# American Institute of Mathematical Sciences

March  2021, 14(3): 1197-1212. doi: 10.3934/dcdss.2020234

## Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media

 1 Division of Mathematical and Physics Science, Kanazawa University, Kakuma, Kanazawa 920-1192, Japan 2 Faculty of Mathematical and Physics, Kanazawa University, Kakuma, Kanazawa 920-1192, Japan 3 Japan Science and Technology Agency, PRESTO, Kawaguchi 332-0012, Japan

* Corresponding author: Imam Wijaya

Received  January 2019 Revised  May 2019 Published  March 2021 Early access  December 2019

The purposes of this work are to study the $L^{2}$-stability of a Navier-Stokes type model for non-stationary flow in porous media proposed by Hsu and Cheng in 1989 and to develop a Lagrange-Galerkin scheme with the Adams-Bashforth method to solve that model numerically. The stability estimate is obtained thanks to the presence of a nonlinear drag force term in the model which corresponds to the Forchheimer term. We derive the Lagrange-Galerkin scheme by extending the idea of the method of characteristics to overcome the difficulty which comes from the non-homogeneous porosity. Numerical experiments are conducted to investigate the experimental order of convergence of the scheme. For both simple and complex designs of porosities, our numerical simulations exhibit natural flow profiles which well describe the flow in non-homogeneous porous media.

