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ABSTRACT

THE ANALYSIS AND SIMULATION OF WATER TRANSFER
THROUGH THE SOIL-ROOT DOMAIN

BY

Namik Kemal Kilic

The role of soil water in controlling the growth

and development of a plant has been shown in various ways.

However, a quantative analysis of the water absorption and

transfer process by a continuously growing root system of

a plant has been neglected. It is the purpose of this

study to investigate the transfer of water through the
soil-root domain with an approach based on the principles

of unsaturated water flow.

A deterministic non—stationary mathematical model
is developed based on the following assumptions;

i) The effective root zone may be determined by
the vertical and horizontal extension of root system.

ii) The density of the root system is based on the
optimum utilization of soil with an overlapping coefficient
and represented as the root surface per unit volume of soil.

iii) The irreversible resistance against diffusion

of water through the suberized tissue of the root surface

is defined as the degree of suberization. The magnitude

of suberization is based on the existence of a water

 

 

 

 

Namik Kemal Kilic

potential gradient from root xylem into soil as the end of
the period of water transfer through the root system from

moist soil into dry soil is approached due to the relaxation

of root potential during the night.
iv) The absorption of water by the plant roots is

defined with a source term, i.e., positive for water

release, negative for water absorption, in the analysis of

the soil water flow equation.

v) There exists a plant root system with a water
uptake pattern to satisfy the experimental transpiration,

evaporation, and soil water potential distribution.

An environmental growth chamber was modified to

simulate field condition and measure the controlling

parameters. Based on the experimental transpiration and

evaporation, the movement of water through the developing

root system of a kidney bean plant in Hillsdale sandy

loam was simulated on the computer. It was found that

the maximum absorption is limited to the lower part of
the root zone as a conical shell and moving downward as

the growth progressed. The rate of maximum water absorp—

tion takes place where the growth of root density is
optimum rather than where the density of the root system

is maximum due to suberization of older roots. The maximum
rate of water uptake also coincides with the development of

an optimum ratio between soil and root water potential

based on the analysis of water uptake limitation.

 

Namik Kemal Kilic

The development of suberization shows that the
transfer of water through the root system from moist soil
into dry soil could not bring the root zone into an

equilibrium condition as far as the soil water potential

distribution is concerned. It appears that the suberiza—

tion of the root surface is essential to optimize the
absorption of water as well as protect the roots from an

unfavorable environment. The amount of water absorption

by the root system drops with the depletion of soil water

in the root zone. The root water potential fluctuations

remain steady during the growth of root zone into a soil

with low water tension. As the soil dries, the root water

potential drops to the wilting point during the day in
order to absorb more water for transpiration.

The simulated soil water potential and sum of
water uptake by the proposed root model were consistent

with the experimental results for the first three weeks

of growth. Then, the simulated water potential distribu—

tion deviates from the observed potential distribution.

Therefore, further investigation of this model for the

remainder of growth season is required. HOpefully, the

development and density of the root system can be used to
complete the model of a plant enVironment, while the
process and pattern of water uptake can be used to increase

the efficiency of irrigation and drainage systems.

XW {xiv} QM‘VW

 

 

THE ANALYSIS AND SIMULATION OF WATER TRANSFER

THROUGH THE SOIL-ROOT DOMAIN

BY

Namik Kemal Kilic

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Agricultural Engineering

1973

 

 

(9 Copyright by
NAMIK KEMAL KILIc

1973

 

 

 

ACKNOWLEDGEMENTS

The author expresses deep appreciation to the
Department of Agricultural Engineering, especially to
Dr. George Merva, under whose inspiration and constant
supervision this investigation was undertaken.

The author also wishes to express his sincere
thanks to Professor E. H. Kidder of the Department of
Agricultural Engineering, Dr. R. J. Kunze and Dr. W. M.
Adams of the Department of Crop and Soil Science for their

assistance and suggestions.

ii

LIST OF
LIST OF
LIST OF
Chapter
I.

II.

III.

IV.

V.

VI.

VII.

TABLE OF CONTENTS

TABLES . . . . . .
FIGURES . . . . . .

SYMBOLS . . . . . .

INTRODUCTION . . . .
REVIEW OF LITERATURE .

Plant Growth Models .
Root Development . .

Water Absorption and Transfer

the Soil- -Root Domain

 

Processes in

ANALYSIS AND DEFINITION OF THE PROBLEM .

MATHEMATICAL MODELING OF WATER TRANSFER THE

SOIL-ROOT DOMAIN . .

The Rate of Root Growth

0 O O O O

The Geometry of the Root System . . . .

Root Density .

Suberization of Root Surface

Limitation to Water Uptake by Plant Roots .
Derivation of Governing Equation for Water
Transfer Through the Soil—Root Domain .

EXPERIMENTAL DESIGN AND PROCEDURE . . . .

Plant Environments .
Soil Properties .

Instrumentation and Data Acquisition System

Experimental Procedure
Analysis of the Data .

COMPUTER SIMULATION OF THE PROBLEM . . . .

DISCUSSION OF EXPERIMENTAL AND COMPUTER

SIMULATION RESULTS .

The Growth of Plant Leaves

The Weight Loss from the Soil— Plant System
The Distribution of Soil Water Potential .

iii

Page

U190

18

26

26
29
30
34
37

41
48
48
52
57
60
62

65

69

69
71
76

 

 

Page

Title
The Development and Distribution of

Root System . . . . . . . . . . - 85
The Development of Suberization . . . . . 89
The Distribution of Source Term . . . . . 95
VIII. CONCLUSIONS . . . . . . . . . . . 105
Recommendation for Future Work . . . . . 107
REFERENCES . . . . . . . . . . . . . . 110
APPENDIX I . . . . . . . . . . . . . . . 115
. . . 125

APPENDIX II . .

iv

 

Table

l.

The Result of the

LIST OF TABLES

Psychrometer Calibration with

KCL Solution of 0.5 Molarity at Constant 25°C

Temperature

Page

61

 

Figure

10.

ll.

12.

13.

14.

15.

LIST OF FIGURES

Block Diagram of Soil-Plant—Atmosphere System

Root Growth Function . .

The Rate of Root Density Increase Versus Soil
Water Potential in the Hillsdale Sandy Loam

The Radius of Influence When the Degree of
Overlapping is Equal to One . . . . . .

The Radius of Influence When the Degree of
Overlapping is Optimum . . . . .

The Development of Root Water Potential When
the Initial Soil Water Potential is Equal
to Field Capacity . . . . . . . .

The Rate of Water Absorption per Unit Length of
Root in Hillsdale Sandy Loam when the DevelOp—
ment of Root Water Potential as in Figure 6.

The Development of Root Water Potential When the
Initial Soil Water Potential is Equal to -2.3
Bars . . . . . . . . . . . . .

The Rate of Water Absorption per Unit Length of
Root in a Hillsdale Sandy Loam When the
Development of Root Water Potential as
in Figure 8 . . . . . . . . . . .

Schematic of the Environmental Control System

Temperature and Relative Humidity of Air in the
Upper Part of the Growth Chamber . . . . .

Observed and Calculated Influx Curves for
Hillsdale Sandy Loam

Relationship Between Water Potential and Water
Content on a Volume Basis for Hillsdale
Sandy Loam . . . . .

Relationship Between Water Potential and Hydraulic
Conductivity for Hillsdale Sandy Loam

Relationship Between Water Potential and

Diffusion Co-efficient for Hillsdale
Sandy Loam . . . .

Vi

56

56

Page

28

31

31

39

39

40

4o

51

53

56

 

 

Figure Page

16. The Relationship Between the Leaf Area of a
Bean Plant and Growth Time . . . . . . 7O

17. The Rate of Weight Loss from the Soil—Plant
System due to Evapotranspiration . . . . 72

18. The Rate of Weight Loss from the Soil—Plant
System During the Nighttime due to

Evaporation . . . . . . . . . . . 74
l9. The Rate of Evaporation and Transpiration from

the Soil-Plant System over the Time of

Growth . . . . . . . . . . . . . 75
20. The Average Soil Water Potential Distribution

in the Root Zone for 5, 6, and 7 a.m. of a

lS-day old Bean Plant . . . . . . . . 77

21. The Average Soil Water Potential Distribution
in the Root Zone for 2, 3, and 4 p.m. of a
15—day old Bean Plant . . . . . . . . 78

22. The Average Soil Water Potential Distribution
in the Root Zone for 5, 6, and 7 a.m. of a
16-day old Bean Plant . . . . . -. . 79

23. The Average Soil Water Potential Distribution
in the Root Zone for 2, 3, and 4 p.m. of a
30-day old Bean Plant . . . . . . . . 8O

24. The Average Soil Water Potential Distribution
in the Root Zone for l, 2, and 3 p.m. of a
45—day old Bean Plant . . . . . . 8l

25. The Simulated Soil Water Potential Distribution
at 6 p.m. of a 15—day old Bean Plant . . . 83

‘26. The Simulated Soil Water Potential Distribution
at 6 p.m. of a 30—day old Bean Plant . . . 84

27. The Simulated Soil Water Potential Distribution
at 6 p.m. of a 45-day old Bean Plant . . . 85

28. The Simulated Boundary of the Root System at
Different days of Growth . . . . . . . 88

29. The Simulated Distribution of Root Density at
6 p.m. of a 15—day old Bean Plant . . . . 9O

30. The Simulated Degree of Superization at 6 p.m.
of a 15—day old Bean Plant . . . . . . 90

vii

 

 

Figure Page

31. The Simulated Distribution of Root Density

at 6 p.m. of a 30-day old Bean Plant . . . . 92
32. The Simulated Degree of Superization at 6 p.m.

of a 30-day old Bean Plant . . . . . . . 92
33. The Simulated Distribution of Root Density at

6 p.m. of a 45-day old Bean Plant . . . . . 93
34. The Simulated Degree of Superization at 6 p.m.

of a 45—day old Bean Plant . . . . . . . 93

35. The Development of Water Potential in the Root
System of a 15—day old Bean Plant . . . . . 94

36. The Rate of Water Absorption by the Root System
of a 15-day old Bean Plant with and without
Superization . . . . . . . . . . . . 94

37. The Simulated Water Uptake Pattern at 6 p. m.
of a 15— —day old Bean Plant . . . . . . 96

38. The Simulated Water Absorption and Release
at 6 a.m. of a 16—day old Bean Plant . . . . 96

39. The Simulated Water Uptake Pattern at 6 p.m. of
a 30—day old Bean Plant . . . . . . . . 97

40. The Simulated Water Absorption and Release at
6 a.m. of a 31-day old Bean Plant . . . . . 97

41. The Simulated Water Uptake Pattern at 6 p.m. of
a 45-day old Bean Plant . . . . . . . . 98

42. The Simulated Water Absorption and Release at
6 a.m. of a 46-day old Bean Plant . . . . . 98

43. The Rate of Observed and Simulated Transpiration
from a Bean Plant . . . . . . . . . . 102

44. The Development of the Root Water Potential at the
Root Surface of a Bean Plant . . . . . . . 103

45. The Space Grid System . . . . . . . . . . 116

viii

 

LIST OF SYMBOLS

The density function of the root system (cmZ/cm3)
The area of a bean leaves (dm2)

Cross section of a root (cm2)

The degree of suberization (cm)

The water capacity of soil (l/cm)

The diffusion coefficient of soil (cmZ/day)
Evapotranspiration (cm/day)

The hydraulic conductivity of soil (cm/day)
The upper limit of z—axis

Constant

The soil water flux (cm/day)

Radius of influence (cm)

The ideal gas constant

Source term (cm3/day)

The surface of root per unit length (cm2)
The absolute temperature

The volume of cylindrical soil ring corresponding
to the rth increment (cm3)

The volume of influence (cm3)
The molar volume of water

The daily weight loss from the soil—plant system
(gr/day)

ix

 

The radius of root (cm)

The reciprocal of the soil conductivity

The degree of overlapping

Evaporation (cm/day)

The growth function (dimensionless)

Constant

Transpiration (cm/day)

The exponential value of the soil characteristic curve
The actual vapor pressure

The vapor pressure of pure water

The rate of water uptake (cm/day)

r-axis in cylindrical coordinates

Time (day)

The rate of root extension (cm/day)

z—axis in cylindrical coordinates

The ratio of evaporation to evapotranspiration
Water content of the soil on a volumetric basis
The soil water potential (cm)

The water potential of the plant root (cm)
Euler's constant

The flux of water (cm/day)

 

I. INTRODUCTION

A knowledge of the development and distribution of
plant roots is very important to both researchers and
farmers, especially in extensively cropped areas where
irrigation and drainage systems are required. The place*
ment of fertilizer and the methods of tillage are influ-
enced by the activity of plant roots and their distribu—
tion in the soil. The amount of water and nutrients
available to a plant in a given soil is determined largely
by the volume of soil in contact with it. The volume of
soil depends on the amount of branching and extension of
the root system. Gardner (1960) and Cowan (1965) con—
cluded that water movement toward the root through the’
soil is relatively slow and consequently, the only water
immediately available is that occurring within a few

centimeters of the root. Thus, the horizontal and verti—

cal extent of the root system and the density of roots
are important to the growth and development of plants.

Most of the efforts to model plant behavior have

 

 

been confined to studies of the relationship of yield A

1 F:-

with weather and other environmental variables using
correlation and regression methods. The biological pro—

cesses occurring in the plant world are complex and highly

— - —i‘.‘1"I—- n—u-F- kw-"W_.rw -1-.-— .1

 

 

interactive with nonlinear response to environmental

changes. The analytical solutions of nonlinear differential

equations describing the response of plants has been vir-

tually impossible. However, the development of high-speed

digital computers and dynamic system stimulation languages
has made it possible to deal with such phenomena. Recently,
researchers have developed the growth model of a single
plant by using simulation techniques. Peters (1969)
concluded that although the growth processes of a plant

are dependent on plant water status, a quantitative

analysis of water absorption and transfer by plant root
system has been neglected.

Gardner (1960) and Cowan (1965) studied the absorp—
tion of water by stationary single roots. Whisler, et_a1.
(1968) developed a mathematical model to study the water
absorption and transfer through the root system in a soil
column. However, the absorption and transfer of water by
a continuously growing root system has not been studied.
Hence, the goals of the project reported in this thesis

were:

i) to study the development and distribution of

a plant's root system in soil,

ii) to develop a mathematical model to study the

absorption and transfer of water in the soil—root domain.

 

 

 

II. REVIEW OF LITERATURE

2.1 Plant Growth Models

The development of a plant growth simulator has
received considerable attention recently. The modeling
effort has included levels of organization from the
individual leaf to the full—foliage canopy of the crop
in the field. Examples of outstanding contributions to
the modeling effort will be reviewed.

Waggoner (1969) developed a model for assimila—
tion and respiration of CO2 in a single leaf. The model
has been used to explore: (a) the steady-state response
of net photosynthesis to variations in light and dark; A
(b) respiration of the leaf; (c) stomatal resistance to 1
CO2 penetration into the leaf; and (d) CO2 compensation
point and maximum attainable photosynthesis.

