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

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.