A Dual-Porosity-Stokes Model and Finite Element Method for Coupling Dual-Porosity Flow and Free Flow

Hou, Jiangyong; Qiu, Meilan; He, Xiaoming; Guo, Chaohua; Wei, Mingzhen; Bai, Baojun

doi:10.1137/15m1044072

Cited by 67 publications

(39 citation statements)

References 68 publications

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“…In this section, we recall the nonstationary dual‐porosity‐Stokes fluid flow model with four physically valid interface conditions …”

Section: The Dual‐porosity‐stokes Modelmentioning

confidence: 99%

“…In the subdomain Ω d , to find the microfracture flow pressure φ f ( x , t ), microfracture flow velocity

{\overset{u}{true}}_{f} false(x, t false)

, matrix flow pressure φ m ( x , t ), and matrix flow velocity

{\overset{u}{true}}_{m} false(x, t false)

in the dual‐porous region, we can consider the following system of equations:

η_{f} C_{f t} \frac{\partial φ_{f}}{\partial t} + \nabla \cdot {\overset{u}{true}}_{f} + \frac{σ k_{m}}{μ} false(φ_{f} - φ_{m} false) = f_{d} in {normalΩ}_{d} \times false(0, T false],

{\overset{u}{true}}_{f} = \frac{- k_{f}}{μ} \nabla φ_{f} in {normalΩ}_{d} \times false(0, T false],

η_{m} C_{m t} \frac{\partial φ_{m}}{\partial t} + \nabla \cdot {\overset{u}{true}}_{m} + \frac{σ k_{m}}{μ} false(φ_{m} - φ_{f} false) = 0 in {normalΩ}_{d} \times false(0, T false],

{\overset{u}{true}}_{m} = - k

…”

Section: The Dual‐porosity‐stokes Modelmentioning

confidence: 99%

“…Since none of the existing Stokes‐Darcy type models consider the dual‐porosity medium when they coupled the porous media flow with free flow, Hou et al proposed and numerically solved a coupled dual‐porosity‐Stokes multiphysics interface system where dual‐porosity equations were considered to describe the porous media flow and the Stokes equation was utilized to model the free flow in the conduits/macrofractures. In general, the naturally fractured reservoir consists of two separate but connected continua where the network of fractures is associated with fissure, macropores, or interaggregate pores with the faster flow.…”

Section: Introductionmentioning

confidence: 99%

“…The large difference between the permeability of the matrix and that of the microfractures makes the microfractures an easy pathway for the flow in the dual‐porosity media to the conduits/macrofractures, while the matrix system provides the main storage of the fluid due to its much larger porosity . Therefore, in the work of Hou et al, four physically valid interface conditions are utilized to couple the two separate systems through the interface between the dual‐porosity media and conduits/macrofractures. Moreover, the dual‐porosity medium flow is modeled by the dual‐porosity equations, which is interconnected with a mass exchange term.…”

Section: Introductionmentioning

confidence: 99%

“…Based on the traditional coupled finite element method in the work of Hou et al, in this article, we first propose and analyze a coupled stabilized mixed finite element method to lay a solid ground. Then, in order to significantly improve the computational efficiency, a decoupled stabilized mixed finite element method is constructed by using a two‐level decoupling idea.…”

Section: Introductionmentioning

confidence: 99%

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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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“…In this section, we recall the nonstationary dual‐porosity‐Stokes fluid flow model with four physically valid interface conditions …”

Section: The Dual‐porosity‐stokes Modelmentioning

confidence: 99%

“…In the subdomain Ω d , to find the microfracture flow pressure φ f ( x , t ), microfracture flow velocity

{\overset{u}{true}}_{f} false(x, t false)

, matrix flow pressure φ m ( x , t ), and matrix flow velocity

{\overset{u}{true}}_{m} false(x, t false)

in the dual‐porous region, we can consider the following system of equations:

η_{f} C_{f t} \frac{\partial φ_{f}}{\partial t} + \nabla \cdot {\overset{u}{true}}_{f} + \frac{σ k_{m}}{μ} false(φ_{f} - φ_{m} false) = f_{d} in {normalΩ}_{d} \times false(0, T false],

{\overset{u}{true}}_{f} = \frac{- k_{f}}{μ} \nabla φ_{f} in {normalΩ}_{d} \times false(0, T false],

η_{m} C_{m t} \frac{\partial φ_{m}}{\partial t} + \nabla \cdot {\overset{u}{true}}_{m} + \frac{σ k_{m}}{μ} false(φ_{m} - φ_{f} false) = 0 in {normalΩ}_{d} \times false(0, T false],

{\overset{u}{true}}_{m} = - k

…”

Section: The Dual‐porosity‐stokes Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Mahbub

Nasu

et al. 2019

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

An artificial compressibility ensemble algorithm for a stochastic Stokes‐Darcy model with random hydraulic conductivity and interface conditions

Jiang

Qiu

2019

Numerical Meth Engineering

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We propose and analyze an efficient ensemble algorithm with artificial compressibility for fast decoupled computation of multiple realizations of the stochastic Stokes-Darcy model with random hydraulic conductivity (including the one in the interface conditions), source terms, and initial conditions. The solutions are found by solving three smaller decoupled subproblems with two common time-independent coefficient matrices for all realizations, which significantly improves the efficiency for both assembling and solving the matrix systems. The fully coupled Stokes-Darcy system can be first decoupled into two smaller sub-physics problems by the idea of the partitioned time stepping, which reduces the size of the linear systems and allows parallel computing for each sub-physics problem. The artificial compressibility further decouples the velocity and pressure which further reduces storage requirements and improves computational efficiency. We prove the long time stability and the convergence for this new ensemble method. Three numerical examples are presented to support the theoretical results and illustrate the features of the algorithm, including the convergence, stability, efficiency and applicability.

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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

show abstract

A Dual-Porosity-Stokes Model and Finite Element Method for Coupling Dual-Porosity Flow and Free Flow

Cited by 67 publications

References 68 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

An artificial compressibility ensemble algorithm for a stochastic Stokes‐Darcy model with random hydraulic conductivity and interface conditions

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

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