Influential factors on the repose angle of desert sands

JianBao Liu , NaiAng Wang , HongYi Cheng , WenTao Yang , ZhiQiang Guo

1. Research School of Arid Environment and Climate Change, Lanzhou University, Lanzhou, Gansu 730000, China 2. College of Resources and Environment, Lanzhou University, Lanzhou, Gansu 730000, China

*Correspondence to: Dr. NaiAng Wang, Professor of College of Resources and Environment, Lanzhou University, Lanzhou, No.222, Tianshuinan Road, Lanzhou, Gansu 730000, China. Email: wangna06@lzu.cn

Influential factors on the repose angle of desert sands

JianBao Liu1,2, NaiAng Wang2*, HongYi Cheng2, WenTao Yang2, ZhiQiang Guo2

1. Research School of Arid Environment and Climate Change, Lanzhou University, Lanzhou, Gansu 730000, China 2. College of Resources and Environment, Lanzhou University, Lanzhou, Gansu 730000, China

*Correspondence to: Dr. NaiAng Wang, Professor of College of Resources and Environment, Lanzhou University, Lanzhou, No.222, Tianshuinan Road, Lanzhou, Gansu 730000, China. Email: wangna06@lzu.cn

The repose angle is one of the most significant macroscopic parameters in describing the behavior of granular materials. Under a static condition, the repose angle is the steepest angle at which sediment particles can rest without motion. In this paper, we use existing data and aeolian physics to analyze the main factors that influence the repose angle of sand dunes, and we investigate different repose angles involving various states and types of materials. We have determined that different factors have differential influence on the magnitude of the repose angle. Our results show that for powdery (＜400-μm diameter) desert sands, the main influential factor on the magnitude of repose angle is the molecular force among particles. Particle size does not influence the repose angle of desert sands directly, but has an indirect impact by affecting the grit sphericity and surface roughness, of which the grit sphericity acts as a major factor. Even at the same average particle size, the repose angle differs with different grain compositions.Furthermore, with increasing unevenness in grain composition, the repose angle increases correspondingly. Sand texture also has a direct influence on the repose angle of desert sands. In two sand samples having the same grain composition but different textures,the repose angles may be different. Water content has a stronger influence on the repose angle than any other factor. However, the relationship between the repose angle and water content is not a simple direct proportion. In fact, with increasing water content,the repose angle first increases and then decreases. These research results will be useful for understanding the mechanisms of dune transport, variations of dune morphology, and the stability and fluidity of dune sands.

repose angle; desert sands; plastic sands; uniformity coefficient; water content

1. Introduction

The repose angle is an important parameter to scale the fluidity of desert sands, which may significantly affect the movement of dunes (Bagnold, 1941; Miller and Byrne, 1966;Anderson, 1988; Craig, 2000). Accordingly, it is important to understand the influential factors on the repose angle of desert sands because it is an important physical parameter that aids in the exploration of bedload movement theory, as well as in a better understanding of the morphology, stability,and vegetative protection of sand dunes (Jaegeret al., 1989;Barabasiet al., 1999).

There are many factors that affect the repose angle of desert sands. Many countries began researching this topic around the 1940s (Bagnold, 1941), but investigations stopped in the 1980s (Chiew and Parker, 1994; Yalin, 1997;Barry, 2008; Chen and Wang, 2008) and research re-focused on the underwater repose angle of sands and sand models(Lee and Jeng, 2007). Some early scholars, such as Bagnold and Miller, generated equations and empirical formulae involving particle size, sphericity, surface roughness, and grain composition of desert sands, based on the perspective of the mechanism of a single granule. These formulae have been used to reverse calculate the repose angle of desert sands(Bagnold, 1941). However, there are still some flaws in this approach, as highlighted in the following four areas:

(1) Does the particle size influence the repose angle of desert sands directly? If not, what factors directly affect the repose angle through the particle size? And what are the differences among these factors?

(2) Previous investigations mostly focused on particle sizes greater than powder, and research on the repose angle of powdery desert sands (＜400-μm diameter) is rare.

(3) Does the mineral composition of desert sands influence their repose angle? If the mineral composition of desert sands is variable among sites and the particle size distribution is even, will the repose angle of desert sands change?

(4) Our research has shown that, the water content also can affect the repose angle of desert sands, but previous research seems to have ignored it, or it is rarely investigated.

In view of these points, we sampled aeolian sands in the Badain Jaran Desert, which is 49,000 km2and spans the provinces of Gansu and Ningxia in China as well as Inner Mongolia. We analyzed the factors that influence the repose angle, including particle size, grit sphericity, surface roughness, and grain composition. Furthermore, the molecular force among particles, the mineral composition, and the water content of these sands were also investigated.

