Advertisement

Transport in Porous Media

, Volume 98, Issue 2, pp 377–400 | Cite as

A Comparison Study Between an Adaptive Quadtree Grid and Uniform Grid Upscaling for Reservoir Simulation

  • Masoud BabaeiEmail author
  • Ahmed H. Elsheikh
  • Peter R. King
Article

Abstract

An adaptive quadtree grid generation algorithm is developed and applied for tracer and multiphase flow in channelized heterogeneous porous media. Adaptivity was guided using two different approaches. In the first approach, wavelet transformation was used to generate a refinement field based on permeability variations. The second approach uses flow information based on the solution of an initial-time fine-scale problem. The resulting grids were compared with uniform grid upscaling. For uniform upscaling, two commonly applied methods were used: renormalization upscaling and local-global upscaling. The velocities obtained by adaptive grid and uniformly upscaled grids, were downscaled. This procedure allows us to separate the upscaling errors, on adaptive and uniform grids, from the numerical dispersion errors resulting from solving the saturation equation on a coarse grid. The simulation results obtained by solving on flow-based adaptive quadtree grids for the case of a single phase flow show reasonable agreement with more computationally demanding fine-scale models and local-global upscaled models. For the multiphase case, the agreement is less evident, especially in piston-like displacement cases with sharp frontal movement. In such cases a non-iterative transmissibility upscaling procedure for adaptive grid is shown to significantly reduce the errors and make the adaptive grid comparable to iterative local-global upscaling. Furthermore, existence of barriers in a porous medium complicates both upscaling and grid adaptivity. This issue is addressed by adapting the grid using a combination of flow information and a permeability based heuristic criterion.

Keywords

Adaptive quadtree grid Permeability- and flow-based gridding  Renormalization upscaling Local-global upscaling  Downscaling 

Notes

Acknowledgments

The authors are grateful to one of the anonymous reviewers for suggesting to implement transmissibility-based upscaling. This suggestion has made the numerical comparison more consistent. We also, would like to thank the editor and the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

