Influences of hydrate decomposition on submarine landslide

Abstract

Natural gas hydrate reservoirs on the sea floor have characteristics of a shallow overburden depth, weakened rock formation, and easy deformation. Thus, the mining of hydrate may cause geological disasters such as submarine landslides. To clearly learn about the failure process of submarine landslides induced by hydrate decomposition and influence the laws of various factors on submarine slope stability, this study established a thermal-fluid-solid-stress coupling model for hydrate. In combination with a strength reduction method, submarine slope stability in the mining of hydrates was analyzed. By utilizing the orthogonal experimental design method, this study analyzed the influence laws of these factors, including the hydrate decomposition range, submarine slope angle, overburden depth of hydrates, thickness of hydrate layers, seawater depth, and initial hydrate saturation on submarine slope stability, and evaluated the sensitivity to these factors. The result shows that hydrate mining may result in settlement of the seafloor and slippage of submarine sedimentary layers to the mining center, thus probably inducing submarine landslides. According to the significance of influences on submarine slope stability, the factors were ranked in descending order as follows: submarine slope angle, hydrate decomposition scale, thickness of hydrate layer, overburden depth of hydrates, initial saturation of hydrate layer, and seawater depth. Among the above factors, except for the overburden depth of hydrates and the seawater depth, the increases of which promoted submarine slope stability, the increase in the values of the other factors reduced submarine slope stability to different degrees. Therefore, in the mining of hydrates, the effects of each factor on submarine slope stability should be evaluated comprehensively to prevent disasters such as submarine landslides.

Appendix. Multifield coupling model in porous media

ABAQUS can solve the heat flow-solid-stress field coupling in porous media. Based on the principle of effective stress, the constitutive behavior of each phase in the porous medium is analyzed by discretizing the porous medium. According to the continuity equation, the deformation behavior of each phase is coupled with the thermal behavior. The following are the key solutions. The detailed solution process can be found in the ABAQUS 2016 Documentation.

The effective stress principle for porous media is as follows:

$$ {\overline{\sigma}}^{\ast }=\left(1-{n}_t\right)\overline{\sigma}-{n}_t\overline{p_t}\mathbf{I} $$

$$ {n}_t\stackrel{\scriptscriptstyle\mathrm{def}}{=}\frac{d{V}_t}{dV} $$

• 1. Discretized equilibrium statement for a porous medium

Equilibrium is expressed by writing the principle of virtual work for the volume under consideration in its current configuration at time t:

$$ {\int}_V\boldsymbol{\sigma} :\delta \boldsymbol{\epsilon} dV={\int}_St\bullet \delta \boldsymbol{v} dS+{\int}_V\boldsymbol{f}\bullet \delta \boldsymbol{v} dV+{\int}_V\left( sn+{n}_t\right){\rho}_wg\bullet \delta \boldsymbol{v} dV $$

where f are all body forces except the weight of the wetting liquid.

The virtual velocity field is interpolated by

$$ \delta \boldsymbol{v}={N}^N\delta {v}^N $$

where N^N(S_i) are interpolation functions defined with respect to material coordinates, S_i.

The virtual rate of deformation is interpolated as

$$ \updelta \upvarepsilon ={\beta}^N\delta {v}^N $$

where, in the simplest case,

$$ {\beta}^N=\mathit{\operatorname{sym}}\left(\frac{\partial \delta {N}^N}{\partial x}\right) $$

Although, more general forms are used in some of the elements in ABAQUS. The virtual work equation is thus discretized as follows:

$$ \delta {v}^N{\int}_V{\beta}^N:\boldsymbol{\sigma} dV=\delta {v}^N\left[{\int}_S{N}^N\bullet tdS+{\int}_V{N}^N\bullet \boldsymbol{f}\delta \boldsymbol{v} dV+{\int}_V\left( sn+{n}_t\right){\rho}_w{N}^N\bullet gdV\right] $$

• 2. Constitutive behavior in a porous medium

Liquid response:

For the liquid in the system (the free liquid in the voids and the entrapped liquid), we assume the following:

$$ \frac{\rho_w}{\rho_w^0}\approx 1+\frac{u_w}{K_w}-{\varepsilon}_w^{th} $$

where ρ_w is the density of the liquid, ${\rho}_w^0$ is its density in the reference configuration, K_w(θ) is the liquid’s bulk modulus, and

$$ {\left.{\varepsilon}_w^{th}=3{\alpha}_w\left(\theta -{\theta}_w^0\right)-3{\alpha}_w\right|}_{\theta^I}\left({\theta}^I-{\theta}_w^0\right) $$

is the volumetric expansion of the liquid caused by the temperature change. Here, α_w(θ) is the thermal expansion coefficient of the liquid, θ is the current temperature, θ^I is the initial temperature at this point in the medium, and ${\theta}_w^0$ is the reference temperature for thermal expansion. Both $\raisebox{1ex}{$u$}\!\left/ \!\raisebox{-1ex}{${K}_w$}\right.$ and ${\varepsilon}_w^{th}$ are assumed to be small.

