Convergence of nonlinear numerical approximations for an elliptic linear problem with irregular data

The accuracy of the Darcy velocity, flux, and stream function computed from lowest‐order, triangle‐based, control volume and mixed finite element approximations to the two‐dimensional pressure equation is considered. The control volume finite element method, similar to integrated finite difference methods and analogous to the interpolation of Galerkin finite element results over “control volumes,” is shown to yield a conservative velocity field and smooth streamlines. The streamlines and fluxes through the system computed with the control volume finite element approach are compared to those computed from the mixed finite element method, which approximates the pressure and velocity variables separately. It is shown that for systems with only moderate degrees of heterogeneity, the control volume finite element method is the more computationally efficient alternative; i.e., it provides more accurate flow results for a given number of unknowns. For more variable or discontinuous permeability fields, by contrast, such as sand/shale systems, the mixed finite element method is shown to approximate flow variables more accurately and more realistically than the control volume method with the same number of unknowns.

<p>Control volume finite element (CVFE) methods provide flexible framework for modelling flow and transport in complex geological features such as faults and fractures. They combine the finite element method that captures complex flow characteristics with the control volume approach known for its stability and mass conservative properties. The general approach of CVFE methods maps the physical properties of the system onto the element mesh (element-wise properties) while the node centred control volumes span element boundaries. In the presence of abrupt material interfaces between elements which are often encountered in fractured models, the method suffers from non-physical leakage in the saturation solution as the result of control volume discretization used for advancing the transport solution. In this work, we present a discontinuous pressure formulation based on control volume finite element (CVFE) method for modelling coupled flow and transport in highly heterogeneous porous media. We propose the element pair P<sub>(1,DG)</sub>-P<sub>(0,DG)</sub>, a discontinuous first order velocity approximation combined with a discontinuous low order pressure approximation. The approach circumvents the non-physical leakage issue by incorporating a discontinuous, element-based approximation of pressure. Hence, the resultant control volume representation directly maps to the element mesh as well as to the projected physical properties of the system. Due to the low order nature of the formulation, low computational requirement per element and the improved control volume discretization, the presented formulation is proven more robust and accurate than classical CVFE methods in the presence of highly heterogeneous domains.</p>

In this work the control volume finite element (CVFE) method is used to discretise a two-dimensional non-linear transport model on an unstructured mesh. First and second order temporal weighting, combined with various flux limiting techniques (spatial weighting) are analysed in order to identify the most accurate and efficient numerical scheme. An inexact Newton method is used to resolve the underlying discrete non-linear system. In computing the Newton step the performance of the preconditioned iterative solvers BiCGSTAB and GMRES, in conjunction with a two-node Jacobian approximation is also examined. A linear benchmark problem that admits an analytical solution is used to assess the accuracy and computational efficiency of the numerical model. The results show that the flux limited second order temporal scheme substantially reduce numerical diffusion on relatively coarse meshes. The low temperature wood drying non-linear case study highlights that the first order temporal scheme combined with flux limiting achieves good accuracy on a relatively coarse mesh and improves the overall computational efficiency.

Interface control volume finite element method for modelling multi-phase fluid flow in highly heterogeneous and fractured reservoirs

