# mathematical modeling for water uptake by plant root in saturated

```Antarctica J. Math., 11(3)(2014), 265-277
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MATHEMATICAL MODELING FOR WATER UPTAKE
BY PLANT ROOT IN SATURATED AND
UNSATURATED ZONE
P. S. AVHALE
[Email : [email protected]]
[District], Maharastra, India.
&amp;
S. B. KIWNE
Department of Mathematics, Deogiri College, Aurangabad, Maharastra, India.
The root system of the plants uptake the water and nutrients from the soil. The
present paper explains systematic description of root functioning for water uptake
from the saturated and unsaturated soil and transport through the stem to the leaves.
We use basic principles of physics and fluid-dynamics, tiny xylems tubes as water
transport channel in the stem and develop simple mathematical model for water
transport. In this paper we write mathematical model for water profile at root
surface, at the xylem and leaves. We solve the mathematical model using root surface
boundary conditions by analytical method.
1. INTRODUCTION
The primary physiological function of root is up taking the water as well as
nutrients and transport to leaves for photosynthesis. Investigations and observation of
the uptake of water and nutrient in plant root and stem can be traced back to many
years ago, it possesses importance in point of view of agricultural production and
economical development. In traditional farming like planting and agricultural the
----------------------------------------------------------------------------------------------------------------Subject classification (AMS) : 35XX-35K xx , 76XX-76Rxx, 76XX-7605, 76XX-76Zxx76Z05 and 92XX-92B
265
266
P. S. AVHALE and S. B. KIWNE
Key words &amp; Phrases:Plant root, Root water uptake, Poiseuille flow, Mathematical modelXX
mechanism of water and nutrients is invaluable for utilizing water and fertilizer for
increasing production. Now a new trend of planting inedible plant emerges on
industrial basis. The view of planting inedible plant are prevent the salinization,
desertification of soil, to clean pollution of heavy metals, radioelement and plant's
mining. To collect the valuable metals, like gold, in soil by planting some plants
whose roots possess a special capability of absorbing the valuable metals. The plants
of genus Bauhina have many species out of which Bauhinia variegata plant extract is
analyzed and found it contain micro-particles of gold [1]. Since ancient times
Bauhinia racemosa Lam. family: Caesalpinaceae has been an integral part of life in
India. Leaves of Bauhinia racemosa are traditionally used on occasion of Dashera
festival as symbol of gold in India [2]. Recently proved that Bauhina racemosa extract
also contain micro particles of gold. Water enters into root through root surface from
the soil. Then it flow through micro-tube of xylem in the stem to rich leaves and being
vaporized into air from the pores on leaves. The transport of water in plant can affect
the water flow in soil, and vice versa. Then study of Mathematical model for plant
water uptake divides into the parts: (1) the flow of water in the porous soil; (2) water
flow into root through root surface; (3) flow of water through micro-tube of xylem;
(4) vaporization of water through leaves. In recent years, a number of researchers
from various fields, such as physics, applied mathematics and plant physiology, paid
more attention to develop mathematical model for water and nutrient uptake. The
outstanding work in this field is done by T. Roose and proposed a mathematical
model for uptake of water and nutrient. Roose work is development of Nye, Tinker
and Barber model for water and nutrient uptake assuming that the root is an infinitely
long cylinder [7][15]. Before this model many root water uptake model like, a linear
root water uptake model (Prasad, Rama 1988) [3], CERES (Ritche, 1985) [4] and
SWAP (Van Dam et al, 1997) [5] was developed .
2. FLOW OF WATER IN POROUS MEDIUM
In 1856 Henry Darcy make experiment and concluded that, rate of flow ( i. e.
volume of water per unit time), π is (a) proportional to the cross-sectional area π΄, (b)
proportional to (β1 − β2 ), the lengths β1 and β2 are measured with respect to some
arbitrary datum level (c) inversely proportional to the length πΏ [6].
