Apparent Gas Permeability Research Articles

GSIS: An efficient and accurate numerical method to obtain the apparent gas permeability of porous media

The apparent gas permeability (AGP) of a porous medium is an important parameter to predict production of unconventional gas. The Klinkenberg correlation, which states that the ratio of the AGP to the intrinsic permeability is approximately a linear function of reciprocal mean gas pressure, is one of the most popular estimations to quantify AGP. However, due to the difficulty in defining the characteristic flow length in complex porous media where the rarefied gas flow is multiscale, the slope in the Klinkenberg correlation varies significantly for different geometries such that a universal expression seems impossible. In this paper, by solving the gas kinetic equation using the general synthetic iterative scheme (GSIS), we compute the AGP in porous media that are represented by Sierpinski fractals and pore body/throat systems. With the abilities of fast convergence to steady-state solution and asymptotic preserving of Navier-Stokes limit, it is shown that GSIS is a promising tool to simulate low-speed rarefied gas flow through complex multiscale geometries. A new definition of the characteristic flow length is proposed as a function of porosity, tortuosity and intrinsic permeability of porous media, which enables to find a unique slope in the Klinkenberg correlation for all the considered geometries. This research also shows that the lattice Boltzmann method using simple wall scaling for the effective shear viscosity is not able to predict the AGP of porous media.

Apparent gas permeability behaviour in the near critical region for real gases

Gas apparent permeability is an important parameter for gas flow in the reservoir rocks. It is higher than the absolute permeability measured using liquid. The Klinkenberg equation corrects the liquid permeability to the apparent gas permeability allowing for the pressure dependent effects of interactions between gas molecules and pore walls. However, in this paper it is shown that Klinkenberg equation does not apply for gas flow in the near critical region for real gases. This has important implications for a number of reservoir management applications including carbon geosequestration and gas condensate reservoirs. In this paper, a series of permeability measurements with respect to pressure and temperature are presented on sandstone core samples using carbon dioxide (CO2), ethane (C2H6) and helium (He). The experimental results show that the apparent permeability is enhanced in the near-critical region. CO2 apparent permeability at 311 K (critical temperature of 304.25 K) increases about 2.24 times from 7.18 MPa to 8.6 MPa (critical pressure of 7.37 MPa) on the Sandstone Sample M1 and 1.83 times on the Sandstone Sample M2 for the same pressure change. C2H6 apparent permeability at 311 K (critical temperature of 305.35 K) increases 1.83 times from 4.6 MPa to 5.4 MPa (critical pressure of 4.9 MPa) on the Sandstone Sample M1 and 1.59 times on the Sandstone Sample M2 for the same pressure change. These results suggest that the permeability enhancement is stronger for the lower permeability sample. Moreover, the experimental results show that enhancement decreases as temperature increases. Modelling work shows that applying Klinkenberg equation is invalid for describing the apparent gas permeability in near critical region, even with attempts to use different equations to calculate the mean free path required in the Klinkenberg equation. A dual mechanism model, which simultaneously considers convection flow and molecular diffusion, is modified allowing the gas diffusivity to be a function of gas density and it can well describe the gas apparent permeability in the near critical region. It is suggested that the flow is in the viscous flow regime or weak slip flow regime for the near critical gas on the sandstone samples studied in this work, and the enhancement of permeability in the near critical region is attributed to the higher density (concentration) gradient with respect to pressure.

Study on Two Component Gas Transport in Nanopores for Enhanced Shale Gas Recovery by Using Carbon Dioxide Injection

