Study on Parameters of Two Widely Used Cohesive Soils Erosion Models

Abstract

The erosion rate of cohesive soils was typically modeled with the excess shear stress model and the Wilson model. Several kinds of research have been conducted to determine the erodibility parameters of the two models, but few attempts have been made hitherto to investigate the general trends and range of the erodibility parameter values obtained by the commonly used Erosion Function apparatus. This paper collected a database of 177 erosion function apparatus tests to indicate the variability of all erodibility parameters; the range of erodibility parameters is determined by data statistics and parameter theoretical value derivation. The critical shear stress (τ_c) and erodibility coefficient (Z₀) in the over-shear stress model have a positive proportional relationship when the data samples are sufficient. However, there is no such relationship between the erodibility coefficient (b₀) and erodibility coefficient (b₁) in the Wilson model. It is necessary to express the soil erosion resistance by considering all erosion parameters in the erosion model. Equations relating erodibility parameters to water content, plasticity index, and median particle size were developed by regression analysis.

1. Introduction

Erosion models represent the constitutive law of soils for erosion problems, much like a stress–strain curve represents the constitutive law of soils for settlement problems. The most commonly used erosion models up to date are the excess shear stress model and the Wilson model. The excess stress shear model can be expressed as [1]:

$$\dot{z}=k_D{(\tau -{\tau }_c)}^{\alpha }$$

(1)

where $\dot{z}$ is the erosion rate (m s⁻¹), k_D is the coefficient of erodibility ($m^3/\left(N\times S\right)$), τ is the applied shear stress (Pa), τ_c is the critical shear stress (Pa), and α is an empirical exponent sometimes assumed to be unity [2,3,4]. Some scholars proposed dimensionless nonlinear excess shear stress models to calculate the erosion rate [1,5,6,7]. The typical dimensionless excess shear stress model is defined as:

$$\dot{z}=Z_0{(\frac{\tau }{{\tau }_c}-1)}^{\alpha }$$

(2)

where Z₀ is the erodibility parameter of the dimensionless excess shear stress model (mm/h). However, it is difficult to predict the values of erodibility parameters due to the erodibility parameters (Z₀ and τ_c) changing with the season [8,9].

Wilson (1993a, 1993b) proposed a mechanistic detachment model to provide a general framework for studying soil and fluid characteristics [10,11]. The advantage of the Wilson model over the over-shear stress model is that the factors affecting erosion can be considered and evaluated in greater depth; this has been studied extensively by related scholars, incorporating factors such as turbulence, roughness, seepage forces, material soil orientation, root effects, and negative pore water pressure effects into the Wilson model [12,13,14,15,16]. The Wilson model was based on the balance of all the forces and moments driving and resisting detachment that would be preferred for modeling the range of possible environmental conditions, which is expressed as:

$$\dot{z}=\frac{b_0\sqrt{\tau }}{{\rho }_b}\left[1-\exp \left\{-\exp \left(3-\frac{b_1}{\tau }\right)\right\}\right]$$

(3a)

$$b_0={\rho }_s\frac{k_r}{K_e}\sqrt{\frac{K_n}{k_{dd}\left({\rho }_s-{\rho }_w\right)}}$$

(3b)

$$b_1=\left(\frac{\pi }{e_v\sqrt{6}}\right)\frac{k_r\left(K_{ls}+f_c\right)}{K_d}g\left({\rho }_s-{\rho }_w\right)d$$

(3c)

where ${\rho }_b$ is the bulk density of soil (g m⁻³),${\rho }_s$ and ${\rho }_w$ are the densities of particle and water (1000 kg m⁻³), respectively, K_e is the exposure of lower particle parameter, K_n is a combination and fluid factors, k_r is the ratio of k_v and k_a (k_r = k_v/k_a = 2/3), k_dd is detachment distance parameter, K_d is dimensionless hydraulic drag parameter, K_ls is the dimensionless bed and slope parameter that depends on particle (or aggregate) size, f_c is the dimensionless cohesive force parameter, and e_v is the coefficient of variation. Wilson suggested that the critical stress can be calculated by drag forces equaling time-average drag forces. The critical stress is expressed as:

$${\tau }_c=\frac{k_r}{K_d}\left({\rho }_s-{\rho }_w\right)dg\left(K_{ls}+f_c\right)$$

