The Description and Quantification of the Truncation Errors Produced by Local-Grid Refinement in Reservoir Simulation
- Paul Tijink (MMbbls Limited) | Juan Cottier (MMbbls Limited)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- May 2019
- Document Type
- Journal Paper
- 660 - 672
- 2019.Society of Petroleum Engineers
- Grid Orientation Effects, Discretization errors, History Matching, Two-Point Flux Approximation, Local Grid Refinement
- 25 in the last 30 days
- 138 since 2007
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Geologically realistic 3D models yield nonorthogonal simulation grids as a result of grid distortion at faults, steeply dipping horizons, and zonal truncations. Finite-difference (FD) simulators applying discretization principles and volumetric flux calculation use two-point flux approximation (TPFA). On nonorthogonal grids, the TPFA causes numerical errors in the solution of pressure and saturation such that simulation results become inaccurate. These inaccuracies are known as grid-orientation effects.
This paper will demonstrate that TPFA applied to grids containing local-grid refinement (LGR) causes identical numerical errors. In the case of a vertically partial LGR, the numerical errors become so severe that they invalidate the simulation results. These results include unphysical oscillations in the fluxes within the LGR and unphysical baffles to flow at the interface between the coarse grid and the LGR.
The invalidation of simulation results is demonstrated by two case studies involving the history matching of well tests. To avoid invalid simulation results, any LGR applied in a grid must be extended over the entire vertical extent of the grid such that there exists no TPFA between the coarse- and the refined-grid cells in the vertical direction.
|File Size||1 MB||Number of Pages||13|
Aarnes, J. E., Gimse, T., and Lie, K.-A. 2007. An Introduction to the Numerics of Flow in Porous Media Using Matlab. In Geometric Modelling, Numerical Simulation, and Optimization, ed. G. Hasle, K.-A. Lie, E. Quak, 265–306, Oslo, Norway: Applied Mathematics at SINTEF.
Aavatsmark, I. 2007a. Interpretation of a Two-Point Flux Stencil for Skew Parallelogram Grids. Computat. Geosci. 11 (3): 199–206. https://doi.org/10.1007/s10596-007-9042-1.
Aavatsmark, I. 2007b. Multipoint Flux Approximation Methods for Quadrilateral Grids. Oral presentation given at the 9th International Forum on Reservoir Simulation, Abu Dhabi, 9–13 December.
Aavatsmark, I., Barkve, T., Bøe, Ø. et al. 1994. Discretization on Non-Orthogonal, Curvilinear Grids for Multi-Phase Flow. Oral presentation given at ECMOR IV–4th European Conference on the Mathematics of Oil Recovery, Røros, Norway, 7–10 June.
Aavatsmark, I., Eigestad, G. T., Heimsund, B.-O. et al. 2007. A New Finite-Volume Approach to Efficient Discretization on Challenging Grids. Presented at SPE Reservoir Simulation Symposium, Houston, 26–28 February. SPE-106435-MS. https://doi.org/10.2118/106435-MS.
Aavatsmark, I., Reiso, E., and Teigland, R. 2001. Control-Volume Discretization Method for Quadrilateral Grids With Faults and Local Refinements. Computat. Geosci. 5 (1): 1–23. https://doi.org/10.1023/A:1011601700328.
Adamson, G., Crick, M., Gane, B. et al. 1996. Simulation Throughout the Life of a Reservoir. Oilfield Rev. 8 (2): 16–27.
Arbogast, T., Pencheva, G., Wheeler, M. F. et al. 2007. A Multiscale Mortar Mixed Finite Element Method. Multiscale Model. Simul. 6 (1): 319–346. https://doi.org/10.1137/060662587.
Aziz, K., and Settari, A. 1979. Petroleum Reservoir Simulation. London: Applied Science Publishers Ltd.
Bastian, P., Kraus, J., Scheichl, R. et al. eds. 2013. Simulation of Flow in Porous Media: Applications in Energy and Environment. Berlin: De Gruyter.
Beckner, B., Haugen, K., Maliassov, S. et al. 2015. General Parallel Reservoir Simulation. Presented at the Abu Dhabi International Petroleum Exhibition and Conference, Abu Dhabi, 9–12 November. SPE-177532-MS. https://doi.org/10.2118/177532-MS.
Carlson, M. 2006. Practical Reservoir Simulation. Tulsa: PennWell.
Chen, Z., Huan, G., and Ma, Y. 2006. Computational Methods for Multiphase Flows in Porous Media. Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics.
