical Research, 82, 5719–5735, doi: 10.1029/JB082i036p05719.CrossrefWeb of ScienceGoogle Scholar
  • Pimienta, L., J. Fortin, and Y. Gueguen, 2015, Bulk modulus dispersion and attenuation in sandstones: Geophysics, 80, no. 2, D111–D127, doi: 10.1190/geo2014-0335.1.AbstractWeb of ScienceGoogle Scholar
  • Quintal, B., R. Jänicke, J. Rubino, H. Steeb, and K. Holliger, 2014, Sensitivity of S-wave attenuation to the connectivity of fractures in fluid-saturated rocks: Geophysics, 79, no. 5, WB15–WB24, doi: 10.1190/geo2013-0409.1.AbstractWeb of ScienceGoogle Scholar
  • Quintal, B., J. Rubino, E. Caspari, and K. Holliger, 2016, A simple hydromechanical approach for simulating squirt-type flow: Geophysics, 81, no. 4, D335–D344, doi: 10.1190/geo2015-0383.1.AbstractWeb of ScienceGoogle Scholar
  • Quintal, B., H. Steeb, M. Frehner, and S. Schmalholz, 2011, Quasi-static finite-element modeling of seismic attenuation and dispersion due to wave-induced fluid flow in poroelastic media: Journal of Geophysical Research, 116, B01201, doi: 10.1029/2010JB007475.CrossrefWeb of ScienceGoogle Scholar
  • Rubino, J., L. Guarracino, T. Müller, and K. Holliger, 2013, Do seismic waves sense fracture connectivity?: Geophysical Research Letters, 40, 692–696, doi: 10.1002/grl.50127.CrossrefWeb of ScienceGoogle Scholar
  • Subramaniyan, S., B. Quintal, C. Madonna, and E. H. Saenger, 2015, Laboratory-based seismic attenuation in Fontainebleau sandstone: Evidence of squirt flow: Journal of Geophysical Research, 120, 7526–7535.Google Scholar
  • Tillotson, P., M. Chapman, J. Sothcott, A. Best, and X.-Y. Li, 2014, Pore fluid viscosity effects on P- and S-wave anisotropy in synthetic silica-cemented sandstone with aligned fractures: Geophysical Prospecting, 62, 1238–1252, doi: 10.1111/1365-2478.12194.CrossrefWeb of ScienceGoogle Scholar
  • Vinci, C., J. Renner, and H. Steeb, 2014, On attenuation of seismic waves associated with flow in fractures: Geophysical Research Letters, 41, 7515–7523, doi: 10.1002/2014GL061634.CrossrefWeb of ScienceGoogle Scholar
  • White, J., 1975, Computed seismic speeds and attenuation in rocks with partial gas saturation: Geophysics, 40, 224–232, doi: 10.1190/1.1440520.AbstractWeb of ScienceGoogle Scholar
  • Zhang, Y., and M. Toksöz, 2012, Computation of dynamic seismic responses to viscous fluid of digitized three-dimensional Berea sandstones with a coupled finite-difference method: The Journal of the Acoustical Society of America, 132, 630–640, doi: 10.1121/1.4733545.CrossrefWeb of ScienceGoogle Scholar