# Computing of Permeability Tensor and Seepage Flow Model of Intact Malan Loess by X-ray Computed Tomography

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}. The indoor seepage experiment was consistent with the simulation experiment, which verifies the reliability of the calculated model.

## 1. Introduction

_{3}) of the Fourth Series in arid and semi-arid regions; it is one of the most common materials in the Loess Plateau, where Malan loess plays an important role in various infrastructures such as housing, arable land, roads, and railroads [1,2,3,4,5]. During its long formation process, Malan loess has formed different types of pores that affect its hydrogeological geotechnical properties. Because loess pores are characterized by high porosity and vulnerability, it is important for the identification and quantitative characterization of loess pores. Geological hazards are mostly developed in the Malan loess layer [6,7,8]. In order to ensure engineering construction and people’s life and property safety, Malan loess has become the focus of relevant research by researchers in the field of hydrogeology and engineering geology [3,9].

## 2. Study Area and Data Preparation

#### 2.1. Collection and Preparation of Malan Loess Samples

_{1}Malan loess. Malan loess is light yellow in color, with developed vertical pore space and visible soil agglomerates. Figure 1 shows the location of the intact Malan loess sampling site and the intact Malan loess layer in the field.

^{3}. When making samples, the excess soil samples were cut out and divided into three parts and some basic physical properties of the loess samples were measured by a Malven Mastersizer 2000 laser analyzer (photoelectric liquid limit and plastic limit combined tester Inmalvern Instruments Co., Ltd., Worcestershire, UK); then, the average value was found. Around the sampling point, three samples were taken by a ring knife with a diameter of 61.8 mm and a height of 40 mm; the permeability coefficient of the soil samples was measured by TST-55 variable-head permeameters (Nanjing Ningxi Soil Instrument Co., Ltd., Nanjing, China) and the average value was obtained. The results are shown in Table 1.

#### 2.2. CT Scan and CT Image Calibration

## 3. Methods

#### 3.1. CT Image Processing

#### 3.1.1. Image Filtering Analysis (Bilateral Filtering)

#### 3.1.2. Image Segmentation

#### 3.2. Intact Loess Seepage Experiment

_{1}. Empty the water from beaker 8 and beaker 13.

_{4}. The volume of Malan loess specimen 5 is recorded as V

_{0}. The sample specific yield is calculated as shown in Equation (1) and the sample specific yield is calculated as shown in Table 2:

#### 3.3. Theoretical Approach

#### 3.3.1. Absolute Penetration Rate

^{2}); 1d = 0.9869233 μm

^{2}. Absolute permeability indicates the intrinsic property of porous media material, which is only related to the pore structure of porous media material and has nothing to do with other external conditions [38,39,40,41].

^{3}·s

^{−1}) through the porous medium of the intact Malan loess; S is the cross-sectional area (unit: m

^{2}) of the fluid flowing through the intact Malan loess sample; K is the permeability coefficient (unit: m

^{2}·s

^{−1}) [42,43,44].

^{2}); μ is the viscosity of the fluid (unit: Pa·s); ΔP is the pressure difference (unit: Pa) applied around the intact Malan loess sample; L is the length of the sample in the direction of flow (unit: m); Q/S is called the flow velocity v, which indicates the average velocity or Darcy velocity of the fluid through the surface of the intact Malan loess sample.

#### 3.3.2. Solving Stokes Equation Based on Volume Average

#### 3.3.3. Boundary Conditions

#### 3.3.4. Equation Solving

## 4. Results and Analysis

#### 4.1. Analysis of Seepage Experiment Results

#### 4.2. 3D Structure Reconstruction

^{3}, the volume of the smallest pore was 2.92 × 10

^{−6}cm

^{3}, and the average volume of the pores was 0.001777 cm

^{3}.

#### 4.3. Seepage Simulation and Calculation of Absolute Permeability Coefficient Tensor

#### 4.3.1. Experimental Simulation of Absolute Permeability

^{2}.

^{2}. The results of the seepage simulation are shown in Figure 14. The water flow entered from the seepage boundary set at the bottom and first passed through the connected pores in the loess sample in the vertical direction. Seepage in the horizontal direction occurred during the seepage process as the volume of water entering the sample increased. The seepage simulation process was highly consistent with the actual water infiltration process.

