APPENDIX A. CONSTITUENT PROPERTIES

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 K^d and shear modulus G) are well modeled using the following variant of the Walton (1987) theory [c.f., Pride (2003) for details]

$$\begin{eqnarray} K^d \!\!&=& \frac{1}{6} \left[\frac{4 (1-\phi_o)^2 n_o^2 P_o}{\... ...{\left\{1 + [16 P_e/(9 P_o)]^4\right\}^{1/24}}, \\ G &=& 3 K^d/5, \end{eqnarray}$$ (133) (134)

where P_e 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 P_o is the effective pressure at which all grain-to-grain contacts are established. For P_e < P_o, the coordination number n (average number of grain contacts per grain) is increasing as (P_e/P_o)^1/2. For P_e > P_o, the coordination number remains constant n=n_o. The parameter P_o 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 P_e = 0). The porosity of the grain pack is $\phi_o$ and the compliance parameter C_s is defined

$$C_s = \frac{1}{4 \pi} \left(\frac{1}{G_s} + \frac{1}{K_s + G_s/3}\right)$$ (135)

where K_s and G_s 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 < n_o < 11.

For consolidated sandstones, the frame moduli are modelled in the present paper as [c.f., Pride (2003) for details]

$$\begin{eqnarray} K^d &=& K_s \frac{1-\phi}{1 + c\phi}, \\ G &=& G_s \frac{1-\phi}{1 + 3c\phi /2}. \end{eqnarray}$$ (136) (137)

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 K^u 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

$$\begin{eqnarray} B &=& \frac{1/K^d - 1/K_s}{1/K^d -1/K_s +\phi(1/K_f -1/K_s)},\\ K^u &=& \frac{K^d}{1-B(1-K^d/K_s)}, \end{eqnarray}$$ (138) (139)

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  

$$\frac{k(\omega)}{k_o} = \left[\sqrt{1- i \frac{4}{n_J} \frac{\omega}{\omega_c} } - i \frac{\omega}{\omega_c}\right]^{-1},$$ (140)

where the relaxation frequency $\omega_c$, which controls the frequency at which viscous-boundary layers first develop, is given by  

$$\omega_c = \frac{\eta}{\rho_f F k_o}.$$ (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 n_J is, for convenience, taken to be 8 (cylinder model of the porespace).