Lower bound limit analysis of unsaturated soil slope stability under fluctuating water levels using finite elements and convex optimization

Advances in Water Resources, 212:105297, 2026.

J. Zhou and Q. Zhang contributed equally to this work.

• Manuscript

Extensive evidence indicates that slope failure is frequently associated with intensive rainfall, but the coupled effects of matric suction and fluctuating water levels under such events have not been sufficiently investigated. This work develops a lower bound limit analysis framework, based on finite elements and convex optimization, to assess the stability of unsaturated slopes under fluctuating water levels. Within Bishop’s effective stress framework, a novel convex formulation of the load multiplier $\lambda_F$ considering the mixed-form Richards equation and finite element discretization is proposed. A rigorous lower bound solution for the Factor of Safety (FoS) of an unsaturated slope is derived by coupling the strength reduction technique with CVX, a MATLAB package for disciplined convex programming. Through model validation, the proposed convex formulation greatly outperforms the original Linear Program (LP) formulation, and the performance of CVX is highly competitive compared with other specific solvers in the literature. The model application section systematically summarizes the relative magnitudes and variation patterns of FoS under four typical slope seepage scenarios: slow reservoir drawdown, non-trivial steady-state reservoir drawdown considering external water supply, rapid transient reservoir drawdown, and post-drawdown rainfall. Results indicate that neglecting matric suction leads to an underestimation of slope stability, and when matric suction is incorporated, the $\chi = S_e$ model yields a more conservative estimate of the FoS compared to other $\chi$ models. During reservoir drawdown, the asynchronous response between the internal and external phreatic surfaces significantly influences the FoS. Consequently, two bounding scenarios, i.e., slow drawdown FoS and the initially sharply declining FoS during rapid drawdown, can be used to establish a reasonable prediction range for the FoS in practice. These results provide valuable guidance for the safety assessment of slopes subjected to fluctuating water levels, as well as for reservoir management.