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Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 6(6):387- 398 (ISSN: 2141-7016)

387

Geoelectric Appraisal of Aquifer Contamination Risks within the Campus of Federal University of Technology, Akure Southwestern

Nigeria

Oladapo, Michael Ilesanmi and Omotola, Olubunmi Oluwatoyin

Department of Applied Geophysics,

Federal University of Technology, Akure, Nigeria

Corresponding Author: Oladapo, Michael Ilesanmi _________________________________________________________________________________________

Abstract

One hundred and fourteen (114) Schlumberger geoelectric soundings were undertaken within The Federal

University of Technology, Akure campus with the aim of appraising the risk of aquifer contamination. The

campus is underlain by the Precambrian Crystalline Basement Complex rocks. Interpretations of the data

indicate subsurface sequence comprising the topsoil, variably thick weathered/fractured basement and the

resistive basement. The weathered and fractured basement rock columns constitute aquifer units in the campus.

Weathered basement aquifers were delineated on the northwestern, north central and eastern areas while

fractured basement aquifers were delineated on the western, northern and southeastern areas with the major

aquifer units situated on the western and north central flanks. The total longitudinal unit conductance values

(0.0242 ≤ S ≤ 1.0297) utilized for contamination risk rating show that the aquifers underlying the northwest (Vice Chancellor’s lodge), northeast (Junior Staff Quarters) and southeast (Oba-Kekere) areas are at risk of low

contamination (0.7 ≤ S ≤ 1.029 mhos). Kachalla village on the western flank and North Gate/Sports Complex in

the north central zone are characterized by 0.2 ≤ S ≤ 0.69 mhos indicative of some contamination risk. High

levels of contamination risks (0.1 ≤ S ≤ 0.19) are prevalent in the central and southern areas. Cathodic protection

units should constitute integral part of buried metallic utilities in areas of high corrosivity (13 < ρ2 < 180 -m) identified in the central, southern and northwestern flanks. The coefficient of anisotropy variation (1.001 ≤ λ ≤

1.96) enhanced the delineation of lithologic contacts that are hydrogeologically favourable but may constitute

migratory pathways for contaminants.

__________________________________________________________________________________________

Keywords: geoelectric soundings, aquifer, contamination risk, protective capacity, corrosivity

INTRODUCTION

The Federal University of Technology Akure relies

largely on boreholes for potable water supply thus

necessitating the need to evaluate the level of

protection of the groundwater from contamination. This study ascertains the hydrogeologic setting for

the purpose of delineating the aquifer units and

identifies areas considered vulnerable to

contamination/pollution due to activities on the

ground surface.

The recharge areas for the aquifers delineated in this

work may not be fully covered in this study as such

aquifer recharge areas may be located far from the

university campus and may only be identifiable on a

regional scale study. The aquifer contamination risk study in this research is therefore limited to

identifying geoelectric filtering characteristics of the

regolith to vertically percolating contaminants

occasioned by environmentally hazardous activities

within the campus. This study has also not considered

direct contamination at any of the existing wellheads

via faulty, or poorly maintained, well or borehole

construction.

In order to achieve the objective of this study, the

protective capacity of the underlying regolith

materials is established using non-intrusive (dc

resistivity) methodology. One of the most effective

ways of evaluating the environment without interfering with the hydrogeologic system is through

dc resistivity studies (Okiongbo, et al., 2011; Ehirim

and Nwankwo, 2010; Braga et al., 2006; Goes and

Meekes, 2004; Oladapo et al., 2004; Kelly, 1977;

Henriet, 1975). There is a moderately large body of

scientific literature on the use of geophysical

techniques for ground-water investigations that dates

back to the late 1930s (Boulding, 1993). The

electrical resistivity method adopted for this study is

non-intrusive.

LOCATION DESCRIPTION

The Federal University of Technology (FUT), Akure

(Figure 1), occupies about 6.4 km2 of land area on the

northwestern outskirts of Akure (Adesida and

Omosuyi, 2005). The campus lies within latitude N7°

17’ and N7° 18’ and lies within longitudes N7° 08’ E

and 7° 09’E.