π=πΎ
π΄(β1 −β2 )
πΏ
(2. 1)
where πΎ is a coefficient of proportionality, which is known as soil hydraulic
conductivity. Equation (2. 1) is known as Darcy formula. Specific discharge, π as the
volume of water owing per unit time through a unit cross-sectional area normal to the
πΈ
direction of flow, i. e. , π = π¨, we obtain
π = πΎπ½ (2. 2)
where π½ =
(β1 −β2 )
πΏ
is in one dimension. In three dimension, the generalization
MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT
IN SATURATED AND UNSATURATED ZONE
267
of equation (2. 2) is given by
π = πΎπ½ = − τπΎπππππ = τ − ∇∅ (2. 3)
The average areal porosity is equal to the volumetric porosity π, the portion of
the area π΄ available to flow is ππ΄. The average velocity π of flow is given by
π=
π
π
= (2. 4)
ππ΄ π
Another form of Darcy formula is in term of specific discharge, i. e. , Darcy flux is
written as
π
π = − (πππππ π − ππ) (2. 5)
π
where =
πππ
π
πΎ
, π is dynamic viscosity of fluid, π = π is the density of the fluid, k is the
medium's permeability and π = −πππ is the vector gravity acceleration directed
downward. If fluid is not compressible then ∇π’ = 0 . The equation for the
conservation of water in the soil with no water sink is given by [7],
π∅π
+ ∇. π = 0 (2. 6)
ππ‘
where π is given by,
π
π = − [∇π − πππΜ] (2. 7)
π
The π soil permeability is just differ from hydraulic conductivity K by scaling
factor which is given by,
πΎ=
πππ
, (2. 8)
π
where ππ present the density of water and gravity. If soil is saturated then we have
π
πππ = 0 therefore the Darcy flux is given by = − π [∇p] . However, the
conductivity of unsaturated soil is determined from experiments in terms of effective
water saturation, where effective water saturation π is defined as
π =
∅π − ∅π,π
(2. 9)
∅π,π  − ∅π,π
where ∅π is the moisture content of the soil, ∅π,π is the residual (minimum) moisture
content of the soil and ∅π,π  is the saturated moisture content of the soil, i. e. the soil
268
P. S. AVHALE and S. B. KIWNE
porosity. Van Genuchten drive a formula for the soil hydraulic conductivity as a
function of soil moisture content and porosity which is given by [14]
πΎ = πΎπ  π 1/2 [1 τ − (1 τ − π π )π ]2 πππ 0 &lt; π &lt; 1; (2. 10)
where πΎπ  is the hydraulic conductivity of fully saturated soil. In addition to deriving
the soil hydraulic conductivity formula (2. 10), he also presents a formula for the
suction characteristic. He uses the pressure head β suction −π &gt; 0 can be written in
terms of pressure head, . i. e. , −π = −ππβ. Hence we have
π
= [
1
πΌ
1+( )
β
π]
π
1
1
=[
, 0 &lt; π &lt; 1; (2. 11)
π] , π = 1 −
π
πΌ
1 + [ππ (−π)]
where πΎπ  is a fitting parameter that can be inversely linked to the bubbling pressure.
