Modelling of multiscale structures in flow simulations for petroleum reservoirs

Geometric Modelling, Numerical Simulation, and Optimization: Applied Mathematics at SINTEF

Volumn , Issue , 2007, Pages 307-360

  Lie, Knut Andreas ^a   Aarnes, Jorg E ^a   Kippe, Vegard ^a,b   Rustad, Alf Birger ^b  

^a SINTEF ICT   (Norway)

^b STATOIL   (Norway)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84889970583     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-3-540-68783-2_10     Document Type: Chapter

References (65)

1. J.E. Aarnes. On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation. Multiscale Model. Simul., 2(3):421-439, 2004.
2. J.E. Aarnes, T. Gimse, and K.-A. Lie. An introduction to the numerics of flow in porous media using Matlab. In this book.
3. J.E. Aarnes, V. Kippe, and K.-A. Lie. Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels. Adv. Water Resour., 28(3):257-271, 2005.
4. J. E. Aarnes, S. Krogstad, and K.-A. Lie. A hierarchical multiscale method for two-phase flow based upon mixed finite elements and nonuniform coarse grids. Multiscale Model. Simul., 5(2):337-363, 2006.
5. J. E. Aarnes, S. Krogstad, and K.-A. Lie. Multiscale mixed/mimetic methods on corner-point grids. Comput. Geosci., submitted.
6. J.E. Aarnes and K.-A. Lie. Toward reservoir simulation on geological grid models. In Proceedings of the 9th European Conference on the Mathematics of Oil Recovery. EAGE, Cannes, France, 2004.
7. O. J. Arntzen. Higher-order terms in the capillary limit approximation of viscous-capillary flow. Transp. Porous Media, 57:17-34, 2004.
8. J.W. Barker and P. Dupouy. An analysis of dynamic pseudo relative permeability methods. In Proceedings of the 5th ECMOR conference, Leoben, Austria. 1996.
9. J.W. Barker and F.J. Fayers. Transport coefficients for compositional simulations with coarse grids in heterogeneous media. SPE Adv. Tech Series, 2(2):103-112, 1994.
10. J.W. Barker and S. Thibeau. A critical review of the use of pseudorelative permeabilities for upscaling. SPE Reservoir Eng., 12(2):138-143, 1997.
11. S.H. Begg, R.R. Carter, and P. Dranfield. Assigning effective values to simulator gridblock parameters for heterogeneous reservoirs. SPE Reservoir Eng., 4(4):455-463, 1989.
12. A. Benesoussan, J.-L. Lions, and G. Papanicolaou. Asymptotic Analysis for Periodic Structures. Elsevier Science Publishers, Amsterdam, 1978.
13. D. Braess. Finite elements: Theory fast solvers and applications in solid mechanics. Cambridge University Press, Cambridge, 1997.
14. S.C. Brenner and L.R. Scott. The mathematical theory of finite element methods. Springer-Verlag, New York, 1994.
15. F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Computational Mathematics. Springer-Verlag, New York, 1991.
16. Y. Chen and L.J. Durlofsky. Adaptive local-global upscaling for general flow scenarios in heterogeneous formations. Transp. Porous Media, 62(2):157-185, 2006
17. Y. Chen, L.J. Durlofsky, M. Gerritsen, and X.H. Wen. A coupled local-global upscaling approach for simulationg flow in highly heterogeneous formations. Adv. Water Resour., 26:1041-1060, 2003.
18. Z. Chen and T.Y. Hou. A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comp., 72:541-576, 2003.
19. M. A. Christie and M. J. Blunt. Tenth SPE comparative solution project: A comparison of upscaling techniques. SPE Reservoir Eval. Eng., 4(4):308-317, 2001. url: www.spe.org/csp.
20. M.A. Christie. Upscaling for reservoir simulation. JPT J. Pet. Tech., 48:1004-1010, 1996.
21. L.J. Durlofsky. Numerical calculations of equivalent gridblock permeability tensors for heterogeneous porous media. Water Resour. Res., 27(5):699-708, 1991.
