Predicting the Permeability of Sandy Soils From

Home , Permeability of soils, Sieve analysis, Water content

PREDICTING THE PERMEABILITY OF SANDY SOILS FROM

GRAIN SIZE DISTRIBUTIONS

A thesis submitted to Kent State University in partial

fulfillment of the requirements for the

Degree of Master of Science

Emine Mercan Onur

May, 2014

Thesis written by

Emine Mercan Onur

B.S., Middle East Technical University, 2009

M.S., Kent State University, 2014

Approved by

______, Advisor

______, Chair, Department of Geology

______, Dean, College of Arts and Sciences

TABLE OF CONTENTS

LIST OF FIGURES ...... vi

LIST OF TABLES ...... xiii

ACKNOWLEDGEMENT ...... xiv

ABSTRACT ...... 1

CHAPTER 1: INTRODUCTION ...... 3

1.1 Background ...... 3

1.2 Factors affecting Permeability ...... 4

1.2.1 Effect of Grain Size and Grain Size Distribution ...... 5

1.2.2 Effect of Density and Void Ratio ...... 7

1.2.3 Effect of Soil Texture and Structure ...... 7

1.3 Previous Investigations ...... 7

1.4 Objectives of the Study ...... 11

CHAPTER 2: RESEARCH METHODS ...... 13

2.1 Sample Collection and Preparation ...... 13

2.2 Laboratory Investigations ...... 13

2.2.1 Grain Size Distribution Test ...... 13

2.2.2 Compaction Test ...... 15

2.3 Data Analysis ...... 18

2.3.1 Statistical Analysis ...... 18

2.3.2 Modeling of Data ...... 20

CHAPTER 3: DATA PRESENTATION ...... 22

3.1 Grain Size Distribution ...... 22 iii

3.1.1 Grain Size Distribution by Sieve Analysis ...... 22

3.1.2 Grain Size Distribution by the Camsizer Video Grain Size Analyzer ...... 24

3.1.3 Comparison of Sieve Analysis and the Camsizer Video Grain Size Analyzer ...... 26

3.2 Compaction Test Data ...... 32

3.3 Permeability Data...... 32

CHAPTER 4: EFFECT OF GRAIN SIZE DISTRIBUTION AND DENSITY ON

PERMEABILITY ...... 41

4.1 Statistical Analysis ...... 41

4.1.1 Statistical Evaluation of Data ...... 41

4.1.2 Bivariate Analysis ...... 44

4.1.3 Multiple Regression Analysis ...... 51

4.2 New Permeability Index ...... 57

4.3 3-D Prediction Model ...... 64

CHAPTER 5: DISCUSSION ...... 68

CHAPTER 6: CONCLUSIONS AND RECOMENDATIONS ...... 766

6.1 Conclusions ...... 76

6.2 Recomendations ...... 777

REFERENCES ...... 788

APPENDIX A: GRAIN SIZE DISTRIBUTION PLOTS ...... 81

APPENDIX B: COMPACTION AND PERMEABILITY DATA ...... 922

APPENDIX C: HISTOGRAMS OF NON-TRANSFORMED DATA ...... 944

APPENDIX D: HISTOGRAMS AND P-P PLOTS OF TRANSFORMED DATA ...... 99

APPENDIX E: BIVARIATE ANALYSIS PLOTS OF NON-TRANSFORMED DATA

...... 1088

APPENDIX F: CALCULATION OF NEW PERMEABILITY INDEX ...... 1166

APPENDIX G: GRAIN SIZE DISTRIBUTION, DENSITY, PERMEABILITY, AND

NEW PERMEABILITY INDEX DATA FOR THE THREE SOILS (#7 AND 8) USED

FOR VALIDATION PURPOSE ...... 119

LIST OF FIGURES

Figure 1.1: Typical grain size distribution curve with commonly used grain size indices ..6

Figure 1.2: Particle shape characterization: (a) chart for visual estimation of roundness

and sphericity (from Krumbein and Sloss, 1963). (b) Examples of particle

shape characterization (from Powers, 1953) ...... 8

Figure 1.3: Diagram showing horizontal flow through grains (From Das, 2008) ...... 8

Figure 2.1: Six sandy soils used in the study ...... 14

Figure 2.2: (a) Camsizer video grain size analyzer for grain size distribution analysis; (b)

close up view of grains before falling ...... 16

Figure 2.3: Standard compaction test apparatus ...... 17

Figure 2.4: Constant head permeability test set up ...... 17

Figure 3.1: Grain size distribution of six soil samples by sieve analysis ...... 23

Figure 3.2: Grain size distribution curves by Camsizer video grain size analyzer. Soil 3,

results of Brooks study, was not available to test by camsizer ...... 25

Figure 3.3: Comparison of D10 values by sieve analysis and Camsizer video grain size

analyzer ...... 27

Figure 3.4: Comparison of D30 values by sieve analysis and Camsizer video grain size

analyzer ...... 27

Figure 3.5: Comparison of D60 values by sieve analysis and Camsizer video grain size

analyzer ...... 28

Figure 3.6: Comparison of Cu values by sieve analysis and Camsizer methods ...... 28 vi

Figure 3.7: Comparison of Cc values by sieve analysis and Camsizer video grain size

analyzer ...... 29

Figure 3.8: Comparison of %C values by sieve analysis and Camsizer video grain size

analyzer ...... 29

Figure 3.9: Comparison of %M values by sieve analysis and Camsizer video grain size

analyzer ...... 30

Figure 3.10: Comparison of %F values by sieve analysis and Camsizer video grain size

analyzer ...... 30

Figure 3.11: Comparison of Hazen permeability (kHazen) values by using D10 values both

fromsieve analysis and the Camsizer video grain size analyzer ...... 31

Figure 3.12: Permeability (top) and compaction (bottom) curves for soil 1 ...... 34

Figure 3.13: Permeability (top) and compaction (bottom) curves for soil 2 ...... 35

Figure 3.14: Permeability (top) and compaction (bottom) curves for soil 3 ...... 36

Figure 3.15: Permeability (top) and compaction (bottom) curves for soil 4 ...... 37

Figure 3.16: Permeability (top) and compaction (bottom) curves for soil 5 ...... 38

Figure 3.17: Permeability (top) and compaction (bottom) curves for soil 6 ...... 39

Figure 4.1: Matrix of bivariate scatterplots of non-transformed data. Numbers on left of

the matrix represent the row numbers and numbers at the bottom of the

matrix are the column numbers. Pearson’s correlation coefficient (r) values

are shown in red on the top of the individual plots ...... 45

Figure 4.2: Matrix of bivariate scatter plots of transformed data with Pearson’s

correlation coefficients (r) value shown on the top of each plot ...... 46 vii

Figure 4.3: (a) Permeability vs. percentage of medium sand size (%M) at different

densities. Red square represents loose state permeability; (b) at maximum dry

density (MDD) ...... 48

Figure 4.4: (a) Permeability vs. D10 at different densities. Red square represents loose

state permeability; (b) at maximum dry density (MDD) ...... 49

Figure 4.5: Transformed data of permeability vs. percentage of medium sand size(%M).50

Figure 4.6: Transformed data of permeability vs. D10 ...... 51

Figure 4.7: Frequency distribution of standardized residual values ...... 55

Figure 4.8: Scatter plot of standardized residual and predicted values...... 55

Figure 4.9: Measured versus predicted values of permeability ...... 56

Figure 4.10: Grain size distribution of soil 5 for calculation of new permeability index ..58

Figure 4.11: Relationship between permeability index and permeability with a prediction

equation; Ln (Permeability) =1.19 – 0.22 (Permeability Index) + 0.006

(Permeability Index) 2 ...... 60

Figure 4.12: Measured versus predicted values of permeability for all density based on

Equation 8. Soil 7 and 8 used for validation are represented by the red points.

...... 62

Figure 4.13: Measured versus predicted values of permeability for average density. Soil 7

and 8 used for validation are represented by the red point...... 63

Figure 4.14: 3-D plot of permeability index, density, and permeability. The color scale

represents permeability values in 10-3 cm/sec units ...... 64

viii

Figure 4.15: Measured versus predicted values of permeability for all density based on

Equation 9. Soil 7 and 8 used for validation are represented by the red points.

