Stability of infinite slopes under transient partiallysaturated seepage conditions

Jonathan W. Godt,1 Bas�ak S�ener-Kaya,2,3 Ning Lu,2 and Rex L. Baum1

Received 21 September 2011; revised 12 March 2012; accepted 18 March 2012; published 3 May 2012.

[1] Prediction of the location and timing of rainfall-induced shallow landslides is desiredby organizations responsible for hazard management and warnings. However, hydrologicand mechanical processes in the vadose zone complicate such predictions. Infiltratingrainfall must typically pass through an unsaturated layer before reaching the irregular andusually discontinuous shallow water table. This process is dynamic and a function ofprecipitation intensity and duration, the initial moisture conditions and hydrologicproperties of the hillside materials, and the geometry, stratigraphy, and vegetation of thehillslope. As a result, pore water pressures, volumetric water content, effective stress, andthus the propensity for landsliding vary over seasonal and shorter time scales. We apply ageneral framework for assessing the stability of infinite slopes under transient variablysaturated conditions. The framework includes profiles of pressure head and volumetricwater content combined with a general effective stress for slope stability analysis. Thegeneral effective stress, or suction stress, provides a means for rigorous quantification ofstress changes due to rainfall and infiltration and thus the analysis of slope stability over therange of volumetric water contents and pressure heads relevant to shallow landslideinitiation. We present results using an analytical solution for transient infiltration for a rangeof soil texture and hydrological properties typical of landslide-prone hillslopes and show theeffect of these properties on the timing and depth of slope failure. We follow by analyzingfield-monitoring data acquired prior to shallow landslide failure of a hillside near Seattle,Washington, and show that the timing of the slide was predictable using measured pressurehead and volumetric water content and show how the approach can be used in a forwardmanner using a numerical model for transient infiltration.

Citation: Godt, J. W., B. S�ener-Kaya, N. Lu, and R. L. Baum (2012), Stability of infinite slopes under transient partially saturated

seepage conditions, Water Resour. Res., 48, W05505, doi:10.1029/2011WR011408.

1. Introduction[2] Shallow landslides are important erosion processes

and are potentially lethal hazards in mountainous areas.They are typically translational slope failures that involvethe upper few meters of unconsolidated surficial materials(i.e., soil or regolith) and dominate sediment transport insome hillslope environments [e.g., Dietrich and Dunne,1978; Caine and Swanson, 1989; Trustrum et al. 1999].Because shallow landslides can initiate potentially destruc-tive debris flows [e.g., Iverson et al., 1997] they present ahazard to human activities [e.g., Sidle and Ochiai, 2006].

[3] Change in pore water pressure resulting from rainfallonto hillslopes is a dynamic physical process that can pro-voke slope failure in hillside materials. Field monitoring ofrainfall and pore water response as well as theoretical andphysical models have provided insight into the timing

and mechanisms of landslides induced by rainfall [e.g.,Reid et al., 1997; Torres et al., 1998; Ebel et al., 2007;Montgomery et al., 2009; Borja and White, 2010]. Analysisof slope stability under positive pore water conditions inwhich the soil or regolith is saturated is fundamental to soilmechanics and geotechnical engineering. However, theo-retical and analytical results have advanced the hypothesisthat in some hillslope environments shallow landslidingis possible absent positive pore water pressures [e.g.,Morgenstern and de Matos, 1975; Wolle and Hachich,1989; Rahardjo et al., 1996; Springman et al., 2003;Collins and Znidarcic, 2004; Lu and Godt, 2008]. Recentfield monitoring results seem to confirm that shallow land-slides can be initiated by changes in pore water pressures inthe unsaturated regime [Godt et al., 2009]. Assumption ofpositive pore water pressures is necessary to apply Terzaghi’s[1943] effective stress principle. This restrictive assumptionhinders assessment of slope stability leading to overly con-servative estimates of shallow landslide susceptibility.

[4] Instability under partially saturated conditions is morelikely to occur on steep slopes that have been disturbed bynatural or human activities that have altered the strength ofhillside materials or increased the inclination of the slopes.In such settings, the effect of disturbance is to reduce the

1U.S. Geological Survey, Denver, Colorado, USA.2Department of Civil and Environmental Engineering, Colorado School

of Mines, Golden, Colorado, USA.3Now at Itasca Denver, Lakewood, Colorado, USA.

Copyright 2012 by the American Geophysical Union0043-1397/12/2011WR011408

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WATER RESOURCES RESEARCH, VOL. 48, W05505, doi:10.1029/2011WR011408, 2012

change in effective stress needed to cause instability. Afterdisturbance, steep hillslopes that were previously stable evenwhen heavy rainfall generated positive pore water pressuresmay now be susceptible to instability under partially satu-rated conditions. Examples of disturbance that alter slopestability are wildfire [e.g., Jackson and Roering, 2009],industrial forest practices [Montgomery et al., 2000], conver-sion of forests to pasture [Glade, 2003], and large-scaleslope deformation, which may initiate smaller-scale instabil-ities [e.g., Imaizumi et al., 2006]. Slopes oversteepened byactive uplift, coastal erosion and bluff retreat [e.g., Godtet al., 2009] or excavated for road construction [e.g., Higginsand Modeer, 1996] may also be susceptible to slope failureunder partially saturated conditions.

