Skip to main content
SpringerLink
Log in

Upscaled Unstructured Computational Grids for Efficient Simulation of Flow in Fractured Porous Media

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

Discrete fracture modeling (DFM) is currently the most promising approach for modeling of naturally fractured reservoirs and simulation of multiphase fluid flow therein. In contrast with the classical double-porosity/double permeability models, in the DFM approach all the interactions and fluid flow in and between the fractures and within the matrix are modeled in a unified manner, using the same computational grid. There is no need for computing the shape factors, which are crucial to the accuracy of the double-porosity models. We have exploited this concept in order to develop a new method for the generation of unstructured computational grids. In the new approach the geological model (GM) of the reservoir is first generated, using square or cubic grid blocks. The GM is then upscaled using a method based on the multiresolution wavelet transformations that we recently developed. The upscaled grid contains a distribution of the square or cubic blocks of various sizes. A map of the blocks’ centers is then used with an optimized Delauney triangulation method and the advancing-front technique, in order to generate the final unstructured triangulated grid suitable for use in any general reservoir simulator with any number of fluid phases. The new method also includes an algorithm for generating fractures that, contrary to the previous methods, does not require modifying their paths due to the complexities that may arise in spatial distribution of the grid blocks. It also includes an effective partitioning of the simulation domain that results in large savings in the computation times. The speed-up in the computations with the new upscaled unstructured grid is about three orders of magnitude over that for the initial GM. Simulation of waterflooding indicates that the agreement between the results obtained with the GM and the upscaled unstructured grid is excellent. The method is equally applicable to the simulations of multiphase flow in unfractured, but highly heterogeneous, reservoirs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Adler P.M., Thovert J.-F.: Fractures and Fracture Networks. Kluwer, Dordrecht (1999)

    Google Scholar 

  • Amado, L.C.W., Pedrosa, O.A.: A finite volume approach with triangular grid for flow problems in reservoirs. SPE Paper 23633 (1992)

  • Aziz K.: Reservoir simulation grids: opportunities and problems. J. Pet. Technol. 45(7), 658 (1993)

    Google Scholar 

  • Bogdanov I.I., Mourzenko V.V., Thovert J.-F., Adler P.M.: Two-phase flow through fractured porous media. Phys. Rev. E 68, 026703 (2003a)

    Article  Google Scholar 

  • Bogdanov I.I., Mourzenko V.V., Thovert J.-F., Adler P.M.: Pressure drawdown well tests in fractured porous media. Water. Resour. Res. 39, 1021 (2003b). doi:10.1029/2000WR000080

    Article  Google Scholar 

  • Bogdanov I.I., Mourzenko V.V., Thovert J.-F., Adler P.M.: Effective permeability of fractured porous media in steady state flow. Water. Resour. Res. 39, 1029 (2003c). doi:10.1029/2001WR000756

    Article  Google Scholar 

  • Castellini, A., Edwards, M.G., Durlofsky, L.J.: Flow based modules for grid generation in two and three dimensions. In: Proceeding of 7th European Conference on the Mathematics of Oil Recovery, Baveno, Lago Maggiore, Italy, September 2000

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

    Article  Google Scholar 

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

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

    Article  Google Scholar 

  • Daubechies I.: Orthonormal basis of compactly supported wavelets. Commun. Pure Appl. Math. 41, 901 (1988)

    Google Scholar 

  • Daubechies I.: Ten Lecture on Wavelets. SIAM, Philadelphia (1992)

    Google Scholar 

  • Du Q., Wang D.: Constrained boundary recovery for three dimensional Delaunay triangulation. Int. J. Numer. Methods Eng. 61, 1471 (2004)

    Article  Google Scholar 

  • 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 

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

    Article  Google Scholar 

  • Ebrahimi F., Sahimi M.: Multiresolution wavelet scale up of unstable miscible displacements in flow through heterogeneous 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 transformation. Mech. Mater. 38, 772 (2006)

    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)

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

  • Geiger S., Roberts S., Matthäi S.K., Zoppou C., Burri A.: Combining finite volume and finite element methods for efficient multiphase flow simulation in highly heterogeneous and structurally complex geologic media. Geofluid 4, 284 (2004)

    Article  Google Scholar 

  • George, J.A.: Computer implementation of the finite element method. Ph.D. Thesis, Stanford University (1971)

  • Granet S., Fabrie P., Lemonnier P., Quintard M.: A two-phase flow simulation of a fractured reservoir using a new fissure element method. J. Pet. Sci. Eng. 32, 35 (2001)

