Gas Rarefaction Research Articles

3D Multi-scale Reconstruction of Fractured Shale and Influence of Fracture Morphology on Shale Gas Flow

Quantitative characterization of shale gas flow in fractured shale is a key problem in shale gas extraction. In this study, 3D multi-scale structures in fractured shale were reconstructed, firstly, by X-ray micro-computerized tomography (CT), high-resolution scanning electron microscope (SEM), and fractal function. Then, a REV (representative elementary volume)-scale lattice Boltzmann (LB) model, considering Klinkenberg’s effect and gas absorption, was built, and the effects of fracture complexity and gas pressure on shale gas flow and shale permeability were analyzed and studied quantitatively. The simulation results indicate that the gas flow behaviors in the fractured shale are related strongly to fracture morphology and gas pressure. The decreased fracture fractal dimension or increased complexity of the fracture network leads to increase in permeability of the fractured shale. The gas velocity in the shale matrix decreases with increasing fracture roughness. Increased fracture network connectivity contributes to the formation of local high-intensity velocity fields and the improvement of gas flow in the shale matrix. The gas rarefaction effect has a significant influence on gas flow in the fractured shale, and permeability of the fractured shale must be regarded as a dynamic shale reservoir parameter. The gas flow behavior in the fractured shale is more sensitive at low-pressure condition whereas the gas rarefaction effect is more sensitive to the fractured shale with highly rough fracture or low fracture connectivity.

Oscillatory pressure-driven rarefied binary gas mixture flow between parallel plates.

The rarefied, oscillatory, pressure-driven binary gas mixture flow between parallel plates is computationally investigated in terms of the mixture molar fraction and molecular mass ratio of the species, in a wide range of gas rarefaction and oscillation frequency. Modeling is based on the McCormack kinetic model. The output quantities are in dimensionless form and include the flow rate, wall shear stress and pumping power of the mixture, as well as the velocity and shear stress distributions and flow rates of the species. The presented results are for He-Xe and Ne-Ar. The heavier species are affected more drastically than the lighter ones from the inertial forces, resulting to large differences between the flow rate amplitudes of the species, which are increased as the flow becomes less rarefied, provided that the oscillation frequency is adequately high. At very high frequencies the ratio of the flow rate amplitudes of the light over the heavy species tends to the inverse of their molecular mass ratio in the whole range of gas rarefaction. The velocity overshooting effect becomes more pronounced as the molecular mass is increased. The mixture flow rate amplitude is larger, while its phase angle is smaller, than the corresponding ones of single gas, and they both vary nonmonotonically with the molar fraction. The effect of the mixture composition on the wall shear stress and pumping power is small. The present work may be useful in the design of gas separation devices, operating at moderate and high frequencies in rarefied and dense atmospheres.

Pressure and temperature driven fully-developed rarefied gas flow in a channel with uniform injection/suction through its permeable walls

The pressure and temperature driven fully-developed rarefied gas flow between two parallel permeable plates, with uniform gas injection and suction from the bottom and top plates respectively, is investigated, based on the linearized Shakhov (S) model and the linearized Boltzmann equation (BE). Both flow configurations are characterized by the gas rarefaction parameter and the injection/suction velocity magnitude. Computational results for the so-called kinetic coefficients, as well as for the macroscopic quantities are provided in a wide range of the two parameters. The Onsager-Casimir reciprocity relation between the mechanocaloric and thermal creep coefficients has been proven to hold for arbitrary values of the injection velocity and has been used to validate the accuracy of the obtained results. It is shown that, the kinetic coefficients and macroscopic quantities are significantly affected as the injection velocity is increased. In addition, the kinetic results properly recover the corresponding analytical solutions in the free molecular and slip regimes. The examined prototype rarefied injection/suction flows may be useful to the investigation of more complex and realistic flow configurations in channels of various cross-sections with permeable walls where the injection velocity is not constant in the whole flow domain.

