Equilibrium of a crack in a porous medium with injection of the filtering liquid

In connection with the exploitation of petroleum deposits, the article discusses the equilibrium of a porous medium with a crack under conditions of plane deformation, with the steady-state filtration of a liquid injected into the porous medium through a crack. It is assumed that the crack, which has initial zero dimensions, can become wider and longer with a rise in the pressure. The displacement of the sides of the crack is determined on the basis of the theory of elasticity, taking account of the deformation properties of a saturated porous medium. The stress and the displacement are expressed in terms of two analytical Muskhelishvili functions and the complex filtration potential. A change in the volume of the porous medium leads to a discontinuity of the displacements at the feed contour, and to distortion in the filtration region. For a circular stratum, the dimensions of the crack and the mass flow rate of the liquid are determined in the first approximation. The region of values of the pressure in which there exists a stable equilibrium state of the open crack and a steady-state flow of the liquid is found.     

Filtration of a liquid with free boundaries in unbounded regions

The variational method is used to solve problems of filtration of a liquid in unbounded regions (inflow of a liquid to a drain, filtration of a liquid through a plain earth dam on a permeable base, etc.). Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 188–195, September–October, 2000.     

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.     

Motion of a viscoplastic liquid in a porous inhomogeneous medium

Equations of motion are derived for a viscoplastic liquid in a nonuniform medium of type 2 (piecewise uniform) or type 3 (with a variable filtration coefficient) [1] on the assumption that the motion is of steady-state type. Solutions are presented for a parallel flow and a flow with axial symmetry.     

Centrally symmetrical filtration of liquid in cracked porous media with a hemispherical supply contour

The motion of a homogeneous liquid in a well with a hemispherical face is studied for the case of transient, spherically radial filtration in cracked porous media comprising mutually superposed hemispherical regions with different crack permeabilities, having a supply contour in the outer hemispherical region. Using a Laplace integral transformation with respect to the time variable, the systems of differential equations describing the filtration of liquid in these media are solved for zero initial and corresponding boundary conditions. Exact solutions are obtained for the reduction in stratal pressure with time and distance, and also for the changes taking place in the output of a well operating under conditions of specified face pressure. On the basis of corresponding numerical calculations, the influence of the parameters of the cracked porous strata and the radius of the surface containing the supply contour on the indices of the production process is established.     

Numerical solution of the problem of two-phase influx to a stratum sampler on a cable

An investigation is made of the process of the two-phase influx to a stratum sampler on a cable in a gas-water stratum by solution of the initial boundary-value problem of the filtration of a gasified liquid [1].     

Plane steady-state filtration in a stratum divided by a rectilinear crack

This article discusses the problem of the complex potential for the plane steady-state filtration of a homogeneous liquid in a homogeneous infinite stratum, divided into two half-planes by a rectilinear crack or a difficultly permeable curtain. The corresponding boundary-value problem of the theory of functions is formulated and solved. The example used is the filtration field formed by a source. A general principle is formulated by the use of which, in conjunction with the solution of the problem of a crack, a solution can be obtained in the case where the crack is replaced by a difficultly permeable curtain.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 4, pp. 173–177, July–August, 1973.     

Boiling of a liquid in a porous medium under a pressure drawdown influence

The problem of the boiling of a liquid which initially completely saturates a porous medium under a pressure drawdown influence is considered in a planar one-dimensional formulation and a radial self-similar formulation. The influence of the initial state of the medium and pressure drawdown on the filtration boiling process is analyzed. Boiling regimes are identified, and a criterion for distinguishing between these regimes is determined.     

Exponential filtration of a liquid in the presence of sources

A study is made of the plane exponential filtration of an incompressible liquid under the action of two sources (sinks). The solution is based on an S. A. Chaplygin transformation, the possibility of whose use in the investigation of nonlinear filtration was first noted in [1]. In [2–5] this transformation was used in a consideration of filtration with a limiting gradient. In the present article, another nonlinear law of resistance, an exponential law, is used to construct an exact solution. The use of S. A. Chaplygin variables makes it possible to transform the starting system of equations to a Helmholtz equation, which then reduces to a functional relation which is solvable by the Wiener-Hopf method. The results obtained point to the possibility of using the proposed method to solve other problems of plane exponential filtration, generated by sources or sinks, particularly when they are arranged symmetrically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 91–96, September–October, 1973.     

