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Transport in Porous Media

, Volume 68, Issue 1, pp 91–105 | Cite as

Low-frequency dilatational wave propagation through unsaturated porous media containing two immiscible fluids

  • Wei-Cheng Lo
  • Garrison Sposito
  • Ernest Majer
Original Paper

Abstract

An analytical theory is presented for the low-frequency behavior of dilatational waves propagating through a homogeneous elastic porous medium containing two immiscible fluids. The theory is based on the Berryman–Thigpen–Chin (BTC) model, in which capillary pressure effects are neglected. We show that the BTC model equations in the frequency domain can be transformed, at sufficiently low frequencies, into a dissipative wave equation (telegraph equation) and a propagating wave equation in the time domain. These partial differential equations describe two independent modes of dilatational wave motion that are analogous to the Biot fast and slow compressional waves in a single-fluid system. The equations can be solved analytically under a variety of initial and boundary conditions. The stipulation of “low frequency” underlying the derivation of our equations in the time domain is shown to require that the excitation frequency of wave motions be much smaller than a critical frequency. This frequency is shown to be the inverse of an intrinsic time scale that depends on an effective kinematic shear viscosity of the interstitial fluids and the intrinsic permeability of the porous medium. Numerical calculations indicate that the critical frequency in both unconsolidated and consolidated materials containing water and a nonaqueous phase liquid ranges typically from kHz to MHz. Thus engineering problems involving the dynamic response of an unsaturated porous medium to low excitation frequencies (e.g., seismic wave stimulation) should be accurately modeled by our equations after suitable initial and boundary conditions are imposed.

