Vertical integration of three-phase flow equations for analysis of light hydrocarbon plume movement

A mathematical model is derived for areal flow of water and light hydrocarbon in the presence of gas at atmospheric pressure. Vertical integration of the governing three-dimensional, three-phase flow equations is performed under the assumption of local vertical equilibrium to reduce the dimensionality of the problem to two orthogonal horizontal directions. Independent variables in the coupled water and hydrocarbon areal flow equations are specified as the elevation of zero gauge hydrocarbon pressure (air-oil table) and the elevation of zero gauge water pressure (air-water table). Constitutive relations required in the areal flow model are vertically integrated fluid saturations and vertically integrated fluid conductivities as functions of air-oil and air-water table elevations. Closed-form expressions for the vertically integrated constitutive relations are derived based on a three-phase extension of the Brooks-Corey saturation-capillary pressure function. Closed-form Brooks-Corey relations are compared with numerically computed analogs based on the Van Genuchten retention function. Close agreement between the two constitutive models is observed except at low oil volumes when the Brooks-Corey model predicts lower oil volumes and transmissivities owing to the assumption of a distinct fluid entry pressure. Nonlinearity in the vertically integrated constitutive relations is much less severe than in the unintegrated relations. Reduction in dimensionality combined with diminished nonlinearity, makes the vertically integrated water and hydrocarbon model an efficient formulation for analyzing field-scale problems involving hydrocarbon spreading or recovery under conditions for which the vertical equilibrium assumption is expected to be a satisfactory approximation.     

Upscaling in Vertically Fractured Oil Reservoirs Using Homogenization

Flow modeling in fractured reservoirs is largely confined to the so-called sugar cube model. Here, however, we consider vertically fractured reservoirs, i.e., the situation that the reservoir geometry can be approximated by fractures enclosed columns running from the base rock to the cap rock (aggregated columns). This article deals with the application of the homogenization method to derive an upscaled equation for fractured reservoirs with aggregated columns. It turns out that vertical flow in the columns plays an important role, whereas it can be usually disregarded in the sugar cube model. The vertical flow is caused by coupling of the matrix and fracture pressure along the vertical faces of the columns. We formulate a fully implicit three-dimensional upscaled numerical model. Furthermore, we develop a computationally efficient numerical approach. As found previously for the sugar cube model, the Peclet number, i.e., the ratio between the capillary diffusion time in the matrix and the residence time of the fluids in the fracture, plays an important role. The gravity number plays a secondary role. For low Peclet numbers, the results are sensitive to gravity, but relatively insensitive to the water injection rate, lateral matrix column size, and reservoir geometry, i.e., sugar cube versus aggregated column. At a low Peclet number and sufficiently low gravity number, the effective permeability model gives good results, which agree with the solution of the aggregated column model. However, ECLIPSE simulations (Barenblatt or Warren and Root (BWR) approach) show deviations at low Peclet numbers, but show good agreement at intermediate Peclet numbers. At high Peclet numbers, the results are relatively insensitive to gravity, but sensitive to the other conditions mentioned above. The ECLIPSE simulations and the effective permeability model show large deviations from the aggregated column model at high Peclet numbers. We conclude that at low Peclet numbers, it is advantageous to increase the water injection rate to improve the net present value. However, at high Peclet numbers, increasing the flow rate may lead to uneconomical water cuts.     

