Method For Multiphase Flows Research Articles

Adaptive Multiscale Finite-Volume Method for Multiphase Flow and Transport in Porous Media

We present a multiscale finite-volume (MSFV) method for multiphase flow and transport in heterogeneous porous media. The approach extends our recently developed MSFV method for single-phase flow. We use a sequential scheme that deals with flow (i.e., pressure and total velocity) and transport (i.e., saturation) separately and differently. For the flow problem, we employ two different sets of basis functions for the reconstruction of a conservative fine-scale total velocity field. Our basis functions are designed to have local support, and that allows for adaptive computation of the flow field. We use a criterion based on the time change of the total mobility field to decide when and where to recompute our basis functions. We show that at a given time step, only a small fraction of the basis functions needs to be recomputed. Numerical experiments of difficult two-dimensional and three-dimensional test cases demonstrate the accuracy, computational efficiency, and overall scalability of the method.

A projection scheme for incompressible multiphase flow using adaptive Eulerian grid: 3D validation

A three-dimensional finite element method for incompressible multiphase flows with capillary interfaces is developed based on a (formally) second-order projection scheme. The discretization is on a fixed (Eulerian) reference grid with an edge-based local h-refinement in the neighbourhood of the interfaces. The fluid phases are identified and advected using the level-set function. The reference grid is then temporarily reconnected around the interface to maintain optimal interpolations accounting for the singularities of the primary variables. Using a time splitting procedure, the convection substep is integrated with an explicit scheme. The remaining generalized Stokes problem is solved by means of a pressure-stabilized projection. This method is simple and efficient, as demonstrated by a wide range of difficult free-surface validation problems, considered in the paper. Copyright © 2005 John Wiley & Sons, Ltd.

NUMERICAL PREDICTION FOR INCIPIENT MOTION OF BODIES UNDER SHIELDING EFFECTS WITH MICS

The applicability of the computational method for multiphase flows, MICS (Multiphase Incompressible flow solver with Collocated grid System), is investigated for the incipient motion of a cylinder under the shielding effects in a duct flow. In order to understand the actual phenomena, hydraulic experiments are conducted with two cylinders, one fixed and another movable. In the experiments, the distance of the two cylinders are widely changed and critical flow rates that rotate the movable cylinder are measured. As a result, resistance force and sheltering coefficients are derived and the defects of the usual evaluation method to derive these values are pointed out. The incipient motion of the cylinder is numerically investigated with MICS. As a result of the computations, it is shown that the behaviors of the movable cylinder are reasonably predicted with MICS, while MICS needs no empirical coefficients regarding the resistance force and sheltering effects.

Use of slopelimiter techniques in traditional numerical methods for multi-phase flow in pipelines and wells

The aim of this paper is to show how simple and traditional methods for simulating multi-phase flow can be improved by introducing higher order accuracy. Numerical diffusion is reduced to a minimum by using slopelimiter techniques, and better predictions of flow rates and pressure are obtained. Slopelimiter techniques, originally developed to achieve higher order of accuracy in Godunov's method, is applied to a method following a finite element approach and a predictor–corrector shooting technique. These methods are tested and compared to a Godunov-type scheme recently developed for multi-phase flow. Implementation of Godunov-type schemes for multi-phase flow tends to be a complicated and challenging task. Introducing the slopelimiter techniques in the finite element approach and the predictor–corrector shooting technique is however simple, and provides an overall reduction of the numerical diffusion. The focus is on using these techniques to improve the mass transport description, since this is the main concern in the applications needed. The presented schemes represent different semi-implicit approaches for simulating multi-phase flow. An evaluation of higher order extensions, as well as a comparison by itself, is of large interest. We present a model for two-phase flow in pipelines and wells, and an outline of the numerical methods and the extensions to second order spatial accuracy. Several examples motivated by applications in underbalanced drilling are presented, and the advantages of using higher order schemes are illustrated. Copyright © 2005 John Wiley & Sons, Ltd.

A multi-phase flow method with a fast, geometry-based fluid indicator

We present a novel methodology for incompressible multi-phase flow simulations in which the fluid indicator is a local signed distance (level set) function, and front-tracking is used to evaluate accurately geometric interfacial quantities and forces. Employing ideas from Computational Geometry, we propose a procedure in which the level set function is obtained at optimal computational cost without having to solve the level set equation and its associated re-initialization. This new approach is robust and yields an accurate and sharp definition of the distinct bulk phases at all times, irrespective of the geometric complexity of the interfaces. We illustrate the proposed methodology with an example of surface tension-mediated Kelvin–Helmholtz instability.

