Introduction

Groundwater is the most reliable and primary source of freshwater in many parts of the world. It constitutes a major source for irrigation, with approximately 40% of the global irrigated areas relying on groundwater (Wyrwoll 2012). However, exploitation of groundwater greater than it’s natural replenishment has led to unremitting depletion of groundwater resources in many parts of the world and India is no exception. According to the United Nations World Water Development Report (WWAP 2015), India is the largest consumer of groundwater in the world. Groundwater accounts for over 65% of irrigation water supply and 85% of drinking water supply in India (World Bank 2012). Overutilization of groundwater in the northwestern states of India (e.g., north Gujarat, western Rajasthan and parts of Haryana) has caused drastic lowering of groundwater levels (CGWB 2013). A GRACE-based study revealed that groundwater has been exploited for irrigation and other purposes at a rate of 17.7 ± 4.5 km3/year in the northwestern states of Rajasthan, Punjab and Haryana (Rodell et al. 2009). In fact, the problem of groundwater depletion prevails in different parts of India and there is an urgent need for managing this resource in an efficient and sustainable manner.

Hydrogeologic investigation provides fundamental information about the existence of aquifer and nonaquifer formations, which in turn is helpful for proper development and management of groundwater resources. Although some researchers have highlighted the importance of hydrogeologic investigation in the recent past (e.g., Arnold et al. 2001; Maclear 2001; Soupios et al. 2010; Elwaseif et al. 2012), it is not a common practice in developing countries. Accurate estimation of aquifer hydraulic parameters such as transmissivity (T), hydraulic conductivity (K), storage coefficient (S) and leakage factor (B) is indispensable for the successful planning and management of vital groundwater resources. Among these parameters, K (and thereby T) is the key parameter that controls groundwater recharge and plays a vital role in groundwater-flow analysis (Fetter 2000; Schwartz and Zhang 2003). In recent years, researchers have highlighted the importance of field-pumping tests for estimating hydraulic parameters of aquifer systems (e.g., Samuel and Jha 2003; Li and Qian 2013; Li et al. 2014; Machiwal et al. 2017). Besides the pumping test and slug test (most reliable and widely used techniques), there exist other field methods like the floodwave-response technique and tidal-response technique for estimating aquifer parameters (e.g., Reynolds 1987; Jha et al. 2004, 2008; Jha and Singh 2014). These methods, however, require specific infrastructure as well as significant resources, time and labor; hence, a relatively simple and cost-effective method such as grain-size analysis (GSA) can be used for estimating aquifer parameters. A couple of studies have been carried out in recent years, wherein the values of K estimated by the GSA method, pumping-test method and permeameter method are compared (Cheng and Chen 2007; Song et al. 2009; Pliakas and Petalas 2011; Lu et al. 2012; Rosas et al. 2015). Lu et al. (2012) compared the K of shallow streambed sediments estimated by 12 GSA methods with K determined by the permeameter method. They concluded that the Saurebrei, Terzaghi and Shepherd GSA methods yielded K values reasonably close to those obtained by the permeameter test. On the other hand, Cheong et al. (2008) estimated K by pumping test, slug test, numerical modeling and grain-size analysis. It was found that the geometric mean of K estimates from GSA methods is 3–4 times higher than that obtained from the pumping test; however, the geometric mean of the K value obtained by the slug test is 1–2 times lower than that obtained by the pumping test and GSA methods.

Past studies confirmed that K of the aquifer material is a function of characteristic grain diameter and grain-size distribution of porous media (Freeze and Cherry 1979). Various empirical relationships have been developed based on grain-size distribution characteristics for estimating hydraulic conductivity (Hazen 1892; Slichter 1898; Beyer 1964; Vukovic and Soro 1992; Alyamani and Şen 1993; Barr 2001; Chapuis 2004). These grain-size analysis (GSA) methods vary from simple to somewhat complex calculations depending upon the type and number of input data required to estimate hydraulic conductivity. The attractiveness of GSA methods lies in their simplicity, cost effectiveness and ease of use. As such, the K values estimated by GSA methods are less reliable than in situ methods such as the pumping test or slug test because disturbed geologic samples form the basis for GSA methods. In addition, the GSA methods can estimate only aquifer hydraulic conductivity (K) and transmissivity (T), and fail to estimate other hydraulic parameters of the aquifer systems such as storage coefficient (S), specific yield (Sy) and leakage factor (B). Grain-size analysis, however, provides important information about subsurface hydraulic conductivity, which is very useful for groundwater studies, such as the design of pumping tests or any preliminary studies in the absence of field-measured hydraulic conductivity values. Further, in the absence of pumping test data, GSA methods provides an inexpensive source of supporting data, which represent an attractive starting point for hydrogeologic investigation, especially where sand and gravel subsurface formations are of concern (Devlin 2015). This is the reason that the overall GSA method is still used despite inherent limitations and has been employed by many researchers in the recent past (e.g., London et al. 2001; Hwang and Powers 2003; Cronican and Gribb 2004; Odong 2007; Cheong et al. 2008; Ishaku et al. 2011; Vienken and Dietrich 2011; Lu et al. 2012; Rosas et al. 2014; Rosas et al. 2015; Devlin 2015; Machiwal and Jha 2016).

Based on the review of literature, it can be inferred that hydrogeologic investigations in developing countries are mostly done by using indirect approaches such as geophysical methods or by visual interpretation of randomly collected geologic samples. Unlike earlier studies on hydrogeologic investigation, in this study, an attempt has been made to demonstrate a scientifically sound methodology for exploring hydrogeology of a study area. On the other hand, the estimation of aquifer parameters by a pumping test is something of a daunting task in developing countries because of economic constraints, inadequate infrastructure and limited knowledge. As a result, the unavailability of aquifer hydraulic parameters is a serious problem in many basins of the world. Given these constraints, this study also evaluates the efficacy of salient GSA methods for estimating aquifer parameters in a cost-effective way. The objectives of the study are: (1) to conduct test drilling and perform grain-size analysis of the collected geologic samples, (2) to construct well logs and analyze stratigraphy, (3) to estimate hydraulic conductivity of subsurface formations (aquifer and nonaquifer layers), and (4) to evaluate the performance of the GSA methods. Thus, this study demonstrates a comprehensive scientific methodology for hydrogeologic and hydraulic investigations at a basin or sub-basin scale. Additionally, this research identifies suitable GSA methods for estimating the hydraulic conductivity of subsurface formations in the lateritic terrains.

