MODELING OF PRODUCING PRESSURE IN HETEROGENEOUS OIL-BEARING RESERVOIRS

Numerical modeling of the distribution of the reservoir pressure drop in the vicinity of an operating well was carried out taking into account the inhomogeneous distribution of filtration characteristics (permeability and oil viscosity) in the near and distant zones of the well operation in order to study the practical aspects of filtration in heterogeneous oil-bearing formations based on a combined finite-element-difference method for non-stationary problem of piezoconductivity. The use of the combined finite-element-difference method enables to combine the advantages of the finite-element method and the finite difference method: to model geometrically complex areas, to find the value at any point of the object under study, while the implicit difference scheme. It is shown that the intensity of filtration processes in the vicinity of the operating well depends mainly on the permeability, and, to a lesser extent, on the viscosity of the oil. Moreover, the influence of the permeability of the oil phase in the remote zone (Rd < 5 m) is greater than the effect in the close zone (Rd > 5 m) of the operating well. In the case of low permeability of the oil phase in the vicinity of the existing well, to maintain stable oil production, it is necessary to place an injection well near the production well. Using the method suggested, it is possible to predict the effect of the injection well on the formation pressure distribution in the formation. The scientific novelty of the work lies in the study of the influence of the heterogeneous permeability and oil viscosity distribution on the reservoir pressures distribution around the wells by modeling filtration processes based on a combined finite-element-difference method. The practical significance of the research results comes down to confirming the close relationship between the heterogeneity of the porous medium and the reservoir pressures distribution around an operating producing well. The combined finite-element-difference method used in this work can be used to solve other filtration problems (for example, to calculate the gas saturation of a reservoir, create a method for calculating well flow rates, assess the effect of injection wells on filtration processes).

Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary

The chemotaxis system u t = Δ u − ∇ ⋅ ( u ∇ v ) , v t = Δ v − u v , is considered under the boundary conditions ∂ u ∂ ν − u ∂ v ∂ ν = 0 and v = v ⋆ on ∂Ω, where Ω ⊂ R n is a ball and v ⋆ is a given positive constant. In the setting of radially symmetric and suitably regular initial data, a result on global existence of bounded classical solutions is derived in the case n = 2, while global weak solutions are constructed when n ∈ {3, 4, 5}. This is achieved by analyzing an energy-type inequality reminiscent of global structures previously observed in related homogeneous Neumann problems. Ill-signed boundary integrals newly appearing therein are controlled by means of spatially localized smoothing arguments revealing higher order regularity features outside the spatial origin. Additionally, unique classical solvability in the corresponding stationary problem is asserted, even in nonradial frameworks.

Exact Solutions of Boundary Layer Equations in Polymer Solutions

The paper presents new exact solutions of equations derived earlier. Three of them describe unsteady motions of a polymer solution near the stagnation point. A class of partially invariant solutions with a wide functional arbitrariness is found. An invariant solution of the stationary problem in which the solid boundary is a logarithmic curve is constructed.

Numerical simulation of a stabilizing Poiseuille-type polymer fluid flow in the channel with elliptical cross-section

This work is devoted to the numerical analysis of stabilization of the incompressible viscoelastic polymer fluid flow in the channel with elliptical cross-section. To describe the flow, mesoscopic rheological relations are used, and resolving non-stationary equations are derived. For solving them a special pseudo-spectral method is developed and implemented. As time increases, under certain conditions on the parameters of flow the solution to the non-stationary problem stabilizes and converges to the one of three branches of the solution to the corresponding stationary problem. It is shown that the variation of the parameters describing polymer microstructure leads to the switch of stabilized solution between these branches. The work provides the results of simulation of the flow stabilization and the analysis of the threshold values of parameters at which the switching occurs.

