# Impacts of Fracture Roughness and Near-Wellbore Tortuosity on Proppant Transport within Hydraulic Fractures

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Buildup

#### 2.1. Experiment Setup

^{3}/h), as shown in Figure 1. The transparent fracture-flow model is used to observe the sand transport in a narrow and flat fracture, whose results can be used to validate the numerical model in this study. In the transparent fracture-flow experiments, silica sand and tap water with a certain sand volume fraction are continuously injected into a fracture with dimensions of 1.5 m (length) × 0.3 m (height) × 0.004 m (width).

^{3}/min liquid inflows. In order to ensure the maximum similarity of the flow, the velocity used is shown in Table 1. Other parameters can be referred in Table 2, shown below.

_{e}is Reynolds number. The ρ and v are the density and velocity of fluid. The d and μ are the hydraulic diameter of the fracture and the viscosity.

#### 2.2. Simulation Setup

_{s}is the diameter of the particle. C

_{d}is the drag coefficient.

_{s}and C

_{d}are expressed by (Gidaspow) [28]

_{s}is the Reynolds number of solid.

_{s}) contains kinetic viscosity (μ

_{s,kin}), collisional viscosity (μ

_{s,col}), and fractional viscosity (μ

_{s,fri}).

_{0}is the radial distribution function, which represents the resistance of particles to deformation and can be obtained by [32]

#### 2.3. Rough and Tortuous Model Development

_{va}), power spectral density (D

_{psd)}, and triangular prism analyses (D

_{tp}) to describe the roughness of a fracture. In this study, the integral of surface heights is used, as defined in Equation (23). In this study, conglomerate and shale fractures are used, which have Ra values of 0.39 and 0.74, respectively, as shown in Figure 5.

## 3. Results and Discussion

#### 3.1. Validation of Numerical Model

#### 3.2. Sand Transport in Rough Fractures

#### 3.2.1. Effect of Slurry Velocity

#### 3.2.2. Effect of Sand Volume Fraction

#### 3.2.3. Effect of Sand Diameter

#### 3.2.4. Effect of Surface Roughness

#### 3.2.5. Effect of the Tortuous Fractures

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$\overline{\overline{{\tau}_{f}}}$ | the stress tensors of the fluid |

$\overline{\overline{{\tau}_{s}}}$ | the stress tensors of the solid |

$\underset{g}{\to}$ | the acceleration of gravity |

${v}_{f}$ | velocity of fluid |

${v}_{s}$ | velocity of solid |

${\Theta}_{S}$ | granular temperature |

${\alpha}_{f}$ | volume fraction of fluid |

${\alpha}_{s}$ | volume fraction of solid |

${\rho}_{f}$ | density of the fluid |

${\rho}_{s}$ | density of the solid |

Cd | the drag coefficient |

d | hydraulic diameter |

ds | the diameter of the particle |

g0 | radial distribution function |

P | pressure of all the phases |

Re | Reynolds number |

Res | Reynolds number of solid |

v | velocity |

β | the Gidaspow drag force coefficient |

μ | viscosity |

μs,col | collisional viscosity |

μs,fri | fractional viscosity |

μs,kin | kinetic viscosity |

μs | shear viscosity of solid |

ρ | density |

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**Figure 8.**The results comparison between experiments and simulation at t = 50 s. (

**a**) v = 12 m/min, (

**b**) v = 24 m/min, and (

**c**) v = 36 m/min.

**Figure 10.**Sand distribution at the end of simulation (black arrow points to the farthest location of sand settlement).

**Figure 12.**Changes in propagation rate of the propped fractures with slurry injection rate and sand volume fraction.

**Figure 16.**Changes in sand bank equilibrium height with slurry velocity and sand diameter. (The y-axis represents the percentage of sand bank height in the fracture.).

**Figure 17.**Changes in sand bank equilibrium height ratio (left column; the y-axis represents the percentage of sand bank height in the fracture) and propagation rate of propped fractures (right column) with slurry velocity, sand diameter, and sand volume fraction.

**Figure 18.**Changes in sand bank equilibrium height and propagation rate of propped fractures with Ra (roughness of fractures, slurry velocity 12 m/min, and 20/40 mesh sand; 7.3% sand.).

**Figure 19.**Settling velocity with different diameters (the legend means the slurry velocity, and the unit is m/min).

Volumetric Flow Rate (m ^{3}/min) | Fracture Height (m) | Fracture Width (mm) | Flow Rate (m/min) | Particle Reynolds Number | Fluid Reynolds Number | |
---|---|---|---|---|---|---|

Field Scale | 5–10 | 10–60 | 2–4 | 3.5–167 | 30–5337 | 457–21,929 |

Lab Scale | 0.01–0.25 | 0.3 | 4 | 8.3–208 | 75–6590 | 1096–27,412 |

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

Slurry velocity, m/min | 12, 24, 36 |

Sand diameter, mm | 0.208, 0.325, 0.425, 0.739 |

Fracture width, mm | 4 |

Fluid viscosity, Pa·s | 0.001 |

Real sand density, kg/m^{3} | 2600 |

Sand bulk density, kg/m^{3} | 1500 |

Volume fraction of sand | 7.3% |

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

slurry velocity, m/min | 12, 24, 36, 66, 96, 126 |

sand diameter, mm | 0.208, 0.325, 0.425, 0.739 |

fracture width, mm | 4 |

sand volume fraction | 4.7%, 7.3%, 10% |

Ra (fracture roughness) | 0.39, 0.59, 0.79 |

Straight Facture | Tortuous Facture 1 | Tortuous Facture 2 | |
---|---|---|---|

length, m | 1.5 | 1.5 | 1.5 |

width, m | 0.004 | 0.004 | 0.004 |

height, m | 0.3 | 0.3 | 0.3 |

tortuous angle, degree | 120 | 120 | |

tortuous length, m | 0.06 | 0.06 | |

tortuous position, m | 0.75 | 0.2 | |

tortuous width, m | 0.00346 | 0.00346 |

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**MDPI and ACS Style**

Wang, D.; Bai, B.; Wang, B.; Wei, D.; Liang, T.
Impacts of Fracture Roughness and Near-Wellbore Tortuosity on Proppant Transport within Hydraulic Fractures. *Sustainability* **2022**, *14*, 8589.
https://doi.org/10.3390/su14148589

**AMA Style**

Wang D, Bai B, Wang B, Wei D, Liang T.
Impacts of Fracture Roughness and Near-Wellbore Tortuosity on Proppant Transport within Hydraulic Fractures. *Sustainability*. 2022; 14(14):8589.
https://doi.org/10.3390/su14148589

**Chicago/Turabian Style**

Wang, Di, Bingyang Bai, Bin Wang, Dongya Wei, and Tianbo Liang.
2022. "Impacts of Fracture Roughness and Near-Wellbore Tortuosity on Proppant Transport within Hydraulic Fractures" *Sustainability* 14, no. 14: 8589.
https://doi.org/10.3390/su14148589