Engineering Geology 105 (2009) 32–43

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Engineering Geology

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Infiltration analysis to evaluate the surficial stability of two-layered slopesconsidering rainfall characteristics

Sung Eun Cho ⁎Korea Institute of Water and Environment, 462-1, Jeonmin-Dong, Yusung-Gu, Daejon 305-730, Republic of Korea

⁎ Tel.: +82 42 870 7632; fax: +82 42 870 7639.E-mail address: drsecho@hanmail.net.

0013-7952/$ – see front matter © 2008 Elsevier B.V. Adoi:10.1016/j.enggeo.2008.12.007

a b s t r a c t

a r t i c l e i n f o

Article history:

Shallow slope failures in r Received 21 July 2008Received in revised form 7 December 2008Accepted 19 December 2008Available online 3 January 2009

Keywords:InfiltrationUnsaturated soilSlope stabilitySeepageRainfallSuction

esidual soil during periods of prolonged infiltration are common throughoutthe world. Using a one-dimensional infiltration model and an infinite slope analysis, this study examines anapproximate method of determining how infiltration influences the surficial stability of two-layered slopes.The method extends Moore's infiltration model, which is based on the Green–Ampt model, to cover moregeneral situations, including those where water moves upward from a perched water table in decreasinglypermeable soil. The method has also been used to evaluate the likelihood of a shallow slope failure beinginduced by a particular rainfall event. In making this evaluation, the method takes into accounts the rainfallintensity and duration of various return periods in a two-layered soil profile. A comparison of the results ofthe infiltration model with the results of numerical analyses shows that, with the use of properly estimatedinput parameters, the proposed model compares reasonably well with other model that rely on morerigorous finite element method.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Rainfall leads to the development of a perched water table, a rise inthe main groundwater level, and surface erosion; and, as the moisturecontent increases, the unit weight increases. Moreover, the instabilityof unsaturated residual soil slopes during wet periods is commonthroughout the world (Morgenstern and de Matos, 1975; Fukuoka,1980; Brand, 1984; Vargas et al., 1986; Kim et al., 1991). These failuresare generally shallow and the failure surfaces are usually parallel tothe slope surface. Therefore, this paper reviews an infinite slopeanalysis to estimate the influence of infiltration on surficial stability ofslopes by the limit equilibrium method.

A better understanding of the process of rain infiltration and itseffects on unsaturated soils is closely related to the assessment ofslope stability. To estimate the hydraulic response, continual fieldmonitoring of the groundwater conditions is needed to avoid dif-ficulties in obtaining the parameters required for theoretical analyses.In most cases, however, it is not feasible to continually monitorchanges in field conditions (Sun et al., 1998).

On the other hand, during the past few decades, many analyticaland numerical solutions to the unsaturated flow equation have beendeveloped and have become a necessary tool. Although a considerableamount of research has been conducted to clarify the various failuremechanisms based on these solutions (Collins and Znidarcic, 2004;Zhan and Ng, 2004), it is still difficult to develop a theoretical frame-work that incorporates these various failure mechanisms.

ll rights reserved.

One way to evaluate the effect of infiltration is to use one-dimensional infiltration models. These models are based on widelyaccepted concepts of soil physics and are relatively easy to use.However, for simplification, most of these models are limited to caseswith uniform initial conditions and homogeneous soils. Because thenatural soil profile is typically nonhomogeneous, the approximationsdeveloped for homogeneous soils have been adapted to show howinfiltration behaves in layered soils.

The main objective of this study, mainly focuses on a decreasingconductivity soil profile, is to propose an approximate method ofevaluating the likelihood that a particular rainfall will induce a shallowfailure in a two-layered slope. The method takes into account therainfall intensity and the duration of various return periods. To capturethe principalmechanisms involvedwhenwater infiltrates two-layeredsoil slopes, the results of a series of finite element analyses of a one-dimensional seepage flow are presented. On the basis of these results,this paper presents an infiltration model that is an extended formof Moore's infiltration model to evaluate how water infiltrates two-layered slopes under various intensities of rainfall.

2. Analysis of the slope stability in relation to infiltration

2.1. Slope failures due to rain infiltration

According to previous studies on the way rain infiltration affectsthe stability of slopes, rain induces a rise in the groundwater leveland an increase in pore pressure, both of which decrease the effectivestress and shear strength, thereby resulting in slope failures. Hence,most traditional analyses of slope stability incorporate rainfall

Fig. 1. Infinite slope failure in a two-layered soil profile.

33S.E. Cho / Engineering Geology 105 (2009) 32–43

influences; that is, they change the groundwater flow patterns withincreasing pressure heads and they often assume that the phreaticsurface rises to coincide with the surface of the slope (Collins andZnidarcic, 2004). However, it is now well known that slope failuresalso occur when the shear strength provided by matric suctiondecreases enough to trigger the failures (Brooks and Richards, 1994;Rahardjo et al., 1994; Terlien, 1998; Van Asch et al., 1999).

When rainwater infiltrates through an unsaturated zone, theadvancement of the wetted zone near the slope surface may lead tofailure during periods of prolonged rainfall. These failures are usuallycharacterized by shallow failure surfaces that develop parallel to theslope surface (Rahardjo et al., 1994; Fourie et al., 1999). A simpleinfinite slope analysis method can therefore be used to estimate thefactor of safety.

