2. Permeability of Soil and Seepage Force 31
2.4 Determination of Permeability Coefficient
September 3, 2020 17:14 Soil Mechanics - 9in x 6in b3878-ch02 page 38
38 Soil Mechanics
Hansbo (1960) reported the results of four undisturbed natural clays (Fig. 2.3(b)). On the basis of his results, it is seen that
v=k(i−i0), i≥i0, and
v=kin, i < i0.
The value of n for the four Swedish clays was about 1.6. There are several studies, however, that refute the preceding conclusion.
For gravel or other coarse-grained soils, the discharge velocity bears a linear relation with the hydraulic gradient just when the hydraulic gradient is small; when the hydraulic gradient is quite large, turbulent flow of water can be expected. Discharge velocity and hydraulic gradient will not present a linear relationship (Fig. 2.3(c)).
It must be pointed out that the velocityvgiven by Eq. (2.3) is the discharge velocity calculated on the basis of the gross cross-sectional area. Since water can flow only through the interconnected pore spaces, the actual velocity of seepage through soil is v. Assuming the rate of seepage is Q, the cross-sectional area of the soil is equal to A. Hence, the actual cross-water area
A =nA.
According to the continuity of water, Q = vA = vA, then we have
v =v×A
A =vn=v e
1 +e, (2.4)
where eis the void ratio of the soil andnis the porosity of the soil.
2.4 Determination of Permeability Coefficient
The two most common laboratory methods for determining the coefficient of permeability of soils are constant-head test and falling- head test. In both cases, water flows through a soil sample and the rates of flow and the hydraulic gradients are measured.
It needs to be noted that the values of the coefficient of perme- ability measured in laboratory permeameter tests are often highly inaccurate, for a variety of reasons such as anisotropy (i.e., values of k different for horizontal and vertical flow) and small samples being unrepresentative of large volumes of soil in the ground, but in practice, the values ofkmeasured fromin situtests are much better.
2.4.1.1 Constant-head test
The constant-head test is suitable for more permeable granular materials. The basic laboratory test arrangement is shown in Fig. 2.4.
The soil specimen is placed inside a cylindrical mold, and the constant-head loss h of water flowing through the soil is maintained by adjusting the supply. The outflow water is collected in a measuring cylinder, and the duration of the collection period is recorded. From Darcy’s law, the total quantity of flowQ in timet can be given by
Q=vAt=kiAt,
where A is the area of cross-section of the specimen. However, i =h/L, where L is the length of the specimen, so Q =k(h/L)At.
Fig. 2.4. Constant-head laboratory permeability test.
September 3, 2020 17:14 Soil Mechanics - 9in x 6in b3878-ch02 page 40
40 Soil Mechanics
Rearranging gives
k= QL
Aht. (2.5)
Once all the variables on the right-hand side of Eq. (2.5) have been determined from the test, the coefficient of permeability of the soil can be calculated.
2.4.1.2 Falling-head test
The falling-head permeability test is more suitable for fine-grained soils. Figure 2.5 shows the general laboratory arrangement for the test. The soil specimen is placed inside a tube, and a standpipe is attached to the top of the specimen. Water that comes from the standpipe flows through the specimen. The initial head differenceh1 at time t=t1 is recorded, and then water is allowed to flow through the soil such that the final head difference at time t=t2 is h2.
The rate of flow through the soil is Q=kiA=kh
LA=−adh
dt, (2.6)
wherehis the head difference at any timet,Ais the area of the spec- imen, ais the area of the standpipe,Lis the length of the specimen.
Fig. 2.5. Falling-head laboratory permeability test.
From Eq. (2.6),
k= aL
A(t2−t1)lnh1
h2, (2.7)
or
k= 2.3 aL
A(t2−t1)lgh1
h2. (2.8)
The values of a,L,A,t1,t2,h1, and h2 can be determined from the test, and the coefficient of the permeability kfor a soil can then be calculated from Eq. (2.7) or Eq. (2.8).
There are several factors affecting the coefficient of permeability, such as soil types, gradation, void ratio, and water temperature.
Hence, in order to make a precise calculation of the coefficient of permeability, we must try to keep the original state of a soil and eliminate any possible influences of artificial interference. The reference values of the coefficient of permeability of some soil types are listed in Table 2.1.
2.4.2 Effective coefficient of permeability for stratified soils
In general, natural soil deposits are stratified. If the stratification is continuous, the effective coefficients of permeability for flow in the horizontal and vertical directions can be readily calculated.
Table 2.1. Reference values of the coefficient of permeability.
Basic soil type Coefficient of permeability Degree of k/(cm s−1) permeability
Pure gravels >10−1 High
Mix of pure gravels and 10−3−10−1 Middle other types of gravels
Finest sand 10−5−10−3 Low
Mix of silt, sand, and clay 10−7−10−5 Very low
Clay <10−7 Almost impermeable
September 3, 2020 17:14 Soil Mechanics - 9in x 6in b3878-ch02 page 42
42 Soil Mechanics
2.4.2.1 Flow in the horizontal direction
Figure 2.6 shows several layers of soil with horizontal stratification.
