**4. Coupled mathematical model in macroscopic-scale shale matrix**

In this section, the experimental results of full-scale pore size distributions of real shale samples from a gas field are combined with the coupled flow model in nanopores to realize the upscaling transformation of the flow model into that in macroscopic-scale shale matrix by integration.

In the unitary model which is widely used for the flow estimation on a macroscopic scale [12, 18–22], indirect averaging methods are applied, e.g., the pore space of shale is assumed to be composed of a certain number of isodiametric pores, regardless of the pore size distributions. Some research [15, 47] used specific functions to characterize the probability density function of shale pore size distributions, with, however, assumed parameters for the purpose of conducting parameter sensitivity analysis. Here, the fitting parameters needed for the macroscopic form of the derived coupled flow model in nanopores are obtained by performing the experiments of pore size distributions of real shale samples from a gas field.

Michel et al. [15] and Xiong et al. [47] described the probability density function of shale pore size distributions as logarithmic normal distribution. Enlightened by their studies, the following expression is used to fit the experimental data of full-scale shale pore size distributions:

$$f(r\_{in}) = \frac{1}{r\_{in}\sigma\sqrt{2\pi}}e^{-0.5\left(\frac{\text{lw}\_{in} - \eta}{\sigma}\right)^2} \tag{6}$$


*Mechanism, Model, and Upscaling of the Gas Flow in Shale Matrix: Revisit DOI: http://dx.doi.org/10.5772/intechopen.91821*

#### **Table 2.**

*Fitting results of η and σ.*

where *η* = normal mean, dimensionless.

*σ* = variance, dimensionless.

Three kinds of experiments, i.e., the high-pressure mercury intrusion experiment, the liquid nitrogen adsorption experiment, and the low-temperature carbon dioxide adsorption experiment, were performed, and the full-scale pore size distribution data of the three shale samples from the Well "Ning 203", Longmaxi formation of Changning-Weiyuan district, Sichuan Basin of China, were obtained by stitching the three results together according to the effective range of each experiment, where the total volume of pores greater than 100 nm is attributed to the pore whose radius is closest to 100 nm in the experiments allowing for the difficulty of curve fitting caused by the severe fluctuations of the pore size data [42]. The values of *η* and *σ* are listed in **Table 2**. Because the samples "Ning 203-219", "Ning 203-240", and "Ning 203-250" are all taken from a depth interval of 2300-2400 m, the three groups of data in **Table 2** are averaged, i.e., *η* = 0.8055 and *σ* = 0.9060, to represent the typical shale pore size distribution in this depth range.

The number of single pipes in shale with the radius range of *rin* to *rin* + d*rin* is expressed in Eq. (7). By integrating in the entire pore size range, the flow rate in shale is obtained as Eq. (8):

$$\frac{10^{18} \phi\_{co} V\_{co}}{\pi r\_{in}^2 L\_{co}} f(r\_{in}) \mathrm{d}r\_{in} \tag{7}$$

$$Q = \int\_{r\_{in,\min}}^{r\_{in,\max}} \frac{10^{18} N \phi\_{co} V\_{co}}{\pi r\_{in}^2 L\_{co}} f(r\_{in}) \mathrm{d}r\_{in} \tag{8}$$

where *Vco* = apparent volume of a core sample, m<sup>3</sup> .

*Lco* = length of a core sample, m.

*rin,*min = lower limit of integration, which should be larger than 0.19 nm because the diameter of methane molecules is 0.38 nm [48].

*rin,*max = upper limit of integration.

