Dimensional model reduction for flow through fractures in poroelastic media
Bukač, Martina1 ; Yotov, Ivan2 ; Zunino, Paolo3, 4
1 Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Indiana 46556, USA.
2 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA.
3 MOX, Department of Mathematics, Politecnico di Milano, 20133 Milano, Italy.
4 Department of Mechanical Engineer.ing and Materials Science, University of Pittsburgh, PA 15260, USA.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1429-1471.

We study the interaction between a poroelastic medium and a fracture filled with fluid. The flow in the fracture is described by the Brinkman equations for an incompressible fluid and the poroelastic medium by the quasi-static Biot model. The two models are fully coupled via the kinematic and dynamic conditions. The Brinkman equations are then averaged over the cross-sections, giving rise to a reduced flow model on the fracture midline. We derive suitable interface and closure conditions between the Biot system and the dimensionally reduced Brinkman model that guarantee solvability of the resulting coupled problem. We design and analyze a numerical discretization scheme based on finite elements in space and the Backward Euler in time, and perform numerical experiments to compare the behavior of the reduced model to the full-dimensional formulation and study the response of the model with respect to its parameters.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016069
Classification : 76S05, 76D07, 74F10, 65M60, 65M12
Mots-clés : Reduced model, fracture flow, poroelasticity
Bukač, Martina 1 ; Yotov, Ivan 2 ; Zunino, Paolo 3, 4

1 Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Indiana 46556, USA.
2 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA.
3 MOX, Department of Mathematics, Politecnico di Milano, 20133 Milano, Italy.
4 Department of Mechanical Engineer.ing and Materials Science, University of Pittsburgh, PA 15260, USA. 
@article{M2AN_2017__51_4_1429_0,
     author = {Buka\v{c}, Martina and Yotov, Ivan and Zunino, Paolo},
     title = {Dimensional model reduction for flow through fractures in poroelastic media},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1429--1471},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {4},
     year = {2017},
     doi = {10.1051/m2an/2016069},
     zbl = {1372.76100},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2016069/}
}
TY  - JOUR
AU  - Bukač, Martina
AU  - Yotov, Ivan
AU  - Zunino, Paolo
TI  - Dimensional model reduction for flow through fractures in poroelastic media
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2017
SP  - 1429
EP  - 1471
VL  - 51
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2016069/
DO  - 10.1051/m2an/2016069
LA  - en
ID  - M2AN_2017__51_4_1429_0
ER  - 
%0 Journal Article
%A Bukač, Martina
%A Yotov, Ivan
%A Zunino, Paolo
%T Dimensional model reduction for flow through fractures in poroelastic media
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2017
%P 1429-1471
%V 51
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2016069/
%R 10.1051/m2an/2016069
%G en
%F M2AN_2017__51_4_1429_0
Bukač, Martina; Yotov, Ivan; Zunino, Paolo. Dimensional model reduction for flow through fractures in poroelastic media. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1429-1471. doi : 10.1051/m2an/2016069. http://www.numdam.org/articles/10.1051/m2an/2016069/

P. Angot, Analysis of singular perturbations on the Brinkman problem for fictitious domain models of viscous flows. Math. Methods Appl. Sci 22 (1999) 1395–1412. | DOI | Zbl

P. Angot, F. Boyer and F. Hubert, Asymptotic and numerical modelling of flows in fractured porous media. Math. Model. Numer. Anal. 43 (2009) 239–275. | DOI | Numdam | Zbl

T. Arbogast, L. C. Cowsar, M.F. Wheeler and I. Yotov, Mixed finite element methods on non-matching multiblock grids. SIAM J. Numer. Anal. 37 (2000) 1295–1315. | DOI | Zbl

S. Badia, A. Quaini and A. Quarteroni, Coupling Biot and Navier–Stokes equations for modelling fluid–poroelastic media interaction. J. Comput. Phys. 228 (2009) 7986–8014. | DOI | Zbl

G.S. Beavers and D.D. Joseph, Boundary conditions at a naturally impermeable wall. J. Fluid. Mech. 30 (1967) 197–207. | DOI

M.A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26 (1955) 182–185. | DOI | Zbl

D. Boffi, F. Brezzi and M. Fortin, Mixed finite element methods and applications. Vol. 44 of Springer Series in Computational Mathematics. Springer, Heidelberg (2013). | MR | Zbl

M. Bukač, I. Yotov, R. Zakerzadeh and P. Zunino, Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsches coupling approach. Comput. Methods Appl. Mech. Engrg. 292 (2015) 138–170. | DOI | Zbl

M. Bukač, I. Yotov and P. Zunino, An operator splitting approach for the interaction between a fluid and a multilayered poroelastic structure. Numer. Methods Partial Differ. Equ. 31 (2015) 1054–1100. | DOI | Zbl

A.P. Bunger and E. Detournay, Experimental validation of the tip asymptotics for a fluid-driven crack. J. Mech. Phys. Solids 56 (2008) 3101–3115. | DOI

P. Ciarlet, The finite element method for elliptic problems. Vol. 4. North Holland (1978). | Zbl

C. D’Angelo and A. Scotti, A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. ESAIM: M2AN 46 (2012) 465–489. | DOI | Numdam | Zbl

