# A New Model for Determining the Effective Permeability of Tight Formation

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## Abstract

Unlike conventional reservoirs, tight reservoirs have complex pore structures and severe boundary-layer effect. The pore throat of tight reservoirs is in nanoscale and the boundary layer cannot be ignored because the boundary layer has an important effect to the fluid flow and its influence increases with the reduction in the pore throat radius. These are the main reasons for the ultra-low permeability and low oil recovery for these reservoirs. However, previous studies have paid limited attention to the influences of the boundary-layer effect and the pore size distribution. In this paper, a new model was built to determine the effective permeability for the tight gas and oil reservoirs by taking into account the boundary-layer effect and the pore distribution. The results from this new model show good agreement with the experiment data, and the main factors that impact the effective permeability were analyzed in the study. It is found that the fluid type, means and standard deviation of pore radius, and displacement pressure gradient are the main factors influencing effective permeability. The relationship of air permeability and liquid permeability is also analyzed for tight reservoirs.

## Keywords

Tight formation Effective permeability Pore size distribution Boundary layer Gas slippage## List of symbols

*r*Pore throat radius (\(\upmu \mathrm{m}\))

- \(\nu \)
Mean pore radius (\(\upmu \mathrm{m}\))

- \(\sigma \)
Standard deviation (\(\upmu \mathrm{m}\))

- \(f(r_i )\)
The proportion of micro-tubes of radius \(r_i \) (%)

*N*The total number of the micro-tubes

*A*The cross-sectional area of the rock (\(\upmu \mathrm{m}^{2}\))

- \(\mu \)
Oil viscosity (\(\mathrm{mPa}\,\mathrm{s}\))

- \(\tau \)
Pore structure coefficient

- \(\Delta P\)
Pressure difference (MPa)

- \(\phi _m \)
Porosity (%)

- \(p_o \)
Barometric pressure (MPa)

- \(\mu /\varepsilon \)
Slip coefficient \((1/{\upmu } \mathrm{m})\)

## 1 Introduction

Considerable efforts have been made on the recovery of the tight oil as part of developing unconventional resources in the last decade or so, and oil from tight reservoirs represent an increasing share of the world’s hydrocarbon source. Tight oil reservoirs have ultra-low permeability, which presents greater challenges for field development in general and for oil recovery in particular. The main reasons for low permeability of tight oil reservoirs include the complex distribution of the pore structures and boundary-layer effect in the pore throats of these reservoirs. In tight oil reservoirs, most pore throats are highly rugose. As the rugosity inevitably increases the length of the path for fluid flow, the resistance would be extremely high for the fluid flow. Moreover, the boundary-layer effect is a problem. The interaction between the solid particles and the liquid is very strong so that a layer of liquid will be attached on the inner surface of the pore and restricts the flow.

For conventional reservoirs, pore throats are relatively uniform, and in fact, uniform pore radius is often used to describe the distribution of pore throats. A number of permeability models have been proposed.

Kozeny (1927) and Carman (1937) built the equivalent capillary model for matrix, which is a simplified representation of the rock and plays a fundamental role in characterizing reservoir permeability. Purcell (1949) derived an equation showing the relationship of permeability to capillary pressure curve. Burdine (1953) calculated the relative permeability from pore size distribution data and used the tortuosity factor in the permeability equation as a term of correction for the difference between the theoretical model and the actual rock. Brooks and Corey (1966) modified the representation of capillary pressure function to a more general form and obtained a new formation of permeability function because of the limitation of Corey’s model (1954). However, for tight oil reservoirs, the pore throat distribution is distinct from the conventional reservoirs and displays the feature of heterogeneity. Many experiments have shown that the pore distribution of the tight oil reservoirs exhibits a wide range (Youssef et al. 2007; Riepe et al. 2011; Li et al. 2012; Wang et al. 2009; Nelson 2009; Desbois et al. 2011; Camp 2011). Thus, high deviations may occur when calculating the permeability of tight oil reservoirs while assuming a uniform distribution (Bernabé and Bruderer 1998; Latour et al. 1995; Banavar and Johnson 1987) and the micro-tube model based on uniform radius is not suitable to the determination of permeability for tight reservoirs.

