Filtration of two miscible liquids in a fractured porous medium in a gravity field

The article discusses the one-dimensional problem of the displacement of a liquid (petroleum), saturating a fractured porous medium, by another liquid miscible with it in all ratios, in a gravity field. A system of two nonlinear equations in partial derivatives, of the hyperbolic type, is obtained for the concentration of one of the liquids in fractures and blocks. The system is integrated numerically by the method of characteristics. On the basis of the results universal dependences are obtained for the distribution of the concentrations over the stratum and the yield of petroleum.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 166–168, March–April, 1973.The authors axe grateful to V. M. Entov for his evaluation of the work.     

MHD flows of a second grade fluid between two side walls perpendicular to a plate through a porous medium

Exact analytical solutions for magnetohydrodynamic (MHD) flows of an incompressible second grade fluid in a porous medium are developed. The modified Darcy's law for second grade fluid has been used in the flow modelling. The Hall effect is taken into account. The exact solutions for the unsteady flow induced by the time-dependent motion of a plane wall between two side walls perpendicular to the plane has been constructed by means of Fourier sine transforms. The similar solutions for a Newtonian fluid, performing the same motion, appear as limiting cases of the solutions obtained here. The influence of various parameters of interest on the velocity and shear stress at the bottom wall has been shown and discussed through several graphs. A comparison between a Newtonian and a second grade fluids is also made.     

Parametric oscillations of liquid in communicating vessels

It is well known that a periodic change in the equilibrium or flow parameters of an incompressible liquid exerts a material influence on the hydrodynamic stability. As an example we may quote the parametric excitation of surface waves (gravitational-capillary [1], electrohydrodynamic [2], magnetohydrodynamic [3]) and the oscillations of liquid in communicating vessels [4, 5]. The chief object of the foregoing experimental investigations was that of determining the boundaries of the regions of unstable equilibrium with respect to small perturbations. In the present investigation we made an experimental study of the parametric resonance and finite-amplitude parametric oscillations arising in a liquid-filled U-tube subject to alternating vertical overloadings. We shall describe two forms of oscillations in the liquid, and we shall determine the corresponding ranges of unstable equilibrium with respect to small random perturbations (self-excitation) and also to finite-amplitude perturbations. We shall study nonlinear modes of excitation and mutual transitions between the two forms of oscillations. We shall find the ranges of existence of steady-state oscillations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 36–42, March–April, 1976.The authors wish to thank G. I. Petrova and the participants in his seminar for useful discussions, and S. S. Grigoryan for valuable advice.     

Filtration of a three-component mixture in a nonhomogeneous porous medium

We shall consider the problem of injecting a mixture of two incompressible fluids having different viscosities into an infinite nonhomogeneous porous stratum which is initially filled with a third fluid. The filtration rate of each of the phases depends basically on its concentration and viscosity, and therefore in the displacement process in the general case their rates of movement will be different, and as a result of this, zones of three-, two-, and single-phase flow are formed. These zones will be separated from one another by moving interfaces (fronts) at which there are jumps of the corresponding concentration levels.We shall assume for simplicity that in the entire region where there is combined flow of several fluids they are incompressible and insoluble, and outside this region, in the external zone, a homogeneous elastic fluid moves.     

Flow through a porous annulus

Summary The problem of laminar flow through a porous annulus with constant velocity of suction at the walls and with swirl is reduced to the solution of four non-linear differential equations. The significance of each of these equations is discussed. By taking the swirl to be zero series solutions are obtained for (i) small suction or blowing (ii) when the total flow into the channel through the walls is small. Finally the asymptotic behaviour of the flow for large suction or blowing is discussed.     

Three dimensional magnetohydrodynamic flow between two porous disks

This is a study of conducting flow in the gap between two parallel co-axial nonconducting disks of which one is rotating and the other stationary in the presence of a uniform axial magnetic field. The effect of uniform suction or injection on the velocity distribution is investigated and asymptotic solutions are obtained for R≪M ². Expressions for the average normal force and the torque on the disks are obtained. We find that all components of velocity are affected by uniform suction or injection and in particular we note that the effect of suction or injection on the radial component of velocity predominates over the effect of rotation for a given Hartmann number.     

