A state-dependent hypoplastic model for methane hydrate-bearing sands

Abstract

The mechanical behaviors of methane hydrate-bearing sands (MHBS) are largely affected by the presence of methane hydrate, temperature, and pore pressure. In this study, we present a simple hypoplastic model for MHBS. Methane hydrate saturation is included as a state parameter affecting the mechanical behaviors of MHBS. A new phase parameter is introduced to account for the coupled effects of temperature and pore pressure on the mechanical behaviors of MHBS. The phase parameter can be determined by a simple function of temperature and pore pressure. Comparison of the predictions with experiments shows that the model is able to capture the salient behaviors of MHBS.

Appendices

Appendix 1

The complete hypoplastic equations for MHBS are summarized as follows:

$$\begin{aligned} \mathring{\varvec{\sigma }}= & {} f_{\mathrm{s}}\Big [C_1({\text {tr}}\varvec{\sigma }_{\mathrm{T}})\dot{\varvec{\epsilon }}+(1-f_{\mathrm{d}})C_2({\text {tr}}\varvec{\sigma }_{\mathrm{T}})({\text {tr}}\dot{\varvec{\epsilon }})\mathbf{I }+C_3 \frac{{\text {tr}}(\varvec{\sigma }_{\mathrm{T}}\dot{\varvec{\epsilon }})}{{\text {tr}}\varvec{\sigma }_{\mathrm{T}}}\varvec{\sigma }_{\mathrm{T}}\nonumber \\&+f_{\mathrm{d}} C_4(\varvec{\sigma }_{\mathrm{T}}+\varvec{\sigma }_{\mathrm{T}}^*)\Vert \dot{\varvec{\epsilon }} \Vert \Big ] \end{aligned}$$

where

$$\begin{aligned} \left\{ \begin{array}{lr} \varvec{\sigma }_{\mathrm{T}} =\varvec{\sigma }+S\mathbf{I } &{} \\ S=aS^{b}_{\mathrm{MH}}L^\prime &{} \\ f_{\mathrm{s}}=\frac{\text {exp}[\beta (e_{\mathrm{crt}}-e)]}{(1+r)^2} &{} \\ f_{\mathrm{d}}=\left( \frac{e}{e_{\mathrm{crt}}}\right) ^\alpha &{} \\ e_{\mathrm{crt}}=e_{\Gamma (\text {MH},L^\prime )}-\lambda \text {ln}(p^\prime /p_{\mathrm{a}}) &{} \\ e_{\Gamma (\text {MH},L^\prime )}=e_{\Gamma }+{\xi }S^{\chi _1}_{\mathrm{MH}}{L^\prime }^{\chi _2} \end{array} \right. \end{aligned}.$$

In the above equations, $\varvec{\sigma }_{\mathrm{T}}$ is the transformed stress tensor, $\mathring{\varvec{\sigma }}$ is the Jaumann stress rate, $\varvec{\sigma }$ is the Cauchy stress tensor, ${\dot{\varvec{\epsilon }}}$ is the stretching tensor, I is the second-order unit tensor, S is the internal variable, e is the current void ratio, $e_{\mathrm{crt}}$ is the critical state void ratio, r is the stress ratio $\Vert \varvec{\sigma }^* \Vert /({\text {tr}}\varvec{\sigma })$, $L'$ is the phase state parameter, $e_{\Gamma (\text {MH},L^\prime )}$ is the critical state void of MHBS at the atmospheric pressure, $e_\Gamma$ is the critical state void of sand at the atmospheric pressure $p_{\mathrm{a}}$, and $p'$ is the effective mean stress. The parameters of the model are $C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$, $e_\Gamma$, $\lambda$, $\alpha$, $\beta$, a, b, $\xi$, $\chi _1$, $\chi _2$, and $L'$.

