Permeability
Permeability of soil
Q: what is permeability? Explain Darcy’s law.
Permeability is the property of soil which permits seepage through interconnecting voids.
Darcy’s Law:
Darcy's law is an equation that describes the flow of a fluid through a porous medium.
The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of soil.
Statement: In saturated soil, for laminar flow conditions, rate of flow is directly proportional to hydraulic gradient.
v=qA=k i
Where, v is velocity of flow, q is discharge per unit time, A is total cross sectional area, i is hydraulic gradient, k is Darcy’s coefficient of permeability
If Q is quantity of water flows in time ‘t’ then
q=Qt
q=A k i=Qt
k =Qt 1A 1i
here i is hydraulic gradient for length of soil sample L and hydraulic head h,
i = h/L
thus
k =Qt Lh 1A
Validity of Darcy’s law
Darcy's law is valid for laminar flow. In fine-grained sediments, the dimensions of interstices are small and thus flow is laminar. Coarse-grained sediments also behave similarly but in very coarse-grained sediments the flow may be turbulent. Hence Darcy's law is not always valid in such sediments.
Q: Explain factors affecting permeability
Following are factors effecting permeability of soils.
Size of Soil Particle
Permeability varies according to size of soil particle.
If the soil is coarse grained, permeability is more and if it is fine grained, permeability is low.
The relation between coefficient of permeability (k) and particle size (D) can be shown from equation (1) as follows.
k α D102
k=100 D102
This formula is known as Allen Hazen formula
Specific Surface Area of Particles
Specific surface area of soil particles also effects the permeability. Higher the specific surface area lower will be the permeability.
k α 1Specific surface area
Shape of Soil Particle
Rounded Particles will have more permeability than angular shaped.
It is due to specific surface area of angular particles is more compared to rounded particles.
Void Ratio
In general, Permeability increases with void ratio.
But it is not applicable to all types of soils. For example, Clay has high void ratio than any other types of soil but permeability for clays is very low.
This is due to, the flow path through voids in case of clays is extremely small such that water cannot permit through this path easily.
The relation between coefficient of permeability and void ratio can be expressed from equation as below
For Clay, k α C e31+e
Where, C = Shape of the flow path, e = Void ratio.
For coarse grained soil, “C” can be neglected. Hence
k α e31+e
Soil Structure
Structure of any two similar soil masses at same void ratio need not be same.
It varies according to the level of compaction applied.
If a soil contains flocculated structure, the particles are in random orientation and permeability is more in this case.
If the soil contains dispersed structure, the particles are in face to face orientation hence, permeability is very low. The permeability of stratified soil deposits also varies according to the flow direction.
If the flow is parallel, permeability is more.
If it is perpendicular, permeability is less.
Degree of Saturation
Partially saturated soil contains air voids which are formed due to entrapped air or gas released from the percolating fluid or water.
This air will block the flow path thereby reduces the permeability.
Fully saturated soil is more permeable than partially saturated soil.
Water Properties/Properties of pore fluid
Various properties of water or fluid such as unit weight and viscosity also effects the permeability.
However, unit weight of water will not affect much since it does not change much with temperature.
But when temperature is increased viscosity decreases rapidly. From equation, permeability increase when viscosity decreases.
k α w
Temperature
Temperature also affects the permeability in soils.
From equation, permeability is inversely proportional to the viscosity of the fluid.
It is known that viscosity varies inversely to the temperature.
Hence, Permeability is directly related to temperature.
Greater the temperature, higher will be the permeability.
That is the reason, seepage is more in summer seasons than in winter.
k α 1 α temperature
Adsorbed Water
Adsorbed water is the water layer formed around the soil particle especially in the case of fine-grained soils.
This reduces the size of the void space by about 10%. Hence, permeability reduces.
Organic matter
Presence of organic matter decreases the permeability.
This is due to blockage of voids by the organic matter.
