# Thermo-Hydro-Mechanical Coupled Modeling of Methane Hydrate-Bearing Sediments: Formulation and Application

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Components, Phases and Partial Saturations

#### 2.2. Multiphysical Coupled System

#### 2.3. Governing Equations

**(i) Balance equations**

- Mass balance of mineral grains:The mineral grains coincide with the permanent solid phase and define the skeletal structure of the porous medium. The mass conservation of this component can be written as:$$\frac{\partial}{\partial t}\left({\rho}_{s}(1-{\varphi}_{p})\right)+\nabla \left({\mathbf{j}}_{s}^{gr}\right)=0$$$${\mathbf{j}}_{s}^{gr}={\rho}_{s}(1-{\varphi}_{p})\frac{\partial \mathbf{u}}{\partial t}$$Applying the chain rule for all the derivatives, Equation (3) can be rewritten as:$$(1-{\varphi}_{p})\frac{\partial {\rho}_{s}}{\partial t}+(1-{\varphi}_{p})(\frac{\partial \mathbf{u}}{\partial t}\nabla {\rho}_{s})-{\rho}_{s}\frac{\partial {\varphi}_{p}}{\partial t}-{\rho}_{s}(\frac{\partial \mathbf{u}}{\partial t}\nabla {\varphi}_{p})+{\rho}_{s}(1-{\varphi}_{p})\nabla \frac{\partial \mathbf{u}}{\partial t}=0$$Neglecting the gradients of density and porosity convected by the solid phase and under the assumption of small strain, Equation (5) can be rewritten as:$$\frac{\partial {\varphi}_{p}}{{\partial}_{t}}=\frac{1}{{\rho}_{s}}\left((1-{\varphi}_{p})\frac{\partial {\rho}_{s}}{{\partial}_{t}}\right)+(1-{\varphi}_{p})\nabla \frac{\partial \mathbf{u}}{\partial t}$$$$\nabla \frac{\partial \mathbf{u}}{\partial t}=\frac{{\partial}^{2}{u}_{x}}{\partial x\partial t}+\frac{{\partial}^{2}{u}_{y}}{\partial y\partial t}+\frac{{\partial}^{2}{u}_{z}}{\partial z\partial t}=\frac{\partial}{\partial t}({\epsilon}_{x}+{\epsilon}_{y}+{\epsilon}_{z})=\frac{\partial {\epsilon}_{v}}{\partial t}$$
- Mass balance of methane:Methane component is present in liquid, gas and hydrate phases, and its total mass balance is expressed as:$$\frac{\partial}{\partial t}\left(\left({\theta}_{l}^{m}{S}_{l}+{\theta}_{g}^{m}{S}_{g}\right){\varphi}_{a}+{\theta}_{h}^{m}{S}_{h}{\varphi}_{p}\right)+\nabla ({\mathbf{j}}_{l}^{m}+{\mathbf{j}}_{g}^{m}+{\mathbf{j}}_{h}^{m})={f}^{m}$$The mass flux terms ${\mathbf{j}}_{l}^{m}$ and ${\mathbf{j}}_{g}^{m}$ are the relative motion of methane in the liquid and gas phases, respectively, with respect to the solid phase. These terms are obtained as the sum of advective and diffusive flux terms as follows:$${\mathbf{j}}_{l}^{m}={\mathbf{i}}_{l}^{m}+{\theta}_{l}^{m}{\mathbf{q}}_{l}+{\varphi}_{a}{S}_{l}{\theta}_{l}^{m}\frac{\partial \mathbf{u}}{\partial t}$$$${\mathbf{j}}_{g}^{m}={\mathbf{i}}_{g}^{m}+{\theta}_{g}^{m}{\mathbf{q}}_{g}+{\varphi}_{a}{S}_{g}{\theta}_{g}^{m}\frac{\partial \mathbf{u}}{\partial t}$$The mass flux term ${\mathbf{j}}_{h}^{m}$ denote the relative motion of methane in the hydrate phase with respect to the solid phase as a result of the medium deformation:$${\mathbf{j}}_{h}^{m}={\varphi}_{p}{S}_{h}{\theta}_{h}^{m}\frac{\partial \mathbf{u}}{\partial t}$$The term ${f}^{m}$ is the external sink/source of methane per unit volume. Please note that because of the compositional approach adopted in the formulation, this term do not include methane mass changes from hydrate kinetics.
- Mass balance of water:Water is present in liquid, ice and hydrate phases but it is neglected as vapour in the gas phase. Thus, the water total mass balance of water is expressed as:$$\frac{\partial}{\partial t}\left({\theta}_{l}^{w}{S}_{l}{\varphi}_{a}+\left({\theta}_{i}^{w}{S}_{i}(1-{S}_{h})+{\theta}_{h}^{w}{S}_{h}\right){\varphi}_{p}\right)+\nabla ({\mathbf{j}}_{l}^{w}+{\mathbf{j}}_{i}^{w}+{\mathbf{j}}_{h}^{w})={f}^{w}$$The mass flux term of water in liquid (${\mathbf{j}}_{l}^{w}$) is computed similarly as in Equation (9), while the flux terms in hydrate (${\mathbf{j}}_{h}^{w}$) and ice phases (${\mathbf{j}}_{i}^{w}$) are computed similarly as in Equation (11) for ${S}_{h}$ and ${S}_{i}$, respectively.The ${f}^{w}$ is an external sink/source of water per unit volume and do not include water mass changes from hydrate kinetics.
- Mass balance of salt:Salt is present as a dissolved component in the liquid phase, and it is not allowed to precipitate as a solid phase. Its concentration modifies the liquid density, influences the solubility of methane and can inhibit hydrate stability conditions. The total mass balance of salt is expressed as:$$\frac{\partial}{\partial t}\left({\theta}_{l}^{st}{S}_{l}{\varphi}_{a}\right)+\nabla {\mathbf{j}}_{l}^{st}={f}^{\mathit{st}}$$The flux term of salt in the liquid phase (${\mathbf{j}}_{l}^{st}$) is the relative motion of salt in the liquid phase with respect to the solid phase and is computed similarly as in Equation (9). Finally, the term ${\mathit{f}}^{st}$ is the external sink/source term of salt.

