A Globally Coupled Pressure Method for the Discretization of the Tensor-Pressure Equation on Non-K-orthogonal Grids

SPE Journal ◽  
2016 ◽  
Vol 22 (02) ◽  
pp. 679-698 ◽  
Author(s):  
Wenjuan Zhang ◽  
Mohammed Al Kobaisi
Keyword(s):  
Boundary Conditions ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Unstructured Grids ◽  
Control Volume ◽  
Dirichlet Boundary Conditions ◽  
Complex Permeability ◽  
Pressure Equation ◽  
Flow Equations ◽  
Flux Approximation

Summary Complex permeability tensors together with general nonorthogonal and unstructured grids pose great challenges to reservoir simulation. The widely used two-point flux approximation (TPFA) is inadequate for a rigorous discretization of the flow equations on such challenging grids. Multipoint flux approximation (MPFA) methods have been proposed to meet the challenges and are currently being deployed in next-generation simulators. In this work, we propose an alternative flux-continuous cell-centered finite-volume method called the globally coupled pressure (GCP) method to discretize the pressure equation on general grids with full permeability tensors. To accurately construct fluxes through control-volume interfaces, pressure at the centroid of all interfaces is introduced as auxiliary unknowns. Flux continuity across each interface gives one equation. Assembling all the flux-continuity equations together gives a system of linear equations that can be solved simultaneously for all the auxiliary unknowns. Flux across control-volume interfaces can then be approximated with the pressure values at control-volume centers only. The fundamental difference between the GCP method and MPFA methods is that, in the latter, auxiliary unknowns are locally coupled within an interaction region and then eliminated in a local stencil by imposing flux-continuity conditions, whereas in the former, all the auxiliary unknowns are globally coupled and can only be eliminated in a global stencil. Consequently, control volumes in our GCP method are directly associated with the edges of the original grid and not by means of a dual grid overlaid and allied with the centers of the grid. Two variants of the GCP method are presented here, and extensive numerical experiments are conducted to test the performance of the GCP methods. The results show that both variants of our GCP method are in good agreement with the classical MPFA-O method on non-K-orthogonal grids for less-challenging problems. Convergence studies reveal that the first variant of our GCP method has slower convergence rates than the MPFA-O method for some problems. However, the second variant of GCP has comparable, and in some cases, better convergence properties compared with the MPFA-O method. With numerical experiments, we further investigate monotonicity properties of our GCP method on highly anisotropic media. For Dirichlet boundary conditions, our GCP methods also suffer from nonphysical oscillations, with some degrees of improvement over the MFPA-O method. When no-flow boundary conditions are used, our GCP method is much more robust and does not produce spurious boundary extrema as MPFA methods do. Finally, we extend our GCP method to fully unstructured grids, and the results show that it is also more robust than the MPFA-O method on unstructured grids.

Tracing Streamlines on Unstructured Grids From Finite Volume Discretizations

SPE Journal ◽  
2008 ◽  
Vol 13 (04) ◽  
pp. 423-431 ◽  
Author(s):  
Sebastien F. Matringe ◽  
Ruben Juanes ◽  
Hamdi A. Tchelepi
Keyword(s):  
Finite Element ◽  
Velocity Field ◽  
Finite Difference ◽  
Finite Volume ◽  
Unstructured Grids ◽  
Control Volume ◽  
Next Generation ◽  
Flux Approximation ◽  
Streamline Tracing ◽  
Tracing Method

