Assessment of natural groundwater reserve of a morphodynamic system using an information-based model in a part of Ganga basin, Northern India

Assessment of morphodynamic groundwater reserves is important for the sustainable management of water resources. It is a truth that groundwater resource evaluation is anxious with the ambiguity of its several factors and employing methods. Thus, an information-based model has been hypothesized to assess natural groundwater reserves in a morphodynamic system in a part of the Ganga basin of Northern India, where the spatial variability in natural groundwater reserve exists. Marginal information of rainfall data, and transinformation among the rainfall, and monthly depth to groundwater level measurement at 50 wells in a dense monitoring network were used for evaluating natural groundwater reserve. The results indicate that an average recharge rate is about 246 mm/monsoon and or 32.65% of the seasonal rainfall, and its values are well-correlated with the soil infiltration rate. It has been found that the estimated recharge rates are about 54.08, 45.85, 33.77, 32.48, and 32.14% of the seasonal rainfall in an active flood plain, back swamp, natural levees, flood plain, and palaeochannel, respectively. The calculated annual rainfall input to groundwater reserve is found about 127.98 MCM/monsoon rainfall, which could be employed for sustainable management of groundwater resources in the morphodynamic system of the Ganga river basin.

method, which has been applied first in the granitic aquifer of Southern India 27 and then in a basaltic terrain of Central India 28 . Here we used this method to quantify natural recharge in a morphodynamic groundwater system in a part of the Ganga basin, Northern India (Fig. 1). Thus, our objectives of this work are in three folds (1) to assess natural groundwater reserves using the measurement of groundwater level corresponding to rainfall event with the help of an entropy-based model, (2) compare the results with the soil infiltration rate and monsoon groundwater reserve (MGWR) obtained from the water table fluctuation (WTF) method, and (3) also studied its spatial variability in the geomorphological setting.

Materials and methods
Description of the study area. The geomorphic system covers an area of 521 km 2 spreading in the parts of 7-blocks of Patna district, Bihar in Middle Ganga Plain (MGP) in Northern India. The area lies between longitudes: 84°49 / 12 // E to 85°13 / 12 // E and latitudes:25°25 / 12 // N to 25°40 / '48 // N and falls in the Survey of India Topo-sheets 72G/2 and 72C/14,15. Topography varies from about 45.45 m to 69.00 m, above the mean sea level (amsl) with a general slope from S-W to N-E, and N-directions with minor variations. The mighty River Ganga forms the northern boundary, and other rivers (i.e., Son and Punpun) are draining just outside the area (Fig. 1c). Information theory. Information (entropy) theory was firstly coined by Shannon 30 , who established a quantitative measure of the uncertainty or the information content of a random variable 31 . This information is mainly calculated by the entropy indices such as marginal entropy, joint entropy, and transformation. If it follows for the discrete random variables X = {x i }, i = 1, 2, 3…, n, and Y = {y j }, j = 1,2, 3,…, m 32,33 , then marginal entropy, H(X) for the variable, X defined as.
where, p(x i ): the probability of ith random variable, X = {x i }, n: is the number of observations, and the value of total probabilities = n i = 1 p(x i ) = 1.
The H(X) of the Eq. (1) provides the amount of information or uncertainty that X has, and similarly, the marginal entropy, H(Y) shows the information or uncertainty that Y has 32 . If both the X and Y variables are stochastically dependent, then the joint entropy, H (X, Y) is expressed as.
