In this paper, by using the SEEP/W module in Geostudio2004 developed by the Geo-slope company (Yang, 1992), the finite element numerical calculation of one-dimensional transient saturated–unsaturated seepage for an isotropic homogeneous soil column with a height of 5 m and a width of 0.4 m is carried out, considering rainfall conditions. The influence of the rainfall intensity I and the infiltration characteristic parameters on the infiltration characteristics of unsaturated soil columns is studied.

The underground water level of the soil column model is located at its bottom and does not change during the whole rainfall process. Therefore, a total head boundary with constant 0 is set at the bottom of the soil column. The left and right sides of the soil column are impermeable, and the boundary condition of 0 flow is set at both sides. The surface runoff during rainfall and the evaporation of surface water after rain stop are not be considered in the model test. The seepage calculation and analysis of seven different rainfall intensities with a rainfall duration of 24 h are carried out for the soil column. The design values of rainfall intensity I and soil parameters under each working condition are shown in Table 1 .

In Table 1 , the rainfall intensity I is selected according to the multiple of saturated permeability coefficient k_s . For example, 0.5×k_s indicates that the rainfall intensity I (surface rainwater flow) is equal to 50% of the saturated permeability coefficient of soil. In this way, the permeability coefficient of the soil is linked to the amount of rainfall flow applied. The rainfall time step indicates that the time from the beginning of rainfall is the starting point of timing. The rainfall duration of 24 h indicates that the rainfall stops at 24 h, and 24 h to 240 h indicate the time period after the rainfall stops.

In the finite element calculation, the mesh division of the soil column model, the water head boundary condition, and the flow boundary condition are shown in Figure 1A , and the pore water pressure u_w distribution is shown in Figure 1B .

The data calculated by the soil column of each working condition will be extracted and sorted out. The variation rules of volumetric water content, infiltration characteristic parameters, and rainfall intensity in the process of rainfall duration will be analyzed.

The analysis of the law of the soil column in the rainfall duration introduces three parameters to describe the infiltration characteristics: infiltration front depth WF, suction reduction depth MR_n , and section infiltration rate IR. These three parameters are defined in detail as follows.

At present, the saturated permeability function and soil–water characteristic curve are mostly used to estimate the permeability function of unsaturated soil.

The SEEP/W module of the finite-element seepage analysis software Geostudio2004 developed by Geo-slope International Ltd (2004) used in this paper contains Fredlund–Xing formula [correction coefficient C(ψ) = 1] (Fredlund and Xing, 1994), as shown in Eq. 4. In this paper, this model (Fredlund et al., 1997) is used to describe the soil–water characteristic curve of unsaturated soil column.

where k_{w y} is the permeability coefficient of water in the y direction (mm/h), q is the flow of rainwater applied on the surface (mm/h), H_w is the total water head (m), m₂ ^w is the slope corresponding to the suction at any point in the soil–water characteristic curve, γ _w is the bulk density of water (kN/m³), and t is the time of rainfall (h).

The transient seepage analysis of unsaturated soil includes two hydraulic characteristic functions (i.e., soil–water characteristic curve and permeability function). The soil–water characteristic curve and permeability function used in this paper are the experimental curve data obtained by Professor Zhang Zhao (2009) according to the prediction method of Fredlund et al. (1994), as shown in Figure 5 .

Firstly, the distribution curve of the volumetric water content θ of the soil column along its height at each time under different rainfall intensity I is sorted out, as shown in Figure 6 .

As can be seen from Figure 6 , during the continuous rainfall process, the volumetric water content θ gradually increases with time. However, when the rainfall stops (i.e., T = 24 h), the volumetric water content θ at the top of the soil column continues to decrease, while the θ at the bottom of the soil column continues to increase. Even 9 days after the rainfall stops (i.e., T = 240 h), the θ still does not return to the initial state. Analysis shows that with the passage of time, the water continues to penetrate downward gradually under the action of gravity and the matric suction. If the evaporation of water is not considered, the volumetric water content in the soil column will take a long time to return to the initial state.

It also can be seen from Figure 6 that under different rainfall intensities, the infiltration of rainwater in the soil column first increases the volumetric water content θ of the soil near the surface, and the θ section line gradually approaches the groundwater level line. Furthermore, in order to determine the value of WF, the distribution map of the volumetric water content θ along the depth of soil column is drawn, as shown in Figure 7 .

From Figure 7 , we can see that with the extension of rainfall duration, the WF is more and more large. With further analysis, from the beginning to the end of rainfall, the volume water content θ of the soil near the surface is most affected by the rainfall, and that of the soil at a greater depth is less affected. With the time extension, the rain continues to penetrate downward; thus, even if the rainfall stops for a long time, WF continues to increase.

Similarly, the curves of pore water pressure u_w at different heights of soil column under different rainfall intensity I and different rainfall time T are sorted out, as shown in Figure 8 .

It can be seen from Figure 8 that the pore water pressure u_w increases with the extension of rainfall time gradually, and the matrix suction (u_a − u_w ) gradually decreases correspondingly. After the rain stops, the pore water pressure u_w at the top of the soil column begins to decrease in the direction of the initial hydrostatic pressure. However, after 10 days of rain stop (i.e., = 240 h), the pore water pressure is still not restored to the initial state, and the pore water pressure at the bottom of the soil column continues to increase slowly, that is, MR_n is still increasing. This analysis shows that the downward migration of rainwater in the soil column increases the u_w and reduces the (u_a − u_w ). The increase of MR_n indicates that the infiltration and suction reduction in the soil column continue after the rain stops. After the rain stops, the u_w gradually reduces to the initial hydrostatic pressure, which is the redistribution process of pore water pressure after the rain stops. This phenomenon is consistent with the in situ monitoring law of pore water pressure during rainfall by Rahardjo et al., (2007). The whole movement process is very slow, indicating that the pore water pressure in the soil column needs a long time to restore to the initial water pressure value.

