References
A. V. Azevedo, A. de Souza, F. Furtado, D. Marchesin, and B. Plohr (2010) “The solution by the wave curve method of three-phase flow in virgin reservoirs,” Transp. Porous Media 83: 99–125.
S. E. Buckley and M. C. Leverett (1942) “Mechanism of fluid displacements in sands,” Transactions of the AIME 146: 107–116.
J. Bruining (2007) Multiphase Flow in Porous Media. TU-Delft, Lecture notes. (94 pages.)
P. Castañeda, F. Furtado, and D. Marchesin (2013) “The convex permeability three-phase flow in reservoirs.” IMPA Preprint Série E - 34: 1–35.
P. Castañeda, E. Abreu, F. Furtado, and D. Marchesin “On a universal structure for immiscible three-phase flow in virgin reservoirs.” (Submitted.)
O. A. Olei˘nik (1957) “Discontinuous solutions of non-linear differential equations,” Uspekhi Mat. Nauk 12: 3–73; English translation: AMS Transl. (1963) 26(2): 95–172.
M. Shearer (1988) “Loss of strict hyperbolicity of the Buckley-Leverett equations in three-phase flow in a porous medium”, Numerical Simulation in Oil Recovery IMA Vol. Math. Appl. 11: 263–283.
A. Spivak, H.S. Price, and A. Settari (1977) “Solution of the equations for multidimensional, two-phase, immiscible flow by variational methods,” Soc. Pet. Eng. J. 17: 27–41.
H. Wahanik, A. A. Eftekhari, J. Bruining, D. Marchesin, and K. H. Wolf (2010) “Analytical solutions for mixed CO2-water injection in geothermal reservoirs,” CSUG/SPE 138154.
Acknowledgments
I thank Professor Furtado (University of Wyoming) and Marchesin (IMPA) for many enlightening conversations, their friendship, and the homework to do aboard the “4-O-9.” I am indebted to the reviewer for his fine comments and also to Gustavo Hime for his friendship. This work was supported by the Department of Mathematics at ITAM and by FAPERJ under Grant E-26/110.017/2014.
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Castañeda, P. Dogma: S-shaped . Math Intelligencer 38, 10–13 (2016). https://doi.org/10.1007/s00283-015-9611-4
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DOI: https://doi.org/10.1007/s00283-015-9611-4