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CASRP Publisher CASRP Journal of Applied Sciences 1(3) (2015) 104–112

Original Article

Open Access

Mathematical Modelling of Mineral Oil Contamination for Obite to Ede by Application of Finite Element Method U. Chinedua*, L. Nwaogaziea, C. Peterb a

P.G.,Department of Civil Engineering, University of Port Harcourt P. M. B.5232, Nigeria

b

Department of chemical / Petrochemical Engineering Rivers State University of Science and Technology Nkpolu P. M. B. 5080, Port Harcourt, Nigeria.

CASRP Journal of Applied Sciences

Abstract Mathematical model was developed to monitor, predict and simulate the rate of mineral oil contamination from Obite to Ede Community Using the application of finite element method. The result obtained from the investigation revealed decrease in the concentration of mineral oil contamination upon the influence of distance travelled by contaminant. The numerical method of polynomial was established from the experimental data obtained during the investigation as presented in this paper. The application of finite element model by Galerkin’s was found useful in monitoring and predicting the rate of reduction of mineral oil contamination upon the influence travelled from Obite to Ede in Egi Community of Ogba/Egbema/Ndoni Local Government Area of Rivers State in Niger Delta Region of Nigeria.

© 2015 Published by CASRP Ltd. Selection and/or peer-review under responsibility of Center of Advanced Scientific Research and Publications Ltd Keywords: Mathematical, mineral oil, contamination, Obite, Ede, finite element.;

*Corresponding author. P.G.,Department of Civil Engineering, University of Port Harcourt P. M. B.5232, Nigeria

Received 09 July 2015

Accepted 05 August 2015

Available online 22 September 2015

105

Chinedu et al. / CASRP Journal of Applied Sciences 1(3) (2015) 104–112

ASRP Journal of Applied Sciences

1.Introduction The high exploitation, exploration, production and processing of mineral oil in Niger Delta area of Nigeria has been observed to create heavy amount of hazardous substance that are not environmentally friendly to human health and other organism that make us of water as a source of energy for their body system. It is observed that this mineral oil has posed a great impact on the groundwater of different water table or aquifer and this high level of contamination was found to have reduced as a result of diffusion and dispersion of the mineral oil from one point to the other. The characteristics of mineral oil movement within the groundwater zone was studied in this paper as well as the governing equation of groundwater mass transport model concept. The aim of this research is to examine the presence of mineral oil & concentration transport for Obite – Ede community groundwater, in Ogba / Egbema / Ndoni Local Government Area, Rivers State, Nigeria. It involved the determination of mineral oil concentration of the groundwater of a point and applying finite element model by Galerkiin’s to monitor mineral oil concentration migration within the research area. Contaminated water contains significant levels of contaminants above WHO guideline for drinking water quality standards and can generate problems when ingested by humans (Gajendran etal. 2013; Ukpaka and Chuku, 2012; Mensah, 2011; Cunningham, 1999; Oden, 1969). Refined mineral oil these include petrol aviation fuel, diesel fuel and heating oil of various grades they range in viscosity, but have densities less than of water and a heterogeneous composition dominated by pure hydrocarbons (Ashley and Misstear, 1990). In ground water they cause taste and odour problems. Contamination arises from petrol stations, oil storage depots and spillages during operation, failure or devices. Ground water supplies face a growing threat from a wide range of synthetic organic chemicals as a result of their casual deposal leakage or spillage (Lawrence, 1990) or these, the chlorinated solvent which are widely used in industry appear to be most commonly occurring contaminants and pollutant transport (Ify, 1992). Hemant (2012), applied multiple linear regression and mathematical equation model in evaluating the ground water quality of Bhaiansa village, it is a complementary tool to multiple linear regression testing the potential interaction between variables. Mathematical equation model reviews that the ground water quality in the village is dominated by (SO4), (HCO3), Cl and Mg result of water rock interaction (carbonate and halite dissolution). Water table is generally not horizontal, and has high and low points in it or it is not in equilibrium. In order that the equilibrium is approached, water moves inside the ground from the high point on water table to the points, lower down and the rate of the movement is dependent upon the ability of the porous medium to pass water though it (permeability) and the driving force or hydraulic gradient (Garg, 2007). The mechanism of pollutant transport depends on hydraulic conductivity of the soil / aquifer. If the hydraulic conductivity is very low as in some aquifers and clays, then the transport mechanism may be primarily by diffusion. For high conductivity, advection is the dominant transport mechanism (Gerard, 1998 and Ify, 2008). In this research work, Obite town is located in Ogba / Egbema / Ndoni Local Government Area, Rivers State ground water will be monitored for mineral oil contaminant transport using the approach of finite element method to simulate the one – dimensional governing equation for ground water. This evaluation is based on the fact that the community of concerned in some rounded by oil / gas industrial activities. 2.Maerials and Methods 1.1 Governing Equation of Groundwater Mass Transport Model The governing differential equation for one – dimension flow – state mass transport model is given by: =

