Well Modeling in the Multiscale Finite Volume Method for Subsurface Flow Simulation
Abstract
A multiscale method for effective handling of wells (source/sink terms) in the simulation of multiphase flow and transport processes in heterogeneous porous media is developed. The approach extends the multiscale finite volume (MSFV) framework. Our multiscale well model allows for accurate reconstruction of the fine‐scale pressure and velocity fields in the vicinity of wells. Accurate and computationally efficient modeling of complex wells is a prerequisite for field applications, and the ability to model wells within the MSFV framework makes it possible to solve large‐scale heterogeneous problems of practical interest. Our approach consists of removal of the well singularity from the multiscale solution via a local change of variables and the computation of a smoothly varying background field instead. The well effects are computed using a separate basis function, which is superposed on the background solution to yield accurate representation of the flow field. The multiscale well treatment accounts for both types of well constraints: fixed pressure and fixed rate. The details of modeling wells with one or multiple completions (perforations) are also presented. The accuracy of the method is assessed using a variety of examples including highly heterogeneous permeability fields.
MSC codes
Keywords
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
J. E. Aarnes, On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation, Multiscale Model. Simul., 2 (2004), pp. 421–439.
2.
T. Arbogast, Implementation of a locally conservative numerical subgrid upscaling scheme for two‐phase Darcy flow, Comput. Geosci., 6 (2002), pp. 453–481.
3.
T. Arbogast and S. L. Bryant, A two‐scale numerical subgrid technique for waterflood simulations, SPEJ, 7 (2002), pp. 446–457.
4.
K. Aziz and A. Settari, Petroleum Reservoir Simulation, Applied Science Publishers, London, England, 1979.
5.
G. F. Carey and S. S. Chow, Well singularities in reservoir simulation, SPE Res. Eng., 2 (1987), pp. 713–719.
6.
A. Chen and T. Y. Hou, A mixed finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 72 (2003), pp. 541–576.
7.
Z. Chen and X. Yue, Numerical homogenization of well singularities in the flow transport through heterogeneous porous media, Multiscale Model. Simul., 1 (2003), pp. 260–303.
8.
M. A. Christie and M. J. Blunt, Tenth SPE comparative solution project: A comparison of upscaling techniques, SPERE, 4 (2001), pp. 308–317.
9.
G. Dagan, Flow and Transport in Porous Formations, Springer‐Verlag, New York, 1989.
10.
C. Deutsch and A. G. Journel, GSLIB: Geostatistical Software Library and User’s Guide, 2nd ed., Oxford University Press, Reading, MA, 1998.
11.
Y. Ding and L. Jeannin, New numerical schemes for near‐well modeling using flexible grids, SPEJ, 9 (2004), pp. 109–121.
12.
L. J. Durlofsky, W. J. Milliken, and A. Bernath, Scale up in the near‐well region, SPEJ, 5 (2000), pp. 110–117.
13.
T. Hou and X. H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), pp. 169–189.
14.
P. Jenny, S. H. Lee, and H. A. Tchelepi, Multi‐scale finite‐volume method for elliptic problems in subsurface flow simulation, J. Comput. Phys., 187 (2003), pp. 47–67.
15.
P. Jenny, S. H. Lee, and H. A. Tchelepi, Adaptive multiscale finite‐volume method for multi‐phase flow and transport in porous media, Multiscale Model. Simul., 3 (2005), pp. 50–64.
16.
P. Jenny, S. H. Lee, and H. A. Tchelepi, An adaptive fully implicit multi‐scale finite‐volume algorithm for multi‐phase flow in porous media, J. Comput. Phys., to appear.
17.
S. H. Lee, H. Tchelepi, P. Jenny, and L. J. DeChant, Implementation of a flux‐continuous finite difference method for stratigraphic, hexahedron grids, SPEJ, 7 (2002), pp. 267–277.
18.
D. W. Peaceman, Interpretation of wellblock pressures in numerical reservoir simulation, SPEJ, (June 1978), pp. 183–94.
19.
M. Slodicka, Mathematical treatment of point sources in a flow through porous media governed by darcy’s law, Transp. Porous Media, 28 (1997), pp. 51–67.