We write equation (2. 6) of conservation of water in the soil in terms of relative
moisture content S using the full van Genuchten formulas, and then we get the
following nonlinear diffusion-convection equation
(∅π,π  − ∅π,π )
ππ
ππ ππ ππ
(2. 12)
= ∇. (π·(π)∇π) −
ππ‘
π ππ ππ§
where D(S) is called the soil water diffusivity and it is defined as
π·(π) =
π(π) ππ
| |
π ππ
ππ  ππ
π
(1 − π) 1/2−1/π
−π
π
&times;
π
[(1 − π 1/π ) + (1 − π 1/π ) − 2] (2. 13)
∝π
=
and
1 π 2
ππ(π)
1 −1
= ππ  [ π 2 [1 − (1 − π π ) ]
ππ
2
1 1
1 π
1 (π−1)
+ 2π (−2+π) [1 − (1 − π π ) ] (1 − π π )
] (2. 14)
RANGE OF VALIDITY
In the flow through conduits, the Reynolds number, ππ , which is
dimensionless number expressing the ratio of inertial to viscous forces acting on the
fluid, is used as a criterion to distinguish between laminar flow occurring at low
velocities and turbulent flow occurring at higher velocities. By analogy, a Reynolds
number is defined also for flow through porous media [6]
MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT
IN SATURATED AND UNSATURATED ZONE
ππ =
269
ππ
(2. 15)
π£
where π is some representative length of the porous matrix and π£ is the kinematic
viscosity of the fluid. Practically all evidence indicates that Darcy's law is valid as
long as the Reynolds number does not exceed some value between 1 and 10. There
the equation (2. 12) valued as long as Darcy's law obeys, . i. e. , for Reynolds number
value less than flow will be at very low pressure gradients through low-permeability
sediments, also for Reynolds number value greater than 10 there will be large flows
3. WATER FLOW THROUGH ROOT SURFACE INSIDE THE
ROOT
Water flow through root surface, then there will be change in water pressure,
which serves as boundary condition in equation of water conservation in the root. The
root surface membrane work as semi permeable membrane represents the water
movement path across the cortex to the xylem. The rate of water uptake by pant roots
per unit surface area π½π€[ππ3 ππ−2 τπ τ−1 ] is given by [7][11]
π½π€ = ππ (∇π − πΔ∏) (3. 1)
where ππ is the speed of water flow through the root surface membrane per unit
difference in water pressure on either side of the membrane, ∇π is the hydraulic water
pressure across the membrane, π is the reflection which give relation between osmotic
pressure to hydrostatic and Δ∏ is the osmotic pressure difference across the
membrane. Neglecting the osmotic pressure effects on water uptake by plant roots and
π· is the water diffusivity in cortical tissues, then for a root surface membrane with the
inner boundary at ππ₯ and an outer boundary at the root surface π, the water pressure
distribution inside this membrane is given by a solution to the equation
πππ π· π
πππ
=
(π
) (3. 2)
ππ‘
π ππ
ππ
with boundary conditions given for example by ππ = π on π = π and ππ = ππ at
π = ππ₯ , where ππ is the water pressure in the root cortex tissues, π is the water
pressure at the root surface and ππ is the water pressure in the xylem. Nondimensionalising the above equation with typical timescale [π‘] and root radius a, we
find that the dimensionless problem becomes
π2 πππ 1 π
πππ
=
(π
) (3. 3)
π·[π‘] ππ‘
π ππ
ππ
With
ππ = π on π = 1 and ππ = ππ at π = ππ₯ /π = ππ₯ (3. 4)
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P. S. AVHALE and S. B. KIWNE
For small order radius, water pressure distribution profile is at pseudo steady
state, i. e. , the time derivative term in the equation (3. 3) is small, and the steady state
solution is given by [8]
ππ =
≈
π − ππ
ππ(π) + π (3. 5)
−ππ(ππ₯ )
π − ππ
(π − 1) + π πππ ππ₯ &lt; π &lt; 1
1 − ππ₯
Hence, in dimensional terms the ux of water through the unit of root surface
area is given approximately by
π·
where ππ =
π·
(π−ππ₯ )
πππ
ππ
π·
|π=π = π−π (π − ππ ) (3. 6)
π₯
is generally known as a root surface membrane hydraulic
conductivity. Using boundary condition equation (3. 6) at the root surface, we can be
written as
π(π) ππ
π ππ
= ππ (π − ππ ) ππ‘ π = π (3. 7)
where π(π) is the soil water permeability, ππ is the radial conductivity of root surface
membrane, π is the water pressure in the soil, ππ₯ is the water pressure in the root
xylem and π is the radius of the root.
4. WATER FLOW IN THE XYLEM LONGITUDINALLY UPWARDS INTERIOR OF ROOT
Interior parts of root are made of macro tubes of xylem with average radius
ππ . Assume that the number of the micro-tubes is N. The total area cross-section of
the micro tubes can be written as π∅ππ2 = ∅π 2 . We assume that the water path
effectively by a single tiny circular tube with the radius π . Water motion in the
interior of root is described by one dimensional Poiseuille flow along the pipe with
variable cross section area of radius r [7][9].
