Keywords

1 Introduction

Methane hydrate (MH) is an ice-like crystalline compound widely spreading in the sediments on marine continental margins where low temperature and high pressure are present. Natural or anthropologic perturbations could cause massive hydrate dissociation that releases a large amount of methane gas, thereby triggering or priming seafloor instability due to pore pressure buildup in addition to reduction in shear strength of the sediments. The connection of oceanic hydrates dissociation and submarine landslides has been a scientific concern that attracts increasing attentions.

An accurate quantification of expected excess pore pressure levels is necessary to properly predict the occurrence and position of the failure surface of submarine landslides triggered or primed by hydrate dissociation. Different models have been proposed for this purpose [1,2,3]. For instance, Xu and Germanovich [1] related the volume expansion during hydrate dissociation to density difference and compressibility of the system, and theoretically formulated the excess pore pressure for confined or interconnected pore space. Nixon and Grozic [2] quantified the excess pore pressure assuming undrained conditions for a given amount of dissociating hydrate, and proposed a conservative approach for assessing seafloor instability by neglecting pressure diffusion during progressive hydrate dissociation. Kwon and Cho [3] established a sequentially coupled model that decouples hydrate dissociation (with pressure buildup) from consolidation (with pressure diffusion) in each time step. Since oceanic hydrate reservoirs are usually seated in marine sediments with more or less interconnected pore space, as addressed by Xu and Germanovich [1], the magnitude of excess pore pressure primarily depends on dissociation rate and pressure diffusion rate (controlled by the sediment permeability) at similar or miscellaneous time scales. Meanwhile, the excess pore pressure impacts the pressure-dependent dissociation process and in turn affects pore connectivity. It is still challenging and however desired to quantify excess pore pressure associated with hydrate dissociation under different submarine settings where multi-physics coupling process can be considered.

This paper develops a simple thermo-hydro-chemically (THC) coupled model to quantify evolving excess pore pressure during hydrate dissociation by considering thermodynamic chemical action, heat transfer, pressure diffusion, and their interplay at different time scales. The model is incorporated into limit equilibrium analysis in order to assess the stability of an idealized submarine slope with a hydrate layer under rising temperature.

2 A Theoretical Model

Figure 1 illustrates a simplified infinite submarine slope with a MH-bearing layer. The stability of the slope can be quantified within the framework of the limit equilibrium method using a safety factor computed from:

Fig. 1.
figure 1

Schematic illustration of an infinite submarine slope with a MH bearing layer

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$$ F_{s} = \frac{c}{{\gamma^{{\prime }} H\,\sin \,\beta \,\cos \,\beta }} + (1 - \frac{{u_{\text{e}} }}{{\gamma^{{\prime }} H}})\frac{\tan \,\varphi }{\tan \,\beta } $$
(1)

where H is the depth of the potential slip surface below the seafloor; c and φ are the cohesion and the friction angle of the soil at the potential slip surface, respectively; γ′ is the submerged unit weight of the soil; β is the slope angle; and u e is the excess pore pressure:

$$ u_{\text{e}} = P - P_{static}^{{}} $$
(2)

where P static is the hydrostatic pressure; and P is the total pore pressure weighted by the pore water pressure P w and pore gas pressure P g [4]:

$$ P = \frac{{S_{w} }}{{S_{w} + S_{g} }}P_{w} { + }\frac{{S_{g} }}{{S_{w} + S_{g} }}P_{g} $$
(3)

where S w and S g are water and gas saturations, respectively.

This slope could become unstable (i.e. F s  < 1) because of hydrate dissociation. For simplicity, we assume that the cohesion is correlated with hydrate saturation through a linear relationship once MHs dissociate and focus on the effect of excess pore pressure on stability of the slope. To quantify the excess pore water pressure, we develop a THC coupled model to be elaborated on below.

