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128

CHAPTER 4

CAPILLARITY

4.1 YOUNG-LAPLACE EQUATION

4.1.1 Three-Dimensional Meniscus

The capillary tube models discussed in the previous chapter provide a usefulphysical interpretation to facilitate understanding of the relationships amongfluid pressure, relative humidity, and vapor pressure at an air-water-solid in-terface. In soil pores, however, the geometry of the pores and fluid menisciare far more complicated, particularly at a scale greater than the largest poredimension. At a scale less than the largest pore dimension, the air-water-solidinterface may be approximated by using simple geometric configurations,

including parallel plates, cylinders, ellipsoids, or spheres.A double-curvature model may be developed on the basis of analytical

geometry and mechanical equilibrium to represent the complicated geometryof the air-water-solid interface. The Young-Laplace equation employs thisdouble-curvature concept, providing a general relationship between matricsuction and the interface geometry. The Young-Laplace equation may be writ-ten as

1 1

u u T (4.1)

a w s R R

1 2

where ua and uw are the air and water phase pressures, respectively, the dif-ference ua uw is the matric suction, Ts is the surface tension of the waterphase, and R

1and R

2are the two principal radii of curvature of the interface

near the area of interest.

Copyright 2004 John Wiley & Sons Retrieved from: www.knovel.com

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4.1 YOUNG-LAPLACE EQUATION 129

B

B

r2

A

Ts

Ts

ds

r1

A

O

z

ds

ds

ds

Figure 4.1 Mechanical equilibrium of a three-dimensional double-curvature air-waterinterface.

P.S. Laplace first derived eq. (4.1) in 1806 on the basis of potential theory,not surface tension. Interestingly, T. Young introduced the concept of mac-roscopic surface tension in 1805, which was employed by others to proveLaplaces equation on the basis of mechanical equilibrium. The surface ten-sion approach provides an extremely useful means to interpret many interfacephenomena. A derivation of eq. (4.1) follows.

Consider mechanical equilibrium near a point O on any arbitrary air-waterinterface (see Fig. 4.1). Cut an infinitesimal circular element having radius with an axis at point O. The segments AA and BB are pairs of any orthogonallines on the element that pass through point O. The small segments ds atpoints A, A , B, and B are subjected to a force arising from surface tensionequal to T

sds with projections along the vertical direction (z) equal to 2T

sds

sin at points A and A and 2Ts ds sin at points B and B. Since is small, is also small, which leads to the following:

2T ds sin 2T ds 2T ds (4.2)s s s

r1

Similarly, the total vertical force on the segments ds at points B and B is

2T ds sin 2T ds 2T ds (4.3)s s s

r2

and the total vertical force on segments along A, A , B, and B is


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130 CAPILLARITY

1 12T ds (4.4) s

r r1 2

The values of r1

and r2

generally vary from any one pair of lines AA andBB to any other pair but can be uniquely linked to the principal radii ofcurvature R

1and R

2by a theorem of Euler as

1 1 1 1 (4.5)

R R r r1 2 1 2

Thus, the total vertical force on the segments along A, A , B, and B becomes

1 1 1 12T ds 2T ds (4.6) s s

R R r r1 2 1 2

Since the choice of A, A, B, and B is completely arbitrary, eq. (4.6) canbe integrated along the entire circumference of the meniscus to obtain thetotal vertical force due to surface tension. Because eq. (4.6) represents theforce on four segments on the circumference, the integration requires only aquarter rotation along the circumference, leading to

1 12F T (4.7) z s

R R1 2

At mechanical equilibrium, a force provided by matric suction acting overthe projected area of the interface will balance the vertical force Fz:

1 12 2 (u u ) T (4.8a) a w s R R1 2

or

1 1u u T (4.8b) a w s

R R1 2

which is the familiar form of the Young-Laplace equation.By introducing the mean meniscus curvature Rm, the Young-Laplace

equation can be considered a generalized form of the mechanical equilibriumequation for a capillary tube containing a perfectly wetting material. For athree-dimensional meniscus, the mean curvature is


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4.1 YOUNG-LAPLACE EQUATION 131

(a) (b) (c)

21

22

r2

r1

21

Figure 4.2 Representation of air-water-solid interface by an ellipsoid geometry: (a)

in a cylindrical tube, (b) finite ellipsoid interface, and (c) an example in soil pores.

1 1 1 1 (4.9)

R 2 R Rm 1 2

which allows eq. (4.8) to be simplified to the form introduced in previouschapters:

2Tsu u (4.10)a w

Rm

Thus, if the geometry of the air-water-solid interface in an unsaturated soil-water system can be represented by an ellipsoidal shape with principal radii

1 and 2, it can be shown (as illustrated in Fig. 4.2) as

r cos r cos (4.11)1 1 2 2

Substituting eqs. (4.5) and (4.11) into eq. (4.8) results in

1 1u u T cos (4.12) a w s

r r1 2

4.1.2 Hydrostatic Equilibrium in a Capillary Tube

The negative pore water pressure resulting from interfacial surface tensionleads to the redistribution of water in a capillary tube or unsaturated soil.Figure 4.3 demonstrates this capillary rise phenomenon for a series of dif-


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132 CAPILLARITY

, ua

Air

uw0

uw1

uw2

uw3

Water

h1

h2

h3

r1

r2

r3

z

uw

uw3

uw2

uw1

uw= 0

()

(+)

Negative Pressure

Positive Pressure

Figure 4.3 Rise of water in capillary tubes of various sizes at hydrostatic equilibrium.

ferent sized capillary tubes at hydrostatic equilibrium. Because the air-waterinterface in the large tank containing the tubes is flat, the radius of curvaturetends to infinity and the matric suction in the bulk fluid tends to zero:

2T cos su u 0 (4.13a)a w0

or

u u (4.13b)a w0

On the other hand, mechanical equilibrium near the air-water interface in the

capillary tubes requires

2T cos su u (4.14a)a wi

ri

or

2T cos 2T cos s su u u (4.14b)wi a w0r r

i i

where the subscript i runs from 1 to 3, corresponding to the three capillarytubes in the figure. As shown in the pressure profile on the right-hand sideof Figure 4.3, the water pressure is equal to zero at the water table, increaseshydrostatically below the water table, and decreases hydrostatically above thewater table.


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4.2 HEIGHT OF CAPILLARY RISE 133

At mechanical equilibrium, the pore water pressure at the air-water inter-face uwi is equal to the unit weight of water w multiplied by the height ofthe capillary rise h

i

:

2T cos 2T cos s su u u h (4.15)wi a w0 i w

r ri i

or

2T cos sh (4.16)i

ri w

The above equation states that the height of capillary rise in a capillarytube is directly proportional to surface tension and contact angle, but inverselyproportional to the tube radius. In unsaturated soil, the hydrostatic equilibriumposition can be inferred from eq. (4.16) if the principal radii of curvature areestimated.

4.2 HEIGHT OF CAPILLARY RISE

4.2.1 Capillary Rise in a Tube

Capillary rise in soil describes the upward movement of water above the watertable resulting from the gradient in water potential across the air-water inter-face at the wetting front. Simple capillary tube models for predicting theultimate height and rate of capillary rise in soil have been developed basedon assumptions of ideal pore geometries and permeability. These models pro-

vide excellent insight into the physics of capillary rise and in some casesprovide reasonable semiquantitative predictions.

