# Modeling of Distributed Control System for Network of Mineral Water Wells

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{3}(norm is 2–3 g/dm

^{3}), dissolved carbon dioxide-to 0.40 g/dm

^{3}(norm is 1.0–2.5 g/dm

^{3})) [1]. The dynamics of changes in mineralization and dissolved carbon dioxide in Dolomite Narzan and in Sulfate Narzan is given in [1]. Existing problems and ways to solve them in relation to groundwater in sanatorium-resort areas are described in detail in [1,2,3]. There were also studies of the industrial facilities influence at water quality, in particular gravel pits [4] and waste disposal sites [5]. Works [6,7,8] are devoted to the study of the factors that determine the quality and change in groundwater reserves. Works [9,10] are devoted to the problems of the risk of polluted water for health. The problems of transboundary groundwater resources management are considered in [11,12].

- determination of the optimal number of production wells;
- monitoring the current state of the hydrogeological system;
- forecasting processes in the hydrogeological system for short-term (up to 10 years) and long-term (up to 100 years) perspectives.

## 2. Materials and Methods

_{gw}is the water column height in the free aquifer, m; H

_{a}is the water column height in the confined aquifer, m; k

_{gw,x}, k

_{gw,y}, k

_{gw,z}are the hydraulic conductivities along spatial coordinates in the free aquifer, m/day, k

_{a,x}, k

_{a,y}, k

_{a,z}are the hydraulic conductivities along spatial coordinates in the confined aquifer, m/day; x

_{pw,i}, y

_{pw,i}are coordinates of the location of production wells, m; η

_{a}is the coefficient of elastic capacity, m

^{−1}, in this case η

_{a}= 1.6 × 10

^{−3}m

^{−1}; x, y, z are the spatial coordinates, m; N is number of wells, [–]; i is an order number of the production well, i = 1…N, [–]; t is time, day; a function δ(x

_{pw,i}, y

_{pw,i}, z

_{pw,}

_{1}≤ z ≤ z

_{pw,}

_{2}) = 1, if x = x

_{pw,i}, y = y

_{pw,i}, z

_{pw,}

_{1}≤ z ≤ z

_{pw,}

_{2}(that corresponds to an imperfect water intake from an aquifer), and δ(x

_{pw,i}, y

_{pw,i}, z

_{pw,i}≤ z ≤ z

_{pw,i}) = 0 in all other cases, [–]; V is a decrease in the water column height due to the impact of the production rate Q

_{i}of the i-th well, m/day; K

_{i}is a coefficient determined experimentally, in this case K

_{i}= 0.187 m

^{−2}; Q

_{i}(t) are production well rates, m

^{3}/day. In modeling, the condition was accepted that production rates are evenly distributed over the interval z

_{pw,}

_{1}≤ z ≤ z

_{pw,}

_{2}.

_{gw,x}= 0.099; k

_{gw,y}= 0.097; k

_{gw,z}= 0.095; k

_{a,x}= 0.199; k

_{a,y}= 0.197; k

_{a,z}= 0.108.

_{o}= 3 × 10

^{−5}day

^{−1}is a leakage parameter.

_{x}= 150 m; L

_{y}= 120 m; L

_{z}

_{1}= 55 m; L

_{z}

_{2}= 85 m.

## 3. Results

#### 3.1. Study of the Characteristics of the Hydrogeological Process

_{1}= 100 m

^{3}/day. This value was used to determine the static transfer coefficients K

_{p}and K

_{c}, respectively, for the production and control wells. The static transfer coefficient is equal to the steady-state level change in the corresponding well measured at the midpoint z = (z

_{pw,1}+ z

_{pw,2})/2 m, divided by the pumping rate (input action) Q

_{1}:

_{p}= −0.512/(−100) = 5.12 × 10

^{−3}m

^{−2}; K

_{c}= −0.202/(−100) = 2.02 × 10

^{−3}m

^{−2}.

_{ap}is the transfer function of the approximating link; s is differential operator; K, B are parameters whose values are determined using experimental data or numerical modeling.

_{p}, K

_{c}, as well as geometric data: the radius of the well Δy

_{1}= 0.2 m [1] and the distance between the production well 1 and the control well Δy

_{2}= 10 m.

^{−4}m

^{−2}; B = 9.007 × 10

^{−3}[–].

