DE’E‘EEMMTEGN OF CAPELMRY
CONEE‘C?‘§VE?? é}? ENSfiUEé’E‘EE WEGEES
REE-EA FEW BeiGESTEERE
CHARACTERISTICS

Thesis {or fire Deg?“ of pit. D.
EECSEGAS SEfi‘E {JEWEESEW
Abdal Razziq Qazi

2927-3-

LIBRARY

TH 55:13 ' Michigan State
Lhuyennqr

 

This is to certify that the

 

thesis entitled

Determination of Capillary CwnductiVity of
Unsaturated Porous Media From
Moisture Characteristics

presented by

Abdul Raviq qui

has been accepted towards fulfillment
of the requirements for

__P_'£L._D_.._ degree in M5 .

fALKLWMW

Major professorzl—s

Datel7AV? 70

0-169

ABSTRACT

DETERMINATION OF CAPILLARY CONDUCTIVITY OF
UNSATURATED POROUS MEDIA FROM

MOISTURE CHARACTERISTICS

BY
Abdul Raziq Qazi

Conductivity function is very essential for pre-
diction of moisture movement in unsaturated porous mater-
ials. Some mathematical models have been prOposed in
previous studies for analytical determination of unsat-
urated capillary conductivity for porous materials from
their moisture characteristics.

Moisture characteristics used in these models were
obtained by static method, whereas, in nature the varia-
tion of moisture and capillary pressure in soil are
gradual and dynamic. Therefore, in this study a dynamic
method was develOped and used to determine moisture char—
acteristics of three natural soils. Gamma ray attenua-
tion technique has been studied and used to determine

the varying moisture content of the soils. The mass

Abdul Raziq Qazi

attenuation coefficient of water has been found to vary
with thickness of water.

The curves of static and dynamic moisture char-
acteristics have been found to differ from each other
appreciably.

Capillary conductivities calculated by various
models from both dynamic and static moisture character-
istic are compared with the experimentally determined
capillary conductivities. It is observed that no model
can accurately predict the conductivity for adsorption
Brooks and Corey's model does not seem to work for fine
textured soils and for other soils at higher moisture
range without modification. A modification has been sug-
gested. Kunze's equation seems to have the tendency of
giving lower conductivity values at low moisture contents.
For glass beads of Topp and Miller, it tends to predict
higher values of conductivity at all moisture contents
using dynamic data. It gives fairly accurate results
for static data for the soils studied. Direct numerical
integration of the original Burdine equation using dynamic
capillary pressure and saturation data gives conductivity
values that approximate the experimental results better

than other mathematical models.

   

Approved

 

r f r

ajo
Approvedfi W
.He of Department

 

s

DETERMINATION OF CAPILLARY CONDUCTIVITY OF
UNSATURATED POROUS MEDIA FROM

MOISTURE CHARACTERISTICS

BY

Abdul Raziq Qazi

A THESIS

Submitted to
Michigan State University
in partial fullfilment of requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Agricultural Engineering

1970

ACKNOWLEDGMENTS

The author expresses deep appreciation to the
department of Agricultural Engineering, especially to
Professor E. H. Kidder, Committee chairman, members of
the Committee, Dr. Merle Esmay, Professor L. V. Nothstine
and particularly Dr. R. J. Kunze for his guidance. The
author is also grateful to the department of Soil Science
for the use of their facilities.

The author is alSo thankful to his wife, Lorna,

for all the efforts put forth by her during the period of
this study.

ii

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . .
LIST OF FIGURES O O O O O I O O O O O I O O I 0
INTRODUCTION 0 O O O O O O O O O O O O O O O 0

Need for Study 0 O O O O O O I O O O O O O O
Sc0pe of Research . . . . . . . . . . . . .
REVIEW OF LITERATURE . . . . . . . . . . . . .
Use of Functions K(6), D(9) and C(6) in the
Diffusion Equation and its Application .
Methods of Determination of K(6), D(6)

Functions . . . . . . . . . . . . . . . .
Transient methods . . . . . . . . . . . .

Computational techniques for K(6)
measurements . . . . . . . . . . . . .

Gamma Ray Attenuation Technique for
Moisture Measurements . . . . . . . . . .

Hydraulic Gradient and its Application to
D'Arcy's Equation in Unsaturated Flow . .

Measurement of Moisture Characteristics and
conductiVity O O O O O I O O O O O O O 0

EXPERIMENTAL DESIGN AND PROCEDURE . . . . . . .
Materials, Equipment and Methods . . . . . .

Materials 0 O O O O O O O O O I O O O O 0

Equipment 0 O O O O O O O O O O O I O O 0

Calibration and response of tensiometer-

transducer and recorder system . .

Moisture content determination equipment

iii

Page

ii

U1 UH I-‘ <

11
11

20

32

38

41
44
44

44
49

53
54

Page

Determination of mass absorption coeffi-

cient . . . . . . . . . . . . . . . . . ~ 58

Methods . . . . . . . . . . . . . . . . . . . 60
Moisture characteristic determination . . 61
Conductivity measurements . . . . . . . . 64
PRESENTATION AND ANALYSIS OF DATA . . . . . . . . . 68
Theory . . . . . . . . . . . . . . . . . . . . . 68

Theory of gamma radiation, attenuation

teChnique I O O O O O O I O O I O O O O O 68
Mathematical models used in the computation

of conductivity function . . . . . . . . . 74

Presentation of Data and Analysis . . . . . . . 76

Dynamic method . . . . . . . . . . . . . . . 76
Dynamic method compared with static method . 78
Head loss between tensiometers and

boundary I O O I O O I O O O O O O O O I O 78
K, e, and PC data. 0 o o o o o o o o o o o o o 98
Comparison of theoretical and experimental

conductivity results ... . . . . . . . . . 118

DISCUSSION 0 O C O O O O I O I O I O O O I O O O I 133

Gamma Radiation Attenuation Technique . . . . . 133
Static and Dynamic Methods . . . . . . . . . . . 134
Experimental and Calculated Conductivity . . . . 135

CONCLUSIONS 0 I O O O O O O O O O O O I O O O O I O 14 2

REFERENCES 0 O O O O O O O O O . . . . . . . . . . 144

APPENDIX I o o o o o o o o o o o o o o o o 0 O 0 0 149

APPENDIX II o o o o o o o o o I O ' . . . . . . . . 153

APPENDIX III e o o o o o o o o o 0 ° ' . ' . . . . 157

iv

LIST OF FIGURES

Figure Page

1. Grain size distribution of porous materials
used 0 O O O O O O O O O O I I O O O O O O O O 45

2. (a) Cross sectional view of the tensio-
meter assembly
(b) Cross sectional view of one end
section and one tensiometer assembled
in the pressure cell . . . . . . . . . . . . . 46

3. (a) Schematnzdiagram of the equipment
(b) Side View of the pressure cell . . . . . . 50

4. Strain gage bridge of pressure transducer . . 52

5. Relationship between moisture contents
and count rate . . . . . . . . . . . . . . . . 71

6. Verification of exponential law by plotting
experimental values of moisture contents
against calculated values . . . . . . . . . . 72

7. Variation of mass attenuation coefficient
of water with increase in thickness of
water 0 O O O O O O O I O I O O O O O O O O O 73

8. Variation of non-wetting, wetting and
capillary pressures with time in medium
sand for air pressure application rate
of 1.65 mm./cm. H20. . . . . . . . . . . . . . 79

9. Variation of P Pw and PC . in medium
sand for P = “35. 5w min. /cm. and AH = 25.0

cm. H20 CAB O O O O O O O O O O O O O O O O O 80

10. Variation of P and P in medium sand
for R = 26. 7 nKin. Pycm. H2 5 and AH = 31.0

cmHgB 81

Figure Page

11. Variation of Pn' and P in medium sand

for p = 37.1 fiin. Pycm. H2 8 and AH = 25. 0

cm. £8 0 O I O O O O O O O O I I I I O I O O O 82
12. Variation of P and P with time in fine

sand for AH= 3?. OP ”R -$9. 6 min. /cm. H20 . . 83
13. Variation of PRW, Pw and PC in fine sand for

RAP = 1. 795 m1 /cm. H20 . . . . . . . . . . . . 84
14. Variation of PPn, and P with time in fine

sand for RA 8. 625 min. /8m. H20 . . . . . . . 85
15. Variation of P P and P with time for

sandy loam withR w: 18.15 min. /cm. H20

and AH= 18 cm. H98 . . . . . . . . . . . . . . 86

16. Moisture characteristics by dynamic method

and lag between P and Pn nw in medium sand for

various RAP and AH values . . . . . . . . . . . 87
17. (a) Moisture characteristics by dynamic method

and lag between PC and in for RAP=lo55 min./cm.

(b) Comparison of moisture characteristics

obtained by dynamic and static equilibrium

method for medium sand . . . . . . . . . . . . . 88

18. Moisture characteristics by dynamic method and

lag between Pc and in in fine sand for RAP =

19.6 min. /cm. H20 and AH = 16 cm. H20 . . . . . 89
19. Moisture characteristics by dynamic method and

lag between PC and in in fine sand for R =

1.6 minO/cm. H20 0 O O O O I O O O O O O @P. O O 90

20. Moisture characteristic by dynamic method and
lag between PC and in in fine sand for RAP =
37 sec./cm. H2

0 O O O O O O I O O O C I I O O O 91
21. Comparison of moisture characteristics by
dynamic method, at various RAP values, with
static equilibrium method for fine sand . . . . 92

22. Comparison of moisture characteristics by static
and dynamic method for sandy loam . . . . . . . 93

23. Variation of hydraulic gradient between tensio-
meters with flow velocity in medium sand . . . . 95

vi

Figure

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

Variation of hydraulic gradient between

tensiometers with flow velocity in fine sand

Variation of hydraulic gradient between
tensiometers with flow velocity in sandy
loam O O O O O O O O O O O O O O O O O O 0

Experimental capillary conductivity versus
moisture contents for medium sand . . . .

Experimental capillary conductivity versus
moisture contents for fine sand . . . . .

Experimental capillary conductivity versus
moisture content for sandy loam . . . . .

Experimental capillary conductivity versus
capillary pressure for medium sand . . . .

Experimental capillary conductivity versus
capillary pressure for fine sand . . . .

Experimental capillary conductivity versus
capillary pressure for sandy loam . . . .

Variation of hydraulic conductivity of
mono-dispersed beads with capillary
pressure in rewet-redry loop . . . . . . -

Variation of capillary conductivity with
capillary pressure for aggregated beads
for redry and rewet loop . . . . . . . . .

Saturation versus capillary pressure for
medium sand for desorption and adsorption

Saturation versus capillary pressure

for fine sand for desorption and adsorption

Saturation versus capillary pressure for

sandy loam desorption and adsorption process

Saturation versus capillary pressure for

mono-dispersed glasstxnxks rewet-redry loop

Saturation versus capillary pressure for
aggregated glass beads rewct-redry loop .-

vii

Page

96

100

101

102

103

104

105

106

108

109

110

111

112

Figure

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

Relationship between effective saturation and
capillary pressure for medium sand . . . . . .

Relationship between effective saturation and
capillary pressure for fine sand . . . . . .

Relationship between effective saturation and
capillary pressure for sandy loam . . . . .

Relationship between effective saturation and
capillary pressure for mono-dispersed glass
beads I I I I I I I I I I I I I I I I I I I

Relationship between effective saturation and
capillary pressure for aggregated glass beads.

Area under the curve for relative conductivity
computation for medium sand . . . . . . . .

Area under the curve for relative conductivity
computations for fine sand . . . . . . . . . .

Area under the curve for sandy loam for
relative conductivity computations . . . . . .

Area under curve for mono-dispersed glass
beads for relative conductivity computations

Area under curve for aggregated glass beads
for relative conductivity computations

Conductivity comparison for medium sand
desorption . . . . . .

Conductivity comparison for medium sand
adsorption . . . . . . . . . . .

Conductivity comparison for fine sand
desorption . . . . . . . . . . . . . . . .

Conductivity Comparison for sandy loam
desorption . . . . . . . . . . .

Conductivity Comparison of mono-dispersed
glass beads desorption . . . . . . . . . . . .

Conductivity comparison of mono-dispersed
glass beads adsorption . . . . . .

viii

Page

113
114

115

116
117
119
120
121
122
123
125
126
127
128
129

130

Figure Page

55.

56.

Conductivity comparison for aggregated
glass beads desorption . . . . . . . . . . . . 131

Conductivity comparison for aggregated
glasstxxxks adsorption , , , , . . , . . . . . 132

ix

INTRODUCTION

Need for Study

 

In order to understand, control and accurately pre—
dict the water movement in soils due to drainage, evap-
oration and infiltration after rainfall or irrigation
one must understand the factors which control this move—
ment. In agriculture this is important from the point
of View of plant growth. In construction work it is
important from the point of view of drainage of highways
and seepage through walls, foundations and dams. To
the petroleum industry it is important for removing oil
from porous materials by displacement with either wetting
or non-wetting fluids.

This complex process of water movement in porous
materials needs more and more attention in order to
increase agricultural production for the growing popu-
lation of the world. The increasing loss of agricultural
land to buildings and roads makes it more crucial to
make use of these unused lands. Deserts or marshlands
both offer problems of water movement in soil--in one
case the supplemental irrigation is needed and in the

other case the drainage is important.

Since D'Arcy discovered the relationship, over
one hundred years ago, between flow velocity, hydraulic
"gradient and hydraulic conductivity of porous material,
many investigators have been attempting to develop
mathematical models describing flows in saturated soils.
They attempted first to describe the one dimensional
flow and then the two dimensional flow and finally the
unsaturated flow with limited boundary conditions.
D'Arcy's equation is basically for steady state laminar
incompressible flow. Flow of liquids in soils is 1am-
inar and incompressible but not always steady. Richards
in 1931 suggested a second order non-linear differential
equation for general case by combining the D'Arcy law
(V=KI) and the continuity equation V.V = de/dt.

These kind of equations are converted into a
system of linear equations in the finite difference
form. Knowing soil moisture characteristics, the con—
ductivity or diffusivity function and the initial and
boundary conditions, the solution of these equations
through a process of iteration with the help of a com—
puter gives a relationship between water content, ten-
sion, space and time. Thus the moisture conditions
can be predicted at a given point at a certain time for
certain problems.

When the medium is unsaturated and conductivity

K becomes a function of moisture content, 6, it is no

longer a constant. The same is true for diffusivity

0(6) and water capacity C(G) which are related as K(e) =
C(B)-D(6). Attempts have been made to determine K(6)
from soil moisture characteristics but soil moisture
characteristics exhibit hysteresis when the medium goes
through the process of desorption and then adsorption.
Thus the conductivity function computed from soil mois-
ture characteristics will be affected by this phenomenon.
Moisture characteristics which characterize the pore size
distribution of the porous material depend upon the pro-
cess used to empty or fill the pores. Usually pressure
steps are applied successively to drain or fill the
pores, achieving equilibrium state after each step at
which point the tension or the capillary pressure cor-
responds to the air pressure. Saturation of the media

is accomplished by releasing the pressure in steps. In
nature, however, the porous material is not saturated

or desaturated by successive pressure steps but is in
fact a gradual process which seems to differ from the

step method.

Scope of Research

 

In this study, therefore, a gradual and dynamic
method has been developed to vary the capillary pressure
and moisture contents continuously, but gradually. The

moisture contents are obtained by nondestructive method

of gamma ray attenuation. The unsteady state of flow
was maintained and capillary conductivity values corres-
ponding to measured values of moisture contents and
capillary pressure were determined. Results of the dy-
namic data compared with the standard step method data
are presented. Two mathematical models for determining
the K(9) from moisture characteristics were investigated
to see if the K(6) values predicted by these models fit
the experimental K(6) values. Effects of the hysteresis
on these models was analyzed.

Effect of moisture content variations on the mass
absorption coefficient of the water was studied. The
exponential law for the attenuation of gamma radiation

137 was verified for a fixed geometry. A regression

from Cs
technique to correlate moisture contents with the count
rate is developed and analyzed. The results from the
study of effects of external hydraulic gradients and
air pressure application rates on the wetting phase
pressure and capillary pressure are presented. The
effects of the head loss in the barriers as related to

the suction gradient between two points in the soils are

discussed.

REVIEW OF LITERATURE

Use of Functions K(e), D(9), and C(6) in the

 

Diffusion Equation and its Application

 

The heat diffusion equation and its application
to unsaturated flow problems appear in early literature.
The flow parameters—-capi11ary conductivity K(6), dif-
fusivity D(6), and water capacity C(G) have been defined
by the following relationships: Volume rate of flux

per unit area is

39
'1‘”) is? “ 'me) “a? (1)

where D(6) = K(6)/C(6), C(e) = ae/aH, H is hydraulic head
(cm), L is length of soil column (cm) and 6 is volumetric
moisture contents of the soil (cc/cc).

By combining the continuity equation

_ 36
Vov _' T
with D'Arcy's equation
V = K(0)I

Richard (34) in 1939 obtained the following second order

non-linear differential equation:

v. [K(0)VH] = 13% (2)

where (V.V) is the divergence of velocity vector V, t
is time in seconds and I is the hydraulic gradient

(cm/cm). The hydraulic gradient can be expressed as

V¢
= V = —
I H g (3)

where ¢ is the total potential (cmZ/secz), and ¢ = w +
gz, w is the capillary potential (cmZ/secz), gz is the
gravitational potential (cmz/secz), g being the acceler-
ation due to gravity and z the distance in the soil
column along the vertical axis (cm). The intrinsic
permeability k(6) (cm2) is defined as the property of
the media for transmitting fluid irrespective of the
nature of the fluid passing and is related to capillary

hydraulic conductivity K(6) as

K(6) = (pg/u) k(6) (4)

where p is the density and u is the absolute viscosity
of the fluid respectively. Substituting the relation-
ships (3) and (4) in equation (2), results in the equa-
tion for vertical flow as

30

_=3_
3t 32

x E k(e) %% + %% k(6) - (5)

‘C

If the change in the w is small and hence the
change in moisture content is small, for horizontal flow

the above expression reduces to

2
36 _ g 3 w

e- [w 2:21
Equation (6) implies one dimensional horizontal flow in
isotr0pic material with k(6) constant for a small change

in moisture content. For vertical flow H = ¢/g, for

horizontal flow H = W/g,

D 1 3w _ 86
'3 me) §_3x - me) B—X (7)
and
2
ae _ a e
'at ' ”‘9’ 5? ‘8)

which is similar to the diffusion equation for one di-
mensional heat flow.

In order to predict moisture distribution after
infiltration into the soil, Ashcroft gt_§1. (1) developed
a technique for solving an implicit difference analogue
of the diffusion equation. The diffusivity values were
chosen to convert the nonlinear diffusion equation into
a system of linear equations which were finally solved
by Gaussian elimination technique. The equation used

in this method was

B — EL 22
'3? — 8x (D(Q) 3x)

Q)

which was converted to finite difference form as

 

99+1 —69
J 3
At =
D(9?:i;§) ( §:{_ Og+l> /Ax -o(6?fij§) (03+1— 9?ti>/AX

 

Ax

where superscripts and subscripts denote time and space
respectively. Further, substituting the initial and
boundary cond1tions and approximating

n+l/2
311/2

n=1/2 .* n
0 and D(8 jil/Z

ej_+_l/2 by 311/2

by D(0* ) = D(6 )

j:1/2

where a is an arbitrarily chosen small constant and

n . . .
8j+1/2 is an average of two succe551ve m01sture con—

tents at x and x + Ax, the following equation results:

* .n+l 1* n+1
-_ D 0. ). + E. ; .
r ( 3:1/2) t'j-l (1+r D()j—1/2)+rD(tj+1/2)) 0]
_ 2* ,n+l_ n
where r I (At)/([x)2. The above system of linear equa-

tIODS forms a coefficient matrix of the form

 

 

bl cl W
a2 b2 C2
a b
m m
6* e ‘*
where al — -rD( 1—1/2)’ bl — 1 + rD( 1-1/2) +rD(Bi+l/2)
*
and c1 = —rD(Bi+l/2). The above tri-diagonal system of

equations was solved by Gau551an elimination technique
for 8?. The solution so obtained was compared by them

to the solution obtained by using Boltzmann transforma-
tion. They reported that the two solutions gave similar
results for infiltration into horizontal columns of semi-
infinite uniform media as obtained experimentally.

Hanks and Bowers(l6) used Similar techniques for
flow upwards and downwards in vertical columns. They
compared their results with other investigators and
claim excellent agreement. However, the results were
not verified experimentally. It was p01nted out by
them that the soil mOisture characteristics and a rela-
tionship between soil moisture content and diffu51vity
must be known to use their method.