**(ii) Constitutive equations and equilibrium constraints**

**Hydrate and ice phase boundaries**

#### 2.4. Numerical Solution Strategy

## 3. Results and Discussion

#### 3.1. Thermo-Hydraulic Validation

#### 3.2. THM Modelling of Synthetic Experimental Tests

#### 3.3. THM Modelling of the Experimental Tests of Li et al. (2018)

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MHBS | Methane hydrate-bearing sediments |

NETL | National Energy Technology Laboratory |

P | Pressure |

T | Temperature |

TH | Thermo-hydraulic |

THM | Thermo-hydro-mechanical |

USGS | United States Geological Survey |

## Appendix A. Development of Hydrate and Ice Mass Balance Equations

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**Figure 1.**Volumetric relationships and pore-scale phase distribution in an elementary volume of MHBS. Please note that the formulation considers the existence of unfrozen water below freezing temperature due to capillary effects.

**Figure 4.**Pressure-Temperature equilibrium relationship (${P}_{eq}$ − T) in the methane hydrate system at different salinity. Please note that the presence of salt inhibits hydrate stability and shifts the hydrate phase equilibrium curve towards higher pressure and lower temperature conditions.

**Figure 5.**(

**a**) Initial conditions and mesh applied in the simulation. Simulated (

**b**) gas pressure, (

**c**) temperature, (

**d**) dissolved methane mass fraction and (

**e**) hydrate saturation distributions along the domain at 1, 100 and 1000 days of simulation.

**Figure 7.**Simulated evolution of the (

**a**,

**b**) deviatoric stress, (

**c**,

**d**) potential porosity, (

**e**,

**f**) temperature, (

**g**,

**h**) liquid pressure and (

**i**,

**j**) hydrate and (

**j**,

**k**) gas saturation of the host and hydrate-bearing specimens subjected to thermal stimulation and depressurization during shear. Dotted lines represent the behavior of the host sediment specimens while solid lines refer to hydrate-bearing specimens. Please note that variations in hydrate and gas saturation before and after dissociation respectively, are due to the volumetric deformation of the specimen.

