# Numerical Simulation of the Wormhole Propagation in Fractured Carbonate Rocks during Acidization Using a Thermal-Hydrologic-Mechanics-Chemical Coupled Model

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

#### 2.1. Chemical System

#### 2.2. Darcy Scale Model

#### 2.3. Pore Scale Model

#### 2.4. Energy Transport

## 3. Dimensionless Model

## 4. Numerical Method

- (1)
- Generate randomly distributed discrete fractures.
- (2)
- Calculate the starting point and end point coordinates of each fracture.
- (3)
- Loop all faces of the grid cell to check whether the fracture segment intersects the face segment.
- (4)
- If the fracture intersects the face segment of the grid cell, set the two cells sharing the face as the fracture cell, and change its porosity to 0.99.

## 5. Effect of Stress

## 6. Results and Discussions

#### 6.1. Effect of Fracture Parameters

#### 6.1.1. Effect of Fracture Orientation

#### 6.1.2. Effect of Fracture Length

#### 6.1.3. Effect of Fracture Density

#### 6.2. Effect of Reaction Temperature

#### 6.3. Effect of Stress Sensitivity

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

THmC | Thermal-Hydrologic-mechanics-Chemical |

TSC | Two-Scale Continuum |

TVD | Total Variation Diminishing |

EOR | Enhanced Oil Recovery |

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**Figure 1.**Porosity fields show the impact of fracture orientation on wormhole structures (

**A**) Linear case, (

**B**) Radial case and at different fracture dips: (

**a**) ${30}^{\circ}$, (

**b**) ${60}^{\circ}$, (

**c**) ${90}^{\circ}$, (

**d**) ${120}^{\circ}$ ($-{60}^{\circ}$), (

**e**) ${150}^{\circ}$ ($-{30}^{\circ}$).

**Figure 2.**The effect of fracture dip on dimensionless breakthrough volumes corresponds to the dissolution structures in Figure 1 column (A).

**Figure 6.**Influence of fracture density on wormhole structures, the number of fractures: (

**A**) 12; (

**B**) 17; (

**C**) 30.

**Figure 7.**Influence of fracture density on wormhole structures, the number of fractures: (

**A**) 20; (

**B**) 35; (

**C**) 50.

**Figure 8.**Comparison of dissolution structures for non-isothermal and isothermal cases in linear acidizing simulation.

**Figure 9.**Comparison of dissolution structures for non-isothermal and isothermal cases in radial acidizing stimulation.

**Figure 10.**Comparison of dissolution structures at different effective stress: (

**A**) $\sigma =0$ MPa, (

**B**) $\sigma =15$ MPa, (

**C**) $\sigma =30$ MPa, (

**D**) $\sigma =45$ MPa and at different injection rates: $1/Da$ = (

**a**) ${10}^{-6}$ (

**b**) $2\times {10}^{-6}$, (

**c**) ${10}^{-4}$, (

**d**) $0.005$, (

**e**) $0.05$.

**Figure 11.**Comparison of dissolution structures at different effective stress: (

**A**) $\sigma =0$ MPa, (

**B**) $\sigma =15$ MPa, (

**C**) $\sigma =30$ MPa, (

**D**) $\sigma =45$ MPa and at different injection rates: $1/Da$ = (

**a**) $0.00075$, (

**b**) $0.0225$, (

**c**) $0.075$, (

**d**) $0.75$.

Parameters | Values |
---|---|

L | 5 cm |

H | 2 cm |

${r}_{e}$ | 10 cm |

${r}_{w}$ | 0.5 cm |

${\varphi}_{0}$ | 0.35 |

${K}_{0}$ | 100 md |

${r}_{0}$ | $5\times {10}^{-6}$ |

${k}_{s}$ | 0.2 |

$\upsilon $ | 0.0286 |

$S{h}_{\infty}$ | 3 |

$Sc$ | 1000 |

${N}_{ac}$ | 0.1 |

$\eta $ | $1\times {10}^{-6}$ |

${v}_{0}$ | 0.004 cms |

${\alpha}_{os}$ | 0.5 |

${\lambda}_{T}$ | 0.1 |

${\lambda}_{X}$ | 0.5 |

m | 1 |

$\beta $ | 1 |

${h}_{T}^{2}$ | 0.07 |

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**MDPI and ACS Style**

Liu, P.; Huang, C.; Jia, L.; Ji, W.; Zhang, Z.; Zhang, K.
Numerical Simulation of the Wormhole Propagation in Fractured Carbonate Rocks during Acidization Using a Thermal-Hydrologic-Mechanics-Chemical Coupled Model. *Water* **2022**, *14*, 4117.
https://doi.org/10.3390/w14244117

**AMA Style**

Liu P, Huang C, Jia L, Ji W, Zhang Z, Zhang K.
Numerical Simulation of the Wormhole Propagation in Fractured Carbonate Rocks during Acidization Using a Thermal-Hydrologic-Mechanics-Chemical Coupled Model. *Water*. 2022; 14(24):4117.
https://doi.org/10.3390/w14244117

**Chicago/Turabian Style**

Liu, Piyang, Chaoping Huang, Lijing Jia, Weijing Ji, Zhao Zhang, and Kai Zhang.
2022. "Numerical Simulation of the Wormhole Propagation in Fractured Carbonate Rocks during Acidization Using a Thermal-Hydrologic-Mechanics-Chemical Coupled Model" *Water* 14, no. 24: 4117.
https://doi.org/10.3390/w14244117