Equations Of Fluid Motion In A Wide Range Of Knudsen Numbers

Due to massive exploitation of unconventional natural gas, we need to research fluidflows in tight porous media, e.g., shale, tight sandstone and coal. The initial problem tobe solved in this research is to determine the intrinsic permeability of tight porous media.This problem is attributed to the Klinkenberg efect which occurs in the situation of fluidflows in tight porous media. The Klinkenberg efect is in fact a phenomenon that fluid-s slip on the boundary of pores. This efect renders the permeability measured from theDarcy test is only an apparent permeability but not the intrinsic one. The reason for this isthe theoretical foundation of Darcy’s law, the Navier-Stokes equations, is no longer validwhen the Knudsen number is high. To find new governing equations that can replace theNavier-stokes equations and describe the high-Knudsen-number flows, we investigate thehigher-order approximation of the Boltzmann equation, like the Burnett equations, supperBurnett equations and regularized13moment equations. However, these equations aretoo complicated to solve our problems. We then learn a simpler set of governing equa-tions, i.e., volume difusion hydrodynamics that is newly proposed and seems to have thepotential of describing the flow in a wide range of Knudsen numbers. However, it is alsoimperfect. We find the limit of it is ignoring the micro-boundary efect of micro-fluids.Thus, we introduce a concept of efective transport coefcients, and derive a new set ofgoverning equations for fluid flows. To be distinguished from the volume difusion hydro-dynamics, the newly derived governing equations are named efective volume difusionhydrodynamics. We validate the present theory by solving the gas flows in micro-tubes.After compering the present theory to this problem with other solutions and experimentaldata, we find the present solution is suitable in the entire flow regimes. Then we em-ploy this theory to solve the problem of gas flows in micro-channels. By combining animproved general slip boundary condition with the efective volume difusion hydrody-namics, we obtain an analytical solution to this problem. Compared with other solutionsand experimental data, the present solution shows significant improvements that it pre-cisely predicts the mass flow rate up to the Knudsen number of50. We also find why thetraditional solutions cannot pass the Knudsen number1. The reason is they are indeedmade up of efective parts and inefective parts which will become significant when theKnudsen number increase above1. Because the present theory discards the inefective parts, it improves the prediction ability of its solutions. This is also the intrinsic meaningof the present theory’s name: efective volume difusion hydrodynamics. Finally, to solveour initial problem, i.e., determining the intrinsic permeability of tight porous media, wederive a new permeability correction for tight porous media by using the efective volumedifusion hydrodynamics. The present correction is simple and beautiful, having only oneparameter which we reasonably fix as a constant to make this correction simpler. Aftercomparing the present correction with other corrections and experimental data, we findthe present theory is valid in the entire flow regime. Thereby, in conclusion, we think theefective volume difusion hydrodynamics is suitable for describing fluid flows in a widerange of Knudsen numbers.