Amoo

TITLE PAGE

ANALYSIS OF FLOW THROUGH POROUS MEDIA: A CASE STUDY OF OSUN RIVER

SAND, OSOGBO OSUN STATE.

BY

AMOO, AFEEZ OLADEJI(CVE/06/7931)

A Thesis submitted to the School of Post Graduate Studies, The Federal University of

Technology Akure, Nigeria, in partial fulfillment of the requirements for the award of Masters

Degree (M.Eng.) in Civil and Environmental Engineering (Water Resources and Environmental

Engineering Option)

June, 2015

i

DECLARATION

I hereby declare that this Thesis was written by me and is a correct record of my own research

work. It has not been presented in any previous application for any degree of this or any other

university. All citations and sources of information are clearly acknowledged by means of

references.

AMOO, AFEEZ OLADEJI

CVE/06/7931

SIGN: …….…………

DATE: ………………

ii

CERTIFICATION

I certify that this Thesis entitled “Analysis of Flow through Porous Media: A Case Study

of Osun River sand Osogbo, Osun state” is the outcome of the research carried out by Amoo,

Afeez Oladeji (CVE/06/7931) in the Department of Civil and Environmental Engineering,

Federal University of Technology, Akure.

………………….… ………………….

Dr. C.S. Okoli Date

(Major Supervisor)

iii

ABSTRACT

A study was conducted to determine the index properties and hydraulic characteristics of

the porous media. The porous media was dug from river bed of Osun river at eight (8) different

locations at five (5) metres interval during the November period. Specific gravity, moisture

content, particle size distribution (PSD) and permeability test of the porous media were

determined in accordance with the International Standard IS: 2720. Data from the above-

mentioned tests were analysed using Statistical Package for Social Sciences (SPSS). The specific

gravity of the sand (2.65 to 2.67), coefficient of curvature (0.54 to 1.35) and uniformity

coefficient (1.25 to 2.77) show that the sand samples are well graded and uniformly distributed.

The models generated for the locations were

f f =−1.314 ln ℜ , ℜ=59.755 Q−1.627 e7Q 2−4.642 e−5∧i=55.919 e3 v−120.821 e6 v2−4.666.

The hydraulic conductivity of the river sand sample ranges from 1.14¿1.43 ×10−2cm / sfor the

various locations considered and location A at upstream of the river bed which is about 1m depth

from sea level has the highest value (1.43 ×10−2cm / s¿ is the best location and location G at

downstream of the river bed which is about 1.5 m depth from sea level has the lowest value (

1.14 × 10−2 cm /s¿ is the worst location. It is recommended that proper attention should be given

to the friction factor, velocity of flow, flow rate and the Reynolds number as these are the main

products that result in flooding of embankment thereby resulting in environmental hazard and

seepage of water through dams.

iv

DEDICATION

This Thesis is dedicated to ALMIGHTY ALLAH, the one who makes ocean to flow in

the desert, and to my parents ALH. & MRS. AMOO for the love and care they show to me

during the project.

v

ACKNOWLEDGEMENTS

To begin with, I thank the ALMIGHTY GOD the unchangeable changer for His infinity

mercy and grace upon my life during my stay in The Federal University of Technology, Akure.

My sincere gratitude goes to my supervisor, Dr. C.S. Okoli for his guidance, commitment

and encouragement throughout the entire period of the research project. His immeasurable

support assisted me to remain focused on the research investigation from the beginning of the

research work to the very end, for all the time spent on coordinating and supervising the whole

thesis.

I also appreciate my parents Alh. & Mrs. Amoo without whom my academic pursuit

would have been unaccomplished. Your prayers, encouragement and financial support are

appreciated. May you live long to reap the fruits of your labour in the name of Allah.

I can’t but express my sincere appreciation to my beloved fiancé Onifade Soliat for her

prayer, encouragement and financial support during the course of this work, may God bless and

keep us together in love.

I also acknowledge the support of my siblings Olanrewaju Yusuf and Alabi Ibrahim. It’s

really being a blessing having you around, you are the best.

I can’t fail to acknowledge my family, most especially Mr & Mrs A.A. Amoo, Mr & Mrs

M.G. Amoo, Surv. & Mrs N.B. Amoo, Mr & Mrs Sokoya, Mr & Mrs Olaitan, Amoo Basirat,

Amoo Abdul_azeem & Fathiat, for their support morally, financially and prayerful during the

course of this work. I cannot say how much am indebted to them but pray that the ALMIGHTY

GOD continue to bless them.

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Above all, I remain deeply indebted to all my friends who have made my stay in the

University environment a memorable one, they are: Engr. Adavi, Laoye Rahmon, Engr.

Tanimola, Akinola Tunde, Chris, Jerry, Ilori Boluwatife, Gabriel Busayo, Ajayi Olawale,

Popoola Tosin, Oladipupo Kola, Uduebor Micheal, Mayowa, Kayode, Khadijat, Azeezat, Idayat,

Fatimah, Usman, Nafizat, Memunat, Ilelaboye Adeleke, Mama Alarobo, Oyebade, NASFAT

Akure branch, MSSN FUTA, I love you all.

vii

TABLE OF CONTENTS

TITLE PAGE....................................................................................................................................i

DECLARATION.............................................................................................................................ii

CERTIFICATION..........................................................................................................................iii

ABSTRACT...................................................................................................................................iv

DEDICATION.................................................................................................................................v

ACKNOWLEDGEMENTS............................................................................................................vi

TABLE OF CONTENTS.............................................................................................................viii

LIST OF TABLES........................................................................................................................xii

LIST OF FIGURES.......................................................................................................................xv

LIST OF PLATES........................................................................................................................xvi

NOTATION................................................................................................................................xvii

CHAPTER ONE..............................................................................................................................1

1.0 INTRODUCTION................................................................................................................1

1.1 BACKGROUND..............................................................................................................1

1.2 PROBLEM STATEMENT...............................................................................................3

1.3 AIM AND OBJECTIVES.................................................................................................4

1.4 PROJECT JUSTIFICATION............................................................................................4

1.5 DESCRIPTION OF THE STUDY AREA........................................................................4

CHAPTER TWO.............................................................................................................................8

viii

2.0 LITERATURE REVIEW.....................................................................................................8

2.1 GENERAL........................................................................................................................8

2.2 OVERVIEW OF FLOW THROUGH POROUS MEDIA...............................................8

2.2.1 Darcy Law and Laminar Flow...................................................................................9

2.2.2 Validity of Darcy’s Law..........................................................................................11

2.2.3 Comparison between Darcy’s Law and Stokes’ Law..............................................13

2.2.4 Permeability and Porosity........................................................................................17

2.3 POST-DARCY FLOW...................................................................................................19

2.3.1 Steady Flow Experiment..........................................................................................19

2.3.2 Flow.........................................................................................................................20

2.3.3 Flow Regimes..........................................................................................................21

2.3.4 Flow in Conduits......................................................................................................22

2.3.5 Non Stationary Laminar Flow.................................................................................23

2.4 POROUS MEDIA...........................................................................................................26

2.4.1 Types of Porous Media............................................................................................27

2.4.2 PROPERTIES OF POROUS MEDIA.....................................................................28

CHAPTER THREE.......................................................................................................................31

3.0 RESEARCH METHODOLOGY AND MATERIALS......................................................31

3.1 MATERIALS..................................................................................................................31

3.2 EXPERIMENTAL PROCEDURE.................................................................................31

ix

3.2.1 Preliminary Test.......................................................................................................32

3.2.2 Engineering Property Test.......................................................................................38

3.3 SOFTWARE USED........................................................................................................42

3.3.1 Statistical Packages for the Social Sciences (SPSS)................................................42

3.3.2 Regression Analysis.................................................................................................42

CHAPTER FOUR.........................................................................................................................43

4.0 RESULTS AND DISCUSSION.........................................................................................43

4.1 NATURAL MOISTURE CONTENT RESULT............................................................43

4.2 SPECIFIC GRAVITY RESULT....................................................................................47

4.3 PARTICLE SIZE DISTRIBUTION (PSD) RESULT....................................................52

4.4 PERMEABILITY OF SAND.........................................................................................63

4.5 SUMMARY OF THE CLASSIFICATION TEST ON SAMPLES...............................68

4.6 SUMMARY RESULT OF THE HYDRAULIC PROPERTIES OF SAMPLES...........70

4.7 OBSERVATIONS OF THE HYDRAULIC PROPERTIES..........................................71

4.8 ANALYTICAL MODEL IN POROUS MEDIA............................................................71

4.8.1 The Model Summary for the relationship between Friction factor and Reynolds

Number (Re) in Flow through Porous Media.........................................................................71

4.8.2 The Model Summary for the relationship between Reynolds Number (Re) and

Flow rate (Q) in Flow through Porous Media........................................................................74

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4.8.3 The Model Summary for the relationship between Hydraulic gradient (ἰ) and

Velocity of flow (v) in Flow through Porous Media..............................................................77

CHAPTER FIVE...........................................................................................................................79

5.0 CONCLUSION AND RECOMMENDATIONS...............................................................79

5.1 CONCLUSION...............................................................................................................79

5.2 CONTRIBUTION TO KNOWLEDGE..........................................................................80

5.3 RECOMMENDATIONS................................................................................................80

REFERENCES..............................................................................................................................81

APPENDICES...............................................................................................................................86

xi

LIST OF TABLES

Table Page

1 Natural moisture content result of sample from location A……………… 43

2 Natural moisture content result of sample from location B……………… 44

3 Natural moisture content result of sample from location C……………… 44

4 Natural moisture content result of sample from location D………………. 45

5 Natural moisture content result of sample from location E……………… 45

6 Natural moisture content result of sample from location F………………. 46

7 Natural moisture content result of sample from location G……………… 46

8 Natural moisture content result of sample from location H……………… 47

9 Specific gravity result of sample from location A……………………….. 48

10 Specific gravity result of sample from location B……………………….. 48

11 Specific gravity result of sample from location C………………………. 49

12 Specific gravity result of sample from location D………………………. 49

13 Specific gravity result of sample from location E……………………….. 50

14 Specific gravity result of sample from location F……………………….. 50

15 Specific gravity result of sample from location G………………………. 51

xii

16 Specific gravity result of sample from location H………………………. 51

17 Particle size distribution result of sample from location A……………… 53

18 Particle size distribution result of sample from location B……………… 54

19 Particle size distribution result of sample from location C……………… 55

20 Particle size distribution result of sample from location D……………… 56

21 Particle size distribution result of sample from location E………………. 57

22 Particle size distribution result of sample from location F……………….. 58

23 Particle size distribution result of sample from location G………………. 59

24 Particle size distribution result of sample from location H………………. 60

25 Coefficient of permeability result of sample from location A……………. 64

26 Coefficient of permeability result of sample from location B……………. 64

27 Coefficient of permeability result of sample from location C……………. 65

28 Coefficient of permeability result of sample from location D……………. 65

29 Coefficient of permeability result of sample from location E……………. 66

30 Coefficient of permeability result of sample from location F……………. 66

31 Coefficient of permeability result of sample from location G…………… 67

32 Coefficient of permeability result of sample from location H…………… 67

xiii

33 Standard values of range for Specific gravity of soil……………………. 68

34 Standard values of range for Uniformity coefficient and Coefficient of

curvature…………………………………………………………………..

