Eliminating Noise of Pumping Test Data Using the Theis Solution Implemented in the Kalman Filter

Abstract

This study presents a novel approach that integrates the Kalman filter and genetic algorithms to obtain the hydraulic parameters of a confined aquifer with precision, eliminating noise that is not normally considered in traditional procedures; these parameters are necessary for the design of wells, the calculation of water balances and the numerical modeling of aquifers. The Theis solution for horizontal radial groundwater flow to an extraction well within a confined aquifer is implemented in the Kalman filter to calibrate the hydraulic transmissivity and the storage coefficient, minimizing the differences between drawdown estimates and the Theis solution by means of genetic algorithms. The estimate error variances provided by the method allowed for the quantification of an approximate average drawdown measurement error of 0.12 m and 0.02 m, respectively, during the execution of two pumping tests.

1. Introduction

Water is an essential substance for regulating the Earth’s temperature, maintaining ecosystem balance and supporting human development. In particular, groundwater plays a vital role in agriculture, industry and domestic water supply, especially in regions with limited surface water availability [1]. The costs associated with groundwater exploitation are generally lower than those associated with the construction and operation of reservoirs and canals. Additionally, aquifers have a lower vulnerability to droughts compared to surface reservoirs [2].

Effective groundwater management is essential to prevent over-extraction, pollution, depletion, scarcity and stress on aquifers. It involves monitoring water levels, implementing sustainable extraction limits and protecting recharge areas, to ensure its availability for future generations [1].

Knowledge of hydraulic parameters in aquifers (such as the hydraulic conductivity, specific yield, storage coefficient or specific storage) is of great importance for the development of any groundwater project. Pumping tests are the usual procedure to determine these parameters at a local scale under field conditions. Alternatively, the performance of oscillatory pumping tests has been promoted [3].

The specialized literature presents various procedures for determining the spatial distribution of hydraulic parameters within a limited area of an aquifer, a method known as hydraulic tomography, using hydraulic head data from pumping tests and additional data from the study area [4,5,6,7]. Inverse modeling of the hydraulic head [8,9], the incorporation of a geophysical data set with a joint scheme [10,11] and the Ensemble Kalman filter [12,13] are commonly used for this purpose; however, these procedures are limited to areas with a high density of observation wells near a pumping well or the availability of geophysical equipment, a condition that is uncommon in many regions of the world.

Traditional methods for interpreting pumping tests have relied on analytical solutions for specific flow conditions that idealize the behavior of water within an aquifer. These methods typically involve graphical techniques to manually match drawdown data [14,15,16,17], meaning that hydraulic parameters are determined by finding the best match between a reference curve and plotted drawdown measurements from observation wells [18]. This approach was a cutting-edge technique at the time and is still used in some technical reports today.

However, with the advancement in modern computational resources, numerical methods have been developed to overcome the limitations of traditional and quasi-analytical solutions, for example, numerical models that solve the groundwater flow equation to both single-layer and two-layered media [19,20].

It has been recognized that drawdown measurement errors, field heterogeneities and other conditions that do not align with the assumptions of analytical solutions used in the interpretation of pumping tests can lead to hydraulic parameter values that do not accurately reflect real aquifer conditions [21,22]. In these cases, the Kalman filter represents an alternative for obtaining reliable values of the hydraulic parameters, as it accounts for both measurement and model errors [6,23]. In these approaches, the hydraulic parameters are updated each time a new drawdown measurement is incorporated into the Kalman filter implementation. The resulting parameter estimates can vary depending on the initial guess provided. It is important to note that, in the cases presented, measurement errors are considered very small or are neglected.

In an alternative approach to interpreting pumping tests, the hydraulic parameters are calibrated using filtered drawdown data. To achieve this, the Cooper–Jacob solution was applied within the Kalman filter [24]. However, a limitation of this method is that the solution is not valid for drawdown measurements taken shortly after the pumping test begins.

In this paper, that limitation is addressed by incorporating the Theis solution as the model used to filter all measured drawdown data collected during a pumping test. The hydraulic parameters are determined using a calibration process that involves a genetic algorithm-based optimization procedure that compares estimated drawdown data and the Theis solution. Furthermore, estimate error variances are used to assess the quality of the drawdown measurements during the pumping test.

The proposed method is intended for practical use in field pumping tests, aiming to improve the accuracy of hydraulic parameter estimation, particularly in confined aquifers. This approach enables the quantification of measurement errors in drawdown observations while also accounting for deviations from the field conditions assumed in the Theis solution.

2. Methods

2.1. Equations That Describe Groundwater Movement

Differential equations describing flow in porous media are derived by combining Darcy’s law with mass balance principles. Mass balance accounts for input and output flows as well as changes in storage. Darcy’s law is strictly valid under laminar flow conditions, and its application in a turbulent regime can result in significant quantitative errors.

Well hydraulics, as a specialized branch of groundwater dynamics, focuses on studying the movement of water toward wells due to the gradients created by water extraction. It aims to relate the extracted flow rate to the spatial and temporal distribution of drawdowns in the aquifer, considering the key hydrogeological parameters of the region.

