# Modified Convergent Flow Tracing Method for Evaluating Advective Velocity and Effective Porosity in Fractured Rock Aquifers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{a}= KI/n

_{e},” or EP, “n

_{e}= KI/v,” where Q is the flow rate, K is the hydraulic conductivity, I is the hydraulic gradient, and A is the cross-sectional area. In aquifers, where both Darcy’s equation and the velocity equation (with the consideration of the effects of the regional velocity during the tracer tests) are valid. The two equations can be substituted, one on the other, and rearranged to yield algebraic expressions for velocity and porosity. In particular, Leap and Kaplan (1988) [4] reported that the single-well drift-and-pumpback tracer test is useful for estimating the groundwater velocity. Hall et al. (1991) [5] suggested an equation using the drift-and-pumpback tracer test for determining two independent functional relationships between AV and EP. Stephens et al. (1998) [6] compared estimates of EP derived from soil–water characteristic curves and particle size, and those obtained from laboratory or field tracer tests. Although they estimated AV using the equation of Leap and Kaplan (1988), the reliabilities of the two parameters were not evaluated, particularly at fractured rock aquifers. Fernàndez-Garcia et al. (2002) [7] suggested EP to be assumed as the geometric mean of the principal components of the apparent porosity tensor in order to obtain a meaningful EP value estimated from radial flow tracer tests. Neuman (2005) [8] suggested that EP, a quantity relating AV to the Darcy flux, may show directional variations in field tracer tests. The difference between the AV of a conservative solute and the Darcy flux may indicate that the hydraulic properties and mass transport process are far more complex and sophisticated than are typically understood. Moreover, the measurement of an accurate AV, as opposed to that of the Darcy flux, is a crucial factor in the predictive evaluation method of a waste disposal site in terms of safety in the storage of the waste pollutants.

## 2. Field Study

#### 2.1. Field Site

#### 2.2. Hydraulic Characteristics

^{−5}–1.18 × 10

^{−5}m

^{3}/s was employed at the extraction well (SP-05) for approximately 8 h. The hydraulic conductivity was calculated using the equation for slab-shaped blocks, as reported by Barker (1988) [11]. Table 1 summarizes the hydraulic parameters of the zonal pumping tests. Q is the pumping rate, K is the hydraulic conductivity, b is the thickness of test section and I is the hydraulic gradient.

#### 2.3. Aquifer Characteristics

## 3. Theory and Experiment

#### 3.1. Overview

#### 3.2. Modified Method

_{a}is the advective velocity, n

_{e}is the effective porosity, P is the time elapsed from the start of the tracer extraction until the peak concentration arrival time at the extraction well, E is the total length of time from the start of the tracer injection to the first concentration arrival time, and from the start of the tracer injection to the peak concentration arrival time (from the start of tracer injection to the first concentration arrival time (FCAT) plus P), T is the transmissivity, K is the hydraulic conductivity, b is the saturated thickness of the aquifer, and I is the forced gradient resulting from the pumping conditions.

#### 3.3. Experimental Methods

#### 3.3.1. Convergent Flow Tracer Test

#### 3.3.2. Push-Pull Test

^{−5}m

^{3}/s and approximately 10 h, respectively.

## 4. Field Results and Interpretation

#### 4.1. Analysis of Breakthrough Curves in CFTT and PPT

#### 4.2. Advective and Effective Porosity

#### 4.3. Comparative Analysis of Experimental Results

^{−4}to 1.0

^{−3}, which was similar to the effective porosity values obtained in the rock aquifer of this study site [28,29,30].

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Tracer well location, groundwater flow direction, and zonal convergent flow tracer test using double packers in study area.

**Figure 2.**Schematic of the location of double packer and fracture distribution in mainly fractured zone.

**Figure 3.**Analysis of the fracture densities at intervals of 1.12 m ((

**a**) 3D spatial distribution; (

**b**) section view of SP-10, 02, 05, and 08 wells; (

**c**) section view from SP-01 to 03 wells, 04 to 06, and 07 to 09 wells; and (

**d**) section view of SP-03, 05, and 07 wells).

**Figure 4.**Analysis of the permeability fracture densities at intervals of 1.12 m ((

**a**) 3D spatial distribution; (

**b**) section view of SP-10, 02, 05, and 08; (

**c**) section view from SP-01 to 03, 04 to 06, and 07 to 09 wells; and (

**d**) section view of SP-03, 05, and 07 wells).

**Figure 5.**Transport processes because of (

**a**) pulse tracer injection in groundwater and (

**b**) fractured rock mass.

**Figure 6.**Tracer pathways and tracer responses of breakthrough curves according to the fracture density of (

**a**) high, (

**b**) moderate and (

**c**) low values (injection well—red circle, extraction well—blue circle), which was modified from Suzuki et al. (2012) [15].

**Figure 7.**Schematic of tracer pulse history in convergent flow tracer test: (

**a**) typical breakthrough curve owing to two-well tracing test, (

**b**) during pulse pumping in an extraction well, pulse position of first concentration arrival is pulse drift between two wells, and (

**c**) pulse history including first drift (from injection to first concentration arrival time (FCAT)), second drift (from injection to PCAT), and apparent radial position; ${r}_{1}$ and ${r}_{2}$ are radial displacements owing to the steady-state flow of constant pumping, and ${r}_{3}$ is the true radial displacement owing to constant volumetric flow rate of constant pumping in extraction well (where, ${r}_{1or3}$ is a true radial position).

