# Investigating Formation Factor–Hydraulic Conductivity Relations in Complex Geologic Environments: A Case Study in Taiwan

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data Sources

## 3. Methods

#### 3.1. Data Processing and Classification

#### 3.2. Theory of Formation Factor–Hydraulic Conductivity Relation for Clay-Free Formations

_{o}) to the resistivity of the water saturating the pores (R

_{w}). Archie [17] also proposed the relation between F and porosity n. Since 1942, the F-n relation has been confirmed empirically in a variety of singular formations. The relation is commonly expressed in the following form [21].

_{w}is the dynamic viscosity; g is the acceleration gravity; μ is the fluid density; d is the mean grain size; n is the porosity. Equation (2) is developed by the concept of the capillary tubes and often estimates the saturated K for most soils [30].

#### 3.3. Data Clustering for Eliminating Data Containing Clay Content

- (a)
- Natural Gamma Ray Threshold

- (b)
- Modified Archie’s law

_{a}is the apparent formation factor; F

_{c}is the corrected formation factor; R

_{w}is the resistivity of the water saturating the pores; Q

_{v}is the cation exchange capacity; and B is the equivalent conductivity of each cation. After proper rearrangement, Equation (5) can be presented as a linear relationship between 1/F

_{a}and R

_{w}.

_{v}/F

_{c}stands for the gradient [26]; the data of R

_{w}(x-axis) and 1/F

_{a}(y-axis) are plotted, and then the simple linear regression formula is obtained by using regression analysis. The intercept of the regression formula is 1/F

_{c}, and its value is calculated by inversion to obtain F

_{c}.

_{a}/F

_{c}) is calculated by dividing the uncorrected apparent formation factor (F

_{a}) and the corrected formation factor (F

_{c}) calculated by the correction method addressed above. A clay-free formation may be expected if F

_{a}/F

_{c}is greater than or equal to 0.9 [26]. The criterion indicated by Worthington [26] can be used to separate the collected samples into clay-bearing and clay-free sample groups for data clustering. The clay-free sample group can meet the original Archie’s law theory hypothesis, and the samples can be utilized to establish relations between the formation factor and hydraulic conductivity.

## 4. Results and Discussion

#### 4.1. Data Processing Result

#### 4.2. Correlation Analysis for Various Well-logging Signals with Hydraulic Conductivity

_{w}), as shown in Equation (1). It is then combined with SHN to calculate F.

#### 4.3. Data Clustering Results

#### 4.3.1. Outcomes from the Natural Gamma Threshold Method

#### 4.3.2. Outcomes from the Modified Archie’s Law Method

_{a}) using formation resistivity (R

_{o}) and the inverse of formation fluid conductivity (R

_{w}). Then, the modified formation factor (F

_{c}) was computed employing Equations (5) and (6) in Section 3.3. The F

_{a}and F

_{c}values are divided to obtain a dimensionless parameter F

_{a}/F

_{c}, and data samples where the F

_{a}/F

_{c}value falls in the [0.9,1.0] interval as the clean segments [26]. Finally, conducting a correlation analysis between the corrected formation factor (F

_{c}) and hydraulic conductivity (K) within the clean zones allows this study to understand the benefits of the improved correlation between F

_{c}and K resulting after being corrected by this clustering approach.

_{c}values for each sample under a single lithologic classification using the techniques described in Section 3.3, a scatter plot with 1/F

_{a}and R

_{w}values was created using 150 downhole signal data points for each sample section (1.5 m double-packer test interval). Furthermore, simple linear regression analysis was applied to determine the regression equation’s intercept (representing F

_{c}) and slope. However, if the obtained intercept or slope from the data points is negative, it would contradict the theoretical expectations. Such samples from the analysis will be excluded from the subsequent correlation analysis between F

_{c}and K. Based on the analysis as mentioned above criteria, each sample’s (F

_{a}, F

_{c}) data was obtained for each individual lithological type. Simultaneously, the F

_{a}/F

_{c}ratio was calculated for each sample. The correlation between F

_{c}and K was then analyzed for different ranges of the F

_{a}/F

_{c}ratio. This analysis aimed to investigate the relationship between the ratio and the correlation of F

_{c}vs. K. Theoretically, as the F

_{a}/F

_{c}ratio increases, the sample’s mud content decreases, aligning better with Archie’s law’s theoretical assumptions.

