Critical state soil constitutive model for methane hydrate soil - Wiley

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, B03209, doi:10.1029/2011JB008661, 2012

Critical state soil constitutive model for methane hydrate soil S. Uchida,1 K. Soga,1 and K. Yamamoto2 Received 7 July 2011; revised 15 January 2012; accepted 27 January 2012; published 16 March 2012.

[1] This paper presents a new constitutive model that simulates the mechanical behavior of methane hydrate-bearing soil based on the concept of critical state soil mechanics, referred to as the “Methane Hydrate Critical State (MHCS) model”. Methane hydrate-bearing soil is, under certain geological conditions, known to exhibit greater stiffness, strength and dilatancy, which are often observed in dense soils and also in bonded soils such as cemented soil and unsaturated soil. Those soils tend to show greater resistance to compressive deformation but the tendency disappears when the soil is excessively compressed or the bonds are destroyed due to shearing. The proposed model represents these features by introducing five extra model parameters to the conventional critical state model. It is found that, for an accurate prediction of ground settlement, volumetric yielding plays an important role when hydrate soil undergoes a significant change in effective stresses and hydrate saturation, which are expected during depressurization for methane gas recovery. Citation: Uchida, S., K. Soga, and K. Yamamoto (2012), Critical state soil constitutive model for methane hydrate soil, J. Geophys. Res., 117, B03209, doi:10.1029/2011JB008661.

1. Introduction [2] Methane hydrate-bearing soil is a natural soil deposit that contains methane hydrate inside its pores. Methane hydrate is a metastable solid material that consists of methane gas (CH4) and water molecules. The water molecules form a structure in which a floating methane gas molecule is encaged. Typically, 1 m3 of methane hydrate can release 164 m3 of methane gas and 0.87 m3 of water when the hydrate dissociates (i.e. becomes unstable) under standard conditions of temperature and pressure. [3] Methane hydrate-bearing soil can exist under conditions of high pressure and low temperature (e.g. 100 kPa at 80 C and 2.5 MPa at 0 C). Therefore, natural methane hydrate-bearing soil is usually found in deep water sediments (e.g. 1–2 km below sea level) or permafrost regions (e.g. 1–2 km below the ground surface) [Collett, 2002; Kvenvolden, 1999]. [4] Deep water sediments with a high concentration of methane hydrate in the pore space are considered to be a possible energy resource for future exploitation. This has triggered numerous researchers to investigate the physical, chemical and mechanical properties of methane hydratebearing soil. Two methods are generally accepted to be feasible for dissociating methane hydrate and thus extracting methane gas in situ: (1) the use of depressurized wells, which reduce the pressure in the soil around the wells; and (2) the use of thermal injection wells, which increase the temperature in the surrounding soil by injecting hot fluid. 1

Department of Engineering, University of Cambridge, Cambridge, UK. Japan Oil, Gas and Metal National Corporation, Chiba, Japan.

2

Copyright 2012 by the American Geophysical Union. 0148-0227/12/2011JB008661

[5] Methane hydrate-bearing soil is generally categorized into two types based on hydrate morphology, in relation to the location where hydrate is formed in the pore space: [6] 1. Pore filling - hydrates exist inside pore space. The hydrates may bridge neighboring grains, the type also known as “load bearing”, when the hydrate saturation in volume fraction exceeds 25–40% [e.g., Yun et al., 2007; Lee and Waite, 2008]. As a result, the hydrate-bearing soil is effectively denser (i.e. less pore space) than that without hydrate and is likely to behave as an overconsolidated or dense soil. [7] 2. Cementing - hydrates exist at soil grain contacts and act as bonding agent. Thus, the soil is likely to behave as a bonded soil. [8] Based on the review of the mechanical properties of methane hydrate-bearing soil by Soga et al. [2006] and Waite et al. [2009], the following characteristics are apparent for both pore filling and cementation types: (1) the peak strength of hydrate-bearing soil depends on hydrate saturation (Sh); (2) the hydrate contribution to its shear behavior is of a cohesive nature rather than frictional, meaning that the peak strength may vary but the critical strength at large shear strain is similar regardless of hydrate saturation; (3) the dilation angle increases with hydrate saturation; and (4) the stiffness of hydrate-bearing soil tends to be greater than that of soil without hydrate. [9] Note that the degree of these changes depend highly on hydrate morphology. For example, the rate of increase in small strain stiffness with increase in hydrate saturation is much greater in the cementing type compared to that in the pore filling type [e.g., Clayton et al., 2010; Priest et al., 2009]. The detailed explanation on modeling of mechanical behavior at larger strains with different morphologies is shown later.

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UCHIDA ET AL.: MHCS MODEL

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Figure 1. Conventional drained triaxial compression test on bonded and dense soil. [10] Once the hydrate dissociates either through a decrease in pore pressure or increase in temperature, the hydrates inside the pore space disappear and the soil may behave as less dense soil or unbonded soil. The loss of shear resistance and increase in pore space due to hydrate dissociation may lead to some geomechanical problems. For example, the depressurization process used for methane gas extraction significantly increases the effective stress state by the reduction in pore pressure and may change the mechanical properties of hydrate soils, which may lead to potential geomechanical hazards such as excessive ground settlement, submarine landslide or wellbore collapse, which have been considered theoretically by many researchers [e.g., Settari, 2002; Sultan et al., 2004; Xu and Germanovich, 2006; Freij-Ayoub et al., 2007]. [11] This paper presents a new constitutive model that incorporates the effect of hydrate on the stress-strain behavior of soils. A simple example of the application of the model is presented by evaluating the possible magnitudes of ground deformation of hydrate-bearing sediments when methane gas is recovered by the depressurization method.

