Fractal Characterization on Three-Dimensional Tortuosity of Fault Tectonic

Abstract

Faults, as a kind of fracture tectonics, play a role in reservoir closure or provide oil and gas transportation channels. The accurate understanding of the distribution characteristics of faults is significant for oil and gas exploration. The traditional fractal dimension for fault number (D_f3) cannot comprehensively characterize the complexity and heterogeneity of fault network distribution. In this paper, a fractal characterization method on three-dimensional (3D) tortuosity of fault tectonics is proposed based on 3D seismic exploration. The methodology is described in detail to establish the model on the fractal dimension for the 3D tortuosity of fault tectonics. The results show the proposed method of estimation of the D_T3 displaying high accuracy and rationality. Compared with the traditional fractal dimension D_f3, the proposed D_T3 can comprehensively characterize the fractal characteristics of faults network systems in the 3D space. This study achieves a breakthrough in the fractal characterization of the 3D tortuosity of fault tectonics. It is worth further study for establishing an analytical fractal equation based on the D_T3 and oil or gas transfer, which can provide the theoretical foundation and technical support for oil and gas exploration.

1. Introduction

Geological tectonics are crucial in the generation and distribution of hydrocarbons in oil and gas basins [1]. Faults, as classical fracture tectonics [2], can promote the gathering of hydrocarbons in their vicinity to form oil and gas reservoirs [3]. The strike–slip faults can connect the source rock with the oil recovery layer [4], providing paths for the aggregation of hydrocarbons in a wider range [5]. Therefore, an accurate understanding of the fault distribution characteristics is significant for the exploration of fracture zones in oil and gas basins [6,7].

Recently, mathematical analysis methods have facilitated the progress of fracture network systems from qualitative analysis to quantitative description [8,9]. As a tool for assessing measures of spatial occupation and irregularity of complex forms, the fractal theory has been widely used in the fields of geophysics, geology [10,11], minerals [12], etc. Scholars adopted fractal theory to explore the fault network system [13], a self-similar fractal system consisting of faults with different levels [14]. The complexity of the fault network system can be quantitatively characterized by extracting the fault parameters, specifying its fractal structure, and calculating the fractal dimension. Li et al. analyzed the mechanical properties of fissured coal under different gas pressures using crack length–quantity fractal dimensions [15]. Delphine Roubinet et al., combining fracture density and fractal dimension, created an inversion method that enables the deduction of the statistical characteristics of fracture networks from cross-borehole thermal experiments (CBTEs) [16]. Zhang et al. carried out quantitative statistical analyses of parameters such as the number of pores, pore throats, and coordination numbers and used fractal dimensions to analyze in depth the characteristics of microporous connectivity and the degree of development of coal [17]. Oynakov E and his team explored the distribution of these faults by analyzing earthquakes and their activity in the Balkans [18]. Xu et al. developed an analytical model of the pore and capillary structure of porous media based on fractal theory [19]. However, most previous studies have used the fractal dimension for fracture numbers to characterize the complexity of faults. Actually, faults not only have length, width, and openness in 3D space [20] but also form a series of curved channel networks as the tortuosity changes, as shown in Figure 1 [21]. Due to the high degree of opening of fracture activity, the equilibrium state of primary oil and gas reservoirs is often destroyed, and the fault becomes a channel for oil and gas transport [22,23]. Therefore, it is worth exploring a fractal characterization method on three-dimensional tortuosity of fault tectonics to accurately and comprehensively capture the complexity of their internal tortuosity.

In this study, the 3D seismic exploration technique is employed to derive the three-dimensional structure parameters of fault tectonics, by which the model on the fractal dimension for the 3D tortuosity of fault tectonics (D_T3) is established, proposing a fractal characterization method on three-dimensional (3D) tortuosity of fault tectonics. The accuracy and rationality of the proposed method on estimation of the D_T3 are verified. A comparative analysis of the proposed D_T3 with the traditional fractal dimension D_f3 is discussed. This study achieves a breakthrough in the fractal characterization of the 3D tortuosity of fault tectonics, which can provide the theoretical foundation and technical support for oil and gas exploration.

