A basis for a theory of nonlocal elastic filtration using the equations of elasticity

The proposed [1–3] nonlocal formulation of the hypothesis that the ground pressure is constant in nonstationary pressure filtration in a deep elastic stratum is derived from the equilibrium equations for the stratum-roof system. The roof is considered to be a flat plate [4] and the floor of the stratum is assumed to be rigid. An equation is established for the scale of the region of influence on the stress and pressure distributions at a point.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 82–86, July–August, 1971.The author wishes to thank V. N. Nikolaevskii for a number of useful observations.     

Unsteady-state flow induced by a well in a deformable stratum interacting with the adjacent rocks

Unsteady-state plane radial flow induced by a well in a thin deformable stratum is studied taking into account the stratum interaction with the adjacent rocks. The stratum permeability is assumed to depend on the lateral deformation. The behavior of the well productivity properties for harmonic reservoir pressure oscillations and after pressure drawdown in a bilayered stratum is analyzed. Steady-state well behavior under these conditions was studied in [1]. A qualitative estimate of the well productivity variation due to a stepwise pressure change in an adjacent stratum was previously derived in [2].Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 85–90, January–February, 1996.     

Flow in a stratum in the case of pumping from an imperfect well near a basin

A study is made of the three-dimensional stationary problem of the flow of ground water to a well of the type of a point sink in a stratum of unbounded thickness in one direction. The stratum is bounded at the top by the bottom of the basin and a stratum of impermeable ground. The problem is investigated in the framework of potential flow theory based on Darcy's law [1, 2], and the solution is obtained in the form of quadratures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 161–164, September–October, 1980.     

Advance of the interface of liquids with pressure filtration

The article is a continuation of the investigations of a number of authors [1–3] on the advance of the interface between different liquids: it gives a solution of the problem in the case of one-dimensional filtration of liquids with different viscosities in distorted layers of variable permeability; it indicates a method for determining the interface between multicolored liquids, and the method is generalized for the case of a series of two-dimensional flows, connected by a conformai transformation. The article discusses problems that reduce to formation of the functions determining the flow, and to calculation of the integrals.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 185–191, September–October, 1971.     

Filtration of a three-phase liquid (Buckley-Leverett model)

Buckley and Leverett [1] formulated the problem of the displacement of immiscible liquids in a porous medium and obtained a very simple one-dimensional solution for a two-phase flow. Different generalizations of it are known [2]. In [3, 4], a method of characteristics is proposed for numerical solution of the problem of three-phase flow. Articles [5, 6] consider the problem of the injection (at a given pressure) of two incompressible liquids into a porous stratum previously saturated with a third, elastic liquid. The authors started from the assumption of the existence, for this problem, of zones of three-, two-, and single-phase flow, separated by unknown mobility gradients. The present work investigates the solution for a three-phase flow, analogous to the Buckley-Leverett solution for two phases. It is shown that the character of the degrees of saturation depends essentially on the initial saturation of the porous stratum and on the phase composition of the mixture being injected.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 39–44, January–February, 1972.     

Unsteady-state filtration of a liquid in a fractured-porous stratum to a well with a hemispherical end-face

A study is made of the motion of a liquid toward a well with a hemispherical end-face with unsteady-state spherical-radial filtration in a fractured-porous stratum consisting of hemispherical regions with different values of the permeability of the system of cracks, superposed one on another. A Laplace transform is used to find exact solutions to the problem of the lowering of the stratum pressure as a function of time and distance as well as of the output of a well working with a constant end-face pressure. The article discusses partial cases corresponding to the exploitation of closed and bounded open fractured-porous strata by a central well with a hemispherical end-face. On the basis of numerical calculations, the effect of the parameters of fractured-porous strata on the change in the indices of the process of their exploitation is established. It is established that, with the exploitation of fractured-porous strata, the process of the lowering of the end-face pressure of the well and its output become stabilized with sufficiently large values of the time.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 115–123, November–December, 1973.     

