numerical study of soil conditions under which matric suction can be maintained

Numerical study of soil conditions under whichmatric suction can be maintained

L.L. Zhang, D.G. Fredlund, L.M. Zhang, and W.H. Tang

Abstract: The effect of negative pore-water pressure is often ignored in slope stability studies. There is a perceptionamong geotechnical engineers that negative pore-water pressures will dissipate with rainfall infiltration and cannot berelied upon in design considerations. The objective of this paper is to illustrate that under certain conditions soil suc-tion can be maintained. Based on the theory of infiltration and seepage through a saturated–unsaturated soil system,steady state and transient finite element seepage analyses were conducted using Seep/W on a 20 m high slope inclinedat 30°. The results of the analysis showed that under steady state conditions, the most important factor influencing thepermanency of matric suction in the soil is the magnitude of rainfall flux expressed as a percentage of the saturatedcoefficient of permeability of soil. For the analysis under transient seepage conditions, the results showed that the pore-water pressure profile depends on the magnitude of the rainfall flux, the saturated coefficient of permeability, the soil-water characteristic curve, and the water storage function. For a soil with a low coefficient of permeability and a largewater storage capacity, the matric suction needs a substantial amount of time to dissipate and thus may be maintainedover a longer time period than the rain is likely to fall, even if the ground surface flux is equal to or greater than thesaturated coefficient of permeability. Engineers should address more appropriate engineering design assumptions thatcan be related to the permanence of matric suction in soil slopes based on the numerical analysis. Measures such asslope cover or surface recompaction can be taken into consideration to minimize the rainfall infiltration and thus main-tain active matric suction in slopes.

Key words: unsaturated soils, slope, rainfall infiltration, matric suction, permeability.

Résumé : On ne tient souvent pas compte de l’effet de la pression interstitielle négative dans les études de stabilité detalus. Il existe une certaine perception parmi les ingénieurs géotechniciens à l’effet que les pressions interstitielles né-gatives vont se dissiper avec l’infiltration de la pluie et qu’on ne peut donc pas compter sur elles pour fins de calcul.Le but de cet article est d’illustrer que dans certaines conditions la succion peut être maintenue dans le sol. Sur la basede la théorie de percolation et d’infiltration à travers un système de sols saturés-non saturés, on a conduit des analysesd’infiltration par éléments finis dans les états permanent ou transitoire au moyen de Seep/W sur un talus de 20 m dehauteur incliné à 30 degrés. Les résultats des analyses ont montré que dans des conditions d’état permanent, le plusimportant facteur influençant la permanence de la succion matricielle dans le sol est l’amplitude du flux de la chute depluie exprimée en pourcentage du coefficient de perméabilité du sol saturé. Pour l’analyse dans des conditions transi-toires, les résultats ont montré que le profil de pression interstitielle dépend de l’amplitude du flux de la chute depluie, le coefficient de perméabilité du sol saturé, la courbe caractéristique sol-eau et la fonction d’entreposage del’eau. Pour un sol avec un faible coefficient de perméabilité et une grande capacité d’entreposage, la succion matri-cielle requiert un temps appréciable pour se dissiper et ainsi peut être maintenue pour une période de temps pluslongue que la durée vraisemblable de la pluie, même si le flux à la surface du terrain est égal ou plus grand que lecoefficient de perméabilité du sol saturé. Pour fins de calcul basé sur l’analyse numérique, les ingénieurs devraient uti-liser des hypothèses plus appropriées pouvant être reliées avec la permanence de la succion matricielle dans un talus desols. Des mesures telles que le couvert du talus ou le recompactage de la surface peuvent être prises en compte pourminimiser l’infiltration des chutes de pluie et ainsi maintenir la succion matricielle dans les talus.

Mots clés : sols non saturés, talus, infiltration des chutes de pluie, succion matricielle, perméabilité.

[Traduit par la Rédaction] Zhang et al. 582

Can. Geotech. J. 41: 569–582 (2004) doi: 10.1139/T04-006 © 2004 NRC Canada

569

Received 25 October 2002. Accepted 8 January 2004. Published on the NRC Research Press Web site at http://cgj.nrc.ca on5 August 2004.

L.L. Zhang,1 D.G. Fredlund,2 L.M. Zhang, and W.H. Tang. Department of Civil Engineering, Hong Kong University of Scienceand Technology, Clear Water Bay, Kowloon, Hong Kong.

1Corresponding author (e-mail: [email protected]).2Present address: Department of Civil Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada.

