^{1}

^{1}

^{2}

^{1}

^{1}

^{1}

^{2}

Permeability of soil plays an important role in geotechnical engineering and is commonly determined by methods combining measurements with theory. Using the double-scale asymptotic expansion method, the Navier-Stokes equation is numerically solved to calculate the permeability, based on the homogenization method and the assumption that the homogeneous microstructure of the relevant porous media is represented accurately as the Representative Elemental Volume (REV). In this study, the commonly used square model is tested in the calculation of sea clay permeability. The results show large deviations. It is suspected that the square model could not represent the flattened shape of the clay particles and the bound water film wrapping around them. Hence, the Rectangle Particle-Water Film Model (i.e., the R-W model) is proposed. After determining the horizontal and vertical characteristic length of the unit cell using two pairs of initial data, the permeabilities of other different void ratios could be inversely calculated. The results of three types of clay obtained using the R-W model agree well with the experimental data. This shows the efficient feasibility and accuracy of the R-W model by providing a good representation of the clay particles when using the double-scale asymptotic expansion method to calculate clay permeability.

Permeability of soil is of fundamental importance in geotechnical engineering. Experimental methods, such as the classical Darcy law, are commonly used to obtain the permeability of soil. These methods are convenient to be conducted but always require much time and effort. Moreover, the experimental methods focus only on the external phenomenon from the macroscopic view and ignore the seepage processes inside the porous materials at the microscopic scale. Actually, as a type of porous material, soils can be distinguished at three different scales: the microscopic scale, the Darcy scale, and the macroscopic scale [

The homogenization method [

In geotechnical engineering, Wang et al. [^{2} at the largest void ratio ^{2}; also, the calculated value was ^{2} when the smallest void ratio was determined to be ^{2}.

In this paper, the existing square model, which was used in the inversed calculation by the double-scale asymptotic expansion method, was tested to find the defects of the model. Additionally, a new model would be proposed based on the real features of clay particles aiming to improve the calculated accuracy of clay permeability.

Previous studies have shown that, for a given type of statistically homogeneous soil, an equivalent REV always exists to represent the structure of the soil. A similar description could be used for nonhomogeneous porous materials [

Sketch of porous media and unit cell.

The Stokes flow through the porous media is governed by the Navier-Stokes equation, the continuity equation, and the no-slip boundary condition:

Introduce two-scale coordinates

Averaging over

Equations (

For a given unit cell, as shown in Figure

The specific procedure is as follows: according to the degrees of freedom of the REV, take one or more pairs of test data (void ratio

For the calculation of clay permeability, the REVs are commonly set as regular polygon particles. Sun et al. [

Firstly, the REV with square particle, which Sun et al. had used in their study, was tested again to help us to find the defects. As Figure

Sketch of unit cell of the square model.

For an equitable comparison, the Champlain sea clay of Louiseville [^{2}), which is the largest void ratio and the corresponding permeability tested in the reference, was used as the initial data point; these allowed the characteristic length of the square model to be obtained:

Similarly, the measured value that has the smallest void ratio and the corresponding permeability (^{2}) was used to determine the characteristic length of the square model again:

Calculated and measured permeabilities of sea clay using the square model.

| ^{2} | ^{2} | |
---|---|---|---|

| | ||

1.91 | 12.60 | 12.60 | 4.72 |

1.82 | 10.06 | 11.43 | 4.25 |

1.73 | 7.93 | 10.22 | 3.80 |

1.64 | 6.48 | 9.07 | 3.37 |

1.55 | 5.29 | 7.99 | 2.97 |

1.49 | 4.59 | 7.30 | 2.71 |

1.37 | 3.37 | 6.02 | 2.24 |

1.31 | 2.85 | 5.43 | 2.02 |

1.25 | 2.39 | 4.86 | 1.81 |

1.19 | 1.93 | 4.33 | 1.61 |

1.13 | 1.63 | 3.83 | 1.42 |

1.07 | 1.37 | 3.36 | 1.25 |

1.01 | 1.08 | 2.93 | 1.08 |

Using the measured void ratio

Comparison of the calculated and measured permeabilities of sea clay using the square model.

From the documents published by ASTM [

Theoretically, if the accuracy of the REV is sufficient enough, the results of the characteristic length will not vary observably, irrespective of whichever the pair of data with the largest or smallest void ratio is chosen as the initial value. Comparing (^{2} was significantly larger than ^{2}. Additionally, the values of ^{2}; these deviations increased along with the differences between the used void ratio and the initial void ratio. Similarly, the values of ^{2}; the deviation also increases with the magnitude of the void ratio difference. Fitting the calculated results into straight lines, the slopes of the lines were

The inaccuracies described above should be attributed to the lack of representativeness of the square model. Generally speaking, clay is composed of particles with flattened shape [

As Figure

SEM results of clay particles.

Additionally, a thin bound water film exists wrapping around the soil particles for both sand and clay at the microscopic scale. The water film acts as a solid due to its low energy level and its inability to move freely and dissolve solutes. During permeation, the water film can be regarded as part of the soil particles. However, during the measurement of the void ratio, the water film would be evaporated and could not be considered in the void ratio. Thus, for the same void ratio, the existence of the water film reduces the passage of permeation and decreases the permeability accordingly.