#### 4.3.2. Absolute Permeability Tensor Calculation

^{2}. The absolute permeability tensor calculation module in AVIZO software was run, as shown in Figure 15. After 65,540 iterations to finally reach the iteration accuracy, the vertical absolute permeability coefficient tensor of the loess in the simulated area was finally calculated, as shown in Table 3. The absolute permeability coefficient of the Malan loess sample was 0.2848 μm

^{2}in the X-direction, 0.03156 μm

^{2}in the Y-direction, and 0.07378 μm

^{2}in the Z-direction. X-direction (vertical) absolute permeability coefficient was 9.02 times higher than the Y-direction (horizontal) absolute permeability coefficient and 3.86 times higher than the Z-direction (horizontal) absolute permeability coefficient. The absolute permeability coefficients along the Z-direction were on the same order of magnitude as those along the Y-direction.

^{2}), ${\rho}_{w}$ is the density of water (unit: g/cm

^{3}), and $\mu $ is the dynamic viscosity of the liquid (unit Pa·s).

^{−4}cm/s, Y-direction (horizontal) permeability coefficient was 3.093 × 10

^{−5}cm/s, and Z-direction (horizontal) permeability coefficient was 7.23 × 10

^{−5}cm/s. The absolute permeability of Malan loess was calculated from 0.13 to 7.81 μm

^{2}by the pore network model method by Xin Li et al. The permeability coefficients of Malan loess ranged from 1.28 × 10

^{−4}cm/s to 7.65 × 10

^{−3}cm/s. In this study, the absolute permeability coefficients and permeability coefficients of Malan loess samples were calculated within this interval using this method of volume averaging, which indicates that it is feasible to calculate the absolute permeability of Malan loess using the volume averaging method.

## 5. Discussion

^{−4}cm/s to 1.67 × 10

^{−4}cm/s [8,31,47]. The permeability coefficient in the vertical direction calculated by using the CT scan image of intact Malan loess combined with the volume averaging method was 2.791 × 10

^{−4}cm/s and the horizontal permeability coefficients were 3.093 × 10

^{−5}cm/s and 7.23 × 10

^{−5}cm/s, respectively, while the permeability coefficient of the soil samples around the samples measured with the ring knife sampling was 1.42 × 10

^{−4}cm/s, which was slightly smaller than that of the permeability coefficients calculated by the simulation, because the ring knife may smash into the loess during the sampling process, thus compacting the loess and resulting in the permeability coefficient to become smaller. Calculating the permeability coefficient of Malan loess by sampling using the method mentioned in this paper can effectively avoid the measurement error due to the disturbance of the loess during the sampling process.