Elasticoplastic conditions of the filtration of a liquid in porous media

The article discusses questions in the theory of filtration in porous media, taking account of elastic, elasticoplastic, and plastic deformations. Parameters are introduced to evaluate irreversible effects in petroleum- and water-bearing strata, i.e., coefficients of the change in the porosity and the permeability. Equations are derived for filtration under unsteady-state and steady-state working conditions of wells and galleries. Two limiting cases, which allow analytical solutions, are separated out. In the general case, the equations of elasticoplastic filtration conditions are solved on an electronic computer. The numerial calculations show that the predominating effect results from taking account of the irreversible change in the permeability, depending on the change of the pressure in the stratum.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 84–90, September–October, 1973.The author is grateful to V. N. Nikolaevskii for his evaluation of the work.     

Stability of the solutions of inverse mixed boundary-value problems such as those encountered in filtration theory

The stability of the solutions of a group of problems regarding the construction of an underground contour on the basis of the distribution of filtration pressure is considered for bounded and unbounded depths of the water-permeable layer and also for the case of an inclined water-resistant boundary.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 54–63, March–April, 1977.In conclusion, the author wishes to thank L. A. Aksent'ev for constant interest in the work and valuable comments.     

On the connection between the principal components of the stress and permeability tensors of porous media

The dependence of the flow properties of rocks on the conditions under which they are loaded is one of the least studied questions in the general problem of the connection between the physical properties of porous systems and the loads acting on them. The experimental [l] and theoretical [2, 3] studies so far made have dealt primarily with the dependence of the permeability of oil and gas collector rocks on the hydrostatic compression to which they are subjected. In the present paper, a connection is established between the parameters of a nonuniform anisotropic load and the permeability tensor of rock.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 173–177, January–February, 1984.     

Sound absorption near the boundary dividing two liquids

It is well known that sound absorption in finite media is caused mainly by fluid viscosity and thermal conductivity. Kirchhoff [1] developed a general theory describing the mechanism of such absorption and applied it to the particular case of sound propagating in tubes. Rayleigh [2] used Kirchhoff's theory to study sound absorption by a porous wall with normal incidence of the sound wave. Konstantinov [3] also used Kirchhoff's theory to solve the problem of sound absorption by a rigid, isothermal (with infinite thermal conductivity) and a thermally insulating plane wall with arbitrary angle of sound-wave incidence. A natural extension of these efforts is a study of sound absorption on the boundary dividing two liquids. Aside from its scientific interest, such a problem is of practical significance, for example, in hydroacoustics or in creating methods for visualization of sound in gases and liquids [4]. The present study will attempt to solve this problem. The results can be applied to both liquid and solid (resinlike) materials.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 6–9, January–February, 1984.The author thanks T. P. Zhizhina for much assistance in the study.     

Investigation of convective motions of a gas due to propagation of a temperature discontinuity along the boundary of a closed region

The Navier-Stokes equations are used in a numerical study of the two-dimensional motions of a compressible gas in a closed rectangular region in a gravity field. The motion of the gas is due to the propagation of a temperature discontinuity along the lower boundary of the region. The mechanism of formation of eddy structures is followed in detail for different velocities of the discontinuity and different ratios of the sides of the region. The method of stabilization is used to obtain different stationary solutions to the problem of convection in a rectangular region heated below. The realization of a particular stationary solution depends on the history of the process. Problems of the convective motion of liquids and gases in closed regions heated below, including questions relating to the nonuniqueness of stationary solutions, are considered in the monograph [1] and the review [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 87–92, September–October, 1980.We thank V. B. Librovich, L. A. Chudov, and G. M. Makhviladze for guidance and helpful discussions.     

