Geofluids / 2020 / Article
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Multiscale and Multiphysical Approaches to Fluids Flow in Unconventional Reservoirs

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Review Article | Open Access

Volume 2020 |Article ID 8877777 |

Bo-ning Zhang, Xiao-gang Li, Yu-long Zhao, Cheng Chang, Jian Zheng, "A Review of Gas Flow and Its Mathematical Models in Shale Gas Reservoirs", Geofluids, vol. 2020, Article ID 8877777, 19 pages, 2020.

A Review of Gas Flow and Its Mathematical Models in Shale Gas Reservoirs

Academic Editor: Wei Wei
Received13 Jul 2020
Revised04 Oct 2020
Accepted04 Nov 2020
Published01 Dec 2020


The application of horizontal wells with multistage hydraulic fracturing technologies has made the development of shale gas reservoirs become a worldwide economical hotspot in recent years. The gas transport mechanisms in shale gas reservoirs are complicated, due to the multiple types of pores with complex pore structure and special process of gas accumulation and transport. Although there have been many attempts to come up with a suitable and practical mathematical model to characterize the shale gas flow process, no unified model has yet been accepted by academia. In this paper, a comprehensive literature review on the mathematical models developed in recent years for describing gas flow in shale gas reservoirs is summarized. Five models incorporating different transport mechanisms are reviewed, including gas viscous flow in natural fractures or macropores, gas ad-desorption on shale organic, gas slippage, diffusion (Knudsen diffusion, Fick diffusion, and surface diffusion), stress dependence, real gas effect, and adsorption layer effect in the nanoshale matrix system, which is quite different from conventional gas reservoir. This review is very helpful to understand the complex gas flow behaviors in shale gas reservoirs and guide the efficient development of shale gas. In addition to the model description, we depicted the type curves of fractured horizontal well with different seepage models. From the review, it can be found that there is some misunderstanding about the essence of Knudsen/Fick diffusion and slippage, which makes different scholars adopt different weighting methods to consider them. Besides, the contribution of each mechanism on the transport mechanisms is still controversial, which needs further in-depth study in the future.

1. Introduction

The reserves of unconventional gas reservoirs, such as shale gas, coalbed methane, tight gas reservoirs, and natural gas hydrate, are rich around the world. The annual production of them has become increasingly important to the global energy supply. As a kind of unconventional natural gas, shale gas is trapped in the source rock with self-accumulation, which is very tight with the permeability range from nD to mD [17]. The shale gas storage in shale reservoirs are mainly with the status of free gas, adsorbed gas, and dissolved gas. The adsorbed gas can account for up to 85% of the total. Shale gas has become an increasingly important source of natural gas since the success of exploitation in the United States, and interest has spread to potential shale gas reservoirs around the world [13]. The horizontal wells and the multistage hydraulic fracturing technique have proven to be the key for the cost-effective development of such tight reservoirs [1, 811].

While shale gas reservoirs have the characteristic of multiscaled space from ultramicropores (pore  nm), micropores (0.7 nm~2 nm), and mesopores (2 nm~50 nm) to macropores (>50 nm), the small-scale nanopores of less than 10 nm dominate in shale [1, 2, 5, 6]. Due to the complex pore size distribution and different gas storage mechanisms in shale gas reservoirs, which dominated by the viscous flow, slippage flow, Fick diffusion, Knudsen diffusion, surface diffusion, etc., gas transport mechanisms in shale gas reservoirs are very complicated and vastly different from those in conventional reservoirs. Although many models have been proposed and used to analyze the gas flow behaviors in nanoscale, some of which are extended to the application of numerical simulation and well testing, there is no unified model that can be acceptable in the industry so far. The related studies on such topics are being widely reported. The objective of this paper is to review the progress of gas transport mechanisms and some mathematical models developed for shale reservoirs [1], which is of great significance to establish a shale gas reservoir development strategy that is completely different from the conventional gas reservoir. Only by revealing the complex transmission mechanism of shale gas in the micro-nanopore system can an accurate mechanism model be provided for the numerical simulation and production dynamic evaluation of shale gas.

