**3. 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:

$$N\_D = -\frac{10^{-36}\pi\rho\_{avg}}{8\mu} \left( r\_{in} - \frac{pd\_m}{p\_L + p} \right)^4 \frac{\mathrm{d}p}{\mathrm{d}l} \tag{1}$$

$$N\_b = N\_F = -\frac{10^{-9} M k\_B}{3 R \mu d\_m} \left( r\_{in} - \frac{p d\_m}{p\_L + p} \right)^2 \frac{\mathrm{d}p}{\mathrm{d}l} \tag{2}$$

$$N\_K = -\frac{2 \times 10^{-27}}{3} \left(\frac{8\pi M}{RT}\right)^{0.5} \left(r\_{in} - \frac{pd\_m}{p\_L + p}\right)^3 \frac{\mathrm{d}p}{\mathrm{d}l} \tag{3}$$

$$N\_t = -0.016 \times 10^{-22} \times \exp\left(-\frac{0.45q}{RT}\right) \frac{\rho\_s M}{pV\_{std}} \frac{q\_L p}{p\_L + p} \cdot \frac{1 - \phi\_{co}}{\phi\_{co}} \pi r\_{in}^2 \frac{\text{d}p}{\text{d}l} \tag{4}$$

where *ND* = viscous mass flow in a pipe, kg�s �1 .

*Nb* = mass flow of bulk diffusion, kg�s �1 .

*NF* = mass flow of Fick diffusion, kg�s �1

. *NK* = mass flow of Knudsen diffusion, kg�s �1 .


<sup>d</sup>*p*/d*<sup>l</sup>* = pressure gradient, Pa�m�<sup>1</sup> .


*<sup>ρ</sup><sup>s</sup>* = density of shale matrix, kg<sup>m</sup><sup>3</sup> .

*Vstd* = molar volume of gas under standard conditions, m<sup>3</sup> mol<sup>1</sup> .

*qL* = Langmuir volume, m<sup>3</sup> kg<sup>1</sup> .

*Φco* = porosity of a core sample, dimensionless.

The expression of Fick diffusion (2) is referred to as bulk diffusion and represented by *Nb*.

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%. The coupling coefficients of viscous flow, bulk diffusion, Knudsen diffusion, and surface diffusion are represented by *f*1(Kn), *f*2(Kn), *f*3(Kn), and *f*4(Kn) respectively, which are the functions of Kn. The expressions of the coupling coefficients are set according to the characteristics of flow regimes, where the following assumptions are used:


Based on the above narrations, it physically defines that *f*1(Kn) = e*<sup>α</sup>*Kn, *<sup>f</sup>*2(Kn) = 1/(1 + Kn) <sup>e</sup>*<sup>α</sup>*Kn, *<sup>f</sup>*3(Kn) = e*<sup>β</sup>*/Kn, and *<sup>f</sup>*4(Kn) = Kn/(1 + Kn) <sup>e</sup>*<sup>β</sup>*/Kn, where *α* and *β* are dimensionless constants determining the bump levels of the variation curves. *α* and *β* are set as 5 and 1.8 [42], respectively, to further realize the compliance of the coupling coefficients with the narrated flow regime characteristics.

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

Hence, the mass flow in nanopores can be expressed as:

$$N = e^{-5\text{Kn}}N\_D + \left(\frac{\text{1}}{\text{1} + \text{Kn}} - e^{-5\text{Kn}}\right)N\_b + e^{-1.8/\text{Kn}}N\_K + \left(\frac{\text{Kn}}{\text{1} + \text{Kn}} - e^{-1.8/\text{Kn}}\right)N\_s \tag{5}$$

where *N* = mass flow in a pipe, kg�s �1 .

The variation curves of the four coupling coefficients and *f*1(Kn) � *ND*, *f*2(Kn) � *Nb*, *f*3(Kn) � *NK*, and *f*4(Kn) � *Ns* 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:


#### **Figure 3.**

*Variation curves of the coupling coefficients (dimensionless) of viscous flow, bulk diffusion, Knudsen diffusion, and surface diffusion with Kn (dimensionless) [42].*

**Figure 4.**

*Variations of viscous flow and diffusion with Kn (dimensionless) after introducing coupling coefficients for the gas flow in pores of (a) 5 nm, (b) 10 nm, (c) 20 nm, and (d) 40 nm at 353 K. f1\*ND, f2\*Nb, f3\*NK, and f4\*Ns denote the results of viscous flow, bulk diffusion, Knudsen diffusion, and surface diffusion, respectively [42].*