There are still rare reports about the inclusion of pore size heterogeneity in permeability model. It is possible to use a function to represent the distribution, and such a function has the potential advantage of integrating all the variables related to the pore throat and permeability.

Another factor influencing the permeability of the tight reservoirs is the boundary-layer effect, and the boundary layer will reduce the effective flowing space of the micro-tubes and constrain fluid flow. The conception of boundary-layer effect for flow in porous media was first proposed by Mapxacин И Л (1977). He supposed there was a thin layer of liquid on inner surface of the pore because of the interaction of the solid and the liquid. Li et al. (2011) and Mala and Li (1999) conducted micro-tube experiment with deionized water, which proved the existence of boundary layer. Some previous experiment results (Li and He 2005; Zhang et al. 2008; Xu and Yue 2007) showed that the proportion of the boundary layer was controlled by several factors, such as radius of the micro-tube and liquid viscosity.

For tight reservoirs, the scale of the pore throat is mainly in microscale and nanoscale so that the effect of boundary layer on fluid flow in tight reservoirs is obvious. The comparison between gas permeability and liquid permeability is a good case for the impact of boundary layer on permeability measurement of the same reservoir. There exists gas slippage effect (Sampath and Keighin 1982; Daixun 2002; Carman 1956) when gas transports in the pore throats so that the boundary-layer effect is negligible. Gas permeability is normally larger than liquid permeability (Klinkenberg 1941; Dabbous et al. 1974; Li and Horne 2001, 2004). In general, people use gas permeability to represent the permeability of tight reservoirs for its convenience to obtain. However, this is usually not recommended for its inaccuracy. Boundary-layer effect has been ignoring in the permeability expression, which may develop deviation in application especially for tight reservoirs.

The objective of this paper is to propose a new method for calculating the effective permeability of tight formation considering the pore distribution, which plays crucial roles for the tight formation. And also, the boundary-layer effect, which is obvious in nano- or micropores, is included in the calculation of the effective permeability. This paper aims to introduce a conception that pore distribution and boundary-layer effect, which are rarely considered in the previous work, are really essential to permeability of tight formation and cannot be neglected. We use the experimental results to verify the accuracy of this model and discuss the influential variables based on factor analysis with the new model. The results show this model has enough accuracy and can be used in practice for its convenience and simplicity.

## 2 Modeling

- 1.
The model consists of a bunch of micro-tubes in serial radii and there is sealed volume between micro-tubes;

- 2.
Fluid and rock are incompressible;

- 3.
The flow in reservoirs conforms to Darcy’s law and the flow in the micro-tubes satisfies Poiseuille Equation;

- 4.
The throat size distribution can be represented by a function, such as Gaussian distribution;

- 5.
The pore throat distribution is based on the mercury experiment and can be used to represent the heterogeneity of the rock.

### 2.1 Boundary Layer Effect on Effective Pore Radius and Permeability

### 2.2 Expression of Boundary Layer from Micro-tube Experiments

#### 2.2.1 Experimental Setup

Results and parameters of the micro-tube experiment (Data digitized from Li et al. 2011)

Case1 | Case2 | Case3 | Case4 | Case5 | |||||
---|---|---|---|---|---|---|---|---|---|

\(r=2.5\,\upmu \hbox {m}\) | \(\upmu =1\,\hbox {mPa}\,\hbox {s}\) | \(r=5.0\,\upmu \hbox {m}\) | \(\mu =1\,\hbox {mPa}\,\hbox {s}\) | \(r=7.5\,\upmu \hbox {m}\) | \(\mu =1\,\hbox {mPa}\,\hbox {s}\) | \(r=10\,\upmu \hbox {m}\) | \(\mu =1\,\hbox {mPa}\,\hbox {s}\) | \(r=5.0\,\upmu \hbox {m}\) | \(\mu =1.5\,\hbox {mPa}\,\hbox {s}\) |

