Cm5 Unidimensional English

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UNIVERSIDAD NACIONAL DE COLOMBIABOGOT

SOIL MECHANICS

FLOW IN POROUS MEDIA

JULIO ESTEBAN COLMENARES MONTAEZ TITULAR PROFFESOR

BOGOT D.C2015 - II


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In the last session stress physical concepts were approached, also effective stress and interstitial prconcepts were introduced

Effective stress of a saturated soil is defined as:

Underground Water can be found in three states: Hygroscopic, capillary and free or gravity:

In the saturated zone, where is located gravity water, the interstitial pressure is equal hydrostatic pre

Only if water is static condition , otherwise interstitial pressure decrease or increase in funct

direction

REVIEW


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1. INTRODUCTION

2. DARCYS LAW

3. HYDRAULIC CONDUCTIVITY

4. HYDRAULIC CONDUCTIVITY IN MULTILAYER MEDIA

5. EFFECTIVE STRESSES DUE TO FLOW

6. TEST TO DETERMINE HYDRAULIC CONDUCTIVITY

CONTENT


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1 INTRODUCTION

In this session the basic concepts about Unidimensional flow are studied, the concept

about infiltration force and hydraulic conduc

tivity are defined and also the variaeffective stresses due to flow is evaluated.

In the professional job is necessary to know the flow principals in porous mediaunderstand and to solve problematic related to:

I. Caudal of infiltration (Leak in ground dam, diggings and tunnels)

II. Variation of fluid volume (Consolidation Abatement of water table)

III. Stability of structures (Embankments - slopes)


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1 INTRODUCTIONBetween the particles which form the soil, due to its geometric features, there are aamount of holes which are releases. Even in the soils of the fine fraction and its lami

particles, pores are connected to form a ductwork where fluids can move.

Taken from Lambe (1991)

Taken from http://coleccion.educ.ar/coleccion/CD23/contenidos/familia/index

Micrograph of a sample of Sandstone in which the dispositionof rocks pores is shown.


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1 INTRODUCTION

Due to complicated path across porous media, friction between fluids and rough surfacparticles lets the main energy loss in the flow.

Basic theory about soil mechanics study that phenomenon with the following assumptions

I. The flow can be considerate incompressible.

II. The flow across soil is laminar

III. Darcys Law is valid for all kind of soil


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1 INTRODUCTION

When considering a permanent and incompressible flow, Bernoullis principle says the total energy peunit weight that has the flow is given by:


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1 INTRODUCTION

Using the Bernoullis equation in the soil topic, the speed load term can be despised due tothe magnitude of speeds which are following.

Using this equation in a soil element, the following is obtained:

( )

Hydraulic Gradient i is defined as total energy loss per unit length:


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2 DARCYS LAW

Henry Darcy did a study about water flow across a sand bed in 1850.

Varying the pressure and length of bed, Darcy found that caudal which passes through the soil isproportional to hydraulic gradient applied. In math terms:

The proportionality constant of Darcys Lawis known as:

Hydraulic Conductivi ty of soil


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2 DARCYS LAW

Discharge Speed is the speed of moving of a raindrop from Position 1 to Position 2. (fig 1).

Infiltration Speed is the average flow speed through soil, it means, the raindrop speed from

Position 3 to Position 4. (fig 1).

Infiltration speed takes intoeffective area where the fluid mmeans, relates flow speematerials porosity.

Using the continuity equation:

A

L


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3 HYDRAULIC CONDUCTIVITY

Kinematic Viscosity of fluid.

Is accepted to say that hydraulic conductivity has two components:

Hydraulic Conductivity or permeability coefficient k describes the ease of a fluid to move in a porous medand its units are

Hydraulic conductivity of soil depends on following factors

I. Fluid viscosityII. Void ratio and its distribution in the soil massIII. Rugosity of mineral particlesIV. Degree of saturation

Absolute Permeability depends on pores

structure and its geometric. .


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3 HYDRAULIC CONDUCTIVITY

Hydraulic conductiv ity can be expressed as a function of soil moisture conditions.

Is important to say that hydraulic conductivity decreasesquickly with decrease of degree saturation of soil. Thefollowing are some reasons:

1. In the drying, liquid phase is not continuous.

2. Decrease the effective area such that exist flow.

Taken from: Analysis of underground water contaminatiosystemshttp://www.bvsde.ops-oms.org/eswww/fulltext/repind46


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4 HYDRAULIC CONDUCTIVITY IN AN ANISOTROPIC MEDIA

Due to formation of soils, variation of its properties can be important in small distances(depth and area).

Model the soil as an unique layer with equivalent parameters can be advantageous, it meato suppose that the effects in a real soil can be represented by a fictitious soil with averageproperties.

In the case of moving of fluid in porous media, is possible to determinate:

I. Equivalent hydraulic conductivity for horizontal flow

II. Equivalent hydraulic conductivity for vertical flow


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4.1 HORIZONTAL FLOW

Using the continuity equation:

=

( )(

( ) 1=( )

In this conditions the hydraulic conditio

the same in all layers:



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4.1 VERTICAL FLOW The condition of problem is:

= + (From continuity equation:

=

= +

From Darcys equation:



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5 EFFECTIVE STRESSES DUE TO FLOW

Method: Unit weight and peripheral forces of water Flow up

Lh Lh L z

wt

wt w

'

'


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Method: Unit weight submerged and infiltration forces Flow up

Li L

h L

L z

wb

wb

t w

'

'

'