_{c}) and hydraulic conductivity (K) for different lithological types. According to the study by Worthington [26], the F

_{a}/F

_{c}value of the clean and mud-free formation interval should be greater than 0.9. However, the analysis results in the table show that very few samples of different lithological types meet the criterion of F

_{a}/F

_{c}greater than 0.9. Only a few sandstone samples can exceed this 0.9 threshold. Using this approach to separate samples from muddy and clean sections of the formation, it is clear that relatively few lithological samples in the Taiwan region meet the requirements of Archie’s law well.

_{a}/F

_{c}ratio, the higher the probability of containing mud. Table 7 also shows sample groups for different ranges of F

_{a}/F

_{c}ratios along with the correlation analysis results between F

_{c}and K. Taking sandstone as an example, the correlation between Fc and K improves as the F

_{a}/F

_{c}ratio increases. Similar trends were observed for most other lithological types. However, upon comparing these trends with the results of the correlation coefficient trends for each lithological type, this clustering approach suggests that sedimentary rock types have a better chance of yielding mud-free samples. On the other hand, slates in metamorphic rocks are more challenging to screen for mud-free samples. This conclusion aligns with the findings in the analysis of Section 4.3.1.

_{c}in this study, the estimated quality of F

_{c}data deserves discussion, as it determines the reliability of the correlation analysis results between F

_{c}and K. Figure 6 displays scatter plots and regression curves for three different sample sets of 1/F

_{a}against R

_{w}. The corresponding regression coefficients (R

^{2}) for the three sample sets are 0.9969, 0.8557, and 0.0099, respectively. The first sample set exhibits the highest R

^{2}value and a concentrated point distribution, indicating a strong matching degree and concentrated point set. The estimated F

_{c}value from this kind of well-matching and concentrated point distribution samples may belong to higher-quality data. Conversely, the point distribution in the last sample set is more scattered, suggesting lower-quality data for the estimated F

_{c}value. Thus, the quality of F

_{c}estimation has a specific impact on the quality of subsequent correlation analysis results between F

_{c}and K. Therefore, there is room for improvement in the feasibility of this clustering method for determining mud-containing intervals.

#### 4.4. Establishment of Hydraulic Conductivity Estimation Models

^{2}values of regression analyses between original data points and checked data points showed significant differences between the two sets of analysis results. The accuracy of hydraulic conductivity estimation models can be greatly enhanced by re-evaluating the rationality of data signals. Regression analysis results with the removal of outliers revealed R

^{2}values of 0.63, 0.60, and 0.83 for sandstone, schist, and slate regression models, respectively. For these three lithological types, the hydraulic conductivity estimation models best matched the power law model. Equations (7)–(9) show the estimation equations for sandstone, schist, and slate, respectively. Through the established estimation models, the steps for estimating hydraulic conductivity are (1) selecting the appropriate lithological estimation model; (2) collecting short normal resistivity and fluid conductivity data from borehole electrical logging; (3) obtaining the formation factor (F) using Archie’s law theory; and (4) estimating hydraulic conductivity using the calculated F value and Equations (7)–(9).

_{a}/F

_{c}≥ 0.9 after clustering. This clustering method is relatively stringent and theoretically grounded for identifying mud-containing formation intervals. After undergoing this clustering process, the sample count was reduced from 41 to only 3 that met the clustering criterion. Finally, regression analysis between K and F

_{c}and the establishment of hydraulic conductivity estimation models was performed for samples conforming to theoretical signal values. The regression analysis results for sandstone indicated that this lithological type’s hydraulic conductivity estimation model exhibited the best match with the power law model, with an R

^{2}value of 0.98. Equation (10) presents the estimation equation for sandstone.

