Introduction

For many communities worldwide, especially in arid and semi-arid areas, groundwater is an essential source of water supply since it offers a dependable source for drinking, agriculture, and industrial uses (Mohammed et al. 2022b). The petrophysical and hydrogeological properties of groundwater aquifers are the primary factors influencing the potentiality of groundwater aquifers; hence, they must be thoroughly understood to manage groundwater water resources effectively. The ability of water to flow through an aquifer is determined by its hydraulic conductivity (Khalil et al. 2022), which is one of the fundamental properties used to evaluate the productivity of groundwater aquifers. A higher hydraulic conductivity means that water can flow more easily through the aquifer, which can benefit water supply sustainability. The hydraulic conductivity is dependent not only on rock matrix properties but also on the properties of the saturated fluid. The hydraulic conductivity is typically measured by analysis of pumping tests which are controlled experiments designed for determination of high producing areas (Oli et al. 2022). Estimating hydraulic conductivity using pumping tests is costly and time-consuming; besides that, the obtained conductivity is an average estimation for the whole aquifer thickness. Alternatively, geophysical well logging methods give a continuous estimate of hydraulic conductivity which is more beneficial for groundwater modeling (Szabó et al. 2015a, b). The use of geophysical well logging as a tool for determining the hydraulic conductivity of groundwater aquifers will be discussed in this research.

Geophysical well logging is a technique that uses various instruments to measure the physical properties of rocks and fluids in a borehole (Szabó 2018). It is commonly used in groundwater prospecting to delineate and characterize groundwater aquifers. Geophysical well logs such as spontaneous potential (SP), natural gamma ray (GR) and electrical resistivity (RS) are common tools used in hydrogeological investigation. Spontaneous potential and natural gamma rays are mainly used to study the lithological variation and estimation of the volume of shale presented in the aquifer, while the electrical resistivity log is used to predict the water saturation and salinity. In this research, Csókás (1995) method is used to estimate the hydraulic conductivity, primarily based on the porosity and resistivity of water and formation (Szabó et al. 2015b). Based solely on well-logging data, Csókás (1995) method is sophisticated approach developed to estimate the hydraulic conductivity of unconsolidated aquifer units in eastern Hungary. Additionally, this method can determine critical flow velocity and provide information about the optimal discharge rate. In this study, the hydraulic conductivity obtained for geophysical well logs are interpolated with surface electrical sounding to reveal the vertical and horizontal distribution of hydraulic conductivity. Vertical electrical sounding technique is widely employed to estimate the hydraulic conductivity based on the analogy between groundwater and electrical current movement (Heigold et al. 1979; Niwas and De Lima 2003; Mohammed et al. 2023b, a). The calculated parameters using this approach offer useful data for groundwater investigation and groundwater supply sustainability.

This research further estimated the hydraulic conductivity using exploratory factor analysis. Factor analysis is a statistical technique commonly used to interpret geophysical well-logging data (Asfahani 2014). It allows the identification of patterns or relationships in the data that may not be immediately apparent and reduces the complexity of the data by identifying a smaller number of underlying factors or variables that explain the majority of the variation in the data. It helps to better understand the subsurface and develop more effective exploration and production strategies. In geophysical well-logging, factor analysis can identify the most important geophysical parameters related to subsurface rock and fluid properties (Szabó et al. 2021). This includes parameters such as resistivity, porosity, and permeability, which are important for understanding the hydrogeological characteristics of the subsurface.

The estimation of hydraulic conductivity using the conventional approach, for instance, pumping test or laboratory analysis, is often expensive and time-consuming. Furthermore, the data coverage is of poor quality since the petrophysical and hydrogeological parameters change by order of magnitudes in a relatively short distance. As a result, this research aims to detect the vertical and horizontal variation of hydraulic conductivity using an integrated geophysics-based approach using Csókás (1995) method, multivariate factor analysis, and vertical electrical sounding technique. This is the first international application of Csókás (1995) method since it is only applied in Hungarian testing sites. This approach is beneficial for regional hydrogeological investigation. It gives a reliable estimation of the hydrogeological parameters to be used for groundwater management.

