In one dimension, water flows through a fully saturated soil in accordance with Darcy’s empirical law:
vd=ki (2.3)
or
q v A Aki= d =
where q is the volume of water flowing per unit time (also termed flow rate), A the cross-sectional area of soil corresponding to the flow q, k the coefficient of permeability (or hydraulic conductiv‑
ity), i the hydraulic gradient and vd the discharge velocity. The units of the coefficient of perme- ability are those of velocity (m/s).
The coefficient of permeability depends primarily on the average size of the pores, which in turn is related to the distribution of particle sizes, particle shape and soil structure. In gen- eral, the smaller the particles, the smaller is the average size of the pores and the lower is the coefficient of permeability. The presence of a small percentage of fines in a coarse-grained soil results in a value of k significantly lower than the value for the same soil without fines. For a given soil, the coefficient of permeability is a function of void ratio. As soils become denser (i.e. their unit weight goes up), void ratio reduces. Compression of soil will therefore alter its permeability (see Chapter 4). If a soil deposit is stratified (layered), the permeability for flow parallel to the direction of stratification is higher than that for flow perpendicular to the direction of stratification. A similar effect may be observed in soils with plate-like particles (e.g. clay) due to alignment of the plates along a single direction. The presence of fissures in a clay results in a much higher value of permeability compared with that of the unfissured material, as the fissures are much larger in size than the pores of the intact material, creat- ing preferential flow paths. This demonstrates the importance of soil fabric in understanding groundwater seepage.
The coefficient of permeability also varies with temperature, upon which the viscosity of the water depends. If the value of k measured at 20°C is taken as 100%, then the values at 10 and 0°C are 77% and 56%, respectively. The coefficient of permeability can also be represented by the equation:
k= g K h
w w
where γw is the unit weight of water, ηw the viscosity of water and K (units m2) an absolute coef- ficient depending only on the characteristics of the soil skeleton.
The values of k for different types of soil are typically within the ranges shown in Table 2.1.
For sands, Hazen (1911) showed that the approximate value of k is given by
k CD= 102 (m/s) (2.4)
where D10 is in mm. This is applicable for 0.1 <D10< 3.0 mm. C varies between 0.4 and 0.8 × 10−2 for well-graded sands and sands with appreciable fines, 0.8–1.2 × 10−2 for coarse and poorly graded sands and 1.2–1.5 × 10−2 for very coarse, very poorly graded and gravelly sands, although it is also common to take a value of C ≈ 1.0 × 10−2 for all sands.
On the microscopic scale the water seeping through a soil follows a very tortuous path between the solid particles, but macroscopically the flow path (in one dimension) can be considered as a smooth line. The average velocity at which the water flows through the soil pores is obtained by dividing the volume of water flowing per unit time by the average area of voids (Av) on a cross-section normal to the macroscopic direction of flow: this velocity is called the seepage veloc- ity (v′). Thus
¢ = v q
Av
The porosity of a soil is defined in terms of volume as described by Equation 1.9. However, on average, the porosity can also be expressed as
n A
= Av Hence
¢ = = v q
nA v
n
d
or
¢ = v ki
n (2.5)
Alternatively, Equation 2.5 may be expressed in terms of void ratio, rather than porosity, by sub- stituting for n using Equation 1.11.
Determination of coefficient of permeability
Laboratory methods
The coefficient of permeability for coarse soils can be determined by means of the constant‑head permeability test (Figure 2.2(a)). The soil specimen, at the appropriate density, is contained in a Table 2.1 Coefficient of permeability (m/s)
1 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8 10−9 10−10
Desiccated and fissured clays (10−2–10−7) Clean gravels Clean sands and sand – gravel
mixtures
Very fine sands, silts and clay-silt laminate
Unfissured clays and clay-silts (>20% clay)
Seepage
Perspex cylinder of cross-sectional area A and length l: the specimen rests on a coarse filter or a wire mesh. A steady vertical flow of water, under a constant total head (h), is maintained through the soil, and the volume of water flowing per unit time (q) is measured. Tappings from the side of the cylinder enable the hydraulic gradient (i = h/l) to be measured. Then, from Darcy’s law:
k ql
= Ah
A series of tests should be run, each at a different rate of flow. Prior to running the test, a vacuum is applied to the specimen to ensure that the degree of saturation under flow will be close to 100%. If a high degree of saturation is to be maintained, the water used in the test should be de-aired.
