# Numerical Modeling on Dissociation and Transportation of Natural Gas Hydrate Considering the Effects of the Geo-Stress

^{1}

^{2}

^{3}

^{*}

_{2}Transformation and Storage in Deep Formations)

## Abstract

**:**

## 1. Introduction

_{2}replacement method [7,8,9]. Current thermal stimulation methods in mining technology are inefficient and can only achieve local heating [10]. The chemical reagents huff-n-puff method is high-cost and low-efficiency, and has a potential risk of environmental pollution [11]. The efficiency of the carbon dioxide replacement method is also low [12]. Consequently, the depressurization method is regarded as the most economical and environmentally friendly way for NGH exploitation [13,14,15]. By depressurization, the solid NGH decomposes into gas and water, and, then, gas escapes from the reservoir. Stable NGH serves as the cementing material for the mineral grains of rocks and improves the strength of the rock mass. The dissociation of the NGH may cause potential geo-hazards. Thus, a study on the dissociation and transportation mechanism of NGH, considering the effects of the geo-stress, contributes to the economic and safe development of NGH.

## 2. Methods

#### 2.1. Computer Model

^{2}. The gas phase followed the Peng–Robinson equations. The Euler model was adopted to simulate the multiphase flow in the porous rock containing NGH. This study turned the original three-dimensional model into a two-dimensional axisymmetric model for processing using ICEM software, as in Figure 1. The symmetry axis was chosen to be the bottom edge, and the calculation area was divided into 120 × 20 grids. FLUENT was then used to import the mesh model. In Masuda’s experiment, the left side was the outlet, while the surrounding and right side were the non-slip and free convection walls with external heat flow through the rubber casing around the sandstone. In this study, the casing thickness in the experiment was 10 mm, and the appropriate range of h was 1.90–272.7 W/(m

^{2}·K), and 50 W/(m

^{2}·K) was employed. The temperature monitoring points were P1 (7.5, 2.5), P2 (15, 2.5), and P3 (22.5, 2.5) in Figure 1. The initial conditions, and boundary conditions used in this simulation followed the experiment of Masuda et al. [43], as shown in Table 1 The properties of the fluids in the simulation referred to the standard database at the same temperature and pressure conditions as the experiment. The material properties of the porous sandstone referred to average values of the Berea sandstones in literature [44], as shown in Table 2.

#### 2.2. Mathematical Model

_{v}is volumetric strain, G is the shear modulus, K

_{T}is bulk modulus, δ is a unit matrix, H′ is the constitutive constant (stress-strain coupling of fluid), σ is confining pressure, and P

_{T}is pore pressure.

_{v}= 0) [52]:

_{V}is volumetric strain.

## 3. Results and Discussions

#### 3.1. Model Validation

^{2}K), this study holds that the result was mostly caused by the boundary’s uneven heat transmission. Considering the gas production at the outlet, the far-field boundary pressure and the temperature at the monitoring point, the curve fits of this study were good, compared with most previous research.

#### 3.2. Effect of Initial Gas Saturation

_{90%}(referring to the required time of 90% NGH dissociating in the core), decreased with increase of the initial gas saturation, as shown in Figure 6, which also showed that the higher the initial gas saturation, the faster the NGH decomposed.

#### 3.3. Effect of Outlet Pressure

_{90%}, decreased with a decrease in the outlet pressure, as shown in Figure 8, which also showed that the lower the outlet pressure, the faster the natural gas hydrate decomposed.

#### 3.4. Effect of Initial Temperature

_{90%}, decreased with increasing initial temperatures, as shown in Figure 10, which also showed that the higher the initial temperatures, the faster the natural gas hydrate decomposed. As shown in Figure 11, it was discovered that, under adiabatic conditions, the gas generation rate rose instantly in the beginning and then gradually fell until it reached zero. As the final cumulative gas production was 2800 Scm

^{3}, no new gas was generated, so it could be concluded that the hydrate would not decompose under adiabatic conditions (insufficient energy).

