# Coil System Design for Multi-Frequency Resistivity Logging Tool Based on Numerical Simulation

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## Abstract

**:**

_{OI}). By analyzing the electromagnetic attenuation and phase difference of induced signals at different frequencies, the complex resistivity can be jointly inverted. The coil systems were designed with four D

_{OI}options (0.3 m, 0.5 m, 1 m, and 1.5 m) and six measurement frequencies (250 kHz, 500 kHz, 1 MHz, 2 MHz, 4 MHz, and 8 MHz). Their detection performance was evaluated using the finite element method on the COMSOL platform. For higher frequencies or a deeper D

_{OI}, a coil system with a larger source-receiver distance was selected. These designed coil systems can provide qualitative identification of formations with thicknesses greater than 0.05 m and quantitative identification of formations with thicknesses greater than 1.5 m. In the single-transmission, dual-reception coil system, response signals are distorted at the formation boundary, and this distortion increases with the source-receiver distance. Adding a secondary transmission coil can reduce the distortion of response signals at the formation interface without increasing the overall length of the coil system. This research enriches the theoretical framework of complex resistivity spectrum (CRS) logging and contributes to the commercial development of CRS logging tools.

## 1. Introduction

_{OI}) by increasing the number of measurement frequencies. However, due to the invasion of mud, the formation medium near the wellbore exhibits significant heterogeneity with increasing radial depth, rendering the real spectrum measurement unattainable for these tools. While the dielectric scanner by Schlumberger measures the conductivity and permittivity of formations at four frequencies from 20 MHz to 1 GHz [18,19,20,21,22], other logging tools utilizing a complex resistivity spectrum have not emerged. Consequently, many research results have not been effectively utilized in geophysical logging.

## 2. Principle and Method

#### 2.1. Measuring Principle

_{1}and R

_{2}, are used to detect the electromagnetic signals induced by the transmission coil.

_{1}and R

_{2}are collected, and the electromagnetic attenuation (EATT) and phase difference (Δφ) can be calculated using the following formulas [26]:

_{1}, and $\left|{V}_{2}\right|$ and ${\phi}_{2}$ are the amplitude and phase of the induced voltage on the reception coil R

_{2}. EATT is primarily influenced by the conductivity of the medium along the electromagnetic wave propagation path, while the phase difference signal is mainly affected by the dielectric constant of the medium.

_{1}and L

_{2}are the distance between the transmission coil with the two reception coils, where L

_{2}> L

_{1}.

_{OI}). The resistivity spectrum can be obtained by using impedance dispersion models [8,17] based on measured data from all six frequencies [23].

_{OI}values at different frequencies. As a result, multiple transmission coils with different source-receiver distances are necessary to maintain the same D

_{OI}when measuring at different frequencies. It is important to investigate how source-receiver distances and measurement frequency affect the detection performance of the coil system to optimize its design structure.

#### 2.2. Simulation Method

#### 2.3. Determine the Source-Receiver Distance

_{OI}) for the resistivity logging tool is typically evaluated using pseudo-geometric factor theory [29]. The medium model comprises only the invasion zone and the intact zone when the effects of the borehole are ignored. There should be a correlation between the apparent conductivity and the conductivity of each component, which can be expressed as follows:

_{i}is the depth of the invasion, and g(r) is the radial differential geometric factor.

_{OI}of the tool.

_{OI}) of the coil system in this study:

_{OI}that is as accurate as possible in actual logging environments. $\Delta {\phi}_{a}$, $\Delta {\phi}_{i}$, and $\Delta {\phi}_{t}$ all contain the response signal of drilling mud.

_{OI}of a resistivity logging tool, it is typically necessary to vary the radius of the invasion zone in the model and observe how the pseudo integral geometric factor changes. However, since the purpose of this study is to design a coil system with a specific DOI, we have set the radius of the invasion zone as our fixed target D

_{OI}, with a ΔL of 0.2 m. The key parameters of the model are listed in Table 1.

_{1}to investigate the impact of changing it on the pseudo integral geometric factor. Through these simulations, we could determine the change in both the pseudo integral geometric factor and Δφ with respect to L

_{1}. Using Equation (11), we calculated the change in the pseudo integral geometric factor with respect to L

_{1}. Finally, we selected an appropriate value of L

_{1}as the design outcome when the pseudo integral geometric factor was equivalent to the target value. This process significantly expedites the optimization of the coil system.

## 3. Result and Discussion

#### 3.1. Structure of Coil Systems

_{1}for the target D

_{OI}of 0.3 m, 0.5 m, 1 m, and 1.5 m. When L

_{1}is fixed, higher measurement frequencies correspond to larger values of ${G}_{\Delta \phi}({d}_{i})$. Therefore, to ensure that different frequencies correspond to the same D

_{OI}, a high measurement frequency should be paired with a transmission coil that has a larger L

_{1}. By examining the L

_{1}value associated with each curve in Figure 5 when ${G}_{\Delta \phi}({d}_{i})=0.5$, Table 2 provides the values of L

_{1}and L

_{2}for single-transmission dual-reception coil systems that meet the requirements for each target D

_{OI}and measurement frequency. Based on the information from Table 2, six-transmission dual-reception coil systems can be designed to conduct measurements at six frequencies with the same D

_{OI}, as presented in Table 3.

