Transport in Porous Media

, Volume 127, Issue 3, pp 661–684 | Cite as

Upscaling of Geological Models of Oil Reservoirs with Unstructured Grids Using Lifting-Based Graph Wavelet Transforms

  • Amin Rezapour
  • Antonio Ortega
  • Muhammad SahimiEmail author


Wavelet transforms (WTs) constitute a class of powerful tools for spectral and local analysis of data, such as time series, geophysical data, and well logs, as well as modeling of various problems in oil reservoirs, and in particular upscaling their geological models. The application to upscaling has, however, been limited to the reservoir models that are represented by regular computational grids with blocks or cells with regular shapes, such as squares and cubes. The most natural structure of the computational grid is, however, one with irregular and unequal cells, distributed spatially with stochastic orientations. Such a grid is also the ideal computational model for, for example, fractured reservoirs as it allows inclusion of fractures with spatially distributed orientations. In this paper, we propose a generalization of the WT approach to upscaling by developing a new model of a reservoir based on irregular graphs that make it possible to use the WTs for upscaling the geological model highly efficiently. To do so, we first define a computational grid representing a reservoir as a graph and its adjacency matrix and, then, introduce graph WTs using the concept of lifting, utilized in classical signal processing and its extension to graphs. The application of the lifting-based graph WT to upscaling is then developed. The result is an algorithm that may be applied to upscaling of any unstructured geological model represented by a computational grid in which the multiresolution graph WT is applied directly to the spatial distribution of the permeabilities (or other suitable properties). Examples in which the geological model is represented by the Voronoi tessellations are described, and simulation of waterflooding with such graph networks is carried out in order to demonstrate the accuracy and efficiency of the new method.


Oil reservoirs Geological model Upscaling Graph wavelet transformation Lifting Waterflooding 



A.R. thanks Hassan Dashtian for help to generate the permeability field by the fractional Brownian motion.


  1. Alpak, F.O.: Quasiglobal multiphase upscaling of reservoir models with nonlocal stratigraphic heterogeneities. SPE paper 170245-PA (2015)Google Scholar
  2. Athichanagorn, S., Horne, R.N., Kikani, J.: Processing and interpretation of long-term data acquired from permanent pressure gages. SPE paper 56419 (1999)Google Scholar
  3. Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6, 405 (2002)CrossRefGoogle Scholar
  4. Aavatsmark, I.: Interpretation of a two-point flux stencil for skew parallelogram grids. Comput. Geosci. 11, 199 (2007)CrossRefGoogle Scholar
  5. Aavatsmark, I., Barkve, T., Bøe, O., Mannseth, T.: Discretization on nonorthogonal, quadrilateral grids for inhomogeneous, anisotropic Media. J. Comput. Phys. 127, 14 (1996)CrossRefGoogle Scholar
  6. Aavatsmark, I., Barkve, T., BØe, O., Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: Derivation of the methods. SIAM J. Sci. Comput. 19, 1700 (1998)CrossRefGoogle Scholar
  7. Aavatsmark, I., Eigestad, G.: Numerical convergence of the MPFA O-method and U-method for general quadrilateral grids. Int. J. Numer. Methods Fluids 51, 939 (2006)CrossRefGoogle Scholar
  8. Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Scientific Publishing, London (1979)Google Scholar
  9. Babaei, M., King, P.R.: A comparison between wavelet and renormalization upscaling methods and iterative upscaling–downscaling scheme. SPE Reserv. Simul. Symp. 1, 469 (2011)Google Scholar
  10. Barker, J.W., Thibeau, S.: A critical review of the use of pseudo relative permeabilities for upscaling. SPE Reserv. Eng. 12, 138 (1997)CrossRefGoogle Scholar
