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

Abstract

A deep understanding of the dissociation and transportation mechanism of natural gas hydrate (NGH), taking into account the effects of geo-stress, contributes to optimizing the development strategy and increases the exploitation efficiency of NGH. In this paper, the mathematical model, coupled with fluid heat and mass transfer, multiphase flow mechanics, and reaction kinetics with phase change in the process of hydrate decomposition was established. An axisymmetric two-dimensional model was developed to simulate the depressurization decomposition process of natural gas hydrate in the Berea sandstones. FLUENT software was used to solve the fundamental governing equations of the multi-phase flow, and UDF programming was employed to program the hydrate decomposition model and the modified permeability model in the dissociation and transportation of NGH. The simulation results were then validated by Masuda’s experimental data. The effects of gas saturation, outlet pressure, temperature, absolute permeability and geo-stress on the decomposition of natural gas hydrate were studied. The results indicated that a higher absolute permeability, higher initial gas saturation, lower outlet pressure, and higher initial temperature advance the decomposition rate of hydrate. Thus, an optimized production plan is essential to promote the extraction efficiency of the NGH. The geo-stress causes a decrease in the porosity and permeability of the porous rock, which restricts the efficiency of the heat and mass transfer by the fluid flow, leading to a slower dissociation and transportation rate of the NGH. Thus, it is important to take geo-stress into consideration and balance the extracting efficiency and the well pressure, especially when the NGH is developed by depressurization.

1. Introduction

The increasing global demands for energy and environmental protection urge governments and scientists to exploit unconventional oil and gas resources, but with tremendous reservation, such as coalbed methane, shale gas, tight sandstone gas, and natural gas hydrate, among others [1,2]. Natural gas hydrate (NGH), which accounts for the largest proportion of unconventional natural gas resources, is a kind of high-quality, efficient, and clean energy with great potential [3]. NGH is a non-fixed stoichiometric cage-like crystalline compound formed by the reaction of one or several small molecular gases with water at a certain temperature and pressure [4]. One cubic meter of NGH can release 164 cubic meters of methane in standard conditions, which is 10 times the energy density of other unconventional gas source rocks (such as coalbed methane), and 2~5 times the energy density of conventional natural gas [5]. Therefore, the commercial exploitation of NGH is crucial to worldwide energy supply and economic development.

NGH requires low temperature and high-pressure conditions to maintain a stable state, and is generally distributed in deep sea or continental permafrost. Up to now, the short-term field tests of hydrate exploitation in the world have been carried out in sandy (or partially sandy) sediments [6]. The exploitation methods of NGH mainly include the thermal stimulation method, depressurization method, chemical reagent injection method and CO₂ 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.

To date, extensive studies have been performed to investigate the dissociation and transportation process of NGH in porous media by numerical modeling. Zhao et al. [16] simulated the distribution of temperature, pressure, and hydrate saturation as natural gas hydrate decomposed in a two-dimensional axisymmetric model. Deng et al. [17] established a coupling model of two-phase flow and heat transfer for hydrate dissociation. Zhang et al. [18] adopted the dynamic grid algorithm to analyze the effects of permeability, lithology, and reservoir compression modulus on the recovery efficiency of natural gas hydrate in the core. Based on the implicit pressure–explicit saturation (IMPES) method, Liang et al. [19] created a two-dimensional axisymmetric model and analyzed the effects of outlet pressure, initial hydrate saturation and other parameters on hydrate decomposition and gas production rate in porous media. Ruan et al. [20] investigated the responses of hydrate dissociation to permeability and hydrate saturation of the reservoir. Two-dimensional axisymmetric models were established in FLUENT software to investigate the impact of temperature, permeability, pressure, phase distribution and other factors on the rate of NGH dissociation [21,22,23,24,25,26,27,28]. However, the geo-stress was not considered in these studies. The effective pressure is believed to be one of the key factors for efficiency and stability in the formation of NGH [29]. White et al. [30] integrated the geomechanical calculation into the existing thermal and hydrological coupling model framework to simulate the process of NGH exploitation by using IGHCCS2 codes. Cheng et al. [31] developed a gas–water two-phase fluid mechanics coupling model to investigate the effects of the interaction of the fluid and solid on the gas production of NGH. Lee et al. [32] developed a 3D geological model to investigate the geo-stress on gas production of NGH by using the cyclic depressurization method. Liu et al. [33] utilized the thermal-hydrological-mechanical-chemical (THCM) coupling model to study the geomechanical issues during the exploitation of NGH. Wu et al. [34] investigated the effects of the effective stresses on the permeability and porosity of natural gas hydrate reservoirs with different NGH saturations. Sun et al. [35] developed a fully coupled thermal-hydrological-mechanical-chemical (THCM) model. The impacts of compressibility on the rate of gas production, pore pressure, temperature, permeability, and other parameters were investigated. Jang et al. [36] studied the effects of geo-stress on fluid flow in NGH reservoirs. Kimoto et al. [37] conducted a numerical simulation based on a chemical-thermal-mechanical coupling model and an updated Lagrangian equation, and investigated the influencing factors of ground deformation caused by NGH dissociation. Sun et al. [38,39] used COMSOL to simulate the mechanical behavior of NGH by integrating the Mohr–Coulomb model into the fully coupled thermal-water-mechanics model. Cheng et al. [40] proposed a thermal-hydro-mechanical (THM) coupling model and studied the distribution of pressure, saturation of each phase, temperature, and mechanical parameters around the wellbore. Wan et al. [41] proposed a thermal-water-mechanical-chemical (THMC) coupling model based on the CVFEM framework, which was then validated by Masuda’s triaxial compression experiment [42] and sandstone core experiment [43]. However, the mechanical parameters used in most of these studies were estimated empirically, which limited the validation and applicability in engineering applications.

