Mechanism, Model, and Upscaling of the Gas Flow in Shale Matrix: Revisit

Shale gas accounts for an increasing proportion in the world ’ s oil and gas supply, with the properties of low carbon, clean production, and huge potential for the compensation for the gradually depleted conventional resources. Due to the ubiq-uitous nanopores in shale matrix, the nanoscale gas flow becomes one of the most vital themes that are directly related to the formulation of shale gas development schemes, including the optimization of hydraulic fracturing, horizontal well spac-ing, etc. With regard to the gas flow in shale matrix, no commonly accepted consensus has been reached about the flow mechanisms to be considered, the coupled flow model in nanopores, and the upscaling method for its macroscopic form. In this chapter, the propositions of wall-associated diffusion, a physically sound flow mechanism scheme, a new coupled flow model in nanopores, the upscaling form of the proposed model, and the translation of lab-scale results into field-scale ones aim to solve the aforementioned issues. It is expected that this work will contribute to a deeper understanding of the intrinsic relationship among various flow mechanisms and the extension of the flow model to full flow regimes and to upscaling shale matrix, thus establishing a unified model for better guiding shale gas development.


Introduction
Shale gas refers to a kind of self-generating and self-preserving natural gas, which gathers mainly in a free or adsorbed state in the organic-rich dark shale or high-carbon mud shale [1]. With vast reserves and the potential to offset the gradually depleted conventional resources worldwide and cut down carbon emissions at the same time, shale gas is playing an increasingly important role in ensuring global energy safety. Because shale matrix is characterized by various nanopores, where the gas flow is of high nonlinearity and complexity, an in-depth study of the mathematical model for the gas flow capacity in shale matrix is in urgent demand.
The mechanisms considered in different literature are listed in Table 1. It is obvious that opinions vary greatly on the flow mechanism scheme applied. The noteworthy aspects include the following: what the relationship among the various flow mechanisms of shale gas, e.g., slippage, Fick diffusion, Knudsen diffusion, etc., is; whether there is a repeated superposition of these mechanisms for specific flow calculation; and how to deal with the relationship among the various flow mechanisms, etc. There is no clear answer to these problems in current literature. Figure 1 shows the common research methodology of the flow models used in different literature. It indicates that because the method of the continuum model with a boundary condition based on the molecular one is considered inconsistent and the limitations and drawbacks of first-order, second-order, and 1.5-order slip models are described, some studies, which are listed in Figure 1, are inclined to add related flow mechanisms linearly. Furthermore, the mathematical models of viscous flow and various types of diffusion do not fully agree with common flow cognition as these theories and models were experimentally verified or developed for a limited range of conditions [27]. For this reason, coupling coefficients are introduced to rectify this kind of limitation, so as to enhance the correspondence between the flow model and Knudsen number (Kn). Finally, because the secondhand average method, e.g., assuming the pore space of shale to be composed of a certain number of isodiametric pores regardless of the pore size distribution, is widely used in the research of shale gas flow, more explicit means, like taking the existence of various pore sizes in shale into account, should be adopted for transforming the flow model in nanopores to that in macroscopic-scale shale matrix. Table 1. Different flow mechanism schemes in literature.
Based on the literature survey for shale gas flow in shale matrix, we know that the flow mechanism scheme with its corresponding coupling method is very crucial and has not yet been solved. In addition, although the integration method using specific functions has been proposed to facilitate the consideration of various pore sizes in shale matrix, real shale experiments are rarely involved to realize this point with definitely determined parameters.
Firstly, in this chapter, the concept of wall-associated diffusion is presented to clarify the relationship between slippage effect and several types of diffusion. Secondly, a physically sound flow mechanism scheme, which considers both division of mechanical mechanisms in nanopores and partition of flow space, has been proposed by virtue of the proposition of wall-associated diffusion. Thirdly, the coupling coefficients corresponding to the flow mechanisms considered are deduced to comply with the basic flow regime cognition, so as to establish a new coupled flow model in nanopores. Fourthly, the pore size distribution experiments for real shale samples from a gas field are utilized to realize the upscaling transformation of the flow model in nanopores into that in the macroscopic-scale shale matrix, with definitely determined fitting parameters for the establishment of a unified model for the gas flow prediction in shale matrix. Finally, a case study is presented to show how the lab-scale results are translated into field-scale ones.