2. Objects and Methodology of Research

2.1. Methodology

The tortuosity τ is the ratio of the seepage channel’s distance to the porous medium’s length in which it is located [24]. It depicts the actual path the mass point follows within the opening while the seepage fluid moves a one-unit distance through the medium [25]. As shown in Figure 2, tortuosity is one of the key parameters for studying the fracture structure of rocks, and it can also be applied to fault fracture channels [26]. It is usually defined as Equation (1).

$$\tau =L_t/L$$

(1)

The porosity ϕ of a two-dimensional fault can be determined by comparing the total area (A_fracture) of the fault to the mining zone’s area (A_media), as shown in Equation (2) [27,28].

$$\varphi ={\displaystyle \frac{A_{fracture}}{A_{media}}}$$

(2)

In 3D space, fault porosity ϕ can be determined by comparing the volume of the fault (V_fault) to the total volume of the zone it occupies (V_total), based on the definition of the 3D fracture porosity [29,30]. This means that the porosity of a fault in 3D space can be represented as the proportion of the fault’s overall volume to the volume of the geological unit, as illustrated in Equation (3). Both V_fault and V_total can be obtained through the geological data of the 3D seismic exploration.

$$\varphi ={\displaystyle \frac{V_{fault}}{V_{total}}}$$

(3)

The average fault tortuosity τ_av is a key parameter reflecting the tortuosity of 3D faults, which can be used to estimate by ϕ in Equation (3), based on the definition of the average fault tortuosity of the fracture [31].

$${\tau }_{\mathrm{av}}={\displaystyle \frac{1}{2}}\left[1+{\displaystyle \frac{1}{2}}\sqrt{1-\varphi }+\sqrt{1-\varphi }{\displaystyle \frac{\sqrt{{(\frac{1}{\sqrt{1-\varphi }}-1)}^2+\frac{1}{4}}}{1-\sqrt{1-\varphi }}}\right]$$

(4)

where ϕ is the fault fissure degree; A_fracture and A_media are the areas of the fault and the area of the mining area in 2D space, respectively; V_fault and V_total are the volume of the fault and the volume of the mass unit where the fault is located in 3D space. In addition, the four parameters can be derived from the 3D seismic exploration data [32,33].

The faults’ length in each unit (l) is measured and the count of faults is equal to or larger than l, represented by N(l). The fault network has a certain regularity [34,35]. The probability density function’s range of the fracture network f(l) can be represented by Equation (5) [36].

$$f(l)=D_f{l}_{min}^{Df}{l}^{-(Df+1)}$$

(5)

The relationship between l and N(l) is shown in Equation (6) [37].

$$N(l)\propto l^{-D_f}$$

(6)

Taking the logarithm on both sides gives Equation (7).

$$\log N(l)=-D_f\log l+C$$

(7)

where N(l) represents the faults’ lengths that are equal to or greater than l; l indicates the fault length; D_f is the fractal dimension of the fault count, while C is a constant. The values of l and N(l) can be obtained from the geological exploration data of the 3D seismic exploration.

Fracture channels in faults can be characterized by the tortuosity fractal dimension D_T3, as shown in Equation (8).

$$\tau (l)={\displaystyle \frac{L_t(l)}{L}}={\left({\displaystyle \frac{L}{l}}\right)}^{D_T-1}$$

(8)

The average fault tortuosity in three dimensions can be calculated by Equation (9) [38].

$${\tau }_{\mathrm{av}}={\displaystyle {\int }_{l_{\min }}^{l_{\max }}\tau }(l)f(l)dl={\displaystyle \frac{D_{f3}}{\left(D_{f3}+D_{T3}\right)-1}}{({\displaystyle \frac{L}{l_{\min }}})}^{D_{T3}-1}$$

(9)

