Mixed finite element approximation for a coupled petroleum reservoir model
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 2, p. 349-376

In this paper, we are interested in the modelling and the finite element approximation of a petroleum reservoir, in axisymmetric form. The flow in the porous medium is governed by the Darcy-Forchheimer equation coupled with a rather exhaustive energy equation. The semi-discretized problem is put under a mixed variational formulation, whose approximation is achieved by means of conservative Raviart-Thomas elements for the fluxes and of piecewise constant elements for the pressure and the temperature. The discrete problem thus obtained is well-posed and a posteriori error estimates are also established. Numerical tests are presented validating the developed code.

DOI : https://doi.org/10.1051/m2an:2005015
Classification:  35Q35,  65N15,  65N30,  76S05
Keywords: petroleum reservoir, thermometrics, porous medium, mixed finite elements, a posteriori estimators
@article{M2AN_2005__39_2_349_0,
     author = {Amara, Mohamed and Capatina-Papaghiuc, Daniela and Denel, Bertrand and Terpolilli, Peppino},
     title = {Mixed finite element approximation for a coupled petroleum reservoir model},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {2},
     year = {2005},
     pages = {349-376},
     doi = {10.1051/m2an:2005015},
     zbl = {1130.76045},
     mrnumber = {2143952},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2005__39_2_349_0}
}
Amara, Mohamed; Capatina-Papaghiuc, Daniela; Denel, Bertrand; Terpolilli, Peppino. Mixed finite element approximation for a coupled petroleum reservoir model. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 2, pp. 349-376. doi : 10.1051/m2an:2005015. http://www.numdam.org/item/M2AN_2005__39_2_349_0/

[1] C. Abchir, Modélisation des écoulements dans les réservoirs souterrains avec prise en compte des interactions puits/réservoir. Thèse de doctorat, Université de Saint-Etienne (1992).

[2] M. Amara, D. Capatina, B. Denel and P. Terpolilli, Modelling, analysis and numerical approximation of flow with heat transfer in a petroleum reservoir, Preprint No. 0415, Université de Pau (2004) (http://lma.univ-pau.fr/publis/publis.php).

[3] G. Bourdarot, Well testing: Interpretation methods. Editions Technip, Paris (1998).

[4] S. Brenner and R. Scott, The mathematical theory of Finite Element Methods. Springer Verlag, New York (1994). | MR 1278258 | Zbl 0804.65101

[5] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Verlag, New York (1991). | MR 1115205 | Zbl 0788.73002

[6] G. Chavent and J.E. Roberts, A unified physical presentation of mixed, mixed-hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems. Adv. Water Resources 14 (1991) 329-348.

[7] P.G. Ciarlet, The finite element method for elliptic problems error analysis. North Holland, Amsterdam (1978). | MR 520174 | Zbl 0383.65058

[8] R.E. Ewing, J. Wang and S.L. Weekes, On the simulation of multicomponent gas flow in porous media. Appl. Numer. Math. 31 (1999) 405-427. | Zbl 0940.76033

[9] P. Grisvard, Elliptic problems on non-smooth domains. Pitman, Boston (1985). | Zbl 0695.35060

[10] F. Maubeuge, M. Didek, E. Arquis, O. Bertrand and J.-P. Caltagirone, Mother: A model for interpreting thermometrics. SPE 28588 (1994).

[11] D.Y. Peng and D.B. Robinson, A new two-constant equation of state. Ind. Eng. Chem. Fundam. 15 (1976) 59-64. | Zbl 0332.20008

[12] J.E. Roberts and J.-M. Thomas, Mixed and Hybrid Methods, in Handbook of Numerical Analysis Vol. II. North Holland, Amsterdam (1991) 523-639. | Zbl 0875.65090

[13] R. Verfürth and D. Braess, A posteriori error estimator for the Raviart-Thomas element. SIAM J. Numer. Anal. 33 (1996) 2431-2444. | Zbl 0866.65071