Beavers Joseph Interface Condition Research Articles

An improved partitioned time stepping method based on modified characteristic FEM for the evolutionary dual-porosity-Navier–Stokes model

In this paper, we study an improved partitioned time stepping method based on the modified characteristic finite element method for the evolutionary dual-porosity-Navier–Stokes coupling model with the Beavers–Joseph interface condition. By referring to the idea of partitioned time stepping, we lead into the half-time steps to separate the original coupling problem into three subproblems, implement two-level decoupling to simplify the complexity of solving coupling systems. Before that, the matrix and microfracture equations on the half-time steps in the dual-porosity media are calculated first, with the aim of replacing the initial values of the decoupling equations. Besides, for the nonlinear convection term in Navier–Stokes equations, the modified characteristic finite element method is chosen to avoid the problem of low computational efficiency. The convergence of the decoupling algorithm is rigorously derived by using the idea of mathematical induction. Finally, the reliability of the theoretical analysis and the rationality of the scheme are verified by numerical experiments.

Local and parallel finite element methods based on two-grid discretizations for the unsteady mixed Stokes-Darcy model with the Beavers-Joseph interface condition

Local and parallel finite element methods based on two-grid discretizations for the unsteady mixed Stokes-Darcy model with the Beavers-Joseph interface condition

A modified local and parallel finite element method for the coupled Stokes-Darcy model with the Beavers-Joseph interface condition

A modified local and parallel finite element method for the coupled Stokes-Darcy model with the Beavers-Joseph interface condition

A Parallel Robin–Robin Domain Decomposition Method based on Modified Characteristic FEMs for the Time-Dependent Dual-porosity-Navier–Stokes Model with the Beavers–Joseph Interface Condition

A Parallel Robin–Robin Domain Decomposition Method based on Modified Characteristic FEMs for the Time-Dependent Dual-porosity-Navier–Stokes Model with the Beavers–Joseph Interface Condition

Modeling and a Robin-type decoupled finite element method for dual-porosity–Navier–Stokes system with application to flows around multistage fractured horizontal wellbore

In this article, we present a time-dependent dual-porosity–Navier–Stokes model with four interface conditions, including Beavers–Joseph interface condition, to describe a coupling system of complex porous media and conduit networks. This system has many applications, such as the flow simulation problems for a multistage fractured horizontal wellbore with suitable boundary/interface conditions and complex geometries. For this coupling system, a decoupled finite element method is proposed based on Robin type transmission conditions. In order to avoid the iteration for the traditional domain decomposition, the interface information, which is needed for the Robin type transmission conditions at the current time step, is predicted directly from the numerical solution of the previous time steps. The stability and convergence of this Robin-type decoupled finite element method are rigorously analyzed. A series of analysis techniques are utilized to analyze various components of this complicated multi-physics problem term by term, especially for the Beavers–Joseph interface condition, the interaction terms of dual-porosity model, and the nonlinear advection of Navier–Stokes equation. Moreover, a pair of Robin parameters are considered in the scheme and analysis, and their effects on the robustness for the case with low permeability and low storativity are numerically investigated. Numerical experiments are provided to validate the convergence of the proposed algorithm and illustrate the features of the application to flow problems around multistage fractured horizontal wellbore.

A novel convergence analysis of Robin–Robin domain decomposition method for Stokes–Darcy system with Beavers–Joseph interface condition

In this paper, we demonstrate the convergence analysis of Robin–Robin domain decomposition method with finite element discretization for Stokes–Darcy system with Beavers-Joseph interface condition, with particular attention paid to the case which is convergent for small viscosity and hydraulic conductivity in practice. Based on the techniques of the discrete harmonic extension and discrete Stokes extension, the convergence is proved and the almost optimal geometric convergence rate is obtained for the case of γf>γp. Here γf and γp are positive Robin parameters introduced in Cao et al., 2011, which was not able to show the analysis for γf>γp but only numerically illustrated its importance to the convergence for the practical situation with small viscosity and hydraulic conductivity. The analysis result provides a general guideline of choice on the relevant parameters to obtain the convergence and geometric convergence rate. The numerical results verify the theoretical conclusion.

Decoupled modified characteristic FEMs for fully evolutionary Navier–Stokes–Darcy model with the Beavers–Joseph interface condition

In this paper, we develop the numerical theory of decoupled modified characteristic FEMs for the fully evolutionary Navier–Stokes–Darcy model with the Beavers–Joseph interface condition. Based on lagging interface coupling terms, the system is decoupled, which means that the Navier–Stokes equations and the Darcy equation are solved in each time step, respectively. In particular, the Navier–Stokes equations are solved by the modified characteristic FEMs, which overcome the computational inefficiency and analytical difficulties caused by the nonlinear term. Then we prove the optimal L2-norm error convergence order of the solutions by mathematical induction, whose proof implies the uniform L∞-boundedness of the fully discrete velocity solution. Finally some numerical tests are presented to show high efficiency of this method.