Injecting carbon dioxide to enhance shale gas recovery (CO2-EGR) is a useful technique that has raised great research interests. Clear understanding of the two-component gas transport mechanisms in shale nanopores is the foundation for the efficient development of shale gas reservoir (SGR) and also the long-term geological storage of CO2. Although extensive studies on single-component gas transport and corresponding models in shale nanopores have been carried out in recent years, limited studies have been conducted on two-component or even multi-component gas transport models in shale nanopores. In this work, the shale nanopores were classified into inorganic and organic nanopores. The corresponding models for two-component gas transport were constructed. Mechanisms including Knudsen diffusion, slip flow, viscous flow, and molecular diffusion are considered in the inorganic pores. In the organic pores, due to existence of adsorption gas, surface diffusion is further considered besides the aforementioned mechanisms. Effects of pressure, temperature, fraction of organic nanopores, and gas concentration were analyzed. Results show that gas apparent permeability is negatively correlated with pressure, and positively correlated with temperature and organic nanopore fraction. As the concentration of CH4 decreases, the apparent permeability of CH4 increases continuously, while the apparent permeability of CO2 decreases. The permeability ratio of CH4 in the total permeability is negatively correlated with pressure and gas concentration ratio. Additionally, the contribution of transport mechanisms to the total gas apparent permeability has been analyzed. It is found that the surface diffusion contributes up to 5.68% to gas apparent permeability under high pressure. The contribution of molecular diffusion can reach up to 88.83% in mesopores under low pressure. Under high pressure and macropores, it contributes less than 1.41%. For all situations, the contribution of viscous flow is more than 46.36%, and its contribution can reach up to 86.07%. Results of this study not only can improve the understanding of two-component gas transport in nanochannels, but also can lay the foundation for more reliable reservoir simulation of CO2-EGR.

Viscous Dissipation and Apparent Permeability of Gas Flow in Nanoporous Media

AbstractThe nanopore confinement effect can increase the mobility for gas flow in shale rocks, and the apparent permeability is usually adopted to account for this effect. However, most of the apparent permeability calculation models are derived based on straight channels (capillaries) and neglect the influence of end effect. The end effect is caused by the bending of streamlines which yields more viscous dissipation and additional flow resistance for gas flow in nanoporous media. In this paper, for the first time the viscous dissipation for gas flow in nanoporous media is analyzed. In addition, an improved apparent permeability calculation model is proposed based on the Beskok‐Karniadakis (B‐K) model by introducing the influence of end effect, which is characterized by the ratio of the length to the width of a short channel (L/H). The accuracy of this model is validated by lattice Boltzmann simulations of gas flow in three different geometries: an infinite‐length straight channel, a finite‐length straight channel, and nanoporous media. For the infinite‐length straight channel, there is no end effect therefore both the B‐K model and the improved model can well predict the apparent permeability. However, for the finite‐length straight channel and nanoporous media, the original B‐K model overestimates the gas apparent permeability as it neglects the influence of end effect, while the improved model can still well predict the gas apparent permeability. In summary, the improved apparent permeability calculation model is more reasonable in predicting gas mobility in nanoporous media.

A FRACTAL PERMEABILITY MODEL FOR REAL GAS IN SHALE RESERVOIRS COUPLED WITH KNUDSEN DIFFUSION AND SURFACE DIFFUSION EFFECTS

A better comprehension of the behavior of shale gas transport in shale gas reservoirs will aid in predicting shale gas production rates. In this paper, an analytical apparent permeability expression for real gas is derived on the basis of the fractal theory and Fick’s law, with adequate consideration of the effects of Knudsen diffusion, surface diffusion and flexible pore shape. The gas apparent permeability model is found to be a function of microstructural parameters of shale reservoirs, gas property, Langmuir pressure, shale reservoir temperature and pressure. The results show that the apparent permeability increases with the increase of pore area fractal dimension and the maximum effective pore radius and decreases with an increase of the tortuosity fractal dimension; the effects of Knudsen diffusion and surface diffusion on the total apparent permeability cannot be ignored under high-temperature and low-pressure circumstances. These findings can contribute to a better understanding of the mechanism of gas transport in shale reservoirs.

Influences of test method and loading history on permeability of tight reservoir rocks

Steady-state and unsteady-state (downstream pressure build-up) gas permeability tests were conducted on low permeability siltstone at a series of upstream pressures during the loading and unloading processes. The characteristics of downstream pressure build-up curves are analysed in detail, and the permeability is calculated based on the data in the stabilization stage. Analysing approaches with and without consideration of the sample pore volume are used to obtain the unsteady-state permeability for both the real pressure and the pseudo pressure. Findings suggest that the permeability based on the pseudo pressure is generally lower than that based on the real pressure, with their ratio ranging from 0.75 to 0.98. The sample pore volume corrected permeability is 1.42–1.51 times of that without the consideration of sample pore volume. The apparent steady-state gas permeability is higher than the sample pore volume corrected permeability due to slip flow, while its intrinsic permeability is lower as the pore pressure is smaller in the steady-state test. The permeabilities decrease with the confining pressure in the loading path especially at lower confinements, while only part of the reductions is recovered during the unloading process. Increase of pore pressure enhances permeability under low confinement conditions. Water permeability is lower than gas permeability in steady-state test due to the water-rock interaction and the residual gas inside the sample.