Duncan, et a1. (1967) computed photosynthesis in

a foliage canOpy divided into many horizontal layers,

 

 

each defined in regard to optical properties including the
angular distribution of leaf elements. The simulator
first calculates the direct and diffuse illumination of
each leaf element for given canopy, solar, and sky condi—

tions. From this, hourly photosynthesis at various sun

 

 

 

 

angles is computed and summed to obtain daily totals. This
simulation has many useful applications under which breed-
ing for erect-leafed plant types would be a useful strategy
for improving agricultural yields.

DeWit and Brouwer (1968) developed a simulation
model of plant growth for corn. Their objective was to
model the processes of photosynthesis, respiration, trans—
piration, and growth at the tissue and organ levels of
plant organization. Growth is simulated by taking into
account the influence of temperature, carbonhydrate
reserves, age of tissues, and degree of water stress. Field
experiments with corn were conducted in California, Iowa,
and the Netherlands, and appropriate weather data was
supplied to the model. In all three cases, the growth rate
was predicted quite well.

Chen and Curry (1971) also developed a simulation .

model of plant growth for corn. They concluded that the

 

 

model can be used to test various plant growth parameters,
both physiological and environmental, to determine which

ones might be a key factor in increasing the efficiency of

 

plant production.

None of the existing models are capable of simula-
ting the growth of a plant root system. There is only
limited information available concerning the amount of
dry matter incorporated in roots as compared with shoots,
largely because of the difficulty of measuring the entire

root system.

 

 

 

DeWit and Brouwer (1968) observed that the relation
between leaf dry weight and root dry weight of a bean plant
growing in an optimum environment is linear. However, when
light intensity is reduced, the overall growth rate
decreased but root growth decreased more than leaf growth,
resulting in a higher leaf to root ratio. A reduced supply
of nitrogen or CO2 also reduced the overall growth rate,
but leaf growth decreased more than root growth. Roots and
shoots are dependent on each other in several ways since
root growth depends on the supply of water and minerals
from the soil via the roots. DeWit and Brouwer (1968)
observed the effect of carbohydrate supply during the
development of fruits and seeds at which time root growth
is reduced. Apparently, the leaf to root ratio depends on
those internal and external conditions which influence the

activity of the supplying organ and the requirements of the

dependent organ.

2.2 Root Develgpment

Investigations of root systems have been limited in
number and scope largely because of the difficulty of
observation and the labor required. Most measurements of
rooting behavior have been obtained in situ by digging
soil away from plant roots. Weaver (1926) and Dittmer
(1937) studied the anatomical features and development of
crop roots. They concluded that the development of a plant
root system is controlled by the plant's innate hereditary

potentialities as well as by their environments.

 

 

 

 

 

Kramer (1969) discussed the importance of hereditary
factors in controlling the root development of a plant.
Some species always develop fibrous root systems while
others will always develop a tap root system. Russell and
Mitchell (1971) studied root development and rooting
patterns of soybeans under field conditions. They observed
that the first root is the radicle which grows straight
downward as it emerges from the seed. Lateral roots then
appeared three to seven days after germination at ninety
degree angles around the radicle. They concluded that the
remainder of the root development is the result of secondary
and tertiary branching which fills the soil volume between
the taproots as growth progresses.

The relationship of various soil factors to root
growth and development has been discussed in detail by
Shaw (1952), Danielson (1967), and Kramer (1969). The
successful growth and functioning of root systems as absorb—
ing surfaces depends on many factors in the soil environment
including soil moisture, soil aeration, and soil temperature.

1. Soil moisture: Either an excess or deficiency
of soil water limits root growth and functioning. Newman
(1966) observed a marked reduction in the growth of flax
roots at a soil water potential of —7.0 bars. At -15.0
bars, root growth was 20 per cent or less of control rates.
It also appeared that the growth of an individual root is
independent of other roots. As root growth in the upper

layer of the soil was diminishing due to water stress, root

 

 

growth in the lower and moist soil layer was progressing
normally. Kramer (1969) concluded that not only is root
elongation stopped by a lack of water, but roots tend to
become suberized up to their tips under water deficiency.

2. Soil aeration is also a limiting factor for the
growth and functioning of plant roots. The reSpiration of
roots and soil organisms tends to reduce the oxygen and
increase the carbon dioxide concentration in the soil.
However, considerable gas exchange takes place by diffusion
between the soil surface and the air above. The effective—
ness of such gas exchange in maintaining favorable oxygen
and carbon dioxide levels depends largely on soil texture,
structure, and moisture. Wooley (1966) pointed out that
there are many contradictory reports concerning the levels
of oxygen and carbon dioxide which limit root growth. He
estimated that the diffusion process alone might supply the
required oxygen to a depth of one meter if as much as 4
per cent of the soil volume consists of interconnected gas
filled pores.

3. Soil temperature: Root growth and development
is often limited or stOpped by low and/or high temperatures.
DeWit and Brouwer (1968) studied the effects of soil
temperature on plant growth. They observed that the maxi-
mum root growth of bean plants occurred between temperatures
of 200 and 300 C. Outside of this temperature interval,
root growth was reduced and the external surface of roots

heavily suberized.

‘41!

 

 

4. Concentration of ions: Kramer (1969) discussed
the effects of ion concentration on root growth and func-
tioning. An abundance of certain elements, particularly
phosphorous and nitrogen, stimulates root growth. However,
an excess of salt and other minerals will reduce cell
division and elongation of roots. The reduction in growth
is due to the development of higher osmotic potentials in
the zone of root growth.

2.3 Water Absorption and Transfer Processes
in the Soil-Root Domain

 

 

Continuous absorption of water is essential to the
growth and the survival of plants. Water absorption is
not an independent process, rather, it is related to the
rate of water loss by transpiration. Absorption and
transpiration are linked by water transport in the xylem
tissue of plants. The first complete model of the trans—
piration process was proposed by Vanden Honert (1948). He
treated the movement of water through the soil-plant
system as a catenary process under steady-state conditions.

Recently, the dynamic aspects of water transpiraa
tion in the soil-plant—atmosphere continuum have been dis—
cussed by several authors. The analysis might be classified
into two groups according to the type of modeling approach.
The first group developed a model based on an individual
root as a line sink. The second group developed a model
based on the transfer of water through the soil—plant“

atmosphere continuum.

 

The absorption of water occurs along a gradient
created by decreasing water potential from soil to roots.
The cause of the water potential gradient, however,
differs in active and passive absorption of water accord—
ing to Kramer (1969). Active absorption is due to the
reduction of xylem water potential by the accumulation of
solutes, whereas passive absorption is due to suction
pressure which is developed by transpiration. The absorp-
tion of water by transpiring plants is assumed to be

passive absorption in the following models:

A. Analysis of water absorption by a single root.

Gardner (1960) developed a model based on the
assumption that an individual root acts as a cylindrical
sink. He assumed that the root is stationary with a
uniform radius in an infinite soil medium. The soil water
flow equation in a cylindrical coordinate system with
appropriate boundary conditions for constant K, D, and q

may be written as:

_ l
A—‘EAr “DE ”'1‘”
Q = $0 at t = O for a S r 5 oo (2.1b)
GA >
2naK 5? = q at r = a for t — 0 (2.1c)
where Q = the soil water potential (cm. of water),
D = the diffusion coefficient of soil (cmg/day),
K = the hydraulic conductivity of soil (cm./day),

 

.lqzsgw II

 

 

10

q the rate of water uptake (cm.3/day,

a the radius of the root (cm.).
Gardner (1960) obtained the following solution for this

boundary problem:

T - = ___w 4Dt
AQ — ® — @o 4HK (ln——§ — 6) (2.2)
a
where 6 is Euler's constant (0.57722). An inspection of

equation (2.2) shows that the soil water potential
gradient is:

i) proportional to the rate of water uptake,

ii) inversely prOportional to the hydraulic
conductivity of the soil.

The calculation of the water potential distribution
as a function of distance from the root surface indicates
that the radius of influence is finite and equal to
R:2%PB_} Peters (1969) calculated that the radius of
influence is less than a few centimeters after one day,
when the radius of the root is one millimeter.

Cowan (1965) has explored this approach more fully
by taking into account the diurnal variation of the
tranSpiration rate due to stress-induced stomatal closure.
He assumed that a root is an infinitely long cylinder of
radius a, extracting water within the region a S r E R =
Z/EE, at a rate of q (cm3/day) per centimeter of root with
a one to one relationship between the capillary conducti—
vity and the soil water potential. A comparison of

Cowan's solution with Gardner's work shows the influence

 

 

ll

of a variable soil conductivity on water potential distri-
bution as a function of distance from the root surface.
The radius of influence, R is reduced to half of Gardner's
approximation for the same soil. The analysis shows that
the dependence of the hydraulic conductivity of soil to
the water potential change may limit the flux of water
uptake. Gardner and Lang (1970) studied this point and
concluded that large decreases in water potential of the
root surface may increase the flux of water uptake only
slightly beyond a certain point depending on the conduc~
tivity—water content relationship.

The above models are for a stationary-infinite
root system. However, root lengths are finite and increase
with time. Also, the location and extension of the absorb-
ing zone varies with age and type of species as well as
with the rate of transpiration. Brouwer (1965) observed
that the water absorption zone of growing bean roots
extends maximum of six to eight centimeters behind the
root tips, then decreases sharply toward the base. When
the rate of transpiration is increased, the zone of water
absorption moves toward the base. Wolf (1970) examined
this problem theoretically by simulating simultaneous
water movement and root growth on a computer. He assumed
that a root is extending downward at a constant rate v (cm/

day), and defined the amount of water uptake as:

q = Arv (6i - 90) (2.3)

 

 

12

where q = the rate of water uptake by one root—tip

Ar = cross section area of that root

0i = water content of the soil prior to root
entry

00 = water content of the soil in equilibrium

with the root.
He concluded that root growth will increase the zone of

water absorption and the amount of moisture supplied.

B. The dynamics of water transportation through the
soil—plant—atmosphere continuum.

In nature, evaporative losses from the soil surface
change the distribution of water potential in the root zone
of a plant with time and space. Thus, plants must adapt
their root systems in such a way that they will be pro“
tected from water stress and still be able to absorb the
necessary amount of water and nutrients from the soil.
Therefore, it has been difficult to model and study the
process of water absorption and transfer by the root system
of a plant with complex boundary conditions. Philip (1958)
suggested that the process of water absorption and transfer
in the soil-plant domain may be considered in the liquid
phase and can be analyzed using the flow equation of water

in the soil:

ae_ _ ,
S—E — V (13(0) V9) +

8K(e)

82 + S (2.4)

 

 

13
where 0 = the volumetric water content of soil
D(0) = the diffusivity of soil
K(0) = the conductivity of soil
S = source term per unit volume of soil
Gardner and Ehlig (1962) assumed that water uptake
by a root system can be represented by a continuous source
function with negative values. Equation (2.4) for a one-
dimensional steady—state flow condition in which gravity

is neglected may be written:

3Q
0 = —_
where Q = the soil water flux (cm/day)
S = source function for water uptake by root.

Applying this equation to a vertical soil column of length
L and integrating from z = L to a given position 2,

Gardner and Ehlig obtain:

Q(z) = Q(-L) + f: S(z) dz (2.6)

According to equation (2.6), one can delineate the flow

in the soil at a given position as the sum of the flux
across the bottom of the column and the integral of the
source function from the bottom of the column to the given
depth. They calculated the strength of the sink term for
cotton, sorghum, and pepper plants on pachappa sandy loam
and found that these plants possess a localized uptake of

water.

 

 

 

 

 

 

 

14

Gardner (1964) applied equation (2.4) to finite
difference form and neglected the flow of soil water from
layer to layer in the soil to analyze the effect of root
distribution on water uptake and availability. He defined
the source term as the sum of the source terms for each
layer of soil rather than integrated over the root zone.
In Gardner's analysis, the source term for the ith layer

is expressed as

S. = N (Ar - Cb- — 2.) K- A- (2.7)

where S. = the rate of water uptake per unit across

section of a layer of soil

Ar = the suction of plant roots,
@i = the suction of soil at ith layer,
21 = the distance from the soil surface to

ith layer,
K- = the conductivity of the soil,
A1,: the length of root per unit volume of soil,

N a constant.

ll

Gardner (1964) concluded that calculated root
distributions fit well with experimental observation of
root density in the upper part of the soil. He suggested
that a discrepancy in the lower part of the root zone was
due to the assumption of a negligible impedance in the
root xylem.

Whistler et_al. (1968) used equation (2.4) for

steady—state conditions with a source function given by

 

 

 

15

S (2) = A (2) K (4)) (<1>p - $8) (2.8)

where S(z) = the amount of water uptake per unit

volume of soil,

N

A(z) the root density function,
K(®) = the conductivity of the soil (cm/day),

@p = water potential of the plant root (cm),

$5 = water potential of the soil (cm).
The root density function A(z) was expressed in terms of a
length of root per unit volume of soil. Evaporation,
transportation, and their ratio were assumed constant. A
relationship was obtained between evapotranspiration and

water uptake by assuming transpiration in equal to a source

term, viz:

 

E = i + e and d = e/i (2.9a)
0
— J A(z) K(®) (©p — ®s) = — i = - (l-Q)E, (2.9b)
-L
where E = the rate of evapotranspiration (cm/day),
i = the rate of transpiration (cm/day),
e = the rate of evaporation (cm/day),
d = the ratio of evaporation to evapotranSpiration.

By solving this equation for @p Whistler, et al. (1968)

obtained

 

App (2.10)

 

 

16

Equation (2.10) can be solved simultaneously with equation
(2.4) using numerical techniques for steady~state flow
conditions to obtain soil water potential distribution as
well as the distribution of the source function with depth.
Using this technique, Whistler, et_al. (1968) found that
the magnitude of the source was greatest at the bottom of
the rooting zone as expected. An interesting part of the
result is the transfer of water from the lower part of the
soil profile via the root system to the upper part of the
profile. The transfer of water through a root system from
moist soil to dry soil may occur, although it is contradic—
tory if one considers the suberization of older roots.
McWilliam (1968) observed that a living root of Mediterranean
grasses (Phalaris Tuberosa Li) in dry soil is surrounded by
a thin layer of moist soil, indicating the transfer of water
from moist soil into the surrounding dry soil. However,
the magnitude and duration of the "shorting" effect was not
made clear. McWilliam and Kramer (1968) reported that
cutting the deep roots of a plant, which is growing in a
soil with dry surface layers, caused the shoots to die
immediately. This experiment shows that the absorption of
water is reduced by suberization of the root system which
has been subjected to water stress in a dry soil layer.
Therefore, one might conclude that there must be some
relation between the shorting effect and the suberization

of a root system. Peters (1969) suggested that the processes

 

 

 

 

 

17

of water uptake and its "shorting" effect might be under~
stood better by improving the source term of equation

(2.4).