**Figure 9.**Classification of breakthrough curves because of heterogeneous aquifers in convergent flow tracer test (CFTTs): Group A is (

**a**) SP-02 to 05 and (

**b**) SP-10 to 05, Group B is (

**c**) SP-03 to 05, (

**d**) SP-06 to 05, and Group C is (

**e**) SP-04 to 05 and (

**f**) SP-08 to 05.

Well ID | Separation Distance (m) | Q (m^{3}/s) | K (m/s) | b (m) | I | Note |
---|---|---|---|---|---|---|

02 to 05 | 1.50 | 1.18 × 10^{−5} | 2.17 × 10^{−6} | 1.12 | 0.31 | Two-well test |

03 to 05 | 2.21 | 1.18 × 10^{−5} | 2.27 × 10^{−6} | 1.12 | 0.28 | Two-well test |

04 to 05 | 1.50 | 1.18 × 10^{−5} | 3.38 × 10^{−6} | 1.12 | 0.28 | Two-well test |

05 to 05 | 0 | 1.18 × 10^{−5} | 2.17 × 10^{−6} | 1.12 | 0.14 | Single well test |

06 to 05 | 1.50 | 1.17 × 10^{−5} | 3.08 × 10^{−6} | 1.12 | 0.30 | Two-well test |

08 to 05 | 1.50 | 1.18 × 10^{−5} | 3.67 × 10^{−6} | 1.12 | 0.31 | Two-well test |

10 to 05 | 3.33 | 1.18 × 10^{−5} | 2.24 × 10^{−6} | 1.12 | 0.16 | Two-well test |

**Table 2.**Results of fracture and hydraulic fracture densities in zoning tracer tests (unit: m

^{−1}).

Well ID | Fractures ^{(1)} | Permeability Fractures | Well ID | Fractures ^{(1)} | Permeability Fractures |
---|---|---|---|---|---|

SP-01 | 2.68 | 0.89 | SP-06 | 5.36 | 2.68 |

SP-02 | 4.46 | 4.46 | SP-07 | 4.46 | 1.79 |

SP-03 | 2.68 | 1.79 | SP-08 | 4.46 | 1.79 |

SP-04 | 2.68 | 1.79 | SP-09 | 3.57 | 1.79 |

SP-05 | 4.46 | 4.46 | SP-10 | 7.14 | 5.36 |

^{(1)}is sum of impermeability and permeability fractures.

Well ID | Separation Distance (m) | Accumulative Recovery Rate of Tracer (%) | Note |
---|---|---|---|

02 to 05 | 1.50 | 92.83 | CFTT |

03 to 05 | 2.21 | 77.01 | CFTT |

04 to 05 | 1.50 | 88.64 | CFTT |

06 to 05 | 1.50 | 90.98 | CFTT |

08 to 05 | 1.50 | 96.25 | CFTT |

10 to 05 | 3.33 | 91.06 | CFTT |

**Table 4.**Results of nine tests for determining advective velocity (AV) and effective porosity (EP) by the sequence of emplacement, drift, and pumping.

Type of Tests | Convergent Flow Tracer Tests | Push-Pull Test | ||||||
---|---|---|---|---|---|---|---|---|

Trial Well (SP) | 02 to 05 | 03 to 05 | 04 to 05 | 06 to 05 | 08 to 05 | 10 to 05 | 05 | |

Travel Time (s) | FCAT | 300 | 360 | 180 | 210 | 570 | 480 | 1535 |

P | 1260 | 720 | 1200 | 720 | 1440 | 1800 | 1192 | |

E ^{(1)} | 1560 | 1080 | 1380 | 930 | 2010 | 2280 | 2637 | |

${v}_{a}$ (m/s) | 2.6 ×${10}^{-3}$ | 3.3 ×${10}^{-3}$ | 2.2 ×${10}^{-3}$ | 3.0 ×${10}^{-3}$ | 1.0 ×${10}^{-3}$ | 3.3 ×${10}^{-3}$ | 2.4 ×${10}^{-3}$ | |

${n}_{e}$ | 2.6 ×${10}^{-4}$ | 1.9 ×${10}^{-4}$ | 4.3 ×${10}^{-4}$ | 3.1 ×${10}^{-4}$ | 1.1 ×${10}^{-3}$ | 3.6 ×${10}^{-4}$ | 1.3 ×${10}^{-4}$ |

^{(1)}are P plus FCAT.

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**MDPI and ACS Style**

Kim, B.-W.; Lee, H. Modified Convergent Flow Tracing Method for Evaluating Advective Velocity and Effective Porosity in Fractured Rock Aquifers. *Water* **2020**, *12*, 3565.
https://doi.org/10.3390/w12123565

**AMA Style**

Kim B-W, Lee H. Modified Convergent Flow Tracing Method for Evaluating Advective Velocity and Effective Porosity in Fractured Rock Aquifers. *Water*. 2020; 12(12):3565.
https://doi.org/10.3390/w12123565

**Chicago/Turabian Style**

Kim, Byung-Woo, and Hangbok Lee. 2020. "Modified Convergent Flow Tracing Method for Evaluating Advective Velocity and Effective Porosity in Fractured Rock Aquifers" *Water* 12, no. 12: 3565.
https://doi.org/10.3390/w12123565