### Well Hydraulics

#### Specific yield (S_{y})

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).

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 m^{3}/s

h_{1} = Height of water table of 1^{st} observation well

h_{2} = Height of water table of 2^{nd} observation well

s_{1} = Drawdown of 1^{st} test well

s_{2} = Drawdown of 2^{nd} test well.

r_{1} = and r_{2} 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**

**Spherical flow through well**

where, r = Radius of well

S = Drawdown

q_{s} = Rate of flow through spherical well in m^{3}/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'

h_{1} = Position of water table of t = 0

h_{2} = Position of water table of t = T

**Value of Permeability**

### Compressibility and Consolidation 5

#### Coefficient of Compressibility (a_{v})

e_{1} = Void ratio at effective stress

e_{2} =Void ratio at effective stress

Δ*V* = Change in volume in m^{3}, or cm^{3}

V_{0} = Initial volume in m^{3} or cm^{3}.

Δ*H* = Change in depth in 'm' or 'cm'.

H_{0} = original depth in 'm' or 'cm'.

**Coefficient of Compression (C _{c})**

**A. **

↓

B.

For undisturbed soil of medium sensitivity.

W_{L} = % liquid limit.

C.

For remolded soil of low sensitivity

D.

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

E. For remoulded soil of low sensitivity.

*C _{c}* = 1.15(e

_{0}-0.35)

*F. C _{c}* = 0.115

*w*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, e_{0} = 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.

U_{1} = = Initial excess pore pressure

at

at

(ii) where,

e_{f} = Void ratio at 100% consolidation.

i.e. of t = ∞

e = Void ratio at time 't'

e_{0} = Initial void ratio i.e., at t = 0

(iii) where,

Δ*H *= Final total settlement at the end of completion of primary consolidation i.e.,

at t = ∞

Δ*h* = Settlement occurred at any time 't'.

**Time factor**

where, T_{V} = Time factor

C_{V }= Coeff. of consolidation in cm^{2}/sec.

d = Length of drainage path

t = Time in 'sec'

For 2-way drainage

*d* = *H*_{0 }For one-way drainage.

where, H_{0} = Depth of soil sample.

(i) if u ≤ 60% T_{50} = 0.196

(ii) if u > 60%

#### Method to find 'C_{v}'

**(i) Square Root of Time Fitting Method**

where,

T_{90} = Time factor at 90% consolidation

t_{90} = Time at 90% consolidation

d = Length of drainage path.

**(ii) Logarithm of Time Fitting Method**

where, T_{50} = Time factor at 50% consolidation

t_{50} = Time of 50% consolidation.

#### Compression Ratio

**(i) Initial Compression Ratio**

where, R_{i} = Initial reading of dial gauge.

R_{0} = Reading of dial gauge at 0% consolidation.

R_{f} = Final reading of dial gauge after secondary consolidation.

**(ii) Primary Consolidation Ratio**

where, R_{100} = 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/m^{2}

H_{0} = 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,

H_{100} = Thickness of soil after 100% primary consolidation.

e_{100} = Void ratio after 100% primary consolidation.

t_{2} = Average time after t_{1} 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 m^{3}/sec.

K = Coefficient of permeability in m/s

I = Hydraulic gradient

A = Area of cross-section of sample

where, H_{L} = Head loss = (H_{1} – H_{2})

**(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 m^{3}.

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 m^{2}

A = Area of sample in m^{2}

t = time in 'sec'

L = length in 'm'

h_{1} = level of upstream edge at t = 0

h_{2} = 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/m^{2})

= unit weight of water in kN/m^{3}

^{}

**Allen Hazen Equation**

Where, D_{10} = 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 cm^{2}/sec

m_{v} = Coefficient of volume Compressibility in cm^{2}/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 (k _{0})**

### 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*where, P

_{ω}_{s}= Seepage pressure

*γ _{ω }*= 9.81 kN/m

^{3}

Here, h = head loss and z = length

(ii) *F _{S}* =

*hAγ*where, F

_{ω}_{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 sin^{2} ∝

(ii) For ∝ > 30°

q = k a sin ∝ tan ∝ and

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