  11. Brewer, K.E., Wheatcraft, S.W.: Including multi-scale information in the characterization of hydraulic conductivity distributions. In: Wavelets in Geophysics, p. 213 . Academic Press, San Diego (1994)Google Scholar
  12. Cao, J., Kitanidis, P.K.: Adaptive-grid simulation of groundwater flow in heterogeneous aquifers. Adv. Water Resour. 22, 681 (1999)CrossRefGoogle Scholar
  13. Cao, Y., Helmig, R., Wohlmuth, B.I.: Geometrical interpretation of the multipoint flux approximation l-method. Int. J. Numer. Methods Fluids 60, 1173 (2009)CrossRefGoogle Scholar
  14. Castellini, A., Edwards, M.G., Durlofsky, L.J.: Flow based modules for grid generation in two and three dimensions. In: Proceedings of Seventh European Conference on the Mathematics of Oil Recovery (2000)Google Scholar
  15. Caumon, G., Levy, B., Castanie, L., Paul, J.-C.: Visualization of grids conforming to geological structures: a topological approach. Comput. Geosci. 31, 671 (2005)CrossRefGoogle Scholar
  16. Cescotto, S., Wu, Z.D.: A variable-density mesh generation for planar domains. Commun. Appl. Numer. Methods 5, 473 (1989)CrossRefGoogle Scholar
  17. Chappelear, A., Hirasaki, G.J.: A model of oil-water coning for two dimensional areal reservoir simulation. SPE paper 4980 (1976)Google Scholar
  18. Chen, H., Bishop J.: Delaunay triangulation for curved surfaces. In: Proceedings of the 6th International Meshing Roundtable, Sandia National Laboratories, p. 115 (October 1997)Google Scholar
  19. Chen, T., Gerritsen, M.G., Lambers, J.V., Durlofsky, L.J.: Global variable compact multipoint methods for accurate upscaling with full-tensor effects. Comput. Geosci. 14, 65 (2010)CrossRefGoogle Scholar
  20. Chew, L.P.: Constrained Delaunay triangulations. Algorithmica 4, 97 (1989)CrossRefGoogle Scholar
  21. Christie, M.A., Blunt, M.J.: Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE paper 66599 (2001)Google Scholar
  22. Dashtian, H., Jafari, G.R., Koohi Lai, Z., Masihi, M., Sahimi, M.: Analysis of cross correlations between well logs of hydrocarbon reservoirs. Transp. Porous Media 90, 445 (2011a)CrossRefGoogle Scholar
  23. Dashtian, H., Jafari, G.R., Sahimi, M., Masihi, M.: Scaling, multifractality, and long-range correlations in well log data of large-scale porous media. Phys. A 390, 2096 (2011b)CrossRefGoogle Scholar
  24. Dashtian, H., Yang, Y., Sahimi, M.: Non-universality of the Archie exponent due to multifractality of the resistivity well logs. Geophys. Res. Lett. 42, 10655 (2015)CrossRefGoogle Scholar
  25. Ding, Y.: Upscaling on distorted gridblocks for simulation of advanced wells. J. Pet. Sci. Eng. 43, 87 (2004)CrossRefGoogle Scholar
  26. Durlofsky, L.J., Chen, Y.: Uncertainty quantification for subsurface flow problems using coarse-scale models. Lect. Notes Comput. Sci. Eng. 83 (2012)Google Scholar
  27. Durlofsky, L.J., Jones, R.C., Milliken, W.J.: A non-uniform coarsening approach for the scale up of displacement processes in heterogeneous porous media. Adv. Water Resour. 20, 335 (1997)CrossRefGoogle Scholar
  28. Durlosfky, L.J., Milliken, W.J., Bemath, A.: Scaleup in the near-well region. SPE paper 61855 (2000)Google Scholar
  29. Ebrahimi, F.: Ph.D. Dissertation, Ferdowsi University of Mashhad, Mashhad, Iran (2002)Google Scholar
  30. Ebrahimi, F., Sahimi, M.: Multiresolution wavelet coarsening and analysis of transport in heterogeneous porous media. Phys. A 316, 160 (2002)CrossRefGoogle Scholar
  31. Ebrahimi, F., Sahimi, M.: Multiresolution wavelet scale up of unstable miscible displacements in flow through porous media. Transp. Porous Media 57, 75 (2004)CrossRefGoogle Scholar
  32. Ebrahimi, F., Sahimi, M.: Grid coarsening, simulation of transport processes in, and scale-up of heterogeneous media: application of multiresolution wavelet transformations. Mech. Mater. 38, 772 (2006)CrossRefGoogle Scholar
  33. Edwards, M.G.: Elimination of adaptive grid interface errors in the discrete cell centered pressure equation. J. Comput. Phys. 126, 356 (1996)CrossRefGoogle Scholar
