Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 303-330.

This paper presents a finite volume discretization of two-phase Darcy flows in discrete fracture networks taking into account the mass exchange between the matrix and the fracture. We consider the asymptotic model for which the fractures are represented as interfaces of codimension one immersed in the matrix domain, leading to the so called hybrid dimensional Darcy flow model. The pressures at the interfaces between the matrix and the fracture network are continuous corresponding to a ratio between the normal permeability of the fracture and the width of the fracture assumed to be large compared with the ratio between the permeability of the matrix and the size of the domain. The discretization is an extension of the Vertex Approximate Gradient (VAG) scheme to the case of hybrid dimensional Darcy flow models. Compared with Control Volume Finite Element (CVFE) approaches, the VAG scheme has the advantage to avoid the mixing of the fracture and matrix rocktypes at the interfaces between the matrix and the fractures, while keeping the low cost of a nodal discretization on unstructured meshes. The convergence of the scheme is proved under the assumption that the relative permeabilities are bounded from below by a strictly positive constant. This assumption is needed in the convergence proof in order to take into account discontinuous capillary pressures in particular at the matrix fracture interfaces. The efficiency of our approach compared with CVFE discretizations is shown on two numerical examples of fracture networks in 2D and 3D.

Reçu le :
DOI : 10.1051/m2an/2014034
Classification : 65M08, 65M12, 76S05
Mots clés : Finite Volume Scheme, discrete fracture network, two-phase darcy flow, discontinuous capillary pressure
Brenner, K. 1 ; Groza, M. 1 ; Guichard, C. 2 ; Masson, R. 1

1 Laboratoire de Mathématiques J.A. Dieudonné, UMR 7351 CNRS, University Nice Sophia Antipolis, and team COFFEE, INRIA Sophia Antipolis Méditerranée, Parc Valrose, 06108 Nice cedex 02, France
2 Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, CNRS, Laboratoire Jacques-Louis Lions, 75005 Paris, France
@article{M2AN_2015__49_2_303_0,
     author = {Brenner, K. and Groza, M. and Guichard, C. and Masson, R.},
     title = {Vertex {Approximate} {Gradient} {Scheme} for {Hybrid} {Dimensional} {Two-Phase} {Darcy} {Flows} in {Fractured} {Porous} {Media}},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {303--330},
     publisher = {EDP-Sciences},
     volume = {49},
     number = {2},
     year = {2015},
     doi = {10.1051/m2an/2014034},
     zbl = {1311.76078},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2014034/}
}
TY  - JOUR
AU  - Brenner, K.
AU  - Groza, M.
AU  - Guichard, C.
AU  - Masson, R.
TI  - Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2015
SP  - 303
EP  - 330
VL  - 49
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2014034/
DO  - 10.1051/m2an/2014034
LA  - en
ID  - M2AN_2015__49_2_303_0
ER  - 
%0 Journal Article
%A Brenner, K.
%A Groza, M.
%A Guichard, C.
%A Masson, R.
%T Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2015
%P 303-330
%V 49
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2014034/
%R 10.1051/m2an/2014034
%G en
%F M2AN_2015__49_2_303_0
Brenner, K.; Groza, M.; Guichard, C.; Masson, R. Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 303-330. doi : 10.1051/m2an/2014034. http://archive.numdam.org/articles/10.1051/m2an/2014034/

C. Alboin, J. Jaffré, J. Roberts and C. Serres, Modeling fractures as interfaces for flow and transport in porous media. In vol. 295 of Fluid flow and transport in porous media, edited by Chen, Ewing. American Mathematical Society (2002) 13–24. | Zbl

O. Angelini, K. Brenner and D. Hilhorst, A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation. Numer. Math. 123 (2013) 219-257. | DOI | Zbl

P. Angot, F. Boyer and F. Hubert, Asymptotic and numerical modelling of flows in fractured porous media. ESAIM: M2AN 23 (2009) 239–275. | DOI | Numdam | Zbl

S.N. Antontsev, A.V. Kazhikhov and V.N. Monakhov, Boundary value problems in mechanics of nonhomogeneous fluids, vol. 22 of Stud. Math. Appl. North-Holland Publishing Co., Amsterdam (1990). Translated from Russian. | Zbl

B.R. Baliga and S.V. Patankar Sv, A control volume finite-element method for two dimensional fluid flow and heat transfer. Numerical Heat Transfer 6 (1983) 245–261. | Zbl

K. Brenner and R. Masson, Convergence of a Vertex centred Discretization of Two-Phase Darcy flows on General Meshes, Int. J. Finite Volume Methods (2013).

