Study on fractal characteristics of particle size of soil-rock mixture on the right bank of Longpan in Hutiaoxia

(Key Laboratory of Engineering Geomechanics, Institute of Geology and Geophysics, China Academy of Sciences, Beijing 100029)

Based on fractal theory, the fractal law of particle size distribution of soil-rock mixture on the right bank of Longpan in Hutiaoxia is studied and analyzed, and a quantitative relationship model between average particle size and fractal dimension is established. The research shows that the soil-rock mixture has good statistical self-similarity. Because it is poorly graded soil, it presents double fractal distribution on fractal curve. This special fractal distribution is related to the origin and formation process of soil-rock mixture.

Average particle size of soil-rock mixture; Fractal particle size analysis

Earth-rock mixture is generally a special engineering geological body with crushed stone or block stone as aggregate and clay or sand as filler (Figure 1). Its causes are generally complicated, mainly including slope deposit, collapse, alluvial deposit, ice deposit and artificial accumulation. It has the characteristics of complex material composition, extremely irregular structure distribution and regional, and its mechanical properties are between soil and rock mass. Soil-rock mixture is a typical granular body, and its mechanical and engineering properties are closely related to its structural characteristics, which determine the mechanical and engineering properties of soil-rock mixture to some extent. However, due to the formation of soil-rock mixture, its structure is highly nonlinear, which brings us great difficulties in studying its structural characteristics.

Figure 1 Earth-rock mixture on the right bank of Longpan, Hutiaoxia

In recent years, catastrophe theory, chaos theory, fractal geometry and other new theories, new ideas and new methods related to nonlinear complex phenomena have penetrated into the research field of geotechnical mechanics to varying degrees. Especially, the theory of fractal geometry has been widely used in the field of geotechnical mechanics since it was formed in the 1980s, which makes it possible to solve problems that we thought were difficult to explain or describe in the past. At the end of 1980s, fractal method was introduced into the study of soil structure, and many scholars at home and abroad conducted in-depth discussion and research on it, including E.Perfect[ 1]+0, V. Rasiab [2], D. L. Turcotte [3], and Liu Songyu [4] and Hu Ruilin [5] in China. These theories and experiences open up a new way for us to study the structure of soil-rock mixture.

In this paper, the fractal geometry theory is used to analyze the particle composition of soil-rock mixture on the right bank of Longpan in Hutiaoxia, and some useful conclusions are obtained.

1 basic particle size composition of earth-rock mixture

According to the exploration data, the distribution thickness of surface soil-rock mixture on the right bank of Longpan is between 5.0 m and 40.0 m, and the particle size of gravel is mostly 1 ~ 5 cm. Gravel aggregate is mainly sandstone, containing a small amount of slate weathered debris, with extremely rough surface, clear water chestnut and irregular shape (Figure 1). The filler is clay with little content. The main causes of soil-rock mixture in the study area are slope accumulation and ice accumulation.

In order to meet the needs of research, we chose the earth-rock mixture near Longpan 1# branch tunnel of Hutiaoxia hydropower exploration project as the sampling point of this experimental study, and its elevation was about1892.0m. During the test, * * * selected 7 sampling points, which were located in the deformation body on the right bank of Longpan (Figure 2). See table 1 for the screening sampling quality of each test point. For these samples, we conducted on-site particle size screening and obtained the mass percentage content of each particle group (Table 2).

Fig. 2 Schematic diagram of particle size analysis point arrangement

Table 1 sampling quality of each test point

Table 2 Particle size fractal analysis results of each sample

Note: R stands for particle size in the table, and all particle size units are centimeters.

Fractal dimension calculation model of particle size of soil-rock mixture

Fractal geometry is a new branch of mathematics, which is mainly used to describe irregular and chaotic phenomena and behaviors in nature. At present, linear fractal is widely used, that is, fractal with self-similarity The so-called self-similarity refers to the property that the part is proportional to the whole, and the parameter that quantitatively describes this self-similarity is fractal dimension. However, natural fractal is not as pure and "clean" as theoretical fractal, and there is a scale area, so the standard to distinguish whether the research object is fractal is scale-free [4].

Since the French mathematician Mandelbrot put forward the fractal theory, people's research on the fractal properties of geological phenomena is increasing day by day. The particle size distribution of soil is a widely studied geological phenomenon. If the studied soil is regarded as a system, it is obviously an open self-organizing system. The so-called self-organization is a spatial, temporal or functional structure obtained by the system without diplomatic intervention. Such a system has a degree of freedom that plays a leading role in its evolution, and its size represents the degree of order of the system. Particle size distribution essentially describes the spatial structure (fractal structure) of the material composition of this system. Because the fractal dimension is closely related to the evolution environment and mechanical properties of soil, it can be used as a time series parameter to describe the degree of self-organization of this system [4]. Soil-rock mixture belongs to the advanced evolution stage in the whole evolution process of soil self-organization system, and it is of great significance to describe its particle size distribution with fractal dimension for the structural study of soil-rock mixture.

