Effective Knudsen Diffusion Research Articles

Modified experimental method to investigate micro transport in the shale matrix for shale gas production

Matrix permeability is the primary transport property for shale gas recovery, due to the special production process of gas from matrix to wellbore. With the micro pore size of shale matrix, it is necessary to consider micro transport mechanisms for the obtained permeability from macro experimental techniques. In this study, the macro experimental permeability was investigated in micro way, by the analyzation of pressure drop curves of shale matrix at high- and low-pressure experimental conditions, with modified numerical model containing micro transport mechanisms. As selected parameters, porosity, pore radius and tortuosity were regressed to obtain apparent permeabilities of three samples. The analysis shows that, tortuosity is much higher than macro one, and has been proved to be necessary and reasonable. The Knudsen diffusion and slippage effect control the micro transport of shale matrix at low and high pressure, separately, which is the main difference of macro and micro transports. Slippage flow weight factor helps to increase of slippage flow proportion with pressure. Higher pore radius and tortuosity both could weaken the effect of Knudsen diffusion, and slippage flow appears positive relationship with pore size at high pressure. This research provides an inspiration to shale transport study and a micro perspective theoretical guidance for the macro recovery of in-site shale gas reservoir.

Enhancement of multi-gas transport process in coalbed methane reservoir by oxidation treatment: Based on the change of the interaction force between coal matrix and gas molecules and knudsen number

Anthracite coal of Zhijin Block was taken as the coal samples and NaClO as the oxidants to investigate enhancement of multi-gas transport process in coalbed methane reservoir by oxidation treatment. Industrial microscope observation, SEM observation, low-temperature N2 sorption measurement and NMR test were used to comprehensively characterize the multiscale pore structure of coal. The change of macromolecular structure of coal before and after oxidation treatment was tested by 13C NMR test, and the force between coal matrix and gas molecules was estimated. Based on the pore size distribution of coal obtained by NMR analysis, the Knudsen number was calculated. Oxidation treatment effectively promoted the desorption of methane in coal, increased the content of free methane in coal. It also led to the CBM gas well entering the stable production period earlier, and shortens the gas breakthrough time of the gas well. Oxidation treatment improved micropores to larger pores by dissolving the coal matrix, and the porosity and permeability of coal were increased. Oxidation treatment led to the widening of pore diameter, and the effect of Knudsen diffusion on the core seepage of the matrix was gradually enhanced. Macroscopically, the gas slippage effect was more significant and the permeability was enhanced. The mechanism of oxidation treatment on multi-gas transport process was mainly reflected in two aspects: increasing the content of free methane and enhancing the gas supply capacity of coal.

Numerical Simulation of Well Type Optimization in Tridimensional Development of Multi-Layer Shale Gas Reservoir

Aimed at the development of shale gas reservoirs with large reservoir thickness and multiple layers, this paper carried out a numerical simulation study on the optimization of three different well types: horizontal well, deviated well, and vertical well. To make the model more in line with the characteristics of shale gas reservoirs, a two-phase gas–water seepage mathematical model of shale gas reservoirs was established, considering the adsorption and desorption of shale gas, Knudsen diffusion effect, and stress sensitivity effect. The embedded discrete fracture model was used to describe hydraulic fracture and natural fracture. Based on Fortran language, a numerical simulator for multi-layer development of shale gas reservoirs was compiled, and the calculation results were compared with the actual production data of Barnett shale gas reservoirs to verify the reliability of the numerical simulator. The spread range of hydraulic fractures in the reservoir with different natural fracture densities is calculated by the simulation to determine well spacing and fracture spacing. The orthogonal experimental design method is then used to optimize the best combination of well spacing and fracture spacing for different well types. The results show that the well productivity of the high-density (0.012 m/m2) natural fractures reservoir > the well productivity of the medium-density (0.006 m/m2) natural fractures reservoir > the well productivity of the low-density (0.001 m/m2) natural fractures reservoir. According to the design of the orthogonal test, it can be seen that the most significant factor affecting the productivity of horizontal wells is the fracture spacing in the Y direction. For deviated wells and vertical wells, the X-direction well spacing has the greatest impact on its productivity.

