Vol 10, No 1 (2018)

We consider the propagation of a shock wave in a mixture of a gas and fine solid particles with allowance for the difference in their velocities and the availability of the proper pressure of the phase of particles; here, equations of the Anderson type and others are used. We propose an approximate mathematical model of the flow; in this model, the dependence of the pressure of the first (gaseous) phase from the particles’ volume-concentration can be ignored, but the terms that present the phase volume-concentration multiplied by the pressure gradient of the gas are taken into account. It turns out that with this representation of the equation of state, the mathematical model has the hyperbolic type. For this system of equations of mechanics of heterogeneous media, we carry out the classification of the types of shock waves implemented in the considered mixture. The presented statements about the types are illustrated by numerical computations in stationary and nonstationary formulations; for this purpose, the numerical method of the TVD type is developed.

A technique for local upscaling of absolute permeability is proposed intended for the superelement modeling of petroleum reservoir development. The upscaling is performed for every block of an unstructured superelement grid based on solving a series of stationary one-phase flow in reservoir problems on a refined grid with the initial permeability field under various boundary conditions reflecting the characteristic structural variants of the filtrational flow and taking into account the presence or absence of wells inside the block. The resulting components of the effective permeability tensor in each superelement are sought from the solution of the problem on minimizing the deviations of the normal flows through the faces of the superelement averaged on a refined computational grid from those approximated on a coarse superelement grid. The application of the method is demonstrated by examples of the reservoir of the periodic and nonperiodic structure. The method is compared with the traditional techniques for local upscaling.

We study a three-dimensional coupled problem of the creeping flows of a viscous fluid in a hydraulic (and magmatic) fracture and the strain and flow in the external poroelastic medium induced by them. The process is governed by injection fluid into a well. The flow in the fracture is described by the Stokes hydrodynamic equations in the approximation of the lubricating layer. The external problem is described by the equations of poroelasticity. An ordered sequence of interdependent geomechanical processes occurring under hydraulic (and magma) fracturing is established. In other words, the solution of the coupled problem of hydraulic fracturing is reduced to the sequential solution of three incompletely coupled problems representing the motion in the fracture, as well as the elastic and flow processes in the host rock. The proposed transformation is of practical importance for petroleum geophysics; it allows conducting a more profound in-depth study of the nonisothermal and physicochemical phenomena for occurring in the above-mentioned fractures; and it provides a better insight into the physics of the associated processes.

The work is devoted to modeling the phase transformations of gas hydrate inclusions in porous media. Studying these problems makes it possible to elaborate various technologies for the development of gas hydrate deposits. A two-block mathematical model of the dissociation of gas hydrates in a porous medium based on splitting with respect to physical processes is proposed and studied. An absolutely stable difference scheme is constructed and implemented in the spatially onedimensional case to numerically analyze the model. The water saturation, thawing, and the thermodynamic parameters (pressure and temperature) are calculated based on this difference scheme. An analysis of the data obtained by the calculations has confirmed the possibility of solving a number of typical problems of gas hydrate fluid dynamics using the above approach, including the problem of the complex dynamics of water and hydrate saturation of a formation.

A numerical technique based on the application of the boundary element method is proposed for studying the axially symmetric dynamics of a bubble in a liquid near a solid wall. It is assumed that the liquid is ideally incompressible and its flow is potential. The process of expansion and contraction of a spheroidal bubble is considered, including the toroidal phase of its movement. The velocity and pressure fields in the liquid surrounding the bubble are evaluated along with the shape of the bubble surface and the velocity of its displacement. The numerical convergence of the algorithm with an increase in the number of boundary elements and a refinement of the time step is shown, and comparison with the experimental and numerical results of other authors is performed. The capabilities of the technique are illustrated by solving a problem of collapse of a spheroidal bubble in water. The bublle is located a short distance from the wall.

An algorithm is presented, which enables us to use the iterative Richardson method for solving a system of linear algebraic equations with the matrix corresponding to a sign-definite selfadjoint operator, in the absence of information about the lower boundary of the spectrum of the problem. The algorithm is based on the simultaneous operation of two competing processes, the effectiveness of which is constantly analyzed. The elements of linear algebra concerning the spectral estimates, which are necessary to understand the details of the Richardson method with the Chebyshev set of parameters, are presented. The method is explained on the example of a one-dimensional equation of the elliptic type.