2. Gas Flow and Transport Mechanisms

2.1. Pore Types in Shale Gas Reservoirs

Shale gas reservoirs are typical unconventional oil and gas reservoirs, which consist of fine-grained and organic-rich sedimentary rock. Shale is both the reservoir and the source of oil and gas [1, 912]. According to the International Union of Pure and Applied Chemistry (IUPAC), classification of pores in shale can be divided into organic intragranular pores, inorganic pores, and natural fractures [1].

In general, organic intragranular pores are formed only when thermal maturity (Ro) reaches 0.6% or above. When Ro is smaller than this value, few or no organic intragranular pores are developed. Organic pores in shale reservoirs are mainly developed during the thermal cracking of the hydrocarbon generation phase, and the sizes of organic pores range from 5 nm to 700 nm (as shown in Figure 1). Such organic nanometer pores are well developed in shale reservoirs, providing tremendous surface areas for shale gas adsorption as well as flow paths for gas flow [1, 6, 7].

The inorganic pores in shale matrix can be categorized as residual primary intergranular pores (as shown in Figure 2(a)), intercrystalline pores (as shown in Figure 2(b)), and secondary dissolved pores formed by dissolution of unstable minerals (e.g., calcite and feldspar, as shown in Figure 2(c)).

There are lots of natural fractures that are developed in shale gas reservoirs (as shown in Figure 3). Most natural fractures in shale reservoir developed during the organic evolution of hydrocarbon source rocks, and they are intermittently open or closed with the changes of reservoir pressure. Therefore, for those shale gas reservoirs with mass microfractures, the permeability of the fracture system is sensitive to stress caused by closing of the microfractures during reservoir development. Similar to conventional dual-porosity reservoirs, natural fractures are considered to the main flow channel for shale gas [1].

2.2. Gas Adsorption and Desorption

The physical properties, pore types, and accumulation mechanisms of shale gas reservoirs are different from those of conventional gas reservoirs, resulting in natural gas to exist in diverse states in shale. Although a small amount of gas is dissolved in kerogen, asphaltene, liquid hydrocarbons, and formation water, the majority of natural gas exists in a free or an adsorbed state. Adsorbed shale gas is mainly adsorbed on the surface of organic matter with a single multimolecular layer, accounting for 20% to 85% of total reserves (as shown in Figure 4). Free gas is mainly stored in the pores of microfractures, inorganic and organic pores [1416]. When the quantity of gas present in the reservoir is greater than the saturated adsorption capacity, free gas can exist [17, 18]. The relative proportion of adsorbed and free gas varies with temperature, pressure, organic matter content and quality, degree of microfracture development, and formation water content (the gas storage status of shale gas reservoirs is shown in Figure 4) [1, 13, 19].

As mentioned above, the amount of adsorbed gas can account for up to 85% of the gas reserve, which will be desorbed from the organic particle surface into free gas when the pressure is lower than the desorption pressure [1]. Therefore, the adsorption model is very important to describe the amount of gas in a given formation under certain conditions. According to the previous studies, there are two types of gas adsorption and desorption models—monomolecular layer model including the single layer adsorption model, such as Langmuir isothermal adsorption (L model); multimolecular layer model including the Freundlich adsorption model (F model), BET model, bi-parameter BET model (B-BET model), Toth adsorption model (T model), Langmuir-Freundlich adsorption model (L-F model), extended Langmuir model (E-L model), three-parameter BET model (T-BET model), and Dubinin-Radushkevich volume filling model (D-R model). These models can be used to calculate gas adsorption rate, pore size distribution, and desorption pressure. The formulas for these models are shown in Table 1 [17, 20, 21].