2. Summary of Mechanical Behavior of Hydrate Soils 2.1. The Effect of Hydrate on Stress-Strain Behavior [12] Bonded soils such as cemented soil (i.e. cement bonding) and unsaturated soil (i.e. meniscus bonding) exhibit greater stiffness, strength and dilatancy compared to unbonded soils with the equivalent soil skeleton structure under the same confining stress. Although different in magnitudes, it is also known that denser soils (i.e. less pore space) exhibit similar mechanical enhancements under the same confining stress due to greater interlocking of soil grains. The mechanical behavior of such soils is well studied

[e.g., Lade and Overton, 1989; Consoli et al., 1998; Miura et al., 2001; Asghari et al., 2003; Wang and Leung, 2008] for cemented soils [e.g., Cui and Delage, 1996; Rampino et al., 1999; Toll and Ong, 2003; Tarantino and Tombolato, 2005; Zhan and Ng, 2006], for unsaturated soils [e.g., Lade, 1977; Been and Jefferies, 1985; Bolton, 1986; Burland, 1990], for dense soils. Figure 1 shows idealized geomechanical behavior of bonded soils and dense soils. The formation of bonds at grain contacts or the interlocking of grains increases the contact stiffness and hence the macroscopic stiffness, which is illustrated by the greater gradient of deviator stress q to deviatoric strain d in Figure 1. The enhanced adhesion at grain contacts and interlocking of grains increase the shear resistance and hence the peak strength, represented by greater values of q in Figure 1. However, once the bonds or the interlocking break due to shearing, the soil exhibits strain softening behavior, decreasing its shear resistance. With increased bonding in the soil pores or increased density, more dilation (i.e. volume increase when the soil is sheared) is observed, which can be seen as an increase in the specific volume v (i.e. 1 + e, where e is the void ratio) in Figure 1. This occurs because of two mechanisms: (1) the bonding aggregates the soil grains and thus creates larger sized grains with more kinematic constraints; and (2) soil grains with dense arrangement need to be lifted upwards when the soil skeleton is sheared. [13] The degree of bonding and soil density also influence the compressive response of the soil [e.g., Burland, 1990; Cui and Delage, 1996; Rotta et al., 2003; Consoli et al., 2005; Futai and Almeida, 2005; Chiu et al., 2009; Sivakumar et al., 2010]. Figure 2 shows a typical relationship between mean effective stress p′ (i.e. mean value of three normal effective stresses) and the change in the 1þe ) of bonded soils and dense soils specific volume (i.e. 1þe 0

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Figure 2. A typical compressive response of bonded and dense soil under isotropic loading. under an isotropic loading condition (i.e. zero shear loading). As bonding or interlocking strengthens the structure of the soil skeleton, such soils tend to resist compressive deformation. However, once the skeleton structure is destroyed by excessive compression (i.e. volumetric yielding), greater deformation occurs, which can be seen as an abrupt change in slope of the compression curve in Figure 2. [14] Soil straining may deteriorate the bonds and the degree of interlocking, which in turn reduces its contribution to the enhancement of stiffness and strength. This is particularly evident in unconsolidated weak soils with bonding [e.g., Leroueil and Vaughan, 1990; Cuccovillo and Coop, 1997; Malandraki and Toll, 2001; Sharma and Fahey, 2003; DeJong et al., 2006; Hamidi and Haeri, 2008]. [15] As the hydrates in the pore space make the soil “effectively” denser or more bonded, it is natural to assume that hydrate-bearing soil behaves similarly to such soils. This is confirmed by results of drained triaxial tests on artificially created hydrate-bearing soils [e.g., Hyodo et al.,

2005; Masui et al., 2005; Miyazaki et al., 2008; Yun et al., 2007]. Some of the data are replotted in Figure 3, showing the effect of hydrate saturation (defined as the ratio of the hydrate volume to the pore volume) on maximum tangent stiffness (i.e. initial gradient of deviator stress against axial strain; Figure 3a), peak strength (i.e. maximum deviator stress; Figure 3b) and dilation angle (Figure 3c). All the samples presented are tested at an effective confining stress of 1 MPa. These mechanical values at different hydrate saturations are normalized by those without hydrate. From Figure 3, it is clear that hydrate-bearing soil exhibits greater stiffness, strength and dilatancy compared to soils without hydrate. [16] Some researchers investigated the mechanical behavior of tetrahydrofuran (THF) hydrate soil. Yun et al. [2007] showed the degradation of tangent stiffness of synthetic THF hydrate-bearing soil due to shearing. Dai et al. [2010] showed that the compressibility of THF hydrate soil increases with a decrease in hydrate saturation.