Productivity model with mechanisms of multiple seepage in tight gas reservoir

The productivity model with unilateral threshold pressure gradient, real gas effect or non-Darcy effect has been documented in the research of tight gas reservoir. However, the multiple seepage mechanisms are poorly addressed in a productivity model, especially the application of gas-water relative permeability curve changing with the displacement pressure. A novel productivity model of multi-stage fractured horizontal wells with mechanisms of multiple seepage was proposed the present study. Results show that the stress effect has a negative effect on the effective permeability, while the Knudsen diffusion effect of gas has a positive effect. The water flow can inhibit the gas flow in the tight gas reservoir, whose effect is more significant with larger displacement pressure differences. The higher initial water saturation and reservoir temperature, whereas the lower gas well productivity, contribute to a high gas well productivity in the reservoir of high pressure.

An Extensive Study on Desorption Models Generated Based on Langmuir and Knudsen Diffusion

Although gas desorption is a known phenomenon, modeling fluid flow in tight gas reservoirs often ignores the governing desorption effect, assuming that viscous transport is the predominant controller, resulting in an erroneous prediction of mass transport and fluid flow calculations. Thus, developing a new model accommodating all the major contributing forces in such a medium is essential. This work introduces a new comprehensive flow model suitable for tight unconventional reservoirs, including viscous, inertia, diffusion, and sorption forces, to account for fluid transport. Based on Langmuir law and Knudsen diffusion effect, three models were generated and compared with different known models using synthetic data. The model was solved and analyzed for different scenario cases, and parametric studies were conducted to evaluate the desorption effect on different reservoir types using MATLAB. Results show that the contribution of the sorption mechanism to the flow increases with the reducing permeability of the medium and lower viscosity of the flowing fluid and an additional pressure drop up to 10 psi was quantified.

Asymmetric behavior of solid oxide cells between fuel cell and electrolyzer operations

The electrochemical performance of solid oxide cells (SOCs) is investigated under both fuel cell and electrolyzer operations to understand their asymmetric behavior between the two operation modes. The current–voltage and electrochemical impedance characteristics of a hydrogen-electrode-supported cell are experimentally analyzed. Also, a numerical model is developed to reproduce the cell performance and to understand the internal resistances of the cell. Partial pressures of supplied gas and load current are varied to evaluate their effects on the cell performance. The gas partial pressures of hydrogen and steam supplied to the hydrogen electrode are kept equivalent so that the cell performance can be fairly compared between the two operation modes when the same current is applied. It is found that the origin of the asymmetry is mostly from the hydrogen electrode; both activation and concentration overpotentials show asymmetric behavior particularly at high current densities. A numerical experiment is also conducted by deliberately changing parameters in the model. Asymmetry in the activation overpotential is found to be originated from the non-identical charge-transfer coefficients in the Butler–Volmer equation and also from the non-uniform gas concentration formed in the hydrogen electrode under current-biased conditions. On the other hand, asymmetry in the concentration overpotential is associated with the non-equimolar counter diffusion of hydrogen and steam caused by the effect of Knudsen diffusion. Therefore, enhancing gas transport in the hydrogen electrode and reducing the contribution of Knudsen diffusion are effective approaches to reduce asymmetry not only in the concentration overpotential but also in the activation overpotential.

Non-isothermal pore change model predicting CO2 adsorption onto consolidated activated carbon

Accurate simulation and detailed description of the dynamics of the adsorption process play a significant role in forecasting the performance of new materials when used in various adsorption systems, like cooling/heating. The activated carbon (AC) consolidation allows improving the heat transfer rate inside the adsorption/desorption bed and compacting the systems. There are numerous mathematical models in literature for gas adsorption onto granular AC. But for consolidated AC, because of the absence of macropores, most assumptions that work well for granular AC may lead to significant discrepancies. Therefore, the present research proposes a new mathematical model for gas (CO2) adsorption onto consolidated adsorbent, a non-isothermal pore change model. The model takes into account the porosity and permeability changes due to the adsorption. The validation of the developed model is performed via comparison with the results obtained experimentally and numerically using an isothermal model. The effective Knudsen diffusion coefficient for the working pair is evaluated from porosity data. The rate of adsorption or mass transfer coefficient is estimated using the van't Hoff plot. The study results could be applied in the development of waste heat-driven cooling systems employing consolidated composite material as the adsorbent. The proposed mathematical model is also applicable for many other working pairs.

