Numerical study of filtration process of ground and pressure waters in multilayer porous media

A article discusses the process of forecasting changes in the level of ground and pressure water. A brief analysis and computational experiments of scientific papers on mathematical and numerical modeling of the object under study are given. For a comprehensive study of the problem under consideration, a mathematical model was developed that takes into account the external source, evaporation, filtration coefficients, active porosity, filtration rate and two-way boundary conditions. An effective numerical algorithm has been developed for predicting changes in the ground water level using a combination of finite-difference schemes and run-through methods. It has been studied that changes in the level of ground and pressure water, filtration permeability, water loss coefficient and filtration rate associated with the water level can have a serious impact on the environmental process.


Introduction
The main tasks of hydrogeology, including the tasks related to land development, reclamation and irrigation construction, assessment of groundwater reserves and resources, and many other aspects, are to provide a prediction of hydrodynamic and hydromechanical regime of groundwater, which is a closely interconnected element of a single geofiltration system.
In conditions of acute shortage of water resources, water supply issues for the population, especially in ecologically disadvantaged zones of Central Asia, Khiva, Bukhara, including in the territories of Karakalpakstan, are especially relevant. One of the main sources of household and drinking water supply to the population in such conditions is groundwater formed by the construction of groundwater intakes.
Groundwater intakes are designed by performing numerous computational experiments using mathematical models of geofiltration and salt transfer in the underground hydrosphere, in order to determine the main indicators and parameters of the object of study.
The main objective in the issues of stable development of the agricultural sector is to increase crop yields and the quality of the output product, subject to significant savings in labor and energy resources, environmental requirements, etc. Which, in turn, is associated with solving the problems of substantiating the intensity of water reclamation of agrolandscapes, optimizing the calculation of agricultural drainage and managing the water regime of agricultural land. It should be noted that the MPCPE 2020 IOP Conf. Series: Materials Science and Engineering 896 (2020) 012069 IOP Publishing doi: 10.1088/1757-899X/896/1/012069 2 volume of drainage and waste water in many irrigation systems in Central Asia, the Caucasus and other areas reaches water intake.
In particular, a decrease in the Aral Sea level caused significant changes in the exploitation of aquifers, groundwater in the coastal zone, and had a negative impact on the environment. In fact, fisheries have been eliminated, groundwater exploitation conditions have worsened, the fauna has become poor, and the surface freed from the sea is subjected to aeolian processes, which entail a decrease in the productivity of coastal pasture lands adjacent to the Aral Sea, etc.
In forecasts of hydrogeological and ameliorative situation for the study of hydrodynamic regime of ground and pressure waters and the salt regime of soils, the following methods are mainly used: the balance, correlation and regression, analytical methods, the methods of identification, of numerical and analog modeling.
To conduct a comprehensive study, forecast and to make managerial decisions on the above mentioned issues, a number of problems have been solved, where the core is a mathematical model, a numerical algorithm and a software-instrumental complex for conducting computer experiment.
The basics of the science of groundwater movement (hydrogeology) are associated with the names of A. Darcy, J. Dupuis, N.E. Zhurkovsky, F. Forchheimer and others. A major role in the development of mathematical methods with the intensive development of the theory and practice of groundwater Jacob for the first time, apparently, pointed out the need to take into account the elastic regime in a poorly permeable layer.
To conduct a comprehensive study, forecast and to make managerial decisions on the above mentioned issues, a number of problems have been solved, where the core is a mathematical model, a numerical algorithm and a software-instrumental complex for conducting computer experiment.
In [1], the hydrogeological conditions of construction site of an underground complex and a mathematical model of soil base geofiltration were described. The results of computational studies of hydrogeological regime change in construction site when fencing the pit with a wall in the ground were considered.
A stationary model of groundwater filtration developed in [2], was used to quantify and analyze the underground hydrodynamics in the Akaki catchment, paying particular attention to the borehole field that supplies with fresh water the city of Addis Ababa. The simulation was performed in a two-layer unlimited aquifer with a spatially variable recharge and hydraulic conductivity under well-defined boundary conditions. The model was used to predict the pattern of groundwater flow, the interaction of groundwater and surface-water, and the effect of pumping on the borehole field under various scenarios.
In [3], a one-dimensional mathematical model of dissolved substances transport in finite aquifers was considered. The basic equation of dissolved substances transport by an unsteady-state flow of groundwater was solved analytically by the Laplace transform method. Initially, the aquifer was subjected to a spatially dependent concentration of the source with zero order formation. One end of the aquifer receives the concentration of the source and is represented by a mixed-type boundary condition in the time domain of solution. The concentration gradient at the other end of the porous medium is assumed to be zero.
