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Highlights • Construct special techniques to preserve two bounds without using the maximum-principle-preserving technique. • Treat the time derivative of the pressure as a source of the concentration equation. • Apply the algorithm on unstructured meshes. • In the flux limiter, use the second-order flux as the lower-order one. • Use L2-projection of the porosity and construct special limiters that suitable for multi-component fluid mixtures. Abstract In this paper, we develop high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the j th component, c j , should be between 0 and 1. There are three main difficulties. Firstly, c j does not satisfy a maximum-principle. Therefore, the numerical techniques introduced in Zhang and Shu (2010) [44] cannot be applied directly. The main idea is to apply the positivity-preserving techniques to all c j ′ s and enforce ∑ j c j = 1 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure d p / d t as a source in the concentration equation and choose suitable fluxes in the pressure and concentration equations. Secondly, it is not easy to construct first-order numerical fluxes for interior penalty DG methods on triangular meshes. One of the key points in the high-order BP technique applied in this paper is the combination of high-order and lower-order numerical fluxes. We will construct second-order BP schemes and use the second-order numerical fluxes as the lower-order one. Finally, the classical slope limiter cannot be applied to c j. To construct the BP technique, we will not approximate c j directly. Therefore, a new limiter will be introduced. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel discrete fracture model for flow simulation of fractured porous media containing flow blocking barriers on non-conforming meshes. The methodology of the approach is to modify the traditional Darcy's law into the hybrid-dimensional Darcy's law where fractures and barriers are represented as Dirac- δ functions contained in the permeability tensor and resistance tensor, respectively. As a natural extension of the previous discrete fracture model [21] for highly conductive fractures, this model is able to account for the influence of both highly conductive fractures and blocking barriers accurately on non-conforming meshes. The local discontinuous Galerkin (LDG) method is employed to accommodate the form of the hybrid-dimensional Darcy's law and the nature of the pressure/flux discontinuity. The performance of the model is demonstrated by several numerical tests. • Introduce line Dirac delta functions into the discrete fracture model for fracture and barrier networks. • Construct discontinuous Galerkin methods for the discrete fracture model on noncon- forming meshes. • Apply the model to steady state single phase flow and contaminant transportation in fractured porous media. [ABSTRACT FROM AUTHOR]

We propose a novel hybrid-dimensional model for the Darcy–Forchheimer flow in fractured rigid porous media, with a natural applicability to non-conforming meshes. Motivated by the previous work on the reinterpreted discrete fracture model (RDFM) for Darcy flows, we extend its key idea to non-Darcy flows. Coupling the Darcy's law in the matrix and the Forchheimer's law in the fractures through the introduction of the Dirac- δ functions to characterize the fractures, we derive a relationship between the total flow velocity in the porous media and the fluid velocity in the fractures. With this relation, it is natural to model the Darcy–Forchheimer flow in the whole computational domain by one equation. The local discontinuous Galerkin (LDG) method is applied for numerical discretization of the steady-state single-phase flow problem and a time-marching method is adopted to find the solution of the resulting nonlinear system. Besides, we construct a direct solver for the nonlinear equation of the fluid velocity in fracture to save computational cost. As an application, we discuss a simple transport model coupled with the flow equation. Several numerical experiments validate the performance of the model with its effectiveness on non-conforming meshes. We observe that the Darcy–Forchheimer model effectively reduce the flow rates and make the predictions more realistic in case of excessive fracture velocities. • Construct a hybrid-dimensional Darcy–Forchheimer model in fractured porous media. • The model is applicable on non-conforming meshes. • Two different derivations were provided. • Add pseudo time variable in the numerical scheme. [ABSTRACT FROM AUTHOR]

• Combine the reinterpreted discrete fracture model with contaminant transportation. • The degree of freedom keeps the same as the number of fractures increases. • Construct enriched Galerkin methods to preserve the local mass conservation for the pressure equation. • Design the bound-preserving technique to enforce the physical bounds of the concentration. The discrete fracture model (DFM) has been widely used to simulate fluid flow in fractured porous media. Traditional DFM is considered to be limited on conforming meshes, hence significant difficulty may arise in generating high-quality unstructured meshes due to the complexity of the fracture networks. Recently, Xu and Yang reinterpreted DFM and demonstrated that it can actually be extended to non-conforming meshes without any essential changes. However, the continuous Galerkin (CG) method was applied and the local mass conservation was missing. This paper is a follow-up work, and we apply the interior penalty discontinuous Galerkin (IPDG) method and enriched Galerkin (EG) method for the pressure equation. With the numerical fluxes, the local mass is conservative. As an application, we combine the reinterpreted DFM (RDFM) with the incompressible miscible displacements in porous media. The bound-preserving techniques are applied to the coupled system. We can theoretically guarantee that the concentration is between 0 and 1. Finally, several numerical experiments are given to demonstrate the good performance of the RDFM based on the above two methods on non-conforming meshes and the effectiveness of the bound-preserving technique. [ABSTRACT FROM AUTHOR]