Q: Explain constant head laboratory test
Q: Explain laboratory method for determination of coefficient of permeability for coarse grained soil with equation and sketch
Objective and Scope: The objective of constant head permeability test is to determine the coefficient of permeability of a coarse grained soil.
Apparatus: Permeameter mould, internal diameter = 100mm, effective height =127.3 mm, capacity = 1000ml.Detachable collar, porous disc, Compaction equipment, Constant head water supply reservoir, Constant head collecting chamber, Stop watch, Filter paper.
Specimen Preparation
Remove the collar of the mould. Measure the internal dimensions of the mould. Weigh the mould with dummy plate to the nearest gram.
Apply a little grease on the inside to the mould. Clamp the mould between the base plate and the extension collar and place the assembly on a solid base.
Take about 2.5kg of the soil sample, from a thoroughly mixed wet soil, in the mould. Compact the soil at the required dry density using a suitable compacting device.
Take a small specimen of the soil in a container for the water content determination.
Remove the collar and base plate. Trim the excess soil level with the top of the mould.
Clean the outside of the mould and the dummy plate. Find the mass of the soil in the mould.
The mould with the sample is now placed over the permeameter. This will have drainage and cap discs properly saturated
Test Procedure
Through the top inlet of the constant head reservoir, the specimen is connected.
The bottom outlet is opened and a steady flow is established
For a particular time interval, the quantity of flow can be collected.
Measure the difference of head (h) in levels between the constant head reservoir and the outlet in the base.
For the same interval, this is repeated three times.
Formula
k= Qt Lh 1A
Where, Q is total quantity of flow, t is time for discharge, L is length of soil sample, h is head causing flow, A is cross sectional area of soil sample,
Q: Explain variable/falling head laboratory test
Q: Explain laboratory method for determination of coefficient of permeability for fine grained soil with equation and sketch
Objective and Scope: Variable head permeability test is one of several techniques by which the permeability of soil is determined. It is used to evaluate the permeability of fairly less previous soil. Permeability is the measure of the ability of soil to allow water to flow its pores or voids.
Apparatus: Permeameter mould, internal diameter = 100mm, effective height =127.3 mm, capacity = 1000ml.Detachable collar, porous disc, Compaction equipment, Constant head water supply reservoir, Constant head collecting chamber, Stop watch, Filter paper, stand pipe.
Specimen Preparation
Remove the collar of the mould. Measure the internal dimensions of the mould. Weigh the mould with dummy plate to the nearest gram.
Apply a little grease on the inside to the mould. Clamp the mould between the base plate and the extension collar and place the assembly on a solid base.
Take about 2.5kg of the soil sample, from a thoroughly mixed wet soil, in the mould. Compact the cohesive soil at the required dry density using a suitable compacting device.
Take a small specimen of the soil in a container for the water content determination.
Remove the collar and base plate. Trim the excess soil level with the top of the mould.
Clean the outside of the mould and the dummy plate. Find the mass of the soil in the mould.
The mould with the sample is now placed over the permeameter. This will have drainage and cap discs properly saturated
Test Procedure
Connect the specimen to the selected stand-pipe through the top inlet.
Open the bottom outlet and record the time interval required for the water level to fall from a known initial head to a known final head as measured above the center of the outlet.
Refill the stand-pipe with water and repeat the test till three successive observations give nearly same time interval; the time intervals being recorded for the drop in head from the same initial to final values, as in the first determination.
Alternatively, after selecting the suitable initial and final heads, h1, and h2, respectively, observe the time intervals for the head to fall from h1 to (h1*h2)^0.5, and similarly from (h1*h2)^0.5 to h2.
The time intervals should be the same; otherwise the observation shall be repeated after refilling the stand-pipe.
Formula
k=2.3 a LA t h1h2
Where, L is length of soil sample, h is head causing flow, A is cross sectional area of soil sample, a is cross sectional area of stand pipe, h1 & h2 are top and bottom heads measured on stand pipe in time ‘t’
Q: Explain flow net. Derive Laplace equation in two dimensional flow used for flow net construction.
