Modeling the Mechanical Behavior of Methane Hydrate-Bearing Sand Using the Equivalent Granular Void Ratio

Abstract

For the safe extraction of methane from hydrate reservoirs, modeling the mechanical behavior of the methane hydrate-bearing soil properly is crucial in order to enable designers to analysis hydrate-dissociation-induced geotechnical failures. Hydrate morphology is one of major factors affecting the mechanical behavior of soil containing hydrate. This paper presents a new constitutive model for methane hydrate-bearing sand (MHBS) using the equivalent granular void ratio as a state variable, which can quantify the effects of the pore-filling and load-bearing hydrate morphology under a unifying framework. The proposed model is a combination of generalized plasticity and an elastic damage model so as to take into account the observed frictional and bonding aspects of MHBS, respectively. By using the concept of state-dependent dilatancy, the equivalent granular void ratio is formulated and adopted in the generalized plasticity model. In addition, a nonlinear damage function is implemented to elucidate the degradation of hydrate bonds with respect to shearing. Compared with the basic generalized plasticity model for host sand, only three additional parameters are required to capture key mechanical behaviors of MHBS. By comparing the triaxial test results of MHBS synthesized from a range of host sands with a predicted behavior by the proposed model, it is demonstrated that the new model can satisfactorily capture the stress–strain and volumetric behavior of MHBS under different hydrate saturations, confining pressures, and void ratios.

1. Introduction

Methane hydrate is an ice-like crystalline clathrate composed of methane encaged in a lattice of hydrogen-bonded water molecules. It is formed and preserved in locations where there is an ample supply of methane, as well as where the temperature and pressure requirements have met the stability conditions [1,2]. By increasing temperature and reducing the pressure, methane hydrate can decompose into methane gas and water. The occurrence of natural methane hydrate has been verified through core drilling at locations all over the world, such as in the Mallik–Mackenzie Delta, Gulf of Mexico, Nankai Trough, Northern South China Sea, and Ulleung Basin. The large number of reserves of methane hydrate has impelled it to be a potential future energy source [3,4,5,6,7]. However, geological disasters and engineering accidents could occur from the production of methane hydrate, such as subsea landslides, seabed settlement, mining hole, and production platform collapse [8,9,10]. Thus, understanding the mechanical behavior of methane hydrate-bearing soil is essential for the safe extraction of methane hydrate. Considering the economic costs and technical feasibility, the hydrate-bearing coarse-grained reservoir should be the ideal sediment for the effective recovery of methane gas from the deep subsurface, such as those found in the Mallik permafrost site and Nankai Trough, where hydrate saturations can reach as high as 80% and 50%, respectively [5,11]. Therefore, the present study focuses on modeling the constitutive behavior of methane hydrate-bearing sand (MHBS).

Waite et al. described the three most common types of hydrate morphology in the pore space of sand: (a) pore-filling, (b) load-bearing, and (c) cementing [12]. For the pore-filling type, hydrates nucleate on the surface of soil grains and extend to the pore space. As a result, “hydrate primarily affects the bulk stiffness and conduction properties of pore fluid” [13]. When hydrate saturation exceeds a certain value, say 25%~40%, the hydrate morphology naturally turns into the load-bearing type [14]. Under this morphology, some hydrates that bridge neighboring soil grains contribute to the mechanical stability of the granular skeleton [15,16]. For the cementing type, hydrates act as bonding agents to cement soil grains at intergranular contacts, leading to strengthening of the granular skeleton, even with a small amount of hydrates [17].