Effect of temperature on the nonlinear dynamic behavior of two-lobe non-circular gas-lubricated micro-bearings

In this paper, the molecular gas lubrication model was used to analyze the nonlinear dynamic behavior of two-lobe non-circular gas-lubricated micro-bearings. The effects of temperature rise are taken into account. At high temperatures, in addition to gas rarefaction, its viscosity and friction will also change, and slip across boundaries will occur. The rarefaction of the lubricating gas film caused by the microscale effect at high temperatures was considered. The effects of temperature and rotation speed (with and without rarefaction effect) on the dynamic behavior of the non-circular micro gas bearing were studied. The nonlinear equation governing the gas behavior is discretized using the finite-element method and then solved simultaneously with the dynamic equations of rotor motion using the fourth-order Runge–Kutta method. Center orbit diagrams, phase portraits, Poincare maps, power spectrum, and bifurcation diagrams are used to investigate the dynamic behavior of two-lobe non-circular gas-lubricated micro-bearings. Some results show that with increasing temperature, the rotor behavior changes from T-periodic to quasi-periodic. It was also observed that at high temperatures, with increasing rotational speed, the behavior of the system changes from T-periodic to quasi-periodic, but if the gas is rarefied, this change occurs at a slower speed.

Quantitative visualization and characteristics of gas flow in 3D pore-fracture system of tight rock based on Lattice Boltzmann simulation

The geometric complexity of the 3D pore-fracture system of tight rock at the pore-scale is a primary challenge for the prediction of gas flow. In this paper, the 3D pore-fracture systems of tight rock are reconstructed based on the representation unit volume (REV) of the intact tight sandstone and fractal fracture. A 3D regularized lattice Boltzmann model that considered the gas slippage effect is employed to simulate gas flow in the 3D pore-fracture systems. Effects of the micro-fracture morphology and gas rarefaction effect on gas flow and permeability of the pore-fracture systems are examined and investigated. The simulation results indicate that the gas flow behaviors in the pore-fracture system are strongly related to the above two factors. Gas flow behaviors in the pore-fracture structure is more sensitive at the low-pressure condition and gas rarefaction effect are more sensitive to the pore-fracture system with the highly rough fracture or the low fracture connectivity.

Oscillatory Couette flow of rarefied binary gas mixtures

The oscillatory Couette flow of binary gas mixtures is numerically investigated on the basis of the McCormack model. The dependence of the velocity and shear stress amplitudes and the penetration depth on the gas rarefaction and the oscillation parameters is studied numerically. Two typical mixtures of noble gases, i.e., a neon–argon (Ne–Ar) mixture with a molecular mass ratio less than 2 and a helium–xeon (He–Xe) mixture with a molecular mass ratio of about 32, are considered to explore the influences of the molecular mass ratio and molar concentration. It is found that the Ne–Ar mixture exhibits similar behavior with a single gas, while significant deviations can be observed between a single gas and the He–Xe mixture. Particularly when the gases are in the transitional and near-continuum regimes and the oscillation frequency is high, the amplitudes of velocity and shear stress for the He–Xe mixture vary non-monotonically between the plates as the molar concentration of the light species He exceeds 50% due to the oscillation superposition of the two species. These findings are helpful to design the structure of micro-electromechanical devices.

Heat transfer in binary polyatomic gas mixtures over the whole range of the gas rarefaction based on kinetic deterministic modeling

In the present work, the problem of heat conduction through binary mixtures of non-vibrating polyatomic gases is studied over the whole range of the gas rarefaction in a deterministic manner. This is achieved by applying a recently proposed kinetic model, which takes into account the internal degrees of freedom of the gas molecules. The cross and self-collision frequencies are determined based on the thermal conductivity formulas proposed in the literature for the polyatomic gas mixtures. Numerical investigation is carried out for several binary mixtures consisting of linear and nonlinear gases in a wide range of all involved parameters. The validity of the present kinetic modeling is demonstrated by performing comparisons with the corresponding numerical and experimental data. Analytical solutions for the heat flux in the free molecular and hydrodynamic regimes are formulated showing excellent agreement with the numerical estimations. The effect of the internal degrees of freedom on the heat flux, as a function of the mole fraction, is also studied. The numerical results show that the relative deviation between monatomic and polyatomic heat fluxes varies linearly with the mole fraction when the difference between the molecular masses of the species is small. However, in the case of a mixture being composed of gases with quite different molecular masses, a nonlinear behavior is observed. It is clearly shown that the heat flux problem through polyatomic gas mixture cannot be captured by monatomic modeling. Furthermore, an approximate formula for the heat flux over the whole range of the Knudsen number is examined.

