A New Unified Gas-Transport Model for Gas Flow in Nanoscale Porous Media

Summary A new unified gas-transport model has been developed to characterize single-component real-gas flow in nanoscale organic and inorganic porous media by modifying the Bravo (2007) model. More specifically, a straight capillary tube is characterized by a conceptual layered model consisting of a viscous-flow zone, a Knudsen-diffusion zone, and a surface-diffusion zone. To specify the contributions of the viscous flow and the Knudsen diffusion to the gas transport, the virtual boundary between the viscous-flow and Knudsen-diffusion zones is first determined using an analytical molecular-kinetics approach. As such, the new unified gas-transport model is derived by integrating the weighted viscous flow and Knudsen diffusion, and coupling surface diffusion. The model is also comprehensively scaled up to the bundles-of-tubes model considering the roughness, rarefaction, and real-gas effect. Nonlinear programming methods have been used to optimize the empirical parameters in the newly proposed gas-transport model. Consequently, the newly proposed gas-transport model yields the most accurate molar fluxes compared with the Bravo (2007) model and four other analytical models. One of the advantages of the new unified gas-transport model is its great flexibility, because the Knudsen number is included as an independent variable, which also endows the newly proposed model with the capability to cover the full-flow regimes. In addition, the apparent permeability has been mathematically derived from the new unified gas-transport model. A series of simulations has been implemented using methane gas. It is found through sensitivity analysis that apparent permeability is strongly dependent on pore size, porosity, and tortuosity, and weakly dependent on the surface-diffusivity coefficient and pore-surface roughness. The increased viscosity can reduce the total molar flux in the inorganic pores up to 66.0% under the typical shale-gas-reservoir conditions. The viscous-flow mechanism cannot be neglected at any pore sizes under reservoir conditions, whereas the Knudsen diffusion is found to be important when pore size is smaller than 2 nm and pressure is less than 35.0 MPa. The contribution of surface diffusion cannot be ignored when the pore size is smaller than 10 nm and the pressure is less than 15.0 MPa.

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

Fractals ◽

10.1142/s0218348x20500176 ◽

2020 ◽

Vol 28 (01) ◽

pp. 2050017 ◽

Cited By ~ 2

Author(s):

TAO WU ◽

SHIFANG WANG

Keyword(s):

Fractal Dimension ◽

Surface Diffusion ◽

Shale Gas ◽

Gas Transport ◽

Fractal Theory ◽

Knudsen Diffusion ◽

Apparent Permeability ◽

Permeability Model ◽

Real Gas ◽

Shale Reservoirs

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.

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

Petroleum Science ◽

10.1007/s12182-019-00358-4 ◽

2019 ◽

Vol 17 (1) ◽

pp. 168-181 ◽

Cited By ~ 3

Author(s):

Qi Zhang ◽

Wen-Dong Wang ◽

Yilihamu Kade ◽

Bo-Tao Wang ◽

Lei Xiong

Keyword(s):

Pore Pressure ◽

Surface Diffusion ◽

Pore Radius ◽

Gas Transport ◽

Gas Permeability ◽

Knudsen Diffusion ◽

Permeability Model ◽

Real Gas ◽

Apparent Gas Permeability ◽

Transport Behavior

Abstract 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.

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Multiscale Apparent Permeability Model of Shale Nanopores Based on Fractal Theory

Energies ◽

10.3390/en12173381 ◽

2019 ◽

Vol 12 (17) ◽

pp. 3381 ◽

Cited By ~ 10

Author(s):

Qiang Wang ◽

Yongquan Hu ◽

Jinzhou Zhao ◽

Lan Ren ◽

Chaoneng Zhao ◽

...