In the literature, a mass of constitutive models has been proposed to predict the mechanical behavior of hydrate-bearing soil [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. Miyazaki et al. proposed a non-linear elastic model based on the Duncan–Chang model by introducing the influence of hydrate saturation on the peak strength and elastic modulus [18]. However, non-linear elastic models cannot capture the strain-softening and dilatancy of hydrate-bearing soil. On the other hand, several models have been developed from the Mohr–Coulomb model for hydrate-bearing soil, which consider the stiffness, shear strength, and dilation angle as functions of hydrate saturation [19,20,21]. However, these models cannot (1) capture strain-softening behavior, (2) consider effects of hydrate debonding, and (3) predict the satisfactorily volume change of hydrate-bearing soil. Sultan and Garziglia considered the influence of hydrate on the compressibility and pre-consolidation pressure based on the Cam-Clay model [22]. The performance of the model is satisfactory compared with the experimental data; however, it still cannot be used to smoothly describe the strain-softening behavior. Uchida et al. and Lin et al. further proposed extended Cam-Clay models for hydrate-bearing soil by taking into account the degradation of hydrate bonding [23,24]. These models predict satisfactorily predict the triaxial test results of laboratory-synthesized and intact specimens extracted from the Nankai Trough. However, the associated flow rule is adopted in these models, resulting in inaccurate volume change predictions of hydrate-bearing soil. Shen et al. and Ng et al. adopted the state-dependent dilatancy into a new critical state-based model, leading to a better prediction of the volume change of hydrate-bearing soil over a wide range of confining stress and void ratios [25,26,27]. Sánchez et al. developed a constitutive model for hydrate-bearing soil based on the concept of partition stress and hierarchical single surface framework by modeling the behaviors of the soil skeleton and methane hydrate separately [28,29]. The model can simulate behavior of hydrate-bearing soil upon hydrate dissociation well. Iwai et al. recently proposed a new elasto-plastic constitutive model by taking into account the different hardening rules corresponding to the different types of hydrate morphology [30]. The model can reproduce the different stress–strain relations and dilatancy behaviors by only giving consideration to the different morphology distributions and not changing the fitting parameters.

One of the assumptions adopted in the existing constitutive models of hydrate-bearing soil is that the formation of hydrate does not change the void ratio of soil. In other words, hydrates only fill up the pore space of the soil, but the load-bearing hydrate morphology cannot be considered. Thus, the void-ratio-dependent behavior of the hydrate-bearing soil is modeled empirically as a function of hydrate saturation, leading to too many model parameters. Recent studies on the behavior of sand with fines have shown that a new equivalent granular void ratio can be used to characterize different roles of fines in sand–fines mixtures [33,34]. Hence, the equivalent granular void ratio may be a more appropriate state variable to quantify the void-ratio-dependent behavior of sand–fines mixtures.

In this study, the concept of the equivalent granular void ratio is first presented and its rationality for capturing the behavior of MHBS is verified through reported experimental data. Then, a new constitutive model formulated using the concept of the equivalent granular void ratio is introduced. The newly proposed model consists of (1) a generalized plasticity model for uncemented granular skeleton and (2) an elastic damage model for hydrate bonding. The calibration of parameters and prediction ability of the model are demonstrated by comparing the simulations with the triaxial test results of MHBS under various hydrate saturations, confining pressures, and void ratios.

2. Equivalent Granular Void Ratio

The void ratio (e) is commonly used as a state variable in soil mechanics for the interpretation of the soil behavior. However, past studies have shown that the addition of a small amount of fines in the sand decreases its void ratio, but the stress–strain behavior essentially remains the same, implying that the void ratio is no longer a suitable state variable to represent the stress–strain response of sand–fines mixtures. The main reason for such behavior is the inactive contribution of fines to the force chain of sand–fines mixtures. To properly describe the mechanical behavior of sand–fines mixtures, a new density parameter, the intergranular void ratio e_g, has been proposed by considering fines as voids [35,36,37]:

$$e_g=\frac{e+f_c}{1-f_c}$$

(1)

where f_c is the fines content.

However, with the increase in the fines content, fines may contact each other and bridge the neighboring sand grains, participating in the force chain of the sand–fines mixtures. Some initially inactive fines may also become part of the force chain as a result of the movements of soil during shearing [38]. To take such a mechanism into consideration, Thevanayagam et al. further proposed an equivalent granular void ratio (e*) accounting for the “actual” density of the sand-–fines mixture [33], which is defined as follows:

$$e^{*}=\frac{e+\left(1-b\right)f_c}{1-\left(1-b\right)f_c}$$

(2)

where b is the fraction of fines that actively take part in the force chain of the sand skeleton. It should be noted that Equation (2) is only applicable for sand–fines mixtures in which sand is the major component of the mixture. If the fines content exceeds a threshold value such that the fines becomes the major component of the sand–fines mixture, another density parameter will be adopted [33]. The successful application of the equivalent granular void ratio as a state variable has been verified in the studies conducted by Thevanayagam et al., Ni et al., Yang et al., and Chiu and Fu [33,39,40,41]. A single critical state line (or steady state line) expressed in terms of the equivalent granular void ratio can be identified for sand–fines mixtures over a wide range of fines contents. Furthermore, the equivalent granular void ratio can be used in conjunction with the corresponding critical state line to predict the behavior of sand–fines mixtures in the framework of critical state soil mechanics [34].

