J. Nig. Soc. Phys. Sci. 3 (2021) 354–359

Journal of the
Nigerian Society

of Physical
Sciences

Mathematical Models and Comparative Analysis for Rice and
Soya Bean Irrigation Crop Water Needs: A Case Study of Bida

Basin Niger State, Nigeria

A. Abdulrahim, M. D Shehu, E Yisa, Z. A. Ishaq

Department of Mathematics, School of Physical Science, Federal University of Technology, Minna, Niger State, Nigeria.

Abstract

In this manuscript, mathematical models for cropping water need (C.W.N) and the size of land for irrigation (S.L.I) were formulated. The solutions
of the models for Crop water need for Soya beans and Rice, and the size of land for irrigation (S.L.I) of the two crops was obtained. We fill the
gap by considering the size of the irrigation land which is not considered by the Food and Agriculture Organization (F.A.O). The computational
Method of solutions is carried out to get effective results. The climatic data of the study area (Bida Basin) under which our research is based
includes: Rainfall, Humidity, Sunshine hours, minimum and maximum temperature, evapotranspiration were secondary data collected from
Nigeria Metrological Society (NIMET). We compared the results of CROPWAT 8.0 software developed by the Food and Agriculture Organization
(F.A. O) and our computational method so that we can arrive at a new finding and better results. The results for the computational method with the
size of Land for irrigation shows that there is an increase in crop water need for the crops than the results of CROPWAT 8.0 software developed
by the Food and Agriculture Organization (F.A. O) in which the size for the land is not considered. We therefore, recommended that the integral
calculus can be used to estimate the irregular shape of the size of the land if the land shape is not in rectangular form before solutions are given
for accuracy and effective results.

DOI:10.46481/jnsps.2021.149

Keywords: Bida Basin, Crop water Coefficient, Evapotranspiration

Article History :
Received: 01 February 2021
Received in revised form: 01 March 2021
Accepted for publication: 07 June 2021
Published:29 November 2021

c©2021 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: O. J. Abimbola

1. Introduction

Bida Basin lies in the sedimentary terrain of the middle part of
Nigeria. It has an area of coverage of about 27,000 km2. The
area falls under the middle climatic belt which is mainly tropi-
cal with an average rainfall of about 1250m. We are therefore
considering the crop water need of Rice and Soya Bean on the
aquifer of two lithological groups: unconfined and semi - con-
fined aquifer in our selected study area.

Email address: m.shehu@futminna.edu.ng (Z. A. Ishaq)

Cropwat 8.0 software is software developed by Food and Agri-
cultural Organisation (F.A.O), used to evaluate farmer’s irriga-
tions, irrigation practices and to estimate crop performance un-
der both rainfall and irrigated condition. Our computational
method is used as a tool to solve the models and obtain the so-
lutions to get effective results. The weakness of the cropwat 8.0
software developed by the Food and Agriculture Organization
(F.A.O) is that the irrigator farmers do not know the size of the
Land the results of the software is given as the size of land for
irrigation (S.L.I) is not considered by the cropwat 8.0 software

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Abdulrahim et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 354–359 355