2.2. Theis Solution

Most of the existing methods to interpret pumping tests start from emulating or modifying the work of C. V. Theis [14], which laid the foundations of modern well hydraulics in transient regime. The method is only valid for ideal confined aquifers with elastic water release. The general equation of the radial flow to a well in a transient regime, considering two dimensions and the non-existence of vertical recharge is as follows:

$${\displaystyle \frac{{\partial }^{2}h}{{\partial r}^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial h}{\partial r}}={\displaystyle \frac{S}{T}}{\displaystyle \frac{\partial h}{\partial t}},$$

(1)

where $h$ is the hydraulic head, $S$ is the storage coefficient (dimensionless), $t$ is the time elapsed from the start of the pumping, $r$ is the distance between the pumping and the observation well and $T$ is the aquifer transmissivity. Theis equation was derived assuming strong limitations on the physical reality of the tested aquifer:

• Homogeneous, isotropic and infinite extension;
• Radial flow and laminar regime;
• Absence of external recharges;
• Fully penetrating, zero diameter well;
• Constant pumping flow, which produces an immediate drop in level [14].

It is used for the determination of hydraulic parameters of a confined aquifer. Following Theis’ formulation [14], these parameters are obtained employing Equations (2)–(4) as presented below:

$$s={\displaystyle \frac{Q}{4\pi T}}W\left(u\right)$$

(2)

where $s=h_0-h\left(t\right)$ is the water level drawdown at a distance $r$ from the pumping well, $Q$ is the constant pumping flow rate and $W\left(u\right)$ is the well function. Setting

$$u={\displaystyle \frac{r^{2}S}{4Tt}},$$

(3)

the well function will be

$$W\left(u\right)={\int }_u^{\infty }{\displaystyle \frac{e^{-u}}{u}}du,$$

(4)

which is the well-known integral function. This integral has no analytical solution, so it is solved by approximate methods:

$$W\left(u\right)=-\gamma -\ln \left(u\right)+u-{\displaystyle \frac{u^2}{2·2!}}+{\displaystyle \frac{u^3}{3·3!}}-{\displaystyle \frac{u^4}{4·4!}}+\dots$$

(5)

where $\gamma \approx 0.577216$ is the Euler–Mascheroni constant. For this work, no truncation or approximation of the well function is employed; we use the complete well function taking advantages of numerical packages.

2.3. Kalman Filter

The Kalman filter is a mathematical algorithm for data assimilation that operates through a mechanism to predict future data and correct possible errors. The filter is sequential since it calculates the solution each time a new measurement is obtained without reusing previous data. With this process, a new estimate of a state is obtained from its previous estimate by adding a correction term that incorporates the information provided by new data, so that the prediction error is statistically minimized. The Kalman filter is constituted of three models and two phases:

System model. The state vector, expressed as ${\mathbf{x}}_{\mathbf{n}+1}$, describes the evolution in time of the quantity to be estimated. The transition between states ${\mathbf{x}}_{\mathbf{n}} \mathrm{a}\mathrm{n}\mathrm{d} {\mathbf{x}}_{\mathbf{n}+1}$ is characterized by the transition matrix ${\mathbf{A}}_{\mathbf{n}+1}$ and the addition of Gaussian white noise ${\mathbf{w}}_{\mathbf{n}+1}$ as follows:

$${\mathbf{x}}_{n+1}={\mathbf{A}}_{n+1}{\mathbf{x}}_n+{\mathbf{w}}_{n+1}, {\mathbf{w}}_{n+1}~\mathrm{N}\left(0,{\mathbf{Q}}_{n+1}\right),$$

(6)

with a covariance matrix ${\mathbf{Q}}_{\mathrm{n}+1}$, which shall not be confused with the discharge $Q.$

Measurement model. Relates the measurement vector ${\mathbf{z}}_n$ to the state of the system ${\mathbf{x}}_n$ through the measurement matrix ${\mathbf{H}}_n$ and the addition of a Gaussian white noise vector ${\mathbf{v}}_n$ with covariance matrix ${\mathbf{R}}_n$.

$${\mathbf{z}}_n={\mathbf{H}}_n{\mathbf{x}}_n+{\mathbf{v}}_n ,{\mathbf{v}}_n~\mathrm{N}\left(0,{\mathbf{R}}_n\right).$$

(7)

Prior model. Describes prior knowledge about the state vector at the initial time ${\mathbf{x}}_n^0$ in terms of the expected value and the covariance matrix ${\mathbf{P}}_n^0$. Process and sensor noises are assumed to be uncorrelated.

$$\begin{array}{ccc}\hfill \mathrm{E}\left[{\mathbf{x}}_n^0\right]& =& {\widehat{\mathbf{x}}}_n^0\hfill \\ \hfill {\mathbf{P}}_n^0& =& \mathrm{E}\left[\left({\mathbf{x}}_n^0-{\widehat{\mathbf{x}}}_n^0\right){\left({\mathbf{x}}_n^0-{\widehat{\mathbf{x}}}_n^0\right)}^{\mathrm{T}}\right]\hfill \\ \hfill \mathrm{E}\left[{\mathbf{w}}_n\right]& =& \mathrm{E}\left[{\mathbf{v}}_n\right]=0\hfill \\ \hfill \mathrm{E}\left[{\mathbf{w}}_n{\mathbf{v}}_n^{\mathrm{T}}\right]& =& \mathrm{E}\left[{{\mathbf{v}}_n\mathbf{w}}_n^{\mathrm{T}}\right]=0\hfill \\ \hfill \mathrm{E}\left[{\mathbf{w}}_n{\mathbf{w}}_n^{\mathrm{T}}\right]& =& {\mathbf{Q}}_n\hfill \\ \hfill \mathrm{E}\left[{\mathbf{v}}_n{\mathbf{v}}_n^{\mathrm{T}}\right]& =& {\mathbf{R}}_n.\hfill \end{array}$$