Owing to fabric anisotropy, the coefficient of permeability of each soil layer may vary depending on the direction of flow. Therefore, let us assume that k1, k2, . . . , kn are the coefficients of permeabil- ity for layers 1,2, . . . , n, respectively, for flow in the horizontal direction.
Considering unit width of the soil layers as shown in Fig. 2.6, the rate of seepage in the horizontal direction can be given by
Qx =Q1x+Q2x+· · ·+Qnx= n i=1
Qix, (2.9)
where Q is the flow rate through the combined stratified soil layers and Q1x, Q2x, . . . , Qnx are the rates of flow through soil layers 1,2, . . . , n, respectively. Note that for flow in the horizontal direction (which is the direction of stratification of the soil layers), the
Fig. 2.6. Flow in horizontal direction in stratified soil.
hydraulic gradient is the same for all layers (i1 = i2 = · · ·in = i= ΔhL ). So,
Q1x = k1iH1 Q2x = k2iH2
· · ·
Qnx = kniHn, (2.10)
whereiis the hydraulic gradient,H1,H2, . . . ,Hnare the thicknesses of layers 1, 2, . . . ,n, respectively, and
Qx=kxiH, (2.11)
where kx is the effective coefficient of permeability for flow in the horizontal direction
H=H1+H2+· · ·+Hn.
Substituting Eqs. (2.10) and (2.11) into Eq. (2.9) yields kxH=k1H1+k2H2+· · ·+knHn.
Hence, kx = 1
H(k1H1+k2H2+· · ·knHn) = 1 H
n i=1
kiHi. (2.12)
From Eq. (2.12), for flow in the horizontal direction, it can be seen that if the thickness of each soil layer is close and the coefficient of permeability is quite different, then the value ofkx depends on the most permeable soil layer (k andH are the permeability coefficient and thickness of the most permeable soil layer, respectively). Thus, the value of kx is approximately equal to kH/H.
September 3, 2020 17:14 Soil Mechanics - 9in x 6in b3878-ch02 page 44
44 Soil Mechanics
2.4.2.2 Flow in the vertical direction
For flow in the vertical direction for the soil layers shown in Fig. 2.7, Qy =Q1y =Q2y =· · ·=Qny (2.13) where Q1y, Q2y, . . . , Qny are the discharge velocities in layers 1,2, . . . , n, respectively.
Note that for flow in the vertical direction, the head loss of each soil layer is Δhi and the hydraulic gradientii is Δhi/Hi, so
Qiy=k1i1=k2i2 =· · ·=knin=kiΔhi
Hi A, (2.14) where k1, k2, . . . , kn are the coefficients of permeability for layers 1,2, . . . , n, respectively, for flow in the vertical direction,i1, i2, . . . , in are the hydraulic gradients in soil layers 1,2, . . . , n, respectively, A is the area of cross-section of the soil perpendicular to the direction of flow.
The head loss of the entire soil mass h is
Δhi, the overall average hydraulic gradient iis h/H, so
Qy =ky h
HA, (2.15)
where ky is the effective coefficient of permeability for flow in the vertical direction.
Fig. 2.7. Flow in vertical direction in stratified soil.
Substitution of Eqs. (2.14) and (2.15) into Eq. (2.13) yields ky = H
n i=1(Hki
i)
. (2.16)
From Eq. (2.16), for flow in the vertical direction, it can be seen that if the thickness of each soil layer is similar and the coefficient of permeability is quite different, then the value of ky depends on the most impermeable soil layer (k and H are the permeability coefficient and thickness of the most impermeable soil layer, respectively). Thus, the value of ky is approximately equal to kH/H.
2.4.3 Factors affecting the coefficient of permeability The coefficient of permeability depends on several factors, most of which are listed as follows:
(1) Shape and size of the soil particles:Permeability increases when the soil particles are much coarser and rounded.
(2) Void ratio: Permeability increases with an increase in void ratio. In soil with good gradation, fine soil particles fill in coarse soil particles, causing the void ratio to decrease and eventually leading to a decrease in permeability.
(3) Degree of saturation:Permeability increases with an increase in the degree of saturation.
(4) Composition of soil particles: It is not an important factor for sands and silts. However, for soils with clay minerals, this is one of the most important factors. Permeability depends on the thickness of water surrounding the soil particles, which is a function of the cation exchange capacity, valency of the cations, and so forth. Other factors remaining the same, the coefficient of permeability decreases with increasing thickness of the diffuse double layer.
(5) Soil structure: Fine-grained soils with a flocculated structure have a higher coefficient of permeability than those with a dispersed structure.
(6) Viscosity of the fluid.
(7) Density and concentration of the fluid.
September 3, 2020 17:14 Soil Mechanics - 9in x 6in b3878-ch02 page 46
46 Soil Mechanics
2.5 Flow Nets