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This study presents a probabilistic methodology for solving the one-dimensional (1D) consolidation equation using the Feynman-Kac formula incorporating the randomness of coefficient of consolidation ( \(c_v\) ). The Feynman-Kac formula establishes a connection between the expected value of a stochastic differential equation and the numerical solutions of the associated partial differential equation. A set of \(c_v\) values following a normal distribution is selected and employing the Feynman-Kac framework, Monte Carlo simulations are executed till the exit time of the process. As a result, probabilistic solutions depicting excess pore water pressure (EPWP) profiles are generated which are then compared with the established analytical solutions of 1D consolidation under single drainage boundary conditions. The simulated pore-pressure profiles highlight that in contrast to the conventional analysis, the maximum EPWP need not occur at the farthest depth from the drainage boundary; rather, the location of maximum EPWP would be guided by the random variation of \(c_v\) .

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Deciphering Uncertainty in Excess Pore Water Pressure Profile in Single Drainage 1D Consolidation Through Feynman-Kac Formulation

  • 会议论文

  • 发表日期：

    2025年

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  • Naina Deb,
  • Budhaditya Hazra,
  • Arindam Dey

摘要

This study presents a probabilistic methodology for solving the one-dimensional (1D) consolidation equation using the Feynman-Kac formula incorporating the randomness of coefficient of consolidation ( \(c_v\) ). The Feynman-Kac formula establishes a connection between the expected value of a stochastic differential equation and the numerical solutions of the associated partial differential equation. A set of \(c_v\) values following a normal distribution is selected and employing the Feynman-Kac framework, Monte Carlo simulations are executed till the exit time of the process. As a result, probabilistic solutions depicting excess pore water pressure (EPWP) profiles are generated which are then compared with the established analytical solutions of 1D consolidation under single drainage boundary conditions. The simulated pore-pressure profiles highlight that in contrast to the conventional analysis, the maximum EPWP need not occur at the farthest depth from the drainage boundary; rather, the location of maximum EPWP would be guided by the random variation of \(c_v\) .