FLUID DRAINAGE IN THE NIGER DELTA REGION OF NIGERIA: A MODELLING APPROACH

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1 International Journal of Physics and Research (IJPR) ISSN Vol. 3 Issue 2, Jun 213, TJPRC Pvt. Ltd. FLUID DRAINAGE IN THE NIGER DELTA REGION OF NIGERIA: A MODELLING APPROACH NGIANGIA ALALIBO 1 & AMADI OKECHUKWU 2 1 Department of Physics and Department of Mathematics, Pout Harcourt, Nigeria 2 Ignatius Ajuru University of Education, Rumuolumeni, Pout Harcourt, Nigeria ABSTRACT A mathematical model for analysis of fluid drainage in the Niger Delta region of Nigeria is made. An experimental validation of the model was also carried out and comparison of the mathematical model and experiment showed that more drainage is needed to reduce the time of run off of fluid (water) from the surface of soil in some location in the region. KEYWORDS: Fluid, Niger Delta, Modeling, Differential Equation INTRODUCTION Fluid drainage is the natural or artificial removal of surface and subsurface fluid from a given area,it is expected that in any planned location in a given environment, prediction can be made on when a given volume of fluid (water) can be drained away. Drainage can either be natural or artificial (manmade). The Niger Delta region is the lowland part of Nigeria where the River Niger and River Benue empty its contents. Because of its topography, it is faced with occasional overflow of its banks by sea water and also flooding from rainfall. Rainfall in the coastal belt of the Niger Delta is heavy due to the closeness of the Delta region to the equator. The long raining season starts in March and last to the end of July with a peak period in June over most parts of the southern Nigeria, It is a period of thick clouds and is excessively wet particularly in the Niger Delta and the coastal lowlands, it is marked by humidity with values hardly below 85% in most parts of the forested south (Omogbai (21)). According to Wikipedia.org free online report mean annual rainfall along the coast vary from 24mm to 4mm as shown in some Niger Delta cities., Warri 273mm, Forcados 487mm, Port Harcourt 24mm, Calabar 37mm, and Bonny 42mm. Odjugo(25), also posit that the rainfall pattern in Nigeria has enhanced wind erosion/desertification, soil erosion and coastal flooding with the coastal areas experiencing slight increase. Many areas have somenatural drainage, this means that excess water flows from the environment to swamps or to lakes and rivers. Natural drainage however, is often inadequate and artificial drainages are required. About 7 years before now in most of the Niger Delta areas natural drainages are sources of evacuating outlets or point for rainfall. As a result of poor planning and implementation of developmental strides, many of these outlets and valleys have either been blocked or entirely covered and the attendant effect is flooding of the surrounding areas and very long time taken for the water to drain off. A simple explanation to this is that, as a result of the infrastructural development that has taken place within the period under consideration, the existing few drainage can no longer cope with the volume of water neededto be drained away in time hence the need to study the existing situation which is assuming an alarming dimension.

2 78 Ngiangia Alalibo & Amadi Okechukwu However, an accurate mathematical solution to physical fluid flow problems is difficult and depends on experimental data which must cover a wide range of conditions (Giles 1962). According to Ledder (21) most researches in fluid drainage are concentrated on open channel or conduit in which the fluid flows with a free surface subjected to atmospheric pressure. The studies only make provision for fluid to drain away notwithstanding the time it takes the fluid to dry up and not putting into consideration the time lag for fluid (water) to dry up from a given geographical location has often resulted to seasonal flooding. Since flooding is associated with economic and human losses as well as constitutes environmental problems, timely removal of fluid (water) from the surface of a given area is of necessity. Our aim is to study the timely removal of fluid (water) from the surface of the Niger Delta region to avoid possible flooding. FORMALISM We consider a cylindrical tank of constant cross-sectional area A, with an orifice of cross-sectional area a, as shown below Figure 1: A Tank Filled with Water to Depth Ho Although, fluid flow models are complicated (Ledder 21) we simplify our study by applying the idea that fluid flow is a response to differences in pressure between the top and bottom of the water column. This pressure difference ( p ) is related to velocity (u) of the fluid(water) column according to Bernoulli s equation as 1 u 2 p 2 (1) where is the density of water and u is the velocity. The pressure difference is also given by the weight per unit area of the water column as p gh (2) where g is the acceleration due to gravity and h is the height of the water column. Combining equations (1) and (2), we obtain the velocity of the water exiting from the tank (Torricelli s law) as u 2gh (3) We determine the flow rate (volume/time) by the relation

