Linear stability analysis of the explicit treatment of mobilities in non-Newtonian and non-Darcy porous media flow simulation
A von Neumann stability analysis of the discretized conservation equation for single-phase porous media flows is performed, where non-Newtonian and non-Darcy effects are accounted for using a velocity (or mass flux)-dependent mobility factor. Comprehensive results in three dimensions for two low-ord...
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| Published in | Computational geosciences Vol. 18; no. 2; pp. 185 - 209 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.04.2014
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1420-0597 1573-1499 |
| DOI | 10.1007/s10596-013-9395-6 |
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| Abstract | A von Neumann stability analysis of the discretized conservation equation for single-phase porous media flows is performed, where non-Newtonian and non-Darcy effects are accounted for using a velocity (or mass flux)-dependent mobility factor. Comprehensive results in three dimensions for two low-order finite-volume discretizations typically encountered in reservoir simulation are provided, based on edge-centered and upstream cell-centered mobility calculations. It is found that common semi-implicit schemes, where the pressure gradient driving the flow is taken implicitly while the velocity-dependent mobility is evaluated explicitly, are subject to restrictions on the logarithmic derivative of mobility with respect to velocity. A remarkable new result is nevertheless obtained: for any physically acceptable strength of non-Newtonian and non-Darcy effects, there exists a stable and explicit method to evaluate the mobility, rendering the need to implement costly fully implicit schemes more difficult to justify. |
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| AbstractList | A von Neumann stability analysis of the discretized conservation equation for single-phase porous media flows is performed, where non-Newtonian and non-Darcy effects are accounted for using a velocity (or mass flux)-dependent mobility factor. Comprehensive results in three dimensions for two low-order finite-volume discretizations typically encountered in reservoir simulation are provided, based on edge-centered and upstream cell-centered mobility calculations. It is found that common semi-implicit schemes, where the pressure gradient driving the flow is taken implicitly while the velocity-dependent mobility is evaluated explicitly, are subject to restrictions on the logarithmic derivative of mobility with respect to velocity. A remarkable new result is nevertheless obtained: for any physically acceptable strength of non-Newtonian and non-Darcy effects, there exists a stable and explicit method to evaluate the mobility, rendering the need to implement costly fully implicit schemes more difficult to justify.[PUBLICATION ABSTRACT] A von Neumann stability analysis of the discretized conservation equation for single-phase porous media flows is performed, where non-Newtonian and non-Darcy effects are accounted for using a velocity (or mass flux)-dependent mobility factor. Comprehensive results in three dimensions for two low-order finite-volume discretizations typically encountered in reservoir simulation are provided, based on edge-centered and upstream cell-centered mobility calculations. It is found that common semi-implicit schemes, where the pressure gradient driving the flow is taken implicitly while the velocity-dependent mobility is evaluated explicitly, are subject to restrictions on the logarithmic derivative of mobility with respect to velocity. A remarkable new result is nevertheless obtained: for any physically acceptable strength of non-Newtonian and non-Darcy effects, there exists a stable and explicit method to evaluate the mobility, rendering the need to implement costly fully implicit schemes more difficult to justify. |
| Author | de Loubens, Romain Patacchini, Leonardo |
| Author_xml | – sequence: 1 givenname: Leonardo surname: Patacchini fullname: Patacchini, Leonardo email: leonardo.patacchini@total.com organization: Total S.A – sequence: 2 givenname: Romain surname: de Loubens fullname: de Loubens, Romain organization: Total S.A |
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| Cites_doi | 10.1137/080745390 10.2118/18417-MS 10.1007/978-3-642-97651-3 10.1007/s11242-013-0132-3 10.1007/978-94-011-3044-8 10.2118/1243-A 10.1063/1.1699639 10.2118/36715-MS 10.2118/84924-PA 10.2118/163627-MS 10.1023/A:1021291114475 10.2118/123971-MS 10.2118/18089-MS 10.1007/s10596-005-5665-2 |
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| DOI | 10.1007/s10596-013-9395-6 |
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| Keywords | Semi-implicit Stability analysis Non-Newtonian von Neumann Non-Darcy |