Synthesizing the flattened shape of the clay particles and the bound water film, the Rectangle Particle-Water Film Model (R-W model) was proposed, as shown in Figure

Sketch of the unit cell of the R-W model.

Figure

The vertical and horizontal lengths of the unit cell

Similarly, the vertical and horizontal thicknesses of the water lamina

From the hygroscopicity index [^{3} of hygroscopic water is a typical average for every 1 cm^{3} soil particle for sea clay. This means that, in the progress of permeation, the solid part

Calculated and measured permeabilities of sea clay using the R-W model.

| ^{2} | ^{2} |
---|---|---|

1.91 | 12.60 | 12.68 |

1.82 | 10.06 | 10.73 |

1.73 | 7.93 | 8.99 |

1.64 | 6.48 | 7.44 |

1.55 | 5.29 | 6.09 |

1.49 | 4.59 | 5.18 |

1.37 | 3.37 | 3.90 |

1.31 | 2.85 | 3.22 |

1.25 | 2.39 | 2.63 |

1.19 | 1.93 | 2.11 |

1.13 | 1.63 | 1.67 |

1.07 | 1.37 | 1.30 |

1.01 | 1.08 | 0.98 |

Similarly, using

Comparison of the calculated and measured permeabilities of sea clay using the rec. model and the R-W model.

From Figure

To verify the applicability of the R-W model, the permeabilities of kaolin [^{2}.

Calculated and measured permeabilities of kaolin clay using the R-W model.

| ^{2} | ^{2} |
---|---|---|

2.05 | 4.97 | 4.86 |

1.93 | 4.18 | 4.14 |

1.84 | 3.30 | 3.64 |

1.73 | 2.75 | 3.08 |

1.57 | 2.09 | 2.34 |

1.53 | 1.85 | 2.18 |

1.41 | 1.77 | 1.71 |

1.38 | 1.48 | 1.60 |

1.31 | 1.35 | 1.37 |

1.26 | 1.23 | 1.21 |

1.21 | 0.80 | 1.07 |

1.16 | 0.94 | 0.93 |

1.11 | 0.77 | 0.81 |

1.00 | 0.55 | 0.57 |

0.97 | 0.50 | 0.51 |

Calculated and measured permeabilities of illite clay using the R-W model.

| ^{2} | ^{2} |
---|---|---|

3.72 | 11.53 | 11.42 |

3.28 | 7.25 | 8.65 |

2.87 | 4.56 | 6.36 |

2.60 | 3.42 | 5.02 |

2.43 | 11.28 | — |

2.28 | 3.57 | 3.61 |

2.17 | 2.10 | 3.18 |

1.85 | 1.48 | 2.05 |

1.45 | 0.71 | 0.97 |

1.36 | 0.40 | 0.77 |

1.25 | 0.38 | 0.56 |

1.15 | 0.36 | 0.39 |

1.12 | 0.27 | 0.35 |

0.98 | 0.22 | 0.16 |

0.91 | 0.12 | 0.08 |

Using

Comparison of the calculated and measured permeabilities using R-W model.

Kaolin clay

Illite clay

From Figure

It should be pointed out that the measured permeability of the illite clay was indirectly derived by the consolidation curve of Terzaghi’s Consolidation Theory. This result is shown to have poor stability in the literature [

The double-scale asymptotic expansion method used in this study can numerically solve the Navier-Stokes equation and determine the soil permeability based on the assumption that the porous media are homogeneous, and the microstructure of the media is known to be an incompressible Newtonian fluid with a small Reynolds number. Conversely, using several measured values of the void ratio and the corresponding permeabilities of the media, the characteristic length of a reasonable REV can be calculated. Then, the permeability of the other void ratio can be calculated by changing the distance between the unit cells of the REV.

For the calculation of sea clay permeability, the existing square model produced significant deviations. Using the measured value with the largest void ratio as the initial data would lead to a larger characteristic length of the REV and larger calculated results, while using the smallest void ratio as the initial data would lead to smaller results. The straight lines which were fitted by the calculated results had a large slope, which means the deviations would increase more significantly along with the differences between the used void ratio and the initial void ratio. It was speculated that the square model could not represent the flattened shape of the clay particle and ignored the bound water film wrapping around the clay particles.

Thus, the R-W model was proposed, which could represent the flattened shape of the clay particles and the water film around them. Using the measured values with both the largest and the smallest void ratio along with the corresponding permeability and the known hygroscopicity index, the horizontal and vertical characteristic lengths of R-W model can be confirmed. Then, the permeabilities of the other void ratios can be inversely calculated by changing the distance between the unit cells based on the same ratio of the horizontal and vertical characteristic length. The example, where the same sea clay was used, shows higher accuracy and a smoother slope of the fitted straight line using the R-W model. Additionally, the permeabilities of two other types of clay, which were kaolin clay and illite clay, were also calculated. The calculations using R-W model also show satisfactory results, which prove the feasibility and accuracy of the R-W model for the calculation of clay permeabilities using the double-scale asymptotic expansion method.

The authors declare that they have no competing interests.

The authors would like to express sincere gratitude to the National Natural Science Foundation of China (Grant no. 51179168) and the Cultural Relics Protection Project of Zhejiang (Grant no. 2013010) for their financial support of this study.