A model for real gas flow in shale gas matrices is proposed and consists of two main steps: (a) developing a microscopic (single pore) model for a real gas flow by generalizing our previously reported Extended Navier-Stokes Equations (ENSE) method and (b) by using fractal theory concepts, up-scaling the single pore model to the macroscopic scale. A prominent feature of the up-scaled model is a predictor for the apparent permeability (AP). Both models are successfully validated with experimental data. The impact of the deviation of the gas behavior from ideality (real gas effect) on the gas transport mechanisms is investigated. The effect of the structural parameters (porosity Ф, the maximum pore diameter D_max, and the minimum pore diameter D_min) of the shale matrix on the apparent permeability is studied and a sensitivity analysis is performed to evaluate the significance of the parameters for gas transport. We find that (1) the real gas transport models for a single pore and porous shale matrix are both reliable and reasonable; (2) the real gas effect affects the thermodynamic parameters of the free gas and the adsorption and transport capacity of the adsorbed gas; (3) the real gas effect decreases the effective permeability for convective flow and surface diffusion; i.e., the derivation degree of the effective permeability for bulk diffusion and Knudsen diffusion increases with increasing pressure but presents a bathtub shape when the pore diameter is smaller than 10 nm; and (4) the apparent permeability increases with Ф, D_max, and D_min. It is more sensitive to D_max, followed by the porosity. D_min has a minor impact.

Natural gas flow in shale matrices consisting of both organic and inorganic components was modeled using fractal concepts. The expression of the apparent permeability (AP) of such shale systems was derived following three main steps: (a) modeling real gas flow in a single pore by generalizing our previously reported Extended Navier-Stokes Equations method, then (b) using fractal theory concepts to obtain the apparent permeability for both the organic and inorganic-matter cells, and finally (c) upscaling the AP model to the sample scale, accounting for the heterogeneous distribution of the organic matter. The up-scaled AP model is more realistic, because it considers not only the varying cross-section shapes and tortuosity of the pores, but also the characteristic flow mechanisms and multi-scale pore size distribution in heterogeneous shale matrix systems. The model was successfully validated with experimental data from real samples. The effects of the organic matter and shape of the pores on the AP are investigated. The sensitivity of the AP to the total organic carbon (TOC), porosity, pore shape, and structural parameters of both organic and inorganic systems is studied. The results indicated that the AP would be overestimated by up to 24.1% if the characteristics of the organic matrix are ignored, and the AP proves more sensitive to the effect of the pore shape in the inorganic matrix. In addition, the top three key parameters affecting the AP are pore shape, maximum pore diameter in the inorganic matrix, and porosity. The AP is more sensitive to the pore size within the inorganic matrix than within the organic matter. The proposed model provides some theoretical and technical support for shale gas simulations.

A model for gas flow in nano-pores was developed based on the extended Navier–Stokes equations with the assumption of neglecting adsorption and desorption. The model describes multiple flow regimes, including continuum regime, slip flow regime, transition regime, as well as molecular regime. The total mass flux includes a convective motion term and a diffusion mass transport term. The latter was obtained as a weighted superposition of bulk and Knudsen diffusion. The mass flux, contributed by different transport mechanisms, was analyzed by varying the Knudsen number (ratio between mean free path of the molecules and the pore diameter). The effect of the ratio coefficient (the power-law exponent in the relation between bulk and Knudsen diffusion), the pore size and the pressure on the gas transport was investigated using the proposed model. The predictions of the newly developed model are in good agreement with the Direct Simulation Monte Carlo (DSMC) method and with the results of the experiments. Our results show that: (1) As the Knudsen number increases, the contribution of viscous flow to gas transport decreases monotonically, bulk diffusion increases to a peak, and then decreases, whereas the Knudsen diffusion increases monotonically. (2) For a larger ratio coefficient, the bulk diffusion changes more rapidly and the peak is higher. The changing trend of the bulk diffusion is opposite to the Knudsen diffusion. (3) The pressure range in which the viscous flow dominates becomes larger and the peak of bulk diffusion is smaller in larger pores. When the pore pressure is higher, the viscous flow and the bulk diffusion tend to dominate.