For two-layered soil slopes with a surface layer of thickness L1, thelimit equilibriummethod can be readily applied to calculate the factorof safety as shown in Fig. 1. The shear resistance that is associated withthe net normal stress and the matric suctionwithin the slice mass canbe characterized as follows by the modified Mohr–Coulomb failurecriterion (Fredlund et al., 1978) for unsaturated soil:

Fs =τfτm

=cV+ σn − uað Þ tan/V+ ua − uwð Þ tan/b

τm

=2cV

γ1L1 + γ2 zw − L1ð Þf g sin2α +tan/Vtanα

− 2ua tan/Vγ1L1 + γ2 zw − L1ð Þf g sin2α

� �

+2 ua − uwð Þ tan/b

γ1L1 + γ2 zw − L1ð Þf g sin2α

ð1Þ

where τm is the shear stress at any point along the slip surface, τf isthe shear strength at the corresponding point, α is the slope angle, Wis the weight of a slice with unit width, c′ is the cohesion intercept, σn

is the total normal stress, ua is the pore air pressure, uw is the porewater pressure, (σn−ua) is the net normal stress, (ua−uw) is thematric suction, ϕ′ is the effective angle of friction, and ϕb is the anglethat defines how the shear strength increases with the increase inmatric suction. If the soil is fully saturated at a certain depth, then theair pressure becomes equal to the water pressure (that is, ua=uw).This approach uses two independent stress variables that enable aconsistent approach to be used when analyzing stability, irrespectiveof whether the pore pressures are positive or negative. In thisequation, the first term is related to the cohesion, the second term isrelated to the effective angle of friction, and the last term is related tothe matric suction.

When an advancingwet front from rainfall infiltration saturates theupper soil layer, if the rainfall intensity is greater than the hydraulic

conductivity of the sublayer, a rise in the groundwater tablemayoccur.The rise in the ground water table induces the condition of a flowparallel to the slope and hydrostatic state, resulting in the followingexpression for the pore pressure at a depth of zw from the water table:

uw = γwzw cos2 α ð2Þ

where γw is the unit weight of water. Then Eq. (1) can be reduced tothe classical solution for infinite slope in the saturated slope.

When the effective cohesion of soil is zero (c′=0), and the slopeangle (α) is greater than or equal to the effective internal friction angleof the soil (ϕ′), the unsaturated soil slope fails from a loss of apparentcohesion upon saturation of the soil by the infiltrating wetting front.In this case, failure from a reduction in the shear strength because of arise in the ground water table or the occurrence of a perched watertable is unlikely since the slope can only be stable with the shearstrength due to the matric suction that fully disappears beforesaturation is achieved. However, if the slope angle does not exceed theeffective friction angle of the soil, slopes are not susceptible to failurefrom the loss of matric suction since the slopes remain stable withoutthe additional shear strength due to the matric suction; yet the slopeswill fail from a reduction in the effective stress in the saturatedcondition that results in the reduction in the shear strength of soil.

When a slope possess an effective cohesion component (c′), andthe effective cohesion of the soil is adequate, even a partially saturatedslope with a slope angle that is greater than the effective friction anglecan remain stable in spite of a complete loss of apparent cohesion.However, even these types of slopes would fail in the saturatedcondition if an increase in pore pressure reduces the effective stress(Sudhakar, 1996).

3. Infiltration model

A key factor that dominates slope stability is the hydrologicalresponse associated with infiltration. Hence, the soil–water profilemust be reviewed during rainfall infiltration of unsaturated soil toevaluate the effect of infiltration on slope stability. Due to thecomplexity of the problem, numerical methods, such as the finiteelement method, are commonly used to solve the partial differentialequation that governs the seepage. However, to get solutions forvarious boundary conditions is laborious and computationally expen-sive as the numerical solution requires an implicit iterative techniquewith fine spatial and time discretisation.

Another way to evaluate the effect of infiltration is to use a one-dimensional infiltration model. Green and Ampt (1911) first deriveda physically based model by describing the infiltration capacity of a soilfor ponded surfaces. The Green–Ampt model has received considerable

Fig. 2. Conceptual water content profile for two-layered soil.

34 S.E. Cho / Engineering Geology 105 (2009) 32–43

attention in recent years and, although it is an approximate equation,it has been shown to have a theoretical basis, as well as measurableparameters.

Mein and Larson (1973) developed a simple two-stage modelfor predicting infiltration before and after surface ponding; theybased their model on the Green–Ampt model by assuming the initialmoisture content was uniform under rainfall with a constant intensity.

On the basis of the one-dimensional solution, Morgenstern and deMatos (1975), Lumb (1975) and Gavin and Xue (2008) calculated theadvancement of the wetting front, and Vargas et al. (1986) studied theimportance of the rate of rainfall on slope stability. In the method ofPradel and Raad (1993), which is based on the Green–Ampt model,two conditions must be satisfied to saturate the soil to the criticaldepth at which the slope can fail. These conditions are, firstly, that therain intensity must be greater than the infiltration capacity of the soiland, secondly, that the rainfall must be longer than the critical timenecessary to saturate the soil to the critical depth. Fourie et al. (1999)applied the method of Pradel and Raad to a field problem.

