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A Robust and Efficient Continuous‐Differentiable Seepage Face Boundary Condition for Dynamic Groundwater Modeling.

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Title: A Robust and Efficient Continuous‐Differentiable Seepage Face Boundary Condition for Dynamic Groundwater Modeling.
Authors: Park, Young‐Jin1 (AUTHOR), Hwang, Hyoun‐Tae1,2 (AUTHOR) hthwang@aquanty.com, Tanaka, Tatsuya3 (AUTHOR), Ozutsumi, Takenori3 (AUTHOR), Morita, Yutaka4 (AUTHOR), Mori, Koji5 (AUTHOR), Berg, Steven J.1,2 (AUTHOR), Illman, Walter A.1 (AUTHOR)
Source: Water Resources Research. Feb2026, Vol. 62 Issue 2, p1-18. 18p.
Subjects: Groundwater flow, Differentiable functions, Groundwater analysis, Hydrogeology, Boundary value problems, Neumann boundary conditions
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 are often unknown a priori and difficult to predict. The standard mathematical representation of seepage boundaries defines head as equivalent to elevation only when groundwater pressure exceeds atmospheric pressure, which is a mixed conditional Dirichlet and Neumann boundary condition. While this representation has been widely implemented in groundwater models, it is rarely noted that convergence is guaranteed only when both the efflux and zero‐pressure conditions are simultaneously satisfied, often requiring unnecessarily small timestep sizes, resulting in low computational efficiency. This study presents a continuous‐differentiable seepage face (CDSF) equation that replaces the conventional mixed boundary condition (or traditional seepage face, TSF) with a head‐dependent Robin boundary condition, improving numerical stability and computational performance. It is a refined adaptation of an existing seepage boundary condition approach previously used in integrated surface‐subsurface hydrologic models, specifically optimized for saturated flow simulations. Through a series of verification models, we demonstrate that the refined method provides robust and efficient solutions for seepage boundary conditions in saturated flow models. The results suggest that this CDSF approach improves accuracy and computational performance compared to TSF methods, offering a more stable alternative for groundwater modeling. These findings contribute to the advancement of subsurface hydrology by providing a practical framework for handling seepage boundary conditions in groundwater simulations. Plain Language Summary: In groundwater flow simulations, a seepage boundary condition is used to represent areas where water may seep out toward a boundary of atmospheric pressure; however, convergence may not be guaranteed unless certain conditions are satisfied. This study proposes a refined and optimized formulation using a continuous‐differentiable equation derived by analogy to the first‐order exchange equation, commonly used in integrated hydrologic simulations. The approach is demonstrated to be both robust and efficient through a series of verification models, suggesting it can improve the accuracy and efficiency of groundwater flow simulations. The proposed approach can also advance the understanding of subsurface hydrology and related processes. Key Points: A seepage boundary condition using a continuous‐differentiable equation is suggestedThis boundary condition is numerically robust and efficient even for complicated problemsThis approach can significantly improve the accuracy and efficiency of groundwater flow simulations [ABSTRACT FROM AUTHOR]
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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 are often unknown a priori and difficult to predict. The standard mathematical representation of seepage boundaries defines head as equivalent to elevation only when groundwater pressure exceeds atmospheric pressure, which is a mixed conditional Dirichlet and Neumann boundary condition. While this representation has been widely implemented in groundwater models, it is rarely noted that convergence is guaranteed only when both the efflux and zero‐pressure conditions are simultaneously satisfied, often requiring unnecessarily small timestep sizes, resulting in low computational efficiency. This study presents a continuous‐differentiable seepage face (CDSF) equation that replaces the conventional mixed boundary condition (or traditional seepage face, TSF) with a head‐dependent Robin boundary condition, improving numerical stability and computational performance. It is a refined adaptation of an existing seepage boundary condition approach previously used in integrated surface‐subsurface hydrologic models, specifically optimized for saturated flow simulations. Through a series of verification models, we demonstrate that the refined method provides robust and efficient solutions for seepage boundary conditions in saturated flow models. The results suggest that this CDSF approach improves accuracy and computational performance compared to TSF methods, offering a more stable alternative for groundwater modeling. These findings contribute to the advancement of subsurface hydrology by providing a practical framework for handling seepage boundary conditions in groundwater simulations. Plain Language Summary: In groundwater flow simulations, a seepage boundary condition is used to represent areas where water may seep out toward a boundary of atmospheric pressure; however, convergence may not be guaranteed unless certain conditions are satisfied. This study proposes a refined and optimized formulation using a continuous‐differentiable equation derived by analogy to the first‐order exchange equation, commonly used in integrated hydrologic simulations. The approach is demonstrated to be both robust and efficient through a series of verification models, suggesting it can improve the accuracy and efficiency of groundwater flow simulations. The proposed approach can also advance the understanding of subsurface hydrology and related processes. Key Points: A seepage boundary condition using a continuous‐differentiable equation is suggestedThis boundary condition is numerically robust and efficient even for complicated problemsThis approach can significantly improve the accuracy and efficiency of groundwater flow simulations [ABSTRACT FROM AUTHOR]
ISSN:00431397
DOI:10.1029/2025WR041547