Application of Dynamic Programming Models for Improvement of Technological Approaches to Combat Negative Water Leakage in the Underground Space

Abstract

The article solves an urgent problem, which is to develop a new approach to finding solutions to improve technological methods to combat negative water leakage in underround spaces. We propose the use of dynamic programming methods to select the optimal technology to secure such spaces. In accordance with the algorithm proposed in this paper, the problem was broken into a number of stages. At each stage, an optimal solution was sought (organisation of transport, delivery of materials to the destination, selection of materials, etc.). Thus, we applied a decomposition approach that allowed us to take into account the variety of parameters that affect the efficiency of the process. All these stages and their corresponding technological solutions were formalised by building network models. In these network models, vertices corresponded to solutions, and the distances between vertices (edges) corresponded to the value of the optimisation parameter. Thus, the shortest route from the initial to the final vertex corresponded to the optimal technological solution to combat negative water leakage in underground spaces. Based on the systematisation of data on technologies to combat water inflow into underground spaces, basic and refined models were developed. These models allowed us to take into account the risks associated with water breakthroughs into underground spaces. To minimise the risks, additional measures to combat water inflows are envisaged. In the practical part of this study, the results of the selection of a method with which to control water inflows are presented. This method involves the use of anchoring to reduce water filtration. According to the results of field observations, no water breakthroughs into the underground space were recorded.

1. Introduction

Throughout its history, mankind has undertaken the construction of structures in underground spaces. At first, it was caves, then tunnels, vaults, labyrinths. Today, underground structures are used not only in production activities, but also in public construction (for example, subways). Regardless of the stage of development of science and technology, there was always a problem in dealing with negative water manifestations. Negative water effects include water filtration, the soaking of rocks surrounding the contour of an underground structure, and a high probability of water inflow into underground spaces. All these phenomena can be called unfavourable hydrogeological conditions. The negative phenomena associated with water infiltration seriously complicate the aforementioned processes, significantly increasing the cost and slowing the pace of the construction of underground structures, and sometimes making it impossible altogether. To solve these problems, mankind has created technological methods and means that make it impossible for water to enter the underground space. Such means include lining underground spaces with different types of materials, applying surfactants to minimise water filtration, etc. However, not every solution is optimal. Perhaps the most reliable technology reduces the risk of water leaks, but it is significantly more expensive or requires more human resources. That is why we set ourselves the task of developing decision-making approaches that would allow us to obtain the most rational solution. By “rationality”, we mean obtaining a solution regarding the technology intended to prevent water seepage into underground spaces that will meet two conditions: first, it will be optimal from the point of view of minimising the optimisation parameter (for example, the cost of constructing a support); second, it will cause the least harm to the environment, and, if possible, will allow for the disposal of production waste. One way or another, in the first stage, it all comes down to developing decision-making tools.

In this article, we propose a new methodology and decision-making tools for the design of measures to minimise the negative impact of groundwater in underground spaces.

However, first of all, it is necessary to analyse existing approaches to decision-making. There are several classifications of decision-making methods and tools in production. These classifications can be divided into three types: classification based on economic and mathematical models (this classification was popular until the 1980s), classification based on the results of decisions, i.e., quantitative or qualitative characteristics (this classification is still popular, but it became most popular in the 1990s), and classification based on methods of obtaining decisions and the duration of forecasting (2010—present).

The authors of [1] described economic and mathematical models, as well as the scope of the models. It is worth noting that the difference in the use of mathematical modelling methods lies solely in the limitations and types of parameter changes, as well as the requirements for the resulting solution. As already mentioned, in [2,3], a retrospective analysis of decision-making methods in production was carried out. According to these works, classifications can be divided according to the type of economic and mathematical model used [4], with the main differences being the nature of the parameter change and the design time. For example, if the process includes several stages that change over time, dynamic models are used. Another classification approach is based on the type of decision obtained [5], i.e., will we get a quantitative or qualitative answer as a result of the decision? For example, in the organisation of transport deliveries, a quantitative solution is appropriate, because we will get an answer to the question “Which of the alternatives in the transport chain has the lowest delivery cost, route length, etc”. When applying hierarchical analysis methods, a qualitative characteristic is obtained, that is, we will get an answer to the question “Which of the obtained solutions is the most satisfactory for our needs?” For example, when solving the problem of concentrator location, the connection with the enterprise, location, community interests, etc. are taken into account. The last classification type [6] takes into account the duration of the forecast (long-term or short-term), as well as the process of obtaining a solution (software, modelling on equivalent materials, simulation modelling, finite element method, etc.).

