ARID ZONE JOURNAL OF ENGINEERING, TECHNOLOGY & ENVIRONMENT
AZOJETE December 2019. Vol. 15(4):847-858

Published by the Faculty of Engineering, University of Maiduguri, Maiduguri, Nigeria.
Print ISSN: 1596-2490, Electronic ISSN: 2545-5818

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847

ORIGINAL RESEARCH ARTICLE

A NEW FORMULA FOR GROUNDWATER RECHARGE ESTIMATION IN YOLA, NIGERIA

A. B. Umaru1*, S. Lukman2, I. A. Oke3, and B. G. Umara4
(*1Department of Agricultural and Environmental Engineering, Modibbo Adama University of Technology, Yola

Nigeria
2 Civil Engineering Department, University of Hafr Al-Batin, Hafr Al-Batin, Saudi Arabia

3Department of Civil Engineering, Obafemi Awolowo University, Ile-Ife, Nigeria
4Department of Agricultural and Environmental Resources Engineering, University of Maiduguri, Nigeria)

* Corresponding author’s email address: ababayi234@yahoo.com

1.0 Introduction
Moisture of soil and soil – water are important ingredients in fertility of the soil and food
security. Groundwater recharge includes the drive of water from the unsaturated zone into the
saturated region of the soil below the water table. The water runs together with the associated

ARTICLE INFORMATION ABSTRACT

In this paper,relationship between rainfall and groundwater recharge
estimation was evaluated. A standard numerical formula that relates
rainfall to groundwater recharge estimation was proposed with the
aim of providing error-free groundwater estimate from rainfall.
Rainfalls measured in a selected location in Yola, Nigeria and standard
formula were used to fix the constants (A,B and c) in the new
numerical formula using Microsoft Excel Solver. The constants were
used to establish new numerical formula. The new numerical formula
was used to estimate groundwater recharge from the rainfall. The
accuracy of the new numerical formula was evaluated statistically
using Analysis of Variance (ANOVA), relative error; the degree of
accuracy, numerical reliability, Model of Selection Criterion (MSC) and
Akaike Information Criterion (AIC) and compared with the previous
formulae in use for field groundwater recharge estimation with Uttar
Pradesh (UP) as the reference value of groundwater recharge.
The study revealed that groundwater recharge estimated using the
formula   814.0019.1621.0  PRgr was similar to groundwater
recharge estimated using standard formulae (UP, Modified
Chaturvedi, Kumar and Seethapathi and Rao). In all cases there were
significant differences between the groundwater recharge estimated
using all the formulae. The new formula provided the lowest relative
error of 0.887%, the highest MSC of 4.911; the degree of accuracy of
99.113 % and the lowest AIC of 436.306. The accuracy of the
formulae was in the order of new formula greater than Chaturvedi
formula greater than Kumar and Seethapathi formula greater than
Rao formula. It was concluded that modelling of groundwater
recharge using the numerical formula is a promising tool for
estimating groundwater recharge with minimum error in water
resources management.

© 2019 Faculty of Engineering, University of Maiduguri, Nigeria. All rights reserved.

Submitted 11 April, 2018
Revised 01 September, 2018
Accepted 05 September, 2018

Keywords:
Groundwater recharge
soil –moisture
numerical solution
Microsoft excel solver
statistical methods.

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Umaru, et al: A New Formula for Groundwater Recharge Estimation in Yola, Nigeria. AZOJETE, 15(4):847-858. ISSN
1596-2490; e-ISSN 2545-5818, www.azojete.com.ng

Corresponding author’s e-mail address: ababayi234@yahoo.com 848

flow pattern away from the water table within the saturated region (Freeze and Cherry, 1979;
Yeh et al., 2008). Groundwater recharge occurs at a point when water containing dissolved
minerals flows beyond the groundwater level and infiltrates into the saturated zone.
Groundwater recharge is a tremendously important water component of the hydrological cycle
in nature. Modifications in groundwater storage comprise various recharge and discharge
progressions. Most important recharge sources are rainfall, recharge from rivers, recharge from
ponds, recharge from irrigation fields. The crucial discharge processes are evapotranspiration,
pumping, baseflow to rivers. There are several factors that can affect the existence and
discrepancy of groundwater level in a region. The major and significant factors include
topography, lithology, geological structures, depth of weathering, the extent of fractures, the
primary and secondary porosity of the soil, slope and drainage patterns of the region, landform
and land cover, environmental factors and climate (Mukherjee 1996; Jaiswal et al., 2003).