The microscopic investigation showed that the average particle size of the hydrates was about 0.04 mm, which was far less than that of the sand grains. Thus, hydrates may be considered as fine particles [42]. The results of the core drilling revealed that the average particle size of the coarse-grained sediments extracted from the Nankai Trough was about 0.2 mm and 0.3 mm for that from the Mallik–Mackenzie Delta [43,44], which is five and eight times larger than that of the hydrates, respectively. Therefore, it is reasonable to draw an analogy between MHBS and sand–fines mixtures, where the roles of hydrates in pore-filling and the load-bearing hydrate morphology of MHBS are similar to those of the fines in sand–fines mixtures. As a result, it is postulated that the equivalent granular void ratio may be adopted as a state variable to describe the stress–strain behavior of MHBS.

$$e^{*}=\frac{e+\left(1-\chi S_h\right)}{1+\chi e{S}_h}$$

(3)

where S_h is the hydrate saturation and χ is the fraction of hydrates actively participating in the force chain of MHBS. It should be noted that in the context of MHBS, the void ratio (e) in Equation (3) is calculated by considering the hydrates as voids, i.e., equal to e_g in Equation (1) for sand–fines mixtures. The value of χ is set between 0 and 1. When χ = 0, Equation (3) is reduced to e* = e, which assumes that all hydrates are inactive in the force chain of MHBS. By setting χ = 1, e* becomes e(1 − S_h)/(1 + eS_h), in which all hydrates are assumed to be part of the skeleton of MHBS.

Rahman and Lo found out that parameter b in Equation (2) is a function of both the fines content and particle size ratio based on the experimental data of McGeary conducted on binary mixture of spheres [45,46]. Using several other published data sets, Rahman et al. proposed the following semi-empirical equation for parameter b [34,47]:

$$b=\left[1-\exp \left(-0.3\frac{f_c/f_{thre}}{1-r^{0.25}}\right)\right]\times {\left(r\frac{f_c}{f_{thre}}\right)}^r$$

(4)

where r is the particle size ratio, d₅₀/D₁₀, d₅₀ is the average size of fines, D₁₀ is the effective size of sand, and f_thre is the threshold fines content, denoting a change of soil fabric from “fines-in-sand” to “sand-in-fines” [33]. Generally, an approximate value of 0.30 may be assumed for f_thre [34].

Considering a fixed size for the hydrate, it is reasonable to assume that parameter χ in Equation (3) is related to the hydrate saturation and the grain size of the sand. Then, the following empirical equation of parameter χ is proposed:

$$\chi =1-\exp \left(-a{S}_h\right)$$

(5)

where a is a parameter that depends on the grain size of the sand. Ni et al. suggested that the effective size of the sand, D₁₀, may be used to estimate its average pore size [39]. The larger the value of D₁₀, the more room hydrates have to move around and thus not participate in the force chain. Therefore, it is expected that the value of χ (or parameter a) decreases with the increase in D₁₀.

Hyodo et al. conducted a series of triaxial tests on MHBS synthesized from Toyoura sand [48]. All MHBS specimens were sheared to large strains. The critical state was approximately reached at the end of each drained triaxial test. Figure 1a shows the variation of the critical state lines (CSLs) of MHBS with hydrate saturation in the plane. It can be seen that CSL shifted towards the upper side when increasing the hydrate saturation. However, as shown in Figure 1b, a single CSL expressed in terms of the equivalent granular void ratio was identified by setting parameter a = 0.7. The single CSL for MHBS can be expressed as follows:

$$e_c^{*}=e_{\Gamma }-{\lambda }_c\ln \left(\frac{p^{\prime }}{p_a}\right)$$

(6)

where e_Γ and λ_c are two parameters, and p_a is the atmospheric pressure, taken as 100 kPa. Thus, it is postulated that the equivalent granular void ratio may be adopted as an alternative state variable to model the mechanical behavior of MHBS, which is incorporated in the following constitutive model.