In which, p (x i, y j ) is the joint probability between x i, and y j. Transinformation (called mutual information) between two variables describes the amount of information, which is common in both two stochastically dependent variables, X and Y. It is defined as 33 . where, p (x i , y j ): the joint probability of x i and y j , and p(x i ) and p(y j ): both discrete probability occurrence of x i and y j , respectively. The unit of information measure depends on the logarithmic base used. We had used a base '2' in our computation, then the unit is 'bit' 34 . Entropy-based model development. Depth to groundwater table (WT) fluctuates due to the rainfall (R) recharge. Both the WT and R are random variables. Then the marginal entropies [i.e., H(R) and H(WT)] are estimated in the probability distribution, and categorized as the potential information of these variables 27,28 . Joint entropy, H (R, WT) is calculated as the total information having in the measurements of WT and R, as shown in Fig. 2. Then the transinformation [T (R, WT)] gain measures the reduction of entropy by splitting the dataset according to a probability space of random variable, which provides the physical process of the hydrogeological system. A larger information gained suggests a lower entropy of the analysed data and/ gaining the information of the physical processes. It is calculated as the reduction in the original uncertainty of the measurement of WT due to the knowledge gained in the measurement of rainfall (R). The discrete forms of these entropies are described as 27,28 . www.nature.com/scientificreports/ where, R and p(R i ): discrete variable of rainfall and probability of R, respectively. WT and p(WT j ): discrete variable of depth to water level and probability of WT, respectively. p (R i , WT j ): joint probability of R i and WT j . R i , i = 1, 2, …, n; and WT j , j = 1, 2, …, m, defined in the same probability space.
Then, the percentage of rainfall recharge, R e (%) is the ratio of T (R, WT) and H(R) multiplying 100, which is represented the natural groundwater reserve due to the rainfall as described in the following equation.
A 2-D contingency table (Table S1) Table S1. To construct this table, rainfall data comprised the u-class intervals, whereas the depth to water table data was assumed to be the v-class intervals. Then, the joint frequency of the rainfall and depth to water level data categorised by (i, j) is denoted by f ij , i = 1, 2, . . . , u; and j = 1, 2, . . . , v, where 'i' considered the column, and 'j' considered the row. In deciding the number of classes, the 2 k rule [2 k ≥ N] was utilized in the characterizing data sets 38 . This rule was used to reduce the error of recharge percentage by changing the number of classes (k) and class intervals of the data (N) at the well site of the study area.
Monsoon groundwater reserve. Monsoon groundwater reserve is estimated using the water table fluctuation (WTF) method. It is based on the principle that rises in water level in shallow (unconfined) aquifer is due to groundwater recharge arriving at the water table 8,39 . Monsoon groundwater reserve (MGWR) was obtained from the following equation.   www.nature.com/scientificreports/ Then the MGWR was estimated using the above equation at 50 monitoring well sites (mainly at each Thiessen polygon) during the years 2012 and 2013. Because the monthly groundwater level data in a dense monitoring well network were available, as shown in Fig. 1c. The idea of the Thiessen polygon method was that the zones addressing every groundwater measure point were characterized by drawing lines between neighbouring points on a map. The perpendicular bisectors were developed to lines joining each measurement point with individuals proximately surrounding it. These bisectors formed a succession of polygons, each polygon containing one measuring point 40 . Groundwater reserves had been estimated at all observation well sites, and the area of influence of each well site was calculated using the polygon method in ArcGIS (10.4). In total, 50 polygons were created. The minimum area of ~ 3.19 km 2 was influenced by the well (W-40) located in the flood plain, and the maximum area of 20.12 km 2 by the well (W-47), as shown in Figs. 1c, 4. The estimated natural groundwater resources at each polygon had been also utilized to compare with the results obtained from the information-based model. The data. Primary data. Monthly groundwater level at 50 observation wells of a dense monitoring network from February 2012 to March 2014 were accessed and utilized for the analysis. The monthly rainfall data were also collected from January 2010 to December 2019 (http:// hydro. imd. gov. in/ hydro metweb/). It had been observed that the rainy season, in general, continues from mid-June to the end of September, which receives the Southwest monsoon and accounts for about 86.2% of the total average rainfall (~ 872.8 mm). It gets an average normal monsoon rainfall of about 752.3 mm/year from the year 2010-19 (Fig. S1a). The seasonal rainfall of the year 2010-2019 had been calculated as around 752 mm, as presented in Table S2, and used for our analysis.
The transformation value, T (R, WT) of the rainfall and WT, along with the marginal entropy, H(WT) of the groundwater level at the individual well, had been estimated with the aid of Eqs. (4,5,6,7). Then the percentage of rainfall as a natural groundwater recharge was calculated with the help of Eq. (8). In addition, the specific yield values of 8-10% were considered for the sandy loam and sandy clay loam, whereas 7-8% for the clay loam of the study area ( Fig. S2) as per the GEC norms 41 . These values were considered to estimate the MGWRs at these well areas using the Theissen polygon method 40 .