Cordazzo, J., Maliska, C., and Carvalho da Silva, A. 2002. Interblock Transmissibility Calculation Analysis for Petroleum Reservoir Simulation. Oral presentation given at the 2nd Meeting on Reservoir Simulation, Buenos Aires, 5–6 November.
Edwards, M. G. 1996. Elimination of Adaptive Grid Interface Errors in the Discrete Cell Centered Pressure Equation. J. Comput. Phys. 126 (2): 356–372. https://doi.org/10.1006/jcph.1996.0143.
Ewing, R. E. 1983. The Mathematics of Reservoir Simulation. Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics.
Forsyth, P. A. and Sammon, P. H. 1985. Local Mesh Refinement and Modeling of Faults and Pinchouts. Presented at the Reservoir Simulation Symposium, Dallas. SPE-13524.
Götka, B. and Ertekin, T. 1999. Implementation of a Local Grid Refinement Technique in Modeling Slanted, Undulating Horizontal and Multi-Lateral Wells. Presented at the SPE Annual Technical Conference and Exhibition, Houston, 3–6 October. SPE-56624-MS. https://doi.org/10.2118/56624-MS.
Letkeman, J. and Ridings, R. L. 1970. A Numerical Coning Model. SPE J. 10 (4): 418–424. SPE-2812-PA. https://doi.org/10.2118/2812-PA.
Lipnikov, K., Morel, J., and Shashkov, M. 2004. Mimetic Finite Difference Methods for Diffusion Equations on Non-Orthogonal Non-Conformal Meshes. J. Comput. Phys. 199 (2): 589–597. https://doi.org/10.1016/j.jcp.2004.02.016.
MacDonald, R. C. and Coats, K. 1970. Methods for Numerical Simulation of Water and Gas Coning. Presented at the Second Symposium on Numerical Simulation of Reservoir Performance, Dallas. SPE-2796.
Moog, G. 2013. Advanced Discretization Methods for Flow Simulation Using Unstructured Grids. PhD dissertation, Stanford University, Stanford, California (June 2013).
Nikitin, K., Terekhov, K., and Vassilevski, Y. 2013. A Monotone Nonlinear Finite Volume Method for Diffusion Equations and Multiphase Flows. Computat. Geosci. 18 (34): 311–324. https://doi.org/10.1007/s10596-013-9387-6.
Peaceman, D. W. 1977. Fundamentals of Numerical Reservoir Simulation. New York City: Elsevier.
Ponting, D. K. 1989. Corner Point Grid Geometry in Reservoir Simulation. Presented at ECMOR I—1st European Conference on the Mathematics of Oil Recovery, Cambridge, 25–27 July. https://doi.org/10.3997/2214-4609.201411305.
Quandalle, P. and Besset, P. 1985. Reduction of Grid Effects Due to Local Sub-Gridding in Simulations Using a Composite Grid. Presented at the SPE Reservoir Simulation Symposium, Dallas, 10–13 February. SPE-13527-MS. https://doi.org/10.2118/13527-MS.
Ringrose, P. and Bentley, M. 2015. Reservoir Model Design—A Practitioner’s Guide. Dordrecht, The Netherlands: Springer.
Sammon, P. H. 2000. Calculation of Convective and Dispersive Flows for Complex Corner Point Grids. Presented at the SPE Annual Technical Conference and Exhibition, Dallas, 1–4 October. SPE-62929-MS. https://doi.org/10.2118/62929-MS.
Schlumberger. 2015. ECLIPSE Technical Description—Version 2015.2.
Stopa, J., Wojnarowski, P., and Janiga, D. 2014. Integrated Model of Hydraulic Fracturing and Hydrocarbon Production. AGH Drill. Oil Gas 31 (1): 49–56. https://doi.org/10.7494/drill.2014.31.1.49.
Thom, J. and Höcker, C. 2009. 3-D Grid Types in Geomodeling and Simulation—How the Choice of the Model Container Determines Modeling Results. Oral presentation given at the AAPG Annual Convention and Exhibition, Denver, 7–10 June.
Wesseling, P. 2001. Principles of Computational Fluid Dynamics. Heidelberg, Germany: Springer.
Wheeler, J. A., Wheeler, M. F., and Yotov, I. 2002. Enhanced Velocity Mixed Finite Element Methods for Flow in Multiblock Domains. Computat. Geosci. 6 (3–4): 315–332. https://doi.org/10.1023/A:1021270509932.
Yasin, I. B. E. 2012. Pressure Transient Analysis Using Generated Well Test Data From Simulation of Selected Wells in Norne Field. Master’s thesis, Norwegian University of Science and Technology, Trondheim, Norway (May 2012).