A comparative analysis of MLR, MARS, and GMDH techniques for predicting the recharge rate of a sand-mixed storm runoff filtration system Open Access

In recent periods, the population growth rate of urban areas of India has been increased by eight times. In the first three decades of the 20th century, the urban area level rose by 11–12% due to population growth, reaching 17.3% by 1951 and 25.7% in 1991. The growth of the urban area population is higher than the overall population growth rate of 2%. The urban population is expected to increase to 658 million by 2025. The increasing population trend is attributed to rural-to-urban migration seeking better livelihoods (CGWB 2014; Kumar & Singh 2021, 2023a; Pir 2021).

In India, groundwater plays a significant role in meeting the requirements of Indian cities. 60% of irrigation and 80% of drinking water demands are met from groundwater (Raheja et al. 2022). So, the average water table is declining by 15% annually. In the northern states of India (Haryana and Punjab), water tables have been falling at an alarming rate of 25–70 cm for the past three decades, affecting agriculture and increasing the cost of pumping (CGWBD 2020). The groundwater decline can be slowed by increasing groundwater recharge using rainwater harvesting and stormwater management. These techniques aim to recharge groundwater using stormwater runoff at its source simply and naturally (Yuan et al. 2017; Osheen & Singh 2019, 2020; Environmental 2020). However, artificial groundwater recharging must ensure that groundwater does not get polluted. It necessitates the filtration of stormwater runoff, which artificial groundwater recharge is a process where the groundwater reservoir is augmented at a rate higher than natural recharge. Problems and questions arising from field operations indicate the need for a thorough investigation into the use of filtering media to achieve high infiltration rates in recharging wells. The filtration unit must perform effectively to realize the potential benefits of installed recharge structures. The most critical issue with the efficiency of the filtering unit is clogging, i.e., a decrease in the permeability of the filtering medium due to governing physical processes (Bouwer 2002). Additionally, there are no well-defined criteria for designing the thickness of different layers of filter material (Dillon et al. 2009; Kumar & Singh 2023a, b, 2024). Coarse sand (CS) or medium sand (MS) is the finest filter material and is first exposed to runoff water to retain suspended particulates. Therefore, the particle size of CS or MS plays an important role, but it is not standardized. It leads to uncertainty in achieving adequate recharge rates and causes frequent filter clogging, which may contain suspended sediment.

The literature review revealed that experimental work to assess recharge rate with various input parameters is time-consuming, costly, and energy-intensive. As a result, predictive models were reviewed to minimize the need for extensive experimental efforts in estimating recharge rate. These models play a crucial role in reducing energy costs, optimizing biological processes, maintaining regulatory compliance, and ensuring the long-term sustainability of operations in water treatment, aquaculture, and other industries reliant on aeration. Recently, soft computing techniques have been implemented by the various researchers in the field of civil and water resources engineering (Singh et al. 2017, 2021, 2022, 2023; Arora et al. 2019; Sihag et al. 2019, 2021, 2022; Pandhiani et al. 2020; Singh 2020; Aradhana et al. 2021; Bhoria et al. 2021; Sepahvand et al. 2021; Nivesh et al. 2022; Arora et al. 2024; Singh & Minocha 2024a, b, c). In this study author evaluated and compared the performance of linear regression, Multi-Layer Perceptron (MLP), and Long Short-Term Memory (LSTM) models in predicting groundwater recharge. Linear regression performed the worst, while the MLP model improved predictions on the other side LSTM achieved the best performance (Huang et al. 2019). In this study various soft computing methods to identify the most accurate model for predicting groundwater recharge rates, critical for resource management. Experimental data were used to evaluate the Gaussian process (GP), M5P tree, random forest (RF), and empirical models. While all methods demonstrated good performance, the RF model outperformed others (Sihag et al. 2020). This study evaluates the recharging rate of a stormwater filter system using GP and Support Vector Machines (SVM) with four kernel functions: normalized poly, polynomial, Pearson VII (PUK), and radial basis (RBF). Results showed that the Pearson VII kernel-based GP regression performed best among the other models (Sihag et al. 2018). Mogaji et al. (2015), developed a multiple linear regression (MLR) model to estimate groundwater recharge rates in southern Perak, Malaysia, by correlating rainfall recharge rate with geophysical parameters (Anthony et al. 2015). Considering the recent implementation of soft computing in water resource engineering, this study aims to advance groundwater recharge prediction methods by leveraging modern soft computing approaches such as MLR, MARS, and GMDH. These models not only enable accurate recharge rate estimation but also allow for analyzing the influence of multiple input parameters. This combination of soft computing techniques to predict recharge rates has not been explored in previous research. Before predicting the recharge rate with the help of the observed data set, this study also aims to evaluate the recharge rate of CS when used as the top layer in the filter medium of groundwater recharging wells with sediment-laden runoff. Simultaneously, the impact of various input parameters on output parameters was identified.

