# Stability Analysis of Methane Hydrate-Bearing Soils Considering Dissociation

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Figure 1.**Schematic view of possible hazards in marine sediments induced by gas hydrates dissociation.

**Figure 2.**Illustrative view of stable and unstable regions of methane hydrate-bearing sediments with and without dissociation.

## 2. One-Dimensional Instability Analysis of Methane Hydrate Bearing Viscoplastic Material

#### 2.1. Governing Equations

#### 2.1.1. General Settings

_{F}is given by:

^{α}is defined as the local ratio of the volume element with respect to the total volume given by:

^{W}will be denoted as s:

^{α}, and the total phase ρ is denoted by:

^{α}is the mass of each phase α.

#### 2.1.2. Stress Variables

^{W}and P

^{G}are the pore water pressure and the pore gas pressure, respectively. Tension is positive for the stresses. For simplicity, we assume that the soil phase and the hydrate phase are in the same phase, namely, the solid phase. Thus, the partial stress of the solid phase is defined as:

^{S}and n

^{H}are the volume fractions of the soil phase and the hydrate phase, respectively, and P

^{F}is the average fluid pressure given by:

#### 2.1.3. Conservation of Mass

#### 2.1.4. Balance of Momentum

#### 2.1.5. Darcy Type of Law

#### 2.1.6. Conservation of Energy

#### 2.1.7. Dissociation Rate of Methane Hydrates

_{H}is the moles of hydrates in the volume V, N

_{H0}is the moles of hydrates in the initial state, P

^{F}is the average pore pressure and P

^{e}is an equilibrium pressure at temperature θ. When the dissociation occurs, the dissociation rate is negative, i.e., ${\dot{N}}_{H}<0$. The rates of generation of water and gas are given by:

_{H}, M

_{W}, and M

_{G}are the molar mass of the methane hydrates, the water, and the methane gas, respectively.

#### 2.1.8. Simplified Viscoplastic Constitutive Model

#### 2.2. Perturbed Governing Equations

#### 2.3. Conditions of Onset of Material Instability

#### 2.3.1. Sign for the Coefficients ${a}_{5}$ and ${a}_{0}$

- (A)
- $\text{\epsilon}<0$: compressive strain
- (1)
- When parameter $H$ is positive, that is, the viscoplastic hardening, the term in ${a}_{0}$ is always positive. The sign for ${a}_{0}$ always becomes positive:$$\epsilon <0,\text{\hspace{0.17em}}H>0\text{\hspace{0.17em}\hspace{0.17em}}\iff \text{\hspace{0.17em}\hspace{0.17em}}{a}_{0}>0$$
- (2)
- When the parameter $H$ is negative, that is, the viscoplastic softening, ${a}_{0}$ becomes negative, if it satisfies the following inequality:$$\{\begin{array}{l}\mathrm{\epsilon}<0,\text{\hspace{0.17em}}H<0\\ nH<\mathrm{\epsilon}{H}_{SH}{S}_{r}^{H}<0\end{array}\text{\hspace{0.17em}\hspace{0.17em}}\Rightarrow \text{\hspace{0.17em}\hspace{0.17em}}{a}_{0}<0$$

- (B)
- $\text{\epsilon}>0$: expansive strain
- (3)
- When parameter $H$ is positive, that is, the viscoplastic hardening, the term in ${a}_{0}$ becomes negative, if it satisfies the following inequality:$$\{\begin{array}{l}\mathrm{\epsilon}>0,\text{\hspace{0.17em}}H>0\\ 0<nH<\mathrm{\epsilon}{H}_{SH}{S}_{r}^{H}\end{array}\text{\hspace{0.17em}\hspace{0.17em}}\Rightarrow \text{\hspace{0.17em}\hspace{0.17em}}{a}_{0}<0$$
- (4)
- When parameter $H$ is negative, that is, viscoplastic softening, the term $H-\text{\epsilon}{H}_{SH}{n}^{H}/{n}^{2}$ is always negative. Thus, the sign for ${a}_{0}$ is always negative. This may lead to the material instability, because it does not satisfy the first condition of the Routh-Hurwitz criteria:$$\{\begin{array}{l}\text{\epsilon}>0,\text{\hspace{0.17em}}H<0\\ nH<0<\text{\epsilon}{H}_{SH}{S}_{r}^{H}\end{array}\text{\hspace{0.17em}\hspace{0.17em}}\Rightarrow \text{\hspace{0.17em}\hspace{0.17em}}{a}_{0}<0$$

#### 2.3.2. Sign for the Coefficients ${a}_{5}$ and ${a}_{4}$

- In the case of large values for ${k}^{W}$, ${k}^{G}$, and ${k}^{\text{\theta}}$, ${a}_{4}$ can become positive more easily. In contrast, low permeabilities for water and gas make the material system unstable.

