The purpose is the development of a mathematical apparatus that makes it possible to calculate both the amount of possible increase in the irrigated area due to the recycling of drainage and waste water, and the amount of natural water savings provided that the irrigation area remains unchanged, as well as determining the possible annual additional profit from the recycling of drainage and waste water on irrigation systems. Materials and methods. The systematic analysis and, as a result, a balance method, the key factor of which is the observance of the annual soil salt balance during irrigation are used, together with various options for recycling drainage and waste water. Results. To achieve the research goal, the following was carried out: modeling the determination of the maximum allowable salinity of drainage and waste water suitable for irrigation; modeling of drainage and waste water recycling without dilution and with their successive dilution; modeling of drainage and waste waters recycling. As a result of the studies carried out for the option of drainage and waste water recycling with their successive dilution, the possibility of obtaining an annual additional profit from the recycling of such water in irrigation systems under conditions of insufficient water supply, which will be about 18 %, is revealed; the payback period for modernization of the irrigation system will be about 6 years. Conclusion. Recycling of drainage and waste water during irrigation is a promising direction for increasing water supply and improving the efficiency of water resources protection. With the help of the ratios obtained, the issues of optimal water distribution between irrigated plots and determination of their areas can be solved, taking into account the availability of natural waters and drainage and waste water recycling.

The purpose of the research was to create an adequate mathematical apparatus to establish the interrelation between surface and drainage waters, since the need of interconnection between surface and drainage waters in the field of environmental safety, reclamation and hydrology arises regularly. At the same time, conducting field studies that determine this interrelation is a very costly and time-consuming event, therefore, the development of a mathematical apparatus that adequately describes the relationship between surface and drainage waters is an urgent task. Materials and methods. The developed mathematical apparatus is based on the compilation of the classical theory, namely: the Chezy, Darcy formulas, the water table, the water balance equation and the expression for the water surface slope. Results. As a result of the research, the drainage water runoff zone was divided into a finite number of elements, and the average velocity of the filtration flow interacting between the canal and drainage water was determined. Numerical verification of the modeling results was carried out on the basis of data obtained during field studies at the inter-farm canal section near the village Swamp Belevsky district Tula region. In the process of research, an algorithm and a program were developed for calculating the drainage water flow rate for the considered section of the inter-farm canal. Based on the simulation results, a hydrograph of drainage water flow rate was constructed in the inter-farm canal near the village Swamp, while the hydrograph of drainage water flow, built on the basis of modeling results, practically coincides with the hydrograph, built on the basis of data provided from the results of field studies. Conclusions. Approbation of the results of the developed mathematical apparatus and mathematical model showed the correlation between theoretical and experimental studies with a determination coefficient value of 90.22 %.