1 Introduction

Researchers have been working tirelessly to establish a relationship between hydraulic parameters and current flow in a given formation with respect to groundwater studies. The flow of electric current and water in porous media are controlled by the same mechanisms [1]. The groundwater flow follows Darcy’s law while electric current flow follows Ohm’s law and both are influenced by the same factor, i.e., the aquifer porosity saturated with water. Theoretically, both laws deal with the conservation of mass and charge for water and electric current, respectively [2]. Thus, researchers employ many empirical equations to determine hydraulic parameters from the geophysical measurements [3]. One of the early breakthroughs in this subject matter is the work of Archie (1942), in which he suggested an empirical equation to determine the porosity of a given media [4]:

$$ {\uprho } = {\upalpha }.{\uprho }_{\text{w}} .{\upvarphi }^{{ - {\text{m}}}} $$
(1)

where ρ and ρw represent the aquifer’s bulk and water resistivity, respectively, \( {\upvarphi } \) represents the porosity and α and m are given coefficients, which represent saturation and cementation factors linked to the medium and grain-shape or pore-shape factors, respectively. In the case of m, the more compacted the sediment layers are, the higher the value of m [5]. Archie’s equations [4] have set the practical foundation for understanding and calculating liquid saturation in a medium [6]. However, the equation was designed on the assumption that the aquifer in question is clean and clay-free. Should there be any change in respect to that assumption on which the original equation was based, this would render Archie’s equation null and void [1, 7]. Researchers use only the one dimension (1D) vertical electric sound (VES) in the determination of hydraulic parameters from electrical resistivity measurement [1,2,3]. The major limitation of the 1D approach is its disregard for horizontal changes in the resistivity [8]. It will be difficult to satisfy the requirement of real world geological conditions with this assumption, especially in areas with complex geology as demonstrated by research that 2D and 3D techniques provide more reasonable results [8, 9]. Therefore, this research aims at determining the aquifer’s hydraulic parameters on Kapas Island (Malaysia) using 2D resistivity technique.

2 Materials and Methods

It has been established that the hydraulic conductivity and permeability of rocks, as well as the groundwater velocity, strongly depend on porosity [2, 10] while porosity depends on the intrinsic formation factor [4] (Eq. 2).

$$ F = \frac{{\rho_{sat} }}{{\rho_{w} }} $$
(2)

where the formation factor is represented by F (dimensionless); \( {\uprho }_{\text{sat}} \) and \( {\uprho }_{\text{w}} \) represent the resistivity of aquifer’s bulk (Ωm) and the resistivity of water (Ωm), respectively [5]. To reduce the error in porosity estimation, bulk resistivity values that were measured using 2D resistivity were utilized based on Eq. (3) [1]:

$$ \rho_{w} = \frac{{10^{4} }}{{\sigma_{w} }} $$
(3)

where \( \rho_{w} \) represents the resistivity of water (Ωm); while σw is the electrical conductivity of water (μmhos/cm) [15]. The results were then used to calculate the formation factor. The results were then added in the straight line form of a simplified Bussian equation [11] for calibration. These outputs were afterward integrated with the actual resistivity data and the water resistivity both measured from the field to calculate the porosity of the profile area through Eqs. (4)–(6):

$$ {\upsigma }_{0} \cong {\upsigma }_{\text{w}} {\upvarphi }^{{{\text{m}}/\left( {1 - {\text{m}}} \right)}} $$
(4)
$$ \frac{1}{\text{F}} = {\upsigma }_{0} /{\upsigma }_{\text{w}} = {\upvarphi }^{{{\text{m}}/\left( {1 - {\text{m}}} \right)}} $$
(5)
$$ {\text{F}} = \frac{1}{{{\upvarphi }^{{{\text{m}}/\left( {1 - {\text{m}}} \right)}} }} $$
(6)