## 6. Conclusions

^{2}and the absolute permeability coefficient in X-direction (vertical) of Malan loess sample is 9.02 times and 3.86 times of the absolute permeability coefficient in Y- and Z-directions (horizontal). This also indicates that the Malan loess has developed pore space in the vertical direction and has better connectivity. Using this method can effectively avoid the error caused by the disturbance of loess soil samples in the sampling process by the ring knife method; the calculated results are closer to the permeability coefficient of the soil samples in the natural state.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Li, X.A.; Hong, B.; Wang, L.; Li, L.; Sun, J. Microanisotropy and preferred orientation of grains and aggregates (POGA) of the Malan loess in Yan’an, China: A profile study. Bull. Eng. Geol. Environ.
**2019**, 79, 1893–1907. [Google Scholar] [CrossRef] - Li, X.-A.; Wang, L.; Yan, Y.-l.; Hong, B.; Li, L.-C. Experimental study on the disintegration of loess in the Loess Plateau of China. Bull. Eng. Geol. Environ.
**2018**, 78, 4907–4918. [Google Scholar] [CrossRef] - Li, Y.; Mo, P.; Wang, Y.; Zhang, T.; Zhang, H. Strength anisotropy of Malan loess and the implications for the formation of loess walls and columns. Catena
**2020**, 194, 104809. [Google Scholar] [CrossRef] - Wei, T.; Fan, W.; Yuan, W.; Wei, Y.-N.; Yu, B. Three-dimensional pore network characterization of loess and paleosol stratigraphy from South Jingyang Plateau, China. Environ. Earth Sci.
**2019**, 78, 333. [Google Scholar] [CrossRef] - Wei, Y.-N.; Fan, W.; Yu, B.; Deng, L.-S.; Wei, T. Characterization and evolution of three-dimensional microstructure of Malan loess. Catena
**2020**, 192, 104585. [Google Scholar] [CrossRef] - Lu, Q.; Qiao, J.; Peng, J.; Liu, Z.; Liu, C.; Tian, L.; Zhao, J. A typical Earth fissure resulting from loess collapse on the loess plateau in the Weihe Basin, China. Eng. Geol.
**2019**, 259, 105189. [Google Scholar] [CrossRef] - Yuan, K.; Ni, W.; Lü, X.; Zhu, M.; Wang, H.; Nie, Y. Mechanical properties and microstructure evolution of Malan loess. Quat. Int.
**2022**, 637, 74–84. [Google Scholar] [CrossRef] - Zhang, X.; Lu, Y.; Li, X.; Lu, Y.; Sun, J.; Pan, W. Multilevel Collapsibility of Loess under Irrigation in Jinya Town, Gansu Province, China. Adv. Civ. Eng.
**2019**, 2019, 1–13. [Google Scholar] [CrossRef] [Green Version] - Li, X.; Lu, Y.; Zhang, X.; Fan, W.; Lu, Y.; Pan, W. Quantification of macropores of Malan loess and the hydraulic significance on slope stability by X-ray computed tomography. Environ. Earth Sci.
**2019**, 78, 522. [Google Scholar] [CrossRef] - Garcia Giménez, R.; Vigil de la Villa, R.; González Martín, J.A. Characterization of loess in central Spain: A microstructural study. Environ. Earth Sci.
**2011**, 65, 2125–2137. [Google Scholar] [CrossRef] - Li, Y.; He, S.; Deng, X.; Xu, Y. Characterization of macropore structure of Malan loess in NW China based on 3D pipe models constructed by using computed tomography technology. J. Asian Earth Sci.
**2018**, 154, 271–279. [Google Scholar] [CrossRef] - Zhou, J.Q.; Chen, Y.F.; Wang, L.; Cardenas, M.B. Universal Relationship Between Viscous and Inertial Permeability of Geologic Porous Media. Geophys. Res. Lett.
**2019**, 46, 1441–1448. [Google Scholar] [CrossRef] - Battiato, I.; Ferrero, V.P.T.; O’ Malley, D.; Miller, C.T.; Takhar, P.S.; Valdés-Parada, F.J.; Wood, B.D. Theory and Applications of Macroscale Models in Porous Media. Transp. Porous Media
**2019**, 130, 5–76. [Google Scholar] [CrossRef] - Blunt, M.J.; Bijeljic, B.; Dong, H.; Gharbi, O.; Iglauer, S.; Mostaghimi, P.; Paluszny, A.; Pentland, C. Pore-scale imaging and modelling. Adv. Water Resour.