The approximate method of Cho and Lee (2002) is a modified formof the Pradel and Raad method. Based on the Mein and Larson model,themethod of Cho and Lee takes into account the rainfall intensity andduration of various return periods in order to evaluate the likelihoodthat a particular rainfall will induce a shallow slope failure.

Although thesemethods are limited by the assumption that rainfallinfiltrates homogeneous soil, the approximate methods provide apractical tool for assessing the hydraulic response to infiltration.

3.1. Hydraulic characteristics

There are two important hydraulic properties associated withinfiltration into unsaturated soils, namely, the soil–water characteristiccurve and unsaturated hydraulic conductivity. A complete description ofthe infiltration process entails the characterization of these properties.

The soil–water characteristic curve relates the water content of asoil to the matric suction, and a number of empirical and semi-empirical functions have been proposed in the past to represent thesoil–water characteristic curves.

The unsaturated hydraulic conductivity that varies with the soilwater content is usually predicted on the basis of the measured soil–water characteristic curve due to the difficulty associated with themeasurement of unsaturated hydraulic conductivity. The variationof unsaturated hydraulic conductivity is commonly represented bythe relative hydraulic conductivity function, which represents a ratioof the unsaturated hydraulic conductivity to the saturated hydraulicconductivity.

4. Infiltration of two-layered soils

To model the infiltration process in a layered soil under anunsaturated condition has been a great concern in many engineeringand scientific fields related to soils (Srivastava and Yeh, 1991; Chooand Yanful, 2000). Although some analytical and conceptual modelsfor transient vertical infiltration in layered soils have been presented,the solutions are too complex for practical applications.

Extensions of the Green–Ampt approach to describe the infiltra-tion in nonuniform soils have been made (Bouwer, 1969; Childs andBybordi, 1969; Fok, 1970; Ahmed et al., 1980; Moore and Eigel, 1981;Moore,1981; Flerchinger et al., 1988). However, few have been appliedto the problem of slope stability analysis.

Moore (1981) modified the Mein and Larson model to describeinfiltration for delayed ponding with nonuniform soils when the con-stant intensity of the rainfall is greater than the saturated hydraulicconductivity of the soil. According to Moore and Eigel (1981), themodel shows good to excellent agreement with the numerical model,thereby indicating that Moore's model closely represents the perfor-mance of the Richards' equation.

Although the natural soil profile is typically nonhomogeneousand the hydraulic conductivity may progressively change with depthaccording to the degree of weathering, few applications of the Green–Ampt equation to consider the progressive change in the hydraulicconductivity have been developed due to the complexity of the pro-blem and the basic assumption of the Green–Ampt model. Only afew researches that conceptualize the soil profile by dividing it intomultiple internally homogeneous layers have been conducted(Bouwer, 1969; Childs and Bybordi, 1969).

In this study, the proposed model based on the Green–Ampt modelassumes a piston-typewater contentprofilewith awell-definedwettingfront; therefore, the approximations developed for homogeneoussoils have also been adapted to showhow infiltration behaves in layeredsoils.

4.1. Moore's model

Fig. 2 shows the conceptual two-layered soil profile used in thederivation of Moore's model. The soil consists of a surface layer ofthickness L1, with an initial moisture deficit, Δθ1, and the soil ischaracterized by hydraulic conductivity, K1. Below this layer, the soil ishomogeneous and semi-infinite with an initial moisture deficit, Δθ2.The soil is also characterized by hydraulic conductivity in the wettedzone of K2.

When the rainfall has a constant intensity, i, that is greater than thesaturated hydraulic conductivity of the soil, the governing equationscan be expressed as follows: f= i until t= tp, where tp=Fp/i and Fp isgiven by either

Fp =Δθ1ψf1

i = K1ð Þ− 1; zp = Fp =Δθ1 for zp V L1 or ð3Þ

Fp =H − E i= K2ð Þi= K2ð Þ− 1

+ F1; zp =Fp − L1 Δθ1 − Δθ2ð Þ

Δθ2for zp > L1; ð4Þ

where f=the infiltration rate, F=the cumulative infiltration, Fp=the cumulative infiltration at the time of the surface ponding,H=Δθ2(L1+ψf2), E=L1Δθ2(K2/K1), F1=L1Δθ1, zp=the depth ofthe wetting front at the time of the surface ponding, tp=the time of thesurface ponding,ψf1=the suction head at thewetting front in the surface,and ψf2=the suction head at the wetting front in the subsurface layer.

Fig. 3. Conceptual water content profile for two-layered soil (K2b ibK1).

35S.E. Cho / Engineering Geology 105 (2009) 32–43

For t> tp the infiltration capacity is given by either

f = K1 1 +Δθ1ψf1

F

� �for zw V L1or ð5Þ

f = K2H + F − F1E + F − F1

� �for zw > L1; ð6Þ

where zw=the depth of the wetting front. The correspondingcumulative infiltrations are given by either

F = Fp + K1 t − tp� �

+ ψf1Δθ1 lnF + ψf1Δθ1Fp + ψf1Δθ1

!for zw V L1 or ð7Þ

F = Fp + K2 t − tp� �

− E − Hð Þ ln F + H − F1Fp + H − F1

!for zw > L1: ð8Þ

Eqs. (7) and (8) are implicit, and can therefore be solved by aniterative procedure or by the method presented in Appendix A.

