A method for calculating rainfall infiltration and runoff of slope suitable for any initial water content distribution

Abstract

The application provides a kind of slope rainfall infiltration-runoff calculation method suitable for any initial water content distribution, comprising step 1, according to the initial water content profile of soil, equal interval segmentation, a discrete soil infiltration capacity curve is obtained by calculation;Step 2, based on the discrete curve, using binomial slope infiltration formula regression fitting, obtain continuous infiltration capacity curve and cumulative infiltration curve;Step 3, using continuous infiltration curve, by the change data of rainfall intensity with time, the waterlogging occurrence time and runoff production are obtained by calculation.The application deduces the analytical solution that can accurately describe the homogeneous soil slope water infiltration under constant water head boundary conditions, and develops a kind of rainfall infiltration-runoff calculation method that can quickly calculate any initial water content distribution.The method is stable, fast, and ensures water balance, while avoiding complex time and space discretization operation.

Description

Technical Field

[0001] This invention relates to a method for calculating rainfall infiltration-runoff on slopes applicable to any initial water content distribution, belonging to the fields of soil physics and hillside hydrology. Background Technology

[0002] When raindrops fall onto the soil surface, they infiltrate into the soil until the rainfall intensity exceeds the soil's limited infiltration capacity. At this point, water accumulates on the soil surface, leading to runoff and erosion. Accurately estimating the timing of water accumulation and the amount of runoff is crucial for hydrological forecasting, agricultural irrigation design, and soil and water conservation.

[0003] Rainfall infiltration is a complex dynamic process, primarily influenced by factors such as the temporal variation of rainfall intensity, the hydraulic properties of the soil profile, the initial water content distribution along the profile, and slope. Currently, the finite element method and finite difference method are mainly used to numerically solve the Richards equation to estimate the timing of water accumulation and runoff generation. However, numerical simulation methods often require high levels of discretization in both time and space to accommodate complex rainfall intensity variations at the upper boundary. Furthermore, solving a highly nonlinear Richards equation consumes significant computational resources, leading to problems such as numerical instability and high computational costs. As an improved approach, empirical, analytical, and semi-analytical infiltration equations have been discovered and applied to the simulation of infiltration-runoff processes due to their simplicity, lack of need for spatiotemporal discretization, and preservation of water balance. However, due to the complexity of rainfall infiltration, existing infiltration equations either ignore the natural characteristic of rainfall intensity changing over time (CN 105547957; CN 112685874; CN 109898489), or their inherent assumption of uniform initial moisture content distribution cannot meet the condition of non-uniform initial moisture content distribution in the field (Chen and Young, 2006; Wang et al., 2018), making them difficult to apply to real field environments. Therefore, there is currently a lack of a slope infiltration-runoff calculation method that can be used in the field to simulate non-uniform initial moisture content distribution under unsteady rainfall conditions, which will greatly reduce the timeliness and accuracy of hydrological forecasts. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art and provide a slope rainfall infiltration-runoff calculation method applicable to any initial water content distribution.

[0005] The model parameters that need to be determined beforehand for this calculation method include: 1. The initial soil moisture content distribution profile. i i (z); 2. Brooks-Corey Soil hydraulic parameters described by the modeln , h d saturated hydraulic conductivity of soil K s and saturated water content i s The specific calculation method includes the following steps: Step 1, for the known initial soil moisture content distribution profile i s - i i ( z Divide the sample into N equal-width strips, and take the center point of each strip as the marker point of the division. x j ,but (11) (12) In the formula, Z max It is the maximum depth of the known soil moisture profile; Z 0、 Z 1 represents the positions of the upper and lower ends of the first equal-width strip in the soil moisture profile; Z N It is the position of the lower end of the Nth equal-width strip in the soil moisture profile; Step 2: Based on the segmented profile from Step 1, calculate the cumulative soil infiltration. I Represented in the form of Riemann sum: (13) In equation (13), J w0 ( z fe = x j ) is the length of the virtual piston-shaped wetted area. z fe Arrive at the marker x j Surface soil infiltration rate at that time; △ t j and △ I j They are z fe Experience △ Z j The time required for the distance and the increased infiltration rate; i i ( x j () is the depth of the soil moisture profile. Z = x j The initial soil moisture content at the location.

