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ISSN: 2381-8719
Journal of Geology & Geophysics
OPEN ACCESS Freely available online
Research Article
1
J Geol Geophys, Vol. 9 Iss. 1 No: 470
Empirical Relationship between Gravimetric and Mechanical Properties
of Basement Rocks in Ado-Ekiti, Southwestern Nigeria
Ajayi CA1*, Akintorinwa OJ2 and Ademilua LO1
1Department of Geology, Ekiti State University Ado Ekiti (EKSU)
2Department of Applied Geophysics, Federal University of Technology Akure (FUTA)
ABSTRACT
Gravimetric and mechanical parameters of Basement rocks in Ado-Ekiti, South-western Nigeria were correlated for
engineering foundation studies with the aim of establishing an empirical relationship between the two parameters.
Field operations revealed Charnockite, Migmatite, Granite Gneiss and Quartzite as principal basement rocks in
the study area. Fresh rock samples were taken from thirty (30) locations cutting across the geology of the study area.
Simple pendulum principle and Archimede’s principles were employed to determine the gravity and the specific
gravity of the rock specimens repectively. The mechanical analyses (uniaxial compressive strength, shear strength,
Young’s modulus, Bulk modulus and Poisson’s ratio) for the thirty rock samples were determined employing
standard method. This is applicable to all engineering foundation studies to determine the compence of such areas
for engineering developments. The engineering studies revealed the reliability, stiffness, soundness and resistance of
the subsurface rocks to the prevailing overhead loads.
The results indicated that the gravity and specific gravity values ranged from 935055.46 mgal to 1038167.647 mgal
and 2.61 to 2.83 respectively. The values of Uniaxial Compressive Strength (UCS), Young’s modulus (E), Shear
modulus (µ), Bulk modulus (K) and Poisson’s ratio (Ѵ) ranged from 49–107 mpa, 1003–3321 mpa, 416–1310
mpa,707–2728 mpa and 0.232-0.316 respectively. The cross plots of the mechanical parameters with gravity and
specific gravity showed good correlation with coefficient of correlation (R) ranging from 0.52 to 0.84 and 0.52 to
0.81 respectively. Results validation exercise also indicated that some of the Uniaxial Compressive Strength and
Poisson’s ratio have good representation in the derived empirical equation with the two geophysical parameters
in this study. The established relationship between the gravimetric and the mechanical parameters revealed that;
the mechanical strength of rock is a function of the gravitational pull effect on the rocks and that migmatitic and
granitic rocks possessed more mechanical strength than the gneissic and quarzitic rocks that characterised the study
area. Some of the equations generated has been found reliable and useful in the determination of the mechanical
properties. The physical methods adopted being faster, cheaper, proven and more comprehensive would solve some
engineering problems in examining the engineering properties of these basement rocks related terrains.
Keywords: Gravity; Specific gravity; Correlation; Physical properties; Mechanical strength; Young’s modulus; Bulk
modulus; Shear modulus; Poisson’s ratio; Rocks; Empirical relationship
*Correspondence to: Ajayi Christopher Ayodele, Department of Geology, Ekiti State University, Nigeria, Tel: +2348033505138; E-mail:
christopher.ajayi eksu.edu.ng
Received: October 23, 2019; Accepted: December 21, 2019; Published: January 2, 2020
Citation: Ajayi CA, Akintorinwa OJ and Ademilua LO (2020) Empirical Relationship between Gravimetric and Mechanical Properties of
Basement Rocks in Ado-Ekiti, Southwestern Nigeria. J Geol Geophys 9:470. 10.35248/2381-8719.19.9.470
Copyright: © 2019 Ajayi CA, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are
credited.
INTRODUCTION
Geophysical methods have been embraced over the years by most
technologically advanced countries as a vital tool in engineering
site investigations for estate development and management.
Geophysics has been used to solve many civil engineering problems
that had hitherto proved costly, complex or unattainable by other
civil engineering methods. Methods employed in geophysical
investigation are considered to be non-destructive, time-saving, less-
expensive and very effective in site probing for engineering studies
[1]. Over the years, geophysical prospecting method coupled with
geotechnical analysis has been successfully helpful in determining
the condition of the subsurface for civil engineering investigation
[2]. Mineralogical alteration of rocks contributes to changes in their
physical and mechanical properties [3]. The finer-grained rocks are
usually stronger than coarse grained varieties as a result of higher
grain to grain contacts in fine-grained samples [4].
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J Geol Geophys, Vol. 9 Iss. 1 No: 470
850000
845000
842000
838000
834000
738000 742000 746000 750000
750000
738000 742000 746000
850000
845000
842000
838000
834000
Figure 1: Map of Nigeria showing relief, morphology and the road
networks within the study area.
composed of a mafic portion, made up of biotite, hornblende and
opaque minerals while the felsic portion is quartzofeldspatic [7].
The charnockitic rocks outcrops within the study area are massive,
dark-greenish in colour with medium to coarse grained texture.
The charnockites in Ado-Ekiti fall within those that occur along
the margins of Older Granites bodies especially the porphyritic
granites [6]. Petrological studies reveal that charnockite contains
quartz, alkali feldspar, plagioclase and biotite as major mineral.
The charnockitic rocks outcropped as pavement and oval or semi-
circular hills of between five and ten meters (10 m) high with a lot
of boulders at some outcrops. They are generally massive, dark-
greenish in colour with medium to coarse grained texture. The fresh
outcrops with little or no sign of weathering have a lot of quartz,
aplite and pegmatite intrusions occurring in it [7]. The basement
rocks are believed to have evolved as a result of at least four major
orogenic cycles of deformation, metamorphism and re-mobilization
corresponding to the Liberian (2700Ma), Eburnean (25000Ma),
Kiberian (1100Ma) and lastly, the Pan-African Orogeny (650Ma).