Study area

The study area a part of the Khartoum sub-basin and is located in the northern periphery of the Nile rift basin in Bahri city, Sudan (Fig. 1). The region has a semi-arid to arid climate where the temperatures can exceed 40 degrees Celsius in the summer season from May to September (Mohammed et al. 2022a, b). The rainy season, however, is limited to the remaining few weeks of summer. The study area is a component of the Pan-African series that supervised the genesis of various rock units and geological structures. The Pan-African Basement Complex confines the Khartoum sub-basin to the northeast and southwest and defines its bottom limit at a depth of more than 500 m (Köhnke et al. 2017). The basement complex consists of gneisses, schists, and granites, and the depth varies between zero when it crops at the surface, mainly to the north and eastern sides of the area, and reaches up to 500 m in the southern part (Hussein and Awad 2006). The Cretaceous Nubian formation overlies the Precambrian basement rocks. This formation consists of conglomerates, sandstone, and mudstone (Schrank and Awad 1990). The Nubian formation, with considerable thicknesses, is the primary groundwater aquifer in the Khartoum basin which covers more than 30% of the landmass of Sudan. This formation is intruded on by basaltic volcanic rocks. The recent deposits in the study area include windblown and alluvium wadi deposits. The lithology of these recent deposits comprises sand, gravel, and silt of depths ranging from 3 up to 15 m (Haggaz and Kheirallah 1988). Figure 2 shows the geological map of the study area in which the main rock units are presented. In the Khartoum basin, groundwater occurs in the weakly cemented sandstone beds of the Nubian formation under confined to semiconfined conditions. This condition is due to the presence of thin to relatively thick aquitards and aquicludes composed of silts and clays.

Fig. 1
figure 1

The geographical location of the study area in Sudan

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Fig. 2
figure 2

The geological map of the study area showing the main geological units

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Methodology

Csókás method

Csókás (1995) method is a deterministic approach suggested for the estimation of hydraulic conductivity, critical groundwater velocity, and optimal discharge rate solely from well logs. This method can be effectively applied to measure these parameters in the absence of pumping test data and core samples. Csókás method is an experimentally modified version of Kozeny (1927) and Carman (1937) (Kozeny–Carman) method. Kozeny–Carman method has become one of the most widely accepted formulas for the calculation of hydraulic conductivity based on water density (\({\rho }_{w}\)) (g/cm3), porosity (\(\varphi\)) viscosity (\(\mu\)), acceleration of gravity (\(g\)) (cm/s2), and dominant grain size (d) (cm). In the Kozeny–Carman method, the Navier–Stokes equation is satisfied by treating a rock with primary porosity as an assemblage of capillaries and the hydraulic conductivity (K) (cm/s) is measured by (Eq. 1). The Kozeny–Carman method is based on rock sampling and analysis; as a result, the dominant grain size (d) can be obtained from the grain size distribution curve using Eq. 2. In Eq. 2 the d10 and d60 represent the grain diameter at 10% and 60% of the cumulative frequency during the sieve analysis respectively.

$$K= \frac{{\rho }_{w}\mathcalligra{g}}{\mu } \frac{{d}^{2}}{180} \frac{{\varphi }^{3}}{{(1- \varphi )}^{2}}$$
(1)
$$d= \frac{{d}_{10}+{d}_{60}}{2} \sqrt{\frac{{d}_{10}}{{d}_{60}}}$$
(2)

Csókás method is mostly valid in lossy and saturated formations based on the connection between the grain size and the formation factor (F). The formation factor is the ratio of the resistivity of rocks to the resistivity of water in the saturated formations and is calculated by Eq. 3.

$$F= \frac{{R}_{0}}{{R}_{w}}$$
(3)

In the lab, Alger (1966) discovered a direct correlation between the formation factor and grain size of water-saturated sediments and the formation factor of loose sediments was experimentally matched to the effective grain size (d10) derived by sieve analysis as

$${d}_{10}= {C}_{d} log \,F$$
(4)

Cd is the site constant and proposed to be 5.22 * 10–4 for medium to well-sorted sediments with F of less than 10, which is likely to be fulfilled in clastic aquifers. This formula (Eq. 4) is a foundation for linking the well-logging data and hydrogeological parameters.

The dominant grain size is measured using Eq. 2 and the grain size distribution curve. However, several researchers have suggested equations for the calculation of d. The coefficient of uniformity (U = d60/d10), which defines the grain distribution and uniformity of the clastic materials, is connected to the dominant grain size (Eq. 5) as

$$\frac{d}{{d}_{10}}=1.919*lgU+1$$
(5)

The amount of U is negatively correlated with the logarithm of hydraulic conductivity. For poorly sorted materials, U is greater than 5. While for not very poorly sorted sand, the condition of 2 ≤ U ≤ 2.5 is fulfilled. As a result, Eq. 4 takes the following form (Eq. 6)

$$d=1.671* {C}_{d}\mathrm{log}\,F$$
(6)

The volume of shale highly influences the hydraulic conductivity and effective porosity of groundwater aquifers; therefore, the distribution of these parameters must first be understood for an effective estimation of hydraulic conductivity. Archie (1942) proposed an empirical formula (Eq. 7), which was created using laboratory measurements of several sedimentary samples for the estimation of porosity.