An Experimental Study of Three-Phase Trapping in Sand Packs

The trapped saturations of oil and gas are measured as functions of initial oil and gas saturation in water-wet sand packs. Analogue fluids—water, octane and air—are used at ambient conditions. Starting with a sand-pack column which has been saturated with brine, oil (octane) is injected with the column horizontal until irreducible water saturation is reached. The column is then positioned vertically and air is allowed to enter from the top of the column, while oil is allowed to drain under gravity for varying lengths of time. At this point, the column may be sliced and the fluids analyzed by gas chromatography to obtain the initial saturations. Alternatively, brine is injected through the bottom of the vertical column to trap oil and gas, before slicing the columns and measuring the trapped or residual saturations by gas chromatography and mass balance. The experiments show that in three-phase flow, the total trapped saturations of oil and gas are considerably higher than the trapped saturations reported in the literature for two-phase systems. It is found that the residual saturation of oil and gas combined could be as high as 23 %, as opposed to a maximum two-phase residual of only 14 %. For very high initial gas saturations, the residual gas saturation, up to 17 %, was also higher than for two-phase displacement. These observations are explained in terms of the competition between piston-like displacement and snap-off. It is also observed that less oil is always trapped in three-phase flow than in two-phase displacement, and the difference depends on the amount of gas present. For low and intermediate initial gas saturations, the trapped gas saturation rises linearly with initial saturation, followed by a constant residual, as seen in two-phase displacements. However, at very high initial gas saturations, the residual saturation rises again.     

A Mathematical Model for Hysteretic Two-Phase Flow in Porous Media

We develop a mathematical model for hysteretic two-phase flow (of oil and water) in waterwet porous media. To account for relative permeability hysteresis, an irreversible trapping-coalescence process is described. According to this process, oil ganglia are created (during imbibition) and released (during drainage) at different rates, leading to history-dependent saturations of trapped and connected oil. As a result, the relative permeability to oil, modelled as a unique function of the connected oil saturation, is subject to saturation history. A saturation history is reflected by history parameters, that is by both the saturation state (of connected and trapped oil) at the most recent flow reversal and the most recent water saturation at which the flow was a primary drainage. Disregarding capillary diffusion, the flow is described by a hyperbolic equation with the connected oil saturation as unknown. This equation contains functional relationships which depend on the flow mode (drainage or imbibition) and the history parameters. The solution consists of continuous waves (expansion waves and constant states), shock waves (possibly connecting different modes) and stationary discontinuities (connecting different saturation histories). The entropy condition for travelling waves is generalized to include admissible shock waves which coincide with flow reversals. It turns out that saturation history generally has a strong influence on both the type and the speed of the waves from which the solution is constructed.     

A first order relaxation model for the prediction of the local interfacial area density in two-phase flows

Many energy production and chemical processes involve vapor/liquid two-phase flows. Mass and energy are often exchanged between the vapor and the liquid phases, and the fluid mechanics of the two-phase system is strongly influenced by the exchange of momentum between each phase. Significantly, the transport of mass, energy and momentum between the phases takes place across interfaces. Therefore the interfacial area density (i.e. the interfacial area per unit volume) has to be accurately known in order to make reliable predictions of the interfacial transfers. Indeed, the interfacial area density must be known for both steady and transient two-phase flows. It is the purpose of this paper to present a first order relaxation model which is derived from the Boltzmann transport equation, and which accurately describes the evolution of interfacial area density for bubbly flows. In particular, the local, instantaneous interfacial area densities and volume fractions are predicted for vertical flow of a vapor/liquid bubbly flow involving both bubble clusters and individual bubbles.     

Network Modeling of EOR Processes: A Combined Invasion Percolation and Dynamic Model for Mobilization of Trapped Oil

A novel concept for modeling pore-scale phenomena included in several enhanced oil recovery (EOR) methods is presented. The approach combines a quasi-static invasion percolation model with a single-phase dynamic transport model in order to integrate mechanistic chemical oil mobilization methods. A framework is proposed that incorporates mobilization of capillary trapped oil. We show how double displacement of reservoir fluids can contribute to mobilize oil that are capillary trapped after waterflooding. In particular, we elaborate how the physics of colloidal dispersion gels (CDG) or linked polymer solutions (LPS) is implemented. The linked polymer solutions consist of low concentration partially hydrolyzed polyacrylamide polymer crosslinked with aluminum citrate. Laboratory core floods have shown demonstrated increased oil recovery by injection of linked polymer solution systems. LPS consist of roughly spherical particles with sizes in the nanometer range (50–150 nm). The LPS process involve mechanisms such as change in rheological properties effect, adsorption and entrapment processes that can lead to a microscopic diversion and mobilization of waterflood trapped oil. The purpose is to model the physical processes occurring on pore scale during injection of linked polymer solutions. A sensitivity study has also been performed on trapped oil saturation with respect to wettability status to analyze the efficiency of LPS on different wettability conditions. The network modeling results suggest that weakly wet reservoirs are more suitable candidates for performing linked polymer solution injection.     