20.
C. Wolfsteiner, L. J. Durlofsky, and K. Aziz, Calculation of well index for nonconventional wells on arbitrary grids, Comput. Geosci., 7 (2003), pp. 61–82.
Information & Authors
Information
Published In
Copyright
Copyright © 2006 Society for Industrial and Applied Mathematics.
History
Submitted: 20 September 2005
Accepted: 30 May 2006
Published online: 10 October 2006
MSC codes
Keywords
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- A Local Grid-Refined Numerical Groundwater Model Based on the Vertex-centred Finite-Volume MethodAdvances in Water Resources, Vol. 173 | 1 Mar 2023
- Linear Solvers for Reservoir Simulation Problems: An Overview and Recent DevelopmentsArchives of Computational Methods in Engineering, Vol. 29, No. 6 | 11 April 2022
- An algebraic dynamic multilevel and multiscale method with non-uniform mesh resolution and adaptive algebraic multiscale solver operator for the simulation of two-phase flows in highly heterogeneous petroleum reservoirsJournal of Computational Physics, Vol. 462 | 1 Aug 2022
- An algebraic multiscale solver for the simulation of two-phase flow in heterogeneous and anisotropic porous media using general unstructured grids (AMS-U)Applied Mathematical Modelling, Vol. 103 | 1 Mar 2022
- Scalable Graphics Processing Unit–Based Multiscale Linear Solvers for Reservoir SimulationSPE Journal, Vol. 27, No. 01 | 2 December 2021
- Striving to translate shale physics across ten orders of magnitude: What have we learned?Earth-Science Reviews, Vol. 223 | 1 Dec 2021
- Adaptive generalized multiscale approximation of a mixed finite element method with velocity eliminationComputational Geosciences, Vol. 25, No. 5 | 1 July 2021
- Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous MediaFluids, Vol. 6, No. 8 | 23 August 2021
- Velocity postprocessing schemes for multiscale mixed methods applied to contaminant transport in subsurface flowsComputational Geosciences, Vol. 24, No. 3 | 20 February 2020
- A singularity removal method for coupled 1D–3D flow modelsComputational Geosciences, Vol. 24, No. 2 | 20 December 2019
- Mathematical analysis, finite element approximation and numerical solvers for the interaction of 3D reservoirs with 1D wellsGEM - International Journal on Geomathematics, Vol. 10, No. 1 | 20 January 2019
- A Massively Parallel Algebraic Multiscale Solver for Reservoir Simulation on the GPU ArchitectureDay 1 Wed, April 10, 2019 | 29 March 2019
- A mixed generalized multiscale finite element method for planar linear elasticityJournal of Computational and Applied Mathematics, Vol. 348 | 1 Mar 2019
- Successful application of multiscale methods in a real reservoir simulator environmentComputational Geosciences, Vol. 21, No. 5-6 | 10 March 2017
- Multiscale finite volume method for discrete fracture modeling on unstructured grids (MS-DFM)Journal of Computational Physics, Vol. 351 | 1 Dec 2017
- Use of Multiple Multiscale Operators To Accelerate Simulation of Complex GeomodelsSPE Journal, Vol. 22, No. 06 | 14 December 2017
- Zonal Multiscale Finite-Volume frameworkJournal of Computational Physics, Vol. 337 | 1 May 2017
- A Dual-Grid Method for the Upscaling of Solid-Based Thermal Reactive Flow, With Application to the In-Situ Conversion ProcessSPE Journal, Vol. 21, No. 06 | 14 December 2016
- Multiscale model reduction for shale gas transport in fractured mediaComputational Geosciences, Vol. 20, No. 5 | 18 May 2016
- Algebraic multiscale method for flow in heterogeneous porous media with embedded discrete fractures (F-AMS)Journal of Computational Physics, Vol. 321 | 1 Sep 2016
- Investigating Darcy-scale assumptions by means of a multiphysics algorithmAdvances in Water Resources, Vol. 95 | 1 Sep 2016
- Monotone multiscale finite volume methodComputational Geosciences, Vol. 20, No. 3 | 16 August 2015