$$F=a {\varphi }^{-m}$$
(7)

where a is the coefficient of tortuosity (a ≈ 1) and m is the cementation exponent, which is nearly constant along the geological formations and assumed to be 1.8 to 2.6 for loose sediments. The research conducted by Alger (1966) demonstrated that in aquifers saturated with low mineralized water, besides the porosity, the formation factor also relies on the resistivity of the saturated water. The porosity and formation factors are linked with the tortuosity factor in Eq. 8 (Ogbe and Bassiouni 1978) as

$${a}^{2}= {\left(\frac{{R}_{0}}{{R}_{w}} \varphi \right)}^{1.2}$$
(8)

The effective porosity (\({\varphi }_{e}\)) is calculated using (Schlumberger 1991) equation (Eq. 9) in which the porosity is measured using Archie formula. The shale volume is calculated using a nonlinear equation (Eq. 10) suggested by Larionov (1969) by employing the natural gamma ray intensity (I) log. The gamma intensity is calculated using the linear formula (Eq. 11) of Schlumberger (1984).

$${\varphi }_{e}=\varphi *(1- {V}_{sh})$$
(9)
$${V}_{sh}=0.33 \left({2}^{2*{I}_{\gamma }}- 1\right)$$
(10)
$${I}_{\gamma }= \frac{{GR}_{log}- {GR}_{min}}{{GR}_{max}- {GR}_{min}}$$
(11)

Kovacs suggested an equation for the estimation of hydraulic conductivity by modification of Kozeny-Carman equation (Eq. 1), and the resulting formula (Eq. 12) appeared as

$$K= \frac{1}{5}\frac{\mathcalligra{g}}{\upsilon } \frac{{\varphi }^{3}}{{\left(1- \varphi \right)}^{2}} {\left(\frac{d}{\alpha }\right)}^{2}$$
(12)

where \(\upsilon\) is the kinematic viscosity (m2/s) of groundwater and expressed as a function of formation temperature; therefore, the ratio of acceleration of gravity and kinematic viscosity (\(\mathcalligra{g}/\upsilon\)) appeared in Eq. 12 is equal to 5.517 * 104 Ct (m/s), and Ct is a constant depending on formation temperatures. It can be calculated as Ct = 1 + 3.37 * 10–2 T + 2.21 * 10–4 T2, where T is the formation temperature (C°). Another form of KC equation (Eq. 13) is published by Pirson (1963) to calculate the permeability of the formation by taking the specific surface of the rocks (S, m2/m3) into account as

$$K= \frac{1}{5}\frac{\mathcalligra{g}}{\upsilon } \frac{{\varphi }^{3}}{{\left(1- \varphi \right)}^{2}} {\left(\frac{1}{a*S}\right)}^{2}$$
(13)

In the comparison of Eq. 12 and Eq. 13, an identical equation (Eq. 14) can be derived as

$${\left(\frac{d}{\alpha }\right)}^{2}= {\left(\frac{1}{a*S}\right)}^{2}$$
(14)

Gálfi and Liebe (1981) compiled several empirical relationships between the hydraulic conductivity of sands and gravels and their particle's electric resistance. Electric current is primarily transmitted on the surfaces of particles rather than via the voids between grains in freshwater-bearing sediments. Thus, the relationship between the resistivity and the particle surface is inverse. The specific surface can be estimated by making the assumption that the sedimentary rock is made up of spherical particles as

$$S=6 \frac{\left(1- \varphi \right)}{d}$$
(15)

The integration of Eq. 6 and Eq. 15 provides the following equation (Eq. 16) as

$${S}^{2}= \frac{36 {\left(1-\varphi \right)}^{2}}{{\left(1.671 {C}_{d}*log \frac{{R}_{0}}{{R}_{w}}\right)}^{2}}$$
(16)

The Csókás method is based on the assumptions of Eq. 8, Eq. 12, Eq. 14, and Eq. 16. Consequently, the hydraulic conductivity can be measured (Eq. 17) as

$$K= {C}_{k }\frac{{\varphi }^{3}}{{\left(1- \varphi \right)}^{4}} \frac{{\left(log \frac{{R}_{0}}{{R}_{w}}\right)}^{2}}{{\left(log \frac{{R}_{0}}{{R}_{w}}\right)}^{1.2}}$$
(17)

where Ck is the constant of the proportionality and Ck = 855.7 CtCd2. The hydraulic conductivity is measured in m/s. The distinctiveness of Csókás method resulted from the direct estimation of the parameters in Csókás equation only from geophysical well logs; as a result, a continuous estimate of hydraulic conductivity along a borehole can be provided.

