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This work investigates and analyzes the performance of conventional slip models among various regimes of Knudsen number and developes a new multicoefficient slip-velocity model, by using Taguchi quality control techniques and numerical analysis. A modified Reynolds equation is also derived based on the new slip-flow model. The multicoefficient slip...
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... applying the Taguchi method, we need to set sev- eral monitor points so that to produce the calculated results. Here the three points are selected for the wider possible range of flow rate and defined as shown in Figure 2. In the first case (case 1), three monitoring points are set at Kn = 0.8, Kn = 8.0, and Kn = 80. ...
Citations
... Beskok and Karniadakis 38 also developed physics-based empirical higher-order BCs using asymptotic analysis, but the empirical parameters need to be determined either by experimental measurement, by solving the linear Boltzmann equation, or using DSMC data. Ng and Liu 44 presented a new form of multicoefficient high-order BCs based on the Taguchi quality control techniques. Xue and Fan 45 substituted directly the hyperbolic tangent function of the Knudsen number into the first-order dimensionless MS BCs to construct a new type of higher-order BCs containing only first-derivative terms, which give results close to the DSMC data at relatively high Knudsen numbers. ...
A newly heuristic form of second-order slip/jump boundary conditions (BCs) for the Navier–Stokes–Fourier (NSF) equations is proposed from the viewpoint of generalized hydrodynamic equations (GHE) to extend the capability of the NSF equations for moderately rarefied gas flows. The nonlinear Rayleigh–Onsager dissipation function appearing in the GHE, which contains useful information about the nonequilibrium flow fields of interest, is introduced into the proposed BCs named the simplified generalized hydrodynamic (SGH) BCs as a correction parameter. Compared with the classical Maxwell/Smoluchowski (MS) BCs, the SGH BCs may be more sensitive to capture the nonequilibrium information of flows adaptively and produce physically consistent solutions near the wall. Subsequently, the SGH BCs are implemented in the NSF equations for planar micro-Couette gas flows over a wide range of Knudsen numbers. The results indicate that the SGH BCs make impressive improvements against the MS BCs for diatomic and monatomic gases at the slip region and early transition regime, particularly in terms of capturing precisely the temperature and normal heat flux profiles in the flow and the temperature jump on the wall. More importantly, the SGH BCs conducted in NSF equations with less computational cost still can obtain well-pleased results comparable to the non-Newton–Fourier equations, such as several Burnett-type equations and regularized 13-moment equations, and even perform better than these models near the wall compared with direct simulation Monte Carlo data for the Couette flows to some extent.
... Considering both the slip flow and the Knudsen diffusion, Javadpour (2009) developed an apparent permeability model to examine the dynamic effect on gas transport in mudrocks. Based on the above two models, various apparent permeability models have been put forward to further investigate the dynamic effect of gas flow by considering different slip velocity models (Ng and Liu 2005), appropriate parameter correlation (Civan 2010), proper weight of different flow regimes (Chen et al. 2015), the shape of pore cross-section (Wu et al. 2015a), the anisotropic permeability (Huang et al. 2017) and so on. In general, dynamic effect can promote the gas flowability in and between micro/nanoscale pores. ...
Pore bulk modulus in continuum apparent permeability models was generally treated as an independent parameter or deduced from the Betti-Maxwell reciprocal theorem neglecting the sorption effect, which is not appropriate for the unconventional reservoirs with strong gas sorption. In this work, pore bulk modulus was firstly derived from the Betti-Maxwell theorem with the consideration of the gas sorption strain and was incorporated into the solid–fluid coupling apparent permeability model. Numerical simulation was then conducted under the uniaxial strain condition and the stress confined condition to investigate the gas transport characteristics. The results showed that when taking into account for the sorption effect in the Betti-Maxwell theorem, the pore bulk modulus dramatically decreased with the falling pore pressure during the gas transport process under two loading conditions. Moreover, gas transport ability under the stress confined condition was greatly degraded while it was hardly changed under the uniaxial strain condition. Both the pore bulk modulus and the stress distribution in the reservoirs were found responsible for the variation of gas flow capacity. Finally, the effects of the parameters associated with the pore bulk modulus were examined to further discuss the impact of involving the sorption effect in the Betti-Maxwell theorem.
