# The Theoretical Approach to the Modelling of Gully Erosion in Cohesive Soil

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Gully Erosion by Water

_{s}= Q C is the sediment transport rate (m

^{3}s

^{−1}), Q is the water discharge (m

^{3}s

^{−1}); X is the longitudinal co-ordinate (m); C is the mean volumetric sediment concentration; C

_{w}is the sediment concentration of the lateral input; q

_{w}is the specific lateral discharge (m

^{2}s

^{−1}); E is the erosion rate or the mean particle detachment rate (m s

^{−1}); E

_{b}—bank erosion rate (m s

^{−1}); W is the flow width (m); d is the bank height (m); and U

_{f}is the sediment particles’ falling velocity in the turbulent flow (m s

^{−1}).

_{b}is related to the rate of erosion E of the gully bed [10]. Therefore, the main term in Equation (1) is the erosion rate E.

_{0}or the increase in volume ΔV eroded from the area S during the time interval T.

_{ai}with correction on soil porosity ε:

_{i}and the duration of particle detachment τ

_{i}, Formula (2) takes the form:

_{ai}/s

_{i}τ

_{i}represents the instant and local rate of particle detachment α, and the term s

_{i}τ

_{i}/ST represents the probability density of this particle detachment. Therefore [11]:

_{α}is the spatial-temporal probability density function (PDF) for α. Equations (4) and (5) show that the mean soil erosion rate is the result of spatial (on the area S) and temporal (during the period T) averaging of instant and local detachment rates α. The detachment rate α possess all positive values, including zero (“detachment” of stable particles).

#### 2.2. Instant and Local Rate of Soil Particle Detachment

_{dr}exceed the maximum of the resisting forces F

_{res}and the resultant force Θ is more than zero [12]:

_{s}is soil density.

#### 2.3. Probability Density Function for the Rate of Detachment

_{1}, x

_{2}, … x

_{n}) [14]:

_{1}, … x

_{n}) is equal to the product of the PDFs of each of the random variables.

#### 2.4. The Main Resistance and Driving Forces

_{b}.

_{d}) and lift (F

_{l}) forces, which are parameterizations of longitudinal and vertical pressure gradients:

_{R}is the coefficient of drag resistance; C

_{y}is the coefficient of uplift; U is the instant near-bed flow velocity; ρ

_{s}and ρ are the soil aggregate density and water density, respectively; S

_{d}is the area of aggregate cross-section exposed perpendicular to flow; and S

_{a}is the cross-section area of the soil aggregate parallel to the flow (vertical projection). The direction of the sum of drag and lift forces (the angle with the mean soil surface) is:

_{U}is $\frac{\rho D}{4{\rho}_{s}{V}_{a}}\left({S}_{a}{C}_{y}+{S}_{d}{C}_{R}\right)$, k

_{D}is $\frac{g\left({\rho}_{s}-\rho \right)}{2{\rho}_{s}}\mathrm{cos}\gamma \mathrm{sin}\beta $ and k

_{C}= $\frac{D{S}_{b}}{2{\rho}_{s}{V}_{a}}$. If the expression under the square root is equal to zero or negative, then α = 0. For this case, Equation (16) is a generalized form (with cohesion) of a well-known expression for incipient motion criteria (see, for example, [16]).

_{U}, k

_{D}and k

_{C}. In the case of turbulent flow, C

_{y}, C

_{R}and k

_{U}are constants [15]. If we assume a simple shape and composition of soil particles (aggregates), then the coefficient k

_{D}is also constant. Then, Equation (16) contains three independent stochastic variables: flow velocity U, soil particle size D and force of cohesion k

_{C}C, and Equation (10) takes the form:

_{U}U), p(k

_{D}D) and p(k

_{C}C) are probability density functions for flow velocity, soil particles vertical size and cohesion, respectively. In numerical calculations, the continuous quantities are replaced by discrete one, infinitesimal by finite, and integrals by sums: ${{\displaystyle \int}}_{Xmin}^{Xmax}X\mathrm{d}X\to \Delta X{\displaystyle \sum}_{iX=1}^{nX}{X}_{iX}$.

#### 2.5. Probability Density Function for the Factors of Soil Erosion

_{mean}is the mean velocity and σ is the standard deviation of velocity fluctuations. For small mean velocities, the function p(U) can be asymmetrical.