Effect of gas adsorption-induced pore radius and effective stress on shale gas permeability in slip flow: New Insights

Abstract Shale, a heterogeneous and extremely complex gas reservoir, contains low porosity and ultra-Low permeability properties at different pore scales. Its flow behaviors are more complicated due to different forms of flow regimes under laboratory conditions. Flow regimes change with respect to pore scale variation resulting in change in gas permeability. This work presents new insights regarding the change of pore radius due to gas adsorption, effective stress and impact of both on shale gas permeability measurements in flow regimes. From this study, it was revealed that the value of Klinkenberg coefficient has been affected due to gas adsorption-induced pore radius thickness impacts and resulting change in gas permeability. The gas permeability measured from new proposed equation is provides better results as compare to existing equation. Adsorption parameters are the key factors that affect radius of shale pore. Both adsorption and effective stress have an effect on the pore radius and result gas permeability change. It was found that slip effect enhances the apparent gas permeability and also changes with effective stress; therefore, combine impact of slip flow and effective stress is very important as provides understanding in evolution of apparent permeability during shale gas production.

A new model for the transport of gaseous hydrocarbon in shale nanopores coupling real gas effect, adsorption, and multiphase pore fluid occupancies

A new gaseous hydrocarbon transport model, considering pore geometry, multiphase pore fluid occupancy, and porous deformation, has been developed to characterize the real gas flow in both nanoscale organic and inorganic porous media. To do so, Navier-Stokes equations were solved with modified second-order slip boundary conditions in nanocapillaries of organic matter (OM) and nanoslits of inorganic matrix (iOM). Due to gas adsorption, both surface diffusion and bulk gas flow affect the apparent gas permeability (AGP) of OM. Particularly, the Langmuir-slip model is used for the calculation of boundary velocity of bulk gas. In contrast, gas flow in a single nanoslit of iOM accounts for the impact of adsorbed water film with a certain thickness quantified by Li's model [15]. Additionally, porous deformation and real gas effect are included in both models of OM and iOM. As such, a unified AGP model of shale matrix is formulated based on the total organic carbon (TOC) content including contributions of both OM and iOM. Results show that, under depressurization condition, the AGP of OM/iOM with a large pore size distribution (PSD) presents a similar shape of “V”, while the AGP of iOM with a small PSD increases monotonously. The total flow capacity of OM is contributed by competitive mechanisms of surface diffusion, induced- and pure- bulk gas flow under various PSDs and pressure. For gas flow capacity in iOM with a small PSD, it can be weakened by adsorbed water film but compensated by nanoscale effect considerably at low pressures. Moreover, a higher relative humidity (RH) leads to a thicker water film resulting in a lower gas conductance. The AGP of shale matrix can be reduced or enhanced by the TOC content when accounting for gas adsorption and adsorbed water film. Besides, the real gas effect enhances flow capacity significantly in small pores with a high pressure. The proposed model provides a comprehensive understanding of the gas flow mechanism in shale nanopores under reservoir conditions.

Direct Simulation Monte Carlo investigation of fluid characteristics and gas transport in porous microchannels

The impetus of the current research is to use the direct simulation Monte Carlo (DSMC) algorithm to investigate fluid behaviour and gas transport in porous microchannels. Here, we demonstrate DSMC’s capability to simulate porous media up to 40% porosity. In this study, the porous geometry is generated by a random distribution of circular obstacles through the microchannel with no interpenetration between the obstacles. The influence of the morphology along with rarefaction and gas type on the apparent permeability is investigated. Moreover, the effects of porosity, solid particle’s diameter and specific surface area are considered. Our results demonstrate that although decreasing porosity intensifies tortuosity in the flow field, the tortuosity reduces at higher Knudsen numbers due to slip flow at solid boundaries. In addition, our study on two different gas species showed that the gas type affects slippage and apparent gas permeability. Finally, comparing different apparent permeability models showed that Beskok and Karniadakis model is valid only up to the early transition regime and at higher Knudsen numbers, the current data matches those models that take Knudsen diffusion into account as well.