The following economic and mathematical methods are currently known: linear programming [7], quadratic programming [8], integer programming [9] (for example, when choosing mechanisation means by combinatorial optimisation, the number of equipment types should be an integer), stochastic programming [10] (in which nonlinear changes in parameters are possible) dynamic programming [11] (in which the process changes in time and includes several stages), geometric programming [12], inventory management [13] (in which the problem of allocation is solved), game theory [14] (when it is necessary to solve the problem of ensuring interests), network models and optimisation algorithms on network models and graphs [15] (which search for the shortest route corresponding to the optimal solution), models based on establishing dependencies and predicting changes in parameters [16], service theory [17], study of reliability as the ability of an object to perform its functions [18] (for example, the productivity of a technological chain is the ability of an object to reproduce a given level of productivity), forecasting and building correlation models [19], and finally, the application of simulation modelling methods using electrical circuits, information systems, etc. [20].

It is worth noting that the vast majority of the described models and methods [7,8,9,10,11,12,13,14,15,16,17,18,19,20] are used in short-term planning. In addition, these models contain quantitative rather than qualitative dimensions of the decision. However, the decision-making process is influenced not only by economic, technological, and social factors, but also by qualitative ones. If the objective function is, for example, the unit cost or transportation costs [21], the decision will be optimal in terms of a certain parameter, but there is a high probability that it will not meet the ‘quality’ characteristic, i.e., a set of features that characterises the object (e.g., for a mineral, the degree of depletion, ash content, etc.; and for an enterprise, infrastructure, location, communication with suppliers and consumers, etc.) [22]. It is also necessary to take into account production risks associated with unfavourable production conditions [23,24]. Water-related risks are also considered to be risks, but in addition to complicating production, the quality of water resources deteriorates [25]. All of this affects the quality of life of the population in regions where underground spaces are developed [26,27]. As noted, the methods [7,8,9,10,11,12,13,14,15,16,17,18,19,20] provide an answer about quantitative characteristics, not qualitative ones. If the transportation problem is solved [28,29], the solution will be optimal and quantitative, for example, in terms of specific transportation costs, the length of the transportation chain, etc. However, this does not take into account the wishes of suppliers, consumers, etc. That is why a classification system has been developed that takes into account the types of solutions and models used [30,31]. These models are divided into information and optimisation models. Optimisation models provide a quantitative answer, while information models reflect the qualitative characteristics of expert groups. Informational models include mathematical programming methods [32,33,34,35,36,37,38], as well as game theory and decision-making criteria under conditions of uncertainty (Wald, Laplace, Hurwitz, Savage, Maximin). Information models are based on the AHP hierarchy analysis method [39] developed by T. Saaty; they comprise variations of the AHP method [40,41,42,43,44,45,46,47,48,49,50,51], a method that involves the use of fuzzy sets [52,53]. In these methods, hierarchies are built and priorities are set. As a result, we get a high-quality solution that takes into account the wishes of the expert group. The disadvantages of these methods include situations in which it is impossible to build a connectivity matrix, i.e., when one of the parameters significantly outweighs the others (for example, the cost of purchasing equipment) [54]. Another disadvantage is the need to use expensive software [55,56]. In [57], a separate group of approaches based on simulation modelling methods is identified [58]. Simulation modelling is used to model groups of production parameters by breaking larger problems into local ones and applying appropriate algorithms to find the optimal solution for each group [59,60,61].

Based on the analysis of recent works [62,63,64,65,66,67,68], the following generalisation can be made, which forms trends in the development of approaches to process design: the optimality criterion is “quality”, but it has different quantitative indicators; the method of breaking down larger problems into local ones should be applied; but only after finding the optimal solution in the previous stage should one proceed to finding a solution in the next stage, in other words, applying the principle of ‘Belman optimality’. The following questions need to be answered for the task of preventing water leakage in underground spaces:

What technology should be chosen? The chosen technology should not only prevent water from entering the underground space but should also be inexpensive. The cost can be expressed not only in monetary terms, but also in terms of the amount of resources required. Of course, the use of concrete-based materials is cheaper than technologies that use composite materials, but the former are more resource-intensive and cause more environmental damage.

How to minimise the negative impact on the environment? Or, which technology is environmentally preferable? Here, we analyse not only the amount of resources required to prevent water spills, but also the environmental damage caused by the materials and technologies used. Nowadays, it is no longer enough to analyse and develop economic strategies; environmental strategies should be developed as well [69,70].

Is it possible to use production waste to prevent water leakage? This possibility can be applied not only in the production process, when the technology involves leaving materials in the underground space, but also on the basis of the comprehensive utilisation of production waste.

The answers to these questions are contained in the present study. We have proposed not only approaches to decision-making, but also paid considerable attention to the possibility of minimising the technogenic impact on the environment.

2. Materials and Methods

2.1. Study Area

The approaches presented in this paper and the proposed dynamic programming model are valid for countries where minerals are extracted underground at considerable depths, i.e., when there is a need to secure the mined space. An additional condition is the likelihood of water inflows into such underground spaces, i.e., the presence of waterlogged layers in the rock mass. The above technological operations are described for conditions with unstable or medium stability of roof rocks. The rocks of the workings’ base are prone to uplift. In the absence of unfavourable hydrogeological conditions, as well as when there is no need to secure the excavated space (the rocks are stable), the above models are not applicable. For the conditions of Eastern Europe and countries where mining is carried out under unfavourable mining and geological conditions, this approach is appropriate. In addition, it is worth noting that the main area of application is the extraction of seam minerals (coal, potash, manganese, etc.). For iron ore deposits, this approach can be applied only in capital workings when there is a need to secure the mined space.

