A generalized model for gas flow prediction in shale matrix with deduced coupling coefficients and its macroscopic form based on real shale pore size distribution experiments

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

“…where 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.…”

“…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: [42].…”

“…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.…”

“…Based on the above narrations, it physically defines that f 1 (Kn) = e ÀαKn , f 2 (Kn) = 1/(1 + Kn) À e ÀαKn , f 3 (Kn) = e Àβ/Kn , and f 4 (Kn) = Kn/(1 + Kn) À e Àβ/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.…”

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