Keyword(s):

Shale Gas ◽

Gas Flow ◽

Gas Transport ◽

Fractal Theory ◽

Knudsen Diffusion ◽

Slip Flow ◽

Clear Understanding ◽

Apparent Permeability ◽

Permeability Model ◽

Shale Gas Transport

Based on fractal geometry theory, the Hagen–Poiseuille law, and the Langmuir adsorption law, this paper established a mathematical model of gas flow in nano-pores of shale, and deduced a new shale apparent permeability model. This model considers such flow mechanisms as pore size distribution, tortuosity, slippage effect, Knudsen diffusion, and surface extension of shale matrix. This model is closely related to the pore structure and size parameters of shale, and can better reflect the distribution characteristics of nano-pores in shale. The correctness of the model is verified by comparison with the classical experimental data. Finally, the influences of pressure, temperature, integral shape dimension of pore surface and tortuous fractal dimension on apparent permeability, slip flow, Knudsen diffusion and surface diffusion of shale gas transport mechanism on shale gas transport capacity are analyzed, and gas transport behaviors and rules in multi-scale shale pores are revealed. The proposed model is conducive to a more profound and clear understanding of the flow mechanism of shale gas nanopores.

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A non-empirical model for gas transfer through circular nanopores in unconventional gas reservoirs

Journal of Petroleum Exploration and Production Technology ◽

10.1007/s13202-021-01145-x ◽

2021 ◽

Vol 11 (5) ◽

pp. 2217-2232

Author(s):

Jiangtao Li ◽

Jianguang Wei ◽

Liang Ji ◽

Anlun Wang ◽

Gen Rong ◽

...

Keyword(s):

Gas Flow ◽

Gas Transport ◽

Transport Model ◽

Knudsen Diffusion ◽

Slip Flow ◽

Gas Reservoirs ◽

Permeability Model ◽

Proposed Model ◽

Unconventional Gas Reservoirs ◽

Empirical Coefficients

AbstractIt is difficult to predict the flow performance in the nanopore networks since traditional assumptions of Navier–Stokes equation break down. At present, lots of attempts have been employed to address the proposition. In this work, the advantages and disadvantages of previous analytical models are seriously analyzed. The first type is modifying a mature equation which is proposed for a specified flow regime and adapted to wider application scope. Thus, the first-type models inevitably require empirical coefficients. The second type is weight superposition based on two different flow mechanisms, which is considered as the reasonable establishment method for universal non-empirical gas-transport model. Subsequently, in terms of slip flow and Knudsen diffusion, the novel gas-transport model is established in this work. Notably, the weight factors of slip flow and Knudsen diffusion are determined through Wu’s model and Knudsen’s model respectively, with the capacity to capture key transport mechanism through nanopores. Capturing gas flow physics at nanoscale allows the proposed model free of any empirical coefficients, which is also the main distinction between our work and previous research. Reliability of proposed model is verified by published molecular simulation results as well. Furthermore, a novel permeability model for coal/shale matrix is developed based on the non-empirical gas-transport model. Results show that (a) nanoconfined gas-transport capacity will be strengthened with the decline of pressure and the decrease in the pressure is supportive for the increasing amplitude; (b) the greater pore size the nanopores is, the stronger the transport capacity the nanotube is; (c) after field application with an actual well in Fuling shale gas field, China, it is demonstrated that numerical simulation coupled with the proposed permeability model can achieve better historical match with the actual production performance. The investigation will contribute to the understanding of nanoconfined gas flow behavior and lay the theoretical foundation for next-generation numerical simulation of unconventional gas reservoirs.

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Nanoconfined Water in Shales and Its Impact on Real Gas Flow

Geofluids ◽

10.1155/2021/8092331 ◽

2021 ◽

Vol 2021 ◽

pp. 1-18

Author(s):

Yudan Li ◽

Shaohua Gu ◽

Cheng Dai

Keyword(s):