(4)

Al-Madhhachi (2013) tested the appropriateness of the excess stress shear model and Wilson model for cohesive sediment using flume tests, “mini” JET tests, and JET tests [12]. It was found that the Wilson model fitted the observed data from the original and “mini” JET better than the excess shear stress model, while it was consistent with the excess shear stress model in fitting the observations from the flume tests. Then, the seepage force was introduced into the Wilson model to improve the prediction effect of cohesive soil [17,18]. The recent Erosion Function Apparatus (EFA) test by Gao X et al. (2019) showed that the Wilson and shear stress models could also be applied to the test [19].

In the application of the scouring model, the determination of its parameters is a significant problem. Hanson and Cook (1997) and Hanson et al. (2002) developed analytical procedures to directly estimate k_D and τ_c based on diffusion principles of the circular jet scour test using an Excel spreadsheet [20,21]. Daly et al. (2013) proposed a new spreadsheet tool to solve for k_D and τ_c values simultaneously [22]. Al-Madhhachi et al. (2017) used the solver routine in Microsoft Excel to derive erodibility parameters b₀ and b₁, which utilized the generalized reduced gradient method [23]. Hanson and Simon (2001) found inversely proportional between the critical shear stress and the erodibility parameter k_D based on 83 jet-tests on cohesive streambeds in the Midwestern U.S., implying that one parameter can fully describe the soil erosion resistance [24]. An alternative relationship was proposed by Simon et al. (2011) based on hundreds of JETs on stream banks across the U.S [25]. However, other scholars found no correlation between erodibility parameter k_D and critical flow shear stress [26,27]. These studies have different views on the relationship between each parameter, which needs further confirmation. In addition, for the data set of the Erosion Function Apparatus test, whether the above relationship is applicable still needs further verification.

Erosion resistance of soil is closely related to its parameters such as soil gradation, compaction, clay content, etc. [28,29]; Briaud (2005) found that soil porosity ratio, water temperature, soil temperature, sodium adsorption ratio, and other parameters are directly proportional to soil’s erosion resistance ability [30], while soil specific gravity, plastic index, particle proportion with particle size <0.075 mm and other parameters are inversely proportional to soil’s erosion resistance ability. The study shows that soil’s physical and mechanical parameters closely affect its scour characteristics [29]. Therefore, soil properties play an essential role in soil erosion resistance. For the parameters in the scour model, it is also vital to determine their relationship with soil parameters. Hasan and Al-Madhhachi (2018) investigated the effect of crude oil on erosion parameters (b₀ and b₁) [13]. For clean soil and polluted soil, parameter b₁ increases with increasing water content. However, compared with clean soil, polluted soil has a more significant effect on parameter b₀. Al-Madhhachi (2013) proposed that the scouring parameter Z₀ is inversely proportional to water content in the scouring experiment of fine-grained soil [12]. In the scour experiment of silty clay and sand mixture, Z₀ and α are positively proportional to dry density and inversely proportional to the void ratio [19,31]. Although there has been some advance in studying erosion in cohesive soils, further research is needed to estimate erodibility parameters accurately.

In this study, a database of 177 cohesive soil erosion function apparatus tests with soil properties was collected from the literature review. The Wilson model and the dimensionless over-shear stress model were further extended for applications in EFA devices. The range of scouring parameters in the EFA data set is determined based on the experimental data and the theoretical derivation of erosion parameters. The values of the erosion parameters under EFA equipment are summarized. The general trend of erosion parameters and the relationship between erosion parameters are further illustrated. The variability of the parameters and the importance of each parameter are revealed. The relationship between soil erodibility parameters (b₀, Z₀, and α) and soil engineering properties is proposed by regression analysis, which could be helpful for computer simulation analysis of Scouring failure.

