# An Improved Rock Resistivity Model Based on Multi-Fractal Characterization Method for Sandstone Micro-Pore Structure Using Capillary Pressure

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Derivations of a Resistivity Model Based on Multi-Fractal Characterization Method for Sandstone Micro-Pore Structure Using Capillary Pressure

#### 2.1. Multi-Fractal Characterization Method for Sandstone Micro-Pore Structure Using Capillary Pressure

#### 2.2. An Improved Rock Resistivity Model Considering Pore Structure

## 3. Model Validation

_{1}, D

_{2}, S

_{1}, P

_{cmin}, c, and d are calculated using the least square method with the experimental capillary pressure and resistivity according to Equation (6).

#### 3.1. Multi Fractal Characterization of Pore Structure Using Capillary Pressure

#### 3.2. An Improved Rock Resistivity Model Considering Pore Structure

#### 3.3. The Relationship between Capillary Pressure Curve and Resistivity Index Curve Based on Fractal Theory

## 4. Discussion and Future Work

#### 4.1. Multi-Fractal Based Modeling of Capillary Pressure Curves

- (1)
- According to Equation (6), the fractal ${D}_{v,1}$ and parameter c can be calculated from permeability and porosity measured in actual formation evaluation. Further, pore structure can be classified according to the fractals ${D}_{v,1}$, ${D}_{v,2}$, and parameter c, capillary pressure curve and pore structure characterization can be achieved.
- (2)
- Figure 11 analyzes the affection of ${D}_{v,1}$, ${D}_{v,2}$, c, and ${S}_{1}$ on capillary pressure curves according to Equation (6). The simulated capillary pressure curves with different c (${D}_{v,2}$) and fixed ${D}_{v,1}=2.7$, ${S}_{1}=0.2$ v/v, ${P}_{cmin}=0.05$ Mpa is depicted in Figure 11a. The result shows that as c increases (${D}_{v,2}$ decreases), ${P}_{c}$ under specific water saturation decreases, which is an indication of pore structure improvement, it is identical with that c has a positive correlation with $\sqrt{K/\varphi}$. In addition, the capillary pressure curve exhibits non-power function features (power function exhibits linear feature in a logarithmic coordinate system). The simulated capillary pressure curves with different ${S}_{1}$ and fixed ${D}_{v,1}=2.7$, $c=5$, ${P}_{cmin}=0.05$ Mpa is depicted in Figure 11b. The result shows that as ${S}_{1}$ increases at large c value, ${P}_{c}$ under specific water saturation increases, it indicates that the pore structure gets poor.

#### 4.2. Multi-Fractal Based Modeling of Resistivity Index Curves

#### 4.3. Multi-Fractal Features of Pore Structure

#### 4.4. Future Work

- The capillary pressure curve supplies fundamental data for pore structure characterization methods [43]. According to Equation (6), the pore structure characterization accuracy is improved for porous rock with complex pore structure, for instance, tight rock and shales.
- Reservoir flow unit division research based on capillary pressure curves is an important way for reservoir pattern studies [44]. According to Equation (6), more accurate reservoir flow unit division can be achieved.
- Resistivity models are crucial for oil and gas saturation calculation in practical applications [45]. In this paper, the accuracy of Equation (15) is improved than the Archie model. Equation (15) can be further utilized for reservoir estimation and shale organic carbon assessments.
- Equation (16) describes the relationship between the capillary pressure curve and rock resistivity based on multi-fractal theory. It provides a new idea and an effective way for studying the effect of pore structure on rock resistivity [46], especially for rocks with complex micro-pore structures, such as rock within fractures, shale, and carbonate rocks.
- As the heterogeneity of pore size and morphology of the study areas increases, single-dimension fractal theory is unable to meet the accuracy requirements. Multi-fractal theory can provide an effective way for pore structure characterization [47], which will be a research focus.

## 5. Conclusions

- (1)
- Based on multi-fractal theory, a multi-fractal characterization method for sandstone micro-pore structure using capillary pressure is developed, and its accuracy is improved than the commonly used power function model for the fitting of experimental capillary pressure curves. Based on the multi-fractal characterization method for sandstone micro-pore structure using capillary pressure, a rock resistivity model considering pore structure is developed. The new model is proven to have higher accuracy than the Archie model; it can accurately describe the rock conductivity characteristics and calculate the oil saturation of complex pore structure reservoirs.
- (2)
- A distinct interrelationship between fractal dimensions of capillary pressure curves (${D}_{v}$) and resistivity index curves (${\mathrm{D}}_{t}$ and ${\mathrm{D}}_{r}$) is obtained. The capillary pressure curve can be directly converted to the resistivity index when d is determined. The fractal feature parameters ${D}_{v,1}$, ${D}_{v,2}$, c strongly depend on pore structure properties. Parameters c, d, and ${D}_{v,1}$ have a good relationship with $\sqrt{K/\varphi}$, the pore structure typing result by ${D}_{v,1}$ and c is accordance with that according to the morphology of capillary pressure curve.
- (3)
- According to the multi-fractal-based analysis of capillary pressure curves and resistivity increase rate, the multi-fractal feature of pore structure is the main reason for non-Archie resistivity and non-power function capillary pressure relationships. As the difference between the two types of pores becomes stronger, non-Archie and non-power function features become obvious. Therefore, the multi-fractal method can improve the accuracy of pore structure characterization when rock pore structure complexity increases.
- (4)
- This study provides new ideas to improve the accuracy of pore structure characterization and oil saturation calculation; it has good application prospects and guiding significance in reservoir evaluation and rock physical characteristics research.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Thin section photomicrographs of dominant lithology in the study area. Q: quartz, F: feldspar, M: matrix, P: pore space. (

