# Influence of 3D Fracture Geometry on Water Flow and Solute Transport in Dual-Conduit Fracture

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}[8,9]. Current research on the non-Darcy and non-Fickian effects of solute transport in rock fracture flows focuses on the effects of fracture geometry, which facilitates access to the flow and transport behavior in the fracture [10,11,12]. Accordingly, as a result of these experiments on cross-fracture models, some fundamental mechanisms governing the flow and transport in rock fractures have been revealed [13,14]. In addition, the flow rate at the fracture intersections matters considerably in determining the fluid flow through a fracture network. However, an in-depth understanding of fluid flow and solute transport in multicrossing fractures is still lacking.

## 2. Theoretical Background

_{c}) separates high-velocity non-Darcian flow from Darcy flow, which is the proportion of linear to nonlinear pressure gradients according to Forchheimer’s law. While applying Forchheimer’s law, the critical value of BQ/A = 0.11 was typically used, showing that when the nonlinear pressure gradient accounted for 10% of ΔP the well-known Darcy–Weisbach equation was used to illustrate how the pressure head loss is caused by flow acceleration and wall friction during the flow process:

_{f}is the friction factor, which is essential to computational fluid dynamics and is frequently used to describe pressure head losses in rock fractures, e

_{m}is the arithmetic mean point-to-point distance between the two walls of a fracture, L is the total length of the fracture, ρ is the fluid density, and V is the average velocity calculated by Q/(e

_{m}w).

_{i}denotes the dispersion coefficient (L

^{2}T

^{−1}); C

_{i}, the outlet concentration (ML

^{−3}); u

_{i}, the flow velocity (LT

^{−1}); x, the space coordinate (L); and t, time (T).

_{x}, D

_{y}, and D

_{z}are the transverse dispersion coefficients. For the convenience of discussion in this study, the diffusion coefficient is considered only in the x-direction.

_{1}is flow velocity in the main fracture (LT

^{−1}); v

_{2}, the flow velocity in diversion fracture (LT

^{−1}); and w

_{i}, the volume fraction of region. Based on the equation, only one of the two fractions (w

_{1}, w

_{2}) is a free parameter. To fit the experimental BTCs, the parameter set (v

_{1}, v

_{2}, D

_{1}, D

_{2}, w

_{1}) was calibrated.

## 3. Numerical Models’ Setup

^{3}and a dynamic viscosity of 0.001 Pa·s. Generally, different Q was applied to the inflow boundary associated with the Reynolds number Re = ρq/(wμ) between 1 and 300, which could be thought of as being within the linear laminar flow regime with minimal turbulent effects. Due to the modest amount of the injected solute and the adoption of the case of diluted solute transport (C

_{0}= 0.1 mol/m

^{3}), the initial flow properties would not be affected when the water diffusion coefficient of the solute was set to 1 × 10

^{−7}m

^{2}/s. Considering both advection-dominating and diffusion-dominating transport, the transport behaviors in the 4 fracture cases were evaluated under different flow velocities, and the resulting Peclet number Pe = Q/(wD) ranged from 3 to 30 for the given Q. Injecting a known quantity of tracer at an upstream position and monitoring the variations in the tracer concentration at a downstream location yielded a BTC.

## 4. Results and Discussion

#### 4.1. Water Flow Test

^{2}> 0.99). Variations in fracture geometry lead to changes in hydraulic parameters. With increases in the value of H, the coefficient A representing the cohesive force increases while the coefficient B representing the inertial force decreases in 2D fractures. The viscosity coefficient increases from 3.75 × 10

^{5}to 3.92 × 10

^{5}, while the inertia coefficient decreases from 7.63 × 10

^{7}to 6.94 × 10

^{7}, and the coefficients A and B are both essentially constant in 3D fractures. The viscosity coefficient ranges from 3.88 × 10

^{5}to 3.92 × 10

^{5}, while the inertia coefficient ranges from 11.6 × 10

^{7}to 11.7 × 10

^{7}. Thus, the Re

_{c}varies with geometric parameters in 2D fracture but is not significantly different in 3D. The non-Darcy phenomena are more pronounced the closer the branching in 2D. Moreover, branch fracture geometry controls the flow rate and seepage channel at the intersection.

