Unstructured Cut-Cell Grids for Modeling Complex Reservoirs
- Bradley Mallison (Chevron Energy Technology Company) | Charles Sword (Chevron Energy Technology Company) | Thomas Viard (Chevron Energy Technology Company) | William Milliken (Chevron Energy Technology Company) | Amy Cheng (Chevron Energy Technology Company)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- April 2014
- Document Type
- Journal Paper
- 340 - 352
- 2014.Society of Petroleum Engineers
- 5.1.1 Exploration, Development, Structural Geology, 4.1.2 Separation and Treating, 5.1.2 Faults and Fracture Characterisation, 5.1.5 Geologic Modeling
- polyhedra, unstructured gridding, grid generation, gridding, mesh generation
- 1 in the last 30 days
- 421 since 2007
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Effective workflows for translating Earth models into simulation models require grids that preserve geologic accuracy, offer flexible resolution control, integrate tightly with upscaling, and can be generated easily. Corner-point grids and pillar-based unstructured grids fail to satisfy these objectives; hence, a truly 3D unstructured approach is required. This paper describes unstructured cut-cell gridding tools that address these needs and improve the integration of our overall reservoir-modeling workflows. The construction of simulation grids begins with the geologic model: a numerical representation of the reservoir structure, stratigraphy, and properties. Our gridding utilizes a geochronological (GeoChron) map from physical coordinates to an unfaulted and unfolded depositional coordinate system. The mapping is represented implicitly on a tetrahedral mesh that conforms to faults, and it facilitates accurate geostatistical modeling of static depositional properties. In the simplest use case, we create an explicit representation of the geologic model as an unstructured polyhedral grid. Away from faults and other discontinuities, the cells are hexahedral, highly orthogonal, and arranged in a structured manner. Geometric cutting operations create general polyhedra adjacent to faults and explicit contact polygons across faults. The conversion of implicit models to explicit grids is conceptually straightforward, but the implementation is nontrivial because of the limitations of finite precision arithmetic and the need to remove small cells formed in the cutting process. In practice, simulation grids are often constructed at coarser resolutions than Earth models. Our implementation of local grid coarsening and refinement exploits the flexibility of unstructured grids to minimize upscaling errors and preserve critical geologic features. Because the simulation grid and the geologic model are constructed by use of the same mapping, fine cells can be nested exactly inside coarse cells. Therefore, flow-based upscaling can be applied efficiently without resampling onto temporary local grids. This paper describes algorithms and data structures for constructing, storing, and simulating cut-cell grids. Examples illustrate accurate modeling of normal faults, y-faults, overturned layers, and complex stratigraphy. Flow results, including a field-sector model, show the suitability of cut-cell grids for simulation.
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Aavatsmark, I., Barkve, T., and Mannseth, T. 1998. Control Volume Discretization Methods for 3D Quadrilateral Grids in Inhomogeneous, Anisotropic Reservoirs. SPE J. 3 (2): 146−154. http://dx.doi.org/10.2118/38000-PA.
Aavatsmark, I., Eigestad, G.T., Heimsund, B.-O. et al. 2010. A New Finite-Volume Approach to Efficient Discretization on Challenging Grids. SPE J. 15 (3): 658−669. http://dx.doi.org/10.2118/106435-PA.
Aavatsmark, I., Reiso, E., and Teigland, R. 2001. Control-Volume Discretization Method for Quadrilateral Grids With Faults and Local Refinements. Comput. Geosci. 5 (1): 1−24.
Aftosmis, M.J. 1997. Solution Adaptive Cartesian Grid Methods for Aerodynamic Flows With Complex Geometries. VKI Lecture Series, 2.
Aftosmis, M.J., Berger, M.J., and Melton, J.E. 1995. Adaptation and Surface Modeling for Cartesian Mesh Methods. AIAA Paper 99-1725-CP.
Ahmadi, M. 2012. Modeling and Quantification of Structural Uncertainties in Petroleum Reservoirs Assisted by a Hybrid Catresian Cut Cell/Enriched Multipoint Flux Approximation Approach. Doctoral dissertation. Heriot-Watt University, UK.
