Area of specialization: 010900 Applied Mathematics and Physics
Course: 12 Computational Physics
Course director: Prof. S.L. Yakovlev
Department of Computational Physics
Scientific adviser: Prof. S.L. Yakovlev
Reviewer: Head Specialist. E.R. Nugumanov

Modeling of boreholes and pipelines systems with 
using implicit TVD scheme
A.	S. Galkin

There are quite little modeling tools in the problems, which relate to the calculation of wells unlike tools calculation for pipeline. Despite the fact that the differential equations in the case of one-dimensional performances are identical. The difference is in the magnitude of the angle of inclination of the borehole / tube. This approval can be illustrated with big problems in the calculation of bottom hole pressure in inclined wells. In the solution of these problems, many engineers are turning to the empirical correlations, where error can achieve tens of percent. The issue of the calculation of multiphase is studied in scientific articles, with no practical implementation. The Department of Computational Physics has papers on this topic. In these papers, difference equations were solved by Newton-type methods, which are sensitive to the initial approximation. To improve the convergence, we have to reduce the time step, and calculations become very long. 

Numerical scheme Implicit TVD is devoid of this disadvantage (because of its stability in the Courant number larger than 1), and thus belongs to a class of implicit difference schemes and is reduced to a scalar / matrix sweep. However, this scheme is described only for a generalized statement of hyperbolic equations and systems.

This paper contains a study on the applicability of Implicit TVD-schemes for hydro-simulation processes both in a vertical well and in a horizontal pipe

We obtained the following results:
1.	The method for use Implicit TVD-schemes for the equations with a nonzero right-hand part is described. 
2.	The universal program-solver of hyperbolic equations in the general form is designed and developed.
3.	Three physical models for the movement of the medium in the pipeline / well are modeled.
4.	The task about the stop in the pumping station in the pipeline is modeled.
5.	The problem of the ascent of a gas bubble in the well is modeled.