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A mathematical model of unsaturated filtration in swelling soils has been developed under the conditions of constancy of the entire volume of the medium. On the basis of the developed model, a solution to the problem on capillary rise of moisture in swelling soils has been obtained and characteristic features of the process have been established. A series of experiments on capillary rise of moisture in swelling and unswelling soils has been conducted. A comparison of solutions to the capillary-rise problem with experimental data has been made. Good agreement has been shown between the experimental data and the solutions to corresponding boundary-value problems.

The work is dedicated to the experimental and numerical study of a coupled erosion and deposition process in deformable porous media. The mathematical model is based on the theory of flow in deformable porous media with mass-variable skeleton. The experiment involves clogging of samples with clay solution and subsequent washing by injection of water. 3-D numerical simulation of water injection into the layer is performed. It is shown that precipitation of particles which were eroded from the solid phase by fluid flow near a well (suffosion) causes clogging of porous space at some distance from the injecting well where the flow velocity is lower. Parallel AMGCL solver library is used for calculations on a GPU. New block matrix preconditioner is developed to achieve fast solution of the coupled displacements-pressure linear system.

This paper systematically investigates the impact of porous media disorder and its coupling with flow rates, favorable and unfavorable viscosity ratios, as well as surface tensions on the dynamics of interfaces development during two-phase drainage flow. A special attention is paid to establishing relationship between the dynamics of fluid–fluid and fluid–solid interfacial lengths, the pore selectivity and the displacement efficiency using imaging of fluids distribution in porous media. As samples of study, we used artificially generated models of porous media with different disorder parameters and with two-types of pore-channels systems "— hexagonal and square. In our methodology, the disorder defines the range of grain size distribution and is applied to control the pore size range. For two-phase flow simulation, the lattice Boltzmann equations and the color-gradient model are applied. It was established the linear relationships between fluid–fluid and fluid–solid interfacial length and saturation of the invaded fluid. During numerical simulations at different disorders, the lack of disorder effect on the fluid–fluid interface dynamics and negative disorder impact on the fluid–solid interface dynamics was found. When varying the flow parameters, it was identified that the increase in the fluid–fluid interface dynamics is accompanied by a decrease in the fluid–solid interface dynamics. For all displacement mechanisms considered in this paper, except capillary fingering, an inverse relationship between pore selectivity and pore number, involved in displacement, was detected. We found a shift of pore selectivity towards higher values with increasing disorder which negatively impacts on the displacement efficiency. In capillary fingering regime, a strong tendency to minimize fluid–fluid interfacial length with surface tension explains the lack of relationship between pore selectivity and pore number which leads to bad predictable displacement efficiency in this regime.

A mathematical model of unsaturated filtration in swelling soils under conditions of constant volume of the medium as a whole has been developed. On the basis of the developed model, a solution to the problem of capillary rise of moisture in swelling soils was obtained, and the characteristic features of the process were revealed. A series of experiments on the capillary rise of moisture in swelling and non-swelling soils was carried out. The comparison of solutions of the capillary rise problem with experimental data is carried out. Good agreement of experimental data with solutions of the corresponding boundary value problems is shown.

A study is made of the influence of surface tension on two-phase drainage flows in a porous medium. Computational filtration experiments are conducted in a three-dimensional digital microtomographic model of sandstone. To simulate two-phase flows, use is made of lattice Boltzmann equations; interfacial phenomena and wetting effects are described using the color-gradient-based method. Calculations are carried out at the same rate of injection of a nonwetting fluid and at a ratio of the viscosities of nonwetting and wetting phases of 1:10. Results are obtained according to which growth in the interfacial tension contributes to the rise in the efficiency of displacement of the wetting fluid, which is recorded at the moment the injected phase breaks through, and to the decrease in the efficiency after its breakthrough. It is established that distinctive features of flows with capillary fingering that arise at high surface tensions are the stepwise character of travel of the leading front and the growth of displacement channels in directions different from the hydrodynamic pressure difference. It is shown that interfacial tension is a parameter whose variation enables us to change the type of two-phase flow. Increase in the interfacial tension contributes to the transition of the flow with viscous fingering to a flow with capillary fingering. The efficiency of displacement of the wetting fluid in the transition crossover zone is the lowest.

