Effect of flows between formations and capillary forces on the process of oil displacement in a stratified inhomogeneous bed

The results of a numerical investigation of the process of oil displacement in a stratified inhomogeneous formation on the basis of the two-phase flow model with account for capillary forces are presented. It is shown that in many cases the vertical inhomogeneity of oil reservoirs may not be a cause of nonuniform displacement and the non-recovery of large oil reserves by the time of water breakthrough to the extraction surface. The action of the capillary forces is an additional factor leading to equalization of the water propagation front in the inhomogeneous formation, water breakthrough delay, and intensification of the mass transfer between the layers with different permeabilities. Analysis of the contribution of the interlayer flows to the water flooding of low-permeability formation intervals calls into question the practicability of blocking high-permeability inclusions in the neighborhood of pumping wells.     

On the stability of a thermally stratified conducting fluid under the action of aligned magnetic field

The stability of a thermally stable stratified viscous electrically conducting shear flow is investigated in the presence of an impressed uniform aligned magnetic field. Only two-dimensional disturbances are studied in this paper because Squire's theorem does not apply in general, owing to the presence of the aligned magnetic field. The analysis is partly analytical and partly numerical. The asymptotic solutions for non-viscous fluid are first obtained analytically and they are then improved by introducing viscous and thermal diffusion terms (but only for =1) to get a uniformly valid solution. The neutral stability curves are numerically computed for a range of values of Richardson and Stuart numbers, which show that the flow is completely stabilized when a Stuart number exceeds a certain value for a given R _i>0. It is shown that the combined effects of magnetic field and stratification is to make the system stable to two-dimensional disturbances at lower Stuart number than the one given by Stuart (1954) in the absence of thermal stratification.     

Hydrodynamic Effects in Two-Phase Multicomponent Mixture Flow through a Porous Medium in Oil Reservoirs with a Complex Structure

The features of the hydrodynamic processes in stratified inhomogeneous oil reservoirs are investigated using a numerical solution of the equations of two-phase multicomponent flow through a porous medium. The structures of the two-phase flows caused by the reservoir structure and the hydrodynamic interaction between the phases are analyzed in relation to problems of the displacement of oil by water in ordinary flooding and in the presence of moving thickener slugs.     

大气绕山体的运动及其输移特性的研究

对分层大气山体绕流的流动模式及扩散输移特性进行了数值模拟。采用隐式时间离散方法在贴体网格系统下求解雷诺平均的N-S方程,计算结果描述出大气流动的特征,证实了分层(以Froude数为特征参数,定义为F=U/NH,U为来流风速,N为Brunt-Vaisala频率,H为山体高度)变化对山体绕流流态的影响。数值结果表明:当Froude数大于4.0时,山体绕流的流动不再依赖于大气分层的变化。当Froude数介于4.0和1.0之间时,流场中出现了Lee波,并随着Froude数的进一步减小,流动分离发生及Lee波破碎现象。同时模型也预测了在各种流动模式下大气中夹杂着的污染物绕山体的传输特性,表明大气的分层现象对污染物的分布有着非常重要的影响。     

The solitary waves in stratified shear flow

In this paper, the basic equation of internal long waves in stratified shear flow is derived under Boussinesq assumption, the first order approximation solution is given for solitary waves with the effects of slowly varying topograph at the sea bottom, weak stratification and basic shear flow. The Project Supported by the National Natural Science Foundation of China.     

Liquid and gas entrainment to a small break hole from a stratified two-phase region

A theoretical and experimental study has been conducted for liquid and gas entrainment to a small break hole from a stratified two-phase region. Theoretical correlations previously obtained for top, side and bottom entrainment were modified to express the relation between break flow rate, break quality and bulk water level so that they can be used easily for any break geometry. The modified correlations were assessed with experimental data obtained under room temperature and low pressure conditions using air and water. The experiment results were predicted well with the present model without using any adjustment coefficient when the upstream flow was symmetrical around the break. The effects of vortex, crossflow and wavy flow, observed in the experiment but not taken into account in the model, were empirically correlated based on the present correlation. By using the empirically modified correlations, the data in the literature, including high-pressure steam-water conditions, were reasonably predicted.     

Oscillations of a cylinder piercing a linearly stratified fluid layer

The plane problem of the small steady-state oscillations of a horizontal cylinder arbitrarily located in a three-layer fluid whose upper and lower layers are homogeneous and whose middle layer is linearly stratified is considered in the linear formulation using the Boussinesq approximation. The fluid is assumed to be ideal and incompressible. The method of mass sources distributed along the body contour is used in the internal wave generation regime and an integral equation for the fluid pressure is derived in the non-wave regime. The hydrodynamic load acting on the body is calculated as a function of the oscillation frequency of the cylinder and its location. The results are compared with experimental data.     