Method of constructing estimates of the solution of equations of filtration of an aerated liquid

The exact solutions of the nonlinear equations of filtration of an aerated liquid have been obtained in [1–3]. In [4] the system of equations of an aerated liquid have been reduced to the heat-conduction equation under certain assumptions. An approximate method of computing the nonsteady flow of an aerated liquid is given in [5], where the real flow pattern is replaced by a computational scheme of successive change of stationary states. In [6] the same problem is solved by the method of averaging. In the present article estimates of the solution of the equations for nonstationary filtration of an aerated liquid in one-dimensional layer are constructed under certain conditions imposed on the desired functions. These estimates can be used as approximate solutions with known error or for the verification of the accuracy of different approximate methods. We note that the use of comparison theorem for the estimate of solutions of equations of nonlinear filtration is discussed in [7–9]. The methods of constructing estimates of solutions of various problems of heat conduction are given in [10, 11]     

Liquid Filtration in an Unbounded Water‐Bearing Stratum with an Inclined Confining Bed

PolubarinovaKochina, Numerov, and other authors paid much attention to filtration problems of a heavy incompressible liquid in inclined waterbearing strata. In this work, therefore, classical schemes of liquid filtration on inclined confining beds are considered along with the general problem of filtration for arbitrary polygonal impermeable walls of a waterbearing stratum. In doing so, we also consider direct problems of physical and geometrical parameters of filtration flows.     

Steady-state filtration in a dome-shaped nonhomogeneous stratum

The article gives a solution to the problem of the three-dimensional filtration of a homogeneous incompressible liquid in a thick curved stratum. It discusses different variants of the change in the permeability and the porosity of the medium in a vertical section. In known work [1, 2], studies have been made of the filtration flows in homogeneous thin (i.e., with a vanishingly small thickness) curved strata.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 168–172, March–April, 1973.     

Fluidization of a bed of solid particles

As is known, fluidization of a bed of solid particles by liquid or gas filtration takes place for certain critical values of the parameters of the filtration regime. The determination of these critical values and the nature of the transition is of interest in connection with the development of fluidization technology in many branches of industry, and also in connection with certain other questions, among which we note the problem of the suspension of a sand plug in an oil well.The two-dimensional fluidization problem has been examined previously [1] as the problem of the limiting equilibrium of a medium which cannot withstand arbitrarily small tensile stresses. This model describes well the behavior of many types of bulk media encountered in practice. However, many cases lie beyond the limits of this model because of the presence of bonding forces between the particles. Bonding may be due to the adhesive forces which arise during the fluidization of fine powders [2, 3], and/or to magnetic and electrostatic forces [3, 4]. Another example is the agglomeration of particles during gas fluidization when small amounts of liquid are injected [5]; still another is the case in which sand particles are surrounded by thin films of oil when a sand plug is suspended in an oil well.In the present paper an extension of the results obtained in [1] is used to examine fluidization of a bed with account taken of the bonding forces between the particles. The two- and three-dimensional problems are studied.     

Quasi-one-dimensional model of transcapillary filtration

The main physiological function of the capillary blood vessels consists in sustaining the exchange processes between blood and tissues. The transport of water and substances dissolved in it occurs by filtration through the capillary walls. The filtered liquid flows through the tissues (cells and other structures) in the zone of influence of the capillary (“tissue cylinder”) and later almost all of it returns into the capillary; only a small part of the liquid drains into the lymphatic capillaries. Insufficiency of such drainage leads to edema, i.e., filling of water into the tissues. The walls of the capillaries are almost opaque for large albuminous molecules. Therefore the motion of the liquid through the walls is determined not only by the pressure drop, but also by osmosis effects. In the tissue the motion of the liquid is similar to filtration through a porous medium and in some cases it is accompanied by reverse attachment of water, its absorption by tissues, and other phase reversal-type phenomena. The theoretical investigation of transcapillary transport of liquid is of interest for solving a number of physiological problems, in particular, for understanding the mechanisms of regulating exchanges in the tissues. Below, a quasi-one-dimensional model is proposed which describes the hydrodynamics of transcapillary filtration. The basic equations are formulated in Sec. 1 and the subsequent sections contain a discussion of the model including its comparison with the results of other authors (Sec. 3).     