Solid response:

The solid matter in the porous medium is assumed to have the local mechanical response under pressure:

$$ \frac{\rho_g}{\rho_g^0}\approx 1+\frac{1}{K_g}\left(s{u}_w+\frac{\overline{p}}{1-n-{n}_t}\right)-{\varepsilon}_g^{th} $$

where K_g(θ) is the bulk modulus of this solid matter, s is the saturation in the wetting fluid, and

$$ {\left.{\varepsilon}_g^{th}=3{\alpha}_g\left(\theta -{\theta}_g^0\right)-3{\alpha}_g\right|}_{\theta^I}\left({\theta}^I-{\theta}_g^0\right) $$

is its volumetric thermal strain. Here, α_g(θ) is the thermal expansion coefficient for the solid matter, and ${\theta}_g^0$ is the reference temperature for this expansion; $1-{\rho}_g/{\rho}_g^0$ is assumed to be small.

• 3. Continuity statement for the wetting liquid phase in a porous medium

The total mass of wetting liquid in the control volume is

$$ {\int}_V{\rho}_w\left[d{V}_w+d{V}_t\right]={\int}_V{\rho}_w\left({n}_w+{n}_t\right) dV $$

where ρ_w is the mass density of the liquid.

The time rate of change of this mass of wetting liquid is

$$ \frac{d}{dt}\left[{\int}_V{\rho}_w\left({n}_w+{n}_t\right) dV\right]={\int}_V\frac{1}{J}\frac{d}{dt}\left(J{\rho}_w\left({n}_w+{n}_t\right)\right) dV $$

Constitutive behavior:

Introducing the flow constitutive law allows the mass continuity equation to be written:

$$ {\int}_V\left[\delta {u}_w\left(\left(\frac{\rho_w}{\rho_w^0}\left( sn+{n}_t\right)\right)-\frac{1}{J}{\left(\frac{\rho_w}{\rho_w^0}J\left( sn+{n}_t\right)\right)}_t\right)+\Delta t\frac{k_s}{\rho_w^0\left(1+\beta \sqrt{{\mathbf{v}}_w\bullet {\mathbf{v}}_{\mathrm{w}}}\right)}\frac{\partial \delta {u}_w}{\mathrm{\partial x}}\bullet \mathbf{k}\bullet \left(\frac{\partial \delta {u}_w}{\mathrm{\partial x}}-{\rho}_wg\right)\right] dV\Delta t{\int}_S\delta {u}_w\frac{\rho_w}{\rho_w^0} sn\mathbf{n}\cdotp {\mathbf{v}}_w dS=0 $$

Volumetric strain in the liquid and grains:

$$ \frac{\rho_w}{\rho_w^0}n\approx 1-\frac{1}{J}\left(1-{n}^0-{n}_t^0+{h}_t\right)+\frac{\overline{p}}{K_g}+{u}_w\left(\frac{1}{K_w}+\frac{1-{n}^0-{n}_t^0}{J}\left(\frac{1}{K_g}-\frac{1}{K_w}\right)\right)-{\varepsilon}_w^{th}+\frac{1}{J}\left(1-{n}^0-{n}_t^0\right)\left({\varepsilon}_w^{th}-{\varepsilon}_g^{th}\right) $$

• 4. Solution strategy for coupled diffusion/deformation

The coupled system of equations to be solved is as follows:

$$ {K}^{MN}{\overline{c}}_{\delta}^N-{L}^{MP}{\overline{c}}_u^P={P}^M-{I}^M $$

and pore fluid flow:

$$ -{\left({B}^{MQ}\right)}^T{\overline{c}}_{\delta}^M-\Delta t{H}^{QP}{\overline{c}}_u^P={R}^Q $$

where

$$ {R}^Q=\Delta t\left[-{Q}_{t+\Delta t}^Q+{\left({\hat{B}}^{MQ}\right)}^T{\overline{v}}_{t+\Delta t}^M+{\hat{H}}^{QP}{\overline{u}}_{t+\Delta t}^P\right] $$

• 5. Heat transfer solution

Energy balance: The basic energy balance is as follows:

$$ {\int}_V\rho \dot{U} dV={\int}_S qdS+{\int}_V rdV $$

This relationship is usually written in terms of a specific heat, neglecting the coupling between mechanical and thermal problems:

$$ c\left(\theta \right)=\frac{d U}{d\theta} $$

Heat conduction is assumed to be governed by the Fourier law,

$$ \mathbf{f}=-\mathbf{k}\frac{\partial \theta }{\partial x} $$

Boundary conditions can be specified as prescribed temperature, θ = θ(x, t).

Spatial discretization

A variational statement of the energy balance, together with the Fourier law, is obtained directly by the standard Galerkin approach as follows:

$$ {\int}_V\rho \dot{U}\delta \theta dV+{\int}_V\frac{\partial \delta \theta}{\mathrm{\partial x}}\bullet \mathbf{k}\bullet \frac{\partial \theta }{\mathrm{\partial x}}={\int}_V\delta \theta rdV+{\int}_{S_q}\delta \theta qdS $$

The thermal equilibrium equation for a continuum in which a fluid is flowing with velocity v is as follows:

$$ \int \delta \theta \left[\rho c\left(\frac{\partial \theta }{\partial t}+\mathbf{v}\cdotp \frac{\partial \theta }{\mathrm{\partial x}}\right)-\frac{\partial }{\mathrm{\partial x}}\bullet \left(\mathbf{k}\cdotp \frac{\partial \theta }{\mathrm{\partial x}}\right)-q\right] dV+{\int}_{S_q}\delta \theta \left[\mathbf{n}\cdotp \mathbf{k}\bullet \frac{\partial \theta }{\mathrm{\partial x}}-{q}_s\right] dS=0 $$