It is well known and established that ground water recharge estimation methods are numerous.
It has been documented that none of these methods will give the same groundwater recharge
rates when used for the same area. The use of these groundwater recharge eastimate methods
are limited to particular environments (humid or arid), and the availability of data for evaluation
(Adams et al., 2004). The groundwater recharge methods and models in the determination of
the groundwater recharge are characterized into three groups (physical model, calculated from
the base flow, chemical model, measurement of water-soluble substances and numerical model,
using numerical method). The numerical approaches include the computer programs such as
HELP, RORA, PULSE, PART, HY SEP and Wells Water-table fluctuations (Risser et al., 2008; Seyed
et al., 2013). Chandra (1979) reported that the methods and techniques that are prevalently
applied to estimate the natural groundwater recharge are soil water balance, zero flux plane;
one-dimensional soil-water flow model, inverse modeling technique, ground water level
fluctuation, hybrid water fluctuation, ground water balance method; isotope and solute profile
techniques. Kumar (2000) grouped the methods and the techniques of estimation of
groundwater recharge into four groups as empirical methods; groundwater resource estimation;
groundwater balance approach and soil moisture data based methods (Seyed et al., 2013; Islam
et al., 2014).

The empirical methods are the following equations modified Chaturvedi, Kumar and
Seethapathi, UP and Rao formulae. Chaturvedi (1973) derived an empirical equation which
expresses recharge as a function of annual precipitation as follows ((Ala-aho et al., 2015; Oke et
al., 2017; Natarajan et al., 2018)):

  4.0150.2  PRr (1)
where: Rr is the net recharge due to precipitation during the year (inches), and P is the annual
precipitation(inches).
The Chaturvedi formula was restructured and modified by further work at the U.P. Irrigation
Research Institute, Roorkee and the modified Chaturvedi (UP) form of the formula is as follows
(Oke et al., 2017):

  5.01435.1  PRr (2)
Kumar and Seethapathi (2002) established an empirical relationship between groundwater
recharge and the corresponding values of rainfall in the monsoon season using the non-linear
regression equation of (Oke et al., 2017):

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  76.028.1563.0  mrm PR (3)
where: Rrm is the groundwater recharge from rainfall in monsoon season (inch), and Pm is the
mean rainfall in monsoon season (inch).
Rao derived an empirical relationship to determine the groundwater recharge in limited
climatological homogeneous areas as follows (Ala-aho et al., 2015; Oke et al., 2017; Natarajan et
al., 2018):

 XRKRr  (4)
where Rr is the recharge (mm); K is constant; R is the precipitation(mm), and X is the number
of point rainfall. The following boundary empirical equations were applied to different parts of
Karnataka as follows:
Rr = 0.20 (P - 400); for regions with annual normalrainfall (P) between 400 and 600 mm;
Rr = 0.25 (P - 400); for zones with P between 600 and 1000 mm
Rr = 0.35 (P - 600); for regions with P above 2000mm, (Ala-aho et al., 2015; Oke et al., 2017;
Natarajan et al., 2018).
Islam et al. (2014) gave an empirical relationship between recharge and the corresponding
values of rainfall. The relationship is as follows (Oke et al., 2017):

  56.051.2785.0  PRr (5)
Several other studies conducted on groundwater recharge estimate include Ichwana and
Summono (2013), Kung et al. (2013), Rana et al. (2014), Adeleke et al. (2015), Islam et al. (2015),
Sabri et al. (2015), and Oke et al. (2015; 2016). Estimating recharge in Nigeria has been a difficult
issue due to many factors which comprise the paucity of data, non-availability of mathematical
and numerical formulae applicable in the country. Literature (Ala-aho et al., 2015; Oke et al.,
2017; Natarajan et al., 2018) stressed the prominence of groundwater recharge in the national
development and economics. These discoveries show that there is the need to develop a
numerical formula for groundwater estimate in Nigeria. The key objective of this study,
therefore was to develop a numerical formula for groundwater estimation that will be applicable
in Nigeria, and evaluate its accuracy and applicability using relative error; total error; degree of
accuracy, numerical reliability, Model of Selection Criterion (MSC) and Akaike Information
Criterion (AIC).