The deformation of an interface between two fluids in a porous medium

Summary The interface between two moving fluids in a porous medium will, in general, deform under the influence of gravity and drag forces. An example of some importance is the formation of so-called gravity tongues in oil reservoirs. This paper deals with the displacement of oil by water in a homogeneous non-horizontal oil stratum. The deformation of such an interface can be deduced by numerical procedures based upon exact methods. The use of these methods is limited, however, owing to the fact that in oil reservoirs the dip is usually smaller than 10 to 20 degrees. In such cases, where the interface is initially horizontal, the computation of the form of the interface as a function of time becomes so enormous, even when a fast electronic computer is used, that an approximative method is more useful. In this paper two approximate solutions are presented. The first one is obtained by using a simplified form of the dynamic interface condition, in which the flow velocity component perpendicular to the dip direction of the reservoir is neglected. This simplification has previously been used by Dietz, who gave a first-order approximation with respect to time. More complicated results are obtained by using the second approximation where, in accordance with the dynamic boundary condition, this velocity component is more or less taken into account. In both methods, the form of the interface as a function of time is expressed in a parametric representation. Moreover, the amount of water that has passed a given cross-section and the flow of water at this section are obtained as a function of time and the parameter used. Results of both methods are compared with each other and with those obtained by an exact method. Both approximations are found to be good in those cases where the dip of the reservoir is not too high, but this is precisely when exact methods are impracticable.Nomenclature d thickness of the idealised reservoir (see fig. 1) - f function of y as given by (2.7) - f, f, f first, second and third derivative of f with respect to y - F(y, ) function of y and as given in the appendix - G dimensionless quantity - G* dimensionless quantity {= G cos /(1–G sin )} - H(y, ) function of y and as given in the appendix - M dimensionless quantity _{2 1}/ _{1 2} - p pressure - q _w the flow of water at a given cross-section - Q _w the total amount of water that has already passed a given cross-section at a certain time - S ₀ oil saturation in the oil region - S _w water saturation in the water region - r integration variable - s the co-ordinate along the interface (positive direction as given in fig. 1) - t time - t _w time at which water breaks through at a given cross-section - u ₁ mean velocity component of fluid 1 in x-direction in the pores of the porous medium (water) - u ₂ mean velocity component of fluid 2 in x-direction in the pores of the porous medium (oil) - U _r the relative deformation velocity of the interface {=(x _i –W ₀ t)/t} _y - the mean fluid velocity vector in the pores of the porous medium - v ₁ mean velocity component of fluid 1 in y-direction in the pores of the porous medium (water) - v ₂ mean velocity component of fluid 2 in y-direction in the pores of the porous medium (oil) - v _n mean velocity component of the fluids normal to the interface (positive direction from fluid 1 to fluid 2) - W ₀ mean velocity of fluid 1 (water) when x –, where the velocity component in y-direction is equal to zero - x co-ordinate, parallel to the boundaries of the reservoir (see fig. 1) - x _e value of x for a given cross-section - x _i , y _i values of the x and y co-ordinates corresponding to the points of the interface - x ₀(y) initial value of the x co-ordinate of the points of the interface (at t=0) - y co-ordinate, perpendicular to the boundaries of the reservoir (see fig. 1) - y _e (t) time-dependent value of the y co-ordinate of the interface if the value of the x co-ordinate is equal to x _e - y _i , x _i values of the y and x co-ordinates corresponding to the points of the interface - z vertical co-ordinate (positive direction as given in fig. 1) - the angle between the horizon and the boundaries of the reservoir (see fig. 1) - the angle between the x axis and the normal to the interface (see fig. 1) - _e the angle if the value of x _i is equal to x _e - ₀(y) initial value of the angle (at t=0) - effective permeability of the porous medium divided by the product of the porosity and fluid saturation - ₁ effective permeability of the porous medium to fluid 1 divided by the product of the porosity and the saturation of fluid 1 - ₂ effective permeability of the porous medium to fluid 2 divided by the product of the porosity and the saturation of fluid 2 - fluid viscosity - ₁ viscosity of fluid 1 (water) - ₂ viscosity of fluid 2 (oil) - fluid density - ₁ density of fluid 1 (water) - ₂ density of fluid 2 (oil) - porosity of the porous medium Formerly with Koninklijke/Shell Exploratie en Produktie Laboratorium, Rijswijk, The Netherlands.     