22. L.J. Durlofsky. Use of higher moments for the description of upscaled, process independent relative permeabilities. SPE J., 2(4):474-484, 1997.
23. L.J. Durlofsky. Coarse scale models of two-phase flow in heterogeneous reservoirs: Volume averaged equations and their relation to existing upscaling techniques. Comput. Geosci., 2:73-92, 1998.
24. L.J. Durlofsky, R.C. Jones, and W.J. Milliken. A nonuniform coarsening approach for the scale-up of displacement processes in heterogeneous porous media. Adv. Water Resour., 20:335-347, 1997.
25. Y. Efendiev. The Multiscale Finite Element Method and its Applications. PhD thesis, California Institute of Technology, 1999.
26. Y. Efendiev and L.J. Durlofsky. Numerical modeling of subgrid heterogeneity in two phase flow simulations. Water Resour. Res., 38:1128, 2002.
27. Y. Efendiev and L.J. Durlofsky. A generalized convection-diffusion model for subgrid transport in porous media. Multiscale Model. Simul., 1:504-526, 2003.
28. Y. Efendiev, L.J. Durlofsky, and S.H. Lee. Modeling of subgrid effects in coarse scale simulations of transport in heterogeneous porous media. Water Resour. Res., 36(8):2031-2041, 2000.
29. Y. Efendiev, V. Ginting, T. Hou, and R. Ewing. Accurate multiscale finite element methods for two-phase flow simulations. J. Comp. Phys., 220(1):155-174, 2006.
30. Y. Efendiev and B. Popov. On homogenization of nonlinear hyperbolic equations. Commun. Pure and Appl. Anal., 4:297-311, 2005.
31. S. Ekrann and J. O. Aasen. Steady-state upscaling. Transp. Porous Media, 41(3):245-262, 2000.
32. M.H. Garcia, A.G. Journel, and K. Aziz. Automatic grid generation and adjustment method for modeling reservoir heterogeneities. Technical report, Stanford Center for Reservoir Forecasting, 1990.
33. Y. Gautier, M. J. Blunt, and M. A. Christie. Nested gridding and streamline-based simulation for fast reservoir performance prediction. Comput. Geosci., 3:295-320, 1999.
34. J.J. Ǵomez-Herńandez. A Stochastic Approach to the Simulation of Block Conductivity Conditioned Upon Data Measured at a Smaller Scale. PhD thesis, Stanford University, CA, 1991.
35. R.E. Guzman, D. Giordano, F.J. Fayers, A. Godi, and K. Aziz. Evaluation of dynamic pseudo functions for reservoir simulation. In SPE Annual Technical Conference and Exhibition, 6-9 October, Denver, Colorado, SPE 35157, 1996.
36. T.A. Hewett and R.A. Behrens. Scaling laws in reservoir simulation and their use in a hybrid finite difference/streamtube approach to simulating the effects of permeability heterogeneity. In L.W. Lake, H.B. Caroll, and T.C. Wesson, editors, Reservoir Characterization II. Academic Press, San Diego, Ca, 1991.
37. T.A. Hewett, K. Suzuki, and M.A. Christie. Analytical calculation of coarse-grid corrections for use in pseudofunctions. SPE J., 3(3):293-304, 1998.
38. L. Holden and B.F. Nielsen. Global upscaling of permeability in heterogeneous reservoirs; the output least squares (ols) method. Trans. Porous Media, 40(2):115-143, 2000.
39. U. Hornung. Homogenization and Porous Media. Springer-Verlag, New York, 1997.
40. T.Y. Hou and X-H. Wu. A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys., 134:169-189, 1997.
41. H. Jacks, O.J.E. Smith, and C.C. Mattax. The modeling of a three-dimensional reservoir with a two-dimensional reservoir simulator-the use of dynamic pseudo functions. SPE J., pages 175-185, 1973.
42. P. Jenny, S. H. Lee, and H. A. Tchelepi. Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys., 187:47-67, 2003.
43. P. Jenny, S. H. Lee, and H. A. Tchelepi. Adaptive multiscale finite-volume method for multiphase flow and transport in porous media. Multiscale Model. Simul., 3(1):50-64, 2004.
