# Geometric and Fractal Characterization of Pore Systems in the Upper Triassic Dolomites Based on Image Processing Techniques (Example from Žumberak Mts, NW Croatia)

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}or CO

_{2}[18,20,25,27,30,31,32,33,34,35,36], nuclear magnetic resonance [37], and X-ray methods (SANS/USANS) [12,18,27,33,38] although these methods can estimate porosity values but do not give information about pore morphology. CT scans [12,30,39,40], scanning electron microscopy [16,19,25,26,29,30,33,41,42,43,44,45], and thin-section analysis [29,37,46] are methods that can quantify geometrical parameters of the rock pore structure. These methods in themselves cannot define heterogeneity and pore shape complexity of analyzed rocks [26,47,48,49]. Researchers found that fractal theory can be very effective in describing and quantifying irregular and complex pore geometry. The complexity of pore structures can be quantified by the fractal dimension’s complex fractal characteristics [16,17,25,28,39,50,51,52,53,54,55,56].

## 2. Study Area and Geological Setting

#### 2.1. Study Area

#### 2.2. Geological Setting

_{3}) are the most important lithological unit in the geological structure of Žumberak, with a total thickness of 1570 m [2,3] (Figure 1 and Figure 2).

## 3. Materials and Methods

#### 3.1. Porosity Determination Using Saturation and Buoyancy Techniques

#### 3.2. Microphotograph Analysis

## 4. Results

_{3}dolomites (without detail stratigraphy determination). The Posinak formation was not sampled due to its similarity with the Slapnica formation and representative outcrop absence comparing to Slapnica and the Main Dolomite Formation. As all formations have similar properties resulting from a similar depositional environment, we decided to present results by lithofacies.

#### 4.1. Porosity Measurements

#### Effective and Microphotograph Porosity

#### 4.2. Geometrical Pore Characterization and Distribution of Pore Characteristics

**Area**: minimum value is 0.001 mm^{2}, and the maximum is 957 mm^{2}, with a median of 0.007 mm^{2}and a mean of 2.46 mm^{2}(Table 1). Due to the limit on the image resolution, and the fact that it is proportional to the square of a very small quantity and the automatic rounding, a loss of significant digits has occurred (the data has one significant digit, and at least three are needed). Therefore, no distribution was found to fit the data sufficiently well. The histogram in Table 1 presents pores with an area less than 1 mm^{2}for better visibility since only 27 pores have an area between 1 and 957 mm^{2}.**Perimeter:**minimum value is 0.015 mm, and the maximum is 242.5 mm, with a median of 0.34 mm and a mean of 1.36 mm. It is best fit by the exponential power distribution (p = 0.99), or the power log-normal distribution (p = 0.61) (Table 1).**AR**: minimum value is 1, and the maximum is 16.91, with a median of 1.75 and a mean of 2.047. It is best fit by the Laplace distribution (p = 0.4) or the exponential power distribution (p = 0.22) (Figure 7). These results mean that most of the pores have a lon axis 2–3× longer than the short axis.

- Feret AR: minimum value is 1.10 and maximum is 12.50, with a median of 1.64 and a mean of 1.84. It is best fit by the chi-squared distribution (p = 0.21), or the gamma distribution (p = 0.17) (Figure 7). Feret AR results confirm AR results, and the elongation of the pores is generally between 2 and 3 in one direction.
- Circularity: minimum value is 0.03 and maximum is 1, with a median of 0.56 and a mean of 0.54. It is best fit by the generalized gamma distribution (p = 0.75), or noncentral F-distribution (p = 0.14) (Figure 8).
- Roundness: minimum value is 0.059, and the maximum is 1, with a median of 0.57 and a mean of 0.57. It is best fit by the beta distribution (p = 0.37), or the cosine distribution (p = 0.33) (Figure 8).
- Solidity: minimum value is 0.348, and the maximum is 1, with a median of 0.80 and a mean of 0.78. It is best fit by the cosine distribution (p = 0.91), or the power-law distribution (p = 0.24) (Figure 8).
- Compactness: minimum value is 0.24, and the maximum is 1, with a median of 0.757 and a mean of 0.75. It is best fit by the hypergeometric distribution (p = 0.3), or the exponential distribution (p = 0.19) (Figure 8).

