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The non-equilibrium transport of inhomogeneous and dense gases highly confined by surface is encountered in many engineering applications. For example, in the shale gas production process, methane is extracted from ultra-tight pores under high pressure so the gas is inhomogeneous and dense. Currently, the complex non-equilibrium transport of inhomogeneous and dense gases where gas surface interactions play a key role is commonly investigated by molecular dynamics or on a continuum-assumption basis. Here, a tractable kinetic model based on the generalized Enskog equation and the mean-field theory is employed to couple the effects of the volume exclusion and the long-range intermolecular attraction forces. The interactions between gas molecules and confined surface are modelled by a 10-4-3 Lennard-Jones potential, which can capture gas surface adsorption. The cross-sectional density profiles of methane under different confinements are in good agreement with the molecular dynamics results reported in the literature, and the transport behaviors are validated by the non-equilibrium molecular dynamics. The velocity of methane flow in shale matrix is plug-like due to its dense characteristics in nanopores. The influence of pressure, temperature, pore size and shale composition on density and velocity profiles is analyzed quantitatively. Our results show that the Klinkenberg correction is not applicable to model shale gas flow in the production process; the Navier-Stokes model using the second-order slip boundary condition cannot produce the proper velocity profiles, and consequently fails to predict the accurate flow rate in nanopores. This study sheds new light on understanding the physics of non-equilibrium dense gas flows in shale strata.