The three first cycles were characterized by intense deformation and
isoclinal folding accompanied by regional metamorphism, which
was further followed by extensive migmatization, granitisation,
and gneissification which produced syntectonic granites and
homogenous gneisses [8]. Late tectonic emplacement of granites
and granitoids are associated with contact metamorphism which
accompanied end stages of the last deformation.
The Older Granites comprise of felsic and mafic minerals. The
felsic minerals include quartz, orthoclase, plagioclase feldspar and
muscovite while the mafic group comprise of the black coloured
biotite and the dark green to black hornblende of the amphibole
group [9]. The granites are distinguishably unique because of their
visible minerals, lack of foliation, fine-medium grained texture
and compact interlocking crystals that developed during the
crystallisation of magma [7]. A plutonic complex containing both
charnickitic and Non-charnockitic granite rocks (Older Granites)
occurs within the amphibolite facies rocks of gneisses and migmatites
in Ado Ekiti, Southwestern Nigeria [10]. Mineralogical alteration
of rocks contributes to changes in their physical and mechanical
properties [3]. Geophysical methods were employed over the years
to investigate geologic structural features, Basement disposition,
delineation of rock types, depth to competent bedrock etc. Remote
sensing and aeromagnetic as geophysical method can be integrated
to delineate geologic structural features and hydrothermal
Adapted civil engineering methods for rock strength investigation
consume time and money which can be reduced drastically by
applying specific geophysical method. Therefore, establishing
empirical relationships between the geophysical properties of
rocks and their mechanical properties, can serve as complementary
measure in determining the mechanical strength of rock from the
geophysical data. This study is aimed at evaluating the empirical
relationship between geophysical parameters (gravity) and some
mechanical properties of Basement rocks in Ado-Ekiti.
Ado-Ekiti been the capital of Ekiti-State is witnessing rapid
structural development such as fly-over bridges, high rising
building etc. The durability and stability of these structures depend
on the mechanical strength of the underlying rock/subsoil. The
conventional ways of determining the mechanical properties of the
parent rocks which weathered into subsoil is time consuming and
not cost effective. These challenges are not limited to Ado-Ekiti as
construction activities are on continuous basic. Hence, this study
focused on establishing empirical relationship between gravity
and some mechanical properties of basement rocks. Mechanical
properties of basement rocks can be evaluated directly from the
empirical equation thereby reducing cost and time waste.
Location of study
Ado-Ekiti lies between Longitude 736000 to 754000 and Latitude
832000 to 854000 Universal Traverse Mecartum (Figure 1),
covering a total area of 346.5 km2. The study area is accessible
through major and minor roads (Figure 1). Ado-Ekiti and its
environs are dominated by crystalline rocks (Figure 2) which consist
mainly of migmatite-gneiss-quartzite complex, older granites,
quartzite, charnockites, and fine to medium grained granites [5]. In
the study area, there is a close association between the charnockites
and granitic rocks due to their field relationship as documented
in the Basement complex rocks of Nigeria [6]. Migmatite covers
over 50% of the study area (Figure 1 and Figure 2) which host
intrusion of other rocks. Migmatite rock exposures occur as highly
denuded hills of essentially fine texture while the pegmatites are
very coarse-grained with phenocrysts of feldspar over 2500 mm
in length, usually of granitic composition and forming at a late
stage of crystallization. In the study area, the migmatite-gneiss rocks
Figure 2: Geology map of Ado-Ekiti study area [4].
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J Geol Geophys, Vol. 9 Iss. 1 No: 470
was hung to the ceiling of the laboratory with the aid of the clips
nailed against the ceiling. The height of the ceiling to the floor
was 3.6 m. Each of the samples was set at lengths of 3.2 m tied
on the rope tightly. The sample on the pendulum was held at an
angle of about 600 to the perpendicular axis against the ceiling. The
pendulum was set in motion until it completes fifty (50) to and fro
oscillations. Time taken to make fifty (50) oscillations was recorded
twice as ‘t1’ and ‘t2’ in seconds. An average time-taken (t) for fifty
(50) oscillations was recorded in seconds, while the period (T) was
calculated by dividing the total time (t) for the fifty oscillations by
50 (no of oscillations) (Plate 3.1). The square of the period (T2) was
also calculated. The length ‘L’ of the rope was further varied to 3.0
m, 2.8 m, 2.6 m, 2.4 m and 2.2 m with the sample attached.
The square of the period (T2) was also calculated at the varying
lengths. Values of swing rope length (L) were plotted against square
of period (T2). The gradient was then determined from the plot.
Using the Galileo equation of simple pendulum motion, which
states “The period (T) for a simple pendulum does not depend on
the mass or the initial angular displacement, but depends only on
the length (L) of the string and the value of the gravitational field
strength (g),” where;
2 /gTL
π
=
(1)
2 (n ) / gTL
π
= −
(2)
Where, T = period,
L=length of the rope,
h=distance between the floor and the sample before swinging and
g=acceleration due to gravity.
Square both sides of equation 1:
2
24 (h L)
Tg
π
−
= (3)
22
244hL
Tgg
ππ
= −
(4)
22
244Lh
Tgg
ππ
=−+
(5)
Where
2
4g
π
−
is the gradient (m) of the linear graph. The negative
sign signifies deceleration during the pendulum motion. The
values of gradient (m) calculated from the graph was equated with
the gradient
2
24
Tg
π
= −
of the linear equation (equation 5) without
the negative sign to obtain the gravity (g) values.
