Technical Paper by J.P. Giroud, R.C. Bachus and R. Bonaparte - IGS ...

Technical Paper by J.P. Giroud, R.C. Bachus and 

R. Bonaparte 

INFLUENCE OF WATER FLOW ON THE 

STABILITY OF GEOSYNTHETIC-SOIL 

LAYERED SYSTEMS ON SLOPES 

ABSTRACT: This paper presents an analysis of the stability of geosynthetic-soil layered 

systems constructed on slopes when water is flowing along the slope. Equations 

are presented for the factor of safety of infinite slopes as well as slopes of finite height. 

In the case of an infinite slope, only the interface shear strength along the slip surface 

contributes to the stability. In the case of a slope of finite height, the contributions of 

the buttressing at the toe of the slope and of geosynthetic tension are taken into account, 

in addition to interface shear strength. The analysis shows that the influence of water 

flow on the stability of a geosynthetic-soil layered system can be very significant if the 

slip surface is above the geomembrane. In this case, the factor of safety of a layered 

system with water flow can be as low as one half of the factor of safety without water 

flow. The analysis shows that the influence of water is small, and in some cases negligible 

or even zero, if the slip surface is below the geomembrane. 

KEYWORDS: Geosynthetic, Geomembrane, Slope, Stability, Water flow, Liner system, 

Layered system. 

AUTHORS: J.P. Giroud, Senior Principal, GeoSyntec Consultants, 621 N.W. 53rd 

Street, Suite 650, Boca Raton, Florida 33487, USA, Telephone: 1/407-995-0900, 

Telefax: 1/407-995-0925; R.C. Bachus, Principal, and R. Bonaparte, Principal, 

GeoSyntec Consultants, 1100 Lake Hearn Drive N.E., Suite 200, Atlanta, Georgia 

30342, USA, Telephone: 1/404-705-9500, Telefax: 1/404-705-9400. 

PUBLICATION: Geosynthetics International is published by the Industrial Fabrics 

Association International, 345 Cedar St., Suite 800, St. Paul, Minnesota 55101, USA, 

Telephone: 1/612-222-2508, Telefax: 1/612-222-8215. Geosynthetics International is 

registered under ISSN 1072-6349. 

DATES: Original manuscript received 7 September 1995, accepted 27 October 1995. 

Discussion open until 1 July 1996. 

REFERENCE: Giroud, J.P., Bachus, R.C. and Bonaparte, R., 1995, “Influence of 

Water Flow on the Stability of Geosynthetic-Soil Layered Systems on Slopes“, 

Geosynthetics International, Vol. 2, No. 6, pp. 1149-1180. 

GEOSYNTHETICS INTERNATIONAL S 1995, VOL. 2, NO. 6 

1149

GIROUD, BACHUS AND BONAPARTE D Influence of Water on Layered Systems on Slopes 

1 INTRODUCTION 

A geosynthetic-soil layered system constructed on a slope typically includes one or 

more fluid barriers (e.g. geomembranes, geosynthetic clay liners, and compacted clay 

layers), often referred to as liners, and one or more drainage layers (e.g. geosynthetic 

materials such as geonets and geocomposite drains, and granular layers). As a result of 

the layered structure, water may flow downslope parallel to the slope direction, thereby 

exerting drag forces on the layers within which the flow takes place. These drag forces, 

being oriented downslope, are detrimental to the stability of the geosynthetic-soil layered 

system. 

The purpose of this paper is to evaluate the influence of water flow on the stability 

of geosynthetic-soil layered systems on slopes. It will be seen that the action of water 

flow is complex and must be carefully analyzed. For example, in some cases the factor 

of safety of a layered system when water is flowing is approximately one half of the 

factor of safety when water is not flowing, whereas in other cases water flow along a 

slope has no influence on the factor of safety. 

2 ANALYSIS OF THE EFFECT OF WATER FLOW 

2.1 Overview of Water Flow in a Drainage Layer 

When water flows in a drainage layer placed on a low-permeability material (i.e. a 

liner) located on a slope of finite height and subjected to a supply of water, the phreatic 

surface generally is not parallel to the slope (Figure 1). In other words, the thickness 

of the portion of the drainage layer where water flows (hereinafter referred to as the flow 

Figure 1. 

Water flow in a drainage layer. 

1150 GEOSYNTHETICS INTERNATIONAL S 1995, VOL. 2, NO. 6


thickness) is generally not uniform. The determination of the phreatic surface in a drainage 

layer is beyond the scope of this paper. As shown by Giroud and Houlihan (1995), 

an average value of the flow thickness, t wavg , can be calculated as a function of the water 

supply rate, the slope and length of the drainage layer, and the hydraulic conductivity 

of the drainage layer material. In this paper, a uniform flow thickness, t w , is considered 

and it is reasonable to assume that t w = t wavg when using the equations presented herein. 

The presence of a liner (i.e. a low-permeability material) below the drainage layer 

is essential in the study presented herein. The liner separates the medium above the liner, 

where water flows, from the medium below the liner, where it is assumed that there 

is no flow. Since geomembranes are now used extensively in liner systems and they are 

generally placed immediately below a drainage layer, it will be assumed that there is 

a geomembrane below the drainage layer. 

As indicated in Section 1, drainage layers can be constructed with either granular or 

geosynthetic materials. Geosynthetic materials used in drainage layers are typically 

only a few millimeters thick. For properly designed drainage layers, the thickness of 

the flow is less than the thickness of the drainage layer. Therefore, in the case of properly 

designed geosynthetic drainage layers, the flow thickness is very small. It will be 

shown hereinafter that the effect of water flow on slope stability is proportional to flow 

thickness. Accordingly, water flow has virtually no effect on the stability of layered systems 

incorporating properly designed geosynthetic drainage layers. Therefore, this paper 

will focus on layered systems incorporating granular drainage layers. 

Notwithstanding the foregoing comments, the equations presented herein are applicable 

to geosynthetic drainage layers provided that: 1) the considered physical properties 

of the granular drainage material (such as porosity and the various densities) are 

replaced by the corresponding properties of the geosynthetic material; and 2) it can be 

assumed that the structure of the geosynthetic is such that, even under stress, the contact 

areas between solid constituents of the geosynthetics or between these solid constituents 

and adjacent materials are small. The latter requirement is necessary to ensure the 

applicability to the geosynthetic of the effective stress principle which describes the 

way stresses are distributed between the solid phase and the liquid phase of a saturated 

soil. 

The equations presented in this paper can easily be extended to the case where the 

considered drainage layer, granular or geosynthetic, does not have sufficient flow capacity 

and, as a result, a fraction of the flow occurs in a layer of soil above the drainage 

layer. In this case, the effect of water flow on slope stability is evaluated considering 

the flow in the entire depth of the drainage layer, plus the flow up to a certain depth in 

the soil layer located above the drainage layer. 

Although the liquid considered in this paper is water, the equations presented are valid 

without any modification for aqueous solutions, such as leachates. The equations can 

also be used for any other liquid provided that the density (or the unit weight) of water 

is replaced by the density (or the unit weight) of that liquid. 

The equations presented in this paper are not applicable if water does not flow parallel 

to the slope. This situation may occur in some cases, especially under certain transient 

flow conditions. However, in most cases, under steady-state as well as transient conditions, 

water flows parallel to the slope in layered systems. 


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2.2 Effects of Water on Soil 

2.2.1 Overview 

The term soil, in this paper, encompasses all materials made of mineral particles. The 

term encompasses, in particular, granular materials used in drainage layers such as 

sand, gravel, and aggregate. The term also encompasses materials made of fine particles 

such as silts and clays that may be placed above a geomembrane or a granular 

drainage layer as part of a layered system. 

It is known from soil mechanics, and the theory of flow in porous media, that water 

has two major mechanical effects on soils. Water (whether it flows or not) tends to uplift 

soil particles due to a buoyancy effect and flowing water tends to drag particles in the 

direction of flow. (The same two effects explain why a barge both floats on a river and 

moves with the current.) In general, soil particles subjected to these two effects do not 

move because of their high density, and because they are interlocked within a stable soil 

structure. However, these two effects, alone or combined with others, can cause soil instability 

in some cases. 