2.0 Materials and Methods
A non-linear regression formula that represents generalization of the simple formulae that
relate rainfall to groundwater recharge estimate (depth) was proposed as follows (Oke et al.,
2017):

  cgr BPAR  (6)
where; Rgr is the groundwater estimate (depth, mm); A, B and c are the constants for the model
and P is the depth of rainfall (mm). The formula was proposed and selected based on the
simplicity, nature of the soil, soil cover, infiltration rate, previous studies from literature (Ala-aho
et al., 2015; Oke et al., 2017; Natarajan et al., 2018) and a minimum number of fixed parameters
to be determined. Monthly rainfall data between 1931 and 2010 for Yola, Nigeria was obtained
from literature (Akintola, 1986) and Upper Benue River Basin Development Authority (UBRBDA)
meteorological station and used. The rainfall data (annual and monthly) were analysed statistical
using Analysis of Variance (ANOVA) and used to calculate the recharge from rainfall using

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Umaru, et al: A New Formula for Groundwater Recharge Estimation in Yola, Nigeria. AZOJETE, 15(4):847-858. ISSN
1596-2490; e-ISSN 2545-5818, www.azojete.com.ng

Corresponding author’s e-mail address: ababayi234@yahoo.com 850

standard empirical formulae (equations 1 to 5). The constants in the proposed new numerical
formula (A, B and c) were determined using Microsoft Excel Solver and groundwater estimated
from rainfall. Microsoft Excel Solver was selected based on availability (at no additional cost)
and accuracy in numerical solutions. The new formula was used to estimate groundwater
recharge from the rainfall in Yola, Nigeria. Relationship between groundwater recharge and year
was established.
The accuracy of the developed formula was evaluated using analysis of variance (ANOVA),
relative error; a model of selection (MSC) and Akaike Information Criterion (AIC) and compared
with selected formulae in use (Rao, Islam et al, 2014, Chaturvedi and modified Chaturvedi
formulae). Although, Chaturvedi and the Kumar and Seethapathi formulae were developed
using the imperial unit, the groundwater recharge estimated was converted to system
international (SI) unit to ensure consistency in the units of measurement(Ala-aho et al., 2015;
Oke et al., 2017; Natarajan et al., 2018). Analysis of Variance was conducted using Microsoft
Excel package. Procedures employed in the computations of model parameters using Microsoft
Excel Solver are as follows (Oke et al., 2017):
Microsoft Excel Solver was added in on the toolbar of Microsoft Excel;
Target (limit, 0, equation (7)) value of the iteration was set for the software based on square of
difference as

 
2

1 1
0

cn n

t t
t t
R A P B

 

 
   

  
 

(7)
Changing cells of the iterations were selected, number of iterations, degree of accuracy and
maximum time for the iteration were set for the software to meet the target; and
The iteration started through Microsoft Excel Solver as presented in Figure 1.

The Model of Selection Criterion (MSC) is interpreted as the proportion of expected
groundwater recharge variation that can be explained by the obtained groundwater recharge.
The higher the value of MSC indicates a higher accuracy, validity and the good fitness of the
method. MSC can be computed using equation (8) as follows (Oke, 2007):

 

 
n
p

YY

YY
MSC

n

i
caliobsi

n

i
obsobsi

2ln 2

1

2

1 















(8)
where: Yobsi is the groundwater recharge estimated using UP formula; obsY is the average
groundwater recharge estimated using UP formula; p is the total number of fixed constants to
be estimated in the equation; n is the total number of groundwater recharge estimated, and
Ycali is the groundwater recharge estimated using new formula.

The AIC was derived from the Information Criterion of Akaike (1976). It allows a direct
comparison among models with a different number of parameters. The AIC presents the
information on a given set of parameter estimates by relating the coefficient of determination
to the number of parameters. The AIC and relative errors (RErr) were determined using Eqns 9
and 10 respectively as follows:

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  pYYnAIC
n

i
caliobsi 2ln

2

1















 

 (9)

n
Y
YY

RErr

n

i obsi

caliobsi









 

 1
100

(%)
(10)

3.0 Results and Discussion
Results from this study are discussed in the following categories: rainfall and statistical
evaluation, the new formula, estimated groundwater recharges and performance evaluation of
the developed formula.

