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une « bonne administration de la justice »

a. objectifs de la délégation

2. une « bonne administration de la justice »

Podemos mencionar as seguintes sugestões de trabalhos futuros:

 Melhorar a monotonicidade da matriz de transmissibilidade na malha grossa, re- tirando termos que acarretam oscilações;

 Pesquisar e implementar diferentes condições de contorno para melhorar a acu- rácia do operador de prolongamento;

 Implementar outras formulações MPFA (ex: MPFA-D, MPFA- Enriched,etc) ;

 Fazer um estudo detalhado, sobre a influência das formulações MPFA na matriz de transmissibilidade da malha grossa primal;

 Desenvolver novas formas de correção de fluxo, buscando métodos mais efici- entes;

 Implementar outros suavizadores , entre eles o método Multigrid e avaliar o seu efeito na taxa de convergência.

REFERÊNCIAS

AAVATSMARK, I; BARKVE, T; Bøe Ø; MANNSETH, T.. Discretization on un- structured grids for inhomogeneous, anisotropic media. Part I. Derivation of the methods.

SIAM Journal on Scientific Computing. v. 19, p. 765–781,1998.

ARBOGAST, T., BRYANT, S. L., A two-scale numerical subgrid technique for wa- terflood simulations. Soc. Petrol. Eng. J.. p. 446-457, 2002.

ARBOGAST, T., Implementation of a locally conservative numerical subgrid upscal- ing scheme for two phase Darcy flow. J. Comput. Geosci. v. 6, p. 453-481, 2002.

BERALDO, V. T. Simulação por Linhas de Corrente com Compressibilidade e

Variação Espacial e Dinâmica de Composição de Óleo. Tese de Doutorado. Universidade

estadual de Campinas, 2015.

BLAZEK, J. Computational Fluid Dynamics: Principles and Applications. Elser- vier Science. Oxford. 2001.

CARVALHO, D. K. E., Uma Formulação do Método dos Volumes Finitos em Ma-

lhas Não-estruturadas com Estrutura de Dados por Arestas para a Solução de Escoa- mentos em Meios Porosos. Tese de doutorado. Universidade Federal de Pernambuco, 2005.

CARVALHO, D.K.E; WILLMERSDORF, R.B; LYRA P.R.M. A node-centered finite volume formulation for the solution of two-phase flows in non-homegeneous porous media.

International Journal for Numerical Methods in Fluids. v. 53, p. 1179-1219, 2006.

CHEN, Q; WAN, J; YANG, Y; MIFFLIN, R. T. Enriched Multi-Point Flux Approxi- mation for General Grids. Journal of Computational Physics. v. 227, p. 1701–1721, 2008.

CHEN, Z. M., Hou, T. Y., A mixed multiscale finite element method for elliptic prob- lems with oscillating coefficients. J. Math. Comput. v. 72, p. 541-576, 2003.

CHRISTIE, M. A., & BLUNT, M. J. Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques. Society of Petroleum Engineers. v. 4, p. 308- 317,2001.

CHUEH C.C., SECANELL M., BANGERTH W., & DJILALI N. Multi-level adap- tive simulation of transient two-phase flow in heterogeneous porous media. Computers &

Fluids. v. 39, p. 1585–1596, 2010.

COMPUTER MODELLING GROUP LTD. (CMG), IMEX (RESERVOIR SIMULA- TON SOTWARE), http://www.cmgl.ca/software/imex2014.

CORTINOVIS, D., JENNY, P., Iterative Galerkin-enriched multiscale finite-volume method. J. Comp. Phys. v. 277, p. 248-267, 2014.

CORTIS, A., GALLO, C., SCHER, H., BERKOWITZ, B., Numerical simulation of non-Fickian transport in geological formations with multiple-scale heterogeneities. Water

Resources Research. v. 40. doi:10.1029/2003WR002750. 2004.

COTAS, K. H., DEMPSEY, J. R., HENDERSON, J. H., The use of vertical equilibri- um in two dimensional simulation of three-dimensional reservoir performance, Society of

Petroleum Engineers Journal. v.11, p. 63-71, 1971.

CUSHMAN, J.H., Hu, B.X., e Deng, F. W., Nonlocal reactive transport with physical and chemical heterogeneity: localization erros. J. Water Resource Resources. v. 31, p. 2219–2237, 1995.

DAGAN, G. Flow and transport in porous formations, New York City, New York: Springer-Verlag. 1989.

DEHKORDI, M. M., MANZARI, M.T., Effects of using altered coarse grids on the implementation and computational cost of the multiscale finite volume method. Adv. Water

Resour. v. 59, p. 221-237, 2013.

EDWARDS, M. G.; ZHENG, H. A quasi-positive family of continuous Darcy-flux fi- nite- volume schemes with full pressure support. Journal of Computational Physics, 2008; 227:

EDWARDS, M. G; ROGERS, C. F; Finite Volume Discretization with Imposed Flux Continuity for the General Tensor Pressure Equation. J. Computational Geoscience, v. 2, 259–290, 1998.

EFENDIEV, Y., DURLOFSKY, L. J., LEE, S. H., Modeling of subgrid effects in coarse-scale simulations of transport in hetereogeneous porous media. Water Resources Re-

search. v. 36, p. 2031-2041, 2000.

EFENDIEV, Y., DURLOFSKY, L. J., Numerical modeling of subgrid heterogeneity in two phase flow simulations, Water Resources Research. vol. 38, 2002.

ERMAKOV K. On the Multiscale Finite Volume Method for the flow in porous

media. MSc Dissertation. TU Kaiserslautern, 2010.

EWING, R. E. The Mathematics of Reservoir Simulation. Philadelphia, SIAM, 1983.