Specific gravity determination
The densities of samples of the thirty rock samples were determined
in the laboratory by adopting the bulk density and buoyancy
methods. Small sizes of the sample were first weighed on the
weighing balance to determine the weight in air ‘Wa’, which ranges
from 26 gm to 79 gm. The weight of the beaker half-full with water
was also weighed as ‘Wb’. Rock sample was then hung on the clip
of the tripod stand with aid of the thread and suspended into the
water and then weighed as ‘Wc’. Weight of the sample in water
‘Ww’ was determined by subtracting the weight of the beaker with
water ‘Wb’ from weight of the beaker with water and the suspended
sample ‘Wc’. Bulk density (ρ) is then determined by:
a
aw
W
WW
ρ
=− (6)
alterations [11]. Aeromagnetic data was also used for enhancing
geologic features applying co-occurrence matrices [12]. In their
work were able to evaluate brittleness of rock using ultrasonic pulse
velocity [13]. Rock mechanics properties were characterized using
correlated laboratory test and numerical interpretations of well logs
[14]. This research is aimed at evaluating the empirical relationship
between gravity as a physical parameter and some mechanical
properties of Basement rocks within Ado-Ekiti, Southwestern
Nigeria. To achieve this aim, the objectives of the study are: To
determine the mechanical properties, density, specific gravity of
the sampled rocks, to determine the gravity values of each rock type
employing simple harmonic motion (simple pendulum principle)
method in the laboratory, use above to establish an empirical
equations from which mechanical parameters can be determined
using the measured gravity values; and validate the established
empirical equations.
METHODS OF STUDY
This research employed the use of gravity geophysical method vis-
à-vis mechanical method involving determination of the uniaxial
compressive strength, Young’s modulus, Bulk modulus, Poisson’s
ratio and shear modulus of each of the rock sample. The specific
gravity and gravity were determined and correlated with each of the
mechanical properties.
Gravity values determination
Simple harmonic motion of Galile Galileo principle was adopted
to determine the gravity of each sampled rock [15]. The materials
used for the determination of the Gravity values are: swing rope,
roof clip, electric weighing balance, meter rule, hammer and stop-
watch. Thirty samples from the different rock types (charnockite,
migmatite, granite, gneiss and quartzite) were taken from different
locations within the study area (Figure 3). Six samples of each rock
unit with weights varying from 3-4 kg were taken with the aid of
sledge hammer. The samples were further broken into smaller
sizes of equal weight of approximately 60 gm. The pendulum rope
Figure 3: Gravity values map of the study area.
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Ajayi CA, et al.
OPEN ACCESS Freely available online
J Geol Geophys, Vol. 9 Iss. 1 No: 470
The Specific gravity of each sample was then calculated by
multiplying the bulk density (ρ) obtained with the density of water
(ρw) which is equal to 0.9986 g/cm3. Then,
SG=ρ × ρw (7)
SG=ρ × 0.9986 (8)
By applying the buoyancy method, the weight of the rock sample
in air was determined and the volume of water displaced (Vs) in
the beaker was measured. The bulk density of the rock sample was
calculated by dividing the weight of the sample in air (Wa) by the
displaced volume of water (Vs) and multiplying by the density of
water (0.9986 g/cm3).
a
s
W
V
ρ
= (9)
SG = ρ x ρw (10)
SG = ρ x 0.9986 (11)
Determination of mechanical properties of rocks
Fresh rock samples of rocks were collected from outcrop in each of
the locations within the study area (Figure 2). A total of thirty (30)
rock samples were taken to the laboratory and cut with the aid of
rock cutting machine into cuboid shape of 2 cm by 1.5 cm by 6 cm
dimension. Determination of the Uniaxial Compressive Strength
(UCS) entails measuring and recording the actual dimension of
each of the prepared rock sample. Then each of the prepared
samples was mounted on the Uniaxial Compressive Strength
machine. The dial gauge and the load gauge of the machine were
standardized to zero reading prior to use.
The dial gauge reads the strain on the rock sample; while the load
gauge reads the stress. The coarse adjustment load roller is then
turned until the rock breaks (as a sign of failure). The plunger
was made to touch the surface of the specimen, and the load
and penetration measuring dial was set to zero. The plunger was
made to penetrate the prepared rock sample at constant rate of
1 mm per minute. The deformation readings were taken at every
25 deformation dial reading until the compacted rock specimen
breaks or deforms. The sample stress and strain were computed
and the normal stress was plotted against the axial strain. The
peak of the resultant curve was taken as the Uniaxial Compressive
Strength (MPa).
From the uniaxial compression test curve, Mohr circle was
generated with the aid of Microsoft-Excel equation. Shear stress
and the corresponding strain were obtained from the Mohr-circle
and uniaxial compression test curve to obtain the shear modulus.
Employing Mavko et al., (2003) formula, Bulk Modulus (k) was
obtained as stated in equation 12.
3(3 E)
E
K
µ
µ
=−
(12)
Where, E =Young’s Modulus, µ =Shear Modulus and k =Bulk
Modulus
Poisson’s Ratio (v) was obtained by applying Mavko formula that
relate the Poisson’s Ratio (Ѵ) with the Young’s Modulus (E) and
the Shear Modulus (µ) in equation 13), i.e.
1
2
E
V
µ
= −
(13)
RESULTS AND DISCUSSION
Gravity results
The gravity values map generated from the laboratory data is as
shown in Figure 3. Table 1 shows the values of the gravity for each
of the sample analyzed. The gravity distribution within the study
area ranges from 935000-104000 mGal (Figure 3). Relatively low
gravity values (<985000 mGal) were observed at the central and
the southeastern part of the study area. This falls within the region
underlain by gneiss and quartzite rock. However, relatively high
gravity values (>985000 mGal) were observed within the areas
underlain by migmatite, charnockite and granite. This shows that
migmatite, granite and charnockite are denser than other rock types
within the study area. This may reflect in their weathering-end. The
gravity values reflect how dense the subsurface basement rocks are
which can be used as a parameter to determine their durability in
withstanding surface loads from engineering structures (Table 2).
Specific gravity
Table 3 shows the result of the specific gravity test within the study
area. Specific gravity values in the study area range from 2.59–2.84
(Figure 4). It shows relative low values (<2.71) in the areas mostly
Table 1: Gravity values of the sampled rocks.