Unsuitability of the Beavers–Joseph interface condition for filtration problems

Coupled free-flow and porous-medium systems appear in a variety of industrial and environmental applications. Fluid flow in the free-flow domain is typically described by the (Navier–)Stokes equations while Darcy’s law is applied in the porous medium. The correct choice of coupling conditions on the fluid–porous interface is crucial for accurate numerical simulations of coupled problems. We found out that the Beavers–Joseph interface condition, which is widely used not only for fluid flow parallel to the porous layer but also for filtration problems, is unsuitable for arbitrary flow directions. To validate our statement, we provide several examples and compare numerical simulation results for the coupled Stokes–Darcy problems to the pore-scale resolved models. We show also that the Beavers–Joseph parameter cannot be fitted for arbitrary flow directions.

A domain decomposition method for the time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition and defective boundary condition

In this article a domain decomposition method is proposed to solve a time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition and defective boundary condition. Robin boundary conditions between the Navier-Stokes domain and Darcy domain are constructed by directly re-organizing the terms in the three interface conditions, including the Beavers-Joseph condition. In order to avoid the traditional iteration for the domain decomposition method at each time step, the interface information, which is needed for the Robin type transmission conditions at the current time step, is directly predicted based on the numerical solution of the previous time steps. Backward Euler scheme is first utilized for the temporal discretization while finite elements are used for the spatial discretization. The convergences of this domain decomposition method are rigorously analyzed for the time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition. The major difficulties in the analysis arise from nonlinear terms and Beavers-Joseph interface condition, including a series of technical treatments and the final special norm used in the discrete Gronwall's inequality for the analysis of full discretization. Based on the above preparation, we further develop a Lagrange multiplier method under the framework of the domain decomposition method to overcome the difficulty of non-unique solutions arising from the defective boundary condition. One interesting finding of this paper is that the Lagrange multipliers are time dependent functions instead of constants. In order to improve the accuracy order for the temporal discretization, a three-step backward differentiation scheme is used to replace the backward Euler scheme. Compared with the first scheme, the second one allows us to use the relative larger time step to reduce the computational cost while keeping the same accuracy. Numerical examples are provided to illustrate the features of the proposed method.

On the Beavers–Joseph Interface Condition for Non-parallel Coupled Channel Flow over a Porous Structure at High Reynolds Numbers

A channel flow coupled with a transversal stream through porous structures is investigated numerically in this study. Velocity profiles are obtained on the pore scale and averaged to the macroscale in order to evaluate the validity of the Beavers–Joseph interface condition. For this purpose, different ratios between the velocity at the inlet of the channel and the velocity at the base of the porous structure are considered. The effects of Reynolds number, velocity ratio, and geometrical arrangement of the porous structure on the Beavers–Joseph constant are then examined. A critical velocity ratio is found where, for lower ratios, the interface momentum transfer is mainly affected by the channel flow and, for higher ratios, the flow in the porous structure governs. The uniformity of the Beavers–Joseph constant at the interface is found to decrease with increasing velocity ratios. The present results have then been compared with simulations from a coupled Navier–Stokes/Darcy–Forchheimer model.

On the solution of coupled Stokes/Darcy model with Beavers–Joseph interface condition

In this paper, we first get the uniqueness result of the possible solution to the steady-state coupled Stokes/Darcy model with Beavers–Joseph interface condition for any physical parameters, especially for any α>0. Then we show the existence of solutions for any α>0 by using Galerkin method. Furthermore, we analyze the error of the corresponding coupled finite element scheme and derive the optimal error estimates.

A multi-grid technique for coupling fluid flow with porous media flow

In this paper, we consider the coupling of fluid flow with porous media flow. A multi-grid finite element method for the coupled Stokes–Darcy problem with the Beavers–Joseph interface condition is proposed and discussed. The optimal error estimates are obtained. Numerical experiment is given to verify the theoretical analysis and indicate the accuracy and efficiency of the multi-grid method.

Numerical analysis for the mixed Navier–Stokes and Darcy Problem with the Beavers–Joseph interface condition

In this article, we consider the coupled Navier–Stokes and Darcy problem with the Beavers–Joseph interface condition. With suitable restrictions of physical parameters α and ν, we prove the existence and local uniqueness of a weak solution. Then we propose a coupled finite element scheme and a decoupled and linearized scheme based on two‐grid finite element. Under suitable further restrictions, their optimal error estimates are obtained. Finally numerical experiments indicate the validity of the theoretical results as well as the efficiency and effectiveness of the decoupled and linearized two‐grid algorithm. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1009–1030, 2015

Analysis of the Effect of the Beavers-Joseph Interface Condition on Flow in Karst Conduits1

