Multifractal Characteristics of Grain Size Distributions in Braided Delta-Front: A Case of Paleogene Enping Formation in Huilu Low Uplift, Pearl River Mouth Basin, South China Sea

Abstract

Multifractal analysis has been used in the exploration of soil grain size distributions (GSDs) in environmental and agricultural research. However, multifractal studies regarding the GSDs of sediments in braided delta-front are currently scarce. Open-source software designed for the realization of this technique has not yet been programmed. In this paper, the multifractal parameters of 61 GSDs from braided delta-front in the Paleogene Enping Formation in Huilu Low Uplift, Pearl River Mouth basin, are calculated and compared with traditional parameters. Multifractal generalized dimension spectrum curves are sigmoidal and decrease monotonically. Multifractal singularity spectrum curves are asymmetric, convex, and right-hook unimodal. The entropy dimension and singularity spectrum width ranges of silt-mudstones and gravelly sandstones are wider than those of fine and medium-coarse sandstones. The symmetry degree scopes from different lithologies are concentrated in distinguishing intervals. With the increase of grain sizes, the symmetry degree decreases overall. Both the symmetry degree and mean of GSDs are effective to distinguish the different lithologies from various depositional environments. A flexible and easy-to-use MATLAB (2021b)^® GUI (graphic user interface) package, MfGSD (Multifractal of GSD, V1.0), is provided to perform multifractal analysis on sediment GSDs. After raw GSDs imported into MfGSD, multifractal parameters are batch calculated and graphed in the interface. Then, all multifractal parameters can be exported to an Excel file, including entropy dimension, singularity spectrum, correlation dimension, symmetry degree of multifractal spectrum, etc. MfGSD is effective, and the multifractal parameters outputted from MfGSD are helpful to distinguish depositional environments of GSDs. MfGSD is open-source software that can be used to explore GSDs from various kinds of depositional environments, including water or wind deposits.

1. Introduction

In sedimentology and geology, grain size distribution (GSD) is defined as the frequency of occurrence of different-diameter particles. The grain sizes of sediments contain information on multiple depositional factors, and the GSD is an informative measurement in understanding transport dynamics and depositional environment and process [1,2]. It is one aspect of the basic data used to reveal modern and ancient depositional environments in rivers, lakes, oceans, deserts, loess, etc. [3,4]. In fact, superposed by multi-subpopulations from different depositional processes, the corresponding frequency curves of GSDs could be bimodal or multimodal [5,6]. Different size fractions may explain the various scaling domains observed in GSDs. Traditional GSD analytical methods just adopt mean, standard deviation, skewness, and kurtosis to describe the features of depositional processes and environments, which presents difficulties. Especially in multimodal GSDs, these traditional parameters cannot reflect nonuniformity accurately [7,8].

The consistency frequency data structure of GSDs is the one-dimensional vector X = (x₁, x₂, …, x_n), with ${\sum }_{k=1}^n{x}_k=100\mathrm{\%}$. Several mathematical and statistical methods, such as clustering analysis, principal component analysis (PCA), sediment trend analysis (STA), and the multifractal method, have been designed to find untraditional GSD parameters to offer more information. Unsupervised clustering algorithms calculate the similarity of GSD frequency or cumulative frequency curves [9,10]. PCA extracts a meaningful sedimentary first principal component from high-dimensional GSD data [11,12]. STA uses statistical parameters to reflect time and space differentiation in large amounts of GSDs [13,14]. Widely used in signal processing and image identification, the multifractal method meticulously characterizes the complexity and nonuniformity of one- or two-dimensional data. Multifractal methods have been used to characterize grain properties from soil particle size distributions in environmental and agricultural science [15,16,17]. It is proven that the multifractal method may be best suited to representing the multiplicative action of the various depositional processes acting on various given distributions [18,19]. The grain size of sediments is coarser than soil particle size; however, they are essentially statistical distributions. In sedimentology, the multifractal method may be best suited to represent the multiplicative action of the various depositional processes acting on various given GSDs [2]. Chang et al. [20] used multifractal theory to analyze the sand particle size of land surface sediments along the banks of the Ulanbuh Desert, in reach of the Yellow River. Li et al. [21] applied a multifractal structure to study the GSDs of suspended sediments in the Three Gorges Reservoir in different seasons.