In general, a mass of MH-bearing sediment can be viewed as a porous system composed of a soil skeleton and three components (i.e. water, methane and hydrate) in pores of the skeleton. For simplification, we assume: (1) temperature is the same cross all phases at the same locality; that is, thermal equilibrium between phases is assumed; (2) each phase contains only one single component, that is, water, methane and hydrate are present as liquid, gas, and solid phase, respectively; (3) hydrates are immobile in pores and only dissociate in the same locality; (4) pore water and hydrate are incompressible; (5) soil skeleton does not deform.

Under one-dimensional conditions as illustrated in Fig. 1, the mass and energy conservation equations of the slope are written as:

$$ \frac{\partial }{\partial t}\left( {\phi S_{\alpha } \rho_{\alpha } } \right){ = } - \frac{{\partial q_{\alpha } }}{\partial z}{ + }\frac{{\partial m_{\alpha }^{d} }}{\partial t},\quad \alpha \equiv w,g ,h $$
(4)
$$ \left[ {\left( {1 - \phi } \right)\rho_{s} c_{s} + \sum\limits_{\alpha \equiv w,g,h} {\left( {\phi S_{a} \rho_{\alpha } c_{\alpha } } \right)} } \right]\frac{\partial \theta }{\partial t} = - \frac{\partial h}{\partial z} - \frac{\partial }{\partial z}\sum\limits_{\alpha \equiv w,g} {\left( {q_{\alpha } c_{\alpha } \theta } \right)} + \frac{{\partial Q_{h}^{d} }}{\partial t} $$
(5)

where \( \phi \) is the porosity; S α , ρ α , q α and c α are the saturation, density, mass flux and specific heat capacity of the component α (i.e., w: water; g: methane gas; h: hydrate), respectively; θ and h are temperature and heat flux, respectively; the subscript s represents soil; \( \partial m_{\alpha }^{d} /\partial t \) is the mass rate of component α per unit soil volume; and \( \partial Q_{h}^{d} /\partial t \) is the dissociation heat rate.

By introducing Darcy’s law, Eq. (4) is re-written as

$$ \frac{\partial }{\partial t}\left( {\phi S_{\alpha } \rho_{\alpha } } \right){ = } - \frac{\partial }{\partial z}\left[ { - \frac{{kk_{r\alpha } \rho_{\alpha } }}{{\mu_{\alpha } }}\left( {\frac{{\partial P_{\alpha } }}{\partial z} - \rho_{\alpha } g\,\cos \,\beta } \right)} \right]{ + }\frac{{\partial m_{\alpha }^{d} }}{\partial t},\quad \alpha \equiv w,g,h $$
(6)

where k rα and μ α are the relative permeability and viscosity of component α respectively. Note that k rh is zero since hydrate is immobile and the relative permeability k rw and k rg depend on water and gas saturations (see Table 1).

Table 1. Parameters of the hydrate layer
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By introducing Fourier’s law, Eq. (5) is re-written as

$$ \begin{aligned} \hfill \left[ {\left( {1 - \phi } \right)\rho_{s} c_{s} + \sum\limits_{\alpha \equiv w,g,h} {\left( {\phi S_{\alpha } \rho_{\alpha } c_{\alpha } } \right)} } \right]\frac{\partial \theta }{\partial t} = - \frac{\partial }{\partial z}\left( { - \lambda_{\Theta } \frac{\partial \theta }{\partial z}} \right) \\ \hfill - \frac{\partial }{\partial z}\sum\limits_{\alpha \equiv w,g} {\left[ { - \frac{{kk_{r\alpha } \rho_{\alpha } }}{{\mu_{\alpha } }}\left( {\frac{{\partial P_{\alpha } }}{\partial z} - \rho_{\alpha } g\,\cos \,\beta } \right)c_{\alpha } \theta } \right]} + \frac{{\partial Q_{h}^{d} }}{\partial t} \\ \end{aligned} $$
(7)

where λΘ is the composite thermal conductivity of the system, of which the calculating model is given in Table 1.

