1. The 6th BIOT Conference on poromechanics, Paris, 2017
Propagation of Elastic Waves in a Gas-Filled Poroelastic Medium in the Slip-Flow Regime.
Mikhail Markov, and Anatoly Markov
1 Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, CP 07730, México, D.F.
2 Instituto Tecnológico y de Estudios Superiores de Monterrey, Campus Estado de México, México C.P , México
The 6th BIOT Conference on poromechanics, Paris, 2017

2. The 6th BIOT Conference on poromechanics
Introduction 1
The basic concepts of the theory of propagation of elastic waves in saturated porous media was done in the pioneering works of Frenkel (1944) and Biot (1956). Then, the theory was generalized in a series of fundamental works (Bourbié et al., 1987; Coussy, 2004). The main conclusion of the Biot theory about the existence of two types of compressional waves was proven experimentally (Plona, 1980).
The further development of this theory for gas-saturated porous media with thin pores is of considerable interest, because in this case physical effects related to the discrete nature of gases are observed.
The 6th BIOT Conference on poromechanics

3. The 6th BIOT Conference on poromechanics
Introduction 2
A gas may be described as a classic continuous medium only for a limited range of pore sizes, in the case of sufficiently small Knudsen number 𝐾𝑛=𝜆/𝑙, where 𝜆 is the mean free path of gas molecules; 𝑙 is the characteristic size of inclusion. In the range of Knudsen number much higher than zero, in order to adequately describe the gas behavior, methods of statistical physics and rarefied gas dynamics should be used (Kogan, 1969, Cercignani, 1975).
In this work we have considered so-called slip flow regime, 0<Kn<<1. We have obtained the expressions for drag and added mass coefficients for the Biot equations as functions of Knudsen number and characteristics of the solid grains (accommodation coefficients).
The 6th BIOT Conference on poromechanics

4. The 6th BIOT Conference on poromechanics
The Knudsen number Kn
The Knudsen number Kn is a principal parameter used to characterize the gas flow regime.
If the Knudsen number is large (Kn>>1), we can neglect the collisions between the molecules (so-named free molecular regime or ballistic approximation).
When Kn=0, we have the standard hydrodynamic regime.
If 0<Kn<1, we have so-named slip flow regime. In this case it is necessary to use the methods of rarefied gas dynamics.
The 6th BIOT Conference on poromechanics

5. Gas behaviour for different Knudsen numbers
The 6th BIOT Conference on poromechanics

6. The Boltzmann equation
In physical kinetics the behavior of rarefied gas (only pair collisions of molecules occur) is described by the Boltzmann equation [Kogan, 1969; Cercignani, 1975] for one-particle velocity distribution function f(t,r,v):
f(r,v,t)drdv is the number of molecules in drdv;
v’ and v’* are pre-collision molecular velocities;
v and v* are post-collision molecular velocities;
n(t; r) =∫f(t; r; v)dv is the number density;
The 6th BIOT Conference on poromechanics

7. The boundary conditions for the Boltzmann equation
The most popular model is the Maxwell model (Kogan, 1969). Maxwell supposed that the part α of incident molecules was diffusely reflected and the part (1 - α) was specularly reflected. Here α is the accommodation coefficient of the tangential impulse.
where is the distribution function of incident molecules.
The 6th BIOT Conference on poromechanics

8. The 6th BIOT Conference on poromechanics
The slip flow regime
If 0<Kn<<1 we have so-named slip flow regime. In this case, we can use the classical mechanics equations of continuum media in the all gas-filled region with the exception of a thin layer near the solid.
To describe the behavior of gas in thin layer near the solid we have to solve the Boltzmann equation. This solution gives us the modified boundary conditions for classical equations of hydrodynamics and heat conduction. δU
Gas
Solid λ
is the tangential component of the viscous stress tensor in gas
The 6th BIOT Conference on poromechanics

9. The 6th BIOT Conference on poromechanics
Calculation of the drag and virtual mass coefficients in the equations of a poroelastic medium
U, Uf, are the displacement vectors of the skeleton and the fluid in the pores, respectively; T and S are the stress tensors in the elastic skeleton and in the fluid; b and c are the drag coefficient and the additional mass coefficient, respectively
The 6th BIOT Conference on poromechanics

