Shale formation is featured in nanopores and much gas adsorptions. Gas flow in the shale matrix is not a singular viscous flow, but a combination of multiple mechanisms. Much work has been carried out to analyze apparent permeability of shale, but little attention has been paid to the effect of unique gas behavior in nanopores at high pressure and adsorbed layer on apparent permeability. This work presents a new model considering multiple transport mechanisms including viscous flow (without slip), slip flow, Knudsen diffusion, and surface diffusion in the adsorption layer. Pore diameter and mean free path of gas molecules are corrected by considering the adsorption layer and dense gas effect, respectively. Then the effects of desorption layer, surface diffusion, and gas behavior on gas apparent permeability in nanopores of shale are analyzed. The results show that surface diffusion is the dominant flow mechanism in pores with small diameter at low pressure and that the effect of adsorbed layer and dense gas on apparent permeability is strongly affected by pressure and pore diameter. From the analysis results, the permeability value calculated with the new apparent permeability model is lower than in the other model under high pressure and higher than in the other model under high pressure, so the gas production calculated using the new permeability model will be lower than using the other model at early stage and higher than using the other model at late stage.

Much attention has been paid to shale due to the considerable volume of natural gas trapped in it. Over the past decades, technology advances in horizontal drilling and hydraulic fracturing have enabled profitable production of shale gas. However, because of the unique deposit character and flow mechanisms of shale gas, controversy still exists on how much gas can be produced from shale [

Compared with conventional reservoirs, shales are characterized with pores between 1 and 100 nm, of which the dominant diameter is in nanoscale [

The first characteristic of shale formation affecting apparent permeability is that there are a lot of organics and clay minerals with much gas adsorption in the shale matrix [

This paper presents a comprehensive apparent permeability model in which all the flow mechanisms, including viscous flow, slip flow, Knudsen diffusion, adsorption layer, surface diffusion, and dense gas effect, are taken into account. Then the effects of the desorption layer, surface diffusion, and dense gas in shale nanopores with different diameters are analyzed.

The Langmuir isothermal adsorption equation is widely used to describe the adsorption and desorption of coal bed methane [

Langmuir adsorption isotherm.

As the radius of shale nanopores is in the same order with that of methane molecules, the effect of the adsorbed layer on gas retention and flow capacity is not negligible. For example, a pore whose diameter is 4 nm cannot allow 10 methane molecules at most to pass at the same time. In a single capillary, the adsorbed layer reduces the capacity of fluid flow, as shown in Figure

The effect of the adsorbed layer on the effective sectional flow area of pores of different diameters.

The adsorbed layer is taken into account in this paper with the assumption of single-layer adsorption. As adsorption molecules do not exist on all adsorption sites, coverage of adsorption molecules is introduced to calculate the thickness of the adsorbed layer; the effective pore diameter is [

The effective pore radius is

The hopping model is used in this paper to characterize surface diffusion in shale gas reservoirs, as shown in Figure

A local equilibrium exists between bulk gas and adsorbed gas.

Surface transport takes place by activated diffusion, that is, site hopping.

The surface diffusion coefficient at higher pressure can be corrected by that at lower pressure with coverage of adsorbed gas defined in (

Hopping model for surface diffusion.

A surface diffusion coefficient at higher pressure is offered by correcting the coefficient at lower pressure, considering the effect of gas coverage on surface diffusion [^{2}/s; ^{2}/s;

Eqs. (

The intrinsic size of molecules and the interaction between molecules are negligible for ideal gas. In this scenario, the mean free path of gas molecules can be expressed as [^{−23} J/K;

However, the size of the gas molecule is not negligible in actual shale formations. On the other hand, due to high pressure and short distance between gas molecules, the gas should be viewed as dense gas. According to the dense gas theory presented by Enskog [

According to the dense gas theory of Enskog [

As the collision rate increases, a tertiary collision between molecules and the blocking effect are introduced. The rectified collision rate [

Eq. (

Figure

The effect of dense gas on the mean free path of gas molecules (

There are three transport mechanisms for gas flow in shale nanopores (see Figure

Three gas transport mechanisms in shale nanopores.

Based on the surface diffusion coefficient in ^{3}, which is a fitting parameter of experiment data based on the Langmuir theory, which can be calculated with SLD models [^{3}; and

According to Wu et al. [

However, surface diffusion is the transport regime for adsorbed gas and is irrelevant to free gas in the bulk phase. Therefore, in order to get total apparent permeability, surface diffusion permeability of the adsorption layer should be corrected and combined with apparent permeability of the bulk phase. For surface diffusion, the flow section is made up by the molecules in the adsorption layer and should be converted to the effective flow section for the whole pore, so the weighting coefficient of surface diffusion

In summary, surface diffusion permeability in a single capillary can be expressed as

For the viscous/slip flow and Knudsen diffusion of the bulk phase, the weighting coefficients are [

For a single capillary, the viscous flow and slip flow permeability of the bulk phase can be expressed as

Despite the fact that the generalized model cannot cover all the flow regimes, it is completely applicable to viscous flow and slip flow when

The viscous flow and slip flow permeability without considering the weighting coefficient can be expressed as [

The viscous flow and slip flow permeability in a single capillary driven by pressure difference considering the weighting coefficient defined in (

The Knudsen diffusion permeability of the bulk phase is [^{3}/mol.