1. Introduction

The effect of negative pore-water pressure is often ignoredin geotechnical engineering practice, particularly when it re-lates to slope stability studies. For example, in slope stabilityanalyses the contribution to the shear strength of soil bymatric suction is often ignored. The main reason for this ap-pears to be related to the perception that infiltration of rain-fall will always produce a wetting front that gradually movesdownwards and consequently causes a reduction in thematric suction profile. It is assumed that negative pore-waterpressures will eventually dissipate with rainfall infiltration.Consequently, it is prematurely concluded that matric suc-tion cannot be relied upon to provide long-term stability tothe slope. However, the infiltration of rainfall in an initiallyunsaturated soil slope depends not only on the rainfall inten-sity and duration, but also on the saturated coefficient of per-meability, the permeability function, and the water storagecapacity of the soil.

An unsaturated soil may have a coefficient of permeabilitythat is greatly reduced from that of a saturated soil and ahigh capacity for water storage. To eliminate matric suctionfrom the soil, the rainfall needs to be sustained over a sub-stantial time period and the rainfall intensity needs to ap-proach the saturated coefficient of permeability of the soil atthe ground surface.

In situ measurements of matric suction illustrate that neg-ative pore-water pressures may or may not disappear follow-ing a rainstorm. Sweeney (1982) presented in situ suctionmeasurements from a slope instrumented with tensiometersin Hong Kong. The slope was inclined at 60° with an aver-age height of 30 m. The soil from the slope surface to 10 mdepth consisted of a completely weathered rhyolite. The rangeof values of the saturated coefficient of permeability for insitu completely decomposed rhyolite is 10–5 to 10–7 m/s ac-cording to the Geotechnical Engineering Office of the Gov-ernment of Hong Kong (GEO 1993). The next stratum was a5–10 m thick, completely to highly weathered rhyolite withan underlying layer of slightly weathered rhyolite. Freshbedrock was 20–30 m below the ground surface.

Suction measurements were made throughout the year1980 and a few observations can be made from the resultspresented in Fig. 1. First, matric suction showed a gradualreduction during the rainy season. The pore-water pressuresremained negative, even during the rainy season. Second,matric suctions between 5 and 17 m depth remained constantthroughout the year. The groundwater table, which was at agreater depth, rose and fell about 9 m throughout the year.These results demonstrate the water storage capacity of thesoil and its role in maintaining matric suction in a zone be-low the ground surface. Water continued to flow downwardthrough the soil and matric suction decreased at the shallowdepths in the slope while the matric suction throughout theintermediate depths remained almost constant.

Anderson (1983) presented in situ suction measurementsin a slope consisting of colluvium materials in Hong Kongas shown in Fig. 2. The saturated coefficient of permeabilityof colluvium ranges from 10–4 to 10–7 m/s (GEO 1993). Thematric suctions measured were relatively low and essentiallydisappeared after a heavy rainstorm.

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570 Can. Geotech. J. Vol. 41, 2004

Fig. 1. Suction measurements in a weathered rhyolite in HongKong (after Sweeney 1982).

Fig. 2. Suction measurements in a colluvium in Hong Kong (af-ter Anderson 1983).

The objective of this paper is to illustrate, by means of arainfall infiltration and saturated–unsaturated seepage model,the matric suction behavior in an unsaturated soil slope. Theconditions under which matric suction can be maintained ina slope are investigated through the use of a finite elementnumerical study. The range of rainfall and soil conditionsunder which soil suction can be maintained are presentedalong with a rationale for using the infiltration–seepagemodel in geotechnical engineering practice.

2. Brief review of matric suctions duringrainfall infiltration

The physical processes of infiltration of rainfall and seep-age through a saturated–unsaturated soil system have beenstudied by many researchers. The “wetting front” or “wet-ting band” concept was introduced by Lumb (1962) in rela-tion to the investigation of slope failures in Hong Kong.Figure 3 shows the variation of degree of saturation withdepth during rainfall. The soil is assumed to become satu-rated near the ground surface and nearly saturated down to adepth h under infiltration conditions. The wetting front isassumed to have a sharp separation between the initial con-dition and the wetted zone. Under prolonged and heavy rain-fall, the depth h of the wetting front is defined as

[1] hk t

n S S=

−sat

f( )0

where ksat is the saturated coefficient of permeability; Sf andS0 are the final and initial degrees of saturation, respectively;n is the porosity of the soil; and t is the time.

Sun et al. (1998) proposed a generalized wetting bandequation based on Lumb’s (1962) wetting band approachand presented the results of a series of one-dimensional fi-nite element analyses. Figure 4 shows a typical variation ofsoil suction with depth in an unsaturated soil. For a givenground surface moisture flux q0 less than ksat under steadystate conditions, the matric suction is u0. If the infiltrationrate is increased to q1, a new infiltration zone with thematric suction u1 will be formed that gradually progressesdownwards with time. The depth of the wetting front can becalculated using the following equation:

[2] hk k t= −

−( )1 0

1 0θ θ

where k0 = q0; k1 = q1; and θ0 and θ1 are the initial and newvolumetric water contents, respectively.