Figure 3. The effect of hydrate on (a) stiffness, (b) strength and (c) dilatancy. 3 of 13


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Figure 4. Isotropic loading in critical state models. 2.2. Constitutive Soil Model Incorporating the Effect of Hydrate [17] Constitutive modeling of the mechanical behavior of bonded soils and dense soils has been researched for more than three decades. Most of the models adopt the critical state concept [Roscoe et al., 1958] due to its capability of modeling both shear and volumetric yielding, the latter of which Mohr-Coulomb model cannot capture. Figure 4 shows the volumetric yielding in the critical state model framework. Under isotropic loading, the soil becomes fully plastic when the mean effective stress p′ reaches the volumetric yield stress p′cs, and the volumetric response then dramatically changes. It is usually defined by two material parameters: k which represents the slope of the line in v:lnp′ space while the soil is elastic, also known as the slope of the swelling line; and l represents the slope of the line while

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plastic, also known as the slope of the normal compression line. Critical state models also give both strain softening and hardening behavior depending on the stress state and soil density, which avoids the explicit modeling of strain softening used in Mohr-Coulomb model. Further details of critical state models can be found elsewhere [e.g., Schofield and Wroth, 1968; Wood, 1990; Schofield, 2005]. [18] For modeling of bonded soils, critical state models are further extended by introducing new state parameters such as water saturation and bonding strength that change the size of the yield surface [e.g., Alonso et al., 1990; Wheeler and Sivakumar, 1995; Rampino et al., 2000; Gallipoli et al., 2003] for unsaturated soils; [e.g., Gens and Nova, 1993; Vatsala et al., 2001; Liu and Carter, 2002; Nova et al., 2003; Lee et al., 2004] for cemented soils. The increased stiffness by bonding has been modeled by Alonso et al. [1990], Gens and Nova [1993], and Liu and Carter [2002]. Degradation of bonding due to shearing has been modeled by Liu and Carter [2002], Lee et al. [2004], and Vatsala et al. [2001]. [19] Ideally, a constitutive model for hydrate-bearing soil should include the mechanical characteristics of bonded soils and dense soils, which are enhanced stiffness, strength and dilatancy; limited compressibility; and degradation of hydrate cementing and interlocking. Table 1 summarizes the constitutive models currently available for simulating hydrate-bearing soil behavior. An extension of MohrCoulomb (MC) model has been a popular choice as shown in Table 1. Typically, the stiffness and strength parameters are a function of hydrate saturation. Klar et al. [2010] acknowledge the importance of enhanced dilation characteristics of methane hydrate-bearing soil and used their model to simulate wellbore stability problems. Most hydrate-bearing soil models focus on shear deformation but ignore volumetric yielding. Such behavior may be of engineering importance during the depressurization process as the effective stress increases due to pore pressure reduction. Kimoto et al. [2010] extended the original Cam-clay (OCC) model [Roscoe et al., 1958] for hydrate-bearing soil but did not include the stiffness enhancement by the presence of hydrate. To date, none of the hydrate-bearing soil models considers the degradation of hydrate cementing and interlocking by plastic straining. Such behavior at large strains may be important for wellbore stability and landslide problems.

3. Methane Hydrate Critical State Model [20] In this section, a new constitutive model for the mechanical behavior of methane hydrate-bearing soil,

Table 1. Hydrate Constitutive Model Mohr-Coulomb Model Based

Critical State Model Based

Parameter

Freij-Ayoub et al. [2007]

Rutqvist and Moridis [2007]

Klar et al. [2010]

Kimoto et al. [2010]

Methane Hydrate Critical State

Stiffness Strength Dilation Softening Vol. yield Bond deg.

Yes Yes No explicit No No

Yes Yes No explicit No No

Yes Yes Yes No No No

No Yes Yes Yes Yes No

Yes Yes Yes Yes Yes Yes

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associated flow rule and hence the plastic potential g is the same as the yield function f: f ¼ g ¼ q2 þ M 2 p′ðp′ p′cs Þ

Figure 5. Dilation enhancement by hydrate. named as the methane hydrate critical state (MHCS) model, is derived. 3.1. Volume Change Behavior [21] In critical state models, the stress-dependent elastic soil bulk modulus K′ is given by [Roscoe et al., 1958; Roscoe and Burland, 1968] K′ ¼

v p′ k

ð1Þ

where v is the specific volume (=1 + e), k is the slope of a swelling line and p′ is the mean effective stress (compression positive). [22] Under isotropic conditions, full plastic volumetric deformation occurs when the isotropic stress reaches the volumetric yield stress p′cs as shown in Figure 4. In this study, the volumetric yield stress is used as a hardening parameter of the yield surface (cf. equation (4)) and increases with plastic volumetric strain (i.e. hardening law). Using Figure 4, the following conventional evolution law used by many critical state models is adopted in this study: vp′cs d p dp′cs ¼ lk v

where q is the deviator stress and M is the material property that relates to a frictional behavior of the soil. [25] Methane hydrate-bearing soil exhibits both enhanced dilation and cohesion due to hydrate bonding (cementing) and less pore space (pore filling) (cf. Figure 3). As mentioned before, it is known that hydrate morphology dominates the cohesive or dilative nature of the hydrate-bearing soils [e.g., Masui et al., 2005; Soga et al., 2006; Waite et al., 2009]. In the proposed model, the two mechanisms are separately modeled so that the effect of morphology can be produced by adjusting two additional model parameters (shown later). Figure 5 illustrates the mechanism of dilation enhancement and the mathematical representation of the yield surface expansion in the p′ : q space. The additional hardening parameter p′cd is added to the yield function in such a way that the yield surface expands to the right hand side of the original ellipse. As a result, using an associative flow rule, greater dilation is given by a greater normal angle to the tangent of the surface when the soil yields at a given mean effective stress p′ at the dry side of the critical state. [26] Figure 6 shows the mechanism of cohesion enhancement and the mathematical representation of yield surface expansion in the p′ : q space. The additional hardening parameter p′cc is added to the yield function to enlarge the yield surface uniformly both the left and right hand sides of the original oval. This allows an increase in cohesive strength but the dilatancy characteristic upon yielding at a given mean effective stress is not affected. [27] Adding the enhanced dilation (cf. equation (5)) and cohesion (cf. equation (6)), equation (3) becomes f ¼ q2 þ M 2 ðp′ þ p′cc Þ½p′ ðp′cs þ p′cd þ p′cc Þ

