In this paper, we discuss virtual element method (VEM) approximation of optimal control problem governed by Brinkman equations with control constraints. Based on the polynomial projections and variational discretizati...In this paper, we discuss virtual element method (VEM) approximation of optimal control problem governed by Brinkman equations with control constraints. Based on the polynomial projections and variational discretization of the control variable, we build up the virtual element discrete scheme of the optimal control problem and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L<sup>2</sup> and H<sup>1</sup> norm are derived. The theoretical findings are illustrated by the numerical experiments.展开更多
Cavities and fractures significantly affect the flow paths in carbonate reservoirs and should be accurately accounted for in numerical models.Herein,we consider the problem of computing the effective permeability of r...Cavities and fractures significantly affect the flow paths in carbonate reservoirs and should be accurately accounted for in numerical models.Herein,we consider the problem of computing the effective permeability of rock samples based on high-resolution 3DCT scans containingmillions of voxels.We use the Stokes-Brinkman equations in the entire domain,covering regions of free flow governed by the Stokes equations,porous Darcy flow,and transitions between them.The presence of different length scales and large(ten orders of magnitude)contrasts in permeability leads to highly ill-conditioned linear systems of equations,which are difficult to solve.To obtain a problem that is computationally tractable,we first analyze the relative importance of the Stokes and Darcy terms for a set of idealized 2D models.We find that,in terms of effective permeability,the Stokes-Brinkman equations are only applicable for a special parameter set where the effective free-flow permeability is less than four orders of magnitude different from the matrix permeability.All other cases can be accurately modeled with either the Stokes or the Darcy end-member flows,depending on if there do or do not exist percolating free-flow regions.The insights obtained are used to perform a direct computation of the effective permeability of a rock sample model with more than 8 million cells.展开更多
The problem of steady rotation of a composite sphere located at the centre of a spherical container has been investigated. A composite particle referred to in this paper is a spherical solid core covered with a permea...The problem of steady rotation of a composite sphere located at the centre of a spherical container has been investigated. A composite particle referred to in this paper is a spherical solid core covered with a permeable spherical shell. The Brinkman＇s model for the flow inside the compos- ite sphere and the Stokes equation for the flow in the spheri- cal container were used to study the motion. The torque ex- perienced by the porous spherical particle in the presence of cavity is obtained. The wall correction factor is calculated. In the limiting cases, the analytical solution describing the torque for a porous sphere and for a solid sphere in an un- bounded medium are obtained from the present analysis.展开更多
We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman(CHB)system,which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brin...We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman(CHB)system,which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow.The CHB system is a generalization of the Cahn-Hilliard-Stokes model and describes two phase very viscous flows in porous media.The scheme is based on a convex splitting of the discrete CH energy and is semi-implicit.The equations at the implicit time level are nonlinear,but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable,regardless of time step size.Owing to energy stability,we show that the scheme is stable in the time and space discrete ℓ^(∞)(0,T;H_(h)^(1))and ℓ^(2)(0,T;H_(h)^(2))norms.We also present an efficient,practical nonlinear multigrid method–comprised of a standard FAS method for the Cahn-Hilliard part,and a method based on the Vanka smoothing strategy for the Brinkman part-for solving these equations.In particular,we provide evidence that the solver has nearly optimal complexity in typical situations.The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium,as well as to the more general problems of buoyancy-and boundary-driven flows.展开更多
