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Dispersion of solids in fracturing flows of yield stress fluids

Published online by Cambridge University Press:  29 September 2017

S. Hormozi Open the ORCID record for S. Hormozi [Opens in a new window]  and
I. A. Frigaard Open the ORCID record for I. A. Frigaard [Opens in a new window]
S. Hormozi*
Affiliation:
Department of Mechanical Engineering, Ohio University, 251 Stocker Center, Athens, OH 45701, USA
I. A. Frigaard
Affiliation:
Departments of Mathematics and Mechanical Engineering, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2
*
†Email address for correspondence: hormozi@ohio.edu
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Abstract

Solids dispersion is an important part of hydraulic fracturing, both in helping to understand phenomena such as tip screen-out and spreading of the pad, and in new process variations such as cyclic pumping of proppant. Whereas many frac fluids have low viscosity, e.g. slickwater, others transport proppant through increased viscosity. In this context, one method for influencing both dispersion and solids-carrying capacity is to use a yield stress fluid as the frac fluid. We propose a model framework for this scenario and analyse one of the simplifications. A key effect of including a yield stress is to focus high shear rates near the fracture walls. In typical fracturing flows this results in a large variation in shear rates across the fracture. In using shear-thinning viscous frac fluids, flows may vary significantly on the particle scale, from Stokesian behaviour to inertial behaviour across the width of the fracture. Equally, according to the flow rates, Hele-Shaw style models give way at higher Reynolds number to those in which inertia must be considered. We develop a model framework able to include this range of flows, while still representing a significant simplification over fully three-dimensional computations. In relatively straight fractures and for fluids of moderate rheology, this simplifies into a one-dimensional model that predicts the solids concentration along a streamline within the fracture. We use this model to make estimates of the streamwise dispersion in various relevant scenarios. This model framework also predicts the transverse distributions of the solid volume fraction and velocity profiles as well as their evolutions along the flow part.

JFM classification

Low-Reynolds-number flows: Hele-Shaw flows Non-Newtonian Flows: Plastic materials Complex Fluids: Suspensions

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Journal of Fluid Mechanics , Volume 830 , 10 November 2017 , pp. 93 - 137
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© 2017 Cambridge University Press 

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References

Ahnert, T., Münch, A. & Wagner, B.2014 Models for the two-phase flow of concentrated suspensions. Weierstrass Institute for Applied Analysis and Stochastics, Preprint 2047 http://www.wias-berlin.de/preprint/2047/wias_preprints_2047.pdf.Google Scholar
Andreotti, B., Forterre, Y. & Pouliquen, O. 2013 Granular Media: Between Fluid and Solid. Cambridge University Press.CrossRefGoogle Scholar
Asmolov, E. S. 1999 The inertial lift on a spherical particle in a plane Poiseuille flow at large channel Reynolds number. J. Fluid Mech. 381, 63–87.CrossRefGoogle Scholar