#### 3.2. Determination of the Optimal Number of Production Wells

_{1}= 0; y = y

_{1}= b); then the angle π/2 rad is divided into n

_{f}parts. The canonical ellipse equation and the straight line equation are used to determine the coordinates of the second well:

_{f}and i provides the coordinates of the point i (x

_{i}, y

_{i}). The coordinates of the other points of the ellipse were defined in the similar way. A total of 4n

_{f}points were obtained.

_{f}) that provide the highest profit for the period of operation of the deposit, taking into account the restrictions described below.

_{w}= 0.2 m; MW wells operate 24 h per day for 10 years (3650 days); the cost of 1 m

^{3}of extracted mineral water p

_{1}is 350 rubles; expenses for the maintenance of buildings and equipment and personnel for 10 years of operation C

_{p}is 520 million rubles, the cost of drilling, construction, equipment and maintenance of each well for 10 years C is 45 million rubles; subsoil use tax is 7.5% [43].

_{k}), m; r

_{w}is the radius of the well, m; ∆r

_{i,j}is the distance from the i-th to the j-th interacting wells, m; Q

_{i}, Q

_{j}are the pumping rates of the i-th and j-th wells, m

^{3}/day.

_{1}(1 − 0.075) − C

_{p}− 4 × C × n

_{f}.

^{3}/day.

#### 3.3. Synthesis of a Distributed Controller

_{j}is the spatial frequency; γ

_{i}is the distance between two considered wells; Δγ is the distance between adjacent wells.

_{pw,i}, y

_{pw,i}) in the space {x, y} (at the points of location of the production wells). If the calculated value U

_{i}(i = 1… 16) is displayed in the space {x, y}, then the following expression can be obtained:

_{1}and K

_{2}) and the generalized coordinate (G

_{1}and G

_{2}) were determined as:

_{1}= 775.0 m

^{−2}; G

_{1}= ψ

_{1}

^{2}= (2 × π × 1/L)

^{2}= 1.003 × 10

^{−3}[–]; K

_{2}= 583.3 m

^{−2}; G

_{2}= ψ

_{2}

^{2}= (2 × π × 2/L)

^{2}= 4.01 × 10

^{−3}[–].

_{γ}are the defined parameters.

_{1}= 775.0 m

^{−2}; G

_{1}= 1.003 × 10

^{−3}[–]; K

_{2}= 583.3 m

^{−2}; G

_{2}= 4.01 × 10

^{−3}[–]; ∆φ = −0.797 rad, an approximation model of the control object was determined:

_{a}, E

_{i}, E

_{d}are the overall gains of the spatially amplifying, spatially integrating and spatially differentiating links, respectively; γ is the spatial coordinate; s is differential operator; ${\nabla}^{2}$ is Laplacian; n

_{a}, n

_{i}, n

_{d}are weighting factors of the spatially amplifying, spatially integrating, and spatially differentiating links.

_{a}= 0.22999 m

^{2}, E

_{i}= 0.00108 m

^{2}, E

_{d}= 7.71 m

^{2}

_{,}n

_{a}= 1.065 [–], n

_{i}= 1.0859 [–], n

_{d}$\to $ ∞ [–].

## 4. Discussion

_{a}(i,t) is sent to the input of the controller, this signal is a mismatch between the desired value of the level H

_{ad}(i,t) in the well location zones and the current value of the level H

_{a}(i,t):

_{a}(i,t) = H

_{ad}(i,t) − H

_{a}(i,t), i = 1, 2, …, 16.

_{i}(pumping rate of the i-th production well):

^{3}/day.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Pershin, I.M.; Papush, E.G.; Kukharova, T.V.; Utkin, V.A.
Modeling of Distributed Control System for Network of Mineral Water Wells. *Water* **2023**, *15*, 2289.
https://doi.org/10.3390/w15122289

**AMA Style**

Pershin IM, Papush EG, Kukharova TV, Utkin VA.
Modeling of Distributed Control System for Network of Mineral Water Wells. *Water*. 2023; 15(12):2289.
https://doi.org/10.3390/w15122289

**Chicago/Turabian Style**

Pershin, Ivan M., Elena G. Papush, Tatyana V. Kukharova, and Vladimir A. Utkin.
2023. "Modeling of Distributed Control System for Network of Mineral Water Wells" *Water* 15, no. 12: 2289.
https://doi.org/10.3390/w15122289