# Effect of Rainfall Intensity on Infiltration into Partly Saturated Slopes

## Authors

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s10706-007-9157-0

- Cite this article as:
- Xue, J. & Gavin, K. Geotech Geol Eng (2008) 26: 199. doi:10.1007/s10706-007-9157-0

- 20 Citations
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## Abstract

This paper describes the development of a model to analyse the rate of infiltration and run-off experienced by a partly saturated soil slope during rainfall. The paper first reviews some of the most popular infiltration models used in geotechnical analysis, and highlights some of the problems associated with their application. One particular model, the Horton Equation is extended to include rainfall intensity directly in its formulation. The new model is shown to predict infiltration responses, which agree with field measurements. In the final section the influence of the rainfall intensity and pattern of rainfall (variation of rainfall intensity) on the infiltration response of a soil is investigated using the new model.

### Keywords

Water flowUnsaturated soil slopesRainfall patternRunoffCumulative infiltration## 1 Introduction

When rain falls on an unsaturated soil slope, a portion of the total rainfall infiltrates into the soil, whilst the deficit (total rainfall minus total infiltration) will run-off the surface. Water that percolates into the slope increases the water content of the soil and reduces in-situ suction, thereby decreasing the infiltration capacity of the soil. For this reason, the proportion of the total rainfall that results in infiltration and run-off changes continuously during a rainfall event. The process of the reduction of in-situ suction, results in a decrease in the effective stress (and therefore strength) in the near surface soils, and may result in slope failure (Fourie et al. 1999). Whilst geotechnical engineers are therefore exclusively concerned with predicting the amount of infiltration that will result under a given rainfall intensity, hydrologists are primarily interested in the amount of run-off for catchment flooding studies. This paper considers the effect of rainfall intensity on the proportion of rainfall which will infiltrate into an unsaturated soil slope during a given rainfall event.

## 2 Classical Rainfall-runoff Models

Three infiltration models currently used in geotechnical analyses are reviewed in this section. These include the Green–Ampt (1911) model, Mein–Larson (1973) model and Horton’s equation (Jury and Horton 2004).

### 2.1 Green–Ampt model

*t*can be calculated:

In which: *i* is the infiltration capacity; *Z*, the depth of wetting front; *H*, the depth of ponded water; ψ, the suction head at the wetting front; *K*_{s}, saturated soil permeability.

In rainfall induced slope failures, the wetting front and the slip surface are coincident, as reduced suction due to infiltrating water is the trigger for failure to occur. Forensic investigations of slope failures caused by rainfall Olivares and Picarelli (2003), Springman et al. (2003) reveal that the soil above the slip surface is often partly saturated at failure. The assumption that suction in the wetted zone is zero is therefore questionable. The Green–Ampt model has been shown to greatly under-predict the time for a wetting front depth to form in partly saturated soil Gavin and Xue (2007). This is predominantly due to the use of *K*_{s}, in the formulation. Since the soil in a slope fails prior to reaching full saturation, the operational permeability (*K*) is lower than *K*_{s}, with Bouwer (1966) suggesting that *K* ≈ 0.5*K*_{s}.

### 2.2 Mein–Larson Model

At the start of a rainfall event, because of the large suctions present in the unsaturated soil the infiltration capacity of a partly saturated slope is initially high. It is common during the early stages of a rainfall event for the infiltration capacity to exceed the rainfall intensity. Therefore the amount of water, which can infiltrate into the soil, is controlled by the rainfall intensity. As rainfall continues the near surface suctions, and therefore the infiltration capacity, reduce. At some point the rainfall intensity may exceed the infiltration capacity, and the rate of flow into the soil becomes controlled by the infiltration capacity of the soil.

