# Relationship between Creep Property and Loading-Rate Dependence of Strength of Artificial Methane-Hydrate-Bearing Toyoura Sand under Triaxial Compression

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Testing Method

#### 2.1. Specimen Preparation

^{3}on average, corresponding to a relative density of 96%, indicating that the sand particles in the host specimens were well compacted. Each host specimen was 50 mm in diameter and 100 mm in length.

^{−4}m

^{3}of water passed through the specimen during the water substitution process. We hereafter refer to a water-saturated specimen of the densely packed sand sediment containing synthesized methane hydrate prepared by the above procedure as a hydrate-sand specimen.

#### 2.2. Axial Loading

_{a}was calculated by dividing the axial displacement, measured with two 25 mm linear variable differential transformers (LVDTs), by the initial height of the specimen. In this paper, a positive strain value denotes compression. In the constant-strain-rate tests, the axial strain ε

_{a}was increased at a rate C

_{e}ranging from 0.001 to 0.1%/min. In the constant-stress-rate tests, the differential stress Δσ (= σ

_{a}− σ

_{r}) was increased at a rate C

_{s}ranging from 0.01 to 1 MPa/min, where σ

_{a}and σ

_{r}are the axial and radial stresses, respectively. In the creep tests, the differential stress Δσ was increased to a predetermined creep stress σ

_{cr}. In this paper, constant-strain-rate and constant-stress-rate tests are inclusively referred to as constant-loading-rate tests.

_{h}of the specimen, was calculated from the volume of released methane measured by a gas flow meter. The S

_{h}for each specimen used in the constant-strain-rate tests, constant-stress-rate tests, and creep tests are also shown in Table 1, Table 2 and Table 3, respectively.

## 3. Review of Test Results

#### 3.1. Constant-Loading-Rate Tests

_{a}as shown in Figure 2. Masui et al. [12] noted that the strength (maximum differential stress) σ

_{fe}increases with the methane hydrate saturation S

_{h}due to the cementation effect of hydrate between sand particles. Miyazaki et al. [27] obtained the following approximate formula by least-squares regression:

_{fe}

_{0.1}= σ

_{fe}

_{0}+ 42.2 × S

_{h}

^{3.24},

_{fe}

_{0.1}is a function of S

_{h}expressing the strength at an axial-strain rate C

_{e}of 0.1%/min, and σ

_{fe}

_{0}is 3.75 MPa, which is the average strength of the sand specimens. Note that some values of S

_{h}in Table 1 are in parentheses; Miyazaki et al. [27] used these results to derive Equation (1).

_{a}curve for the hydrate-sand specimens depends on C

_{e}[18,21,23,24]. Figure 3 clearly shows that, as C

_{e}increases, the strength σ

_{fe}increases and the axial strain at the peak strength ε

_{fe}decreases. As noted by Miyazaki et al. [18], the time-dependence of a hydrate-sand specimen is stronger than that of a non-hydrate-sand specimen, because the axial-strain-rate dependence of the mechanical properties of the non-hydrate-sand specimens was hardly observed. Miyazaki et al. [18] also reported that the time-dependence of a hydrate-sand specimen is as strong as that of frozen sand and is stronger than those of most geomaterials such as rocks and soils. Parameswaran [31] presumed that the strain-rate dependence of frozen sand is governed by the liquid phase around the sand grains—namely, a quasi-liquid layer—and that this phase is associated with the melting of ice under pressure at the points where grains are in contact. Miyazaki et al. [18] suggested that the strong time-dependence of hydrate-sand specimens was caused by the local dissociation of hydrate occurring at the points where hydrate grains are in contact, similar to the pressure melting of ice.

_{a}curve for the hydrate-sand specimens depends on the differential-stress rate C

_{s}. Figure 5 shows that, as C

_{s}increases, the strength (differential stress at the final rupture) σ

_{fs}increases and the axial strain at the final rupture ε

_{fs}decreases [22].

