# Settling of Spherical Particles in High Viscosity Friction Reducer Fracture Fluids

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experiment Description

#### 2.1. Experimental Materials

^{−1}) to evaluate how the fluids governed the proppant transport. This rheometer can measure the temperature effect until 200 °C. The viscosity measurements included obtaining the consistency index k and the flow behavior index n. The elastic measurements studied included the relaxation time parameter and the loss factor (G″/G′), where G″ is the viscous modulus and G′ is the elastic modulus. The fluid density of the HVFR concentrations was measured by calculating the weight and volume of the friction reducer using an accurate weight scale and graduated lab glasses.

#### 2.2. Spherical Particles Settling Description

^{−1}) to avoid any damage to the fracture fluid properties that could occur using a mechanical blender at a high mix rate. Then, the friction reducer was tested in two models: unconfined and confined.

#### 2.2.1. Unconfined Fracture Model

#### 2.2.2. Confined Fracture Model

#### 2.3. Settling Velocity Procedure

## 3. Results and Analysis of Rheology Measurements

#### 3.1. Viscosity Measurements

^{−1}, the viscosities of the HVFR were 334, 136, 70, and 37 mPa·s for 8, 4, 2, and 1 gpt, respectively. However, at a high shear rate of 100 s

^{−1}, the viscosities of the HVFRs reduced to 63, 31, 17, and 11 mPa·s for 8, 4, 2, and 1 gpt, respectively. Additionally, the significant reduction in the viscosities of the HVFRs at a high shear rate might imply that the HVFR viscosity did not significantly impact proppant transport near the wellbore. Therefore, at high shear rates, the reliability of the fluid viscosity might not provide good sand transport, and the fluid would have to depend on its elasticity to hold the proppant in the fracture networks. Moreover, the insignificant changes in the viscosities of the HVFRs that were observed at higher shear rates (Figure 4) suggest that increasing the concentration of the HVFRs above a certain level would not continue to improve the HVFR viscosity.

#### 3.2. Temperature Effect on the HVFR Viscosity

#### 3.3. Time-Dependent Temperature Effect on the HVFR Viscosity

^{−1}. Figure 6 illustrates the influence of increasing the temperature using an HVFR concentration of 4 gpt. At a shear rate of 10 s

^{−1}, the results showed that the HVFR viscosity at 50 °C remained relatively stable, but after approximately 2000 s, the viscosity of the HVFR started to drop gradually. At the same shear rate, increasing the temperature to 75 °C caused the viscosity of the HVFR to decrease significantly after only 500 s. At higher temperature values of 100 °C, the HVFR started to lose its viscosity, and a portion of the solution started to evaporate after a very short period. These results could imply that HVFRs cannot maintain their viscosity at temperatures above 75 °C. The viscoelastic parameters of the fluids exhibit degradation in their rheological structure due to the strength of the interconnection between fluid’s molecules that reduce as temperature increases [29,30,31]. As a result, poor proppant transport would be observed inside the network fractures if this type of HVFR were used for high-temperature formations. Interestingly, when increasing the shear rate to 100 s

^{−1}, similar trends were observed, but a larger reduction in the fluid viscosity was also noticed for all temperature ranges compared to the viscosity measurements at low shear rates.

#### 3.4. Elasticity Measurements

#### 3.5. Loss Factor or Damping Factor

## 4. Results and Analysis of the Spherical Particles Settling

#### 4.1. Settling Particle Velocities into Unconfined Fracture Systems

#### 4.2. Reynolds Number and Drag Coefficient Calculations

_{PL}) was determined using Equation (1) [33], while the drag coefficient was calculated using Equation (2). The drag coefficient equation should be used if the Reynolds numbers are below 10

^{6}[34]. The fluid resistance that prevented the spherical particle from falling is represented by the drag coefficient.

#### 4.3. Settling Particle Velocities in Confined Fracture Systems

#### 4.4. Fracture Width Effect

#### 4.5. Particle Concentration Effect

#### 4.6. Fracture Orientation Effect

## 5. Conclusions

- ➢
- The viscosity and elastic properties of HVFRs provided excellent efficiency to transport particles for a significant time, which can provide a better distribution of the proppant in fracture networks.
- ➢
- At a high shear rate, the viscosity of HVFRs decreased sharply, so the viscosity would not provide good sand transport, and the fracture fluid may depend on its elasticity to hold the proppant.
- ➢
- Increasing the temperature cut the friction reducer efficiency to suspend the spherical particles, and that was observed clearly at temperatures that reached 75 °C.
- ➢
- The settling velocity of the spherical particles in the unconfined model was faster than the settling velocity in the confined model due to the absence of interference by the fracture wall.
- ➢
- As the ratio of the spherical particle diameter to the fracture width increased, so did the wall effect; however, the wall effect can be reduced by increasing the HVFR concentration.
- ➢
- The fracture angulation had a large impact on the particle settling, where the spherical particles had more contact with the fracture wall at a lower fracture angulation of 45° compared to vertical positions (90°), which may cause a reduction in the settling rate of the spherical particles.

