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EQUATIONS OF POROELASTICITY

For long-wavelength disturbances ($\lambda \gt\gt h$, where h is a typical pore size) propagating through such 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 ${\bf w} = \phi({\bf u} - {\bf u}_f)$, where $\phi$ is the porosity. For small strains, the frame dilatation is
   \begin{eqnarray}
e = e_x + e_y + e_z = \nabla\cdot{\bf u},
 \end{eqnarray} (1)
where ex,ey,ez are the Cartesian strain components. Similarly, the average fluid dilatation is
   \begin{eqnarray}
e_f = \nabla\cdot{\bf u}_f
 \end{eqnarray} (2)
(ef 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} (3)
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} (4)
and similarly for $\delta\tau_{yy}$,$\delta\tau_{zz}$,
   \begin{eqnarray}
\delta\tau_{zx} = 
\mu \left(\frac{\partial u_x}{\partial z} + \frac{\partial u_z}{\partial x}\right)
,
 \end{eqnarray} (5)
and again similarly for $\delta\tau_{yz}$,$\delta\tau_{xy}$, and finally
   \begin{eqnarray}
\delta p_f = 
M\zeta - Ce.
 \end{eqnarray} (6)
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 (4)-(6) are of the form
      \begin{eqnarray}
-\omega^2(\rho{\bf u} + \rho_f{\bf w}) = 
(H-\mu)\nabla e + \mu...
 ...ho_f{\bf u} + q{\bf w}) = -M\nabla\zeta + C\nabla e,\qquad\qquad
 \end{eqnarray} (7)
(8)
where $\rho = \phi\rho_f + (1-\phi)\rho_s$, with $\rho_f$ being the fluid density, $\rho_s$ being the solid density, and $\rho$ being the overall bulk-density of the material, while
   \begin{eqnarray}
q(\omega) = \rho_f\left[\alpha/\phi + iF(\xi)\eta/\kappa_0\omega\right]
 \end{eqnarray} (9)
is the frequency dependent effective density of the fluid in relative motion. This expression is also the one that we study later in order to introduce the concepts of dynamic permeability and tortuosity (Johnson et al., 1987). So we want to emphasize now how important this equation for $q(\omega)$ will be to our later analyses. The kinematic viscosity of the liquid is $\eta$;the low frequency permeability of the porous frame is $\kappa_0$;the 3D dynamic viscosity factor [as it was first called by Biot (1956b)] 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. (For a model calculation we discuss later, h = a, the radius of a cylindrical pore.) 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 (1980). We discuss $\alpha$ and its various interpretations in greater detail later in this review.

The coefficients H, C, and M are given (Gassmann, 1951; Biot and Willis, 1957; Biot, 1962) by
   \begin{eqnarray}
H = K + {4\over3}\mu + (K_s-K)^2/(D-K),
 \end{eqnarray} (10)

 
<I>CI> = <I>KI><I>sI>(<I>KI><I>sI>-<I>KI>)/(<I>DI>-<I>KI>),      (11)
and
 
<I>MI> = <I>KI><I>sI>2/(<I>DI>-<I>KI>),      (12)
where
   \begin{eqnarray}
D = K_s[1+\phi(K_s/K_f - 1)],
 \end{eqnarray} (13)
with Kf being the fluid bulk modulus and Ks being the solid bulk modulus. The frame (porous solid without liquid in the pores) constants are K for bulk and $\mu$ for shear. Equations (10)-(13) are correct as long as the porous material may be considered homogeneous on the microscopic scale as well as the macroscopic scale.

Eq. (7) is essentially the equation of elastodynamics of the solid frame with coupling terms (involving ${\bf w}$ and $\zeta$)to the fluid motion. Eq. (8) reduces exactly to Darcy's equation when the solid displacement ${\bf u}$ and frame strain e are zero, since the right hand side of the equation is just $-\nabla p_f$.

To decouple (and subsequently solve) the wave equations in (7) and (8) into Helmholtz equations for the three modes of propagation, 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} (14)
where $\Upsilon$, $\psi$ are scalar potentials and $\mbox{\boldmath$\beta$}$, $\mbox{\boldmath$\chi$}$ are vector potentials. Substituting (14) into (7) and (8), the two equations are solved 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} (15)
and
   \begin{eqnarray}
(\nabla^2 + k_\pm^2)A_\pm = 0.
 \end{eqnarray} (16)
The wavenumbers in (15) and (16) are defined by
   \begin{eqnarray}
k_s^2 = \omega^2(\rho-\rho_f^2/q)/\mu
 \end{eqnarray} (17)
and
   \begin{eqnarray}
k_\pm^2 = {\textstyle {{1}\over{2}}}\left\{b + f \mp
[(b-f)^2 + 4cd]^{1\over2}\right\},
 \end{eqnarray} (18)
   \begin{eqnarray}
b = \omega^2(\rho M - \rho_f C)/\Delta, \,\, c = \omega^2(\rho_...
 ...ho_fH - \rho C)/\Delta, \,\, f = \omega^2(qH - \rho_f C)/\Delta,
 \end{eqnarray}
(19)
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} (20)
With the identification (20), the decoupling is complete.

Equations (15) and (16) are valid for any choice of coordinate system, not just Cartesian coordinates, and they are therefore very useful in all applications of the theory.


next up previous print clean
Next: INDUCED MASS EFFECT Up: Berryman: Dynamic permeability Previous: INTRODUCTION
Stanford Exploration Project
7/8/2003