Skip to main content
SpringerLink
Log in

Voronoi grids conforming to 3D structural features

  • ORIGINAL PAPER
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

Flow simulation in a reservoir can be highly impacted by upscaling errors. These errors can be reduced by using simulation grids with cells as homogeneous as possible, hence conformable to horizons and faults. In this paper, the coordinates of 3D Voronoi seeds are optimized so that Voronoi cell facets honor the structural features. These features are modeled by piecewise linear complex (PLC). The optimization minimizes a function made of two parts: (1) a barycentric function, which ensures that the cells will be of good quality by maximizing their compactness; and (2) a conformity function, which allows to minimize the volume of cells that is isolated from the Voronoi seed w.r.t., a structural feature. To determine the isolated volume, a local approximation of the structural feature inside the Voronoi cells is used to cut the cells. It improves the algorithm efficiency and robustness compared to an exact cutting procedure. This method, used jointly with an adaptive gradient solver to minimize the function, allows dealing with complex 3D geological cases. It always produces a Voronoi simulation grid with the desired number of cells.

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

  1. Branets, L.V., Ghai, S.S., Lyons, S.L., Wu, X.H.: Efficient and accurate reservoir modeling using adaptive gridding with global scale up. In: Proceedings of SPE Reservoir Simulation Symposium. The Woodlands (2009). 11p SPE118946 doi:10.2118/118946-MS.

  2. Cao, H.: Development of Techniques for General Purpose Simulators, p. 202. Ph.D. thesis, Stanford University (2002)

  3. Chen, Y., Wu, X.: Upscaled modeling of well singularity for simulating flow in heterogeneous formations. Comput. Geosci. 12(1), 29–45 (2008). doi:10.1007/s10596-007-9059-5

    Article  Google Scholar 

  4. Colletta, B., Letouzey, J., Pinedo, R., Ballard, J., Balé, P.: Computerized x-ray tomography analysis of sandbox models: examples of thin-skinned thrust systems. Geology 19, 1063–1067 (1991)

    Article  Google Scholar 

  5. Courrioux, G., Nullans, S., Guillen, A., Boissonnat, J.D., Repusseau, P., Renaud, X., Thibaut, M.: 3D volumetric modelling of cadomian terranes (northern brittany, france): an automatic method using Voronoi diagrams. Tectonophysics 331(1–2), 181–196 (2001). doi:10.1016/S0040-1951(00)00242-0

    Article  Google Scholar 

  6. Du, Q., Faber, V., Gunzburger, M.: Centroidal Voronoi tesselations: Applications and algorithms. SIAM Rev. 41, 637–676 (1999)

    Article  Google Scholar 

  7. Durlofsky, L.J.: Upscaling and gridding of fine scale geological models for flow simulation. In: 8th International Forum on Reservoir Simulation. Iles Borromees, Stresa, Italy (2005). 59p

  8. Evazi, M., Mahani, H.: Unstructured coarse grid generation using background-grid approach. SPE J. 15(2), 326–340 (2010)

    Article  Google Scholar 

  9. Lévy, B., Liu, Y.: Lp centroidal Voronoi tessellation and its application. ACM Trans. Graph. 29(4) (2010). doi:10.1145/1778765.1778856. 11p

  10. Liu, Y., Wang, W., Lévy, B., Sun, F., Yan, D.M., Lu, L., Yang, C.: On centroidal Voronoi tesselation—energy smoothness and fast computation. ACM Trans. Graph. 8(4) (2009) doi:10.1145/5597551559758 . 30p

  11. Merland, R., Lévy, B., Caumon, G.: Building PEBI grids conforming to 3D geological features using centroidal Voronoi tessellations. In: Proceedings IAMG. Salzburg (2011). 12p

  12. Merland, R., Lévy, B., Caumon, G., Collon-Drouaillet, P.: Building centroidal Voronoi tessellations for flow simulation in reservoirs using flow information. In: Proceedings of SPE Reservoir Simulation Symposium. The Woodlands (2011). SPE141018-PP, 11p. doi:10.2118/141018-MS.

  13. Mlacnik, M., Durlofsky, L.: Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients.J. Comput. Phys. 216(1), 337–361 (2006). doi:10.1016/j.jcp.2005.12.007

    Article  Google Scholar 

  14. Mlacnik, M., Durlofsky, L., Heinemann, Z.: Sequentially adapted flow-based PEBI grids for reservoir simulation. SPE J. 11(3), 317–327 (2006)

    Article  Google Scholar 

  15. Nivoliers, V.: échantillonnage pour l’approximation de fonctions sur des maillages ch. 4. Ph.D. thesis, Université de Lorraine (2012).

  16. Palagi, C.L., Aziz, K.: Modeling vertical and horizontal wells with Voronoi grid. SPE Reserv. Eng. 9(1), 15–21 (1994). doi:10.2118/24072-PA

    Article  Google Scholar 

  17. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes: The Art of Scientific Computing, Chap. 10, 2nd edn. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  18. Prevost, M., Lepage, F., Durlofsky, L., Mallet, J.L.: Unstructured 3D gridding and upscaling for coarse modelling of geometrically complex reservoirs. Pet. Geosci. 11(4), 339–345 (2005)

    Article  Google Scholar 

  19. Samet, H.: Foundations of multidimensional and metric data structures. Computer Graphics and Geometric Modeling, Morgan Kaufmann Series p. 993 (2006)

  20. Souche, L.: Generation of unstructured 3D streamline pressure-potential-based k-orthogonal grids. In: Proceedings of 9th ECMOR. Cannes (2004). 8p

  21. Verma, S.: Flexible Grids for Reservoir Simulation. Ph.D. thesis. Standford University (1996). 247p

  22. Verma, S., Aziz, K.: A control volume scheme for flexible grids in reservoir simulation. In: Proceedings of SPE Reservoir Simulation Symposium. Dallas (1997) doi:10.2118/37999-MS. 13p

  23. Yan, D.M., Lévy, B., Liu, Y., Sun, F., Wang, W.: Isotropic remeshing with fast and exact computation of restricted Voronoi diagram. Comput. Graph. Forum 28(5), 1445–1454 (2009). doi:10.1111/j.1467-8659.2009.01521.x

    Article  Google Scholar 

  24. Yan, D.M., Wang, W., Lévy, B., Liu, Y.: Efficient computation of 3D clipped Voronoi diagram. In: Advances in Geometric Modeling Processing, pp. 269–282 (2010)

Download references

Author information

Authors and Affiliations

  1. Géoressources (UMR 7356), Université de Lorraine-ENSG / CNRS / CREGU, Vandœuvre-lès-Nancy, France

    Romain Merland, Guillaume Caumon & Pauline Collon-Drouaillet

  2. Institut Nationale de Recherche en Informatique, Le Chesnay, France

    Bruno Lévy

Authors
  1. Romain Merland
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guillaume Caumon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bruno Lévy
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Pauline Collon-Drouaillet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Romain Merland.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Merland, R., Caumon, G., Lévy, B. et al. Voronoi grids conforming to 3D structural features. Comput Geosci 18, 373–383 (2014). https://doi.org/10.1007/s10596-014-9408-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-014-9408-0

Keywords

Search

Navigation