Study on Lateral Water Migration Trend in Compacted Loess Subgrade Due to Extreme Rainfall Condition: Experiments and Theoretical Model

Abstract

Water migration occurs in unsaturated loess subgrade due to extreme rainfall, making it prone to subgrade subsidence and other water damage disasters, which seriously impact road safety and sustainable development of the Loess Plateau. The study performed a rainfall test using a compacted loess subgrade model based on a self-developed water migration test device. The effects of extreme rainfall on the water distribution, wetting front, and infiltration rate in the subgrade were systematically explored by setting three rainfall intensities (4.6478 mm/h, 9.2951 mm/h, and 13.9427 mm/h, namely J1 stage, J2stage, and J3 stage), and a lateral water migration model was proposed. The results indicated that the range of water content change areas constantly expands as rainfall intensity and time increase. The soil infiltration rate gradually decreased, and the ratio of surface runoff to infiltration rainfall increased. The hysteresis of lateral water migration refers to the physical phenomenon in which the internal water response of the subgrade is delayed in time and space compared to changes in boundary conditions. The sensor closest to the side of the slope changed first, with the most significant fluctuations. The farther away from the slope, the slower the response and the smaller the fluctuation. The bigger the rainfall intensity, the faster the wetting front moved horizontally. The migration rate at the slope toe is the highest. The migration rate of sensor W3 increased by 66.47% and 333.70%, respectively, in the J3 stage compared to the J2 and J1 stages. The results of the model and the measured data were in good agreement, with the R² exceeding 0.90, which verifies the reliability of the model. The study findings are important for guiding the prevention and control of disasters caused by water damage to roadbeds in loess areas.

1. Introduction

Highway construction in the Loess Plateau is driving local economic development. This is due to the implementation of Western development and the “Belt and Road” strategy. The Loess Plateau is covered by a large amount of loess [1]. The shallow loess on the surface is the main material of the subgrade. Loose structure, macropores, and susceptibility to erosion are loess’s characteristics [2,3,4]. This leads to water damage disasters such as subgrade subsidence, which are prone to occur after use of the highway under the harsh local natural climate conditions [5]. This can also cause chain disasters such as ecological degradation, which have a serious impact on the sustainable development of the Loess Plateau. Water migration is the main reason for subgrade water damage. In addition, global warming has led to an increase in extreme rainfall events in the Loess Plateau region [6]. Under extreme rainfall conditions, water in the subgrade soil is continuously increased due to water infiltration. This leads to instability of the soil after saturation, exacerbating the occurrence of water damage disasters such as roadbed subsidence [7,8]. This seriously affects the safe operation and service life of highways and restricts regional economic construction and development.

Many scholars have studied soil water migration through indoor or field tests and obtained the law of water migration. Tang et al. [9] found that the dominant channels of rainfall infiltration improve soil permeability. This caused the wetting front to move downwards overall. Zhang et al. [10] developed a model device to test water migration. It was observed that the rainfall infiltration lines of the loess column rose in dry density and took the shape of a “Y”, “D”, or “∧” under light, heavy, and rainstorm conditions. Liu et al. [11] created an unsaturated infiltration model. The infiltration depth and rate of slopes under rainfall infiltration can be derived by analyzing the initial water content changes. Sun et al. [12] performed field experiments on slopes. The results showed that the maximum infiltration depth was 0.6 m following two consecutive days of heavy rainfall. The faster infiltration and seepage velocities were observed at the top and foot of the slope. Zhang et al. [13] believe that the water content of loess decreases with rising depth. Water content change lags over time. Lin et al. [14] conducted one-dimensional vertical seepage simulation experiments in loess columns to analyze the influence of water and salt transport patterns on loess. The water migration process on loess slopes under freeze–thaw cycles was studied by Xu et al. [15]. Ren et al. [16] found that cracks at the slope crest significantly affect the infiltration speed of the wetting front. A strong linear correlation was observed between the average migration depth of the horizontal wetting front at the slope crest and the parallel wetting front on the slope surface. Shen et al. [17] explored the infiltration law of water in in situ loess under multiple heavy rainfall events and established a wet front movement model based on monitoring data.

Currently, research on water migration of subgrades during rainfall mainly focuses on vertical infiltration and normal rainfall environments [12,13]. However, research on the water migration of subgrades in extreme rainfall environments and horizontal infiltration directions is limited. The study performed a rainfall test using a compacted loess subgrade model to explore the effects of extreme rainfall on the water distribution, wetting front, and infiltration rate in the subgrade based on a self-developed water migration test device. The spatiotemporal laws of lateral water migration were summarized, and a lateral water migration model was proposed and validated. The study findings are important for guiding the prevention and control of disasters caused by water damage to roadbeds in loess areas.

