A fully-coupled upwind discontinuous Galerkin method for incompressible porous media flows: High-order computations of viscous fingering instabilities in complex geometry

Scovazzi, Guglielmo; Huang, Hao; Collis, S. Scott; Yin, Jichao

doi:10.1016/j.jcp.2013.06.012

Cited by 21 publications

(19 citation statements)

References 48 publications

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“…This is because, as opposed to the case of a stable physical system, in an unstable flow there is a natural tendency for slight differences and discrepancies between solutions to grow uncontrollably over time. Grid orientation effects can become more severe in multi-dimensional problems with large mobility ratios and heterogeneous coefficients, as discussed in [156,157,158,159,160,144].…”

Section: Grid Dependence and Grid Orientation Effects In Numerical Comentioning

confidence: 99%

Analytical and variational numerical methods for unstable miscible displacement flows in porous media

Scovazzi

Wheeler

Mikelić

et al. 2017

Journal of Computational Physics

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International audienceThe miscible displacement of one fluid by another in a porous medium has received considerable attention in sub-surface, environmental and petroleum engineering applications. When a fluid of higher mobility displaces another of lower mobility, unstable patterns-referred to as viscous fingering-may arise. Their physical and mathematical study has been the object of numerous investigations over the past century. The objective of this paper is to present a review of these contributions with particular emphasis on variational methods. These algorithms are tailored to real field applications thanks to their advanced features: handling of general complex geometries, robustness in the presence of rough tensor coefficients, low sensitivity to mesh orientation in advection dominated scenarios, and provable convergence with fully unstructured grids. This paper is dedicated to the memory of Dr. Jim Douglas Jr., for his seminal contributions to miscible displacement and variational numerical methods

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Section: Grid Dependence and Grid Orientation Effects In Numerical Comentioning

confidence: 99%

Analytical and variational numerical methods for unstable miscible displacement flows in porous media

Scovazzi

Wheeler

Mikelić

et al. 2017

Journal of Computational Physics

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“…Many flavors of DG have been analyzed in terms of error estimates and convergence properties, and it is hard to do justice to the full scope of this work (the following papers provide an overview of pioneering and recents efforts in the analysis of DG methods: [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52]).…”

Section: Introductionmentioning

confidence: 99%

Mixed-hybrid and vertex-discontinuous-Galerkin finite element modeling of multiphase compositional flow on 3D unstructured grids

Moortgat

Firoozabadi

2016

Journal of Computational Physics

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Problems of interest in hydrogeology and hydrocarbon resources involve complex heterogeneous geological formations. Such domains are most accurately represented in reservoir simulations by unstructured computational grids. Finite element methods accurately describe flow on unstructured meshes with complex geometries, and their flexible formulation allows implementation on different grid types. In this work, we consider for the first time the challenging problem of fully compositional three-phase flow in 3D unstructured grids, discretized by any combination of tetrahedra, prisms, and hexahedra. We employ a mass conserving mixed hybrid finite element (MHFE) method to solve for the pressure and flux fields. The transport equations are approximated with a higher-order vertex-based discontinuous Galerkin (DG) discretization. We show that this approach outperforms a face-based implementation of the same polynomial order. These methods are well suited for heterogeneous and fractured reservoirs, because they provide globally continuous pressure and flux fields, while allowing for sharp discontinuities in compositions and saturations. The higherorder accuracy improves the modeling of strongly non-linear flow, such as gravitational and viscous fingering. We review the literature on unstructured reservoir simulation models, and present many examples that consider gravity depletion, water flooding, and gas injection in oil saturated reservoirs. We study convergence rates, mesh sensitivity, and demonstrate the wide applicability of our chosen finite element methods for challenging multiphase flow problems in geometrically complex subsurface media.

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“…The discrete form of the LBB condition provides the necessary and sufficient criterion for a stable finite element formulation. To satisfy this condition, several approaches including stabilized methods, 4,[31][32][33][34][35][36] discontinuous Galerkin, [37][38][39][40][41][42][43] special elements such as Taylor-Hood elements 44 or MINI elements, 45 and mixed finite elements 28,46,47 have been proposed in the literature. As discussed in Section 3.2, in a three-field finite element formulation for poroelasticity, the LBB condition must be satisfied in two limiting cases-undrained limit and rigid skeleton limit.…”

mentioning

confidence: 99%

Mixed finite element formulation for dynamics of porous media

Lotfian

Sivaselvan

2018

Numerical Meth Engineering

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Summary This paper presents a numerical approach for computing solutions to Biot's fully dynamic model of saturated porous media with incompressible solid and fluid phases. Spatial discretization is based on a three‐field (u‐w‐p) formulation, employing a lowest‐order Raviart‐Thomas mixed element for the fluid Darcy velocity (w) and pressure (p) fields, and a nodal finite element for skeleton displacement field (u). The discretization is constructed based on the natural topology of the variables and satisfies the Ladyženskaja‐Babuška‐Brezzi stability condition, avoiding locking in the incompressible and undrained limits. Since fluid acceleration is not neglected, the three‐field formulation fully captures dynamic behavior even under high‐frequency loading phenomena. The importance of consistent initial conditions is addressed for poroelasticity equations with the mass balance constraint, a system of differential algebraic equations. Energy balance is derived for the porous medium and used to assess accuracy of time integration. To demonstrate the performance of the proposed approach, a variety of numerical studies are carried out including verification with analytical and boundary element solutions and analyses of wave propagation, effect of hydraulic conductivity on damping and frequency content, energy balance, mass lumping, mesh pattern and size, and stability. Some discrepancies found in dynamic poroelasticity results in the literature are also explained.

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A fully-coupled upwind discontinuous Galerkin method for incompressible porous media flows: High-order computations of viscous fingering instabilities in complex geometry

Cited by 21 publications

References 48 publications

Analytical and variational numerical methods for unstable miscible displacement flows in porous media

Analytical and variational numerical methods for unstable miscible displacement flows in porous media

Mixed-hybrid and vertex-discontinuous-Galerkin finite element modeling of multiphase compositional flow on 3D unstructured grids

Mixed finite element formulation for dynamics of porous media

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