Furthermore, according to the data of Figures 8D–G , the relationship curve of change of the normalized suction reduction depth MR_n ^nor with the suction reduction rate n% under four higher rainfall intensities (i.e., I = 2.5k_s , I = 5.0k_s ,I = 10.0k_s , and I = 25.0k_s , respectively) is sorted out, as shown in Figure 9 .

Figure 9 shows that the MR_n ^nor of the four rainfall intensities decreases with the increase of n%, which indicates that the shallower the depth in the soil column is, the more obvious the matrix suction reduction. In addition, the four curves show a good exponential function (R ^2 = 0.9767). The four groups of data can be fitted to a curve, and the curve fitting formula is as Eq. 4. This also reflects that there is a limit of the effect of rainfall intensity I on the reduction of matric suction; that is, MR_n has a limit value. Beyond this limit value, the effect of I on the reduction of matric suction is not obvious.

where a and b are the curve fitting parameters, respectively, a = −0.1361, b = 0.664.

Then, the distribution law of suction reduction at different depths of soil column during rainfall is sorted out, as shown in Figure 10 .

According to Figure 10 , the suction reduction caused by rainfall is unevenly distributed along the depth of the soil column. Under the condition of 24 h of rainfall duration, the maximum normalized suction reduction depth MN_n ^nor is approximately 0.35; that is, the maximum depth of matrix suction affected by rainfall is approximately 35% of the total depth.

In order to analyze the variation of section infiltration rate IR during rainfall, we extracted the relationship curve of section infiltration rate IR with time corresponding to different normalized heights H^nor under condition I = 1.0k_s , as shown in Figure 11 .

As can be seen from Figure 11 , under each H^nor , the IR at the top of the soil column gradually increased with time during the rainfall process, but after the rain stops, it began to decrease again. The IR at the bottom of the soil column begins to increase again after a long period of rain stop. According to the analysis, the infiltrated rainwater moves slowly from the top of the soil column to the bottom, resulting in a lag in time. Therefore, as the rainfall continues, the IR of the surface gradually increases to the saturated permeability coefficient of the soil. After rain stops, the IR of the soil near the surface (the top of the soil column) decreases to zero, and at this time, the IR of the soil near the bottom of the soil column begins to increase. It can also be seen from Figure 11 that the maximum infiltration rate IR_max at the bottom of the soil column is significantly smaller than that of the top of the soil column. This is because the infiltration of rainwater into the bottom of the soil column becomes much slower due to the reduction of the hydraulic gradient. During the whole process of rainfall, the maximum infiltration rate IR_max near the surface (i.e., H^nor = 1.0) is the largest, and the IR_max decreases with the decrease of H^nor .

The curves in Figure 11 also show that the distribution of IR in the soil column along the depth direction is very uneven. Then, the relationship curves between the normalized height H^nor and the section infiltration rate IR under the conditions of I ≤ 1.0k_s and I > 1.0k_s are drawn, as shown in Figure 12 . In the figure, T ₁ and T ₂ are the different moments during the rain, and T ₁< T ₂, T ₃ is a certain moment after the rain stops.

From the previous article, we can find that when the rainfall intensity I is different, the two parameters of the calculated normalized infiltration front depth WF^nor and the normalized suction reduction depth MR_n ^nor are also different. We also sorted out the change of the WF^nor and the MR_n ^nor of the soil column under different rainfall intensities I and different times T during the rainfall process, as shown in Figure 13 .

Figure 13A shows the variation of WF^nor with the rainfall duration under different rainfall intensities. It can be seen from the figure that WF^nor generally increases exponentially with time, and with the increase of rainfall intensity I, the depth of infiltration front WF gradually approaches the depth of groundwater level WT (i.e., WF = WT). In addition, it can be seen that when the rainfall began, the movement of the infiltration front is very rapid, but it gradually slowed down over time, especially after the rain stops, where the movement of the infiltration front became slower. When the rainfall duration is 1 day under different rainfall intensities, the normalized infiltration front depth WF^nor under each working condition has a maximum value; that is, the influence depth of rainwater has a limit, and as the rainfall intensity increases, this limit value also increases. When I ≥ 2.5 k s , the individual curves almost coincide; it can be considered that I = 2.5k_s is an eigenvalue.

Figure 13B shows the variation of the depth of normalized suction reduction MR_n ^nor with rainfall intensity I under different suction reduction rates n%. It can be seen from the figure that the matric suction in the soil column is not completely lost after the rainfall: the area where the suction reduction in the soil column is more than 10% only accounts for approximately 35% of the height of the soil column. In other words, from the groundwater level up, the suction reduction of the soil column in the range of approximately 65% height is very small (i.e., the reduction is less than 10%). Therefore, only a small part of the soil column’s shear strength is reduced.

It also can be seen from Figure 13 that the movement speed of the infiltration front increases with the increase of rainfall intensity I. Similarly, the amplitude of suction reduction and the section infiltration rate IR also increase with the increase of I. However, there is a limit to the influence of rainfall intensity on these parameters, that is, I = 2.5k_s .

All these phenomena indicate that there is indeed a “special” rainfall intensity when the soil column is affected by rainfall infiltration. In a certain range, increasing the rainfall intensity I will accelerate the infiltration of rainwater in the soil column, but there is a maximum limit (“special”) rainfall intensity. At this time, the influence of I on the infiltration of the soil column reaches the maximum. When I exceeds this value, its effect on the infiltration of the soil column will be less obvious.