For fine soils, the falling‑head test (Figure 2.2(b)) should be used. In the case of fine soils undisturbed specimens are normally tested (see Chapter 6), and the containing cylinder in the test may be the sampling tube itself. The length of the specimen is l and the cross-sectional area A. A coarse filter is placed at each end of the specimen, and a standpipe of internal area a is connected to the top of the cylinder. The water drains into a reservoir of constant level.
The standpipe is filled with water, and a measurement is made of the time (t1) for the water level (relative to the water level in the reservoir) to fall from h0 to h1. At any intermediate time t, the water level in the standpipe is given by h and its rate of change by –dh/dt. At time t, Figure 2.2 Laboratory permeability tests: (a) constant head, and (b) falling head.
the difference in total head between the top and bottom of the specimen is h. Then, applying Darcy’s law:
q=Aki
- =
\ - =
\ =
=
ò ò
a h t Akh
l a dh
h Ak
l dt k al
At h h al At
h
h t
d d
0
1 1
0
1 0 1
1
2 3 ln . loghh
h
0 1
(2.6)
Again, precautions must be taken to ensure that the degree of saturation remains close to 100%.
A series of tests should be run using different values of h0 and h1 and/or standpipes of different diameters.
The coefficient of permeability of fine soils can also be determined indirectly from the results of consolidation tests (see Chapter 4). Standards governing the implementation of laboratory tests for permeability include BS EN ISO 17892-11 (UK), CEN ISO 17892–11 (Europe) and ASTM D5084-16a (US). The reliability of laboratory methods depends on the extent to which the test specimens are representative of the soil mass as a whole. More reliable results can generally be obtained by the in-situ methods described below.
Well pumping tests
This method is most suitable for use in homogeneous coarse soil strata. The procedure involves continuous pumping at a constant rate from a well, normally at least 300 mm in diameter, which penetrates to the bottom of the stratum under test. A screen or filter is placed in the bottom of the well to prevent ingress of soil particles. Perforated casing is normally required to support the sides of the well. Steady seepage is established, radially towards the well, resulting in the water table being drawn down to form a cone of depression. Water levels are observed in a number of boreholes spaced on radial lines at various distances from the well. An unconfined stratum of uniform thickness with a (relatively) impermeable lower boundary is shown in Figure 2.3(a), the water table being below the upper surface of the stratum. A confined layer between two imperme- able strata is shown in Figure 2.3(b), the original water table being within the overlying stratum.
Frequent recordings are made of the water levels in the boreholes, usually by means of an electri- cal dipper. The test enables the average coefficient of permeability of the soil mass below the cone of depression to be determined.
Standards governing the implementation of field pumping tests for permeability include BS EN ISO 22282–4 (UK), CEN ISO 22282–4 (Europe) and ASTM D4043-17 (US).
Analysis is based on the assumption that the hydraulic gradient at any distance r from the cen- tre of the well is constant with depth and is equal to the slope of the water table, i.e.
i h
r dr
= d
Seepage
where h is the height of the water table at radius r. This is known as the Dupuit assumption, and is reasonably accurate except at points close to the well.
In the case of an unconfined stratum (Figure 2.3(a)), consider two boreholes located on a radial line at distances r1 and r2 from the centre of the well, the respective water levels relative to the bot- tom of the stratum being h1 and h2. At distance r from the well, the area through which seepage takes place is 2πrh, where r and h are variables. Then, applying Darcy’s law:
q= Aki q rhk h
r
q r
r k h h
q r
r k h h
r r
h h
=
∴ =
∴
=
(
−∫ ∫
2 2
1 2
1 2
2
1 22
12
p p p
d d
d d
ln
))
∴ =k 2 3.pq
(
hlog( / )22−r rh212)
1(2.7) Figure 2.3 Well pumping tests: (a) unconfined stratum, and (b) confined stratum.