68

35 Summary of the results for the classification test on all sample location… 69

36 Summary of the results of the hydraulic properties on all sample

location........................................................................................................

70

37 Model summary relationship between friction factor and reynolds

number………………………………………………………..……..

71

38 Model summary relationship between reynolds number and flow rates… 74

39 Model summary relationship between hydraulic gradient and velocity of

flow………………………………………………………………………

77

xiv

LIST OF FIGURES

Figure Page

1 Map of Nigeria showing Osun state………………………………….. 5

2 Map of Osun state showing all the local government………………… 6

3 Morphological map showing the sample location on Osun River in

Osogbo…………………………………………………………………

7

4 Porous medium & Sphere in Unbounded fluid………………………... 14

5 Porous medium elemental volume in E………………………………... 16

6 Period-dependent Permeability………………………………………… 26

7 Falling head test permeability arrangement…………………………… 39

8 Combined particle size distribution curve of sample location A, B, C &

D………………………………………………………………………..

61

9 Combined particle size distribution curve of sample location E, F, G &

H……………………………………………………………….………..

62

10 Regression curve between friction factor and reynolds number

curve……………………….…………………………………….…........

73

xv

11 Regression curve between reynolds number and flow rates curve……... 76

12 Regression curve between hydraulic gradient and velocity of flow

curve……………………………………………………….......................

78

LIST OF PLATES

Plate Page

1 Sample location for A………………………………………………….... 86

2 Sample location for B………………………………………………....... 86

3 Sample location for C………………………………………………....... 86

4 Sample location for D………………………………………………...… 86

5 Sample location for E………………………………………………..…. 87

6 Sample location for F………………………………………………….... 87

7 Sample location for G…………………………………………….…….. 87

8 Sample location for H……………………………………………….….. 87

9 Permeability test in the laboratory…………………………………….… 88

10 Specific gravity test in the laboratory………………………………….... 88

11 Placement of sample A – H……………………………………………… 88

12 Sample inside sack bag in the laboratory before test……………………. 88

xvi

13 Sieve analysis test in the laboratory……………………………………... 89

NOTATION

The following symbols are used in this research:

Symbols Notation

D, dc Inside diameter of capillary;

∆H Head loss through medium;

∆L Depth of the river bed;

L Length of the capillary;

K Darcy coefficient;

Q Flow rate through medium;

A Area of filter bed in the plan;

µ Dynamic viscosity of the fluid;

ρ Density of the fluid;

g Acceleration due to gravity;

Vc Velocity of fluid in capillary;

xvii

η Bulk porosity of bed of the porous media;

V Volume flux across a unit area of the porous media in unit

time;

ζ Tortuosity of flow path in porous media;

Vv Volume of voids;

Pp Particle density;

Mp Mass of voids;

Vp Volume of particles;

s Distance in the direction of flow;

κ Permeability of the medium;

z Vertical coordinate;

δp/δs Pressure gradient along s at the point to which v refers;

P Pressure gradient of the medium;

η Viscosity of water;

ρw Density of water;

d Diameter of pipe;

v Velocity of flow;

ἰ Slope of the energy line or hydraulic gradient;

xviii

h1 Head loss at initial;

h2 Head loss at final;

l Length of soil;

fk Friction factor using square root of permeability as

characteristics length;

Re Reynolds number;

Gradient operator;

AT Cross sectional flow area;

c Dimensionless constant;

R Hydraulic radius of pore spaces;

ϕ Porosity of porous media;

S Specific surface of the flow tube;

t Time taken;

n Friction manning coefficient;

Fr Froude number;

F Total force;

α Coefficient of proportionality;

β Volumetric shape factor;

xix

λ Factor incorporating effects on particles;

C1 Acceleration coefficient;

ἰ0 Magnitude of the gradient;

CD Drag coefficient;

θ Phase shift induced between the velocity and energy

gradient;

Rp Total resistance force;

ϒ Random variable;

X Non-random variable;

β0 Concept or Intercept team;

β1 Coefficient or Slope parameter;

U Un-observed random variable;

Da Diameter of average particles; and

Φa Some factors related to dimensional property ‘a’

xx

CHAPTER ONE

1.0 INTRODUCTION

1.1 BACKGROUND

Flow through porous media is a subject of interest in many branches of

science, i.e., hydrogeology, chemical engineering, and in the field of

petroleum extraction. The investigation of its features plays a major role in

the comprehension of many phenomena, as the subsidence caused by water

shortage, or the process of crystallization of the ores in a well thermal exit,

which makes them unusable for the extraction of the heat. Moreover, it is

important to investigate both the correlation between seismo-genesis and

the introduction of fluids in the subsoil, studied in the Rangeley Colorado

experiment (Raileigh et al., 1976), and the link between the increase in the

seismic activity and the growth of the water level in wells (Bell and Nur

1978). The steady flow through filter having sand bed of various thickness and under various

pressure is directly proportional to the hydraulic gradient and the Darcy’s equation in the

following commonly used (except at high velocities when turbulence occurs) is:

∆ H∆ L

= 1K

QA

(1)

where, H= head loss through medium (L), L= depth of filter bed (L), Q= flow rate through

medium (L3/T), A= area of the filter bed in plan (L2) and κ = Darcy’s coefficient (L/T).

Poiseuille equation can be applied to estimate the head loss from the velocity of flow in

each individual capillary as follow:

1

Δ HΔ L

=32( μρg ) Vc

d c2(2)

VcQ

nAb=V .

n(3)

n=∀ v∀

(4)

where,

L= length of capillary (L),

µ= dynamic viscosity (M/L.T),

= density (M/L.T),

g = standard acceleration of gravity, (9.81 m/s2),

dc= inside diameter of capillary (L),

Vc= velocity of fluid in capillary (L/T),

n = bulk porosity of bed of porous media (L3/L3),

v = volume of the voids (L3),

= total volume of the bed (L3), and

V = velocity of fluid

ʄ = tortuosity (L/L).

Control of the movement of water and prevention of the damaged caused by the

movement of water in soils are vital aspects of soil engineering (Leonard, 1962). The study of

seepage patterns in cross section with soils having more than one permeability is one of the most

worthwhile and rewarding applications, especially in selecting a protective filter or seepage

control in man-made constructions (Elsayed and Lindly, 1966). Excessive seepage is caused by

2

high permeability or short seepage path. Its permeability can be reduced by a proper selection of

materials, for example, mixing a small amount of clay with the sand (protective filter) used for

construction can reduce the permeability greatly (Sower and Sower, 1970). A filter or protective

filter is any porous material whose opening is small enough to prevent movement of the soil into

the drains and which is sufficiently pervious to offer little resistance to seepage (Jacob, B. 2001).

The permeability is the most important physical property that determines the porosity of a

medium, which is a measure of the ability of a material to transmit fluid through it. Frequently, a

soil is employed as a filter and in preparing a good filter, knowledge of permeability of

homogeneous and heterogeneous media is very essential. A medium is homogeneous if the

permeability is constant from point to point over medium while it is heterogeneous if

permeability changes from point to point in the medium. The permeability can be determined or

computed from hydraulic conductivity (Domenico and Schwartz, 2008).

Since porous media can either be natural or artificial and the natural type can readily be

available, this research will be based on the laboratory test of determining the hydraulic

properties of sand (an example of natural porous media).

1.2 PROBLEM STATEMENT

The design of dams and hydraulic structures depend on the flow through porous media

i.e., the soil. Since soil are porous and the presence of voids in the soil allow the flow of fluids

through the soil particles, the construction of roads and structures is affected by the flow through

the media. Failures of structures are also caused by the impermeability of the soil to water.

In order to avoid this menace of failures on these hydraulic structures, there is need to

analyse flow through porous media, so that the hydraulics characteristics of the soil can be

3

determined for the stability of these structures. The nature of the flow is also determined and

models are also generated to give the relationship between the flow parameters.

1.3 AIM AND OBJECTIVES

The aim of this research is to determine the index properties and hydraulic characteristics

of the porous media (Osun river sand).

The specific objectives are to:

a) determine the flow rates of the porous media (Osun river sand);

b) determine the hydraulic properties of the Osun river sand; and

c) compare the relationship between the obtained flow rates and the hydraulic properties of

the porous media in accordance with international standards of soil classification.

1.4 PROJECT JUSTIFICATION

The hydraulic properties of soil are of great importance to civil engineers and the rate of

flow of water through a sand medium is a function of permeability, porosity, hydraulic gradient

and hydraulic conductivity of the soil governed by Darcy’s law.

This research will help to determine the:

a) rate of recharge of wells, underground of flows and its characteristics;

b) design and construction of hydraulics structures such as dam and spillways;

c) possible way of preventing or controlling environmental hazards; and

d) water losses from the river due to seepage.

4

1.5 DESCRIPTION OF THE STUDY AREA

Osogbo is the capital city of Osun state, south western Nigeria. It is some 88 kilometers by

road Northeast of Ibadan. It is also 100 kilometers by road south of Ilorin and 115 kilometers

Northwest of Akure. It is situated on latitude on 9.7 N and longitude 4.5 E. The soil samples is

gotten from eight (8) different location at five (5) metres interval at the river bed of Osun river

beside Old Governor office along Gbongan-Ibadan expressway, Osogbo, Osun State.

Figure 1: Map of Nigeria Showing Osun StateSource: OSRBDA 2014

5

Figure 2: Map of Osun state showing all the Local Government Source: Google Maps, 2014

6

Figure 3: Morphological map showing the sample location of Osun riverSource: Amoo Afeez Research 2015

7

CHAPTER TWO

2.0 LITERATURE REVIEW

2.1 GENERAL

For the past 150 years or so, investigations into porous media flow have yielded a great

deal of understanding of the phenomenon. Much of the work has been empirical in nature,

similar to the well-known contribution of Darcy law. Such investigations have identified the

parameters relevant to the phenomenon and have provided some useful relationships between

them. The majority of the analytical work has been directed towards deriving such empirical

relationships from the equations of motion and continuity (with appropriate simplifications and

approximations), thereby isolating the effects of, and the relative importance of, individual terms

(Rumer & Drinker 1966).

The goal of this research is of course to produce a reliable predictive description of the

phenomenon of flow through porous media (river sand). It is therefore useful to review previous

theoretical and empirical results in order to better understand the nature of the present

investigation and this begins with a general overview of the concepts of porous flow followed by

a brief review of Darcy's law, including a description of the parameters comprising this law, its

upper limit of validity and the reasons for its failure after this limit.

