Three-phase Flow Model Research Articles

Capillary instability in models for three-phase flow

Standard models for immiscible three-phase flow in porous media exhibit unusual behavior associated with loss of strict hyperbolicity. Anomalies were at one time thought to be confined to the region of nonhyperbolicity, where the purely convective form of the model is ill-posed. However, recent abstract results have revealed that diffusion terms, which are usually neglected, can have a significant effect. The delicate interplay between convection and diffusion determines a larger region of diffusive linear instability. For artificial and numerical diffusion, these two regions usually coincide, but in general they do not.Accordingly, in this paper, we investigate models of immiscible three-phase flow that account for the physical diffusive effects caused by capillary pressure differences among the phases. Our results indicate that, indeed, the locus of instability is enlarged by the effects of capillarity, which therefore entails complicated behavior even in the region of strict hyperbolicity. More precisely, we demonstrate the following results. (1) For general immiscible three-phase flow models, if there is stability near the boundary of the saturation triangle, then there exists a Dumortier-Roussarie-Sotomayor (DRS) bifurcation point within the region of strict hyperbolicity. Such a point lies on the boundary of the diffusive linear instability region. Moreover, as we have shown in previous works, existence of a DRS point (satisfying certain nondegeracy conditions) implies nonuniqueness of Riemann solutions, with corresponding nontrivial asymptotic dynamics at the diffusive level and ill-posedness for the purely convective form of the equations. (2) Models employing the interpolation formula of Stone (1970) to define the relative permeabilities can be linearly unstable near a corner of the saturation triangle. We illustrate this instability with an example in which the two-phase permeabilities are quadratic.Results (1) and (2) are obtained as consequences of more general theory concerning Majda-Pego stability and existence of DRS points, developed for any two-component system and applied to three-phase flow. These results establish the need for properly modelling capillary diffusion terms, for they have a significant influence on the well-posedness of the initial-value problem. They also suggest that generic immiscible three-phase flow models, such as those employing Stone permeabilities, are inadequate for describing three-phase flow.

Mass partition during collapsing and transitional columns by using numerical simulations

Pyroclast dispersal and mass partition between convective and collapsing transport systems represent a major issue in the understanding of the dynamics of quasi-steady explosive eruptions and the interpretation of their deposits. The spatial and temporal dispersal of pyroclasts during the initial 15 min of collapsing and transitional columns was investigated by using a transient, two-dimensional and three-phase (one gas phase and two solid phases representative of pyroclasts of varying size and density) flow model. The different behaviors of the column were simulated by using different water contents of the eruptive mixture and assuming both pressure-balanced and overpressured conditions at the vent. For each simulation results allowed the quantification of the mass of pyroclasts of different size forming the pyroclastic density current, the co-ignimbrite column and the convective plumes rising from the proximal area and above the fountain. For collapsing columns, the total mass transported in the convective system (composed of the plume above the fountain and the co-ignimbrite column) ranges from about 10 to 30 wt% of the total erupted mass according to the specific water content of the mixture at the vent. Fine particles are dominant in this convective system. With the increase of the water content the character of the column shows a collapsing/buoyant transitional behavior. Such a regime is characterized by greater collapse height, generation of more dilute density currents, shorter flow runout, less steady behavior of the column, and intermittent feeding of the flows. For these columns, the total mass forming the convective portions reaches values of up to 50 wt% of the erupted pyroclasts with a significant amount of coarse particles entrained in the convective plume above the fountain. At higher water contents the column becomes fully buoyant with the whole mass feeding the convective column. Simulations assuming overpressured conditions at the vent are characterized by more unsteady dynamics of the dispersal process with significant variations in the collapse height of the column, mass flow-rate of the flow and amount of mass elutriated in the convective system. Simulation results suggest that Plinian columns that undergo a transition from convective to collapsing regimes, as a consequence, in our case, of a water content decrease in the magma (but not necessarily limited to this case), would be characterized by a gradual decrease of the carrying capacity of the convective column above the fountain and by the emplacement of pyroclastic density currents which evolve from dilute to dense, gradually increasing their runout.