2. Experimental design

We used a repose angle detector to test the repose angle(Figure 1). After the leak flap was closed, the selected sand samples were placed in the conical hopper, which was adjusted to a height (H) and fastened to a supporting bar with screws so as to fix the hopper at the appropriate height above the coordinate paper which was laid on the leveled table. Then the lower plate was moved horizontally by the gradienter, and the leak flap was opened so that the desert sands fell on the lower plate. The leak flap was not closed until the top of the accumulated sand was about to make contact with the bottom of the hopper. The cone bottom diameter was measured six times and its average radius (R)was calculated. Finally, we obtained a repose angle value by calculating the ratio of the fixed cone roof height (H) and the cone radius (R) as a tangent value. The test was performed six times on each sand sample to obtain an average value of the ratio for the repose angle of the desert sands.

We chose certain principal factors as variables and controlled other factors as constants at different points in the experiments. Moreover, the sands used in the experiments were selected carefully from the field to make the samples including every different particle sizes. After the sand samples were dried in an oven, we increased the humidity and maintained them hermetically isolated for about 20 hours.We air-dried the samples to produce sand samples of different moisture contents. To measure each level of water content, we chose the optimal time to take a rapid measurement and monitored the ambient environment to maintain room temperature and humidity so as to reduce any environmental impact on testing precision. The moisture content of the samples was measured by the weight method and denoted as wet bases. To obtain different particle sizes, we separated the sand experimentally by passing samples through sieves of different apertures. Thus,each size class interval is identified by the average value within that interval. During the experiment, we used the same conical hopper and used the "even adding" method to add sand to it. Before the experiment, we laid the same sands on the lower plate whose plane was wide enough(with the same thickness) to reduce the influence of sand collision and bounce.

Figure 1 Sketch of the experiment device (repose angle detector)

3. Analysis of impacting factors on the repose angle of desert sands from the perspective of aeolian physics

Generally, particles with a diameter less than 400 μm are termed "powder" (Geldartet al., 1990; Orband and Geldart,1997). Under dry conditions, the powdery particles are likely to gather together because the force of cohesion is larger than gravity, resulting from Van der Waals and electrostatic forces (Hamaker, 1937; Adamet al., 2000; Eichenlaubet al., 2004). Thus, an increase in powdery particle diameter results in a decrease in the contact among particles and a decrease of adhesion, cohesion, and repose angle.

In contrast, particles larger than 400 μm in diameter are called "granules". Granular mixtures usually accumulate sand, and the molecular force is likely to be zero (Geldartet al., 2006). The main factors influencing the repose angle of such particles are outlined as follows.

Based on the definition of the repose angle of desert sands and the equilibrium condition of forces on the repose slope surfaces, the repose angle reflects the magnitude of the friction coefficient on the repose slope surface. According to classical physics, the friction coefficient on the repose slope surface is determined by the value of slope roughness. Thus, the value of the repose angle depends on the slope roughness. Since slope roughness is controlled by both particle sphericity and surface roughness, a comprehensive understanding of these two factors will determine the value of the repose angle.

3.1. Particle sphericity

Smooth surfaces of spherical particles can reduce the contact area, leading to lower values of friction and repose angles, and vice versa (Aguirreet al., 2000; Cleary and Sawley, 2002; Dorbolo, 2005). For desert sands, under the flow function of the wind-drift sand flow, larger-sized particles are not easily carried away (Arbogaset al., 2002; Baas,2007; Edwardet al., 2009). Therefore, the larger the particle size, the lower the degree of psephicity and sphericity, which finally leads to more anomalously shaped granules and therefore a high value of repose angle. As a result, irregular particle shapes correspond to larger repose angles, which were observed in our experiments. In non-uniform desert sands, large and small particles are intermixed and interact with each other. Besides the typical particle size, particle sphericity is related to the grain composition of the sand samples (Bertrandet al., 2005).

3.2. Particle surface roughness

The surface roughness of the repose slope is directly related to particle surface roughness (Barker and Mehta, 2000;Zhouet al., 2003) which, in turn, is determined by mineral components and the degree of surface abrasion. Mineral components determine the degree of particle surface roughness. For two sand samples that contain different mineral components, their particle surface roughness also will be different; this difference leads to different repose angles. For example, plastic and desert sands contain completely different mineral components. Although these sands might have the same grain composition, the repose angle of plastic sands is always smaller than that of desert sands because the surface roughness of plastic sand is smoother than that of desert sand. The degree of surface abrasion refers to the smoothness of particles under collision and friction during sand flow. Generally, larger-sized particles more readily experience collision and abrasion. Therefore, the surface roughness of larger-sized particles is smoother. Furthermore,the larger the particle size of desert sands, the greater the surface roughness of the repose slope.