... Flows in a microgap between two bodies under relative motion have an important application in the lubrication between the magnetic disk and the head slider in hard disk drives. [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] In Ref. 10, exploiting the slowly varying characteristic of the flow, a generalized Reynolds equation or the molecular gas-film lubrication (MGL) equation, which is valid for an arbitrary gap size, has been derived from the Boltzmann equation; a more systematic derivation is found in Ref. 23. Currently, the MGL equation and its variations are perceived as the basic equations of micro lubrication and have been widely applied to the design of hard disk drives. ...
... Substituting the expansions (20) and (21) into Eqs. (8), (19), and (22) and arranging the same order terms in ε, we obtain a series of boundary value problems that determinesf (0) ,f (1) , . . . from the lowest order as follows. ...
... Our final task is to determine the unknown C orp (0) , which is accomplished by the mass conservation. Integrating the BGKW equation (19) with respect to ζ and then with respect to x 2 from 0 to h(χ) and applying the diffuse reflection boundary conditions (8) and (22), we obtain the mass conservation law dM/dχ = 0. Here, M = ∫ h 0ρv1 dx 2 is the dimensionless mass flow rate through a cross section χ = const per unit depth. Substituting Eq. (49) into Eq. ...
Micro lubrication of a gas between two walls with an arbitrary temperature difference is studied on the basis of the Bhatnagar–Gross–Krook–Welander model of the Boltzmann equation. Applying the slowly varying approximation, the kinetic equation is studied analytically when the Knudsen number based on the gap size is large. The leading order approximation, which ought to be the solution of the nonlinear heat transfer problem, is replaced by its free molecular solution. Due to this crude approximation, a macroscopic lubrication model of Reynolds-type equation is derived in a closed form. For an assessment of the model, a direct numerical analysis of the kinetic equation is also conducted. The lift calculated using our model approximates that of the direct numerical analysis within the error of 4% uniformly in the range of the temperature ratio between 0.75 and 2 and the Knudsen number Kn between 0.1 and 10. A heating of the moving wall reduces the lift acting on the other wall when Kn is sufficiently large, whereas it is enhanced when Kn is sufficiently small.
... Another type of slip model is based on slip boundary conditions, including the first-and second-order modifications ( Table 1). The Maxwell slip model 13 is a first-order approximation from kinetic theory, and second-order [14][15][16][17][18] slip models have been successively proposed to improve calculation precision. In summary, slip velocity models can uniformly be expressed as 19 : Table 1 shows the different slip velocity models and the disadvantages when these models are applied in shale system. ...
Multiflow mechanisms coexist in shale gas reservoirs (SGRs) due to the abundant nanopores and the organic matter as a medium of gas souring and storage. The gas transport mechanisms in nanopores including bulk gas transfer and adsorption‐gas surface diffusion were already investigated in pore‐scale models, but their effects on actual gas production of multistage fractured horizontal wells in SGRs are not clearly understood, which are crucial for the economic development of unconventional resources. Therefore, a comprehensive apparent permeability (AP) model which couples the surface diffusion of adsorbed gas, slippage flow considering the additional flux generated by surface diffusion based on Langmuir's theory, and Knudsen diffusion is established. The presented model is validated with the experimental data and lattice Boltzmann method (LBM) simulation results. Then, we propose a numerical model which combines multiflow mechanisms in microscale pores and a multistage fractured horizontal well (MSFHW) in macroscale shale gas reservoirs together. The effects of different transport mechanisms on both AP of nanopores and gas production are analyzed thoroughly. The results show that the effect of surface diffusion on the apparent permeability of nanopores is much greater than that on the actual gas production of MSFHW, and the influence of high‐pressure condition must be considered when calculating the surface diffusion coefficient. The presented numerical model has important implications for accurate numerical simulation and efficient development of shale gas reservoirs.
... The discrepancies magnify and polarize in transition flow regime, where the models of Lockerby and Reese [124] and Karniadakis and Beskok [125] with negative second order slip coefficients produce a obvious downward trend, while other second order slip models show an upward tendency. During free molecular flow regime, the dimensionless flow rate of most second order slip models keep the upward trend with similar speed, except the model of Ng et al. [126], which mainly keeps the same during this flow regime. Among the second order slip models in Fig. 27, the models of Kim et al. [127] with C 1 = (6/p) 0.5 and C 2 = 4/p and Shamberg [128] with C 1 = 1 and C 2 = 5p/12 generate similar and two largest prediction results in free molecular flow regime. ...