Validation of a novel algorithm for the adaptive calculation of transient stratified flow of gas, oil and water in pipelines

In this paper, a mathematical model for three-phase flow in pipelines is developed and a novel algorithm for solving the resulting system of partial differential equations is presented. The numerical solution is based on an adaptive mesh refinement technique. Significant gains in the numerical performance have been achieved by using this technique combined with high-resolution schemes like FCT. This approach is highly beneficial in terms of speed while preserving accuracy. An explicit adaptive numerical solver adaptive conservation law solver (ACLS) has been developed in fortran90 using object-oriented methods, which give a modular and flexible code. Some numerical tests are performed to show the good behaviour of this solver and to compare the numerical results against the experimental data. Copyright © 2001 John Wiley & Sons, Ltd.

Dimensionless groups for three-phase gravity drainage flow in porous media

The downward displacement of oil by gas (either through gas cap expansion or by gas injection) at the crest of the reservoir is an attractive method of oil recovery. The drainage of oil under gravity forces is a potentially efficient method as it can reduce the remaining oil saturation to below that obtained after waterflooding. This paper describes a series of experiments of gas invasion under gravity-dominated conditions with special attention to the effects of wettability and water saturation on three-phase flow. The experiments were performed in bead-pack models by spontaneous gas invasion at both low and high water saturations with a spreading oil. Different oil recovery rates were observed depending on the wettability of the beads and initial water saturation. At irreducible water saturation, the process appeared to be less efficient for the oil-wet conditions, while similar oil recoveries are observed for both oil-wet and water-wet media at residual oil saturation. Different recovery rates occur with different fluid morphology, which depend on the matrix wettability and the balance between gravity, viscous and capillary forces. The results have been analysed using dimensionless groups. The Bond ( N B) and capillary numbers ( N C) were modified to include the 3-phase effects of gas, oil and water. However, for these cases the Bond and capillary numbers alone were insufficient to fully describe the dynamics of oil recovery by gravity drainage. Therefore, a new dimensionless group combining the effects of gravity and viscous forces to capillary forces was defined as: N= N B+ A( μ d/ μ g) N C, where A is a scaling factor (in all our experiments A=−17225) and ( μ d/ μ g) is the viscosity ratio between the displaced and displacing phase. A linear relationship was found between this new group and the total recovery for all the scenarios tested. The slope was approximately 40 for three cases, i.e., water-wet case at irreducible water saturation, and water-wet and oil-wet cases at residual oil saturation. The oil-wet case at irreducible water saturation has a larger slope, probably due to the blocking effect of water. These experimental results may be used as a benchmark to test theoretical models of three-phase flow under gravity dominated conditions. The new dimensionless group should improve the understanding of the pore scale mechanisms so that these processes can be included in the development of network models and in the processes of upscaling laboratory results.

Analysis of a Compositional Model for Fluid Flow in Porous Media

In this paper we consider a compositional model for three-phase multicomponent fluid flow in porous media. This model consists of Darcy's law for volumetric flow velocities, mass conservation for hydrocarbon components, thermodynamic equilibrium for mass interchange between phases, and an equation of state for saturations. These differential equations and algebraic constraints are rewritten in terms of various formulations of the pressure and component-conservation equations. Phase, weighted fluid, global, and pseudoglobal pressure and component-conservation formulations are analyzed. A numerical scheme based on the mixed finite element method for the pressure equation and the Eulerian--Lagrangian localized adjoint method for the component-conservation equations is developed. Numerical results are reported to show the behavior of the solution to the compositional model and to investigate the performance of the proposed numerical scheme.

Numerical pore-scale modeling of three-phase fluid flow: Comparison between simulation and experiment

It has been recently shown experimentally by \O{}ren and Pinczewski [SPE Form. Eval. 7, 70 (1992); 9, 149 (1994)] that on the pore scale the presence of continuous wetting and spreading films is vitally important in describing the mobility of the various fluid phases in three-phase fluid flow. In this paper we put in place a numerical, pore-scale model to describe three-phase drainage dominated flow. Our model incorporates all the mechanisms observed in glass micromodel experiments. The pressures of all phases are calculated explicitly so that no ad hoc rules need be introduced to describe fluid flow. In doing this we are able to reproduce many of the experimentally observed characteristics of three-phase flow, i.e., multiple, simultaneous displacements, flow through thin wetting and spreading films, and double drainage events. The numerical simulations are carried out for a variety of wetting and spreading conditions, i.e., oil-wet, water-wet, positive spreading, and negative spreading, and in all cases we see good agreement with micromodel experiments.