(8)

Propagation phase. The new value of the quantity to be estimated is predicted using the system model. For this, the estimate of the previous state ${\mathbf{x}}_n$ and its covariance matrix ${\mathbf{P}}_n$ are extrapolated to form the predicted state vector ${\mathbf{x}}_{n+1}^{-}$ and its covariance matrix ${\mathbf{P}}_{n+1}^{-}$, where

$${\mathbf{x}}_{n+1}^{-}={\mathbf{A}}_{n+1}{\mathbf{x}}_n,$$

(9)

and

$${\mathbf{P}}_{n+1}^{-}={\mathbf{A}}_{n+1}{\mathbf{P}}_n{\mathbf{A}}_{n+1}^{\mathrm{T}}+{\mathbf{Q}}_{n+1}.$$

(10)

Update phase. In this phase, the new state vector ${\mathbf{x}}_{n+1}$ and its covariance matrix ${\mathbf{P}}_{n+1}$ are calculated. For this, the predicted covariance is used to calculate the Kalman gain ${\mathbf{K}}_{n+1}$. The new state vector ${\mathbf{x}}_{n+1}$ is calculated adding to the predicted state vector ${\mathbf{x}}_{n+1}^{-}$ the measurement residual ${\mathbf{z}}_{n+1}-{\mathbf{H}}_{n+1}{\mathbf{x}}_{n+1}^{-}$ scaled with the Kalman gain.

$${\mathbf{K}}_{n+1}={\mathbf{P}}_{n+1}^{-}{\mathbf{H}}_{n+1}^{\mathrm{T}}{\left({\mathbf{H}}_{n+1}{\mathbf{P}}_{n+1}^{-}{\mathbf{H}}_{n+1}^{\mathrm{T}}+{\mathbf{R}}_{n+1}\right)}^{-1},$$

(11)

$${\mathbf{x}}_{n+1}={\mathbf{x}}_{n+1}^{-}+{\mathbf{K}}_{n+1}\left({\mathbf{z}}_{n+1}-{\mathbf{H}}_{n+1}{\mathbf{x}}_{n+1}^{-}\right),$$

(12)

$${\mathbf{P}}_{n+1}=\left(\mathbf{I}-{\mathbf{K}}_{n+1}{\mathbf{H}}_{n+1}\right){\mathbf{P}}_{n+1}^{-}.$$

(13)

The Kalman gain is constructed to obtain minimum variance estimates. After each update, the process is repeated, taking as a starting point the new estimates of the state and of the matrix error covariance.

2.4. Implementation of the Theis Solution Within the Kalman Filter

The partial derivative with respect to time of Equation (3) is

$${\displaystyle \frac{\partial u}{ \partial t}}=-{\displaystyle \frac{r^{2}S}{4T{t}^2}}=-{\displaystyle \frac{u}{t}}.$$

(14)

On the other hand, the partial derivative with respect to time of Equation (5) is obtained applying the chain rule as

$${\displaystyle \frac{\partial \left[W\left(u\right)\right]}{\partial t}}={\displaystyle \frac{\partial \left[W\left(u\right)\right]}{\partial u}}{\displaystyle \frac{\partial u}{\partial t}}={\displaystyle \frac{e^{-u}}{t}},$$

(15)

where the derivative of exponential integral function is used. Therefore, the derivative of Equation (2) with respect to time is as follows:

$${\displaystyle \frac{\partial s}{\partial t}}={\displaystyle \frac{Q}{4\pi T}}{\displaystyle \frac{\partial }{\partial t}}\left[W\left(u\right)\right].$$

(16)

Solving ${\displaystyle \frac{Q}{4\pi T}}$ from Equation (16) leads to the following:

$${\displaystyle \frac{Q}{4\pi T}}={\displaystyle \frac{1}{\frac{\partial }{\partial t}\left[W\left(u\right)\right]}}\left({\displaystyle \frac{\partial s}{\partial t}}\right),$$

(17)

since

$$s_2-s_1={\displaystyle \frac{Q}{4\pi T}}\left[W\left(u_2\right)-W\left(u_1\right)\right],$$

(18)

we solve for $s_2$ from Equation (18), substituting (17) for $s_1$, obtaining

$${ s}_2=s_1+{\displaystyle \frac{W\left(u_2\right)-W\left(u_1\right)}{\frac{\partial }{\partial t}\left[W\left(u_1\right)\right]}}\left({\displaystyle \frac{\partial s_1}{\partial t}}\right).$$

(19)

To calculate the value of $W\left(u_2\right)$, it is necessary to know the time in which the subsequent drawdown measurement will be taken. From Equation (16), it follows that

$${\displaystyle \frac{\partial s_n}{\partial t}}={\displaystyle \frac{\frac{\partial }{\partial t}\left[W\left(u_n\right)\right]}{\frac{\partial }{\partial t}\left[W\left(u_{n-1}\right)\right]}}\left({\displaystyle \frac{\partial s_{n-1}}{\partial t}}\right).$$

(20)

