Permeability of Soils

Permeability of soil is an engineering property of soil that allows the flow of water through its interconnected voids. It is the ease with which water can flow through it. Soil is considered impervious if its permeability is very low and vice versa.

The permeability of soil is determined for various soil engineering problems like the settlement of buildings, the yield of wells, the design of filters to prevent piping in hydraulic structures, etc.

The article discusses the features of the permeability of the soil, Darcy's law, the coefficient of permeability (k), and their relationship with seepage velocity.

Interconnected Voids in Permeability of Soil

The permeability of soil is affected not by the total volume of voids present in the soil medium but by the way they are connected. This can be explained by conceptualizing the structure of gravel and clay medium. 

Consider the volume of a single void in gravel and clay. Then,

(Volume of Single Void)_Gravel > (Volume of Single Void) _Clay

When we consider the total number of voids in gravel and clay medium,

(Number of Voids)_Gravel < (Number of Voids) _Clay

Then it is clear that,

(Total Volume of Voids)_Gravel < (Total Volume of Voids)_Clay

Clay does have a greater volume of voids compared to gravel. But, the voids in gravel are interconnected so that they allow water passage compared to clay medium. This is why gravel is more permeable than clay. From this concept, the following conclusions can be made:

Permeability of Coarse Grain Soil > Permeability of Fine Grain Soil

Hence, the permeability of the following soil is arranged in increasing order.

Clay < Silt < Sand < Gravel

Darcy's Law of Permeability

Darcy demonstrated experimentally that for a laminar flow in a homogeneous soil, the velocity of flow (v) is directly proportional to the hydraulic gradient (i).

v α i

v = ki

Here, v is the discharge velocity or the superficial velocity and k is the coefficient of permeability or hydraulic conductivity.

The coefficient of permeability is a measure of how permeable or impermeable the soil is. The higher the value of k, the higher the permeability. Its unit is mm/sec or cm/sec. The expected values of the coefficient of permeability (k) of different soil types are explained in the table below:

Soil Type	Coefficient of Permeability (k) mm/sec	Drainage Properties
Clean gravel	101 to 10+2	Very Good
Coarse and Medium Sands	10-2 to 10+1	Good
Fine Sand, Loose Silt	10-4 to 10-2	Fair
Dense Silt and Clayey Silts	10-5 to 10-4	Poor
Silty Clay, Clay	10-8 to 10-5	Very Poor

The concept of Darcy’s law and related theories are better explained by the following experimental study followed by Darcy in 1856.

Consider an experimental setup where a sample soil medium as shown is arranged for water seepage through it. The pressure head at point A and point B of the sample medium is measured as shown in the apparatus. The length of the soil sample medium is ‘L’.

Water flows freely from a high head (high energy) to a low head (lower energy) by means of gravity. For example, in the apparatus shown above, the pressure head at A must be higher than at B for the water seepage to be possible.

Let’s consider the pressure head and elevation head (potential head) at A and B as PH_A, Z_A and PH_B, Z_B respectively. While applying Bernoulli’s Equation for a pipe flow for this arrangement, it is given as;

Total Head = Pressure Head + Velocity Head + Elevation Head

TH = PH +EH + Z

In Bernoulli’s equation, the velocity considered is the velocity of the flow of water through a pipe of reasonable size. In the case of soil, the water movement or seepage occurs through the interconnected pores of very small size compared to a pipe. Hence, the velocity head in the case of soil medium is neglected (Velocity Head = 0).

Hence, TH = P/ρg + Z

This relation can be derived for point A and point B as:

TH_A = PH_A + Z_A

TH_B = PH_B + Z_B

and, TH_A > TH_B

The relations convey the theory that there is a difference between the heads at A and B that in turn results in seepage. The direction of the flow of water depends on the type of head difference and it always moves from high to lower energy.

Head Loss or Head Difference (ΔH)

From the above explanations:

ΔH = TH_A - TH_B

The head difference or head loss creates a flow path from the high head to the low head. Higher the head loss, the higher the seepage in the soil.

Hydraulic Gradient (i)

The hydraulic gradient is given by:

i = ΔH/L;

Here ‘L’ is the length of flow path. In the given example, the water undergoes seepage from point A to B through the sample medium, whose length is ‘L’. 

The hydraulic gradient (i) is a dimensionless quantity whose value shows the presence of flow of water in the soil.

If the value of the hydraulic gradient increases, the discharge velocity (v) in the soil also increases. This means the value of ‘v’ is a function of ‘i'.

i.e. v = f(i);

An experiment conducted to analyze the increase in discharge velocity with an increase in hydraulic gradient gives results that can be plotted as shown in the graph below.

Zone I – Laminar Flow

Darcy’s law is defined for laminar flow, and in this zone, the discharge velocity is directly proportional to the hydraulic gradient.

v α i

v = ki

Features of Darcy’s Law for Soil Permeability

• Darcy’s law is valid only for soil types that permit the laminar flow of water or liquid. Example: fine sand and coarse silts.
• It is not applicable for very fine particles like cement as the velocity is very low.
• It is not applicable for very coarse aggregates like boulders, brick aggregates, etc as a high velocity of flow occurs within the particles.

Seepage Velocity (v_s) in Soils

The discharge velocity (v) is not the actual velocity through which the water moves through the interstices of the soil. Actually, it is a fictitious velocity (v = total discharge / cross-sectional area) i.e. given by Q/A;

The total cross-sectional area in the case of soil medium consists of both solids and voids. As the flow happens only through the voids, the actual velocity is very much less than the discharge velocity. Hence, the actual velocity of flow in the macroscopic scale in the soil is called seepage velocity (v_s).

The figure above shows the longitudinal section through a soil medium, where the total voids and Vv and Vs are segregated  [ For understanding the concept]. Av is the cross-section of the voids through which actual seepage occurs with a seepage velocity of v_s.

From the equation of continuity:

A₁v₁ = A₂v₂

Av = A_vv_s

v_s = [A/A_v ] v = [(A . L )/(A_v . L) ] v  [ Multiplying and dividing by ‘L’]

A.L = V (Volume); Av . L = Vv (Volume of Voids)

v_s = (V/V_V) v

During seepage, the voids is filled with water. Hence, volume of voids Vv = Volume of water Vw;

v_s = (V/Vw ) v

v_s = [ (V/Vv) . (Vv/Vw) ]v

Vv/V = porosity (n) ; Vw/Vv = Degree of Saturation (s)

That gives,

v_s = v/ns;

When the soil is saturated, the degree of saturation is s=1;

v_s = v/n

As the velocity of flow increases with the decrease in the cross-section of flow. Av is small compared to total area A. Hence, v_s is always greater than v.

Relationship Between Seepage Velocity (v_s)& Hydraulic Gradient (i)

From Darcy's law, it is clear that:
v = ki;

The seepage velocity v_s = v/n;

This means seepage velocity is a function of discharge velocity. i.e;

v_s = f(v)

v α i;

Therefore, v_s α i

v_s = ko . i

Here, ko is the coefficient of percolation whose unit is mm/s or cm/sec

Relation Between Coefficient of Percolation & Coefficient of Permeability

v_s = v/n;

ko . i = ki/n;

Which gives,

ko = k/n

Read More: Terzaghi's Bearing Capacity Equation | Bearing Capacity of Soil for Foundation