Ahmadi, M., Christie, M., and Gerritsen, M. 2013. Structural Uncertainty Quantification With Immersed Interface Methods. Paper SPE 163606 presented at the 2013 SPE Reservoir Simulation Symposium, The Woodlands, Texas, 18−20 February. http://dx.doi.org/10.2118/163606-MS.
Alumbaugh, T. and Jiao, X. 2005. Compact Array-Based Mesh Data Structures. In Proceedings of the 14th International Meshing Roundtable, pg. 485−503.
Arbogast, T., Cowsar, L., Wheeler, M. et al. 2000. Mixed Finite Element Methods on Nonmatching Multiblock Grids. SIAM J. Numer. Anal. 37 (4): 1295−1315.
Aziz, K. 1993. Reservoir Simulation Grids: Opportunities and Problems. J. Pet Tech 45 (7): 658−663. http://dx.doi.org/10.2118/25233-PA.
Baumgart, B. 1972. Winged Edge Polyhedron Representation. Technical report, DTIC Document.
Baumgart, B. 1975. A Polyhedron Representation for Computer Vision. In Proceedings of the 1975 National Computer Conference and Exhibition, pg. 589−596. ACM.
Branets, L.V., Ghai, S.S., Lyons, S.L. et al. 2009. Challenges and Technologies in Reservoir Modeling. Commun. Comput. Phys. 6 (1): 1−23.
Cao, H., Crumpton, P.I., and Schrader, M.L. 2009. Efficient General Formulation Approach for Modeling Complex Physics. Paper SPE 119165 presented at the 2009 SPE Reservoir Simulation Symposium, The Woodlands, Texas, 2−4 February. http://dx.doi.org/10.2118/119165-MS.
Cao, H., Tchelepi, H.A., Wallis, J. et al. 2005. Parallel Scalable Unstructured CPR-Type Linear Solver for Reservoir Simulation. Paper SPE 96809 presented at the 2005 SPE Annual Technical Conference and Exhibition, Dallas, Texas, 9−12 October. http://dx.doi.org/10.2118/96809-MS.
Caumon, G., Lévy, B., Castanié, L. et al. 2005. Visualization of Grids Conforming to Geologic Structures: A Topological Approach. Computers and Geosciences 31 (6): 671−680.
Christie, M. and Blunt, M. 2001. Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques. SPE. Res Eval & Eng 4 (4): 308–317. http://dx.doi.org/10.2118/72469-PA.
Conreaux, S., Lévy, B., Mallet, J.-L. 1999. Modélisation Volumique de 3-Variétés. In Proceedings of the Association Française d’Informatique Graphique.
DeBaun, D., Byer, T., Childs, P. et al. 2005. An Extensible Architecture for Next Generation Scalable Parallel Reservoir Simulation. Paper SPE 93274 presented at the 2005 SPE Reservoir Simulation Symposium, The Woodlans, Texas, 31 January−2 February. http://dx.doi.org/10.2118/93274-MS.
Edwards, M.G. and Rogers, C.F. 1998. Finite Volume Discretization With Imposed Flux Continuity for the General Tensor Pressure Equation. Comput. Geosci. 2: 259−290.
Evazi, M. and Mahani, H. 2010. Unstructured-Coarse-Grid Generation Using Background-Grid Approach. SPE J. 15 (2): 326−340. http://dx.doi.org/10.2118/120170-PA.
Gringarten, E., Arpat, B., Haouesse, A. et al. 2008. New Grids for Robust Reservoir Modeling. Paper SPE 116649 presented at the 2008 SPE Annual Technical Conference and Exhibition, Denver, Colorado, 21−24 September. http://dx.doi.org/10.2118/116649-MS.
Gringarten, E., Haouesse, A., Arpat, B. et al. 2009. Advantages of Using Vertical Stair Step Faults in Reservoir Grids for Flow Simulation. Paper SPE 119188 presented at the 2009 SPE Reservoir Simulation Symposium, The Woodlands, Texas, 2−4 September. http://dx.doi.org/10.2118/119188-MS.
Hale, D. 2001. Atomic Meshes—From Seismic Imaging to Reservoir Simulation. In Proceedings of the 8th European Conference on the Mathematics of Oil Recovery.