#### 3.2. Creep Tests

_{cr}. We designate the point when Δσ reaches σ

_{cr}as the starting point of creep. We hereafter refer to an increase in ε

_{a}from the starting point of creep and the temporal derivative of ε

_{a}as the creep strain ε

_{cr}and creep-strain rate (dε

_{cr}/dt), respectively. The relationship between creep strain ε

_{cr}and elapsed time t for the hydrate-sand specimen with S

_{h}= 41% and σ

_{cr}= 4.5 MPa is shown in Figure 6 [22,24]. The creep strain ε

_{cr}monotonically increased with the elapsed time t, while (dε

_{cr}/dt), i.e., the slope of the ε

_{cr}-t curve, changed with t as follows: first, (dε

_{cr}/dt) decreased; second, it remained almost constant; and third, it increased. These three phases of creep are often referred to as primary creep, secondary creep, and tertiary creep, respectively. During secondary creep, (dε

_{cr}/dt) did not remain exactly constant; it reached a minimum at a point during secondary creep, as indicated with a triangle in Figure 6. At the end of tertiary creep, as indicated with a square in Figure 6, the specimen could not withstand the creep stress σ

_{cr}and finally ruptured. Note that not all the hydrate-sand specimens exhibited final rupture before 200,000 s had elapsed, as shown in Table 3.

_{cr}*, which is the ratio of creep stress to the strength σ

_{fe}

_{0.1}, is often useful in comparing creep properties between geomaterials with different strengths. In this paper, since each specimen has a different value of S

_{h}, we decided to calculate σ

_{cr}* using the following equation:

_{cr}* = σ

_{cr}/σ

_{fe}

_{0.1}.

_{e}= 0.1%/min, σ

_{fe}

_{0.1}depends on S

_{h}. For example, as represented in Figure 6, σ

_{cr}of 4.5 MPa for the hydrate-sand specimen with S

_{h}= 41% corresponds to σ

_{cr}* = 75%. Miyazaki et al. [22,24] noted that ε

_{cr}of the hydrate-sand specimen is larger than that of the non-hydrate-sand specimen at nearly equal σ

_{cr}*, suggesting that the time-dependence of a hydrate-sand specimen is stronger than that of a non-hydrate-sand specimen as described above.

_{cr}/dt) plotted against t for the test result shown in Figure 6. The (dε

_{cr}/dt) decreased with t during primary creep, reached a minimum during secondary creep, and then increased with t during tertiary creep. As shown in earlier works [22,24,26,27], the slope of the log(dε

_{cr}/dt)-log(t) relationship in primary creep varied with σ

_{cr}in the range of −1 to −0.4; the slope approached −1 as σ

_{cr}decreased. Hereafter, m is the slope of the log(dε

_{cr}/dt)-log(t) relationship in primary creep.

_{cr}decreased for both frozen Ottawa sand and frozen Manchester fine sand under triaxial compression. In this sense, the primary creep behavior of hydrate-sand specimens closely resembles that of frozen sand. However, Sales & Haines [35] reported that m for Hanover silt and Suffield clay ranged from −1.1 to −0.9 and was nearly independent of σ

_{cr}. It was reported that m is nearly equal to −1 for unfrozen sand [36], in agreement with our result for non-hydrate-sand specimens [22,24]. For reference, it is also reported that m is nearly equal to −1 for many types of rocks [37,38].

_{mcr}. The elapsed time until final rupture is hereafter referred to as the creep life t

_{fcr}. Figure 8 shows that the minimum creep-strain rate (dε

_{cr}/dt)

_{min}decreases with increasing t

_{mcr}and t

_{fcr}, with a linear relationship between log((dε

_{cr}/dt)

_{min}) and both log(t

_{mcr}) and log(t

_{fcr}), and a slope of approximately −0.9. The ratio of t

_{mcr}to t

_{fcr}for the specimens that exhibited final rupture was approximately 0.5. The slope of the log((dε

_{cr}/dt)

_{min})-log(t

_{mcr}) relationship of a hydrate-sand specimen is almost equal to those of frozen Manchester fine sand (−0.8 to −1.2) [33] and Fairbanks silt (−1.1) [39]. As shown in Figure 9, (dε

_{cr}/dt)

_{min}increased with σ

_{cr}*. The slope of the log((dε

_{cr}/dt)

_{min})-log(σ

_{cr}*) relationship of a hydrate-sand specimen was 9.6, which is nearly equal to that of frozen Manchester fine sand (9.0) [40]. It appears that hydrate sand and frozen sand have many common time-dependent properties.

## 4. Discussion

#### 4.1. Differential Stresses Versus Axial Strains in Three Tests

_{a}relationships are shown: σ

_{fe}-ε

_{fe}obtained from the constant-strain-rate tests (unfilled triangles), σ

_{fs}-ε

_{fs}obtained from the constant-stress-rate tests (unfilled squares), and σ

_{cr}-ε

_{mcr}(filled triangles) and σ

_{cr}-ε

_{fcr}(filled squares) obtained from the creep tests, where ε

_{mcr}and ε

_{fcr}are the axial strains ε

_{a}corresponding to the minimum creep-strain rate and final rupture, respectively. Although a large variation can be seen in each Δσ-ε

_{a}relationship, Δσ appears to be negatively correlated with ε

_{a}. For some rocks, it has been reported that both ε

_{fe}and σ

_{fe}increase with C

_{e}and that ε

_{mcr}increases with σ

_{cr}[38,41]. Thus, in this sense, the hydrate-sand specimens exhibited the opposite trend to these rocks. It appears that the σ

_{fe}versus ε

_{fe}plots are close to the σ

_{cr}versus ε

_{mcr}plots, and that the σ

_{fs}versus ε

_{fs}plots are an extension of the σ

_{cr}versus ε

_{fcr}plots. These quantitative agreements suggest that the creep property of the hydrate-sand specimens is closely related to the loading-rate dependence of strength, which is a typical time-dependent behavior.

#### 4.2. Loading-Rate Dependence of Strength

_{fe}* and σ

_{fs}* and the normalized differential-stress rate C

_{s}* are calculated as follows for each hydrate-sand specimen with a different value of S

_{h}:

_{fe}* = σ

_{fe}/σ

_{fe}

_{0.1},

_{fs}* = σ

_{fs}/σ

_{fe}

_{0.1},

_{s}* = σ

_{fe}*/(σ

_{fe}/C

_{e}), for constant-strain-rate tests,

_{s}* = C

_{s}/σ

_{fe}

_{0.1}, for constant-stress-rate tests.

_{s}* for the constant-strain-rate tests is the normalized strength σ

_{fe}*, given by Equation (3), divided by the time to the peak strength (σ

_{fe}/C

_{e}), which corresponds to the average normalized differential-stress rate (differential-stress rate divided by σ

_{fe}

_{0.1}) until the peak strength is reached at the axial-strain rate C

_{e}. Realignment using the normalized differential stresses given by Equations (3)–(6) made it possible to quantitatively compare the loading-rate dependencies obtained in the constant-strain-rate tests and constant-stress-rate tests. As shown in Figure 11, plots of σ

_{fe}* and σ

_{fs}* versus C

_{s}* are close to each other. The approximate curves in Figure 11 were calculated using the following expressions obtained by least-squares regression:

_{fe}* = 1.77 × (C

_{s}*)

^{0.0732},

_{fs}* = 1.54 × (C

_{s}*)

^{0.0562}.

#### 4.3. Creep Life

_{fcr}and normalized creep stress σ

_{cr}* in Figure 12 [21,22] as is often the case with a geomaterial, t

_{fcr}tends to decrease with increasing σ

_{cr}*.

_{fe}

_{0.1}and n is a parameter expressing the time-dependence of a hydrate-sand specimen, and that the final rupture of a specimen occurs when a state quantity D reaches a value D

_{f}, where D is the time integral of (σ*)

^{n}given by

_{s}*, by substituting

_{s}* × t.

_{fe}* and σ

_{fs}* expressed by Equations (7) and (8) as σ

_{f}* in Equation (13), we obtain n and D

_{f}. Using Equation (7) derived from the results of the constant-strain-rate tests, we obtain

_{f}= 180.

_{f}= 121.

_{cr}* in the creep tests, by substituting

_{cr}*

_{cr}*)

^{n}× t.

_{cr}* and creep life t

_{fcr}:

_{f}= (σ

_{cr}*)

^{n}× t

_{fcr},

_{fcr}= D

_{f}× (σ

_{cr}*)

^{−n}.

_{fcr}-σ

_{cr}* on the whole, suggesting that the results of the constant-strain-rate tests, constant-stress-rate tests, and creep tests are in reasonable agreement with the simple hypothesis described above. A lot of tests will be required to verify the hypothesis exactly.

#### 4.4. Review of Constitutive Models Proposed for Methane-Hydrate-Bearing Sediments

_{e}= 0.1%/min [12]. Uchida et al. [29] developed a constitutive model based on the concept of critical-state soil mechanics, and showed that predictions by their model fitted well with the stress–strain relationships and volumetric behaviors observed in constant-strain-rate tests at C

_{e}= 0.1%/min for both artificial and natural hydrate-bearing soil [12]. The above-mentioned models are thought to be unable to express the loading-rate dependencies or the creep behaviors without modification, because they do not currently consider the strong time-dependence of hydrate-bearing soil. When these models are improved in the future by incorporating the time-dependent terms (viscous terms) of hydrate-bearing soil into their mathematical equations, the results presented in this and earlier studies [18,21,22,23,24,26,27] are expected to be used.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 5.**Strength σ

_{fs}and axial strain at the final rupture ε

_{fs}versus differential-stress rate C

_{s}in constant-stress-rate tests [22].

**Figure 8.**Minimum creep-strain rate (dε

_{cr}/dt)

_{min}versus elapsed time until creep-strain rate reached minimum t

_{mcr}(triangles) and creep life t

_{fcr}(circles).

**Figure 10.**Differential stress Δσ versus axial strain ε

_{a}at the peak strength in constant-strain-rate tests (unfilled triangles), at the final rupture in constant-stress-rate tests (unfilled squares), at the minimum creep-strain rate in creep tests (filled triangles), and at the final rupture in creep tests (filled squares).

**Figure 11.**Normalized strengths σ

_{fe}* and σ

_{fs}* versus normalized differential-stress rate C

_{s}* in constant-strain-rate and constant-stress-rate tests.

Axial-Strain Rate C_{e} | Methane Hydrate Saturation S_{h} |
---|---|

0.1%/min | 39%, 40%, 41%, 41% (15%, 16%, 21%, 31%, 34%, 35%, 48%) * |

0.05%/min | 37%, 37%, 45% |

0.01%/min | 43%, 43% |

0.005%/min | 43% |

0.001%/min | 42%, 41% |

_{h}values in parentheses were used to derive Equation (1).

Differential-Stress Rate C_{s} | Methane Hydrate Saturation S_{h} |
---|---|

1 MPa/min | 39%, 42%, 42% |

0.1 MPa/min | 44%, 50% |

0.01 MPa/min | 44%, 48%, 50% |

Creep Stress σ_{cr} | Methane Hydrate Saturation S_{h} |
---|---|

1 MPa | 42% *, 42% * |

2 MPa | 48% *, 48% * |

3 MPa | 40% *, 45% *, 47% * |

4 MPa | 39% *, 50% * |

4.5 MPa | 41%, 42%, 42% * |

5 MPa | 36%, 41%, 45%, 48% |

5.5 MPa | 36%, 39%, 48% |

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**MDPI and ACS Style**

Miyazaki, K.; Tenma, N.; Yamaguchi, T. Relationship between Creep Property and Loading-Rate Dependence of Strength of Artificial Methane-Hydrate-Bearing Toyoura Sand under Triaxial Compression. *Energies* **2017**, *10*, 1466.
https://doi.org/10.3390/en10101466

**AMA Style**

Miyazaki K, Tenma N, Yamaguchi T. Relationship between Creep Property and Loading-Rate Dependence of Strength of Artificial Methane-Hydrate-Bearing Toyoura Sand under Triaxial Compression. *Energies*. 2017; 10(10):1466.
https://doi.org/10.3390/en10101466

**Chicago/Turabian Style**

Miyazaki, Kuniyuki, Norio Tenma, and Tsutomu Yamaguchi. 2017. "Relationship between Creep Property and Loading-Rate Dependence of Strength of Artificial Methane-Hydrate-Bearing Toyoura Sand under Triaxial Compression" *Energies* 10, no. 10: 1466.
https://doi.org/10.3390/en10101466