## 5. Conclusions

- 1.
- For results concerning data processing and classification, statistical analysis showed a high degree of geological heterogeneity in the study site has been found. While developing hydraulic conductivity estimation models, well-logging signals were suggested to be categorized by lithological types to establish effective relationships with hydraulic conductivity.
- 2.
- For results regarding correlation analysis for various single well-logging signals with hydraulic conductivity, the three types of resistivity (LON, SHN, and SPR) and fluid conductivity (COND) signals with hydraulic conductivity for most of the lithological cases had better correlation performance than SP and NGAM signals. This better performance confirmed that the resistivity and fluid conductivity parameters were required to be composed of the formation factor (F). Nevertheless, a single signal alone is insufficient for constructing a model to estimate hydraulic conductivity.
- 3.
- To improve electrical–hydraulic relationships in response to the effect of clay mineralogy, the natural gamma ray threshold clustering and modified Archie’s law clustering methods successfully play an important role in filtering clayed data. However, to satisfy Archie’s law’s theoretical requirements, many data entries for various rock types needed to be removed, indicating that Taiwan’s mountainous rock formations are complex and often contain significant clay content. Therefore, careful consideration of clay-related issues in formation layers is essential in practical engineering applications in mountainous regions.
- 4.
- With the assistance of two mud clustering techniques, this study has successfully established four permeability estimation models for three rock types (sandstone, schist, and slate). The R
^{2}values are at least 0.6. However, the issue of limited data during model development is worth noting. - 5.
- During the exploration of electrical well-logging data, it was found that the clay effect is present in most rock formations in Taiwan. To enhance the utilization of a mathematical model for estimating hydrogeological parameters of individual rock types using single resistivity signals, more data collection is required to ensure the reliability of the model. Furthermore, for hydrogeological parameter estimation models applicable to multiple rock types, it is recommended to consider recombining the collected signals. This approach could yield novel signal indicators, enabling the construction of new relationships between indicators and different hydrogeological parameters.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**The short normal resistivity profiles at two different double-packer intervals. (

**a**) Abnormal resistivity profile. (

**b**) Normal resistivity profile.

**Figure 4.**Box plots for demonstrating the distributions of various signals with three major lithologies.

**Table 1.**Statistics of lithology for the data samples collected from the double-packer hydraulic test.

Main Lithology | Sub-Lithology | Amount |
---|---|---|

Sedimentary rock | Sandstone | 93 |

Shale | 31 | |

Sandy Shale | 3 | |

Sandstone interbedded with Argillite | 2 | |

Mudstone | 8 | |

Siltstone | 14 | |

Silty Sandstone | 20 | |

Alternations of Sandstone and Shale | 44 | |

Argillaceous Siltstone | 5 | |

Quartz Sandstone | 6 | |

Igneous rock | Andesite | 7 |

Metamorphic rock | Volcanic Agglomerate | 6 |

Phyllite | 6 | |

Slate | 61 | |

Schist | 41 | |

Marble | 6 | |

Gneiss | 5 | |

Argillite | 18 | |

Metasandstone | 3 | |

Argillite interbedded with some Sandstone | 10 | |

Quartzite | 7 | |

Total | 396 |

Signal Type | SP (V) | NGAM (cps) | COND (ohm.m) | |||||

Category | N | μ | S.D. | μ | S.D. | μ | S.D. | |

All lithologic types | 388 | 141.81 | 155.80 | 121.02 | 38.61 | 598.91 | 704.94 | |

Sedimentary Rock | 230 | 127.49 | 129.31 | 114.77 | 30.93 | 634.64 | 608.68 | |

Sandstone | 90 | 97.59 | 147.39 | 105.02 | 31.42 | 238.39 | 479.28 | |

Shale | 30 | 147.15 | 97.09 | 129.37 | 20.23 | 810.53 | 393.21 | |

Sandy Shale | 3 | 112.41 | 2.73 | 135.09 | 5.46 | 357.70 | 10.84 | |

Sandstone interbedded with Argillite | 10 | 182.97 | 88.13 | 150.42 | 13.17 | 475.71 | 433.32 | |

Mudstone | 8 | 153.16 | 12.01 | 75.20 | 43.42 | 660.81 | 177.84 | |

Siltstone | 14 | 127.85 | 81.40 | 125.69 | 14.60 | 1430.85 | 1018.14 | |

Silty Sandstone | 20 | 179.10 | 168.03 | 102.14 | 22.45 | 324.68 | 129.65 | |

Alternations of Sandstone and Shale | 44 | 135.32 | 104.48 | 124.28 | 27.52 | 468.51 | 374.78 | |