Flow mechanisms in gas-shale matrix
There are many types of flow mechanisms in shale matrix, including slippage effect, Fick diffusion, transition diffusion, Knudsen diffusion, surface diffusion, etc. It can be seen from the literature survey in Section 1 that different flow mechanism schemes have formed aiming at establishing a calculation model to properly characterize the nanoscale shale gas flow. There may be views that the more flow mechanisms are taken into account, the more precise the established models are. However, this is not the opinion in this chapter.
As is known, Klinkenberg [33] first discovered in 1941 the phenomenon that, when measuring the gas permeability of rock, not only the measurement result is higher than the liquid measurement value but also it has strong pressure dependence and attributed it to the slippage behavior of gas in the rock pores. Specifically, gas slippage refers to the phenomenon that the near-wall gas molecules move relative to the wall surface when flowing through the medium channels [34]. In essence, the gas slip flow results from the interaction of gas molecules and pore walls, so the gas molecules in the vicinity of walls are in motion and contribute an additional flux, which is macroscopically characterized by the non-zero gas velocities on channel walls, thus resulting in slip flow [35,36]. The jump model assumes that the adsorbed gas molecules jump from one adsorption site to the adjacent adsorption site on the pore surface, which is considered to be suitable for the research on the surface diffusion of the adsorbed gas in shale nanopores [37]. Meanwhile, when the molecular mean free path is obviously larger than the pore diameter, the gas-wall collision dominates, and the collision between gas molecules is secondary, which is characterized by Knudsen diffusion [9,38,39].
In brief, both Knudsen diffusion and surface diffusion lead to non-zero moving speeds of the gas molecules around walls. Furthermore, from the viewpoint of microscopic motion mechanisms, they are both related to gas-solid interactions, which is consistent with slippage phenomenon in essence. Therefore, a new concept named "wall-associated diffusion" [40] is proposed, which characterizes the overall role of surface diffusion and Knudsen diffusion, as shown in Figure 2.
The proposition of wall-associated diffusion has practical significance and multiple research significance as follows [40].
To begin with, in terms of mechanical mechanisms, since wall-associated diffusion describes the diffusion mechanisms of shale gas related to gas-wall interactions, it bridges the relationship between slippage effect and several types of diffusion, which prevents reduplicated superposition of shale gas flow mechanisms in nanoscale pores. This is where the practical significance lies. Besides, wall-associated diffusion can be regarded as a detailed form of slippage effect, dividing slippage effect into two distinct parts, i.e., surface diffusion and Knudsen diffusion. The two parts differ obviously in their mechanical mechanisms and motion patterns. Accordingly, the research significance of wall-associated diffusion involves not only the function of morphological descriptions but also the possibility of slip phenomenon research by different mechanical mechanisms. Lastly, another research significance is that wallassociated diffusion breaks through the limitation that the concept of slippage does not apply for high Knudsen number, with, however, the fact that wall effects still contribute to gas flow for high Knudsen number. Therefore, in extremely small nanopores, for example, where slip flow regime is not applicable, the wall-associated diffusion derived from physical morphology can well be used to explore the so-called slip phenomenon in other flow regimes apart from slip flow regime.
By virtue of the concept of wall-associated diffusion, the flow mechanism scheme used in this work is to be discussed next.
There is no doubt that all the mechanisms, such as continuum flow, slip flow, Knudsen diffusion, bulk diffusion, etc., have been studied in previous literature for the exploration of shale gas flow. However, it is a determinative flow mechanism scheme that is vital. According to the literature survey, apart from combining the Navier-Stokes solution with slip boundary condition whose deficiency has been mentioned in Section 1, there is also a trend in literature to assume a combination of certain flow mechanisms and check the consistency of the model results with experimental data. This method is favorable from an engineering point of view but meanwhile leads to the status that coincidence often exists and no commonly accepted consensus has formed currently. In this work, we discuss the issue physically. Firstly, due to the multiple advantages of wall-associated diffusion over the concept of slippage effect, slippage effect is replaced with wall-associated diffusion in the following discussion. On the one hand, the flow space in nanopores can be divided into two parts: the bulk phase region and the Knudsen layer [41]. On the other hand, the microscopic mechanical mechanisms can be divided into the gasgas and gas-wall interactions. If a new comprehensive flow scheme, including viscous flow and bulk diffusion which belong to bulk phase flow and surface diffusion and Knudsen diffusion which are associated with gas-wall interactions causing non-zero flow velocities near pore walls, is proposed, the considerations of the division of flow space and mechanical mechanisms can be both realized. Relationship between wall-associated diffusion and slippage effect [40].
It should be noted that with the help of the methodology applied here, some flow mechanisms that are easily omitted are now included, such as bulk diffusion, an important diffusion process which is controlled by a mechanical mechanism obviously different from Knudsen diffusion. Furthermore, because the individual flow expressions, e.g., those for viscous flow and diffusion, were experimentally verified or developed for a limited range of conditions [27], the proposed physical flow mechanism scheme avoids unnecessary attempts to fit the mathematical models to experimental data so as to determine which flow mechanisms should be considered, laying a solid foundation for the research on the coupled flow model in nanopores discussed below.
To conclude, taking both division of mechanical mechanisms in nanopores and partition of flow space into account, viscous flow and bulk diffusion, which belong to bulk phase flow and result from gas-gas interactions, and surface diffusion and Knudsen diffusion, which are associated with gas-solid interactions and result in non-zero flow velocities near pore walls, are included in the proposed flow mechanism scheme.