... Among the second order slip models in Fig. 27, the models of Kim et al. [127] with C 1 = (6/p) 0.5 and C 2 = 4/p and Shamberg [128] with C 1 = 1 and C 2 = 5p/12 generate similar and two largest prediction results in free molecular flow regime. The model of Wu and Bogy [129] with C 1 = 2/3 and C 2 = 1/4 predicts the smallest dimensionless flow rate for Knudsen number 1 < Kn < 2 and Ng et al. [126] predicted the smallest for Kn> 2. The difference among these models may be caused by different assumptions during theoretical derivation or different manipulating procedures during experiments. Overall, it is not appropriate to take a negative value for second order slip coefficient, since the predicted dimensionless flow rates are below zero in high Knudsen number conditions. ...
... The slip models from the studies of Maxwell [94], Porodnov [132], Sreekanth [97] underestimates the modification function for Knudsen number larger than 0.2, while the second order slip model from Yamaguchi [101] fits the data well until Knudsen number reaches 0.52. Overall, the models of Maurer et al. [100], Kazemi et al. [7], Ng et al. [126], Gibeli model [133] can fit the experimental data on a relatively good performance, while Li et al. model [134] and Deissler model [96] severely overestimates the results at large Knudsen numbers. ...
... Noted that C 1 and C 2 is the first-and second-order slip coefficient, respectively. Ng et al. [63] through Taguchi quality control techniques and numerical analysis proposed a slip-corrected Reynolds equation that are suitable to a wide range of Knudsen numbers covering slip to transition flow regimes. Later, this model is proved to be more suitable for applications in real reservoir conditions compared with other modified models [64]. ...
... Maxwell [28] (2 − σ v )/σ v 0 Hsia et al. [30] (2 − σ v )/σ v −1/2 Hwang et al. [34] 0.01807 −0.676775(4/π) 0.58734 Kn −0.82532 Beskok et al. [31] 1/(1 − b Kn) 0 Aubert et al. [35] (2 − σ v )/σ v −9/8 Bahukudumbi et al. [36] [1.2977 + 0.71851tan −1 (−1.17488 Kn 0.58642 )] (2 − σ v )/σ v 0 Hadjiconstantinou [37] 1.1466 −0.647 Ng et al. [38] 1.15 −0.25 Kn −0. 65 Zhang HW et al. [32] [ ...
Nanopores are extremely developed and randomly distributed in shale gas reservoirs. Due to the rarefied conditions in shale strata, multiple gas transport mechanisms coexist and need further understanding. The commonly used slip models are mostly based on Maxwell slip boundary condition, which assumes elastic collisions between gas molecules and solid surfaces. However, gas molecules do not rebound from solid surfaces elastically, but rather are adsorbed on them and then re-emitted after some time lag. A Langmuir slip permeability model was established by introducing Langmuir slip BC. Knudsen diffusion of bulk phase gas and surface diffusion of adsorbed gas were also coupled into our nanopore transport model. Considering the effects of real gas, stress dependence, thermodynamic phase changes due to pore confinement, surface roughness, gas molecular volume, and pore enlargement due to gas desorption during depressurization, a unified gas transport model in organic shale nanopores was established, which was then upscaled by coupling effective porosity and tortuosity to describe practical SGR properties. The bulk phase transport model, single capillary model, and upscaled porous media model were validated by data from experimental data, lattice Boltzmann method or model comparisons. Based on the new gas transport model, the equivalent permeability of different flow mechanisms as well as the flux proportion of each mechanism to total flow rate was investigated in different pore radius and pressure conditions. The study in this paper revealed special gas transport characteristics in shale nonopores and provided a robust foundation for accurate simulation of shale gas production.
... Introduction of slip velocity on the boundaries as a function of Knudsen number is also used in generalized models. Multi-coefficient slip-corrected Reynolds equation in comparison with slip models up to the second order is presented in the work of Ng and Liu [21]. Cited model is claimed suitable at wide range of Knudsen number covering slip to transition regimes. ...