[0006] Using the first-order Taylor series expansion and the central difference method, i i ( x j )and J w0 ( z fe = ξ j ) is represented as: (14) (15) △ is calculated by combining formulas (13) and (14). I j (j = 1,2,···N), the calculation expression is: (16) For the known △ I j (j = 1,2,···N), combining formula (13) and formula (15), we can calculate △ t j (j = 2, ..., N), the calculation expression is: (17) For the known △ t j (j = 1,2,···N), z fe To reach a defined soil moisture profile depth Z j The time taken is calculated using the following formula. (19) Furthermore, for a given initial soil moisture content distribution profile... i s - i i ( z A discrete soil infiltration capacity curve can be calculated. J w0 ( z fe ) ~ t ( z fe ); Step 3: For the discrete soil infiltration capacity curves obtained in Step 2, use... Philip The parameters of the two slope infiltration formulas are used. α , β By fitting the data, a continuous surface soil infiltration rate curve can be obtained. J w0 (t ), (20) In the formula, c Given the slope, and based on the principle of mass conservation, a continuous cumulative infiltration curve is obtained. I ( t ) is represented as, (twenty one); Step 4: Based on the rainfall monitoring resolution, divide the non-steady-state rainfall event into segmented steady-state rainfall intervals, and measure the rainfall rate within the nth steady-state rainfall interval. r n ( t ) = r n cos c It can be represented as, (26) In the formula, t n and t n-1 These are the rainfall durations at the end of the nth and (n-1)th segments, respectively. R ( t n )and R ( t n-1 ) represent the cumulative rainfall at the end of the nth and (n-1)th segments, respectively; △ t n The duration of the rainfall interval in the nth segment; Step 5, for the first segment of steady-state rainfall, the rainfall rate is r 1cos c Time of water accumulation within the interval t p Virtual Time t s And the cumulative infiltration amount when water accumulates under steady-state rainfall. I p It is calculated using the following formula. (twenty three) (twenty four) (25) The soil infiltration capacity at this point is described by the following formula: (twenty two) Cumulative infiltration I ( t 1) Obtained by simultaneously solving formula (21) and the following formula. (27) Cumulative runoff F ( t 1) is represented as, (28); Step 6: Based on the time compression method, calculate the soil infiltration capacity at the end of segment n-1 (n > 2). J w0 ( t sn-1 ), In equation (29), I ( t n-1 () represents the cumulative soil infiltration at the end of the n-1 segment of steady-state rainfall; Step 7, compare soil infiltration capacity J w0 ( t sn-1 ) and rainfall rate r n cos c The magnitude of the value determines the timing of water accumulation and the amount of runoff generated within the nth steady-state rainfall interval. a). When J w0 ( t sn-1 ) ≤ r n cos c This indicates that during the nth steady-state rainfall interval, water accumulation will occur instantaneously or continue to occur. The soil infiltration capacity at this time is described by formula (22). t p = t n-1 , t s = t sn-1 Cumulative infiltration I ( t n Calculated using the following formula: (31) Cumulative runoff F ( t n Calculated using the following formula: (32) b) When J w0 ( t sn-1 ) > r n cos c This indicates that during the nth steady-state rainfall interval, water accumulation will not occur instantaneously, and the duration of water accumulation within the nth steady-state rainfall interval is [not specified]. t pn Calculated using the following formula: (33) if t pn ≥ △ t n This indicates that no water accumulation occurs in this area, so the cumulative infiltration rate at this time is... I ( t n and cumulative runoff F ( t n ) are respectively represented as: (35) (36) Conversely, if t pn <△ t n The soil infiltration capacity at this point is described by the following formula ( t p = t n-1 + t pn ) (twenty two) Cumulative infiltration I ( t n and cumulative runoff F ( t n ) are respectively represented as: (37) (38).

[0007] As a preferred technical solution of the present invention: in step 2, the formula (17) J w0 ( z fe = Z j First, follow the formula below: (18) Perform fixed-point iteration to obtain the length of the saturated region within the wet region. z fw (cm), will z fw Substitute into the following formula, (8) Obtain J w0 ( z fe = Z j ); The unknown parameters in the formula are defined as follows: (39).

[0008] As a preferred technical solution of the present invention: In step 2, since at the initial moment of infiltration, the surface soil infiltration rate in formula (17) is... J w0 ( z fe = Z 0) is infinite, therefore △ t 1. Calculate the initial water content using the following formula, where the initial water content is... i i = [ i i ( Z 0) + i i ( Z 1)] / 2, t = △ t 1 and J w0 = J w0 ( z fe = Z 1), (6) In the formula, △ H = H f + h d =- hh d The definitions of the remaining parameters are the same as above.

[0009] As a preferred technical solution of the present invention: In step 6, based on the saturated hydraulic conductivity of the soil K s and infiltration parameters α , β， According to the following formula: (30) Execute the fixed-point iterative algorithm to obtain the virtual time at the end of the (n-1)th rainfall interval. t sn-1 (min), will t sn-1 Substituting into formula (20), we get J w0 ( t sn-1 ).

[0010] As a preferred technical solution of the present invention: In step 7, by substituting formula (20) into formula (21) and eliminating the time variable t, the following calculation is obtained: (34) The above formula I Replace with I ( t n-1 )+ r n cos γt pn , J w0 Replace with r n cos c Perform a fixed-point iterative algorithm to calculate and obtain t pn .