Dx

2

Ux

(1)

where, C = concentration of non – reactive contaminants, (mg/l), Dx = hydrodynamic dispersion, 2 (m2/day), Ux = average fluid velocity in x-direction, (m/day).

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Galerkins Finite Element Formulations Applying the finite element method of obtaining a solution to equation (1) the mass transport functions and the domains are discretized into elements. A linear shape function was chosen for this research work as given: Step 1: Linear element approach C (x) = Nei ci + Nei+1 ci+1 = [N] [C] In which,

(2)

Nie = 1 +

(3)

and Ne i+1 =

(4)

Given that the linear shape or approximation functions of element, e at nodes i and i + 1 Step 2: Derivation of Element Equation The Galerkins Weighted Residual Method is the basis of the element derivation equations of the governing equation (Equation (1)). Applying this principle as follows: T

2

(Dx

2

- Ux

+

) dx = 0

(5)

Step 3: Assemble Element Equation into Global equation Assuming one – dimensional stretch, the divided element generated were assemble to produce the element assemblage for the governing Equation (1) which gives contaminant concentration at each the node. Darcy’s Law for Determining Groundwater Velocity The velocity in the governing Equation (1) was determined using Darcy’s Law concept. The hydraulic gradient of Obite – Ede was studied and determined. Applying Darcy’s discharge velocity equation, which is given as; V = KI (6) Where, k = strata permeability, (m/day), v = groundwater velocity, (m/day), I = strata hydraulic gradients, (m/m) Discharge velocity is not the actual flow velocity through the soil medium, since the flow occurs through the voids or cross – sectional area Av and not in A itself .Actual velocity of flow of the studied area Va =

(7)

Where, 𝜼 = porosity, (%) Determination of Hydrodynamic Dispersion (D) Tracer experiment was conducted at a closed range and properly monitored and the dispersion value of the aquifer were determined. Water Quality Examination Water samples were collected using plastic bottles and the various samples collected was then transported to Chemical Department Laboratory of University of Port Harcourt for onward analysis on the mineral oil concentration on each of the sampled point.

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3.Results and Discussion Galerkin’s Finite Element Computation The individual terms of Equation (6) can be calculated by the application of the linear shape functions or finite element method as follows: 1st Term Evaluation T

D 2

dx =

[N] [c] dx

T

(9)

2 1

=

2

(10) 2nd Term Evaluation T

Ux

T

dx =

Ux

[N] [C]

(11)

1

=

(12)

2

3rd Term Evaluation T

dx =

(13)

Assembling each of the evaluated terms, yields 1

-

1

+ 1

2

=0

(14)

2


2

Considering s1 =

and s2 =

(15)

