Pure and Applied Geophysics

, Volume 174, Issue 3, pp 849–863 | Cite as

The Gassmann–Burgers Model to Simulate Seismic Waves at the Earth Crust And Mantle

  • José M. CarcioneEmail author
  • Flavio Poletto
  • Biancamaria Farina
  • Aronne Craglietto


The upper part of the crust shows generally brittle behaviour while deeper zones, including the mantle, may present ductile behaviour, depending on the pressure–temperature conditions; moreover, some parts are melted. Seismic waves can be used to detect these conditions on the basis of reflection and transmission events. Basically, from the elastic–plastic point of view the seismic properties (seismic velocity and density) depend on effective pressure and temperature. Confining and pore pressures have opposite effects on these properties, such that very small effective pressures (the presence of overpressured fluids) may substantially decrease the P- and S-wave velocities, mainly the latter, by opening of cracks and weakening of grain contacts. Similarly, high temperatures induce the same effect by partial melting. To model these effects, we consider a poro-viscoelastic model based on Gassmann equations and Burgers mechanical model to represent the properties of the rock frame and describe ductility in which deformation takes place by shear plastic flow. The Burgers elements allow us to model the effects of seismic attenuation, velocity dispersion and steady-state creep flow, respectively. The stiffness components of the brittle and ductile media depend on stress and temperature through the shear viscosity, which is obtained by the Arrhenius equation and the octahedral stress criterion. Effective pressure effects are taken into account in the dry-rock moduli using exponential functions whose parameters are obtained by fitting experimental data as a function of confining pressure. Since fluid effects are important, the density and bulk modulus of the saturating fluids (water and steam) are modeled using the equations provided by the NIST website, including supercritical behaviour. The theory allows us to obtain the phase velocity and quality factor as a function of depth and geological pressure and temperature as well as time frequency. We then obtain the PS and SH equations of motion recast in the velocity–stress formulation, including memory variables to avoid the computation of time convolutions. The equations correspond to isotropic anelastic and inhomogeneous media and are solved by a direct grid method based on the Runge–Kutta time stepping technique and the Fourier pseudospectral method. The algorithm is tested with success against known analytical solutions for different shear viscosities. An example shows how anomalous conditions of pressure and temperature can in principle be detected with seismic waves.


Brittle ductile Burgers model Gassmann theory seismic-wave simulation attenuation Fourier method 