The effects of buoyancy and drag are discussed below. Other potential effects of water 

are not considered in this paper. These effects include, for example, changes in soil 

properties induced by water and dissolution of some soil components by water. 

2.2.2 Buoyancy Force 

Water exerts a pressure on the surface of a submerged soil particle. The magnitude 

of this pressure varies linearly with elevation, increasing from the uppermost to the lowermost 

point of the considered particle. This pressure results in an upward vertical force 

applied to the particle. A classical demonstration shows that this force (known as Archimedes 

thrust) is equal to the weight of water that would fill the volume occupied by the 

particle. 

A soil is composed of particles and voids. The proportion of particles relative to the 

combined volume of particles and voids is 1-n (where n is the soil porosity) and, since 

the Archimedes thrust is applied only to particles, the soil buoyant density, ρ b ,is: 

ρ b = (1 − n) (ρ s − ρ w ) 

(1) 

where: n = soil porosity; ρ s = density of soil particles; and ρ w = density of water. 

Simple geometric considerations give: 

ρ d = (1 − n) ρ s (2) 

(3) 

ρ sat = (1 − n) ρ s + n ρ w 

where: ρ d = “dry density” of soil (i.e. the density of soil calculated considering only the 

particles, which is equal to the density of dry soil if the soil volume does not change 

as the soil dries); and ρ sat = “saturated density” of soil (i.e. the density of soil calculated 



assuming that all voids are filled with water, which is equal to the density of saturated 

soil if the soil volume does not change as the soil is being saturated with water). 

Combining Equations 1 and 3 gives: 

ρ b = ρ sat − ρ w 

(4) 

Finally, the density of a soil which contains a certain amount of water retained by capillarity 

is: 

ρ t = ρ d (1 + w) 

(5) 

where: ρ t = “total density” of soil (i.e. the density of soil that contains a certain amount 

of water); and w = water content defined as the mass of water in a given volume of soil 

divided by the mass of dry soil (i.e. mass of particles) in the same volume of soil. 

In the above equations, the densities ρ t , ρ s , ρ sat , ρ b , ρ d and ρ w can be replaced by unit 

weights γ t , γ s , γ sat , γ b , γ d and γ w , respectively. The general relationship between unit 

weight and density is: 

γ = ρ g 

(6) 

where g is the acceleration due to gravity (g = 9.81 m/s 2 ). 

In examples presented hereinafter, a typical soil will be used; this soil is defined by 

a porosity n = 0.35 and a density of particles ρ s = 2700 kg/m 3 . Based on the above values 

of n and ρ s , and considering a water content of 8%, the typical soil used in the examples 

has the characteristics presented in Table 1, which are consistent with Equations 1 to 

6. 

Table 1. 

Typical soil used for examples and comparisons. 

Densities 

Unit weights 

ρ s = 2700 kg/m 3 

γ s = 26.49 kN/m 3 

ρ sat = 2105 kg/m 3 

γ sat = 20.65 kN/m 3 

ρ b = 1105 kg/m 3 = 0.525 ρ sat 

γ b = 10.84 kN/m 3 =0.525γ sat 

ρ d = 1755 kg/m 3 

γ d = 17.22 kN/m 3 

for w =8%: 

for w =8%: 

ρ t = 1895 kg/m 3 = 0.900 ρ sat γ t = 18.59 kN/m 3 = 0.900 γ sat 

Note: The soil porosity is n = 0.35. The densities and unit weights were calculated using Equations 1 to 6. 


1153


2.2.3 Drag Force 

Flowing water exerts drag forces on soil particles. A classical demonstration of the 

theory of flow in porous media shows that the resultant of drag forces on soil particles 

is in the direction of the flow and can be expressed as a force per unit volume of soil 

as follows: 

f w = γ w i 

(7) 

where: f w = drag force per unit volume of soil; γ w = unit weight of water; and i = hydraulic 

gradient that characterizes the flow at the point where the drag force is calculated. 

It is noteworthy that the drag force per unit volume is independent of the soil hydraulic 

conductivity, k.Ifk increases, the drag force tends to increase because flow velocity 

increases; at the same time, however, the drag force tends to decrease because the medium, 

being more permeable, offers less resistance to flow. The two mechanisms balance 

each other exactly. 

From Equation 7, the drag force, F w , applied to a volume, V w , of soil where water 

is flowing is: 

F w = f w V w = γ w iV w 

(8) 

Although the drag force per unit volume, f w , does not depend on the soil hydraulic 

conductivity, k, of the drainage layer material, the drag force, F w , generally depends 

on k because the volume of soil occupied by water flow, V w , generally depends on k 

(e.g. in Figure 1 the phreatic surface rises if the hydraulic conductivity, k, of the drainage 

layer material decreases). 

2.3 Stresses in Materials Located on Slopes 


Having presented a discussion on the effects of water on soil, the next logical step is 

to discuss the effect of water on stresses in materials located on slopes. Three cases are 

considered: the case where water flows over the entire thickness of the soil layer (“full 

flow”); the case where there is no water flow (“no flow”) either because the soil is dry 

or it contains only water retained by capillarity; and the case where water flows in a 

fraction of the thickness of the soil layer (“partial flow”). These three cases are analyzed 

below, starting with the full flow case which provides an opportunity to present as simply 

as possible the concepts involved in the analysis. The simple no flow case will then 

be presented and the more complex partial flow case will be derived from the two other 

cases. Finally, a comparison between the three cases will lead to conclusions on the effect 

of water flow on stresses. 

It is assumed that flow occurs only in areas where the soil is saturated. In areas where 

the soil is not saturated it is assumed that water is retained by capillarity. Herein, capillary 

suction is neglected and the pore water pressure is assumed to be zero when the soil 



is not saturated. It is also assumed that the soil below the geomembrane is never saturated. 

Therefore, it is assumed that the pore water pressure is zero and that there is no 

flow below the geomembrane. 

In subsequent analyses, conditions above and below the geomembrane will be considered. 

It is important to note that, in the discussions, the phrase “above the geomembrane” 

refers to a potential slip surface located either along the upper face of the geomembrane 

or at any location above the geomembrane. Similarly, the phrase “below the 

geomembrane” refers to a potential slip surface located either along the lower face of 

the geomembrane or at any location below the geomembrane. However, all equations 

are written for the case where the slip surface is at, or very close to, the geomembrane 

(upper or lower face). If the slip surface is not along the geomembrane, the term t in 

the equations (which represents the thickness of soil above the geomembrane) should 

be replaced by a term that represents the thickness of soil above the slip surface. 

2.3.2 Case of Full Water Flow 

Evaluation of Applied Forces. A layer of saturated soil resting on a geomembrane on 

a slope is considered (Figure 2). It is assumed that there is a sufficient supply of water 

to ensure steady-state flow of water in the entire soil thickness. The thickness of the soil 

layer is t and a vertical slice of soil, of width b in the direction of the slope, is considered. 

All forces considered in the analyses presented below are, in fact, forces per unit length, 

the unit length being perpendicular to the plane of the considered figure or cross section. 

However, for the sake of simplicity, the phrase “per unit length” will not be repeated 

after the words “force” or “weight”. 

Figure 2. 

Water flowing over the entire thickness, t, of a soil layer on a slope. 


1155


If the considered slice is relatively far from the toe or the top of the slope, the forces 

applied by the soil on the right side and on the left side of the vertical slice are balanced. 