**Figure 8.**(

**a**) Initial test conditions. (

**b**) Boundary conditions adopted for dissociation simulation. The monitoring location shows the location of the profiles in Figure 9.

**Figure 9.**Experimental data and modeling results from drained triaxial tests performed on synthetic MHBS subjected to dissociation via depressurization: (

**a**) Deviatoric stress, (

**b**) volumetric strain, (

**c**) liquid pressure, and (

**d**) hydrate saturation relationships with axial strain. Experimental data from [21].

Model Reference | Mechanical Approach |
---|---|

Kimoto et al. (2010) [30] | Viscoplasticity with ${S}_{h}$ dependency |

Rutqvist (2011) [31] | Mohr-Coloumb elastoplasticity with ${S}_{h}$ dependency |

Kim et al. (2012) [32] | Mohr-Coloumb elastoplasticity with ${S}_{h}$ dependency |

Klar et al. (2013) [33] | Mohr-Coloumb elastoplasticity with ${S}_{h}$ dependency |

Gupta et al. (2016) [34] | Poroelasticity with ${S}_{h}$ dependency |

Sun et al. (2018) [35] | Thermodynamics-based elastoplastic model with ${S}_{h}$ dependency |

Sánchez et al. (2018) [36] | Elastoplasticty with ${S}_{h}$ dependency + Damage model |

Bulk volume | ${V}_{t}={V}_{s}+{V}_{h}+{V}_{i}+{V}_{l}+{V}_{g}$ |

Potential void space | ${V}_{p}={V}_{t}-{V}_{s}$ |

Potential porosity | ${\varphi}_{p}={V}_{p}/{V}_{t}$ |

Available void space | ${V}_{a}={V}_{p}-{V}_{h}-{V}_{i}$ |

Available porosity | ${\varphi}_{a}={V}_{a}/{V}_{t}$ |

Hydrate saturation | ${S}_{h}={V}_{h}/{V}_{p}$ |

Ice saturation | ${S}_{i}={V}_{i}/({V}_{p}-{V}_{h}$) |

Gas saturation | ${S}_{g}={V}_{g}/{V}_{a}$ |

Liquid saturation | ${S}_{l}={V}_{l}/{V}_{a}$ |

**Table 3.**Nomenclature used to define the governing equations for MHBS. Please note that superscript $\alpha $ correspond to any component of the system and subscript $\beta $ refers to the phase wherein the component is partitioned. Bold symbols denote vectors and tensors.

Roman Symbols | |||

$a\left(T\right)$ | Peng-Robinson EoS parameter | M | Slope of critical state line in p’-q space |

${A}_{\beta}$ | Viscosity parameter | ${M}_{\beta}$ | Molecular mass of phase $\beta $ |

$A\left(T\right)$ | Viscosity thermal function | n | Stress-state coefficient: yield surface shape parameter |

${A}_{h}$ | Hydrate surface area | ${n}_{h}$ | Hydration number |

$\mathbf{b}$ | Body forces | ${p}^{\prime}$ | Mean effective stress |

b | Peng-Robinson EoS constant | ${P}_{0}$ | Van Genuchten parameter |

${B}_{\beta}$ | Viscosity parameter of phase $\beta $ | ${p}_{0}$ | Isotropic yield stress of hydrate-free sediment |

${c}_{\beta}^{\alpha}$ | Specific heat of component $\alpha $ in phase $\beta $ | ${p}_{0h}$ | Isotropic yield stress of MHBS |

$d{m}_{\beta}$ | Mass change of phase $\beta $ | ${P}_{eq}$ | Hydrate phase equilibrium pressure |

${D}_{\beta}^{\alpha}$ | Diffusion coefficient of component $\alpha $ in phase $\beta $ | ${P}_{\beta}$ | Pressure of phase $\beta $ |

e | Void ratio of hydrate-free sediment | ${P}_{{\beta}_{0}}$ | Reference pressure for phase $\beta $ |

${e}_{ah}$ | Available void ratio of the MHBS | ${P}_{c}$ | Critical pressure |

${e}_{h}$ | Hydrate void ratio | ${P}_{p}$ | Pore pressure |

${E}_{\beta}$ | Energy of phase $\beta $ | q | Deviatoric stress |

${E}_{\beta}^{\alpha}$ | Energy of component $\alpha $ in phase $\beta $ | ${\mathbf{q}}_{\beta}$ | Advective fuid flow |

f | Hydrate-CASM yield function | r | Yield surface spacing ratio |

${f}^{\alpha}$ | External mass supply of component $\alpha $ | R | Subloading ratio |