## 6. Future Work

- ➢
- The roughness of the surface of fracture walls has a significant impact on the settling velocity of particles where the smooth surface of the walls used in this current study could have the lowest impact on the particle settling velocity compared to roughened surfaces.
- ➢
- The effect of the temperature in this work was investigated only in rheology measurements. The settling velocity of spherical particles using a static model was measured at 25 °C, which has not been studied beyond this room temperature. Therefore, it is recommended to determine the particles settling at high-temperature ranges.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

C_{D} | Drag coefficient |

D_{P} | Spherical particle diameter, m |

G′ | Storage modulus, Pa |

G | Loss modulus, Pa |

K | Flow consistency index, Pa.s^{n} |

n | Flow behavior index |

Re_{PL} | Reynold’s number of proppants settling |

Velocity in a power low liquid | |

V | Setting proppant velocity, m/s |

ρf | Fluid density, kg/m^{3} |

μ | Viscosity, mPa·s |

ω | Angular frequency, rad/s |

$\lambda $ | Relaxation time of the friction reducer |

Fw | wall factor |

Re | Reynold’s number |

We | Weissenberg number |

${V}_{\infty EL}$ | Settling velocity in viscoelastic fluids, m/s |

${V}_{\infty INEL}$ | Settling velocity in inelastic fluids, m/s |

$W{e}_{\infty INEL}$ | Dimensionless number relating the relaxation time to particle diameter and unconfined settling velocity |

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**Figure 9.**Terminal spherical particle settling in the unconfined model: (

**a**) HVFR concentrations and (

**b**) particle sizes impact.

**Figure 11.**Velocity ratio using: (

**a**) friction reducer concentrations and (

**b**) spherical particle sizes.

**Figure 12.**Terminal spherical particle settling in the confined model using (

**a**) 6 mm particle size, and (

**b**) 4 gpt HVFR concentration.

**Figure 13.**Terminal settling velocity of spherical particles using two fracture widths and two friction reducer concentrations.

**Figure 16.**Terminal settling velocity of spherical particles using (

**a**) single particle and (

**b**) multi particles.

**Figure 18.**Effect of the fracture angulation on the settling velocity using (

**a**) 4 gpt concentration, and (

**b**) 6 mm particle size.

**Figure 19.**Fracture angulation effect on (

**a**) Re and (

**b**) CD using a 6 mm particle size and different HVFR concentrations.

Glass Bubble Diameter, mm | Density, g/cm^{3} |
---|---|

2 | 2.6253 |

4 | 2.6261 |

6 | 2.626 |

Fluid Concentration, gpt | Fluid Density, g/cm^{3} | Relaxation Time, s | Consistency Index, k | Flow Behavior Index, n | R^{2} |
---|---|---|---|---|---|

1 | 0.996 | 4.464 | 0.1125 | 0.4916 | 0.9968 |

2 | 0.997 | 9.346 | 0.3077 | 0.3772 | 0.9929 |

4 | 0.997 | 21.322 | 0.7858 | 0.3062 | 0.9823 |

8 | 0.998 | 39.8406 | 1.9291 | 0.2601 | 0.9883 |

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**MDPI and ACS Style**

Biheri, G.; Imqam, A.
Settling of Spherical Particles in High Viscosity Friction Reducer Fracture Fluids. *Energies* **2021**, *14*, 2462.
https://doi.org/10.3390/en14092462

**AMA Style**

Biheri G, Imqam A.
Settling of Spherical Particles in High Viscosity Friction Reducer Fracture Fluids. *Energies*. 2021; 14(9):2462.
https://doi.org/10.3390/en14092462

**Chicago/Turabian Style**

Biheri, Ghith, and Abdulmohsin Imqam.
2021. "Settling of Spherical Particles in High Viscosity Friction Reducer Fracture Fluids" *Energies* 14, no. 9: 2462.
https://doi.org/10.3390/en14092462