Whisler and Klute (44) used the equation of the type

8h _ EL . ah 8K(h,z)
C(h,Z) t _ 82 K(hrZ) 3‘2- + ———BZ

10

to develop a numerical solution to determine time and
depth distribution of water content and pressure head
during the infiltration process in vertical columns
when both wetting and drying is taking place in some
parts of the soil column. Thus hysteresis effects had
to be considered. However, moisture characteristics
used by them were arbitrary and C(h,z) values were ob-
tained by differentiating the moisture characteristic
curve; K(6) values were obtained by use of Millington
and Quirk (29) formula and moisture characteristic curves.
Whisler and Klute concluded that the position of wetting
front was over estimated or under estimated if hysteresis
was ignored depending upon whether the drainage or wett-
ing curve was used.

Staple (37) conducted a similar experiment and
used a numerical technique to compute infiltration and
redistribution of water in vertical soil columns.
Hysteresis was incorporated by using conductivity and
soil moisture tension data obtained from the desorption
curve for the upper portion of the column which was
desaturating; at the lower end of the column the adsorp-
tion curve filled this need. Staple points out that use
of diffusion equation involving D in the finite difference
form predicted moisture profile for wetting which agreed
better with experimental data than the equation using K.

However, when accompanied by the hysteretic effects, the

11

error in the use of an equation involving K was small

for drying part of the profile.

Methods for Determination of K(6) and D(9) Functions

 

It has been the goal of physicists, soil physicists
and engineers to develop some efficient method for deter-
mination of the flow parameters used in unsaturated flow
studies. Experimental methods which have been developed
to determine these parameters indirectly are less time
consuming than direct methods. Some of the indirect
methods involve determining the advance of a wetting
front in a soil column and the moisture contents at
various points along the axis of flow. Other methods in-
volve determination of outflow data or moisture charac-
teristics. Some methods involve both. These methods are

briefly discussed.

Transient methods

 

1/2

By using Boltzmann transformation 1 = xt_ Bruce
and Klute (4) avoided the assumption of constant K(6) or

0(6) and equation (8) becomes

38 __ a as

with the initial and boundary conditions as

12

where Si and as are initial and saturation moisture con-

tents of the soil. Solution of Eris expression yields
x

Mex) = — 2%; ($99,: ei xde (9)

which is evaluated from the plot of 9 versus x graphi-
cally.

Their experiments indicated that D(6) increases
with moisture contents with a maximum value before sat-
uration.

Gardner (12) used a pressure plate outflow method
to determine capillary conductivity K(6). He assumed
that K(6) was constant for small changes in soil moisture
tension and hence also for small changes in 6. It was
also assumed that 6 is a linear function of soil moisture
tension for a small increment in soil moisture tension

-P i.e.
6=d+bP (10)

Neglecting gravity, the one-dimensional flow equation

(2) results in

66 _ l 6 3P
5? “ 35 ‘52 We) '3?
which by equation (10) reduces to
6? pg 2

13

with boundary conditions, P(O,t) = 0, (BP/Bz)z=L = O,
P (2L,t) = 0, P(z,0) = AP. Equation (11) was solved by
method of separation of variables to yield
A 00
£EE 2 Sin (nflz) exp (—a2Dt)
2L

P(z,t) =

II till-4

n l

where a = nfl/ZL. Substituting 6 from equation (10) and

defining

_ ae _ as c _
b-W—Wthenb—E&K-ngD

9(z,t) = d +

 

4b$P 1 % Sin (nflz ) exp (—a2Dt)
n=l 2L

total moisture content is obtained by integrating the

above expression which yields

00

L
W(t) =jr A. e(z,t) dz=dV 8bA§V 2 $7 exp (-a2Dt) (12)

0 N n=1 n

 

where A is cross sectional area and V is sample volume.
The series converge rapidly with time. The initial

water content of the soil becomes Wi = dV + bVAP, and

final moisture content at t = w is W = dV. Total out-

f
flow for the process is

Q = (wi «wf) = bVAP
(13)

 

14

thus b can be calculated from experimental data. The

equation for cumulative outflow at time t is

_ _§_ .1. -2
Q(t) — QO 1 “2 n2 exp ( a Dt)]

:3
ll M3

1

neglecting all but the first term of the series and tak-

ing the logarithm yields

_ _ 8Q _ 2
1n(QO Qt) — 1n (759) a Dt.

Thus diffusivity D and the capillary conductivity
K can be obtained by plotting the experimental data.
Gardner (12) reported that the lower boundary condition
P(6,t) was not always met since the resistance to flow
between the sample and the porous plate was not negligible
especially at higher moisture contents when large amounts
of water were released. A logarithmic relationship be—
tween conductivity and tension was suggested by Gardner

as
Log K = Log (a') - b' Log (P). (14)

Kunze and Kirkhan1(23) gave a method of determin-
ing D(6) and K(6) from experimental data without compli-
cated computations. The technique takes into account
the impedance of the porous plate and contact between
the sample and the plate, and uses the initial outflow
data for each pressure step thus reducing errors which

may arise with time (due to air bubbles diffusing into

15

the plate). The technique makes use of the theoretical
curves (Q/QO versus alth/LZ) plotted by using the

equation

0_ = l — 2.. 2 “EN-0% Dt/Lz (15)
2 0?
Q0 n=l q§(1 +a+an n

where
volumetric cumulative outflow at time t (Cm3)

Q:
QO = final cumulated outflow at equilibrium (Cm3)

G“ = is obtained by solution of equation aah =
Cota
n
D = diffusivity (sz/min)
t = time (min)
L = length of soil sample (Cm)
a = ratio of plate impedance to soil slab impedance.

Appropriate values of t when aiDt/L2 =1, give D = Lz/di t
and K = (DAB/PZ-Pl), where change in moisture content
corresponds to pressure change of (PZ-Pl). By these in—
vestigations some doubts were cast on the application of
diffusion theory, firstly, because piston action was anti-
cipated and remedied by placing a capillary tube in the
sample to convey high pressure from upper surface of sam-
ple to the porous plate to remove moisture from the lower
end first instead of the upper end and, secondly, because
at low pressure the same values of K could not be obtained
by using millipore filter and ceramic plates in spite of

accounting for the impedance of the plates.

l6

Nielsen gt_al. (30) experimentally investigated
the applicability of diffusion equation by using oil or
water as fluid entering the horizontal soil columns at
different negative pressures. He assumed: (1) that
D'Arcy's law is valid, i.e., the flux is proportional
to water content gradient or pressure gradient for iso-
thermal condition and (2) that there exists 1(6) such

that 1(6) = xt'l/2

, where A is a single valued function
of e, and x is the distance of the wetting front at time
t from the starting end. The results of the above study
question the validity of one of the above assumptions

1/2 was curvilinear

because the relation between x and t
for negative entry pressures larger than -2mb. Also,
the predicted relationship between 9 and x was different
from the measured values for higher negative pressures.
They concluded that values of diffusivity calculated
depend upon the boundary conditions at which the water
enters the soil column and therefore those values cannot
be used for the solution of diffusion equation for other
boundary conditions.

Gardner (13) and Doering (10) simplified the cal-
culation of diffusivity by using one large pressure step
instead of several small steps thus eliminating the need
of the assumption of constant diffusivity by using in-
stantaneous outflow from the pressure cell apparatus.

The relationship used by them has the same theoretical

foundation, equation (12), and is given as:

 

17

2
D = _—__.-4L gfl

fl2(W-Wf) dt (16)
where dW/dt is instantaneous outflow rate at time t, W
is volume water content and Wf is the final equilibrium
moisture content. Their results compared well with
other methods. They found that the membrane impedance
was not a significant factor except at saturation mois-
ture content but boundary impedance had some effect when
small pressure steps were used. For the multistep method
the diffusivity values for negligible and non-negligible
boundary impedance were not the same. Scatter of the
points for diffusivity and conductivity versus moisture
content was noticed in the case of multiple step method
whereas in the one step method a smooth curve was ob-
tained thus pointing out that the flow properties are
affected by the applied pressure gradient.

Many methods for determination of capillary dif-
fusivity have been suggested as mentioned earlier but
this parameter is still not completely understood.

Kunze (21) believed that besides the loss of moisture
content there are some other factors such as size and
rate of the applied pressure which are time dependent

and affect this parameter. He observed that in order

to determine moisture status in a draining soil, moisture
history and relationships of these flow parameters with

soil moisture tension must be known.

18

Skaggs et al. (35) used the modified form of
equation (14) of Gardner to relate conductivity K(h)

with tension —h as

K(h) = [(h/hl)a + b1'1 (17)
with boundary and initial conditions
11:0 6:68 X=0 tio
h = h. 6 = 6. x > O t = 0
1 1
h = h. 6 = 6. x = L t < t
1 1 - e
where a, hl' and b are constants to be determined. 6S
is saturated moisture contents, 6i and hi are initial

moisture contents and tension respectively in the soil
column, and te is time at which the wetting front
arrives at the bottom of the column. The constant b

is determined at the end of the experiment when steady
flow is reached at which time h is assumed to be zero
whereby equation (17) gives b = l/KO, and a and hl are
determined by trial and error. Value of K(h) so deter-

mined is used to solve the equation

6 3h K
a—x [K(h) 8E] ' 3x . (18)

|

0
EE
ll

C(h) in equation (18) is water capacity determined from
the moisture characteristics for absorption. The results
of equation (18) are compared with the experimental re—

sults from flow into the vertical soil column. The

19

difference in the experimental results and that of the
equation (18) is reduced by choosing another set of
values for hl and a, until the difference is minimum.
As a first approximation the value of a is taken as

a = n, where n is the slope of a plot of log K(h) ver-
sus log (-h). The basis for this choice of value of a

is the equation
K(h) = K $12)” for -h>Pb (19)

given by Brooks and Corey (3) which will be discussed
later in this chapter. KS is saturated conductivity,
Pb is bubbling pressure and n is defined as pore size
distribution index. According to equation (19) the
slope of the plot of log K(h) versus log (-h) is a
straight line with slope = n; equation (17) gives a
straight line also with slope = a whose first value is
thus taken as n.

This method is described by the authors as approxi-
mate since it uses equation (17) which is not universal
in the sense that it does not hold true for all the
soils. Secondly, two separate experiments are needed.
The errors from both these experiments may accumulate
as compared to some of the methods discribed later which

only use the moisture characteristics.

 

 

20

Computational techniques for K(6) measurements
Parallel model

Navier Stokes equations are applied to the flow
through the unsaturated porous media. The flow is as-
sumed to be laminar and incompressible for Newtonian
fluid. The Navier-Stokes equation for creeping fluid

is VP = uVZZ. For flow in a capillary tube the equation

gives

E = 8% VP (20)
and

E = 5%:- VP (21)

for thin film flow on a flat plate and

_ -b2
u = 12? VP (22)

for flow between parallel plates, where H is the one
dimensional average velocity, r is the radius of the
capillary tube, u is the coefficient of absolute vis-
cosity, AP is the drop in piezometric pressure, d is
the thickness of the film and b is the distance between
two plates.

When the above equations are applied to flow in

porous materials they can be written as

u = -——- VP (23)

21

where R is the hydraulic radius of the pore and is de-
fined as cross sectional area divided by wetted peri-
meter. If in equations (20), (21), and (22) the lengths
r, d, and b are replaced by R, then values of kf become
2 for equation (20) and 3 for equations (21) and (22).
Carman (7), from this observation, concluded that the
value of the shape factor kf must lie between 2 and 3.
However, the value of hydraulic radius R would differ
considerably for the two cases, hence a mean value of
R5 was suggested instead of R2. Another parameter
called tortuosity T = (Le/L)2 was incorporated in the
equation (23); Le is the distance of the tortuous path

and L is the direct distance. Thus the equation (23)

becomes

 

The volume flux is

uT L (24)

where ¢ is the porosity and s is saturation defined by
the relation S = 6/¢ where 6 is volumetric moisture con-
tent. According to the theories advanced by Purcell (33)
and later used by Burdine (6) the mean of R2 over the
entire range of saturation can, when applicable, be ex-

pressed in terms of the capillary rise equation

22

r = 2ygos
c

where Pc is the capillary pressure at saturation S,
9 is contact angle, Y is surface tension and r is pore

radius. Then R the hydraulic radius is

 

2
Area = fir = r
WEtted perimeter 2flr 2
R = Y cos 9 5
PC , s = If ds
0
then R25 = (1—%9§—9)2.]: ds
c
s
2 2
2_Yc056 Ids
therefore R - ————§———- 0 P_7 (25)
c

Equation (24) resembled D'Arcy's Law V = Ki, therefore,

the intrinsic permeability is given by

2
_ ¢SR (s)
k " kaTs (26)

T and R are.functions of saturation. Burdine (6) ob-

tained an empirical expression relating tortuosity and

saturation
(W) - (175—) ‘27)

T1 0 is tortuosity at S = l and Sr is the residual sat-
uration, i.e., saturation point on the moisture charac-
teristic curve after which no appreciable amounts of

liquid will drain. (L/Le)2 = l/Ts varies from zero at

23

S = Sr to 1.0 at S = 1.0, because Le becomes infinitely
long at low saturation, Sr' This was verified by Corey
(9). Equation (27) gives

2
l-S
T(S) = T(l.0) (FE?)

substituting T(s) and R2 in equation (26) Burdine (6)

obtained the expression for intrinsic permeability as

8-5 2 s
k(s) = “F522" Y2 .0529 f 01—35
EfT(1.0) 0 PC (28)

for unsaturated flow, and
l

_ 2 2
k ‘ +ka1.0)fl YC°S 8 f0 9‘37 (29)

for saturated flow in isotropic media. The relationship

__ s-s
se ‘ H's: (30)

is called effective saturation. All terms including
tortuosity can be eliminated if relative conductivity

kr is considered as the ratio kS/k = kr resulting in

_ 2 3 ds 1 ds
kr ‘ (53) f :3 :7 (31)
0 ? O C

This equation can be integrated graphically or numberi-
cally. However, Brook and Corey (3) obtained the follow-

ing results

24

1
Se = (gh) for PC > Pb (32)
c

where A is the slope of the plot of log (Pc/pg) versus
log (Se) and is termed the pore size distribution index
of the medium, and Pb is defined as bubbling pressure
of the medium at which the desaturation starts. 1Sub-

stituting equation (32) expressed as PC = Pb(Se)x and

the relation
ds = (l-Sr)dse

(obtained from equation (30)) in equation (28) and (29)

 

 

 

we get
s 2/1
k(s) = ¢y2 c0326 (Se)2 (1-sr) j‘e ase
ka(1.0) ;;2‘ 0 se
2 2 _ 2+3)
¢Y C°s e (1 gr) (xég)(Se)‘Y“ (33)
ka(1.0) Pb
and
gyz‘cosze (l-S ) A
k = _2.I. (-—)
ka(l.0) Pb 1+2 (34)
2+3)
kr = (se) ‘T"' (35)
or
kr = $11)” (36)

25

where 0 = (2+3X/X) is the slope of the plot of log (kr)
versus log (Po/pg). They observed good prediction of
permeability by the above relationship except for uncon-
solidated fine sand,GE No.13, and Touchet silt, GE NO. 3,
and consolidated Berea Sandstone. They used the step
method for determination of the soil moisture charac-
teristics for desorption only. For nonconsolidated
porous material they had to perform two separate experi-
ments using different samples of the same materials, one
to determine the moisture characteristic and the other
for conductivity. They used hydrocarbons instead of

water to avoid the swelling effects of water on porous

 

 

media.
'Laliberte (25), (26) modified equation (38) by
substituting
_ S'Sr 2
T(1.0) - T(S) (113;)
to obtain
2 2 1:2.
k(s) = ¢Yk §?:)9 <1'§r> (1:2) (Se) 4 . (37)
f Pb

Where ¢ (l-Sr) may be denoted by ¢e and equation (37)
may be written as

k _ 12 COS2e 1+2

¢e 1 ___
(s) ' kgfi(syv 3-7 (fi2) (Se) 1 (33)
b

 

26

for saturated condition equation (38) becomes

Y2 c0528 ¢e

A
k = ka(s)P137 (747%

 

Laliberte assumed cos 8:1 and used values of Carman (7)

for k = 2.5 and T = 2.0 to attain a dimensionless form

f
of equation (38) as

2
¢e A _
:‘L'f (x17) 5 0
13%

He used this equation to determine the effects of changes
in ¢e or bulk density and noted that the decrease in
bulk density or increase in ¢e was followed by an in-
crease in permeability and decrease in bubbling pressure

and the pore size distribution index.
Series parallel model

Child et al. (8), Marshall (28) and Millington (29)
have given methods of calculating the capillary conduc-
tivity from soil moisture characteristics. Child and Collis

— George (8) used the equation for intrinsic permeability
R
X o f(o) 6rf(o) Gr (39)

Which is based upon the probability of continuity of
lPores. In this equation f(p) Sr is the cross sectional
‘area corresponding to the range of pore p to p + Sr and

f(cJ 5r is the area corresponding to the range of pore

27

o to c + 6r. R is the largest pore size of interest
that remains full of water, m is a matching factor ob-
tained by matching calculated and experimental values
of the permeability at a certain point. The computa-
tion: are based on Table l of reference (8).
Marshall (28) uses equation of the type
¢2n—2 2

_ 2 2 _____ _ 2
k - 8 (rl +3r2 + 5r3+ + (2n l)rn ) (40)

 

where r1, r ----rn are radii of n equal classes of por—

2
osity of interest in decreasing order and ¢ is the por-
osity of the porous material. If in equation (40) r =

2Y/pgh is substituted in terms of h, where Y is surface

tension and r1 corresponds to hl’ hl<h2 etc., then equa-
I

tion (40) becomes

2 - —2 —2
k = (ggoz ¢ n (hl +3h2 +5h3 + ——- +2(n—1)hn

8
(41)
The above equations give intrinsic conductivity (cm2,)
for various suction values. The value of capillary con-
ductivity K(6) is obtained as K(e) = pgk/u (cm/sec). Thus
K becomes
¢2n-2

= L9312___ ’
K ‘qug’ 8 (hl

2 -2 -2
+3h2 +--- 2(n-l)hn

).
(42)
‘Marshall (28) pointed out that accuracy of these calcu—

lations depends upon the accuracy of measuring pore size

28

distribution and will be affected in case of swelling
materials.

Millington and Quirk (29) further improved the
computational equation by taking ¢4/3 instead of ¢2 in
equation (41) and n as total number of porosity classes.

Jackson e£_al. (18) compared the results of the
above three computational equations with the experimental
results for capillary conductivity for graded sand. He
concluded that the Child and Collis - George (8) method
matched at saturated conductivity but did not predict
the general shape of experimental curve. Marshall's
(28) method predicted only the saturated conductivity
and Millington and Quirk (29) method reasonably pre-
dicted the measured values for adsorption as well as
desorption when calculated values were multiplied by a
matching factor.

Kunze §£_2l. (24) modified the equation (42) and
used ¢ (instead of $2 or ¢4/3) as water filled porosity
and not the-total porosity, and n as the total number of
Pore classes. This modified form of equation is written

as:

 

-2 2
K(6)i = 55— 30 n Y 9

C . l
2 +1—21 43)
Ksc pgu ( 3 ) (

hzj

I-‘- Mt!

j:

i = 1,2 ---— n.

29

where K(8)i is calculated conductivity for a specified
moisture content or pressure class (cm/min), Ks/Ksc = m
is the matching factor which is the ratio of measured
saturated conductivity to calculated saturated conduc—
tivity, Y is surface tension of water (dynes/cm), p is
density of water (gm/cm3), g the acceleration due to
gravity (cm/secz), ufis fluid viscosity (gm/cm sec) and
h is the pressure or suction head (cm).

By not raising the ¢ to any power, Kunze observed
a better fit between experimental and calculated con—
ductivities at lower moisture contents. He investigated
the effects of a number of pore classes n and concluded
that larger n reduced calculated conductivity and a
larger m was required. But after m > 32 this effect
was negligible. He also observed that the accuracy of
the conductivity function depended upon the range of the
moisture characteristic. A more complete moisture char-
acteristic curve resulted in larger m values; m was
smaller for fine textured soils as compared to coarse
textured soils. The conductivities obtained from ad—
sorption and desorption did not agree in all cases.

Kunze (22) also suggested a more economical com-
;puter method of solving the equations of the type (43),
iby'adding the terms of the summation series backwards

from r: the smallest radius to r 2 the largest radius.

1
The calculation of Kn' the largest value of calculated

30

conductivity, by conventional computer programs requires
(n-l) multiplications and additions; for K(n-l), (n-2)
multiplications and additions are required, and so on.
But if addition is started from the smallest radius
going to the next higher radius, only four additions

are needed for each K value. Thus the total number of
additions and multiplications is reduced from n(n-l)

to 4n.