Assuming that the rainfall intensity is constant throughout a rainfall event, they describe a range of possible soil responses: Case A: In this scenario the rainfall intensity (*R*_{i}) is lower than the saturated permeability (*K*_{s}) of the soil. All rainfall is assumed to percolate into the soil, and the infiltration rate remains constant (equal to the rainfall intensity) throughout the event. Cases B and C: In these scenarios *R*_{i} is greater than *K*_{s}. During the initial stage of the rainfall event, the infiltration capacity exceeds *R*_{i}, and all water infiltrates into the soil. At some point (*T*_{p}) the near surface soils become saturated and run-off begins. For a given initial condition, the time *T*_{p}, will depend on the rainfall intensity, with more severe events (higher *R*_{i}) resulting in rapid saturation of near surface soil. Once the near surface soils become saturated the rainfall intensity exceeds the infiltration capacity (*R*_{i} ≥ *i* > *K*_{s}), ponding begins and is followed by runoff over the ground surface. The infiltration rate starts to decrease when *t* > *T*_{p}. Case D: *R*_{i} > *I* > *K*_{s}. In situations where the rainfall intensity is higher than the infiltration capacity at the start of the rainfall event, the response will be capacity controlled throughout the rainfall event. Surface run-off and decrease of the infiltration rate begin once rainfall commences.

^{−6}m/s, overlying a 4–5 m deep layer of purple clayey silt with

*K*

_{s}= 1.67 × 10

^{−7}m/s. A simulated rainfall event, with an intensity of 47 mm/h (flow rate = 13 × 10

^{−6}m/s or 2.5

*K*

_{s}) and duration of 73.3 min was applied to the slope. The proportion of infiltration and run-off recorded is shown in Fig. 3, where the infiltration rate is seen to decrease to a limiting value of 2 × 10

^{−6}m/s (which corresponds to 0.4

*K*

_{s}) after approximately 65 min.

^{−6}to 1 × 10

^{−5}m/s. The authors recorded the infiltration rate due to rainfall over a number of days in August and September 2001. The rainfall intensity during this period ranged from 2.8 × 10

^{−7}m/s to 2.3 × 10

^{−6}m/s, i.e. close to, or lower than

*K*

_{s}. The infiltration coefficient (defined as the ratio of the infiltration rate to rainfall intensity) measured during this period is shown in Fig. 4, together with the ratio of the rainfall intensity to the saturated soil permeability (assuming

*K*

_{s}= 1 × 10

^{−6}m/s). Both data sets contradict the assumption of the Mein–Larson model in showing that significant run-off occurs when the rainfall intensity is below

*K*

_{s}.

### 2.3 Horton Equation

*t*), where:

*i*

_{0}is the initial infiltration capacity at

*t*= 0;

*i*

_{f}, the steady state

*final*infiltration capacity; β, a constant which describes the rate of decrease of the infiltration capacity;

*t*, time.

*R*

_{i}is the rainfall intensity and S

_{m}is the maximum storage capacity of soil in the layer above the location to be examined.

- (i)
The initial moisture content is zero.

- (ii)
The infiltration rate is high enough to maintain the rate of infiltration at the infiltration capacity.

- (iii)
The rate of percolation is proportional to the moisture content of the near surface soil.

## 3 Proposed Method

### 3.1 Infiltration Capacity

The problem of modelling complex material response, such as infiltration into unsaturated soil, arises due to the large number of variables, which affect the solution. The parameter β in Hortons’ equation is controlled by many variables, including the soil type, the initial water content, surface conditions (e.g. whether the slope is vegetated), the slope, rainfall characteristics, location on the slope and sub-surface drainage, amongst other factors. It is not possible (or in some cases desirable) to include every possible variable in the formulation of a model because of the lack of experimental data and the need for simplicity.

In the model, four parameters are used to describe the variation of infiltration capacity with time: rainfall intensity (*R*_{i}), saturated soil permeability (*K*_{s}), initial and final infiltration capacity (*i*_{0} and *i*_{f}). The time is in units of hours, while the initial and final infiltration capacity and saturated permeability and are in m/s.

The last three parameters (*K*_{s}, *i*_{0} and *i*_{f}), are material parameters, with the saturated soil permeability being unique for the soil, and *i*_{0} and *i*_{f} being dependent primarily on the soil type, water content and the surface condition (Miyazaki et al. 1993). Whilst *K*_{s} can be measured using routine laboratory tests, *i*_{0} can be related to the in-situ suction or water content monitored using in-situ devices and *i*_{f} can be established in the laboratory or on site by establishing the residual permeability of a sample of soil with a constant water supply and a head differential equal to that which might reasonably be expected in-situ.