Flownet
A flownet is a graphical representation of two-dimensional steady-state groundwater flow through aquifers.
Construction of a flownet is often used for solving groundwater flow problems where the geometry makes analytical solutions impractical.
The method is often used in soil mechanics as a first check for problems of flow under hydraulic structures like dams or sheet pile walls.
As such, a grid obtained by drawing a series of equipotential lines is called a flownet.
The flownet is an important tool in analyzing two-dimensional seepage/flow problems.
Laplace’s Equation:
The equation is used in the construction of flow net.
Assumptions:
1. Flow is two dimensional.
2. Water and soil are incompressible.
3. Soil is isotropic and homogeneous.
4. The soil is fully saturated.
5. The flow is steady, i.e., flow conditions do not change with time.
6. Darcy’s law is valid.
Let us consider an element of soil of size dx, dz through which flow is taking place. The third dimension along y-axis is large. For convenience, it is taken as unity.
As the flow is steady and the soil is compressible, the discharge entering the element is equal to that leaving the element.
Equation (1) is the continuity equation for two-dimensional flow.
Let h be the total head at any point. The horizontal and vertical component of the hydraulic gradient are, respectively.
Equation (2) is the Laplace’s equation in terms of head h. Sometimes, the Laplace’s equation can be represented in terms of velocity potential ɸ, given by –
the Laplace’s Eqn. 3 in terms of velocity potential.
Laplace’s Eqn. 3 can be solved if the boundary conditions at the inlet and exit are known.
Properties of Flow net:
Flow lines and equal potential lines intersect each other at 90 degrees.
The areas bounded by the flow lines and equal potential lines form approximate squares.
Flow nets must satisfy the boundary conditions of flow field.
Quantity of water flowing through each flow channel is the same.
The potential drop in any two consecutive equal potential lines is same/constant.
Flow lines and equal potential lines are smooth curves.
Flow lines do show refraction at the interface between two soils having different coefficient of permeability.
Applications of flownet
Determination of Seepage discharge
Determination of seepage pressure
Determination of seepage velocity (& discharge velocity)
Determination of pore pressure
Determination of exit gradient
Determining Seepage Discharge:
Let b and L be the dimensions of the field and Δh be the head drop through this field.
Also, let Δq represent the discharge passing through the flow channel, per unit length of structure (perpendicular to paper).
Then from Darcy’s law, we can write –
Where k = permeability of the soil.
Δh = head loss in the field, i.e., potential drop between two equipotential forming the field.
b x 1 = cross-sectional area of the field, considering 1 m length perpendicular to paper.
Nd represents the total number of potential drop in the complete flow net, then –
Δh – Nd = HL …(5)
Where HL is the total hydraulic head causing flow, and is equal to the difference of the upstream and the downstream heads.
Substituting the value of Δh form Eqn. (5), we get –
From the drawn flow net, Nf and Nd can be easily counted, and hence, the seepage discharge can be easily computed by using Eqn. (9).
Determination of seepage pressure
The change in the effective pressure due to flow of water is known as seepage pressure. The seepage pressure at any point can be determined by
Ps= H.Yw
where, H= hydraulic potential at any point = h - n.h/Nd = (Nd-n).h/Nd
n = numbers of potential drops.
Yw = unit weight of water = 9.81
Pore pressure Uplift Pressure (Pu)
The uplift pressure at any point within the soil mass can be found using the undermentioned formula. It is also called as hydrostatic pressure.