Well interaction in strata with dual porosity

The basic equations describing the process of the filtration of a homogeneous liquid in media with dual porosity were obtained in [1]. Analogous equations are used in the study of filtration of a homogeneous liquid in strata separated by a low-permeability membrane [2]. In the present study we solve the problem of the interference of wells in such media for an areal system of well locations. Particular attention is devoted to the study of the solution for periodic boundary conditions, which are characteristic for the cyclic methods of oil extraction.The author wishes to thank V. L. Danilov and A. A. Bokserman for posing the problem and for suggestions during discussions of the results.     

Steady filtration through foam

The process of filtration of a liquid through a deformable porous medium with the structure of a gas-liquid foam is considered. A basis is provided for a model of fluid flow through foam. Expressions are obtained for the differential law of steady filtration and the integral laws of filtration through a vertical column of foam under the influence of body and surface forces.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 98–103, May–June, 1986.     

Steady-state filtration of gas-condensate mixtures

In the present article, in an analysis of the results of an investigation of gas-condensate wells by the method of steady-state samples, use is made of known solutions to the problem of the steady-state filtration of a stratified liquid (gas) with constant properties [1, 2]. However, the filtration of a gas-condensate mixture to a well with the reduction of the pressure below the pressure of the start of condensation p₊ is accompanied by considerable changes in the composition and properties of the gas and liquid phases which, in turn, have an effect on the filtration characteristics of a porous medium. Among the many communications devoted to this problem (see, for example, the reviews [3, 4]), there are investigations which take account of the effect of a change in the composition and properties of the phases on the characteristics of steady-state filtration [5–7]. Here, it is proposed to model real gas-condensate mixtures by pseudobinary or pseudoternary systems with experimentally determined phase ratios. However, there are still no sufficiently well-justified examples of the reduction of multicomponent mixtures to the above-mentioned model systems. The present article, within the framework of the theory of the filtration of multicomponent systems, discusses the problem of the steady-state filtration of a real gas-condensate mixture to a central well in a round stratum.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 69–75, September–October, 1973.The authors thank V. N. Nikolaevskii for his invaluable evaluation of the work, and R. G. Zotov for his aid in making the calculations.     

Filtration convection in a high-frequency vibration field

The effect of high-frequency translational vibrations on the occurrence of filtration convection in a plane horizontal layer of a viscous incompressible liquid saturating a porous medium is studied. Constant temperature is maintained at the boundaries of the layer. It is established that for any vibration direction different from the vertical (transverse) direction, convection in gravity and thermal gravitational convection under both heating from above and heating from below can arise. In the case of reduced gravity, values of the vibration parameter that lead to transition to zero gravity are established. The results are obtained from an analysis of the averaged equations of filtration convection, derived for an arbitrary region. This work was presented at the joint X European and VI Russian Symposium on Physical Sciences in Microgravity (St. Petersburg, June 15–20, 1997). Rostov State university, Rostov-on-Don 344090. Rostov State Academy of Building, Rostov-on-Don 344022. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 22–29, May–June, 1999.     

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.     

Stabilization of solutions of the nonlinear equation of filtration of a two-phase liquid

The asymptotic behavior (with unlimited increase in time) of solutions of boundary-value problems for the filtration equation for a two-phase liquid that describe the displacement of immiscible incompressible liquids from a bed is studied. Convergence of these solutions to the unique solution of the steady problem (stabilization) is established, and, under additional assumptions, the rate of convergence is evaluated. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 30–36, May–JJune, 1999.