Determination of Incompressibility (Bulk Modulus), Elasticity (Young’s Modulus) and Rigidity (Shear Modulus) of Uyo and Its Environ, Southeastern Nigeria ()

Ubong E. Essien^{1}, Akaninyene Okon Akankpo^{2*}, Magnus U. Igboekwe^{3}, Emmanuel B. Umoren^{2}

^{1}Department of Science Technology, Akwa Ibom State Polytechnic, Ikot Osurua, Nigeria.

^{2}Department of Physics, University of Uyo, Uyo, Nigeria.

^{3}Department of Physics, Michael Okpara University of Agriculture, Umudike, Nigeria.

**DOI: **10.4236/gep.2023.111008
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In this work, seismic refraction was used to obtain elastic properties
(shear modulus (μ), Young modulus (*E*),
Bulk modulus (*K*) and lithological
information in Uyo and its environ as an aid to engineering foundation. Using
seismic refraction method, the top and weathered layer of the engineering
foundation in the study area was investigated to determine the elastic
parameters of top soil and also assess the strength of engineering foundation based
on the parameter distribution. A 24-channel signal enhancement seismograph,
geophones, sledge hammer and a metal plate (source) for generating seismic waves were
used. The study area lies between latitudes 4˚45' and
5˚15'N and between longitudes 7˚45' and
8˚30'E in the Niger Delta region of southern Nigeria. Geologically, the area
is located in the Tertiary to Quaternary
Coastal Plain Sands (CPS) (otherwise called the Benin Formation) and Alluvium
environments of the Niger Delta region of southern Nigeria. Shear Modulus had
average values of 0.43 × 10^{8} N/m^{2} and 1.40 × 10^{8} N/m^{2} for layers 1 and 2 respectively. The average values of the
Young Modulus for layers 1 and 2 were determined as 2.32 × 10^{8} N/m^{2} and 3.84 × 10^{8} N/m^{2} respectively. The average values of
the bulk Modulus for layers 1 and 2 were estimated as 1.52 × 10^{8} N/m^{2} and 4.93 × 10^{8} N/m^{2} respectively.

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Essien, U. , Akankpo, A. , Igboekwe, M. and Umoren, E. (2023) Determination of Incompressibility (Bulk Modulus), Elasticity (Young’s Modulus) and Rigidity (Shear Modulus) of Uyo and Its Environ, Southeastern Nigeria. *Journal of Geoscience and Environment Protection*, **11**, 127-138. doi: 10.4236/gep.2023.111008.

1. Introduction

Rock/soil elastic properties are sources of valuable information for most projects in rock/soil mechanics as the knowledge of deformational characteristics of rocks/soils are vital in locating and extracting mineral resources and designing and constructing any structure on the rock or soil. Geotechnical testing has increasingly been used for geotechnical investigation to identify subsurface irregularities, such as fill, cavities and variable strata (Budhu & Al-Karni, 1993). It can also be used to obtain quantitative information that is useful for foundation assessment and design.

Due to the incessant failure of roads and collapse of buildings in Nigeria and Uyo in Akwa Ibom State in particular, the need to find a lasting solution to these problems led to the undertaking of this research work. In this work, seismic refraction was used to obtain the elastic properties and lithological information as an aid to engineering foundation. The earth model is assumed to be spherically symmetric non-rotating, elastic and isotropic in nature (Ogagarue & Asor, 2010). Using seismic refraction method, the top and weathered layer of the engineering foundation in Uyo senatorial district of Akwa Ibom State, Nigeria was investigated to determine the elastic parameters of top soil and also assess the strength of engineering foundation based on the parameter distribution. The results obtained will help in the development of the geotechnical aspect of Geophysics.

2. Elastic Moduli and Their Characteristics

The application of external forces to an elastic body produces a balance in internal body focus within a body. The force in this case is elastic, produced by expansion or compression as the case may be when a wave propagates through the body. Therefore the whole elastic force arising from the passage of wave is given by ∆*T _{xx}S*, where ∆

$\frac{\Delta {T}_{xx}\Delta {X}_{s}}{\Delta X}=\rho \Delta XS\frac{{\text{d}}^{2}u}{\text{d}{t}^{2}}$ (1)

The equation of motion for the wave is

$\rho \frac{{\text{d}}^{2}u}{\text{d}{t}^{2}}=\frac{\text{d}{T}_{xx}}{\text{d}x}$ (2)