In this paper, a mathematical model, considering fluid heat and mass transfer, multiphase flow mechanics, and reaction kinetics with phase change in the process of hydrate decomposition, was established. In this model, the mechanical properties of the Berea sandstone and its stress sensitivity were adopted. The THMC codes were programmed in C language and used as a subroutine in FLUENT software. The Masuda’s experiments on the Berea sandstone were simulated and used as benchmark data for model verification. The effects of initial gas saturation, outlet pressure, initial temperature, absolute permeability and geo-stress on the decomposition of natural gas hydrate were studied.

2. Methods

2.1. Computer Model

This work employed Masuda’s experimental benchmark data [43] to validate the numerical model, and, on that basis, addressed and investigated the effect of effective stress on hydrate decomposition and gas production in sandy sediments. Masuda’s experiment involved distributing a specified volume of water, methane gas, and hydrate in the core of a cylindrical sandstone. The core had a cross-sectional size of 20.3 cm². 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/(m²·K), and 50 W/(m²·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. 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

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. Material properties in the simulation.

	Density (kg/m3)	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

.

2.2. Mathematical Model

Mathematical models of the multiphase flow and heat transfer include mass balance equations, heat balance equations, additional source terms of mass and energy, and initial and boundary conditions. The basic governing equations for the multiphase flow in the porous rock and the reaction kinetics of the NGH dissociation are presented in

Table 3. Different models of porous media methane hydrate dissociation flow.