...... 66

Figure 4.16: Measured versus predicted values of permeability for average density. Soil 7

and 8 used for validation are represented by the red point ...... 67

Figure 5.1: Measured versus predicted values of permeability based on Equation 1 ...... 73

Figure 5.2: Measured versus predicted values of permeability based on Equation 2 ...... 73

Figure 5.3: Measured versus predicted values of permeability based on Equation 3 ...... 74

Figure 5.4: Measured versus predicted values of permeability based on Equation 4 ...... 74

Figure A-1: Grain size distributions of soil 1 replicates ...... 82

Figure A-2: Grain size distributions of soil 2 replicates ...... 83

Figure A-3: Grain size distributions of soil 4 replicates ...... 84

Figure A-4: Grain size distributions of soil 5 replicates ...... 85

Figure A-5: Grain size distributions of soil 6 replicates ...... 86

Figure A-6: Comparison of grain size distributions of soil 1 by sieve analysis and

Camsizer video grain size analyzer ...... 87

Figure A-7: Comparison of grain size distributions of soil 2 by sieve analysis and

Camsizer video grain size analyzer ...... 88

Figure A-8: Comparison of grain size distributions of soil 4 by sieve analysis and

Camsizer video grain size analyzer ...... 89

Figure A-9: Comparison of grain size distributions of soil 5 by sieve analysis and

Camsizer video grain size analyzer ...... 90

Figure A-10: Comparison of grain size distributions of soil 6 by sieve analysis and

Camsizer video grain size analyzer ...... 91

Figure C-1: Frequency distribution of permeability and density ...... 95

Figure C-2: Frequency distribution of D10 and Cu, ...... 96

Figure C-3: Frequency distribution of Cc and %C ...... 97

Figure C-4: Frequency distribution of %M and %F ...... 98

Figure D-1: Frequency histograms (left) and p-p plots (right) of transformations of

permeability with r2, skewness (Sk), D’Agostino-Pearson test score (DP), and

Kolmogorov-Smirnov test score (KS) ...... 100

Figure D-2: Frequency histograms (left) and p-p plots (right) of transformations of

density with r2, skewness (Sk), D’Agostino-Pearson test score (DP), and

Kolmogorov-Smirnov test score (KS) ...... 101

Figure D-3: Frequency histograms (left) and p-p plots (right) of transformations of D10

with r2, skewness (Sk), D’Agostino-Pearson test score (DP), and

Kolmogorov-Smirnov test score (KS) ...... 102

Figure D-4: Frequency histograms (left) and p-p plots (right) of transformations of

2 coefficient of uniformity (Cu) with r , skewness (Sk), D’Agostino-Pearson

test score (DP), and Kolmogorov-Smirnov test score (KS) ...... 103

Figure D-5: Frequency histograms (left) and p-p plots (right) of transformations of

2 coefficient of curvature (Cc) with r , skewness (Sk), D’Agostino-Pearson test

score (DP), and Kolmogorov-Smirnov test score (KS) ...... 104

Figure D-6: Frequency histograms (left) and p-p plots (right) of transformations of

percent of coarse sand size (%C) with r2, skewness (Sk), D’Agostino-Pearson

test score (DP), and Kolmogorov-Smirnov test score (KS) ...... 105

Figure D-7: Frequency histograms (left) and p-p plots (right) of transformations of

percent of medium sand size (%M) with r2, skewness (Sk), D’Agostino-

Pearson test score (DP), and Kolmogorov-Smirnov test score (KS) ...... 106

Figure D-8: Frequency histograms (left) and p-p plots (right) of transformations of

percent of fine sand size (%F) with r2, skewness (Sk), D’Agostino-Pearson

test score (DP), and Kolmogorov-Smirnov test score (KS) ...... 107

Figure E-1: (a) Permeability vs. all measured density. Red square represents loose state

permeability; (b) permeability vs. maximum dry density (MDD) ...... 109

Figure E-2: (a) Permeability vs. D10 at all density. Red square represents loose state

permeability; (b) at maximum dry density (MDD) ...... 110

Figure E-3: (a) Permeability vs. coefficient of uniformity (Cu) at all density. Red square

represents loose state permeability; (b) at maximum dry density (MDD) ...111

Figure E-4: (a) Permeability vs. coefficient of curvature (Cc)at all density. Red square

represents loose state permeability; (b) at maximum dry density (MDD) ...112

Figure E-5: (a) Permeability vs. percentage of coarse sand size (%C) at all density. Red

square represents loose state permeability; (b) at maximum dry density

(MDD) ...... 113

Figure E-6: (a) Permeability vs. percentage of medium sand size (%M) at all density. Red

square represents loose state permeability; (b) at maximum dry density

(MDD) ...... 114

Figure E-7: (a) Permeability vs. percentage of fine sand size (%F) at all density. Red

square represents loose state permeability; (b) at maximum dry density

(MDD) ...... 115

Figure G-1: Grain size distribution of soil 7 for calculation of new permeability index. 120

Figure G-2: Permeability (top) and compaction (bottom) curves for soil 7 ...... 121

Figure G-3: Grain size distribution of soil 8 for calculation of new permeability index .121

Figure G-4: Permeability (top) and compaction (bottom) curves for soil 8 ...... 123

xii

LIST OF TABLES

Table 3.1: Grain size distribution indices based on results of sieve analysis ...... 23

Table 3.2: Grain size distribution indices based on Camsizer grain size analyzer. Note

grain size index data for soil 3, tested by Brooks (2001), are not available ....25

Table 3.3: Optimum water content and maximum dry density of soil samples with

corresponding permeability values ...... 33

Table 4.1: Descriptive statistics of all variables ...... 42

Table 4.2: Transformation functions...... 43

Table 4.3: Model summary and Anova of stepwise multiple regression analysis ...... 52

Table 4.4: Coefficients of independent variables for the three models ...... 53

Table 4.5: An example of permeability index calculation for soil 5 ...... 58

Table 4.6: Residuals of predicted vs. measured permeability values for average density.63

Table 4.7: Residuals of predicted vs. measured permeability values for average density.67

Table B-1: Compaction and permeability data ...... 93

Table F-1: Calculation of permeability index values ...... 117

Table G-1: New permeability index calculation for soil 7 ...... 120

Table G-2: New permeability index calculation for soil 8 ...... 122

xiii

ACKNOWLEDGEMENT

First and foremost, I would like to express my gratitude to my supervisor Dr.

Abdul Shakoor for his guidance, patience, help and support in every stages of this thesis.

Without the support, encouragement and advices of him this thesis would not be possible.

I would also like to thank to my committee members Dr. Ortiz and Dr. Hacker for their valuable advices, suggestions and comments.

I am grateful to Dr. Kazim Khan for contributions in mathematical modeling. My sincere thank goes to Dr. Neil Well for his time and helping me to improve my background in statistical analysis. I would like to thank to Merida for her help on technical issues.

Funding for this project was generously provided by the Turkish Petroleum

Pipeline Corporation. I am indebted to the company for their financial supports in this master’s project.

At last but not the least, I would like to express to my gratitude to my parents

Fatos Mercan and Koksal Onur, to my sister Selcan Onur and to my friend Yinal Huvaj.

My parents have been my best friends all my life and thank them for all their advice, encouragements, and support.

xiv

ABSTRACT

Permeability is one of the most important and frequently used properties of soils.

Grain size distribution and density are known to influence the permeability of sandy soils.

Although the relationships between grain size distribution and permeability has been quantified in previous studies, the influenced of density has not been quantified. The objective of this research was to investigate the quantitative relationships between permeability and grain size distribution indices such as effective particle size (D10), coefficient of uniformity (Cu), coefficient of curvature (Cc), percentage of coarse sand fraction by weight of sample (%C), percentage of medium sand fraction by weight of sample (%M), and percentage of fine sand fraction by weight of sample (%F) to determine whether these relationships could be used for reliable estimates of permeability. Six samples of sandy soils, ranging from well graded to poorly graded, were tested in the laboratory to determine their grain size distribution, maximum dry density (MDD), and optimum water content (OWC). The D10, Cu, Cc, %C, %M, and %F values for each soil were calculated from the grain size distribution plots. Based on the compaction curves, five replicate samples of each soil were prepared at varying dry density values and tested for permeability using the constant head permeability test.

Results show that the lowest permeability for sandy soils is achieved at or slightly on the dry side of OWC. To investigate the relationship between permeability and grain size distribution indices, bivariate and step-wise regression analyses were performed.

2 The results show that D10, density, and %M have the strongest correlation (Adjusted R =

0.67) with permeability, explaining 67% of the variability in permeability.

Permeability depends on the sizes and shapes of interconnections between adjacent pores which, in turn, are influenced by the entire grain size distribution. This research proposes a new grain size distribution index for predicting permeability, designated as the “new permeability index”. In addition to considering the entire grain size distribution, the new permeability index assigns different weights to different size fractions in the soil with the finest fraction having the maximum weight and the coarsest fraction having the least weight. The new permeability index values for the six soils were correlated with their corresponding permeability values, resulting in a second order quadratic equation with an R2 value of 0.76. This relationship can reliably be used to predict permeability as is indicated by the small amount of residuals between measured and predicted values of permeability.