2 1

0 4

0 1

0 0

0 0

c1 c2

0

1

4

1

0

c3

0 0

0 0

1 0

4 1

1 2

c4 c5

s1 + s2 -s1 + s2 0 0 0

-s1-s2 0 2s1 -s1 –s2 0 -s1+s2 2s1 -s1-s2 0 -s1+s2 2s1 0 0 -s1+s2

+

0 0 c2 0 c3 -s1-s2 s1-s2 c5

0

c1

= 0 c4

Groundwater Velocity The groundwater velocity of the research area is determined as follows:

(16)

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V=K

(17)

Where,

= hydraulic gradient, (m/m)

v=

= 0.045

Actual velocity, (Ux) =

=

2

And D = 0.03m /d Therefore, s1 =

= 0.0001

and s2 =

= 0.1145

(18)

Putting Equation (18) into Equation (16) gives , the following expression presented in matrix form as: Shown in equation 19. 100.11 49.89 0 0 0

49.89

0 200

49.89 0 0

0 49.89

200 49,89 0

0 0

49,89 200 49.89

c1 0

0 49.89 99.89

c2 = 0

c3 c4 c5

(19)

Initial mineral oil concentration at node 1, is c1 = 8.45mg/l Then, applying the upstream boundary condition to the matrix equation, yields the expression in equation.

1 49.89 0 (21) 0 0

0

0 200

0 49.89

0 0

c1 0

49.89

200

49,89

0

c3

0 0

49,89 0

200 49.89

49.89 99.89

c4 c5

8.45 c2 = 0

Resolving Equation (21) by the application of matrix gives the following solution, thus; c1 8.45 c2 -2.26 c3 c4 c5

=

0.61 -0.17 0.09

0 0

0

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Table 1: Concentration of Mineral Oil at various distances S/N Distance (m) Concentration of o mineral oil (mg/l) 1 0 8.45 2 300 2.26 3 600 0.61 4 900 0.17 5 1200 0.09

Figure 1: Graph of Concentration of Mineral Oil against distance travelled by contaminant and its polynomial plot. The relationship between concentration mineral oil and distance travelled by contaminant was shown in Figure 1.The result presented in Table 1 was used to illustrate the behavior of concentration of mineral oil upon distance. The empirical model in terms of polynomial concept was applied, which yield the mathematical expression of y= 0.00001X2-0.019X+7.996 and its square root of the best fit given and R2=0.963. It was observed that a high concentration of mineral oil at the initial distance decreases as the contaminant migrate along the entire distance. Therefore, neglecting the negative, the concentration of the mineral oil at the nodes are C 1 = 8.45mg/l, C2 = 2.26mg/l, C3 = 0.61mg/l, C4 = 0.17mg/l and C5 = 0.09mg/l

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Table 2: Theoretical concentration of Mineral Oil at various distances S/N Distance (m) Concentration of o mineral oil (mg/l) 1 0 7.996 2 300 3.196 3 600 0.196 4 900 -1.004 5 1200 -0.404

Figure 2: Graph of theoretical concentration of Mineral Oil against distance travelled by contaminant and its polynomial plot. Figure 2 illustrate the relationship between the theoretical values of the concentration of mineral oil upon the distance travelled by the contaminant. The result in Table 2 was used show the behavior of theoretical concentration of mineral oil upon the distance travelled. The theoretical empirical model in terms of quadratic polynomial concept was also applied, which give the mathematical equation of y = 0.00001x2 – 0.019x + 7.996 and its square root of the best fit given as R2 = 1, indicating the reliability or perfect match of the model. It was also observed that a high concentration of mineral oil at the initial point of reference decreases upon the distance examined. Indicating a wide spread of pollutant.