Evolution of the diffusion-induced flow over a sphere submerged in a continuously stratified fluid

The problem of the stabilization of the diffusion-induced flow over a sphere submerged in a continuously stratified fluid is solved using both asymptotic and numerical methods. The analytical solution describes the structure of the main convective cells, including thin meridional jets flowing along the surface and plumes spreading from the flow convergence regions above the upper and lower poles of the sphere which gradually return the fluid particles to the neutral buoyancy horizon. The total width of the flows adjacent to the surface exceeds the thickness of the salinity deficit layer or the density boundary layer. The numerical solution of the complete problem in the nonlinear formulation describes the main convective cells and two systems of unsteady integral waves formed in the vicinity of the sphere poles. At large times, out of the entire system of internal waves only those nearest to the neighborhood of their horizon of formation remain clearly defined. The calculated flow patterns are in agreement with each other and the data of shadow visualization of the stratified fluid structure near a submerged obstacle at rest.     

Effect of a periodic action on average flow in an inhomogeneous liquid-saturated porous medium

The problem of the average flow of a viscous incompressible fluid saturating a stationary porous incompressible matrix under a periodic action is considered. It is shown that a spatial inhomogeneity of the medium porosity leads to an average fluid flow, quadratically dependent on the action amplitude, in the direction of increase in porosity. In particular, this means that wave action on an oil reservoir could lead to fluid flow on the interfaces from low-porosity,weakly permeable collector regions into high-porosity regions, for example, to flow from blocks to fractures in fractured porous reservoirs, which makes it possible to enhance oil production. It is shown that in the presence of a constant pressure gradient the flow component generated by a periodic action can provide a substantial proportion of the total flow, especially on the boundaries with low-porosity strata or blocks. With increase in amplitude this may significantly exceed the component associated with the constant pressure gradient.     

Two-phase flow in heterogeneous porous media III: Laboratory experiments for flow parallel to a stratified system