The proposed methodology can be applied to other processes that take place in underground spaces. These are tasks that are not only related to securing the mined space to prevent negative manifestations of water; dynamic programming models can be used to develop transport chains and emergency response plans. It is worth noting that we have presented the application of the dynamic programming method in terms of minimising the cost of construction of workings (the forward problem), but it is also possible to apply and solve the reverse problem. If a mineral is being extracted and there is a need to extract the maximum amount of it, i.e., to maximise the cost of 1 tonne of mineral, then the inverse problem is solved.

When used in regions with different hydrogeological characteristics, existing technologies for the control of water inflows should be taken into account. For example, in Ukraine, Polesia, and the Czech Republic, anchoring is a very common method in underground spaces; it ensures the monolithicity of the waterlogged rock mass. At the same time, for the conditions in the Republic of Kazakhstan, where the water content is lower, the use of frame supports is more traditional. Also, if such projects are not related to the extraction of minerals, for example, in the construction of subways or underground storage facilities, concrete supports are traditionally used. The only differences are in the existing technologies to control water inflows into underground spaces, in accordance with the country’s conditions.

Dynamic programming has several advantages:

Reduced execution time, i.e., by breaking down a task into smaller subtasks to avoid repetitive calculations, thereby reducing the overall execution time.

The efficient use of resources, i.e., the memorisation of results avoids repeated calculations and reduces resource requirements.

• The optimal solution: dynamic programming usually provides an optimal solution to a problem.

Limitations of dynamic programming:

• Dynamic programming has certain limitations.
• Dependence on the structure of the problem: to use dynamic programming, the problem must have an optimal structure, which may not always be achieved.
• The need for a large amount of memory: the memorisation of results can require a significant amount of memory, especially for large problems.

The time complexity of the Bellman-Ford algorithm depends on the number of vertices in the graph. To reduce the number of calculations, you can use the time-to-memory ratio, and with more memory, the execution time decreases.

Part of the data calculated by the algorithm about the distances to the vertices of the graph can be stored in memory. Then, if you need to find a path from the same node, the path can be loaded from memory instead of having to be calculated again.

The data calculated by the Bellman-Ford algorithm can be partially stored, which will allow you to choose the optimal time–memory ratio for a particular system. Unlike the Dijkstra algorithm, the Bellman-Ford algorithm allows negative edge weights to occur in the graph. In addition, the complexity of the Bellman-Ford algorithm is O(VE) (V is the number of vertices, E is the number of edges), while the complexity of the Dijkstra algorithm in the classical implementation is O(V2E). Thus, the Bellman-Ford algorithm was chosen to implement data transmission in a multi-parameter system.

2.2. Dynamic Programming Model

To solve this problem, we used dynamic programming methods.

Dynamic programming models can be applied in less controlled environments, for example, to design mining, processing, and beneficiation processes. In this case, not only quantitative indicators should be addressed (the reduction of production costs and expenses, the maximisation of profits), but also qualitative indicators should be taken into account. The main purpose of dynamic programming models is to make decisions in multi-stage processes, i.e., to find the optimal/extreme (largest or smallest) value of the parameter to be optimised. In this case, “multistage” means a certain process, technology, process structure, which contains a certain sequence and the possibility of its division into stages. The term “dynamic” indicates that the design process is time-consuming. However, it is worth noting that dynamic programming models, due to software complexities in terms of implementation, should only be used for long, multi-stage processes, such as the consolidation of excavated spaces under complex hydrogeological conditions. This is because at the decision-making stage, a “compromise” decision is made, i.e., either to reduce the cost of mining or to not apply the measures, in which case the risks of water inflow and subsequent water filtration will increase significantly. The amount of environmental damage is much greater than the economic effect. Thus, dynamic programming models are used in several cases:

-

When the process is multi-stage and all stages are interconnected;

-

when the process is time-consuming;

-

when the efficiency of the process is influenced not only by quantitative but also by qualitative indicators.

In particular, the decision-making process can be considered using the following functional equation [71]

$$\frac{\partial x(t)}{\partial t}=f(x(t),u(t),t);x(\tau )=y;\tau \le t\le t_1,$$

(1)

• τ—an arbitrary parameter;
• y—is an arbitrary vector.

The direct immersion invariant method allows us to consider the production process in time t with a limited amount of resources. All possible solutions are considered, and each subsequent solution (alternative) must be no worse than the previous one (Figure 1).

To make a decision, we will present the process of finding the final solution as a set of procedures for finding optimal solutions at each stage (Figure 2). For this purpose, we will present Equation (1) in the form

$$x(t+1)=f(x(t),u(t),t);x(\tau )=y;\tau \le t\le t_1,$$

(2)

Figure 2. Schematic of the computational procedure.

