Sources
This chapter is made up of verbatim extracts from the following sources for which copyright permissions have been obtained as listed in the Acknowledgements.
• Fredlund, D.G. (2000). The 1999 R.M. Hardy Lecture: The implementation of unsaturated soil mechanics into geotechnical engi-neering. Can. Geotech. J., 37, 963–986.
• Fredlund, D.G. (2002). Use of soil–water characteristic curves in the implementation of unsaturated soil mechanics. Proc. 3rd Int. Conf. on Unsat. Soils, Recife, Brazil, Vol. 3, Balkema, Rotterdam, pp. 887–902.
• Fredlund, D.G. and Rahardjo, H. (1993). Soil Mechanics for Unsatu-rated Soils. John Wiley & Sons, Inc., New York, p. 517.
• Fredlund, D.G., Fredlund, M.D. and Zakerzadeh, N. (2001b). Predict-ing the permeability function for unsaturated soils. Proc. Int. Symp.
on Suction, Swelling, Permeability and Structured Clays, IS-Shizuoka, Shizuoka, Japan, pp. 215–222.
• Fredlund, D.G., Rahardjo, H., Leong, E.C. and Ng, C.W.W. (2001a).
Suggestions and recommendations for the interpretation of soil–water characteristic curves. Proc. 14th Southeast Asian Geotech. Conf., Hong Kong, Vol. 1, Balkema, Rotterdam, pp. 503–508.
• Hillel, D. (1998). Introduction to Environmental Soil Physics. Academic Press, San Diego, CA, USA, p. 364.
• Khanzode, R.M., Vanapalli, S.K. and Fredlund, D.G. (2002). Measure-ment of soil–water characteristic curve for fine-grained soils using a small centrifuge. Can. Geotech. J., 39(5), 1209–1217.
• Lam, L., Fredlund, D.G. and Barbour, S.L. (1987). Transient seepage model for saturated–unsaturated soil systems: a geotechnical engineer-ing approach. Can. Geotech. J., 24, 565–580.
• Ng, C.W.W. and Pang, Y.W. (2000a). Experimental investigations of the soil–water characteristics of a volcanic soil. Can. Geotech. J., 37, 1252–1264.
• Ng, C.W.W. and Shi, Q. (1998). A numerical investigation of the sta-bility of unsaturated slopes subject to transient seepage. Computers and Geotechnics, Vol. 22, 1–28.
• Ng, C.W.W., Wang, B. and Gong, B.W. (2001a). A new triaxial appara-tus for studying stress effects on soil–water characteristics of unsaturated soils. Proc. 15th Conf. of Int. Soc. Soil Mech. and Geotech. Engrg., Istanbul, Vol. 1, Balkema, Rotterdam 611–614.
• Zhan, L.T. and Ng, C.W.W. (2004). Analytical analysis of rainfall infiltration mechanism in unsaturated soils. Int. J. Geomech., ASCE.
Vol. 4, No. 4, 273–284.
Flow laws for water and air (Fredlund and Rahardjo, 1993)
Introduction
In the pores of an unsaturated soil, there are two fluid phases: water and air.
The analysis of fluid flow requires a law to relate the flow rate to driving potential using appropriate coefficients. The air in an unsaturated soil may be in an occluded form when the degree of saturation is relatively high. At lower degrees of saturation, the air phase is predominantly continuous. The form of the flow laws may vary for each of these cases. In addition, there may be the movement of air through the water phase, which is referred to as air diffusion through the pore water in Figure 3.1.
A knowledge of the driving potentials that cause air and water to flow or to diffuse is necessary for understanding the flow mechanisms. The driving
Flow systems common to unsaturated soils
Occluded air bubbles (Compression pore fluid flow) Continuous air phase
(Two phase flow)
Water Air
Air diffusion through water
Figure 3.1 Flow systems common to unsaturated soils (after Fredlund and Rahardjo, 1993).
Flow laws, seepage and state-dependent soil–water characteristics 95 potentials of the water phase are given in terms of ‘heads’ in this chapter.
Water flow is caused by a hydraulic head gradient, where the hydraulic head consists of an elevation head plus a pressure head. A diffusion process is usually considered to occur under the influence of a chemical concentration or a thermal gradient. Water can also flow in response to an electrical gradient (Casagrande, 1952).
The concept of hydraulic head and the flow of air and water through unsat-urated soils are presented here. A brief discussion on the diffusion process is also presented, together with its associated driving potential. Flows due to chemical, thermal and electrical gradients are beyond the scope of this book.