Asmolov, E. S., Lebedeva, N. A. & Osiptsov, A. A. 2009 Inertial migration of sedimenting particles in a suspension flow through a Hele-Shaw cell. Fluid Dyn. 44, 405–418.CrossRefGoogle Scholar
Asmolov, E. S. & Osiptsov, A. A. 2009 The inertial lift on a spherical particle settling in a horizontal viscous flow through a vertical slot. Phys. Fluids 21, 063301.CrossRefGoogle Scholar
Auradou, H., Boschan, A., Chertcoff, R., Gabbanelli, S., Hulin, J.-P. & Ippolito, I. 2008 Enhancement of velocity contrasts by shear thinning solutions flowing in a rough fracture. J. Non-Newtonian Fluid Mech. 153, 53–61.CrossRefGoogle Scholar
Auradou, H., Drazer, G., Boschan, A., Hulin, J.-P. & Koplik, J. 2006 Flow channeling in a single fracture induced by shear displacement. Geotherm. 35, 576–588.CrossRefGoogle Scholar
Auradou, H., Drazer, G., Hulin, J.-P. & Koplik, J. 2005 Permeability anisotropy induced by the shear displacement of rough fracture walls. Water Resour. Res. 41, W09423.CrossRefGoogle Scholar
Balhoff, M. T. & Thompson, K. E. 2004 Modeling the steady flow of yield-stress fluids in packed beds. AIChE J. 50, 3034–3048.CrossRefGoogle Scholar
Barbati, A. C., Desroches, J., Robisson, A. & McKinley, G. H. 2016 Complex fluids and hydraulic fracturing. Annu. Rev. Chem. Biomol. Engng 7, 415–453.CrossRefGoogle ScholarPubMed
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Boronin, S. A. & Osiptsov, A. A. 2014 Effects of particle migration on suspension flow in a hydraulic fracture. Fluid Dyn. 49, 208.CrossRefGoogle Scholar
Boronin, S. A., Osiptsov, A. A. & Desroches, J. 2015 Displacement of yield-stress fluids in a fracture. Intl J. Multiphase Flow 76, 47–63.CrossRefGoogle Scholar
Boschan, A., Auradou, H., Ippolito, I., Chertcoff, R. & Hulin, J.-P. 2007 Miscible displacement fronts of shear thinning fluids inside rough fractures. Water Resour. Res. 43, W03438.CrossRefGoogle Scholar
Boschan, A., Auradou, H., Ippolito, I., Chertcoff, R. & Hulin, J.-P. 2009 Experimental evidence of the anisotropy of tracer dispersion in rough fractures with sheared walls. Water Resour. Res. 45, W03201.CrossRefGoogle Scholar
Boschan, A., Ippolito, I., Chertcoff, R., Auradou, H., Talon, L. & Hulin, J.-P. 2008 Geometrical and Taylor dispersion in a fracture with random obstacles: an experimental study with fluids of different rheologies. Water Resour. Res. 44, W06420.CrossRefGoogle Scholar
Boyer, F., Guazzelli, E. & Pouliquen, O. 2011 Unifying suspension and granular rheology. Phys. Rev. Lett. 107, 188301.CrossRefGoogle ScholarPubMed
Brady, J. F.2015 Reviving the suspension balance model. In Symposium on Multi-Phase Continuum Modelling of Particulate Flows, University of Florida, Gainesville, Florida.Google Scholar
Brady, J. F. & Morris, J. F. 1997 Microstructure of strongly sheared suspensions and its impact on rheology and diffusion. J. Fluid Mech. 348, 103–139.CrossRefGoogle Scholar
Brown, S. R. 1987 Fluid flow through rock joints: the effect of surface roughness. J. Geophys. Res. 92, 1337–1347.CrossRefGoogle Scholar
Buka, A., Kertesz, J. & Vicsek, T. 1986 Transitions of viscous fingering patterns in nematic liquid crytals. Nature 324, 424–425.CrossRefGoogle Scholar
Buka, A. & Palffy-Muhoray, P. 1987 Transitions of viscous fingering patterns in nematic liquid crytals. Phys. Rev. A 36, 1527–1529.CrossRefGoogle Scholar
