A Novel Intelligent Inversion Method of Hydrogeological Parameters Based on the Disturbance-Inspired Equilibrium Optimizer

Abstract

Accurate and quick acquisition of hydrogeological parameters is the critical issue for groundwater numerical simulation and sustainability of the water sources. A novel intelligent inversion method of hydrogeological parameter, based on the global optimization algorithm called the disturbance-inspired equilibrium optimizer (DIEO), is developed. Firstly, the mathematical model and the framework of DIEO are reported. Several types of mathematical benchmark functions are used to test the performance of the DIEO. Furthermore, the intelligent inversion of hydrogeological parameters of pumping tests is transformed into the global optimization problem, which can be solved by meta-heuristic algorithms. The objective function for hydrogeological parameter inversion is constructed, and the novel inversion method based on DIEO is finally proposed. To further validate the competitiveness and efficiency of the proposed intelligent inversion method, three types of case studies are carried out. The results show that the proposed intelligent inversion method is reliable for obtaining the hydrogeological parameters accurately and quickly, providing a reference for the inversion of parameters in other fields.

1. Introduction

Determination of the hydrogeological parameters is one of the most important stages of scientific evaluation of the sustainability of groundwater resources and numerical simulation [1,2]. The hydrogeological parameters of aquifers are usually inversed using pumping tests in engineering [3,4]. Therefore, efficiently and reasonably inverting the hydrogeological parameters by time-depth data is a critical and meaningful problem [5,6,7,8,9].

Conventional methods for determining hydrogeological parameters are trial-and-error (manual adjustment of parameters) [10], type curve [11], and graphics [12], and the trial-and-error method requires more manual intervention. In addition, the type curve method and the graphics method are prone to be affected by artificial factors in the calculation process, resulting in a reduction of solution accuracy [13,14]. Another category is gradient-based optimization methods [13], which rely on the selection of initial values. The gradient-based methods are easy to become trapped in local optima when the initial value is unreasonable [15], resulting in a failure in the inversion of hydrogeological parameters. In summary, the above conventional methods rely on people’s experience, which may lead to low accuracy results.

A meta-heuristic algorithm is sometimes called an intelligent optimization algorithm. It solves complex optimization problems based on high-level strategies. The optimization process of the meta-heuristic algorithm is often inspired by natural behavior and social behavior. Recently, the meta-heuristic algorithms-based inversion method of aquifer parameters has been proposed, with the aim of reducing the influence of subjective factors and initial values [16,17,18]. The advantages of meta-heuristics include their simplicity, automaticity, and gradient-free nature [19], which obviously reduces the dependence on subjective experience [20,21].

The performance of meta-heuristics depends on the balance between exploration and exploitation. Exploration is the ability to search the entire domain. Conversely, exploitation is the ability to increase the quality of promising solutions. It should be noted that the traditional meta-heuristics such as particle swarm optimization is easy to become trapped in the local optima [22,23], leading to unreasonable hydrogeological parameters in the solution [24]. Therefore, the high-performance meta-heuristics-based inversion method needs to be developed [25].

In this study, a novel intelligent inversion method of hydrogeological parameters is developed based on the global optimization algorithm called DIEO. In Section 2, the mathematical model and the performance of DIEO are reported. In Section 3, the general process of the intelligent inversion method of hydrogeological parameters is proposed. The effectiveness and superiority of the proposed intelligent inversion method are validated by several case studies in Section 4, which include three different application scenarios. The discussion and main conclusions can be found in Section 5 and Section 6.

2. Global Optimization Algorithm: Disturbance-Inspired Equilibrium Optimizer

Inspired by a generic mass balance equation for a control volume, Faramarzi [26] et al. proposed a novel meta-heuristic algorithm named the equilibrium optimizer (EO). The basic equation is as follows:

$$V\frac{dC}{dt}=Q{C}_{eq}-QC+G$$

(1)

$$\overrightarrow{C}={\overrightarrow{C}}_{eq}+\left(\overrightarrow{C}-{\overrightarrow{C}}_{eq}\right)\overrightarrow{F}+\frac{\overrightarrow{G}}{\overrightarrow{\lambda }V}\left(1-\overrightarrow{F}\right)$$

(2)

In Equation (1),$V$ represents control volume, $C$ is the concentration inside the control volume, which is regarded as an n-dimensional vector in the algorithm, representing the particle position; $Q$ is the volumetric flow rate; $C_{eq}$ represents equilibrium concentration. The $C_{eq}$ is treated as the position vector of the particle corresponding to the optimal fitness; $G$ is the mass generation rate. In Equation (2), $\overrightarrow{\lambda }$ is a random vector uniformly distributed in [0, 1], $F$ represents the exponential term. The detailed information of standard EO can be further referred to in Faramarzi’s contribution [26].

While standard EO has the advantages of fewer parameters and higher optimization efficiency compared to traditional meta-heuristic algorithms, it still has the shortcomings of stagnation in local optima and low accuracy during practical application. Therefore, the proposed intelligent inversion method is based on the newly developed meta-heuristic named the disturbance-inspired equilibrium optimizer (DIEO). In this section, the framework and performance of DIEO will be reported.