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Table 3: Theoretical and modelled mineral oil concentration S/No Distance(m) Theoretical Concentration of mineral oil values(mg/l) 1 0 7.996 2 300 3.196 3 600 0.196 4 900 -1.004 5 1200 -0.404

Modelled concentration of mineral oil values(mg/l) 8.45 2.26 0.61 0.17 0.09

Figure 3: Graph of theoretical and modelled mineral oil concentration versus distance. The result presented in Figure 3 illustrates the relationship between mineral oil concentration and distance travelled from Obite to Ede community. In this case mathematical model was developed using finite element method by Galerkin’s approach. The general solution of the model was simulated and the results obtained were compared with the polynomial theoretical data as presented in Figure 3. The result obtained shows a good match indicating the reliability of the developed model using Galerkin’s method of finite element concept. Decrease in mineral oil concentration was observed upon the distance travelled by contaminant. The data used in plotting the graph of Figure 3 is shown in Table 3 . 4. Conclusion

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In this investigation, mineral oil contaminant of Obite community is capable of migrating to Ede community within the aquifer examined as determined by Galerkin’s finite element method. However, the assessment reviewed the groundwater supply of Obite – Ede Community are polluted beyond acceptable limited by World Health Organization (WHO). The research reveals the following characteristics such as; 1. Decrease in mineral oil concentration upon the influence of dispersion from one aquifer to the other. 2. Constant deposit of the mineral oil will increase the level of pollution in future. 3. The rate of dispersion within Obite to Ede can be monitored and predicted using finite element concept and other groundwater dispersion. 4. The comparison of the theoretical data with the modelled data shows a good match revealing the reliability of the developed model. References Ashley, R.P. and Misstear, B.D. (1990): Industrial development: the threat to groundwater quality, paper presented to Institution of Water and Environmental Management, East Anglia Branch. Chukwuemeka Peter (2015): Personal Communication, Rivers State University of Science and Technology, Nkpolu, Rivers State, Nigeria. Gajendran, C., Jayapriya, C., Diana Yohannan, Oshin Victor, Christina Jacob (2013): Assessment of groundwater quality in Tirunelveli District,Tamil Nadu, India,International Journal of Environmental Sciences,Vol 3, no.6, pp.71 -80 Garg, S.K. (1998): Hydrology and Water Resources Engineering, Fourteenth Revised Edition,Khonna published, Dedhi, pp. 700-703. Gerad, K. (1998): Environmental Engineering, International Edition, McGraw-Hill, New York,PP. 200 – 929. Hemant Pathak (2012): Evaluuation of groundwater quality using multiple linear regression and mathematical equation modeling. pp. 304 – 307. Ify, L. Nwaogazie (2008): Finite Element Modelling of Engineering Systems with Emphasis in Water Resources,2nd Edition, pp. 115-177. Ify, L. Nwaogazie (1992):River mass Transport Linear-Quadratic Element Model (RIMTRA),Environmental Software, Vol. 7, pp103-121. Lawrence, A. (1990) ‘Groundwater Pollution threat from Industrial Solvents’ NERC News, No. 13, pp. 1819. Mensah, M.K. (2011): Assessment of drinking water quality in Ehi Community in the ketu-north District of the volta region of Ghana, pp 1 – 65. Oden, J.T.(1969):”A GeneralTheory of Finite Elements,2,Applications,Int. J. Numerical Engineering, Vol. 1, pp 247-259. Ukpaka, C.P and Chuku J. (2012). “Investigating the physiochemical parameters and the portability of some rivers water in Rivers State of Nigeria”, Journal of research in Environmental science toxicology, Vol. 7, pp. 168 – 185 Ukpaka Chinedu (2015): Mathematical Modeling of groundwater interaction prediction and assessment, (A Case Study of Egi Clan in Ogba/Egbema/Ndoni L.G.A, Rivers State, Nigeria), Master Degree Thesis research work ongoing, University of Port Harcourt, Nigeria. How to cite this article: Ukpaka Chinedu, L. Nwaogazie, Chukwuemeka Peter, 2015. Mathematical Modelling of Mineral Oil Contamination for Obite to Ede by Application of Finite Element Method. CASRP Journal of Applied Sciences 1(3), p. 104-112.