Numerical modeling of thermocapillary convection in a liquid layer with a nonlinear dependence of the surface tension on temperature

The principal characteristics of thermocapillary convection in a rectangular channel with one of the boundaries heated to a temperature higher and the other to a temperature lower than T₀ are investigated numerically on the basis of the Navier-Stokes equations. Certain convection characteristics corresponding to normal and anomalous thermocapillary effects are qualitatively compared. The conditions under which self-similar solutions of the type obtained in [10] can be used to describe the flow in a bounded region are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 138–143, January–February, 1991.     

Transition flow of a fibrous suspension in a pipe as the flow of an anisotropic fluid

The transition flow is considered of a fibrous suspension in a pipe. The flow region consists of two subregions: at the center of the flow a plug formed by interwoven fibers and fluid moves as a rigid body; between the solid wall and the plug is a boundary layer in which the suspension is a mixture of the liquid phase and fibers separated from the plug [1–3]. In the boundary region the suspension is simulated as an anisotropic Ericksen—Leslie fluid [4, 5] which satisfies certain additional conditions. Equations are obtained for the velocity profile and drag coefficient of the pipe, which are both qualitatively and quantitatively in good agreement with the experimental results [6–8]. Within the framework of the model, a mechanism is found for reducing the drag in the flow of a fibrous suspension as compared to the drag of its liquid phase.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 91–98, September–October, 1985.     

Numerical simulation of flow of multicomponent mixtures in porous media

A study is made of the isothermal flow of multicomponent mixtures in a porous medium, accompanied by phase transitions, interphase mass exchange, and change in the physicochemical properties of the phases [1–3], It is assumed that at each point of the flow region, phase equilibrium is established instantaneously and the flow velocities of the separate phases conform to Darcy's law. Approximate solutions of problems of displacing oil by high-pressure gas were obtained in [1]. By generalizing the theory developed in [4], a study is made in [5] of the structure of the exact solutions of the problems of the flow of three-component systems which describe the displacement of oil by different reactants (gases, solvents, micellar solutions). The numerical solutions of the problems of multicomponent system flow are considered in [2, 3, 6, 7]. This paper presents a numerical method which is distinguished from the well-known ones [2, 3, 6, 7] by the following characteristics. The flow equations are approximated by a completely conservative finite-difference scheme of the implicit pressure-explicit saturation type, the calculation being carried out using Newton's method of iteraction with spect to both the pressure and the composition of the mixture. The minimum derivative principle [8] is used in the approximation of the divergence terms of the equations. The phase equilibrium is calculated using the equation of state.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 101–110, July–August, 1985.     

Numerical finding of the equilibrium interface of liquids in an axisymmetric vessel

A study is made of the problem of numerical determination of doubly connected axisymmetric equilibrium shapes of the interface of two immiscible liquids in an arbitrary axisymmetric vessel under conditions of reduced gravity. Problems of this kind are of practical interest in connection with the study of the behavior of fuel in the tanks of spacecraft [1]. It is assumed that the liquids are homogeneous and incompressible and have a temperature constant throughout the complete volume. In the investigation of the equilibrium of the interface of liquids under conditions of complete or partial weightlessness, it is necessary to take into account the forces of surface tension, which play a decisive part in such a situation. The equilibrium of liquids in a vessel with allowance for surface tension is described by a system of nonlinear equations [2]. In the present paper, the problem is solved by the Kantorovich— Newton iterative method [3], which makes it possible to reduce the nonlinear problem to a succession of linear problems solvable by the method of finite differences.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 134–137, September–October, 1981.I thank I. E. Tarapov and I. I. Ievlev for constant interest in the work and helpful comments.     

One method for solving the plane problem of pressure filtration to boreholes in nonhomogeneous strata

The article discusses one of the modifications of the method of S. Bergman [1] with application to problems of the pressure filtration of a liquid in horizontal strata with a permeability depending continuously on the coordinates of the points of the plan of the flow. A complex representation of the pressure of artesian waters is constructed in the form of an integral operator containing one arbitrary analytical function. The article discusses examples of the determination of the pressure drop in a round stratum with point-type boreholes with a linear change of the permeability. The results obtained are compared with previously known results [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 71–77, January–February, 1973.     