2.1. Initialization Strategy

Optimization results of population evolutionary-related algorithms are directly corresponding to the quality of the initial population. The randomness of initial population distribution can be regarded as from completely random to completely deterministic; both have advantages and disadvantages [27].

The initialization strategy of DIEO contains deterministic components and stochastic components. The deterministic component is based on good nodes set theory [28], while the chaotic sequence is employed as the stochastic component [29]. The function relation is shown as Equation (3):

$${\overrightarrow{X}}_{io}=\overrightarrow{GN}+d_{cof}\overrightarrow{TN}$$

(3)

where ${\overrightarrow{X}}_{io}$ represents the position vector of the initial population, $\overrightarrow{GN}$ and $\overrightarrow{TN}$ are the initial position vectors generated by the good nodes set theory and the chaotic sequence, respectively; $d_{cof}$ is the disturbance factor, and the value is 0.05. In this study, tent chaotic maps are adopted, and their basic form is shown in Equation (4).

$$x^{\left(k+1\right)}=\{\begin{array}{c}{x}^{\left(k\right)}/\mu 0<x^{\left(k\right)}\le \mu \\ (1-x^{\left(k\right)})/\left(1-\mu \right) \mu <x^{\left(k\right)}\le 1\end{array}$$

(4)

It is worth noting that $\mu$ is a user-defined constant. Eventually, based on the initialization strategy, the final initial position vector ${\overrightarrow{C}}_i$ of particle i can be written as Equation (5):

$${\overrightarrow{C}}_i=\overrightarrow{Lb}+{\overrightarrow{X}}_{io,i}\otimes \left(\overrightarrow{Ub}-\overrightarrow{Lb}\right)$$

(5)

where, $\overrightarrow{Lb}$ and $\overrightarrow{Ub}$ represent the lower and upper boundaries of the solution domain in a certain dimension, respectively.

Figure 1 shows the distribution of initial particles using different initialization strategies. It can be found that the new population initialization strategy makes the distribution of populations uniform and ergodic without the loss of being stochastic (Figure 1b). We found that the novel initialization strategy makes the population almost throughout the solution space, and there was no large “blank” in Figure 1b, enabling the particles to explore the entire solution space. The population distribution also considers randomness, which makes the initial population have the ability of random exploration in a small range.

2.2. Update Rule

The update of particle’s position in DIEO is based on “update pool” and “equilibrium pool”. It should be noted that the DIEO inherited the “memory saving mechanism” of the standard EO. Opposition-based learning (OBL) [24] is employed to disturb the initial position of particles before updating, which enables the algorithm to maintain continuous exploration. The function relation of OBL is shown as Equation (7). The equilibrium pool ${\overrightarrow{C}}_{eq,pool}$ is the four historical optimal solutions ${\overrightarrow{C}}_{eq\left(n\right)}$ and their average value ${\overrightarrow{C}}_{eq\left(\mathrm{ave}\right)}$ found in the iteration of the algorithm. They are randomly selected with equal probability to guide the update direction of the population. The function relation of the update pool can be seen in Equations (6)–(8), and the equilibrium pool and the randomly selected mathematical expressions can be seen in Equations (9) and (10).

$${\overrightarrow{C}}_{else}=\overrightarrow{C}$$

(6)

$${\overrightarrow{C}}_{ob}=\overrightarrow{Lb}+\overrightarrow{Ub}-\overrightarrow{C}$$

(7)

$${\overrightarrow{C}}_{upd,pool}=\left\{{\overrightarrow{C}}_{else},{\overrightarrow{C}}_{ob}\right\}=\left\{{\overrightarrow{C}}_{upd}\right\}$$

(8)

$${\overrightarrow{C}}_{eq,pool}=\left\{{\overrightarrow{C}}_{eq\left(1\right)},{\overrightarrow{C}}_{eq\left(2\right)},{\overrightarrow{C}}_{eq\left(3\right)},{\overrightarrow{C}}_{eq\left(4\right)},{\overrightarrow{C}}_{eq\left(\mathrm{ave}\right)}\right\}$$

(9)

$${\overrightarrow{C}}_{eq}=randi\left\{{\overrightarrow{C}}_{eq,pool}\right\}$$

(10)

where, ${\overrightarrow{C}}_{else}$ represents the particle that does not perform OBL, and ${\overrightarrow{C}}_{ob}$ represents the particle that performs OBL; ${\overrightarrow{C}}_{upd}$ represents the position vector stored in the update pool; ${\overrightarrow{C}}_{eq,pool}$ represents the equilibrium pool. ${\overrightarrow{C}}_{eq}$ represents an optimal position vector or average vector.