[0011] As a preferred technical solution of the present invention: in step 7, the parameters in formula (22) t s Calculate using the following formula, (twenty four).

[0012] As a preferred embodiment of the present invention: in step 1, the value of N is at least 101 or higher.

[0013] The beneficial effects of this invention are as follows: 1. Compared with existing technologies, this invention takes into account real field conditions such as the non-steady-state nature of rainfall intensity and the non-uniform distribution of initial soil moisture content. Based on the slope infiltration equation under water accumulation conditions, an analytical calculation method for rainfall infiltration under arbitrary initial moisture content distribution conditions is proposed, which accurately estimates the water accumulation time and runoff generation in homogeneous soil. This provides an effective calculation tool for fields such as hydrological forecasting, agricultural irrigation design, and soil and water conservation. 2. Simple calculation method. This invention only requires known hydraulic properties of the soil profile to be calculated, the distribution of initial soil moisture content, and the data on the change of rainfall intensity at the upper boundary over time to calculate the time of water accumulation and runoff generation. The calculation process does not require complex time and space discretization, the calculation is stable and fast, and it ensures water balance. It is a simple and accurate calculation method. Attached Figure Description

[0014] Figure 1 This is a flowchart of the calculation process of the present invention; Figure 2 A conceptual diagram of slope infiltration profile under water accumulation conditions; Figure 3 In the diagram, a) represents a segmentation concept diagram for an arbitrary initial water content profile. Figure 3 b) is a conceptual diagram of infiltration of a virtual piston-type profile in an arbitrary initial water content profile; Figure 4The results of slope infiltration-runoff simulation under unsteady rainfall conditions in two typical soil types are shown in the figure. (a) is a typical V-shaped initial water content distribution profile in arid and semi-arid areas, (b) is a comparison of rainfall infiltration in sandy loam under the V-shaped initial water content distribution profile, (c) is a typical initial water content distribution profile in humid areas with shallow groundwater, and (d) is a comparison of rainfall infiltration in silty sand under the initial water content distribution profile with shallow groundwater. Detailed Implementation

[0015] The following description, in conjunction with the accompanying drawings, further illustrates the content of the present invention, but should not be construed as limiting the invention. Modifications and substitutions made to the methods, steps, or conditions of the present invention without departing from the spirit and essence of the invention are all within the scope of the invention. Unless otherwise specified, the technical means used in the following embodiments are conventional means well known to those skilled in the art. Figure 1 The calculation process of this invention is explained.

[0016] Example 1

[0017] First, we analyze and describe the conditions under water accumulation. Richards The governing equations for slope flow and their boundary conditions are as follows: (1) In the formula: i Soil moisture content (cm) 3 min -3 ), z It represents the spatial coordinates (cm) of the slope normal direction. t It is the infiltration time (min). h It is the soil matrix potential (cm). i i It is the initial moisture content of the soil (cm). 3 min -3 ), h p It is the water head pressure (cm) on the soil surface. c It is the slope (°).

[0018] And analyze the Brooks-Corey model to describe the unsaturated hydraulic conductivity of soil. K ( h The expressions for the relationship between soil matric potential and soil moisture content are as follows: (2) (3) In the formula: S It is the effective soil water saturation. i s It is the saturated soil moisture content (cm). 3 min -3 ), i r It is the residual soil moisture content (cm) 3 min -3 ), K s Soil saturated hydraulic conductivity (cm min) -1 ), n It is the soil pore size distribution index. h d It is the soil air suction value (cm). m = 3 n+ 2.

[0019] Using the principle of least action, the flux assumption, and the mean value theorem for integrals, a profile equation describing soil water infiltration is obtained. The flux assumption and the mean value theorem are used to simplify the functional form of the profile equation. The expression for the flux assumption is as follows: (4) In the formula: J w0 The water flow rate at the inlet (cm min) -1 ), J w Soil water flow rate (cm min) -1 ), K i This corresponds to the unsaturated hydraulic conductivity (cm min) at the initial soil moisture content. -1 ), S i This is the effective soil water saturation corresponding to the initial soil moisture content; the other parameters have the same meaning.

[0020] Based on the principle of mass conservation and the profile equation, a complete analytical model (5) – (9) is derived to describe the slope water infiltration under constant head boundary conditions with a virtual piston profile. Figure 2 ).

[0021] (5)

[0022] (6)

[0023] (7)

[0024] (8)

[0025] (9) In the formula, (10) In the formula, z f It is the length of the wetted area (cm).z fe It is a virtual piston-shaped wetted zone length (cm). z fw It is the length (cm) occupied by the saturated zone within the moist zone. I It is the cumulative infiltration volume (cm). J w0 It is the infiltration rate of the topsoil (cm min).

