Relation between Discharge velocity and Seepage velocity in soil mass

Before I explain to you the relation between discharge velocity and seepage velocity in soil mass you must have some knowledge of permeability and percolation of soil.
Because there is the use of the term coefficient of permeability and percolation in derivation. So let me make you clear about these terms.
The phenomena in which the movement and filtering of fluids occur through porous materials are known as percolation.
The movement of rainwater down on a slope under the earth’s surface is the example of percolation. It is denoted by (Kp).

What is permeability?

The property of soil by virtue of which the soil mass allows water (or any fluid) to flow through it is known as permeability.
It is very important to calculate its value to determine for the study of soil engineering problems involving the flow of water through soil, such as seepage through the body of earth dam and settlement of foundations. It is denoted by (k).
Relation between Discharge velocity and Seepage velocity in soil mass
Fig:- Discharge and Seepage velocity

Now, let’s derive the relation. The relation between discharge velocity and seepage velocity will be initiated from Darcy’s law.

Relation between discharge velocity and seepage velocity

According to Darcy’s law, for laminar flow condition, the velocity of flow (v) is directly proportional to the hydraulic gradient (i).

              i.e v ∝ i
Therefore, v = K X i
The proportionality constant, k between v and i is called Darcy’s coefficient of permeability.
When ( i=1 ), we have (K=v).
Thus, the coefficient of permeability can also be defined as the velocity of flow through soil under a unit hydraulic gradient and has the same unit as that of velocity.
It is usually expressed in mm/sec, m/hr, m/day.
Further, If (q) be the rate of flow or discharge per unit time and (A) be the area of the cross-section of flow perpendicular to the direction of flow then we have,
q = A X v
                   = (A X K X i)
The velocity of flow of water, (v) through soil mass is obtained from Darcy’s law assuming that the flow takes place through the total cross-sectional area, (A) of soil mass perpendicular to the direction of flow.
This velocity is referred to as discharge velocity or theoretical velocity.
Thus, discharge velocity,
v = q/A
  Again, The total area (A) is composed of area of voids, (Av ) and area of solids, (As) But flow can take place only through an area of voids, (Av).
The actual velocity of flows referred to as seepage velocity and denoted by (Vs) is thus greater than the theoretical velocity obtained from Darcy’s law.
Seepage velocity,
Vs = q/(Av)
So,  q = Av = Av X Vs
Vs = V X (A/Av) X (L/L)             [multiply by (L) to both Area]
  = V X {(V/Vv)}
       Vs = V / n

  So, this is the required relation.

      Where (n) is the porosity of soil mass.
V = K i
Or, Vs = Kp i
Kp is referring to as the coefficient of percolation.
(Vs/V) = (Kp^2 / K i)
= (Kp/ Ki)
Vs/V = 1/ n
Kp/K = 1/n
           Or, Kp = (k/n)
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