Perhaps the best-known analytical model to quantify the pressure dropacross an air-water-solid interface for a nonzero contact angle is the Young-Laplace equation, which was derived in the previous section as

1 1u u T cos (4.17) a w s

r r1 2

In an ideal cylindrical capillary tube with a diameter d, r1

r2

d/2 andeq. (4.17) becomes

4T cos su u (4.18)a w

d


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134 CAPILLARITY

ua

uw

Ts Ts

hc

ua

d

Figure 4.4 Mechanical equilibrium for capillary rise in small-diameter tube.

As described in Chapter 3, the contact angle reflects the ability of waterto wet the solid surface at the air-water-solid interface. A contact angle equalto zero describes a perfectly wetting material; 90 describes neutral wettingability; and an angle greater than 90 describes the interaction between waterand a water repellent material. For soil such as sands under drying conditions,the contact angle is often assumed to be equal to 0.

A simple analysis of mechanical equilibrium can confirm eq. (4.18). Con-sider the free-body diagram in the area of the small dashed circle shown inFig. 4.4. Vertical force equilibrium considering ua uw acting over the areaof meniscus and the vertical projection of Ts acting over the circumferenceof the meniscus leads to

2(u u ) d Td cos (4.19)a w s4

which can be directly reduced to eq. (4.18).If the air pressure is set to a reference value of zero, water pressure uw has

a negative value, representing a positive matric suction. The smaller the di-ameter of the capillary tube d, the greater the matric suction. The greater thewetting ability of the solid surface (i.e., very small contact angle ), the

greater the matric suction.The ultimate height of capillary rise, hc, can be evaluated by consideringmechanical equilibrium in the area of the large dashed circle in Fig. 4.4. Here,the total weight of the water column under the influence of gravity is balancedby surface tension along the water-solid interface as


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0

1000

2000

3000

0.0001 0.001 0.01 0.1

Capillary Tube Diameter dor Sphere Diameter D(cm)

MaximumC

apillaryRise(cm)

Capillary TubeSpheres (SC packing)Spheres (TH packing)

Clay Silt Fine Sand Med.Sand

Figure 4.5 Maximum height of capillary rise in capillary tube or idealized soil com-

prised of uniform spherical particles. Particle diameter is delineated in terms of soil

type for comparison.

2h g d Td cos (4.20)c w s4

or simply

4T cos sh (4.21)c

d gw

Imposing values of water density w as 1 g/cm3, gravitational acceleration

g 980 cm/s2, Ts 72 mN/m at 25C, and a zero contact angle, a simplerelationship between capillary rise and capillary tube diameter can be written

as

0.3h (cm) (4.22)c

d (cm)

The solid curve in Fig. 4.5 shows a plot of eq. (4.22) in terms of themaximum capillary rise versus tube diameter.


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136 CAPILLARITY

Ground Surface

Water Table

hc

(a) (b)

Air-Entry Head,ha

VolumetricWater Content,

ha

z

Maximum Capillary Rise, hc

Capillary Fingers

Suction

Head

(z)

r sCapillary Fringe

Figure 4.6 Capillary rise and associated pore water retention in an unsaturated soil

profile: (a) conceptual illustration and (b) corresponding soil-water characteristic curve.

The upper limit of eq. (4.21) or (4.22) in a glass capillary tube is about10 m, corresponding to a negative water pressure of about 100 kPa or 1atm at sea level. As discussed previously in Section 2.5, free water tends tocavitate below this pressure. In soil, however, where the pore water may beunder the influence of short-range physicochemical interaction effects at thewater-solid interface that lower its chemical potential and alter its physicalproperties, cavitation may not occur at the same pressure as that for free water.As described in Section 1.6, the intensity of these liquid-solid interactioneffects in an unsaturated soil system is a function of the specific surface andsurface charge properties of the soil mineral. In clayey soil, for example,which possesses both a very large surface area and a highly active surface,capillary rise may be as high as several tens of meters.

4.2.2 Capillary Finger Model

The uniform capillary tube model is often used to describe capillary rise inunsaturated soil and the associated pore water retention characteristics. Al-though the concept of perfectly uniform tubes in soil is unrealistic, the con-tinuous water fringes or fingers that develop above the water table can beconceptualized as bundled tubes of various diameters. This conceptualizationis illustrated in Fig. 4.6a, where the rising fingers of water are shown withdifferent average diameters and heights at the equilibrium condition. The as-sociated pore water retention curve is shown as Fig. 4.6b in terms of volu-


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metric water content versus suction head (i.e., the height above the watertable).

As illustrated in the figure, pore water rises above the water table undercapillary suction. The soil remains essentially saturated, described by the sat-urated water content s, until the suction head reaches the air-entry head,designated ha. The air-entry head may be defined as the suction head at whichair initially begins to displace water from the soil pores. The saturated zoneextending from the water table up to the air-entry head is commonly referredto as the capillary fringe. Above the air-entry head, the water content de-creases with increasing height, reflecting the fact that fewer and smaller cap-illary fingers are present for a given cross section of the soil column withincreasing elevation. Following the principles developed in the previous sec-tion, the narrowest capillary fingers rise to a maximum height hc, whereas thelargest fingers are restrained to relatively low elevations. At relatively largevalues of suction head, therefore, very little water is retained by the soil. Porewater within this regime is primarily in the form of thin films surroundingthe particle surfaces or disconnected pendular water menisci. The watercontent within this regime is commonly referred to as the residual watercontent, or r.

If the soil column above the water table is initially dry, a head gradientexists between the continuous capillary fingers and the overlying soil, whichis approximately equal to (hc z) /z, where z is the height of the advancingwetting front and hc is the driving head described by eq. (4.21). As the fingersmove into higher elevations, the gradient decreases. Eventually, the wettingfront reaches a point where hc z, thus satisfying the requirement for me-chanical equilibrium, and the capillary rise ceases. When coupled with anappropriate description for hydraulic conductivity, consideration of the chang-ing driving head as the wetting front advances allows the rate of capillaryrise to be evaluated. Two such developments are presented in Section 4.3.

4.2.3 Capillary Rise in Idealized Soil

Equations (4.21) and (4.22) provide a means to estimate the height of capillaryrise in a uniform capillary tube. Given this theoretical basis, the upper andlower bounds of capillary rise in idealized soil comprised of uniform sphericalparticles may be evaluated by considering simple cubic (SC) packing (i.e.,loosest possible packing) and tetrahedral (TH) packing (i.e., densest possible

packing) as two limiting cases.Figures 4.7a and 4.7b show simple cubic and tetrahedral packing geome-tries in plan view for uniform spheres of diameter D, respectively. Minimumpore diameters across these sections corresponding to SC and TH packing aredenoted d

scand d

th. The relationship between particle size and minimum pore

diameter for the case of SC packing is described by:


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138 CAPILLARITY

dsc

D D

dth

45 30

(a) (b)

Figure 4.7 Plan view illustration of (a) simple (SC) cubic and (b) tetrahedral (TH)packing for uniform spherical particles.