and we fill the gap by considering the size of land for irrigation.
Many irrigation farmers faced the challenges of what the crop
water need of irrigation crops would be before embarking on
irrigation water planning. This research aims are to develop a
mathematical model for crop water need problems and deter-
mine the exact amount of water need for Rice and Soya beans
crops by considering the size of the land and shape. Our ob-
jective is to use Cropwat 8.0 software developed by the Food
and Agricultural Organisation (F.A.O) and our computational
method to determine the crop water need of two crops and com-
pared the two results to unravel crop water need problems.
Irrigated Agriculture in South Africa has not been profitable
over the years. Even though, it is the highest user of total con-
sumptive water [1]. Its economic returns have not been im-
pressive. The sustainable management of irrigation water re-
sources is therefore, a necessity. A common procedure for es-
timating crop water use is to first determine the daily reference
crop Evapotranspiration (ET0) and then multiply it by a spe-
cific crop coefficient(Kc), as given by the Food and Agriculture
Organisation [2].
[3] determine the crop water requirements for Maize in Ab-
shege Woreda, Gurage zone in Ethiopia, they determine crop
water requirement which is the major food crop of the area,
they used the climatological records of Sunshine, maximum
and minimum temperature, humidity and wind speed were used
as secondary data. Penman - Monteith method was used and
crop water requirement was estimated using CROPWAT 8.0 the
results show that a Maize variety with a growing period of 140
days to maturity would require 423 mm depth of water, while
101mm of water would be required as supplementary irrigation.
[4] came up with a quantitative analysis of hydraulic interac-
tion process in the stream – aquifer systems. It revealed both
the theoretical and laboratory tests have demonstrated that, the
hydraulic connectedness of the stream aquifer system can reach
a critical disconnection state.
[5] researched on the crop water requirement for Agriculture in
a typical River Basin of India. The results show the crop wa-
ter requirements is much below the available rainfall and even
available groundwater at various location of the river basin, the
crop water requirement is required to be increased by increas-
ing the crop production with multiple crops and by using more
Agriculture land for crop production. With these, there is a
need to develop a Mathematical model for solving crop water
need with the specific land size. [6] develop estimating aquifer
hydraulic properties in Bida Basin Central Nigeria, using the
Empirical method. He determined aquifer properties such as
hydraulic conductivity, porosity and effective porosity, and co-
efficient of uniformity. [7] the understanding of crop water need
becomes necessary, as it enables efficient use of water and bet-
ter irrigation practices like scheduling as the supply of water
through rainfall is limited in these areas.
[8] Carried out a study to determine the crop water need of few
selected crops for the commanded area, the outcomes of the
study are capable of planners of the water resources for future
planning and helps save water in satisfying the crop water need.
[9] Cropwat 8.0 software helps irrigator planner in allocating
the water resources in the future.

2. Mathematical Model for Crop Water requirement.

2.0.1. Metrological Data
To calculate this, the respective climatic data was collected from
the Nigeria meteorological station. The data used for ET0 com-
putation was the meteorological data obtained from the station;
for instance, minimum and maximum temperatures (C), wind
speed in km per day, the relative humidity (maximum and min-
imum, in %) and the hours of sunshine, and the physical data
such as altitude, latitude and longitude. The climatic records
obtained were then adjusted into the format accepted by CROP-
WAT 8.0. The rainfall data collection was also obtained from
the meteorological station. Rainfall records from a range of
years (10–15) were collected to allow for a calculation of Crop
water need.

2.0.2. Soil Data
The data utilized on the soil characteristics were acquired through
laboratory soil analyses done on the soil samples collected. Af-
ter the collection of all these data, it was entered into the CROP-
WAT 8.0 program and saved. Crop and irrigation water needs
were then calculated using the model for the majorly observed
high - value crops including Rice and Soya Bean. The weakness
of the Crop water need developing by the food and Agricultural
Organisation (F.O.A) is that the results for the amount of water
need for crops are given without the size of the land, thus. The
size of land for irrigation have to consider.

2.1. Mathematical Derivation of Reference Crop Water Need.
Considering an energy balance at the earth surface equates all
incoming and outgoing energy flux. The following governing
equation is considered;

Rn = H + λE + G (1)

Where Rn=energy flux density net incoming radiation (w/m2)
H= flux density of latent heat into the air(w/m2)
λE= flux density into the water body (w/m2)
G = heat flux density into the water body (w/m2)
λ= the latent heat of vaporization of water
E = the vapour flux density in kg/m2 s
Where

H = C1
(T s − Ta)

ra
(2)

and

λE = C2
(es − ed )

ra
(3)

where,
C1, C2= Constant
T s= temperature at a certain height above the surface (k pa)
ea= Prevailing vapour pressure at the same height as Ta (k pa)
ra= aerodynamic diffusion resistance.
Applying the similarity of transport heat and water vapour, we
have a Bowen ration yield as;

H
λE

=
C1(T s − Ta)
C2(es − ed )

(4)

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Abdulrahim et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 354–359 356

Equation (4) becomes;

H
λE

= γ

(
T s − Ta
es − ed

)
(5)

C1
C2

= γ

}
(6)

γ =
C pρa
λε

(7)

making C p the subject of the relation in (7) we have:

C p =
γλε

ρa
(8)

where,

ρa =
Pa

TvkR
(9)

where,
Tkv = the virtual temperature
R = specific gas constant
ρa = mean air density at constant pressure kgm−3

C p = specific heat at constant pressure mj k/g/ 0C
Equation (7) is referred to as Psychometric constant (k pa/oC)
In determining the surface Temperature, we considered the Pen-
man - Monteith equation which is given as

es − ea = ∆(T s − Ta) (10)

From equation (10)

T s − Ta =
es − ea

∆
(11)

Substituting equation (11) into equation (5), we have:

H
λE

=
γ

∆

(
es − ea
es − ed

)
(12)

Replacing es − eaby es − ed − ea + ed

H
λE

=
γ

∆

(
1 −

ea − ed
es − ed

)
(13)

Considering isothermal evaporation λEa given as:

λEa = C2
es − ed

ra
(14)

Setting λ = 1in equation (14) ,then

Ea = C2
es − ed

ra
(15)

replacing C2 by
ερa
p in equation (15) we have:

Ea =
ερa
p

(
es − ed

ra

)
(16)

Dividing equation (16) by equation (3) we have:

Ea
E

=
(ea − ed )
(es − ed )

(17)

Substituting equation (17) into equation (12) we have:

H
λE

=
γ

∆

(
1 −

Ea
E

)
(18)

Replacing E by ET0in equation (18) and make H the subject of
relation, we have:

H =
λγET0

∆

(
1 −

Ea
ET0

)
(19)

Substitute equation (19) for H in equation (1) and simplify, we
have:

ET0 =
1
λ

(Rn − G) ∆ + γEa
∆ + γ

(20)

where,
ET0=open water evaporation rate (kg/m2)
∆= proportionality constant (k pa/oc)
Rn= net radiation (J/kg)
γ= Psychometric constant (k pa/oc)
Ea= isothermal evaporation rate (kg/m2 s)

The term
1
λ

(Rn−G)∆
∆+γ

in equation (20) is called radiation term

The term γEa
∆+γ

in equation (20) is called aerodynamic term
Substitute equation (7) and (16) into equation (20). Simplifying
further to obtain

ET0 =
1
λ

(
(Rn − G)∆ +

C pρa
ra

(es − ed )
)

(γ + ∆)
(21)

where,
ε= ration of molecular masses of water vapour and dry air (-)
pa= density of moist air (kg/m3)
ρa= atmospheric pressure (kpa)
cp=Specific heat of dry air at constant pressure (J/Kg.k)
Considering vapour diffusion rate then equation (16) is express
as:

Ea =
εpa
pa

.
ea − ed

ra
=
εpa
pa

ea − ed
rc

=
εpa
pa

e0 − ed
rc + ra

(22)

Then,
εpa
pa

ea − ed
ra

=
εpa
pa

e0 − ed
rc + ra

(23)

Simplifying equation (23) we have:

ea − ed =

 e0 − ed1 + rcra
 (24)

where,
Ea= Isothermal evapotranspiration rate from canopy
eo= Internal saturated vapour pressure at (cpa)
ea= Saturated vapour pressure at the leaf surface
ed = Vapour pressure in the external air
ra= aerodynamic resistance (s/m)
rc= Canopy diffusion resistance (s/m)
Substitute equation (24) into equation (20) we have:

ET0 =
1
λ

(
(Rn − G)∆ +

C pρa
ra

e0 − ed
)

∆ + γ
(
1 + rcra

) (25)
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Abdulrahim et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 354–359 357

Let

γ

(
1 +

rc
ra

)
= γ∗ (26)

then equation (25) becomes:

ET0 =
1
λ

(
(Rn − G)∆ +

C pρa
ra

e0 − ed
)

∆ + γ∗
(27)

where,
ETo=evaporation rate from dry surface (kg/m2 s)
γ∗ = Modified Psychometric constant (k pa/oc)
Substitute equations (8) and (9) into equation (27) we have;

ET0 =

1
λ

(
(Rn − G)∆ +

γελ
Pa

Pa
Tvk R

ra
e0 − ed

)
∆ + γ∗

(28)

Simplifying equation (28) we have:

ET0 =

(
1
λ
(Rn − G)∆ +

γε
ra Tvk R

e0 − ed
)

∆ + γ∗
(29)

ra can be expressed as:

ra =
ln (zm−d)zom ln

(zh−d)
zoh

k2u2

=
ln

[
2−0.08
0.01476

]
ln

[
2−0.08

0.001476

]
(0.41)2 u2

=
208
u2

(30)

with the following standard values:

d = 0.67h, zom = 0.123h, and
zov = 0.120m, k = 0.41, h = 0.12. (31)

where
Z = height at which wind speed is measured (m)
d = displacement height (m)
zom = roughness length for momentum (m)
zov = roughness length for water vapour (m)
K = Von Karman Constant equal 0.41
u2= wind speed measure at height (m/s)
Tkv =the virtual temperature
R =specific gas constant
ρa = mean air density at constant pressure kgm−3

C p = specific heat at constant pressure m j kg−1 0C−1

From equation (30) we have:

C pρa
ra

=
γε

raTvkR

=
(86400)(0.622)γλ

(Ta + 273)(0.287)(208)
u2

= γ
900

Ta + 273
u2 (32)

with the following standard values:

ra = (208)u2, Tvk = (Ta + 273),
R = (0.287)

(33)

From equation (30) we have:

1
λ

=
1

2.45
= 0.408 (34)

with standard value of λ = 2.45
from equation (27)

γ∗ = γ

(
1 +

rc
ra

)
(35)

we let,

rc =
ri

LAIactive
= 100(0.5)(24)(0.12) = 70sm

−1 (36)

Substituting equation (31) and (37) into equation (27), we have;

γ∗ = γ (1 + 0.34u2) (37)

Substitute equation (31), (33), (37) and (38) into equation (30)
we have:

ET0 =
0.408∆(Rn − G) + γ

900
T a+273 u2(e0 − ed )

∆ + γ(1 + 0.34u2)
(38)

2.2. Mathematical Formulation of Crop Coefficient

Considering crop coefficient, we let:
Kcb tab= is the value for (Kcb)mid or (Kcb)end

Ke = Kr (Kc − Kcb) (39)

Simplifying the equation, we have

Kc = Ke + Kcb (40)

where,

Kr = 1 (41)

Ke = soil evaporation coefficient
Kcb = Basal crop coefficient
kc= crop coefficient value of Kc following rain or irrigation
Kr = Dimensionless evaporation reduction coefficient depen-
dent on the cumulative depth of water depleted (evaporated)
from the top so

2.3. Mathematical Formulation of Irrigated Area of Land (Ai)

We consider the Area of the irrigated land as a rectangular sur-
face;
Let L = length of the farm
B = Breadth of the farm
l = length of the spacing on the farmland
b = breadth of the spacing on the farmland
The spacing area of the farm land is considered to be;

lb =
(

LB
Pn

)
N (42)

where,
Pn = number of plants on the farmland.
N = number of seeds per stand.

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Abdulrahim et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 354–359 358

Table 1. The comparison of result Cropwat 8.0 software and the computational method for Rice Crops
Month ETo(F.A.O) ETo Kc (FAO) Kc ETcwn (FAO) ETcwnMm Ai (F.A.O) Ai Hectares

November 4.44 4.92 4.27 3.93 1689 9571 Nill 569
December 4.45 4.32 3.3 3.12 1515 6721 Nill 569
January 4.42 4.25 4.4 3.67 1521 1826 Nill 569
February 4.73 4.21 3.24 3.84 1431 1646 Nill 569
March 5.17 4.08 5.2 1.29 7560 627 Nill 569

Table 2. The comparison of result Cropwat 8.0 software and the computational method for Soya Bean
MONTH ET0F.A,O ET0 Kc F.A.O Kc= Kcb + Ke ETcwnF.A.O ETcwn Ai F.A.O Ai

November 4.44 4.33 0.8 0.23 19.6 12.7 Nill 569
December 4.45 4.32 2.47 0.3 114.4 128 Nill 569
January 4.42 4.25 3.18 0.82 146.5 347 Nill 569
February 4.73 4.08 1.14 1.64 36.7 666 Nill 569

Also, loss of plants on two adjacent rows is;

S =
( L

l
+

B
b

+ I
)

N (43)

The accurate plant population formula becomes:

Pn =
(

lb
LB

)
N + S (44)

Substitute (43) into (44) which further simplified to;

Pn=
(

LB + Lb + lB + lbI
lb

)
N (45)

Furthermore,

LB = lbPn − (Lb + lB + lbI) (46)

Replacing LBwith Ai; then

Ai = lbPn − (Lb + lB + lbI) (47)

Where;
Ai = irrigated area of land.