Multiphase flow with a simplified model for oil entrapment

A computationally simple procedure is described to model effects of oil entrapment on three-phase permeability-saturation-capillary pressure relations. The model requires knowledge of airwater saturation-capillary pressure relations, which are assumed to be nonhysteretic and are characterized by Van Genuchten's parametric model; scaling factors equal to the ratio of water surface tension to oil surface tension and to oil-water interfacial tension; and the maximum oil (also referred to as nonwetting liquid in a three-phase medium) saturation which would occur following water flooding of oil saturated soil. Trapped nonwetting liquid saturation is predicted as a function of present oil-water and air-oil capillary pressures and minimum historical water saturation since the occurrence of oil at a given location using an empirically-based algorithm. Oil relative permeability is predicted as a simple function of apparent water saturation (sum of actual water saturation and trapped oil saturation) and free oil saturation (difference between total oil and trapped oil saturation), and water relative permeability is treated as a unique function of actual water saturation. The proposed method was implemented in a two-dimensional finite-element simulator for three-phase flow and component transport, MOFAT. The fluid entrapment model requires minimal additional computational effort and computer storage and is numerically robust. The applicability of the model is illustrated by a number of hypothetical one- and two-dimensional simulations involving infiltration and redistribution with changes in water-table elevations. Results of the simulations indicate that the fraction of a hydrocarbon spill that becomes trapped under given boundary conditions increases as a nonlinear function of the maximum trapped nonwetting liquid saturation. Dense organic liquid plumes may exhibit more pronounced effects of entrapment due to the more dynamic nature of flow, even under static water table conditions. Disregarding nonwetting fluid entrapment may lead to significant errors in predictions of immiscible plume migration.     

Effect of Low Rotation Rate on Steady Convection During the Solidification of a Ternary Alloy

We consider the problem of steady convective flow during the directional solidification of a horizontal ternary alloy system rotating at a constant and low rate about a vertical axis. Under the limit of large far-field temperature, the flow region is modeled to be composed of two horizontal mushy layers, which are referred to here as a primary layer over a secondary layer. We first determine the basic state solution and then carry out linear stability analysis to calculate the neutral stability boundary and the critical conditions at the onset of motion. We find, in particular, that there are two flow solutions and each solution exhibits two neutral stability boundaries, and the flow can be multi-modal in the low rotating rate case with local minima on each neutral boundary. The critical Rayleigh number and the wave number as well as the vertical volume flux increase with the rotation rate, but the flow is found to be less stabilizing as compared to the binary alloy counterpart flow. The effects of low rotation rate increase the solid fraction and the liquid fraction at certain vertically oriented fluid lines, and the highest value of such increase is at a horizontal level close to the interface between the two mushy layers.     

Effect of vertical grid variability on a free surface flow model

A previously developed numerical model that solves the incompressible, non‐hydrostatic, Navier–Stokes equations for free surface flow is analysed on a non‐uniform vertical grid. The equations are vertically transformed to the σ‐coordinate system and solved in a fractional step manner in which the pressure is computed implicitly by correcting the hydrostatic flow field to be divergence free. Numerical consistency, accuracy and efficiency are assessed with analytical methods and numerical experiments for a varying vertical grid discretization. Specific discretizations are proposed that attain greater accuracy and minimize computational effort when compared to a uniform vertical discretization. Published in 2007 by John Wiley & Sons, Ltd.     