2.2 OVERVIEW OF FLOW THROUGH POROUS MEDIA

A porous medium acts as a resistance to flow; it is the goal of research to describe the

form of the resistance coefficient. Fluid particles flowing through the pore spaces pass through

expansions and constrictions to the flow and experience other convectional inertia effects caused

by the curvilinear flow paths. Given that resistance coefficients for pipe flow through

8

expansions, contractions, etc., are determined by empirical means, the complexity of defining

resistance coefficients for a generalized, random porous medium is readily apparent.

A porous medium is generally visualized as a continuum having properties of dimension

and porosity (Shih, 1990). The permeability of the porous medium is usually described in terms

of directly measurable quantities, most commonly the porosity and a large body of work has

been (and is still being) directed towards relating permeability and porosity. The permeability is

however, obviously dependent upon other properties including particle size, shape, orientation

and surface roughness. Analytically, the continuum approach requires averaging of the terms in

the equations of motion and continuity, as these quantities cannot be used directly owing to the

complex boundary conditions of flow through the pore spaces of the medium. Thus, the

properties of velocity and pressure must be averaged over a volume which is large enough for the

averaging procedure to be valid and yet small enough so as to be considered infinitesimal with

respect to the total sample volume (Scheidegger, 1960; Bear, 1972; Le Mehaute, 1976). This

requires that the magnitude of the flow be much greater than the pore volume. Therefore, flow

through large pores (or past large obstructions), such as waves passing through the armour layer

of a breakwater, cannot be validly described by this approach. Gray and O'Neill (1976) described

such a technique of "local averaging" to obtain generalized porous flow equations and Le

Mehaute (1976) illustrated how such an averaging of the terms in the Navier-Stokes equations

can result in Darcy's law.

2.2.1 Darcy Law and Laminar Flow

Darcy's experiments yielded the results that over a limited range of flowrates (Q),

Q=K AT

(h2−h1)l

(5)

where;

9

AT = cross-sectional flow area,

l = length of the sample,

h1and h2 are the piezometric heads at locations 1 and 2 at elevations ‘z’, i.e.

h1

P1

ρg

+Z1(6)

= the density of water,

g = acceleration due to gravity and;

k = constant of proportionality which Darcy called the permeability of the material.

Expressing (5) in terms of pressure and noting that the average or "bulk" or "superficial" velocity

is q= QAT

Darcy's law can be written as:

q=K .∇( Pρg

+Z)=κἰ(7)

= gradient operator,

ἰ = slope of the energy grade line (i = dh/dx), commonly termed the hydraulic gradient,

fluid density,

acceleration due to gravity and the permeability,

k = function of the fluid and the porous medium; these two aspects can be separated yielding.

κ= μρg

K (8)

where, κ is defined as the intrinsic permeability of the material (because it depends only on

properties of the material) and has dimensions of (Length)2 and the dynamic viscosity of the

fluid.

10

Hence, Darcy's law states that the energy loss across a porous medium due to friction is

directly proportional to the averaged or "bulk" velocity. However, this law applies only to a

limited range of flowrates where effects of inertia are negligible compared to those due to

viscous forces (Wright, 1968; Scheidegger, 1960; Philip, 1970; Dybbs and Edwards, 1982).

2.2.2 Validity of Darcy’s Law

It was discovered that Darcy’s law of linear dependency between velocity of flow (v) and

hydraulic gradient (ἰ) is only valid for laminar flow conditions in soil. Reynolds number found

that the flow is laminar as far as the velocity of flow is less than a lower critical velocity (vc)

expressed in terms of Reynolds number by the expression;

V C

ηgdρw=2000(9)

where;

V C = lower critical velocity in the pipe (cm/sec)

d = diameter of pipe (cm)

ρw = Density of water (g/cm3)

Viscosity of water (gsec/cm2); and

g = acceleration due to gravity (cm/sec2).

Based on the analogy above, the flow through soils is assumed to depend on the

dimensions of the pore spaces. It was seen that in coarse grained soils, where the pore

dimensions are larger, there is a possibility of flow being turbulent. Francher et al (1993)

11

demonstrated that flow through sands remain laminar and that Darcy’s law remain valid as the

Reynolds number as seen in the expression below, is equal to or less than unity, that is:

V D a ρѡ

ηg≤ 1(10)

where;

V = velocity of flow (cm/sec)

Da = diameter of average particle that is effective size mostly D10 (cm)

From Darcy’s concept it is seen that;

V=κἰ(11)

ἰ=h1−h2

l(12)

V=κ (h1−h2)

l(13)

It is established that in laminar flow “f” depends only on the Reynolds number and it is given by

f =64Re

(14)

where;

f = friction factor,

Re = Reynolds number,

h1= head loss at initial,

12

h2 = head loss at final,

l = length of the soil sample and;

k = permeability of the soil.

2.2.3 Comparison between Darcy’s Law and Stokes’ Law

Similarities between Darcy's law and Stokes' law for flow past a single obstruction have

been noted by many researchers (Wright, 1968; Scheidegger, 1960; Philip, 1970). Both are

solutions to the Navier-Stokes' equation for an incompressible fluid and negligible inertia. Both

Darcy's and Stokes' laws fail when the effects of inertia cease to be small; this occurs when the

Reynolds' number (q.D/v, with D chosen as some characteristic grain diameter) attains values of

about 1 and the departure from the linear law is a gradual one. The connection can be illustrated

by the use of dimensional analysis as follows: Figure 4(a) and (b) shows, graphically, the flow

patterns being compared. Figure 4(a) is a homogeneous, isotropic porous medium comprised of

equal sized spheres of diameter D (which is small compared to the container dimension H) with

porosity (n). A uniform steady laminar flow of magnitude VP is passing through the medium.

Figure 4(b) shows a single circular cylinder of diameter (D) in an unbounded laminar flow of

magnitude (U).

For the porous medium, the phenomenon can be completely described by the

characteristic parameters (D. n. q) along with the fluid viscosity and density (Yalin and Franke,

1961). Expressing these parameters in terms of dimensionless variables by using the

"Buckingham-Pi Theorem", (Yalin, 1971), and choosing basic quantities D, q and p), any given

dimensionless property (Ya), can be described by the relation:

13

Y a=∅ a( qDv

×n)(15)

where Φa represents some function related to the dimensional property "a" and the geometry of

the medium under consideration. For the case of filtration flow the velocity heads are neglected

as q2/2g d 0. Although this is not the case for flow through rockfill structures, the velocity heads

remain small compared to other energy losses, even in prototype (Parkin et al 1966; Hall, 1990).

For the case of waves plunging on a breakwater the velocity heads in the armour layers are likely

not negligible. In addition, other effects such as air entrainment and fully turbulent flow require a

different method of solution to be adopted. Therefore; the influence of the velocity heads will not

be considered in this analysis of porous media flow.

Figure 4: (a) Porous medium (b) Sphere in Unbounded fluid

If the flow is laminar (small Re) viscous forces dominate over the convective inertial

forces, which are represented by ρ. If the inertial forces are not to be considered then ρ (thus R e)

cannot be included in the functional relation but the viscosity, μ, must remain. If the quantity (a)

14

is the pressure gradient (dH/dx = ρgἰ = 1), then application of dimensional analysis (by choosing

basic quantities as D, q, μ) provides the relationship:

Y 1=D2

μqI=∅ 1 (n )(16)

Equation (16) can be written in terms of the hydraulic gradient, ἰ, as

ρg D 2

μqἰ=∅ 1 (n )(17)

Or equally,

ἰ=f r

Re

∅ 1 (n )=( vg D2 ∅ 1 (n ))q (18)

(where Fr is the Froude number) which is identical to Darcy’s law if 1/K = v/gD2 Φ1(n). In

addition, the quantity i/Fr is proportional to 1/Re. The term i/Fr is a common expression for the

friction factor. If the property under investigation is the total force (F) on the porous medium

then, as above, for small Reynolds numbers,

F=μ2

ρ∅ F( ρqD

μ∙ n)(19)

where, for laminar flow, p is not a parameter and must vanish as above. To accomplish this; Fr

must be linearly proportional to Re and therefore;

F=μqDα ∙∅ (n )(20)

where α is the coefficient of proportionality; and equation (20) is seen to be identical to Stokes'

law for a single particle, that is except that for a porous medium a term for the porosity (which is

commonly used to represent the permeability) must be included.

F=CD μD ∙q (21)

For the flow situation in Figure 4b, the analysis proceeds identically to that given above

except that the porosity term is not present. Therefore, any property under investigation becomes

15

a function of the Reynolds number only (other properties such as surface roughness are

incorporated into the functional relationship for all geometrically similar configurations). For the

case of the total force, F, on the particle, this becomes

F=μ2

ρ∅ F( ρUD

μ )(22)

and for laminar flow the density term is neglected, leaving the result (Stokes' law)

F=μD CD U (23)

where CD is the proportionality constant commonly called the drag coefficient.

Rumer and Drinker (1966) used this similarity to show that Darcy's law can be derived from a

simple force balance if Stokes' law can be assumed to apply to a porous medium (with

appropriate modifications). They considered a cylindrical element (E), (Figure 5) with porosity (

n), length ds and inclined at an angle θ to the vertical so that cosθ = dz/ds. The force balance

(stationary flow) is then

−δHδs

−RF

ρn δAT δs=0(24)

16

Figure 5: Porous medium Elemental Volume ESource: Gregory. M. Smith 1991

where, H is the pressure head and Rp is the total resistance force of all the grains within the

volume (dA)(ds). For the case of laminar flow all energy is dissipated by friction, as shown

above, and the resistance of the solid particles can be described by Stokes' law, modified to

account for the effects of many particles, i.e.,

f p=λμD vp (25)

where fp is the drag force per unit volume acting on one particle, D is a characteristic length,

diameter for a sphere, Vp is the actual fluid pore velocity, and λ is a factor incorporating effects

of the neighboring particles (λmax = 3π for a single sphere in an unbounded fluid). Considering N

particles within the volume element E, then

N=(1−n)δAδs

β D3 (26)

where β is a volumetric shape factor (π/6 for a sphere). Substituting equations (25) and (26) into

(24) and assuming the relationship vp = q/n, then

q= −β n2

λ(1−n)D2 ρg

μδHδs

(27)

The term βn2/(λ(1-n) is a function of the pore system only and can thus be replaced by the

dimensionless coefficient “c”. Recalling that the (constant) intrinsic permeability k has

dimensions of L2, as does the constant cD2, then the product cD2 can be replaced by the constant

k and equation (27) becomes identified with Darcy's law, equation (17) as

q=−kρgμ

δHδs

(28)

Note that here the permeability k is a function of the term (n2/(1-n)). This is the form of the

pororsity function Φ(n) derived on a semi-theoretical basis.

17

2.2.4 Permeability and Porosity

The intrinsic permeability term (κ) in Darcy's law (equation 8) is dependent upon the

configuration of the granular matrix; the size and shape of the particles, their size distribution

and orientation and the porosity of the sample. Owing to the fact that these properties are

difficult to measure and control in laboratory and prototype, most effort has been concentrated

on the last factor, porosity, as a descriptor of the permeability of the sample. In addition, most

studies assume an isotropic porous medium so that (κ) is a constant; for anisotropic media (κ) is

a tensor quantity.

