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



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.


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




\(\omega \)

Angular frequency


Apparent phase velocity


Wave number


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


Shear modulus of solid grains


Shear modulus of porous frame


Bulk modulus of porous frame


Bulk modulus equivalent of capillary pressure


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


Internal pressure of pore-gas


Internal pressure of pore-liquid


Elastic coefficients

\(\delta _{ij}\)

Kronecker symbol

\(\chi \)

Capillary parameter for liquid-gas saturation


Consolidation parameter for incompressibility


Consolidation parameter for shear

\(\xi \)

Non-dimensional frequency parameter



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


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Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Department of MathematicsKurukshetra UniversityKurukshetraIndia

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