Probing and Interpreting the Porosity and Tortuosity Evolution of Li-O2 Cathodes on Discharge through a Combined Experimental and Theoretical Approach.

Li-O2 batteries offer a high theoretical discharge capacity due to the formation of light discharged species such as Li2O2, which fill the porous positive electrode. However, in practice, it is challenging to reach the theoretical capacity and completely utilize the full electrode pore volume during discharge. With the formation of discharge products, the porous medium evolves, and the porosity and tortuosity factor of the positive electrode are altered through shrinkage and clogging of pores. A pore shrinks as solid discharge products accumulate, the pore clogging when it is filled (or when access is blocked). In this study, we investigate the structural evolution of the positive electrode through a combination of experimental and computational techniques. Pulsed field gradient nuclear magnetic resonance results show that the electrode tortuosity factor changes much faster than suggested by the Bruggeman relation (an equation that empirically links the tortuosity factor to the porosity) and that the electrolyte solvent affects the tortuosity factor evolution. The latter is ascribed to the different abilities of solvents to dissolve reaction intermediates, which leads to different discharge product particle sizes: on discharging using 0.5 M LiTFSI in dimethoxyethane, the tortuosity factor increases much faster than for discharging in 0.5 M LiTFSI in tetraglyme. The correlation between a discharge product size and tortuosity factor is studied using a pore network model, which shows that larger discharge products generate more pore clogging. The Knudsen diffusion effect, where collisions of diffusing molecules with pore walls reduce the effective diffusion coefficients, is investigated using a kinetic Monte Carlo model and is found to have an insignificant impact on the effective diffusion coefficient for molecules in pores with diameters above 5 nm, i.e., most of the pores present in the materials investigated here. As a consequence, pore clogging is thought to be the main origin of tortuosity factor evolution.

Evaluating the effective diffusion coefficient of reactant gas in the catalyst layer of PEMFC using the fractal method considering the pore size distribution

AbstractThe pore size distribution of the catalyst layer (CL) significantly influences the effective diffusion coefficient (EDC) of the reactant gas due to the Knudsen diffusion effect in the proton exchange membrane fuel cell (PEMFC). In this study, a novel fractal method is proposed to evaluate the Knudsen diffusion based on the pore size distribution of the CL, which is then incorporated in the calculation of the EDC of oxygen in the CL. It suggests that the EDC of oxygen obtained from the proposed method considering the pore size distribution is greatly smaller than that obtained from the conventional Bruggeman correlation, hence resulting in much smaller maximum current density of PEMFC.

Multicomponent landfill gas transport in soil cover: column tests and numerical modelling

A soil column test was carried out to investigate the mechanism of multicomponent landfill gas transport in loess. Four gas components, namely methane (CH4), carbon dioxide (CO2), oxygen (O2) and nitrogen (N2), were considered. Based on the dusty-gas model and mass balance equation, a one-dimensional transient model for multicomponent gas transport in landfill layered cover soils was then developed. Methane oxidation in the soil is also considered in the model. The numerical model was solved by the finite-element method-based program Comsol Multiphysics v5.1. The numerical results agree well with those observed from the soil column tests. The parameter analysis shows that ordinary diffusion plays an important role in the transport of multicomponent gases. For methane and carbon dioxide, ordinary diffusion contributes to 97% of the total transport flux in the shallow zone. The effect of ordinary diffusion decreases with the increase in the depth. Advection contributes to 37% of the flux at the bottom of the cover soil. The effect of Knudsen diffusion is relatively weak, which only contributes 0.5–12% to the total flux. This is due to the relatively high gas permeability of the soil. The contribution of advection is comparable to that of diffusion when the gas permeability increases up to 3 × 10−13 m2. The maximum methane oxidation rate is found to increase by a factor of 1.8, when influent flux increases from 1 × 10−5 to 1 × 10−4 (mol/m2)/s. The numerical solution can be used for the analysis of multicomponent landfill gas transport in the soils and the design of landfill cover systems with respect to gas pollution control.