Early analytical attempts at defining the permeability led to the development of models

that represented the porous medium as a series of capillary tubes, and common pipe flow laws

were applied to describe the hydraulics of the system. Scheidegger (1960) gives a good review of

these models and most are based upon the Hagen-Poiseuille equation for laminar flow in straight,

circular pipes, (which is also a solution to the Navier-Stokes equation for these specific boundary

conditions) i.e.

d p

d x

=−32μ V p

D 2 (29)

For porous media flow ‘Vp’ is replaced by the bulk velocity q, and the diameter D, is

usually replaced by a typical grain diameter, D50 for example, because of the difficulty in

assigning a typical pore size. This approach is limited because the nature of flow through a

porous medium is quite different from that in pipes. At Reynolds numbers of about one

convective acceleration terms in the Navier-Stokes equation can no longer be neglected as the

fluid must follow curvilinear paths through the granular matrix (Scheidegger 1960; Wright 1968;

Dudgeon 1964; Philip 1970). When applied to flow in straight, circular pipes, equation (21) is an

exact linear equation (non-linear convective terms are identical to zero) which applies to a

18

specific geometry and breaks down suddenly at the onset of turbulence, commencing at

Reynolds numbers of about 2000 (Streeter and Wylie, 1981). In a porous medium, however, the

linear flow law (Darcy's law) is conditional upon the non-linear convective terms being "small",

i.e. at Reynolds numbers less than about 1 to 10 (Philip 1970; Dybbs and Edwards 1982; Yalin

and Franke, 1961). One application of pipe flow analogies to a porous medium is the series of

models named hydraulic radius theories that make use of the fact that the intrinsic permeability

has dimensions and is described by a length term, the hydraulic radius, R. The hydraulic radius is

defined as the ratio of sample volume to the surface area of the pores (again, difficult to assess

for most porous media). The basic form of the permeability relation is:

k=cR2

∅ (η)(30)

where, c is a dimensionless constant and represents some function of the porosity. Of the

hydraulic radius theory and that of Kozeny (1993) is the most widely used description of

permeability. Kozeny's theory couples the steady-state, Navier-Stokes equation (neglecting

inertia terms) with Darcy's law and describes the permeability as:

k= cn2

S2 (31)

where, c is a dimensionless shape factor, analytically derived to be approximately 0.5, and S is

the specific surface of the flow tube - a measure of the hydraulic radius.

2.3 POST-DARCY FLOW

2.3.1 Steady Flow Experiment

Darcy's law is only valid over a limited range of conditions and is commonly referred to

as the Darcy or linear laminar regime of flow. Outside of this range the relation between flowrate

and energy loss is not linear. It is the "Post-Darcy" regime that is relevant for consideration in

19

this law. Two forms are commonly quoted, the series type and the exponential type. The

exponential type is of the form:

ἰ=a' q f (32)

where,

ἰ = hydraulic gradient, = h/L

q = bulk velocity, = Q/A nvp and

ɑ´,f = coefficients.

Note that if f = 1 and a'=μ/ρgk then Darcy's law is expressed. This form of flow law is preferred

by some (Barends, 1980) because of its similarity to pipe flow equations and thus compatibility

with standard measures such as drag coefficient. Muskat (1989), using dimensional analysis,

showed that (f) has an upper limit of 2 for gravity flows, signifying fully developed turbulent

flow in all pores in analogy to the pipe flow law. From the result of the dimensional analysis,

Muskat has shown that:

a '=ρ D f−3 v2−f (33)

The resistance of the medium must be inversely proportional to the viscosity of the fluid

hence a' must be proportional to viscosity. For physically meaningful results (f) must therefore

be less than 2. A drawback to this formulation is that the coefficients (a' & f) vary continuously

over flow regimes and are therefore difficult to parameterize.

2.3.2 Flow

Flow is defined as the quantity of fluid (gas, liquid or vapor) that passes a point per unit

time. A simple equation to represent this is:

20

Flow=QUOTEQuantity (Q)

Time(t )(34)

Flow is sometimes written as Q (rate of change of mass or volume). Due to the number of

different fluids that are given to our patients during a routine an aesthetic, flow is obviously an

important area of mechanics to understand. Industrial flow measurements include measuring of

flow rate of solids, liquids and gases. There are two basic ways of measuring flow; one on

volumetric basis and the other on weight basis. Solid materials are measured in terms of either

weight per unit time or mass per unit time. Very rarely solid quantity is measured in terms of

volume. Liquids are measured either in rate or in weight rate. Gases are normally measured in

volume rate.

2.3.3 Flow Regimes

Flow can be classified as laminar or turbulent. Laminar flow is characterized by parallel

streamlines, due to viscous forces that are dominant over shear stresses. On the other hand, for

turbulent flow conditions, the streamlines do not have a clear pattern. Rather, the flow pattern is

more randomly because shear stresses are dominant. Between the laminar and turbulent flow

there is a transition region in which the flow may switch between these two conditions in an

apparently random fashion (Young et al., 2004).

The Reynolds number expresses the ratio between inertial and viscous forces and is used

to differentiate between turbulent and laminar flow.

Re=ρvlμ

(35)

where,

Re= Reynold’s number,

density of the fluid,

21

fluid velocity,

µ = dynamic viscosity, and

characteristics linear dimension.

2.3.4 Flow in Conduits

Specific discharge and the head loss by friction in tubes are proportional. Pressure drop in

conduits depend also on its geometry, roughness and flow velocity. To quantify the roughness is

commonly used a friction factor. The Darcy-Weisbach equation describes the relationship

between the friction factor (f), head loss (h), discharge (Q), area (a), diameter (D) and length (L).

Losses occur mostly at a few isolated constrictions or collapses into the conduit system.

Q=( 2 Dg a2

f ) 12

hl

1 /2

(36)

Under the laminar flow regimen roughness has not a significant effect, in this case the

Darcy-Weisbach friction factor depends only on the Reynolds number and its formula was

derived by Poiseuille (Tullis, 1989).

f =64ℜ (37)

Then for laminar flow is used the Hagen-Poiseuille equation

Q= πD4 ρg128 μ

.hl(38)

In fully turbulent flow conditions the value of the friction factor is given by the eqn (38).

As the velocity increases, “f ” becomes independent of Reynolds number and depends only on

pipe roughness.

1f=−2 log( e

3.7D+ 2.51

R e√ f)(39)

22

where e is the absolute roughness in the conduit “Darcy-Weisbach “f ” values determined in

karst hydrology investigations lie in the range 0.039 to 0.340 (Ford, 2007). An empirical formula

also used for this type of calculation is Manning in equation (40). This equation is valid for

turbulent flow.

Q=A1η

.D4

0.667√ hl(40)

where,

friction Manning coefficient.

In the equation (10) for circular conduits ℓ is equal to the pipe diameter. Through pipe

flow experiments, Reynolds found two critical numbers. One, RC1, for the transition from laminar

to turbulent flow, when steady motion changes into eddies and the second critical number RC2 for

the opposite process, the transition from turbulent to laminar flow. RC1 was determined with the

color band method and is equal to 12830 and using the pressure loss method RC2 as 2030.

Without precautions to eliminate disturbances for general purposes in pipes the flow is

considered laminar when Reynolds number is smaller than 2100 and turbulent when is larger

than 4000. Between this two values flow can be classified as transitional (Young et al., 2004).

The critical Reynolds number could be importantly reduced in tubes with variable diameter or

curvature (Hillel and Hillel, 2004). In natural conduits the critical value can be set as greater than

1000 (Shoemaker et al., 2007).

White (2002) stated that in fractures when the Reynolds number is in the range of 500,

the flow is not longer laminar, this occurs when the aperture has at least 1 cm, at which starts

being considered conduit.

23

2.3.5 Non Stationary Laminar Flow

Few studies have addressed the case of accelerated or cyclic flow. In this case an

additional external force must be required to accelerate the mass of water (Dean and Dalrymple,

1984; den Adel, 1987). This extra force is M.dvp/dt where M is mass of accelerated water and vp

is the actual velocity of water in the pore space. Then the total force balance per unit volume (in

terms of bulk velocity q = nvp) results in

ἰ= δHδx

= μρgk

q+ 1g

δqδt

(41)

The general form of this equation is

ἰ=aq+C1 δqδt

(42)

where the coefficient C1 is called the acceleration coefficient and is thought to be a constant for

any given media (as is the coefficient a). If the velocity q is described by a sine function with

period T, i.e.:

q=q0 sin( 2 πtT )(43)

then from equation (42):

ἰ=q0 (asinωt +C' ωcosωt ) (44)

with ω=2 π /T or if θ is defined by:

ἰ0 cosθ=a q0(45 a)

csinθ=C' ωq0(45 b)

where ἰ0is the magnitude of the gradient, then the hydraulic gradient can be best described by

ἰ=ἰ0 sin (ωt+θ )(46)

24

where θ is a phase shift induced between the velocity and energy gradient.

Den Adel (1987) wrote equation (42) in terms of the potential H = P +ρgh, then assumed a

sinusoidal applied hydraulic gradient ἰ = ἰ0sin(2πt/T) so that;

ἰ= qK

+ 1ng

δqδt

=ἰ0 sinωt (47)

The analytical solution to equation (47) is q=c1 sinωt+c2cosωt with

c1=ἰ0

K1

( 1K )+( ω

ng)(48 a)

and

c1=ἰ0 ω

K1

( 1K )+( ω

ng)(48 b)

The velocity, q, may be written as

q=q0 sin (ωt−θ )(49)

If q0 = (c12 + c2

2)1/2 and θ = tan-1(-c2/c1). Thus for laminar flow, if either the velocity or

gradient is described by a sine function then both will be sine functions with a phase shift

between them.

From equations (48) and (49) Den Adel defined a period-dependent cyclic permeability K0(T) as:

( 1K0 (T ) )2=

ἰ02

q02 =( 2 π2

ngT )+( 12

K )(50)

For applied hydraulic gradients with small periods, the permeability K0(T) becomes much

smaller (i.e. increased resistance) than the stationary Darcy permeability, K, (Figure 6). It

appears that as the period of the applied gradient decreases less pore water can be brought into

motion, and the flow becomes less dependent on the applied pressure gradient.

25

The concept of permeability of a porous medium has yet to be well defined, for it is

dependent upon, and sensitive to, many parameters that are difficult to control even in a

laboratory environment. To this day the permeability coefficient must still be determined

indirectly in laboratory permeameter tests, for no reliable general predictive formulae have been

produced.

