A MODIFIED FLOW ORIENTATION SCHEME COUPLED WITH A ROBUST MPFA-DIAMOND FOR THE SOLUTION OF TWO-PHASE FLOW IN HIGHLY ANISOTROPIC PETROLEUM RESERVOIRS
DOI:
https://doi.org/10.26512/ripe.v2i21.21705Palavras-chave:
Oil and Water displacements. Anisotropic Porous Media. MPFA-D. MLP. Modified Flow Oriented Scheme.Resumo
In this paperwe simulate two-phase flow in anisotropic petroleum reservoirs. The IMPES procedure is used to solve the coupling between pressure and saturation equations. The pressure equation is discretized by a robust Multipoint Flux Approximation Method with a Diamond-type support. This formulation is capable of reproducing piecewise linear solutions exactly and deals with anisotropic media. To solve the saturation equation a Modified Flow Oriented Scheme (M-FOS) is proposed. This alternative computes the multidimensional numerical fluxes using higher order accuracy in space. This formulation explicitly takes into account the angular distortion of the computational mesh by means of an adaptive weight that tunes the multidimensional character of the formulation according to the grid distortion. A recently devised Multidimensional Limiting Process is adopted in this paper to control the spurious oscillations in higher order approximation. This strategy guarantees monotone solutions and can be used with any polygonal mesh. Finally, an efficient entropy fix strategy, originally proposed in magneto-dynamics context, is also employed in order to produce convergent solutions. The performance of this set of numerical schemes is verified by solving some relevant benchmark problems, where we observe that the Grid Orientation Effects are clearly diminished by using this M-FOS framework.
Downloads
Referências
AZIZ, K. and SETTARI, A. 1979. Petroleum Reservoir Simulation Applied. Science
Publishers Ltd., London, England.
BASTIAN, P. 2002. Higher order discontinuous Galerkin methods for flow and transport in
porous media. In E. Bänsch, editor, Challenges in Scientific Computing ”“ CISC 2002, volume
of Lecture Notes in Computational Science and Engineering, pages 1”“22.
COLELLA, P. 1990. Multidimensional upwind methods for hyperbolic conservation laws
Journal of Computational Physics; 87: 171”“200.
CONTRERAS, F. R. L.; LYRA, P. R. M.; SOUZA, M. R. A.; CARVALHO, D. K. E. 2016.
A cell-centered multipoint flux approximation method with a diamond stencil coupled with a
higher order finite volume method for the simulation of oil”“water displacements in
heterogeneous and anisotropic petroleum reservoirs; Computers & Fluids, 127, 1-16.
DELIS, A. I.; NIKOLOS, I. K. 2012. A Novel Multidimensional Solution Reconstruction and
EdgeBased Limiting Procedure For Unstructured CellCentered Finite Volumes with
Application to Shallow Water Dynamics. Intern. Journ. Num. Methods Fluids; 71: 584-633.
GAO, Z. M.; WU, J. M. 2010. A Linearity-Preserving Cell-Centered Scheme for the
Heterogeneous and Anisotropic Diffusion Equations on General Meshes. International
Journal for Numerical Methods in Fluids; 67: 2157-2183.
GOOCH, O. 1997. Quasi-ENO Schemes for Unstructured Meshes Based on Unlimited Data-
Dependent Least-Squares Reconstruction, J. of Computational Physics; 133: 6”“17.
HERMELINE, F. 2007. Approximation of 2-D and 3-D diffusion operators with variable full
tensor coefficients on arbitrary meshes. Computer meth. App. mech eng; 196: 2497-2526.
HURTADO, F. S. V., MALISKA, A. F., da SILVA, A. F., CORDAZZO, J. 2007. A
Quadrilateral Element-Based Finite-Volume Formulation for the Simulation of Complex
Reservoir. In: SPE paper 107444-MS presented at the SPE Latin American and Caribbean
Petroleum Engineering Conference held in Buenos Aires, Argentina; 15-18.
KOZDON, J. E; MALLISON, B. T.; GERITSEN, G. T. 2011. Multidimensional Upstream
Weighting for Multiphase Transport in Porous Media. Comp. Geoscience; 15: 399”“419.
LAMINE; EDWARDS, M. 2010. Multidimensional Convection Schemes for Flow in Porous
Media on Structured and Unstructured Quadrilateral Grids. J. C. A. Math; 234: 2106-2117.
LAMINE, S.; EDWARDS, M. 2013. Higher Order Cell-Based Multidimensional Upwind
Schemes for Flow in Porous Media on Unstructured Grids. C. M. A. M. Eng.; 259: 103-122.
LEVEQUE, R. J. 2002. Finite Volume Methods for Hyperbolic Problems. Cambridge
University Press-London.
PARK, J. S.; YOON, S. H.; KIM, C. 2010. Multi-Dimensional Limiting Process for
Hyperbolic Conservation Laws on Unstructured Grids. J. of Comput. Physics; 229: 788”“812.
SCHNEIDER, G. E. e RAW, M. J. 1986. A Skewed, Positive Influence Coefficient
Upwinding Procedure for Control-Volume-Based Finite-Element Convection-Diffusion
Computation. Numerical Heat Transfer; 9: 1-26.
SERNA, S. 2009. A Characteristic-Based Nonconvex Entropy-Fix Upwind Scheme for the
Ideal Magnetohydrodynamic Equations. Journal of Computational Physics; 228: 4232-4247.
SHU, C. W.; OSHER, S. 1989. Efficient Implementation Of Essentially Non-Oscillatory
Shock-Capturing Schemes 2, Journal of Computational Physics; 83: 32”“78.
SOUZA, M. 2015. Simulação Numérica de Escoamento Bifásico em Reservatórios de
Petróleo Heterogêneos e Anisotrópicos Utilizando um Método de Volumes Finitos
“Verdadeiramente” Multidimensional com Aproximação de Alta Ordem. Tese de doutorado.
UFPE, Recife.
TRAN, D.; MASSON, C.; SMAÃLI, A. 2006. A stable secondorder massweighted upwind
scheme for unstructured meshes. Int. journal for numerical methods in fluids; 51: 749-771.
VAN ALBADA, G. D.; VAN LEER, B. ROBERTS Jr, W. W. 1982. A Comparative Study
of Computational Methods in Cosmic gas Dynamics. Astron. and Astrophysics; 32: 76-84.
VAN LEER, B. 1979. Towards the ultimate conservative difference scheme V: A Secondorder
sequel to Godunov’s method, Journal of Computational Physics.; 32: 101.
VENKATAKRISHNAN, V. 1995. Convergence to Steady-State Solutions of the Euler
Equations on Unstruckred Grids with Limiters. J. Computational Physics. 118: 120-130.
Downloads
Publicado
Como Citar
Edição
Seção
Licença
Autores que publicam nesta revista concordam com os seguintes termos:
Autores mantém os direitos autorais e concedem à revista o direito de primeira publicação, sendo o trabalho simultaneamente licenciado sob a Creative Commons Attribution License o que permite o compartilhamento do trabalho com reconhecimento da autoria do trabalho e publicação inicial nesta revista.
Autores têm autorização para assumir contratos adicionais separadamente, para distribuição não-exclusiva da versão do trabalho publicada nesta revista (ex: publicar em repositório institucional ou como capítulo de livro), com reconhecimento de autoria e publicação inicial nesta revista.
Autores têm permissão e são estimulados a publicar e distribuir seu trabalho online (ex: em repositórios institucionais ou na sua página pessoal) a qualquer ponto antes ou durante o processo editorial, já que isso pode gerar alterações produtivas, bem como aumentar o impacto e a citação do trabalho publicado.
