Relationship between Discharge Velocity and Seepage Velocity

The relationship between discharge velocity (v) and seepage velocity (vs) can be explained as follows:

1. Definition of Discharge Velocity and Seepage Velocity:
- Discharge velocity (v) is the rate of flow of water through a porous medium per unit cross-sectional area.
- Seepage velocity (vs) is the speed at which water moves through soil or rock due to gravitational force or pressure gradient.

2. Darcy's Law:
- Darcy's law states that the discharge velocity (v) is proportional to the hydraulic gradient (i) and the hydraulic conductivity (k) of the porous medium.
- Mathematically, it can be expressed as v = ki.

3. Relationship between Seepage Velocity and Hydraulic Gradient:
- Seepage velocity (vs) is related to the hydraulic gradient (i) by the equation vs = ki/n, where n is the porosity of the porous medium.
- This equation is derived from Darcy's law by substituting the expression for v in terms of i and k, and simplifying.

4. Comparison of Seepage Velocity and Discharge Velocity:
- From the above equations, it can be seen that seepage velocity (vs) is related to discharge velocity (v) by the equation vs = v/nb.
- This equation implies that seepage velocity is always lower than the discharge velocity, since n and b are both greater than 1.
- Therefore, the relationship between discharge velocity and seepage velocity is that seepage velocity is a fraction of the discharge velocity, and this fraction is equal to 1/nb.

Conclusion:
- The relationship between discharge velocity and seepage velocity is given by vs = v/nb, where n is the porosity and b is the thickness of the porous medium.
- This relationship can be derived from Darcy's law and the definition of seepage velocity.
- Seepage velocity is always lower than discharge velocity, since it represents the speed at which water moves through the pores of the medium, while discharge velocity represents the overall rate of flow through the medium.