A projection scheme for incompressible multiphase flow using adaptive Eulerian grid

AbstractThis paper presents a finite element method for incompressible multiphase flows with capillary interfaces based on a (formally) second‐order projection scheme. The discretization is on a fixed Eulerian grid. The fluid phases are identified and advected using a level set function. The grid is temporarily adapted around the interfaces in order to maintain optimal interpolations accounting for the pressure jump and the discontinuity of the normal velocity derivatives. The least‐squares method for computing the curvature is used, combined with piecewise linear approximation to the interface. The time integration is based on a formally second order splitting scheme. The convection substep is integrated over an Eulerian grid using an explicit scheme. The remaining generalized Stokes problem is solved by means of a formally second order pressure‐stabilized projection scheme. The pressure boundary condition on the free interface is imposed in a strong form (pointwise) at the pressure‐computation substep. This allows capturing significant pressure jumps across the interface without creating spurious instabilities. This method is simple and efficient, as demonstrated by the numerical experiments on a wide range of free‐surface problems. Copyright © 2004 John Wiley & Sons, Ltd.

A numerical study of the statistics of a two-dimensional Rayleigh–Taylor mixing layer

A two-dimensional Rayleigh–Taylor mixing layer was numerically simulated to determine the growth rate of the mixing layer for an ensemble of initial conditions. The numerical algorithm used is a recently developed lattice Boltzmann method for multiphase flow. Small variations in the initial conditions of the fluid interface are observed to yield large variations in the growth rate of the mixing-layer width. Consequently, an ensemble of simulations has been generated to provide a statistical basis for determining the mixing-layer growth rate. The results for this ensemble indicate that the mixing-layer attains an approximate state of self-similarity and yields a distribution of mixing-layer growth rates.

７．流体力学 非圧縮性混相流場に対する数値解法と水‐シリコンオイル成層流への適用性に関する考察

This paper describes a computational method for incompressible multiphase flows, consisting of immiscible two fluids with different density. A robust unified procedure proposed in this paper enables us to deal with the two-fluid system as a single-phase flow with a density distribution. In this method, consideration was given to the treatment of the pressure Poisson equation as well as the numerical accuracy of the convection terms in the Navier-Stokes equations, and to the advection method of the normalized density. This method was applied to the stratified two-fluid flows arising in a cavity. As a result, it was shown that the flow patterns and the profiles of the interface between two fluids are reasonably predicted with the computational method.

An efficient parallel-computing method for modeling nonisothermal multiphase flow and multicomponent transport in porous and fractured media

This paper presents an efficient massively parallel scheme for modeling large-scale multiphase flow, multicomponent transport and heat transfer in porous and fractured reservoirs. The new scheme is implemented into a parallel version of the TOUGH2 code and the numerical performance is tested on a Cray T3E-900 (a distributed-memory parallel computer with 692 processors) and IBM RS/6000 SP (a distributed-memory parallel computer with 3328 processors). The efficiency and robustness of the parallel-computing algorithm are demonstrated by completing three over-million-cell simulations using site-specific data for site characterization. The first application is the development of a three-dimensional unsaturated zone numerical model simulating flow of moisture in the unsaturated zone of Yucca Mountain, Nevada; the second problem is the modeling of flow of moisture, gas, and heat at the same site. The third application is the study of flow-focusing phenomena through fractures for the same site. Simulation results show that the parallel-computing technique enhances modeling capability by several orders of magnitude in speedup of computing times for large-scale modeling studies.

