Model Selection and Design of Storm Water Drainage Systems at Oleh , Delta State , Nigeria

DOI: http://dx.doi.org/10.24018/ejeng.2022.7.1.2728 Vol 7 | Issue 1 | February 2022 70 Abstract — Rainfall Intensity-Duration-Frequency curves and equations present the exceedance probability of a given rainfall intensity and storm duration expected to occur at a particular location, for the design of storm water drainage systems and hydraulic structures. The aim of this study was to develop IDF curves/equations aim for Oleh, a municipality situated in the low-lying and flood prone region of Nigeria. The study was conducted using annual maxima rainfall series (ARMS) of 28 years obtained from Nigeria Meteorological Agency (NiMet). The AMRS were extracted into 15 Nos. storm durations between 10minutes and 600 minutes. The AMRS of each duration was fitted to Gumbel (EV1), LN2 and Normal (N) distributions using the K-S and A-D goodness-of-fit module of Easyfit software, Version 5.6. The graphical plots, development of IDF models and computations of performance measures of R NSE and RSR, were executed in Microsoft Excel, 2010. The results of GOF tests show that for K-S tests, Gumbel (EVI) is best-fit distribution in four durations, LN2 was best –fit in Nine out of fifteen durations, and Normal distribution durations scored 2. Similarly, for A-D GOF test, Gumbel (EVI) scored three (3), LN2 scored eleven (11), while Normal distribution scored one (1). Consequently, LN2 is the best-fit distribution, seconded by Gumbel (EVI). Accordingly, LN2 and Gumbel (EVI) distributions have been adopted in the development of the IDF models for Oleh municipality, while the performances of the IDF models may be presented the inequality; 0.984 ≤ R ≤ 0.998; 0.990 ≤ NSE ≤ 0.998 and 0.045 ≤ RSR ≤ 0.096. The graphical plots, IDF curves and models exhibited the peculiar attributes of IDF curves/equations reported in literature. The performance measures show that the equations are robust for practical application in the design of storm water drainage systems and hydraulic structures.


I. INTRODUCTION 1
Rainfall is the primary component of the hydrologic cycle, and its role is crucial to the survival of natural processes. Rainfall either in extreme or scarcity has detrimental impact on society and stability of eco-systems. In the extreme, excess rainfall cause floods. Flood disaster account for about a third of all natural disasters throughout the world and they responsible for more than half of the fatalities Berz [1]. Rainfall in prolonged scarcity is responsible for drought with concomitant effects of poor agricultural productivity, poverty, and health related problems. Submitted  Engineers cannot stop the occurrence of floods, they should seek structural and non-structural measures to reduce the consequences of floods and the associated economic losses, social vulnerability, environmental damages, and loss of life. Accordingly, engineers must develop tools to confront the problems posed by excess rainfall causing excess runoff and urban flooding.
The Rainfall Intensity -Duration -Frequency curves and equations are standard tools for planning, designs, and operation of water resources engineering against floods. An efficient storm water drainage system is vital to evacuating excess surface runoff from urban areas, caused by intense rainfall during raining season. Other practical uses of IDF curves and equations include; (i) Design of hydraulic structures (such as culverts and bridges), (ii) Land-use planning and soil conservation studies, (iii) Management of municipal infrastructure including sewers, storm water management ponds and street curb (iv) Design of safe and economical structures for the control, storage, and routing of storm-water and surface drainage (v) Risk assessment of dams and bridges (vi) Design of roof and storm water drainage systems (vii) Floodplain management (viii) Soil conservation studies (ix) Water-resources management. The IDF curves and equations provide the required quantile estimates which serves as input into rainfallrunoff models used for computations of runoff rates at selected times of concentration. That is rainfall intensityfrequency (IDF) relationship provides the quantile rainfall intensity for runoff estimation.
Furthermore, due to the growing awareness of the role that atmospheric and climatic processes play on the hydrologic cycle. There is a compelling need for updating and reassessment of the IDF curves and equations on regular basis. A review of previous IDF studies at Oleh and it environ include Metibaye et al. [2], Ologhadien and Nwaogazie [3], Oyegoke et al. [4] and Nwaogazie et al. [5]). The review shows that IDF curves/equation for Oleh municipality are not available and consequently will be developed. In the light of the current climate change occasioned by frequent flooding and high intensity rainfall, the IDF curves and equation need to be updated. There has not been any IDF studies carried out at Oleh and its environs, leading to development of IDF curves and equations as input into rainfallrunoff models for design of storm water drainage systems mitigate the natural environmental hazards of urban flooding at Oleh Municipality. The present study is a niche for design of stormwater drainage systems to mitigate the effect of flooding in the area. An important step in the development of IDF relations is the selection of a suitable probability distribution function (PDF) that best fit the rainfall data.
Consequently, three PDFs namely Gumbel (EV1), Lognormal (LN2) and Normal are evaluated and the best distribution will be selected through goodnessoffit tests and performance measures. The IDF curves and equations developed this study will be recommended to the local government corporate organizations to enable them cope with the environment hazards of urban flooding which include severe traffic develop, damage to properties and infrastructure.