Factor analysis

Factor analysis as a multivariate statistical method is also employed to verify CSM. Factor analysis is a dimensionality reduction technique in which the big dataset is simplified to a comparatively small number of factors that can be used to characterize the formation by developing connections between these factors and the petrophysical and hydrogeological parameters. In this research, the original data was the logging data of spontaneous potential (SP), natural gamma ray (GR), and deep resistivity (R), and the intended parameters to be estimated are the shale volume and hydraulic conductivity. In the first stages, due to the fact that the logging was conducted using different probes and thus different measurements unit, the well logs are standardized (Eq. 20), integrated into a data matrix (D) (Eq. 21), and decomposed by the model of factor analysis (Eq. 22)

$$D_{il} \,\, = \,\,\frac{{\left( {D_{il} - \overline{D}_{il} } \right)}}{{\sqrt {\frac{1}{N}\,\sum\nolimits_{i - 1}^{N} {\left( {D_{il} - \overline{D}_{il} } \right)} } }}$$
(20)
$$D= \left[\begin{array}{ccc}{SP}_{1}& {GR}_{1}& {RS}_{ 1}\\ {SP}_{2}& {GR}_{2}& {RS}_{2}\\ \vdots & \vdots & \vdots \\ {SP}_{i}& {GR}_{i}& {RS}_{i}\\ \vdots & \vdots & \vdots \\ {SP}_{n}& {GR}_{n}& {RS}_{n}\end{array}\right]$$
(21)
$$\boldsymbol{\hat{D} }=F{W}^{T}+E$$
(22)

where \({\boldsymbol{\hat{D} }}_{{\varvec{i}}{\varvec{l}}}\) is the n-th scaled data of the l-th observed well log,\(\overline{D}_{i}\) is the average value of raw data of the l-th well log (L is the number of borehole geophysical tools and N is the number of measuring points in the processed depth interval). F stands for the N by M factor score matrix (M is number of the extracted factors), W for the L by M factor loading matrix, E for the N by L matrix of residuals, T is the matrix transpose operator. The first factor accounts for the largest variance in the data, and the other subsequent components account for a comparatively smaller variance. The matrix of factor loadings, which essentially measures the level of correlation between the factors and actual information, provides the specific weights of each data type. Given the fact that the factors are statistically independent, the correlation matrix of the measured data can be expressed (Eq. 23) as

$$R= \frac{1}{N} {D}^{T}D=L{L}^{T}+\Psi$$
(23)

where Ψ is the diagonal matrix of particular variances. When Ψ = 0, the problem can be resolved by solving an eigenvalue problem, which is the same as principal component analysis; if not, the factor scores are measured by the maximum likelihood method and the following objective function (Eq. 24) is optimized to jointly estimate L and Ψ (Jöreskog 2007) as

$$\Omega \left(L,\Psi \right)=tr {(R,L{L}^{T}-\Psi )}^{2}=min$$
(24)

An orthogonal transformation of factor loadings is typically used to clearly interpret factor (Lawley and Maxwell 1962). Using the varimax technique, the factors in this study were rotated (Kaiser 1958). The factors scores can be measured based on the solution of Eq. 25 by assuming linearity (Bartlett 1937) as

$${F}^{T}={({L}^{T}{\Psi }^{-1}L)}^{-1}{L}^{T}{\Psi }^{-1}{D}^{T}$$
(25)

Pumping test analysis

The hydraulic conductivity estimated from geophysical well logs in three groundwater wells is validated using pumping test data and exploratory factor analysis. The pumping test analysis was performed based on the assumption of Cooper Jr and Jacob (1946) to calculate the transmissivity (T) and hydraulic conductivity (K) of the groundwater aquifers. Cooper Jr and Jacob (1946) method is frequently applied to both multi and single wells assuming that the flow is transient, and the aquifer is confined, isotropic and homogenious (Gomo 2019). Consequently, it is applied in this study in the partially penetrated single well setting since the aquifer is fulfilling the boundary conditions of the method. This method involves fitting a straight line in a semi-logarithmic scale of a plot between the time since pumping started versus drawdown (s) to determine the average drawdown (∆s) and transmissivity. In this research, the test duration lasted from 150 to 200 min until the steady-state condition was fulfilled. The transmissivity and hydraulic conductivity using Cooper Jr and Jacob (1946) method are measured by Eq. 26 and Eq. 27 as

$$\Delta s= \frac{2.3 Q}{4 \pi T}$$
(26)
$$T=K*b$$
(27)

Vertical electrical sounding

In this research, 9 VES measurements were conducted using Schlumberger configuration with maximum electrode spacing (AB) of 900 m. In the Schlumberger configuration, the depth of penetration increases as the electrode spacing increases. The apparent resistivity is acquired by multiplying Schlumberger geometric factor and the electrical resistance. The obtained apparent resistivity is processed using IPI2WIN software. This software applies one-dimensional geophysical inversion using the damped least square method through which the observed data is compared to a synthetic model. The acceptability of the resulting model is based on the fitness criteria between the observed and calculated apparent resistivity curves (Bobachev 2002). In this study, geological maps and reports from previous surveys were integrated with geophysical data for reliable interpretation of the vertical electrical sounding curves. The primary goal of conducting VES measurements is to interpolate the petrophysical and hydrogeological parameters obtained from the geophysical well logging.