In this manner, from Equations (10) and (11), the propagation phase of the Kalman filter is expressed as follows:

$${\mathbf{x}}_{n+1}^{-}=\left[\begin{array}{c}{s}_{n+1}^{-}\\ {\displaystyle \frac{\partial s_{n+1}^{-}}{\partial t}}\end{array}\right]=\left[\begin{array}{cc}1& {\displaystyle \frac{W\left(u_{n+1}\right)-W\left(u_n\right)}{\frac{\partial }{\partial t}\left[W\left(u_n\right)\right]}}\\ 0& {\displaystyle \frac{\frac{\partial }{\partial t}\left[W\left(u_{n+1}\right)\right]}{\frac{\partial }{\partial t}\left[W\left(u_n\right)\right]}}\end{array}\right]\left[\begin{array}{c}{s}_n\\ {\displaystyle \frac{\partial s_n}{\partial t}}\end{array}\right]$$

(21)

and

$${\mathit{P}}_{n+1}^{-}=\left[\begin{array}{cc}1& {\displaystyle \frac{W\left(u_{n+1}\right)-W\left(u_n\right)}{\frac{\partial }{\partial t}\left[W\left(u_n\right)\right]}}\\ 0& {\displaystyle \frac{\frac{\partial }{\partial t}\left[W\left(u_{n+1}\right)\right]}{\frac{\partial }{\partial t}\left[W\left(u_n\right)\right]}}\end{array}\right]\left[\begin{array}{cc}var\left(s\right)& cov\left(s,{\displaystyle \frac{\partial s}{\partial t}}\right)\\ cov\left(s,{\displaystyle \frac{\partial s}{\partial t}}\right)& var\left({\displaystyle \frac{\partial s}{\partial t}}\right)\end{array}\right]\left[\begin{array}{cc}1& 0\\ {\displaystyle \frac{W\left(u_{n+1}\right)-W\left(u_n\right)}{\frac{\partial }{\partial t}\left[W\left(u_n\right)\right]}}& {\displaystyle \frac{\frac{\partial }{\partial t}\left[W\left(u_{n+1}\right)\right]}{\frac{\partial }{\partial t}\left[W\left(u_n\right)\right]}}\end{array}\right]+{\mathit{Q}}_{n+1}$$

(22)

where $var\left(s\right)$ is the variance of drawdowns, $var\left({\displaystyle \frac{\partial s}{\partial t}}\right)$ is the variance of drawdown rates and $cov\left(s,{\displaystyle \frac{\partial s}{\partial t}}\right)$ is the covariance of drawdowns and drawdown rates. In the proposed methodology, these last three terms are defined by the analysis.

To determine the solution to the problem, with the proposed approach, the search technique known as genetic algorithms (GAs) was used, since these types of metaheuristic technique, versatile and easy to implement, have demonstrated their usefulness as tools for optimization in complex problems, especially those in which it is difficult to guarantee the determination of the optimal solution analytically. Thus, GAs have been used as effective tools in various areas of engineering, particularly in the management and monitoring of water resources [25,26,27,28].

GAs, originally proposed by Holland [29], imitate the process of natural evolution such as selection, crossover and mutation. Generally, they begin by randomly generating an initial set of individuals who are candidates to be solutions, that are evaluated using metrics corresponding to the problem. The result of its evaluation is known as fitness, whose values allow a set of candidate solutions to be classified or selected. Subsequently, offspring are produced through reproduction or mutation mechanisms. It is possible to add these offspring to the existing population, or replace the population entirely. This process is repeated iteratively until a certain stopping condition is met.

In order to obtain the $T$ and $S$ values that calibrate the Theis solution, the following optimization problem was posed:

$$\begin{array}{cc}\mathrm{minimize}& f={‖{\mathit{s}}^{\mathrm{est}}-{\mathit{s}}^{\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{s}}‖}^2\\ \mathrm{subject} \mathrm{to}& l_S\le S\le u_S,\\ & 0<T\le u_T, \\ & s_1^{\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{s}}>0\end{array}$$

(23)

where ${\mathit{s}}^{\mathrm{est}}$ and ${\mathit{s}}^{\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{s}}$ are the estimates for the drawdown vectors at times $\mathit{t}$ using the Kalman filter and the Theis solution, respectively.

Thus, for optimization using GAs, an initial population ${\mathsf{\Omega }}_1$ with $N_{\mathsf{\Omega }}$ candidate individuals are created through the pair $\mathit{x}=\left(T,S\right)$, generated randomly considering the limits established by restrictions. Subsequently, the fitness $f$ of each individual is determined by calculating the metric corresponding to the quadratic norm of the difference of the estimated drawdown using the Kalman filter and the Theis solution, as indicated in Algorithm 1. Then, they are selected the best $N_{sel}$ candidates according to this metric, forming the set ${\mathsf{\Omega }}_{sel}$. From a subset of individuals of this selection, an offspring ${\mathsf{\Omega }}_{cross}$ is produced using a crossover operator, consisting of a weighted average. Some elements of this offspring are mutated by randomly replacing elements of the real-coded string of characters that form it, constituting the set ${\mathsf{\Omega }}_{mut}$. Also, a subset of individuals ${\mathsf{\Omega }}_{rand}$ is randomly generated in order to reduce the possibility of premature convergence to a suboptimal solution. Finally, the union of the sets ${\mathsf{\Omega }}_{sel}$, ${\mathsf{\Omega }}_{cross}$, ${\mathsf{\Omega }}_{mut}$ and ${\mathsf{\Omega }}_{rand}$ form a new population, with $N_{\mathsf{\Omega }}$ individuals, which replaces the previous generation. This process is repeated iteratively until the stopping criterion is met, which is a function of the convergence of the last $\xi$ generations or a maximum number of iterations is reached (see Algorithm 2).