An outlet is provided at the bottom of the column to drain out filtrate water. A measuring bucket is provided at the bottom of the column to measure filtrate water from the outlet. The column is first filled with the pebble (2–4 cm diameter) in a 20 cm thick layer. Then, a layer of 22 cm of gravel with a 0.8–2 cm diameter is placed over the pebble layer. A layer of 3 cm of marble chips having 0.475–0.8 cm diameter is put on over gravel because of more stability of sand formation on gravel to reduce sand passing through gravel for all the experiments, the thickness of the pebble, gravel, marble chips are kept constant. Then, a layer of sand with variable thickness and diameter filled over the gravel pack is tried. Before each experiment, clean tap water was put through a filtering medium to remove any soluble contaminants for 10 min. Dry fine sand was sieved by passing through a 0.150 mm sieve and retained from 0.075 mm to create synthetic sediment loads with a sediment load of 250–3,000 ppm for laboratory testing. All experiment runs employ the synthetic sediment load with a constant head of 30 cm. As shown in Table 2, different CS media were maintained at varying thicknesses as long as they met the minimal thickness standards necessary to successfully reduce the sediment content in the outflow filtrate water from 3,000 to 0 ppm. Experiments are conducted with varied concentrations of sediment loads of 250; 500; 1,000; 1,500; 2,000; 2,500; and 3,000 ppm in the recharging water to simulate field conditions of runoff. Gravels and boulders' sizes as well as their thicknesses are kept constant for all treatment combinations. Clean tap water is passed through a filtering medium for 10 min before each experiment run to drain any soluble materials. Tables 1 and 2 summarise the experimental plan.

The recharging rates are affected by the sediment deposition at the upper layer of CS, which causes the filter media to clog. The time between the start of the experiment and the point at which the recharging rate is constant is defined as the clogging time. After each trial, the top 10 cm of filter material and deposited sediments were meticulously removed to determine removal effectiveness. The material was dried, sieved (size 0.150 mm) to remove the adhering fine sand particles, and weighed again. The amount of material to be deposited throughout the experiment run was determined by comparing the weight of the fine sand before and after sieving to create a synthetic sediment load. Then the whole CS material from the rectangular column is drawn out and dried in the open air, and new sand of a specific size and thickness is replaced in the column. The above procedure is repeated for the next experiment by varying the CS sizes and thickness.

Soft computing is a paradigm in computational intelligence that employs approximate solutions to handle complex real-world problems where precision is not strictly required. It integrates techniques such as MLR, deep learning, boosting method, Gaussian processes, fuzzy logic, neural networks, and evolutionary algorithms, enabling systems to mimic human decision-making and adapt to uncertainty. In this study, we utilized three techniques: MLR, MARS, and GMDH. The combination of these methods has not been explored and is notably absent in the existing literature.