A 3-D model was developed to show the combined effect of the new permeability index and density on permeability.

CHAPTER 1

INTRODUCTION

The purpose of this study is to investigate the effect of grain size distribution on the permeability of sandy soils and to evaluate grain size distribution indices as predictors of permeability.

1.1 Background

Permeability, also known as hydraulic conductivity, is the property that represents the ease with which water flows through porous media (Jabro, 1992; Alyamani and Sen,

1993; Holtz et al., 2011; Salarashayeri and Siosemarde, 2012). It is one of the most important physical properties of soil used in geotechnical engineering. The rate of settlement of saturated soils under load, the stability of slopes and retaining structures, the design of filters made of soil, and the design of earth dams are some of the examples of applications of permeability in geotechnical engineering (Das, 2008). Additionally, information about permeability is necessary to estimate the quantity of seepage that will occur through earth dams and levees and through their foundations, to solve problems involving pumping seepage water from construction excavations, to determine spacing and depth of drains for lowering the water table under roads and highways, and to conduct stability analyses of earth structures and earth retaining walls when they are subjected to seepage forces (Das, 2008). However, permeability is also one of the most

variable properties, varying in both horizontal and vertical directions (Jabro, 1992). This is particularly true for glacial soils which are heterogeneous in nature. In a laboratory, permeability is usually measured on small samples which do not represent the heterogeneity of soils in the field (Holtz et al., 2011). No matter how many samples are tested in the laboratory, one cannot reliably estimate permeability. In addition, reliability of laboratory test results depends on the quality of undisturbed soil samples collected in the field (Holtz et al., 2011). Since undisturbed samples cannot be obtained for granular soils, the accuracy of permeability test results for such soils depends on how well the soil structure and density of laboratory samples represent the natural state of soil in the field

(DeGroot et al., 2012). To overcome this problem, field pumping tests are generally used for major engineering projects. However, performing a series of field pumping tests is both expensive and time consuming (Shepherd, 1989; Jabro, 1992). Also, in situ methods usually measure horizontal permeability (DeGroot et al., 2012). Because of these limitations of laboratory and field methods, many researchers (Hazen, 1892; Kozeny,

1927 and Carmen, 1956; Terzagi and Peck, 1964; Kenney et al., 1984; Alyamani and

Sen, 1993) have attempted to develop empirical equations for predicting permeability from grain size distribution parameters.

1.2 Factors affecting Permeability

Permeability is a complex property that is controlled by physical properties of both the soil and the permeating fluid (DeGroot et al., 2012). At a constant temperature of

20°C, the common room temperature, the viscosity and unit weight of water remain

constant. Therefore, physical properties such as grain size distribution, density, void ratio, and soil texture and structure affect the magnitude of permeability.

1.2.1 Effect of Grain Size and Grain Size Distribution

Grain size distribution of granular soils affects their permeability (Freeze and

Cherry, 1979). There are several ways to characterize grain size distribution of a granular

D soil. Commonly used indices include coefficient of uniformity ( 60 ), coefficient of Cu  D10

D 2 curvature ( 30 ), particle sizes, D , D and D , where D , D , and D are Cc  10 30, 60 10 30 60 D10D60 particle sizes, in mm, of 10%, 30%, and 60%, by weight of soil, passing the respective sieve sizes (Figure 1.1). Cu is an important shape factor that represents the degree of sorting of a soil and indicates the slope of the grain size distribution curve (Mitchell and

Soga, 2005). Larger Cu values indicate well-graded soils and smaller Cu values indicate uniformly-graded soils (Holtz et al., 2011). Poorly-graded soils have higher porosity and permeability values than well-graded soils in which smaller grains tend to fill the voids between larger grains. Cc is another important shape factor representing the grain size distribution that takes into account three points on the grain size distribution curve, reducing the possibility of considering a gap-graded soil as well-graded.

Figure 1.1: Typical grain size distribution curve with commonly used grain size indices.

1.2.2 Effect of Density and Void Ratio

Dry density (ρ) is the ratio of the mass of the solids in a soil to its total volume, the sum of volume of solids and volume of voids. Void ratio (e) is defined as the ratio of the volume of voids to the volume of solids (Das, 2008). Density and void ratio are inversely related. Permeability decreases as density increases or void ratio decreases.

1.2.3 Effect of Soil Texture and Structure

Texture and structure relate to size, shape, and arrangement of particles in a soil mass. Particle shape has an important effect on permeability as it influences the size and shape of interconnection between particles (Figure 1.2). The more angular the grains are, the smaller the voids and more tortuous the flow paths will be (Figure 1.3). This is because edges and corners of angular grains can fit into voids; i.e. there is a greater degree of interlocking (Holtz et al., 2011).

1.3 Previous Investigations

In addition to laboratory and field measurements, permeability can be empirically predicted from grain size distribution indices. Some researchers (Hazen, 1892; Kozeny,

1927 and Carmen, 1956; Terzagi and Peck, 1964; Kenney et al., 1984; Alyamani and

Sen, 1993) have attempted to correlate various indices of grain size distribution with the permeability of granular soils. Hazen’s formula (Hazen, 1892) given below, is most frequently encountered in the literature and is the most commonly used empirical relationship to determine permeability:

Figure 1.2: Particle shape characterization: (a) chart for visual estimation of roundness and sphericity (from Krumbein and Sloss, 1963). (b) Examples of particle shape characterization (from Powers, 1953).

Figure 1.3: Diagram showing horizontal flow through grains (From Das, 2008).

2 k  C(D10 ) (1)

Where:

k = coefficient of permeability (cm/sec)

C = constant ranging from 0.4 to 1.2, typically assumed to be 1.0.

D10 = grain size corresponding to 10% by weight passing, also referred to as the

effective size (mm).

The advantage of Hazen’s formula is that D10 from a large number of samples at a given site can be quickly and easily determined to compute permeability. This helps evaluate the variability of permeability at a given site in a quick and cost effective manner. However, a major limitation of Hazen’s formula is that it is more reliably valid for clean sands with D10 ranging from 0.1 to 3.0 mm (Holtz et al., 2011). Additionally, this method is based on only one size fraction, D10, which represents the percentage of fine material in a granular soil.

Another empirical equation for predicting permeability from grain size distribution, originally proposed by Kozeny (1927) and modified by Carman (1937,

1956), to become the Kozeny-Carman equation is given below. This equation is not appropriate for soils with effective particle size (D10) greater than 3 mm or for clayey soils (Carrier, 2003).

g  n3  k  8.3x103 D 2  2  10 v (1 n)  (2)

Where:

k = permeability (cm/sec)

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

v = kinematic viscosity (mm2/sec)

n = porosity

D10 = grain size corresponding to 10% by weight passing (mm).

Terzaghi and Peck (1964) developed the following empirical equation for

predicting permeability of course grained sands (Cheng and Chen, 2007).

2 g  n  0.13  k  C   D 2 t 1/3 10 (3) v (1 n) 

Where:

k = permeability (cm/sec).

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

v = kinematic viscosity (mm2/sec).

-3 -3 Ct = sorting coefficient, ranging between 6.1x10 and 10.7x10 .

n = porosity.

D10 = grain size corresponding to 10% by weight passing (mm).

Kenney et al. (1984) proposed the following equation for estimating permeability

using only a single point from the grain size distribution curve of the soil.

2 k  (0.005)D5 (4)

Where,

k = permeability (cm/day).

D5 = grain size corresponding to 5% by weight passing (mm).

Alyamani and Sen (1993) proposed the following equation which is more applicable to well-graded soils (Odong, 2007).

2 k 1.5046I0  0.025(D50  D10) (5)

Where:

k = permeability (m/day)

I0 = the x intercept of the slope of the line formed by D50 and D10 of the grain-size distribution curve (mm)

D50 = grain size corresponding to 50% by weight passing (mm).

D10 = grain size corresponding to10% by weight passing (mm).

None of the equations presented above considers the effect of the entire grain size distribution on the permeability of soils. Since grain size distribution controls the nature of interconnections between pores, the entire grain size distribution, rather than a single point on the grain size distribution curve, needs to be considered to reliably estimate the permeability of granular soils. Furthermore, none of the previously developed equations considers the effect of soil density on permeability.

1.4 Objectives of the Study

The objectives of this research are as follows:

1. To investigate the relationships between permeability of sandy soils and their

corresponding values of D10, Cu, and Cc to determine if these grain size

distribution indices can be used to reliably predict permeability while considering

the effect of density.

2. To develop a prediction model relating permeability to grain size distribution and

density.

