## 3 Results and Discussion
### Solution Methods
\begin{array}{l}x^{\circ}\ =\ \mu_{c}\chi/C_{c};\ t^{\circ}\ =\ \mu_{c}t;\,\,\sigma_{c}^{\circ}\ =\ \sigma_{c}/\sigma_{max};\\ v_{c}^{\circ}\ =\ \upsilon_{c}/\upsilon_{max};\ \varepsilon_{c}^{\circ}\ =\ \varepsilon_{c}/\varepsilon_{max};\\ \sigma_{g}^{\circ}\ =\ \sigma_{g}/\sigma_{max};\ \upsilon_{g}^{\circ}\ =\ \
u_{g}/\upsilon_{max};\ e_{g}^{\circ}\ =\ \varepsilon_{g}/\varepsilon_{max};\\ v_{max}\ =\ -\sigma_{max}/C_{c}\rho_{0c};\ \varepsilon_{max}\ =\ \sigma_{max}/E_{g};\\ \gamma_{c}\ =\ E_{Dc}/E_{Sc};\ \gamma_{g}\ =\ E_{Dg}/E_{Sg}\end{array} \right\} \tag{36}\]  
After passing to dimensionless variables \(x^{\circ},t^{\circ},\sigma^{\circ},\varepsilon^{\circ},\) and \(v^{\circ}\), Equations (18)-(35) take the following form for underground pipeline  
\[\left. \begin{array}{l}\frac{\partial x^{\circ}_{c}}{\partial t^{\circ}}+ \frac{\partial x^{\circ}_{c}}{\partial x^{\circ}}-\chi\sigma_{\tau c}^{\circ} \ =\ 0\\ \frac{\partial x^{\circ}_{c}}{\partial x^{\circ}}-\frac{\partial x^{\circ}_{c}}{ \partial x^{\circ}}\ =\ 0\\ \frac{\partial x^{\circ}_{c}}{\partial t^{\circ}}+\varepsilon_{c}^{\circ}\ =\ \frac{ \partial x^{\circ}_{c}}{\partial t^{\circ}}+\gamma_{c}\sigma_{c}^{\circ} \end{array} \right\} \tag{37}\]  
\[\left.\begin{array}{l}d\sigma_{c}^{\circ}+d\upsilon_{c}^{\circ}\ =\ (\varepsilon_{c}^{\circ}-\gamma_{c}c_{c}^{\circ})dt^{\circ}+ \chi\sigma_{c}^{\circ}dt^{\circ}dt^{\circ}\frac{dx^{\circ}}{dt^{\circ}\ =\ +1}\\ d\sigma_{c}^{\circ}-d\upsilon_{c}^{\circ}\ =\ (\varepsilon_{c}^{\circ}-\gamma_{c}c_{c}^{\circ})dt^{ \circ}-\chi\sigma_{c}^{\circ}dt^{\circ}\frac{dx^{\circ}}{dt^{\circ}\ =\ -1}\\ d\sigma_{c}^{\circ}-d\varepsilon_{c}^{\circ}\ =\ (\varepsilon_{c}^{\circ}-\gamma_{c}c_{c}^{\circ})dt^{ \circ}dt^{\circ}\frac{dx^{\circ}}{dt^{\circ}\ =\ -1 for soil  
\[\left. \begin{array}{c}r_{g}\frac{\partial\sigma_{k}^{\circ}}{d\tau}+\frac{ \partial\sigma_{g}^{\circ}}{\partial x}+\chi\sigma_{\tau g}^{\circ}\ =\ 0\\ \frac{\partial\sigma_{g}^{\circ}}{\partial x^{\circ}}+\frac{\partial\sigma_{g} ^{\circ}}{\partial x}=\ 0\\ l_{g}\frac{\partial\sigma_{g}^{\circ}}{d\tau^{\circ}}+l_{g}m_{g}\varepsilon_{g}^ {\circ}\ =\ \frac{\partial\sigma_{g}^{\circ}}{d\tau^{\circ}}+m_{g}\gamma_{g}\sigma_{g}^{ \circ}\end{array} \right\} \tag{39}\]  
\[\left. \begin{array}{c}d\sigma_{g}^{\circ}+C_{g}f_{g}d\sigma_{g}^{\circ}\ =\ m_{g}\left(l_{g} \varepsilon_{g}^{\circ}-\gamma_{g}\sigma_{g}^{\circ}\right)dt^{\circ}-\chi \sigma_{\tau g}^{\circ}dt^{\circ}dt\frac{d\sigma^{\circ}}{d\tau^{\circ}}=+C_ {g}c\\ d\sigma_{g}^{\circ}-l_{g}f_{g}dv_{g}^{\circ}\ =\ m_{g}\left(l_{g} \varepsilon_{g}^{\circ}-\gamma_{g}\sigma_{g}^{\circ}\right)dt^{\circ}+\chi \sigma_{\tau g}^{\circ}dt^{\circ}dt\frac{d\sigma^{\circ}}{dt^{\circ}}=-C_{g}c \\ d\sigma_{g}^{\circ}-l_{g}d\varepsilon_{g}^{\circ}\ =\ m_{g}\left(l_{g} \varepsilon_{g}^{\circ}-\gamma_{g}\sigma_{g}^{\circ}\right)dt^{\circ}dt \frac{d\sigma^{\circ}}{dt^{\circ}}=0\end{array} \right\} \tag{40}\]  
Boundary conditions take the form (39):  
\[\left. \begin{array}{c}\sigma^{\circ}\ =\ sin\left(\frac{\pi t^{\circ}}{ \mu_{c}t}\right)\ at\ x^{\circ}\ =\ 0,\ 0\leq t^{\circ}\leq\mu_{c}\theta\\ \sigma^{\circ}\ =\ 0\ at\ x^{\circ}\ =\ 0,\ t^{\circ}\geq\mu_{c}\theta\\ \sigma_{c}^{\circ}\ =\ 0,\ \varepsilon_{c}^{\circ}\ =\ 0,\ v_{c}^{\circ}\ =\ 0\ at\ x^{\circ}\ =\ t^{\circ}\\ \sigma_{g}^{\circ}\ =\ 0,\ \varepsilon_{g}^{\circ}\ =\ 0,\ v_{g}^{\circ}\ =\ 0\ at\ x^{\circ}\ =\ C_{g}c^{\circ}\end{array} \right\} \tag{41}\]  
where \(r_{g}\ =\ \frac{\rho_{\Omega}}{\rho_{\alpha}}\); \(l_{g}\ =\ \frac{E_{Dg}}{E_{\Sigma g}}\); \(m_{g}\ =\ \frac{\mu_{g}}{\mu_{c}}\); \(C_{gc}\ =\ \frac{C_{g}}{C_{c}^{\circ}}\); \(\sigma_{\tau c}^{\circ}\ =\ \sigma_{\tau}\frac{C_{c}}{\sigma_{\tau cmax^{\circ} \tau g}\frac{c_{g}}{\sigma_{\tau cmax}}}\). Equations (18) and (24) in partial derivatives are thus reduced to ordinary differential equations in dimensionless form (38) and (40) at boundary conditions (41). The initial conditions, as noted above, are zero.