Exponential term F will assist DIEO in having a reasonable balance between global exploration and local exploitation. Its function relation is shown in Equation (11):

$$\overrightarrow{F}= a_1\mathrm{sign}(\overrightarrow{r} - 0.5)[{\mathrm{e}}^{-\overrightarrow{\lambda }t} - 1]$$

(11)

where, $a_1$ is a constant taken as 2; $\overrightarrow{r}$ is a random vector uniformly distributed between 0 and 1, $t$ represents the time factor, which is related to iteration steps. It is worth noting that $t$ is directly related to iteration step $Iter$ and maximum iteration step $Maxiter$. The time factor $t$ is shown in Equation (12):

$$t=1-\sin ({\left(\frac{Iter}{Maxiter}\right)}^3\times \frac{\pi }{2})$$

(12)

The updated rules of DIEO can be seen in Equation (13).

$$\overrightarrow{C}=\{\begin{array}{cc}{\overrightarrow{C}}_{eq}+\left({\overrightarrow{C}}_{upd}-{\overrightarrow{C}}_{eq}\right)\overrightarrow{F}+\frac{\overrightarrow{G}}{\overrightarrow{\lambda }V}(1-\overrightarrow{F}) ,\hfill & \hfill r_2\ge GP\\ {\overrightarrow{C}}_{eq}+\left({\overrightarrow{C}}_{upd}-{\overrightarrow{C}}_{eq}\right)\overrightarrow{F},\hfill & \hfill r_2<GP \mathrm{and} r_{\sin }\le 0.5\mathrm{and} {\overrightarrow{C}}_{upd}\ne {\overrightarrow{C}}_{eq}\\ {\overrightarrow{C}}_{upd}\left|\sin \left(2\pi r_{s1 }\right)\right|-\left(\pi r_{s1 }\sin \left(2\pi r_{s1 }\right)\right)\left|\left(1-2S_{g }\right)\pi {\overrightarrow{C}}_{eq}-\left(2S_{g }-1\right)\pi {\overrightarrow{C}}_{upd}\right|) ,\hfill & \hfill \mathrm{else}\end{array}$$

(13)

where $V$ is set to 1, $r_2$ and $r_{\sin }$ represent two random numbers between [0, 1], respectively. The $GP$ is defined as the generated rate control parameter, and it is equal to 0.5. The $r_{s1 }$ is a random number from 0 to 1; $S_{g }$ is a constant, which is set to $\frac{\sqrt{5}-1}{2}$. The function relation of $\overrightarrow{G}$ is as follows:

$$\overrightarrow{G}=\left({\overrightarrow{C}}_{eq}-\overrightarrow{\lambda }\overrightarrow{C}\right)\overrightarrow{F}$$

(14)

Equation (14) is valid only when $r_2$ is greater than 0.5.

The expressions of boundary check strategy built in DIEO are shown in Equation (15):

$$C_i^j=\{\begin{array}{cc}U{b}^j or L{b}^j \hfill & \hfill r_b<1/3\\ C_{eq\left(1\right)}{}^j\hfill & \hfill 1/3\le r_b<2/3\\ L{b}^j+r_{br}\left(U{b}^j-L{b}^j\right)& \hfill r_b\ge 2/3\end{array}$$

(15)

where $r_b$ and $r_{br}$ are the random numbers from 0 to 1.

2.3. Adaptive Global Disturbance Update Mechanism

The strategy of the global position disturbance in DIEO is based on the particle’s best-so-far positions. This similarity of populations [30] is employed as the global disturbance probability, which can be written as Equation (16).

$$P_{gd}=\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n(1-\frac{ d_{i,ceq }}{ d_{max }})$$

(16)

$$\overrightarrow{C}=\overrightarrow{C}+d_{per }\left({\overrightarrow{C}}_i-{\overrightarrow{C}}_j\right),r_p\le P_{gd}$$

(17)

where $d_{i,ceq}$ represents the Euclidean distance from particle i to the optimal particle, $d_{max}$ represents the maximum distance between populations; $d_{per }$ is uniformly distributed from 0.4 to 0.9. ${\overrightarrow{C}}_i$ and ${\overrightarrow{C}}_j$ are two randomly selected particles. In addition, the $r_p$ is a random number from 0 to 1.

2.4. Implementation of DIEO

The framework and pseudocode of the proposed DIEO are shown in Figure 2. The first step of the implementation of DIEO is the optimization problem definition, followed by parameter initialization. After performing these procedures, the maximum number of iterations should be determined. Then, the objective function is optimized automatically by DIEO, and the iterations are increased. The optimization process will be iterated until the maximum number of iterations is reached. Furthermore, in order to evaluate the performance of DIEO, the mathematical benchmark functions, including the unimodal function (UF), the multimodal function (MF), the fixed-dimensional multimodal function (FMF), the hybrid function (HF), and the composite function (CF), were optimized using DIEO.

2.5. Performance Tests of DIEO and Comparison with Other Optimizers

The global exploration and local exploitation performance of DIEO are evaluated by benchmark functions (

Table 1. Description of benchmark functions (F1–F6).