- Fractal dimension: minimum value is 1.21 and maximum is 2, with a median of 1.66 and a mean of 1.65. It is best fit by the Maxwell distribution (p = 0.98), or the normal distribution (p = 0.67) (Figure 8).

## 5. Discussion

- Parameters that quantify size: area and perimeter;
- Parameters that quantify elongation: aspect ratio and Feret aspect ratio;
- Parameters that quantify the object’s overall shape without considering the roughness of the object’s surface: circularity and roundness;
- Parameters that quantify roughness or complexity of the surface (small irregularities on the object’s surface): solidity, compactness, and fractal dimension.

^{2}. Additionally, we were not able to quantify pores smaller than 0.001 mm

^{2}due to image resolution limitations. We found a strong linear correlation between area and perimeter (r = 0.95), which indicates the pores’ elongation.

## 6. Conclusions

- Dolomicrite microfacies had very low sedimentation and intergranular porosity reduced or remained the same in the marine diagenetic environment, recrystallization, and dolomitization processes. The effective porosity of dolomicrite facies is from 0.657 to 2.548%. 2D porosity for these facies is in the range of 0.12 to 1.31%. Increased porosity values in the dolomicrite facies result from microcrack formation that causes microcracks in the measured samples, so the upper part of the interval should be taken with caution.
- Microfacies of fenestral dolomites had a high sediment porosity, over 30%. Porosity is significantly reduced by diagenesis in the vadose, meteoric, marine environment, geopetal filling processes, crystallization, and dolomitization. The effective porosities of these facies range from 0.35 to 2% and correspond to diagenetic porosity. 2D porosity for these facies is in the range of 0.05 to 5%.
- Microfacies of stromatolite dolomites generally have the highest porosities. Depositional porosity was also quite high but was significantly reduced by diagenesis in the vadose, meteoric, and marine environments. The amounts of effective porosity range from 1.85 to 4.38% and generally correspond to diagenetic porosity. 2D porosity for these facies is in the range of 0.027 to 9.03%.
- Both porosity measurements (effective type and 2D) indicated that non-fracture porosity could not be ignored in permeability calculations, so a double-porosity model should be applied.
- Geometric parameters of pores can be subdivided into four groups: a) size quantification parameters (area, perimeter); elongation parameters (aspect ratio, Feret aspect ratio); overall shape parameters (circularity, roundness) in the sense of comparison of an object to a sphere; roughness parameters (solidity, compactness, fractal dimension).
- To quantify the pore geometry, it is necessary to use parameters from all four groups since each group and each parameter quantify the pore geometry differently. Therefore, analyzed parameters are an effective way to reflect the complexity of the pore structure. Fractured aquifers are very valuable geothermal and groundwater resources regarding quantity and quality. Hence, their sustainable management and protection are of the highest priority. Therefore, every piece of knowledge about the porosity and permeability of these systems is crucial for later modeling of fluid/contaminant flow.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Pore area (mm**—the area occupied by a two-dimensional object.^{2})**Perimeter (mm)**—the total length of the boundary of a two-dimensional object.**Circularity (−)**—the degree to which the particle is similar to a circle, regarding the curvature and the smoothness of the boundary. It takes values from [0,1], where a circle achieves the maximum of 1. Circularity is a measure of particle shape and roughness.

**AR (**−**)**—the ratio of the particle’s circumscribed ellipse’s axes. The values of AR are greater or equal to 1.

**Roundness (**−**)**—the inverse of aspect ratio. Measures how closely the shape of an object approaches that of a circle regarding the curvature of the object’s boundary. Its values are from [0,1], with the circle having a roundness of 1 and, for instance, an ellipse with axes a and b (a > b) has a roundness of b/a < 1.

**Solidity (**−**)**—the fraction of the region as compared to its convex hull. It measures the convexity of an object. Objects containing all lines between any two of its points (in particular, without holes and with no boundary irregularities) have a solidity value of 1. Otherwise, the solidity value is less than 1.

**Feret AR (**−**)**—elongation index, the values of Feret AR are greater or equal to 1.

**Compactness (**−**)**—ratio of the area of an object to the area of a circle with same diameter. A circle is the most compact object with value 1, and all other objects have compactness less than 1.