Figure 6: Period-dependent Permeability

2.4 POROUS MEDIA

A porous medium (or a porous material) is a material containing pores (voids). The

skeletal portion of the material is often called the "matrix" or "frame". The pores are typically

filled with a fluid (liquid or gas). The skeletal material is usually a solid, but structures like

foams are often also usefully analyzed using concept of porous media. A porous medium is most

often characterized by its porosity. Other properties of the medium (e.g., permeability, tensile

strength, electrical conductivity) can sometimes be derived from the respective properties of its

26

constituents (solid matrix and fluid) and the media porosity and pores structure, but such a

derivation is usually complex. Even the concept of porosity is only straight forward for a

poroelastic medium. Often both the solid matrix and the pore network (also known as the pore

space) are continuous, so as to form two interpenetrating continua such as sponge.

Porous materials are measured by the amount of empty or void space within the object,

and these materials are capable of absorbing gas or liquids like a sponge within the void space.

Porous materials occur naturally in nature and are also manufactured for a multitude purposes.

2.4.1 Types of Porous Media

2.4.1.1 Natural porous materials

These are the kind of porous media that are naturally occurring in nature state such as

rocks and soil (e.g., aquifers, petroleum reservoirs), zeolites, biological tissues (e.g. bones, wood,

cork). Soil and rock are naturally occurring porous materials; the pores in soil make space for

roots and insects to retain water and nutrients to nourish life. Soils are highly permeable and will

sort because they need to allow water to flow well to protect plant life from drowning. Also, the

pores in rock have several types of occurring porosity due to fractures and chemical alterations

caused by leeching minerals. Vuggy porosity, for example is the dissolution of a rocks larger

features, which creates large holes.

2.4.1.2 Manufactured porous materials

These are the kind of porous materials that occurred artificially or man-made use. The

most common purpose of porous material are manufactured for liquid filtration, such as cements

and ceramics can be considered as porous media. Ceramics filters, for instance have millions of

micro pores that trap dirt, bacteria or living organisms too large to fit through, thus making the

27

water safe to drink. Other pores materials include manufacturing metals with an increasing

number of pores to reduce weight, save materials and increase heat retention. Sinter paper, for

example is made out of powdered copper or stainless steel. The concept of porous media is used

in many areas of applied science and engineering: filtration, mechanics (acoustics, geo-

mechanics, soil mechanics, rock mechanics), engineering (petroleum engineering, bio-

remediation, construction engineering), geosciences (hydrogeology, petroleum geology,

geophysics), biology and biophysics, material science, etc.

2.4.2 PROPERTIES OF POROUS MEDIA

The properties of porous media include density, porosity and permeability.

2.4.2.1 Density

This is the ratio of mass of the void to the total volume. For unconsolidated media,

consider the density of individual particles (particle density):

Pp=M p

V p

(51)

For the bulk material (i.e., in a quantity at least equal to the REV), we consider the bulk density:

Pp=M p

V T

(52)

2.4.2.2 Porosity

Porosity (n) is defined as the ratio of void space to the total volume of media; it is

measured through a simple calculation that divides void volume by the material volume.

η=V p

V T

(53)

where;

28

Vv = volume of void space (L3)

VT = total volume (volume of solids plus volume of voids) (L3)

Porosity is not a function of grain size, but rather grain size distribution. Spherical models

comprised of different sized balls will always yield a lower porosity than the uniform model

arranged in a similar packing arrangement. Primary porosity in a material is due to the properties

of the soil or rock matrix, while secondary porosity is developed in the material after its

emplacement through such processes as solution and fracturing.

The porosity of fluid depends on the degree of compaction, a material with high

compaction is significantly reduces porosity by shrinking the sizes of the pores or filling them

with a finer sediment. The naturally occurring or human-induced sorting of a porous material

greatly affects porosity. In sediment, for example soil naturally sorts itself into layers; a well

sorted layer of sediment will contain grains of the same or similar sizes, which greatly increases

porosity. Badly sorted soil contains grains of a wide range of size, greatly reducing porosity. This

is usually in the range of 0.1 - 0.5.

The void volume is equal to the total volume minus the volume of the particles:

V V =V T−V P(54 )

Porosity, particle density, and bulk density are then related as follows:

η=(V T−V P )

V T

(55)

2.4.2.3 Permeability

This is refers to as the amount of air, water or gas that the porous material can absorb at a

given time and how quickly it flows. When water flows from a soil of low permeability into a

soil of higher permeability, less area is required to accommodate the same quantity of water and

29

lower gradients are needed. If the flow is from high permeability into lower permeability, steeper

(or higher) gradient are required and a relatively more area is needed to accommodate the flow

(Cedergren, 1976). If layers of beds of porous media of different porosity are considered and it is

assumed that each layer is homogeneous and isotropic, then each layer is however characterized

by a different hydraulic conductivity rendering the sequence as a whole heterogeneous. It was

found that for horizontal flow, the most permeable unit dominates the system. For vertical flow

the least permeable unit dominates the system. Under the same hydraulic gradient, horizontal

flow is of the order of six orders of magnitude faster than vertical flow (Domenico and Schwartz,

2008). Fluid flow through a porous material of permeability κ, by Darcy is generally written as

(Frick and Taylor, 1978; Olowofela and Adegoke, 2005).

V= κμ∇ ( P− ρgz )(56)

This can be expressed as;

V=− κ

μ(dp)

ds−ρg

dzds

(57)

where,

s = distance in the direction of flow,

v = volume flux across a unit area of the porous medium in unit time along the flow path,

z = vertical coordinate, considered downward,

= density of the fluid,

g = acceleration of gravity,

dpds

= pressure gradient along s at the point to which v refers,

µ = viscosity of the fluid,

30

κ = permeability of the medium.

P = pressure gradient of the medium.

CHAPTER THREE

3.0 RESEARCH METHODOLOGY AND MATERIALS

3.1 MATERIALS

The materials for this test are sand and water. The clean sand sample were collected from

eight (8) different points at five metres (5m) interval in the riverbed of Osun River in Osogbo, by

the help of the local sand dredgers present in the locality. The samples are labelled accordingly

for easy identification. After the sand sample is collected it is allowed to dry for some days

before placing into the sack and transported to the laboratory for the necessary tests to be

performed on it. The water that is used for the test is the one sourced from the dug well around

the geotechnical laboratory of Federal University of Technology Akure, as this is a good source

of water around since it is clean. Immediately the samples are taken to the laboratory a few

quantity is taken for moisture content test to determine the natural moisture content of the sand,

after that the sand is then placed into the oven so as to remove all the water present in the sand to

get the right condition needed for carrying out the necessary tests.

3.2 EXPERIMENTAL PROCEDURE

The laboratory tests performed were carried out in the Geotechnical laboratory of Civil

Engineering Department, Federal University of Technology, Akure. Two categories of tests were

carried out namely; Preliminary test and Engineering test. The preliminary test is the initial test

carried out on the sample to identifying and classifying the soil. The tests include; Natural

moisture content, particle size distribution and specific gravity (test were conducted based on

31

procedures outlined in BS 1377(1990). The Engineering test is the strength test, which is used to

investigate the hydraulic properties of the sample.

Laboratory tests were conducted, recorded and the results were analyzed graphically to

ascertain the suitability of the river bed sand material satisfying standard for flow of fluid to pass

is pores. Test carried out on the sample is permeability test and the results were discussed.

3.2.1 Preliminary Test

The preliminary test include: natural moisture content, specific gravity and particle size

distribution.

Atterberg’s limit tests cannot be used to classify the sand because there is no clay content in the

soil; hence the basic test that can be use is the particle size distribution (analysis) to be able to

determine the gradation of the soil.

The procedures for the various tests are carried out in accordance with stipulated in BS 1377 –

1990: 1- 8.

3.2.1.1 Natural Moisture Content

The moisture content of a soil is assumed to be the amount of water within the pore space

between the soil grains which is removable by oven drying at a temperature not exceeding

110°C. The moisture content has a profound effect on soil behavior.

Apparatus

i. a drying oven, capable of maintaining a temperature 105oC to 110oC.

ii. a balance readable to 0.01g

iii. a corrosion – resistant container.

32

Procedure

Clean and dry the metal container and weigh to the nearest 0.1g. This is taken as m 1, the

sample is collected and placed in the container, and the lid is then replaced. The weight of the

container and it content are taken and recorded to the nearest 0.1g (m2). The lid is removed, and

the container with it lid and contents are then placed in the oven and allowed to dry at a

temperature of 105oC to 110oC.

After drying, the container and its content are removed from the oven and allowed to

cool. The lid is replaced and then weighed to the nearest 0.1g (m3).

Calculations and Expression of Results

Calculate the moisture content of the soil specimen, w, as a percentage of the dry soil

mass to the nearest 0.1%, from the equation:

w=m2−m3

m3−m1

×100 (58)

where;

m1 = mass of container

m2 = mass of container and wet soil

m3 = mass of container and dry soil.

33

3.2.1.2 Specific Gravity Test

The specific gravity of solid particles is the ratio of the mass density of solids to that of

water.

Apparatus

i. 50ml density bottle with stopper.

ii. Oven (105oC to 110oC).

iii. Constant temperature water bath (27oC).

iv. Vacuum desiccators.

v. Vacuum pump.

vi. Weighing balance, accuracy 0.001g.

vii. Spatula

viii. Wash bottle containing air – free distilled water.

Procedure

The sample of about 100g is collected and sieved through a 2mm test sieve. Two

specimens, each of 50g are then obtained by rifling, and then oven dried at 105 oC to 110oC, the

samples are then stored in air – tight containers.

The gas jar is cleaned, along with its round cover. The weight of the gas jar with the cover is

measured and recorded to the nearest 0.1g (m1). The first soil specimen is transferred to the gas

jar. The weight of the gas jar, content and the cover is measured and recorded to the nearest 0.1g

34

(m2). Sufficient air – free water is added to cover the soil in the gas jar say one third the depth of

the jar and then stirred with the help of a stirrer. The stirred content in the gas jar is allowed to

stand for 30mins. Then more water is then added to the top of the gas jar and also stirred so as to

remove any entrapped air in the soil, the sample with the water is then left to stand for 24 hours

to allow for the best settling of all particles in the soil. Note that during this period the cover is

placed. The gas jar is then weighed with all the contents and the value is recorded as (m 3). After

taking the weight, the content in the gas jar is emptied and the gas jar is wiped clean to remove

any particle of the soil. Then the water is then placed inside the jar without bubbles and the cover

attached and the weight of the gas jar, water and the cover is taken and recorded as (m4).

The procedure is then repeated with the second specimen. Then the calculations are made

to determine the average specific gravity of the sand, the formula is given in the next session.

Calculations

Calculate the relative density from the equation:

Gs=m2−m1

( m4−m1 )−(m3−m2)(59)

where,

m1 = mass of the density bottle (g)

m2 = mass of bottle and dry soil (g)

m3 = mass of bottle, soil and water in (g)

m4 = mass of bottle, when full only of water in (g)

35

The average of the two results is then calculated.

3.2.1.3 Particle Size Determination (PSD)

Two methods of sieving are known. Wet sieving is the definitive method that is

applicable to cohesion-less soils. Drying sieving is suitable only for soils containing insignificant

quantities of silt and clay.