#### 3.5. Effect of Absolute Permeability

_{90%}, decreased with an increase in the absolute permeability, as shown in Figure 13, which also showed that the higher the absolute permeability, the faster the natural gas hydrate decomposed.

#### 3.6. Effect of Geo-Stress

_{90%}rose as effective stress increased. This was because the increase of effective stress led to the volume shrinkage of pore space, thus reducing the permeability and porosity of sediments. Low permeability slowed thermal convection, which slowed down the pressure drop from one end of the core to the outlet and reduced mass transfer efficiency. This slowed down hydrate decomposition, which increased the time it took to produce gas from natural gas hydrates and reduced their exploitation efficiency. Therefore, the natural gas hydrate production would be less effective as effective stress rose, under various effective stresses.

## 4. Conclusions

- The established mathematical model and the simulation scheme were validated by historical matching with the experimental benchmark data.
- The sensitivity analysis of the parameters revealed that a higher absolute permeability, higher initial gas saturation, lower outlet pressure, and higher initial temperature advanced the decomposition rate of hydrate. Thus, an optimized production plan is essential to promote the extraction efficiency of the NGH.
- Geo-stress caused a decrease of the porosity and permeability in the porous rock, which restricted the efficiency of the heat and mass transfer by the fluid flow, leading to a slow dissociation and transportation rate of the NGH. Thus, it is essential to take geo-stress into consideration and balance the extracting efficiency and the well pressure, especially when the NGH is developed by depressurization.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

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**Figure 2.**Gas production vs. time in the simulation experiment and Masuda’s experiment [43].

**Figure 3.**Temperature (T1, T2, T3) vs. time in the simulation experiment and Masuda’s experiment [43].

**Figure 4.**Pore pressure vs. time in the simulation experiment and Masuda’s experiment [43].

**Figure 5.**Gas production rate and cumulative gas production vs. time under different initial gas saturations.

**Figure 7.**Gas production rate and cumulative gas production vs. time under different outlet pressure.

**Figure 9.**Gas production rate and cumulative gas production vs. time at different initial temperature.

**Figure 11.**Gas production rate and cumulative gas production vs. time under adiabatic boundary conditions.

**Figure 14.**Fitted curves of porosity and permeability of Berea sandstone vs. effective stress in Zhu and Wang’s experimental data [56].

**Figure 19.**Gas production rate and cumulative gas production vs. time under different effective stresses.

**Figure 22.**Temperature of monitoring points vs. time under different effective stress conditions and without considering stress. Subgraph (

**a**–

**c**) show that the temperature of monitoring points vs. time under the effective stress of 5 MPa, 15 MPa, 25 MPa and without considering stress.

**Table 1.**Initial conditions, boundary conditions and properties of sandstone cores in Masuda’s experiment.

Properties | Value | Properties | Value |
---|---|---|---|

Average saturation of initial hydrate | 0.501 | Initial permeability of sandstone core | 97.98 mD |

Average saturation of initial water | 0.199 | Sandstone core porosity | 0.182 |

Average saturation of initial methane gas | 0.3 | Joule-Thomson throttling coefficient | −1.5 × 10^{−4} |

initial temperature | 275.45 K | Critical pressure of methane | 4.599 MPa |

initial pressure | 3.75 MPa | Critical temperature of methane | 190.56 K |

outlet pressure | 2.84 MPa | ambient temperature | 274.15 K |

Density (kg/m^{3)} | Fluid Viscosity (cP) | Thermal Conductivity (w·m^{−1}·k^{−1}) | Thermal Capacity (J·kg^{−1}·k^{−1}) | |
---|---|---|---|---|

water | 1001.5 | 1 | 0.6 | 4180 |

hydrate | 913 | - | 0.393 | 2010 |

methane | PR equation | 0.01 | 0.00332 | 2190 |

Berea sandstone | 2030 | - | 5 | 800 |

Sun et al. (2005) | Nazridoust and Ahmadi. (2007) | Ruan et al. (2012) | Chen et al. (2016) | This Study | ||
---|---|---|---|---|---|---|

