A Model for the Apparent Gas Permeability of Shale Matrix Organic Nanopore Considering Multiple Physical Phenomena

The flow of shale gas in nano scale pores is affected by multiple physical phenomena. At present, the influence of multiple physical phenomena on the transport mechanism of gas in nano-pores is not clear, and a unified mathematical model to describe these multiple physical phenomena is still not available. In this paper, an apparent permeability model was established, after comprehensively considering three gas flow mechanisms in shale matrix organic pores, including viscous slippage Flow, Knudsen diffusion and surface diffusion of adsorbed gas, and real gas effect and confinement effect, and at the same time considering the effects of matrix shrinkage, stress sensitivity, adsorption layer thinning, confinement effect and real gas effect on pore radius. The contribution of three flow mechanisms to apparent permeability under different pore pressure and pore size is analyzed. The effects of adsorption layer thinning, stress sensitivity, matrix shrinkage effect, real gas effect and confinement effect on apparent permeability were also systematically analyzed. The results show that the apparent permeability first decreases and then increases with the decrease of pore pressure. With the decrease of pore pressure, matrix shrinkage, Knudsen diffusion, slippage effect and surface diffusion effect increase gradually. These four effects will not only make up for the permeability loss caused by stress sensitivity and adsorption layer, but also significantly increase the permeability. With the decrease of pore radius, the contribution of slippage flow decreases, and the contributions of Knudsen diffusion and surface diffusion increase gradually. With the decrease of pore radius and the increase of pore pressure, the influence of real gas effect and confinement effect on permeability increases significantly. Considering real gas and confinement effect, the apparent permeability of pores with radius of 5 nm is increased by 13.2%, and the apparent permeability of pores with radius of 1 nm is increased by 61.3%. The apparent permeability model obtained in this paper can provide a theoretical basis for more accurate measurement of permeability of shale matrix and accurate evaluation of productivity of shale gas horizontal wells.

Gas separation processes in a diffusion module based on anodic aluminum oxide membrane elements

Anodic alumina membranes with an ordered microstructure have been synthesized and investigated. It was found that Knudsen diffusion is the predominant mechanism for gas penetration through the obtained membranes. The technology made it possible to obtain porous membranes with specified structural characteristics for the separation of gas mixtures. Designs of a diffusion element and a gas separation module based on membranes made of anodic aluminum oxide have been developed, and the features of mass transfer under various operating conditions have been studied. The membrane module without recirculation made it possible to concentrate the heavy component from the model helium-methane mixture (99 % / 1 %) up to 18 %. The membrane module with recirculation made it possible to concentrate a light component from a model helium-methane mixture (1 % / 99 %) up to 40 %.

Image-based modelling of coke combustion in a multiscale porous medium using a micro-continuum framework

Non-isothermal reactive transport in complicated porous media is diverse in nature and industrial applications. There are challenges in the modelling of multiple physicochemical processes in multiscale pore structures with various length scales ranging from nanometres to micrometres. This study focuses on coke combustion during in situ crude oil combustion techniques. A micro-continuum model was developed to perform an image-based simulation of coke combustion through a multiscale porous medium. The simulation coupled weakly compressible gas flow, species transport, conjugate heat transfer, heterogeneous coke oxidation kinetics and structural evolution. The unresolved nanoporous coke region was treated as a continuum, for which the random pore model, permeability model and species diffusivity model were integrated as sub-grid models to account for the sub-resolution reactive surface area, Darcy flow and Knudsen diffusion, respectively. A Pe–Da diagram was provided to present five characteristic combustion regimes covering the ignition temperature and air flux in realistic field operations and laboratory measurements. The present model proved to achieve more accurate predictions of the feasible ignition temperature than previous models. Compared with the air flux of $\phi \sim O\textrm{(1) s}{\textrm{m}^\textrm{3}}(\textrm{air})\;{({\textrm{m}^\textrm{2}}\ \textrm{h})^{ - 1}}$ in the field, the increasing air flux in the laboratory transformed the combustion regime from diffusion-limited to convection-limited, which led to an overpredicted burning temperature. Reactive fingering combustion was analysed to understand the potential risks in some experimental measurements. The findings provide a better understanding of coke combustion and can help engineers design sustainable combustion methods. The developed image-based model allows other types of multiscale and nonlinear reactive transport to be simulated.

EDFM-based Multi-Continuum Shale Gas Simulation with Low Velocity Non-Darcy Water Flow Effect

The exploitation of shale gas has attracted extensive attention in industry and academia. Multi-scale gas transportation mechanisms in matrix and fractures have been well studied. However, due to the presence of water originating from both fracking fluids and connate water, shale gas production is also greatly affected by water imbibition and flowback, of which the processes have not been thoroughly analyzed. This paper aims at presenting a comprehensive multi-continuum multi-component model to characterize the complicated shale gas flow behaviors as well as the impact of non-Darcy water flow on shale gas production. A two-phase numerical simulator is built up with multi-continuum settings. Shale matrix is split into organic and inorganic matters while natural and hydraulic fractures are modeled using an embedded discrete fracture model (EDFM). Fracture closure and elongation are modeled using a dynamic gridding approach. Different transportation mechanisms are considered to describe gas flow in shale, including Knudsen diffusion, ab/desorption, and convection. The low-velocity non-Darcy flow of water is used in inorganic pores to analyze the effect of water flow. A pre-stage model based on pumping history is simulated firstly before production starts. This serves as an initialization step to model fracking fluid imbibition and early-stage water flowback. This pre-stage simulation gives out more precise pressure and saturation profiles than the conventional non-equilibrium initialization method, especially in enhanced pore volumes and fractures. Based upon simulation results from the production period, Langmuir isotherm absorption has shown a massive impact on gas flow in shale, and Knudsen diffusion weights highest among transport mechanisms. Water non-Darcy flow better benefits in simulating both early-stage water flowback and production process compared with Darcy flow, which gives us a new explanation on the low flowback efficiency in real shale gas operations. Studies on early-stage water flowback also show that the flowback affects saturation distribution, which has a strong relationship with gas production and shall not be ignored. This work establishes a novel method to simulate and analyze shale gas production. It considers multiple and complex flow mechanisms and gives out better estimates of water flux. It is also used to initialize a model for pumping water imbibition and early-stage flowback, which can be used as technical resources for analyzing and simulating unconventional plays.