 

 

III. ANALYSIS AND DEFINITION OF THE PROBLEM

The multi~level character of a biological system,
such as a plant, presents a difficult problem to model and
study. A plant is made of three main components; leaves,
stem, and roots. Living leaves contain a variety of cells
and membranes in which complex biological processes such
as photosynthesis, respiration, and transpiration take
place. The stem provides a connection between the leaves
and the root system for the transportation of water and
carbohydrates. The roots of a plant grow downward into
the soil and serve to anchor the plant as well as absorb
water and minerals from the soil. The three parts are
inter-related into a complex organization of control over
one another as shown in Figure l. DeWit and Brouwer (1968)
assumed that such interdependence can be characterized
conveniently in terms of a state of functional equilibrium.
The equilibrium is governed by the activities of the organs
involved. Thus, the response of a plant system to any
environmental change depends on the response of the
individual components plus the arrangement of the compo~
nents and the paths of communication between them. There—

fore, the integration of available knowledge related with

18

 

 

 

 

 

l9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

""""""" 1
Photosyn— l
0 *~-—- thesis Respiration l
'53 l
E53 |
g I 4 l l m
<2 1 I | l [I]
0. 1— —. i I V l 3,;
O 33
E: Translo— .
§§——~—. cation of Leaves Transpi-
f3 Carbohyd- “AF- Growth *- ration
CD‘F”__ rates _“ __
4 I 1
' I
4 I
i I .._ -- l
l Translo- Water and
l cation of Stem ' minerals E3
I Carbohyd— 3... Growth ._. transport. {74,
| rates
I l P I A
I g i l
1. A :— J 1'
(£3 E 1? t l l l
——~—-- A ' A '
é; Translo- l, . Water and U‘
a . “r . 8
(L‘P"-—- cation of Root minerals EA
.4 Carbohyd- "7’ Growth absorption ‘
g ___. rates .4-.. __

 

 

 

 

 

 

 

 

 

Figure l.—-Block Diagram of Soil—Plant-Atmosphere System.

 

 

 

20

the response of plant components to their environment to
model and study the responses of an individual plant is
highly desirable for the plant scientist involved in
breeding and improving crop production.

Under certain circumstances, one can study a part
of a system by isolating it from other parts using justi—
fied constraints. On this basis, the growth process of
plants have been modeled and studied extensively.

Although the role of water in controlling the physiological
processes of the plant is recognized, a quantative analysis
of water absorption and transfer processes has been
neglected (Peters, 1969). In this analysis, we are
following the above logic, attempting to model and study
water absorption and transfer processes in the soil—root
domain.

A detailed review of water transfer processes in
the plant—soil continuum can be found in Slatyer (1967)
and Kramer (1969). The energy status of water in plants
and soils is represented by the water potential which is
the difference between the partial free energy of soil or
plant water, and pure water at atmospheric pressure and
the same temperature. The water potential function is
uniquely related to the water content of plant tissue or
soil. In nature, plant leaves lose water vapor to the air
when their stomatas are open to allow efficient carbon
dioxide for photosynthesis. This process is called trans—

piration. The rate of transpiration is proportional to

 

21

the difference between the water potential inside and
outside the leaf as well as to the degree of stomatal
opening. The stomatal openings are controlled by the
guard cells which are sensitive to environmental para—
meters such as sunlight, and operate in response to turgor
pressure of the cells. Sunlight not only affects the
stomata's opening, but also exerts strong physical effects
on the transpiration rate. This combined influence of
sunlight causes daily fluctuations in the rate of trans—
piration.

The water loss from leaf cells will be absorbed
from the xylem of the leaves. The removal of water reduces
the water potential of xylem. Thus, a potential gradient
will exist between the leaf and root xylem to transfer
water from the root system to the leaves of the plant.

The amount of water transferred has to be provided by the
absorption of water from the soil.

We should review briefly the structure and develop—
ment of plant roots to understand the mechanics of water
absorption. A more detailed account of root structure
can be found in Esau (1965) and Street (1966). They
regarded an extending root in four regions; root cap,
meristemic region, the region of elongation, and the region
of differentiation and maturation. In the meristemic
region, growth and cell division both occur until they
reach the elongation zone. The growth of the cell occurs

mainly lengthwise with little lateral expansion in the

 

 

22

elongation zone. This zone extends into the differentiation
and maturation zone, where the epidermal of the root cell
develops root hairs. Kramer (1969) concluded that as the
new root hairs develop,cutinization. and suberization of
the epidermis occurs and older root hairs tend to disappear.
The pathway of water movement from the root surface
to the xylem tissue takes place through the epidermis, the
root cortex and the endodermis. The rate of water absorpe
tion is directly proportional to the water potential
gradient between root xylem and soil, and inversely propor«
tional to the resistance encountered in the pathway. Gard—
ner (1960) and Cowan (1965) treated the absorption of water
by individual roots as a diffusion process and neglected

the resistance of the root to the flow of water from the

 

root surface into root xylem tissue. However, Kuiper (1963)
observed that the resistance of root tissues has a signifi—
cant effect on the absorption and release of water. He
claimed that the main barrier to the flow of water is at
the endodermis due to the development of a casparian strip.
Brouwer (1965) observed that maximum water absorption takes
place in the root hair zone where there is least resistance
to the diffusion of water. The least resistance in the
root hair zone may be associated with unsuberized root
epidermis rather than with the development of the casparian
strip. When the rate of transpiration is increased, the
zone of water absorption extends along the root, where the

surface of the root is supposed to be suberized. Kramer

23

(1969) concluded that the suberized layer of the root sur—
faCe is not impervious and uniform along the root, but
merely presents a higher resistance to the diffusion of
water. The question can be raised concerning the advantages
of suberization of the older root surface, when the primary
function of roots is to absorb water. Wolf (1970) con—
cluded that the advantage of suberization is to prevent
water loss into the soil that has dried. Gardner (1960)
assumed that root water potential is constant throughout the
root system. However, the soil water potential distribution
in the root zone is not constant due to evaporation and
diurnal fluctuations of water uptake by the root system.
Thus, the root system must adapt by suberization to protect
water loss due to shorting of water flow along the root.

The transfer of water through a root system from
moist soil to dry soil has been reported by Hunter and
Kelley (1946) and McWilliam and Kramer (1968). The shorting
of water flow along the root might be explained if one
considers the dynamic aspects of water potential distribu~
tion in the soil-root domain. The soil water potential
around the roots might not relax as fast as the water potena
tial of the root xylem when active'transpiration stops
during the night. The xylem may then release water accord~
ing to the magnitude of the potential gradient and the
degree of suberization along the root, while the absorption
of water is continued at the tip of the root to keep the

system in equilibrium. Therefore, it may be necessary to

 

 

 

 

 

24

model the whole soil—root system to study the process of
water absorption and transfer phenomena.

The root system of a plant is branching and grows
into new soil from which it may obtain adequate water for
the transpiration demand of the growing plant canopy.
Existing growth models do not describe the growth and
distribution of a root system in the soil. ‘Most researchers
have equated growth and distribution of a root system to
the dry weight of the root. However, the distribution of
a root system in terms of dry weight has no meaning as far
as water and nutrient absorption is concerned. A more
meaningful model would appear to be one based on the
absorbing surface of the root system rather than dry weights
of roots.

Gardner (1960) and Cowan (1965) show that the rate
of water flow'toward the root surface is controlled by the
hydraulic conductivities of the soil, and the only water
available is that occurring within a few centimeters of the
root. Gardner and Lang (1970) reported that the rate of
water uptake reaches a limiting value as the potential
gradient increases between soil and root xylem. Therefore,
an equilibrium condition may exist between total transpiraw
tion and root surface with a minimum root potential and
optimum water uptake under a normal environment.

Whenever transpiration demand increases over the
state of dynamic equilibrium, plant leaves cannot force

the root system to absorb more water even by increasing

 

 

 

 

25

the water potential gradient between the soil and root xylem.
At this point, plant leaves apparently close their stomatas
to control the rate of transpiration. The state at which
the soil can no longer supply sufficient water to the plant
is called the wilting point. Therefore, water transfer
through the soil—plant—atmosphere continuum is controlled
by plant and soil as well as by climatic variables. Fur—
thermore, the closed stomates stop the diffusion of carbon
dioxide, which is a basic ingredient of the photosynthetic
process. This analysis shows the importance of the soil—
plant water status in plant growth models.

One can conclude from the preceding analysis that
the development of a complete dynamic model for a plant
system requires quantative analysis of water absorption
and transfer processes which take into consideration the
following factors:

i) The development and geometry of a root system,

ii) The density of the root system,

iii) The degree of the suberization process,

iv) The limiting effect of soil conductivities
on water uptake,

v) The development of water potential distribution

due to evaporation and water uptake by the root

system.

 

 

 

 

IV. MATHEMATICAL MODELING OF WATER TRANSFER

THROUGH THE SOIL-ROOT DOMAIN

It is desirable to set up a determinate non—
stationary model for water absorption and transfer through
the soil and root system. The types of information needed
to set up a model of this nature may be discussed as

follows.

4.1 The Rate of Root Growth

 

The growth rate of a root system and its functioning
as an absorbing surface is controlled largely by parameters
of the soil environment such as soil moisture, soil aeration,
soil temperature, as well as the translocation rate and
carbohydrate accumulation. It is impossible to set up a
mathematical relationship between the rate of root growth
and these variables due to a lack of information about
their interaction. Therefore, one must eliminate some of
the variables by making the following assumptions:

1) The soil temperature is constant at 250C.

Brouwer (1965) observed that optimum growth occurs at

this temperature.

ii) The lower boundary of root growth is due to lack

of aeration, a condition which occurs in the zone of near

saturation.

26

 

 

 

 

 

 

 

 

 

27

iii) The upper limit of root growth is due to water
stress. In the model, it will be assumed that the
limiting value of water stress is -15.0 bars water potential.

iv) The soil is free of salt concentration, i.e.,
no osmotic contribution to the soil water potential.

Under these assumptions, one can represent the

effects of soil water potential on the rate of root growth
by defining a dimensionless growth function. The growth
function can be derived by using the observations on root
extension of Newman (1966), and Ratliff and Taylor (1969)

as follows in empirical form:

 

-(<I>) e(<b+l) for§D<O
g(®) = (4.1)
0 otherwise
where g(©) = the growth function

Q the soil water potential (bars)

It appears that the rate of root growth is optimum
at —1. bar soil water potential as shown in Figure 2. The
actual rate of root extension at different locations with

correspondingly different water potentials can be calculated

by multiplying the growth function by the maximum rate of

root extension. The root extension function may be written
as:

v(®,t) = va(t)g(®) (4.2)
where v(@,t) = the actual rate of root extension

(cm/day)

 

 

 

 

 

28

1.0 ‘7

.6J

soil-day

 

.1.—1

/cm

.2-

 

cm

 

 

0.0. . . .1

.JJ

xo—J
5.1

Figure 2.—-Root Growth Function.

.2-

cine/am3 soil-day

 

OJ)

 

I I I T I l 1 I

23 1.56 7 8 910

H“

Negative water potential (bars)

Figure 3.--The Rate of Root Density Increase Versus
Soil Water Potential in the Hillsdale
Sandy Loam.

 

 

 

 

 

 

29

va(t) = the maximum rate of root extension
(cm/day).
The maximum rate of root extension is dependent
on the age and type of plant. Russell and Mitchell (1971)
observed that the average rate of root extension was between

2.0 and 5.0 cm/day for soybeans.

4.2 The Geometry of the Root System

 

The geometry of the root-soil interfaces is very
complicated since it changes as the root system branchesv
and grows with time. A model which neglects the time
dependence of the root system geometry may be appropriate
for a perennial plant. However, root extension is probably
very important for annual plants such as agricultural crOps.
It may be possible to simulate the process of branching by
assuming the root system has to branch to satisfy the demand
of transpiration. However, the determination of the loca—
tion and growth direction of too many branches in three—
dimensional space is not feasible. Although the importance
of root geometry in the water absorption and transfer process
is recognized, it must be simplified to reduce the mathemati—
cal complication of the model to a point where one can hope
to make some progress with it.

It is thought that it is feasible to use the obser—
vations of Russell and Mitchell (1971) to represent the
geometry of a root system for a bean plant. They observed
that four to six lateral roots appeared after germination

with an almost 90.0 degree separation from the radicle,

 

 

 

 

3O

i.e., the first root which grows downwards from the seed.
The remainder of the root development is the result of
secondary and tertiary branching to fill the soil between
these taproots as growth progresses. The shape of an
effective root zone for our purposes may be assumed to be
a conical prism whose outer surface can be determined by

the vertical and horizontal extension of the root system.

4.3 Root Density

 

In nature, the competitive root tips may be growing
in all directions with some average separation. It has
been difficult to measure and express such growth in a
meaningful way. Whisler §E_al. (1968) proposed that the
density function of root systems is a function of the area
of the absorbing surface per unit volume and the effective
distance of water flow to the root surface. Thus, the
maximum value of R, the radius of influence of a single
root, can be correlated with the density function of the
root system by assuming that it is one—half the average
distance between neighboring roots, as shown in Figure 4.
However, it is possible that the radius of influence of
neighboring roots might be overlapped by the formation of a
new root branch as the growth of plant progresses. There—
fore, we should define a factor, c to represent the degree
of overlapping in the root zone. The best combination of
overlapping, where optimum utilization of soil occurs, will
be a hexagon as shown in Figure 5, and the optimum degree

Of overlapping may be calculated as:

 

 

 

 

 

 

 

 

 

 

 

 

31

 

Figure 4.-—The Radius of Influence when
the Degree of Overlapping is
Equal to One.

 

Figure 5.-—The Radius of Influence when the
Degree of Overlapping is Optimum.

 

 

 

 

 

 

32

H R2
2

2.59808 R

= 1.21 (4.3)

If one defines the density function of a root
system as the area of root surface per unit volume, then

one obtains:

Sr (r, z, t)

 

A (r, z, t) = Vs (r, z, t c (4.4)
where A (r, z, t) = the root density function,
Sr (r, z, t) = the surface of root per unit length,
Vs (r, z, t) = the volume of influence

II

c the degree of overlapping.
The surface of the root per unit length is a function

of the root radius, a(r,z,t), and may be written as:
Sr (r, z,t) = 2Ha(r,z,t) (4.5)

The volume of influence is a function of the radius
of influence, i.e., the distance from the center of the
root at which the water potential gradient is zero. The

volume of influence may be written as:
2
Vs (r,z,t) = H(R(I}Z,t)) , (4.6)

The radius of influence can be calculated from equation
(2.2), which is the steady—state solution of water absorption
for a stationary root by Gardner (1960). By setting equation

(2.2) equal to zero and solving for R, we obtain:

 

 

 

 

33

(4.7)

 

where D(®) = the soil diffusivity (cm/day),

H-
II

time of active absorption (assumed 12 hours),

09
ll

Euler's constant.
By substituting the equations (4.5), (4.6), and
(4.7) into equation (4.4), one obtains:

: 2 a(rrzrt) C
2.25 D(¢) t

 

A (r, z, t) (4.8)

where A (r, z, t) is the optimum root density function,
when c is equal to 1.21. The actual density function of
the root system may be calculated as the product of the
growth and optimum density function, since the rate of root
extension is controlled by the growth function as explained
in previous section. The actual density function of the

root system may be written as:

.A (r, z, t) =.A (r, z, t-l) +
(4.9)
[A (If, 2, t) - A (:r, z, t—l)] g(®)

An inspection of equation (4.8) and (4.9) shows that
the density function of the root system is:
i) proportional to the radius of the root,
ii) inversely proportional to the diffusion
coefficient of the soil,

iii) controlled by soil water potential.