Wet Sieving Method

This method encompasses the quantitative determination of the particle size distribution

in an essentially cohesion-less soil, down to the fine sand size. The combined silt and clay

fraction can be obtained by the difference (ASCE, 2002).

The procedure given involves the preparation of the sample by wet sieving to remove silt

and clay – sized particles, which are rejected, followed by dry – sieving of the remaining coarser

material.

Apparatus

i. Test sieves having the following sizes: 4.36mm, 2.36mm,1.70mm, 1.18mm, 600µm,

500µm, 425µm, 212µm, 150µm, 75µm and appropriate receiver.

ii. Drying oven capable of maintaining a temperature of 105°C to 110°C.

iii. A balance readable to 0.1g

iv. Sieve brushes, and a wire brush.

36

v. Mechanical sieve shaker.

Sample Preparation

The sample collected is passed through a 4.75mm I.S sieve. The fraction retained on the

4.75mm I.S sieve is discarded, and that passing through the sieve for the fine sieve analysis. The

fine sample is then soaked in a water, to ensure the dispersion of the particles. The sample is

soaked for some hours.

After soaking, the sample is then washed to remove any silt content. The weighing is

done using the 75µm sieve and the pan. The washing is done by using water, till the sample is

clean. The cleaned sample is then dried in the drying oven for 24hours and at a temperature of

110°C.

Procedure

250g of the sample is washed and dried. The dried sample is removed from the oven and

allowed to cool under atmospheric temperature. After cooling, the sieves are arranged in the

order; 4.36mm, 2.38mm, 1.70mm, 1.18mm, 600µm, 500µm, 425µm, 212µm, 150µm, 75µm and

the receiver. The sample that is washed is then poured on the top sieve, that is the 4.36mm sieve

and then shaken for some minutes.

The mass retained on each sieve is then weighed using the balance. The mass on the pan is then

calculated, together with the washed mass as the mass of fines.

Calculation

37

After the weight retained is gotten, the weight passing is then calculated. This is calculated using

the equation; weight retained is equal to total weight minus weight passing.

W R=W T−W P (60)

The percentage passing and the percentage retained are then calculated. The sieve graph is then

plotted of percentage passing versus sieve size. The percentage passing the 75µm sieve is taken

as the percentage fines.

3.2.2 Engineering Property Test

Engineering property test are test used to determine the strength of soil, or some other

parameters that will help the engineer to be able to give useful information in regards to the soil.

The basic engineering test for this work is the permeability test as it helps to determine the

hydraulic conductivity of the soil.

3.2.2.1 Falling Head Permeability Test

The falling head permeability test is another experimental procedure to determine the

coefficient of permeability of sand. A schematic diagram of a falling head permeameter is shown

in Fig. 7. This consists of a specimen tube essentially the same as that used in the constant head

test. The top of the specimen tube is connected to a burette by plastic tubing. The specimen tube

and the burette are held vertically by clamps from a stand. The bottom of the specimen tube is

connected to a plastic funnel by a plastic tube. The funnel is held vertically by a clamp from

another stand.

38

Figure 7: Falling head permeability arrangement

Equipment

i. Falling head permeameter

ii. Balance sensitive to 0.1 g

iii. Thermometer

iv. Stop-watch

v. Burette

vi. Tripod stand

vii. Funnel

viii. Porous stone

39

Procedure

The permeameter (mould) with base plate and gasket attached is weighed. The inside

diameter (d) of the permeameter is measured also. The area (a) of the standpipe is also measured

and the values are calculated and recorded. Since the burette is graduated in volume, measuring

the distance between graduations will yield as simple in direct computation. A small portion of

the soil sample is taken for water content determination. The air-dried soil sample is placed into

the permeameter and compacted to the desired density. The permeameter with base plate and

gasket attached plus compacted soil is weighed and also the length of the specimen is measured

in centimeters. The dry density and void ratio of the specimen is determined. A piece of porous

disc is placed on the top of the specimen and a spring on the porous disc. The permeameter rim is

carefully cleaned. With its gasket in place, press down the top plate against the spring and attach

it securely to the top of the permeameter cylinder, making an air-tight seal. The spring should be

compressed and should apply a pressure to the compacted soil specimen to keep it in place when

it is saturated with water. Ensure that the outlet pipe is open so that water can back up through

the specimen. This procedure is done so as to saturate the sample with a minimum amount of

entrapped air. When water in the plastic inlet tube on top of the mould reaches equilibrium with

water in the sink (allowing for capillary rise in the tube), the specimen may be assumed to be

saturated. When the water level is stable in the inlet tube of the permeability mould, take a hose

clamp and clamp the exit tube. Remove the permeameter from the sink and attach it to the rubber

tube at the base of the burette, which has been fastened to a ring stand. Fill the burette with water

from a supply, which should be temperature-stabilized (and de-aired if desired). Now de-air the

lines at the top of the specimen by opening the hose clamp from the burette and opening the

petcock on top of the cover plate. Allow water to flow (but keep adding water to the burette so it

40

does not become empty) from the petcock. When no more air comes out, close the pet-cock. Do

not close the inlet tube from the burette. Remember that the exit tube is still clamped shut. The

burette is filled to a convenient height, and measure the hydraulic head across the sample to

obtain h1. The exit tube (and petcock) is opened and simultaneously start timing the test. The

water is allowed to flow through the sample until the burette is almost empty. Simultaneously

record the elapsed time and clamp only the exit tube. Measure the hydraulic head across the

sample at this time to obtain h2. Take the temperature each time. The burette is refilled with

water and the test is repeated three additional times. Take the temperature each time. To check

on whether the sample is saturated, one may collect the water coming out of the exit tube and

compare this volume with that entering the sample.

Calculation

From the above parameters gotten the hydraulic conductivity can be calculated using the

formula;

κ=2.303 aLAt

logh1

h2

(61)

where;

a = area of cross section of the burette;

A = area of cross section of the permeameter;

L = length of the sample in the permeameter;

h1 = head at the start time;

h2 = head at the end time;

t = time difference when the head falls from h1 to h2; and

κ = coefficient of hydraulic conductivity in cm/s.

41

3.3 SOFTWARE USED

3.3.1 Statistical Packages for the Social Sciences (SPSS)

All the hydraulics properties of the river sand were analyzed using the regression model

of the Statistical Package for Social Sciences (SPSS). It is a comprehensive system for analyzing

data. This can take data from almost any type of file and use them to generate tabulated reports,

charts, plots of distributions and trends, descriptive statistics and complex statistical analysis

(Reynald, 2006).

3.3.2 Regression Analysis

Regression is a statistical technique to determine the linear relationship between two or

more variables and is primarily used for prediction and causal inference (Cambell & Cambell,

2008). In its simplest (bivariate) form, regression shows the relationship between one

independent variable (X) and a dependent variable (Y), as in the formula below:

Y=β0+β1 X+U (62)

The magnitude and direction of that relation are given by the slope parameter (β 1), and the status

of the dependent variable when the independent variable is absent is given by the intercept

parameter (β 0). An error term (u) captures the amount of variation not predicted by the slope and

intercept terms. The regression coefficient (R2) shows how well the values fit the data.

Y is an observed random variable;

X is an observed non-random or conditioning variable;

β0 is an unknown parameter, known as the constant or intercept term;

β1is an unknown parameter, known as the coefficient or slope parameter; and

u is an unobserved random variable, known as the error or disturbance term.

42

CHAPTER FOUR

4.0 RESULTS AND DISCUSSION

4.1 NATURAL MOISTURE CONTENT RESULT

The moisture content of a soil is assumed to be the amount of water within the pore space

between the soil grains which is removable by oven drying at temperature not exceeding 110oC.

The moisture content has a profound effect on soil behavior. Table 1 – 8 shows the results of

eight samples for the moisture content A - H.

The table gives the analysis of the result of natural moisture content for the sample

location.

Table 1: Natural Moisture Content of Sample from Location A

Container No A B C

Wet soil and container (g) 109.1 168.3 111.0

Dry soil and container (g) 107.9 166.6 110.2

Mass of container (g) 46.9 120.4 46.6

Mass of dry soil (g) 61 46.2 63.6

Mass of moisture loss (g) 1.2 1.7 0.8

Moisture content (%) 1.97 3.68 1.26

Average moisture content 2.30

43

Table 2: Natural Moisture Content of Sample from Location B

Container No A B C




Mass of dry soil (g) 55.5 66.5 66.0




Table 3: Natural Moisture Content of Sample from Location C

Container No A B C








44

Table 4: Natural Moisture Content of Sample from Location D

Container No A B C








Table 5: Natural Moisture Content of Sample from Location E

Container No A B C







45


Table 6: Natural Moisture Content of Sample from Location F

Container No A B C








Table 7: Natural Moisture Content of Sample from Location G

Container No A B C







46


Table 8: Natural Moisture Content of Sample from Location H

Container No A B C








4.2 SPECIFIC GRAVITY RESULT

The true specific gravity of a soil is actually the weighted average of the specific gravities

of all the mineral particles present in the soil. The result shows that the sample is sand due to the

fact that its specific gravity ranges from 2.65 to 2.67 as stated by Krishna (2002) and Table 9 –

16, shows the result of the soil sample of specific gravity.

47

The table gives the result of the entire specific gravity test as performed in the laboratory.

Table 9: Specific Gravity of Sample from Location A

Sample No A B

Weight of jar + water (full) (m4) g 615.9 612.9

Weight of jar + soil +water (m3) g 647.1 644.2

Weight of jar + soil (m2) g 286.9 286.1

Weight of jar (m1) g 236.9 236.1

Gs 2.65 2.67

Average Gs 2.66

Table 10: Specific Gravity of Sample from Location B

Sample No A B





Gs 2.68 2.66

48

Average Gs 2.67

Table 11: Specific Gravity of Sample from Location C

Sample No A B





Gs 2.64 2.66

Average Gs 2.65

Table 12: Specific Gravity of Sample from Location D

Sample No A B





Gs 2.65 2.65

Average Gs 2.65

49

Table 13: Specific Gravity of Sample from Location E

Sample No A B





Gs 2.86 2.46

Average Gs 2.66

Table 14: Specific Gravity of Sample from Location F

Sample No A B





Gs 2.65 2.69

50

Average Gs 2.67

Table 15: Specific Gravity of Sample from Location G

Sample No A B





Gs 2.66 2.64

Average Gs 2.65

Table 16: Specific Gravity of Sample from location H

Sample No A B




51


Gs 2.65 2.67

Average Gs 2.66

4.3 PARTICLE SIZE DISTRIBUTION (PSD) RESULT

This was carried out to determine the fineness of the sample i.e. to have an idea of the

quality of fine particle contained in the sample. Determining the mass of soil sample left on each

sieve, the following calculations can be made:

% retained on sieve=mass of soil retainedTotalmass

×100(63)

Cumulative percentage retained on any sieve = sum of percentages retained on all coarse sieves.