[5] The close link between rainfall characteristics andshallow landslide initiation is well documented [e.g., Caine,1980; Godt et al., 2006] and quantitative understanding ofthis link is gained by analyzing the transient pore waterresponse to rainfall and consequent changes in stability [e.g.,Reid, 1994]. Steady state flow models [e.g., Montgomery andDietrich, 1994; Lu and Godt, 2008] may provide insight intothe pore water conditions at the time of slope failure, but lit-tle information on the rainfall needed to reach that point.Reduction of the problem to steady conditions yields a com-parison of a steady infiltration flux to hydraulic conductivity,K, but requires neglecting the transient storage of water inthe unsaturated zone. Thus, the nonlinear dependence of thepore water response to rainfall is ignored making quantitativeassessment of rainfall patterns and slope stability unreliable.

[6] In partially saturated soils, changes in volumetricwater content � resulting from rainfall are accompanied bychanges in the pressure head , which is described by thesoil water characteristic curve �ð Þ. The slope of thiscurve, or specific moisture capacity, Cð Þ ¼ d�=d , is alsopressure head dependent and describes the available storagein the partially saturated soil. Soil water diffusivity,Dð Þ ¼ Kð Þ=Cð Þ is the relative amount of storage as afunction of pressure-dependent hydraulic conductivity[Freeze and Cherry, 1979]. Iverson [2000] showed that forthe limiting case of nearly saturated hillslopes, the domi-nant direction of strong pore pressure transmission result-ing from rainfall can be assessed using two time scales.Assuming a reference soil water diffusivity, D0, in an iso-tropic, homogeneous hillslope, the time scale for strongpore pressure diffusion normal to the slope is H2/D0 whereH is the slope normal depth. The time scale for strongpore pressure diffusion parallel to the slope to some pointlocated below a catchment with contributing area A isA/D0. For hillslopes where the ratio of these two timescales, " ¼ H=

ffiffiffiAp� 1, the long- and short-term pressure

head responses to rainfall can be adequately described by1-D linear and quasi-linear approximations to Richardsequation. Typically, for locations near the failure surface ofexisting or potential landslides where contributing area, A,may range from �100 m2 to 104 m2, slope normal depths ofa few to tens of meters, and soil water diffusivities from 10�2

to 10�5 m2 s�1, slope-normal response times range from tensof minutes to hundreds of days. In contrast, time scales forslope-parallel pressure diffusion typically range from severalhours to years [Iverson, 2000; Baum et al., 2010].

[7] In what follows, we describe the form of effectivestress in variably saturated materials and present results for

four hypothetical hillslopes using an analytical solutionfor transient infiltration [Srivastava and Yeh, 1991; Baumet al., 2010] to examine the variation in timing and depthof potential instability above the water table with materialhydrologic properties. Here, we use the term ‘‘infiltration’’to describe the flow of water across the ground surface andinto the underlying soil [Freeze and Cherry, 1979, p. 211].The four hillslopes are representative of the range of mate-rials for which shallow landslides may occur under par-tially saturated conditions. We then apply the framework toa case study where pressure heads and water contents weremonitored leading up to the occurrence of a landslide at thefield site. These data are then used to calibrate a numericalsolution to the governing equations of variably saturatedflow to illustrate how the approach could be applied to fore-cast landslides.

2. Effective Stress in Variably SaturatedHillslopes

[8] Effective stress in partially saturated soil differs fromeffective stress in saturated soils [e.g., Terzaghi, 1943] inthat in partially saturated soils, effective stress is not thedifference between total stress and pore water pressure, butrather the difference between total stress and some functionof negative pore water pressure head or soil suction. Oneform of this function proposed by Lu and Likos [2004,2006] is the suction stress characteristic curve. The suctionstress characteristic curve describes the saturation-depend-ent macroscopic stress that tends to hold particles togetherin partially saturated soil. This stress can vary from a fewkPa in sands to several hundred kPa in silts and clays [Luand Godt, 2008]. The generalized form of effective stress�0 for variably saturated soils is given by [Lu et al., 2010]:

�0 ¼ ð�� uaÞ � �s (1)

where ua is the pore air pressure, � is the total stress, andsuction stress, �s is given by

�s ¼ � �� �r

�s � �rðua � uwÞ ¼ �Seðua � uwÞ (2)

where � is the volumetric water content, �r is the residualvolumetric water content, �s is the saturated volumetricwater content, Se is the effective degree of saturation, anduw is pore water pressure. Lu and Likos [2004] and Lu et al.[2010] show that suction stress can be written solely asfunctions of the effective degree of saturation or matricsuction ðua � uwÞ :

�s ¼ � Se

�S

n1�ne � 1

� �1n (3)

�s ¼ � ðua � uwÞ�1þ ½�ðua � uwÞ�n

�ðn�1Þ=n (4)

where � and n are parameters defined in two commonlyused soil water characteristic models [van Genuchten,1980]. Note that Terzaghi’s effective stress �0 ¼ �� uw is

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recovered from equation (1) for saturated materials. Below,we use these equations to extend the infinite slope stabilityanalysis to variably saturated conditions.

3. Infinite Slope Stability[9] Infinite slope stability analyses [Taylor, 1948] are

statically determinate and are therefore convenient forexamining the role of infiltration and consequent transientchanges in the distribution of soil moisture on stability[e.g., Iverson, 2000]. Slope failure is modeled as the ratioof resisting Coulomb basal friction to gravitationally drivendownslope stress. This ratio, F, is called the factor ofsafety. Instability is predicted when the driving stressexceeds the resisting friction. Infinite slope stability analy-sis is appropriate for landslides in which the failure andground surfaces are parallel and the slide thickness is rela-tively small compared to the slide length, and where thelocal topographic curvature is small. For transient infiltra-tion conditions, the factor of safety F at depth Z below theground surface is given by [Lu and Godt, 2008]