Vertical electric sounding locations were chosen

based on consideration for coverage of the geologic

Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 6(6): 387- 398

© Scholarlink Research Institute Journals, 2015 (ISSN: 2141-7016)

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388

units underlying the campus. The study area is

accessible through Ilesha – Ife road from the west

and Akure from the east

7 17 ' N0

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Sports Complex

SIMES

Library

Auditorium

Abiola & Jibowu Halls

Akindeko Hall

FUTA Staffs' Secondary Schl.

Senate Building

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Junior Staff Quarters

Reserved area/Zoological garden

Outcrop

Stream

Study Area

NIGERIA

0 300Scale

Km

Figure 1: The administrative map of The Federal University of Technology, Akure. Inset is the map of Nigeria

GEOLOGY AND GEOMORPHOLOGY OF THE

STUDY AREA

The university campus is underlain by the

Precambrian Basement Complex rocks of

southwestern Nigeria. The rock units recognized in

the area are migmatite gneiss, charnockite, granites

and quartzites (Figure 2). The migmatite gneiss underlies the central (academic area) portions while

the charnockite rock underlies the eastern flank (Oba

Kekere campus). The granites that are more dominant

occupy the western half (undeveloped part) and

sizable part of the eastern half of the area. The

quartzites form a distinctive topographic high in the

north central part (university sports complex area).

There are two major seasons in this area namely:

rainy season and dry season. Rainy season starts from

the month of April to November while the dry season begins from December to March. The main rain

bearing system affecting the study area is the easterly

wind current with average annual rainfall of 1333.2

mm while the relative humidity ranges from 50 to

90% depending on the season (Owoyemi, 1997). The

mean maximum temperature is 33°C while mean

minimum temperature is 18°C.

Elevation within the area varies between 340 and 400

m above mean sea level. Elevation of the northern

flank of the university campus (Sports Complex and

the hill adjoining Oduduwa Road) is higher than in

the other areas. The campus is drained by three

streams that flow southwards.

HYDROGEOLOGY

Basement terrains generally have regolith and

fractured bedrock (Odusanya and Amadi, 1990).

Weathering processes create superficial layers having

varying degree of porosity and permeability in

tropical and equatorial regions and studies have

shown that the unconsolidated overburden could

constitute reliable aquifer if significantly thick

(Olorunfemi and Olorunniwo, 1985). The concealed

basement rocks may contain highly faulted and

tightly folded areas, incipient joints and fracture systems derived from multiple tectonic events they

have experienced. These structures may have

abundant groundwater in a typical basement setting.

Detection of such structural features may facilitate

the location of groundwater prospect zones in the

study area.


389

Granite

Charnockite

Quarzite

Migmatite/Gniess

Untarred/Undeveloped road

Tarred/Developed road

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Figure 2: Geological Map of The Federal University of Technology, Akure showing the location of the VES

points

MATERIALS AND METHOD OF STUDY

R50 DC Resistivity meter was utilized for the field data acquisition. The Vertical Electrical Sounding

(VES) technique was adopted. One Hundred and

Fourteen (114) soundings (VES) were conducted

using the Schlumberger electrode configuration with

minimum spread of 2 m and maximum spread that

was varied from 32 to 100 m. Traverse orientation of

north–south directions were ensured to align with the

general strike of rocks in the area.

The principal procedures of data processing adopted

are smoothing of noisy field data, accurate computation of apparent resistivity models and

inversion of resistivity data in an iterative procedure

(based on modified “Marquardt-Levenberg”

technique to prevent convergence problems) which

includes a priori information of the model parameters

that have been supplied using results of partial curve

matching. RESISTTM (version 1.0 software) that has

been designed to process sounding data according to flexibility criteria (Vander Velpen, 2004; Vander

Velpen and Sporry, 1993) was utilized. Data

smoothing involved single point correction, curve

branch shifting, and eccentricity correction (for

Schlumberger arrays).

The interpreted data are presented as sounding curves

(Figure 3). The curves obtained range from simple 3-

layer H, A, K to complex 6-layer HAKH curves. First

order layer (geoelectric) parameters obtained from

the curve interpretation were utilized in generating sections and maps while aquifer protective capacity

ratings were derived from second order (Dar Zarrouk)

parameters of the curve interpretation (Oladapo et al.,

2004)


390

Figure 3: Typical VES curve obtained from the study area.


