Cell-Centered Nonlinear Finite-Volume Methods With Improved Robustness

SPE Journal ◽  
2019 ◽  
Vol 25 (01) ◽  
pp. 288-309 ◽  
Author(s):  
Wenjuan Zhang ◽  
Mohammed Al Kobaisi
Keyword(s):  
Finite Volume ◽  
Real Life ◽  
Finite Volume Methods ◽  
Optimal Order ◽  
Test Case ◽  
Correction Algorithm ◽  
Optimal Order Of Convergence ◽  
Industry Standard ◽  
Nonnegative Coefficients ◽  
Volume Method

Summary We present a nonlinear finite-volume method (NFVM) that is either positivity-preserving or extremum-preserving with improved robustness. The key ingredient of the method is the construction of one-sided fluxes, which involves decomposition of conormal vectors by introducing harmonic-averaging points as auxiliary points. The original NFVM using harmonic-averaging points is not robust in the sense that decomposition of conormal vectors with nonnegative coefficients can easily run into difficulties for heterogeneous and anisotropic permeability tensors on general nonorthogonal meshes. To improve NFVM robustness, we first present an alternative derivation of harmonic-averaging points and give a different formula that shows more clearly a point's location. On the basis of the derivation of the new formula, a correction algorithm is proposed to make modifications to those problematic harmonic-averaging points so that all the conormal vectors can be decomposed with nonnegative coefficients successfully. As a result, the resulting NFVM can be applied to more-challenging problems when conormal decomposition with nonnegative coefficients is not possible without correction. The correction algorithm is a compromise between robustness and accuracy. While it improves the robustness of the resulting NFVM, results of numerical convergence tests show that the effect of our correction algorithm on accuracy is problem-dependent. Optimal order of convergence is still maintained for some problems, and the convergence rate is reduced for others. Monotonicity and extremum-preserving properties are verified by numerical experiments. Finally, a field test case is used to demonstrate that the NFVM combined with our correction algorithm can be applied to simulate real-life reservoirs of industry-standard complexity.

Numerical simulation of two-dimensional and three-dimensional axisymmetric advection–diffusion systems with complex geometries using finite-volume methods

2010 ◽  
Vol 466 (2118) ◽  
pp. 1621-1643 ◽  
Author(s):  
J. M. A. Ashbourn ◽  
L. Geris ◽  
A. Gerisch ◽  
C. J. S. Young
Keyword(s):  
Finite Volume ◽  
Three Dimensional ◽  
Finite Volume Methods ◽  
Two Dimensional ◽  
Corner Singularities ◽  
Bone Fracture Healing ◽  
Curved Boundaries ◽  
Advection Diffusion ◽  
Diffusion Systems ◽  
Volume Method

A finite-volume method has been developed that can deal accurately with complicated, curved boundaries for both two-dimensional and three-dimensional axisymmetric advection–diffusion systems. The motivation behind this is threefold. Firstly, the ability to model the correct geometry of a situation yields more accurate results. Secondly, smooth geometries eliminate corner singularities in the calculation of, for example, mechanical variables and thirdly, different geometries can be tested for experimental applications. An example illustrating each of these is given: fluid carrying a dye and rotating in an annulus, bone fracture healing in mice, and using vessels of different geometry in an ultracentrifuge.

Finite Volume Methods for Well-Driven Flows in Anisotropic Porous Media

2014 ◽  
Vol 14 (4) ◽  
pp. 473-483 ◽  
Author(s):  
Milan Dotlić
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Finite Volume ◽  
Finite Volume Methods ◽  
Hydraulic Head ◽  
Anisotropic Porous Media ◽  
Flow Simulations ◽  
First Order ◽  
Anisotropic Porous Medium ◽  
Volume Method

AbstractWe consider a finite volume method for flow simulations in an anisotropic porous medium in the presence of a well. The hydraulic head varies logarithmically and its gradient changes rapidly in the well vicinity. Thus, the use of standard numerical schemes results in a completely wrong total well flux and an inaccurate hydraulic head. In this article we propose two finite volume methods to model the well singularity in an anisotropic medium. The first method significantly reduces the total well flux error, but the hydraulic head is still not even first-order accurate. The second method gives a second-order accurate hydraulic head and at least first-order accurate total well flux.