Brutsaert (5) gives a theoretical discussion of
various models advanced for permeability calculations.
Some of them have been discussed earlier in this chapter.
Some of his views are quoted below:

From the point of view of probability theory there
is an interesting difference between the parallel and
the series parallel model. In the later it is assumed
by cutting and random rejoining, that the sizes of
the pores in sequence are completely independent of
one another. On the other hand, in the simple par-
allel model it is assumed that the sizes of pores in
sequence are completely dependent on one another; as
a matter of fact each flow channel is assumed to have
a uniform cross section over its whole length.

Therefore the simple parallel model tends to over
estimate the flow rate and the concept of tortuosity
had to be introduced. In the parallel model the non-
uniformity of the pores in the direction of the flow
is taken care of by "tortuosity" while in series
parallel model it is done by "cutting and rejoining."

Intuitively it would appear that the sizes of the
pore sequences are not completely independent either.
Therefore, the cutting and rejoining may yield an
underestimation-of the flow rate. Moreover, it is
assumed that there is no bypassing of sequences of
several pores, that the smaller pores in the sequence
governs the flow rate, and that it remains uniformly
narrow over its whole length.

However, there are also several assumptions which
tend to yield an overestimate of permeability and
which may thus cancel the effect of above. The tubes

31

are assumed perfectly fitting, except in the model of
Millington and Quirk, and straight without tortuosity.-
They are assumed to have a regular or even circular
cross section while they are in fact highly irregular.
There are also many dead—end pores which do not con-
duct water even though they are filled. In some cases
the permeability is further over predicted because the_
porous medium has strong secondary structure. Often
there is the possibility of non-Newtonian flow espec-
ially in small pores when the medium has a high clay
content.

Another fact is that, while the size of the large
pores in the sequence is assumed to govern the empty-
ing suction, the size of the smallest is assumed to
govern the flow rate. As mentioned, this may yield
an underestimation of permeability at saturation.

But when sequences consisting of large pores are
emptied, the result is a large decrease in saturation
and a disproportionately small decrease in permeabil-
ity. The sequence of small pores with small pores have
a permeability which is relatively higher, at least
when one takes the average pore size of a sequence

as reference.

The net effect of all these assumptions seems to
be that, as was shown in most experiments, the rela-
tive permeability is overestimated at lower moisture
contents by models of Childs and Collis-George and
Marshall. Because Millington and Quirk assumed that
the flow area in the individual pores decreases as
the moisture content decreases, their method produces
better results.

He further indicated that for most of the models
the relationship between the effective saturation Se
and relative permeability kr can be simplified in the

form

c
kr = (Se)a+ B

where the parameters a, c, B vary for different models.
For Brooks and Corey (3) model a = 3, c = 2 and B = A,
which is the pore size distribution index. Thus the

equation becomes

32

2;.
B (44)

For Child and Collis-George's model

2
2+ —
kr = (Se) B (45)

the exponent is one less than the exponent in Brooks and

Corey's equation. For a uniform material B = A = w,
I”
thus Brooks and Corey's model reduces to f
__ 3 L
Kr — (S )

From this and from other models Brutsaert (5) concluded
that the exponent depends on the pore size range and
tortuosity. He also indicated that the two equations,

(44) and (45), give different results because they

represent different models.

Gamma Ray Attenuation Technique

for Moisture Measurements

The attenuation of gamma rays has proved to be a
very useful discovery in the soil moisture studies es-
pecially for rapidly varying moisture contents. Its
:major advantages are that it is nondestructive to the
Sch.system, is very fast as compared to gravimetric
Inethods, is very accurate and has high resolution.

A beam of gamma ray energy is directed towards a

spot in the soil column, part of which is attenuated

33

depending upon the density of the soil—water system.
Some of the photons are scattered after collisions and
the rest of the photons go through the soil. A detector
picks up these photons and after amplification the re-
sulting signal pulses are counted on a scaler.

The source of gamma photons usually used is C5137,
which has a peak value of .661 mev.l The source is con-
tained in a lead shield centered against a collimating
hole in the shield. The detector consists of a NaI

crystal photoelectrically connected to a photomultiplier

and a preamplifier. The detector is also collimated and

 

1C5137 has an half life of 30 years and emits both

gamma and beta radiation. The specific gamma radiation
constant in Roentgen/millicuries-hour at 1 cm. is 3.0.
Exposure rate mR/hr at lm=.003. One Roentgen = 2.58 x
10"4 Coulomhs/K gm of dry air, and is unit of available
radiation concentration at certain distance from the
source. Rad. is unit of absorbed radiations, i.e. energy
absorbed/ gm. of a material. One Rad. is equivalent of
100 ergs absorbed / gm., or 0.01 Jouls per K gm. Rem.
(radiation equivalent man), represents the biological
effectiveness of different kinds of radiation and includes
the quality factor of the radiation. The quality factor
for x,E3&y'rays is 1.0 and that of fast neutrons and pro-
tons up to 10 mev. and a particles is 10.0. Larger qual-
ity factor decreases the biological effectiveness. A
millicurie represents the rate of atom disintegration-

l mci = 107 x 3.7/sec. Allowable safe dosage for the
human body should not exceed 100 m Rem/week or 5 Rem/
year. For gamma radiation for persons above 18 years 1
Rem is equivalent to l Rad. .03 m Rad/hour is well within
the A.E.C. background radiation regulation. This amount
of radiation is absorbed by a man facing a 100 mc source
.in a 10 cm. lead shielding and standing at a distance of
.10 cm. from the surface of the lead shielding.

34

shielded to prevent the counting of scattered photons.
The scaler is set to measure a certain section of the
energy spectrum usually very near the peak energy value
(.611 to .711 mev.). The purpose is to discriminate
against all lower energy photons outside of desired
range which might have lost their energy slightly by
collision with H atoms in the soil but were not absorbed
completely.

The photomultiplier is energized by a high voltage;
this of course varies from detector to detector. The
high voltage is supplied to the detector through the
scaler.

The equation

I = IOe'D“X (46)

is used to determine the moisture contents. I is the
signal pulse count number measured in a certain time.
I0 is the count number if an attenuating medium is ab-
sent, 9 is the density of the medium, x the thickness
of the medium and u is called the mass absorption co-
efficient of the medium.

Equation (46) can be adopted to the geometry.
and the procedure followed in the experiment to be pre-
sented later.

Topp (38), (39) used a somewhat different technique

VVitm.200 mc (millicure) source in hysteresis studies of

35

glass bead media. He used ionization chambers one of
which acted as a detector chamber and the other a moni—
tor chamber. Both these chambers converted the energy
from the gamma rays into current. The difference of the
two currents was amplified and recorded. The absorption
by the sample was directly compared with a standard ab-
sorber. The correction for any drift was applied to the
recorder reading. This standard absorber was frequently
used to check the drift and make corrections.

Gardner (14) gives a description of the gamma
radiation attenuation method. The relationship

= ln(Nm/Nd)

(47)
-uws

 

is used to determine moisture contents 6, where Nm are
counts in a certain time coming out of the sample at
any moisture contents, Nd are counts coming out of a
dry sample in the same time, “w is mass attenuation co-
efficient of the water for gamma rays and S is the thick-
ness of the soil column. He also gives general speci-
fication for setup and design of the geometry of the
system.

Considering a normal distribution for the gamma
ray emission, randomness of the emission can be checked
Jay the fact that the area under a normal curve covering

<>ne standard deviation on either side of the mean value

36

is 67.8%. One standard deviation as estimated by Gard-
ner (14) is approximately equal to square root of the

counts.
I m. (48)

therefore the value of the counts measured in a certain
time should be within i/Iglof the mean value 68% of the
time or roughly 7 out of 10 readings on the scaler
should be lat/f; where Ia is mean value. Gardner (14)
also gives a rough estimate of 06, the standard devia-

tion of moisture contents measured by this method as»

_ 1
° " WI: ‘4”

which can be reduced by increasing either Im or count-
ing time.

Nutter (31) and Smith (36) used similar methods
for moisture measurements. Smith mentions some investi-
gators who achieved a precision of 0.006 gm. per cu. cm.
for water contents from 0.05 to 0.40 gm. per cc. at dry
density of 1.3 gm. per cc.

Ligon (27) applied the same technique for field
measurements of moisture contents variation with per—
cipitation and drainage. He reports some drift in the
System indicated by the check on standard Mg and plexi-
EJlass absorbers. These checks were made before and

after series of counts were taken. A mass absorption

37

coefficient of u = 0.0775 cmZ/gm was used for both soil
and water; individual values of “w were not determined.

The equation

 

Isoil
1n _EH__-
0 = 103.81 - 32.21 I 31 (so)
ln IBl
mg

for soil density measurement was used which required
three readings, one for soil and two for standard absorbers.
If the two standard absorber readings, taken before and
after the soil reading at a particular depth, differed
from each other by more than 5%, the scaler was recali-
brated and the readings were taken again. The change in

the moisture contents of the soil was calculated by the

 

 

 

 

 

 

equation
7 _ I soil
ln Isoil ln
w = .4734 Ilmg = Img
1n Ipl 1n Ipl (51)
._ mg .4 l L mg -J 2

 

 

Klock (19) used the gamma ray attenuation method for
moisture determinations in his conductivity studies.

The theory and general technique was the same as used
.by’other investigators except that he used 229 me. of
1Un241 for the reason that 60% of its radiation has 0.061
nuev. energy which required less shielding and less soil

'tflickness for proper attenuation.

38

Holland (17) investigated the possibilities of
constructing calibration curves for moisture contents and
radiation counts by use of regression technique. A beta
guage was used for moisture measurements in leaves and a
neutron guage was used in the soil moisture measurements.

It was found that for fitting the regression equation
y = a + bx

by least squares method to the beta gauge data the de-
pendent variate y (radiation counts) needed logrithmic
transformation to fit the experimental data while neutron
gauge data fitted better for a limited moisture range in

soil with no transformations required.

Hydraulic Gradient and Its Application to

D'Arcy's Equation in Unsaturated Flow

Vachaud (41) and Olsen gt_al. (32) studied the re-
lationship between the flow velocities and suction grad-
ients. Even though the techniques followed by both of
these investigators are quite different from this study,
their results are interesting to note. Vachaud used a
transient flow technique. Initially dry soil was wetted
in.a horizontal position, the moisture contents at various
Imoints of the column were determined by the gamma ray

attenuation technique. The distance and the time of

aiérvance of the wetting front were noted. However, the

39

moisture contents versus suction relationship for wetting
were determined in a separate experiment on the same soil
by a suction plate method (step procedure). Moisture con-
tent 6 versus time t relationship was plotted for each
distance x, thus obtaining a family of curves. Equation

of continuity for horizontal direction is

3 -36
2% = TE ‘52)

and for a fixed moisture content an, den = 0, i.e.

_ Ben Ben _
den—Wdt+—3—tdt—O. (53)
Here q is the flow velocity; other variables are the
same as used earlier. Combining equations (52) and (53)

gives

12:39. 91‘.
3x 8x ' dt (54)

where dx/dt = V is defined as the advancing velocity of
a water content 6n and is given by the slope of the re-
lationship x = f(0n,t).

By integrating (54) one gets

AV (6 - e (55)

qk+l ' qk = k+l k)

*wdiere qk=1 and qk correspond to the water contents ek+l

and 0k. AV is the small change assumed in V. If 6i is

40

the initial water contents of the soil and 80 the final,
then 90 is calculated by the total intake of water by
the soil volume and 6i is determined before starting the
flow. At t = o, qi = o; q0 corresponds to 90 and is
determined at the end of the experiment by achieving a
steady saturated flow.

At a fixed point x, an increment AG was assumed
.and corresponding At and Ax were calculated from the 0
versus t curves with x as parameter (figure 4, refer-
ence (41).) Thus Aq was calculated from equation (55).
The data of the family of curves (figure 4 of reference)
of 9 versus t with x as parameter was replotted as 0
versus x with time as parameter (figure 5 of reference).
Corresponding values of suctions obtained from a separate
experiment were inserted along the 6 axis. Suction
gradients were thus obtained from this family of curves
for each tn at different values of 6. Lastly the suc-
tion gradients were plotted against the flow velocity
for each value of 6 (6 = 40% to 25%). At each 6 the re-
lationship was indicated as a straight line proving that
for K = constant, Val and hence D'Arcy's law is valid
for unsaturated flow.

Olson gt_gl. (32) carried similar studies for

srteady state flow conditions for desaturation only.

They also give a graphical solution for determining

41

flow velocity versus suction relationship. Their re-
sults are very similar to the transient flow results
discussed above. However, no moisture characteristics
are reported in this report.

Both the above mentioned studies, however, indi-
cate that with decreasing moisture contents and flow
velocity, the suction gradient increases but for fixed
(6) or (-h) V is directly prOportional to hydraulic

gradient.

Measurements of Moisture Characteristics

and Conductivity

As mentioned earlier, most methods employed in the
measurement of moisture characteristic use pressure steps.
T0pp (39) in his hysteresis study of glass beads media
designed two water pressure regulators to keep the water
pressure atmospheric at the level of these regulators.
Since no air pressure was used, the level of the above
regulators was changed to change the capillary pressure
and moisture contents of the soil samples. The whole
procedure and equipment was quite elaborate and several
trial runs were necessary to acquaint himself with some
0f the unpredictable occurrences. Capillary pressure
and pressure gradients were directly recorded by two
transducers and flow measurement was measured by mea-

suring the pressure drOp in a capillary tube for both

42

inflow and outflow and then applying Poiseuille's law
to the average dr0p. In fact, the average drOp was
calibrated for flow velocity and recorded directly.

Moisture contents were measured by gamma radia-
tion method as mentioned earlier. Thus 0, V, I, Pc and
K were computed. His results and the results of other
investigators have shown hysteresis in the 0 versus Pc
and PC versus K relationship but only a small hystere-
sis was encountered in 6 versus K relationship. The
independent domain theory when applied did not predict
the desirable results.

Topp gE_§l. (39) investigated the difference in
shapes of moisture characteristics of a sand for desat-
uration by several methods of obtaining the desaturation
curve. The results reported show that the curve obtained
by applying static pressure steps and achieving equilib—
rium after every stOp, crosses the curves obtained by
applying small pressure steps (of the order of a cm)
while not achieving any equilibrium (semi dynamic method).
These curves differed from each other considerably.

The lepe of static equilibrium method curve became
steeper than the curve of the other method at tension
of -30 cm. It flattened at about -50 cm. of tension and
appeared to cross the other curve again at tension of

-57 cm. This indicated that the desaturation curves of

Semddynamic method seem to retain more moisture initially

43

than the curves of the static equilibrium method. Finally,
however, the dynamic curve becomes steeper and tends to
join the static equilibrium curve. TOpp et al. also re-
ported the results of the rate of the pressure application
on the shape of the curves. Three rates were compared,

two of the curves were similar but the third was slightly
different.

Another curve was obtained by achieving steady
state of flow in the soil after each large pressure step.
This curve lies near the static equilibrium curve.
Apparently large steps in both cases gave similar shape

of curve.

EXPERIMENTAL DESIGN AND PROCEDURE

Materials! Equipment and Methods

Materials

Three types of soils were used. Thegrain size

distribution of these soils is given in Figure (1). Soil

one is medium sand (Sphinx series), soil two is a fine

sand (Ottawa series) and soil three is a sandy loam

(Hillsdale series). The moisture characteristics of

these materials as obtained by the static equilibrium
method (pressure steps) and dynamic method (gradual

pressure application and release) are given in Figures

(17): (21) and (22).

The aluminum pressure cell (H), Figure (2b) used

in the experimental work had an inside diameter of 12.8
cm. and a height of 11.65 cm. with an air inlet (E) and

two tensiometers (C) and (D) set 4 cm. apart.

The cell had two end sections (G); each section

consists of a NRWP millpore 1 micron filter (K)l

‘

lMillpore filter NRWP 142 is made of nylon, has
ii pore size of 1 micron, porosity of 63%, suggested bub-
ling pressure of 12 psi., thickness of 150 + 10 microns.
1“llters were obtained from Millpore Corporation, Bedford,

Massachusetts .

44

‘ercent Finer

D

45

 

100 --""‘
L-
L 1
L
‘ -o—o—o—o— Medium Sand
“*‘°—‘*—<*'Fine Sand
’ ~0—0—0—O- Sandy Loam

 

 

 

 

 

 

 

10.0 b J
l.() _
L I l L l l L l l i l
0 .1 . 2 . 3 .4 . 5 C 6 .7 . 8 . 9 l. 0

Diameter m.m.

Figure 1.-- Grain size distribution of porous materials
used.

 

46

Figure 2. (a) Cross sectional view of the tensiometer

assembly.

(b) Cross sectional view of one end section

A,B
C,D

BR

and one tensiometer assembled in the
pressure cell.

Outlets for end sections
Tensiometers‘l outlets

TOp and bottom ring of tensiometer
Air pressure inlet

Top ring of end section

End sections

Bottom plate of end section
Rubber gaskets

Aluminum cylinder

Bolts

Millipore filter

Porous plate

Space to receive water
Screws

O-ring seals

Porcelain ring

Center of soil solumn

T1,T2 Tens iometers

47

manEwmm¢ HHOU wusmmmum Any

 

1HU| u

umumaoflmcwe Amy

#13-..

 

 

 

 

 

 

 

 

48

supported on a very porous polyethylene disc (pore size
120 micron) 0.5 cm. thick.1 This combination of filter
and porous disc rests against a collar in a corrosion re-
sistant brass bottom plate (61). In the back of the
porous plate is a small space (M) with two outlets (A,A).
The plate and filter combination are secured between the
tOp brass ring (F) and the bottom brass plate (G) by six
metal screws (N). A rubber gasket (GS) serves as an air
seal. The two end sections (G) of the cell can be easily
assembled together by four bolts and wing nuts (J). The
saturated conductivity of the filter and porous disc
combination varies between 0.01 to 0.04 cm./min. The air
entry value of this combination was 11 cm. of Hg. It was
observed that the saturated conductivity of porous disc-
filter combination decreased with time as fine soil parti-
cles settled on the surface of the filter. However, this
combination proved much better than a plain porcelain
plate of the same air entry value, but a much lower con-
ductivity, or a porvic filter and porcelain combination
that were used in earlier trial runs.

Details of tensiometers (T1 and T2) are shown in

Figures 2a and b. The brass holder, which holds the

 

lPolyethylene porous discs, Cat. No. F-1255, 120
mm. diameter, 5/10" thickness, pore size of 120 microns,
were ground to smooth surface with 0.5 cm. thickness.
These were obtained from Bel Art Products, Pequannock,
New Jersey.

49

porcelain ring (Q),} has an upper (BRl) and a lower part
(BR2)' O-ring (P) is wedged between the porcelain ring
and BRl' BR2 pushes against two O-rings, one in contact
with the porcelain ring and the other in contact with the
lower part. The brass holder with porcelain and O-rings
in place is held together with six screws. Water moving
through the porcelain ring may be withdrawn from the
tensiometers through copper tubing (C and D). The tubing
(C and D) goes out of the aluminum cylinder wall and is

fastened to the wall with a nut and a rubber gasket.

Equipment

 

The experimental equipment is shown in the sche-
matic diagram Figure (3). The tensiometer outlet tubes
are connected to a transducer (T) through a 3-way valve
(U). The transducer used is a strain gage type Dynisco
Model PT 14-01 with a maximum range of 15 psi. The
liquid pressure and electrical connections of the trans-
ducer are shown in Figures (3) and (4). Four l.5v. long
life telephone batteries (Y) in series provide a six

volt excitation voltage to the bridge of the strain gage.

 

lThe porcelain rings were cut from porcelain plates
five inches in diameter having air entry value (factory
suggested) of 3.5-4.6 psi. Actual tested value was 8-9
cm. of Hg, pore diameter of 9.2-12 microns, porosity of
39.4% and flow rate for water of 3-5 cc./sec/in under
20psi. head. These plates were obtained from Coors Por-
celain Co., 600 9th St., Golden, Colo.

50

Figure 3. (a) Schematic diagram of the equipment.

K x 2: <3 c: 0% m H

N

(b) Side view of the pressure cell.

Inflow Mariotte burrete

Outflow Mariotte burrete

Tensiometer connections to 3-way valve
Air pressure inlet

Constant temperature and humidity chamber
Jacks

Pressure transducer

3-way valve

Detector

Recorder

Sealer/analyzer

Batteries

Lead shielding

Source holder

Source

Pressure regulator

Air inlet

Multispeed transmission

Driving motor

Manometer

Collimation plug

51

Hamu whammwum ADV

 

D:

 

 

ucmsflummxm may no smnwmsv oflumsmsom Amy

 

w

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

52

.Hmospmcmup whommmum mo mmoflun mmmm cflmuum

 

 

"I
I
l
l
l
l
|
I
I
l
l
I

 

 

Hmpuoomu OB

 

J————--_--————

 

.v musmflm

53

The strain gage is attached to the dry side of the trans-
ducer diaphram, i.e., the side which is open to atmos-
pheric pressure. Full range sensitivity of the trans-
ducer is .918 uv/v. Recorder (W) used was Sargent recorder
model S-72150. It is a versatile recorder as it has many
ranges (1.25 millivolts or microamperes to 2500 volts or
milliamperes.) Accuracy is .1% or 20 uv. whichever is
smaller. Chart speed can be varied from 1/3 inch to 12
inches/min. Pen speed is 1.8 seconds full scale. D.C.
power is supplied by duracell mercury batteries supported
by two dry cells placed inside the recorder. The useful
features of this recorder are: (1) easy standardization
at any time, (2) quick zero setting at any point of the
chart by a displacement knob and (3) provision for rolling
the chart in a backward direction for quick scanning of
the data.

