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EQUATIONS FOR A POROUS CYLINDER

To work most easily in cylindrical geometry, we rewrite the stress-strain relations (06)-(12) in cylindrical coordinates. If z is the coordinate along the cylinder axis while rand $\theta$ are the radial and azimuthal coordinates, it is not difficult to show that
         \begin{eqnarray}
\delta\tau_{rr} &=& He -
2\mu(e_\theta+e_z) - C\zeta,
\
\delta\...
 ...r}\over{\partial z}}
+ {{\partial u_z}\over{\partial r}}\right),
 \end{eqnarray} (17)
and (12) for $\delta p_f$ remains unchanged. The stress increments $\delta\tau_{zz}$, $\delta\tau_{\theta\theta}$,and $\delta\tau_{\theta z}$ are not of direct interest in the present application. The dilatations are given by
   \begin{eqnarray}
e = e_r + e_\theta + e_z,
 \end{eqnarray} (18)
where
   \begin{eqnarray}
e_r = {{\partial u_r}\over{\partial r}}, \quad
e_\theta = {{u_r...
 ...partial \theta}}, \quad
e_z = {{\partial u_z}\over\partial z{}}.
 \end{eqnarray} (19)

We redefine potential $\mbox{\boldmath$\beta$}$ in terms of two scalar potentials according to
   \begin{eqnarray}
\mbox{\boldmath$\beta$}= \hat{\bf z}\beta_1 + \nabla\times (\hat{\bf z}\beta_2),
 \end{eqnarray} (20)
where both $\beta_i$ satisfy
   \begin{eqnarray}
(\nabla^2 + k_s^2)\beta_i = 0 \qquad\hbox{for}\qquad i = 1,2.
 \end{eqnarray} (21)

For the problem of interest here, we will have two distinct regions: The first region is a cylinder centered at the origin, within which solutions of (25) and (39) must be finite at the origin. Results take the form
         \begin{eqnarray}
A_{\pm} &=& \alpha_\pm J_0(j_\pm)\exp i(k_z z-\omega t),
\
\bet...
 ...,
\
\beta_2 &=& (\alpha_s/ik_z)J_0(j_s)\exp i(k_z z - \omega t),
 \end{eqnarray} (22)
where
   \begin{eqnarray}
j_\pm = k_{\pm r} r, \qquad j_s = k_{sr}r
 \end{eqnarray} (23)
and
   \begin{eqnarray}
k_{\pm r}^2 = k_\pm^2 - k_z^2, \qquad k_{sr}^2 = k_s^2 - k_z^2.
 \end{eqnarray} (24)
J0 is the zero-order Bessel function of the first kind. The coefficients $\alpha_\pm$, $\alpha_s$, $\gamma_s$, are constants to be determined from the boundary conditions.

The second region is a cylindrical shell around the first region. In this region, the factors $k_\pm$ and ks take different values from the those in the central region, indicated by $k_\pm^*$ and ks* (where * means air-filled, and does not ever mean complex conjugate in this paper). Furthermore, two linearly independent solutions of the equations are allowed, i.e., both J0 and Y0 (the Bessel function of the second kind, sometimes known as the Neumann function). In the outer shell, we have four coefficients apiece for J0 and Y0, all of which must also be determined by the boundary conditions.

Noting that
   \begin{eqnarray}
\Upsilon &=& (A_+-A_-)/(\Gamma_+-\Gamma_-),\nonumber\
\psi &=& (A_+\Gamma_--A_-\Gamma_+)/(\Gamma_--\Gamma_+)
 \end{eqnarray}
from the definitions of $A_\pm$,and substituting (40)-(42) and (45) into (23), and the result into (12) and (32)-(34), we finally obtain
            \begin{eqnarray}
\delta\tau_{r\theta} &\!\!=\!\!& m_{11}\gamma_s \equiv -\mu k_{...
 ...a\tau_{rz} &=& a_{31}\alpha_+ + a_{32}\alpha_- + a_{33}\alpha_s,
 \end{eqnarray} (25)
where
   \begin{eqnarray}
(\Gamma_+ - \Gamma_-)a_{11} = \left[(C\Gamma_- -H)k_+^2 +
2\mu k_z^2\right]J_0(j_+) \nonumber\
+ 2\mu k_{+r}J_1(j_+)/r,\qquad
 \end{eqnarray}
   \begin{eqnarray}
(\Gamma_+ - \Gamma_-)a_{12} = - 2\mu k_{-r}^2J_1(j_-)/j_-\qquad\nonumber\
+\left[(H-C\Gamma_+)k_-^2 - 2\mu k_z^2\right]J_0(j_-),
 \end{eqnarray}
   \begin{eqnarray}
a_{13} = 2\mu k_{sr}^2J_2(j_s),
 \end{eqnarray} (26)
   \begin{eqnarray}
(\Gamma_+ - \Gamma_-)a_{21} = (M\Gamma_- - C)k_+^2J_0(j_+),
 \end{eqnarray} (27)
   \begin{eqnarray}
(\Gamma_+ - \Gamma_-)a_{22} = (C-M\Gamma_+)k_-^2J_0(j_-),
 \end{eqnarray} (28)
   \begin{eqnarray}
(\Gamma_+ - \Gamma_-)a_{31} = -2i\mu k_zk_{+r}J_1(j_+),
 \end{eqnarray} (29)
   \begin{eqnarray}
(\Gamma_+ - \Gamma_-)a_{32} = 2i\mu k_zk_{-r}J_1(j_-),
 \end{eqnarray} (30)
   \begin{eqnarray}
ik_z a_{33} = -\mu(k_s^2-2k_z^2)k_{sr}J_1(j_s),
 \end{eqnarray} (31)
and a23=0. There is an implicit factor of $\exp i(k_z z - \omega t)$on the right-hand side of (46)-(49).

Berryman (1983) has shown that a11, a13, a31, and a33 reduce in the limit $\phi \to 0$ to the corresponding results for isotropic elastic cylinders by Pochhammer (1876), Chree (1886, 1889), Love (1941), and Bancroft (1941), as they should.


next up previous print clean
Next: BOUNDARY CONDITIONS Up: Berryman and Pride: Cylinder Previous: EQUATIONS OF POROELASTICITY
Stanford Exploration Project
11/11/2002