Analytical Solutions to the One-Dimensional Coupled Seepage and Deformation of Unsaturated Soils with Arbitrary Nonhomogeneous Boundary Conditions

Abstract

In one-dimensional coupled seepage and deformation problems, permeability coefficients are a function of matric suctions in unsaturated soils. Due to the presence of permeability coefficient and the coupling effect in one dimension, coupled equations are nonlinear. Additionally, different boundary conditions can also make coupled seepage and deformation difficult to get its analytical solution. This study presents solutions in the one-dimensional coupled seepage and deformation of unsaturated soils with arbitrary nonhomogeneous boundary conditions. Analytical solutions were derived for a finite thickness and confined lateral directions in unsaturated soils. Furthermore, the coupled seepage and deformation equation in one dimension was derived from the mass conservation principle integrated with Dakshanamurthy’s constitutive relationships, in which the water and air flows follow Darcy’s law in unsaturated soils. To linearize a governing equation, an exponential transform and several dimensionless variables were selected. Solutions were obtained by applying a Laplace integral transform to a dimensionless linear equation under arbitrary initial and boundary conditions. After performing inverse Laplace transforms and using the residue theorem, analytical solutions were obtained in the time domain. Finally, three typical examples of nonhomogeneous boundary conditions were considered as case studies. It can be noted that the analytical solutions are consistent with the numerical results by a finite difference method. The result also indicates that the boundary conditions and the coupling effect have a significant influence on the seepage in unsaturated soils.

Appendices

Appendices

1.1 Appendix 1: $\tilde{F}_L (s)$

$$\begin{aligned}&\tilde{F}_B (s)=\frac{ \tilde{\varphi }_1^B \left( {\hbox {e}^{\frac{1}{2} \eta S}\hbox {e}^{-\frac{1}{2} S}-\hbox {e}^{\frac{1}{2}S}\hbox {e}^{-\frac{1}{2}\eta S}} \right) +\tilde{\varphi }_2^B e^{\frac{1}{2}C_3 }\left( {\hbox {e}^{-\frac{1}{2}\eta S}-\hbox {e}^{\frac{1}{2}\eta S}} \right) }{\left( {\hbox {e}^{-\frac{1}{2} S}-\hbox {e}^{\frac{1}{2} S}} \right) } \\&\quad +\frac{C_1 \hbox {e}^{\frac{1}{2}\eta C_3 }\hbox {e}^{\frac{1}{2} \eta S} \left( {\hbox {e}^{-\frac{1}{2} S}\int _0^1 \hbox {e}^{\frac{1}{2} x C_3 }\hbox {e}^{\frac{1}{2} xS}\hbox {e}^{\frac{C_4 \phi \left( x \right) }{u_0 }}\mathrm{d}x-\hbox {e}^{\frac{1}{2} S}\int _0^1 \hbox {e}^{\frac{1}{2} x C_3 }\hbox {e}^{-\frac{1}{2} x S}\hbox {e}^{\frac{C_4 \phi \left( x \right) }{u_0 }}\mathrm{d}x} \right) }{sS\hbox {e}^{\eta C_3 }\left( {\hbox {e}^{-\frac{1}{2} S}-\hbox {e}^{\frac{1}{2} S}} \right) } \\&\quad +\frac{C_1 \hbox {e}^{\frac{1}{2} \eta C_3 }\hbox {e}^{-\frac{1}{2} \eta S}\left( {\hbox {e}^{\frac{1}{2} S}\int _0^1 \hbox {e}^{\frac{1}{2} xC_3 }\hbox {e}^{-\frac{1}{2}xS}\hbox {e}^{\frac{C_4 \phi \left( x \right) }{u_0 }}\mathrm{d}x- \hbox {e}^{-\frac{1}{2} S}\int _0^1 \hbox {e}^{\frac{1}{2} x C_3 }\hbox {e}^{\frac{1}{2} xS}\hbox {e}^{\frac{C_4 \phi \left( x \right) }{u_0 }}\mathrm{d}x} \right) }{sS\hbox {e}^{\eta C_3 }\left( {\hbox {e}^{-\frac{1}{2} S}-\hbox {e}^{\frac{1}{2} S}} \right) } \\&\quad +\frac{C_1 \hbox {e}^{\frac{1}{2} \eta C_3 }\hbox {e}^{\frac{1}{2} \eta S}\left( {\hbox {e}^{\frac{1}{2}S}\int _0^\eta \hbox {e}^{\frac{1}{2}xC_3 }\hbox {e}^{-\frac{1}{2} x S}\hbox {e}^{\frac{C_4 \phi \left( x \right) }{u_0 }}\mathrm{d}x-\hbox {e}^{-\frac{1}{2} S}\int _0^\eta \hbox {e}^{\frac{1}{2} x C_3 }\hbox {e}^{-\frac{1}{2}x S}\hbox {e}^{\frac{C_4 \phi \left( x \right) }{u_0 }}\mathrm{d}x} \right) }{sS\hbox {e}^{\eta C_3 }\left( {\hbox {e}^{-\frac{1}{2} S}-\hbox {e}^{\frac{1}{2} S}} \right) } \\&\quad +\frac{C_1 \hbox {e}^{\frac{1}{2} \eta C_3 }\hbox {e}^{-\frac{1}{2} \eta S}\left( {\hbox {e}^{-\frac{1}{2} S}\int _0^\eta \hbox {e}^{\frac{1}{2}x C_3 }\hbox {e}^{\frac{1}{2} x S}\hbox {e}^{\frac{C_4 \phi \left( x \right) }{u_0 }}\mathrm{d}x-\hbox {e}^{\frac{1}{2} S}\int _0^\eta \hbox {e}^{\frac{1}{2}x C_3 }\hbox {e}^{\frac{1}{2}xS}\hbox {e}^{\frac{C_4 \phi \left( x \right) }{u_0 }}\mathrm{d}x} \right) }{sS\hbox {e}^{\eta C_3 }\left( {\hbox {e}^{-\frac{1}{2} S}-\hbox {e}^{\frac{1}{2} S}} \right) } \end{aligned}$$

1.2 Appendix 2: $W_L (\eta ,\tau )$ ($I$ is the imaginary root)

where $\beta _m$ satisfies $C_3 \sin (\beta )=2\beta \cos (\beta )$.

where $\beta _m$ satisfies $C_3 \sin (\beta )=-2\beta \cos (\beta )$.