Consolidation of Elastic Porous Media Saturated by Two Immiscible Fluids
Publication: Journal of Engineering Mechanics
Volume 122, Issue 11
Abstract
A theory is presented to simulate the consolidation of elastic porous media saturated by two immiscible Newtonian fluids. The macroscopic equations, including mass and momentum balance equations and constitutive relations, are obtained by volume averaging the microscale equations. The theory is based on the small deformation assumption. In the microscale, the grains are assumed to be linearly elastic and the fluids are Newtonian. The bulk and shear moduli of the solid matrix are introduced to obtain the macroscopic constitutive equations. Momentum transfer terms are expressed in terms of intrinsic and relative permeabilities assuming the validity of Darcy's law. In one dimension, the governing equations reduce to two coupled diffusion equations in terms of the pore pressures of the fluid phases. An analytical solution is obtained for a column with a fixed impervious base and a free drainage surface. Results are presented for cases of practical interest, i.e., columns saturated by oil-water and air-water phases. Results indicate that the presence of a second fluid phase affects pore water pressure and total settlement.
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References
1.
Adkins, J. E.(1963). “Nonlinear diffusion—I. diffusion and flow of mixtures of fluids.”Trans., Royal Society London, 255, 607–634.
2.
Anderson, T. B., and Jackson, R.(1967). “A fluid mechanical description of fluidized beds.”I&EC Fundamentals, 6, 527–539.
3.
Bear, J. (1972). Dynamics of fluids in porous media . American Elsevier, New York, N.Y.
4.
Bear, J., Corapcioglu, M. Y., and Balakrishna, J.(1984). “Modeling of centrifugal filtration in unsaturated deformable porous media.”Adv. Water Resour., 7(4), 150–167.
5.
Biot, M. A. (1941). “General theory of three-dimensional consolidation.”J. Appl. Phys., 12(Feb.), 155–164.
6.
Biot, M. A.(1956). “The theory of deformation of a porous viscoelastic anisotropic solid.”J. Appl. Phys., 27(5), 459–467.
7.
Biot, M. A., and Willis, D. G. (1957). “The elastic coefficients of the theory of consolidation.”J. Appl. Mech., 24(Dec.), 594–601.
8.
Bishop, A. W.(1959). “The principle of effective stress.”Teknik Ukeblad, 39, 859–863.
9.
Bishop, A. W., and Blight, G. E.(1963). “Some aspects of effective stress in saturated and partly saturated soils.”Geotechnique, 13, 177–197.
10.
Bowen, R. M.(1982). “Compressible porous media models by the use of the theory of mixtures.”Int. J. Engrg. Sci., 20(6), 697–735.
11.
Brutsaert, W., and El-Kadi, A. I.(1984). “The relative importance of compressibility and partial saturation in unconfined groundwater flow.”Water Res. Resour., 20(3), 400–408.
12.
“Clark's Tables.” (1971). Science Data Base . Oliver and Boyd, Edinburgh, U.K.
13.
Coleman, J. D.(1962). “Stress-strain relations for partly saturated soils.”Geotechnique, 12, 348–350.
14.
Corapcioglu, M. Y., and Bear, J.(1983). “A mathematical model for regional land subsidence due to pumping—3. integrated equations for a phreatic aquifer.”Water Resour. Res., 19(4), 895–908.
15.
Corapcioglu, M. Y., and Hossain, M. A.(1990). “Ground-water contamination by high density immiscible hydrocarbon slugs in gravity-driven gravel aquifers.” Ground Water, 28(3), 403–412.
16.
de la Cruz, V., and Spanos, T. J. T.(1985). “Seismic wave propagation in a porous medium.”Geophysics, 50, 1556–1565.
17.
de la Cruz, V., and Spanos, T. J. T.(1989). “Thermomechanical coupling during seismic wave propagation in a porous medium.”J. Geophys. Res., 94, 637–642.
18.
Fredlund, D. G., and Hasan, J. U.(1979). “One-dimensional consolidation theory: unsaturated soils.”Can. Geotech. J., 16(3), 521–531.
19.
Fredlund, D. G., and Morgenstern, N. R.(1976). “Constitutive relations for volume change in unsaturated soils.”Can. Geotech. J., 13(3), 261–276.
20.
