The Collapse Mechanism of Slope Rill Sidewall under Composite Erosion of Freeze-Thaw Cycles and Water

Abstract

The composite erosion of freeze-thaw and water flow on slope rills is characterized by periodicity and spatial superposition. When revealing the collapse mechanism of slope rill sidewalls under the composite erosion of freeze-thaw and water flow, it is necessary to fully consider the effect of water migration and its impact on the stability of the rill sidewall. In this paper, we placed the self-developed collapse test system in an environmental chamber to carry out model tests on rill sidewall collapse on slopes under the composite erosion of freeze-thaw and water flow. We utilized three-dimensional reconstruction technology and the fixed grid coordinate method to reproduce the collapse process of the rill sidewall and precisely locate the top crack. We obtained the relationship between the water content of the specimen and mechanical indexes through the straight shear test. The main conclusions are as follows: The soil structure of the rill sidewall is significantly affected by the freeze-thaw cycle, which benefits capillary action in the soil. One freeze-thaw cycle has the most serious effect on the soil structure of the rill sidewall, and the change in the moisture field is more intense after the soil temperature drops below zero. The friction angle of the soil increases with the number of freeze-thaw cycles and tends to stabilize gradually. The effect of the freeze-thaw cycle on the rate of change of the water content of the soil at each position of the wall can be accurately described by a logarithmic function. The expression of the two-factor interaction effect on the rate of change of water content of soil at each position of the rill sidewall can be accurately fitted. We propose a calculation system for locating cracks at the top of the rill sidewall and determining the critical state of instability and collapse of the rill sidewall during the process of freeze-thaw and water flow composite erosion. The results of this research can help improve the accuracy of combined freeze-thaw and water flow erosion test equipment and the development of a prediction model for the collapse of the rill sidewall under compound erosion. This is of great significance for soil and water conservation and sustainability.

1. Introduction

The process of sidewall expansive erosion in rills unfolds during the later stages of rill development, exhibiting a complex, variable, and unpredictable nature attributable to numerous factors including flow dynamics, slope gradients, soil attributes, and management practices [1,2,3,4,5,6,7]. As the developmental trajectory of erosion features, spanning from rills to river gullies, presents a cohesive continuum, the underlying principles governing the expansion of rill sidewalls remain consistent across these various scales, particularly evident in the erosive action of lateral water flow along the rill sidewalls, concentrated scouring at the rill base, mid-flow erosion within loamy substrates, internal soil pipeline erosion, and bank erosion triggered by soil stability-related avalanches [8,9,10,11,12,13,14]. Following the scouring action of concentrated water flow at the rill base, tension fissures are inclined to develop along the inner surface of the rill sidewall once the driving force, primarily exerted by soil gravity, surpasses the resistance arising from inter-particle bonding and friction [15,16,17,18,19]. Subsequent to further scouring by concentrated flow at the rill base, the accumulation of overhanging soils escalates, initiating random collapses along the rill walls, thereby augmenting rill width and intensifying sidewall erosion rates, especially in response to slope gradients and the confluence of upstream flows [20,21,22,23,24,25]. Current investigations into pivotal processes governing rill sidewall expansive erosion, notably focusing on rill base scouring and the inception and evolution of tension fissures, remain notably deficient, necessitating extensive in situ observations and model simulations to elucidate their underlying mechanisms and characteristics [26,27,28,29,30,31,32,33,34,35,36].

