Calculation of limiting configuration of stagnant zones when viscoplastic oil is displaced by water

A study is made of the problem of determining the position of the limiting equilibrium portions of unrecovered viscoplastic oil displaced by water from a porous stratum in a many-well system. This problem was formulated by Bernadiner and Entov [1] and is of interest in connection with the obtaining of estimates of the volume of displaced oil. For two-dimensional isothermal flow in a homogeneous undeformed stratum and certain restrictions on the geometry of the flow region, the problem can be investigated by the methods of the theory of analytic functions [1–3]. An approximate solution of one problem with complicated flow geometry has been obtained [4] by means of potential theory. In the present paper the methods of the theory of jets are used to construct and analyze an exact analytic solution to the problem for three possible flow schemes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 77–81, March–April, 1991,We thank M. M. Alimov for discussing the work.     

The spherical problem of the filtration of a liquid and a gas with a change in the permeability of rocks in accordance with an exponential law

The article considers the problem of the filtration of liquids (or gases), pumped through a borehole at a constant rate with elastic filtration conditions. The permeability of the stratum is assumed to be an exponential function of the coordinates. The viscosities of the injected and displaced liquids are assumed to be different. To increase the capacity of strata, i.e., of collectors used for the burial of industrial waste flows and gases, various methods are employed to increase the fracturing and the permeability of the rocks (hydro-pulse techniques, explosions, and other methods). As a result of this, a spherical region is formed in the rocks, in which the permeability varies along the radius. The character of this change is well described by an exponential function. The pumping of waste flows or industrial gases into such a cavity leads to the displacement of the stratum liquid (or gas). The problem of the displacement of one liquid by another liquid not miscible with it under rigid filtration conditions was first discussed in [1–5]. Here a study was made of a region of finite dimensions, bounded by two boundaries, with given pressures or mass flow rates (the linear and axisymmetric flow problems). The permeability of the stratum was assumed to be independent of the coordinates. A special characteristic of these problems is the fact that it is impossible to consider unbounded or semi-bounded filtration conditions in them since, under rigid filtration conditions, the condition of bounded character of the pressure (the head) is not satisfied at infinity. Elastic filtration conditions for two immiscible liquids were first discussed in [6], and later in [7, 8] and other reports. Here an investigation was made of the linear and axisymmetric problems for an unbounded region. In [9, 10] solutions are given to some problems with spherical symmetry for an unbounded region, with rigid filtration conditions and a jumpwise change of the permeability along the radius. In the problems of [6–10] the condition of the bounded character of the pressure is satisfied. In [11] the case of a hyperbolic change in the permeability of the rocks is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 42–51, November–December, 1974.     

Solution of a problem of flow in a porous medium with limiting gradient

For the law of flow in a porous medium with limiting gradient studied previously in [1], an exact solution is found for the problem formulated in [2] of the plane steady motion of an incompressible fluid in a channel with a rectangular step. Particular cases of the solution obtained are given; these represent the solutions of the problem of flow past a broken wall and of motion from a point source in a strip.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 76–78, January–February, 1985.     

Self-similar solution to a problem of axisymmetric motion of gas through a porous medium in the case of a quadratic law of resistance

The results of solution of the self-similar problem of planar flow of gas through a porous medium in the case of a quadratic law of resistance [1] are generalized to the case of axisymmetric motion. The equation in similarity variables for the velocity of isothermal gas flow is reduced to an equation having cylindrical functions as solution. Analytic dependences of the pressure and the gas velocity on the coordinate and time are obtained for a given flow rate of the gas at the coordinate origin and for zero Initial gas pressure in the porous medium.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 4, pp. 168–171, July–August, 1982.     

Displacement of oil by solutions of an active additive from a uniform stratum with allowance for gravity

Self-similar solutions describing the displacement of oil by solutions of an adsorbed active additive have been obtained and investigated [1–3] in the framework of a one-dimensional flow model with neglect of diffusion, capillary, and gravity effects. In the present paper, a self-similar solution is constructed for the problem of oil displacement by an aqueous solution of an active additive from a thin horizontal stratum with allowance for gravity under the assumption that there is instantaneous vertical separation of the phases. This makes it possible to estimate the effectiveness of flooding a stratum by solutions of surfactants and polymers in the cases when gravitational segregation of the phases cannot be ignored.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 87–92, January–February, 1984.     

