# A Fractal Permeability Model of Tight Oil Reservoirs Considering the Effects of Multiple Factors

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fractal Geometry Theory

## 3. Fractal Permeability Model of Tight Oil Reservoirs

^{2}/s); ${k}_{B}$ denotes the Boltzmann’s constant (J/K); $T$ is the temperature (K); and ${n}_{L}$ is the number of molecules per unit volume (1/m

^{3}).

^{2}).

^{3}/s).

^{3}/s).

^{3}/s).

^{−15}; $A$ is the cross-section of cores (m

^{2}); and $K$ is the reservoir permeability considering the effects of multiple factors (10

^{−3}μm

^{2}).

## 4. Model Verifications and Analysis

#### 4.1. Fractal Permeability Model Verification

^{−3}μm

^{2}to 0.049 × 10

^{−3}μm

^{2}when the effective stress was larger than 35 MPa (Table 3). Figure 3b shows the relative error between the permeability calculated by the proposed model and that published by Liu et al. [48]. When the effective stress was 20 MPa or 29 MPa, the relative error was more than 5%, but the absolute error was small and ranged from just 0.035 × 10

^{−3}μm

^{2}to 0.069 × 10

^{−3}μm

^{2}when the effective stress was larger than 20 MPa (Table 4). The reason for the scenario where the relative error is great when the effective stress increases is that the classic Hertzian contact theory is no longer applicable due to the rock nonlinear deformation. From the analysis above, we could conclude that our proposed model can accurately calculate the tight reservoir permeability under a small effective stress condition. For the scenario where the effective stress was great, although the relative error was larger than 5%, the absolute error was tiny at less than 0.069 × 10

^{−3}μm

^{2}, which meets the field requirement.

#### 4.2. The Effects of Multiple Factors on Tight Reservoir Permeability

^{−3}μm

^{2}to 1.33 × 10

^{−3}μm

^{2}.

^{−3}μm

^{2}to 2.19 × 10

^{−3}μm

^{2}, and the increase rate was 10%.

^{−3}μm

^{2}to 0.71 × 10

^{−3}μm

^{2}when the rock elastic modulus decreased from 6 GP to 1 GP.

## 5. Conclusions

- (1)
- A decreasing porous fractal dimension results in a decrease in permeability. The permeability decreases with the increase in tortuosity fractal dimension and irreducible water. When the irreducible water increased from 0.35 to 0.55, the permeability decreased from 2.78 × 10
^{−3}μm^{2}to 1.33 × 10^{−3}μm^{2}, and reduced by more than half the permeability. - (2)
- An increase in slip length results in an increasing actual flow channel, which makes permeability increase. The initial permeability linear increases with an increasing slip lengths. When the slip length increased from 1 nm to 600 nm, the initial permeability increased from 1.98 × 10
^{−3}μm^{2}to 2.19 × 10^{−3}μm^{2}, and the increase rate was 10%. - (3)
- The permeability decreases with the increase of effective stress. The porous fractal dimension, tortuosity fractal dimension, slip length, and irreducible water have a tiny effect on the permeability changing rate with effective stress. When the parameters of the rock’s mechanical property (the rock elastic modulus and Poisson’s ratio) increased, the permeability’s decreasing rate with effective stress became small.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Flow Rate of Fluid Flowing through a Pore considering Multiple Factors’ Effects

^{3}/s); and ${q}_{1}(R)$ and ${q}_{2}(R)$ are the flow rate flowing through the bulk region and that flowing through the interface region (m

^{3}/s), respectively. The expressions of ${q}_{1}(R)$ and ${q}_{2}(R)$ are as follows:

## Appendix B. The Detailed Derivation of the Flow Rate of Fluid Flowing through a Core considering the Effects of Multiple Factors

^{3}/s).

^{3}/s).

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**Figure 1.**Schematic representation of oil flow in tight reservoirs: (

**a**) oil flow in a core with numerous tortuous pores; (

**b**) oil velocity profile in a pore considering the effects of multiple factors, including oil slip, irreducible water, and the effect of the wall.

**Figure 2.**The comparison between initial permeability calculated by the proposed model and that found in experiments.

**Figure 3.**The relative errors of the permeability calculated by the proposed model and that found in experiments under various effective stress conditions; ((

**a**) the relative error when compared with the experimental data published by Zhong et al. [49]; (

**b**) The relative error when compared with the experimental data published by Liu et al. [48]).

**Figure 4.**The comparison between the results of our model and that of Lei’s model [32].

**Figure 5.**The effect of PSD on permeability under various effective stresses. ((

**a**) Pore fractal dimensions; (

**b**) tortuosity fractal dimensions).

**Figure 7.**The effect of slip lengths on permeability: ((

**a**) various effective stresses; (

**b**) initial permeability).

**Figure 8.**The effect of rock’s mechanical property on permeability: ((

**a**) rock elastic modulus; (

**b**) Poisson’s ratio).

Sources | Core’s No. | Porosity,% | Permeability, 10^{−3} μm^{2} | Core’s Length, cm | Core’s Diameter, cm | The Maximum Pore Diameter, μm | The Minimum Pore Diameter, μm |
---|---|---|---|---|---|---|---|