[0026] Based on the aforementioned infiltration analytical model, a method for calculating slope water infiltration under conditions of non-uniform initial water content distribution was established. The key to the method's validity lies in: the length of the virtual piston-shaped wetting zone... z fe As the infiltrating water moves downwards, a piston-shaped profile always exists, regardless of whether the initial water content distribution in the water profile is uniform in the early stages (see...). Figure 3 (b)). For a specific moment, the existing piston-shaped profile can be regarded as a profile generated by water infiltration under the condition of uniform initial water content distribution. The water infiltration process can be solved by an analytical infiltration model, and the magnitude of its initial water content is... i i ( z fe Therefore, for the provided initial soil moisture content profile... i i ( z By segmenting and summing the moisture profile, slope infiltration under non-uniform initial moisture content distribution conditions can be simulated based on the infiltration analytical model. Example

[0027] Based on Example 1, a slope rainfall infiltration-runoff calculation method applicable to arbitrary initial moisture content distribution is proposed. The specific technical solution mainly includes two parts: 1. For a given soil profile with non-uniform initial moisture content distribution, a method for calculating slope water accumulation infiltration under non-uniform initial moisture content distribution conditions is used to obtain a continuous soil infiltration rate curve. J w0 ( t ) and cumulative infiltration curve I ( t ); 2. Combining time compression methods, and J w0 ( t )and I ( t The cumulative infiltration and runoff of the soil were obtained by simulating the slope rainfall infiltration-runoff method under unsteady rainfall conditions.

[0028] 1. Method for calculating slope water infiltration under conditions of non-uniform initial moisture content distribution The model parameters required for the calculation method include: 1. Initial soil moisture content distribution profile. i i ( z );2、 Brooks- Corey Soil hydraulic parameters described by the model n , h d saturated hydraulic conductivity of soil K s and saturated water content i s The specific calculation method includes the following steps: First, provide an arbitrary distribution of initial soil moisture content profiles. i i ( z ),Will i s - i i ( z )The cross-section is divided into n A long strip of equal width ( n (The value must be large enough), and the center point of each bar is taken as the marker point for that segmentation. x j (See Figure 3 (a) (11) (12) In the formula, Z max It is the maximum depth of the known soil moisture profile.

[0029] Secondly, based on the aforementioned segmented profile, the cumulative soil infiltration is expressed in the form of Riemann sums. (13) In the formula, J w0 ( z fe = x j ) is when z fe Arrive at the marker x j Surface soil infiltration rate at that time; △ t j and △ I j They are z fe Experience △ Z j The time required for the distance and the increased infiltration rate. Using a first-order Taylor series expansion and the central difference method, i j ( x j )and J w0 ( z fe = ξ j ) is represented as, (14)

[0030] (15) By combining formulas (13) and (14), △ can be calculated. I j (j = 1,2,···n), the calculation expression is, (16) For the known △ I j (j = 1,2,···n), and then substituting formula (15) into formula (13), we can calculate △ t j (j = 2, ..., n), the calculation expression is: (17) In the formula J w0 ( z fe = Z j The initial water content required for the calculation is obtained by simultaneously solving formulas (8) and (9). i i ( z fe The specific steps are as follows: express formula (9) as follows, where z fe Replaced with Z j , (18) Perform fixed-point iteration on the above expression to obtain the result. z fw ,Will z fw Substituting into formula (8), we obtain J w0 ( z fe = Z j ).

[0031] Because at the initial moment of infiltration, the infiltration rate of the surface soil is... J w0 ( z fe = Z0) is infinite, therefore △ t 1. The initial moisture content will be calculated separately using formula (6), and set as [ i i ( Z 0) + i i ( Z 1)] / 2. Then z fe Reach a certain depth Z j The time taken was, (19) In summary, for a specific initial water content distribution profile i i ( z A discrete soil infiltration capacity curve can be calculated. J w0 ( z fe ) ~ t ( z fe ).

[0032] Finally, for the discrete soil infiltration capacity curves obtained in the above steps, the following method is used: Philip Parameter fitting of the two slope infiltration formulas (fitting parameters) α , β This allows us to obtain a continuous surface soil infiltration rate curve. (20) Based on the principle of conservation of mass, integration yields a functional expression for the cumulative infiltration rate versus time. (twenty one).

[0033] 2. Slope Rainfall Infiltration-Runoff Simulation Method under Unsteady Rainfall Conditions The cumulative infiltration curve calculated based on the above method I ( t ) and soil infiltration rate curve J w0 ( t A method for simulating slope rainfall infiltration-runoff under unsteady rainfall conditions is proposed, as follows: First, for a steady-state rainfall event, the rainfall rate is... r cos c (cm min) -1 The infiltration rate of the surface soil after water accumulation is expressed using the time compression method as follows: (twenty two) In the formula, t pThe time (in minutes) required for water to accumulate under steady-state rainfall. t s The virtual time (min) represents the time when the surface soil infiltration rate reaches the required level during water infiltration. r cos c (cm min) -1 The time required. According to the definition of time compression methods, J w0 ( t s The infiltration capacity (SCI) represents the soil's ability to absorb water at a certain rate. Using formulas (19), (20), and a given steady-state rainfall intensity, t p , t s The cumulative infiltration amount when water accumulates under steady-state rainfall. I p It is represented as, (twenty three) (twenty four)