Freezing and thawing occur as a consequence of volume expansion and contraction of soil pore water at low temperatures, altering the interconnections among soil particles, thereby inducing damage to soil structure and changes in physical properties. Consequently, this phenomenon impacts soil erodibility and infiltration capacity [37,38,39]. The interaction of freezing and thawing with water erosion, gravity erosion, and other factors either concurrently or alternatively leads to a soil erosion process distinct from that driven by singular forces, consequently exacerbating soil erosion rates [40,41,42,43,44,45,46,47,48]. The influence of freezing and thawing extends to soil erodibility and critical shear, potentially leading to modifications in soil erosion rates [49,50]. Indoor simulations of freeze-thaw and snowmelt erosion tests conducted on slopes demonstrated that runoff from thawed slopes exhibited a higher maximum sand content compared to slopes devoid of freeze-thaw action [51,52,53,54]. Freeze-thaw cycling resulted in a 36.5% increase in the erodibility of black soils, with the influence of freeze-thaw cycling on soil erosion closely associated with the initial soil moisture content [38]. Freezing and thawing episodes during winter and spring significantly impact soil erodibility, resulting in the development of loose topsoil vulnerable to erosion [55]. Key factors influencing soil erosion resistance in the context of freezing and thawing include the initial soil water content pre-freezing, the frequency of freeze-thaw cycles, soil temperature, aggregate stability, and soil shear strength [56,57]. While the alternating freeze-thaw action per se does not induce soil erosion, it instigates alterations in numerous soil properties encompassing bulk density, aggregate stability, porosity, shear strength, cohesion, penetration resistance, and permeability, among others [58,59,60,61,62,63,64,65,66,67,68]. These interrelations subsequently impact soil erosion dynamics, underscoring the imperative of elucidating the potential ramifications of freeze-thaw action on soil erosion processes.

There is insufficient research on the formation and evolution of cracks at the top of the rill sidewall under the composite erosion of freeze-thaw and water flow, as well as insufficient consideration of water migration and its influence on the stability of the rill sidewall in the analysis of the collapse mechanism. This study introduces a self-developed collapse test system specifically designed for assessing the stability of eroded slope rill sidewalls. This system is employed to conduct collapse expansion erosion tests on rill sidewalls along slopes. Environmental chamber equipment facilitates the replication of freeze-thaw cycles, enabling controlled temperature variations within the collapse test system. Additionally, precise localization and quantification of slope erosion-induced rill sidewall collapses are achieved through advanced techniques such as three-dimensional reconstruction and fixed grid coordinate methods. Moisture-temperature monitoring equipment, alongside direct shear testing, elucidates the interplay between soil moisture distribution, temperature variation, and mechanical properties. Subsequently, based on calculations utilizing the rill sidewall instability criterion, the precise location and critical state of rill sidewall cracks leading to instability collapse are identified, culminating in the proposal of a calculation system adept at accurately predicting rill sidewall collapses along slopes. These findings will facilitate the proposal of soil and water conservation measures and sustainability in the study area.

2. Materials and Methods

2.1. Materials

The natural volumetric water content of arsenic sandstone in Ordos, China, is 0.13, the maximum dry density is 2.05 g/cm³, and the content of particle size <0.075 mm is 5.1%; according to the classification standard of soil body in the “Specification for Geotechnical Engineering Investigation GB50021-2001” (Ministry of Construction, PRC, 2002) [69], it can be known that the arsenic sandstone in Ordos, China, is fine sand.