2. Materials and Methods

2.1. Test Materials

Loess was used (Chang’an District, Xi’an City, Shaanxi Province) in the tests. Loess’s physical properties are shown in

Table 1. Loess physical properties.

Dry Density (g/cm3)	Maximum Dry Density (g/cm3)	Water Content w (%)	Optimal Water Content (%)	Specific Gravity	Liquid Limit (%)	Plastic Limit (%)	Permeability Coefficient (cm/s)
1.42	1.62	15.87	17.00	2.71	31.10	17.46	1.12 × 10−5

.

2.2. Test Device

The water migration testing device consists of a model box, rainfall device, rainwater collection device, and data acquisition device. The device is shown in Figure 1. The model box outer size is 1050 mm × 330 mm × 455 mm (length × width × height), made of polymethyl methacrylate (PMMA). It is highly transparency and lightweight. The bottom of the box is equipped with drainage holes with a diameter of 20 mm. The rainfall device consists of rain racks, rotating atomizing nozzles, water pumps, Pu hoses, water storage tanks, etc. The rotary atomizing nozzle can rotate to adjust the water output, and the rainfall uniformity reaches 86%. The rainwater collection device is located at the base of the box for collecting runoff. The data acquisition device consists of MTD-05 soil water sensors, a CR1000X data acquisition instrument, a computer, etc. The MTD-05 water sensor was used to monitor water migration in the test, with external dimensions of 95 mm × 20 mm × 11 mm, a range of 0–100%, and an accuracy of ±3%. The sensor indirectly obtained the soil volumetric water content through the Frequency Domain Reflectometry (FDR) method [18].

2.3. Test Method

The prototype had a subgrade slope height of 4.5 m and width of 15 m. Geometric dimensions (I), dry density (ρ), and gravitational acceleration (g) were selected as basic physical quantities [19], considering the limitations of indoor sites and reducing the difficulty of sample preparation [19,20,21]. The model is a homogeneous loess subgrade, and the similarity ratios of I, ρ, and g between the subgrade prototype and model were set to 15:1, 1:1, and 1:1, namely C_I = 15, C_ρ = 1, and C_g = 1 [21]. The similarity relationship of model experiments was derived according to the dimensional analysis theory and similarity criteria, as shown in

Table 2. Similarity relationship test.

Parameters	Physical Quantity	Dimension	Similarity Relation	Similarity Ratio
physical dimension	I	[L]	$C_I$	15
permeability coefficient	k	[L][T]−1	$C_k=C_I^{{\displaystyle \frac{1}{2}}}{C}_g^{{\displaystyle \frac{1}{2}}}$	$\sqrt{15}$
water content	ω	-	$C_{\omega }=1$	1
rainfall intensity	q	[L][T]−1	$C_q=C_I^{{\displaystyle \frac{1}{2}}}\times C_g^{{\displaystyle \frac{1}{2}}}$	$\sqrt{15}$
rainfall duration	t	[T]	$C_t={\displaystyle \frac{C_I^{\frac{1}{2}}}{C_g^{\frac{1}{2}}}}$	$\sqrt{15}$

. The optimal water content and the corresponding dry density are set as the initial values of the model, and the compaction degree of the model is 92% [20,22]. The model is a homogeneous loess subgrade with an initial volumetric water content of 25% and slope angle of 34° [23,24]. According to the distance between the soil and the top of the slope, the model is divided into three layers: L1 (200~300 mm), L2 (100~200 mm), and L3 (0~100 mm). There are 12 sensors, with their burial locations shown in Figure 2. The model parameters are shown in

Table 3. Model parameters.

Maximum Dry Density (g/cm3)	Optimal Water Content (%)	Compaction Degree (%)	Slope Ratio	Size (Length × Width × Height) (mm)
1.62	17.00	92	1:1.5	950 × 300 × 300

. Extreme rainfall was defined by the threshold method [25]. The threshold value of extreme daily rainfall is determined to be 30 mm based on references and meteorological data of Xi’an [25]. Rainfall intensity was designed to be 72 mm/24 h, 144 mm/24 h, and 216 mm/24 h, respectively. Based on the characteristics and duration of extreme rainfall in Xi’an city [25,26,27,28,29] the rainfall duration is set to 4 h, and the standing duration is set to 20 h. The model rainfall intensities are 4.6478 mm/h, 9.2951 mm/h, and 13.9427 mm/h. We investigated three intermittent extreme rainfall working conditions (4.6478 mm/h, 9.2951 mm/h, and 13.9427 mm/h), namely J1, J2, and J3. Each working condition takes 72 h, and the cycle of “4 h rainfall period + 20 h static period” is repeated three times. The experimental rainfall is only on one side of the slope. The test plan is shown in

Table 4. Test plan.

