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

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Abstract

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.

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References

  • Alpak, F.O.: Quasiglobal multiphase upscaling of reservoir models with nonlocal stratigraphic heterogeneities. SPE paper 170245-PA (2015)

  • Athichanagorn, S., Horne, R.N., Kikani, J.: Processing and interpretation of long-term data acquired from permanent pressure gages. SPE paper 56419 (1999)

  • Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6, 405 (2002)

    Article  Google Scholar 

  • Aavatsmark, I.: Interpretation of a two-point flux stencil for skew parallelogram grids. Comput. Geosci. 11, 199 (2007)

    Article  Google Scholar 

  • Aavatsmark, I., Barkve, T., Bøe, O., Mannseth, T.: Discretization on nonorthogonal, quadrilateral grids for inhomogeneous, anisotropic Media. J. Comput. Phys. 127, 14 (1996)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Applied Scientific Publishing, London (1979)

    Google Scholar 

  • 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 

  • Barker, J.W., Thibeau, S.: A critical review of the use of pseudo relative permeabilities for upscaling. SPE Reserv. Eng. 12, 138 (1997)

    Article  Google Scholar 

  • 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)

  • Cao, J., Kitanidis, P.K.: Adaptive-grid simulation of groundwater flow in heterogeneous aquifers. Adv. Water Resour. 22, 681 (1999)

    Article  Google Scholar 

  • Cao, Y., Helmig, R., Wohlmuth, B.I.: Geometrical interpretation of the multipoint flux approximation l-method. Int. J. Numer. Methods Fluids 60, 1173 (2009)

    Article  Google Scholar 

  • 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)

  • Caumon, G., Levy, B., Castanie, L., Paul, J.-C.: Visualization of grids conforming to geological structures: a topological approach. Comput. Geosci. 31, 671 (2005)

    Article  Google Scholar 

  • Cescotto, S., Wu, Z.D.: A variable-density mesh generation for planar domains. Commun. Appl. Numer. Methods 5, 473 (1989)

    Article  Google Scholar 

  • Chappelear, A., Hirasaki, G.J.: A model of oil-water coning for two dimensional areal reservoir simulation. SPE paper 4980 (1976)

  • Chen, H., Bishop J.: Delaunay triangulation for curved surfaces. In: Proceedings of the 6th International Meshing Roundtable, Sandia National Laboratories, p. 115 (October 1997)

  • 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)

    Article  Google Scholar 

  • Chew, L.P.: Constrained Delaunay triangulations. Algorithmica 4, 97 (1989)

    Article  Google Scholar 

  • Christie, M.A., Blunt, M.J.: Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE paper 66599 (2001)

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Ding, Y.: Upscaling on distorted gridblocks for simulation of advanced wells. J. Pet. Sci. Eng. 43, 87 (2004)

    Article  Google Scholar 

  • Durlofsky, L.J., Chen, Y.: Uncertainty quantification for subsurface flow problems using coarse-scale models. Lect. Notes Comput. Sci. Eng. 83 (2012)

  • 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)

    Article  Google Scholar 

  • Durlosfky, L.J., Milliken, W.J., Bemath, A.: Scaleup in the near-well region. SPE paper 61855 (2000)

  • Ebrahimi, F.: Ph.D. Dissertation, Ferdowsi University of Mashhad, Mashhad, Iran (2002)

  • Ebrahimi, F., Sahimi, M.: Multiresolution wavelet coarsening and analysis of transport in heterogeneous porous media. Phys. A 316, 160 (2002)

    Article  Google Scholar 

  • Ebrahimi, F., Sahimi, M.: Multiresolution wavelet scale up of unstable miscible displacements in flow through porous media. Transp. Porous Media 57, 75 (2004)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Edwards, M.G.: Elimination of adaptive grid interface errors in the discrete cell centered pressure equation. J. Comput. Phys. 126, 356 (1996)

    Article  Google Scholar 

  • Edwards, M.G.: M-matrix flux splitting for general full tensor discretization operators on structured and unstructured grids. J. Comput. Phys. 160, 1 (2000)

    Article  Google Scholar 

  • Edwards, M.G.: Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci. 6, 433 (2002)

    Article  Google Scholar 

  • Edwards, M.G., Agut, R., Aziz, K.: Quasi k-orthogonal streamline grids: gridding and discretization. SPE Paper 49072 (1998)

