Staff Research Highlight - A Continuous Differentiable Formulation for Seepage Face Boundary Conditions in Dynamic Groundwater Systems
Park, Y.-J., Hwang, H.-T., Tanaka, T., Ozutsumi, T., Morita, Y., Mori, K., Berg, S. J., & Illman, W. A. (2024). A Continuous Differentiable Formulation for Seepage Face Boundary Conditions in Dynamic Groundwater Systems. Wiley. https://doi.org/10.22541/essoar.172108341.19668507/v2
“The steady-state hillslope simulation was conducted using the integrated surface-subsurface feature in HydroGeoSphere. The simulation results show that precipitation infiltrates into subsurface at the high elevation area, the infiltrated water percolates down through vadose zone, and groundwater flows downhill toward the low elevation discharge area.”
CLICK HERE TO READ THE ARTICLE.
We’re pleased to highlight this publication, co-authored by Aquanty’s Hyoun-Tae Hwang and Steven Berg, which introduces a continuously differentiable formulation for seepage face boundary conditions in dynamic groundwater systems. Traditional approaches often model seepage faces with abrupt boundary transitions, leading to numerical instabilities, convergence issues, and computational inefficiencies in transient groundwater simulations. This research presents a novel approach that ensures smooth transitions between saturated and unsaturated zones, improving the stability and accuracy of numerical groundwater models.
Seepage faces occur when groundwater emerges at the surface due to changes in hydraulic pressure, influencing water table fluctuations, stream-aquifer interactions and groundwater discharge into surface water bodies. Standard numerical models often represent seepage faces using step-function-based boundary conditions, which introduce discontinuities that can cause artificial oscillations in pressure head values and difficulties in model convergence. These limitations are particularly problematic in transient groundwater flow simulations where rapid changes in hydraulic conditions require high numerical stability. This study addresses these challenges by developing a continuously differentiable seepage face formulation that provides a gradual transition between saturated and unsaturated conditions, reducing computational errors and improving the physical realism of simulated flow dynamics.
Figure 1. Schematic of (a) a hydrologic system and the development of a seepage face in a 2-D cross sectional hillslope and (b) surface water depth (red line) and exchange flux distributions (blue line).
The research integrates this new seepage face boundary formulation into HydroGeoSphere (HGS), a physics-based model that simulates fully coupled surface and subsurface hydrological processes. HGS is well suited for variably saturated flow problems, making it an ideal platform for testing and validating the proposed improvements. The study applies the new formulation to multiple test cases, including benchmark problems, laboratory-scale seepage experiments, and field-scale simulations with complex topography and fluctuating boundary conditions. Model performance is evaluated by comparing numerical results with analytical solutions and experimental observations to assess improvements in stability, convergence, and computational efficiency.
The results demonstrate that the continuously differentiable formulation eliminates artificial pressure oscillations commonly observed in traditional step-function seepage face models. The approach enhances numerical stability, allowing simulations to run with fewer iterations while maintaining accuracy in predicting seepage dynamics. This refinement also improves the representation of groundwater-surface water interactions, enabling more reliable predictions of seepage face development under dynamic hydrological conditions. By reducing the numerical challenges associated with transient seepage modelling, this work advances groundwater modelling capabilities and offers significant benefits for applications such as contaminant transport, aquifer recharge assessment, and groundwater management in engineered and natural systems. The findings highlight the importance of incorporating smooth boundary condition transitions in numerical models to achieve more realistic and computationally efficient hydrological simulations.
“Using the seepage face boundary condition suggested in this study, we implemented this feature within a watershed-scale model designed to capture intricate hydrologic and hydrogeologic dynamics.”
Abstract:
Seepage boundary conditions are commonly used in groundwater simulations to allow groundwater to discharge at the upper surface of the model when groundwater head exceeds atmospheric pressure. However, the extent and transient behavior of the seepage zone is often unknown a priori and is difficult to predict. A mathematical description of the boundary condition is straightforward, such that head is equivalent to elevation only when groundwater flow indicates a seepage condition, which is a mixed conditional Dirichlet and Neumann boundary condition. This standard representation of the boundary condition has been successfully implemented and applied in a real-world context by most groundwater models. However, it is rarely reported that convergence is only guaranteed when both the efflux and zero pressure conditions are simultaneously satisfied, often requiring unnecessarily small timestep sizes, which results in low computational efficiency. This study suggests a continuous differentiable equation as an alternative to model the seepage boundary. The new formulation is derived by analogy to the first-order exchange equation, which is commonly used to represent the interactions between surface water and groundwater flow in integrated hydrologic simulations. The results of this study suggest that mixed Dirichlet and Neumann boundary conditions can be effectively converted into a Robin boundary condition, which is a head-dependent flux condition that incorporates appropriate physical considerations. This new approach has the potential to significantly improve the accuracy and efficiency of groundwater flow simulations and can help to advance the understanding of subsurface hydrology.