theoretical solution for coefficient of permeability

NPTEL- Advanced Geotechnical Engineering

Dept. of Civil Engg. Indian Institute of Technology, Kanpur, 1

Module 2

Lecture 7

Permeability and Seepage -3

Topics

1.1.6 Determination of Coefficient of Permeability in the Field

Pumping from wells

Packer test

1.1.7 Theoretical Solution for Coefficient of Permeability

1.1.8 Variation of Permeability with Void Ratio in Sand

1.1.6 Determination of Coefficient of Permeability in the Field

It is sometimes difficult to obtain undisturbed soil specimens from the field. For large construction projects,

it is advisable to conduct permeability tests in situ and compare the results with those obtained in the

laboratory. Several techniques are presently available for determination of the coefficient of permeability in

the field, such as pumping from wells, borehole tests, etc., and some of these methods will be treated briefly

in this section.

Pumping from wells: Gravity wells. Figure 2.12 shows a permeable layer underlain by an

impermeable stratum. The coefficient of permeability of the top permeable layer can be determined by

pumping from a well at a constant rate and observing the steady state water table in nearby observation

wells. The steady state is established when the water level in the test well and the observation wells become

constant. At the steady state, the rate of discharge due to pumping can be expressed as

From Figure 2.12 is approximately equal to (this is referred to as Dupuit’s assumption), and

. Substituting these in the above equation for rate of discharge, we get



Figure 2.12 Determination of coefficient of permeability by pumping from wells-gravity well

So,

(1.33)

If the values of are known from field measurements, the coefficient of permeability can

be calculated from the simple relation given in equation. (1.23)

According to Kozeny (1933), the maximum radius of influences, R (Figure 2.12), for drawdown due to

pumping can be given by

(1.34)

Where

Also note that if we substitute then

(1.35)

Where H is the depth of the original groundwater table from the impermeable layer. The depth h at any

distance r from the well thus



Or

(1.36)

It must be pointed out that Dupuit’s assumption (i.e., that does introduces large errors regarding

the actual phreatic line near the wells during steady state pumping. This is shown in Figure 2.12. For

greater than to , the phreatic line predicted by equation (1.36) will coincide with the actual phreatic

line.

The relation for the coefficient of permeability given is equation (1.33) has been developed on the

assumption that the well fully penetrates the permeable layer. If the well partially penetrates the permeable

layer as shown in Figure 2.13, the coefficient of permeability can be better represented by the following

relation (Mansur and Kaufman 1962):

(1.37)

The notations used in the right-hand side of equation (1.37) are shown in Figure 2.13.

Figure 2.13 Pumping from partially penetrating gravity wells

Artesian wells. The coefficient of permeability for a confined aquifier can also be determined from well

pumping tests. Figure 2.14 shows an artesian well penetrating the full depth of an aquifier from which water

is pumped out at a constant rate. Pumping is continued until a steady state is reached. The rate of water

pumped out at steady is given be

(1.38)

Where T is the thickness of the confined aquifier, or

(1.39)

Solution of equation (1.39) gives

(1.40)

Hence, the coefficient of permeability k can be determined by observing the drawdown in two observation

wells, as shown in Figure 2.14.



Figure 2.14 Determination of coefficient of permeability by pumping from wells-confined aquifier case

If we substitute we get

(1.41)

Bore hole test recommended by USBR. Well pumping tests are costly, and so for economic reasons bore

holes are used in many cases to estimate the coefficient of permeability of soils and rocks (U. S. Bureau of

Reclamation, 1961).

Open-end test. Figure 2.15 shows the schematic diagram for determination of the permeability of soils by

means of the open-end test. Casings are inserted in the bore holes, and they extend to the soil layers whose

permeability needs to be determined. The groundwater table can be above or below the bottom of the

casings.

Figure 2.15 Open-end test for soil permeability in the field. (Redrawn after U. S. Bureau of Reclamation,

1961, by permission of the United States Department of the Interior, Water and Power Resources Service.)



Before conducting the permeability test, the hole is cleaned out. The test begins by adding clear water to

maintain gravity flow at a constant head. The coefficient of permeability can be determined by the equation

(1.42)

Where

To ensure that equation (1.42) is used correctly, the following points should be noted:

In Figure 2.15a and b, the constant water level maintained inside the casing coincides with the top of the

casing.

In Figure 2.15a, h is measured up to the bottom of the casing.

In soils that have low permeability, it may be desirable to apply some pressure to the water. This is the case

in Figure 2.15c and d. for these cases, the value of h used in equation (1.42) is given by

(1.43)

Any consistent set of units may be used in equation (1.42). If q is expressed in gal/min, then

(1.44)

The values of for various sizes of standard casings used for field exploration are given in table 1.3. The

inside and outside diameters of these casings are given in table 1.4.