3. Model Formulation

3.1. Conceptual Framework

According to the above discussion, by using the concept of the equivalent granular void ratio, the effects of pore-filling and load-bearing hydrate morphology on the behavior of MHBS can be uniquely described. Furthermore, regarding the cementing hydrate morphology, MHBS is treated as a composite material of the uncemented granular skeleton and hydrate bonding. As shown in Figure 2, the uncemented granular skeleton contains coarse soil grains as well as uncemented hydrate particles existing in the intergranular pore space, in which hydrates present pore-filling and the load-bearing morphology. Hydrate bonds are hydrates at intergranular contacts, in which hydrates present cementing morphology. Therefore, simulating the behavior of MHBS can be carried out by separately modeling the response of the uncemented granular skeleton and hydrate bonding. This kind of modeling approach is based on the framework first proposed by Chazallon and Hicher, and further developed by Chazallon and Hicher, and Vatsala et al. [49,50,51]. The idea is to model the behavior of the cemented geomaterial using two separate models—one to capture the behavior of uncemented soil skeleton and the other to simulate the behavior of cementation bonds. The overall behavior of the cemented geomaterial is the sum of the above two individual behaviors. It is assumed that “a cemented soil is a combination of uncemented soil skeleton at the same void ratio and microfabric compared to the cemented soil, and cementation bonds at interparticle contacts” [51]. If a cemented soil is subject to a given strain ε, the corresponding stress σ will be carried by an uncemented soil skeleton with an amount of σ_s and cementation bonds with an amount of σ_b.

Under this framework, when MHBS is subject to a certain value of strain ε, deformations of uncemented granular skeleton (ε_s) and hydrate bonding (ε_b) are assumed to be consistent with MHBS, which is mathematically expressed as follows:

$$\epsilon ={\epsilon }_s={\epsilon }_b$$

(7)

The corresponding stress exhibited on MHBS (σ) is the summation of stresses of uncemented granular skeleton (σ_s) and hydrate bonding (σ_b), and is written as follows:

$$\sigma ={\sigma }_s+{\sigma }_b$$

(8)

3.2. Generalized Plasticity Model for the Uncemented Granular Skeleton

In this study, the framework of the generalized plasticity was adopted for the constitutive modeling of the mechanical behavior of the uncemented granular skeleton. This framework was first proposed by Zienkiewicz and Mroz [52]. It has been successfully used in modeling the behavior of sand and rockfill materials under monotonic or cyclic loading [53,54,55,56]. Under the framework of generalized plasticity, no plastic potential, yield surface, or hardening rule is explicitly defined, but loading and plastic flow directions, as well as the plastic modulus, are specified instead. As a result, some complexities associated with the classical plasticity can be avoided and a simpler modeling of experimental results is permitted [53].

In the generalized plasticity, the stress increment vector (dσ_s) and the strain increment vector (dε) are related through an elastoplastic matrix D^s_ep, expressed as follows:

$$d{\sigma }_s=D_{ep}^{s}d\epsilon$$

(9)

The formulas of the proposed model are presented in the p′ – q stress plane. The mean effective stress and the deviator stress are defined as p’ = (σ_1′ + 2σ_3′) and q = σ_1′ − σ_3′, respectively, where σ_1′ and σ_3′ are the major and the minor principal stresses, respectively. The work conjugate strain rates for p’ and q are dε_v and dε_q, respectively (volumetric strain increment dε_v = dε₁ + 2dε₃ and shear strain increment dε_q = 2(dε₁ − dε₃)/3, where dε₁ and dε₃ are the principal strain increments).

Similar to the classical elastoplasticity, the elastoplastic matrix under generalized plasticity is expressed as follows [55]:

$$D_{ep}^s=D_e^s-\frac{D_e^s{n}_g{n}_f^T{D}_e^s}{H+n_f^T{D}_e^s{n}_g}$$

(10)

where D_e^s, H, n_f, and n_g are the elastic matrix, plastic modulus, loading direction vector, and plastic flow direction vector, respectively.