D/2cos 45 (4.23)

D/2 d /2sc

which leads to the simple relationship

d 0.41D (4.24)sc

Similarly, the relationship between particle size and minimum pore diam-eter for the case of TH packing is

D/2cos 30 (4.25)

D/2 d /2th

or

d 0.15D (4.26)th

A more realistic system of spherical particles would likely have a packinggeometry that falls somewhere between these two limiting cases. Substitutingeqs. (4.24) and (4.26) into eq. (4.21), therefore, the following bounds for theultimate height of capillary rise in such a system may be defined:

9.76T cos 26.67T cos s s h (cm) (4.27a)c

D (cm) g D (cm) gw w

Assuming values for w 1 g /cm3, g 980 cm/s2, Ts 72 mN/m, and

zero contact angle leads to


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0.73 2 h (cm) (4.27b)c

D (cm) D (cm)

Equation (4.27b) implies that for the same soil, packing can affect thecapillary height by a factor of about 2.75. For example, a sand column pre-pared in the lab using uniform Ottawa sand with particle diameter of 0.1 cmmight have capillary rise ranging anywhere between 7.3 and 20 cm. Themaximum capillary rise corresponding to the bounds described by eq. (4.27b)for a wide range of particle diameter is included in Fig. 4.5.

4.2.4 Capillary Rise in Soil

Because real soil is comprised of a range of different particle sizes fallingwithin some size distribution and complex packing geometry, analytical eval-uation of the height of capillary rise is extremely difficult. To overcome thisdifficulty, empirical equations have been developed to relate the height ofcapillary rise to more easily measured soil properties. These properties mostcommonly include particle or pore size distribution parameters, void ratio,and air-entry head. In general, hysteresis effects are not considered in theempirical relationships. Most of the empirical equations assume an initially

dry soil undergoing a wetting process from a stationary water table.Peck et al. (1974), for example, describe an empirical equation expressing

the height of capillary rise as an inverse function of the product of void ratio,e, and the 10% finer particle size, D

10, as

Ch (4.28)c

eD10

where hc and D10 are in units of millimeters and C is a constant varying

between 10 and 50 mm2 depending on surface impurities and grain shape.Because an increase in either void ratio or D

10reflects an increase in the

average pore diameter of the soil, the corresponding maximum height of cap-illary rise decreases.

Analysis of capillary rise experiments conducted by Lane and Washburn(1946) for eight different soils indicates that the maximum height of capillaryrise may be described by a linear function of D

10as

h 990(ln D ) 1540 (4.29)c 10

where both D10

and hc are in units of millimeters and D10 ranges from 0.006to 0.2 mm. Equations (4.28) and (4.29) indicate that the 10% finer particlefraction may adequately describe the effective diameter of the smallest con-tinuous capillary fingers in soil.


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140 CAPILLARITY

Perhaps the most reliable method to determine the height of capillary riseis by direct measurement through open-tube capillary rise tests conducted inthe laboratory. Numerous experimental programs in this regard have beendescribed in the literature (e.g., Lane and Washburn, 1946; Malik et al., 1989;Kumar and Malik, 1990). Table 4.1, for example, shows a summary of resultsfrom laboratory capillary rise experiments for several different types of soil.Maximum capillary rise hc was determined in each case by observing theequilibrium wetting front manually. Air-entry head ha was determined fromthe soil-water characteristic curve, measured either using representative spec-imens or by measuring the final equilibrium water content of the soil columnas a function of height from the water table. The final column on Table 4.1shows the dimensionless ratio of maximum capillary rise to air-entry head,hc/ha.

The data in Table 4.1 supports the notion of an empirical relationshipbetween air-entry head and the maximum height of capillary rise. For thewide range of soil tested, the ratio hc/ ha varies from 2 to 5 with only a fewexceptions. Thus, if the air-entry head is estimated from independent mea-surements of grain size distribution or the soil-water characteristic curve, itappears that the upper and lower limits for maximum height of capillary risemay be reasonably estimated.

Kumar and Malik (1990) also found that the difference between the heightof capillary rise and the height of capillary fringe is a decreasing function ofthe square root of an equivalent pore radius r. One such relationship wassuggested in the form

h h 134.84 5.16 r (4.30)c a

where hc and ha are in centimeters and r is in micrometers.

4.3 RATE OF CAPILLARY RISE

4.3.1 Saturated Hydraulic Conductivity Formulation

As early as 1943, Terzaghi formulated a simple theory to predict the rate ofcapillary rise in a one-dimensional column of soil. To quantify the rate ofcapillary rise, Terzaghi made two major assumptions: (1) Darcys law forsaturated flow is applicable to unsaturated flow, and (2) the upward hydraulicgradient i responsible for capillary rise at the wetting front can be approxi-

mated as

h zci (4.31)

z

where z is a distance measured positive upward from the elevation of thewater table (see Fig. 4.8). Physically, the maximum capillary height hc rep-


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141

TABLE 4.1 Experimental Capillary Rise Parameters for Several Different Soi

Test No.a Soil

Gravel

(%)

Sand

(%)

Silt/Clay

(%)

1

23

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

2021

22

23

24

Class 5

Class 6Class 7

Class 8

Ludas sand

Rawalwas sand

Rewari sand

Bhiwani sand

Tohana loamy sand 1

Hisar loamy sand 1

Barwala sandy loam 1Rohtak sandy loam 1

Hisar sandy loam 1

Pehwa sandy clay loam

Hansi clayey loam 1

Ambala silty clay loam 1

Tohana loamy sand 2

Hissar loamy sand 2

Barwala sandy loam 2

Rohtak sandy loam 2Hissar sandy loam 2

Pehowa sandy clay loam

Hansi clayey loam 2

Ambala silty clay loam 2

25.0

0.020.0

0.0

68.0

47.060.0

5.0

89.0

82.5

75.0

63.063.0

55.0

30.2

15.0

7.0

53.020.0

95.0

6.0

11.5

13.5

23.024.0

27.0

26.5

49.0

a14 (Lane and Washburn, 1946), 516 (Malik et al., 1989), and 1724 (Kumar and Malik, 19

Copyright 2004 John Wiley & Sons Retrieved fro

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142 CAPILLARITY

Figure 4.8 System geometry for analytical prediction of rate of capillary rise.

resents the drop in pressure head across the air-water interfaces in the soilpores.

Terzaghis other assumption, Darcys law, can be expressed in familiarform as

dzq k i n (4.32)sdt

where q is the discharge velocity, ks is the saturated hydraulic conductivityof the soil column, and n is the porosity.

Solving eqs. (4.31) and (4.32) and imposing an initial condition of a zerocapillary rise at zero time, Terzaghi arrived at a solution describing the lo-cation of the capillary wetting front z as an implicit function of time t:

nh h zc ct ln (4.33a) k h z hs c cwhich can be rearranged in a compact form by introducing dimensionlesstime T kst/nhc and dimensionless distance Z z /hc as

1T ln Z (4.33b)

1 Z

4.3.2 Unsaturated Hydraulic Conductivity Formulation

Subsequent experimental investigations of capillary rise (e.g., Lane and Wash-burn, 1946; Krynine, 1948) have shown that Terzaghis original analyticalsolution (4.33) significantly overpredicts the rate of rise. The assumption ofconstant (saturated) hydraulic conductivity had been identified by Terzaghi


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4.3 RATE OF CAPILLARY RISE 143

(1943) and Krynine (1948) as the cause for the discrepancies. In some cases,a reduction of the saturated hydraulic conductivity by more than 2 orders ofmagnitude is required to yield a reasonable match between the theory andexperimental data.