${f}^{Q}$ | Internal/external energy supply | ${R}_{g}$ | Regnault constant |

${f}_{w}$ | Peng-Robinson EoS function | ${R}_{h}$ | Rate of hydrate mass change |

g | Gravity forces | S | Salinity |

${\mathbf{i}}_{\beta}^{\alpha}$ | Diffusive flux of component $\alpha $ in phase $\beta $ | ${S}_{\beta}$ | Saturation of phase $\beta $ |

${\mathbf{i}}_{c}$ | Conductive heat flow | ${S}_{e}$ | Effective liquid saturation |

${\mathbf{i}}_{{E}_{\beta}}$ | Energy dispersivity in phase $\beta $ | ${S}_{ls}$ | Maximum liquid saturation |

$\mathbf{I}$ | Identity matrix | ${S}_{rl}$ | Residual liquid saturation |

${\mathbf{j}}_{\beta}^{\alpha}$ | Mass flux of component $\alpha $ in phase $\beta $ | t | Time |

${\mathbf{j}}_{{E}_{\beta}}$ | Advective flux of energy of phase $\beta $ | ${T}_{0}$ | Reference temperature |

${\mathbf{k}}_{0}$ | Intrinsic permeability of hydrate-free sediment | T | Temperature |

$\mathbf{k}$ | Intrinsic permeability of MHBS | ${T}_{c}$ | Critical temperature |

${k}_{{r}_{\beta}}$ | Relative permeability of phase $\beta $ | ${T}_{eq}$ | Hydrate phase equilibrium temperature |

${K}_{C{H}_{4}}$ | Solubility of methane in water | ${T}_{r}$ | Reduced temperature |

${K}_{d}$ | Hydrate dissociation constant | $\mathbf{u}$ | Displacement |

${K}_{f}$ | Hydrate formation constant | u | Subloading parameter controlling the |

${L}_{h}$ | Latent heat of hydrate dissociation | plastic deformations before yielding | |

${L}_{i}$ | Latent heat of ice melting | v | Molar volume |

m | Van Genuchten parameter | V | Unitary volume |

Greek Symbols | |||

${\alpha}_{B}$ | Biot’s coefficient | ∇ | Differential operator = ($\frac{\partial}{\partial x};\frac{\partial}{\partial y};\frac{\partial}{\partial z}$) |

${\beta}_{t}$ | Thermal expansion coefficient of the liquid phase | $\nu $ | Poisson’s ratio |

∂ | Partial derivative | ${\omega}_{\beta}^{\alpha}$ | Mass fraction of component $\alpha $ in phase $\beta $ |

$|{\mathit{\epsilon}}^{p}|$ | Norm of the incremental plastic strain | ${\mu}_{\beta}$ | Viscosity of phase $\beta $ |

${\epsilon}^{v}$ | Volumetric strain | ${\varphi}_{p}$ | Potential porosity |

${\epsilon}_{(x,y,z)}$ | Cartesian strains | ${\varphi}_{a}$ | Available porosity |

$\gamma $ | Solute variation | ${\rho}_{\beta}$ | Density of phase $\beta $ |

$\kappa $ | Slope of swelling line of hydrate-free sediment | ${\rho}_{{\beta}_{0}}$ | Reference density of phase $\beta $ |

${\kappa}_{h}$ | Slope of swelling line of MHBS | $\mathit{\sigma}$ | Cauchy total stress (compression positive) |

${\kappa}_{rf}$ | Swelling line slope reduction factor | ${\mathit{\sigma}}^{\prime}$ | Effective stress (compression positive) |

$\lambda $ | Slope of normal compression line | ${\sigma}_{c}$ | Confining stress (compression positive) |

${\lambda}_{c}$ | Composite thermal conductivity | ${\theta}_{\beta}^{\alpha}$ | Volumetric mass of component $\alpha $ in phase $\beta $ |

${\lambda}_{dry}$ | Dry thermal conductivity | $\tau $ | Tortuosity coefficient |

${\lambda}_{sat}$ | Liquid saturated thermal conductivity | $\vartheta $ | Peng-Robinson acentric factor |

**Table 4.**Specific energy expression and representative values of specific and latent heat for each phase of the system.