Equations [1] and [2] are similar, with the later equationapplying to a flux change from one steady state condition toanother steady state condition. The wetting front conceptprovides a simplified methodology for considering changesin soil saturation (or matric suction) under a change in rain-fall conditions. However, there will not always be a distinctdifference between the infiltration zone and the zone wherethe negative pore-water pressures have been maintained. Theresults from a numerical study conducted by Sun et al.(1998) show that under the same initial pore-water pressuredistributions, the wetting fronts are less distinct for the case

with an infiltration rate of 5 mm/h than those of the casewith an infiltration rate of 80 mm/h (Fig. 5).

The matric suction in an unsaturated soil will not neces-sarily be destroyed, even under long-term conditions of rain-fall infiltration. Kasim (1997) and Kasim et al. (1998)studied the relationship between steady state rainfall, thecoefficient of permeability function, and the soil-water char-acteristic curve on the equilibrium matric suction conditionsfor both a horizontal ground surface and a sloping groundsurface. The study showed that steady state rainfall does notnecessarily eliminate matric suction in the soil. When thesteady state rainfall is one or more orders of magnitude lessthan the saturated coefficient of permeability, the long-termmatric suction in the soil is largely maintained. It appearsthat matric suction will not disappear unless the steady staterainfall flux approaches the saturated coefficient of perme-ability of the soil near the ground surface.

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Zhang et al. 571

Fig. 3. Variation of degree of saturation with depth during infil-tration (after Lumb 1962).

Fig. 4. Transient infiltration in an unsaturated soil (modifiedfrom Sun et al. 1998).

3. Theory of saturated–unsaturatedseepage

3.1. Water flow through an unsaturated soilWater flow can be assumed to be governed by Darcy’s law

in saturated and unsaturated soils (Childs and Collis-George1950). According to Darcy’s law, the discharge velocity ofwater is proportional to the gradient of hydraulic head:

[3] v khy

= − ∂∂

where v is the discharge velocity of water (m/s), k is the co-efficient of permeability with respect to the water phase(m/s), h is the hydraulic head (m), and ∂h/∂y is the hydraulicgradient in the y direction.

The hydraulic head h is comprised of an elevation head yand a pore-water pressure head [uw/(ρwg)]:

[4] h yu

g= + w

wρ

where uw is the pore-water pressure, ρw is the density ofwater, and g is the gravitational acceleration.

The coefficient of permeability for a saturated soil canusually be assumed to be a constant value. For an unsatu-rated soil, however, the coefficient of permeability is a func-tion of matric suction (or negative pore-water pressure) anddecreases with an increase in the matric suction of the soil.The relationship between the coefficient of permeability ofan unsaturated soil and soil suction is called the permeabilityfunction.

Based on continuity considerations and Darcy’s law, thegoverning partial differential equation for two-dimensionaltransient flow through an unsaturated soil can be formulatedas follows:

[5]∂∂

∂∂

∂∂

∂∂

∂θ∂x

khx y

khy t

x y

+

= − w

where kx is the coefficient of permeability in the x direction,ky is the coefficient of permeability in the y direction, andθw is the volumetric water content.

Fredlund and Morgenstern (1976) proposed the followingwater phase constitutive relationship for an unsaturated soil:

[6] d d dww

aw

a wθ σ= − + −m u m u u1 2( ) ( )

where σ is the total stress; ua is the pore-air pressure; m1w is

the slope of the water volume versus σ − ua relationshipwhen d a w( )u u− is zero; and m2

w is the water storage coeffi-cient, which is the slope of the water volume versus (u ua w− )relationship when d a( )σ − u is zero.

For a transient seepage analysis, it can be assumed thatthe total stress remains constant and the pore-air pressureremains at atmospheric conditions. Then a change of volu-metric water content can be related to a change in pore-water pressure:

[7] d dww

wθ = −m u2 ( )

Substituting eqs. [4] and [7] into eq. [5] leads to the fol-lowing governing partial differential equation for water flowthrough an unsaturated soil:

[8]∂∂

∂∂

∂∂

∂∂

ρ ∂∂x

khx y

khy

m ght

x y

+

= 2

ww

Under steady state conditions, the time dependence of thematric suction profile disappears and eq. [8] can be simpli-fied as follows:

[9]∂∂

∂∂

∂∂

∂∂x

khx y

khy

x y

+

= 0

In eq. [8], the three soil parameters kx, ky, and m2w are not

constant values in an unsaturated soil but are functions ofnegative pore-water pressure, which is part of the hydraulichead. The meaning of the parameters and their relationshipto the soil-water characteristic curve are explained in the fol-lowing section.