Algorithm 1. Fitness function $f(\mathit{x},\mathit{t},{\mathit{s}}^{obs},N,Q,r).$

Input: $\mathit{x}=\left(T, S\right),\mathit{t}, {\mathit{s}}^{obs}, N, Q,r$ Output: $f$ Prior estimate vector ${\mathit{x}}_1={\left[\begin{array}{cc}{s}_1& {\displaystyle \frac{\partial s_1}{\partial t}}\end{array}\right]}^T={\left[\begin{array}{cc}{s}_1^{obs}& {\displaystyle \frac{Q}{4\pi T}}\left({\displaystyle \frac{1}{t_1}}\right)\end{array}\right]}^T$ for $n=1$to $N-1$ do      Propagation phase ${\mathit{x}}_{n+1}^{-},{\mathit{P}}_{n+1}^{-}$      Update phase ${\mathit{K}}_{n+1}, {\mathit{x}}_{n+1}, {\mathit{P}}_{n+1}$      Theis solution ${\mathit{s}}^{\mathrm{Theis}}$      Fitness function $f={‖{{\mathit{s}}^{\mathrm{est}}-\mathit{s}}^{\mathrm{Theis}}‖}^2$      return $f$

Although Algorithm 2 depends on the creation of random individuals, the results are reproducible when a large population number is employed. Nevertheless, we compare to other algorithms [30,31], to verify the veracity of the results.

Algorithm 2. Genetic algorithm optimization.

Input: $\mathit{t},{\mathit{s}}^{\mathrm{obs}},N, Q,r, l_S, u_S, u_T,N_{\mathsf{\Omega }}, N_{\mathrm{s}\mathrm{e}\mathrm{l}}, N_{\mathrm{cross}}, N_{\mathrm{mut}}, N_{\mathrm{rand}},\xi , Tol,i_{max};N_{\mathsf{\Omega }}=N_{\mathrm{s}\mathrm{e}\mathrm{l}}+N_{\mathrm{cross}}+N_{\mathrm{mut}}+N_{\mathrm{rand}}$ Output: F $\mathrm{Set} i=1$ Initial Population ${\mathsf{\Omega }}^1=\left\{{\mathit{x}}_k^1|{\mathit{x}}_k^1=\left(T_k^1,S_k^1\right)={\mathrm{rand}}_{\mathbb{R}}\left(u_T,l_S, u_S\right); k=1, 2,\dots N_{\mathsf{\Omega }}\right\}$ while $i\le i_{max}$ do     update phase     fitness evaluation ${\mathit{F}}^{\mathit{i}}=\left\{F_k^i|F_k^i=f\left({\mathit{x}}_k^i,\mathit{t}, {\mathit{s}}^{\mathrm{obs}}, N, Q,r\right); k=1, 2,\dots N_{\mathsf{\Omega }}\right\}$     ${\mathsf{\Omega }}_{\mathrm{sort}}=\mathrm{sort}\left({\mathsf{\Omega }}^i,{\mathit{F}}^{\mathit{i}}\right)$     selection ${\mathsf{\Omega }}_{sel}^i=\left\{{\mathit{x}}_k^{\mathrm{sel}}|{\mathit{x}}_k^{\mathrm{sel}}\in {\mathsf{\Omega }}_{\mathrm{sort}};k=1, 2,\dots N_{\mathrm{s}\mathrm{e}\mathrm{l}}\right\}$     if $i>\xi$ then         if ${\sum }_{j=i-\xi }^{i-1}‖{\mathsf{\Omega }}_{\mathrm{sel}}^{\mathit{i}}-{\mathsf{\Omega }}_{\mathrm{sel}}^{\mathit{j}}‖\le Tol$ then         return $\left({\mathit{x}}_1^{\mathrm{sel}}\right)$ stop     crossover ${\mathsf{\Omega }}_{\mathrm{cross}}^i=\left\{{\mathit{x}}_k^{\mathrm{cross}}|{\mathit{x}}_k^{\mathrm{cross}}=\mathrm{cross}({\mathit{x}}_{k_1}^{\mathrm{sel}},{\mathit{x}}_{k_2}^{\mathrm{sel}}, u_T,l_S, u_S); k_1={\mathrm{rand}}_{\mathbb{Z}}\left(1,N_{\mathrm{s}\mathrm{e}\mathrm{l}}\right);k_2={\mathrm{rand}}_{\mathbb{Z}}\left(1,N_{\mathrm{s}\mathrm{e}\mathrm{l}}\right); k=1, 2,\dots N_{\mathrm{cross}}\right\}$     mutation ${\mathsf{\Omega }}_{\mathrm{mut}}^i=\left\{{\mathit{x}}_k^{\mathrm{mut}}|{\mathit{x}}_k^{\mathrm{mut}}=\mathrm{mut}({\mathit{x}}_{k_1}^{\mathrm{cross}},u_T,l_S, u_S); k_1={\mathrm{rand}}_{\mathbb{Z}}\left(1,N_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}\right); k=1, 2,\dots N_{\mathrm{mut}}\right\}$     random individuals ${\mathsf{\Omega }}_{\mathrm{rand}}^i=\left\{{\mathit{x}}_k^{\mathrm{rand}}|{\mathit{x}}_k^{\mathrm{rand}}={\mathrm{rand}}_{\mathbb{R}}\left(u_T,l_S, u_S\right); k=1, 2,\dots N_{\mathrm{rand}}\right\}$     new population ${\mathsf{\Omega }}^{i+1}={\mathsf{\Omega }}_{\mathrm{sel}}^i\cup {\mathsf{\Omega }}_{\mathrm{cross}}^i\cup {\mathsf{\Omega }}_{\mathrm{mut}}^i\cup {\mathsf{\Omega }}_{\mathrm{rand}}^i$ $i=i+1$