[0034] (25) Secondly, based on the resolution of rainfall monitoring, non-steady-state rainfall events can be divided into segmented steady-state rainfall intervals (Chu, 1978). The rainfall rate within the nth steady-state rainfall interval... r n cos c (cm min) -1 ) can be represented as, (26) In the formula, t n and t n-1 These are the rainfall durations (in minutes) at the end of the nth and (n-1)th segments, respectively. R ( t n )and R ( t n-1 ) are the cumulative rainfall (cm) at the end of the nth and (n-1)th segments, respectively.

[0035] Within the first steady-state rainfall interval, the rainfall rate is r 1cos c Time of water accumulation within the interval t p Virtual Time t s And the cumulative infiltration amount when water accumulates under steady-state rainfall. I pThe soil infiltration capacity curve can be calculated using formulas (22), (23), and (24), and the cumulative infiltration amount can be calculated using formula (21). I ( t 1) and cumulative runoff F ( t 1) Calculate using the following formula: (27) (28) Based on the time compression method, the soil infiltration capacity at the end of the (n-1)th segment (n > 2) J w0 ( t sn-1 ) is represented as, (29) In the formula, I ( t n-1 The value represents the cumulative soil infiltration (cm) at the end of the n-1 segment of steady-state rainfall. (Virtual time) t sn-1 The iterative equation is obtained by using the fixed-point iteration method with formula (20). (30) and then J w0 ( t sn-1 It can be calculated using formula (19).

[0036] Finally, based on the comparison of soil infiltration capacity J w0 ( t sn-1 ) and rainfall rate r n cos c The magnitude of the value determines the timing of water accumulation and the amount of runoff generated within the nth steady-state rainfall interval.

[0037] a). When J w0 ( t sn-1 ) ≤ r n cos c This indicates that during the nth steady-state rainfall interval, water accumulation will occur instantaneously or continue. The soil infiltration capacity at this time can be described by formula (21). t p = t n-1 , t s = t sn-1), cumulative infiltration I ( t n )Calculated using formula (20), (31) Cumulative runoff F ( t n ) is represented as, (32) b) When J w0 ( t sn-1 ) > r n cos c This indicates that at the beginning of the nth steady-state rainfall interval, water accumulation will not occur instantaneously. Therefore, by combining formulas (20) and (21), the water accumulation time within the nth steady-state rainfall interval can be calculated. t pn It can be calculated using the following formula: (33) The specific calculation steps are as follows: Substitute formula (20) into formula (21), eliminate the time variable t, and calculate: (34) if t pn ≥ △ t This indicates that no water accumulation occurs in this area, so the cumulative infiltration rate at this time is... I ( t n and cumulative runoff F ( t n ) are respectively represented as, (35) (36) Conversely, if t pn <△ t Then, the soil infiltration capacity at this time can be described by formula (21). t p = t n-1 + t pn , t s The cumulative infiltration volume is calculated using formula (23). I ( t n and cumulative runoff F ( tn ) are respectively represented as, (37) (38).

[0038] Example 3 The following soil textures (soil hydraulic properties are shown in Table 1), different unsteady rainfall events, and two typical field soil initial moisture content distribution profiles (see Table 1) were used. Figure 4 (a) Figure 4 (c) further illustrates the slope rainfall infiltration-runoff calculation method disclosed in this invention, which is applicable to any initial water content distribution.

[0039] Table 1. Hydraulic properties of two typical soil textures

[0040] Note: The parameters in the table have the same meaning as above.

[0041] 1. Figure 4 (a) shows a V-shaped initial water content profile. Figure 4 (b) shows the data on the variation of non-steady-state rainfall intensity over time, with the soil type being sandy loam and the slope set to 30°.

[0042] Step 1: Divide the V-shaped initial water content profile into equal intervals according to formulas (a) and (b), and its Δ Z j =0.2cm, n = 501; Step 2: Calculate the segmented water volume profile according to formulas (c), (f), (g), and (h) to obtain a discretized soil infiltration capacity curve. J w0 ( z fe ) ~ t ( z fe ); Step 3: Use formula (i) to... J w0 ( z fe ) ~ t ( z fe Regression analysis was performed to obtain the parameters. α = 0.36685, β = -0.58651 Continuous soil infiltration capacity curve; Step 4: Based on the rainfall monitoring resolution, non-steady-state rainfall events are divided into steady-state rainfall intervals with a time resolution of one hour, see... Figure 4 (b) in the middle.

