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
v α iv = ki
Soil Type |
Coefficient
of Permeability (k) mm/sec |
Drainage
Properties |
Clean gravel |
10^{1} 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 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.
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.
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.
v
α i
v
= ki
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:
A_{1}v_{1} = A_{2}v_{2}
Av = A_{v}v_{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.
v_{s} = ko . i
ko = k/n
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