_{OI}) by matching different measuring frequencies with their corresponding transmission coils. If more transmission coils are integrated into a single tool, such as a 12- or 18-transmission dual-reception coil system, it becomes feasible to conduct measurements with multiple frequencies and multiple D

_{OI}s using a single tool. This expanded capability allows for more comprehensive data acquisition and analysis, providing a deeper understanding of the subsurface formation properties.

#### 3.2. Vertical Resolution

_{OI}of 0.5 m, using a measurement frequency of 8 MHz. These curves represent the behavior of the coil system when different thicknesses of the target layer are present. It can be observed that both EATT and Δφ values in response to the target layer are smaller compared to the surrounding layer.

_{OI}is set to 0.3 m. Remarkably, even when the thickness of the target layer is as small as 0.05 m, the curves corresponding to all frequencies can still qualitatively identify the presence of the target layer.

_{1}− L

_{2}, of 0.2 m. The consistent reception coil configuration ensures that the measurements are made under similar conditions, allowing for reliable and comparable results.

#### 3.3. Structure Improvement

_{1}are both positioned in the surrounding layer above the target layer, effectively placing the entire coil system within the surrounding layer. However, when the recording point is at the bottom interface of the target layer, as shown in Figure 10b, the transmitting coil T and receiving coil R

_{1}are both positioned within the target layer, potentially including some upper rounding layer depending on the source-receiver distance of the coil system. Therefore, for symmetric geological models, the asymmetry in the response curve around the middle of the target layer is mainly attributed to the asymmetric structure of the coil system.

_{2}) can be added adjacent to the original transmission coil (T

_{1}) near the reception coil. The improved coil system under a single frequency has been changed to a dual-transmit dual-receive coil system, as shown in Figure 11. The distance between T

_{2}and R

_{2}is equal to that between T

_{1}and R

_{1}. With this dual-transmission and dual-reception coil system for signal D

_{OI}measurement and single frequency, the degree of curve distortion at the formation interface can be reduced.

_{1}and received by R

_{1}and R

_{2}to obtain EATT

_{R}and Δφ

_{R}, and the recording point is the center of the two reception coils. Subsequently, the same frequency signal is transmitted by T

_{2}and received by R

_{2}. The displacement of the tool can be ignored due to the short time difference between the two reception processes. Based on the induction signals received by R

_{2}corresponding to the two transmission coils, respectively, another electromagnetic attenuation EATT

_{T}and phase difference Δφ

_{T}can be obtained at the center of the two transmission coils.

_{R−x}, Δφ

_{R−x}, EATT

_{T−x}, and Δφ

_{T−x}, and then records EATT

_{R−x−L}

_{1}, Δφ

_{R−x−L}

_{1}, EATT

_{T−x−L}

_{1}, and Δφ

_{T−x−L}

_{1}at the logging depth point H = x

_{−}L

_{1}after lifting L

_{1}. EATT

_{T−x}and Δφ

_{T−x}have the same recording point depth as EATT

_{R−x−L}

_{1}and Δφ

_{R−x−L}

_{1}. By averaging two sets of data at the same recording point depth, the response curve distortion caused by the asymmetry of the coil system can be weakened.

_{OI}of 1 m and a measurement frequency of 8 MHz. The EATT

_{T}curve and Δφ

_{T}curve with subscript T are the measurement results of T

_{1}-T

_{2}-R

_{2}, while the EATT

_{R}curve and Δφ

_{R}curve with subscript R are the measurement results of T

_{1}-R

_{1}-R

_{2}. Averaging the two sets of data yields the measurement results of the dual-transmission and dual-reception coil system, which are shown as the red curves with subscript A in Figure 12.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Some low-porosity and low-permeability rocks have dispersion frequency bands beyond 500 kHz [11].

**Figure 3.**Comparison of analytical and numerical solutions of: (

**a**) EATT and (

**b**) Δφ. ΔL is set to 5 cm, and L

_{1}and L

_{2}are set to 30 cm and 25 cm.

**Figure 5.**Pseudo radial integral geometric factor changes with L

_{1}at measurement frequencies and different D

_{OI}: (

**a**) 0.3 m; (

**b**) 0.5 m; (

**c**) 1 m; and (

**d**) 1.5 m. ΔL is set to 0.2 m, and the L

_{1}corresponding to the intersection of the curve. The straight line when G

_{Δ}

_{φ}(d

_{i}) = 0.5 is the L

_{1}that meets the design requirements.

**Figure 6.**The three-layer formation model with a wellbore. The logging depth of the target layer center is 0 m.

**Figure 7.**The response signals of the single-transmission dual-reception coil system to the three-layer formation model with different target layer thickness: (

**a**) EATT and (

**b**) Δφ. The D

_{OI}of the coil system is 0.5 m, and the measurement frequency is 8 MHz.