Summary The accuracy of streamline reservoir simulations depends strongly on the quality of the velocity field and the accuracy of the streamline tracing method. For problems described on complex grids (e.g., corner-point geometry or fully unstructured grids) with full-tensor permeabilities, advanced discretization methods, such as the family of multipoint flux approximation (MPFA) schemes, are necessary to obtain an accurate representation of the fluxes across control volume faces. These fluxes are then interpolated to define the velocity field within each control volume, which is then used to trace the streamlines. Existing methods for the interpolation of the velocity field and integration of the streamlines do not preserve the accuracy of the fluxes computed by MPFA discretizations. Here we propose a method for the reconstruction of the velocity field with high-order accuracy from the fluxes provided by MPFA discretization schemes. This reconstruction relies on a correspondence between the MPFA fluxes and the degrees of freedom of a mixed finite-element method (MFEM) based on the first-order Brezzi-Douglas-Marini space. This link between the finite-volume and finite-element methods allows the use of flux reconstruction and streamline tracing techniques developed previously by the authors for mixed finite elements. After a detailed description of our streamline tracing method, we study its accuracy and efficiency using challenging test cases. Introduction The next-generation reservoir simulators will be unstructured. Several research groups throughout the industry are now developing a new breed of reservoir simulators to replace the current industry standards. One of the main advances offered by these next generation simulators is their ability to support unstructured or, at least, strongly distorted grids populated with full-tensor permeabilities. The constant evolution of reservoir modeling techniques provides an increasingly realistic description of the geological features of petroleum reservoirs. To discretize the complex geometries of geocellular models, unstructured grids seem to be a natural choice. Their inherent flexibility permits the precise description of faults, flow barriers, trapping structures, etc. Obtaining a similar accuracy with more traditional structured grids, if at all possible, would require an overwhelming number of gridblocks. However, the added flexibility of unstructured grids comes with a cost. To accurately resolve the full-tensor permeabilities or the grid distortion, a two-point flux approximation (TPFA) approach, such as that of classical finite difference (FD) methods is not sufficient. The size of the discretization stencil needs to be increased to include more pressure points in the computation of the fluxes through control volume edges. To this end, multipoint flux approximation (MPFA) methods have been developed and applied quite successfully (Aavatsmark et al. 1996; Verma and Aziz 1997; Edwards and Rogers 1998; Aavatsmark et al. 1998b; Aavatsmark et al. 1998c; Aavatsmark et al. 1998a; Edwards 2002; Lee et al. 2002a; Lee et al. 2002b). In this paper, we interpret finite volume discretizations as MFEM for which streamline tracing methods have already been developed (Matringe et al. 2006; Matringe et al. 2007b; Juanes and Matringe In Press). This approach provides a natural way of reconstructing velocity fields from TPFA or MPFA fluxes. For finite difference or TPFA discretizations, the proposed interpretation provides mathematical justification for Pollock's method (Pollock 1988) and some of its extensions to distorted grids (Cordes and Kinzelbach 1992; Prévost et al. 2002; Hægland et al. 2007; Jimenez et al. 2007). For MPFA, our approach provides a high-order streamline tracing algorithm that takes full advantage of the flux information from the MPFA discretization.

Basic Boundary Conditions for Incompressible and Compressible Flows Using Unstructured Meshes

2003 ◽  
Author(s):  
Branislav Basara
Keyword(s):  
Convergence Rate ◽  
Boundary Conditions ◽  
Compressible Flows ◽  
Unstructured Grids ◽  
Control Volume ◽  
Unstructured Meshes ◽  
Control Volume Method ◽  
Polyhedral Cells ◽  
Volume Method ◽  
Basic Boundary

The paper compiles the basic and frequently used boundary conditions in CFD calculations. Regardless of the type of boundary conditions, Dirichlet or Neumman, there are very important differences in the implementation procedure depending on the solved equations as well as on variables which are updated on the boundaries. Boundary conditions in the frame of the control volume method presented here, are adopted for the unstructured grids consisting of arbitrary polyhedral cells. There are no limitations on the employment of boundary conditions regarding mesh type. Some special treatments to improve results and the convergence rate are proposed. The emphasis is on the wall and the pressure boundaries.

A Multiscale Control Volume framework using the Multiscale Restriction Smooth Basis and a non-orthodox Multi-Point Flux Approximation for the simulation of two-phase flows on truly unstructured grids

2020 ◽  
Vol 188 ◽  
pp. 106851
Author(s):  
Artur Castiel Reis de Souza ◽  
Lorena Monteiro Cavalcanti Barbosa ◽  
Fernando Raul Licapa Contreras ◽  
Paulo Roberto Maciel Lyra ◽  
Darlan Karlo Elisiário de Carvalho
Keyword(s):  
Unstructured Grids ◽  
Control Volume ◽  
Two Phase Flows ◽  
Two Phase ◽  
Flux Approximation ◽  
Smooth Basis

Finite Volume Methods with Multi-Point Flux Approximation with Unstructured Grids for Diffusion Problems

2010 ◽  
Vol 297-301 ◽  
pp. 670-675
Author(s):  
Jaime Ambrus ◽  
C.R. Maliska ◽  
F.S.V. Hurtado ◽  
A.F.C. da Silva
Keyword(s):  
Finite Volume ◽  
Unstructured Grids ◽  
Control Volume ◽  
Finite Volume Methods ◽  
Porous Media Flow ◽  
Diffusion Operator ◽  
Flux Approximation ◽  
Grid Points ◽  
Volume Method ◽  
Diffusion Problems