 

 

 

 

34

Assume that the radius of a root is constant; then
one might expect from equation (4.8) a higher root density
in the older and upper part of root zone due to reduction
in soil diffusivity with decreasing soil water potential.
This situation might represent the growth of root diameter
and the formation of new root branches as it happens in
nature. However, the growth rate of root density is con—

trolled by soil water potential as shown in Figure 3.

4.4 Suberization of Root Surface

 

The root system of a plant presents structural and
physiological adaptations which appear to be inconsistent
at first. For instance, the development of suberiZation
in the older part of growing roots is contradictory to
the primary root functions which are the absorption of
water and minerals as well as the anchoring of plants.
Although the resistance of the suberized layer against
movement of water through the soil—root boundary, there
has been no attempt to investigate the development of
suberization and the resulting resistance. The present
knowledge of the suberization was described nicely by

Kramer (1969):

Resistance to water movement through suberized
roots must be located in the layer of suberized
tissue around the outside of the root, in the
cork and vascular cambia, and in the phloem.
The relative importance of these t1ssues as
sources of resistance might be established by
removal of one layer at a time while water 18
moving through the roots under pressure, but

this has never been done.

 

 

 

 

 

 

 

35

Wolf (1970) suggested that the advantage of suberi—
zation lies in preventing water loss along the root, conse~
quently minimizing the expenditure of energy to absorb the
required amount of water for transpiration. Based on
Wolf's (1970) suggestion, one might understand the need
and development of suberization by investigating the
absorption and transfer of water by a plant root system
with and without suberization.

Gardner (1960) assumed that the development of water
potential in the root xylem is constant throughout the root
system. However, the water potential of soil is not con-
stant in the root zone due to evaporation and cyclic water
absorption by the root system. Furthermore, the older parts
of the root system has been in situ for some time, and
consequently, they may have dried out the soil in their
Vicinity. Under these circumstances, the root system
without suberization serves as a short to transfer water
from moist soil to dry soil and thus keep the entire root
zone in an equilibrium condition. Such a condition does not
exist according to the observation of Gardner (1964).
Therefore, the plant is forced to develop a water potential
in the root system lower than minimum water potential in
the vicinity of the root system in order to absorb water,
when active transpiration begins in the early part of the
day.

The transfer of water through the root system from

moist soil to dry soil will continue until the morning or

 

 

——i—i

36

at a time such that the water potential of the root xylem
decreases to a value sufficient to keep the system in an
equilibrium condition as far as water absorption and release
are concerned. Now, assume that the root system develops

a suberized layer at the root surface where a potential
gradient from root xylem to soil exists during the night.
When transpiration begins, the plant does not have to

develop a lower water potential in the xylem to overcome

 

the lowest soil water potential in the root zone in order
to absorb water. Rather the zone of water absorption will
move along the root from the tips to the base as the water
potential of the root xylem decreases with increasing
transpiration demand.

In comparing these two situations, it can be seen

 

that the suberization of older roots will minimize the
energy expenditure of the plant and is part of plant
adaptation. Therefore, one can deduce the following assump—
tions to model the development of suberization and its
effect on water absorption:

i) The surface of older roots will become

 

suberized to prevent the shorting effect of water flow
along the root tissues.

ii) The degree of suberization is equal to the
maximum potential gradient from root to soil which develops
during the nighttime due to transfer of water from moist

soil to dry soil.

 

 

 

 

 

————

37

iii) Shorting of water flow will take place whenever
the degree of suberization (i.e., the maximum nighttime
potential gradient) is overcome by the existing potential
gradient.

4.5 Limitation to Water Uptake
by Plant Roots

 

 

Gardner and Lang (1970) reported that the rate of

water uptake reached to a limiting value as the potential

 

gradient increased between soil and root xylem. They
assumed that the flow of water into the root surface is
radial and the governing equation for water absorption by
a single root with initial and boundary conditions is

written as:

 

ST 8®

 

_ l 3 __
‘75 — 3? ‘3? (r 9“") 3r) (4'10”
©==©O at t = ()for a S r S X (4.10b)
84 _
2 H a K(®) 3? = (fit) at r — a for t > 0 (4.10c)
84 _
‘51:- = O at r - X for t > 0 (4.10d)

Gardner and Lang (1970) solved equation (4.10a) by
numerical techniques to determine the limiting water uptake
from the boundary condition, equation (4.10b). They con—
cluded that decreasing water potential within a plant does
not necessarily increase the flow from the soil to the roots.

Hence, the limiting value of water uptake and the corresponding

 

IIIIIIIIIIIIIIIIIII-llll---————______i

38

water potential of the root xylem may be used in our model
to optimize the relation between the root surface and the
transpiration demand.

The absorption of water by the plant root is a
dynamic process, since the flux of water is a function of
the water potential gradient and soil hydraulic conduc~
tivity, which changes with time. In the present model, we
assume that the water potential of the root xylem decreases
linearly from some initial value at dawn to a low at 2 p.m.,

at which time the transpiration demand reaches a peak value.

 

This assumption enables us to estimate the minimum root
potential corresponding to maximum water absorption by the
root system.

Equation (4.10a) with the boundary condition

 

(4.10b) through (4.10d) is solved by numerical techniques
over twelve hours for different initial soil water poten-
tials to determine the temporal behavior of water absorp—
tion at the root surface. The root potential is increased
linearly to a maximum value at the end of twelve hours.
The selection of maximum water potential at the root sur—
face is based on the ratio of the difference between root
and soil water potentials, to the initial water potential
of the soil. The resulting water flux is plotted against
time as shown in Figure 7 and 9. It appears that the
maximum water absorption is obtained at lower soil tension
when the ratio is about 1.5. However, the ratio increases

to about 2.0 as the initial water potential of the soil

 

 

Negative water potential (bars)

 

1011 I
1.2“

H

o

O
l

39

2.5

1.5
1J0

 

 

I 1“' r T n I I 1 I
2 h 6 8

Time (hours)

Figure 6.——The Development of Root Water Potential

"1

The rate of water uptake (mmJ/hr)

 

 

when the Initial Soil Water Potential is
Equal to Field Capacity.

 

2J3 <“IIIIIIII-—_
140
0.5
a j I 1 r 1 ’r T I 3 l 1
2 h 6 8 10 12

Time (hours)

Figure 7.—-The Rate of Water Absorption per Unit Length

 

of Root in Hillsdale Sandy Loam when the
Development of Root Water Potential as in

Figure 6.

 

 

40

CD
0
O
_I

 

 

Negative water potential (bars)

[\3
o
O

 

I I I I I I I I I '_I 1
0.0 2. h. 6. 8. 10.- 12.

Time (hours)

Figure 8.——The Development of Root Water Potential when
the Initial Soil Water Potential is Equal to

 

 

 

 

—2o3 Bars.
500‘}
7?: 2.5 -———\
(“\1'0‘“ 1.5 ‘
E .0
V30“:
(3
'8 005
4,120.—
:3
i
4310.~
g
90*:(IAIAAAHIA‘I
0.0 2. LI». 60 8o 10. 12.

Time (hours)

Figure 9.——The Rate of Water Absorption per Unit Length
of Root in a Hillsdale Sandy Loam when the
Development of Root Water Potential as in
Figure 8.

 

 

 

 

 

 

 

41

decreases. The rate of water uptake is almost negligible
when the initial soil water potential drops below the
wilting point.

The pattern of water uptake with time depends upon
the development of the root water potential. As the gradi—
ent of water potential between root xylem and soil increases
rapidly, the rate of water uptake reaches a peak value in
a few hours and then drops suddenly. This situation is
contradictory to the observation of Weatherly (1963) on
daily fluctuation of transpiration. Therefore, one must
consider not only the optimization of water absorption in
finding a relation between soil and root water potential
but one should also select the type of water absorption

pattern which would represent the actual transpiration

losses.

4.6 Derivation of Governing Equation
for Water Transfer Through
the Soil-Root Domain
Childs (1940) suggested that the flow of water in
porous material can be analyzed as a diffusion process,

and he applied Darcy's (1856) equation to solve the

unsaturated water flow in soils.

V. = - K (T) V T (4.11)
n n

the volume of water passing through a unit

where v
n

cross-section of soil per unit time

the soil hydraulic conductivity

N
'9
ll

 

42

4 = the soil water potential gradient in the
nth direction.

By imposing the conservation of mass principle, he

obtained:

89

fi=‘VVn+S (4.12)
where O = the volumetric water content of soil,

t = time,

S — the source term per unit volume.

In the original derivation of equation (4.12) it
was assumed that the relation between water content and
water potential of soil is unique. Buckingham (1907)
defined the change in moisture content with potential as
the specific water content (or water capacity,C3). If
one assumes that water capacity is constant for a small
time increment, the left side of equation (4.12) may be

written as:

O)
(D
o;
G
o;
0
Q)
0

(4.13)

”I
‘1.

|
”I
’9
°’|
rt.

|

0
O)
F'-

By substituting equation (4.11) and equation (4.13) into

equation (4.12), one obtains:

Q)
'9'

<——§(q’) v A) + (S: (4.14)

Al
I
<

when the diffusion coefficient, which is equal to the ratio

Of soil conductivity to water capacity, as defined by

 

43

Childs and Collis-George (1950) is used one

obtains:
3—3 = — v- (MA) A (b) +§ (4.15)

Whisler §t_§I. (1968) solved equation (4.12) by
numerical techniques for a steady—state, one dimensional
unsaturated water flow in a soil column. Philip (1966)
compared the transient and steady— state solution of evaporation
from soil and suggested that the steady— state calculations
are ill fitted to yield the distribution of potential,
especially near the absorbing surfaces. Therefore, it is
desirable to solve equation (4.15) for the transient case.

Assume that a single plant is growing in a uniform
and isotropic soil with a root zone in the shape of a
symetric conical prism around the stem as a vertical axis.
Then, one is able to analyze the process of water absorption
and transfer in the soil~root domain by expressing equation

(4.15) in cylindrical coordinates, as

39:;13— [rD(¢) §—¢I+§—z [13W 3‘31

t 3r r
(4.16)
8K(¢) S(r,z,t)
+ 32 + C

Equation (4.16) is the describing equation for
radially symmetric and vertical water flow in the soil—root
domain. It expresses the water potential distribution as a
function of the time and space coordinates. Since it is

Second—order in the space variable and first-order in the

 

 

 

 

44

time variable, two boundary and one initial condition must
be specified.

Assume that the water potential or water content
of the soil medium is constant at some initial value. The

initial condition may be writted as:

4 = 4 at t 0 for 0 E z E L

O

(4.17)

the depth of the soil system,

H

where L

X the radius of the soil system.

H

The upper boundary condition of the soil-root system
is determined by the soil surface where evaporation takes

place. The upper boundary condition may be written as:

SQ + K(®) at z = ()for t > 0

- K(¢) 8;

e(t)

(4.18)

the rate of evaporation.

II

where e(t)

The lower and lateral boundary conditions of the
soil-root system are determined by assuming they coincide
with the distance where the influence of the root zone

ended. Hence, the lower boundary conditions may be written

as:

L for t > 0

I0)
*9:
H
O
Q.)
('1'
N

II

(4.19)

0
IA
H
l/\
ix“.

 

45
The side boundary condition may be written as:

84

8; r = X = 0 for t > 0

(4.20)

The source term, S(r,z,t) may be defined as done by
Gardner (1960), Cowan (1965), and Whisler et_al. (1968),
where it was assumed that the source term is a function of:

i) the density of the root system,

ii) the hydraulic conductivity of the soil,

iii) the potential gradient between the soil and
the root system.

They assumed that the resistance of a suberized root surface
against the diffusion of water is zero. However, experi—
mental observation and deductive analysis indicate that the
resistance to water uptake due to suberization of the root
surface has a significant contribution in the process of
water uptake and transfer by the root system. Therefore,
the source term is also a function of the degree of
suberization. The degree of suberization, B(r,z,t), is
defined as the required water potential between soil and
absorbing root surface to overcome the resistance of the
suberized layer.

Thus, the source term, S(r,Z,t), will be used to
represent the uptake of water, i.e., a negative source,

and also to describe the release of water by the plant

root, i.e., a positive source. The source term may be

written as:

 

 

I.

 

 

46

S(r,z,t) = A(r,z,t) K(<D) V<I>(r,z,t), (4.21)

H

where S(r,z,t) the source term per unit volume,
A(r,z,t) = the density function of the root system,

K(®)

the hydraulic conductivity of the soil,

ll

V©(r,z,t) the effective water potential gradient

between soil and root xylem.

The calculation of the effective water potential
gradient between soil and root xylem requires knowledge of
the value of the root potential and the degree of suberizav
tion. It is assumed that the resistance to the flow of
water within the root is negligible. Hence, there will be
one value for root potential throughout the entire root
system. The optimum value of root potential may be esti—
mated from the analysis of water uptake limitations as we
stated previously. Then, the effective water potential

gradient between soil and root xylem will be

V<I>(r, Z,t) = Tr " ©S(r’ Z,t) “I: B(rr-zr t) (4022)
where ®rg= the water potential of root xylem,
®S(r,z,t) = the water potential of soil,
B(r,z,t) = the degree of suberization, which is

negative for water release and pos1t1ve

for water absorption.

Inspection of equation (4.22) reveals that whenever

the effective water potential gradient is zero, i.e., the

actual water potential gradient between soil and root xylem

 

 

 

 

 

47

is less than or equal to the degree of suberization, there

is no water absorption or release at the root surface.
Assume that the sum of source terms throughout the

entire root zone is equal to the rate of transpiration,

then one obtains:

i(t)

II

I
ll MIT“
ll Mix!

S(r,z,t) V(r) (4.23)
z .

where i(t) the rate of transpiration,

V(r) the volume of cylindrical soil ring corres—
ponding to the rth increment.
By combining equation (4.21) with equation (4.22) and sub—
stituting the resulting equation into equation (4.23), one
obtains:

L X

i(t) = - 2 Z A(r,z,t) K(<I>)
z=0 r=0

(4.24)

[Ar — cps + B(r,z,t)] V(r)

This equation will be solved simultaneously with
equation (4.16) by using numerical techniques to determine
the distribution of water potential, root density function,
source function, the degree of suberization and the
development of the root system at different soil and

moisture conditions.