Linear instability of the electrified free interface between two cylindrical shells of viscoelastic fluids through porous media

In this paper, we have discussed the linear stability analysis of the electrified surface separating two coaxial Oldroyd-B fluid layers confined between two impermeable rigid cylinders in the presence of both interfacial insoluble surfactant and surface charge through porous media. The case of long waves interfacial stability has been studied. The dispersion relation is solved numerically and hence the effects of various parameters are illustrated graphically. Our results reveal that the influence of the physicochemical parameter β is to shrink the instability region of the surface and reduce the growth rate of the unstable normal modes. Such important effects of the surfactant on the shape of interfacial structures are more sensitive to the variation of the β corresponding to non-Newtonian fluids-model compared with the Newtonian fluids model. In the case of long wave limit, it is demonstrated that increasing β, has a dual role in-fluence (de-stabilizing effects) depending on the viscosity of the core fluid. It has a destabilizing effect at the large values of the core fluid viscosity coefficient, while this role is exchanged to a regularly stabilizing influence at small values of such coefficient.     

Drag difference between steady and shocked gas flows passing through a porous body

Experimental and numerical investigations of gas flows through porous materials have been carried out. We have investigated steady and unsteady processes occurring when the gas flow interacts with porous materials. Densities and porosities of the four open-cell-type polyurethane foams which were investigated are kg/m and , with the foams having different structures. Experiments were conducted to determine the steady drag coefficient of the porous material at low Reynolds numbers, evaluated from the pressure drop. The Forchheimer equation was applied to determine the drag. Values of permeability coefficients () in the Forchheimer equation were estimated by comparing computed and experimental results. Results show that the drag coefficient is largely affected by the internal structure of the foam, and has almost no effect on the stress history, while the value of dominates the stress history variation. Differences of 1000 times exist between the steady flow and unsteady shock tube flow values. Received 15 May 1998/ Accepted 15 March 1999     

Numerical investigations of asymmetric flow of a micropolar fluid between two porous disks

Numerical solution is presented for the two- dimensional flow of a micropolar fluid between two porous coaxial disks of different permeability for a range of Reynolds number Re （-300≤ Re 〈 0） and permeability parameter A （1.0≤A ≤2.0）. The main flow is superimposed by the injection at the surfaces of the two disks. Von Karman＇s similarity transformations are used to reduce the governing equations of motion to a set of non-linear coupled ordinary differential equations （ODEs） in dimensionless form. An algorithm based on the finite difference method is employed to solve these ODEs and Richardson＇s extrapolation is used to obtain higher order accuracy. The results indicate that the parameters Re and A have a strong influence on the velocity and microrotation profiles, shear stresses at the disks and the position of the viscous/shear layer. The micropolar material constants cl, c2, c3 have profound effect on microrotation as compared to their effect on streamwise and axial velocity profiles. The results of micropolar fluids are compared with the results for Newtonian fluids.     

Axial dispersion in fluid flow through porous plates in parallel and through a porous tube

Summary Effects of axial diffusion on liquid-liquid displacement in fluid flow through porous plates in parallel and through a porous tube are considered as problems of two zones in unsteady state mass transfer. The solutions of the differential equations of the system in terms of the Laplace transformed variable contain an infinite number of essential singularities in a complicated form. Therefore approximate solutions are obtained by numerical inversion of the Laplace transform. Some of the numerical results are presented and discussed.Nomenclature C ₁ concentration of solute in Zone 1 - C ₂ concentration of solute in Zone 2 - C ₀ initial concentration of solute in Zone 2 - D _e effective diffusivity - D* axial dispersion (mixing) coefficient - K ratio D*/D _e - P _e Péclet number, Xv/D _e, Rv/D _e - P _e * longitudinal Péclet number, Xv/D*, Rv/D* - R inner radius of a porous tube - t time - v average velocity of fluid flow through Zone 1 - W width of a porous plate - Y length of a porous plate (tube) - porosity - ₁ dimensionless concentration of solute in Zone 1, C ₁/C ₀ - ₂ dimensionless concentration of solute in Zone 2, C ₂/C ₀ - Laplace transform of ₁ - Laplace transform of ₂ - ₁ dimensionless distance in porous plate, x/X - ₂ dimensionless distance in a porous tube, r/R - ₁ dimensionless axial distance in porous plate, y/X - ₂ dimensionless axial distance in a porous tube, y/R - ₁ dimensionless time in porous plate, tD _e/X ² - ₂ dimensionless time in a porous tube, tD _e/R ² - Units CGS system