# Multi-Scale Insights on the Threshold Pressure Gradient in Low-Permeability Porous Media

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## Abstract

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## 1. Introduction

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^{3}in 2020 and the proportion of unconventional gas will then exceed 30% of the total gas production [2]. Due to the low permeability, high water saturation, and complex pore structure of shale gas reservoirs, the gas flow in a low-pressure gradient zone is always slow and non-Darcy [3,4,5,6]. This low-velocity non-Darcy flow has been emphasized in low-permeability reservoirs for the following two issues [3,7,8], which are still unsolved:

_{T}, $TPG~{k}^{-{D}_{T}/(1+{D}_{T})}$. Ye et al. [26] proposed a fractal model for the TPG in tight oil reservoirs. In this model, the residual water saturation was treated as connate water by forming a boundary layer fluid, which could hinder the gas flow. Their results indicated that a higher residual water saturation would greatly increase the TPG, especially when the permeability was less than 0.01 mD. Although the above literature presented different TPG models, the coupling of multi-scale geometry and physical mechanisms in the TPG is still unclear. Therefore, it is necessary to investigate the flow mechanism of the TPG with complex pore structure and to explore its application in numerical simulations.

## 2. A Novel Semi-Empirical Formula for the Threshold Pressure Gradient (TPG)

#### 2.1. Effects of Macroscopic Parameters on TPG

#### 2.1.1. Effect of Boundary Layer

#### 2.1.2. Effect of Capillary Pressure

#### 2.1.3. Effect of Stress Sensitivity

#### 2.2. Verification of the Proposed TPG Model Using Experimental Data

#### 2.2.1. Case 1: Correlation of the TPG with Permeability and Water Saturation

#### 2.2.2. Case 2: Correlation of the TPG with Water Saturation at Different Permeabilities

#### 2.2.3. Case 3: Correlation of the TPG with Pore Pressure

## 3. A Fractal Model for the Threshold Pressure Gradient

#### 3.1. Derivation of Threshold Pressure Gradient Based on Fractal Theory

#### 3.2. Model Validation and Analysis

#### 3.3. Effect of Fractal Dimensions

#### 3.4. Effect of Residual Water Saturation

#### 3.5. Effect of Capillary Pressure

## 4. Conclusions

- The gas flow in low-permeability reservoirs exhibits a strong non-Darcy behavior. The boundary layer effect and fluid plasticity play the main role in this nonlinear behavior.
- The threshold pressure gradient is directly related to permeability, water saturation, and pore pressure. An approximate power function can describe the relationship between permeability and the TPG. An exponential relationship is more consistent with the effects of water saturation and pore pressure.
- Both pore-size distribution in the fractal dimension and tortuosity fractal dimension can increase the TPG by characterizing the complexity of the pore structure. However, the tortuosity fractal dimension has a more significant enhancement effect.
- Complex pore structure has a significant effect on the TPG only in the range of low porosity, but water saturation and yield stress have a wider influence range. A linear relationship between yield stress and TPG can be observed.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A: Derivation of Fractal Parameters

## References

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**Figure 3.**(

**a**) Verification of the TPG correlation with different permeabilities, (

**b**) Verification of the TPG correlation with different water saturations.

**Figure 4.**(

**a**) The variation of the TPG with water saturation at the permeability of 0.687, 1.51, and 9.02 mD, (

**b**) The variation of the TPG with water saturation at the permeability of 9.02 mD.

**Figure 9.**The evolution of the TPG with yield stress predicted by our fractal model and the classic model of Equation (2).