Coupled model of shale gas flow in nanopores
Based on the flow scheme proposed in Section 2, the flow mechanisms considered include viscous flow, bulk diffusion, surface diffusion, and Knudsen diffusion. Considering the influence of adsorption layers, in which the system is assumed to reach dynamic adsorption equilibrium state instantaneously, the mass flow of the four mechanisms can be expressed, respectively, as: (4) where N D = viscous mass flow in a pipe, kgÁs À1 . N b = mass flow of bulk diffusion, kgÁs À1 . N F = mass flow of Fick diffusion, kgÁs À1 . N K = mass flow of Knudsen diffusion, kgÁs À1 . N s = mass flow of surface diffusion, kgÁs À1 . r in = inner radius of a pipe, nm. ρ avg = density of gas at average pressure of inlet and outlet, kgÁm À3 . μ = gas viscosity, PaÁs. d m = diameter of gas molecules, nm. p L = Langmuir pressure, Pa. dp/dl = pressure gradient, PaÁm À1 . M = molecular weight, kgÁmol À1 . R = universal gas constant, =8.314 JÁmol À1 ÁK À1 . k B = Boltzmann constant, =1.38 Â 10 À23 JÁK À1 . T = ambient temperature, K.  (2) is referred to as bulk diffusion and represented by N b .
The case study in literature [42] shows that although the equations of viscous flow and diffusion already contain variables varying with temperature, pressure, and other factors, they make sense within only a certain range of flow regimes and deviate from the actual situation within other range that is not taken into account. Introducing coupling coefficients to different flow mechanisms can help modify the correspondence between the mathematical models (i.e., those of viscous flow and diffusion) and Knudsen number and establish generalized models without segment processing as Kn varies.
In contrast to the coupling coefficients reported in published literatures [29,31,43,44], the derivation of new coupling coefficients corresponding to the proposed flow mechanism scheme is performed, and the coupling coefficient of one certain flow mechanism will not be optionally set as 100%. 3. It is transition flow when 10 À1 < Kn < 10, and several diffusion processes play roles at the same time ( [31,46]; thus, f 1 (Kn) is assumed to be negligible at the logarithmic median of this range ( [29,43]), i.e., f 1 (Kn) is close to 0 when Kn > 1.
Hence, the mass flow in nanopores can be expressed as: (5) where N = mass flow in a pipe, kgÁs À1 . The variation curves of the four coupling coefficients and f 1 (Kn) Â N D , f 2 (Kn) Â N b , f 3 (Kn) Â N K , and f 4 (Kn) Â N s with Kn are depicted in Figures 3 and 4 [42].
The benefits of introducing the above coupling coefficients to viscous flow and diffusion are significant: 1. It is clear that because f 1 (Kn), f 2 (Kn), f 3 (Kn), and f 4 (Kn) are all nonnegative, the segment processing of mathematical models can be avoided, i.e., Eq. (5) can be uniformly used for the coupling calculation in the scope of 0 < Kn < ∞, without the need to change the functional forms by reason of the limited applicability of coupling coefficients.
2. Eq. (5) bridges the gaps between different flow regimes, i.e., the jumps of flow rates at the critical points between different regimes have vanished. Furthermore, the mathematical models are further constrained by virtue of the molecular collision theory to better reflect the basic flow regime knowledge.
3. Taking the viewpoints of Refs. [30,32] as examples for comparison with this work, it should be noted that slip flow refers to the enhanced flow, including the part of original viscous flow and the other part called slippage effect which is represented by the non-zero velocities of the near-wall molecules due to gaswall interactions. Therefore, it is more suitable to regard the ratio of gas-gas collision frequency to total collision frequency as the total coupling coefficient of viscous flow and bulk diffusion rather than that of the slip flow [30,32].