2.2. Methods

Drawing upon the characterization of the spatial geometric morphology development of rills in the morphological evolution tests of rills on slope erosion conducted by numerous researchers, including Zhu Xianmu et al. [70,71], who highlighted that rill depths on slopes typically do not exceed 20 cm, this paper autonomously devises a collapse test system, as depicted in Figure 1. The collapse test system comprises a water source module, a specimen module, a rill water flow module, an image and video acquisition module, a slope inclination control module, a monitoring module, a loss acquisition module, and a data processing and analysis module. The water source module, constructed from acrylic material, features a drum with a 50 cm diameter, devoid of a top cover, possessing sidewalls with a thickness of 5 mm. It is equipped with four 2 cm diameter overflows positioned on the sidewall, spaced at intervals of 10 cm, 10 cm, and 20 cm, respectively, each accompanied by a water flow control valve. Additionally, a 3 cm diameter water outlet is oriented at a 90° angle to the adjacent overflow, also equipped with a water flow control valve set at the same height as the lowest overflow. A continuous water pipeline supplies test water to the water source module, facilitating adjustment of the head height via the water flow control valve to examine the impact of varying water potentials on rill water flow velocity and the erosive effects on the sidewall bottom. Constructed from acrylic material, the specimen module measures 50 cm × 25 cm × 25 cm with a thickness of 15 mm, lacking a roof box. One side of the plate (with dimensions of 50 cm × 25 cm) can be easily installed and disassembled, facilitating its use as a template during the soil modeling process and adjustment to conform to the actual constraints on the rill sidewall during testing. The rill water flow module, crafted from acrylic material, features a rectangular groove with dimensions of 80 cm × 4 cm × 4 cm and a thickness of 15 mm. One end is sealed adjacent to the water module, while the other end serves as an outlet without closure. The seamless contact between the specimen module and the module facilitates the analysis of rill water flow at the bottom of the sidewall, accounting for the rill erosion phenomenon and the capillary effect of moisture leading to upward migration within the specimen. Employing a Canon EOS 70D SLR camera paired with a Canon EF-S 18–135 mm zoomable lens, the image acquisition module captures various angles of the rill sidewall collapse process manually. The setup maintains an average distance of 50 cm between the camera and the target object, with a focal length of 18 mm. Fifteen shooting control points are established at an average spacing of 10 cm, ensuring an overlap of more than 60% between adjacent image pairs to facilitate successful image matching during subsequent processing. The slope inclination control module employs a hydraulic system to facilitate adjustment within the 0° to 50° range for the specimen module’s slope angle. The monitoring module employs a moisture-temperature sensor, TEROS12 (METER Group, Inc., Pullman, WA, USA), and a data collector, ZL6 (METER Group, Inc., USA), to continuously monitor moisture-temperature variations at various locations within the specimen module during the entirety of the test. The loss acquisition module captures soil specimen particle losses during the test via a collection device positioned at the outlet of the water flow module. The data processing and analyzing module utilizes a Lenovo IdeaPad U410 laptop (Lenovo, Beijing, China) to establish a connection to the data collector within the monitoring module, enabling real-time acquisition of monitoring data for comprehensive analysis regarding the mechanism underlying the collapse factor of the rill sidewall on the slope.

The specimens were filled and compacted in layers to achieve a site-specific soil compaction of 63%. The total thickness of the soil in the specimen module was 15 cm, with a volumetric moisture content of 0.13, and a slope inclination a of 4°. The placement of temperature and moisture sensors is depicted in Figure 2. Following the covering of the specimen module with a plastic sheet and a 24-h standing period, the test commenced.

For the slope rill sidewall expansion erosion test under freeze-thaw cycles, temperature control was achieved using an environmental chamber. Based on previous studies [72,73,74], the average extreme maximum and minimum temperatures recorded in the Ordos, China, arsenic sandstone test area over the past 38 years were +32.8 °C and −22.6 °C, respectively. The temperature variation during the test follows a cosine curve with a 16-h period. The extreme maximum and minimum values, +32.8 °C and −22.6 °C, respectively, were applied for 0, 1, 3, 5, 7, and 10 cycles. Freeze-thaw tests were conducted at different minimum temperatures, ranging from −8 °C to −24 °C. The collapse test system and straight shear test specimens were placed in the environmental chamber in accordance with the test requirements. They underwent the prescribed number of freeze-thaw cycles before proceeding to the slope rill sidewall expansion erosion test and straight shear test.

Based on the variation in water content across different sections of the specimen obtained from the monitoring module, the change in internal friction angle of the specimen soil is determined through straight shear tests conducted under varying water content and freeze-thaw cycle numbers. This serves as a foundational analysis for investigating the mechanisms behind rill sidewall collapse.

The image processing software Agisoft Metashape 2.1.0 (Agisoft LLC, St. Petersburg, Russia) was used to process the acquired high-definition images to generate three-dimensional graphics of the different stages of rill sidewall collapse, combined with the fixed grid coordinate method, i.e., the specimen surface is drawn with a fixed grid to accurately locate the morphology of the different stages of the rill sidewall collapse process and carry out analysis of the spatial and temporal evolution of geometric forms at different stages of erosion and rill sidewall collapse.

Figure 3 shows the high-definition captured photographs and 3D reconstructed graphics; comparison of the spatial geometry, fixed grid pattern, and collapse of the rill sidewall on the slope in the figure shows that the morphology parameters of the rill sidewall in the high-definition photographs and the 3D reconstructed graphs are identical. It shows that the three-dimensional graphics generated by image processing software Agisoft Metashape 2.1.0 can accurately reflect the evolution of spatial geometry in the collapse process of the rill sidewall, which is helpful for the extraction of subsequent test data and the analysis of spatial geometry evolution in the collapse process of the rill sidewall.