$\rho {\text{d}}^{{}^{2}}u=\frac{\text{d}{T}_{xx}}{\text{d}{Y}_{xx}}=\frac{\text{d}{Y}_{xx}}{\text{d}x}$ (3)

Substituting Equations (2) into (3) gives

$\frac{\rho {\text{d}}^{2}U}{\text{d}{t}^{2}}=\frac{\text{d}{T}_{xx}}{\text{d}{Y}_{xx}}=\frac{X{\text{d}}^{2}U}{\text{d}{x}^{2}}$ (4)

$\frac{\text{d}T}{\text{d}Y}=\text{constant}$ (5)

where d*T* is stress and d*Y* is strain. This quantity d*T*/d*Y* for isotropic body is constant and is known as elastic modulus (parameter). The direct relationship between stress and strain in the elastic field is unique for any material by its different elastic moduli, each of which expresses the ratio of a particular type of stress to a particular strain (Domenico, 2012).

2.1. Young’s Modulus (*Y*)

If the body is stretched with a lateral force from stress, the constant would be the Young’s modulus. Mathematically, Young’s Modulus (*E*) =
$\frac{\text{LongitudinalStress}\left(\frac{f}{A}\right)}{\text{LongitudinalStrain}\left(\frac{\Delta L}{L}\right)}$ expressed in N/m^{2}.

Young’s modulus (*E*) can be calculated using Equation (6) below

$E=2\mu \left(1+\sigma \right)$ (6)

where µ is shear modulus and σ is the Poisson’s ratio.

2.2. Bulk Modulus (*K*)

The bulk modulus (*K*) expresses the stress-strain ratio as in simple hydrostatic pressure *P*, the resultant volume strength being the change in volume ∆*V* divided by the original volume. That is

$K=\frac{\text{VolumeStress}\left(\rho \right)}{\text{VolumeStrain}\left(\Delta V\right)}$ (7)

Bulk Modulus (*K*) can be calculated using Equation (8) below

$K=\frac{2\mu \left(1+\sigma \right)}{3\left(1-2\sigma \right)}$ (8)

where µ is shear modulus and σ is the Poisson’s ratio.

2.3. Shear Modulus (µ)

It is the measure of an ability of an object to withstand or oppose the shape from being deformed under a tangential stress condition. The tangential forces due to seismic wave propagation in a medium produce an angle of shear (θ). Therefore shear modulus is defined as the ratio of shear stress to the resultant shear strain.

Mathematically, $\text{shearmodulus}\left(\mu \right)=\frac{\text{shearstress}}{\text{shearstrain}}=\frac{\text{Force}/\text{Area}}{\text{Extension}/\text{Originallength}}$

The formula below is used to calculate the Shear Modulus (µ)

$\mu =\rho {V}_{s}^{2}$ (9)

or

$\mu =\frac{E}{2\left(1+\sigma \right)}$ (10)

where *E*, μ and ρ are Young’s modulus, Shear modulus and average density (2200 kg/m^{3}) respectively.

3. Location and Geology of the Study Area

The study area shown in Figure 1, lies between latitudes 4˚45' and 5˚15'N and between longitudes 7˚45' and 8˚30'E in the Niger Delta region of southern Nigeria. It covers an area of about 1110.1 km^{2}. It is located in an equatorial climatic region that is characterised by two major seasons: the rainy season (March-October) and dry season (November-February) (Evans et al., 2010; George et al., 2010a, 2010b). The dry season is a period of extreme aridity characterized by excruciating high temperatures that could climb to 35˚C. The area has been severely affected by the current global climatic changes in such a way that there have been shifts in both the upper and lower boundaries of these climatic conditions (Martínez et al., 2008; Rapti-Caputo, 2010; Riddell et al., 2010; Wagner & Zeckhauser, 2011; Farauta et al., 2012).