_{1}–a

_{4}and the other conditions of Routh-Hurwitz criteria due to the complexity, the material instability will be studied numerically. In the next section, the results of various numerical simulations of the dissociation-deformation problem using the one-dimensional finite element mesh will be presented in order to study the material instability by using the chemo-thermo-mechanically coupled model proposed by Kimoto et al. [9,10]. The results will be compared to those of the instability analysis.

## 3. Numerical Simulation of Instability Analysis by an Elasto-Viscoplastic Model Considering Methane Hydrate Dissociation

#### 3.1. One-Dimensional Finite Element Mesh and Boundary Conditions

#### 3.2. Initial and Simulation Conditions

_{ini}; the degree of depressurization ΔP varies from 20% to 80% with increments of 10%, as shown in Figure 5. By changing the magnitude of the depressurization, it becomes possible to control the MH dissociation. The depressurizing rate for each case is the same, that is 0.116 kPa/s (10 MPa/day); thus, the time when the pore pressure at the depressurization source reaches the target value is different for the different cases. The total time of the simulation is 100 h for each case. In the simulations, the total calculation time (100 h) is determined by considering the depressurization rate and depressurization level. The depressurization rate is 0.116 kPa/s (10 MPa/day), which is almost the same as that of offshore methane hydrate production trial conducted by Japan Oil, Gas, and Metals National Corporation (JOGMEC) in March 2013 [25]. According to the rate, the depressurization will finish at about 6.2 h in the case of 20% of depressurization level, and even in the case of the maximum depressurization level, i.e., 80%, it will end at about 25 h. Another reason is that in each case the computation time takes more than 10 h, and we need vast amounts of h to calculate total 42 cases and more. Thus, the total simulation time is set to be 100 h.

Name | Paremeter | Value |
---|---|---|

Initial Void Ratio | ${e}_{0}$ | 1.00 |

Initial water saturation | ${s}_{r0}$ | 1.0 |

Initial hydrate saturation | ${S}_{r0}^{H}$ | 0.51 |

Name | Parameter | Value |
---|---|---|

Compression Index | $\text{\lambda}$ | 0.185 |

Swelling index | $\text{\kappa}$ | 0.012 |

Initial shear elastic modulus (kPa) | ${G}_{0}$ | 53,800 |

Viscoplastic parameter (1/s) | $C$ | 1.0 × 10^{−10} |

Viscoplastic parameter | ${m}^{\prime}$ | 23.0 |

Stress ratio at critical state | ${M}_{f}^{*}$ | 1.09 |

Compression yield stress (kPa) | ${{\text{\sigma}}^{\prime}}_{mbi}$ | 1882 |

Degradation parameter | ${{\sigma}^{\prime}}_{maf}/{{\sigma}^{\prime}}_{mai}$ | 1.0 |

Degradation parameter | $\text{\beta}$ | 0.0 |

Parameter for suction effect (kPa) | ${P}_{i}^{C}$ | 100 |

Parameter for suction effect | ${S}_{I}$ | 0.2 |

Parameter for suction effect | ${s}_{d}$ | 0.25 |

Parameter for hydrate effect | ${S}_{ri}^{H}$ | 0.51 |

Parameter for hydrate effect | ${n}_{m}$ | 0.6 |

Parameter for hydrate effect | ${n}_{d}$ | 0.75 |

Thermo-viscoplastic parameter | $\text{\alpha}$ | 0.15 |

Permeability coefficient for water (m/s) | ${k}_{0}^{W}$ | variable |

Permeability coefficient for gas (m/s) | ${k}_{0}^{G}$ | $10\times {k}_{0}^{W}$ |

Permeability reduction parameter | $N$ | 7 |

^{W}and μ

^{G}are the viscosity for water and gas, ρ

^{W}and ρ

^{G}are the density of water and gas, respectively. In the simulation, the seabed ground at 1300 m water depth is modelled, and the pore pressure at the initial state is around 13 MPa. Considering that the gas is treated as an ideal gas, the dynamic viscosity for gas phase v

^{G}becomes about 0.1 times that of the water phase, although it varies depending on the temperature. Consequently, the permeability coefficient for the gas phase becomes about 10 times larger than that of the water phase.

**Figure 6.**Relationship between ${\left(1-{S}_{r}^{H}\right)}^{N}$ and the hydrate saturation with different values of N.

^{−i}(m/s) and the depressurization level is j (%). In the following section, the results of the numerical simulation and the discussion which intends to show a trend in the material instability will be presented.

Name | Values | Permeability ${k}_{0}^{W}$ (m/s) | |||||
---|---|---|---|---|---|---|---|

1.0 × 10^{−3} | 1.0 × 10^{−4} | 1.0 × 10^{−5} | 1.0 × 10^{−6} | 1.0 × 10^{−7} | 1.0 × 10^{−8} | ||

Degree of depressurization | 20% | Case-3-20 | Case-4-20 | Case-5-20 | Case-6-20 | Case-7-20 | Case-8-20 |

30% | Case-3-30 | Case-4-30 | Case-5-30 | Case-6-30 | Case-7-30 | Case-8-30 | |

40% | Case-3-40 | Case-4-40 | Case-5-40 | Case-6-40 | Case-7-40 | Case-8-40 | |

50% | Case-3-50 | Case-4-50 | Case-5-50 | Case-6-50 | Case-7-50 | Case-8-50 | |

60% | Case-3-60 | Case-4-60 | Case-5-60 | Case-6-60 | Case-7-60 | Case-8-60 | |

70% | Case-3-70 | Case-4-70 | Case-5-70 | Case-6-70 | Case-7-70 | Case-8-70 | |

80% | Case-3-80 | Case-4-80 | Case-5-80 | Case-6-80 | Case-7-80 | Case-8-80 |

#### 3.3. Simulation Results

#### Results of the Stable-Unstable Behavior during MH Dissociation

**Figure 7.**Stable and unstable regions of permeability and the depressurization level during the MH dissociation process.

^{W}= 1.0 × 10

^{−7}and k

^{W}= 1.0 × 10

^{−8}, it becomes more stable for the large depressurization levels than in the cases of k

^{W}= 1.0 × 10

^{−5}and k

^{W}= 1.0 × 10

^{−6}. The reason why the lower permeability makes the material system more stable is that the low permeability may limit the spreading of the depressurization; consequently, the area where the MHs dissociates becomes smaller and the production of the pore gas pressure is reduced. The balance between the permeability and the depressurization is one of the important factors in material instability. In order to investigate the details for the onset of material instability, several cases are selected, namely, two stable cases and two unstable cases. For the stable cases, we choose Case-4-30 and Case-7-30, which have the same depressurized level and different permeabilities, and for the unstable cases, Case-4-40 and Case-7-40 are chosen. In Case-4-30 and Case-7-30, the depressurization finishes after about 9.4 h, while in Case-4-40 and Case-7-40, it ends after 12.5 h.

^{G}(MPa) in elements-1, 2, and 3 for each case. The pore gas pressure is calculated in the elements where the MHs begin to dissociate. In Case-4-30, which is illustrated in Figure 8a, the pore gas pressure decreases with the progress of the depressurization. The production of pore gas pressure in elements-2 and 3 is initiated soon after that in element-1. This is because the depressurization spreads to the next element easily due to the large permeability. After that, the pore gas pressure in each element becomes the same value, which is consistent with the depressurized one. In Case-4-40, on the other hand, the pore gas pressure diverges just after 9.4 h, and the calculation stops as shown in Figure 8b. The large depressurization level may enhance gas production, and the permeability is not enough for the pore gas pressure to be allowed to dissipate. The time profiles of the pore gas pressure are the same as Case-4-30 until 9.4 h; the pore gas pressure in each element consists of the depressurized value due to the larger permeability.

**Figure 8.**Time profiles of the pore gas pressure P

^{G}(MPa) (

**a**) Case-4-30; (

**b**) Case-4-40; (

**c**) Case-7-30; (

**d**) Case-7-40.

^{−4}(m/s), which can be estimated from the gradient of the pore pressure and the permeability indicated in Equation (90). As for a diameter of the groundwater flow, an average grain size ${D}_{50}$ is often used in the geomechanics field. Therefore, we use ${D}_{50}=0.15\text{\hspace{0.17em}\hspace{0.17em}}(\text{mm})$, which is the value of fine sand or silt. Dynamic viscosity ${\text{\nu}}^{W}$ for water is 1.52 × 10

^{−6}(m

^{2}/s), at a temperature of 5 degrees. The Reynolds number parameters are listed in Table 4. Substituting those values into the Reynolds number equations, we obtain ${Re}_{water}=8.29\times {10}^{-4}<1~10$ for the water flow.

Variable | Value |
---|---|

v^{W} | 8.4 × 10^{−6} m/s |

D_{50} | 0.15 mm |

v^{W} (5 °C, 10 MPa) | 1.52 × 10^{−6} m^{2}/s |

Variable | Value |
---|---|

v^{G} | 8.4 × 10^{−5} m/s |

${D}_{50}$ | 0.15 mm |

v^{G} (5 °C, 10 MPa) | 1.72 × 10^{−7} m^{2}/s |

**Figure 9.**Time profiles of the remaining MH ratio $100{N}_{H}/{N}_{H0}$ (%). (

**a**) Case-4-30; (

**b**) Case-4-40; (

**c**) Case-7-30; (

**d**) Case-7-40.

**Figure 10.**Time profiles of the average pore pressure P

^{F}(MPa). (

**a**) Case-4-30; (

**b**) Case-4-40; (

**c**) Case-7-30; (

**d**) Case-7-40.

**Figure 11.**Time profiles of the mean skeleton stress ${{\text{\sigma}}^{\prime}}_{m}$(MPa). (

**a**) Case-4-30; (

**b**) Case-4-40; (

**c**) Case-7-30; (

**d**) Case-7-40.

^{F}is defined by Equation (14), and the mean skeleton stress ${{\text{\sigma}}^{\prime}}_{m}$ are defined as follows:

**Figure 12.**Time profiles of the volumetric strain ${\text{\epsilon}}_{v}$ (%). (

**a**) Case-4-30; (

**b**) Case-4-40; (

**c**) Case-7-30; (

**d**) Case-7-40.

## 4. Conclusions

- The parameters which have a significant influence on the material instability are the viscoplastic hardening-softening parameter, its gradient with respect to hydrate saturation, the permeability of the water and the gas, and the strain.
- Material instability may occur in both the viscoplastic hardening region and the softening region regardless of whether the strain is compressive or expansive. However, when the strain is expansive, material instability can occur even if it is in the viscoplastic hardening region. The expansive strain makes the possibility of the instability higher in the model.
- Permeability is one of the most important parameters associated with material instability. The larger the permeability for the water and the gas become, the more stable the material system becomes. In other words, the lower the permeability is, the higher the possibility is for material instability to occur. These results are consistent with the results obtained from the experimental studies.

- 4.
- Basically the simulation results become more stable with increases in permeability. However, they also become stable in the region of the lower permeability. This was because the depressurized area is limited due to the low permeability; and consequently, the amount of MH dissociation is also reduced.
- 5.
- When the calculation became unstable, the pore gas pressure diverged, and then the mean skeleton stress was decreased drastically. The larger expansive volumetric strain was also observed. These results are consistent with those obtained from the linear stability analysis.
- 6.
- In the case of a higher permeability and a larger depressurization level, the divergence occurred during depressurization and MH dissociation. On the other hand, in the case of the lower one, the instability was observed around the end part of the simulation when the MH dissociation almost converged. It is important to consider the material instability over the long term, that is, even after the dissociation calms down.
- 7.
- The compressive volumetric strain kept increasing after the depressurization finished and the changes in the pore pressure and the mean skeleton stress became small. It also proves the importance of considering the long term stability.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

## Appendix B

_{k}> 0, for k = 1, …, n.

## References

- Kvenvolden, K.A. Natural gas hydrate occurrence and issues. Ann. N. Y. Acad. Sci.
**1994**, 715, 232–246. [Google Scholar] [CrossRef] - Nisbet, E.G.; Piper, D.J.W. Giant submarine landslides. Nature
**1998**, 392, 329–330. [Google Scholar] [CrossRef] - Rothwell, R.G.; Thomson, J.; Kahler, G. Low-sea-level emplacement of a very large Late Pleistocene “megaturbidite” in the western Mediterranean Sea. Nature
**1998**, 392, 377–380. [Google Scholar] [CrossRef] - Sultan, N.; Cochonat, P.; Canals, M.; Cattaneo, A.; Dennielou, B.; Haflidason, H.; Laberg, J.S.; Long, D.; Mienert, J.; Trincardi, F.; et al. Triggering mechanisms of slope instability processes and sediment failures on continental margins: A geotechnical approach. Mar. Geol.
**2004**, 213, 291–321. [Google Scholar] [CrossRef] - Hyodo, M.; Li, Y.; Yoneda, J.; Nakata, Y.; Yoshimoto, N.; Nishimura, A. Effects of dissociation on the shear strength and deformation behavior of methane hydrate-bearing sediments. Mar. Pet. Geol.
**2014**, 51, 52–62. [Google Scholar] [CrossRef] - Hyodo, M.; Yokoyama, N.; Nakata, Y.; Kato, A.; Yoshimoto, N. Shear behaviour of methane hydrate bearing sand. In Proceedings of the 13th Japan Symposium on Rock Mechanics & 6th Japan-Korea Joint Symposium on Rock Engineering, Okinawa, Japan, 9–11 January 2013; pp. 987–992.
- Miyazaki, K.; Masui, A.; Sakamoto, Y.; Aoki, K.; Tenma, N.; Yamaguchi, T. Triaxial compressive properties of artificial methane-hydrate-bearing sediment. J. Geophys. Res.
**2011**, 116. [Google Scholar] [CrossRef] - Zhou, M.; Soga, K.; Xu, E.; Yamamoto, K. Effects of methane hydrate gas production on mechanical responses of hydrate bearing sediments in local production region at Eastern Nankai Trough. In Proceedings of the 14th IACMAG; Oka, F., Murakami, A., Uzuoka, R., Kimoto, S., Eds.; CRC Press: Kyoto, Japan, 2014; pp. 1707–1712. [Google Scholar]
- Kimoto, S.; Oka, F.; Fushita, T. A chemo-thermo-mechanically coupled analysis of ground deformation induced by gas hydrate dissociation. Int. J. Mech. Sci.
**2010**, 52, 365–376. [Google Scholar] [CrossRef] - Kimoto, S.; Oka, F.; Fushita, T.; Fujiwaki, M. A chemo-thermo-mechanically coupled numerical simulation of the subsurface ground deformations due to methane hydrate dissociation. Comput. Geotech.
**2007**, 34, 216–228. [Google Scholar] [CrossRef] - Rice, J.R. On the Stability of Dilatant Hardening for Saturated Rock Masses. J. Geophys. Res.
**1975**, 80, 1531–1536. [Google Scholar] [CrossRef] - Anand, L.; Kim, K.H.; Shawki, T.G. Onset of shear localization in viscoplastic solids. J. Mech. Phys. Solids
**1987**, 35, 407–429. [Google Scholar] [CrossRef] - Zibib, H.M.; Aifantis, E.C. On the Localization and Postlocalization Behavior of Plastic Deformation.1. On the Initiation of Shear Bands. Res Mech.
**1988**, 23, 261–277. [Google Scholar] - Loret, B.; Harireche, O. Acceleration waves, flutter instabilities and stationary discontinuities in inelastic porous media. J. Mech. Phys. Solids
**1991**, 39, 569–606. [Google Scholar] [CrossRef] - Benallal, A.; Comi, C. Material instabilities in inelastic saturated porous media under dynamic loadings. Int. J. Solids Struct.
**2002**, 39, 3693–3716. [Google Scholar] [CrossRef] - Oka, F.; Adachi, T.; Yashima, A. A strain localization analysis using a viscoplastic softening model for clay. Int. J. Plast.
**1995**, 11, 523–545. [Google Scholar] [CrossRef] - Higo, Y.; Oka, F.; Jiang, M.; Fujita, Y. Effects of transport of pore water and material heterogeneity on strain localization of fluid-saturated gradient-dependent viscoplastic geomaterial. Int. J. Numer. Anal. Methods Geomech.
**2005**, 29, 495–523. [Google Scholar] [CrossRef] - Kimoto, S.; Oka, F.; Kim, Y.; Takada, N.; Higo, Y. A Finite element analysis of the thermo-hydro-mechanically coupled problem of a cohesive deposit using a thermo-elasto-viscoplastic model. Key Eng. Mater.
**2007**, 340–341, 1291–1296. [Google Scholar] [CrossRef] - Garcia, E.; Oka, F.; Kimoto, S. Instability analysis and simulation of water infiltration into an unsaturated elasto-viscoplastic material. Int. J. Solids Struct.
**2010**, 47, 3519–3536. [Google Scholar] [CrossRef] [Green Version] - Terzaghi, K. Theoretical Soil Mechanics; John Wiley: Hoboken, NJ, USA, 1943. [Google Scholar]
- Jommi, C. Remarks on the constitutive modelling of unsaturated soils. In Experimental Evidence and Theoretical Approaches in Unsaturated Soils; Tarantino, A., Mancuso, C., Eds.; Balkema: Rotterdam, The Netherlands, 2000; Volume 153, pp. 139–153. [Google Scholar]
- Bear, J. Dynamics of Fluids in Porous Media; Elsevier: New York, NY, USA, 1972; pp. 176–177. [Google Scholar]
- Sato, K.; Iwasa, Y. Groundwater Hydraulics; Springer: Berlin, Germany, 2003; pp. 20–21. [Google Scholar]
- Kim, H.C.; Bishnoi, P.R.; Rizvi, S.S. H.; Engineering, P. Kinetics of methane hydrate decomposition. Chem. Eng. Sci.
**1987**, 42, 1645–1653. [Google Scholar] [CrossRef] - Yamamoto, K. Offshore methane hydrate resource development; from a viewpoint of geomechanics. In Proceedings of the 14th IACMAG; Oka, F., Murakami, A., Uzuoka, R., Kimoto, S., Eds.; CRC Press: Kyoto, Japan, 2014; pp. 1725–1730. [Google Scholar]
- Oka, F.; Kimoto, S. An Elasto-Viscoplastic Constitutive Model and Its Application to the Sample Obtained from the Seabed Ground at Nankai Trough. J. Soc. Mater. Sci. Jpn.
**2008**, 57, 237–242. [Google Scholar] [CrossRef] - Wu, L.; Grozic, J.L.H.; Eng, P. Laboratory Analysis of Carbon Dioxide Hydrate-Bearing Sands. J. Geotech. Geoenviron. Eng.
**2008**, 134, 547–550. [Google Scholar] [CrossRef] - Iwai, H. Behavior of Gas Hydrate-Bearing Soils during Dissociation and its Simulation. Ph.D. thesis, Kyoto University, Kyoto, Japan, 2015. [Google Scholar]

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Iwai, H.; Kimoto, S.; Akaki, T.; Oka, F.
Stability Analysis of Methane Hydrate-Bearing Soils Considering Dissociation. *Energies* **2015**, *8*, 5381-5412.
https://doi.org/10.3390/en8065381

**AMA Style**

Iwai H, Kimoto S, Akaki T, Oka F.
Stability Analysis of Methane Hydrate-Bearing Soils Considering Dissociation. *Energies*. 2015; 8(6):5381-5412.
https://doi.org/10.3390/en8065381

**Chicago/Turabian Style**

Iwai, Hiromasa, Sayuri Kimoto, Toshifumi Akaki, and Fusao Oka.
2015. "Stability Analysis of Methane Hydrate-Bearing Soils Considering Dissociation" *Energies* 8, no. 6: 5381-5412.
https://doi.org/10.3390/en8065381