However, multifractal studies regarding the GSDs of sediments in a braided delta-front sedimentary environment and comparisons of the multifractal method with traditional parameters are currently scarce. Moreover, practicing the multifractal analysis of GSD is computationally consuming. Available software designed for GSD multifractal realization has not yet been programmed. In this paper, a case study from the Paleogene Enping Formation in Huilu Low Uplift, Pearl River Mouth basin, South China Sea, is applied to explore the multifractal features of GSDs in a braided delta-front. Several multifractal parameters are analyzed and compared with traditional parameters. A flexible MATLAB^® GUI (graphic user interface) package, named MfGSD (Multifractal of GSD), is provided to facilitate multifractal work on sediment GSDs.

2. Geological Setting

The Pearl River Mouth basin, in the South China Sea, is a Cenozoic continental margin rift basin. Its formation and evolution are controlled by the collision between the India-Australia Plate and the Eurasian Plate, the compression of the ancient Pacific Plate and the Philippine Plate, and the expansion of the South China Sea [22,23]. From north to south, the Pearl River Mouth basin is divided into six structural units: Northern Uplift, Northern Depression, Central Uplift, Central Depression, Southern Uplift, and Southern Depression. Each structural zone is divided into several sags and low uplifts. The Huilu Low Uplift is between the Huizhou Sag and Lufeng Sag in the Northern Depression [24,25] (Figure 1).

The Cenozoic in the Pearl River Mouth Basin can be divided into the Rifting Period, Depression Period, and Fault Block Lifting Period. The Rifting Period included the Paleocene and Eocene, and the formations were river and lake sediments. The Depression Period was from the Oligocene to Middle Miocene. In this period, the marine-continental transitional sands and muds were deposited. The Fault Block Lifting Period covered from the Late Miocene to Quaternary, and marine clays were the dominated sediments. In the Huilu Low Uplift, the sedimentary environment of the Paleogene Enping Formation was braided delta-front [26,27] (Figure 1).

3. Materials and Methods

3.1. GSD Samples

In the Huilu Low Uplift, several petroleum wells have been drilled into the Enping Formation. Conventional natural gamma ray logging (GR, API) was run in all boreholes, with low values in sandy interval and high values in muddy intervals. Cores of 52.95 m were obtained in the LF-A well. The lithologies of cores were observed and recorded carefully, from mudstones to gravelly sandstones. In order to analyze the grain size characteristics of different depositional environments, 61 samples were taken in different lithologies: 9 silt-mudstone samples in the interdistributary bay, 11 fine sandstone samples in the sheet sand, and 24 medium-coarse sandstone and 17 gravelly sandstone samples in the distributary channel (Figure 2). These bimodal or multimodal GSDs from different lithologies can be used to analyze the multifractal characteristics of braided delta-front.

3.2. GSD Experiment

Essential sample pretreatment is firstly performed before GSD measurement. All samples were crushed. A total of 10 g of sediment was weighed from each sample and treated with 10 mL of 10% hydrogen peroxide (H₂O₂) to heat and still to remove organic matter completely. Each sample was then treated with 10 mL of 10% hydrochloric acid (HCl) to heat and still to eliminate calcium carbonate fully. After washing with distilled water repeatedly, 10 mL of 0.02 mol/L Calgon ((NaPO₃)₆) was added to separate the sediment particles totally [28,29]. Secondly, a laser particle-size analyzer, the LS13 320, produced by Beckman Coulter, USA, was used to measure the grain size composition. The measuring range was set as 1.95–2000 μm. Measurement was repeatedly until the error was less than 2%. Finally, frequency data of GSDs were exported (Figure 2). In addition, mean, sorting, skewness, and kurtosis of GSDs were calculated by the graphical method of Folk and Ward [30].

3.3. GSD Interval Reconstruction

The grain size interval of the laser diffraction analyzer is closed to I = [0.02 μm, 2000 μm] theoretically, which is a logarithmic increase. To analyze the GSD characteristics by multifractal theory, a dimensionless interval must be established with an equal length subinterval. The dimensionless interval could be defined by J = [lg(0.02/0.02), lg(2000/0.02)] = [0, 5]. The interval J is then divided into equal-length subintervals of N(ε) = 2^K (ε = 5 × 2^−K). To ensure that all subintervals cover the grain size interval and each subinterval contains at least one measured value, K is just set as 1, 2, 3, 4, 5, and 6. Therefore, 64 dyadic scaling subintervals J_i (i = 1, 2, …, 64) are obtained, and the first subinterval is J₁ = [0.02 μm, 0.024 μm], while the last subinterval is J₆₄ = [1670.725 μm, 2000 μm] [16,18].