Calibration and response of tensiometer-transducer

 

and recorder system.--This system had to be calibrated
quite often by directly connecting a column of water to

the transducer and raising or lowering it and observing

the change in signals on the recorder. It was seen that
although the results were exactly alike when positive or
negative pressure was applied, yet after some time because
of the weakening of batteries, the signal voltage decreased;
therefore, the variation had to be checked occasionally

and the system recalibrated. The variation was 7 mv/10 cm.

54

of water pressure for new batteries to 6.2 mv/lO cm. of
water pressure when batteries were replaced after approxi-
mately six months.

The time constant and response of the system was
determined by applying a sudden air pressure step to the
pressure cell with saturated soil inside while inflow and
outflow were closed. A pressure of 29.3 cm. of water was
applied and was recorded as 27.5 cm. of water on the chart
in 1.2 minutes giving a time constant of 0.43 minutes.1
Experimentally tC corresponding to AP = 63.2% of P was .4
minute.

The volume of water required for a displacement
of the transducer diaphram was measured as lcc/250 cm. of
water pressure.

Moisture content determination equipment.--One

137

hundred mc. of CS was used as gamma ray source. It

was shielded in a cylindrical lead bucket 8 inches in
diameter and 8 inches high. It was designed according

to the equation

A exp(-ut)

a
D(R,t) = 2.134 13w. t) 3‘3— Bo MR;

1This follows the equation AP/P = (1 - eEE) where
A? is the recorded change in pressure in time t, th is
time constant and P is the applied pressure step. If t =
t0 is put in the ab ve equation the following equation re-
sults: Ap = P(l - 2) = .632 P. Similarly for t equal to
2' 3: ahd 4 time constants the recorded pressure will be
86'5%: 95% and 98.2% of the applied pressure respectively.

HOWever' the reSponse becomes much slower at lower moisture
contents .

55

given by Gardner (14).1 In this equation D(R,t) is dose
rate in millirad per hour at a distance R in cm. from the
source of strength A in mc. and energy E0 in mev; t is
the thickness of shielding in cm. and u the attenuation
coefficient in cm-l. B(u,t) is buildup factor; “a is
energy absorption mass-attenuation coefficient for the
source energy and the material which is to absorb the
dosage; and p is density of the shielding material.
Source (No. 2 in Figure 3) is held by a bolt (No.
1 in Figure 3) screwed in a sleeve in the shielding.
Thus it can be raised or lowered for minor adjustments
of the source against the collimation plug (No. 8) which
is 4 1/2 inches long with a 6 mm. circular hole drilled
through it. The plug was first molded then machined.
This size of collimation gave better results than other
collimation sizes. The collimated gamma rays were di-
rected at a spot (R) on the pressure cell in between the
tensiometers. The detector (V) was shielded against
scattered radiation by a 4 1/2-inch thickness of lead in
all directions. A 3/16-inch hole was drilled in the lead

for collimation in front of the detector and the two

 

1For 0.661 mev. gamma radiation from C3137, ua/E
as measured in tissues is 0.317 cmZ/gm., u is 1.134 cm’ ,
B(u,t) for lead thickness of 3 to 20 cm. is 1.2 + 0.13t.
When A is 100 and R = 10, D(R,t) = 0.1 mrad at the sur-
face of the lead and 0.03 mrad at 10 cm. from the surface
of the lead. Four inches radius and 8 inches height for
the bucket was thus a safe value.

56

collimation holes were aligned so that a maximum count
was obtained. This was achieved by rotating (Z), lower-
ing or raising jack (8) and bolt (No. l).

The detector used was a modified Nuclear Chicago
model D8 100 with a 1 1/2 inch NaI crystal connected to a
photomultiplier and preamplifier. The detector was con-
nected to a Nuclear Chicago model 8725 scalar-analyser.

The scaler was calibrated to represent one mev.
for full scale of 10 volts baseline. The base was set at
0.611 mev. A narrow differential counting mode was used
in which case the full scale setting of 10 on the window
scans only 10% of the base. Since the energy scale was
calibrated to represent 1 mev. for full scale baseline of
10 volts, the window width corresponded to 0.1 mev. Thus,
a window setting of 10 scanned the energy Spectrum from
.611 to .711 mev. and rejected all other energies. The
high voltage required to excite the scintillation NaI crys-
tal was of the order of 910 volts. This voltage had to
be readjusted occasionally in order to maintain a peak
value of counts. The readjustment was necessitated by
the drift in the baseline and/or amplifier or window.
Need of this readjustment was determined by obtaining
counts at several voltages higher and lower than the volt-
age in use previously. If the counts thus obtained were
higher at a different voltage, this voltage was used in

subsequent moisture measurements.

57

In order to account for the randomness of the
gamma ray emission, ten one-minute counts were taken for
slow runs of conductivity measurements and 3 to 4 one-
minute counts were taken for fast runs of moisture charac-
teristic determinations. Maximum c/m (counts per minute)
for the fixed geometry were 351,000. The fixed geometry
in this case was 4 1/2-inch thickness of lead shielding
around the source with a 6-mm. collimation hole followed
by a 6-inch air space and then another 4 1/2-inch lead
shield for the detector with a 3/16-inch collimation hole.
When the cell replaced the air space the c/m were of the
order of 60,000/minute. However, the last digit is drop-
ped in calculations as it varied considerably for con-
stant moisture content. This does not affect the accur—
acy of results appreciably. The values of 0 are calculated
from the equation (64), Chapter IV:

I

l m
m
s uws IS

'where am is moisture content corresponding to Im and as is
saturated moisture content corresponding to 15' “w and S
are mass attenuation coefficient of water and thickness

of soil, respectively. This equation involves the ratio
(Im/Is) which will give correct results up to 4 decimal
places even if the last digit in c/m is drOpped. Plotting

the graphs for more than 3 decimal places for 6 is not

58

very practical anyway. Theoretical standard deviation
for counts of the order of 59,000 to 60,000 at saturation
is of the order of i 243 (Oe= /Is). Actual variation was
well within this range. Theoretical standard deviation

in moisture content at saturation is

 

1
o = ,
e uws7f;
or
o = 1 = + 00444

or i 0.44% which is quite reasonable.

Determination of mass absorption coefficient uw.--

 

In literature different values of uw are reported; it is
therefore necessary to determine the values of “w for the
fixed geometry to be used. Four cubical cells of dimen-
sions 2.85 x 4.80 x 5 cm. each were constructed from a
1/8 inch thick plexiglass plate. The cells were placed
in between the two collimators at fixed positions. Seven
one—minute counts were taken with the empty cells in the
normal position. Then the first cell was filled with
water and seven more one-minute counts were taken. Simi-
larly, counts per minute for the remaining cells were

taken after filling each. Using equation

1 _0

s:
m
'o
H

59

the mass attenuation coefficient of water was determined.
In this equation 3 is the dimension 2.85 cm. through
which the gamma rays pass. IO and II are the average of
seven one-minute counts taken before and after filling a
cell. The effect of the order in which the cells were
filled did not seem to effect the “w value. But higher
values of “w resulted when the number of cells filled was
increased, (uw varied from 0.072 to .075), thus indicating
that “w increases with thickness of water. The value of
“w to be used in equation (56) was therefore obtained by
using the aluminum cell (H, Figure 3), and taking counts
before and after filling the cell with water. Here 5 is
12.8 cm.; “w obtained was 0.078 cm.2/gm.

The counter of the sealer/analyser is insensitive
during the short interval in which the pulse builds up and

this time is called dead time. A correction may be applied

to obtain the true counts by equation

Imot
I = _ (57)
m (t Imodt)

 

where dt is the dead time per pulse, Imo are the observed
counts in time t and Im is the corrected value. The analy-
ser used had a 1.5 u sec. pulse pair resultation time and

total dead time for minimum moisture contents (maximum c/m)

60

never exceeded Imodt = 0.111 sec. Therefore, correc-

tions were not necessary.

Methods

The pressure cell was packed with dry soil in
layers of one inch. It was then tapped to settle the
soil. When the cell became full it was wetted by imbibi-
tion with deaerated water containing .1% phenol for
several hours after which some additional soil was needed
to fill the cell. The soil level in the cell was kept
slightly above the rim of the cell before assembling the
three parts of the cell together to preclude the possi-
bility of further settlement or consolidation of the soil
later. After clamping all outlets, counts were taken at
different levels of the cell to check packing irregulari-
ties. No appreciable difference was observed in count
rates. Both end sections were flushed and then positive
water pressure was applied to the lower end of the pres-
sure cell. Outflows from the end sections were shut off
while maintaining positive water pressure at the lower
end section. When water started flowing out of the ten-
Siometers, the cell was laid on its side with tensiometer
outlets pointing upwards, then the tensiometers were con-
nected to a suction device to remove air. The tensio-
meters were clamped and the cell was connected as shown

in Figure (3).

61

All other connections were made as shown in Fig-
ure (3). The equipment enclosed in the rectangle (I) was
set inside a constant humidity and temperature chamber1
and kept at a dry bulb temperature of 72°F. and wet bulb
temperature'of 66°F. resulting in a relative humidity of
77 per cent.

Initially saturated conductivities were deter-
mined in a stainless steel permeameter. A sample from
this permeameter was used to determine the saturated
moisture content by oven drying. Saturated moisture
content of the same soil used in the pressure cell were
also determined at the end of the series of experiments
by taking samples from the middle of the cylinder and
oven drying them. This moisture content was lower than
the saturated moisture content obtained from the permea-
meter sample. However, the saturated moisture content
as obtained from the actual pressure cell sample was used
in equation (56) to determine 6m, since IS corresponding
to as was available. Saturated conductivities obtained
in the pressure cell were also lower than those obtained
by permeameter.

Moisture characteristics determination.--At first

a water column was connected to one of the two outlets (C)

*

. lAminco Air, Cat. No. 4-5478D and 4-5479D. Sup-
Eilied by American Instrument Co., Inc., 8030 Georgia Ave.,
31 lversprings , Md .

62

of the transducer. The other end of this water column
was placed in level with the center of the cell corres-
ponding to spot (R) in Figure (3). At this point the
recorder pen was centered on the recorder chart estab-
lishing a zero gage pressure or a reference line on the
chart. The tensiometer (C) was reconnected, and a mar-
riot burrette (Al) was brought to the level of (R) and
inflow was allowed. All other outlets were clamped.
Valve (U) was so positioned that both tensiometers were
connected to the transducer giving average pressure be-
tween two tensiometers. The air pressure inlet E was
left open to let any air in the cell escape.- These con-
ditions were maintained until the pen reached the center
of the chart, a position originally established to indi-
cate zero gage pressure or zero suction. After achieving
equilibrium state, inflow from (Al) was clamped and the
dripping point of (Bl) was raised to the level (R) and
outlet (B1) was unclamped. Then the air pressure hose
was connected at (E). Now the air pressure could be ap-
plied to the soil either in steps (static method) or
gradually (dynamic method) to change the moisture con-
tent. For static equilibrium, steps of approximately 10
cm. of water pressure were applied with an air pressure
regulator and were noted on the manometer (7). When
equilibrium was reached, as indicated by the fact that

the recorder pen came back to the center of the chart and

63

outflow stopped, another step was applied. For the first
few steps the equilibrium could be achieved in a few
hours but for higher pressure, equilibrium state was
reached after several days depending upon the kind of
soil and range of pressures. The same process was fol—
lowed for rewetting the soil. Several gamma counts were
taken after each equilibrium state to calculate 0m. Cap—
illary pressure values were assumed at the corresponding
values at equilibrium.

For the dynamic experiments the regulator stem
was rotated at a controlled rate by the device shown by
numbers (3), (5) and (6) Figure (3). Number (6) is a D.C.
motorl controlled by a speed control.2 The motor has two
shafts. The main shaft is geared to the second shaft
reducing its speed from 10 to one. This speed can be
further controlled by the minarik speed control which can
reduce the speed of the second shaft from a maximum of
173 RPM to zero RPM continuously. Through this control
the direction of rotation of the motor can also be re-

versed to reduce air pressure in the cell. The motor is

 

1The motor is type NSH-12 R, 115 v. D.C., 0.36
amps., 1726 RPM max. 1/50 H.P. reducer motor output shaft
173 RPM speed reduction 10:1, torque 49 in lbs. Supplied
by Bodine Electric Co., Chicago.

2Speed control and converter from A.C. 60 C. 115
v, to D.C. 115 v. Model SH-12. Supplied by Minarik Elec-
tric Co., Los Angeles, California 90013.

64

further geared to a multispeed transmissionl (No. 5, Fig-
ure 3) with a choice of 12 speeds, the fastest equaling
the speed of the motor shaft. It can be reduced to 1/5000
by choosing the slowest gear speed. In between each step
of gear speed any other speed can be chosen with the help
of the Minarik speed control. Theoretically it is possible
to obtain almost any RPM desired. Each revolution of the
regulator shaft corresponds to 51.4 cm. of water pressure.

Once a speed is chosen, it is kept constant for
that entire saturation and desaturation process. Several
such speeds were selected by trial and error for the

three $0113. The equation, PC = Pw - in, was used to

A

, given by a manometer and

compute capillary pressure Pc from the air pressure P or

non wetting phase pressure, in

P absolute pore water or wetting phase pressure recorded

WI

by the tensiometer—transducer-recorder system. PA and

in are used interchangeably. Values of 8 were computed
from the average of several gamma—ray counts, half of them
taken before and half of them taken after Pw and PA were
recorded.

Conductivity measurements.--Experiments were also

 

run for three soils to determine the experimental values

 

lMultispeed transmission has 12 gear speeds and a
nuetral. Neutral is used by applying pressure steps by
manually rotating the regulator stem to get the desired
pressure step indicated by the manometer. Supplied by
Harvard Apparatus Co., Dover, Mass.

65

of K(Pc) along with 0 and PC for each soil. The values
for the conductivity functions were determined as follows:
initial saturated conductivity was determined for zero

Pc and (g by keeping both inflow and outflow levels above
the soil level R and establishing a saturated steady flow
in the soil. Steady state flow was achieved when inflow
and outflow were equal. Conductivity could be calculated
by noting the head loss between tensiometers. A certain
RPM value was chosen to affect an air pressure change.

As the pressure started increasing, the inflow decreased
and the outflow increased depending upon the pressure
rate and the external levels of inflow and outflow points
or the external hydraulic gradient. If this external
gradient was too low,outflow could occur from both inflow
and outflow outlets. If the gradient was too large,
K(Pc) could not be measured accurately because of the
dependency of K on Pc‘ Thus the external hydraulic
gradient was chosen so that it took into account the
above considerations and gave inflow and outflow values
relatively close to each other (within lcc/min. in this
study). The average flux was used in D'Arcy's equation
to determine K(PC). The choice of the pressure rates

was governed by: (1) maximum time needed for each set

of readings, (2) maximum desirable -PC and (3) differ-

ence between inflow and outflow values. If the pressure

rate was too fast, Pw became larger and eventually would

66

go out of range of the recorder. The maximum value of PA
itself was limited by the air entry value of the filter
disc combination and the air entry value of the tensio-
meters. If either of these air entry values was exceeded,
air bubbles were forced into the water systems resulting
in erroneous measurements. Rapid pressure changes caused
larger differences in inflow and outflow quantities.
Therefore, pressure rates for conductivity experiments
were kept much lower than in other experiments in which
only the moisture characteristic determination was the

objective.

P,P

w inflow, outflow and gamma counts were re-

A’
corded for an interval At. The internal head loss in
soil APC was determined by positioning valve (U) to first
connect one and then the other tensiometer. These values
were averaged over the period At to correspond to a time
t1 + At/2, where tl was time of previous averaged read-
ings. K(e) was calculated from K(e) = V/I, I - APC/CAl
where APC is the head loss in terms of chart reading and
C is calibration constant. Al is the distance between
the tensiometers and V is the average of inflow and out-
flow velocities. Pw was given by the average of the two

tensiometer readings divided by calibration constant.

Short programs were written to solve the equations

A
A

O
|-'

K(e) = %E« (cm/min)

:3”

1 Im
0 = 0 --——— ln —— cc/cc
m s uws Is
and
PC = %§ - 2(MR-50) cm of water

on an electronic computer. Here Q is average inflow and
outflow in time t (min.). A is crossectional area of
flow. CR is chart reading CR/C = Pw’ MR is manometer
reading. Initially both legs of a manometer were set at

50.

PRESENTATION AND ANALYSIS OF DATA

Theory

Theory of gamma radiation attenuation technique.

 

When count rate per minute was plotted against ex-
perimentally determined values of moisture content of a
soil (contained in similar containers and saturated to
different degrees of moisture content) on a semi-log
paper, a straight line resulted as shown in Figure (5).
This indicates that a logarithmic relationship of the

form

0 = -K 1n I (58)

exists between 0, the moisture content, and I, the count
rate. Since the value of I varies with geometry, source
strength, type of soil and collimation, a reference mois-
ture content as of known value corresponding to Is for a
fixed geometry is necessary. Saturated conditions can be

used for this purpose when
0 = -K 1n Is. (59)

S

By subtracting equation (59) from (58) the equation

68

69

8 = BS - K(ln I -1nIS) (60)

or

S

results, where K is the slope of the plot.

Examination of the exponential law

I = ID e"me (61)

for attenuation of gamma rays shows that K is the recipro-
cal of the product of S, the thickness of the soil, and

u the mass attenuation coefficient, as follows:

w,

For wet soil contained in an aluminum cylinder

I = I exp-s @ (62)

O + Guw)-ZS

soil usoil AuApA

where p, u, s are density, mass attenuation coefficient
and thickness of the respective material. I is the
count rate recorded with attenuation and I0 is the count
rate recorded without any attenuation.

When the soil is saturated, equation (61) can be

written as

I = I exp-s (p (63)

S O U

+ 0 uw)-ZS

soil soil 5 AuApA

Dividing equation (62) by(63)and taking logarithm gives

I
1n IE. 1n exp s(9uw + esuw) (64)

= -suw (6 - 65)

70

or
_1_

U)

(.0
I:

H

Therefore, if §%; = K, the exponential law can be used

for moisture content determination. It was further veri-
fied by plotting the experimentally determined values of
6 against the calculated values as shown in Figure (6).
Disregarding possible experimental errors, the plot is
practically a straight line.

The mass attenuation coefficient for water can be
determined for the same fixed geometry by taking count
rates before and after filling the experimental cylinder
with water. For the empty cylinder the exponential law

can be written as

I = I exp-ZtDA A- (55)

For the cylinder full of water the same law is expressed
as

-2tDAU

I2 = I exp A - Spwuw - (66)

Dividing equation (65) by (66) and taking logarithm re-

sults in:

71

+4

 

 

3X10 P

ZXl0+4F

.5x10+4-

c3

"-1

E

‘\

2

+4

8 10 '

o

I

H
l e l
10 20 30

0 - % Volumetric moisture contents

Figure 5.--Relationship between moisture contents and
count rate.

Pi

72

30 —

20-

10

umetric calculated moisture contents

6 -Vol
c

 

io 20 3o 46
ee-% Volumetric experimental moisture contents
Figure 6.--Verification of exponential law by plotting

experimental values of moisture contents
against calculated values.

“guns; “0 UCQNfiUJ-umwmwfidannv :nJ_UlN~J~.~nVUU1 nannyp: .l 3~a

73

.076

.075

.074

.073

.072

- mass attenuation coefficient of water

 

2.8 5.6 8.4 11.2

 

Ll

Thickness of water interposed (cm.)

Figure 7.-—Variation of mass attenuation coefficient of
water with increase in thickness of water.

74

The fact that a change in the sample thickness results in
the change of the mass attenuation coefficient is demon-
strated by Figure (7). The mass attenuation coefficient
was determined by increasing the thickness of water by
successively filling four plastic cells placed between
the source and detector and computing “w after each addi-
tional cell was filled.