 

 

 

 

 

 

 

 

V. EXPERIMENTAL DESIGN AND PROCEDURE

This study is conducted in the plant-water physio«
engineering laboratory located in the Agricultural
Engineering Department of Michigan State University in
East Lansing, Michigan. The basic equipment used in this
experiment had been described by Merva and Kilic (1971).

A brief description of this apparatus as related to this

research will be discussed.

5.1 Plant Environments

 

Agricultural scientists have been studying the
physiological response of plants under various environ—
mental conditions in the greenhouses where it was impos—
sible to control and monitor all environmental variables.
Thus, the need for more accurate control of environmental
parameters led to the development of growth chambers.

The plant—water physioengineering laboratory has a

Percival Model MB 60 growth chamber with 56"x26"x50"

 

internal dimensions. The radiation source of growth chamber

is provided by a set of cool white fluorescent and incan—
descent lights withvthree timers. The chamber is divided

by two inch styrofoam into two compartments to separate

48

 

 

 

49

plant crown from the soil—root system as shown in Figure 10.
The lower compartment of the growth chamber is insulated
with two inch styrofoam to obtain a uniform temperature
control in the soil—root system. The front part of
insulation has a door with double plexiglass viewing ports
and an access port fitted with a glove to adjust the soil
weighing mechanism without disturbing the environmental
condition.

The average temperature and relative humidity of the
Lansing area from May 15 through July 15 was simulated by a
one horsepower Aminco Aire unit with a Taylor Time Schedule
Recording fullscope controller at the upper compartments
of the growth chamber. The temperature is varied between
20°C and 27°C, while relative humidities is changing between
58 per cent and 78 per cent, as shown in Figure 11. The
relative humidity and temperature of the lower chamber is
controlled by a second Aminco Aire unit which maintained a
temperature of 25°C and relative humidity of approximately
90 per cent. The air flow rate through the Aminco Aire
unit was adjusted to provide 300 ft3/min.

A drum with 50 cm.diameter and 54 cm deep is used
to provide the necessary volume of soil for a single plant.
The interior and exterior of the drum was coated with epoxy
paint to prevent contamination of the soil. The soil water
was supplied or released through perforated tygon tubing
laid in the bottom of the drum and covered with approxima-

tely five centimeters of coarse sand.

 

 

 

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43.—.th001 much; mom
mmwkmiomxo> mm

.33 98.9 Swim 5.5%

NH?) 32qu smotja\\ -

 

 

 

E9221 3:33

X #:231sz

fill... WI. €64.18 It. .IIJ

 

 

 

 

 

 

 

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L: 48.
I 52.3.28

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“54:00.33...“ , mm-<IOOZ§3
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545.242 E821 _ . I II I :3

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E9221 53m :3. _
”$.2de / I u “ E9221 35.1mm
mmDEmmasE \ .1. 335185»

womDOm 20F<5<m

 

 

 

 

 

 

 

51

may no name some: men ca nae no snaeassm m>apmHmm

C) K) C)
'0 N N

Oo'BHHIVHBdWBI
I IO
N)

 

 

 

 

 

.Hmneoso £D3OHU
paw musumsmemBII.HH magmam

 

 

 

 

mmDOI .MEE.
2002 50.2
N_ O. mw m ¢_ N N. O. m m we N _Io;‘
A Ia Ia _ q A m u a m a _
Mu
I + 1..+ + 1. + lion 1.
.+ + 1. .I
Hv
+ + .+ o o + Ii
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0 0+ +o++o+o+o+o o o .M
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.. I om m.
M
mmDF<mm¢ZUP .+ I 0m:

TEQZDI m>_._.<.._mm o . o/o

———7'

52

5.2 Soil Properties

In this experiment a Hillsdale sandy loam is used.
The moisture characteristic curve of this soil as shown
in Figure 13 is obtained from Qazi (1970) who used the
static equilibrium method. The conductivity and diffu—
sivity of the soil is determined by using an approximate
method which is developed by Huggins et_al. (1970). The
method is based on the assumption that the conductivity~
pressure head relationship could be effectively represented
by an empirical three~parameter equation presented by

Gardner (1958):

[<©/h)n + bJ'l ‘ (5.1)

H

K(¢)

 

 

where K(¢)

II

the soil conductivity (cm/day),

¢ = the soil potential (cm)

n = the exponential value of soil charac—
teristic curve,

b = the reciprocal of the soil conductivity

 

at saturation (day/cm),

h = empirical constant (cm).
Huggins et_aI. (1970) developed a computer program to
estimate empirical parameter, h, by comparing experimental
and predicted infiltration rateetime relationships of a
given soil in a soil column. The infiltration rate-time
relationship is estimated by numerical techniques for a
first trial value of the empirical parameter. The trial-

error method is continued until the absolute differences

 

 

 

 

.Ewoq Ntcmm mamcwaaflm How mm>Hsu xsaqu tmumasoaoo can tm>umeoII.mH musmwm

emcee: SE.

 

 

 

 

.8 .ma .2 .m 0.0
w p u . _ _ _ p _ _ F 1 . P a _ r 1 _ a .o
0|: U m m. u. MI I w u r. ..
llanlllm-lll-o w...) 0. ON
a.oe
H
Ioow m
mm
I o Ow WM
3 L09. m
5 h.
. 1.02 m.
r o \..I/_
. Qenm
. mr
/( 1. ooa_mw
. A .03
$89643 ullu 11.08
omtemmmo oulo .. .98
.oem

 

 

IIIIIIIIIIIIIIIII-Illlll---——______1

54

between the experimental and predicted infiltration rate-
time relationships are a minimum. The basic experimental
determination required in this method is the measurement
of influx curves for infiltration into soil columns from
shallow ponded surface conditions.

The soil was packed into a 12-inch cylindrical
column which was constructed from plexiglass tubing with
0.20 inch wall thickness and five inch internal diameter.
The base was constructed so that a constant head could be
maintained at the end of the columns for saturated condi—

tions. The Hillsdale sandy loam was air dried in the

 

laboratory and screened through an ASTM No. 30 sieve. The
soil was packed in the column until the desired mean bulk

density was 1.39 gr/cm3. The top and bottom ends of the

 

soil column were supported by a hardware cloth. The bottom
end was kept open to the atmosphere during the unsteady—
state portion of the test. After the wetting front had

reached the bottom of the column, the soil column was

 

placed into a constant head overflow reservoir. Flow was
maintained for 12 hours, as recommended by_Huggins €24a1.
(1970), to establish equilibrium conditions in the column.
Then the steady state inflow rate was measured, and the
hydraulic conductivity of saturated soil was calculated.
Water was supplied by a siphon from a Mariotte-
type water supply reservoir. The weight of infiltrated
water into the soil column was continuously measured by

placing the reservoir on a Daytronics 152A load cell which

 

 

 

 

 

 

——-—f—

55

was coupled to a Speedomax G Model recorder. To determine
the infiltration rate—time relationship, the volume of
infiltrated water was read from the recorder chart at
specified time intervals. Then the rate of infiltration
was computed by dividing the rate of infiltrated water
volume by the cross-sectional area of the soil column.
Observed and calculated influx curves for infiltra-
tion into initially dry columns of Hillsdale sandy loam are
shown in Figure 12. A mean value of the soil conductivity

for saturated conditions was calculated as 3.60 cm/hr. and

 

used to estimate b = 1000, sec./cm in equation (5.1). From
the computer analysis, the empirical parameter was approxi—
mated as -l.57 cm to yield the absolute differences between

the computed and observed influx curves of .95 cm. The

 

relationship between the conductivity and the water poten~

tial of soil as shown in Figure 14 was determined to be:

K(¢) 2 86400. (5.2)

((c)/-1.57) 3 + 1000)

 

where K(¢) is the hydraulic conductivity of soil in

 

cm/day.
The diffusion coefficient of soil water was calculated from
the relationship between the water capacity and hydraulic
conductivity. The resulting diffusion coefficient is
plotted against soil water potential in Figure 15. This

relationship may be written as:

 

 

 

 

.amoa steam mamemaaam How namaoammm
Ioo coem5mmflo tam Hmflucmpom Henna
consuom mflnchADmammII.mH mHsmHm

h

_I

mmampv.coepodm
.H

—.h..p—I b P

 

 

H 0
02m
G
I. 1
m I
S I.
W n
6 o I
5 n. AM
o
o
m J
I I
o
a 1
w .1.
\Il/ l
m n
/../v0 OOHIL
m
K 1
(
L
m.
.ooQH

 

.Ewoq

seamm mamemaaam non spa>
Ifluostcou oeadmutmm tam
Hmflpcmuom Hmpmz ammzumm
mHSmQOADmamMII.wH musmflm cmozpmm QflnchHumammII

.Emoq Spnmm
mantmaaflm mom memmm mesao> m no

Dampcoo septa can Htflpcmuom scum:
.ma mummam

Amswnv soapoSm Amsdaob mv pumpcoo hopes
Ho OOH mo .40 m. N. H
n n _ . .CLIL w , _ _ L h p _ F _ p HO.

 

 

H00.

1

l

JIJJILI
IAWIT ] 1

O
o

J

L

ITIIITI

JJLILJ 1

'1
o
(sang) notions

(flap/mo) thAtqonpuoo ortnerKH

 

 

IIITII

OH

 

 

 

 

 

57

= 236758.9

D(¢)
(_¢)1.5

(5.3)

where D(¢) is the diffusion coefficient for the Hills-
dale sandy 1oam in cmZ/day.
5.3 Instrumentation and Data
AcquISition System
The objective of this study was to analyze the
processes of water absorption and transfer by developing
root system in the soil. It is thought that the measure—
ment of soil water potential distribution in situ may be
correlated with the water uptake pattern, and the root
development. There are numerous techniques for soil water
potential measurement, but most of them have an important
limitation in obtaining continuous measurement without
disturbing the soil environment. The development of
peltier type psychrometer by Spanner (1951) makes it possiv
ble to measure the soil water potential in situ without

disturbing the soil and root environments.

 

The Spanner instrument utilizes the relationship
between water potential and the ratio of actual and

saturated vapor pressures. This relationship may be

written:
+ P
v 0
Where ¢ = the water potential,
Ri = the ideal gas constant,

 

T = the absolute temperature,

 

 

 

 

—7—

58
p = the actual vapor pressure,
‘po== the vapor pressure of pure water,

V = the molar volume of water.

When a current is passed through the thermocouple junction
in an appropriate direction, dew formation occurs at the
junction due to cooling of the junction by the peltier
effect. If the current is cut off, the condensed water
evaporates at a rate depending on the humidity in the air.
Thus, the junction becomes a sensitive wet—bulb therometer.
The difference in temperature between the cooled junction
and a reference junction produces an electromotive force
(emf), which may be amplified and measured with a sensitive
microvoltmeter. By calibrating the psychrometer against

a range of salt solutions of known vapor pressures at a
constant temperature, a relationship between water potential
and thermocouple output can be obtained.

Twenty-four Spanner type psychrometers were brought
from Lepco, a division of Block Engineering, Inc., Logan,
Utah. The psychrometers were connected through an automatic
stepping switch for cycling so that all of them could be
read sequentially on a Keithly micro voltmeter, Model 1508.
The switching unit consisted of a stepping switch and a
solid state stepping relay. At each step, the system auto—
matically read the initial value of the psychrometer out“
put, then cooled with a pre—selected cooling current over a
preset time interval. After cooling, the final reading of

the psychrometer was taken, and the system was switched

 

 

 

 

 

—____(__

59

automatically to the next step. The same process was
continued for each step in the system.

The amplified output of the psychrometers obtained
from the Keithly microvoltmeter, was fed into a six«channel
analog-to-digital converter with a paper tape punch unit,
which punched the output on paper tape in a binary coded
decimal form. The first channel on the Data Acquisition
SyStem contained either a positive or negative sign depend-
ing upon whether the readings of the psychometer recorded
in the next channel corresponded to initial, or final
readings following cooling. The second and third channels
contained the readings of the psychrometer. The fourth and
fifth channels were reserved for relative humidity readings
at the outlet of the upper and lower chambers. The sixth
channel contained the readings of a load cell which detected
changes in mass of the plant soil system. A Daytronics
152A load cell was mounted on the lower platform of the

scale to measure the weight loss due to evaporation and

 

transpiration.

The psychrometers were calibrated over standard
solutions of KCL with 0.5 molality corresponding to ~22.3
bars water potential to determine the response of the
psychrometers in terms of micro—volts per bar as recom«
mended by the manufacturer.

The psychrometers enclosed in ceramic cups were
immersed directly into a flask containing a standard

solution of KCL for calibration. The flask containing the

 

 

 

 

 

 

 

 

 

‘ I
6 0

solution and the psychrometers were in turn immersed in a
constant temperature bath at 25°C, and the psychrometer
outputs were recorded hourly for a day after temperature
equilibrium was reached. After calibration, the psychro-
meters were washed for a few hours in several changes of
distilled water to remove all traces of solutes. The
analysis and the result of the calibration are given in

Table l.

5.4 Experimental Procedure

 

The soil barrel was cleaned and placed on the
modified platform scale. Four hardware cloth cylinders with
0.5 cm mesh spaced concentrically 5.0 cnuapart in radius
from the vertical axis of the barrel were fitted into the
soil container to observe the development of the root
system at the end of the experiment. The hardware cloths
were sprayed with epoxy to prevent the interaction of the
plant roots with the metal. Psychrometers were installed
in a horizontal position at the concentric hardware cloth
in such a way that they made two vertical planes intersect—
ing at the vertical axis of the barrel. The psychrometers
were spaced 5.0 centimeters apart on each plane. Room

dried Hillsdale sandy loam soil was screened and placed in

 

the barrel. Water was supplied from the bottom of the soil
container until the soil reached saturation. The soil was
then drained until flow ceased prior to planting of a bean

plant at the center of the soil drum. The chamber was left

 

 

 

 

 

61

TABLE l.—-The Result of the Psychrometer Calibration with
KCL Solution of 0.5 Molarity at Constant 25°C

 

 

Temperature.
Average Standard
Psychrometer Response Deviation
Numbers MV/bar MV/bar

1 .442 .022
2 .434 .058
3 .458 .074
4 .465 .072
5 .490 .043
6 .477 .044
7 .397 .037
8 .393 .038
9* -~ «-

10* n- ——

11 .390 .050
12 .300 .084
13 .364 .063
14 .436 .053
15 .423 .099
16 .459 .059
17 .434 .047
18 .456 .076
19 .446 .074
20 .414 .091
21 .404 .096
22 .433 .128
23 .439 .075
24 .412 .060

 

*These psychrometers failed to give consistent readings.
Thus, they were ignored during the reduction of data
obtained from the experiment.