Cassar, C., Nicolas, M. & Pouliquen, O. 2005 Submarine granular flows down inclined planes. Phys. Fluids 17, 103301.CrossRefGoogle Scholar
Chapman, B.1990 Shear-induced migration phenomena in concentrated suspensions. PhD thesis, University of Notre Dame.Google Scholar
Chateau, X., Ovarlez, G. & Luu Trung, K. 2008 Homogenization approach to the behavior of suspensions of noncolloidal particles in yield stress fluids. J. Rheol. 52, 489–506.CrossRefGoogle Scholar
Coussot, P., Tocquer, L., Lanos, C. & Ovarlez, G. 2009 Macroscopic versus local rheology of yield stress fluids. J. Non-Newtonian Fluid Mech. 158, 85–90.CrossRefGoogle Scholar
Dagois-Bohy, S., Hormozi, S., Guazzelli, E. & Pouliquen, O. 2015 Rheology of dense suspensions of non-colloidal spheres in yield-stress fluids. J. Fluid Mech. 776, R2.CrossRefGoogle Scholar
Dbouk, T., Lobry, L. & Lemaire, E. 2013 Normal stresses in concentrated non-Brownian suspensions. J. Fluid Mech. 715, 239–272.CrossRefGoogle Scholar
Dontsov, E. V. & Peirce, A. P. 2014 Slurry flow, gravitational settling and a proppant transport model for hydraulic fractures. J. Fluid Mech. 760, 567–590.CrossRefGoogle Scholar
Dougherty, T. J.1959 Some problems in the theory of colloids. PhD thesis, Case Institute of Technology.Google Scholar
Drazer, G. & Koplik, J. 2000 Permeability of self-affine rough fractures. Phys. Rev. E 62, 8076–8085.CrossRefGoogle ScholarPubMed
Drazer, G. & Koplik, J. 2002 Transport in rough self-affine fractures. Phys. Rev. E 66 (2), 026303.CrossRefGoogle ScholarPubMed
Drew, D. A. 1983 Mathematical modeling of two-phase flow. Annu. Rev. Fluid Mech. 15, 261–291.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
Eskin, D. & Miller, M. J. 2008 A model of non-Newtonian slurry flow in a fracture. Powder Technol. 182, 313–322.CrossRefGoogle Scholar
Fang, Z., Mammoli, A., Brady, J. F., Ingber, MZ. S., Mondy, L. A. & Graham, A. L. 2002 Flow-aligned tensor models for suspension flows. Intl J. Multiphase Flow 28, 137–166.CrossRefGoogle Scholar
Firouznia, M., Metzger, B., Ovarlez, G. & Hormozi, S.2016 The hydrodynamic interaction of two small freely-moving particles in a Couette flow of a yield stress fluid. 69th Annual Meeting of the APS Division of Fluid Dynamics, Vol. 61, No. 20.Google Scholar
Frigaard, I. A. & Ryan, D. 2004 Flow of a visco-plastic fluid in a channel of slowly varying width. J. Non-Newtonian Fluid Mech. 123, 67–83.CrossRefGoogle Scholar
Frigaard, I. A. & Ryan, D.2016 The hydrodynamic interaction of two small freely-moving particles in a Couette flow of a yield stress fluid. 69th Annual Meeting of the APS Division of Fluid Dynamics, Vol. 61, No. 20.Google Scholar
Gadala-Maria, F. & Acrivos, A. 1980 Shear-induced structure in a concentrated suspension of solid spheres. J. Rheol. 24, 799–814.CrossRefGoogle Scholar
Gholami, M., Lenoir, N., Hautemayou, D., Ovarlez, G. & Hormozi, S. 2017 Time-resolved 2D concentration maps in flowing suspensions using X-ray. J. Rheol. (submitted).Google Scholar
Gidaspow, D. 1994 Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. Academic.Google Scholar
Gillard, M., Medvedev, O., Pena, A., Medvedev, A., Penacorada, F. & d’Huteau, E. 2010 A new approach to generating fracture conductivity. In SPE Annual Technical Conference and Exhibition held in Florence, Italy, September, SPE Paper 135034.Google Scholar
