# Permeability of Soil and Darcys Law.

Permeability of soil

• Permeability

It is defined as the property of a porous material which permits the passage or seepage of water (or other fluid) through its interconnecting voids.

A material having continuous void is called permeable.

Gravels are highly permeable, while stiff clay is least permeable but for practical purposes clay is considered as impermeable.

• Darcy’s law

The percolation of water through soil was first studied by “Darcy” in (1856), who demonstrated experimentally that for flow missing condition in a saturated soil, the rate of flow i.e. discharge is proportional to hydraulic gradient.

i.e. q = KiA

or, v = q/A = Ki  ——-(i)

[Where, q = Rate of flow

A = Cross Sec. Area of soil mass, perpendicular to the direction of flow

v = Avg. discharge velocity

K = Darcy coefficient of permeability]

From equation (i)

If i = 1 then v = K

Thus coefficient of permeability may be defined as the average velocity of flow that will occur through the total cross sec. Area of soil under unit hydraulic gradient.

Unit of K is cm/sec or m/sec.

• Note

“v” is called superficial velocity (apparent). Actual velocity of water flowing in the voids is called seepage velocity (vs).

vs = v/n

• Validity of Darcy’s law

Flow of water may be laminar or turbulent depending upon the mode of travel of water particles. If all water particle follow definite path which never intersect one another, the flow is termed as laminar.

If the particle paths are haphazard and irregular, it is turbulent flow.

Darcy’s law is valid only for laminar flow. Again the soil must be saturated.

• Factors affecting Permeability

From Poiseuille Equation,    Q = cde2xxiA

Comparing the equation with Darcy’s law –    K = cde2x

Thus the following factors affecting permeability –

(i) Grain size

(ii) Properties of pore fluid

(iii) Void ratio of soil

(iv) Structural arrangement of soil particles and stratification

(v) Entrapped air and foreign matter

• Grain size

Permeability varies approximately as the square of the grain size.

As per Allen Hazen formula, for clean sand with particle size between 0.1 mm and 3 mm,

K = CD102

[CD10 = Effective grain size in cm

K = Coefficient of permeability in cm/sec

C = Constant ≈ 100

• Properties of pore fluid

The permeability is directly proportional to the unit weight of percolating water and inversely proportional to its viscosity

i.e. K ∞ 𝛾w/𝜂

[𝛾w = Unit weight of water

𝜂 = Viscosity of water]

• Void ratio of soil

The variation of permeability with void ratio (e) has been empirically established from laboratory investigations and the equations are –

K ∞ e3/(1+e)

Sometimes it may be – K ∞ e2

[Where, e = void ratio of soil]

• Structural arrangement of soil particles and stratification

For the same soil at the same void ratio, the permeability may vary with different methods of placement or compaction resulting in different arrangement and shape of voids.

It is much pronounced in fine grained soils because their natural structure when once disturbed can never be reconstructed.

Stratified soil masses have marked variation in their permeability in the direction parallel and perpendicular to stratification, the permeability parallel to stratification being always greater.

• Entrapped air and foreign matter

Permeability is greatly reduced if air entrapped in the voids thus reducing its degree of saturation as we know the theory of permeability where relations have been experimentally established on soils with 100% degree of saturation.

It will also be affected if organic impurities are present in the pores of a soil.

[Thickness = 10 to 15 Ao where 1 Ao = 10-10 m

The adsorbed water surrounding the fine soil particles is not free to move and hence it causes an obstruction to the flow of free water by reducing the effective pore space, thus affecting permeability.

[ As per Casagrande 0.1 may be taken as void ratio occupied by adsorbed water and accordingly

K ∞ (e – 0.1)2 ]

• Determination of permeability in the laboratory by constant head method

• Object

To determine the coefficient permeability of coarse grained soil.

• Apparatus

Constant head permeameter with all accessories, Stop watch, Graduated measuring jar etc.

• Materials

Coarse grained soil, water.

• Theory

As per Darcy’s law –

q = KiA

Or,

Or, K =     —–(i)

[ Where, Φ = Volume of water collected in the measuring jar in time ‘t’.

h = Constant head of water.

L = Length of soil sample.

A = Cross sectional area of soil sample perpendicular to the flow of water.

q = Discharge.

K = Coefficient of permeability of soil ]

• Procedure

(i) The soil sample is placed in the vertical cylinder between two porous plates, as shown in fig. and the bottom tank is filled with water missing

(ii) The outlet tube of the constant head tank is connected to the inlet of the permeameter after removing the air.

(iii) The hydraulic head will be adjusted such that ‘h’ will be constant during the test.

(iv) The test will be started now and stop watch will be ‘on’. Test will be continued for some convenient time during which water collected in the measuring jar. The time is recorded.

The test should be repeated at least twice more under the same head and for the same time interval.

From this, we will get average value of ‘Φ’

Knowing Φ, L, h, A and t, the coefficient of permeability of soil sample can be obtained from equation(i)

• Conclusion

To avoid large error, it is necessary that quantity of water collected should be large. Hence this method is suitable for pervious soil or coarse grained soil.

• Determination of permeability in the laboratory by falling head test

• Object

To determine the coefficient of permeability of fine grained soil.

• Apparatus

Falling head permeameter with all accessories, stop watch etc.

• Materials

Fine grained soil, water

• Theory

In falling head test, a stand pipe is fitted on the top of the permeameter and the change in hydraulic head with time is recorded.

Let ‘h’ be the head of water at any intermediate time ‘t’, from where head drops by ‘dh’ in time ‘dt’

Now, discharge in time dt,

q =

[ -ve sign is used as head decreases when time increases]

Again as per Darcy’s law –

q = KiA

• Procedure

(i) The soil sample is placed in vertical cylinder between two porous plates. The inside area of the cylinder is measured which gives cross sectional area of soil sample (A) and length of the soil sample (L) is measured between the porous plates.

(ii) The permeameter mould assembly is placed in the bottom tank and the bottom tank is filled with water.

(iii) The permeameter is connected to the stand pipe having cross sectional area ‘a’, water is permitted to flow for some reasonable time.

(iv) With the help of stop watch, the time interval (T) required for the water level in the stand pipe to fall from some convenient initial head (h1) to final head (h2) is noted.

(v) Knowing the values of A, L, a, T, h1 and h2, coefficient of permeability (k) will be obtained from the above equation.

• Conclusion

In this test, quantity of water collected is small and hence this method is suitable exclusively for fine grained soil.