391

RESULTS AND DISCUSSION

Twelve curve types obtained from the VES data

acquired within the university campus are A, H, K,

AA, KH, HA, QH, KQH, HKH, QHA, AKH and

HAKH. The KH and HKH in many instances indicate fracturing and may present favourable zones for

groundwater accumulation and abstraction. Area

exhibiting such favourable hydro-geoelectric

characteristics is the hilly north central part of the

campus around the Sports Complex which is

underlain by quartzite rock unit.

The VES data interpretation enabled the delineation

of four geoelectric layering units comprising the

topsoil, the weathered basement, fractured basement

and the fresh basement. In an attempt to correlate the

geoelectric sequence across the study area, geoelectric sections were generated across four

directions (Figures 4-7) thus evolving the

hydrogeologic settings beneath the campus. Fractured

basement rock columns were delineated in few

locations especially around the quartzite region. The

topsoil is characterized by resistivity and thickness

values ranging from 40 to 2450 -m and 0.5 m to 2.2 m respectively. The weathered layer resistivity and

thickness values range from 13 to 689 -m and 3.4 m to 52 m respectively. The fractured basement has

resistivity values which range from 462 to 779 -m. Depth to bedrock values vary from 1.8 m to 72.8 m

across the campus.

Studies have shown that the unconsolidated

overburden could constitute reliable aquifer, if

significantly thick (Olorunfemi and Olorunniwo,

1985). The overburden thickness values derived from

the geoelectric parameters are presented spatially in Figure 8. The map enhanced the classification of the

study area into three groundwater potential zones.

Areas of thin overburden (< 15 m) are classified as

low groundwater potential zones. Areas of moderate

overburden thickness with values ranging from 15 to

30 m are classified as medium groundwater potential

zones while areas having thick overburden with

thickness above 30 m are classified as high

groundwater potential zones. Previous hydro-

geophysical studies in the same geologic terrain

(Omosuyi et al, 2003) have identified areas with thick

overburden as high groundwater potential zones, areas with thin overburden as areas with less

hydrogeologic appeal. The overburden thickness map

of the study area shows thick overburden at VES 40,

VES 41 and VES 42 around the Sports Complex and

VES 80 in Oba-Kekere (mini-campus). The areas

constituting northwestern (Senior Staff Quarters and

Kachalla Village), north central (Usman dan Fodio

Road and northwest flank of Oduduwa Road),

northeast (west of junior staff quarters) and southeast

(Oba kekere) have moderate overburden thickness.

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938

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4542

497165

595204

3870

20462

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4305124

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Fractured Basement

Fresh Basement

0 80 160

Distance (m)

0

10

20

Ele

vati

on

(m

)

Ohm-m

Ohm-m

Figure 4: Geoelectric section combining VES 24, 56, 38, 75, 74, 50, 86 and 79


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TIO

N (m

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Fresh Basement0 80 160

Distance (m)

0

10

20

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vati

on

(m

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Ohm-m

Ohm-m

Figure 5: Geoelectric section combining VES 22, 23, 55, 36, 58, 38, 70, 79, 85 and 78

310

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VAT

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Weathered Layer

Fractured Basement

Fresh Basement0 80 160

Distance (m)

0

10

20

Ele

vati

on (m

)

Ohm-mOhm-m

Figure 6: Geoelectric section combining VES 43, 41, 66, 13 and 46


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Fractured Basement

Fresh Basement

0 80 160

Distance (m)

0

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20

Elev

atio

n (m

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Ohm-m

Ohm-m

Figure 7: Geoelectric section combining VES 102, 106, 107, 69, 71, 83 and 91

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V VES Location

200 Contour line

Tarred road

Untarred road

m

Thin overburden

Moderate overburden

Thick overburden

Figure 8: Overburden Thickness Map of the Study Area.

Areas with thin overburden thickness are the south

central and the central portions of the study area

(west of Center for Research and Development

building, the longer straight stretch of Oduduwa road,

junior staff quarters and senior staff quarters).

Second order (Dar Zarrouk) parameters were

obtained from the first order geoelectric parameters

(Maillet, 1947; Zohdy et al., 1974; Oladapo et al.,

2004). The parameters are total longitudinal unit conductance (S), total transverse unit resistance (T),

longitudinal resistivity (ρL), average transverse

resistivity (ρt) and coefficient of anisotropy () (Table 1).