ð2Þ

where p′cs is the volumetric yield stress (also known as preconsolidation stress), l is the slope of the normal compression line and vp is the plastic volumetric strain (compression positive). [23] At low confining stress, it is known that the normal compression line of sand is nonlinear in v:lnp′ space. At high confining pressure (e.g. above 1–10 MPa), however, the slope becomes constant [Been et al., 1991]. At such stress levels, sand grains start crushing and the compression behavior becomes greater [e.g., McDowell and Bolton, 1998; Bolton, 2000; Cheng et al., 2003]. The effective confining stress of 1–10 MPa is approximately equivalent to 100–1000 meters below the seabed, where hydrate commonly exists. Therefore, a critical state model with constant l is considered to be appropriate for typical hydrate-bearing sediments.

ð3Þ

ð4Þ

[28] This yield surface is shown in Figure 7. 3.3. Degradation of Mechanical Properties [29] The two hardening parameters defined in section 3.2 are modeled as a function of hydrate saturation:

3.2. Dilation and Cohesion Enhancement by Hydrate [24] For the mechanical behavior of sand without hydrate, the proposed model utilizes the following yield function f of the modified Cam-clay model [Roscoe and Burland, 1968]. Note that the plastic strain development is modeled using the 5 of 13

b p′cd ¼ a Shmec

ð5Þ

d p′cc ¼ c Shmec

ð6Þ

Figure 6. Cohesion enhancement by hydrate.


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very high hydrate saturation conditions, very limited experimental data (with the exception of Yun et al. [2007] in which THF hydrate is used) exists for such conditions and further work is needed. [34] For the elastic shear stiffness, however, the presence of hydrate inside pores may contribute to shear resistance of hydrate-bearing soil as shown in Figure 3. Thus, the elastic shear modulus of hydrate-bearing soils Ghs is assumed to be the summation of the stress-dependent shear modulus of soil skeleton Gs and the shear stiffness increase due to presence of hydrate Gh. For simplicity, the increase is modeled to be a linear function of hydrate saturation: Ghs ¼ Gs þ Gh ¼ 3K′

Figure 7. Yield surface of MHCS model.

where a, b, c and d are material constants that describe the degree of hydrate contribution to the two hardening parameters, which are hydrate morphology dependent and determined from the experimental data, and Shmec is the “mechanical” hydrate saturation. [30] In equations (5)–(6), the “mechanical” hydrate saturation is used rather than the actual hydrate saturation Sh in order to model the behavior of hydrate bonding degradation as the soil is sheared. This is done by introducing a degradation factor c:

1 2n þ m2 Shmec 2ð1 þ n Þ

ð11Þ

where n is the Poisson’s ratio of soil skeleton and m2 is a parameter that gives the degree of increase in shear modulus with hydrate saturation, which is hydrate morphology dependent and is determined from experimental data. 3.4. Yield Function and Plastic Strains [35] It is a well-known fact that irrecoverable plastic strains develop even when the stress state is inside the yield surface [Jardine, 1992; Mitchell and Soga, 2005]. In order to provide a smooth transition from elastic to plastic behavior, a subloading surface ratio R [Hashiguchi, 1989] can be used such that equation (4) becomes f ¼ q2 þ M 2 ðp′ þ p′cc Þ½p′ Rðp′cs þ p′cd þ p′cc Þ

ð12Þ

where Sh is the hydrate saturation. [31] The evolution of the degradation factor c is modeled as follows:

1 þ p′cd þ p′cc ln R jd p j dR ¼ u p′cs

ð13Þ

dc ¼ mcddp ð0 ≤ c ≤ 1Þ

where p is the plastic strain vector and u is the material constant that controls the plastic deformation while the soil is elastic, which represents the development of plastic strain when the stress state is inside the yield surface. A smaller value u generates more plastic strain. dR > 0 represents the plastic state and dR < 0 when the soil is elastic. [36] The yield function of the proposed model is a function of six variables: q, p′, p′cs, p′cd, p′cc and R. The consistency equation becomes

Shmec ¼ cSh

ð7Þ

ð8Þ

where m is the material parameter that gives the rate of mechanical degradation upon shearing and dp is the plastic deviatoric strain. [32] This implies the change in the mechanical hydrate saturation can be achieved by either shearing, hydrate dissociation or a combination of the two as shown by the differential form of equation (7): dShmec ¼ dcSh þ cdSh ¼ mcddp Sh þ cdSh

