Transport in Porous Media

, Volume 93, Issue 3, pp 597–607 | Cite as

Poroelasticity-III: Conditions on the Interfaces

Article

Abstract

In this, the third part of our paper, we continue consideration of the major elements of the poroelastic theory which we started in Parts I and II (in Lopatnikov and Gillespie, Transp Porous Media, 84:471–492, 2010; Transp Porous Media, 89:475–486, 2011). This third part is devoted to considering the general interfacial conditions, consistent with the governing differential equations of the theory. Specifically, we will consider associated mass and momentum conservation laws. Because we developed the theory by construction, general boundary conditions obtained can be applied to the arbitrary interfaces: boundaries between different materials or, for example, moving interfaces of the shock fronts. We do not consider here the last group of conservation laws: the energy conservation laws, which we are going to introduce and investigate in the special part, devoted to the shock wave propagation. In the meantime, special attention is devoted to discussing the problem of “partial permeability” of the interfaces reflected in the literature. Particularly we show, that in the stationary case, the general theory allows only two conditions: either the interface is completely penetrable, or the interface is completely impenetrable. Thus, “partial permeability” solution always appears as only an approximation of an exact dynamic problem, which includes either thin low-permeable interfacial layer (with permeable boundaries), or a non-homogeneous boundary containing permeable and non-permeable patterns.

Keywords

Poroelastics Interfacial conditions Conservation laws Wave propagation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Biot M.A.: The 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, 168–191 (1956)CrossRefGoogle Scholar
  2. Biot M.A.: Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 33, 1482–1498 (1962a)CrossRefGoogle Scholar
  3. Biot M.A.: Generalized theory of acoustic propagation in porous dissipative media. J. Acoust. Soc. Am. 34, 1254–1264 (1962b)CrossRefGoogle Scholar
  4. Bourbie T., Coussy O., Zinszner B.: Acoustics of Pourous Media. Technip, Paris (1987)Google Scholar
  5. de la Cruz V., Spanos T.J.T.: Seismic boundary conditions for porous media. J. Geophys. Res. 94, 3025–3029 (1989)CrossRefGoogle Scholar
  6. Deresiewicz H.: The effect of boundaries on wave propagation in a liquid-filled porous solid: I. Reflection of plane waves at a free boundary (non-dissipative case). Bull. Seismol. Soc. Am. 50, 599–607 (1960)Google Scholar
  7. Deresiewicz H.: The effect of boundaries on wave propagation in a liquid-filled porous solid: II. Reflection of plane waves at a free plane boundary (general case). Bull. Seismol. Soc. Am. 51, 17–27 (1961)Google Scholar
  8. Deresiewicz H.: The effect of boundaries on wave propagation in a liquid-filled porous solid: IV. Surface waves in a halfspace. Bull. Seismol. Soc. Am. 52, 627–638 (1962)Google Scholar
  9. Deresiewicz H., Skalak R.: On uniqueness in dynamic poroelasticity. Bull. Seismol. Soc. Am. 53, 783–788 (1963)Google Scholar
  10. Frenkel Ya.: On the theory of seismic and seismoelectric phenomena in moist soil. J. Phys. 8, 230–241 (1944)Google Scholar
  11. Guiroga-Goode, G., Carcione, J.M.: Heterogeneous modeling behavior at an interface in porous media. Comput. Geosci. 1, 109–125 (1996). European Association of Geoscientists and Engineers Extended Abstract Paper C005Google Scholar
  12. Gurevich B., Schoenberg M.: Interface conditions for Biot’s equations of poroelasticity. J. Acoust. Soc. Am. 105(5), 2585–2589 (1999)CrossRefGoogle Scholar
  13. Lopatnikov S.L., Cheng A.H.D.: Macroscopic Lagrangian formulation of poroelasticity with porosity dynamics. J. Mech. Phys. Solids 52(12), 2801–2839 (2004)CrossRefGoogle Scholar
  14. Lopatnikov S., Gillespie J.W. Jr: Poroelasticity-I: governing equations of the mechanics of fluid-saturated porous materials. Transp. Porous Media 84, 471–492 (2010)CrossRefGoogle Scholar
  15. Lopatnikov S., Gillespie J.W. Jr: Poroelasticity-II: On the Equilibrium State of the Fluid-Filled Penetrable Poroelastic Body. Transp. Porous Media 89, 475–486 (2011)CrossRefGoogle Scholar
  16. Nikolaevskiy, V., Basniev, K., Gorbunov, A., Zotov, G.: Mechanics of Saturated Porous Media. Nedra, Moscow (1970) (in Russian)Google Scholar
  17. Rasolofosaon P.N.J.: Importance of the interface hydraulic condition on the generation of second bulk compressional wave in porous media. Appl. Phys. Lett. 52, 780–782 (1988)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Center for composite MaterialsUniversity of DelawareNewarkUSA
  2. 2.Department of Materials ScienceUniversity of DelawareNewarkUSA
  3. 3.Department of Civil and Environmental EngineeringUniversity of DelawareNewarkUSA

Personalised recommendations