FðZ; tÞ ¼ tan�0ðZÞtan �

þ c0 þ �sðZ; tÞtan�0ðZÞ�sZsin�cos �

(5)

where �0ðZÞ is a depth-dependent angle of internal frictionand c0 is the soil cohesion, �s the soil unit weight, and � theslope angle (Figure 1). We follow Lu and Godt [2008] anddescribe the variation of the friction angle, �0ðZÞ, in weath-ered surficial materials as a function of a depth-dependentporosity m [e.g., Rowe, 1969; Cornforth, 2005]

�0 ¼ �00 þ��0

�mðm0 � mÞ (6)

where �00 and m0 are the friction angle and porosity at theground surface and ��0 and �m are the range of friction

angle and porosity, respectively, within the weatheringzone zw (Figure 2). The decrease in porosity with depth Zresults mainly form mechanical compaction and consolida-tion of the soil under its own weight and can be describedas a power reduction equation:

m ¼ m0 ��m

1þ zw

Z

(7)

Substituting equation (7) into (6) yields a form for thechange in friction angle with depth:

�0 ¼ �00 þ��0

1þ zw

Z

(8)

4. Analytical Solution for Transient Flow[10] The one-dimensional form of Richards equation is

used to describe flow of water in the vertical direction inthe unsaturated zone [Freeze and Cherry, 1979]

@�

@t¼ @

@zKð Þ @ð � zÞ

@z

� �� �(9a)

where z is the vertical coordinate, t is time, is the pressurehead, and �ð Þ and Kð Þ are the pressure head–dependentvolumetric water content and hydraulic conductivity, respec-tively. Equation 9a can be linearized using the exponentialmodel of Gardner [1958] to provide an analytical solutionfor transient infiltration above a fixed water table [Srivastavaand Yeh, 1991; Baum et al., 2008, 2010]. The hydraulic con-ductivity function and soil water characteristic curve aregiven by

Kð Þ ¼ KS expð� Þ (9b)

�ð Þ ¼ �r þ ð�s � �rÞ expð� Þ (9c)

Figure 1. Schematic cross section and definitions for avariably saturated infinite slope.

Figure 2. Relative change in dimensionless friction angleas a function of depth for a weathered zone of varyingthickness, zw.

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respectively, where Ks is the saturated hydraulic conductiv-ity and � is the inverse of the vertical height of the capil-lary fringe.

[11] Substitution of (9b) and (9c) into the one-dimen-sional form of Richards equation (9a) yields a linear partialdifferential equation in terms of K(z,t)

�ð�s � �rÞKS

@K

@t¼ @

2K

@z2� �@K

@z(9d)

We use solutions for a constant surface flux and a station-ary water table developed by Srivastava and Yeh [1991]with the method that is recapitulated below from thatdescribed by Baum et al. [2008, 2010].

[12] The flux, q, at the base of the unsaturated zone,obtained by solving the linearized Richards equation, is

qðdu; tÞ ¼

IZ � 4 IZ � IZLTð Þ exp�1du

2

� �exp �D

t

4

� �

X1m¼1

�m sinð�m�1duÞ

1þ �1du

2þ 2�2

m�1du

exp½��2mD t�

8>>>>><>>>>>:

9>>>>>=>>>>>;(10a)

where

D ¼�1Ks

ð�s � �rÞ(10b)

is the decay constant, and D=�2 is the soil water diffusivity[Freeze and Cherry, 1979]. D t is equivalent to the nondi-mensional time used by Srivastava and Yeh [1991], du isthe vertical depth to the top of the capillary fringe, �1du isequivalent to the nondimensional depth used by Srivastavaand Yeh [1991], and the values of �m are the roots of thecharacteristic equation (12b), Iz is the surface flux of agiven intensity, and IZLT is the steady (initial) surface flux.

[13] The hydraulic conductivity in the unsaturated zonevaries with depth according to the following formula. Theinitial condition, K(Z,0), is

KðZ; 0Þ ¼ IZLT � ½IZLT � Ks expð�1 0Þ� exp½��1ðdu � ZÞ� (11)

which reduces to Ks expð�1 0Þ at depth Z ¼ du (or Ks atthe water table if d is substituted for du so that 0 ¼ 0) andapproaches IZLT at the ground surface as du becomes large.The initial condition given in equation (11) is obtained bysolving the steady part of equation (9d) with specified flux,IZLT, at the ground surface and specified pressure head, 0,at the water table. The hydraulic conductivity solution forthe unsaturated zone is given as

KðZ; tÞ ¼ InZ � ½InZ � Ks expð�1 0Þ� exp½��1ðdu � ZÞ�

� 4ðInZ � IZLT Þ exp�1Z

2

� �exp �D

t

4

� �

�X1m¼1

sin½�m�1ðdu � ZÞ� sinð�m�1duÞ

1þ �1du

2þ 2�2

m�1du

exp½��2mD t�

(12a)

The values of �m are the positive roots of the pseudoperi-odic characteristic equation,

tanð��1duÞ þ 2� ¼ 0 (12b)

For the values of �m, a combination of bracketing, bisec-tion, and Newton-Raphson iteration is used [Baum et al.,2008, 2010].

5. Timing of Slope Failure Above the WaterTable in Steep Hillslopes

[14] We examine the timing of potential instability of asteep (45�) hillslope composed of four hypothetical materi-als using the analytical solution for transient infiltrationdescribed in section 4. The analytical solution allows us toisolate the effects of the material hydrological properties onthe timing and depth of potential instability. The materials(coarse, medium and fine sand, and silt) have soil waterand suction stress characteristics (Figure 3) and materialstrength properties that cover a range typical for soils thatmantle steep hillsides and that are prone to shallow land-sliding under partially saturated conditions (Table 1). In thehypothetical examples described below the hydrologicalproperties of the soil are constant and the internal frictionangle varies with depth according to equation (8).