Li, Guangquan, Yuan Cheng, and Bei Zhao, 2012. Analysis of the Effect of the Beavers-Joseph Interface Condition on Flow in Karst Conduits. Journal of the American Water Resources Association (JAWRA) 1-8. DOI: 10.1111/j.1752-1688.2012.00683.x Abstract: In this study, we derive an approximate analytic solution for the distribution of flow velocity in a cylindrical conduit and the surrounding media, and analyze the effect of differing parameters (e.g., conduit radius) on the velocity of conduit flow. The solution is then employed to estimate the thickness of the boundary layer inside the media. The results reveal that when conduit radius is large, the Beavers-Joseph condition has only a minor effect on the velocity of conduit flow (such that the nonslip condition on the conduit wall still works pretty well), and the boundary layer is so thin that the wall can still be treated as the interface between fast water in the conduit and slow water in the media. The solution indicates that the velocity of conduit flow is the superposition of the velocity profile in the nonslip situation onto the slip velocity on the wall. Our study theoretically shows that the coupled continuum pipe flow model in MODFLOW-2005 constructed by the U.S. Geological Survey is reasonable in that there is no need to consider the Beavers-Joseph condition when simulating flow in karst conduits. The role of the boundary layer in transport and its effect on the hyporheic zone is not clear, which is a suitable topic for future study.

Non-iterative domain decomposition methods for a non-stationary Stokes–Darcy model with Beavers–Joseph interface condition

In order to solve a non-stationary Stokes–Darcy model with Beavers–Joseph interface condition, two non-iterative domain decomposition methods are proposed. At each time step, results from previous time steps are utilized to approximate the information on the interface and decouple the two physics. Both of the two methods are parallel. Numerical results suggest that the first method has accuracy order O(h3+Δt). In order to improve the accuracy and efficiency, a three-step backward differentiation is used in the second method to achieve an accuracy order O(h3+Δt3), which is illustrated by a numerical example.

Experimental and computational validation and verification of the Stokes–Darcy and continuum pipe flow models for karst aquifers with dual porosity structure

AbstractIn our previous study, we developed the Stokes–Darcy (SD) model was developed for flow in a karst aquifer with a conduit bedded in matrix, and the Beavers–Joseph (BJ) condition was used to describe the matrix–conduit interface. We also studied the mathematical well‐posedness of a coupled continuum pipe flow (CCPF) model as well as convergence rates of its finite element approximation. In this study, to compare the SD model with the CCPF model, we used numerical analyses to validate finite element discretisation methods for the two models. Using computational experiments, simulation codes implementing the finite element discretisations are then verified. Further model validation studies are based on the results of laboratory experiments. Comparing the results of computer simulations and experiments, we concluded that the SD model with the Beavers–Joseph interface condition is a valid model for conduit–matrix systems. On the other hand, the CCPF model with the value of the exchange parameter chosen within the range suggested in the literature perhaps does not result in good agreement with experimental observations. We then examined the sensitivity of the CCPF model with respect to the exchange parameter, concluding that, as has previously been noted, the model is highly sensitive for small values of the exchange parameter. However, for larger values, the model becomes less sensitive and, more important, also produces results that are in better agreement with experimental observations. This suggests that the CCPF model may also produce accurate simulation results, if one chooses larger values of the exchange parameter than those suggested in the literature. Copyright © 2011 John Wiley & Sons, Ltd.

Forced Convection in a Channel Partly Occupied by a Bidisperse Porous Medium: Symmetric Case

An analytic solution is obtained for the problem of fully developed forced convection in a channel between parallel plane walls, partly occupied by a bidisperse porous medium (BDPM) and partly by a fluid clear of solid material, the distribution of material being symmetrical with the BDPM forming either the core or the sheath portion of the channel. The case of uniform flux boundaries is considered. For the porous medium, a two-velocity two-temperature model based on Darcy’s law is employed and the Beavers–Joseph interface condition is imposed. In each case, Nusselt number values are obtained in terms of various parameters.

Robin–Robin domain decomposition methods for the steady-state Stokes–Darcy system with the Beavers–Joseph interface condition

Domain decomposition methods for solving the coupled Stokes–Darcy system with the Beavers–Joseph interface condition are proposed and analyzed. Robin boundary conditions are used to decouple the Stokes and Darcy parts of the system. Then, parallel and serial domain decomposition methods are constructed based on the two decoupled sub-problems. Convergence of the two methods is demonstrated and the results of computational experiments are presented to illustrate the convergence.

Asymptotic analysis of the differences between the Stokes–Darcy system with different interface conditions and the Stokes–Brinkman system

We consider the coupling of the Stokes and Darcy systems with different choices for the interface conditions. We show that, comparing results with those for the Stokes–Brinkman equations, the solutions of Stokes–Darcy equations with the Beavers–Joseph interface condition in the one-dimensional and quasi-two-dimensional (periodic) cases are more accurate than are those obtained using the Beavers–Joseph–Saffman–Jones interface condition and that both of these are more accurate than solutions obtained using a zero tangential velocity interface condition. The zero tangential velocity interface condition is in turn more accurate than the free-slip interface boundary condition. We also prove that the summation of the quasi-two-dimensional solutions converge so that the conclusions are also valid for the two-dimensional case.