Unstructured K-orthogonal PEBI grids with permeability defined on triangles (or tetrahedra) have been used successfully by previous authors for mildly anisotropic systems. This paper presents two unstructured K-orthogonal grid systems, in which permeability is defined on cells. The first is the previously mentioned K-orthogonal PEBI grids; the other is the dual of a PEBI grid constructed by aggregating triangles (or tetrahedra), which is termed a composite tetrahedral grid. Such grids, carefully generated, enable the accurate modeling of highly anisotropic and heterogeneous systems. Good K-orthogonal grids for highly anisotropic systems can be generated by transforming the physical space into an isotropic computational space in which an orthogonal grid is generated. The steps involved in generating K-orthogonal grids and their application to reservoir simulation, namely, anisotropy scaling, point distribution, triangulation (or tetrahedralization), triangle aggregation, cell generation, transmissibility calculation, grid smoothing, well connection factors and cell renumbering for linear algebra, are described. This paper also describes how independently generated multiple domains are merged to form a single grid. The paper presents 2D and 3D simulation results for single phase and multi-phase problems in well test and full field situations. The grids are tested under high anisotropy, high mobility ratios, complex geometries and grid orientations, in order to establish the true limitations of K-orthogonal grids. The error due to non-orthogonality is reported for each cell, suggesting regions where multi-point flux approximations may be of advantage. Relative merits of PEBI and composite tetrahedral grids are also discussed. The grids are applicable to multi-layered, multi-phase well test and full field simulations, with full heterogeneity and anisotropy limited to a spatially varying kv/kh ratio. Technical contributions in the paper include, definition and generation of composite tetrahedral grids, the process of generating good K-orthogonal PEBI and composite tetrahedral grids, algorithms for computing volumes, transmissibilities, well connections and cell renumbering for general K-orthogonal grids. Introduction Flow simulations on grids based on triangles have been used by various authors inside and outside the petroleum industry. Winslow used control volumes formed around each node of a triangular grid by joining the edge midpoints to the triangle centroids for solving a 2D magnetostatic problem. This technique was applied to reservoir simulation by Forsyth, and is commonly known as the control volume finite element (CVFE) method. Cottrell et al. used control volumes formed by joining the perpendicular bisectors of triangle edges of a Delaunay triangulation for solving semiconductor device equations. Heinemann et al. applied this technique to reservoir simulation, which is known as the PEBI or the Voronoi method. Further work on the CVFE method was presented by Fung and on the PEBI method by Palagi and Gunasekerat. Both Forsyth and Fung handled heterogeneous problems by defining permeability to be constant over a triangle. Aavatsmark and Verma derived an alternative difference scheme based on the CVFE method in which permeabilities are defined as constant within control volumes. This approach handles boundaries of layers with large permeability differences better than the traditional CVFE method and, as with the traditional method, it leads to a multi-point flow stencil, hence referred to as an MPFA scheme. By contrast, the PEBI method reduces to a two point flow stencil. Heinemann et al. and Amado et al. extended the PEBI method to handle anisotropic heterogeneous systems by defining permeability to be constant within a triangle and by adjusting the angle between triangle edges and cell boundaries. This approach has two problems: firstly, handling layers of contrasting permeabilities is poor, secondly in highly anisotropic systems the angle condition between triangle edges and cell boundaries may become so severe that cells begin to overlap, as shown in Verma. P. 199^

In this paper we systematically reviewthe control volume finite element (CVFE) methods for numericalsolutions of second-order partial differential equations.Their relationships to the finite differenceand standard (Galerkin) finite element methods are considered.Through their relationshipto the finite differences, upstream weightedCVFE methods and the conditions on positivetransmissibilities (positive flux linkages)are studied. Through their relationshipto the standard finite elements,error estimates forthe CVFE are obtained. These estimates arecomparable to those for the standard finite element methodsusing piecewise linear elements. Finally, anapplication to multiphase flows in porousmedia is presented.

Simulation of thermal debinding: effects of mass transport on equivalent stress

A new control volume finite element (CVFE) method is formulated from integral form of conservation laws, i.e. conservation of mass, for overland flows under kinematic wave assumption. With an approximation m v ha = , the system becomes a nonlinear system with unknowns such as flow velocity and flow depth. The differential system of error equations is obtained by linearization of conservation laws in their discrete form for a control volume and an error analysis is obtained using Fourier or von Neumann method. Four nodal quadrilateral or bilinear rectangular shape functions are used to evaluate local and global matrices in the resulting control volume finite element (CVFE) formulation. The nodal amplification factors using coefficient method are compared with those obtained using exact method or eigen-value analysis. The nodal amplification factors show a straight line behavior for all of the wavenumbers for explicit, semi-implicit, and implicit control volume finite element schemes.

Summary Control volume finite element (CVFE) methods are commonly used for modeling flow and transport in geometrically complex porous media domains with unstructured meshes. The CVFE approach is based on the finite element method to approximate the pressure and velocity fields, and uses the finite volume method to model saturation ensuring mass conservation. Control volumes are constructed by spanning element boundaries, leading to an artificial smearing of the numerical solution in the presence of sharp material interfaces. Recently, a CVFE method based on discontinuous pressure was introduced that enabled the construction of discontinuous control volumes, thus preventing control volumes from spanning element boundaries. This modification of the method provides accurate solutions but is computationally very expensive due to the discontinuous approximation which incorporates additional degrees of freedom per element. In this work, we propose using hybrid finite element pressure approximations to capture flow and transport in highly heterogeneous porous media. The CVFE element pair $P_{0,DG}-P_{1,H}$ denotes a constant, element-wise, discontinuous Galerkin velocity vector approximation and a hybrid (continuous/discontinuous) Galerkin first-order pressure scalar approximation of the flow model. The method exploits the efficient continuous CVFE method in most of the model domain while the discontinuous CVFE approach is applied exclusively along material discontinuities. We demonstrate that this hybrid scheme outperforms the classical CVFE continuous approach as well as the discontinuous Galerkin modification by incorporating the best of both approaches. The presented hybrid approach computational requirements are comparable to the continuous approach while the accuracy of the transport solution corresponds to that of the discontinuous pressure method. We validate the presented hybrid approach and discuss the convergence of the method. The effectiveness of the new scheme is demonstrated with several numerical experiments for highly heterogeneous subdomains."