In reality, capillary rise above the air-entry head is no longer governed bythe saturated hydraulic conductivity. As described in Chapter 8, the hydraulicconductivity of soil decreases dramatically with decreases in the degree ofsaturation, following what is commonly referred to as the unsaturated hy-draulic conductivity function. By the time the wetting front approaches themaximum height of capillary rise, the degree of saturation may be as low asa few percent, and the hydraulic conductivity may be reduced by 5 to 7 ordersof magnitude from its value at saturation. This significant reduction in con-ductivity, together with the reduction in the available driving head (hc z) /

z as the wetting front moves upward, leads to a significant decrease in therate of rise. Consequently, the discrepancies between Terzaghis theoreticalequation and the actual height of capillary rise propagate as time elapses.

The characteristic dependence of hydraulic conductivity with respect tosuction, water content, or degree of saturation has been a focus of intensiveresearch since Terzaghis original work. Numerous models for describing theunsaturated conductivity function have been developed, with the majority ac-counting for the drastic reduction in conductivity using either exponential,power, or series functions. Several of these models are described in detail inChapter 12.

Lu and Likos (2004) developed an alternative solution for the rate of cap-illary rise by incorporating the Gardner (1958) one-parameter model to esti-mate the unsaturated hydraulic conductivity function. As described in Chapter12, Gardners model may be expressed as an exponential function in termsof the saturated hydraulic conductivity and suction head as

k(h ) k exp(h ) (4.34)m s m

where k is the unsaturated hydraulic conductivity at suction head hm (cm) and is a pore size distribution parameter (cm1) representing the rate of decreasein hydraulic conductivity with increasing suction head. As illustrated on Fig.4.8, the inverse of can be interpreted as the air-entry head, or equivalently,as the height of the saturated portion of capillary rise, that is, the capillaryfringe.

The general behavior and performance of eq. (4.34) is demonstrated in Fig.

4.9a

by comparison with experimental data for sand (Richards, 1952) andclay (Moore, 1939). Figure 4.9b provides a more general illustration of themodels behavior for parameters (ks and ) representative of three differentsoil types.

Incorporating the Gardner model to represent hydraulic conductivity at thewetting front, a governing equation for the rate of capillary rise can be writtenas


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144

0.1 1 10 100 1000Suction Head (cm)

HydraulicConductivity(cm/s)

Yolo Light Clay:(Moore, 1939)

ks = 1.23 10-5 cm/s

Superstition Sand:(Richards, 1952)

ks = 1.83 10-3 cm/s

= 0.08

= 0.025

10-8

10-7

10-6

10-5

10-4

10-3

10-2

(a)

(b)

0.1 1 10 100 1000 10000Suction Head (cm)

HydraulicConductivity(cm/s)

Sand:

ks= 1 10-3 cm/s

= 0.1

Silt:

ks= 1 10-5 cm/s

= 0.01

Clay:

ks= 1 10-7 cm/s

= 0.001

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Figure 4.9 Unsaturated hydraulic conductivity function according to the Gardner

(1958) one-parameter model: (a) comparison with experimental data and (b) general

pattern for three representative soil types.


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4.3 RATE OF CAPILLARY RISE 145

dz k h zs c exp(z) (4.35)

dt n z

Analytical solution of eq. (4.35) can be written in series form:

m jj s j1sn h h zc cj1t h ln (4.36a) ck j! h z j 1 sj0 s0s c

If the nonlinearity in hydraulic conductivity is ignored by setting the seriesindex m to zero, eq. (4.36a) reduces to Terzaghis original analytical solution[eq. (4.33)]. Convergent solutions are typically obtained by setting m equal

to 10. In applying eq. (4.36a), the material parameter can be determined ifeither the hydraulic conductivity function or soil-water characteristic curve ismeasured a priori. Given the former, can be determined in conjunction withGardners (1958) model to find the value giving a best fit to the data. Giventhe latter, can be determined by estimating the air-entry head ha and byrecognizing that may be interpreted as its inverse. The practical range of for most soil reported in the literature varies from 1.0 cm1 for coarse-grainedmaterials, to 0.001 cm1 or lower for relatively fine-grained materials. Theultimate height of the capillary rise for use in eq. (4.36a) may be approxi-

mated using a capillary tube analogy and applying the Young-Laplace equa-tion or by applying the empirical relationships described in the previoussection.

Equation (4.36a) may also be written in terms of the dimensionless vari-ables T and Z as

m jj j1s(h ) 1 ZcT ln

j! 1 Z j 1 sj0 s0

k t zsT Z (4.36b)nh hc c

By writing the solution in dimensionless space and time, arrival time con-tours can be predicted if the soil parameters hc, , n, and ks are known. Figure4.10 shows a series of such contours for T

50, T

60, T

70, T

80, and T

90. The arrival

time T50

, for example, is defined as the dimensionless time required to ad-vance the wetting front to the position half of the total height of the maximumcapillary rise, that is, z 0.5 hc.

4.3.3 Experimental Verification

In 1946, Lane and Washburn reported a systematic experimental study on theheight and rate of capillary rise using open-tube column tests. Soils were


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146 CAPILLARITY

0.1

1

10

100

1,000

10,000

0 2 4 6 8 10 12

hc= hc/ha

Dimensio

nlessTIme,

T=k

st/h

cn

T50

T60

T70

T80

T90

Figure 4.10 Solution for rate of capillary rise in dimensionless time and space.

prepared from natural sandy gravel that was graded and remixed in desired

proportions to create eight classes of soils representing a wide range ingrain size and grain size distribution. Figure 4.11 shows grain size distribution

curves for four of these classes. Direct measurements were obtained for sat-

urated hydraulic conductivity ks, porosity n, soil-water characteristic curves,total height of capillary rise hc, rate of capillary rise (e.g., the elevation of

wetting front as a function of time), and in some cases, the height of the

capillary fringe ha.

Figure 4.12 (page 148) shows height of capillary rise as a function of time

from the experimental measurements for class 2 (Fig. 4.12a) and class 4 (Fig.

4.12b) materials, a poorly graded coarse sand and poorly graded fine sand,respectively. Experimental data for class 5 and 6 soils, a well-distributed

coarse sand with fines and a sandy silt, respectively, are shown in Figs. 4.13aand 4.13b (page 149). Theoretical solutions for the rate of capillary rise based

on the saturated hydraulic conductivity formulation [eq. (4.33)] and the un-

saturated hydraulic conductivity formulation [eq. (4.36)] are included for

comparison. Note the significant improvement in the prediction when theunsaturated nature of the soil is considered.


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4.4 CAPILLARY PORE SIZE DISTRIBUTION 147

0

20

40

60

80

100

0.010.1110100

Grain Size (mm)

PercentFinerbyMass(%) Class 5

Class 2

Class 4

Class 6

Figure 4.11 Grain size distributions for Lane and Washburn (1946) capillary rise

tests.

4.4 CAPILLARY PORE SIZE DISTRIBUTION

4.4.1 Theoretical Basis

The size, shape, and distribution of the pore spaces in soil comprise a criticalelement of soil fabric and play principal roles in governing the overall engi-neering behavior of the bulk soil mass. Methodologies to measure or estimatephysical properties of the pore space provide significant insight in predictingstrength, compressibility, and permeability behavior. This section describesthe theoretical basis for evaluating relationships among pore size, pore sizedistribution, and capillary pressure in unsaturated soil. A step-by-step list ofcomputational procedures and a series of example problems are provided todemonstrate use of the soil-water characteristic curve for estimating pore sizedistribution.

