Abstract
We investigate the acoustics of saturated porous media with a double porosity constituted by pores and fractures. This work is the direct extension of earlier papers by Auriault and Boutin, where the quasi-static behaviour was studied. The different macroscopic descriptions of the acoustics are shown to be the quasi-static ones, completed by classical inertial terms and with a generalized seepage law for the fractures. Therefore, when the three scales, i.e. the pore, the fracture and the macroscopic scales are equally separated, the medium exhibits memory effects. Finally, we investigate the interpretation of laboratory experiments on single porosity medium under an acoustic excitation. It is shown that the viscoelastic effects which are observed when the frequency is about a few kHz have their origins in the same phenomenon. But the macroscopic description now depends on the size and the shape of the sample, and therefore it is nonspecific for the porous medium.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- a :
-
elastic tensor of the skeleton material
- c′, c″ :
-
effective elastic tensors of the skeleton
- D :
-
rate of deformation tensor
- e :
-
deformation tensor
- f:
-
subscript for the fractures
- k p,k f :
-
particular solutions for the velocity fieldv in the pores and the fractures, respectively
- K p ,K f :
-
filtration tensors of the pores and the fractures, respectively
- l :
-
unit tensor
- l, l′, l″ :
-
characteristic lengths of the pore scale, the fracture scale and the macroscopic medium, respectively
- n, n′ :
-
pore porosity and fracture porosity, respectively
- N :
-
normal unit vector
- p p, pf :
-
pressures in the pores and the fractures, respectively
- p:
-
subscript for the pores
- Q :
-
dimensionless number
- Q a :
-
quality factor
- s :
-
subscript for the solid
- T p, Tf :
-
characteristic times for the pore and fracture flows, respectively
- u s ,u p ,u f :
-
solid, pore and fracture fluid displacements, respectively
- v s ,v p ,v f :
-
solid, pore and fracture fluid velocities, respectively
- x, x′, x″ :
-
space variables for the pore, fracture and macroscopic scales, respectively
- α′, α″, γ′, γ′ :
-
coupling effective tensors entering the behavior relations of the porous medium
- Β′, Β″ :
-
coupling effective scalars entering the behavior relations of the porous medium
- г, г′:
-
boundaries of the pores and the fractures, respectively
- δ :
-
Laplace operator
- ∇ :
-
gradient operator
- ε:
-
small parameter of the homogenization process
- η′ :
-
particular solutions for the displacement field
- Ν′:
-
particular solutions for the pressure field
- Ω :
-
pulsation
- ρ s,ρF :
-
density of the solid and the fluid, respectively
- Μ :
-
viscosity
- ξ′ :
-
particular solutions for the displacement field
- σ s , σ p , σ f :
-
solid, pore and fracture fluid stress tensors, respectively
- Τ′:
-
particular solutions for the pressure field
- Ω, Ω′ :
-
periods at the pore and fracture scales, respectively
- Ω s ,Ω p ,Ω′ sp :
-
solid, pore and fracture fluid stress tensors, respectively
- Τ′:
-
particular solutions for the pressure field
- Ω, Ω :
-
periods at the pore and fracture scales, respectively
- Ω s ,Ω p ,Ω′ sp ,Ω′ f :
-
parts of the periods occupied by the solid, the pores, the solid plus the pores and the fractures, respectively
- 〈Φ〉Ω, 〈Φ〉Ω', 《Φ》ΩΩ' :
-
volume averages of the quantityФ onΩ, Ω′, andΩ plusΩ′, respectively
References
Auriault J.-L., 1980, Dynamic behaviour of a porous medium saturated by a newtonian fluid,Int. J. Engng. Sci. 18, 775–785.
Auriault J.-L., 1991a; Heterogeneous medium. Is an equivalent macroscopic description possible?Int. J. Engng. Sci. 29, 7, 785–795.
Auriault J.-L., 1991b, Dynamic behaviour of porous media, in J. Bear and M. Y. Corapcioglu (eds),Transport Processes in Porous Media, Kluwer Academic Publishers, Dordrecht.
Auriault J.-L., 1992, Acoustics of three phase porous media, in M. Quintard and M. Todorovic (eds),Heat and Mass Transfer, Elsevier, New York, pp. 221–224.
Auriault J. L. and Boutin C., 1992, Deformable porous media with double porosity. Quasi-statics. I,: Coupling effects,Transport in Porous Media 7, 63–82.
Auriault J. L. and Boutin C., 1993, Deformable porous media with double porosity. Quasi-statics. II: Memory effects,Transport in Porous Media,10, 153–169.
Biot M. A., 1941, General theory of three-dimensional consolidation,J. Appl. Physics 12, 155–164.
Biot M. A., 1956, Theory of propagation of elastic waves in a fluid-saturated porous solid. II Higher frequency range,J. Acous. Soc. Am. 28, 2, 179–191.
Borne L., 1983, Contribution à l'étude du comportement dynamique des milieux poreux saturés déformables. Etude de la loi de filtration dynamique, Doctorat thesis, Grenoble.
Bourbié T., Coussy O., and Zinszner B., 1986,Acoustique des milieux poreux, Editions Technip, Paris.
Jones T. and Nur A., 1983, Velocity and attenuation in sandstone at elevated temperatures and pressures,Geophys. Res. Lett. 10(2), 140–143.
Levy T., 1979, Propagation of waves in a fluid-saturated porous elastic solid,Int. J. Engng. Sci. 17, 1005–1014.
Murphy W. F., 1982, Effects of partial water saturation on attenuation in Massilon sandstone and Vycor porous glass,J. Acoust. Soc. Amer. 71, 6, 1458–1468.
Winkler K. W. and Nur A., 1982, Seismic attentuation: effects of pore fluids and frictional sliding,Geophysics 47, 1–15.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Auriault, J.L., Boutin, C. Deformable porous media with double porosity III: Acoustics. Transp Porous Med 14, 143–162 (1994). https://doi.org/10.1007/BF00615198
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00615198