The results of the particle size analysis done on 250grams on each samples is also used to

determine, if the soil uniformity graded or well graded, (Hazen 1991) proposed the following

equations as;

i. Uniformity coefficient

Cu=D60

D10

(64)

ii. Coefficient of curvature

C c=D30

2

D10 × D60

(65)

D10 = effective grain size with 10% finer particles.

52

D30 = the size of particle at 30% finer on the gradation curve.

D60 = Diameter of the particle at 60% finer on the grain size distribution curve.

Table 17: Particle Size Distribution Result of Sample from Location A

Sieve No Diameter (mm) Mass Retained % Mass Retained % Passing

4 4.76 0.00 0.00 100

8 2.36 0.00 0.00 100

12 1.70 0.10 0.05 99.95

16 1.18 0.20 0.10 99.85

30 0.600 2.20 1.10 98.75

35 0.500 2.90 1.45 97.30

40 0.425 0.40 0.20 97.10

70 0.212 148.5 74.25 22.85

100 0.150 18.6 9.30 13.55

200 0.075 9.3 4.65 8.90

Pan 17.8 8.90 0.00

53

Table 18: Particle Size Distribution Result of Sample from Location B


4 4.76 1.46 0.56 99.44

8 2.36 3.70 1.48 97.96

12 1.70 6.30 2.52 95.44

16 1.18 23.50 9.40 86.04

30 0.600 102.10 40.84 45.20

35 0.500 16.70 6.68 38.52

40 0.425 11.10 4.44 34.08

70 0.212 60.60 24.24 9.84

100 0.150 14.70 5.88 3.96

200 0.075 8.60 3.44 0.52

Pan 1.30 0.52 0.00

54

Table 19: Particle Size Distribution Result of Sample from Location C


4 4.76 1.20 0.48 99.52

8 2.36 1.60 0.64 98.88

12 1.70 1.20 0.48 98.40

16 1.18 1.20 0.48 97.92

30 0.600 25.70 10.28 87.64

35 0.500 5.20 2.08 85.56

40 0.425 20.30 8.12 77.44

70 0.212 176.00 70.40 7.04

100 0.150 11.00 4.40 2.64

200 0.075 5.60 2.24 0.40

Pan 1.00 0.40 0.00

Table 20: Particle Size Distribution Result of Sample from Location D


4 4.76 0.00 0.00 100

55

8 2.36 0.40 0.16 99.84

12 1.70 0.20 0.08 99.76

16 1.18 0.10 0.04 99.72

30 0.600 0.70 0.28 99.44

35 0.500 0.30 0.12 99.32

40 0.425 1.30 0.52 98.80

70 0.212 237.7 95.08 3.72

100 0.150 5.20 2.08 1.64

200 0.075 3.70 1.48 0.16

Pan 0.40 0.16 0.00

Table 21: Particle Size Distribution Result of Sample from Location E

Sieve No Diameter (mm) Mass Retained %Mass Retained % Passing

4 4.76 1.30 0.52 99.48

8 2.36 3.50 1.40 98.08

56

12 1.70 2.00 0.80 97.28

16 1.18 1.60 0.64 96.64

30 0.600 106.10 42.44 54.20

35 0.500 4.40 1.76 52.44

40 0.425 2.30 0.92 51.52

70 0.212 105.50 42.20 9.32

100 0.150 14.00 5.60 3.72

200 0.075 8.20 3.28 0.44

Pan 1.10 0.44 0.00

Table 22: Particle Size Distribution Result of Sample from Location F


4 4.76 2.00 0.80 99.20

8 2.36 3.80 1.52 97.68

12 1.70 2.20 0.88 96.80

16 1.18 2.10 0.84 95.96

57

30 0.600 7.50 3.00 92.96

35 0.500 10.40 4.16 88.80

40 0.425 12.60 5.04 83.76

70 0.212 165.10 66.44 17.32

100 0.150 27.30 10.92 6.40

200 0.075 14.20 5.68 0.72

Pan 1.80 0.72 0.00

Table 23: Particle Size Distribution Result of Sample from Location G


4 4.76 0.00 0.00 100

8 2.36 0.00 0.00 100

12 1.70 0.00 0.00 100

16 1.18 12.00 4.80 95.20

30 0.600 117.00 46.80 48.40

35 0.500 5.90 2.36 46.04

58

40 0.425 4.90 1.96 44.08

70 0.212 97.00 38.80 5.28

100 0.150 8.20 3.28 2.00

200 0.075 4.30 1.72 0.28

Pan 0.70 0.28 0.00

Table 24: Particle Size Distribution Result of Sample from Location H


4 4.76 0.70 0.28 99.72

8 2.36 1.40 0.56 99.16

12 1.70 1.60 0.64 98.52

16 1.18 3.90 1.56 96.56

30 0.600 114.40 45.76 51.20

35 0.500 16.80 6.72 44.48

40 0.425 14.30 5.72 38.76

70 0.212 85.00 34.00 4.76

100 0.150 8.00 3.20 1.56

59

200 0.075 3.30 1.32 0.24

Pan 0.60 0.24 0.00

60

Particle Size Distribution ChartBritish Standard Sieve Sizes

CLAY

FINE MEDIUM TO COARSE SAND WITH LITTLE SILT

DESCRIPTION

FINE MEDIUM TO COARSE SAND


FINE TO MEDIUM SAND

Legend

FineSAND

CoarseMediumGRAVELMediumFine

Sample №& Depth

CoarseMediumFineSILT

BOULDERSCOBBLESCoarse

Clay (%)Soil Composition

Gravel Sand (%) Silt (%)

1.04 8.90

84.11

91.21

99.41

0.52

0.40

0.16

B

C

D

90.06

0.00

0.00

0.00

15.37

8.39

0.43 0.00

A

0.053

0.075

0.15 0.25 0.3

0.425

0.6

1.18 1.7

22.36

4.75 6.7

9.5

13.2

20.0

26.5

37.5

53

600.0200.0

60.0

7563

14100.2

0.06

0.02

0.006

0.002

6

0

10

20

30

40

50

60

70

80

90

100

0.001 0.01 0.1 1 10 100 1000

Cum

ulat

ive

% P

assi

ng

Sieve Size (mm)

Sieve Size (mm)

Figure 8: Combined Particle Size Distribution Curve of Sample Location A, B, C & D

61

Particle Size Distribution ChartBritish Standard Sieve Sizes

0.00

E

H

84.27

0.00

0.00

0.00

6.33

14.28

15.06

0.44

92.95

85.44

84.70

0.72

0.28

0.24

F

G

BOULDERSCOBBLESCoarse

Clay (%)Soil Composition

Gravel Sand (%) Silt (%)

15.29

Fine

Sample №& Depth

CoarseMediumFineSILT

Fine CoarseMediumGRAVELMediumCLAY


DESCRIPTION

FINE TO MEDIUM SAND



Legend

0.053

0.075

0.15 0.25 0.3

0.425

0.6

1.18 1.7

22.36

4.75 6.7

9.5

13.2

20.0

26.5

37.5

53

600.0200.0

60.0

7563

14100.2

0.06

0.02

0.006

0.002

6

0

10

20

30

40

50

60

70

80

90

100

0.001 0.01 0.1 1 10 100 1000

Cum

ulat

ive

% P

assi

ng

Sieve Size (mm)

Sieve Size (mm

Figure 9: Combined Particle Size Distribution Curve of Sample Location E, F, G & H

62

4.4 PERMEABILITY OF SAND

The coefficient of permeability can be computed using Darcy’s Law. Discharge in unit

time (Q).

Q=κἰA=κ ∙hl

∙ A (66)

Therefore, κ=2.3 aLAt

× logh1

h2

Permeability steadily decreases with increase in sand content. Due to the values of k

obtained and compared to the classification of soils stated by Terzaghi and Peck (1967) in their

coefficients of permeability (κ), soil over 10-1 k is Gravel and the degree is high and k between

the ranges of 10-1 to 10-3 is fine sand, clean sand, sandy gravel and the degree is medium. Hence

the soil material is fine sand. The transport of water through a soil will be faster if the soil has a

higher coefficient of permeability than if it has a lower value (Craig, 1992).

The results were the values of the test obtained from tests performed in the laboratory.

Description OSUN RIVER SAND

Length of Soil Sample (L) 11.4 cm

Diameter of Soil Sample (D) 10 cm

Area of Soil Sample (A) 78.55 cm2

Area of Standpipe (a) 1.00 cm2

Temperature of the water 27

63

Table 25: Coefficient of Permeability Result of Sample for Location A

S/NO Head (h )₁ (cm)

Head (h )₂ (cm)

Time (t) (sec)

log (h /h )₁₀ ₁ ₂ Κ Average κ cm/sec

1 16.40 2.00 10 0.9138 0.03054

1.43 × 2 30.40 16.40 10 0.2680 0.00896 3 43.20 30.40 10 0.1526 0.00509 4 12.00 5.00 10 0.3802 0.01271

Table 26: Coefficient of Permeability Result of Sample for Location B


Head (h )₂ (cm)

Time (t)(sec)

log (h /h )₁₀ ₁ ₂ Κ Average κ (cm/sec)

1 50.00 35.80 10 0.1451 0.00485

1.32 × 2 35.80 13.80 10 0.4140 0.0138 3 50.00 24.20 10 0.3152 0.0105 4 24.20 4.80 10 0.7026 0.0235

Table 27: Coefficient of Permeability Result of Sample for Location C

64


Head (h )₂ (cm)

Time (t) (sec)


1 50.00 25.50 10 0.2890 0.00965

1.25 × 2 25.70 6.50 10 0.5971 0.01994 3 50.00 30.00 10 0.2218 0.00741 4 30.00 12.30 10 0.3872 0.01293

Table 28: Coefficient of Permeability Result of Sample for Location D


Head (h )₂ (cm)

Time (t) (sec)


1 50.00 31.50 10 0.2007 0.00673

1.23 × 2 31.50 15.50 10 0.3080 0.0103 3 50.00 24.40 10 0.3116 0.0104 4 24.40 5.50 10 0.6470 0.0216

Table 29: Coefficient of Permeability Result of Sample for Location E

S/NO Head (h )₁ Head (h )₂ Time (t) log (h /h )₁₀ ₁ ₂ Κ Average κ

65

(cm) (cm) (sec) (cm/sec) 1 50.00 21.00 10 0.3768 0.01258

1.35 × 2 21.00 5.20 10 0.6062 0.02024 3 50.00 23.40 10 0.3298 0.01101 4 23.40 11.50 10 0.3085 0.0103

Table 30: Coefficient of Permeability Result of Sample for Location F

S/NO

Head (h )₁ (cm)

Head (h )₂ (cm)

Time (t) (sec)


1 50.00 39.60 10 0.1014 0.00339

1.32 × 2 39.60 22.40 10 0.2474 0.00826 3 22.40 15.60 10 0.1571 0.00525 4 15.60 1.30 10 1.0792 0.03604

Table 31: Coefficient of Permeability Result of Sample for Location G


Head (h )₂ (cm)

Time (t) (sec)


1 50.00 24.80 10 0.3045 0.010266

1.14 × 2 24.80 8.80 10 0.4499 0.0150 3 50.00 29.90 10 0.2233 0.00746 4 29.90 12.50 10 0.3788 0.0130

Table 32: Coefficient of Permeability Result of Sample for Location H


Head (h )₂ (cm)

Time (t) (sec)

log (h /h )₁₀ ₁ ₂ Κ Average k (cm/sec)

1 50.00 31.20 10 0.2048 0.0068

1.19 × 2 31.20 15.50 10 0.3123 0.0104 3 50.00 24.00 10 0.3188 0.0106 4 24.00 6.20 10 0.5878 0.0196

4.5 SUMMARY OF THE CLASSIFICATION TEST ON SAMPLES

Table 33: Standard values of range for specific gravity of soil

Soils Range

Inorganic soil 2.60 – 2.80

Lateritic soil 2.75 – 3.00

Organic soil < 2.60

67

Sand particles 2.65 – 2.67

Inorganic clay 2.70 – 2.80

Source: ASTM (1999)

Table 34: Standard values of range for Uniformity coefficient and Coefficient of

Curvature

Cu Cc Remark

> 4 – 6 1 – 3 Well graded

< 4 < 1 & > 3 Uniformly graded

Cu ≈ 1 < 1 & > 3 Poorly graded & Gap graded

Source: Krishna (2002); Murthy (2000)

The samples taken from the selected locations were classified using their calculated C u

and Cc values according to the specification given in table 34 and summarized in table 35 as

illustrated below.