Study area

Location and climate

The campus of the Indian Institute of Technology Kharagpur, West Bengal, India, was selected as a study area which is situated at latitude 22°19′10.97″ and longitude 87°18′35.87″ and covers an area of 850 ha (8.5 km2). It is located in West Medinipur district of West Bengal (Fig. 1), which is typically representative of lateritic terrain of eastern India. The experimental farm and its periphery (Fig. 1), occupying an area of 25 ha, were intensively investigated as part of this study for hydrogeologic and hydraulic characterization. Moreover, to verify the findings obtained in the experimental farm and its periphery, similar investigations were carried out at a field located about 6 km south-east of the study area. The field is owned by several farmers and is, thus, named Farmers’ field in this study. The topography of the area is moderately flat with some undulation and the elevation varies from 36 to 58 m above mean sea level (MSL). Soil of the study area is predominantly sandy loam up to 0.5-m depth and sandy clay loam from 0.5 to 2.0 m depth. The climate of the study area is characterized as hot and humid during summer seasons and cold and dry during winter seasons. The minimum temperature usually ranges from 8 to 10 °C in January, whereas the maximum temperature ranges from 37 to 46 °C in May. The relative humidity varies from 20 to 95%. The study area has a mean annual rainfall of about 1,400 mm, most of which (>80%) occurs during the monsoon season (June to early October). The remaining months generally remain dry with some occasional rainstorms.

Fig. 1
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Location map of the study area

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The hydrogeology of West Medinipur district can be divided into two broad groups: (1) fractured/fissured formation, and (2) porous formation. The southern part of this district is predominantly comprised of porous formations having geology of Quaternary alluvium underlain by Tertiary sediments. Tertiary aquifers exit 75–100 m below ground level (bgl). Generally, the aquifers present at depth < 30 m bgl are unconfined in nature and the aquifers present at depth > 60 m are confined in nature (CGWB 2008).

Water problems

The study area’s laterization causes prolonged perching conditions during the rainy season, which triggers significant lateral flow, thereby limiting recharge to groundwater (Garg et al. 2005). Groundwater is the main source of drinking and irrigation water in the study area. Adequate water is available during the monsoon season which is confined to 4–5 months. However, during summer seasons the dug wells usually get dry and the groundwater levels in the relatively deep aquifers go down to 30 m bgl, causing water scarcity in the study area. Thus, adequate water supply in this area has been a major problem particularly during the summer season and it is expected to be acute in the future due to increased population growth and development of this area. Over the years, gradual depletion in the groundwater levels has been of major concern in this region. Hence, hydrogeologic and hydraulic characterization is necessary to manage this fast depleting groundwater resource.

Methodology

Test drilling and lithologic sampling

Test drilling was conducted up to about 90 m depth at eight sites (W1–W8) over the study area (Fig. 1). In Farmers’ field, test drilling was also conducted up to 90 m depth but only at two sites—W9 (observation well) and W10 (pumping well)—due to financial constraints. During drilling, geologic samples were collected at 10-ft (3.05 m) intervals up to the drilling depth at each site over the study area and Farmers’ field; thereafter, nine observation wells (W1–W9) and one pumping well (W10) were installed at these sites. Polyvinyl chloride (PVC) pipes of different diameters (17.78, 7.62 and 5.08 cm) were used for well casings and screens. The well screens (perforated PVC pipes) were placed in the layers having coarser materials. After the completion of installation, all the wells were developed by using an air compressor. Specifications of the observation wells (W1–W9) thus installed, and the two pumping wells (P3, existing) and (W10, installed), are summarized in Table 1.

Table 1 Specifications of the observation wells and pumping wells
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Analysis of geologic samples

Sieve analysis gives information about the grain-size distribution, which affects the porosity and permeability of the aquifer material and, hence, plays an important role in the hydrogeologic and hydraulic characterization of subsurface systems. Sieves of different sizes, varying from 2.00 to 0.05 mm, were used for sieving the geologic samples, identifying percentage of coarse sand, medium sand, fine sand, and silt/clay in each sample (Bouwer 1978). Based on the results of sieve analyses, grain-size distribution curves (GSDCs) were developed for each geologic sample by plotting the ‘particle size’ on the log-scale as abscissa and the ‘percentage finer’ on the arithmetic-scale as ordinate. From these curves, the values of characteristic grain diameters (d10, d17, d20, d50, and d60) and uniformity coefficient (U) were obtained. The value of U indicates the range of grain size present in a geologic sample; the lower value (U ≤ 2) represents uniform or poorly graded geologic samples, whereas a higher value of U indicates nonuniform or well-graded geologic samples (Raghunath 2007). The uniform geologic sample produces an S-shape grain-size distribution curve and has a high and/or low value of d10 and/or d60.

Construction of well logs and stratigraphy analysis

Hydro GeoAnalyst software package developed by Waterloo Hydrogeologic, Inc. (WHI), Canada, was used for constructing well logs for the ten sites. Based on the relative percentage of coarse, medium and fine particles in each geologic sample, the lithologic layers were classified as aquifer layer (AL) and nonaquifer layer (NL). The layer having large percentage of coarse sand and medium sand with/without pea gravel followed by nil or very less percentage of fine sand and/or silt was considered as an aquifer layer. In contrast, the layer having lower percentage of coarse sand and/or medium sand but higher percentage of fine sand and silt with or without murrum (local name of a reddish gravely geologic material) or irregular-shaped gravels was considered as a non-aquifer layer (confining layer). Finally, the stratigraphy analysis of each well log was carried out in order to describe the depth and thickness of different subsurface formations.

Estimation of hydraulic conductivity by grain-size analysis methods

The equation for estimating hydraulic conductivity (K) from grain-size distribution has been obtained from the Darcy-Weissbach equation and its general form can be expressed as follows (Vukovic and Soro 1992):

$$ K=\frac{\rho g}{\mu}\times N\times f(n)\times {d_{\mathrm{e}}}^2 $$
(1)

where K = hydraulic conductivity of the material (cm/s), ρ = density of water (g/ml), g = gravitational constant (cm/s2), μ = dynamic viscosity of water (g/cm/s), N = case-specific constant known as ‘shape factor’, f (n) = function of porosity, and de = effective grain size (cm).

