A High-Order Implicit Bound-Preserving Discontinuous Galerkin Method for Incompressible Miscible Displacement in Porous Media_Vol. 4 No. 3 (JES 2026)_Journal of Engineering System (ISSN: 2959-0604)

Home > Journal of Engineering System (ISSN: 2959-0604) > Vol. 4 No. 3 (JES 2026) >

A High-Order Implicit Bound-Preserving Discontinuous Galerkin Method for Incompressible Miscible Displacement in Porous Media

Download PDF

DOI: https://doi.org/10.62517/jes.202602319

Author(s)

Xiaojuan Gong*

Affiliation(s)

School of General Education, Yantai Institute of Science and Technology, Yantai, China *Corresponding Author

Abstract

A high-order implicit bound-preserving discontinuous Galerkin method is developed for incompressible miscible displacement in porous media. The pressure–velocity system and the concentration equation are discretized in discontinuous finite element spaces, and backward differentiation formulas are used for time integration. Following a predictor–corrector strategy, a standard implicit DG step first produces a provisional concentration. A space-dependent Lagrange multiplier then enforces the Karush–Kuhn–Tucker conditions associated with the interval constraint, while an additional scalar multiplier preserves the porosity-weighted mass supplied by the DG prediction. The corrector reduces to pointwise formulas and one scalar monotone equation. The nodal bound, discrete mass balance, and compatibility of the two-component concentrations are established. Numerical experiments verify third-order accuracy, elimination of nonphysical undershoots and overshoots, and mass conservation in a source-free transport problem. Two field-oriented tests further demonstrate channeling in a strongly heterogeneous five-spot pattern and contaminant removal from an aquifer containing conductive pathways and a low-permeability barrier.

Keywords

Incompressible Miscible Displacement; Discontinuous Galerkin Method; Bound Preservation; Mass Conservation; Lagrange Multiplier; Bdf Method.

References

[1] M.F. Wheeler, B.L. Darlow. Interior penalty Galerkin procedures for miscible displacement problems in porous media. In: Computational Methods in Nonlinear Mechanics, North-Holland, Amsterdam, 1980, pp. 458–506. [2] B. Rivière. Discontinuous Galerkin Finite Element Methods for Solving the Miscible Displacement Problem in Porous Media. PhD thesis, The University of Texas at Austin, Austin, TX, 2000. [3] S. Sun, B. Rivière, M.F. Wheeler. A combined mixed finite element and discontinuous Galerkin method for miscible displacement problem in porous media In: Tony Chan, et al. (Eds.), Recent Progress in Computational and Applied PDEs, Kluwer Academic Publishers, Plenum Press, Dordrecht, New York, 2002, pp. 323–351. [4] S. Sun, M.F. Wheeler. Discontinuous Galerkin methods for coupled flow and reactive transport problems. Applied Numerical Mathematics, 2005, 52: 273–298. [5] H. Guo, Q. Zhang, Y. Yang. A combined mixed finite element method and local discontinuous Galerkin method for miscible displacement problem in porous media. Science China Mathematics, 2014, 57: 2301–2320. [6] S. Bartels, M. Jensen, R. Müller. Discontinuous Galerkin finite element convergence for incompressible miscible displacement problems of low regularity. SIAM Journal on Numerical Analysis, 2009, 47: 3720–3743. [7] B. Cockburn, C.-W. Shu. The local Discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numerical Analysis, 1998, 35(6): 2440–2463. [8] B. Rivière. Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. SIAM, 2008. [9] X. Guo, H. Guo, L. Tian, Y. Yang. High-order bound-preserving local discontinuous Galerkin methods for incompressible and immiscible two-phase flows in porous media. Journal of Scientific Computing, 2024, 99: 71. [10] K. L. A. Kirk, B. Riviere. A combined mixed hybrid and hybridizable discontinuous Galerkin method for Darcy flow and transport. Journal of Scientific Computing, 2024, 100(2): 57. [11] Y. Kang, T. Xiong, Y. Yang. Oscillation-free implicit pressure explicit concentration discontinuous Galerkin methods for compressible miscible displacements with applications in viscous fingering. Journal of Computational Physics, 2024, 516: 113348. [12] M. S. Fabien. A cell centered Galerkin method for miscible displacement in heterogeneous porous media. Computer Methods in Applied Mechanics and Engineering, 2026, 450: 118608. [13] X. Zhang, C.-W. Shu. On maximum-principle-satisfying high order schemes for scalar conservation laws. Journal of Computational Physics, 2010, 229: 3091–3120. [14] X. Zhang, C.-W. Shu. On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. Journal of Computational Physics, 2010, 229: 8918–8934. [15] X. Zhang, C.-W. Shu. Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms. Journal of Computational Physics, 2011, 230: 1238-1248. [16] Y. Zhang, X. Zhang, C.-W. Shu. Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection-diffusion equations on triangular meshes. Journal of Computational Physics, 2013, 234: 295–316. [17] H. Guo, Y. Yang. Bound-preserving discontinuous Galerkin method for compressible miscible displacement in porous media. SIAM Journal on Scientific Computing, 2017, 39: A1969–A1990. [18] Q. Cheng, J. Shen. A new Lagrange multiplier approach for constructing structure preserving schemes, I. Positivity preserving. Computer Methods in Applied Mechanics and Engineering, 2022, 391: 114585. [19] Q. Cheng, J. Shen. A new Lagrange multiplier approach for constructing structure preserving schemes, II. Bound preserving. SIAM Journal on Numerical Analysis, 2022, 60(3): 970-998.