Unstructured Voronoi grids conforming to lower dimensional objects

We present a novel mixed-dimensional method for generating unstructured polyhedral grids that conform to prescribed geometric objects in arbitrary dimensions. Two types of conformity are introduced: (i) control-point alignment of cell centroids to accurately represent horizontal and multilateral wells or create volumetric representations of fracture networks, and ii) boundary alignment of cell faces to accurately preserve lower dimensional geological objects such as layers, fractures, faults, and/or pinchouts. The prescribed objects are in this case assumed to be lower dimensional, and we create a grid hierarchy in which each lower dimensional object is associated with a lower dimensional grid. Further, the intersection of two objects is associated with a grid one dimension lower than the objects. Each grid is generated as a clipped Voronoi diagram, also called a perpendicular bisector (PEBI) grid, for a carefully chosen set of generating points. Moreover, each grid is generated in such a way that the cell faces of a higher dimensional grid conform to the cells of all lower dimensional grids. We also introduce a sufficient and necessary condition which makes it easy to check if the sites for a given perpendicular bisector grid will conform to the set of prescribed geometric objects.

[1]  Dong-Ming Yan,et al.  Efficient Computation of 3D Clipped Voronoi Diagram , 2010, GMP.

[2]  Guillaume Caumon,et al.  Building PEBI Grids Conforming To 3D Geological Features Using Centroidal Voronoi Tessellations , 2011 .

[3]  Chenglei Yang,et al.  On centroidal voronoi tessellation—energy smoothness and fast computation , 2009, TOGS.

[4]  Bernard Mourrain,et al.  Advances in Geometric Modeling and Processing, 6th International Conference, GMP 2010, Castro Urdiales, Spain, June 16-18, 2010. Proceedings , 2010, GMP.

[5]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[6]  Xiao-Hui Wu,et al.  Effect of Grid Deviation on Flow Solutions , 2005 .

[7]  S. Geiger,et al.  Black-Oil Simulations for Three-Component, Three-Phase Flow in Fractured Porous Media , 2009 .

[8]  Alessio Fumagalli,et al.  PorePy: An Open-Source Simulation Tool for Flow and Transport in Deformable Fractured Rocks , 2017, ArXiv.

[9]  Shahid Manzoor,et al.  Correction to: Interior boundary-aligned unstructured grid generation and cell-centered versus vertex-centered CVD-MPFA performance , 2017, Computational Geosciences.

[10]  L. Durlofsky,et al.  An Efficient Discrete-Fracture Model Applicable for General-Purpose Reservoir Simulators , 2004 .

[11]  Xiao-Hui Wu,et al.  Efficient and Accurate Reservoir Modeling Using Adaptive Gridding with Global Scale Up , 2009 .

[12]  R. Helmig,et al.  Black-Oil Simulations for Three-Component -- Three-Phase Flow in Fractured Porous Media , 2007 .

[13]  Khalid Aziz,et al.  Use of Voronoi Grid in Reservoir Simulation , 1994 .

[14]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..

[15]  Olav Møyner,et al.  Unstructured Gridding and Consistent Discretizations for Reservoirs with Faults and Complex Wells , 2017 .

[16]  D.R. Guerillot,et al.  An Interactive 3D Mesh Builder for Fluid Flow Reservoir Simulation , 1993 .

[17]  D. K. Ponting,et al.  Corner Point Geometry in Reservoir Simulation , 1989 .

[18]  D. W. Peaceman.  Interpretation of Well-Block Pressures in Numerical Reservoir Simulation(includes associated paper 6988 ) , 1978 .

[19]  Larry S.K. Fung,et al.  Unconstrained Voronoi Grids for Densely Spaced Complex Wells in Full-Field Reservoir Simulation , 2014 .

[20]  Magne S. Espedal,et al.  Meshing of domains with complex internal geometries , 2006, Numer. Linear Algebra Appl..