	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 Krw and krg are the relative permeability of water and gas, Swr 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 φO is absolute porosity, εv is volumetric strain, kDO absolute permeability of the sandstone without contains hydrate, mD Sh 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 (kd, Ad)	${\dot{m}}_g=k_d{A}_S\left(f_e-f\right),k_d=4.4\times {10}^{-16}$				${\dot{m}}_g=k_d{M}_g{A}_d\left(p_{eh}-p_g\right)=k_d^o{e}^{-\frac{\mathrm{\Delta }E}{RT}}{M}_g{A}_d\left(P_{eg}-P_g\right)$	where Peh is the equilibrium pressure, Pg is the methane pressure, Ad 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 kd (kmol/Pa.s.m2)	$k_d^0=2.75\times {10}^{-12}$	$\begin{array}{l}{k}_d^0=8.06\\ \mathrm{\Delta }E=77.33\times {10}^3 \mathrm{J}\end{array}$	$\begin{array}{l}{k}_d^0=36\\ \mathrm{\Delta }E=81.08\times {10}^3 \mathrm{J}\end{array}$	$\begin{array}{l}{k}_d^0=124\\ \mathrm{\Delta }E=78.15\times {10}^3 \mathrm{J}\end{array}$	$\begin{array}{l}{k}_d^0=36\\ \mathrm{\Delta }E=81.08\times {10}^3 \mathrm{J}\end{array}$
Surface area of hydrate per unit volume Ad	$A_d=\sqrt{\frac{{\phi }_e^3}{2K_D}}$	$\begin{array}{l}{A}_d={\phi }_0{S}_h{A}_{geo},2r=16 \mathsf{\mu }\mathrm{m}\\ A_{geo}=3.75\times {10}^5 {\mathrm{m}}^2/{\mathrm{m}}^3\end{array}$	$A_d=\sqrt{\frac{{\phi }_e^3}{2K_D}}$	$\begin{array}{l}{A}_d={\phi }_0{S}_h{A}_{geo},2r=16 \mathsf{\mu }\mathrm{m}\\ A_{geo}=3.75\times {10}^5 {\mathrm{m}}^2/{\mathrm{m}}^3\end{array}$	$A_d=\sqrt{\frac{{\phi }_e^3}{2K_D}}$	${\phi }_e={\phi }_o(1-s_h)$ where φe 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 Mw and Mh are molecular weights of water and hydrate. where q is boundary heat flux, To is air temperature, λb 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, mk is the mass rate of dissociation formation substance, φ0 is the porosity, S is the saturation, μk 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 uk is the relative permeability of phase k, KD is the absolute permeability of hydrated sandstone, Krk 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. λe 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, ΔHd is the latent heat of hydrate during the dissociation of methane hydrate, σg is the Joule-Thomson throttling coefficient, σg = −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 λR,λh,λw,λg 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)

.

The effects of the geo-stress are also coupled with the transportation and dissociation process of the NGH in the porous media. Based on the assumptions of stress and strain linearity and the reversibility of medium deformation, Biot et al. [45] proposed the pore elastic constitutive equation of fluid saturated porous media. Detournay et al. [46] proposed the pore elastic constitutive equation illustrated in Equation (1), and defined volumetric strain as follows, under isotropic compressive stress:

$${\epsilon }_V=\frac{\sigma }{2G}-(\frac{1}{6G}-\frac{1}{9K_T})\delta \sigma +\frac{1}{3H^{\prime }}\delta P_T$$

(1)

$${\epsilon }_V=3\epsilon =\frac{1}{K}(\sigma -\alpha P_T)$$

(2)

where ε_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.

Usually, the effective stress is calculated, based on the test results of variable confining pressure experiments. There are various forms of effective stress. In 1923, Terzaghi proposed the calculation formula of effective stress:

$${\sigma }^{\prime }=\sigma -p$$

(3)

In this study, the influence of effective stress in sandstone core was studied, and the effective stress coefficient was set to 1 in this study.

$${\sigma }^{\prime }=\sigma -\alpha p$$

(4)

where α is the effective stress coefficient. Fatt et al. [47,48] suggested that α could be set between 0.75 and 1. Knaap et al. [49], Hubbert et al. [50] and Zimmerman et al. [51] believed that the effective stress coefficient was set to 1 through the experimental results. In this study, α = 1.

When the stress and pore pressure are known, the volumetric strain can be obtained by using the linear isotropic theory of pore elasticity. For the case of nonlinear volumetric deformation in porous rock, it was found that the fitting equation of stress–strain data collected in the Berea experiment was as follows (meeting the initial conditions σ′ = 0, ε_v = 0) [52]:

$${\epsilon }_V=A^{\prime }{\sigma }^{\prime }-B^{\prime }{e}^{-{\sigma }^{\prime }/C^{\prime }}+B^{\prime }{C}^{\prime }$$

(5)

where A’, B’, C’ are the fitting constants [53], ε_V is volumetric strain.

Haghi et al. [54] derived the calculation equation of stress-dependent porosity of intact rock without considering the differences between particle compressibility and pore compressibility:

$${\phi }_t=\frac{{\phi }_0-{\epsilon }_V}{1-{\epsilon }_V}$$

(6)

Then the effective porosity under the condition of different geo-stresses is:

$${\phi }_e=\frac{{\phi }_0-{\epsilon }_V}{1-{\epsilon }_V}(1-s_h)$$

(7)

