In laboratory or field experiments for the determination of permeability, usually the total cross-sectional area of the soil A is used for convenience. The velocity of water obtained using the total cross-sectional area of the soil is known as discharge velocity or superficial velocity.
Since water flows only through voids and not through the total cross-sectional area, the actual velocity of water is much higher than the discharge velocity and that is known as seepage velocity (vs). By continuity equation used in fluid mechanics –
q = vA = vs × Av …(10.10)
where vs is the seepage velocity and Av is the area of voids in the cross-section. From Eq. (10.10), we have vs = v A/Av. We know that –
(Av/A) = [(Av x L)/(A x L)] = (Vv/V) = n
where Vv is the volume of voids, V is the total volume of the soil, and n is the porosity of the soil. Hence –
vs = v/n …(10.11)
Seepage velocity is also proportional to the hydraulic gradient –
vs = kp × i …(10.12)
where kp is the coefficient of percolation. Dividing Eq. (10.12) with Eq. (9.1), we have –
Seepage velocity is always more than discharge velocity. The coefficient of percolation is always more than the coefficient of permeability.
Seepage velocity or absolute velocity is also not the true velocity of water, but only an average velocity over the given length over which a hydraulic gradient is considered and the area from which discharge is considered.
It should be noted that the voids are not regular, uniform, or continuous in the flow direction and hence the actual velocity changes in both magnitude and direction from one point to another point in the soil. Water, when flowing in the forward direction, has to change direction by as much as 90°, depending on the availability of the free void space in the succeeding forward points.
The size of the void space and hence the area of flow is also not exactly uniform at different points. Also, water has to flow past solid particles, during which there will be head loss due to frictional resistance. The magnitude of resistance to flow by friction is again variable from point to point in the soil mass, depending on particle size, shape, and texture.
The total head causing the flow at any point in a soil mass is equal to the sum of pressure head, kinetic (velocity) head, and datum head –
h = hp + hv + hd …(10.14)
The pressure head can be measured by inserting a piezometer at the required point, and it is equal to the height of the water above the point in the piezometer. The kinetic or velocity head, given by v2/2g, is very small and hence is usually neglected because the velocity of flow of water through the soil is usually small. The datum head at any point is the elevation of the point above the reference line or datum.
Thus, the total head of water is equal to the sum of the pressure head and the elevation head. For flow problems through the soil, the water level on the downstream (d/s) side of the hydraulic structure is usually taken as the datum.
The pressure exerted by the flowing water on the soil due to viscous friction between the water and soil particles is known as seepage pressure. Mathematically, it is given by –
Seepage pressure, σs = γwh …(10.15)
where h is the hydraulic head lost due to the viscous friction and γw is the density of water. Seepage pressure always acts in the direction of flow. When the flow takes place in the vertical direction –
σs = γwh = γw (h/z). z = γw.i.z …(10.16)
where i is the hydraulic gradient. The vertical effective pressure may be increased or decreased due to the seepage, depending upon whether the flow takes place in the downward or the upward direction, respectively.
Effective pressure, σ’ = γ’z ± γw iz …(10.17)
When the flow takes place in the upward direction, the seepage pressure therefore acts in the upward direction and the effective pressure is reduced. If the flow takes place at high hydraulic gradient, the net effective pressure, defined by Eq. (10.17), is reduced to zero.
In such a case, a cohesionless soil mass becomes weightless and cannot support any load, because its shear strength is reduced to zero. This phenomenon of loss of shear strength of soil in the upward flow condition in cohesionless soils is known as the quick condition, boiling condition, or quicksand.
Quicksand is not a type of sand but a flow condition occurring in cohesionless soils when its effective pressure is reduced to zero due to seepage pressure during an upward flow of water. Thus, under quick condition –
The hydraulic gradient during the quick condition is called critical hydraulic gradient. Thus, critical hydraulic gradient, where the quick condition occurs, is –
ic (G – 1)/(1+ e) …(10.18)
For loose deposits of sand or silt, if the void ratio is taken as 0.67 and G is 2.67, the critical hydraulic gradient works out to be unity.