Flow of water in soils (Fredlund and Rahardjo, 1993)
Several concepts have been used to explain the flow of water through an unsaturated soil. These are
• a water content gradient
• a matric suction gradient
• a hydraulic head gradient.
They have all been considered as driving potentials. However, it is impor-tant to use the form of the flow law that most fundamentally governs the movement of water.
Water content gradient
A gradient in water content has sometimes been used to describe the flow of water through unsaturated soils. It is assumed that water flows from a point of high water content to a point of lower water content. This type of flow law, however, does not have a fundamental basis since water can also flow from a region of low water content to a region of high water con-tent when there are variations in the soil types involved, hysteretic effects, or stress history variations are encountered. Therefore, a water content gradi-ent should not be used as fundamgradi-ental driving potgradi-ential for the flow of water (Fredlund, 1981).
Matric suction gradient
In an unsaturated soil, a matric suction gradient has sometimes been consid-ered to be the driving potential for water flow. However, the flow of water does not fundamentally and exclusively depend upon the matric suction gra-dient. Three hypothetical cases where the air and water pressure gradients
Case 1
Unsaturated soil Air pressure
ua (kPa)
Water pressure uw (kPa) Matric suction (ua – uw) (kPa)
–25 –50
–50 –100 25 50
Case 2
Unsaturated soil ua
uw
(ua – uw)
0 –50
–150 –200
150 150 Case 3
Unsaturated soil ua
uw
(ua – uw)
200 0
–100 –200 300 200
Figure 3.2 Pressure and matric suction gradients across an unsaturated soil element (after Fredlund and Rahardjo, 1993).
are controlled across an unsaturated soil element at a constant elevation are illustrated in Figure 3.2.
In all cases, the air and water pressures on the left-hand side are greater than the pressures on the right-hand side.
The matric suction on the left-hand side may be smaller than on the right-hand side (Case 1), equal to the right-right-hand side (Case 2) or larger than on the right-hand side (Case 3). However, air and water will flow from left to right in response to the pressure gradient in the individual phases, regardless of the matric suction gradient. Even in Case 2, where the matric suction gradient is zero, air and water will still flow.
Flow laws, seepage and state-dependent soil–water characteristics 97
Hydraulic head gradient
Flow can be defined more appropriately in terms of a hydraulic head gradient (i.e. a pressure head gradient in this case) for each of the phases. Therefore, the matric suction gradient is not the fundamental driving potential for the flow of water in an unsaturated soil. In the special case where the air pressure gradient is zero, the matric suction gradient is numerically equal to the pressure gradient in the water. This is the common situation in nature and is probably the reason for the proposal of the matric suction form for water flow. However, the elevation head component has then been omitted.
The flow of water through a soil is not only governed by the pressure gradient but also by the gradient due to elevation differences. The pressure and elevation gradients are combined to give a hydraulic head gradient as the fundamental driving potential. The hydraulic head gradient in a specific fluid phase is the driving potential for flow in that phase. This is equally true for saturated and unsaturated soils.
Driving potential for water phase
The driving potential for the flow of water defines the energy or capacity to do work. The energy at a point is computed relative to a datum. The datum is chosen arbitrarily because only the gradient in the energy between two points is of importance in describing flow.
A point in the water phase had three primary component of energy, namely, gravitational, pressure and velocity. Figure 3.3 shows point A in the water phase which is located at an elevation, y, above an arbitrary datum.
Elevation = 0 Pressure = 0 Velocity = 0 Density = ρw
Assume: ρw = constant g = constant A
x
y y
z
Elevation = y Pressure = uw Velocity = vw
Density = ρw
Arbitrary datum
⎧ ⎪
⎨ ⎪
⎩
Figure 3.3 Energy at point A in the y-direction relative to an arbitrary datum (after Fredlund and Rahardjo, 1993).
The total energy [i.e. gravitational (potential) energy, pressure energy and velocity energy] at point A can be expressed as energy per unit weight, which is called a potential or a hydraulic head. The hydraulic head, hw, at point A is obtained by dividing the energy equation by the weight of water at point A:
hw= y + uw
wg+v2w
2g (3.1)
where
hw= hydraulic head or total head g= gravitational acceleration uw= pore water pressure at point A
vw= flow rate of water at point A (i.e. in the y direction) y= elevation of point A above the datum
w= density of water at point A.