M. Discacciati, E. Miglio and A. Quarteroni, Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43 (2002) 57–74. | DOI | Zbl

A. Ern and J.-L. Guermond, Theory and practice of finite elements. Vol. 159 of Appl. Math. Sci. Springer-Verlag, New York (2004). | MR | Zbl

N. Frih, V. Martin, J.E. Roberts and A. Sada, Modeling fractures as interfaces with nonmatching grids. Comput. Geosci. 16 (2012) 1043–1060. | DOI

N. Frih, J.E. Roberts and A. Saada, Modeling fractures as interfaces: A model for Forchheimer fractures. Comput. Geosci. 12 (2008) 91–104. | DOI | Zbl

A. Fumagalli and A. Scotti, Numerical modelling of multiphase subsurface flow in the presence of fractures. Commun. Appl. Ind. Math. 3 (2012) e–380, 23. | Zbl

B. Ganis, V. Girault, M. Mear, G. Singh and M.F. Wheeler, Modeling fractures in a poro-elastic medium. Oil Gas Sci. Technol. 69 (2014) 515–528. | DOI

B. Ganis, M. E. Mear, A. Sakhaee-Pour, M. F. Wheeler and T. Wick, Modeling fluid injection in fractures with a reservoir simulator coupled to a boundary element method. Comput. Geosci. 18 (2014) 613–624. | DOI | Zbl

V. Girault, D. Vassilev and I. Yotov, Mortar multiscale finite element methods for Stokes-Darcy flows. Numer. Math. 127 (2014) 93–165. | DOI | Zbl

V. Girault, M.F. Wheeler, B. Ganis and M.E. Mear, A lubrication fracture model in a poro-elastic medium. Math. Models Methods Appl. Sci. 25 (2015) 587–645. | DOI | Zbl

E. Gordeliy and A. Peirce, Implicit level set schemes for modeling hydraulic fractures using the XFEM. Comput. Methods Appl. Mech. Engrg. 266 (2013) 125–143. | DOI | Zbl

P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, Boston (1985). | Zbl

F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. | DOI | MR | Zbl

J. Jaffre, 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

J. Jaffre and J.E. Roberts, Modeling flow in porous media with fractures; discrete fracture models with matrix-fracture exchange. Numer. Anal. Appl. 5 (2012) 162–167. | DOI | Zbl

W. J. Layton, F. Schieweck and I. Yotov, Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40 (2002) 2195–2218. | DOI | Zbl

M. Lesinigo, C. D’Angelo and A. Quarteroni, A multiscale Darcy–Brinkman model for fluid flow in fractured porous media. Numer. Math. 117 (2011) 717–752. | DOI | Zbl

J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, vol. 1. Springer-Verlag (1972). | Zbl

V. Martin, J. Jaffre 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. Mikelić and M. F. Wheeler, On the interface law between a deformable porous medium containing a viscous fluid and an elastic body. Math. Models Methods Appl. Sci. 22 (2012) 1250031, 32. | DOI | Zbl

A. Mikelić and M. F. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. (2013) 17 455–461. | DOI | Zbl

A. Mikelic, M.F. Wheeler and T. Wick, A phase-field method for propagating fluid-filled fractures coupled to a surrounding porous medium. Multiscale Model Simul. (2015).

F. Morales and R. E. Showalter, The narrow fracture approximation by channeled flow. J. Math. Anal. Appl. 365 (2010) 320–331. | DOI | Zbl

Fernando Morales and Ralph E. Showalter, Interface approximation of Darcy flow in a narrow channel. Math. Methods Appl. Sci., 35(2):182–195, 2012. | Zbl

E.-J. Park, Mixed finite element methods for generalized Forchheimer flow in porous media. Numer. Methods Partial Differ. Equ. 21 (2005) 213–228. | DOI | Zbl

P.J. Phillips and M.F. Wheeler, A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: The continuous in time case. Comput. Geosci. 11 (2007) 131–144. | DOI | Zbl

P.J. Phillips and M.F. Wheeler, A coupling of mixed and continuous Galerkin finite element methods for poroelasticity II: The discrete-in-time case. Comput. Geosci. 11 (2007) 145–158. | DOI | Zbl

P.J. Phillips and M. F. Wheeler, A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity. Comput. Geosci. 12 (2008) 417–435. | DOI | Zbl

K. R. Rajagopal, On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math. Models Methods Appl. Sci. 17 (2007) 215–252. | DOI | Zbl

J. Rungamornrat and M.E. Mear, SGBEM-FEM coupling for analysis of cracks in 3D anisotropic media. Int. J. Numer. Methods Engrg. 86 (2011) 224–248. | DOI | Zbl

P.G. Saffman, On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50 (1971) 93–101. | DOI | Zbl

L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. | DOI | MR | Zbl

R.E. Showalter, Poroelastic filtration coupled to Stokes flow. In Control theory of partial differential equations. In vol. 242 of of Lect. Notes Pure Appl. Math. Chapman & Hall/CRC, Boca Raton, FL (2005) 229–241. | Zbl

K. Terzaghi, R.B. Peck and G. Mesri, Soil Mechanics in Engineering Practice. Wiley-Interscience publication. Wiley (1996).

Cité par Sources :