We present a new hybrid pressure formulation applied to the control volume finite element (CVFE) method to model multiphase flow and transport in highly heterogeneous porous media. The formulation effectively captures sharp saturation changes in the presence of discontinuous material properties by employing a discontinuous pressure approximation at material interfaces. The heterogeneous porous medium is divided into sub-domains within which material properties are uniform or smoothly varying. By construction, the resultant control volume dual mesh is restricted within a sub-domain. The artificial mass leakage across material property boundaries observed in classical CVFE methods is therefore circumvented. The approach applies the robust continuous pressure approximation in the rest of the computational domain; the discontinuous approximation is applied only at the sub-domain boundaries. The discontinuous parameters necessary to achieve mass conservative solutions, locally and globally, are described. We demonstrate the accuracy and efficiency of the new approach by comparison with the classical continuous CVFE method on various examples of heterogeneous domains as well as establishing the convergence of the numerical method. The proposed hybrid formulation significantly outperforms the accuracy and efficiency of classical CVFE methods that use the same order of approximation for modeling multiphase flow in heterogeneous porous media.

In this paper, we study the computer simulation of gas cycling in a rich retrograde condensate reservoir. Two prediction cases are studied. The first case is gas cycling with constant sales gas removal, and the second case is cycling with some gas sales deferral to enhance pressure maintenance in the early life of this reservoir. In this problem the great majority of cycling takes place at pressure below the dew point pressure, indicating the need for modeling the compositional three-phase, multicomponent flow in the reservoir. This compositional model consists of Darcy's law for volumetric flow velocities, mass conservation for hydrocarbon components, thermodynamic equilibrium for mass interchange between phases, and an equation of state for saturations. The control volume finite element (CVFE) method on unstructured grids is used to discretize the model governing equations for the first time. Numerical experiments are reported for the benchmark problem of the third comparative solution project (CSP) organized by the society of petroleum engineers (SPE). The PVT (pressure-volume-temperature) data are based on a real fluid analysis.

Accurate determination of fluid flow within heap leaching is crucial for understanding and improving performance of the process. Numerical methods have the potential to assist by modelling the process and studying the transport phenomena within the porous medium. This paper presents an adaptive mesh numerical scheme to solve for unsaturated incompressible flow in porous media with applications to heap leaching. The governing equations are Darcy’s law and the conservation of mass. An implicit pressure explicit saturation method is used to decouple the pressure and saturation equations. Pressure is discretized using a control volume (CV) finite element method, while for saturation a node-centred CV method is employed. The scheme is equipped with dynamic anisotropic mesh adaptivity to update the mesh resolution as the leaching solution infiltrates through the heap. This allows for high-fidelity modelling of multiscale features within the flow. The method is verified against the Buckley–Leverett problem where a quasi-analytical solution is available. It is applied for two-phase flow of air and leaching solution in a simplified two-dimensional heap geometry. We compare the accuracy and CPU efficiency of an adaptive mesh against a static mesh and demonstrate the potential to achieve high spatial accuracy at low computational cost through the use of anisotropic mesh adaptivity.

A streamline upwind control volume finite element method for modeling fluid flow and heat transfer problems

Purpose The purpose of this study is to peruse natural convection in a CuO-water nanofluid-filled complex-shaped enclosure under the influence of a uniform magnetic field by using control volume finite element method. Design/methodology/approach Governing equations formulated in dimensionless stream function, vorticity and temperature variables using the single-phase nanofluid model with the Koo–Kleinstreuer–Li correlation for the effective dynamic viscosity and the effective thermal conductivity have been solved numerically by control volume finite element method. Findings Effects of various pertinent parameters such as Rayleigh number, Hartmann number, volume fraction of nanofluid and shape factor of nanoparticle on the convective heat transfer characteristics are analysed. It was observed that local and average heat transfer rates increase for higher value of Rayleigh number and lower value of Hartmann number. Among various nanoparticle shapes, platelets were found to be best in terms of heat transfer performance. The amount of average Nusselt number reductions was found to be different when nanofluids with different solid particle volume fractions were considered due to thermal and electrical conductivity enhancement of fluid with nanoparticle addition. Originality/value A comprehensive study of the natural convection in a CuO-water nanofluid-filled complex-shaped enclosure under the influence of a uniform magnetic field by using control volume finite element method is addressed.

Chapter 2 - The control volume finite element method: application for magnetohydrodynamic nanofluid hydrothermal behavior