A new permeability formula under effective stress can be obtained as follows:

$$k_D=\left\{\begin{array}{l}5.51721\times {\left(\frac{{\phi }_0-{\epsilon }_V}{1-{\epsilon }_V}(1-s_h)\right)}^{0.86},\frac{{\phi }_0-{\epsilon }_V}{1-{\epsilon }_V}(1-s_h)<0.11\\ k_{DO}{(\frac{{\phi }_0-{\epsilon }_V}{1-{\epsilon }_V}(1-s_h))}^{1.2},\frac{{\phi }_0-{\epsilon }_V}{1-{\epsilon }_V}(1-s_h)\ge 0.11\end{array}\right.$$

(8)

3. Results and Discussions

3.1. Model Validation

The gas production vs. time in the simulation and Masuda’s experiment are presented in Figure 2. The simulation results showed the generation of methane gas in this study was slightly delayed, compared to experimental results. The temperature of the inspecting points (P1, P2, P3) vs. time in the simulation and Masuda’s experiment are presented in Figure 3. It can be seen that when the time t was in the range of 0–100 min, closer to the outlet, the faster the temperature dropped, which was generally consistent with Masuda’s experimental results. The minimum temperature of the three monitoring points in the simulation was lower than that of the Masuda experiment. The R-square values of the three temperature monitoring points were 0.882, 0.799, and 0.852, respectively. The pore pressure at the far-field boundary vs. time is presented in Figure 4. The overall trend of the pressure drop at the far end in the simulation experiment was consistent with the results of the Masuda experiment. The R-square of the far end boundary pressure was 0.890. The variations between the simulation and experimental results were caused by the heterogeneous distribution of the different phases, as well as the heterogeneous pore structure in the Berea sandstone. Considering the influence of pore structure, hydrate heterogeneity, and the application of the mathematical model on the experiment, the simulation results were believed to be in good agreement with the Masuda experiment.

As shown in Appendix A, the gas production of methane vs. time varies in literatures and in this study. The reasons lie in the fact that the initial hydrate saturation in some literature [19,20,21,22,23,24,25,43] was 0.443, while it was 0.501 in some others [17,18,26,28,41] and in this study. Zhang et al. [18] and Ruan’s [20] gas production curves were in good agreement with the trials; however, Zhang only simulated gas production. The far-field boundary pressure was not simulated by Ruan et al. Chen et al. [23] simulated the pressure and temperature at the monitoring point without the gas generation. Additionally, the findings in this study and the literature [17,21,22,23,24,25,27] demonstrated that the early-stage gas generation rate was lower. The gas production rate was relatively higher, according to modeling results from Song et al. [44]. Overall, the simulation results in this study agreed well with the experimental benchmark data in far-field boundary pressure, temperature, and gas production, by assuming a homogenous initial distribution of the hydrate, water and methane in Berea sandstone.

As shown in Appendix B, Nazridoust et al. [21] the distal pressure did not significantly change at the start of the simulation, and the distal pressure of Chen et al. [23] was at its lowest point after 125 min of reaction time. Furthermore, Wan et al. [41] found that although the downward pressure trend throughout the earlier time was consistent, the downward pressure trend between 50 and 200 min was rather sluggish, and none of their results included small peaks. Small peaks were visible in the simulation findings from this work. The small peak predicted by Song et al. [44] and the trials were essentially in agreement; however, the pressure value remained high after the small peak arrived. The trials ‘small peak fluctuated as the pressure lowered to roughly 3.2 MPa, and the small peak anticipated in this work exhibited the same behavior, but it occurred about 20 min later. Overall, the simulation results in this study agreed well with the experimental benchmark data in far-field boundary pressure.

Theoretically, the closer to the outlet, the lower the temperature and the earlier the dissociation of the hydrate. However, as shown in Appendix C, Chen et al.’s findings [23] did not support this conclusion. In terms of the temperatures at each monitoring point in the sandstone core during hydrate dissociation, the findings in this study and the literature [17,18,19,20,21,22,24,25,26,41] basically predicted the correct result. That is, the temperature at T1 decreased earlier than that at T2, and the temperature at T2 decreased earlier than that at T3. In this study and in [17,21,26,27,41], the monitoring point temperature was even lower, while in [19,20,23,24] it was higher. Since their boundary heat transfer coefficients were (50, 41.8, 16.6, 45.4, 45, 50) W/(m²K), 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