On the flow regime model for fast estimation of tight sandstone gas apparent permeability in high-pressure reservoirs

ABSTRACT A flow regime-based gas apparent permeability model in high-pressure tight sandstone reservoirs is established by bridging molecular kinetics, gas transport mechanisms, and apparent permeability. This model is well validated against simulation and experimental data. Results indicate that an equal-flow point tends to move to a low-pressure region with an increase in a pore diameter. As an equal-flow line is close to an isoline of a Knudsen number of 1, a flow regime is divided into three Knudsen number-based regions. Both the equal-flow line and the isoline of a Knudsen number indicate that an increasing temperature expands a Knudsen diffusion-dominated region, which further enhances the gas transport permeability. A typical curve of flow-regime gas apparent permeability provides a fast estimation of gas apparent permeability based on a Knudsen number and a pore diameter. The minimum Knudsen number line constrained by a packing fraction in a typical curve also indicates a minimum gas apparent permeability when a pore diameter is less than 100 nm.

Investigation of Shale-Gas-Production Behavior: Evaluation of the Effects of Multiple Physics on the Matrix

Summary Shale gas is a major component of natural-gas supply in the United States. Multistage-fractured horizontal wells significantly improve the production performance of ultralow-permeability shale-gas reservoirs. Researchers have believed that shale-gas-production simulations should take into account the complex-flow behaviors in both fractures and the matrix. However, multiple physics applied on the matrix are generally incomplete in previous studies. In this study, we considered the comprehensive physics that occurred in the matrix including the effective stress, slip flow/pore diffusion, adsorption/desorption, and surface diffusion, as well as the dynamic properties of fractures. We investigated the importance of these features of the physics separately and in an integrated fashion by step-by-step production simulations. Afterward, comprehensive sensitivity analysis was performed with regard to stress dependency of the matrix and fractures. This work shows that natural-fracture spacing is the most prominent factor affecting shale-gas-reservoir performance. The work highlights the importance to gas recovery of mechanical squeezing of the pore volume by the effective stress. Surface diffusion might be essential for gas recovery that depends on surface-diffusivity values. Slip flow and pore diffusion do not significantly contribute to gas recovery even though they increase gas apparent permeability under low pressures.

Analysis of gas transport behavior in organic and inorganic nanopores based on a unified apparent gas permeability model

Different from the conventional gas reservoirs, gas transport in nanoporous shales is complicated due to multiple transport mechanisms and reservoir characteristics. In this work, we presented a unified apparent gas permeability model for real gas transport in organic and inorganic nanopores, considering real gas effect, organic matter (OM) porosity, Knudsen diffusion, surface diffusion, and stress dependence. Meanwhile, the effects of monolayer and multilayer adsorption on gas transport are included. Then, we validated the model by experimental results. The influences of pore radius, pore pressure, OM porosity, temperature, and stress dependence on gas transport behavior and their contributions to the total apparent gas permeability (AGP) were analyzed. The results show that the adsorption effect causes Kn(OM) > Kn(IM) when the pore pressure is larger than 1 MPa and the pore radius is less than 100 nm. The ratio of the AGP over the intrinsic permeability decreases with an increase in pore radius or pore pressure. For nanopores with a radius of less than 10 nm, the effects of the OM porosity, surface diffusion coefficient, and temperature on gas transport cannot be negligible. Moreover, the surface diffusion almost dominates in nanopores with a radius less than 2 nm under high OM porosity conditions. For the small-radius and low-pressure conditions, gas transport is governed by the Knudsen diffusion in nanopores. This study focuses on revealing gas transport behavior in nanoporous shales.

Erratum: Feng, X., et al. Gas Multiple Flow Mechanisms and Apparent Permeability Evaluation in Shale Reservoirs. Sustainability 2019, 11, 2114

The authors did not notice during the proofreading phase that we have accidentally deleted a related reference of Figure 1 [...]

Image-based core-scale real gas apparent permeability from pore-scale experimental data in shale reservoirs