_{c}of the 2D fracture can be approximately 40% less than that of the 3D fractures, which is attributed to the fact that the increase in the length of the diversion fracture causes the flow to become biased towards the main fracture, which causes the decreased pressure drop loss.

_{f}and the flow rate show a negative-correlation linear trend in the case of the power exponent. When the flow velocity increases to a certain value (blue dotted line), the downward trend of the friction coefficient of the diversion fracture becomes gentle and the curve presents a nonlinear change. However, the λ

_{f}of the main fracture is not changed obviously, the main sites responsible for fracture non-Darcy are in the diversion fractures, and the intersection between the main and diversion fracture does not cause significant nonlinearity in the main fracture flow.

#### 4.2. Solute Concentration Distribution

_{1}and D

_{2}in Table 3), and the diffusion is more evident in the path of the diversion fracture (Figure 4a). As the Pe increases, the BTC curves of the fractures become steeper (Figure 4b–d) and the role of convection in solute transport increases while the dispersion decreases. Compared to the 2D fracture, the dispersion coefficient of the 3D fractures increases when the Pe > 6 (Figure 4b–d). Notably, as depicted from the concentration distributions in Figure 4a–d, the distance between the two peaks decreases with an increasing Pe number, thereby resulting in a relatively late arrival of the peak in the diversion fracture. On the other side, the diffusion coefficient of the solute increases with increases in the flow velocity (Table 3).

_{2}) increases and the diffusion coefficient of the main fracture (D

_{1}) decreases. The above deductions are summarized, since BTCs can bring information on underground geometry.

_{1}/Q

_{2}) is the same. Theoretically, when the fracture solute transport is in the streamlined routing model, the distribution of the flow rates leads to the distribution of outlet concentrations. The ratio of w

_{1}/w

_{2}, which represents the concentration ratio between the main fracture and diversion fracture, should be the same as the fracture flow ratio. In addition, by comparing Table 1 and Table 3, it can be found that the ratio of water flow is greater than the concentration between the ratio of w

_{1}/w

_{2}, indicating the essential role of the streamline routing mode in the solute transfer. With increasing flow rates, the ratio of w

_{1}/w

_{2}tends to be closer to the flow ratio; due to this, the effect of the complete mixing mode is weakened in the solute transfer process. In particular, the ratios of w

_{1}/w

_{2}are significantly larger in 3D than those in 2D, indicating that the streamlined routing mode is more important in the 3D fracture.

#### 4.3. Discussion

_{1}/w

_{2}becomes larger as the flow rate increases with increasing H values under the same hydraulic and geometry conditions, and the ratio of w

_{1}/w

_{2}is larger for 3D than that for 2D. It was conventionally considered that the solute distribution in the fracture at the intersection is linearly related to the flow. However, as shown in Table 1 and Table 2, the boundary layer affects the diffusion of solute transport at the same flow velocity.

