Introduction

Most ground inhomogeneity and anisotropy induce strong directional effects on resistivity distributions and sounding curves realized from measurements (Caglar and Avsar 2007). Lithological diversity, a dynamic process, which causes electrical anisotropy of soils/rocks, is an effect of alternating beddings of arenites and argillites within the subsurface. The arenite–argillite sequence can be regarded as a medium of anisotropic diversity characterized by vertical and horizontal resistivities. Such a medium has a conductivity sensitivity, which determines the shapes of geo-electric curves. Conductivity sensitivity is the ease of soil layer/bed to conduct current that passes through them. Specifically, geo-electric curves give insights into the local geology and structure of subsurface units penetrated by the injected current (Ekanem et al. 2019; Thomas et al. 2020). As the current finds penetrates the earth, information on the primary geo-electrical indices, such as layer thickness (h), layer resistivity (ρ) and their corresponding depths (D) can be determined. Layer thickness is generally estimated from the depths of the layer parameters, which are initially gauged from the available VES curves. These primary geo-electric indices are estimates of importance and are used in evaluating transverse resistance \(\left( {\rho \cdot h} \right)\) and longitudinal conductance \(\left( {\frac{h}{\rho }} \right)\) according to Hasan et al. (2019). These secondary geo-electric indices are called Dar Zarrouk parameters (DZP). Natural soils of various types have different electrical properties due to the composition, structure, water content, and temperature (George et al. 2010; Obiora et al. 2015). When the water content is below its optimum, the electrical conductivity of soils increases nonlinearly and the variation rate increases dramatically. However, when the water content is optimum, the degree of saturation, or dry density increases to a certain value and the electrical conductivity tends to be a constant. In addition, soil electrical conductivity increases with the increase in temperature, and it is observed that the electrical conductivity decreases with the increase in the number of wetting–drying cycles (Wei et al. 2013). Anisotropy is simply a condition where the earth resistivity/conductivity and hence the measured apparent resistivity/voltage is a function of the path of measurement. This anisotropy is common in clay, slate and shale, which are marked by divergent lineation or platy fabric (Greenhalgh et al. 2010). Anisotropy can exist in a macroscopic scale (Bala and Cichy 2015). Here, bands or a series of layers of divergent isotropic materials behave as they are single and equivalent anisotropic units. When anisotropy exists in a particular geologic units or minerals, it is referred to as micro-anisotropy or intrinsic anisotropy, which peculiarly depends on the crystal symmetry or texture of the material (Bala 2011). Besides the micro- and macro-anisotropies, rock cleavage, joints and fracturing, can also produce structural anisotropy peculiar to layers with alternating conductivities (rock and joint fill) Hobbs et al. (2009) and Bala and Cichy (2015). Such pseudo-anisotropy arises when the thickness of the individual isotropic bands or units is small relative to the electrode separation used for the measurement (Greenhalgh et al. 2010). The earth electrical specific resistance/resistivity deviates with the course or direction. These deviations are amply necessitated by the diversity of lithological units seen in the degree of electrical anisotropy. Permeability may be anisotropic when it changes in soils/rocks in different directions (Aissaoui et al. 2019). The effect of anisotropy exists in geologic units and if ignored, would lead to error in the interpretation of ground resistivities and the geologic structures. Some publications, which address the effects of anisotropy on surface resistivity measurements, include: Matias and Habberjam (1986), Matias (2002), Ekanem (2020). Borehole information remains the reliable means of constraining resistivity data interpretation (Asten 1974; Olasehinde and Bayewu 2011). This implies that the subsurface exhibits different electric properties in different directions (Karanth 1987; Karnkowski 1999; Yeboah-Forson and Whitman 2013). This study aims at exploring the veracity of electrical technique and geologic information in delineating the electrical conductivity sensitivity of sedimentary layers for effective understanding of the contribution of electrical resistivity in hydrogeophysical studies.

