Finite element modeling of fluid flow in fractured porous media using unified approach
Keywords:finite element method, fractures porous media, transport properties, permeability
Understanding fluid flow in fractured porous media is of great importance in the fields of civil engineering in general or in soil science particular. This study is devoted to the development and validation of a numerical tool based on the use of the finite element method. To this aim, the problem of fluid flow in fractured porous media is considered as a problem of coupling free fluid and fluid flow in porous media or coupling of the Stokes and Darcy equations. The strong formulation of the problem is constructed, highlighting the condition at the free surface between the Stokes and Darcy regions, following by the variational formulation and numerical integration using the finite element method. Besides, the analytical solutions of the problem are constructed and compared with the numerical solutions given by the finite element approach. Both local properties and macroscopic responses of the two solutions are in excellent agreement, on condition that the porous media are sufficiently discretized by a certain level of finesse. The developed finite element tool of this study could pave the way to investigate many interesting flow problems in the field of soil science.
Arbogast T., Brunson D.S., 2007. A computational method for approximating a Darcy–Stokes system governing a vuggy porous medium. Computational Geosciences, 11, 207–218.
Auriaul J.-L., Sanchez-Palencia E., 1977. Etude du comportement macroscopique d’un milieu poreux saturé déformable. Journal de mécanique, 16, 575–603.
Auriault J.L., Boutin C., 1993. Deformable porous media with double porosity. Quasi-statics. II: Memory effects. Transport in porous media, 10, 153–169.
Auriault J.L., Boutin C., 1994. Deformable porous media with double porosity III: Acoustics. Transport in Porous Media, 14, 143–162.
Auriault J.L., Boutin, C., 1992. Deformable porous media with double porosity. Quasi-statics. I: Coupling effects. Transport in porous media, 7, 63–82.
Bachu S., 2008. CO2 storage in geological media: Role, means, status and barriers to deployment. Progress in Energy and Combustion Science, 34, 254–273. https://doi.org/10.1016/j.pecs.2007.10.001
Beavers G.S., Joseph D.D., 1967. Boundary conditions at a naturally permeable wall. Journal of Fluid Mechanics, 30, 197–207.
Bose S., Roy M., Bandyopadhyay A., 2012. Recent advances in bone tissue engineering scaffolds. Trends in Biotechnology, 30, 546–554. https://doi.org/10.1016/j.tibtech.2012.07.005
Burman E., Hansbo P., 2007. A unified stabilized method for Stokes’ and Darcy’s equations. Journal of Computational and Applied Mathematics, 198, 35–51.
Celle P., Drapier S., Bergheau J.-M., 2008. Numerical modelling of liquid infusion into fibrous media undergoing compaction. European Journal of Mechanics-A/Solids, 27, 647–661.
Correa M.R., Loula A.F.D., 2009. A unified mixed formulation naturally coupling Stokes and Darcy flows. Computer Methods in Applied Mechanics and Engineering, 198, 2710–2722.
de Borst R., 2017. Fluid flow in fractured and fracturing porous media: A unified view. Mechanics Research Communications, Multi-Physics of Solids at Fracture, 80, 47–57. https://doi.org/10.1016/j.mechrescom.2016.05.004
Dietrich P., Helmig R., Sauter M., Hötzl H., Köngeter J., Teutsch G., 2005. Flow and transport in fractured porous media. Springer Science & Business Media.
Discacciati M., Miglio E., Quarteroni A., 2002. Mathematical and numerical models for coupling surface and groundwater flows. Applied Numerical Mathematics, 43, 57–74.
Discacciati M., Quarteron A., Vall A., 2007. Robin-Robin domain decomposition methods for the Stokes–Darcy coupling. SIAM Journal on Numerical Analysis, 45, 1246–1268.
Herzi J.P., Lecler D.M., Gof P. Le., 1970. Flow of Suspensions through Porous Media-Application to Deep Filtration. Industrial & Engineering Chemistry, 62, 8–35. https://doi.org/10.1021/ie50725a003.
Hunt A.G., Sahim M., 2017. Flow, transport, and reaction in porous media: Percolation scaling, critical-path analysis, and effective medium approximation. Reviews of Geophysics, 55, 993-1078.
Karper T., Mardal K.-A., Winther R., 2009. Unified finite element discretizations of coupled Darcy-Stokes flow. Numerical Methods for Partial Differential Equations: An International Journal, 25, 311–326.
Layton W.J., Schieweck F., Yotov I., 2002. Coupling fluid flow with porous media flow. SIAM Journal on Numerical Analysis, 40, 2195–2218.
Ly H.-B., Le Droumaguet B., Monchiet V., Grande D. 2015. Designing and modeling doubly porous polymeric materials. The European Physical Journal Special Topics, 224, 1689–1706.
Ly H.B., Le Droumaguet B., Monchiet V., Grande D., 2015. Facile fabrication of doubly porous polymeric materials with controlled nano-and macro-porosity. Polymer, 78, 13–21.
Ly H.B., Le Droumaguet B., Monchiet V., Grande D., 2016a. Tailoring doubly porous poly (2-hydroxyethyl methacrylate)-based materials via thermally induced phase separation. Polymer, 86, 138–146.
Ly H.B., Monchiet V., Grande D., Lewis R.W., 2016b. Computation of permeability with Fast Fourier Transform from 3-D digital images of porous microstructures. International Journal of Numerical Methods for Heat & Fluid Flow, 26.
Miglio E., Quarteroni A., Saleri F., 2003. Coupling of free surface and groundwater flows. Computers & Fluids, 32, 73–83.
Mikelic A., Jäger W., 2000. On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM Journal on Applied Mathematics, 60, 1111–1127.
Monchiet V., Bonnet G., Lauriat G., 2009. A FFT-based method to compute the permeability induced by a Stokes slip flow through a porous medium. Comptes Rendus Mécanique, 337, 192–197.
Monchiet V., Ly H.-B., Grande D., 2019. Macroscopic permeability of doubly porous materials with cylindrical and spherical macropores. Meccanica,54, 1583–1596.
Nguyen T.-K., Monchiet V., Bonnet G., 2013. A Fourier based numerical method for computing the dynamic permeability of periodic porous media. European Journal of Mechanics-B/Fluids, 37, 90–98.
Pan C., Hilpert M., Miller C.T., 2004. Lattice-Boltzmann simulation of two-phase flow in porous media. Water Resources Research, 40.
Rahm D., 2011. Regulating hydraulic fracturing in shale gas plays: The case of Texas. Energy Policy, 39, 2974–2981. https://doi.org/10.1016/j.enpol.2011.03.009.
Royer P., Auriault J.-L., Boutin C., 1996. Macroscopic modeling of double-porosity reservoirs. Journal of Petroleum Science and Engineering, 16, 187–202.
Wang C.Y., 2001. Stokes flow through a rectangular array of circular cylinders. Fluid Dynamics Research, 29, 65–80. https://doi.org/10.1016/S0169-5983(01)00013-2.
Wang C.Y., 2003. Stokes slip flow through square and triangular arrays of circular cylinders. Fluid Dynamics Research, 32, 233–246. https://doi.org/10.1016/S0169-5983(03)00049-2
Whitaker S., 1967. Diffusion and dispersion in porous media. AIChE Journal, 13, 420–427. https://doi.org/10.1002/aic.690130308.
Xie X., Xu J., Xue G., 2008. Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models. Journal of Computational Mathematics, 437–455.