# Development of a Distributed Control System for the Hydrodynamic Processes of Aquifers, Taking into Account Stochastic Disturbing Factors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Topical Area Analysis and Formulation of the Problem

## 3. Study Area Description

_{1}) crystalline schists and Upper Paleozoic (PZ) granites. They were discovered by wells at depths ranging from 80 m in the southern and up to 400 m in the northern parts of the deposit.

_{1}t): Toarcian sediments are found only in the upper southern part of the Berezovskaya Gully. To the north, they are cut off by the pre-Tithonian transgression.

_{3}tt): The sediments of the Tithonian stage hide on the eroded surface of the basement. They were discovered by wells at depths from 100 to 400 m, increasing in a northerly direction. The Tithonian variegated sequence consists of loose sandstones, clays, and granite gruss. The sediments of the Tithonian stage are divided by researchers into three sequences: the lower one, which is dominated by coarse-grained variegated sandstones with rare interlayers of gruss and clays; the middle one, consisting of granite gruss with interlayers of sandstones and clays, and the upper one, composed of red–brown clayey sandstones with gypsum inclusions. The thickness of the sediments of the Tithonian stage is inversely related to the relief of the top of the Paleozoic basement. In areas where the basement surface forms depressions, the thickness of the variegated stratum increases, and, conversely, in the elevated areas of the basement, it is minimal.

_{1}v). Valanginian sediments, represented in the section by dolomitic limestones, overlie the variegated Tithonian sequence. They are exposed on the surface in a small area in the southern part of the deposit, along the valleys of the rivers Berezovaya, Kabardinka and Olkhovka.

_{1}g): The sediments of the Hauterivian stage lie on the dolomitic limestones of the Valanginian and are important in terms of sandy-clay composition. The sediments were formed by a number of shell limestones and argillaceous sandstones with limestone interbeds and outcrops in the main and northern parts of the field, as well as in rarely elevated areas.

_{1}br): The sediments of the Barremian Stage are represented by a sequence of dense oolitic limestones interbedded with limestone–argillaceous sandstones. Above are calcareous sandstones with a small number of thin layers of oolitic limestones. They are overlain by argillaceous siltstones, which in some places change into strongly argillaceous dark brown sandstones and higher up are replaced by grayish-yellow fine-grained, argillaceous sandstones. The section of the Barremian sediments ends with red ferruginous sandstones. Sediment thickness about 40 m.

_{1}a): The Aptian stage is represented by clays (lower substage) overlain by sands and sandstones (upper substage). The total thickness of the sediments of the Aptian stage is 245 m.

_{1}al): The sediments of the Albian stage are found in the north and west of the described territory. In the lower part, they are represented by sands and sandstones (lower substage), and in the upper part, by black calcareous clays (upper substage). The thickness of the Albian Stage is 150–200 m.

_{IV}), which are distributed mainly along the thalwegs of permanent and temporary watercourses. Eluvial sediments are widespread in the watershed areas north and south of the Berezovaya, Alikonovka, Podkumok rivers. They are represented by yellowish-gray compacted loams, containing crushed sandstones mainly in the lower part of the section.

## 4. Research Methods

_{t}with other ones, a resulting correlogram can be presented in the following form in Figure 3. Since the first-order autocorrelation coefficient turned out to be the highest, the time series contains a trend. As the delay increases, the correlation monotonically decreases, which indicates the stationarity of the series, the probabilistic properties of which do not change with time. To confirm the results, the second method was used for estimating the stationarity of the series.

_{i,x}, k

_{i,y}, k

_{i,z}—filtration coefficients; ${\mathrm{x}}_{\mathrm{j}},{\mathrm{y}}_{\mathrm{j}},{\mathrm{z}}_{\mathrm{j}}$—coordinates of the location of producing wells (j = 1,2,3,4); $\mathsf{\delta}({\mathrm{x}}_{\mathrm{j}},{\mathrm{y}}_{\mathrm{j}},{\mathrm{z}}_{\mathrm{j}})$—parameter that is equal to one when x = x

_{j}, y = y

_{j}, z = z

_{j}, otherwise it is equal to zero.

_{p}—proportional gain, T

_{d}—derivative time constant, T

_{in}—integral time constant.

## 5. Results

_{dis}= 20% < σ

_{pid}= 53%), and the system reaches a steady state faster.

## 6. Discussion

## 7. Conclusions

- A mathematical model that was developed for the study area of the Kislovodsk mineral water deposit, which took into account the impact of random factors on the flow speed and water debits in aquifers.
- A closed loop system was developed for controlling the hydrolithospheric process under random influences. In addition, the use of distributed controllers minimized the deviation of a random variable at the output of the control system.
- Software was developed for modeling a distributed control system of the hydrodynamic process in aquifers under random impacts.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Schematic model of the field as a control object. F—flow velocity in the aquifer (disturbance); V—decrease in the piezometric level at the locations of production wells (control action); $\mathrm{F}=\overline{\mathrm{F}}+\stackrel{~}{\mathrm{F}}$, $\mathrm{V}=\overline{\mathrm{V}}+\stackrel{~}{\mathrm{V}}$—sums of deterministic and random components of processes; b

_{i}—overflow parameters (constant coefficients obtained experimentally on a real object, characterizing the rate of fluid overflow through aquicludes between layers); H—head in the aquifer (controlled value at the points of location of control wells).

