Transport in Porous Media

, Volume 126, Issue 3, pp 701–712 | Cite as

Estimation of Sandstone Permeability with SEM Images Based on Fractal Theory

  • Qingyang Yu
  • Zhenxue DaiEmail author
  • Zhien Zhang
  • Mohamad Reza Soltanian
  • Shangxian YinEmail author


Permeability is one of the key parameters for quantitatively evaluating groundwater resources and accurately predicting the rates of water inflows into coal mines. This paper presents an efficient method to estimate the macroscopic permeability by using the scanning electron microscopy (SEM) images. A correlation between the microscopic features of sandstone porosity and the macropermeability is approached by an image identification technique. Firstly, the gray images were transformed into the binary images by using the histogram of the entropy method. Then, the Green and Euler distance methods were applied to calculate the length and area of the pores, and the fractal parameters were estimated according to the slit island method. Based on the theory of microscopic seepage flow, the seepage coefficient and permeability were calculated by fractal parameters. Typical water-bearing sandstone samples in the Kailuan coal field area in North China were selected to demonstrate the methodology. SEM microscopic images of nine groups of sandstone samples collected at different depths (from the outcrop to the deep mines) were analyzed. Based on the theoretical model of micropore structures and the fractal theory, the permeabilities were estimated. The results provide insights for understanding the hydraulic properties of the sandstone.


Sandstone permeability Fractal theory SEM images Entropy method Image identification 

1 Introduction

The quantitative evaluation of water resources in coal mine areas and the prediction of groundwater inflow into mines are challenging. This is because neither the seepage theory nor the accuracy of predictive models is appropriate for the complex hydrogeological conditions in coal mining systems. In addition, estimated aquifer properties (e.g., permeability) are inaccurate resulting in an inconsistency between numerical modeling results with the observed water inflow into the mine. Furthermore, inaccurate results could affect the design and construction of the mine drainage system, leading to serious threats due to safety issue (Yu et al. 2003). Therefore, it is important to develop reliable methods for calculating permeability using the micropore structure parameters.

To obtain rock permeability, many prior works have been conducted, such as experimental, theoretical, or empirical model methods. Balankin et al. (2016) performed experimental and theoretical studies of Newtonian fluid flow through permeable media. They found that the seepage velocity was linearly proportional to the pressure drop, while the absolute permeability clearly increased with the increase in sample length in the flow direction. Behrang and Kantzas (2017) developed a theoretical approach to study the permeability of gas in organic tight porous media. Grader et al. (2010) offered an example in using digital rock physics in determining porosity, permeability, and relative permeabilities of a carbonate sample. Aldana et al. (2014) used empirical models to determine permeability. Habibi et al. (2014) formulated the permeability of two-dimensional (2D) fractured and permeable rock matrix using a distinct element method and proposed a reliable mathematical model. Chen et al. (2015) proposed a model for fluid flow through tight porous media with slippage. Moreover, experimental measurements are affected by scale and disturbance of the sample (Zhang and Yin 2017; Dai and Samper 2004). Therefore, scaling effects should be considered when using small-scale data to estimate large-scale permeability values (Dai et al. 2010; Soltanian and Ritzi 2015; Soltanian et al. 2015a, b), hence conducting the analysis of micropore structures by scanning electron microscope (SEM) as an effective approach for analyzing the pore geometry characteristics and investigating the relationship between the parameters of micropore structures and macroscopic permeability.