[21]  B. Lévy,et al.  Voronoi grids conforming to 3D structural features , 2014, Computational Geosciences.

[22]  Hussein Mustapha,et al.  G23FM: a tool for meshing complex geological media , 2011 .

[23]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[24]  Michael L. Brewer,et al.  Generation of Hybrid Grids for Simulation of Complex, Unstructured Reservoirs by a Simulator with MPFA , 2015, ANSS 2015.

[25]  Per-Olof Persson,et al.  A Simple Mesh Generator in MATLAB , 2004, SIAM Rev..

[26]  Jianlei Sun,et al.  Grid-Sensitivity Analysis and Comparison Between Unstructured Perpendicular Bisector and Structured Tartan/Local-Grid-Refinement Grids for Hydraulically Fractured Horizontal Wells in Eagle Ford Formation With Complicated Natural Fractures , 2016 .

[27]  D. Filippov,et al.  Reservoir Modeling of Complex Structure Reservoirs on Dynamic Adaptive 3D Pebi-Grid (Russian) , 2017 .

[28]  David S. Schechter,et al.  Optimization-Based Unstructured Meshing Algorithms for Simulation of Hydraulically and Naturally Fractured Reservoirs With Variable Distribution of Fracture Aperture, Spacing, Length, and Strike , 2015 .

[29]  J. Boissonnat,et al.  3D volumetric modelling of Cadomian terranes (Northern Brittany, France): an automatic method using Voronoı̈ diagrams , 2001 .

[30]  Khalid Aziz,et al.  A Control Volume Scheme for Flexible Grids in Reservoir Simulation , 1997 .

[31]  Bert Jüttler,et al.  Advances in Geometric Modeling and Processing , 2008 .

[32]  Olav Møyner,et al.  The multiscale restriction smoothed basis method for fractured porous media (F-MsRSB) , 2016, J. Comput. Phys..

[33]  Kazuo Murota,et al.  A fast Voronoi-diagram algorithm with applications to geographical optimization problems , 1984 .

[34]  Masao Iri,et al.  System Modelling and Optimization , 1988 .

[35]  Tamal K. Dey,et al.  Delaunay Mesh Generation , 2012, Chapman and Hall / CRC computer and information science series.

[36]  Larry S.K. Fung,et al.  An Unstructured Gridding Method for Simulating Faulted Reservoirs Populated with Complex Wells , 2015, ANSS 2015.

[37]  Tareq M. Al-Shaalan,et al.  Interior boundary-aligned unstructured grid generation and cell-centered versus vertex-centered CVD-MPFA performance , 2018, Computational Geosciences.

[38]  Xiao-Hui Wu,et al.  Challenges and Technologies in Reservoir Modeling , 2009 .

[39]  Palle Thoft-Christensen,et al.  System modelling and optimization : proceedings of the 11th IFIP conference, Copenhagen, Denmark, July 25-29, 1983 , 1984 .

[40]  Qin Yang,et al.  Local PEBI grid generation method for reverse faults , 2018, Comput. Geosci..

[41]  C. Brand,et al.  Modeling reservoir geometry with irregular grids , 1991 .

[42]  Emmanuel Gringarten,et al.  Advantages of Using Vertical Stair Step Faults in Reservoir Grids for Flow Simulation , 2009 .

[43]  Mohamed Aymen Haouesse,et al.  New Grids for Robust Reservoir Modeling , 2008 .

[44]  Michael G. Edwards,et al.  Boundary Aligned Grid Generation in Three Dimensions and CVD-MPFA Discretization , 2015, ANSS 2015.

[45]  Larry S.K. Fung,et al.  Reservoir Simulation With a Control-Volume Finite-Element Method , 1992 .

[46]  Bradley T. Mallison,et al.  Unstructured Cut-Cell Grids for Modeling Complex Reservoirs , 2013, ANSS 2013.

[47]  Runar L. Berge.  Unstructured PEBI-grids Adapting to Geological Features in Subsurface Reservoirs , 2016 .