_{*}increases from the flow surface to the bottom [19];

_{y}= 0.1 and C

_{R}= 0.42; then k

_{U}= $\left(0.1{S}_{a}+0.42{S}_{d}\right)$.

_{m}and standard deviation σ

_{D}, are

_{D}=$\mathrm{cos}\mathsf{\gamma}\text{}\frac{g\left({\rho}_{s}-\rho \right)}{2{\rho}_{s}}\mathrm{sin}\left[\mathrm{arctan}\left(0.238\frac{{S}_{a}}{{S}_{d}}\right)\right]$.

#### 2.6. The Algorithm of Erosion Rate Calculation

- The probability density p(k
_{C}C) in the part of the cohesion PDF where resistance forces are less than driving forces decreases due to the erosion E_{iC}of soil with particular cohesion C_{iC}$${E}_{iC}=\frac{1}{1-\epsilon}{{\displaystyle \int}}_{Umin}^{Umax}p\left(\sqrt{{k}_{U}}U\right)\mathrm{d}\sqrt{{k}_{U}}U{{\displaystyle \int}}_{Dmin}^{Dmax}p\left({k}_{D}D\right)\mathrm{d}{k}_{D}D\sqrt{{k}_{U}{U}^{2}-{k}_{D}D-{k}_{C}C}$$The initial PDF transforms into intermediate PDF (p*)$${p}^{*}\left[{\left({k}_{C}C\right)}_{iC},{t}_{i+1}\right]\mathrm{d}{k}_{C}C=p\left[{\left({k}_{C}C\right)}_{iC},{t}_{i+1}\right]\mathrm{d}{k}_{C}C\left[1-\frac{{E}_{iC}}{{D}_{m}}\left({t}_{i+1}-{t}_{i}\right)\right]$$ - Simultaneously, the intermediate PDF of cohesion is transformed due to the exposition of fresh initial soil in the “windows” of the eroded surface layer to PDF of armored soil (p
_{a})

_{0}(${k}_{C}$C) is the PDF of cohesion in the initial soil. This process leads to the increase in the proportion of the surface covered by stable soil aggregates, and an increase in the mean soil cohesion. The rate of erosion decreases through time when armoring is the prevailing process.

_{b}between and within soil aggregates over time. The rate of failure of links in soils can be described with the common exponential failure function [25].

_{i+}

_{1}is the product of the contact area at time T

_{i}and the failure function (Equation (26))

#### 2.7. The Materials for Comparison of Calculations with Measurements

_{s}≈ $\sqrt{\frac{4{S}_{a}}{\pi}}$ was 1.83 mm, with a standard deviation σ

_{D}= 0.88 mm. The aggregates were flattened, to a plate or ellipsoidal shape, with $\frac{{D}_{m}}{{D}_{s}}$ ≈ 1/3. The mean soil strength, measured with a tor-vane, was nearly equal for parent loess and topsoil: 52 and 51 kPa, respectively. The soil density ρ

_{s}was 1460 and 1230 kg/m

^{3}, respectively.

_{U}for ellipsoidal aggregates is $\frac{3\rho}{16{\rho}_{s}}\left(0.1+0.42\frac{{S}_{d}}{{S}_{a}}\right)$ and varied in the range 0.026–0.04 for the first plot and 0.031–0.047 for the second (depending on S

_{a}/S

_{d}ratio in the range 2–4). The standard deviation of velocity fluctuations was ${\sigma}_{U}\cong 2.1{U}_{*}$ (according to [19]).

_{D}= $\mathrm{cos}\mathsf{\gamma}\text{}\frac{g\left({\rho}_{s}-\rho \right)}{2{\rho}_{s}}\mathrm{sin}\left[\mathrm{arctan}\left(0.238\frac{{S}_{a}}{{S}_{d}}\right)\right]$ was in the range 0.44–0.64 for the first plot and 0.24–0.35 for the second.

_{C}= $D\frac{{S}_{b}}{2{\rho}_{s}{V}_{a}}=\frac{3}{8{\rho}_{s}}\frac{{S}_{b}}{{S}_{a}}\approx \frac{3\left(1-{\epsilon}_{}^{2/3}\right)}{8{\rho}_{s}}$ was 2.04 × 10

^{−4}for the first plot and 1.54 × 10

^{−4}for the second. The variation coefficient for cohesion was estimated as 0.59 in the experiments with the cutting of the surface of the soil sample with a moving blade [24].