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| References_xml | – reference: ForchheimerPWasserbewegung durch BodenVDIZ19014517821788 – reference: AllaireGAnalyse numérique et optimisation2005PalaiseauLes Editions de l’Ecole Polytechnique – reference: Schlumberger: Eclipse reservoir simulation software Technical Description http://www.software.slb.com/products/foundation/pages/eclipse.aspx (2010) – reference: Al-Hussainy, R., Ramey, H.J., Crawford, P.B.: The flow of real gases through porous media. SPE 1243A (1965) – reference: MichelsenMLMollerupJThermodynamic Models: Fundamentals and Computational AspectsTie-Line Publications20072HolteTie-Line Publications – reference: FerzigerJHPericMComputational Methods for Fluid Dynamics.1996Springer, New YorkSpringer10.1007/978-3-642-97651-3 – reference: PatacchiniLde LoubensRA class of physically stable non-linear models of flow through anisotropic porous mediaTransp. Porous. Media.201397340942910.1007/s11242-013-0132-3 – reference: Coats, K.H.: IMPES stability: selection of stable timesteps. SPE J 84924 (2003) – reference: AavatsmarkIAn introduction to multipoint flux approximations for quadrilateral gridsComput. Geosci.2002640543210.1023/A:1021291114475 – reference: EdwardsMGZhengHQuasi M-matrix multifamily continuous Darcy-flux approximations with full pressure support on structured and unstructured grids in 3-DSIAM J. Sci. Comput.201133245548710.1137/080745390 – reference: DelshadMPopeGASepehrnooriKUTChem Version 2011-7 Technical Documentation.2011AustinCPGE (The University of Texas at Austin) – reference: Kalaydjian, F.J.-M., Bourbiaux, B.J., Lombard, J.M.: Predicting gas-condensate reservoir performance: how flow parameters are altered when approaching production wells. SPE 36715 (1996) – reference: Cannella, W.J., Huh, C., Seright, R.S.: Prediction of Xanthan rheology in porous media. SPE 18089 (1988) – reference: LakeLEnhanced Oil Recovery1989Englewood CliffsPrentice Hall – reference: Mattax, C.C., Dalton, R.L.: Reservoir simulation. SPE Monograph 13 (1989) – reference: NordbottenJMAavatsmarkIMonotonicity conditions for control volume methods on uniform parallelogram grids in homogeneous mediaComput. Geosci.200591617210.1007/s10596-005-5665-2 – reference: de Loubens, R., Patacchini, L.: Adaptive-implicit strategy for treating velocity-dependent mobilities in reservoir simulation. SPE 163627 (2013) – reference: Coats, K.H.: Reservoir simulation: state of the art. SPE Distinguished Author Ser (1982) – reference: HildebrandFBMethods of Applied Mathematics1965Englewood CliffsPrentice-Hall – reference: Von NeumannJRichtmyerRA method for the numerical calculation of hydrodynamic shocksJ. Appl. Phys.19502123223710.1063/1.1699639 – reference: AzizKSettariAPetroleum Reservoir Simulation. Applied Science Publishers1979LondonApplied Science Publishers – reference: SorbieKSPolymer-Improved Oil Recovery1991GlasgowBlackie10.1007/978-94-011-3044-8 – reference: BearJDynamics of Fluids in Porous Media1972New YorkDover – reference: Foulser, R.W.S., Goodyear, S.G.: Improved Stability of the IMPES Formulation for Chemical Flooding. SPE 18417 (1989) – reference: AlSofi, A.M., Blunt, M.J.: Streamline-based simulation of non-Newtonian polymer flooding. SPE J 123971 (2010) – reference: RaghavanRWell test analysis1993Englewood CliffsPrentice Hall – ident: 9395_CR2 – volume: 33 start-page: 455 issue: 2 year: 2011 ident: 9395_CR8 publication-title: SIAM J. Sci. Comput. doi: 10.1137/080745390 – volume-title: Well test analysis year: 1993 ident: 9395_CR24 – ident: 9395_CR12 doi: 10.2118/18417-MS – volume-title: Petroleum Reservoir Simulation. Applied Science Publishers year: 1979 ident: 9395_CR3 – volume-title: Computational Methods for Fluid Dynamics. year: 1996 ident: 9395_CR5 doi: 10.1007/978-3-642-97651-3 – ident: 9395_CR13 – volume: 97 start-page: 409 issue: 3 year: 2013 ident: 9395_CR18 publication-title: Transp. Porous. Media. doi: 10.1007/s11242-013-0132-3 – volume-title: Polymer-Improved Oil Recovery year: 1991 ident: 9395_CR9 doi: 10.1007/978-94-011-3044-8 – ident: 9395_CR19 doi: 10.2118/1243-A – volume-title: UTChem Version 2011-7 Technical Documentation. year: 2011 ident: 9395_CR15 – volume: 21 start-page: 232 year: 1950 ident: 9395_CR16 publication-title: J. Appl. Phys. doi: 10.1063/1.1699639 – ident: 9395_CR11 doi: 10.2118/36715-MS – ident: 9395_CR1 – ident: 9395_CR21 doi: 10.2118/84924-PA – volume-title: Tie-Line Publications year: 2007 ident: 9395_CR4 – volume-title: Enhanced Oil Recovery year: 1989 ident: 9395_CR10 – volume-title: Dynamics of Fluids in Porous Media year: 1972 ident: 9395_CR6 – volume-title: Methods of Applied Mathematics year: 1965 ident: 9395_CR20 – ident: 9395_CR25 doi: 10.2118/163627-MS – volume: 6 start-page: 405 year: 2002 ident: 9395_CR7 publication-title: Comput. Geosci. doi: 10.1023/A:1021291114475 – ident: 9395_CR14 doi: 10.2118/123971-MS – ident: 9395_CR23 doi: 10.2118/18089-MS – volume-title: Analyse numérique et optimisation year: 2005 ident: 9395_CR17 – volume: 45 start-page: 1782 year: 1901 ident: 9395_CR22 publication-title: VDIZ – volume: 9 start-page: 61 issue: 1 year: 2005 ident: 9395_CR26 publication-title: Comput. Geosci. doi: 10.1007/s10596-005-5665-2 |
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| SubjectTerms | Acceptability Derivatives Discretization Earth and Environmental Science Earth Sciences Geotechnical Engineering & Applied Earth Sciences Hydrogeology Linear equations Mathematical analysis Mathematical Modeling and Industrial Mathematics Media Mobility Original Paper Porous materials Porous media Rendering Simulation Soil Science & Conservation Stability analysis Stabilization Three dimensional |
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| Title | Linear stability analysis of the explicit treatment of mobilities in non-Newtonian and non-Darcy porous media flow simulation |
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