  34. Edwards, M.G.: M-matrix flux splitting for general full tensor discretization operators on structured and unstructured grids. J. Comput. Phys. 160, 1 (2000)CrossRefGoogle Scholar
  35. Edwards, M.G.: Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci. 6, 433 (2002)CrossRefGoogle Scholar
  36. Edwards, M.G., Agut, R., Aziz, K.: Quasi k-orthogonal streamline grids: gridding and discretization. SPE Paper 49072 (1998)Google Scholar
  37. Edwards, M.G., Li, B., Aziz K.: Modular mesh generation with embedded streamline potential grids. SPE paper 51911 (1999)Google Scholar
  38. Edwards, M.G., Rogers, C.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2, 259 (1998)CrossRefGoogle Scholar
  39. Fayazi, A., Bagherzadeh, H., Shahrabadi, A.: Estimation of pseudo relative permeability curves for a heterogeneous reservoir with a new automatic history matching algorithm. J. Pet. Sci. Eng. 140, 154 (2016)CrossRefGoogle Scholar
  40. Garcia, M.H., Journel, A.G., Aziz, K.: An automatic grid generation and adjustment method for modeling reservoir heterogeneity. SPE paper 21471 (1992)Google Scholar
  41. Gjika, A.T.: Multi-resolution analysis for lifting transform on general graphs. Master’s Thesis, The University of Southern California, Los Angeles (2012)Google Scholar
  42. Gong, B., Karimi-Fard, M., Durlofsky, L.J.: Upscaling discrete fracture characterizations to dual-porosity, dual-permeability models for efficient simulation of flow with strong gravitational effects. SPE J. 13, 1 (2008)CrossRefGoogle Scholar
  43. Hashemi, A., Shadizadeh, S.R., Zargar, G.: Upscaling of relative permeability using pseudo functions. Energy Resour. A 36, 2227 (2014)Google Scholar
  44. Heinemann, Z.E., Brand, C.W.: Gridding techniques in reservoir simulation. In: Proceedings of the Second International Forum on Reservoir Simulation, Alpbach, Austria (1989)Google Scholar
  45. Heinemann, Z.E., Brand, C.W., Munka, M., Chen, Y.M.: Modeling reservoir geometry with irregular grid. SPE Res. Eng. 6, 225 (1991)CrossRefGoogle Scholar
  46. Jenny, P., Lee, S.H., Tchelepi, H.A.: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187, 47 (2003)CrossRefGoogle Scholar
  47. Jenny, P., Lee, S., Tchelepi, H.A.: Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media. J. Comput. Phys. 217, 627 (2006)CrossRefGoogle Scholar
  48. Jensen, J.L., Lake, L.W., Corbett, P.W.M., Goggin, D.J.: Statistics for Petroleum Engineers and Geoscientists, 2nd edn. Prentice Hall, Amsterdam (2000)Google Scholar
  49. Jin, H., Wiberg, N.E.: Two-dimensional mesh generation, adaptive remeshing and refinement. Int. J. Numer. Methods Eng. 29, 501 (1990)CrossRefGoogle Scholar
  50. Kallmann, M., Bieri, H., Thalmann, D.: Fully dynamic constrained Delaunay triangulations. Comput. Geom. Theory Appl. 2, 55 (1992)CrossRefGoogle Scholar
  51. Karimi-Fard, M., Gong, B., Durlofsky, L.J.: Generation of coarse-scale continuum flow models from detailed fracture characterizations. Water Resour. Res. 42, W10423 (2006)CrossRefGoogle Scholar
  52. Kikani, J., He, M.: Multiresolution analysis of long-term pressure transient data using wavelet methods. SPE paper 48996 (1998)Google Scholar
  53. King, P.R., Snyder, D.E., Obut, T.S., Perkins, R.L.: A case study of the full-field simulation of a reservoir containing bottomwater. SPE paper 21203 (1991)Google Scholar
  54. Landauer, R.: The electrical resistance of binary metallic mixtures. J. Appl. Phys. 23, 779 (1952)CrossRefGoogle Scholar
  55. Lau, T.S., Lo, S.H.: Finite element mesh generation over analytical curved surfaces. Comput. Struct. 59, 301 (1996)CrossRefGoogle Scholar
  56. Li, D., Beckner, B.: Optimal uplayering for scaleup of multimillion-cell geologic models. SPE paper 62927 (2000)Google Scholar
  57. Lo, S.H., Lee, C.K.: Generation of gradation meshes by the background technique. Comput. Struct. 50, 21 (1994)CrossRefGoogle Scholar