Theoretical insights

The diversity of specific resistance is based on 2 intrinsic anisotropic parameters, which are anisotropic coefficient \(\left( \lambda \right)\) and the average resistivity \(\left( {\rho_{{\text{m}}} } \right)\) associated with the units of soils or rock considered (Aissaoui et al. 2019). Anisotropic diversity exists in many sedimentary formations due to permeability diversity (Bala and Cichy 2015). In homogeneously isotropic media of vertical resistivity \(\left( {\rho_{{\text{v}}} } \right)\) and horizontal resistivity \(\left( {\rho_{{\text{h}}} } \right)\) coefficient of anisotropy is equal to 1 as \(\rho_{{\text{m}}} = \rho_{{\text{h}}} = \rho_{{\text{v}}}\) and lies between 1 and 2 for homogeneously anisotropic media characterized by anisotropy (Caglar and Avsar 2007). The coefficient of anisotropy in homogeneous and anisotropic geological unit is greater unity, because resistivity is always greatest in the transverse direction. A homogeneously isotropic unit filled with water, air or rock-field fractures is equally considered to demonstrate anisotropy. Practically, electrical current flowing perpendicularly to the soil bedding planes is largely opposed by high resistance (vertical resistivity) due a scenario that looks like series combination of resistors. Along the bedding planes, opposition to the current flow becomes comparatively reduced due to high resistance (horizontal resistivity) caused by a scenario that looks like a parallel combination of resistors. The former scenario has a higher resistivity than the later and hence explains the reason why vertical resistivity is proudly greater than the horizontal resistivity. In a formation that is homogeneous and anisotropic, with dipping angle, \(\theta\) between the layer and the vertical to the direction of measurement, the electric field generates a potential difference \(\left( U \right)\) at an arbitrary point of the medium, which decreases inversely proportionally to the distance \(\left( r \right)\) from the source of the direct current, I as stated in Eq. 1

$$U = \frac{{I \cdot \rho_{{\text{m}}} }}{{4\pi \cdot r \cdot \left( {1 + \left( {\lambda^{2} - 1} \right) \cdot \cos^{2} \theta } \right)^{\frac{1}{2}} }}$$
(1)

where \(\rho_{{\text{m}}}\) is the geometric mean resistivity, the factor \(\lambda\), is anisotropic coefficient defined by \(\rho_{{\text{h}}}\) and \(\rho_{{\text{v}}}\), respectively, representing horizontal/longitudinal (minimum) resistivity and vertical/transverse (maximum) resistivity. Anisotropic coefficient \(\left( \lambda \right)\) is related with \(\rho_{{\text{h}}}\) and \(\rho_{{\text{v}}}\) in Eq. 2:

$$\lambda = \left( {\frac{{\rho_{{\text{v}}} }}{{\rho_{{\text{h}}} }}} \right)^{\frac{1}{2}}$$
(2)

The geometric mean resistivity \(\left( {\rho_{{\text{m}}} } \right)\) is gauged from the horizontal and vertical resistivities as given in Eq. 3. Practically, \(\rho_{{\text{v}}}\) is the electric resistivity measured vertically to the bedding, while \(\rho_{{\text{h}}}\) is the electric resistivity measured along the bedding.

$$\rho_{m} = \left( {\rho_{v} \cdot \rho_{h} } \right)^{\frac{1}{2}}$$
(3)

Based on the formation values of \(\rho_{m}\) and \(\lambda\) the electric potential difference depends on the strike direction, \(\theta\) along which the potential difference changes with \(r\). In collinear array, Kunz and Moran (1958); Dachnov (1967); Greenhalgh et al. (2009), theorized that a uniform and anisotropic medium is characterized by the measured resistivity \(\rho \left( \theta \right)\) which depends on the anisotropic parameters given in Eq. 4:

$$\rho \left( \theta \right) = \frac{{\lambda \cdot \rho_{{\text{h}}} }}{{\left( {1 + \left( {\lambda^{2} - 1} \right) \cdot \cos^{2} \theta } \right)^{\frac{1}{2}} }} = \frac{{\rho_{{\text{m}}} }}{{\left( {1 + \left( {\lambda^{2} - 1} \right) \cdot \cos^{2} \theta } \right)^{\frac{1}{2}} }}$$
(4)