Heinemann, Z.E., Brand, C.W., Munka, M. et al. 1991. Modeling Reservoir Geometry With Irregular Grids. SPE. Res Eng 6 (2): 225−232. http://dx.doi.org/10.2118/18412-PA.
Hida, Y., Li, X.S., and Bailey, D.H. 2001. Algorithms for Quad-Double Precision Floating Point Arithmetic. Paper presented at the 15th IEEE Symposium on Computer Arithmetic. IEEE Computer Society, 155−162.
Hobby, J.D. 1999. Practical Segment Intersection With Finite Precision Output. Computational Geometry 14 (4): 199−214.
Jenny, P., Wolfsteiner, C., Lee, S.H. et al. 2002. Modeling Flow in Geometrically Complex Reservoirs Using Hexahedral Multiblock Grids. SPE J. 7 (2): 149−157. http://dx.doi.org/10.2118/78673-PA.
Katzmayr, M. and Ganzer, L. 2009. An Iterative Algorithm for Generating Constrained Voronoi Grids. Paper SPE 118942 presented at the 2009 SPE Reservoir Simulation Symposium, The Woodlands, Texas, 2−4 February. http://dx.doi.org/10.2118/118942-MS.
King. M.J., Ballin, P., Bennis, C. et al. 2012. Reservoir Modeling: From RESQUE to RESQML. SPE Res Eval & Eng 15 (2): 127−138. http://dx.doi.org/10.2118/135280-PA.
Lasseter, T.J. and Jackson, S.A. 2004. Improving Integrated Interpretation Accuracy and Efficiency Using a Single Consistent Reservoir Model From Seismic to Simulation. The Leading Edge 23 (11): 1118−1121.
Lee, S.H., Jenny, P., and Tchelepi, H.A. 2002a. A Finite-Volume Method With Hexahedral Multiblock Grids for Modeling Flow in Porous Media. Comput. Geosci. 6: 353−379.
Lee, S.H., Tchelepi, H.A., Jenny, P. et al. 2002b. Implementation of a Flux-Continuous Finite-Difference Method for Stratigraphic, Hexahedron Grids. SPE J. 7 (3): 267−277. http://dx.doi.org/10.2118/80117-PA.
Lee, S.H., Wolfsteiner, C., Durlofsky, L.J. et al. 2003. New Developments in Multiblock Reservoir Simulation: Black Oil Modeling, Nonmatching Subdomains and Near-Well Upscaling. Paper SPE 79682 presented at the 2003 SPE Reservoir Simulation Symposium, Houston, Texas, 3−5 February. http://dx.doi.org/10.2118/79682-MS.
Lévy, B., Caumon, G., Conreaux, S. et al. 2001. Circular Incident Edge Lists: A Data Structure for Rendering Complex Unstructured Grids. In Visualization VIS’01 Proceedings, pg. 191−557.
Lie, K.-A., qKrogstad, S., Ligaarden, I.S. et al. 2012. Open Source Implementation of Consistent Discretizations on Complex Grids. Comput. Geosci. 16 (2): 297−322.
Mallet, J.-L. 2004. Space-Time Mathematical Framework for Sedimentary Geology. Mathematical Geology 36 (1): 1−32.
Mallet, J.-L. and Tertois, A.-L. 2010. Solid Earth Modeling and Geometric Uncertainties. Paper SPE 134978 presented at the 2010 SPE Annual Technical Conference and Exhibition, Florence, Italy, 19−22 September. http://dx.doi.org/10.2118/134978-MS.
Merland, R., Lévy, B., Caumon, G. et al. 2011. Building Centroidal Voronoi Tessellations for Flow Simulation in Reservoirs Using Flow Information. Paper SPE 141018 presented at the 2011 SPE Reservoir Simulation Symposium, The Woodlands, Texas, 21−23 February. http://dx.doi.org/10.2118/141018-MS.
Mlacnik, M.J., Durlofsky, L.J., and Heinemann, Z.E. 2006. Sequentially Adapted Flow-Based PEBI Grids for Reservoir Simulation. SPE J. 11 (3): 317−327. http://dx.doi.org/10.2118/90009-PA.