This paper presents the results of numerical simulations of two-phase flows in porous media under capillary forces dominance. For modeling of immiscible two-phase flow, the lattice Boltzmann equations with multi relaxation time operator were applied, and the interface phenomena was described with the color-gradient method. The objective of study is to establish direct links between quantitative characteristics of the flow and invasion events, using high temporal resolution when detecting simulation results. This is one of the few works where Haines jumps (rapid invasion event which occurs at meniscus displacing from narrow pore throat to its wide body) are considered in three-dimensional natural pore space, but the focus is also on the displacement mechanics after jumps. It was revealed the sequence of pore scale events which can be considered as a period of drainage process: rapid invasion event during Haines jump; finish of jump and continuation of uniform invasion in current pore; switching of mobile interfaces and displacement in new region. The detected interface change, along with Haines jump, is another distinctive feature of the capillary forces action. The change of the mobile interfaces is manifested in step-like behavior of the front movement. It was obtained that statistical distributions of pressure drops during Haines jumps obey lognormal law. When investigating the flow rate and surface tension effect on the pressure drop statistics it was revealed that these parameters practically don’t affect on the statistical distribution and influence only on the magnitude of pressure drops and the number of individual Haines jumps.

The authors have developed a mathematical model of multiphase nonisothermal filtration with chemical reactions and phase transitions in deformable porous media. A modified Biot approach was used to describe filtration through a deformable porous medium. Equations for the stress–strain state have been formulated in terms of displacements, and the elastic rheology of a porous skeleton in nonisothermal form was used in the model. The dependence of the permeability on porosity has been written in the form of a Kozeny–Carman equation. A test problem of combustion in a low-permeability sample of a bituminous reservoir has been solved. Results of the work can be used to numerically substantiate experiments on the technology of interbedding combustion in low-permeability beds.

Processes of mass transfer (primarily swelling and hydration/dehydration metamorphic processes) between underground fluid and solid phase in porous media in various kinds of rocks (primarily, sedimentary rocks) is examined. Penetration of water in swelling porous media and coupled process of consolidation and physical-chemical interaction between fluid and solid phase are analyzed. It is shown that government equation in this case is non-linear equation of “diffusion—reaction” type. The equation describes the influence of mass exchange processes (for example swelling and metamorphic dehydration) on distribution of pressure in pores, stresses and deformations in rocks.

The results of numerical simulation of the processes of two-phase flow through a porous medium in three-dimensional digital models of the porous space of three natural sandstone samples are given. The calculations are carried out using the lattice Boltzmann equations and the digital field gradient model over a wide range of the capillary numbers and the viscosity ratios of injected and displaced fluids. The conditions of flow through a porous medium with capillary fingering, viscous fingering and with stable displacement front are revealed.

Secondary micas after smectite, including mix illite–smectite phases, can form in sediments by three mechanisms, each being specific to particular environments. As the process develops, the newly formed phases undergo structure ordering. Two mechanisms involve transformation of 2:1 mixed-layer structures, and the third is the growth of screw dislocations, with formation of ordered mix phases having a Reichweite parameter of R = 1 or R = 2. We propose methods for identifying such phases when they are present in small amounts or when their XRD patterns lack well-pronounced superperiodic reflections, as well as mathematical formalism for illitization modeling. The theoretical issues are illustrated with field examples, and the illitization mechanisms are discussed in terms of their possible practical applications.

Mathematical model of porous media dissolution coupled with two-phase flow is proposed. The model is based on the conception of dissolvable porous medium with deformable mass-variable porous skeleton. Model can be used for simulation of coupled chemo- and hydrogeomechanical processes which are difficult to examine experimentally. Acidizing of calcite oil reservoir is used as an example of the process. Water solution of hydrochloric acid and oil are two fluid phases of the model with several components. Dissolvable porous media is treated as deformable mass-variable solid phase. Change in mass of the solid phase is caused by hydrochloric acid dissolving the calcite part of the solid phase. Dissolution is supposed to be congruent; kinetics is governed by the Nernst law. Software for numerical solution of the model is developed. It uses AmgCL parallel library for high-performance computing in order to deal with large algebraic systems on the each time step of calculations. The library uses algebraic multigrid methods for preconditioning and parallel iterative solvers. NVidia CUDA framework is used as a backend to perform GPGPU calculations, because it proved to be faster than OpenCL framework on this problem. Numerical experiments on the basis of data set from real reservoirs are conducted with the developed software. Good correlation between field and calculated data is achieved. Numerical experiments for different configurations of heterogeneous layer are performed. Acidizing of layers with highly permeable conduit and with random distribution of permeability is modeled.