The expression above shows that \(\rho {}_{m} = \lambda \cdot \rho {}_{{\text{h}}}\). When \(\theta = 0{^\circ }\) and \(\theta = 90{^\circ }\), respectively, Eqs. 5 and 6 result from Eq. 4

$$\rho_{\theta }^{v} = \frac{{\lambda \cdot \rho_{{\text{h}}} }}{\lambda } = \rho {}_{{\text{h}}}$$
(5)
$$\rho_{\theta }^{h} = \lambda \cdot \rho_{{\text{h}}}$$
(6)

Interestingly, Eqs. 5and 6 indicate that \(\rho_{\theta }^{v} < \rho_{\theta }^{h}\) by a factor of \(\lambda\). This means that the apparent resistivity \(\left( {\rho_{\theta }^{v} } \right)\) measured normal or vertical to the strike orientation is less than apparent resistivity \(\left( {\rho_{\theta }^{h} } \right)\) measured along the strike direction. The observed inference is in opposition to the fact that the true resistivity \(\left( {\rho_{v} } \right)\) of anisotropic formation, normal to its stratification, is greater than the true resistivity \(\left( {\rho_{{\text{h}}} } \right)\), parallel to the plane of stratification. This condition is referred to as “the paradox of anisotropy” (Aissaoui et al. 2019).

Description of location and geology of the study area

The site of the survey has an area coverage of about 435 km2 within the coastal fringe of Akwa Ibom State, southeastern Nigeria. The locations of soundings fall within latitudes 4°45′–4°35′ N and longitudes 7°30′–8°10′ E in the hinterland off the Atlantic coast (Fig. 1). The area has a semi-temperate climate with distinct seasons: wet and dry seasons. The wet season commences in April and ends in September, while the dry season begins from October and ends in March. The temperature during the dry season is about 2 °C–32 °C. Annual rainfall for the area ranges between 200 and 250 cm, and can ascend to ~ 320 cm at the peak period. The area, characteristically noted for its mangrove swamp is economically rich in crude oil and agricultural productions.

Fig. 1
figure 1

Schematic map showing a the location of Akwa Ibom State, which indicate the study area and b the study area showing the local geology, VES points, borehole cored sample points, borehole locations and the local government boundaries

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The area is geologically described by 3 lithostratigraphic formations. The oldest is Akata Formation (Eocene to recent) according to Peters et al. (1989), Akpan et al. (2013) and Ibanga and George (2016). This Formation exists as pro-delta facies and serves as source rock for crude oil (Short and Stauble 1967). It is opined that the shales of this formation were formed during the initial development stages of Niger Delta Basin progradation and they are compacted and over-pressured with diapiric structures such as shale swells and ridges, which intrude into the overlying younger Agbada Formation. The Agbada Formation occurs throughout Niger Delta clastic wedge as the main reservoir and seal for crude oil accumulation in the basin. It is primarily paralic deltaic front facies with maximum thickness of about 13,000 feet or 4 km (Sort and Stauble 1967). The lithologies consist of alternating system of sands, silts and shales, arranged within about 10–90 feet successions and defined by progressive upward changes in grain size and bed thickness. The depositional environment of the Agbada Formation is interpreted to be fluvial–deltaic (Short and Stauble 1967; George et al. 2015). The base of the formation extends beyond 4.5 km or 4600 feet in some areas and is defined as earliest marine shale, while the surficial parts of the formation consist entirely of non-marine sand deposited in alluvial or upper coastal plain environments during progradation of the delta (Doust and Omatsola 1989). The youngest Benin Formation in which geologic units are considered in this work is the top part of the basin clastic wedge, from the Benin–Onitsha area in the north to the coastline in the south (Short and Stauble 1967). The subsurface structure in the Niger Delta Basin spans from simple rollover faults, multiple growth faults, antithetic faults and collapsed crest faults (Onuoha and Dim 2017). These sediments formed during the Late Eocene to Early Oligocene with the reservoirs mainly controlled by pre-and syn-sedimentary tectonic elements that responded to variable rates of subsidence and sediment supply (Doust and Omatsola 1989). The hydrogeological units here are mainly brownish and believed to be developed from moderately coarse textured alluvium. Habitually, the geologic units have grayish brown, and slightly finer texture occasionally intercalated with grits.

