Decoupled, Linear, and Energy Stable Finite Element Method for the Cahn--Hilliard--Navier--Stokes--Darcy Phase Field Model

Gao, Yali; He, Xiaoming; Mei, Liquan; Yang, Xiaofeng

doi:10.1137/16m1100885

Cited by 80 publications

(18 citation statements)

References 103 publications

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“…In recent years, many scientists and engineers have investigated the fluid flow interaction between the conduit and porous media regime . The massive applications, such as karst aquifer subsurface flow system, interaction between the surface flows and subsurface flows, petroleum extraction, industrial filtration, biochemical transport, field flow fractionation for separation and characterization of proteins, blood flow in arteries and veins, etc, attract scientists and engineers to build related fluid dynamical models, including (Navier‐)Stokes‐Darcy model, Stokes‐Darcy‐transport model, Darcy‐Stokes‐Brinkman model, etc . It is not surprising that a great deal of effort has been devoted to develop appropriate numerical methods to solve the (Navier‐)Stokes‐Darcy fluid flow system, including coupled finite element methods, domain decomposition methods, Lagrange multiplier methods, mortar finite element methods, least‐square methods, partitioned time‐stepping methods, two‐grid and multigrid methods, discontinuous Galerkin finite element methods, boundary integral methods, and many others …”

Section: Introductionmentioning

confidence: 99%

Coupled and decoupled stabilized mixed finite element methods for nonstationary dual‐porosity‐Stokes fluid flow model

Mahbub

Nasu

et al. 2019

Numerical Meth Engineering

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In this paper, we propose and analyze two stabilized mixed finite element methods for the dual-porosity-Stokes model, which couples the free flow region and microfracture-matrix system through four interface conditions on an interface. The first stabilized mixed finite element method is a coupled method in the traditional format. Based on the idea of partitioned time stepping, the four interface conditions, and the mass exchange terms in the dual-porosity model, the second stabilized mixed finite element method is decoupled in two levels and allows a noniterative splitting of the coupled problem into three subproblems. Due to their superior conservation properties and convenience of the computation of flux, mixed finite element methods have been widely developed for different types of subsurface flow problems in porous media. For the mixed finite element methods developed in this article, no Lagrange multiplier is used, but an interface stabilization term with a penalty parameter is added in the temporal discretization. This stabilization term ensures the numerical stability of both the coupled and decoupled schemes. The stability and the convergence analysis are carried out for both the coupled and decoupled schemes. Three numerical experiments are provided to demonstrate the accuracy, efficiency, and applicability of the proposed methods. KEYWORDS decoupled numerical methods, dual-porosity-Stokes model, horizontal wellbore, mixed finite elements, stabilization Int J Numer Methods Eng. 2019;120:803-833. wileyonlinelibrary.com/journal/nme 804 AL MAHBUB ET AL.model, It is not surprising that a great deal of effort has been devoted to develop appropriate numerical methods to solve the (Navier-)Stokes-Darcy fluid flow system, including coupled finite element methods, 17-21 domain decomposition methods, 22-31 Lagrange multiplier methods, 3,32,33 mortar finite element methods, 34,35 least-square methods, 36-38 partitioned time-stepping methods, 39,40 two-grid and multigrid methods, 41-44 discontinuous Galerkin finite element methods, 45-50 boundary integral methods, 51,52 and many others. [53][54][55][56][57][58] Although the traditional Stokes-Darcy model has been well studied, it has limitation to describe the heterogeneity of the porous medium which contains multiple porosities. It is worth to notice that the realistic naturally fractured reservoir consists of two coexisting and interacting medium, namely, tight matrix and microfractures. Furthermore, it is more important to characterize the intrinsic properties and accurately modeled the flow interactions between each medium. 59,60 The first multiporosity model was proposed by Barenblatt et al 61 for the naturally fractured reservoir in 1960 where the microfracture and matrix systems are formulated by individual but overlapping continua. Warren and Root 62 developed a homogeneous orthotropic dual-porosity model in 1963 based on the model proposed by Barenblatt. There are many applications for the dual-porosity model such as the geothermal system, hydrogeology, pe...

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Section: Introductionmentioning

confidence: 99%

Coupled and decoupled stabilized mixed finite element methods for nonstationary dual‐porosity‐Stokes fluid flow model

Mahbub

Nasu

et al. 2019

Numerical Meth Engineering

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“…In order to deal with the nonlinear terms, we recall the following inequality, there exists constants C 1 > 0 and C 2 > 0 depending only on Ω s , such that

| v | \leq C_{1} ‖ \nabla v ‖_{0}, ‖ v ‖_{L^{4}} \leq C_{2} | v |,

where |⋅| denote the semi‐norm of space H 1 (Ω s ) and v ∈ Y s . Based on the work in [39, 40], we have the following lemma.Lemma Assume that both u and u h satisfy the following smallness condition

‖ \nabla u ‖_{L^{2}} < \frac{ν}{8 C_{1}^{3} C_{2}^{2}}, \forall t \in false[0, T false] .

