Derive and explain Darcy’s Law in light of its application in design of foundation for the discharge of water through the soil sample.
Solution
ANSWER
Darcy’s law;
Darcy’s law states the
principle which governs the movement of fluid in the given substance. Darcy’s
law equation that describes the capability of the liquid to flow via any porous
media like a rock. The law is based on the fact according to which, the flow
between two points is directly proportional to the pressure differences between
the points, the distance, and the connectivity of flow within rocks between the
points. Measuring the inter-connectivity is known as permeability.
Darcy's
law demonstrat experimently that for leminar flow condition in a saturated soil
mass, the velocity of flow is directly
proportional to hydraulic gradients
i.e V ∝ i
Or V = ki
Where
K is coefficient of permeability
I
is hydraulic gradient
I = Δh/ L
Δ h = change
in head b.w two observation points.
L= length b.w the two points
Now considering applipcation in the design of foundation the equation for discharge can be derived from a flow net as
Δ q = k× Δh/ L
Δh = H/Nd where Nd is number of potential drops
Δ q= Kh/ L × 1/Nd ×b
Or
Δ q= kH/Nd × (b/L)
The
total discharge through flow net will be
I= h1
-h2 /L
Or
v=k
(h1 -h2 ) / L
Now discharge of water through the soil sample
q=v
× area= v×A
or
q=
k (h1-h2) / L ×L
q
= Nf × Δ
q
=
Nf × k H/Nd × b/L
q
= kH Nf/Nd × b/L
The
field is approximate square b= L
q = kH Nf/Nd
or
Total
head = P/w + v²/2g +z
piezometric head = p/w +z
In soil mechanics v²/2g is neglected as it is very small
Flow occurs only when there is difference in total head b.w 2point
Total
head difference is H1 -H2
= hf
hf= H1 -H2
Total
head loss b.w 2 poits in a soil is equal
to the difference in the elevation of water level in the 2 piezometer kept @ those 2
poits
Seepage
length = L
hydraulic
gradient = h/L (Head loss per unit
length)
c/s of section x-x
c/s area of soil = A = b×d
Mutual
area of flow = Av = nA ( AREA of
voids)
discharge velocity/apparent velocity of flow =
Q/A
seepage velocity/ actual velocity Vs = Q/Av
Q=
AV =Av ×Vs
Vs
= A/Av (v)
Vs = v/n Vs > V
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