Langmuir model (L model)
Freundlich empirical formula (F model)
Bi-parameter BET model (B-BET model)
Toth adsorption model (T model)
Langmuir-Freundlich adsorption model (L-F model)
Extended Langmuir model (E-L model)
Three-parameter BET model (T-BET model)
Dubinin-Radushkevich volume filling model (D-R model)

Figure 5 shows the fitting curves of the methane isotherm adsorption data from Barnett formation by different adsorption models, and Figure 6 shows the fitting curves of the test data by samples from the target layer of Longmaxi shale in southern Sichuan Basin. From the fitting results, the following results can be obtained: in general, the single molecular adsorption model is much better for the North American Barnett shale gas reservoirs, and the multimolecular layer adsorption model of T-BET is much better for the Changning shale gas reservoirs of China.

3. Seepage Mechanism Models for Shale Gas Reservoirs

Due to the complex pore space and gas accumulation dynamics in shale gas reservoirs, the movement of shale gas occurs via complex mechanisms, including adsorption-desorption, diffusion, and seepage [23]. The specific flow processes are as follows. With decreasing reservoir pressure, the gas adsorbed on the organic matter is desorbed. The desorbed gas then enters macropores, nanopores, and natural microfractures and becomes free gas. Due to the difference in gas concentration between kerogen/clay and organic nanopores, the gas flows to the low-pressure zone through diffusion (through matrix macropores or microfractures). Finally, free gas stored in matrix macropores and natural microfractures flows into wellbores and artificial hydraulic fractures under the pressure gradient [23, 24]. Gas flow in macropores, fractures, and wellbores follows Darcy’s law [25].

Generally, continuity hypothesis or molecular hypothesis can be used to model fluid flow in nanopores. The continuity hypothesis model can be used to describe the relationship between macroscopic fluid properties and spatial coordinates, which is widely used in fluid flow. Knudsen number usually is used to justify if fluid flow satisfies the continuity hypothesis and then determine the fluid flow regime. Knudsen number is defined as the ratio of gas molecular mean free path to the characteristic length of porous media, representing the relative degree of gas molecule collision with the gas molecules and pore walls. Its expression is where is the gas molecular mean free path of gas (nm) and is the average hydraulic radius of pore media (nm).

As shown in Figure 7, when , the gas molecule velocity of a pore wall is zero, and Darcy’s law is valid. This transport mechanism is also known as continuum flow. When , the gas molecule velocity of a pore wall is no longer zero, and, consequently, the gas flux is increased. Darcy’s law is no longer valid, and this transport mechanism is referred to as rarefied gas transport. The rarefied gas transport is further subdivided into slip flow (), transition flow (), and free-molecular flow (). Under normal reservoir conditions of shale gas reservoirs, Knudsen number ranges from 0.0002 to 6 [1, 26].

By using Equation (1), the Knudsen number of pure methane under different pressure and different radius of pores with a temperature of 350 K can be calculated. And the results can be plotted as shown in Figure 8. We can clearly see that the gas flow in larger-scale pores, such as natural fractures, can be treated as a continuous flow. However, for small-scaled pores, the gas flow covers from slippage flow to transitional flow regimes, which is much more complicated than the continuous flow. The small scale pores always exist in the shale matrix.

To describe the mechanism of nanopore gas transport in continuous flow, slip flow, transition flow, and diffusion flow, scholars have proposed many coupling models considering different mechanisms. Ertekin et al. first established the coupling mechanism model considering continuous flow and Fick diffusion, and these two mechanisms directly adopted linear superposition. In this paper, the transport mechanism of gas in porous media was divided into bulk flow (intermolecular interaction), Knudsen diffusion (gas-solid interface interaction), and surface diffusion of the adsorption layer, among which the gas slip flow was essentially equivalent to Knudsen diffusion [28].

In 2007, Javadpour et al. proposed an apparent permeability model of shale matrix considering the dual mechanism of Knudsen diffusion and slip flow. The model had a similar form to the Darcy equation and was easy to apply [29]. Then, based on the Javadpour model, Darabi et al. introduced the influence of the pore-throat structure characteristics of the shale matrix on the gas flow law (the tortuosity and roughness, etc.) by introducing the fractal theory [30].