Sample Rock Type Gravity
(m/s2)
Gravity
(Gal or cm/s2))
Gravity
(m gal)
C1 Charnockite 9.956706274 995.6706274 995670.6274
C2 Charnockite 10.03359008 1003.359008 1003359.008
C3 Charnockite 9.975060808 997.5060808 997506.0808
C4 Charnockite 9.845036712 984.5036712 984503.6712
C5 Charnockite 9.885468575 988.5468575 988546.8575
C6 Charnockite 10.09229453 1009.229453 1009229.453
M1 Migmatite 10.0832778 1008.32778 1008327.78
M2 Migmatite 10.03180638 1003.180638 1003180.638
M3 Migmatite 10.09538131 1009.538131 1009538.131
M4 Migmatite 10.10934145 1010.934145 1010934.145
M5 Migmatite 10.04737144 1004.737144 1004737.144
M6 Migmatite 10.09538949 1009.538949 1009538.949
G1 Granite 10.36587525 1036.587525 1036587.525
G2 Granite 10.38167647 1038.167647 1038167.647
G3 Granite 10.1116705 1011.16705 1011167.05
G4 Granite 10.13632045 1013.632045 1013632.045
G5 Granite 10.11840488 1011.840488 1011840.488
G6 Granite 10.09719608 1009.719608 1009719.608
GN1 Gneiss 10.00284649 1000.284649 1000284.649
GN2 Gneiss 9.386061732 938.6061732 938606.1732
GN3 Gneiss 9.48774606 948.774606 948774.606
GN4 Gneiss 9.547127686 954.7127686 954712.7686
GN5 Gneiss 9.495718839 949.5718839 949571.8839
GN6 Gneiss 9.83694611 983.694611 983694.611
Q1 Quartzite 9.350734021 935.0734021 935073.4021
Q2 Quartzite 9.772938667 977.2938667 977293.8667
Q3 Quartzite 9.76791535 976.791535 976791.535
Q4 Quartzite 9.728411191 972.8411191 972841.1191
Q5 Quartzite 9.904528559 990.4528559 990452.8559
Q6 Quartzite 9.3505546 935.05546 935055.46
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underlain by quartzite and gneiss. However, relatively high values
(>2.71) of specific gravity are observed around the areas underlain
predominantly by charnockites, migmatite and granite. Specific
gravity is also a function of weight or density which is also a factor
to be considered for a sustainable foundation rock or soils (Table 4).
Mechanical Properties
Uniaxial compressive strength (ucs): The results of the mechanical
properties of rocks underlain the study area are shown in Table 5.
The distribution of the Uniaxial Compressive Strength within the
study area is as shown in Figure 5. The map indicates low Uniaxial
Compressive Strength values (40–70 MPa) within zones that are
characterized by charnockite, quartzite and gneiss rocks. The areas
underlain by granite and migmatite show relatively high Uniaxial
Table 3: Specific gravity values in the study area.
Table 2: Classification of gravity values of the subsurface formation and their implication on the surface engineering structure.
Description Gravity (m gal) Implication on Surface Structure
Very High >1035000 Very Dense
High 1010000 - 1034999 Dense
Medium 985000 - 1009999 Rarely Dense
High
Low 9600000 - 984999 Low Weight
Very Low < 960000 Very Low Weight
S/N Sample Rock Type Density In g/cm3Specific Gravity From
Density (G1)
Specific Gravity From
Buoyancy (G2)
Average Specific Gravity (G)
1 C1 Charnockite 2.594 2.619 2.6153 2.7679
2 C2 Charnockite 2.663 2.706 2.7022 2.7059
3 C3 Charnockite 2.649 2.701 2.6972 2.7143
4 C4 Charnockite 2.612 2.702 2.6982 2.6552
5 C5 Charnockite 2.624 2.651 2.6477 2.7234
6 C6 Charnockite 2.659 2.679 2.6756 2.6954
7 M1 Migmatite 2.831 2.827 2.8519 2.8395
8 M2 Migmatite 2.821 2.8171 2.7778 2.7976
9 M3 Migmatite 2.82 2.8156 2.7914 2.8035
10 M4 Migmatite 2.803 2.7994 2.8412 2.8203
11 M5 Migmatite 2.827 2.8233 2.8344 2.8289
12 M6 Migmatite 2.849 2.8447 2.7654 2.8051
13 G1 Granite 2.664 2.6603 2.6552 2.6578
14 G2 Granite 2.673 2.6693 2.7 2.6847
15 G3 Granite 2.651 2.6473 2.5882 2.6178
16 G4 Granite 2.648 2.6444 2.5537 2.5991
17 G5 Granite 2.657 2.6531 2.7554 2.7043
18 G6 Granite 2.659 2.6549 2.5965 2.6257
19 GN1 Gneiss 2.65 2.6463 2.6957 2.671
20 GN2 Gneiss 2.64 2.6433 2.6714 2.6574
21 GN3 Gneiss 2.658 2.6544 2.6465 2.6505
22 GN4 Gneiss 2.654 2.6498 2.6776 2.6637
23 GN5 Gneiss 2.649 2.6448 2.6934 2.6691
24 GN6 Gneiss 2.66 2.6558 2.6891 2.6745
25 Q1 Quartzite 2.594 2.5904 2.619 2.6047
26 Q2 Quartzite 2.663 2.6596 2.6316 2.6456
27 Q3 Quartzite 2.649 2.6453 2.6667 2.656
28 Q4 Quartzite 2.612 2.6083 2.6154 2.6119
29 Q5 Quartzite 2.624 2.6208 2.6332 2.627
30 Q6 Quartzite 2.659 2.6555 2.6566 2.6561
Table 4: Classification of specific gravity values of the subsurface formation
and their implication on the surface engineering structure.
Description Specific Gravity Implication on Surface Structure
Very High >2.83 Very Dense
High 2.77–2.83 Dense
Medium 2.71–2.76 Moderately Dense
High
Low 2.65–2.70 Low Weight
Very Low <2.65 Very Low Weight
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Figure 4: Specific gravity map of the study area. Figure 5: Uniaxial compressive strength map of the study area.