2. Methods

In the analysis of this paper, mean grain size and bulky soil density are normalized, and the data are converted into dimensionless data between 0 and 1. The remaining three parameters are expressed in decimal form, and then, regression analysis is conducted. The normalization formula is as follows:

$$x^{✱}=\frac{x-x_{\min }}{x_{\max }-x_{\min }}$$

(5)

where $x^{✱}$ is the normalized data, and x_max and x_min, respectively, represent the maximum and minimum values in this data group.

For the regression analysis model, the determination coefficient $R^2$ and $F_{value}$ were used to evaluate the model’s performance and determine the optimal parameter values [32,33]. The determination coefficient $R^2$ and $F_{value}$ are defined as:

$$R^2=1-\frac{{\displaystyle \sum {\left(y_i-{\widehat{y}}_i\right)}^2}}{{\displaystyle \sum {\left(y_i-\overline{y}\right)}^2}}$$

(6)

where $y_i$ is the actual value of the sample, ${\widehat{y}}_i$ is the fitting value, and $\overline{y}$ is the average value. R-square ranges from 0 to 1.0, which closer to 1.0 means better fit.

$$F_{value}=\frac{{{\displaystyle \sum (f_i-\overline{y})}}^2}{{\displaystyle \sum {(y_i-f_i)}^2}}\times (\frac{D}{V}-1)$$

(7)

where D is the number of data points, and V is the number of independent variables. When the F_value is much higher than the critical value of the F test, the result of the regression equation is more reliable. The critical of F_value can be obtained from the F distribution table, and its significance level is set at 0.05.

Therefore, the combinations with higher R² and F_value/F_statistic are chosen as equations relating the erodibility parameters to the soil geotechnical properties.

3. Erosion Function Apparatus Tests Database

The erosion function apparatus is currently one of the most commonly used soil erosion test equipment. It is used to investigate the erodibility of the non-cohesive soil, cohesive soil, and gravel. The EFA can measure the erosion rate curve and critical shear stress of cohesive soil [34].

The study collected a database of 177 tests. The coupling of various soil parameters influences the scour behavior of soil, and the scour mechanism is studied by investigating the influence of soil parameters on scouring behavior. The experimental data selected in this paper include the experimental data of scouring and soil parameters; some data lacking soil parameters are excluded. Among the database, eight tests were from Gao (2019), twelve tests were from ILIT (2006), eighty tests were from Shafii (2018), thirty-eight tests were from Briaud (1999, 2008), twelve tests were from Kwak (2000), six tests were from Park (2007), three tests were from Bernhardt (2011), five tests were from Govin (2009), and the thirteen tests were from Straub (2013) [19,28,35,36,37,38,39,40,41,42].

Table 1. List of the USCS classifications related to 177 texts.

USCS Classification	Number of Samples
Low-plasticity clay (CL)	99
High-plasticity clay (CH)	56
CH with sand	2
CL with sand	20

shows the soil types according to the USCS classifications and the number of tests.

Figure 1 shows the soil samples main properties in the database, including plasticity index, percent fines, soil bulky density, water content, and mean grain size.

4. Results and Discussion

4.1. The Data Range of the Experimental Erodibility Parameters Values

In this study, the Wilson model’s erodibility parameters (b₀ and b₁) were derived using a curve fitting tool based on the principle of minimizing the error between predicted erosion data and measured erosion data. The critical shear stress τ_c is defined for this study as the interface shear stress corresponding to a very low erosion rate of 0.1 mm h⁻¹. The value of the erodibility parameter (Z₀) in equation 2 is equal to th e erosion rate when the hydraulic shear stress (τ) is equal to twice the critical shear stress (τ_c). After the critical shear stress and the erodibility parameter (Z₀) were determined, the exponential term ($\alpha$) was estimated using the curve fitting tool by minimizing the sum of squared errors between the predicted erosion data and the measured erosion data. The erodibility parameters (τ_c, Z₀, and $\alpha$) of the dimensionless excess shear stress model and the erodibility parameters (b₀ and b₁) of the Wilson model are shown in Figure 2 and Figure 3, respectively.