## 5. Conclusions

- Based on the numerical results, the Forchheimer equations can more accurately describe the nonlinear fracture flow characteristics in DCFs. The further apart the branch fractures are from each other, the more the coefficient A representing the cohesive force increases while the coefficient B representing the inertial force decreases in 2D fractures, and the coefficients A and B are essentially constant in 3D fractures.
- Relative to the 2D fracture, the 3D DCFs cause a more pronounced non-Darcy phenomenon. Additionally, the Re
_{c}of 2D fractures can be approximately 40% less than that of 3D fractures. With an increasing flow rate, the friction coefficient in the bypass is nonlinearly related to the flow rate at exponential conditions, demonstrating the diversion fractures as the main scene of the non-Darcy phenomenon. - The BTC in each graph shows a double-peak phenomenon with significant differences in propagation times in the main and diversion fractures due to the variation in flow rates. Dual-peaked BTCs can be reproduced by the WSADE model. Furthermore, the further apart the branch fractures are from each other, the larger the Pe number, when the values of parameter D
_{1}of WSADE decrease and parameter D_{2}and the ratio of w_{1}/w_{2}increase. Moreover, in the same geometry and flow conditions, the BTC of the 2D fracture is steeper than that of the 3D fracture, and this phenomenon becomes more obvious with increases in Pe. - Both the complete mixing model and the streamlined routing model affect the transport of fractured solutes, and the streamlined routing mode plays a major role in the solute transfer. With increasing flow rates, the effect of the complete mixing mode is weakened in the solute transport. From a statistics point of view, the streamlined routing mode is more obvious in 3D than that in 2D.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Variations of the friction factor with flow rates in different H values: (

**a**) 100 mm; (

**b**) 150 mm; (

**c**) 200 mm; and (

**d**) 250 mm.

**Figure 3.**Solute concentration distributions at different times when Pe = 15 and H = 250 mm. (

**a**–

**h**) is when time = 200 s, 700 s, 1200 s, 1700 s, 2200 s, 2700 s, 3200 s and 3700 s, respectively.

Fracture | H (mm) | A (10^{5}) | B (10^{7}) | Re_{c} | R^{2} | Q_{1}/Q_{2} |
---|---|---|---|---|---|---|

2D | 100 | 3.75 | 7.63 | 54.61 | 0.998 | 1.60 |

150 | 3.82 | 7.88 | 53.86 | 0.997 | 1.88 | |

200 | 3.88 | 7.25 | 59.46 | 0.996 | 2.14 | |

250 | 3.93 | 6.94 | 62.92 | 0.999 | 2.41 | |

3D | 100 | 3.92 | 11.6 | 37.55 | 0.999 | 1.63 |

150 | 3.88 | 11.7 | 36.85 | 0.999 | 1.89 | |

200 | 3.88 | 11.7 | 36.85 | 0.998 | 2.17 | |

250 | 3.92 | 11.6 | 37.55 | 0.999 | 2.43 |

Pe | H (mm) | v_{1} (m/s) | v_{2} (m/s) | D_{1} (m^{2}/s) | D_{2} (m^{2}/s) | w_{1}/w_{2} |
---|---|---|---|---|---|---|