4.2. Model parameters

For Moore's model, the following parameters must be estimated:K1, K2, Δθ1, Δθ2, ψf1 and ψf2. The moisture deficit from saturation, Δθ1,is specified as the initial condition when considering the antecedentmoisture contents. If the effect of air entrapment is unimportant, thevalue of the hydraulic conductivity above the wetting front K1 is thesaturated hydraulic conductivity. The suction head at the wettingfront, ψf1, is a function of the soil water content and can be determinedfrom experimental measurements or from the following equation(Mein and Farrel, 1974):

ψf =Z ψi

0Kr ψð Þdψ; ð9Þ

where ψi corresponds to the suction value at the initial moisturecontent and Kr is the relative hydraulic conductivity function.

Estimates of K2, Δθ2 and ψf2 depend on whether the profile of thetwo-layer soil consists of fine-over-coarse stratification or coarse-over-fine stratification. For the coarse-over-fine stratification, the finelower layer restricts the infiltration of the profile, and both layers willbe saturated. Therefore, for this case, K2 is the saturated hydraulicconductivity, Δθ2 is the moisture deficit from saturation and ψf2 isdefined by Eq. (9) for the lower layer (Moore, 1981).

For the fine-over-coarse stratification, the surface layer restrictsinfiltration, and the coarse lower layer is not saturated. A procedure isthen needed to estimate the moisture content and the correspondinghydraulic conductivity in the lower layer. Moore (1981) derived thefollowing equation by considering steady state infiltration in two-layered profiles of fine-over-coarse stratification:

K2

ψ θwð Þ + L1ð Þ =K1

L1; ð10Þ

where ψ(θw)=the matric suction head immediately below thesurface layer.

Using the soil–water characteristic curve (SWCC), the hydraulicconductivity curve and the known values of L1 and K1, the remainingparameters can be evaluated by solving Eq. (10) by trial and error.

4.3. Extension of Moore's infiltration model

Moore's infiltration model can be applied when the constantintensity, i, of the rainfall is greater than the saturated hydraulicconductivities in the profile. However, in the coarse-over-fine stra-tification, the surface ponding can also occur as a result of the upwardmovement of the water because the rainfall intensity may be greaterthan the saturated hydraulic conductivity of the lower layer but

smaller than the saturated hydraulic conductivity of the upper layer.Fig. 3 shows the conceptual moisture profile for the conditionof K2b ibK1. Fig. 3 implies that if the application rate has a valuesomewhere between the lowest and highest hydraulic conductivity inthe profile, then subsurface saturation occurs at the interface. If waterapplication is continued, then the upward movement of the waterfrom the perched water table eventually saturates the surface. Zhanand Ng (2004) referred to this effect of soil heterogeneity on the basisof the analytical solution of Srivastava and Yeh (1991).

In such case, the infiltration processes can be conveniently brokendown into the following four conceptual stages, which is confirmedto be reasonable from the results of numerical analysis in a latersection:

(a) Stage I: Until the time t1 when the wetting front reaches theinterface between the soil layers (t≤ t1)

Because the rainfall intensity is smaller than the infiltration capa-city of the soil, all the rain will infiltrate the soil without any runoff(f= i, F= it). Thus, the time at which the wetting front reaches theinterface can be calculated as follows:

t1 =Δθ V1L1

i; ð11Þ

whereΔθ1′=θ1o−θ1i is defined as the difference between the newandinitial volumetric water content.

The new volumetric water content at the surface differs fromthe saturated volumetric water content in the upper soil, θ1s,because the surface ponding does not occur during this stage.Careful consideration is therefore needed to evaluate the volumetricwater content at the surface. If the rain falls on the surface, a wet-ting front is created in the soil. Thus, according to Darcy's law, thehydraulic gradient at the wetting front abruptly increases to deliverthe applied rainwater because the hydraulic conductivity in theunsaturated soil is far smaller than the intensity of the appliedrainfall. As the rain continues to fall, the hydraulic gradient decreasesand consequently becomes equal to unity or very close to unity for asubstantial portion of the soil (Sun et al., 1998) because the hydraulicconductivity of the soil increases to a value that is numericallyequal to the intensity of the applied rainfall behind the wetting front.The matric suction that corresponds to the value that is numericallyequal to the intensity of the rainfall can be estimated from thehydraulic conductivity function. As a result, the volumetric water

Fig. 4. Example1: Hydraulic properties for analysis.

36 S.E. Cho / Engineering Geology 105 (2009) 32–43

content, θ1o, which corresponds to the matric suction, is determinedfrom the SWCC.