Citation: Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234
##### References:
 [1] M. J. Ahammad and J. M. Alam, A numerical study of two-phase miscible flow through porous media with a Lagrangian model, The Journal of Computational Multiphase Flows, 9 (2017), 127-143.  doi: 10.1177/1757482X17701791. [2] K. Boukir, Y. Maday, B. Métivet and E. Razafindrakoto, A high-order characteristics/finite element method for the incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids, 25 (1997), 1421-1454.  doi: 10.1002/(SICI)1097-0363(19971230)25:12<1421::AID-FLD334>3.0.CO;2-A. [3] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0. [4] H. C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particle, Flow, Turbulence and Combustion, 1 (1949), 27-34.  doi: 10.1007/BF02120313. [5] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, Vol. 4. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. [6] F. Cimolin and M. Discacciati, Navier-Stokes/Forchheimer models for filtration through porous media, Applied Numerical Mathematics, 72 (2013), 205-224.  doi: 10.1016/j.apnum.2013.07.001. [7] M. Choi, G. Son and W. Shim, A level-set method for droplet impact and penetration into a porous medium, Computers & Fluids, 145 (2017), 153-166.  doi: 10.1016/j.compfluid.2016.12.014. [8] D. M. Dolberg, J. Helgesen, T. H. Hanssen, I. Magnus, G. Saigal and B. K. Pedersen, Porosity prediction from seismic inversion, Lavrans Field, Halten Terrace, Norway, The Leading Edge, 19 (2000), 392-399.  doi: 10.1190/1.1438618. [9] S. Ergun, Fluid flow through packed columns, Chemical Engineering Progress, 48 (1952), 89-94. [10] R. E. Ewing and T. F. Russell, Multistep Galerkin methods along characteristics for convection-diffusion problems, Advances in Computer Methods for Partial Differential Equations, IMACS, 4 (1981), 28-36. [11] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5. [12] A. Hazen, Some physical properties of sand and gravels with special reference to their use in filtration, 24th Annual Report, Massachusetts State Board of Health, 2 (1893), 539-556.  doi: 10.4159/harvard.9780674600485.c25. [13] F. Hecht, New development in freefem++, Journal of Numerical Mathematics, 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013. [14] C. T. Hsu and P. Cheng, Thermal dispersion in a porous medium, International Journal of Heat and Mass Transfer, 33 (1990), 1587-1597.  doi: 10.1016/0017-9310(90)90015-M. [15] M. K. Hubbert, Darcy's law and the field equations of the flow of underground fluids, Hydrological Sciences Journal, 2 (1957), 23-59. [16] M. R. Islam, M. E. Hossain, S. H. Mousavizadegan, S. Mustafiz and J. H. Abour-Kassem, Advance Petroleum Reservoir Simulation, 2nd edition, Scrivener, Canada, 2016. [17] G. A. Nasilio, O. Buzzi, S. Fityus, T. S. Yun and D. W. Smith, Upscaling of Navier-Stokes equations in porous media: Theoretical, numerical, and experimental approach, Computers and Geotechnics, 36 (2009), 1200-1206. [18] J. Nečas, Les Méthods Directes en Théories des Équations Elliptiques, Masson, Paris, 1967. [19] D. A. Nield, The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface, International Journal of Heat and Fluid Flow, 12 (1991), 269-272.  doi: 10.1016/0142-727X(91)90062-Z. [20] D. A. Nield, Modeling fluid flow and heat transfer in a saturated porous medium, J. Appl. Math. Decis. Sci., 4 (2000), 165-173.  doi: 10.1155/S1173912600000122. [21] D. A. Nield and A. Bejan, Convection in Porous Medium, 5th edition, Springer, Switzerland, 2016. [22] P. Nithiarasu, K. N. Seetharamu and T. Sundararajan, Natural convection heat transfer in a fluid saturated variable porosity medium, International Journal of Heat and Mass Transfer, 40 (1997), 3955-3967. [23] H. Notsu and M. Tabata, Error estimates of a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equation, Mathematical modeling and numerical analysis., 50 (2016), 361-380.  doi: 10.1051/m2an/2015047. [24] H. Notsu and M. Tabata, Error estimates of a stabilized Lagrange-Galerkin scheme of second-order in time for the Navier-Stokes equations, Mathematical Fluid Dynamics, Present and Future, Springer Proc. Math. Stat., Springer, 183 (2016), 497-530. [25] H. Notsu and M. Tabata, Stabilized Lagrange-Galerkin schemes of first- and second-order in time for the Navier-Stokes equations, Advances in Computational Fluid-Structure Interaction and Flow Simulation, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, (2016), 331-343. [26] W. Sobieski and A. Trykozko, Darcy's and Forchheimer's laws in practice. Part Ⅰ. The experiment, Technical Sciences, 17 (2014), 321-335. [27] Y. Su and J. H. Davidson, Modeling Approaches to Natural Convection in Porous Medium, SpringerBriefs in Applied Sciences and Technology, Springer, New York, 2015. [28] H. Teng and T. S. Zhao, An extension of Darcy's law to non-Stokes flow in porous media, Chemical Engineering Science, 55 (2000), 2727-2735.  doi: 10.1016/S0009-2509(99)00546-1. [29] L. Wang, L.-P. Wang, Z. Guo and J. Mi, Volume-average macroscopic equation for fluid flow in moving porous media, International Journal of Heat and Mass Transfer, 82 (2015), 357-368. [30] S. Whitaker, The transport equations for multi-phase systems, Chemical Engineering Science, 28 (1973), 139-147.  doi: 10.1016/0009-2509(73)85094-8.