Type	No.	Function Equation	Dim	Global Optimal	Range
UF	F1	$\mathbf{Sphere}{\mathit{F}}_{\mathbf{1}}$$:F_1\left(x\right)={\displaystyle \sum }_{i=1}^n{x}_i^2$	30	0	[−100, 100]
MF	F2	$\mathbf{Schwefel}{\mathit{F}}_{\mathbf{2}}$$:F_2\left(x\right)=-{\displaystyle \sum }_{i=1}^n{x}_i\sin \left(\sqrt{\left|x_i\right|}\right)$	30	−12,569.5	[−500, 500]
FMF	F3	$\mathbf{Shekel}\mathbf{10}{\mathit{F}}_{\mathbf{3}}$$:F_3\left(x\right)=-{\displaystyle \sum }_{i=1}^{10}{\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}^{-1}$	4	−10.5363	[0, 10]
HF	F4	$\mathbf{Hybrid}{\mathit{F}}_{\mathbf{4}}$: Happycat, Katsuura, Ackley, Rastrigin, Modified Schwefel and Schaffer	10	2000	[−100, 100]
CF	F5	$\mathbf{Composite}{\mathit{F}}_{\mathbf{5}}$: shifted and rotated Rastrigin, Expanded Scaffer and Lunacek Bi_Rastrigin	10	2900	[−100, 100]
CF	F6	$\mathbf{Composite}{\mathit{F}}_{\mathbf{6}}$: shifted and rotated Rastrigin, Non-Continuous Rastrigin and Levy Function	10	3000	[−100, 100]

). Functions F1–F3 are well-known benchmark functions, and detailed information can be further obtained from Yao’s contribution [31]. In addition, the functions F4–F6 belong to the CEC2017 test set [32].

In order to illustrate the competitiveness and superiority of DIEO, four state-of-the-art meta-heuristic algorithms were used for comparison, including an equilibrium optimizer (EO) [26], arithmetic optimization algorithm (AOA) [33], hunger games search (HGS) [25], and the autonomous group’s particle swarm optimization (AGPSO) [34]. The parameter settings of these algorithms are shown in

Table 2. Key parameter values for the comparative algorithms.

Algorithm	Parameter	Value	References
EO	GP $a_1$ $a_2$	0.5 2 1	Faramarzi et al. [26]
AOA	$\alpha$ $\mu$	5 0.5	Abualigah et al. [33]
HGS	shrink $l$	Decreasing from 2 to 0 0.03	Yang et al. [25]
AGPSO	${\omega }_{max}$ ${\omega }_{min}$ $c_1$ $c_1$	0.9 0.4 2 2	Mirjalili et al. [34]

, which can be found in the original source.

In order to reduce the effect of randomness, the performance of the algorithm is evaluated by average performance (Ave) and standard deviation (Std) in 50 independent runs. In addition, the population size of the algorithm is set to 30 along with 500 iterations.

The results of six benchmark function optimizations are shown in

Table 3. Results of the test functions.

No.		EO	AOA	HGS	AGPSO	DIEO
F1	Ave	5.60 × 10−40	6.21 × 10−15	5.11 × 10−153	0.260929	4.05 × 10−225
	Std	2.98 × 10−39	4.93 × 10−14	3.61 × 10−152	0.4140	0
F2	Ave	−8849.47	−5272.63	−12,260.36	−9503.72	−12,569.44
	Std	555.26	415.41	948.34	539.65	0.05
F3	Ave	−9.61678	−3.67118	−9.77930	−8.70485	−10.53641
	Std	2.54	1.41	1.90	3.01	1.01 × 10−14
F4	Ave	2034.04	2164.21	2045.11	2029.23	2011.91
	Std	35.21	74.71	50.26	14.61	10.44
F5	Ave	3191.00	3406.86	3246.31	3194.25	3160.11
	Std	31.10	130.18	65.48	40.15	14.06
F6	Ave	330,223.20	27,126,774.00	428,689.46	302,804.60	4245.44
	Std	484,276.69	23,488,101.74	428,470.12	419,584.93	4795.29
Average ranking		3	4.833	3	3.167	1

. As can be seen, DIEO shows the best performance in the unimodal function (F1). In the multimodal function (F2) and the fixed-dimensional multimodal function (F3), DIEO shows significant superiority, while the standard EO and AOA fallen into local minimum many times, resulting in the decrease of average performance (Ave). Composite function (F6) is the most challenging, and we find that the DIEO obtained better results compared with other state-of-the-art algorithms.

To further evaluate the optimization performance of the DIEO, we compared the convergence behavior of other algorithms along with the DIEO algorithm (Figure 3). The vertical coordinate represents the best-so-far fitness, and the horizontal coordinate represents the number of iterations. In order to have a reliable and fair comparison, this study selected the best performance obtained by five algorithms in 50 optimizations. As shown in Figure 3, DIEO has better performance than other algorithms in terms of both convergence speed and solution accuracy.

The convergence curve lacks information on the stability of the algorithm and the distribution of results. In order to have a better understanding of the distribution of results, the boxplot of results for F3 and F6 are shown in Figure 4. The box length is positively correlated with the number of times of falling into local optima, and the dotted line denotes the range of the solution. It should be noted that the range of fitness obtained by EO, HGS, AGPSO, and DIEO are significantly different in Figure 4b. To summarize, DIEO showed excellent performance. Therefore, DIEO will be further applied to solve engineering problems. A novel intelligent inversion method of hydrogeological parameters based on DIEO will be developed in the next section.