Then , we have the estimate

| false(false(u \cdot \nabla false) v, w false) | \leq \frac{ν}{8} ‖ \nabla v ‖_{0} ‖ \nabla w ‖_{0}, \forall v, w \in Y_{s} .

Proof

\begin{matrix} | ((u \cdot normal∇) v, w) | & \leq ‖ u ‖_{L^{4}} | v | ‖ w ‖_{L^{4}} \leq C_{2}^{2} | u false‖ v false‖ w | \leq C_{1}^{3} C_{2}^{2} ‖ normal∇ u ‖_{0} ‖ normal∇ v ‖_{0} ‖ normal∇ w ‖_{0} \\ \leq \frac{ν}{8} ‖ normal∇ v ‖_{0} ‖ normal∇ w ‖_{0} .<...> \end{matrix}

…”

Section: Stability Of Decoupled Schemementioning

confidence: 99%

A decoupled stabilized finite element method for the dual‐porosity‐Navier–Stokes fluid flow model arising in shale oil

Gao

2020

Numerical Methods Partial

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In this paper, we consider the decoupled stabilized finite element method for the dual-porosity-Navier-Stokes model coupling the free flow region and the microfracture-matrix system by using four interface conditions on the interface. The stabilized finite element method is decoupled in two levels, and it allows the coupling problem to be divide into three subproblems in a non-iterative manner, which improves the computational efficiency. In addition, the stability and convergence of the decoupling scheme are also analyzed. Finally, the theoretical results are illustrated by some numerical experiments. KEYWORDS convergence, decoupled method, dual-porosity-Navier-Stokes, finite element method, numerical experiments, stability 1 INTRODUCTION Many scientists and engineers have investigated the fluid flow interaction between conduit and porous media region [1-3]. A large number of practical problems, such as groundwater flow system, interaction between surface, and groundwater flow, biochemical transportation, blood flow in artery, vein, and so forth [4-7], have been established relevant hydrodynamic models by scientists. In addition, a lot of efforts have been devoted to develop appropriate numerical methods to solve the Stokes-Darcy fluid system, including the coupled finite element methods [8-10], domain decomposition methods [11-13], the Lagrange multiplier method [14, 15], and so forth. Although the traditional Stokes-Darcy model [16-18] has been well studied, it has some limitations in describing the heterogeneity of porous media. It is worth noting that the real natural fractured

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“…Moreover, the presence of the energy law serves as a guide line for the design of energy stable numerical schemes. Various numerical methods have been developed and analyzed for different phase field models, such as the finite element method [3,30,35,40,45,52,54,100], finite difference method [14,16,99], spectral method [46,67,87,91,102], extended finite element method [18,32], discontinuous Galerkin finite element method [31,66,84], finite volume method [7,106], penalty-projection method [83], lattice Boltzmann method [26,107], and many others [13,50,53,63,71,77,79,80,98,105].…”

Section: Introductionmentioning

confidence: 99%

Discontinuous finite volume element method for a coupled Navier-Stokes-Cahn-Hilliard phase field model

Gao

Chen

et al. 2020

Adv Comput Math

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In this paper, we propose a discontinuous finite volume element method to solve a phase field model for two immiscible incompressible fluids. In this finite volume element scheme, discontinuous linear finite element basis functions are used to approximate the velocity, phase function, and chemical potential while piecewise constants are used to approximate the pressure. This numerical method is efficient, optimally convergent, conserving the mass, convenient to implement, flexible for mesh refinement, and easy to handle complex geometries with different types of boundary conditions. We rigorously prove the mass conservation property and the discrete energy dissipation for the proposed fully discrete discontinuous finite volume element scheme. Using numerical tests, we verify the accuracy, confirm the mass conservation and the energy law, test the influence of surface tension and small density variations, and simulate the driven cavity, the Rayleigh-Taylor instability.

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Decoupled, Linear, and Energy Stable Finite Element Method for the Cahn--Hilliard--Navier--Stokes--Darcy Phase Field Model

Cited by 80 publications

References 103 publications

Coupled and decoupled stabilized mixed finite element methods for nonstationary dual‐porosity‐Stokes fluid flow model

Coupled and decoupled stabilized mixed finite element methods for nonstationary dual‐porosity‐Stokes fluid flow model

A decoupled stabilized finite element method for the dual‐porosity‐Navier–Stokes fluid flow model arising in shale oil

Discontinuous finite volume element method for a coupled Navier-Stokes-Cahn-Hilliard phase field model

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