A Mimetic Finite-Volume-Discretization Operator for Reservoir Simulation

Alpak, Faruk O.

doi:10.2118/106445-ms

Cited by 4 publications

(4 citation statements)

References 49 publications

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“…Focussing on generalized diffusion equations we highlight [13, 69, 74, 76-78, 102, 107, 108, 111, 113, 114] for a finitedifference/finite-volume setting, [23,[36][37][38] for polyhedral discretizations, and [19,21,96,106,121] for a finite element/mixed finite element setting. For applications to Darcy flow equations and reservoir modelling see for example [1,6,7,55,73,83,89].…”

Section: Overview Of Mimetic Discretizationsmentioning

confidence: 99%

“…In finite element methods the direct finite element formulation for the anisotropic diffusion problem is given by: For (K∇p, n n n) = 0 along Γ u and f ∈ H −1 (Ω ), find p ∈ H 1 0,Γ p (Ω ) such that 6 Note that the set of polynomials {h i }, i = 0, . .…”

Section: Expanding P (Direct Formulation)mentioning

confidence: 99%

“…Let f (ξ ) be expanded in terms Lagrange polynomials as in (17), then the derivative 6 of f is given by, [61,82,96]…”

Section: One Dimensional Spectral Basis Functionsmentioning

confidence: 99%

“…The points in the grid shown in Figure1are also 'oriented', in the sense that when we 'move into a point following the integration direction' we assign a positive value and when we 'leave a point' we assign a negative value. That is why we have plus P(B) and minus P(A) in(6). This is just a convention.…”

mentioning

confidence: 99%

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Mimetic Spectral Element Method for Anisotropic Diffusion

Gerritsma

Palha

Jain

et al. 2018

Numerical Methods for PDEs

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This paper addresses the topological structure of steady, anisotropic, inhomogeneous diffusion problems. Two discrete formulations: a) mixed and b) direct formulations are discussed. Differential operators are represented by sparse incidence matrices, while weighted mass matrices play the role of metric-dependent Hodge matrices. The resulting mixed formulations are point-wise divergence-free if the right hand side function f = 0. The method is inf-sup stable and displays optimal convergence on orthogonal and non-affine grids.

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Section: Overview Of Mimetic Discretizationsmentioning

confidence: 99%

Section: Expanding P (Direct Formulation)mentioning

confidence: 99%

“…Let f (ξ ) be expanded in terms Lagrange polynomials as in (17), then the derivative 6 of f is given by, [61,82,96]…”

Section: One Dimensional Spectral Basis Functionsmentioning

confidence: 99%

mentioning

confidence: 99%

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Mimetic Spectral Element Method for Anisotropic Diffusion

Gerritsma

Palha

Jain

et al. 2018

Numerical Methods for PDEs

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Semianalytical Representation of Wells and Near-Well Flow Convergence in Numerical Reservoir Simulation

Kurtoglu

Medeiros

Özkan

et al. 2008

SPE Annual Technical Conference and Exhibition

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This paper presents a new approach to compute wellbore pressures and near-well flow convergence in numerical reservoir simulation. In this semi-analytical approach, the solution of the flow problem in porous media is related to the boundaryelement formulations. To account for reservoir heterogeneity, solution domain is subdivided into internally homogeneous reservoir blocks, which may include wells, and the solutions are coupled with the requirement of flux and pressure continuity at the block boundaries. Knowing the boundary values of the solution domain, this formulation solves for the flux at the block boundaries and flux and pressure on wells. Knowing the flux boundary condition, this formulation also enables the solution of pressure at any point within the reservoir blocks. This feature of the solution serves the purposes of multiscale simulation of flow and transport in porous media. Although the solution can be applied on field scale, this paper focuses on the applications to near-well flow convergence region. When the solution is used for the near-well flow-convergence region, the required boundary values of the solution domain can be obtained from any standard solution techniques, including finitedifference simulation, by accounting for wells as source terms.

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A Strategic Approach for the Development of Next Generation Reservoir Simulators

Jha

Bansal

Sangwai

2012

SPE Oil and Gas India Conference and Exhibition

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In present scenario the applications of numerical reservoir simulators are very wide and extensive. Simulation is the only tool to describe quantitatively multi-phase flow in a heterogeneous reservoir. But, nowadays most of the users of reservoir simulators are becoming the hostages of computer-generated results, even though, many incorrect analysis techniques, mathematical models and computational approaches are used extensively in reservoir simulation study. This paper revealing such incorrect but commonly accepted approaches presents alternate and correct mathematical approaches for the advancement of simulators. First of all authors take privilege of their previous work where, through wide literature survey, various examples (e.g. fracture and shale flow solutions where fine mesh discretization technique may lead to erroneous results due to singular velocities at the fracture and shale tips, fracture flow modeling using discrete point source which was replaced by continuous line source, etc.) of incorrect models were shown and proved to be the source of erroneous results and numerical instability [5]. After revealing such incorrect models, the solutions for eliminating this incorrect techniques and models are presented and proved numerically and analytically. Mathematical techniques and formulations used in this paper are finite element method, finite volume method, advanced conformal mapping, streamfunction and streamline tracing, modern singular integral equation approaches, moving boundary value problems technique, artificial viscosity analysis, generalization of coordinate system, curvilinear grid generation, integral transformations, etc., which stabilize the mentioned simulation models. The recent technological innovations in drilling and production and complicated reservoir studies have challenged the existing simulators so is the key target of this paper is to move in direction of the next generation simulators that can solve the various mentioned inaccuracies. Various accurate mesh generation algorithms are anticipated to develop through this paper to revolutionize the modern reservoir simulation study.

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A Mimetic Finite-Volume-Discretization Operator for Reservoir Simulation

Cited by 4 publications

References 49 publications

Mimetic Spectral Element Method for Anisotropic Diffusion

Mimetic Spectral Element Method for Anisotropic Diffusion

Semianalytical Representation of Wells and Near-Well Flow Convergence in Numerical Reservoir Simulation

A Strategic Approach for the Development of Next Generation Reservoir Simulators

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