Secondary data. In addition, the geomorphic map of the study area (Fig. 1c) was utilized to compare the estimated recharge at the well site. Figure 1c shows five different units such as active flood plain (~ 6 km 2 ), palaeochannel (~ 55 km 2 ), back swamp (~ 65 km 2 ), natural levees (~ 91 km 2 ), and flood plain (~ 304 km 2 ). This map was used to relate the estimated groundwater reserves in the various geomorphic units, as shown in the flowchart (Fig. 2).

Results and discussion
Rainfall. In the study area, about 6.8% of the annual rainfall precipitated in winter (October to February), about 7.0% in summer (March-May), and about 86.2% in the Southwest monsoon period (June-September), as presented in Table S2. It had shown that the rainfall of around 752 mm occurred during the southwest monsoon. This monsoon rainfall was comparatively more (> 220 mm) 17 , and was also found only one stretch for the monsoon period (Fig. S1b). Rangarajan and Athavale 17 studied based on tracer technique to estimate the natural groundwater recharge in different hydrogeological provinces in India. Their results suggest that the linear relation between rainfall and natural recharge exists in the major hydrogeological units of India. The minimum precipitation is required to initiate the rainfall recharge due to the soil characteristics and hydrogeological conditions of the aquifer. It has been illustrated that the minimum values of the rainfall needed for the recharge, are 40 mm for alluvial areas, 220 mm for sediments, 255 mm for granite, and 355 mm for basaltic areas 17 . Thus, the natural groundwater recharge had been calculated during the Southwest monsoon when around 86.2% (~ 752 mm) of the annual rainfall occurred in the study area (in Table S2).  Fig. S3. There was no decline in groundwater level after the monsoon, but the groundwater level raised more than 4.00 after the monsoon in the central northern and southern parts.

Groundwater level and its fluctuation.
In this river basin, where groundwater occurs in unconfined (shallow) aquifers, the rise in the groundwater table is a direct consequence of rainfall, particularly in the rainy season 42 . This rise in groundwater level at the well site is a characteristic feature of the unsaturated zone 43,44 . So, there exists a definite relationship between the rainfall and the depth to water table for a particular region. The well hydrographs corresponding rainfall were made, as shown in Fig. 3. This figure had shown that the groundwater level at the wells located in an active flood plain (W-8), natural levees (W-12), back swamp (W-22), and palaeochannel (W-34) are strongly correlated with the rainfall after 2-month lags having the correlation coefficients of −0.65, −0.76, −0.63, and −0.50, respectively. In comparison, the well (W-1) located at flood plain in the extreme north-western part of the study area responded after a 3-month lag. It was due to the confluence of rivers Son and Ganga. Therefore, cross-correlation among rainfall and groundwater level at the individual well was carried out. It had been observed that the well (W-9) and W-8 were responded after 1-and 2-month lags of rainfall with the cross-correlation coefficients of −0.62 and −0.65, respectively, in an active flood plain located in the north-western part. In the back swamp, the wells were responded to after 1-2 month lags in the range of correlation coefficients of −0.49 to −0.77, with an average Information-based recharge estimation. The shallow groundwater occurs in unconfined aquifers, particularly in a morphodynamic system in a part of the Ganga basin of Northern India. The water level rise in the rainy season was an immediate result of precipitation, when the groundwater draft was comparatively less. The water table rise at a specific place was a typical feature of the saturated zone 39 . Thus, the information-based model had been hypothesized and utilized to assess natural groundwater recharge in this morphodynamic system. For this, the marginal and joint probabilities were estimated separately using a 2-D contingency table 27,28 for both the monthly and SW monsoon data. In total, 8 events of the SW monsoon data, and 26 events of the monthly data were used to construct the contingency tables. An