show all references

##### References:
 [1] M. J. Ahammad and J. M. Alam, A numerical study of two-phase miscible flow through porous media with a Lagrangian model, The Journal of Computational Multiphase Flows, 9 (2017), 127-143.  doi: 10.1177/1757482X17701791. [2] K. Boukir, Y. Maday, B. Métivet and E. Razafindrakoto, A high-order characteristics/finite element method for the incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids, 25 (1997), 1421-1454.  doi: 10.1002/(SICI)1097-0363(19971230)25:12<1421::AID-FLD334>3.0.CO;2-A. [3] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0. [4] H. C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particle, Flow, Turbulence and Combustion, 1 (1949), 27-34.  doi: 10.1007/BF02120313. [5] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, Vol. 4. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. [6] F. Cimolin and M. Discacciati, Navier-Stokes/Forchheimer models for filtration through porous media, Applied Numerical Mathematics, 72 (2013), 205-224.  doi: 10.1016/j.apnum.2013.07.001. [7] M. Choi, G. Son and W. Shim, A level-set method for droplet impact and penetration into a porous medium, Computers & Fluids, 145 (2017), 153-166.  doi: 10.1016/j.compfluid.2016.12.014. [8] D. M. Dolberg, J. Helgesen, T. H. Hanssen, I. Magnus, G. Saigal and B. K. Pedersen, Porosity prediction from seismic inversion, Lavrans Field, Halten Terrace, Norway, The Leading Edge, 19 (2000), 392-399.  doi: 10.1190/1.1438618. [9] S. Ergun, Fluid flow through packed columns, Chemical Engineering Progress, 48 (1952), 89-94. [10] R. E. Ewing and T. F. Russell, Multistep Galerkin methods along characteristics for convection-diffusion problems, Advances in Computer Methods for Partial Differential Equations, IMACS, 4 (1981), 28-36. [11] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5. [12] A. Hazen, Some physical properties of sand and gravels with special reference to their use in filtration, 24th Annual Report, Massachusetts State Board of Health, 2 (1893), 539-556.  doi: 10.4159/harvard.9780674600485.c25. [13] F. Hecht, New development in freefem++, Journal of Numerical Mathematics, 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013. [14] C. T. Hsu and P. Cheng, Thermal dispersion in a porous medium, International Journal of Heat and Mass Transfer, 33 (1990), 1587-1597.  doi: 10.1016/0017-9310(90)90015-M. [15] M. K. Hubbert, Darcy's law and the field equations of the flow of underground fluids, Hydrological Sciences Journal, 2 (1957), 23-59. [16] M. R. Islam, M. E. Hossain, S. H. Mousavizadegan, S. Mustafiz and J. H. Abour-Kassem, Advance Petroleum Reservoir Simulation, 2nd edition, Scrivener, Canada, 2016. [17] G. A. Nasilio, O. Buzzi, S. Fityus, T. S. Yun and D. W. Smith, Upscaling of Navier-Stokes equations in porous media: Theoretical, numerical, and experimental approach, Computers and Geotechnics, 36 (2009), 1200-1206. [18] J. Nečas, Les Méthods Directes en Théories des Équations Elliptiques, Masson, Paris, 1967. [19] D. A. Nield, The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface, International Journal of Heat and Fluid Flow, 12 (1991), 269-272.  doi: 10.1016/0142-727X(91)90062-Z. [20] D. A. Nield, Modeling fluid flow and heat transfer in a saturated porous medium, J. Appl. Math. Decis. Sci., 4 (2000), 165-173.  doi: 10.1155/S1173912600000122. [21] D. A. Nield and A. Bejan, Convection in Porous Medium, 5th edition, Springer, Switzerland, 2016. [22] P. Nithiarasu, K. N. Seetharamu and T. Sundararajan, Natural convection heat transfer in a fluid saturated variable porosity medium, International Journal of Heat and Mass Transfer, 40 (1997), 3955-3967. [23] H. Notsu and M. Tabata, Error estimates of a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equation, Mathematical modeling and numerical analysis., 50 (2016), 361-380.  