3.2. Unsaturated soil property functions related toseepage

The soil-water characteristic curve is a relationship be-tween water content and suction in the soil, either for dryingor wetting conditions. Several mathematical equations have

© 2004 NRC Canada


Fig. 5. Transient infiltration at (a) q = 80 mm/h and (b) q = 5 mm/h from one-dimensional Seep/W analysis (after Sun et al. 1998).

been proposed to describe the soil-water characteristic curve.In this paper, the Fredlund and Xing (1994) equation with acorrection factor C(ψ), equal to 1, is used:

[10] θ θ

ψw

s=

+

ln e

a

nm

where θs is the saturated volumetric water content, e is thenatural base of logarithms, ψ is the soil suction (i.e., matricsuction at low suction values and total suction at high suc-tion values beyond the residual condition), a is the matricsuction value at the inflection point and is closely related tothe air-entry value of the soil (the air-entry value of the soilis the suction beyond which the soil starts to desaturate), n isthe slope of the soil-water characteristic curve at the inflec-tion point, and m is a fitting parameter related to residualwater content.

The water storage coefficient m2w represents the water

storage capacity of the unsaturated soil at any soil suctionlevel. The function of m2

w with respect to the soil suction isdefined as the water storage function. According to the defi-nition of the water storage coefficient (eq. [6]), differentia-tion of the equation of the soil-water characteristic curve onan arithmetic scale yields the water storage function. Fig-ures 6a and 6b illustrate a soil-water characteristic curve andthe corresponding water storage function.

The character of the coefficient of permeability functionbears a relationship to the soil-water characteristic curve asshown in Figs. 6a and 6c. The coefficient of permeability re-mains relatively constant until the air-entry value of the soilis exceeded. At suctions greater than the air-entry value, thecoefficient of permeability decreases rapidly.

The coefficient of permeability function can be satisfacto-rily estimated for most slope stability problems using thesaturated coefficient of permeability and the soil-water char-acteristic curve. In this paper, the coefficient of permeabilityfunction is estimated using the Fredlund et al. (1994) predic-tion method.

3.3. Kisch’s interpretation of the infiltration modelAnalysis of seepage and infiltration in an unsaturated soil

can be readily performed through the use of available finiteelement seepage software. The results from numerical analy-sis are sometimes difficult to interpret, however, becauseseveral parameters are involved. These parameters are re-lated to the soil-water characteristic curve, coefficient of per-meability function, rainfall intensity, and rainfall duration.The fundamental processes associated with infiltration underboth steady state and transient situations need to be consid-ered separately when interpreting the results of a numericalanalysis.

Kisch (1959) derived an equation for one-dimensionalsteady flow in a soil subjected to a constant water flux q atthe ground surface. The magnitude of q can be calculated fora vertical soil column of unit cross section:

[11] q kug y

= +

1

dd

w

wρ

Under steady state conditions, the flux q is a constant, andeq. [11] can be written as

[12]d

dw w( / )

( / )u g

yq k

ρ = −1

If the magnitude of the ground surface flux q approachesthe coefficient of permeability of the soil k, d dw w( / )/u g yρ = 0(i.e., pressure head u gw w/( )ρ tends to be a constant).

Figure 7 (Kisch 1959) shows the steady state pressurehead profiles in a clay liner and an underlying sand layer fora surface water flux q equal to the saturated coefficient ofpermeability of the clay liner, ksclay. The saturated coefficientof permeability of the sand, kssand, is 149ksclay. The thicknessof the clay liner is 0.5 m. The water tables are at depths of2.5 and 4.5 m, respectively, for cases I and II. Under steadystate conditions, the gradient of pressure head is zero at theupper portion of the sand layer for both cases. The pore-water pressures at the top of the sand remain at a constantvalue of –5.5 kPa when the ground surface flux q, which is

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Zhang et al. 573

Fig. 6. Relationship between the soil-water characteristic curve,the water storage function, and the coefficient of permeabilityfunction for an unsaturated soil.

equal to ksclay, is equal to the coefficient of permeability ofthe unsaturated sand.

Figures 8 and 9 show typical pore-water pressure distribu-tions under steady state and transient seepage conditions, re-spectively. Under hydrostatic conditions, there is no groundflux. According to eq. [12], the gradient of the pore-waterpressure is –1, as shown in Fig. 8. Under steady state condi-tions, the water flux in and out of the soil reaches a balance.If the magnitude of the water flux is the same as the coeffi-cient of permeability of the unsaturated soil at a particularvalue of matric suction, the value of the pore-water pressureis constant, as shown in Fig. 8.

Infiltration under transient seepage conditions can be con-sidered as a transitional state between initial state and thefinal steady states (Fig. 9). The time to reach steady state isa function of the ground surface flux, the coefficient of per-meability of the soil, and the water storage of the soil. Whenthe flux is less than the saturated coefficient of permeability,the matric suction in the unsaturated soil can decrease butnot disappear. The matric suction can be eliminated onlywhen the ground surface moisture flux is equal to or greaterthan the saturated coefficient of permeability. The frame-work set forth by Kisch (1959) sets steady state bounds forthe infiltration model for unsaturated soils.

