## 5 Results and Discussions  
The physical quantity distribution in a pore-elastic material was investigated. The thermal properties of saturated sandstone at \(T_{o}=27\ ^{\circ}\mathrm{C}\) have been written as in Singh [11; 53] for the numerical computation:  
\[\mathrm{R}=0.07435\times 10^{11}(\mathrm{dyne})\big{(}\mathrm{cm}^{-2} ),\rho^{s*}=2.6(\mathrm{g})\big{(}\mathrm{cm}^{-3}\big{)},\ \mathrm{K}^{s}=0.4(\mathrm{cal})\big{(}\mathrm{cm}^{-1}\big{)}\big{(}\mathrm{s}^{-1 }\big{)}(^{\circ}\mathrm{C}^{-1})\] \[\mathrm{K}^{f}=0.3(\mathrm{cal})\big{(}\mathrm{cm}^{-1}\big{)} \big{(}\mathrm{s}^{-1}\big{)}(^{\circ}\mathrm{C}^{-1}),\mathrm{Q}=0.99663\times 1 0^{11}(\mathrm{dyne})\big{(}\mathrm{cm}^{-2}\big{)},\ \rho^{ts}=0.82(\mathrm{g}) \big{(}\mathrm{cm}^{-3}\big{)},\] \[\lambda=0.44363\times 10^{11}(\mathrm{dyne})\big{(}\mathrm{cm}^{-2} ),\ \rho_{11}=0.002137(\mathrm{g})\big{(}\mathrm{cm}^{-3}\big{)}\] \[\mathrm{C}_{\mathrm{E}}^{f}=1.9(\mathrm{cal})\big{(}\mathrm{g}^{- 1}\big{)}(^{\circ}\mathrm{C}^{-1}),C_{\mathrm{E}}^{s}=2.1(\mathrm{cal})\big{(} \mathrm{g}^{-1}\big{)}(^{\circ}\mathrm{C}^{-1}),\mu=0.2765\times 10^{11}(\mathrm{ dyne})\big{(}\mathrm{cm}^{-2}\big{)}\] \[\alpha^{f}=\alpha^{s}=\alpha^{fs}=\alpha^{sf}=0.1\big{(}\mathrm{ cm}^{-3}\big{)}\big{(}\mathrm{g}^{-1}\big{)}\]  
The physical quantities calculated numerically for different x values are explained in Figures 2-7, and are based on the previous dataset. Numerical calculations were carried out for the liquid and solid displacements, the liquid and solid temperatures, and the components of the stress distribution concerning the distances under six thermal delay times in the framework of the poro-thermo-elastic theory. Figures 2-7 show the four curves predicted by different levels of porosity (\(\beta=0.15\), \(\beta=0.30\)) using the three-phase lag model (TPL) and the Green\(-\)Naghdi type III model (GNIII) at time (\(t=0.2\)). Figure 2 displays the variation of solid displacement with respect to distance \(x\). It was discovered that the displacement began at zero, satisfying the boundary constraints, and subsequently increased to the maximum values at a specific spot near the surface. Then, based on the values of porosity and the thermal delay times, monotonic declines in displacement to zero at \(1.3\leq x\leq 2\) were observed. Figure 3 depicts the fluid phase displacement changes along distance \(x\). It is obvious that the liquid displacement began at zero, satisfying the boundary constraints, and subsequently increased to maximum values at a specific spot near the surface. According to the values of thermal delay times and porosity, monotonic declines in liquid displacement to zero at \(1.3\leq x\leq 2\) were found. Figure 4 shows the variations of solid temperature via distance \(x\). It was observed that the solid temperature started at its greatest values at \(x=0\) and progressively reduced as the distances \(x\) grew to zero beyond a wavefront, satisfying the problem's theoretical boundary requirements. Figure 5 shows how the liquid temperature changes via distance \(x\). It was discovered that the \(\Theta^{f}\) value was maximum at \(x=0\) and declined with an increasing x distance until they were close to zero at \(x=0.3\), depending on the porosity and model type. The effects of thermal delay times and porosity in liquid and solid stresses throughout the distance \(x\) are shown in Figures 6 and 7 under the three-phase lag model (TPL) and Green\(-\)Naghdi type III model (GNIII). As expected, the porosity and thermal delay times and porosity had the most significant influence on all of the examined fields' values.  
Figure 4: The distributions of solid temperature \(\Theta^{\rm s}\) along the distance \(x\).  
Figure 3: The distributions of fluid displacement U along the distance \(x\).  
Figure 5: The distributions of fluid temperature \(\Theta^{t}\) along the distance \(x\).  
Figure 6: The distributions of solid stress \(\sigma_{\rm xx}\) along the distance \(x\).