Figure 1 presents the flow diagram outlining the steps from Algorithms 1 and 2, which are used to obtain the hydraulic parameters $T$and $S$ by applying the Kalman filter to drawdown data.

3. Case Study

The proposed algorithm was applied to interpret pumping tests conducted in confined aquifers. These data correspond to well-known examples that have become standard exercises for hydrogeologists to evaluate the performance of new techniques.

1.

The “Oude Korendijk” pumping test presented in [32].

This pumping test was carried out in the “Oude Korendijk” aquifer, south of Rotterdam, the Netherlands. The distinguishing features of the impermeable confining layer aquifer is composed of clay, peat and fine clayey-sand for the first 18 m below the surface. Between 18 and 25 m below the surface of the aquifer consists of coarse sand with some gravel, while the bottom of the aquifer, which is considered impermeable, consists of fine sandy and clayey sediments.

The pumping well screen was placed throughout the thickness of the aquifer, and a 20 m deep piezometer was installed 30 m from it. It is assumed that all the extracted water was derived from the aquifer between 18 and 25 m [32].

During the pumping test, the data in

Table 1. Observed drawdown measurements ${\mathit{s}}^{\mathrm{obs}}$ from the “Oude Korendijk” pumping test [32].

Time After Pumping Started (min)	Drawdown (m)	Time After Pumping Started (min)	Drawdown (m)
0.10	0.040	18.00	0.680
0.25	0.080	27.00	0.742
0.50	0.130	33.00	0.753
0.70	0.180	41.00	0.779
1.00	0.230	48.00	0.793
1.40	0.280	59.00	0.819
1.90	0.330	80.00	0.855
2.33	0.360	95.00	0.873
2.80	0.390	139.00	0.915
3.36	0.420	181.00	0.935
4.00	0.450	245.00	0.966
5.35	0.500	300.00	0.990
6.80	0.540	360.00	1.007
8.30	0.570	480.00	1.050
8.70	0.580	600.00	1.053
10.00	0.600	728.00	1.072
13.10	0.640	830.00	1.088

were collected at the piezometer, and the well was pumped at a constant discharge rate of $9.12 \mathrm{L} {\mathrm{s}}^{-1}$ $\left(\mathrm{o}\mathrm{r} 788 {\mathrm{m}}^3{\mathrm{d}}^{-1}\right)$ for almost 14 h.

2.

The “Todd and Mays” pumping test, which is found in [33].

This case study shows an aquifer test in a well that penetrates a confined aquifer; the pumping rate was 2500 ${\mathrm{m}}^3{\mathrm{d}}^{-1}$. The data collected in an observation well 60 m away are presented in

Table 2. Observed drawdown ${\mathit{s}}^{\mathrm{obs}}$ measurements from the “Todd and Mays” pumping test [33].

Time After Pumping Started (min)	Drawdown (m)	Time After Pumping Started (min)	Drawdown (m)
1	0.2	24	0.72
1.5	0.27	30	0.76
2	0.3	40	0.81
2.5	0.34	50	0.85
3	0.37	60	0.9
4	0.41	80	0.93
5	0.45	100	0.96
6	0.48	120	1
8	0.53	150	1.04
10	0.57	180	1.07
12	0.6	210	1.1
14	0.63	240	1.12
18	0.67

. In this case, there is no additional information on the aquifer conditions and the construction characteristics of the pumping and the observation well.

4. Results

To apply the proposed algorithm, starting values of $T=5000 {\mathrm{m}}^2{\mathrm{d}}^{-1}$ and $S=0.0005$ were considered for the two cases, with bounds $100\le T\le \mathrm{10,000}$ and $0.00001\le S\le 0.001$, respectively. Also, a model error covariance matrix

$$\mathit{Q}=\left[\begin{array}{cc}0.00001& 0.00001\\ 0.00001& 0.00001\end{array}\right]$$

(24)

was employed; this implies that model errors are very small.

For the “Oude Korendijk” pumping test, the prior estimate error covariance matrix was

$${\mathit{P}}_{\mathit{n}}^0=\left[\begin{array}{cc}0.2& \sqrt{0.2\times \mathrm{30,000}}\\ \sqrt{0.2\times \mathrm{30,000}}& \mathrm{30,000}\end{array}\right]$$

(25)

Therefore, the initial variance of drawdowns was 0.2 ${\mathrm{m}}^2$ whereas the initial variance of drawdown rates was 30,000 ${\mathrm{m}}^2{\mathrm{d}}^{-2}$. These values were proposed from calculations using measured data.