Guazzelli, E. & Morris, J. F. 2012 A Physical Introduction to Suspension Dynamics, Cambridge Texts in Applied Mathematics. Cambridge University Press.Google Scholar
Hammond, P. S. 1995 Settling and slumping in a Newtonian slurry, and implications for proppant placement during hydraulic fracturing of gas wells. Chem. Engng Sci. 50, 3247–3260.CrossRefGoogle Scholar
Hampton, R. E., Mammoli, A. A., Graham, A. L., Tetlow, N. & Altobelli, S. A. 1997 Migration of particles undergoing pressure-driven flow in a circular conduit. J. Rheol. 41, 621–640.CrossRefGoogle Scholar
Hasegawa, E. & Izuchi, H. 1983 On steady flow through a channel consisting of an uneven wall and a plane wall. Part 1. Case of no relative motion in two walls. Bull. Japan Soc. Mech. Engng 26, 514–520.CrossRefGoogle Scholar
Hogg, A. J. 1994 The inertial migration of non-neutrally buoyant spherical particles in two-dimensional shear flows. J. Fluid Mech. 272, 285–318.CrossRefGoogle Scholar
Jenkins, J. T. & McTigue, D. F. 1990 Transport processes in concentrated suspensions: the role of particle fluctuations. In Two Phase Flows and Waves (ed. Joseph, D. D. & Schaeffer, D. G.), The IMA Volumes in Mathematics and Its Applications, vol. 26, pp. 70–79.CrossRefGoogle Scholar
Krieger, I. M. 1972 Rheology of monodisperse lattices. Adv. Colloid Interface Sci. 3, 111–136.CrossRefGoogle Scholar
Lakhtychkin, A., Vinogradov, O. & Eskin, D. 2011 Modeling of placement of immiscible fluids of different rheology into a hydraulic fracture. Ind. Engng Chem. Res. 50, 5774–5782.CrossRefGoogle Scholar
Lashgari, I., Picanoa, F., Breugemc, W. P. & Brandt, L. 2014 Laminar, turbulent and inertial shear-thickening regimes in channel flow of neutrally buoyant particle suspensions. Phys. Rev. Lett. 113, 254502.CrossRefGoogle ScholarPubMed
Lashgari, I., Picanoa, F., Breugemc, W. P. & Brandt, L. 2016 Channel flow of rigid sphere suspensions: particle dynamics in the inertial regime. Intl J. Multiphase Flow 78, 12–24.CrossRefGoogle Scholar
Lebedeva, N. A. & Asmolov, E. S. 2011 Migration of settling particles in a horizontal viscous flow through a vertical slot with porous walls. Intl J. Multiphase Flow 37, 453–461.CrossRefGoogle Scholar
Lecampion, B. & Garagash, D. I. 2014 Confined flow of suspensions modelled by a frictional rheology. J. Fluid Mech. 759, 197–235.CrossRefGoogle Scholar
Leighton, D. & Acrivos, A. 1987 The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181, 415–439.CrossRefGoogle Scholar
Lemaire, E., Levitz, P., Daccord, G. & Van Damme, H. 1991 From viscous fingering to viscoelastic fracturing in colloidal fluids. Phys. Rev. Lett. 67, 2009–2012.CrossRefGoogle ScholarPubMed
Leveque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press.CrossRefGoogle Scholar
Lhuillier, D. 2009 Migration of rigid particles in non-Brownian viscous suspensions. Phys. Fluids 21, 023302.CrossRefGoogle Scholar
Lindner, A., Bonn, D., Poire, E. C., Amar, M. B. & Meunier, J. 2002 Viscous fingering patterns of silica suspensions in polymer solutions: effects of viscoelasticity and gravity. J. Fluid Mech. 469, 237–256.CrossRefGoogle Scholar
Lindner, A., Coussot, P. & Bonn, D. 2000 Viscous fingering in a yield stress fluid. Phys. Rev. Lett. 85, 314–317.CrossRefGoogle Scholar