3.1 Rainfall
Figure 2a presents monthly and annual rainfall of the location. From the figure the monthly
rainfall varied with the month as well as the year, which indicated that rainfall season in Yola
starts from April and ends in the month of October every year. Although there were fluctuations
in the amount of rainfall, the range for the month remains valid. Table 1 shows summary of
Analysis of Variance (ANOVA) for the monthly and annual rainfalls. The Table revealed that there
was a significant difference between the annual rainfall at 95 % confidence level (F76, 836 =
1.099, p = 0.271). The Table also revealed that there was no significant difference between the
pattern of the monthly rainfall for the selected period and location (F11, 836 = 233.562, p = 3.9
x 10 -246).

Open Microsoft
Excel

Check under Data at the tool bar if Solver is available

No

Yes

At the toolbar click Microsoft logo, open Excel option and select add in. OK

Set the Target ($L$53), operation (minimization or value of zero) and changing cell($k$2: $k$6)

At Solver dialogue set the number of iterations and time. Click on Solver
to solve

Target reached

End (Record the values)
Yes

No

Figure 1: Procedure for using Microsoft Excel Solver in the computation of the
Constants

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Umaru, et al: A New Formula for Groundwater Recharge Estimation in Yola, Nigeria. AZOJETE, 15(4):847-858. ISSN
1596-2490; e-ISSN 2545-5818, www.azojete.com.ng

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3.2 The New Formula
The proposed numerical formula is as presented in equation (11). The values of the parameters
were A = 0.621; B =1.019 and c = 0.814. The value indicates that the new numerical equation for
estimating groundwater recharge can be expressed as follows:

  0.8140.621 1.019grR P  (11)
This result shows that the new formula is similar to previous numerical formulae in literature
such as Oke et al. (2016), Asani et al. (2019). In previous studies, Oke et al. (2017), the numerical
formula for groundwater recharge for Abeokuta, Nigeria is as presented in equation (12) with
the value of the parameters as A = 1.673; B = 7.219 and c = 0. 672.

These constants in the formulae indicated that the numerical equation for estimating
groundwater recharge in Abeokuta can be written as follows:

   672.0219.7673.1  PRgr (12)
Oke et al. (2016) documented that relationship between rainfall and groundwater recharge in
Abeokuta and Ikeja, Nigeria can be expressed as follows:

0.15 102.8grR P  (13)
0.320 258grR P  (14)

From Oke et al. (2016) correlation coefficients (R2) for Abeokuta [equation (13)] and Ikeja
[equation (14)] are 0.180 and 0.393 respectively, which indicates that these relationships are
weak and needs to be improve upon. Lukman et al. (2018) developed relationship between
rainfall and groundwater recharge in Sokoto, Nigeria as

0.5006.814( 355.722)grR P  (15)

These results revealed that no numerical model is the same in values and in magnitude of the
constants.
Figure 2b presents relationship between groundwater recharge and period (year). From the
figure this relationship is weak, which indicated that there was no strong relationship between
year and groundwater recharge.

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Figure 2a: Rainfall data for Yola, Nigeria over a period of 77 years

Figure 2b: Relationship between groundwater Recharge and Period

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Umaru, et al: A New Formula for Groundwater Recharge Estimation in Yola, Nigeria. AZOJETE, 15(4):847-858. ISSN
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Figure 3: Estimated groundwater Recharge using Various Numerical formulae

3.2.1 Estimated Groundwater Recharges
Figure 3 presents the statistical summary of groundwater recharges estimated using the new
numerical formula and formulae that are in use for all the locations. The Figure revealed that
groundwater recharge estimated using the Islam et al. (2014) formula had the lowest minimum
and the mean in all the cases, while groundwater estimated using the Kumar and Seethapathi's
formula had the highest maximum and the mean in all the cases. The implication is that
accuracies of these two formulae are lower than other formulae. The Figure also revealed that
the lowest standard deviation came from Chaturvedi formula in all the case, while the highest
standard deviation came from Rao formula. The result from standard deviation indicates that
variation in the groundwater recharge estimated was higher in Rao formula than any other
formula. The highest median of computed groundwater recharge came from Rao formula, and
the lowest median of computed groundwater recharge came from Islam et al. (2014) formula.