Two-phase flow in stratified porous media is a problem of central importance in the study of oil recovery processes. In general, these flows are parallel to the stratifications, and it is this type of flow that we have investigated experimentally and theoretically in this study. The experiments were performed with a two-layer model of a stratified porous medium. The individual strata were composed of Aerolith-10, an artificial: sintered porous medium, and Berea sandstone, a natural porous medium reputed to be relatively homogeneous. Waterflooding experiments were performed in which the saturation field was measured by gamma-ray absorption. Data were obtained at 150 points distributed evenly over a flow domain of 0.1 × 0.6 m. The slabs of Aerolith-10 and Berea sandstone were of equal thickness, i.e. 5 centimeters thick. An intensive experimental study was carried out in order to accurately characterize the individual strata; however, this effort was hampered by both local heterogeneities and large-scale heterogeneities.The theoretical analysis of the waterflooding experiments was based on the method of large-scale averaging and the large-scale closure problem. The latter provides a precise method of discussing the crossflow phenomena, and it illustrates exactly how the crossflow influences the theoretical prediction of the large-scale permeability tensor. The theoretical analysis was restricted to the quasi-static theory of Quintard and Whitaker (1988), however, the dynamic effects described in Part I (Quintard and Whitaker 1990a) are discussed in terms of their influence on the crossflow.Roman Letters A interfacial area between the -region and the -region contained within V, m² - a vector that maps onto , m - b vector that maps onto , m - b vector that maps onto , m - B second order tensor that maps onto , m² - C second order tensor that maps onto , m² - E energy of the gamma emitter, keV - f fractional flow of the -phase - g gravitational vector, m/s² - h characteristic length of the large-scale averaging volume, m - H height of the stratified porous medium , m - i unit base vector in the x-direction - K local volume-averaged single-phase permeability, m² - K - {K}, large-scale spatial deviation permeability - { K} large-scale volume-averaged single-phase permeability, m² - K ^* large-scale single-phase permeability, m² - K ^** equivalent large-scale single-phase permeability, m² - K local volume-averaged -phase permeability in the -region, m² - K local volume-averaged -phase permeability in the -region, m² - K - {K } , large-scale spatial deviation for the -phase permeability, m² - K ^* large-scale permeability for the -phase, m² - l thickness of the porous medium, m - l characteristic length for the -region, m - l characteristic length for the -region, m - L length of the experimental porous medium, m - characteristic length for large-scale averaged quantities, m - n outward unit normal vector for the -region - n outward unit normal vector for the -region - n unit normal vector pointing from the -region toward the -region (n = - n ) - N number of photons - p pressure in the -phase, N/m² - p ₀ reference pressure in the -phase, N/m² - local volume-averaged intrinsic phase average pressure in the -phase, N/m² - large-scale volume-averaged pressure of the -phase, N/m² - large-scale intrinsic phase average pressure in the capillary region of the -phase, N/m² - - , large-scale spatial deviation for the -phase pressure, N/m² - p_c , capillary pressure, N/m² - p _c capillary pressure in the -region, N/m² - p capillary pressure in the -region, N/m² - {p _c } ^c large-scale capillary pressure, N/m² - q -phase velocity at the entrance of the porous medium, m/s - q -phase velocity at the entrance of the porous medium, m/s - S_wi irreducible water saturation - S /, local volume-averaged saturation for the -phase - S _i initial saturation for the -phase - S _r residual saturation for the -phase - S ^* { }^*/}^*, large-scale average saturation for the -phase - S saturation for the -phase in the -region - S saturation for the -phase in the -region - t time, s - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the -phase, m/s - {v } large-scale averaged velocity for the -phase, m/s - v local volume-averaged phase average velocity for the -phase in the -region, m/s - v local volume-averaged phase average velocity for the -phase in the -region, m/s - v -{v } , large-scale spatial deviation for the -phase velocity, m/s - v -{v } , large-scale spatial deviation for the -phase velocity in the -region, m/s - v -{v } , large-scale spatial deviation for the -phase velocity in the -region, m/s - V large-scale averaging volume, m³ - y position vector relative to the centroid of the large-scale averaging volume, m - {y}^c large-scale average of y over the capillary region, m Greek Letters local porosity - local porosity in the -region - local porosity in the -region - local volume fraction for the -phase - local volume fraction for the -phase in the -region - local volume fraction for the -phase in the -region - {}^* { }^*+{ }^*, large-scale spatial average volume fraction - { }^* large-scale spatial average volume fraction for the -phase - mass density of the -phase, kg/m³ - mass density of the -phase, kg/m³ - viscosity of the -phase, N s/m² - viscosity of the -phase, Ns/m² - V /V , volume fraction of the -region ( + =1) - V /V , volume fraction of the -region ( + =1) - attenuation coefficient to gamma-rays, m^-1 - -      

Formation of a tripolar vortex in a stratified fluid

This paper reports on an experimental study of vortices in a stratified fluid. The vortices were generated by two different stirring devices, viz. a rotating sphere and a rotating bent rod. It was found that the vortices created with the rotating sphere are mostly axisymmetric and stable, whereas the vortices produced with the bent rod generally show instabilities, under certain conditions leading to the formation of a tripolar vortex. This report concentrates on this tripolar structure and presents quantitative information about the flow obtained through streak photography of tracer particles.     

Analysis of wellbore crossflows for nonstationary operation of a well in an inhomogeneous multilayer bed

This paper considers the nonstationary filtration of a homogeneous elastic fluid in a rigid, inhomogeneous, multilayer bed with no hydrodynamic communication between interbeds. The effect of wellbore crossflow between interbeds after stopping of well operation is studied numerically. It is shown that the volume and time of fluid crossflow between the interbeds are comparable to the volume of injection and the time of chemical treatment of wells. It is concluded that crossflows can be used for selective injection of chemicals. A procedure is proposed to control the volume of selective fluid injection into a well by changing the injection rate.     

Semi-Analytic Axisymmetric Pressure Distribution in a Consolidating Porous Medium

A semi-analytic solution of the consolidation problem in a finite hollow axisymmetric elastic porous medium is given. According to Biot's theory, we have rigorously derived the consolidation equations and demonstrated that in the axisymmetric problems, the pore pressure diffusion equation can be uncoupled. In the problem of infinite domain, the uncoupled pressure diffusion equation is homogeneous and only the diffusion coefficient is changed. In the problem of finite domain, the uncoupled pressure diffusion equation is nonhomogeneous. In fact, it is a linear differential-integral equation. We solve it by the variables separation method in the time domain.     

