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Currently, methods and approaches of scientific visualization based on additional data analysis are being intensively developed due to the rapid development of computer technology in geology. The general concept is that the main data field on the day surface and additional conditions are specified at the input. Further, the methods of mathematical geophysics are used for their analysis and processing, as a result of which new information is obtained for deep exploration. Then visualization tools are applied to the obtained information and main data field in an information system. Thus, the information system is based on the synthesis of visual representation methods, methods of mathematical geophysics, computational mathematics, and various branches of science. This paper presents a description of the geoinformation system module for deep forecast search modeling of deposits developed on the base of the methods of intelligent anomalies detection of hidden deposits. Operation of the software module is based on the application of the theory of mathematical geophysics inverse problems using geological data on the Earth’s surface, geophysical measurements and geochemical analyses as input data. The software module for the inverse problem of the continuation of potential fields in the direction of perturbing masses is used for real data of a particular mineral deposit.

The paper studies a model of fluid flow in a fractured porous medium in which fractures are distributed uniformly over the volume. This model includes a nonlinear equation containing several terms with fractional derivatives in the sense of Caputo of order belonging to the interval 1,2. The relevance of studying this problem is determined by its practical significance in the oil industry, since most of the world’s oil reserves are in these types of reservoirs. The uniqueness of the solution to the problem in a differential form and its dependence on the initial data and the right-hand side of the equation is proved. A numerical method is proposed based on the use of the finite difference approximation for integer and fractional time derivatives and the finite element method in the spatial direction. A change of variables is introduced to reduce the order of the fractional derivatives. Furthermore, the fractional derivative is approximated by using the L1-method. The stability and convergence of the proposed numerical method are rigorously proved. The theoretical order of convergence is confirmed by the results of numerical tests for a problem of fluid flow in fractured porous media with a known exact solution.

This article presents an algorithm for the numerical solution of an initial-boundary value problem for a symmetric t-hyperbolic system of partial differential equations. This problem is based on continual filtration model, which describes the propagation of seismic waves in a poroelastic medium saturated with a fluid characterized by such physical parameters as the propagation velocities of longitudinal P- (fast and slow) and transverse S-waves, the density of the medium materials, and porosity. The system of linearized equations of saturated porous media is formulated in terms of physical variables of the velocity–stress tensor of the porous matrix and the velocity–pressure of the saturating fluid in the absence of energy dissipation. The solution is implemented numerically using an explicit finite difference upwind scheme built on a staggered grid to avoid the appearance of oscillations in the solution functions. The program code implementing parallel computing is developed in the high-performance Julia programming language. The possibility of using the approach is demonstrated by the example of solving the problem of propagation of seismic waves from a source located in the formation. Computational experiments based on real data from oil reservoirs have been implemented, and dynamic visualization of solutions consistent with the first waves arrival times has been obtained.

This article is devoted to the construction and study of the finite element method for solving a two-dimensional nonlinear equation of elliptic type. Equations of this type arise in solving many applied problems, including problems of the theory of multiphase filtering, the theory of semiconductor devices, and many others. The relevance of the study of this problem is associated with the need to develop effective parallel methods for solving this problem. To discretize the equation, a mixed finite element method with Brezzi-Douglas-Marini elements is used. The issue of the convergence of the finite element method is investigated. To linearize the equation, the Picard iterative method is constructed. Two classes of basis functions of finite elements are used in the paper. A comparative analysis of the effectiveness of several direct and iterative methods for solving the resulting system of linear algebraic equations is carried out, including the method based on the Bunch-Kaufman LDLt factorization, the method of minimal residuals, the symmetric LQ method, the stabilized biconjugate gradient method, and a number of other iterative Krylov subspace algorithms with preconditioners based on incomplete LU decomposition. The method has been tested on several model problems by comparing an approximate solution with a known exact solution. The results of the analysis of the method error in various norms depending on the diameter of the mesh are presented.

In this paper, we propose an efficient numerical method for solving an initial boundary value problem for a coupled system of equations consisting of a nonlinear parabolic partial integro-differential equation and an elliptic equation with a nonlinear term. This problem has an important applied significance in petroleum engineering and finds application in modeling two-phase nonequilibrium fluid flows in a porous medium with a generalized nonequilibrium law. The construction of the numerical method is based on employing the finite element method in the spatial direction and the finite difference approximation to the time derivative. Newton’s method and the second-order approximation formula are applied for the treatment of nonlinear terms. The stability and convergence of the discrete scheme as well as the convergence of the iterative process is rigorously proven. Numerical tests are conducted to confirm the theoretical analysis. The constructed method is applied to study the two-phase nonequilibrium flow of an incompressible fluid in a porous medium. In addition, we present two examples of models allowing for prediction of the behavior of a fluid flow in a porous medium that are reduced to solving the nonlinear integro-differential equations studied in the paper.