Fractal methods have been widely used to estimate permeability. Song and Yu (2012) developed a fractal permeability model for non-woven fabrics based on fractal characteristics of pores. They reported that the fractal permeability was associated with fractal dimension of pore characteristics. Helwani et al. (2015) used the fractal theory employing a box-counting method to describe hydrogen gas diffusion into membrane pores in the meso-porosity regime and developed a fractal permeability model that reflected different mechanisms of hydrogen diffusion. Jin et al. (2015) analytically derived a permeability model using series–parallel flow resistance mode and a fractal permeability–pore structure relationship. Li et al. (2016) proposed a multiple fractal model by considering fractal properties of both porous matrices and fracture networks for the permeability of dual-porosity media embedded with randomly distributed fractures. Yuan et al. (2016) developed the application of the most frequently used analytical approach for single uniform capillary in a porous media. The fractal theory was applied to mathematically express the capillary diameter distribution and the corresponding tortuosity. Karimpouli and Tahmasebi (2016) applied a general trend based on fractal dimensions of pore space and tortuosity of the Berea sandstone sample. A hierarchical sampling technique was also implemented by two and three steps based on 2D and three-dimensional (3D) stochastic models. Shokri et al. (2016) analyzed fractal-based correlations to clarify a physical relationship among network properties and the correlated parameters. Chen and Yao (2017) proposed a model based on the fractal geometry theory to estimate permeability values of 133 sandstone samples from a gas reservoir. Li and Deng (2017) proposed a permeability model using nuclear magnetic resonance (NMR) based on the fractal theory. The model was more accurate in terms of the permeability calculations and met practical requirements for production. Fu et al. (2017) performed a series of laboratory experiments to analyze characteristics of pore structure in low-rank coal and defined and calculated the fractal features of adsorbed and seepage pores using fractal dimensions.

To sum up, experimental, theoretical, or empirical model methods were used to obtain sandstone permeability, and some permeability estimation models have been proposed. Many scholars have used fractal model and hydraulics-related formula to establish the mathematical model of microcosmic pore parameters and permeability. However, research on the relationship between micropore fractal characteristics of SEM image and macropermeability of sandstone is rarely reported.

Based on the method of microfractal characteristics, in this paper we present a relationship between micropore fractal characteristics obtained from SEM images and macroscopic permeability. In addition, we propose a reliable method for calculating rock and soil permeability using SEM images. We propose a calculation method of permeability based on fractal parameters obtained from image processing. We test this methodology using typical sandstone samples from a coal field in North China.

2 Analytical Method for Calculating Permeability with Fractal Theory

Using an image processing software, we performed detailed analyses of geometrical feature and calculated fractal parameters. The permeability is obtained by analyzing SEM images. The surface microscopic morphology and microstructure (e.g., fissure development situation, mineral composition, mineral arrangement and cementation, and mineral attitude) can be directly observed using SEM. To improve the micropore resolution, a relationship between the pixel and the length is formulated based on magnification of the SEM. To facilitate analysis of image characteristics and adjust the threshold, as an image shows a single-peak histogram sample, threshold value is manually set to realize binary image processing. The SEM image of the threshold value is transformed into the binary image, and the most representative binary SEM image is obtained. Using Green and Euclidean distance equations, geometrical parameters such as pore area, pore perimeter, and fractal dimension D are calculated based on regression analysis method. Based on fractal dimension, fractal coefficient, and microscopic permeability theory, the permeability parameters of rock samples are calculated.

2.1 Image Binarization

Image binarization transforms a SEM image into a binary image. The pore information such as area and perimeter of irregular pores in the image is retained, which is a basis for further analysis and calculation. Global histogram-based binary algorithm is an effective method to solve this problem that can retain all information of the pore. It includes “average method” of grayscale conversion, the threshold value method of percentage, minimum value method, average method based on the twin peaks, the optimal iteration method, and a one-dimensional (1D) maximum entropy classification method. Among them, global binary entropy method based on histogram splits image by statistics and maximum entropy can retain information better than other methods.