[15] The soil water characteristics were calculated usingthe Gardner [1958] equations (9b) and (9c) and suctionstress using equation (2). Peak suction stresses vary amongthe soils from about �1.0 kPa at a pressure head of about�0.3 m for the coarse sand, �3.5 kPa at a pressure head ofabout �1.0 m for the medium sand, �5.1 kPa at a pressurehead of about �1.4 m for the fine sand, and �7.2 kPa at apressure head of about �2.0 m of pressure head for the silt(Figure 3c). The pressure head response to an infiltrationrate, IZ equivalent to the saturated hydraulic conductivityKS was computed for an 2.0 m thick unsaturated zoneabove a stationary boundary with a constant pressure head ¼ 0 in the sand examples and a 5.0 m thick unsaturatedzone in the silt example. Initial conditions were prescribedas hydrostatic for all examples, which leads to the charac-teristic distribution of the effective degree of saturationabove the water table. These four limiting cases demon-strate the progression of instability above the water table.For hillslopes less susceptible to landsliding, such as thoseinclined at shallower angles, composed of stronger materi-als, or subject to lower infiltration rates [Lu and Godt,2008], slope failure would likely occur in the presence ofpositive pore water conditions.

5.1. Coarse Sand Hillslope

[16] This hypothetical example can be considered an ana-log for hillslope environments where surficial materials arecomposed of a coarse sandy regolith with poorly developedsoils lacking organic content. Such a setting is commonwhere alteration of biotite and other minerals in graniticbedrock forms grus [Wahrhaftig, 1965], deposits of whichmay be tens of meters thick at ridge crests [Birkeland,1999]. In such settings, particularly where vegetation issparse or removed by wildfire, discrete shallow landslidesmay occur under both very dry and very wet conditions[e.g., Andersen et al., 1959; Durgin, 1977; Gabet, 2003].

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[17] Under initially hydrostatic conditions with a watertable located 2.0 m below the ground surface (Figure 4a)the effective saturation (Figure 4b) is reduced to zeroat about 1.5 m above the water table. Under hydrostatic

conditions suction stress has a peak of �1.0 kPa at 0.3 mabove the water table, is zero at the water table, and dimin-ishes to �0.2 kPa at 1.2 m above the water table (Figure 4c).The changes in suction stress with height above the water ta-ble under hydrostatic conditions lead to factors of safety ofless than unity for these initial conditions (Figure 4d) giventhe slope geometry, material strength and hydrologic proper-ties selected for this example. Although unstable initial con-ditions are not realistic, for this case, very dry conditionslead to relatively small values of suction stress and instabil-ity compared to conditions when the soil is modestly wet.This example provides a mechanism to explain field obser-vations of shallow landslide occurrence under very dry con-ditions following wildfire from the San Gabriel range frontin southern California [Anderson et al., 1959; Krammes,1963].

[18] After 6 h of rainfall pressure heads increase to nearzero at the ground surface (Figure 4a) with a consequentincrease in effective saturation (Figure 4b). The effect ofinfiltration on the profile of suction stress is complex withlarge relative decrease in suction stress to a minimum ofabout �1.0 kPa over the upper 1.0 m from the hydrostaticinitial condition after 6 h of rainfall (Figure 4c). Thisdecrease in suction stress leads to an increase in the factorof safety and a shift to relatively stable conditions com-pared to the initial state (Figure 4d). The profile is stablethroughout at 12 h, but a zone of factors of safety approach-ing unity develops at about 0.2 m below the ground surfaceat this time (Figure 4d). At later times (24 and 48 h inFigures 4a and 4b), pressure heads are nearly zero through-out the profile and effective saturation is almost 1.0. Suctionstress is diminished nearly completely throughout the profileand the upper 0.8 m is potentially unstable (Figure 4d).

5.2. Medium Sand Hillslope

[19] Here, we consider the transient effects of infiltrationon the stability of a hypothetical hillslope with a surficialcover of medium sand. A comparable real-world settingmight be the Puget Sound region where glacial depositsmechanically weather to form a sandy colluvium a fewmeters thick on steep coastal hillsides [Galster and Laprade,1991; Schulz, 2007; Troost and Booth, 2008].

[20] Under initially hydrostatic conditions with a watertable located 2.0 m below the ground surface (Figure 5a)the effective saturation decreases from 1.0 at the water tableto about 0.1 at the ground surface (Figure 5b). The profileof suction stress for the medium sand (Figure 5c) has a simi-lar form to that for the coarse-sand example (Figure 4c).Suction stress is zero at the water table, peaks at about�3.6 kPa 1.0 m above the water table and increases toabout �2.8 kPa at the ground surface (Figure 5c). In con-trast to the coarse sand, the medium sand is stable underthe same initially hydrostatic conditions (Figure 5d). Fac-tors of safety increase monotonically above the water tablefor the medium sand because of the contribution of suctionstress. Comparison of these profiles for the coarse and me-dium sand cases highlights the influence of the hydrologi-cal properties on slope stability [Lu and Godt, 2008].

[21] After 36 h of infiltration pressure heads are nearlyzero and effective saturation is close to 1.0 throughout theprofile (Figures 5a and 5b). At this time suction stress peaksat �0.2 kPa at 1.0 m above the water table (Figure 5c) and

Figure 3. (a) Effective saturation, (b) hydraulic conductiv-ity, and (c) suction stress as a function of pressure head forfour hypothetical soils. Hydrologic properties were calculatedusing the Gardner [1958] models (equations (9b) and (9c)),and suction stress characteristics were calculated using equa-tion (2). Material hydrologic parameters are shown in Table 1.

