Hydrate formation only in the pores (Model 2)

Another possible means of hydrate formation might be the solidification of the methane in the pore space only, away from the grain contacts (Figure 2). In this case the hydrate is solidified, but does not become part of the dry rock matrix by cementation. The rock grains are held together only by the effect of the confining pressure. An increase in stiffness would result only at considerable hydrate saturation of the pore space, in contrast to the previously-described cementation case. The effective dry moduli of the hydrate-bearing sandstone at critical porosity $\phi_c = 0.36$ can be described by the Hertz-Mindlin contact theory Mindlin (1949):

 

$$K_{HM} \: = \: {\left( {{n^2 \: {(1- \phi_c)}^2 \: G^2} \over {18 \: \pi \: {(1- \nu)}^2} \: P } \right) }^{1 \over 3}$$ (5)

 

$$G_{HM} \: = \: {{5 \: - \: 4 \: \nu } \over {5 \: (2 - \nu)}... ...} \over {2 \: \pi \: {(1- \nu)}^2} \: P } \right) }^{1 \over 3}$$ (6)

where $\nu$ is the grain Poisson's ratio, G is the grain shear moduli and P is the confining pressure.

 

nocement.pore Figure 2 Configuration of the rock matrix after deposition of the hydrate as pore-only cement.

In order to calculate the effective dry moduli at a different porosity, we used the heuristically modified Hashin-Strikman lower bound proposed by Dvorkin and Nur 1995:
 

$$K_{eff} \: = \: { \left( {{\phi_0 /\phi_c} \over {K_{HM} \: ... ... \over 3} G_{HM} }} \right)}^{-1} \: - \: {4 \over 3} \: G_{HM}$$ (7)

$$\begin{eqnarray} G_{eff} & = \: { \left( \: {{ \phi_0 \:/ \: \phi_c} \: \over {G... ...\: + \: 8 \: G_{HM} } \over {K_{HM} \: + \: 2 \: G_{HM}} \right) }\end{eqnarray}$$ (8)

where K is the grain bulk modulus, and $\phi_0$ is the original porosity of the grains.