New non-equilibrium matrix imbibition equation for double porosity model

Andrey Konyukhov; Leonid Pankratov

doi:10.1016/j.crme.2016.02.011

New non-equilibrium matrix imbibition equation for double porosity model

Andrey Konyukhov ^1,² ; Leonid Pankratov ^2,³

¹ Joint Institute for High Temperatures of the Russian Academy of Sciences, Izborskaya 13 Bldg, 2, Moscow, 125412, Russian Federation
² Laboratory of Fluid Dynamics and Seismic, Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russian Federation
³ Laboratoire de mathématiques et de leurs applications, CNRS – UMR 5142, Université de Pau et des pays de l'Adour, av. de l'Université, 64000 Pau, France

Comptes Rendus. Mécanique, Volume 344 (2016) no. 7, pp. 510-520.

Résumé

The paper deals with the global Kondaurov double porosity model describing a non-equilibrium two-phase immiscible flow in fractured-porous reservoirs when non-equilibrium phenomena occur in the matrix blocks, only. In a mathematically rigorous way, we show that the homogenized model can be represented by usual equations of two-phase incompressible immiscible flow, except for the addition of two source terms calculated by a solution to a local problem being a boundary value problem for a non-equilibrium imbibition equation given in terms of the real saturation and a non-equilibrium parameter.

Reçu le : 2015-07-11
Accepté le : 2016-02-22
Publié le : 2016-03-30

DOI : 10.1016/j.crme.2016.02.011

Mots clés : Porous media, Double porosity, Homogenization, Two-phase flow, Non-equilibrium model

Affiliations des auteurs :

Andrey Konyukhov ^{1, 2} ; Leonid Pankratov ^{2, 3}

@article{CRMECA_2016__344_7_510_0,
     author = {Andrey Konyukhov and Leonid Pankratov},
     title = {New non-equilibrium matrix imbibition equation for double porosity model},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {510--520},
     publisher = {Elsevier},
     volume = {344},
     number = {7},
     year = {2016},
     doi = {10.1016/j.crme.2016.02.011},
     language = {en},
}

TY  - JOUR
AU  - Andrey Konyukhov
AU  - Leonid Pankratov
TI  - New non-equilibrium matrix imbibition equation for double porosity model
JO  - Comptes Rendus. Mécanique
PY  - 2016
SP  - 510
EP  - 520
VL  - 344
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crme.2016.02.011
LA  - en
ID  - CRMECA_2016__344_7_510_0
ER  -

%0 Journal Article
%A Andrey Konyukhov
%A Leonid Pankratov
%T New non-equilibrium matrix imbibition equation for double porosity model
%J Comptes Rendus. Mécanique
%D 2016
%P 510-520
%V 344
%N 7
%I Elsevier
%R 10.1016/j.crme.2016.02.011
%G en
%F CRMECA_2016__344_7_510_0

Andrey Konyukhov; Leonid Pankratov. New non-equilibrium matrix imbibition equation for double porosity model. Comptes Rendus. Mécanique, Volume 344 (2016) no. 7, pp. 510-520. doi : 10.1016/j.crme.2016.02.011. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2016.02.011/

Bibliographie
Cité par

[1] A. Konyukhov; L. Pankratov Upscaling of an immiscible non-equilibrium two-phase flow in double porosity media, Appl. Anal. (2015) | DOI

[2] S. Bottero; S.M. Hassanizadeh; P.J. Kleingeld; T. Heimovaara Nonequilibrium capillarity effects in two-phase flow through porous media at different scales, Water Resour. Res., Volume 47 (2011), pp. 1-11

[3] O. Coussy Poromechanics, Wiley, New York, 2004

[4] U. Hornung Homogenization and Porous Media, Springer-Verlag, New York, 1997

[5] M. Panfilov Macroscale Models of Flow Through Highly Heterogeneous Porous Media, Kluwer Academic Publishers, London, 2000

[6] H. Salimi; J. Bruining Improved prediction of oil recovery from waterflooded fractured reservoirs using homogenization, SPE Reserv. Eval. Eng., Volume 13 (2010), pp. 44-55

[7] H. Salimi; J. Bruining Upscaling in vertically fractured oil reservoirs using homogenization, Transp. Porous Media, Volume 84 (2010), pp. 21-53

[8] H. Salimi; J. Bruining Upscaling of fractured oil reservoirs using homogenization including non-equilibrium capillary pressure and relative permeability, Comput. Geosci., Volume 16 (2012), pp. 367-389

[9] B. Amaziane; S. Antontsev; L. Pankratov; A. Piatnitski Homogenization of immiscible compressible two-phase flow in porous media: application to gas migration in a nuclear waste repository, Multiscale Model. Simul., Volume 8 (2010), pp. 2023-2047

[10] B. Amaziane; L. Pankratov Homogenization of a model for water-gas ow through double-porosity media, Math. Methods Appl. Sci., Volume 39 (2016), pp. 425-451

