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In order to use the unified double-porosity framework of the present paper, it is convenient to have models for the various porous-continuum constituent properties.

For unconsolidated sands and soils, the frame moduli (drained bulk modulus Kd and shear modulus G) are well modeled using the following variant of the Walton (1987) theory [c.f., Pride (2003) for details]
K^d \!\!&=& \frac{1}{6} \left[\frac{4 (1-\phi_o)^2 n_o^2 P_o}{\...
 ...{\left\{1 + [16 P_e/(9 P_o)]^4\right\}^{1/24}},
\\ G &=& 3 K^d/5, \end{eqnarray} (133)
where Pe is the effective overburden pressure [e.g., $P_e = (1-\phi)(\rho_s - \rho_f) g h$ where g is gravity and h is overburden thickness] and where Po is the effective pressure at which all grain-to-grain contacts are established. For Pe < Po, the coordination number n (average number of grain contacts per grain) is increasing as (Pe/Po)1/2. For Pe > Po, the coordination number remains constant n=no. The parameter Po is commonly on the order of 10 MPa. As $P_o\rightarrow 0$, the Walton (1987) result is obtained (all contacts in place starting from Pe = 0). The porosity of the grain pack is $\phi_o$ and the compliance parameter Cs is defined
C_s = \frac{1}{4 \pi} \left(\frac{1}{G_s} + \frac{1}{K_s + G_s/3}\right)\end{displaymath} (135)
where Ks and Gs are the mineral moduli of the grains. For unimodal grain-size distributions and random grain packs, one typically has $0.32 < \phi_o < 0.36$ and 8 < no < 11.

For consolidated sandstones, the frame moduli are modelled in the present paper as [c.f., Pride (2003) for details]
K^d &=& K_s \frac{1-\phi}{1 + c\phi}, \\ G &=& G_s \frac{1-\phi}{1 + 3c\phi /2}. 
 \end{eqnarray} (136)
The consolidation parameter c represents the degree of consolidation between the grains and lies in the approximate range 2 < c < 20 for sandstones. If it is necessary to use a c greater than say 20 or 30, then it is probably better to use the modified-Walton theory.

The undrained moduli Ku and B are conveniently and exactly modeled using the Gassmann (1951) theory whenever the grains are isotropic and composed of a single mineral. The results are
B &=& \frac{1/K^d - 1/K_s}{1/K^d -1/K_s +\phi(1/K_f -1/K_s)},\\ K^u &=& \frac{K^d}{1-B(1-K^d/K_s)},
 \end{eqnarray} (138)
from which the Biot-Willis constant $\alpha$ may be determined to be $\alpha = 1-K^d/K_s$. These Gassmann results are often called the ``fluid-substitution'' relations.

The dynamic permeability $k(\omega)$ as modeled by Johnson et al. (1987) is  
\frac{k(\omega)}{k_o} = \left[\sqrt{1- i \frac{4}{n_J} \frac{\omega}{\omega_c} }
- i \frac{\omega}{\omega_c}\right]^{-1},
 \end{displaymath} (140)
where the relaxation frequency $\omega_c$, which controls the frequency at which viscous-boundary layers first develop, is given by  
\omega_c = \frac{\eta}{\rho_f F k_o}.\end{displaymath} (141)
Here, F is exactly the electrical formation factor when grain-surface electrical conduction is not important and is conveniently (though crudely) modeled using Archie's law $F=\phi^{-m}$. The cementation exponent m is related to the distribution of grain shapes (or pore topology) in the sample and is generally close to 3/2 in clean sands, close to 2 in shaly sands, and close to 1 in rocks having fracture porosity. The parameter nJ is, for convenience, taken to be 8 (cylinder model of the porespace).

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