4. Numerical study to illustrate severalcases of infiltration

In this study, analyses for both steady state and transientseepage conditions were conducted on a 20 m high slopeinclined at 30°. The slope is composed of a homogenous,isotropic soil. The finite element seepage analysis softwareSeep/W (Geo-Slope International Ltd. 2001) for saturated–unsaturated soil systems was used in this study. The finite el-ement mesh, along with the boundary conditions, is shown

in Fig. 10. Along the left and right boundaries beneath thegroundwater table, a constant head was applied. A zero fluxboundary was applied along the left and right boundariesabove the groundwater table. To illustrate the pore-waterpressure profiles more clearly, the groundwater table wasfixed by applying a constant pressure head equal to zeroat the groundwater table, on the nodes of the mesh. The

© 2004 NRC Canada


Fig. 7. Variation of pore-water pressure with depth in the clayliner and the underlying sand layer for a surface water flux q =ksclay = 0.0068kssand (modified from Kisch 1959).

Fig. 8. Infiltration into an unsaturated soil under steady stateconditions with various ground surface moisture fluxes.

Fig. 9. Infiltration into an unsaturated soil under transient seep-age conditions with two different ground surface fluxes: (a) q <ksat; (b) q ≥ ksat.

groundwater table in a soil slope may rise during rainfall inreal situations, however. An example with a free groundwa-ter table is presented later in the paper to illustrate the effectof different boundary conditions. In other cases, the ground-water tables are fixed.

It was assumed that the base of the finite element meshwas impermeable. Precipitation was modeled as a moistureflux boundary, q, applied along the slope surface. SectionX–X is in the middle of the slope. The groundwater table atthe selected section is 15.28 m under the sloping surface.The pore-water pressure profiles at section X–X are pre-sented to illustrate conditions under which soil suction canbe maintained. It should be pointed out, however, that the in-filtration behavior in the entire slope controls the stability ofthe slope.

Soils with various a values (e.g., 1, 5, 10, 20, 50, 100, and200) and the same n, m, and saturated coefficient of perme-ability ksat values (i.e., n = 2, m = 1, and ksat = 1 × 10–5 m/s)were studied in detail for the parametric study. The soil-water characteristic curves and the corresponding coefficientof permeability functions for the soils are shown inFigs. 11a and 11b, respectively. The effect of n and ksat isalso presented.

4.1. Pore-water pressure profiles under steady stateconditions

The results in Fig. 12 were obtained for soils having thesame saturated coefficients of permeability of 1 × 10–5 m/s,n = 2, and m = 1. The pore-water pressure profiles shown inFigs. 12a–12f correspond to soils with a equal to 5, 10, 20,50, 100, and 200, respectively. Different ratios of groundsurface flux to saturated coefficient of permeability, q/ksat(i.e., 0.001, 0.01, 0.1, 0.2, and 0.5), were applied to the sur-face of the slope in the numerical analysis.

The results illustrate that as the rainfall flux approachesthe saturated coefficient of permeability of the soil, thematric suction at the surface of the slope approaches zero.There is essentially a vertical matric suction profile (i.e.,d(uw/ρwg)/dy = 0) established under steady state conditionswhen the ground surface flux approaches the saturated coef-

ficient of permeability. The depth of the constant matric suc-tion profile increases when increasing the ratio of steadystate rainfall flux to the saturated coefficient of permeability,q/ksat. It should be noted that these results are based on theboundary condition of a fixed groundwater table. The pore-water pressure profiles can be significantly influenced byvarying boundary conditions.

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Zhang et al. 575

Fig. 10. Finite element mesh and boundary conditions of the slope.

Fig. 11. Soil-water characteristic curves and permeability func-tions with varying a values for the soils used in the study.

The values of matric suction on the vertical portion of thepressure profiles decrease with decreasing a values, and thedepth of the constant matric suction portion increases withdecreasing a values for the soil. The long-term matric suc-tion does not disappear but remains essentially close to orunchanged from the hydrostatic profile when the steady staterainfall flux is two or more orders of magnitude less than thesaturated coefficient of permeability and a is greater than100 (Fig. 12f ). When the a values are smaller, the reductionin matric suction is more significant.

Figure 13 illustrates the long-term matric suction profilesin the slope for the soils with different n values. It showsthat the values of matric suction at the same value of q/ksatdo not decrease or increase monotonically with different nvalues.

