Radial flow of two immiscible fluids: analytical solutions and bifurcations

2003 ◽  
Vol 477 ◽  
Author(s):  
VAKHTANG PUTKARADZE
Keyword(s):  
Analytical Solutions ◽  
Radial Flow ◽  
Immiscible Fluids

An Approximate Method for Transient Radial Flow

1962 ◽  
Vol 2 (03) ◽  
pp. 225-256 ◽  
Author(s):  
G. Rowan ◽  
M.W. Clegg
Keyword(s):  
Infinite Series ◽  
Analytical Solutions ◽  
Radial Flow ◽  
Numerical Solutions ◽  
Basic Theory ◽  
Variable Pressure ◽  
Flow Problems ◽  
Disturbed Zone ◽  
Approximate Analytical Solutions ◽  
Wide Range

Abstract The basic equations for the flow of gases, compressible liquids and incompressible liquids are derived and the full implications of linearising then discussed. Approximate solutions of these equations are obtained by introducing the concept of a disturbed zone around the well, which expands outwards into the reservoir as fluid is produced. Many important and well-established results are deduced in terms of simple functions rather than the infinite series, or numerical solutions normally associated with these problems. The wide range of application of this approach to transient radial flow problems is illustrated with many examples including; gravity drainage of depletion-type reservoirs; multiple well systems; well interference. Introduction A large number of problems concerning the flow of fluids in oil reservoirs have been solved by both analytical and numerical methods but in almost all cases these solutions have some disadvantages - the analytical ones usually involve rather complex functions (infinite series or infinite integrals) which are difficult to handle, and the numerical ones tend to mask the physical principles underlying the problem. It would seem appropriate, therefore, to try to find approximate analytical solutions to these problems without introducing any further appreciable errors, so that the physical nature of the problem is retained and solutions of comparable accuracy are obtained. One class of problems will be considered in this paper, namely, transient radial flow problems, and it will be shown that approximate analytical solutions of the equations governing radial flow can be obtained, and that these solutions yield comparable results to those calculated numerically and those obtained from "exact" solutions. It will also be shown that the restrictions imposed upon the dependent variable (pressure) are just those which have to be assumed in deriving the usual diffusion-type equations. The method was originally suggested by Guseinov, whopostulated a disturbed zone in the reservoir, the radius of which increases with time, andreplaced the time derivatives in the basic differential equation by its mean value in the disturbed zone. In this paper it is proposed to review the basic theory leading to the equations governing the flow of homogeneous fluids in porous media and to consider the full implications of the approximation introduced in linearising them. The Guseinov-type approximation will then be applied to these equations and the solutions for the flow of compressible and incompressible fluids, and gases in bounded and infinite reservoirs obtained. As an example of the application of this type of approximation, solutions to such problems as production from stratified reservoirs, radial permeability discontinuities; multiple-well systems, and well interference will be given. These solutions agree with many other published results, and in some cases they may be extended to more complex problems without the computational difficulties experienced by other authors. THEORY In order to review the basic theory from a fairly general standpoint it is proposed to limit the idealising assumptions to the minimum necessary for analytical convenience. The assumptions to be made are the following:That the flow is irrotational.That the formation is of constant thickness.Darcy's Law is valid.The formation is saturated with a single homogeneous fluid. SPEJ P. 225^

Approximate Analytical Solutions for Marangoni Mixed Convection Boundary Layer

2010 ◽  
Author(s):  
Yan Zhang ◽  
Liancun Zheng ◽  
Jiemin Liu
Keyword(s):  
Boundary Layer ◽  
Mixed Convection ◽  
Analytical Solutions ◽  
External Pressure ◽  
Immiscible Fluids ◽  
Temperature Fields ◽  
Convection Boundary Layer ◽  
Coupling Condition ◽  
Approximate Analytical Solutions ◽  
Convection Boundary

The paper deals with a steady coupled dissipative layer, called Marangoni mixed convection boundary layer, which can be formed along the interface of two immiscible fluids, in surface driven flows. The mixed convection boundary layer is generated besides the Marangoni convection effects induced flow over the surface due to an imposed temperature gradient, there are also buoyancy effects due to gravity and external pressure gradient effects. We shall use a model proposed by Chamkha wherein the Marangoni coupling condition has been included into the boundary conditions at the interface. The similarity equations are first determined, and the approximate analytical solutions are obtained by an efficient transformation, asymptotic expansion and Pade´ approximant technique. The features of the flow and temperature fields as well as the interface velocity and heat transfer at the interface are discussed for some values of the governing parameters. The associated fluid mechanics was analyzed in detail.