Mathematical models used in computation of conductivity
function

Three mathematical equations were used for con-

ductivity function determination. Burdine's equation

2 s l
Kr = (Se) Jr -J¥; j, -§%-for PC > Pb (67)
0 PC 0 PC

was solved by numerical integration (Appendix 1) for both
1
adsorption and desorption. In this equationjr ds/Pc2 re-
0
presents the total area under curves in Figures (44), (45),
s
(46), (47), (48) and] ds/PC2 is the fractional area under
0
the curve corresponding to different values of Pc and S.
The purpose of summing the areas directly is to avoid the
use of relationshi “l§'= ~l-(s )2/A involvin A and P
because A and Pb cannot be correctly estimated for adsorp-
tion curves.
In a different approach instead of using the experi-

mental moisture characteristic data, the functions

75

 

 

.1. _ __1_ S‘Sr 2H for p > p (68)
P 2 - P 2 l-Sr c I
c b
and
d
12 = 1 exp _ c (1:: ) for Pc < PI (69)
PC PS2 1

generated the relationship between the saturation and the
capillary pressure. Brooks and Corey's equation (68)
generates a curve shown dotted in Figure (44) and there—
fore, does not represent the experimental data completely.
Function (69) is, therefore, a necessity because at

s = s and P = P the curve changes its curvature from

I I

concave downwards to concave upwards.

At s = SI
1 _ 1 _ 1 e-c
2 ’ __2 ’ 2
Pc PI PS
PS2
... C = ‘11’1 (——2-) .
PI

The value of d = 0.4 was determined by trial and error.
Theoretically PS is capillary pressure at saturation when

S = l and

76

For adsorption the value of PS is 0 for the three soils
investigated in this study. Therefore, in order to make
the program (Appendix II) function properly, a finite
value of PS greater than zero has to be used. The rela-
tive conductivity is computed by substituting the func—
in

tion (68) for PC > P and function (69) for Pc < P

I I
equation (67). Capillary conductivity is calculated as

K = CSTD. x K
r

by numerical integration, where CSTD is the experimental
value of conductivity corresponding to 05.

Equation (43) of Kunze

K

_ -2 -2

-2 2
s 30 n y ¢
+
pgu (h1 3h2

—2
+--(2n-l)hn )

 

so
(70)

was also used and the results are compared in Figures

(49) to (56).
Presentation of Data and Analysis

Dynamic Method

Figures (8) to (16) represent the dynamic processes
of desorption and adsorption. The air pressure (PA) or non-
wetting phase pressure and the wetting or absolute pore
water pressure (PW) are plotted as functions of time; the

difference of these two pressures, i.e., the capillary

I

77

pressure is also shown. Pore water pressure can be either
positive or negative and is affected both by the externally

applied hydraulic head and also by the time rate of air

pressure application RAP' As RAP increases from 1.65
min./cm. in Figure (8) to RAP = 25.5 min./cm. in Figure

(9) for the same soil , maximum value of Pw decreases from
34 cm. H20 to 11.0 cm. H20. Therefore, to achieve the
same maximum value of Pc' the maximum value of PNw is
larger for faster runs than slower runs. This can be ob-
served by comparing Figures (9) and (11) where AH is the
same but RAP differs. Figure (9) and (10) show the ef-
fect of increasing AH. As a result of increase in AH,

Pw is higher in Figure (10) as compared to Figure (9).

The same general behavior was observed in the
other two soils. In fine sand (Figures (12), (13) and
(14)) Pw decreased as RAP increased from 0.625 min./cm.
to 19.6 min./cm.

It was also observed that Pw does not increase as
much in fine sand as in medium sand and increases the least
in sandy loam (Figure(15)). Figures (16), (17a), (18),

(19) and (20) show the lag between the air pressure and
the capillary pressure. In fast runs it was noted that
in became zero before PC became zero (Figure (8)) and
that it took a long time afterwards for the soil to satu-

rate suggesting that it took a long time to replace the

entrapped air from the pores. The lag between in and Po

78

for sandy loam is negligible and is not shown in Figure

(22).

Dynamic method compared with static method

 

In Figure (17b), (21) and (22) the dynamic data
are compared with the static data. In all cases it is
noted that the two moisture characteristics do not follow
the same path. These results agree with the results of
Topp gE_al. (40). It is significant that when different
RAP are compared, as in Figures (17b) and (21), the de-
sorption process follows approximately the same path. In
Figure (17b) the maximum Pc achieved was the same for all
RAP values; hence, it is not unusual that the adsorption
process also follows the same path. Since maximum Pc

values in Figure (21) were different, adsorption curves

were also different.

Head loss between tensiometers and the boundary

Figures (23), (24), and (25) show the variation
of head loss between the two tensiometers installed in
the soil with flow velocity in the three soils studied.
Even though the curves for adsorption and desorption are
not the same, they do follow a similar pattern. The flow

velocity in the soil can be expressed as

79

.0mm .EO\.EE mm.a Mo mpwu GOMDMOAHQQM unammmnm Ham How pawn Esfltmfi
CH mEHu spas mmusmmmum mumaaflmmo was mcflu903 .msfluum3lsos mo GOHDMHHM>II.m musmflm

 

 

.oH

 

m.
o
A.sHEV wasp
oom omw( com com OOH
q . fi _ _ . 4
3
m .
madmmmnm mnmaaflmmo u om
3
unannoum mmmnm mcfluums u m
unannoum mmmnm msfluumzso: u 3cm
.Eo\.cws mo.a u mfim

3G

om

 

om

om

(02H °mo) ma pus

ov

on

00

 

 

ooaa

80

.so\.cws m.mm

mfl

com

A.:HEV mEHu

com com

.ONm

00¢

.80 o.mN

oom

Ed

cow

was

m How pawn ESwUmE cw mfiflp nvw3 om was 3m .BCm mo SOHDMAHm>aI.

 

 

 

3C

- a

1

O

N

m .Eo\.cwE

m.mm

A.Eov mam>ma onmuso was 3odusfl swmzuun mocouommflo

 

 

wusmwh

O
V‘

O
('1

O
N

O
H

C)

O
H

(.3
N

C) 0
fl' m

0
:n

(OZH 'mD) “a pus ”“a '°a

m4 0.0mm .wo owmm n ma can o~m
.EO\.GHE m.mm u m How comm anome as m was m . m mo soaumanm>ll.ca mnzmflm
l.anev was“

ocov comm cccm comm ccom coma coca com

d d - - , u 5

 

81

 

m.ao\.ans s.m~
.so o.Hm

 

 

 

oNI

om

(cm

82

m .EU\.¢HE H.5m

a .ONm .ao o.m~ noma ccmzom 3a
a mo conumsnm>--.aa musmsm

H mm How psmm Edmoms SH mafia cuss m cam m .

A.assv was»
0004 comm ooom comm ooom coma oooa com

d q - d1 4 J_( A

 

 

.so\.css H.5m u

.80 c.mm

 

 

.om

 

.ONm .so\.cna m.ma

mm 3 .3:

u m .o.mH u 34 How UQMm mean as was» cuss om can m m mo soflumflum>ll.ma musmflm

 

oml

c¢l

cmu

cm:

cal

83

 

A.cssv.mssu
comm . onWm - a comsir///,ooma . com cos
_ 4 —

_ J... i

&

 

 

N

o m .Eo\.cwE m.ma

CEO OImH

 

 

cm

ow

cm

om

(OZH °mo) Md ’Mud ’od

84

Ad

.0

N

m

 

 

 

 

 

.EO\.saE mmh.a u m Mom pawn scam as om was 3m .zsm mo coflumwum>ll.ma musmwm
A.cssv wasp
omm can own son cos oma om ov
a. . — q 3 J - owl
om . .04:
o l
on.
o
. m
.om
o 3cm
0 6 [CV
.so\.cfls mas.s u mam
. .oo

 

 

 

(02H 'm0) Md pus

Mom Ucmm mcflm GH mEHu nuHB cm can m

omH ova

oNH

m .Eo\.:HE mNm.o n m<m

3 .3cm mo soaumwnm>ll.va mnsmflm

.ON

A.GHEV mEHu

ooa cm cm ow cN

 

85

.Eo\.sHE

 

mNm.o

- u u d —

 

'4‘.

ow

-ccH

 

(OZH 'mO) “a pus M“a '°a

86

Comm 080 ma

n me new ems .so\.css

 

 

 

 

 

 

cmm .50 ms n ma

ma.ma n mém £ua3 Emoa mpsmm so“ 08H“ nufl3 om was 3m .3sm mo coHuMHHm>II.mH musmflm
A.csac mans
ooov comm coom comm ooom coma OOOH com . o
T q A . _ W q . NH]
. o O

O o o. o noml Dd
c U5

. .cvl M

o D.

u
['qu (JDf' AU A C} c a p

i

3 d

m... o M
o - IO? ml

33 . m

m o
a 0.10m _CH
. m

can .Eo\.css ma.ma u mam . .

 

NH

 

87

mQSHM> SQ .muflm @4m MDOHHm> HOW Uflwm ESHGGE

Ca 3cm can om comauoo and cam oocume UHEmswo an moflumflumuomumno musumwozll.ma muomflm

3C 0

 

   

 

  

 

 

 

 

Aon .Eov muowmmnm “Hm m can whammmum Nunaaflmmo m
can QNI cm: owI cm: can ONI omu ceI 0H1 ONI omI owl omI
1 - . 4 q q a _ . 1 q 4 . —
NH1 1 (b O Nrc
T NV \ a \ m
\l . \\ \09C
3 m NJCI O m
. VAT Q \C I x I \
\.
m V. ‘\
MW I AV \ \
3 \ o o \ .
mmma. r \. 1 . o
m - \ \ E \ 31 \ 3
k \. r \
t m: x
n . \
e
.m. om: .. j
o I
c 1
mm . ,
u T
in. 1
.m em. I m w
m o m
% . N .Eo\.cflE
o m t
_ was .5 T? I. Hi cmm .2
8 \.GH8 m.mN n mdm N EO\.CHE m.mN n m
0 m
cm N . . . r N . I
O m So am H mo 60 0 mm 0 mo 0 m SO ma I m<
om r

 

 

 

 

 

 

88

 

 

0 - % moisture content (cm.3/cm.3)

 

 

/
/ ./ (b) (a)
/ Z—¢>—o—-o—StatJ.C ‘ #0
///°/ WEAP=%75i RAP = 1.65 min./cm.
fl £"5.AP= ° . 1
¢.// -o—o—o-RAP=25-7 8

 

 

 

1 IL I 1 1 J l
-40 -30 -20 -10 0 —50 -40 -30 ~20 -10
Pc capillary pressure and in air pressure (cm. H20)

 

Figure 17. (a) Moisture characteristics by dynamic method
and lag between PC and in for RAP = 1.65 min./cm.
(b) Comparison of moisture characteristics ob-
tained by dynamic and static equilibrium method
for medium sand.

89

 

 

 

 

 

aH = 16 cm. H20 0 O
R = 19.6 min./cm. 0

AP H20 ° 1

0,14
/ / ‘
/, . / -
// ./
’ .//
/ / .
,4
/
in ,
/ . 1
P .
/ c o //
./ /
,/ _

// /

l_ L l 1 L .L _l

‘50 -50 —40 -30 -20 -10

>26

24

22

20

18

PC capillary pressure and in air pressure (cm. H20)

Figure 18.--Moisture characteristics by dynamic method
and lag between Pc and in in fine sand for

RAP = 19.6 min./cm. H20 and AH =

16 cm.

2

0 - % moisture content (cc/cc)

H O-

90

o .on .Eo\.:HE o.a u mmm now comm ocflm CH 3cm
was m smmzumo oma cam conume UHEmczo mo moflpmflumuomumzo mnopmflozII.mH musmflm

 

 

CHI ONI oml owl oml oml owl
v . . . LJI .Iro.|.olh
\.
ONI . 0m \
\
w \\\\ o \»s
/ . . m
CNN. \0 \
.w \x .\
t \\ o
m. \.. \
m N \ x
V t
w .\ x
e \. _\
u \ ..\
mom-.\ \\
m 1.. \
% \
.mmf 0.
e
. o . . 0mm .EU\.cfls m.a u mum
om.

 

 

 

91
8 - % moisture content (cc/cc)

.on .Eo\.omm mm H mam How pawn was“ as

3cm was om smokumn mud cam oozumfi OHEmsmp an Usumflumuomsmno mHsumHozII.om Gunman

3 0

AONm .Eov muommmnm Ham cm pom ousmmonm wumaaflmmo m

 

OH! ONI oml ovl oml owl OBI owl om! OOHI
q q d _ 1 u 5 q q -
lllollloll-ll\
.4m\o
\
\
\
\
O\
\
\\
\.
\.
ONm .EO\.Omm mm H mflm.

 

 

 

 

92

.osmm scam How ponumfi Edmunflawswm oaumum nuflz .mmsam>

8 - % moisture content (cm.3/cm.3)

mdm msoHHm>

um .conumfi anmsmo an moflumaumuomumco unsumHOE mo sOmHHmmEooII.HN mHsmHm

NH

HN

MN

mN

5N

mN

oHI
_

ACNE .Eov musmmmum mundaflmmo m

omI
q

oMI

U

o¢| owl owl OOHI ONHI
J 3 . a

 

 

 

\\\\\||o
.\\\\
\\w.\
\

   

O m
. .EU\.G«E Hm.um¢m.|4IdIATl€I
0 ONE .EO\.G._..E ofnflmdm IOIOIII
on .EO\.s..nE m.m.numdmulololclol
GOSUOZ mwflm Idl¢llTl¢l

 

 

93

8-% volumetric moisture content

(co/cc)

vH.I

ma

ma

cN

NN

vN

mN

mN

.EMOH Npcmm MOM

oonumE Deannao pom oeumum an moaumflumuomumno musumflofi mo COmHHmmEOUII.NN musmflm

Aon .EOV mndmwmum mumaaflamo om

oaI cNI OMI ch omI owl chI omI omI coal oHHI cNHI omal oval omHI .omHI

 

 

l _ q _ . . _ _ q _ _ _ _ I a .

   

 

 

 

N

O 3 .EO o.ma u m

.EO\.cHe mH.mH u mam

ponumfi OHEMGNU Mom

0mm

conuma UHEmcmo.IIIVIIIOIII

 

 

e059: 033m IIJIIaII.

 

 

 

94

v = K(8)I
or
Ahs
and AH = Ahs + 2Ahb + 2Ahf + 2Ahp + Aht (71)

where AH is the total external applied head, and Ahs,
Ahb, Ahf, Ahp, Aht are losses in soil, boundary between
soil and supporting filter-plate combination, filter,

plate and tubing. Head loss in the tubing can be calcu-

lated from

2
_ EV— 72
Aht — f D 2g ( )
where f is the friction factor for laminar flow. D is

diameter of tubing in ft., v is kinematic viscosity and
L is the length of the tubing. hf in the tubing was
calculated as 0.000032 ft. or .000974 cm. of water which
is negligible; therefore, the main causes of head loss
are the soil, supports and boundary. The head loss de-
termined experimentally in 4 cm. of soil ranged between
0.2 to 3.2 cm. of water, or 0.6 to 9.6 cm. of water in
the full length of soil, the head loss being greater in
finer soils. The total applied head AH ranged from 16.0
to 31.0 cm. of water for different soils.

From the Figures (23), (24) and (25) it is ob-

served that the head loss in the soil between the

95

 

0.04

002 b O

 

.01

 

Vfi
P

0.003 ~ ° A

V - Flow Velocity (cm./min.)

0.002 '

D

—~0-—0- Desorption

-——A—-A——-Adsorption

0.001 '

7L 1 1 1 l I g L L 1

0 .05 .1 .15 .2 .25

Hydraulic gradient I = %%-(cm./cm.)

 

 

 

Figure 23.--Variation of hydraulic gradient between
tensiometers with flow velocity in
medium sand.

96

 

 

 

 

.01 r 0 o
O

"' O
’7 o
a
.,.I I.
E
\.
é o
3 1
>.
.p
-:-l
o
O o
H
8’
3 .001 '- o
o
H
m
I - o
>

G
o
’ o
r -—O——O-—Desorption
-—4F-4>—-Adsorption
.0002 ' 1 l

 

.2 .3 .4
A11

A2 (cm./cm.)

Hydraulic gradient I =

Figure 24.--Variation of hydraulic gradient between
tensiometers with flow velocity in fine
sand.

97

 

 

 

 

 

 

 

b.—
02 L
01 -
’7 L
I:
-a
E
\.
a L
.IJ
ea
0
o
H b
m
>
3
Fl
‘7'
>
001?
L —O— O— Desorption
L _A_A— Adsorption
)-
1 1 J I J 5
O 3 O 4 O 5 O 6 O 7 O 8

Gradient I = %% (cm./cm.)

Figure 25.--Variation of hydraulic gradient between tensio-
meters with flow velocity in sandy loam.

98

tensiometers, and thus in the whole length of the soil,
first increases as the velocity decreases but then de-
creases with a continuing decrease in velocity. The
pattern is similar in adsorption and desorption, if the
direction of the pathway is ignored.

Equation (72) suggests that initially when ve-
locity is high, the head loss in the filter-plate combin-
ation is very high as compared to head loss in the soil.
This is attributed to the very small pore diameters in the
filter-plate. Head loss being proportional to the square
of the velocity, it decreases considerably in the filter-
plate combination as the velocity decreases. The total
head loss AH being constant throughout the run, Ahs in-
creases as large pores are emptied. After the moisture
content is decreased, the boundary between the soil and
the filter-plate combination may also contribute to head
loss. With further decreases in moisture content, it ap-
pears that the head loss in the boundary becomes very
large. Perhaps this is the result of separation of soil

and filter or a loss of water film continuity.

K, 8 and Pc data

 

Figures (26) to (28) present the experimental
data for K(8)-8 relationships and Figures (29) to (33)

present data for K(Pc)-Pc relationships.

99

 

K(8) capillary conductivity tm./min.)

   
 

 

 

 

.04
.02 , .
-—<r—O-— Desorption
_4L_4L—— Adsorption
01 L ‘ 4 4
15 20 25 30
8 - % moisture content (cc./cc.)

Figure 26.--Experimenta1 capillary conductivity versus
moisture contents for medium sand.

(cm./min.)

(8)

K

capillary conductivity

100

 

.00

.02‘

.01

  

 

004
“43—43-Desorption
—11-—A— Adsorption
.002
.001 1 1 1 l l 1 l I l I

 

 

V

19 20 21 22 23 24 25 26 27 28

8 - % moisture content (cc./cc.)

Figure 27.--Experimental capillary conductivity versus
moisture contents for fine sand.

101

 

 

 

 

 

b-
O4»
.5
"'0 .0 b
a Z
\
E
o
5'
"'5 001'—
>
"-1 r
P .
o
:5 I.
'U
c
o I-
o
L
>«
36
H .003).
H
-a
m
m
o
,4 I
(D
".002¥
M
—-o—o— Desorption
.001 —4L—A—-Adsorption
l i L
15 20 25 30
8 - % moisture content (cc./cc.)

Figure 28.--Experimenta1 capillary conductivity versus
moisture content for sandy loam.

102

 

 

 

 

.7 ’

A ' A .
c' - k
E

\, .4I- ’

E
3 o

3‘ A

’S'

-a .2 _ .

.|J

U 0

9

"U

I:

O A

o

if

m .l :

S . ' ‘

-H

m .

m

o

no . L

21.04 -

M ‘ .

.02 '
O
- A —o—o—Desorption
-4L4L—Adsorption
. A
l g l
-30 -20 -10
PC ~ capillary pressure (cm. H20)

Figure 29.—-Experimenta1 capillary conductivity versus
capillary pressure for medium sand.

103

 

 

 

 

 

0.04 P ___o ’0 Lb
' o
P O

0.02 "
c3
'2
\ A
(a) O
-0.01- .
a .
u .
H
>
-r-(
4., )-
o
:3 o
p
8 A
U .004 ~
f: o
m p
H A
H
-:-l
8* A
0.002 * —<>4&—- Desorption
0? A —4hqs—- Adsorption
E .

.001 1 1 e I !

~50 ~40 ~30 ~20 ~10

PC ~ capillary pressure (cm. H20)

Figure 30.~-Experimenta1 capillary conductivity versus
capillary pressure for fine sand.

104

 

 

 

 

0.05 ’
0.04 t O _

’t 0.03 .
c
.a
E
\.
é 0 02, o
3
a
.1)
H
>
W4
‘8
5 0.01
U
c
o
0
>.
u
m
H
T: A
5340.004 . 0
o
”b
9:.
m

0.002 ’

0.001 ’

I J l L l l 1
~120 ~100 ~80 ~60 ~40 ~20
PC ~ capillary pressure (cm. H20)

Figure 31.-~Experimenta1 capillary conductivity versus
capillary pressure for sandy loam.

 

K ~ conductivity (cm./min.)
O
H

.001‘

 

 

 

n J L 1 I l 4

-70 ~60 ~50 ~40 ~30 -20 ~10 0

 

PC ~ capillary pressure (cm. H20)

Figure 32o--VAriation of hydraulic conductivity of mono~
dispersed glass beads with capillary pressure
in rewet and redry loop.

