Compositional flow in porous media: Riemann problem for three alkanes
Authors:
Vítor Matos and Dan Marchesin
Journal:
Quart. Appl. Math. 75 (2017), 737-767
MSC (2010):
Primary 35L65, 76S05, 76T30
DOI:
https://doi.org/10.1090/qam/1477
Published electronically:
July 20, 2017
MathSciNet review:
3686519
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Additional Information
Abstract:
We consider the flow in a porous medium of three fluid compounds such as alkanes with different boiling points; the compounds partition into a liquid and a gaseous phase. Under some judiciously chosen physical assumptions, the flow is governed by a system of conservation laws; we derive the expression for the Rankine-Hugoniot locus, which involves a parameter dependent fifth degree polynomial in two variables. This expression allows us to establish in detail the bifurcation behavior of the locus
Supplemented by the analysis of characteristic speeds and eigenvectors, the bifurcation analysis of the Rankine-Hugoniot locus is the enabling fulcrum for solving the Riemann problem for all data, which should be a prototype for general three component flow of two phases in porous media. Despite the existence of many similarities between this model and earlier models where proofs were not possible, here we managed to prove analytically many features.
This system of conservation laws has three equations yet it leads to a characteristic polynomial of degree two; this peculiar feature has been unveiled recently, and it is typical of flow of fluids that change density upon changing phase.
References
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- J.M. Dumore, J. Hagoort, and A.S. Risseeuw, An analytical model for one-dimensional, three-component condensing and vaporizing gas drives, SPEJ 24 (1984), 169 – 179.
- Eli L. Isaacson and J. Blake Temple, Analysis of a singular hyperbolic system of conservation laws, J. Differential Equations 65 (1986), no. 2, 250–268. MR 861520, DOI https://doi.org/10.1016/0022-0396%2886%2990037-9
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- Vitor Matos, Julio D. Silva, and Dan Marchesin, Loss of hyperbolicity changes the number of wave groups in Riemann problems, Bull. Braz. Math. Soc. (N.S.) 47 (2016), no. 2, 545–559. MR 3514420, DOI https://doi.org/10.1007/s00574-016-0168-4
- Jeffrey Nunemacher, Asymptotes, Cubic Curves, and the Projective Plane, Math. Mag. 72 (1999), no. 3, 183–192. MR 1573393
- O. A. Oleĭnik, On the uniqueness of the generalized solution of the Cauchy problem for a non-linear system of equations occurring in mechanics, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 6(78), 169–176 (Russian). MR 0094543
- Stephen Schecter, Dan Marchesin, and Bradley J. Plohr, Structurally stable Riemann solutions, J. Differential Equations 126 (1996), no. 2, 303–354. MR 1383980, DOI https://doi.org/10.1006/jdeq.1996.0053
- Stephen Schecter, Bradley J. Plohr, and Dan Marchesin, Classification of codimension-one Riemann solutions, J. Dynam. Differential Equations 13 (2001), no. 3, 523–588. MR 1845094, DOI https://doi.org/10.1023/A%3A1016634307145
- Julio Daniel Silva and Dan Marchesin, Riemann solutions without an intermediate constant state for a system of two conservation laws, J. Differential Equations 256 (2014), no. 4, 1295–1316. MR 3145758, DOI https://doi.org/10.1016/j.jde.2013.10.005
- Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779
- W.J. Todd, M.R.; Longstaff, The development, testing, and application of a numerical simulator for predicting miscible flood performance, Journal of Petroleum Technology 24 (1972), 874–882.
- Burton Wendroff, The Riemann problem for materials with nonconvex equations of state. I. Isentropic flow, J. Math. Anal. Appl. 38 (1972), 454–466. MR 328387, DOI https://doi.org/10.1016/0022-247X%2872%2990103-5
References
- Arthur V. Azevedo, Aparecido J. de Souza, Frederico Furtado, Dan Marchesin, and Bradley Plohr, The solution by the wave curve method of three-phase flow in virgin reservoirs, Transp. Porous Media 83 (2010), no. 1, 99–125. MR 2646866, DOI https://doi.org/10.1007/s11242-009-9508-9
- S.E. Buckley and M.C. Leverett, Mechanism of fluid displacements in sands, Transactions of the AIME 146 (1942), 107–116.
- Olav Dahl, Thormod Johansen, Aslak Tveito, and Ragnar Winther, Multicomponent chromatography in a two phase environment, SIAM J. Appl. Math. 52 (1992), no. 1, 65–104. MR 1148319, DOI https://doi.org/10.1137/0152005
- J.M. Dumore, J. Hagoort, and A.S. Risseeuw, An analytical model for one-dimensional, three-component condensing and vaporizing gas drives, SPEJ 24 (1984), 169 – 179.