In this work, we used a global binary entropy method based on histogram to convert grayscale images into black and white images (Kapur et al. 1985). The expressions for conversion are:
$$ T_{\text{opt}} = \arg { \hbox{max} }\left[ {H_{\text{f}} \left( T \right) + H_{\text{b}} \left( T \right)} \right] $$
$$ H_{\text{f}} \left( T \right) = \mathop \sum \limits_{g = 0}^{T} \frac{p\left( g \right)}{P\left( T \right)}\log \frac{p\left( g \right)}{P\left( T \right)} $$
$$ H_{\text{b}} \left( T \right) = - \mathop \sum \limits_{g = T + 1}^{G} \frac{p\left( g \right)}{P\left( T \right)}\log \frac{p\left( g \right)}{P\left( T \right)} $$
where Topt denotes the maximum entropy, \( H_{\text{f}} \left( T \right) \) and \( H_{\text{b}} \left( T \right) \) represent the entropy of foreground and background image, respectively, and p(g) and p(T) are the cumulative probability of the pixels, respectively, when the threshold is g or T.

This method firstly calculates the probability of the foreground and background image pixels at a given threshold. Then, the entropy of the foreground and background image is counted by cumulating the probability of all pixels. Many entropies can be gained by changing the threshold. When the entropy is maximum, the corresponding threshold is splitting value. The pixel whose gray value is greater than the threshold is foreground image, and the others are background image.

2.2 Calculation of Geometrical Parameters

Geometrical parameters are calculated based on pore characteristics in the binary image found in the previous section. Area and perimeter of a pore in an image are mathematically calculated using Image J (2012). Green method is herein used to calculate the pore area:
$$ A = \frac{1}{2}{\oint }\left( {x{\text{d}}y - y{\text{d}}x} \right) $$
$$ A = \frac{1}{2}\mathop \sum \limits_{i = 0}^{n} \left[ {x_{i} y_{i + 1} - x_{i + 1} y_{i} } \right] $$
where \( x_{i} \) and \( y_{i} \) represent coordinates of the ith pixel.
The Euler distance method is used to calculate the pore perimeter:
$$ L_{s} = N_{\text{e}} + \sqrt 2 N_{\text{o}} $$
where Ne denotes the number of even pixels encoded on the boundary line and No is the number of odd pixels.

2.3 Calculation of Fractal Dimension

There are many ways to compute the pore fractal dimension (Cai et al. 2018; Xia et al. 2018; Xie and Wang 1999; Keller et al. 1989). According to the self-similarity concept of fractal theory, the slit island method (Mandelbrot et al. 1984), also known as area–length method, is used to calculate the fractal dimension in this study. The correlation between the perimeter and area of the micropore section of sandstone is calculated by fractal dimension. When the area of the pore cross section is known, the pore has a certain perimeter, that is X = ZSD/2 (Feder 1988). The ratio of area to perimeter is determined, that is, the fractal structure of the perimeter and area of the pore section with statistical significance. The fitting curve of the pore perimeter X and the area S is plotted on the log–log coordinates. According to the power function curve fitting, the fractal dimension of the perimeter–area of the pore cross section is calculated. The formula describing the irregular length and integral dimension of the slit island method is as follows (Mandelbrot et al. 1984):
$$ \left[ {p\left( \lambda \right)} \right]^{1/D} = \frac{{a_{0} \lambda \left( {1 - D} \right)}}{{D\left[ {A\left( \lambda \right)} \right]^{{\frac{1}{2}}} }} = a_{0} \lambda^{1/D} \lambda^{ - 1} \left[ {A\left( \lambda \right)} \right]^{1/2} $$
where a0 represents a constant that is associated with the shape of the island, λ is the measurement dimension, and D is the fractal dimension. With drawing \( {{\log \left[ {p\left( \lambda \right)} \right]} \mathord{\left/ {\vphantom {{\log \left[ {p\left( \lambda \right)} \right]} \lambda }} \right. \kern-0pt} \lambda } \) and \( {{\log \left[ {A\left( \lambda \right)^{1/2} } \right]} \mathord{\left/ {\vphantom {{\log \left[ {A\left( \lambda \right)^{1/2} } \right]} \lambda }} \right. \kern-0pt} \lambda } \) fitting straight line, the reciprocal slope of the line is equal to the D.

