A semi-implicit method for incompressible three-phase flow in porous media

Dong, Justin; Rivière, Béatrice

doi:10.1007/s10596-016-9583-2

Cited by 7 publications

(4 citation statements)

References 17 publications

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“…problems with very high contrasts in heterogeneity, the discretization of model (1) alone may be very hard to solve nu-281 merically due to a large condition number of the arising stiffness matrix. Moreover, the situation in even more intricate 282 for modeling non trivial two- [19,21] and three-phase [2, 1] transport convection dominated phenomena problems 283 for flow through porous media (see also other relevant works [23,4,17,5]). In addition, existence, uniqueness and 284 regularity issues for such problems at a fine level out of reach.…”

Section: Smooth Forcing 272mentioning

confidence: 99%

“…In multi-phase immiscible incompressible flow, p and Λ are the unknown pressure and the given phase mobitity of one of the phases in consideration (water, oil or gas); ( see e.g., [2,3,4,5,6,7]). In general, the forcing term q is due to gravity, phase transitions, sources and sinks, or when we transform a nonhomogeneous boundary condition problem to a homogeneous one.…”

Section: Problemmentioning

confidence: 99%

“…For Darcy-like model problems with very high contrasts in heterogeneity, the discretization of Darcy-like models alone may be very hard to solve numerically due to a large condition number of the arising stiffness matrix. Moreover, the situation in even more intricate for modeling non trivial two- [17,18] and three-phase [7,6] transport convection dominated phenomena problems for flow through porous media (see also other relevant works [19,2,5,20]). Thus, to achieve a sufficiently coupling between the volume fractions (or saturation) and the pressurevelocity, the full problem can be treated along with a fractional-step numerical procedure [7,6]; we point out that we are aware about the very delicate issues linked to the discontinuous capillary-pressure (see [3] and the references therein).…”

Section: Problemmentioning

confidence: 99%

“…The mobility phase in consideration Λ(x) = K(x)k r (S (x))/µ, 6 where K(x) is absolute (intrinsic) permeability, k r is the relative phase permeability and µ the phase viscosity of the 7 fluid.Here Ω is a convex polygonal and two-dimesional domain with boundary ∂Ω. 8 Efficiently and accurately solving the equations like (1) governing fluid flow in oil reservoirs as well as in ground-9 water modeling and simulation of flow linked to advective/convective transport phenomena (e.g., [21,5]) is very 10 challenging because of the complex porous media environment and the intricate properties of fluid phases. A key 11 ingredient on the transport phenomena in porous media and related real-life applications is precisely the well-known 12 Darcy law, in which linked to equations in (1), is a fundamental PDE with a wide spectrum of relevance, of fundamen-13 tal applied mathematics [10,22], fundamental of modeling fluid flow flow through porous media [21,5] as well as 14 of a benchmark prototype model for proof-of-concept, efficient implementation and rigorous analysis for the design 15 and development of new finite element approaches, as the one discussed here, but also for other novel procedures, for 16 1 / Computers & Mathematics with Applications 00 (2016) 1-19 2 instance MsFEM [20], virtual finite elements [6], classical mixed finite elements [8].…”

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confidence: 99%

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On high-order conservative finite element methods

Abreu

Díaz

Galvis

et al. 2018

Computers & Mathematics with Applications

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A new high-order conservative finite element method for Darcy flow is presented. The key ingredient in the formulation is a volumetric, residual-based, based on Lagrange multipliers in order to impose conservation of mass that does not involve any mesh dependent parameters. We obtain a method with high-order convergence properties with locally conservative fluxes. Furthermore, our approach can be straightforwardly extended to three dimensions. It is also applicable to highly heterogeneous problems where high-order approximation is preferred. the phase in consideration (water, oil or gas); ( see e.g., [4,3,18,17,1,2]). The objective of find an approximation for 4 p satisfying the above equation and without loss of generality we assume Dirichlet boundary conditions. In general, 5 the forcing term q is due to gravity, sources or sinks. The mobility phase in consideration Λ(x) = K(x)k r (S (x))/µ, 6 where K(x) is absolute (intrinsic) permeability, k r is the relative phase permeability and µ the phase viscosity of the 7 fluid.Here Ω is a convex polygonal and two-dimesional domain with boundary ∂Ω. 8 Efficiently and accurately solving the equations like (1) governing fluid flow in oil reservoirs as well as in ground-9 water modeling and simulation of flow linked to advective/convective transport phenomena (e.g., [21,5]) is very 10 challenging because of the complex porous media environment and the intricate properties of fluid phases. A key 11 ingredient on the transport phenomena in porous media and related real-life applications is precisely the well-known 12 Darcy law, in which linked to equations in (1), is a fundamental PDE with a wide spectrum of relevance, of fundamen-13 tal applied mathematics [10,22], fundamental of modeling fluid flow flow through porous media [21,5] as well as 14 of a benchmark prototype model for proof-of-concept, efficient implementation and rigorous analysis for the design 15 and development of new finite element approaches, as the one discussed here, but also for other novel procedures, for 16 1 / Computers & Mathematics with Applications 00 (2016) 1-19 2 instance MsFEM [20], virtual finite elements [6], classical mixed finite elements [8]. Indeed, Discontinuous Galerkin 17 (DG) formulations have become an increasingly popular way to discretize the Darcy flow equations, either in the the 18 mixed finite element DG [11] or in the stabilized mixed DG [25] framework, just no name a few of the relevance 19 of model problem (1) from different perspectives. The field of fluid flow simulation in petroleum reservoirs [21] as 20well as the groundwater modeling and simulation of flow [5] linked to several transport phenomena have seen signif-21 icant advances in the last few decades (see, e.g., [28,16,14,13,12, 27]) due to novel discretizations associated do 22 Darcy problem (1), along with the challenges in modeling: flow and transport. We emphasize the challenges in the 23 construction of new methodologies into a reservoir simulation should have into account the following issues: 24• local mass...

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