Matrix permeability measurement from fractured unconventional source-rock samples: Method and application

Shale matrix permeability is one of the most important parameters for characterizing a source rock reservoir and for predicting hydrocarbon production. The low permeability value and the presence of induced fractures during core retrieval and transportation make the accurate measurement of the true permeability values for source rocks a significant challenge for the industry. The steady state flow method and the transient pressure pulse decay method on core plug samples mainly measure the permeability of fractures when fractures are present. While the Gas Research Institute (GRI) method that uses pressure decay on crushed rock samples was designed to overcome this difficulty associated with the induced fractures, its measurement results are reported to be sensitive to the particle size of crushed rock samples and also need correction of Knudsen diffusion effect. Moreover, the GRI method is limited to the unconfined stress condition.This work develops a practical method to measure the matrix permeability values from fractured source rock samples by extending the commonly used pressure pulse decay method. A source rock sample with fractures can be more accurately described by a dual-continuum system consisting of a fracture continuum and a matrix continuum. During the pulse decay test, the initial flow across the source rock sample is dominated by the fracture continuum because it has much higher permeability values than those for the rock matrix. Thus, the initial gas pressure signals from the test are used to estimate the fracture permeability. During the late-stage of a pulse decay test, the flow process within the rock sample is controlled by the rock matrix. The observed pressure signals at this stage are used for estimating matrix permeability. The method is based on the analytical solution to gas flow in the fractured rock sample and relatively simple to apply in practice. Both fracture and matrix permeability's dependence on the effective stress can be assessed with this method.

An efficient laboratory method to measure the combined effects of Knudsen diffusion and mechanical deformation on shale permeability

In a shale gas reservoir, the rock matrix has a relatively large porosity and gas in place, but extremely low permeability. Thus, the rock matrix is a bottleneck for shale gas flow from the reservoir to hydraulic fractures and then to the production well. We speculate that the next big thing after hydraulic fracturing for unconventional resources development is to enhance the matrix permeability in an economically feasible way. Consequently, the efficient and accurate characterization of rock matrix permeability in the laboratory is a critical task. The current laboratory techniques for source rock permeability measurement follow a “point-by-point” approach. They need multiple test runs to obtain a permeability-pressure curve, because they can only measure one permeability data point for one test run, and are thus time consuming. The root cause of this “point-by-point” approach is that these laboratory techniques are based on linearized gas flow theory requiring only small pore pressure disturbances to the experiment system. Liu et al. (2019) and this work introduce a new methodology that is based on the nonlinear gas flow theory and allows for direct measurement of the permeability-pressure curve with a single test run. This makes the approach highly time efficient. The feasibility and validity of the methodology are demonstrated in this work based on laboratory measurement results and their consistency with theoretical expectations and other independent measurements. The observed permeability exhibits a complex relationship with pore pressure as a result of the combined effects of Knudsen diffusion and mechanical deformation. For a given confining pressure, the observed permeability initially decreases with pore pressure because of Knudsen diffusion and then increases with pore pressure owing to the mechanical deformation. A rock sample with a lower permeability corresponds to a stronger Knudsen diffusion effect and weaker mechanical diffusion effect. This complex behavior highlights the need to accurately and efficiently measure the combined effects that may have important impacts on shale gas production.

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.