44. V.V. Jikov, S.M. Kozlov, and O.A. Oleinik. Homogenization of differential operators and integral functionals. Springer-Verlag, New York, 1994.
45. A.G. Journel, C.V. Deutsch, and A.J. Desbarats. Power averaging for block effective permeability. In SPE California Regional Meeting, 2-4 April, Oakland, California, SPE 15128, 1986.
46. M.J. King, D.G. MacDonald, S.P. Todd, and H. Leung. Application of novel upscaling approaches to the Magnus and Andrew reservoirs. In European Petroleum Conference, 20-22 October, The Hague, Netherlands, SPE 50643, 1998.
47. P.R. King. The use of renormalization for calculating effective permeability. Transp. Porous Media, 4:37-58, 1989.
48. P.R. King, A.H. Muggeridge, and W.G. Price. Renormalization calculations of immiscible flow. Transp. Porous Media, 12:237-260, 1993.
49. V. Kippe, J. E. Aarnes, and K.-A. Lie. A comparison of multiscale methods for elliptic problems in porous media flow. Comput. Geosci., submitted.
50. A. T. Kumar and G. R. Jerauld. Impacts of scale-up on fluid flow from plug to gridblock scale in reservoir rock. In SPE/DOE Improved Oil Recovery Symposium, 21-24 April, Tulsa, Oklahoma, SPE 35452, 1996.
51. J.R. Kyte and D.W. Berry. New pseudofunctions to control numerical dispersion. SPE J., pages 269-276, August 1975.
52. P. Langlo and Magne S. Espedal. Macrodispersion for two-phase, immiscible flow in porous media. Adv. Water Resour., 17:291-316, 1994.
53. B. F. Nielsen and A. Tveito. An upscaling method for one-phase flow in heterogeneous reservoirs; a Weighted Output Least Squares (WOLS) approach. Comput. Geosci., 2:92-123, 1998.
54. P.-E. Øren, S. Bakke, and O.J. Arntzen. Extending predictive capabilities to network models. SPE J., 3(4):324-336, 1998.
55. G.E. Pickup and K.S. Sorbie. Development and application of a new two-phase scaleup method based on tensor permeabilities. SPE J., pages 369-382, 1996.
56. P.A. Raviart and J.M. Thomas. A mixed finite element method for second order elliptic equations. In I. Galligani and E. Magenes, editors, Mathematical Aspects of Finite Element Methods, pages 292-315. Springer-Verlag, Berlin-Heidelberg-New York, 1977.
57. P. Renard and G. de Marsily. Calculating equivalent permeability. Adv. Water Resour., 20:253-278, 1997.
58. E.H. Smith. The influence of small scale heterogeneity on average relative permeability. In L.W. Lake, H.B. Carroll, and T.C. Wesson, editors, Reservoar Characterization II. Academic Press, 1991.
59. K. D. Stephen, G. E. Pickup, and K. S. Sorbie. The local analysis of changing force balances in immiscible incompressible two-phase flow. Transp. Porous Media, 45:63-88, 2001.
60. H.L. Stone. Rigorous black-oil pseudo functions. In SPE Symposium on Reservoir Simulation, 17-20 February, Anaheim, California, SPE 21207, 1991.
61. G.W. Thomas. An extension of pseudofunction concepts. In SPE Reservoir Simulation Symposium, 15-18 November, San Francisco, California, SPE 12274, 1983.
62. X.-H. Wen and J.J. Ǵomez-Herńandez. Upscaling hydraulic conductivities in heterogeneous media: An overview. J. Hydrol., 183:ix-xxxii, 1996.
63. O. Wiener. Abhandlungen der Matematisch. PhD thesis, Physischen Klasse der K̈oniglichen S̈achsischen Gesellscaft der Wissenschaften, 1912.
64. X.H. Wu, Y. Efendiev, and T.Y. Hou. Analysis of upscaling absolute permeability. Discrete Contin. Dyn. Syst. Ser. B, 2(2):185-204, 2002.
65. P. Zhang, G. E. Pickup, and M. A. Christie. A new upscaling approach for highly heterogenous reservoirs. In SPE Reservoir Simulation Symposium, 31 January-2 Feburary, The Woodlands, Texas, SPE 93339, 2005.