Semi-analytical solution for the one-dimensional consolidation of multi-layered unsaturated soils with semi-permeable boundary

Abstract

In this paper, a general solution was developed by the Laplace transform applying to the one-dimensional (1D) consolidation equation in a multi-layered unsaturated soil layer, and the analytical solutions of excess pore pressures as well as settlements under a semi-permeable boundary were derived in the Laplace domain employing the transfer matrix technique. Then to acquire the final solution in the time domain, a numerical computation was implemented by Crump’s method. Moreover, the proposed semi-analytical result is in good agreement with that obtained from the finite difference method. It validated that the present mathematical methodology is accessible, efficient, and reliable for deriving a semi-analytical solution of the 1D consolidation equation. A parametric study was conducted to describe the consolidation characteristic of a two-layered unsaturated soil. It is shown that both the boundary permeability and the soil-layer properties play a significant role in the consolidation of multi-layered unsaturated soils.

Appendix

$$\begin{aligned} {\chi _1}= & {} {\eta _{Rt2}}\left( \begin{array}{l} {\mathrm{e}^{2H({\eta ^{(n)}} + {\xi ^{(n)}})}}({m_{13}}{m_{31}} - {m_{11}}{m_{33}}) - {\mathrm{e}^{2H{\eta ^{(n)}}}}({m_{23}}{m_{31}} - {m_{21}}{m_{33}}) \\ + {m_{23}}{m_{41}} - {m_{21}}{m_{43}} - {\mathrm{e}^{2H{\xi ^{(n)}}}}({m_{13}}{m_{41}} - {m_{11}}{m_{43}}) \\ \end{array} \right) \\&+\; {\eta _{Rt1}}\left( \begin{array}{l} {\mathrm{e}^{2H({\eta ^{(n)}} + {\xi ^{(n)}})}}({m_{14}}{m_{31}} - {m_{11}}{m_{34}}) - {\mathrm{e}^{2H{\eta ^{(n)}}}}({m_{24}}{m_{31}} - {m_{21}}{m_{34}}) \\ + {m_{24}}{m_{41}} - {m_{21}}{m_{44}} - {\mathrm{e}^{2H{\xi ^{(n)}}}}({m_{14}}{m_{41}} - {m_{11}}{m_{44}}) \\ \end{array} \right) , \\ {\chi _2}= & {} {\eta _{Rt2}}\left( \begin{array}{l} {\mathrm{e}^{2H({\eta ^{(n)}} + {\xi ^{(n)}})}}({m_{13}}{m_{32}} - {m_{12}}{m_{33}}) - {\mathrm{e}^{2H{\eta ^{(n)}}}}({m_{23}}{m_{32}} - {m_{22}}{m_{33}}) \\ + {m_{23}}{m_{42}} - {m_{22}}{m_{43}} - {\mathrm{e}^{2H{\xi ^{(n)}}}}({m_{13}}{m_{42}} - {m_{12}}{m_{43}}) \\ \end{array} \right) \\&+\; {\eta _{Rt1}}\left( \begin{array}{l} {\mathrm{e}^{2H({\eta ^{(n)}} + {\xi ^{(n)}})}}({m_{14}}{m_{32}} - {m_{12}}{m_{34}}) - {\mathrm{e}^{2H{\eta ^{(n)}}}}({m_{24}}{m_{32}} - {m_{22}}{m_{34}}) \\ + {m_{24}}{m_{42}} - {m_{22}}{m_{44}} - {\mathrm{e}^{2H{\xi ^{(n)}}}}({m_{14}}{m_{42}} - {m_{12}}{m_{44}}) \\ \end{array} \right) , \\ {\chi _3}= & {} {\mathrm{e}^{2H({\eta ^{(n)}} + {\xi ^{(n)}})}}({m_{14}}{m_{33}} - {m_{13}}{m_{34}}) - {\mathrm{e}^{2H{\eta ^{(n)}}}}({m_{24}}{m_{33}} - {m_{23}}{m_{34}}) \\&+ {m_{24}}{m_{43}} - {m_{23}}{m_{44}} - {\mathrm{e}^{2H{\xi ^{(n)}}}}({m_{14}}{m_{43}} - {m_{13}}{m_{44}}), \\ {\chi _4}= & {} {\mathrm{e}^{2H({\eta ^{(n)}} + {\xi ^{(n)}})}}({m_{13}}{m_{32}} - {m_{12}}{m_{33}}) - {\mathrm{e}^{2H{\eta ^{(n)}}}}({m_{23}}{m_{32}} - {m_{22}}{m_{33}}) \\&+ {m_{23}}{m_{42}} - {m_{22}}{m_{43}} - {\mathrm{e}^{2H{\xi ^{(n)}}}}({m_{13}}{m_{42}} - {m_{12}}{m_{43}}), \\ {\chi _5}= & {} {\mathrm{e}^{2H({\eta ^{(n)}} + {\xi ^{(n)}})}}({m_{14}}{m_{32}} - {m_{12}}{m_{34}}) - {\mathrm{e}^{2H{\eta ^{(n)}}}}({m_{24}}{m_{32}} - {m_{22}}{m_{34}}) \\&+ {m_{24}}{m_{42}} - {m_{22}}{m_{44}} - {\mathrm{e}^{2H{\xi ^{(n)}}}}({m_{14}}{m_{42}} - {m_{12}}{m_{44}}), \\ {\chi _6}= & {} {\mathrm{e}^{2H({\eta ^{(n)}} + {\xi ^{(n)}})}}({m_{13}}{m_{31}} - {m_{11}}{m_{33}}) - {\mathrm{e}^{2H{\eta ^{(n)}}}}({m_{23}}{m_{31}} - {m_{21}}{m_{33}}) \\&+ {m_{23}}{m_{41}} - {m_{21}}{m_{43}} - {\mathrm{e}^{2H{\xi ^{(n)}}}}({m_{13}}{m_{41}} - {m_{11}}{m_{43}}), \\ {\chi _7}= & {} {\mathrm{e}^{2H({\eta ^{(n)}} + {\xi ^{(n)}})}}({m_{14}}{m_{31}} - {m_{11}}{m_{34}}) - {\mathrm{e}^{2H{\eta ^{(n)}}}}({m_{24}}{m_{31}} - {m_{21}}{m_{34}}) \\&+ {m_{24}}{m_{41}} - {m_{21}}{m_{44}} - {\mathrm{e}^{2H{\xi ^{(n)}}}}({m_{14}}{m_{41}} - {m_{11}}{m_{44}}), \\ {\chi _8}= & {} {\mathrm{e}^{2H({\eta ^{(n)}} + {\xi ^{(n)}})}}({m_{12}}{m_{31}} - {m_{11}}{m_{32}}) - {\mathrm{e}^{2H{\eta ^{(n)}}}}({m_{22}}{m_{31}} - {m_{21}}{m_{32}}) \\&+ {m_{22}}{m_{41}} - {m_{21}}{m_{42}} - {\mathrm{e}^{2H{\xi ^{(n)}}}}({m_{12}}{m_{41}} - {m_{11}}{m_{42}}), \\ {\chi _{\mathrm{{9}}}}= & {} {\mathrm{e}^{2H({\eta ^{(n)}} + {\xi ^{(n)}})}}\left( {\left( {{m_{13}} - {m_{14}}} \right) \left( {{m_{31}} - {m_{32}}} \right) - \left( {{m_{11}} - {m_{12}}} \right) \left( {{m_{33}} - {m_{34}}} \right) } \right) \\&- {\mathrm{e}^{2H{\eta ^{(n)}}}}\left( {\left( {{m_{23}} - {m_{24}}} \right) \left( {{m_{31}} - {m_{32}}} \right) - \left( {{m_{21}} - {m_{22}}} \right) \left( {{m_{33}} - {m_{34}}} \right) } \right) \\&+ \left( {{m_{23}} - {m_{24}}} \right) \left( {{m_{41}} - {m_{42}}} \right) - \left( {{m_{21}} - {m_{22}}} \right) \left( {{m_{43}} - {m_{44}}} \right) \\&- {\mathrm{e}^{2H{\xi ^{(n)}}}}\left( {\left( {{m_{13}} - {m_{14}}} \right) \left( {{m_{41}} - {m_{42}}} \right) - \left( {{m_{11}} - {m_{12}}} \right) \left( {{m_{43}} - {m_{44}}} \right) } \right) . \\ \end{aligned}$$