Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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
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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.


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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
Email: marchesi@fluid.impa.br, marchesi@impa.br

DOI: https://doi.org/10.1090/qam/1477
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


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