Fredlund, D. G., and Morgenstern, N. R.(1977). “Stress state variables for unsaturated soils.”J. Geotech. Engrg., 103(5), 447–466.
21.
Geraminegad, M., and Saxena, S. K.(1986). “A coupled thermoelastic model for saturated-unsaturated porous media.”Geotechnique, 36, 539–550.
22.
Ishihara, K. (1967). “Propagation of compressional waves in a saturated soil.”Proc., Int. Symp. Wave Propagation and Dyn. Properties of Earth Mat., Univ. of New Mexico Press, Albuquerque, N.M., 451–467.
23.
Jennings, J. E. B., and Burland, J. B.(1962). “Limitations to the use of effective stresses in partly saturated soils.”Geotechnique, 12, 125–144.
24.
Kohgo, Y., Nakano, M., and Miyazaki, T.(1993). “Theoretical aspects of constitutive modelling for unsaturated soils.”Soils and Founds., 33(4), 49–63.
25.
Lloret, A., and Alonso, E. E.(1980). “Consolidation of unsaturated soils including swelling and collapse behavior.”Geotechnique, 30, 449–477.
26.
Marle, C. M.(1967). “Ecoulements monophasiques en milleu poreux.”Rev. Inst. Francais du Petrole, 22, 1471–1509.
27.
Narasimhan, T. N., and Witherspoon, P. A.(1977). “Numerical model for saturated-unsaturated flow in deformable porous media—1. theory.”Water Resour. Res., 13(3), 657–664.
28.
Narasimhan, T. N., and Witherspoon, P. A.(1978a). “Numerical model for saturated-unsaturated flow in deformable porous media—2. the algorithm.”Water Resour. Res., 14(2), 255–261.
29.
Narasimhan, T. N., and Witherspoon, P. A.(1978b). “Numerical model for saturated-unsaturated flow in deformable porous media—3. applications.”Water Resour. Res., 14(6), 1017–1034.
30.
Nur, A., and Byeerle, J. D.(1971). “An exact effective stress law for elastic deformation of rock with fluids.”J. Geophys. Res., 74, 6414–6419.
31.
Pride, S. R., Gangi, A. F., and Morgan, F. D. (1992). “Deriving the equations of motion for isotropic media.”J. Acoust. Soc. Am., 88(Dec.), 3278–3290.
32.
Scott, P. H., and Rose, W.(1953). “An explanation of the Yuster effect.”J. Petr. Technol., 5, 19–20.
33.
Skempton, A. W. (1960). “Effective stress in soils, concrete and rock.”Pore Pressure and Suction in Soils, Butterworths, London, 1–13.
34.
Slattery, J. C.(1967). “Flow of viscoelastic fluids through porous media.”AIChE J., 13, 1066–1071.
35.
Slattery, J. C.(1968). “Multiphase viscoelastic fluids through porous media.”AIChE J., 14, 50–56.
36.
Slattery, J. C. (1981). Momentum, energy and mass transfer in continua . Krieger, New York, N.Y.
37.
Terzaghi, K. (1923). “Die berechnung der durchlaessigkeitsziffer des tones aus dem verlauf der hydrodynamicschen spannungserscheinungen.”Sitzbericht (AbtIIa). Akademia der Wissenschaften, Vienna, Austria, 132 (in German).
38.
Whitaker, S.(1967). “Diffusion and dispersion in porous media.”AIChE J., 13, 420–427.
39.
Wilson, J. L., Conrad, S. H., Mason, W. R., Peplinski, W., and Hagon, E. (1990). “Laboratory investigation of residual liquid organics from spills, leaks and disposal of hazardous wastes in groundwater.”U.S. Envir. Protection Agency Rep. No. EPA/600/6-90/004, Ada, Okla.
40.
Van Genuchten, M. T.(1980). “A closed form equation for predicting the hydraulic conductivity of unsaturated soils.”Soil, Sci. Soc. Am. J., 44, 892–898.
41.
Yuster, S. T.(1953). “Theoretical considerations of multiphase flow in idealized capillary system.”Proc., Third World Petr. Congr., The Hauge, The Netherlands, 2, 436–445.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 122 • Issue 11 • November 1996
Pages: 1077 - 1085
Copyright
Copyright © 1996 American Society of Civil Engineers.
History
Published online: Nov 1, 1996
Published in print: Nov 1996
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