Effect of Environmental Conditions on Quality Factors of MEMS Cantilever Beam Resonator in Gas Rarefaction

Effect of Environmental Conditions on Quality Factors of MEMS Cantilever Beam Resonator in Gas Rarefaction

Numerical Study of Nanoparticle Deposition in a Gaseous Microchannel under the Influence of Various Forces

Nanoparticle deposition in microchannel devices inducing contaminant clogging is a serious barrier to the application of micro-electro-mechanical systems (MEMS). For micro-scale gas flow fields with a high Knudsen number (Kn) in the microchannel, gas rarefaction and velocity slip cannot be ignored. Furthermore, the mechanism of nanoparticle transport and deposition in the microchannel is extremely complex. In this study, the compressible gas model and a second-order slip boundary condition have been applied to the Burnett equations to solve the flow field issue in a microchannel. Drag, Brownian, and thermophoretic forces are concerned in the motion equations of particles. A series of numerical simulations for various particle sizes, flow rates, and temperature gradients have been performed. Some important features such as reasons, efficiencies, and locations of particle deposition have been explored. The results indicate that the particle deposition efficiency varies more or less under the actions of forces such as Brownian force, thermophoretic force, and drag force. Nevertheless, different forces lead to different particle motions and deposition processes. Brownian or thermophoretic force causes particles to move closer to the wall or further away from it. The drag force influence of slip boundary conditions and gas rarefaction changes the particles’ residential time in the channel. In order to find a way to decrease particle deposition on the microchannel surface, the deposition locations of different sizes of particles have been analyzed in detail under the action of thermophoretic force.

Rarefied gas effect in hypersonic shear flows

Recently, as aerodynamics was applied to flying vehicles with very high speed and flying at high altitude, the numerical simulation based on the Navier–Stokes (NS) equations was found that cannot correctly predict certain aero-thermo-dynamic properties in a certain range of velocity and altitude while the Knudsen number indicates that the flow is still in the continuum regime. As first noted by Zhou and Zhang (Science in China, 2015), the invalidity of NS equations for such flows might be attributed to an non-equilibrium effect originating from the combined effects of gas rarefaction and strong shear in the boundary-layer flows. In this paper, we present the scope, physical concept, mathematical model of this shear non-equilibrium effect in hypersonic flows, as well as the way of considering this effect in conventional computational fluid mechanics (CFD) for engineering applications. Several hypersonic flows over sharp bodies and blunt bodies are analyzed by the proposed new continuum model, named direct simulation Monte Carlo (DSMC) data-improved Navier–Stokes (DiNS) model. Recently, as aerodynamics was applied to flying vehicles with very high speed and flying at high altitude, the numerical simulation based on the Navier–Stokes (NS) equations was found that cannot correctly predict certain aero-thermo-dynamic properties in a certain range of velocity and altitude while the Knudsen number indicates that the flow is still in the continuum regime. As first noted by Zhou and Zhang (Science in China, 2015), the invalidity of NS equations for such flows might be attributed to an non-equilibrium effect originating from the combined effects of gas rarefaction and strong shear in the boundary-layer flows. In this paper, we present the scope, physical concept, mathematical model of this shear non-equilibrium effect in hypersonic flows, as well as the way of considering this effect in conventional computational fluid mechanics (CFD) for engineering applications. Several hypersonic flows over sharp bodies and blunt bodies are analyzed by the proposed new continuum model, named direct simulation Monte Carlo (DSMC) data-improved Navier–Stokes (DiNS) model..

Production Model of a Fractured Horizontal Well in Shale Gas Reservoirs

We present a forecasting model for shale gas production based on different gas flow mechanisms and discrete fracture models. The production forecasting model takes into account the gas rarefaction ...

Kinetic modelling of evaporation and condensation phenomena around a spherical droplet

The steady evaporation and condensation phenomena around a spherical droplet of the condensed phase of a vapor are investigated with basis on a kinetic model to the linearized Boltzmann equation. The kinetic equation is solved via a discrete velocity method which takes into account the discontinuity of the distribution function of molecular velocities on the spherical interface. The calculations are carried out in a wide range of the gas rarefaction and evaporation-condensation coefficient. The results obtained via solution of the linearized Navier-Stokes equations with temperature and pressure jump boundary conditions are also presented and compared to those obtained via kinetic equation. A comparison between the linearized and non-linear solutions of the kinetic model is also presented to show the limit of applicability of the linearized approach.