3. Results and Discussion

3.1. Analysis of the Causes of Rill Sidewalls Collapse by Water Erosion under Freeze-Thaw Cycles

The freeze-thaw cycling test was conducted in a controlled environment chamber. The maximum and minimum temperature values within the chamber were 32.8 °C and −22.6 °C, respectively, following a cosine curve pattern. Temperature and moisture variation curves for each monitoring position of the soil specimen during 0, 1, 3, 5, 7, and 10 freeze-thaw cycles are presented in Figure 4. The temperature variation of each monitoring position on the soil specimen tends to approximate the cosine curve representing the changes in test environment temperature. However, as stated in the results of Xu, Wang, Ban, Seyed et al. [41,43,51,56], due to energy transfer loss and hysteresis effects, monitoring positions farther from the specimen’s top surface exhibit diminished responses to ambient temperature fluctuations. Consequently, greater discrepancies emerge between the specimen and ambient temperature, leading to a gradual attenuation of the test environment’s influence on the soil body.

During the freeze-thaw cycle, the volumetric water content of soil specimens at each monitoring position exhibits a cosine curve pattern over time. Notably, monitoring positions 4^# and 7^# are situated closest to the specimen’s top surface, thus experiencing significant impacts from the test environment temperature and displaying more pronounced responses to changes in volumetric moisture content. As the freeze-thaw cycle test was configured as a no-water recharge experiment, the observed variation in volumetric water content at each monitoring location of the soil specimen remained minimal, with a maximum increase of only 0.039.

Soil specimens subjected to 0, 1, 3, 5, 7, and 10 freeze-thaw cycles were utilized in rill water erosion tests employing a collapse test system. The corresponding variation in volumetric water content at each monitoring position is depicted in Figure 5. Erosion from rill flow water at the rill sidewall bottom, combined with the capillary effect within the soil specimen, resulted in a progressive rise in volumetric water content at each monitoring position. Notably, monitoring location 1^#, being in closest proximity to the rill flow, exhibited the most significant response to volumetric water content variations. Freezing and thawing cycles induce fragmentation of larger soil particles, and water migration within the soil drives displacement of smaller soil particles. These processes lead to a redistribution of soil particles within the specimens, alterations in soil structure, reduced soil porosity, and heightened capillary phenomena. For instance, compared to soil specimens subjected to erosion tests without freeze-thaw cycles, those subjected to freeze-thaw cycles exhibit earlier onset of volumetric water content changes at monitoring position 2^#, significant changes at position 3^#, and increased rates of increase at positions 5^# and 6^#. The impact of a single freeze-thaw cycle on soil specimen structure is notable, with significant changes observed in volumetric water content at each monitoring position after one cycle of rill water flow erosion tests. Furthermore, as the number of freeze-thaw cycles increases, their impact gradually diminishes, resulting in a reduced response to volumetric water content changes at each monitoring position.

3.2. Analysis of the Collapse Causes of Rill Sidewalls by Water Erosion under One-Time Freeze-Thaw Cycles Effects of Different Minimum Temperatures

A single freeze-thaw test was conducted within the test environment chamber, with the maximum test environment temperature set at 21 °C and the minimum temperature ranging from −8 °C to −24 °C, following a cosine curve pattern. The soil specimen underwent a single freeze-thaw, and the resulting temperature change curve at each monitoring position is depicted in Figure 6. The temperature variation at each monitoring position of the soil specimen tends to follow the cosine curve representing changes in test environment temperature. However, owing to energy transfer loss and hysteresis effects, monitoring positions located farther from the top surface of the specimen exhibit diminished responses to ambient temperature changes. Consequently, as the distance between the monitoring position and the specimen’s top surface increases, the discrepancy between the temperature of the specimen and the test ambient temperature amplifies, indicating a gradual attenuation of the test ambient temperature’s influence on the soil body.