    Article  Google Scholar 

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

  • 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)

    Google Scholar 

  • Karimi-Fard, M., Durlofsky, L.J., Aziz, K.: An efficient discrete fracture model applicable for general purpose reservoir simulators. SPE Paper 79699, SPE Reservoir Simulation Symposium, Houston, Texas (2003)

  • King P.R.: The use of renormalization for calculating effective permeability. Transp. Porous Media 4, 37 (1989)

    Article  Google Scholar 

  • Koudina N., Gonzalez Garcia R., Thovert J.-F., Adler P.M.: Permeability of three-dimensional fracture networks. Phys. Rev. E 57, 4466 (1998)

    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)

  • Li D., Cullick A.S., Lake L.W.: Global scale-up of reservoir model permeability with local grid refinement. J. Pet. Sci. Eng. 14, 1 (1995)

    Article  Google Scholar 

  • Lo S.H.: A new mesh generation scheme for arbitrary planar domains. Int. J. Numer. Methods Eng. 21, 1403 (1985)

    Article  Google Scholar 

  • 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 

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

    Article  Google Scholar 

  • Molz F.J., Rajaram H., Lu S.: Stochastic fractal-based models of heterogeneity in subsurface hydrology: origins, applications, limitations and future research questions. Rev. Geophys. 42, RG1002 (2004)

    Article  Google Scholar 

  • Pao W.K.S., Lewis R.W.: Three-dimensional finite element simulation of three-phase flow in a deforming fissured reservoir. Comput. Methods Appl. Mech. Eng. 191, 2631 (2002)

    Article  Google Scholar 

  • 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 

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

    Article  Google Scholar 

  • Peraire J., Vahdati M., Morgan K., Zienkiewicz O.C.: Adaptive remeshing for compressible flow computations. J. Comput. Phys. 72, 449 (1987)

    Article  Google Scholar 

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

    Google Scholar 

  • Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P.: Numerical Recipes, 2nd edn. Cambridge University Press, London (1992)

    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 

  • Rozon, B.J.: A generalized finite volume discretization method for reservoir simulation. SPE Paper 18414 (1989)

  • Sahimi M.: Flow and Transport in Porous Media and Fractured Rock. VCH, Weinheim (1995)

    Google Scholar 

  • Sahimi M.: Large-scale porous media and wavelet transformations. Comput. Sci. Eng. 5(4), 75 (2003)

    Article  Google Scholar 

  • Sahimi M., Tajer S.E.: Self-affine distributions of the bulk density, elastic moduli, and seismic wave velocities of rock. Phys. Rev. E 71, 046301 (2005)

    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, SPE Reservoir Simulation Symposium, Houston, Texas (2005)

  • Santos, R.L.A., Pedrosa, O.A.: An efficient finite volume approach for modelling miscible displacement. SPE Paper 23691 (1992)

  • Shan J.-L., Guan Z.-Q., Song C.: A reliable and effective tetrahedral meshing algorithm. Chin. J. Comput. 11, 104 (2007)

    Google Scholar 

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

    Article  Google Scholar 

  • Verma, S.K.: Flexible grid for reservoir simulation. PhD thesis, Stanford University (1996)

  • Verma, S.K., Aziz, K.: A control volume scheme for flexible grids for reservoir simulation. SPE Paper 37999, SPE Reservoir Simulation Symposium, Dallas, Texas (1997)

  • 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 

  • Younis, R.M., Caers, J.: A method for static-base upgridding. In: Proceedings of 8th European Conference on the Mathematics of Oil Recovery, Freiberg, Germany, September 2002

  • Zienkiewicz O.C., Wu J.: Automatic directional refinement in adaptive analysis of compressible flows. Int. J. Numer. Methods Eng. 37, 2189 (1994)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Muhammad Sahimi

  2. Institute for Petroleum Engineering, University of Tehran, 11365-4563, Tehran, Iran

    R. Darvishi & M. Reza Rasaei

  3. The Australian School of Petroleum, University of Adelaide, Adelaide, 5000, Australia

    Manouchehr Haghighi

Authors
  1. Muhammad Sahimi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Darvishi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Manouchehr Haghighi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Reza Rasaei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Muhammad Sahimi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sahimi, M., Darvishi, R., Haghighi, M. et al. Upscaled Unstructured Computational Grids for Efficient Simulation of Flow in Fractured Porous Media. Transp Porous Med 83, 195–218 (2010). https://doi.org/10.1007/s11242-009-9500-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-009-9500-4

Keywords

Search

Navigation