Moyen, R., Mallet, J.-L., Frank, T. et al. 2004. 3D-Parameterization of the 3D Geological Space—The Geochron Model. In Proceedings of the 9th European Conference on the Mathematics of Oil Recovery.
Natvig, J.R. and Lie, K.-A. 2008. Fast Computation of Multiphase Flow in Porous Media by Implicit Discontinuous Galerkin Schemes With Optimal Ordering of Elements. J. Comput. Phys. 227 (24): 10108−10124.
Natvig, J.R., Lie, K.-A., Eikemo, B. et al. 2007. An Efficient Discontinuous Galerkin Method for Advective Transport in Porous Media, Water Injection Optimization Using a Streamline-Based Workflow. Advances in Water Resources 30: 2424−2438.
Nilsen, H.M., Lie, K.-A., and Natvig, J.R. 2012. Accurate Modeling of Faults by Multipoint, Mimetic and Mixed Methods. SPE J. 17 (2): 568−579. http://dx.doi.org/10.2118/149690-PA.
Nilsson, J., Gerritsen, M., and Younis, R. 2005 A Novel Adaptive Anisotropic Grid Framework for Efficient Reservoir Simulation. Paper SPE 93243 presented at the 2005 SPE Reservoir Simulation Symposium, Houston, Texas, 31 January−2 February. http://dx.doi.org/10.2118/93243-MS.
Palagi, C.L. and Aziz, K. 1994. Modeling Vertical and Horizontal Wells With Voronoi Grid. SPE Res Eng 9 (1): 15−21. http://dx.doi.org/10.2118/24072-PA.
Ponting, D.K. 1989. Corner Point Geometry in Reservoir Simulation. In Proceedings of the 1st European Conference on the Mathematics of Oil Recovery.
Prévost, M. 2003. Accurate Coarse Reservoir Modeling Using Unstructured Grids, Flow-Based Upscaling and Streamline Simulation. Doctoral dissertation, Stanford University.
Shahvali, M., Mallison, B., Wei, K. et al. 2011. An Alternative to Streamlines for Flow Diagnostics on Structured and Unstructured Grids. SPE J. 17 (3): 768−778. http://dx.doi.org/10.2118/146446-PA.
Shewchuk, J.R. 1997. Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates. Discrete and Computational Geometry 18: 305−363.
Shook, G.M. and Mitchell, K.M. 2009. A Robust Measure of Heterogeneity for Ranking Earth Models: The F-PHI Curve and Dynamic Lorenz Coefficient. Paper presented at the 2009 SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, 4−7 October. http://dx.doi.org/10.2118/124625-MS.
Skoreyko, F., Sammon, P.H., and Melichar, H. 2003. Use of PEBI Grids for Complex Advanced Processes Simulators. Paper SPE 79685 presented at the 2003 SPE Reservoir Simulation Symposium, Houston, Texas, 3−5 February. http://dx.doi.org/10.2118/79685-MS.
Weiler, K. 1986. Topological Structures for Geometric Modeling. Doctoral dissertation, Rensselaer Polytechnic Institute.
Weiler, K. 1988. The Radial Edge Structure: A Topological Representation for Non-Manifold Geometric Boundary Modeling. Geometric Modeling for CAD Applications, pg. 3−36.
Wheeler, J.A., Wheeler, M.F., and Yotov, I. 2002. Enhanced Velocity Mixed Finite Element Methods for Flow in Multiblock Domains. Comput. Geosci. 6: 315−332.
Wu, X.-H. and Parashkevov, R.R. Effect of Grid Deviation on Flow Solutions. 2009. SPE J. 14 (1): 67−77. http://dx.doi.org/10.2118/92868-PA.
Yang, G., Causon, D.M., Ingram, D.M. et al. 1997. A Cartesian Cut Cell Method for Compressible Flows: Part A. Static Body Problems. Aeronautical J. 101 (1001): 47−56.
Yang, G., Causon, D.M., and Ingram, D.M. 2000. Calculation of Compressible Flows About Complex Moving Geometries by Use of A 3D Cartesian Cut Cell Method. International J. for Numerical Methods in Fluids 33: 1121−1151.