**2013**, 51, 197–216. [Google Scholar] [CrossRef] [Green Version] - Bu, S.S.; Yang, J.; Zhou, M.; Li, S.Y.; Wang, Q.W.; Guo, Z.X. On contact point modifications for forced convective heat transfer analysis in a structured packed bed of spheres. Nucl. Eng. Des.
**2014**, 270, 21–33. [Google Scholar] [CrossRef] - Weigand, T.M.; Schultz, P.B.; Giffen, D.H.; Farthing, M.W.; Crockett, A.; Kelley, C.T.; Gray, W.G.; Miller, C.T. Modeling Nondilute Species Transport Using the Thermodynamically Constrained Averaging Theory. Water Resour. Res.
**2018**, 54, 6656–6682. [Google Scholar] [CrossRef] - Wood, B.D.; Valdés-Parada, F.J. Volume averaging: Local and nonlocal closures using a Green’s function approach. Adv. Water Resour.
**2013**, 51, 139–167. [Google Scholar] [CrossRef] - Yang, C.; Huang, R.; Lin, Y.; Qiu, T. Volume averaging theory (VAT) based modeling for longitudinal mass dispersion in structured porous medium with porous particles. Chem. Eng. Res. Des.
**2020**, 153, 582–591. [Google Scholar] [CrossRef] - Carmignato, S.; Dewulf, W.; Leach, R. Applications of CT for Non-destructive Testing and Materials Characterization. In Industrial X-Ray Computed Tomography; Springer: Berlin/Heidelberg, Germany, 2018; pp. 267–331. [Google Scholar] [CrossRef]
- Ivanov, A.L.; Shein, E.V.; Skvortsova, E.B. Tomography of Soil Pores: From Morphological Characteristics to Structural–Functional Assessment of Pore Space. Eurasian Soil Sci.
**2019**, 52, 50–57. [Google Scholar] [CrossRef] - Samant, P.; Trevisi, L.; Ji, X.; Xiang, L. X-ray induced acoustic computed tomography. Photoacoustics
**2020**, 19, 100177. [Google Scholar] [CrossRef] - Shanti, N.O.; Chan, V.W.L.; Stock, S.R.; De Carlo, F.; Thornton, K.; Faber, K.T. X-ray micro-computed tomography and tortuosity calculations of percolating pore networks. Acta Mater.
**2014**, 71, 126–135. [Google Scholar] [CrossRef] - Wang, Z.Y.; Xu, Q.; Ni, W. Study of undisturbed loess stress-strain relation during CT test. Rock Soil Mech.
**2010**, 31, 387–391+396. [Google Scholar] - Li, X.; Zhang, D.L. Application of C T in Analysis of Structure of Com pacted Soil. Chin. Rock Soil Mech.
**1999**, 20, 62–66. [Google Scholar] - Bird, M.B.; Butler, S.L.; Hawkes, C.D.; Kotzer, T. Numerical modeling of fluid and electrical currents through geometries based on synchrotron X-ray tomographic images of reservoir rocks using Avizo and COMSOL. Comput. Geosci.
**2014**, 73, 6–16. [Google Scholar] [CrossRef] - Li, T.; Shao, M.A.; Jia, Y.; Jia, X.; Huang, L. Using the X-ray computed tomography method to predict the saturated hydraulic conductivity of the upper root zone in the Loess Plateau in China. Soil Sci. Soc. Am. J.
**2018**, 82, 1085–1092. [Google Scholar] [CrossRef] - Chen, Y.; Hao, X.; Xue, D.; Li, Z.; Ma, X. Creep behavior and permeability evolution of coal pillar dam for underground water reservoir. Int. J. Coal Sci. Technol.
**2023**, 10, 11. [Google Scholar] [CrossRef] - Liu, X.; Wei, J.; Wei, G.; Wu, C.; Liu, C.; Ni, X. Combined control of fluid adsorption capacity and initial permeability on coal permeability. Int. J. Coal Sci. Technol.
**2022**, 9, 85. [Google Scholar] [CrossRef] - Wang, K.; Zhang, G.; Wang, Y.; Zhang, X.; Li, K.; Guo, W.; Du, F. A numerical investigation of hydraulic fracturing on coal seam permeability based on PFC-COMSOL coupling method. Int. J. Coal Sci. Technol.
**2022**, 9, 10. [Google Scholar] [CrossRef] - Zou, G.; Zhang, Q.; Peng, S.; She, J.; Teng, D.; Jin, C.; Che, Y. Influence of geological factors on coal permeability in the Sihe coal mine. Int. J. Coal Sci. Technol.
**2022**, 9, 6. [Google Scholar] [CrossRef] - Chen, S.; Ma, W.; Li, G. Study on the mesostructural evolution mechanism of compacted loess subjected to various weathering actions. Cold Reg. Sci. Technol.