ð9Þ

[33] The model assumes that the contribution of hydrates to the mechanical behavior becomes zero when the soil is excessively sheared even though “sheared” hydrates exist in the soil pores, unlike crushed sand grains altering the soil properties [Guimaraes et al., 2007]. This assumption is based on the fact that the stiffness of hydrate crystal is significantly smaller than that of soil grains. This also leads to an assumption that the elastic bulk stiffness of hydratebearing soil is dominated by that of soil skeleton: Khs ¼ K′ þ Kh ≈ K′

ð10Þ

df ¼

∂f ∂f ∂f ∂f ∂f ∂f dp′ þ dp′ þ dp′ þ dR ¼ 0 dq þ dp′ þ ∂q ∂p′ ∂p′cs cs ∂p′cd cd ∂p′cc cc ∂R

ð14Þ

[37] Together with the definition of the plastic strains p ∂g ∂g dpv ¼ L ∂p ′ and dd ¼ L ∂q, solving equation (14) in terms of the plastic multiplier L gives ∂f ∂f dq þ dp′ þ Lh ∂q ∂p L¼ ∂g 1 þ p′cd þ p′cc ∂f vp′cs ∂g ∂g ∂f ln R þ mLh þ u l k p′cs ∂p′cs ∂p′ ∂q ∂R ∂s′

where Khs is the elastic bulk stiffness of hydrate-bearing soil and Kh is the stiffness enhancement due to presence of hydrate. Although this assumption may not be realistic at

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Lh ¼

∂f ∂f abðcSh Þb þ cd ðcSh Þd ∂p′cd ∂p′cc

ð15Þ


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3.5. Stress Relaxation Due to Hydrate Dissociation [38] Natural hydrate-bearing soil is often formed in sandy sediments under confinement with continuous growth of hydrate inside pores. That is, the original soil skeleton carries most of the in situ stresses s′0 . When the additional effective stresses s′ are applied such as by wellbore construction and depressurization, however, both the hydrate and soil skeleton will be loaded. This results in s′ s′0 ¼ Deh eh þ Des es ¼ Dehs e

ð16Þ

where s′ is the effective stress vector of hydrate-bearing soil after deformation from the original in situ stress s′0 (compression positive), e is the elastic strain vector and Dehs is the hydrate-soil combined elastic stiffness matrix (using equations (1) and (11)). This unique behavior of hydratebearing soil was first introduced by Klar et al. [2010] (see the referred paper for more detail). The formulation was further developed by A. Klar et al. (Explicitly coupled thermal-flow-mechanical formulation for gas hydrate sediments, submitted to Society of Petroleum Engineers Journal, 2011), incorporating thermomechanical effects. Herein, a brief summary of the formulation will be presented. [39] Considering the thermomechanical effect, the incremental form of equation (16) can be expressed as ∂s′ ds′ ¼ Dehs de þ dDehs d¼0 e þ dT ∂T

dDehs jd¼0 ¼ dDeh ¼ Deh cdSh 0 4 m B 3 2 B 2 B m B 3 2 B e 2 Dh ¼ B B m2 B 3 B 0 B @ 0 0

2 m2 3 4 m2 3 2 m2 3 0 0 0

2 m2 3 2 m2 3 4 m2 3 0 0 0

thus the term (s′ s′0) remains. It also depends on how much hydrate dissociates and thus the term contains the change in hydrate saturation dSh. [43] The effective stress change due to the change in temperature is caused by the expansion of soil grains and hydrate. The third term of equation (19) can be obtained by ∂s′ d dT ¼ ½ð1 nÞb s þ nSh b h K′ddT ¼ Dehs b ∗ dT ∂T 3 b ∗ ¼ ½ð1 nÞbs þ nSh b h

1 0

0

0

0

0

0

0 m2 0 0

0 0 m2 0

0 0 0 m2

C C C C C C C C C C A

d ¼ de þ d p ¼ de þ L

∂g ∂s′

ð21Þ

[45] Combining equations (15), (20), and (21), equation (19) can be rewritten as " ds′

¼

T

Dehs

∂g ∂f e Dehs ∂s′ ∂s′ Dhs ∂f T e ∂g ∂s′ Dhs ∂s′

∂f ∂k ∂k ∂p

T ∂g ∂s′

# d

2 3 e 1 ∂f T e ∂f ðs′ s′0 Þ þ ∂S 1 ∂s′ Dh c Dhs e ∂g 5 h dSh þ4Deh c Dehs ðs′ s′0 Þ D hs ∂f T ∂f ∂k T ∂g ∂s′ ∂s′ ∂k ∂ p ∂s′ " # T De ∂g ∂f De d ∗ þ Dehs T hs ∂s′ ∂s′ hs T ð22Þ b dT ∂f De ∂g ∂f ∂kp ∂g 3 ∂s′

hs ∂s′

∂k ∂

∂s′

[46] Equation (22) shows that the effective stress change can be caused by soil straining, hydrate dissociation and temperature change, which are represented by each line on the right hand side.