## 3. Results

#### 3.1. General Numerical Experiments

_{E}, where the driving forces were greater than the resistance forces, corresponded with the trends in erosion rate. The influence of the soil particle size on the erosion rate was much smaller than that of cohesion, and is thus not discussed further.

_{E}(Figure 4C). The values of probability density (PD) are very small here (Figure 5B). Therefore, numerical experiments focusing on this segment require precise calculations.

#### 3.2. The Comparison of Calculated Erosion Rates with the Measured

_{C}C and parameters λ in the failure function. Examples of such fields, represented by isolines of erosion rate E, are shown in Figure 6 for two runs in plot 1. The soil erosion rate is characterized by equifinality. The calculation for each pair of k

_{C}C and λ at a given flow velocity leads to a unique value of the resulting erosion rate E. At the same time, calculations for different pairs of initial soil cohesion and parameter λ at a given flow velocity lead to the same resulting erosion rate, forming the isoline in Figure 6. The same measured rate of erosion at a given flow velocity, shown by bold dashed isolines, can also appear with different combinations of soil properties. This set of numerical experiments shows that the result of a single run with measured characteristics of flow and erosion rate cannot be used to find indefinite soil properties (initial soil cohesions k

_{C}C and parameters λ), which are used in the proposed model.

_{C}C and λ. The intersections of erosion rate isolines narrow the range of such combinations for the set of experiments on the same soil with different flow velocities (green borders at Figure 7 and Figure 8). The soil at plot 1 is characterized by a cohesion force k

_{C}C in the range of 12–14 m

^{2}/s

^{2}(or C = 60–70 kPa), a parameter λ in the range of 0.0025–0.004 and the percentage of the area, occupied by erodible soils after 3600 s of the armoring–loosening cycles, in the range of 0.55–0.7%.

_{C}C in the range of 27–31 m

^{2}/s

^{2}(or C = 180–200 kPa), a parameter λ in the range of 0.0015–0.002, and the percentage of the area, occupied by erodible soils after 3600 s of the armoring–loosening cycles, in the range of 0.35–0.55% per second.

## 4. Discussion

_{U}, k

_{D}and k

_{C}. The mathematical methods for the calculation of the PDF of the function of stochastic variables, knowing the PDFs of these stochastic variables [14], are used. The relationships between the main factors of erosion (for example, flow velocity) and erosion rate are theoretically determined within the model for a given combination of input data. This theory opens up a new way for better understanding the experimental results of soil erosion, and demonstrates the direction for future investigations. It is possible to estimate the main soil characteristics used in the model by gully erosion simulation in flume or direct measurements of the gully erosion rate in the field with a variety of flow velocities in the same soil. The results of numerical experiments are consistent with the measurements in the laboratory flumes [15] and show the same stages of erosion development in cohesive soils.

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Algorithm of the calculation of the gully erosion rate with the stochastic model. The explanation is given in the text.

**Figure 2.**Experimental site at the Ballantrae Hill Country Research Station near Woodville, New Zealand.

**Figure 3.**The relationship between flow velocity U and erosion rate E for plot 1 (circles) and plot 2 (triangles) in the experiments in Ballantrae Hill Country.

**Figure 4.**Change through time in calculated specific soil cohesion force k

_{C}C (

**A**), erosion rate E (

**B**) and the percentage of the area occupied by erodible soils after the armoring–loosening cycles P

_{E},% (

**C**) for different flow velocities. The initial ${k}_{C}$ is 17, the parameter of loosening λ = 0.001, and other parameters were obtained from plot 2.

**Figure 5.**Temporal changes in the PDF of soil cohesion for a flow velocity of 2.15 m/s. (

**A**) PDF of driving force (green line) and PDFs of cohesion force, initial (red line) and at the stage of rapid changes (black lines). Two modes show two stage of stabilization. (

**B**) Temporal changes in probability density of cohesion force for k

_{C}C = 4 m

^{2}/s

^{2}at the segment, overlapping with the PDF of driving forces. (

**C**,

**D**) Changes in the probability density of cohesion force for k

_{C}C = 15–20 and 30 m

^{2}/s

^{2}. Numerical experiments for initial ${k}_{C}C$ = 30, parameter of loosening λ = 0.0015, and other parameters obtained from plot 2.