Liu, Y., Gadde, P. B. & Sharma, M. M. 2007 Proppant placement using reverse-hybrid fracs. SPE Prod. Oper. 22 (3), 348–356.Google Scholar
Lyon, M. K. & Leal., L. G. 1998 An experimental study of the motion of concentrated suspensions in two-dimensional channel flow, Part 1. Monodisperse systems. J. Fluid Mech. 363, 25–56.CrossRefGoogle Scholar
Mahaut, F., Chateau, X., Coussot, P. & Ovarlez, G. 2008 Yield stress and elastic modulus of suspensions of noncolloidal particles in yield stress fluids. J. Rheol. 52, 287–313.CrossRefGoogle Scholar
Makino, K., Kawaguchi, M., Aoyama, K. & Kato, T. 2002 Transition of viscous fingering patterns in polymer solutions. Phys. Fluids 7, 455–457.CrossRefGoogle Scholar
Maron, S. H. & Pierce, P. E. 1956 Application of Ree–Eyring generalized flow theory to suspensions of spherical particles. J. Colloid Sci. 11, 80–95.CrossRefGoogle Scholar
Maxey, M. R. & Riley, J. J. Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26 (1983), 883. doi:10.1063/1.864230.CrossRefGoogle Scholar
Merhi, D., Lemaire, E., Bossis, G. & Moukalled, F. 2005 Particle migration in a concentrated suspension flowing between rotating parallel plates: investigation of diffusion flux coeffcients. J. Rheol. 49, 1429–1448.CrossRefGoogle Scholar
Mobbs, A. T. & Hammond, P. S. 2001 Computer simulations of proppant transport in a hydraulic fracture. In SPE Production Facilities, SPE Paper 69212, pp. 112–121.Google Scholar
Morris, J. F. 2009 A review of microstructure in concentrated suspensions and its implications for rheology and bulk flow. Rheol. Acta 48, 909–923.CrossRefGoogle Scholar
Morris, J. F. & Boulay, F. 1999 Curvilinear flows of noncolloidal suspensions: the role of normal stresses. J. Rheol. 43, 1213–1237.CrossRefGoogle Scholar
Nessyahu, H. & Tadmor, E. 1990 Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408.CrossRefGoogle Scholar
Nott, P. R. & Brady, J. F. 1994 Pressure-driven flow of suspensions: simulation and theory. J. Fluid Mech. 275, 157–199.CrossRefGoogle Scholar
Nott, P. R., Guazzelli, E. & Pouliquen, O. 2011 The suspension balance model revisited. Phys. Fluids 23, 043304.CrossRefGoogle Scholar
Oh, S., Song, Y., Garagash, D. I., Lecampion, B. & Desroches, J. 2015 Pressure-driven suspension flow near jamming. Phys. Rev. Lett. 114, 088301.CrossRefGoogle ScholarPubMed
Ovarlez, G., Bertrand, F., Coussot, P. & Chateau, X. 2012 Shear-induced sedimentation in yield stress fluids. J. Non-Newtonian Fluid Mech. 177–178, 19–28.CrossRefGoogle Scholar
Ovarlez, G., Bertrand, F. & Rodts, S. 2006 Local determination of the constitutive law of a dense suspension of noncolloidal particles through MRI. J. Rheol. 50, 259–292.CrossRefGoogle Scholar
Ovarlez, G., Mahaut, F., Deboeuf, S., Lenoir, N., Hormozi, S. & Chateau., X. 2015 Flows of suspensions of particles in yield stress fluids. J. Rheol. 59 (6), 1449–1486.CrossRefGoogle Scholar
Patir, N. & Cheng, H. S. 1978 An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication. Trans. ASME J. Lubr. Technol. 100, 12–17.CrossRefGoogle Scholar
Pearson, J. R. A. 1994 On suspension transport in a fracture: framework for a global model. J. Non-Newtonian Fluid Mech. 54, 503–513.CrossRefGoogle Scholar
Phillips, R., Armstrong, R., Brown, R. A., Graham, A. & Abbott, J. R. 1992 A constitutive model for concentrated suspensions that accounts for shear-induced particle migration. Phys. Fluids A 4, 30–40.CrossRefGoogle Scholar
