Model 2 - Hydrate is part of the solid

In this case, we assume that hydrate becomes part of the solid sediment frame. This has two effects: porosity reduction and change of the elastic frame moduli. The reduced porosity $\phi_r$ can be expressed as:  

$$\phi_r \:=\: \phi\:(1-S_h),$$ (7)

where S_h is the hydrate saturation of the pore space. The bulk and shear moduli of the solid phase are now a mixture of the sediment solid and the hydrate and can be calculated from the Hill average:

$$\begin{eqnarray} &K& =\:{1\over2}(f_h\:K_h\:+\:(1-f_h)\:K_s\:+\:[f_h/K_h\:+\:(1-... ...er2}(f_h\:G_h\:+\:(1-f_h)\:G_s\:+\:[f_h/G_h\:+\:(1-f_h)/G_s]^{-1};\end{eqnarray}$$

where K_s and G_s are the bulk and shear moduli of the sediment without hydrate and f_h is the volume fraction of hydrate in the solid phase. It can be calculated as follows:

$$f_h\:=\:{{\phi\:S_h}\over{1-\phi\:(1-S_h)}}$$ (9)

The dry and saturated moduli can then be determined using equations 3, 4, 5 and 6.