K. Brenner, C. Cancès and D. Hilhorst, Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure. Comput. Geosci. 17 (2013) 573–597. | DOI | Zbl

C. Cancès, Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities. ESAIM: M2AN 43 (2009) 973–1001. | DOI | Numdam | Zbl

Cancès, Clément and Pierre, Michel, An existence result for multidimensional immiscible two-phase flows with discontinuous capillary pressure field. SIAM J. Math. Anal. 44 (2012) 966–992. | DOI | Zbl

G. Chavent and J. Jaffré. Mathematical Models and Finite Elements for Reservoir Simulation, vol. 17 of Stud. Math. Appl. North-Holland, Amsterdam (1986). | Zbl

C.J. Van Duijn, J. Molenaar and M. J. De Neef, The effect of capillary forces on immiscible two-phase flows in heterogeneous porous media. Transp. Porous Media 21 (1995) 71–93. | DOI

R. Eymard, R. Herbin and A. Michel. Mathematical study of a petroleum-engineering scheme. ESAIM: M2AN 37 (2003) 937–972. | DOI | Numdam | Zbl

R. Eymard, C. Guichard and R. Herbin, Small-stencil 3D schemes for diffusive flows in porous media. ESAIM: M2AN 46 (2010) 265–290. | DOI | Numdam | Zbl

R. Eymard, C. Guichard, R. Herbin and R. Masson, Vertex centred Discretization of Two-Phase Darcy flows on General Meshes. ESAIM Proc. 35 (2012) 59–78. | DOI | Zbl

R. Eymard, R. Herbin, C. Guichard and R. Masson, Vertex Centred discretization of compositional Multiphase Darcy flows on general meshes. Comput. Geosci. 16 (2012) 987–1005.

R. Eymard, P. Féron, T. Gallouët, R. Herbin and C. Guichard. Gradient schemes for the Stefan problem. Int. J. Finite Volumes (2013).

R. Eymard, C. Guichard, R. Herbin and R. Masson, Gradient schemes for two-phase flow in heterogeneous porous media and Richards equation, article first published online, ZAMM – J. Appl. Math. Mech. (2013). Doi:. | DOI | Zbl

E. Flauraud, F. Nataf, I. Faille and R. Masson, Domain Decomposition for an asymptotic geological fault modeling. C. R. l’Académie des Sciences, Mécanique 331 (2003) 849-855. | DOI | Zbl

L. Formaggia, A. Fumagalli, A. Scotti and P. Ruffo, A reduced model for Darcy’s problem in networks of fractures. ESAIM: M2AN 48 (2014) 1089–1116. | DOI | Numdam | Zbl

H. Haegland, I. Aavatsmark, C. Guichard, R. Masson and R. Kaufmann, Comparison of a Finite Element Method and a Finite Volume Method for Flow on General Grids in 3D. In Proc. of ECMOR XIII. Biarritz (2012).

J. Hoteit and A. Firoozabadi, An efficient numerical model for incompressible two-phase flow in fracture media. Adv. Water Resources 31 (2008) 891–905. | DOI

R. Huber and R. Helmig, Node-centred finite volume discretizations for the numerical simulation of multi-phase flow in heterogeneous porous media, Comput. Geosci. 4 (2000) 141–164. | DOI | Zbl

J. Jaffré, M. Mnejja and J.E. Roberts, A discrete fracture model for two-phase flow with matrix-fracture interaction. Procedia Comput. Sci. 4 (2011) 967–973. | DOI

M. Karimi-Fard, L.J. Durlovski and K. Aziz, An efficient discrete-fracture model applicable for general-purpose reservoir simulators. SPE journal (2004).

S. Lacroix, Y.V. Vassilevski and M.F. Wheeler, Decoupling preconditioners in the Implicit Parallel Accurate Reservoir Simulator (IPARS). Numer. Linear Algebra Appl. 8 (2001) 537–549. | DOI | Zbl

V. Martin, J. Jaffré and J.E. Roberts, Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26 (2005) 1667–1691. | DOI | Zbl

A. Michel, A finite volume scheme for two-phase immiscible flow in porous media. SIAM J. Numer. Anal. 41 (2003) 1301–1317. | DOI | Zbl

J. Monteagudu and A. Firoozabadi, Control-volume model for simulation of water injection in fractured media: incorporating matrix heterogeneity and reservoir wettability effects. SPE J. 12 (2007) 355–366. | DOI

V. Reichenberger, H. Jakobs, P. Bastian and R. Helmig, A mixed-dimensional finite volume method for multiphase flow in fractured porous media. Adv. Water Resources 29 (2006) 1020–1036. | DOI

R. Scheichl, R. Masson and J. Wendebourg, Decoupling and block preconditioning for sedimentary basin simulations. Comput. Geosci. 7 (2003) 295–318. | DOI | Zbl

X. Tunc, I. Faille, T. Gallouet, M.C. Cacas and P. Havé, A model for conductive faults with non matching grids. Comput. Geosci. 16 (2012) 277–296. | DOI | Zbl

http://tetgen.org.

Cité par Sources :