3.4. GSD Multifractal Parameters Calculation

In each subinterval J_i, grain size represents the relative volume of sediment particles, which is given by probability density functions p_i(ε) as follows [17,18]:

$$u_i\left(q,\epsilon \right)={\displaystyle \frac{p_i{\left(\epsilon \right)}^q}{\sum _{i=1}^{N\left(\epsilon \right)}{p}_i{\left(\epsilon \right)}^q}}$$

(1)

where u_i(q, ε) is the q-order probability of the J_i subinterval and q is an integer in the range [−10, 10].

The multifractal generalized dimensions D(q), reflecting the overall multifractal characteristics, are calculated as follows [18,19]:

$$\begin{array}{c}D\left(q\right)={\displaystyle \frac{1}{q-1}}\underset{\epsilon \to 0}{\lim }{\displaystyle \frac{\log \left(\sum _{i=1}^{N\left(\epsilon \right)}{p}_i{\left(\epsilon \right)}^q\right)}{\mathrm{l}\mathrm{o}\mathrm{g}\epsilon }},(q\ne 1)\\ D_1=\underset{\epsilon \to 0}{\lim }{\displaystyle \frac{\sum _{i=1}^{N\left(\epsilon \right)}{p}_i{\left(\epsilon \right)}^q\mathrm{l}\mathrm{o}\mathrm{g}{p}_i\left(\epsilon \right)}{\mathrm{l}\mathrm{o}\mathrm{g}\epsilon }},(q=1)\end{array}$$

(2)

If q > 0, information of high probability distribution in the GSD is amplified, and if q < 0, information of small probability distribution in GSD is amplified. If q = 0, D₀ is the capacity dimension and reflects the distribution range of sediment grain size. When D₀ is higher, the distribution range is wider. If q = 1, D₁ is the entropy dimension and indicates the concentration of GSD. When D₁ is higher, the size distribution is more dispersive. If q = 2, D₂ is the correlation dimension and suggests the uniformity of the measurement interval of GSD. When D₂ is higher, the measurement interval is more uniform [31,32].

The multifractal singularity index α(q) and singularity spectrum f(α(q)), reflecting the local branch multifractal characteristics, are calculated as follows [18,31]:

$$\alpha \left(q\right)=\underset{\epsilon \to 0}{\lim }{\displaystyle \frac{\sum _{i=1}^{N\left(\epsilon \right)}{u}_i(q,\epsilon )\mathrm{l}\mathrm{o}\mathrm{g}{p}_i\left(\epsilon \right)}{\mathrm{l}\mathrm{o}\mathrm{g}\epsilon }}$$

(3)

$$f\left(\alpha \left(q\right)\right)=\underset{\epsilon \to 0}{\lim }\begin{array}{c}{\displaystyle \frac{\sum _{i=1}^{N\left(\epsilon \right)}{u}_i(q,\epsilon )\mathrm{l}\mathrm{o}\mathrm{g}{u}_i(q,\epsilon )}{\mathrm{l}\mathrm{o}\mathrm{g}\epsilon }}\\ \end{array}$$

(4)

If q = 0, α₀ is the average of the singular spectrum functions and is inversely proportional to the local density of the GSD fractal structures. The difference between the maximum and minimum of α(q), Δα = α(q)_max − α(q)_min, is defined as the width of the multifractal singularity spectrum. It represents the difference and inhomogeneity in the fractal structure of the GSD. When Δα is higher, the GSD is more uneven. The difference between the maximum and minimum of α(q) in f(α(q)), Δf = f(α(q)_max) − f(α(q)_min), is defined as the symmetry degree of the multifractal spectrum. It reflects the shape of the multifractal spectrum of the GSD. If Δf < 0, then the shape of the singularity spectra deviates to the left, which means that small probability subsets of the GSD play a major role in the fractal system. If Δf > 0, then the shape of the singularity spectra deviates to the right, which indicates that large probability subsets of the GSD play a major role in the fractal system [17,18,19].

4. Results

4.1. Multifractal Spectrum

The generalized dimension spectrum, q and D(q), reflects the global multifractal features of GSDs. As illustrated in Figure 3, q (−10 ≤ q ≤ 10) is the X-coordinate. GSDs forming different lithologies is the Y-coordinate, and multifractal generalized dimensions D(q) are the Z-coordinate. The generalized dimension spectrum curves are sigmoidal and decrease monotonically. The decrease range in D(q) for the negative moment (−10 ≤ q < 0) is greater than that for the positive moment (0 < q ≤ 10). It indicates that the multifractal generalized dimension values are more sensitive to the probability subset in the sparse probability area. In other words, the non-uniformity of the GSDs is more pronounced in sparse areas than in dense areas. D(q) values in the negative moment are entirely similar in all GSDs of different lithologies. It suggests that the small probability distributions in GSDs are in the same magnification amplitude. D(q) values in the positive moment can reflect the non-uniformity difference.

The multifractal singularity spectrum, α(q) and f(α(q)), indicates the local multifractal characteristics of GSDs. As shown in Figure 4, the multifractal singularity index α(q) is the X-coordinate, GSDs forming different lithologies is the Y-coordinate, and the multifractal singularity spectrum f(α(q)) is the Z-coordinate. The multifractal singularity spectrum curves are all asymmetric, convex, and right-hook unimodal. This suggests that the multifractal structure of the GSDs of braided delta-front sediments has varying inhomogeneity. Specifically, the small probability subsets of the GSDs, the silty and clay fractions, play a major role in the fractal system.

4.2. Multifractal Parameters

The shapes of multifractal generalized dimension spectrum and singularity spectrum curves of different lithologies are analogous; however, different ranges of multifractal parameters exist, including the entropy dimension D₁, singularity spectrum width Δα, and symmetry degree Δf. The ranges of these multifractal parameters from different lithologies are illustrated in respective violin diagrams (Figure 5).

The D₁ ranges of silt-mudstones and gravelly sandstones are 0.76–1.22 and 0.60–1.17, respectively, while those of fine and medium-coarse sandstones are 0.89–1.09 and 0.90–1.14, respectively. The Δα ranges of silt-mudstones and gravelly sandstones are 7.62–8.21 and 7.75–8.45, respectively, while those of fine and medium-coarse sandstones are 7.90–8.10 and 7.78–8.12, respectively. The D₁ and Δα ranges of silt-mudstones and gravelly sandstones are wider than those of fine and medium-coarse sandstones, indicating that the grain sizes of silt-mudstones and gravelly sandstones show more complexity. Moreover, this suggests more complex depositional processes: silt-mudstones are deposited in low-energy-flow and quiet water, and gravelly sandstones are transported by alternate hydrodynamic forces. The Δf scopes from different lithologies are concentrated in different intervals. Moreover, it is obvious that the Δf values decrease from mudstones to gravelly sandstones. This suggests that the major roles of small-probability silty and clay fractions decrease with the increase of grain sizes. Sandy and gravelly grains are dominant sediments in braided delta-front; however, the silt and clay are sensitive in GSD fractal systems. Therefore, Δf is the most sensitive with regards to the depositional environment.

5. Discussion

5.1. Multifractal vs. Traditional Parameters

The statistical parameters of mean, sorting, skewness, and kurtosis are traditionally used to analyze the characteristics of GSDs [7,8]. In these GSDs of braided delta-front, the means of GSDs increase from silt-mudstones to gravelly sandstones. However, the range of sorting, skewness, and kurtosis of different lithologies are confused, indicating that they are hard to differentiate in terms of the depositional features (Figure 6). Therefore, the symmetry degree of the multifractal spectrum Δf and the mean of GSDs M_z are both effective for distinguishing the different lithologies. In the cross-plot of the symmetry degree and mean of GSDs from different lithologies, it is obvious that the symmetry degree decreases overall with the increase of grain sizes (Figure 7). Quantificationally, the responses of both parameters are as follows: silt-mudstones, Δf 11.37–1.46 and M_z < 60 μm; fine sandstones, Δf 1.34–1.40 and M_z 60–220 μm; medium-coarse sandstones, Δf 1.26–1.32 and M_z 200–350 μm; gravelly sandstones, Δf 1.26–1.32 and M_z > 300 μm (

Table 1. Symmetry degree Δf and mean M_z of GSDs from different lithologies in braided delta-front.

Parameter	Silt-Mudstone	Fine Sandstone	Medium-Coarse Sandstone	Gravelly Sandstone
Δf	1.35–1.5	1.33–1.4	1.27–1.37	1.25–1.32
Mz (μm)	<60	60–220	200–350	>300

). In fact, the medium-coarse and gravelly sandstones may be confused, as they are deposited in the same distributary channel with different hydrodynamic forces. This means that the inhomogeneity of grain sizes from mudstones to gravelly sandstones increases. This characteristic accords with the features of GSD frequency curves: the coarser the grains, the more peaks in the GSD frequency curve (Figure 2).

