Effects of pumping flow rates on the estimation of hydrogeological parameters

Groundwater, as a source of water supply, is of great importance for many rural and urban communities. It is a valuable resource that nearly half of the world's population uses for several activities such as irrigation, consumption and industrial use (Brindha & Elango 2015). Nonetheless, during the last decades, intensive agricultural activities contributed to the quantitative and qualitative degradation of groundwater, as traditional irrigation practices were widely used without consistent management of chemical fertilizers and pesticides (Ncibi et al. 2020). Moreover, groundwater quality and quantity are being threatened by increasing population and changing lifestyles, overconsumption, rapid urbanization, industrial wastewater and abusive farming practices (Gardner & Vogel 2005; Saidi et al. 2011; Hamed et al. 2022; Kirlas et al. 2022). Well-planned management of groundwater requires reasonable estimation of the hydrogeological parameters of an aquifer, such as transmissivity (T) and storativity (S). The calculation of T and S can be used for the modeling of groundwater flow, as well as for the prediction of contaminant transport as a step required for the planning and implementation of groundwater remediation activities (Sanchez-Vila & Fernàndez-Garcia 2016; Demir et al. 2017). Proper evaluation of these parameters, based on drawdown measurements, constitutes the inverse problem of groundwater hydraulics (Yeh 2015). The aim of the inverse modeling approach is to find the aquifer parameter values that minimize an objective function that calculates the differences between the observed and the simulated values of the state variables (Smaoui et al. 2018). Difficulties arising quite often in praxis are scarcity of accurate and sufficient groundwater level data, economically unattainable measurements, as well as the inaccessibility of study areas (Kirlas 2017; Smaoui et al. 2018).

The Theis (1935) and the Cooper & Jacob (1946) methods are still predominant for the evaluation of hydrogeological parameters (Chapuis 1992; Osiensky et al. 2000; Avci et al. 2012; Anomohanran & Iserhien-Emekeme 2014; Kirlas 2021; Pfannkuch et al. 2021; Ali et al. 2022). The basic assumptions underlying the methods are the following: the aquifer is confined, homogenous, isotropic and pumped at a constant flow rate, the pumping well penetrates the total thickness of the aquifer, the piezometric surface is horizontal before pumping and the well diameter is small (Kruseman & de Ridder 2000). Moreover, Boulton (1954, 1963) proposed an analytical method for unconfined aquifers by exhibiting a delayed yield concept, whereas Prickett (1965) suggested a systematic graphical approach based on the type curve methods of Boulton. Moench (1995) proposed a combination of the Boulton and Newman methods for unconfined aquifers.

Notwithstanding, additional techniques have been proposed for the calculation of aquifer parameters, including Newman analytical solution (Neuman 1972; Naderi & Gupta 2020; Gunawardhana et al. 2021), numerical evaluation (Halford et al. 2006; Tumlinson et al. 2006; Lin et al. 2010; Chattopadhyay et al. 2015; Calvache et al. 2016), electrical resistivity tomography (González et al. 2021; Rao & Prasad 2021), hydraulic tomography based on geostatistical inversion (Yin & Illman 2009; Illman et al. 2015), direct push technologies (Dietrich et al. 2008; Bohling et al. 2012) and supervised committee machine with training algorithms (Tabari et al. 2021).

Genetic Algorithms (GAs) are increasingly used in groundwater hydraulics, because of their ability to solve multivariable complex problems, with a known objective function (Katsifarakis & Kontos 2020). They are probabilistic algorithms that mimic the functioning of natural phenomena, such as genetic inheritance and the Darwinian struggle for survival. They have been widely used in optimizing quantitative and qualitative aquifer management (McKinney & Lin 1994; Rauch & Harremoës 1999; Erickson et al. 2002; Kontos & Katsifarakis 2017; Seyedpour et al. 2021).