Materials and methods

The employment of vertical electrical sounding (VES) method whose interpretation is constrained by logged borehole information as geophysical prospecting tool is a standard geophysical prospecting method. Fourteen VES data (Fig. 1) were acquired near borehole in the study area to reduce the non-uniqueness problems associated with its interpretation and at the same time to comfortably assess the deeper subsurface sedimentary electrical resistivity information. The Schlumberger electrode configuration was used to assess vertical and longitudinal electrical conductivity distributions at shallower and deeper depths through the use of IGIS signal enhancement resistivity meter SSP-MP-ATS resistivity meter. Even though the resistivity meter has the capability to average up to 32 cycles of values, measurement cycles were truncated after 4 stacks, provided the reading on the liquid crystal display correlates well with standard deviation < 10% when direct currents were artificially injected into the earth (Yadav and Singh 2007; George 2020). Half of the current electrode separation (AB/2) ranged between 1 and 150 m and that of the potential electrode (MN/2) varied from 0.25 to 30 m. The variation in the potential electrode separation was necessary to enhance the input signal strength. When the VES field survey was completed, the VES data were modeled using the RESIST code developed by Vander Velpen and Sporry (1993) and Vander Velpen (1988). Lithological data from nearby wells were used to constrain the 1D inverse models. After a couple of iterations, a satisfactory variation between the observed field data and theoretical data were represented by means of absolute root-mean square error, which was generally found to be less than 10% (Fig. 2).

Fig. 2
figure 2

Correlation of VES curves and the nearby lithology from borehole data

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Results and discussion

Primary and secondary geo-electrical indices

The results of the VES investigation, gave a 1-D electrical resistivity, thickness and depth of geo-electrical layers given in Table 1. The current at its maximum current electrode separations penetrated 4 sedimentary layers characterized with resistivity ranges of 95.2–3455.5 Ωm and mean of 834.8 Ωm in the first layer. The second layer ranged from 8.8 to 3606.6 Ωm with mean of 1327.8 Ωm. The third and fourth layers have ranges of 72.5–1464.5 Ωm with mean 657.2 Ωm and 117.1–4893.0 Ωm with mean 1551.2 Ωm, respectively. The 4th layer was not adequately delineated as current could not penetrate this layer at the maximum current electrode separation. With the use of lithological logs from nearby boreholes, VES data were manually and electrically modeled. The statistics of each of them showed characteristic four geo-electric layers with A, HA, KQ KH and HK cluster of curves. Noticeably, due to the information from wells, resistivity values and sizable thickness of geologic units for accumulation of water in the geologic aquifers, layers 2 and 3 were identified as potential aquifers. The respective inferred thicknesses and depths vary across the sounding points with ranges 0.5–19.6 m, 1.6–56.7 m, 16.1–89.2 m and 0.5–19.6 m, 2.1–76.3 m, 20.5 m for layers 1, 3 and 3. Table 1 indicates thickness soars with depth as the mean values attest. The geologic units penetrated by currents have noticeable electrical sensitivity of varying magnitudes due to intra and inter-bedded lithological diversity (Ibuot et al. 2013; George and Ekanem 2019). The hypothetical model of the subsurface in Fig. 3 identifies 2 distinct resistivities within the conducting earth units. These are the vertical \(\left( {\rho_{{\text{v}}} } \right)\) resistivity and the horizontal resistivity \(\left( {\rho_{{\text{v}}} } \right)\) estimated using Eqs. 7 and 8:

$$\rho_{{\text{v}}} = \frac{{\sum\nolimits_{{{i}}}^{{{n}}} {h_{{\text{i}}} \rho_{{\text{i}}} } }}{{\sum\nolimits_{{{i}}}^{{{n}}} {h_{{\text{i}}} } }}$$
(7)
$$\rho_{{\text{h}}} = \frac{{\sum\nolimits_{{{i}}}^{{{n}}} {h_{{\text{i}}} } }}{{\sum\nolimits_{{{i}}}^{{{n}}} {\frac{{h_{{\text{i}}} }}{{\rho_{{\text{i}}} }}} }}$$
(8)
Table 1 Summary of geophysics survey in the study
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Fig. 3
figure 3