106

 

6x10"

J4x10-

32x10—

8x10—
4x10-

2x10-

10‘

’6x10-

 

 

 

l L l J L _1_ _1

~80 ~70 ~60 ~50 ~40 ~30 ~20 ~10

 

PC ~ capillary pressure (cm. H20)

Figure 33.-~Variation of capillary conductivity with
capillary pressure for aggregated glass
beads for redry and rewet loop.

(cm./min.)

K ~ capillary conductivity

107

Figures (32) and (33) represent Topp's (38) experiments

on mono-dispersed and aggregated glass beads. Topp's data
was obtained by a dynamic method using tension supplied by
a water column instead of air pressure. One point of
difference between the data obtained from three natural
soils in this study and Topp's glass bead data is that

his adsorption curves for glass beads approach a horizontal
asymptote as the capillary pressure becomes zero. But ad-
sorption curves of the soils used in this investigation
approach zero pressure with a sharp slope. Topp's data

is included in this investigation to broaden the scope of
testing the mathematical models for conductivity.

From these figures it is observed that hysteresis
exists when curves are plotted for K versus Pc but little
hysteresis was exhibited in plots of 9 versus K. In the
case of fine sand the adsorption and desorption curves
even cross each other.

Figures (34) to (38) are plots of moisture char-
acteristics using saturation S = 0/¢ instead of 8. These
plots are needed to make the first estimate of Sr' the
residual saturation by extrapolation as shown. Sr is to
be used later for plotting Se versus Pc on log-log scales

(Figures (39) to (43). Here Se’ called the effective

 

saturation, is equal to (§:§:). From these plots the

values of Pb and )I are estimated and used in equation (68)

for computation of conductivity function.

  
 
   

 

 

108

60

50'
EL ‘ 0 - 0 312
m ¢ — s 7 °
. Residual saturation Sr = .11
E
o
I; 40’ “*>——°‘—‘ Dynamic data
2 -—qG—a*—+- Static data
3’:
m
u
m
>. 30"
H
m
H
H
-r-I
Q
m
o
I 20"
o
m
I

10“

0.1 .

5 ~ saturation (8/0)

Figure 34.~~Saturation versus capillary pressure for
medium sand for desorption and adsorption.

109

 
  
     

100 r
4 ¢ = 8s = 0.29

E; S = 0.32
mNBOI— r
5'
8
m -—0—-0- Dynamic data
3 -A-4L-'Static data
0160"
(I)
m
I.)
m
>.
L4
,‘3
F4 40 ’
-H
04
m
o
l

o
“‘20-
I

~ I l I J

 

 

.3 .4 .5 .6

s - saturation

Figure 35.~~Saturation versus capillary pressure data
for fine sand for desorption and adsorption.

~P

~ capillary pressure

C

160

140

120

100

60

40

20

 

80H-

1

 

110

__o__o——.Dynamic data

-—A-—A-— Static data

 

 

.1 .2 .3 .4 .5 .6 .7 .8 .‘9 1

5 ~ saturation

Figure 36.-~Saturation versus capillary pressure for

sandy loam desorption and adsorption.

111

60’

50’

40

- capillary pressure

_PC

20'

 

 

 

5 ~ saturation

Figure 37.-~Saturation versus capillary pressure for
mono-dispersed glass beadsrewet-redry loop.

 

 

 

100’
190’
o

N
a:
.80'
E
3
0,70-
H
:3
3’. (-
0’60
14
o.
a! 50 F
(U
f...
33 I.
o, 40
(U
U
' 30 r
U
D;
I 20 .
10 L
0

3 ~ saturation

Figure 38.-~Saturation versus capillary pressure for
aggregated glass beads rewet-redry loop.

113

 

 

 

 

 

 

uorndzosea ]; uoradzospv

 

 

 

 

.vcmm Esflvme
Mom musmmmum uumHHHmmo 6cm coflumuduMm w>fluommmm cmm3umn mflcmcoflumammnu.mm musmfim

 

 

 

 

 

 

 

a
Oflumum Intlldallldll
UHEmamo .1: 0'0] ..
HH.
4

r b P 0 b Q p ‘
om om ow om om oa w v m o.H

musmmmnm annaaflmmo I um I ,

e
- S

nexnnes aArqoegga

no;

114

mHSmmmHm mumHHHmmu cam

.Ufimm GGflM HON

 

 

bvh.

   

men.

 

mmmaoo.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hm.H Nm.N
rm.
Nvmoo. Hmmooo. w
H
mm mtnv 1mm
oaumum owfimcha .

 

 

uorquospv

    

    

uorquosaa

 

 

 

 

 

 

 

 

mm . um .
. IN
IM-
oflmum F «
ounamcaalollol Iv .
10.
o is.
o o .
. .. w! W
cm ofi‘ m

 
 

muammmhm aumaafimmo I

coHumusuMm m>Huowmmm cmm3umn maanOHumemII.ov mudmflm

uorneanes 8AT338;J3

115

— PC - capillary pressure (cm. H20)

 

 

$0 29 30 40 5060 70 80 100
-‘ I 1‘1
. ' o

 

 

 

 

 

 

 

 

 

 

 

 

 

' A
. A A O
c
.3 " A '
”06' 'A ‘
m
H v o
s - ~
3.4 S = 0.34
m
.33 A‘ .
é —0—o—Dynamic
3.2- +A—Static
I .
m .
m
.l

8 Dynamic Static

"S Pb 89 122

g* l

8 sz .000127 .000067

3 A 2.43 8.5

c

.3 IE) 5.8 49

u l

8

rd .

¢ A .317 l 63

 

 

Figure 4l.--Relationship between effective saturation

and capillary pressure for sandy loam.

 

- Effective saturation

S

116

 

 

   
   

 

.8!- O O Sr=0003
'6 P Desorption
Pb = 39
.4 - l
P 2 = .000657
b
.3 -
A = 5.04
2 _ Adsorption
Pb = 20.5
L
Pb2 = 00248
01"
A = 2.52
-| L l l I l
10 20 30 4O 6O 80

 

- PC - capillary pressure (cm. H20)

Figure 42.--Relationship between effective saturation
and capillary pressure for mono-dispersed
glass beads.

S

- effective saturation

e

117

 

         
    
  
  

- PC - capillary pressure (cm. H20)
20 30 49 59 7? $00 200
o

.8 '
'6 '- S_. = 0008

Desorption
.4 -

Pb = 33.0
'3? 1

i = .000917

P

b
'2 ” A = 3.17

Adsorption

= 19.4

01 I.

-l—2 = .00266
.08 .

b

006i’ A = 2.12
.04 *

 

 

 

Figure 43.--Relationship between effective saturation
and capillary pressure for aggregated glass
beads.

118

Comparison of theoretical and experimental conductivity
results

The theoretical results for conductivity function
obtained by three different methods are presented separ-
ately for adsorption and desorption to avoid confusion
because for each experimental desorption or adsorption
curve six calculated curves are obtained, three for
static and three for dynamic data. The plots of only K
versus 8 are compared. These plots are shown in Figures
(49) to (56).

Figures (44) to (46) are plots of Eli versus 8
for the three natural soils, for both adsorgtion and de-
sorption. Figure (47) and (48) are plots for Topp's glass
beads experiments. In the adsorption process (arrows up-

wards) it may be noted that the area under the adsorption

curves is greater than the area under desorption curves.

In Figures (49) to (56) CSTD. is the measured value
of capillary conductivity near saturation. (FMT)D and(FMT)S
are the matching factors for dynamic and static processes
respectively. The matching factor is the ratio of the
measured conductivity to calculated conductivity near
saturation. (FMB) is the measure of the deviation of
the matched calculated conductivity from the measured

CODdUCtiVity at minimum moisture content, that is:

Measured conductivit
(ME) = ( v) ,

(calculated conductivity).(FMT)
Therefore (FMBXX‘FMT) gives matching factor at 8min.

 

 

 

 

 

.0084 11%
s
2 ds
(S ) ‘—‘Z
.007‘ e j; PC
K = ___—_________
r l d
s
I
f .2
0 c
.006-
_1_2 = _1_2 we)”
Pc Pb
.005 - .
D o
N
H o
9' .004
\7
. Medium sand
.003 -“ ,
—o-—0- Dynamic
-H—o— Static
,I~‘ '
1
i X
i
i x
.001 - E \
i \K
a \
J5 g \
“I5i ' *
0 0 c 7
l. .8 .6 .4 .2 0.0

 

S - saturation

Figure 44.--Area under the curve for relative conductivity
computation for medium sand.

120

 

 

 

 

.007: T)
I, I.
A
.006:
Fine sand
-—o——o——Dynamic
.00 -—A——A——Static
.00 ‘
I" i
.4 o
m
.1
.00,
I
i
.00:l’ o

 

 

‘.\O
0.0 . \‘ L
1.0 .8 .6 .4

S - saturation

Figure 45.--Area under the curve for relative conductivity
computation for fine sand.

121

.007 ' A

.006 '

.005 ’

Sandy loam
—o—o-— Dynamic

'_A_A_ .
.004 . Static

 

 

 

O 003

 

.002

 

 

 

 

.001

 

00.

 

 

.4

S - saturation

Figure 46.--Area under the curve for sandy loam for
relative conductivity computations.

122

 

 

 

 

 

 

.007 »
mono-dispersed
0 glass beads
0006 "
.005 -
.004 —
N
r4 0
m
.003 v
.002 F
.001 -
o
0-0 s : s 2e
1.0 .8 .6 .4 .2

S - saturation

Figure 47.--Area under curve for mono-dispersed glass
beads for relative conductivity computations.

123

.007 .

.006 ‘

-005 ‘ Aggregated
glass beads

 

0002 '-

 

 

 

1.0 .8 .6 .4 .2

S - saturation

Figure 48.——Area under curve for aggregated glass beads
for relative conductivity computations.

124

Experimental data ‘{*<*<>'

 

Model la Burdine Equation

2 3 ds 1 ds
K = CSTD x (s )
.10—2 47

PC C

Dynamic -4n4s—-
Static -¥—-X—

 

 

 

 

1b For P > PI Brooks and Corey equation
C Z/A
s S-Sr
K - CSTD s 2) JC -Sr ~ d8
‘ x ( e 1 *2/1
Jf s—sr
ds
0 l-Sr
For PC < PI suggested modification
8 > S
I S 1-8 0.4
f exp {-C) Iz—S— dS
K = CSTD x (s 2) ° I as
e l l-S 0.4
f exp ('C) -l—_-S— d8
0 I

Dynamic -—A—Ar-
Static -H—

2 Millington and Quirk equation as modified by Kunze.

K 2 -2
K = 33 307 n ¢ (h. 2+3h
KSC pgu 1

 

-2 -l -2
2 ---(2n ) hn )

Dynamic I-CF{}-
Static —1|—1l—

 

Legend for Figures (49) to (55) for the comparison of
experimental conductivity with theoretical conductivity.

  

 

 

*I 125
.4-
.2r
.5
«4
E
\\
E 01 p
8
L
>‘.08
4.!
'9
'H .06 p
4.1
o
s
'2 04 I-
8 ’ Medium sand
I . Desorption
M O CSTD = 0.8
(FMT)D = .143
.02 - °‘ _
‘ / (FMB)D — 4.31
000 J I (FMT)S = .143
/' (FMB)S = .877
.fi
1 1 l 1
10 20 30 40
8 - % volumetric moisture content

Figure 49.--Conductivity comparison for medium
desorption.

sand

 

(cm./min.)

K - conductivity

 

 

126

C
§..

 

Figure 50.--Conductivity
adsorption.

.8 h
o
i 0
°/ 0
/
.4 r ., O
o
I 2 p
o o
.1 — 0 °
0 0
Medium Sand
0 o Adsorption
° 0 CSTD = 0.8
' (FMT) == .058
.04 - g” .2 D
(FMB)D = 12.22
c (FMT)S = .033
° (FMB)S = 15.18
0 a
o
.02 —
P0
.01
l :9 l 1 l
15 20 25 30
6 - % volumetric moisture content

comparison for medium sand

 

127

 

 

.O4L
fine sand
desorption

»~ L
é '02 CSTD = 0.06
'2 (FMT)D = .170
\.
é (FMB)D = 1.23
.8 (FMT)S = .133
31'01” (FMB)S = 2.82 ’
"-4
>
-a
43
U
:3
c
c
o
0

..004-
M

.OOZF

O
.001 I I I l I 4 ‘L I l __l

 

 

19 20 21 22 23 24 25 26 27 28

8 - % volumetric moisture content

Figure 51.--Conductivity comparison for fine sand
desorption.

128

     

 

 

 

.08 .
.06 P fl
0‘8
.04 r G i
sandy loam
desorption .
CSTD = 0.06
.02 '-
,~ (FMT)D = .574
8 (FMB) = 1.198 0
.H D
E (FMT)S = .210
E .01 — (FMB)S = 2.
m
JJ
H
>
-H
4J
o
5
U
8
o .004 —
I
M
. o
.002 - ' o
’08 O
o
.001
.0007 E . a
20 25 30
8 - % volumetric moisture content

Figure 52.--Conductivity comparison for sandy loam
desorption.

 

129

 
 
  
  
   
      

 

.4r
mono-dispersed
.2 glass beads
desorption
CSTD = 0.9
l L- FMT = .611
FMB .217
.06 -
A 004 .-
c5
"-1
E.
\.
E .02 _
3
>3
3'?
S .01.
H
4.)
o
:3
U L-
g .008
o
' .004'”
m
.002 .
.001 - l
.0006 L O
o
.0004 -
A
a 21 l l ! 4
5 10 15 20
6 — % volumetric moisture contents
Figure 53.-—Conductivity comparison of mono-dispersed

glass beads desorption.

Eh.

(cm./min.)

- conductivity

r.

.01 '

.04 h

.01

.006 '

.004 '

.002 b

.001 “

 

Figure

  
 
 
 
 
 
  
   
   

130

mono—dispersed

glass beads

adsorption
CSTD = 0.9
FMT = .302

FMB = .633

l 1 I 1 I

5 10 1'5 20 25
0 - % volumetric moisture contents

54.--Conductivity comparison of mono-dispersed
glass beads adsorption.

 

131

 

 

1'0“ aggregated beads ///
desorption -
’6 CSTD = 2.0
.4F FMT = .572
FMB = .400
02"
,fi .1_
E
\ .
E3 .06»
>1 .04~
4J
-r-{
>
'3
o .02p
5
c
c
o
0 .01-
I
x
.006* 0 '
.004» ”
o ”I
.002.
o
.001- ‘
o
.0006’ '
.0004-
P g L I I
0 20 30 40 50
6 - % volumetric moisture contents

Figure 55.—-Conductivity comparison for aggregated glass

beads desorption.

.06.

.02-

.01

K - conductivity (cm./min.)

.006'

.004"

.002’

.001_

.0006.

.0004,

 

Figure

132

 
 
  
  
     

aggregated beads

adsorption o
CSTD = 2.0 ‘
FMT = .259

  

FMB = .933

 

l

;; I I 1
10 20 30 40 50

8 - % volumetric moisture contents

56.--Conductivity comparison for aggregated glass
beads adsorption.

DISCUSSION

Gamma Radiation Attenuation Technique

 

Although this technique is very useful for the
determination of varying moisture contents, there are
some factors which must be considered for its success-
ful application.

The detector must be shielded with lead as much
as the source itself. Whereas the shielding of the
source is necessary for personal safety, the shielding
of the detector is necessary in order to prevent the
counting of scattered radiation.

When a strong CS137 source was used (lOOmci)
and the shielding was not adequate around the detector,
it was noticed that even though the moisture content
of the soil changed, the change in the count rate was
not appreciable; therefore, accurate calculation of the
moisture content was not possible. Additional shield-
ing of the detector solved this problem. This factor
must be considered in using this technique in the field.

Another factor is prOper collimation of the gamma beam

both at the source and at the detector.

133

134

Static and Dynamic Methods

 

As in the studies of Topp et_§l, (40) it has been
observed in these investigations also that moisture char-
acteristics obtained by dynamic and static methods (Fig-
ures (17b), (21),(22)) differ from one another. It seems
that more water drains in the beginning of the desorption
process by the static method than by the dynamic method.
The possible reason for this could be that when a certain
-P

pressure step P 1 is applied, all pores of radii r2 to

2

r1 corresponding P to P tend to drain simultaneously,

2 1
whereas when pressure is changed gradually, only the
largest pore starts draining and in doing so creates dis-
continuities in the porous material making it difficult
for the next smaller pore to drain. Therefore, the amount
of water drained is smaller in the dynamic process initially.
After a few pressure steps the same kind of discontinu-
ities caused a slow down of drainage by the static method
and therefore, the slope of the moisture characteristic
curve tends to decrease.

Different rates of pressure application did not
seem to affect the draining process and give approximately
the same desorption curve for a particular soil. It was
also observed that the adsorption curves did not become
horizontal near saturation as reported by Topp (38). It

approached saturation very steeply.

135

The plots in Figures (9), (12) and (15) for medium
sand, fine sand and sandy loam show that a higher value
of maximum Pc is achieved for sandy loam than for fine
sand and similarly maximum PC is higher in the case of
fine sand than in medium sand. The reason for this is
that Pw did not build up as fast in sandy loam as in
medium sand. Pw built up fastest in medium sand. This
indicates that Pw builds up faster in coarse material
than in fine material. In the case of medium sand (Fig-
ure (9)), it was observed that after in = 42 cm. H20,
the increase in Pw was approximately the same as the in-
crease in in. The reason for this appears to be that
in fine textured soils the increase in Pw is alleviated
by drainage of the excess water. Apparently in the
coarser textured soils the film contact is lost, and
hence there is no pathway out of the tensiometer in
order to relieve this pressure.

Higher values of the external hydraulic gradient
AH, are necessary in coarse material than in fine material
to be able to measure Ah, the internal head loss in

soil.

Experimental and Calculated Conductivity

 

The experimental values of conductivity plotted

against moisture content (Figures (26)—(28)) show some

 

136

scatter of data for adsorption and desorption processes.
This scatter has been observed by Topp (38) but was not
attributed to hysteresis. This observation is further
strengthened in this study. In the case of medium sand
the adsorption curve is below the desorption curve, in
fine sand the two curves cross each other and in sandy
loam the adsorption curve is above the desorption curve.
The scatter seems to be due to experimental errors.
Dotted lines in these figures show the average of the
adsorption and desorption conductivity values. P16ts
in Figures (26), (27) and (28) are reproducible with
the same degree of accuracy. Three runs for medium and
fine sands produced similar results. For sandy loam
(Figure (28)), only one run was made.

The experimental conductivity data when repro-
duced for comparison with the calculated values in Figures
(49) to (56) are shown by open circles for both desorption
and adsorption. Figures (29) to (33) show the plots of
conductivity versus the capillary pressure. In these
plots hysteresis is well marked. This has been shown by
several investigators previously. Conductivity K(6) may
be assumed as a unique function of moisture content but
not of capillary pressure, because for each value of Pc

there are two values of 0, one for desorption and another

 

137

for adsorption; therefore, for each value of Pc there are
two values of conductivity K.

Figures (49) to (56) give a comparison of experi-
mental and calculated conductivities of five materials.
Comparisons for the adsorption process of fine sand and
sandy loam are not included because the calculated values
compared very poorly with the experimental values. CSTD,
the saturated conductivity, and matching factors for dy-
namic and static data are given on these plots. (FMT)D
is the matching factor for the dynamic process and is the
ratio of the experimental to calculated conductivities
near saturation ; (FMB) is a similar ratio at 5min after
multiplying the calculated conductivity values with (FMT).

For the adsorption process none of the equations
gave accurate results. By using Burdine or Brooks and
Corey models, the calculated values are much smaller than
the experimental values. The total area f ds/PC2
shown in Figures (44) to (48) is much greater for adso p-
tion than desorption; therefore, the ratio fog ds/PC 2
.[1 ds/Pc2 is smaller for adsorption than.desorption and
the resulting calculated conductivity K is smaller. The
Brooks and Corey equation Se = (;E)A or —£— = -l— (Se)2/A

c

PCZ P02

assumes a unique relationship between Se and Pc but
actually the same equation can not represent both the
adsorption and desorption processes. If the above rela-

tionship is to be used for the adsorption process, a

138

different value of 1 will have to be used. The use of
Brooks and Corey's technique for the determination of 1
for the adsorption process is not possible because the
plot of log (PC) versus log (Se) is not a straight line
but is a curve (Figures (39) to (43)). Even if a straight
line is fitted through the adsorption data, it gives a
different value of Pb' Theoretically, there should be
only one value of bubbling pressure for a particular soil.
Therefore, this approach does not hold true for the ad—
sorption process.

Millington and Quirk's equation predicted higher
values of conductivities for adsorption than the equa-
tions discussed above; however, the predicted values
still differed very much from the experimental values.