Keywords

Dilatational waves Immiscible fluid flow Poroelastic behavior 

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References

  1. Avraam D.G., Payatakes A.C. (1995) Generalized relative permeability coefficients during steady-state two-phase flow in porous media, and correlation with the flow mechanisms. Transp. Porous Media 20, 135–168CrossRefGoogle Scholar
  2. Bear J. (1988) Dynamics of Fluids in Porous Media. Dover, New YorkGoogle Scholar
  3. Beresnev I.A., Johnson P.A. (1994) Elastic-wave stimulation of oil production: a review of methods and results. Geophysics 59(6): 1000–1017CrossRefGoogle Scholar
  4. Berryman J.G. (1983) Dispersion of extensional waves in fluid-saturated porous cylinders at ultrasonic frequencies. J. Acoust. Soc. Am. 74(6): 1805–1812CrossRefGoogle Scholar
  5. Berryman J.G., Thigpen L., Chin R.C.Y. (1988) Bulk elastic wave propagation in partially saturated porous solids. J. Acoust. Soc. Am. 84(1): 360–373CrossRefGoogle Scholar
  6. Biot M.A. (1956) Theory of propagation of elastic waves in a fluid-saturated porous solid, I. Low-frequency range, II. Higher frequency range. J. Acoust. Soc. Am. 28(2): 168–191CrossRefGoogle Scholar
  7. Biot M.A., Willis D.G. (1957) The elastic coefficients of the theory of consolidation. J. Appl. Mech. 24, 594–601Google Scholar
  8. Biot M.A. (1962) Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 33(4): 1482–1498CrossRefGoogle Scholar
  9. Brutsaert W. (1964) The propagation of elastic waves in unconsolidated unsaturated granular mediums. J. Geophys. Res. 69(2): 243–257Google Scholar
  10. Brutsaert W., Luthin J.M. (1964) The velocity of sound in soils near the surface as a function of the moisture content. J. Geophys. Res. 69(4): 643–252Google Scholar
  11. Chandler R.N., Johnson D.L. (1981) The equivalence of quasistatic flow in fluid-saturated porous media and Biot’s slow wave in the limit of zero frequency. J. Appl. Phys. 52(5): 3391–3395CrossRefGoogle Scholar
  12. Chen J., Hopmans J.W., Grismer M.E. (1999) Parameter estimation of two-fluid capillary pressure-saturation and permeability functions. Adv. Water Resour. 22(5): 479–493CrossRefGoogle Scholar
  13. Cowin S.C. (1999) Bone poroelasticity. J. Biomech. 32, 217–238CrossRefGoogle Scholar
  14. Donskoy D.M., Khashanah K., McKee T.G. (1997) Nonlinear acoustic waves in porous media in the context of Biot’s theory. J. Acoust. Soc. Am. 102(5): 2521–2528CrossRefGoogle Scholar
  15. Dullien F.L.A. (1992) Porous Media: Fluid Transport and Pore Structure. Academic Press, San DiegoGoogle Scholar
  16. Dutta N.C., Ode H. (1979) Attenuation and dispersion of compressional waves in fluid-filled porous rocks with partial gas saturation (White model)—Part I: Biot theory. Geophysics 11, 1777–1788CrossRefGoogle Scholar
  17. Dvorkin J., Nur A. (1993) Dynamic poroelasticity—a unified model with the squirt and the Biot mechanisms. Geophysics 58(4): 524–533CrossRefGoogle Scholar
  18. Ehrlich R. (1993) Viscous coupling in two-phase flow in porous media and its effect on relative permeabilities. Transp. Porous Media 11, 201–218CrossRefGoogle Scholar
  19. Garg S.K., Nayfeh A.H. (1986) Compressional wave propagation in liquid and/or gas saturated elastic porous media. J. Appl. Phys. 60(9): 3045–3055CrossRefGoogle Scholar
  20. Gray W.G. (1983) General conservation equations for multi-phase systems: 4 Constitutive theory including phase change. Adv. Water Res. 6, 130–140CrossRefGoogle Scholar
  21. Jerauld G.R., Salter S.J. (1990) The effect of pore-structure on hysteresis in relative permeability and capillary pressure: pore level modeling. Transp. Porous Media 5, 103–151CrossRefGoogle Scholar
  22. Johnson D.L. (1986) Recent developments in the acoustic properties of porous media. In: Sette D. (ed) Proceedings of the International School of Physics ≪Enrico Fermi≫ Course XCIII, Frontiers in Physical Acoustics. North Holland, Amsterdam, pp. 255–290Google Scholar
  23. Kamon M., Endo K., Katsumi T. (2003) Measuring the kSp relations on DNAPLs Migration. Eng. Geol. 70, 351–363CrossRefGoogle Scholar
  24. Kearey P., Brooks M., Hill I. (2002) An Introduction to Geophysical Exploration. Oxford, Blackwell ScienceGoogle Scholar
  25. Kouznetsov O.L., Simkin E.M., Chilingar G.V., Katz S.A. (1998) Improved oil recovery by application of vibro-energy to waterflooded sandstones. J. Pet. Sci. Eng. 19(3–4): 191–200CrossRefGoogle Scholar
  26. Lake L.W. (1989) Enhanced Oil Recovery. Prentice-Hall, Englewood CliffsGoogle Scholar
  27. Lewis R.W., Schrefler B.A. (1998) The finite element method in the static and dynamic deformation and consolidation of porous media. John Wiley & Sons, ChichesterGoogle Scholar
  28. Li X., Zhong L., Pyrak-Nolte L.J. (2001) Physics of partially saturated porous media: residual saturation and seismic-wave propagation. Annu. Rev. Earth Planet. Sci. 29, 419–460CrossRefGoogle Scholar
  29. Lo W.-C., Sposito G., Majer E. (2002) Immiscible two-phase fluid flows in deformable porous media. Adv. Water Resour. 25(8–12): 1105–1117CrossRefGoogle Scholar
  30. Lo, W.-C., Sposito, G., Majer, E.: Wave propagation through elastic porous media containing two immiscible fluids. Water Resour. Res. 41, W02025 (2005)Google Scholar
  31. Lo W.-C., Sposito G., Majer E. (2006) Low-frequency dilatational wave propagation through fully-saturated poroelastic media. Adv. Water Resour. 29(3): 408–416CrossRefGoogle Scholar
  32. Mei C.C., Foda M.A. (1981) Wave-induced responses in a fluid-filled poro-elastic solid with a free-surface—a boundary-layer theory. Geophys. J. Roy. Astronom. Soc. 66(3): 597–631Google Scholar
  33. Mualem Y. (1976) A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12(3): 513–522CrossRefGoogle Scholar
  34. Rice J.R., Cleary M.P. (1976) Some basic stress diffusion solutions for fluid-saturated elastic porous-media with compressible constituents. Rev. Geophys. 14(2): 227–241Google Scholar
  35. Roberts P.M., Sharma A., Uddameri V., Monagle M., Dale D.E., Steck L.K. (2001) Enhanced DNAPL transport in a sand core during dynamic stress stimulation. Environ. Eng. Sci. 18(2): 67–79CrossRefGoogle Scholar
  36. Santos J.E., Corbero J.M., Douglas J. (1990) Static and dynamic behavior of a porous solid saturated by a two-phase fluid. A model for wave propagation in a porous medium saturated by a two-phase fluid. J. Acoust. Soc. Am. 87(4): 1428–1448CrossRefGoogle Scholar
  37. Stoll R.D. (1974) Acoustic waves in saturated sediments. In: Hampton L. (ed) Physics of Sound in Marine Sediments. Plenum Press, New York, pp. 19–39Google Scholar
  38. Truesdell C. (1984) Rational Thermodynamics. Springer-Verlag, New YorkGoogle Scholar
  39. van Genuchten M.T. (1980) A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44(5): 892–898CrossRefGoogle Scholar
  40. Wartenberg D., Reyner D., Scott C.S. (2000) Trichloroethylene and cancer: epidemiologic evidence. Environ. Health Perspect. 108(Suppl. 2): 161–176Google Scholar
  41. Yuster, S.T.: Theoretical considerations of multiphase flow in idealized capillary systems. In: World Petroleum Congress Proceeding, Section II, The Hague, pp. 436–445 (1951)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Wei-Cheng Lo
    • 1
    • 2
  • Garrison Sposito
    • 2
    • 3
  • Ernest Majer
    • 2
  1. 1.Department of Hydraulic and Ocean EngineeringNational Cheng Kung UniversityTainanTaiwan
  2. 2.Department of GeophysicsLawrence Berkeley National LaboratoryBerkeleyUSA
  3. 3.Department of Civil and Environmental EngineeringUniversity of CaliforniaBerkeleyUSA

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