According to the discussion on the effects of water on soil in Section 2.2, the two forces 

applied to the soil particles comprised in the considered slice are the buoyant weight, 

W b , and the drag force, F w . Since the volume (per unit length) of the slice is bt, the 

buoyant weight is: 

W b = γ b bt 

(9) 

and the drag force is, according to Equation 8: 

F w = γ w bti 

(10) 

The hydraulic gradient for flow with a phreatic surface parallel to the slope is: 

i = sin β 

(11) 

Combining Equations 10 and 11 gives: 

F w = γ w btsin β 

(12) 

Stresses on the Upper Face of the Geomembrane. Since F w is parallel to the geomembrane, 

the effective normal stress, σ ′ n , on the geomembrane (i.e. the stress applied by 

the soil particles on the geomembrane) results only from the buoyant weight, W b ,and 

it is obtained by projecting W b (given by Equation 9) on the normal to the geomembrane 

and dividing by b: 

W b (cos β)∕b = σ ′ n = γ b t cos β 

(13) 

According to the effective stress principle, which is applicable to materials made of 

low-deformability particles with small contact areas, such as soils, the total normal 

stress applied on the geomembrane is the normal stress applied by the particles (i.e. the 

effective normal stress) plus the pressure applied by water (i.e. the pore water pressure): 

σ n = σ ′ n + u 

(14) 

where: σ n = total normal stress; σ ′ n = effective normal stress; and u = pore water pressure. 

The pore water pressure on the geomembrane is derived as shown in Figure 3 by considering 

an equipotential line MN, which is perpendicular to the flow lines according 

to the theory of flow in porous media. By definition of the hydraulic potential: 



Figure 3. 

Use of equipotential surfaces to determine the pressure in a flowing liquid. 

u = u M = u N + ρ w g (z N − z M ) 

(15) 

where: u M = pore water pressure at M; u N = pore water pressure at N; z M = elevation of 

M; and z N = elevation of N. 

The pore water pressure is zero at the phreatic surface if the atmospheric pressure is 

used as the zero reference. Therefore, u N = 0. Also, from simple geometric considerations 

in Figure 3: 

z N − z M = z Q − z M = t cos β 

(16) 

where z Q is the elevation of Q (which is equal to z N ). 

From the above considerations and Equation 6, the pore water pressure on the geomembrane 

is: 

u = ρ w gt cos β = γ w t cos β 

(17) 

The total normal stress on the geomembrane can then be calculated by combining 

Equations 13, 14 and 17 as follows: 

σ n = (γ b + γ w ) t cos β 

(18) 


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Combining Equations 4, 6 and 18 gives: 

σ n = γ sat t cos β 

(19) 

Equation 19 could have been obtained directly, as Equation 13 was obtained, but with 

the saturated weight, W sat , instead of the buoyant weight: 

W sat (cos β)∕b = σ n = γ sat t cos β 

(20) 

The shear force applied on the geomembrane by the slice shown in Figure 2 is obtained 

as follows by adding the projection on the geomembrane of the buoyant weight 

of the slice and the drag force, which is parallel to the geomembrane: 

S = W b sin β + F w 

(21) 

Combining Equations 9, 12 and 21, and dividing by b, give the shear stress as follows: 

S∕b = τ = γ b t sin β + γ w t sin β 

(22) 


τ = γ sat t sin β 

(23) 

Stresses on the Lower Face of the Geomembrane. As assumed in Section 2.3.1, the 

pore water pressure, u, is zero below the geomembrane. Therefore, according to Equation 

14, σ n = σ ′ n below the geomembrane. The total normal stress, σ n , on the lower face 

of the geomembrane is equal to the total normal stress, σ n , on the upper face of the geomembrane, 

since they both result from the weight of soil and water located above the 

geomembrane, as shown in Equation 20 (the weight of the geomembrane itself being 

neglected). Consequently, the stresses on the lower face of the geomembrane, when 

there is “full flow” above the geomembrane, are: 

σ n = σ ′ n = γ sat t cos β 

(24) 

τ = γ sat t sin β (25) 

Comparing Equations 19, 23, 24 and 25 shows that the total normal stresses and the 

shear stresses are the same above and below the geomembrane, whereas the effective 

normal stresses on either side of the geomembrane are different. However, it should be 

noted that the shear stresses are the same above and below the geomembrane only if 

the continuum is not disrupted, i.e. if there is no slip surface above the geomembrane 



and if the geomembrane tension is not mobilized. If there is a slip surface above the 

geomembrane or if the geomembrane tension is mobilized, only a fraction of the shear 

stress is transmitted through the slip surface. 

2.3.3 Case of No Water Flow 

If there is no flow, there is no drag force, F w , and the pore water pressure, u, is zero 

above the geomembrane. Also, as indicated in Section 2.3.1, the pore water pressure 

is zero below the geomembrane. Therefore, according to Equation 14, σ n = σ ′ n . In this 

case the soil above the geomembrane is either dry or, more generally, retains some water 

by capillarity. Its unit weight is then γ t (defined by Equations 5 and 6) and the stresses 

on the geomembrane are: 

σ n = σ ′ n = γ t t cos β 

τ = γ t t sin β 

(26) 

(27) 

In this case, since there is no water above and below the geomembrane, the conditions 

are the same above and below the geomembrane and Equations 26 and 27 express the 

stresses on the lower face as well as on the upper face of the geomembrane (assuming 

that there is no slip surface above the geomembrane). 

2.3.4 Case of Partial Water Flow 

The case where there is partial flow is shown in Figure 4. In this case, the thickness 

of water flow is t w , which is less than the thickness, t, of the soil layer. The equations 

Figure 4. 

t. 

Water flowing in a portion of uniform thickness, t w , in a soil layer of thickness 


1159


for stresses are then obtained by combining the equations for the case where there is 

full flow over the thickness t w , and the case where there is no flow over the thickness 

t - t w . The equations thus obtained are presented below. 

Stresses on the Upper Face of the Geomembrane. The stresses on the upper face of 

the geomembrane for the case of partial flow (Figure 4) are as follows: 

S Pore water pressure on the upper face of the geomembrane: 

u = γ w t w cos β 

(28) 

S Effective normal stress on the upper face of the geomembrane: 

σ ′ n = γt t − t w 

+ γ b t w 

cos β 

(29) 

S Total normal stress on the upper face of the geomembrane: 

σ n = γt t − t w 

+ γ sat t w 

cos β 

(30) 

S Shear stress on the upper face of the geomembrane: 

τ = γt t − t w 

+ γ sat t w 

sin β 

(31) 

Stresses on the Lower Face of the Geomembrane. The stresses on the lower face of 

the geomembrane for the case of partial flow (Figure 4) are as follows: 

S Pore water pressure on the lower face of the geomembrane: 

u = 0 

(32) 

S Effective and total normal stress on the lower face of the geomembrane: 

σ ′ n = σ n = γ t (t − t w ) + γ sat t w 

cos β 

(33) 

S Shear stress on the lower face of the geomembrane: 

τ = γ t (t − t w ) + γ sat t w 

sin β 

(34) 



The eight equations given above are applicable to all three cases: full flow, partial 

flow, and no flow. These equations give the corresponding equations for the case of full 

flow, with t w = t, and for the case of no flow, with t w =0. 

2.3.5 Effects of Water Flow on Stresses 

The equations for the three cases studied above are summarized in Tables 2 and 3. 

The following comments can be made regarding the stresses on the lower face of the 

geomembrane: 

S The pore water pressure is zero under the geomembrane, as discussed in Section 

2.3.1. Therefore, under the geomembrane, the effective normal stress is identical to 

the total normal stress. 

S The total normal stress is the same on the lower face and on the upper face of the geomembrane. 

S The shear stress is the same on the lower face and on the upper face of the geomembrane 

if there is no slip surface above the geomembrane. 

The effects of water flow on the stresses at the geomembrane level can be evaluated 

by comparing the equations given in Tables 2 and 3 for the case with full flow and the 

case where there is no flow. These effects can be summarized as follows: 

S The shear stress, τ, increases from γ t t sinβ to γ sat t sinβ. For the typical soil presented 

in Table 1 the increase is 11%. 

Table 2. 

Stresses on the upper face of the geomembrane. 

Stress No flow Full flow Partial flow 

Pore water pressure, u 

0 

γ w t cosβ 

γ w t w cosβ 

Effective normal stress, 

σ ′ n 

γ t t cosβ 

γ b t cosβ 

[γ t (t - t w )+γ b t w ]cosβ 

Total normal stress, σ n 

γ t t cosβ 

γ sat t cosβ 

[γ t (t - t w )+γ sat t w ]cosβ 

Shear stress, τ 

γ t t sinβ 

γ sat t sinβ 

[γ t (t - t w )+γ sat t w ]sinβ 

Table 3. 