Geologically, the study area is located in the Tertiary to Quaternary Coastal Plain Sands (CPS) (otherwise called the Benin Formation) and Alluvium environments of the Niger Delta region of southern Nigeria as shown in Figure 1. The sediments of the Benin Formation consist of interfringing units of lacustrine and fluvial loose sands, pebbles, clays and lignite streaks of varying thicknesses while the alluvial units comprise tidal and lagoonal sediments, beach sands and soils (Emujakporue & Ekine, 2009; Reijers et al., 1997; Nganje et al., 2007) mostly found in the southern parts and along the river banks. The CPS is covered by thin lateritic overburden materials with varying thicknesses at some locations but is massively exposed near the shorelines. The CPS constitutes the engineering foundations in the area. It comprises poorly sorted continental (fine-medium-coarse) sands and gravels that alternate with lignite streaks, thin clay horizons and lenses at some locations (Essien & Akankpo, 2013; Essien et al., 2014). The coastal plain sand covers 80 percent of the area and forms the major aquiferous and foundation zones of the study area. Thin clay horizons and

Figure 1. Map showing the study area location and general geology of Akwa Ibom State of Nigeria (a) and the nine (9) central Local Government Areas in Uyo Senatorial district that the study area situates (b).

lenses disturb the horizontal and vertical systems that make up the subsurface (Emujakporue & Ekine, 2009). The area is generally porous and permeable and this is usually interrupted by clay-sand sequence at different depths (Okwueze, 1991; Ekwueme & Onyeagoda, 1985).

4. Materials and Methods

In this study, a 24-channel signal enhancement seismograph, geophones, sledge hammer and a metal plate (source) for generating seismic wave were used. The electromagnetic geophone which were in direct contact with the earth, transformed the seismic energy generated by the source to electrical voltage which is a function of velocity. The mechanically generated seismic disturbances sensed by the geophones were received and recorded by a seismograph cascaded with the geophones (Reynolds, 1997). The double seismic source, in which one of them was for shear wave source and the other, compressional wave source, has two set of geophones for the S-wave and P-wave respectively (Kesavula, 1993). The generated energy penetrated into the subsurface and refracted off at various interfaces corresponding to the geological boundaries and consequently returned to the surface at later time to be picked up by the geophone (Kearey & Brooks, 1991). The seismic wave received by the geophone was converted into electrical pulse and was amplified by the preamplifier.

This plot was printed out from the seismograph from which arrival times were obtained. The refraction time-distance measurement at the surface of the ground led to the determination of V_{p}/V_{s} ratio and other principal properties of the near surface rocks. P-wave and S-wave velocities were obtained from seismic refraction survey covering a spread line of 50 m, with 2 m geophones spacing in the foundation layer of Uyo and its environs of Akwa Ibom State, Southern Nigeria. The arrival times of recorded signal (seismogram) were picked and plotted against the offset distance using IX Refrax and Pickwin software programmes.

5. Results and Discussion

The summary of the geoelastic parameters such as shear modulus (μ), Young modulus (*E*), Bulk modulus (*K*) is presented in Table 1, while the detailed parameters and the geographic coordinates taken from global positioning system (GPS) radar are presented in Appendix Table A1. The estimation of these parameters was necessary in order to evaluate the geotechnical strength of the foundation layers. Shear modulus (μ) values ranged from 0.21 × 10^{8} to 0.63 × 10^{8}

Table 1. Summary of layer parameters in the study area.

N/m^{2} with an average of 0.43 × 10^{8} N/m^{2} for layer 1 and 0.78 × 10^{8} to 2.55 × 10^{8} N/m^{2} with an average of 1.40 × 10^{8} N/m^{2} for layer 2. The higher values of Shear moduli increases the cohesion of the topsoil.

Using 3D contour maps in Figure 2(a) and Figure 2(b), the distribution of the shear modulus or modulus of rigidity in the study area was examined. Generally, the topsoil under study has shear modulus ranging from 21,200 kPa - 255,000 kPa with an average value of 93,200 kPa. Comparing these to the table of shear modulus (Table 2) generated by Sawangsuriya (2012), the geoelastic parameters of the engineering foundation fall within dense sands and gravels as well as silty sands. The Ultimate Bearing capacity depends on the soil type, moisture content, compaction and the amount of uniformity of the formation.

(a)(b)

Figure 2. 3-D blanked contour map of layer 1 shear modulus (a) and layer 2 shear modulus (b) showing their distributions in the study area.

Table 2. Typical ranges of values of shear modulus for different types of soil formations (after Sawangsuriya, 2012).

Soils with high arenaceous formations have a higher bearing capacity than soil with high argillaceous materials (Atat et al., 2013). The range indicates that the topsoil under study can support load that is being subjected to shear stress, provided the materials within the layer are well compressed. The considered foundation layers are cohesionless, gritty and therefore not susceptible to creep, erosion and failures provided proper compaction is done during road construction.