The hydraulic head consists of three components, namely, the gravitational head, y, the pressure head uw/wg and the velocity head v2w/2g. The velocity head in a soil is negligible in comparison with the gravitational and the pressure heads. The above equation can therefore be simplified to yield an expression for the hydraulic head at any point in the soil mass:
hw= y + uw
wg (3.2)
The heads expressed in this equation have the dimension of length. Hydraulic head is a measurable quantity, the gradient of which causes flow in satu-rated and unsatusatu-rated soils. To illustrate how water flow through a soil mass, Figure 3.4 considers two arbitrary points A and B at which a tensiometer and a piezometer are used to measure its in situ pore water pressure, respectively.
The tensiometer at A is used to measure the pore water pressure when the pres-sure is negative, whereas the piezometer at B is used to meapres-sure the pore water pressure when the pore water pressure is positive.
The distance between the elevation of the point under consideration and the datum indicates the elevation head (i.e. yA and yB).
The water level in the measuring device will rise or drop, depending upon the pore water pressure at the point under consideration. For example, the water level in the piezometer rises above the elevation of point B at a distance equal to the positive pore water pressure head at point B. Alternately, the water level in the tensiometer drops below the elevation of point A to a distance equal to the negative pore water pressure head at point A. The distance between the water level in the measuring device and the datum is the sum of the gravitational and pressure heads (i.e. the hydraulic head).
Flow laws, seepage and state-dependent soil–water characteristics 99
uw
hw = y + ρwg Tensiometer
Piezometer
Datum Ground surface
uw
ρwg (negative)
uw ρwg (positive) yB
yA
hw(A) hw(B)
A
B
Figure 3.4 Concept of potential and head for saturated soils (after Fredlund and Rahardjo, 1993).
In Figure 3.4, point A has a higher total head than point B [i.e. hwA >
hwB]. Water will flow from point A to point B due to the total head gradient between these two points. The driving potential causing flow in the water phase has the same form for both saturated (i.e. point B) and unsaturated (i.e. point A) soils (Freeze and Cherry, 1979). Water will flow from a point of high total head to a point of low total head, regardless of whether the pore water pressures are positive or negative.
Osmotic suction has sometimes been included as a component in the total head equation for flow. However, it is better to visualize the osmotic suction gradient as the driving potential for the osmotic diffusion process (Corey and Kemper, 1961). Osmotic diffusion is a process where ionic or molecular con-stituents move as a result of their kinetic activity. For example, an osmotic gradient across a semi-permeable membrane causes the movement of water through the membrane. On the other hand, the bulk flow of solutions (i.e. pure water and dissolved salts) in the absence of a semi-permeable mem-brane is governed by the hydraulic head gradient. Therefore, it would appear superior to analyse the bulk flow of water separately from the osmotic diffu-sion process since two independent mechanisms are involved (Corey, 1977).
Darcy’s law for unsaturated soils
The flow of water in a saturated soil is commonly described using Darcy’s law (1856). He postulated that the rate of water flow through a soil mass was proportional to the hydraulic head gradient:
vw= −kw
hw
y (3.3)
where
vw= flow rate of water
kw= coefficient of permeability with respect to the water phase
hw/y= hydraulic head gradient in the y-direction, which can be designated as iwy.
The coefficient of proportionality between the flow rate of water and the hydraulic head gradient is called the coefficient of permeability, kw. The coefficient of permeability is relatively constant for a specific saturated soil.
The above equation can also be written for the x- and z-directions. The negative sign in the equation indicates that water flows in the direction of a decreasing hydraulic head.
Darcy’s law also applies for the flow of water through an unsaturated soil (Buckingham, 1907; Richards, 1931; Childs and Collis-George, 1950).
In a saturated soil, the coefficient of permeability is a function of the void ratio (Lambe and Whitman, 1979). However, the coefficient of permeability of a saturated soil is generally assumed to be a constant when analysing problems such as transient flow. In an unsaturated soil, the coefficient of permeability is significantly affected by combined changes in the void ratio and the degree of saturation (or water content) of the soil. Water flows through the pore space filled with water; therefore, the percentage of the voids filled with water is an important factor. As a soil becomes unsaturated, air first replaces some of the water in the large pores, and this causes the water to flow through the smaller pores with an increased tortuosity to the flow path. A further increase in the matric suction of the soil leads to a further decrease in the pore volume occupied by water. In other words, the air–water interface is drawn closer and closer to the soil particles as shown in Figure 3.5. As a result, the coefficient of permeability with respect to the water phase decreases rapidly as the space available for water flow reduces.