Soil specimens underwent a single freeze-thaw cycle at temperatures ranging from −8 °C to −24 °C, with the highest temperature set at 21 °C. Subsequently, the rill erosion test was performed using the collapse test system, and the variation in volumetric water content at each monitoring position during the test is illustrated in Figure 7. The erosion caused by rill flow water at the bottom of the rill sidewall, coupled with the capillary effect within the soil specimen, results in a gradual increase in volumetric water content at each monitoring position. Notably, monitoring position 1^# exhibits the most pronounced response. A lower minimum temperature exacerbates the impact of freezing and thawing on soil structure and enhances capillary phenomena within the soil specimen. For instance, the onset of volumetric water content change at monitoring positions 3^#, 5^#, and 6^# occurs earlier as the minimum temperature decreases, resulting in a higher rate of increase in volumetric water content. In the freeze-thaw cycles ranging from −12 °C to −24 °C, the internal temperature of the soil specimen dropped below zero, indicating the onset of moisture freezing within the specimen. This effect on the soil specimen’s structure is more pronounced compared to freeze-thaw cycles at −8 °C, leading to a more significant change in volumetric water content at each monitoring position after freeze-thaw cycles at temperatures between −12 °C and −24 °C during the rill water flow erosion test.

3.3. Analysis of the Collapse Mechanism of Rill Sidewall under Freeze-Thaw and Water Flow Action

(1)

Influence of different water contents and freeze-thaw cycles on the friction angle in the soil:

Straight shear tests were performed on soils with varying water contents, as well as on soils subjected to 0–10 freeze-thaw cycles, thereby yielding the corresponding curves depicting changes in the internal friction angle, as illustrated in Figure 8. The internal friction angle of soil diminishes with escalating water content, eventually reaching a stabilizing tendency, as deduced from data fitting computations used to derive the equation representing the relationship between the soil’s internal friction angle and its water content, as presented below:

$$\varphi \left(\omega \right)=8.694-0.16732\omega +0.00225{\omega }^2 ({\mathrm{R}}^2>0.99)$$

(1)

where:

• φ—the soil’s internal friction angle (°);
• ω—the water content of sample soil (m³/m³).

The freeze-thaw cycle facilitates the rearrangement of soil particles and enhances soil compactness. Consequently, the internal friction angle of soil tends to elevate and stabilize post freeze-thaw cycles. Through data fitting calculations, the relationship between the internal friction angle and the number n of freeze-thaw cycles in the soil was established as:

$$\varphi \left(n\right)=0.09645\ln n+7.09781 ({\mathrm{R}}^2>0.99)$$

(2)

where:

• φ—the soil’s internal friction angle (°);
• n—number of freeze-thaw cycles.

The interaction between soil moisture content and the number of freeze-thaw cycles on the internal friction angle of rill sidewall soils, in comparison to the impact of each individual factor, exhibited substantial disparities. The relationship:

$$\varphi (\omega ,n)=a_1{\omega }^2+b_1\omega +c_1\ln n+p_1$$

(3)

where:

• φ—the soil’s internal friction angle (°);
• n—number of freeze-thaw cycles;
• ω—the water content of sample soil (m³/m³);
• a₁, b₁, c₁, p₁—coefficients of the equation;
• elucidates the influence of the two-factor interaction on the internal friction angle within the soil of the rill sidewall, as depicted in Figure 9, with an R² value exceeding 0.99.

(2)

Fitting a rill sidewall moisture field expression:

Following the freeze-thaw cycle treatment of the specimens in the rill sidewall expansion erosion test, variations in the water content across different positions within the soil layer exhibit an initial increase followed by a tendency to stabilize over time, as depicted in Figure 5. It is hypothesized that during the initial phase of increase, the water content of the soil layer increases linearly with the duration of the test until reaching a plateau. Subsequently, the water content of the soil layer in the rill sidewall after the freeze-thaw cycle can be mathematically described by the following equation in the expansion erosion test:

$$\omega =\left\{\begin{array}{l}kt+m,t\in (t_0,t_1]\\ {\omega }_1,t\in (t_1,t_2)\end{array}\right.$$

(4)

where:

• ω—the water content of sample soil (m³/m³);
• t—expansive erosion test duration (min);
• t₀—onset time of change in soil moisture content (min);
• t₁—end time of change in soil moisture content (min);
• t₂—expansive erosion test end time (min);
• ω₁—final water content of the soil layer (m³/m³);
• k—the change rate of soil moisture content;
• m—coefficient of the equation; m is approximately equal to the initial water content of the soil.