 

 

 

 

 

”—

62

unseparated until the plant shoot was long enough to seal
off. The seal between the upper and lower parts of the
chamber was done on the fifteenth day of planting by using
foam rubber as shown in Figure 10. After the test termi—
nated in 60 days, the soil barrel was taken out of the
growth chamber and the soil was washed off by sprinkling
water to observe the root system as held by the concentric

hardward cloth cylinders.

5.5 Analysis of the Data

 

Plant growth is represented in terms of leaf area.
The leaf area was computed by measuring the major and minor
axis of a leaf and using the average of this measurement
as the diameter of an equivalent circle. The total area of
the leaves were calculated by summing up the area of each
individual leaf. The measurements were taken daily at
approximately noon.

The rate of transpiration from the plant and the

rate of evaporation from the soil surface were calculated

 

by measuring the temperature and the relative humidity of
air entering and leaving the growth chamber. The validity
of this method can be checked with the weight loss of the
soil-plant system. .The weight loss of the system was
calculated from the readings of load cell taken on an
hourly basis every fifth day after separation of the plant
crown from the soil-root system. At other times, the read—
ings of load cells were taken on a daily basis. The hourly

readings were taken to calculate the rate of evaporation

 

 

 

 

’——

63

during the night time and the rate of evaporatranspiration
during the day time, from the slope of the weight loss in
time for the plant-soil system.

The distribution of the soil water potential was
determined from the response of Spanner psychrometers
buried in the soil. The initial and final readings of
each psychrometer were punched on the paper tape by the
Data Acquisition System. The data on the paper tape was
analyzed by computer as follows: First, the average of
the initial readings were calculated to obtain a zero
point for a given psychrometer.h The final reading values
were searched by the computer program and the peak value
used as the final reading. The difference between the
initial and the final readings was divided by a calibration
value to obtain the soil water potential corresponding to
the location of the psychrometer. The soil water potential
for each psychrometer was calculated and punched out on
cards by computer. The lines of equal water potential in

each plane of the soil were plotted for 15 days, 30 days,

 

and 45 days of the plant growth.

The water absorption and transfer processes were
studied in the soil—root domain by applying the consequa—
tive water potential distribution of the soil to equation
(4.16). Equation (4.16) is solved for the source term
according to the boundary conditions of the soil-root

system in our experiment. The distribution of the source

 

 

 

 

 

 

‘—

64

term is associated with the root development and root

distribution as an absorbing surface in the soil.

 

 

 

 

VI. COMPUTER SIMULATION OF THE PROBLEM

The purpose of the computer Simulation was to
present a quantative analysis of the water absorption and
transfer in the soil—root domain using the mathematical
model which was developed in the previous chapter. Accord-
ing to the mathematical model, equation (4.26) must be
solved simultaneously with equation (4.16) by using numeriv
cal techniques to determine the distribution of the water
potential, root density function, source function, the
degree of suberization and the development of the root
System at the given soil and moisture conditions.

The initial and boundary conditions of the soil-root
system are based on the experimental design. In the
experiment, the soil—root system was defined as the cylin-

drical soil mass with 50 cm diameter and 50 cm depth. Thus

 

in the computer simulation, the bottom and side boundaries
of the soil—root system were chosen to be finite and drying
continuously while the upper boundary of the soileroot
system is a soil surface possessing the experimental
evaporation rate. The initial water potential of the soil
was assumed decreasing linearly from —330.0 cm of water at

the top to —130.0 cm of water at the bottom of the soil

65

 

 

 

 

66

barrel. Based on these boundary and initial conditions, the
soil water flow equation (4.16), in which the developing
plant roots are treated as a continuous source term, was
solved by the alternating implicit method as shown in
Appendix I.

The solution of the describing equation (4.24) for
the amount of water absorption required the development of
a root zone, the determination of a root density, and the
development of root water potential.

A kidney bean was assumed to have been planted at
a 5 cm depth from the soil surface in the center of the
soil barrel. The volume of the root zone was estimated
by first predicting the vertical extension, and then the
horizontal extension for each increment of the root zone.
The root density for each grid point in the root zone was
calculated by using the optimum overlapping coefficient,

1.21, and a constant radius of one millimeter in equation

 

(4.8). The resulting equation was
_ .216
A( r,z,t) - D(¢) (6.1)

The root water potential during the day time was
determined from the optimization of water uptake by the
plant root system. For the calculated root zone and root
density, the root potential was iterated until the maximum
water absorption was obtained. To reduce the number of

iterations, the initial root water potential was multiplied

 

 

 

 

 

 

 

IIIIIIIIIIIIIIIIIll-IIII::*———————————*fi

67

by 1.5 using as-a basis the concept of water uptake limita—
tion. The average rate of root extension was adjusted
until the sum of water absorption was equal to the corres~
ponding experimental transpiration.

The water potential of the root system during the
nighttime was relaxed with the iteration technique until
the amount of water absorbed was almost equal to the amount
of water released from the older parts of the root system.
To eliminate unnecessary iteration, a counter and a criterion
were assigned to the program. The value of the criterion
was based on the assumption of Gardner (1960) that the amount
of water uptake during the nighttime should be equal to 20
per cent of the previous day's transpiration loss to recover
the water stress of a plant. Whenever the absolute value of
water absorption was negative and less than the criterion,
the iteration was stopped. The resulting water potential
of the plant root system was then used to determine the
degree of suberization by comparing with the relaxed soil
water potential distribution in each grid point of the soil~
root system.

The calculated source terms for each time increment
were substituted into equation (4.16) and solved with other
boundary conditions to determine the new soil water poten-
tial distribution. The simulated soil water potential
distribution was compared with the experimental soil water

potential distribution to check the consistency of the

 

mathematical model.

 

 

 

 

 

 

 

IIIIIIIIIIIIIIIIII-llll------—————

68

In solving the finite difference form of the des—
cribing equations, it was necessary to select a proper
value for the space increment AX and time increment At.
The smaller increment for both variables tended to produce
slightly better results. The best solution with a reason“
able computation time was obtained when AX was set equal

to 1.25 cm while At was chosen to be 0.5 day. The computer

 

program for the simulation of the problem is shown in
Appendix II, was written in FORTRAN IV and processed on a
CDC 6500 computer. Approximately 41,000 core memory is
required to process this program. Therefore, it was neces-
sary to minimize the required computer time, since the
analysis of water absorption and transfer in the soil—root
domain required many solutions of the describing equations

over the growth season of the plant.

 

   

 

 

 

 

 

 

 

VII. DISCUSSION OF EXPERIMENTAL AND COMPUTER

SIMULATION RESULTS

7.1 The Growth of Plant Leaves
The growth of the bean plant was measured by the
increase in area of plant leaves, and the resulting values
were plotted against time for two replications under the
same environment. It was found that the growth rate of
plant leaves is exponential as shown in Figure 16. The
data for both replications is fitted with an exponential

model,

.1374 t

Al 14.51 e (7.1a)

.1122 t

A2 15.12 e (7.1b)

A1 and A2 are the areas of bean leaves for two replications,
respectively and t is the time in days. The correlation
coefficient of the models are .996 and .987 respectively.
The analysis of plant growth in terms of dry weight is not
attempted. However, the mass of bean leaves and stem was
determined after harvest at the end of 58 days, and found
to be 280. and 260. grams wet and 51. and 46. grams dry.
The moisture content of the bean plants based upon dry mass

was approximately 460. and 450. per cent respectively.

69

 

 

7O

 

 

 

100.1
.J
i
I /
. /
j 0 Bean plant I /
A Bean plant II /
10::
1
—I
n. A
N
:5, .
«I
93
"’ '1
(H
m
.3
1u~
.J
1
1
001 I I f 1 T I I I —I
o 5 10 15 20 25 30 35 1:0 145
Time (days)
Figure l6.——The Relationship Between the Leaf Area of a
Bean Plant and Growth Time.

 

 

 

 

 

 

 

—i—

71

7.2 The Weight Loss from the
Soil-Plant System

 

The rate of transpiration from the plant canopy and
the rate of evaporation from the soil surface could not be
calculated from the measurement of temperature and the
relative humidity of air entering and leaving the growth
chamber due to high fluctuation in the measurement of
relative humidity. The high fluctuation was due to the fact
that the aminco unit cycles to maintain the temperature and
relative humidity of the air. Even a small fluctuation is
magnified in the estimation of the amount of moisture
carried by such a high air flow rate. However, the rate of
transpiration and evaporation were calculated from the weight
loss of the soil plant system.

The daily weight loss from the soil—plant system for
both replications were combined and plotted against time as
shown in Figure 17. The resulting values were represented
by an exponential model and the following equation was

obtained with a correlation coefficient of 0.982

W(t) = 640.9 e'°0219 t (7.2)

where t is the time in days. The total amount of water used
by the soil-plant system can be estimated by integrating
'the equation (7.2). It was found that approximately 18.0
kilograms of water was used by the system in 45 days of

growth as shown in Figure 17.

 

The amount of evaporation from the soil surface was

estimated from the slope of the soil—plant system.weight

 

 

 

 

 

 

 

 

 

72
700.1 P210
A $00“ P18.
a: .
'U 0
E 500.4 0 L15.
v D 0 O D
g; hOO.-I D on P12.
0 o o
H O D n 0 D O D D
:3 300.4 D 8 no 0 can. " 9‘
4” n o
200.“4 I- 0
° Bean plant I
lOOc-I ... 30
u Bean‘plant II
0’0 I T I I I I I I T I 0’

 

1
0 L; 8 12 16 2021; 28 3236 no hth
Time (days)

Figure l7.—-The Rate of Weight Loss from the Soil-Plant
System due to Evapotranspiration.

Cumulative mass loss (Kgr)

 

 

 

 

 

”—

73

loss during the night time by assuming that the evapora—
tive losses of the plant canopy were negligible. The
resulting values from both replications are plotted against
time and fitted to an exponential model as shown in Figure
18 yielding the following equation with a correlation

coefficient of .998

-.O302 t

E(t) = 640.9 e (7.3)

where t is time in days. The sum of the evaporative losses
were calculated by using the equation (7.3) and found to

be almost 15.0 kilograms in 45 days of growth as shown in

 

Figure 18.
The rate of evaporation per unit area of soil was

calculated by dividing the evaporative loss In? the soil 1

 

area. The resulting values are plotted against time as
shown in Figure 19, and represented with the following

equation:

e(t) = 0.3175 e“°03°2 t (7.4)

where e(t) is the rate of evaporation in cm/day and t is

 

 

time in days. It appears that the rate of evaporation
decreases with time as proposed by Philip (1957). However,
the percentage of drop in the rate of evaporation with
time is less than that observed by Philip (1957). A
possible explanation for the inconsistency is the presence

of active roots in our soil.

 

 

 

 

 

 

 

74
700.? P21.
6000J fl ~18.
85
A 500.—I ° -15.:4,
‘3 h00.-I n 1.12. g
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5
o
0‘0 I “I I I r I I I I l ‘T 0'
0 LI 8 12 16 20 2h 28 32 36 no uh LIB

Time (days)

 

Figure 18.—~The Rate of Weight Loss from the Soil-Plant
System During the Nighttime due to Evaporation.

 

 

75

 

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The differences between the daily weight loss from
the soil—plant syscem and the amount of evaporation was
assumed to be equal to the transpiration losses from the
plant canopy during the day time. Under this assumption
the sum of the transpiration loss in 45 days of growth is
equal to 3.0 kilograms of water. The transpiration rate in
centimeters of water per unit area of leaves, is calculated
by dividing the daily transpiration loss by the corres~
ponding leaf area. The resulting data is plotted against
time as shown in Figure 19 and represented by a linear

model. The corresponding equation is

i(t) = 0.378 - 0.00782 t (7.5)

where i(t) is the rate of transpiration in cm/day and t is
time in days. The reduction in the rate of transpiration
with time may be explained by considering the ageing effects
of the plant leaves and continuous drying of the soil in

the root zone.

7.3 The Distribution of Soil Water Potential

 

The distribution of water potential was determined
from the readings of Spanner type psychrometers buried in
the root zone. The soil water potentials corresponding to
the location of the psychrometers in the xz and yz plane
are plotted for 15, 30 and 45 days of plant growth as shown
in Figure 20 through 24. As expected, the soil water
potential decreases from top to bottom of the soil barrel.

The equipotential lines of the root zone are concave shaped

 

 

 

 

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82

with a slight symetry. The shape of equipotential lines
indicates the direction of soil moisture movement upward
and toward the center where the denser root system is
assumed to be located. It also appears that the equipo—
"tential lines fluctuate in a diurnal fashion according to
the amount of water removal as shown in Figure 20 through
22.

The simulation of water potential distribution for
15 days of growth fits the experimental results as shown
in Figure 25. This shows that the development of the root
system and the pattern of water absorption is consistent
with our model, since the amount of water absorption and
the soil water potential distribution matches with the
experimental result.

The equipotential lines for 30 and 45 days shows
that the surface of the soil is dried rapidly to form a
soil crust, whereas the lower portion of soil was still
quite uniformly damp with a water potential less than one
bar as shown in Figure 23 and 24. The simulation of soil
water potential for the same days does not agree with the
experimental result as shown in Figures 26 and 27. This
inconsistency might be explained if the Peltier type
pschrometer does not respond to the potential changes when
it has been buried in the soil more than 20 days. Note
that the one bar equipotential line is almost stationary
from 15 to 45 days, although the soil-plant system lost

almost 10 kilograms of water during the same period.

 

 

 

h

 

 

 

83
X-AXIS (cm)
0 0 5 10 15 20 2.
”:xf‘ffp/
5 1;“); /
10 fl/1
15 4/
20 —
’5
:D’ 25 -
ii
N'
30 ..
35 ~
In -
85 ~
50

 

 

 

 

Figure 25.-—The Simulated Soil Water Potential Distribution
at 6 p.m. of a lS-day old Bean Plant.

 

 

 

 

 

 

 

 

 

 

84
X4fll$<bm)

O 5 10 15 20 25
0. EEééEEEL—~’j::f,l.pr—t ;,,.F_
5 - ~6i:::::::::::::

._______.————----"""'""S '
M38
.15 ..

25. {3-2 /
7° Z-l/

24fl18(cm)

 

 

 

 

Figure 26.--The Simulated Soil Water Potential Distribution
at 6 p.m. of a 30—day old Bean Plant.

 

 

 

 

 

85
X—AXIS (cm)
0 5 10 15 20 25
0 W
<~——-—-—————-———“" -1
S -7
10 a -
15 1
20 _
25 - '10"
75?
v 30 .-
E
1?
N -9 ’
35 -
Lo 1 —8-‘ /
us 4 "‘7 6/
//--S/
A
50 / )LI

 

 

 

Figure 27.-—The Simulated Soil Water Potential Distribution
at 6 p.m. of a 45—day old Bean Plant.