3 | 250 | 8.06 × 10^{−5} | 2.72 × 10^{−5} | 0.0121 | 0.0688 | 2.39 |

200 | 7.79 × 10^{−5} | 3.21 × 10^{−5} | 0.0132 | 0.0656 | 2.06 | |

150 | 7.65 × 10^{−5} | 3.82 × 10^{−5} | 0.0197 | 0.0621 | 1.79 | |

100 | 7.53 × 10^{−5} | 4.61 × 10^{−5} | 0.0191 | 0.0598 | 1.52 | |

6 | 250 | 1.67 × 10^{−4} | 5.45 × 10^{−5} | 0.0083 | 0.13 | 2.44 |

200 | 1.59 × 10^{−4} | 6.37 × 10^{−5} | 0.0096 | 0.138 | 2.12 | |

150 | 1.48 × 10^{−4} | 7.58 × 10^{−5} | 0.0111 | 0.149 | 1.84 | |

100 | 1.40 × 10^{−4} | 9.98 × 10^{−5} | 0.0122 | 0.162 | 1.55 | |

15 | 250 | 4.21 × 10^{−4} | 1.35 × 10^{−4} | 0.0065 | 0.233 | 2.73 |

200 | 4.02 × 10^{−4} | 1.58 × 10^{−4} | 0.0074 | 0.174 | 2.22 | |

150 | 3.88 × 10^{−4} | 1.89 × 10^{−4} | 0.0083 | 0.141 | 1.9 | |

100 | 3.76 × 10^{−4} | 2.27 × 10^{−4} | 0.0102 | 0.143 | 1.53 | |

30 | 250 | 7.75 × 10^{−4} | 2.70 × 10^{−4} | 0.0041 | 0.272 | 2.84 |

200 | 7.65 × 10^{−4} | 3.13 × 10^{−4} | 0.0083 | 0.162 | 2.30 | |

150 | 7.59 × 10^{−4} | 3.7 × 10^{−4} | 0.0076 | 0.129 | 1.93 | |

100 | 7.51 × 10^{−4} | 4.55 × 10^{−4} | 0.012 | 0.097 | 1.52 |

Pe | H (mm) | v_{1} (m/s) | v_{2} (m/s) | D_{1} (m^{2}/s) | D_{2} (m^{2}/s) | w_{1}/w_{2} |
---|---|---|---|---|---|---|

3 | 250 | 7.75 × 10^{−5} | 2.7 × 10^{−5} | 0.142 | 0.113 | 3.50 |

200 | 7.65 × 10^{−5} | 3.16 × 10^{−5} | 0.144 | 0.107 | 3.22 | |

150 | 7.59 × 10^{−5} | 3.75 × 10^{−5} | 0.149 | 0.113 | 2.92 | |

100 | 7.51 × 10^{−5} | 4.05 × 10^{−5} | 0.156 | 0.111 | 2.67 | |

6 | 250 | 1.64 × 10^{−4} | 5.37 × 10^{−5} | 0.117 | 0.132 | 3.57 |

200 | 1.61 × 10^{−4} | 6.32 × 10^{−5} | 0.118 | 0.134 | 3.35 | |

150 | 1.53 × 10^{−4} | 7.47 × 10^{−5} | 0.124 | 0.138 | 3.08 | |

100 | 1.45 × 10^{−4} | 9.02 × 10^{−5} | 0.135 | 0.162 | 2.82 | |

15 | 250 | 4.28 × 10^{−4} | 1.35 × 10^{−4} | 0.097 | 0.164 | 3.93 |

200 | 4.16 × 10^{−4} | 1.56 × 10^{−4} | 0.153 | 0.152 | 3.64 | |

150 | 4.01 × 10^{−4} | 1.85 × 10^{−4} | 0.178 | 0.164 | 3.39 | |

100 | 3.89 × 10^{−4} | 2.39 × 10^{−4} | 0.213 | 0.143 | 3.06 | |

30 | 250 | 8.78 × 10^{−4} | 2.7 × 10^{−4} | 0.0334 | 0.224 | 4.59 |

200 | 8.69 × 10^{−4} | 3.13 × 10^{−4} | 0.0184 | 0.232 | 4.14 | |

150 | 8.62 × 10^{−4} | 3.7 × 10^{−4} | 0.0100 | 0.221 | 3.69 | |

100 | 8.54 × 10^{−4} | 4.54 × 10^{−4} | 0.0056 | 0.191 | 3.15 |

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**MDPI and ACS Style**

Li, Y.; Chen, L.; Shi, Y.
Influence of 3D Fracture Geometry on Water Flow and Solute Transport in Dual-Conduit Fracture. *Water* **2023**, *15*, 1754.
https://doi.org/10.3390/w15091754

**AMA Style**

Li Y, Chen L, Shi Y.
Influence of 3D Fracture Geometry on Water Flow and Solute Transport in Dual-Conduit Fracture. *Water*. 2023; 15(9):1754.
https://doi.org/10.3390/w15091754

**Chicago/Turabian Style**

Li, Yubo, Linjie Chen, and Yonghong Shi.
2023. "Influence of 3D Fracture Geometry on Water Flow and Solute Transport in Dual-Conduit Fracture" *Water* 15, no. 9: 1754.
https://doi.org/10.3390/w15091754