Test Stage	Rainfall Intensity R (mm/h)	Duration of Rainfall D (h)	Duration of Standing S (h)
J1-1	4.6478	4	20
J1-2	4.6478	4	20
J1-3	4.6478	4	20
J2-1	9.2951	4	20
J2-2	9.2951	4	20
J2-3	9.2951	4	20
J3-1	13.9427	4	20
J3-2	13.9427	4	20
J3-3	13.9427	4	20

.

The soil water content is collected by using data acquisition instruments. The data were collected every 30 min. The infiltration rate of rainfall was calculated using Formulas (1) and (2):

$$Q_i=Q-Q_{\mathrm{r}}$$

(1)

$$A={\displaystyle \frac{Q_i}{T\times S^{\prime }}}$$

(2)

where Q_i—the infiltration water volume, kg; Q—the rainfall amount, kg; Q_r—the runoff water volume, kg; A—the water injection infiltration rate per unit area per unit time, kg/h·m² (when water density is 1000 kg/m³, kg/h·m² is equal to mm/h); T—the duration of rainfall, h; and S′—the rainfall area, m².

The wet front is the interface between the wet and dry zones within the soil. It can cause a change in the water content when the wet front moves to a certain location. The horizontal movement rate of the wet front in the test was calculated using Formula (3)

$$v={\displaystyle \frac{\Delta d}{\Delta t}}$$

(3)

where Δd—the horizontal migration distance of the wetting front (distance between two adjacent sensors in the horizontal direction), mm; and Δt—the time of wetting front migration (the absolute value of the difference in the time when two adjacent sensors start to change in the horizontal direction), s.

3. Results

3.1. Extreme Rainfall Effect on the Water Distribution

The rainfall intensity directly affects the soil water content during infiltration. The water will redistribute its profile during infiltration, which has an important impact on the study of water migration in roadbed soil [30]. Figure 3 shows the spatiotemporal variation in soil water content under three different rainfall intensities. The impact range of a single rainfall event on the water content is small. The range of water content change areas constantly expands as rainfall intensity and time increase. Water content increases from the initial value to the peak value, and the peak water content eventually stabilizes at a certain value. The peak water content of sensor W4 increased by 1.42% and 13.12%, respectively, as the rainfall intensity was 13.9427 mm/h, after the third rainfall compared to the second and first rainfalls. The hysteresis of lateral water migration refers to the physical phenomenon in which the internal water response of the subgrade is delayed in time [31] and space compared to changes in boundary conditions. The main manifestation is that (a) the sensor closest to the side of the slope changes first, and the water fluctuation is most obvious. The farther the distance, the slower the response and the smaller the fluctuation. (b) multiple rainfall has a significant influence on the soil water content far from the slope compared to a single rainfall event. At the same location, there is a situation where the peak water content occurs after the rain stops. The cause of this is clear: the shallow soil of the slope is still relatively dry. It has a strong infiltration ability in the early stage of the first rainfall. Once rainwater begins to infiltrate, a high suction gradient will be formed, causing a sharp increase in the water content near the slope side. There is a lack of water supply when the rainfall stops, and coupled with surface evaporation from the atmosphere, the water gradually decreases. On the other hand, the soil layer in the saturated or transient saturated zone cannot retain all the water. A portion of rainwater will continue to infiltrate into the slope [30], resulting in a gradual decrease in water. Horizontal water migration is affected by water supply, original water content, and distance. The degree of hysteresis effect varies at different locations. The farther away from the slope, the more difficult it is for water to reach. Therefore, in some areas, the peak value is only reached after a period of rest after the rainfall is completed.

3.2. Extreme Rainfall Effect on the Wetting Front

Table 5. Horizontal migration rate of wet front.

