Novel Framework for Solving the Two-Dimensional Consolidation Problem Using the Random Feynman–Kac Formulation

This paper presents a novel framework for solving the plane strain two-dimensional (2D) consolidation problem of saturated soils using the random Feynman–Kac (RF–K) formulation. The proposed framework addresses the inherent spatial variability of soils by modeling the coefficients of horizontal (ch) and vertical (cv) consolidation as 2D random fields, generated using the Karhunen–Loève (K–L) expansion. Unlike conventional grid-based numerical methods, this mesh-free approach formulates stochastic differential equations (SDEs) as generators of stochastic processes corresponding to the governing partial differential equation (PDE) of the 2D consolidation problem. These SDEs are simulated using Monte Carlo simulations (MCS) to trace the trajectories of excess pore water pressure (EPWP) dissipation until the exit time (Te). The EPWP solutions are calculated by taking the expectation over the ensemble of simulated trajectories under various drainage boundary conditions with uniform initial conditions. In addition to the RF–K framework, a random field finite difference method (RF-FDM) incorporating random fields of ch and cv is developed for comparison to demonstrate the stability, accuracy, and computational efficiency of the proposed RF–K framework. To further validate the accuracy of the RF–K framework, it is applied to the simple case of the one-dimensional (1D) consolidation problem, and the resulting solutions are benchmarked against both the FDM and existing analytical solutions. By simulating trajectories of the Brownian motion particles in two directions, the proposed RF–K framework captures more closely the underlying physics of EPWP dissipation and settlement of the 2D consolidation problem of saturated soils. Considering the spatial variability of soils in both 1D and 2D consolidation scenarios, the proposed mesh-free and computationally efficient RF–K framework offers a significant advancement in the probabilistic analysis of the consolidation of saturated soils under different drainage boundary scenarios.