2.3 Elastic Modulus Model
High-porosity marine sediment can be considered as a granular system where the elastic wave velocities are linked to porosity；effective pressure；mineralogy；elastic properties of the pore-filling material；and water，gas，and gas-hydrate saturation of the pore space［78］.To estimate the effect of gas hydrate on the sediment's elastic moduli，two models are used［5］.In model A，gas hydrate is assumed as part of the pore fluid and affects its compressibility.In model B，gas hydrate is assumed as part of the solid frame and reduces the porosity and alter the elastic properties of the mineral phase，but does not significantly stiffen the frame.
The bulk and shear elastic moduli of the dry frame of sediment without gas hydrate can be computed from the Hashin-Shtrikman-Hertz-Mindlin theory［7］：
whereφis porosity；φcis the porosity of a pack of dense random spheres（about 40%）；K and G are the bulk and shear moduli of the mineral phase，respectively.They can be computed from the moduli of m mineral constituents using Hill's average：
where fi is the volumetric fraction of the ith component in the solid phase.KHM and GHM are calculated from the Hertz-Mindlin theory［7］as
where v is the Poisson's ratio of the mineral phase calculated from K and G；n is the average number of contacts per grain taken as 8.5；P is the effective pressure
whereρs andρf are the densities of the solid and fluid phase，respectively；g is gravity acceleration；and h is depth below seafloor.
The saturated sediment bulk and shear moduli are calculated from the Gassmann equation：
where Kf is the bulk modulus of the pore fluid which，at saturation S，is the isostress average of those of water Kw and gas Kg：
In model A of sediments with gas hydrate the hydrate is part of pore fluid.In this case，the bulk modulus of the fluid is the isostress average of those of water and hydrate Kh：(https://www.daowen.com)
In model B of sediments with gas hydrate the hydrate is part of the dry sediment frame.In this case，gas hydrate reduces the porosity and change the elastic properties of the solid phase.The reduced porosity
where Sh is gas-hydrate saturation，and the elastic moduli of the changed solid phase are calculated using Hill's average as
where Ks and Gs are the solid bulk and shear moduli of the original sediment's solid phase，respectively，and Kh and Gh are those of pure gas hydrate.Those altered porosity and elastic moduli are used as input to the above model for the moduli of the dry frame.
Finally，the elastic wave velocities are calculated from the elastic moduli and bulk densityρB as
Elastic moduli in table 3are used to calculate the relationships between the velocity of gas hydrate-bearing sediments and the concentrations of gas hydrate，and the calculated results are shown in table 4and Fig.1.Elastic parameters of gas hydrate-bearing sediments and free gasbearing sediments calculated from elastic modulus model are shown in Fig.1.P wave velocities and S wave velocities of gas hydrate-bearing sediments of model A and model B and free gasbearing sediments with different concentrations are presented in Fig.1（a）.P wave velocities of both gas hydrate models increase with the concentration，but velocity of model B increases faster than that of model A.S wave velocity of model B increases with the concentration too，but S wave velocity of model A does not vary with different concentrations.P wave velocity of free gas-bearing sediments decreases with the concentration especially in the concentration range of 0%～10%.S wave velocity increases slightly with the concentration.Poisson's ratio of free gas-bearing sediments decreases with the concentration〔Fig.1（b）〕.There is difference of the relationships between Poisson's ratio and the concentration for two gas hydrate models.Poisson's ratio of model A increases linearly with concentration.While Poisson's ratio of model B decreases linearly with concentration slowly in the range of low porosity and rapidly in the range of high porosity.
Table 3 Calculation elastic modulus parameters of sediments
Table 4 Elastic parameters based on the elastic modulus model
Fig.1 Relationships between elastic properties and gas hydrate/free gas concentration（a）Velocity①P wave velocity of gas hydrate model A，②P wave velocity of gas hydrate model B，③P wave velocity of free gas-bearing sediment④S wave velocity of gas hydrate model A，⑤S wave velocity of gas hydrate model B，⑥S wave velocity of free gas-bearing sediment.（b）Poisson's ratio of①gas hydrate model A，②gas hydrate model B，③free gas-bearing sediment.