Kelvins equation provides the thermodynamic basis to relate relative hu-midity or matric suction to pore size. As introduced in Section 3.3, capillaryradius r can be expressed as a function of surface tension Ts, contact angle

, and relative humidity RH as

2T cos s wr (4.37a)

RT ln(RH)

or in terms of matric suction ua uw as


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148

0

5

10

15

20

25

30

0.0001 0.001 0.01 0.1 1 10

Time (days)

CapillaryRise(cm)

Lane and Washburn (1946)Class 2, poorly graded coarse sand

ks= 1.6 10-2cm/s

hc= 28.4 cmn= 0.31

Eq. (4.36), hc= 5

1

2

Terzaghi (1943)

1

2

0

20

40

60

80

100

120

0.01 0.1 1 10 100Time (days)

CapillaryR

ise(cm)

Lane and Washburn (1946)Class 4, poorly graded fine sand

ks= 4.6 10-4 cm/s

hc= 106 cmn= 0.31

1 2

3

Eq. (4.36), hc= 4

1

2

3

Terzaghi (1943)

Eq. (4.36), hc= 5

(a)

(b)

Figure 4.12 Comparison of eq. (4.33), eq. (4.36), and experimental data for the rate

of capillary rise in (a) coarse sand and (b) fine sand.


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149

0

20

40

60

80

100

0.01 0.1 1 10 100 1000Time (days)

CapillaryRise(cm)

Lane and Washburn (1946)

Class 5, coarse sand with finesks= 1.1 10

-4 cm/s

hc= 82 cm

ha= 41 cmn= 0.21

1 2

Eq. (4.36), hc= hc/ha= 2.0

1

2

Terzaghi (1943)

0

50

100

150

200

250

0.1 1 10 100 1000

Time (days)

CapillaryR

ise(cm)

Lane and Washburn (1946)Class 6, sandy silt

ks= 6.2 10-5cm/s

hc= 239.6 cm

ha= 175 cmn= 0.40

Eq. (4.36), hc= hc/ha= 1.4

1

2

Terzaghi (1943)

1

2

(a)

(b)

Figure 4.13 Comparison of eq. (4.33), eq. (4.36), and experimental data for the rate

of capillary rise in (a) coarse sand with fines and (b) sandy silt.


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150 CAPILLARITY

2T cos sr (4.37b)

u ua w

Analyses based on eqs. (4.37a) and (4.37b) have been extensively exploredto evaluate pore size and pore size distribution in porous media (e.g., Lowell,1979). Pore fluids commonly used for such analyses most commonly includewater, water vapor, nitrogen, and mercury. Water vapor sorption isotherms foruse with eq. (4.37a), whereby the relationship between relative humidity andpore size may be established, and soil-water characteristic curves (SWCC)for use with eq. (4.37b), whereby the relationship between matric suction andpore size may be established, are typically considered along drying (desorp-

tion or drainage) paths. A zero contact angle is typically assumed. A relatedtype of analysis involves the intrusion of a nonwetting pore fluid (most com-monly mercury) into an initially evacuated specimen under externally appliedpositive pressure. In this case, a more general form of the pore sizecapillarypressure relationship can be written in terms of applied intrusion pressure upas

2T cos sr (4.37c)

up

where the contact angle is greater than 90, or about 130 to 150 formercury. Diamond (1970) and Sridharan et al. (1971) provide detailed de-scriptions of mercury intrusion porosimetry (MIP) and its application to theevaluation of pore size and pore size distribution in soil.

By definition, capillary pore size analysis is applicable over the range ofpore size for which capillarity remains the dominant pore fluid retentionmechanism. As described in Chapter 3, this range is approximately 109 to104 m in terms of pore radius, which corresponds to matric suction ranging

from approximately 144,000 to 0 kPa, or relative humidity ranging from ap-proximately 35 to 100%. Below relative humidity of about 35%, pore wateradsorption and retention are controlled primarily by surface hydration mech-anisms, which cannot be directly described by eq. (4.37). Application of con-ventional pore size distribution analyses to clayey soil, particularly expansiveclay, is also limited because adsorption mechanisms other than capillarity(e.g., hydration and osmotic effects) dominate over an extremely wide andpoorly understood range of suction and because the pore fabric is not a con-stant but rather may radically deform as a function of water content.

4.4.2 Pore Geometry

Several relationships are required to conceptualize the geometry of the soilpores in order for analysis based on capillary pressure measurements to bepossible. These include pore volume, average pore radius, the thickness of


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the adsorbed water film on the soil solids, and the ratio of pore volume tosurface area. Because the computational procedures for estimating pore sizedistribution involve numerical integration, it is convenient to define thesequantities in incremental form as functions of relative humidity or matricsuction. Each may then be quantified at incremental steps along the sorptionisotherm or soil-water characteristic curve under consideration.

The change in the air-filled pore volume or the water-filled pore volumeper unit mass of solid, V (m3/kg), for the ith increment of relative humidityipor suction can be defined as

iwiV (4.38a)p

w

or in an integral form

iwiV (4.38b)p

w

The gravimetric water content wi in the above equations can be directlyobtained from the sorption isotherm or soil-water characteristic curve. The

density of water, w, may be considered essentially constant within the cap-illary adsorption regime, assuming that solid-liquid interaction effects willonly cause significant density changes in the thin films located adjacent tothe particle surfaces (see Section 2.1.3).

The ratio of pore volume to surface area for a given pore depends on thepore geometry. However, because pore shapes in soil are highly irregular, anexact mathematical expression of the volume-to-area ratio is practically im-possible. Alternatively, simple shapes such as cylinders, parallel plates, andspheres may be assumed to provide estimates or bounds on such ratios. The

volume-to-area ratio for a cylinder, pair of parallel plates, and sphere arer/2, r/2, and r/3, respectively, where r is the cylinder radius, sphere radius,or the separation distance between parallel plates (Fig. 4.14). The geometryof the air-filled pores under a given relative humidity, matric suction, or de-gree of saturation can also be conceptualized as the simple shapes depictedin Fig. 4.14 and can be calculated using eq. (4.37).

The Kelvin radius rk (air-filled pore radius) can be evaluated from eqs.(4.37a) or (4.37b) as

2T 2Ts w sir (4.39)kRT ln(u /u ) u u

v v0 a w

The actual pore radius r is the Kelvin radius plus the thickness of theipwater film, ti, adsorbed on the particle surface at the prevailing relative hu-midity or matric suction, and thus may be written as


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152 CAPILLARITY

L

r

L

r

H

r

Volume: r2LSurface: 2rL

Volume/Surface: r/2

Volume: LHrSurface: 2LH

Volume/Surface: r/2

Volume: 4/3r3

Surface: 4r2

Volume/Surface: r/3

(a) (b) (c)

Figure 4.14 Idealized geometries for soil pores: (a) cylinder, (b) parallel plates, and

(c) sphere.

i i ir r t (4.40)p k

Several methods have been proposed to estimate adsorbed film thickness

t. The Halsey equation (1948) is commonly used for pore size distributionanalyses as it has been shown to provide a close fit to experimental data formany porous media and because it is independent of porous media type forrelative humidity greater than 30%. The Halsey (1948) equation is written as

1/3

5it (4.41) iln(RH )

where ti is the thickness of the water layer on the surface of the soil solid at

the ith increment in relative humidity, and is the effective diameter of thesorbate molecule. The effective diameter of an adsorbed water molecule maybe calculated by considering the area and volume occupied by one mole ofwater if it were spread over a surface to a depth of one molecular layer.Assuming the occupied cross-sectional area of a liquid water molecule isapproximately A 10.8 A2 (Livingston, 1949), and given the molar volumeof water w 18 10

6 m3/mol, and Avogadros number NA 6.02 1023

mol1, the effective diameter for an adsorbed water molecule may be esti-mated as

6 3 18 10 m /molw 2.77 A (4.42)2 23AN (10.8 A )(6.02 10 1/mol)A

Figure 4.15 shows a plot of adsorbed water film thickness as a function ofrelative humidity calculated using eqs. (4.41) and (4.42) for an effective di-ameter of water molecules equal to 2.77 A.