Table 1.3 Values of for various sizes of casings

Size of casings

EX 204,000

AX 160,000

BX 129,000

NX 102,000

After U. S. Bureau of Reclamation (1961)

Packer test : The arrangement for carrying out permeability tests in a bore hole below the casing is

shown in Figure 2.16. These tests can be made above or below the ground water table. However, the soils

should be such that the bore hole will stay open without any casing. The test procedure is illustrated in

Figure 2.16. The equations for calculation of the coefficient of permeability are as follows:



Figure 2.16 The packer test for soil permeability. (Redrawn after U. S. Bureau of Reclamation, 1961, by

permission of the United States Department of the Interior, Water and Power Resources Service.)

(1.45)

And

(1.46)

Where

Constant rate of flow into bore hole

Length of test hole

Radius of bore hole

Differential head of water

For convenience,

(1.47)

The values of are given in table 1.5.

Table 1.4 diameters of standard casings

Casing Inside diameter, in Outside diameter, in

EX

AX

BX



NX 3

Note: 1 in = 25.4 mm.

Variable-head tests by means of piezometer observation wells. The U. S. Department of the Navy (1971) has

adopted some standard variable-head tests for determination of the coefficient of permeability by means of

piezometer observation wells. These methods are described in table 1.6. Careful attention should be paid to

the notations. Figure 2.17 gives the shape factor coefficient S’ used for condition (A). Figure 2.18 gives the

shape factor coefficient used for condition (F-1). Figure 2.19 shows the analysis of permeability by

variable-head tests with reference to table 1.6.

Table 1.6 computation of permeability from variable head tests (for observation well of constant cross

section

Example 1.1 Refer to Figure 2.20 for the pumping test from a gravity well. If

and the pumping rate from the well is determine the coefficient of

permeability .



Example 1.2 refer to Figure 2.21. For the steady state-condition, . The coefficient of permeability of the layer is 0.03 mm/s. for the steady-state pumping condition,

estimate the rate of discharge .

Solution from equation (1.37),

So,

Figure 2.22 Shape factor coefficient used for condition (A) of table 1.6. (after U. S. Navy 1971, based on

Figure 11-11 from Soil Engineering, 3d ed, by Merlin G. Spangler and Richard L. Hardy. Copyright 1951

1963, 1973 by Harper and Row, Publishers, Inc. Reprinted by permission of the publisher.)



Figure 2.23 Shape factor coefficient, - condition (F-1) of table 1.6 (After U. S. Navy, 1971)

Observation well in isotropic soil: Piezometer in isotropic soil: Test in anisotropic soil:

Obtain shape factor from table 1.6 Radius of intake point, R, differs

from radius of stand-pipe, r

Estimate ratio of horizontal to vertical

permeability and divide horizontal

dimensions of the intake by:

For condition (C)

to compute mean

permeability . For

condition (C), table 1.6:



For this equation for q, we can construct the following table:

25 0.5

30 0.48

40 0.45

50 0.43

100 0.37

From the above table, the rate of discharge is approximately .

Table 1.5 values of [equation (1.47)]

EX AX BX NX

1 31,000 28,500 25,800 23,300

2 19,400 18,100 16,800 15,500

3 14,00 13,600 12,700 11,800

4 11,600 11,000 10,300 9,700

5 9,800 9,300 8,800 8,200

6 8,500 8,100 7,600 7,200

7 7,500 7,200 6,800 6,400

8 6,800 6,500 6,100 5,800

9 6,200 5,900 5,600 5,300

10 5,700 5,400 5,200 4,900

15 4,100 3,900 3,700 3,600

20 3,200 3,100 3,000 2,800

After U. S. Bureau of Reclamation (1961) noted: 1 ft = 0.3048 m.



1.1.7 Theoretical Solution for Coefficient of Permeability

It was pointed out earlier in this chapter that the flow through soils finer than coarse gravel is laminar. The

interconnected voids in a given soil mass can be visualized as a number of capillary tubes through which

water can flow (Figure 2.24).