CHAPTER 2

RESEARCH METHODS

2.1 Sample Collection and Preparation

Five sandy soils, exhibiting different grain size distribution curves, were collected for this research from locations around Kent, Ohio. A sixth sandy soil, previously tested by Brooks (2001), was added to the samples used in this study. Figure 2.1 shows the six soils used in the study. All soil samples were oven dried at 105°C for 24 hours. The oven- dried samples were stored in five gallon plastic buckets with covers.

2.2 Laboratory Investigations

Laboratory tests performed on the six soils included grain size distribution, standartd Proctor, and constant head permeability tests. All tests were conducted according to American Society for Testing and Materials (ASTM) specifications (ASTM,

1996).

2.2.1 Grain Size Distribution Test

This test was used to determine the percentages of different grain sizes present in each of the six soils in order to establish the grain size distribution curves and determine the grain size distribution indices such as effective grain size (D10), coefficient of uniformity (Cu), and coefficient of curvature (Cc). These indices are shown in Figure 1.1

Figure 2.1: Six sandy soils used in the study.

of Chapter 1. Sandy soils are classified as well-graded when the Cu values are greater than 6 and the Cc falls between 1 and 3. If these criteria are not met, the sandy soils are classified as poorly-graded. Using these criteria, the six sandy soils were classified according to Unified Soil Classification System (Holtz et al., 2011).

In addition to sieve analysis, the Camsizer video grain size analyzer was used to determine grain size distribution to evaluate the importance of a different method for determining grain size distribution indicies. The results obtained by the two methods were compared. Sieve analysis is the ASTM procedure for determining grain size distribution; the Camsizer technique is not a standardized procedure. Camsizer video grain size analyzer is a digital imaging process that has two cameras: a zoom video that analyzes smaller grains and a wide angle video that analyzes larger grains (Figure 2.2.).

2.2.2 Compaction Test

The standard Proctor test (ASTM D 698); (ASTM, 1996) was performed on all soil samples to establish their compaction curves and to determine their maximum dry density (MDD) and optimum water content (OWC) values. The standard Proctor test equipment is shown in Figure 2.3.

2.2.3 Constant Head Permeability Test

The constant head permeability test (ASTM D 2434); (ASTM, 1966) was used to determine the permeability of the six soils (Figure 2.4). Five or six samples of each sandy soil were compacted at different density values and tested for permeability. The quantity

(a)

(b)

Figure 2.2: (a) Camsizer video grain size analyzer; (b) close up view of grains before falling.

Figure 2.3: Standard compaction test apparatus.

Figure 2.4: Constant head permeability test set up.

of water passing through the sample in 5 minutes (300 seconds) was collected in a graduated cylinder to compute permeability in accordance with Darcy’s law (Holz, et al.,2011). The test was repeated five times for each sample and average permeability values were computed and reported in cm/sec.

2.3 Data Analysis

Microsoft Excel 2010 was used to generate smooth line plots showing grain size distribution curves, density versus water content relationships, and density versus permeability relationships for each soil.

2.3.1 Statistical Analysis

An unpublished computer program, written by Dr. Neil Wells, Department of

Geology, Kent State University, and SPSS program were used for statistical analysis.

Statistical analyses were performed in three steps: univariate, bivariate, and stepwise regression analyses. In the first step, distribution properties of each variable were analyzed by univariate analysis. In the second and third steps, bivariate and step-wise regression analyses were perfomed to investigate the relationships between permeability as the dependent variable and density and various grain size distribution indices as the independent variables.

Multiple regressions can be simultaneously or hierarchically performed. In simultaneous multiple regression analysis, all independent variables are entered into the equation at one time, whereas, in hierarchical multiple regression, independent variables

are entered in different steps. The important difference between these two types of regression analysis is that hierarchical regression is helpful in understanding the effect of each variable. Variations of hierarchical regression analysis include backward elimination, forward selection, and stepwise regression analysis (Dielman, 2001). Step- wise regression, including both backward elimination and forward selection, was used to evaluate the importance of the independent variables by adding the variable according to their partial F statistic (Dielman, 2001). Regression started with the variable that had the largest partial F statistic. When a new variable was entered to the model, partial F statistic values with other variable were recalculated. The importance of each variable was evaluated. Based on their importance, variables were re-entered or removed from the model. The significance level (partial F value) of a variable should be less than 0.05 for it to enter the model and the significance level of a variable should be more than 0.1 to remove it from the model to the model and the significance level of a variable should be more than 0.1 to remove it from the model. When the regression procedure was finalized, the most important variables contributing to variation in dependent variable stayed in the model whereas the least important variables were excluded from the model (Dielman,

2001).

Two important assumptions of multiple regression analysis are that the relationship is linear and is based on a Gaussian (normal) distribution (Kokoska, 2011;

Ghasemi, and Zahediasl, 2012). Therefore, it was important to evaluate the normality of data both visually and by various statistical tests. Frequency distribution histograms, Q-Q plots, and P-P plots were used as visual methods whereas the D’Agostino-Pearson (D’A-

P) test and the Kolmogorov-Smirnov (KS) test were used as statistical means of testing normality. In D’Agostino-Pearson test, both skewness and kurtosis deviations from

Gaussian are used to calculate the D’A-P score. A deviation of zero is desirable for both skewness and kurtosis but a score of less than 6 is considered acceptable for 0.05 significance level. If the score exceeds this value at a significance level of 0.05, the distribution will not be Gaussian (Wells 2013, [unpublished]). In the Kolmogorov–

Smirnov test, the largest vertical difference between the data and a Gaussian curve is measured on a cumulative curve. A zero value is also the best for Kolmogorov-Smirnov value. If the deviation in the KS test exceeds 0.238, the data distribution may differ markedly from Gaussian distribution at a significance level of 0.05 (Wells 2013,

[unpublished]). Since the K-S test is sensitive to extreme values, it should not be the first choice for testing normality.

Although the importance of the effect of fine grains on permeability is known, the entire grain size distribution should be taken into account to predict permeability. For this purpose, the use of a new index, designated as “permeability index”, representing the entire grain size distribution of a soil, was investigated. Non-linear regression analysis between permeability and the new index was performed to establish the relationship and to develop a prediction equation.

2.3.2 Modeling of Data

The relationships between permeability, density, and the new index were investigated in three dimensions by using the Matlab software program (MathWorks,

2012). The program generated a combined surface among the three mutually perpendicular axes representing permeability, density, and the new index.

CHAPTER 3

DATA PRESENTATION

This chapter presents the laboratory test data for the properties given below for the six soils used in the study: effective grain size (D10), coefficient of uniformity (Cu), coefficient of curvature (Cc), percentage of coarse sand fraction by weight of sample

(%C), percentage of medium sand fraction by weight of sample (%M), percentage of fine sand fraction by weight of sample (%F), compaction water content (w), dry density (ρd), and permeability (k) values.

3.1 Grain Size Distribution

Grain size distribution tests of the soil samples were used to classify the soils according to the Unified Soil Classification System (USCS) and to determine grain size distribution indices for each soil.

3.1.1 Grain Size Distribution by Sieve Analysis

The average grain size distribution curves for the six soils, based on seive analysis data, are shown in Figure 3.1. Replicates of grain size distribution for each sample are shown in Appendix A . Table 3.1 provides a summary of the grain size distribution indices including D10, D30, D60, Cc, Cu, %C, %M, and %F. According to USCS (Holtz et al.).

Figure 3.1: Grain size distribution of six soil samples by sieve analysis.

Table 3.1: Grain size distribution indices based on results of sieve analysis.

Soil D10 D30 D60 Cu Cc %C %M %F 1 0.19 0.34 0.93 4.89 0.65 18.18 43.81 36.37 2 0.17 0.22 0.33 1.94 0.86 0.35 31.13 67.65 3 0.19 0.30 0.80 4.21 0.59 13.64 39.09 40.91 4 0.34 0.42 0.46 1.35 1.13 0 64.83 35.17

Sieve Analysis Sieve 5 0.14 0.40 1.25 8.93 0.91 21.29 42.08 27.21 6 0.31 0.80 2.35 7.33 0.97 26.48 40.49 13.88

2011), well-graded sands have Cu values greater than 6 and Cc values falling between 1 and 3. Based on the grain size distribution curves shown in Figure 3.1 and the Cc and Cu data provided in Table 3.1, Soils 1, 2, 3, and 4 can be classified as poorly-graded sands

(SP) and soils 5 and 6 can be classified as close to being well-graded sands (SW), with their Cc values being just below 1 instead of 1.

