The effects of control-volume distributed multi-point flux approximation (CVD-MPFA) on upscaling-A study using the CVD-MPFA schemes

2010 ◽  
Vol 68 (1) ◽  
pp. 18-35
Author(s):  
Mayur Pal ◽  
Michael G. Edwards
Keyword(s):  
Control Volume ◽  
Flux Approximation

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.

Acute PEBI Grid Generation for Reservoir Geometries

2021 ◽  
Author(s):  
Shahid Manzooor ◽  
Michael G. Edwards ◽  
Ali H. Dogru
Keyword(s):  
Unstructured Grid ◽  
Control Volume ◽  
Control Point ◽  
Vertical Direction ◽  
Grid Generation ◽  
Geometric Constraints ◽  
Reconstruction Technique ◽  
Delaunay Triangulations ◽  
Flux Approximation ◽  
Mesh Reconstruction

Abstract An unstructured grid generation method is presented that automates control-volume boundary alignment to geological objects and control point alignment to complex wells. The grid generation method is coupled with an iterative acute mesh reconstruction technique, to construct essentially acute triangulations, while satisfying quite general geometric constraints. For well aligned grids control points are constrained to the well trajectory and protection circles are used, whereas for boundary aligned grids halo construction is performed. Unstructured Delaunay triangulations (DT) have the desirable locally orthogonal perpendicular bisectional (PEBI) property, required by the industry standard two-point flux approximation for consistency for isotropic fields. The PEBI property can only be exploited if such grids are comprised of acute simplexes, with circumcentres located inside their respective elements. The method presented enables acute DT layered mesh generation while honoring internal boundaries and wells in a two dimensional space. A dual (Voronoi) grid derived from a feature honored simplicial mesh is then projected in the vertical direction generating 2.5D PEBI grids. Effectiveness of the method to construct acute PEBI grids honoring geological objects and complex wells is demonstrated by meshing representative reservoir geometries. Numerical results are presented that verify consistency of the two-point flux on the resulting boundary-aligned acute PEBI grids. Development of an unstructured grid generation method which 1) Automates interior boundary alignment, 2) Honors features with respect to control point and/or control volume, and 3) Generates acute PEBI grids for reservoir geometries is presented. A unique workflow is presented to generate boundary aligned acute PEBI grids for complex geometries. Development of boundary aligned grids that honor both geological objects and multilateral complex wells, together with mesh reconstruction to ensure circumcenter containment is presented. Further, 3D PEBI grid generation method which can limit refinement to well perforations and geological objects is also described.

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.

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.

Control-Volume Distributed Multi-Point Flux Approximation Coupled with a Lower-Dimensional Fracture Model

2015 ◽  
Author(s):  
R. Ahmed ◽  
M.G. Edwards ◽  
S. Lamine ◽  
B.A.H. Huisman ◽  
M. Pal
Keyword(s):  
Control Volume ◽  
Fracture Model ◽  
Flux Approximation ◽  
Lower Dimensional

scholarly journals Control-volume distributed multi-point flux approximation coupled with a lower-dimensional fracture model

2015 ◽  
Vol 284 ◽  
pp. 462-489 ◽  
Author(s):  
R. Ahmed ◽  
M.G. Edwards ◽  
S. Lamine ◽  
B.A.H. Huisman ◽  
M. Pal
Keyword(s):  
Control Volume ◽  
Fracture Model ◽  
Flux Approximation ◽  
Lower Dimensional

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.

Sign in / Sign up

Export Citation Format

Share Document