2D Unstructured Mesh Finite Volume Method for Simulating Structural Dynamics

2013 ◽  
Vol 376 ◽  
pp. 345-348
Author(s):  
Miao Yu Hai ◽  
Xiao Hui Su ◽  
Yao Cao ◽  
Yong Zhao ◽  
Jian Tao Zhang
Keyword(s):  
Finite Volume ◽  
Unstructured Mesh ◽  
Solid Mechanics ◽  
Elastic Solid ◽  
Test Case ◽  
Time Step ◽  
Step Algorithm ◽  
Volume Method ◽  
Matrix Free ◽  
Fluidstructure Interaction

A novel procedure for calculating the dynamic response of elastic solid structures is presented. The ultimate aim of this study is to develop a consistent set of finite volume (FV) methods on unstructured meshes for the analysis of dynamic fluidstructure interaction (FSI). This paper describes a two-dimensional (2D) FV cell-vertex based method for dynamic solid mechanics. A novel matrix-free implicit scheme was developed using the Newmark method and dual time step algorithm and the model is validated with a 2D cantilever test case as well as a 2D plate one.

Application of Finite Volume Method for Solid Mechanics

2003 ◽  
Author(s):  
Bryce L. Fowler ◽  
Raymond K. Yee
Keyword(s):  
Finite Element ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Solid Mechanics ◽  
Preliminary Test ◽  
Test Case ◽  
Element Analysis ◽  
Vibration Isolators ◽  
Incompressible Materials ◽  
Volume Method

Polymers constitute a large class of nearly incompressible solid materials (i.e., Poisson’s Ratio near 0.5). These materials are often used as passive vibration isolators. Accurately modeling vibration isolators made of nearly incompressible materials has been extremely difficult with standard finite element analysis. This paper provides an alternative to the specialized finite element formulations currently used to model incompressible materials. The finite volume methodology of computational fluid dynamics is employed in this paper to solve the Hooke’s Law equations in solid mechanics. Test cases have been performed to evaluate the performance of finite volume method applied to solid mechanics problems. The formulation has been coded in Matlab for practical use. Based on the preliminary test case results, the finite volume formulation compares favorably to finite element method.

scholarly journals Postprocessing of standard finite element velocity fields for accurate particle tracking applied to groundwater flow

2020 ◽  
Vol 24 (4) ◽  
pp. 1605-1624
Author(s):  
Philipp Selzer ◽  
Olaf A. Cirpka
Keyword(s):  
Finite Element ◽  
Velocity Field ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Particle Tracking ◽  
Mass Conservation ◽  
Velocity Fields ◽  
Test Case ◽  
Hydraulic Gradients ◽  
Volume Method

Abstract Particle tracking is a computationally advantageous and fast scheme to determine travel times and trajectories in subsurface hydrology. Accurate particle tracking requires element-wise mass-conservative, conforming velocity fields. This condition is not fulfilled by the standard linear Galerkin finite element method (FEM). We present a projection, which maps a non-conforming, element-wise given velocity field, computed on triangles and tetrahedra, onto a conforming velocity field in lowest-order Raviart-Thomas-Nédélec ($\mathcal {RTN}_{0}$ R T N 0 ) space, which meets the requirements of accurate particle tracking. The projection is based on minimizing the difference in the hydraulic gradients at the element centroids between the standard FEM solution and the hydraulic gradients consistent with the $\mathcal {RTN}_{0}$ R T N 0 velocity field imposing element-wise mass conservation. Using the conforming velocity field in $\mathcal {RTN}_{0}$ R T N 0 space on triangles and tetrahedra, we present semi-analytical particle tracking methods for divergent and non-divergent flow. We compare the results with those obtained by a cell-centered finite volume method defined for the same elements, and a test case considering hydraulic anisotropy to an analytical solution. The velocity fields and associated particle trajectories based on the projection of the standard FEM solution are comparable to those resulting from the finite volume method, but the projected fields are smoother within zones of piecewise uniform hydraulic conductivity. While the $\mathcal {RTN}_{0}$ R T N 0 -projected standard FEM solution is thus more accurate, the computational costs of the cell-centered finite volume approach are considerably smaller.