  • Edwards, M.G., Li, B., Aziz K.: Modular mesh generation with embedded streamline potential grids. SPE paper 51911 (1999)

  • Edwards, M.G., Rogers, C.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2, 259 (1998)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Garcia, M.H., Journel, A.G., Aziz, K.: An automatic grid generation and adjustment method for modeling reservoir heterogeneity. SPE paper 21471 (1992)

  • Gjika, A.T.: Multi-resolution analysis for lifting transform on general graphs. Master’s Thesis, The University of Southern California, Los Angeles (2012)

  • 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)

    Article  Google Scholar 

  • Hashemi, A., Shadizadeh, S.R., Zargar, G.: Upscaling of relative permeability using pseudo functions. Energy Resour. A 36, 2227 (2014)

    Google Scholar 

  • Heinemann, Z.E., Brand, C.W.: Gridding techniques in reservoir simulation. In: Proceedings of the Second International Forum on Reservoir Simulation, Alpbach, Austria (1989)

  • Heinemann, Z.E., Brand, C.W., Munka, M., Chen, Y.M.: Modeling reservoir geometry with irregular grid. SPE Res. Eng. 6, 225 (1991)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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 

  • Jin, H., Wiberg, N.E.: Two-dimensional mesh generation, adaptive remeshing and refinement. Int. J. Numer. Methods Eng. 29, 501 (1990)

    Article  Google Scholar 

  • Kallmann, M., Bieri, H., Thalmann, D.: Fully dynamic constrained Delaunay triangulations. Comput. Geom. Theory Appl. 2, 55 (1992)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Kikani, J., He, M.: Multiresolution analysis of long-term pressure transient data using wavelet methods. SPE paper 48996 (1998)

  • 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)

  • Landauer, R.: The electrical resistance of binary metallic mixtures. J. Appl. Phys. 23, 779 (1952)

    Article  Google Scholar 

  • Lau, T.S., Lo, S.H.: Finite element mesh generation over analytical curved surfaces. Comput. Struct. 59, 301 (1996)

    Article  Google Scholar 

  • Li, D., Beckner, B.: Optimal uplayering for scaleup of multimillion-cell geologic models. SPE paper 62927 (2000)

  • Lo, S.H., Lee, C.K.: Generation of gradation meshes by the background technique. Comput. Struct. 50, 21 (1994)

    Article  Google Scholar 

  • Loze, M.K., Saunders, R.: Two simple algorithms for constructing a two-dimensional constrained Delaunay triangulation. Appl. Numer. Math. 11, 403 (1993)

    Article  Google Scholar 

  • Lu, P., Horne, R.N.: A multiresolution approach to reservoir parameter estimation using wavelet analysis. SPE paper 62985 (2000)

  • Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, San Diego (2008)

    Google Scholar 

  • Mehrabi, A.R., Rassamdana, H., Sahimi, M.: Characterization of long-range correlation in complex distributions and profiles. Phys. Rev. E 56, 712 (1997)

    Article  Google Scholar 

  • Mehrabi, A.R., Sahimi, M.: Coarsening of heterogeneous media: application of wavelets. Phys. Rev. Lett. 79, 4385 (1997)

    Article  Google Scholar 

  • Mehrabi, A.R., Sahimi, M.: Analysis and simulation of long-range correlations in curved space. Int. J. Mod. Phys. C 20, 1211 (2009)

    Article  Google Scholar 

  • 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)

  • Moog, G.J.E.A.: Advanced discretization methods for flow simulation using unstructured grids. Ph.D. Thesis, Stanford University (2013)

  • 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)

    Article  Google Scholar 

  • 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)

  • Neivergelt, Y.: Wavelets Made Easy. Birkhäser, Boston (1999)

    Book  Google Scholar 

  • Neupauer, R.M., Powell, K.L.: A fully-anisotropic Morlet wavelet to identify dominant orientations in a porous medium. Comput. Geosci. 31, 465 (2005)

    Article  Google Scholar 

  • Nordbotten, J.M., Eigestad, G.T.: Discretization on quadrilateral grids with improved monotonicity properties. J. Comput. Phys. 203, 744 (2003)

    Article  Google Scholar 

  • Palagi, C.L., Aziz, K.: The modeling of vertical and horizontal wells with Voronoi grid. SPE paper 24072 (1992)