The effects of initial gas saturation (0.1, 0.2, 0.25, 0.3, 0.4) on hydrate dissociation were investigated in this section. Other simulation conditions, except the initial gas saturation, were maintained. As shown in Figure 5, natural gas production with different initial gas saturations fluctuated over time. When the outlet valve was opened, gas production first increased sharply and then dropped steadily to zero over time. The fundamental cause for the first peak was that the gas tended to flow out when the outlet valve opened, due to the pressure difference between the inside and outside of the core. When the hydrate gradually decomposed, the difference between gas fugacity and equilibrium fugacity grew, promoting hydrate decomposition and increasing the gas production rate to the second peak. The value, t_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

The effects of outlet pressure (2.52 MPa, 2.68 MPa, 2.84 MPa, 3.00 MPa, 3.16 MPa) on hydrate dissociation were investigated in this section. Other simulation conditions, except the outlet pressure, were maintained. As shown in Figure 7, gas production and cumulative production fluctuated over time as the outlet pressure changed. The study found that natural gas production had four stages and two peaks, which were basically the same characteristics as mentioned earlier. The NGH in the core did not entirely dissociate in the first 400 min when the outlet pressure exceeded 3.16 MPa. Additionally, it was evident that when outlet pressure fell, a greater overall amount of gas was generated. The residual gas in the core varied under various pressure differences, which led to different cumulative gas production. The value,t_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

The effects of initial temperatures (273.45 K, 274.45 K, 275.45 K, 276.45 K, 277.45 K) on hydrate dissociation were investigated in this section. Other simulation conditions, except the initial temperatures, were maintained. As shown in Figure 9, at higher initial temperatures, natural gas production and cumulative natural gas production fluctuated over time as the initial temperature changed, and as the initial temperature increased, the rate of natural gas hydrate dissociation accelerated. The total production of gas did not change much, though. The value, t_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 Scm³, 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

The effects of absolute permeability (10 mD, 50 mD, 98.97 mD, 150 mD, 200 mD) on hydrate dissociation were investigated in this section. Other simulation conditions, except the absolute permeability, were maintained. As shown in Figure 12, variation of absolute permeability caused gas production rate and cumulative gas production to fluctuate with time. The gas production rate and cumulative gas production for 10 mD absolute permeability were different from the other four data, and the gas production rate was very low. This was because reduced absolute permeability limited fluid flowability and decreased the pressure driving force of hydrate dissociation, resulting in a lengthier hydrate dissociation process. The value, t_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

As early as the 1940s, some researchers in other countries began to investigate the fluctuation of porosity and permeability with confining pressure. Fatt et al. [47,48], Gray et al. [53], and Mclatchie et al. [55] investigated the effect of reservoir stress fluctuations on rock permeability and porosity, and the researchers confirmed that porosity and permeability steadily declined as effective stress increased. In the late 1980s, China conducted a substantial study on reservoir stress sensitivity. Research [47,48,51,52,53,54,55,56] showed that the porosity and permeability of the reservoir decreased when effective stress increased. Furthermore, the stress sensitivity effect was particularly noticeable in low permeability reservoirs.

Zhu and Wang et al. [56] reported the stress-related porosity data of five different sandstones under isotropic effective stress conditions. As shown in Figure 14, the effective stress range of 0–30 MPa appeared in the data and the fitting curves of volumetric strain and stress-related porosity of Equations (5) and (6) to assess the validity of the aforesaid fitting correlation equation for Berea sandstone. The certain value of the fitting constants are displyed in

Table 4. Fitting constants in Zhu and Wang experiments.

Rock	${\mathit{A}}^{\prime }$	${\mathit{B}}^{\prime }$	${\mathit{C}}^{\prime }$
Berea sandstone	0.183	0.01859	9.842

.

Based on Masuda’s hydrate core decomposition experiment, the effective stress (5 MPa, 15 MPa, and 25 MPa) was changed by varying the confining pressure of the sandstone core, and numerical models were built for the evolution of physical properties of hydrate sandstone cores under different stresses.