Application of Multiphase Flow Methods to Horizontal Underbalanced Drilling

Abstract Multiphase flow can be present in all aspects of underbalanced drilling. This paper outlines the ways in which multiphase flow pressure loss calculations can be used by drilling engineers in the design and optimization of underbalanced drilling operations. Then, detailed field measurements for several horizontal wells drilled underbalanced with coiled tubing are used to evaluate the application of existing pressure loss calculation methods to this unique application. Introduction Underbalanced drilling (UBD) is rapidly gaining popularity in the oil and gas industry. The Alberta Energy and Utilities Board ID94–3(1) defines underbalanced drilling as follows: "When the hydrostatic head of a drilling fluid is intentionally designed to be lower than the pressure of the formation being drilled, the operation will be considered underbalanced drilling. The hydrostatic head of the drilling fluid may be naturally less than the formation pressure or it can be induced. The induced state may be created by adding natural gas, nitrogen, or air to the liquid phase of the drilling fluid. Whether induced or natural, this may result in an influx of formation fluid which must be circulated from the well and controlled at surface." Benefits of Underbalanced Drilling Maintaining the pressure in the wellbore below the reservoir pressure allows reservoir fluids to enter the wellbore while UBD operations proceed, thus preventing flow of drilling fluids (and associated solids) into the formation, thereby minimizing or even eliminating formation damage. This is of particular importance in the drilling of horizontal wells as the formation is exposed to the drilling fluids for an extended period of time. Although formation damage reduction is the most widely recognized benefit of underbalanced drilling, several additional benefits are outlined below: Increased Penetration Rates Underbalanced drilling can achieve higher rates of penetration due to reduced "chip holdown" and decreased hydrostatic pressure at the bit face. Reduction in "chip holdown" refers to easier removal of drilled solids from the vicinity of the drill bit due to the flow of drilling and reservoir fluids, thus allowing the bit to drill into fresh rock continuously. The decreased hydrostatic pressure at the bit face reduces stress in the rock being drilled, allowing it to fail more easily. The experience of drilling engineers(2) familiar with UBD is that the rate of penetration can be increased by between three and ten times that of conventional drilling. Minimal Lost Circulation Underbalanced drilling gives better control in situations where fractured, low pressure, or high permeability formations may lead to the loss of drilling fluids and the associated problems that can cause. Evaluation While Drilling Data acquired in real time during underbalanced drilling operations allow for both the short term on site optimization of the UBD operation, and the longer term assessment of the well's potential. On site, the data acquired can be used to optimize drilling parameters such as the well's horizontal length, vertical depth, and orientation. Other data that can be obtained, useful both on site and long term, include fluid properties, productivity, and geological interpretations of the formation.

A Galilean invariant model of the lattice Boltzmann method for multiphase fluid flows using free-energy approach

A Galilean invariant model of the lattice Boltzmann method (LBM) for multiphase fluid flows using free-energy approach is proposed. The asymptotic theory proposed by Sone (1971) is applied to the LBM model for multiphase fluid flows using the free-energy approach developed by Swift et al. [Phys. Rev. Lett. 75 (1995) 830] and governing equations for macroscopic variables of the model are obtained. From the governing equations the condition of a Galilean invariant LBM model is identified. In two problems, moving droplet and drop deformation and breakup in a shear flow, Galilean invariance of the proposed model is demonstrated.

M-Matrix Flux Splitting for General Full Tensor Discretization Operators on Structured and Unstructured Grids

Diagonal tensor flux approximations are commonly used in fluid dynamics. This approximation introduces an O(1) error in flux whenever the coordinate system is nonaligned with the principal axes of the tensor which is particularly common when employing curvilinear gridding. In general a consistent full tensor flux approximation leads to a significant increase in support and consequent size of the Jacobian matrix. After decomposition of a general full tensor flux into a diagonal tensor flux together with cross terms, time-split semi-implicit, stable, full tensor flux approximations are introduced with in a general finite volume formalism, enabling the standard diagonal tensor Jacobian matrix structure to be retained for single phase flow, IMPES, and standard block fully implicit formulations while ensuring spatial consistency of the discretization for both structured and unstructured grids. Stability of the scheme is proven for constant elliptic coefficients. The results presented demonstrate the benefits of the method for multiphase flow within a fully implicit framework on structured and unstructured grids.

The Ghost Fluid Method for Deflagration and Detonation Discontinuities

The level set method for multiphase compressible flows is simple to implement, especially in the presence of topological changes. However, this method was shown to suffer from large spurious oscillations. A new Ghost Fluid Method (GFM) was developed to remove these spurious oscillations by minimizing the numerical smearing in the entropy field with the help of an Isobaric Fix technique. The GFM was designed for traditional contact discontinuities where the interface moves with the fluid velocity only. In this paper, the GFM is extended to treat multimaterial interfaces where the interface velocity includes a regression rate due to the presence of chemical reactions converting one material into another. Specifically, interface models for deflagration and detonation discontinuities are considered. The resulting numerical method is robust and easy to implement.

