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Implicit Tetrahedral and Surface Mesh Generation for 3D Reservoir Structural Modeling

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Abstract

This paper presents a method for tetrahedral mesh generation for subsurface reservoir structural modeling based on sequentially cutting an input tetrahedral grid with geological surfaces represented by implicit functions. Most of the existing literature on meshing to implicit surfaces mainly focuses on independent closed surfaces, making it hardly applicable to subsurface structural modeling where, in addition to occasional closed surfaces (e.g., detached salt diapirs), the subsurface model is typically made of a mixture of multiple intersecting finite surfaces (e.g., faults) and infinite surfaces (e.g., horizons). This peculiar setting sometimes makes it challenging to build a sealed structural surface model (i.e., a boundary representation model), which is the starting point of traditional, but also challenging, constrained Delaunay solid mesh generation. The method presented herein bypasses the creation of a sealed surface model and generates a solid mesh directly from point sets and/or a set of topologically disconnected surfaces; the sealed surface model is obtained as a byproduct, if needed.

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References

  • Alliez P, Cohen-Steiner D, Yvinec M, Desbrun M (2005) Variational tetrahedral meshing. ACM Trans Graph 24:617–625

    Article  Google Scholar 

  • Bagley B, Sastry SP, Whitaker RT (2016) A marching-tetrahedra algorithm for feature-preserving meshing of piecewise-smooth implicit surfaces. Procedia Engineering 163:162–174

    Article  Google Scholar 

  • Caumon G, Lepage F, Sword CH, Mallet JL (2004) Building and editing a sealed geological model. Math Geosci 36:927–945

    Google Scholar 

  • Caumon G, Collon P, de Veslud CLC, Viseur S, Sausse J (2009) Surface-based 3D modeling of geological structures. Math Geosci 41:927–945

    Article  Google Scholar 

  • Caumon G, Gray G, Antoine C, Titeux MO (2013) Three-dimensional implicit stratigraphic model building from remote sensing data on tetrahedral meshes: theory and application to regional model of La Popa Basin, NE Mexico. IEEE Trans Geosci Remote Sens 51(3):1613–1621

    Article  Google Scholar 

  • Cavalcante Neto JB, Wawrzynek PA, Carvalho MTM, Martha LF, Ingraffea AR (2001) An algorithm for three-dimensional mesh generation for arbitrary regions with cracks. Eng Comput 17:75–91

    Article  Google Scholar 

  • Chauvin B, Lovely P, Stockmeyer J, Plesch A, Caumon G, Shaw J (2018) Validating novel boundary conditions for three-dimensional mechanics-based restoration: An extensional sandbox model example. AAPG Bull 102:245–266

    Article  Google Scholar 

  • Chen L (2004) Mesh smoothing schemes based on optimal delaunay triangulations. In: IMR

  • Chen L, Holst M (2011) Efficient mesh optimization schemes based on optimal delaunay triangulations. Comput Methods Appl Mech Eng 200:967–984

    Article  Google Scholar 

  • Cheng SW, Dey TK, Shewchuk JR (2012) Delaunay mesh generation. CRC Press, Cambridge

    Google Scholar 

  • Cherpeau N, Caumon G, Lévy B (2010) Stochastic simulations of fault networks in 3D structural modeling. CR Geosci 342:687–694

    Article  Google Scholar 

  • Cherpeau N, Caumon G, Caers J, Lévy B (2012) Method for stochastic inverse modeling of fault geometry and connectivity using flow data. Math Geosci 44:147–168

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Cutler B, Dorsey J, McMillan L (2004) Simplification and improvement of tetrahedral models for simulation. In: Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on geometry processing, SGP ’04. Association for Computing Machinery, Nice, France, pp 93–102

  • Dapogny C, Dobrzynski C, Frey P (2014) Three-dimensional adaptive domain remeshing, implicit domain meshing, and applications to free and moving boundary problems. J Comput Phys 262:358–378

    Article  Google Scholar 

  • Erten H, Üngör A, Zhao C (2009) Mesh smoothing algorithms for complex geometric domains. In: IMR

  • Freitag L, Ollivier-Gooch C (1997) Tetrahedral mesh improvement using swapping and smoothing. Int J Numer Methods Eng 40:3979–4002

    Article  Google Scholar 

  • Gravouil A, Moës N, Belytschko T (2002) Non-planar 3D crack growth by the extended finite element and level sets-part II: level set update. Int J Numer Methods Eng 53:2569–2586

    Article  Google Scholar 

  • Hilton A, Illingworth J (1997) Marching triangles: delaunay implicit surface triangulation. Technical report, University of Surrey

  • Hilton A, Stoddart AJ, Illingworth J, Windeatt T (1996) Marching triangles: range image fusion for complex object modelling. In: Proceedings of 3rd IEEE international conference on image processing, vol 2, pp 381–384

  • Hoffman KS, Neave JW (2007) The fused fault block approach to fault network modelling. Geological Society of London 292

  • Irakarama M (2022) Subsurface tetrahedral mesh generation without a sealed surface model. In: 83rd EAGE conference and exhibition, Madrid, Spain

