A modification of the Quasi-Newton Successive Substitution method is presented, which is used to calculate phase equilibrium with a capillary pressure jump. To check the correctness, a calculation results of the considered method were compared with similar calculations using the Successive Substitution method. A low error of the method of Quasi-Newton Successive Substitution was obtained when calculating phase equilibrium with a capillary pressure jump.

This article is devoted to the study of the effect of the capillary pressure jump (CPJ) on the phase equilibrium between the liquid and gas phases, which are described by the Peng-Robinson equation of state. A numerical analysis the form of phase diagrams (PD) of a gascondensate mixture at various CPJ is carried out. Based on the specifics of the problem, the PD is constructed in the gas pressure - liquid pressure coordinates. The boundary of the two-phase region is defined as the region of existence of the two-phase state of the mixture, without additional studies on the stability of the single-phase state. The analysis is carried out without reference to any specific porous medium, and it is based on the conditions of phase equilibrium at different CPJ only. The obtained results demonstrate the importance of CPJ effects in computing the phase equilibrium of a gas-condensate mixture, when modeling the flow in a porous medium. The described computational and theoretical technique is applicable to two-phase multicomponent systems with an arbitrary number of components and is easily generalized to other equations of state, such as the Redlich-Kwong equation, equations of state of gas condensate systems.

A direct numerical simulation at the pore level of the transport of a multicomponent aqueous solution, which includes salt ions and a surfactant, is carried out. The mathematical model takes into account the formation of a double electric layer (DEL) and the adsorption of surfactants at the liquid–solid interfaces. Flows in real numerical models of porous media are considered. Based on the analysis of the results of calculations at the pore level (microscale), a theory is introduced and an integral computational and theoretical model is constructed to describe the transport and thermodynamic properties on large scales (macroscale); i.e., the problem of thermodynamic and transport upscaling is solved.

The problem of stability of a mixture of two liquids between two conductive plates separated by characteristic distances about several nanometers is considered. In this case, an additional term appears in the expression for the Helmholtz energy, which depends on the distance between the plates and also on the dielectric permittivity of the liquids. The thermodynamic properties of such a system differ from those in the bulk of the system. The influence of the additional term is demonstrated by numerical solutions of two problems: 1) computation of the parameters of the mixture between the plates, when there is a thermodynamic equilibrium with the same mixture in the bulk; 2) computation of two states of the mixture that can coexist in thermodynamic equilibrium between the plates.

The turbulence caused by the Rayleigh–Taylor instability represents a complicated phenomenon. It is usually related to the major hydrodynamic activities, the tangling of the media contact boundary, merging, separation and intermixing of originally smoothed initial structures. An important role in the theory of the Rayleigh–Taylor instability is played by the discontinuity of density on a contact interface between two homogeneous (in terms of density) fluids. A numerical modeling of the intermixing of two fluids with different rheology whose densities differ twice as a result of the Rayleigh–Taylor instability has been carried out. The coefficients of turbulent intermixing in a multimode statement of the problem for the Bingham, dilatant and pseudo-plastic fluids have been obtained.

The Rayleigh–Taylor and Richtmyer–Meshkov instabilities of a visco-plastic fluid are discussed. The Bingham model is used as an effective rheological model which takes into account plastic effects. For the purposes of numerical simulation a one-mode disturbance of the contact surface between two fluids is considered. The main goal of this work is to construct numerical 2D and 3D models and to obtain the relationship between yield stress and the development of instability.