The mass rate in Eq. (4) is computed as:

$$ \frac{{\partial m_{w}^{d} }}{\partial t} = - \eta M_{w} \chi ;\quad \frac{{\partial m_{g}^{d} }}{\partial t} = - M_{g} \chi ;\quad \frac{{\partial m_{h}^{d} }}{\partial t} = M_{h} \chi $$
(8)

where M w , M g and M h are the molar masses of water, methane and hydrate, respectively; η represents the hydrate number; χ (mol/s/m3) is the reaction rate given by Kim-Bishnoi model [5]:

$$ \chi = - k_{0} \exp ( - \frac{{\Delta E_{a} }}{R\theta })A_{s} (P_{e} - P_{g} ) $$
(9)

where k0 is the intrinsic dissociation constant; ∆E a is the activation energy of hydrate dissociation; R is the universal gas constant; A s is the hydrate action area; P g is pressure of gas phase; P e is the equilibrium pressure at temperature θ, given by Moridis’s model [6].

The dissociation heat rate in Eq. (5) is computed as:

$$ \frac{{\partial Q_{h}^{d} }}{\partial t} = \chi \Delta H $$
(10)

where ΔH is the latent heat, given by the Kamath equation [7]:

$$ \begin{array}{*{20}l} {\Delta H = A + \frac{B}{\theta }} \hfill \\ {\left\{ {\begin{array}{*{20}l} {A = 56552,\;B = - 16.8137;\quad 273.15 < \theta \le 298.15} \hfill \\ {A = 27329,\;B = - 48.2748;\quad 248.151 \le \theta < 273.15} \hfill \\ \end{array} } \right.} \hfill \\ \end{array} $$
(11)

In summary, Eqs. (4) and (5) form the governing equations of the system that involve four primary variables: gas pressure P g , temperature θ, water saturation S w and gas saturation S g . These equations are closed by the following auxiliary equations:

$$ S_{w} + S_{g} + S_{h} = 1 $$
(12)
$$ P_{g} = P_{w} + P_{c} $$
(13)

where P c is the capillary pressure to be determined by the van Genuchten function of S w and S g [8], as shown in Table 1.

No analytical solution is available for Eqs. (6) and (7). Instead, a numerical solution was coded in MATLAB by linearizing the governing equations with the implicit finite-difference method. The central and forward difference approximation was used for the spatial and time derivatives, respectively. Then the linearized difference equation set was solved with the Newton-Raphson iteration method. The proposed model and the numerical implementation have been validated via experimental data [9] and have not been included in this paper due to length limit.

3 A Case Study of an Overburden-Free Slope T

The proposed model is applied to a case study addressed by Reagan and Moridis [10] that represents a scenario of gentle submarine slopes with a shallow hydrate-bearing layer (hydrate layer for short here). Below a water depth of 570 m, the hydrate layer extends vertically from the seafloor to a depth of 16 m, and the initial hydrate saturation is 3%. Initially, the temperature is 6 °C at the seafloor and linearly increases with depth at a geothermal gradient of 28 °C/km. It is assumed that a temperature rise occurs at the seafloor at a rate of 0.03 °C/year due to climate change. This triggers thermal dissociation in the hydrate layer.

To verify our model and code, we adopted the boundary and initial conditions, and parameters as the same as those in [10]. Note that salt and methane dissolution in water are ignored in our simulation. Table 1 lists the major parameters.

In this setting-up, the dissociation front propagates from the top to the bottom of the hydrate layer. Figures 2a and b compare our results with those given by Reagan and Moridis. Our simulation agrees with the published data except that the dissociation lags for about 60 years. This lag is due to the fact that the effect of salinity on the phase equilibrium, considered by Reagan and Moridis, is ignored in our simulation. Under a pressure of 5.7 MPa, the equilibrium temperature for MH in pure water is 1.7 °C higher than that in sea water with a salinity of 0.035 (considered by Reagan and Moridis). Given a temperature rising rate of 0.03 °C/year, hydrate dissociate 60 years later in our simulation than that provided by Reagan and Moridis. Nevertheless, the consistent profiles of hydrate and gas saturation as illustrated in Figs. 2a and b verified our numerical code.