  58. Loze, M.K., Saunders, R.: Two simple algorithms for constructing a two-dimensional constrained Delaunay triangulation. Appl. Numer. Math. 11, 403 (1993)CrossRefGoogle Scholar
  59. Lu, P., Horne, R.N.: A multiresolution approach to reservoir parameter estimation using wavelet analysis. SPE paper 62985 (2000)Google Scholar
  60. Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, San Diego (2008)Google Scholar
  61. Mehrabi, A.R., Rassamdana, H., Sahimi, M.: Characterization of long-range correlation in complex distributions and profiles. Phys. Rev. E 56, 712 (1997)CrossRefGoogle Scholar
  62. Mehrabi, A.R., Sahimi, M.: Coarsening of heterogeneous media: application of wavelets. Phys. Rev. Lett. 79, 4385 (1997)CrossRefGoogle Scholar
  63. Mehrabi, A.R., Sahimi, M.: Analysis and simulation of long-range correlations in curved space. Int. J. Mod. Phys. C 20, 1211 (2009)CrossRefGoogle Scholar
  64. Merland, R., Caumon, G., Levy, B., Collon-Drouaillet, P.: Building centroidal Voronoi tessellations for flow simulation in reservoirs using flow information. In: SPE Reservoir Simulation Symposium, Woodlands, Texas (2011)Google Scholar
  65. Moog, G.J.E.A.: Advanced discretization methods for flow simulation using unstructured grids. Ph.D. Thesis, Stanford University (2013)Google Scholar
  66. Moslehi, M., de Barros, F.P.J., Ebrahimi, F., Sahimi, M.: Upscaling of solute transport in heterogeneous porous media by wavelet transformations. Adv. Water Resour. 96, 180 (2016)CrossRefGoogle Scholar
  67. Narang, S.K., Ortega, A.: Lifting based wavelet transforms on graphs. In: Proceedings of the Asia-Pacific Signal and Information Processing Association, 2009 Annual Summit and Conference, p. 441 (2009)Google Scholar
  68. Neivergelt, Y.: Wavelets Made Easy. Birkhäser, Boston (1999)CrossRefGoogle Scholar
  69. Neupauer, R.M., Powell, K.L.: A fully-anisotropic Morlet wavelet to identify dominant orientations in a porous medium. Comput. Geosci. 31, 465 (2005)CrossRefGoogle Scholar
  70. Nordbotten, J.M., Eigestad, G.T.: Discretization on quadrilateral grids with improved monotonicity properties. J. Comput. Phys. 203, 744 (2003)CrossRefGoogle Scholar
  71. Palagi, C.L., Aziz, K.: The modeling of vertical and horizontal wells with Voronoi grid. SPE paper 24072 (1992)Google Scholar
  72. Pancaldi, V., Christensen, K., King, P.R.: Permeability up-scaling using Haar wavelets. Transp. Porous Media 67, 395 (2007)CrossRefGoogle Scholar
  73. Panda, M.N., Mosher, C., Chopra, A.K.: Application of wavelet transforms to reservoir data analysis and scaling. SPE paper 3656 (1996)Google Scholar
  74. Pazhoohesh, E., Hamzehpour, H., Sahimi, M.: Numerical simulation of ac conduction in three-dimensional heterogeneous materials. Phys. Rev. B 73, 174206 (2006)CrossRefGoogle Scholar
  75. Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulations. Elsevier, New York (1977)Google Scholar
  76. Peraire, J., Morgan, K.: Unstructured mesh generation including directional refinement for aerodynamic flow simulation. Finite Elem. Anal. Des. 25, 343 (1997)CrossRefGoogle Scholar
  77. Portella, R.C.M., Hewett, T.A.: Upscaling, gridding, and simulation using streamtubes. SPE J. 5, 315 (2000)CrossRefGoogle Scholar
  78. Prevost, M., Lepage, F., Durlofsky, L.J., Mallet, J.-L.: Unstructured 3D gridding and upscaling for coarse modelling of geometrically complex reservoirs. Pet. Geosci. 11, 339 (2005)CrossRefGoogle Scholar
  79. Qi, D., Zhang, S.: Major challenges for reservoir upscaling. Pet. Sci. Technol. 27, 1985 (2009)CrossRefGoogle Scholar
  80. Qian, Y.Y., Dhatt, G.: A simple adaptable 2D mesh generation package. Comput. Struct. 53, 801 (1994)CrossRefGoogle Scholar
  81. Rasaei, M.R., Sahimi, M.: Upscaling and simulation of waterflooding in heterogeneous reservoirs using wavelet transformations: Application to the SPE-10 model. Transp. Porous Media 72, 311 (2008)CrossRefGoogle Scholar
  82. Rasaei, M.R., Sahimi, M.: Upscaling of the permeability by multiscale wavelet transformations and simulation of multiphase flows in heterogeneous porous media. Comput. Geosci. 13, 187 (2009a)CrossRefGoogle Scholar
