Semi-Analytical Solution for the Wave Motion of Unsaturated Viscoelastic Porous Media Based on Fractional Poynting-Thomson Model

The wave equations for unsaturated porous media in a Biot-type form are modified by incorporating a fractional-order Poynting-Thomson viscoelastic model in place of the original linear elastic model. Utilizing the elastic-viscoelastic correspondence principle, the state-space method is employed to derive a semi-analytical solution in the time domain for the one-dimensional transient response of unsaturated viscoelastic porous media. The validity of the presented solutions is confirmed through comparison with existing solutions. The results indicate that fluid saturation ${S_w}$Sw significantly affects the hysteresis of the compressional waves and the wave velocity; the viscosity coefficient ${\eta _0}$eta 0 introduces a notable effect in the ${P_2}$P2 wave behavior.