This article investigates the effect of anisotropy on two-phase flows in porous media (both drainage and imbibition) under various wetting angles and capillary numbers. The focus is on the applied results useful to petroleum engineering. Mathematical modeling based on lattice Boltzmann equations in combination with multi-relaxation time and color-gradient model is used as a research tool. This paper proposes a new method for creating digital models of porous media, based on a combination of a quartet structure generation set and Monte-Carlo algorithms. The advantage is the ability to control both heterogeneity and anisotropy. This paper presents visual phase diagrams of the invasion patterns in anisotropic samples as a function of wetting angle and capillary number. The results show that the ratio between capillary and viscous forces is highest for the sample with favorable anisotropy, which has the biggest permeability, and decreases on going to unfavorable anisotropy (lowest permeability). This effect leads to the finding that boundary capillary numbers, which determine the transition between crossover mode and invasion patterns (viscous and capillary fingering, and compact displacement), are shifted towards lower values as anisotropy increases. The images of fluid distributions are numerically described using fractal dimension and displacement efficiency. It was found that the anisotropic effect is most pronounced at capillary fingering and does not significantly affect the invasion characteristics in the imbibition mode. The sensitivity of the fractal dimension and displacement efficiency to a change in the wetting angle is maximal at favorable anisotropy and decreases on going to unfavorable anisotropy. During drainage, the fractal dimensions, estimated for favorable and unfavorable anisotropy, show opposite trends depending on the capillary number. In the imbibition mode, there is an increase in the fractal dimension with an increase in the capillary number for all porous structures. In addition, this paper investigates the anisotropy effect on the «injected fluid - skeleton » interfacial length under different wetting and dynamic conditions. It was shown that the anisotropy has no significant effect on this parameter, and the best interaction efficiency was found with a compact displacement, and the least with a viscous fingering. Also, the influence of the no-flow and periodic boundary conditions on the invasion characteristics is examined. The results show that boundary conditions do not significantly affect the invasion in the sample with favorable anisotropy for any wetting angles. Periodic boundary conditions negatively influence the displacement efficiency for samples with isotropic and unfavorable structures at drainage, but no noticeable effect was found during imbibition.

This paper investigates the wettability effect on the characteristics of two-phase immiscible flows in porous media at different capillary numbers. This issue presents both applied results useful to engineering disciplines and explores the fundamentals of displacement mechanisms, especially in the mode of compact displacement. The main findings of this study are based on numerical flow simulations in a digital model of heterogeneous porous structure, for which the lattice Boltzmann equations are used in combination with the multi-relaxation time and color-gradient models. This work is one of the few which studies the wettability effect in porous structures with a random arrangement of particles. The influence of the contact angle and the capillary number is presented in the form of images of the fluid distribution at various contact angles and capillary numbers, which were associated with the phase-diagrams for displacement efficiency and fractal dimension, as well as fluid-fluid interfacial length and finger width. It was found that increasing capillary number suppresses the wettability effect on displacement characteristics. This article first identifies four crossover zones between capillary and viscous fingering and compact displacement. It was shown that the critical contact angle, which determines the appearance of compact displacement, increases with the capillary number. Particular attention is paid to the displacement mechanisms for compact displacement. Analysis of the pressure signature at various contact angles and capillary numbers clarifies the role of touches and overlaps that stabilize interfacial front during compact displacement and determines the real driving force for this invasion pattern. This article is the first to establish a relationship between pore-filling events and fluctuations in the pressure signature during compact displacement. It was found that an increase in the pressure signature is characterized by a uniform displacement of all menisci along with the entire fluid-fluid interface, while invasion with a sharp decrease in pressure is determined by the burst-like displacement of a concave meniscus in narrowly localized areas of the pore space.