illustration of a 2-D contingency table for the monthly data of well W42 (village: Tishkhora) had been prepared, as presented in Table 1. The rainfall data had been categorized into 4 class intervals in the ranges of 0-100, 100-200, 200-300, and 300-410 mm, whereas the groundwater level data into 5 class intervals in the ranges of 0-2, 2-4, 4-6, 6-8, and 8-10 m, bgl based on the 2 k rule 38 to reduce the error of Re (%) estimation at the well site. The joint frequency (i, j) denoted by f i,j (where i = 1,2,…,4; and j = 1,2,.., 5), were considered as the column (rainfall, i) and the row (water level, j). The f i . and f j . were referred to as the marginal frequencies of these two variables, respectively. The marginal entropies of rainfall (1.290 bits) and water level (0.961 bits), joint entropy (2.144 bits), and transinformation (0.107 bits) were estimated, as presented in Table 1. During the SW monsoon, the natural recharge at the well (W-42) site was calculated with the help of Eq. (8), and found about 8.30% of the rainfall. Similarly, it was estimated at the remaining well sites, as presented in Table S4. It indicates that the wells located at Phulwari (W-28), Dalluchak (W-29), and Raghopur (W-40) were not inconsiderable response due to the precipitation. These wells were independent of natural groundwater recharge, but the water level fluctuation might be responded to due to the lateral flows, and human intervention. Except at these well sites, the transinformation was measured an average of 0.238 bits in the monthly data, whereas average information of 0.662 bits in the SW monsoon season. It shows that the precipitation in the SW monsoon season is more dominant in influencing the natural groundwater recharge in the study area. It had been observed that during the monsoon season, the average information gained was around 1.031, 0.874, 0.610, 0.619, and 0.643 bits in the active flood plain, back swamp, palaeochannel, flood plain, and natural levees, respectively. Annual average rainfall recharge was estimated at around 18.49% (Table S4), whereas it was an average of 34.73% of the SW monsoon season. It had been found that the natural recharge was comparatively more in the active flood plain (~ 38.41%) than the natural levees (17.03%) in the monsoon season. The spot measurements of natural recharge using the tracer techniques at a few alluvium sites located in Tamil Nadu, Andhra Pradesh, West Bengal, Bihar, Uttar Pradesh, Gujrat, Rajasthan, Haryana, and Punjab had been carried out by several researchers 17,[45][46][47] . Their results indicate that the range of natural recharge varied from 10.90 to 19.70% of seasonal rainfall, with an average of 13.86%. In the study area, the recharge zones had been identified considering the soil texture 48,49 . It had been found that the monthly recharge of rainfall estimated ~ 20% of the infiltration factor was used for the groundwater flow model, which had shown reasonably matched with www.nature.com/scientificreports/ the calibrated well hydrographs 50 . The estimated average natural recharge in the study area using informationbased theory was an average of 18.49% of the rainfall (Table S4), which is quite acceptable. Because it was also estimated in the range of 10.90 to 19.70% of seasonal rainfall at the same typical hydrogeological conditions in the states of Uttar Pradesh, Bihar, and West Bengal 17,46,47 . It was noted that the estimated joint entropies at the well sites were not systematic due to the non-uniformity of rainfall and various morphological units in the study area. Therefore, the measurement of natural recharge using this entropy-based model will be more accurate, if the precipitation is recorded at each nearby well site. Cross plot of soil infiltration rate and natural recharge. The soil infiltration experiment was carried out at nine sites in barren lands of the study area (Fig. 1c). The infiltration rates ranged from 12.0 to 45.0 cm/ hr, with an average of 23.9 cm/hr. A cross plot of the estimated natural groundwater recharge with the infiltration rate had been made. A good positive correlation between the infiltration rate and the percentage of natural groundwater recharge was obtained with R 2 = 0.75 in the monsoon period, as shown in Fig. 5.