According to Devlin (2015), the values of ρ and μ in Eq. (1) can be calculated as:

$$ \rho =3.16\times {10}^{-8}\times {T}^3-7.0\times {10}^{-6}\times {T}^2+4.19\times {10}^{-5}\times T+0.99985 $$
(2)
$$ \mu =-7.0\times {10}^{-8}\times {T}^3+1.002\times {10}^{-5}\times {T}^2-5.7\times {10}^{-4}\times T+0.0178 $$
(3)

where T = water temperature (°C). For this study, the average temperature of groundwater was found to be 25 °C from the water level logger installed in the observation wells.

The values of N, f (n) and de are dependent on the methods used for grain-size analysis. Vukovic and Soro (1992) formulated an empirical relationship between porosity (n) and coefficient of grain uniformity (U), which can be used to calculate the value of n and is expressed as:

$$ n=0.255\times \left(1+{0.83}^U\right) $$
(4)

where U = coefficient of grain uniformity and is expressed as:

$$ U=\frac{d_{60}}{d_{10}} $$
(5)

Here, d60 indicates the diameter of the soil particles for which 60% particles are finer. Likewise, d10 represents diameter of the soil particles for which 10% particles are finer.

Eight well-known GSA methods (Hazen, Haze Simplified, Slitcher, Beyer, Saurebrei, Kozeny, USBR and Harleman method) were considered for estimating the hydraulic conductivity of geologic samples and their relative performances were evaluated. These methods are concisely described in Table 2. An Excel-based tool was also developed to prepare GSDCs to estimate characteristic grain diameter and to calculate the hydraulic conductivities of geologic samples by GSA methods with the required inputs.

Table 2 Salient empirical formulae for estimating hydraulic conductivity of subsurface formations from grain-size distribution
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Field pumping test

The field pumping test is considered to be a standard procedure for obtaining the aquifer and well parameters. In this study, a pumping test was conducted at the P3 pumping well (on 20 March 2016) and at W10 well (on 28 January 2017) located at the study area and Farmers’ field, respectively. The P3 pumping well was pumped at a constant rate for a duration of 5 h and 40 min and drawdown in the eight observation wells (W1, W2, W3, W4, W5, W6, W7 and W8) were simultaneously measured at a regular time interval until the end of the pumping test. Among these eight observation sites, appreciable drawdown was observed at five sites (W4, W5, W6, W7 and W8), and hence the time-drawdown data of only five sites were used for estimating aquifer parameters of the study area. Moreover, the W10 pumping well located in Farmers’ field was pumped at a constant rate for 7 h and 10 min and the time-drawdown was measured at the nearby observation well W9 (about 71 m away from the pumping well). Therefore, to estimate the aquifer hydraulic parameters at Farmers’ field time-drawdown data obtained at site W9 was used. Discharge of the W10 pumping well was measured with the help of an installed flow meter connected to the discharge pipe. However, the discharge of the P3 pumping well was measured with the help of a calibrated 90° V-Notch fitted with the conveyance channel through which pumped water is distributed into the experimental farm. The discharge of the P3 pumping well was calculated by the following equation (Garg 2011):

$$ Q=\frac{8}{15}\times {C}_{\mathrm{d}}\times \sqrt{2g}\times \tan \left(\frac{\theta }{2}\right)\times {H}^{\frac{5}{2}} $$
(6)

where Q = discharge (m3/s), Cd = coefficient of discharge, g = acceleration due to gravity (m2/s), θ = angle of V-Notch at center (degree), and H = head over the notch (m).

Analysis of pumping-test data

Field measured time-drawdown data obtained from the six sites (five sites at the study area and one site at Farmers’ field) were analyzed to determine hydraulic parameters of the underlying aquifer systems following the standard procedure (e.g., Roscoe and Moss Company 1990). Several nonlinear models such as the Theis model (for confined aquifers), corrected Theis model (for unconfined aquifers without delayed yield), Hantush-Jacob model (for leaky confined aquifers without aquitard storage), Hantush model (for leaky confined aquifers with aquitard storage) and Neuman model (for unconfined aquifers with delayed yield) are available to analyze the time-drawdown pumping-test data. Selection of the appropriate model is crucial in the interpretation of pumping-test data. In this study, initially, Theis-Type Curve method was used to analyze the six sets of pumping-test data; however, matching of the field data with the Theis Type Curve was not found to be satisfactory for all the sites. Thereafter, the Hantush-Jacob model was selected for analyzing the six sets of time-drawdown data using the Walton-Type Curve method, which provided a better match between field-data curves and the Walton-Type Curve. The Hantush-Jacob model is commonly adopted for analyzing the time-drawdown data of leaky confined aquifers, which is expressed as (Hantush and Jacob 1955):

$$ s\left(r,t\right)=\frac{Q}{4\Pi T}\times W\left(u,\frac{r}{B}\right) $$
(7)

where s(r, t) = drawdown at a distance r and at a time t [L], T = transmissivity of the leaky confined aquifer [L2/T], Q = constant pumping rate from the aquifer [L3/T], B = leakage factor [L], and \( W\left(u,\frac{r}{B}\right) \) = well function for the leaky aquifer (Hantush-Jacob well function) which is given by:

$$ W\left(u,\frac{r}{B}\right)=\underset{u}{\overset{\infty }{\int }}\frac{1}{\gamma}\exp \left(-\gamma \frac{r^2}{4{B}^2\gamma}\right) d\gamma $$
(8)

wherein

$$ u=\frac{r^2S}{4 Tt} $$
(9)

where S = storage coefficient of the leaky confined aquifer.

Furthermore, leakage factor (B) is given as:

$$ B=\sqrt{T\times C} $$
(10)

where C = hydraulic resistance [T] of the aquitard and it is given as:

$$ C=\frac{b^{\prime }}{K^{\prime }} $$
(11)

where b′ = thickness of the aquitard [L] and K′ = hydraulic conductivity of the aquitard layer [L/T].

