Solution of the Enhanced Third Order Water Seepage Equation in Earth Dams Using Crank Nicholson Scheme

Earth dams serve critical infrastructure functions but pose significant risks when failures occur. Water seepage and internal erosion account for approximately 35% of earth dam failures worldwide, causing devastating losses. Traditional seepage analysis relies exclusively on diffusion-based models derived from Darcy’s law, which assumes water movement occurs solely through concentration gradient-driven transport. However, this approach systematically neglects advective transport mechanisms that become significant in heterogeneous dam materials with preferential flow pathways, layered soil systems, and fractured foundations. Previous research by Nyachwaya et al. developed third-order seepage equations incorporating temporal-spatial coupling but omitted advection terms, resulting in incomplete modeling. This research addresses this limitation by developing and solving an enhanced third-order advection-seepage equation incorporating advection, diffusion, and third-order temporal- spatial coupling. The Crank-Nicolson finite difference scheme was selected for numerical solution due to its second-order temporal accuracy, favorable stability characteristics, and computational efficiency. Comprehensive stability analysis using matrix eigenvalue methods established necessary convergence conditions relating dimensionless mesh ratios, while consistency verification through Taylor series expansion confirmed scheme convergence. Computational case studies with varying mesh parameters demonstrated numerical convergence and revealed exponential spatial decay from upstream boundaries, monotonic temporal growth, and attenuation length scales of 2-6 meters. Comparative analysis against diffusion-only models showed the enhanced formulation predicts seepage rates 30-150% higher, with enhancement effects increasing with distance from boundaries, having critical safety implications since traditional analyses may significantly underestimate seepage in zones where internal erosion initiates. This research represents the first comprehensive application of finite difference methods to the complete third-order advection-seepage equation with rigorous stability analysis, enabling more accurate prediction of dam performance and supporting improved drainage design, refined safety assessment, and better risk management for aging infrastructure.