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Non-Darcy effects in fracture flows of a yield stress fluid

Published online by Cambridge University Press:  16 September 2016

A. Roustaei ,
T. Chevalier ,
L. Talon  and
I. A. Frigaard
A. Roustaei
Affiliation:
Department of Mechanical Engineering, University of British Columbia, 6250 Applied Science Lane, Vancouver, BC V6T 1Z4, Canada
T. Chevalier
Affiliation:
Laboratoire FAST, Univ. Paris-Sud, CNRS, Université Paris-Saclay, F-91405, Orsay, France
L. Talon*
Affiliation:
Laboratoire FAST, Univ. Paris-Sud, CNRS, Université Paris-Saclay, F-91405, Orsay, France
I. A. Frigaard
Affiliation:
Department of Mechanical Engineering, University of British Columbia, 6250 Applied Science Lane, Vancouver, BC V6T 1Z4, Canada Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada
*
†Email address for correspondence: talon@fast.u-psud.fr
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Abstract

We study non-inertial flows of single-phase yield stress fluids along uneven/rough-walled channels, e.g. approximating a fracture, with two main objectives. First, we re-examine the usual approaches to providing a (nonlinear) Darcy-type flow law and show that significant errors arise due to self-selection of the flowing region/fouling of the walls. This is a new type of non-Darcy effect not previously explored in depth. Second, we study the details of flow as the limiting pressure gradient is approached, deriving approximate expressions for the limiting pressure gradient valid over a range of different geometries. Our approach is computational, solving the two-dimensional Stokes problem along the fracture, then upscaling. The computations also reveal interesting features of the flow for more complex fracture geometries, providing hints about how to extend Darcy-type approaches effectively.

JFM classification

Low-Reynolds-number flows: Lubrication theory Non-Newtonian Flows: Plastic materials Low-Reynolds-number flows: Porous media

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 805 , 25 October 2016 , pp. 222 - 261
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© 2016 Cambridge University Press 