4.
The same examples [30,32] are used for comparison. It is continuum flow when Kn approximates to 0. However, the coupling coefficient of slip flow is 1 when Kn = 0 in papers [30,32], implying slip flow dominates in continuum flow regime, which contradicts the flow regime knowledge. This issue has been solved in this chapter.

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][19][20][21][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: 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 r in to r in + dr in is expressed in Eq. (7). By integrating in the entire pore size range, the flow rate in shale is obtained as Eq. (8): r in, min 10 18 Nϕ co V co πr in 2 L co f r in ð Þdr in (8) where V co = apparent volume of a core sample, m 3 . L co = length of a core sample, m. r in,min = lower limit of integration, which should be larger than 0.19 nm because the diameter of methane molecules is 0.38 nm [48]. r in,max = upper limit of integration. The macroscopic-scale mathematical model of shale gas flow can be obtained by substituting Eqs. (5) and (6) (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][19][20][21][22] as already mentioned.
Method (2): the sum method of calculating the permeability of every straight capillary tube [27]. Make the consideration of various pore sizes happen; easy for calculation Negligence of pore structure, e.g., different pore shapes, pore connectivity, etc.
Total addition model [27] q ¼ P i q i (q, total flow, m 3 Ás À1 ; q i , flow rate of the ith single pipe, m 3 Ás À1 ) Consider the flow rate in every single pipe Impractical to implement; negligence of pore structure, e.g., different pore shapes, pore connectivity, etc.
Model of statistical sum of permeability from each shape type [49,50] k app,pm À Á eff ¼ k (k app,pm , apparent permeability modified for ultratight porous media, m 2 ; x/100, percentage of rectangular slits pores; (1Àx/100), percentage of cylindrical pores) Pore shapes, i.e., rectangular slits and cylindrical tubes, are taken into account The quantification of the percentages of different pore types using image analysis tools is hard to implement; negligence of various pore sizes 3D fractal model [51] Please refer to Eqs. (24)- (27) in literature [51] for the specific expressions where the formulas are complex Multi-scale pore size distribution and tortuous flow line in 3D space of shale matrix are characterized Many parameters to be determined; negligence of different pore shapes Homogenization model [52,53] The homogenization method is used to upscale gas flow through two distinct constituents, a mineral matrix and organic matter. A gas flow in a twoconstituent composite The constituents, i.e., mineral matrix and organic matter, in shale are taken into account Multiple assumptions; redundant processing for model establishment and solution Method (3): the statistical sum method of the individual permeability from each shape type [49,50].
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

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 gas flow models 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 Table 3.
Comparison of upscaling methods from microscopic to macroscopic scale. 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.