\(\nabla P\) (MPa/m) | | \(\nabla P\) (MPa/m) | | \(\nabla P\) (MPa/m) | | \(\nabla P\) (MPa/m) | | \(\nabla P\) (MPa/m) | |

0.01 | 0.001 | 0.013 | 0.023 | 0.011 | 0.069 | 0.011 | 0.127 | 0.015 | 0.015 |

0.03 | 0.005 | 0.026 | 0.034 | 0.022 | 0.103 | 0.022 | 0.231 | 0.029 | 0.022 |

0.05 | 0.011 | 0.04 | 0.056 | 0.034 | 0.149 | 0.033 | 0.335 | 0.044 | 0.037 |

0.08 | 0.02 | 0.053 | 0.079 | 0.045 | 0.195 | 0.044 | 0.462 | 0.058 | 0.052 |

0.099 | 0.038 | 0.067 | 0.113 | 0.055 | 0.253 | 0.055 | 0.601 | 0.073 | 0.075 |

0.149 | 0.072 | 0.08 | 0.148 | 0.066 | 0.322 | 0.067 | 0.728 | 0.088 | 0.097 |

0.199 | 0.115 | 0.094 | 0.159 | 0.077 | 0.391 | 0.078 | 0.866 | 0.103 | 0.104 |

0.241 | 0.145 | 0.106 | 0.205 | 0.098 | 0.576 | 0.09 | 1.005 | 0.117 | 0.135 |

0.28 | 0.186 | 0.121 | 0.25 | 0.11 | 0.656 | 0.101 | 1.155 | 0.133 | 0.165 |

0.635 | 0.855 | 0.146 | 0.319 | 0.121 | 0.772 | 0.123 | 1.444 | 0.161 | 0.21 |

1.252 | 1.282 | 0.174 | 0.387 | 0.142 | 0.922 | 0.146 | 1.733 | 0.191 | 0.255 |

1.851 | 1.709 | 0.2 | 0.467 | 0.153 | 1.002 | 0.168 | 2.022 | 0.22 | 0.308 |

2.45 | 1.709 | 0.214 | 0.513 | 0.165 | 1.095 | 0.179 | 2.195 | 0.236 | 0.338 |

3.049 | 2.137 | 0.241 | 0.64 | 0.198 | 1.36 | 0.202 | 2.507 | 0.265 | 0.421 |

3.784 | 2.564 | 0.268 | 0.72 | 0.208 | 1.452 | 0.225 | 2.831 | 0.294 | 0.474 |

4.356 | 2.991 | 0.389 | 3.487 | 0.544 | 3.205 | 0.266 | 3.846 | 0.718 | 2.294 |

4.927 | 3.856 | 1.443 | 7.051 | 0.98 | 5.769 | 0.381 | 7.051 | 1.312 | 4.639 |

1.824 | 8.333 | 1.497 | 8.974 | 0.653 | 10.256 | 1.658 | 5.482 | ||

2.532 | 11.539 | 2.015 | 11.539 | 0.98 | 14.103 | 2.302 | 7.591 | ||

2.995 | 13.462 | 3.022 | 17.308 | 1.388 | 20.513 | 2.722 | 8.856 | ||

3.457 | 16.026 | 3.92 | 23.077 | 1.661 | 23.718 | 3.143 | 10.543 | ||

4.165 | 17.949 | 4.546 | 26.282 | 2.559 | 35.256 | 3.787 | 11.808 | ||

4.927 | 21.154 | 4.927 | 28.846 | 3.838 | 54.487 | 4.479 | 13.917 | ||

5.771 | 25.082 | 5.635 | 32.692 | 5.172 | 72.436 | 5.247 | 16.447 |

#### 2.2.2 Representation of the Thickness of Boundary Layer

### 2.3 PTR Distributions for Tight Reservoirs

Characteristic values of different pore throat distributions

Pattern | Maximum throat radius \((\upmu \hbox {m})\) | Mean pore radius \((\upmu \hbox {m})\) |
---|---|---|

