The reservoir simulator CMG STARS^TM was used for numerical simulations on gas production and consolidation based on MH dissociation and elasticity function of MH saturation. In the simulations, MH was defined as solid phase that saturation in MH reservoir porosity reduces absolute permeability with power function [3] . According to Singh et al. (2008) [8] , the MH dissociation rate was calculated by the MH formation-decomposi- tion equilibrium curve [pressure; P(MPa), temperature: T(˚C)] of for phase transition from solid to fluid phases in sea water (3% NaCl) was given by the following equation;

Figure 2 shows the MH equilibrium curve for salt water with comparison to the experimental results presented by Sloan (1998) [9] . The equilibrium temperature of the water is almost 1.0˚C is lower than that of sea water. Thus, the equilibrium equation line for water, which is shown in Figure 2, is derived by setting 105.25 instead of the value of 106.25 in Equation (1). Other thermal quantity and MH decomposition heat were given based on compositions (solid, gas and water) in MH reservoir blocks.

The increase of MH saturation (solid phase) results in sharply decrease of relative permeability due to decrease of apparent porosity in the MH reservoir. Conversely, apparent porosity and permeability increase rapidly by MH dissociation. In addition, porosity depended on congenital compressibility of MH reservoir is decreased by increase of effective stress with depressurization. In this numerical simulation, effective porosity, which depends on MH saturation, compressibility and depressurization, is defined by following Equations (2) and (3).

where

ϕ_v: Porosity [-]

ϕ_i: Initial porosity [-]

κ: Compressibility [MPa⁻¹]

p: Reservoir pressure [MPa]

p_i: Initial reservoir pressure [MPa]

φ_e: Effective porosity [-]

S_MH: MH saturation [-]

The initial permeability of MH reservoir is remarkably low at the initial condition which has high MH saturation over 50%, however the apparent permeability of MH reservoir is improved rapidly with MH dissociation [3] . On the other hand, porosity of MH reservoir is decreased by the consolidation depended on depressurization, therefore absolute permeability of MH reservoir is decreased simultaneously. In this research, following Equation (4) based on Kozeny-Carman equation which represents the relationship of permeability-porosity.

where

k: Apparent permeability [m²]

k_ab: Absolute permeability [m²]

N: Permeability reduction index (=6) [-]

In present research, the experimental results by [10] were referred for constructing MH dissociation and consolidation models to check a model of permeability-porosity in MH reservoir during depressurization process. Their laboratory experiments were done to study MH dissociation, consolidation, temperature and permeability characteristic in a sand-pack including synthetic MH (hereinafter call as “MH core”) as shown in Table 1. The temperature distribution was measured by thermocouples set with 5 cm intervals in radial and axial directions in the MH core.

The relative permeabilities were presented by Sakamoto et al. (2010) [11] for the MH core by following Equations (5) and (6).

where

k_rg: Gas relative permeability [-]

k_rw: Water relative permeability [-]

c: End point for gas relative permeability [-]

b: End point for water relative permeability [-]

: Normalized water saturation [-]

m: Index of gas relative permeability k_rg [-]

n: Index of water relative permeability k_rw [-]

The numerical block models in cylindrical coordinate were made for the laboratory experiments. Outer radius and height of the stainless cell are 89.4 mm and 144 mm (see Figure 3). The total number of cylindrical-grid blocks was 1612 with 31 in z-direction and 52 in r direction. The stainless section was set as a constant temperature and closing state. The initial conditions of the numerical model are listed in Table 1. The equilibrium curve for MH formation and dissociation was given by the same equation presented by Sasaki et al. (2014) [6] . The history matching simulations on the experimental results were carried out by using the STARS.

Equations (4) and (5) presented by [11] were used for the relative permeabilities in the numerical history matching. In the case of high water saturation in the MH core, water is produced selectively and gas is remained

in pores by capillary pressure. The values of m and n were set as 10 and 3 in the Equations (5) and (6) respectively in order to represent that water has higher mobility than gas in the region of high water saturation.