Moreover, GAs have been used to estimate the transmissivity of non-homogenous aquifers under a steady flow (Karpouzos et al. 2001). Applications include coastal aquifers, as well (Smaoui et al. 2018). Ha et al. (2020) used a GA combined with the Levenberg–Marquardt algorithm and with the Neuman and Witherspoon model and ratio method to accurately estimate aquifer parameters from pumping tests. Thomas et al. (2018) proposed a new simulation–optimization model for the estimation of aquifer parameters by coupling the radial point collocation meshfree method with cat swarm optimization. GAs are combined with Theis equation in this paper to solve the inverse groundwater problem. Their results are compared with the results obtained from Cooper–Jacob for a variety of pumping flow rates. The next section briefly presents the framework of this analysis.

During the second approach:

• 1.

  GAs are used to create populations and generations of possible T, S solutions.

• 2.

  Hypothetical drawdown curves are created using these T, S values and Theis equation.

• 3.

  RMSE is used to estimate the difference between the actual and the hypothetical drawdowns.

• 4.

  The set of T, S that minimizes the RMSE between the actual and the hypothetical drawdowns is considered the solution to the inverse problem. In other words, RMSE is the fitness value of the genetic algorithm. This set is the solution of the optimization.

• 5.

  The process is repeated for a new pumping flow rate hypothesis.

For their method to be applicable, Cooper & Jacob (1946) recommended that u values should not exceed 10⁻². Many authors, such as Freeze & Cherry (1979), Schwartz & Zhang (2003), Todd & Mays (2005) follow their suggestion, while Fetter (2001) and Sterrett (2007) affirm that a maximum value of u_max = 0.05 is satisfactory. Nevertheless, Alexander & Saar (2011) propose a significantly higher value of u_max = 0.2.

Gomo (2019) demonstrated that in Cooper–Jacob method there might be some inaccuracies in transmissivity and storativity values, irrespective of the u value. For this reason, instead of using u, another objective criterion was proposed, namely the Infinite Acting Radial Flow (IARF) condition, in order to determine the applicability of Cooper–Jacob method (Spane 1993; Renard et al. 2009; Gomo 2020).

Furthermore, Kirlas & Katsifarakis (2020) investigated the accuracy of T and S values and showed that the precise recording of pumping initiation and shutdown time is of crucial importance, because when pumping is estimated to start earlier than it actually does, transmissivity is underestimated, while storativity is overestimated. Additionally, they concluded that when the residual drawdown is substantial, the transmissivity value might be overestimated.

It is important to mention that RMSE indicates that the results are accurate in case the input data are accurate. In this paper, we use inaccurate data (testing a number of possible pumping flow rates), therefore this inaccuracy is inserted in the outputs of the algorithm. In other words, low RMSE values indicate that the inverse problem algorithm-method operates correctly. However, this does not guarantee that the outcomes should be trusted. This point is better explained in the following sections.

As mentioned in the previous sections, GAs have been used to estimate T and S values, based on field measurements. In this case, the decision variables, which are included in the chromosomes, are T and S. RMSE as defined above, can serve as an evaluation function. RMSE has been already efficiently combined with GAs (Bastani et al. 2010; Amaranto et al. 2018). RMSE is a useful tool, when combined with GAs, because it can be used to calculate the fitness values of the chromosomes. Bastani et al. (2010) use RMSE in groundwater flow modeling to compare how close observed and simulated values are and Amaranto et al. (2018) use RMSE to test the accuracy of forecasting. One recent study uses RMSE to find the characteristics of aquifers (T, S values) and the time schedules of pumping, when more than one wells pump simultaneously (Nagkoulis 2021). The main idea behind that is that the aquifer behaves like a conduit and the well-piezometer as a receiver. For each aquifer, there is a unique hydraulic drawdown curve ‘received’ by the well-piezometer. When the hypothetical hydraulic drawdown, created for a hypothetical set of T and S values, matches to the actual hydraulic drawdown measured at the well-piezometer, the hypothetical T and S values correspond to the actual aquifer characteristics. Consequently, the objective of GAs is to find the aquifer characteristics (T₀, S₀) that minimize RMSE_(T,S). In Figure 4, we can see that after a number of generations, RMSE stabilizes, reaching its minimum values. The script of the aforementioned paper is applied in this one too, using actual drawdown data. The algorithm is written in R studio, using the GA package (Scrucca 2013).