[11] A. Bourgeat; S. Luckhaus; A. Mikelić Convergence of the homogenization process for a double-porosity model of immicible two-phase flow, SIAM J. Math. Anal., Volume 27 (1996), pp. 1520-1543

[12] L.M. Yeh Homogenization of two-phase flow in fractured media, Math. Methods Appl. Sci., Volume 16 (2006), pp. 1627-1651

[13] B. Amaziane; J.P. Milišić; M. Panfilov; L. Pankratov Generalized nonequilibrium capillary relations for two-phase flow through heterogeneous media, Phys. Rev. E, Volume 85 (2012)

[14] A. Bourgeat; M. Panfilov Effective two-phase flow through highly heterogeneous porous media: capillary nonequilibrium effects, Comput. Geosci., Volume 2 (1998), pp. 191-215

[15] G.I. Barenblatt; T.W. Patzek; D.B. Silin The mathematical model of non-equilibrium effects in water-oil displacement, SPE J., Volume 8 (2003), pp. 409-416

[16] S.M. Hassanizadeh; W.G. Gray Thermodynamic basis of capillary pressure in porous media, Water Resour. Res., Volume 29 (1993), pp. 3389-3405

[17] Upscaling Multiphase Flow in Porous Media (D.B. Das; S.M. Hassanizadeh, eds.), Springer, Dordrecht, The Netherlands, 2005

[18] X. Cao; I.S. Pop Two-phase porous media flows with dynamic capillary effects and hysteresis: uniqueness of weak solutions, Comput. Math. Appl., Volume 69 (2015), pp. 688-695

[19] J. Koch; A. Rätz; B. Schweizer Two-phase flow equations with a dynamic capillary pressure, Eur. J. Appl. Math., Volume 24 (2013), pp. 49-75

[20] V.I. Kondaurov A non-equilibrium model of a porous medium saturated with immiscible fluids, J. Appl. Math. Mech., Volume 73 (2009), pp. 88-102

[21] A. Konyukhov; A. Tarakanov On two approaches in investigation of non-equilibrium effects of filtration in a porous medium, ASCE 2013 (2013), pp. 2307-2316

[22] S.N. Antontsev; A.V. Kazhikhov; V.N. Monakhov Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland, Amsterdam, 1990

[23] G. Chavent; J. Jaffré Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam, 1986

[24] C. Galusinski; M. Saad Weak solutions for immiscible compressible multifluid flows in porous media, C. R. Acad. Sci. Paris, Sér. I, Volume 347 (2009), pp. 249-254

[25] M. Jurak; L. Pankratov; A. Vrbaški A fully homogenized model for incompressible two-phase flow in double porosity media, Appl. Anal. (2015) | DOI

[26] N. Bakhvalov; G. Panasenko Homogenisation: Averaging Processes in Periodic Media, Kluwer Academic Publishers, Dordrecht, 1989

[27] A. Bensoussan; J.-L. Lions; G. Papanicolaou Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978

[28] E. Sanchez-Palencia Non-homogeneous Media and Vibration Theory, Springer-Verlag, Berlin, 1980

[29] J.G. Richardson Flow through porous media (V.L. Streeter, ed.), Handbook of Fluid Dynamics, McGraw Hill, 1961, pp. 16-65

[30] T. Arbogast A simplified dual-porosity model for two-phase flow, Denver, CO, 1992 (T.F. Russell; R.E. Ewing; C.A. Brebbia; W.G. Gray; G.F. Pindar, eds.) (Mathematical Modeling in Water Resources), Volume vol. 2, Comput. Mech, Southampton, UK (1992), pp. 419-426

[31] B. Amaziane; M. Panfilov; L. Pankratov Homogenized model of two-phase ow with local non-equilibrium in double porosity media, J. Math. Fluid Mech. (2015) (submitted for publication)

[32] A.E. Berger; H. Brezis; J.C.W. Rogers A numerical method for solving the problem $u_{t} - Δ f (u) = 0$ , RAIRO. Anal. Numér., Volume 13 (1979), pp. 297-312

[33] B. Amaziane; L. Pankratov; A. Piatnitski The existence of weak solutions to immiscible compressible two-phase flow in porous media: the case of fields with different rock-types, Discrete Contin. Dyn. Syst., Ser. B, Volume 18 (2013), pp. 1217-1251

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Symmetry for extremal functions in subcritical Caffarelli–Kohn–Nirenberg inequalities

Jean Dolbeault; Maria J. Esteban; Michael Loss; ...

C. R. Math (2017)

Spectral properties of Schrödinger operators on compact manifolds: Rigidity, flows, interpolation and spectral estimates

Jean Dolbeault; Maria J. Esteban; Ari Laptev; ...

C. R. Math (2013)

Some preliminary observations on a defect Navier–Stokes system

Amit Acharya; Roger Fosdick

C. R. Méca (2019)