[0043] Step 5: For the first 0-60 minutes of a non-steady-state rainfall event, its rainfall intensity r 1cos c = 7.77 cm / h, the water accumulation rate was calculated according to formulas (l) and (m) respectively. t p = 21.19 min, virtual time t s = 10.55 min, then the soil infiltration capacity and cumulative infiltration amount for this steady-state rainfall period were calculated using formulas (o) and (p). At 60 min, the cumulative soil infiltration amount... I ( t 1) = 6.29cm, cumulative runoff F ( t 1) = 1.48cm.

[0044] For the remaining three steady-state rainfall intervals, steps 6 and 7 will be executed 3 times in a loop to calculate the cumulative soil infiltration. I ( t ) and cumulative runoff F ( t ).

[0045] ① For the second phase (60-120 minutes) of a non-steady-state rainfall event, the rainfall intensity is: r 2cos c = 3.88 cm / h, calculated according to formula (r) J w0 ( t 2) = 4.47 cm / h, that is J w0 ( t 2) > r 2cos c Then, according to formula (v), we can calculate the result. t p2 = 36.45 min, which is less than △ t = 60 minutes, then the time between water accumulation and water accumulation is... t p = 96.46 min, virtual time t s = 83.55 min, and then the soil infiltration capacity and cumulative infiltration amount for this steady-state rainfall period were calculated using formulas (y) and (z). At 120 min, the cumulative soil infiltration amount was... I ( t 1) = 10.13cm, cumulative runoff F ( t1) = 1.53cm.

[0046] ② For the third segment of the non-steady-state rainfall event, from 120 to 180 minutes, the rainfall intensity is: r 3cos c = 7.77 cm / h, calculated according to formula (r) J w0 ( t 3) = 3.66 cm / h, that is J w0 ( t 3) < r 3cos c Then water accumulation occurs. t p =120 min, virtual time t s = 107.11 min, and then the soil infiltration capacity and cumulative infiltration amount for this steady-state rainfall period were calculated using formulas (t) and (u). At 180 min, the cumulative soil infiltration amount was... I ( t 1) The cumulative runoff is 13.61 cm. F ( t 1) It is 5.81cm.

[0047] ③ For the fourth segment of the non-steady-state rainfall event, from 180 to 240 minutes, the rainfall intensity is: r 4cos c = 10.36 cm / h, calculated according to formula (r) J w0 ( t 4) = 3.37 cm / h, that is J w0 ( t 4) < r 4cos c Then water accumulation occurs. t p =180 min, virtual time t s = 167.1 min, and then the soil infiltration capacity and cumulative infiltration amount for this steady-state rainfall period were calculated using formulas (t) and (u). At 180 min, the cumulative soil infiltration amount was... I ( t 1) = 16.85cm, cumulative runoff F ( t 1) = 12.93cm.

[0048] 2. Figure 4(c) shows the initial water content profile when shallow groundwater is present. Figure 4 (d) in the figure shows the data on the change of non-steady-state rainfall intensity over time, with the soil type being silty sand and the slope set to 30°.

[0049] Step 1: Divide the V-shaped initial water content profile into equal intervals according to formulas (a) and (b), and its Δ Z j =0.2cm, n = 501; Step 2: Calculate the segmented water volume profile according to formulas (c), (f), (g), and (h) to obtain a discretized soil infiltration capacity curve. J w0 ( z fe ) ~ t ( z fe ); Step 3: Use formula (i) to... J w0 ( z fe ) ~ t ( z fe Regression analysis was performed to obtain the parameters. α = 0.36685, β = -0.58651 Continuous soil infiltration capacity curve; Step 4: Based on the rainfall monitoring resolution, non-steady-state rainfall events are divided into steady-state rainfall intervals with a time resolution of one hour, see... Figure 4 (d) in the middle.

[0050] Step 5: For the first 0-60 minutes of a non-steady-state rainfall event, its rainfall intensity r 1cos c =6.80 cm / h, the water accumulation rate was calculated according to formulas (18) and (19) respectively. t p = 4.82 min, virtual time t s = 2.23 min, and then the soil infiltration capacity and cumulative infiltration amount for the subsequent steady-state rainfall period were calculated using formulas (o) and (p). At 60 min, the cumulative soil infiltration amount was... I ( t 1) = 2.75 cm, cumulative runoff F ( t 1) = 4.04cm.

[0051] For the remaining three steady-state rainfall intervals, steps 6 and 7 will be executed 3 times in a loop to calculate the cumulative soil infiltration. I ( t ) and cumulative runoff F ( t ).

[0052] ① For the second phase (60-120 minutes) of a non-steady-state rainfall event, the rainfall intensity is: r 2cos c = 1.36 cm / h, calculated according to formula (r) J w0 ( t 2) = 1.59 cm / h, that is J w0 ( t 2) > r 2cos c Then, according to formula (v), we can calculate the result. t p2 = 36.45 min, which is less than △ t = 60 minutes, then water accumulation will occur t p = 114.78 min, virtual time t s =92.3 min, and then the soil infiltration capacity and cumulative infiltration amount for this steady-state rainfall period were calculated using formulas (y) and (z). At 120 min, the cumulative soil infiltration amount was... I ( t 1) = 4.09 cm, cumulative runoff F ( t 1) = 4.07 cm.