- Eli L. Isaacson and J. Blake Temple, Analysis of a singular hyperbolic system of conservation laws, J. Differential Equations 65 (1986), no. 2, 250–268. MR 861520, DOI https://doi.org/10.1016/0022-0396%2886%2990037-9
- Thormod Johansen and Ragnar Winther, The solution of the Riemann problem for a hyperbolic system of conservation laws modeling polymer flooding, SIAM J. Math. Anal. 19 (1988), no. 3, 541–566. MR 937469, DOI https://doi.org/10.1137/0519039
- Barbara L. Keyfitz and Herbert C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal. 72 (1979/80), no. 3, 219–241. MR 549642, DOI https://doi.org/10.1007/BF00281590
- Barbara L. Keyfitz and Herbert C. Kranzer, The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy, J. Differential Equations 47 (1983), no. 1, 35–65. MR 684449, DOI https://doi.org/10.1016/0022-0396%2883%2990027-X
- L.W. Lake, Enhanced oil recovery, Prentice Hall, 1989.
- Wanderson Lambert and Dan Marchesin, The Riemann problem for multiphase flows in porous media with mass transfer between phases, J. Hyperbolic Differ. Equ. 6 (2009), no. 4, 725–751. MR 2604254, DOI https://doi.org/10.1142/S0219891609001988
- Wanderson Lambert, Dan Marchesin, and Johannes Bruining, The Riemann solution for the injection of steam and nitrogen in a porous medium, Transp. Porous Media 81 (2010), no. 3, 505–526. MR 2599978, DOI https://doi.org/10.1007/s11242-009-9419-9
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 0093653, DOI https://doi.org/10.1002/cpa.3160100406
- Tai Ping Liu, The Riemann problem for general $2\times 2$ conservation laws, Trans. Amer. Math. Soc. 199 (1974), 89–112. MR 0367472, DOI https://doi.org/10.2307/1996875
- Arthur V. Azevedo, Cesar S. Eschenazi, Dan Marchesin, and Carlos F. B. Palmeira, Topological resolution of Riemann problems for pairs of conservation laws, Quart. Appl. Math. 68 (2010), no. 2, 375–393. MR 2663005, DOI https://doi.org/10.1090/S0033-569X-10-01154-7
- V. Matos, A. V. Azevedo, J. C. Da Mota, and D. Marchesin, Bifurcation under parameter change of Riemann solutions for nonstrictly hyperbolic systems, Z. Angew. Math. Phys. 66 (2015), no. 4, 1413–1452. MR 3377695, DOI https://doi.org/10.1007/s00033-014-0469-7
- Vítor Matos and Dan Marchesin, Large viscous solutions for small data in systems of conservation laws that change type, J. Hyperbolic Differ. Equ. 5 (2008), no. 2, 257–278. MR 2419998, DOI https://doi.org/10.1142/S0219891608001477
- Vitor Matos, Julio D. Silva, and Dan Marchesin, Loss of hyperbolicity changes the number of wave groups in Riemann problems, Bull. Braz. Math. Soc. (N.S.) 47 (2016), no. 2, 545–559. MR 3514420, DOI https://doi.org/10.1007/s00574-016-0168-4
- Jeffrey Nunemacher, Asymptotes, Cubic Curves, and the Projective Plane, Math. Mag. 72 (1999), no. 3, 183–192. MR 1573393
- O. A. Oleĭnik, On the uniqueness of the generalized solution of the Cauchy problem for a non-linear system of equations occurring in mechanics, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 6(78), 169–176 (Russian). MR 0094543
- Stephen Schecter, Dan Marchesin, and Bradley J. Plohr, Structurally stable Riemann solutions, J. Differential Equations 126 (1996), no. 2, 303–354. MR 1383980, DOI https://doi.org/10.1006/jdeq.1996.0053
- Stephen Schecter, Bradley J. Plohr, and Dan Marchesin, Classification of codimension-one Riemann solutions, J. Dynam. Differential Equations 13 (2001), no. 3, 523–588. MR 1845094, DOI https://doi.org/10.1023/A%3A1016634307145
- Julio Daniel Silva and Dan Marchesin, Riemann solutions without an intermediate constant state for a system of two conservation laws, J. Differential Equations 256 (2014), no. 4, 1295–1316. MR 3145758, DOI https://doi.org/10.1016/j.jde.2013.10.005
- Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779
- W.J. Todd, M.R.; Longstaff, The development, testing, and application of a numerical simulator for predicting miscible flood performance, Journal of Petroleum Technology 24 (1972), 874–882.
- Burton Wendroff, The Riemann problem for materials with nonconvex equations of state. I. Isentropic flow, J. Math. Anal. Appl. 38 (1972), 454–466. MR 0328387, DOI https://doi.org/10.1016/0022-247X%2872%2990103-5
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Additional Information
Vítor Matos
Affiliation:
Centro de Matemática, Faculdade de Economia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-464 Porto, Portugal
Email:
vmatos@fep.up.pt
Dan Marchesin
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110,22460-320 Rio de Janeiro, RJ, Brazil
MR Author ID:
119555
Email:
marchesi@fluid.impa.br, marchesi@impa.br
Received by editor(s):
September 12, 2016
Received by editor(s) in revised form:
June 14, 2017
Published electronically:
July 20, 2017
Additional Notes:
This work was partially supported by CNPq under Grants 402299/2012-4, 304264/2014-8, 470635/2012-6, and 170135/2016-0 as well as supported by FAPERJ under Grants E-26/110.658/2012, E-26/110.114/2013, E-26/201.210/2014, and E-26/210.738/2014.
Article copyright:
© Copyright 2017
Brown University