One-dimensional displacement of petroleum by aqueous solutions of sorption agents taking account of interlayer return flows

Experimental and theoretical investigations of the process of the displacement of petroleum by aqueous solutions of surface-active substances (surfactants), polymers, micelle solutions, and their combinations show that sorption phenomena can exert a considerable effect on the process of flooding [1, 2]. In some cases, they can lead to an appreciable lag of the front of the dissolved substance behind the carrier liquid and to the appearance of a second jump in the saturation of the displacing liquid [1, 3]. In [4] a mathematical model is proposed for the displacement of petroleum from a laminar stratum with isolated intercalculations of aqueous solutions of surfactants and polymers, taking account of sorption phenomena. In the present article, a study is made of the displacement of petroleum from a stratum by aqueous solutions of sorption agents, taking account of interlayer return flows of liquids.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 38–45, March–April, 1977.     

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.     

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.     

Drag of porous cylinders in a viscous fluid at low reynolds numbers

The problem of flow past a permeable cylinder at low Reynolds numbers is of interest for the solution of a number of problems in chemical technology in, for example, the design of porous electrodes and porous catalysts and in the calculation of nonstationary filtration of aerosols by fibrous filters. In the present paper, we solve the problem of transverse flow of a viscous fluid past a continuous cylinder in a porous shell and, in particular, in the case of a porous cylinder under conditions of constrained flow (system of cylinders) and an isolated cylinder at arbitrary permeability. The analogous problem of Stokes flow past permeable spheres has been solved in a number of papers [1–3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 122–124, November–December, 1979.     

Flow processes with change of regime in fractured reservoirs

Experimental and industrial observations indicate a strong nonlinear dependence of the parameters of the flow processes in a fractured reservoir on its state of stress. Two problems with change of boundary condition at the well — pressure recovery and transition from constant flow to fixed bottom pressure — are analyzed for such a reservoir. The latter problem may be formulated, for example, so as not to permit closure of the fractures in the bottom zone. For comparison, the cases of linear [1] and nonlinear [2] fractured porous media and a fractured medium [3] are considered, and solutions are obtained in a unified manner using the integral method described in [1]. Nonlinear elastic flow regimes were previously considered in [3–6], where the pressure recovery process was investigated in the linearized formulation. Problems involving a change of well operating regime were examined for a porous reservoir in [7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 67–73, May–June, 1991.     

Percolation model of two-phase equilibrium flow in a porous medium with microheterogeneous wettability

The percolation model of two-phase flow described in [1, 2] is used as a basis for examining the problem of the behavior of the characteristics of two-phase equilibrium flow in a porous medium when the capillaries have a radius distribution and differ with respect to the wettability properties of their surfaces. Analytic expressions describing the dependence of the relative phase permeability coefficients on the saturation of the medium by the displacing phase and the microinhomogeneous wettability parameters are obtained. A qualitative comparison shows the theoretical results to be consistent with the data of a direct numerical computer calculation of a grid model [3]. The effect of the microinhomogeneity parameters and the form of the capillary radius distribution function on the phase permeabilities is analyzed within the framework of the approach developed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 86–93, September–October, 1989.     

Shear flow over a porous particle

Linear axisymmetric Stokes flow over a porous spherical particle is investigated. An exact analytic solution for the fluid velocity components and the pressure inside and outside the porous particle is obtained. The solution is generalized to include the cases of arbitrary three-dimensional linear shear flow as well as translational-shear Stokes flow. As the permeability of the particle tends to zero, the solutions obtained go over into the corresponding solutions for an impermeable particle. The problem of translational Stokes flow around a spherical drop (in the limit a gas bubble or an impermeable sphere) was considered, for example, in [1,2]. A solution of the problem of translational Stokes flow over a porous spherical particle was given in [3]. Linear shear-strain Stokes flow over a spherical drop was investigated in [2].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 113–120, May–June, 1995.     

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.     

The zonal linearization method for multidimensional mass transfer problems in porous media

A number of methods have been proposed in recent years for calculating the combined flows of immiscible and miscible liquids in strata to systems of boreholes. We propose a method which can naturally be called the zonal linearization method [1]. It is more compact than the usual finite-difference method and has high accuracy, in particular, in the neighborhood of a borehole, since it is closely similar to the method of characteristics. The method can be applied to both continuous and discontinuous flows and in principle makes it possible to investigate the formation and breakdown of discontinuities. As distinct from the method of characteristics, it is well suited to programming and implementation on a computer, and it also makes it possible to obtain an approximate analytic solution of the problem in many cases and to estimate the accuracy of the solution. The method is based on the zonal linearization of the equation for mass conservation in the total flow between chosen surfaces or contour lines (lines of equal saturation or concentration). Determination of the dynamics of the contour surfaces leads to a Cauchy problem for a system of integrodifferential equations involving partial derivatives. The zonal linearization method is a development of the scheme described in [2–4], and the method of solving the Cauchy problem is a generalization of the methods described in [4–13]. The essence of the method and its convergence are illustrated by means of two-dimensional problems in two-phase filtration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 66–80, July–August, 1973.     