Leakance can be expressed as:

$$ \mathrm{Leakance}=\frac{1}{C} $$
(12)

For analyzing the time-drawdown pumping-test data, field-data curves for individual sites were prepared and they were matched with the Walton-Type Curve using AquiferTest Version3.5 software (WHI 2002). While matching the curves, both the automatic-fit method and professional judgment were employed to obtain site-specific aquifer parameters such as transmissivity (T), storage coefficient (S) and leakage factor (B). This procedure was followed for all the six sets of time-drawdown data.

Furthermore, leakage coefficient (leakance) was also calculated using Eq. (12) to quantify the leakage through the aquitard layer; thereafter, hydraulic conductivities of the aquifer layers (K) were calculated from the transmissivity values obtained from pumping-test data analysis and the thickness of aquifer layers obtained from well logs. In addition, values of K were also estimated by using the aforementioned eight GSA methods at the ten sites. As the estimation of hydraulic conductivity was carried out by GSA methods at a depth interval of 3.05 m (10 ft), the equivalent values of K were computed to evaluate the performance of GSA methods against the K estimates obtained by pumping-test data analysis.

Results and discussion

Composition of lithologic layers

Figure 2a–j illustrate the well logs prepared for the nine observation wells (W1–W9) and one pumping well (W10) located in the study area and Farmers’ field. It is apparent from the well logs (Fig. 2a–h) that a mixture of murrum with coarse sand and medium sand exists up to about 12 m depth and their relative proportion varies with depth and location. Exceptionally, at site W4, presence of murrum was found up to 9 m depth. A high proportion of murrum (>50%) exists at sites W3 (3.05–12.19 m), W4 (3.05–9.14 m), W6 (3.05–6.10 m) and W2 (0–3.05 and 6.10–12.19 m) as compared to other sites. It is also evident from the well logs (Fig. 2i,j) that the murrum layer with coarse and medium sand is found up to 15 and 12 m depths at sites W9 and W10, respectively, in Farmers’ field. It is worth mentioning that murrum denotes reddish-brown-colored hard-rock-like geologic material, which aggregates when dry and acts as a semi-pervious or an impervious layer to percolating water. This type of geologic material is very common in the lateritic terrains.

Fig. 2
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a–j Well logs of the ten sites

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Table 3 summarizes the proportions of coarse, medium and fine sand in the aquifer layers (AL) at the ten sites located in the study area and Farmers’ field. Among eight sites (W1–W8), it is evident from this table that the first shallow aquifer layers (AL1) consist of 56–75% coarse sand and 9–28% medium sand. The second shallow aquifer layers (AL2) are comprised of 58–72% coarse sand and 10–25% medium sand. Overall, it can be inferred that the shallow aquifer layers have a high percentage of coarse and medium sand (> 80%) at sites W5 and W6 with a lower percentage of fine sand (< 15%). This suggests that shallow aquifer layers situated towards the north-east portion of the study area are expected to be high yielding, and this zone is suitable for constructing shallow pumping wells.

Table 3 Percentage of coarse sand, medium sand and fine sand in the aquifer layers of ten sites
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On the other hand, the deepest aquifer layers of sites W4 (AL6) and W1 (AL6) have the highest proportion (85%) of coarse and medium sand, whereas the lowest proportion of coarse and medium sand is found at site W3 (30%) for layer AL4 (Table 3). In contrast, a relatively high proportion of fine sand is apparent at sites W3 (67%), W5 (29%), W8 (28%), W2 (17%) and in the deepest aquifer layer. This high proportion of fine sand at site W3 (AL4) might be due to some error during sampling. The presence of a relatively high percentage of coarse sand followed by medium sand, in the deep aquifer system situated in the north-west portion of the study area (sites W4 and W1), indicates high yielding deeper aquifer layers in this zone compared to other sites. A detailed geologic investigation of the Farmers’ field (Table 3) revealed that the percentage of coarse and medium sand in the shallow aquifer layers varies from 69–93%, whereas the deeper aquifer layers consists of 56–93% coarse and medium sand.

Existence of aquifer layers

Practically, ‘aquifer’ is a relative term and the aquifer layers are expected to be present at shallow and deeper depths. In the present study, aquifer layers up to 40 m depth are considered as ‘shallow aquifers’, whereas aquifer layers beyond 40 m depth are considered as ‘deeper aquifers’. The well screens were placed at depths ranging from 37 to 79 m with a varying length of 3–22 m at the nine sites (W1–W8 and P3), which basically signifies the potential shallow and deep aquifer layers of the study area (Table 1). The well logs of the eight sites (W1–W8) revealed that the first shallow aquifer layer exists respectively at 12, 12, 18, 9, 18, 21, 15 and 12 m depths with thicknesses of 3–6 m (Fig. 2a–h). The second aquifer layer exists at depths ranging from 15 (at site W4) to 40 m (at site W8) with the thickness varying from 3 to 6 m. Interestingly, at site W4, the shallowest aquifer layer exists just below the murrum layer compared to other sites and four consecutive shallow aquifer layers exist from depth 9 to 27 m. Besides the shallow aquifer, aquifer layers also exist at relatively deeper depths (Fig. 2a–h), which are tapped by the observation wells installed at the eight sites. The deepest aquifer layers exist at depths ranging from 43 m (at site W6 and W7 with thickness 9 m each) to 79 m (at site W1 with thickness 6 m). Moreover, it is found that the thickness of deeper aquifer layers is greater than that of the shallow aquifer layers at most sites.

It is worth mentioning that although sites W5, W6 and W7 are situated relatively close to each other (distance between two consecutive wells varying from 50 to 100 m), the lithologic layers of these three sites are not similar. This finding indicates variability of lithology at a smaller scale in the study area.

In Farmers’ field (sites W9 and W10), the well screens were placed at depths 37 and 73 m with a respective length of 6 and 12 m, to tap the potential aquifer layers (Table 1). The well logs of sites W9 and W10 (Fig. 2i,j) indicate that the first shallow aquifer layer exists at depths 15.24 m (site W9) and 12.21 m (site W10) with respective thicknesses of 6 and 9 m, whereas the deepest aquifer layer exists at 73 m depth with a thickness of 12 m at both the sites (W9 and W10).