Number	J1 Stage		J2 Stage		J3 Stage
	Start Change Time (h)	Rate (mm/s)	Start Change Time (h)	Rate (mm/s)	Start Change Time (h)	Rate (mm/s)
W1	1.5	2.78 × 10−2	0.5	8.33 × 10−2	0.5	8.33 × 10−2
W2	3.5	2.08 × 10−2	1.0	8.33 × 10−2	1.0	8.33 × 10−2
W3	10.0	6.41 × 10−3	3.5	1.67 × 10−2	2.5	2.78 × 10−2
W4	25	2.78 × 10−3	10.5	5.96 × 10−3	9.5	5.96 × 10−3
W5	50.0	1.67 × 10−3	27.0	2.53 × 10−3	24.5	2.78 × 10−3
W6	1.5	2.78 × 10−2	0.5	8.33 × 10−2	0.5	8.33 × 10−2
W7	4.0	1.67 × 10−2	1.5	4.17 × 10−2	1.5	4.17 × 10−2
W8	27.5	1.77 × 10−3	6.0	9.26 × 10−3	5.5	1.04 × 10−2
W9	51.0	1.77 × 10−3	25.5	2.14 × 10−3	24.5	2.19 × 10−3
W10	2.5	1.67 × 10−2	1.0	4.16 × 10−2	0.5	8.33 × 10−2
W11	28.5	2.63 × 10−3	6.5	7.58 × 10−3	3.5	1.39 × 10−2
W12	52.0	1.74 × 10−3	28.0	1.94 × 10−3	24.5	1.98 × 10−3

shows the horizontal migration rate of the wetting front under three different rainfall intensities. A sketch map of the migration trajectory of wetting front is plotted based on Table 5, as shown in Figure 4. It can be seen that each sensor starts to change at different times. Sensors with the same depth but different positions start to change at different times, indicating that the horizontal migration rate of the wetting front is different. The migration rate increases with the increase in rainfall intensity. The migration rate of sensor W3 increased by 66.47% and 333.70%, respectively, in the J3 stage compared to the J2 and J1 stages. Among them, the horizontal migration rate in the middle and foot of the slope is higher. The lowest is at the top of the slope. This is because only the slope receives rainfall. The infiltration gradually occurs from the slope surface to inside the subgrade. Sensors near the slope are the first to sense water fluctuations and changes. The further away from the slope, the slower the start time of the change [32]. Therefore, the wetting front moves more slowly. The soil on the slope surface becomes saturated, resulting in surface runoff as the intensity and times of rainfall increase. At the slope toe, surface runoff is collected, which leads to water accumulation and a higher infiltration rate. Therefore, the horizontal infiltration rate in the middle slope and slope toe is relatively high, indicating a faster migration rate of the wetting front.

3.3. Extreme Rainfall Effect on Infiltration Rate

The infiltration rate is the rainwater injection infiltration rate per unit area per unit time, as represented by Formula (2). The intensity and time of rainfall have a significant impact on the soil infiltration rate. Figure 5 shows the changes in the infiltration rate under different rainfall intensities. The infiltration rate decreases gradually as rainfall time and intensity increase. Under the J1 stage, two hours before the first rainfall, the infiltration rate was 4.6478 mm/h. At this time, soil infiltration was controlled by rainfall intensity, which was higher than the infiltration rate [17]. The rainfall was completely absorbed by the soil. As rainfall continues, the surface soil gradually becomes saturated. Rainwater is unable to infiltrate continuously, and the runoff is formed on the slope surface. This results in a decrease in the soil’s infiltration rate. The infiltration rates of the second and third rainfalls decreased by 53.99% and 89.89%, respectively, compared to the first rainfall when it rained for 3 h under the J1 condition.

3.4. Extreme Rainfall Effect on the Ratio of Runoff Rainfall to Infiltration Rainfall

Figure 6 shows the relationship between runoff rainfall and seepage rainfall under different rainfall intensities. As shown in the figure, the ratio of surface runoff to infiltration rainfall increases with the increase of rainfall time and intensity [17], whether it is a single rainfall event or cyclical rainfall. There was no surface runoff at the beginning of the first rainfall. This indicated that the rainwater had completely infiltrated into the soil, and the infiltration rate was greater than the rainfall intensity. As the frequency of rainfall increases, the time when surface runoff begins to appear becomes earlier and earlier. The ratio of runoff rainfall to seepage rainfall increased by 55.71% and 872.17% at 2 h of the third rainfall compared to the J2 and J1 stages when the rainfall intensity was 4.6478 mm/h. The reason for the above phenomenon is that the soil remains at its initial water content, being relatively dry during the initial rainfall, and it belongs to the unsaturated stage. Rainwater is immediately absorbed and infiltrated by the soil after it falls on the slope. The pores in the surface soil of the slope are gradually filled with rainwater as the duration and frequency of rainfall increase. The soil gradually transitions from an unsaturated state to a saturated state. This results in the inability of rainwater to continuously infiltrate and only form runoff on the slope surface.

4. Lateral Water Migration Model

4.1. Modeling

Lateral water migration of the subgrade can be predicted using a lateral water migration model, based on relevant research [17,33,34] and the development trends of the wetting front under three working conditions, as shown in Formula (4):

$$s=\alpha \cdot I^{\beta }\cdot t^{\gamma }$$

(4)

where s—lateral migration distance of the wetting front, mm; I—rainfall intensity, mm/h; t—start time of the test, h; and α, β, γ—parameters, dimensionless.

4.2. Fitting Parameters

Empirical parameters α, β, and γ and R² values at different soil layers under three working conditions were calculated using the least squares method, based on Formula (4). Table 5 summarizes the model parameter values. The lateral water migration model is shown in Formula (5). The lateral migration distance of the wetting front was related to α, β, γ, t, and I, based on

Table 6. Lateral water migration model parameters.

Layer	Parameters			Fit Degree
	α	β	γ	R2
L1	67.500	0.328	0.408	0.915
L2	55.800	0.315	0.348	0.908
L3	63.200	0.285	0.338	0.896

. The s increased as I and t grew. α was directly proportional to s. The larger the α, the larger the s under the same I and t conditions. β was the index of I, which determines the sensitivity of s to changes in I. The larger the β, the more sensitive s is to changes in I. γ is the exponent of t, which determines the sensitivity of s to changes in t. The larger the γ, the more sensitive s is to changes in t. The movement of the wetting front of the L1 layer is the fastest, and α was the biggest among the three layers of soil, with the highest sensitivity to rainfall intensity I and t.

$$s=\left\{\begin{array}{cc}\begin{array}{c}67.5\times {{\rm I}}^{0.328}\times t^{0.408}\\ 55.8\times {{\rm I}}^{0.315}\times t^{0.348}\\ 63.2\times {{\rm I}}^{0.285}\times t^{0.338}\end{array}& \begin{array}{c}200<\mathrm{L}\le 300\\ 100<\mathrm{L}\le 200\\ 0<\mathrm{L}\le 100\end{array}\end{array}\right.$$

(5)

Here, s—lateral migration distance of the wetting front, mm; I—rainfall intensity, mm/h; t—start time of the test, h; 200 < L ≤ 300—L1 soil layer, mm; 100 < L ≤ 200—L2 soil layer, mm; and 0 < L3 ≤ 100—L3 soil layer, mm.

4.3. Model Validation

Figure 7 shows the comparison of lateral water migration between the test results and predicted results. The test results were in good agreement with the predicted results. The growth rate of the power function is too fast when t and q are small, as shown in Figure 7a, resulting in a larger predicted value of s. Overall, the R² of the model exceeds 0.90, indicating its suitability for simulating the development of a lateral water migration model in the subgrade under different working conditions. Based on model prediction of water migration patterns, subgrade drainage facilities can be designed to reduce water damage and promote sustainable development.

5. Conclusions

The study performed a rainfall test using a compacted loess subgrade model to explore the effects of extreme rainfall on the water distribution, wetting front, and infiltration rate in the subgrade based on a self-developed water migration test device. The spatiotemporal laws of lateral water migration were summarized, and a lateral water migration model was proposed and validated. The main conclusions are as follows:

(1)

The impact range of a single rainfall event on the soil water content is small. The water content change area continues to expand with the improvement in rainfall intensity and time. The water content increases from the initial value to the peak value, and the peak water content eventually stabilizes at a certain value.

(2)

The lateral migration of water has a time hysteresis. The main manifestation is that (a) the sensor closest to the side of the slope changes first, and the water fluctuation is most obvious. The farther the distance, the slower the response and the smaller the fluctuation. (b) Multiple rainfall events have a significant influence on the soil water content far from the slope compared to a single rainfall event. At the same location, there is a situation where the peak water content occurs after the rain stops.

(3)

Sensors with the same depth but different positions start to change at different times. This indicated that the horizontal migration rate of the wetting front is different. The migration rate increases with the increase in rainfall intensity. The migration rate of sensor W3 increased by 66.47% and 333.70%, respectively, in the J3 stage compared to the J2 and J1 stages. The horizontal migration rate in the middle and slope toe is higher. The lowest is the slope top.

(4)

The infiltration rate gradually decreases and the ratio of surface runoff to infiltration rainfall increases with the increase in rainfall frequency and intensity. As the times of rainfall increase, the time when surface runoff begins to appear becomes earlier and earlier.

(5)

A lateral water migration model was proposed and validated. The predicted results of the model are in good agreement with the test, with R² over 0.90. This indicated that the model was suitable for simulating the development of a lateral water migration model in the subgrade under different working conditions.

(6)

In the future, the lateral water migration trend in compacted loess subgrade under extreme rainfall will be further studied using numerical simulation methods.