The database was analyzed to obtain 177 Z₀, τ_c, α, b₀, and b₁ values. Figure 2 shows that the erodibility parameters Z₀, τ_c, and α of cohesive soils ranged from 0.10 mm h⁻¹ to 117.23 mm h⁻¹, from 0.11 Pa to 50.25 Pa, and from 0.26 to 2.75, respectively. Briaud (2017) analyzed 84 EFA tests carried out by Texas A&M University and TxDOT and obtained the range of critical shear stress values for various soil types [43]. The range of critical shear stress of cohesive soils is 0.15–20 Pa obtained from Briaud (2017), which is consistent with the range of critical shear stress obtained in this paper. Several studies have shown that α varies between 1.0 and 6.8 [43,44,45], consistent with the range of α values obtained in the current paper.

Figure 3 shows that the erodibility parameters b₀ and b₁ values of cohesive soils ranged from 0.09 g m⁻¹ s⁻¹ N^−0.5 to 80.29 g m⁻¹s⁻¹N^−0.5 and from 0.56 Pa to 267.26 Pa, respectively. Based on equation 3b, it can be shown that the erodibility parameter b₀ is related to the bulk density of soil (ρ_b), the densities of the particle (ρ_s), the ratio of volume/area parameters (k_r), the exposure of lower particle parameter (K_e), a combination and fluid factors (K_n), and detachment distance parameter (k_dd). Wilson recommended that Wilson recommended that kr = 2/3 for spherical particles, k_dd = 2. Wilson (1993b) defines a combination and fluid factors (K_n) as [11]:

$$K_n=K_t{K}_d/k_r$$

(8)

where K_t is the cumulation of instantaneous force. Wilson suggested that K_t = 2.5 and the dimensionless hydraulic drag parameter (K_d) is defined as:

$$K_{\mathrm{d}}=\frac{C_d{k}_f{\left[\ln \left(\phi \right)/\kappa +B\right]}^2}{2}$$

(9)

where C_d is the drag coefficient, Einstein and El-Samni (1949) suggested that the value of C_d is 0.2 [46]; k_f is the fraction of exposed area, and Wilson recommended a value of 0.92; $\phi$ is the drag velocity fractional height, Einstein and El-Samni (1949) suggested $\phi$ = 0.35, while Yang (1973) suggested $\phi$ = 1 [46,47]; $\kappa$ is the von Kannon constant, Wilson recommended $\kappa$ = 0.4; B is the Log-velocity intercept parameter which ranges from 6.5 to 9.8 [11]. The maximum value of K_d is 8.84, and the minimum value is 1.38. Therefore, the maximum value of K_n is 33.15. Kothyari (2008) suggested that the density value of clay is 2.7 g cm⁻³ [48]. The influence of the exposure factor k_e on the erosion parameter b₀ is critical, but little information is available directly on the parameter k_e [11,12]. The method of determination of this parameter needs to be further studied.

To obtain the maximum value of parameter b₁, combine Equations (3c) and (4):

$$b_{1(\max )}=\frac{\pi {\tau }_{c(\max )}}{e_v\sqrt{6}}$$

(10)

The value of e_v in this study is 0.35. Due to the maximum value of τ_c being 50.25 Pa obtained from the database, the theoretical maximum value of b₁ is 184.04 Pa. We can find that the range of theoretical values of the erodibility parameters (b₀ and b₁) includes the experimental values.

A comparison of erodibility parameters between high-plasticity clay (CH: PI ≥ 0.73(w_L-20), PI ≥ 7, and w_L ≥ 50) and low-plasticity clay (CL:PI ≥ 0.73 (w_L-20), PI ≥ 7, and w_L < 50) is shown in Figure 4 and Figure 5. Compared with the erodibility parameter b₀ of high-plasticity clay, the erodibility parameter b₀ of low-plasticity clay is higher. The range of erodibility parameters b₀ of high-plasticity clay and low-plasticity clay mainly concentrates in 0.44–2.77 g m⁻¹ s⁻¹ N^−0.5 and 1.35–4.6 g m⁻¹ s⁻¹ N^−0.5, respectively (Figure 4a). The erodibility parameter b₀ decrease with the water content increase due to the rise of soil cohesion (which increases K_e). Similar observations are reported for Z₀. The range of erodibility parameters Z₀ of high-plasticity clay and low-plasticity clay is mainly concentrated in 0.5–3.31 mm h⁻¹ and 1–8.94 mm h⁻¹, respectively (Figure 5b). The range of erodibility parameters Z₀ of low-plasticity is higher than the range of erodibility parameters Z₀ of high-plasticity. Therefore, b₀ has a similar relationship with z₀ relative to the liquid limit (Figure 4a and Figure 5b).