As shown in Figure 4, Masui et al. (2005) [12] and Miyazaki et al. (2005) [13] have presented the relationship between MH saturation and elastic modulus, E by the tri-axial compression test using the MH cores. It means that E value is increased linearly with increasing MH saturation in MH cores. In this research, it was assumed that elastic modulus and compressibility can be given by the following equation.

; (7)

where

κ: Compressibility [MPa⁻¹]

ν: Poisson’s ratio [-]

E: Elastic modulus of MH reservoir () [MPa]

E₀: Elastic modulus of sands (Rock matrix) () [MPa]

S_MH: MH saturation [-]

β: Increase rate of elastic modulus vs. [MPa]

Seabed subsidence is induced by decrease of porosity of sediment layers and the MH reservoir consolidation. In this research, the amount of seabed subsidence was evaluated as summation of vertical compaction in a grid at each distance from the seabed to MH reservoir given by

; (8)

where

Δh: Displacement of seabed [m]

z: distance from seabed [m]

α: Subsidence ratio [-]

ζ: Distance index of subsidence [1/m]

The subsidence ratio (α), that has the value from 0 to 1, is contributing ratio of each grid’s displacement at z on the seabed subsidence at z = 0 based on Aoki et al. (1991) [14] and Nishida et al. (1981) [15] . In the case of the laboratory experiments on sand pack consolidation discussed in the next section α was set as α = 1.0, because z is very short comparing to a field.

Authors have done the comparisons of cumulative methane gas and water productions between the laboratory experiments and present numerical simulation (see Figure 5). In the both results, water is produced rapidly in early stage after depressurizing, and gas is produced after water. However, it is clear that there are differences between numerical and experimental results. The reason of the difference was made from the models on gas and water relative permeabilities presented by Equations (5) and (6). Sakamoto et al. [11] also carried out the numerical history matching for the laboratory experiment, and they showed better matching results than present results. However, we have succeeded to get better matching results by considering effects of consolidation in the MH core as described.

Figure 6(a) shows the comparison of displacements at the point of 4.75 cm from center in the MH core. The numerical result on the displacement at the end of experiments showed almost same value with the experiment, however, displacement shows different behavior at the early stage of depressurization. It was assumed that MH has an effect to make larger elastic modulus (Young’s modulus) with cohesive strength by connecting unconsolidated sand grains.

The displacement behavior was recalculated by using modified compressibility considering cohesive strength of MH with varying initial elastic modulus of rock matrix E₀ = 100 to 200 MPa, Poisson’s ratio ν = 0.2 to 0.6

and increase rate of elastic modulus β = 600 to 1000 MPa based on experimental results presented by Masui et al. (2005) [12] and Miyazaki et al. (2005) [13] . When the values, that are E₀ = 200 MPa, ν = 0.217 and β = 700 MPa, were used, the numerical simulation result showed the best matching with the experiments by [13] . Figure 6(a) shows the result of history matching of displacement behavior using modified elastic modulus and compressibility that is the function of MH defined by Equation (7). As shown in Figure 6(a), the final cumulative displacement shows same value when MH dissociation finished. However, it is clear that matching degree of the numerical simulation with Equation (7), E = E₀ + βS_MH, is better than that with E = E₀ = constant.

The comparison of temperature distributions between the laboratory experiment and present numerical simulation is shown in Figure 6(b). Temperature in the MH core was decreased from 10.7˚C to 2˚C by endothermic reaction during MH dissociation. The temperature of 2˚C is equal to equilibrium temperature for pressure of 3.3 MPa. After finishing MH dissociation, temperature of the MH core was increased from 2˚C to 10.7˚C by heat supply from the stainless cell. The numerical results also showed almost the same temperature distribution measured by the experiments.

As shown in Figure 7, the cumulative gas production for the initial MH core temperature of 2˚C was approximately 18% of that of 10.7˚C in the present numerical simulation. In the case of 2˚C, MH dissociation was suppressed due to small heat supply by small temperature difference between initial core temperature and MH equilibrium temperature after depressurization.

It was confirmed that present numerical simulation model can be used for simulating MH dissociation process and consolidation behavior in laboratory scale based on the history matching results presented in [10] .

Comparative studies on numerical MH model were done by [3] to investigate the validity of present numerical simulation model for gas production from field scale of MH sediment layers by depressurization method. Numerical simulations were carried out by using MH21-HYDRES (Kurihara et al., 2011) [16] developed by MH21 reserch consortium.