The hydraulic conductivity is measured using the product of the 1D inversion of the VES curves, namely, the true resistivity of the subsurface layers. In this investigation, an experimental equation developed by Heigold et al. (1979) (Eq. 28). The estimation of hydraulic conductivity is based on the resistivity of the sandy aquifer layer (Raq) as

$$K=386.4 {R}_{aq}^{-0.93283}$$
(28)

Results and discussion

Hydraulic conductivity by Csókás method

The hydraulic conductivity is measured using Csókás method in three boreholes scattered in the study area. Csókás method is based on the analysis of geophysical well logs, and in this study, three well logs, including Spontaneous potential (SP), natural gamma ray (GR), and electrical resistivity logs (RS). The primary goal of the well logging investigation was to design the groundwater wells. These logs are obtained and digitized to fulfil the aim of the recent investigation. The obtained logs are combined to estimate the layer parameters and give insight into the zone parameters. The layer parameters, including porosity (\(\varphi\)), shale volume (Vsh) vary dramatically along the boreholes, while the zone parameters are almost constant. The logging data is linked to the layer parameters by means of experimental equations. For instance, Archie (1942) formula is used to measure the formation factors and then linked to formation porosity. The resistivity of the pore fluid used for the estimation of formation factor was obtained from the hydrochemical analysis of groundwater samples obtained after the well design. Larionov's (1969) equation is used to detect the effective porosity based on the gamma-ray intensity to avoid the overdetermination of the shale volume. The aquifer in the study area is confined and fully saturated; consequently, the water saturation is set to unity (100%). The groundwater occurs in the weakly cemented sandstone layer of the Cretaceous Nubian formation. The previous investigation indicated that the depth of groundwater in northern Khartoum state varies between 60 to 150 m. The hydraulic conductivity is validated by the pumping test method in which Cooper-Jacob (1946) is applied under the given boundary conditions.

Borehole KH1 is located in the northwestern part of the study area in the vicinity of the Nile River, and the aquifer ranges in depth between 95 to 133 m. The total depth of the borehole is 133 m, thus, it is partially penetrating the groundwater aquifers since the aquifer thickness is up to 200 m (Schrank and Awad 1990). The lithology description of the borehole shows that the aquifer is in confined conditions overlain by clay and dry sand. Figure 3 shows the well logs and parameters used to estimate the vertical variation of the hydraulic conductivity along the borehole. The aquifer showed a homogeneous SP, RS, formation factor and porosity. This is likely due to the homogeneous nature of the aquifer. The measured hydraulic conductivity of the aquifer using Csókás method ranged from 0.5 to 1.7 m/day with an average value of 1.1 m/day. The hydraulic conductivity gradually changed from the upper to the lower zone of the aquifer, which is attributed to the increase in rock compaction. The variation in hydraulic conductivity depends on the rocks' petrophysical parameters and the groundwater's hydrochemical properties. However, in this study, the groundwater samples represented the hole thickness of the aquifer. The hydraulic conductivity of Csókás method is validated with pumping test data. The data is analyzed using Cooper-Jacob (1946) method. The aquifer is pumped for 100 min with a discharge rate of till the study state is reached. A transmissivity of 123 m2/day is obtained, and as the aquifer thickness is 38 m, the hydraulic conductivity is approximated as 3.2 m/day. The result of the pumping test is shown in Fig. 4. The influence of the partial peneration of the well is clearly observed on the pumping data, resulted in the deviation of the drawdown measurements after 20 min of pumping. The diviation is due to the difference in the hydraulic properties of the penetrated and non-pentrated parts of the aquifer (Wu et al. 2017). Cooper-Jacob (1946) method does not directely account for partiall pentration, however, the results are minimally affected by this condition (Halford et al. 2006). Since the hydraulic conductivity generally varies by orders of magnitude in short distances, the agreement between Csókás method and pumping test is acceptable considering that the pumping test measures the overall hydraulic conductivity while Csókás method gives a continuous estimation along the borehole.

Fig. 3
figure 3

The observed well logs in KH1 borehole in which SP stands for spontaneous potential, GR is the gamma-ray intensity, RS represents the electrical resistivity log, F is the formation factor, and K is the hydraulic conductivity measured with Csókás method

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Fig. 4
figure 4

The analysis of the pumping test data in KH1 borehole using Cooper-Jacob (1946) method