4. Verification of the MHCS Model ð18Þ

[41] Thus, equation (17) becomes ∂s′ dT ∂T 1 ∂s′ dT ¼ Dehs de þ Deh cdSh Dehs ðs′ s′0 Þ þ ∂T

ð20Þ

where n is the porosity, b is the thermal expansion coefficient, d is the Kronecker’s delta vector = (1, 1, 1, 0, 0, 0)T and the subscripts s and h are soil and hydrate, respectively. [44] The definition of strain is given by

ð17Þ

[40] Because only hydrate elastic stiffness changes due to hydrate dissociation (i.e. soil elastic stiffness is independent of hydrate, dDes ¼ 0), the following stiffness relation can be obtained using equation (11):

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ds′ ¼ Dehs de þ Deh cdSh e þ

ð19Þ

where T is the temperature. [42] The first term of the right hand side of equation (19) is the conventional incremental stress-strain relationship, the second term is the stress relaxation term due to hydrate dissociation under zero straining and the third term is the stress change caused solely by the temperature change. Stress relaxation occurs because “stressed” hydrate disappears due to hydrate dissociation. The magnitude depends on how much additional stress is carried by the hydrate and

4.1. Synthetic Hydrate-Bearing Soil [47] Hydrate morphology (i.e. pore filling and cementing) plays an important role on the mechanical behavior of hydrate-bearing soil. Masui et al. [2005] created both pore filling and cementing types of synthetic hydrate-bearing soils with Toyoura sands and conducted drained triaxial compression tests. Figure 8 shows that stress-strain relationship and volumetric behavior of hydrate-bearing Toyoura sands by Masui et al. [2005] and the simulations made by the Methane Hydrate Critical State (MHCS) model. It is clear that the cementing case exhibits more enhancement in stiffness, strength and dilatancy than the pore filling case. In order to produce the differences, the hydrate dependent parameters such as additional hardening parameters for cohesion p′cc and for dilation p′cd and the elastic shear stiffness Gh are adjusted accordingly as shown in Table 2. The degradation parameter m is also dependent on hydrate morphology and the cementing case has a greater degrading effect of shearing on mechanical behavior. Conventional critical state parameters such as the slopes of critical state line, the normal compression line, the swelling line are M = 1.07, l = 0.16 and k = 0.004, respectively. These parameters are soil material properties and are independent of hydrate morphology. The measured porosity is

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Figure 8. Drained triaxial tests on hydrate-bearing Toyoura sands by Masui et al. [2005] (normal lines) and the MHCS model (bold lines). approximately 0.37, resulting in the specific volume v of 1.59. [48] Using the calibrated parameters for the synthetic hydrate-bearing Toyoura sand samples (cf. Table 2), the MHCS model gives the change in the mechanical properties with different hydrate saturations. Figure 9 shows the variation of peak strength with hydrate saturation for the cementing and pore filling cases presented by Masui et al. [2005]. The model predicts the rate of increase in the peak strength to be greater in the cementing case than that in the pore filling case, as demonstrated by the experimental data. As discussed before, the model is intended to be used for methane hydrate-bearing soil samples with hydrate saturation of up to 70%, where the experimental data exists. Further experimental data and model modification are needed for very high hydrate saturation conditions. 4.2. Natural Hydrate-Bearing Soil [49] Masui et al. [2006] performed drained triaxial compression tests on natural methane hydrate-bearing soil samples that were recovered from Eastern Nankai Trough, Japan. These experimental data are simulated by using the Methane Hydrate Critical State (MHCS) model and the Methane Hydrate (MH) Mohr-Coulomb model by Klar et al. [2010]. Table3 summarizes the soil properties calibrated for the Nankai methane hydrate-bearing soil. For the MH Mohr-Coulomb model, the elastic stiffness K′ and G, dilation angle y and cohesion c′ are dependent on hydrate saturation as suggested by Klar et al. [2010], Soga et al. [2006], and Waite et al. [2009]. For the MHCS model, the slope of the normal compression line and the slope of the swelling line are l = 0.15 and k = 0.01, respectively, for the Nankai methane hydrate-bearing soil. The experiments were conducted at the initial effective stress of 1 MPa (i.e. p′0 ¼ 1 MPa). The porosity is assumed to be 0.35, equivalent to the initial specific volume of 1.54. [50] Figure 10 shows the results of the model predictions as well as the experimental data. The dotted line, the dashed line and the normal line represent the experimental data, the

MH Mohr-Coulomb model and the MHCS model, respectively. It is clear that the natural hydrate-bearing soils exhibit greater stiffness, strength and dilatancy with increasing hydrate saturation. The MH Mohr-Coulomb model is an elastic-perfectly plastic model and thus the deviator stress develops linearly during the elastic state and then becomes constant after it fails in shear. This model therefore cannot capture nonlinear elasticity and softening behavior of the natural methane hydrate-bearing soil specimens. However, the volumetric response of the soil upon shearing is captured by introducing the hydrate dependent dilation angle. [51] The MHCS model gives the variation of peak strength using the calibrated model parameters (cf. Table 3), which is shown by the solid line in Figure 9. The model prediction fits well with the experimental data presented by Masui et al. [2006].