2.4 Calculation of Permeability and Hydraulic Conductivity

Based on fractal dimension, fractal coefficient, and micropermeation theory, permeability of rock samples can be calculated as (Dorner and Dec 2007):
$$ k = \frac{{2\pi n^{4} \mathop \sum \nolimits_{i = 1}^{y} A_{i}^{4 - 2D} }}{{A_{k} }} $$
where k is the permeability (mD), \( A_{i} \) represents the area of a single pore, \( A_{k} \) is the effective total area (m2), y denotes the number of pores in the scanning region \( A_{k} \), and \( n \) is the fractal coefficient.

3 Results and Discussions

Sandstone samples were classified according to their lithologic compositions and depths. The spatial distribution of characteristics of lithic facies and lithologic compositions were analyzed in 1–2 dominant directions using statistical method. The layered surface and end face of sandstone were selected for observation. The rock was prepared into a sample with a natural section of about 1 cm3. After removing impurities, drying water and gold plating on sandstone surface, the samples were put into the SEM. We firstly observe areas in low-magnification images and then enlarge and record the images.

3.1 Acquisition and Processing of Micro-SEM image

Kailuan coal formation in Hubei province, China, is dominated by a set of sandstone–mudstone clastic rock series which consist of mud–limestone, sandy mudstone, argillaceous sandstone and coarse sandstone. In this paper, from the surface outcrop to the depth in the Kailuan mining area, 19 core samples of representative sandstone on the vertical and horizontal cover depth were selected. We used SEM to study the microscopic characteristics of sandstone particles and pore space distribution and established a relation between permeability and the structural parameters including porosity, pore and grain size, and shape. A photograph of our samples is shown in Fig. 1.
Fig. 1

Photograph of sandstone samples

The samples were first sprayed with gold to make their surface conductive and make them ready for the electronic imaging. Gold injection was carried out by sputtering apparatus. SEM images were attempted to study the distribution microscopic sandstone particles and pore space. An analytical model was developed that correlates sandstone permeability with the characteristics of microscopic sandstone particle and pore.

In this paper, using a SEM (S-4800FE, Hitachi, Tokyo, Japan), 114 microimages of 19 rock samples were obtained by SEM. Three test regions were randomly selected for each sample surface. The images with clear pore boundaries with uniform distribution and the same magnification were taken. These images of zoom in 2000 times and 10,000 times were selected to calculate microfractal characteristics. According to the threshold of the gray histogram for each image, the grayscale image is transformed into a binary image. The results of grayscale and their corresponding binary image are presented in Fig. 2.
Fig. 2

Binary image of sandstone samples created from grayscale images. a SEM figure of sample 1LA2_B_10000, b binarization figure of sample 1LA2_B_10000, c SEM figure of sample 2ML_B_10000, d binarization figure of sample 2ML_B_10000, e SEM figure of sample 3GQ5a2_A_2000, f binarization figure of sample 3GQ5a2_A_2000, g SEM figure of sample 4LA2_C_2000, h binarization figure of sample 4LA2_C_2000, i SEM figure of sample 5LA3_D_2000, j binarization figure of sample 5LA3_D_2000, k SEM figure of sample 6LA4_B_2000, l binarization figure of sample 6LA4_B_2000, m SEM figure of sample 7LA5_A_2000, n binarization figure of sample 7LA5_A_2000, o SEM figure of sample 8TL2_B_2000, p binarization figure of sample 8TL2_B_2000

3.2 Calculation of Fractal Parameters

The area and perimeter of the pores in a binary image were calculated by using Green and Euler distance methods. The numerical values of area and perimeter are obtained, and the power function is fitted to obtain D and n parameters.