10. The 6th BIOT Conference on poromechanics
Calculation of the drag and virtual mass coefficients in the equations of a poroelastic medium - 2
Let the solid phase performs homogeneous spatial oscillations
Then, the fluid will also perform spatially homogeneous oscillations with the displacement
The 6th BIOT Conference on poromechanics

11. The 6th BIOT Conference on poromechanics
Drag and virtual mass coefficients for a simple model of the pore space U Un Uτ
where Un is the average displacement in the direction normal to the cylinder axes and Uτ is the average displacement in the axial direction.
The 6th BIOT Conference on poromechanics

12. The 6th BIOT Conference on poromechanics
Numerical results 1
The capillary radius a = 3 × 10-6 m,
the kinematic viscosity v = 0.12 cm2/s,
and the gas density is 1.2 × 10 g/cm3.
The frequency is equal to 50 kHz.
In the low frequency range the drug coefficient b0 has the form:
Drag and virtual mass coefficients, b and c, versus the tangential momentum accommodation coefficient α.
The 6th BIOT Conference on poromechanics

13. The 6th BIOT Conference on poromechanics
Numerical results 2
Velocity and attenuation of compressional wave of the second kind as a function of the tangential momentum accommodation coefficient α. The calculations are presented for different Knudsen numbers. The frequency f is 50 kHz.
The 6th BIOT Conference on poromechanics

14. The 6th BIOT Conference on poromechanics
Numerical results 3
Velocity and attenuation of compressional wave of the second kind as a function of the tangential momentum accommodation coefficient α. The calculations are presented for different frequencies. The Knudsen number Kn is 0.25.
The 6th BIOT Conference on poromechanics

15. The 6th BIOT Conference on poromechanics
Conclusion
In this paper we consider the propagation of elastic waves in gas-filled porous media at small but not zero Knudsen numbers 0 < 𝐾𝑛 ≤ 0.3 (the so named slip regime). In this case, it is possible to apply the classic equations of hydrodynamics with modified boundary conditions.
We have calculated the drag and additional mass coefficients in the equations of poromechanics taking into account the effect of the interfacial slip.
The calculations showed that the effect of interfacial slip cannot explain the high attenuation coefficients observed in real porous media (for example in rocks) for compressional waves of the first kind and shear waves. At the same time, for the slow and rapidly attenuating compressional waves of the second kind, the effect of interfacial slip on their kinematic and dynamic parameters is considerable and can be estimated experimentally.
The 6th BIOT Conference on poromechanics

16. The 6th BIOT Conference on poromechanics
References
Allard J., Atalla N. (2009). Propagation of sound in porous media: Modelling sound absorbing materials. J. Willey and Sons Ltd.
Biot M.A. (1956). “Theory of propagation of elastic waves in a fluid – saturated porous solid. “ J. Acoust. Soc. Am., 28(2),
Bedford A., Costley R.D., Stern M. (1984). “On the drug and virtual mass coefficients in Biot’s equations.” J. Acoust. Soc. Am., 76,
Bourbié T., Coussy O., Zinzner B. (1987). Acoustics of porous media. Houston, Gulf Publishing Co.
Cercignani C. (1975). Theory and Applications of the Boltzmann Equation. Scottish Academic Press, Edinburgh.
Chastanet J, Royer P., and Auriault J. L., (2004). “Acoustics with wall-slip flow of gas saturated porous media,” Mech. Res. Commun. 31, 277–286, DOI: /j.mechrescom
Coussy, O., 2004, Poromechanics, John Wiley & Sons.
Frenkel Ya. I. (1944), “On the theory of seismic and seismoelectric phenomena in a moist soil” Izv. Akad. Nauk USSR, Ser. Geogr. I Geofiz., 8,
Klinkenberg L.J. (1941). “The Permeability of Porous Media to Liquids and Gases.” API Drilling and production practice
Kogan M. N. (1969). Rarefied Gas Dynamics. Plenum Press, New York, US. Trans. R. Soc. London, 170, (1879), 231–256.
Plona T. J. (1980) “Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies.” Appl. Phys. Lett. 36, , DOI: /
The 6th BIOT Conference on poromechanics

17. Thank you for your attention
The 6th BIOT Conference on poromechanics