Abstract
We present a new method for generating pure hexahedral meshes for reservoir simulations. The grid is obtained by extruding a quadrangular mesh, using ideas from the latest advances in computational geometry, specifically the generation of semi-structured quadrangular meshes based on global parameterization. Hexahedral elements are automatically constructed to smoothly honor the geometry of input features (domain boundaries, faults, and horizons), thus making it possible to be used for multiple types of physical simulations on the same mesh. The main contributions are as follows: the introduction of a new semi-structured hexahedral meshing workflow producing high-quality meshes for a wide range of fault systems, and the study and definition of weak verticality on triangulated surface meshes. This allows us to design better and more robust algorithms during the extrusion phase along non-vertical faults. We demonstrate (i) the simplicity of using such hexahedral meshes generated using the proposed method for coupled flow-geomechanics simulations with state-of-the-art simulators for reservoir studies, and (ii) the possibility of using such semi-structured hexahedral meshes in commercial structured flow simulators, offering an alternative gridding approach to handle a wider family of fault networks without recourse to the stair-step fault approximation.
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References
Bommes D, Zimmer H, Kobbelt L (2009) Mixed-integer quadrangulation. ACM Trans Graph 28(3):77. https://doi.org/10.1145/1531326.1531383
Bommes D, Zimmer H, Kobbelt L (2012) Practical mixed-integer optimization for geometry processing. In: Proceedings of the 7th international conference on curves and surfaces. Springer, Berlin, Heidelberg, pp 193–206
Botella A, Lévy B, Caumon G (2016) Indirect unstructured hex-dominant mesh generation using tetrahedra recombination. Comput Geosci 20:437–451. https://doi.org/10.1007/s10596-015-9484-9
Campen M, Bommes D, Kobbelt L (2015) Quantized global parametrization. ACM Trans Graph 10(1145/2816795):2818140
Crane K (2019) The n-dimensional cotangent formula. https://api.semanticscholar.org/CorpusID:203578769
Desobry D, Protais F, Ray N, Corman E, Sokolov D (2022) Frame Fields for CAD models. Lect Notes Comput Sci 13018:421–434. https://doi.org/10.1007/978-3-030-90436-4_34
Ebke H-C, Bommes D, Campen M, Kobbelt L (2013) QEx: robust quad mesh extraction. ACM Trans Graph 4:168–116810. https://doi.org/10.1145/2508363.2508372
Fremming N (2002) 3d geological model construction using a 3d grid. https://doi.org/10.3997/2214-4609.201405917
Garanzha V, Kaporin I, Kudryavtseva L, Protais F, Ray N, Sokolov D (2021) Foldover-free maps in 50 lines of code. ACM Trans Graph 10(1145/3450626):3459847
Gringarten E, Haouesse A, Arpat B, Nghiem L (2009) Advantages of using vertical stair step faults in reservoir grids for flow simulation. In: Proceedings of SPE reservoir simulation conference. SPE, pp SPE-119188
Gringarten E, Lecuyer JD, Villarubias E, Cosson C, Li W-C (2017) Optimized grids for accurately representing geology in geomechanical simulations. In: Proceeding of SPE annual technical conference and exhibition
Gross H, Mazuyer A (2021) GEOSX: a multiphysics, multilevel simulator designed for exascale computing. In: SPE Reservoir Simulation Conference. SPE, On-Demand, pp 011-010007. https://doi.org/10.2118/203932-MS. https://onepetro.org/spersc/proceedings/21RSC/1-21RSC/D011S010R007/470761. Accessed 28 July 2022
Hoffman K, Neave J, Klein R (2003). Streamlining the workflow from structure model to reservoir grid. https://doi.org/10.2118/84280-MS
Kälberer F, Nieser M, Polthier K (2007) Quadcover—surface parameterization using branched coverings. Comput Graph Forum 26(3):375–384. https://doi.org/10.1111/j.1467-8659.2007.01060.x
Klemetsdal ØS, Berge RL, Lie K-A, Nilsen HM, Møyner O (2017) Unstructured gridding and consistent discretizations for reservoirs with faults and complex wells. In: Proceedings SPE reservoir simulation conference. SPE, p D031S009R005
Lévy B, Petitjean S, Ray N, Maillot J (2002) Least squares conformal maps for automatic texture atlas generation. In: ACM (ed) ACM SIGGRAPH conference proceedings. http://www.loria.fr/publications/2002/A02-R-065/A02-R-065.ps
Mallet J-L (2002) Geomodeling/Jean–Laurent Mallet. Applied Geostatistics Series. Oxford University Press, Oxford
Mallison B, Sword C, Viard T, Milliken W, Cheng A (2013) Unstructured cut-cell grids for modeling complex reservoirs. SPE J 19:340
Merland R, Lévy B, Caumon G (2011) Building PEBI grids conforming to 3D geological features using centroidal voronoi tessellations. In: Proceeding of IAMG
Palagi CL, Aziz K (1994) Use of Voronoi grid in reservoir simulation. SPE Adv Technol Ser 2:69–77
Pellerin J, Caumon G, Julio C, Mejia-Herrera P, Botella A (2015) Elements for measuring the complexity of 3d structural models: connectivity and geometry. Comput Geosci 76:130–140. https://doi.org/10.1016/j.cageo.2015.01.002
Ray N, Vallet B, Li WC, Lévy B (2008) N-symmetry direction field design. ACM Trans Graph 10(1145/1356682):1356683
Santoshini S, Harris S, Kashem S, Levannier A, Benabbou A, Viard T, Macé L (2018) Depogrid: Next generation unstructured grids for accurate reservoir modeling and simulation. https://doi.org/10.2118/191615-18RPTC-MS
Si H (2015) TetGen, a delaunay-based quality tetrahedral mesh generator. ACM Trans Math Software 41:1–36
Vaxman A, Campen M, Diamanti O, Bommes D, Hildebrandt K, Technion MB-C, Panozzo D (2017) Directional field synthesis, design, and processing. In: ACM SIGGRAPH 2017 Courses. SIGGRAPH ’17. Association for Computing Machinery, New York, NY, USA. https://doi.org/10.1145/3084873.3084921
Wellmann F, Caumon G (2018) Chapter one-3d structural geological models: concepts, methods, and uncertainties. Adv Geophys 59:1–121. https://doi.org/10.1016/bs.agph.2018.09.001
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Lopez, D., Coudert-Osmont, Y., Desobry, D. et al. 2.5D Hexahedral Meshing for Reservoir Simulations. Math Geosci (2024). https://doi.org/10.1007/s11004-023-10106-5
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DOI: https://doi.org/10.1007/s11004-023-10106-5