5. Soil Compaction Due to Depressurization and Hydrate Dissociation [52] In this section, a hydrate-bearing formation that is subjected to depressurization is modeled. The depressurization increases effective stress in the formation and hence the soil will be compacted. As shown in Figure 11, two extreme geometric cases are considered. The first case is at a location Table 2. Soil Properties for the Toyoura Specimens Properties

Pore Filling

Cementing

l k p′cs (MPa) M n u p′cd (MPa) p′cc (MPa) G (MPa) m

0.16 0.004 12.0 1.07 0.37 15 14Sh1:6 0.8Sh 0.75K′ + 250 1

0.16 0.004 12.0 1.07 0.37 15 42Sh1:6 0.1Sh 0.75K′ + 850 3

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Figure 9. Change in peak strength with different hydrate saturations. close to the wellbore, where the pore pressure reduction by depressurization causes an increase in mean effective pressure. It is assumed that a soil element at this location experiences large volumetric compression in an isotropic manner due to large change in pore pressure. The second case is at a location relatively far away from the wellbore, where the soil deformation is more likely to be one dimensional in the vertical direction (i.e. K0 condition). The actual deformation mechanism by depressurization is more complicated and a proper numerical simulation is required (S. Uchida et al., Geomechanical study of the mallik methane gas production field trials, submitted to BulletinGeological Survey of Canada, 2010). However, the solutions obtained from these two cases are indicative of general soil compaction behavior due to depressurization. [53] For the simulation, the in situ vertical and horizontal effective stresses are 3 MPa (i.e. K0 = 1.0) and the initial pore water pressure is 13 MPa. The model parameters are the same as those of Nankai methane hydrate-bearing soil, presented in section 4.2 (cf. Table 3). The temperature T0 is set to be 285.23 K and kept constant throughout the dissociation process. At this temperature, methane hydrate starts to dissociate at a pore pressure of 10MPa. The well is depressurized from 13 MPa to 4 MPa to simulate the methane gas production process and then pressurized back to 13 MPa to simulate the pressure recovery process. As the total “overburden” vertical stress is kept constant at 16 MPa, the soil element therefore undergoes effective stress loading from 3 MPa to 12 MPa and then unloading back to 3 MPa. Such loading and reloading process will produce permanent soil deformation due to volumetric yielding. The hydrate starts to dissociate when the effective stress is 6 MPa (i.e. total stress 16 MPa - pore pressure 10 MPa). 5.1. Isotropic Compression [54] Figure 12 shows the volumetric strain v against the mean effective stress p′ under isotropic compression and unloading of the Nankai hydrate-bearing soil with the MHCS model and the MH Mohr-Coulomb model. There are two cases considered: Case 1: initial hydrate saturation of

Sh = 50%; and Case 2: no hydrate Sh = 0%. Each case model undergoes both depressurization and pressure recovery stages. [55] Before hydrate dissociation takes place at the effective stress of 6 MPa, both MHCS and MH Mohr-Coulomb models show that the initial loading curve of Case 1 is stiffer than that of Case 2 because of additional resistance by the presence of hydrate. However, once the dissociation initiates at 6 MPa, the compressibility increases and the loading curve of Case 1 moves toward that of Case 2. In the MHCS model (Figure 12a), the loading curve of Case 2 follows the normal compression line l as the volumetric yield stress is p′cs ¼ 3:6 MPa, which is slightly larger than the initial state p′ = 3 MPa. Thus, the soil immediately experiences plastic deformation. In Case 1, the initial part of the loading curve is almost elastic (i.e. k) due to the strength enhancement by the two hardening parameters p′cc and p′cd . After dissociation, however, the contribution of these two hardening parameters disappears and the soil exhibits large plastic deformation. When unloading occurs by the recovery process, the soil swells and the deformation is elastic. Hence, a permanent residual volumetric strain develops at the end and the magnitude is approximately 12% in this case. [56] In the MH Mohr-Coulomb model (Figure 12b), the stress state is always elastic due to isotropic stress loading Table 3. Soil Properties for the Nankai Hydrate-Bearing Specimens Properties

MH MC

MHCS

n K′ (MPa) G (MPa) Dilation Coh. (MPa) Friction l p′cs m u

0.2 417Sh + 72 313Sh + 54 ∘ y ¼ 24Sh0:6 3 c′ ¼ 10Sh + 0.15 f′cri = 33.9∘

0.2 1.54p′/0.01 200Shmec + 0.75K′ p′cd ¼ 20Shmec (MPa) p′cc ¼ 0:1Shmec

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M ¼6sinf′cri 3sinf′cri

0.15 3.6 MPa 2 30

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Figure 10. Drained triaxial tests on Nankai hydrate-bearing soil specimens. conditions. Hence, the loading curve of both Case 1 and 2 are linear because of the use of linear elastic bulk modulus (i.e. Young’s modulus with constant value of Poisson’s ratio), which is a function of hydrate saturation (cf. Table 3). When unloading occurs by the recovery process, the soil swells and the deformation is again elastic. Hence, there is no development of permanent residual volumetric strain using this model. This highlights the deficiency of this model to reproduce soil compaction due to loading and unloading when the change in the effective stress is significant (as in the case of depressurization). 5.2. One-Dimensional Compression [57] Figure 13 shows the vertical strain against the effective vertical stress during one-dimensional compression and unloading with the MHCS model and the MH MohrCoulomb model. The simulated behavior is similar to that observed in the isotropic loading cases. During the depressurization stage, the hydrate-bearing soil layer consolidates, producing vertical settlement. The MHCS model predicts vertical strain of 11%, whereas the MH Mohr-Coulomb model predicts vertical strain of 6%. The greater vertical strain prediction of the MHCS model is due to its capability of modeling shear induced volume contraction in normally consolidated K0 conditions, which is a unique feature of soil behavior that cannot be captured by the MH Mohr-Coulomb model. The computed residual vertical strain by the MHCS model is approximately 10%, which is equivalent to 2 meters of vertical displacement for a 20 meter thick