Table 5: Results of the rock mechanical test in the study area.
S/N Sample Rock Type Uniaxial Compressive
Strength (MPa)
Young Modulus
(MPa)
Poisson's Ratio Bulk Modulus
(MPa)
Shear Modulus
(MPa)
1 C1 Charnockite 65.5 1709.72 0.251 1144.39 683.341
2 C2 Charnockite 74.2 2411.66 0.262 1688.837 955.491
3 C3 Charnockite 65.6 1881.57 0.252 1264.495 751.426
4 C4 Charnockite 63.4 1668.47 0.248 1103.485 668.458
5 C5 Charnockite 63.5 1677.11 0.262 1174.446 664.465
6 C6 Charnockite 77. 7 2528.29 0.273 1856.306 993.044
7 M1 Migmatite 76 2172 0.271 1580.786 854.445
8 M2 Migmatite 72.3 3321.83 0.267 2376.13 1310.904
9 M3 Migmatite 81. 5 2370.4 0.288 1863.522 920.186
10 M4 Migmatite 91.3 2677.6 0.297 2198.357 1032.228
11 M5 Migmatite 71.2 1791.5 0.247 1180.171 718.324
12 M6 Migmatite 87.1 2677.6 0.288 2105.031 1039.441
13 G1 Granite 107.3 2766.08 0.307 2388.67 1058.179
14 G2 Granite 107.8 2364.78 0.316 2142.011 898.473
15 G3 Granite 92.1 2070.89 0.292 1659.367 801.428
16 G4 Granite 93.8 3105.3 0.294 2512.379 1199.884
17 G5 Granite 104.8 2705.6 0.314 2424.373 1029.528
18 G6 Granite 90.2 3359.05 0.294 2717.678 1297.933
19 GN1 Gneiss 69 1044.68 0.254 707.778 416.539
20 GN2 Gneiss 49.3 1311.49 0.242 8 47. 216 527.975
21 GN3 Gneiss 55.8 1757.78 0.245 1148.876 705.936
22 GN4 Gneiss 57.8 2181.23 0.251 1459.993 871.795
23 GN5 Gneiss 57 1063.03 0.252 714.402 424.533
24 GN6 Gneiss 63.2 1426.46 0.259 986.487 566.505
25 Q1 Quartzite 48.8 2076.1 0.235 1305.723 840.526
26 Q2 Quartzite 59.8 1767. 3 4 0.232 1099.092 717.265
27 Q3 Quartzite 59.4 1766.75 0.233 1102.84 716.444
28 Q4 Quartzite 58.3 2044.28 0.234 1280.877 828.314
29 Q5 Quartzite 40.2 1539.25 0.228 943.168 626.73
30 Q6 Quartzite 47. 9 2091.88 0.235 1315.648 846.915
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Compressive Strength values (85–115 MPa). This indicates that,
the weathering products of granite and migmatite may be of high
strength than other rocks in the study area. The UCS can be used
to determine the soundness of the subsurface rock for the purpose
of engineering constructions (Table 6).
Young’s modulus: Figure 6 shows Young’s modulus map of the
study area. The Young’s modulus of the study area ranges from
1000–3400 MPa. Relatively low values (1000–2120 MPa) were
observed around the area characterized by gneiss, quartzite and
charnockite rocks. Relatively high Young’s composed of granite
and migmatite rocks.
This is confirming the high strength nature of migmatite and
granite relative to other rock types in the study area. Young’s
Modulus determines the stiffness of the subsurface rock against
the overhead load of the engineering structures (Table 7). Highly
yielding rock formation with low Young’s Modulus values are
susceptible to imminent failure if proper engineering precautions
are not well envisaged.
Shear modulus: Shear modulus is parameter that can be used
to determine the resistance of material to shearing stresses. The
shear modulus distribution in the study area is as shown in Figure
7. It ranges from 400–1350 MPa. The map shows relatively low
values (400–860 MPa) within the areas underlain by quartzite and
charnockite rocks. The area characterised with relatively high shear
mdulus values (>860 MPa) are underlain by migmatite and granite
rocks. The shear modulus of the study shows that, area underlain
by charnockite, gneiss and quartzite are of comparatively of lower
strengths than areas underlain by migmatite and granite rocks.
Shear Modulus reveals the resistance the underlying rocks possess
against the shearing forces (Table 8).
Bulk modulus: Bulk modulus is a geomechanical parameter that
best represents the mechanical behavior of rock mass. It describes
how resistive a material can be to compressive forces. The bulk
modulus (k) map of the study area is as shown in Figure 8. The
value ranges from 700 to 2900 MPa. The map reveals relatively
low values (<1700 MPa) of bulk modulus within the areas that
are underlain by quartzite, charnockite and gneissic rocks. The
relatively high values (>1700 MPa) were observed within the
underlain by migmatite and granitic rocks. This also confirmed the
high strength nature of migmatite and granite rocks. Bulk modulus
can be used to estimate the reliability of the foundation rocks
under all round pressure (Table 9)
Poisson’s ratio
Poisson ratio describes the ratio of the longitudinal displacement
to the axial displacement under compressive stresses. The Poisson’s
ratio value in the study area ranges from 0.225 to 0.320 (Figure 9).
It indicates higher rate of axial displacement than the longitudinal
dislocation under distressing forces that delimits the mechanical
strength of the rock. Relatively low values (<0.270) were obtained
Figure 6: Young’s modulus map of the study area.
Figure 7: Shear modulus map of the study area.
Table 7: Classification of Young’s modulus of the subsurface formation
and their implication on the surface engineering structures.
Description Young’s Modulus (mpa) Implication on
Surface Structure
Very High >3240 Very Stiff
High 2680–3239 Stiff
Medium high 2120–2679 Medium Stiffness
Low 1560–2119 Low Stiffness
Very Low <1559 High Yielding
Table 6: Classification of UCS of the subsurface formation and their
implication on the surface engineering structures.
