Soil Mechanics & Foundation Engg. : Sepage stress and permeabilty of soil

Seepage & Permeability of Soil

Seepage Pressure and Seepage Force

Seepage pressure is exerted by the water on the soil due to friction drag. This drag force/seepage force always acts in the direction of flow.

The seepage pressure is given by

P_S = hγ_ω where, P_s = Seepage pressure

γ_ω = 9.81 kN/m³

Here, h = head loss and z = length

(ii) F_S = hAγ_ω where, F_s = Seepage force

(iii)  where, f_s = Seepage force per unit volume.

i = h/z where, I = Hydraulic gradient.

Quick Sand Condition

It is condition but not the type of sand in which the net effective vertical stress becomes zero, when seepage occurs vertically up through the sands/cohesionless soils.

Net effective vertical stress = 0

where, i_c = Critical hydraulic gradient.

2.65 ≤ G≤ 2.70 0.65 ≤ e ≤ 0.70

• To Avoid Floating Condition

 and

Laplace Equation of Two Dimensional Flow and Flow Net: Graphical Solution of Laplace Equation

(i) 

where, ∅ = Potential function = kH

H = Total head and k = Coefficient of permeability

(ii)  

… 2D Laplace equation for Homogeneous soil.

where, ∅ = k_X H and ∅ = k_y H for Isotropic soil, k_x= k_y

Seepage discharge (q)

where, h = hydraulic head or head difference between upstream and downstream level or head loss through the soil.

• Shape factor = 
• 

where, N_f = Total number of flow channels

 = Total number of flow lines.

where, N_d = Total number equipotential drops.

 = Total number equipotential lines.

• Hydrostatic pressure = U = 

where, U = Pore pressure h_w = Pressure head

h_w = Hydrostatic head – Potential head

• Seepage Pressure

P_s = h'γ_w where, 

• Exit gradient,

where, size of exit flow field is b x b.

and  is equipotential drop.

Phreatic Line

It is top flow line which follows the path of base parabola. It is a stream line. The pressure on this line is atmospheric (zero) and below this line pressure is hydrostatic.

(a) Phreatic Line with Filter

Phreatic line (Top flow line).

 ↓

Follows the path of base parabola

CF = Radius of circular arc = 

C = Entry point of base parabola

F = Junction of permeable and impermeable surface

S = Distance between focus and directrix

= Focal length.

FH = S

(i) q = ks where, q = Discharge through unit length of dam.

(ii) 

(iii) 

(b) Phreatic Line without Filter

(i) For ∝ < 30°

q = k a sin² ∝ 

(ii) For ∝ > 30°

q = k a sin ∝ tan ∝ and 

Permeability

Permeability of Soil

The permeability of a soil is a property which describes quantitatively, the ease with which water flows through that soil.

Darcy's Law : Darcy established that the flow occurring per unit time is directly proportional to the head causing flow and the area of cross-section of the soil sample but is inversely proportional to the length of the sample.

(i) Rate of flow (q)

Where, q = rate of flow in m³/sec.

 K = Coefficient of permeability in m/s

 I = Hydraulic gradient

 A = Area of cross-section of sample

where, H_L = Head loss = (H₁ – H₂)

(ii) Seepage velocity

where, V_s = Seepage velocity (m/sec)

n = Porosity & V = discharge velocity (m/s)

(iii) Coefficient of percolation

where, K_P = coefficient of percolation and n = Porosity.

Constant Head Permeability Test

 where, Q = Volume of water collected in time t in m³.

Constant Head Permeability test is useful for coarse grain soil and it is a laboratory method.

Falling Head Permeability Test or Variable Head Permeability Test

a = Area of tube in m²

A = Area of sample in m²

t = time in 'sec'

L = length in 'm'

h₁ = level of upstream edge at t = 0

h₂ = level of upstream edge after 't'.

Konzey-Karman Equation

Where, C = Shape coefficient, ∼5mm for spherical particle

S = Specific surface area = 

For spherical particle.

R = Radius of spherical particle.

When particles are not spherical and of variable size. If these particles passes through sieve of size 'a' and retain on sieve of size 'n'.

e = void ratio

μ = dynamic viscosity, in (N-s/m²)

 = unit weight of water in kN/m³

Allen Hazen Equation

Where, D₁₀ = Effective size in cm. k is in cm/s C = 100 to 150

Loudens Equation

Where, S = Specific surface area

 n = Porosity.

a and b are constant.

Consolidation equation 

Where, C_v = Coefficient of consolidation in cm²/sec

m_v = Coefficient of volume Compressibility in cm²/N

Capillary Permeability Test

where, S = Degree of saturation

K = Coefficient of permeability of partially saturated soil.

where h_c = remains constant (but not known as depends upon soil)

 = head under first set of observation,

n = porosity, h_c = capillary height

Another set of data gives,

 = head under second set of observation

• For S = 100%, K = maximum. Also, k_u ∝ S.

Permeability of a stratified soil

(i) Average permeability of the soil in which flow is parallel to bedding plane,

(ii) Average permeability of soil in which flow is perpendicular to bedding plane.

(iii) For 2-D flow in x and z direction

(iv) For 3-D flow in x, y and z direction 

Coefficient of absolute permeability (k₀)