In Equation (1), f(t) represents the recharge rate, variable, c₀, c₁, c₂, c₃, … , c_n denotes the regression coefficients, and x₁, x₂, x₃, … , x_n are the independent variables.

The MARS approach can identify interactions among various variables. It employs a divide-and-conquer strategy by partitioning the training data sets into distinct regions, where each region is assigned its regression line. As demonstrated by the work of Jin and Chen in 2000, MARS has been successfully utilized in Engineering (Jin et al. 2001). In a study conducted by Kim et al. in 2003, researchers adopted the MARS technique to simulate the transport of pesticides in soils (Kim et al. 2003). They highlighted that MARS could accurately simulate complex physicochemical phenomena even with limited training data. Furthermore, MARS has been employed to predict the distribution of freshwater diadromous fish in New Zealand by establishing relationships between fish species and various environmental variables. In another investigation by Zhou and Leung in 2007, MARS was used to forecast the maintainability of object-oriented software (Zhou & Leung 2007). The researchers concluded that MARS performed with comparable accuracy to other models examined in their study.

The following relationships have been derived between recharge rates (Q in lpm) as an output parameter and the inflow parameters sediment load (S in ppm), size of CS (D in mm), and thickness of CS (T in cm). In this study, 63 observations were used to predict the recharge rate of the rectangular filtration system. 75% of the data was used as a training data set, and the remaining 25% was used as a testing data set.

Based on statistical measures such as coefficient of determination (R²), root mean square errors (RMSE), and Nash–Sutcliffe efficiency (NSE) model coefficients, the performance of MLR, MARS, and GMDH was assessed. The relationships employed in the study are listed in the following.

• R²: It can be calculated as Equation (4):

  

  (4)
• RMSE: The Equation (5) of the RMSE is:

  

  (5)
• NSE model efficiency coefficient: The NSE can be computed as Equation (6):

  

  (6)

  where N is the number of observations of rate; ⁠; ⁠, and ⁠.

The mobility of suspended particles within the CS beds is influenced by both the size of the CS particles and the sediment load of the input water. When sediment-filled water has reached the CS beds, sediments are trapped at different depths. The number of entrapped sediments reduces when moving deeper into the bed, with the largest concentration occurring in the first few centimetres. The top 10 cm of all CS beds collect more than 60% of the suspended particles. The sediment from the inflow load deposits at the top surface of the CS, causing a significant impact on the recharging rates by clogging the filter media. The duration from the start of the experiment until the recharging rate is constant is considered the clogging time.

Interestingly, bigger particle CS beds (T3) allow sediments to penetrate deeper layers than smaller particles (T1 and T2), independent of the sediment load entering the CS beds. It shows that thicker CS beds are required to successfully lower the sediment content in the influent water to the necessary level. According to varied sediment concentrations in the input water, empirical relationships have been created to connect the sediment load with the distance passed through the CS medium.

The sediment load of the recharging water increases when the sediment load in the outflow should increase. However, it has been shown that the sediment load in the outflow becomes insignificant due to the addition of marble pieces beneath the CS layer. Regression models have been built for all treatments to connect the input sediment loads to the observed effluent recharging rate. For all treatments, the filter bed thickness of 70 cm yielded the best-fitting regression equations.

As the sediment load increased, there was a subsequent decrease in the recharge rate. The instantaneous obstruction of flow routes by the recharging water caused a considerable drop-in recharge rate in all three CS beds with greater sediment levels.

During the experiment, at the 250-ppm turbidity level, the turbidity remained constant, and the recharge rate remained relatively constant. However, at the turbidity level of 500 ppm, the recharge rate initially remained constant for a certain period after the start of the experiment. After this initial period, within just a few minutes, the recharge rate dropped to the minimum level.

The turbidity remained consistent throughout the trial at the turbidity level of 250–250 ppm, and the recharging rate remained relatively constant. However, at the turbidity level of 500 ppm, the recharge rate initially remained constant for a certain period after the start of the experiment. After that, within a few minutes, the recharge rate reaches the minimum level. On top of the CS, suspended particles create a clogging layer. This layer slowed the pace of recharging.