Hypothetical geophysical model of a homogeneous and anisotropic earth due vertically dipping beds or fractures. Transverse/vertical resistivity (\(\rho_{{\text{v}}}\)) is perpendicular to the bedding planes or fractures, while longitudinal/horizontal resistivity (\(\rho_{{\text{h}}}\)) is parallel to the bedding planes or fractures; θ, strike direction

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where \(h_{{\text{i}}}\), and \(\rho_{{\text{i}}}\) are the layer thickness and resistivity, respectively, for a given nth layer. These 2 directional parameters, defined by resistivity and thickness are directly proportional to each other. For homogeneous and anisotropic media, they are linearly related to each other by the square of average coefficient of anisotropy (Fig. 4). For the volume of homogenous and anisotropic medium that current passed through, the adjusted gradient \(\left( {\tan \theta = 0.9509; \, \theta \approx 44^{ \circ } } \right)\) for the line in Fig. 4 is characterized by a highly correlated \(\left( {R = 0.8844} \right)\) expression given below:

$$\rho_{{\text{v}}} = 0.9508\rho_{{\text{h}}} + 340.78$$
(9)
Fig.4
figure 4

A graph showing the transverse/vertical resistivity against longitudinal/horizontal resistivity

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Coefficient of anisotropy

Although the medium has adjusted average approaching unity, the geological formation is not adjudged isotropic, because the estimated values of \(\lambda\) ranged from values 1.00 to 2.12 with 1.22 as the mean value. Besides, the scattering in Fig. 4 shows that the formation in the layers involved is inhomogeneous and anisotropic. The disparity as noted in Fig. 5, is a valid suggested reason for departure of \(\lambda\) from unity. Due to homogeneous and anisotropic disparities in the assessed geological units, classification of anisotropy in the area comparatively spans from low anisotropy \(\left( {\lambda < 1.2} \right)\), moderate anisotropy \(\left( {1.2 < \lambda \le 1.3} \right)\) and high anisotropy \(\left( {\lambda > 1.3} \right)\) with percentage of 64.3, 14.3 and 21.3%, respectively. The lower disparity between the values of \(\rho_{{\text{v}}}\) and \(\rho_{{\text{h}}}\) is tended toward lower values of \(\lambda\) (Fig. 5). Hence, it can be inferred that on the average, anisotropy in the area is intrinsically or microscopically induced by virtue of the consequent sequences of argillaceous intercalations with arenaceous sedimentary units (Greenhalgh et al. 2010). Due to sensitivity of \(\lambda\) to geological diversity, very narrow and typical values of \(\lambda\) are associated with divergent geologic units. Based on the low coefficient of anisotropy \(\left( {\lambda < 1.2} \right)\), which covers 64.3%, the geologic units that current passed through are mainly alluvia (intercalation of clay, silt or gravel of sedimentary origin) according to the range provided by (Hill 1972; Asten 1974; Greenhalgh et al. 2010). Comparatively, moderate range of coefficient of anisotropy \(\left( {1.2 < \lambda \le 1.3} \right)\) and elevated range \(\left( {\lambda > 1.3} \right)\), with respective 14.3 and 21.3% coverage also define the study area. These geologic units are, respectively, described as inter-bedded shale and sandstone as well as shale and slate by Greenhalgh et al. (2010) and are all minor formations as their percentages comparatively reflect. The absence of extremely high values of coefficient of anisotropy \(\left( {\lambda > 2.1} \right)\) indicates that there is no intrusion of metasediments into the young sedimentary age from the deeply seated basement (Greenhalgh et al. 2010). The degree of anisotropy within the unit area is well expressed in the shape of longitudinal conductance versus transverse resistance curves for layers 1–3, respectively, as revealed in Fig. 6a–c. As observed in Fig. 6a–c, the values of DZP soar with depth and this showcases that thickness increases with depth. In 6a, the longitudinal conductance versus transverse resistance distribution is completely discordant with Fig. 6b and c, which are fairly consistent. The fair consistency accounts for cohesivity which is a function of depth of burial of geologic formation (George 2020).

