Equations of Biot's Single-Porosity Poroelasticity

For long-wavelength disturbances ($\lambda \gt\gt h$, where h is a typical pore size) propagating through a single-porosity 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 e, while the increment of fluid content is defined by

$$\begin{eqnarray} \zeta = -\nabla\cdot{\bf w} = \phi(e-e_f). \end{eqnarray}$$ (1)

With time dependence of the form $\exp(-i\omega t)$, the coupled wave equations that follow in the presence of dissipation are

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

where $\mu_d$ is the drained shear modulus, H, C, and M are bulk moduli,

$$\begin{eqnarray} \rho = \phi\rho_f + (1-\phi)\rho_m, \end{eqnarray}$$ (3)

and

$$\begin{eqnarray} q = \rho_f\left[\alpha/\phi + iF(\xi)\eta/\kappa\omega\right]. \end{eqnarray}$$ (4)

The kinematic viscosity of the liquid is $\eta$;the permeability of the porous frame is $\kappa$;the dynamic viscosity factor is given approximately [or see Johnson et al. (1987) for more discussion], for our choice of sign for the frequency dependence, by

$$\begin{eqnarray} F(\xi) = {\textstyle {{1}\over{4}}} \{\xi T(\xi)/[1+2T(\xi)/i\xi]\}, \end{eqnarray}$$ (5)

where

$$\begin{eqnarray} T(\xi) = {{\hbox{\small ber}'(\xi) - i\hbox{\small bei}'(\xi)} \over{\hbox{\small ber}'(\xi) - i\hbox{\small bei}'(\xi)}} \end{eqnarray}$$ (6)

and

$$\begin{eqnarray} \xi \equiv (\omega/\omega_0)^{1\over2} = (\omega\alpha\kappa/\eta\phi)^{1\over2} = (\omega h^2/\eta)^{1\over2}. \end{eqnarray}$$ (7)

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 (Johnson et al., 1982) and has also been estimated theoretically (Berryman, 1980a; 1983a).

The coefficients H, C, and M are given by (Gassmann, 1951; Geertsma, 1957; Biot and Willis, 1957; Geertsma and Schmidt, 1961; Stoll, 1974)

$$\begin{eqnarray} H = K_d + {4\over3}\mu_d + (1-K_d/K_m)^2M, \end{eqnarray}$$ (8)

 

<I>CI> = (1-<I>KI>_<I>dI>/<I>KI>_<I>mI>)<I>MI>, (9)

where

$$\begin{eqnarray} M = 1/[(1-\phi-K_d/K_m)/K_m + \phi/K_f]. \end{eqnarray}$$ (10)

The constants are drained bulk and shear moduli K_d and $\mu_d$,mineral bulk modulus K_m, and fluid  bulk modulus K_f. Korringa (1981) showed equations (8)-(10) to be correct as long as the porous material may be considered homogeneous on the microscopic scale as well as the macroscopic scale. Also, see a recent tutorial on Gassmann's equations (Gassmann, 1951) by Berryman (1999).

To decouple the wave equations (2) into Helmholtz equations for the three modes of propagation, we note that the displacements ${\bf u}$ and ${\bf w}$ can be decomposed as

$$\begin{eqnarray} {\bf u} = \nabla\Upsilon + \nabla\times\vec{\beta},\qquad {\bf w} = \nabla\psi + \nabla\times\vec{\chi}, \end{eqnarray}$$ (11)

where $\Upsilon$, $\psi$ are scalar potentials and $\vec{\beta}$, $\vec{\chi}$ are vector potentials. Substituting (11) into (2), we find (2) is satisfied if two pairs of equations are satisfied:

$$\begin{eqnarray} (\nabla^2 + k_s^2)\vec{\beta} = 0,\qquad \vec{\chi} = -\rho_f\vec{\beta}/q \end{eqnarray}$$ (12)

and

$$\begin{eqnarray} (\nabla^2 + k_\pm^2)A_\pm = 0. \end{eqnarray}$$ (13)

The wavenumbers in (12) and (13) are defined by

$$\begin{eqnarray} k_s^2 = \omega^2(\rho-\rho_f^2/q)/\delta\mu \end{eqnarray}$$ (14)

and

$$\begin{eqnarray} k_\pm^2 = {\textstyle {{1}\over{2}}}\left[b + f \mp \left[(b-f)^2 + 4cd \right]^{1\over2}\right], \end{eqnarray}$$ (15)

$$\begin{eqnarray} b &=& \omega^2(\rho M - \rho_f C)/\Delta,\qquad c = \omega^2(\r... ..._fH - \rho C)/\Delta, \qquad f = \omega^2(qH - \rho_f C)/\Delta, \end{eqnarray}$$ (16)

with

$$\begin{eqnarray} \Delta = HM - C^2. \end{eqnarray}$$ (17)

The linear combination of scalar potentials has been chosen to be

$$\begin{eqnarray} A_\pm = \Gamma_\pm\Upsilon + \psi, \end{eqnarray}$$ (18)

where

$$\begin{eqnarray} \Gamma_{\pm} = d/(k_\pm^2 - b) = (k_\pm^2 - f)/c. \end{eqnarray}$$ (19)

With the identification (19), the decoupling is complete.

  

Figure 1: Thin layering of isotropic materials produces an effective transversely isotropic medium at low frequencies of propagation. Overall permeability $\kappa_{eff}$ normal to the layering depends most strongly on the most impermeable layers since $1/\kappa_{eff} = \int_0^L \kappa^{-1}(z) dz/L$, being the harmonic mean. In contrast, the seismic attenuation (in the usual band from 1-100 Hz) ordinarily depends most strongly on the ones that are most permeable, since $1/Q(z) \propto \kappa(z)$. The character of this relationship between attenuation and permeability changes significantly at higher frequencies as described in the text.