Figure 15a,b show the volume of gas hydrate in the cores considering, or not considering, the effect of ground stress. The average volume fraction in the core, considering the geo-stress, was higher than that without considering the geo-stress at the same time point, which indicated that the effect of geo-stress reduced the decomposition rate of natural gas hydrate in sandstone core. The volume fraction of methane in the core without considering, or considering, the effect of geo-stress is shown in Figure 16a,b. The average volume fraction in the core, considering the geo-stress, was lower than that without considering the geo-stress at the same time, which indicated that the effect of geo-stress reduced the gas production rate in sandstone core. Figure 17 also illustrates this point. The temperature in the core without considering, or considering, the effect of geo-stress is shown in Figure 18a,b. In the process of the temperature dropping, caused by endothermic reaction, the average core temperature considering geo-stress effect was higher than that without considering geo-stress. In the process of the temperature rising, caused by heat transfer, the average core temperature considering geo-stress was lower than that without considering geo-stress, which indicated that the effect of geo-stress slowed down the efficiency of the heat and mass transfer in the sandstone core, leading to slower dissociation and transportation rates of the NGH.

As shown in Figure 19, natural gas production and cumulative natural gas production fluctuated with time under different effective stresses. The gas production rate graph shows that, at 125 min, before the reaction, the instantaneous gas production rate, without taking the effect of effective stress into account, was higher than that under the effect of effective stress. The rate of gas production decreased as effective stress increased. In addition, as the effective stress increased, the time to drop to zero for the corresponding gas production rate increased, which also meant that the time for the hydrate dissociation increased. This was because the effective stress decreased the amount of pore space, which lowered the core’s permeability and porosity. The permeability and porosity in the core decreased as the effective stress increased. The slower the heat convection, the lower the mass transfer efficiency, and the longer it took for the hydrate to dissociate, the smaller the permeability, so, consequently, the longer the completion date for gas production from the use of natural gas hydrates. The cumulative gas output showed that, under various effective stress levels, the final cumulative gas output was equal. The time it took to reach the final cumulative gas output increased with the effective stress. This indicated that the effective stress decreased the permeability of the core, hence decreasing the rate of hydrate decomposition and mining efficiency. However, it had no impact on the hydrate decomposition’s overall cumulative gas generation. As shown in Figure 20, t_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.

Under various effective stress levels, pore pressure at the far-field boundary varied over time as shown in Figure 21. The diagram shows that when the effective stress was low, the duration and size of the small peak were close and the pore pressure test curve was more like the experimental results of Masuda. The pore pressure test curve was significantly different from the experimental data of Masuda. With a rise in effective stress, the small peak value also gradually declined, and the small peak value’s appearance time arrived later and later. Additionally, there was a small peak in several conditions, and this small peak in the pore pressure curve at the far-field boundary was not a coincidence; rather, it was one of the characteristics of the two-phase flow in porous media that was brought on by hydrate dissociation. In the case of hydrate dissociation, for instance, the inability of the gas to spread quickly led to an increase in local pore pressure. This demonstrated that the impact of effective stress caused a reduction in pore space volume, which caused a reduction in the core’s porosity and permeability. As a result, the gas produced by hydrate decomposition could not be discharged quickly and accumulated in the core, increasing core pressure and further impeding hydrate decomposition. Additionally, hydrate breakdown was slowed down by increasing effective stress.

Three monitoring points’ temperatures over time, and under various effective stress levels, are shown in Figure 22. The minimum temperature considering effective stress was higher than it was without considering effective stress 125 min before decompression, as shown in Figure 22. The increasing rate of temperature recovery (the heat transfer-induced temperature recovery process) was lower than it would be without taking effective stress into account. This was due to the decrease in effective stress, which lowered the core’s porosity and permeability, and, moreover, slowing down the pace of hydrate dissociation to achieve a late minimum temperature. Additionally, it can be seen from a comparison of Figure 22a–c that the greater the effective stress, the later the minimum temperature appeared. This was because the core’s porosity and permeability decreased as the effective stress increased. The low permeability restricted the efficiency of the heat and mass transfer by the fluid flow, leading to a slow dissociation and transportation rate of the NGH.

4. Conclusions

In this study, an axisymmetric two-dimensional model was developed to simulate the depressurization decomposition process of natural gas hydrate in Berea sandstones. A mathematical model that considered fluid heat, mass transport, multiphase flow mechanics, and the kinetics of hydrate decomposition, was established. FLUENT was employed to solve the basic governing equations of multi-phase flow and UDF was used to program the hydrate dissociation model and modified permeability model during NGH dissociation and transport. The simulation results were then validated by Masuda’s experimental data, and multi-parametric analysis was performed. The following conclusions could be achieved:

• 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.

In this paper, the numerical modeling on the core scale dissociation and transportation of NGH was conducted. Future study will focus on the development of NGH in its formation.