It is known that much of the technology aimed at intensifying fluid inflow by means of hydraulic fracturing involves the use of proppant. In order to transport and position grains in the fracture, a uniform supply of proppant with a given concentration into the fracturing fluid is ensured. The aim of the operation is to eliminate the occurrence of distortions in the injection program of proppant HF. A mathematically accurate linear increase of concentration under given conditions is possible only if the transient concentration is correctly defined. The proposed approach allows to correctly form a proppant HF work program for both linear and non-linear increase in proppant concentration. The scientific novelty of the work lies in application of a new mathematical model for direct calculation of injection program parameters, previously determined by trial and error method. A mathematical model of linear and non-linear increase of proppant concentration during HF was developed. For the first time, an analytical solution is presented that allows direct calculation of parameters of the main HF stages, including transient concentrations for given masses of the various types of proppant. The application of the mathematical model in formation of a treatment plan allows maintaining correct proppant mass distribution by fractions, which facilitates implementation of information and analytical systems, data transfer directly from a work program into databases. It is suggested to improve spreadsheet forms used in production, which would allow applying mathematical model of work program formation at each HF process without additional labour costs. The obtained mathematical model can be used to improve the software applied in the design, modelling and engineering support of HF processes.

The article presents a mathematical model of linear and nonlinear increase in proppant concentration during hydraulic fracturing. For the first time, an analytical solution using direct calculation for determining the parameters of the main hydraulic fracturing stages including the transition concentrations for given masses of proppant of various types is presented. It is shown that the use of a mathematical model in the development of a processing plan provides the correct distribution of the proppant mass by fractions, which facilitates the introduction of information and analytical systems and transferring data directly from the operating program to the database. The spreadsheet forms used in the production have been elaborated, which enables to apply the mathematical model of the operating program at each hydraulic fracturing without additional labor costs. The obtained mathematical model can be applied to improve the software used in the design, modeling and engineering support of hydraulic fracturing.