Figure 2. Schematic of the computational procedure.

The process of finding a solution consists of examining possible solution vectors x for compliance with optimal solution domain F0. If the value of vector x at time t is F ≤ F0, you need to memorise the result and postpone it (memorise) on phase grid x; otherwise, the search should be interrupted. After that, based on responses x, x1, x2, we calculated the value of the optimisation parameter.

By using graphs and their corresponding network models, information can be presented in a compact and informative way. Dynamic programming methods can be used to select measures to combat water leakage in underground spaces. The idea is as follows: these stages and corresponding decisions are represented as vertices in a graph, and the distances between the vertices are the values of the optimisation parameter (e.g., cost). The shortest route will correspond to the optimal solution. It should be noted that the model is divided into stages and contains only real connections. That is, if there is a solution, we calculate the value of the optimisation parameter; if not, we do not eat the vertices or take the distance to be infinity.

The use of dynamic programming has a number of advantages:

Firstly, when making a decision, a multi-parameter problem is broken down into smaller ones, i.e., a decomposition approach is applied. The essence of this approach is as follows: at each stage, one “priority” control parameter is selected. It is the optimisation of this parameter that allows us to identify the optimal solution at this stage, which shapes the overall efficiency at the subsequent stages. For example, when choosing a technology for preventing water inflows into underground spaces, a number of tasks should be solved: to ensure the stability of the rocks surrounding the contour of the underground structure (to do this, we reduce filtration by using anchoring [72,73,74]); to ensure insulation against water ingress; to organise the delivery of materials, etc. However, in parallel, we solve the problems of material delivery, work organisation, and waste disposal—all these issues are important, but within a certain time frame. In other words, we take the general technology of combating negative water leakage into the underground space and break it down into stages: the delivery of materials to the warehouse, the delivery of materials underground, the delivery of materials to the place of construction of the support, work to minimise negative water leakage, etc. In each stage, we analyse alternative solutions. Without finding the optimal solution at a certain stage, we do not proceed to the next stage.

Dynamic programming models are based on the principle of Belman optimality [75]. According to Belman’s optimality principle, “If a control is optimal, then whatever the initial state of the system and the control of the system at an initial time, the subsequent control is optimal with respect to the state that the system will take as a result of the initial control”. Thus, the resulting solution will be optimal in terms of a certain optimisation parameter.

The search for a solution can be carried out in both forward and backward order. For the task of choosing the optimal technological solution to combat water inflows, the direct task is to minimise the cost of preventing water seepage into the underground space. The inverse problem is to minimise the waste that will be generated. In other words, the inverse problem involves an input condition that determines the quality of the solution.

After describing the principles of dynamic programming, we should focus on the decision-making procedure for improving technological solutions to combat negative water leakage in underground spaces.

2.3. The Development of a Model for Finding an Optimal Technological Solution in Complex Hydrogeological Conditions

Groundwater has a negative impact on the condition of underground structures and technological processes. In accordance with our objectives, we have established the relationship between hydrogeological factors and technological processes (

Table 1. Relationship between hydrogeological and technological factors in the construction and operation of workings.

Negative Impact of Groundwater	Hydrogeological Factor	Technological Factor and Risk
reduced productivity	specific water inflow	reduced equipment performance
	the degree of watering of rocks and water permeability	increased construction time and costs
	groundwater head, aquifer thickness, water permeability	flooding of workings, additional costs for drainage
reduced stability of products	filtration, degree of water saturation	additional costs for the support of the workings, measures to combat the heaving of the sole of the workings
reduced labour safety	water inflow, water permeability, filtration	destruction of temporary or permanent fasteners
deterioration in the operation of transport equipment	water saturation	complicated transport of materials, increased construction time

).

Table 1 shows that the problem of choosing a technological solution comprises justifying not only the safest and most reliable method, but also the most resource-saving one. In addition to cost, the time factor should be taken into account.

Regardless of the chosen optimisation parameter (cost of construction of 1 m of workings, costs of ensuring stability of 1 m³ of underground space, etc.), this indicator depends not only on the cost of resources but also on the consistency of technological processes at each stage of construction and operation (

Table 2. Costs for the construction and operation of the support.

Cost Component of the Total Cost	Construction and Operation Phase	Risks Associated with Water Inflows
materials from which the fasteners will be made to prevent water leakage	initial stage	None
transport and warehousing costs	loading, transport and unloading of fasteners	are present, it is necessary to take into account water permeability
tools, time costs	preparatory stages (drilling holes for anchors, etc.), transport of materials	are present, it is necessary to have an idea of the thickness of the aquifers of the massif
materials, labour intensity	construction of temporary support	are present, it is necessary to analyse the filtering
maintenance costs, labour intensity	construction of a permanent support and further operation	are present, the water inflow into the workings is analysed

).