The authors propose a mathematical model of two-phase filtration that occurs under the conditions of dissolution of a porous medium. The model can be used for joint description of complex chemical-hydrogeomechanical processes that are of frequent occurrence in the oil-and-gas producing and nature conservation practice. As an example, consideration is given to the acidizing of the bottom zone of the injection well of an oil reservoir. Enclosing rocks are represented by carbonates. The phases of the process are an aqueous solution of hydrochloric acid and oil. A software product for computational experiments is developed. For the numerical experiments, use is made of the data on the wells of an actual oil field. Good agreement is obtained between the field data and the calculated data. Numerical experiments with different configurations of the permeability of an oil stratum are conducted.

A study has been made of the problem on the process of formation of a zone of higher-than-average permeability with a moving boundary in an initially low-permeability porous medium (problem on hydraulic fracturing of a bed) in a three-dimensional formulation. A characteristic feature of the three-dimensional problem was the taking into account the existence of two zones (zone with a regular permeability and the destruction zone) in the porous medium, whose contact region was determined using the condition of mass balance on the moving boundary. Special features fundamental to the process of mass transfer in such filtration-inhomogeneous porous media have been revealed and analyzed.

This paper presents a new mathematical model of coupled processes of filtration, deformations of porous skeleton and mass transfer between solid and liquid phases in sedimentary and igneous rocks. A numerical solution is obtained and processes which can be described by the model are analyzed. Examples of such processes are: formation of a fluid layer along metamorphism of sedimentary rocks, formation of a structure and filtration characteristics of the Bazhenov Shale, appearance of intrusive igneous body (lopolith).

Mathematical model and experiment on suffosion/clogging process are presented. The model is based on the theory of flow in deformable porous media with mass-variable skeleton. It is shown that clogging occurs at a distance from injecting borehole. It is caused by precipitation of particles which were eroded by grounwater flow during the suffosion of the solid skeleton near the borehole.

The behavior of unsaturated swelling porous media under load is investigated. An equation for unsaturated filtration in swelling porous media and an equation relating the volume deformation of the medium to the magnitude of applied load have been obtained. The results of calculations by these equations are compared with corresponding experimental data.

The subject of an article is the mathematical modeling of the rainwater runoff along the surface catchment taking account the transport of pollution which permeates into the water flow from a porous media of soil at the certain areas of this surface. The developed mathematical model consists of two types of equations: the equations for calculating of the water layer thickness over the slope surface given the precipitation and evaporation, and equation of the mass transfer of impurity coming into the surface water during its filtration in zone of incomplete saturation of soil. The model also takes into account a reverse process – adsorption of impurity in soil with its low concentration or in the uncontaminated soil. Water content in the zone of incomplete saturation is determined within the approximate approach based on the model of capillary impregnation. The principal features of nonlinear differential equations are theoretically studied. In particular, it is shown that the equation for water content in this zone can become a differential equation with lagging argument. Keywords: Mathematical Modeling, Runoff, Transport of Contaminant, Catchment, Sorption, Desorption, Incomplete Saturation

This paper proposes a method for characterization of the two-phase displacement mechanisms (viscous fingering, capillary fingering, stable front, and crossover) in porous media. The approach is based on separate estimation of the capillary and viscous forces and subsequent calculation of the ratio between them (designated as W). The value W determines the dominance of capillary force and is essentially the inverse capillary number. The advantage of this approach consists in taking into account the scale of the samples and predicting the flow mechanisms using parameters measured in laboratory conditions. The balance between viscous and capillary forces is controlled by surface tension and fluid viscosities. For flow simulation, the lattice Boltzmann equations and the color-gradient model are applied. The values of W equal to 0.5 and 2, which define the boundary of the crossover zone and separate it from the flow mechanisms, were determined and the proposed method was successfully validated on three two-dimensional samples of porous media with different permeability coefficients. It revealed a directly proportional effect of permeability on W. As a result, the surface tensions, which determine the boundaries of viscous fingering, capillary fingering, stable front, are shifted towards smaller values with increasing permeability. Also, the effect of the permeability and displacement mechanisms on the displacement efficiency before, after and at quasi-steady regime was investigated. The tendency to efficiency decrease with increasing surface tension and decrease in viscosity ratio was revealed for quasi-steady regime. The absence of the mobile wetting fluid after breakthrough is detected for crossover and capillary fingers. No-displacement regime after breakthrough and, consequently, effectiveness drop to minimal values in quasi-steady regime occurs at lower surface tensions with increasing permeability.