According to the fractal theory, the soil-rock mixture in the test area is screened by a sieve with the aperture of r, and the number of particles with the particle size ≤r is N(r), which satisfies the following relationship with the particle size r:

N(r)∝ r-D ( 1)

Where: d is the fractal dimension of particle distribution of soil-rock mixture.

Because of the large sample size in the particle size analysis of soil-rock mixture, it is quite difficult or even impossible to directly calculate the particle number. In order to be practical and convenient, the formula (1) must be improved.

By taking the derivative of (1), we can get

dN(r)∝ r-D- 1dr (2)

In the geological bodies of the same genetic type and distributed in the same study area, the particle density can be considered as a constant. In this case, the mass of a particle is directly proportional to the cube of its particle size, that is, the following relationship is satisfied:

dM(r)∝ r3dN(r) (3)

Where: M(r) is the total mass of particles with particle size ≤ r.

Substitute formula (3) into formula (2)

dM(r)∝ r2-Ddr (4)

Through the integral formula (4)

M(r)∝ r3-D (5)

Since the cumulative mass percentage of particles with a particle size of ≤r is directly proportional to M(r), that is, p (r) ≤ m (r), it is obtained by substituting it into formula (5).

P(r)∝ r3-D (6)

Therefore, according to the cumulative curve of the particle distribution of soil-rock mixture, draw the curve of P (r) ~ R in double logarithmic coordinates, and calculate the slope n of the straight line in the scale-free area, so that the fractal dimension of the particle distribution of soil-rock mixture can be easily calculated:

D= 3 - n (7)

Data processing of particle size fractal of soil-rock mixture

According to the particle screening results of each sample in the field, the corresponding mass percentage of each particle group is obtained, and the cumulative mass percentage of particles smaller than the particle size r can be obtained by accumulation. According to the obtained P(r) and the corresponding R, the fractal dimension of soil-rock mixture in the study area can be analyzed. According to the calculation results, the fractal dimension d and regression correlation coefficient r of each sample are obtained (see Table 2), as well as the particle fractal dimension distribution and particle content accumulation curve of each sample (Figure 3).

Fractal distribution and particle content accumulation curve of each sample.

From the fractal analysis structure and particle size fractal curve in Table 2, it can be seen that although the particle size difference of the soil-rock mixture in the test area is as high as 2 ~ 160 mm, the particle size scale of each sample is divided into two sections, r≤20mm and r > 20 mm, and then the fractal dimensions in each measurement scale range are calculated and analyzed. It is found that there is a good linear correlation between lgP(r) and lgr in each measurement scale, and the regression coefficient r is 0.

Fractal characteristics analysis of particle size of soil-rock mixture

4. 1 Fractal dimension curve characteristics of soil-rock mixture in the study area

It can be seen from the fractal distribution curve of particle size of each sample in the study area that the particle size distribution of soil-rock mixture in this area has two dimensions, that is, double fractal distribution. This is related to the extremely uneven particle size distribution of soil-rock mixture. For example, the soil-rock mixture in this area, with Cu of 50 and Cc greater than 5, belongs to poorly graded soil.

It can also be seen from the fractal distribution curve that although each sample has the characteristics of double fractal dimensions, for each research sample, both fractal dimensions are divided by r = 20 mm Two fractal spaces are formed, namely, a "fine-grained" region containing sand, silt and clay with r≤20mm and a "coarse-grained" region containing gravel and rubble with r > 20 mm. Each fractal space corresponds to different dimensions D 1 and D2, and satisfies D 1 < D2.

The phenomenon of double fractal dimension or even multiple fractal dimension of soil-rock mixture particle size should be explained by the spatial causes of its particle size fractal dimension. The soil-rock mixture distributed in the study area is mainly caused by slope deposition or ice deposition. Because of the short transportation distance, its provenance is mostly composed of blocks with extremely uneven particle size and poor sorting, which constitute the skeleton of the present soil-rock mixture-"coarse particles"; However, under tens of thousands of years of weathering and groundwater scouring, some large particles are decomposed into "fine particles", that is, the filling components of soil-rock mixture. Due to the differences in the causes, these "fine particles" have better sorting performance than the "coarse particles" as the skeleton, and the fractal dimension curve presents a segmentation phenomenon (multiple fractal dimension phenomenon), while the corresponding fractal dimension presents the aforementioned D 1 < D2.