Table 3.1 shows that soil 4 has the highest D10 value of 0.34 mm and soil 5 has the lowest D10 value of 0.14 mm. Soil 6 has the highest percentage (26.5%) of coarse sand and soil 2 has the highest percentage (67.7%) fine sand. Percentages of coarse and fine sand as well as D10 are expected to influence the permeability of sandy soils.

Additionally, well-graded soils with high Cu values are expected to have low permeability values compared to poorly-graded soils with low Cu values.

3.1.2 Grain Size Distribution by the Camsizer Video Grain Size Analyzer

In addition to sieve analysis, Camsizer video grain size analyzer was used to determine grain size distribution of the six soils and the results were compared with the sieve analysis method. The purpose was to evaluate the influence of test method on grain size distribution indices. Figure 3.2 shows grain size distributions of the six soils by the

Camsizer video grain size analyzer method and Table 3.2 provides the values of various grain size distribution indices.

Figure 3.2: Grain size distribution curves by the Camsizer video grain size analyzer. Soil

3, results of Brooks study, was not available to test by camsizer.

Table 3.2: Grain size distribution indices based on the Camsizer video grain size analyzer. Note grain size index data for soil 3, tested by Brooks (2001), are not available.

Soil D10 D30 D60 Cu Cc %C %M %F 1 0.27 0.48 1.17 4.29 0.71 17.50 50.00 23.90 2 0.20 0.31 0.46 2.29 1.06 0.40 46.60 52.90 3 ------4 0.58 0.67 0.78 1.35 1.01 0 98.90 1.10 5 0.31 0.53 1.25 4.03 0.73 19.40 52.20 19.90 Camsizer Analysis Camsizer 6 0.42 0.80 2.67 6.37 0.57 21.80 44.00 10.50

3.1.3 Comparison of Sieve Analysis and the Camsizer Video Grain Size Analyzer

The grain size distribution curves obtained by the two methods for the five soils were compared (Appendix A). The plots show that the two methods yield similar results for the coarse and medium size fractions of the soil samples. Plots of measured values both from seieve analysis and the Camsizer video grain size analyzer for all grain size distribution indicies including D10, D30, D60, Cu, Cc, %C, %M, and %F, are shown in

Figures 3.3 to Figures 3.10. Although the Camsizer video grain size analyzer overestimate D10, D30, and D60 values, differences between sieve analysis and the

Camsizer values decrease from D10 to D60. The Camsizer video grain size analyzer underestimates %C and %F values and overestimate %M values. The highest differances are seen in D10 and %F with a high scatter. The two methods provide different percentages of the finer fractions. For example, D10 values by the two methods are quite different. Thus, when Hazen’s formula (Hazen, 1892) was used to predict permeability, the differences in permeability values by the two methods, seen in Figure 3.11, are significant. The comparison shows that the Camsizer video grain size analyzer plots generally underestimate the percent passing for a given grain size, as compared to the sieve analysis. However, this differences can be attributed to the fact that sieve analysis measures the percentage of particles, by weight, passing a given size where as, Camsizer video grain size analyzer digitially measures percentage of number of particles of a given size compare to the total number of particles.

Figure 3.3: Comparison of D10 values by sieve analysis and the Camsizer video grain size analyzer.

Figure 3.4: Comparison of D30 values by sieve analysis and the Camsizer video grain size analyzer.

Figure 3.5: Comparison of D60 values by sieve analysis and the Camsizer video grain size analyzer.

Figure 3.6: Comparison of Cu values by sieve analysis and the Camsizer video grain size analyzer.

Figure 3.7: Comparison of Cc values by sieve analysis and the Camsizer video grain size analyzer.

Figure 3.8: Comparison of %C values by sieve analysis and the Camsizer video grain size analyzer.

Figure 3.9: Comparison of %M values by sieve analysis and the Camsizer video grain size analyzer.

Figure 3.10: Comparison of %F values by sieve analysis and the Camsizer video grain size analyzer.

Figure 3.11: Comparison of Hazen permeability (kHazen) values by using D10 values both from sieve analysis and the Camsizer video grain size analyzer.

3.2 Compaction Test Data

Appendix B provides the compaction test data for the six soils. Figures 3.12 to

3.17 show the compaction curves for all six soils. The compaction curves were used to determine the maximum dry density (MDD) and optimum water content (OWC) values for each soil as summarized in Table 3.3. The table shows that soils 5 and 6, being nearly well-graded (Figure 3.1), have the highest maximum dry density values of 1.92 Mg/m3 and 1.89 Mg/m3, respectively, whereas soils 2 and 4, being uniformly-graded (Figure

3.1), have the lowest maximum dry density values of 1.62 Mg/m3 and 1.64 Mg/m3, respectively. As will be demonstrated later, density has a significant influence on the permeability of sandy soils.

3.3 Permeability Data

Permeability values for the six soils were measured on samples compacted to different states of density during the compaction test discussed above. Figures 3.12 to

3.17 show the variation of permeability with varying density values. Since a given state of density is achieved at a specific water content, the plots in Figures 3.12 to 3.17 show the relationships between compaction water content and permeability for different samples of the same soil. In these figures, a comparison of the permeability curves on top with the compaction curves at the bottom shows that permeability of sandy soils decreases with increasing density, reaching its minimum value at the OWC or slightly on the dry side of OWC. After reaching its minimum value, the permeability increases as the density decreases on the wet side of the OWC. It should be noted that soil 4 is a

uniformly graded soil with nearly round grains. Therefore, the compaction curve for this soil is relatively flat, showing negligible variations in density with varying water content.

The flat nature of the compaction curve for soil 4 can be attributed to the fact that the smaller grains required to fill the voids between larger grains are not present in this soil.

Also, the interlocking of grains that reduces pore space, is absent due to the rounded nature of the grains.

Table 3.3: Optimum water content and maximum dry density of soil samples with corresponding permeability values.

Max Dry Optimum Water Permeability at MDD* x 10-3 Soil Density Content (%) (cm/sec) (Mg/m3) 1 8.40 1.81 1.66 2 6.00 1.62 0.32 3 11.40 1.85 0.59 4 3.80 1.62 3.00 5 8.50 1.92 0.27 6 7.30 1.80 1.37 *MDD stands for maximum dry density.

The permeability versus density relationships shown in Figures 3.12 to 3.17 clearly demonstrate the influence of density on permeability. Therefore, any equations used for predicting permeability from grain size distribution indices alone, such as those proposed by Hazen (1892), Kozeny (1927), Kozeny and Carmen (1956), Terzagi and

Peck (1964), Kenney et al. (1984), and Alyamani and Sen (1993), ignoring the role of density, are of limited application.

Figure 3.12: Permeability (top) and compaction (bottom) curves for soil 1.

Figure 3.13: Permeability (top) and compaction (bottom) curves for soil 2.

Figure 3.14: Permeability (top) and compaction (bottom) curves for soil 3.

Figure 3.15: Permeability (top) and compaction (bottom) curves for soil 4.

Figure 3.16: Permeability (top) and compaction (bottom) curves for soil 5.

Figure 3.17: Permeability (top) and compaction (bottom) curves for soil 6

As stated previously, Figures 3.12 to 3.17 show that the minimum permeability for granular soils is achieved at or slightly on the dry side of the OWC. This finding is contrary to the permeability behavior of fine-grained soils which exhibit minimum permeability values at water contents on the wet side of the OWC (Holtz et al., 2011).

CHAPTER 4

EFFECT OF GRAIN SIZE DISTRIBUTION AND DENSITY ON PERMEABILITY

This chapter focuses on investigating the effect of grain size distribution and density of granular soils on their permeability. Various statistical analyses were performed to investigate any relationship permeability might have with grain size distribution indices and density as individual, independent variables. The software program, Matlab, was used to study the combined effect of grain size distribution and density on permeability.

A new grain size distribution index, designated as “new permeability index”, is introduced as a possible predictor of permeability.

4.1 Statistical Analysis

4.1.1 Statistical Evaluation of Data

In many statistical investigations, data distributions are assumed to be Gaussian

(Kokoska, 2011; Ghasemi and Zahediasl, 2012). For statistical analysis to be more meaningful, it is important to evaluate the normality of data either visually or by various statistical tests. Results of descriptive statistics for all variables can be seen in Table 4.1.