methane hydrate-bearing soil layer with hydrate saturation of 50%. Again, there is no residual strain in the MH MohrCoulomb model as the compression behavior is modeled by elastic theory, as discussed previously. 5.3. Effect of Initial Hydrate Saturation, Hydrate Dissociation and Drawdown Pressure [58] Results from sections 5.1 and 5.2 highlight the importance of modeling plastic volumetric behavior when simulating ground deformation by depressurization. The plastic behavior is generated by a significant increase in vertical effective stress (3 to 12 MPa). The degree of hydrate dissociation also determines the amount of the plastic deformation. Therefore, the magnitudes of vertical strain during depressurization and after pressure recovery depend on initial hydrate saturation, the degree of hydrate dissociation as well as the magnitude of drawdown pressure. [59] Figure 14 shows the maximum vertical strain under one dimensional depressurization with different drawdown pressures (i.e. DPw = 4, 7 and 10 MPa) and different degrees of hydrate dissociation (i.e. 0, 25, 50, 75 and 100%) using the MHCS model. [60] Hydrate-bearing soils with greater hydrate saturation exhibit more elastic behavior. In the case of no dissociation (i.e. 0% lines in Figure 14), the plastic behavior is solely caused by the increase in the effective stresses. The greater depressurization causes the volumetric yielding at greater initial hydrate saturation, which are denoted as arrows in Figure 14 (e.g. Sh = 20% for DPw = 4 MPa, Sh = 45% for

Figure 11. Isotropic and one-dimensional compression on wellbore model. 10 of 13

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Figure 12. Isotropic compression simulation with (a) MHCS model and (b) MH Mohr-Coulomb model. DPw = 10 MPa). This implies that even strong hydratebearing soils with high hydrate saturation may cause large settlement under substantial depressurization. [61] Figure 14 also shows that vertical settlement is produced by hydrate dissociation. The more dissociation occurs, the more vertical strain develops as the additional effective stress carried by the hydrate is transferred to the soilskeleton. This can be seen as each curve is shifted downward with increasing dissociation. When full dissociation is achieved, the magnitude of vertical strain converges with that of pure soil (Sh = 0), denoted as the dotted line in Figure 14. [62] Although the results are preliminary, Figure 14 may be used to predict a settlement caused by depressurization

and pressure recovery in the Eastern Nankai Trough, provided the magnitude of pressure drawdown, the initial hydrate saturation and the final hydrate saturation are known. Suppose 20 meters of hydrate bearing sediments that contain alternating unit layers (Layer 1: Sh = 40% and Layer 2: Sh = 60%) are depressurized by 7 MPa and eventually the hydrate saturation in each layer becomes 10% (75% dissociation) in Layer 1 and 30% (50% dissociation) in Layer 2, respectively, then the vertical settlement can be estimated by using the chart shown in Figure 14. From Figure 14, the values of the maximum vertical strain of the layers can be estimated as z = 7% in Layer 1 and z = 2% in Layer 2. As a result, the expected vertical settlement is 1.8 meters for a

Figure 13. 1D compression with (a) MHCS model and (b) MH Mohr-Coulomb model. 11 of 13


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Figure 14. Maximum vertical strain induced under 1D compression with different drawdown pressures and different degrees of dissociation. hydrate-bearing sediment with a summed thickness of 20 meters.

6. Summary [63] In this paper, a new constitutive model for hydratebearing soil, Methane Hydrate Critical State (MHCS) model, was presented. The model incorporated (1) volumetric yielding; (2) enhanced cohesion, dilation and stiffness by the presence of hydrate in pores; (3) strain softening due to shearing deformation; (4) smooth nonlinear stress-strain relationship; and (5) stress relaxation due to hydrate dissociation. [64] In addition, a simple model of wellbore in hydratebearing sediments that was subjected to depressurization was simulated. Several key issues were noted: [65] 1. Under isotropic loading condition, the MHCS model and the MH Mohr-Coulomb model exhibited similar maximum volumetric strain when the models were calibrated with triaxial test data. [66] 2. After pressure recovery, residual strain remained in the MHCS model, whereas the MH Mohr-Coulomb model exhibited zero residual strain due to full elastic recovery. The permanent strain may be important to be considered when the soils undergo significant change in the effective stresses. [67] 3. Under one dimensional compression, greater vertical strain was generated in the MHCS model compared to that in the MH Mohr-Coulomb model. This is because of its capability of modeling shear induced volumetric yielding. Thus, modeling plastic volumetric deformation is important for predicting the amount of soil compaction due to depressurization. [68] 4. Greater depressurization induced more plastic deformation in the Eastern Nankai Trough scenario. [69] 5. Greater amount of hydrate dissociation caused more plastic deformation. [70] 6. Vertical strain can be estimated using the proposed chart (Figure 14), provided the magnitude of depressurization, the initial hydrate saturation and the final hydrate saturation are known.

[71] As described in this paper, the proposed model is a simple extension of critical state model. Although the model provides an accurate stress-strain relationship for methane hydrate-bearing soils, it may be necessary to test its capability with more experimental data and field data. The proposed wellbore model is considered at the element level and thus a numerical model with more realistic geometry and hydrate saturation profile needs to be analyzed for better prediction of the geomechanical behavior of hydrate-bearing sediments during depressurization (e.g., S. Uchida et al., submitted manuscript, 2010). [72] Acknowledgments. This research was funded by the Research Consortium for Methane Hydrate Resources in Japan (MH21).

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