In the aforementioned paper, it has been noticed that even though 1,000 generations were used, the solution was usually found before generation 300. The variable inputs are inserted in the algorithm in binary form using 20 digits for T and 17 digits for S. These digits are chosen so that the algorithm can search in a wide space from 10⁻¹ to 10⁻⁸ approximately for possible T and S solutions. A typical ‘rank selection’ is used, following the package's initial settings. The computational time was approximately 10 h using a typical i7 CPU. It should be mentioned however that this time can even be reduced in 1 h in case that less generations are used. We have used the following parameters in our code: Number of generations: 1,000; population size: 100, crossover probability: 0.85, mutation probability: 0.45. Running the same code for 30 min (using 100 generations) for some specific cases, resulted in very similar results. The mutation probability was chosen after a number of tests. For low probabilities, the algorithm was often trapped in local minima. The selection process included elitism, preserving the three best chromosomes of the current generation for the new one.

In most cases, the pumping flow rate is unsteady in practice. There can be many technical reasons for unintentional variation in pumping flow rates, such as pumping well's diameter (attachment of a smaller than suggested hose), intake line obstruction (a common problem is debris blockage and slurry flow) and improperly connected motor (due to incorrectly electrical connections to the electric motor) (Farokhzad et al. 2012; Derakhshan & Bashiri 2018). In the aforementioned cases, the error increases in time. The more time that a well operates, the higher the drawdowns get, the more difficult it is for a pumping system that operates insufficiently to pump water. Nonetheless, there is one more situation that has critical differences from the previous ones and therefore it should be considered a separate case, outside of the scopes of this paper. There can also be variations in the pumping flow rate, due to nearby pumping wells, which might start pumping during the pumping phase of the examined well. Whereas, in the previous cases the energy loss increases with time, in this case, there can be radical increases in drawdowns in the examined well, which cannot be modeled (without additional information) from the tools used.

In Equation (8), HP_pump(W) represents the power that the pump provides to the system, γ (N/m³) is the specific weight of water, Q (m³/s) is the pumping flow rate and s (m) is the hydraulic drawdown. For HP_pump(t)=constant we get Q(t)s(t)=constant. This way, as the hydraulic drawdown increases, the pumping flow rate necessarily decreases. The practical solution to this problem is using an inverter to increase the energy offered by the pump to the system so that HP_pump(t₁)<HP_pump(t₂).

Nonetheless, the main equations used in praxis (Theis and Cooper–Jacob) are derived under the assumption that pumping flow rates are constant. In order to deal with the fact that the flow rate in many cases decreases with time as the water level drops, Kruseman & de Ridder (2000) suggested checking and, if necessary, even adjusting the well flow rate on an hourly basis.

In this paper, on the one hand, we propose RMSE as a parameter that can be used to find out if Theis equation is still valid (under unsteady pumping flow rates) and on the other hand, we prove that errors in pumping flow rates estimation can result in minor errors in T and S evaluation.

Firstly, we applied the Cooper–Jacob method to three groundwater level datasets (drawdown data), in order to determine the hydrogeological parameters, such as transmissivity T and storativity S. To create the diagrams and calculate the transmissivity and storativity values, we used MS Excel, considering that the deviations between MS Excel calculations and other computational tools, regarding the T and S values, are negligible (Kirlas 2017; Kirlas & Katsifarakis 2020).

We can see that the RMSE takes low values and varies from 0.6916 to 0.8980. Finally, it can be seen that the first dataset which contains fewer drawdown measurements results in higher RMSE than the next two datasets. This is because Cooper–Jacob method can solve more accurately the inverse problem when there are more data available.

From Figures 11 and 14, we can see that RMSE is not affected bythe pumping flow rate variations. The nearly-constant RMSE line means the methods solving the inverse problem operate ‘normally’ (Cooper–Jacob and GA–Theis). This means that there exists a logarithmic line or curve that can be used to ‘reconstruct’ the hydraulic drawdowns using these characteristics.