Coefficient of permeability with respect to the water phase
The coefficient of permeability with respect to the water phase, kw, is a measure of the space available for water to flow through the soil. The coefficient of permeability depends upon the properties of the fluid and the properties of the porous medium. Different types of fluid (e.g. water and oil) or different types of soil (e.g. sand and clay) produce different values for the coefficient of permeability, kw.
Flow laws, seepage and state-dependent soil–water characteristics 101
Soil particles
Air–water interface 1 2 2
3 4 4 3
5 5
5
5
Figure 3.5 Development of an unsaturated soil by the withdrawal of the air–water interface at different stages of matric suction or degree of saturation (i.e. stages 1–5) (from Childs, 1969; after Fredlund and Rahardjo, 1993).
FLUID AND POROUS MEDIUM COMPONENTS
The coefficient of permeability with respect to the water phase, kw, can be expressed in terms of the intrinsic permeability, K:
kw=wg
wK (3.4)
where
w= absolute (dynamic) viscosity of water K= intrinsic permeability of the soil.
The above equation shows the influence of the fluid density, w, and the fluid viscosity, w, on the coefficient of permeability, kw. The intrinsic permeability of a soil, K, represents the characteristics of the porous medium and is independent of the fluid properties.
The fluid properties are commonly assumed to be constant during the flow process. The characteristics of the porous medium are a function of the volume–mass properties of the soil. The intrinsic permeability is used in numerous disciplines. However, in geotechnical engineering, the coefficient of permeability, kw, is the most commonly used term.
RELATIONSHIP BETWEEN PERMEABILITY AND VOLUME–MASS PROPERTIES
The coefficient of permeability, kw, is a function, f , of any two of three possible volume–mass properties (Lloret and Alonso, 1980; Fredlund, 1981):
kw= f1S e or kw= f2e w or kw= f3w S
where
S= degree of saturation e= void ratio
w= water content
f= a mathematical function.
EFFECT OF VARIATIONS IN DEGREE OF SATURATION ON PERMEABILITY
The coefficient of permeability of an unsaturated soil can vary considerably during a transient process as a result of changes in the volume–mass proper-ties. If the change in void ratio in an unsaturated soil may be assumed to be small, its effect on the coefficient of permeability may be secondary. How-ever, the effect of a change in degree of saturation may be highly significant.
As a result, the coefficient of permeability is often described as a singular function of the degree of saturation, S, or the volumetric water content, w. A change in matric suction can produce a more significant change in the degree of saturation or water content than can be produced by a change in net normal stress. The degree of saturation has been commonly described as a function of matric suction. The relationship is called the matric suction versus degree of saturation curve, as shown in Figure 3.6.
Numerous semi-empirical equations for the coefficient of permeability have been derived using either the matric suction versus degree of satura-tion curve or the soil–water characteristic curve (SWCC). In either case, the soil pore size distribution forms the basis for predicting the coefficient of permeability. The pore size distribution concept is somewhat new to geotechnical engineering. The pore size distribution has been used in other disciplines to give reasonable estimates of the permeability characteristics of a soil (Fredlund and Rahardjo, 1993).
The prediction of the coefficient of permeability from the matric suction versus degree of saturation curve is discussed first, followed by the coefficient of permeability prediction using the SWCC.
RELATIONSHIP BETWEEN COEFFICIENT OF PERMEABILITY AND DEGREE OF SATURATION
Coefficient of permeability functions obtained from the matric suction ver-sus degree of saturation curve have been proposed by Burdine (1952) and Brooks and Corey (1964). The matric suction versus degree of saturation curve exhibits hysteresis. Only the drainage curve is used in their derivations.
In addition, the soil structure is assumed to be incompressible.
There are three soil parameters that can be identified from the matric suction versus degree of saturation curve. These are the air entry value of the soil, ua− uwb, the residual degree of saturation, Sr, and the pore size distribution index, . These parameters can readily be visualized if
0 20 40 60 80 100 15
10
5
0
Fine sand
Degree of saturation, S (%) (a)
Matric suction (ua – uw) (kPa) First estimate of Sr = 15%
Fine sand
1 4 10 15 20
Matric suction (ua – uw) (kPa) (b)
1.0
0.1
Effective degree of saturation, Se0.01
S – Sr
Se = 1 – Sr
(ua – uw)b
Computed values of Se using Sr = 15%
λ
2nd estimate of Sr obtained by fitting computed point on straight line
Figure 3.6 Determination of the air-entry value ua− uwb, residual degree of satu-ration Sr and pore air size distribution index . (a) Matric suction ver-sus degree of saturation curve; (b) effective degree of saturation verver-sus matric suction curve (from Brooks and Corey, 1964; after Fredlund and Rahardjo, 1993).
the saturation condition is expressed in terms of an effective degree of saturation, Se, (Corey, 1954) (see Figure 3.6b):
Se=S− Sr
1− Sr
(3.5)
where
Se= effective degree of saturation Sr= residual degree of saturation.