The Knudsen diffusion permeability of the bulk phase, considering the weighting coefficient defined in (

The total apparent permeability of shale nanopore is

In the apparent permeability model shown in (

To validate the model proposed in this paper, experimental data for CO_{2} from [_{2} is used as absorbent gas. Therefore, the experiment data is applied to validate the model with surface diffusion considered in this study. The detail of the experiment is presented in [

The parameters are shown in Table ^{∗}) are matched with the experimental data. The density of bulk gas is calculated in (

Parameters for the validation case.

Parameter | Symbol | Unit | Value |
---|---|---|---|

Formation temperature | K | 300 | |

Porosity | Dimensionless | 0.048 | |

Tortuosity | Dimensionless | 1.35 | |

Gas type | CO_{2} |
— | — |

Gas molecule diameter | m | 4.64 × 10^{−10} | |

Gas molar mass | M | kg/mol | 4.4 × 10^{−2} |

Universal gas constant | J/(mol·K) | 8.3145 | |

Boltzmann constant | J/K | 1.3805 × 10^{−23} | |

Molar volume of gas under standard condition | m^{3}/mol |
0.0224 | |

Avogadro’s constant | 1/mol | 6.02 × 10^{23} | |

Langmuir pressure | MPa | 1.8 | |

J/mol | 203,000 | ||

Dimensionless | 0.21 | ||

nm | 2.8 |

The parameters highlighted with asterisk (

In this study, three models are used to match the experiment data as shown in Figure

Validation of the proposed model with experimental data (from [

The apparent permeability model of shale nanopores presented in this paper is compared with previous models, as shown in Figure

The effect of different factors on total apparent permeability (pore

Parameters for modeling results and discussion.

Parameter | Symbol | Unit | Value |
---|---|---|---|

Formation temperature | K | 368 | |

Porosity | Dimensionless | 0.05 | |

Tortuosity | Dimensionless | 3.5 | |

Gas type | CH_{4} |
— | — |

Gas molecule diameter | m | 3.8 × 10^{−10} | |

Gas molar mass | kg/mol | 1.6 × 10^{−2} | |

Universal gas constant | J/(mol·K) | 8.3145 | |

Boltzmann constant | K_{B} |
J/K | 1.3805 × 10^{−23} |

Molar volume of gas under standard condition | m^{3}/mol |
0.0224 | |

Avogadro’s constant | 1/mol | 6.02 × 10^{23} | |

Langmuir pressure | MPa | 5.82 | |

Isosteric adsorption heat at the gas coverage of zero | J/mol | 16,000 | |

The ratio of the rate constant for blockage to the rate constant for forward migration | Dimensionless | 0.5 | |

Pore diameter | nm | 2/4/10/20/50/100/200 |

Figure

Figure

The effect of different factors on total apparent permeability (pore

Parameters used in this section are shown in Table

Surface diffusion permeability and its percentage in total apparent permeability.

Surface diffusion permeability

Percentage of surface diffusion permeability in total apparent permeability

Meanwhile, in smaller pores, the percentage of the adsorption layer in pore space is larger and the surface diffusion effect is more significant: when the pore diameter is no more than 10 nm, the contribution of surface diffusion to total apparent permeability is more than 60% even when formation pressure is up to 40 MPa; when the pore diameter is more than 50 nm, the percentage of surface diffusion permeability is smaller than 10% even when pressure is as low as 5 MPa.

Figure

Viscous flow and slip flow permeability and their percentage in total apparent permeability.

Viscous flow and slip flow permeability

Percentage of viscous flow and slip flow permeability in total apparent permeability

The viscous flow and slip flow permeability and surface diffusion permeability are in the same magnitude and decrease with increasing pressure, but the decline rate of the former is significantly smaller than that of the latter, and the percentage of viscous flow and slip flow in total apparent permeability increases with increasing pressure. For pores whose diameter is larger than 50 nm, viscous flow and slip flow permeability is dominant and even when pressure decreases to 5 MPa, the contribution is more than 70%; for pores whose diameter is no more than 10 nm, even when pressure reaches 40 MP, the percentage of viscous flow and slip flow in total apparent permeability is smaller than 15%.

Figure

Knudsen diffusion permeability and its percentage in total apparent permeability.