Wettability heterogeneities in gas injection; experiments and modelling

We carried out experiments to study the effect of wettability heterogeneities on: (1) displacement mechanisms, (2) sweep efficiency and (3) trapped oil quantities and fluid distribution in three-phase gas injection and developed a theoretical simulator. Secondary gas injection experiments was conducted in transparent glass micromodels of heterogeneous wettability. Oil-wet patches in a water-wet matrix were obtained by selective silane grafting on the glass surface. Different heterogeneity patterns were considered for the same oil-wet over water-wet surface ratio. Displacement sequences were video-recorded and fluid saturations determined by image analysis. A theoretical model of three-phase flow in a porous structure was developed. In this model the porous medium is simulated as a network of interconnected pores. The model permits an imposition of heterogeneous wettability by assigning different water-oil contact angles according to the desired wettability pattern. The calculation of flow within the network takes into account the flow of oil through wetting and spreading films and the displacement mechanisms observed in the transparent micromodels. Comparison between experimental results and simulations shows that the size and distribution of wettability heterogeneities strongly affects microscopic and macroscopic behaviour during gas injection.

Phreatic explosion hazard assessment by numerical simulation

Phreatic explosions (or eruptions) represent very common and dangerous phenomena occurring in active volcanic and geothermal regions. In spite of this, the understanding of their dynamics and the assessment of their hazards are still scarcely known and quantified. In the present work we try to simulate the dynamics of a phreatic explosion by using a transient, two-dimensional, three-phase flow model. Simulations predict the effects of an explosion of a mixture of compressed gases and ground particles into the atmosphere. The governing transport equations of mass, momentum, and energy are solved by a finite difference multifield scheme for both solids and gas phases. We performed several simulations, assuming different initial conditions and system parameters. The model allows analysis of the timewise and spatial distribution of the groundhugging surge generated by the explosion and estimation of its potential hazards in terms of dynamic pressure, ash-in-air concentration, and temperature.

A New Model to Calculate Three-Phase Relative Permeabilities: Application and Validation for a Sandstone

Three-phase flow in a porous medium is usually described by the Stone model (1970). This model is based on two-phase data and relies on empirical correlations. It is valid only under strong water wettability and recognized as being a poor predictor. The goal of the present study is to develop a mathematical model for three-phase flow avoiding any empirical correlations. In this paper, only strong water-wet and spreading conditions are considered. However the model could be relatively easily extended to oil-wet or even mixed-wet conditions. The model is based on a physically relevant description of phase distribution and flow mechanisms at the pore scale. The porous medium is described as a set of fractal pores, whose linear fractal dimension and size distributions are derived from a mercury intrusion capillary pressure curve. The fluids are allowed to flow together in the same fractal pore, gas in the center, water near the walls and oil in an intermediate phase. The relative permeabilities are evaluated by calculating the flow of each fluid applying Poiseuille's law. The model results are compared to relative permeabilities obtained by history matching of gas injection experiments. The same experiments are also simulated using Stone's model and laboratory measured two-phase data.

Network Modeling of Three-Phase Flow in Porous Media

SummaryWe present a network model of three-phase flow in water-wet porous media. To explain the high oil recoveries in gas injection and gravity drainage experiments, we show that the mechanism for oil recovery is flow through connected oil layers in the pore space that are on the order of a micron thick. We then describe a simple model for the configuration of oil, water, and gas in a single pore and present an approximate expression for the conductance of oil layers. We use this expression to derive the oil relative permeability when flow is dominated by layer drainage. We show that for low oil and water saturations kro∼So2, consistent with the results of several experiments. To predict kro for the full range of oil saturation we use a capillary equilibrium-based network model that can simulate any sequence of oil, water, and gas injection. We introduce a self-consistency procedure to ensure that the correct sequence of saturation changes is used in the network model to compute relative permeability. We then present relative permeabilities and oil recoveries for gas injection into different initial oil saturations, and for waterflooding a reservoir containing gas and oil. We show that the relative permeabilities are strongly affected by the fluid properties and by the type of displacement process.