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.

Continuum-Scale Gas Transport Modeling in Organic Nanoporous Media Based on Pore-Scale Density Distributions

This paper presents a continuum-scale diffusion-based model informed by pore-scale data for gas transport in organic nanoporous media. A mass transfer and adsorption model is developed by considering multiple transport and storage mechanisms, including bulk diffusion and Knudsen diffusion for free phase, surface diffusion for sorbed phase, and multilayer adsorption. The continuum-scale diffusion-based governing equation is developed solely based on free phase concentration for the overall mass conservation of free and sorbed phases, carrying a newly-defined effective diffusion coefficient and a capacity factor to account for multilayer adsorption. Diffusion of free and sorbed phases is coupled through the pore-scale simplified local density method based on the modified Peng-Robinson equation of state for confinement effects. The model is first utilized to analyze pore-scale adsorption data from the krypton (Kr) gas adsorption experiment on graphite. Then we implement the model to conduct sensitivity analysis for the effects of pore size on gas transport for Kr-graphite and methane-coal systems. The model is finally used to study Kr diffusion profiles through a coal matrix obtained through X-ray micro-CT imaging. The results show that the sorbed phase occupies most of the pore space in organic nanoporous media due to multilayer adsorption, and surface diffusion contributes significantly to the total mass flux. Therefore, neglecting the volume of sorbed phase and surface diffusion in organic nanoporous rocks may result in considerable errors. Furthermore, the results reveal that implementing a Langmuir-based model may be erroneous for an organic-rich reservoir with nanopores during the early depletion period when the reservoir pressure is high.

Transport Reagents through the Pore Structure of a Membrane Catalyst under Isothermal and Non-Isothermal Conditions

The article presents the results of an experimental comparison of methane transport in the pore structure of a membrane catalyst under isothermal and non-isothermal Knudsen diffusion conditions. It is shown that under the conditions of non-isothermal Knudsen diffusion in the pore structure of the membrane catalyst, there is a coupling of dry reforming of the methane (DRM) and gas transport, which leads to the intensification of this process. The reasons for the intensification are changes in the mechanism of gas transport, an increase in the rate of mass transfer, and changes in the mechanism of some stages of the DRM. The specific rate constant of the methane dissociation reaction on a membrane catalyst turned out to be an order of magnitude (40 times) higher than this value on a traditional (powder) catalyst.

An Apparent Gas Permeability Model for Real Gas Flow in Fractured Porous Media with Roughened Surfaces

The investigation of gas transport in fractured porous media is essential in most petroleum and chemical engineering. In this paper, an apparent gas permeability model for real gas flow in fractured porous media is derived with adequate consideration of real gas effect, the roughness of fracture surface, and Knudsen diffusion based on the fractal theory. The fractal apparent gas permeability model is obtained to be a function of micro-structural parameters of fractured porous media, relative roughness, the pressure, the temperature, and the properties of gas. The predictions from the apparent gas permeability model based on the fractal theory match well with the published permeability model and experimental data, which verifies the rationality of the present fractal apparent gas permeability model.

A Novel Approach To Predict Gas Flow in Entire Knudsen Number Regime through Nanochannels with Various Geometries

Summary Various unified gas flow (UGF) and apparent permeability models have been proposed to characterize the complex gas transport mechanisms in shale formations. However, such models are typically expressed as combinations of multiple gas flow mechanisms so that they cannot predict gas velocity profile. In this study, we develop a novel approach to predict the gas velocity profile in the entire Knudsen number (Kn) regime for circular and noncircular (i.e., square, rectangular, triangular and elliptical) nanochannels and investigate the effects of cross-sectional geometry on gas transport in nanochannels. To this end, a new UGF model is proposed to describe the gas flow behaviors in the entire Kn regime, considering the effects of gas slippage, bulk diffusion, Knudsen diffusion, surface diffusion, and cross-sectional geometry of flow channel. In addition, the boundary condition of the semianalytical second-order slip model applicable to various cross-sectional geometries is modified by adjusting the slip coefficients through the comparison between the proposed UGF model and the Navier-Stokes (N-S) equation with second-order slip boundary condition. As a result, the velocity profile of free gas in the entire Kn regime for the nanochannel with a specific cross section can be determined by solving the second-order slip model with adjusted slip coefficients via the finite element method. The results indicate that the geometry of the cross section has a significant influence on the mass flow rate and gas velocity profile in nanochannels. The predicted mass flow rates for the nanochannels with identical hydraulic diameter decrease with the cross-sectional geometry in the sequence as ellipse > equilateral triangle > rectangle > square > circle. However, the ranking of velocity profiles for such nanochannels, which is governed by the cross-sectional geometry, also varies with Kn. These findings indicate that the developed approach can predict the synergetic gas transport (i.e., gas slippage, bulk diffusion, Knudsen diffusion, and surface diffusion) and gas velocity profile in nanochannels with different cross-sectional geometries for a wide range of Kn, which gives insight into the characterization of gas flow behaviors in nanoporous shale.