  1. Aleotti, L., Poletto, F., Miranda, F., Corubolo, P., Abramo, F., & Craglietto, A. (1999). Seismic while-drilling technology: use and analysis of the drill-bit seismic source in a cross-hole survey. Geophysical Prospecting, 47, 25–39.CrossRefGoogle Scholar
  2. Carcione, J. M., 2015. Wave fields in real media: Wave propagation in anisotropic, anelastic, porous and electromagnetic media. Handbook of Geophysical Exploration, vol. 38, Elsevier (3nd edition, revised and extended).Google Scholar
  3. Carcione, J. M., & Gurevich, B. (2011). Differential form and numerical implementation of Biots poroelasticity equations with squirt dissipation. Geophysics, 76, N55–N64.CrossRefGoogle Scholar
  4. Carcione, J. M., Helle, H. B., & Gangi, A. F. (2006). Theory of borehole stability when drilling through salt formations. Geophysics, 71, F31–F47.CrossRefGoogle Scholar
  5. Carcione, J. M., Herman, G., & ten Kroode, F. P. E. (2002). Seismic modeling. Geophysics, 67, 1304–1325.CrossRefGoogle Scholar
  6. Carcione, J. M., Morency, C., & Santos, J. E. (2010). Computational poroelasticity—a review. Geophysics, 75, A229–A243.Google Scholar
  7. Carcione, J. M., & Poletto, F. (2013). Seismic rheological model and reflection coefficients of the brittle-ductile transition. Pure and Applied Geophysics,. doi: 10.1007/s00024-013-0643-4.Google Scholar
  8. Carcione, J. M., Poletto, F., Farina, B., & Craglietto, A. (2014). Simulation of seismic waves at the Earth’s crust (brittle-ductile transition) based on the Burgers model. Solid Earth Discuss, 6, 1371–1400.CrossRefGoogle Scholar
  9. Carter, N. L., & Hansen, F. D. (1983). Creep of rocksalt. Tectonophysics, 92, 275–333.CrossRefGoogle Scholar
  10. Castro, R. R., Gallipoli, M. R., & Mucciarelli, M. (2008). Crustal Q in southern Italy determined from regional earthquakes. Tectonophysics, 457(2), 96–101.CrossRefGoogle Scholar
  11. Dragoni, M. (1990). Stress relaxation at the lower dislocation edge of great shallow earthquakes. Tectonophysics, 179, 113–119.CrossRefGoogle Scholar
  12. Dragoni, M., & Pondrelli, S. (1991). Depth of the brittle-ductile transition in a transcurrent boundary zone. Pure and Applied Geophysics, 135, 447–461.CrossRefGoogle Scholar
  13. Engelder, T., 1993. Stress regimes in the lithosphere, Princeton University Press.Google Scholar
  14. Gangi, A. F. ,1981. A constitutive equation for one-dimensional transient and steady-state flow of solids. Mechanical Behavior of Crustal Rocks, Geophysical Monograph 24, AGU, 275–285.Google Scholar
  15. Gangi, A. F. (1983). Transient and steady-state deformation of synthetic rocksalt. Tectonophysics, 91, 137–156.CrossRefGoogle Scholar
  16. Hegret, G. (1987). Stress assumption for underground excavation in the Canadian Shield. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 24, 95–97.Google Scholar
  17. Jaya, M. S., Shapiro, S. A., Kristinsdóttir, L. H., Bruhn, D., Milsch, H., & Spangenberg, E. (2010). Temperature dependence of seismic properties in geothermal rocks at reservoir conditions. Geothermics, 39, 115–123.CrossRefGoogle Scholar
  18. Kaselow, A., & Shapiro, S. A. (2004). Stress sensitivity of elastic moduli and electrical resistivity in porous rocks. Journal of Geophysics and Engineering, 1, 1–11.CrossRefGoogle Scholar
  19. Lemmon, E. W., McLinden, M. O., Friend, D. G., 2005. Thermophysical properties of fluid systems. In: Lindstrom, P.J., Mallard, W.G. (eds.), NIST Chemistry Webbook 69, NIST Standard Reference Database, Gaithersburg, MD, USA,
  20. Mainardi, F., & Spada, G. (2011). Creep, relaxation and viscosity properties for basic fractional models in rheology. The European Physical Journal, Special Topics, 193, 133–160.CrossRefGoogle Scholar
  21. Manzella, A., Ruggieri, G., Gianelli, G., & Puxeddu, M. (1998). Plutonic-geothermal systems of southern Tuscany: a review of the crustal models. Mem. Soc. Geol. It., 52, 283–294.Google Scholar
  22. Mavko, G., & Nur, A. (1975). Melt squirt in the asthenosphere. Journal of Geophysical Research, 80, 1444–1448.CrossRefGoogle Scholar
  23. Meissner, R., & Strehlau, J. (1982). Limits of stresses in continental crusts and their relation to the depth-frequency distribution of shallow earthquakes. Tectonics, 1, 73–89.CrossRefGoogle Scholar
  24. Montesi, L. G. J. (2007). A constitutive model for layer development in shear zones near the brittle-ductile transition. Geophysical Research Letters, 34, L08307. doi: 10.1029/2007GL029250.CrossRefGoogle Scholar
  25. Poletto, F., Corubolo, P., Schleifer, A., Farina, B., Pollard, J., & Grozdanich, B. (2011). Seismic while drilling for geophysical exploration in a geothermal well. Transactions, Geothermal Resources Council, 35(2), 1737–1741.Google Scholar
  26. Poletto, F., & Miranda, F. (2004). Seismic while drilling. Pergamon, Amsterdam: Fundamentals of drill-bit seismic for exploration.Google Scholar
  27. Popp, T., & Kern, H. (1994). The influence of dry and water saturated cracks on seismic velocities of crustal rocks—a comparison of experimental data with theoretical model. Surveys in Geophysics, 15, 443–465.CrossRefGoogle Scholar
  28. Schön, J. H., 2011. Physical properties of rocks:A workbook, Handbook of Petroleum Exploration and Production, vol. 8.Google Scholar
  29. Singh, S. C., Taylor, M. A. J., and Montagner, J. P., 2000. On the presence of liquid in Earths inner core, Science 287, March Issue.Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • José M. Carcione
    • 1
    Email author
  • Flavio Poletto
    • 1
  • Biancamaria Farina
    • 1
  • Aronne Craglietto
    • 1
  1. 1.Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (OGS)TriesteItaly

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