It appears, therefore, that the theory behind all of
these approaches is not applicable to the adsorption
process because the mechanics of the adsorption process
differ from that of the desorption process. Adsorption
of liquids by porous media is affected by air entrapped
in the pores but this is not a significant factor in de-
sorption.

Use of Millington and Quirk's equation with
matching factors (as shown in Figures (49) to (56)) re-
sulted in different matching factors for the upper and lower
parts of the conductivity curves which indicate that the

curves obtained from calculated values are not parallel

139

to the experimental curves. Generally, the curves matched
better at low moisture content with the exception of the
desorption curves for Topp's glass beads. The calculated
values of conductivity are larger than the experimental
values at saturation and when multiplied by the matching
factor the calculated values became smaller than the mea-
sured values at low moisture contents.

For the desorption process, the difference between
the experimental values and the calculated values of con-
ductivity, even though not negligible, were not as much as

for adsorption.

_1_

sz

used to estimate 3&7 and this was substituted in Burdine's
c

. l _ 2/1
Brook's and Corey's equation 5;: — (Se) was

equation (67) to compute conductivity. The approximation

_l_.= .1? (se)2/A is shown by a dotted line in Figure (44).

Pc2 Pb
It does not exactly represent the experimental data. This

approximation may give good results for those soils which
have plots of Fifi-versus S concave downwards. For soils
having these plots either partially or completely concave
upwards as shown in Figures (44) to (48) this approxima-
tion did not yield good results. For such soils approxi-
mation by equation (69) yielded better results (See Figure
(44) Open squares) for dynamic process.

For the above reasons conductivity values calcu-

lated by use of the original Burdine equation (67) did

not match the values calculated by substituting equation

 

140

(68) and (69) in equation (67) (See Figure (49), Open
triangles and crossed circles). However, when the use of
equation (67) is reduced, i.e., in plots of SEE-versus S,
when the curves became concave upwards (See dark circles
for static desorption (Figure (44), triangles and circles
for desorption in Figures (45) to (48)), results of
Burdine equation and that of equation (69) substituted

in equation (67) gave similar results. These results are
shown in Figure (49) dark triangles and crosses for static
desorption and open triangle and crossed circles in Fig-
ures (52), (53) and (55) for dynamic desorption. It,
therefore, seems more desireable either to use equation
(67) directly or when the curves of Figures (44) to (49)
are concave upwards, equation (69) may be substituted in
equation (67) and then integrated for the conductivity
calculation.

From the above analysis it may be concluded that
the direct numerical integration of Burdine's equation,
which is now possible with the aid of a computer, gave
better results for dynamic desorption process. Equation
(70) gave better conductivity values from static desorp-
tion data than from the dynamic desorption data (see
closed squares compared with open squares in Figures (49)
to (52)). In the case of glass beads it overestimated
the conductivity values at all moisture contents. This

probably was due to assumptions on which the development

141

of Millington and Quirk's equation is based, i.e., cutting
the physical flow model at planes perpendicular to the
direction of flow and then random rejoining, thus, over-
estimating the flow area and hence the conductivity.

This is contrary to what had been anticipated since
Millington and Quirk assumed that the flow area decreased
with a decrease in moisture content.

In spite of these limitations, equation (70) in-
volves no such parameters as Sr' 1, Pb’ PS etc. and,
therefore, is easy to use. Sr is important in the use
of Burdine's equation and in equation (68). Values of
Sr cannot be correctly estimated unless complete moisture
characteristics are available. Lambda (1) can be estimated
only if the relationship log PC versus log (Se) is a
straight line.

From Figures (34) to (38) it may be observed
that the moisture characteristic curves are extended be-
yond the experimental values to properly estimate the

values of Sr'

CONCLUSIONS

1. Dynamic moisture characteristics can be sat-
isfactorily obtained by gamma radiation attenuation tech-
nique with proper shielding and collimation.

Mass attenuation coefficient of water varies with
the thickness of water interposed between the source and
detector.

Logarithmic relationship between the count rate
and moisture content of the porous material is valid for
gamma radiation attenuation technique.

2. The moisture characteristics obtained by the
dynamic method differ from those obtained by the static
method. The use of different pressure application rates
in the dynamic method gave very similar results.

Moisture characteristics were obtained for
medium sand by raising air pressure as fast as 1 cm. of
water in 1.65 min. For finer soils such as fine sand and
sandy loam this rate can be higher. In coarse soils such
as medium sand the drainage of water from the soil is
considerably impeded after a capillary pressure of 30 cm.
H20; whereas in sandy loam higher capillary pressure

values can be reached before the drainage slows down.

142

 

143

3. The existing theories for the determination
of capillary conductivity from moisture characteristics
cannot predict the capillary conductivity values for the
adsorption process. For the desorption process Burdine's
equation gives better results for the dynamic process.
Brooks and Corey's approximation for determination of Pc
from Se does not match experimental data; therefore, its
use does not produce the same results as direct integra-
tion of Burdine's equation. However, when this approxima-
tion is replaceable by equation (69) (when Eii-versus S
curves are concave upwards for desorption), the calculated
values of conductivity by Burdine's equation and by using
equation (69) along with equation (67) give similar re-
sults.

Equation (70) is more convenient to use, but the
results obtained by employing dynamic data do not match
experimental conductivity values. Using static data in
equation (70) gives results which fit better with the ex-
perimental conductivity data. Generally, the conductivity
values predicted for low moisture content by this equation
require a smaller matching factor, showing that the calcu—
lated conductivities are higher than the experimental

conductivity values.

10.

REFERENCES

Ashcroft, G. L. 1962 Gaussian elimination tech-
nique for solving the diffusion equation for
moisture movement in unsaturated soil. Ph.D.
Thesis, Oregon State University. Microfilm copy
available from University Microfilms Inc., Ann
Arbor, Mich. 79 p.

Bauer, L. D. 1956 Soil Physics. 3d ed. New York.
John Wiley & sons, Inc. 489 p.

Brooks, R. H., and A. T. Corey. 1964 Hydraulic
properties of porous media. Hydrology paper no.
3. Colorado State University. 27 p.

Bruce, R. R. and A. Klute. 1956 Measurement of
soil moisture diffusivity. Soil Sci. Soc. Amer.
Proc. 20:458-462.

Brutsaert, Wilfried. 1967 Some methods of cal-
culating unsaturated permeability. Am. Soc. Ag.
Eng. Trans. 10, 3:400-404.

Burdine, N. T. 1953 Relative permeability cal-
culations from pore size distribution data.
Am. Inst. Mining Metullurgical Engrs. Petro
Trans. 198:71-78.

Carman, D. C. 1937 Fluid flow through grannular
beds. Inst. Chem. Engrs. (London) 15:150-156.

Child, E. C., and N. Collis-George. 1950 Perme-
ability of porous materials. Proc. Roy. Soc.
(london) A201:392-405.

Corey, A. T. 1954 Interrelation between gas and
oil relative permeabilities. Producers Monthly,
Vol. XIX, NO. 1.

Doering, E. G. 1964 Soil water diffusivity by one
step method. Soil Science 90:322-326.

144

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

145

Ferguson, H., and W. H. Gardner. 1962 Water con-
tent measurement in soil columns by Gamma-ray
adsorpsion. Soil Sci. Soc. Amer. Proc. 26:
11-14.

Gardner, W. R. 1956 Calculation of capillary con-
ductivity from pressure plate outflow data.
Soil Sci. Soc. Amer. Proc. 20:317—321.

Gardner, W. R. 1962 Note on the separation and
solution of diffusion type equations. Soil Sci.
Amer. Proc. 29:485-487.

Gardner, W. R. 1965 Water contents by gamma-ray
and neutron attenuation. Methods of Soil Analy-
sis, C. A. Black, ed. p. 114-125.

Hanks, R. J. 1968 A numerical method for estimat-
ing infiltration, redistribution, drainage and
evaporation of water from soil. Paper no. 68-
214 presented at 1968 annual meeting of Am. Soc.
Ag. Eng. 17 p.

Hanks, R. J., and S. A. Bowers. 1962 Numerical
solution of moisture flow equation for infil-
tration into layered soil. Soil Sci. Soc. Amer.
Proc. 26:530-534.

Holland, D. A. 1969 Construction of calibration
curve for determining water contents from radi—
ation counts. Jour. Soil Sc. 20:132-140.

Jackson, R. D., R. J. Reginato, and C. M. H. von
Bavel. 1963 Examination of pressure-plate out-
flow method for measuring capillary conductivity.
Soil Sci. 96:249-256.

Klock, G. O. 1968 Pore size distributions as
measured by mercury intrusion method and their
use in predicting permeability. Ph.D. Thesis,
Oregon State University. Microfilm copy avail-
able from University Microfilms Inc., Ann Arbor,
Mich. 91 p.

Klute, A. 1965 Laboratory measurements of hydraulic
conductivity of unsaturated soils. Methods of
Soil Analysis, C. A. Black, ed. p. 253-261.

Kunze, R. J. 1965 Unsaturated flow in drainage
problem. Drainage for efficient crop production
conference, Chicago. Published by ASAE. p. 40-
45.

 

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

146

Kunze, R. J. 1969 A rapid method for summing
series of terms in calculating hydraulic con-
ductivity. Soil Sc. no. 2, 108:149-150.

Kunze, R. J., and D. Kirkham. 1962 Simplified
accounting for membrane impedance in capillary
conductivity determinations. Soil Sci. Soc.
Amer. Proc. 26:421-426.

Kunze, R. J., G. Uehara and K. Graham. 1968
Factors important in the calculation of hydraulic
conductivity. Soil Sc. of Amer. Proc. no. 6,
32:761-765.

Laliberte, G. E., R. H. Brooks and A. T. Corey.
1968 Permeability calculated from desaturation
data. Am. Soc. Civil Eng. Proc., Journal of
the Irrigation and Drainage Division, IRI,
March 1968, 57-71.

 

Laliberte, G. E. 1966 Properties of unsaturated
porous media. Ph.D. Thesis, Colorado State
University, Fort Collins, Colorado. Microfilm
copy available from University Microfilms Inc.,
Ann Arbor, Mich. 121 p.

Ligon, J. T. 1968 Evaluation of gamma transmis-
sion method for determining soil water balance
and evapotransporation. Paper no. 68-220, pre-
sented at 1968 annual meeting of Am. Soc. Agr.
Eng., Utah State University, Logan, Utah. 28 p.

Marshall, T. J. 1968 A relation between perme—
ability and size distribution of pores. Soil
Science 9:1-8.

Millington, R. J. and J. P. Quirk. 1960 Perme-
ability of porous solids. Trans. Faraday Soc.
57:1200-1207.

Nielsen, D. R., J. W. Biggar, and J. M. Davidson.
1962 Experimental consideration of diffusion
analysis in unsaturated flow problems. Soil
Sci. Soc. Amer. Proc. 26:107-111.

Nutter, L. W. 1967 Field measurements of soil
water contents in sito by gamma attenuation.
Presented to Forestry Section, Michigan Academy
of Science, Arts and Letters. 6 p.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

147

Olsen, T. C. and D. Swartzendruber. 1967 Velocity-
gradient relationships for steady—state unsat-
urated flow of water in nonswelling artificial
soils. Paper presented at Division S—l, Soil
Sc. Soc. of Amer., Nov. 9, 1967, Washington D.

C. 28p.

Purcell, W. R. 1949 Capillary pressures—-their
measurements using mercury and calculation of
permeability therefrom. Am. Inst. Mining
Metullurgical Engrs. Petr. Trans. 186:39-48.

Richard, L. A. 1931 Capillary Conduction of
liquids through porous medium. Physics 1:318-
333.

Skaggs, R. W., E. J. Monk and L. F. Hugins. 1969
An approximate method for defining the hydraulic
conductivity-pressure potential relationship for
soils. Paper no. 69-742 presented at 1969
winter meeting of Am. Soc. Ag. Eng., Chicago,

16 p.

Smith, E. M., T. H. Taylor, and S. W. Smith. 1967
Soil moisture measurement using gamma trans-

mission techniques. Am. Soc. of Ag. Eng.
Trans. 10, no. 2:205-208.

Staple, N. J. 1966 Infiltration and redistribu-
tion of water in verticle columns of loam soil.
Soil Sci. Soc. Amer. Proc. 30:553-558.

Topp. G. C. 1964 Hysteretic moisture character-
istics and hydraulic conductivities for glass
bead media. Ph.D. Thesis, University of Wis-
consin. Microfilm copy available from Univer-
sity Microfilms, Inc., Ann Arbor, Mich. 86 p.

Topp, G. C. 1969 Soil water hysteresis measured
in a sandy loam and compared with the hysteretic
domain model. Soil Sc. Soc. Amer. Proc. no. 5,
Sept.-Oct., 1969, 30:645-651.

Topp. G. C., A. Klute and D. B. Peters. 1967
Comparison of water'content pressure-head data
obtained by equilibrium steady-state and un-
steady-state methods. Soil Sci. Soc. Amer.
Proc. 31:312—314.

Vachaud, Georges. 1967 Determination of hydraulic
conductivity of unsaturated soils from analysis
of transient flow data. Water Resource Research,
1967. 3:697—705.

42.

43.

44.

45.

148

Van, J. C. Improved outflow barriers for perme-
ability measurements. Can. J. Soil Sc. 49:
261-262, Je 1969.

Watson, K. K. 1967 Response behavior oftensio-
meter pressure transducer system under condi-
tions of changing pore air pressure. Soil Sc.
104:439-443. '

Whisler, F. D., and A. Klute. 1964 The numerical
analysis of infiltration, considering hystere-
sis, into a vertical column at equilibrium under
gravity. Soil Sci. Soc. Amer. Proc. 29:489-
494.

Wyllies, M. R. J., and M. B. Spangler. 1952
Application of electrical resistivity measure-
ments to problems of fluid flow in porous
media. Bulletin, Am. Assoc. Petr. Geol., 36:
359-403.

APPENDIX I

Computer Program and Data for Numerical Integration

of Burdine Equation

2 3 ds 1 ds
K = CSTD- (s ) .
e ’4; PC2 /fo PC2

149

l"

13228200
2215.0 +..~In.a Ixnca St.,.nIn.» I10.» .\1.~In.e I.~.uze»..u.~._0c
. 032.8200
0 or 00 .._.uz~».bc.knch .0»
Z.~nn doc
~J.$n~ m 00
uozeezcu 00
czecrm+2~.uzekuae+ecozek
«4.20“ 00 CO
m.o*0220hm+.2.ku.zcozek
2+2uz a 22I24.\..2.kI.2.».uozhokm m ~+uz.okm\.achus.
. ~4\.2.»noz~o»m K ~J\omu~2
.\.c<0.xom.*o#. #42000 00
m»m.qpm.mem.mkm._bm.km.co F2200 ca
.o.cum~.»<zacu 022
a2.~u~..~.a ..o~_ Cquc .
20.0umeckazccu r22
a2.~u~..~.h..r- 040a
20.021m.meckaeccu r22
au.CF¢U._In.Z.c—~ Lqma hm
00.0..cccuomcue
.m<0.n~.quaou c.“
mkm.q»m.mkm.mem."hm.»m.14.¢12 o<uc mm
0200 .o>I mm< u<u.m**UQ\mo 23m «0000 .>»_>L»unozoo m>~kaumsuao
.awIL.\AamImcumm.>e~moaou\maoemaozum 003mmman zo.knaomuouu
>e~>~kuoozoo omeaaohqm «cemo . >F~mcaonnala
.Oznk w>—kumamma aou waDmmmau ZOHhaaomwo ome<uoaam»z~u~ao
.qkwrk awhzmsmaoze 0022» .mwam Fzmzwaoza oaqozqhmuozeohm
.men <k<o bonzu no .0202 ..<»wrr. PZMFZOU muosmSOZu» .mzqz qhqo u m.kw
.oc_.azmk.aoc~.<u..c0~cocoa.
...oo~.au.aoo~.oz~_.2002.200..ooucmm..oo~.m..oo~.0.koo~.» zoemzuzac
.pDQFDOu~0mn<».hoakoo.hauzu«comn<».k302~. <wa<u 2<eooau

<11.”

UUUUUU

.

.003.

203.2»1
.ma.cccvczo.cc».~u<c

150

02m
uDZ~F2CU 0
cm CF C0
mm CF 06 “cm .FJ.HJ.uH
0*nuu_4
ah.c_u.*u.~J.ocau*.xr.~
c.0u.*00hmu*.xm.comu.*u~Ia#.xm.mobu.*uam*.xm.r~.*uZ*.*o*quzacu «0
A"JVOOHQ.OPmUo~IO.QmoZ.Qm FZHGQ
. mozekzco cm
“m.mguo.~oo~u.boo~u.m~.* *.k<§aou 00
.~.<U.x~.au..~cooaa..~.aswh...cwm..~cm.2_.200..~cuz~h.~.mm»zaaa
_J.2u~ on C0
2\.<¥Imoxm~.&¥Im.xo..dUn\mo.23wI.H.Xb.m_
00\nOI®.XO~.mnIN.anomlaoxm—.UQIK.XF.<IPMIFI®.X6.ZI~.Xm.*0*vh<iacu C
0 F2» 00
uDZCFZCu (r
Ckmu*a~vacuanvdu
H4.zu_ 0m 00.
wDZ~FZCU a!
n.~Jvcoaa\2~.ooaa.*m**x~.mwu.~.au
"J.Zn~ Hm co
mDZLFZOO (m
.m**a~._no*HInc\UZ~ka+.«lecooauunn.0000
“0*toaV~Qc*~IQC\UZ~okmu.~cuimk
”Join” mm CC
oua~vccou cw
oc~._u~ cu co
“cocuu.*u—Iu*.xm.c.onu.*u02~0kw*.*c*cquacu h
~IQ.OZ~okm.th~&u
wDZePZCO hr
“am300~v\.amta~vmvunuvwm m ~IQ\A~CUZ—Fu»~vm .
.J.20~ hm 00
a or 00 ao.wz.ncun
filinzmnlnunmuzeokaAECOznku."Izouzuk
22.200*.S+zcaao\kzeanonkuIzcaao a
~2\nuo

151

09¢”! CoCN! Cocml

mm. mm. QFN.

m00| maul m.ml
ON. .mmN. 0N.

Comm! 0.0ml Oomnl

me. 00m. NwNo

o.~I o.mI o.mI
mom. 0N. mm.

“.0l m.ml Cow”!
60m. mom. FON.

~00!
Nam.

Coccl Oocml C9C0l C.CF! cecal C.COIC.CC~IC.C~plC.Cn~|C.00—l
an.
Ohm. EN. NON. NmN. New. mmm. mum. mow. wk". be.
cm.o $0.0 IN. 8—
zowhnaomwo UNEIZ>O “mudomqu. £404 >OZ<m ~N<c UK
0.0! m.hwl m.r_l b.0ml v.00! b.0ml C.b®l b.0ml
maom. HON. mom. hem. 0mm. mum. Nm_. «ha.
mm.0 $0.0 00.0 m
zouhuaommo U~h<km.. 3<FFOC ozqm mzwu ~N<c UK
m.m: 0b.ml .Nul m.mul o.CNl c.0ml F.LMI m.mml c.0cl c.0ml
bN. 0N. mN. «N. mm. mm. «N. N. ofi. mm".
. um.c $0.0 00.0. m”
ZO~FuaOmO< ~2<Z>O A<3<hboc 024m mzuu ~N<C UK
0.0 0.0.! 0.00!
c.0cc c.mcl m.mel m.h¢| «.00! c.~w1 c.mml c.dUI m.UUI c.¢KI
«on. CA. Ca.
NbN. ON. mm. 6N. mm. mm. "N. N. ow. mm".
mm.o mo.o om.o cw
ZO~FnaOmmo O_S<Z>o .434Fb0. oz<m w2~u ~N<C Km
«.0! m.ml $.0I C.-| o.m~l C.wwl c.0NI m.cml C.ccl 0.001
cm. 0N. NN. N. w—. 0~. N". 1 a. fimc. «to.
. —H. m. mum. me
ZO~FDHOmO< U~F<Fm0x2~omc 02<m EDMOMZ ~N<C Un
N.w~I m.mNI m.mml m.mmt o.ch c.mmI c.hmI m.omI c.ch c.00I
0m. mm. mm. on. «H. «2. ~. 00. me. «be.
H“. m. mam. my
zohhaaomwo AX~2Hum~ 024m SD~OME ~N<C U0
m.l o.~l c.ml N.¢I c.ml «.anl C.m~l m.m_8 m.mrs H.003
com. «on. m. ON. ON. mNN. om”. om”. me». mm".
"a. m. Nam. 0H
Zoukuaomoq U~2<Z>Ohx~2~0mcozqm ED_owZ ~N<c rm
o.mul m.bfil 0.0HI c.~NI momml ".mml m.le m.cml m.cml U.O&I
um. mom. ON. ON. mm. N. b". cm“. an. Km".
"a. .m. Nun. «—
20~h000mw0 U~E<Z>oaxuzeamvoz<m Ebnomi “N40 Um

152

.
c.c~I m.mmI c.0«I c.00I 0.00I 0.60I u.qu
"mam. 0000. 000m. emu”. the”. meoc. .trc.

mo. 0._ Oct. 0

zo_kaaomoq m0<m0.~cocq awsnes 02< aaor Km
0.0—I c.0NI c.mmI c.0mI.c.KQI v.00I 0.60I u.ch
“mam. 2000. “rem. «mom. Noam. 20m". 0NSH. 2060.