As shown conceptually in Fig. 8 and further illustrated byFigs. 12 and 13 from the numerical modeling, the matricsuction profiles under steady state conditions consist of asection of constant pore-water pressure, a transition section,

and a section of the hydrostatic condition. Kasim (1997)proposed a method to estimate the value of matric suction(ua – uw)1 at the constant pore-water pressure section, associ-ated with the steady state rainfall flux q1, using the coeffi-cient of permeability function for the case of a horizontalground surface (Fig. 14). Figure 15 presents the soil-watercharacteristic curves and the corresponding permeabilityfunctions for soils with varying n values. Applying the esti-mation method proposed by Kasim, the theoretical matricsuction (ua – uw)1 for soils with varying n values does notdecrease or increase monotonically at the same value of q1.Therefore, the pore-water pressure profiles shown in Fig. 13are reasonable.

4.2. Pore-water pressure profiles under transientseepage conditions

The effects of air-entry value, water storage capacity, satu-rated coefficient of permeability, and groundwater boundaryconditions on pore-water pressure profiles are illustrated in

© 2004 NRC Canada


Fig. 12. Pore-water pressure profiles in slopes with various a values subjected to various rainfall fluxes under steady state conditions.

this section. The initial state is taken to be the hydrostaticcondition for all cases. It should be noted that varying initialsoil moisture conditions could significantly influence therainfall infiltration and pore-water pressure profiles in soilslopes, as has been studied by Freeze (1969), Tsaparas et al.(2002), and others.

4.2.1. Effect of air-entry value on the wetting frontFigures 16a–16f show the pore-water pressure profiles un-

der transient seepage conditions with various a values. Therainfall flux is equal to the saturated coefficient of perme-ability, and so the ground surface is subjected to the maxi-mum moisture flux that it can absorb.

The results illustrate the different patterns createdthroughout the pore-water pressure profile. For the soil witha equal to 1, the wetting front is sharp and distinct. The

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Zhang et al. 577

Fig. 13. Comparison of pore-water pressure profiles for soilswith various n values.

Fig. 14. Determination of the matric suction values at a horizon-tal ground surface for a soil subjected to a steady state rainfallflux q1 (after Kasim 1997).

Fig. 15. Soil-water characteristic curves and permeability func-tions with varying n values for the soils used in the study.

© 2004 NRC Canada


matric suction near the ground surface decreases with time,but the rate of downward movement is small. After 50 daysof rainfall with a flux equal to the saturated coefficient ofpermeability, the depth of the wetting front is only about1 m below the ground surface.

For a soil with a equal to 5, the transition zone betweenthe infiltration zone and the unaffected zone is still quitesharp and distinct. At shallow depths into the slope, thematric suction decreases but remains essentially constant fordeeper soils. It takes only 1 day for the wetting front tomove to a depth of about 1 m. After 3 days the infiltrationdepth is about 4 m. The negative pore-water pressure almostdisappears after 6 days of rainfall.

The pore-water pressure profiles of the soil with a equal

to 10 resemble the profiles of the soil with a equal to 5.However, the negative pore-water pressures almost disap-pear after 4 days of rainfall infiltration. For a values greaterthan 10, the transition between the infiltration zone and theunaffected zone becomes less distinct. The remaining matricsuction in the soil decreases more rapidly with an increase inthe a value.

The results show that as the a value increases, the wettingfront becomes less distinct. According to eq. [12], the gradi-ent of the pore-water pressure depends on the ratio of flux tothe coefficient of permeability of the unsaturated soil. Giventhe same initial matric suctions (e.g., all the cases are inhydrostatic conditions initially in this study), the soil with asmaller air-entry value corresponds to a smaller initial coef-

Fig. 16. Pore-water pressure profiles for soils with various air-entry values and ksat = 10–5 m/s subjected to a flux of q = 10–5 m/s un-der transient seepage conditions.

ficient of permeability, and the soil with a larger air-entryvalue has a larger coefficient of permeability (see Fig. 11b).If the moisture flux q is the same for both cases, then thevalue of q/k is larger for the soil with a smaller air-entryvalue. Consequently, the gradient of the pore-water pressureis greater for the soil with the smaller air-entry value.

The previous observations can be further illustrated by theexamples of pore-water pressure profiles when the rainfallflux is 10–6 m/s, which is 10% of the saturated coefficientof permeability of the soils (Fig. 17). Comparing the twographs in Fig. 17, the most significant difference is the shapeof the wetting front. For the soil with a equal to 10(Fig. 17a), the wetting front is approximately horizontal,which means the infiltration rate is much greater than theunsaturated coefficient of permeability according to eq. [12].For the soil with a equal to 100 (Fig. 17b), however, the ini-tial coefficient of permeability in the soil is comparable tothe flux rate and the pore-water pressure gradient ap-proaches zero.