Description Ucs Strenght (mpa) Implication
(foundation)
Very High >100 Sound
High 85-99 Good for any structure
Moderately High 70-84 Good for any structure
except large dam
Low 45-69 Variable
Very Low <44 Unreliable
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Figure 8: Bulk modulus map of the study area.
Figure 9: Poisson’s ratio map of the study area.
Figure 10: Crossplot of the gravity (G) and uniaxial compressive strength
(UCS).
in the areas underlain by quartzite, charnockite and gneiss.
Relatively higher values (>0.270) characterize the area underlain
by magmatic and granitic rocks. Poisson’s ratio shows the strength
of subsurface rock formation under the influence of the overhead
prevailing stresses (Table 10).
Comparative analysis of the geophysical and mechanical
results
The regression plots of the gravity and specific gravity values as
physical parameters against each of the determined mechanical
parameters can be represented by an empirical equation of the
form;
Y=MX+C (15)
‘Y’ represents the mechanical parameters, ‘X’ represent the physical
parameters, ‘M’ represent the gradient of the trend line, and ‘C’ is
the intercept on the mechanical parameter (vertical) axis. From the
plot, the relationship between the mechanical parameter and the
physical parameters is best described by linear relationships, where
Table 8: Classification of shear modulus of the subsurface formation and
their implication on the surface engineering structures.
Description Shear Modulus
(mpa)
Implication
on Surface
Engineering
Structure
(to shearing forces)
Very High >1320 Highly Resistive
High 1090–1319 Resistive
Medium high 860–1089 Medium Resistance
Low 630–859 Yielding
Very Low <629 Very Yielding
Table 9: Classification of bulk modulus of the subsurface formation and
their implication on the surface engineering structures.
Description Bulk Modulus
(mpa)
Implication on Surface
Structure
Very High >2700 Sound
High 2200–2699 Good for any structure
Medium high 1700–2199 Good for any structure except
large dam
Low 1200–1699 Variable
Very Low <1200 Unreliable
Table 10: Classification of Poisson’s ratio of the subsurface formation and
their implication on the surface engineering structures.
Class Description Poisson’s Ratio Implication
on Surface
Structure
A Very High >0.31 Very Strong
B High 0.29–0.30 Strong
C Medium high 0.27–0.28 Medium Strong
DLow 0.25–0.26 Weak
E Very Low < 0.24 Very Weak
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the mechanical parameter is taken as the dependent variable and
the physical parameter as independent variable. i.e. the determined
mechanical parameter of the rock samples varies with the physical
parameter of the underlying rocks within the study area.
Relationship between the Gravity(G) and Uniaxial
Compressive Strength (UCS)
The regression plot of gravity (G) against the Uniaxial Compressive
Strength (UCS) of the rock samples is presented in Figure 10.
The trend line with 0.84 coefficient correlation (R) show a good
correlation between the gravity and the Uniaxial Compressive
Strength.
The empirical equation for the relationship between Gravity and
the Uniaxial Compressive Strength is given as;
UCS=(0.00005)G–469.06 (16)
Where UCS=Uniaxial Compressive Strength
G=Gravity
Relationship between the Gravity(G) and Young’s
modulus (E)
The cross plot of gravity (G) against the Young’s Modulus (E) of
the rock samples is as shown in Figure 11. The trend line shows
a direct relationship; which shows that, the higher the gravity of
the rock, the higher the Young’s Modulus of the rocks. The trend
line equation for the cross plot gives coefficient of correlation (R)
of 0.52 which indicates moderately good correlation between the
gravity and the Young’s Modulus.
The empirical equation representing the relationship between
Gravity and the Young’s Modulus is given as;
E=0.0111G–8921.6 (17)
Where E=Young’s Modulus
G=Gravity
Relationship between the Gravity(G) and Poisson’s ratio
(Ѵ)
The cross plot of gravity (G) against the Poisson’s Ratio (Ѵ) of the
rock samples is as presented in Figure 12. The cross plot shows a
direct relationship between them, which shows that, the higher the
gravity of a rock, the higher the Poisson’s Ratio of the rock. The
equation relating the two parameters together gives 0.76 coefficient
of correlation (R), indicating relatively good relationship between
the two parameters.
The empirical equation relating gravity and the Poisson’s Ratio is
given as;
Ѵ=(7 × 10-7)G-0.4223 (18)
Where Ѵ=Poisson’s Ratio
G=Gravity
Relationship between the Gravity(G) and Shear modulus
(µ)
Figure 13 shows cross plot of gravity (G) and the Shear Modulus
(µ) of the rock samples. The trend line shows a direct relationship,
which shows that the higher the gravity of a rock, the higher the
Shear Modulus of the rocks. The trend line equation for the cross
plot gives coefficient of correlation (R) equal to 0.48, indicating
relatively weak correlation between the two parameters.
The empirical equation representing the relationship between
gravity and the Shear Modulus is given as;
µ=0.0039G -3031.7 (19)
Where µ=Shear Modulus
G=Gravity
Relationship between the Gravity(G) and Bulk modulus
(K)
The cross plot of gravity (G) against the Bulk Modulus (K) of the rock
samples is as presented in Figure 14. The trend line shows a direct
relationship between the two parameters with 0.72 coefficient of
correllation (R) which shows that the higher the gravity of the rock,
the higher the Bulk Modulus of the rocks. This indicates relatively
good correlation between the gravity and the Bulk Modulus.
Figure 11: Crossplot of the gravity (G) and Young’s modulus (E).
Figure 12: Crossplot of the gravity(G) and Poisson’s ratio (V).
Figure 13: Crossplot of the gravity(G) and shear modulus (µ).