Determination of relative permeabilities in a three-phase flow in a porous medium

A study is made of the problem of determining the parameters of flow described by the Buckley-Leverett system of equations by using functions that admit direct measurement. The well-known solution to the analogous problem for two-phase flow [1–3] is generalized. In contrast to [4], the general case is considered, when the fractions of the phases in the flow and the phase permeabilities depend on two variables.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 187–189, September–October, 1984.The author wishes to thank B. V. Shalimov for his helpful advice.     

Flow of a viscous fluid in an annular gap with intensive blowing from the walls

Self-similar solutions are obtained in [1, 2] to the Navier-Stokes equations in gaps with completely porous boundaries and with Reynolds number tending to infinity. Approximate asymptotic solutions are also known for the Navier-Stokes equations for plane and annular gaps in the neighborhood of the line of spreading of the flow [3, 4]. A number of authors [5–8] have discovered and studied the effect of increase in the stability of a laminar flow regime in channels of the type considered and a significant increase in the Reynolds number of the transition from the laminar regime to the turbulent in comparison with the flow in a pipe with impermeable walls. In the present study a numerical solution is given to the system of Navier-Stokes equations for plane and annular gaps with a single porous boundary in the neighborhood of the line of spreading of the flow on a section in which the values of the local Reynolds number definitely do not exceed the critical values [5–8]. Generalized dependences are obtained for the coefficients of friction and heat transfer on the impermeable boundary. A comparison is made between the solutions so obtained and the exact solutions to the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–24, January–February, 1987.     

Motion of a free axisymmetric viscous liquid film

Axisymmetric free-film flows are encountered in connection with the atomization of liquids and the collision of jets [1, 2]. In [3] steady motion with transverse symmetry is examined and its inviscid instability is studied. Here, steady flow with an arbitrary velocity profile is investigated numerically by the collocation method. The study of the stability of the steady flow under the assumption of local plane-parallelism leads to the formulation of a sixth-order eigenvalue problem which is solved numerically. The existence of unstable disturbances of two types is demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 23–29, July–August, 1990.     

Electrization of dielectric liquids flowing in tubes

The methods of the mechanics of continuous media [1] are used to consider the problem of electrization of dielectric liquids flowing in tubes [2–6]. According to modern ideas [2–6], there is always dissolved in such liquids a slight admixture of an electrolyte, whose molecules in such a dilute solution dissociate to a certain extent into positively and negatively charged ions. On the walls, oxidizing and reducing reactions take place, as a result of which the negative and positive ions, respectively, give up to the wall surplus electrons or take missing electrons from it. Thus, a positive (respectively, negative) total electric charge is induced in the liquid by the flow. We consider in this paper the electrization of a dielectric liquid in laminar flow in a circular cylindrical tube. We find the distribution of the electric charge in the liquid, the maximal electric current, and the dependence of the length over which the distribution of the electric charge in the tube is established on the tube radius, the Debye radius of the liquid, and the Péclet diffusion number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 41–47, November–December, 1979.We thank V. V. Gogosov for helpful comments made in a discussion of thwe work.     

Self-similar problems of propagation of an extended hydrofracture in a permeable medium

The propagation of an extended hydrofracture in a permeable elastic medium under the influence of an injected viscous fluid is considered within the framework of the model proposed in [1, 2]. It is assumed that the motion of the fluid in the fracture is turbulent. The flow of the fluid in the porous medium is described by the filtration equation. In the quasisteady approximation and for locally one-dimensional leakage [3] new self-similarity solutions of the problem of the hydraulic fracture of a permeable reservoir with an exponential self-similar variable are obtained for plane and axial symmetry. The solution of this two-dimensional evolution problem is reduced to the integration of a one-dimensional integral equation. The asymptotic behavior of the solution near the well and the tip of the fracture is analyzed. The difficulties of using the quasisteady approximation for solving problems of the hydraulic fracture of permeable reservoirs are discussed. Other similarity solutions of the problem of the propagation of plane hydrofractures in the locally one-dimensional leakage approximation were considered in [3, 4] and for leakage constant along the surface of the fracture in [5–7].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 91–101, March–April, 1992.     

Filtration of a piecewise-homogeneous liquid

The nonsteady filtration of two immiscible liquids, in the absence of pressure, in a homogeneous and isotropic porous medium, is considered. A nonlinear hydrodynamic problem is formulated, and a number of its features are noted. The solution of the corresponding linearized problem is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 62–69, November–December, 1976.