Percentage of fine sand (%F) exhibits the highest standard deviation, 16.91. Density, Cc, and %C have negative skewness values, indicating that data distributions for these

parameters are skewed to the right where mean is smaller than median. Percentage of medium sand (%M) has the highest skewness value, 1.09. Frequency histograms of permeability, density, and grain size distribution indices (D10, Cu, Cc, %C, %M, %F) were plotted to examine the data distribution and are provided in Appendix C. Based on visual evaluation of the frequency histograms, the data show only a small deviation from normal distribution. Permeability, density, and %F have D’Agostino-Pearson (D’A-P) scores less than 6 and Kolmogorov-Smirnov (KS) scores less than 0.24 indicating that their distributions may be Gaussian. D’A-P scores of D10, Cu, Cc, %C, and %M are more than 6. Therefore, based on D’A-P test, distributions of D10, Cu, Cc, %C, and %M are non-Gaussian. However, K-S scores of D10, %C, and %M indicates non-Gaussian distribution whereas distribution of Cu and Cc are evaluated as Gaussian based on their K-

S scores (Table 4.1).

Table 4.1: Descriptive statistics of all variables.

Permeability Density D 10 C C %C %M %F x10-3 cm/sec Mg/m3 mm u c

Maximum 0.29 1.92 0.34 8.93 1.13 26.48 64.83 67.65 Minimum 3.05 1.50 0.14 1.35 0.59 0 31.13 13.88 Median 1.32 1.77 0.19 4.21 0.86 13.64 40.49 36.37 Mean 1.47 1.73 0.22 4.68 0.85 12.90 43.17 37.86 Std. Deviation 0.89 0.11 0.074 2.71 0.18 10.15 10.38 16.91 Skewness 0.50 -0.27 0.61 0.26 -0.074 -0.19 1.09 0.53 Kurtosis -1.10 -1.16 -1.40 -1.41 -1.25 -1.62 0.19 -0.62 D'A-P score 5.26 5.38 13.93 12.6 6,83 27.23 6.42 1.97 K-S score 0.10 0.13 0.34 0.20 0.19 0.24 0.31 0.23

Once distributions of variables were evaluated, transformations were applied as needed to provide more Gaussian distribution appropriate for further statistical tests.

Table 4.2 summarizes the equations used for transformations. Frequency plots and z score plots of reciprocal (inverse), logarithm, square root, and z transformations (the rescaled original data; mean = 0 and standard deviation = 1) of each variable are shown in Appendix D. Square root transformation resulting in highest R2 of 0.94, lowest skewness of 0.27, and lowest Kolmogorov–Smirnov test score of 0.08, is the best transformation for permeability data. The reflected reciprocal transformation is the best for density data with highest R2 of 0.96 and lowest skewness of -0.016. Reciprocal

2 (inverse) transformation is the best for D10 with highest R of 0.81, and lowest skewness of 0.54. The best transformation for Cu, %M, and %F is the square root transformation whereas the reflected square root transformation is best for Cc and the reflected z transformation is best for %C.

Table 4.2: Transformation functions

Transformation Equations

Inverse 2 -[1 /(X+1-Min)]

Straight Logarithm log(X+10-Min)

Square root (X+1-Min)1/2

Inverse 1-[1/(1+Max-Min)]+[1/(1+Max-X)]

Reflected Logarithm 1+log(10+Max-Min)-log(10+Max-X)

Square root 1+ (1+Max-Min)1/2 - (1+ Max-X)1/2

4.1.2 Bivariate Analysis

The effects of each grain size distribution indices and density on the permeability of the soils tested were analyzed separately using bivariate analysis. The matrix shown in

Figure 4.1 represents all bivariate relations between non-transformed variables. The independent variables representing the horizontal axes of the plots are shown along the diagonal of the matrix. The plot in the first row and second column shows the permeability (k) versus density relationship where permeability is plotted along the vertical axis and density is plotted along the horizontal axis. The plot of k versus D10 is shown in the first row, third column, and so on. The plots below the diagonal are the same as those above the diagonal, except that the axes are switched. The red numbers on top of the plots are Pearson’s correlation coefficient (r) values for raw (non-transformed) data. The first row of the matrix shows the correlations grain size distribution indices and density have with the property of permeability. As can be seen from the plot between permeability and density, permeability is inversely correlated with density with an r value of -0.37. The low correlation coefficient value can be attributed to soil 2 acting as an outlier even though it shows the same correlation trend as other soils. %M and D10 exhibit the strongest correlations with permeability with r values of 0.78 and 0.75, respectively. The correlation plots of permeability and grain size distribution indices in

Figure 4.1 (row 1, columns 3-8) show vertical distributions. These vertical variations indicate the effect of density. Effects of grain size distribution and density also were evaluated at maximum dry density and optimum water content. Figure 4.2 shows the scatter plot matrix for transformed variables; square root of permeability, reflected

Figure 4. 1: Matrix of bivariate scatterplots of non-transformed data. Numbers on left of the matrix represent the row numbers and numbers at the bottom of the matrix are the column numbers. Pearson’s correlation coefficient (r) values are shown in red on the top of the individual plots.

Figure 4.2: Matrix of bivariate scatter plots of transformed data with Pearson’s correlation coefficients (r) value shown on the top of each plot.

inverse of density, inverse of D10, square root of Cu, reflected square root of Cc, square root of %M, and square root of %F, where the distributions are more Gaussian. The plots of permeability versus %M and D10, exhibiting the strongest correlations, are shown in

Figures 4.3a-b and Figures 4.4a-b, respectively. Figure 4.5 and Figure 4.6 confirm these relationships for the transformed data. Large-size plots of all other bivariate analyses of non-transformed and transformed variables, both at maximum density and at all density values, are provided in Appendix E.

The plots show that D10, Cc, and %M are directly related to permeability whereas density, Cu, %C, and %F are inversely related to permeability. The presence of a correlation between permeability and Cu, Cc, and %F is expected. The strong positive correlation between D10 and permeability, both for non-transformed and transformed variables, also is expected. This is because the fines fill the pore spaces between larger particles and thus reduce permeability. Generally, one would expect that the larger the proportion of coarser sand, the higher would be the permeability. However, it is not clear why %C exhibits a negative correlation with permeability. Such unexpected trends indicate the complex nature of permeability and suggest that sizes and shapes of interconnections between pores, which control permeability, are influenced by more than one parameter. This further indicates the need to use the entire grain size distribution of a soil to predict its permeability, rather than a given point on the grain size distribution curve or only a portion of the curves, such as %C or %F.

(a)

(b)

Figure 4.3: (a) Permeability vs. percentage of medium sand size (%M) at different densities. Red square represents loose state permeability; (b) at maximum dry density

(MDD).

(a)

(b)

Figure 4.4: (a) Permeability vs. D10 at different densities. Red square represents loose state permeability; (b) at maximum dry density (MDD).

Figure 4.5: Transformed data of permeability vs. percentage of medium sand size (%M).

Figure 4.6: Transformed data of permeability vs. D10.

4.1.3 Multiple Regression Analysis

The percent coarse sand (%C) variable had a considerably high correlation with density and Cu. Therefore, to avoid a colinearity problem, %C was not included in stepwise regression analysis. Step-wise regression analysis between permeability, density, D10, Cu, Cc, %M, and %F was performed. The square root of permeability was used as the dependent variable. The independent variables are: reflected inverse of density, inverse of D10, square root of Cu, reflected square root of Cc, square root of %M, and square root of %F. Three different models generated during analysis are shown in

2 Table 4.3. D10 was evaluated in model 1 and resulted in an R value of 0.54. Addition of

%M and density increased the R2 value increased by 0.11 and 0.06, in models 2 and 3 respectively. Adjusted R2 value of 0.67 for model 3 shows that it explains about 67% of the variance in permeability. Significance levels of D10, %M, and density were found to be less than 0.05 whereas Cu, Cc, and %F had higher significance levels. Therefore, Cu,

Cc, and %F did not contribute to the stepwise regression. D10 explains the most variability in permeability.

Table 4.3: Model summary and Anova of stepwise multiple regression analysis

Summaries of Models R Adjusted R Std. Error of the Model R Square Square Estimate 1 .736a .54 .52 .26 2 .807b .65 .62 .23 3 .840c .71 .67 .22 a. Predictors: (Constant), D10 (inverse) b. Predictors: (Constant), D10 (inverse), %M (square root) c. Predictors: (Constant), D10 (inverse), %M (square root), Density (reflected inverse)

ANOVAa Sum of Mean Model df F Sig. Squares Square Regression 2.37 1 2.37 34.27 .000b 1 Residual 2.00 29 0.07 Total 4.37 30 Regression 2.85 2 1.42 26.13 .000c 2 Residual 1.53 28 0.05 Total 4.37 30 Regression 3.08 3 1.03 21.49 .000d 3 Residual 1.29 27 0.05 Total 4.37 30 a. Dependent Variable: Permeability (square root) b. Predictors: (Constant), D10 (inverse) c. Predictors: (Constant), D10 (inverse), %M (square root) d. Predictors: (Constant), D10 (inverse), %M (square root), Density (reflected inverse)

Table 4.4: Coefficients of independent variables for the three models.

Unstandardized Standardized Collinearity Coefficients Coefficients Statistics Model t Sig. Std. B Beta Tolerance VIF Error (Constant) 2.10 0.17 12.44 0.00 1 D 10 -0.19 0.03 -0.74 -5.85 0.00 1.00 1.00 (Inverse) (Constant) .457 0.57 0.79 0.433 D 10 -0.128 0.04 -0.50 -3.59 0.00 0.65 1.53 2 (Inverse) %M (Square 0.204 0.07 0.41 2.96 0.01 0.65 1.53 Root) (Constant) 1.35 0.67 2.01 0.05 D 10 -0.08 0.04 -0.32 -2.09 0.05 0.47 2.12 (Inverse) %M 3 (Square 0.26 0.07 0.51 3.73 0.00 0.58 1.73 root) Density (Reflected -1.28 0.58 -0.27 -2.22 0.04 0.72 1.38 Inverse)

As discussed in Chapter 2, assumptions of the step-wise regression model should be met. Frequency plot of regression standardized residuals was used to evaluate the normality of disturbance. Figure 4.7 shows that the residuals have a normal frequency distribution. Moreover, the scatter plot of residuals also shows the model assumptions were met. It does not show any clear trend (Figure 4.8).

The following regression equation, based on model 3, can be used to predict permeability from D10, %M, and density.

Sqrt(k) 1.35 0.085(InvD10 )  0.256(Sqrt%M) 1.28(RInvDensity) (6)

Where:

Sqrt(k) = Square root of permeability x10-3 (cm/sec).

InvD10 = Inverse of D10.

Sqrt%M = Square root of %M.

RInvDensıty = Reflected inverse of density.

Figure 4.9 shows the plot of measured versus predicted values of permeability for the six soils at varying density values. The residuals (differences between measured and predicted values) appear to be large, with the maximum residual value being 0.00136 cm/sec which amounts to 74% of the measured value. This large variation in the predicted value is due to the complex nature of the factors that control permeability.

Additional data from future research can help improve the prediction equation and reduce the residuals between measured and predicted values.

Figure 4.7: Frequency distribution of standardized residual values.

Figure 4.8: Scatter plot of standardized residual and predicted values.

Figure 4.9: Measured versus predicted values of permeability.

4.2 New Permeability Index

As stated in Chapter 1, permeability is a property that is controlled by the entire grain size distribution of a soil. Therefore, there is a need to find a new index that represents the effect of the entire grain size distribution on permeability. Furthermore, the new index needs to take into account the effect of the silt and clay size fraction (material passing sieve #200) present in the soil as it has the greatest effect on permeability. A new index considering the relative effect of different size fractions on permeability was developed in this study and is designated as the “New Permeability Index”. It is defined as follows:

n Fj New Permeability Index   7(MS 0.075) (7) j1100 j

Where:

n = Total number of sieve used in grain size distribution test.

Fj = Percentage of the amount of material passing the sieve j.

MSj = Middle size between sieve j and j-1.

In the above equation, Fj is for the percentage of the amount of material passing the sieve j and dominator of the equation represents the relative weights assigned to different size fraction. For example, six sieves were used to establish the grain size distribution curve for soil 5. Thus, an n value is 6 and j ranges from 1 to 6. Value of j is 1 for sieve #200; j is 2 for sieve #100 and so on until one reaches 3/8 inch sieve for which j is 6. Thus, for soil 5, F(1), percent passing through #200 sieve, is 4.67. MSj in the new permeability index represent the “Midsize”, the middle size of j and j-1 sieves.

Figure 4.10: Grain size distribution of soil 5 for calculation of new permeability index.

Table 4.5: An example of the calculation of new permeability index for soil 5.

Middle size Sieve Percent Sieve (MS ) MS - Permeability size passing j J j F [7*(Mj-0.075)] # [(M + M ) 0.075 j / 100 index (mm) (%) F j j-1 j /2] 0.375 9.51 100 6 7.13 7.06 1.70E-97 4 4.75 95.25 5 3.38 3.30 6.01E-45 10 2 73.96 4 1.21 1.14 8.79E-15 40 0.425 31.88 3 0.29 0.21 0.03 16.53 100 0.149 10.48 2 0.11 0.04 3.18 200 0.075 4.67 1 0.04 -0.03 13.31 0.01 0 Sum 16.53

For instance, MS(1) is the average size between #200 (0.075 mm) and minimum sieve sizes (0.01 mm). Figure 4.10 shows the grain size distribution of soil 5 for a new permeability index calculation. Table 4.5 summarizes the calculation of a new permeability index for soil 5. A hydrometer test was not performed to measure the exact minimum size of each soil; therefore, minimum size is considered as 0.01mm because most sandy soils do not contain particles smaller than 0.01 mm (Holtz et al., 2011). MS(2) represents the average of the 0.15 and 0.075 (sieves #100 and #200, respectively). Tables showing details of new permeability index calculations for the 6 samples can be found in

Appendix F.

It is important to consider not only the entire grain size distribution but also the relative importance of different grain sizes. The proposed new permeability index pays attention to this relative importance. The higher the percentage of finer grains, the lower the permeability value because the finer fraction fills the pore space between the coarser particles. For instance, percentage of fine sand is important for the grains retained on sieve #40 (medium sand size). Also, the effect of each finer fraction is greater than the fraction that is just coarser than the finer fraction. That is, the effect of silt and clay is greater than the effect of fine sand. This effect of relative sizes is incorporated in the new permeability index equation by the exponent 7 in the dominator. The effect from one sieve to the next larger sieve decreased by a factor of 1007 for the soils studied.

Figure 4.11 shows the correlation between the new permeability index and permeability. The data points along the vertical lines represent different density values at

Figure 4.11: Relationship between new permeability index and permeability with a prediction equation; Ln (Permeability) =1.19 – 0.22 (Permeability Index) + 0.006

(Permeability Index) 2.

which permeability values were measured. Regression between ln of permeability and the new permeability index, with a quadratic equation, was performed and the correlation coefficient was calculated. The regression equation, a second order quadratic equation, has a statistically significant R2 value of 0.76 (Johnson, 1984);

Ln (Permeability) =1.19 – 0.22 (Permeability Index) + 0.006 (Permeability Index) 2 (8)

Predicted versus measured values of permeability were plotted to evaluate reliability of the equation. Plots showing predicted versus measured values of permeability considering all densities and average densities are shown in Figure 4.12 and

4.13, respectively. In these plots the vertical distances of data points from 1:1 line represent the residuals. Figure 4.13 shows that the predicted values are close to the measured values, with a maximum residual value of 0.37 below the 1:1 line (Table 4.6).

Although the entire grain size distribution is taken into account, the effect of density on permeability can be seen in Figure 4.11. The vertical distribution of data illustrates the density effect. Therefore, this effect needs to be considered to predict the permeability.

For this purpose, data distribution of permeability, density, and new permeability index were modeled in 3-D as discussed in the following section.

Figure 4.12: Measured versus predicted values of permeability for all density based on

Equation 8. Soil 7 and 8 used for validation are represented by the red points.

Figure 4.13: Measured versus predicted values of permeability for average density. Soil 7 and 8 used for validation are represented by the red point.

Table 4.6: Residuals of predicted vs. measured permeability values for average density.

Predicted Measured Deviation Soil Permeability x10-3 Permeability x10- Residuals (%) (cm/sec) 3 (cm/sec) 1 1.39 1.76 -0.37 21.02 2 0.81 0.88 -0.07 7.95 3 1.06 1.04 0.02 1.92 4 3.26 3.00 0.26 8.67 5 0.45 0.70 -0.25 35.71 6 1.25 1.50 -0.25 16.67 7* 0.55 0.55 0.00 0.00 8* 2.20 1.99 0.21 10.55 *Soil 7 and 8 are used for validation.

4.3 3-D Prediction Model

As discussed earlier, density has considerable effect on permeability.

Permeability index, representing grain size distribution, density, representing state of compaction, and permeability were plotted along three mutually perpendicular axes to generate a 3-dimensional plot. The relationship between density and permeability is linear (first order quadratic) and the relationship between permeability index and permeability is nonlinear (second order quadratic). The following equation was generated for the regression surface by Matlab software.

Ln (Permeability) = 1.65 - 0.27 (Density) – 0.22 (Permeability index) + 0.006

(Permeability Index) 2 (9)

Regression surface given by the above equation is plotted and shown in Figure

4.14. The color scale in Figure 4.14 represents the permeability values for different color ranges. Red areas represent the lower permeability whereas blue and pink areas represent higher permeability values. The relationship between new permeability index and permeability in Figure 4.11 can also clearly be seen on a curvature of the surface in

Figure 4.14. The higher the new permeability index, the lower is the permeability. The colors change as one move towards increasing density on the surface. For example, yellow changes into red with increasing density values indicating a decrease in permeability.