Stresses on the lower face of the geomembrane. 

Stress No flow Full flow Partial flow 

Pore water pressure, u 

0 

0 

0 

Effective normal stress, 

σ ′ n 

γ t t cosβ 



Total normal stress, σ n 

γ t t cosβ 



Shear stress, τ 

γ t t sinβ 

γ sat t sinβ 

[γ t (t - t w )+γ sat t w ]sinβ 


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S The total normal stress, σ n , increases from γ t t cosβ to γ sat t cosβ. The increase is also 

11% for the typical soil presented in Table 1. 

S The effective normal stress, σ ′ n , on the lower face of the geomembrane increases 

from γ t t cosβ to γ sat t cosβ. The increase is also 11% for the typical soil presented in 

Table 1. 

S The effective normal stress, σ ′ n , on the upper face of the geomembrane decreases 

from γ t t cosβ to γ b t cosβ. For the typical soil presented in Table 1 the decrease is 

42%. 

It is important to note that the main effect of water flowing along the slope is not the 

increase in shear stress but the decrease in effective normal stress above the geomembrane, 

which causes a significant decrease in shear strength above the geomembrane, 

hence a significant decrease in stability above the geomembrane, as discussed below. 

Also, it is interesting to note that the percentage changes indicated above are independent 

of the slope angle. 

3 EFFECT OF WATER FLOW ON THE STABILITY OF A 

GEOSYNTHETIC-SOIL LAYERED SYSTEM ON AN INFINITE SLOPE 

3.1 Introduction to Infinite Slopes 

3.1.1 Definition and Assumption 

An infinite slope is a slope which has a uniform inclination over an infinite length and, 

therefore, an infinite height. Calculations are simpler for the case of an infinite slope 

than for the case of a slope of finite height. When there is no water flow, an approximate 

evaluation of the stability of a layered system on a slope of finite height is often performed 

by calculating the factor of safety of the same layered system on an infinite slope 

with the same inclination as the slope of finite height. The factor of safety thus obtained 

is smaller than the factor of safety which would be obtained using the actual slope geometry, 

which is conservative. 

If an infinite slope were subjected to a water supply distributed over its entire length 

(like the slope of finite height in Figure 1), the rate of water flow along the slope would 

be infinite. This would not correspond to any real situation. To use an infinite slope for 

an approximate evaluation of the stability of a layered system where water is flowing, 

it is necessary to assume that the flow thickness is uniform in the layered system 

installed on the infinite slope (which implies the presence of a constant source of water 

at the “top” of the infinite slope). In other words, it must be assumed that the flow rate 

is the same at any elevation along the slope, which is not the case with a slope of finite 

height exposed to a uniform supply of water (Figure 1). 

Even though the assumption described above is not consistent with the flow rate along 

a slope of finite height, it is an appropriate assumption because it provides a good 

approximation of the flow thickness, which is the parameter that influences stability as 

discussed hereafter. To ensure that the approximate evaluation of stability provided by 

the use of an infinite slope is as accurate as possible, the flow thickness, t w , in the infinite 



slope should be equal to the average flow thickness, t wavg , for the actual slope geometry, 

calculated as indicated in Section 2.1. 

3.1.2 Factor of Safety 

In the case of an infinite slope, there is no mechanism at the toe or at the top of the 

slope that may contribute to stability. Therefore, the stability depends only on the shear 

strength along the potential slip surface. Accordingly, the factor of safety of an infinite 

slope is defined as follows: 

FS = s τ 

(35) 

where: s = interface shear strength along the slip surface; and τ = interface shear stress 

along the slip surface. This equation represents a limit state wherein shear stress and 

shear strength are assumed to be independent of slope deformation. 

The interface shear strength, s, of a soil-geosynthetic or a geosynthetic-geosynthetic 

interface may be expressed as follows using Coulomb’s law: 

s = a + σ ′ n tan δ 

(36) 

where: a = interface adhesion along the slip surface; and δ = interface friction angle 

along the slip surface. 

The remainder of Section 3 presents a discussion on how the factor of safety is affected 

by water flow. 

3.2 Case of No Water Flow 

If there is no water flow, the effective normal stress on the geomembrane, σ ′ n ,isgiven 

by Equation 26. Therefore, Equation 36 becomes: 

s = a + γ t t cos β tan δ 

(37) 


FS = a + γ t t cos β tan δ 

γ t t sin β 

(38) 

hence: 

FS = tan δ 

tan β + a 

γ t t sin β 

(39) 


1163


The discussion presented in Section 3.3 will show that the effect of water flow on stability 

is different whether the slip surface is above or below the geomembrane. Consequently, 

if there is no water flow and the slip surface is above the geomembrane, it is 

convenient to write the factor of safety as follows: 

a A 

FS A = tan δ A 

tan β + γ t t sin β 

(40) 

where: δ A = interface friction angle along a slip surface located above the geomembrane; 

and a A = interface adhesion along a slip surface located above the geomembrane. 

Similarly, the factor of safety is written as follows if there is no water flow and the 

slip surface is below the geomembrane: 

a B 

FS B = tan δ B 

tan β + γ t t sin β 

(41) 

where: δ B = interface friction angle along a slip surface located below the geomembrane; 

and a B = interface adhesion along a slip surface located below the geomembrane. 

It should be noted that it has been implicitly assumed that there is only one slip surface. 

In particular, Equation 41 is only valid if the only slip surface is below the geomembrane. 

If there was another slip surface above the geomembrane, the shear stress 

due to the weight of the soil layer above the geomembrane would not be entirely transmitted 

through the slip surface, and the equation would not be correct. 

3.3 Case of Full Water Flow 

3.3.1 Effect of Water Flow on the Stability Above the Geomembrane 

It is assumed that the slip surface is at, or close to, the upper face of the geomembrane, 

i.e. the thickness of soil above the slip surface is t, as defined in Figure 2. To calculate 

the factor of safety defined by Equation 35, it is necessary to evaluate the interface shear 

strength, s, and the shear stress, τ. The shear stress, τ, is given by Equation 23. The interface 

shear strength must be calculated using the effective normal stress, σ ′ n , expressed 

by Equation 13. Combining Equations 13 and 36, and using the subscript A for “above”, 

give: 

s = a A + γ b t cos β tan δ A 

(42) 


FS A = γ b 

γ sat 

tan δ A 

tan β + a A 

γ sat t sin β 

(43) 



This factor of safety is to be compared with the factor of safety expressed by Equation 

40 for the case of no water flow. The comparison shows that, for the typical soil presentedinTable1: 

S FS A full-flow / FS A no-flow =0.52ifa A =0; 

S FS A full-flow / FS A no-flow =0.90ifδ A =0;and 

S FS A full-flow / FS A no-flow is between 0.52 and 0.90 if a A ≠ 0andδ A ≠ 0. 

Based on these results, it is clear that, for slip surfaces located above the geomembrane, 

the factor of safety can significantly decrease if water is flowing. 

The ratios presented above are based on the assumption that the interface shear 

strength properties, δ A and a A , are not influenced by the presence of water. If the presence 

of water reduced the interface shear strength parameters, the effects noted in the 

above comparison would be even more substantial. 

3.3.2 Effect of Water Flow on the Stability Below the Geomembrane 

It is assumed that the slip surface is located at, or close to, the lower face of the geomembrane 

and it is assumed that there is no other slip surface above the geomembrane. 

(The case of a dual slip surface is theoretically possible, but in this case only a fraction 

of the shear stress would be transmitted through the upper slip surface and reach the 

lower slip surface.) Therefore, it is assumed that the totality of the shear stress is transmitted 

to the considered slip surface that is located below the geomembrane. 