Quasi–incompressible Cahn–Hilliard fluids and topological transitions

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

Computational methods for multiphase flow and reactive transport problems arising in subsurface contaminant remediation

A mathematical formulation and some numerical approximation techniques are described for a system of coupled partial differential and algebraic equations describing multiphase flow, transport and interactions of chemical species in the subsurface. A parallel simulator PARSIM has been developed based on these approximation techniques and is being used to study contaminant remediation strategies. Numerical results for a highly complex geochemistry problem involving strontium disposal in a pit at Oak Ridge National Laboratory are presented.

Correction to “An efficient finite element method for modeling multiphase flow” by J. J. Kaluarachchi and J. C. Parker

In the paper An efficient finite element method for modeling multiphase by J. J. Kaluarachchi and J. C. Parker (Water Resources Research, 25(1), 43-54, 1989), a finite element model for water and oil flow with gas at constant pressure was presented. The governing equations were expressed in terms of water and oil pressures with mass storage t rms expanded as the products of pressure-time and saturation-pressure derivatives. The split time derivative formulation used by Kaluarachchi and Parker [1989] has the

Flow through emergency relief devices and reaction forces

The choice of calculation method for single-phase liquid or vapour flow through emergency pressure relief device systems is quite straightforward and well-defined. By contrast, the choice among the many methods for multiphase flow requires proper anticipation and treatment of the fluid and relief device behaviour in the particular piping system. Computational models of various degrees of applicability have been developed for most types of fluids and installations under the usual emergency pressure relief conditions. Reaction forces are adequately defined for known flow rates of the phases.

Design of Offshore Gas Pipelines Accounting For Two-Phase Flow (includes associated papers 6401 and 6402 )

In the past, offshore gas pipelines with simultaneous flow of liquids havebeen undersized. Many offshore designs have been performed using thePanhandle flow equation with an efficiency factor that is adjusted with gasflow rate. Correlations are used to show that efficiency does not changewith flow rate for horizontal lines, but does change with liquid loadingand underwater terrain. Introduction This paper describes new techniques and conceptsrequired to obtain the most efficient design of offshoregas pipelines with simultaneous flow of liquids. Recentdevelopments in multiphase flow methods can besuccessfully applied to "wet gas" design problems.In the past, offshore gas pipelines were designedwithout cognizance of several important problems.The first problem is the effect of liquid loading ofthe gas stream. The liquid may be introduced into theflow stream from several sources, but in almost all cases, offshore gas pipelines are transporting a wetgas. Even though operators may consider the line dry, there is usually some condensation of either water orhydrocarbon components as the gas is moved towardshore. The most prevailing source of liquids is fromthe formation. The formation will tend to graduallydry itself out around the wellbore over a period oftime by producing small amounts of connate fluidfrom even so-called dry gas pools. In addition, waterconing and retrograde condensation of hydrocarbonsin the reservoir may also account for liquidproduction along with the gas.Substantial liquid production is usually separatedfrom the gas stream at the platform. However, lowrates of liquid production are usually passed directlyinto the pipeline to shore. Indeed, many pipeline contracts are written with the assumption that a certainliquid volume will be moved along with the gas. Thesimultaneous flow of gas with small liquid loadingsis an accepted practice. However, a problem arisesin the initial design of these pipelines becausesingle-phase gas flow equations are used. Even the bestdesigns merely adjust the "flow efficiently" to accountfor the presence of liquid in a gas line.The second problem often neglected in offshorepipeline design is that of inclination. When gaspipelines are viewed as dry systems, it is perfectlyreasonable to neglect the change of elevation since it usuallyis no more than several hundred feet. However, theeffects of inclination, and specifically terrain, arecrucial in determining the performance of a wetsystem; i.e., gas and liquid flowing simultaneously. Ina wet system that is inclined upward in the directionof flow, the liquid will tend to accumulate and let the gas "slip" by. This concept of slip implies that gas istraveling at a different velocity than the liquid in thepipe. The degree of slip or "liquid holdup" is afunction of diameter, flow rates, flow regimes, and other variables. The magnitude of this liquid holdup hasbeen shown by Beggs and Brill and others to be afunction of the inclination of the system. Objectives The first objective is to show the relationship betweenmultiphase flow models and the single-phase gasequations used in current practice. Although there aremany single-phase gas-flow formulas available in the literature this paper will concentrate on thePanhandle A and modified Panhandle equations. JPT P. 366^