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the profile of the factor of safety indicates potential insta-bility in a zone that extends from 0.1 to 0.5 m below theground surface (Figure 5d). Pressure head, effective satura-tion, and suction stress continue to increase with continuedinfiltration and at 48 h the upper half of the profile is poten-tially unstable.

5.3. Fine Sand Hillslope

[22] In this hypothetical example we examine the effectof transient infiltration on the potential for landslide occur-rence in a fine sand hillslope such as the well-drained collu-vium from Pleistocene age sandstone of the Boso Peninsulain Japan where shallow landslides have resulted from heavyrainfall [Matsushi et al., 2006]. Under initially hydrostaticconditions with a water table located 2.0 m below theground surface (Figure 6a) effective saturation decreasesfrom 1.0 at the water table to about 0.35 at the ground sur-face (Figure 6b). Suction stress decreases from zero at thewater table to a broad peak of about �5.2 kPa in a zonebetween 1.3 and 1.6 m above the water table (Figure 6c).Under initially hydrostatic conditions, factors of safetyincrease monotonically above the water table (Figure 6d)similar to the medium sand example described in section 5.2.

[23] After 36 h of infiltration pressure heads increase tomore than �0.2 m in the upper 1.5 m of the profile (Figure6a) and saturations are greater than 0.9 (Figure 6b). Suctionstress is less than �1.0 kPa in this same zone (Figure 6c)and stability holds (Figure 6d). At 36 h pressure heads arenearly zero and the profile is almost saturated (Figures 6aand 6b), but suction stress of about �0.5 kPa persists (Fig-ure 6c) and although factors of safety approach unity, theslope is still stable. Not until 48 h are pressure heads andsaturation sufficiently great (Figures 6a and 6b) to diminishsuction stress (Figure 6c) leading to factors of safety lessthan unity and potential instability of the upper 0.5 m of thehillslope (Figure 6d).

5.4. Silt Hillslope

[24] Finally, we examine the timing of potential instabil-ity in a steep hillslope composed of hypothetical silt. Land-slides in loess mantled landscapes, such as those that occurduring heavy monsoonal rain on the Loess Plateau in north-ern China present a significant hazard to human activities[Derbyshire, 2001]. For this example we assume the unsat-urated zone is 5 m thick above a stationary boundary with aconstant pressure head ¼ 0. Initial conditions are hydro-static and in equilibrium with the lower boundary condition(Figure 7a). The upper boundary is a prescribed flux IZ atthe ground surface equivalent to the saturated hydraulicconductivity of the silt (Table 1).

[25] Under the hydrostatic initial conditions (Figure 7a)the effective saturation decreases from 1.0 at the water ta-ble to 0.08 at the ground surface (Figure 7b). Suction stressdecreases with distance above the water table to about�7.2 kPa at about 2.0 m above the water table andincreases to �4.0 kPa at the ground surface (Figure 7c) andthe soil profile is initially stable (Figure 7d). After 20 daysof infiltration pressure heads are more than �0.2 m andeffective saturation is greater than about 0.9 throughout theprofile (Figures 7a and 7b). Suction stress increases to�1.1 kPa at the ground surface with a minimum of about�2.0 kPa 2.5 m above the water table (Figure 7c). Thischange in suction stress in response to infiltration leads topotential instability in a zone near the ground surface extend-ing from 3.6 to 4.4 m above the water table (Figure 7d).

5.5. Summary of Hypothetical Examples

[26] Table 2 provides a summary of the simulated timeof failure and the failure depth for the hypothetical exam-ples. In general, failure depth increases for the finergrained soils. This results in part from the influence of the� parameter (i.e., inverse of the air entry head) in theGardner [1958] formulation of the soil water characteristiccurve (equations 9b and 9c) on both the hydrologic andmechanical properties of the materials. The maximum con-tribution to stability from suction stress is relatively smallfor the coarse sand (Figure 4c), which has a large � com-pared to that for the fine sand (Figure 6c). Because the �parameter also controls the hydraulic conductivity function(Figure 3b) it influences the simulated shape of the wettingfront during infiltration. The simulated degree of saturationfrom the coarse-sand example (Figure 4b) shows a moreabrupt change with depth compared to that for the fine-sand example (Figure 6b), which in turn leads to a greaterreduction in suction stress and stability near the groundsurface. The presence of soil cohesion also leads to anincrease in the failure depth. The silty sand example (notshown) has the same hydrological properties and responseto rainfall as the medium sand case but the friction angleand cohesion used in the case study described in section 6(Table 1). The addition of 1.1 kPa of soil cohesion resultsin an increase in failure depth of about 1 m over that simu-lated for the medium sand example without any cohesion.

[27] Because we applied a rainfall flux at the ground sur-face equivalent to the saturated conductivity Ks of thematerials, the variation in the timing of failure for the hypo-thetical examples is largely a function of the variation inhydraulic conductivity among the different materials. Forexample, the saturated hydraulic conductivity of the coarsesand is about five times that of the fine sand, and the timeto failure is longer by about the same amount. Variation in

Table 1. Hydrologic and Shear Strength Properties for the Hypothetical Soilsa

Soil Type � (m�1) Ks (m s�1) �s �r c0 (kPa) �0 (deg) D�0 (deg) � (deg) ZW (m)

Coarse sand 3.5 1.0 � 10�5 0.41 0.05 0.0 40 7 45 0.5Medium sand 1.0 7.5 � 10�6 0.41 0.05 0.0 40 7 45 0.5Silty sand 1.0 7.5 � 10�6 0.41 0.05 1.1 36 8 45 0.5Fine sand 0.7 5.0 � 10�6 0.41 0.05 0.0 40 7 45 0.5Silt 0.5 9.0 � 10�7 0.45 0.10 1.7 30 15 45 1.5

aSee Gardner [1958] for hydrologic properties.