To calculate the factor of safety defined by Equation 35, it is necessary to evaluate 

the interface shear strength, s, and the shear stress, τ. The shear stress is expressed by 

Equation 25, which is identical to Equation 23. It is interesting to note that the shear 

stress acting below the geomembrane is the same as the shear stress acting above the 

geomembrane (provided that there is no dual slip surface and that no geomembrane tension 

is mobilized, as discussed in Section 2.3.2). 

The interface shear strength must be calculated using the effective normal stress below 

the geomembrane, which is given by Equation 24. Combining Equations 24 and 

36, and using the subscript B for “below”, give: 

s = a B + γ sat t cos β tan δ B 

(44) 


a B 


tan β + γ sat t sin β 

(45) 

This factor of safety is to be compared with the factor of safety expressed by Equation 

41 for the case of no water flow. The comparison shows that, for the typical soil presentedinTable1: 

S FS B full-flow / FS B no-flow =1ifa B =0; 

S FS B full-flow / FS B no-flow =0.90ifδ B =0;and 


1165


S FS B full-flow / FS B no-flow is between 0.9 and 1 if a B ≠ 0andδ B ≠ 0. 

Based on these results, it is clear that, for slip surfaces located below the geomembrane, 

the factor of safety is not greatly affected if water is flowing above the geomembrane, 

provided that there is no pore water pressure under the geomembrane. This result 

is a logical extension of the conclusions presented in Section 2.3.5 where it was shown 

that water flowing above the geomembrane has only a small influence on the shear 

stress and the effective normal stress below the geomembrane. 

3.4 Case of Partial Water Flow 

The process used in Section 3.3 is used again below. Therefore, no detailed explanation 

is given in Section 3.4. 

3.4.1 Effect of Water Flow on the Stability Above the Geomembrane 

Combining Equations 29 and 36 gives the interface shear strength as follows: 

s = a A + γ t (t − t w ) + γ b t w 

cos β tan δ A 

(46) 

Combining Equations 31, 35 and 46 gives the factor of safety as follows: 

FS A = γ t (t − t w ) + γ b t w tan δ A 

γ t (t − t w ) + γ sat t w tan β + 

a A ∕ sin β 

γ t (t − t w ) + γ sat t w 

(47) 

3.4.2 Effect of Water Flow on the Stability Below the Geomembrane 

Combining Equations 33 and 36 gives the interface shear strength as follows: 

s = a B + γ t (t − t w ) + γ sat t w 

cos β tan δ B 

(48) 

Combining Equations 34, 35 and 48 gives the factor of safety as follows: 


tan β + a B ∕ sin β 


(49) 

3.5 Conclusions Regarding the Influence of Water Flow on Infinite Slopes 

3.5.1 Summary of Equations 

Equations giving the factor of safety for the case of an infinite slope are regrouped 

in Table 4. It is possible to represent all cases by one general equation: 



FS = γ RESIST 

γ DRIVE 

tan δ a∕ sin β 

+ 

tan β γ DRIVE t 

(50) 

where the values of γ RESIST and γ DRIVE are given in Table 5. As the subscripts indicate, 

γ RESIST and γ DRIVE are the unit weights associated with the resisting and driving forces, 

respectively, that govern stability. It should be noted that, consistent with the definition 

of the factor of safety, γ DRIVE appears in both denominators, whereas, consistent with 

Coulomb’s law, γ RESIST appears only in the numerator with tanδ. 

Table 4. 

Factor of safety for an infinite slope. 

No flow 

FS A = tan δ A 

tan β + a A 

γ t t sin β 


tan β + a B 

γ t t sin β 

Full flow 

FS A = γ b tan δ A 

γ sat tan β + a A 



tan β + a B 


Partial flow 


γ t (t − t w ) + γ sat t w tan β + a A ∕ sin β 





Notes: Subscripts A for “above the geomembrane” and B for “below the geomembrane”. The flow thickness, 

t w , and the drainage layer thickness, t, are defined inFigure 4.The expressionsfor partialflow becomeidentical 

to the expressions for no flow if t w = 0 and identical to the expressions for saturated flow if t w = t. 

Table 5. 

General stability equation for an infinite slope. 

General equation 

Type 

of 

flow 

FS = γ RESIST 

γ DRIVE 

Above 

(δ = δ A , a = a A ) 

tan δ a∕ sin β 

+ 

tan β γ DRIVE t 

Values of the parameters 

Below 

(δ = δ B , a = a B ) 

γ RESIST γ DRIVE γ RESIST = γ DRIVE 

No flow γ t γ t 

Full flow γ b γ sat 

Partial flow γ t 1 − t w 

t 

+ γ b 

t wt 

γ t 1 − t w 

t 

+ γ sat 

t wt 

Notes: The flow thickness, t w , and the drainage layer thickness, t, are defined in Figure 4. The expressions 

for partial flow become identical to the expression for no flow if t w = 0 and identical to the expressions for 

saturated flow if t w = t. 


1167


Equation 50 is essentially of academic interest for infinite slopes because it can only 

be used with Table 5, which makes it more cumbersome than the equations summarized 

in Table 4. However, Equation 50 is useful within the scope of this paper because it provides 

the model that will be used to establish with minimum effort the equation for 

slopes of finite height. 

3.5.2 Discussion of the Effects of Water Flow on Infinite Slopes 

As indicated in Sections 3.3 and 3.4, the effect of water flow on the stability of a geosynthetic-soil 

layered system on a slope is much greater if the slip surface is above the 

geomembrane than if it is below. The reasons for this can be summarized as follows: 

S The main effect of water flowing in a geosynthetic-soil layered system on a slope is 

the significant decrease in the effective normal stress above the geomembrane. 

S Other effects of water flowing in a geosynthetic-soil layered system are a slight increase 

in the effective normal stress below the geomembrane and a slight increase in 

the shear stress above and below the geomembrane. 

S As a result of the changes in effective normal stress, the shear strength (soil and interface 

shear strength) significantly decreases above the geomembrane and slightly increases 

below the geomembrane. 

S As a result of the changes in shear strength and the slight increase in shear stress, the 

factor of safety is significantly affected above the geomembrane and only mildly affected 

below the geomembrane. 

This is confirmed by the simple examples presented in Tables 6 and 7. Table 6 shows 

that if there is no interface adhesion, which is a case frequently considered in analyses, 

the stability below the geomembrane is not affected at all by water flow above the geomembrane, 

yet the stability above the geomembrane is significantly influenced by water 

flow. 

Table 6. Example of factor of safety calculation for an infinite slope in the case where there 

is no interface adhesion, and where the slope angle, β, equals the interface friction angle, δ. 

Assumptions: δ A = δ B = β a A = a B =0 

No flow 

Full flow 

Case 

Partial flow 

(half full) 

(t w =0.5t) 

Factor of safety 

FS A =1.00+0=1.00 

FS B =1.00+0=1.00 

FS A = (0.525) (1.00) + 0 = 0.525 

FS B =1.00+0=1.00 

(0.900)(0.5) + (0.525)(0.5) 

FS A = (1.00) + 0 = 0.750 

(0.900)(0.5) + (1.0)(0.5) 

FS B =1.00+0=1.00 

Notes: The meaning of the subscripts is as follows: A = above the geomembrane; and B =belowthe 

geomembrane. The following numerical values are from Table 1: γ b /γ sat = 0.525, and γ t /γ sat =0.900.The 

equations used in this table are presented in the same order in Table 4. 



Table7. Exampleoffactorofsafetycalculationforaninfiniteslopeinacasewherethereis 

interface adhesion. 

Assumptions: tanδ A =tanδ B =0.75tanβ a A = a B =0.25γ t tsinβ 

Case 

No flow 

Full flow 

Partial flow 

(half full) 

(t w =0.5t) 

(0.900)(0.5) + (0.525)(0.5) 

FS A = 

(0.900)(0.5) + (1.0)(0.5) 

Factor of safety 

FS A = 0.75 + 0.25 = 1.00 

FS B = 0.75 + 0.25 = 1.00 

FS A = (0.525) (0.75) + (0.900) (0.25) = 0.619 

FS B = 0.75 + (0.900) (0.25) = 0.975 

(0.900)(0.25) 

(0.75) + 

(0.900)(0.5) + (1.0)(0.5) = 0.799 

(0.900)(0.25) 

FS B = 0.75 + 

(0.900)(0.5) + (1.0)(0.5) = 0.987 

Notes: The meaning of the subscripts is as follows: A = above the geomembrane; and B =belowthe 

geomembrane. The following numerical values are from Table 1: γ b /γ sat = 0.525, and γ t /γ sat = 0.900. The 

equations used in this table are presented in the same order in Table 4. 