From the RMSE charts, we can see that there is no unique set of (Q, T, S) that minimizes that RMSE. If the pumping flow rate is not accurately obtained from the field survey studies, an error will be inserted in finding the T, S parameters. The variability of the pumping flow rate (which is assumed to be constant in the aforementioned equations), results in errors. This variability might be due to intense heterogeneity of aquifer's geological formations as well as technical issues of the water pumping system (e.g. mechanical jamming in pump, unsuitable pump selection, pump efficiency, power failure and short blackouts with zero pumping flow and intake line obstruction) (Trabucchi et al. 2018). Having tested both Cooper–Jacob and GA–Theis methods we can see that the GA–Theis approach results in lower RMSE values. However, the differences between the RMSE values are not strong. What is the most important is that in both cases the RMSE values present the same characteristics in terms of pumping flow rate variations

One of the most important results of this paper is the linear form of Q–T and Q–S graphs. In the inverse groundwater problem, researchers are usually interested in the order of magnitude of T and S. Thence, the linear relationship between T, S and Q indicates that errors in Q do not strongly affect the results. For instance, using data from the third dataset, for Q = 30 m³/h we get T1.1 × 10⁻⁵ m²/s and for Q = 25 we get T × 10⁻⁵m²/s. An error of 16% in Q will result in an error of 18% in T. This variation is usually inconsiderable in practice. The main reason that makes hydrogeologists interested in T and S values is that they can be used to find hydraulic drawdowns for different pumping flow rates. The logarithmic form of the relationships of groundwater has led the scientific community to target to the order of magnitude of the T and S values. On the other hand, 16% flow variation (5/30) is a large number and should not be considered common. An error of 18% for T should be considered mirror, whereas a 16% miscalculation of Q should be considered major. Schematically, a major miscalculation of Q results in minor miscalculations of T and S, because of the linear relationships between the errors and the non-linear requirements for T and S.

In this paper, we investigate the validity of hydrogeological parameters calculation using the Cooper–Jacob method, the RMSE and the GA. In particular, we examine how pumping flow rate errors affect the hydrogeological parameters estimation, considering a confined aquifer and using actual hydraulic drawdown measurements. Low values of RMSE mean that the calculated values of transmissivity (T) and storativity (S) can accurately reproduce the actual hydraulic drawdown curve. As indicated in the literature review, the GA combined with Theis resulted in lower RMSE that the graphical Cooper–Jacob method. This means that GA can be used to accurately reproduce the actual hydraulic drawdown and therefore the results of T and S obtained by GA should be considered more accurate than those obtained from Cooper–Jacob. We have found out that:

• The RMSE is not affected by inaccurate pumping flow rate estimations. This means that both Cooper–Jacob and GAs can be successfully used to reproduce the actual hydraulic drawdown, regardless the accuracy of the pumping flow rate value.

• The transmissivity and storativity values decrease as the pumping flow rate decreases, in a linear way. Hence, errors in pumping flow rate estimation should be considered minor by researchers who are interested in finding the order of magnitude of an aquifer's T and S values.

Overall, we have conducted 48 GAs and Cooper–Jacob tests using three drawdown datasets. The main challenge in terms of future research is to apply this method using more datasets. This way it will be possible to find the characteristics of the linear relationship between Q and T, S errors (slope and constant values). At the same time, it is interesting to use GA to find out how accurately they operate (in terms of solving the inverse problem) when external parameters are included (e.g. simultaneous pumping from a system of wells, geological faults, recharging act). Applying the proposed methodology to a range of aquifers will help in generalizing the results about the pumping flow rates’ effect in transmissivity and storativity errors’ generation. It is also interesting to test how RMSE varies according to the range of values chosen as acceptable (e.g. removing the last drawdowns). The results can be used as a rule of thumb to practitioners, indicating that they should not reject measurements obtained from pumping tests when there are reasonable uncertainties about flow rate estimations.

The authors are indebted to Prof. N. Theodossiou, professor at the Aristotle University of Thessaloniki, Greece, for providing the field data and to Prof. K.L. Katsifarakis for his insightful comments. The genetic algorithm script used is available upon request.

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

The authors declare there is no conflict.