4.2. Geotechnical Model
The coefficient of volume compressibility ( is defined as follows:
The coefficient is defined as the compression of a soil layer per unit of original thickness per unit increase in effective stress on the load.
Chen et al. [63] researched the effective thermal conductivity of hydrate-bearing sediments with initial water saturation ranging from 20 to 60% and hydrate saturation ranging from 7.43 to 48.74%. In their study, the effective thermal conductivity was measured in the steady state using a high-pressure experimental setup, and the results demonstrated an increase with increasing hydrate saturation, attaining a maximum value in the saturation range of 25–36%. Therefore, we use mean values for water and hydrate saturations of 40% and 28% respectively. We calculate free gas saturation as follows:
Using the effective medium theory, the effective Young modulus is calculated as follows [64]:
Using value of bulk moduli as 7.9 GPa, 2.17 GPa, and 0.1 GPa for hydrate, formation water, and gas, respectively [62], the effective bulk modulus is calculated based on Equation (18) as 0.24 GPa.
The pore volume fraction ( is given as follows [65]:
The value of porosity calculated in Equation (19) falls within those reported by Wang et al. [62].
We use a medium value for Biot’s pore pressure coefficient of 0.7 [66]. The value of Skempton’s pore pressure coefficient ( was calculated using the following equation [67]:
In Equation (21), is Biot’s pore pressure coefficient, is the effective bulk modulus [Pa], is the fluid modulus [Pa], and is the frame modulus [Pa].
Following the research work of Qadrouh et al. [68], the elastic modulus of the solid skeleton of the gas hydrate-bearing sediments is calculated using the Krief model (Krief et al.) [69], where the bulk modulus Kdry and shear modulus Gdry of the solid skeleton are expressed as follows:
In Equation (23), is the frame modulus () in Equation (21) [Pa], is the solid modulus [Pa], and is an empirical constant set to 3 [70].
Based on Equation (23) we calculate the solid modulus using effective medium theory as follows [70]:
where m is the number of solid components, fi is the volume fraction of the i-th component, and Ki is the bulk modulus of the i-th component.
In this paper, we assume a gas hydrate-bearing marine sediment to consist solid-wise of clay and quartz, with bulk moduli given as 20.9 GPa and 36 GPa, [70] respectively, which are non-variable solid components. Assuming the volume contents of quartz and clay to be 0.8 and 0.2, respectively [70], is calculated as 32 GPa. Therefore, from Equation (21), and using a porosity value of 0.6 as calculated earlier, = 16 GPa. Based on the procedure, we calculate Stefan’s pore pressure coefficient as 0.91.
In this paper, we consider gas hydrate sediments in the Bay of Fundy, Canada. Fern [71] has provided the stratigraphy of marine sediments in the Bay of Fundy (see Figure 8). In this regard, we assume glacimarine sediments with an average thickness of 30 m underlying a water depth of 150 m to host gas hydrates. For simplicity, we assume the total hydrostatic pressure at the center of the sediment to consist of 2 contributions, namely that due to the water column and that due to the weight of the sediment above the center of the sediment. We assume the submerged unit weight of the glaciomarine silt to be 11.39 kNm−3. The total hydrostatic pressure in this regard is 3.22 MPa.
The compresssibility of water associated with natural gas systems within the pressure range of 100 MPa is constant and close to 4.3 MPa−1 [72]. Therefore, for the systems under consideration in this paper, the pressure is 4.3 × 10−10 Pa−1.
In geophysical literature, the normalized permeability describes the ratio of the permeability of hydrate-bearing sediments to that of hydrate-free sediments. Using the bundle of parallel capillary tubes model, the permeability of the hydrate-bearing sediments is calculated as follows [73]:
in which is the normalized permeability.
Substituting the hydrate saturation calculated earlier (0.4) into Equation (25) yields a normalized permeability of 0.107.
We assume the permeability of silty sand to be 10−10 ms−1 (Stranne et al.) [74], which is equivalent to 1.39 × 10−17 m2.
Thus, using Equations (24) and (25), the permeability of gas hydrate-bearing sediments can be calculated. The approach sets the intrinsic permeability to .
