Mixed finite element approximation for a coupled petroleum reservoir model
Amara, Mohamed ; Capatina-Papaghiuc, Daniela ; Denel, Bertrand ; Terpolilli, Peppino1
1 Total, CST Jean Feger, Avenue Larribau, 64018 Pau Cedex, France.
ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 2, pp. 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 : 10.1051/m2an:2005015
Classification : 35Q35, 65N15, 65N30, 76S05
Mots-clés : petroleum reservoir, thermometrics, porous medium, mixed finite elements, a posteriori estimators
Amara, Mohamed  ; Capatina-Papaghiuc, Daniela  ; Denel, Bertrand  ; Terpolilli, Peppino 1

1 Total, CST Jean Feger, Avenue Larribau, 64018 Pau Cedex, France. 
@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: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {349--376},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {2},
     year = {2005},
     doi = {10.1051/m2an:2005015},
     mrnumber = {2143952},
     zbl = {1130.76045},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an:2005015/}
}
TY  - JOUR
AU  - Amara, Mohamed
AU  - Capatina-Papaghiuc, Daniela
AU  - Denel, Bertrand
AU  - Terpolilli, Peppino
TI  - Mixed finite element approximation for a coupled petroleum reservoir model
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2005
SP  - 349
EP  - 376
VL  - 39
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an:2005015/
DO  - 10.1051/m2an:2005015
LA  - en
ID  - M2AN_2005__39_2_349_0
ER  - 
%0 Journal Article
%A Amara, Mohamed
%A Capatina-Papaghiuc, Daniela
%A Denel, Bertrand
%A Terpolilli, Peppino
%T Mixed finite element approximation for a coupled petroleum reservoir model
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2005
%P 349-376
%V 39
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an:2005015/
%R 10.1051/m2an:2005015
%G en
%F 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: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 2, pp. 349-376. doi : 10.1051/m2an:2005015. http://archive.numdam.org/articles/10.1051/m2an:2005015/

[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 | Zbl

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

[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 | Zbl

[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

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

[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

[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

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

Cité par Sources :