BOUNDARY CONDITIONS

Appropriate boundary conditions for use with Biot's equations have been considered by Deresiewicz and Skalak (1963), Berryman and Thigpen (1985), and Pride and Haartsen (1996) and we make use of these results here.

At the external surface r=R₂ where the outer porous material contacts the surrounding air, it is appropriate to use the free surface conditions

$$\begin{eqnarray} -\delta p_f = 0, \,\, \delta \tau_{rr} = 0, \,\, \delta \tau_{r\theta} = 0, \,\, \delta \tau_{rz} = 0,\end{eqnarray}$$ (216)

for the deviations from static equilibrium. If the cylinder is sealed on r=R₂, then the first of these needs to be replaced by w_r=0.

The internal interface at r=R₁ needs more precise definition. We assume that all the meniscii that are separating the inner fluid from the outer fluid are contained within a thin layer (shell) of thickness $\delta h$ (a few grain sizes in width) straddling the surface r=R₁. All fluid that enters this interface layer goes into stretching the meniscii since as Pride and Flekkoy (1999) have shown, it is reasonable to assume that the contact lines of the meniscii remain pinned under seismic stressing. The locally incompressible flow conserves fluid volume so that the rate at which the inner fluid enters the interface layer is equal to the rate at which the outer fluid leaves the layer thus requiring

$$\begin{eqnarray} \dot{w}_r (r=R_1 + \delta h/2)= \dot{w}_r^* (r=R_1 - \delta h/2).\end{eqnarray}$$ (217)

This and the following conditions are to be understood in the limit where $\delta h/R_1 \rightarrow 0$.It is also straightforward to obtain the standard results

$$\begin{eqnarray} \tau_{rr} = \tau_{rr}^*,\,\,\,\, \tau_{r\theta} = \tau_{r\theta}^*, \,\,\,\, \tau_{rz} = \tau_{rz}^*, \end{eqnarray}$$ (218)

and

$$\begin{eqnarray} \dot{u}_r = \dot{u}_r^*,\, \, \, \, \dot{u}_\theta = \dot{u}_\theta^*, \,\,\,\, \dot{u}_z = \dot{u}_z^*. \end{eqnarray}$$ (219)

The final condition to establish on r=R₁ is that involving the fluid pressure.

The rate at which energy fluxes radially through the porous material is given by $\tau_{ri} \dot{u}_i - p_f \dot{w}_r$ with implicit summation over the index i. The difference in the rate at which energy is entering and leaving the interface layer is due to work performed in stretching the meniscii. Each meniscus has an initial mean curvature H_o that is determined by the initial fluid pressures (those that hold before the wave arrives) as $p_{f0} - p_{f0}^* = \sigma H_o$ where $\sigma$ is the surface tension. As the wave passes, the ratio between the actual mean curvature H and H_o is a small quantity on the order of the capillary number $\epsilon= \eta \vert\dot{w}_r\vert /\sigma$ [see Pride and Flekkoy (1999)] where $\vert\dot{w}_r\vert$ is some estimate of the induced Darcy flow and that goes as wave strain times wave velocity ($\vert\dot{w}_r\vert < 10^{-3}$ m/s). Since $\sigma \gt 10^{-2}$ Pam for air-water interfaces, we have $\epsilon < 10^{-4}$, which can be considered negligible. By integrating the energy flux rate over a Gaussian shell that straddles r=R₁, it is straightforward to obtain

$$\begin{eqnarray} &&[\tau_{ri} \dot{u}_i - (p_{f0} + \delta p_f) \dot{w}_r] - \ &... ... \dot{w}_r^*] \nonumber = \sigma H_o \dot{w}_r [1 + O(\epsilon)].\end{eqnarray}$$

Thus, since all components here except fluid pressure are continuous, we find that, when $\epsilon$ is small compared to unity,

$$\begin{eqnarray} \delta p_f = \delta p_f^* .\end{eqnarray}$$ (220)

In other words, to the extent that the capillary number can be considered small (always the case for linear wave problems), the wave-induced increments in fluid pressure are continuous at r=R₁.

To apply the boundary conditions () and (), we need in addition to (48) the result

$$\begin{eqnarray} w_r = a_{41}\alpha_+ + a_{42}\alpha_- + a_{43}\alpha_s, \end{eqnarray}$$ (221)

where

$$\begin{eqnarray} (\Gamma_+ - \Gamma_-)a_{41} &=& k_{+r}J_1(j_+)\Gamma_-, \ (\Gam... ...& - k_{-r}J_1(j_-)\Gamma_+, \ a_{43} &=& k_{sr}J_1(j_s)\rho_f/q. \end{eqnarray}$$ (222)

The remaining stress conditions (62) are determined by (47) and (49).

To apply the boundary conditions (63), we need the explicit expressions for the displacement which follow from (23). The results are of the form

$$\begin{eqnarray} u_r = a_{51}\alpha_+ + a_{52}\alpha_- + a_{53}\alpha_s, \end{eqnarray}$$ (223)

where

$$\begin{eqnarray} (\Gamma_+-\Gamma_-)a_{51} &=& -k_{+r}J_1(j_+), \ (\Gamma_+-\Gamma_-)a_{52} &=& k_{-r}J_1(j_-), \ a_{53} &=& k_{sr}J_1(j_s), \end{eqnarray}$$ (224)

and

$$\begin{eqnarray} u_z = a_{61}\alpha_+ + a_{62}\alpha_- + a_{63}\alpha_s, \end{eqnarray}$$ (225)

where a₆₁ = a₆₂ = 0, and

 

<I>a₆₃I> = <I>kI>_<I>srI>²<I>J₀I>(<I>jI>_<I>sI>)/<I>ikI>_<I>zI>. (226)

Both (ur) and (uz) are needed for extensional waves, while the remaining component,

$$\begin{eqnarray} u_\theta = m_{21}\gamma_s \equiv k_{sr}J_1(j_s)\gamma_s, \end{eqnarray}$$ (227)

is needed only for torsional waves. As before, there is an implicit factor of $\exp i(k_z z - \omega t)$ on the right-hand side of (66)-(68), (a51)-(a53), and (a63).

It follows from (46)-(49), (65), and (utheta) that $\gamma_s$ (for the inner cylinder) and the corresponding coefficients for the cylindrical shell are all completely independent of the other mode coefficients and, therefore, relevant to the study of torsional waves, but not for extensional waves. Pertinent equations for the torsional wave dispersion relation are continuity of the angular displacement, $u_\theta$, and stress, $\tau_{r\theta}$, at the internal interface, and vanishing of the stress, $\tau_{r\theta}$, at the external surface.

The final set of equations for the extensional wave dispersion relation involves nine equations with nine unknowns. The nine unknowns are: $\alpha_+$, $\alpha_-$, $\alpha_s$ (coefficients of J₀ in the central cylinder), plus three $\alpha^*$'s (coefficients of J₀) and three $\eta^*$'s (coefficients of Y₀) for region of the cylindrical shell. The nine equations are: the continuity of radial and one tangential stress as well as radial and one tangential displacement at the interfacial boundary (totaling four conditions), continuity of fluid pressure and normal fluid increments across the same boundary (two conditions), and finally the vanishing of the external fluid pressure, radial and one tangential stress at the free surface (three conditions). The extensional wave dispersion relation is then determined as in Berryman (1983) by those conditions on the wavenumber k_z that result in vanishing of the determinant of the coefficients of this $9\times 9$ complex matrix.