Effect of residual oil saturation on the flow through a porous medium in the neighborhood of an injection well

Models of the residual oil saturation and models of its effect on the flow in injection wells are proposed. The threshold nature of the dependence of the residual oil saturation on the capillary number determines a change in the flow regimes in the neighborhood of the injection well. The cases of pure, contaminated, and compressible reservoirs are considered. The dependences of the basic problem parameters on the displacement conditions and the state of the reservoir are obtained, together with formulas for the pressure distribution and well injectivity. The topicality of such a simulation for field calculations is demonstrated.     

A non-penetrating well in a confined aquifer. An approximate closed solution

The non-steady flow towards a non-penetrating well in a closed aquifer with storativity is studied theoretically. A closed solution can be derived, if the boundary condition at the interface between the well and the aquifer is properly approximated. The analysis shows that if the thickness of the aquifer is much bigger than the radius of the well, the approxiamtion is good and that the flow in the aquifer is concentrated to the top of the aquifer, leaving the water at the bottom effectively stagnant, if the distance from the well far exceeds the diameter of the well but is far below the thickness of the aquifer. The solution together with the well-known solution for a complete well can serve as limits for the practically important case of a partially penetrating well.     

Gas-dynamic analogy in the theory of stratified liquid flows with a free boundary

Novel approximate mathematical models of long-wave theory describing flows of a density-stratified liquid with a free boundary are proposed. It is shown that in certain cases the equations of the novel models coincide with either the equations of nonisentropic gas dynamics with a polytropic equation of state for γ = 2 or the equations describing the dynamics of a mixture of two perfect gases.     

Prediction of liquid level distribution in horizontal gas-liquid stratified flows with interfacial level gradient

A one-dimensional momentum equation has been derived based on a two-fluid model and used to predict the axial distribution of liquid level or void fraction in steady, cocurrent, gas-liquid stratified flows in horizontal circular pipes and rectangular channels. The equation is carefully formulated to incorporate the effect of interfacial level gradient. Two different critical liquid levels are found from the momentum equation and are adopted as a boundary condition to calculate the liquid level or void fraction distribution upstream of the channel exit. The predicted void fraction distributions are compared with the experimental data obtained in a rectangular channel in this work and other data reported for large-diameter pipes. Good agreement is shown for air-kerosene, air-water and stream-water stratified flows with a smooth gas-liquid interface.     

Mixing in a stably stratified medium by horizontal shear near vertical walls

Stratified environmental flows near boundaries can have a horizontal mean shear component, orthogonal to the vertical mean density gradient. Vertical transport, against the stabilizing force of gravity, is possible in such situations if three-dimensional turbulence is sustained by the mean shear. A model problem, water with a constant mean density gradient flowing in a channel between parallel vertical walls, is examined here using the technique of large eddy simulation (LES). It is found that, although the mean shear is horizontal, the fluctuating velocity field has significant vertical shear and horizontal vorticity, thereby causing small-scale vertical mixing of the density field. The vertical stirring is especially effective near the boundaries where the mean shear is large and, consequently, the gradient Richardson number is small. The mean stratification is systematically increased between cases in our study and, as expected, the buoyancy flux correspondingly decreases. Even so, horizontal mean shear is found to be more effective than the well-studied case of mean vertical shear in inducing vertical buoyancy transport as indicated by generally larger values of vertical eddy diffusivity and mixing efficiency.     

Simulation of non‐hydrostatic,density‐stratified flow in irregular domains

A numerical model has been developed for simulating density‐stratified flow in domains with irregular but simple topography. The model was designed for simulating strong interactions between internal gravity waves and topography, e.g. exchange flows in contracting channels, tidally or convectively driven flow over two‐dimensional sills or waves propagating onto a shoaling bed. The model is based on the non‐hydrostatic, Boussinesq equations of motion for a continuously stratified fluid in a rotating frame, subject to user‐configurable boundary conditions. An orthogonal boundary fitting co‐ordinate system is used for the numerical computations, which rely on a fourth‐order compact differentiation scheme, a third‐order explicit time stepping and a multi‐grid based pressure projection algorithm. The numerical techniques are described and a suite of validation studies are presented. The validation studies include a pointwise comparison of numerical simulations with both analytical solutions and laboratory measurements of non‐linear solitary wave propagation. Simulation results for flows lacking analytical or laboratory data are analysed a posteriori to demonstrate satisfaction of the potential energy balance. Computational results are compared with two‐layer hydraulic predictions in the case of exchange flow through a contracting channel. Finally, a simulation of circulation driven by spatially non‐uniform surface buoyancy flux in an irregular basin is discussed. Copyright © 2000 John Wiley & Sons, Ltd.