Well № | Exploitation Reserves, m^{3}/Days | Drilling Year |
---|---|---|

23 | 3 | 1955 |

8 | 2 | 1963 |

**Table 2.**The initial series of level dynamics (in meters) with shifted series for statistical processing.

Y_{t} | Y_{t-1} | Y_{t-2} | Y_{t-3} | Y_{t-4} | Y_{t-5} | Y_{t-6} | Y_{t-7} |
---|---|---|---|---|---|---|---|

821.05 | |||||||

821.12 | 821.05 | ||||||

821.29 | 821.12 | 821.05 | |||||

820.96 | 821.29 | 821.12 | 821.05 | ||||

820.89 | 820.96 | 821.29 | 821.12 | 821.05 | |||

820.89 | 820.89 | 820.96 | 821.29 | 821.12 | 821.05 | ||

820.89 | 820.89 | 820.89 | 820.96 | 821.29 | 821.12 | 821.05 | |

820.75 | 820.89 | 820.89 | 820.89 | 820.96 | 821.29 | 821.12 | 821.05 |

820.75 | 820.75 | 820.89 | 820.89 | 820.89 | 820.96 | 821.29 | 821.12 |

820.75 | 820.75 | 820.75 | 820.89 | 820.89 | 820.89 | 820.96 | 821.29 |

821.22 | 820.75 | 820.75 | 820.75 | 820.89 | 820.89 | 820.89 | 820.96 |

821.67 | 821.22 | 820.75 | 820.75 | 820.75 | 820.89 | 820.89 | 820.89 |

822.05 | 821.67 | 821.22 | 820.75 | 820.75 | 820.75 | 820.89 | 820.89 |

822.44 | 822.05 | 821.67 | 821.22 | 820.75 | 820.75 | 820.75 | 820.89 |

822.08 | 822.44 | 822.05 | 821.67 | 821.22 | 820.75 | 820.75 | 820.75 |

821.55 | 822.08 | 822.44 | 822.05 | 821.67 | 821.22 | 820.75 | 820.75 |

821.58 | 821.55 | 822.08 | 822.44 | 822.05 | 821.67 | 821.22 | 820.75 |

822.22 | 821.58 | 821.55 | 822.08 | 822.44 | 822.05 | 821.67 | 821.22 |

822.91 | 822.22 | 821.58 | 821.55 | 822.08 | 822.44 | 822.05 | 821.67 |

823.07 | 822.91 | 822.22 | 821.58 | 821.55 | 822.08 | 822.44 | 822.05 |

823.18 | 823.07 | 822.91 | 822.22 | 821.58 | 821.55 | 822.08 | 822.44 |

823.52 | 823.18 | 823.07 | 822.91 | 822.22 | 821.58 | 821.55 | 822.08 |

823.94 | 823.52 | 823.18 | 823.07 | 822.91 | 822.22 | 821.58 | 821.55 |

824.06 | 823.94 | 823.52 | 823.18 | 823.07 | 822.91 | 822.22 | 821.58 |

824.06 | 824.06 | 823.94 | 823.52 | 823.18 | 823.07 | 822.91 | 822.22 |

823.85 | 824.06 | 824.06 | 823.94 | 823.52 | 823.18 | 823.07 | 822.91 |

822.86 | 823.85 | 824.06 | 824.06 | 823.94 | 823.52 | 823.18 | 823.07 |

823.2 | 822.86 | 823.85 | 824.06 | 824.06 | 823.94 | 823.52 | 823.18 |

822.97 | 823.2 | 822.86 | 823.85 | 824.06 | 824.06 | 823.94 | 823.52 |

822.25 | 822.97 | 823.2 | 822.86 | 823.85 | 824.06 | 824.06 | 823.94 |

822.15 | 822.25 | 822.97 | 823.2 | 822.86 | 823.85 | 824.06 | 824.06 |

821.79 | 822.15 | 822.25 | 822.97 | 823.2 | 822.86 | 823.85 | 824.06 |

821.91 | 821.79 | 822.15 | 822.25 | 822.97 | 823.2 | 822.86 | 823.85 |

822.25 | 821.91 | 821.79 | 822.15 | 822.25 | 822.97 | 823.2 | 822.86 |

822.34 | 822.25 | 821.91 | 821.79 | 822.15 | 822.25 | 822.97 | 823.2 |

822.29 | 822.34 | 822.25 | 821.91 | 821.79 | 822.15 | 822.25 | 822.97 |

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**MDPI and ACS Style**

Ilyushin, Y.V.; Asadulagi, M.-A.M.
Development of a Distributed Control System for the Hydrodynamic Processes of Aquifers, Taking into Account Stochastic Disturbing Factors. *Water* **2023**, *15*, 770.
https://doi.org/10.3390/w15040770

**AMA Style**

Ilyushin YV, Asadulagi M-AM.
Development of a Distributed Control System for the Hydrodynamic Processes of Aquifers, Taking into Account Stochastic Disturbing Factors. *Water*. 2023; 15(4):770.
https://doi.org/10.3390/w15040770

**Chicago/Turabian Style**

Ilyushin, Yury Valeryevich, and Mir-Amal Mirrashidovich Asadulagi.
2023. "Development of a Distributed Control System for the Hydrodynamic Processes of Aquifers, Taking into Account Stochastic Disturbing Factors" *Water* 15, no. 4: 770.
https://doi.org/10.3390/w15040770