Discharge modeling of a DC magnetron

A two-dimensional simulation of a glow discharge of a dc magnetron operating in an argon is performed in this work. The “Plasm” and “Magnetic Fields/No Currents” modules of the COMSOL Multiphysics software package were used for modeling. All models are constructed within the framework of a stationary problem. Obtained results correspond to the physical concepts of the glow discharge of a balanced magnetron

UPPER MANTLE CONVECTION RELATED TO SUBDUCTION ZONE AND APPLICATION OF THE MODEL TO INVESTIGATE THE CRETACEOUS-CENOZOIC GEODYNAMICS OF CENTRAL EAST ASIA AND THE ARCTIC

A geodynamic model of upper mantle convection related to the Pacific subduction zone is mathematically substantiated and applied to investigate the Cretaceous-Cenozoic evolution of Central East Asia (CEA) and the Arctic. We present a solution for the two-dimensional stationary problem of thermal convection in the upper mantle layer, considering different Rayleigh numbers and taking into account the influence of the subduction process and lithospheric movements along the upper mantle base. We describe the results of 3D modeling of nonstationary upper mantle convection in a subduction zone. Our data give grounds to propose explanations for the entire spectrum of tectonic-magmatic processes developing within CEA in the Cenozoic and the Arctic in the Upper Cretaceous and Cenozoic. We discuss the reasons why the lithosphere in CEA and the Arctic is generally shifting towards the Pacific subduction zone, considering the presence of separate magmatic provinces and rift zones. In our opinion, this is due to the existence of a large horizontally elongated convective cell, which interior is composed of smaller isometric cells. This long cell creates the effect of conveyor dragging of the lithosphere, while its internal cells produce the effect of upper mantle plumes.

The inverse non-stationary problem of identification of defects in an elastic rod

Non-stationary inverse problems of deformed solid mechanics are among the most underexplored due to, inter alia, increasing dimension of non-stationary problems per unit as compared with stationary and static problems, as well as necessity to consider the initial conditions. In the context of the continuing progress of the aviation and aerospace industries, the question arises about technical condition monitoring of aircraft for the purposes of their safe operation. A large proportion of an aircraft structure consists of beam and rod elements exposed to various man-made and natural effects which cause defects inaccessible for visual inspection and required to be identified well in advance. It is well known that defects (such as cracks, cavities, rigid and elastic inclusions) are concentrators of stresses and largely cause processes, which lead to the destruction of elastic bodies. Therefore, the problem of identification of such defects and their parameters, i.e. the problem of identification, represents a great practical interest. Mathematically, the problem of identification represents a non-linear inverse problem. The development of methods of solving such problems is currently a live fundamental research issue.

ANALYSIS OF THE CAPABILITIES OF THE LOGOS SOFTWARE PACKAGE FOR CALCULATING THE SEISMIC IMPACT ON A STRUCTURE

The possibilities of the LOGOS software package for calculating seismic vibrations of buried structures, considering the contact interaction with the ground and the gravity field, are investigated and expanded. To reduce computational costs, the LOGOS software package includes a method for modeling non-reflective boundary conditions that was developed earlier and implemented in the “Dynamics-2” software package, which allows reducing the size of the computational domain. The results of numerical simulation in the LOGOS of shear wave propagation in an elastic bounded subdomain of a continuous medium are presented, demonstrating the effectiveness of non-reflecting boundary conditions for a three-dimensional formulation. The dynamic relaxation technique used in the dynamic strength module of the LOGOS software package for calculating the initial static stresses from the action of the gravity field with the subsequent solution of the non-stationary problem is investigated. Numerical estimates of the expected accuracy of the dynamic solution are obtained, depending on the specified accuracy of the static calculation. Thus, the relative error in the change in kinetic energy when using the dynamic relaxation algorithm 10–4 gives a relative error of 10–3 in velocities and 10–2 in stresses, reducing the specified calculation error by 2 orders of magnitude reduces the relative error in calculating velocities by 3 times, stresses – by 5 times. The calculation of the initial stress-strain state of the building-ground system from the action of the gravity field is compared using the dynamic relaxation procedure and using the stationing procedure implemented in the “Dynamics-2” software package. In all problems in the three-dimensional formulation, 8-node hexahedra with one-point integration are used, in the two-dimensional formulation – equivalent 4-node finite elements, for the integration of the defining system of equations, an explicit “cross” scheme is used. Between the subdomains, variants of contact with gluing and contact with friction are implemented. The capabilities of LOGOS for conducting multiprocessor calculations allowed us to make estimates of the convergence of the problem under consideration based on a series of computational experiments.

Study of unsteady thermal conductivity in a multilayer rubber-metal product during post-vulcanization cooling and after-vulcanization

The article considers analytical solutions for a non-stationary problem of heat and mass transfer in a multilayer elastomeric material. Determined are the criteria, influence the process of temperature change in the treated material. A functional relationship between the main criteria of heat transfer and the temperature of the processed material was established, which is the basic relation during the development of an engineering method for calculating an industrial installation.