Figure 10 illustrates a fitting analysis of the change rate, denoted as k, of soil moisture content at various locations along the rill sidewall subjected to different numbers of freeze-thaw cycles in the rill water flow expansive erosion test. The correlation between the change rate k of soil moisture content and the number n of freeze-thaw cycles, treated as the independent variable, can be represented as:

$$k(n)=a_2+b_2\ln (n+c_2) ({\mathrm{R}}^2>0.99)$$

(5)

where:

• k—the change rate of soil moisture content;
• n—number of freeze-thaw cycles;
• a₂, b₂, c₂—coefficients of the equation.

Likewise, the relationship between the change rate k of soil moisture content and the distance z of the soil layer from the bottom of the rill sidewall can all be expressed as an exponential function:

$$k\left(z\right)=c_3{e}^{-p_{2}z} ({\mathrm{R}}^2>0.99)$$

(6)

where:

• k—the change rate of soil moisture content;
• z—distance of the rill sidewall soil layer from the bottom (cm);
• c₃, p₂—coefficients of the equation.

The interaction between the number of freeze-thaw cycles and the distance from the bottom of each soil layer of the rill sidewall on the change rate of soil water content exhibited significant disparities compared to the impact of each individual factor. Based on the results obtained from fitting the one-factor equation, the relationship:

$$k(n,z)=a_2+b_2\ln \left(n+c_2\right)+c_3{e}^{-p_{2}z}$$

(7)

where:

• k—the change rate of soil moisture content;
• n—number of freeze-thaw cycles;
• z—distance of the rill sidewall soil layer from the bottom (cm);
• a₂, b₂, c₂, c₃, p₂—coefficients of the equation;
• can aptly depict the influence of two-factor interaction on the change rate of soil water content within the rill sidewall, as illustrated in Figure 11, with an R² value exceeding 0.9.

Utilizing the fitting equations derived from the number of freeze-thaw cycle actions and the respective distances of the rill sidewall soil layers from the bottom, in conjunction with the change rate of the rill sidewall soil water content, the post-freeze-thaw cycle water content of the rill sidewall soil layer in the rill water flow expansive erosion test can be elaborated upon as follows:

$$\omega =\left\{\begin{array}{l}\left[a_2+b_2\ln \left(n+c_2\right)+c_3{e}^{-p_{2}z}\right]t+m,t\in (t_0,t_1]\\ {\omega }_1,t\in (t_1,t_2)\end{array}\right.$$

(8)

where:

• ω—the water content of sample soil (m³/m³);
• n—number of freeze-thaw cycles;
• z—distance of the rill sidewall soil layer from the bottom (cm);
• t—expansive erosion test duration (min);
• t₀—onset time of change in soil moisture content (min);
• t₁—end time of change in soil moisture content (min);
• t₂—expansive erosion test end time (min);
• ω₁—final water content of the soil layer (m³/m³);
• a₂, b₂, c₂, c₃, p₂, m—coefficients of the equation; m is approximately equal to the initial water content of the soil.

(3)

Mechanism analysis of the collapse causation of rill sidewall after freeze-thaw cycles in the water erosion test:

The rill sidewall, subjected to varying numbers of freeze-thaw cycles, gradually develops a cantilever structure due to the ongoing erosion caused by the rill water flow. The number of freeze-thaw cycles influences both the soil structure and the internal friction angle of the cantilevered section. Additionally, the gravity and center of gravity position of this cantilevered part undergo continuous changes owing to the erosion and capillary action of the water flow. The capillary action leads to water migration from the bottom of the rill sidewall upward, resulting in alterations in the water content of the rill sidewall soil, subsequently affecting the pore water pressure of the soil, which in turn is the effective stress. Consequently, the internal friction angle undergoes constant reduction. The internal friction angle change function φ(t, ω, n) is intricately linked to the variables of time (t), water content (ω), and the number of freeze-thaw cycle actions (n), as depicted in Figure 12. The moment action function M(t, G, D, ω) generated by the cantilever structure is dependent on time (t), gravity (G), moment (D), and water content (ω). Due to a significant increase in bending moments during the water erosion test, visible cracks emerged at the top position of the rill sidewall subjected to different numbers of freeze-thaw cycles, continuing to expand until eventual instability and collapse, as illustrated in Figure 13.