**2019**, 167, 102846. [Google Scholar] [CrossRef] - Huo, A.; Wang, X.; Zhao, Z.; Yang, L.; Zhong, F.; Zheng, C.; Gao, N. Risk Assessment of Heavy Metal Pollution in Farmland Soils at the Northern Foot of the Qinling Mountains, China. Int. J. Env. Res. Public Health
**2022**, 19, 14962. [Google Scholar] [CrossRef] - Huo, A.; Yang, L.; Luo, P.; Cheng, Y.; Peng, J.; Nover, D. Influence of landfill and land use scenario on runoff, evapotranspiration, and sediment yield over the Chinese Loess Plateau. Ecol. Indic.
**2021**, 121, 107208. [Google Scholar] [CrossRef] - Huo, A.; Zhao, Z.; Luo, P.; Zheng, C.; Peng, J.; Abuarab, M.E.L.S. Assessment of Spatial Heterogeneity of Soil Moisture in the Critical Zone of Gully Consolidation and Highland Protection. Water
**2022**, 14, 3674. [Google Scholar] [CrossRef] - Chen, K.H.; Hwang, C.; Chang, L.C.; Tsai, J.P.; Yeh, T.C.J.; Cheng, C.C.; Ke, C.C.; Feng, W. Measuring Aquifer Specific Yields With Absolute Gravimetry: Result in the Choushui River Alluvial Fan and Mingchu Basin, Central Taiwan. Water Resour. Res.
**2020**, 56, e2020WR027261. [Google Scholar] [CrossRef] - Malík, P.; Coplák, M.; Švasta, J.; Černák, R.; Bajtoš, P. Recharge, delayed groundwater-level rise and specific yield in the Triassic karst aquifer of the Kopa Mountain, in the Western Carpathians, Slovakia. Hydrogeol. J.
**2020**, 29, 499–518. [Google Scholar] [CrossRef] - Pendiuk, J.E.; Guarracino, L.; Reich, M.; Brunini, C.; Güntner, A. Estimating the specific yield of the Pampeano aquifer, Argentina, using superconducting gravimeter data. Hydrogeol. J.
**2020**, 28, 2303–2313. [Google Scholar] [CrossRef] - Azizmohammadi, S.; Matthäi, S.K. Is the permeability of naturally fractured rocks scale dependent? Water Resour. Res.
**2017**, 53, 8041–8063. [Google Scholar] [CrossRef] - Hamzehpour, H.; Khazaei, M. Effective Permeability of Heterogeneous Fractured Porous Media. Transp. Porous Media
**2016**, 113, 329–344. [Google Scholar] [CrossRef] - Sedaghat, M.; Azizmohammadi, S.; Matthäi, S.K. Does the symmetry of absolute permeability influence relative permeability tensors in naturally fractured rocks? J. Pet. Explor. Prod. Technol.
**2019**, 10, 455–466. [Google Scholar] [CrossRef] [Green Version] - Szewczyk, R. Generalization of the Model of Magnetoelastic Effect: 3D Mechanical Stress Dependence of Magnetic Permeability Tensor in Soft Magnetic Materials. Materials
**2020**, 13, 4070. [Google Scholar] [CrossRef] - Elliot, T.R.; Reynolds, W.D.; Heck, R.J. Use of existing pore models and X-ray computed tomography to predict saturated soil hydraulic conductivity. Geoderma
**2010**, 156, 133–142. [Google Scholar] [CrossRef] - Hong, B.; Li, X.A.; Wang, L.; Li, L.; Xue, Q.; Meng, J. Using the Effective Void Ratio and Specific Surface Area in the Kozeny–Carman Equation to Predict the Hydraulic Conductivity of Loess. Water
**2019**, 12, 24. [Google Scholar] [CrossRef] [Green Version] - Liu, P.; Zhang, X.; Zhang, M.; Yang, X. Effect of Admixture on the Hydraulic Conductivity of Compacted Loess: A Case Study. Adv. Civ. Eng.
**2020**, 2020, 1–12. [Google Scholar] [CrossRef] - Brzeźniak, Z.; Dhariwal, G.; Le Gia, Q.T. Stochastic Navier–Stokes Equations on a Thin Spherical Domain. Appl. Math. Optim.
**2020**, 84, 1971–2035. [Google Scholar] [CrossRef] - Bear, J. Dynamics of Fluids in Porous Media; Elsevier: Amsterdam, The Netherlands, 1972; Volume 7, pp. 174–175. [Google Scholar]
- Zhang, X.; Lu, Y.; Li, X.; Lu, Y.; Pan, W. Microscopic structure changes of Malan loess after humidification in South Jingyang Plateau, China. Environ. Earth Sci.
**2019**, 78, 287. [Google Scholar] [CrossRef] - Hu, W.; Liu, G.; Zhang, X. A pore-scale model for simulating water flow in unsaturated soil. Microfluid. Nanofluid.
**2018**, 22, 71. [Google Scholar] [CrossRef] - Wildenschild, D.; Sheppard, A.P. X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems. Adv. Water Resour.
**2013**, 51, 217–246. [Google Scholar] [CrossRef]