**a**) Thin section photomicrographs of sample 1, (

**b**) Thin section photomicrographs of sample 2, (

**c**) Thin section photomicrographs of sample 5, (

**d**) Thin section photomicrographs of sample 7.

**Figure 6.**Experimental formation factor F and resistivity index I. (

**a**) Experimental formation factor F, (

**b**) Experimental resistivity index I.

**Figure 7.**Comparison between capillary pressure curve and resistivity index curve for four rock types. (

**a**) Comparison of type 1, (

**b**) Comparison of type 2, (

**c**) Comparison of type 3, (

**d**) Comparison of type 4.

**Figure 10.**Comparison between the calculated resistivity by Equation (16) and the experimental data.

**Figure 11.**Affection of ${D}_{v,1}$, ${D}_{v,2}$, c and ${S}_{1}$ on capillary pressure curves ((

**a**) c effects on capillary pressure curves, (

**b**) ${S}_{1}$ effects on capillary pressure curves).

**Figure 12.**Affection of ${\mathrm{D}}_{t,1}$, d on resistivity index curves. (

**a**) ${\mathrm{D}}_{t,1}$ effects on resistivity index curves, (

**b**) d effects on resistivity index curves.

Type | No. | Depth m | Porosity % | Permeability md | F | n | D_{v1} | D_{v2} | S_{1} | P_{cmin} | c | d |
---|---|---|---|---|---|---|---|---|---|---|---|---|

I | 1 | 317.46 | 30.2 | 1910 | 3.08 | 1.889 | 2.82 | 2.29 | 0.14 | 0.015 | 3.95 | 0.9 |

2 | 316.82 | 32.59 | 2020 | 2.98 | 1.869 | 2.79 | 2.31 | 0.15 | 0.028 | 3.285 | 0.98 | |

II | 3 | 319.23 | 30.55 | 556 | 2.95 | 1.981 | 2.76 | 2.42 | 0.18 | 0.038 | 2.42 | 1.02 |

4 | 321.55 | 32.5 | 530 | 2.63 | 1.841 | 2.77 | 2.49 | 0.27 | 0.023 | 2.22 | 1.34 | |

III | 5 | 322.69 | 25.86 | 115 | 3.6 | 2.149 | 2.68 | 2.59 | 0.44 | 0.13 | 1.28 | 1.28 |

IV | 6 | 314.55 | 13.35 | 14.7 | 7.27 | 1.932 | 2.67 | 2.66 | 0.5 | 0.41 | 1.03 | 1.56 |

7 | 318.77 | 15.91 | 52 | 10.07 | 1.746 | 2.69 | 2.65 | 0.59 | 0.69 | 1.13 | 1.71 |

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**MDPI and ACS Style**

Xie, W.; Yin, Q.; Zeng, J.; Yang, F.; Zhang, P.; Yan, B.
An Improved Rock Resistivity Model Based on Multi-Fractal Characterization Method for Sandstone Micro-Pore Structure Using Capillary Pressure. *Fractal Fract.* **2024**, *8*, 118.
https://doi.org/10.3390/fractalfract8020118

**AMA Style**

Xie W, Yin Q, Zeng J, Yang F, Zhang P, Yan B.
An Improved Rock Resistivity Model Based on Multi-Fractal Characterization Method for Sandstone Micro-Pore Structure Using Capillary Pressure. *Fractal and Fractional*. 2024; 8(2):118.
https://doi.org/10.3390/fractalfract8020118

**Chicago/Turabian Style**

Xie, Weibiao, Qiuli Yin, Jingbo Zeng, Fan Yang, Pan Zhang, and Binpeng Yan.
2024. "An Improved Rock Resistivity Model Based on Multi-Fractal Characterization Method for Sandstone Micro-Pore Structure Using Capillary Pressure" *Fractal and Fractional* 8, no. 2: 118.
https://doi.org/10.3390/fractalfract8020118