Regardless of the type of technological solutions to combat water leakage in underground spaces, the cost components shown in Table 2 are identical, but different at intermediate stages, i.e., when comparing spray concrete and anchor supports, the amount of materials for the construction of support, time costs, labour intensity are analysed, but the level of mechanisation and technological stages are different.

As noted earlier, regardless of the type of support that prevents water inflows into underground structures, the stages of construction and operation of structures are identical, so the life cycle can be represented as a network model that takes into account alternative options (Figure 3).

Each vertex, from 1 to 17, corresponds to a separate solution, except for vertex 1, the start, and vertex 17, the end. These vertices are needed to connect the intermediate vertices (2–16) that correspond to the process solutions. In this case, the connections between the vertices are mutually exclusive, i.e., if the fastening is sprayed concrete (2) or anchored (3), there is no possibility of reuse (15), and there are no dismantling costs. The model can also be considered in reverse order: if it is not possible to dismantle the support (16) from the excavated space, then anchor supports (3) are not used.

As mentioned earlier, the model is structured by levels. To make an optimal decision, the shortest route from the first (1) to the last vertex (17) should be found. For this purpose, dynamic programming algorithms can be applied to network models and graphs; in our case, the Bellman-Ford algorithm [76]. For the algorithm to work, condition (3) must be met—the process must be uninterrupted in time. That is, to find the shortest route from node 1 to node j, some edge (arc) must be the final one.

$$f_i=\underset{i(i,j)}{\min }(f_i+t_{ij}),$$

(3)

Additional conditions are reflected in [77]. According to these requirements, only realistic alternatives should be reflected in the model, and the parameters should be managed centrally; without finding an optimal solution in one stage, it is impossible to proceed to the search for an optimal solution in the next.

In this case, depending on the stage of construction of the support, the parameters to be taken into account will be different.

The developed approach allows us to take into account hydrogeological parameters. This is done in the following way: knowing the degree of watering of the massif and its hydrogeological characteristics, it is possible to provide measures to improve the stability of the workings. For example, knowing that there is a risk of water breakthrough, it is necessary to use anchoring, which will reduce the water permeability of the workings and cover the cost of making grooves for drainage. Additional technological stages can be combined with existing ones. To do this, the network model (Figure 3) can be adjusted for the degree of hydrogeological risk, and a certain cost increase factor (shaded vertices) can be added to the intermediate stage (wavy line).

Conversely, when considering the model in the opposite case, having an idea of the support used (e.g., anchoring), it is possible to weaken the connections in the model (Figure 4), i.e., the probability of water inflows into the workings will be reduced, which will reduce the cost of auxiliary operations.

Thus, at the beginning of the decision-making process, a network model will be proposed with which to choose the optimal technological solution. In the absence of information on unfavourable hydrogeological factors, the search for a technological solution is carried out according to the parameter “costs of water inflow control per 1 m³ of underground space”, etc. If it is known that the massif is waterlogged, then certain links in the network model should be strengthened, which will result in an increase in costs, while others should be weakened.

In the following section, we present the results of studies on the use of dynamic programming methods to select optimal technological solutions to combat water leakage in underground spaces.

3. Results

3.1. Systematisation of the Stages of Construction of Support to Combat Negative Water Leakage: Basic Model

We considered the conditions in Western Donbas (Ukraine) as the object of this study. The Western Donbas geological and industrial region is characterised by complex mining, geological, and hydrogeological conditions. The structure of the region’s productive strata contains a large number of aquifers, which leads to wetting and collapse of the surrounding rocks, threatens safety, and increases the complexity of excavations. The effectiveness of existing methods to combat water inflows is insufficient, as they bring about large financial costs and do not take into account changes in the stress–strain state of rocks around the underground space and its impact on the permeability of the massif [78,79].

Our goal was to prevent water seepage into underground spaces in the conditions of the Samarska coal mine. Underground workings are used to transport coal and auxiliary materials. However, a complicating factor is the location of the mine in the area of the Bogdanovsky dump geological disturbance. This results in frequent water breakthroughs into the underground space.

Surface and groundwaters are located within the mine area. The main water inflows come from the widespread Buchak aquifer, which is composed of lightly clayey sands. Water from overburden deposits does not contribute to the watering of the mine workings. In the coal deposits, the water-bearing layers are sandstones and limestones, the thickness of which varies from 7–10 m to 100.7 m. The rocks of the upper part of the strata have increased water-bearing capacity. The water-bearing capacity of rocks in tectonic fault zones does not differ from the water-bearing capacity of faulted water-bearing rocks. However, this does not exclude the temporary inflow of water to mine workings when they are close to tectonic faults.

The average annual water inflow to the mine is 41 m³/h, and the salinity of mine water is 20–35 g/L. The water is corrosive, with a large amount of solid sediment, and has a pH of 6.5–6.8. The water hardness ranges from 21 to 40 mg equivalent per litre.

Therefore, our task was to choose the most rational technology for the construction of workings to prevent water leakage in the underground space.

