Model 1 - Hydrate is part of the fluid

In this first model, the hydrate is treated as part of the pore fluid (Figure 2A). The bulk modulus of the fluid can be calculated as an average of the water and hydrate moduli:

$$K_f \:=\: {[S_w\: / \: K_w \: + \: (1-S_w)\:/\:K_h]}^{-1},$$ (1)

where K_h is the bulk modulus of the hydrate, K_w the bulk modulus of water and S_w the water saturation. In this first model, the hydrate does not affect the moduli of the solid sediment frame. Those solid moduli can be calculated using the modified Hashin-Shtrikman-Hertz-Mindlin theory of Dvorkin and Nur Dvorkin and Nur (1996).

First, the effective moduli are calculated at the critical porosity($\phi_c \: \simeq \: 0.4$) using the Hertz-Mindlin theory Mindlin (1949):

   $$\begin{eqnarray} & K_{HM}& = \: {\left [{{n^2\:(1-\phi_c)^2 \: G^2}\over{18\:\pi... ...\phi_c)^2\:G^2}\over{2\:\pi^2\:(1-\nu)^2}}\:P \right ]}^{1\over3},\end{eqnarray}$$

where K and G are the bulk and shear moduli of the solid phase, in this case the sediment. The Poisson's ratio is given by $\nu$, P is the effective pressure and n is the average number of grain contacts, taken to be 8.5 Murphy (1982). Subsequently, the dry moduli of the solid phase can be calculated for porosities above and below the critical porosity $\phi_c$using a modified Hashin-Strikam upper and lower bound Dvorkin and Nur (1996); Ecker et al. (1996).

Porosity $\phi$ below critical porosity $\phi_c$:

For sediment porosities below the critical porosity, the dry moduli are determined by:

   $$\begin{eqnarray} &K_{dry}& = {\left [ {{\phi/\phi_c}\over{K_{HM}\:+\:{4\over3}\:... ...HM} \: + \: 8 \: G_{HM}}\over{K_{HM}\: + \: 2 \: G_{HM}}} \right);\end{eqnarray}$$

Porosity $\phi$ above critical porosity $\phi_c$:

If the porosity is above the critical porosity of 0.4, the dry moduli can be calculated as follows:

   $$\begin{eqnarray} &K_{dry}& =\:{\left [ {{(1-\phi)/(1-\phi_c)}\over{K_{HM}\:+\:{4... ...}}\: +\:{{(\phi-\phi_c)/(1-\phi_c)}\over{Z}} \right ]}^{-1} - \:Z;\end{eqnarray}$$

Saturated Sediment Properties

After having determined the dry properties of the solid phase and the moduli of the fluid phase, the saturated rock properties can be determined using Gassman's equations:

   $$\begin{eqnarray} &K_{sat}& =\: K\:{{\phi\:K_{dry}\:-\:(1+\phi)\:K_f\:K_{dry}/K\:... ...\:\phi\:K\:-\:K_f\:K_{dry}/K}}, \nonumber \\ &G_{sat}& =\:G_{dry};\end{eqnarray}$$

The elastic velocities v_p and v_s and the bulk density$\rho_B$ are then given by:

   $$\begin{eqnarray} &\rho_B& \:=\: (1-\phi)\: \rho_s \: +\: \phi\:\rho_f,\nonumber ... ...:G_{sat})/\rho_B}, \nonumber \\ &v_s& \:=\: \sqrt{G_{sat}/\rho_B};\end{eqnarray}$$

where $\rho_s$ is the bulk density of the solid phase and $\rho_f$ the density of the pore fluid.