The elastic matrix in the p’ – q stress plane is written as:

$$D_e^s=\left[\begin{array}{cc}{K}_s& 0\\ 0& 3G_s\end{array}\right]$$

(11)

where K_s and G_s are the elastic bulk and shear moduli for the uncemented granular skeleton, respectively. It is assumed that the elastic shear modulus G_s is a function of stress and the equivalent granular void ratio, which is expressed as follows:

$$G_s=G_0\frac{{\left(2.97-e^{*}\right)}^2}{1+e^{*}}\sqrt{p^{\prime }{p}_a}$$

(12)

The elastic bulk modulus is defined as follows:

$$K_s=\frac{2\left(1+{\nu }_s\right)}{3\left(1-2{\nu }_s\right)}{G}_s$$

(13)

where G₀ denotes a model parameter and ν_s is the Poisson’s ratio for the uncemented granular skeleton.

For the sake of simplicity, the loading direction vector in the stress plane is defined as follows:

$$n_f={\left(n_f^v,n_f^s\right)}^T={\left(\frac{-\eta }{\sqrt{1+{\eta }^2}},\frac{1}{\sqrt{1+{\eta }^2}}\right)}^T$$

(14)

where η is the stress ratio, defined as η = q/p.

The plastic flow direction vector is expressed as follows:

$$n_g={\left(n_g^v,n_g^s\right)}^T={\left(\frac{d}{\sqrt{1+d^2}},\frac{1}{\sqrt{1+d^2}}\right)}^T$$

(15)

where d is the dilatancy, defined as d = dε_v^p/dε_s^p, and dε_v^p and dε_s^p are the plastic volumetric and shear strain increments, respectively.

By substituting Equations (11), (14), and (15) into Equation (10), the elastoplastic matrix D^s_ep in the p′ – q stress plane is then written as

$$D_{ep}^s=\frac{1}{H+n_v^f{n}_v^{g}K+3G{n}_s^f{n}_s^g}\left[\begin{array}{cc}K\left(H+3G{n}_s^f{n}_s^g\right)& -3G{n}_s^f{n}_v^{g}K\\ -3G{n}_v^f{n}_s^{g}K& 3G\left(H+n_v^f{n}_v^{g}K\right)\end{array}\right]$$

(16)

A general form of the state-dependent dilatancy proposed by Li and Dafalias can be expressed as [57]:

$$d=\frac{d_0}{M}\left(M^d-\eta \right)$$

(17)

where d₀ is a model parameter. M^d is the stress ratio at the phase transformation state, generally expressed as M^d = Mexp(mψ), in which m is a model parameter, and ψ is a state parameter defined as a measure of how far the material state is from the critical state in terms of the void ratio. Using the equivalent granular void ratio as the state variable, the equivalent granular state parameter ψ*, which is expressed as ψ* = e* − e_c*, is introduced. In the proposed model, M^d is expressed as a function of ψ*.

The plastic modulus, H, can be expressed as

$$H=\frac{H_{0}G\exp \left(n{\psi }^{*}\right)}{\eta }\left(M^b-\eta \right)$$

(18)

where n and H₀ are two model parameters. M^b is the stress ratio at the peak strength state, which is expressed as M^b = Mexp(−nψ*).

By deriving Equation (3), the increment of the equivalent granular void ratio is expressed as follows:

$$d{e}^{*}=\frac{1-\chi S_h}{{\left(1+\chi e{S}_h\right)}^2}de-\frac{\left(e+e^2\right)\left(\chi +a{S}_h\exp \left(-a{S}_h\right)\right)}{{\left(1+\chi e{S}_h\right)}^2}d{S}_h$$

(19)

Equation (19) indicates that the change in the equivalent granular void ratio can be caused by the change in void ratio and hydrate saturation. In this study, hydrate dissociation is not considered (i.e., a constant hydrate saturation). Thus, Equation (19) can be simplified as follows:

$$d{e}^{*}=\frac{1-\chi S_h}{{\left(1+\chi e{S}_h\right)}^2}de$$

(20)

