12...Fluid flow through soils - Motivation The ability of engineers to understand and predict the...

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

12


Permeability and Seepage -1


No. Module Contents

Permeability and Seepage

Permeability; Seepage force and effectivestress during seepage; Measurement ofpermeability in the laboratory and field;Laplace equations of fluid flow for 1-D, 2-Dand 3D seepage, Flow nets, Anisotropic andnon-homogeneous medium, Confined andUnconfined seepage.

M2


Fluid flow through soils - Motivation

The ability of engineers to understand and predict theflow of fluids (usually water) in soils is essential formany application in civil engineering.


Environmental Engineering

Toxic liquid

Holding lagoon

At what rate is toxic liquid escaping the holding lagoon?

How long might it take the liquid to reach the ground water table?

What can be done to slow down the rate of escape of the pollutant?


Construction Engineering

Temporary sheet-pile wall cofferdam structure around the site and to pump the water out….


Typical example of a Cofferdam section


Construction Engineering

Important questions that would need to be answered are:

What will be the rate of water inflow to the site ?

Is it possible that the soil will liquify and endanger construction workers?

Our objective is to gain an understanding of themechanics of fluid flow in soils so that engineeringproblems of this type can eventually be addressed.


Fluid flow through soils

Soils are assemblages of solid particles withinterconnected voids through which water can flowfrom a point of high energy to a point of low energy.



The study of the flow of water through porous soil media is important in soil mechanics:

1) Involving the rate at which water flows through soil (i.e., determination of rate of leakage through an earth dam)

2) Involving rate of settlement of a foundation

3) Involving strength (I.e. the evaluation of factor of safety of an embankment)


Flow path – microscopic length

Flow path – macroscopic scale

AFluid flow through soils

The Water does not flow from point A to point B in a straight line at constant velocity, but rather in a winding path from pore to pore.

Winding path or tortuous path

B



According to Bernoulli’s equation, the total head at a point in water under motion can be given by the sum of pressure, velocity and elevation heads.

zg

vphw

w ++=2

2

γPw/γw represents the pressure head of the fluid and has unit of length


Z represents the elevation w.r.t an arbitrary datum. The value isthe distance of the point at which head is being measuredabove the datum. This can be either positive if the point isabove the datum, or negative if the point is below the datum.

Fluid flow through soils V2/2g represents the kinetic or velocity head of the fluid andalso has units of length. Since water flowing in typically has verysmall velocities, the kinetic head or velocity head is typicallynegligible compared to that of the pressure and elevationheads. For this reason the velocity head is neglected in soilmechanics.

zphw

w +=γTherefore


Flow of water through soilsWater flows in soils only when thereis a gradient in head h. Lack of agradient in head implies that water isnot flowing

Whenever there is water flow in soils,there is energy dissipation occurs

In soils, water or (permeant) alwaysflow down the gradient. That is, waterflow from high energy regions to lowenergy regions.

Water flow in soils

Energy dissipation

Gradient in head h


Flow of water through soilsExample 1:

Reference datum

1

2

h1h2

z2

z1

H

Uniform head distribution in a soil deposit

As the total head H is identical flow wont take place either from 1 to 2 or 2 to 1.


Flow of water through soilsExample 2: Heads in static water in capillary tube

hc

1

2Datum

Elev. Head

Pressure Head

TotalHead

1 hc -hc 0

2 0 0 0

Therefore, there is no flow of water in this situation.


Fluid at rest in soil (No flow condition)

Zone of capillary rise (saturated)

Moist or dry soil

GWT

zPw(z)=γwz

Saturated soil

Pw = 0

Pw < atmospheric pressure ⇒-ve in the capillary zone


Flow of water through soils Fluid flows down the hydraulic

gradient, not necessarily downhill

Observe that H1 > H2. Therefore water flows down the hydraulic gradient from point 1 toward point 2.

Confined aquifer

Reference datum

1

2

p1/γw p2/γw

z1

z2

H1

H2


Flow of water through soils

Assumptions:

i) Soil is fully saturated,

ii) Frictionless boundaries

iii) Flow is laminar (i.e., Reynolds Number Re < 1)

w

we

vdRµ

ρ 10=Where:

v = discharge velocity;

d10 = effective particle size;

µw = dynamic viscosity of water

Flowdirection


ZB

pA/γw

ZAL

C/S area A

∆h =ha -hb

Datum

hApB/γw

hB

ha > hb



The head loss between two points can be given by:

∆h = hA – hB = (pA/γw + zA) – (pB /γw +ZB)

Lhi ∆

=

The head loss ∆h can be expressed in a non dimensional form as:

i = hydraulic gradient

L = Length of flow over which the loss of head occurred.



Variation of discharge velocity with hydraulic gradient ZONE – II:

Transient zone

ZONE – I: Laminar Flow zone

ZONE – III: Turbulent zone

i

v

When i ↑ (gradually)

Flow remains in laminar inZone I and II and v bears alinear relationship with i.

At higher i, flow becomesturbulent.

