Effective Stresses and Permeability and Well Hydraulics

By Sachin Singh|Updated : October 28th, 2016

Well Hydraulics

Specific yield (Sy)

The specific yield of an unconfined aquifer is the ratio of volume of water which will flow under saturated condition due to gravity effect to the total volume of aquifer (v).

image001 where, Vwy = 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).

image002 where, VWR = 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 1st observation well

h2 = Height of water table of 2nd observation well

s1 = Drawdown of 1st test well

s2 = Drawdown of 2nd test well.

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

B. Dupits Theory


image008 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 image012 where, b = width
  • Dupits theory image013

Spherical flow through well

image014 where, r = Radius of well

S = Drawdown

qs = Rate of flow through spherical well in m3/s image015



A. Open end test

image017 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

image018 …when L > 10 r

where, L = Length of perforated section of pipe

image019 …when L < 10 r

r = Radius of pipe

h = Head of which water is added.

Open well (Recuperation test)


Where, image021

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 (av)


e1 = Void ratio at effective stress image026

e2 =Void ratio at effective stress image027



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 (Cc)

A. image032




B. image034  

For undisturbed soil of medium sensitivity.

WL = % liquid limit.

C. image035

For remolded soil of low sensitivity

D. image036

For undisturbed soil of medium sensitivity e0 = Initial void ratio


E. For remoulded soil of low sensitivity.

Cc = 1.15(e0-0.35)

F. Cc = 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

image040 where, u = Excess pore pressure.

image041 = Rate of change of pore pressure

Cv = Coefficient of consolidation

image042 = Rate of change of pore pressure with depth.

Coefficient of volume compressibility image043 where, e0 = Initial void ratio

mv = Coefficient of volume compressibility

Compression modulus

image044 where, Ec =Compression modulus.

Degree of consolidation

(i) image045 where,

%U = % degree of consolidation.

U = Excess pore pressure at any stage.

U1 = image046 = Initial excess pore pressure

at image047

at image048

(ii) image049 where,

ef = 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) image050 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

image051 where, TV = Time factor

CV = Coeff. of consolidation in cm2/sec.

d = Length of drainage path

t = Time in 'sec'

image052 For 2-way drainage

d = H0 For one-way drainage.

where, H0 = Depth of soil sample.

(i) image053 if u ≤ 60% T50 = 0.196

(ii) image054 if u > 60%

Method to find 'Cv'

(i) Square Root of Time Fitting Method

image055 where,

T90 = Time factor at 90% consolidation

t90 = Time at 90% consolidation

d = Length of drainage path.

(ii) Logarithm of Time Fitting Method

image056 where, T50 = Time factor at 50% consolidation

t50 = Time of 50% consolidation.

Compression Ratio

(i) Initial Compression Ratio


where, Ri = Initial reading of dial gauge.

R0 = Reading of dial gauge at 0% consolidation.

Rf = 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 = Si + Sp + Ss where, Si = Initial settlement

Sp = Primary settlement

Ss = Secondary settlement

(i) Initial Settlement


For cohesionless soil.

where, image062

where, Cr = Static one resistance in kN/m2

H0 = Depth of soil sample image063 For cohesive soil.

where, It = Shape factor or influence factor

A = Area.

(ii) Primary Settlement

  • image064
  • image065
  • image066
  • image067image068 

image069 = Settlement for over consolidated stage

image070 = Settlement for normally consolidation stage


(ii) Secondary Settlement


where, image073

100 = 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 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

image076 where, HL = Head loss = (H1 – H2)


(ii) Seepage velocity

image078 where, Vs = Seepage velocity (m/sec)

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

(iii) Coefficiency of percolation

image079 where, KP = coefficiency of percolation and n = Porosity.

Constant Head Permeability Test

image080 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 = image086

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)

image089 = unit weight of water in kN/m3


Allen Hazen Equation

image091 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 image093

Where, Cv = Coefficient of consolidation in cm2/sec

mv = Coefficient of volume Compressibility in cm2/N

Capillary Permeability Test


image095 where, S = Degree of saturation

K = Coefficient of permeability of partially saturated soil.


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

image099 = head under first set of observation,

n = porosity, hc = capillary height

Another set of data gives,


image099 = head under second set of observation

  • For S = 100%, K = maximum. Also, ku ∝ 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 image107

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

PS = ω where, Ps = Seepage pressure

γω = 9.81 kN/m3

Here, h = head loss and z = length

(ii) FS = hAγω where, Fs = Seepage force

(iii) image122 where, fs = 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

image129 where, ic = Critical hydraulic gradient.

2.65 ≤ G≤ 2.70 0.65 ≤ e ≤ 0.70

  • To Avoid Floating Condition

image130 and image131

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

(i) image132

where, ∅ = Potential function = kH

H = Total head and k = Coefficient of permeability

(ii) image133 … 2D Laplace equation for Homogeneous soil.

where, ∅ = kX H and ∅ = ky H for Isotropic soil, kx= ky

Seepage discharge (q)

image134 where, h = hydraulic head or head difference between upstream and downstream level or head loss through the soil.

  • Shape factor = image135
  • image136

where, Nf = Total number of flow channels

image137 = Total number of flow lines.

  • image138

where, Nd = Total number equipotential drops.

image139 = Total number equipotential lines.

  • Hydrostatic pressure = U = image140

where, U = Pore pressure hw = Pressure head

hw = Hydrostatic head – Potential head

  • Seepage Pressure

Ps = h'γw where, image142


  • Exit gradient,

image144 where, size of exit flow field is b x b.

and image145 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 = image148

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) image149

(iii) image150

(b) Phreatic Line without Filter


(i) For ∝ < 30°

q = k a sin2 ∝ image152

(ii) For ∝ > 30°

q = k a sin ∝ tan ∝ and image153


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