The residual degree of saturation, Sr, is defined as the degree of saturation at which an increase in matric suction does not produce a significant change in the degree of saturation. The values for all degree of saturation variables used in the above equation are in decimal form.
The effective degree of saturation can be computed by first estimating the residual degree of saturation (see Figure 3.6b). The effective degree of saturation is then plotted against the matric suction, as illustrated in Figure 3.6b. A horizontal and a sloping line can be drawn through the points. However, points at high matric suction values may not lie on the straight line used for the first estimate of the residual degree of saturation.
Therefore, the point with the highest matric suction must be forced to lie on the straight line by estimating a new value of Sr(see Figure 3.6b). A second estimate of the residual degree of saturation is then used to recompute the values for the effective degree of saturation. A new plot of matric suction versus effective degree of saturation curve can then be obtained. The above procedure is repeated until all of the points on the sloping line constitute a straight line. This usually occurs by the second estimate of the residual degree of saturation.
The air entry value of the soil, ua− uw, is the matric suction value that must be exceeded before air recedes into the soil pores. The air entry value is also referred to as the ‘displacement pressure’ in petroleum engineering or the ‘bubbling pressure’ in ceramics engineering (Corey, 1977). It is a measure of the maximum pore size in a soil. The intersection point between the straight sloping line and the saturation ordinate (i.e. Se= 10) in Figure 3.6b defines the air entry value of the soil. The sloping line for the points having matric suctions greater than the air entry value can be described by the following equation:
Se=
ua− uwb
ua− uw
for ua− uw≥ ua− uwb (3.6)
where
= pore size distribution index, which is defined as the negative slope of the effective degree of saturation, Se, versus matric suction, ua−uw, curve.
Flow laws, seepage and state-dependent soil–water characteristics 105
Fine sand
Glass beads
Volcanic sand
Degree of saturation, S (%) (a)
Matric suction (ua – uw) (kPa) (b)
Matric suction (ua – uw) (kPa)
18
15
10
5
00 20 40 60 80 100 Touchet silt loam
0.1 1 10 50
Volcanic sand Glass beads Fine sand Touchet silt loam
Effective degree of saturation, Se
1.0
0.1
0.01
0.001
λ = 1.82 λ = 2.29
λ = 3.70
λ = 7.30
Figure 3.7 Typical matric suction versus degree of saturation curves for various soils with their corresponding values. (a) Matric suction versus degree of saturation curves; (b) effective degree of saturation versus matric suction (from Brooks and Corey, 1964; after Fredlund and Rahardjo, 1993).
Soils with a wide range of pore sizes have a small value for . The more uniform the distribution of the pore sizes in a soil, the larger is the value for . Some typical values for various soils which have been obtained from matric suction versus degree of saturation curves are shown in Figure 3.7.
The coefficient of permeability with respect to the water phase, kw, can be predicted from the matric suction versus degree of saturation curves as follows (Brooks and Corey, 1964):
kw= ks for ua− uw≤ ua− uwb (3.7) kw= ksSe for ua− uw≥ ua− uwb (3.8) where
ks= coefficient of permeability with respect to the water phase for the soil at saturation (i.e. S= 100 per cent)
= an empirical constant.
The empirical constant, , is related to the pore size distribution index:
=2+ 3
(3.9)
Table 3.1 presents several values and their corresponding pore size distri-bution indices, , for various soil types.
RELATIONSHIP BETWEEN WATER COEFFICIENT OF PERMEABILITY AND MATRIC SUCTION
The coefficient of permeability with respect to the water phase, kw, can also be expressed as a function of the matric suction by substituting the effective degree of saturation, Se, into the permeability function (Brooks and Corey, 1964). Several other relationships between the coefficient of permeability and matric suction have also been proposed (Gardner, 1958a;
Arbhabhirama and Kridakorn, 1968) and these are summarized in Table 3.2.