Putz, A., Frigaard, I. A. & Martinez, M. D. 2009 On the lubrication paradox and the use of regularisation methods for lubrication flows. J. Non-Newtonian Fluid Mech. 163, 62–77.CrossRefGoogle Scholar
Quemada, D. 1982 Stability of Thermodynamic Systems (ed. Casas-Vazquez, J. & Lebon, J.), Lecture Notes in Physics, vol. 164, pp. 210–247. Springer.CrossRefGoogle Scholar
Ramachandran, A. 2013 A macro transport equation for the particle distribution in the flow of a concentrated non-colloidal suspension through a circular tube. J. Fluid Mech. 2013, 219–252.CrossRefGoogle Scholar
Roustaei, A., Chevalier, T., Talon, L. & Frigaard, I. A. 2016 Non-Darcy effects in fracture flows of a yield stress fluid. J. Fluid Mech. 805, 222–261.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.CrossRefGoogle Scholar
Roustaei, A. & Frigaard, I. A. 2015 Residual drilling mud during conditioning of uneven boreholes in primary cementing. Part 2: Steady laminar inertial flows. J. Non-Newtonian Fluid Mech. 226, 1–15.CrossRefGoogle 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.CrossRefGoogle Scholar
Sangani, A. S., Mo, G. B., Tsao, H. K. & Koch, D. L. 1996 Simple shear flows of dense gas–solid suspensions at finite Stokes numbers. J. Fluid Mech. 313, 309–341.CrossRefGoogle Scholar
Savins, J. G. 1969 Non-Newtonian flow through porous media. Ind. Engng Chem. 61, 18–47.CrossRefGoogle Scholar
Segre, G. & Silberberg, A. 1961 Radial particle displacements in Poiseuille flow of suspensions. Nature 189, 209–210.CrossRefGoogle Scholar
Shahsavari, S. & McKinley, G. 2016 Mobility and pore-scale fluid dynamics of rate-dependent yield-stress fluids flowing through fibrous porous media. J. Non-Newtonian Fluid Mech. 235, 76–82.CrossRefGoogle Scholar
Skjetne, E., Hansen, A. & Gudmondson, J. S. 1999 High velocity flow in a rough fracture. J. Fluid Mech. 383, 1–28.CrossRefGoogle Scholar
Snook, B., Butler, J. E. & Guazzelli, E. 2016 Dynamics of shear-induced migration of spherical particles in oscillatory pipe flow. J. Fluid Mech. 786, 128–153.CrossRefGoogle Scholar
Sultanov, B. I. 1960 Filtration of visco-plastic fluids in a porous medium. Isv. Akad. Nauk AzSSR, Ser. Fiz. Mat. Tekh. Nauk 5, 820–833.Google Scholar
Tabuteau, H., Coussot, P. & de Bruyn, J. R. 2007 Drag force on a sphere in steady motion through a yield-stress fluid. J. Rheol. 51, 125–137.CrossRefGoogle Scholar
Trulsson, M., Andreotti, B. & Claudin, P. 2012 Phys. Rev. Lett. 109, 118305.CrossRefGoogle Scholar
Vu, T.-S., Ovarlez, G. & Chateau, X. 2010 Macroscopic behavior of bidisperse suspensions of noncolloidal particles in yield stress fluids. J. Rheol. 54, 815–833.CrossRefGoogle Scholar
Wierenga, A. M. & Philips, A. P. 1998 Low-shear viscosity of isotropic dispersions of (Brownian) rods and fibres; a review of theory and experiments. Colloids Surf. A 137, 355–372.CrossRefGoogle Scholar
Wylie, J. J., Koch, D. L. & Ladd, A. J. C. 2003 Rheology of suspensions with high particle inertia and moderate fluid inertia. J. Fluid Mech. 480, 95–118.CrossRefGoogle Scholar
Yeo, I. W., De Freitas, M. H. & Zimmerman, R. W. 1998 Effect of shear displacement on the aperture and permeability of a rock fracture. Intl J. Rock Mech. Min. Sci. 35, 1051–1070.CrossRefGoogle 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. Sci. Geomech. Abstr. 28, 335–341.CrossRefGoogle Scholar