4 EFFECT OF WATER FLOW ON THE STABILITY OF A 

GEOSYNTHETIC-SOIL LAYERED SYSTEM ON A SLOPE 

OF FINITE HEIGHT 

4.1 Factor of Safety for a Slope of Finite Height with No Water Flow 

Giroud et al. (1995) have shown that the factor of safety of a geosynthetic-soil layered 

system (such as a liner system) constructed on a slope of finite height is expressed by 

the following equation when there is no water flow: 

FS = tan δ 

tan β + a 

γt sin β + t h 

sin Ô 

2sinβ cos β cos(β + Ô) + c 

γh 

cos Ô 

sin β cos(β + Ô) + T 

γht 

(51) 

where: δ = interface friction angle; a = interface adhesion; Ô = internal friction angle 

of soil above the geomembrane; c = cohesion of soil above the geomembrane; γ = unit 

weight of soil above the geomembrane; t = thickness of soil above the geomembrane; 

h = height of slope (as defined in Figure 5); β = slope angle; and T = geosynthetic tension 

above the slip surface. 

This equation can also be written as follows: 

FS = tan δ 

tan β + a 

γt sin β + t h 

tan Ô∕(2 sin β cos 2 β) 

+ c 

1 − tan β tan Ô γh 

1∕(sin β cos β) 

1 − tan β tan Ô + T 

γht 

(52) 

Equation 52 is slightly more complex than Equation 51, but it better shows the influence 

of Ô through the term tanÔ. 


1169


Figure 5. Definition of a geosynthetic-soil layered system on a slope of finite height. 

(Note: The slip surface considered in the development of Equation 51 or 52 is DABC.) 

Equation 51 (or Equation 52) comprises five terms. These terms can be characterized 

as follows: 

S The first term quantifies the contribution of the interface friction angle to stability. 

S The second term quantifies the contribution of the interface adhesion to stability. 

S The third and fourth terms quantify the contribution of the toe buttressing effect, 

which results from the shear strength of the soil located at the toe of the slope above 

the slip surface. Both terms depend on the soil internal friction angle, whereas only 

the fourth term depends on the soil cohesion. 

S The fifth term quantifies the contribution to the factor of safety of any tension in the 

geosynthetics located above the slip surface (which may include one or more geosynthetics 

specifically used as reinforcement). 

A detailed discussion of the five terms may be found in the paper by Giroud et al. 

(1995). 

4.2 Factor of Safety for a Slope of Finite Height with Water Flow 

The approach followed below to develop the factor of safety equation with water flow 

for the case of a slope of finite height is similar to the approach used for the case of an 

infinite slope to develop the general equation presented in Section 3.5.1 and Table 5. 

This approach consists of writing a unique equation that identifies the relevant values 

of the unit weight (see Equation 50). The method used by Giroud et al. (1995) to establish 

Equation 51 (or Equation 52, which is equivalent) using the two-wedge approach 

shown in Figure 6 was followed step-by-step. The calculations originally performed by 

Giroud et al. (1995) were repeated using the following three different values for the unit 



Figure 6. Definition of the two wedges used in the development of the factor of safety 

equation for a geosynthetic-soil layered system on a slope of finite height. 

weights, depending on the considered wedge and the role of the considered force (resisting 

or driving): 

S γ 1RESIST is the unit weight associated with the internal friction angle, Ô, that contributes 

to the portion of the resisting force related to Wedge 1 (i.e. the toe buttressing effect); 

S γ 2RESIST is the unit weight associated with the interface friction angle, δ, that contributes 

to the portion of the resisting force related to Wedge 2 (i.e. the interface shear 

strength); and 

S γ 2DRIVE is the unit weight that contributes to the driving force due to the weight of 

Wedge 2. 

The following equation, which is similar to Equation 51, but which also includes the 

general stability approach presented in Equation 50 and Table 5, has thus been obtained: 

FS = γ 2RESIST 

γ 2DRIVE 

tan δ 

tan β + a 

γ 2DRIVE t sin β + γ 1RESIST 

γ 2DRIVE 

t 

h 

sin Ô 

2sinβ cos β cos(β + Ô) 

+ c 

γ 2DRIVE 

h 

cos Ô 


γ 2DRIVE ht 

(53) 

It should be noted that, as a result of the definition of the factor of safety, the two unit 

weights related to resisting forces must appear in two of the numerators of Equation 53, 

and the unit weight related to the driving force must appear in all of the denominators 

of Equation 53. It should also be noted that, consistent with the two-wedge approach 

used to develop Equation 51, the unit weight of Wedge 1, γ 1 , only contributes to resist- 


1171


ing forces, whereas the unit weight of Wedge 2, γ 2 , contributes to both resisting and 

driving forces. 

The unit weights of the soil overlying the geomembrane used in Equation 53 are consistent 

with those given in Table 5 and are as follows: 

γ 2DRIVE = γ t 

1 − t w 

t 

+ γ sat 

t wt 

(54) 

γ 1RESIST = γ t 1 − t* w 

t 

+ γ b 

t * w 

t 

(55) 

The third unit weight used in Equation 53 depends on the considered interface. For 

a slip surface located above the geomembrane: 

γ 2RESIST = γ t 

1 − t w 

t 

+ γ b 

t wt 

(56) 

and for a slip surface located below the geomembrane: 

γ 2RESIST = γ t 

1 − t w 

t 

+ γ sat 

t wt 

(57) 

where: t w = thickness of flow in Wedge 2; and t * w = thickness of flow in Wedge 1. If 

the flow thickness is not uniform in Wedge 2, an average value between A and B (Figure 

7) should be used, t w = t wavg , as indicated in Section 2.1. In Wedge 1, the flow thickness 

is rarely uniform and an average value should be selected by the design engineer. The 

average value may be small if there is effective drainage at the toe of the slope (e.g. a 

Figure 7. 

Flow thickness for the case of a slope of finite height. 



collector pipe that works) or may be large (up to t * w = t) if the toe of the slope is saturated 

or flooded. (Note that γ 2DRIVE and γ 1RESIST each have a unique value, regardless of whether 

the slip surface is above or below the geomembrane, because they are always acting 

above the geomembrane, whereas γ 2RESIST acts at the location of the slip surface and 

therefore has a different value depending on whether the slip surface is above or below 

the geomembrane.) 

Combining Equations 53, 54, 55 and 56 gives the factor of safety for a slip surface 

above the geomembrane as follows: 


γ t (t − t w ) + γ sat t w tan β + 

+ γ t (t − t * w) + γ b t * w 


t 

h 

a A ∕ sin β 


sin Ô 


ct∕h 

+ 


cos Ô 

sin β cos(β + Ô) 

T∕h 

+ 

(58) 


Combining Equations 53, 54, 55 and 57 gives the factor of safety for a slip surface 

below the geomembrane as follows: 



+ γ t (t − t * w) + γ b t * w 

γ t (t − t w ) + γ sat t w γ t (t − t w ) + γ sat t w 

t 

h 

sin Ô 


+ 

ct∕h 

cos Ô 

γ t (t − t w ) + γ sat t w sin β cos(β + Ô) + T∕h 


(59) 

It should be noted that only the first two terms of Equations 58 and 59 are different. 

In other words, the contribution of the toe buttressing effect (third and fourth terms) and 

of the geosynthetic reinforcement (fifth term) is the same whether the slip surface is 

above or below the geomembrane. 