The number of aquifer and nonaquifer layers along with their depth and thickness are summarized in Table 4. The number of aquifer layers varies from 3 (site W7) to 6 (sites W1, W2 and W4) in the study area, whereas the number of nonaquifer layers varies from a minimum of 6 at site W4 to a maximum of 16 at sites W1 and W5. At sites W9 and W10, the number of aquifer layers varies from 4–6 and the number of nonaquifer layers varies from 7–10.

Table 4 Depth and thickness of aquifer and nonaquifer layers at eight sites over the study area
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A comparison between well logs (Fig. 2i–j) and groundwater level at corresponding sites reveals that although the deeper aquifers exist at depths of >40 m from the ground level (bgl), the maximum groundwater-level decline in the deeper aquifers is 23–27 m for sites W1–W8 and 18–19 m for sites W9 and W10 from the ground surface. As the groundwater level in the deeper aquifer is always above the upper confining layer, the aquifer is confined in nature. The upper confining layers of the deeper aquifers are composed of mostly fine particles (fine sand, silt and/or clay) along with a small percentage of coarse and medium sand. Therefore, these confining layers are most likely to behave as leaky confining layers (aquitards), and hence they can contribute significant leakage to the underlying deeper aquifers.

Moreover, Fig. 3 reveals that the elevation of the deepest aquifer layers tapped by the eight observation wells (W1–W8) over the study area is below mean sea level (MSL). It is also evident from Fig. 3 that the observation well W9 and pumping well W10 located at Farmers’ field have the deepest aquifer layers existing below MSL. Consequently, controlled pumping is necessary to ensure safe and long-term utilization of available groundwater resource in the study area and Farmers’ field. If the groundwater level declines below MSL, it will pose a threat of seawater intrusion into the aquifer because the sea is located at about 83 km east of the study area and about 81 km east of the Farmers’ field.

Fig. 3
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Location of deepest aquifer layers at different sites of the study area and Farmers’ field

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Grain-size distribution curves of geologic formations

Grain-size distribution curves (GSDCs) along with the values of uniformity coefficients (U) for aquifer layers are shown in Fig. 4a–j. It is discernible from Fig. 4a–h that the shallow aquifer layers (AL1 and AL2) are coarser in nature having a relatively higher proportion of coarse grains (high d10 and d50 values) as compared to the deep aquifer layers for all the sites. The values of U for the aquifer layers vary from a minimum of 0.19 (AL5 of site W2) to a maximum of 11.4 (AL6 of site W2) over the study area. Among the eight sites (W1–W8), the aquifer layers at site W6 have the maximum number of uniform ‘S’ shape GSDCs among all the sites, which represent uniformly distributed or poorly graded layers. In contrast, the shape of GSDCs for AL4 of site W3 (U4 = 2.99), AL4 of site W8 (U4 = 7.07) and AL5 of site W2 (U5 = 11.4) shows an appreciable deviation from the uniform ‘S’ shape. These findings indicate remarkable heterogeneity in the grain-size distribution of subsurface formations at different depths.

Fig. 4
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a–j Grain-size distribution curves for the aquifer layers at ten sites. Note: U1–U6 indicate uniformity coefficient values for aquifer layers 1–6, respectively

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Furthermore, sites W4, W1 and W6 have nearly uniform ‘S’ shape GSDCs along with the low value of U for maximum number of aquifer layers. This indicates existence of more uniformly graded geologic formations having relatively high porosity compared to other sites; therefore, these three sites have aquifer layers with greater storage capacity.

In Farmers’ field, the shallow aquifer layers are also found to be coarser than the deeper aquifer layers for site W9 (AL1 and AL2) and W10 (AL1 and AL3; Fig. 4i,j). The values of U vary from 2.44 (AL5 of site W10) to 4.68 (AL3 of site W9) with uniform GSDCs at both the sites. Considering the range of U values and shapes of GSDCs, it is worth mentioning that the underlying aquifer systems of Farmers’ field are found to be more uniform as compared to the experimental farm of the study area.

Subsurface hydraulic conductivity by grain-size analysis

Depth-wise variation of hydraulic conductivity

The range of K estimates obtained from the eight grain-size-analysis (GSA) methods for eight sites at a depth interval of 3.05 m (10 ft) is presented in Table 5. The minimum range of K values—0.67–3.41, 0.75–3.33, 0.69–3.30 and 0.81–3.55 m/day—are obtained for sites W3, W6, W7 and W8, respectively, for the topmost geologic samples (first 3.05 m depth) over the study area. Further, at 3.05 m depth, the highest K value, varying from 0.87–20.23 m/day, is found at site W2, followed by sites W4 (1.23–14.75 m/day) and W5 (2.87–13.75 m/day). However, at deeper depths ranging from 64.01 to 88.4 m, the lowest range of K values are obtained at sites W1 (0.97–4.14 m/day), W2 (0.71–4.09 m/day), W4 (1.13–5.57 m/day) and W5 (0.27–1.77 m/day). Among all the sites, the maximum range of K value is found for site W4 (100.17–578.87 m/day) at 57.9 m depth followed by sites W2 (104.77–395.51 m/day) and W3 (60.52–216.38 m/day) at 15.24 and 30.5 m depth, respectively. At Farmers’ field, the minimum range of K values is 0.99–4.09 m/day at site W9 (at 48.77 m depth) and 1.01–4.32 m/day at site W10 (at 45.72 m depth). The maximum range of K values are obtained at sites W9 and W10, and are 16.4–64.28 m/day (at depth 30.5 m) and 58.16–196.44 m/day (at depth 76.2 m), respectively.

Table 5 Depth-wise variation of hydraulic conductivity values estimated by the eight GSA methods for the eight observed sites
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An abrupt rise in the K values is observed in the geologic samples at shallow (<18 m) and deeper depths (≥18 m) for all the sites. At shallow depth, this is attributed to the presence of 55 and 72% of irregularly-shaped gravel (murrum) at sites W3 and W4, respectively, at a depth of 3.05 m. However, a relatively higher percentage of coarse sand is responsible for high values of K at deeper depths—for example, sites W5 and W4 have higher values of K at depths 21.35 and 58 m, respectively, because of the higher percentage of coarse sand (respectively 79 and 90%). As the GSA methods used in this study are mainly a function of effective grain diameter and/or uniformity coefficient of the geologic sample, high K estimates are logical. Interestingly, the same range of K values (0.85–4.15 m/day) is found at site W5 at 39.7 and 42.75 m depths, which indicates nearly the same proportion of different grain sizes at these depths. Further, it is evident that the K values for the geologic samples at relatively deeper depths (>30.48 m) are lower as compared to shallow depths for all the sites except sites W4, W6 and W7. On the whole, it can be concluded that there is a large variation in subsurface hydraulic conductivities at different depths and locations over the study area, which is attributed to variability in the GSA methods and heterogeneity in the subsurface formations.