3. Intelligent Inversion Method of Hydrogeological Parameters Based on DIEO

The potential of the DIEO algorithm in solving practical engineering problems was verified in the optimization of benchmark functions. Firstly, the intelligent inversion of hydrogeological parameters of pumping tests is transformed into an optimization problem, and the form of the objective function is introduced. Then, the intelligent inversion method of hydrogeological parameters based on DIEO is proposed, and the procedures of the intelligent inversion method are reported.

3.1. Transform Intelligent Inversion into Optimization Problem

Without losing generality, the intelligent inversion of hydrogeological parameters using pumping tests can be written as Equations (18) and (19).

$$s_i=g(x^1,x^2\dots x^n)$$

(18)

$$\mathrm{Minimize}f\left(x^1, x^2\dots x^n\right)=\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{\left(s_i-s_i^{obs}\right)}^2$$

(19)

where, $s_i^{obs}$ is the water level drawdown data obtained by pumping tests, and Equation (18) means that the analytical model $g(x^1$, $x^2$ …$x^n)$ is used to calculate the water level drawdown $s_i$. In addition, the unknown variable (need to be inversed or optimized) is $x^n$.

Accordingly, the intelligent inversion of hydrogeological parameters is transformed into the optimization problem, i.e., finding a set of reasonable hydrogeological parameters to minimize the $f\left(x^1, x^2\dots x^n\right)$ as much as possible. It is worth noting that the calculation of $s_i$ in Equation (18) is flexible, and different theoretical models $g\left(x\right)$ can be selected according to the application scenarios, such as Theis and Hantush–Jacob well functions. In addition, necessary constraints can be added to the equation. The variable $x$ can be written as a multi-dimensional vector in the proposed inversion method, and its dimension represents the number of hydrogeological parameters to be inverted. For example, in the Theis equation, the dimension of $x$ is 2, which represents the transmissivity coefficients and storage coefficient of aquifers, respectively. It should be mentioned that $g\left(x\right)$ may involve multi-dimensional variables and complex nonlinear forms, resulting in multiple local optima and increasing the difficulty of inversion.

3.2. Implementation of the Proposed Intelligent Inversion Method

The framework of the proposed intelligent inversion method is shown in Figure 5. The input parameters of this method are maximum iterations, hydrogeological parameters which need to be inverted, and the population size; Output parameters are the minimum value of objective function $f\left(x\right)$ and the optimal hydrogeological parameters $x^1$, $x^2$ …$x^n$. It is worth noting that the whole process of acquiring hydrogeological parameters is automatic. The specific procedures of the inversion method are as follows:

Step 1.

Conduct the hydrogeological tests, such as pumping tests, to obtain test data $s_i^{obs}$;

Step 2.

Initialize the DIEO’s parameters, including the population size and the maximum iteration;

Step 3.

Determine the theoretical model of $s_i$ in the objective function according to the application scenario, such as the Theis well flow function shown in Equation (20), and further construct the objective function shown in Equation (19);

Step 4.

DIEO performs optimization of $f\left(x\right)$ automatically;

Step 5.

DIEO outputs the optimal solution of hydrogeological parameters ${\overrightarrow{C}}_{eq\left(1\right)}$ and the minimum value of objective function $f({\overrightarrow{C}}_{eq\left(1\right)})$.

4. Engineering Application of Intelligent Inversion Method for Hydrogeological Parameters

In order to further evaluate the reliability and efficiency of the proposed DIEO-based intelligent inversion method, three case studies of hydrogeological parameters inversion were carried out. The inversion of the parameters of the Theis well flow model was firstly conducted due to its simpler objective function form and lower dimension. Then, the inversion of leaky aquifer parameters with higher dimensions was taken as another case. Finally, to further evaluate the advancement of the proposed method, inversion for multiple monitoring wells pumping test with more complex objective functions was achieved. The potential of the DIEO algorithm in solving practical engineering problems was verified by comparing the results with other state-of-the-art intelligent algorithms. In order to obtain fair and reliable test results, all the hydrogeological parameters inversion problems are conducted using the same number of iterations (50) and population size (30).

4.1. Case 1: Parameters Inversion of the Theis Well Flow Model

The Theis well flow model assumes that a fully penetrating well pumps water from an isotropic, homogenous, and infinite confined aquifer without inflow from surrounding formations [5]. The drawdown of water level $s$ at the moment $t^{\prime }$ can be written as Equation (20),

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

(20)

where, $Q$ represents pumping rate, $T$ represents transmissivity. In addition, the radial distance is written as $r$, and the $W\left(u\right)$ is called the Theis well function, which is written as:

$$W\left(u\right)={{\displaystyle \int }}_u^{\infty }\frac{e^{-y}}{y}\mathrm{d}y$$

(21)

where u is a dimensionless variable, and it can be written as:

$$u=\frac{r^2\mu }{4T{t}^{\prime }}$$

(22)

where $\mu$ is the storage coefficient. It can be seen from Equations (18)–(20) that the dimension of the variable $x$ in the Theis well flow model is 2, which is $T$ and $\mu$, respectively.