Total Longitudinal Unit Conductance

(Overburden Protective Capacity of Aquifers)

The total longitudinal unit conductance values can be

utilized in evaluating overburden protective capacity

in an area (Oladapo et al., 2004) and consequently be

utilized in evaluating risk of aquifer

contamination/pollution. The ability of the earth to

retard and filter percolating fluid is a measure of its

protective capacity (Olorunfemi et al, 1998). Henriet

(1976) described the protective capacity of an

overburden overlying an aquifer as being

proportional to its hydraulic conductivity.

Environmental impact assessment can be enhanced

using overburden protective capacity rating in an area

of proposed project(s). It enables the identification of

areas where aquifers are well protected from pollution prior to making a decision on the location

of project(s) that could bring about some

contaminants/pollutants. The classification of Henriet

(1976) as modified by Oladapo et al., (2004) in Table

2 to suite a Crystalline Precambrian Basement

Complex environment is adopted in evaluating the

protective capacity in the university campus. The

modification involves the increase of protective

capacity rating owing to the geologic and geoelectric

complexity characterizing the basement rocks and

their associated pediments.


394

The total longitudinal unit conductance map (Figure

9) generated from the calculated values presented

shows the protective capacity rating variation (0.0242

≤ S ≤ 1.0297) within the study area. The map shows

that the overburden materials in the areas around the northwest (Vice Chancellor’s lodge), northeast

(Junior Staff Quarters) and southeast (Oba-Kekere)

demonstrate good protective capacity (0.7 ≤ S ≤

1.029 mhos). The area around Kachalla village on the

western flank and North Gate/Sports Complex in the

north central zone of the campus are characterized by

moderate protective capacity rating (0.2 ≤ S ≤ 0.69

mhos).

Table 1: Second Order (Dar Zarrouk) Parameters

Obtained from First Order Parameters

VES

No

Long Unit

Conductance

(S)

(mhos)

Coefficient

of

Anisotropy

λ

Corrosivity

(2nd layer)

(Ω-m)

Overburden

Thickness (m)

1 0.091852 1.058283 88.8 9

2 0.034074 1.015667 231.9 11.4

3 0.057376 1.044519 108.5 6.7

4 0.050461 1.451380 65.9 4.2

5 0.098488 1.054885 89.4 9.6

6 0.057268 1.011854 140.9 7.5

7 0.107665 1.702488 26.8 3.8

8 0.041853 1.083566 280.9 5.7

9 0.076289 1.028599 89.1 7.4

10 0.060363 1.106995 82.6 5.5

11 0.118076 1.209105 349.8 11.7

12 0.034772 1.250632 104.5 15.9

13 0.069822 1.003966 110.0 7.9

14 0.054392 1.030726 72.7 4.5

15 0.033484 1.717265 1246.8 11.1

16 0.074041 1.292046 30.8 3

17 0.130253 1.354926 92.1 13.6

18 0.210855 1.155628 200.1 12.4

19 0.111647 1.084143 114.5 13.8

20 0.104736 1.110162 124.0 14.8

21 0.419258 1.036292 44.4 19.6

22 0.251447 1.095431 198.3 17.8

23 0.214127 1.043015 68.9 16

24 0.332836 1.021371 136.6 26.5

25 0.264549 1.141935 200.0 16.9

26 0.339439 1.082442 176.7 26.1

27 0.438385 1.228841 204.0 22.3

28 0.050212 1.110768 352.3 6.8

29 0.335410 1.066893 51.4 18.2

30 0.160680 1.096269 55.5 9.6

31 0.190082 1.855092 30.8 6.9

32 0.366770 1.074581 32.3 13.4

33 0.213852 1.318361 96.4 14.1

34 0.170315 1.147913 116.7 21.3

35 0.223347 1.016096 84.8 19.6

36 0.041114 1.795999 52.5 11

37 0.104083 1.250950 81.9 9.2

38 0.097392 1.082274 78.9 8.2

39 0.044954 1.024186 80.0 3.9

Table 1: Longitudinal unit conductance/Protective

capacity rating (Oladapo et al., 2004) continuation

VES

No

Long Unit

Conductance

(S)

(mhos)

Coefficient

of

Anisotropy

λ

Corrosivity

(2nd layer)

(Ω-m)

Overburden

Thickness (m)