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DR2 borehole is situated in the eastern part of the study area with a total thickness of 220 m. The lithological log of the borehole indicated that the sandstone aquifer lies over a highly resistive layer of silicified sandstone and is overlain by a reasonably thick mudstone layer. The thickness of the aquifer is 141 m and ranges from a depth of 66 to 207 m. Figure 5 shows the well logs and parameters used to estimate the vertical variation of the hydraulic conductivity along the borehole. In this borehole, the aquifer shows heterogenous layer parameters due to the variation in the sandstone properties. This was indicated by fluctuation in GR, SP, RS logs. As a result, the variation in the petrophysical properties is observed. The aquifer is intercepted by shaly sandstone intervals, which are likely the main cause of the variation of petrophysical and hydrogeological parameters. In this log, the hydraulic conductivity ranged from 0.6 to 4 m/day, with an average value of 2.3 m/day. The vertical variation in the hydraulic conductivity is because the hydraulic conductivity is highly sensitive to the variation in petrophysical parameters such as porosity and shale volume. The lower part of the aquifer shows low conductivity since it is associated with high shale volume. As usual, the hydraulic conductivity is validated with pumping test analysis. The result of the pumping test indicated that the transmissivity of the aquifer is 683 m2/day (Fig. 6); consequently, the hydraulic conductivity is 4.8 m/day, indicating acceptable agreement with Csókás hydraulic conductivity Fig. 7.

Fig. 5
figure 5

The observed well logs in DR2 borehole in which SP stands for spontaneous potential, GR is the gamma-ray intensity, RS represents the electrical resistivity log, F is the formation factor, and K is the hydraulic conductivity measured with Csókás method

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Fig. 6
figure 6

The analysis of the pumping test data in DR2 borehole using Cooper-Jacob (1946) method

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Fig. 7
figure 7

The observed well logs in HA3 boreholes in which GR is the gamma-ray intensity, RS represents the electrical resistivity log, F is the formation factor, and K is the hydraulic conductivity measured with Csókás method

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HA3 borehole is located in the southwestern part of the area with a total depth of 270 and an aquifer thickness of 160 m. In this borehole, the aquifer is in confined condition overlaid by mudstone layer, and within the aquifer, an exchange between mudstone and sandstone is observed from a depth of 160 to 175 m. The vertical variation in the petrophysical and hydrogeological parameters is illustrated in Fig. 6. The small variation in the vertical conductivity is observed in this borehole. However, a noticeable decrease is observed at a depth of 160 m due to the mudstone layer. The hydraulic conductivity varied between 4.9 to 1.6 m/day, with an average of 3.3 m/day. The results are also validated with the pumping test and illustrated in Fig. 8. The transmissivity of the aquifer obtained from the pumping test is 733 m2/day; consequently, a hydraulic conductivity of 4.6 m/day is recorded.

Fig. 8
figure 8

The analysis of the pumping test data in HA3 borehole using Cooper-Jacob (1946) method

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It can be indicated that the measured hydraulic conductivity of the pumping test shows a close agreement with that of Csókás method. However, higher values are obtained by the pumping test since it measures the overall hydraulic conductivity or the amount of transmitted per unit distance of the aquifer without giving due account to the variation in the petrophysical properties of the aquifer. The main advantage of Csókás method over the pumping test is that it can give a continuous estimation of the petrophysical and hydrogeological parameters, which can be useful in the identification of the groundwater high-producing zones. The use of this approach is beneficial in groundwater prospecting and management. For instance, modeling the vertical distribution of the hydrogeological parameters in groundwater flow and contaminant transport is crucial since it highly influences the resolution and accuracy of the resulting models. Furthermore, this geophysical approach is cheaper compared to the pumping test methods. With the integration of the surface geophysical method, the Csókás approach can effectively map the vertical and horizontal variation of the hydrogeological parameters. In conclusion, Csókás method is highly advised for hydrogeologists to delineate the potential groundwater zones or use it as part of complex hydrogeological modeling efforts.

Hydraulic conductivity by factor analysis

This research uses multivariate statistical factor analysis to measure the hydraulic conductivity from geophysical well logging data. All the measured logs are combined in a data matrix, and the derived factors are linked to the petrophysical and hydrogeological parameters. The first factor shows the greatest variance in the data set and thus can be linked to the most influential parameters in all geophysical logs. Consequently, by regression analysis, Szabó et al. (2014) linked the first factor (F1) resulting from factor analysis and the shale volume of the aquiferous layer, and the results showed a strong exponential connection. As a result, and due to the fact that the hydraulic conductivity is inversely proportional to the shale volume in the sedimentary aquifers with primary porosity, Szabó (2015) proved a direct linear relationship between the logarithm of hydraulic conductivity and the first factor and suggested the following equation (Eq. 29) for estimation of hydraulic conductivity based of the factor analysis as

$$-lg \left(K\right)=a{F}_{1}+b$$
(29)

where \({F}_{1}\) is the first factor obtained from factor analysis, while a and b are the site-specific constants obtained for the linear regression between the first factor and the logarithm of the hydraulic conductivity. The values of these constants vary from formation to another based on variations in the petrophysical parameters.