As shown in Equation (9), τ_av is a function of D_f3, D_T3, and l_min. Thus, taking the logarithm of Equation (9) yields an implicit expression for computing D_T3, as shown in Equation (10). Equation (10) can be transformed into Equation (11).

$$D_{T3}=1+{\displaystyle \frac{\ln \left(\frac{{\tau }_{\mathrm{av}}(D_{f3}+D_{T3}-1)}{D_{f3}}\right)}{\ln \left(\frac{L}{l_{\min }}\right)}}$$

(10)

$$1+{\displaystyle \frac{\ln \left(\frac{{\tau }_{\mathrm{av}}\left(D_{f3}+D_{T3}-1\right)}{D_{f3}}\right)}{\ln \left(\frac{L}{l_{\min }}\right)}}-D_{T3}=0$$

(11)

where τ_av can be derived from Equation (4); D_f3 can be fitted by Equation (7); L and l can be measured by the geological exploration data of the 3D seismic exploration.

For the calculation of D_T3 for each geological unit, we divided it into the following three steps: (a) Basic data were measured by a 3D seismic exploration. Basic data include the length (l) and dip (θ) of each fault, the number of faults with a length longer than or equal to l, and the minimum length (l_min) and maximum length (l_max) of each fault, which were derived by comparison; (b) The key parameters are calculated from the derived model. Key parameters include fault volume (V_fault), geological unit volume (V_total), fault porosity (ϕ), average fault tortuosity (τ_av), and fractal dimension of the number of faults (D_f3). We drew Figure 3 for easy understanding of the calculation process of D_T3.

2.2. Research Object

The research object is from Tianchi Coal Mine in Shanxi Province of China, and its geographic coordinates are 113°28′14″~113°29′14″ East and 37°14′45″~37°15′53″ North. The whole mine is circled by 26 coordinate points, with a length of 7.5 km, a width of 3.5 km, and an area of 17.9103 km².

Tianchi Coal Mine is in the eastern part of Qingshui Basin, east of Taihangshan Fracture and Taihangshan Uplift Zone. Since the beginning of the Indo-Chinese movement at the end of the Triassic period, the region entered a period of tectonic activity and subjected the Paleozoic coal seams to relatively strong modification. However, there are some differences in the characteristics of tectonic action and the nature of tectonic movement during different periods of tectonic movement. Thereby, tectonics of different nature, direction, and period have also been formed within the well field, which tends to complicate the faults in each geological unit. The result is that the porosity of faults varies within different geological units.

2.3. Geological Prospecting by the 3D Seismic Exploration

Three-dimensional seismic exploration is a high-density area exploration technique, which uses the flexible combination of gun points and wave detectors to obtain a grid of evenly distributed CDP points. This time, our exploration work was divided into the following three main parts: (a) Exploration and acquisition of field seismic data: The data were acquired using a seismic exploration data collector, and the target layer for acquisition was near the coal seam. Data collection requires staff to carefully plan and manage the location, depth, and quantity of explosives for each borehole. Explosives were placed in pre-planned locations while each location was recorded; (b) Processing of data exploration information: Data processing in the region was carried out in our team’s computing center, using the seismic data processing software in version 7.1 of CGG’s Geovecteur Plus. To ensure the high resolution and fidelity of the seismic results, it is important to focus on the following steps: Establishment of spatial properties, surface coherence, amplitude compensation, speed analysis, etc; (c) explanation of data and information: Interpretation of 3D data is based on 3D stereoscopic data bodies, and the interpretation process is realized on workstations. The known geological information and logging information are combined for a comprehensive interpretation. Figure 4 displays the arrangement of the gun checkpoints for this 3D seismic exploration.

During 3D seismic exploration, the receiver line distance affects the number of explorations covered and the density of data. In complex near-surface conditions, the selection of an appropriate receiver line distance is particularly important for improving imaging quality. The size of the shot interval affects the imaging accuracy of the subsurface target body. Smaller shot intervals improve the lateral resolution of the data and help to delineate subsurface structures more finely. Imaging accuracy has a direct impact on the accuracy of exploration data acquisition and processing, which in turn has an impact on the accuracy of D_T3 calculations.