4.2.2. Effect of varying the saturated coefficient ofpermeability

Figures 18a–18c show the pore-water pressure profiles forsoils with the same soil-water characteristic curve (i.e., a =100, n = 2, m = 1) but different saturated coefficients of per-meability (i.e., ksat = 10–7, 10–5, and 10–3 m/s, respectively).The rainfall fluxes are equal to the saturated coefficient ofpermeability for all three cases (i.e., the ratios of flux versusthe saturated coefficient of permeability are unity for thethree cases). Comparing the patterns of the pore-water pres-sure profiles, it can be seen that the shapes of the profilesare similar but the rates of downward movement of the wet-ting fronts are distinct, which indicates that the behavior ofrainfall infiltration under transient seepage conditions shouldbe related to the absolute intensity of the rainfall and the soilproperties.

4.2.3. Influence of water storage on the pore-waterpressure profile

Figure 19 can be used to explain the behavior of infiltra-tion for soils with varying air-entry values. The soil-watercharacteristic curves and the corresponding water storagefunctions are shown on the graph. The desaturation rate (n =2) is the same for both soils, but the air-entry values of thetwo soil-water characteristic curves are different. The maxi-mum water storage coefficient for the soil with a equal to 1is 0.1445, whereas that for the soil with a equal to 10 is only0.0145. These values are reasonable because the desaturationrate n represents the slope at the inflection point of the soil-water characteristic curve, which is expressed on a logarithmscale of matric suction, whereas the water storage coeffi-cient, m2

w, is the arithmetic slope of the soil-water character-istic curve. As the n values of the two soil-watercharacteristic curves are the same, the same change of volu-metric water content, ∆θw, is associated with the samechange of matric suction on a logarithmic scale (i.e., ∆ ln(ua –uw). However, the change of matric suction, ∆(ua – uw), ismuch smaller for the case of a equal to 1 than for the case ofa equal to 10 because the desaturation part is in the lowmatric suction portion for a equal to 1 and consequentlyyields a larger water storage coefficient.

The difference in the water storage functions can help ex-plain why the rate of movement of the wetting front for soilswith different air-entry values is distinctly different, asshown in Fig. 16. For soils with the same coefficient of per-meability and the same desaturation rate, subjected to thesame magnitude of rainfall flux, the soil with a lower air-entry value has a greater water storage capacity. Thus, themovement of the wetting front is much slower in the soilwith the lower air-entry value.

On the other hand, if both soils have the same soil-watercharacteristic curve (i.e., the same water storage function),the soil with a smaller saturated coefficient of permeabilityallows less infiltration than the soil with a larger coefficientof permeability, although both have the same water storagecapacity. That is why the rate of downward movement of thewetting front in a soil with ksat equal to 10–7 m/s is muchsmaller than that in a soil with ksat equal to 10–3 m/s, asshown in Figs. 18a and 18c.

4.2.4. Effect of varying the groundwater boundaryconditions

In all the previous cases illustrating the infiltration behav-ior in an saturated–unsaturated soil system, it was assumedthat the groundwater table in the slope did not vary duringthe rainstorms. In reality, however, the pore-water pressureprofiles and the groundwater tables in soil slopes can be sig-nificantly influenced by varying boundary conditions.

Figure 20 shows the pore-water pressure profiles when thegroundwater table is allowed to rise (i.e., the constant pres-sure boundary condition for the groundwater table inside the

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Zhang et al. 579

Fig. 17. Examples of pore-water pressure profiles in a soil sub-jected to a surface flux q = 10–6 m/s.

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slope in Fig. 10 is removed). Comparing Fig. 20a withFig. 16c and Fig. 20b with Fig. 16e, it can be observed thatthe rates of downward movement of the wetting front arecomparable. The rising of the groundwater table is moremarked for the soils with a larger air-entry value because thesoil has a smaller water storage capacity.

All the pore-water pressure profiles presented earlier are

at section X–X in the middle part of the slope as shown inFig. 10. The pore-water pressure distribution in the entireslope needs to be analyzed, however, to better understandthe overall permanency of matric suction. Figure 21 illus-trates an example of pore-water pressure contours in a soilslope subjected to a rainfall flux q equal to 10–5 m/s. Al-though absolute pore-water pressure values vary from onecross section to another, the change of pore-water pressurewith time in the entire slope follows trends similar to thosepresented for section X–X.

5. Geotechnical engineering implications

Numerous research studies have been undertaken thatshow the changes in pore-water pressure profiles in a soilslope subjected to various surface moisture flux conditions.Much has been learned about the response of the soil in theunsaturated soil zone to rainfall conditions. A wide varietyof factors can be considered using numerical studies duringthe design of a soil slope. Using the soil-water characteristiccurves and the saturated coefficients of permeability for thematerials involved, the times required for the dissipation ofnegative pore-water pressure under various rainstorm condi-tions can be estimated.

The long-term matric suction changes in a slope are con-trolled by factors related to the ground surface moisture fluxand the hydraulic properties of the soils near the ground sur-face. The ratio of the ground surface moisture flux to thesaturated coefficient of permeability of the soil near theground surface is the primary variable to be considered

Fig. 18. Pore-water pressure profiles in a slope subjected to thesame ratio of q/ksat = 1 for soils with a = 100, n = 2, andm = 1.