A. Study Area and Climate
Oleh is a town in Isoko South Local Government Area of Delta State. The area covers a low-lying section of the larger Niger Delta Basin. The area is located one latitude of 50 271 42N and longitude of 60 121 2211 E on the geographical map.
The Niger Delta, comprising Oleh municipality and it environ display climatic characteristics that may be as Semi-Hot Equatorial type with heavy rainfall, Gobo [6].
The wet season lasts from March to October while the dry season lasts from November to February. Studies have shown that the area experience heavy rainfall and humidity from May to December and that it has short dry season from December to March. Flooding is the most common natural environmental hazard experienced in this area. Fig. 1 shows the map of Oleh municipality.

B. Collection of Data
The rainfall data for Oleh municipality was procured from the Nigeria Meteorological Agency (NiMet). The data length is 28 years  and were extracted for rainfall duration ranging between 10minute (0.17 hours) to 600 minute (26 hours). The point rainfall data is therefore representative of one particular location on the catchment area. The rainfall duration refers to the time period (D) in minutes over which the rainfall occurred. However, duration is not necessary the time period for the whole storm as any event can be subdivided and analyzed for a range of durations. It is common to represent the frequency or exceedance probability of the rainfall as return period (T).  Preliminary data screening to ascertain the basic assumptions of independence, randomness and stationarity of the rainfall data was performed by NiMet in-house consultant. The in-house consultant adopted correlation coefficient at lag-1 and Wald-Wolfowitz tests for independence and randomness respectively. While stationarity was checked using Mann-Kendal test. The results of the above tests indicate that the data is adequate for frequency analysis.

C. Methods and Procedures
The derivation of a preliminary intensity duration frequency relationship is important as it enables us to assign a return period that a flood storm will occur and also allow to analysis the intensities for the short duration rainfall for a range of return periods which could subsequently be used in the flood protection works.

a) IDF Quantile Estimation
The intensity-duration-frequency (IDF) is a frequency relationship among rainfall depth, intensity, and storm duration. They are used in the design of storm management facilities and flooding reservations. The IDF may be constructed by adopting the following steps: -gather time series records of different durations (e.g., 5, 10, 15, 20 minutes etc.); -extract annual extremes form the record of each duration; -fit the annual extreme data to a probability distribution in order to estimates rainfall depths for different return periods. The Gumbel's extreme value distribution is used to fit the annual extreme rainfall data. The Gumbel probability distribution has the following form: where XT represents the magnitude of the T-year event, z and z are the mean and standard deviation of the annual maximum series, and KT is a frequency factor depending on the return period (T) and also distributionspecific. The frequency factor KT for Gumbel's extreme value distribution is given by: Tabulation of annual maximum series for different durations and application of (1) to each duration.
Calculation of average intensity as follows: where D is duration.

b) Log Normal (LN2) Distribution
Using the direct method of moments without logarithm transform of the rainfall data, the population parameters are computed using the mean and standard deviation of the sample's moments: where y  and y  are population parameters of LN2 distribution, x and σ are mean and standard deviation of the sample moments respectively. The frequency factor (KT) and quantile estimate (QT) are shown in (6)- (10). Calculate the frequency factor KT is calculated: when p > 0.5, 1p is substituted for p in (7) and the value of z computed by (7) is given a negative sign. KT is approximated by Kite [7] as: where k = Cs/6.
The T-year flood quantile is obtained by (9) and (10).