Table 1. Main parameters of the observing system in the 3D seismic exploration.

Item	Parameter
Type of observing system	8-wire, 8-gun, harnessed
Receiving channels/channel	384 (8 × 48 = 384)
Number of receiving wires (bar)	8
Motivation method	Midpoint excitation
Receiver line distance (m)	40
Receiver channel distance (m)	20
Excitation line distance (m)	20
Shot interval (m)	80
CDP grid	10 m × 10 m
Fold (time)	24 (vertical 6 × horizontal 4)
Offsetmax (m)	523
Offsetmin (m)	0

presents the specific parameters of the system. The 408 UL digital seismometer is used, and the instrument parameters are set as follows: Sampling interval: 0.5 ms, recording length: 1.5 s, pre-amplification gain: 18 db, recording band: full-band reception. The whole area is stimulated by a single well with a depth of 10 m and a dosage of 1.5 kg/well. Four 100 Hz geophones are used for the receiving device, and a total of 3749 physical points of seismic production are designed for this exploration. The seismic data were processed by the GEOVECTEURPLUS software package [3].

3. Results and Discussion

3.1. Analysis of Faults by 3D Seismic Exploration

According to the geological exploration data of the 3D seismic exploration, the total number of faults is 37 in the mine. Among them, there are 12 faults with a drop of less than 5 m, 9 faults with a drop of 5–10 m, and 16 faults with a drop of more than 10 m. The section strike is mainly NE direction, partly NNE direction, NS direction, and rarely NW direction. The tendency is variable, but mainly NW or SE. The length of the faults with drops greater than 10 m ranges from 140 to 940 m with an average of 405 m, and the drop ranges from 10 to 25 m with an average of 13.9 m.

As shown in Figure 4, we divided three mining area ranges as the study objects, which are recorded as geological units Ⅰ, Ⅱ, and Ⅲ, respectively. The whole mining field range is recorded as geological unit Ⅳ. Using the length l, drop H, and dip θ of each fault measured by geological exploration, the horizontal break is obtained by using the relationship between dip and drop. The steps for deriving the volume of the faults are as follows: (a) The horizontal break of the fault is obtained by using the relationship between drop and dip; (b) the volume of the faults is obtained by multiplying the drop, the horizontal break, and the length of the interval; (c) the measured volume of the faults within each geological unit is added up, and the total volume of the faults within geological units I, II, III, and IV is 779,193.9221, 281,160.8316, 42,179.90738, and 1,102,534.66108 m³, respectively. Then we can obtain that the volumes of the units in which the faults are located are 14,430,150, 10,612,800, 2,324,118, and 27,367,068 m³ by using the maximum difference between the area of the geological unit and the elevation of the fault. Finally, the values of fault porosity ϕ can be calculated by Equation (8).

The tectonic distribution of the mine is generally in the form of monoclinic tectonics with a northeast direction and northwestward inclination with little undulation, on which there are small folds, faults, and trap columns developed, as shown in Figure 5.

3.2. Key Parameters of Faults Derived by 3D Seismic Exploration

The geological exploration data were used to derive the key parameters for calculating the fractal dimension D_T3 of the 3D fault tortuosity in the mine. The l_min and l_max of the geological units can be derived from the measurements. As shown in

Table 2. Basic analysis of the geological unit.