Knudsen diffusion permeability

Percentage of Knudsen diffusion permeability

The percentage of Knudsen diffusion permeability in total apparent permeability depends on both formation pressure and pore diameter. For pores whose diameter is larger than 50 nm, the percentage of

Figure

Apparent permeability of shale nanopores of different diameters.

In this section, the effect of the adsorbed layer, Knudsen diffusion, dense gas effect, and pore size distribution on apparent permeability is analyzed; the parameters of the model are shown in Table

The effect of the adsorption layer on total apparent permeability of shale (pore

The effect of the adsorbed layer on apparent permeability of shale (pore

Effect on surface diffusion permeability

Effect on viscous flow and slip flow permeability

Effect on Knudsen diffusion permeability

Effect on total apparent permeability

The effect of the adsorption layer on total apparent permeability of shale (pore

The effect of adsorbed layer on apparent permeability of shale (pore

Effect on surface diffusion permeability

Effect on viscous flow and slip flow permeability

Effect on Knudsen diffusion permeability

Effect on total apparent permeability

In order to characterize the effect of the adsorption layer on total apparent permeability, the improvement factor of total apparent permeability due to adsorption layer is defined as

Figure

The effect of the adsorbed layer on total apparent permeability (formation

Figure

The effect of gas density on apparent permeability of shale (pore

Effect on surface diffusion permeability

Effect on viscous flow and slip flow permeability

Effect on Knudsen diffusion permeability

Effect on total apparent permeability

Figure

The effect of gas density on apparent permeability of shale (pore

Effect on surface diffusion permeability

Effect on viscous flow and slip flow permeability

Effect on Knudsen diffusion permeability

Effect on total apparent permeability

Similarly, in order to characterize the effect of dense gas on total apparent permeability, the improvement factor of total apparent permeability due to dense gas effect is defined as

Figure

The effect of gas density on total apparent permeability (20 MPa).

Gas retention and transport mechanism in shale nanopores is closely related to the pore diameter. However, pore size distribution of the shale core is in a wide range, the effect of the adsorption layer on pores of different diameters varies, and consequently the Knudsen number at the same temperature and pressure is different. Therefore, pore size distribution should be considered in the characterization of total apparent permeability of actual shale formation to make it more feasible.

Based on the transport mechanism for a single capillary in shale gas reservoirs, the storage space in shale is simplified as ideal rock which is made up by a bundle of capillaries of various diameters, as shown in Figure

Schematic of capillary bundle.

Pore size distribution curves of typical shale in Barnet (from [

In the capillary bundle model, the tortuosity of each capillary is 3.5, which is an assumption in this case. Actually, this value may be higher than its truth, and different capillaries are of different tortuosity.

The model is used to analyze the apparent permeability of typical shale at different pressure, as shown in Figure

Total apparent permeability of type shale.

If a certain shale field is given, the apparent permeability can be calculated with the procedure shown in Figure

Procedure for apparent permeability calculation.

However, there are still some limitations for this model. The pore size will change at high pressure because of the reduction of effective stress [

An apparent permeability model is presented in this paper, in which multiple flow mechanisms, including viscous flow, slip flow, Knudsen diffusion, adsorption layer, surface diffusion, and dense gas effect are taken into account. The main findings can be summarized as follows:

The effect of the adsorption layer and dense gas on apparent permeability of shale nanopores is characterized and surface diffusion of adsorbed gas is introduced to establish an apparent permeability model for the shale matrix.

Apparent permeability of the shale matrix is composed of three parts: surface diffusion permeability of the adsorbed phase, Darcy flow and slip flow permeability of the bulk phase, and Knudsen diffusion permeability of the bulk phase. The percentage of permeability caused by various mechanisms in total apparent permeability depends on pore diameters: when the pore diameter is larger than 50 nm, Darcy flow and slip flow of the bulk phase are dominant transport mechanisms; when the pore diameter is no more than 10 nm, surface diffusion of the adsorbed phase is the dominant transport mechanism; when the pore diameter is between 10 and 50 nm, especially when the pore diameter is around 20 nm, the percentage of surface diffusion of the adsorbed phase, Darcy flow, and slip flow of the bulk phase and Knudsen diffusion of the bulk phase is equivalent.

If pressure is more than 20 MPa, for pores whose diameter is smaller than 20 nm, the adsorption layer improves total apparent permeability; for pores whose diameter is larger than 20 nm, the adsorption layer reduces total apparent permeability.

If pressure is less than 20 MPa, for pores whose diameter is smaller than 20 nm, the dense gas effect makes the declining rate of total apparent permeability increase with pore diameter; for pores whose diameter is larger than 20 nm, the dense gas effect makes the declining rate of total apparent permeability decrease with pore diameter.

This study was once presented in “China Shale Gas 2015 (International Conference)”.

The authors declare that they have no conflicts of interest.

This study is supported by the National Science and Technology Major Project (no. 2017ZX05037001) and National Natural Science Fund of China (nos. 41672132 and U1762210).