Argillaceous Siltstone | 5 | 61.72 | 6.04 | 106.40 | 7.05 | 476.60 | 10.33 | |

Quartz Sandstone | 6 | 182.97 | 216.57 | 123.38 | 41.24 | 2100.29 | 1459.38 | |

Igneous Rock | 13 | 238.67 | 168.08 | 43.35 | 26.55 | 359.45 | 96.49 | |

Andesite | 7 | 124.80 | 66.79 | 59.53 | 7.32 | 329.46 | 63.19 | |

Vocanic Agglomerate | 6 | 371.52 | 152.05 | 24.48 | 28.88 | 394.43 | 121.77 | |

Metamorphic Rock | 145 | 155.86 | 186.87 | 137.85 | 39.11 | 563.72 | 859.02 | |

Phyllite | 6 | 458.57 | 17.38 | 155.20 | 4.95 | 861.69 | 55.67 | |

Slate | 57 | 133.41 | 165.13 | 157.13 | 26.16 | 782.87 | 1213.49 | |

Schist | 41 | 155.60 | 202.61 | 119.14 | 35.60 | 375.59 | 120.84 | |

Marble | 6 | 130.69 | 15.25 | 46.27 | 15.65 | 244.82 | 46.86 | |

Gneiss | 5 | −55.84 | 22.01 | 143.68 | 5.69 | 269.90 | 142.64 | |

Argillite | 18 | 203.31 | 169.42 | 157.10 | 23.74 | 277.10 | 225.91 | |

Metasandstone | 3 | 463.68 | 4.05 | 136.56 | 34.48 | 131.96 | 12.33 | |

Argillite interbedded with some Sandstone | 2 | 49.03 | 70.90 | 153.44 | 18.23 | 168.15 | 2.17 | |

Quartzite | 7 | 30.03 | 127.06 | 96.60 | 42.01 | 1186.50 | 1339.26 | |

Signal | SHN (ohm.m) | LON (ohm.m) | SPR (ohm.m) | |||||

Category | N | μ | S.D. | μ | S.D. | Μ | S.D. | |

All lithologic types | 388 | 288.70 | 596.58 | 243.23 | 446.49 | 182.30 | 260.95 | |

Sedimentary Rock | 230 | 124.45 | 254.45 | 123.47 | 214.02 | 106.12 | 140.89 | |

Sandstone | 90 | 207.92 | 366.58 | 176.30 | 289.81 | 161.32 | 193.35 | |

Shale | 30 | 35.69 | 33.42 | 56.02 | 48.89 | 40.64 | 28.14 | |

Sandy Shale | 3 | 30.40 | 7.35 | 38.85 | 6.28 | 58.89 | 7.84 | |

Sandstone interbedded with Argillite | 10 | 277.11 | 228.22 | 330.14 | 247.94 | 176.69 | 100.96 | |

Mudstone | 8 | 14.27 | 4.93 | 17.66 | 2.94 | 28.19 | 11.50 | |

Siltstone | 14 | 27.87 | 11.44 | 47.34 | 27.40 | 31.32 | 13.51 | |

Silty Sandstone | 20 | 60.82 | 82.72 | 104.34 | 148.53 | 73.49 | 58.88 | |

Alternations of Sandstone and Shale | 44 | 63.51 | 79.04 | 87.46 | 135.64 | 82.45 | 69.08 | |

Argillaceous Siltstone | 5 | 9.54 | 1.15 | 12.09 | 1.21 | 21.00 | 1.22 | |

Quartz Sandstone | 6 | 225.83 | 222.09 | 105.54 | 85.37 | 148.34 | 173.97 | |

Igneous Rock | 13 | 375.77 | 590.63 | 254.86 | 373.40 | 217.36 | 209.74 | |

Andesite | 7 | 618.32 | 739.84 | 399.87 | 474.51 | 330.52 | 232.63 | |

Volcanic Agglomerate | 6 | 92.78 | 43.96 | 85.68 | 25.45 | 85.35 | 42.46 | |

Metamorphic Rock | 145 | 541.43 | 846.39 | 433.46 | 627.32 | 300.01 | 352.76 | |

Phyllite | 6 | 523.33 | 120.23 | 513.80 | 79.44 | 190.98 | 42.20 | |

Slate | 57 | 314.15 | 250.56 | 290.28 | 261.60 | 188.74 | 93.86 | |

Schist | 41 | 379.55 | 417.83 | 289.07 | 305.18 | 238.53 | 139.27 | |

Marble | 6 | 3973.34 | 1417.58 | 2864.96 | 1171.03 | 1606.99 | 850.91 | |

Gneiss | 5 | 1329.79 | 327.79 | 1031.72 | 195.14 | 737.45 | 118.70 | |

Argillite | 18 | 315.24 | 316.01 | 226.36 | 171.24 | 291.30 | 186.95 | |

Metasandstone | 3 | 373.64 | 134.02 | 384.41 | 125.77 | 312.53 | 39.43 | |

Argillite interbedded with Sandstone | 2 | 211.48 | 141.82 | 216.72 | 81.60 | 205.53 | 75.00 | |