In this investigation, a linear regression between hydraulic conductivity obtained from Csókás method and the first factor is performed to obtain the site-specific constant of the central Sudan hydrogeological system. The analysis is performed in three boreholes. In KH1 borehole (Fig. 9), the correlation coefficient (R2) of − 0.77 is indicated between the logarithm of hydraulic conductivity and first factor. For DR2 borehole (Fig. 10), a high negative correlation (-0.84) is revealed, while for HA3 borehole, a relatively high correlation of -0.76 is indicated. As a result, three equations (Eq. 30, 31, and 32) for the estimation of hydraulic conductivity based on factor analysis are obtained as

Fig. 9
figure 9

The regression between the first factor and logarithm of the hydraulic conductivity of Csókás method in KH1 borehole

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Fig. 10
figure 10

The regression between the first factor and logarithm of the hydraulic conductivity of Csókás method in DR2 borehole

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$$-lg \left(K\right)=-0.29{F}_{1}+0.04$$
(30)
$$-lg \left(K\right)=-0.4{F}_{1}+0.25$$
(31)
$$-lg \left(K\right)=-0.03{F}_{1}+1.1$$
(32)

This method can be used as an alternative method when it is validated with different approaches, such as core and laboratory measurements. The core data is unavailable on our borehole; however, the obtained equations can be efficiently used to continuously estimate hydraulic conductivity along the aquifer. Fig. 11.

Fig. 11
figure 11

The regression between the first factor and logarithm of the hydraulic conductivity of Csókás method in HA3 borehole

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Interpolation of hydraulic conductivity

In this study, the hydraulic conductivity measured with Csókás method is interpolated with the VES measurement to comprehensively characterize the groundwater aquifer in the study area. The VES curves (Fig. 12) revealed five geoelectrical layers interpreted with the aid of geological and lithological logs. A profile trending east–west consists of four VES stations (VES 3, 4, 5, and 6) is constructed to map the subsurface lithology of the study area and illustrated in Fig. 13. The top layer is interpreted as superficial deposits with an average thickness of 7 m and resistivity of 120 Ωm. The second layer with low resistivity and average thickness of 35 m is referred to as clay followed by fine sand layer of thickness of 20 m. The fourth layer is interpreted as mudstone layer followed by thick saturated sandstone. The aquifer layer has an average resistivity of 160 Ωm under confined conditions.

Fig. 12
figure 12

Examples of the 1D inversion of the VES data in (a) S1 and (b) S3 locations

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Fig. 13
figure 13

Geological cross section obtained from the interpreteation of the VES data

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Using the results of VES inversion, the hydraulic conductivity is measured using the formula suggested by Heigold et al. (1979). This equation is based on the analogy of groundwater flow and the movement of electric current in porous formations. The hydraulic conductivity is measured in different VES stations. Unlike Csókás method, the hydraulic conductivity with VES data is similar to that of the pumping test since it gives an average estimation of the properties. The mesured hydraulic conductivity ranged from 3 m/day in S3 location to 3.74 m/day in S9 with an average value of 3.37 m/day. Similar to the result of Csókás approach, the hydraulic conductivity gradually decreases from the northern to the southern part of the study area. The results show close agreement with the result of Csókás method and pumping test analysis.

In this study, to have a consistent estimation of hydraulic conductivity throughout the study area, in this study, the hydraulic conductivity obtained from Heigold et al. (1979) approach is connected to that obtained from Csókás method by means of regression analysis. This is achieved by estimation of hydraulic conductivity using Heigold et al. (1979) from geophysical well logs and comparing it to the hydraulic conductivity of Csókás method. The hydraulic conductivity is measured by employing the resistivity log since Heigold et al. (1979) is based only on the resistivity of the aquifer layer. As a result, a continuous estimation of hydraulic conductivity is obtained. The regression between the hydraulic conductivity between the two approaches revealed a high exponential correlation (Fig. 14). A correlation coefficient of 0.9997 between the two conductivities is obtained in KH1 borehole (Fig. 14a), while for DR2 a coefficient of 0.9986 is obtained (Fig. 14b). A lower correlation coefficient of 0.732 (Fig. 14c) is indicated in HA3 borehole, which is likely due to the influence of the shale intervals within the aquifer. The main disadvantage of Heigold et al. (1979) approach is that it is highly sensitive to the presence of shales, leading to the overdetermination of the hydraulic conductivity. However, the obtained results are consistent with that of Csókás method and pumping test. The regression analysis between Csókás and Heigold conductivities suggested three local equations that can be used to have a consistent estimation and interpolate the result to reveal the horizantal of the hydraulic conductivity within the aquifer. The equations for KH1 (Eq. 33), DR2 (Eq. 34), and HA3 (Eq. 35) borehole are

$${K}_{CS}=0.0084{{K}_{HG}}^{2.7857}$$
(33)
$${K}_{CS}=0.0258{{K}_{HG}}^{2.8918}$$
(34)
$${K}_{CS}=0.0187{{K}_{HG}}^{2.801}$$
(35)

where KCS and KHG are the hydraulic conductivity from Csókás and Heigold methods respectively.

