Derive and explain Darcy’s Law in light of its application in design of foundation for the discharge of water through the soil sample.

Q#:Derive and explain Darcy’s Law in light of its application in design of foundation for the discharge of water through the soil sample.     

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 =  h_f

                                                          h_f= H_{1 -}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