В работе проанализирован процесс проектирования и создания интегрированной распре- деленной информационной системы для хранения оцифрованных трудов ученых научно- исследовательских институтов Академгородка города Алматы. Проведен сравнительный ана- лиз двух систем хранения данных, Ceph и GlusterFS. Представлено описание программной ча- сти информационной системы, которая состоит из четырех подсистем: хранилища цифровых объектов, подсистемы управления текущей исследовательской информацией, подсистемы ин- теграции распределенных информационных ресурсов, подсистемы доступа к распределенным информационным ресурсам на основе веб-технологий. Описана связь между подсистемами и их интеграция. Определены требования к хранилищу цифровых объектов; проведен сравни- тельный анализ программного обеспечения с открытым исходным кодом, используемого для этих целей. Система в полной мере обеспечивает необходимыми вычислительными ресурса- ми проводимые научные исследования и образовательные процессы, упрощая перспективу дальнейшего его развития, и позволяет построить передовую IT-инфраструктуру управления интеллектуальным капиталом, электронной библиотекой, в котором будут храниться все книги и научные труды Казахстанского инженерно-технологического университета и научно- исследовательских институтов Академгородка города Алматы.

The paper is devoted to the numerical solution of a two-dimensional partial differential equation with a fractional derivative in the sense of Riemann-Liouville. This equation is of great applied importance and underlies many multiphysical simulators for modeling the distribution of pollutants in the atmosphere, fluid motion in a fractured porous medium with fractal fracture geometry, and many others. A two-step implicit computational algorithm for solving the problem is proposed, which leads to the need to solve a large number of systems of linear algebraic equations with dense matrices. To solve the latter, three iterative Krylov subspace algorithms are considered, namely, generalized method of minimal residuals with restarts, quasiminimal residual method, and induced dimension reduction method. A method for organizing parallel computations using CPU threads is proposed. The proposed method is tested on a two-dimensional model problem for an equation with fractional derivatives. An equation of this type typically arises when considering non-Darcian fluid flow in a fractured porous medium with a fractal geometry of fractures. The results of calculations for different values of the fractional derivative exponent and mesh configurations are presented. It is shown that this approach is able to increase the performance of the algorithm by more than 3.6 times.

In this paper, we study numerical methods for solving the Navier-Stokes equations in doubly connected domains. Two methods for solving the problem are considered. The first method is based on constructing a difference problem in variables of the stream function and the vortex of velocity using the uniqueness condition for pressure. The numerical solution of the elliptic equation for stream functions is found as the sum of the solutions of two simple problems of an elliptic type. One problem is with homogeneous boundary conditions, and the other is with a homogeneous equation. An alternative approach to solving the problem is the fictitious domain method with the continuation of the least coefficient. This method does not require satisfying the pressure uniqueness condition, and is simple to implement. An important direction in the development of numerical simulation methods is the study of approximate methods for solving problems of mathematical physics in complex multidimensional areas. To solve many applied problems in irregular areas, the fictitious domain method is widely used, which is characterized by a high degree of automation of programming. The main idea of the fictitious domain method is that the problem is solved not in the original complex domain, but in some other, simpler domain. This allows to create software immediately for a fairly wide class of problems with arbitrary computational domains. The possibilities of applying the fictitious domain method to the problems of hydrodynamics in the variables “stream function, vortex of velocity” are considered in many works. In this paper, we study a numerical method for solving the Navier-Stokes equations in doubly connected domains. A computational finite difference algorithm for solving an auxiliary problem of the fictitious domain method has been developed. The results of numerical modeling of the two-dimensional Navier-Stokes equations by the fictitious domain method with continuation by the lowest coefficient are presented. For this problem, a parallel algorithm was developed using the CUDA architecture, which was tested on various grid dimensions.

In this paper, we study an initial boundary value problem for a two-dimensional differential equation with a fractional time derivative in the sense of Caputo. This equation is of great applied importance in modeling flow processes and anomalous dispersion. The uniqueness and continuous dependence of the solution on the input data in differential form is proved. A computationally effective implicit scheme with weights is proposed. A priori estimates are obtained for the solution of the problem under the assumption that a solution exists in the class of sufficiently smooth functions. These estimates imply the uniqueness of the solution and the stability of the scheme with respect to the initial data and the right-hand side of the equation. The convergence of the approximate solution to the solution of the differential problem with the second order both in time and space variables is proved. The results of computational experiments confirming the reliability of theoretical analysis are presented.