For a confined stratum of thickness H (Figure 2.3(b)) the area through which seepage takes place is 2πrH, where r is variable and H is constant. Then
q rHk h r
q r
r Hk h
q r
r Hk h h
h h r
r
=
∴ =
∴
= −
∫
∫
2 2
2
1 2 1
2
2
1 2 1
p p p
d d
d d
ln ( ))
. log( / )
( )
∴ =k q r r− H h h 2 3
2
2 1
2 1
p
(2.8)
Borehole tests
The general principle is that water is either introduced into or pumped out of a borehole which terminates within the stratum in question, the procedures being referred to as inflow and outflow tests, respectively. A hydraulic gradient is thus established, causing seepage either into or out of the soil mass surrounding the borehole, and the rate of flow is measured. In a constant-head test, the water level above the water table is maintained throughout at a given level (Figure 2.4(a)). In a falling-head test, the water level is allowed to fall or rise from its initial position and the time taken for the level to change between two values is recorded (Figure 2.4(b)). The tests indicate the permeability of the soil within a radius of only 1–2 m from the centre of the borehole. Careful boring is essential to avoid disturbance in the soil structure.
A problem in such tests is that clogging of the soil face at the bottom of the borehole tends to occur due to the deposition of sediment from the water. To alleviate the problem the borehole may be extended below the bottom of the casing, as shown in Figure 2.4(c), increasing the area through which seepage takes place. The extension may be uncased in stiff fine soils, or supported by perforated casing in coarse soils.
Expressions for the coefficient of permeability depend on whether the stratum is unconfined or confined, the position of the bottom of the casing within the stratum, and details of the drainage face in the soil. If the soil is anisotropic with respect to permeability and if the borehole extends below the bottom of the casing (Figure 2.4(c)), then the horizontal permeability tends to be mea- sured. If, on the other hand, the casing penetrates below soil level in the bottom of the borehole (Figure 2.4(d)), then vertical permeability tends to be measured. General formulae can be written, with the above details being represented by an intake factor (Fi).
For a constant-head test:
k q
= Fh
i c
(2.9a) For a falling-head test:
k A
F t t h
= h - 2 3
2 1
1 2
.
( )log
i
(2.9b) where k is the coefficient of permeability, q the rate of flow, hc the constant head, h1 the vari- able head at time t1, h2 the variable head at time t2, and A the cross-sectional area of casing or
Seepage
standpipe. Values of intake factor were originally published by Hvorslev (1951), and are also given in Cedergren (1989). For the case shown in Figure 2.4(b):
F R
i=11 2
where R is the inner radius of the casing, while for Figure 2.4(c):
F L
i= L R
(
2p)
ln / and for Figure 2.4(d)
F R
R L
i= + 11
2 11
p 2
p
Figure 2.4 Borehole tests: (a) constant-head, (b) falling-head, (c) extension of borehole to prevent clogging, (d) measurement of vertical permeability in anisotro- pic soil, and (e) measurement of in-situ seepage.
The coefficient of permeability for a coarse soil can also be obtained from in-situ measurements of seepage velocity, using Equation 2.5. The method involves excavating uncased boreholes or trial pits at two points A and B (Figure 2.4(e)), seepage taking place from A towards B. The hydraulic gradient is given by the difference in the steady-state water levels in the boreholes divided by the distance AB. Dye or any other suitable tracer is inserted into borehole A, and the time taken for the dye to appear in borehole B is measured. The seepage velocity (v') is then the distance AB divided by this time. The porosity of the soil can be determined from density tests (Chapter 1).
Then
k v n
= i¢
Standards governing the implementation of borehole tests for permeability include BS EN ISO 22282–2 (UK), CEN ISO 22282–2 (Europe) and ASTM D6391-11 (US). Further information on the implementation of in-situ permeability tests in general may be found in Clayton et al. (1995).