Modle | 1-D | 2-D | 2-D | 2-D | 2-D | |

Flow model | Darcy’s law | Darcy’s law | Darcy’s law | Darcy’s law | Darcy’s law | |

Relative permeability | Corey’s model (1954) | Corey’s model (1954) | Corey’s model (1954) | Corey’s model (1954) | Adapted Corey’s model (1954) | |

Permeability model | $\begin{array}{l}{k}_{rw}=k{\left(\frac{\frac{{s}_{w}}{{s}_{w}+{s}_{g}}-{s}_{wr}}{1-{s}_{wr}-{s}_{gr}}\right)}^{nw},{k}_{rg}={\left(\frac{\frac{{s}_{g}}{{s}_{w}+{s}_{g}}-{s}_{gr}}{1-{s}_{wr}-{s}_{gr}}\right)}^{ng}\\ {n}_{w}=4,{n}_{g}=2,{s}_{wr}=0.2,{s}_{gr}=0.3;\end{array}$ | $\begin{array}{l}{k}_{D}={k}_{D0}{\left(1-{s}_{h}\right)}^{N}\\ N=15\end{array}$ | $\begin{array}{l}{k}_{D}={k}_{D0}{\left(1-{s}_{h}\right)}^{N}\\ N=11\end{array}$ | $\begin{array}{l}{k}_{D}={k}_{D0}{\left(1-{s}_{h}\right)}^{N}\\ N=15\end{array}$ | $\begin{array}{l}{k}_{rw}={\left(\frac{{s}_{w}-{s}_{wr}}{1-{s}_{wr}}\right)}^{{n}_{w}},{k}_{rg}={\left(\frac{{s}_{g}}{1-{s}_{wr}}\right)}^{{n}_{g}}\\ {n}_{w}=0.6,{n}_{g}=1.42;\end{array}$ | where K_{rw} and k_{rg} are the relative permeability of water and gas, S_{wr} is the irreducible saturation of water. |

${k}_{D}=\left\{\begin{array}{l}5.51721\times {\left({\phi}_{e}\right)}^{0.86},{\phi}_{e}<0.11\\ 4.84653\times {10}^{8}\times {\left({\phi}_{e}\right)}^{9.13},{\phi}_{e}\ge 0.11\end{array}\right.$ | ${k}_{D}=\left\{\begin{array}{l}5.51721\times {\left(\frac{{\phi}_{0}-{\epsilon}_{V}}{1-{\epsilon}_{V}}(1-{s}_{h})\right)}^{0.86},\frac{{\phi}_{0}-{\epsilon}_{V}}{1-{\epsilon}_{V}}(1-{s}_{h})\ge 0.11\\ {k}_{DO}{(\frac{{\phi}_{0}-{\epsilon}_{V}}{1-{\epsilon}_{V}}(1-{s}_{h}))}^{1.2},\frac{{\phi}_{0}-{\epsilon}_{V}}{1-{\epsilon}_{V}}(1-{s}_{h})\ge 0.11\end{array}\right.$ | where φ_{O} is absolute porosity,ε _{v} is volumetric strain,k _{DO} absolute permeability of the sandstone without contains hydrate, mDS _{h} is saturation of hydrate. | ||||

Kim et al. (1987) | Kim et al. (1987) | Kim et al. (1987) | Kim et al. (1987) | Kim et al. (1987) | ||

Dissociation rate (k_{d}, A_{d}) | ${\dot{m}}_{g}={k}_{d}{A}_{S}\left({f}_{e}-f\right),{k}_{d}=4.4\times {10}^{-16}$ | ${\dot{m}}_{g}={k}_{d}{M}_{g}{A}_{d}\left({p}_{eh}-{p}_{g}\right)={k}_{d}^{o}{e}^{-\frac{\mathrm{\Delta}E}{RT}}{M}_{g}{A}_{d}\left({P}_{eg}-{P}_{g}\right)$ | where P_{eh} is the equilibrium pressure,P _{g} is the methane pressure,A _{d} is reacting surface of hydrate, ${K}_{d}^{0}$ is the intrinsic constant,R is the universal gas constant, ΔE is an activation energy. | |||