(b) Stage II: From t1 to t2 when subsurface ponding occurs(t1b t≤ t2)

All the rain will infiltrate the lower soil layer because the intensityof the rainfall is still smaller than the infiltration capacity of the soil(f= i, F= it). Subsurface saturation eventually occurs at the interface,however, because the application rate is greater than the saturatedhydraulic conductivity of the lower layer. The time of subsurfaceponding can then be calculated as follows:

t2 =K2Δθ2ψ2f

i i − K2ð Þ + t1: ð12Þ

The corresponding depth of the wetting front is

zw = L1 +it2 − L1ΔθV1

Δθ2: ð13Þ

(c) Stage III: From t2 to t3 when surface ponding occurs (t2b t≤ t3)

Once the subsurface saturation occurs, excess water starts to accu-mulate in the upper layer because the infiltration capacity of the lowerlayer monotonically decreases with time until the minimum infiltrationcapacity is reached. If the water application continues, the upward watermovement from the perched water table finally reaches the surface andsurface saturation occurs. Before the surface ponding, the infiltration rateof the two-layered soil is equal to the application rate (f=i, F=it).

The excess water that accumulates in the upper layer, R, during theperiod from t2 to t3 is calculated as

R =Zt3t2

i − f� �

dt = i t3 − t2ð Þ− ΔF; ð14Þ

where f ̄ is the infiltration capacity through the less permeable lowerlayer and ΔF

—is the cumulative infiltration between the time of t2 and

t3 through the lower layer (see Appendix A). Moreover, an additionalamount of water, S, which can be absorbed by the upper layer after thesubsurface ponding, is calculated as

S = ΔθW1 L1; ð15Þwhere Δθ1″=θ1s−θ1o is the moisture deficit from the saturationexpressed as the difference between the volumetric water contentsbefore and after the upward wetting in the surface layer.

Because the surface ponding occurs when R becomes equal to Sand the pressure head at the surface becomes zero, the time of theincipient surface ponding can be computed as

t3 = ΔθW1 L1 + ΔF + it2i:

ð16Þ

The corresponding depth of thewetting front at the surface ponding is

zp = L1 +it3 − L1Δθ1

Δθ2: ð17Þ

(d) Stage IV: After t3 (t3b t)

Eqs. (6) and (8) can be applied after the surface ponding becausethe infiltration capacity decreases with time as the wetting front goesdeeper into the soil.

5. Application of the infiltration model to slope stability

To illustrate the procedure for estimating the infiltration process intwo-layered slopes, the extendedmodel was applied to two examples.As shown in the following equations, the SWCC is represented byusing the van Genuchten function (van Genuchten, 1980), and the

relative hydraulic conductivity is represented by the Gardner equation(Gardner, 1958):

θ = θr + θs − θrð Þ 1 + aua−uw

γw

� �� �n� �−m

ð18Þ

Kr = 1 + αVua−uw

γw

� �β� �−1

; ð19Þ

where θr and θs=the residual and saturated volumetric watercontent, respectively; a, n, and m=constants; and α′ andβ=constants.

5.1. Example 1: Application to a two-layered slope

In this example, two types of soil, namely sand and silt wereconsidered. Fig. 4 shows the SWCCs and the relative hydraulic con-ductivity functions for each soil. The properties have been selected asa representative for fine-grained and coarse-grained soils since two-layer soil systems that consist of fine-grained and coarse-grained soilshave been widely studied due to the contrasting hydraulic character-istics of the soils. Soil–water characteristic curves for fine-grained soilsmay be relatively flat, while for coarse-grained soils the function maybe quite steep, such that as soon as the soil desaturates, the hydraulicconductivity drops dramatically.

Table 1Example 1: Parameters for SWCC and hydraulic conductivity function.

Soil type Hydraulicconductivityfunction

SWCC

α′ β θr θs a n m

Fine sand 11.31 4.03 0.185 0.346 1.841 6.442 0.229Silt 9.88×10−4 7.38 0.0 0.373 0.426 40.057 0.041

37S.E. Cho / Engineering Geology 105 (2009) 32–43

Table 1 tabulates the parameters for the SWCC and the relativehydraulic conductivity function. It was assumed that the initial matricsuction was 30 kPa through the depth and that the thickness of thesurface layer was 0.5 m. Next, because the infiltration behavior in thelayered soil depends on whether the profile is increasingly permeableor decreasingly permeable, both the coarse-over-fine stratificationand the fine-over-coarse stratification were examined. Table 2 pre-sents the estimated input data for the infiltration model.

The mechanical properties of the soils to calculate the factor ofsafety were also presented in Table 2.

5.1.1. Analysis for a decreasing conductivity profileFig. 5 shows the typical results from a one-dimensional finite

element analysis using SEEP/W (Geo-Slope, 2003) under the condi-tion of K1b i. The results show that the rainfall infiltrates the soil andthat eventually surface ponding occurs (Fig. 5(a) and (b)). If the raincontinues to fall after the ponding, positive pore pressure (a perchedwater table) develops in the upper layer (Fig. 5(a)). Fig. 5(c) showsthat the hydraulic gradient increases sharply at the wetting front, andthat the largest gradient occurs when the wetting front passes theinterface. The gradient in the lower layer is not so high because thehydraulic conductivity of the silt changes more gradually.

Results for the condition of K2b ibK1 show that the conceptualmoisture profile presented in Fig. 3 is reasonable (Fig. 6(b)). That is,the upward saturation starts when subsurface ponding occurs at theinterface and eventually surface ponding occurs (Fig. 6(a)). Fig. 6(c)indicates that the hydraulic gradients behind the wetting front in theupper layer are near to unity before the wetting front passes theinterface, as mentioned in the previous section (Section 4.3).

