Entropy solution of motion model in planar environment of two-phase fluid

During the economic development of countries, the discovery of new energy sources and their optimal usage processes need to be examined on a scientific basis. This in turn requires the creation of novel mathematical and engineering methods. In the thesis, a method has been developed to find a numerical solution of the system of equations that models the co-movement of two-phase fluids in the porous medium, which is a special field of hydrodynamics. Equation systems that represent the co-movement of two-phase fluids (eg water and oil) are reduced to finding the solution of the first order nonlinear partial-differential differential equation based on the water saturation function. One of the specific features of this equations or system of equations is that they have discontinuities at unknown locations in the solution. This feature presents some difficulties in the application of well-known classical methods in the literature. Therefore, in the first part of the thesis, the necessary framework has been created for the formation of the concept of weak (generalized) solution. In the second part, in order to explain the dynamics of the subject, movement models of incompressible, then compressible, fluids on one dimension has been examined. To examine the properties of the solutions of Buckley-Leverett model, Cauchy model has been examined mathematically, then this examination method has been applied to the problem of compressing oil using water as in practice. Moreover, a method has been proposed to separate the complex equation system into separate equations based on the physical parameters; a saturation function and a pressure function. Lastly, a finite difference method has been developed to find the solution of the equation system that describe the compression and extraction of oil using chemically active water with the appropriate initial and boundary conditions. In the third part, considering the flow dynamics and capillary pressure of fluids in a microscopic environment, the process of compressing oil with water has been examined. In the last part, the movement equations of two-phased fluids in the planar layer has been examined. Using the methods developed by M. Rasulov, that describe the complete physical properties of the physical phenomenon, some improvements have been suggested and applied to the solutions of a few practical problems.