40 0.117316 1.397921 1666.3 29

41 0.146310 1.140255 183.9 72.8

42 0.164641 1.347390 977.0 36.2

43 0.181234 1.723389 1644.0 20.3

44 0.100694 1.033719 382.7 25

45 0.106060 1.129728 230.1 16.2

46 0.024199 1.234714 837.4 9.1

47 0.073458 1.094725 277.3 9.8

48 0.111420 1.427987 774.4 13.6

49 0.070891 1.283014 99.7 11.6

50 0.178143 1.031413 44.8 8.3

51 0.128992 1.001211 83.6 10.9

52 0.131439 1.035943 36.3 5.3

53 0.200965 1.179110 266.6 16

54 0.130043 1.002253 111.8 14.8

55 0.156948 1.255715 235.7 19.5

56 0.169279 1.021003 127.2 22.3

57 0.065931 1.056643 267.4 12.7

58 0.058776 1.375523 60.3 4.4

59 0.068443 1.101043 384.8 10.7

60 0.270444 1.315586 14.0 4.7

61 0.056582 1.170526 418.4 19.2

62 0.081388 1.010802 154.7 11.9

63 0.046998 1.028037 119.1 6.2

65 0.131204 1.076842 41.8 11.6

66 0.033706 1.440539 360.6 4.9

67 0.144951 1.002588 159.8 20.8

68 0.083518 1.036470 310.5 23.4

69 0.184505 1.103439 403.8 30.4

70 0.028192 1.078712 422.0 9.2

71 0.047415 1.000857 143.6 6.7

72 0.099325 1.056247 57.1 6.4

73 0.090097 1.555129 13.2 1.8

74 0.046392 1.360272 62.4 6.6

75 0.036597 1.154664 165.1 9.8

76 0.026480 1.494753 449.4 3.5

77 0.102944 1.021363 83.7 7.4

78 0.319096 1.472053 336.1 16.9


395


capacity rating (Oladapo et al., 2004) continuation

VES

No

Long Unit

Conductance

(S)

(mhos)

Coefficient

of

Anisotropy

λ

Corrosivity

(2nd layer)

(Ω-m)

Overburden

Thickness

(m)

79 0.714670 1.041185 15.3 11.3

80 0.420108 1.175445 321.0 35.2

81 0.067474 1.197134 71.3 5.6

82 0.075099 1.504241 73.7 5.2

83 0.094814 2.106342 118.5 24.2

84 0.025330 1.061066 476.6 6.7

85 0.173801 1.068243 91.2 16.8

86 0.086975 1.012032 506.4 35.2

87 0.473412 3.468908 69.8 4.5

88 1.029741 1.781336 90.8 9.3

89 0.389136 2.321153 45.3 14.3

90 0.215968 1.261673 55.0 14.1

91 0.485445 1.955817 55.1 12.8

92 0.180037 1.163272 58.6 11.8

93 0.241665 1.217754 41.3 11.9

94 0.085059 1.074368 32.1 3.1

95 0.724072 1.591157 146.3 16.9

96 0.661218 1.579335 98.0 17.5

97 0.279618 1.024001 54.9 15.8

98 0.125679 1.065121 83.2 11.7

99 0.415978 1.107310 145.8 20.1

100 0.443357 1.023048 52.3 23.5

101 0.030823 1.122139 397.2 5

102 0.291177 1.532410 462.0 16.5

103 0.296171 2.216284 1112.3 9.8

104 0.031744 1.083622 376.2 5.2

105 0.033170 1.125221 432.9 4.8

106 0.198180 1.554075 569.7 15.1

107 0.190363 1.799346 605.9 12

108 0.186705 1.828353 650.8 11.7

109 0.251542 1.624254 563.3 14

110 0.035365 1.073746 480.2 6.7

111 0.079339 1.741474 19.4 2

112 0.070389 1.759121 17.8 1.8

113 0.072881 1.235905 33.2 2.9

114 0.076230 1.362828 23.4 2.4


capacity rating (Oladapo et al., 2004) Longitudinal Conductance

(mhos)

Protective Capacity

Rating

> 10

5 - 10

0.7 - 4.9

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Figure 9: Total Longitudinal Unit Conductance (S)/Overburden Protective Capacity Rating Map of the Study

Area

Other areas of the campus (central to southern zones)

present poor to weak protection for the underlying

aquifers (0.1 ≤ S ≤ 0.19) and are thus vulnerable to

contamination in case of any accidental hazardous

spill.