**Figure 6.**The fields in isolines of erosion rate E (m/s), calculated for varying initial cohesion ${k}_{C}C$ and the parameter of loosening λ for two flow velocities from plot 1: (

**A**) 1.4 m/s and (

**B**) 1.79 m/s. The measured rate of erosion at a given flow velocity is shown by a bold dashed isoline.

**Figure 7.**The possible combinations of initial specific soil cohesion force (k

_{C}C, m

^{2}/s

^{2}) with parameter λ in the failure function, which lead to the same resulting erosion rate at a given flow velocity (

**A**), and of the percentage of the area, occupied by erodible soils after 3600 s of the armoring–loosening cycles P

_{E},% (

**B**). The green box shows the possible range of soil properties, which fits the measured flow velocity and erosion rate in plot 1.

**Figure 8.**The possible combinations of initial specific soil cohesion force (k

_{C}C, m

^{2}/s

^{2}) with parameter λ in the failure function, which lead to the same resulting erosion rate at a given flow velocity (

**A**), and of the percentage of the area, occupied by erodible soils after 3600 s of the armoring–loosening cycles P

_{E},% (

**B**). The green box shows the possible range of soil properties, which fits the measured flow velocity and erosion rate in plot 2.

N Plot–Soil–Run | Q, l/s | U, m/s | W, m | d, m | U_{*} | σ_{U} | E, m/s | Re | Fr |
---|---|---|---|---|---|---|---|---|---|

1–b2–3 | 3.13 | 1.19 | 0.46 | 0.0057 | 0.12 | 0.26 | 2.29 × 10^{−7} | 6020 | 5.0 |

1–b2–4 | 4.01 | 1.30 | 0.46 | 0.0067 | 0.13 | 0.28 | 2.33 × 10^{−7} | 7670 | 5.1 |

1–b2–5 | 5.12 | 1.40 | 0.47 | 0.0077 | 0.14 | 0.30 | 2.29 × 10^{−7} | 9510 | 5.1 |

1–b2–6 | 6.86 | 1.53 | 0.49 | 0.0092 | 0.15 | 0.33 | 5.49 × 10^{−7} | 12,340 | 5.1 |

1–b2–7 | 11.13 | 1.79 | 0.51 | 0.0121 | 0.17 | 0.38 | 1.66 × 10^{−6} | 19,070 | 5.2 |

2–b4–3 | 1.48 | 1.20 | 0.28 | 0.0043 | 0.14 | 0.31 | 3.13 × 10^{−8} | 4590 | 5.8 |

2–b4–4 | 2.22 | 1.56 | 0.29 | 0.0049 | 0.15 | 0.33 | 4.8 × 10^{−8} | 6750 | 7.1 |

2–b4–5 | 3.04 | 1.66 | 0.29 | 0.0062 | 0.17 | 0.38 | 8.54 × 10^{−8} | 9100 | 6.7 |

2–b4–6 | 4.01 | 1.91 | 0.30 | 0.0070 | 0.18 | 0.40 | 1.4 × 10^{−7} | 11,810 | 7.3 |

2–b4–7 | 5.92 | 2.15 | 0.31 | 0.0089 | 0.21 | 0.46 | 3.49 × 10^{−7} | 16,830 | 7.3 |

_{*}—kinematic velocity, σ

_{U}—velocity fluctuation standard deviation (from Equation (19)), E—erosion rate, Re—Reynolds number, Fr—Froude number.

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Sidorchuk, A.
The Theoretical Approach to the Modelling of Gully Erosion in Cohesive Soil. *Earth* **2022**, *3*, 228-244.
https://doi.org/10.3390/earth3010015

**AMA Style**

Sidorchuk A.
The Theoretical Approach to the Modelling of Gully Erosion in Cohesive Soil. *Earth*. 2022; 3(1):228-244.
https://doi.org/10.3390/earth3010015

**Chicago/Turabian Style**

Sidorchuk, Aleksey.
2022. "The Theoretical Approach to the Modelling of Gully Erosion in Cohesive Soil" *Earth* 3, no. 1: 228-244.
https://doi.org/10.3390/earth3010015