Fig. 7(a) shows the infiltration diagram for different rainfallintensities that are greater than K1. If the applied rainfall intensityis smaller than the infiltration capacity of the soil, then all the rainwillinfiltrate the soil. Once the surface saturation occurs, the runoff startsand the infiltration capacity decreases monotonically with time untilthe minimum infiltration capacity, K2, is reached. Fig. 7(a) also com-pares the infiltration model and the numerical model for differentrainfall intensities. Given the basic assumptions of both models, thecomparative performance of the infiltration model is acceptable.

Fig. 7(b) shows the infiltration diagram for different rainfallintensities that are greater than the saturated hydraulic conductivityof the lower layer but smaller than the saturated hydraulic conductivityof the surface layer, that is, when K2b ibK1. In the diagram, the time

Table 2Example 1: Input parameters for the analysis

Parameters for the infiltration model

K1 (cm/h) K2 (cm/h) Δθ1 Δθ2 ψf1 (cm) ψf2 (cm)

Coarse/fine 1.55(4.31×10−6 m/s) 0.09(2.50×10−7 m/s) 0.15 0.14 60.64 251.78Fine/coarse 0.09(2.50×10−7 m/s) 0.22 (6.11×10−7 m/s) 0.14 0.08 251.78 76.96

Mechanical properties

γt (kN/m3) c′ (kPa) ϕ′ (°) ϕb (°)

Sand 19 0 35 15Silt 19 10 25 10

Fig. 5. Example 1: Results of numerical analyses for coarse-over-fine stratifications (K1b i).

Fig.6.Example1:Results ofnumerical analyses forcoarse-over-fine stratifications (K2b ibK1).

Fig. 7. Example 1: Infiltration diagram of uniform rainfall conditions for coarse-over-finestratifications.

38 S.E. Cho / Engineering Geology 105 (2009) 32–43

when thewetting front reaches the interface (t1), the time of subsurfaceponding at the interface (t2), and the time of surface ponding (t3) areindicated, respectively. The results also agreewell with the results of thenumerical model.

The relationship between the time of surface ponding and thecorresponding depth of the wetting front for rainfall intensities can bedetermined by using the proposed infiltration model as summarizedin Fig. 8. Curve B in Fig. 9 shows the time of surface ponding as afunction of the rainfall intensity, along with the intensity–duration–frequency curve. The curve can be obtained by calculating the time ofsurface ponding for rainfall intensities larger than the saturatedhydraulic conductivity K2. If K1b i, then the time of surface pondingand the corresponding depth of the wetting front can be calculatedfrom Eqs. (3) and (4). If K2b i≤K1, then the time of surface pondingand the corresponding depth of the wetting front can be calculatedfrom Eqs. (16) and (17). Each point on the curve provides a minimumrequirement for saturation (imin, tmin) to the corresponding depth zp.That is, to saturate the soil to the corresponding depth zp, the rainfallintensity must be greater than imin and the rainfall must last longerthan tmin. Fig. 9 therefore indicates that only the rainfall above curve Bcan saturate the soil to the corresponding depth of the wetting frontindicated in Fig. 10.

The corresponding depth of the wetting front at incipient surfaceponding is a function of the rainfall intensity. This is shown in Fig. 10,with the shallower depth being associated with the higher intensity,and, therefore, the earlier time of ponding.

The factor of safety associated with various depths of the wettingfront can be calculated easily by using Eq. (1). The detailed equationsderived from Eq. (1) are presented in Fig. 11. The results are

Fig. 8. Flow diagram to determine the time of surface ponding and the corresponding depth of the wetting front for uniform rain.

39S.E. Cho / Engineering Geology 105 (2009) 32–43

presented in Fig. 11 for depths from 0 to 3 m for a slope angle of 35°.From the figure, the effect of infiltration on slope stability can beseen to reduce the factor of safety after sufficient advancement of thewetting front.

According to the analysis of slope having slope-parallel layers byCho and Lee (2001), a saturated slope-parallel flow in the upper layerof the slope was formed due to a perched water table caused by thepresence of an underlying less permeable layer. They showed that thehydrological response can induce concentrated mechanical responsessuch as shear strain and shear stress above the interface, which finallyleads to failure. Therefore, when the lower layer is less permeable, anyfailure along the interface must be also checked in relation to thepositive pore water pressure (Eq. (2)), which usually leads to themostcritical condition as shown in Fig. 11. As an illustration, the minimum

Fig. 9. Example 1: Minimum requirements for saturation to a depth zp.

requirement of rainfall for saturation to a depth of 0.5 m by theproposed method is shown in Figs. 9 and 10. The limiting conditionplotted as point A that moves on the curve B was Tmin=2.19 h andimin=3.425 cm/h from Eq. (3) (Fig. 9). This statistically means thatrainfall intensities greater than imin=3.425 cm/h have to last at least2.19 h in order to saturate up to a depth of 0.5 m (point C in Fig. 10).

5.1.2. Analysis for an increasing conductivity profileFig. 12 shows the typical results of a one-dimensional finite ele-

ment analysis for a fine-over-coarse stratification under the conditionof K1b i. The results show that the sublayer does not reach saturationbecause the surface layer is less pervious than the subsoil (Fig. 12(a)and (b)). The hydraulic gradient behind the wetting front in the lowerlayer drops because the hydraulic conductivity of the soil increases to

Fig. 10. Example 1: Depth of wetting front in relation to ponding time.