- Spatiotemporal adaptive multiphysics simulations of drainage-imbibition cyclesComputational Geosciences, Vol. 20, No. 3 | 27 August 2015
- Adaptive algebraic multiscale solver for compressible flow in heterogeneous porous mediaJournal of Computational Physics, Vol. 300 | 1 Nov 2015
- Constrained pressure residual multiscale (CPR-MS) method for fully implicit simulation of multiphase flow in porous mediaJournal of Computational Physics, Vol. 299 | 1 Oct 2015
- Local–global splitting for spatiotemporal-adaptive multiscale methodsJournal of Computational Physics, Vol. 280 | 1 Jan 2015
- Iterative Galerkin-enriched multiscale finite-volume methodJournal of Computational Physics, Vol. 277 | 1 Nov 2014
- Compositional Multiscale Finite-Volume FormulationSPE Journal, Vol. 19, No. 02 | 20 November 2013
- Algebraic multiscale solver for flow in heterogeneous porous mediaJournal of Computational Physics, Vol. 259 | 1 Feb 2014
- Auxiliary variables for 3D multiscale simulations in heterogeneous porous mediaJournal of Computational Physics, Vol. 238 | 1 Apr 2013
- Near-well upscaling for three-phase flowsComputational Geosciences, Vol. 16, No. 1 | 6 September 2011
- Uncertainty Quantification for Subsurface Flow Problems Using Coarse-Scale ModelsNumerical Analysis of Multiscale Problems | 24 August 2011
- Adaptive Strategies in the Multilevel Multiscale Mimetic (M3) Method for Two-Phase Flows in Porous MediaMultiscale Modeling & Simulation, Vol. 9, No. 3 | 19 July 2011
- An iterative multiscale finite volume algorithm converging to the exact solutionJournal of Computational Physics, Vol. 230, No. 5 | 1 Mar 2011
- Adaptive iterative multiscale finite volume methodJournal of Computational Physics, Vol. 230, No. 3 | 1 Feb 2011
- Accurate Representation of Near-well Effects in Coarse-Scale Models of Primary Oil ProductionTransport in Porous Media, Vol. 83, No. 3 | 10 October 2009
- Multiscale finite-volume method for parabolic problems arising from compressible multiphase flow in porous mediaJournal of Computational Physics, Vol. 228, No. 14 | 1 Aug 2009
- Multiscale Mixed-Finite-Element Modeling of Coupled Wellbore/Near-Well FlowSPE Journal, Vol. 14, No. 01 | 15 March 2009
- Modeling complex wells with the multi-scale finite-volume methodJournal of Computational Physics, Vol. 228, No. 3 | 1 Feb 2009
- An Operator Formulation of the Multiscale Finite-Volume Method with Correction FunctionMultiscale Modeling & Simulation, Vol. 8, No. 1 | 22 October 2009
- Iterative multiscale finite-volume methodJournal of Computational Physics, Vol. 227, No. 19 | 1 Oct 2008
- Multiscale finite-volume formulation for multiphase flow in porous media: black oil formulation of compressible, three-phase flow with gravityComputational Geosciences, Vol. 12, No. 3 | 10 January 2008
- A locally conservative variational multiscale method for the simulation of porous media flow with multiscale source termsComputational Geosciences, Vol. 12, No. 3 | 20 February 2008
- Multiscale finite-volume method for density-driven flow in porous mediaComputational Geosciences, Vol. 12, No. 3 | 15 January 2008
- Upscaled modeling of well singularity for simulating flow in heterogeneous formationsComputational Geosciences, Vol. 12, No. 1 | 4 January 2008
- Well modeling in the multi‐scale finite‐volume contextPAMM, Vol. 7, No. 1 | 6 October 2008
- Adaptive Multiscale Finite-Volume Framework for Reservoir SimulationSPE Journal, Vol. 12, No. 02 | 19 June 2007
- Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous mediaJournal of Computational Physics, Vol. 217, No. 2 | 1 Sep 2006
- Multiscale finite-volume method for compressible multiphase flow in porous mediaJournal of Computational Physics, Vol. 216, No. 2 | 1 Aug 2006