Quasi-implicit treatment of velocity-dependent mobilities in underground porous media flow simulation

Quasi-implicit schemes for treating velocity-dependent mobilities in underground porous media flow simulation, occurring when modeling non-Newtonian and non-Darcy effects as well as capillary desaturation, are presented. With low-order finite-volume discretizations, the principle is to evaluate mobi...

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Vydané v:	Computational geosciences Ročník 25; číslo 1; s. 119 - 141
Hlavní autori:	Patacchini, Leonardo, de Loubens, Romain
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Cham Springer International Publishing 01.02.2021 Springer Nature B.V
Predmet:	Aquatic reptiles Components Desaturation Earth and Environmental Science Earth Sciences Flow simulation Geotechnical Engineering & Applied Earth Sciences Hydrogeology Mathematical Modeling and Industrial Mathematics Multiphase flow Numerical stability Original Paper Porous media Pressure gradients Simulation Soil Science & Conservation Stability analysis Three dimensional flow Two dimensional flow Velocity Semi-implicit Capillary desaturation Quasi-implicit Stability analysis Non-Newtonian Non-Darcy
ISSN:	1420-0597, 1573-1499
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Abstract	Quasi-implicit schemes for treating velocity-dependent mobilities in underground porous media flow simulation, occurring when modeling non-Newtonian and non-Darcy effects as well as capillary desaturation, are presented. With low-order finite-volume discretizations, the principle is to evaluate mobilities at cell edges using normal velocity components calculated implicitly, and transverse velocity components calculated explicitly (i.e., based on the previously converged time-step); the pressure gradient driving the flow is, as usual, treated implicitly. On 3D hexahedral meshes, the proposed schemes require the same 7-point stencil as that of common semi-implicit schemes where mobilities are evaluated with an entirely explicit velocity argument. When formulated appropriately, their higher level of implicitness however places them, in terms of numerical stability, closer to “real” fully implicit schemes requiring at least a 19-point stencil. A von Neumann stability analysis of these proposed schemes is performed on a simplified pressure equation, representative of both single-phase and multiphase flows, following an approach previously used by the authors to study semi-implicit schemes. Whereas the latter are subject to stability constraints which limit their usage in certain cases where the logarithmic derivative of mobility with respect to velocity is large in magnitude, the former are unconditionally stable for 1D and 2D flows, and only subject to weak restrictionsfor 3D flows.
AbstractList	Quasi-implicit schemes for treating velocity-dependent mobilities in underground porous media flow simulation, occurring when modeling non-Newtonian and non-Darcy effects as well as capillary desaturation, are presented. With low-order finite-volume discretizations, the principle is to evaluate mobilities at cell edges using normal velocity components calculated implicitly, and transverse velocity components calculated explicitly (i.e., based on the previously converged time-step); the pressure gradient driving the flow is, as usual, treated implicitly. On 3D hexahedral meshes, the proposed schemes require the same 7-point stencil as that of common semi-implicit schemes where mobilities are evaluated with an entirely explicit velocity argument. When formulated appropriately, their higher level of implicitness however places them, in terms of numerical stability, closer to “real” fully implicit schemes requiring at least a 19-point stencil. A von Neumann stability analysis of these proposed schemes is performed on a simplified pressure equation, representative of both single-phase and multiphase flows, following an approach previously used by the authors to study semi-implicit schemes. Whereas the latter are subject to stability constraints which limit their usage in certain cases where the logarithmic derivative of mobility with respect to velocity is large in magnitude, the former are unconditionally stable for 1D and 2D flows, and only subject to weak restrictionsfor 3D flows.
Author	de Loubens, Romain Patacchini, Leonardo
Author_xml	– sequence: 1 givenname: Leonardo orcidid: 0000-0001-7705-9180 surname: Patacchini fullname: Patacchini, Leonardo email: lpatacchini@stoneridgetechnology.com organization: Total S.A., Stone Ridge Technology – sequence: 2 givenname: Romain surname: de Loubens fullname: de Loubens, Romain organization: Total S.A
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Cites_doi	10.1007/s10596-017-9624-5 10.2118/161615-MS 10.2118/163627-MS 10.2118/123971-MS 10.1007/s11242-013-0132-3 10.1080/00029890.1991.11995702 10.1023/A:1021291114475 10.2118/18089-MS 10.2118/18417-MS 10.2118/1243-A 10.2118/36715-MS 10.2118/129200-MS 10.2118/182638-PA 10.3997/2214-4609.201802119 10.1063/1.1699639 10.1088/0965-0393/12/5/016 10.1007/BF00141261 10.2118/14098-PA 10.2118/84924-PA
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References	Patacchini, L., de Loubens, R., Trouillaud, A.: New insights on the treatment of velocity-dependent mobilities in reservoir simulation Paper 830 presented at the 2013 ADIPEC held in Abu Dhabi (2013) de Loubens, R., Patacchini, L.: Adaptive-implicit strategy for treating velocity-dependent mobilities in Reservoir simulation. SPE 163627 (2013) Al-Hussainy, R., Ramey, H.J., Crawford, P.B.: The flow of real gases through porous media. SPE 1243A (1965) Cannella, W.J., Huh, C., Seright, R.S.: Prediction of Xanthan rheology in porous media. SPE 18089 (1988) Delshad, M., Pope, G.A., Sepehrnoori, K.: UTChem version 2011-7 technical documentation. CPGE (The University of Texas at Austin) (2011) IdrisZOrgéasLGeindreauCBlochJ-FAuriaultJ-LMicrostructural effects on the flow law of power-law fluids through fibrous mediaModel Simul Mater Sci Eng200412995101510.1088/0965-0393/12/5/016 Foulser, R.W.S., Goodyear, S.G.: Improved stability of the IMPES formulation for chemical flooding. SPE 18417 (1989) Raghavan, R.: Well-test analysis for multiphase flow. SPE Formation Evaluation 14098-PA (1989) AavatsmarkIAn introduction to multipoint flux approximations for quadrilateral gridsComput. Geosci.2002640543210.1023/A:1021291114475 PatacchiniLde LoubensRA class of physically stable non-linear models of flow through anisotropic porous mediaTransp. Porous Media201397340942910.1007/s11242-013-0132-3 WhitakerSThe Forchheimer equation: a theoretical developmentTransp. Porous Media199625276110.1007/BF00141261 Schlumberger: Eclipse reservoir simulation software Technical Description (2017) Seright, R.S., Fan, T., Wavrik, K., de Carvalho Balaban, R.: New insights into polymer rheology in porous media. SPE-129200-PA (2011) 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) AzizKSettariAPetroleum Reservoir Simulation1979LondonApplied Science Publishers LakeLEnhanced Oil Recovery1989Englewood CliffsPrentice Hall ForchheimerPWasserbewegung durch BodenVDIZ19014517821788 Moncorgé, A., Patacchini, L., de Loubens, R.: Multi-phase, multi-component simulation framework for advanced recovery mechanisms. SPE 161615 (2012) GilbertGTPositive definite matrices and Sylvester’s criterionAm. Math. Mon.1991981444610.1080/00029890.1991.11995702 de Loubens, R., Léon, L., Patacchini, L.: Application of a quasi-implicit scheme to the simulation of non-Newtonian flows through porous media. In: ECMOR XVI-16th European Conference on the Mathematics of Oil Recovery (2018) Schlumberger: Intersect reservoir simulation software Technical Description (2018) Von NeumannJRichtmyerRA method for the numerical calculation of hydrodynamic shocksJ. Appl. Phys.19502123223710.1063/1.1699639 de Loubens, R., Vaillant, G., Regaieg, M., Yang, J., Moncorgé, A., Fabbri, C., Darche, G.: Numerical modeling of unstable waterfloods and tertiary polymer floods into highly viscous oils. SPE J. https://doi.org/10.2118/182638-PA (2018) BaoKLieKAMøynerOFully implicit simulation of polymer flooding with MRSTComput. Geosci.2017211219124410.1007/s10596-017-9624-5 Halliburton: Nexus Technical Reference Guide (2014) PatacchiniLde LoubensRLinear stability analysis of the explicit treatment of mobilities in non-Newtonian and non-Darcy porous media flow simulationComput. Geosci.201418185209 BearJDynamics of Fluids in Porous Media1972New YorkDover Publications Allaire, G: Analyse numérique et optimisation Les Editions de l’Ecole Polytechnique (2005) AlSofi, A.M., Blunt, M.J.: Streamline-based simulation of non-newtonian polymer flooding. SPE J. 123971 (2010) Coats, K.H.: IMPES stability: selection of stable timesteps. SPEJ 84924 (2003) L Lake (9990_CR3) 1989 K Aziz (9990_CR1) 1979 S Whitaker (9990_CR9) 1996; 25 9990_CR20 L Patacchini (9990_CR10) 2014; 18 L Patacchini (9990_CR6) 2013; 97 9990_CR22 9990_CR21 9990_CR27 9990_CR7 J Von Neumann (9990_CR19) 1950; 21 9990_CR29 9990_CR24 9990_CR23 9990_CR26 J Bear (9990_CR2) 1972 9990_CR25 9990_CR4 K Bao (9990_CR14) 2017; 21 9990_CR30 9990_CR11 P Forchheimer (9990_CR5) 1901; 45 Z Idris (9990_CR8) 2004; 12 9990_CR17 I Aavatsmark (9990_CR12) 2002; 6 9990_CR16 GT Gilbert (9990_CR28) 1991; 98 9990_CR18 9990_CR13 9990_CR15
References_xml	– reference: ForchheimerPWasserbewegung durch BodenVDIZ19014517821788 – reference: de Loubens, R., Patacchini, L.: Adaptive-implicit strategy for treating velocity-dependent mobilities in Reservoir simulation. SPE 163627 (2013) – 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: Al-Hussainy, R., Ramey, H.J., Crawford, P.B.: The flow of real gases through porous media. SPE 1243A (1965) – reference: Patacchini, L., de Loubens, R., Trouillaud, A.: New insights on the treatment of velocity-dependent mobilities in reservoir simulation Paper 830 presented at the 2013 ADIPEC held in Abu Dhabi (2013) – reference: de Loubens, R., Vaillant, G., Regaieg, M., Yang, J., Moncorgé, A., Fabbri, C., Darche, G.: Numerical modeling of unstable waterfloods and tertiary polymer floods into highly viscous oils. SPE J. https://doi.org/10.2118/182638-PA (2018) – reference: Coats, K.H.: IMPES stability: selection of stable timesteps. SPEJ 84924 (2003) – reference: PatacchiniLde LoubensRA class of physically stable non-linear models of flow through anisotropic porous mediaTransp. Porous Media201397340942910.1007/s11242-013-0132-3 – reference: GilbertGTPositive definite matrices and Sylvester’s criterionAm. Math. Mon.1991981444610.1080/00029890.1991.11995702 – reference: BaoKLieKAMøynerOFully implicit simulation of polymer flooding with MRSTComput. Geosci.2017211219124410.1007/s10596-017-9624-5 – reference: Halliburton: Nexus Technical Reference Guide (2014) – reference: AavatsmarkIAn introduction to multipoint flux approximations for quadrilateral gridsComput. Geosci.2002640543210.1023/A:1021291114475 – 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: Allaire, G: Analyse numérique et optimisation Les Editions de l’Ecole Polytechnique (2005) – reference: Delshad, M., Pope, G.A., Sepehrnoori, K.: UTChem version 2011-7 technical documentation. CPGE (The University of Texas at Austin) (2011) – reference: BearJDynamics of Fluids in Porous Media1972New YorkDover Publications – 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: PatacchiniLde LoubensRLinear stability analysis of the explicit treatment of mobilities in non-Newtonian and non-Darcy porous media flow simulationComput. Geosci.201418185209 – reference: Moncorgé, A., Patacchini, L., de Loubens, R.: Multi-phase, multi-component simulation framework for advanced recovery mechanisms. SPE 161615 (2012) – reference: AzizKSettariAPetroleum Reservoir Simulation1979LondonApplied Science Publishers – reference: Von NeumannJRichtmyerRA method for the numerical calculation of hydrodynamic shocksJ. Appl. Phys.19502123223710.1063/1.1699639 – reference: IdrisZOrgéasLGeindreauCBlochJ-FAuriaultJ-LMicrostructural effects on the flow law of power-law fluids through fibrous mediaModel Simul Mater Sci Eng200412995101510.1088/0965-0393/12/5/016 – reference: Schlumberger: Intersect reservoir simulation software Technical Description (2018) – reference: de Loubens, R., Léon, L., Patacchini, L.: Application of a quasi-implicit scheme to the simulation of non-Newtonian flows through porous media. In: ECMOR XVI-16th European Conference on the Mathematics of Oil Recovery (2018) – reference: Raghavan, R.: Well-test analysis for multiphase flow. SPE Formation Evaluation 14098-PA (1989) – reference: WhitakerSThe Forchheimer equation: a theoretical developmentTransp. Porous Media199625276110.1007/BF00141261 – reference: Schlumberger: Eclipse reservoir simulation software Technical Description (2017) – reference: Seright, R.S., Fan, T., Wavrik, K., de Carvalho Balaban, R.: New insights into polymer rheology in porous media. SPE-129200-PA (2011) – ident: 9990_CR17 – volume-title: Petroleum Reservoir Simulation year: 1979 ident: 9990_CR1 – volume: 21 start-page: 1219 year: 2017 ident: 9990_CR14 publication-title: Comput. 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Geosci. doi: 10.1023/A:1021291114475 – ident: 9990_CR29 doi: 10.2118/18089-MS – ident: 9990_CR11 doi: 10.2118/18417-MS – ident: 9990_CR26 doi: 10.2118/1243-A – ident: 9990_CR4 doi: 10.2118/36715-MS – volume: 45 start-page: 1782 year: 1901 ident: 9990_CR5 publication-title: VDIZ – ident: 9990_CR20 – ident: 9990_CR30 doi: 10.2118/129200-MS – ident: 9990_CR23 doi: 10.2118/182638-PA – ident: 9990_CR22 doi: 10.3997/2214-4609.201802119 – volume: 21 start-page: 232 year: 1950 ident: 9990_CR19 publication-title: J. Appl. Phys. doi: 10.1063/1.1699639 – ident: 9990_CR7 – volume-title: Enhanced Oil Recovery year: 1989 ident: 9990_CR3 – volume-title: Dynamics of Fluids in Porous Media year: 1972 ident: 9990_CR2 – volume: 18 start-page: 185 year: 2014 ident: 9990_CR10 publication-title: Comput. Geosci. – ident: 9990_CR18 – volume: 12 start-page: 995 year: 2004 ident: 9990_CR8 publication-title: Model Simul Mater Sci Eng doi: 10.1088/0965-0393/12/5/016 – volume: 25 start-page: 27 year: 1996 ident: 9990_CR9 publication-title: Transp. Porous Media doi: 10.1007/BF00141261 – ident: 9990_CR16 doi: 10.2118/163627-MS – ident: 9990_CR25 doi: 10.2118/14098-PA – ident: 9990_CR27 doi: 10.2118/84924-PA
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SubjectTerms	Aquatic reptiles Components Desaturation Earth and Environmental Science Earth Sciences Flow simulation Geotechnical Engineering & Applied Earth Sciences Hydrogeology Mathematical Modeling and Industrial Mathematics Multiphase flow Numerical stability Original Paper Porous media Pressure gradients Simulation Soil Science & Conservation Stability analysis Three dimensional flow Two dimensional flow Velocity
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Title	Quasi-implicit treatment of velocity-dependent mobilities in underground porous media flow simulation
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