3.3. Elastic Damage Model for the Hydrate Bonds

In MHBS, hydrate bonds are randomly distributed at intergranular contacts of soil grains that can be assumed as homogeneous, isotropic, and continuous elastic mediums. Experimental observations indicated that cementation bonds in the soil degrade gradually with plastic deformation. When bonds are completely damaged, the soil may reach the same critical state as for the uncemented state [58,59,60]. Under the above assumption and observations, hydrate bonding is considered as an elastic material with a brittle behavior governed by damage spreading inside the material. Therefore, an elastic law with damage is introduced to model the behavior of hydrate bonding. According to the elasticity theory, the stress-strain relationship of hydrate bonding is written as

$${\sigma }_b=D_e^b\epsilon$$

(21)

where D_e^b is the elastic matrix. Considering the damage of hydrate bonding, the elastic matrix in the p′ – q stress plane is given as:

$${\left[D\right]}_e^b=\left(1-\omega \right)\left[\begin{array}{cc}\frac{2\left(1+{\nu }_b\right)}{3\left(1-2{\nu }_b\right)}{G}_b& 0\\ 0& 3G_b\end{array}\right]$$

(22)

where G_b and ν_b are the elastic shear modulus and the Poisson’s ratio for hydrate bonding, respectively, and ω denotes a damage variable describing the evolution of damage for hydrate bonding.

Past studies have shown that the increase in stiffness of MHBS induced by the cementing hydrate morphology is much larger than that induced by the pore-filling or load-bearing hydrate morphology [61,62]. Based on the resonant column test results for MHBS reported by Clayton et al. [62], Shen et al. proposed the following modified equation of the elastic shear modulus for MHBS, considering the influence of hydrate saturation, pressure level, and void ratio [25]:

$$G=\left(G_0-\zeta S_h\right)\frac{{\left(2.97-e\right)}^2}{1+e}\sqrt{p^{\prime }{p}_a}$$

(23)

where ζ is a model parameter. In Equation (23), the second term in the bracket on the right-hand side presents the contribution of hydrates to the stiffness enhancement of MHBS. Considering the predominant effects of cementing over those of uncemented granular skeleton, Equation (23) is further modified as:

$$G_b=k{S}_h\frac{{\left(2.97-e^{*}\right)}^2}{1+e^{*}}\sqrt{p^{\prime }{p}_a}$$

(24)

where k is a model parameter.

Regarding the Poisson’s ratio, past experimental studies have indicated that the Poisson’s ratio of MHBS may be independent of the hydrate saturation and the hydrate morphology for both laboratory-synthesized and field samples [61,63]. Therefore, in this study, the Poisson’s ratio of hydrate bonding is assumed to be the same as that of the uncemented granular skeleton, i.e., ν_b = ν_s.

The damage variable ω controls the variation of damage from hydrate bonding with shearing. According to the triaxial test results for MHBS conducted by Hyodo et al., the volume change of MHBS is negligible under a confining pressure ranging from 1 to 5 MPa [48]. Thus, it is reasonable to assume that the damage from hydrate bonding is not induced by isotropic compression under a confining pressure of interest. Herein, the damage from hydrate bonding only depends on shearing. The following relationship between the damage variable and the shear strain is proposed:

$$\omega =1-e^{-\xi {\epsilon }_q}$$

(25)

where ξ is a positive model parameter. Figure 3 shows that the damage variable ω varies from 0 to 1 and parameter ξ controls the rate of damage with respect to the shear strain. When there is no shear deformation, ω is equal to zero by setting ε_q = 0 in Equation (25), denoting the intact state of hydrate bonding. With the continuous development of the shear strain, ω gradually increases and reaches the upper limit value of 1, implying complete damage of hydrate bonding. For shear strain below 0.2 (i.e., strain commonly encountered in the triaxial specimens), Figure 3 depicts that the nonlinearity increases with increasing ξ.

Considering the degradation of stiffness with shearing, the incremental stress–stain relationship for hydrate bonding is written as

$$d{\sigma }_b=D_e^{b}d\epsilon +\epsilon d{D}_e^b$$

(26)

By combining Equations (8), (9), and (26), the incremental stress–strain relationship for MHBS is summarized as follows:

$$d\sigma =D_{ep}^{s}d\epsilon +D_e^{b}d\epsilon +\epsilon d{D}_e^b$$

(27)

When there is no hydrate in MHBS, the proposed model will be simplified into a generalized plasticity model for sand by setting the hydrate saturation to zero.

4. Parameter Calibration

4.1. Parameter Calibration for Sand

There are twelve parameters in the proposed model, of which nine are used to describe the mechanical behavior of sand, including elastic moduli parameters G₀ and ν_s, critical state parameters M, e_Γ and λ_c, dilatancy parameters d₀ and m, and plastic modulus parameters H₀ and n. Most of these nine parameters have physical meanings and can be calibrated through conventional laboratory tests. The specific methods for determining these nine parameters can refer to Li and Dafalias, Xiao et al., and Liu and Zou [56,57,64].

