Vertical electrical sounding was carried out in twenty-one (21) locations of the study area employing Schlumberger electrode configuration to obtain the apparent resistance and other field data. Apparent resistivity (ρa) values were calculated from the measured field data. Manual and computer modeling techniques help in reducing the field data to its suitable geological model [32]. WinResist Software was employed to generate the values of the geoelectric layers resistivity, thickness, and depth. The thickness and resistivity values were used to estimate some of the hydraulic properties which were employed in the analysis of aquifer flow units.
Flow unit determination
The field data obtained from the study area was analyzed using the stratigraphic modified Lorenz plot (SMLP) method. This method is a graphical tool which uses various data including the geological framework, storage capacity, and flow capacity. SMLP is a cross-plot of the cumulative flow capacity and cumulative storage capacity of the aquifer, derived from the aquifer geophysical properties. The stratigraphic modified Lorenz (SML) plot is one of the most important techniques that are applied for flow unit (FU) discrimination. It is based on core data, porosity, and permeability which are multiplied by their representative bed thicknesses (h), and the obtained results are called storage capacity (φh) and flow capacity (Kph), respectively [13, 23, 25]. The cumulative flow capacity and storage capacity are calculated using the Maglio [23] mathematical models as shown in Equation 1 and 2.
$${\left({K}_ph\right)}_{cum}={k}_{p1}\left({h}_1-{h}_0\right)+{k}_{p2}\left({h}_2-{h}_1\right)+\dots +{k}_{pi}\left({h}_i-{h}_{i-1}\right)/\sum {k}_{pi}\left({h}_1-{h}_{i-1}\right)$$
(1)
$${\left(\varphi h\right)}_{cum}={\varphi}_1\left({h}_1-{h}_0\right)+{\varphi}_2\left({h}_2-{h}_1\right)+\dots +{\varphi}_i\left({h}_i-{h}_{i-1}\right)/\sum {\varphi}_i\left({h}_i-{h}_{i-1}\right)$$
(2)
where kp is permeability (mD), h is thickness, and (Kph)cum is cumulative flow capacity.
φ = fractional porosity, (φh)cum = cumulative storage capacity
Some of the hydraulic properties, estimated which enhanced the determination of the flow units, include hydraulic conductivity, porosity, permeability, and tortuosity. Values of hydraulic conductivity were estimated using Equation 3 according Heigold et al. [14].
$$K=\frac{386.40}{{\rho_a}^{0.93283}}$$
(3)
Porosity is a property that depends on the grain composition of the soil, and pressure to which it is exposed. Porosity and tortuosity values were determined using Marotz [24] and The Netherland Organisation [31] equations respectively as shown in Equations 4 and 5.
$$\varphi =25.5+4.5 InK$$
(4)
where K is hydraulic conductivity
$$\tau ={\left( F\varphi \right)}^{\frac{1}{2}}$$
(5)
For an aquifer to be productive, it must be porous and permeable. Permeability of the aquifer layer was estimated following Kozeny [20] and Carman [2] equations shown in Equation 6.
$${K}_p=1014\frac{\varphi^3}{{\left(1-\varphi \right)}^2}\left(\frac{1}{F_s{\tau}^2{S_{gv}}^2}\right)=1014\frac{\varphi^3}{{\left(1-\varphi \right)}^2}\ast \left(\frac{1}{2{\tau}^2{S_{gv}}^2}\right)$$
(6)
S
gv is surface area per unit pore volume ,τ is tortuosity, and Fs is shape factor and it equals 2 for a circular cylinder.
Heterogeneity determination
There are varieties of statistical techniques such as coefficient of variation, Dykstra-Parsons coefficient, and the Lorenz coefficient employed in the quantification of heterogeneity. Dykstra-Parsons coefficient is a permeability model and can be considered as a more statistically robust technique though requiring additional application of statistical methodologies [9]. This study employed Dykstra-Parsons coefficient in quantifying heterogeneity through heterogeneity measures. Heterogeneity measures provide a single value for quantifying samples variability and also provide the ability to compare this variability between different reservoirs. Jensen et al. [19] suggest that heterogeneity measures provide a simple way to assess a reservoir and guide investigations towards more detailed analysis of spatial arrangement and internal structure of a reservoir.
Dykstra-Parsons coefficient (VDP) enables the measurement of water flow performance in layered reservoir by providing the degree of stratification (vertical permeability heterogeneity) and sweep efficiency. VDP values vary from 0 to 1 as classified based on the Dykstra-Parsons coefficient. From the classification, 0 indicates a homogeneous system, between 0 and 0.6 represents small heterogeneity, while values from 0.7 to 1 indicates high to extremely high heterogeneities [21]. The Dykstra-Parsons coefficient was determined using Dykstra and Parsons [5] equation as shown in Equation 7.
$${V}_{DP}=1-{e}^{-\sigma }$$
(7)
where σis the standard deviation.
Figure 3 is a flow chart showing the research methodology.