*R*

_{i}/

*i*

_{f}) which is a measure of the ratio of the rainfall intensity to the final infiltration capacity was varied from 0.5 to 2, whilst two initial conditions were considered. In the first the ratio of the initial to the final infiltration capacity (

*i*

_{0}/

*i*

_{f}) was 5, and in the second this ratio was doubled (to model the effect of a prolonged dry period with high initial infiltration capacity). A number of trends are noteworthy:

- (i)
The model clearly captures the effect of varying rainfall intensity on the degradation of the infiltration capacity. For example in the case where

*i*_{0}/*i*_{f}= 10, when the relative rainfall intensity*R*_{i}/*i*_{f}= 2, the infiltration capacity equals i_{f}after just 2 h. In contrast, when the rainfall intensity is halved*R*_{i}/*i*_{f}= 1, the time required for the infiltration capacity to reach equilibrium increases to in excess of 5 h. - (ii)
The exponential form of the infiltration capacity decay curves shown in Fig. 6 highlights that the analysis is more sensitive to assumptions made about the relative rainfall intensity, than it is to the value

*i*_{0}/*i*_{f}(which reflects the initial condition). For example, comparing the response at a time of 2 h, in the analyses where*i*_{0}/*i*_{f}= 10, we note that the range of current infiltration capacity is very sensitive to*R*_{i}/*i*_{f}, with*i*/*i*_{f}increasing from 1 to 5, as*R*_{i}/*i*_{f}increases from 0.5 to 2. In contrast if we compare the response of samples at a given relative rainfall intensity*R*_{i}/*i*_{f}= 2, and*i*_{0}/*i*_{f}of 5 and 10, we see that the predicted response converge quickly.

### 3.2 Time to Runoff

*R*

_{i}<

*K*

_{s}). At the start of a rainfall event, when the infiltration capacity is at a maximum, run-off will not occur unless the rainfall intensity is extremely high (

*R*

_{i}>

*i*). As percolation of water into the slope continues, the infiltration capacity reduces (as near surface suction reduces). Runoff will occur when the rainfall intensity exceeds the infiltration capacity. The time at which the slope moves from the condition of full infiltration, to when runoff begins is clearly the point at which the rainfall intensity is equal to the infiltration capacity:

*T*

_{p}):

We see from Eq. 6 that *T*_{p} is a function of the rainfall intensity, saturated soil permeability, initial and final infiltration capacity, and that as the rainfall intensity increases, *T*_{p} reduces. If the rainfall intensity is equal to the initial infiltration capacity, then *T*_{p} = 0, and run-off occurs immediately (such behaviour is observed in severe storm events with high initial rainfall intensities).

## 4 Validation of the Model

### 4.1 Boundary Conditions

- (i)If the rainfall intensity is equal to, or lower than the final infiltration capacity then runoff will not occur. In this situation the time for runoff to occur is infinite (
*T*_{p}= ∞). Setting*R*_{i}=*i*_{f}in Eq. 6, we get:$$ \begin{aligned}{} T_p & = \frac{{(i_f )^{3/2} \ln \left[ {(i_0 - i_f )/(i_f - i_f )} \right]}} {{i_f (K_s )^{1/2} }} \\ & = \infty \end{aligned} $$ - (ii)If the rainfall intensity, at the start of a rainfall event is equal to or exceeds
*i*_{0}, runoff starts at the beginning of rainfall and*T*_{p}is zero (*T*= 0). Accordingly, in Eq. 6, for*R*_{i}≥*i*_{0}we have:$$ \begin{aligned}{} T_p & \le \frac{{(i_f )^{3/2} \ln \left[ {(i_0 - i_f )/(i_0 - i_f )} \right]}} {{i_0 (K_s )^{1/2} }} \\ & = 0 \end{aligned} $$ - (iii)During a prolonged dry period where no precipitation occurs, and assuming that changes in moisture content due to evaporation are negligible, the infiltration capacity should remain unchanged. So for
*R*_{i}= 0 we have:$$ \begin{aligned}{} i & = i_f + (i_0 - i_f )\exp (0) \\ & = i_0 \end{aligned} $$

### 4.2 Model Validation using In situ Test Results

#### 4.2.1 Infiltration Capacity Decay Curves

Rahardjo et al. (2005) report field measurements of the infiltration response of a cutting, with a slope of 2:1, located on the campus of Nanyang Technological University, Singapore. The soil conditions comprise an upper 1.5 m deep layer of orange silty clay, with the saturated permeability at about 5.18 × 10^{−6} m/s, underlain by a lower permeability (*K*_{s} = 1.67 × 10^{−7} m/s), 4–5 m deep layer of purple clayey silt.