Quartzite | 7 | 598.84 | 467.85 | 491.71 | 393.95 | 270.94 | 181.78 |

Signal Type | All Lithologic Type | Sedimentary Rock | Igneous Rock | Metamorphic Rock | Sandstone | Slate | Schist |
---|---|---|---|---|---|---|---|

Sample quantity | 391 | 230 | 13 | 148 | 90 | 60 | 41 |

SP | - | - | −0.613 | - | - | - | - |

SHN | 0.343 | 0.575 | 0.732 | - | 0.442 | - | - |

LON | −0.277 | 0.480 | - | - | 0.397 | - | - |

SPR | −0.378 | 0.549 | 0.765 | 0.245 | 0.490 | - | - |

NGAM | - | - | - | - | - | - | - |

COND | −0.308 | −0.281 | - | −0.355 | −0.338 | - | 0.345 |

**Table 4.**Correlation variations with various filtered data sets for the lithological group of sandstone.

F-Sandstone | ||||||||
---|---|---|---|---|---|---|---|---|

K | Data Sets | All Data | [30–187] | [30–167] | [30–148] | [30–128] | [30–109] | [30–89] |

r | 0.180 | 0.180 | 0.213 | 0.217 | 0.254 | 0.377 | 0.554 | |

Sig. | 0.089 | 0.094 | 0.049 | 0.050 | 0.030 | 0.006 | 0.002 | |

N | 90 | 88 | 86 | 82 | 73 | 52 | 28 |

**Table 5.**Correlation variations with various filtered data sets for the lithological group of slate.

F-Slate | ||||||||
---|---|---|---|---|---|---|---|---|

K | Data Sets | All Data | [94–210] | [94–193] | [94–177] | [94–160] | [94–144] | [94–127] |

r | −0.125 | −0.089 | −0.101 | −0.161 | −0.283 | −0.321 | −0.455 | |

Sig. | 0.392 | 0.546 | 0.501 | 0.327 | 0.161 | 0.285 | 0.365 | |

N | 49 | 48 | 47 | 39 | 26 | 13 | 6 |

**Table 6.**Correlation variations with various filtered data sets for the lithological group of schist.

F-Schist | |||||
---|---|---|---|---|---|

K | Data Sets | All Data | [23–156] | [23–137] | [23–118] |

r | −0.102 | −0.102 | −0.461 | −0.643 | |

Sig. | 0.526 | 0.550 | 0.015 | 0.01 | |

N | 41 | 37 | 27 | 15 |

**Table 7.**Correlation analysis of corrected formation factor (F

_{c}) and hydraulic conductivity (K) of different lithologies under different F

_{a}/F

_{c}ratio ranges.

Rock Types | Sandstone | Slate | Schist | |||
---|---|---|---|---|---|---|

F_{a}/F_{c} | r | No. of samples | r | No. of samples | r | No. of samples |

0–1 | −0.1 | 41 | 0.168 | 32 | −0.085 | 16 |

0.2–1 | 0.003 | 21 | −0.115 | 22 | −0.297 | 10 |

0.4–1 | 0.245 | 12 | −0.291 | 13 | 0.353 | 6 |

0.6–1 | 0.406 | 10 | −0.066 | 8 | 0.936 | 3 |

0.8–1 | −0.800 | 4 | 3 | N.A. | 1 | |

0.9–1 | −1.000 | 3 | N.A. | 2 | N.A. | 1 |

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**MDPI and ACS Style**

Hsu, S.-M.; Liu, G.-Y.; Dong, M.-C.; Liao, Y.-F.; Li, J.-S.
Investigating Formation Factor–Hydraulic Conductivity Relations in Complex Geologic Environments: A Case Study in Taiwan. *Water* **2023**, *15*, 3621.
https://doi.org/10.3390/w15203621

**AMA Style**

Hsu S-M, Liu G-Y, Dong M-C, Liao Y-F, Li J-S.
Investigating Formation Factor–Hydraulic Conductivity Relations in Complex Geologic Environments: A Case Study in Taiwan. *Water*. 2023; 15(20):3621.
https://doi.org/10.3390/w15203621

**Chicago/Turabian Style**

Hsu, Shih-Meng, Guan-Yu Liu, Ming-Chia Dong, Yi-Fan Liao, and Jia-Sheng Li.
2023. "Investigating Formation Factor–Hydraulic Conductivity Relations in Complex Geologic Environments: A Case Study in Taiwan" *Water* 15, no. 20: 3621.
https://doi.org/10.3390/w15203621