Regardless of the support material (frame-load-bearing, sprayed concrete, anchoring, etc.), we found that there were common stages that involved solving a number of local problems, and we will focus on them below (

Table 3. Formalisation of the task of erecting a support in the underground space to prevent water leakage.

Stage	Stage Name	Objectives	The Essence of the Task Solution
I	organisation of loading and unloading operations on the surface	build a rational structure of the cycle for the organisation of surface loading operations	a balance should be found between resource flows to speed up cargo operations
II	delivery of fastening materials in the underground space, taking into account the cost of storing materials in the underground space	a rational structure of the transport chain should be built with the condition of maximum preservation of the quality and quantity of materials for the construction of workings	a balance should be struck between transport operations, suppliers (warehouses) and delivery to the site of construction
III	carrying out preparatory work required for the construction of temporary support (drilling holes, constructing grooves, stripping, etc.)	minimising time spent on preparatory operations and reducing costs	it is necessary to select drilling, assembly and loading equipment for these operations
IV	erection of a temporary fixture	organisation of the technology for the construction of workings and temporary consolidation of the excavated space	organisation of work on securing the excavated space. main requirements: ease of dismantling, minimisation of the amount of fastening materials, possibility of removing waste generated as a result of previous operations (I–III)
V	carrying out work on the construction of a permanent support	minimisation of the amount of materials required to securely fix the produced space	organisation of works on the construction of permanent support. it is necessary to minimise equipment downtime and reduce the consumption of fastening materials. at the same time, it is necessary to ensure the appropriate reliability of the support
VI	cleaning the underground space from the rock mass formed as a result of the construction of the support	minimisation of manual labour, reduction of waste	it is necessary to organise a technology that will keep the maximum amount of waste in the underground space. at the same time, the number of human resources involved in cleaning the produced space should be minimised
VII	dismantling of fasteners in the underground space	organisation of technology with the least time and financial costs	the technology of dismantling the support in the underground space should be organised. safe working conditions should be ensured. the number of human resources involved in these operations should be minimal

).

Table 3 shows that regardless of the type of support, construction technology, or materials used, all stages are identical (I–VII). This allowed us to formalise the task of selecting the technology for the construction of workings. To do this, it was necessary to represent the cycle of roofing the excavated space as a network model. The vertices of the network model corresponded to certain stages, and the distances between the vertices were the values of the optimisation parameter (e.g., cost, time, quantity of materials, etc.).

Let us depict a network model of the process of the working face construction in Figure 5.

As previously mentioned, the vertices are taken as technology options, and the distance between the vertices is the value of the optimisation parameter.

Table 4. Network formalisation of the problem of mine construction.

Stage Designation in Figure 5	Stage Name	Starting Point	Final Peak (Peaks at Intermediate Stages)	Interpretation
I	loading of fasteners on the surface	1	2–4	1—the starting point; 2–4 options for transport technology
II	delivery of fasteners in the underground space, taking into account warehouse costs	2	5–8	2—the best option after the 1st stage; 5–8 tops corresponding to the transport technologies
III	preparatory work	6	9–11	6—optimal solution after 2 stages; 9–11 variants of borehole drilling technologies
IV	construction of temporary support	9	12–15	9—optimal technology after three stages; 12–15 variants of temporary support construction technology
V	construction of permanent support	12	16–17	12—optimal technology after four stages; 16—fastening technologies
VI	removal and transport of waste generated during construction	16	18–20	16—optimal solution after 5 stages; 18–20 technologies for transporting production waste
VII	removal of fastening materials	18	21–24	18—optimal technology after six stages; 21–24 dismantling technologies
VIII	completion	24	25	21—optimal technology after completion of seven stages; 25—completion of the cycle for selecting the optimal technology

provides an explanation of the network model shown in Figure 5.

Then, to find a solution that corresponds to the optimal technology for the construction of workings, it came down to finding the shortest route from vertex 1 (the start vertex required to execute the algorithm) to vertex 25 (the end vertex required to complete the algorithm). With the help of optimisation algorithms, the route could be found automatically. In Figure 5, for clarity, the optimal route is highlighted with a thickened line; it is the shortest and optimal route. As noted earlier, only actual links were analysed.

Based on the results of the study, we proposed the use of anchoring [80,81]. Based on studies of the geomechanical parameters of the rock mass, it was found that the use of anchoring would reduce the value of the heterogeneity by 25–30% and the permeability in the roof of the mine workings by 25–35% when tampering with the coal aquifer. Anchoring prevents the development of fissures, preserves the host rocks in a natural, monolithic state, and increases the stability of the mine workings, even if there is a waterlogged coal seam in the roof. A significant reduction in the size of the filtration area and a decrease in permeability within it leads to a decrease in the intensity of the filtration movement of fluid and to the prevention or reduction of water inflow into the anchored workings, which makes it possible to use anchoring as a technological method to reduce water inflow into the mine workings [82,83,84]. Figure 6 shows the results of the implemented technological solution to combat the negative effects of water in underground spaces.

Figure 7 shows a close-up of the technological solution. Also, Figure 7 shows that there are no signs of water in the area of the geological fault according to the observations.