 

 

 

”—

86

Furthermore, half of the soil barrel should not be less than
one bar if one makes a rough calculation based on the char—
acteristic curve of Hillsdale sandy loam. The initial water
content of 100 liter sandy loam would be 24 kilograms if

one assumes that the soil water potential drOps to an
average value of ~120.0 centimeters of water tension after
24 hours draining under 60 centimeters suction. It was
observed that the soil—plant system lost 18 kilograms of
water in 45 days of growth. The water remaining in the

soil barrel should be less than or equal to 6 kilograms.
However, investigation of the experimental soil water
potential distribution at 45 days of growth shows that more
than half of the soil barrel is less than —l.0 bar which
corresponds to a water holding capacity of 12 kilograms.
Therefore, there is a definite experimental error in the
measurement of soil water potential distribution for 30 and
45 days of growth.

7.4 The Development and Distribution
of Root System

 

 

 

At the end of each experiment, the soil in the root
zone was washed away with a sprinkler water hose to observe
the development and distribution of the root system held by
the concentric hardware cloth. It was observed that six
taproots extended from top to bottom of the soil barrel
where their tips were full of tertiary branches indicating
absorption of water from the bottom of the soil barrel. As

Russell and Mitchel (1971) concluded, most of the small

 

 

 

”—

87

branches hanging on the tap roots in the upper part of root
zone may have decayed due to high water stress and these
were washed away with the soil. It was interesting to note
that most of the roots were brown almost up to their tips.
This was interpreted as indicating suberization of the root
system.

The rate of root extension was calibrated by fit—
ting the simulated water uptake and soil water potential
distribution with the experimental results for 15 days of
growth. The rate of root extension was found to be
increasing exponentially to provide the necessary amount
of water for increasing transpiration loss. The resulting
values of root extension are fitted with an exponential

model and represented by

av(t) = 1.8 e'01 t (7.6)

where av(t) is the rate of root extension in cm/day and t
is the time in days. The simulation of root extension can

be compared to the observations of Russell and Mitchell

 

(1971) during the first three weeks of growth as shown in
Figure 28. After three weeks of growth the horizontal
extension of the root system was limited due to the vertical
boundary. During the first two weeks of growth, the hori-
zontal extension was greater than the vertical extension

due to the favorable condition of the soil in the upper
layer. As the soil water potential of the root zone in

the upper part of soil dropped, the horizontal extension

 

 

 

 

 

 

 

 

 

88
X-AXIS (cm)

0 5 10 15 20 25
0 ' I
5 , .

7///

/ 10
10 j

/ 15 /
15 7/ 20 /
20 d/ 25
25 _- /

’8 30 1
m 30 _
35
35 J _
ho - /
LIO

LI5 —
SO

 

Figure 28.--The simulated Boundary of the Root System at
Different days of Growth.

   

 

 

89

slowed down and the root boundary became more hyperbolical.
The root system of the bean plant filled the soil barrel
after about 43 days of growth.

The density of the root system decreased linearly
with depth and horizontal distance at 15 days of growth
as shown in Figure 29. The equal density lines at 30 and
45 days of growth show that the density of the root system
decreased from the center to the outside as shown in Figure
30 and 31. The conversion of maximum root density values
from the root surface into the unit length of root per unit
volume of soil gave values of 0.6 cm/cm3 at 15, 1.3 cm/cm3
and 30 and 2.2 cm/cm3 at 45 days of growth, which are

consistent with the observation of Gardner (1964).

7.5 The Development of Suberization

 

The degree of suberization follows the pattern of
soil water potential distribution as it was defined.
Almost half of the root zone in terms of volume is suberized
at 15 days of growth. The maximum degree of suberization
occurs in the older part of the root zone and is equal to.
1.5 bars as shown in Figure 30. The validity of the
assumptions on the suberization process is analyzed for 15
days of growth, since the water potential and transpiration
loss satisfy the experimental results. Assuming the same
root density and soil water potential distribution exists,
one can examine the rate of water uptake during the daytime
for the same root system with and without suberization. If

the dynamic equilibrium value of the root potential in the

 

 

 

 

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91

morning is taken as an initial root potential, the root
system without suberization continues to pump water in

the first half of the day from moist soil into the dry
root zone rather than absorbing water to satisfy the
transpiration demand of the plant as shown in Figure 36.
If one increases the root potential from the initial value
to the maximum soil water potential in the root zone with
a step input, the rate of water uptake reaches a peak
value in a few hours and decreases sharply, a condition
which is contradictory to the observation of Weatherley
(1963). According to the observation of Weatherley (1963),
the transpiration loss from a plant canopy reaches a peak
value sometime between 2 p.m. and 3 p.m.

If one changes the root water potential between
these initial and maximum values to get maximum water
uptake by the root system, it was observed that the total
water uptake without suberization is almost half of the
total water uptake with suberization, furthermore, the
unsuberized root system requires a root potential twice
the root potential of suberized root system. From this
analysis, one might conclude that the suberization process
of the root system is essential for the root system to
absorb maximum water with minimum water potential.

The degree of suberization is equal to 1.75 bars
and almost uniform in the center part of the root zone
for 30 days of growth as shown in Figure 31. The upper

part increases to 3.0 bars quickly as the lower part

 

 

 

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I I I I l
2 3 I- 5 6 '7 8 9 10 11 12
Time (hours)

.1

Figure 35.-—The Development of Water Potential in the
Root System of a 15-day old Bean Plant.

 

)4. l '
. . o . . . . . 0 I
3. ‘1 o ' , .
2. "' - . I I
l. - . . II I ‘ .
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0. "I 0 With superization
-1... , ' 0 Without suberization
-o ‘
“ W I I I I I ‘I I I 1 I I

 

0 1 2 3 u 5 6 7 8 9 10 11 12
Time (days)

Figure 36.—-The Rate of Water Absorption by the Root
System of a 15-day Bean Plant With and
Without Superization.

 

 

 

 

 

 

 

 

95

decreases to zero. In the 45 days of growth the degree of
suberization decreases almost linearly from a maximum 3.5
bars at the top to zero at the right corner of the soil
barrel.

The maximum degree of suberization from 30 to 45
days of growth is not significant, whereas the soil water
potential decreases from ~12.0 bars to -l6.0 bars. This
situation might be attributed to an increase in the root

potential as shown in Figure 44.

7.6 The Distribution of Source Term

 

The distribution of the source term could not be
determined from the solution of equation (4.16) for the
experimentally determined soil water potential distributions.
This is probably due to the fact that the amount of water
absorption per unit volume is not large enough to detect,
and the distribution of soil water potential is not suffi—
ciently accurate because of the interpolation process
between the five cm grid. points. The simulations of
the source term for 15, 30, and 45 days are mapped as
shown in Figures 37 through 42. Maximum water absorption
moves from the center to the tip of the root zone in a
conical shell form. The magnitude of maximum source terms
are 4.0, 3.0, and 1.1 mm3 water/per unit volume of soil at
15, 30, and 45 days of growth respectively.

The rate of water uptake decreases from the maximum

value in the center to zero around the root zone. The

 

 

 

 

 

 

 

 

 

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99

reduction of water uptake at the tip of the root system may
be attributed to the smaller root density. However, the
reduction of water at the upper part of the root system
where the root density is higher may be attributed to the
combined effects of lower soil water potential and a higher
degree of suberization. Another interesting observation is
that the maximum water uptake occurs in the zone of soil
water potential from —l.5 to -4.0 bars. This is not a
coincidence because the root growth model and the root
density shows that optimum root growth is occurring when
soil water potential is between -l.5 and -4.0 bars. There—
fore, one can conclude that the rate of maximum water
absorption takes place where the growth of root density is
optimum rather than where the density of the root system is
maximum. Usually, the maximum root density is at the older
part of the root zone where most of the roots are suberized
or decayed due to water stress as observed by Russell and
Mitchell (1971).

During the night time, the water potential of the
root system was relaxed until the amount of water absorp-
tion was equal to or less than 20 per cent of the previous
day's water absorption for the recovery of the plant under
stress. The tip of the root zone absorbs water as the
upper part of the root zone was releasing part of the
absorbed water into the surrounding dry soil. If one super—
imposes the distribution of soil water potential and the

degree of suberization, it can be seen why the water

 

 

IIIIIIIIIIIIIIIIll-ll----——________l

100

absorption and release process in 30 days of growth was
limited to the tip of the root zone, as shown in Figure 40.
The absorption of water at 45 days of growth was limited
to the right lower corner of the soil barrel where the
unsuberized root system has a low density and low soil
water potential. However, the release of water from the

roots into the surrounding dry soil is taking place at

 

the upper part of root zone rather than immediately above

the zone of water absorption as in the 15 and 30 days of

growth. A possible explanation for a zone of zero source

term between water absorption and release would be the

nullification of root water potential by the combined effects

of the degree of suberization and the soil water potential.
The sum of daily water uptake by the whole root '

system was plotted against the growth time as shown in

Figure 43. The comparison of simulated water uptake with

experimental transpiration shows that there is a reasonable

correlation between the theory and mechanics of water

 

 

absorption. The only discrepancy between the observed and
simulated transpiration occurs for a few days after 32

days of growth. A possible explanation for this discre—
pancy is the limitation of the boundary condition on the
volume and density of the root system. In nature, root tips
would continue to grow at the surface of the boundary and
increase the density of the root system. In the proposed
model, the density of the root system was based on a

continuous density function rather than on the number of

   

IIIIIIIIIIIIIIIIll-ll----——________l

101

root tips. Consequently, the generation of the density
function takes a few days to reach equilibrium with the
development of soil water potential at the boundary.

The rate of water uptake decreases after 40 days
of growth as the experimental transpiration decreased.
This situation may be explained when one considers the
distribution of soil water potential and the development
of root water potential. As the soil dries, the amount
of water uptake drops with the decreasing unsaturated
hydraulic conductivity, and the root water potential
increases to absorb optimum water in dried soil as shown
in Figure 44. According to Figure 44, the root potential
dropped to —15.0 bars during the day and relaxed to —8.0
bars during the night at 45 days of growth. This portrays
the development of the wilting point in the plant as the
soil water is depleted.

In order to study the effect of transpiration demand
and the development of root potential upon the wilting of a

plant, we shall assume that wilting occurs when the trans—

 

 

piration exceeds the amount of water absorption. According
to the proposed model, the daily fluctuations of root water
potential, which are shown in Figure 44, are the maximum
root water potential for optimum water absorption. If the
transpiration demand increases the leaf and root potential
have to increase correspondingly in order to maintain the
rate of transpiration. However, any increase in root

potential does not increase the amount of water uptake.

   

 

 

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104

Therefore, whenever the transpiration demand exceeds the
amount of water absorption, plant leaves have to control
the rate of transpiration by closing their stomataes.

An inspection of Figure 44 shows that the root water
potential does not have to drop to —15.0 bars to wilt the
plant at the early stage of growth. This observation shows
the sensitivity of a plant against water stress at the
early stage of growth and the effect of soil water statue

on the rate of transpiration.

 

 

VIII. CONCLUSIONS

A deterministic non-stationary mathematical model
was developed to investigate the movement of water through
the developing root system of a bean plant in a Hillsdale
sandy loam. The development and density of the root
system, the rate and pattern of water absorption, and the
development of root and soil water potential in the root
zone were simulated on the computer. From the analysis
of the mathematical model and experimental observation,
the following conclusions are obtained:

1. The rate of root extension as a function of
soil water potential is consistent with the observation
of Russell and Mitchell (1971).

2. The geometry of root system can be represented

by the vertical and horizontal extension of root system.

 

3. The density of the root system based on the
definition of Gardner (1960) with a new optimum over—
lapping coefficient is reasonable and at least in the same
order as Gardner's (1964) observation.

4. There exists a relation between the initial
soil and root water potentials to obtain optimum water

absorption with a pattern which fits the pattern of

105

 

 

 

 

106

experimental transpiration during the day time from the
analysis of Gardner and Lang's (1970) suggestions on
limitations to water uptake by plant roots. This relation
is defined as the ratio of difference between maximum root
and initial soil water potential to initial soil water
potential. The maximum rate of water absorption with a
pattern of actual transpiration is obtained when the ratio
is between 1.5 at low soil tension and 2.0 at higher
negative soil water potential.

5. The rate of maximum water absorption takes
place where the increase of root density is optimum rather
than where the density of root system is maximum due to
suberization of older roots. The location of maximum
absorption coincides with the development of optimum ratio
between soil and root potential. The absorption of water
is almost limited to the lower part of the root zone as a
conical shell which is moving downward as growth progressed.

6. The development of suberization shows that the
transfer of water through the root system from moist soil
into dry soil could not bring back the root zone in an
equilibrium condition as far as the soil water potential
distribution is concerned. It appears that the suberization
of the root surface is necessary to optimize the absorption
of water as well as protect the roots from unfavorable

environments.

 

 

 

107

7. The amount of water absorption by the root
system drops with the depletion of soil water in the root
zone. The root water potential fluctuations reach to a
steady-state value during the growth of root zone into a
soil with low water potential. As the soil dries, the
root water potential drops to the wilting point during the
day in order to absorb more water for transpiration.

8. The experimental rate of evaporation from soil
surface decreases exponentially with time. However, the
rate of reduction is less than that observed by Philip
(1957). A possible explanation for this result is that
the rate of evaporation is higher from a soil surface with
an active root system.

9. The experimental rate of transpiration from
plant leaves decreases linearly over the time of growth.

It is probably due to the combined effects of the aging
Of plant leaves and the depletion of soil water.

10. The simulated soil water potential distribution
is consistent with experimental soil water distribution in
the root zone for the first three weeks of growth. Then,
the simulated soil water potential distribution deviates

from the observed potential distribution.

Recommendation for Future Work
1. The investigation of the root development and
density should be restricted to a single plant under dif—

ferent soil with variable moisture and temperature conditions.

 

 

 

 

108

2. The formation and degree of suberization should
be determined experimentally.

3. The leaf water potential should be measured
with the soil water potential to check the development of
root water potential and determine the resistance of plant

to movement of water.

 

 

 

REFERENCES

109

 

 

 

 

REFERENCES

Brouwer, R. (1965). Water movement across the root.
Symposium Society of Experimental Biology.
19:131-149.

Buckingham, E. (1907). Studies of the movement of soil
moisture. United States Department of Agriculture,
Bulletin No. 38.

Carnahan, B.; H. A. Luther; and J. O. Wilkes. (1969).
Applied Numerical Methods. New York: John Wiley
& Sons, Inc.

Childs, E. C. (1940). The use of soil moisture character—
istics in soil studies. Soil Science. 50:239—252.

Childs, E. C. and N. Collis-George. (1950). The permea—
bility of porous materials. Proceedings of Royal
Society Association. 201:392-405.

Chen, L. H. and R. B. Curry. (1971). Dynamic simulation
of plant growth, Part II, incorporation of actual
daily weather data and partitioning of net photo—

synthate. ASEA Paper No. :71—541, St. Joseph,
Michigan.

Cowan, I. R. (1965). Transport of water in the soil-plant—
atmosphere system. Journal of Applied Ecology.
2:221—239.

