Effective Stresses and Permeability and Well Hydraulics

where, V_wy = Volume of water yielded under gravity effect and V = total volume of water.

Specific retention

The specific retention of an unconfined aquifer is the ratio of volume of water retained against gravity effect to the total volume of aquifer (v).

where, V_WR = Volume of water retained under gravity effect.

Coefficient of transmissibility

T = kH where, H = Thickness

k = Coefficient of permeability

Unconfined Aquifer

A. Theims Theory

where, q = Rate of flow in m³/s

h₁ = Height of water table of 1^st observation well

h₂ = Height of water table of 2^nd observation well

s₁ = Drawdown of 1^st test well

s₂ = Drawdown of 2^nd test well.

r₁ = and r₂ are radius of 1^st and 2^nd observation wells respectively.

B. Dupits Theory

and S = H - h

Where, S = Drawdown in the well

k = Permeability coefficient in m/s.

R = Radius of influence in 'm'

150m ≤ R ≤ 300m

r = Radius of tes well in 'm'.

Results of dupits theory are not accurate because 'R' is based on empirical relation.

Confined Aquifer

Theims theory where, b = width
Dupits theory

Spherical flow through well

where, r = Radius of well

S = Drawdown

q_s = Rate of flow through spherical well in m³/s

Pumping-In-Test

A. Open end test

where, r = Radius of pipe

h = Head of water above the base of pipe, it may include gravity head and pressure head.

B. Tacker test

…when L > 10 r

where, L = Length of perforated section of pipe

…when L < 10 r

r = Radius of pipe

h = Head of which water is added.

Open well (Recuperation test)

Where,

Volume = A.H A = area of well

C/A = Specific yield or specific capacity of an open well.

T = Time in 'sec'

h₁ = Position of water table of t = 0

h₂ = Position of water table of t = T

Value of Permeability

Compressibility and Consolidation 5

Coefficient of Compressibility (a_v)

e₁ = Void ratio at effective stress

e₂ =Void ratio at effective stress

ΔV = Change in volume in m³, or cm³

V₀ = Initial volume in m³ or cm³.

ΔH = Change in depth in 'm' or 'cm'.

H₀ = original depth in 'm' or 'cm'.

Coefficient of Compression (C_c)

↓

For undisturbed soil of medium sensitivity.

W_L = % liquid limit.

For remolded soil of low sensitivity

For undisturbed soil of medium sensitivity e_{0 = Initial void}ratio

E. For remoulded soil of low sensitivity.

C_c = 1.15(e₀-0.35)

F. C_c = 0.115w where, w = Water content

Over consolidation ratio

O.C.R > 1 For over consolidated soil.

O.C.R = 1 For normally consolidated soil.

O.C.R < 1 For under consolidated soil.

Differential Equation of 1-D Consolidation

where, u = Excess pore pressure.

= Rate of change of pore pressure

C_v= Coefficient of consolidation

= Rate of change of pore pressure with depth.

Coefficient of volume compressibility where, e₀ = Initial void ratio

m_v = Coefficient of volume compressibility

Compression modulus

where, E_c =Compression modulus.

Degree of consolidation

(i) where,

%U = % degree of consolidation.

U = Excess pore pressure at any stage.

U₁ = = Initial excess pore pressure

(ii) where,

e_f = Void ratio at 100% consolidation.

i.e. of t = ∞

e = Void ratio at time 't'

e₀ = Initial void ratio i.e., at t = 0

(iii) where,

ΔH = Final total settlement at the end of completion of primary consolidation i.e.,

at t = ∞

Δh = Settlement occurred at any time 't'.

Time factor

where, T_V = Time factor

C_V= Coeff. of consolidation in cm²/sec.

d = Length of drainage path

t = Time in 'sec'

For 2-way drainage

d = H₀For one-way drainage.

where, H₀ = Depth of soil sample.

(i) if u ≤ 60% T₅₀ = 0.196

(ii) if u > 60%

Method to find 'C_v'

(i) Square Root of Time Fitting Method

where,

T₉₀ = Time factor at 90% consolidation

t₉₀ = Time at 90% consolidation

d = Length of drainage path.

(ii) Logarithm of Time Fitting Method

where, T₅₀ = Time factor at 50% consolidation

t₅₀ = Time of 50% consolidation.

Compression Ratio

(i) Initial Compression Ratio

where, R_i = Initial reading of dial gauge.

R₀ = Reading of dial gauge at 0% consolidation.

R_f = Final reading of dial gauge after secondary consolidation.

(ii) Primary Consolidation Ratio

where, R₁₀₀ = Reading of dial gauge at 100% primary consolidation.

(iii) Secondary Consolidation Ratio

Total Settlement

S = S_i + S_p + S_s where, S_i = Initial settlement

S_p = Primary settlement

S_s = Secondary settlement

(i) Initial Settlement

For cohesionless soil.

where,

where, C_r = Static one resistance in kN/m²

H₀ = Depth of soil sample For cohesive soil.

where, I_t = Shape factor or influence factor

A = Area.

(ii) Primary Settlement

= Settlement for over consolidated stage

= Settlement for normally consolidation stage

(ii) Secondary Settlement

where,

H₁₀₀ = Thickness of soil after 100% primary consolidation.

e₁₀₀ = Void ratio after 100% primary consolidation.

t₂ = Average time after t₁ in which secondary consolidation is calculated

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) Coefficiency of percolation

where, K_P = coefficiency 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

Lioudens 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)

Effective Stress, Capilarity, Seepage

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