Dissociation constant k_{d} (kmol/Pa.s.m^{2}) | ${k}_{d}^{0}=2.75\times {10}^{-12}$ | $\begin{array}{l}{k}_{d}^{0}=8.06\\ \mathrm{\Delta}E=77.33\times {10}^{3}\mathrm{J}\end{array}$ | $\begin{array}{l}{k}_{d}^{0}=36\\ \mathrm{\Delta}E=81.08\times {10}^{3}\mathrm{J}\end{array}$ | $\begin{array}{l}{k}_{d}^{0}=124\\ \mathrm{\Delta}E=78.15\times {10}^{3}\mathrm{J}\end{array}$ | $\begin{array}{l}{k}_{d}^{0}=36\\ \mathrm{\Delta}E=81.08\times {10}^{3}\mathrm{J}\end{array}$ | |

Surface area of hydrate per unit volume A_{d} | ${A}_{d}=\sqrt{\frac{{\phi}_{e}^{3}}{2{K}_{D}}}$ | $\begin{array}{l}{A}_{d}={\phi}_{0}{S}_{h}{A}_{geo},2r=16\mathsf{\mu}\mathrm{m}\\ {A}_{geo}=3.75\times {10}^{5}{\mathrm{m}}^{2}/{\mathrm{m}}^{3}\end{array}$ | ${A}_{d}=\sqrt{\frac{{\phi}_{e}^{3}}{2{K}_{D}}}$ | $\begin{array}{l}{A}_{d}={\phi}_{0}{S}_{h}{A}_{geo},2r=16\mathsf{\mu}\mathrm{m}\\ {A}_{geo}=3.75\times {10}^{5}{\mathrm{m}}^{2}/{\mathrm{m}}^{3}\end{array}$ | ${A}_{d}=\sqrt{\frac{{\phi}_{e}^{3}}{2{K}_{D}}}$ | ${\phi}_{e}={\phi}_{o}(1-{s}_{h})$ where φ _{e} is the effective porosity of porous media. |

The heat transfer Enthalpy, internal energy (J/Kg) | $\mathrm{\Delta}{H}_{d}=446.12\times {10}^{3}-132.638T$ | $\mathrm{\Delta}{H}_{d}=473.63\times {10}^{3}-140.117T$ | Sun et al. (2005) | $\mathrm{\Delta}{H}_{d}=473.63\times {10}^{3}-140.117T$ | ${\dot{m}}_{w}={M}_{W}{N}_{H}{{\dot{m}}_{g}/M}_{g},-{\dot{m}}_{h}={M}_{h}{{\dot{m}}_{g}/M}_{g}$ $q={\lambda}_{b}\left({T}_{o}-T\right)$ $\mathrm{\Delta}{H}_{d}=\left[\begin{array}{l}215.59\times {10}^{3}-394.945T,248K<T<273K\\ 446.12\times {10}^{3}-132.638T,273K<T<298K\end{array}\right.$ | where M_{w} and M_{h} are molecular weights of water and hydrate.where q is boundary heat flux, T _{o} is air temperature,λ _{b} is the boundary heat transfer coefficient determined by the heat transfer coefficient of the rubber sleeve and the ambient convection intensity. |

Sun et al. (2005) | Nazridoust and Ahmadi. (2007) | Ruan et al. (2012) | Chen et al. (2016) | This Study | ||

