Generalized Thermo-poroelasticity Equations and Wave Simulation

We establish a generalization of the thermoelasticity wave equation to the porous case, including the Lord–Shulman (LS) and Green–Lindsay (GL) theories that involve a set of relaxation times ( $$\tau _i, \ i = 1, \ldots , 4$$ ). The dynamical equations predict four propagation modes, namely, a fast P wave, a Biot slow wave, a thermal wave, and a shear wave. The plane-wave analysis shows that the GL theory predicts a higher attenuation of the fast P wave, and consequently a higher velocity dispersion than the LS theory if $$\tau _1 = \tau _2 > \tau _3$$ , whereas both models predict the same anelasticity for $$\tau _1 = \tau _2 = \tau _3$$ . We also propose a generalization of the LS theory by applying two different Maxwell–Vernotte–Cattaneo relaxation times related to the temperature increment ( $$\tau _3$$ ) and solid/fluid strain components ( $$\tau _4$$ ), respectively. The generalization predicts positive quality factors when $$\tau _4 \ge \tau _3$$ , and increasing $$\tau _4$$ further enhances the attenuation. The wavefields are computed with a direct meshing algorithm using the Fourier pseudospectral method to calculate the spatial derivatives and a first-order explicit Crank–Nicolson time-stepping method. The propagation illustrated with snapshots and waveforms at low and high frequencies is in agreement with the dispersion analysis. The study can be useful for a comprehensive understanding of wave propagation in high-temperature high-pressure fields.