Figure 2 demonstrates powerfully that, for all treatments, the recharge rate declines as the sediment load rises, which used a CS medium of various thicknesses and discovered the lowest recharge rate compared to the other treatments. In addition, treatment T2 is outperforming T1 in terms of performance. Across all treatments and levels of sediment load, T2 had a greater recharge rate than T1. It was observed that treatments T3 had higher recharge rates than T1 and T2 due to their increased thickness, regardless of the sediment load in the inflow, due to intrinsic pores in the CS layer of the filtration system in T3, which facilitated increased recharge rates. However, as the sediment load in the inflow water increased, there was a difference in recharge rates between T1, T2, and T3. However, the disparity in recharge rates between the treatments gradually decreases. The difference in recharge rates between the treatments eventually disappeared as the sediment load in the incoming water rose. Notably, a substantial reduction in recharge rate occurred in T3 due to the blockage of intrinsic pores by dispersed particles, which resulted when the sediment load reached 1,500 ppm or higher.

Comparing treatments with different thicknesses of CS media, it was observed that treatments with 65 cm thickness indicated slightly higher recharge rates than those with 60 cm thickness. In comparison, treatments with 70 cm thickness indicated slightly higher recharge rates than those with 65 cm thickness. The increase in recharge rate for T3 with greater thickness can be attributed to lower head loss associated with increased thickness of the CS filter media. Treatment T3 consistently produced the highest recharge rates, averaging 4.0, 4.80, and 5.20 lpm for inflow sediment load levels during a 30-minute experimental run, with thicknesses of 60, 65, and 70 cm, respectively. The thickness of the CS layer in the filtration unit played a significant role in controlling the recharge rate of the effluent. However, the change in the outflow rate did not align proportionally with the variations in the thickness of the CS layers.

Performance in terms of recharge rate yielded the following conclusions from the conducted experiment:

• 1. About 60% of suspended solids were trapped in the upper 10 cm CS layer for all three CS sizes. The accumulation in this upper layer increased as the influent sediment concentration rose.

• 2. The recharge rate of CS of size 0.425–0.600 mm reached a constant level later than 0.150–0.300 mm and 0.300–0.425 mm, regardless of influent sediment concentration. These findings suggest that using larger CS particles in the filtering unit can enhance the performance of CS media.

• 3. Recharge rates were high for all CS thickness layers when the inflow sediment concentration was 1,000 ppm or less. However, a sharp decline in recharge rate occurred at higher sediment concentrations of all sizes due to rapid clogging of flow pathways. These results indicate that the performance of filter media in practical applications can be improved by implementing measures to reduce the initial sediment load in the inflow water before it reaches the filter media, i.e. pre-sedimentation tanks.

• 4. Treatments with a greater thickness of CS media exhibited higher recharge rates than treatments with a lower thickness.

• 5. The predicted values of MLR, MARS, and GMDH recharge rates lie between error lines of ± 25%.

• 6. For validation, the values of R², RMSE (l/s), and NSE for MLR, MARS, and GMDH were obtained 0.874, 0.014, and 0.860 respectively, 0.958, 0.008, and 0.957 and 0.924, 0.012, and 0.907 respectively

• 7. The performance of the MARS model is the best.

The model of recharge rate using MARS shows potential and merits further attention in the field of water resources engineering.

S.K. and K.K.S. conceptualized the study; did formal analysis, and carried out investigation; S.K. prepared and wrote the original draft; K.K.S. wrote, reviewed, edited, and supervised the study.

The National Institute of Technology (NIT) Kurukshetra Director and the Ministry of Education (MOE), Government of India, together provide financial support for the present research project in the form of a PhD scholarship award with reference number 2K19/NITK/PHD/61900082.

All the authors have been reviewed and approved, who have provided their consent for publication.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.