ETcwn = ET0 × Kc × Ai (48)

Where,
ETcwn =crop water need
ET0 =reference crop evapotranspiration
Kc =crop water coefficient
Ai =crop irrigated area
Combining equation (39), (41) and (48), the crop water need
equation becomes;

ETcwn =
(

0.408∆(Rn−G)+γ
900

T a+273 u2 (es−ea )
∆+γ(1+0.34u2 )

)
× (Kcb + Ke) × (lbPn − (Lb + lB + lbI))

(49)

3. Result and Discussion

Table 4.1 shows the comparison of the result analysis of
the outcome of Cropwat 8.0 Software developed by F.A.O and
the computational method for Rice irrigation crop water needs,

reference evapotranspiration (ETo) and crop coefficient (Kc) in
Bida Basin Irrigation Sites. It is observed from the table that,
reference evapotranspiration (ETo) results in computational method
is decreasing as the dry season is biting harder, this would en-
able the irrigator farmers to know the quantity of water need
for crops provided the size of the land is known. while it alter-
nates in Cropwat 8.0 Software results developed by F.A.O. the
crop coefficient (Kc) results in our computational method main-
tains the same behaviour except in March where we have 1.29
(Kc) this is due to the harvesting period which is when crops
need little or no water, comparing to Cropwat 8.0 Software de-
veloped by F.A.O which is 5.2 (Kc) in march. The crop water
needs result in computing method is higher than the Cropwat
8.0 Software developed by F.A.O, this is own to the fact that
the quantity of crop water need in each month of the dry season
is known to the irrigator farmers within the available land size
of 569 hectares

Table 4.2 shows the comparison of the result analysis of
the outcome of Cropwat 8.0 Software developed by F.A.O and
the computational method for Soya Bean irrigation crop water
needs, reference evapotranspiration (ETo) and crop coefficient
(Kc) in Bida Basin Irrigation Sites. It is observed from the table
that the crop (Soya Bean) is grown from November to February
which is four months, the crop water need (ETcwn) in our com-
putational method is greater than crop water need (ETcwn) in
the cropwat 8.0 software, this is own to the fact that the size of
the Land for irrigation is considered in the former method. The
reference evapotranspiration (ETo) results in our computational
method are closer to that of cropwat 8.0 Software results de-
veloped by F.A.O. The crop coefficient (Kc) results in our com-
putational method maintain the same behaviour which shows
that the crops need little or no water during their harvesting pe-
riod. The crop water needs result in our computational method
is higher than the Cropwat 8.0 Software developed by F.A.O.
The challenges of crop water need (ETcwn) faced by the irriga-
tor farmers can be unraveled if the size of the land for irrigation
and plant population is known.

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Abdulrahim et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 354–359 359

4. Conclusion

The Penman - Monteith Mathematical model for the crop wa-
ter need and Mathematical model for the size of land for ir-
rigation were solved and the results obtained are compared to
CROPWAT 8.0 software results, the climatic data of the study
area (Bida Basin) under which our research is based which in-
clude: Rainfall, Humidity, minimum and maximum tempera-
ture, evapotranspiration were collected from NIMET and used
for the both CROPWAT 8.0 software and the computational
method so that comparative analysis can be made from the two
methods. Our computational method results show that crop wa-
ter need for crops can better be estimated if the size of the land
for irrigation is considered. The irrigator Farmers can inextri-
cably know of the estimated crop water need for a given size of
the Land before commencing irrigation exercise. We therefore,
recommended that integral calculus can be used to estimate the
irregular shape of the size of the land if the land shape is not
in rectangular form before solutions are given for accuracy and
effective results.

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