An Organizing Center for Wave Bifurcation in Multiphase Flow Models

We consider a one-parameter family of nonstrictly hyperbolic systems of conservation laws modeling three-phase flow in a porous medium. For a particular value of the parameter, the model has a shock wave solution that undergoes several bifurcations upon perturbation of its left and right states and the parameter. In this paper we use singularity theory and bifurcation theory of dynamical systems, including Melnikov's method, to find all nearby shock waves that are admissible according to the viscous profile criterion. We use these results to construct a unique solution of the Riemann problem for each left and right state and parameter value in a neighborhood of the unperturbed shock wave solution; together with previous numerical work, this construction completes the solution of the three-phase flow model. In the bifurcation analysis, the unperturbed shock wave acts as an organizing center for the waves appearing in Riemann solutions.

深海底鉱物揚鉱用エアリフトポンプの流動特性に及ぼす諸因子の影響

The purposes of this study are to obtain designing data and safety operational conditions for air-lift pumps of lifting marine mineral resources such as maganese nodules from deep-sea floor of about 5, 000m to a mining ship. A large amount of air is absorbed into water, because the air is injected in a deeper position of conveying pipe at high pressure; therefore, the numerical procedure based on the drift flux model for steady gas-liquidsolid three-phase flow takes account of the mass transfer between gas-phase and liquid-phase.In this paper, we discuss effects of several factors, such as coefficients of the constitutive equations; air absorption into water; inner diameter, wall roughness and length of each section of conveying pipe; pressure drop coefficients; physical properties and delivered concentration of solid particles on flow characteristics. The results of this numerical simulation serve a basic performance of a large scale air-lift pump.

Pore-scale network model for drainage-dominated three-phase flow in porous media

Drainage displacements in three-phase flow under strongly wetting conditions are completely described by a simple generalisation of well understood two-phase drainage mechanisms. As in two-phase flow, the sequence of throat invasions in three-phase flow is determined by fluid connectivity and threshold capillary pressure for the invading interface. Flow through wetting and intermediate spreading films is important in determining fluid recoveries and the progress of the displacement in three-phase flow. Viscous pressure drops associated with flow through films give rise to multiple filling and emptying of pores. A three-phase, two-dimensional network model based on the pore-scale fluid distributions and displacement mechanisms reported by Oren et al. and which accounts for flow through both wetting and intermediate fluid films is shown to correctly predict all the important characteristics of three-phase flow observed in glass micromodel experiments.

Simulation of immiscible multiphase flow in porous media: A focus on the capillary fringe of oil-contaminated aquifers

This paper deals with the analysis of some aspects of the vertical and lateral migration of oil spills in the unsaturated and the capillary zone of a phreatic aquifer. Our motivation stems from the fact that such contamination represents a severe danger for ground-water resources all over the world and from the present acute problem of jet-fuel contamination in some location of Israel. In the present study, we shall focus our efforts on the analysis of the upper layers of the aquifer which are often subjected to the most significant oil contamination. Neglecting coupled processes effects such as dilution, adsorption and volatilization, also adopting Richard's assumption, a three-phase flow model is introduced with capillary heads of the water and the oil as variables. The resulting model which is coupled and strongly non-linear is solved using a vertical two-dimensional Finite-Element procedure together with a quasi-Newton optimization algorithm. Applying that scheme, various scenarios of oil migration in the unsaturated and the capillary zone were simulated. Some migration characteristics prediced by the numerical simulations are discussed. In particular, the dynamics of the water and oil phases during the migration process is discussed.