a | 22 | 10 |

b | 11 | 5 |

c | 1.1 | 0.5 |

Because of the boundary layer, the effective pore radius for fluid flow is smaller than the actual pore radius. The effective pore radius can be obtained by deducting the thickness of the boundary layer from the actual pore radius. Therefore, the distribution of the effective pore radius will vary from that of the total pore radius. For conventional and some low-permeability reservoirs such as shown in Fig. 6a and b, the differences between the two curves are small. This is because the proportion of the boundary layer can be neglected when the pore radius is relatively large. For tight reservoirs, such as shown in Fig. 6c, the distinctions between the two curves are obvious, which is because the boundary layer occupies a large proportion of the pore radius. Therefore, the effective pore radius curve is quite different from the total pore radius curve for tight reservoirs.

### 2.4 Permeability Model

*A*is the cross-sectional area of the rock. For a section with a bunch of tubes in serial radii, the area could be calculated by

## 3 Model Validation

Reservoir parameters of Yanchang formation

Parameters | No. 1 | No. 2 | No. 3 | No. 4 | No. 5 | No. 6 | No. 7 | No. 8 |
---|---|---|---|---|---|---|---|---|

Porosity (%) | 8.20 | 6.20 | 12.70 | 8.40 | 6.80 | 6.00 | 5.10 | 12.10 |

Air Perm \((10^{-3}\times \upmu \hbox {m}^{2})\) | 0.40 | 0.16 | 0.12 | 0.18 | 0.10 | 0.10 | 0.05 | 0.23 |

Oil Perm \((10^{-3}\times \upmu \hbox {m}^{2})\) | 0.014 | 0.006 | 0.004 | 0.005 | 0.003 | 0.003 | 0.001 | 0.01 |

Data of pore distributions of Yanchang Formation

Pore size \((\upmu \hbox {m})\) | No. 1 (%) | No. 2 (%) | No. 3 (%) | No. 4 (%) | No. 5 (%) | No. 6 (%) | No. 7 (%) | No. 8 (%) |
---|---|---|---|---|---|---|---|---|

0–0.01 | 38.41 | 27.42 | 5.52 | 52.03 | 46.86 | 38.69 | 66.50 | 7.31 |

0.01–0.02 | 9.12 | 20.21 | 12.61 | 9.70 | 8.34 | 15.58 | 9.04 | 16.23 |

0.02–0.05 | 15.84 | 17.42 | 29.65 | 17.41 | 11.35 | 21.59 | 8.78 | 17.56 |

0.05–0.10 | 12.62 | 10.43 | 19.56 | 10.94 | 11.21 | 13.09 | 5.98 | 10.25 |

0.10–0.20 | 12.17 | 9.31 | 15.5 | 2.95 | 11.05 | 4.07 | 3.09 | 12.34 |

0.20–0.30 | 6.73 | 8.12 | 9.51 | 1.66 | 5.54 | 2.11 | 1.77 | 11.21 |

0.30–0.40 | 2.78 | 3.84 | 4.76 | 0.91 | 1.64 | 0.91 | 0.86 | 8.65 |

0.40–0.50 | 1.11 | 2.22 | 1.52 | 0.57 | 0.36 | 0.48 | 0.42 | 8.64 |

0.50–0.60 | 0.51 | 1.01 | 0.78 | 0.32 | 0.21 | 0.25 | 0.28 | 4.76 |

0.60–1.00 | 0.31 | 0.37 | 0.65 | 0.26 | 0.14 | 0.13 | 0.22 | 3.05 |

Based on the pore distributions and other related reservoir parameters, the effective permeability of the core samples can be calculated with the new model (Eq. 9). The comparison between experimental data and the calculated data from the model are shown in Fig. 7. The error line gives the differences between experimental data and calculated data. The results show that the calculated permeability has a good match to the permeability from experiment. The accurate match suggests a high reliability of the model for calculating the effective permeability of tight oil reservoirs.