  • Pancaldi, V., Christensen, K., King, P.R.: Permeability up-scaling using Haar wavelets. Transp. Porous Media 67, 395 (2007)

    Article  Google Scholar 

  • Panda, M.N., Mosher, C., Chopra, A.K.: Application of wavelet transforms to reservoir data analysis and scaling. SPE paper 3656 (1996)

  • Pazhoohesh, E., Hamzehpour, H., Sahimi, M.: Numerical simulation of ac conduction in three-dimensional heterogeneous materials. Phys. Rev. B 73, 174206 (2006)

    Article  Google Scholar 

  • Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulations. Elsevier, New York (1977)

    Google Scholar 

  • Peraire, J., Morgan, K.: Unstructured mesh generation including directional refinement for aerodynamic flow simulation. Finite Elem. Anal. Des. 25, 343 (1997)

    Article  Google Scholar 

  • Portella, R.C.M., Hewett, T.A.: Upscaling, gridding, and simulation using streamtubes. SPE J. 5, 315 (2000)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Qi, D., Zhang, S.: Major challenges for reservoir upscaling. Pet. Sci. Technol. 27, 1985 (2009)

    Article  Google Scholar 

  • Qian, Y.Y., Dhatt, G.: A simple adaptable 2D mesh generation package. Comput. Struct. 53, 801 (1994)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Ron, D., Safro, I., Brandt, A.: Relaxation-based coarsening and multiscale graph organization. Multiscale Model. Simul. 9, 407 (2011)

    Article  Google Scholar 

  • Sahimi, M.: Heterogeneous Materials I, Chapters 4 & 5. Springer, New York (2003)

    Google Scholar 

  • Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock, 2nd edn. Wiley-VCH, Weinheim (2011)

    Book  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Sahimi, M., Hashemi, M.: Wavelet identification of the spatial distribution of fractures. Geophys. Res. Lett. 28, 611 (2001)

    Article  Google Scholar 

  • Sahimi, M., Rasaei, M.R., Ebrahimi, F., Haghighi, M.: Upscaling of unstable displacements and multiphase flows using multiresolution wavelet transformation. SPE paper 93320 (2005)

  • 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)

    Article  Google Scholar 

  • 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)

  • 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)

    Article  Google Scholar 

  • Sibson, R.: Locally equiangular triangulations. Comput. J. 21, 243 (1978)

    Article  Google Scholar 

  • Soliman, M.Y., Ansah, J., Stephenson, S., Mandal, B.: Application of wavelet transform to analysis of pressure transient data. SPE paper 71571 (2001)

  • Stanford University website. https://earth.stanford.edu/researchgroups/supri-b/research/research-areas/gprs

  • Thiele, M.R., Rao, S.E., Blunt, M.J.: Quantifying uncertainty in reservoir performance using streamlines. Math. Geol. 28, 843 (1996)

    Article  Google Scholar 

  • Thomas, C.W.: Principles of Hydrocarbon Reservoir Simulation. International Human Resources Development Corporation, Boston (1982)

    Google Scholar 

  • 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)

  • Vetterli, M., Kovacevic, J.: Wavelets and Subband Coding. Prentice Hall, New York (1995)

    Google Scholar 

  • 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)

  • Watson, D.F.: Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes. Comput. J. 24, 167 (1981)

    Article  Google Scholar 

  • Wavelet Toolbox. http://matlab.izmiran.ru/help/toolbox/wavelet/ch06_a12.html

  • Wen, X.-H., Durlofsky, L.J., Chen, Y.: Efficient 3D implementation of local-global upscaling for reservoir simulation. SPE J. 11, 443 (2006)

    Article  Google Scholar 

  • 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)

  • 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)

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Acknowledgements

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

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  1. Department of Electrical Engineering and Signal and Image Processing Institute, University of Southern California, Los Angeles, CA, 90089-2564, USA

    Amin Rezapour & Antonio Ortega

  2. Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, CA, 90089-1211, USA

    Muhammad Sahimi

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Rezapour, A., Ortega, A. & Sahimi, M. Upscaling of Geological Models of Oil Reservoirs with Unstructured Grids Using Lifting-Based Graph Wavelet Transforms. Transp Porous Med 127, 661–684 (2019). https://doi.org/10.1007/s11242-018-1219-7

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