3 Fluid Drainage in the Niger Delta Region of Nigeria: A Modelling Approach 79 dv dt 2gh a (4) Since v Ah, we rewrite equation (4) as dh dt k h a k 2g A (5) Dimensionless Analysis We use the following dimensionless parameters k 2gt u a, A h At h, t u tu u (6) Using equation (6) we can rewrite equation (5) as dh dt k h (7) METHOD OF SOLUTION Equation (7) is a first order ordinary differential equation which we can seek explicit solution using the variable separabletechnique as dh h kdt (8) we get Integrating both sides and imposing the initial condition h v, c.5.5 2h kt 2vc (9) RESULTS AND DISCUSSIONS Values of h, k as the cross-sectional area (a) of the orifice is increased within a given interval of time g=9.8,a=1.v.5 c =1 are shown on the graph below Figure 1: The Dependence of Height of Water Column on Time

4 8 Ngiangia Alalibo & Amadi Okechukwu As the cylindrical surface area of the orifice is increased as shown on graph 1, the height of the fluid (water) decreases. This is also applicable in a given area that the increase in the number of drainages will help in timely removal of surface fluid (water) and this will prevent flooding. Theoretically, for h, t 2vc k Now, if h after 144s then flooding has set in. EXPERIMENTAL SET UP FOR MODEL VALIDATION A 2 litres capacity cylindrical tank is procured in addition to plugging five more holes of.7685m 2 each at the same horizontal line with the orifice at the base of the tank and filled with water. METHODS A stopwatch is started as one plug is removed and the water begins to leak out of the hole until the tank is empty and the time recorded. The same volume of water is refilled and the two plug opened. The process is continued until the remaining plugs are opened and the time recorded for the water to completely drain from the tank. Number of Plugs(m 2 )Removed Table 2 Time(s) Recorded for Water to Leak Out Completely The results on table 2 show that increasing the numbers of channels as demonstrated by increasing the numbers of holes decreases the time for the fluid (water) to drain away. COMPARISON The experimental and theoretical study reveal that increase in the number of drainages decreases the time for the fluid(water) to drain away. If other factors particularly the amount of rainfall (that varies in so many regions of the Niger Delta) can be determined, precise time of evacuation of water from the surface can be determined by providing the required number and sizes of drainages in the region under consideration. CONCLUSIONS The study is primarily aimed at ensuring that fluid (water) is evacuated within four hours after rainfall but environmental factors such as topography weather and amount of rainfall can vary the timely evacuation of fluid (water) from the surface of the soil. Percolation and evaporation and the rate at which they occur may also vary the evacuation time, however, the model is worth considering, although a proper environmental impact assessment should also be carried out to ensure that creating more drainage outlets do not create problems in the given location or nearby area.

5 Fluid Drainage in the Niger Delta Region of Nigeria: A Modelling Approach 81 REFERENCES 1. Giles, R. V (1962). Fluid Mechanics and Hydraulics.(Second Edition).Schaum s Outline Series McGraw-Hill Book Company. 2. Ledder, G (25).Differential Equation.A modelling Approach. McGraw-Hill Higher Education Companies, Inc. 3. Odjugo, P A (25): An analysis of rainfall pattern in Nigeria, Global Journal of Environmental Sciences. 4(2), Omogboi B.E (21). An Empirical Prediction of Seasonal rainfall Nigeria.Journal of human Ecology. 32(1), Rainfall pattern in Nigeria. Wikipedia.org. Retrieved 14 th April 213.

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