Porous Media ◽

Gas Flow ◽

High Pressures ◽

Film Flow ◽

Gas Transport ◽

Transport Model ◽

Water Film ◽

Real Gas ◽

The Real ◽

Nanoconfined Water

The presence of water, i.e., connate or hydraulic fracturing water, along with the gaseous hydrocarbons in shale nanopores is largely overlooked by previous studies. In this work, a new unified real gas-transport model has been developed for both organic and inorganic porous media accounting for the nanoconfined water film flow. More specifically, a gas core flows in the center of the organic/inorganic pore surrounded by a water film which can be further divided into an interfacial region (near-wall water) and bulk region (bulk water). We differentiate the varying water viscosity between the two regions and consider disparate slip boundaries; that is, the near-wall water can slip along the hydrophobic organic pore surface while it is negligible in hydrophilic inorganic pores. Incorporating modified boundary conditions into the Navier-Stokes equations, gas transport model through single organic/inorganic pore is derived. The model is also comprehensively scaled up to the porous media scale considering the porosity, tortuosity, and total organic carbon (TOC) contents. Results indicate that the gas flow capacity decreases in moist conditions with mobile or nonmobile water film. A mobile water film, however, compensates its negative effect up to 50% by enhancing gas flow compared with static water molecules. The real gas flow is dominated by the gas slippage and water film mobility which are dependent upon pore-scale parameters such as pore sizes, topology, pressure, and surface wettability. Compared with inorganic pores, gas transport in organic pores is greatly enhanced by the water film flow due to the strong water slip. Moreover, the contribution of water film mobility is remarkable in small pores with large contact angles, especially at high pressures. At moist conditions, the real gas effect enhances gas flow by improving both gas slippage and water film mobility, which is more prominent in smaller pores at high pressures. The presented model and its results will further advance our understanding of the mechanisms responsible for the water and gas transport in nanoporous media, and consequently, the hydrocarbon exploration of shale reservoirs.

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Gas Multiple Flow Mechanisms and Apparent Permeability Evaluation in Shale Reservoirs

Sustainability ◽

10.3390/su11072114 ◽

2019 ◽

Vol 11 (7) ◽

pp. 2114 ◽

Cited By ~ 6

Author(s):

Xuelei Feng ◽

Fengshan Ma ◽

Haijun Zhao ◽

Gang Liu ◽

Jie Guo

Keyword(s):

Pore Size ◽

Gas Flow ◽

Knudsen Diffusion ◽

Stress Sensitivity ◽

Gas Production ◽

Apparent Permeability ◽

Matrix Shrinkage ◽

Gas Seepage ◽

Flow Mechanisms ◽

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.

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NANOSCALE PORE SIZE DISTRIBUTION EFFECTS ON GAS PRODUCTION FROM FRACTAL SHALE ROCKS

Fractals ◽

10.1142/s0218348x19501421 ◽

2019 ◽

Vol 27 (08) ◽

pp. 1950142

Author(s):

JINZE XU ◽

KELIU WU ◽

RAN LI ◽

ZANDONG LI ◽

JING LI ◽

...

Keyword(s):

Fractal Dimension ◽

Pore Size Distribution ◽

Pore Size ◽

Size Distribution ◽

Gas Transport ◽

Gas Production ◽

Lower Sensitivity ◽

Apparent Permeability ◽

Transport Efficiency ◽

Nanoscale Pore

Effect of nanoscale pore size distribution (PSD) on shale gas production is one of the challenges to be addressed by the industry. An improved approach to study multi-scale real gas transport in fractal shale rocks is proposed to bridge nanoscale PSD and gas filed production. This approach is well validated with field tests. Results indicate the gas production is underestimated without considering a nanoscale PSD. A PSD with a larger fractal dimension in pore size and variance yields a higher fraction of large pores; this leads to a better gas transport capacity; this is owing to a higher free gas transport ratio. A PSD with a smaller fractal dimension yields a lower cumulative gas production; this is because a smaller fractal dimension results in the reduction of gas transport efficiency. With an increase in the fractal dimension in pore size and variance, an apparent permeability-shifting effect is less obvious, and the sensitivity of this effect to a nanoscale PSD is also impaired. Higher fractal dimensions and variances result in higher cumulative gas production and a lower sensitivity of gas production to a nanoscale PSD, which is due to a better gas transport efficiency. The shale apparent permeability-shifting effect to nanoscale is more sensitive to a nanoscale PSD under a higher initial reservoir pressure, which makes gas production more sensitive to a nanoscale PSD. The findings of this study can help to better understand the influence of a nanoscale PSD on gas flow capacity and gas production.