Figure 4.15 shows a plot of predicted values of permeability, based on equation 9, versus the measured permeability values considering all density values. Figure 4.16 shows a similar plot considering average density.

Figure 4.14: 3-D plot of permeability index, density, and permeability. The color scale represents permeability values in 10-3 cm/sec units.

Figure 4.15: Measured versus predicted values of permeability for all density based on

Equation 9. Soil 7 and 8 used for validation are represented by the red points.

Figure 4.16: Measured versus predicted values of permeability for average density. Soil 7 and 8 used for validation are represented by the red point.

Table 4.7: Residuals of predicted vs. measured permeability values for average density.

Predicted Measured Deviation Soil Permeability x10-3 Permeability x10-3 Residuals (%) (cm/sec) (cm/sec) 1 1.39 1.76 -0.37 21.02 2 0.81 0.88 -0.07 7.95 3 1.06 1.04 0.02 1.92 4 3.26 3.00 0.26 8.67 5 0.45 0.70 -0.25 35.71 6 1.25 1.50 -0.25 16.67 7* 0.54 0.55 -0.01 1.82 8* 2.22 1.99 0.23 11.56 *Soil 7 and 8 are used for validation.

CHAPTER 5

DISCUSSION

Permeability is a complex property that depends upon the sizes and shapes of interconnection between particles in a soil mass. It is the sizes and shapes of interconnections that control the rate of movement of groundwater. The factors which influence the nature of interconnections include grain size distribution, particle shape, and density (degree of compaction). Because of the complex nature of permeability and a number of factors influencing it, it is not possible to develop a predictive equation that can explain 100% variability of permeability among different sandy soils.

The effect of density on permeability is clear from Figures 3.12 to Figure 3.17.

These figures show that permeability of sandy soils decreases as density increases, with the minimum permeability occurring at either maximum dry density or slightly on the dry side (1-2 %) of the optimum water content. The relationship between density and permeability of granular soils is different than that of cohesive soils which exhibit their lowest permeability values on the slightly wet side (1-2 %) of the optimum water content

(Holtz et al.,2011).

The effect of grain size distribution on permeability is intuitively obvious. Coarse- grained soils have larger pores, as well as larger interconnections between the pores, whereas fine-grained soils have smaller pores and narrower interconnections between the

pores. Moreover, smaller grains fill the pore spaces between larger grains in well-graded soils thereby reducing the void sizes and the associated interconnections. For example, soil 5 has the lowest permeability because of its well-graded nature as compared to soil 4 which is uniformly-graded. Although the entire grain size distribution controls the permeability of a sandy soil, the findings of this research demonstrate that the amount of fines (silt and clay size particles) present has the most significant effect on permeability.

Thus, the weight of different size fractions in influencing permeability cannot be considered equal. Therefore, in order to develop an equation for predicting permeability, it is essential to have a grain size distribution index that takes into account the influencing weights of different size fractions present within the overall grain size distribution of a soil. The “new permeability index”, discussed in Chapter 4, section 4.2, is a preliminary attempt, the first of its kind, at developing such an index.

Figure 4.11 shows the relationship between permeability index and permeability.

Density variations are included in the plot in Figure 4.11, although the relative effect of density is less than the relative effect of grain size distribution, especially the amount of fines presents. The regression curve in the figure passes through data points representing average density values for the soils studied. The prediction equation, a second order quadratic equation, takes into account the range of density values falling above the curve.

The equation 7 can be expressed as follows:

Ln (Permeability) =1.19 – 0.22 (Permeability Index) + 0.006 (Permeability Index) 2

In spite of the above listed limitations, the prediction equation developed in this study can be used with a reasonable degree of confidence as indicated by the small values

of residuals between predicted and measured values (Table 4.6). The equation was further validated by using two additional soils, not included in the six soils used for developing the equation. The grain size distribution, permeability index, and permeability of these soils (soil 7 and soil 8), at varying density values, were determined using the same procedures as for the other six soils. Appendix G provides the grain size distribution curves and the new permeability index calcualtions for the two soils. The new permeability index values were then used to predict permeabilities for the two soils. The predicted values of permeability for the two soils at varying density values are shown in

Figure 4.12, whereas Figure 4.13 shows the predicted permeability values at average density values. As can be seen from Figure 4.12, the predicted permeability values for the two soils used for validation purpose fall on both side of 1:1 line, reflecting the variations in density values. However, the predicted permeability values computed at average density values are close to the measured permeability values as seen in Figure 4.13.

The best way to check the combined influence of grain size distribution and density on permeability of a sandy soil is to develop a 3-D model incorporating all three parameters. Figure 4.14 shows such a model where density, permeability index, and permeability are plotted along three mutually perpendicular axes. Color changes in Figure

4.14 represent variations in permeability. The color changes from blue to red as permeability decreases with increasing permeability index. The effect of changing density on permeability can also be seen from the gradual alteration of one color into another color. For example, for a new permeability index of 8, change of light orange into dark orange or red indicates decreasing permeability with increasing density.

Equations proposed in this study (Equations 6, 8, and 9) have two main limitations as listed below;

1. Equations 6, 8, and 9 are based on a small population of six soils. They are

subject to modifications as additional data become available in future research.

2. The proposed equations need to be validated further by testing additional soils.

It is important to check the aplicability of same of the previously established equations using the data from this study. Figures 5.1 – 5.4 shows the measured versus predicted values of permeability based on Equations 1, 2, 3, and 4 for the six soils used in this study. As we can see from figures, Equation 1,2, and 3 overestimate the permeability and equation 4 underestimates the permeability. The large deviations from 1:1 lines in these figures may be because of the respective limitations of these equations, because of different types of soil used, and because of the lack of validations.

A number of previous studies (Hazen, 1892; Kozeny, 1927 and Carmen, 1956;

Terzagi and Peck, 1964; Kenney et al., 1984; Alyamani and Sen, 1993) have attempted to develop equations for predicting permeability from grain size distribution indices. The research presented herein is different from those previous studies in the following respects:

3. It demonstrates that density has a significant influence on permeability which

cannot be ignored in developing a prediction equation.

4. It shows that the influence of density on permeability of sandy soils is different

than that of cohesive soils.

5. It proposes a new grain size distribution index designated as the “permeability

index” for predicting permeability. This index assigns different weights to

different size fractions within the range of grain size distribution for a given soil

because finer fractions play a greater role in influencing permeability than coarser

fractions.

Figure 5.1: Measured versus predicted values of permeability based on Equation 1.

Figure 5.2: Measured versus predicted values of permeability based on Equation 2.

Figure 5.3: Measured versus predicted values of permeability based on Equation 3.

Figure 5.4: Measured versus predicted values of permeability based on Equation 4.

6. It generates a 3-D surface showing the relationship between density, grain size

distribution, and permeability.

CHAPTER 6

CONCLUSIONS AND RECOMENDATIONS

6.1 Conclusions

The results of this study can be summarized as follows;

1. Based on bivariate and step-wise multivariate analyses, effective particle size

(D10), density, and percentage of medium sand by total weight of sample (%M)

show the best correlation with permeability, explaining 67% of the variability in

permeability.

2. The relationship between density and permeability shows that sandy soils achieve

their minimum permeability values at water contents on the dry side of the

optimum water content, i.e. at a density value of slightly less than the maximum

dry density values. Any predictive model regarding permeability must consider

the effect of density.

3. Permeability index, a parameter accounting for the entire grain size distribution of

sandy soils and emphasizing the importance of finer fractions of the soil, can be

used to predict permeability. The relationship between permeability index and

permeability can be expressed in the form of a second order quadratic equation.

4. A 3-D model, showing the relationship between the entire grain size distribution,

density, and permeability, can be used to simultaneously evaluate the effect of

both grain size distribution and density on the property of permeability.

6.2 Recomendations

1. This study should be validated using additional sandy soils from different

locations.

2. The concept of permeability index should be verified and refined using additional

soils.

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APPENDIX A

GRAIN SIZE DISTRIBUTION PLOTS

Figure A-1: Grain size distributions of soil 1 replicates.

Figure A-2: Grain size distributions of soil 2 replicates.

Figure A-3: Grain size distributions of soil 4 replicates.

Figure A-4: Grain size distributions of soil 5 replicates.

Figure A-5: Grain size distributions of soil 6 replicates.

Figure A-6: Comparison of grain size distributions of soil 1 by sieve analysis and