References

  1. Aarnes, J.E., Efendiev, Y.: An adaptive multiscale method for simulation of fluid flow in heterogeneous porous media. Multiscale Model. Simul. 5(3), 918–939 (2006)CrossRefGoogle Scholar
  2. Aarnes, J.E., Krogstad, S., Lie, K.A.: 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)CrossRefGoogle Scholar
  3. Babaei, M., King, P.R.: A modified nested-gridding for upscaling downscaling in reservoir simulation. Transp. Porous Med. 93(3), 753–775 (2012)CrossRefGoogle Scholar
  4. Batycky, R.P., Blunt, M.J., Thiele, M.R.: A 3D field-scale streamline-based reservoir simulator. SPE Reserv. Eng. 12(4), 246–254 (1997)Google Scholar
  5. Begg, S., Carter, R., Dranfield, P.: Assigning effective values to simulator gridblock parameters for heterogeneous reservoirs. SPE Reserv. Eng. 4(4), 455–463 (1989)Google Scholar
  6. Castellini, A.: Flow based grids for reservoir simulation. Stanford University, Stanford, CA, Master’s thesis (2001)Google Scholar
  7. Castellini, A., Edwards, M.G., Durlofsky, L.J.: Flow based modules for grid generation in two and three dimensions. In: Proceedings of 7th European Conference on the Mathematics of Oil Recovery, Baveno, Italy (2000).Google Scholar
  8. Chen, Y., Durlofsky, L.J.: Adaptive local-global upscaling for general flow scenarios in heterogeneous formations. Transp. Porous Med. 62(2), 157–185 (2006a)CrossRefGoogle Scholar
  9. Chen, Y., Durlofsky, L.J.: Efficient incorporation of global effects in upscaled models of two-phase flow and transport in heterogeneous formations. Multiscale Model. Simul. 5, 445–475 (2006b)CrossRefGoogle Scholar
  10. Chen, Y., Li, Y.: Local-global two-phase upscaling of flow and transport in heterogeneous formations. Multiscale Model. Simul. 8, 125–153 (2009)CrossRefGoogle Scholar
  11. Chen, Y., Durlofsky, L., Gerritsen, M., Wen, X.H.: A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations. Adv. Water Resour. 26(10), 1041–1060 (2003)CrossRefGoogle Scholar
  12. Christie, M.: Upscaling for reservoir simulation. J. Petroleum Technol. 48(11), 1004–1010 (1996)Google Scholar
  13. Christie, M., Blunt, M.: Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reserv. Eval. Eng. 4(4), 308–317 (2001)Google Scholar
  14. Corey, A.: The interrelation between gas and oil relative permeabilities. Producers Mon. 19(1), 38–41 (1954)Google Scholar
  15. Darman, N., Durlofsky, L.J., Sorbie, K., Pickup, G.: Upscaling immiscible gas displacements: quantitative use of fine-grid flow data in grid-coarsening schemes. SPE J. 6(1), 47–56 (2001)Google Scholar
  16. de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications. Springer, Berlin (2000)Google Scholar
  17. Donoho, D.L.: De-noising by soft-thresholding. IEEE Trans. Inf. Theory 41(3), 613–627 (1995)CrossRefGoogle Scholar
  18. Donoho, D.L., Johnstone, I.M.: Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90, 1200–1224 (1995)CrossRefGoogle Scholar
  19. Durlofsky, L.J.: Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resour. Res. 27(5), 699–708 (1991)CrossRefGoogle Scholar
  20. Durlofsky, L.J.: Use of higher moments for the description of upscaled, process independent relative permeabilities. SPE J. 2(4), 474–484 (1997)Google Scholar
  21. Durlofsky, L.J.: Upscaling and gridding of fine scale geological models for flow simulation. In: Proceedings of 8th International Forum on Reservoir Simulation, Iles Borromees, Stresa, Italy (2005).Google Scholar
  22. Durlofsky, L.J., Behrens, R., Jones, R., Bernath, A.: Scale up of heterogeneous three dimensional reservoir descriptions. SPE J. 1(3), 313–326 (1996)Google Scholar
  23. Durlofsky, L.J., Jones, R.C., Milliken, W.J.: A nonuniform coarsening approach for the scale-up of displacement processes in heterogeneous porous media. Adv. Water Resour. 20(5–6), 335–347 (1997)CrossRefGoogle Scholar
  24. Ebrahimi, F., Sahimi, M.: Multiresolution wavelet coarsening and analysis of transport in heterogeneous media. Physica A 316(1–4), 160–188 (2002)CrossRefGoogle Scholar
  25. 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(8–10), 772–785 (2004)Google Scholar
  26. Edwards, M.G.: Elimination of adaptive grid interface errors in the discrete cell centered pressure equation. J. Comput. Phys. 126(2), 356–372 (1996)CrossRefGoogle Scholar
  27. Edwards, M.G.: Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci. 6(3), 433–452 (2002)CrossRefGoogle Scholar