The considered basic model allowed us to choose the optimal technological solutions to combat water inflows in the underground space. However, there are situations when hydrogeological conditions are unfavourable and there are risks associated with water breakthroughs into underground spaces. Therefore, technological solutions should be provided to minimise these risks.

3.2. Model for Finding Optimal Technological Solutions to Combat Water Inflows in Difficult Hydrogeological Conditions

The procedure is identical to the one described in Section 3.1. However, measures related to the prevention of negative phenomena associated with water inflows are taken into account.

Table 5. Stages of roof support construction, taking into account hydrogeological conditions.

Stage	Stage Name	Complicating Hydrogeological Conditions		Procedure for Taking into Account the Hydrogeological Factor at the Design Stage
		Factor	Hydrogeological Parameter
I	loading of fasteners on the surface	None	None	none
II	delivery of fasteners in products, taking into account warehouse costs	material sticking to the delivery means	filtration coefficient	it is necessary to allow for additional downtime for cleaning the delivery vehicles (as a result, the speed of the workings is recalculated), or an additional stage for strengthening the rock mass
		caking of materials
III	preparatory work (drilling holes, etc.)	destruction of boreholes	filtration coefficient, degree of watering of rocks	it is necessary to include additional stages for the treatment of the contact surfaces of the array, the application of surface-active substances (surfactants)
		unsatisfactory working conditions (waterlogged workings)	location and characteristics of waterstops, degree of watering of rocks
IV	erection of temporary support	unsatisfactory characteristics of the solution, destruction of boreholes	water absorption coefficient, air absorption coefficient	additional funds should be provided for strengthening the rock mass, additional costs for drainage ditches
V	construction of permanent support	unsatisfactory characteristics of the solution, unfavourable working conditionsi	location and characteristics of water stops, filtration coefficient, hydrostatic and hydrodynamic head	additional measures to strengthen the rock mass, application of surfactants, injection of strengthening solutions into boreholes, drainage grooves
VI	excavation and transport of rock mass in the course of work	unfavourable working conditions, rock mass sticking	filtration coefficient, degree of watering of rocks	construction of drainage grooves, additional time spent on cleaning delivery vehicles
VII	removal fasteners	destruction of the fastening material, inability to remove the structure (due to destruction)	chemical aggressiveness of water, location and characteristics of karsts and quicksand	accounting for the time spent on removing deformed support elements, additional measures to minimise water breakthroughs into the workings (injecting the massif with solutions, resins, etc.)

shows the main stages of the support construction and describes possible hydrogeological factors that affect the efficiency of the process. Based on the analysis of these factors, additional technological stages are envisaged to increase the stability of the rock mass. Knowledge of the technologies make it possible to develop a network model of alternatives to support work under difficult hydrogeological conditions.

Table 5 shows that all the stages are universal, i.e., they do not depend on the technology and type of support used. This allowed us to find the optimal technology for controlling water inflows into the underground space based on dynamic programming.

Let us depict in Figure 8 a network model of the process of building a working face. It should be noted that Figure 8a shows the basic variant and Figure 8b a variant taking into account complex hydrogeological conditions. We note that the consideration of complex hydrogeological conditions can be taken into account in different ways: Figure 8b shows a simplified version, when the network model is corrected (loaded) by a certain coefficient equal to the probability of occurrence of an unfavourable hydrogeological factor.

For example, in the baseline case, the preparatory work technology (7–10) corresponds to a cost value of 10 c.u. However, we know that there are additional costs for the application of surfactants and possibly re-drilling holes; these additional costs are 50% for this type of support, so the value of the optimisation parameter for this technology should be multiplied by 1.5.

As noted earlier, the vertices are taken as technology options, and the distance between the vertices is the value of the optimisation parameter.

It should be noted that the accounting of amendments can be applied for express analysis, even with limited resources, but the design process can be carried out more accurately by introducing additional stages to the basic model (Figure 8a) to combat the negative impact of various factors, e.g., strengthening the massif, the construction of drainage grooves, etc. Figure 9 shows a parameter optimisation model which takes into account complex hydrogeological conditions. It is worth noting that by neutralising the negative impact of water on the massif condition, it is possible to minimise costs in subsequent stages, i.e., preliminary measures (technology option) affect the efficiency of the process as a whole.

As shown in Figure 9, the number of main stages is identical, but in difficult hydrogeological conditions, it is necessary to provide intermediate stages which are directly related to measures to strengthen the state of the rock mass. If the provided technology is able to control water inflows, then an additional stage is not provided; otherwise, it should be provided (

Table 6. Network formalisation of the problem of construction of workings in difficult hydrogeological conditions.