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Diffusion-Based Modeling of Gas Transport in Organic-Rich Ultratight Reservoirs

SPE Journal ◽

10.2118/201393-pa ◽

2021 ◽

pp. 1-26

Author(s):

Zizhong Liu ◽

Hamid Emami-Meybodi

Keyword(s):

Diffusion Coefficient ◽

Adsorption Capacity ◽

Surface Diffusion ◽

Diffusion Coefficients ◽

Gas Transport ◽

Knudsen Diffusion ◽

Gas Production ◽

Hydrocarbon Transport ◽

The Impact ◽

And Storage

Summary The complex pore structure and storage mechanism of organic-rich ultratight reservoirs make the hydrocarbon transport within these reservoirs complicated and significantly different from conventional oil and gas reservoirs. A substantial fraction of pore volume in the ultratight matrix consists of nanopores in which the notion of viscous flow may become irrelevant. Instead, multiple transport and storage mechanisms should be considered to model fluid transport within the shale matrix, including molecular diffusion, Knudsen diffusion, surface diffusion, and sorption. This paper presents a diffusion-based semianalytical model for a single-component gas transport within an infinite-actingorganic-rich ultratight matrix. The model treats free and sorbed gas as two phases coexisting in nanopores. The overall mass conservation equation for both phases is transformed into one governing equation solely on the basis of the concentration (density) of the free phase. As a result, the partial differential equation (PDE) governing the overall mass transport carries two newly defined nonlinear terms; namely, effective diffusion coefficient, De, and capacity factor, Φ. The De term accounts for the molecular, Knudsen, and surface diffusion coefficients, and the Φ term considers the mass exchange between free and sorbed phases under sorption equilibrium condition. Furthermore, the ratio of De/Φ is recognized as an apparent diffusion coefficient Da, which is a function of free phase concentration. The nonlinear PDE is solved by applying a piecewise-constant-coefficient technique that divides the domain under consideration into an arbitrary number of subdomains. Each subdomain is assigned with a constant Da. The diffusion-based model is validated against numerical simulation. The model is then used to investigate the impact of surface and Knudsen diffusion coefficients, porosity, and adsorption capacity on gas transport within the ultratight formation. Further, the model is used to study gas transport and production from the Barnett, Marcellus, and New Albany shales. The results show that surface diffusion significantly contributes to gas production in shales with large values of surface diffusion coefficient and adsorption capacity and small values of Knudsen diffusion coefficient and total porosity. Thus, neglecting surface diffusion in organic-rich shales may result in the underestimation of gas production.

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Apparent Permeability Model for Gas Transport in Multiscale Shale Matrix Coupling Multiple Mechanisms

Energies ◽

10.3390/en13236323 ◽

2020 ◽

Vol 13 (23) ◽

pp. 6323

Author(s):

Xiaoping Li ◽

Shudong Liu ◽

Ji Li ◽

Xiaohua Tan ◽

Yilong Li ◽

...

Keyword(s):

Pore Size ◽

Water Distribution ◽

Gas Transport ◽

Water Saturation ◽

Gas Permeability ◽

Apparent Permeability ◽

Permeability Model ◽

Proposed Model ◽

Irreducible Water Saturation ◽

The Impact

Apparent gas permeability (AGP) is a significantly important parameter for productivity prediction and reservoir simulation. However, the influence of multiscale effect and irreducible water distribution on gas transport is neglected in most of the existing AGP models, which will overestimate gas transport capacity. Therefore, an AGP model coupling multiple mechanisms is established to investigate gas transport in multiscale shale matrix. First, AGP models of organic matrix (ORM) and inorganic matrix (IOM) have been developed respectively, and the AGP model for shale matrix is derived by coupling AGP models for two types of matrix. Multiple effects such as real gas effect, multiscale effect, porous deformation, irreducible water saturation and gas ab-/de-sorption are considered in the proposed model. Second, sensitive analysis indicates that pore size, pressure, porous deformation and irreducible water have significant impact on AGP. Finally, effective pore size distribution (PSD) and AGP under different water saturation of Balic shale sample are obtained based on proposed AGP model. Under comprehensive impact of multiple mechanisms, AGP of shale matrix exhibits shape of approximate “V” as pressure decrease. The presence of irreducible water leads to decrease of AGP. At low water saturation, irreducible water occupies small inorganic pores preferentially, and AGP decreases with small amplitude. The proposed model considers the impact of multiple mechanisms comprehensively, which is more suitable to the actual shale reservoir.

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