π∇2 π’ = ∇π − πππ (4. 1)
where π’ is water velocity profile with non-slip boundary condition which is given by
π’(π, π§) =
π 2 −π 2 πππ
4π
[ ππ§ − ππ] (4. 2)
The total flux of water passing upwards through the cross section at the level π§
is given by
MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT
IN SATURATED AND UNSATURATED ZONE
π=−
let ππ₯ =
∅π 4
8π
271
∅π 4 πππ
8π
[ ππ§ − ππ] (4. 3)
then equation (4. 3) becomes
ππ
π = −ππ₯ [ ππ§π − ππ] (4. 4)
5. MODEL FOR CONSERVATION OF WATER INSIDE THE
ROOT
We consider a unit length of the root and assume that there is balance between
water flow through the root surface membrane and upward in the xylem tubes. Then
mass conservation law yields [7][9][16],
2ππππ (π − ππ ) =
ππ
(5. 1)
ππ§
where π is the radius of the root and π is the longitudinal flux of water. Substituting
the value of (4. 4) in equation (5. 1), then we get an ordinary differential equation for
ππ as a function of π§, i. e. ,
2ππππ (π − ππ ) = −ππ₯
π 2 ππ
(5. 2)
ππ§ 2
which we can solve, subject to boundary conditions
ππ = π ππ‘ π§ = 0, and
πππ
ππ§
= 0 at π§ = πΏ (5. 3)
where π is the average pressure at the top of the root, i. e, at the root-shoot boundary
and L is the length of the root.
6. MODEL FOR CONSERVATION OF WATER INSIDE OF THE
ROOT AT CONSTANT SOIL PRESSURE
Suppose the root surrounded by water in porous medium are at constant
pressure, i. e. , π = π then water pressure equation in the root system is
2ππππ (π − ππ ) = −ππ₯
π 2 ππ
(6. 1)
ππ§ 2
Non-dimensionalise the equation (6. 1) with
π§ = πΏπ§ ∗ and ππ = πππ∗ (6. 2)
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P. S. AVHALE and S. B. KIWNE
where π§ ∗ and ππ∗ are the dimensionless length and pressure respectively. The
dimensionless equation (6. 2) becomes with dropping ∗ becomes
π 2 (π0
π 2 ππ
− ππ ) = − 2 (6. 3)
ππ§
with dimensionless boundary condition (5. 3) becomes
ππ = 1 at π§ = π and
π
where π0 = π and π 2 =
2ππππ πΏ2
ππ₯
πππ
ππ§
= 0 at π§ = 1 (6. 4)
are the dimensionless water uptake coefficients.