Natural and artificial rainfall was applied to an instrumented section of the slope, which covered a plan area of 20–25 m^{2}. The rainfall intensity was recorded using a tipping bucket rain gauge, whilst the run-off was measured using a flume at the base of the slope. Infiltration was not measured directly but was taken as the difference between the rainfall intensity and run-off.

- (i)
At the start of each test the infiltration rate was equal to the rainfall intensity, and no run-off occurred.

- (ii)
The time (

*T*_{p}) at which runoff began, decreased with increasing rainfall intensity. Once runoff began (*t*>*T*_{p}), the infiltration rate decreased, and run-off increased with time in all tests. - (iii)
With the exception of the first test (47 mm/h rainfall intensity), run-off continued after the end of the application of rainfall to the slope, indicating that subsurface (lateral) flow was occurring. This suggests that the assumption that the infiltration can be calculated by subtracting the run-off (which occurred during rainfall) from the rainfall intensity, resulted in an overestimate of the actual infiltration for the three high intensity rainfall events.

*K*

_{s}and

*R*

_{i}which were used in the analysis. The value of

*i*

_{f}was taken directly from measured field data for the 47 mm/h rainfall event shown in Fig. 7a, where no run-off was measured after the end of the rainfall event. However, the continuation of run-off after the end of the heavier rainfall (shown in Fig. 7b), suggests that the shape of the measured infiltration capacity decay curve, and particularly the

*i*

_{f}value measured is incorrect (the

*i*

_{f}value would be expected to be too high). Rahardjo et al. (2005) note that the initial and final volumetric content near the toe of the slope always appeared to be higher than those at the crest, suggesting that some water that infiltrates near the crest of the slope may then flow downslope, parallel to the slope surface, thus resulting in higher water content and lower suction near the toe of the slope than near the crest. The same

*i*

_{f}value was therefore adopted for both analyses. The final parameter needed was

*i*

_{0}. Although this was not quoted in the paper (and was naturally variable as it is affected by antecedent rainfall), it was noted that at the start of each experiment the infiltration rate was equal to the rainfall intensity, the initial infiltration capacity exceeded the rainfall intensity in all cases. Therefore, analyses were carried out for a range of assumed

*i*

_{0}values varying from 80 (which is just above the highest rainfall intensity) to 200 mm/h.

The prediction for the 47 mm/h rainfall event is seen to be quite accurate, and as noted, is relatively insensitive to the value of *i*_{0} (which was the only unknown parameter) assumed in the analysis, although the time calculated for runoff to occur time is quite sensitive to the *i*_{0} value assumed. For the higher rainfall intensity rainfall pattern shown in Fig. 7b, the model clearly over-predicts the rate of decay of the infiltration capacity, however, this is primarily due to the differences between the assumed and measured *i*_{f} value.

#### 4.2.2 Total Infiltration

*I*

_{c}) into an unsaturated soil. The total infiltration can be calculated by integrating Eq. 4:

*T*

_{D}is the total duration of the rainfall event and

*i*is the infiltration capacity from Eq. 4.

*i*

_{0}) decreases.

## 5 Effect of Rainfall Intensity and Pattern on Infiltration Response

In this section the effects of rainfall intensity and rainfall pattern on the infiltration response of a partly saturated slope are examined using the new model.

### 5.1 Influence of Rainfall Intensity

One of the drawbacks identified with the original Horton Equation was that the infiltration capacity decay curve was not influenced by the rainfall intensity in a transparent way. The β term, which controls the rate of decay of the infiltration capacity, is usually determined based on the measured field response. A value derived in this way will reflect only the average rainfall during the field-monitoring period. Horton’s equation cannot, therefore account for variations in rainfall intensity, during an infiltration event.

*R*

_{1}) is high. At time

*t*

_{1}, the rainfall intensity reduces (

*R*

_{2}<

*R*

_{1}) and the slope of the infiltration capacity decay curve reduces.

### 5.2 Complex Rainfall Patterns

During a natural rainfall event, the rainfall intensity will not be constant. Given that the rate of decay of the infiltration capacity decay curve is affected by the rainfall intensity, analyses were carried out to study the infiltration and runoff characteristics of soil subjected to complex rainfall patterns.