**Figure 10.**(

**a**) The effect of pore size distribution fractal dimension on the TPG, (

**b**) The effect of tortuosity fractal dimension on the TPG.

TPG Correlations | Core Data | Reference |
---|---|---|

$TPG=16{\left(\frac{k}{\mu}\right)}^{-0.8}$ | Core flooding using brine | Prada and Civan [16] |

$TPG=0.5{\left(\frac{k}{\mu}\right)}^{-0.9813}$ | Core flooding using oil | Li et al. [19] |

$TPG=0.0747{k}^{-1.117}$ | Core flooding using oil | Li et al. [19] |

$TPG=0.4{k}^{-0.9348}$ | Core flooding using formation water | Zeng et al. [20] |

$TPG=0.00965{k}^{-0.9738}$ | Core flooding using oil | Yang et al. [21] |

Fitting Coefficients | a | b | c | m | n | ||
---|---|---|---|---|---|---|---|

Case 1 (permeability effect [10]) | Figure 3a | 0.08464 | 0.42046 | ||||

Figure 3b | $2.542\times {10}^{-4}$ | −2.86125 | 27.63959 | ||||

Case 2 (water saturation effect [7]) | Figure 4a | Core 1 | 0.1165 | −8.89949 | 7.41895 | ||

Core 2 | 0.00248 | 8.00076 | 8.40124 | ||||

Figure 4b | Core 3 | 1 | 5.35414 | 6.2152 | |||

Case 3 (pore pressure effect [9]) | Figure 5 | Core 5 | 0.25791 | 0.04397 | 0.2733 | −0.05898 | 0.00107 |

Core 6 | 0.16632 | 0.06577 | 0.24045 | −0.05544 | 0.00116 | ||

Core 7 | 0.07173 | 0.05531 | 0.14122 | −0.03052 | $6.0346\times {10}^{-4}$ | ||

Figure 6 (Core 5) | S_{w} 0.6 | 0.30482 | 0.05371 | 0.25039 | −0.0687 | 0.00107 | |

S_{w} 0.5 | 0.25943 | 0.05661 | 0.31669 | −0.07261 | 0.00114 | ||

S_{w} 0.4 | 0.18415 | 0.09078 | 0.635 | −0.07849 | 0.00133 | ||

S_{w} 0.3 | 0.16559 | 0.0973 | 0.90743 | −0.08621 | 0.00154 |

Case 1 (permeability effect [10]) | Range of permeability (mD) | Average permeability (mD) | Range of porosity (%) | Average porosity (%) | |

0.011–0.47 | 0.11 | 2.62–14.21 | 6.16 | ||

Case 2 (water saturation effect [7]) | Core Number | Permeability (mD) | Length (cm) | Diameter (cm) | Driving pressure (MPa) |

Core 1 | 0.687 | 5 | 2.5 | 0.504 | |

Core 2 | 1.51 | 5 | 2.5 | 0.401 | |

Core 3 | 9.02 | 5 | 2.5 | 0.053 | |

Case 3 (pore pressure effect [9]) | Core Number | Permeability (mD) | Length (cm) | Diameter (cm) | Porosity (%) |

Core 5 | 0.061 | 5.962 | 2.534 | 6.15 | |

Core 6 | 0.193 | 6.28 | 2.538 | 8.32 | |

Core 7 | 0.317 | 6.286 | 2.54 | 12.08 |

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**MDPI and ACS Style**

Wang, H.; Wang, J.; Wang, X.; Chan, A.
Multi-Scale Insights on the Threshold Pressure Gradient in Low-Permeability Porous Media. *Symmetry* **2020**, *12*, 364.
https://doi.org/10.3390/sym12030364

**AMA Style**

Wang H, Wang J, Wang X, Chan A.
Multi-Scale Insights on the Threshold Pressure Gradient in Low-Permeability Porous Media. *Symmetry*. 2020; 12(3):364.
https://doi.org/10.3390/sym12030364

**Chicago/Turabian Style**

Wang, Huimin, Jianguo Wang, Xiaolin Wang, and Andrew Chan.
2020. "Multi-Scale Insights on the Threshold Pressure Gradient in Low-Permeability Porous Media" *Symmetry* 12, no. 3: 364.
https://doi.org/10.3390/sym12030364