The numerical solution of the thin film flow surrounding a horizontal cylinder resulting from a vertical cylindrical jet

The numerical solution of the thin film flow surrounding a horizontal cylinder resulting from a single vertical cylindrical jet is obtained. This is effected by transforming the domain of the flow, which contains a free surface, onto a rectangular parallelepiped and using a marching strategy to solve the ensuing parabolic equations. The flow terminates at a finite distance along the cylinder, its position depending on the velocity and mass flux of the jet. A comparison with the usual two-dimensional model in which the jet is replaced by a vertical sheet shows that such a representation is valid provided the overall width of the flow is not too large. In particular, the differences in heat transfer characteristics amount to a few per cent, thus validating the use of the two-dimensional model when applied to heat exchanger tubes. A comparison with the more usual multicolumn case is also considered.     

垂直上升管内油—气—水三相弹状流压力降的研究

给出一种垂直上升油－气－水三相弹状流压力降的计算模型。该模型考虑弹状流中Ｔａｙｌｏｒ气泡周围下降液膜的变化历程。通过油－气－水弹状流的实验研究发现，该模型的数值模拟结果与低压工况下的实验值符合得较好。本模型是计算垂直油－气－水三相弹状流中液相的连续相为水相时的压力降的有效方法。     

THE TRANSIENT ELLIPTIC FLOW OF POWER-LAW FLUID IN FRACTAL POROUS MEDIA

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Experimental investigation of a free-surface turbulent jet with Coanda effect

The deviation of a jet from the straight direction due to the presence of a lateral wall is investigated from the experimental point of view. This flow condition is known as Coanda jet (from the Romanian aerodynamicist Henry Marie Coanda who discovered and applied it at the beginning of XXth century) or offset jet. The objective of the work is to detail the underlying mechanisms of such a phenomenon aiming to use it as a flow control method at polluted river flows mouth. To do this, a large laboratory free-surface tank with an incoming channel has been set up and velocity field measurements are performed by Optical Flow methods (namely Feature Tracking). Preliminary tests on the well-known free jet configuration without any marine structure (i.e. lateral wall) are performed to allow comparison with free jet scaling and self-similar solutions. The presence of the free-surface gives rise to centerline velocity decay which is lower than in free unbounded plane or circular jets due to the vertically limited ambient fluid entrainment. In the second part of the paper, the effect of a lateral wall on the jet configuration is examined by placing it at different lateral distances from the jet outlet. The resulting velocity fields clearly show an inclined Coanda jet with details which seems to depend on the lateral wall distance itself. The analysis of self-similarity along the inclined jet direction reveals that for wall distances larger than 5 jet widths this dependence almost disappears.     

Incompressible flow in a rapidly rotating cylinder with a source/sink distribution in the lateral wall and a free inner boundary

The flow of an incompressible fluid in a rapidly rotating right circular cylinder is considered. A source/sink mass distribution at the lateral wall, which is azimuthally uniform and symmetric across the midplane, causes a deviation from wheel flow. The container is only partially full and the inner free surface is allowed to deviate slightly from the vertical. A finite-difference solution of the full axisymmetric, non-linear governing equations was used to obtain the flow field. A special implicit technique for the Coriolis terms which maintains geostrophy was developed and is described. The results obtained for a low Rossby number flow compare quite favourably with the linearized solution. Results are also presented for a case wherein the non-linear terms are important.     

Simulations of vertical jet penetration using a filtered two-fluid model in a gas–solid fluidized bed

The influence of a vertical jet located at the distributor in a cylindrical fluidized bed on the flow behavior of gas and particles was predicted using a filtered two-fluid model proposed by Sundaresan and coworkers. The distributions of volume fraction and the velocity of particles along the lateral direction were investigated for different jet velocities by analyzing the simulated results. The vertical jet penetration lengths at the different gas jet velocities have been obtained and compared with predictions derived from empirical correlations; the predicted air jet penetration length is discussed. Agreement between the numerical simulations and experimental results has been achieved.     