Source of Variation
Sum of
Squares

Degree of
freedom

Mean
Sum of
Square

F- Value P-value

Years 179140.2 76 2357.108 1.098801 0.270962

Months 5511304 11 501027.6 233.5615 3.9 x 10 -246

Error 1793356 836 2145.163

Total 7483800 923

Table 1: ANOVA of the Rainfall data in Yola over a period of 77 years

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These results and observed in term of median of computed groundwater recharge indicates
that Rao formula gave a higher value of groundwater recharge computed than other standard
formulae used in the computation of computed groundwater recharge, while Islam et al. (2014)
gave the lowest groundwater recharge computed. The implication is that in design and
computation of infiltration from rainfall in water budget selection of these two formulae should
be done with caution. Results from computed groundwater recharges revealed that there are
three empirical formulae exhibit some similarities and closeness in term of computed
groundwater recharge. The three empirical formulae are the new formula, modified Chaturvedi
and Chaturvedi formulae. The Figure also showed that the estimated groundwater recharges
varied with the location as well as the empirical formulae used. This result indicates that
accuracy of these formulae is not the same. Table 2 presented result of Analysis of variance of
computed groundwater recharge. Analysis of variance of this computed groundwater recharges
revealed that there was a significant difference between the estimated groundwater recharges
within the formulae and within the year (Table 2) at 99 % confidence level. The F – values range
from 14.59 to 106.06 and p- value ranges between 1.49 x 10-66 and 1.29 x 10-54. This result of
ANOVA indicated that is a significant factor in the magnitude of groundwater recharge
computation

3.2.2 Performance Evaluation of the New formula
Table 3 presented the performance evaluation of the standard formulae and the new formula.
The Table revealed that groundwater recharge computed using Islam et al. (2014) formula
provided the lowest MSC and accuracy, as well as the highest AIC and errors (relative error, total
and root square errors). From the Table, it was revealed that the highest MSC and Accuracy;
lowest errors (total, relative and root square) and AIC were from the new formula. Modified
Chaturvedi and Chaturvedi were the next accurate formulae after the new formula, which
indicated that the new formula is among accurate formulae. The Table also revealed that there
was a slight reduction in the accuracy of these two formulae with reference to the new formula.
This slight reduction in accuracy may be attributed to the development of the two formulae
(Modified Chaturvedi and Chaturvedi) as an imperial unit’s formulae and conversion of the unit
from imperial unit to system international unit(SI). The Table revealed that there were three
empirical formulae that exhibit some similarities and closeness in respect of computed
groundwater recharge. The three empirical formulae are the new formula, modified Chaturvedi
and UP formulae. The Table showed the values of MSC were as negative for three standard
formulae (Kumar and Seethapathi formula, Rao formula and Islam et al. formula), which
indicated that utilization of these three formulae with negative MSC in the groundwater
recharge computation should be at the lowest level.

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Conclusion
It can be concluded that
The new formula and other two empirical formulae are more accurate than other standard
formulae used in this study.
There was a significant difference in the computed groundwater recharges.
Performance evaluation of the formulae revealed that care should be taken in the use of Islam
et al, Rao Kumar and Seethapathi formulae in the computation of groundwater recharge based
on the value of MSC, AIC and their accuracies.
The new formula is among the best for groundwater recharge estimate based on MSC, AIC and
relative error

References
Adams, S., Rian, T. and Yongxin, X. 2004. Groundwater Recharge Assessment of the Basement
Aquifers of Central Namaqualand. Report to the Water Research Commission. South Africa.
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Numerical
Model

Chaturvedi Kumar Rao
Islam et al.,
(2014)

Relative Error (%) 0.887 6.82 8.80 36.31 59.34
Total Error 267.305 8789.63 34009.90 287532.10 661215.22

Root Square Error 3.47 114.15 441.69 3734.18 8587.21
AIC 436.306 705.26 809.45 973.82 1037.94
MSC 4.911 1.42 0.07 -2.07 -2.90

Accuracy 99.113 93.185 91.198 63.686 40.662

Table 2: ANOVA of the Estimated Groundwater Recharges (Yola)

Source of
Variation

Sum of
Squares

Degree of
freedom

Mean Sum of
Square F- Value P-value

Years 418215.2 76 5502.832 14.58899 1.49 x 10 -66
Methods 152478.9 4 38119.72 101.0622 1.29 x 10 -54
Error 114666 304 377.1908
Total 685360.1 384

Table 3: Results of statistical Evaluations of the models

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Umaru, et al: A New Formula for Groundwater Recharge Estimation in Yola, Nigeria. AZOJETE, 15(4):847-858. ISSN
1596-2490; e-ISSN 2545-5818, www.azojete.com.ng

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