Fig. 14
figure 14

The correlation between the hydraulic conductivity of Csókás and Heigold methods in (a) KH1, (b) DR2, and (c) HA boreholes

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Based on the regression-based equations, a universal equation is suggested to have a unified estimation of the hydraulic conductivity. The universal equation based on the average values of regression equations can be used for the estimation of hydraulic conductivity of the shaly aquifer in the central Sudan hydrogeological system. The estimated conductivities based on the suggested approach are shown in Table 1. The average hydraulic conductivity ranged from 0.39 to 3.3 m/day. The lowest value is indicated in S3, while the highest conductivity is observed in HA3 log. The overall estimated values of the hydraulic conductivity show a high consistency with the previous hydrogeological investigations conducted by Algafar et al. (2011) and Elkrail and Adlan (2019). The results of the two approaches are combined, and the horizontal variation of the hydraulic conductivity is revealed. The areal variation in hydraulic conductivity is illustrated in Fig. 15. The distribution of hydraulic conductivity shows a gradual change from the northern to the southern part of the study area. This is likely due to the change in the lithological facies since the northern parts are considered to be a transition zone between the Precambrian basement rocks basin and the Cretaceous sedimentary basin (Zeinelabdein and Elsheikh 2014; Mohammed et al. 2023c). The aquifer in the northern part of the study area is nearly isotropic compared to the southern part of the study area where the horizontal to vertical hydraulic conductivity is variable. This likely due to to variations in the petrophysical and geological characteristics of the subsurface which may cause systematic aquifer pattern change (Bonsor et al. 2017). Nevertheless, the hole aquifer system can be described as moderately anisotropic (Mohammed et al. 2023a).

Table 1 The result of VES inversion and their correspondence hydraulic conductivities using Heigold and Csókás methods
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Fig. 15
figure 15

The spatial variation of hydraulic conductivity in the study area

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According to (Csókás 1995) aquitards are associated with a hydraulic conductivity of 0.0009 m/day, while good aquifers have hydraulic conductivity higher than 0.09 m/day. As a result, the groundwater aquifer in the study area can be considered a highly productive aquifer and optimal for groundwater development. However, Csókás method proved its superiority over Heigold method because of complex criteria that accounts for the variation of the petrophysical properties of the groundwater aquifers. Table 2 gives a detailed comparison of these methods based on the results of the recent investigation. Overall, this integrated geophysics-based approach using Csókás and Heigold methods is valuable for hydrogeologists since it provides a quick and reliable estimate of hydraulic conductivity which is important for sustainable water resource management.

Table 2 Comparison of Heigold and Csókás methods for estimation of hydraulic conductivitybased on the recent investigation
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Conclusions

This research explored the possibility of using a geophysics-based approach to detect the vertical and horizontal variation of hydraulic conductivity in Bahri city, Sudan. This approach is based on geophysical well logging, direct current geoelectrical method, and exploratory factor analysis validated by analysis of the pumping test data. The conclusions of this research can be summarized in the following.

  • Csókás method is successfully used to estimate the vertical variation of hydraulic conductivity based on the geophysical well logging data, and the results showed a close agreement with pumping test data. The Csókás method is useful in areas where it is difficult to obtain direct measurements of hydraulic conductivity or when the hydraulic conductivity changes rapidly with depth. This method is suited for well-sorted shallow aquifers with a formation factor of less than 10.

  • Multivariate factor analysis is furthermore used to investigate the hydraulic conductivity based on the available well logs. The principle of using factor analysis for aquifer characterization is based on the correlation between the extracted factor, which explains the highest variance (i.e. first factor) in the data set and the petrophysical and hydrogeological parameters. In this research, an inverse relationship between the first factor and hydraulic conductivity is revealed. Thus, local equations are suggested to give a continuous estimation of hydraulic conductivity throughout the aquifer. This approach proved efficient in detecting the vertical distribution of hydraulic conductivity. However, in this study, another independent method, for instance, core data and laboratory measurements, is required to calibrate and validate the results of factor analysis.

  • Electrical resistivity method employing vertical electrical sounding (VES) technique is employed to interpolate the hydraulic conductivity obtained from Csókás method. In the first stage, the hydraulic conductivity is measured based on the VES data by employing the analogy between groundwater flow and electrical current movement. Further, the obtained hydraulic conductivity is upscaled to the hydraulic conductivity of Csókás method by regression analysis to have a consistent estimation throughout the study area. As a result, the spatial distribution of hydraulic conductivity is revealed.

  • Overall, the obtained results are reliable and consistent with the previous investigation in the study area. Therefore, it can be concluded that this integrated geophysics-based approach is valuable for hydrogeological studies since it reduces the time and expense of the regional hydrogeological investigation. This approach gives a continuous estimation of hydraulic and petrophysical parameters and is thus highly advised to be used as inputs for groundwater modeling efforts.