Phase | Specific Energy | Specific and Latent Heat |
---|---|---|

Gas (g) | ${E}_{g}={c}_{g}(T-{T}_{0})$ | ${c}_{g}$ = 2500 J (Kg K)^{−1} |

Hydrate (h) | ${E}_{h}={L}_{h}+{c}_{h}(T-{T}_{0})$ | ${c}_{h}$ = 2108 J (Kg K)^{−1}; ^{†}${L}_{h}$ = 3.39 e^{5} J Kg^{−1} |

Ice (i) | ${E}_{i}={L}_{i}+{c}_{i}(T-{T}_{0})$ | ${c}_{i}$ = 3144 J (Kg K)^{−1}; ^{†}${L}_{i}$ = 3.34 e^{5} J Kg^{−1} |

Liquid (l) | ${E}_{l}^{w}={c}_{l}^{w}(T-{T}_{0})$ | ${c}_{l}^{w}$= 4184 J (Kg K)^{−1} |

${E}_{l}^{st}=1.42{e}^{5}$ + ${c}_{l}^{st}(T-{T}_{0})$ | ${c}_{l}^{st}$ = 2200 J (Kg K)^{−1} | |

Solid (s) | ${E}_{s}={c}_{s}(T-{T}_{0})$ | ${c}_{s}$ = 874 J (Kg K)^{−1} |

^{†}${L}_{h}$ and ${L}_{i}$ are positive for hydrate dissociation and ice melting respectively.

**Table 5.**Constitutive equations for MHBS system and dependent variables computed using each constitutive law. Please note that values for the parameters used in the TH simulation are also given.

Constitutive Law | Equation | Dependent Variable |
---|---|---|

Advective fluid flow | ||

Darcy’s law | ${\mathbf{q}}_{\beta}=-\frac{\mathbf{k}{k}_{{r}_{\beta}}}{{\mu}_{\beta}}(\nabla {P}_{\beta}-{\rho}_{\beta}\mathbf{g})\phantom{\rule{56.9055pt}{0ex}}\beta =l,g$ | Advective fluid flow, ${\mathbf{q}}_{l}$ and ${\mathbf{q}}_{g}$ |

Kozeny’s model | $\mathbf{k}={\mathbf{k}}_{0}\frac{{\varphi}_{a}^{3}}{{(1-{\varphi}_{a})}^{2}}\frac{{(1-{\varphi}_{p})}^{2}}{{\varphi}_{p}^{3}}$ | |

${\mathbf{k}}_{0}=1$ × ${10}^{-13}$ m^{2} | ||

${k}_{{r}_{l}}=\sqrt{{S}_{e}}{\left(1-{(1-{S}_{e}^{1/m})}^{m}\right)}^{2}$ | ||

${k}_{{r}_{g}}=1-{k}_{{r}_{l}}$ | ||

${m}^{\u2021}=0.645$ | ||

${\mu}_{\beta}={A}_{\beta}$exp$\left(\frac{{B}_{\beta}}{273.15+T}\right)$ | ||

${A}_{l}=2.1$ × ${10}^{-12}$ MPa s; ${B}_{l}$ = 1808.5 K | ||

${A}_{g}=1.48$ × ${10}^{-12}$ MPa s; ${B}_{g}$ = 119.4 K | ||

Retention curve [46] | ${S}_{e}=\frac{{S}_{l}-{S}_{rl}}{{S}_{ls}-{S}_{rl}}={\left(1+{\frac{{P}_{g}-{P}_{l}}{{P}_{0}}}^{\frac{1}{1-m}}\right)}^{-m}$ | Saturation of mobile phases, ${S}_{l}$ and ${S}_{g}$ |

${P}_{0}^{\u2021}=0.075$ MPa | ||

Liquid density [37] | ${\rho}_{l}={\rho}_{{l}_{0}}\left(1+{\beta}_{t}({P}_{l}-{P}_{{l}_{0}})+A\left(T\right)+\gamma {\omega}_{l}^{st}\right)$ | Liquid density, ${\rho}_{l}$ |