Continuity equation of different fluid phases are: Momentum equation | $\begin{array}{l}\frac{\partial}{\partial t}\left({\phi}_{0}{\rho}_{k}{S}_{k}\right)+\frac{\partial}{\partial x}\left({\rho}_{k}{S}_{k}\right)={\dot{m}}_{k}\\ \left(k=h,g,w\right)\end{array}$ | $\begin{array}{l}-\nabla .{\rho}_{k}\overrightarrow{{u}_{k}}+{\dot{m}}_{k}=\frac{\partial}{\partial t}\left({\phi}_{0}{\rho}_{k}{S}_{k}\right)\\ \left(k=h,g,w,i\right)\end{array}$ | $\begin{array}{l}-\frac{1}{r}\frac{\partial}{\partial r}\left(r{\rho}_{k}{v}_{kr}\right)+\frac{\partial}{\partial x}\left({\rho}_{k}{v}_{kx}\right)+{\dot{q}}_{k}\\ +{\dot{m}}_{k}=\frac{\partial}{\partial t}\left({\phi}_{0}{\rho}_{k}{S}_{k}\right)\left(k=g,w\right)\\ {\dot{m}}_{h}=\frac{\partial}{\partial t}\left({\phi}_{0}{\rho}_{h}{S}_{h}\right)\end{array}$ | Nazridoust and Ahmadi. (2007) | $-\nabla .{\rho}_{k}\overrightarrow{{u}_{k}}+{\dot{m}}_{k}=\frac{\partial}{\partial t}\left({\phi}_{0}{\rho}_{k}{S}_{k}\right)$ $\left(k=h,g,w\right)$ where ρ is the density, m _{k} is the mass rate of dissociation formation substance, φ_{0} is the porosity, S is the saturation, μ_{k} is the fluid velocity. The subscripts h, g, w corresponds to hydrate, gas and water in multiphase systems, respectively.${u}_{k}=-\frac{{k}_{D}{k}_{rk}}{{\mu}_{k}}\nabla {p}_{k,}\left(k=g,w\right)p$ where u _{k} is the relative permeability of phase k, K_{D} is the absolute permeability of hydrated sandstone, K_{rk} is the relative permeability of phase k, and P is fluid pressure $\frac{\partial}{{\partial}_{t}}\left[\begin{array}{l}\left(1-{\phi}_{o}\right){\rho}_{R}{C}_{R}T+{\phi}_{O}{S}_{h}{\rho}_{h}{C}_{h}T\\ +{\phi}_{O}{S}_{w}{\rho}_{w}{C}_{w}T+{\phi}_{O}{S}_{g}{\rho}_{g}{C}_{g}T\end{array}\right]{C}_{K}$ $+\nabla T.\left({\rho}_{w}{C}_{w}{\overrightarrow{u}}_{w}+{\rho}_{g}{C}_{g}{\overrightarrow{u}}_{g}\right)-\nabla .\left({\lambda}_{e}\nabla T\right)={\dot{Q}}_{h}$ where C is the heat capacity, T is the temperature, $\overrightarrow{u}$ is the internal energy, Subscript R represents rock, h is the enthalpy. λ _{e} is the effective thermal conductivity, ${\dot{Q}}_{h}$ is the source term of endothermic reaction based on hydrate dissociation.${\dot{Q}}_{h}=-{\dot{m}}_{h}\mathrm{\Delta}{H}_{d}-\varphi {\rho}_{g}{S}_{g}{\sigma}_{g}\frac{\partial {p}_{g}}{\partial t}-{\rho}_{g}{u}_{g}\nabla {p}_{g}$ where ${\dot{m}}_{h}$ is the mass dissociation rate for methane hydrate, ΔH _{d} is the latent heat of hydrate during the dissociation of methane hydrate, σ_{g} is the Joule-Thomson throttling coefficient, σ_{g} = −1.5 × 10 ^{−4}${\lambda}_{e}=\left(1-{\phi}_{0}\right){\lambda}_{R}+{\phi}_{0}\left({s}_{h}{\lambda}_{h}+{s}_{w}{\lambda}_{w}+{s}_{g}{\lambda}_{g}\right)$ where λ _{R},λ_{h},λ_{w},λ_{g} are thermal conductivity of rock, hydrate, water, gas, respectively. | |