Geological Unit	lmin (m)	lmax (m)	ϕ	τav
Ⅰ	55	940	0.05399	9.64131
Ⅱ	35	670	0.02649	19.25248
Ⅲ	59	598	0.01815	27.91459
Ⅳ	35	940	0.04029	12.7903

, the minimum fault lengths l_min of geological units Ⅰ, Ⅱ, Ⅲ, and Ⅳ are 55, 35, 59, and 35 m, respectively. The value of l_min of geological unit Ⅲ is the largest, followed by that of the geological units Ⅰ, Ⅱ, and Ⅳ, where those of geological units Ⅱ and Ⅳ are equal. Similarly, the maximum fault lengths l_max are 940, 670, 598, and 940 m, respectively. The values of l_max of geological units Ⅰ and Ⅳ are equal and largest, followed by that of geological units Ⅱ and Ⅲ. The ϕ is calculated from Equation (3) to obtain the 3D fault porosity of geological units Ⅰ, Ⅱ, Ⅲ, and Ⅳ. As shown in Table 2, the values of the 3D fault porosity (ϕ) of geological units Ⅰ, Ⅱ, Ⅲ, and Ⅳ are 0.05399, 0.02649, 0.01815, and 0.04029, respectively. The value of ϕ of geological unit Ⅰ is the largest, followed by that of the geological units Ⅳ, Ⅱ, and Ⅲ in that order. The average fracture torsion (τ_av) of each geological unit can be calculated from Equations (1) and (4). As shown as Table 2, the values of τ_av of geological units Ⅰ, Ⅱ, Ⅲ, and Ⅳ are 9.64131, 19.25248, 27.91459, and 12.79030, respectively. The value of the average fracture torsion (τ_av) of geological unit Ⅲ is the largest, followed by that of geological units Ⅱ, Ⅳ, and Ⅰ.

3.3. Estimation on D_f3

According to Equation (7), D_f3 of each geological unit can be fitted. As shown in Figure 6, the D_f3 of geologic unit Ⅰ, geologic unit Ⅱ, geologic unit Ⅲ, and geologic unit Ⅳ are 1.97167, 1.76996, 1.26900, and 2.02397, respectively. This indicates that geological unit IV has the most complex distribution of faults, while geological unit III has the simplest distribution of faults, from the perspective of the number of faults developed.

3.4. Estimation on D_T3

According to the basic parameters of faults, we can determine the fractal dimension for 3D fault tortuosity (D_T3) in the geological units by Equations (10) and (11). The detailed calculating process for the values of D_T3 in the geological units is shown in Figure 7: (a) The first column is set with a series of D_T values, which are 1, 1.1, 1.2, …, 2.8, 2.9, and the second column edits Equation (11) to bring in the known parameters from Table 1, leaving only the D_T for the variables, and then edits the D_T from the first column into Equation (11). There must be two D_T3 values such that the equation has a value between one positive and one negative, and then the solution to D_T3 must lie between the two D_T3 values; (b) find the two D_T values, e.g., between 1.9 and 2.0, and set up a series of columns of data from 1.90, 1.91, 1.92, …, 1.99, 2.00, and again find the two D_T values that make the above formula between positive and negative, e.g., between 1.90 and 1.91, and then set up another series of data columns from 1.350, 1.351, 1.352, …,1.359, 1. 360, and again find the two D_T values; (c) make the above equation between positive and negative, and then repeat the iteration until the required number of decimal places have been calculated to obtain the value of D_T3. By iterating the calculation, the D_T3 for geological units I, II, III, and IV are 1.90737, 1.96026, 2.10998, and 1.69579, respectively.

3.5. Validation of the Calculation Method on D_T3

Previous studies have demonstrated the classical correlation between fractal dimension for fracture tortuosity and fracture porosity [39]. Inspired by them, we plotted the variation curves of the fractal dimension for fault tortuosity (D_T3) versus the fault porosity (ϕ). As shown in Figure 8, D_T3 decreases with ϕ increasing, and the values of D_T3 marked with red dots coincide with the classical correlation between fractal dimension for fracture tortuosity and fracture porosity from the previous studies. Therefore, the above analysis supports the evidence on the validity of the proposed estimating method on D_T3.