$A\left(T\right)=\frac{(T+288.9414){(T-3.9863)}^{2}}{50,8929.2(T+68.12963)}$ | ||

${\rho}_{{l}_{0}}$ = 1002.6 kg/m^{3} | ||

${\beta}_{t}=4.5\times {10}^{-4}$ MPa^{−1} | ||

${P}_{{l}_{0}}=0.1$ MPa | ||

$\gamma =0.6923$ | ||

${\omega}_{l}^{st}=0$ | ||

Gas density [42] | ${P}_{g}=\frac{{R}_{g}T}{v-b}-\frac{a\left(T\right)}{v(v+b)+b(v-b)}$ | Gas density, ${\rho}_{g}$ |

$a\left(T\right)=0.45724{R}_{g}^{2}{T}_{c}^{2}/{P}_{c}{[1+{f}_{w}(1-{T}_{r}^{0.5})]}^{2}$ | ||

${f}_{w}=0.37464+1.54226\vartheta -0.26992{\vartheta}^{2}$ | ||

$\vartheta =0.0015$ | ||

$b=0.0778R{T}_{c}/{P}_{c}$ | ||

${R}_{g}=8.3144598$ J/mol K | ||

${P}_{c}=4.60$ MPa | ||

${T}_{r}=T/{T}_{c},\phantom{\rule{5.69054pt}{0ex}}{T}_{c}=190.4$ K | ||

${\rho}_{g}=\frac{{M}_{g}{P}_{g}}{{R}_{g}T}$ | ||

${M}_{g}=0.016042$ Kg/mol | ||

Non-advective fluid flow | ||

Fick’s law | ${\mathbf{i}}_{\beta}^{\alpha}={\varphi}_{a}\tau {\rho}_{l}{S}_{l}{D}_{l}^{\alpha}\mathbf{I}\nabla {\omega}_{l}^{\alpha}\phantom{\rule{56.9055pt}{0ex}}\alpha =m,st$ | Diffusive flux, ${\mathbf{i}}_{l}^{m}$ and ${\mathbf{i}}_{l}^{st}$ |

${D}_{l}^{m}=5.9\times {10}^{-6}$ exp$\left(\frac{{(273.15+T)}^{2.3}}{{P}_{g}}\right)$ | ||

${D}_{l}^{st}=1.1\times {10}^{-4}$ exp$\left(\frac{-24530}{{R}_{g}(273.15+T)}\right)$ | ||

$\tau =1$ | ||

Fourier’s law | ${\mathbf{i}}_{c}=-{\lambda}_{c}\nabla T$ | Conductive heat flow, ${\mathbf{i}}_{c}$ |

^{†}${\lambda}_{c}={\lambda}_{sat}\sqrt{{S}_{l}}+{\lambda}_{dry}(1-\sqrt{{S}_{l}})$ | ||

${\lambda}_{sat}=2$ W/mK, ${\lambda}_{dry}=2.18$ W/mK | ||

Stress-strain behavior | ||

Effective stress [47] | ${\mathit{\sigma}}^{\prime}=\mathit{\sigma}-{\alpha}_{B}{P}_{p}\mathbf{I}$ | Effective stress tensor, ${\mathit{\sigma}}^{\prime}$ |

${\alpha}_{B}=1$ | ||

Hydrate-CASM [41] | Subloading yield function | Stress tensor, $\mathit{\sigma}$ |

$f={\left(\frac{q}{M{p}^{\prime}}\right)}^{n}+\frac{1}{ln\left(r\right)}ln\left(\frac{{p}^{\prime}}{R{p}_{0h}^{\prime}}\right)$ | ||

$dR=-ulnR|d{\mathit{\epsilon}}^{p}|$ | ||

Densification mechanism | ||

${e}_{ah}=e-{e}_{h};\phantom{\rule{28.45274pt}{0ex}}{e}_{h}=e({S}_{h}+{S}_{i})$ | ||