Some Approximate Solutions of Radial Flow Problems Associated With Production at Constant Well Pressure

1967 ◽  
Vol 7 (01) ◽  
pp. 31-42 ◽  
Author(s):  
M.W. Clegg
Keyword(s):  
Porous Media ◽  
Infinite Series ◽  
Analytical Solutions ◽  
Laplace Transformation ◽  
Radial Flow ◽  
Approximate Solutions ◽  
Time Interval ◽  
Pressure Buildup ◽  
Flow Problems ◽  
The Times

Abstract The application of the Laplace transformation to problems in the flow of compressible fluids in porous media has provided a large number of exact solutions. For plane radial flow, however, these solutions are either complex integrals or infinite series and are of little value to the field engineer. !n the case of production at constant well pressure, the available approximate solutions are valid for large times only. In this paper it is shown that an approximate inversion formula for the Laplace transform, developed for the solution of viscoelastic problems, is applicable to radial flow problems and provides simple analytical solutions to constant terminal pressure problems. The method may be used to obtain approximate solutions to many problems, including media with radial permeability discontinuities, multi-layer formations and pressure buildup in wells after shut-in. The results are compared with the few available computer solutions as well as the large time solutions, and it is shown that this approximate method greatly extends the time interval over which a simple analytical solution is acceptable. INTRODUCTION The study of transient problems in the flow of fluids through porous media has benefited greatly from the application of transform methods. The use of the Laplace transformation for solving parabolic equations has been widely discussed in the field of heat conduction and diffusion as well as in the petroleum literature. Removal of the time variable with the Laplace transformation generally reduces the problem to a boundary value problem which may be solved by standard techniques. A much more formidable problem then faces the engineer, however, for frequently the transform does not possess a simple inverse. The result is that the general inversion integral must be used and this leads to either an infinite integral or an infinite series, both of which are difficult to handle from a computational standpoint. Asymptotic approximations for the inverse have been known for some time and these yield approximate inverse functions that are valid for very large or very small times - but frequently the times of interest lie somewhere between these two extremes. Therefore, some acceptable approximation valid over a larger interval of time is desirable. During the past few years a number of methods for achieving this have been developed and some of these are discussed briefly in this paper. The relative merits of the various methods are not evaluated here, but some general conclusions reached by other authors are given. One of these methods has been applied to problems associated with the radial flow of compressible liquids to producing wells. In the case of production at constant well pressure, the method leads to simple analytical solutions for a number of standard problems; e.g., homogeneous formation, permeability discontinuities, pressure buildup. These solutions greatly extend the range of validity of the asymptotic ones (valid for large times only) and should be of value in studying the behavior of wells producing under constant pressure conditions.

scholarly journals The free compressible viscous vortex

1991 ◽  
Vol 230 ◽  
pp. 45-73 ◽  
Author(s):  
Tim Colonius ◽  
Sanjiva K. Lele ◽  
Parviz Moin
Keyword(s):  
Mach Number ◽  
Radial Velocity ◽  
Analytical Solutions ◽  
Radial Flow ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Axisymmetric Flow ◽  
Tangential Velocity ◽  
Uniform Density ◽  
Heat Conducting