The data of the pumping tests are shown in

Table 4. Pumping test data of Theis well flow model.

${\mathit{t}}^{\prime }/\mathbf{m}\mathbf{i}\mathbf{n}$	${\mathit{s}}^{\mathit{o}\mathit{b}\mathit{s}}/\mathbf{m}$	${\mathit{t}}^{\prime }/\mathbf{m}\mathbf{i}\mathbf{n}$	${\mathit{s}}^{\mathit{o}\mathit{b}\mathit{s}}/\mathbf{m}$
10	0.16	210	1.55
20	0.48	270	1.70
30	0.54	330	1.83
40	0.65	400	1.89
60	0.75	450	1.98
80	1.00	645	2.17
100	1.12	870	2.38
120	1.22	990	2.46
150	1.36	1185	2.54

[35,36]. In addition, detailed information can be found in Xue and Tan’s contribution. The pumping rate $Q$ is 60 m³/h, the pumping time is 1185 min, and the distance $r$ is 140 m. The approach of calculation of Equations (20)–(22) is coded using MATLAB, and the objective function is constructed based on Equation (19). The intelligent inversion results are shown in

Table 5. Comparison of parameters inversion results of Theis well flow model.

Algorithm	Optimal	Average	Worst	Standard Deviation	Inversion Parameters
EO	0.00159496487	0.03443647	0.26865975	0.0731	$\mu :$ 0.000250, $T:$ 193.3793
AOA	0.01193552809	0.29540144	0.33198542	0.0849	$\mu :$ 0.000177, $T:$ 234.9177
HGS	0.00159496487	0.03401261	0.33159432	0.0958	$\mu :$ 0.000250, $T:$ 193.3796
AGPSO	0.00159496512	0.01479637	0.33159432	0.0653	$\mu :$ 0.000250, $T:$193.3849
DIEO	0.00159496487	0.00159496	0.00159496	6.83 × 10−12	$\mu :$ 0.000250, $T:$193.3796

.

Table 5 shows the results of inversion of hydrogeological parameters of the Theis well flow model in 50 independent runs. We present the best fitness, the average performance, the worst fitness, the standard deviation, and the hydrogeological parameters (i.e., the best solution). The results show that the proposed inversion method can find a reliable solution. Since the objective function of the Theis equation is simple, EO and HGS obtained similar results of the best fitness with DIEO. It can also be seen from Table 5 that the meta-heuristic algorithms show better adaptability for the intelligent inversion of hydrogeological parameters.

Comparison of convergence curves (Figure 6) was used to further reveal the performance of each algorithm in the process of hydrogeological parameter inversion. Figure 6 shows that AOA has the fastest convergence efficiency, followed by DIEO. However, the performance of AOA is unstable because of its entrapment of the local optimal many times, and the worst fitness is about 28 times the best fitness in optimization. Hence, the superiority and efficiency of the proposed inversion method were verified in the Theis well flow model.

4.2. Case 2: Parameters Inversion of Leaky Aquifer

In this section, we further considered the drawdown data of leaky aquifers obeying the Hantush model [37],

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

(23)

$$W\left(u,\frac{r}{B}\right)={{\displaystyle \int }}_u^{\infty }\frac{e^{-y\frac{r^2}{4By}}}{y}\mathrm{d}y$$

(24)

$$B=\sqrt{T/\left(K^{\prime }/b^{\prime }\right)}$$

(25)

where, $B$ represents the leakage factor, and $b\prime$ represents the thickness of the adjacent aquitard; $K^{\prime }$ represents the permeability coefficient of the adjacent aquitard. The calculation method of dimensionless variable $u$ is shown in Equation (22). It can be seen from Equations (23)–(25) that the calculation method of leaky aquifer function is more complicated compared with the Theis well flow function. According to Equations (23) and (24), the parameters inversion problem of leaky aquifers is transformed into a global optimization problem with three-dimensional variables. The three dimensions of optimization variable $x$ are $\mu$, $T$ and $B$, respectively.

The data of pumping tests are derived from Li [38] and Zhao’s contribution [39]. The pumping flow rate $Q$ is 69.1 m³/h in the leaky aquifer, and the measuring well is 197 m away from the pumping well. The obtained $s^{obs}$ is shown in

Table 6. Pumping test data of leaky aquifer.

${\mathit{t}}^{\prime }$/min	${\mathit{s}}^{\mathit{o}\mathit{b}\mathit{s}}/\mathbf{m}$	${\mathit{t}}^{\prime }$/min	${\mathit{s}}^{\mathit{o}\mathit{b}\mathit{s}}/\mathbf{m}$	${\mathit{t}}^{\prime }$/min	${\mathit{s}}^{\mathit{o}\mathit{b}\mathit{s}}/\mathbf{m}$
1	0.05	75	0.62	360	0.772
4	0.054	90	0.64	390	0.785
7	0.12	120	0.685	420	0.79
10	0.175	150	0.725	450	0.792
15	0.26	180	0.735	480	0.794
20	0.33	210	0.755	510	0.795
25	0.383	240	0.76	540	0.796
30	0.425	270	0.76	570	0.797
45	0.52	300	0.763	600	0.798
60	0.575	330	0.77	660	0.8

.