This paper presents a new upscaling method to derive the core-scale apparent gas permeability from an improved pore-scale permeability model and experimental data, with more rigorous incorporation of varying gas storage/transport mechanisms in nano/micro pores. First, in use of SEM images of a gas-rich shale field example in Sichuan Basin from our lab, pore network models of inorganic-matter (IOM) and organic-matter (OM) are characterized by using a digital-core technique. Next, an improved pore-scale real gas apparent permeability is modeled rigorously for both IOM/OM, respectively, with 1) bulk gas transport, gas adsorption, surface diffusion, pore-size confined phase behavior, and stress-dependent rock properties and 2) an additional reduction in inorganic pore sizes by water film adhered on pore surfaces. Core-scale permeability is then derived by assembling the permeabilities of stochastically distributed IOM/OM patches with different pore network models properties using the Monte Carlo sampling method. The new core-scale permeability model is validated by pulse-decay permeability experiment. Moreover, the representative elementary volume (REV) size is determined by analyzing the relative standard deviation of apparent gas permeability in cases with different sample sizes. The contributions of different gas transport mechanisms are discussed, and the impacts of stress-dependence for several field examples (i.e., Sichuan, Pierre and Barnett Basins) and water film with varying relative humidity (RH) on core-scale apparent permeability are analyzed. This work provides an effective approach to determine the core-scale shale permeability by directly using pore-scale experimental data, which is a common challenge in the unconventional resources.

A dusty gas model-direct simulation Monte Carlo algorithm to simulate flow in micro-porous media

A new efficient direct simulation Monte Carlo (DSMC) method is proposed for the simulation of microporous media based on the dusty gas model (DGM). Instead of simulating gas flow through a microporous medium with a complex geometry of micropores that mimics the physical pore morphology, the DGM-DSMC method replaces it with the gas flow through a system of randomly distributed motionless virtual particles with simple spherical shapes confined in the considered domain. In addition, the interactions of gas molecules with the porous particles are simulated stochastically. For the aim of our study, the DGM is implemented in Bird’s two-dimensional DSMC code. The obtained results for the average velocity of gas flow through microscale porous media with given porosity are verified for different pressure gradients with those reported in the literature where porous particles are modeled physically in the domain. Thereafter, the effective parameters in porous media such as porosity, particle diameter, and rarefaction on flow behavior including velocity profile, apparent gas permeability, and mass flow rate are investigated. A comparison with the results predicted by the Open source Field Operation and Manipulation (OpenFOAM) software suggests that the employed DGM-DSMC is more accurate in highly porous media and its computational cost is considerably low.

Shale gas transport in wedged nanopores with water films

With the consideration of a mobile high-viscosity water film and a bulk water layer, an analytical model is proposed to predict gas flow behaviors and gas apparent permeability in nanopores. Physical molecular forces are analyzed to determine the properties and formation of the high-viscosity water film and the bulk water layer. Boundary conditions at a solid-liquid interface and a liquid-gas interface are modified to account for water slippage and gas slip. How mobile bulk water molecules influence gas transport capability in pores of different geometry is investigated with different pore sizes and pressure for the whole range of water saturation, which has been validated with experimental data and simulation results. Besides, the effect of a wedged pore structure on multiphase flow behaviors is also investigated to figure out the influences of pore shape.

Gas Multiple Flow Mechanisms and Apparent Permeability Evaluation in Shale Reservoirs

Gas flow mechanisms and apparent permeability are important factors for predicating gas production in shale reservoirs. In this study, an apparent permeability model for describing gas multiple flow mechanisms in nanopores is developed and incorporated into the COMSOL solver. In addition, a dynamic permeability equation is proposed to analyze the effects of matrix shrinkage and stress sensitivity. The results indicate that pore size enlargement increases gas seepage capacity of a shale reservoir. Compared to conventional reservoirs, the ratio of apparent permeability to Darcy permeability is higher by about 1–2 orders of magnitude in small pores (1–10 nm) and at low pressures (0–5 MPa) due to multiple flow mechanisms. Flow mechanisms mainly include surface diffusion, Knudsen diffusion, and skip flow. Its weight is affected by pore size, reservoir pressure, and temperature, especially pore size ranging from 1 nm to 5 nm and reservoir pressures below 5 MPa. The combined effects of matrix shrinkage and stress sensitivity induce nanopores closure. Therefore, permeability declines about 1 order of magnitude compare to initial apparent permeability. The results also show that permeability should be adjusted during gas production to ensure a better accuracy.