It should also be noted that in Equations 58 and 59, the two terms that contain Ô can 

be replaced by the following equivalent expressions derived from a comparison between 

Equations 51 and 52: 

sin Ô 

2sinβ cos β cos(β + Ô) = tan Ô∕(2 sin β cos2 β) 

1 − tan β tan Ô 

cos Ô 1∕(sin β cos β) 

= 

sin β cos(β + Ô) 1 − tan β tan Ô 

(60) 

(61) 


1173


When there is full flow in Wedge 1 (t w = t)aswellasinWedge2( t * w = t), Equation 

58 gives the following equation for the factor of safety for a slip surface above the geomembrane: 

FS A = γ b tan δ A 

γ sat tan β + 

+ c 

γ sat h 

a A 

γ sat t sin β + γ b 

γ sat 

cos Ô 


γ sat th 

t 

h 

sin Ô 


(62) 

and Equation 59 gives the following equation for the factor of safety for a slip surface 

below the geomembrane: 

a B 


tan β + γ sat t sin β + γ b 

+ c 

γ sat h 

γ sat 

t 

h 

cos Ô 


γ sat th 

sin Ô 


(63) 

When there is no water flow, t w = t * w = 0. Equation 58 then gives Equation 51 with 

δ = δ A , a = a A and γ = γ t . Similarly, Equation 59 gives Equation 51 with δ = δ B , a = 

a B and γ = γ t . 

4.3 Discussion of the Effects of Water Flow on Slopes of Finite Height 

The following two comments can be made: 

S Inspection of Equations 58 and 62 (i.e. the equations that give the factor of safety for 

a slip surface located above the geomembrane) and Equations 59 and 63 (i.e. the 

equations that give the factor of safety for a slip surface located below the geomembrane) 

shows that the difference between the two sets of equations occurs only in the 

first two terms of the equations, i.e. the two terms that give the factor of safety of an 

infinite slope (see Table 4). 

S In most cases of practical interest, the magnitudes of the first two terms of Equations 

58, 59, 62 and 63 are far greater than the magnitudes of the three other terms. Therefore, 

the impact that water flow may have on the factor of safety is essentially through 

the first two terms of the equation. 

From the above two comments, it may be concluded that the influence of water flow 

on the stability of geosynthetic-soil layered systems on slopes of finite height is similar 

to the influence of water flow on the stability of geosynthetic-soil layered systems on 

infinite slopes. In particular, water flow significantly reduces the factor of safety for a 

slip surface located above the geomembrane, but has a relatively small effect on the factor 

of safety for a slip surface located below the geomembrane. 



4.4 Design Example 

4.4.1 Presentation of the Design Example 

A simple geosynthetic-soil layered system is considered. It consists of a 0.55 m thick 

layer of granular soil having the unit weights shown in Table 1, resting on a geomembrane, 

which in turn rests on a 1V:4H slope (i.e. β =14_)thatis6mhigh(i.e.h =6m 

as defined in Figure 5). The relevant materials and interface properties are: 

S granular soil shear strength: internal friction angle, Ô =33_, and cohesion, c =0; 

S interface shear strength between the geomembrane and the overlying granular soil: 

interface friction angle, δ A =16_, and interface adhesion, a A =0. 

S interface shear strength between the geomembrane and the underlying soil: interface 

friction angle, δ B =11_, and interface cohesion, a B =2.5kPa. 

According to a preceding design step, not described herein, the maximum expected 

water flow thickness during the worst case precipitation is 0.30 m. It will be conservatively 

assumed that t w = 0.30 m. This design example is depicted schematically in Figure 

8. 

What is the factor of safety of the considered geosynthetic-soil layered system against 

instability? 


The factor of safety will be calculated first for the case where there is no flow, then 

for the case where water flows. In both cases, slip surfaces above and below the geomembrane 

will be considered. 

β =14_ 

Figure 8. 

Illustration of the design example. 


1175


4.4.3 Case of No Water Flow 

Whenthereisnoflow,Equation51or52canbeused.Equation52isusedbelow. 

Slip Surface Above the Geomembrane. If the slip surface is above the geomembrane, 

Equation52isusedwithδ A and a A as follows: 

hence: 

FS A = 

tan 16_ 1∕4 + 0 + 0.55 (tan 33 _)∕(2 sin 14_ cos 2 14 _) 

6 1 − (1∕4) tan 33_ 

FS A = 1.147 + 0 + 0.156 + 0 + 0 = 1.303 

+ 0 + 0 

Slip Surface Below the Geomembrane. If the slip surface is below the geomembrane, 

Equation52isusedwithδ B and a B as follows: 

tan 11_ FS B = 

1∕4 + 2.5 

(18.59)(0.55) sin 14 + 0.55 (tan 33 _)∕(2 sin 14_ cos 2 14 _) 

+ 0 + 0 

_ 6 1 − (1∕4) tan 33_ 

hence: 

FS B = 0.778 + 1.011 + 0.156 + 0 + 0 = 1.945 

4.4.4 Case of Water Flow 

As indicated in Section 4.4.1, the flow thickness to be considered along the slope (i.e. 

in Wedge 2 in Figure 7) is t w = 0.30 m (Figure 8). To be conservative, poor drainage will 

be considered at the toe (Figure 8). Simple geometric considerations in Figure 8 show 

that, with the assumption of a horizontal phreatic surface in Wedge 1, 79% of Wedge 

1 is saturated. Therefore, the following approximate value of t * w will be used: 

t * w = (0.79) (0.55) = 0.44 m 

The equations to be used to calculate the factor of safety when there is water flow are: 

Equation 58 for a slip surface located above the geomembrane and Equation 59 for a 

slip surface located below the geomembrane. These equations will be used with the expressions 

given by Equations 60 and 61 (to familiarize the reader with the fact that there 

are two equivalent expressions). 

Slip Surface Above the Geomembrane. If the slip surface is above the geomembrane, 

Equation 58 (written with the expressions defined by Equations 60 and 61) gives the 

calculated factor of safety as follows: 



FS A = 

hence: 

(18.59)(0.55 − 0.30) + (10.84)(0.30) 

(18.59)(0.55 − 0.30) + (20.65)(0.30) 

(18.59)(0.55 − 0.44) + (10.84)(0.44) 

+ 

(18.59)(0.55 − 0.44) + (20.65)(0.44) 

+ 0 + 0 

tan 16_ 

1∕4 + 0 

0.55 

6 

FS A = (0.729)(1.147) + 0 + (0.612)(0.156) + 0 + 0 = 0.932 

(tan 33 _)∕(2 sin 14_ 

cos 2 14 _) 

1 − (1∕4) tan 33_ 

The first and third terms which carry over from the case of no flow (i.e. 1.147 and 

0.156) have been significantly reduced by the effect of water flow and, as a result, the 

calculated factor of safety above the geomembrane is less than one. 

Slip Surface Below the Geomembrane. If the slip surface is below the geomembrane, 

Equation 59 (written with the expressions defined by Equations 60 and 61) gives the 

calculated factor of safety as follows: 

tan 11_ 

FS B = 

1∕4 + 2.5∕ sin 14_ 

(18.59)(0.55 − 0.30) + (20.65)(0.30) 

hence: 

(18.59)(0.55 − 0.44) + (10.84)(0.44) 

+ 

(18.59)(0.55 − 0.44) + (20.65)(0.44) 

+ 0 + 0 

0.55 

6 

FS B = 0.778 + 0.953 + (0.612)(0.156) + 0 + 0 

(tan 33 _)∕(2 sin 14_ 

cos 2 14 _) 

1 − (1∕4) tan 33_ 

hence: 

FS B = 0.778 + 0.953 + 0.095 = 1.826 

4.4.5 Discussion of the Design Example 

The results of the above design example illustrate that, in the case of a slope of finite 

height, the calculated factor of safety for a slip surface located above the geomembrane 

is significantly affected by water flow, whereas the calculated factor of safety for a slip 

surface located below the geomembrane is not significantly affected by water flow. This 

is consistent with the comments made in Section 3.5.2 for infinite slopes. 