The macroscopic-scale mathematical model of shale gas flow can be obtained by substituting Eqs. (5) and (6) into Eq. (8) as:

$$Q = \int\_{r\_{\rm in,in}}^{r\_{\rm in,max}} \frac{10^{\rm lb/8} \left[ \varepsilon^{-\rm K\rm in} N\_D + \left( \frac{1}{1+\rm K\alpha} - \varepsilon^{-\rm K\rm in} \right) N\_b + \varepsilon^{-1.8/\rm K\alpha} N\_K + \left( \frac{\rm K\alpha}{1+\rm K\alpha} - \varepsilon^{-1.8/\rm I\alpha} \right) N\_i \right] \phi\_{\alpha} V\_{\alpha}}{\pi r\_{\rm in} r\_{\rm in}^3 \sigma \sqrt{2\pi}} \epsilon^{-0.9 \left( \frac{\rm I\alpha}{\rm or} \right)^2} d\tau\_{\rm in} \tag{9}$$

Literature survey shows that there are several main upscaling methods of flow models from microscopic to macroscopic scale, i.e.:

Method (1): the commonly used unitary model [12, 18–22] as already mentioned. Method (2): the sum method of calculating the permeability of every straight capillary tube [27].


**Method Description/equation Advantages Shortcomings** porous medium is considered, in which a microscopic unit cell is periodically repeated Pore network model [54, 55] Generate pore network models by extracting pore structure information from real images or generate porous media by simulating the sedimentation and diagenesis processes and then incorporate relevant flow mechanisms into the Pore size distribution, anisotropy and low connectivity of the pore structure, etc. can be taken into account Substantial work for model establishment; representativeness and verisimilitude of pore network models to the real pore structures remain a challenge

*Mechanism, Model, and Upscaling of the Gas Flow in Shale Matrix: Revisit DOI: http://dx.doi.org/10.5772/intechopen.91821*

#### **Table 3.**

*Comparison of upscaling methods from microscopic to macroscopic scale.*

gas flow models

Method (3): the statistical sum method of the individual permeability from each shape type [49, 50].

Method (4): the 3D fractal model with variable pore sizes [51].

Method (5): the homogenization method to upscale gas flow through two distinct constituents, a mineral matrix and organic matter [52, 53].

Method (6): the pore network model including pore size distribution, anisotropy, and low connectivity of the pore structure, etc. in shale [54, 55].

The comparison among them is summarized in **Table 3**.

After reviewing the upscaling methods in **Table 3**, it is obvious that the method used in this work is not a bad compromise when compared to method (1) which is too simple and coarse, methods (2) and (3) where it is impractical and daunting to count the size/shape of every single pore with huge computational efforts, method (5) where complex processing for the model establishment and solution is needed, and methods (4) and (6) where redundant parameters/information about pore structure need to be assumed or obtained from multiple ways. Therefore, on the one hand, only the pore size distribution experiment is needed for the determination of the upscaling parameters in this chapter to make the consideration of various pore sizes happen. On the other hand, the derived model in this chapter is practical to operate, and the results can thus be readily obtained. However, it does not necessarily mean that there is no drawback for the upscaling method used. For example, although SEM images of the shale samples show that the pores in the organic matter are mostly circular [56], various types of pore shapes, e.g., cylindrical, triangular, rectangular shaped, etc., can be detected in shale samples [50, 57]. Singh et al. [50] concluded that the geometry of pores significantly influences apparent permeability of shale and diffusive flux. The study of effective liquid permeability in a shale system by Afsharpoor and Javadpour [58] confirmed that the assumption of simplified circular pore causes apparent permeability to be significantly overestimated and the discrepancy between the realistic multi-geometry pore model and the simplified circular pore model becomes more pronounced when pore sizes reduce and liquid slip on the inner pore wall is taken into account. Xu et al. [59] developed a model for gas transport in tapered noncircular nanopores of shale rocks and found the following: (1) pore proximity induces faster gas transport, and omitting pore proximity leads to the enlargement of the adsorbed gas-dominated region; (2) increasing taper ratio (ratio of inlet size to outlet size) and aspect ratio weakens real gas effect and lowers free gas transport; (3) moreover, it lowers the total transport capacity of the nanopore, and the tapered circular nanopore owns the greatest

transport capacity, followed by tapered square, elliptical, and rectangular nanopores. To conclude, there is still much room for improvement of the upscaling method in this work in multiple aspects in future research.