Modeling and numerical simulations of fractured,vuggy,porus media is a challenging problem which occurs frequently in reservoir engineering.The problem is especially relevant in flow simulations of karst reservoirs wh...Modeling and numerical simulations of fractured,vuggy,porus media is a challenging problem which occurs frequently in reservoir engineering.The problem is especially relevant in flow simulations of karst reservoirs where vugs and caves are embedded in a porous rock and are connected via fracture networks at multiple scales.In this paper we propose a unified approach to this problem by using the StokesBrinkman equations at the fine scale.These equations are capable of representing porous media such as rock as well as free flow regions(fractures,vugs,caves)in a single system of equations.We then consider upscaling these equations to a coarser scale.The cell problems,needed to compute coarse-scale permeability of Representative Element of Volume(REV)are discussed.A mixed finite element method is then used to solve the Stokes-Brinkman equation at the fine scale for a number of flow problems,representative for different types of vuggy reservoirs.Upscaling is also performed by numerical solutions of Stokes-Brinkman cell problems in selected REVs.Both isolated vugs in porous matrix as well as vugs connected by fracture networks are analyzed by comparing fine-scale and coarse-scale flow fields.Several different types of fracture networks,representative of short-and long-range fractures are studied numerically.It is also shown that the Stokes-Brinkman equations can naturally be used to model additional physical effects pertaining to vugular media such as partial fracture with fill-in by some material and/or fluids with suspended solid particles.展开更多
The molecular coating on the surface of microvascular endothelium has been identified as a barrier to transvascular exchange of solutes. With a thickness of hundreds of nanometers, this endothelial surface layer （ESL...The molecular coating on the surface of microvascular endothelium has been identified as a barrier to transvascular exchange of solutes. With a thickness of hundreds of nanometers, this endothelial surface layer （ESL） has been treated as a porous do- main within which fluid shear stresses are dissipated and transmitted to the solid matrix to initiate mechanotransduction events. The present study aims to examine the effects of the ESL thickness and permeability on the transmission of shear stress throughout the ESL. Our results indicate that fluid shear stresses rapidly decrease to insignificant levels within a thin transition layer near the outer boundary of the ESL with a thickness on the order of ten nanometers. The thickness of the transition zone between free fluid and the porous layer was found to be proportional to the square root of the Darcy permeability. As the per- meability is reduced ten-fold, the interfacial fluid and solid matrix shear stress gradients increase exponentially two-fold. While the interracial fluid shear stress is positively related to the ESL thickness, the transmitted matrix stress is reduced by about 50% as the ESL thickness is decreased from 500 to 100 nm, which may occur under pathological conditions. Thus, thickness and permeability of the ESL are two main factors that determine flow features and the apportionment of shear stress- es between the fluid and solid phases of the ESL. These results may shed light on the mechanisms of force transmission through the ESL and the pathological events caused by alterations in thickness and permeability of the ESL.展开更多
文摘In this paper, we discuss virtual element method (VEM) approximation of optimal control problem governed by Brinkman equations with control constraints. Based on the polynomial projections and variational discretization of the control variable, we build up the virtual element discrete scheme of the optimal control problem and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L<sup>2</sup> and H<sup>1</sup> norm are derived. The theoretical findings are illustrated by the numerical experiments.
基金funded in part by Shell Norge AS and the Research Council of Norway through grants No.175962 and 186935Lie also acknowledges partial funding from the Center of Mathematics for Applications,University of Oslo.The Pipe Creek CT-scan data was originally collected by the Bureau of Economic Geology at The University of Texas at Austin with funding from the Industrial Associates of the Reservoir Characterization Research Laboratory.The authors are grateful to Bob Loucks,Chris Zahm,and Jim Jennings for assistance in accessing the data.
文摘Cavities and fractures significantly affect the flow paths in carbonate reservoirs and should be accurately accounted for in numerical models.Herein,we consider the problem of computing the effective permeability of rock samples based on high-resolution 3DCT scans containingmillions of voxels.We use the Stokes-Brinkman equations in the entire domain,covering regions of free flow governed by the Stokes equations,porous Darcy flow,and transitions between them.The presence of different length scales and large(ten orders of magnitude)contrasts in permeability leads to highly ill-conditioned linear systems of equations,which are difficult to solve.To obtain a problem that is computationally tractable,we first analyze the relative importance of the Stokes and Darcy terms for a set of idealized 2D models.We find that,in terms of effective permeability,the Stokes-Brinkman equations are only applicable for a special parameter set where the effective free-flow permeability is less than four orders of magnitude different from the matrix permeability.All other cases can be accurately modeled with either the Stokes or the Darcy end-member flows,depending on if there do or do not exist percolating free-flow regions.The insights obtained are used to perform a direct computation of the effective permeability of a rock sample model with more than 8 million cells.