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the time to failure also results from differences in the hy-draulic conductivity function at pressure heads less thanzero (Figure 3b) and the failure and overall profile depths.

6. Case Study Application of the SuctionStress Concept

[28] We apply the suction stress concept to assess thepotential for instability using numerical modeling resultscalibrated with hydrologic data collected at an instru-mented hillslope along the Puget Sound near Edmonds,WA about 15 km north of Seattle (Figure 8a) where a shal-low landslide occurred in the apparent absence of positivepore water pressures under partially saturated soil condi-tions [Godt et al., 2009]. Steep hillslopes in this area aresubject to frequent shallow landslides during the winter wetseason [Galster and Laprade, 1991; Schulz, 2007] whenextended rainy periods lasting several days may induceshallow slides [Godt et al., 2006; Chleborad et al., 2008].Pleistocene age glaciation, shoreline wave attack, and massmovement processes have formed steep (>30�) 50 to 100m high coastal bluffs above many areas along the PugetSound shoreline [Shipman, 2004]. In this area, shallowlandslides typically involve the loose, sandy, colluvialdeposits derived from the glacial and nonglacial sedimentsthat form the bluffs [Galster and Laprade, 1991].

[29] The instrumented hillslope is a steep (45�) coastalbluff covered by a thin, sandy (<2.0 m) colluvium, and thesurface morphology is generally planar (Figure 8b). Thesurficial materials on the bluff were initially instrumentedwith water content instruments and open-tube piezometersequipped with pressure transducers and an automated dataacquisition system to collect hourly readings in Septemberof 2001 [Baum et al., 2005]. However, no positive porewater pressures were observed during 2 years of operationand the piezometers were abandoned in September of 2003.In October of 2003, two water content profilers and twonests of 6 tensiometers were installed. During the 16 monthperiod this instrument array was operational, no positivepore water pressures or volumetric water contents in excessof 0.40 were recorded in the hourly data [Godt et al.,2009]. Measurements of the bulk density performed in boththe lab and field indicate that the saturated volumetricwater content or porosity of the colluvium is about 0.40.

[30] On 14 January 2006 a 25 m long by 11 m wide shal-low landslide initiated in colluvium near the instrumentarray [Godt et al., 2009]. The depth of the failure surfacewas between 1.0 and 2.0 m below and roughly parallel tothe original ground surface. The failure surface was appa-rently coincident with the contact between the loose sandycolluvium and the underlying, better consolidated glacialoutwash sand (Figure 8c). The slide exposed a thin (<2 m)silt bed near the headscarp and a shallow zone (<0.8 mbelow the ground surface) of small diameter (<2 mm)blackberry and grass roots. Only a few larger diameter,

Figure 4. Profiles of (a) pressure head, (b) effective satu-ration, (c) suction stress, and (d) factor of safety for coarsesand with � ¼ 3.5, saturated hydraulic conductivity Ks ¼1 � 10�5 m s�1, and material strength properties in Table1. Prescribed flux at the ground surface is equivalent to Ks.

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partially decayed Alder tree roots penetrated the failure sur-face. The landslide deposit did not mobilize as a debrisflow [Godt et al., 2009].

[31] The contributing area A above the site is betweenabout 32 and 64 m2 estimated from a 1.86 m (6 ft) digitalelevation model derived from airborne laser swath map-ping. Taking a reference soil water diffusivity D0 of1 � 10�4 m2 s�1 [Baum et al., 2010], the characteristictimes for slope-normal and slope-parallel pressure diffusion[Iverson, 2000] are 1.4–5.4 and 89–178 h, respectively.

6.1. Numerical Modeling of Transient Flow

[32] We applied a numerical, one-dimensional, finiteelement solution [�Simu°nek et al., 2008a, 2008b] to equation(9a) to simulate the pore water response to rainfall at theEdmonds site. The numerical solution allows us to takeadvantage of available inverse parameter estimation rou-tines, and a more accurate description of the soil watercharacteristics of the hillside materials and boundary condi-tions of the field setting. The relation between pressurehead and volumetric water content � is given by the vanGenuchten [1980] formulation

�ð Þ ¼ �r þ�s � �r

ð1þ j� jnÞ! (13a)

! ¼ 1� 1

n(13b)

where �r and �s are the residual and saturated volumetricwater contents, respectively, and �, n, and ! are curve fit-ting parameters. The parameter � is considered to be theapproximate inverse of the air entry pressure head [vanGenuchten, 1980]. The pressure head–dependent hydraulicconductivity Kð Þ is predicted using the statistical poresize distribution model of Mualem [1976]:

Kð Þ ¼ KsS0:5e ½1� ð1� S1=!

e Þ!�2 (14a)

Se ¼�� �r

�s � �r(14b)

where Se is the effective saturation.[33] The numerical solution allows us to use a split-sam-

ple, inverse-modeling approach [Klemeıs, 1986] describedin section 6.2 to estimate the soil water characteristics fromfield monitoring information. The model domain was aone-dimensional 3.76 m deep homogeneous profile above ano-flow boundary. The infiltration flux at the upper bound-ary was taken from measured rainfall (Figure 9a). Initialconditions were assumed to be hydrostatic and in equilib-rium with a water table coincident with the lower boundary

Figure 5. Profiles of (a) pressure head, (b) effective satu-ration, (c) suction stress, and (d) factor of safety for me-dium sand with � ¼ 1.0, saturated hydraulic conductivityKs ¼ 7.5 � 10�6 m s�1, and material strength properties inTable 1. Prescribed flux at the ground surface is equivalentto Ks.