CORROSIVITY The resistivity values obtained from the second layer

(ρ2) at all the VES locations (13 Ω-m ≤ ρ2 ≤ 1666 Ω-

m) were utilized in the evaluation of the corrosivity

of the soil within the campus (Figure 10). Studies

carried out in similar geologic terrain (Oladapo et. al.,

2004) have shown that resistivity values of the

weathered layer can be utilized in evaluating the

corrosivity of the soil. This is because burial of

utilities and underground storage tanks are restricted

to shallow depths. Soil resistivity has been classified (Baeckmann and Schweak; 1975; Agunloye, 1984) in

terms of the degree of soil corrosivity using Table 3.


396

Areas presumably corrosive in this study are

exhibiting low ρ2 values (ρ2 < 180 -m). Areas showing corrosivity propensity are the central

and southern portions (Center for Research and

Development (CERAD); Oduduwa - Niger Delta

roads) and the northwestern areas (Senior Staff

Quarters) of the study area.

Table 3: Classification of Soil Resistivity in terms of

the Corrosivity (Baeckmann and Schweak, 1975;

Agunloye, 1984) SOIL RE SISTIVITY (Ω-m) SOIL CORROSIVITY

Up to 10

10 – 60

60 – 180

180 and above

Very Strongly Corrosive

Moderate Corrosive

Slightly Corrosive

Practically Non- Corrosive

Coefficient of Anisotropy (λ)

Delineation of lithological contacts can be facilitated

especially in areas of basement rocks concealment by

the use of coefficient of anisotropy (λ) values

(Oladapo et al., 2004; Olorunfemi and Okhue, 1992).

Values of λ in the study area vary between 1.001 and 1.956 (Figure 11) while four rock types identified are

quartzite, migmatite gneiss and charnockite. Granitic

environment of the campus is observed to be

characterized by high λ values (1.4 ≤ λ ≤ 3.4)

indicative of anisotropic characteristics of weathered

granite rocks. Migmatite gneiss is characterized by

low to moderate values (1.001 ≤ λ ≤ 1.6) showing

that the weathering end product are variably

anisotropic. Quartzite is characterized by slightly

high values (1.4 ≤ λ ≤ 1.6) showing that the

weathering end product is moderately anisotropic. Charnockite is characterized by low values (< 1.2)

showing that the weathering end product of

charnockite is fairly homogenous. An attempt is

therefore made in this study at modifying the geology

map of the university campus using coefficient of

anisotropy values obtained (Figure 12). The contact

zones identified in the map are target zones for

groundwater abstraction because such zones

constitute weak planes favourable for groundwater

accumulation.

CONCLUSION This study was carried out to evaluate the risk of

contamination/pollution of the underlying aquifers

within the campus of The Federal University of

Technology, Akure occasioned by activities within

the environment that may involve casual handling of

hazardous spill(s) or wastes. The study has shown

that favourable hydrogeologic structures (aquifers

units) within the campus are situated on the western

(Vice Chancellor’s lodge and Senior Staff Quarters),

north central (Sports Complex) and southeastern

(Oba-Kekere) areas of the campus. Overburden

protection capacities of rock materials overlying the delineated aquifers are rated good over the aquifer

units in the northwestern (Vice Chancellor’s lodge

and Senior Staff Quarters) and eastern locations

(Oba-Kekere) while generally poor to weak

protection rating were obtained beneath the north

central, central and southern parts of the campus

(Sports Complex). Thus location of any industrial

unit, workshop, waste dumps or disposal facilities

(including septic tanks) should be discouraged within

areas underlain by poor to weak protective capacity

constituents.

Buried metallic facilities (pipes and storage tanks) in

the areas of corrosion tendencies (Center for

Research and Development and Senior Staff

Quarters) should be considered for the installation of

cathodic protection devices to counter rapid

oxidization and consequent facilities decay. Future

expansion schemes of the university should take

cognizance of identified contamination facilitators in

this research.

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Figure 10: Corrosivity Map of the Study Area.


397

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Figure 11: Coefficient of Anisotropy () Map of the Study Area

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Figure 12: Modified Geological Map of FUTA based on Coefficient of Anisotropy ()

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