c) Construction of IDF curves/Equations
The relationship between estimated rainfall intensity and duration may be expressed according to the Wisner equation given by (11). The Ontario Drainage Management Manual (MTO) [8] recommends fitting the IDF data to the three parameter function (Wisner's formula): where I is the rainfall intensity (mm/her); D is the rainfall duration (min) and a, b and C are coefficients. After selecting a reasonable value of parameter a, method of least squares is used to estimate the values of C and b. The calculation is repeated for a number of different values of "a" in order achieve the closest possible fit to the data. The best fit value of coefficient "a" is one with the least error sums of squares. The constants were obtained as follows: Taking Logarithms in both sides of (11), gives a linear equation of the form: The best values of a, b and C are those for which the sum of the squares of these deviations as minimum: The partial differentiation of SEE with respect is b and C gives: where n is the number of observational and all the summations are over all n-value. After obtaining the required summations, Equations 12 and 14 were solved simultaneously to obtain the best values of b and C for any assumed values of constant "a" and the best value of "a" was found by trial and error.

D. Goodness of Fit Tests
The performance of the selected distribution fits is ranked using two goodness of fit tests namely, Kolmogorov -Smirnov test and Anderson Darling statistic. Two tests were carried out using Easyfit software, available at a) Kolmogorov-Smirnov (K-S) Test: The K-S test is a nonparametric test for measuring the differences between cumulative distribution functions. The test hypothesis is rejected if the AD statistics is greater than a critical value of 2.5018 at a given significance level = 0.05. For further reading on the Easy fit software. The quantile equation (10) was used to calculate maximum precipitation for return periods 10,15,20,25, 50 and 100 years in Oleh metropolis.

E. Evaluation of IDF Models Efficiency
The indices used are coefficient of determination (R2); Nash-Sutchiffe Efficiency (NSE),) and RMSEobservations standard deviation ratio (RSR). The computational forms of the above indices are given below: where N is the sample size, Io is the observed rainfall intensity discharge, Ip is the predicted rainfall intensity. I is the average observed rainfall intensity. R2 statistics is an indication of the explanatory power of the IDF equation, in terms of how the IDF equations approximate or fit the data points. The higher the value of R2, the more successful the fit or the explanatory power, if R2 is small, it indicates a poor fit, possibly to search for an alternative model. R2 lies between 0(no correlation) and 1 (perfect fit). The Nash-Sutchiffe Efficiency (NSE) lies between 1.0 (perfect fit) and -∞. RMSEobservations standard deviation ratio (RSR) is calculated as the ratio of the RMSE, and standard deviation of measured data as shown in (19). RSR varies from the optimal value of 0, which indicates zero RMSE or residual variation and therefore perfect model simulation, to a large positive value. The lower the RSR, the lower the RMSE, and the better the model simulation performance [9].