Appendix 2

The parameters $C_{1}$, $C_{2}$, $C_{3}$, and $C_{4}$ can be determined as follows. Considering a triaxial compression tests with constant confining pressure, the basic constitutive model (Eq. (1)) can be decomposed as:

$$\begin{aligned} \dot{\sigma _1}= & {} C_1(\sigma _1+2\sigma _3)\dot{\epsilon _1}+C_2(\dot{\epsilon _1}+2\dot{\epsilon _3})(\sigma _1+2\sigma _3)\\&+C_3\frac{\sigma _1\dot{\epsilon _1}+2\sigma _3\dot{\epsilon _3}}{\sigma _1+2\sigma _3}\sigma _1+C_4\frac{5\sigma _1-2\sigma _3}{3}\sqrt{\dot{\epsilon _1}^2+2\dot{\epsilon _3}^2} \\ \dot{\sigma _3}= & {} C_1(\sigma _1+2\sigma _3)\dot{\epsilon _3}+C_2(\dot{\epsilon _1}+2\dot{\epsilon _3})(\sigma _1+2\sigma _3)\\&+C_3\frac{\sigma _1\dot{\epsilon _1}+2\sigma _3\dot{\epsilon _3}}{\sigma _1+2\sigma _3}\sigma _3+C_4\frac{4\sigma _3-\sigma _1}{3}\sqrt{\dot{\epsilon _1}^2+2\dot{\epsilon _3}^2}. \end{aligned}$$

Taking $C_{1}$, $C_{2}$, $C_{3}$, and $C_{4}$ as unknowns, the above equations contain two linear equations. Provided the stress rates ($\dot{\sigma _1}$, $\dot{\sigma _3}$) and strain rate ($\dot{\epsilon _1}$, $\dot{\epsilon _3}$) are known for two arbitrary stress states ($\sigma _1$, $\sigma _3$), a system of four linear equations can be obtained by setting these quantities into the above equations. For hydrostatic stress state and critical state, with R being the stress ratio and $R_{\mathrm{crt}}$ being the critical state stress ratio, the following parameters are introduced: the initial tangential modulus, $E_{\mathrm{i}}=[(\dot{\sigma _1}-\dot{\sigma _3})/\dot{\epsilon _1}]_{R=1}$; the initial Poisson ratio, $\nu _{\mathrm{i}}=[\dot{\epsilon _3}/\dot{\epsilon _1}]_{R=1}$; and the critical state Poisson ratio, $\nu _{\mathrm{crt}}=[\dot{\epsilon _3}/\dot{\epsilon _1}]_{R=R_{\mathrm{crt}}}$. Setting these quantities into the above equations, we obtain the following equation:

$$\begin{aligned} \left[ \begin{array}{cccc} 1 &{} 1+2\nu _{\mathrm{i}} &{} \frac{1+2\nu _{\mathrm{i}}}{9} &{} \frac{\sqrt{1+2\nu _{\mathrm{i}}}}{3} \\ \nu _{\mathrm{i}} &{} 1+2\nu _{\mathrm{i}} &{} \frac{1+2\nu _{\mathrm{i}}}{9} &{} \frac{\sqrt{1+2\nu _{\mathrm{i}}}}{3} \\ R_{\mathrm{crt}}+2 &{} \frac{1+2\nu _{\mathrm{crt}}}{R_{\mathrm{crt}}+2} &{} \frac{R_{\mathrm{crt}}+2\nu _{\mathrm{crt}}}{R_{\mathrm{crt}}+2}R_{\mathrm{crt}} &{} \frac{(5R_{\mathrm{crt}}-2)\sqrt{1+2\nu _{\mathrm{i}}}}{3} \\ R_{\mathrm{crt}}+2 &{} \frac{1+2\nu _{\mathrm{crt}}}{R_{\mathrm{crt}}+2} &{} \frac{R_{\mathrm{crt}}+2\nu _{\mathrm{crt}}}{R_{\mathrm{crt}}+2} &{} \frac{(4-R_{\mathrm{crt}})\sqrt{1+2\nu _{\mathrm{i}}}}{3} \end{array} \right] \left[ \begin{array}{c} C_1 \\ C_2 \\ C_3 \\ C_4 \end{array} \right] = \left[ \begin{array}{c} \frac{E_{\mathrm{i}}}{3\sigma _3} \\ 0 \\ 0 \\ 0 \end{array} \right]. \end{aligned}$$

The values of $C_1$, $C_2$, $C_3$, and $C_4$ can be obtained by solving the above equation.