Compared with the erodibility parameter b₁ of high-plasticity clay, the erodibility parameter b₁ of low-plasticity clay is lower. The erodibility parameter b₁ decreased as the liquid limit increased due to increasing soil cohesion (which increases fc). The range of erodibility parameters b₁ of high-plasticity is mainly concentrated in 20.33–97.82 Pa, whereas the range of parameters b₁ of low-plasticity clay is concentrated in 0.44–50.2 Pa (Figure 4b). Similar observations are reported for τ_c. The τ_c increase as the liquid limit increase due to increasing cohesion. The range of the τ_c of high-plasticity clay experimental data is mainly concentrated in 1.23–8.78 Pa, whereas the range of the critical shear stress of low-plasticity clay is primarily concentrated in 0.5–4.43 Pa (Figure 5a). We can find that the range of erodibility parameters τ_c of low-plasticity is lower than the range of erodibility parameters τ_c of high-plasticity. Therefore, b₁ has a similar relationship with τ_c relative to the liquid limit (Figure 4a and Figure 5b).

4.2. Relation between z₀ and τ_c and between b₀ and b₁

As it is meaningful to explore whether there is a relationship between Z₀ and τ_c and between b₀ and b₁, all datasets from each author have been plotted individually (Figure 6 and Figure 7). When the research data samples are small for the over-shear stress model, the discreteness between parameter Z₀ and over-shear stress τ_c is significant (Figure 6a,b). In the case of sufficient data, the results of Shafii (2018) and Briaud (1999, 2008) showed that there was a direct proportional linear relationship between the parameter Z₀ and τ_c. Additionally, this result has been confirmed in the research results of Hanson and Simon (2001), as shown in Figure 6c,d.

When the Wilson model is studied, this trend does not exist between b₀ and b₁. As shown in Figure 7a, b₁ and b₀ are highly discrete. While the number of data increases, the study samples of Gao (2019), Briaud (1999, 2008), and ILIT (2006) show an inverse proportional linear between the two parameters (Figure 7b). Criswell et al. (2016) also found an inverse relationship between b₀ and b₁ for gravels. However, this relationship is not confirmed in the studies of Shafii (2018), Straud (2013), and Govin (2009) (Figure 7c), which proves that there is no such relationship between parameters b₀ and b₁.

4.3. Multiple Nonlinear Regression Analysis

The multiple nonlinear model is used to study equations relating the erodibility parameters b₀ and Z₀ to the soil geotechnical properties in this research. The multiple no-nlinear model is shown in Equation (11):

$$Y=A\times {\left(P_1\right)}^{{\alpha }_1}\times {\left(P_2\right)}^{{\alpha }_2}\times {\left(P_3\right)}^{{\alpha }_3}\times \cdot \cdot \cdot \times {\left(P_n\right)}^{{\alpha }_n}$$

(11)

where Y is the dependent variable; P_n is the soil geotechnical properties, including plasticity index, percent fines, wet density, water content, and mean grain size in this case. Many combinations between the erodibility parameters and the soil geotechnical properties can be selected to generate regression equations.

The correlation analysis of soil parameters and scour parameters in the database is carried out. The analysis results are shown in the following table. The correlation coefficient between soil parameter PI, WC, D₅₀, and scour parameters is significant, so these three parameters can be used to analyze the relationship with scouring parameters.

According to the results of correlation analysis, we conducted regression analysis on WC, PI, and D₅₀, with R² and F test indexes as evaluation indexes.

Table 2. Correlation analysis results of scouring parameters and soil parameters.