Liu et al. [48] | S161-1 | 13.95 | 1.00 | 3 | 2.5 | 32.6482 | 0.0123 |

S148-2 | 10.54 | 0.55 | 3 | 2.5 | 0.6701 | 0.0021 | |

Z41-9 | 6.16 | 0.20 | 3 | 2.5 | 4.9643 | 0.0001 | |

Zhong et al. [49] | M132-1 | 15.01 | 1.99 | 3.11 | 2.51 | 12.9105 | 0.0067 |

M217-1 | 6.21 | 0.95 | 3.12 | 2.51 | 7.2083 | 0.0068 | |

M23-1 | 3.06 | 0.22 | 2.97 | 2.52 | 2.3986 | 0.0068 |

Sources | Core’s No. | The Pore Fractal Dimension | The Tortuosity Fractal Dimension |
---|---|---|---|

Liu et al. [48] | S161-1 | 1.8916 | 1.1374 |

S148-2 | 1.8300 | 1.2100 | |

Z41-9 | 1.8851 | 1.1632 | |

Zhong et al. [49] | M132-1 | 1.8912 | 1.1358 |

M217-1 | 1.8267 | 1.2298 | |

M23-1 | 1.7418 | 1.3408 |

**Table 3.**The comparison between the dimensionless permeability calculated by the proposed model and that found in experiments by Zhong et al. [49].

The Effective Stress, MPa | Core’s Number: M132-1 | Core’s Number: M217-1 | Core’s Number: M23-1 | ||||||
---|---|---|---|---|---|---|---|---|---|

Experimental Results | Calculated Results | Relative Errors | Experimental Results | Calculated Results | Relative Errors | Experimental Results | Calculated Results | Relative Errors | |

0 | 1.000 | 1.000 | 0.000% | 1.0000 | 1.0000 | 0.000% | 1.000 | 1.000 | 0.000% |

5 | 0.731 | 0.759 | 3.818% | 0.6906 | 0.6629 | 4.014% | 0.629 | 0.640 | 1.741% |

15 | 0.596 | 0.591 | 0.802% | 0.4487 | 0.4710 | 4.984% | 0.430 | 0.445 | 3.484% |

25 | 0.502 | 0.489 | 2.622% | 0.3651 | 0.3668 | 0.469% | 0.357 | 0.340 | 4.801% |

35 | 0.435 | 0.414 | 4.846% | 0.3284 | 0.2949 | 10.208% | 0.298 | 0.269 | 9.806% |

40 | 0.410 | 0.383 | 6.443% | 0.3160 | 0.2665 | 15.651% | 0.263 | 0.242 | 8.230% |

**Table 4.**The comparison between the dimensionless permeability calculated by the proposed model and that found in experiments by Liu et al. [48].

The Effective Stress, MPa | Dimensionless Permeability | ||||||||
---|---|---|---|---|---|---|---|---|---|

Core’s Number: S161-1 | Core’s Number: S148-2 | Core’s Number: Z41-9 | |||||||

Experimental Results | Calculated Results | Relative Errors | Experimental Results | Calculated Results | Relative Errors | Experimental Results | Calculated Results | Relative Errors | |

0 | 1.000 | 1 | 0.000% | 1.000 | 1.000 | 0.00% | 1.000 | 1.000 | 0.000% |

10 | 0.709 | 0.674 | 4.937% | 0.585 | 0.558 | 4.62% | 0.531 | 0.515 | 3.107% |

20 | 0.462 | 0.508 | 10.043% | 0.264 | 0.314 | 18.75% | 0.170 | 0.239 | 40.588% |

29 | 0.415 | 0.450 | 8.386% | 0.219 | 0.265 | 21.05% | 0.110 | 0.162 | 47.364% |

Parameters | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 |
---|---|---|---|---|---|---|

The pore fractal dimension | 1.5–1.7 | 1.8912 | 1.8912 | 1.8912 | 1.8912 | 1.8912 |

The tortuosity fractal dimension | 1.1358 | 1.1–1.5 | 1.1358 | 1.1358 | 1.1358 | 1.1358 |

Irreducible water | 0.45 | 0.45 | 0.35–0.55 | 0.45 | 0.45 | 0.45 |

The length of slipage, nm | 23 | 23 | 23 | 1–200 | 23 | 23 |

Poisson’s ratio | 0.15 | 0.15 | 0.15 | 0.15 | 0.15–0.35 | 0.15 |

The rock elastic modulus, GPa | 1.8 | 1.8 | 1.8 | 1.8 | 1.8 | 1–6 |

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**MDPI and ACS Style**

Wu, Z.; Cui, C.; Yang, Y.; Zhang, C.; Wang, J.; Cai, X.
A Fractal Permeability Model of Tight Oil Reservoirs Considering the Effects of Multiple Factors. *Fractal Fract.* **2022**, *6*, 153.
https://doi.org/10.3390/fractalfract6030153

**AMA Style**

Wu Z, Cui C, Yang Y, Zhang C, Wang J, Cai X.
A Fractal Permeability Model of Tight Oil Reservoirs Considering the Effects of Multiple Factors. *Fractal and Fractional*. 2022; 6(3):153.
https://doi.org/10.3390/fractalfract6030153

**Chicago/Turabian Style**

Wu, Zhongwei, Chuanzhi Cui, Yong Yang, Chuanbao Zhang, Jian Wang, and Xin Cai.
2022. "A Fractal Permeability Model of Tight Oil Reservoirs Considering the Effects of Multiple Factors" *Fractal and Fractional* 6, no. 3: 153.
https://doi.org/10.3390/fractalfract6030153