In the comparative studies, typical properties of MH reservoir at Eastern Nankai Trough given in Table 2 were used, since these data were presented by [3] for their numerical simulations on gas production by depressurization method. Figure 8 shows cylindrical coordinate (r, z) of MH reservoir model in field scale. The reservoir consists of 44 units of mud, silt and sand layers, that structure assumes a turbidite sediment layer. The

pressure difference 10.2 MPa between initial reservoir pressure 13.2 MPa and bottom-hole pressure 3.0 MPa was applied for depressurization in the MH reservoir.

The numerical simulations in considering field-scale sand consolidation were carried out by the considering model presented in the previous section based on matching with the laboratory experiments. Hereinafter simulations without consolidation or porosity change by setting E₀ = ∞ are called as “base case” that was compared with ones considering consolidation in MH reservoir where E₀ = 200, 140 and 80 MPa were given for sand layer, silt layer and mud layer, respectively, while ν = 0.217 was given for three layers that were presented by Yoneda (2013) [17] .

As shown in Figure 9, gas production behaviors by present and Kurihara et al.’s numerical simulation results [3] [17] are compared for of the base case without applying consolidation effects on them. Both numerical simulations show favorable matching except little differences in, peak production rate, declining trend after the peak rate and cumulative gas production that are 3.6 × 10⁷ and 3.8 × 10⁷ m³ respectively. It is concluded that

present model has almost applicability for gas production from MH reservoirs in field scale, because little differences in modelling of relative permeability, heat capacities and pressure-temperature equilibrium between two models made effects on pressure propagation, MH dissociation and gas production.

Figure 9 also shows the effect of the elastic modulus, E₀ in the consolidation model on gas production behaviors. It was confirmed that time showing the peak rate was approximately 530 days that is longer than 290 days of the base case, while the gas peak rate decreased to 45% of the base case. The decrease of absolute permeability by sand consolidation makes delay of pressure propagation and gas peak rate, however, the cumulative gas productions at 1825 days were approximately same value of 32 × 10⁶ m³. After 1600 days, bottom-hole pressure propagated to whole reservoir region, therefore gas production rate or MH dissociation rate depends on reservoir permeability and heat supply to MH reservoir from upper and lower sediment layers. The peak times of gas production rate delayed with decreasing value of E₀. The time showing the peak production rate was around 400 to 930 days for E₀ = 100 to ∞ MPa. However, the differences of cumulative gas productions after 2200 days are much smaller than that of early stage after start of depressurization. Based on calculation results of absolute permeability and pressure distributions in radial direction, it was confirmed that pressure propagation has strong relation with MH dissociation process and gas production behavior.

Numerical simulation of seabed subsidence was done for the MH reservoirs with two models of elastic modulus expressed by E = E₀ and E = E₀ + βS_MH. To decide the subsidence ratio α, field measurement data are required. In this research, the value of water-dissolved gas field, ζ = 0.0012 which was reported by [14] [15] was used for present numerical simulations.

Seabed subsidence behaviors were simulated for two models of elastic modulus to predict the largest subsidence and the largest gradient of subsidence. Figure 10 shows predictions of seabed subsidence for different

values of E₀ at 50 days from start of depressurization. In both of the models, the maximum amount of the subsidence becomes larger with decreasing E₀. For the model of E = E₀, values of the largest subsidence, Δh_max = 1.4, 1.6 and 2.0 m for E₀ = 100, 200 and 400 MPa, respectively, and radius boundaries, r_B showing the value of subsidence Δh = 0.1 m are calculated as r_B = 31.8, 40.0 and 47.0 m for E₀ = 100, 200 and 400 MPa, respectively, while for the model of E = E₀ + βS_MH, Δh_max = 0.50, 1.0 and 1.85 m and r_B = 30.2, 34.8 and 36.4 m for E₀ = 100, 200 and 400 MPa, respectively. The differences of two models are caused by existence of residual MH biding sand grains. It is estimated that the residual MH increases appearance elastic modulus of MH reservoir, then the seabed subsidence is controlled.