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A mathematical model of two-phase capillary-nonequilibrium isothermal flows of incompressible phases in a double porosity medium is constructed. A double porosity medium is considered, which is a composition of two porous media with contrasting capillary properties (absolute permeability, capillary pressure). One of the constituent media has high permeability and is conductive, the second is characterized by low permeability and forms an disconnected system of matrix blocks. A feature of the model is to take into account the influence of capillary nonequilibrium on mass transfer between subsystems of double porosity, while the nonequilibrium properties of two-phase flow in the constituent media are described in a linear approximation within the Hassanizadeh model. Homogenization by the method of formal asymptotic expansions leads to a system of partial differential equations, the coefficients of which depend on internal variables determined from the solution of cell problems. Numerical solution of cell problems for a system of partial differential equations is computationally expensive. Therefore, a thermodynamically consistent kinetic equation is formulated for the internal parameter characterizing the phase distribution between the subsystems of double porosity. Dynamic relative phase permeability and capillary pressure in the processes of drainage and impregnation are constructed. It is shown that the capillary nonequilibrium of flows in the constituent subsystems has a strong influence on them. Thus, the analysis and modeling of this factor is important in transfer problems in systems with double porosity.

In this paper, a mathematical model of cellular tissue dynamics is considered. The first part gives the conclusion of the model, the main provisions and the formulation of the problem. In the second part, the final system is investigated numerically and the simulation results are presented. It is postulated that cellular tissue is a three-phase medium that consists of a solid skeleton (which is an extracellular matrix), cells and extracellular fluid. In addition, the presence of nutrients in the tissue is taken into account. The model is based on the equations of conservation of mass, taking into account mass exchange, the equations of conservation of momentum for each phase, as well as the diffusion equation for nutrients. The equation describing the cellular phase also takes into account the term describing the chemical effect on the tissue, which is called chemotaxis — the movement of cells caused by a gradient in the concentration of chemicals. The initial system of equations is reduced to a system of three equations for finding porosity, cell saturation and nutrient concentration. These equations are supplemented by initial and boundary conditions. In the one-dimensional case, the distribution of porosity, concentration of the cell phase and nutrients is set at the initial moment of time. A constant concentration of nutrients is set on the left border, which corresponds, for example, to the supply of oxygen from the vessel, as well as the flow of cell concentration on it is zero. Two types of conditions are considered at the right boundary: the first is the condition of impermeability of the right boundary, the second is the condition of constant concentration of the cell phase and zero flow of nutrient concentration. In both cases, the conditions for the matrix and extracellular fluid are the same, it is assumed that there is a source of nutrients (blood vessel) on the left border of the modeling area. As a result of modeling, it was revealed that chemotaxis has a significant effect on tissue growth. In the absence of chemotaxis, the compaction zone extends to the entire modeling area, but with an increase in the effect of chemotaxis on the tissue, a degradation area is formed in which the concentration of cells becomes lower than the initial one.

The paper presents an optimization approach to microstructure simulation. Porosity function was optimized by numerical method, grain-size model was optimized by complex method based on criteria of model quality. Methods have been validated on examples. Presented new regression model of model quality. Actual application of proposed method is 3D reconstruction of core sample microstructure. Presented results suggest to prolongation of investigations.