Table 35: Summary of the Results for the Classification tests on all Samples Location

Location

s

Natural

Moisture

Content

Specific

Gravity

%

Passing

Sieve

200

D10

(mm)

D30

(mm)

D60

(mm)

Uniform

Coefficient

(Cu)

Coefficient

of

Curvature

(Cc)

Remarks

A 2.30 2.66 8.90 0.32 0.32 0.4

0

1.25 0.80 Uniformly

graded

68

sand

B 9.56 2.67 0.52 0.30 0.40 0.8

3

2.77 0.64 Uniformly

graded

sand

C 14.47 2.65 0.40 0.30 0.32 0.4

0

1.33 0.85 Uniformly

graded

sand

D 6.81 2.65 0.16 0.30 0.32 0.4

0

1.33 0.85 Uniformly

graded

sand

E 12.50 2.66 0.44 0.30 0.35 0.7

5

2.50 0.54 Uniformly

graded

sand

F 14.62 2.67 0.72 0.20 0.32 0.3

8

1.90 1.35 Uniformly

graded

sand

G 11.15 2.65 0.28 0.31 0.38 0.7

7

2.48 0.61 Uniformly

graded

sand

H 12.53 2.66 0.24 0.31 0.40 0.7

7

2.48 0.67 Uniformly

graded

sand

It can be seen that the uniformity coefficients of the samples got from the locations are

less than 4 and the coefficients of curvature ranges from (Cc < 1 & > 3) hence they are all of

uniformly graded sand from different locations (Krishna, 2002).

69

4.6 SUMMARY RESULT OF THE HYDRAULIC PROPERTIES OF SAMPLES

These are the result of the falling head permeability tests of the sand samples, the

Reynolds number, flow-rate, velocity of flow, area, hydraulic gradient and friction factor as

calculated from the formula applied in the determination.

Table 36: Summary result of the hydraulic properties of all samples location

Properties

Locations

A B C D E F G H

Area (A) cm2 78.55 78.55 78.55 78.55 78.55 78.55 78.55 78.55

D10 (mm) 0.32 0.30 0.30 0.30 0.30 0.20 0.31 0.31

Velocity of flow

(cm/s) 1.57 2.41 2.18 2.11 2.41 1.60 1.92 2.01

Flow rates (Q) x

102 cm3/sec 1.23 1.89 1.71 1.66 1.89 1.26 1.51 1.58

Reynolds number

1.39 8.53 7.75 7.51 8.58 3.79 7.037 7.39

Friction Factor

45.85 7.5 8.26 8.52 7.46 16.90 9.09 8.67

Hydraulic

Gradient (ἰ) 1.09 1.83 1.74 1.72 1.78 1.21 1.68 1.69

Hydraulic

conductivity (κ)

cm/sec

1.43 1.32 1.25 1.23 1.35 1.32 1.14 1.19

Remarks High

medium

sand

High

medium

sand

High

medium

sand

High

medium

sand

High

medium

sand

High

medium

sand

High

medium

sand

High

medium

sand

4.7 OBSERVATIONS OF THE HYDRAULIC PROPERTIES

It was also observed that the hydraulic conductivities of the samples being in the range of

1.14 to 1.43 cm/sec which implies that all the samples are clean sand type and

70

the drainage condition of the sand is good, that is, it can allow the easy flow of fluid through its

pores.

4.8 ANALYTICAL MODEL IN POROUS MEDIA

From the result obtained from the above the model for the relationship between the

friction factor and the Reynolds Number as generated from the statistical analysis is seen below.

4.8.1 The Model Summary for the relationship between Friction factor and

Reynolds Number (Re) in Flow through Porous Media

Table 37: Model summary relationship between Friction factor (ff) and Reynolds

number (Re)

Model Summary

R R Square Adjusted R Square

Std. Error of the Estimate

.999 .997 .997 .942

The independent variable is Reynolds number (Re)

ANOVA

Sum of Squares

df Mean Square F Sig.

Regression 1782.386 1 1782.386 2009.799 .000Residual 5.321 6 .887Total 1787.707 7

The independent variable is Re.

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Coefficients

Unstandardized Coefficients

Standardized Coefficients

T Sig.

B Std. Error Beta

ln(Re) -1.314 .029 -.999 -44.831 .000

The dependent variable is ln(ff).

The equation relating the Reynolds number and the friction factor is given by

f f =−1.314 InRe

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Figure 10: Regression curve between Friction factor (ff) and Reynolds number (Re) inporous media

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4.8.2 The Model Summary for the relationship between Reynolds Number (Re)

and Flow rate (Q) in Flow through Porous Media

Table 38: Model summary relationship between Reynolds number (Re) and flow rate (Q)

Model Summary

R R Square Adjusted R

Square

Std. Error of

the Estimate

.974 .948 .927 .000

The independent variable is flow-rate (Q).

ANOVA

Sum of

Squares

Df Mean Square F Sig.

Regression .000 2 .000 45.485 .001

Residual .000 5 .000

Total .000 7

The independent variable is flow rates (Q).

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Coefficients

Unstandardized Coefficients Standardized

Coefficients

T Sig.

B Std. Error Beta

Q 59.755 15.696 5.891 3.807 .013

Q ** 2

-

16265934.00

9

5048762.713 -4.986 -3.222 .023

(Constant) -4.642E-005 .000 -3.876 .012

The equation for the relationship between the Reynolds number and the Flow rate is given by

ℜ=59.755 Q−1.627 e7Q 2−4.642 e−5

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Figure 11: Regression curve between flow rate (Q) and Reynolds number (Re) in porous

media.

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4.8.3 The Model Summary for the relationship between Hydraulic gradient (ἰ) and

Velocity of flow (v) in Flow through Porous Media

Table 39: Model summary relationship between Hydraulic gradient (ἰ) and Velocity of

flow (v)

Model Summary

R R Square Adjusted R Square

Std. Error of the Estimate

.990 .980 .972 .046

The independent variable is v.

ANOVA

Sum of Squares

Df Mean Square F Sig.

Regression .526 2 .263 123.333 .000Residual .011 5 .002Total .537 7

The independent variable is v.

Coefficients

Unstandardized Coefficients Standardized Coefficients

T Sig.

B Std. Error Beta

V 55919.058 8218.440 6.508 6.804 .001

v ** 2-

120821359.524

20630835.377

-5.601 -5.856 .002

(Constant) -4.666 .804 -5.807 .002

The equation relating hydraulic gradient and the velocity of flow is given by

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i=55.919 e3 v−120.821 e6 v2−4.666

Figure 12: Regression curve between Hydraulic gradient (ἰ) and Velocity of flow (v) in

porous media.

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CHAPTER FIVE

5.0 CONCLUSION AND RECOMMENDATIONS

5.1 CONCLUSION

The study was conducted to determine the flow of liquid but in this case, water through

porous media. The porous media used is River sand from Osun River. Based on the laboratory

tests conducted and the model equation derived the following conclusions were drawn:

i. From the classification tests on the samples collected from the river the natural moisture

content ranges from 2.30% to 14.62%, the specific gravity of the sand are of the range

2.65 to 2.67, the uniformity coefficient and coefficient of curvature of the sand samples

are of the range 1.25 to 2.77 and 0.54 to 1.35 respectively. Since the uniformity

coefficients of the samples is less than (3) and coefficients of curvature is less than one

(1) and not greater than three (3), then the sand from the Osun River is of uniformly sized

particles.

ii. The hydraulic properties of the test show that the range of the hydraulic conductivity of

the sand is from 1.14 to 1.43 cm/s. From these results, it is seen that the

sand samples according to hydraulic classification standards have a degree of

permeability that is of high medium class. That is, there is a larger space between the soil

particles.

iii. The Reynolds numbers are less than one which shows that the flow through the sand

particles is laminar.

iv. The model used to generate the equations relating the Friction factor and Reynolds

number, Flow rate with Reynolds number and Hydraulic gradient with the Velocity of

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flow, is based on the Statistical Package for Social Sciences (SPSS) using the regression

model module to determine the curve estimation. The equations generated are:

f f =−1.314 InRe

ℜ=59.755 Q−1.627 e7Q 2−4.642 e−5

i=55.919 e3 v−120.821 e6 v2−4.666

5.2 CONTRIBUTION TO KNOWLEDGE

This research have provided us with the knowledge that, the flow of liquid (water) can be

determined so as to be able to:

i. develop models for design of hydraulic structures,

ii. know the rate of seepage of water,

iii. know the rate of extraction and abstraction of fluids under the earth, and

iv. avoid failure of hydraulics structures.

5.3 RECOMMENDATIONS

From the results of this study, it is seen that the flow properties of porous media is key to

the design of hydraulic structures and hence it is recommended that proper attention should be

given to the friction factor, velocity of flow, flow rate and the Reynolds number as these are the

main products that result in flooding of embankment thereby resulting in environmental hazard

and seepage of water through dams.

80

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APPENDICES

These plates describe all the sample location and laboratory work as performed for each samples.

Plate 1: Sample location for A Plate 2: Sample location for B

Plate 3: Sample location for C Plate 1: Sample location for D

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Plate 5: Sample location for E Plate 6: Sample location for F

Plate 7: Sample location for G Plate 8: Sample location for H

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Plate 9: Permeability test in the laboratory. Plate 10: Specific gravity test in the laboratory

Table 11: Placement of sample A-H Table 12: Sample inside sack bag in the lab.before test

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Plate 13: Sieve analysis test in the laboratory

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