文摘The problem of steady rotation of a composite sphere located at the centre of a spherical container has been investigated. A composite particle referred to in this paper is a spherical solid core covered with a permeable spherical shell. The Brinkman＇s model for the flow inside the compos- ite sphere and the Stokes equation for the flow in the spheri- cal container were used to study the motion. The torque ex- perienced by the porous spherical particle in the presence of cavity is obtained. The wall correction factor is calculated. In the limiting cases, the analytical solution describing the torque for a porous sphere and for a solid sphere in an un- bounded medium are obtained from the present analysis.
基金support from the NSF grant DMS-095066AFOSR grant FA9550-11-1-0328+1 种基金support from the NSF through the grants DMS-1115390 and DMS-0818030funding through NIMBioS at the University of Tennessee.
文摘We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman(CHB)system,which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow.The CHB system is a generalization of the Cahn-Hilliard-Stokes model and describes two phase very viscous flows in porous media.The scheme is based on a convex splitting of the discrete CH energy and is semi-implicit.The equations at the implicit time level are nonlinear,but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable,regardless of time step size.Owing to energy stability,we show that the scheme is stable in the time and space discrete ℓ^(∞)(0,T;H_(h)^(1))and ℓ^(2)(0,T;H_(h)^(2))norms.We also present an efficient,practical nonlinear multigrid method–comprised of a standard FAS method for the Cahn-Hilliard part,and a method based on the Vanka smoothing strategy for the Brinkman part-for solving these equations.In particular,we provide evidence that the solver has nearly optimal complexity in typical situations.The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium,as well as to the more general problems of buoyancy-and boundary-driven flows.
基金The authors would like to thank the China Petroleum&Chemical Corporation(SINOPEC),for supporting this work.
文摘Modeling and numerical simulations of fractured,vuggy,porus media is a challenging problem which occurs frequently in reservoir engineering.The problem is especially relevant in flow simulations of karst reservoirs where vugs and caves are embedded in a porous rock and are connected via fracture networks at multiple scales.In this paper we propose a unified approach to this problem by using the StokesBrinkman equations at the fine scale.These equations are capable of representing porous media such as rock as well as free flow regions(fractures,vugs,caves)in a single system of equations.We then consider upscaling these equations to a coarser scale.The cell problems,needed to compute coarse-scale permeability of Representative Element of Volume(REV)are discussed.A mixed finite element method is then used to solve the Stokes-Brinkman equation at the fine scale for a number of flow problems,representative for different types of vuggy reservoirs.Upscaling is also performed by numerical solutions of Stokes-Brinkman cell problems in selected REVs.Both isolated vugs in porous matrix as well as vugs connected by fracture networks are analyzed by comparing fine-scale and coarse-scale flow fields.Several different types of fracture networks,representative of short-and long-range fractures are studied numerically.It is also shown that the Stokes-Brinkman equations can naturally be used to model additional physical effects pertaining to vugular media such as partial fracture with fill-in by some material and/or fluids with suspended solid particles.
基金supported by the National Basic Research Program of China(Grant No.2012CB934101)the National Natural Science Foundation of China(Grant Nos.51175282 and 51375254)
文摘The molecular coating on the surface of microvascular endothelium has been identified as a barrier to transvascular exchange of solutes. With a thickness of hundreds of nanometers, this endothelial surface layer （ESL） has been treated as a porous do- main within which fluid shear stresses are dissipated and transmitted to the solid matrix to initiate mechanotransduction events. The present study aims to examine the effects of the ESL thickness and permeability on the transmission of shear stress throughout the ESL. Our results indicate that fluid shear stresses rapidly decrease to insignificant levels within a thin transition layer near the outer boundary of the ESL with a thickness on the order of ten nanometers. The thickness of the transition zone between free fluid and the porous layer was found to be proportional to the square root of the Darcy permeability. As the per- meability is reduced ten-fold, the interfacial fluid and solid matrix shear stress gradients increase exponentially two-fold. While the interracial fluid shear stress is positively related to the ESL thickness, the transmitted matrix stress is reduced by about 50% as the ESL thickness is decreased from 500 to 100 nm, which may occur under pathological conditions. Thus, thickness and permeability of the ESL are two main factors that determine flow features and the apportionment of shear stress- es between the fluid and solid phases of the ESL. These results may shed light on the mechanisms of force transmission through the ESL and the pathological events caused by alterations in thickness and permeability of the ESL.
