EQUATIONS OF POROELASTICITY

For long-wavelength disturbances ($\lambda \gt\gt h$, where h is a typical pore size) propagating through a porous medium, we define average values of the (local) displacements in the solid and also in the saturating fluid. The average displacement vector for the solid frame is $\bf u$ while that for the pore fluid is ${\bf u}_f$. The average displacement of the fluid relative to the frame is ${\bf w} = \phi({\bf u} - {\bf u}_f)$. For small strains, the frame dilatation is

$$\begin{eqnarray} e = e_x + e_y + e_z = \nabla\cdot{\bf u}, \end{eqnarray}$$ (185)

where e_x,e_y,e_z are the Cartesian strain components. Similarly, the average fluid dilatation is

$$\begin{eqnarray} e_f = \nabla\cdot{\bf u}_f \end{eqnarray}$$ (186)

(e_f also includes flow terms as well as dilatation) and the increment of fluid content is defined by

$$\begin{eqnarray} \zeta = -\nabla\cdot{\bf w} = \phi(e-e_f). \end{eqnarray}$$ (187)

With these definitions, Biot (1962) obtains the stress-strain relations in the form

$$\begin{eqnarray} \delta\tau_{xx} = He - 2\mu(e_y + e_z) - C\zeta , \end{eqnarray}$$ (188)

and similarly (with permutations) for the other compressional components $\delta\tau_{yy}$,$\delta\tau_{zz}$, while

$$\begin{eqnarray} \delta\tau_{zx} = \mu \left(\frac{\partial u_x}{\partial z} + \frac{\partial u_z}{\partial x}\right), \end{eqnarray}$$ (189)

and again for $\delta\tau_{yz}$,$\delta\tau_{xy}$ for the other shear components. And finally, for the fluid pressure,

$$\begin{eqnarray} \delta p_f = M\zeta - Ce. \end{eqnarray}$$ (190)

The $\delta\tau_{ij}$ are deviations from equilibrium of average Cartesian stresses in the saturated porous material and $\delta p_f$ is similarly the isotropic pressure deviation in the pore fluid.

With time dependence of the form $\exp(-i\omega t)$, the coupled wave equations that incorporate (06)-(12) are of the form

$$\begin{eqnarray} \omega^2(\rho{\bf u} + \rho_f{\bf w}) = C\nabla\zeta - (H-\mu)... ...rho_f{\bf u} + q{\bf w}) = M\nabla\zeta - C\nabla e,\qquad\qquad \end{eqnarray}$$

where $\rho = \phi\rho_f + (1-\phi)\rho_m$ is the bulk-density of the material and $q = \rho_f\left[\alpha/\beta + iF(\xi)\eta/\kappa\omega\right]$ is the effective density of the fluid in relative motion. The kinematic viscosity of the liquid is $\eta$;the permeability of the porous frame is $\kappa$;the dynamic viscosity factor is given, for our choice of sign for the frequency dependence, by $F(\xi) = {\textstyle {{1}\over{4}}} \{\xi T(\xi)/[1+2T(\xi)/i\xi]\}$,where $T(\xi) = {{\hbox{\small ber}'(\xi) - i\hbox{\small bei}'(\xi)} \over{\hbox{\small ber}'(\xi) - i\hbox{\small bei}'(\xi)}}$and $\xi = (\omega h^2/\eta)^{1\over2}$.The functions $\hbox{ber}(\xi)$ and $\hbox{bei}(\xi)$ are the real and imaginary parts of the Kelvin function. The dynamic parameter h is a characteristic length generally associated with and comparable in magnitude to the steady-flow hydraulic radius. The tortuosity $\alpha \ge 1$ is a pure number related to the frame inertia which has been measured by Johnson et al. (1982) and has also been estimated theoretically by Berryman (1980a,b).

The coefficients H, C, and M are given by [see Gassmann (1951), Geertsma (1957), Biot and Willis (1957), Geertsma and Smit (1961), and Stoll (1974)]

$$\begin{eqnarray} H = K + {4\over3}\mu + (K_m-K)^2/(D-K), \end{eqnarray}$$ (191)

 

<I>CI> = <I>KI>_<I>mI>(<I>KI>_<I>mI>-<I>KI>)/(<I>DI>-<I>KI>), (192)

and

 

<I>MI> = <I>KI>_<I>mI>²/(<I>DI>-<I>KI>), (193)

where

$$\begin{eqnarray} D = K_m[1+\phi(K_m/K_f - 1)]. \end{eqnarray}$$ (194)

Equations (19)-(22) are correct as long as the porous material may be considered homogeneous on the microscopic scale as well as the macroscopic scale.

To decouple the wave equations (13) into Helmholtz equations for the three modes of propagation, we note that the displacements $\bf u$ and ${\bf w}$ can be decomposed as

$$\begin{eqnarray} {\bf u} = \nabla\Upsilon + \nabla\times\mbox{\boldmath$\beta$},\qquad {\bf w} = \nabla\psi + \nabla\times\mbox{\boldmath$\chi$}, \end{eqnarray}$$ (195)

where $\Upsilon$, $\psi$ are scalar potentials and $\mbox{\boldmath$\beta$}$, $\mbox{\boldmath$\chi$}$ are vector potentials. Substituting (23) into (13), we find (13) is satisfied if two pairs of equations are satisfied:

$$\begin{eqnarray} (\nabla^2 + k_s^2)\mbox{\boldmath$\beta$}= 0,\qquad \mbox{\boldmath$\chi$}= -\rho_f\mbox{\boldmath$\beta$}/q \end{eqnarray}$$ (196)

and

$$\begin{eqnarray} (\nabla^2 + k_\pm^2)A_\pm = 0. \end{eqnarray}$$ (197)

The wavenumbers in (24) and (25) are defined by

$$\begin{eqnarray} k_s^2 = \omega^2(\rho-\rho_f^2/q)/\mu \end{eqnarray}$$ (198)

and

$$\begin{eqnarray} k_\pm^2 = {\textstyle {{1}\over{2}}}\left\{b + f \mp [(b-f)^2 + 4cd]^{1\over2}\right\}, \end{eqnarray}$$ (199)

$$\begin{eqnarray} b = \omega^2(\rho M - \rho_f C)/\Delta, \,\, c = \omega^2(\rho_... ...ho C)/\Delta, \,\, f = \omega^2(qH - \rho_f C)/\Delta,\nonumber\ \end{eqnarray}$$

with $\Delta = MH - C^2$.The linear combination of scalar potentials has been chosen to be $A_\pm = \Gamma_\pm\Upsilon + \psi$,where

$$\begin{eqnarray} \Gamma_{\pm} = d/(k_\pm^2 - b) = (k_\pm^2 - f)/c. \end{eqnarray}$$ (200)

With this identification (31) of the coefficients $\Gamma_{\pm}$, the decoupling is complete.

Equations (24) and (25) are valid for any choice of coordinate system. They may be applied to boundary value problems with arbitrary symmetry. Biot's theory will therefore be applied to porous elastic cylinders in the next section.