Numerical Analysis of Filtration Flows in Inhomogeneous Media Using the R-Functions Method

• Дата: 2018
• Автор: Подгорний, Олексій Русланович
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2018
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/159390
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The problem of the stationary porous media flow theory in an isotropic inhomogeneous media is considered in the assumption that the Darcy law is fulfilled. The mathematical model of this problem is the elliptic equation for the stream function, supplemented by second kind boundary conditions at the reservoir boundaries, and the first kind boundary conditions in regions that are impenetrable to the liquid. At the same time, the unknown value of the full liquid flow enters the problem statement and for its finding an additional integral ratio was formulated. The structural-variational method (the R-functions method) is proposed to be used for numerical analysis and that will allow taking into account all the geometric and analytical information from the problem statement most fully. The transition from the original problem to a boundary value problem with known boundary conditions was made. According to the R-functions method for the constructed solution structure, which accurately takes into account all boundary conditions of the obtained problem, the use of the variational Ritz method for the approximation of an indefinite component is substantiated. After that, an approximate solution of the initial problem was found from an additional integral relation. A computational experiment was conducted for different values of filtration coefficients in an area that has the form of the lower half of the ring. Also, the coordinate functions were constructed on the bases of Legendre polynomials. The approximate solution of the problem was compared with the exact solution in the case of a stable filtration coefficient. It is found that the error of determining the full liquid flow and the approximate solution of the problem decreases with the increasing number of coordinate functions. Also, the cases, where the filtration coefficient increases with depth, were considered. It is established that with the increase in the number of coordinate functions, the value of total costs tends to converge. Consequently, the proposed method of numerical analysis proved its effectiveness and can be used for practical problems solving. The advantages of the developed method are obtaining a solution of the boundary value problem in the form of the single analytical expression and the exact satisfaction of all boundary conditions of the problem.