The comparison of algorithms for hydrogeologic parameter inversion is shown in

Table 7. Comparison of parameters inversion results of leaky aquifer.

Algorithm	Optimal	Average	Worst	Standard Deviation	Inversion Parameters
EO	0.000122029	0.000126083	0.000160158	7.26 × 10−6	$\mu :$ 0.0001387, $T:$ 411.9885, $B:$ 566.0923
AOA	0.000503493	0.011101486	0.043756259	0.0108	$\mu :$ 0.0001062, $T:$ 577.77545, $B:$ 1000
HGS	0.000122045	0.000314263	0.002301503	0.0003	$\mu :$ 0.0001385, $T:$ 413.2295, $B:$ 568.6260
AGPSO	0.000122031	0.000187938	0.000441459	0.0001	$\mu :$ 0.0001386, $T:$ 412.3328, $B:$ 566.8948
DIEO	0.000122027	0.000122045	0.000122258	3.52 × 10−8	$\mu :$ 0.0001386, $T:$ 412.1454, $B:$ 566.4333

. It can be indicated that AOA was trapped into local optima due to the traditional boundary check strategy, whereas DIEO achieves a better performance. The comparison results show that the proposed inversion method exhibits competitiveness and superiority in terms of accuracy, success rate, and stability compared to other inversion methods.

The convergence curves of the inversion of the leaky aquifer are presented in Figure 7. It can be observed that the algorithms exhibit the global exploration behavior in the first 20 iterations. Then, the convergence behaviors were changed into local exploitation. Based on the convergence curves and results of inversion shown in Table 7, it can be seen that the proposed inversion method is more competitive than other methods. Therefore, the proposed DIEO-based inversion method can be given priority in the parameter inversion of leaky aquifers.

4.3. Case 3: Parameters Inversion of Pumping Test with Multiple Monitoring Wells

To further challenge the performance of the proposed inversion method, the hydrogeological parameters inversion of pumping test with multiple monitoring wells was carried out. The dimension and the objective function are more complex than in cases one and two. In addition, we further considered the drawdown data during the recovery periods. The calculation model of leaky confined aquifers and the weighted multi-curve fitting method proposed by Wang [40] are used to construct the objective function. The whole process fitting variance ${\mathsf{\Delta }}_i$ of a single observation well is:

$${\mathsf{\Delta }}_i=\frac{1}{N+N^{\prime }}\left[{{\displaystyle \sum }}_{j=1}^N{\left(s_{i,j}-s_{i,j}^{obs}\right)}^2+{{\displaystyle \sum }}_{l=1}^{N^{\prime }}{\left(s_{i,l}^{\prime }-s_{i,l}^{{}^{\prime }obs}\right)}^2\right]$$

(26)

The $s_{i,l}^{\prime }$ and $s_{i,l}^{{}^{\prime }obs}$ in Equation (26) represent the drawdown data during the recovery periods. The weight ${\omega }_i$ in the weighted multi-curve fitting method can be determined by ${\mathsf{\Delta }}_i$, and the weight of a single observation well can be defined as

$${\omega }_i=\frac{1/{\mathsf{\Delta }}_i}{{{\displaystyle \sum }}_{i=1}^{M}1/{\mathsf{\Delta }}_i}$$

(27)

Furthermore, Wang [40] introduced a weighted multi-curve fitting method which could make full use of the drawdown data from multiple monitoring wells in the pumping and recovery periods, which can be written as Equation (28),

$$f\left(x\right)={{\displaystyle \sum }}_{i=1}^M{\omega }_i\left[{{\displaystyle \sum }}_{j=1}^N{\left(s_{i,j}-s_{i,j}^{obs}\right)}^2+{{\displaystyle \sum }}_{l=1}^{N^{\prime }}{\left(s_{i,l}^{\prime }-s_{i,l}^{{}^{\prime }obs}\right)}^2\right]$$

(28)

Therefore, the inversion problem of hydrogeological parameters is transformed into the optimization problem of Equation (28). In addition, we further considered the drawdown data during the recovery periods, and the equation for the remaining water level drawdown during recovery periods are as follows [41]:

$$s^{\prime }=s_{const}-\frac{Q_n}{4\pi T}W\left(u_n^{\text{'}},\frac{r}{B}\right)$$

(29)

$$u_n^{\text{'}}=\frac{r^2\mu }{4T{t}^{\prime }}$$

(30)

where, $s^{\prime }$ represents the remaining water level drawdown during recovery periods, $t^{\prime }$ represents the duration during recovery periods; $s_{const}$ represents the consistent drawdown; $Q_n$ represents the last stage flow before consistent. $u_n^{\text{'}}$ is a dimensionless variable.

The planar graph of the pumping test is shown in Figure 8 [31]. In this case, the pumping flow rate $Q$ is 80 m³/h, and the pumping time is 2880 min, the first 1440 min for the pumping, followed by the recovery periods. The data from the three observation wells are shown in

Table 8. Pumping test data with multiple observation wells.