The gravity of rill sidewall soils, subjected to various numbers of freeze-thaw cycles in expansion erosion tests, undergoes alteration due to the scouring and capillary action induced by the flow of rill water. This is inferred from the fitting equation derived for the water content of the rill sidewall soil, with the assumption that the length of the rill sidewall is 1. Consequently, the gravity of the rill sidewall soil can be expressed as follows:

$$G\left(D,\omega ,\alpha ,z\right)={\displaystyle {\int }_{z_0}^{H}2}D{\rho }_d\left\{\left.1+\omega \left(n,z\right)\right\}\right.\cdot {10}^{-3}\cdot g\cos \alpha dz$$

(9)

where:

• G—the gravity of rill sidewall soils (N);
• H—rill sidewall heights (cm);
• z₀—depth of the rill water flow (cm);
• 2D—length of destabilized zone (cm), calculated from the location of cracking;
• ρ_d—soil dry density (g/cm³);
• ω—the water content of sample soil (m³/m³);
• α—slope inclination (°);
• g—gravity acceleration (N/kg).

The bending moment within the destabilization zone is denoted as:

$$M=G\left(D,\omega ,\alpha ,z\right)D$$

(10)

where:

• M—the bending moment within the destabilization zone (N·cm);
• G—the gravity of rill sidewall soils (N);
• D—perpendicular distance from the fulcrum the line of force (cm).
• while the bending moment resisting crack formation in this zone is designated as:

$$M^{\prime }=E_a{D}^{\prime }$$

(11)

where:

• M′—the bending moment resisting crack formation in this zone (N·cm);
• E_a—the soil pressure (N);
• D′—the height of the soil pressure action location (cm).

Furthermore, the soil pressure, denoted as E_a, can be represented by:

$$E_a={\displaystyle {\int }_{z_0}^H{k}_0{\rho }_d\left(1+\omega \right)}g\cdot {10}^{-3}\cdot \left(H-z\right)dz$$

(12)

where:

• E_a—the soil pressure (N);
• H—rill sidewall heights (cm);
• z₀—depth of the rill water flow (cm);
• k₀—the static earth pressure coefficient;
• ρ_d—soil dry density (g/cm³);
• ω—the water content of sample soil (m³/m³);
• g—gravity acceleration (N/kg);
• z—distance of the rill sidewall soil layer from the bottom (cm);

where k₀ represents the static earth pressure coefficient (typically ranging from 0.25 to 0.33 for sandy soils) [75]. The height $D^{\prime }$ of the soil pressure action location is approximately equivalent to $\left(H-z_0\right)/3$. The precise position of crack initiation at the top of the rill sidewall can be determined based on $M=M^{\prime }$.

The interfacial shear within the instability zone is denoted as:

$$T={\displaystyle {\int }_{z_0}^{H-h_l}\tau dz}$$

(13)

where:

• T—the interfacial shear within the instability zone (N);
• H—rill sidewall heights (cm);
• z₀—depth of the rill water flow (cm);
• h_l—the depth of the crack at the top of the rill sidewall (cm);
• τ—the shear stress (N/cm²);
• z—distance of the rill sidewall soil layer from the bottom (cm);

where h_l represents the depth of the crack at the top of the rill sidewall, and the shear stress is indicated by:

$$\tau ={\rho }_d\left(1+\omega \right)g\cdot {10}^{-3}\cdot \left(H-z\right)\tan \varphi \left(\omega ,n\right)$$

(14)

where:

• τ—the shear stress (N/cm²);
• ρ_d—soil dry density (g/cm³);
• ω—the water content of sample soil (m³/m³);
• g—gravity acceleration (N/kg).
• H—rill sidewall heights (cm);
• z—distance of the rill sidewall soil layer from the bottom (cm);
• φ—the soil’s internal friction angle (°).