Using the values of Biot’s pore pressure coefficient, the effective bulk modulus, and Skempton’s pore pressure coefficients determined earlier, Equation (4) can be used to calculate the epsilon parameter. Thus, assuming the viscosity of seawater at an average salinity of 35 g/kg, the viscosity of water is assumed to be 0.0014 Pa·s. The value of the parameter in Equation (4) is calculated using these data to be 1.47 × 10−6 m2s−1.
In the calculation of the epsilon parameter, the quantity has a negative value, which makes the value meaningless, given that it has a unit of diffusivity. Therefore, the value of the product of Biot’s pore pressure coefficient and Skempton’s pore pressure coefficient multiplied by 2 was reduced by multiplying by the porosity to arrive at a meaningful value. This approach was adopted because in the classical radial diffusion equation, the denominator of the diffusivity parameter contains a porosity multiplier [75].
The Young’s modulus E50 is defined as the Young’s modulus at 50% of the maximum deviator stress. Such a Young’s modulus increases remarkably with the increase in the hydrate saturation, which is governed by the effective confining pressure. To determine the Young’s modulus, we assume an overburden pressure equal to the pressure due to the column of seawater over the sediments (see Figure 8). Assuming the density of seawater to be 1030 kgm−3, the pressure is 1.5 MPa. Using the Poisson [76] effect for calculating lateral stress from axial stress, the confining stress is 0.5 MPa. The calculated value enables us to determine approximately the required Young’s modulus, using Appendix A [77]. The value is deduced approximately using the plot for 0.78 MPa and assuming the hydrate concentration of 0.4 to be 120 MPa.
For most solids, the Poisson ratio is 0.25 [78]. Using Equation (16) and the value for the compressibility of water in the presence of methane gas, the value of the coefficient of volume compressibility is 6.94 × 10−9 Pa−1.
4.3. Hydrological Parameters
Based on the research findings of Li et al. [79], the effect of waves generally decreases from southwest to northeast in the Bay of Fundy, and the mean significant wave height is the greatest in the outer bay and the Gulf of Maine (1–1.6 m), decreasing to 1.0–0.5 m in the mid-bay, and being reduced further to less than 0.5 m in the upper bay. Wave periods reach 6 s in the outer bay, decreasing to 4–5 s in the mid-bay and to less than 4 s in the upper bay. In another study by Amos and Asprey [80], a mean wavelength of approximately 0.3 m and a mean period of 4 s were reported. Consequently, based on the findings of the cited studies, a mean wave period of 2.50 s was used.
Using the outlined methodology, the beta parameter defined by Equation (17) is calculated to be 1.01. Using the mean wave period of 2.5 s, the and parameters in Equations (14) and (15) assume the following forms:
Equation (9) can be used to calculate the maximum pressure (P0) due to the amplitude of the wave, using the density of seawater (1030 kgm−3) and a wave amplitude of 1.6 m [79]. The calculation yields a value equal to 16,150.40 Pa.
Equation (17) permits the calculation of the beta parameter with the variable compressibility of water to determine the effect of methane gas hydrate on the oscillatory excess pore water pressure defined by Equations (14) and (15).
The stiffness of uncemented sediments is determined by the effective stress regime [81], and sediments with low stiffness will experience considerable deformation, resulting in low excess pore pressure generation in the hydromechanical coupling sense. Xu [82] published experimental data on the effect of the gas hydrate volume fraction of dissociation on excess pore pressure generation as a function of effective stress (see Appendix B). In this paper, we use Appendix B to determine the effect of initial excess pore pressure C in Equation (14) on oscillatory excess pore water pressure. Accordingly, we calculate the effective stress for our system with reference to the total stress at the midpoint of the sediment of 30 m thickness (see Figure 6). In this regard, the total stress consists of the surcharge due to a water column of 150 m and the weight of sediment above the midpoint. Effective stress is then calculated as the difference between total stress and pore pressure. The approach provides effective stress as 1.533 MPa based on a submerged unit of sediment equal to 11.39 kNm−3. Accordingly, the initial excess pore pressure is 17 MPa based on a 0.001 volume fraction of dissociated gas hydrate. The following sections sum up the results of the application of the detailed methodology.
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