Darcy’s Law and Coefficient of Permeability | Factors Affecting Permeability

Permeability of Soils

Permeability is a property of soil by virtue of which the soil mass allows water (or any other fluid) to flow through it. It is an engineering property, which is required to be determined for the study of soil engineering problems involving the flow of water through soils, such as seepage through the body of the earth dam and settlement of foundations.

Darcy’s Law and Coefficient of Permeability

According to Darcy’s law, for laminar flow conditions, the velocity of flow is directly proportional to the hydraulic gradient ‘i’.

V α i  

.’.      V = Ki 

The proportionality constant, k between v and i is called Darcy’s coefficient permeability.

When i = 1 we have K= V. Thus, the coefficient of permeability can also be defined as the velocity of flow through soil under unit hydraulic gradient and has the same unit as that of velocity. It is usually expressed in mm/sec, m/hr or m/ day.

Further, q= Av = Aki

where, q= Rate of flow or discharge per unit time

              A=Total area of cross-section of flow perpendicular to the direction of flow.

Discharge Velocity and Seepage Velocity

The velocity of flow of water, v through soil mass is obtained from Darcy’s law assuming that the flow takes place through the total cross-sectional area, A of soil mass perpendicular to the direction of flow. This velocity is referred to as discharge velocity or theoretical velocity.

Thus, discharge velocity, v= q/A

The total area A is composed of area of voids, Av and area of solids, As But flow can take place only through area of voids, Av. The actual velocity of flow, referred to as seepage velocity and denoted by vs is thus greater than the theoretical velocity obtained from Darcy’s law.

Seepage velocity, Vs = q/Av     q = Av = AvVs

Where, n is the porosity of soil mass.

Further, V = ki 

.’. Vs = kp

Kp is referred as coefficient of percolation.  

Kp = k/n

Factors Affecting Permeability 

The general expression for the coefficient of permeability of soil is given by,

…………..(1)

Based on above relation, the various factors affecting permeability can be listed as,

i) Particle size

As it is evident from equation (1), the coefficient of permeability of soil is proportional to the square of the particle size (D). The permeability of coarse-grained soil is very large as compared to that of fine-grained soils. Further, according to Allen Hazen (1911),

k = CD210

where, D10 = effective grain size (cm)

C = Constant which may be taken as 100 

K= Coefficient of permeability in cm/sec.

ii) Structure of soil mass

The coefficient C in equation (1) takes into account the shape of the flow passage which depends on the structural arrangement. For the same void ratio, the permeability is more in the case of flocculated structure as compared to that in the dispersed structure.

Stratified soil deposits have greater permeability parallel to the plane of stratification than that perpendicular to this plane.

iii) Void ratio 

Equation (1) indicates that the coefficient of permeability varies as,

For a given soil the greater the void ratio, the higher is the value of the coefficient of permeability. 

Based on another concept, it has been established that all soils have an e versus log k plot as a straight line. 

iv) Properties of pore fluid

As indicated in equation (1), the coefficient of permeability is directly proportional to the unit weight of water 𝛾 

w and is inversely proportional to its viscosity μ.  

v) Degree of saturation

If the soil is not fully saturated, it contains air pockets formed due to entrapped air, which causes blockage of passage reducing the permeability of partially saturated soils. Permeability test is always conducted on fully saturated soil specimen.

vi) Adsorbed water

The fine-grained soils have a layer of adsorbed water strongly attached to their surface. The adsorbed water layer is not free to move under gravity. It causes an obstruction to the flow of water in the pores reducing the permeability of soils.

vii) Presence of foreign matter

Any foreign matter in water has a tendency to plug the flow passage and reduce the effective voids and hence the permeability of soils.

Happy Learning – Civil Concept

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