As can be seen from Figure 3, the absolute value of the difference between D 1 and D2 (shown as the included angle of two regression fitting straight lines) is related to its corresponding particle accumulation curve. The smaller the absolute value (that is, the smaller the angle between two straight lines), the smoother the corresponding accumulation curve, the worse the sorting performance of earth-rock mixture and the better the gradation. When D 1 and D2 tend to be equal, it corresponds to well-graded soil.

4.2 Fractal characteristics of particle size of soil-rock mixture in the study area

As can be seen from the layout of test sites, the whole test site can be roughly divided into three test site partitions, namely: partition 1, including test sites 1#, 3# and 4 #; Sub-area 2, including 2# and 5# test points; Partition 3, including 6# and 7# test points. From the analysis in Table 2, it can be seen that the fractal dimension of each test point in each sub-region is not much different, but the fractal dimension of each test point in each sub-region is relatively different. For example, the D 1 of the 6# and 7# measuring points in "Zone 3" are 2.39 and 2.34 respectively, and the d 2 is 2.77 and 2.7 1. This is related to the uneven distribution of soil-rock mixture due to various conditions (such as topography, landforms, etc.). ) in the process of its formation. For example, in the field investigation, we can see that there are two huge stones at the entrance of the Hutiaoxia 1 exploration adit, which separates it from the surrounding 1 subarea and No.3 subarea, resulting in the difference in fractal dimension. However, from the whole test site, its fractal dimensions are distributed in the areas of D 1 = 2.34 ~ 2.57 and D2 = 2.67 ~ 2.95438+0. On the other hand, all these reflect that the spatial distribution of soil-rock mixture, a special geological body, is extremely uneven, but the granularity fractal dimension of the same genetic type or a certain part of the same genetic type (for example, the research area is located in the lower part of the deformation body on the right bank of Longpan) is similar or equal, which provides us with a quantitative index for the division of soil-rock mixture.

4.3 Relationship between Average Particle Size and Particle Size of Earth-rock Mixture

For different screening samples, the same P(r) may correspond to different r values and different fractal dimensions. Now let's analyze the corresponding relationship between r and fractal dimension d for different screening samples when P(r) takes a certain value.

Equation (6) can be used.

P(r)= Kr3-D+ C (8)

Where: k and c are constants.

The logarithm on both sides of formula (8) is

Soil-rock mixture

According to formula (9), we can find that the average particle size (that is, the particle size d50 when p (d50) = 50) and the fractal dimension d satisfy the following relationship:

Soil-rock mixture

According to Figure 3, we can get the d50 of each sample in the test area and its corresponding dimension d (Table 3), and then draw a curve in semi-logarithmic fractional coordinates (Figure 4). Through regression analysis, the sum of d50 and d can be obtained.

(1 1), and the corresponding regression correlation coefficient r = 0.9337, which well describes the quantitative relationship between the average particle size of soil-rock mixture and the corresponding dimension.

Soil-rock mixture

Table 3 Average particle size and corresponding fractal dimension of each sample

Fig. 4 Relationship curve between average particle size and corresponding size

From the formula (1 1), it can be seen that the fractal dimension of the sample reflects the average particle size of the soil-rock mixture to a certain extent. With the increase of the average particle size, the fractal dimension tends to decrease, which can be used as an index to describe the thickness of the soil-rock mixture.

5 conclusion

(1) Gravel blocks destroyed by various internal and external dynamics have been transported and deposited, and then weathered and filled for tens of thousands of years, forming various earth-rock mixtures today. Although the genesis of this special geological body is complex and diverse, its particle size distribution still has good statistical self-similarity. Fractal dimension can be used as a quantitative parameter of soil-rock mixture zoning, which is of great significance to further study the genesis and distribution of soil-rock mixture.

(2) Soil-rock mixture belongs to poorly graded soil with extremely uneven particle size distribution, but it has the characteristics of multifractal dimension. The soil-rock mixture in this study area has a double fractal structure (corresponding to two fractal dimensions D 1 and D2), which is related to the genetic types of "coarse particles" and "fine particles" in the soil-rock mixture. The absolute value of the difference between D 1 and D2 reflects the smoothness of the particle accumulation curve of earth-rock mixture and its sorting quality.

(3) Fractal dimension reflects the average particle size of soil-rock mixture to some extent. By regressing the average particle size d50 with its corresponding dimension d, the research results show that the sum of lgd50 has a good linear relationship.

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