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and consistent with the measured pressure head profile inthe middle of September 2005 (Figure 9c).

6.2. Comparison of Model Results WithMonitoring Observations

[34] Soil water characteristics were estimated using aninverse modeling procedure that systematically minimizedthe difference between observed and modeled pressureheads and volumetric water contents [Hopmans et al.,2002; �Simu°nek et al., 2008a]. The differences are expressedusing an objective function [Hopmans et al., 2002] andminimized with the Levenberg-Marquardt optimizationalgorithm [Marquardt, 1963; �Simu°nek et al., 2008a]. Meas-ured rainfall (Figure 9a), volumetric water contents (Figure9b), and pressure heads (Figure 9c) at a depth of 1.5 mfrom the downslope monitoring array (Figure 8d) for theperiod from 28 October 2005 to 21 November 2005 wereused to inversely estimate the saturated hydraulic conduc-tivity, saturated and residual volumetric water contents,and the fitting parameters for the van Genuchten [1980]form of the soil water characteristic (Table 3). The choiceof the October–November period ensured that we obtainedsoil hydraulic parameters that represent wetting conditions,which is likely the case when landslides are initiated [Ebelet al., 2010]. Table 4 provides the root-mean-square error(RMSE) for each of the simulated and derived results forthe forward modeling period after 21 November 2005.

[35] Figure 9b compares observed and volumetric watercontents simulated using the numerical model. The agree-ment between observations and model results between 28October 2005 and 21 November 2005 is best at the 1.5 mdepth as this was the target for the inverse modeling proce-dure. At both the 1.0 and 1.5 m depths, the timing ofchanges in volumetric water content match quite well(Table 4), although in general, modeled volumetric watercontents decrease more rapidly following the cessation ofrainfall. This may result because we ignore any lateraldownslope flow or hysteresis in the soil water characteris-tics in the one-dimensional model, both of which wouldtend to attenuate any decrease in volumetric water contentduring drainage. Comparison of the contributing area andlandslide failure depths at the site indicates that 1-D analy-sis of infiltration over the time scale of typical rainstormsthat initiate landslides at this location are likely of suffi-cient accuracy [Godt et al., 2006], but that longer-term sim-ulations (e.g., seasonal) may be improved by consideringdownslope redistribution of soil water. However, the direc-tion of flow in the unsaturated zone will tend to be verticalor normal to the slope surface as the soil wets and onlydownslope under drying conditions [e.g., Lu et al., 2010].

[36] Compared to the 1.5 m depth, the agreementbetween the modeled and observed volumetric water con-tents is not as close at 1.0 m depth. Initially the simulatedvolumetric water content is about 0.08 too large comparedto the observations. This apparent bias toward higher

Figure 6. Profiles of (a) pressure head, (b) effective satu-ration, (c) suction stress, and (d) factor of safety for finesand with � ¼ 0.7, saturated hydraulic conductivity Ks ¼5 � 10�6 m s�1, and material strength properties in Table 1.Prescribed flux at the ground surface is equivalent to Ks.

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volumetric water contents at the 1.0 m depth is evidentthroughout the simulation (Figure 9b). This discrepancylikely results from minor differences in the hillside materi-als and thus differences in soil water characteristics at thetwo depths.

[37] The timing and magnitude of the modeled andobserved pressure head variation are in general agreementat both the 1.0 and 1.5 m depths (Figure 9c). The arrival ofthe wetting front at the 1.0 m depth in the end of October2005 is well captured by the simulation. The simulated ar-rival of the wetting front at 1.5 m depth is less abrupt anddelayed compared to the observation (Figure 9c). However,differences between observed and modeled pressure headdecrease as the soil becomes wetter. Modeled pressureheads are greater than the observation at the 1.0 m depthafter the beginning of November and greater at both depthsduring the wet period leading to the landslide at the site be-ginning in the middle of December 2005 (Figure 9c). Mod-eled pressure heads also show a greater response to rainfallthan observations during the period from the end ofDecember to 14 January. This may result from limitationsof pressure transducers and tensiometers to resolve pressureheads near zero.

[38] Suction stress was calculated using equation 2 andis shown for both the modeled and observed volumetricwater contents and pressure heads assuming a saturatedvolumetric water content of 0.4 (Figure 9d). At the 1.0 mdepth, suction stress calculated from the monitoring dataincreases from about �12 kPa to �4 kPa after the rainy pe-riod in late October 2005 and fluctuates around �5 kPa forthe remaining period (Figures 9a and 9d). At 1.5 m the cal-culated suction stress has a similar pattern to that calculatedfor the 1.0 m depth, but the increase from about �18 kPa toabout �4 kPa follows the increase at 1.0 m by 2.6 days asthe wetting front moves through the soil (Figure 9c) andremains at about �4 kPa.

[39] The differences in timing between modeled changesin suction stress compared to those calculated from themonitoring data mimic the differences between the mod-eled and observed pressure heads (Figures 9d and 9c). Thedifferences between the modeled and calculated magni-tudes are generally less than a few kilopascals, but have asubstantial impact on modeled factors of safety (Table 4).

[40] Factors of safety calculated from the monitoringdata are greater than 1.5 at both 1.0 and 1.5 m depth priorto the onset of rainfall and consequent increases in volu-metric water content, pressure head, and suction stress thatoccur in the beginning of November 2005 (Figure 9).