Hydraulic conductivity of aquifer and nonaquifer layers

The hydraulic conductivities for the aquifer and nonaquifer layers based on the eight GSA methods are shown in Table 6. For the first aquifer layers, which are at shallow depths (Table 4), the highest K values are found at site W2 (81.33–292.86 m/day) followed by sites W5 (48.84–192.12 m/day) and W4 (48.16–150.14 m/day). However, the highest (60.52–216.38 m/day) and the lowest (3.62–18.59 m/day) values of K for the second aquifer layer are observed at sites W3 and W5, respectively. These high K values are attributed to the higher percentage of coarse sand, medium sand and gravels as well as high values of characteristic grain diameters (d10, d17, d20 and d60) in these aquifer layers. As the GSA methods are directly proportional to the square of these characteristic grain diameters, high K estimates are obtained for these aquifer layers. It is also evident from Table 6 that the deeper aquifer layer (AL5 of site W4), which is the thickest aquifer layer, has the highest K values (52.34–209.93 m/day) compared to other deeper aquifer layers. These high values of K are due to the presence of a higher relative proportion of coarse and medium sand (86%) with high value of the characteristic grain diameter (0.3 mm) compared to other aquifer layers. Moreover, the range of K values for the deeper aquifer layers (≥ 40 m) varies from 1.87 (site W8) to 209.93 m/day (site W4), whereas for the shallow aquifer layers (< 40 m), K values range from 0.97 (site W2) to 292.86 m/day (site W2) over the study area. Thus, it can be concluded that the shallow aquifer layers have a higher range of K value than the deeper aquifers layers. This result is supported by low values of uniformity coefficient and high values of effective grain diameter for the geologic samples of the shallow aquifer layer as compared to the geologic samples for deeper aquifer layers.

Table 6 Range of hydraulic conductivity estimates (K, m/day) for aquifer and nonaquifer layers at the eight sites
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Moreover, Table 6 further reveals that overall, the range of hydraulic conductivities of the nonaquifer layers is lower compared to the aquifer layers at most sites. This is due to a higher proportion of fine-grained particles in the nonaquifer layers. However, the highest range of K values is found for NL2 of site W4 (30.74–292.83 m/day) followed by NL3 of site W2 (33.02–126.52 m/day). The reason behind these high K values is the presence of a significant proportion of murrum in these nonaquifer layers. Based on the above results, it can be inferred that there exists considerable variation in K estimates with depth and location, which indicates significant heterogeneity of geologic formations underlying the study area.

In Farmers’ field, the values of K for shallow aquifer layers range from a minimum of 14.26 m/day (site W9) to 148.06 m/day (site W10), whereas the K values range from 3.62 m/day (site W9) to 193.30 m/day (site W10) for the deeper aquifer layers. Furthermore, the nonaquifer hydraulic conductivity varies from 0.9 m/day (site W10) to 165.41 m/day (W10). This high value of hydraulic conductivity is attributed to the high percentage of murrum (>60%) in these nonaquifer layer.

Relative performance of grain-size analysis methods

The salient basic statistics for the eight GSA methods are illustrated in Figs. 5 (a,b). It can be seen that there is a large variation in K estimates by the eight GSA methods. The highest median value of K is obtained for the US Bureau Reclamation (USBR) method (16.67 m/day), whereas the lowest median value of K is found for the Slichter method (3.09 m/day) over the study area (Fig. 5a). However, the Saurebrei method (21.59 m/day) and Slichter method (6.4 m/day) estimated highest and lowest median values, respectively, at Farmers’ field (Fig. 5b). The position of the median and length of whiskers specify the nonsymmetric (skewed) nature of the K estimates by the eight GSA methods. The box plots for all the GSA methods follow a typical pattern of shorter ‘lower whiskers’ and longer ‘upper whisker’ for the K estimates. This typical pattern indicates least variability in K estimates by all the GSA methods in the first quartile (0–25%) compared to maximum variability in K estimates in the fourth quartile (25–75%) of the data series (Fig. 5a,b). Overall, among all the GSA methods, the maximum variation in K estimates is attributed for the Sauerbrei and Hazen methods, whereas minimum variation is depicted for the Slichter and Harleman methods.

Fig. 5
figure 5

Variation of the hydraulic conductivity of subsurface formations estimated by eight GSA methods a over the study area and b at Farmers’ field. HS indicates Hazen Simplified method

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Figure 6 shows the bar graphs of mean hydraulic conductivity along with the standard errors of K values estimated by the eight GSA methods for individual sites. It is evident that the values of mean K and standard error differ greatly with GSA methods and locations. The Kozeny and Sauerbrei methods yield highest mean K values for the sites W1–W8, whereas at sites W9 and W10 the highest mean K values are estimated by the Saurebrei method. On the other hand, the Slichter, Harleman and USBR methods yield lower mean K values compared to other GSA methods for all the sites. These findings are consistent with those of Vukovic and Soro (1992), Cheng and Chen (2007), Song et al. (2009) and Ishaku et al. (2011). Furthermore, results of an ANOVA test indicated that the difference in mean K values estimated by different GSA methods are found to be strongly significant (at 1% level of significance) at a given location. Similarly, the variation in mean K values at different locations by a given method is also found to be highly significant (at 1% level of significance).

Fig. 6
figure 6

Comparative performance of the eight GSA methods in estimating the mean hydraulic conductivity of geologic formations at ten sites, along with standard errors bars

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The largest values of standard error are for the Kozeny and Sauerbrei methods for most of the sites (Fig. 6), which indicates higher sensitivity of these two methods (Kozeny and Saurbrei) to the variation in effective grain size of the geologic samples. In contrast, the values of K estimated by the Slichter, Harleman and USBR methods show the lowest deviation from their mean values at almost all sites, thereby indicating that these three methods are somewhat consistent in estimating K of different geologic formations.