In [4], the mechanism of artificial recharge affecting the groundwater reservoir was considered. Various scenarios of location of the infiltration basin and replenishment intensity model have been 3 developed based on a generalized groundwater reservoir in a two-dimensional sand reservoir in order to study how to increase the efficiency of artificial replenishment of groundwater reservoir.
The authors in [5] have simulated the process of groundwater filtration taking into account the nonuniform distribution and rarefaction of the aquifer with insufficient data on the object and poor knowledge of their properties. The problem was solved by taking into account the vertical stratification of aquifers of equal thickness.
In [6,18], a mathematical model was developed for predicting groundwater levels in two-layer formations. The authors of the paper consider a two-layer medium consisting of two layers: soil (with low permeability) and water as a mathematical model of the geofiltration process.
The papers [7][8] are devoted to numerical modeling of water and salt transfer process in soil. To conduct a comprehensive study, a mathematical model was proposed taking into account the colmatage of soil pores with fine particles over time; changes in soil permeability coefficient, water loss and filtration coefficient; changes in initial porosity and the porosity of settled mass; an effective numerical algorithm based on the Samarsky-Fryazinov vector scheme with a second order of approximation of differential operators to finite difference one was considered. To derive a mathematical model of salt transfer it was assumed that the pressure gradient in the channel is constant and equal to atmospheric pressure. The calculation results for the proposed algorithms were presented in the form of graphical objects; a detailed analysis of these results was given.
In [9], the computational filtering schemes from canals and water distributors in soil, underlain by a highly permeable pressure aquifer or waterproof base were developed; they allowed a comparative assessment of the role of cross-sectional profile of water source channel and water level in it, water support from underlying horizon or confining layer.
In [10], a model was proposed that allowed obtaining reliable information on groundwater level change and on the intensity of water reclamation of agro-landscapes, to optimize the calculation of agricultural drainage and to adjust the water regime of agricultural lands.
Mathematical modeling proposed in [11], was based on the data of complex studies carried out within the Ararat and Aparan intermountain basins; the issues related to the prevention of environmental consequences caused by large groundwater intakes were considered. As a result of data analysis by the mathematical modeling method, the problems of predicting the regime of groundwater level changes were solved while maintaining a constant load on the existing water intakes.
In [12], a model was given that correctly simulated the behavior of a free aquifer. During the periods of model operation, the values of groundwater level were calculated in a model with piezometers, in accordance with the groundwater level recorded by the piezometers.
A modeling system based on soil moisture content and groundwater dynamics was considered in [13]; numerical solutions were obtained for modeling hydrological processes of replenishment of sediments, surface-water, groundwater, and soil moisture content.
In [14], a method was proposed for numerical solution of non-stationary problems on two-component fluid flow in a porous medium, simulating transport of salt dissolved in groundwater.
In [15], an analytical method for predicting groundwater sequence regimes was considered. In [16], an urgent problem related to the process of change in underground water level and mineral salt transfer in soils is solved in the paper. The problem is described by a system of partial differential equations and the corresponding initial, internal and boundary conditions of various kinds. To derive a mathematical model of the process under consideration, a detailed review of scientific papers devoted to various aspects and software of the object of study is given. To conduct a comprehensive study of the process of filtration and change in salt regime of groundwater, mathematical models and an effective numerical algorithm are proposed taking into account external sources and evaporation. Since the process is described by a nonlinear system of partial differential equations, it is difficult to obtain an analytical solution.

Problem Statement
In mathematical modeling of monitoring and predicting the groundwater level and hydrodynamic processes occurring in them, considering the interaction of external factors: evaporation and infiltration, the studied object is presented schematically in the form shown in Fig. 1. According to the results of hydrogeological conditions analysis, the territory by groundwater renewal (GWR) in geofiltration relation should be considered as a two-layer one in medium, consisting of two aquifers.

Figure 1. Schematic representation of the object of study
The conditions accepted for predicting the changes in groundwater level (ground and pressure aquifers) during the filtration process give reason to present a mathematical model of the object in the form of a system of nonlinear partial differential equations: where ( , , ) h x y t , ( , , ) H x y t -are the groundwater and pressure water levels; System (1) is solved under the following initial and boundary conditions: Later, for simplicity, we omit the "*" sign in equations and problem (9) -(15) in dimensionless variables is written as follows: Problem (16) Under boundary conditions:

Solution Method
To solve problem (16) After some transforms and grouping similar terms, the finite-difference system (23) is rewritten in the form: