# A Geomechanical Model for Gas Hydrate Bearing Sediments Incorporating High Dilatancy, Temperature, and Rate Effects

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## Abstract

**:**

## 1. Introduction

^{3}of hydrate produces during dissociation approximately 164 m

^{3}of free gas and around 800 L of water) triggers significant increments in the fluid pressures, resulting in effective stress reduction, with the associated sediment deformations and changes in both soil porosity and permeability. Furthermore, because hydrate dissociation is a strong endothermic reaction, the reduction of the sediment temperature may freeze the pore water, blocking the sediment permeability. Moreover, methane production is generally accompanied with sand migration, which impact on both borehole mechanical stability and fluids flow. Sand production is considered as one of the major limitations for the commercial exploitation of gas hydrate (e.g., [11,12]). It is then apparent that the profound perturbations in the sediment condition and the multiphysics nature of this problem, with the strong couplings between the different thermo-hydro-mechanical and chemical (THMC) phenomena that control the sediment behavior require the development of advanced and robust models.

_{2}-CH

_{4}exchange). It has been proven in the laboratory that this replacement is energetically favorable and effective (e.g., [15,16,17]). The CO

_{2}-CH

_{4}hydrate exchange will solve two problems simultaneously, i.e., stability of the hydrate bearing sediment during methane production, and capture of carbon dioxide in the sediment. However, the practical implementation of this technique in the field still requires further research and in-situ testing.

## 2. Methodology

_{H}: ratio between the volume of hydrates and the volume of voids) has a strong influence on the mechanical response of MHBS. It has been shown that stiffness, peak deviatoric stress, and dilation of the MHBS specimens increase with S

_{H}(e.g., [18,75]). However, the impact of methane-hydrates on sediment behavior not only depend on the amount of hydrates, but also on its morphology. Gas hydrates are generally found in three main form types in the sediment structure (Figure 1): (a) cementation, (b) pore-filling, and (c) supporting matrix (e.g., [1,76]). In the first pore-habit type, the methane-hydrates are mainly present at the contact between the grains and they act as a bonding material. In the pore filling form, the hydrates generally tend to grow freely in the pore space, without bridging particles together. Hydrates in the supporting matrix form are part of the solid skeleton. For a similar hydrate concentration, the cementing type hydrate morphology provides the maximum stiffness, strength and dilatancy (e.g., [18]).

#### 2.1. Basic Model for MHBS—Brief Introduction

_{c}is the effective pre-consolidation mean stress (which control the size of the elastic domain), and p

_{d}controls the increase of the sediment strength associated with the presence of hydrates.

_{v}

^{p}):

_{H}and hydrate morphology, more details can be found in Gai and Sanchez [67].

#### 2.2. Model Upgrade to Consider MHBS Exhibiting High Dilatancy

#### 2.3. Model Upgrade to Incorporate the Effect of Temperature on MHBS Behavior

_{c}(Equation (1)) on temperature, as suggested in Laloui and Cekerevac [93] for other type of soils:

_{0}is the reference temperature, p

_{c}

_{0}is the preconsolidation mean stress at T

_{0}, and r

_{T}is a model parameter that considers the effect of temperature on the preconsolidation pressure. Figure 3 illustrates the effect of temperature on the MHBS yield. The validation of this model is presented in Section 3.2.

#### 2.4. Model Upgrade to Consider Rate-Dependent Effects on the Behavior of MHBS

_{S}), and the homothetic yield surface passing through the current (predictor) stress state during yielding (i.e., outside F

_{S}) is often called the ‘dynamic yield surface’ (F

_{D}) (e.g., Hinchberger and Rowe [111]). In our model F

_{S}and F

_{D}are given by Equation (1), depending on whether we use the stresses and internal variables at the beginning of the time step (identified with the subscript S), or the stresses and internal variables associated with the predicted stresses (identified with the subscript D), respectively.

_{D}) calculated with the following equation (assuming associated plasticity):

_{0}a reference value (i.e., such that the expression is non-dimensional); and n

_{f}is a model parameter. Note that in this initial model we do not propose any dependence of the flow function on hydrate concentration, but it can be incorporated if needed. The over-stress index is calculated using the internal (static) variables at ‘S’ and the predicted (dynamic) stresses ‘D’, as follows:

## 3. Results and Discussions

#### 3.1. Model Application Involving MHBS Exhibiting Large Dilation

_{3}). Figure 5, Figure 6 and Figure 7 present the experimental results (symbols) of the tests sheared at $\sigma $

_{3}= 1 MPa (S

_{H}= 0%; S

_{H}= 34%; and S

_{H}= 41%), $\sigma $

_{3}= 2 MPa (S

_{H}= 0%; S

_{H}= 31%; and S

_{H}= 43%), and $\sigma $

_{3}= 3 MPa (S

_{H}= 0%, S

_{H}= 27%, and S

_{H}= 42%), respectively. These figures also present the modeling results obtained with the original MHBS-HISS model [67] (dash lines, identified as M

_{B}), and the upgraded model considering Equation (3) for the strain-hardening law (solid lines, identified as M

_{U}). The model parameters are listed in Table A1. Note that p

_{c}= 7.7 MPa, 12.1 MPa and 16.1 MPa were adopted for the cases associated with $\sigma $

_{3}= 1 MPa, 2 MPa, and 3 MPa, respectively.

_{H}, a more marked post-peak softening behavior with the increase of S

_{H}, and increase of MHBS strength with the increase of confinement. The main difference between the two models is apparent when comparing the volumetric behavior (Figure 5b, Figure 6b and Figure 7b). The new model reproduces noticeably much better the large dilation observed in these tests, particularly for the MHBS samples with high S

_{H}. The new model also captures well the decrease in soil dilation with the confinement increase.

_{H}= 0%, S

_{H}= 39%, and S

_{H}= 54%. The samples were sheared at $\sigma $

_{3}= 3 MPa. We modeled these tests using both, the original MHBS-HISS model and the upgrade version proposed in this work. Table A2 lists the model parameters. As in the previous case, the effect of hydrate saturation on MHBS behavior is characterized by a marked influence of S

_{H}on material stiffens, strength (Figure 8a), and volumetric response (Figure 8b). It can be observed that the proposed model is able to reproduce more accurately the large dilation observed in these experiments and overcomes the shortcomings experienced by previous approaches to model this critical feature of MHBS behavior. These results suggest that the proposed model will be able to properly predict the sediment volume changes (with the associated settlements and ground subsidence) in engineering problems involving MHBS.

#### 3.2. Model Application Considering the Effect of Temperature on MHBS Behavior

_{3}= 5 MPa), same hydrate saturation (S

_{H}= 30%), same loading rate, and they were shared at three constant temperatures, T = −5 °C, T = −10 °C, T = −20 °C. Table A4 lists the adopted parameters. The updated model captures well both the increase of strength and the change in material stiffness with temperature (Figure 10).

_{H}= 53.1% and S

_{H}= 51.6%, and sheared at T = 1 °C and at T = 10 °C, respectively. The two samples were tested at the same effective confining pressure ($\sigma $

_{3}′ = 3 MPa). Figure 11a,b present the experimental results together with the modeling outputs. Figure 11a shows that the model predicts very well, the impact of temperature on soil stiffness, strength, and post-peak behavior. The volumetric behavior of the MHBS (Figure 11b) at different temperatures is also well capture by the upgraded model. The adopted parameters are listed in Table A5.

#### 3.3. Model Application Considering the Effect of Loading Rate on MHBS Behavior

_{H}~40%. Given the difficulties to form hydrate in experimental conditions and to determine its saturation, in the examined samples S

_{H}varies between 35~45%. Here we adopt a unique set of parameters (Table A6) to model all the experiments. Three loading rates were investigated, LR = 0.1%/min, LR = 0.05%/min, and LR = 0.01%/min. Three samples were tested per each experiment (i.e., per each S

_{H}and LR). Figure 12a–c presents the comparisons between the experimental (symbols) and modeling (lines) results for the tests conducted at LR = 0.1%/min, LR = 0.05%/min, and LR = 0.01%/min, respectively.

_{H}of the tested specimens, we adopted a unique set of parameters to model all the experiments.

_{H}= 35% and S

_{H}= 48%. The tests considered different loading rates and different axial strain intervals ($\Delta e$). This type of experiment generally contemplates the following test protocol: (i) preparation of the hydrate-sand or sand (S

_{H}= 0%) specimen by the water saturation method in the test apparatus, (ii) application of the loading at a low loading rate (LR

_{L}) until the axial strain reached a target value (e.g., 0.25%); (iii) modification of the loading rate to the high loading rate (LR

_{H}) until the target axial strain interval (i.e., until e.g., another 0.25% is reached). This sequence is repeated successively until the total axial strain is reached (i.e., until shear failure). The three tests were conducted at the same LR

_{H}= 0.1%/min but considering different low loading rates and axial strain intervals. In test (I) LR

_{L}was 0.05%/min, $\Delta e$ = 0.5%, and S

_{H}= 48% (Figure 13a); in test (II) LR

_{L}was 0.05%/min, $\Delta e$ = 0.25%, and S

_{H}= 48% (Figure 13b); and in test (III) LR

_{L}was 0.01%/min, $\Delta e$ = 0.25%, and S

_{H}= 35% (Figure 13c).

_{H}(curve 2) and LR

_{L}(curve 1) are plotted using green dash lines in Figure 13a.

_{H}and LR

_{L}were smaller than in tests (I) and (II). Comparing Figure 13b, c, one can conclude that (for the same $\Delta e$), the larger the difference between the LR

_{H}and LR

_{L}, the larger the difference between the deviatoric stresses associated with these loading rates. Comparing Figure 13a, b, we can see that (for the same loading rates and hydrate saturation), the change in $\Delta e$ does not significantly impact on the deviatoric stresses associated with the LR

_{H}and LR

_{L}. The proposed model is able to capture qualitatively well the main features of MHBS behavior discussed above. The results obtained with the viscoplastic model are also satisfactory in quantitative terms.

_{L}was 0.01%/min, LR

_{H}= 0.05%/min, and S

_{H}~80% (Figure 14a); and in the other test LR

_{L}was 0.001%/min, LR

_{H}= 0.01%/min, and S

_{H}~65% (Figure 14b).

_{H}(Figure 14b). The parameters adopted to model these two experiments are listed in Table A7. In addition, in this case, it can be observed that the model captures well the global behavior observed in these experiments, reproducing very satisfactorily both the stress developed at the different loading rates and the volumetric behavior displayed by these specimens during shearing. These are critical features of MHBS behavior that the proposed approach is able to properly reproduce, enabling a more reliable modeling of engineering problems involving this type of soil.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

S_{H} | Hydrate saturation degree |

$\mathrm{a},\gamma $, n | Model yield function parameter |

p’ | Mean effective stress |

q | Deviatoric stress |

M | Slope of critical line in the q—p’ space |

p_{c} | Mean pre-consolidation effective stress |

p_{d} | Strength increment with presence of hydrates |

R | Evolution variable |

${\u03f5}_{v}^{p}$ | Volumetric plastic strain |

${\u03f5}_{d}^{p}$ | Deviatoric plastic strain |

e | Void ratio |

κ | Slope for elastic isotropic paths in the e—(ln)p’ plane |

λ | Slope for plastic isotropic paths in the e—(ln)p’ plane |

D_{s} | Nova parameter |

p_{c0} | Pre-consolidation mean stress at T_{0} |

T_{0} | Reference temperature |

r_{T} | Temperature effect parameter on the pre-consolidation mean stress |

$\Phi $ | Scalar function |

F_{S} | Yield surface (YS) function for the static yield surface |

F_{D} | Yield surface function for the dynamic yield surface |

G_{D} | Plastic potential function at predicted stress point |

p’_{D} | Mean effective stress at predicted stress point |

q_{D} | Deviatoric stress at predicted stress point |

${\gamma}_{f}$ | Fluidity |

${\sigma}_{D}$ | Cauchy’s stress tensor of predicted stress point |

f | Over-stress index |

f_{0} | Reference value in flow function |

n_{f} | Model parameter in flow function |

p_{cS} | Mean pre-consolidation effective stress for static YS |

p_{dS} | Strength increment with presence of hydrates for static YS |

R_{S} | Evolution variable for static YS |

${\epsilon}_{D}^{vp}$ | Viscoplastic strain associated with the dynamic yield surface |

$\Delta e$ | Axial strain interval |

K’ | Bulk modulus |

$\Lambda $ | Plastic multiplier |

$\alpha $$,$ | Parameters accounting for hydrate contribution |

$\chi $ | Damage variable |

$\mu $ | Parameter controlling rate of mechanical damage |

$\eta $ | Subloading parameter |

## Appendix A

_{s}).

**σ**’ is the effective stress tensor.

^{p}|dεp is the norm of the (total) plastic strain vector and $\eta $ is a sub-loading parameter related to plastic deformations developed inside the initial yield surface.

_{o}, and M) can be determined following the procedure typically adopted in soils mechanics to estimate critical state model constants, as described in [67]. The main parameters that control the effect of hydrates (i.e., hydrate saturation and morphology) on sediment behavior are α β μ and the variable χ. These parameters are mainly related to the increase of preconsolidation pressure and sediment strength in MHBS. β and μ consider the effect of S

_{H}on HBS behavior (i.e., for a given hydrate morphology), and the parameter α can be used to account for the effect of pore habit (i.e., for a given S

_{H}). The parameter μ also controls the rate of mechanical damage. We assume that the rate of mechanical damage increases with S

_{H}, and that the rate of damage is higher for cementing morphology than for pore-filling. These parameters are generally indirectly calibrated from experiments [67].

## Appendix B

Properties | Upgraded Model M_{U} | Basic Model M_{B} |
---|---|---|

M | 1.28 | 1.28 |

λ | 0.2 | 0.2 |

κ | 0.004 | 0.004 |

n | 1.3 | 1.3 |

a | 3 | 3 |

γ | −1/9 | −1/5 |

α | 35 | 20 |

β | 1 | 1 |

μ | 7 | 7 |

υ | 0.1 | 0.1 |

Ds | 0.8 | 0.0 |

**Table A2.**Parameters adopted for Model Validation in Figure 8.

Properties | Upgraded Model M_{U} | Basic Model M_{B} |
---|---|---|

M | 1.28 | 1.28 |

λ | 0.24 | 0.24 |

κ | 0.005 | 0.005 |

p_{c} (MPa) | 9.6 | 9.6 |

n | 0.95 | 0.95 |

a | 3 | 3 |

γ | −1/10 | −1/10 |

α | 8 | 6 |

β | 0.7 | 0.7 |

μ | 4.5 | 4.5 |

υ | 0.35 | 0.35 |

Ds | 0.1 | 0.0 |

**Table A3.**Parameters adopted for simulation in Figure 9.

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

M | 1.25 |

λ | 0.25 |

κ | 0.015 |

p_{c} (MPa) | 3 |

n | 1 |

a | 2 |

γ | −1/9 |

α | 20 |

β | 0.5 |

μ | 5 |

υ | 0.15 |

**Table A4.**Parameters adopted for simulation in Figure 10.

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

M | 1.2 |

λ | 0.5 |

κ | 0.02 |

p_{c} (MPa) | 5 |

n | 1 |

a | 6 |

γ | −1/9 |

α | 15 |

β | 1.2 |

μ | 1 |

υ | 0.15 |

**Table A5.**Parameters adopted for simulation in Figure 11.

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

M | 1.25 |

λ | 0.25 |

κ | 0.015 |

p_{c} (MPa) | 3 |

n | 1 |

a | 3 |

γ | −1/9 |

α | 20 |

β | 1 |

μ | 1 |

υ | 0.15 |

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

M | 1.22 |

λ | 0.2 |

κ | 0.017 |

p_{c} (MPa) | 9 |

n | 1 |

a | 5 |

γ | −1/9 |

α | 20 |

β | 1 |

μ | 20 |

υ | 0.2 |

γ_{f} | 5 × 10^{−6} |

n_{f} | 1 |

**Table A7.**Model parameters adopted for Case 3 in Figure 14.

Parameter | Borehole 16B-4P | Borehole 17C-9P |
---|---|---|

M | 1.2 | 1.2 |

λ | 0.28 | 0.28 |

κ | 0.025 | 0.025 |

p_{c} (MPa) | 10 | 6.8 |

n | 1 | 1 |

a | 5 | 4.3 |

γ | −1/12 | −1/12 |

α | 10 | 10 |

β | 10 | 10 |

μ | 5 | 5 |

υ | 0.2 | 0.2 |

γ_{f} | 2 × 10^{−6} | 2 × 10^{−6} |

n_{f} | 1 | 1 |

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**Figure 1.**Different types of morphology for methane hydrate bearing sediments [1]. (

**a**) Cementation; (

**b**) pore filling; (

**c**) supporting matrix.

**Figure 5.**Modeling the tests reported in Miyazaki et al. [82] at $\sigma $

_{3}= 1 MPa. (

**a**) Deviatoric stress versus axial strains; (

**b**) volumetric strains versus axial strains.

**Figure 6.**Modeling the tests reported in Miyazaki et al. [82] at $\sigma $

_{3}= 2 MPa. (

**a**) Deviatoric stress versus axial strains; (

**b**) volumetric strains versus axial strains.

**Figure 7.**Modeling the tests reported in Miyazaki et al. [82] at $\sigma $

_{3}= 3 MPa. (

**a**) deviatoric stress versus axial strains; (

**b**) volumetric strains versus axial strains.

**Figure 8.**Modeling the tests reported in Hyodo et al. [6] at $\sigma $

_{3}= 3 MPa. (

**a**) Deviatoric stress versus axial strains; (

**b**) volumetric strain versus axial strains.

**Figure 9.**Modeling the tests reported in Hyodo et al. [31] at different temperatures. (

**a**) Deviatoric stress versus axial strains; (

**b**) volumetric strain versus axial strains.

**Figure 10.**Modeling the tests reported in Li et al. [96] at different temperatures, deviatoric stress versus axial strains.

**Figure 11.**Modeling the tests reported in Hyodo et al. [6] at different temperatures. (

**a**) Deviatoric stress versus axial strains; (

**b**) volumetric strain versus axial strains.

**Figure 12.**Modeling the tests reported in Miyazaki et al. [33]. (

**a**) Constant loading rate LR = 0.1%/min; (

**b**) constant LR = 0.1%/min; and (

**c**) constant LR = 0.1%/min.

**Figure 13.**Case 2 considering effect of loading rate on MHBS behavior Miyazaki et al. [33]. (

**a**) Test (I); (

**b**) test (II); (

**c**) test (III).

**Figure 14.**Case 3 considering effect of loading rate on MHBS behavior Yoneda et al. [41]. (

**a**) ‘Specimen one’ from Borehole 16B-4P; (

**b**) ‘Specimen two’ from Borehole 17C-9P.

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**MDPI and ACS Style**

Zhou, B.; Sanchez, M.; Oldecop, L.; Santamarina, J.C. A Geomechanical Model for Gas Hydrate Bearing Sediments Incorporating High Dilatancy, Temperature, and Rate Effects. *Energies* **2022**, *15*, 4280.
https://doi.org/10.3390/en15124280

**AMA Style**

Zhou B, Sanchez M, Oldecop L, Santamarina JC. A Geomechanical Model for Gas Hydrate Bearing Sediments Incorporating High Dilatancy, Temperature, and Rate Effects. *Energies*. 2022; 15(12):4280.
https://doi.org/10.3390/en15124280

**Chicago/Turabian Style**

Zhou, Bohan, Marcelo Sanchez, Luciano Oldecop, and J. Carlos Santamarina. 2022. "A Geomechanical Model for Gas Hydrate Bearing Sediments Incorporating High Dilatancy, Temperature, and Rate Effects" *Energies* 15, no. 12: 4280.
https://doi.org/10.3390/en15124280