Translation of lab-scale results into field-scale ones
With the properties of multi-scale pore structures and various reservoir modes, the shale gas reservoir is complex in reservoir space and occurrence modes, which in turn leads to different flow mechanisms in multi-scale spaces. Therefore, adopting single-scale equations and flow simulation methods will not accurately reveal the flow mechanism in complex shale gas reservoirs [60]. Jiao et al. [61] established an effective conversion relation between physical simulation parameters and field parameters based on similarity criterion to better simulate gas reservoir development. The ideas in literature [61] are narrated as follows.
First, considering the flow mechanism of shale gas in the reservoir, the selected characteristic physical parameters are permeability K, porosity ϕ, pore radius r, length L, original pressure p i , flow rate of gas production q, gas compression factor Z, reservoir temperature T, standard temperature T sc , and standard atmospheric pressure p sc . According to the π theory, there are four basic dimensions named length dimension [L], mass dimension [M], time dimension [T], and temperature dimension [K]. Therefore, each of π is obtained, and field parameters are analyzed to deduce physical simulation parameters in the experiment according to the similarity criterion, as shown in Table 4.
Second, based on the similarity criterion, the conversion relation between physical simulation parameters and field parameters can be established, which is expressed as:  (10) where m indicates that the parameters inside the brackets are for the physical simulation.
Finally, choose the core sample "Ning 211-1" for an example to conduct dynamic physical experiment under different conditions, which is used to verify the rationality of the similarity criterion. The related parameters, values of physical simulation (q m ), converted values of field (q g ), and actual values of reservoir (q) are presented in Table 5. Figure 5 displays the curves of actual values of reservoir and predicted field results based on similarity conversion, the latter of which are calculated from the physical experiment. The results calculated by similarity criterion are basically consistent with the on-site tested data. It is expected that applying the similarity translation from physical simulation of gas reservoirs is capable of predicting the development performance effectively, showing the rationality of the translation method.

Conclusions
Based on our study in this chapter, the following conclusions have been reached: 1. A new concept "wall-associated diffusion" was introduced to the study of gas flow in shale nanopores, which has practical significance and multiple research  Table 5.
Parameters for application. significance. By virtue of this concept, viscous flow, bulk diffusion, surface diffusion, and Knudsen diffusion were considered in the proposed flow mechanism scheme for nanoscale shale gas flow, with both division of mechanical mechanisms in nanopores and partition of flow space taken into account. Viscous flow and bulk diffusion belong to the bulk phase flow, which result from gas-gas interactions. In addition, surface diffusion and Knudsen diffusion are of boundary layer flow, which are associated with gas-wall interactions.
2. An easy-to-operate coupling method of the flow mechanism scheme containing four coupling coefficients and thus a coupled shale gas flow model in nanopores, which applies within the scope of full flow regimes and avoids segment processing, was proposed.
3. Based on the experimental data of pore size distributions of real shale samples from a gas field, a new coupled upscaling flow model in macroscopic-scale shale matrix with the experimentally determined fitting parameters was established. The model uses smooth functions to fit the full-scale pore size distribution results to facilitate the upscaling transformation of the model in nanopores into that in the macroscopic matrix.

4.
A case study was presented to show how the lab-scale results are translated into field-scale ones, revealing the rationality of the translation method used.
In summary, sounder in theoretical bases and better in application effects, the proposed model is expected to be of practical significance for evaluating the gas flow capacity in shale matrix and guiding gas reservoir development in gas fields.