320 research outputs found
Numerical Computations with H(div)-Finite Elements for the Brinkman Problem
The H(div)-conforming approach for the Brinkman equation is studied
numerically, verifying the theoretical a priori and a posteriori analysis in
previous work of the authors. Furthermore, the results are extended to cover a
non-constant permeability. A hybridization technique for the problem is
presented, complete with a convergence analysis and numerical verification.
Finally, the numerical convergence studies are complemented with numerical
examples of applications to domain decomposition and adaptive mesh refinement.Comment: Minor clarifications, added references. Reordering of some figures.
To appear in Computational Geosciences, final article available at
http://www.springerlink.co
Modelling binary alloy solidification with adaptive mesh refinement
The solidification of a binary alloy results in the formation of a porous mushy layer, within which spontaneous localisation of fluid flow can lead to the emergence of features over a range of spatial scales. We describe a finite volume method for simulating binary alloy solidification in two dimensions with local mesh refinement in space and time. The coupled heat, solute, and mass transport is described using an enthalpy method with flow described by a Darcy-Brinkman equation for flow across porous and liquid regions. The resulting equations are solved on a hierarchy of block-structured adaptive grids. A projection method is used to compute the fluid velocity, whilst the viscous and nonlinear diffusive terms are calculated using a semi-implicit scheme. A series of synchronization steps ensure that the scheme is flux-conservative and correct for errors that arise at the boundaries between different levels of refinement. We also develop a corresponding method using Darcy's law for flow in a porous medium/narrow Hele-Shaw cell. We demonstrate the accuracy and efficiency of our method using established benchmarks for solidification without flow and convection in a fixed porous medium, along with convergence tests for the fully coupled code. Finally, we demonstrate the ability of our method to simulate transient mushy layer growth with narrow liquid channels which evolve over time
Analytical and Numerical Aspects of Porous Media Flow
The Brinkman equations model fluid flow through porous media and are particularly interesting in regimes where viscous shear effects cannot be neglected. Two model parameters in the momentum balance function as weights for the terms related to inter-particle friction and bulk resistance. If these are not in balance, then standard finite element methods might suffer from instabilities or error estimates might deteriorate. In particular the limit case, where the Brinkman problem reduces to a Darcy problem, demands for special attention. This thesis proposes a low-order finite element method which is uniformly stable with respect to the flow regimes captured by the Brinkman model, including the Darcy limit. To that end, linear equal-order approximations are combined with a pressure stabilization technique, a grad-div stabilization, and a penalty-free non-symmetric Nitsche method. The combination of these ingredients allows to develop a robust method, which is proven to be well-posed for the whole family of problems in two spatial dimensions, even if any Brinkman parameter vanishes. An a priori error analysis reveals optimal convergence in the considered norm. A convergence study based on problems with known analytic solutions confirms the robust first order convergence for reasonable ranges of numerical (stabilization) parameters. Further, numerical investigations that partly extend the theoretical framework are considered, revealing strengths and weaknesses of the approach. An application motivated by the optimization of geothermal energy production completes the thesis. Here, the proposed method is included in a multi-physics discrete model, appropriate to describe the thermo-hydraulics in hot, sedimentary, essentially horizontal aquifers. An immersed boundary method is adopted in order to allow a flexible, automatic optimization without regenerating the computational mesh. Utilizing the developed computational framework, the optimized multi-well arrangements with respect to the net energy gain are presented and discussed for different geothermal and hydrogeological setups. The results show that taking into account heterogeneous permeability structures and variable aquifer temperatures might drastically affect the optimal configuration of the wells
A numerical study of fluids with pressure dependent viscosity flowing through a rigid porous medium
In this paper we consider modifications to Darcy's equation wherein the drag
coefficient is a function of pressure, which is a realistic model for
technological applications like enhanced oil recovery and geological carbon
sequestration. We first outline the approximations behind Darcy's equation and
the modifications that we propose to Darcy's equation, and derive the governing
equations through a systematic approach using mixture theory. We then propose a
stabilized mixed finite element formulation for the modified Darcy's equation.
To solve the resulting nonlinear equations we present a solution procedure
based on the consistent Newton-Raphson method. We solve representative test
problems to illustrate the performance of the proposed stabilized formulation.
One of the objectives of this paper is also to show that the dependence of
viscosity on the pressure can have a significant effect both on the qualitative
and quantitative nature of the solution
A Generalized Multiscale Finite Element Method for the Brinkman Equation
In this paper we consider the numerical upscaling of the Brinkman equation in
the presence of high-contrast permeability fields. We develop and analyze a
robust and efficient Generalized Multiscale Finite Element Method (GMsFEM) for
the Brinkman model. In the fine grid, we use mixed finite element method with
the velocity and pressure being continuous piecewise quadratic and piecewise
constant finite element spaces, respectively. Using the GMsFEM framework we
construct suitable coarse-scale spaces for the velocity and pressure that yield
a robust mixed GMsFEM. We develop a novel approach to construct a coarse
approximation for the velocity snapshot space and a robust small offline space
for the velocity space. The stability of the mixed GMsFEM and a priori error
estimates are derived. A variety of two-dimensional numerical examples are
presented to illustrate the effectiveness of the algorithm.Comment: 22 page
A uniform and pressure-robust enriched Galerkin method for the Brinkman equations
This paper presents a pressure-robust enriched Galerkin (EG) method for the
Brinkman equations with minimal degrees of freedom based on EG velocity and
pressure spaces. The velocity space consists of linear Lagrange polynomials
enriched by a discontinuous, piecewise linear, and mean-zero vector function
per element, while piecewise constant functions approximate the pressure. We
derive, analyze, and compare two EG methods in this paper: standard and robust
methods. The standard method requires a mesh size to be less than a viscous
parameter to produce stable and accurate velocity solutions, which is
impractical in the Darcy regime. Therefore, we propose the pressure-robust
method by utilizing a velocity reconstruction operator and replacing EG
velocity functions with a reconstructed velocity. The robust method yields
error estimates independent of a pressure term and shows uniform performance
from the Stokes to Darcy regimes, preserving minimal degrees of freedom. We
prove well-posedness and error estimates for both the standard and robust EG
methods. We finally confirm theoretical results through numerical experiments
with two- and three-dimensional examples and compare the methods' performance
to support the need for the robust method
Finite element methods for flow in porous media
This thesis studies the application of finite element methods to porous flow problems. Particular attention is paid to locally mass conserving methods, which are very well suited for typical multiphase flow applications in porous media. The focus is on the Brinkman model, which is a parameter dependent extension of the classical Darcy model for porous flow taking the viscous phenomena into account. The thesis introduces a mass conserving finite element method for the Brinkman flow, with complete mathematical analysis of the method. In addition, stochastic material parameters are considered for the Brinkman flow, and parameter dependent Robin boundary conditions for the underlying Darcy flow. All of the theoretical results in the thesis are also verified with extensive numerical testing. Furthermore, many implementational aspects are discussed in the thesis, and computational viability of the methods introduced, both in terms of usefulness and computational complexity, is taken into account
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