Fig. 19. Comparison of water storage functions for soils witha = 1 and a = 10.

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Zhang et al. 581

when assessing the potential permanence of matric suction.To maintain negative pore-water pressures in a slope, it isimportant to reduce the infiltration flux through the use of asuitable type of cover system at the ground surface. A com-mon practice in Hong Kong is to provide a soil–cement–lime plaster cover called “chunam” on the soil slopes. Limet al. (1996) carried out a field instrumentation program tomonitor negative pore-water pressures in residual soil slopesin Singapore that were protected by different types of sur-face covers. The changes in matric suction due to changes inground surface moisture flux were found to be least signifi-cant under a canvas-covered slope and most significant in abare slope. Other relatively impermeable surface covers canbe adopted depending on the saturated coefficients of per-meability of the surface materials on the slope.

It is also effective to maintain negative pore-water pres-sures in a slope through reducing the saturated permeabilityof the surface soil. After the 1976 Sau Mau Ping slope fail-ure, the investigation panel (Government of Hong Kong1976) recommended that a minimum stabilization require-ment of a loose slope should consist of removing the loosesurface soil by excavating a vertical depth of not less than3 m and recompacting at 95% of standard compaction den-sity. For the Sau Mau Ping fill material, the average satu-rated coefficient of permeability decreased by three timeswhen the relative compaction of the fill was increased from82% to 95%.

Variability of in situ hydraulic properties of a soil may in-

fluence the distribution of matric suction in a soil slope. Forexample, the average annual rainfall in Hong Kong is about2200 mm. If the rainfall is averaged over 1 year, the rainfallflux intensity is approximately 7 × 10–8 m/s. For a soil slopecomprising completely decomposed rhyolites, this averagedrainfall flux intensity would correspond to 0.7%–70% of thesaturated permeability that might vary from 10–5 to 10–7 m/sdue to spatial variability of soil properties. With better mea-surements and monitoring of the field conditions, numericalstudies provide an opportunity to improve the design of thesoil slope by quantifying the permanency of the matric suc-tion.

6. Summary of findings

Based on the theory of infiltration and seepage insaturated–unsaturated soil systems, several series of numeri-cal modeling studies were conducted. The key observationscan be summarized as follows.

Under steady state conditions, the most important factorthat affects the matric suction near the ground surface is the

Fig. 20. Pore-water pressure profiles in a slope with a free watertable subjected to a rainfall flux of q = 10–5 m/s.

Fig. 21. Pore-water pressure contours (in kPa) in a slope sub-jected to a surface flux of q = 10–5 m/s with a soil of ksat =10–5 m/s, a = 100, n = 2, and m = 1.

magnitude of the ground surface moisture flux expressed asa percentage of the saturated coefficient of permeability. Thelong-term matric suction does not disappear and remains es-sentially close to the hydrostatic profile for the steady staterainfall fluxes that are two or more orders of magnitude lessthan the saturated coefficient of permeability when the avalue is greater than 100.

There is essentially a vertical matric suction profile (i.e.,d(uw/ρwg)/dy = 0) established under steady state conditionsabove the water table. The values of matric suction in thevertical portion of the pore-water pressure profile decreasewith increasing q/ksat ratios and decreasing a values. Thematric suction that can be maintained under long-term rain-fall infiltration can be estimated by considering the coeffi-cient of permeability function and the ratio q/ksat.

Under transient seepage conditions, the pore-water pres-sure profile after a rainfall depends on the magnitude ofrainfall flux, the saturated coefficient of permeability, andthe water storage function.

When the ground surface flux under transient conditionsis equal to or greater than the saturated coefficient of perme-ability, the saturated coefficient of permeability essentiallybecomes the upper limit of the infiltration rate. With thesame saturated coefficient of permeability, the soil with alarger water storage coefficient has a larger water storagecapacity. Consequently, the downward movement of the wet-ting front in the soil is slower than that in a soil with asmaller water storage capacity. On the other hand, given thesame water storage function, the matric suction in the soilwith a larger saturated coefficient of permeability will disap-pear more easily.

Engineers may formulate more appropriate engineeringdesign questions that can be addressed relevant to the per-manence of matric suction in soil slopes based on the nu-merical analysis. Measures can also be taken to maintainsuctions in slopes in an active manner. Reduction of perme-ability of fill materials by recompaction and reduction ofsurface flux by cover systems are examples of such activemeasures.

Acknowledgement

The work described in this paper was substantially sup-ported by grants from the Research Grants Council of theHong Kong Special Administrative Region, China (ProjectNo. CA99/00.EG01 and Project No. HKUST 6229/01E).

References

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numerical study of soil conditions under which matric suction can be maintained

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