Figure 2.24 Flow of water through tortuous channels in soil

According to the Hagen-Poiseuille’s equation, the quantity of flow of water in unit time, q, through a

capillary tube of radius R can be given by

(1.48)

Where

Unit weight of water

Absolute coefficient of viscosity

Area cross section of tube

Hydraulic gradient

The hydraulic radius of the capillary tube can be given by

(1.49)

From equation (1.48) and (1.49),

(1.50)



For flow through two parallel plates, we can also derive

(1.51)

So, for laminar flow conditions, the flow through any cross section can be given by a general equation:

(1.52)

Where is the shape factor. Also, the average velocity of flow is given by

(1.53)

For an actual soil, the interconnected void spaces can be assumed to be a number of tortuous channels

(Figure 2.24), and for these the term S in equation (1.53) is equal to now,

(1.54)

If the total volume of soil is , the volume of voids is where n is porosity. Let be equal to the

surface area per unit volume of soil (bulk). From equation (1.54),

(1.55)

Substituting equation (1.55) into equation (1.53) and taking is the actual seepage

velocity through soil), we get

(1.56)

It must be pointed out that the hydraulic gradient used for soils is the macroscopic gradient. The factor S in

equation (1.56) is the microscopic gradient for flow through soils. Referring to Figure 2.24, . So,

(1.57)

Or

(1.58)

Where T is tortuosity, .

Again, the seepage velocity in soils is

(1.59)

Where is the discharge velocity. Substitution of equation (1.59) and (1.58) into equation (1.56) yields



Or

(1.60)

In equation (1.60), is the surface area per unit volume of soil. If we define as the surface area per unit

volume of soil solids, then

(1.61)

Where is the volume of soil solids in a bulk volume that is,

So,

(1.62)

Combining equation (1.60) and (1.62), we obtain

(1.63)

Where e is the void ratio. This relation is the Kozeny-Carman (Kozeny, 1927; Carman, 1956). Comparing

equation (1.4 and 1.63), we find that the coefficient of permeability is

(1.64)

This absolute permeability was defined by equation (1.6) as

Comparing equation (1.6 and 1.64),

(1.65)

The Kozeny-Carman equation works well for describing coarse-grained soils such as sand and some silts.

For these cases, the coefficient of permeability bears a linear relation to . However, serious

discrepancies are observed when the Kozeny-Carman equation is applied to clayey soils.

For granular soils, the shape factor is approximately 2.5 and the tortuosity factor T is about .

1.1.8 Variation of Permeability with Void Ratio in Sand

Based on equation (1.64), the coefficient of permeability can be written as

(1.66)

Or

(1.67)



Where are the coefficients of permeability of a given soil at void ratios of respectively.

Several other relations for the coefficient of permeability and void ratio have been suggested. They are of

the form

(1.68)

(1.69)

A.Hazen (1911) gave an empirical relation for permeability of filter sands as

(1.70)

Where is in and is the effective size of the soil in cm.

Equation (1.70) was obtained from the test results of Hazen where the effective size of soils varied from 0.1

to 3 mm and the uniformity coefficient for all soils was less than 5. The coefficient 100 is an average value.

The individual test results showed a variation of the coefficient from 41 to 146. Although Hazen’s relation is

approximate, it shows a similarity to equation (1.69).

A. Casagrande has also given an empirical relation for k for fine or medium clean sands with bulky grain as

(1.71)

Where is the coefficient of permeability at a void ratio of 0.85.

Variation of Permeability with Void Ratio in Clay

The Kozeny-Carman equation does not successfully explain the variation of the coefficient of permeability

with void ratio for clayey soils. The discrepancies between the theoretical and experimental values are

shown in Figures 2.25 and 2.26. These results are based on consolidation permeability tests (Olsen, 1961,

1962). The marked degree of variation between the theoretical and experimental values arise from several

factors, including deviations from Darcy’s law high viscosity of the pore water, and unequal pore sizes.

Olsen has developed a model to account for the variation of permeability due to unequal pore sizes.



Example 1.3 results of a permeability test are drawn in Figure 2.26(a) Calculate the “composite shape

factor,” , of the Kozeny-Carmen equation, given poise. (b) If

determine . Compare this value with the theoretical value for a sphere of diameter

Solution Part (a): from equation (1.64)

The value of is the slope of the straight line for the plot of against k (Figure

2.26). So

Part (b): (Note the units carefully.)

For

Figure 2.25 Coefficient of permeability for sodium illite. (After H. W. Olsen. Hydraulic Flow

through Saturated Clays, SC. D. Thesis, Massachusetts Institute of Technology, 1961)



This value of agrees closely with the estimated value of .

Figure 2.26 Ratio of the measured flow rate to that predicted by the Kozeny-Carman equation

for several clays. Curve 1: Sodium illite, . Curve 3: Natural kaolinite, Distilled water

. Curve 4: Sodium Boston blue clay, . Curve 5: Sodium Kaolinite, 1% (by Wt.)

sodium tetraphosphate. Curve 6: Calcium Boston blue clay, . (After H. W. Olsen,

Hydraulic Flow through Saturated Clays, Sc. Thesis Massachusetts Institute of Technology

1961)

theoretical solution for coefficient of permeability

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