## 4 Influential Variables and Sensitivity Analysis

### 4.1 Influential Variables

When the pore distribution is represented by Gaussian distribution, two parameters determine the pore throats distribution, the mean and the standard deviation of pore radius. Variations in these two parameters will change pore constitution and also the permeability. Moreover, displacement pressure gradients will also affect the thickness of boundary layer and thus affect the effective permeability.

#### 4.1.1 Effects of Standard Deviation of Pore Radius

In tight reservoirs, pore throats distribute in a quite wide range. Since we use a Gaussian distribution, the mean and standard deviation are used to describe the range of the pores. For example, assuming a mean of \(0.5\,\upmu \hbox {m}\, (\upsilon =0.5\,\upmu \hbox {m})\) for the pore radius, the Gaussian distributions for various standard deviations (\(\sigma =0.05\), 0.15, 0.25, 0.35 and \(0.45\,\upmu \hbox {m}\)) of the pore radius are shown in Fig. 8.

#### 4.1.2 Effects of the Mean Pore Radius

The mean pore radius is one of the most important factors to characterize a pore distribution, and it plays a fundamental role for determining the permeability of tight oil reservoirs. We constructed several pore distributions with different mean pore radius and constant standard deviation (Fig. 10) and then calculated the effective permeability using our model (Eq. 9).

The permeability is calculated with different mean pore radii and a constant standard deviation (Fig. 11). As seen, the permeability increases with the mean pore radius for a given standard deviation. The smaller the standard deviation is, the faster the increasing rate of permeability will be. This is simply because when the mean pore radius increases, the overall pores will become larger so that the permeability will increase.

#### 4.1.3 Effects of Displacement Pressure Gradient

As seen (Fig. 12), the permeability will increase with the increase in pressure gradient; however, the permeability becomes stable under high pressure gradient. This is because the thickness of the boundary layer will decrease with the increasing displacement pressure gradient, and the effective pores radius will increase. Moreover, the boundary layer almost changes to the solid boundary layer under high displacement pressure gradient and the effective permeability will become nearly constant eventually.

### 4.2 Effects of Fluids

As seen (Fig. 13), with the increase in displacement pressure gradient, the parameter \(\alpha \) will decrease rapidly and eventually become almost constant. When the displacement pressure gradient is low, liquid flow only happens in the larger pore throats, while gas can flow through all pore throats, and the value of \(\alpha \) is large. With the increase in displacement pressure gradient, liquid flow can also happen in narrower pore throats. As there will be more pore throats contributing to the permeability, the permeability to liquid will increase and the value of \(\alpha \) will decrease. However, \(\alpha \) is still greater than 1 even when the displacement pressure gradient reaches 1.0 MPa/m. This is because the solid boundary layer always exists and restricts the flow. The effective flow radius to liquid will always be narrower than that to gas.

Figure 14 shows that with the increase in the mean pore radius, the parameter \(\alpha \) will decrease drastically and become almost constant when the pores are large enough. The tendency for \(\alpha \) is similar under different displacement pressure gradients. When the mean pore radius is small, liquid flow is extremely difficult in comparison with the gas flow, and the value of \(\alpha \) is large. The liquid flow will become much easier when the mean pore radius increases so that the oil permeability will become similar to air permeability and \(\alpha \) is small.

## 5 Conclusions

- 1.
For tight oil reservoirs, the boundary-layer effect plays an important role for determining the reservoir permeability, especially when the pore size is similar to the thickness of boundary layer. The boundary layer will drastically reduce the effective flow radius of pore throats.

- 2.
The mean and standard deviation of pore radius are two basic parameters that describe the distributions of the pore throats. The permeability will change as a function of these two parameters.

- 3.
The air permeability is mainly affected by gas slippage, and it is confirmed by experimental data. The ratio of air permeability and liquid permeability for tight reservoirs was obtained with different pore parameters in the study.

## Notes

### Acknowledgments

We acknowledge that this study was partially funded by the Beijing Natural Science foundation (No. 3144033), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130007120014), the National Natural Science Foundation of China (E040351304220) and China University of Petroleum, Beijing (No.2462015YQ0206).

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