Saturated elastic moduli for Model 1 and Model 2

The effective dry moduli of the different hydrate frameworks have to be related to the effective moduli of the same rock-containing fluid. The effective moduli of a saturated rock can be calculated at seismic frequencies by means of Gassmann's formulas:
 

$${K_{sat} \over {K_{solid} \: - \: K_{sat}}} \: = \: {K_{dry}... ...}}} + {K_{fluid} \over {\phi \: (K_{solid} \: - \: K_{fluid})}}$$ (9)

 

$$G_{sat} \: = \: G_{dry}$$ (10)

where K_sat and G_sat are the effective bulk and shear moduli of the saturated rock; K_dry and G_dry the moduli of the dry rock; K_fluid and G_fluid the moduli of the fluid; and K_solid the bulk modulus of the mineral material making up the rock. $\phi$ is the porosity of the saturated rock.

The effective moduli of the mineral K_solid can be calculated using the Voigt-Reuss-Hill average, which is an arithmetic average of the Voigt upper bound (K_V) and the Reuss lower bound (K_R):  

$$K_{solid} \: = \: {{K_V \: + \: K_R } \over 2}$$ (11)

with  

$$K_V \: = \: f \: K \: + \: f_h \: K_h$$ (12)

 

$${1 \over K_R} \: = \: {f \over K} + {f_h \over K_h}$$ (13)

where f and f_h are the volume fractions of the grains and the hydrate, respectively. K is the grain bulk modulus and K_h is the hydrate bulk modulus.

In order to calculate the effective fluid bulk modulus K_fluid for partially-saturated rocks, we used the following equation:
 

$${1 \over K_{fluid}} \: = \: \sum_{i=1}^{N} {f_i \over K_{fluid_{i}}}$$ (14)

where f_i is the volume fraction of the fluid, N is the number of fluids, and K_{fluid<<188>>i} is the bulk modulus of the fluid.

After having derived the effective moduli of the saturated rock, we used the following relationships to find the seismic velocities in saturated rock:
 

$$v_{p_{sat}} \: = \: \sqrt{{{K_{sat} \: + \: {4 \over 3} \: G_{sat}} \over \rho_{sat}}}$$ (15)

 

$$v_{s_{sat}} \: = \: \sqrt{ {G_{sat} \over \rho_{sat}}}$$ (16)

where $\rho_{sat}$ is the density of the saturated rock,  

$$\rho_{sat} \: = \: (1 - \phi) \: \rho_{solid} \: + \: \phi \: \rho_{fluid}$$ (17)

$\rho_{solid}$ is the density of the solid phase, and $\rho_{fluid}$ is the density of the fluid phase which can be obtained from:  

$$\rho_{fluid} \: = \: \sum_{i=1}^{N} {f_i \rho_{fluid_i}}$$ (18)