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.     

On nonlinear evolution equations of a three-dimensional capillary-gravity wave packet at the interface of two superposed fluids

Assuming amplitudes as slowly varying functions of space and time and using a perturbation method, two coupled nonlinear partial differential equations are derived that give the nonlinear evolution of the amplitude of a three-dimensional capillary-gravity wave packet at the interface of two superposed incompressible fluid layers of finite depths, including the effect of its interaction with a long gravity wave. Starting from these two coupled equations, a balanced set of modulation equations, both at nonresonance and at resonance, is derived. The balanced set of modulation equations, at nonresonance, reduces to a single nonlinear Schrödinger equation, if it is assumed that space variation of the amplitudes depends only on variation along an arbitrary fixed horizontal direction. Modulational instability conditions, both at resonance and at nonresonance, are also deduced. The advantage of the perturbation method adopted in the present problem, over the reductive perturbation method, is noticed.     

Motion of the interface of two fluids in a porous medium

In this study we obtain the Integro-differential equation for the motion of the interface of two incompressible fluids in various well areal arrangement systems. The solution of the equation is presented for a five-point system in the form of a power series with respect to time. Formulas are assumed which describe the motion of the particles belonging to the interface along invariant streamlines for five-point, seven-point, and nine-point well arrangement systems. The stratum sweeping coefficients for the fluid which is displacing the stratum oil are calculated (under conditions of the five-point system) at the instant when the fluid breaks through into the operation wells. The results of the calculations are compared with experimental data [1].The author wishes to thank V. L. Danilov for valuable counsel and comments.     

Flow of two stably stratified fluids in an open channel with addition of fluids along the channel length

It is shown that two stably stratified fluids flowing in an open channel have two critical flow conditions. The one at higher flowrates is equivalent to the choked flow condition of a single fluid over a broad-crested weir, when the Froude number is unity. The lower critical condition imposes restrictions, which define the system if fluids are added progressively along the channel length and the flowrates increase from low to high values. However, if the flowrate does not become sufficiently large to pass through the lower critical condition, this condition will then define a form of choking, which again determines the system.It is shown that an important special case, with the proportional flowrates of the two fluids kept constant, has an analytical solution in which the relative depths of the fluids is a constant along the channel. Other systems must be solved numerically.     

Instability of a plane interface between two immiscible conducting viscous fluids in a normal electrostatic field

The dispersion relation for motions of a charged plane interface between two viscous incompressible immiscible conducting fluids is analyzed numerically for finite values of all the parameters involved. It is shown that in addition to the well-known aperiodic (Tonkes-Frenkel’ type) instability for certain values of the physical parameters an oscillatory instability with periodically growing amplitude may be realized in the system. Yaroslavl’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 116–123, November–December, 1998.     

Unsteady nonlinear problem of the horizontal motion of a contour under the interface between two liquids

A method of solving the initial boundary-value problem of the horizontal motion of a circular cylinder under the interface between two liquids is developed within the framework of nonlinear theory and implemented numerically. Profiles of generated waves and hydrodynamic loads are calculated for the problem of the acceleration of a circular cylinder under the free surface of a heavy liquid. The phenomenon of wave breaking is considered in detail. Omsk Department, Sobolev Institute of Mathematics, Siberian Division, Russian Academy of Sciences, Omsk 644099. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 37–43, May–June, 1999.     

Modeling of long nonlinear waves on the interface in a horizontal two-layer viscous channel flow

The dynamics of two-dimensional waves of small but finite amplitude are theoretically studied for the case of a two-layer system bounded by a horizontal top and bottom. It is shown that for relatively large steady-state flow velocities and at certain fluid depth ratios the vertical velocity profile is nonlinear. An evolutionary equation governing the fluid interface disturbances and allowing for the long-wave contributions of the layer inertia and surface tension, the weak nonlinearity of the waves, and the unsteady friction on all the boundaries of the system is derived. Steady-state solutions of the cnoidal and solitary wave type for the disturbed flow are determined without regard for dissipation losses. It is found that the magnitude and the direction of the flow can alter not only the lengths of the waves but also their polarity.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 143–158. Original Russian Text Copyright © 2005 by Arkhipov and Khabakhpashev.     

Motion of the interface of two fluids with formation of a region of two-phase flow

A study is made of the problem of the displacement of one fluid by another with the formation of a region of combined flow in the case of an elastic flow regime in the region of the displaced fluid. A self-similar solution is constructed for the flow equations averaged with respect to the vertical coordinate. A numerical algorithm is developed for determining the saturation in the region of the mixture, the pressure, and also the position and shape of the interface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 8, pp. 78–83, October–December, 1981.We thank A. A. Barmin for a helpful discussion of the work.     

Propagation of weakly nonlinear disturbances of the interface between two layers of fluid

The dynamics of internal waves of small but finite amplitude in a two-layer fluid system bounded by rigid horizontal surfaces at bottom and top is investigated theoretically. For linear disturbances of the fluid interface the authors propose a polynomial approximation of the dispersion relation which has the same asymptotics as the exact formula in the limiting situations of very long and short waves. In the case of three-dimensional, weakly nonlinear disturbances of slowly varying shape (in the coordinate system moving with the wave) an equation like the wave equation is derived. This equation has Stokes solutions coinciding with the well-known results for infinitely deep layers. For fairly long disturbances solitary solutions of the model wave equation which fit the experimental data are determined. Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 125–131, January–February, 1994.     

Stability of the wave motion on the charged interface between two immiscible fluids in the presence of a tangential velocity field discontinuity

In the second-order approximation in the dimensionless wave amplitude, the problem of nonlinear periodic capillary-gravity wave motion of the uniformly charged interface between two immiscible ideal incompressible fluids, the lower of which is perfectly electroconductive and the upper, dielectric, moves translationally at a constant velocity parallel to the interface, is solved analytically. It is shown that on the uniformly charged surface of an electroconductive ideal incompressible fluid the positions of internal nonlinear degenerate resonances depend of the medium density ratio but are independent of the upper medium velocity and the surface charge density on the interface. All resonances are realized at densities of the upper medium smaller than the density of the lower medium. In the region of Rayleigh-Taylor instability with respect to density there is no resonant wave interaction.     

Evolution of long nonlinear waves on the interface of a stratified viscous fluid flow in a channel

The dynamics of disturbances of the interface between two layers of incompressible immiscible fluids of different densities in the presence of a steady flow between the horizontal bottom and lid is studied analytically and numerically. A model integrodifferential equation is derived, which takes into account long-wave contributions of inertial layers and surface tension of the fluids, small but finite amplitude of disturbances, and unsteady shear stresses on all boundaries. Numerical solutions of this equation are given for the most typical nonlinear problems of transformation of both plane waves of different lengths and solitary waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 49–61, July–August, 2007.     

Spatial motion of interface of two heavy viscous fluids in a porous medium

The integrodifferential equation for the spatial motion of the interface of two fluids of differing weights and differing viscosities in a porous medium is presented. Results of its numerical solution are given. The general solution is given for the problem of the dynamics of a gently sloping surface.The author wishes to thank V. L. Danilov for his interest in the study and valuable advice.     

Seiches in a channel with a sharp variation in the bottom relief

A mathematical model of seiches is developed for the case of sharp bottom elevation or depression. An effective high-precision numerical and analytical method is applied to determine the natural frequencies and shapes of the lower modes of oscillations. New important hydrodynamic effects of the bottom relief are revealed. The main features of standing waves in a narrow vessel in the presence of sharp bottom elevation are confirmed by laboratory experiments.