The motivation of the current paper is to provide a detailed review of crucial topics in micro- and nanoscale gas flows, i.e., velocity slip and temperature jump phenomena and their modeling, Knudsen minimum, and thermally-driven flows with application in Knudsen pumps. Non-equilibrium gas flows at the micro and nano scales exhibit peculiarities different from the intuition of macroscopic continuum fluid dynamics and heat transfer. Rarefied gas flows experience unique features such as slip and jump boundary effects, Knudsen paradox, pumping effect, , anti-Fourier heat transfer, thermal critical Knudsen number, and rarefied gas shock polars. This paper reviews the latest advancements and investigations on the gas–surface velocity and temperature inequalities, named velocity slip and temperature jump, in rarefied gas flows. Thermodynamic non-equilibrium effects are localized in the Knudsen layer within a width of a few gas mean free path near the surfaces. These particular features result in inequalities between the velocity and temperature of the gas concerning the surface. Experimental investigations of the slip and jump conditions from the first observation to the recent studies are reviewed. Next, a comprehensive review of the different categories of velocity slip models from the Maxwell model up to the recently developed models will be performed. For the first time, all the current models on the viscosity of rarefied gas are classified and compared in the present paper. Progress in the derivation of the models describing the temperature jump of rarefied gas over a solid surface will be performed. Next, a unique behavior of rarefied gas named the Knudsen paradox or Knudsen minimum is introduced and reviewed based on the different viewpoints. Different possible reasons for the origin of the Knudsen paradox phenomena are reviewed. A complete set of experimental and numerical works which detected the Knudsen paradox feature will be historically reviewed. Then, the closed-form relations obtained from the numerical simulations and experimental measurements indicating and predicting the occurrence of the Knudsen paradox phenomenon are reviewed and compared. The end section of this paper focuses on deemed physical and real-world applications of the rarefied flows at small scales utilizing thermally driven gas flows, i.e., the pumping effect of the rarefied gas. The pumping effect of the rarefied gas under temperature gradient leads to the Knudsen pump or Knudsen compressor concept. The main advantage over macro-scale pumps is that these sorts of pumps do not suffer from reduced life cycles due to various modes of mechanical failure because they do not have any rotating and moving parts. Moreover, the pumps mentioned above could find applications in gas chromatography, spectroscopy, moving gases on a chip, forced convection cooling, and micro-nano scale actuators. Different classes of thermally-induced flows with their origins are proposed and discussed, containing the recently detected thermally-driven flows. Based on that, a critical and comprehensive review of the different types and configurations of Knudsen pumps will be presented in detail. The newest published research containing new designs of Knudsen pumps has been considered in this review. Different numerical techniques developed or employed to simulate Knudsen pumps will be extensively reviewed. Besides, all published experimental investigations on Knudsen pumps will be reviewed and classified. Overall, this review will provide insights into the emerging physical science and engineering application of micro- and nanoscale gas flows.
Shale gas reservoirs (SGRs) have been exploited for commercial natural gas production for the past decade and has become a major source of energy in today’s world. Due to their rising importance, description and characterization of these types of reservoirs are inevitable, especially from the flow modeling point of view. Considering their unique features such as ultra-tight porous structure, depth, and large lateral extent, their petrophysical and reservoir properties are different and distinctive compared to conventional reservoirs. This difference enforces particular mechanisms of flow for the gas through these low-permeable beds, which has made scientists develop a variety of models to represent the fluid flow. In this article, a comprehensive review of existing flow models for SGRs is presented while their specific features including phase behavior of confined gas in tight pores, adsorption/desorption of the gas, flow mechanisms (slip flow, Knudsen diffusion, and surface diffusion), and stress sensitivity is covered. In addition, to declare the effect of porous structure of shale rocks on gas flow, pore network models (PNMs) as representatives of realistic shale porous structure are introduced and most outstanding studies in this area are reviewed as well. All models that have been reviewed here are the most recent literature, while its attempted that earlier ones that are still mainstream are not ignored. Also, the authors made comparisons against experimental data in most cases and implemented sensitivity analysis between models in order to make sure the audience get a better understanding of which one to use in their studies. Ultimately, it was the goal to make this review article a guidebook for future researches in SGR flow and pressure/rate transient analysis and modeling.