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0

5

10

15

20

25

30

0 10 20 30 40 50 60 70 80 90 100

Relative Humidity (%)

ThicknessofAdsorbedWaterFilm()

Figure 4.15 Thickness of adsorbed water film as function of relative humidity.

The change in the specific surface area, S, over the ith increment of relativehumidity or suction can be determined by the volume-to-area ratio for a givenpore geometry. For example, if a cylinder or pair of parallel plates is assumed,the incremental specific surface area is

i2 VpiS (4.43a)irp

If a spherical pore geometry is assumed, then the incremental specific surfacearea is

i3 VpiS (4.43b)irp

4.4.3 Computational Procedures

Numerical integration procedures for calculating pore size distribution froma sorption isotherm or soil-water characteristic curve are summarized in thefollowing steps.


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154 CAPILLARITY

1. Select data from the sorption isotherm or soil-water characteristiccurve for relative humidity greater than about 35% or matric suctionless than about 144,000 kPa.

2. Convert volumetric water content to gravimetric water content if thesoil-water characteristic curve or vapor sorption isotherm is obtainedin terms of volumetric water content.

3. Convert matric suction to relative humidity if the SWCC is obtainedin terms of suction.

4. Convert gravimetric water content to the water-filled pore volume perunit mass of solid by dividing the water content by water density, eq.(4.38b).

5. Calculate the Kelvin radius using eq. (4.39).6. Calculate the thickness of the water film using eq. (4.41).

7. Calculate the pore radius using eq. (4.40).

8. For a given change in relative humidity (i.e., decrement along thedesorption curve under consideration), calculate the decrement in thepore volume per unit mass of solid.

9. Calculate the average Kelvin radius during the decrement.

10. Calculate the average pore radius during the decrement.

11. Calculate the incremental surface area for the assumed pore geometryusing eq. (4.43).

12. Calculate the cumulative pore volume per unit mass by summing theprevious incremental pore volumes.

13. Plot the decrement in pore volume per unit mass versus the averagepore radius and plot the cumulative pore volume versus the pore radius.

Example Problem 4.1 Figure 4.16 shows a soil-water characteristic curvein the form of matric suction versus gravimetric water content, (w), for apulverized specimen of Georgia kaolinite. Given that the surface tension ofwater, Ts, is 72 mN/m, the gas constant R is 8.314 J/mol K, and the molarvolume of liquid water, w, is 0.018 m

3/kmol, develop the pore size andcumulative pore size distribution functions for the clay. Assume the ambienttemperature corresponding to the soil-water characteristic curve is 25C.

Solution The worksheet shown as Table 4.2 was created to follow the gen-eral computational procedures described above. Figure 4.17a illustrates the

resulting pore size distribution for the kaolinite in terms of pore volume perunit mass versus average pore size. Figure 4.17b illustrates the pore sizedistribution in terms of cumulative pore volume versus average pore size. Thecalculated specific surface area is 19.83 m2/g, which is within the typicalrange for Georgia kaolinite of about 10 to 20 m2/g (e.g., Klein and Hurlbut,1977). The total pore volume calculated for the kaolinite is 0.396 cm3/g. Notefrom Fig. 4.17a that pore sizes between about 100 and 10,000 A dominate


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0.0 0.1 0.2 0.3 0.4 0.5

Gravimetric Water Content, w(g/g)

MatricSuction(kPa)

100

101

102

103

104

105

106

Figure 4.16 Soil-water characteristic curve for Georgia kaolinite.

the total pore volume. Since most of the grain sizes for typical kaolinite areless than 2 m, it follows that the majority of pores fall within the range of0.1 m (1000 A) and 1 m (10,000 A). The valley occurring at about 700A reflects the rapid change in matric suction noted in the soil-water charac-teristic curve at water content between 0.16 and 0.22 g/g.

Example Problem 4.2 Figure 4.18a shows grain size distribution curves fortwo sandy soil specimens: poorly graded sand with silt (SP-SM) and siltysand (SM). Soil-water characteristic curves for the sands (Fig. 4.18b) wereobtained in the laboratory along drying paths using a Tempe pressure cellapparatus (Section 10.3). Develop the pore size and cumulative pore sizedistribution functions for each material from this data.

SolutionFigure 4.19

ashows the pore size distribution for each sand interms of pore volume per unit mass versus average pore size. Figure 4.19b

illustrates pore size distributions in terms of cumulative pore volume versusaverage pore size. Note that the relatively narrow grain size distribution ofthe SP-SM specimen is reflected in its poorly graded, or steep, grain sizedistribution curve, its relatively flat soil-water characteristic curve, and bythe distinct maximum on the pore size distribution function (Fig. 4.19a) oc-


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156

TABLE 4.2 Computational Worksheet for Determining Pore Size Dis(Fig. 4.16)

ua uw(kPa)

w

(g/g)

RH

(%)

Vp(cm3/g)

rk(A)

t

(A)

rp

(A)

126

158

200

398631

1,778

3,162

3,548

3,981

5,012

7,943

25,11956,234

100,000

125,893

0.395

0.355

0.312

0.2750.225

0.190

0.170

0.145

0.105

0.085

0.050

0.0300.020

0.018

0.016

99.91

99.89

99.86

99.7199.54

98.72

97.73

97.46

97.15

96.43

94.40

83.3366.48

48.38

40.09

0.396

0.356

0.313

0.2760.225

0.190

0.170

0.145

0.105

0.085

0.050

0.0300.020

0.018

0.016

11,438.3

9,085.8

7,217.1

3,617.12,282.2

809.8

455.4

405.8

361.7

287.3

181.3

57.325.6

14.4

11.4

48.8

45.2

41.9

33.328.5

20.2

16.7

16.0

15.4

14.3

12.3

8.46.4

5.3

4.9

11,487

9,131

7,259

3,6502,310

830

472

421

377

301

193

6532

19

16

Copyright 2004 John Wiley & Sons Re

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157

0.00

0.01

0.02

0.03

0.04

0.05

0.06

1 10 100 1,000 10,000 100,000

Average Pore Size ()

PoreVolumeperUnitMass(cm

3/g)

0.00

0.10

0.20

0.30

0.40

1 10 100 1,000 10,000 100,000

Average Pore Size ()

CumulativePoreVolumeperUnitMass(cm3/g)

(a)

(b)

Figure 4.17 Pore size distribution functions for Georgia kaolinite: (a) pore volume

per unit mass versus average pore size and (b) cumulative pore volume per unit mass

versus average pore size.