Solution of equation (6. 3) with boundary conditions (6. 4) is given by
1
1
ππ = π0 + 2 [(π0 − 1)π‘ππβ(π) − π0+ + 1]π ππ§ + 2 [−(π0 − 1) tan β(π) − π0 +
1]π =ππ§ (6. 5)
7. MODEL FOR WATER FLOW IN THE SOIL WITH ROOT
SURFACE BOUNDARY CONDITION SUBJECTED TO
CONSTANT ROOT INTERNAL PRESSURE
Water flow in soil given by equation (2. 12) can be write in radial co-ordinates
form with boundary condition (3. 7) is given by [10]
(∅π,π  − ∅π,π )
ππ 1 π
ππ
=
(π·(π)π ) (7. 1)
ππ‘ π ππ
ππ
with boundary condition
π·(π)
ππ
= ππ [π(π) − ππ ] ππ‘ π = π (7. 2)
ππ
where π π is the constant pressure inside the root and
π·(π) =
ππ  ππ (1 − π) 1/2−1/π
−π
π
&times;
π
[(1 − π 1/π ) + (1 − π 1/π ) − 2] (7. 3)
π
πΌπ
ππ −1/π
1/π
π(π) = −
(π
− 1) (7. 4)
πΌ
We non-dimensionalise this model with choice of time, radius and pressure in
new scales as follows
π‘ = [π‘]π‘ ∗ =
(∅π,π  − ∅π,π )πΌππ2 π ∗
ππ ∗
π‘ , π = ππ ∗ , π =
π (7. 5)
(1 − π)ππ  ππ
πΌ
and the dimensionless model with boundary condition becomes by dropping
273
MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT
IN SATURATED AND UNSATURATED ZONE
ππ 1 π
ππ
=
(π·(π)π ) (7. 6)
ππ‘ π ππ
ππ
π·(π)
where
π·(π) =
1 1
π 2−π
[(1 −
ππ
= ππ€ [π(π) − ππ ] ππ‘ π = 1 (7. 7)
ππ
1 −π
π
π )
+ (1 −
1 π
π
π )
1/π
− 2] , π(π) = −(π −1/π − 1)
(7. 8)
and dimensionless parameters are
ππ€ =
ππ ππππ
ππ  ππ
ππ πΌ
π€ππ‘β πΎπ  =
πππ ππ =
(7. 9)
(1 − π)πΎπ
π
ππ
Equation (7. 1) can also written in term of relative water saturation in the soil
[12]. The equation for conservation of water in the soil is thus
π
ππ
+ ∇. π = −πΉπ (7. 10)
ππ‘
where ∅ is the (constant) porosity of the soil, π is the relative water saturation in the
soil i. e. , π = ∅π /∅ where ∅π is the volumetric water fraction, π is the volume flux of
water, i. e. , Darcy flux and πΉπ€ is the uptake of water by plant roots. Consider value of
u given in equation (2. 7) with equation (2. 10). Also the water pressure in the soil
pores can be linked to the relative water saturation via suction characteristic [14],
1−π
ππ − π = ππ π(π), π(π) = (π −1/π − 1)
(7. 11)
where ππ is atmospheric pressure, and ππ is a characteristic suction pressure. It is
common to take ππ = 0.
Then the Darcy-Richards equation in terms of relative water saturation only, it
is written as
∅
ππ
= ∇. [π·0 π·(π)∇π − πΎπ  πΎ(π)π − πΉπ ] (7. 12)
ππ‘
π ππ
where the water diffusivity in the soil is π·0 π·(π) = π ππ , π·0 =
1 1
1 −π
π·(π) = π 2−π [(1 − π π )
Also equation (2. 8) can written as follows
1 π
ππ ππ  1−π
π
(
π
+ (1 − π π ) − 2] (7. 13)
) and
274
P. S. AVHALE and S. B. KIWNE
πΎπ  =
ππππ
(7. 14)
π
Equation (7. 12) is studied with boundary condition at soil surface and at the
base of soil layer. If amount of water is flowing due to rainfall, then we take boundary
−π·0 π·(π)
ππ
= ππ  , ππ‘ π§ = 0, (7. 15)
ππ§
where ππ  is the volume flux of water per unit soil surface area per unit time. If surface
pounding occur, then the boundary condition will be π = 1 at π§ = 0. The boundary
condition at base of soil layer is written as
−π·0 π·(π)
ππ
+ πΎπ  πΎ(π) = 0, ππ‘ π§ = ππ€ , (7. 16)
ππ§
The model for water uptake of a single cylindrical root in cylindrical radial coordinates is written as
∅
ππ 1 π
ππ
=
(π·0 π·(π)π ) (7. 17)
ππ π ππ
ππ
with boundary conditions
ππ
π·0 π·(π) ππ = ππ [π(π) − ππ ] ππ‘ π = π
ππ
= 0 ππ‘ π = ππππ‘ (7. 18)
ππ
where
1−π
p(S) = -ππ (π −1/π − 1)
(7. 19)
π is the radius of the root, ππππ‘ is a measure of inter branch distance, ππ is root radial
conductivity and ππ is the root internal pressure, π ≤ ππ ≤ π(π). We take the far-field
soil water content, i. e. , effective water saturation is π∞ which is constant. Then the
far-field boundary condition is given by
π → ∞ ππ  π → ∞
Non-dimensionalizing this model with π = π; π‘ = ∅π2 /π·0 and ππ = |π|
ππ
ππ‘
with boundary conditions
1 π
ππ
= π ππ (ππ·(π) ππ) (7. 20)
MATHEMATICAL MODELING FOR WATER UPTAKE BY PLANT ROOT
IN SATURATED AND UNSATURATED ZONE
π·(π)
275
ππ
= π[−π0 π(π) − ππ ]ππ‘ π = 1
ππ
ππ
ππππ‘
= 0 ππ‘ π =
(7. 21)
ππ
π
where π =
ππ π|π|
π·0
and π = ππ /|π|
The far-filed boundary condition is still
π → ∞ as π → ∞.