On the other hand, the variance of the drawdown measurement error (VDME) = 0.015 ${\mathrm{m}}^2$ (this means that an approximate average drawdown measurement error of 0.12 m was made during the pumping test). For this value, the estimate error variances obtained with the Kalman filter are adequate to represent the differences between the estimated values and the Theis equation.

For the “Todd and Mays” pumping test, the prior estimate error covariance matrix was

$${\mathit{P}}_{\mathit{n}}^0=\left[\begin{array}{cc}0.1& \sqrt{0.1\times 3000}\\ \sqrt{0.1\times 3000}& 3000\end{array}\right]$$

(26)

and a VDME = 0.0004 m² (this means that during the pumping test an approximate average drawdown measurement error of 0.02 m was made).

4.1. Interpretation of the “Oude Korendijk” Pumping Test

The values of the parameters obtained with the proposed algorithm were $T=430.92 {\mathrm{m}}^3{\mathrm{d}}^{-1}$ and $S=0.00016$. In Figure 2, it is observed that estimated drawdowns approximate adequately the Theis solution by filtering the measured data with the proposed model and measurement errors.

To evaluate the adequacy of the selected VDME, it was verified that the ${\mathit{s}}^{\mathrm{est}}$−${\mathit{s}}^{\mathrm{Theis}}$ values fall within the range of the standard deviation for the estimation error (SDEE) computed with the Kalman filter. In Figure 3 is shown the reduction of the drawdown estimation errors each time a new drawdown measurement is incorporated in the proposed procedure; these errors fall amply inside the range of the SDEE obtained with the Kalman filter. The selection of a large VDME would overestimate the SDEE; meanwhile, the ${\mathit{s}}^{\mathrm{est}}$−${\mathit{s}}^{\mathrm{Theis}}$ values would go beyond the range of the SDEE for a low value of VDME.

In

Table 3. Parameters estimated by different methodologies [24] for the “Oude Korendijk” pumping test.

Parameter	Theis Procedure	Cooper–Jacob Procedure	Aquifer Win (Theis Solution)	Sen Procedure	Cooper–Jacob and Kalman Filter Procedure (VDME = 0.01 m2)	Theis and Kalman Filter-Based Proposed Procedure (VDME = 0.015 m2)
$T\left({\mathrm{m}}^2{\mathrm{d}}^{-1}\right)$	342–418	401	480.67	342–420	505.76	430.92
$S$	0.00017	0.00017	0.00011	0.00016–0.0002	0.000088	0.00016

is presented a comparison of the $T$ and $S$ values obtained for the “Oude Korendijk” pumping test with the proposal and other methodologies. It can be observed that $T$ obtained with the proposal falls outside the range found by Sen (also using the Kalman filter); the advantage of the proposal is that an approximation of the measurement errors can be estimated. The Cooper–Jacob and Kalman filter procedure seems to overestimate $T$ and underestimate $S$ since it uses a more limited flow solution.

4.2. Interpretation of the “Todd and Mays” Pumping Test

Analogously to the previous case study, the values $T=1140.57 {\mathrm{m}}^2{\mathrm{d}}^{-1}$ and $S=0.00019$ were obtained. In Figure 4, it is observed that estimated drawdowns are very approximated to the Theis solution. The way that measured data are filtered reflects low model and measurement errors.

In Figure 5, it is observed that with the selected VDME for the “Todd and Mays” pumping test, the estimation errors of the drawdowns are well represented by the standard deviations for the estimation errors obtained with the Kalman filter. This reflects that measurement errors for this pumping test are very small.

In

Table 4. Parameters estimated by different methodologies [24] for the “Todd and Mays” pumping test.

Parameter	Theis Procedure	Cooper–Jacob Procedure	Aquifer Win (Theis Solution)	Cooper–Jacob and Kalman Filter Procedure (VDME = 0.01 m2)	Theis and Kalman Filter-Based Proposed Procedure (VDME = 0.004 m2)
$T\left({\mathrm{m}}^2{\mathrm{d}}^{-1}\right)$	1110	1144	1138.17	1180.43	1140.57
$S$	0.00021	0.00019	0.00019	0.00017	0.00019

, it is observed that the T and S values obtained for the “Todd and Mays” pumping test are very approximate to those using the optimization tool of the software Aquifer Win for the Theis equation and the average values obtained in [12] ($T=1142 {\mathrm{m}}^2{\mathrm{d}}^{-1}$ and $S=0.00019$). This is due to the high precision in the drawdowns measured during the pumping test and the good compliance of the Theis solution hypothesis with the field conditions. In these cases, measured drawdowns are easily adjusted with a type curve such as that used in the Theis method. However, for large deviations from a type curve, it takes relevance to consider model and measurement errors in the interpretation of pumping tests. The Cooper–Jacob and Kalman filter procedure seems to overestimate T and underestimate S for the same reasons exposed in the previous case.

4.3. Interpretation of the “Todd and Mays” Pumping Test Adding Noise to the Data

To evaluate the proposal for data with a known measurement error, we generated drawdown data using the Theis solution, applying the hydraulic parameters determined in the previous section for the “Todd and Mays” pumping test. To this synthetic data, we added random Gaussian noise with a mean of zero and a standard deviation of 3 cm. The generated data are presented in

Table 5. Modified drawdown measurements from the “Todd and Mays” pumping test with added Gaussian noise.