Effect of subsurface heterogeneity on free-product recovery from unconfined aquifers

Free-product record system designs for light-hydrocarbon-contaminated sites were investigated to evaluate the effects of subsurface heterogeneity using a vertically integrated three-phase flow model. The input stochastic variable of the areal flow analysis was the log-intrinsic permeability and it was generated using the Turning Band method. The results of a series of hypothetical field-scale simulations showed that subsurface heterogeneity has a substantial effect on free-product recovery predictions. As the heterogeneity increased, the recoverable oil volume decreased and the residual trapped oil volume increased. As the subsurface anisotropy increased, these effects together with free- and total-oil contaminated areas were further enhanced. The use of multiple-stage water pumping was found to be insignificant compared to steady uniform pumping due to reduced recovery efficiency and increased residual oil volume. This observation was opposite to that produced under homogeneous scenarios. The effect of subsurface heterogeneity was enhanced at relatively low water pumping rates. The difference in results produced by homogeneous and heterogeneous simulations was substantial, indicating greater attention should be paid in modeling free-product recovery systems with appropriate subsurface heterogeneity.

エアリフトポンプの揚固特性に関する数値解析

The purposes of this work are to develop a method of numerical simulation for gas-liquid-solid threephase flow systems and to confirm a method of estimating performance of air-lift pump for deep sea mining. Finite difference equations for the numerical analysis were derived by integration of the partial differential equations based on the drift flux model for three-phase flow with control volumes.The method of the numerical procedure is based on the so-called SIMPLER algorithm proposed by Patankar.To prove the present numerical procedure, calculations were compared with experimental resultswhich had been obtained with the relatively large air-lift pump by Saito et al. and Weber et al. The present method gave good predictions.

Finite element modelling of three-phase flow in deforming saturated oil reservoirs : Lewis, R W; Sukirman, Y Int J Num Anal Meth GeomechV17, N8, Aug 1993, P577–598

Finite element modelling of three-phase flow in deforming saturated oil reservoirs : Lewis, R W; Sukirman, Y Int J Num Anal Meth GeomechV17, N8, Aug 1993, P577–598

Global behavior of compressible three-phase flow in heterogeneous porous media

The homogenization method is used to analyze the equivalent behavior of a compressible three-phase flow model in heterogeneous porous media with periodic microstructure, including capillary effects. Asymptotic expansions lead to the definition of a global or effective model of an equivalent homogeneous reservoir. The resulting equations are of the same type as the points equations, with effective coefficients. The method allows the determination of these effective coefficients from a knowledge of the geometrical structure of the basic cell and its heterogeneities. Numerical computations to obtain the homogenized coefficients of the entire reservoir have been carried out via a finite element method.

Stable hyperbolic singularities for three-phase flow models in oil reservoir simulation

We study the characteristic speeds of systems of two conservation laws representing three-phase flow in a porous medium with gravity taken into account. Generically hyperbolicity fails on open regions (elliptic regions) where the characteristic speeds assume complex values. The presence of such regions creates difficulties such as multiple solutions which indicate a modeling problem, according to some authors. The hyperbolicity of the models we study depends on the relative permeability functions. It is customary in oil engineering studies to suppose that the water and gas permeabilities depend only on their respective saturation, while the oil relative permeability changes with the gas and water saturations. Such a hypothesis on the oil relative permeability generically leads to elliptic regions. We define a set of three curves that surround elliptic regions of any model. By studying these curves, we indicate a procedure to locate the singularities and prove that for any choice of gravitational and viscosity parameters such regions shrinks to points where the characteristic speeds are real and equal, provided it is assumed that each relative permeability depends on its respective saturation only. Our results, together with a paper of Trangenstein, lead to the conclusion that in order to insure real wave speeds, such an assumption is necessary and sufficient when gravitational effects are considered in three-phase models.

Loss of real characteristics for models of three-phase flow in a porous medium

In this paper we examine the generalized Buckley-Leverett equations governing threephase immiscible, incompressible flow in a porous medium, in the absence of gravitational and diffusive/dispersive effects. We consider the effect of the relative permeability models on the characteristic speeds in the flow. Using a simple idea from projective geometry, we show that under reasonable assumptions on the relative permeabilities there must be at least one point in the saturation triangle at which the characteristic speeds are equal. In general, there is a small region in the saturation triangle where the characteristic speeds are complex. This is demonstrated with the numerical results at the end of the paper.