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References

Al-Fariss, T. & Pinder, K. L. 1987 Flow through porous media of a shear-thinning liquid with yield stress. Can. J. Chem. Engng 65 (3), 391–405.CrossRefGoogle Scholar
Ambari, A., Benhamou, M., Roux, S. & Guyon, E. 1990 Distribution des tailles de pores d’un milieu poreux déterminée par l’écoulement d’un fluide á seuil. C. R. Acad. Sci. 311 (11), 1291–1295.Google Scholar
Balhoff, M., Sanchez-Rivera, D., Kwok, A., Mehmani, Y. & Prodanović, M. 2012 Numerical algorithms for network modeling of yield stress and other non-Newtonian fluids in porous media. Trans. Porous Med. 93 (3), 363–379.CrossRefGoogle Scholar
Balhoff, M. T. & Thompson, K. E. 2004 Modeling the steady flow of yield-stress fluids in packed beds. AIChE J. 50 (12), 3034–3048.CrossRefGoogle Scholar
Balmforth, N. J. & Craster, R. V. 1999 A consistent thin-layer theory for Bingham plastics. J. Non-Newtonian Fluid Mech. 84 (1), 65–81.CrossRefGoogle Scholar
Balmforth, N. J, Frigaard, I. A. & Ovarlez, G. 2014 Yielding to stress: recent developments in viscoplastic fluid mechanics. Annu. Rev. Fluid Mech. 46, 121–146.Google Scholar
Barenblatt, G. I., Entov, V. M. & Ryzhik, V. M. 1989 Theory of Fluid Flows Through Natural Rocks, Kluwer Academic.Google Scholar
Bittleston, S. H., Ferguson, J. & Frigaard, I. A. 2002 Mud removal and cement placement during primary cementing of an oil well – laminar non-Newtonian displacements in an eccentric annular Hele-Shaw cell. J. Engng Maths 43 (2–4), 229–253.Google Scholar
Bleyer, J. & Coussot, P. 2014 Breakage of non-Newtonian character in flow through a porous medium: evidence from numerical simulation. Phys. Rev. E 89, 063018.Google Scholar
Borouchaki, H., George, P. L., Hecht, F., Laug, P. & Saltel, E. 1997 Delaunay mesh generation governed by metric specifications. Part I. Algorithms. Finite Elem. Anal. Des. 25 (1–2), 61–83.Google Scholar
Bouchaud, E. 1997 Scaling properties of cracks. J. Phys.: Condens. Matter 9 (21), 4319–4344.Google Scholar
de Castro, A. R., Omari, A., Ahmadi-Sénichault, A. & Bruneau, D. 2014 Toward a new method of porosimetry: principles and experiments. Trans. Porous Med. 101 (3), 349–364.Google Scholar
Chase, G. G. & Dachavijit, P. 2003 Incompressible cake filtration of a yield stress fluid. Sep. Sci. Technol. 38 (4), 745–766.CrossRefGoogle Scholar
Chase, G. G. & Dachavijit, P. 2005 A correlation for yield stress fluid flow through packed beds. Rheol. Acta 44 (5), 495–501.CrossRefGoogle Scholar
Chen, M., Rossen, W. & Yortsos, Y. C. 2005 The flow and displacement in porous media of fluids with yield stress. Chem. Engng Sci. 60 (15), 4183–4202.Google Scholar
Chevalier, T., Chevalier, C., Clain, X., Dupla, J. C., Canou, J., Rodts, S. & Coussot, P. 2013a Darcy’s law for yield stress fluid flowing through a porous medium. J. Non-Newtonian Fluid Mech. 195, 57–66.Google Scholar
Chevalier, T., Rodts, S., Chateau, X., Boujlel, J., Maillard, M. & Coussot, P. 2013b Boundary layer (shear-band) in frustrated viscoplastic flows. Europhys. Lett. 102 (4), 48002.Google Scholar
Chevalier, T. & Talon, L. 2015 Generalization of Darcy’s law for Bingham fluids in porous media: from flow-field statistics to the flow-rate regimes. Phys. Rev. E 91, 023011.Google Scholar
Christianovich, S. A. 1940 Motion of groundwater which does not conform to Darcy’s law. Prikl. Mat. Mekh. 4, 33–52.Google Scholar
Clain, X.2010 Étude expérimentale de l’injection de fluides d’Herschel–Bulkley en milieux poreux. PhD thesis, Université Paris-Est.Google Scholar
Coussot, P. 2005 Rheometry of Pastes, Suspensions, and Granular Materials: Applications in Industry and Environment. Wiley.Google Scholar
Coussot, P. 2014 Yield stress fluid flows: a review of experimental data. J. Non-Newtonian Fluid Mech. 211, 31–49.Google Scholar
Drazer, G. & Koplik, J. 2000 Permeability of self-affine rough fractures. Phys. Rev. E 62, 8076–8085.Google Scholar
Duvaut, G. & Lions, J. L. 1976 Inequalities in Mechanics and Physics. Springer.Google Scholar
El Tani, M. 2012 Grouting rock fractures with cement grout. Rock Mech. Rock Engng 45, 547–561.CrossRefGoogle Scholar
Entov, V. M. 1967 On some two-dimensional problems of the theory of filtration with a limiting gradient. Prikl. Mat. Mekh. 31, 820–833.Google Scholar
Fortin, M. & Glowinski, R. 1983 The Augmented Lagrangian Method. North-Holland.Google Scholar
Frigaard, I. A. & Nouar, C. 2005 On the usage of viscosity regularisation methods for visco-plastic fluid flow computation. J. Non-Newtonian Fluid Mech. 127 (1), 1–26.Google Scholar
Frigaard, I. A. & Ryan, D. P. 2004 Flow of a visco-plastic fluid in a channel of slowly varying width. J. Non-Newtonian Fluid Mech. 123 (1), 67–83.CrossRefGoogle Scholar
Ge, S. 1997 A governing equation for fluid flow in rough fractures. Water Resour. Res. 33 (1), 53–61.CrossRefGoogle Scholar
Glowinski, R. & Le Tallec, P. 1989 Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. vol. 9. SIAM.Google Scholar
Hecht, F. 2012 New development in FreeFem++. J. Numer. Math. 20 (3–4), 251–265.Google Scholar
