# Models and Interpretation Methods for Single-Hole Flowmeter Experiments

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experiments and Interpretation Methods

#### 2.1. Considered Experiments and Methods

- Single-pumping single-log experiments. The vertical flow rates and hydraulic heads are measured along the borehole while the flowmeter is lowered into the well during a single pumping test.
- Single-pumping double-log experiments. As before, the vertical flow rates and hydraulic heads are measured along the borehole during two log experiments that are conducted under the same pumping test.
- Multiple-pumping local-log experiments. A pumping test is performed for each conductive zone that needs to be characterized (except the upper one) with the logging tool localized above this zone.

#### 2.2. Double Flowmeter Test with Transient Flow Rate (DFTTF)

#### 2.3. Transient Flow Rate Flowmeter Test (TFFT)

- Model the data ${Q}_{P}\left(t\right)$ with variable pumping flow rate models, as described in Appendix C.1.
- Estimate the unknowns ${T}_{i}$, ${S}_{i}$, ${\sigma}_{i}$ by inverting the data $({Q}_{i}\left(t\right),{h}_{wS}\left(t\right))$ provided by the multi-aquifer model described in Appendix B using a numerical optimization method. More precisely, the Laplace transform of ${Q}_{i}\left(t\right)$ and ${h}_{wS}\left(t\right)$ are given in expressions (A15) and (A17) and numerically inverted with [37]’s algorithm.

## 3. Examples of Applications on Synthetic Cases

#### 3.1. Considered Configurations

#### 3.2. Single- and Double-Log Flowmeter Experiments

#### 3.3. Series of Local Flowmeter Experiments

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Standard Models and Interpretation Methods

#### Appendix A.1. Single Flowmeter Test (SFT)

#### Appendix A.2. Double Flowmeter Test (DFT)

## Appendix B. Multi-Aquifer Model

## Appendix C. Models for Transient Parameters and Properties

#### Appendix C.1. Pumping Flow Rate Models

#### Appendix C.2. Couples of Equivalent Parameters (S si,σ i )

## References

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**Figure 1.**Estimated values of transmissivities and storativities normalized by the true values (${\mathcal{T}}_{i}$ and ${\mathcal{S}}_{i}$, respectively) for aquifer 1 and 2 ($i=1,2$) along ${\epsilon}_{ij}$ the ratio of diffusivities of aquifer i to that of aquifer j (${\epsilon}_{ij}={\epsilon}_{i}/{\epsilon}_{j}$), for model configurations Config1 (

**first row**), Config2 (

**second row**), and Config3 (

**third row**), and the interpretation methods SFT, DFT, and DFTTF (black, blue, and red symbols, respectively).

**Figure 2.**Transient pumping flow rate (${Q}_{P}$) defined from the exponential model provided in (A18) with ${Q}_{{t}_{1}}=4$ and ${Q}_{{t}_{2}}=3.8$ Lpm at times ${t}_{1}=0$ and ${t}_{2}=300$ min with the fitting coefficient b set to ${10}^{5}$. ${Q}_{P}$ is the pumping flow rate considered in Config3 of Section 3.2 and in all the experiments in Section 3.3. ${Q}_{{t}_{1}}$ and ${Q}_{{t}_{2}}$ correspond to the aquifer flow rates of the reference experiments considered in Section 3.3.

**Figure 3.**(

**a**) Minimum of the objective function (2) obtained for each value of the couple $({\sigma}_{1},{\sigma}_{2})$ for ${\sigma}_{1}$ and ${\sigma}_{2}$ ranging from −2 to 5. The corresponding estimated values of (

**b**) $log\left({T}_{1}\right)$, (

**c**) $log\left({T}_{2}\right)$, (

**d**) $log\left({S}_{1}\right)$, and (

**e**) $log\left({S}_{2}\right)$ obtained with the TFFT method. (

**f**,

**g**) Estimated values of $log\left({S}_{1}^{\prime}\right)$ and $log\left({S}_{2}^{\prime}\right)$ obtained with expression (A23) from the values of ${S}_{1}$ when ${\sigma}_{1}=-2$ (first line in (

**d**)) and ${S}_{2}$ when ${\sigma}_{2}=-2$ (first line in (

**e**)). For all figures, the color cells correspond to increasing values from yellow to red and the log function corresponds to log10 function.

**Table 1.**Considered flowmeter experiments with the following collected data: ${h}_{w,i}$ and ${q}_{i}$ are the hydraulic heads and vertical flow rates, respectively, measured above conductive zone i with ${N}_{aq}$ the number of conductive zones to characterize, and ${h}_{w,i}^{j}$ and ${q}_{i}^{j}$ are their counterpart collected from two logs ($j=1,2$). SFT, DFT, DFTTF, and TFFT are the interpretation methods presented in Appendix A and Section 2.

Experiment Name | Collected Data | Interp. Methods |
---|---|---|

Single-pumping single-log | $({h}_{w,i},{q}_{i})$, $i=1,\cdots ,{N}_{aq}$ | SFT |

Single-pumping double-log | $({h}_{w,i}^{j},{q}_{i}^{j})$, $j=1,2$ | DFT, DFTTF |

Multiple-pumping local-log | $({h}_{w,i}\left(t\right),{q}_{i}\left(t\right))$ | TFFT |

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**MDPI and ACS Style**

Lods, G.; Roubinet, D.
Models and Interpretation Methods for Single-Hole Flowmeter Experiments. *Water* **2023**, *15*, 2960.
https://doi.org/10.3390/w15162960

**AMA Style**

Lods G, Roubinet D.
Models and Interpretation Methods for Single-Hole Flowmeter Experiments. *Water*. 2023; 15(16):2960.
https://doi.org/10.3390/w15162960

**Chicago/Turabian Style**

Lods, Gerard, and Delphine Roubinet.
2023. "Models and Interpretation Methods for Single-Hole Flowmeter Experiments" *Water* 15, no. 16: 2960.
https://doi.org/10.3390/w15162960