This paper addresses the key issue of calculating fluxes at the control-volume interfaces when finite-volumes are employed for the solution of partial differential equations. This calculation becomes even more significant when unstructured grids are used, since the flux approximation involving only two grid points is no longer correct. Two finite volume methods with the ability in dealing with unstructured grids, the EbFVM-Element-based Finite Volume Method and the MPFA-Multi-Point Flux Approximation are presented, pointing out the way the fluxes are numerically evaluated. The methods are applied to a porous media flow with full permeability tensors and non-orthogonal grids and the results are compared with analytical solutions. The results can be extended to any diffusion operator, like heat and mass diffusion, in anisotropic media.

A Tracing Algorithm for Flow Diagnostics on Fully Unstructured Grids With Multipoint Flux Approximation

SPE Journal ◽  
2017 ◽  
Vol 22 (06) ◽  
pp. 1946-1962 ◽  
Author(s):  
Zhao Zhang ◽  
Sebastian Geiger ◽  
Margaret Rood ◽  
Carl Jacquemyn ◽  
Matthew Jackson ◽  
...  
Keyword(s):  
Real Time ◽  
Unstructured Grids ◽  
Hyperbolic Equations ◽  
Control Volume ◽  
Processing Unit ◽  
Flow Diagnostics ◽  
Central Processing ◽  
Reservoir Models ◽  
Flux Approximation ◽  
Tracing Algorithm

Summary Flow diagnostics is a common way to rank and cluster ensembles of reservoir models depending on their approximate dynamic behavior before beginning full-physics reservoir simulation. Traditionally, they have been performed on corner-point grids inherent to geocellular models. The rapid-reservoir-modeling (RRM) concept aims at fast and intuitive prototyping of geologically realistic reservoir models. In RRM, complex reservoir heterogeneities are modeled as discrete volumes bounded by surfaces that are sketched in real time. The resulting reservoir models are discretized by use of fully unstructured tetrahedral meshes where the grid conforms to the reservoir geometry, hence preserving the original geological structures that have been modeled. This paper presents a computationally efficient work flow for flow diagnostics on fully unstructured grids. The control-volume finite-element method (CVFEM) is used to solve the elliptic pressure equation. The flux field is a multipoint flux approximation (MPFA). A new tracing algorithm is developed on a reduced monotone acyclic graph for the hyperbolic transport equations of time of flight (TOF) and tracer distributions. An optimal reordering technique is used to deal with each control volume locally such that the hyperbolic equations can be computed in an efficient node-by-node manner. This reordering algorithm scales linearly with the number of unknowns. The results of these computations allow us to estimate swept-reservoir volumes, injector/producer pairs, well-allocation factors, flow capacity, storage capacity, and dynamic Lorenz coefficients, which all help approximate the dynamic reservoir behavior. The total central-processing-unit (CPU) time, including grid generation and flow diagnostics, is typically a few seconds for meshes with O (100,000) unknowns. Such fast calculations provide, for the first time, real-time feedback in the dynamic reservoir behavior while models are prototyped.

Implementation of a Vertex-Centered Method Inside an Industrial Reservoir Simulator: Practical Issues and Comprehensive Comparison With Corner-Point Grids and Perpendicular-Bisector-Grid Models on a Field Case

SPE Journal ◽  
2017 ◽  
Vol 22 (02) ◽  
pp. 660-678 ◽  
Author(s):  
Pierre Samier ◽  
Roland Masson
Keyword(s):  
Corner Point ◽  
Unstructured Grids ◽  
Control Volume ◽  
Simulation Models ◽  
Work Flow ◽  
Next Generation ◽  
Large Reservoir ◽  
Effective Work ◽  
Flux Approximation ◽  
Reservoir Simulator