Shock-wave thickness influence to the light diffraction on a plane shock wave

This study is devoted to the numerical analysis of the result of light distribution after passing it through a shock wave, depending on the degree of gas rarefaction. The obtained numerical results allowed reproducing the experimental shadowgraph images obtained in our study. The range of shock wave thickness (from 0 mm to 20 mm) allowed considering the qualitative change in the light distribution on the screen during switching from the regime where the wave nature of light has the greatest influence on the distribution of light to the regime of the geometric optics approach. As a result, the criteria for the applicability of the shadowgraphy technique for the experimental description of the shock wave internal structure were obtained.

Two approaches to calculating composition of rarefied gas mixture exposed to chemical reactions at flow through cylindrical channel

Two approaches to calculating the change in the composition of a rarefied gas mixture due to gas-phase and heterogeneous reactions when it flows through a heated cylindrical channel into a vacuum are considered. The first approach is based on the direct simulation Monte Carlo (DSMC) method taking into account gas-phase reactions. The second one consists in combining the DSMC method without gas-phase reactions and the solving of chemical kinetics equations for taking into account gas-phase reactions. An analysis of the approaches is carried out on the example of the hydrogen and methane mixture flow through a heated tungsten channel. The study was conducted for different ratios of the channel length to its diameter, different coefficients of accommodation, and in a wide range of gas rarefaction degrees: from free molecular to transitional regimes. The developed methods provide a possibility of solving similar problems for non-equilibrium flows. The obtained results may be useful for optimization of gas-dynamic sources, based on thin tubes.

Shale gas transport mechanisms in inorganic and organic pores based on lattice Boltzmann simulation

A better understanding on the mechanism and difference of shale gas transport through inorganic and organic pores is critical to accurately predicting shale gas recovery. Due to the complicated gas flow dynamics in such narrow pores, traditional experimental methods or simulation methods is difficult to accurately describe these processes. Here, a regularized lattice Boltzmann (LB) model coupled with the slippage boundary condition and gas adsorption effect is employed to simulate shale gas transport in the inorganic and organic pore. The adsorption parameters of shale gas in the LB model are obtained from the molecular dynamics (MD) simulations. Effects of the adsorption effect, pore size, pore wall roughness and gas rarefaction effect on the shale gas transport are studied. The simulation results show that the shale gas transport mechanism heavily depends on the pore size. Basically, in the nano-scale pore, the gas slippage and Knudsen flow are the dominated transport mechanism, but the gas transport is dominated by the surface diffusion in the organic pore width with less than 5 nm owing to the gas adsorption effect. The surface roughness effect plays an important role in the gas transport, and this effect significantly increases with the decreasing of the Knudsen number. At the same time, the gas adsorption effect can also cause a significant increase of the friction factor and a slight decrease of the mass flow rate, which indicates that the gas adsorption effect must be considered in the organic roughness pores.

Shale gas permeability upscaling from the pore-scale

The effective permeability of large shale samples provides useful insights into shale gas production. However, its determination can only be achieved through upscaling, since the direct pore-scale simulation of gas flows in large rock samples is not feasible due to the high computational cost and absence of pore connectivity in sample images. Although the Brinkman formulation is widely used in the permeability upscaling of conventional rocks, how to choose the effective viscosity in this coarse-scale model is not clear. Moreover, its application in shale rocks, where the rarefaction effects are important so that the conventional Navier–Stokes equations are inadequate, is rare, and its accuracy has not been assessed. This study aims to address the above two problems, by comparing the Brinkman solutions of several two-dimensional and three-dimensional random porous media containing fractures with the fine-scale solutions of the Stokes and Boltzmann equations (for continuum and rarefied gas flows, respectively). It is found that the use of the fluid viscosity in the Brinkman model, instead of the controversial effective viscosity, leads to accurate results for the cases considered. Additionally, the macroscopic quantity in rarefied gas flows in shale rocks is well predicted by the Brinkman model for a wide range of gas rarefaction, since the error is found to be less than 7%, while neglecting the rarefaction effects leads to significant underestimation of effective permeability (up to 90% in the cases studied). Although heterogeneity and anisotropy of the porous medium increase the error of the effective permeability derived from the Brinkman model, generally speaking, the effective permeability extracted from this coarse-scale model compares favorably to its fine-scale counterpart.