Accordingly:

$$T\left(\omega ,H,hl,z,\varphi \right)={\displaystyle {\int }_{z_0}^{H-hl}{\rho }_d\left(1+\omega \right)g\cdot {10}^{-3}\cdot \left(H-z\right)\tan \varphi \left(\omega ,n\right)}dz$$

(15)

where:

• T—the interfacial shear within the instability zone (N);
• H—rill sidewall heights (cm);
• z₀—depth of the rill water flow (cm);
• h_l—the depth of the crack at the top of the rill sidewall (cm);
• ρ_d—soil dry density (g/cm³);
• ω—the water content of sample soil (m³/m³);
• g—gravity acceleration (N/kg).
• z—distance of the rill sidewall soil layer from the bottom (cm);
• φ—the soil’s internal friction angle (°).

The critical state of instability and collapse in the rill sidewall can be computed using $G=T$.

Drawing on the collapse mechanism of the rill sidewall, the calculation formula for determining the top crack location of the rill sidewall was utilized to ascertain the specific position of the top crack with slope inclinations of 2°/4°, a rill sidewall height of 15 cm, water source head height of 0 cm, and arsenic sandstone soils subjected to one-time freeze-thaw cycles in the expansion erosion process caused by rill water flow. The calculated results yielded distances of 5.4 cm/7.7 cm from the end of the rill sidewall, respectively. Additionally, the results of tests conducted using three-dimensional reconstruction techniques to analyze images of rill sidewalls with slope inclinations of 2°/4°, rill sidewall height of 15 cm, water source head height of 0 cm, and arsenic sandstone soils subjected to one-time freeze-thaw cycles during the expansion erosion process induced by rill water flow, as depicted in Figure 14, involved extracting the average distances from the end of the rill sidewall at three locations on the crack at the top of the rill sidewall, resulting in distances of 5.3 cm/7.5 cm from the end of the rill sidewall, respectively. Notably, the maximum difference between the theoretical calculations and the results of tests was 0.2 cm, affirming the exceptional accuracy of the proposed formula for predicting rill sidewall collapse in water flow expansion erosion.

4. Conclusions

Freezing and thawing cycles exert a substantial influence on the soil structure of the rill sidewall. Additionally, the continuous erosion at the bottom of the rill sidewall forms a cantilever structure during the expansion erosion caused by the rill water flow. Furthermore, the capillary action in the soil continuously alters the moisture distribution within the rill sidewall, resulting in cracks at the top of the sidewall that eventually lead to instability and collapse. Revealing the mechanism of fine gully wall collapse under compound erosion is of great significance to soil erosion management and sustainability of study area. Based on an investigation into the collapse mechanism of the rill sidewall under the influence of freeze-thaw cycles and water flow, the following primary conclusions were drawn:

(1)

Freeze-thaw cycles exert a notable influence on the soil structure of rill sidewalls, promoting capillary phenomena within the soils. A single cycle of freezing and thawing exerted the most pronounced impact on the structure of the rill sidewall soil, with the moisture field within the rill sidewall soil undergoing more significant changes under negative soil temperatures;

(2)

Freeze-thaw cycles contribute to enhancing the internal friction angle of the soil, while the internal friction angle tends to decrease with increasing soil water content. Accurately fitted expressions were derived to represent the interaction effect between the number of freeze-thaw cycles and soil water content on the internal friction angle of the soil during expanded erosion by rill water flow;

(3)

The impact of the number of freeze-thaw cycles on the rate of soil water content change at various locations along the rill sidewall can be precisely described using a logarithmic function. Furthermore, an equation was developed to effectively capture the interaction between the number of freeze-thaw cycles and the distance of the soil layer from the bottom of the rill sidewall in influencing the rate of water content change at each location of the rill sidewall;

(4)

By deriving expressions for the bending moment and resistance to cracking within the instability zone of the rill sidewall following freeze-thaw cycles during the expansion and erosion caused by rill water flow, the exact location of the crack at the top of the rill sidewall can be determined based on the equality between them. Furthermore, an expression for the changing gravitational forces and shear forces at the interface within the instability zone of the rill sidewall can be derived, based on their equivalence, enabling the calculation of the critical state of instability and collapse of the rill sidewall.