A difference-spectrum method is proposed for the qualitative assessment of changes of illite–smectite structures accompanying the flooding of oil reservoirs. The method permits one to get an open system and reduce the application of procedures based on Markov’s chain formalism. A computer simulation is made to obtain spectra by subtracting the spectrum of an ethylene glycol-saturated sample from the spectrum of an air-dried preparation throughout the entire range of concentrations of illite and smectite components with a short-range order factor R = 0 or R = 3. It has been established that only in the presence of filtration are the maximum and minimum of the spectra in the range of 12.5-9.4 Å complicated by a number of local extrema, whose position is specified by the structure of intermediate phases. The flooding process first involves mixed-layer phases with R = 0, leading to a partial segregation of the structures into phase with one and two networks of interlayer H2O. When the secondary mica particles break, phases with R = 3 appear along the boundaries of nanoblocks, first only with 1 H2O and then only with 2 H2O in labile interspaces. Their coexistence with the phases R = 0 in the sample proves the existence of percolation effects due to two-phase filtration in the porous medium. The fully flooded reservoir is always dominated by a mechanical mixture of illite–smectite phases of different nature with R = 0 and with different ratios of components. Transformation of mica that can drastically reduce oil production begins long before the appearance of flooding zones, which are revealed by standard logging methods.

A mathematical model of the formation of a clot (thrombosis) has been constructed in which the fibrin of the clot is considered as an elastically deformable solid phase of variable mass and the prothrombin, thrombin, fibrinogen, and fibrin monomers as components of the liquid phase. As a result of numerical solution of the model, data on the influence of various parameters of the process of thrombosis on the rate of clot formation have been obtained.

A mathematical model of the lysis of clots in blood vessels has been constructed on the basis of convectiondiffusion equations. The fibrin clot is considered as an immobile solid phase, and the plasminogen, plasmin, and plasminogen activators — as dissolved liquid phases. As a result of the numerical solution of the model, dependences predicting the development of the lysis process have been obtained. An important role of clot swelling in the process of lysis has been revealed.

The problem of the squeezing out of a solution from aggregates in a porous medium has been considered and investigated. A specific feature of the problem lies in taking account of the existence of two compaction zones in deforming aggregates of a porous medium. Features essential to mass transfer in a deformable aggregated porous medium have been determined and analyzed.

We consider a mathematical model of injection of a pollutant-containing solution into a stratum covered with a weakly penetrable clayey layer (water-confining stratum). The novelty of the statement of the problem lies in taking account of the linear action of pollutants on the buffer (protective) capabilities of the swelling clayey layer covering the stratum.

Consideration is given to a mathematical model of water influx to a perfect well drilling in a homogeneous water-bearing stratum. The novelty of the problem is in the method of allowing for the additional water influx from the overlying swelling clay layer to the stratum. The model of water loss by a swelling clay layer, proposed by the authors earlier, is used for calculation of the water influx. This model is based on the generalization of filtration-consolidation theory to the case where the mass of the solid phase of a porous skeleton changes due to the fluid crossflow during the processes of swelling and shrinkage under the action of osmotic pressure.

It was shown that the fractional dissolution of organic soil components based on elution from a column using an cluent with continuously increasing dissolving power can be used for characterizing the affinity of soil organic matter to mineral soil components. The affinity was estimated from spatial hindrances to the solubilization of macromolecules. The main factors determining the fractional dissolution dynamics of organic matter in the humus profiles of forest-steppe chernozems are the presence of organic residues and texture.

The rheological properties of water-saturated clays have been studied based on the model described in [1, 2]. The problem on shrinkage of a clay layer under strain has been reduced to the well-known problem of N. N. Verigin. The numerical solution of the problem on squeezing of water from a clay layer has been found and analyzed. The distinctive features of the model, which are important for explanation of certain characteristic features of the rheology of clays, have been investigated. It has been shown that the solutions obtained are in qualitative agreement with experimental results.