**Fdim (**−**)**—fractal dimension of an object. The fractal analysis represents a group of methods for quantifying complex patterns [77,78]. Fractals and fractal geometry can be applied to various geological disciplines where classical geometry is not enough to describe complex objects in nature [79,80]. We described only basic elements of fractal geometry which are necessary to understand the application in this paper. Fractals can be defined as “irregular” objects divided into segments that are equal to each other and the whole object; thus they exhibit self-similarity [81,82]. For an object to be fractal, it must have the following properties [81]:- -
- The constituent parts of an object have the same structure as the object as a whole, except that they are slightly deformed in different scales (there are small fluctuations in the measure of fractality between scales)—
**self-similarity**. - -
- Objects are often irregular and fragmented and remain so in all the scales in which they exist.
- -
- Objects are created by an iterative procedure.
- -
- Objects have fractal dimensions.

_{f}) of an object. The D

_{f}is a value that quantifies the complexity of an object.

- N = the number of objects fragments characterized by the linear dimension r;
- C = proportionality constant;
- D
_{f}= fractal (Hausdorff’s) dimension, which is calculated (Equation (A8)):

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**Figure 2.**Representative outcrop and lithofacies types of the Upper Triassic dolomites from Žumberak Mts. (

**a**) The outcrop of the Main Dolomite Formation; (

**b**) sample of fenestral dolomite facies of the Main Dolomite Formation; (

**c**) sample of stromatolite facies of the Main Dolomite Formation; (

**d**) transition from peloid dolomite to fenestral dolomite lithofacies.

**Figure 3.**Samples of test material: (

**a**) field samples from which cored samples were made for testing; (

**b**) part of the prepared samples of regular and irregular geometry.

**Figure 8.**Frequency diagrams for circularity, roundness, solidity, compactness, and fractal dimension parameters of pores.

Name | Equation | Image | Mean | St.dev. |
---|---|---|---|---|

Pore area (mm^{2}) | 2.46 | 37.70 | ||

Perimeter (mm) | 1.36 | 9.17 | ||

Circularity (−) | $\frac{4\times \pi \times area}{pe{r}^{2}}$ | 0.54 | 0.22 | |

Aspect ratio (−) | $\frac{majA}{minA}$ | 2.05 | 1.26 | |

Roundness (−) | $\frac{4\times area}{\pi \times {d}^{2}}$ | 0.57 | 0.18 | |

Solidity (−) | $\frac{area}{conarea}$ | 0.78 | 0.11 | |

Feret AR (−) | $\frac{maxCD}{minCD}$ | 1.84 | 0.85 | |

Compactness (−) | $\frac{\sqrt{\frac{4\times area}{\pi}}}{d}$ | 0.75 | 0.13 | |

Fractal dimension (−) | $\frac{\mathrm{log}\left(\frac{{N}_{i+1}}{{N}_{i}}\right)}{\mathrm{log}\left(\frac{{r}_{i}}{{r}_{i+1}}\right)}$ | 1.65 | 0.07 |

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**MDPI and ACS Style**

Pavičić, I.; Briševac, Z.; Vrbaški, A.; Grgasović, T.; Duić, Ž.; Šijak, D.; Dragičević, I. Geometric and Fractal Characterization of Pore Systems in the Upper Triassic Dolomites Based on Image Processing Techniques (Example from Žumberak Mts, NW Croatia). *Sustainability* **2021**, *13*, 7668.
https://doi.org/10.3390/su13147668

**AMA Style**

Pavičić I, Briševac Z, Vrbaški A, Grgasović T, Duić Ž, Šijak D, Dragičević I. Geometric and Fractal Characterization of Pore Systems in the Upper Triassic Dolomites Based on Image Processing Techniques (Example from Žumberak Mts, NW Croatia). *Sustainability*. 2021; 13(14):7668.
https://doi.org/10.3390/su13147668

**Chicago/Turabian Style**

Pavičić, Ivica, Zlatko Briševac, Anja Vrbaški, Tonći Grgasović, Željko Duić, Deni Šijak, and Ivan Dragičević. 2021. "Geometric and Fractal Characterization of Pore Systems in the Upper Triassic Dolomites Based on Image Processing Techniques (Example from Žumberak Mts, NW Croatia)" *Sustainability* 13, no. 14: 7668.
https://doi.org/10.3390/su13147668