The size of a SEM image is 1280 × 888 pixels, and the resolution is 256 pixels per foot. The actual length and height of the image are 12.7 cm and 8.81 cm, respectively. Representatives of a pixel in an image with actual length are related to the magnification. When the ratio of zoom is 10,000 times, the actual length is 0.127 m, and the corresponding pixels are 1280, we can obtain the length of one pixel by using the actual length divided by the number of pixels. Therefore, one pixel corresponds to the actual length of the rock mass image that is 9.92 × 10−9 m. Then, we can obtain the one-pixel area corresponding to the actual area by using one-pixel length multiply one-pixel length. Thus, one-pixel area is 9.84 × 10−5 μm2. When the ratio of zoom is 2000 times, the corresponding length of one pixel is 4.961 × 10−8 m, and the actual area of one-pixel area is 2.46 × 10−3 μm2. Obtained fractal curves are illustrated in Fig. 3.
Fig. 3

Fractal characteristics of sandstone samples. a 1LA2_B_10000, b 2ML_B_10000, c 3GQ5a2_A_2000, d 4LA2_C_2000, e 5LA3_D_2000, f 6LA4_B_2000, g 7LA5_A_2000, h 8TL2_B_2000

Figure 3 illustrates the logarithmic relationship of the area and perimeter of the pores. Power function is used to fit the data. All curves fit well, and all the R2 is greater than 0.94. According to the fractal fitting results, the fractal dimension and fractal coefficient of the rock samples are listed in Table 1.
Table 1

Fractal dimension, fractal coefficient, and permeability of sandstone samples calculated using SEM image


Sample number

Sample magnification

Fractal dimension D

Fractal coefficient n

Permeability (mD)




2 × 0.6928






2 × 0.6625






2 × 0.7221






2 × 0.7084






2 × 0.7233






2 × 0.7289






2 × 0.7619






2 × 0.7315



3.3 Analysis of Sandstone Permeability

Using parameters of microscopic pore structure as well as a relationship between permeability and fractal coefficient, the fitting coefficient of fractal dimension and fractal coefficient were used in Eq. (8). This results in a relation between sandstone macroscopic permeability and microscopic features of particles.

Table 1 shows the calculated permeability of sandstone samples which are from 0.0066 to 0.2084 mD. Experimental data have found permeability range of 0.002–0.225 mD (Bloch 1991). Our results using fractal theory are in the range of what was reported in experiments, confirming the accuracy of our fractal characteristic analysis and the resulting permeability estimation.

4 Conclusions

Using scanning electron microscopy for 13 set of samples of the upper coal seam of thick-bedded sandstone North China Permo-Carboniferous, the geometrical features, fractal parameters as well as permeability characteristics of the pore were studied. Eight typical microsandstone samples, enlarged 2000 times and 10,000 times, were used for calculation by fractal theory. The relationship between macroscopic permeability of sandstone rock and microscopic features of sandstone particles was developed. Based on the binarization process, geometrical parameters and the fractal fitting parameter of rock samples were calculated, and sandstone permeability were obtained. We found the fractal theory to be reliable in estimating obtain permeability for rock samples with its parameters estimated from microscopic images. We found the image processing and fractal characteristics of porous media to be an efficient method in estimating the macroscopic properties (i.e., permeability).



This work has been funded by Engineering Research Center of Geothermal Resources Development Technology and Equipment of Ministry of Education, Jilin University [Grant Number: 20170035]. Z. Dai thanks Jilin University for the start-up funding and the National Natural Science Foundation of China [Grant Number: 41772253] for supporting this work.