  83. Rasaei, M.R., Sahimi, M.: Upscaling of the geological models of large-scale porous media using multiresolution wavelet transformations. J. Heat Transf. 131, 101007 (2009b)CrossRefGoogle Scholar
  84. Ron, D., Safro, I., Brandt, A.: Relaxation-based coarsening and multiscale graph organization. Multiscale Model. Simul. 9, 407 (2011)CrossRefGoogle Scholar
  85. Sahimi, M.: Heterogeneous Materials I, Chapters 4 & 5. Springer, New York (2003)Google Scholar
  86. Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock, 2nd edn. Wiley-VCH, Weinheim (2011)CrossRefGoogle Scholar
  87. Sahimi, M., Darvishi, R., Haghighi, M., Rasaei, M.R.: Upscaled unstructured grids for efficient simulation of flow in fractured reservoirs. Transp. Porous Media 83, 195 (2010)CrossRefGoogle Scholar
  88. Sahimi, M., Hashemi, M.: Wavelet identification of the spatial distribution of fractures. Geophys. Res. Lett. 28, 611 (2001)CrossRefGoogle Scholar
  89. Sahimi, M., Rasaei, M.R., Ebrahimi, F., Haghighi, M.: Upscaling of unstable displacements and multiphase flows using multiresolution wavelet transformation. SPE paper 93320 (2005)Google Scholar
  90. Sahimi, M., Tsotsis, T.T.: Transient diffusion and conduction in heterogeneous media: beyond the classical effective-medium approximation. Ind. Eng. Chem. Res. 36, 3043 (1997)CrossRefGoogle Scholar
  91. Shirazi, A., Jafari, G.R., Davoudi, J., Peinke, J., Rahimi Tabar, M.R., Sahimi, M.: Mapping stochastic processes onto complex networks. J. Statist. Mech.: Theory Exp. P07046 (2009)Google Scholar
  92. Shuman, D.I., Narang, S.K., Frossard, P., Ortega, A., Vandergheynst, P.: Signal processing on graphs: extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Process. Mag. 30(3), 83 (2013)CrossRefGoogle Scholar
  93. Sibson, R.: Locally equiangular triangulations. Comput. J. 21, 243 (1978)CrossRefGoogle Scholar
  94. Soliman, M.Y., Ansah, J., Stephenson, S., Mandal, B.: Application of wavelet transform to analysis of pressure transient data. SPE paper 71571 (2001)Google Scholar
  95. Thiele, M.R., Rao, S.E., Blunt, M.J.: Quantifying uncertainty in reservoir performance using streamlines. Math. Geol. 28, 843 (1996)CrossRefGoogle Scholar
  96. Thomas, C.W.: Principles of Hydrocarbon Reservoir Simulation. International Human Resources Development Corporation, Boston (1982)Google Scholar
  97. Verma, S., Aziz, K.: Two and three dimensional flexible grids for reservoir simulation. In: Proceedings of the Fifth European Conference on the Mathematics of Oil Recovery, Leoben, Austria (1996)Google Scholar
  98. Vetterli, M., Kovacevic, J.: Wavelets and Subband Coding. Prentice Hall, New York (1995)Google Scholar
  99. Wallstorm, T.C., Hou, S., Durlofsky, L.J.: Application of a new two-phase upscaling technique to realistic reservoir cross sections. SPE Paper 51939 (1999)Google Scholar
  100. Watson, D.F.: Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes. Comput. J. 24, 167 (1981)CrossRefGoogle Scholar
  101. Wen, X.-H., Durlofsky, L.J., Chen, Y.: Efficient 3D implementation of local-global upscaling for reservoir simulation. SPE J. 11, 443 (2006)CrossRefGoogle Scholar
  102. Younis, R.M., Caers, J.: A method for reservoir scale-up by static-based non-uniform gridding. Stanford Center for Reservoir Forcasting Annual Report (2001)Google Scholar
  103. Younis, R.M., Caers, J.: A method for static-base upgridding. In: Proceedings of the Eighth European Conference on the Mathematics of Oil Recovery, Germany (2002)Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Amin Rezapour
    • 1
  • Antonio Ortega
    • 1
  • Muhammad Sahimi
    • 2
    Email author
  1. 1.Department of Electrical Engineering and Signal and Image Processing InstituteUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Mork Family Department of Chemical Engineering and Materials ScienceUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations