Numerical schemes for the Barenblatt model of non-equilibrium two-phase flow in porous media
Authors:
Denise Aregba-Driollet, Gabriella Bretti and Roberto Natalini
Journal:
Quart. Appl. Math. 66 (2008), 201-231
MSC (2000):
Primary 65M06; Secondary 76S05, 35L65
DOI:
https://doi.org/10.1090/S0033-569X-08-01079-0
Published electronically:
February 8, 2008
MathSciNet review:
2416771
Full-text PDF Free Access
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Abstract: We introduce some numerical approximations to a quasilinear problem proposed by G. I. Barenblatt to describe non-equilibrium two-phase fluid flows in permeable porous media, which apply to secondary oil recovery from natural reservoirs. Taking into account the theoretical results of global existence and uniqueness, we approximate the solutions by three numerical schemes, namely, the Diagonal First Order schemes (DFO and DFO2) and the Diagonal Second Order scheme (DSO). For DFO schemes convergence is proved. The schemes’ behaviour is analysed and discussed through some numerical experiments.
References
- G. I. Barenblatt, Flow of two immiscible fluids in homogeneous porous medium (in Russian), Izv. AN SSSR, Mekh. Zhidk. i Gaza 5 (1971), 144-151.
- G. I. Barenblatt, V. M. Entov and V. M. Ryzhik, Theory of fluid flows through natural rocks, Kluwer Academic Publishers, Dordrecht (1990).
- Grigory Isaakovich Barenblatt, Jesús Garcia-Azorero, Arturo De Pablo, and Juan Luis Vazquez, Mathematical model of the non-equilibrium water-oil displacement in porous strata, Appl. Anal. 65 (1997), no. 1-2, 19–45. MR 1674579, DOI https://doi.org/10.1080/00036819708840547
- G. I. Barenblatt, A. P. Vinnitchenko, Non equilibrium seepage of immiscible fluids (in Russian), Uspekhi Mekhaniki 3 (1980), 35-50.
- S. E. Buckley, M. C. Leverett, Mechanism of fluid displacement in sands, Trans. AIME 142 (1942), 107-116.
- M. Muskat and M. Meres, The flow of heterogeneous fluids, Physics, 7 (1936), 346–363.
- Roberto Natalini and Alberto Tesei, On the Barenblatt model for non-equilibrium two phase flow in porous media, Arch. Ration. Mech. Anal. 150 (1999), no. 4, 349–367. MR 1741260, DOI https://doi.org/10.1007/s002050050191
- D. W. Peaceman, Fundamentals of numerical reservoir simulation, Developments in Petroleum Science 6, ed. Elsevier, Amsterdam (1977).
References
- G. I. Barenblatt, Flow of two immiscible fluids in homogeneous porous medium (in Russian), Izv. AN SSSR, Mekh. Zhidk. i Gaza 5 (1971), 144-151.
- G. I. Barenblatt, V. M. Entov and V. M. Ryzhik, Theory of fluid flows through natural rocks, Kluwer Academic Publishers, Dordrecht (1990).
- G. I. Barenblatt, J. Garcia-Azorero, A. De Pablo and J. L. Vazquez, Mathematical Model of the Non-Equilibrium Water-Oil Displacement in Porous Strata, Applicable Analysis 65 (1997), 19-45. MR 1674579 (99j:76121)
- G. I. Barenblatt, A. P. Vinnitchenko, Non equilibrium seepage of immiscible fluids (in Russian), Uspekhi Mekhaniki 3 (1980), 35-50.
- S. E. Buckley, M. C. Leverett, Mechanism of fluid displacement in sands, Trans. AIME 142 (1942), 107-116.
- M. Muskat and M. Meres, The flow of heterogeneous fluids, Physics, 7 (1936), 346–363.
- R. Natalini, A. Tesei, On the Barenblatt model for non-equilibrium two phase flow in porous media, Arch. Rational Mech. Anal. 150 (1999), 349-367. MR 1741260 (2001b:76090)
- D. W. Peaceman, Fundamentals of numerical reservoir simulation, Developments in Petroleum Science 6, ed. Elsevier, Amsterdam (1977).
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Additional Information
Denise Aregba-Driollet
Affiliation:
Mathématiques Appliquées de Bordeaux, Université Bordeaux 1, 351 Cours de la Liberation, F-33405 Talence, France
Email:
aregba@math.u-bordeaux.fr
Gabriella Bretti
Affiliation:
Department of Information Engineering and Applied Mathematics of the University of Salerno, via Ponte don Melillo, 84084 Fisciano (SA), Italy, & Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Viale del Policlinico 137, 00161 - Roma, Italy
Email:
g.bretti@iac.cnr.it
Roberto Natalini
Affiliation:
Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Viale del Policlinico 137, 00161 - Roma, Italy
Email:
r.natalini@iac.cnr.it
Keywords:
Scalar conservation laws,
two-phase flows,
non-equilibrium flows
Received by editor(s):
August 2, 2005
Published electronically:
February 8, 2008
Article copyright:
© Copyright 2008
Brown University
The copyright for this article reverts to public domain 28 years after publication.