00. 0.2 000. a

zoeedaomwo w0<mm._000< cmn4_z oz< 000» mm
0.0—I o.mmI m.omI 0.0mI m.muI c.r¢I
bdrm. mmcm. «mes. 0200. c160. arrc.

mo. 0. 0mm. 0

zoaeaaomcq mcqmm 0202 amnqaz 024 000» mm
0.02I c.0KI o.rmI 0.00I c.qu o.mKI u.mcI c.80I
"rem. “firm. "rpm. «and. 0600. N000. some. mmmc.

m0. 0. 000. m

. zceeaaommo moqwm 0202 amnucz 02< 000» Km
c.mI c.mI m.0~I c.m~I c.0mI 0.88I «.mCI c.00I0._LHIc.qr2I
00m. 0mm. New. mom. mam. m. 6N0“. 00m". 00". ca".

0m.o 00.0 mm. «—

zo~euaomo< unkqem Amnqomnars 2404 >02<m ~N<c ma
0.0I 0.0I b.c~I 0.00I 0.m0I c.20I c.602I0.co~Ic.qr2I
0N. 00m. mrm. 00m. 0m. cam. 002. cm”. cm”.

QC.O $0.0 mm. 0
zonkflflomwo Uukdkm “WUJ<CJ~I. $404 >OZ<W HN<O KN
8.0I 0.2I u.rI c.rl

CoCGI CoCUi CoCQl CoCFI Coca! CoCOlCoCC~ICoC-lCoCfl—IC.FR~I
«bu. bk. cm. FUR.
00". 0m”. cm". or". 0%". Mb". “F". P”. F“. bu.
cm.o mo ..0 mm. ha
zoukaaomoq U~S<Z>OAWJ<OWJ~ICE<OJ >OZ<m ”N46 Kn
Coal

APPENDIX II

Computer Program and Data for the

Solution of Equations

 

 

2 s S-Sr 2/1
CSTD x (S ) Jf '———— ds
e S l-Sr
r
K = for P > P
1 s-s 2/1 C I
f r d
5
Sr I-Sr
and
0.4
CSTD x (S )2 .[S exp(-c) l-S ds
e S I-SI
r
K= '
l l-S 0.4
‘[ exp(-c) (I:§_) ds
'5 I
r
for Pc < PI
CLlC = C
CLZC = 2/1
PB2=._L
P 2
b

DTHETA = 03/10,000

AKSjjsexperimental value of saturated conductivity
corresponding to Saturation ST

153

m0C00|qum

, "00.0m0
00*..Scoaauxncoooa «r
2000.0040.>.u.2u+a”.0000".200000 0
002.200 0 00 b

.fim 0k 00

Am&.U~JU.~m.WVZDu+Auvfloaflu.".0000 Q
cobohnumlmvhh
melmum
oomonun d 00
oCNA~UOOOQ
WOCCC0+FMNV

~COC."¢C a
COHNE
mDZ~PZCU w

~.m.na~0.tnmla2eumcu~

~0.I._IZCOMuazcnm

002.002 2 cc

ch.Ikmna~.0m

NJU\.Ku0040

.m**~a\m**mu.UOJ<IuOHJO

omI.Hu>

pm.my<.Sm.am.<ewreo.mso.uma.m0.20.Honoqma r

m.m.wxzzcue

004.ccm F2200
003.22.002 oqua c

024"

.004.<»wre.u20.00.00.my<.mmm.»<a.oaao<.azmh.ooaa.am 202200

. . Aconcay<m

.200004..00"cqewrk.aoo~cu_c..oofi.00..co~000..oo~.my<.200~.a

«mm..oo~.»<a.2002000004..002cazm».2002cccaa..oo_00m zoemzmseo

.FDQPDOn~0w0<s.»30s30.»302~noomnqb.rbaz~.0200 zqaooaa
.
.004
204.281
.00.000u«20.00».~k<c

154

J¥.~uJ 0m 00
.m.lv*..~+JCUQIAHI40000024000
J¥.NuJ 0” 00

HIZqu
1.2I082m.*2.m.00+2~.00.I.~cu.2.00
.xucazm»\o.~.»aomuancou
m¥<*.nca¥qu.ucwy<
.Jcsqaqucmwmuxqcay<

N**.>\. amI.J.0m..u.J.mmm
X*.Jcoaauqn.4.h<a

z.~n4 01 00

A.CoanO<\.~ux

mozhkzoo

0~.h~.b~aulyou~
.2.0000+.H+xcocao<uxycocao<
leuy

2.1"; 01 00

.scooaauxzvoaaoq
Ho.*.ecazmeu2~cooca

2.0"” m" 00
.mmu.um40.>.2~camczuu._cazmk
2.00. «1 00

mm or 00
.ma.ofiso._m.ancamezouuxncazwe
ma.mu.m~.~mlxncamcue

2.2u0 «a 00
m0*.ncooaauxncooaa

wozakzou
.ma.u~40.”m.mczou+.nsooaau21.oooa
a" o» 00
ANma.UNJU.>.mvzu+.ncooaun“5.0060
0.00.0n22mImcue

. moswum

0".~u~ «a 00

.omnnvooan

m.mmn 0" 0

h V .

0w

(—

bu
Cu

K—
Nm
@—
EH

«a

CH
—~

cm

155

0N0. om. WN.

F0. om. m5.
m0). 0%. 00¢.
moo. ‘om. .80.

"a. "€000Kooow 0mm~0. Com K0¢¢
ZO~FOQOmO< U~F<Fm 024m ZDHCEZ
«“0 ~MCC. Cm.N OCNOC. mom Cocm
ZOhkflaommO Unkdkm OZ<W ED~OMZ
"a. ”MCC0mmmoc mwuco Com CoON
ZODF<QDF<$ U~Z<Z>O OZ<m EDnowZ
a“. unOOo "Dom ONOC. Coma ConN
ZO~F<QDdem0 U~Z<Z>O OZ<W ZDaowZ

02m
“Noaxw*.ms#ma\~.uZDu
¢.0**A.~mi_c\.ml~..*OIuN
amu.U.~m.m.ZDu ZO»FUZDu
02m
40 #* AN I .~0.* ma 0 Zn
ammxlmmI.2.uN
“m0.JU.amm.mm.2u ZO~FUZDE
02m

00km m
0 o»,oo

asomuwmomoo~0.N.Fk.¢omnwmomoou.¢~0F<EQOM
“\.Oluoxfiw.w¥IK.XC~.UQIN.XE.<FMIFIE.XN.&¥IN.XC~.Nwmlmn
mI—.XW.ZI~.XN.OIHVF<EUCK

.K0.O~.(QIm.KF.SDUU<Im.XF.NUO\~IW.XF.

agocuugm

..c.a~u.0.aau.m.\.ura.(0..nrm.xm.nyrm.xm.am1m.xm.am1m.xm.<»mikoro_
.xc..N.Jrc.x0.m.m.maxarm.xcnmarm.x0.earm.xm.oreceqzacu
.m<0.xom.urn.»<zaou
.o.mum..c.mu.o.ru.m.»<zaou
.ma0.m~.»<zaou
.2.~n~.12.u_o.22.my<.12.00....4Fmrk_
..~.a¥<..~.Nmn..~.p<a..ecoauo<..~.02w»..~.0m.~0.nom #2200

UuJU.hm.m¥<.”moflm.<FMIFO.NJU.mmO.

ma.

NCN anflfl
uflonom anflfl

EON

J

N00

~0m
com
«0"
00"

.COCOatadvflmtdkwIkOnnJvdkak en
2.1.; an ac
<»mz»ox.3.00.13.0yqu24.ueo .cm

156

o
o
. I . 2 -
0 c.m ms. 00. 00000.0". mnmoaa no 020 "I
200% - N ooomco. 0.0" 0.Nm
000m04 004wm.~ao - .
0. «cm 00 DC. OCQCCQ 0d QmJJHS OZ< ROOF
201. 61.0 620000. 0.0H c.
.QaOflUG madwmo uamv on
mmm. 0. . I . 04 umnuez 024
00 re. emote. m . 000»
zoeeu - - .- m N mcmcc. o.ma 0.
flowed wadum 02 mm
mm. 0. “b. mc. Ommc.o W2 aw44~2 024 000%
ZO~F I. I c «C K Fmoccc. C.O~ C.
- 030000 00400 02 , «4
moo. m0. 0 . 02 chJ_z 024 00 k
0 cm. GNCCoEQ. . O
200 a #24000. mm. 0.00
mam. m0. Nb. . +aaom04 UHF4Fm 2404 >024m
6m mRCCo m.
400. ms. 00. .8000000 oeh4em 2404 >024m
. QM. QNOCofiumo
203» 0 home. c m. 0.0m"
N00. m0. Nm. . fla0m04 U~E<Z>O EdOJ >OZ<m
4n mmco. m4.
moo. mc. mm. . Foaowmo 0.24z>o 2403 >024m
Nm ~0NCC. Hm.
200» -Hmcoocc. 0.0 0.2m
who. we. 40. mm. N mccmaovmo unk4km 024m 0201
20 Hem C mCOCC. m.“ c.0m
O~FQUOW0< Uh :
n00. mo. 40. mm. Nomcc. m0. e4z>0 024w mz~u
202? a N 00000. o.c2 c.h4
0 ammo uez<z>o 024m mzeu

APPENDIX I II

Computer Program and Data for

Solution of Equation

‘2).

 

-—S— 30m Y ¢ (h '24-3h2'2 + ----- (2n-1)hn

157

00228200

. .N**.~0000*2_02>.\0w00u2~.0 m .~+.orooo.I.*2~0.00u20.2>
.2I500 +22~I500 Ianc0 0*“..LI008 I2708 .\.2~In.8 I.~.0z~80002~.~00
03228200

0 08 00 220.0208.80.2n.8 002

2.205 4 00

"4.200 m 00

03228200

02208m+20002~8022+0002~8

~J.2u~ 0m 00.

m.c*02~08m+.s.8u22.0228

2+2u2 m 22I24.\..2.8I12080002208m m 2+0z~08m\2208u2
m~>\m**.~4\800m0*.2.8*004u0mou a 24\22.8n02_08m m anxomu_z
.\.m40.xom.*o*. 842000

08m.48m.m8m.m8m.28m.8m.oo 82000

20.00m20842000

22.2“".2200 0.022 0400

24.00m208420o0

.z.~u_..~.8..822 0400

_ .o.oe0c.m.o~m.ousc842ac0
208.0208.805m.m2>.004. 02m. “08.2.02, 0400

8m.m..c0.0omc02

. Aeofum.Q<€.F.—.F<ECCO
_828.028.080.48m.m8m.08m.~8m.8m.24.422 0400

0208 zm>20 84 >8Smo0m2>um0> .0208 0m. zo_mzm8 004003mu805m
0000.0000 00384000208 84 .8248m200 442008482>400*>8_mzmo.\.omu004
.0228 m>~8umamw0 000 003mm000 2028000m00 00843000m8zsu200

.48mI8 00820200022 "0228 .wNLm 820200020 0040248mn02~08m
003mm000 200800omwou0 8200 0038m202 00840384w 23304>u02m
.m80.4840 83020 00 .02uz .248018. 8208200 0038mao2u8 .0242 4840 w m.8m
.oo~.2>..ooa.0..oo_00228.20020200.Aooacmmo.2oo~.0..ooac8 zo~m202~0

«PDQFDOI~0NQ<F.FDQFDO.FDQZ~«00w0<F.FDQZ~v amJJQU Edflwcan.

CN

CO
4....
0-

Fan

(2"
ha
4.2
cm

000000

.000
200.280
.ma.oocuqzu.cc8.~xdc

158

uDZ~FZOU 0
0m 08 00
mm 08 00 “om .84.—400_
N*H4u~J
24.00.*u."08*.xn.4.00.*u.z.8* N
.Xm.m.®0.*u828#.xm.m.00.*umz8*.Xm.N~.$ u~Z* a
.X4.4.m0.*002_08m*.xn.4.80.*u 820*.xm.4.b0.*nmz0 #0842000 0
“aeF..Z.8 .828.m28._2. UZ~O8m.820.mzu.m 82~00
00.00.8n0000*.xr N
.o.o~0.*uozou 143*.xn.4.m0.*uuzmx.xm.m.m0.*uA.Ocazw8*.xm.H~.m~0.*u~
0200 Z~2*.xn.m.m0.*u.03n*.xm.m.00.*um_>*.xn.m.00.*u004*.*o*.842000 r
0m00.2m8.02m.QEM8.”m8.8a3m.m~>.004.0 82000
JU\~mFumfiu
Aulfivmmu+22~lfivmm0l.fi.mmuy*02~08m\.A~IDCUZ~8lm280 NJU 0m“
. 002—8200 242
N0" C8 06 .028.80.ADCUZ~800~
"4.200 at” 00
03228200 042
2-.0~0m.~.m~0 .4.m0 .m~.* #0842000 442
an.0.a~.mmu..~0.00.a~002~8.~.40~ 82000 rm”
820*.000000220000
~J.Sn~ 00" C0
DU\ZEPHFS&
.Hlfivmmu+ahulfivmmUlafiommU.#UZ_C8m\.a"ID.UZH818280 "DU €€~
tzkkch Km"
00“ O8 00 2828.80.2DCUZ~8.0~
“A.EMD m0" CC
03228200 02
2~I~cmmu+001+0m>unecmmu e 2000+0010001 w 001000>
Na 08 00 .2.CM.~00H
"J.ZNH m" 00
22.00001u220000
a 08 06 “0.02.000”
. «I2u2 0 "Inns 0 02208mI2200228u22I200228
22.000*."+2.000\.2.~0002~I20~00 m 2200*.~+200\.2.0u.~I200 _
.42\mun

Q

159

ON.
@600

@600

m00!
ON.

.000.

00mN!

me.
OQO.

c.2I
mom.
nmw.

0.0
4cm.
m44.

.mW0.

mc.

ON. OFN. OFN. FN. NON. NmN. NGN.
N0NN 00Nb 00000 GO0000

08m. 082. 200800ommo 00242>0
c.8I m.8HI 0.82I
meow. Sow. 00m.
m.mm c.m8 0000. 403mo.

FmN. Nbu.

mmN. ON. FN. ON. ON. 4N. MN.
N.NN 0.NF 0000. dOOflO.
WON. OOH.

00.
0mm. mum. mo". 08”. 82.
«M. 40I000.0 42
.mq4cm3010 2404 >024m 0N4c 00
8.0mI m.04I 8.8mI c.80I 8.N0I
80m. 0mm. New. 002. «82.
Hr. «OIuCC.a u

ZO~FflflommO Unkdkm a 3<FFOV OZQW wzhu ”N40 00
m.H! m0N! M0MI m00m! .Nu! U0mu! 00CN!

00ON! F0~ml m.mm! 0.00! C.OU!
NN0. "N. N0 0a. mm—0
mm. m0!wm¢.~ M~

.ZonkaflOmad UHE<Z>O “(Bdkkou OZ<m wzuk “N40 WK

C00 0.08! C000!

0.00I o.mMI c.04I 0.04! m.04I 0.84I N.04I 0.20! 0.mmI c.40I 0.0mI c.00I

00m. mmm. .m8m. 0m. mm. 4m. mm.
m.mm 0.N8 0000. 400m0.
800. mm“.

000. 00. 00.
mm. Hm. N. 02. 002.
mm. 00I0m4.2 02

ZonhQGOmwO U~2<Z>O “(BQFFCC 024m wznl ~N<C Kn

c.mI 4.0I 4.8I m.mI m.mI c.2"I c.m~I c.02I 0.4mI m.ch 0.04I ch.
0N. 0N. 0m. 4m. mu. m. me. 02. 02. 2. 200. 00c.
m.mm o.m8 0000. 400mo. 4. mOImco.~ m2

0N. N80. 2008000m04 00848m2x210w. 0240 232002 _N40 00

m.ml 0.0a! m.mul m.le m.mml W0GN! C0cm! 00mm! 00Ffi! m00m! 0.00! (.04!

non. EON. ON. ON. NR. OH. ¢H0
N0NN 00NF 00000 QOCNO.
bON. WNFO0

OOH. 6000 00 ON. ON.
N.NN 0.NF OOCC. dOONO.
m. mN~0

N.NN 0.NF OOCO. €OCMO0
mom. ON".

N”. H. 00. 00. 080.
4. m0I000.~ m"

Zoukflaomwo U~8<PWAXZ~va OZ<m ZDHOUZ ~N<C Kn
m.I c.2I c.0I 4.4: c.mI

N.HH! 0.0—I m.mnl m.le 0.00!
ONN. Omg. omw. m4“. mm".
4. 00I004.~ 0"

2028000m04 00242>02x0220m0024m 230002 2N4c 00
0.0 c.02I 0.82I o.0HI c.2mI 0.00I
man. an. men. 0m. 0w. mm.

H0Fk! U0fim! m0¢ml U0On! U0C0!
N0 Fm. cW—. Fm~. Um—0
Q0 NOIuOQ0— Hm

ZO~FQQOmwO U~2<Z>OAX~Z~QWCOZ<O EDMONZ ~N<C Um

02w

160

F0”

Mb.“

0.

mm.
0.0!

mbN.
660.

CDC“!-

me.
060.

00C~l C0CN!

N0NN

N0NN

N.NN
0."!

OFN.
N.NN

N.NN

QHN.

N.NN

o
.
C0CH! W0C“! C0ON! C0k6! 006$! G0QF! U06d!
"06$. mON6. OOrN. 000". 00c”. 000C. ”CFC.

N.Nb 0000. 40000. 000. 40!000.4 0
cm. FNOO. ZO~FQGOm06 m06wm.~&606 QMJJ~E 026 QQOF 0N
0.0”! c.0N! 0.0K! 0.00! 0.U6! WomU! 0.60! “.60!
~06m. ~06m. H060. 6mOm. Noam. HON”. Omwn. ~Obc.
N.Nb 0000. 60000. OCO. 60!MCN.6 0
6m. FNOO. . ZO~haa0mm0 m06wm.~a006 QMJJ~2 026 uaOk mm
0.0”! 0.mN!.K.Cm! 0.0m! m.mr! (.UOI
FNPN. 0&6N. 606~. mumc. 6aqc. meC.
0.NF OOCC. 60cm0. Oflr. 6OIECN.6 O
FN. ONNO. ZO~kQGCm06 m06mm 0202 QMJJ~E 026 QQOF WA
. C.C~I C.CN! 0.00! c.06! C.06l.C.UU! m.mml C.UO!
“PEN. ~bPN. "FFN. 6~mm. 0000. NWOC. «Omc. EKmC.
0.Nb 0000. 6OCMC. ONm. 60leN.6 0
FN. ONMO. ZOMFQHmeo 006wm 0202 QMJJ~§ 026 QQOP Eu
0.N! 0.0! 0.m! 0.0"! 0.m~! C.mN! K.U6l 6.WO! 0.00!C.-~IC.6E~!
FN. OON. omN. N6N. QMN. mam. N. FNOw. 00m”. 66“. cm“.
0.Nb 0000. 60000. NM. 60lwoc.m mm
OFN. 6Na. ZO~Fanm06 NHF6Fm AMJ60mJulv E604 >026m ~N60 Wu
$.0! C.U! P.C~! 0.06! C.r¢! (.am! C.6C~!0.06~!C.6UH!
0N. mbN. mbN. 06m. 6N. 6HN. 00". or”. 6N".
C.NF 00cc. 60000. Nm. 60leC.m 0
MFN. 6N". zowkaaomwo U~F6bm .mm4604HI. S60J >026m ~N6C KR
, “to... COulu “or! CCU!

00CN! C0Cn! C.C6! C0CU! C0CT! C0CP! C0C¢l C0COIC0CCHIC0C——!C.CQ~!C.rnP!

6FN. FN. OK. FEW.

00m. 00". 000. .40“. 0b.. 0>_. 00.. "by. b". rd. 0“.
0.Nb 0000. 40000. mm. 40|moc.0 b”
orm. b". 200F000m0<u~z<z>0.wdqcmq~x.2404 >02<m “~40 rm

0."!

C000! C006! 0.0m! 000O! C0CF! 00cm! C.CO!00CC~!C.C~HIC0CN~!C.FNWI