Virtual Element Discretization of Optimal Control Problem Governed by Brinkman Equations

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 L2 and H1 norm are derived. The theoretical findings are illustrated by the numerical experiments.

On the Importance of the Stokes-Brinkman Equations for Computing Effective Permeability in Karst Reservoirs

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.

Steady rotation of a composite sphere in a concentric spherical cavity

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.

An Efficient,Energy Stable Scheme for the Cahn-Hilliard-Brinkman System

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.

Multiscale Modeling and Simulations of Flows in Naturally Fractured Karst Reservoirs

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.