Darcy, H. P. G. (1856). Les fontaines publiques de la

ville de Dijon. Victor Delmont, Paris.

Danielson, R. E. (1965). Root systems in relation to "
irrigation. "Irrigation of agricultural lands.
American Society of Agronomy, Madison, WisconSin.

DeWit, C. T. and R. Brouwer. (1968). A simulation model of
plant growth with special attention to root growth
and its consequences. Proceedings of the Fifteenth
Easter School in Agricultural SCience, UniverSity of

Nottingham.

110

 

111

Dittmer, H. J. .(1937). A quantitative study of the roots and
root hairscflfa winter rye plant. American Journal
of Botany. 24:417—420.

Duncan, W. G.; R. S. Loomis; W. A. Williams; and R. Handu.
(1967). A model for simulating photosynthesis in
plant communities. Hilgardia. 38:721—205.

Esau, K. (1965). Plant Anatomy. 2nd. ed. New York:
John Wiley and Sons, Inc.

Gardner, W. R. (1958). Some steady state solutions of the
unsaturated moisture flow equation with application
to evaporation from a water table. Soil Science.
85:228-232.

Gardner, W. R. (1960). Dynamic aspects of water avail—
ability to plants. Soil Science. 89:63e73.

Gardner, W. R. (1964). Relation of root distribution to
water uptake and availability. Agronomy Journal.
56:41-45.

Gardner, W. R. and C. F. Ehlig. (1962).‘ Some observations
on the movement of water to plant roots. Agronomy
Journal. 54:453-456.

Gardner, W. R. and A. R. G. Lang. (1970). Limitation to
water flux from soils to plants. Agronomy Journal.

62:693—695.

Huggins, L. F.; E. J. Monke; and R. W. Skaggs. (1970). .

An approximate method for determining the hydraulic
conductivity function of unsaturated soil. Techni-
cal Report No. 11, Purdue University, Lafayette,
Indiana. .

A. S. and O. J. Kelley. (1946). A new technique

for studying the absorption of moisture and nutrients
from soil by plant roots. Soil Science. 62:441—450.

Hunter,

Kramer, P. J. (1969). Plant and Soil Water Relationships.
New York: McGraw—Hill Book Co., Inc.

Kuiper, P. J. C. (1963). Some considerations on water
transport across living cell membranes. .In I.
Zelithch (ed.), "Stomata and Water Relations in
Plants," pp. 59—58. Conn. Agr. Exp. Sta. Bull.

664, New Haven, Conn.

(1968) . The nature of the perennial
Australian

19:397—409.

McWilliam, J. R. .
response in mediterranean grasses.

Journal of Agricultural Research.

 

112

McWilliam, J. R. and P. J. Kramer. (1968). The nature of
the perennial response in mediterranean grasses.
I. Water relations and summer survival in phalaris.
Australian Journal of Agricultural Research.
19:381—395.

Merva, G. and N. K. Kilic. (1971). A multichannel technique
to study water potential distribution under simulated
field conditions. In "Symposium on Thermocouple
Psychrometers." Utah State University, Utah.

Newman, E. I. (1966). Relationship between root growth
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65:273-283.

Peters, D. B. (1969). Water and oxygen requirements of
plant roots under varying soil water regimes.
Research Committee Great Plains Agricultural Council
Publication No. 34, Kansas State University,
Manhattan, Kansas.

Philip, J. R. (1957). Evaporation, and moisture and heat
fields in the soil. J. Met. 14:354—366.

Philip, J, R. (1958). The osmotic cell, solute diffusi—
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Physiology. 33:264.

Philip, J. R. (1966). Plant water relationships: some
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tivity of unsaturated porous media from moisture
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Michigan State University, East Lansing, Michigan.

Ratliff, L. F. and H. M. Taylor. a969L Root elongation rates
of cotton and peanuts asafunction ofsoil strength and
soil water content. Soil Science. 108(2) : 113—119.

Russel, W. J. and R. L. Mitchell. (1971). Root development
and rooting patterns of soybean evaluated under
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Shaw, B. T. (ed.). (1952). Soil Physical Conditions and
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Slatyer, R. O. (1967). Plant—Water Relationships.
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The Peltier effect and its use in

S ann D. C. 1951).
p er, ( Journal of

the measurement of suction pressure.
Experimental Botany . 11:145—168.

 

 

 

 

113

Street, H. E. (1966). The physiology of root growth.
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Van den honert, T. H. (1948). Water transport in plants
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Waggoner, P. E. (1969). Predicting the effect upon net
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Weatherley, P. E. (1963). The pathway of water movement
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Analysis of steady-state evapotranspiration from
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Wooley, J. T. (1966). Drainage requirements of plants.
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Crop Production. ASAE. St. Joseph, Michigan,

pp. 2—5.

 

 

APPENDIX I

SOLUTION OF THE GOVERNING EQUATION FOR THE
ABSORPTION AND TRANSFER OF WATER IN THE
SOIL ROOT-DOMAIN BY THE IMPLICIT

ALTERNATING DIRECTION METHOD

114

 

 

 

 

 

 

SOLUTION OF THE GOVERNING EQUATION FOR THE
ABSORPTION AND TRANSFER OF WATER IN THE
SOIL ROOT-DOMAIN BY THE IMPLICIT

ALTERATING DIRECTION METHOD

The purpose of this analysis is to present a numerical
solution of the dynamic aspects of water flow in the soil-
root domain. The root systems are considered to be growing
at a specified rate and act as a sink or source, depending
upon the development of water potential gradient between
soil and root. The equation describing this process with

the initial and boundary conditions is defined in Chapter IV,

and can be written as:

M _ 1 8 8o 8 M
a "- 8 85‘1“” a?) + “5:7: (13“?) 32>
(l.a)
L ‘ S(rrzrt)
+ az K(¢) + _CTTT—__
0 5 z E L
o = 80 at t = 0 for (l.b)
O 5 r E X
80 t > O
e(t) = - K(@) 8t + K(¢) at z = 0 for (l.c)

O S r E X

115

 

 

 

 

 

 

8¢ _ > 0
E‘ Z = L -— 0 for <
_ r _ x
t > 0
3Q
—— _ = 0 for
3r r — X _ z E L
1:0 1 2 - I
J=0
AX
l
AX
2 rh
(in)

 

 

 

 

 

L

 

 

Figure 45.--The Space Grid System,

 

(l.d)

(l.e)

 

117

The initial and boundary conditions define the
limits on the space and time variables. One can place a
square mesh over the region with spacing AX, as shown in
Figure 45. If I and J are the numbers of internal mesh

points in the r and z direction, then

X = M ° A x
(2)

L = N ' A X
where, M = I + l and N = J + l are the number of intervals
in the r and 2 directions. The equation (l.a) is a
nonlinear parabolic type of differential equation.

Carnahan §t_al. (1969) have pointed out that either explicit
or implicit finite-difference methods can be used in solv—

ing parabolic type differential equations. Use of the
explicit method is limited by computational restrictions

which must be imposed to insure stability and convergence

of the computations. These difficulties can usually be
eliminated by using the implicit alternating direction method.
In this method, the principle is to employ two different
equations which are used in turn over successive time steps
each of duration At/2.

The first equation is implicit in the r—direction
and the second is implicit only in the z—direction. The
first equation is solved for the intermediate values of ¢
(i.j,11+ %) which are then used in the second equation.
Thus, leading to the solution @(i,j,n + l) at the end of

the whole time interval At. The representations of

 

 

 

ll8

equation (1) by the implicit alternating methods would be
written as:

. . L _ . . 2
“10““) “1'3"” = A @(i.j,n+1/2) + A2®(i,j,n) (3.a)
r
At/2 z

©(i,j,n+l) — <I>(i,j,n+1/2) = A2 <I>(i . n+%)
At/ r I]! 2
2 (3.b)

+ A: ©(i,j,n+l)

Equation (3.a) can be written in full as:

¢(i+l,j,n+%)- ¢(i,j,n+%) ]

Ar

¢(i,j,n+!~§) - d>(i,j,n) : 1 [r(i+%)D(i+l/le)[

At/2

— r(i-l’z)D(i-1uj)l:¢(i’j'n+l’i) - ¢'<i-l,j,n+*2) ]

Ar

A2

+ D(i,j+‘/2) [wi’dl’m _2rp(i""“n)' ]

_ D(i,j-1/2) [WOILLE]

A2
+ [K(i,j+l) - K(i,j-l) + S(i,j) in 1
AZ C(irj)

If one multiplies through by At/2 and assumes Ax = Ar = A2

for simplicity, equation (4) becomes:

A(i) <1>(i-'l,j)* + B(i) <1>(i,j)* + C(i) ©(i+l,j)* = R(i) (5)

 

 

 

 

 

 

 

 

 

 

119
where
._y ._ .
A(i) _ _ A}; Hi 2) MI L2,3)
sz 2
- _ - A: r(i+%) D(i+%,j) At r(i-t) D(i-2,i)
B(J.) -— r(1)+ 2 2 + _2 2
Ax Ax
C(i) = _AEZ r(i+%)2D(i+g,j)
AX
R(. _ . . At . . .1
l) - @(l,j+l,n) *2 r(l) D(1IJ_/Z)/2
Ax
- . . . _ A_t_ r(i) D(i,i+;§) _ A: r(i) D(i j'%)
+ @(l,j,n) [r(i) 2 2 2 2 ’
Ax Ax

+ <I><i.j+1.n> [A—tz (r(i) D<i.j+1>/2>]
AX

+ [R(i,j+l) - K(i,j—lfl /Ax + S(i,j)/C(i,j)

Since the coefficients A, B, C and R depend on the previous
time, n, equation (4) represents a set of linear algebraic
equations. This set of equations is solved to obtain the
values of’ ¢(i,j) at the n+% time. For convenience, the
(n+%) time index has been replaced by an asteric on the
unknowns and the knowns have been lumped together in R.

In the same way, equation (3.b) which is implicit

in z-direction becomes:

 

 

 

 

 

120

 

 

¢(i,j,n+l) — ¢(i,j,n+b) _ l . . ' . ¢(i+l.',n+ ) - ¢ ','. +
At7§i 2 — r(i) [r(1+%) D(l+%,3{1[ J %Ar2 (1 J n %) ]

 

- r(i-%) ram—m“) [“i’j’nw’) E ¢(i-l'j'n+l)]
Ar

 

+

D(irj+;§) [ ¢(i,j+l,n+l) '2¢(irjln+l) ]
Az

 

D(i,j-%) [ ©(1,3,n+l) - ¢él,j-l,n+l) J
A2

+ 1di,j+1) — Kfiqj-l) + S(i,j)
A2 cu,m

Multiplying through by A/2 and rearranging equation (6),
one can obtain:

Al(j) ©(i,j-l)* + Bl(j) ®(i,j>* + Cl(j) ©(i,j+l) = Rl(j) (7)

 

 

 

 

where
Al(j) = _ 932 [r(i) gun-4)]

Ax '

' ' ' g At ' D ','-%)
Bl(j) = r(i) + AP. [r(1)12)(1,j+)] + __2 [r(i) 2‘1 3 J
sz AX
' ’ ' a

Cl”) ___ _ A32 [r(i) 3(1,J+ )]

Ax

 

Rl( ) = ¢(i,j,n+g) AEZ r(i—%)2D(i—%,')

Ax

A_t [r(i+1/2) D(i+%,j)+r(i-}§)D(i-%.j)l ]
2

+ ¢(i,j,n+%) [I(i) -
AX2 2 J

+ K(i,j+l) — K(i.j-l) + S(irj)
Ax C(i,j)

As in the previous case, equation (7) represents a
set of linear equations, since the coefficients Al, Bl, Cl,
and the terms Rl depend on the water potential distribution
at the previous time, n+t. The solution of this set of
equations gives values of ©(i,j) at the n+l time.

Equations (5) and (7) can be applied to each point
of the mesh system in conjunction with the effective

boundary and initial condition. The initial value of water

potential is constant throughout the system, viz.:

i=0,l,2...M

©(i,j) = ¢ at t = 0 for (8)
O i=0,l,2...N

The water potential of the soil surface for each time incre—

ment can be calculated from the first boundary condition due
to evaporative loss as follows:

©(i,0')= <1><i,j> —é’-I§,-el—f%+1 (9)

for i = 0,1,2 . . . M and t > 0. Now one can apply equation

. th .
(5) to each grid point, i = 0,1,2 . M in the J row in

conjunction with the effective boundary conditions. The

 

 

 

 

122

first equation in the system of equations may be written
as:

B(O) ©(0rJ)* + C(O) ©(l,J)* = D(0) (10)

where the mesh system is centered in such a way that ©(0,J)
is the left boundary of the system. The value of ¢(—1,J)
can be represented by ©(1,J) due to symetric projection,

and D(0) can be expressed as:
D(0) = R(O) - A(O) @(1,J) (11)

Similarly, if the mesh is centered so that ¢(M,J)

is the right boundary, then the last equation would be:

A(M-l) ©(M-2,J)* + B(M—l) @(M—l,J)* = D(M-l) (12)
where D(M—l) = R(M~l) — C(M—l) ©(M,J) due to the boundary
condition defined by equation (l.c). Now, one can complete
the set of equations for i = 0,1,2 . . .M and obtain:

13(0) q, (OIJ)* + C(O) (L (lIJ)* _—_ 13(0) = R(O) - A(O) <1> (1,J)

A(1) 1 (0,J)* + B(D i UJJ)* + C(U ¢ QIJ)* = 9(1): R(D

A(2) r» (l,J)* + 13(2) 6» (2,J)* + C<2> <9 <3IJ)* = 9(2) = 12(2)

A(3) 1 (2pm* -FB(3) 4 w,J)* + C(N ¢ MIJ)* = D‘W = R(N

A(M—z) 1:1 (11:3,J)* + B(M—Z) d. (M—2,J)* + C(M—Z) “’9 (M‘lrJ)* = B(M‘Z) = R‘M‘Z)
_ _ - ,J

A(M—l) ¢ (M—2,J)* + B(M-l) ¢ (M-l,J)* = D(M'l) = R‘” 1) C(M 1) ¢ (M )

 

123

As one can see, the elements of this matrix are zero every—
where except on the main diagonal and on two diagonals
parallel and adjacent to it on either side. The system
of equation can be solved for ©(i,j), where i = 0,1,2. . .
M—l, by Gaussian elimination techniques. The procedure
is repeated for successive columns, j = 1,2,3. . . N-l,
until all the ©(i,j) are calculated at the end of the first
half—time step.

The water potential distribution at the end of
second half—time step are calculated similarily by apply—
ing equation (7) to each grid point, of mesh system in

conjunction with boundary condition at Z — O and Z — L.

 

APPENDIX II

COMPUTER PROGRAM FOR THE SIMULATION OF

WATER TRANSFER THROUGH THE

SOIL-ROOT DOMAIN

124

 

 

 

 

 

 

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