# Shape Factor for Analysis of a Slug Test

## Abstract

**:**

## 1. Introduction

^{−1}] is the hydraulic conductivity of the soil, ${r}_{i}$ [L] is the inner radius of the well tubing where the head difference is observed, ${R}_{e}$ [L] is an effective radius, and ${r}_{o}$ [L] is the outer radius of the well screen (including a gravel pack if present). The hydraulic conductivity of the soil can thus be obtained by analyzing the slope of the logarithm of the observed head values against time [1,2] as

^{−1}is the inverse hyperbolic sine function. For a well screen with a large aspect ratio ($L/{r}_{o}\gg 1$), this can be further simplified as

## 2. Materials and Methods

## 3. Results

#### 3.1. Numerical Solution

#### 3.2. Approximate Analytical Solution

#### 3.3. Shape Factor Values

^{−5}. These ${r}_{o}q/{h}_{w}$ values are used to derive the average specific flux value along the well screen and the resulting shape factor using Equation (18).

#### 3.4. Test Case

_{i}= 0.064 m, an outer radius r

_{o}= 0.125 m, and a length L = 1.52 m. The initial head displacement in the well was h

_{0}= 0.671 m and the difference in head h

_{w}was monitored for about 6 min. The observations against time are shown in Figure 3.

^{−1}, as shown in Figure 3. Values for the shape factor obtained with the different methods are given in Table 2. The resulting shape factors are very similar, plausibly because the well screen is located far from the boundaries of the aquifer. The last column of Table 2 gives the resulting estimates of the hydraulic conductivity of the aquifer. All methods more or less agree, but the values obtained by the methods prosed in this study and the method of Zlotnik et al. [13] are very close to each other, while the value obtained with the Hvorslev method [1] is clearly larger and the value obtained with the method of Bouwer and Rice [2] is clearly lower. Hence, this practical example confirms the conclusions from the previous section and clearly demonstrates the applicability and accuracy of the methodology proposed in this study.

## 4. Discussion

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of a slug test conducted in a well with a screened section (shown is a rising head test after a small amount of water is extracted from the well at time zero): r

_{i}is the inner radius of the well casing, r

_{o}is the outer radius of the well screen, L = 2l is the length of the screened section, h

_{0}is the difference in head in the well at time t = 0, h

_{w}is the difference in head in the well at time t > 0, and r and z are the radial and vertical coordinates with origin at the center of the screen.

**Figure 2.**Plot of the shape factor $\mathrm{l}\mathrm{n}\left({R}_{e}/{r}_{o}\right)$ against the aspect ratio $L/{r}_{o}$ of the well screen obtained with (1) numerical approach given by Equation (18), (2) approximate analytical solution given by Equation (24), (3) Hvorslev approximation [1] given by Equation (3), (4) Bouwer and Rice empirical approach [2], and (5) approximate solution of Zlotnik et al. [13].

**Figure 3.**Plot of observed head against time on semilogarithmic paper and slope fitted by linear regression for the falling head slug test performed in Well 4-2, Pratt County Monitoring Site 36, Kansas, U.S [21].

**Figure 4.**Plot of the G-function (Equation (13)) and the approximation for large aspect ratios of the well screen (Equation (20)) against $z/{r}_{o}$: (

**a**) large-scale general plot; and (

**b**) detailed plot near the origin.

**Figure 5.**Plot of ${r}_{o}q/{h}_{w}$ against $z/l$ derived with the numerical solution, Equation (17) (solid lines), and with the approximate analytical solution, Equation (22) (dotted lines), for aspect ratio $L/{r}_{o}$ equal to 1, 10, and 100.

Method | Solution Technique | Flow Distribution at the Well Screen | Aquifer Boundary Conditions |
---|---|---|---|

Hvorslev [1] | Analytical | Uniform | No |

Bouwer and Rice [2] | Empirical | Non-uniform | Yes |

Zlotnik et al. [13]. | Analytical | Uniform | Yes |

This study 1 | Numerical | Non-uniform | No |

This study 2 | Analytical | Non-uniform | No |

**Table 2.**Values obtained with the different methods for the shape factor of the slug test and the hydraulic conductivity of the aquifer.

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**MDPI and ACS Style**

De Smedt, F.
Shape Factor for Analysis of a Slug Test. *Water* **2023**, *15*, 2551.
https://doi.org/10.3390/w15142551

**AMA Style**

De Smedt F.
Shape Factor for Analysis of a Slug Test. *Water*. 2023; 15(14):2551.
https://doi.org/10.3390/w15142551

**Chicago/Turabian Style**

De Smedt, Florimond.
2023. "Shape Factor for Analysis of a Slug Test" *Water* 15, no. 14: 2551.
https://doi.org/10.3390/w15142551