Time After Pumping Started (min)	Drawdown (m)	Time After Pumping Started (min)	Drawdown (m)
1	0.242	24	0.753
1.5	0.283	30	0.768
2	0.260	40	0.824
2.5	0.341	50	0.816
3	0.389	60	0.886
4	0.396	80	0.976
5	0.463	100	0.938
6	0.489	120	0.943
8	0.534	150	0.989
10	0.567	180	1.081
12	0.616	210	1.171
14	0.666	240	1.157
18	0.644

. We applied the proposed interpretation method using the same matrices as in the previous example and set the variance of the drawdown measurement error (VDME) to 0.0009 m², corresponding to an average measurement error of approximately 3 cm, consistent with the added random Gaussian noise.

The parameters obtained after implementing our proposal were $T=1140.97{ \mathrm{m}}^2{\mathrm{d}}^{-1}$ and $S=0.00019$, closely matching those from the previous example. As shown in Figure 6, the drawdown measurements are effectively filtered, enhancing estimation accuracy.

Finally, in Figure 7, it is observed that the estimation errors of the drawdowns are well represented by the standard deviations for the estimation errors obtained with the Kalman filter for the VDME selected for the “Todd and Mays” pumping test with added noise.

In total, 1000 sets of drawdown data with added random Gaussian noise, with a mean of zero and a standard deviation of 3 cm, were generated. The proposed method was applied using the same matrices and VDME value (0.0009 m²) as in the previous case to calibrate the hydraulic parameters, obtaining average values of $T=1141.76{ \mathrm{m}}^2{\mathrm{d}}^{-1}$ and $S=0.00019$. Since these values are very similar to those reported for the “Todd and Mays” pumping test, it is demonstrated that the proposed method can provide a good approximation of the average measurement error, allowing for an adequate filtering of the observed drawdowns.

Sen [23] compares the obtained $T$ and $S$ values with those reported in the literature to assess the performance of his approach, stating that a greater resemblance among them indicates a better estimation. The author highlights the importance of selecting appropriate initial values in the estimation process. In our case, values from previous methodologies serve as reference points, but none can be considered the best option. On the other hand, Leng and Yeh [12] evaluate differences between estimated and observed drawdowns using the mean error and the standard error of estimate. The mean error has the drawback of compensating for large negative and positive values, while the standard error of the estimate is used in that study because estimates are obtained at time steps without observed data, which is not our case.

To evaluate the performance of our proposed method, we selected as the primary metrics the Root Mean Square Error (RMSE) defined as

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}$$

(27)

and the Normalized Root Mean Square Error (NRMSE) percentage as

$$\mathrm{N}\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}={\displaystyle \frac{\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}{\overline{y}}}\times 100$$

(28)

where $y_i$ will be $s_i^{\mathrm{e}\mathrm{s}\mathrm{t}}$, ${\widehat{y}}_i=s_i^{\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{s}}$ and $\overline{y}$ is the mean value for drawdowns obtained with Theis. These metrics are presented in

Table 6. Root Mean Square Error and Normalized Root Mean Square Error (%) for the pumping test data presented in this work.

Pumping Test Data	RMSE (m)	NRMSE (%)
Oude Korendijk	0.0126	1.9335
Todd and Mays	0.0026	0.3825
Todd and Mays with Gaussian noise	0.0068	0.9970

.

Appiah-Adjei et al. [34] reported an average NRMSE value of 8.06% when simulating hydraulic heads in observation wells during a pumping test. In our case studies, the NRMSE values range from 0.38% to 1.93%, demonstrating that the results are within an acceptable range.

5. Discussion

The use of the proposed methodology is recommended for the interpretation of pumping tests in confined aquifers where a poor match between test data and the Theis solution is observed, typically due to large measurement errors or deviations from the assumptions of the Theis model. A similar approach could be applied to filter pumping test data from unconfined, leaky or more complex aquifer systems, whether using curve matching or numerical flow modeling.

Proper filtering of pumping test measurements helps to avoid the underestimation or overestimation of hydraulic parameters in the vicinity of the pumping well. Additionally, estimating the average measurement errors can aid in detecting execution flaws and determining whether the test results are reliable or whether the test should be discarded or repeated.

On the other hand, the proposed methodology does not offer significant advantages when measurement errors in drawdowns during a pumping test are small and the assumptions of the Theis solution are met. In such cases, a traditional curve-matching procedure would be easier to implement and would yield similar results.

6. Conclusions

We present a Kalman filter-based procedure to calibrate the hydraulic transmissivity and storage coefficient using data from a pumping test conducted in a confined aquifer, minimizing the differences between drawdown estimates and the Theis solution using genetic algorithms. Data from the “Oude Korendijk” and “Todd and Mays” pumping tests were used to assess this proposal. For a third case study, Gaussian noise (simulating the presence of measurement errors in the pumping test) was added to the latter pumping test. The results show that the filtered data allow an adequate calibration of the hydrogeological parameters reported in previous works.

The proposed method represents a reliable alternative for interpreting pumping tests conducted in confined aquifers, particularly in cases where large errors occur in drawdown measurements (e.g., when using a manual probe) or where the assumptions of the Theis solution are significantly violated, such as in the presence of heterogeneity near the extraction well or the return of pumped water to the aquifer during the pumping test.

The estimate error variances computed with the Kalman filter can provide a good approximation of the average drawdown measurement errors occurring during a pumping test.