**Figure 8.**The $\Delta \phi $ curves of the single-transmission dual-reception coil systems with the D

_{OI}of 0.3 m to the three-layer formation model at different frequencies: (

**a**) 0.25 MHz; (

**b**) 0.5 MHz; (

**c**) 1 MHz; (

**d**) 2 MHz; (

**e**) 4 MHz, and (

**f**) 8 MHz.

**Figure 9.**EATT curves to three-layer formation model at a frequency of 8 MHz of the single-transmission dual-reception coil systems with different D

_{OI}: (

**a**) 0.3 m; (

**b**) 0.5 m; (

**c**) 1 m; (

**d**) 1.5 m.

**Figure 10.**Schematic diagram of the model when the recording point is located at (

**a**) the top and (

**b**) the bottom interfaces of the target layer.

**Figure 12.**The response curves of the dual-transmission and dual-reception coil system to the three-layer stratum model: (

**a**) EATT and (

**b**) Δφ. In the model, the thickness of the low-resistance target layer is 2 m, and the reception coils spacing is 0.2 m.

Parameters | Borehole | Invasion Zone | Intact Zone |
---|---|---|---|

Inner radius (m) | 0 | 0.054 | D_{OI} |

Outer radius (m) | 0.054 | D_{OI} | 25 |

Relative magnetic permeability | 1 | 1 | 1 |

Relative permittivity | 75 | 50 | 30 |

Conductivity (S/m) | 1 | 0.5 | 0.1 |

**Table 2.**Source-receiver distances for single-transmission dual-reception coil system corresponding to different measurement frequencies and D

_{OI}.

Frequency (MHz) | D_{OI} = 0.3 m | D_{OI} = 0.5 m | D_{OI} = 1 m | D_{OI} = 1.5 m | ||||
---|---|---|---|---|---|---|---|---|

L_{1} (m) | L_{2} (m) | L_{1} (m) | L_{2} (m) | L_{1} (m) | L_{2} (m) | L_{1} (m) | L_{2} (m) | |

0.25 | 0.08 | 0.28 | 0.33 | 0.53 | 1.18 | 1.38 | 2.37 | 2.57 |

0.5 | 0.11 | 0.31 | 0.41 | 0.61 | 1.49 | 1.69 | 2.88 | 3.08 |

1 | 0.16 | 0.36 | 0.53 | 0.73 | 1.83 | 2.03 | 3.34 | 3.54 |

2 | 0.23 | 0.43 | 0.69 | 0.89 | 2.15 | 2.35 | 3.63 | 3.83 |

4 | 0.32 | 0.52 | 0.86 | 1.06 | 2.36 | 2.56 | 3.78 | 3.98 |

8 | 0.42 | 0.62 | 1.02 | 1.22 | 2.47 | 2.67 | 3.83 | 4.03 |

**Table 3.**Parameters of the six-transmission dual-reception coil systems corresponding to different D

_{OI}.

D_{OI}/m | Coil System Structure |
---|---|

0.3 | T_{1}-0.10-T_{2}-0.09-T_{3}-0.07-T_{4}-0.05-T_{5}-0.03-T_{6}-0.08-R_{1}-0.20-R_{2} (m) |

0.5 | T_{1}-0.16-T_{2}-0.17-T_{3}-0.16-T_{4}-0.12-T_{5}-0.08-T_{6}-0.33-R_{1}-0.20-R_{2} (m) |

1 | T_{1}-0.11-T_{2}-0.21-T_{3}-0.32-T_{4}-0.34-T_{5}-0.31-T_{6}-1.18-R_{1}-0.20-R_{2} (m) |

1.5 | T_{1}-0.05-T_{2}-0.15-T_{3}-0.29-T_{4}-0.46-T_{5}-0.51-T_{6}-2.37-R_{1}-0.20-R_{2} (m) |

Parameters | Borehole | Target Layer | Surrounding Layer |
---|---|---|---|

Inner radius (m) | 0 | 0.054 | 0.054 |

Outer radius (m) | 0.054 | 25 | 25 |

Relative magnetic permeability | 1 | 1 | 1 |

Relative permittivity | 75 | 65 | 30 |

Conductivity (S/m) | 1 | 0.001 | 0.1 |

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**MDPI and ACS Style**

Jia, J.; Ke, S.; Rezaee, R.
Coil System Design for Multi-Frequency Resistivity Logging Tool Based on Numerical Simulation. *Water* **2023**, *15*, 4213.
https://doi.org/10.3390/w15244213

**AMA Style**

Jia J, Ke S, Rezaee R.
Coil System Design for Multi-Frequency Resistivity Logging Tool Based on Numerical Simulation. *Water*. 2023; 15(24):4213.
https://doi.org/10.3390/w15244213

**Chicago/Turabian Style**

Jia, Jiang, Shizhen Ke, and Reza Rezaee.
2023. "Coil System Design for Multi-Frequency Resistivity Logging Tool Based on Numerical Simulation" *Water* 15, no. 24: 4213.
https://doi.org/10.3390/w15244213