5.2. Software Package for GSD Multifractal

Multifractal methods have been used in agricultural and environmental analysis; however, there are still no public software packages. To make the multifractal method easy to use, it is necessary to provide a well-designed GUI for users who are not familiar with the algorithm. The MATLAB^® GUI provides point-and-click control of software applications. The apps can be shared within MATLAB and also as standalone installation packages. In this study, MfGSD, a MATLAB-based GUI package, is designed for practicing GSD multifractal analysis specifically. The flexible and efficient MfGSD package contains four functional zones, namely the raw data list showing GSD sample names, function buttons dealing with related operations, the plot window displaying raw GSDs and multifractal graphs, and the result list exhibiting multifractal parameters (Figure 8). The reliability and validity of MfGSD are verified by comparing the output results of GSDs from different depositional environments reciprocally.

Taking 23 GSDs from the modern Ganjiang delta-front in Poyang Lake of China, for example, the application of MfGSD is easy and convenient. Raw GSD data are saved in an Excel file (*.xlsx), as this file type is versatile in data presentation and manipulation. The first row is the grain size scale in micrometers. The second and following rows contain the volume percentage corresponding to each sediment grain size interval. The first column is the name of samples.

Importing the GSD Excel file into MfGSD using the Data Import button, sample names can be displayed in the raw data list. By checking the sample names, GSD frequency curves can be shown using the Draw 1 button. After setting the values of K and max q, multifractal generalized dimension and singularity spectrum curves are shown in the plot window using the Draw 2 and Draw 3 buttons. Then, five parameters, D₀, D₁, D₂, Δα, and Δf, are immediately exhibited in the result list. Multifractal graphs in the plot window can be saved as image files by using the Save Pic1, Save Pic2, and Save Pic3 buttons. After the multifractal analysis is complete, a series of GSD multifractal parameters, including D(q), α(q), f(α(q)), Δα, Δf, and so on, can be saved in a new Excel file by using the Result Export button (Figure 8). Details of the approach can be found in the online documentation at https://github.com/YuanRui-1987/MfGSD (accessed on 1 March 2025).

5.3. Processes of GSD Multifractal Analysis

The analysis processes using the multifractal method to interpret sedimentological information from GSDs require knowledge of the primary depositional environment and comprise four stages. The first stage is acquiring GSD samples in a modern sedimentary or ancient stratum. An understanding of the depositional environment is indispensable when sampling. The second stage is experimenting with GSD data using a grain size analyzer in laboratory. High-resolution GSD using delicate intervals is recommended. The third stage is calculating the multifractal parameters using MfGSD in the MATLAB platform. Traditional GSD parameters, mean, standard deviation, skewness, and kurtosis, are needed as well. The fourth stage is analyzing the sedimentary significance of multifractal parameters and the correlation between multifractal and traditional parameters. Characteristics of multifractal parameters in different depositional environments can be summarized.

MfGSD aims to provide an open-source and comprehensive analysis platform for assessing the multifractal features of sediment GSDs. It can implement many multifractal parameters efficaciously and responsibly, which can be helpful in exploring the distinction of sediment GSDs in various environments. However, to use MfGSD reasonably, a good understanding of the studied sedimentary environment is indispensable. With such background knowledge, MfGSD can be confidently used.

6. Conclusions

Based on GSD multifractal theory, in this study, the multifractal parameters of 61 GSDs from the Paleogene Enping Formation of Huilu Low Uplift, Pearl River Mouth basin, are calculated and compared with traditional parameters. A MATLAB^® GUI package (MfGSD) for the multifractal analysis of GSDs was programmed. The main conclusions comprise the following three points:

(1)

GSDs from silt-mudstones, fine sandstones, medium-coarse sandstones, and gravelly sandstones are obtained in different sedimentary environments of braided delta-front. Generalized dimension spectrum curves are sigmoidal and decrease monotonically. The decrease range of the generalized dimension in the negative moment is greater than that in the positive moment. The multifractal singularity spectrum curves are asymmetric, convex, and right-hook unimodal.

(2)

The entropy dimension and singularity spectrum width ranges of silt-mudstones and gravelly sandstones are wider than those of fine and medium-coarse sandstones. The symmetry degree scopes from different lithologies are concentrated. With the increase of grain sizes, the symmetry degree decreases overall. The symmetry degree and mean of GSDs are effective for distinguishing the different lithologies from various depositional environments in braided delta-front.

(3)

The MfGSD is a flexible, efficient, and easy-to-use tool that can be used to conduct multifractal analysis of sediment GSDs. It generates not only the multifractal spectrum characteristics of GSDs but also outputs many useful multifractal parameters to reveal depositional information. MfGSD is open-source software that may be used to explore GSDs in various kinds of sedimentary environments.