III. RESULTS AND DISCUSSION
The results of the analysis are summarized in Table I-IV, while Fig. 2-6 show their graphical counterpart. Tables I shows the results of fitting Gumbel (EVI), Log-Normal (LN2) and Normal (N) distributions to the rainfall data. The fittings were performed using the Easyfit software version 5.6, while the graphical plots, development of IDF models and computation of performance measures were executed in Microsoft Excel 2010. After the distribution fitting, the best distribution out of each rainfall duration was selected based on K-S and A-D goodness-of-fit tests as shown in Table I. Table I shows that, for K-S GOF test, Gumbel (EVI) is best-fit in four rainfall duration marked with a superscript of one (1), Log-Normal (LN2) is best in Nine (9) durations and Normal scored (2). Similarly, for AD GOF test, Gumbel (EVI) scored 3, LN2 scored eleven (11) and Normal distribution scored one (1). In summary, LN2 is the best fit distribution, seconded by Gumbel (EVI). Fig. 2 Fig. 3. Thus Fig. 2 and 3 confirms that the bestfit distributions actually produce the highest estimated quantiles. Fig. 4 and 5 show the probability density functions of the fitted distributions, which also confirms that LN2 is best fit distribution, seconded by Gumbel (EVI). Tables II and III show the computed rainfall intensities for LN2 and Gumbel (EVI) respectively. Table IV shows  the developed IDF equations using the data in Tables II and  III with the corresponding performance measures for each return period. Fig. 6 and 7 are the graphical equivalent of the IDF models in Table II and III respectively. It may be inferred from Table II and III that rainfall intensity increases with increasing return periods and decreases with increasing storm duration. These observations are consistent with the common IDF behaviour found in literature, e.g., Ewea et al. [10] and Ahmed and Ali [11]. Furthermore, it can be seen from Table II and III, that moderate differences exist between the rainfall intensities predicted by LN2 and Gumbel (EVI), as observed by Elsebaie [12] and Akpen et al. [13]. The predictions by LN2 are higher than those produced by Gumbel (EVI). This is particularly evident for storm durations less or equal to 60minutes. Beyond storm durations of 60 minutes, only slight differences were observed. Storm durations less or equal to 60 minutes (1hr) are commonly used as time of concentration and critical rainfall durations for estimating runoff from highway pavements and urban areas, see Froehlich (14). Table IV presents the derived IDF models  and the corresponding statistical performance measures: R2, NSE and RSR for various return periods. All the IDF models performed very well in R2 index. All the models obtained R2 value greater or equal to 98.4 percent, which implies that 98.4 percent of the original uncertainty has been explained by the IDF models. The RSR index ranges between 0.0 and ∞, with an optimal value of 1.0. In Tables IV, the IDF models performed very good for RSR index, the range of RSR index is between 0.0 and ∞, with optimal value of 0.0. The RSR values for Gumbel (EVI) ranges between 0.082 and 0.096, while the range for LN2 is between 0.045 and 0.496. Generally, it may be observed that the errors associated with each IDF model increase with increasingly return period. Similar observation has been made by Ahilan et al. [15] who reported that the return period of the estimated value in a single site analysis should not exceed the length in years of available record by more than a factor of two. The excellent performances of the IDF models indicate robust models for practical applications.     A dilapidated culvert is to be replaced by a new culvert installation to improve road safety during rainstorms. The drainage above the crossing is 43.7 hectares (ha). The 25year rainfall intensity equation, from Table IV (I25)) =20,644.53/(tc+ 55)1.108 mm/h. Take Manning's n=0.015 and bed slope, so = 0.0015. The current land use consists of 21.8 ha of motor park, 1.5 ha of commercial property (100% impervious), and 20.4 ha residential housing. The principal flow path includes 90 m of short grass at 2 percent slope, 300 m of grassed waterway at 2 percent slope, and 650 m of grassed waterway at 1 percent slope. Determine the maximum discharge that culvert must pass for the indicated design storm and the required diameter of the culvert.
The first part requires computation of peak discharge using Rational Formula Method, followed by the peak discharge (Q), finally determination of the required size of ring culvert using Manning's equation.
Part 1, the following steps are used to compute the peak discharge with the Rational method.
On substitution of Q = 5.42 m 3 /s and n= 0.015. The required diameter (D) of pipe culvert is 0.61 m. Therefore, provide a culvert diameter of D = 900 mm.

IV. CONCLUSION
Intensity duration-frequency curves / equations are needed by hydrologists and Engineers involved in planning and design of hydraulic structures. IDF curves and equations have been developed for many cities in Nigeria. Where they are not available, and consequently, will need to be developed. This is the purpose of this study conducted for Oleh metropolis, Delta state, Nigeria. Three probability distribution functions (PDFs) namely, Gumbel (EV1), 2parameter Lognormal (LN2) and Normal distributions were evaluated using K-S and A-D goodnessoffit tests of Easyfit software, version 5.6. The performances of the three PDFs were evaluated using R2, NSE and RSR. The results show that LN2 is the best fit distribution, seconded by Gumbel (EV1). Consequently, the two distributions have been applied in the development of IDF equations / equations for Oleh metropolis. The IDF curves / equations exhibit the common attributes found in literature. Secondly the equations are robust in view of their high performance scores, therefore recommended for practical applications in the design of engineering projects against floods. The study strongly recommends periodic updating of IDF curves / equations in view of increasing flood risk caused by climate change on the hydrological cycle.