  28. Efendiev, Y., Durlofsky, L.J.: A generalized convection-diffusion model for subgrid transport in porous media. Multiscale Model. Simul. 1, 504–526 (2003)CrossRefGoogle Scholar
  29. Farmer, C., Heath, D., Moody, R.: A global optimization approach to grid generation. Proceedings of SPE Symposium on Reservoir Simulation, Anaheim, California, In (1991)Google Scholar
  30. Forsyth, P., Sammon, P.: Local mesh refinement and modeling of faults and pinchouts. SPE Form. Eval. 1(3), 275–285 (1986)Google Scholar
  31. Garcia, M., Journel, A., Aziz, K.: Automatic grid generation for modeling reservoir heterogeneities. SPE Reserv. Eng. 7(2), 278–284 (1992)Google Scholar
  32. Gerritsen, M., Lambers, J.V.: Integration of local-global upscaling and grid adaptivity for simulation of subsurface flow in heterogeneous formations. Comput. Geosci. 12(2), 193–208 (2008)CrossRefGoogle Scholar
  33. Hauge, V.L., Lie, K.A., Natvig, J.R.: Flow-based coarsening for multiscale simulation of transport in porous media. Comput. Geosci. 16(2), 391–408 (2011)CrossRefGoogle Scholar
  34. He, C.: Structured flow-based gridding and upscaling for reservoir simulation. PhD thesis, Stanford University (2005).Google Scholar
  35. He, C., Durlofsky, L.J.: Structured flow-based gridding and upscaling for modeling subsurface flow. Adv. Water Resour. 29(12), 1876–1892 (2006)CrossRefGoogle Scholar
  36. King, P.R.: The use of renormalization for calculating effective permeability. Transp. Porous Med. 4(1), 37–58 (1989)CrossRefGoogle Scholar
  37. Kippe, V., Aarnes, J.E., Lie, K.A.: A comparison of multiscale methods for elliptic problems in porous media flow. Comput. Geosci. 12(3), 377–398 (2008)CrossRefGoogle Scholar
  38. Lee, S.H., Zhou, H., Tchelepi, H.A.: Adaptive multiscale finite-volume method for nonlinear multiphase transport in heterogeneous formations. J. Comput. Phys. 228(24), 9036–9058 (2009)CrossRefGoogle Scholar
  39. Li, D., Cullick, A.S., Lake, L.W.: Global scale-up of reservoir model permeability with local grid refinement. J. Petroleum Sci. Eng. 14(1–2), 1–13 (1995)CrossRefGoogle Scholar
  40. Mlacnik, M., Durlofsky, L.J., Heinemann, Z.: Sequentially adapted flow-based PEBI grids for reservoir simulation. SPE J. 11(3), 317–327 (2006)Google Scholar
  41. Nilsson, J., Gerritsen, M., Younis, R.: A novel adaptive anisotropic grid framework for efficient reservoir simulation. Proceedings of SPE Reservoir Simulation Symposium, The Woodlands, Texas, In (2005)Google Scholar
  42. Pickup, G.E., Ringrose, P.S., Jensen, J.L., Sorbie, K.S.: Permeability tensors for sedimentary structures. Math. Geol. 26(2), 227–250 (1994)CrossRefGoogle Scholar
  43. Prevost, M., Lepage, F., Durlofsky, L.J., Mallet, J.L.: Unstructured 3D gridding and upscaling for coarse modelling of geometrically complex reservoirs. Petroleum Geosci. 11(4), 339–345 (2005)CrossRefGoogle Scholar
  44. Qi, D., Wong, P., Liu, K.: An improved global upscaling approach for reservoir simulation. Petroleum Sci. Technol. 19(7–8), 779–795 (2001)CrossRefGoogle Scholar
  45. Quandalle, P., Besset, P.: Reduction of grid effects due to local sub-gridding in simulations using a composite grid. Proceedings of SPE Reservoir Simulation Symposium, Dallas, Texas, In (1985)Google Scholar
  46. Rasaei, M.R., Sahimi, M.: Upscaling and simulation of waterflooding in heterogeneous reservoirs using wavelet transformations: application to the SPE-10 model. Transp. Porous Med. 72(3), 311–338 (2008a)CrossRefGoogle Scholar
  47. 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(2), 187–214 (2008b)CrossRefGoogle Scholar
  48. Stein, C.M.: Estimation of the mean of a multivariate normal distribution. Ann. Stat. 9(6), 1135–1151 (1981)CrossRefGoogle Scholar
  49. Wen, X.H., Gómez-Hernández, J.J.: Upscaling hydraulic conductivities in heterogeneous media: an overview. J. Hydrol. 183(1–2):ix–xxxii (1996)Google Scholar
  50. Wen, X.H., Durlofsky, L.J., Edwards, M.G.: Upscaling of channel systems in two dimensions using flow-based grids. Transp. Porous Med. 51(3), 343–366 (2003)CrossRefGoogle Scholar
  51. Younis, R., Caers, J.: A method for static-based upgridding. In: Proceedings of the 8th European Conference on the Mathematics of Oil Recovery (2002)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Masoud Babaei
    • 1
    Email author
  • Ahmed H. Elsheikh
    • 1
  • Peter R. King
    • 1
  1. 1.Department of Earth Science & EngineeringImperial College LondonLondonUK

Personalised recommendations