Where,
Seepage and discharge velocity
Discharge velocity =v=qA
Seepage velocity = vs=qAv
Thus equating for q gives
vsAv=v A=q
Therefore,
vs=AAv v
Multiplying and dividing by unit distance
vs=AAv11 v= VVv v
vs=1n v
Discharge velocity (v) and seepage velocity
v= porosity ×seepage velocity= n ×vs
k=porosity ×coefficient of percolation=n ×ks
Permeability of layered soil
Permeability Parallel to bedding plane
Figure flow Parallel to bedding plane
kx= 1(h1+h2+h3) k1.h1+k2.h2+k3.h3
Permeability perpendicular to bedding plane
Figure flow Perpendicular to bedding plane
1ky= 1(h1+h2+h3) h1k1+ h2k2+ h3k3
Effective stress, total stress and pore pressure
Total stress/pressure (σ)
It is the algebraic sum of effective pressure/stress and pore pressure.
σ= effective pressure+pore pressure=σ'+ u
σ= layer 1h layer 1+ layer 2h layer 2+….
Effective stress/pressure (σ’)
It is the pressure transmitted through particle to particle within soil through direct contact.
σ'= σ- u
Pore pressure (u)
It is the pressure transmitted through water particle within soil.
u= hwater tablew
Different soil and pressure conditions
Figure Submerged soil and effective pressure
Figure Moist soil with surcharge and effective pressure
Figure Dry soil layers and effective pressure
Figure Soil with capillary action and effective pressure
Capillary rise in soil
hc= 4 Ts cos α w d
Where, hc is height of capillary rise, Ts is surface tension, α is angle of incidence, d is diameter of opening
Quick condition
If seepage flow is in upward direction, non-cohesive soil particles lose its shear strength and try to move in direction of flow. This phenomenon of lifting of particles is known as quick condition
Quicksand condition is the floatation of particles of cohesionless soil, like fine gravel and sand, due to vertical upward seepage flow. As sand boiling occurs, the bearing capacity and shear strength of the cohesionless soil decrease and the agitations of soil particles become apparent.
It is also known as boiling condition or quick sand.
When flow is upward and it equals the effective stress of soil, boiling condition occurs
'=effective stress=pore pressure
'=w Hw= ' H
i= 'w=HwH= ic=critical hydraulic gradient
Using fundamental relationship equations we can write
'=sat-w= G+eSr w1+e- w= G+e1+e-1 w ….because Sr=1
ic= G+e1+e-1 ww
ic= G+e1+e-1
ic= G-11+e
Taking G=2.67, and e=0.67, the result of equation is equal to one.
Thus, the effective stress becomes zero for the soil with above values of G and e, when the hydraulic gradient ‘i’ is unity, i.e. head causing the flow is equal to the length of the specimen.
If ic becomes unity, quick condition occur
Ground water table lowering methods
Ditches and sumps
Well point system
Shallow well system
Deep well system
Vacuum method
Elctro-osmosis method
Pumping out test
Measurement of coefficient of permeability – Field tests The coefficient of permeability of the permeable layer can be determined by pumping from a well at a constant rate and observing the steady-state water table in nearby observation wells. The steady-state is established when the water levels in the test well and the observation wells become constant. When water is pumped out from the well, the aquifer gets depleted of water, and the water table is lowered resulting in a circular depression in the phreatic surface. This is referred to as the ‘Drawdown curve’ or ‘Cone of depression’. The analysis of flow towards such a well was given by Dupuit (1863).
Assumptions
1. The aquifer is homogeneous.
2. Darcy’s law is valid.
3. The flow is horizontal.
4. The well penetrates the entire thickness of the aquifer.
5. Natural groundwater regime remains constant with time.
6. Dupuit’s theory is valid that is, i = dz/dr
Let r and z be the radial distance and height above the impervious boundary at any point on the drawdown curve as shown in Figure 4.6. At steady state, the rate of discharge due to pumping can be expressed as, q kiA
Hydraulic gradient at any point is given by Dupuit’s theory
k= q2.72 πb(z2-z1) log10r2r1
Pumping In test
Open end test
k= q5.5 r h
Packer test
k= q2π L h Lr when L ≥ 10r
k= q2π L h L2r when 10r > L ≥ r
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