Parameter	b0	b1	Z0	${\mathit{\tau }}_{\mathit{c}}$	$\mathit{\alpha }$
PI	−0.392 **	0.004	−0.231	0.056	−0.371 **
${\rho }_s$	−0.172	−0.107	−0.316 **	−0.069	−0.085
WC	0.141	0.393 **	0.304 **	0.333 **	0.593 **
Pf	−0.032	0.130	−0.081	0.043	0.077
D50	0.439 **	−0.114	0.391 **	−0.116	0.076

Note: The number in the table represents the correlation between parameters, ** represents significant correlation, and the significance level is 0.01.

summarizes the best regression equations for the erodibility parameters (b₀, α, and Z₀) of cohesive soils. According to

Table 3. Results of multiple nonlinear regression.

Dependent Variable	Model Expression	Number of Data	R2	Fvalue/ Fstatistic
b0 (g m−1 s−1 N−0.5)	b0 = 3.66 × WC1.65 × PI−0.53 × D500.72	54	0.63	10.82
Z0 (mm h−1)	z0 = 0.123 × WC2.37 × PI−0.86 × D500.39	54	0.63	9.23
$\alpha$	$\alpha = 0$.094 × WC1.26 × PI−0.19 × D500.25	54	0.67	92.21

Note: D₅₀ (mm)—mean particle size; WC (%)—water content; PI (%)—plastic index.

, the relationship between parameters b₀, α, and Z₀ with the selected soil parameters is similar. They are inversely proportional to the plasticity index but directly proportional to water content and median grain size, which is consistent with the results of correlation analysis in Table 2. Figure 8a–c shows the comparison of predicted and measured values of the three parameters.

5. Conclusions

A database of 177 EFA tests of cohesive soils with soil properties is present. The database is used to derive the erodibility parameters (Z₀, τ_c, and α) of the dimensionless excess shear stress model and the erodibility parameters (b₀ and b₁) of the Wilson model to obtain a new database of 177 erodibility parameters (Z₀, τ_c, α, b₀, and b₁). The new database is used to determine the range of all erodibility parameters (Z₀, τ_c, α, b₀, and b₁). The parameters Z₀, τ_c, and α values of cohesive soils ranged from 0.10 mm h⁻¹ to 117.23 mm h⁻¹, from 0.11 Pa to 50.25 Pa, and from 0.26 to 2.75, respectively. The parameters b₀ and b₁ of cohesive soils ranged from 0.09 g m⁻¹ s⁻¹ N^−0.5 to 50.25 g m⁻¹ s⁻¹ N^−0.5 and from 0.56 Pa to 267.26 Pa, respectively.

The new databases were categorized according to the USCS (Briaud, 2013) classifications [49]. It found that the range of Z₀, τ_c, b₀ and b₁ values of high-plasticity clay is mainly concentrated in 0.5–3.31 mm h⁻¹, 1.23–8.78 Pa, 0.44–2.77 g m⁻¹ s⁻¹ N^−0.5, and 20.33–97.82 Pa, respectively, whereas the range of Z₀, τ_c, b₀, and b₁ values of low-plasticity clay are mainly concentrated in 1–8.94 mm h⁻¹, 0.5–4.43 Pa, 1.35–4.6 g m⁻¹ s⁻¹ N^−0.5, and 0.44–50.2 Pa. The experimental values obtained from the low-plasticity clay show that the τ_c and b₁ values are lower than the high-plasticity clay values. In comparison, their Z₀ and τ_c values are higher than the high-plasticity clay values. Therefore, erodibility parameters b₀ and b₁ have a similar relationship with Z₀ and τ_c relative to the liquid limit, respectively.

The new databases are also used to develop regression equations that link the erodibility parameters (Z₀ and b₀) to the index soil parameters, such as the mean grain size, the water content, the percent fines, the plasticity index, the bulk density, and the plastic limit. It is found that z₀ and b₀ values consistently decreased when the plasticity index increased, while water content and mean grain size decreased. Therefore, the Wilson model parameter b₀ had a similar relationship but a different magnitude, as parameter Z₀ relative to the water content, the mean grain size, and the plasticity index. These results apply to EFA devices and may not be applicable to other devices such as jet erosion tests and hole erosion tests.