Based on the above comprehensive analysis, it can be concluded that, besides water content, the main factors affecting the repose angle of desert sands are the Van der Waals force among particles, particle size, sphericity, grain composition, texture, and degree of surface abrasion.

4. Experiments on the influential factors of the repose angle of desert sands

In order to investigate how molecular force, particle size,sphericity, grain composition, mineral components, degree of abrasion, and water content of sands influence the repose angle of desert sands, we chose uniform desert sands, uniform plastic sands, non-uniform desert sands, and desert sands with different water contents as our experimental materials. This enabled us to qualitatively analyze different factors that may influence the repose angle of desert sands.In this paper, we define uniform sands as those containing particles of the same size, while non-uniform sands contain particles of different sizes.

4.1. Repose angle of uniform sands

For uniform sands with larger particle sizes (＞400-μm diameter), in which the particle intersection confounds direct experience of molecular force (London, 1937; Donget al., 2004), the main factors affecting repose angle are particle size, mineral components, and degree of particle surface abrasion. Because the degree of particle surface abrasion correlates with particle size, we need only discuss the first two factors. The sands in this study had two distinct components, organic and inorganic materials. In the model experiments, organic materials comprised the lighter sands and inorganic materials comprised the heavier sands.In order to distinguish the influences caused by mineral components, the repose angle is discussed for desert(heavier) and plastic (lighter) sands.

4.1.1 Repose angle of uniform desert sands of different

particle sizes

Using existing measured data, the repose angle of uniform desert sands [φ/(°)] as a function of the particle size [d(mm)] is plotted in figure 2. From this figure it is evident that, for small particle sizes (＜400-μm diameter), the repose angle and particle size show an inverse ratio. Furthermore,the changes of the repose angle with different particle sizes can be calculated by the following expression:

For large particle sizes (＞400-μm diameter), the relation reverses and the expression between the angle of repose and particle size can be established as:

In particular, when the particle size is smaller than 400μm, Van der Waals and electrostatic forces become significant, the particle easily attracts others, and the cohesion among particles both surpasses the particle’s gravity and hinders gravity’s effect upon particles. Thus, for particles with a diameter less than 400 μm (i.e., adhering powdery grains), as the diameter decreases, the number of the contact points become smaller, which triggers the Van der Waals and electrostatic forces, and the repose angle becomes smaller.Conversely, when the particle size is larger than 400 μm(namely, granular), the repose angle of uniform desert sands is only influenced by grit sphericity and particle surface roughness. In circumstances when the particle size increases,the degree of sphericity becomes smaller, as does the surface roughness, but the reverse is the case for the repose angle of uniform desert sands. This is because the repose angle and the degree of sphericity have a proportional relationship,whereas the repose angle and the surface roughness have an inverse relationship. Thus, we can conclude that for uniform desert sands with different particle sizes larger than 400 μm,the repose angle depends primarily on the degree of sphericity and the particle size (mainly through the influence of spherical degree to indirectly affect the value of the repose angle).

Figure 2 Repose angle of uniform desert sands with different particle sizes

4.1.2 Repose angle of uniform plastic sands of different particle sizes

Plastic sands often are used as a light sand counterpart in experimental models (Gennes, 1999). These sands are constructed of spherical polymers of styrene and divinylbenzene(DVB), and their surfaces are much smoother than desert sands. In general, the surface roughness of larger particles of plastic sand is smoother, and therefore the repose angle of larger particles is smaller. Using existing measured data, the repose angle of uniform plastic sands as a function of particle size is plotted in figure 3. Moreover, the changes of the repose angle of uniform plastic sands with different particle sizes can be calculated by the following expression:

From figures 2 and 3 it can be seen that the repose angle of uniform plastic sands is always smaller than that of uniform desert sands when the particle size and other conditions are the same. This is due to the effect of particle surface smoothness.

4.2. Repose angle of non-uniform desert sands

Non-uniform desert sands are made up of an uneven gradation of desert sands. Small particles are concealed by large particles, and large particles are braced by small particles. Thus, the stability of the repose slope of non-uniform desert sands is greater and the repose angle of non-uniform desert sands is larger than that of uniform sands, even when the average particle size is the same.