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158 CAPILLARITY

0

20

40

60

80

100

0.0100.1001.00010.000

Grain Size (mm)

PercentFiner(%)

SP-SM

SM

0

1

10

100

0.0 0.1 0.2 0.3 0.4 0.5

Gravimetric Water Content, w(g/g)

MatricSuction(

kPa)

SP-SM

SM

(a)

(b)

Figure 4.18 (a) Particle size distributions (b) and soil-water characteristic curves (c)

for two sandy soil specimens (data from Clayton, 1996).


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160 CAPILLARITY

(a)

r2r1

R

(b)

Ft

Fe

uw

ua

Figure 4.20 Air-water-solid interaction for two spherical particles and water menis-

cus: (a) toroidal geometry of the air-water-solid interface and (b) free-body diagram

for analysis of interparticle forces.

curring at about 0.02 mm. The average predominant pore sizes for both soilsare marked by values less than the predominant grain size.

4.5 SUCTION STRESS

4.5.1 Forces between Two Spherical Particles

Suction stress refers to the net interparticle force generated within a matrixof unsaturated granular particles (e.g., silt or sand) due to the combined effectsof negative pore water pressure and surface tension. The macroscopic con-sequence of suction stress is a force that tends to pull the soil grains toward

one another, similar in effect and sign convention to an overburden stress orsurcharge loading.

One approach to evaluating the magnitude of suction stress is to considerthe microscale forces acting between and among idealized assemblies ofspherical unsaturated soil particles. Consider, for example, the two-particlesystem shown on Fig. 4.20. At low degrees of pore water saturation, or thependular regime, interparticle forces arise from the presence of the air-water-solid interface defining the pore water menisci between the particles.The magnitude of the capillary force arising from this so-called liquid bridge

between the particles may be analyzed as a function of water content byconsidering the local geometry of the air-water-solid interface as follows.


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4.5 SUCTION STRESS 161

For monosized particles (Fig. 4.20a), it was established in the previouschapter that the water meniscus formed between them may be described bytwo radii r

1and r

2, the particle radius R, and a filling angle . A free-body

diagram for the relevant system forces, which involves contribution from airpressure ua, pore water pressure uw, surface tension Ts, and applied externalforce or overburden Fe, is shown in Fig. 4.20b.

Positive, isotropic air pressure ua will exert a compressive force on the soilskeleton. The total force due to air pressure, Fa, is equal to the product ofthe magnitude of the air pressure and the area of the air-solid interface overwhich it acts:

2 2

F

u (R

r ) (4.44)a a 2

The total force due to surface tension, Ft, acts along the perimeter of thewater meniscus:

F T2r (4.45)t s 2

The projection of total force due to water pressure acting on the water-solidinterface in the vertical direction, F

w, is

2F u r (4.46)w w 2

The resultant capillary force, Fsum

, is the sum of all three of the above forces:

2 2 2F u R u r T2r u r (4.47)sum a a 2 s 2 w 2

Assuming the air pressure is the only contribution to external force leads tothe following:

2 2F u R (u u )r T2r (4.48)e a a w 2 s 2

which is the net interparticle force due to the interfacial interaction. This forceexerts a tensile stress on the soil skeleton as long as the following conditionis met:

2 2(u u )r T2r u R (4.49)a w 2 s 2 a

It was demonstrated in Chapter 3 that matric suction ua uw within thewater lens formed between two spherical particles may be described inde-pendent of contact angle by the spherical radii r

1and r

2and surface tension

Ts as


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162 CAPILLARITY

1 1u u T (4.50) a w s

r r1 2

Substituting the above equation into eq. (4.49) results in

2T(r r )rs 2 1 2 2 T2r u Rs 2 ar r

1 2

and setting air pressure to a reference value equal to zero leads to

T r (r r ) 0 (4.51)s 2 2 1

The above condition will always be satisfied ifr1

0 because r2

is alwaysgreater than or equal to zero. This implies that suction stress in hydrophilicunsaturated soil is always greater than or equal to zero. Therefore, the forceon the soil skeleton will always be tensile, even though r

1and r

2have the

opposite effect on the sign of the pore water pressure, as shown below.

4.5.2 Pressure in the Water Lens

The water pressure in the lens between two spherical particles can be eitherpositive, zero, or negative. The relationship between the sign of the waterpressure and the lens geometry may be illustrated by rearranging eq. (4.50)as

1 1u u T (4.52) w a s

r r1 2

Accordingly, the absolute value of pore water pressure depends on bothair pressure and the interface geometry. For example, if r

1 r

2, a pore water

pressure less than the air pressure will develop within the lens. However, ifr

1 r

2, a pore pressure greater than air pressure will develop within the lens.

For ua equal to zero, eq. (4.52) dictates that a decrease in the menisci radiusr

1results in increasingly negative values of pore water pressure, a reflection

of radius r1s relationship to the concave curvature of the water lens. A de-

crease inr

2, on the other hand, causes the pore water pressure to be lessnegative, a reflection of its relationship to the convex curvature of the waterlens.

Considering the geometry of the contacting spheres and the water lens forzero contact angle, a relationship between R, r

1, and r

2may be written as

follows:


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2 2 2(R r ) R (r r ) (4.53)1 1 2

Ifr1 is equal to r2, which must occur at some value of water content, thepressure in the water lens is equal to the air pressure and the matric suctionis thus equal to zero. Imposing this condition to eq. (4.53) leads to

2 2 2 3(R r ) R (r r ) R r2 2 2 2 2

(4.54)

Considering the geometry shown in Fig. 4.20a, it can be shown that

r r 2r 2r 41 2 2 2tan (4.55a)R R (3/r 2) 3

2

or

53.13 (4.55b)

Therefore, the water content regime corresponding to a negative pore water

pressure corresponds to the range in filling angle described by

0 53.13 r r (4.56)1 2

and the water content regime corresponding to positive pore water pressureis described by

53.13 90 r r (4.57)1 2

For relatively loosely packed particles, such as the simple cubic (SC) order,the filling angle may not be greater than 45 because the adjacent waterlenses start to overlap each other. The condition described by eq. (4.57) isunlikely to occur in unsaturated soil with zero contact angle, indicating thatthe pore water pressure in the water lens is likely to be negative. The conditionwhere the contact angle is not zero is often the case in real soil and will becovered in the next chapter.

4.5.3 Effective Stress due to Capillarity

Effective stress owing to the balance of the interfacial forces described abovecan be evaluated by considering the area over which they act. Figure 4.21illustrates two such areas for analysis: the area over one spherical soil grain,or R2, and a unit area for simple cubic packing order, or 4R2. Considering


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164 CAPILLARITY

R2

4R2

Figure 4.21 Unit areas for analyzing effective stress in simple cubic packing order.

eq. (4.48), the stress contribution due to the capillary interparticle force overthe area R2 is

2 2r 2r r2 2 1 u (u u ) (u u )w a a w a w2 2R R (r r )

2 1

2 2r 2r r2 2 1

u (u u )

a a w2 2

R R (r

r )2 12r r r2 2 1

u (u u ) (4.58)a a w2R r r2 1

and the effective stress under an external total stress is

2r r r2 2 1 u (u u ) (4.59a)w a a w2R r r

2 1

which is in the same form as Bishops (1959) effective stress equation forunsaturated soil, or

2r r r2 2 1

u (u u ) u (u u )w a a w a a w2R r r2 1

(4.59b)

where the effective stress parameter is in this case equal to

2r r r2 2 1 (4.59c)2R r r

2 1

Similarly, for analysis using a cross-sectional area of4R2, the effective stressis


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2 r r r2 2 1 u (u u ) u (u u )w a a w a a w24 R r r

2 1

(4.60a)

where, for this geometry,

2 r r r2 2 1 (4.60b)24 R r r

2 1

Equations (4.59c) and (4.60b) provide a great deal of insight into the nature

of suction stress in unsaturated soil. Physically, the effective stress parameter represents the contribution of matric suction to effective stress. The pa-rameter clearly depends on water content in these equations via r

1and r

2.