  1. Aldana, M., Altamiranda, D., Cabrera, A.: Fractal theory for permeability prediction, Venezuelan and USA wells. Chronobiol. Int. 31, 243–251 (2014)CrossRefGoogle Scholar
  2. Balankin, A.S., Valdivia, J.C., Marquez, J., Susarrey, O., Solorio-Avila, M.A.: Anomalous diffusion of fluid momentum and Darcy-like law for laminar flow in media with fractal porosity. Phys. Lett. A 380, 2767–2773 (2016). CrossRefGoogle Scholar
  3. Behrang, A., Kantzas, A.: A hybrid methodology to predict gas permeability in nanoscale organic materials; a combination of fractal theory, kinetic theory of gases and Boltzmann transport equation. Fuel 188, 239–245 (2017). CrossRefGoogle Scholar
  4. Bloch, S.: Empirical prediction of porosity and permeability in sandstones. AAPG Bull. 75, 1145–1160 (1991)Google Scholar
  5. Cai, J., Zhang, L., Yang, J.U., Pia, G., Zhang, Z.: An introduction to fractal-based approaches in unconventional reservoirs—part I. Fractals Complex Geom. Patterns Scaling Nat. Soc. 26, 1802001 (2018). Google Scholar
  6. Chen, L., Kang, Q., Dai, Z., Viswanathan, H.S., Tao, W.: Permeability prediction of shale matrix reconstructed using the elementary building block model. Fuel 160, 346–356 (2015). CrossRefGoogle Scholar
  7. Chen, X., Yao, G.: An improved model for permeability estimation in low permeable porous media based on fractal geometry and modified Hagen–Poiseuille flow. Fuel 210, 748–757 (2017). CrossRefGoogle Scholar
  8. Dai, Z., Keating, E., Gable, C., Levitt, D., Heikoop, J., Simmons, A.: Stepwise inversion of a groundwater flow model with multi-scale observation data. Hydrogeol. J. 18, 607–624 (2010). CrossRefGoogle Scholar
  9. Dai, Z., Samper, J.: Inverse problem of multicomponent reactive chemical transport in porous media: formulation and applications. Water Resour. Res. 40, 294–295 (2004). CrossRefGoogle Scholar
  10. Dorner, J., Dec, D.: Air permeability and saturated hydraulic conductivity as a tool for the functional characterization of soil pores. J. Soil Sci. Plant Nutr. 7, 1–13 (2007)Google Scholar
  11. Feder, J.: Fractals. Plenum, New York (1988)CrossRefGoogle Scholar
  12. Fu, H., Tang, D., Xu, T., Xu, H., Tao, S., Li, S.: Characteristics of pore structure and fractal dimension of low-rank coal: a case study of lower Jurassic Xishanyao coal in the southern Junggar basin, NW China. Fuel 193, 254–264 (2017). CrossRefGoogle Scholar
  13. Grader, A.S., Nur, A., Baldwin, C., Diaz, E.: Multi-scale imaging process for computations of porosity and permeability from carbonate rocks. Geo 2010 Conference, The European Association of Geoscientists and Engineers (EAGE) Publication (2010)Google Scholar
  14. Habibi, M.J., Mokhtari, A.R., Baghbanan, A., Namdari, S.: Prediction of permeability in dual fracture media by multivariate regression analysis. J. Petrol. Sci. Eng. 120, 194–201 (2014). CrossRefGoogle Scholar
  15. Helwani, Z., Wiheeb, A.D., Shamsudin, I.K., Kim, J., Othman, M.R.: The effects of fractality on hydrogen permeability across meso-porous membrane. Heat Mass Transf. 51, 751–758 (2015). CrossRefGoogle Scholar
  16. Image J User Guide. National Institutes of Health (2012)Google Scholar
  17. Jin, Y., Zhu, Y., Li, X., Zheng, J., Dong, J.: Scaling invariant effects on the permeability of fractal porous media. Transp. Porous Media 109, 433–453 (2015). CrossRefGoogle Scholar
  18. Kapur, J.N., Sahoo, P.K., Wong, A.K.C.: A new method for gray-level thresholding using the entropy of the histogram. Graph Models Image Process 29, 273–285 (1985)CrossRefGoogle Scholar