In addition, a previous study illustrated the relationship between the fractal dimension for the 3D fracture tortuosity and the fractal dimension for the 3D fracture number [40]. Inspired by the studies, we proceeded to compare the D_T3 and D_f3 of the fault by: (a) Deriving fault parameters (l_min, l_max, L) through 3D seismic explorations, calculating the fault volume, and determining the fractal dimension of the number of faults D_f3; (b) Using Equation (3) to compute porosity and obtaining τ_av through the established relationship between porosity and average tortuosity in Equation (4); (c) Calculating D_T3 by inputting the derived parameters (D_f, l_min, L, τ_av) into Equation (11); (d) Employing fitting techniques to establish the correlation between D_T3 and D_f3 as illustrated in Figure 9.

In the process of building the D_T3 model, we have taken D_f3 as one of the important parameters. However, the characteristics of tectonic action and the nature of tectonic movement in different tectonic periods are different, making the tectonic nature, direction, and degree of meandering of each geological unit also different [41]. D_f3 reflects the complexity of faults in the number of faults in each geological unit, and the larger D_f3 is, the more complex the fault system is and the worse the rift connectivity is; D_T3 reflects the complexity of fault tortuosity, the smaller D_T3 is, the less tortuosity the fault system has, and the more favorable it is for oil and gas exploitation [42,43].

Figure 9 displaying D_T3 shows a negative correlation with D_f3. When the value of D_f3 is close to 3.0, the value of D_T3 will reach 1.0. The relationship between D_T3 and D_f3 is marked with red dots in Figure 9, which further proves the validity of the D_T3 estimated. The values of D_T3 with the red dot fit the relation of the D_T3 versus the D_f3 marked with the blue line in Figure 9, further indicating the rationality of the D_T3.

3.6. Implication

The previous studies mainly focused on employing the D_f3 for characterizing the complexity and heterogeneity of the fault network system from the perspective of the number of faults developed [44,45]. Compared with the above conventional fractal characterization method, the proposed fractal characterization method on the 3D tortuosity of fault tectonics can more accurately and comprehensively characterize the complexity and heterogeneity of the fault network system by considering the irregular shape and tortuosity of faults in the 3D space, which achieves a breakthrough in the fractal characterization on the fault network system. But, undoubtedly, D_f3 is still an important parameter to characterize the complexity of the fault network system. Therefore, given the significant impact of fault complexity on the reservoir permeability [46,47] and gas transport [48,49], future studies can further explore the impacts on the transportation and occurrence of oil and gas exploration by using D_T3 in combination with D_f3, which can provide the theoretical foundation and technical support for oil and gas exploration. It should be noted that the proposed model on D_T3 just considers the impact of fault tortuosity, not covering other factors such as the fault type, fault connectivity, fault throw, etc. Therefore, this is worth exploring with further studies.

4. Conclusions

In this study, a fractal characterization method on three-dimensional tortuosity of fault tectonics is proposed based on the 3D seismic exploration. The following conclusions can be summarized.

• We proposed a fractal characterization method on the 3D fault tortuosity (D_T3), and the calculation model on D_T3 is a function of L_min, L_max, τ_av, and D_f3. The key parameters for calculating D_T3 can be obtained through this seismic exploration technique.
• Through the proposed method, the calculated D_T3 values for the four different geological units are determined to be 1.90737, 1.96026, 2.10998, and 1.69579, respectively. The rationality of the proposed model on D_T3 was examined by the classical analytic relationships on the tortuosity and porosity of the fault (fracture) and D_T3 vs. D_f3.
• Compared with the conventional fractal characterization method by D_f3, the proposed fractal characterization method on D_T3 can more accurately and comprehensively symbolize the complexity and heterogeneity of the fault network system in the 3D space, which achieves a breakthrough in the fractal characterization of the fault network system.
• It is worth carrying out further study to explore the impacts on the transportation and occurrence of oil and gas exploration by using D_T3 in combination with D_f3, which can provide the theoretical foundation and technical support for oil and gas exploration.
• Among the first three geological units, the D_T3 of geological unit Ⅰ is the smallest, which indicates that the tortuosity of the fissures in geological unit Ⅰ is the smallest, which is the most favorable for oil and gas collection.