On the other hand, the skewness and kurtosis values, which represent symmetry and peakedness of the K values, respectively, vary markedly according to sites and methods (Fig. 7a,b). Among the eight sites (W1–W8), the skewness values vary from as little as 0.59 for the USBR method (site W6) to 4.78 for the Kozeny method (site W5) over the study area. However, in Farmers’ field, the skewness values vary from a minimum value of 0.4 (site W10) to a maximum value of 1.43 (site W10). Hence, the distribution curve for the estimated K values could be considered positively skewed, thereby indicating that more extreme values are situated towards the right tail of the distribution. Further, among the eight sites (W1–W8), the kurtosis value is found to be negative for all the GSA methods at sites W6 (except Kozeny, Saurebrei and Slichter method), W7 and W10 (except USBR method), which suggests a wider peak and thinner tail of the distribution curve. Additionally, between the eight GSA methods, the lowest values of positive kurtosis are observed for the Hazen Simplified and Harleman methods at most sites (except the USBR method at sites W1, W2, W5 and W8; Fig. 7b). Thus, the kurtosis values indicate that the K values estimated by the Hazen Simplified and Harleman methods could be considered evenly distributed in the central and tail regions of the distribution curve having less possibility of higher peaks that are undesirable.

Fig. 7
figure 7

Variation of a skewness and b kurtosis values for the eight GSA methods at different sites

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Hydraulic parameters of aquifer systems by pumping test

The pre-pumping groundwater level, the groundwater level after the pumping test, and the groundwater level measured next day prior to the initiation of pumping activity for the eight sites (W1–W8) are summarized in Table 7. It is evident from this table that the drawdowns at sites W1, W2 and W3 are very much less after about 6 h of pumping (pumping well P3), which is due to the fact that they are located at relatively large distance from the pumping well. However, appreciable drawdowns were observed at the remaining five sites (sites W4, W5, W6, W7 and W8) during the pumping test and hence, the time-drawdown pumping test data obtained from these sites were used for determining aquifer parameters over the study area. On the other hand, in Farmers’ field the time-drawdown data observed in one observation well (site W9) were used to determine parameters of the deeper aquifer system underlying Farmers’ field. It is interesting to note that the groundwater levels at all the sites on the next day of the pumping test were found to be almost the same as those of the pre-pumping groundwater levels measured just before the start of the pumping test (Table 7). That is, the drawdown that occurred during the pumping test is almost fully recovered during about 16 h, which indicates a rapid recovery of the pumped water and hence, higher transmissivity of the aquifer system.

Table 7 Depth to groundwater before and after pumping test at eight sites
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While analyzing the time-drawdown data of the six sites (sites W4–W8 over the study area and site W9 at Farmers’ field), matching of field-data curves with both Theis and Walton Type Curve was carried out, but it was found that field-data curves better match with the Walton Type Curve. For example, the matched curves are shown for sites W5 and W9 (Fig. 8a,b). The aquifer parameters at the six sites thus obtained are summarized in Table 8. The values of transmissivity (T) are very high for all the five sites (W4–W8) ranging from about 269 m2/day (site W6) to a maximum value of about 694 m2/day (site W4). This finding suggests a high spatial variation of aquifer transmissivity (i.e., significant heterogeneity) in the aquifer systems over the study area. The values of storage coefficient vary from 8.75 × 10−5 (site W8) to 2.13 × 10−4 (site W5), which also indicates significant spatial variation of storage coefficient. Further, the values of leakage factor (B) range from 3552.77 (site W8) to 42,624.64 m (site W5) and the leakance varies from 2.01 × 10−7 (site W6) to 3.96 × 10−5 (site W8), which indicates maximum leakage rate is found at site W8 and minimum leakage rate at site W6. At Farmers’ field, the aquifer transmissivity is found to be lowest (122.68 m2/day) with a maximum leakance value (0.0346 day−1) compared to the study area. Thus, it can be concluded that the deeper aquifer systems underlying the study area and Farmers’ field are a leaky confined aquifers. This finding confirms the related finding obtained from the hydrogeologic analysis.

Fig. 8
figure 8

Matching of field-data curves and Walton-Type curves at sites a W5 and b W9

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Table 8 Aquifer parameters from pumping tests
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The aquifer hydraulic conductivities of the study area (sites W4–W8) obtained from the pumping-test method (Table 8) could be classified as ‘high’ (Todd 1980), which indicates fast groundwater movement in the underlying aquifer system. In fact, a high K value is desirable for large well discharge, but it also implies a condition of less groundwater availability during dry periods. However, in the Farmers’ field (site W9) the aquifer hydraulic conductivity obtained from the pumping-test method (Table 8) could be classified as moderate (Todd 1980).

Validation of grain-size analysis methods

A comparison of aquifer K estimated by eight GSA methods with that obtained by the pumping-test method is shown in Fig. 9 for six sites. Obviously, K values obtained by GSA methods show much larger variation at all the six sites as compared to the K determined by the pumping-test method. This is logical because GSA estimates are based on disturbed geologic samples. Although the pattern of variation in aquifer hydraulic conductivities is the same for all the GSA methods at all the sites, the pattern of variation in aquifer hydraulic conductivities obtained from the pumping-test method is different. Among the eight sites (W1–W8), the estimates of aquifer K from the pumping test varies from 13.65 m/day at site W8 to 25.31 m/day at site W7 (Table 8). It should be noted that the K yielded by the Slichter method (14.98 m/day) is close to that obtained by the pumping test (13.65 m/day) at site W8 followed by the Harleman (20.34 m/day) and USBR (23.32 m/day) methods. However, at site W7, the USBR method yields a somewhat reasonable value of K (29.58 m/day) against 25.31 m/day obtained from the pumping test followed by the Slichter method (34.93 m/day).

Fig. 9.
figure 9

Comparison of aquifer-layer hydraulic conductivities estimated by eight GSA methods with those determined by the pumping-test method over the study area and Farmers’ field.

