Quantifying of viscous fingering instability in porous media

Document Type : Research Article

Authors

1 associate prof./ Mechanical faculty/K.N.Toosi university of technology

2 mechanical faculty/ K.N.Toosi university of Technology

Abstract

In this paper, nonlinear simulation of viscous fingering instability of miscible displacement involving nanofluid is investigated. Using vorticity and stream functions and the spectral method governing equations are obtained. Due to the fractality of fluid-fluid interface in instability phenomena, by using box counting method, its fractal dimension is calculated in different parameters such as deposition rate, mobility ratio and diffusion rates. The results show that increasing the deposition rate reduces the complexity of finger patterns and the diffusion rate of nanofluid has no effect on complexity of finger patterns while increasing the diffusion rate of displaced fluid has significant effect on patterns and makes it more complicated. The fractal analysis also shows that the effect of mobility ratio depends on the deposition rate.  By considering deposition rate, although the mobility ratio has no effect on fractal dimension and effective time is constant and equal to 275, start time of instability is delayed by 25 units. It can be concluded that fractal analysis of viscous fingering phenomena can be considered as one of the instability characteristics.

Keywords

Main Subjects

References

 
[1] G.M. Homsy, Viscous fingering in porous media, Annual review of fluid mechanics, 19(1) (1987) 271-311.
[2] B. Mandelbrot, The fractal geometry of nature WH Freeman San Francisco USA,  (1982).
[3] K. Arakawa, E. Krotkov, Fractal modeling of natural terrain: Analysis and surface reconstruction with range data, Graphical Models and Image Processing, 58(5) (1996) 413-436.
[4] D. Labat, A. Mangin, R. Ababou, Rainfall–runoff relations for karstic springs: multifractal analyses, Journal of Hydrology, 256(3-4) (2002) 176-195.
[5] A. Lagarias, Fractal analysis of the urbanization at the outskirts of the city: models, measurement and explanation, Cybergeo: European Journal of Geography,  (2007).
[6] S. Emamikoupaee, S. Zamani, M.R. Shahnazari, A. Heidarzadeh, A. Saberi, Development and Comparison of Methods Based on Grey and Fractal Models for Predicting Natural Gas Prices, Quarterely Energy Economics Review, 15(62) (2019) 1-18.
[7] M.A. Mohammed, B. Al-Khateeb, A.N. Rashid, D.A. Ibrahim, M.K.A. Ghani, S.A. Mostafa, Neural network and multi-fractal dimension features for breast cancer classification from ultrasound images, Computers & Electrical Engineering, 70 (2018) 871-882.
[8] B. Yu, Analysis of flow in fractal porous media, Applied Mechanics Reviews, 61(5) (2008) 050801.
[9] F. Zhu, S. Cui, B. Gu, Fractal analysis for effective thermal conductivity of random fibrous porous materials, Physics letters A, 374(43) (2010) 4411-4414.
[10] A. Gomez, Thermal Performance of a Double-Pipe Heat Exchanger with a Koch Snowflake Fractal Design,  (2017).
[11] S. Wang, X. Sun, C. Xu, J. Bao, C. Peng, Z. Tang, Investigation of a circulating turbulent fluidized bed with a fractal gas distributor by electrostatic-immune electrical capacitance tomography, Powder Technology,  (2019).
[12] S. Zhang, X. Chen, Z. Wu, Y. Zheng, Numerical study on stagger Koch fractal baffles micromixer, International Journal of Heat and Mass Transfer, 133 (2019) 1065-1073.
[13] B. Mandelbrot, How long is the coast of Britain? Statistical self-similarity and fractional dimension, science, 156(3775) (1967) 636-638.
[14] T. Higuchi, Approach to an irregular time series on the basis of the fractal theory, Physica D: Nonlinear Phenomena, 31(2) (1988) 277-283.