Lattice Boltzmann Method for Fluid-Thermal Systems: Status, Hotspots, Trends and Outlook

Great advances have been made with the lattice Boltzmann (LB) method for complicated fluid phenomena and fundamental thermal processes over the past three decades. This paper presents a systematic overview of the LB method from 1990 to 2018, based on bibliometric analysis and the Science Citation Index Expanded (SCI-E) database. The results show that China took the leading position in this field, followed by the USA and UK. The Chinese Academy of Sciences had the most publications, while the Los Alamos National Laboratory was first as far as highest average citation per paper and h-index are concerned. Physical Review E was the most productive journal and “Mechanics” was the most frequently used subject category. Keyword analysis indicated that recent research has focused on the natural convection and heat transfer of nanofluid or multiphase flow in complex porous media. Hydrothermal treatment of nanofluid with shape factor on the conditions, such as variable magnetic fields, thermal radiation and slipping boundary, were the research hotspots. Further research perspectives mainly explore the multiscale models for coupling multiple transport phenomena, morphology optimization of porous parameters, new nanoparticles with shape factor, multicomponent LB method considering Knudsen diffusion effect, LB-based hybrid methods, radiation performance or boiling-heat transfer of nanofluid, and the active control of droplets, may continue to attract more attention. Moreover, some new applications, such as phase change of metal foam, erosion induced by nanaofluid, anode circulating, 3D modeling in thermal systems with vibration, and magnetohydrodynamics microfluid devices, could be of interest going forward.

Experimental and numerical analyses of apparent gas diffusion coefficient in gas shales

A unified model of apparent gas diffusion coefficient in gas shales has been developed by considering the combined effects of Knudsen diffusion, Fick diffusion, and surface diffusion. Results showed that the model predictions matched the measured apparent gas diffusion coefficient in Longmaxi black shale reasonably well when the correction factor for real surface diffusion was approximately equal to 0.01.Results of sensitivity analyses revealed that Knudsen diffusion and Fick diffusion control the bulk of the methane diffusion in shale nanopores. The governing mechanism of methane molecules transport through shale nanopores varied depending on the overburden pressure, pore pressure, and resultant effective stress (i.e., contributions of Fick diffusion and Knudsen diffusion to the apparent gas diffusion coefficient varied depending on the level of pore pressure and effective stress).The apparent methane diffusion coefficient also varied depending on the size of the pore space. The effective radius of pores in a formation with a relatively high effective stress is smaller than that of the one with a relatively low effective stress. Greater apparent methane diffusion coefficient is, therefore, observed in a relatively lower effective stress formation because of the increasing contributions of Fick and Knudsen diffusion.Considering the combined effects of pore pressure and effective stress, we have found that the overall impact of gas production from shale formations (i.e. the decreasing pore pressure and the corresponding increase in effective stress) was to increase apparent methane diffusion coefficient, which would result into further stimulation of the gas production from shale reservoirs.

Influence of reservoir primary water on shale gas occurrence and flow capacity

In this paper, shale samples of Lower Silurian Longmaxi Fm, taken from the Changning–Weiyuan area in the Sichuan Basin, were selected to figure out the influence of reservoir primary water on the adsorption laws and the flow capacity of shale gas. Experimental samples with different water saturations were prepared using the adsorption equilibrium method. Then, high-pressure isothermal adsorption experiments were carried out, and the isothermal adsorption effects and mechanisms of shale under different water saturations were discussed. Finally, the flow capacity of shale gas under different water saturations was tested using the independently developed steady-state flow test device. And the following research results were obtained. First, the presence of primary water in micron–nanometer pores of shale reservoirs reduces the adsorption capacity of shale. When the water saturation is 40%, the simulated total gas content is 18% lower than that in the conventional calculation result. Second, the apparent shale permeability is a function of pressure. Due to the effect of Knudsen diffusion, the apparent shale permeability increases significantly with the decrease of pressure under low pressure. When the average pressure is 5 MPa and the water saturation reaches 50%, the apparent shale permeability is about 70% lower than that of a dry sample. Third, when the water saturation is lower than the critical value, water is mainly presented as non-movable water in micropores and mesopores, and it has less effect on the flow capacity of shale gas. When the water saturation is greater than the critical value, the lodging point of water is changed, resulting in significant reduction of the shale gas flow capacity. It is concluded that an accurate understanding of the original water saturation and critical water saturation of shale reservoirs helps to calculate shale gas reserves accurately and predict gas well production rate rationally.