Rarefied gas flow into vacuum through a channel with sudden contraction or expansion

Based on the direct simulation Monte Carlo method, we have simulated a flow of rarefied gas into vacuum through a channel with sudden contraction or expansion in cross section. The nonequilibrium effects at the input and output of the channel have been taken into account by including certain pre- and post-channel regions into the geometry under consideration. In a wide range of gas rarefaction, the mass flow rate through the channel and the flow field both inside the channel and in the upstream and downstream regions have been calculated. The results have been compared with the characteristics of a channel with the constant cross section. A significant effect of the channel geometry has been established. The Knudsen minimum has been found in the mass flow rate as a function of gas rarefaction, while a compression zone and a circulation zone have been discovered in the flow field.

The effect of a solid boundary on the propagation of thermodynamic disturbances in a rarefied gas

We study the effect of a rigid boundary on the propagation of thermodynamic disturbances in a gas under non-continuum conditions. We consider a semi-infinite setup confined by an infinite planar wall and introduce initial gas disturbances in the form of density and temperature inhomogeneities. The problem is formulated for arbitrary small-amplitude perturbations and analyzed in the entire range of gas rarefaction rates, governed by the Knudsen (Kn) number. Our results describe the system relaxation to equilibrium, with specific emphasis on the effect of the solid surface. Analytical solutions are obtained in the free-molecular and near-continuum (based on the Navier–Stokes–Fourier and regularized 13 moment equations) regimes and compared with direct simulation Monte Carlo results. The impact of the solid wall is highlighted by comparing between diffuse (adiabatic or isothermal) and specular boundary reflections. Focusing on a case of an initial temperature disturbance, the results indicate that the system relaxation time shortens with increasing Kn. The isothermal boundary consistently reverberates the weakest acoustic disturbance, as the energy carried by the impinging wave is partially absorbed by the surface. The specular and adiabatic wall systems exhibit identical responses in the continuum limit while departing with increasing Kn due to higher-order moment effects. The unsteady normal force exerted by the gas on the surface is quantified and analyzed.

A Radial Two-Sector Gas Bearing with Maximum Load Capacity

A variational problem for a radial gas journal bearing with two sectors is considered. The pressure field in the gas layer can be described by the nonlinear Reynolds equation for arbitrary compressibility numbers. The magnitude of the resultant pressure force vector is used as a functional of the variational problem. The system of necessary conditions of the extremum is qualitatively analyzed and, as a result, a computational procedure is constructed. Based on the analysis of the complete system of necessary conditions, it is shown that the profile on each of the sectors is piecewise linear and single-stage; at the same time, one sector provides only gas rarefaction, while the other provides only gas compression.

Thermally driven pumps and diodes in multistage assemblies consisting of microchannels with converging, diverging and uniform rectangular cross sections

Thermal transpiration pumping in multistage assemblies is computationally investigated. Each stage is formed by combining in series-long microchannels with (a) uniform–uniform (“uni–uni”), (b) converging–uniform (“con–uni”), (c) diverging–uniform (“div–uni”) and (d) converging–diverging (“con–div”) rectangular cross sections. In all four investigated assemblies the generated pressure difference with the associated mass flow rate is fully assessed, in terms of inlet pressure, inclination parameter and number of stages. The analysis is based on linear kinetic modeling and is valid in the whole range of gas rarefaction. It is concluded that the “con–uni” and “div–uni” assemblies provide higher pressure differences and lower mass flow rates than the “uni–uni” assembly and they may be more suitable for specific pumping applications. The characteristics of the “con–uni” and “div–uni” assemblies are very close to each other. The multistage “uni–uni”, “con–uni” and “div–uni” assemblies are more stable when operating at small inlet pressures, where the pressure difference remains almost constant in wide ranges of mass flow rate. It is advisable to add as many stages as possible to increase, depending on the application, either the pressure difference or mass flow rate or both. Furthermore, the “con–div” assembly provides smaller pressure differences and mass flow rates than the other three, but it is suitable for diode applications. It is characterized by the so-called blocking inlet pressure, where the deduced pressure difference in the converging and diverging channels is the same. A detailed parametrization of the blocking inlet pressure in terms of inclination ratio, mean height, temperature difference and gas species, has been performed. It is concluded that multistage “con–div” assemblies may be ideally applied as thermally driven microfluidic diodes to control or block the flow, as well as to separate the species in multicomponent gas mixtures.