[15] M.A. Ibrahim, Multifractal techniques for analysis and classification of emphysema images,  (2017).
[16] S. Hill, Channeling in packed columns, Chemical Engineering Science, 1(6) (1952) 247-253.
[17] D. Peaceman, H. Rachford Jr, Numerical calculation of multidimensional miscible displacement, Society of Petroleum Engineers Journal, 2(04) (1962) 327-339.
[18] M. Christie, D. Bond, Multidimensional flux-corrected transport for reservoir simulation, in:  SPE Reservoir Simulation Symposium, Society of Petroleum Engineers, 1985.
[19] T.F. Russell, M.F. Wheeler, C. Chiang, Large-scale simulation of miscible displacement by mixed and characteristic finite element methods, Mathematical and computational methods in seismic exploration and reservoir modeling,  (1986) 85-107.
[20] C. Tan, G. Homsy, Simulation of nonlinear viscous fingering in miscible displacement, The Physics of fluids, 31(6) (1988) 1330-1338.
[21] C.T. Tan, G. Homsy, Viscous fingering with permeability heterogeneity, Physics of Fluids A: Fluid Dynamics, 4(6) (1992) 1099-1101.
[22] A. De Wit, G. Homsy, Viscous fingering in periodically heterogeneous porous media. II. Numerical simulations, The Journal of chemical physics, 107(22) (1997) 9619-9628.
[23] S. Pramanik, M. Mishra, Nonlinear simulations of miscible viscous fingering with gradient stresses in porous media, Chemical Engineering Science, 122 (2015) 523-532.
[24] S. Sin, T. Suekane, Y. Nagatsu, A. Patmonoaji, Three-dimensional visualization of viscous fingering for non-Newtonian fluids with chemical reactions that change viscosity, Physical Review Fluids, 4(5) (2019) 054502.
[25] D. Moissis, M. Wheeler, C. Miller, Simulation of miscible viscous fingering using a modified method of characteristics: effects of gravity and heterogeneity, SPE Advanced Technology Series, 1(01) (1993) 62-70.
[26] M. Islam, J. Azaiez, Fully implicit finite difference pseudo‐spectral method for simulating high mobility‐ratio miscible displacements, International Journal for Numerical Methods in Fluids, 47(2) (2005) 161-183.
[27] A.I. Maleka, A. Saberi, M. R. Shahnazari, Simulation and nonlinear instability investigation of two miscible fluid flow in homogeneous porous media, Petroleum Research, 27(95) (2017) 147-162. (in Persian)
[28] M. Shahnazari, I. Maleka Ashtiani, A. Saberi, Linear stability analysis and nonlinear simulation of the channeling effect on viscous fingering instability in miscible displacement, Physics of Fluids, 30(3) (2018) 034106.
[29] K. Ghesmat, J. Azaiez, Miscible displacements of reactive and anisotropic dispersive flows in porous media, Transport in porous media, 77(3) (2009) 489.
[30] M.R. Shahnazari, A. Saberi, Fractal Analysis of Viscous Fingering Instability of Two Reactive Miscible Fluids through Homogeneous Porous Media, Journal of Solid and Fluid Mechanics, 9(1) (2019) 265-278. (in Persian)
[31] K. Ghesmat, H. Hassanzadeh, J. Abedi, Z. Chen, Influence of nanoparticles on the dynamics of miscible Hele-Shaw flows, Journal of Applied Physics, 109(10) (2011) 104907.
[32] B. Dastvareh, J. Azaiez, Instabilities of nanofluid flow displacements in porous media, Physics of Fluids, 29(4) (2017) 044101.
[33] K. Zakade, R. Gh, A. Khan, Y.H. Shaikh, Temporal Evolution of Viscous Fingering in Hele Shaw Cell: A Fractal Approach, Int J Sci Res, 6(10) (2015).
[34] S. Tang, Z. Wei, Fractal characteristics of viscous fingering, Fractals, 5(02) (1997) 221-227.
[35] K. Falconer, Fractal geometry: mathematical foundations and applications, John Wiley & Sons, 2004.
[36] B. Dastvareh, Instabilities of Nanofluid Flow Displacements in Porous Media, University of Calgary, 2019.
Amirkabir Journal of Mechanical Engineering
Volume 53, Issue 5
August 2021
Pages 2887-2902

Files

Share

How to cite

Statistics