[0053] ② For the third segment of the non-steady-state rainfall event, from 120 to 180 minutes, the rainfall intensity is: r 3cos c = 2.04 cm / h, calculated according to formula (r) J w0 ( t 3) = 1.27 cm / h, that is J w0 ( t 3) < r 3cos c Then water accumulation occurs. t p =120 min, virtual time t s= 114.81 min, and then the soil infiltration capacity and cumulative infiltration amount for this steady-state rainfall period were calculated using formulas (t) and (u). At 180 min, the cumulative soil infiltration amount was... I ( t 1) The cumulative runoff is 5.29 cm. F ( t 1) It is 4.91cm.

[0054] ③ For the fourth segment of the non-steady-state rainfall event, from 180 to 240 minutes, the rainfall intensity is: r 4cos c = 3.40 cm / h, calculated according to formula (r) J w0 ( t 4) = 1.12 cm / h, that is J w0 ( t 4) < r 4cos c Then water accumulation occurs. t p =180 min, t s = 174.81 min, and then the soil infiltration capacity and cumulative infiltration amount for this steady-state rainfall period were calculated using formulas (t) and (u). At 180 min, the cumulative soil infiltration amount was... I ( t 1) = 6.37cm, cumulative runoff F ( t 1) = 7.23cm.

[0055] Figure 4 The soil infiltration capacity curves simulated by analytical calculation methods in two typical slope rainfall infiltration-runoff scenarios are presented, along with direct comparisons. Richards The numerical solutions to the equations were compared. As can be seen from the figure, the analytical solutions are almost identical to the observation results, with relative errors of 1.81% and 1.65%, respectively. Therefore, the slope rainfall infiltration-runoff calculation model can accurately simulate the time of water accumulation and the amount of runoff under different textures, rainfall intensities, and arbitrary initial water contents.

[0056] Finally, it should be noted that the above specific embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications and substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for calculating slope rainfall infiltration-runoff applicable to arbitrary initial moisture content distributions, wherein the following model parameters are pre-determined:

1. Initial moisture content distribution profile of the soil. i i (z); 2. Brooks-Corey Soil hydraulic parameters described by the model n , h d, in n It is the soil pore size distribution index. h d This represents the soil air suction value and the soil's saturated hydraulic conductivity. K s and saturated water content i s Its characteristics including the following steps: Step 1: Provide an arbitrary distribution of initial soil moisture content profiles. i i ( z ),Will i s - i i ( z )The cross-section is divided into n Divide the rectangle into equal-width rectangles, and take the center point of each rectangle as the marker point for the division. x j ,but (11) (12) In the formula, Z max It is the maximum depth of the known soil moisture profile; Z 0、 Z 1 represents the positions of the upper and lower ends of the first equal-width strip in the soil moisture profile; Z N It is the position of the lower end of the Nth equal-width strip in the soil moisture profile; Step 2: Based on the segmented profile from Step 1, calculate the cumulative soil infiltration. I Represented in the form of Riemann sum: (13) In equation (13), J w0 ( z fe = x j ) is the length of the virtual piston-shaped wetted area. z fe Arrive at the marker x j Surface soil infiltration rate at that time; △ t j and △ I j They are z fe Experience △ Z j The time required for the distance and the increased infiltration rate; i i ( x j () is the depth of the soil moisture profile. Z = x j Initial soil moisture content at the location; Using the first-order Taylor series expansion and the central difference method, i i ( x j )and J w0 ( z fe = ξ j ) is represented as: (14) (15) △ is calculated by combining formulas (13) and (14). I j (j = 1,2,···N), the calculation expression is: (16) For the known △ I j Where j = 1,2,···N, combining formula (13) and formula (15), we can calculate △ t j Where j = 2, ..., N, the calculation expression is: (17) Because at the initial moment of infiltration, the surface soil infiltration rate in formula (17) is... J w0 ( z fe = Z 0) is infinite, therefore △ t 1. Calculate the initial water content using the following formula, where the initial water content is... i i = [ i i ( Z 0) + i i ( Z 1)] / 2, t = △ t 1 and J w0 = J w0 ( z fe = Z 1), (6) In the formula, K i This represents the unsaturated hydraulic conductivity corresponding to the initial soil moisture content; △ H = H f + h d =- hh d， h d This represents the soil air intake suction value. For the known △ t j Where j = 1,2,...N, z fe To reach a defined soil moisture profile depth Z j The time taken is calculated using the following formula. (19) Furthermore, for a given initial soil moisture content distribution profile i s - i i ( z A discrete soil infiltration capacity curve can be calculated. J w0 ( z fe ) ~ t ( z fe ); Step 3: For the discrete soil infiltration capacity curves obtained in Step 2, use... Philip The parameters of the two slope infiltration formulas are used. α , β By fitting the data, a continuous surface soil infiltration rate curve can be obtained. J w0 ( t ), (20) In the formula, c For slope, t This refers to the infiltration time. According to the principle of mass conservation, a continuous cumulative infiltration curve is obtained. I ( t ) is represented as, (twenty one); Step 4: Based on the rainfall monitoring resolution, divide the non-steady-state rainfall event into segmented steady-state rainfall intervals, and measure the rainfall rate within the nth steady-state rainfall interval. r n ( t ) = r n cos c It can be represented as, (26) In the formula, t n and t n-1 These are the rainfall durations at the end of the nth and (n-1)th segments, respectively. R ( t n )and R ( t n-1 ) represent the cumulative rainfall at the end of the nth and (n-1)th segments, respectively; △ t n The duration of the rainfall interval in the nth segment; Step 5, for the first segment of steady-state rainfall, the rainfall rate is r 1cos c Time of water accumulation within the interval t p Virtual Time t s And the cumulative infiltration amount when water accumulates under steady-state rainfall. I p It is calculated using the following formula. (twenty three) (twenty four) (25) The soil infiltration capacity at this point is described by the following formula: (twenty two) Cumulative infiltration I ( t 1) is represented as: (27) Cumulative runoff F ( t 1) is represented as, (28); Step 6: Calculate the soil infiltration capacity at the end of the (n-1)th segment based on the time compression method. J w0 ( t sn-1 ), where n > 2, In equation (29), t sn-1 This represents the virtual time at the end of the (n-1)th steady-state rainfall interval. I ( t n-1 () represents the cumulative soil infiltration at the end of the n-1 segment of steady-state rainfall; Step 7, compare soil infiltration capacity J w0 ( t sn-1 ) and rainfall rate r n cos c The magnitude of the value determines the timing of water accumulation and the amount of runoff generated within the nth steady-state rainfall interval. a). When J w0 ( t sn-1 ) ≤ r n cos c This indicates that during the nth steady-state rainfall interval, water accumulation will occur instantaneously or continue to occur. The soil infiltration capacity at this time is described by formula (22). t p = t n-1 , t s = t sn-1 Cumulative infiltration I ( t n Calculated using the following formula: (31) Cumulative runoff F ( t n Calculated using the following formula: (32) b) When J w0 ( t sn-1 ) > r n cos c This indicates that during the nth steady-state rainfall interval, water accumulation will not occur instantaneously, and the duration of water accumulation within the nth steady-state rainfall interval is [not specified]. t pn Calculated using the following formula: (33) if t pn ≥ △ t n This indicates that no water accumulation occurs in this area, so the cumulative infiltration rate at this time is... I ( t n and cumulative runoff F ( t n ) are respectively represented as: (35) (36) Conversely, if t pn <△ t n The soil infiltration capacity at this time is described by the following formula ( t p = t n-1 + t pn ) (twenty two) Cumulative infiltration I ( t n and cumulative runoff F ( t n ) are respectively represented as: (37) (38).

2. The slope rainfall infiltration-runoff calculation method applicable to arbitrary initial water content distribution as described in claim 1, characterized in that: In step 2, the formula (17) J w0 ( z fe = Z j First, follow the formula below: (18) Perform fixed-point iteration to obtain z fw ,Will z fw Substitute into formula (8). (8) get J w0 ( z fe = Z j ); The unknown parameters in the formula are defined as follows: (39) in, n It is the soil pore size distribution index; i i This represents the initial moisture content of the soil. i r This refers to the residual soil moisture content; i s This represents the saturated water content. K s The saturated hydraulic conductivity of the soil; S i The effective soil water saturation corresponds to the initial soil moisture content; m = 3n + 2.

3. The slope rainfall infiltration-runoff calculation method applicable to arbitrary initial water content distribution as described in claim 1, characterized in that: In step 6, based on the saturated hydraulic conductivity of the soil... K s and infiltration parameters α , β， According to the following formula: (30) Execute the fixed-point iterative algorithm to obtain the virtual time at the end of the rainfall interval of the (n-1)th segment. t sn-1 ,Will t sn-1 Substituting into formula (20), we get J w0 ( t sn-1 ).

4. The slope rainfall infiltration-runoff calculation method applicable to arbitrary initial water content distribution as described in claim 1, characterized in that: In step 7, the parameters in formula (22) t s Calculate using the following formula, (twenty four).

5. The slope rainfall infiltration-runoff calculation method applicable to arbitrary initial water content distribution as described in claim 1, characterized in that: The N value in the described step 1 is at least 101.

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