Malevich, A. E., Mityushev, V. V. & Adler, P. M. 2006 Stokes flow through a channel with wavy walls. Acta Mech. 182 (3–4), 151–182.Google Scholar
Mandelbrot, B. B., Passoja, D. E. & Paullay, A. J. 1984 Fractal character of fracture surfaces of metals. Nature 308 (5961), 721–722.Google Scholar
Mirzadzhanzade, A. K. 1959 Problems of Hydrodynamics of Viscous and Viscoplastic Fluids in Oil Drilling. Aznefteizdat, Baku (In Russian).Google Scholar
Mosolov, P. P. & Miasnikov, V. P. 1966 On stagnant flow regions of a viscous-plastic medium in pipes. J. Appl. Math. Mech. 30 (4), 841–854.Google Scholar
Mosolov, P. P. & Miasnikov, V. P. 1967 On qualitative singularities of the flow of a viscoplastic medium in pipes. Prikl. Mat. Mech. 31 (3), 581–585. J. Appl. Math. Mech. 31 (3) 609–613.Google Scholar
Nelson, E. B. & Guillot, D. 2006 Well Cementing. Schlumberger.Google Scholar
Park, H. C., Hawley, M. C. & Blanks, R. F. 1973 The flow of non-Newtonian solutions through packed beds. Polymer Engng Sci. 15, 761–773.CrossRefGoogle Scholar
Pelipenko, S. & Frigaard, I. A. 2004 Visco-plastic fluid displacements in near-vertical narrow eccentric annuli: prediction of travelling-wave solutions and interfacial instability. J. Fluid Mech. 520, 343–377.CrossRefGoogle Scholar
Putz, A. & Frigaard, I. A. 2010 Creeping flow around particles in a Bingham fluid. J. Non-Newtonian Fluid Mech. 165 (5), 263–280.Google Scholar
Putz, A., Frigaard, I. A. & Martinez, D. M. 2009 On the lubrication paradox and the use of regularisation methods for lubrication flows. J. Non-Newtonian Fluid Mech. 163 (1–3), 62–77.CrossRefGoogle Scholar
de los Reyes, J. C. & González Andrade, S. 2012 A combined BDF–semismooth Newton approach for time-dependent Bingham flow. Numer. Meth. Part. Diff. Equ. 28 (3), 834–860.CrossRefGoogle Scholar
Roustaei, A. & Frigaard, I. A. 2013 The occurrence of fouling layers in the flow of a yield stress fluid along a wavy-walled channel. J. Non-Newtonian Fluid Mech. 198, 109–124.Google Scholar
Roustaei, A., Gosselin, A. & Frigaard, I. A. 2015 Residual drilling mud during conditioning of uneven boreholes in primary cementing. Part 1: rheology and geometry effects in non-inertial flows. J. Non-Newtonian Fluid Mech. 220, 87–98.Google Scholar
Roux, S. & Herrmann, H. J. 1987 Disorder-induced nonlinear conductivity. Europhys. Lett. 4 (11), 1227.Google Scholar
Sahimi, M. 1995 Flow and Transport in Porous Media and Fractured Rock. VCH Verlagsgesellschaft.Google Scholar
Saramito, P. 2016 A damped Newton algorithm for computing viscoplastic fluid flows. J. Non-Newtonian Fluid Mech. Available online 20 May 2016, doi:10.1016/j.jnnfm.2016.05.007.CrossRefGoogle Scholar
Saramito, P. & Roquet, N. 2001 An adaptive finite element method for viscoplastic fluid flows in pipes. Comput. Meth. Appl. Mech. Engng 190 (40), 5391–5412.Google Scholar
Savins, J. G. 1969 Non-Newtonian flow through porous media. Ind. Engng Chem. 61 (10), 18–47.Google Scholar
Scheidegger, A. E. 1957 The Physics of Flow Through Porous Media. University of Toronto Press.Google Scholar
Schmittbuhl, J., Gentier, S. & Roux, S. 1993 Field measurements of the roughness of fault surfaces. Geophys. Res. Lett. 20 (8), 639–641.Google Scholar
Sochi, T. & Blunt, M. J. 2008 Pore-scale network modeling of Ellis and Herschel–Bulkley fluids. J. Pet. Sci. Engng 60 (2), 105–124.Google Scholar
de Souza Mendes, P. R., Naccache, M. F., Varges, P. R. & Marchesini, F. H. 2007 Flow of viscoplastic liquids through axisymmetric expansions–contractions. J. Non-Newtonian Fluid Mech. 142 (1–3), 207–217.Google Scholar
Sultanov, B. I. 1960 Filtration of visco-plastic fluids in a porous medium. Izv. Akad. Nauk. SSSR Ser. Fiz. 5, 820–833.Google Scholar
Talon, L., Auradou, H. & Hansen, A. 2010 Permeability of self-affine aperture fields. Phys. Rev. E 82, 046108.Google Scholar
Talon, L., Auradou, H. & Hansen, A. 2014 Effective rheology of Bingham fluids in a rough channel. Front. Phys. 2, 24.Google Scholar
Talon, L., Auradou, H., Pessel, M. & Hansen, A. 2013 Geometry of optimal path hierarchies. Eur. Phys. Lett. 103 (3), 30003.Google Scholar
Talon, L. & Bauer, D. 2013 On the determination of a generalized Darcy equation for yield-stress fluid in porous media using a lattice-Boltzmann TRT scheme. Eur. Phys. J. E 36 (12), 1–10.Google ScholarPubMed
Treskatis, T., Moyers-Gonzalez, M. A. & Price, C. J.2015 A trust-region SQP method for the numerical approximation of viscoplastic fluid flow. Preprint, arXiv:1504.08057.Google Scholar
Walton, I. C. & Bittleston, S. H. 1991 The axial flow of a Bingham plastic in a narrow eccentric annulus. J. Fluid Mech. 222, 39–60.Google Scholar
Wang, Y. 1997 Finite element analysis of the duct flow of Bingham plastic fluids: an application of the variational inequality. Intl J. Numer. Meth. Fluids 25 (9), 1025–1042.Google Scholar
Zimmerman, R. W., Kumar, S. & Bodvarsson, G. S. 1991 Lubrication theory analysis of the permeability of rough-walled fractures. Intl J. Rock Mech. Min. 28 (4), 325–331.Google Scholar