Diffusion Mass Transfer in Miscible Oil Recovery: Visual Experiments and Simulation

Co-and counter-current type transfers due to diffusion and -free- convection caused by the buoyant forces between fracture and matrix were studied experimentally using 2-D glass-bead models. Mineral oil and kerosene were used as the displaced phase. The model saturated with oil was exposed to solvent phase (pentane) under static conditions (no flow in fracture) to mimic matrix-fracture interaction during gas or liquid solvent injection in naturally fractured reservoirs. Displacement fronts and patterns were analyzed and quantified using fractal techniques to obtain correlations between the fractal properties and displacement type. Displacements resulted in a mixture of bulk diffusion and -free- convection mainly depending on the interaction type (co- or counter-current), oil type, and displacement direction (horizontal and vertical). Conditions yielding different types of displacement patterns were identified. Finally, a stochastic model that was inspired from invasion percolation and diffusion limited aggregation algorithms was developed for the horizontal displacement cases. The experimental observations were matched to the displacement patterns obtained through the stochastic modeling.     

Estimate of the residual volume of viscoplastic oil after flooding

The problem of calculating regions of residual viscoplastic oil with a critical pressure gradient is considered. This problem has hitherto been solved mainly for uniform strata (see, for example, [1–3]) but in the present paper, which follows the work of Entov, Pankov, and Pan'ko [4], the problem is considered not only for uniform strata but also for layered strata and vertically nonuniform strata. It is shown that for sufficiently thin strata allowing free flow between individual seams the formulation of the problem is the same for the piston model and the Buckley-Leverett model and reduces to a two-dimensional boundary value problem with a continuous nonlinear effective flow law for the water. An approximate numerical method of calculating regions of residual oil that is suitable for the three listed types of strata is described and illustrated by examples.     

含油轴承表面供油行为与自润滑机理分析

含油轴承是一类重要的减摩自润滑零件,轴承基体中油液的渗流特性对表面润滑性能有重要影响. 建立含油轴承孔隙渗流与表面油膜润滑的耦合力学模型,分析含油轴承系统中油液的渗流特性,探讨轴承表面油液的供油行为与自润滑机理. 结果表明:在含油轴承的收敛区内同时存在周向旋转流、径向伸缩流和法向渗析流,油液在各方向上的流动状况由该向流体压力梯度决定;受油膜压力影响,油液在接触区向多孔基体渗入,在接触区入口部位向多孔表面析出,由此构成了油液渗入和析出的闭环速度流线,增强摩擦界面间的泵吸效应. 油液在法向上的渗析速度随中心膜厚增加而减小,随转速升高而增大,渗析速度越大,对泵吸效应的增强作用越显著,接触区入口的油液也更易进入摩擦界面,保障含油轴承的良好自润滑效果. 研究结果对揭示含油轴承的供油自润滑机理具有重要意义.      

Integrated numerical model for vegetated surface and saturated subsurface flow interaction

The construction of an integrated numerical model is presented in this paper to deal with the interactions between vegetated surface and saturated subsurface flows. A numerical model is built by integrating the previously developed quasi-three-dimensional (Q3D) vegetated surface flow model with a two-dimensional (2D) saturated groundwater flow model. The vegetated surface flow model is constructed by coupling the explicit finite volume solution of 2D shallow water equations (SWEs) with the implicit finite difference solution of Navier-Stokes equations (NSEs) for vertical velocity distribution. The subsurface model is based on the explicit finite volume solution of 2D saturated groundwater flow equations (SGFEs). The ground and vegetated surface water interaction is achieved by introducing source-sink terms into the continuity equations. Two solutions are tightly coupled in a single code. The integrated model is applied to four test cases, and the results are satisfactory.