Fig. 5
figure 5

A diagram showing the disparity between vertical/transverse and horizontal/longitudinal resistivities within the range of coefficient of anisotropy

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Fig. 6
figure 6

Diagram showing variation of longitudinal conductivity versus transverse resistance for (a) layer 1, (b) layer 2 and (c) layer 3

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Strike-dependent resistivity

The family of fairly collinear lines in Fig. 7 shows the relationship between resistivity, \(\rho_{{\uptheta }}\) estimated as function of strike angle (Eq. 4) against the estimated geometric mean resistivity, \(\rho_{{\text{m}}}\) of homogeneous and anisotropic formation. The line can be summarily represented as Eq. 10:

$$\rho_{\theta } = a\rho_{{\text{m}}} - b$$
(10)
Fig. 7
figure 7

Graphs showing family of lines for strike dependence resistivity variations against mean resistivity

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where a and b are constants representing slope and intercept. Due to consistency of intrinsic or microscopic anisotropy, the constant a appears to converge to unity as shown in the lines in Fig. 7 for strike directions \(\theta\) arbitrarily chosen as 0°, 5°, 10°, 15°, 20°, 25°, 30°, 60°, 90° and 180°. The constant b becomes smaller as a converges to unity and bigger as a marginally departs from unity. Therefore, the convergence and divergence in physical properties is function of anisotropic coefficient, which determines the strike direction. The segment of the lines with non-genetically consistent points indicate unit of non-geologically equivalent units. The contour map in Fig. 8a and b for vertical and horizontal resistivity shows consistency in the spread. This qualitative revelation is consistent with the quantitative analogy given in Fig. 4. The coefficient of anisotropy is Fig. 8c, shows some divergent attributes when compared to Fig. 8a and b. The displays in Fig. 8 also demonstrate that majority of the observed attributes are fairly correlated, while marginal component diverged as indicated in the color code. These graphic displays in contour map give the image of the homogeneous and anisotropic diversity, which gives room to false interpretation when in the absence of borehole information.

Fig. 8
figure 8

Diagram showing contour distributions of a \(\rho_{{\text{v}}}\), b \(\rho_{{\text{h}}}\) and c \(\lambda\) in the stud area

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Conclusion

The VES technique and lithological information from boreholes were engaged to characterize the lithological diversity in homogeneous and anisotropic systems constituting the aquifer units and their overlaying layers in the hydrographic region of Akwa Ibom State. Based on the coefficient of anisotropy, the assessed volume of sedimentary formation is classified into low anisotropy \(\left( {\lambda < 1.2} \right)\), moderate anisotropy \(\left( {1.2 < \lambda \le 1.3} \right)\) and high anisotropy \(\left( {\lambda > 1.3} \right)\) with alluvium (64.3%), inter-bedded shale and sandstone (14.3%) and shale and slate (21.3%). The respective percentage of coverage indicates that alluvium (intercalation of clay, silt or gravel of organic origin) is the dominant formation, while inter-bedded shale and sandstone as well as the shale and slate are minor geologic units. The dominant trend of alluvium indicates that the homogeneous and anisotropic units assessed are intrinsic or microscopic in nature due to the ensuing sequences of argillites intercalating with arenites in the existing hydrogeological units that the impressed current passed through. The sensitivity of current injected into the geologic units is noticeable as the delineation of fairly geologically consistent units is resolved. The absence of extremely high values \(\left( {\lambda > 2.1} \right)\) indicates that there is no intrusion of metasediments into the young sedimentary age (Greenhalgh et al. 2010). The results show that the plot between the strike-dependent resistivity versus geometric mean resistivity can be used as the basis for determining the degree of consistency of homogeneous and anisotropic geologic compositions. From this plot, it was possible to conclude the slopes of families of curves converge to unity with reduced intercepts, while slopes that marginally depart from unity are marked with high intercepts. Both the quantitative (graphic) and qualitative (contour) results portend the thin possibility of having anisotropy free geologic units. Since the effect of anisotropy really exists in geologic units either prominently or obscurely and can falsify the measured ground resistivities and geologic structure during interpretation ignored isotropic homogeneous and isotropic medium only exist in principle but not in practice. The revelations of prominent and obscured anisotropies, which coexist in hydrogeological units as revealed in Figs. 6, 7 and 8, suggest that every ground resistivity measurement and interpretation of geologic structures be accompanied or constrained by borehole information for reliability of results to be guaranteed.