FAROUGHI, S. A., FAROUGHI, S., and MCADAMS, J., A prompt sequential meth- od for subsurface flow modeling using the modified multi-scale finite volume and streamline methods. Int. J. Numer. Anal. Model. v. 4, p. 129-15, 2013.

GAUTIER, Y., BLUNT, M. J., and CHRISTIE, M. A., Nested gridding and stream- line-based simulation for fast reservoir performance prediction. J. Comput. Geosci. v. 3, p. 295–320, 1999.

HAJIBEYGI H., BONFIGLI G., HESSE M. A., Jenny P. Iterative multiscale finite- volume method. J. Comp. Phys. v. 227, p.8604–8621, 2008.

HAJIBEYGI, H. Iterative multiscale finite volume method for multiphase flow in

porous media with complex physics. Tese de Doutorado. Sharif University of Technology.

Tehran.2011.

HESSE, M.A., MALLISON, B.T., TCHELEPI, H.A., Compact multiscale finite vol- ume method for heterogeneous anisotropic elliptic equations, SIAM Multiscale Model. Sim-

ul. v. 7, p. 934–962, 2008.

HOU, T.,and WU, X. H., A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comp. Phys. v. 134, p.169-189, 1997.

JENNY, P, LEE, S. H., TCHELEPI, H. A., Multi-scale finite-volume method for ellip- tic problems in subsurface flow simulation. J. Comp. Phys. v. 187, p. 47-67, 2003.

Jenny, P., Lee, S. H., Tchelepi, H. A., Adaptive multiscale finite volume method for multi-phase flow and transport. Multiscale Model. Simul. v. 3, p. 50-64, 2004.

JENNY, P., LEE, S. H., TCHELEPI, H. A., An adaptive fully implicit multi-scale fi- nite-volume algorithm for multi-phase flow in porous media. J. Comp. Phys. v. 217, pp. 627- 641, 2006.

KIPPE, V., AARNES, J.E., LIE, K.A. A comparison of multiscale methods for elliptic problems in porous media flow. Comput. Geosci. v.12, p. 377-398, 2008.

LEVEQUE, R. J. Numerical Methods for Conservation Laws. Berlin, Birkhauser, 1992.

LI J., RIVIERE B. High order discontinuous Galerkin method for simulating miscible flooding in porous media. Computational Geosciences. v. 19, p. 1251-1268, 2015.

LÖHNER, R. In: Applied DFD Techniques: An Introduction Based on Finite El-

ement Methods. New York, Wiley & Sons, 2001.

LUNATI I., TYAGI, M., LEE, S.H., An iterative multiscale finite volume algorithm converging to the exact solution. J. Comp. Phys. v. 230, p. 1849–1864, 2011.

LUNATI, I., e LEE, S. H. An operator formulation of the Multiscale Finite-Volume Method with correction function. Multiscale Modeling & Simulation. v. 8, p. 96-109, 2009.

LUNATI,I., JENNY, P. Multiscale finite-volume method for compressible multiphase flow in porous media. J. Comp. Phys. v. 216, p. 616-641, 2006.

MØYNER, O. Multiscale Finite Volume Methods: extension to unstructured grids

with applications in reservoir simulation. Dissertação, University of Science and Technol-

ogy, 2012.

MØYNER, O., LIE, K. –A. A Multiscale Two-Point Flux-Approximation Method. download do site da web: http://folk.uio.no/kalie/papers/mstpfa-paper.pdf em 10/03/2014, 2013.

MØYNER, O., LIE, K., A Multiscale Method based on Restriction-Smoothed Ba-

sis Functions Suitable for General Grids in High Contrast Media. SPE Reservoir Simula-

tion Symposium, SPE-173265-MS, Houston, Texas, USA, 2016.

MØYNER, O.; Lie, K. A. A multiscale two-point flux-approximation method. Jour-

nal of Computational Physics, v. 275, p. 273–293, 2014.

RUBIN, Y., A.Y. SUN, R. MAXWELL, AND A. BELLIN. The concept of block- effective macrodispersivity and a unified approach for grid-scale- and plume-scale-dependent transport. Journal of Fluid Mechanics. v. 395. p. 161–180. 1999.

S. H. LEE, M. F. LOUGH, and C. L. JENSEN. Hierarchical modeling o flow in natu- rally fractured formations with multiple length scales. Water Resources Research. v. 37, p.443-455, 2001.

SOUZA, M. R. A. 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 douto-

rado. Universidade Federal de Pernambuco, 2015.

TEIXEIRA, J.C., Simulação por linhas de Fluxos com Acoplamento Geomecânico. Tese de doutorado. Universidade Federal de Pernambuco, 2015.

TENE, M., WANG, Y. , HAJIBEYGI, H., Adaptive algebraic multiscale solver for compressible flow in heterogeneous porous media. Journal of Computational Physics, v.300, p. 679–694, 2015.

van LEER B. TWOARDS. The ultimate conservative difference scheme V. A. second order sequel to Godunov's method. Journal of Computational Physics. vol.45, p. 15-25, 1979.

WANG, Y., HAJIBEYGI, H., TCHELEPI H.A. Monotone multiscale finite volume method. Comput. Geosci. v. 20, p.1-16,2015.

WOODFIELD P. L, KENJIRO S, KAZUYOSHI N. A simple strategy for constructing bounded convection schemes for unstructured grids. International Journal for Numerical

Methods in Fluids, v. 46, p. 1007-1024, 2004.

ZHOU, H. Algebraic Multiscale Finite-Volume for Reservoir Simulation. Tese de doutorado, Stanford University, 2010.

ZHOU, H., e TCHELEPI, H. A., Operator-based multi-scale method for compressible flow. Soc. Petrol. Eng. J., vol. 13, pp. 267-273, 2008.