3.1. One Dimensional Flow

One dimensional flow - Darcy’s law

where,

Q : volumetric water flow rate (volume/time units)

k : coefficient of permeability (or hydraulic conductivity) (length/time units)

h₁: total hydraulic head in the entrance of the cylinder (length units)

h₂: total hydraulic head in the exit of the cylinder (length units)

L: length of the cylinder (length units)

Α: cross sectional area of the flow [(length)² units]

i = (h₁-h₂) / L : the hydraulic gradient

h₁-h₂: hydraulic head difference (length units)

The total hydraulic head has two components: a) the elevation head, h_e, is the height of a point from a random reference level and b) the piezometric head, h_p, is the water pressure at this point divided by the unit weight of water

Flow velocity :

Infiltration rate :

One dimensional flow in stratified soil

i) Flow vertical to bedding planes

ii) Flow parallel to bedding planes

 

Permeameters

1) Constant Head Permeameter

2) Falling Head Permeameter

ho: initial hydraulic head

h: hydraulic head after t time

ln: natural logarithm

Seepage force

As the water flows through the voids, it creates forces that are applied on the soil particles in the direction of its motion.

j: seepage force per unit volume

i: hydraulic gradient of the flow

γ_w: unit weight of water

When the water flows through soil, seepage forces must be taken into account for the calculation of the effective stresses.

When water flows upwards in soil, there is a critical hydraulic gradient value where the weight of the soil particles is counteracted by the seepage force. In that case, soil behaves like a viscous liquid and this phenomenon is called quicksand. The critical hydraulic gradient is given by the equation:

where,

γ_αν: unit weight of buoyancy

γ_w: unit weight of water

Water Temperature Effect in the Coefficient of Permeability

K_20℃= permeability factor at temperature of 20℃

K_T℃ = permeability factor at temperature of T℃

n_20℃ = water viscosity at temperature of 20℃

n_T℃  = water viscosity at temperature of T℃

The diagrams below show the values of the fraction n_T℃  / n_20℃ as a function of temperature that varies from T=12℃ to T=30℃

Empirical Relationships for the Calculation of the Coefficient of Permeability

1) For loose, clean (without fines) sand (Hazen,1930):

k (cm/sec) = c * D²₁₀

where:

c = a constant that varies from 1.0 to 1.5

D₁₀= effective grain size in mm

2) For fine to medium, clean sand (Casagrande, 1967)

 k = 1.4 * e² * k_0.85

where:

k = permeability factor for e void ratio 

k_0.85 = permeability factor for e void ratio equal to 0.85

3) The Kozeny-Carman equation for sandy soils:

Where:

c₁ = a constant

It has been found that: C₁ = C₂ * D^2.32₁₀ * Cu^0.6 

where:

D₁₀ = effective grain size

Cu = uniformity coefficient

C₂ = a constant

4) For normally consolidated clays (1982):

where C₃, n = constants