4.2. Parameter Calibration for MHBS

Besides the nine parameters for sand, three additional parameters are required for MHBS, including an equivalent granular void ratio parameter a, an elastic modulus parameter k, and a damage parameter ξ. These three parameters can also be conveniently calibrated using conventional laboratory tests; preferably, two drained triaxial tests conducted on MHBS with different hydrate saturations.

4.2.1. Equivalent Granular Void Ratio Parameter

The equivalent granular void ratio parameter a can be determined using critical state data (p′ and e_c) obtained from the triaxial tests. By substituting Equations (3) and (5) into (6), a is expressed as

$$a=-\frac{\ln \left(1-\frac{e_c+{\lambda }_c\ln \left(p^{\prime }/p_a\right)-e_{\Gamma }}{\left[e_{\Gamma }+1-{\lambda }_c\ln \left(p^{\prime }/p_a\right)\right]e_c{S}_h}\right)}{S_h}$$

(28)

Considering the scatter of experimental data, the best-fit value of a is determined from the triaxial tests conducted on the specimens with different hydrate saturations.

4.2.2. Elastic Modulus Parameter

Combining Equations (16), (22), and (27) yields

$$dq=\left(\frac{3G_s\left(H+n_v^f{n}_v^g{K}_s\right)}{H+n_v^f{n}_v^g{K}_s+3G_s{n}_s^f{n}_s^g}+3G_b{e}^{-\xi {\epsilon }_q}-3G_{b}b{e}^{-\xi {\epsilon }_q}{\epsilon }_q\right)d{\epsilon }_q-\frac{3G_s{n}_v^f{n}_s^g{K}_s}{H+n_v^f{n}_v^g{K}_s+3G_s{n}_s^f{n}_s^g}d{\epsilon }_v$$

(29)

In the small strain regime, i.e., ε_s ≈ 0 and η ≈ 0, according to Equation (18), the plastic modulus H is nearly infinite, and thus Equation (29) can be approximately written as

$$dq\approx 3\left(G_s+G_b\right)d{\epsilon }_q$$

(30)

By substituting Equation (24) into Equation (30), the elastic modulus parameter k can be expressed as

$$k=\frac{\frac{dq}{3d{\epsilon }_q}-G_s}{\frac{{\left(2.97-e^{*}\right)}^2}{1+e^{*}}\sqrt{p^{\prime }{p}_a}{S}_h}$$

(31)

As G_s has been determined in the parameter calibration for sand, k can then be determined by curve fitting the measured ε_q − q data for MHBS in the small strain regime.

4.2.3. Damage Parameter

Once all of the above parameters have been determined, the damage parameter ξ can be determined by curve fitting the measured ε_q − q data for MHBS in the large strain regime obtained from the drained triaxial tests. As shown in Figure 4, ξ = 2 gives the best-fitted result for the measured stress–strain relationship. Figure 3 shows that ω has a quasi-linear relationship with shear strain for ξ = 2 under the measured strain.

5. Model Verification

The first set of experimental data for model verification are the drained triaxial tests conducted on MHBS synthesized from Toyoura sand [48]. The parameters used in the model predictions are summarized in

Table 1. Parameters for methane hydrate bearing sand (MHBS) synthesized from Toyoura sand.

Parameter	Symbol	Value
Elastic moduli	G0 (MPa)	350
	νb = νs	0.05
	K	30
Critical state	M	1.25
	eΓ	0.98
	λc	0.11
Plastic moduli	H	0.5
	N	1.3
Dilatancy	d0	1.1
	M	1.3
Equivalent granular void ratio	A	0.7
Damage	Ξ	2.0

. It should be noted that in addition to the nine parameters for sand, only three additional parameters (a, k, and ξ) were required to model the behavior of MHBS. In the first series of tests, soil specimens were compacted to an initial void ratio of 0.65, while the hydrate saturation varied between 0% and 53%. Isotropic compression was first conducted to an effective confining pressure of 5 MPa, followed by drained shearing. Figure 5 depicts the model predictions of the drained triaxial tests for MHBS with different hydrate saturations. It can be observed that the model could predict the increase in the initial stiffness, peak shear strength, and dilatancy with increase in hydrate saturation. In the second series of tests, two specimens were compacted to an initial void ratio of 0.662 and 0.803, while maintaining a hydrate saturation of 52%. The specimens were first isotropically compressed to an effective confining pressure of 3 MPa, then sheared in the drained condition.