1177


5 CONCLUSIONS 

The effect of water flow on the stability of geosynthetic-soil layered systems on 

slopes has been analyzed. It has been shown for both infinite slopes and slopes of finite 

height that water flow significantly reduces the factor of safety for slip surfaces located 

above the geomembrane whereas it reduces only slightly the factor of safety for slip 

surfaces located below the geomembrane. 

Equations have been provided that allow design engineers to readily calculate the factor 

of safety of geosynthetic-soil layered systems with water flow. Design examples 

have been provided for an infinite slope and a slope of finite height. 

The authors hope that this paper will be useful to readers who are interested in an analysis 

of the effect of water flow on slope stability, as well as design engineers interested 

in a practical design method. Indeed, the first time the method presented herein was 

used, it was to solve a practical problem: the senior author used the method to analyze 

the failure of a landfill final cover and to demonstrate that the water flowing in the drainage 

layer located above the geomembrane had not significantly contributed to the failure 

since the slip surface was below the geomembrane. 

ACKNOWLEDGMENTS 

The authors are grateful to T. Pelte for his careful review of the manuscript, and G. 

Saunders and S.M. Berdy for their assistance in the preparation of the paper. 

REFERENCES 

Giroud, J.P., Williams, N.D., Pelte, T. and Beech, J.F., 1995, “Stability of Geosynthetic- 

Soil Layered Systems on Slopes”, Geosynthetics International, Vol. 2, No. 6, pp. 

1115-1148. 

Giroud, J.P. and Houlihan, M.F., 1995, “Design of Leachate Collection Layers”, Proceedings 

of the Fifth International Landfill Symposium, Vol. 2, Sardinia, Italy, October 

1995, pp. 613-640. 

NOTATIONS 

The subscripts A and B are used to identify symbols related to slip surfaces located 

“above” and “below” the geomembrane, respectively. Forces and weights are, in reality, 

forces and weights per unit length perpendicular to the plane of the figure, hence 

the unit N/m. Basic SI units are in parentheses. 

a = interface adhesion along the slip surface (Pa) 

a A = interface adhesion along a slip surface located above the geomembrane 

(Pa) 



a B = interface adhesion along a slip surface located below the geomembrane 

(Pa) 

b = width of a vertical slice of soil measured in the direction of the slope (m) 

c = cohesion of soil above the geomembrane (Pa) 

F w = drag force (N/m) 

f w = drag force per unit volume of soil (N/m 3 ) 

FS = factor of safety (dimensionless) 

FS A = factor of safety if the slip surface is above the geomembrane 

(dimensionless) 

FS B = factor of safety if the slip surface is below the geomembrane 


g = acceleration due to gravity (m/s 2 ) 

h = height of slope (m) 

i = hydraulic gradient (dimensionless) 

k = soil hydraulic conductivity (m/s) 

n = soil porosity (dimensionless) 

S = shear force applied on the geomembrane (N) 

s = interface shear strength along the slip surface (Pa) 

T = geosynthetic tension (N/m) 

t = thickness of the soil layer (m) 

t w = water flow thickness (m) 

t * w = water flow thickness in Wedge 1, i.e. in the toe area (m) 

t wavg = average value of water flow thickness (m) 

u = pore water pressure (Pa) 

u M = pore water pressure at M (Pa) 

u N = pore water pressure at N (Pa) 

V w = volume of soil occupied by water flow (m 3 ) 

W b = buoyant weight (N/m) 

W sat = saturated weight (N/m) 

w = water content defined as mass of water divided by mass of dry soil 


z M = elevation of M (Figure 3) (m) 

z N = elevation of N (Figure 3) (m) 

z Q = elevation of Q (= z N )(Figure3)(m) 

β = slope angle (_) 

δ = interface friction angle along the slip surface (_) 

δ A = interface friction angle along a slip surface located above the 

geomembrane (_) 


1179


δ B = interface friction angle along a slip surface located below the 

geomembrane (_) 

Ô = internal friction angle of the soil above the geomembrane (_) 

γ = unit weight (N/m 3 ) 

γ 1RESIST = unit weight of soil associated with the internal friction angle, Ô, that 

contributes to the portion of the resisting force related to Wedge 1 

(i.e. the toe buttressing effect) in the case of a slope of finite height (N/m 3 ) 

γ 2DRIVE = unit weight of soil that contributes to the driving force due to the weight 

of Wedge 2 in the case of a slope of finite height (N/m 3 ) 

γ 2RESIST = unit weight of soil associated with the interface friction angle, δ, that 

contributes to the portion of the resisting force related to Wedge 2 (i.e. 

the interface shear strength) in the case of a slope of finite height (N/m 3 ) 

γ DRIVE = unit weight of soil associated with the driving forces which govern 

stability of an infinite slope (N/m 3 ) 

γ RESIST = unit weight of soil associated with the resisting forces which govern 

stability of an infinite slope (N/m 3 ) 

γ b = “buoyant unit weight” of soil (N/m 3 ) 

γ d = “dry unit weight” of soil (N/m 3 ) 

γ s = unit weight of soil particles (N/m 3 ) 

γ sat = “saturated unit weight” of soil (N/m 3 ) 

γ t = “total unit weight” of soil (N/m 3 ) 

γ w = unit weight of water (N/m 3 ) 

ρ = density (kg/m 3 ) 

ρ b = “buoyant density” of soil (kg/m 3 ) 

ρ d = “dry density” of soil (kg/m 3 ) 

ρ s = density of soil particles (kg/m 3 ) 

ρ sat = “saturated density” of soil (kg/m 3 ) 

ρ t = “total density” of soil (kg/m 3 ) 

ρ w = density of water (kg/m 3 ) 

σ n = total normal stress (Pa) 

σ ′ n = effective normal stress (Pa) 

τ = interface shear stress along the slip surface (Pa) 


Errata 

INFLUENCE OF WATER FLOWONTHE 

STABILITY OF GEOSYNTHETIC-SOIL 

LAYERED SYSTEMS ON SLOPES 

TECHNICAL PAPER FOR ERRATA: Giroud, J.P., Bachus, R.C. and Bonaparte, 

R., 1995, “Influence of Water Flow on the Stability of Geosynthetic-Soil Layered 

Systems on Slopes”, Geosynthetics International, Vol. 2, No. 6, pp. 1149-1180. 

PUBLICATION: Geosynthetics International is published by the Industrial Fabrics 

Association International, 345 Cedar St., Suite 800, St. Paul, MN 55101-1088, USA, 

Telephone: 1/612-222-2508, Telefax: 1/612-222-8215. Geosynthetics International is 

registered under ISSN 1072-6349. 

REFERENCE FOR ERRATA: Giroud, J.P., Bachus, R.C. and Bonaparte, R., 1997, 

“Errata for ‘Influence of Water Flow on the Stability of Geosynthetic-Soil Layered 

Systems on Slopes’”, Geosynthetics International, Vol. 4, No. 2, pp. 209-210. 

The authors would like to make the following corrections to their paper which appeared 

in Geosynthetics International, Vol. 2, No. 6. 

ERRATA FOR SECTION: 

4.4.4 Case of Water Flow 

On page 1177: 

In the denominator of the third term of the first equation for FS A , 0.44 should be replaced 

by 0.30, i.e. the value of t w . (It is important to note that 0.44 in the numerator 

is correct.) As a result, the third term of the second equation for FS A becomes: 

(0.628)(0.156) instead of (0.612)(0.156) 

Consequently: 

FS A = 0.934 instead of FS A = 0.932 

The same corrections apply to the first and second equations for FS B on the same page 

(i.e. p. 1177). Consequently: 


209

ERRATA D Influence of Water on Layered Systems on Slopes 

FS B = 1.829 instead of FS B = 1.826 

It is important to note that Equations 58 to 61 are correct. Equations 58 to 61 are the 

analytical equations which were used to develop the numerical equations of Section 

4.4.4 which contain the errors mentioned above. 

The authors are grateful to R.J. Taylor who pointed out the errors.

Technical Paper by J.P. Giroud, R.C. Bachus and R. Bonaparte - IGS ...

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