Convergence Analysis of Discontinuous Finite Volume Methods for Elliptic Optimal Control Problems

2016 ◽  
Vol 13 (02) ◽  
pp. 1640012 ◽  
Author(s):  
Ruchi Sandilya ◽  
Sarvesh Kumar
Keyword(s):  
Order Of Convergence ◽  
Optimal Control Problems ◽  
Finite Volume Methods ◽  
Optimal Order ◽  
Control Problems ◽  
State Variables ◽  
Optimal Order Of Convergence ◽  
Piecewise Constant ◽  
Adjoint State ◽  
Variational Discretization

In this paper, we discuss the convergence analysis of discontinuous finite volume methods applied to distribute the optimal control problems governed by a class of second-order linear elliptic equations. In order to approximate the control, two different methodologies are adopted: one is the method of variational discretization and second is piecewise constant discretization technique. For variational discretization method, optimal order of convergence in the [Formula: see text]-norm for state, adjoint state and control variables is derived. Moreover, optimal order of convergence in discrete [Formula: see text]-norm is also derived for state and adjoint state variables. Whereas, for piecewise constant approximation of control, first order convergence is derived for control, state and adjoint state variables in the [Formula: see text]-norm. In addition to that, optimal order of convergence in discrete [Formula: see text]-norm is derived for state and adjoint state variables. Also, some numerical experiments are conducted to support the derived theoretical convergence rate.

Simulation of Rotor/Stator Interaction With a 4D Finite Volume Method

2002 ◽  
Author(s):  
Bjo¨rn Laumert ◽  
Hans Ma˚rtensson ◽  
Torsten H. Fransson
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Three Dimensional ◽  
Test Case ◽  
Pressure Distributions ◽  
Time Space ◽  
Rotor Stator Interaction ◽  
Pitch Ratio ◽  
Grid Nodes ◽  
Volume Method

A finite volume method for the computation of rotor/stator interaction for stages with arbitrary rotor/stator pitch ratios is presented and partly validated in this paper. The method which solves the unsteady three-dimensional Euler equations is formulated in the four-dimensional time-space domain. The method of time inclination is utilized to account for unequal pitchwise periodicity by distributing time co-ordinates at the grid nodes such that phase lagged boundary conditions can be employed. Calculated results show excellent agreement with the results of a reference solver for the validation test case. Furthermore the method was applied to the simulation of the unsteady flow field in a transonic test turbine stage with a stator/rotor pitch ratio of 1.875. The results were compared with measurements of the unsteady rotor blade pressure and a reference solver calculation where an approximate pitch ratio of 2.0 with a 6.7% scaled rotor geometry was employed. Both computational cases show satisfactory agreement with the experiments for both time averaged pressure distributions and pressure perturbation amplitudes.

A comparison of finite volume method and sharp model for two dimensional saltwater intrusion modeling

2014 ◽  
Vol 41 (3) ◽  
pp. 191-196 ◽  
Author(s):  
Boutaina Bouzouf ◽  
Zhi Chen
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Saltwater Intrusion ◽  
Mathematical Formulation ◽  
Test Case ◽  
Two Phase ◽  
Closed Form Solutions ◽  
Analytical Test ◽  
Mathematical Equations ◽  
Volume Method

Seawater intrusion in coastal aquifers is a 3-D phenomenon. However, 3-D regional aquifer models are often limited by insufficient geological and hydrological data, the large horizontal to vertical scales ratio, and by numerical constraints. A mathematical formulation and numerical implementation of the model for saltwater intrusion problems are presented in this paper. The mathematical model is based on assumption of two-phase flow between saltwater and freshwater and Dupuit approximation. Finite volume method is used as the numerical method in non-structured grids to have flexibility upon complex configuration domain and was compared to sharp model that uses finite difference method. Both models are based on the same governing mathematical equations. Finite volume method was validated using analytical test case studies with known closed form solutions, and the results showed good agreement. Both models have then been applied to the case of saltwater intrusion into a real study case. The comparison between both methods indicates that the finite volume method provides predictions closer to the observed results.

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