The Stage Designation in Figure 9	Stage Name	Starting Point	Final Peak (Peaks at Intermediate Stages)	Interpretation
I	loading of fasteners on the surface	1	3–5	1—the peak to start from; 3–5 options for transport technology
			1–2, 2–5	2—availability of cleaning equipment, 5—transport technology
II	delivery of fasteners in products, taking into account warehouse costs	3	7–10	3—optimal technology after the first stage; 6—measures for sealing materials; 7–10 transport technologies
III	preparatory work (drilling holes, etc.)	8	12–14	8—optimal technology after two stages; 11—measures for applying surfactants; 12–14 drilling technologies
IV	construction of temporary support	12	16–19	12—optimal technology after three stages; 15—additional costs for processing the massif; 16–19 options for the technology of erecting temporary support
V	construction of permanent support	16	21–22	16—optimal technology after four stages; 20—additional costs for the construction of grooves; 21—22 fastening technologies
VI	excavation and transport of rock mass in the course of work	21	25–27	21—optimal technology after five stages; 23—additional measures for cleaning vehicles, 24—measures for transporting rock mass; 25–27 transport technologies
VII	fastener extraction	25	29–32	25—optimal technology after six stages; 28—additional measures for soil blasting; 29–32 dismantling technologies
VIII	Completion	29	34	29—optimal technology after seven stages; 33—additional measures to reinforce disturbed areas; 34—completion

).

Table 6 shows that to find the optimal solution, we need to find the distance from node 1 to node 34. The optimal solution is the one that corresponds to the shortest path in the network model (Figure 9). A prerequisite is that there is a connection between the technological stages.

Thus, if it is known that the array is waterlogged, then certain links in the network model should be strengthened, which will result in an increase in costs, while others should be weakened.

4. Discussion

In the present study, we proposed the use of dynamic programming methods to improve technological solutions to combat water inflows into underground spaces. Based on our analysis of existing approaches to finding optimal solutions, we found that, currently, the most advanced methods are those that break down the overall problem into partial ones, whereby it is possible to take into account a variety of factors and increase the accuracy of the solution.

Dynamic programming methods can be used to obtain optimal solutions in each stage, which contributes to the maximum probability of achieving the goal. In accordance with the proposed solution, we propose an algorithm with which to find the optimal technological solution to improve the methods of combating negative water leakage in underground spaces. According to this algorithm, it is first necessary to divide the general task of substantiating the technology for combating water leakage into stages that correspond to the search for a specific solution (transport, fastening, dismantling, etc.). After that, alternative solutions are presented in the form of a network model. Optimisation algorithms are used to find the shortest route in the network model that corresponds to the optimal solution.

If there are risks associated with unfavourable hydrogeological conditions, the proposed model should be adjusted. The number of basic stages is identical, but intermediate stages are introduced, which include measures related to the minimisation of hydrogeological risks. However, the construction of a refined network model involves certain time costs. To prevent this, we propose the use of a “simplified” network model, which is identical to the basic one but which makes allowances for negative hydrogeological factors (e.g., water breakthrough into the underground space).

Our approach is universal. It can be applied not only in industrial underground structures but also in civil engineering (subways, tunnels). As an example of this approach, we analysed the conditions in an industrial enterprise—a coal mine. However, this was no coincidence, as coal mines often face risks of water leakage for which there are a large number of alternative solutions, and the structures have a long service life (from 20 to 100 years). This variability allowed us to investigate the effectiveness of the proposed solutions. For the conditions of an enterprise that operates within a geological fault, the use of anchoring is proposed, as this reduces the permeability of roof rocks by 25–30%. According to the results of field observations, no water seepage into the underground space was recorded.

In conclusion, it is worth focusing on further areas of research. Our approaches are basic. We should additionally focus on the search for technologies to combat water inflows that involve the use of industrial waste.

The above models and algorithms can be implemented in software. Therefore, the task in the short term is to create a decision support system in the form of software. This will create a bank of design solutions and significantly increase the dimensionality of the tasks.

5. Conclusions

The utilisation of dynamic programming models represents a significant advancement in the domain of combating negative water leakage in underground spaces. Through our study, we have proposed the application of dynamic programming methods to enhance existing technological solutions, aiming to address water inflows effectively.

To operationalise this approach, we have formulated an algorithm for identifying the optimal technological solution to combat negative water leakage. This algorithm involves segmenting the task into stages corresponding to specific solution components and utilising optimisation algorithms to identify the shortest route in the network model, corresponding to the optimal solution. In scenarios with risks associated with unfavourable hydrogeological conditions, our model can be adjusted to incorporate intermediate stages focused on risk mitigation measures. While constructing refined network models incurs time costs, the proposal of a “simplified” version accounts for negative hydrogeological factors, offering a practical alternative without compromising effectiveness.

By analysing the conditions of a coal mine as an illustrative example, we have showcased the effectiveness of our approach. For instance, in enterprises operating within geological fault zones, anchoring techniques have proven to be effective in reducing roof rock permeability, consequently preventing water seepage into the underground space.

In conclusion, our study underscores the potential of dynamic programming models to improve technological approaches for addressing negative water leakage in underground spaces. By leveraging these advancements and continuing research efforts, we can enhance resilience against water-related challenges within various engineering contexts.