Impact of Pore Geometry and Water Saturation on Gas Effective Diffusion Coefficient in Soil

Gas flow in soil plays a crucial role in terrestrial ecosystems, and numerical simulation of their movement needs to know their effective diffusion coefficients. How pore structure influences the effective diffusion coefficient has been studied intensively for dry porous media, but much remains unknown for unsaturated soils. Here, we employed the X-ray tomography technique at the pore scale to directly obtain the soil structures, the geometry of their pores and the water distribution under different water saturation levels were calculated using a morphological model. The results show that pore structures including porosity, interface area of gas–solid–water and pore diameter are closely related to water saturation. The increase of mean pore diameter with gas saturation can be fitted into a power law. We also investigated the impact of pore geometry and water saturation on the effective diffusion coefficients, which is independent of the molecular mass of gas after normalization. As the normalized effective Knudsen diffusion coefficient increases with average pore diameter following a power law, with the scaling factor related to pore geometry and the exponent is a constant, we explained and proved that the Knudsen diffusion coefficient increases with gas saturation, also following a power law.

Microstructure Effects on Effective Gas Diffusion Coefficient of Nanoporous Materials

In this work, we develop a numerical framework for gas diffusion in nanoporous materials including a random generation-growth algorithm for microstructure reconstruction and a multiple-relaxation-time lattice Boltzmann method for solution of diffusion equation with Knudsen effects carefully considered. The Knudsen diffusion is accurately captured by a local diffusion coefficient computed based on a corrected Bosanquet-type formula with the local pore size determined by the largest sphere method. A robust validation of the new framework is demonstrated by predicting the effective gas diffusion coefficient of microporous layer and catalyst layer in fuel cell, which shows good agreement with several recent experimental measurements. Then, a detailed investigation is made of the influence on effective gas Knudsen diffusivity by many important microstructure factors including morphology category, size effect, structure anisotropy, and layering structure effect. A widely applicable Bosanquet-type empirical relation at the Darcy scale is found between the normalized effective gas diffusion coefficient and the average Knudsen number. The present work will promote the understanding and modeling of gas diffusion in nanoporous materials and also provide an efficient platform for the optimization design of nanoporous systems.

The Effects of Catalyst Layer Microstructure and Water Saturation on the Effective Diffusivity in PEMFC

The effective diffusivity of oxygen in a cathode catalyst layer of PEMFC depends on both the catalyst layer microstructure and the changes in oxygen pathways due to liquid water. In this paper, various aspects of the effective diffusivity are investigated using pore-scale simulation, with emphasis on the effects of Knudsen diffusion. Based on microstructures reconstructed by the sphere-based simulated annealing method, both the higher-order lattice Boltzmann method (LBM) and the continuum diffusion equation with the Bosanquet approximation are used to evaluate the effective diffusivity considering Knudsen diffusion. Results show that the continuum diffusion equation reproduces the results from the higher-order LBM that simulates the effects of Knudsen diffusion based on the kinetic theory, when the mean pore size is evaluated using the erosion-dilation method or the chord length distribution. It is also shown that the change of the pore size with water saturation levels should be considered to predict the effects of Knudsen diffusion accurately. The dependence of the effective diffusivity on the agglomerate size is explained by the effects of Knudsen diffusion. Derjaguin’s correction for Knudsen diffusion is found to have negligible effects on the effective diffusivity for the present cases. The empirical correlations of various parameters are presented.

The effect of Knudsen diffusion and adsorption on shale transport in nanopores

Shale transport properties are investigated by numerical simulations of a simple, physics-based continuum model. Shale samples from the Yanchang Triassic Formation in Ordos Basin are tested to get the pore volume distribution of shale. Due to the existence of nanopores in shale reservoirs, there are great differences in gas transport mechanisms compared to those in conventional gas reservoirs. These mechanisms include Knudsen diffusion and adsorption. A physics-based continuum model is utilized to simulate the gas transport in a channel with nanoscale radius, where these mechanisms are taken into consideration by imposing a general boundary condition with two parameters. The correction factor which is the ratio between apparent permeability and intrinsic permeability is investigated for a wide pressure range, which shows it is always larger than 1, illustrating that Knudsen diffusion always plays a role in shale gas transport.