Fig. 2.
figure 2

Profiles of hydrate saturation, gas saturation, and temperature in comparison with results by Reagan and Moridis [10] at different times (a–b), and profile of the excess pore pressure evolution (c).

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Figure 2c plots the excess pore pressure against time obtained from our simulation. The magnitude of the induced excess pore pressure is in the order of several kilopascals, and is insufficient to cause slope instability (assuming a gentle slope with an inclination angle of several degrees) because of low hydrate saturation and absence of a relatively impermeable overburden above the hydrate layer. The next section will show more vulnerable cases with an overburden.

4 Effect of Overburden Layer

Here we consider the effect of an overburden layer with a thickness ranging from 0 to 80 m. The permeability of the overburden and underburden layers is 1 × 10−17 m2. The thickness of the hydrate layer is 20 m with a hydrate saturation of 30%. The water depth from the sea level to the top of the hydrate layer is constant at 570 m. The temperature at the bottom of hydrate layer is constant at 8 °C. The slope angle is assumed to be 3º, and the cohesion and the internal frictional angle of all the sediments are 200 kPa and 30º, respectively. The other parameters remain the same as the overburden-free case in the preceding section.

Figure 3 illustrates the profile of S h , u e , and F s against the time elapsed from the first onset of hydrate dissociation under different overburden thicknesses. As shown in Fig. 3a, as the dissociation front propagates from the top to the bottom of the hydrate layer, u e continuously builds up and F s decreases. Meanwhile, we assume the cohesion decreases linearly from 200 kPa to zero with the MHs dissociation. As a result, a peak develops on the profile of F s near the dissociation front (see Fig. 3). The failure onset time refers to the time when F s first reaches one at a specific depth, where the failure surface is recognized.

Fig. 3.
figure 3

Profiles of hydrate saturation, excess pore pressure, safety factor at different time periods under a overburden thickness of (a) 5 m; (b) 20 m; and (c) 60 m. The elapsed time starts from the onset of hydrate dissociation at the top of the hydrate layer.

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Figure 4 shows the failure onset time and failure surface position against different overburden thicknesses h o . The presence of the overburden plays two different roles. On one hand, the overburden serves as a barrier that hampers dissipation of the excess pore pressure accumulating in the hydrate layer during dissociation [1]. This tends to endanger the slope. Even though hydrate dissociation is faster at 5 m thick overburden (Fig. 3a) than that at 20 m (Fig. 3b), the pressure buildup is more remarkable for the same amount of dissociated hydrate under the thicker overburden, because the pressure dissipates more slowly. On the other hand, the overburden layer provides overburden pressure that mobilizes additional shear resistance at the potential failure surface [13]. This is beneficial to stabilize the slope. The second mechanism seems dominant in the circumstance under consideration. Figure 4 also provides the position of the failure surface against different overburden thickness. In general, the failure surface is constrained within a narrow band at the top of the hydrate layer. This weak zone could move upwards in the overburden layer if this layer is less cohesive.

Fig. 4.
figure 4

Failure onset time and failure surface depth against different overburden thicknesses.

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5 Conclusions

This paper presents a simple THC coupled model for quantifying the excess pore pressure caused by MH dissociation, and incorporates this model to the stability analysis of an idealized submarine slope with a MH-bearing layer. Through an overburden-free case study, the proposed model is demonstrated to be able to capture the multi-physics coupled processes involved in MH dissociation in porous sediments. The stability analysis of the gentle slope with a relatively impervious overburden layer shows that the presence of the overburden plays two opposite roles in impacting the stability of the slope. It serves as a barrier that hampers excess pore pressure dissipation and therefore endangers the slope. Meanwhile it has beneficial effects by providing overburden pressure that mobilizes additional shear resistance in the slope. For the circumstances under consideration, the potential failure surface of the slope is constrained within a narrow band at the top of the hydrate layer. This could be affected by the strength properties of the overburden layer, which will be further investigated in our ongoing work.