# Surface waves in a cylindrical borehole through partially-saturated porous solid

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## Abstract

Propagation of surface waves is discussed in a cylindrical borehole through a liquid-saturated porous solid of infinite extent. The porous medium is assumed to be a continuum consisting of a solid skeletal with connected void space occupied by a mixture of two immiscible inviscid fluids. This model also represents the partial saturation when liquid fills only a part of the pore space and gas bubbles span the remaining void space. In this isotropic medium, potential functions identify the existence of three dilatational waves coupled with a shear wave. For propagation of plane harmonic waves along the axially-symmetric borehole, these potentials decay into the porous medium. Boundary conditions are chosen to disallow the discharge of liquid into the borehole through its impervious porous walls. A dispersion equation is derived for the propagation of surface waves along the curved walls of no-liquid (all gas) borehole. A numerical example is studied to explore the existence of cylindrical waves in a particular model of the porous sandstone. True surface waves do not propagate along the walls of borehole when the supporting medium is partially saturated. Such waves propagate only beyond a certain frequency when the medium is fully-saturated porous or an elastic one. Dispersion in the velocity of pseudo surface waves is analysed through the changes in consolidation, saturation degree, capillary pressure or porosity.

## Keywords

Cylindrical waves phase velocity dispersion porous solid partial saturation multiphase pore-fluid## Nomenclature

*t*Time

- \(\omega \)
Angular frequency

*c*Apparent phase velocity

*k*Wave number

*f*Total porosity

- \(\sigma \)
Fraction of pore space filled with liquid

- \(\delta _g\)
Volume fraction of gas-filled pores in porous aggregate

- \(\delta _l\)
Volume fraction of liquid-filled pores in porous aggregate

- \(\delta _s\)
Volume fraction of solid grains in porous aggregate

- \(\rho _s, \rho _g, \rho _l\)
Densities of solid grains, pore-gas and pore-liquid, respectively

- \(K_s, K_g, K_l\)
Bulk moduli of solid grains, pore-gas, pore-liquid, respectively

- \(G_s\)
Shear modulus of solid grains

- \(G_p\)
Shear modulus of porous frame

- \(K_p\)
Bulk modulus of porous frame

- \(K_c\)
Bulk modulus equivalent of capillary pressure

*a*Radius of cylindrical borehole

- \(\alpha _j~(j=1,2,3)\)
Velocities of \(P_1, P_2, P_3\) waves, respectively

- \(\beta \)
Velocity of shear wave

- \((r,~\theta ,~z)\)
Cylindrical coordinate system

- \(\mathbf{u}=(u_r,u_\theta ,u_z)\)
Displacement of solid particles

- \(\mathbf{v}=(v_r,v_\theta ,v_z)\)
Displacement of pore-gas particles

- \(\mathbf{w}=(w_r,w_\theta ,w_z)\)
Displacement of pore-liquid particles

- \(\tau _{ij}^{(p)}\)
Stress tensor for porous frame

- \(p^{(g)}\)
Internal pressure of pore-gas

- \(p^{(l)}\)
Internal pressure of pore-liquid

- \(a_{ij}\)
Elastic coefficients

- \(\delta _{ij}\)
Kronecker symbol

- \(\chi \)
Capillary parameter for liquid-gas saturation

- \(c_K\)
Consolidation parameter for incompressibility

- \(c_G\)
Consolidation parameter for shear

- \(\xi \)
Non-dimensional frequency parameter

## Notes

### Acknowledgements

Author is grateful to the unknown reviewers for their contribution in improving the manuscript.

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