  19. Karimpouli, S., Tahmasebi, P.: A hierarchical sampling for capturing permeability trend in rock physics. Transp. Porous Media 116, 1–16 (2016). Google Scholar
  20. Keller, J.M., Crownover, R.M., Chen, S.: Texture description and segmentation through fractal geometry. Comput. Vis. Graph. Image Process. 45, 150–160 (1989). CrossRefGoogle Scholar
  21. Li, B., Liu, R., Jiang, Y.: A multiple fractal model for estimating permeability of dual-porosity media. J. Hydrol. 540, 659–669 (2016). CrossRefGoogle Scholar
  22. Li, H., Deng, S.: Permeability modeling method of NMR in low porosity and low permeability sandstone based on the fractal theory. Int. Geophys. Conf. 17–20, 1060–1063 (2017). Google Scholar
  23. Mandelbrot, B.B., Passoja, D.E., Paulley, A.J.: Fractal character of fracture surfaces of metals. Nature 308, 721–723 (1984)CrossRefGoogle Scholar
  24. Shokri, A.R., Babadagli, T., Jafari, A.: A critical analysis of the relationship between statistical- and fractal-fracture-network characteristics and effective fracture-network permeability. SPE Reserv. Eval. Eng. (2016). Google Scholar
  25. Soltanian, M.R., Ritzi, R.W.: A new method for analysis of variance of the hydraulic and reactive attributes of aquifers as linked to hierarchical and multiscaled sedimentary architecture. Water Resour. Res. 50, 9766–9776 (2015). CrossRefGoogle Scholar
  26. Soltanian, M.R., Ritzi, R., Dai, Z., Huang, C., Dominic, D.: Transport of kinetically sorbing solutes in heterogeneous sediments with multimodal conductivity and hierarchical organization across scales. Stoch. Environ. Res. Risk Assess. 29, 709–726 (2015a). CrossRefGoogle Scholar
  27. Soltanian, M.R., Ritzi, R.W., Dai, Z., Huang, C.: Reactive solute transport in physically and chemically heterogeneous porous media with multimodal reactive mineral facies: the Lagrangian approach. Chemosphere 122, 235–244 (2015b). CrossRefGoogle Scholar
  28. Song, W., Yu, W.: Fractal calculation of air permeability of nonwoven fabrics. J. Text. Inst. Proc. Abstr. 103, 817–826 (2012). CrossRefGoogle Scholar
  29. Xia, Y., Cai, J., Wei, W., Xiangyun, H.U., Wang, X., Xinmin, G.E.: A new method for calculating fractal dimensions of porous media based on pore size distribution. Fractals Complex Geom. Patterns Scaling Nat. Soc. 26, 1850006 (2018). Google Scholar
  30. Xie, H., Wang, J.A.: Direct fractal measurement of fracture surfaces. J. Univ. Sci. Technol. Beijing 36, 3073–3084 (1999). Google Scholar
  31. Yuan, Y., Doonechaly, N.G., Rahman, S.: An analytical model of apparent gas permeability for tight porous media. Transp. Porous Media 111, 193–214 (2016). CrossRefGoogle Scholar
  32. Yu, B., Li, J., Li, Z., Zou, M.: Permeabilities of unsaturated fractal porous media. Int. J. Multiph. Flow 29, 1625–1642 (2003). CrossRefGoogle Scholar
  33. Zhang, J., Yin, S.: Real-time pore pressure detection: indicators and improved methods. Geofluids 2017, 1–12 (2017). Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.College of Construction EngineeringJilin UniversityChangchunChina
  2. 2.Engineering Research Center of Geothermal Resources Development Technology and Equipment, Ministry of EducationJilin UniversityChangchunChina
  3. 3.Key Laboratory of Low-Grade Energy Utilization Technologies and Systems, Ministry of Education of ChinaChongqing UniversityChongqingChina
  4. 4.Department of GeologyUniversity of CincinnatiCincinnatiUSA
  5. 5.North China Institute of Science and TechnologyLangfangChina

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