Fig. 11. Example 1: Variance of the factor of safety with depth of wetting front.

Fig. 12. Example 1: Results of numerical analyses for fine-over-coarse stratifications.

40 S.E. Cho / Engineering Geology 105 (2009) 32–43

a value enough to drain the rainwater infiltrated from the upper layer(Fig. 12(c)). The infiltration rate, as well as the cumulative infiltration,is much lower in the fine-over-coarse stratification than in the coarse-over-fine stratification. The factor of safety can also be calculated forthe depth of the wetting front. In this type of soil profile, the stabilitycondition of slope is more favorable than a decreasing conductivitysoil profile since the unsaturated condition (i.e., negative porewater pressure) is maintained in the lower layer. Therefore, by usinga suitable type of cover system such as compacted clay covers, plasticmembrane and asphalt pavement at the ground surface the stabilitycan be effectively maintained in the slope.

Fig. 13 shows that after the wetting front passes through theinterface the infiltration rate decreases to a constant final value equalto the water flux at the interface. This is because the hydraulic con-ductivity of the sublayer is great enough to drain the infiltrated waterfrom the upper layer. Namely, the entire infiltration process is con-trolled mainly by the less permeable upper layer. Although thediagram shows an abrupt decrease in the infiltration rate, the resultsagree reasonably well with the results of the numerical model. Thereason for the sudden reduction in the modified GAML model arisesfrom the difference between the physics of a true wetting front andthe physics of the piston-type wetting front assumed in the derivationof the GAML equations. In the modified GAML model the piston-typewetting front encounters the interface between the two soil layers ata certain time when the infiltration rate drops to an approximatelyconstant rate. However, the wetting front in the numerical model isnot as sharply defined as that in the modified GAML and so produces agradual reduction in infiltration rate to a nearly constant value, whenthe wetting front completes its passage through the interface (Moore,1981).

Fig. 13. Example 1: Infiltration diagram of uniform rainfall conditions for fine-over-coarsestratifications.

Table 3Example 2: Parameters for SWCC and hydraulic conductivity function.

Soil type Hydraulicconductivityfunction

SWCC

α′ β θr θs a n m

Weathered soil 28.01 1.69 0.0952 0.357 0.447 1.005 0.797Bedrock 28.01 1.69 0.0952 0.357 0.447 1.005 0.797

Table 4Example 2: Input parameters for the infiltration model.

K1 (cm/h) K2 (cm/h) Δθ1 Δθ2 ψf1 (cm) ψf2 (cm)

Weatheredsoil/bedrock

18 (5×10−5 m/s) 0.018 (5×10−8 m/s) 0.10 0.10 23.89 23.89

41S.E. Cho / Engineering Geology 105 (2009) 32–43

5.2. Example 2: Application to a soil-mantled slope with impermeablebedrock

In Korea where the depth of weathering is very shallow, manyslope failures occur in layer of weathered residual soil overlying thebedrock. These failures are characterized by shallow failure surfaces

Fig. 14. Example 2: Hydraulic properties for analysis.

(mostly less than 1 m deep) located near the contact between theweathered soil and the underlying bedrock.

For a two-layered slope in which the hydraulic conductivity ofthe upper layer is much greater than that of the lower layer, if theapplied rainfall intensity is greater than the hydraulic conductivityof the lower layer, the infiltrated water reaches the contact betweenthe soil and the underlying impermeable bedrock, causing a rapid risein pore water pressures, and the formation of a perched water tablethat destabilizes the upper layer of soil (Biavati et al., 2006).Consequently the factor of safety starts to rapidly decrease.

In this example numerical and infiltration models to simulatethe response to rainfall have been applied to a slope with a 1 m thickweathered granite soil layer above the bedrock to study rainfall-induced landslides for a landslide-prone hillslope. A typical weatheredgranite soil, sampled in Seochang, classified as SM by the United SoilClassification System (Kim, 2003) was used. The soil–water character-istic curve (Fig. 14(a)) was obtained through a pressure plate test, andthe relative hydraulic conductivity function (Fig. 14(b)) was measuredby the steady state method involves the measurement of hydraulicconductivity of an unsaturated soil specimen under a constant matricsuction. Table 3 tabulates the parameters for the SWCC and therelative hydraulic conductivity function. The analysis was conductedfor the initial matric suction of 20 kPa through the depth.

Although soil layers are expected to have less water retentioncapacity as they are commonly looser than those from deeper layers,

Fig. 15. Example 2: Infiltration diagram of uniform rainfall conditions for a soil-mantledslope with impermeable bedrock.

Fig. 16. Example 2: Minimum requirements for saturation to a depth zp.

Fig. 18. Example 2: Pressure head response at the contact between the two layers(results of numerical analyses).

42 S.E. Cho / Engineering Geology 105 (2009) 32–43

only soil heterogeneity in terms of the saturated hydraulic conductiv-ity is considered since the two layers share the same parent rock andthe lower bedrock was relatively impermeable. Values for the inputparameters used for the model simulations are presented in Table 4.