${\kappa}_{h}=\kappa {\kappa}_{rf}$ | ||

${\kappa}_{rf}$ = 0 if $({S}_{h}+{S}_{i})=0$ | ||

${\kappa}_{rf}=3{({S}_{h}+(1-{S}_{h}){S}_{i})}^{2}-2.68({S}_{h}+({S}_{h}+(1-{S}_{h}){S}_{i})+0.9934$ | ||

if $0<({S}_{h}+(1-{S}_{h}){S}_{i})\le 0.42$ | ||

${p}_{0h}^{\prime}={e}^{\frac{e({S}_{h}+{S}_{i})}{\lambda -{\kappa}_{h}}}{{p}^{\prime}}_{0}^{\frac{\lambda -\kappa}{\lambda -{\kappa}_{h}}}$ |

^{†}${\lambda}_{c}$ is not computed in terms of ${S}_{h}$ or ${S}_{i}$. This simplification valid for low saturations of ice [48];

^{‡}m and

^{‡}${P}_{0}$ values correspond to those given in the benchmark problem analyzed in Section 3.1. [44]. The m value is included within the range of values reported in the literature for gas hydrate numerical simulation studies (e.g., [49,50]); Please note that ${e}_{ah},{\kappa}_{h}$ and ${p}_{0h}^{\prime}$ recover the hydrate-free parameters $e,\kappa $ and ${p}_{0}^{\prime}$ when $({S}_{h}+{S}_{i})$ = 0; The use of bold symbols denote vectors and tensors.

**Table 6.**Equilibrium restrictions for MHBS system. The dependent variables computed using each of the equilibrium restrictions are also included.

Equilibrium Restriction | Equation | Dependent Variable |
---|---|---|

Hydrate phase change | ||

Hydrate phase boundary [43] | $ln\left({P}_{eq}\right)=-1.644866$ × ${10}^{3}-0.1374178T+5.4979866$ × ${10}^{4}/T+2.64118188$ × ${10}^{2}ln\left(T\right)+S(1.1178266$ × ${10}^{4}+7.67420344T-4.515213$ × ${10}^{-3}{T}^{2}-2.04872879$ × ${10}^{5}/T-2.17246046$ × ${10}^{3}ln\left(T\right))+{S}^{2}(1.70484431$ × ${10}^{2}+0.118594073T-7.0581304$ × ${10}^{-5}{T}^{2}-3.0979619$ × ${10}^{3}/T-33.2031996ln\left(T\right))$ | Equilibrium pressure, ${P}_{eq}$ |

Methane solubility | $ln{\left({K}_{C{H}_{4}}\right)}_{{P}_{g}}=ln{\left({K}_{C{H}_{4}}\right)}_{{P}_{eq}}\frac{ln{\left({K}_{C{H}_{4}}\right)}_{0.1MPa}}{ln{\left({K}_{C{H}_{4}}\right)}_{{T}_{eq}\left({P}_{g}\right)}}$ | Dissolved methane concentration, ${\omega}_{l}^{m}$ |

[43] | $ln{\left({K}_{C{H}_{4}}\right)}_{{P}_{eq}}=-2.5640213$ × ${10}^{5}-1.644805$ × ${10}^{2}T+9.1089042$ × ${10}^{-2}{T}^{2}+4.90352929$ × ${10}^{6}/T+4.93009113$ × ${10}^{4}ln\left(T\right)+S[-5.1628513$ × ${10}^{2}-0.33622376T+1.8819904$ × ${10}^{-4}{T}^{2}+9.76525718$ × ${10}^{3}/T+99.523354ln\left(T\right)]$ | |

[51] | $ln{\left({K}_{C{H}_{4}}\right)}_{0.1MPa}=1$ × ${10}^{-9}\left[-417.5053+599.8626(100/T)+380.3636ln(T/100)-62.0764T/100+S[-0.06423+0.03498(T/100)-0.0052732{(T/100)}^{2}]\right]$ | |

$ln{\left({K}_{C{H}_{4}}\right)}_{{T}_{eq}\left({P}_{g}\right)}=1$ × ${10}^{-9}\left[-417.5053+599.8626(100/{T}_{eq}\left({P}_{g}\right))+380.3636ln({T}_{eq}\left({P}_{g}\right)/100)-62.0764{T}_{eq}\left({P}_{g}\right)/100+S[-0.06423+0.03498({T}_{eq}\left({P}_{g}\right)/100)-0.0052732{({T}_{eq}\left({P}_{g}\right)/100)}^{2}\right]$ | ||