Solution of equation (7. 21) is given by [7] for outer region far away from the root
π2
π = π∞ + π  = π∞ + π΅πΈ1 (4π‘π·(π )) (7. 22)
∞
where S(π; π‘) = π∞ + π (π, π‘), For inner region near the root solution is given by
π = π∞ + π 1 +
where π 1 is π 1 = −
ππ€[π(π∞ −ππ )]
2π·(π∞ )
ππ€ [π(π∞ − ππ )]
πππ (7. 23)
π·(π∞ )
ππ[4π −πΎ π·(π∞ )π‘ + 1]
Also we have the dimensional expression for water uptake by root from soil
given in equation(7. 10) which is written as[8],
πΉπ€ = 2∅πππ ππ [−ππ π(π) − ππ ] (7. 24)
where ππ is the root length density.
8. VAPORIZATION OF WATER THROUGH PLANT LEAVES
Leaf of plant considered as plane at π§ = πΏ from tip of the root. Let ππ is the
number of total opened micro pores of water pathway on the leaves with effective
radius ππ , hence effective area of water pathway in plant is ππ 2 which is actually the
ππ
total areas of all micro pores on leaves, namely ∑π=1
πππ2 [9][13]. The liquid pressure
,
of the opened pore on the leaves πππ
is negative. It is caused by the transpiration on
,
the leaves. There are two well known theories about πππ
.
(1) The liquid in micro pores and air are separated by curved surface due to capillarity
,
negative πππ
is generated. Suppose ππ is air pressure and π is surface tension
π
,
coefficient of liquid, then we have ππ − πππ
= π , where ππ is radius of the vessel.
π
276
P. S. AVHALE and S. B. KIWNE
(2) We start with assumption that liquid and air are separated by a semi permeable
membrane and the fractional pressure of air ππ, depends on the humidity. Suppose the
concentration of water in micro pores is C, according the solution theory, the
thermodynamic equilibrium condition at the membrane is that the chemical potentials
,
on both sides of the membrane must be equal. One may write πππ
= ππ + ππ where ππ
is the osmotic pressure, which depends on the difference of the concentrations (πΆ −
,
πππ
/ππ ) of the water on the both sides of the membrane.
The above condition is applicable for the system that the solutions on the both
sides of the membrane are in the thermodynamics equilibrium. If the solution on the
both sides of the membrane are not in the thermodynamics equilibrium, then there
,
exists a difference of pressures βπ ≠ 0 then for first case we have, βπ = πππ
− ππ −
π
,
and in second case βπ = πππ − ππ + ππ . When βπ &gt; 0 , the water will vaporize
π
π
into air, otherwise water in air will enter leaves. The flux vaporizing into air can be
expressed as ππ = ππ βπ, where ππ is the osmosis coefficient.
9. DISCUSSION
In this paper we try to elaborate the idea of water uptake by plant root which
based on the model discussed by Roose. T. in his dissertation of doctorate and idea
are extended by different angle using many research paper related to this field. We
discussed the mathematical model for plant which is in normal circumstances; the
amount of water within a plant is set by a balance between the water uptake through
the roots and the rate of transpiration through the leaves.
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