Fig. 15 shows the infiltration diagram for rainfall intensities of 1 cm/h and 2 cm/h that are greater than the saturated hydraulic conductivityof the lower layer but smaller than the saturated hydraulic conductivityof the surface layer. The diagram shows that the time of surface ponding(tp) when the soil layer is completely saturated agrees well with theresult of the numerical model.

The time of surface ponding and the corresponding depth ofthe wetting front were plotted varying the rainfall intensity (Figs. 16and 17). Point A in Fig. 16 shows that the time of surface ponding is11.3 h for the constant rainfall intensity of 1 cm/h.

The depth of wetting corresponding to the rainfall intensity 1 cm/hand time of ponding 11.3 h is found to be 113 cm at the point C inFig. 17, which means that the required minimum time of durationto completely saturate the soil layer above the bedrock under theintensity of 1 cm/h is 11.3 h and then the wetting front infiltrate to thedepth of 113 cm for the rainfall condition. The flat portion of the curvein Fig. 17 illustrates that the further infiltration of rainwater into the

Fig. 17. Example 2: Depth of wetting front in relation to ponding time.

bedrock requires much longer time of duration due to the very smallhydraulic conductivity of the bedrock.

Fig. 18 shows the pressure head response with time at the con-tact between the two layers obtained from numerical analyses. Thepressure head continues to increase and reaches the steady statevalue of 1 m after about 11.7 h for the intensity of 1 cm/h and 5.67 hfor the intensity of 2 cm/h as a consequence of the prolongedcontribution of rainfall water from the slope surface. Such aworst casescenario (i.e. a water table at the ground surface and slope-parallelflow above impermeable bedrock) that results its minimum factor ofsafety is often assumed for engineering design purposes (Biavati et al.,2006). Then the factor of safety can be easily calculated from Eq. (1) aspreviously explained in Fig. 11.

The results indicate that the proposed infiltration model to deter-mine the infiltration rate at the surface can provide useful insightfor the stability of a soil-mantled slope with impermeable bedrock torainfall infiltration.

6. Conclusions

During periods of prolonged infiltration, surficial failures occurbecause of the positive pore water pressure or the reduced suctionwhen the pore pressure is still negative. Because infiltration is acomplex process that generally involves an unsaturated flow in avertical direction, one-dimensional finite element analyses were con-ducted to gain a better understanding of the mechanism of seepageflow in a two-layered soil slope.

To assess how the rainfall infiltration affects the slope stability intwo-layered soil, Moore's infiltration model was reviewed. Moore'smodel was then extended to cover more general situations, includingthose in which water moves upward from a perched water tablein decreasingly permeable soil. By taking into account the rainfallintensity and the duration of various return periods in two-layeredsoil, the model was used to evaluate the likelihood of a particularrainfall event inducing a shallow slope failure. The results of theinfiltration model were compared with those of the numericalanalyses. With the use of properly estimated input parameters fromthe SWCCs and the unsaturated hydraulic conductivity functions, theproposed model compared reasonably well with the results of themore rigorous finite element analyses.

Although the proposed model has some limitations with respectto general applicability, it can provide a perspective on the failuremechanism of a two-layered slope under the infiltration of rainfall.In addition, when the proposed framework is properly used it offersa quick and easy way of estimating how infiltration affects a slope's

43S.E. Cho / Engineering Geology 105 (2009) 32–43

stability for the purpose of preliminary analysis and decision-makingin two-layered soil prior to undertaking in-depth analyses.

Appendix A. Computational algorithm for the infiltration model

The simple two-step method proposed by Li et al. (1976) was usedto estimate the cumulative infiltration because the method is simpleand easy to use.

After the ponding but within the period of interest, according toEq. (8) the increment of cumulative infiltration is defined as

ΔF = K2Δt − E − Hð Þ ln 1 +ΔF

F + H − F1

� �for zw > L1; ð20Þ

where F refers to the known cumulative infiltration at the start of timestep and ΔF is the unknown term.

The first estimate of ΔF can be obtained by using a truncated seriesexpansion as

ΔFo =12

K2Δt − 2 F + E − F1ð Þ +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 F + E−F1ð Þ−K2Δtf g2 + 8K2Δt F + H − F1ð Þ

q� �;

ð21Þ

whereΔFo is an initial estimate of the increment of cumulative infiltration.A second-order Newtonmethod based on a Taylor series expansion

can be used to obtain a more precise solution. ΔF must be found suchthat f(ΔF)=0. This equation is expressed as follows:

f ΔFð Þ = ΔF − K2Δt + E − Hð Þ ln 1 +ΔF

F + H − F1

� �: ð22Þ

The second estimate can then be expressed as

ΔF1 = ΔFo − f VΔFoð Þf W ΔFoð Þ +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif VΔFoð Þ

f W ΔFoð Þ� �2

− 2f ΔFoð Þf W ΔFoð Þ

s; ð23Þ

where f VΔFoð Þ = F + E − F1 + ΔFoF + H − F1 + ΔFo

, f W ΔFoð Þ = H − EF + H−F1 + ΔFoð Þ2.

According to Li et al. (1976), the maximum percent error isapproximately 8% for the first estimate and less than 0.003% for thesecond estimate. The procedure is equally applicable when zw≤L1.

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