5 EFFECTIVE STRESSES DUE TO FLOW


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5 1 EFFECTIVE STRESSES DUE TO FLOW UP

hw

LX

soil

z

A

BUsing the Bernoullis equation between point A and a point X inthe soil mass, the pressure in that point is given by:

( )

Hydraulic gradient allow to determine the energy loss in any pointof sample. Using the energy equation between the points A and B:

Reference plane: Plane 1



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5 1 EFFECTIVE STRESSES DUE TO FLOW DOWN

hw

LX

soil

z

A

B

Supposing that the hydrostatic pressures law can be applied inthe model, is possible demonstrate that:

Effective stress in any point in the soil mass:

( ) = = ( )


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flow

hw

Lz

( ) Reference plane: Plane 1

Z Z


Hydraulic gradient allow to determine the energy loss in any pointof sample. Using the energy equation between the points A and B:

Using the Bernoullis equation between point B and a point X inthe soil mass, the pressure in that point is given by:



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flow

hw

Lz

( ) = ( ) =

Effective stress in any point in the soil mass:

Supposing that the hydrostatic pressures law can be applied inthe model, is possible demonstrate that:



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5 3 COMPARISON BETWEEN FLOW CONDITIONS

D e p t h

( m )

Vertical effective stress 'z ( kPa)

Hidrosttica F.Descendente F.Ascendente


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5 4 INFILTRATION FORCE

When water flows through soil, a friction force between soil particles and fluid particlesis produced. The magnitude of that force is related with fluid viscosity.

The infiltration force always acts in flow direction and is equal to:

ww i

LA

hA

suelodel Volumen

on Infiltraci Fuerza j

_ _

_

When the infiltration force is equal to weight of soil mass in flow up, the condition of siphoning or boiling is presented. In that condition, the matrix of soil dont suppallowing the mineral particles move in the flow direction.


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5 4 INFILTRATION FORCE

In order with the last description, the siphoning condition is presented when effectivestress is null.

( )

0 ( )

Is necessary to assess this condition in the projects which involve flow in soils and toverify that the gradient found is the greatest critical gradient of soil. A recommendationfor this is:

23


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In the test, water of burette flows through soil.The initial difference of head, Ho in the timet=to=0 is registered and the flow through soilsample is allowed such that final difference of head in the time t is H.

In the Burette: dt

dH sQ

Through soil: k L H

Q

Is known that the most soils IN SITU are anisotropic,In that order, the horizontal hydraulic conductivityand vertical hydraulic conductivity are different.

6 MEASUREMENT OF HYDRAULIC CONDUCTIVITY

TEST OF VARIABLE HEAD


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PUMPING TEST

The test involves pumping water from aconstant rate (under steady state) recorddecrease in groundwater level.

Analysis of field test is related to the t(the permeable layer of soil)

1. Unconfined Aquifer2. Confined Aquifer



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The process of exploitation of an aquifer or a unconfinedmedium is a bidimentional flow problem, but is preferredin one direction.

For simplicity, assume that the flow is completelyhorizontal and evenly distributed in the medium,obviating the vertical component.

This condition is known as the supposition of Dupuit Forchheimer.

PUMPING TEST



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For simplicity the mathematicapossible to use cylindrical

analyze the problem:

Assuming that the medium iswe can say that abatement ofthe same in all directions.

It means that the abatement odepend only of radius:

0

PUMPING TEST



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From the above, is possible say

Using Darcys law:

Q Q

Q (2) Q

Q( )ln

PUMPING TEST




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Flujo radial

Impermeable

Superficiepiezmetrica

Pozo

Impermeable

Acuferoh1

h2

r 1

r 2

D

qPozos de

observacin

H

ro

Ri

The analysis of problem is doway that the last problem, t

possible conditions of flowaquifer

In the case h1>D for any time t

Q Q

Q (2 )

Q(2 )(

Si h1>D

Q

PUMPING TEST



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1. Flow through soil also can be considered as laminar flow due to its small speed, in thatorder v=k*i in which is a expression of Darcys law.

2. Hydraulic conductivity or permeability coefficient k describes the ease of a flua porous media and is obtained in laboratory test . [L/T]

3. Hydraulic gradient i is the total energy loss h per unit length l l : i= h /l

4. Discharge speed v is the speed of moving of water.

5. Infiltration speed vs is the average speed of flow through soil and is expressed as vs =

REVIEW


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6. There are three important heads in flow through porous media: Head of position, head of

pressure and the total head

7. The principal methods to analyze the stresses in flow condition are: total aforces method and Infiltration forces and weight submerged

8. Infiltration forces per unit volume are expressed as i* w and in isotropic soilflow direction.

9. Siphoning or boiling is a state in which a soil without cohesion losses its strength due toflow . It overrides the effective stress.

REVIEW


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BIBLIOGRAPHY

Holtz, R. (1981). Introduction to Geotechnical Engineering. (2dn Edition)

Craig, R. F. (1997) Soil Mechanics. 6th Edition. Chapman & Hall.

Lambe, T. W., Whitman, R. V. (1979). Soil Mechanics. John Wiley & Sons.

James Cook University, Australia. Engineering and Physical Sciences Rhttp: //eng1.jcu.edu.au/research/compgeo/geores.html

University of Cantabria, Spain. Virtual course of Soil Mechanics.

National Technological University Regional Faculty Santa Fe - Course: Foundations- Verification of safety to siphoning
http://eng1.jcu.edu.au/research/compgeo/geores.htmlhttp://eng1.jcu.edu.au/research/compgeo/geores.htmlhttp://eng1.jcu.edu.au/research/compgeo/geores.html

Cm5 Unidimensional English

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