Summary Corner-point grids (CPGs) and pillar-based unstructured grids do not provide an effective work flow for translating Earth models into simulation models. Such a work flow requires grids that allow an accurate representation of the near-well flow, preserve geological accuracy, and offer flexible resolution control. Hence, a 3D unstructured approach is required. Significant work has been performed for generating unstructured grids, and modeling hydraulic-fracture flow for gas-shale simulation has given a new impulse for unstructured gridding. Recent methods such as vertex-approximate gradient (VAG) or more-mature ones such as multipoint flux approximation (MPFA) provide a numerical scheme dependent on multipoint stencil more physical than two-point flux-approximation (TPFA) methods. This paper presents the implementation of VAG and MPFA schemes inside a next-generation reservoir simulator starting from a source code calculating multipoint flux nonneighbor connections (NNCs) for any polygonal-shaped control volume. The unstructured-scheme approach has been developed as an in-house extension to a next-generation multicompany collaborative reservoir simulator that is designed for handling unstructured grids. The main issues addressed are the introduction of vertices unknowns among the usual cell-center variables, the assignment of vertices properties in the reservoir-simulator model, and the link with the well model. Incidentally, the definition of an exchange format to describe the unstructured geometry (vertices, edges, faces, control volumes) on a large reservoir-simulation model is proposed. Three simulation examples are presented, and we compare results, accuracy, and performance of multipoint-scheme methods such as VAG and MPFA on unstructured grids. Results are compared with TPFA methods on refined structured CPGs and TPFA methods on unstructured Voronoi grids. The two first test cases are academic models, and the third one is a field model.

A New Mixed Finite Element and its Related Finite Volume Discretization on General Hexahedral Grids

2008 ◽  
Author(s):  
Se´bastien F. Matringe ◽  
Ruben Juanes ◽  
Hamdi A. Tchelepi
Keyword(s):  
Finite Element ◽  
Velocity Field ◽  
Finite Volume ◽  
Control Volume ◽  
Mixed Finite Element ◽  
Quadrature Rule ◽  
Mathematical Framework ◽  
Flow Equations ◽  
Hexahedral Elements ◽  
Flux Approximation

Modern reservoir simulation grids are generally composed of distorted hexahedral elements populated with heterogeneous and possibly full-tensor coefficients. The numerical discretization of the reservoir flow equations on such grids is a challenging problem. Finite volume methods based on a two-point flux approximation (TPFA) do not properly account for grid distortion or permeability anisotropy that is misaligned with the grid. Multipoint flux approximation (MPFA) methods have been developed to overcome these shortcomings. Although implemented and used in virtually every commercial reservoir simulator, a proof of convergence for MPFA methods on three-dimensional hexahedral grids has remained elusive. Here, we present a link between MPFA and a new mixed finite element methods (MFEM) on hexahedral grids, which provides a powerful mathematical framework for the analysis of MPFA. First, we introduce a new mixed finite element on 3D hexahedra. The new element defines a velocity field with bilinear normal components through element faces. Thus, the new velocity field is defined by four degrees of freedom per face, which are the normal components of the velocity field at the vertices of each face. The new space is compatible with a piecewise constant pressure discretization and yields a convergent discretization. The application of a vertex-based quadrature rule reduces the new mixed finite element method to a multipoint flux control volume method. For Cartesian grids, this is in fact the classical MPFA O-method. This provides for the first time a direct link between MFEM and MPFA on hexahedral grids, which we use to establish convergence of MPFA for 3D rectangular grids.

scholarly journals General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Eva Llabrés
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Spin Connection ◽  
Departure Point ◽  
Boundary Stress ◽  
Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

scholarly journals Charge algebra in Al(A)dSn spacetimes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Adrien Fiorucci ◽  
Romain Ruzziconi
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Boundary Structure ◽  
Asymptotic Symmetry ◽  
Renormalization Procedure ◽  
Field Dependent ◽  
Odd Dimensions

Abstract The gravitational charge algebra of generic asymptotically locally (A)dS spacetimes is derived in n dimensions. The analysis is performed in the Starobinsky/Fefferman-Graham gauge, without assuming any further boundary condition than the minimal falloffs for conformal compactification. In particular, the boundary structure is allowed to fluctuate and plays the role of source yielding some symplectic flux at the boundary. Using the holographic renormalization procedure, the divergences are removed from the symplectic structure, which leads to finite expressions. The charges associated with boundary diffeomorphisms are generically non-vanishing, non-integrable and not conserved, while those associated with boundary Weyl rescalings are non-vanishing only in odd dimensions due to the presence of Weyl anomalies in the dual theory. The charge algebra exhibits a field-dependent 2-cocycle in odd dimensions. When the general framework is restricted to three-dimensional asymptotically AdS spacetimes with Dirichlet boundary conditions, the 2-cocycle reduces to the Brown-Henneaux central extension. The analysis is also specified to leaky boundary conditions in asymptotically locally (A)dS spacetimes that lead to the Λ-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in n dimensions.

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