doi: 10.1051/m2an/2015047. [24] H. Notsu and M. Tabata, Error estimates of a stabilized Lagrange-Galerkin scheme of second-order in time for the Navier-Stokes equations, Mathematical Fluid Dynamics, Present and Future, Springer Proc. Math. Stat., Springer, 183 (2016), 497-530. [25] H. Notsu and M. Tabata, Stabilized Lagrange-Galerkin schemes of first- and second-order in time for the Navier-Stokes equations, Advances in Computational Fluid-Structure Interaction and Flow Simulation, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, (2016), 331-343. [26] W. Sobieski and A. Trykozko, Darcy's and Forchheimer's laws in practice. Part Ⅰ. The experiment, Technical Sciences, 17 (2014), 321-335. [27] Y. Su and J. H. Davidson, Modeling Approaches to Natural Convection in Porous Medium, SpringerBriefs in Applied Sciences and Technology, Springer, New York, 2015. [28] H. Teng and T. S. Zhao, An extension of Darcy's law to non-Stokes flow in porous media, Chemical Engineering Science, 55 (2000), 2727-2735.  doi: 10.1016/S0009-2509(99)00546-1. [29] L. Wang, L.-P. Wang, Z. Guo and J. Mi, Volume-average macroscopic equation for fluid flow in moving porous media, International Journal of Heat and Mass Transfer, 82 (2015), 357-368. [30] S. Whitaker, The transport equations for multi-phase systems, Chemical Engineering Science, 28 (1973), 139-147.  doi: 10.1016/0009-2509(73)85094-8.
Representative elementary volume (REV)
The order of convergence for scheme (22)
The boundary conditions and the finite element mesh
Time evolution of velocity magnitude
Computation domain and porosity value distribution
Time evolution of magnitude velocity
The unit of important symbols
 No Symbol Unit Name of the symbol [0.5ex] 1 $u$ ${\rm m} \cdot {\rm s}^{-1}$ Darcy velocity 2 $p$ ${\rm kg} \cdot {\rm m}^{-1}\cdot {\rm s}^{-2}$ Pressure 3 $\phi$ - porosity 4 $k_{D}$ ${\rm kg}^{-1} \cdot {\rm m}^3 \cdot {\rm s}$ Hydraulic conductivity 5 $K$ ${\rm m}^2$ Permeability 6 $\mu$ ${\rm kg} \cdot {\rm m}^{-1}\cdot {\rm s}^{-1}$ Dynamic viscosity 7 $\rho$ ${\rm kg} \cdot {\rm m}^{-3}$ Density 8 $d_{p}$ ${\rm m}$ Particle diameter 9 $F$ - Forchheimer constant 10 $B$ ${\rm kg} \cdot {\rm m}^{-2} \cdot {\rm s}^{-2}$ Drag force per unit volume
 No Symbol Unit Name of the symbol [0.5ex] 1 $u$ ${\rm m} \cdot {\rm s}^{-1}$ Darcy velocity 2 $p$ ${\rm kg} \cdot {\rm m}^{-1}\cdot {\rm s}^{-2}$ Pressure 3 $\phi$ - porosity 4 $k_{D}$ ${\rm kg}^{-1} \cdot {\rm m}^3 \cdot {\rm s}$ Hydraulic conductivity 5 $K$ ${\rm m}^2$ Permeability 6 $\mu$ ${\rm kg} \cdot {\rm m}^{-1}\cdot {\rm s}^{-1}$ Dynamic viscosity 7 $\rho$ ${\rm kg} \cdot {\rm m}^{-3}$ Density 8 $d_{p}$ ${\rm m}$ Particle diameter 9 $F$ - Forchheimer constant 10 $B$ ${\rm kg} \cdot {\rm m}^{-2} \cdot {\rm s}^{-2}$ Drag force per unit volume
Values of $Er1$ and $Er2$, their slopes, and CPU times for the problem in Subsection 6.1 by scheme (22)
 $N$ $Er1$ $Er2$ Slope of $Er1$ Slope of $Er2$ CPU time [s] 4 $3.4\times10^{-1}$ $1.6\times10^{-1}$ $-$ $-$ 1.9 8 $7.1\times10^{-2}$ $5.8\times10^{-3}$ 2.26 4.76 16.4 16 $1.4\times10^{-2}$ $1.2\times10^{-3}$ 2.34 2.30 174.8 32 $3.5\times10^{-3}$ $2.9\times10^{-4}$ 2.00 2.05 577.4 64 $1.0\times10^{-3}$ $6.3\times10^{-5}$ 1.81 2.20 5,953.9 128 $2.8\times10^{-4}$ $1.5\times10^{-5}$ 1.84 2.07 58,150.9
 $N$ $Er1$ $Er2$ Slope of $Er1$ Slope of $Er2$ CPU time [s] 4 $3.4\times10^{-1}$ $1.6\times10^{-1}$ $-$ $-$ 1.9 8 $7.1\times10^{-2}$ $5.8\times10^{-3}$ 2.26 4.76 16.4 16 $1.4\times10^{-2}$ $1.2\times10^{-3}$ 2.34 2.30 174.8 32 $3.5\times10^{-3}$ $2.9\times10^{-4}$ 2.00 2.05 577.4 64 $1.0\times10^{-3}$ $6.3\times10^{-5}$ 1.81 2.20 5,953.9 128 $2.8\times10^{-4}$ $1.5\times10^{-5}$ 1.84 2.07 58,150.9
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