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The general empirical equation representing the relationship
between gravity and the Bulk Modulus is given as;
K=0.0132G–11572 (20)
Where K=Bulk Modulus
G=Gravity
Relationship between the Specific Gravity(SG) and
Uniaxial Compressive Strength (UCS)
Figure 15 shows a cross plot of specific gravity (G) and the Uniaxial
Comressive Strength (UCS) of the rock samples. The trend shows
a direct relationship, indicating that the higher the specific gravity
of a rock, the higher the Uniaxial Comressive Strength of the
rocks. The trend line equation for the cross plot gives coefficient
of correlation (R) is equal to 0.81, indicating a relatively strong
correlation between the two parameters.
The empirical equation relating specific gravity and the Uniaxial
Comressive Strength is given as;
UCS=183.25SG-418.16 (21)
Where UCS=Uniaxial Compressive Strength
SG=Specific Gravity
Relationship between the Specific Gravity(SG) and
Young’s modulus (E)
The regression plot of specific gravity (SG) against the Young’s
modulus (E) of the rock samples is as presented in Figure 16. The
trend line shows a direct relationship between the two parameters
with 0.60 coefficient of correlation (R). This indicates a moderately
good correlation between the specific gravity and the Young’s
modulus.
The empirical equation relating specific gravity and the Young’s
Modulus is given as;
E=4734.35G-10604 (22)
Where E=Young’s modulus
SG=Specific Gravity
Relationship between the Specific Gravity(SG) and Bulk
modulus (K)
Figure 17 shows a cross plot of specific gravity (SG) and the Bulk
modulus (K) of the rock samples. The trend line shows a direct
relationship, which shows that the higher the specific gravity of a
rock, the higher the Bulk modulus of rocks. The trend line equation
for the cross plot gives coefficient of correlation (R) equals to 0.70,
indicating a very strong correlation between the two parameters.
The general empirical equation representing the relationship
between specific gravity and the Bulk modulus is given as;
K=5567.8SG-133 (23)
Where K=Bulk modulus
SG=Specific Gravity
Relationship between the Specific Gravity(SG) and Shear
modulus (µ)
The regression plot of specific gravity (SG) against the Shear
Figure 14: Crossplot of the gravity(G) and bulk modulus (µ).
Figure 15: Crossplot of the specific gravity(SG) and uniaxial compressive
strength (UCS).
Figure 16: Crossplot of the specific gravity(SG) and Young’s modulus (E).
Figure 17: Crossplot of the specific gravity(SG) and bulk modulus (K).
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modulus (µ) of the rock samples is presented in Figure 18. The
trend line shows a direct relationship, which demonstrates that
the specific gravity of the rock is directly proportional to the Shear
modulus of the rocks. The two parameters have 0.52 as their
coefficient of correlation (R) showing moderately good correlation
between them.
The empirical equation relating specific gravity and the Shear
modulus is given as;
µ=1679.5SG-3681.5 (24)
Where µ=Shear modulus
SG=Specific Gravity
Relationship between the Specific Gravity(SG) and
Poisson’s ratio (Ѵ)
Figure 19 shows cross plot of specific gravity (SG) and the Poisson’s
ratio (Ѵ) of the rock samples. The trend line shows a direct
relationship the two parameters, with 0.81 as the coefficient of
correlation (R) between them. This shows a very good correlation
between the two parameters.
The empirical equation relating specific gravity and the Poisson’s
ratio is given as;
Ѵ=0.2524SG-0.4098 (25)
Where ѵ=Poisson’s ratio,
SG=Specific Gravity
Data validation
Ten samples were collected from different locations different from
the initials thirty sampling points within the study area for result
validation (Figure 20). Two samples from each of the five rock types
(Granite, Charnockite, Migmatite, Gneiss and Quartz) were taken
for the result validation analysis. The Physical parameters (gravity
and specific gravity) and the mechanical parameters (Young’s
Modulus, Bulk Modulus, Shear Modulus, Poisson’s Ratio and the
Uniaxial Compressive Strength) were also determined following
the same methodology earlier discussed in the study.
The physical parameters (gravity and specific gravity) obtained were
computed into the empirical equations (equation 14–24) generated
from the regression analysis. The predicted results obtained were
compared to the observed results obtained from the laboratory
analysis of the rock samples using linear regression, to check the
reliability of the derived empirical equations. The errors and the
percentage errors were determined from the juxtaposition.
The observed and predicted results were correlated using the
relationship between gravity and the mechanical properties (Figures
21-25). The coefficient of correlation (R) (between the predicted
and the observed) for Uniaxial compressive strength, Young’s
modulus, shear modulus, Poisson’s ratio and bulk modulus are:
0.76, 0.40, 0.37, 0.78 and 0.54 respectively (Table 5) while the
average percentage errors are: 25.75, 9.95, 12.23, 2.07 and 15.97
respectively (Table 6). Therefore, empirical equations gives a good
representation for Uniaxial Compressive Strength and Poisson’s
Figure 18: Crossplot of the specific gravity (SG) and shear modulus (µ).
Figure 19: Crossplot of the specific gravity (SG) and Poisson’s ratio (Ѵ).
738000 742000 748000 750000
850000
845000
842000
838000
834000
738000 742000 748000 750000
G1'
G2'
Q1'
Q2'
M1'
M2'
C1'
C2'
GN1'
GN2'
Sampling Location
Figure 20: Geology map of the study area showing the sampling locations
for the verification data.
Figure 21: Cross plot of the observed and predicted results of uniaxial
compressive strength from result of gravity.
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Figure 22: Cross plot of the observed and predicted results of shear
modulus from result of gravity.
Figure 23: Cross plot of the observed and predicted results of young’s
modulus from result of gravity.
Figure 24: Cross plot of the observed and predicted results of Poisson’s
ratio from result of gravity.
Figure 25: Cross plot of the observed and predicted results of bulk specific-
gravity.
Figure 26: Cross plot of the observed and predicted results of uniaxial
modulus from result of gravity.