The present paper is devoted to the construction and study of numerical methods for solving an initial boundary value problem for a differential equation containing several terms with fractional time derivatives in the sense of Caputo. This equation is suitable for describing the process of fluid flow in fractured porous media under some physical assumptions, and has an important applied significance in petroleum engineering. Two different approaches to constructing numerical schemes depending on orders of the fractional derivatives are proposed. The semi-discrete and fully discrete numerical schemes for solving the problem are analyzed. The construction of a fully discrete scheme is based on applying the finite difference approximation to time derivatives and the finite element method in the spatial direction. The approximation of the fractional derivatives in the sense of Caputo is carried out using the L1-method. The convergence of both numerical schemes is rigorously proved. The results of numerical tests conducted for model problems are provided to confirm the theoretical analysis. In addition, the proposed computational method is applied to study the flow of oil in a fractured porous medium within the framework of the considered model. Based on the results of the numerical tests, it was concluded that the model reproduces the characteristic features of the fluid flow process in the medium under consideration.

For decades, scientists of Almaty Academgorodok research institutes have been conducting enormous research in leading areas of the agro-industrial area producing hundreds of thousands of articles and other documents. However, it should be recognized that the results of these studies remain inaccessible to the vast majority of researchers in the age of the information explosion. One of the reasons for this problem is the lack of a publicly available information repository in the field of agriculture. For this reason, an integrated distributed information system of Academgorodok has been created the main objective of which is not only to preserve the rich heritage of the research institutes, but also to provide access to them and the ability to quickly search for the necessary information. The presented talk describes the architecture of the integrated distributed information system created for storing digitized works of employees of Almaty Akademgorodok research institutes and providing access to them using Web technology. Comparative analysis data storage systems and underlying software for storing digitized works is provided.

The paper studies a model of the gaslift process where the motion in a gas-lift well is described by partial differential equations. The system describing the studied process consists of equations of motion, continuity, equations of thermodynamic state, and concentration equation. A finite-difference scheme is constructed for the numerical solution of the problem. Because of the complexity of the motion equation, two simplified difference analogues for this equation are considered. The stability of these equations are investigated using the method of a priori estimates. The estimates obtained will be used for proving the stability of the whole difference scheme in future works.

The paper focuses on constructing and investigation of cost-effective difference schemes for the numerical solution of the two-dimensional three-phase non-isothermal flow problem without capillary and gravitational forces. In this paper, the finite difference method is used to solve the problem numerically. The method of energy inequalities is applied to examine the stability of the finite difference scheme with respect to the initial data and the right-hand sides of the equations. Three cost-effective difference schemes are constructed on the base of the studied scheme. The efficiency of the proposed algorithms is analyzed on the basis of comparing the average time spent on the numerical implementation of one time layer.

The paper focuses on the numerical implementation of a model optimal control problem governed by equations of three-phase non-isothermal flow in porous media. The objective is to achieve preassigned temperature distribution along the reservoir at a given time of development by controlling mass flow rate of heat transfer agent on the injection well. The problem of optimal control is formulated, the adjoint problem is presented, and an algorithm for the numerical solution is proposed. Results of computational experiments are presented for a test problem.

In this paper, one-dimensional and two-dimensional axisymmetric motion of gas, liquid and a gas-liquid mixture in a gas-lift well is studied. Numerical simulation of the one-dimensional model of gas-lift process is considered where the movement in a gas-lift well is described by partial differential equations of hyperbolic type. Difference schemes for the gas-lift model of the process are developed on a nonuniform grid condensing in subdomains with big gradients of the solution. The results of the proposed algorithm are illustrated on the example of a real well.

The paper is devoted to the problem of optimizing the process of steam stimulation on the oil reservoir by controlling the steam pressure on the injection well to achieve preassigned temperature distribution along the reservoir at a given time of development. The relevance of the study of this problem is related to the need to improve methods of heavy oil development, the proportion of which exceeds the reserves of light oils, and it tends to grow. As a mathematical model of oil displacement by steam, three-phase non-isothermal flow equations is considered. The problem of optimal control is formulated, an algorithm for the numerical solution is proposed. As a reference regime, temperature distribution corresponding to the constant pressure of injected steam is accepted. The solution of the optimization problem shows that choosing the steam pressure on the injection well, one can improve the efficiency of steam-stimulation and reduce the pressure of the injected steam.

An inverse problem is considered for the determination of the parameters, involved in the right-hand side of the system of nonlinear ordinary differential equations by given initial and final conditions. The solution of the problem is reduced to the minimization of the quadratic functional, which indeed is a deviation of the value of the solution from the given values at the end points. Using the quasilinearization method a calculation method is proposed to the solution of the considered problem. The application of this method is demonstrated on the example of the determination of the hydraulic resistance in the tubes.

This paper focuses on modeling of three-phase non-isothermal compressible flow in porous media taking into account capillary effects. A new formulation of the three-phase non-isothermal flow problem using the concept of reduced pressure is proposed. The purpose of the work is to eliminate the gradients of capillary pressure functions, leading to the unbounded growth of the solution, from the equations for temperature and pressure. An algorithm for the numerical implementation of the model based on the finite-difference method is suggested. A study of the developed difference scheme using the method of a priori estimates is conducted. The simulation results on the example of a one-dimensional model problem are presented.