**Figure 1.**Location map of the sampling sites. (

**a**) is the location of the sampling site study area in China; (

**b**) is the location of the sampling site in the Malan loess distribution area on the Loess Plateau.; (

**c**) is the location of the sampling site on the Malan loess profile in Jiuzhoutai, Lanzhou; (

**d**) is the sampling point located at Jiuzhoutai, Lanzhou.

**Figure 2.**(

**a**) is the 308th section of loess samples along Z-direction scanned by CT; (

**b**) is the three-dimensional model of loess samples calibrated and then sliced.

**Figure 5.**(

**a**) is the starting state of the surface of the intact Malan loess sample, (

**b**) is the surface state of intact Malan loess sample when the seepage process lasts 480 s, (

**c**) is the state of the surface of the loess sample at the beginning of 489 s, (

**d**) is the state of the surface of the loess sample at the beginning of 498 s, (

**e**) is the state at the beginning of 545 s, (

**f**) is the state of the surface of the loess sample at the beginning of 680 s of seepage.

**Figure 6.**(

**a**) is the process of threshold selection; (

**b**) after the threshold segmentation, slice 308 along the Z-direction.

**Figure 12.**Seepage simulation model. (

**a**) is the two-dimensional schematic diagram of the seepage simulation. (

**b**)is the 3D model of the 3D seepage boundary.

Dry Density (g/cm ^{3}) | Void Ratio | Clay (%) (d < 5 μm) | Silt (%) (5 < d < 50 μm) | Sand (%) (d > 50 μm) | Liquid Limit | Plastic Limit | Plastic Index | Permeability Coefficient K (cm/s) |
---|---|---|---|---|---|---|---|---|

1.213 | 1.18 | 22.52 | 65.70 | 8.55 | 28.21% | 15.2% | 11.83% | 1.42 × 10^{−4} |

Project | V_{4} (mL) | V_{1} (mL) | V_{0} (mL) | μ |
---|---|---|---|---|

Value | 836 | 735 | 1000 | 10.1% |

k_{x} (μm^{2}) | k_{y} (μm^{2}) | k_{z} (μm^{2}) | Eigen Value |
---|---|---|---|

0.284803 | −0.030083 | −0.124343 | 0.346743 |

−0.029247 | 0.031562 | 0.025184 | 0.034523 |

−0.123694 | 0.28359 | 0.07378 | 0.008879 |

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**MDPI and ACS Style**

Lu, Y.; Lu, Y.; Lu, T.; Wang, B.; Zeng, G.; Zhang, X.
Computing of Permeability Tensor and Seepage Flow Model of Intact Malan Loess by X-ray Computed Tomography. *Water* **2023**, *15*, 2851.
https://doi.org/10.3390/w15152851

**AMA Style**

Lu Y, Lu Y, Lu T, Wang B, Zeng G, Zhang X.
Computing of Permeability Tensor and Seepage Flow Model of Intact Malan Loess by X-ray Computed Tomography. *Water*. 2023; 15(15):2851.
https://doi.org/10.3390/w15152851

**Chicago/Turabian Style**

Lu, Yangchun, Yudong Lu, Ting Lu, Bo Wang, Guanghao Zeng, and Xu Zhang.
2023. "Computing of Permeability Tensor and Seepage Flow Model of Intact Malan Loess by X-ray Computed Tomography" *Water* 15, no. 15: 2851.
https://doi.org/10.3390/w15152851