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.
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
v α iv = 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’.
Let’s consider the
pressure head and elevation head (potential head) at A and B as PHA,
ZA and PHB, ZB 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:
THA
= PHA + ZA
THB
= PHB + ZB
and, THA > THB
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 = THA - THB
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 (vs) 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 (vs).
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 vs.
From the equation of continuity:
A1v1 = A2v2
Av = Avvs
vs = [A/Av ] v = [(A . L )/(Av
. L) ] v [ Multiplying and dividing by ‘L’]
A.L = V (Volume); Av . L = Vv (Volume of Voids)
vs = (V/VV) v
During seepage, the voids is filled with water.
Hence, volume of voids Vv = Volume of water Vw;
vs = (V/Vw ) v
vs = [ (V/Vv) . (Vv/Vw) ]v
Vv/V = porosity (n) ; Vw/Vv = Degree of Saturation
(s)
That gives,
vs = v/ns;
When the soil is saturated, the degree of
saturation is s=1;
vs = 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, vs is always greater than v.
Relationship Between Seepage Velocity (vs)& Hydraulic Gradient (i)
From Darcy's law, it is clear that:v = ki;
vs = ko . i
Relation Between Coefficient of Percolation & Coefficient of Permeability
ko = k/n