Recharge variability in morphodynamic units. Geomorphic features combined with structures and
lithology controls allow the occurrence, movement, and quality of groundwater. It plays an important role in the identification of favourable zone for groundwater recharge. In the study area, the morphological units such were not responding significantly due to the rainfall. It had found that the average percentage of estimated natural groundwater recharge at active flood plain was ~ 38.41(%) of annual rainfall, and ~ 54.08% of monsoon rainfall. In contrast, it was ~ 18.82% and ~ 32.14%, respectively, in palaeochannel, which is the main charter in terms of geomorphology in this river basin. In the flood plain (area: ~ 304 km 2 ), mainly located in the central, southern, eastern, and western parts of the area, the percentages of natural recharge were ~ 16.48% of annual rainfall and ~ 32.48% of monsoon precipitation (Fig. 6). It had been observed that the active flood plain was more dominant for the natural groundwater recharge comparatively the natural levees.
Natural groundwater recharge. The percentage of natural groundwater recharge was estimated using the developed entropy-based model for the data of the year 2012-2014, because monthly groundwater level data were available at 50 well sites within a dense monitoring well network in the research area. It is the general practice when the percentage of recharge is known, used for the estimation of natural groundwater recharge simply measuring the precipitation of that area. This approach was also applied in the tritium injection method by several researchers 17,18 . The calculated natural groundwater recharge using the information-based model varied from 95.25 to 184.11 MCM/monsoon rainfall, with an average of 127.98 MCM for the period of 2010-2019 (Fig. S4), which was almost the same order reported by CGWB and other researchers in the study area 49,50 . Also, it was in the equivalent of about 121.46 MCM/monsoon rainfall estimated by the water level fluctuation  The spatial variability of natural groundwater recharge during the SW monsoon season in 2019 had been estimated by knowing the percentage of recharge and precipitation of that area, as shown in Fig. 7. It had been observed that the recharge varies spatially depending upon the influence factors such as geomorphology, soil, and drains pattern 23 . Its' magnitude was nicely collaborated, with the infiltration rates of clay, sand, and sandy clay 48 . It had been noticed that the natural reserve was negligible at the well areas located at Phulwari (W-26) and Dalluchak (W-29), which lay in natural levees and back swamp, respectively, at the middle-east of the study area. The reserve was also negligible at Raghopur (W-40), located in the extreme western part. Broadly, the natural groundwater reserve was in the range of 150-311 MCM in the northern and southern parts (Fig. 7), whereas it was in the range of 100-250 MCM in the monsoon period of 2019 at the middle part. But it was about 50-100 MCM in the part of Patna city, and the south-western part. This estimated groundwater reserve could be utilized to develop groundwater flow modelling and budgeting of water resources in this area.

Conclusion
The information-based model has been hypothesized and developed to estimate natural groundwater recharge using simply measured groundwater levels in shallow aquifers and rainfall. It has been applied to evaluate groundwater reserve in a morphodynamic system in a part of the Ganga basin, Northern India. The results show that the average natural recharge rate is about 246 mm/ monsoon rainfall, and 32.65% of the seasonal monsoon rainfall. The estimated recharge is well-correlated with the soil infiltration rate, as well as the monsoon groundwater reserve (MGWR) obtained from the water level fluctuation method. It has been found that the natural groundwater recharge rates in an active flood plain, back swamp, natural levees, flood plain, and palaeochannel of the study area are about 54.08, 45.85, 33.77, 32.48, and 32.14% of the seasonal rainfall, respectively. The estimated natural groundwater reserve varies from 95.25 to 184.11 MCM, with an average of 127.98 MCM during the monsoon period from 2010 to 2019. The reserve is nicely manifested with the soil infiltration rate and the geomorphic units. Although other factors such as vegetation, topography, geology, and climate control the recharge, and therefore, there is an impact of the selecting technique for estimating groundwater recharge. But the estimated natural groundwater recharge using this developed information-based model will help to improve groundwater www.nature.com/scientificreports/ management practices such as the construction of artificial recharge structures, and simulating groundwater flow model for the budgeting of water resources on a regional scale at a glance. This approach could be adopted for estimating natural groundwater reserves at any hydrogeological setting under diverse meteorological conditions without any hydrogeological property.

Data availability
The data source: http:// cgwb. gov. in/ AQM/ Pilot/ Patna% 20Dis trict ,% 20Bih ar-Final. pdf and the analyzed information during the current study are available from the corresponding author on reasonable request.