Well production indicators in a deformable reservoir interacting with rocks

Steady-state flow towards a well through a thin porous deformable two-layer reservoir with allowance for deformation of the surrounding rocks is investigated. The permeability of the reservoir is considered to be a function of the displacements of its top and bottom. The effect of deformation on the well production indicators is studied. The results obtained agree qualitatively with the data of full-scale experiments. Earlier, in [1–5], in considering the self-consistent processes of flow through porous media and their deformation attention was concentrated on the analysis of the stress-strain state of the rocks and reservoir and on unsteady problems within the framework of the nonlocally elastic flow regime.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 86–93, January–February, 1995.     

Oscillations of an ideal liquid acted upon by subface-tension forces. Case of a doubly connected free surface

Many articles have appeared on the problems of small oscillations of an ideal liquid acted upon by surface-tension forces. Oscillations of a liquid with a single free surface are treated in [1, 2]. Oscillations of an arbitrary number of immiscible liquids bounded by equilibrium surfaces on which only zero volume oscillations are assumed possible are investigated in [3], We consider below the problem of the oscillations of an ideal liquid with two free surfaces on each of which nonzero volume disturbances are kinematically possible. The disturbances satisfy the condition of constant total volume. A method of solution is presented. The problem of axisymmetric oscillations of a liquid sphere in contact with the periphery of a circular opening is considered neglecting gravity. The first two eigenfrequencies and oscillatory modes are found.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 64–71, May–June, 1976.In conclusion, the author thanks F. L. Chernous'ko for posing the problem and for his attention to the work.     

Development of thermogravitation convection in a two-layer system in the presence of a surface-active material on the boundary

Convective instability of equilibrium in a system of two horizontal layers of immiscible liquids, caused by the Rayleigh instability mechanism, has been studied within the framework of the linear theory in [1–5]. The present study will investigate the effect of a surface-active material (SAM), deposited on the boundary between the liquids, on the development of thermogravitation convection. Calculations were performed for two types of systems, which in the absence of a SAM show instability of a monotonic or an oscillatory character. A new type of oscillatory equilibrium instability was observed, produced by the effect of the SAM. In some region of parameter values the oscillatory instability may prove to be the more dangerous one. The action of the Marangoni effect on thermogravitation oscillations is considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 76–81, September–October, 1986.In conclusion, the authors express their gratitude to E. M. Zhukhovitskii for his helpful evaluation.     

Nonequilibrium effects in the filtration of viscoelastic liquids

An experimental investigation was made of nonequilibrium effects with the filtration of viscoelastic petroleums and polymer solutions. A scheme is proposed to explain the effects obtained. A comparison between theoretical dependences and experimental data confirms the proposed scheme of the phenomenon. The filtration of viscoelastic liquids has recently attracted considerable attention. The interest in it is connected with the discovery of well-expressed elastic properties in several high-viscosity petroleums, as well as with the need to explain the laws governing the movement of solutions and polymer melts in a porous medium, with application to industrial chemical equipment and to the injection of polymer solutions into oil-bearing strata with the aim of increasing the yield of petroleum. Among articles devoted to the filtration of viscoelastic liquids, there should be mentioned the fundamental investigation of Marshall and Metzner [1], me theoretical article of Wissler [2], and me review of Savins [3]. Specifically, it is shown in [1] that, as a result of elastic effects, the filtration resistance in a steady-state flow rises in accordance with the relationship 1+A(V/r)² where V is the filtration rate; r is the mean pore radius; is the relaxation time; A is a constant on the order of magnitude of 10. In [1] this increase in the resistance is explained qualitatively by the action of specific, for elastic liquids, normal stresses in contracting and then expanding pore channels. A corresponding calculation (for a flat channel and a determined nonlinear model of a viscoelastic liquid) was made in [2] using the method of perturbation theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 76–83, September–October, 1973.