Hydrate becomes Part of the Solid

In this case, I assume that hydrate becomes part of the solid frame. This has two effects: porosity reduction and change of the solid bulk and shear moduli, thus slightly stiffening the sediment frame. The reduced porosity $\phi_{\rm r}$ can be calculated from the actual sediment porosity without gas hydrates:  

$$\phi_{\rm r} \:=\: \phi\:(1-\rm S_{\rm h}),$$ (17)

where $\rm S_{\rm h}$ 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:

$$\begin{eqnarray} &\rm K& =\:{1\over2}(\rm f_{\rm h}\:\rm K_{\rm h}\:+\:(1-\rm f_... ...{\rm h}/\rm G_{\rm h}\:+\:(1- \rm f_{\rm h})/\rm G_{\rm s}]^{-1});\end{eqnarray}$$ (18)

where $\rm K_{\rm s}$ and $\rm G_{\rm s}$ are the solid bulk and shear moduli of the sediment without hydrate (as calculated from equation ) and $\rm f_{\rm h}$ is the volume fraction of hydrate in the solid phase. It can be calculated as follows:

$$\rm f_{\rm h}\:=\:{{\phi\:\rm S_{\rm h}}\over{1-\phi\:(1- \rm S_{\rm h})}}$$ (19)

The dry and saturated moduli can then be determined using equations , , , and .