${\mathit{t}}^{\prime }$/min	Drawdown during Pumping/m			${\mathit{t}}^{\prime }$/min	Drawdown during Recovery Periods/m
	Y6	Y5	G1		Y6	Y5	G1
0	0.00	0.00	0.00	1440	0.61	0.81	0.85
3	0.04	0.3	0.23	1443	0.51	0.66	0.54
6	0.15	0.57	0.46	1446	0.41	0.34	0.39
10	0.23	0.62	0.60	1450	0.30	0.22	0.25
15	0.33	0.65	0.65	1455	0.20	0.20	0.21
20	0.39	0.68	0.70	1460	0.14	0.18	0.20
30	0.42	0.70	0.72	1470	0.08	0.16	0.16
60	0.45	0.74	0.75	1500	0.05	0.10	0.11
90	0.46	0.76	0.78	1530	0.02	0.07	0.07
150	0.48	0.77	0.79	1590	0.01	0.04	0.05
210	0.49	0.78	0.80	1650	0.01	0.02	0.03
270	0.51	0.79	0.81	1710	0.00	0.01	0.01
330	0.53	0.79	0.82	1770	0.00	0.00	0.00
390	0.55	0.80	0.83	1830	−0.01	−0.01	0.00
450	0.57	0.80	0.84	1890	−0.01	−0.02	−0.01
510	0.59	0.80	0.84	1950	−0.01	−0.02	−0.01
570	0.60	0.80	0.84	2010	−0.01	−0.02	−0.01
630	0.60	0.80	0.84	2070	−0.01	−0.02	−0.01
690	0.60	0.80	0.84	2130	−0.01	−0.02	−0.01
1440	0.61	0.81	0.85	2880	0.00	−0.02	−0.02

[17].

It should be noted that the objective function was coded as Equation (28) in MATLAB. The hydrogeological parameters were then automatically inversed by the proposed DIEO-based method. The results and comparisons are reported in

Table 9. Comparison of parameters inversion results of pumping test with multiple observation wells.

Algorithm	Optimal	Average	Worst	Standard Deviation	Inversion Parameters
EO	0.061236665	0.075326013	0.110490027	0.01292	$\mu :$ 0.002453, $T:$ 603.6639, $B:$ 88.5730
AOA	0.073441462	0.163632915	0.255037057	0.03976	$\mu :$ 0.003103, $T:$ 505.3558, $B:$ 66.3816
HGS	0.066407693	0.104836371	0.142634242	0.02157	$\mu :$ 0.002004, $T:$ 714.1393, $B:$ 121.5195
AGPSO	0.061231779	0.083531005	0.141905814	0.01721	$\mu :$ 0.002433, $T:$ 605.6198, $B:$ 89.10456
DIEO	0.061229395	0.061603179	0.072460668	0.00168	$\mu :$ 0.002439, $T:$ 605.0028, $B:$ 88.9101

.

It can be seen that the proposed inversion method still achieves the best performance in the parameter inversion of the pumping test with multiple monitoring wells (Table 9). In this case, the best fitness is associated with the local exploitation ability, while the average performance and the worst fitness are associated with the global exploration ability. Figure 9 shows the comparisons of convergence curves, which indicates that the proposed inversion method is very competitive compared to the other algorithms, and the proposed method is also efficient. Therefore, the proposed intelligent inversion method is reliable for obtaining the hydrogeological parameters accurately and quickly, providing a reference for the inversion of parameters in other fields.

5. Discussion

The results of three different problems show that the proposed method is reliable and efficient. In order to reveal the stability of the proposed inversion method, we give the distribution of the results in 50 independent runs shown in Figure 10. It can be seen that the range of solutions obtained by the proposed method is very small compared with other methods. According to the length of the box, the proposed method shows the very success rate of inversion of hydrogeological parameters. However, the other four methods show the obvious boxes, which means they obtained the local minima in some runs leading to large errors.

It is worth mentioning that the proposed inversion method is based on the meta-heuristic algorithm. The efficiency and reliability proposed method are benefited from the new mechanisms built into DIEO, which include the initialization strategy, update rule, and the global disturbance update strategy. However, the application of the proposed inversion method should be further tested because the meta-heuristic algorithm cannot acquire the most accurate solution for all kinds of problems. In addition, some of the inversion problems are multi-objective problems. The multi-objective version of the inversion method will be further studied in the future. In conclusion, the proposed inversion method of hydrogeological parameters can be regarded as a reliable tool, which can be further applied in the field of hydrogeological parameter inversion.

6. Conclusions

In this study, an intelligent hydrogeological parameters inversion method was developed. The reliability and efficiency of the proposed method were demonstrated by three different case studies. The main conclusions can be summarized below.

(1)

The inversion of hydrogeological parameters was transformed as the global optimization problem. A DIEO-based inversion method for hydrogeological parameters was developed.

(2)

The reliability and efficiency of the proposed intelligent inversion method are benefited by the new initialization strategy and the update rules of DIEO.

(3)

The proposed intelligent inversion method was successfully applied to three hydrogeological parameters identification problems. The optimal hydrogeological parameters can be obtained using the proposed inversion method efficiently.