The Gassmann–Burgers Model to Simulate Seismic Waves at the Earth Crust And Mantle JOSE ´ M. CARCIONE, 1 FLAVIO POLETTO , 1 BIANCAMARIA FARINA , 1 and ARONNE CRAGLIETTO 1 Abstract—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 condi- tions; moreover, some parts are melted. Seismic waves can be used to detect these conditions on the basis of reflection and transmis- sion 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 vis- cosity, 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 tech- nique 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. Key words: Brittle, ductile, Burgers model, Gassmann theory, seismic-wave simulation, attenuation, Fourier method. 1. Introduction The seismic characterization of the brittle and ductile parts of the crust and mantle is essential in earthquake seismology and geothermal exploration, since it plays an important role in determining the nucleation depth of earthquakes (Meissner and Strehlau 1982) and the availability of geothermal energy (Manzella et al. 1998). Carcione and Poletto (2013) introduced an elastic–plastic rheology to model ductile behaviour on the basis of variations of the shear modulus as a function of temperature. The ductile medium mainly flows when subject to devi- atoric stress, while it does not show major flow under hydrostatic stress, such that the deformation is mainly associated with the shear modulus of the medium. The criterion by which the shear modulus is affected is based on the octahedral stress, a scalar quantity that is invariant under coordinate transformations. The flow viscosity is a function of temperature and con- fining pressure, determined by the geothermal gradient and the tectonic stresses. Carcione and Poletto (2013) have also modeled the effects of ani- sotropy and seismic attenuation, based on the Burgers model (see Fig. 1) (the Maxwell and Zener model are particular cases of this model) [see Mainardi and Spada (2011)]. The Zener part of the model is used to model the viscoelastic motion with no plastic flow, obtained as the limit of infinite plastic viscosity. Seismic losses are solely due to shear deformations. Carcione et al. (2014) implement the previous model to simulate wave propagation in isotropic and anelastic inhomogeneous media and compute 1 Istituto Nazionale di Oceanografia e di Geofisica Speri- mentale (OGS), Borgo Grotta Gigante 42c, Sgonico, 34010 Trieste, Italy. E-mail: jcarcione@inogs.it Pure Appl. Geophys. Ó 2016 Springer International Publishing DOI 10.1007/s00024-016-1437-2 Pure and Applied Geophysics