The effects of compressibility on free (unsteady) viscous heat-conducting vortices are investigated. Analytical solutions are found in the limit of large, but finite, Reynolds number, and small, but finite, Mach number. The analysis shows that the spreading of the vortex causes a radial flow. This flow is given by the solution of an ordinary differential equation (valid for any Mach number), which gives the dependence of the radial velocity on the tangential velocity, density, and temperature profiles of the vortex; estimates of the radial velocity found by solving this equation are found to be in good agreement with numerical solutions of the full equations. The experiments of Mandella (1987) also report a radial flow in the vortex, but their estimates are much larger than the analytical predictions, and it is found that the flow inferred from the iexperiments violates the Second Law of Thermodynamics for two-dimensional axisymmetric flow. It is speculated that three-dimensionality is the cause of this discrepancy. To obtain detailed analytical solutions, the equations for the viscous evolution are expanded in powers of Mach number, M. Solutions valid to O(M2), are discussed for vortices with finite circulation. Two specific initial conditions – vortices with initially uniform entropy and with initially uniform density – are analysed in detail. It is shown that swirling axisymmetric compressible flows generate negative radial velocities far from the vortex core owing to viscous effects, regardless of the initial distributions of vorticity, density and entropy.

scholarly journals Analytical Solutions of Carbonate Acidizing in Radial Flow

2021 ◽  
Author(s):  
Polyneikis Strongylis ◽  
Euripides Papamichos
Keyword(s):  
Analytical Solutions ◽  
Moving Boundary ◽  
Radial Flow ◽  
Fluid Velocity ◽  
Boundary Problems ◽  
Reactive Infiltration ◽  
Closed Form Solutions ◽  
Engineered Systems ◽  
Carbonate Acidizing ◽  
Reactive Fluids

AbstractThe flow of reactive fluids into porous media, a phenomenon known as reactive infiltration, is important in natural and engineered systems. While most of the studies in this area cover theoretical and experimental analyses in linear acid flow, the present work concentrates on radial flow conditions from a wellbore in the field and on finding exact analytical solutions to moving boundary problems of the uniform dissolution front. Closed-form solutions are obtained for the transient convection–diffusion which allow the demarcation of the range of applicability of the quasi-static limit. The fluid velocity dependency of the diffusion–dispersion coefficient is also examined by comparing results from analytical solutions from constant and velocity-dependent coefficients. These contributions form the basis for linear stability analyses to describe acid fingering encountered in reservoir stimulation.

scholarly journals SIMPLE WAVES IN THE THEORY OF TWO-DIMENSIONAL WATER FLOWS AND THE SCHEME OF THEIR USE FOR FREE FLOW SPREADING

2020 ◽  
Vol 8 (3) ◽  
pp. 47-50
Author(s):  
Maria Aleksandrova
Keyword(s):  
Analytical Solutions ◽  
Radial Flow ◽  
Equations Of Motion ◽  
Experimental Studies ◽  
Free Flow ◽  
Two Dimensional ◽  
Water Flows ◽  
Simple Waves ◽  
Basic Concepts

The basic concepts and equations of motion of a non-stationary radial flow are derived, and the tasks of obtaining analytical solutions are set. The results of experimental studies and their parameters that will be used for verification of the obtained solutions are indicated.

Laminar, Radial Flow of Two Immiscible Fluids in Slender Wedge-Shaped Passages

2017 ◽  
Vol 139 (8) ◽  
Author(s):  
H. M. Soliman
Keyword(s):  
Friction Coefficient ◽  
Void Fraction ◽  
Radial Flow ◽  
Fluid Model ◽  
Reynolds Numbers ◽  
Immiscible Fluids ◽  
Wall Friction ◽  
Practical Significance ◽  
Complex Nature ◽  
Two Fluid Model

The Jeffery–Hamel problem for laminar, radial flow between two nonparallel plates has been extended to the case of two immiscible fluids in slender channels. The governing continuity and momentum equations were solved numerically using the fourth-order Runge–Kutta method. Solutions were obtained for air–water at standard conditions over the void-fraction range of 0.4–0.8 (due to its practical significance) and the computations were limited to conditions where unique solutions were found to exist. The void fraction, pressure gradient, wall friction coefficient, and interfacial friction coefficient are dependent on the Reynolds numbers of both fluids and the complex nature of this dependence is presented and discussed. An attempt to use a one-dimensional two-fluid model with simplified assumptions succeeded in producing a qualitatively similar form of the void-fraction dependence on the two Reynolds numbers; however, quantitatively there are significant deviations between these results and those of the complete model.

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