Gas permeability tests on core plugs from unconventional reservoir rocks under controlled stress: A comparison of different transient methods

Accurate and routinely applicable methods to determine porosities and permeability coefficients are needed in order to ensure effective hydrocarbon recovery in shale and tight sandstone plays. In this study 129 gas uptake measurements (“GRI method”, “inflow” experiments) were performed on core plugs from three unconventional reservoir lithotypes (oil shales, gas shales and tight gas sandstones) under elevated effective stress conditions. The results were compared to those from “flow-through” tests (standard pulse decay) under similar experimental conditions, e.g. the same gas type and pore pressure range. The samples covered a porosity range from 1.3% to 12%. Equilibration times ranged from 102 s to 104 s and permeability coefficients from 10−18 to 10−21 m2. In order to successfully determine apparent gas permeability coefficients and porosities and to reliably interpret fluid dynamic effects from gas uptake data it is necessary to ensure a sufficiently high excess pressure drop during the uptake tests. This can be controlled by adjustment of the reservoir to pore volume ratio and initial differential pressure. Permeability coefficients derived from uptake tests on all six samples do not show any systematic deviations from those obtained from flow-through measurements. Best results were achieved for a core plug from the Lower Palaeozoic Alum Shale (Djupvik, Öland, Sweden), where Klinkenberg regressions of inflow and flow-through differ only by 4% (slope) and 10% (y-axis intercept). Here, the gas storage capacity ratio was ≈2.5 and excess pressure drops ranged from 0.1 to 1.2 MPa. Generally, measurement errors were lowest when excess pressure drops during uptake were at least 0.05 MPa for all samples. Excess pressure drops of less than 0.05 MPa resulted in coefficients of variance of single apparent gas permeabilities of ≈10%–100%, whereas excess pressure drops of >0.05 MPa resulted in coefficients of variance of ≈2%–15%. We show that it is possible to adjust the initial conditions of inflow measurements such, that Klinkenberg-corrected permeability coefficients and gas slippage factors can be readily determined. This will contribute significantly to a better understanding of the effects of anisotropy and/or the pore structure on the transport properties of unconventional lithotypes under elevated stress conditions. However, standard deviations of single apparent permeability coefficients derived from inflow experiments are higher by up to one order of magnitude than for those obtained from flow-through tests under all tested conditions.

Estimation of Shale Apparent Permeability for Multimechanistic, Multicomponent Gas Production Using Rate Transient Analysis

Gas-producing shale and ultratight reservoirs are playing a key role in the energy industry and the global gas market. Compositional simulation of gas production from shale media in the presence of different mechanisms such as viscous flow, slip flow (Klinkenberg effect), Knudsen diffusion, sorption, pore radius variation, and real gas effect is a computational challenge. In this work, we present a model that takes into account all of the noted mechanisms of gas transport in shale media. It is shown that the compositional effect of gas in shale media can be lumped into a single component by introducing an apparent gas permeability, which can be estimated from the conventional rate transient analysis. The main contribution of this study is a workflow incorporating the relevant physics into a single term (apparent permeability) that will substitute the Darcy permeability. This procedure reduces the simulation runtime substantially and will find applications in reservoir characterization and simulation of pr...

Experimental study of gas flow characteristics in micro-/nano-pores in tight and shale reservoirs using microtubes under high pressure and low pressure gradients

Due to abundant micro-/nanoscale pores and throats, gas flow behavior in tight/shale reservoirs always showed strongly nonlinearity which deviated from the conventional Darcy’s law. As micro-/nanoscale flow experiment was a direct approach to investigate flow characteristic, in this paper, we improved the microscale flow experiment which was widely used in the field of MEMS to adapt for high-pressure conditions. By using microtubes with inner diameter ranging from 0.2 to 5 µm, we investigated the low velocity nonlinear flow characteristic of nitrogen especially under high pressures. In addition, we used experimental results to evaluate the accuracy of three typical flow models which were widely used in gas apparent permeability determination. The results indicated that gas flow behavior in microtube under high outlet pressures was opposite to that under low outlet pressures. When the outlet pressure was low, slippage effect had a great influence on the flow. With the increase in the pressure gradient, the slippage effect became weakened. And the slippage effect would disappear with the increase in the outlet pressure as well as tube inner diameter. The minimum outlet pressure for eliminating the slippage effect of nitrogen for the 0.2 µm tube was 10 MPa which decreased with the tube diameter. However, under high outlet pressures, the flow resistance increased in the region with smaller pressure gradients and varied inversely with the pressure gradient. This phenomenon became more significant with the increase in the outlet pressure and the decrease in the tube inner diameter. From the comparison of 3 typical flow models against our experimental results, we can find different models will match with the data measured under different mean pressures. And the applicability of those models also varied with the tube inner diameter. Thus, it was crucial to select an appropriate model according to different reservoir conditions.