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<p>Discrete fracture models with reduced-dimensional treatment of conductive and blocking fractures are widely used to simulate fluid flow in fractured porous media. Among these, numerical methods based on interface models are intensively studied, where the fractures are treated as co-dimension one manifolds with governing equations and interface conditions in a bulk matrix. In this paper, we propose a simple yet effective treatment for modeling the fractures on fitted grids in the interior penalty discontinuous Galerkin (IPDG) methods without introducing any additional degrees of freedom or equations on the interfaces. We conduct stability and <i>hp</i>-analysis for the proposed IPDG method, deriving optimal a priori error bounds concerning mesh size (<i>h</i>) and sub-optimal bounds for polynomial degree (<i>k</i>) in both the energy norm and the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norm. Numerical experiments involving published benchmarks validate our theoretical analysis and demonstrate the method’s robust performance. Furthermore, we extend our method to two-phase flows and use numerical tests to confirm the algorithm’s validity.</p>

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An Interior Penalty Discontinuous Galerkin Method for an Interface Model of Flow in Fractured Porous Media

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月27日

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  • Yong Liu,
  • Ziyao Xu

摘要

Discrete fracture models with reduced-dimensional treatment of conductive and blocking fractures are widely used to simulate fluid flow in fractured porous media. Among these, numerical methods based on interface models are intensively studied, where the fractures are treated as co-dimension one manifolds with governing equations and interface conditions in a bulk matrix. In this paper, we propose a simple yet effective treatment for modeling the fractures on fitted grids in the interior penalty discontinuous Galerkin (IPDG) methods without introducing any additional degrees of freedom or equations on the interfaces. We conduct stability and hp-analysis for the proposed IPDG method, deriving optimal a priori error bounds concerning mesh size (h) and sub-optimal bounds for polynomial degree (k) in both the energy norm and the \(L^2\) L 2 norm. Numerical experiments involving published benchmarks validate our theoretical analysis and demonstrate the method’s robust performance. Furthermore, we extend our method to two-phase flows and use numerical tests to confirm the algorithm’s validity.