Figure 7. Profiles of (a) pressure head, (b) effective satu-ration, (c) suction stress, and (d) factor of safety for siltwith � ¼ 0.5, saturated hydraulic conductivity Ks ¼9 � 10�7 m s�1, and material strength properties in Table 1.Prescribed flux at the ground surface is equivalent to Ks.

Table 2. Summary of Results From the Hypothetical Examples

CoarseSand

MediumSand

SiltySand

FineSand Silt

Failure depth (m) 0.1–0.3 0.2–1.2 1.5–1.7 0.2–1.1 0.4–1.5Failure time (h) 12–4 36 23–29 48 480

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Factors of safety from the monitoring data at 1.5 mdecrease to near unity (indicating the potential for slopefailure) at this time and remain close to 1.0 for the remain-der of the record. At the 1.0 m depth, factors of safety reacha minimum of about 1.2 during several periods at the begin-ning of November, December, and at the end of December.On 14 January 2006, factors of safety calculated from themonitoring data were less than unity at the 1.5 m depthfor about 50 h prior to the occurrence of the landslide [Godtet al., 2009].

[41] Factors of safety calculated from the volumetricwater contents and pressure heads obtained from the

Figure 8. (a) Location of monitoring array and landslideat the Edmonds field site, near Seattle, Washington, (b)hillslope cross section, (c) detailed cross section, and (d)map showing the location of the instrument array and theshallow landslide. Two rain gauges were located at the toeof the bluff about 300 m north of the slide [Godt et al.,2009].

Figure 9. (a) Hourly and cumulative rainfall, (b) modeledand observed volumetric water content, (c) observed andmodeled pressure head, (d) modeled and calculated suctionstress, and (e) modeled and calculated factor of safety forthe period 24 September 2005 to 14 January 2006 at 1.0and 1.5 m depths.

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numerical model are greater than unity at both the 1.0 and1.5 m depths in September and much of October 2005.They decrease at both depths in early November inresponse to the rainfall, increases in volumetric water con-tent, pressure head, and suction stress (Figure 9). In lateDecember the modeled factors of safety at both depthsapproach unity and indicate potential instability beginningon 26 December and remain there for 72 h. After this time,factors of safety at the 1.0 m depth increase to more than1.0, but those at 1.5 m remain less than 1.0 through the endof the period. Figure 10 highlights factors of safety for theperiod from late December 2005 until the onset of the land-slide that damaged the instrumentation on 14 January at thetwo depths calculated from both the monitoring data andmodel results. Factors of safety from the modeling resultsat the 1.5 m depth fluctuates around 0.9 for much of the pe-riod prior to 14 January 2006 in a similar manner as tothose from the field data, but are slightly smaller. However,factors of safety from the model results at the 1.0 m depthare greater than unity, although not as large as those fromthe field data, for the period (Figure 10). On 14 January2006 factors of safety at 1.0 m calculated from the monitor-ing data decrease to 1.0 seven hours prior to landslideoccurrence.

7. Concluding Discussion[42] We have extended a framework for infinite slope

stability analysis for variably saturated materials [Lu andGodt, 2008] from steady state to transient conditions toassess the conditions leading to landslide occurrence abovethe water table. Four hypothetical hillslopes (comprisingcoarse, medium and fine sand, and silt) were examined fortheir representative range of materials for which shallowlandslides may occur under partially saturated conditions.Results show that the depth and timing of the onset ofpotential slope instability are dependent on the soil watercharacteristics of the hillside materials. Given the same soilstrength, the failure depth in the coarse sand hillslope is afew tens of centimeters below the ground surface comparedto about a meter for the fine sand example. This results inpart from the differences in the contribution of suctionstress to stability. For the limiting case of rainfall flux at

the ground surface equivalent to the saturated hydraulicconductivity of the soil, and for hydrostatic initial condi-tions, sand hillslopes can be destabilized in a few hourswhereas in contrast, silt hillslopes are destabilized in tensof days. Less thick deposits, permeability contrasts, accret-ing water tables, and wetter initial conditions all tend toreduce the time needed to achieve potential instability inthe unsaturated zone under a given flux of water at theground surface. Given the lower boundary conditions of afixed water table, rainfall flux much less than the saturatedhydraulic conductivity will generally not provoke instabil-ity [Lu and Godt, 2008].

[43] The case study from the instrumented hillslopealong the Puget Sound is also presented to test the frame-work for assessing the transient conditions leading to varia-bly saturated landslide occurrence. An inverse modelingapproach is used to estimate the soil water characteristicsfrom field monitoring data, although improvements in theprediction of landslide timing and thickness might beachieved using other inverse modeling algorithms [e.g.,Wöhling et al., 2008]. The comparison between modelresults and field observations implies that the model is suit-able for slope stability assessment under transient seepageconditions. The model is also able to predict approximatedepth and timing for possible occurrence of landslides.However, practical application of this approach wouldlikely require interpretation of results in a probabilisticmanner.

[44] Acknowledgments. This work was supported in part by a grantfrom the National Science Foundation (NSF-CMMI-0855783) to N.L. andJ.W.G. Jason Kean, Joshua White, and an anonymous reviewer providedcomments that improved the presentation and content of the paper.

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Figure 10. Modeled and calculated factor of safety forthe period 15 December 2005 to 14 January 2006 at 1.0and 1.5 m depths.

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R. L. Baum and J. W. Godt, U.S. Geological Survey, Box 25046,Denver Federal Center, Denver, CO 80225, USA. (jgodt@usgs.gov)

N. Lu, Department of Civil and Environmental Engineering, ColoradoSchool of Mines, 1500 Illinois St., Golden, CO 80401, USA.

B. S�ener-Kaya, Itasca Denver, Inc., 143 Union Blvd., Lakewood, CO80228, USA.

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