*i*

_{f}and

*K*

_{s}for NSCL = 10 mm/h), and a lower permeability soil (

*i*

_{f}and

*K*

_{s}for CSL = 5 mm/h). In the analyses, three different rainfall intensities: 10, 20 and 40 mm/h were used. The total precipitation in all analyses was equal to 40 mm. The following rainfall patterns were considered:

- 1:
Heavy-Light (H-L) pattern—Rainfall starts at an initial high intensity (20 mm/h), for a period of 1 h, then becomes lighter (10 mm/h), for a 2-h period.

- 2:
Light-Heavy-Light (L-H-L) pattern—Rainfall starts at an initial low intensity (10 mm/h) for a period of 1 h, then becomes heavier (20 mm/h), for 1 h, followed by low intensity (10 mm/h) rainfall for the final hour.

- 3:
Light-Heavy (L-H) pattern—Rainfall starts at an initial low intensity (10 mm/h), for a period of 2 h, this is followed by a 1-h period of high intensity rain (20 mm/h).

- 4:
Heavy (H) pattern—Rainfall is applied at a very high intensity (40 mm/h) for a period of 1 h.

*i*

_{0}) for the soils and an

*i*

_{0}value of 0.29 cm/min (equivalent to 174 mm/h) was assumed for all cases in the original paper.

- (i)
From Fig. 10a–c, which describes the response of NSCL during the 3-h rainfall event, the amount of run-off generated is low. Run-off is only predicted in for the L-H pattern (Fig. 10c), when

*t*> 2.5 h. - (ii)
The responses of CSL during the 3-h rainfall event are shown in Fig. 10e–g. It is clear that the amount of run-off is significantly greater than that generated for the NSCL, this is because of the lower final infiltration capacity of CSL. The total run-off predicted for the two rainfall patterns (L-H and L-H-L), when applied to the CSL are similar, at 34% and 35% of the total rainfall. However, the time at which the run-off occurs is quite different, with the majority of run-off occurring during the period 1–2 h for the L-H-L pattern, and during the 2–3 h period for the L-H pattern. The amount of run-off predicted for the H-L pattern is significantly lower at the period where the rainfall intensity is highest, which coincides with the time when the infiltration capacity of the soil is greatest.

- (iii)
Comparing the response of both soil types to the heavy rainfall pattern (Fig. 10d and h), we see that although the total rainfall, rainfall intensity and initial infiltration capacity are identical, the final infiltration capacity, which is higher for the NSCL, significantly affects both the time to runoff and the proportion of run-off which occurs.

### 5.3 Average Rainfall Intensity

## 6 Conclusion

A model to predict the infiltration response of partly saturated soil slopes is proposed. The model suggests that the amount of infiltration, and run-off, which result from a given rainfall intensity, can be described using the initial and final infiltration capacity, the saturated permeability of the soil and the rainfall intensity. Whilst, the final infiltration capacity and the saturated permeability can be easily obtained in the laboratory, and the rainfall intensity can be assigned based on statistical analysis of rainfall records in the area, the initial infiltration capacity depends on the in-situ conditions and will be strongly affected by antecedent rainfall conditions. The new model considers one-dimensional vertical infiltration only and does not consider the effects of lateral flow or downslope subsurface drainage.

- (i)
Because of the exponential form of the proposed model, predictions were relatively insensitive to

*i*_{0}values used in analyses. Although*i*_{0}does affect the accuracy of the prediction of the time when run-off begins. However, the accurate modelling of run-off response is not critical to geotechnical analysis of infiltration problems. - (ii)
The amount of infiltration (or run-off), which occurs, is particularly sensitive to the rainfall pattern. If the total infiltration is known, the change of the volumetric water content of the near surface soil (reduction in suction) can be calculated and an assessment of the effect of these changes on the stability of the slope can be made. For a given total rainfall, rainfall events which begin at high intensity and end at low intensity generate less run-off than those which start at low intensity and end high. This has significant implications for slope stability analyses. In the case where runoff is low, a greater portion of the total rainfall enters the soil, thereby reducing near surface suction and leading to a greater risk of failure.

## Acknowledgements

This research forms part of a project funded by *Iarnród Éireann.* The authors wish to thank Mr. Brian Garvey, Chief Civil Engineer with *Iarnród Éireann* for technical and financial assistance received. The first author was the recipient of a Geotechnical Trust Fund award from the Geotechnical Society of Ireland.