Figure 6 shows the predicted results of the drained triaxial tests for MHBS compacted to two different initial void ratios. Despite some discrepancies, the model can still capture the effects of the void ratio on the behavior of MHBS qualitatively. As the specimen changes from a loose to dense state, the model can predict the change from strain hardening to strain softening, as well as from a compressive to dilative response. Figure 5a and Figure 6a depict that there are discrepancies between the measured and predicted stress–strain curves. The triaxial test results revealed that the stress ratio at the critical state for MHBS was slightly higher than that for the host sand. However, for simplification, the influence of hydrate saturation on the critical state stress ratio for MHBS was not considered in the proposed model. As a result, there were discrepancies between the measured and predicted stress–strain curves, in particular for those with a higher hydrate saturation (i.e., higher degree of dilatancy).

The second set of experimental data for model verification are drained triaxial tests conducted on MHBS synthesized from Toyoura, No. 7 silica, and No. 8 silica sands [18,63].

Table 2. Parameters for MHBS synthesized from different sands.

Parameter	Symbol	Toyoura Sand	No. 7 Silica Sand	No. 8 Silica Sand
Elastic moduli	G0 (MPa)	600	500	600
	νb = vs	0.05	0.05	0.05
	K	22	20	12
Critical state	M	1.35	1.40	1.50
	eΓ	0.98	1.12	1.26
	λc	0.05	0.10	0.13
Plastic moduli	H	0.10	0.16	0.10
	N	1.0	1.0	1.0
Dilatancy	d0	1.3	1.8	1.6
	M	3.0	1.0	1.0
Equivalent granular void ratio	A	0.2	0.3	0.5
Damage	Ξ	1.0	0.5	1.0

summarizes the parameters adopted for the model predictions. Parameter a in Equation (5) is related to the equivalent granular void ratio, which depends on the grain size of the sand. The effective grain size (D₁₀) of the Toyoura, No. 7 silica, and No. 8 silica sands are 0.175 mm, 0.139 mm, and 0.072 mm, respectively. Based on the critical state data, the calibrated values of parameter a for the Toyoura, No. 7 silica, and No. 8 silica sands are 0.2, 0.3, and 0.5, respectively. The specimens of the Toyoura, No. 7 silica, and No. 8 silica sands were compacted to an initial void ratio of 0.60, 0.63, and 0.70, respectively. Isotropic compression was first carried out to an effective confining pressure of 1 MPa followed by drained shearing. Figure 7, Figure 8 and Figure 9 depict the model predictions of the drained triaxial tests for MHBS synthesized from the Toyoura, No. 7 silica, and No. 8 silica sands, respectively. It is apparent that the proposed model can qualitatively capture the effects of hydrate saturation for MHBS synthesized from different host sands. The model can simulate the increase in brittleness of the stress–strain response and dilatancy with the increase in hydrate saturation.

6. Conclusions

A combined generalized plasticity and elastic damage model is proposed to predict the mechanical behavior of MHBS specimens formed under three different types of hydrate morphology. The generalized plasticity model is formulated using the equivalent granular void ratio as a state variable, from which the contributions of the pore-filling and loading-bearing hydrate morphology on the behavior of the uncemented granular skeleton can be uniquely described. The elastic damage model is adopted to stimulate the behavior of hydrate bonding (i.e., cementing hydrate morphology), in which the degradation of hydrate bonding with shearing is considered. In the new model, the stiffness of the hydrate bonding incorporates the influence of hydrate saturation, pressure level, void ratio, and degradation of bonding. Taking advantage of the equivalent granular void ratio, only three additional parameters are required for modeling the behavior of MHBS formed under different hydrate morphologies compared with those of the basic model without hydrate. By simulating the drained triaxial test results of MHBS synthesized from different host sands, it is demonstrated that the proposed model can satisfactorily capture the stress–strain and volumetric behavior of MHBS under different hydrate saturations, confining pressures, and void ratios.