In order to investigate the effect of the grain composition of non-uniform desert sands on the repose angle, we studied five groups of sand samples with unevenness coefficients (η)of 1.271, 1.480, 1.720, 1.872, and 2.300, and whose median diameters (d) were controlled at about 0.25 mm to facilitate comparisons. These results were used to determine the relationship between the variation coefficient of the repose angle for non-uniform desert sands [(φη-φ)/φ] and the unevenness coefficients (η) (Figure 4). From figure 4, it is evident that the more non-uniform the desert sand, the larger the repose angle. In addition, the changes of (φη-φ)/φwith different unevenness coefficients (η) can be calculated by the following expression:

4.3. The repose angle of desert sands with different water

contents

In order to investigate how different water contents influence the repose angle of uniform desert sands of the same particle size, and how the same water content influences the repose angle of uniform desert sands of different particle sizes, respectively, we selected three desert sand samples—gravel, sand, and silt. We first dried the samples in an oven and then added water to compare the repose angles to those of the air-dried sand samples of different water contents, in which the water content was tested by the weight method.

Using existing measured data, we formulated a mathematical relationship between water content and repose angle of the three desert sand samples (Figure 5). Results indicate that the internal water content does play a great role in determining the angle of repose. For all of the sand samples, an increase in water content increased the repose angle,which subsequently decreased. Furthermore, when water content was low, the repose angle increased slowly. This phenomenon can be explained by the following theoretical analysis.

Figure 3 Repose angle of uniform plastic sands with different particle sizes

Figure 4 Repose angle of non-uniform desert sands

Figure 5 The repose angle of desert sands with different water contents

The four forms of water in desert sands are generally classified as adsorbed, film, capillary, and free water (Yanget al., 2003; Yuuet al., 2005). Those studies showed that when water content was low, the water was adsorbed over the surface of the desert sands, which had little effect on their repose angle. With an increase in water content, film water formed around the adsorbed water, leading to the formation of caking power among the particles. Therefore, the desert sands could not easily move, the fluidity of the entire sand dune was limited, and the angle of repose increased.Moreover, the smaller the particle size, the shorter was the distance between any two particles and the stronger was the caking property of the film water, and thus the more quickly the repose angle increased. When water content exceeded the maximum molecular-bound water, capillary water formed. The negative pressure in capillary water caused sand particles to move closer together, causing the fluidity of the entire sand dune to weaken and even stop; thus, the repose angle of sand samples increased rapidly. When the water content increased further, free water formed and the water content of the desert sands reached saturation. At the same time, the water no longer increased dune viscosity; it created lubricity, which decreased the repose angle.

5. Conclusions

Figure 2 shows that when the particle size of desert sands is less than "powdery", the main influencing factor on the magnitude of repose angle is the molecular force among the particles, and the other factors can be discounted. Also,with increasing particle size the repose angle of desert sands decreases as a result of the decrease in contact points among the particles, which leads to a reduction in molecular forces.With further increases in particle sizes larger than the particle size of powdery desert sands, the molecular force among particles is weakened rapidly and thus can be ignored. At this point, the repose angle of desert sands is mainly determined by the physical properties and gravity behavior of the sands.

However, although particle size is one of the most important physical properties of desert sands, it still does not directly influence the repose angle, but instead affects grit sphericity and surface roughness. Thus, grit sphericity acts as a major factor that leads to the increasing repose angle of desert sands, along with increasing particle size.

Comparing figures 2 and 3, it is evident that the mineral components of desert sands also influence the repose angle of desert sands, because mineralogy influences surface roughness of the sands. Generally speaking, the greater the mineral hardness, the more difficult it is to abrade the grains,thereby increasing the repose angle.

For non-uniform desert sands (mixed and interacting large and small particles), the value of the repose angle is related to the grain composition of the sand sample as well as the typical particle size, grit sphericity, and surface roughness of the particles. From figure 4 it can be seen that the more diverse the grain composition of the sands, the larger the repose angle.

In addition, Figure 5 shows that water content has a stronger influence on repose angle than any other factor.However, the relationship between the repose angle and water content is not a simple direct proportionality. Our results indicate that when water content increases, the repose angle will first increase and then decrease. When the grain compositions are different and the water content is the same,the process by which water content influences the repose angle also will differ. In most cases, when water content is low and the unevenness coefficient of the grain composition is small, the repose angle will be larger.

Due to limitations of the equipment and resulting measurement errors, the data obtained from this research are not precise enough to establish an empirical model to estimate the repose angle of desert sands. Thus, future studies will be needed to more clearly elucidate the response angle problem.

The authors thank Dr. YinZhou Huang, Dr. Jin Zhang,SuFeng He, and YaJun Li for their help with this research.This work was supported by the National Natural Science Foundation of China (Grant No. 50879033) and the National Science Fund for Fostering Talents in Basic Research of the National Natural Science Foundation of China (Grant No.J0730536).

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10.3724/SP.J.1226.2011.00017

23 July 2010 Accepted: 28 October 2010