Plotting the values of log10 F against (1/log10 φ) in a linear coordinate system gives us the relationships shown in Eqs. (7) and (8):

$$ \log_{10} F = 0.5626\left( {\frac{1}{{\log_{10} \varphi }}} \right) + 1.8546 $$
(7)
$$ \log_{10} \varphi = \frac{0.5626}{{\left[ {\log_{10} F - 1.8546} \right]}} $$
(8)

where F is the formation factor as above, \( \varphi \) is the porosity, σ0 and σw represent the bulk conductivity of the aquifer and water conductivity, respectively, and m is the cementation factor, while \( 0.5626 \, and \, 1.8546 \) are constants derived from soil and rock samples used for calibration. The results were used to calculate the permeability via the Kozeny Eq. (9):

$$ {\text{k }} = \left( {\frac{{d^{2} }}{180}} \right)*\left( {\frac{{\varphi^{3} }}{{(1 - \varphi )^{2} }}} \right) $$
(9)

where d is the grain size and φ is the porosity of the aquifer [12]. Subsequently, the hydraulic conductivity values were determined based on Nuttings’ equation by [13] and [14] known as the Kozeny-Carman-Bear Eq. (10):

$$ K = k\frac{\rho g}{\mu } $$
(10)

where k represents the permeability, and fluid dynamic viscosity is represented by μ (kg/ms), while fluid density and gravitation force that influence the movement are represented by ρ (kg/m3) and g (m/s2), respectively [1, 3]. For this research, values of (0.000798 kg/ms) for μ, (995.71 kg/m3) for ρ, and (9.81 m/s2) for g, all at a temperature of 30 °C, were chosen. This is to reflect the actual situation in the field as most of these parameters were believed to be influenced by temperature [15] (Fig. 1).

Fig. 1
figure 1

A straight line graph showing the relationship between the apparent formation factor and fractional porosity based on the Bussian equation

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3 Results

3.1 Results and Discussion

The profile G (Fig. 2 G1) is oriented east to west towards the sea and it is located at the center of an alluvium deposit with mixtures of sand, coral, shale and a little bit of clay. The resistivity of the line ranges from 4 to 300 Ωm, which is within the acceptable limit of alluvium [16], while the porosity section of the profile line (Fig. 2 G2), is mostly within the range of 0.38–0.52. The hydraulic conductivity values of the line mostly fall within the range of 30–65 md−1 (Fig. 2 G3). The area shows low resistivity (200–400 Ωm) which corresponds with higher hydraulic conductivities (20–40 md−1). The aquifer’s depth or thickness from the resistivity image was 26 m on average; therefore, the volume is 7.8 × 106 m3. Thus, the storage capacity was calculated using Eq. (11):

Fig. 2
figure 2

2D profile lines of A to I (1) resistivity; (2) porosity; (3) hydraulic conductivity

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$$ V = V_{\text{T}} \varTheta $$
(11)

where V represents the theoretical exploitable volume (m3); whereas VT is considered to be the total volume of the aquifer (7.8 × 106 m3); and Θ reflects the aquifer’s average porosity which is found to be 40% [17]. Thus, the storage capacity of the main aquifer is (3.12 × 108 m3).

3.2 Validation

The borehole that was used for the pumping test was 6.4 m deep, and the measured water level before the pumping test was found to be 3.16 m. The transmissivity obtained by the pumping test was found to be 123 m2d−1, and the aquifer thickness is 3.24 m (i.e. thickness of water column). The hydraulic conductivity was calculated using

$$ T = K{\text{h}} $$
(12)

where T is the transmissivity; K is the hydraulic conductivity and h is the aquifer thickness [1]. The hydraulic conductivity based on the pumping test is 38 md−1; while the hydraulic conductivity result based on geophysics and the Bussian equation was found to be 40 md−1, based on the calculation of the average hydraulic conductivity via Eq. (13):

$$ K = \frac{{ka_{1} + ka_{2} + ka_{3} \ldots ka_{n} }}{A} $$
(13)

where K is the average hydraulic conductivity, “k” is the hydraulic conductivity of area “a1, a2, a3 … an” and “A” is the total area of the section. The result was found to be very close to that of the pumping test (just a difference of 2 md−1). This value of hydraulic conductivity is also in agreement with the established values by many researches [5, 9, 18].

4 Conclusions

This research aimed at determining the hydraulic properties of an aquifer using 2D resistivity measurements and has successfully achieved its aim. The Bussian equation was used to determine porosity values. The porosity was then used in the Kozeny-Carman-Bear equation to determine hydraulic conductivity. The hydraulic conductivity that was extracted from resistivity data was compared to the pumping test data and the results demonstrate only a difference of 2 md−1. This suggests that the methodology was successfully executed and the results are believed to be promising. Thus, this technique will open a new research window giving scientists an additional advantage when viewing a complex and true hydrogeological system in a 2D or even 3D mode. In this way, this will help to improve the researchers’ ability to come up with a sustainable management plan for any given aquifer system in a way that was not possible before.