Figure 3(a) and Figure 3(b) represent the 3D display of Young’s modulus in the study area. The Young’s modulus (*E*) values for layer 1 ranged from 0.58 × 10^{8} to 9.56 × 10^{8} N/m^{2} with an average of 2.32 × 10^{8} N/m^{2} and 2.15 × 10^{8} to 6.99 × 10^{8} N/m^{2} for layer 2 with an average of 3.84 × 10^{8} N/m^{2}. The higher values of Young’s modulus as seen increases the elasticity of the soil. The contour maps display of Young’s moduli indicates that the topsoil has high degree of rigidity and cannot be subjected to creep and failure in a linearly compressed condition. On the average, Young modulus increases from the north towards the southern part of the study area. In this study, dense and silty sand formations characterised by physical and elastic properties that are nearly homogenous suggest that in many locations, the topsoil does not have the attribute of creeping or failing except compaction is not adequately uniform.

Figure 4(a) and Figure 4(b) represent the 2D blanked contour map of layer 1 and layer 2 bulk moduli in the study area. Bulk modulus (*K*) values for layer 1 ranged from 0.76 × 10^{8} to 2.22 × 10^{8} N/m^{2} with an average of 1.52 × 10^{8} N/m^{2} while that of layer 2 ranged from 2.77 × 10^{8} to 8.95 × 10^{8} N/m^{2} with an average of 4.93 × 10^{8} N/m^{2}. The bulk modulus, describes the elastic properties of a solid or fluid when it is under pressure on all surfaces. The present topsoil in this work has in positive bulk modulus. This signifies that when pressure is imposed and then removed, the formation will not be deformed. The high value of Bulk modulus is very desirable because it will not deform the soil, instead increases the compaction of the soil.

In a similar study conducted in Eket, Akwa Ibom State the following results were obtained: Young’s modulus *E *(−40.772 × 10^{8} to 16.1481 × 10^{8} N/m^{2}), Bulk Modulus *K* (−0.7964 × 10^{8} to 7.6896 × 10^{8} N/m^{2}) and Shear modulus μ (1.3751 × 10^{8} to 7.0209 × 10^{8} N/m^{2}) (Essien et al., 2016). The findings in Eket revealed that a reasonable thickness of the top layer was porous, swampy, air-filled and weak. From the findings, wildcat engineering use of the top soil and weathered soil for construction should be discouraged.

(a)(b)

Figure 3. 3-D blanked contour map of layer 1 Young’s modulus (a) and layer 2 Young’s modulus (b) showing their distributions in the study area.

Figure 4. Blanked 2D contour map of layer 1 bulk modulus (a) and layer 2 bulk modulus (b), showing their distributions in the study area.

6. Conclusion

The results of refraction technique have been used to characterise the cohesionless (friable) topsoil in parts of Uyo and its environ, Akwa Ibom State, Nigeria. Parameters determined were: Shear Modulus whose averages and ranges for layers 1 and 2 were 0.43 × 10^{8} N/m^{2} and 1.40 × 10^{8} N/m^{2}; 0.21 × 10^{8} to 0.63 × 10^{8} N/m^{2} and 0.78 × 10^{8} to 2.55 × 10^{8} N/m^{2} respectively. The averages and ranges of the Young Modulus for layers 1 and 2 were also determined as 2.32 × 10^{8} N/m^{2} and 3.84 × 10^{8} N/m^{2}; 0.58 × 10^{8} to 9.56 × 10^{8} N/m^{2} and 2.15 × 10^{8} to 6.99 × 10^{8} N/m^{2} respectively. The averages and ranges of the bulk Modulus for layers 1 and 2 were estimated as 1.52 × 10^{8} N/m^{2} and 4.93 × 10^{8} N/m^{2}; 0.76 × 10^{8} to 2.22 × 10^{8} N/m^{2} and 2.77 × 10^{8} to 8.95 × 10^{8} N/m^{2} respectively. The higher values of Shear moduli increase the cohesion of the topsoil. The higher values of Young’s modulus as seen increase the elasticity of the soil. The high value of Bulk modulus is very desirable because it will not deform the soil, instead increasing the compaction of the soil.

Appendix

Table A1. Summary of layer parameters and elastic properties in the study area.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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