Figure 27: Cross plot of the observed and predicted results of shear.
Figure 28: Cross plot of the obser ved and predicted results of youngs
modulus from result of specific-gravity.
Ratio, although the percentage error is high, which may be as a
result of other factors that came up during extrapolation, while
weak correlation exist in the cross plots for Young’s Modulus,
shear modulus and bulk modulus. However, the percentage errors
obtained Young’s modulus is less than 10%, which is a good
representation of the developed equation.
The estimated mechanical properties from the developed specific
gravity empirical equations were also correlated with the observed
results to show how symbolic the equations can become in
generating mechanical properties (Figures 25-30). The coefficient
of correlation (R) (between the predicted and the observed) for
Uniaxial compressive strength, Young’s modulus, shear modulus,
Poisson’s ratio and bulk modulus are: 0.79, 0.43, 0.41, 0.41 and
0.50 respectively (Table 5), while the average percentage errors
are: 0.27, 8.02, 8.04, 1.24, and 7.51 respectively (Table 6). The
coefficient of correlation reveals that only the Uniaxial Compressive
Strength developed equation that has a good representation, while
others has very weak correlation coefficients.
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Figure 30: Cross plot of the observed and predicted results of bulk
modulus from result of specific-gravity.
Figure 29: Cross plot of the observed and predicted results of Poisson’s.
Table 12: Coefficients of correlation in the relationships between the physical and mechanical properties of rocks.
Physical and mechanical parameters of basement rocks distributed
within and around parts of Ado-Ekiti, Southwestern, Nigeria were
correlated with the aim of establishing empirical relationship
between the two parameters. The principal Basement rocks in
the study area are; Charnockite, Migmatite, Granite Gneiss and
Quartzite. The largest part of the area is dominated by migmatite.
The gravity values range from 935055.46 mGal to 1038167.647
mGal, while specific gravity values range from and 2.61 to 2.83.
The values of Uniaxial Compressive Strength (UCS), Young’s
modulus (E), Shear modulus (µ), Bulk modulus (K) and Poisson’s
ratio (v) ranging from 49–107 MPa, 1003–3321 MPa, 416–1310
MPa,707-2728 MPa and 0.232-0.316 respectively. Migmatitic rock
was observed to possess highest values of mechanical strength
among the five other rock types.
The cross correlation of the gravity and specific gravity as physical
parameters and all the analyzed mechanical parameters within the
study area show direct relationship i.e. these mechanical parameters
increases with an increase in each of the physical parameters. The
cross plots of the mechanical parameters with the gravity generally
show good correlation with correlation coefficient ranging from
0.52 to 0.84, except for Shear Modulus which show weak (0.50)
correlation with the gravity. Good correlation were obtained in all
the cross plots between the mechanical properties and the specific
gravity, with coefficient of correlation (R) that ranges from 0.52
to 0.81. Since the coefficient of correlation between each of the
established physical parameters and the determined mechanical
properties of the basement rock in the study area are generally
strong, it implies that mechanical properties of basement rocks can
be estimated from physical measurements using the established
empirical equations for each of the determined parameters (Tables
11 and 12). The validation exercise (correlating the observed and
the predicted results supported with the results of the percentage
errors) demonstrates Uniaxial Compressive Strength and Poisson’s
ratio have good representations in their relationships with the
three geophysical parameters (Table 13).
CONCLUSION
The established relationship between the gphysical and the
mechanical parameters reveals that, the mechanical strength of
rock is a function of the gravitational pull effect on the rocks.
Also, The study also reveal that migmatitic and granite possess
more mechanical strength than the other principal rock types that
characterise the study area. The areas where high rock mechanical
properties were observed signify high reliability, stiffness and
mechanical strength for civil engineering developments. The study
is able to establish that physical properties of rocks can be used
to generate mechanical properties as hypothesized from previous
studies. The study is applicable in the study of the mechanical
Table 11: Empirical equations generated from the relationships between the physical and mechanical properties of rocks.
Parameters UCS (MPa) Young’s Modulus
(MPa)
Shear Modulus (MPa) Poisson’s Ratio Bulk Modulus (MPa)
Gravity (mGal) UCS=(0.00005)G - 469.06 E=0.0111G – 8921.6 µ=0.0039G–3031.7 Ѵ=0.0000007G
–0.4223
K=0.0132G – 11572
Specific Gravity UCS=183.25SG - 418.16 E=4734.35SG –10604 µ= 1679.5SG– 3681.5 Ѵ= 0.2524SG–0.4098 K= 5567.8SG –13388
Parameters UCS (MPa) Young’s Modulus
(MPa)
Shear Modulus (MPa) Poisson’s Ratio Bulk Modulus (MPa)
Gravity (m Gal) 0.8409 0.5697 0.5297 0.8546 0.7172
Specific Gravity 0.8133 0.5538 0.5201 0.8125 0.6953
Table 13: Coefficients of correlation of relationships of the predicted mechanical properties from physical investigation and the observed mechanical
properties of rocks.
Parameters UCS (MPa) Young’s Modulus
(MPa)
Shear Modulus (MPa) Poisson’s Ratio Bulk Modulus (MPa)
Gravity (mGal) 0.76 0.4 0.37 0.78 0.54
Specific Gravity 0.79 0.43 0.41 0.41 0.5
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strength of Basement rock as foundation bedrock to withstand
the load to be impacted by any proposed heavy weight civil
engineering structures e.g. high rising buildings, fly-over bridges,
telecommunication mast, tunneling, and rail-lines. The study is
applicable in any region of related geologic terrain. The generated
equations from this research will help civil engineers to acquire
information about any related terrain faster, cheaper and in a
more comprehensive mode. Further studies can also be carried
out in the area using gravimeter equipment for the geophysical
gravity determination and aeromagnetic data to be correlated
with mechanical properties for proper foundation studies. Also,
the study can be conducted distinguishing different rock type
individually.
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