${\omega}_{l}^{m}=1.604$ × ${10}^{-2}{{K}_{C{H}_{4}}}_{{P}_{g}}\phantom{\rule{8.53581pt}{0ex}}Kg/Kg$ | ||

Hydrate kinetic rate [52,53] | ${R}_{h}(T,{P}_{g})={\varphi}_{p}{S}_{h}{A}_{h}\left({K}_{d}\langle {P}_{eq}\left(T\right)-{P}_{g}\rangle -{K}_{f}\langle {P}_{g}-{P}_{eq}\left(T\right)\rangle \right)$ | Hydrate mass change, $d{m}_{h}$ |

${A}_{h}=0.375\phantom{\rule{5.69054pt}{0ex}}\mu /m$ | ||

${K}_{d}=124\times {10}^{3}$exp$(-9400/T\left(K\right))\phantom{\rule{5.69054pt}{0ex}}mol/{m}^{2}Pa\phantom{\rule{2.84526pt}{0ex}}s$ | ||

${K}_{f}=0.5875\times {10}^{-11}\phantom{\rule{5.69054pt}{0ex}}mol/{m}^{2}Pa\phantom{\rule{2.84526pt}{0ex}}s$ | ||

^{†}$d{m}_{h}={M}_{h}{R}_{h}(T,{P}_{g})$ | ||

${M}_{h}=0.018016{n}_{h}+0.016042\phantom{\rule{5.69054pt}{0ex}}Kg/mol$ | ||

Ice phase change | ||

Freezing characteristic function [54] | $1{-}^{\u2020}{S}_{i}=\left(1+{\left(\frac{1-(1-{\rho}_{i}/{\rho}_{l}){P}_{l}-{\rho}_{i}{L}_{i}ln(T/273.15)}{{P}_{0}}\right)}^{\frac{1}{1-m}}\right){)}^{-m}$ | Ice saturation, ${S}_{i}$ |

${\rho}_{i}=0.91\xb7{\rho}_{l}$ |

^{†}See Appendix A for the derivation of hydrate and ice mass conservation equations.

Test Conditions | |||||
---|---|---|---|---|---|

Test Name | Temperature (°C) | Confining Pressure (MPa) | Liquid Pressure (MPa) | Hydrate Saturation (%) | Remarks |

Hs_Temp | 8 → 11.3 | 8 | 6 | 0 | Thermal stimulation at ${\epsilon}_{a}=16\%$ |

Mhbs_Temp1 | 8 → 11.3 | 8 | 6 | 20 | Thermal stimulation at ${\epsilon}_{a}=16\%$ |

Mhbs_Temp2 | 8 → 11.3 | 8 | 6 | 20 | Thermal stimulation at ${\epsilon}_{a}=40\%$ |

Hs_Dep | 8 | 8 | 6 → 4.6 | 0 | Depressurization at ${\epsilon}_{a}=16\%$ |

Mhbs_Dep | 8 | 8 | 6 → 4.6 | 20 | Depressurization at ${\epsilon}_{a}=16\%$ |

© 2019 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 (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

De La Fuente, M.; Vaunat, J.; Marín-Moreno, H. Thermo-Hydro-Mechanical Coupled Modeling of Methane Hydrate-Bearing Sediments: Formulation and Application. *Energies* **2019**, *12*, 2178.
https://doi.org/10.3390/en12112178

**AMA Style**

De La Fuente M, Vaunat J, Marín-Moreno H. Thermo-Hydro-Mechanical Coupled Modeling of Methane Hydrate-Bearing Sediments: Formulation and Application. *Energies*. 2019; 12(11):2178.
https://doi.org/10.3390/en12112178

**Chicago/Turabian Style**

De La Fuente, Maria, Jean Vaunat, and Héctor Marín-Moreno. 2019. "Thermo-Hydro-Mechanical Coupled Modeling of Methane Hydrate-Bearing Sediments: Formulation and Application" *Energies* 12, no. 11: 2178.
https://doi.org/10.3390/en12112178