  • Irakarama M, Laurent G, Renaudeau J, Caumon G (2018) Finite difference implicit modeling of geological structures. In: 80th EAGE conference and exhibition, Copenhagen, Denmark

  • Irakarama M, Laurent G, Renaudeau J, Caumon G (2020) Finite difference implicit structural modeling of geological structures. Math Geosci 53:785–808

    Article  Google Scholar 

  • Irakarama M, Thierry-Coudon M, Zakari M, Anquez P, Caumon G (2021) Implicit 3D subsurface structural modeling by finite elements. In: 82nd EAGE conference and exhibition, Amsterdam, The Netherlands

  • Irakarama M, Thierry-Coudon M, Zakari M, Caumon G (2022) Finite element implicit 3D subsurface structural modeling. Computer-Aided Design

  • Jacquemyn C, Jackson M, Hampson G (2018) Surface-based geological reservoir modelling using grid-free nurbs curves and surfaces. Math Geosci 51:1–28

    Article  Google Scholar 

  • Klingner BM, Shewchuk JR (2007) Aggressive tetrahedral mesh improvement. In: 16th international meshing roundtable, pp 3–23

  • Knupp P (2000) Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Int J Numer Methods Eng 48:1165–1185

    Article  Google Scholar 

  • Labelle F, Shewchuk JR (2007) Isosurface stuffing: fast tetrahedral meshes with good dihedral angles. ACM Trans Graph 26(3):57–es

  • Laurent G, Caumon G, Bouziat A, Jessell M (2013) A parametric method to model 3d displacements around faults with volumetric vector fields. Tectonophysics 590:83–93

    Article  Google Scholar 

  • Lévy B, Petitjean S, Ray N, Maillot J (2002) Least squares conformal maps for automatic texture atlas generation. ACM Trans Graph 21(3):362–371

    Article  Google Scholar 

  • Mallet JL (2002) Geomodeling. Applied geostatistics series. Oxford University Press, Oxford

    Google Scholar 

  • Mallet JL (2014) Elements of mathematical sedimentary geology: the GeoChron model. EAGE

  • Moyen R (2005) Paramétrisation 3D de l’espace en géologie sédimentaire: le modèle GeoChron. Ph.D. thesis, Université de Lorraine-ENSG, Nancy, France

  • Nielson GM, Sung J (1997) Interval volume tetrahedrization. In: Proceedings of the 8th conference on visualization ’97, VIS ’97. IEEE Computer Society Press, Phoenix, pp 221–ff

  • Renaudeau J, Irakarama M, Laurent G, Maerten F, Caumon G (2018) Implicit modeling of geological structures: a Cartesian grid method handling discontinuities with ghost points. In: 41st international conference on boundary elements and other mesh reduction methods, New Forest, UK

  • Ricci A (1973) A constructive geometry for computer graphics. Comput J 16(2):157–160

    Article  Google Scholar 

  • Sethian J, Popovici A (1999) 3d traveltime computation using the fast marching method. Geophysics 64(2):516–523

    Article  Google Scholar 

  • Shewchuk JR (1997) Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discrete Comput Geometry 18(3):305–363

    Article  Google Scholar 

  • Si H (2015) Tetgen, a delaunay-based quality tetrahedral mesh generator. ACM Trans Math Softw (TOMS) 41:1–36

    Article  Google Scholar 

  • Si H, Gärtner K (2005) Meshing piecewise linear complexes by constrained delaunay tetrahedralizations. In: 14th international meshing roundtable, San Diego, California, USA

  • Surazhsky V, Gotsman C (2003) Explicit surface remeshing. In: Eurographics symposium on geometry processing. The Eurographics Association

  • Tertois AL (2007) Création et modification de modèles géologiques par champs de potentiel. Application au modèle GeoChron. Ph.D. thesis, Université de Lorraine-ENSG, Nancy, France

  • Zehner B, Börner JH, Görz I, Spitzer K (2015) Workflows for generating tetrahedral meshes for finite element simulations on complex geological structures. Comput Geosci 79:105–117

    Article  Google Scholar 

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Acknowledgements

I would like to thank IFPEN and C &C Reservoirs for the input data in Fig. 1, TotalEnergies for the data in Fig. 9a, an anonymous source for the data in Fig. 13a, and the RING Consortium for granting me access to these data sets. The data in Fig. 16a is a subset of the open-access geophysical interpretation data of the Volve field published by Equinor. I am thankful to five anonymous reviewers for improving this paper. This work benefits from years of insightful discussions with Guillaume Caumon (RING) on subsurface modeling, and from Shewchuk (1997)’s freely available robust geometric predicates.

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  1. Independent Research, 92160, Antony, France

    Modeste Irakarama

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  1. Modeste Irakarama
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Correspondence to Modeste Irakarama.

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Irakarama, M. Implicit Tetrahedral and Surface Mesh Generation for 3D Reservoir Structural Modeling. Math Geosci 54, 1349–1369 (2022). https://doi.org/10.1007/s11004-022-10016-y

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