In 2012, Shabro et al. established a shale gas flow mechanism model that considers dissolved gas diffusion, slip flow, Knudsen diffusion, and Langmuir desorption in kerogen [31]. In 2014, Mi et al. studied the apparent permeability model of the fracture system and matrix system, respectively, by linear superposition. The methods of establishing these models were based on Javadpour’s model, which was to superimpose the slip flow and the diffusion. The difference was that Mi et al. divided the flow types into Knudsen diffusion, transitional diffusion, and Fick diffusion according to the number [32].

In 2015, based on the transport mechanisms of slip flow and Knudsen diffusion, Wu et al. established a nanopore shale gas transmission model according to the ratio of the collision frequency between molecules and the collision frequency of molecules to the wall of the pores as the weighting factors of the slip flow and the Knudsen diffusion [26]. In the same year, Sheng et al. proposed a comprehensive model for coupling gas viscous flow, slip flow, Knudsen diffusion, and surface diffusion based on the weighted superposition of Wu et al.’s [33]. Subsequently, according to different transport mechanisms and pore structure characteristics of the gas, various multiple apparent permeability models were successively proposed by using different superposition methods [34, 35].

In 2017, Li et al. published their research results in the Journal of Physics that both Nusen diffusion and surface diffusion are related to the gas-solid interaction at the interface [36]. Knudsen diffusion is a diffusion phenomenon that occurs after gas molecules collide with the wall surface. Surface diffusion is a process in which gas molecules continuously jump between adsorption sites on the pore surface. In both cases, the velocity of the gas molecules on the wall surface is not zero, which is consistent with the slip phenomenon. Li et al. believed that it is debatable to superimpose the Knudsen diffusion and slip flow, or to superimpose the slip flow and surface diffusion, or to superimpose these three transport mechanisms in flow models. Since the shale gas mass transport mechanism was introduced from the theory of rarefied gas dynamics, the aerodynamic researcher also thought that, from the point of view of rarefied gas, slip flow was essentially the same as Knudsen diffusion, which was determined by the Boltzmann equation and gas-solid interaction, but the coefficients were different under different [37, 38]. After that, Cai et al. proposed a very interesting apparent permeability model by accounting for three major fluid flow mechanisms in shale stratum, which is modeled as a 3D fractal media. This model can present the gas flow in shale pore media more accurately [39, 40].

Table 2 lists the apparent permeability models proposed by different scholars in recent years for different transport mechanisms of shale gas under different coupling methods. In general, scholars believe that the fluid flow mechanism in the micro-nanopore system of shale gas mainly includes viscous flow, slip flow, Knudsen diffusion, surface diffusion, adsorption layer effect, reservoir pore structure characteristics, and stress sensitivity. However, there are still differences in the essential relations between various mechanisms, which need to be further studied. Meanwhile, the existing mechanism models are not combined with the actual characteristics of pore-throat scale distribution. Whether the same mechanism model is applicable to reservoirs of various types of scales is still debatable and remains to be further studied. In order to better guide the efficient development of shale gas, only by revealing the complex transmission mechanism of shale gas in the micro-nanopore system can we provide an accurate mechanism model for shale gas numerical simulation and production dynamic evaluation.

No.Apparent permeability formulaAuthors

1Ertekin et al. [28]
2Ertekin et al. [28]
3Ertekin et al. [28]
Beskok-Karniadakis [41]
Javadpour et al. [29]
Wu et al. [42, 43]
, ,
Darabi et al. [30]
Xiong et al. [44]
Swami and Settari [45]
Zhao et al. [46]
, ,
Li et al. [25]
Mi et al. [32]
Deng et al. [47]
13Wang and Marongiu-Porcu [48]

Wu et al. [26, 42, 43]
, , ,
Sheng et al. [33]
Ye et al. [49]

Wang et al. [50]
Zhang et al. [35]
19Song et al. [51]
Ren et al. [52]
21Wang et al. [34]
, ,
He et al. [53]
,, ,
Li et al. [36]
Zhang et al. [54]
25 (for inorganic pores)
(for organic pores)
Wang et al. [38]
Cai et al. [55]