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Moreover, a close observation of Fig. 9 reveals that, except for site W5, the estimates of aquifer K by the GSA methods are always larger than those obtained by pumping-test data analysis. At site W5, aquifer hydraulic conductivity estimated by the GSA methods are 2–6-fold less than that determined from the pumping-test method. As the value of effective grain diameter is less (0.093–0.124 mm) for the aquifer material of site W5, all the GSA methods yields lower aquifer hydraulic conductivity compared to other sites. The estimates of K given by the Kozeny, Saurbrei, Hazen, Hazen Simplified and Beyer methods are consistently much higher than the pumping-test results for all the sites over the study area. On the other hand, the K values given by the Harleman and USBR methods are somewhat closer to the K values obtained from the pumping test method (1–3-fold larger than the pumping test). Therefore, it can be inferred that the performance of Slichter, Harleman and USBR methods is better for the underlying aquifer systems of the study area. A general remark for the Hazen Simplified method is that, being the simplest method, it often provides better results as compared to the complicated GSA methods like Kozeny, Saurbrei, Beyer and Hazen.

At site W9, which is located in Farmers’ field, the aquifer K estimates derived from the Slichter method (2.59 m/day) are quite close to those obtained from the pumping-test method (2.50 m/day; Fig. 9). However, the values of K obtained from the Harleman and USBR methods are 2- and 3-fold, respectively, higher than that of the pumping-test estimates. The other five GSA methods are over estimating the aquifer K values. Based on these results, it can be inferred that satisfactory performance is given by the Slichter, Harleman and USBR methods for the aquifer systems of Farmers’ field; it should be noted that the three methods perform better for the study area and Farmers’ field, which have the same geologic settings.

On the whole, all the GSA methods overestimated the values of the aquifer K compared to the pumping test results, except one site (site W5). More or less similar results were reported in past studies conducted in an alluvial hydrogeologic setting (Vukovic and Soro 1992; Cheong et al. 2008). This overestimation of K by the GSA methods is due to the fact that these methods mainly account for effective grain diameter, uniformity coefficient and/or porosity which are not enough for estimating the hydraulic parameter K, which is a function of several other factors. As a result, the performance of GSA methods needs to be evaluated under varying hydrogeologic conditions. Accordingly, suitable GSA methods could be used for estimating K in the absence of field-measured K values. Also, GSA methods offer a useful tool for preliminary hydrogeologic investigation.

Conclusions

The present study demonstrates the scientific approach for hydrogeologic and hydraulic characterization of aquifer and nonaquifer layers in a lateritic terrain of West Bengal, India. In addition, the performance of salient grain-size analysis (GSA) methods has been evaluated. To accomplish this goal, test drilling was carried out at ten sites and geologic samples were collected at 10-ft (3.05 m) depth intervals at individual sites. The collected geologic samples were subjected to sieve analysis, and grain-size distribution curves (GSDC) were prepared for each geologic sample. Thereafter, well logs for individual sites were constructed using state-of-the-art software. Salient GSA methods were used to estimate the hydraulic conductivity of subsurface formations underlying the observed sites. Pumping tests were also conducted to estimate hydraulic parameters of the aquifer systems. Finally, the performance of the GSA methods was evaluated based on the results obtained from the pumping test. Based on the hydrogeologic and hydraulic analysis presented in this paper, the following conclusions are drawn:

  • The hydrogeologic analysis up to 90 m depth indicated that the first aquifer layer in the study area exists at depths ranging from 9–21 m with an average thickness of 5 ± 1.6 m followed by other aquifer layers available at relatively deeper depths (24–79 m) with a significant variation in thickness (3–9 m). In Farmers’ field, the shallow aquifer layers exist at depths varying from 15–37 m and the thickness varies from 3–9 m, whereas deeper aquifer layers exist at depths 37–73 m with the thickness ranging from 6–12 m.

  • The stratigraphic analysis and pumping test results indicated that deeper aquifers tapped by the observation wells and pumping wells are leaky confined aquifers. Greater thicknesses of the aquifer layers are found at 34 m at site W4 (AL5) and 18 m at site W7 (AL3). It was found that the subsurface formations below the high-elevation sites towards the north-west side of the study area consist of a relatively high percentage of coarse and medium sand with low percentage of fine sand in the deepest aquifer layers and hence, they indicate a good potential groundwater resource in this zone.

  • Analysis of grain-size distribution curves indicated that the materials of all the aquifer layers can be characterized as more or less uniform. The materials of the shallow aquifer layers are coarser as compared to deeper aquifer layers at most sites. As the deeper aquifer layers tapped by the observation wells are situated below the mean sea level (MSL), controlled and prudent pumping is suggested to avoid seawater invasion in the future.

  • The average values of K estimated by the GSA methods vary from 3.62–292.86 m/day for shallow aquifer layers and 0.97–209.93 m/day for deeper aquifer layers.

  • The analysis of pumping-test data revealed that the values of transmissivity (T) and storage coefficient (S) of the deeper aquifers range from 268.70–693.79 m2/day and 8.75 × 10−5–2.13 × 10−4, respectively; while the leakance of the aquitard varies from 2.01 × 10−7–3.96 × 10−5 day−1. However, in Farmers’ field, the values of T, S and leakance were found to be 122.69 m2/day, 1.01 × 10− 7 and 34.56 × 10−2 day−1, respectively.

  • The Slichter method yielded aquifer K values much closer to those obtained from the pumping-test method; however, the USBR and Harleman methods yielded K values which are 1–3-fold greater than that determined from the pumping-test method for the aquifer layers at most of the sites. As the GSA methods are dependent on the statistical measures of grain-size distribution and the natural structure of geologic samples, they are not capable of estimating hydraulic conductivity with a higher accuracy. Nevertheless, in the absence of pumping-test data, the GSA methods such as Slichter, Harleman and USBR methods can be used for the preliminary analysis of groundwater systems in the lateritic terrain.

  • The methodology presented in this study is suitable for both basin and sub-basin scales and it could be easily replicated in other regions of India as well as other countries. The outcomes of this study are helpful in formulating efficient strategies for groundwater development and management in the lateritic terrain of eastern India in general and West Bengal in particular.

  • Overall, it is concluded that the study area has multi-layered aquifer systems, with high potential aquifer layers available at relatively deeper depths. However, proper utilization and management of available groundwater resources are essential to avoid groundwater depletion and seawater invasion. In order to ensure long-term sustainability of groundwater resource in the region, it is recommended to expand comprehensive groundwater investigations at a larger scale with adequate monitoring network and to quantify safe yields from available aquifer systems.