Energy equation | $\begin{array}{l}\frac{\partial}{{\partial}_{t}}\left[\begin{array}{l}\left(1-{\phi}_{o}\right){\rho}_{R}{H}_{R}+{\phi}_{O}{S}_{h}{\rho}_{h}{H}_{h}\\ +{\phi}_{O}{S}_{w}{\rho}_{w}{H}_{w}+{\phi}_{O}{S}_{g}{\rho}_{g}{H}_{g}\end{array}\right]\\ +\frac{\partial}{{\partial}_{x}}\left({\rho}_{w}{v}_{w}{H}_{w}+{\rho}_{g}{v}_{g}{H}_{g}\right)\\ =\frac{\partial}{{\partial}_{x}}\left(\lambda \frac{\partial T}{{\partial}_{x}}\right)+q\\ \end{array}$ | $\begin{array}{l}\frac{\partial}{{\partial}_{t}}\left[\begin{array}{l}\left(1-{\phi}_{o}\right){\rho}_{R}{C}_{R}T+{\phi}_{O}{S}_{h}{\rho}_{h}{C}_{h}T\\ +{\phi}_{O}{S}_{w}{\rho}_{w}{C}_{w}T+{\phi}_{O}{S}_{g}{\rho}_{g}{C}_{g}T\end{array}\right]{C}_{K}\\ +\nabla T.\left({\rho}_{w}{C}_{w}{\overrightarrow{u}}_{w}+{\rho}_{g}{C}_{g}{\overrightarrow{u}}_{g}\right)-\nabla .\left({\lambda}_{e}\nabla T\right)={\dot{Q}}_{h}\end{array}$ ${\dot{Q}}_{h}=\frac{-{\dot{m}}_{H}\left(c+dT\right)}{{M}_{H}}$ C = 56.599J/mol, d = −16.744J/mol K | $\begin{array}{l}\frac{1}{r}\frac{\partial}{\partial r}\left(r{k}_{c}\frac{\partial T}{\partial r}\right)+\frac{\partial}{\partial x}\left({k}_{c}\frac{\partial T}{\partial x}\right)\\ -\frac{1}{r}\frac{\partial}{\partial r}\left(r{\rho}_{g}{v}_{gr}{h}_{g}+r{\rho}_{w}{v}_{wr}{h}_{w}\right)\\ -\frac{\partial}{\partial x}\left({\rho}_{g}{v}_{gr}{h}_{g}+{\rho}_{w}{v}_{wr}{h}_{w}\right)\\ +\frac{\partial}{\partial x}\left({\rho}_{k}{v}_{kx}\right)+{\dot{q}}_{g}{h}_{g}+{\dot{q}}_{w}{h}_{w}+{\dot{q}}_{h}+{\dot{q}}_{in}\\ =\frac{\partial}{\partial t}\left[(1-\phi ){\rho}_{r}{h}_{r}+\phi \left({h}_{h}{\rho}_{h}{S}_{h}+{h}_{g}{\rho}_{g}{S}_{g}+{h}_{w}{\rho}_{w}{S}_{w}\right)\right]\\ \end{array}$ | Nazridoust and Ahmadi. (2007) |

Rock | ${\mathit{A}}^{\prime}$ | ${\mathit{B}}^{\prime}$ | ${\mathit{C}}^{\prime}$ |
---|---|---|---|

Berea sandstone | 0.183 | 0.01859 | 9.842 |

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**MDPI and ACS Style**

Song, R.; Duan, Y.; Liu, J.; Song, Y.
Numerical Modeling on Dissociation and Transportation of Natural Gas Hydrate Considering the Effects of the Geo-Stress. *Energies* **2022**, *15*, 9311.
https://doi.org/10.3390/en15249311

**AMA Style**

Song R, Duan Y, Liu J, Song Y.
Numerical Modeling on Dissociation and Transportation of Natural Gas Hydrate Considering the Effects of the Geo-Stress. *Energies*. 2022; 15(24):9311.
https://doi.org/10.3390/en15249311

**Chicago/Turabian Style**

Song, Rui, Yaojiang Duan, Jianjun Liu, and Yujia Song.
2022. "Numerical Modeling on Dissociation and Transportation of Natural Gas Hydrate Considering the Effects of the Geo-Stress" *Energies* 15, no. 24: 9311.
https://doi.org/10.3390/en15249311