When water content for SC packing order approaches saturation, radius r2

approaches the particle radius R and radius r1

approaches zero. Examinationof eq. (4.59c) for a unit area of R2 demonstrates that approaches unityunder these conditions, thus reducing eq. (4.59a) to the classical effectivestress equation for saturated soil:

u (4.61)w

On the other hand, if water content approaches zero (i.e., perfectly dryconditions), then r

2and r

1both approach zero, thus leading to approaching

zero and the condition where the effective stress is equal to the total stressminus the air pressure. Matric suction in this case, no matter its value, hasno contribution to effective stress. For water content values between the com-pletely dry and completely saturated conditions, the effective stress parameter

is dependent on the relationship between r1 and r2. In general, and in realsoil, the relationship between r1

and r2

is complicated and depends on contactangle and the geometric constraints imposed by the soil pores. The analysisbelow illustrates a special case when the contact angle is zero.

4.5.4 Effective Stress Parameter and Water Content

A specific relationship between effective stress parameter and water contentcan be established by considering the geometry of the water lens. As intro-

duced in Section 3.4, Dallavalle (1943) presented the following approxima-tions relating the parameters r

1, r

2, R, and for the case where contact angle

is assumed equal to zero:

1r R 1 r R tan r 0 85 (4.62) 1 2 1cos


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166 CAPILLARITY

0.0

0.5

1.0

1.5

0 10 20 30 40 50

Filling Angle, (deg.)

EffectiveStressParam

eter,

Figure 4.22 Relationship between filling angle and effective stress parameter forspherical particles in simple cubic packing order with 4R 2 unit area.

Substituting the above equation into eq. (4.59c), the effective stress param-eter may thus be described in terms of filling angle for an elementarycross section ofR2:

2(sin cos 1) sin (4.63)

2cos sin 2 2 cos

or considering eq. (4.60b) for an elementary cross section of 4R2:

2 (sin cos 1) sin (4.64)

24 cos sin 2 2 cos

Equation (4.63) or (4.64) can be used to explore a physical interpretationof the effective stress parameter and suction stress, and their dependency onsoil water content in terms offilling angle . For any filling angle , the radiir

1and r

2and the effective stress parameter can be uniquely defined. The

relationship between the effective stress parameter and filling angle is illus-trated in Fig. 4.22 for less than 45 (corresponding to gravimetric watercontent, w 0.063). Interestingly, this relationship is independent of theparticle size R as inferred from equations (4.63) and (4.64).


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0 10 20 30 40 50

Filling Angle, (deg.)

SuctionStress(kPa)

Particle Size R= 1 mm

Particle Size R= 0.1 mm

Particle Size R= 0.01 mm

Ts= 0.072 N/m

10-2

10-1

100

101

102

Figure 4.23 Suction stress as function offilling angle for spherical particles in simplecubic packing order.

Effective stress due to suction stress can also be studied without introduc-ing the concept of the effective stress parameter . Eliminating matric suctionin eq. (4.59b) by substituting eq. (4.50) leads to the effective stress due tosuction stress, c, for the unit area R

2:

2r r r2 2 1 T (4.65)c s2R r r

1 2

and by substituting eq. (4.50) into eq. (4.60a) for a unit area of 4R2:

2

r r

r2 2 1 T (4.66)c s24 R r r1 2

Substituting eq. (4.62) into the above equation to express r1

and r2

in termsof, suction stress can be expressed in terms of filling angle :

T sin cos 1s tan (4.67)c 4R 1 cos


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168 CAPILLARITY

Through eq. (4.67) and as illustrated on Fig. 4.23, one can infer that suctionstress is dependent on particle size R and water content but not directly onmatric suction. The fundamental question stemming from the above analysisis: Is it necessary to use matric suction to represent effective stress in unsat-urated soil? At the present time, this remains an open question.

PROBLEMS

4.1. Compute and compare the equilibrium height of capillary rise in a 5 105 m diameter capillary tube for free water with surface tension of0.072 N/m and soapy water with surface tension of 0.010 N/m. Assumezero contact angle and a fluid density equal to 1 g/cm3 in both cases.

4.2. Water is in a capillary tube at equilibrium. The tube has an inner radiusof 2 105 m, the contact angle is 60, and the surface tension is 0.072N/m. What are the pressure in the water and the relative humidity inthe tube? If the tube were placed in a spacecraft with zero gravity, waterfrom capillary condensation is likely to spread over the inner wall witha uniform water film thickness. Assume the thickness of the water filmat equilibrium is 105 m. What are the pressure in the water and the

relative humidity in the tube?

4.3. Uniform fine sand with particle radius of 0.1 mm is packed in twoarrayssimple cubic packing and tetrahedral closest packingfor anopen-tube capillary rise test. The contact angle is 50 and surface tensionis 0.072 N/m. What is the expected range for height of capillary rise?

4.4. A fine sand specimen was tested for grain size and pore size distributionparameters and the soil-water characteristic curve. Particle size analysisshows D

10 0.06 mm. Pore size analysis shows a mean pore radius of

0.05 cm and a void ratio of 0.4. Soil-water characteristic curve testingindicates an air-entry head of 100 cm. Estimate the maximum height ofcapillary rise for this soil using three different empirical relationships.

4.5. Derive Terzaghis (1943) solution for the rate of capillary rise [eq.(4.33a)].

4.6. Show that eq. (4.36a) can be reduced to eq. (4.33a) if the summationindex m is zero. Reproduce the theoretical curves shown in Fig. 4.12ausing the system parameters shown in the figure. Use a summation index

m 5.

4.7. Data describing the soil-water characteristic curve for a sand specimenis shown in Table 4.3. If the surface tension is 0.072 N/m, the molarvolume of water is 0.018 m3/kmol, and R is 8.314 J/mol K, conducta pore size distribution analysis and provide the following information:specific surface area (m2/g), total pore volume (cm3/g), average pore


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PROBLEMS 169

radius vs. pore volume in an x-y plot, and average pore radius vs. cu-mulative pore volume in an x-y plot.

TABLE 4.3 Soil-Water Characteristic Curve Data

for Problem 4.7

ua uw(kPa) RH

w

(g/g)

10

16

32

63

1581259

12589

125893

0.99993

0.99988

0.99977

0.99954

0.998850.99090

0.91265

0.40092

0.330

0.310

0.250

0.140

0.0700.040

0.035

0.034

4.8. Calculate and plot the interparticle force between two spherical particles(R 0.1 mm) as a function of filling angle from 0 to 30.

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