Abstract
The transmissivity of a variable aperture fracture for flow of a non-Newtonian, purely viscous power-law fluid with behavior index n is studied. The natural logarithm of the fracture aperture is considered to be a two-dimensional, spatially homogeneous and correlated Gaussian random field. We derive an equivalent fracture aperture for three flow geometries: (1) flow perpendicular to aperture variation; (2) flow parallel to aperture variation; (3) flow in an isotropic aperture field. Under ergodicity, results are obtained for cases 1 and 2 by discretizing the fracture into elements of equal aperture and assuming that the resistances due to each aperture element are, respectively, in parallel and in series; for case 3, the equivalent aperture is derived as the geometric mean of cases 1 and 2. When n=1 all our expressions for the equivalent aperture reduce to those derived in the past for Newtonian flow and lognormal aperture distribution. As log-aperture variance increases, the equivalent aperture is found to increase for case 1, to decrease for case 2, and to be a function of flow behavior index n for case 3.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barenblatt, G. I., Entov, V. M., and Ryzhik, V. M.: 1990, Theory of Fluid Flows Through Natural Rocks, Kluwer Acad. Publ., Dordrecht.
Bear, J.: 1972, Dynamics of Fluids in Porous Media, Elsevier, New York.
Bird, R. B., Stewart, W. E., and Lightfoot, E. N.: 1960, Transport Phenomena, Wiley, New York.
Brown, S. R.: 1987, Fluid flow through rock joints: the effect of surface roughness, J. Geophys. Res. B92(2), 1337-1347.
Brown, S. R.: 1989, Transport of fluid and electric current through a single fracture, J. Geophys. Res. B94(7), 9429-9438.
Brown, D. M.: 1984, Stochastic analysis of flow and solute transport in a variable aperture rock fracture, MS Thesis, Mass. Inst. of Technology.
Chabra, R. P.: 1994, Discussion on 'Macroscopic conductivities for flow of Bingham plastics in porous media', by R. P. Vradis and A. L. Protopapas, J. Hydr. Eng., ASCE 120(8), 994-997.
Di Federico, V., and Guadagnini, A.: 1997, Impact of aperture variability on flow and transport properties of a single fracture, Proc. XXVII IAHR Congress, S. Francisco, Theme C, 228-233.
Gelhar, L. W.: 1987, Application of stochastic models to solute transport in fractured rocks, Swedish Nuclear Fuel and Waste Management Company, SKB Tech. Rpt. 87-05, Stockholm, Sweden.
Gelhar, L. W.: 1993, Stochastic Subsurface Hydrology, Prentice-Hall, Englewood Cliffs.
Goldstein, R. V., and Entov, V. M.: 1994, Quantitative Methods in Continuum Mechanics, Wiley, New York.
Gradshteyn, I. S. and Ryzhik, I. M.: 1994, Table of Integrals, Series, and Products. (ed. A. Jeffrey), Academic Press, New York.
Hakami, E., and Larsson, E.: 1996, Aperture measurement and flow experiments on a single natural fracture, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 33, 395-405.
Kutilek, M.: 1972, Non-Darcian flow of water in soils, Laminar region, in Fundamentals of Transport Phenomena in Porous Media, Dev. Soil Sci. 2, IAHR, Elsevier, Amsterdam, pp. 327-340.
James, D. F.: 1984, Non-Newtonian effects in porous media flow, Proc. IX Intnl. Congress on Rheology, Mexico, 279-283.
Johns, R. A., Steude, J. S., Castanier, L. M., and Roberts, P. V.: 1993, Nondestructive measurements of fracture aperture in crystalline rock cores using X ray computed tomography, J. Geophys. Res. B98(2), 1889-1900.
Moreno, L., Tsang, Y. W., Tsang, C. F., Hale, F. V., and Neretnieks, I.: 1988, Flow and tracer transport in a single fracture: a stochastic model and its relation to some field observations, Water Resour. Res. 24(12), 2033-2048.
Neuzil, C. E., and Tracy, J. V.: 1981, Flow through fractures, Water Resour. Res. 17(1), 191-199.
Pop, I., and Nakamura, S.: 1996, Laminar boundary layer flow of power-law fluids over wavy surfaces, Acta Mech. 115, 55-66.
Savins, J. G.: 1969, Non-Newtonian flow through poros media, Ind. Eng. Chem. 6(10), 18-47.
Silliman, S. E.: 1989, An interpretation of the difference between aperture estimates derived from hydraulic and tracer tests in a single fracture, Water Resour. Res. 25(10), 2275-2283.
Snow, D. T.: 1970, The frequency and aperture of fractures in rock, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 7, 23-40.
Tsang, Y. W.: 1984, The effect of tortuosity on fluid flow through a single fracture, Water Resour. Res. 20(9), 1209-1215.
Tsang, Y. W., and Tsang, C. F.: 1987, Channel model of flow through fractured media, Water Resour. Res. 23(3), 467-479.
Tsang, Y. W.: 1992, Usage of 'equivalent apertures' for rock fractures as derived from hydraulic and tracer tests, Water Resour. Res. 28(5), 1451-1455.
Tsay, R. Y., and Weinbaum, S.: 1991, Viscous flow in a channel with cross-bridging fibers: exact solutions and Brinkman approximations, J. Fluid Mech. 226, 125-148.
Vickers, R. A., Neuman, S. P., Sully, M. J., and Evans, D. D.: 1992, Reconstruction and geostatistical analysis of multiscale fracture apertures in a large block of welded tuff, Geophys. Res. Lett. 19, 1029-1032.
Vradis, G. C., and Protopapas, A. L.: 1993, Macroscopic conductivities for flow of Bingham plastics in porous media, J. Hydr. Eng., ASCE 119(1), 95-108.
Zimmerman, R. W., Kumar, S., and Bodvarsson, G. S.: 1991, Lubrication theory analysis of the permeability of rough-walled fractures, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 28(4), 325-331.
Zimmerman, R. W. and Bodvarsson, G. S.: 1996, Hydraulic conductivity of rock fractures, Transport in Porous Media 23(1), 1-30.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Federico, V.D. Non-Newtonian Flow in a Variable Aperture Fracture. Transport in Porous Media 30, 75–86 (1998). https://doi.org/10.1023/A:1006512822518
Issue Date:
DOI: https://doi.org/10.1023/A:1006512822518