In most soils, v ∝ i; In gravel and very coarse sands, turbulent flow conditions may exist and v ∝ i is not valid.



Darcy’s law: After Darcy (1856)

A simple equation for the discharge velocity of waterthrough saturated soils, which may be expressed as:

kiv = k = coefficient of Permeability (m/s)

v = discharge velocity or superficialvelocity, which is the quantity of waterflowing in unit time through a unit C/S areaof soil at right angles to the direction of flow.

Flow is throughpore spaces in soiland not throughentire C/S area.

Formulated based on the observation offlow of water through clean sands.



Darcy’s equation is usually combined with the continuity equation.

ALhkkiAvAq ∆

===

q = Total rate of flow through the C/S area A

k = Darcy’s coefficient of permeability

(which is defined as ease with which flowtakes place through soil)


A

AV

AS

Flow of water through soilsAV

v

vs

A

v

Seepage Velocity vs

For unit width of the sample: e = VV/VS = AV/AS

Using the principle of continuity, q = vA = vsAv

nvv

VVv

AAv

VVs === Since 0 ≤ n ≥ 100 %,

vs always > v

WATER

SOIL SOLIDS



As a particle of water proceeds from A towards B it exerts a frictional drag on soil particles;

⇒

In turn produces a seepage pressurein the soil structure.

Direction of seepage pressure

A

B

Flow direction

Because of the frictional drag, the hydraulic head decreases steadily on every flow line.

Seepage pressure is due to flow of water through voids.


Flow of water through porous media

ws u+′=σσ

When water flows through the soil, it exerts drag forcescalled seepage forces on individual grains of the soil. Thepresence of seepage forces, which causes changesin the direction of flow, will cause changes in the porewater pressure and effective stresses in the soil.

Changes in geostatic stresses with flow of water through soil


Case –I : When no flow takes place through soil (Hydrostatic condition)

H

H1

γwH1+ γsatH

γwH1

σ

γw(H1+ H)

γwH1

u

γ ′H

σ′

No flow; Head loss ∆H = 0; No change in effective stress


Case –II : When flow takes place through soil (Downward flow)

H

H1

γwH1+ γsatH

γwH1

σ

γw(H1+ H)

γwH1

u

γ ′H + h γw

= γ ′H + i H γw

σ′

h

γw(H1+ H - h)Downward flow increases effective stress in soil…


Case –III : When flow takes place through soil (Upward flow)

H

H1

γwH1+ γsatH

γwH1

σ

γw(H1+ H + h)

γwH1

u

γ ′H - h γw

= γ ′H - i H γw

σ′h

γw(H1+ H )Upward flow decreases effective stress in soil…


Effect of seepage on effective stresses

Seepage is the flow of water through soil. It exerts a frictional drag on the soil particles called seepage force, Js which results in head loss. Seepage forces play a very important role in destabilizing geotechnical structures.

• Downward seepage increases the effective stress.

• σ′ = γ ′H + psH

• Upward seepage decreases the effective stress.

• σ′ = γ ′H – psH

Where seepage pressure [kN/m3]

ps = i γw (i.e. Js per unit volume)Js = i γwV


Effect of seepage on Effective Stresses.Seepage forces on the left side increase the effective stresses and lateral thrust on the wall.On the right side the seepage forces are upward and decrease the effective stresses and reduce the resistance by embedment.Seepage stresses play a key role in reducing the stability of a geotechnical structure.

Effective stress

increases

Effective stress decreases

AB


Critical Hydraulic Gradient ic

The hydraulic gradient at which the effectivestress becomes zero is known as CriticalHydraulic gradient.

In the case of upward flow:

0=−′=′ HH wγγσWhen i ic

-Under these circumstances, cohesion-less soils can not support any weight.

-Moreover, as i ic soil becomes much looser and k ↑


At A: σv′ = γsatb - γwh

i = (h-b)/b

h = b( i +1)

σv′ = γsatb - γw b( i +1)

A

Saturated Cohesion-less soil;

γsatb

(h – b)

PWP at A = γwh

DATUM

B

TH at A = h

TH at B = b

Head loss

For quick condition to take place:

σv′ = 0; ⇒ i = ic

eGi s

wC +

−=

′=

11

γγ

Quick condition or Boiling condition in cohesion-less soils


Conditions favourable for the formation quick sandQuick sand is not a type of sand but a flow conditionoccurring within a cohesion-less soil when its effective stressis reduced to zero due to upward flow of water.

Quick sand occurs in nature when water is being forcedupward under pressurized conditions.

In this case, the pressure of the escaping water exceedsthe weight of the soil and the sand grains are forced apart.The result is that the soil has no capability to support a load.

Why does quick condition or boiling occurs mostly in fine sands or silts?


Some practical examples of quick conditionsExcavations in granular materials behindcofferdams alongside rivers

Any place where artesian pressures exist (i.e. wherehead of water is greater than the usual static waterpressure).

-- When a pervious underground structure iscontinuous and connected to a place where head ishigher.

Behind river embankments to protect floods

12...Fluid flow through soils - Motivation The ability of engineers to understand and predict the...

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