Three-phase Flow Problem Research Articles

A new calculation method and model of hydrate slurry flow of the multiphase pipeline in deep water gas field

The transportation environment of multiphase pipelines in deepwater gas fields is likely to cause the problem of hydrate flow safety. Aiming at the gas-liquid-solid three-phase flow problem containing hydrate, a hydrate slurry flow model for a multiphase pipeline in a deep-water gas field was developed based on the two-fluid model coupled with the population balance equation, and a mesh was carried out for the axial distance of pipeline and size difference of hydrate particles to obtain a transient solution. The multiphase flow characteristics of the pipeline are simulated upon engineering scale, the transient flow characteristics of gas and liquid phases, the aggregation and fragmentation characteristics of hydrate particles, and the coupling interaction between them were studied, which lays the foundation for further research on the characteristics of hydrate particle deposition and blockage based on particle size distribution. This study provides support for the safety of hydrate flow in multiphase pipelines in deepwater gas fields.

Existence and convergence of a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media

Abstract This paper presents and analyzes a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media. We use a first-order time extrapolation, which allows us to solve the equations implicitly and sequentially. We show that the discrete problem is well posed, and obtain a priori error estimates. Our numerical results validate the theoretical results, i.e., the algorithm converges with first order.

Improved IMPES Scheme for the Simulation of Incompressible Three-Phase Flows in Subsurface Porous Media

In this work, an improved IMplicit Pressure and Explicit Saturation (IMPES) scheme is proposed to solve the coupled partial differential equations to simulate the three-phase flows in subsurface porous media. This scheme is the first IMPES algorithm for the three-phase flow problem that is locally mass conservative for all phases. The key technique of this novel scheme relies on a new formulation of the discrete pressure equation. Different from the conventional scheme, the discrete pressure equation in this work is obtained by adding together the discrete conservation equations of all phases, thus ensuring the consistency of the pressure equation with the three saturation equations at the discrete level. This consistency is important, but unfortunately it is not satisfied in the conventional IMPES schemes. In this paper, we address and fix an undesired and well-known consequence of this inconsistency in the conventional IMPES in that the computed saturations are conservative only for two phases in three-phase flows, but not for all three phases. Compared with the standard IMPES scheme, the improved IMPES scheme has the following advantages: firstly, the mass conservation of all the phases is preserved both locally and globally; secondly, it is unbiased toward all phases, i.e., no reference phases need to be chosen; thirdly, the upwind scheme is applied to the saturation of all phases instead of only the referenced phases; fourthly, numerical stability is greatly improved because of phase-wise conservation and unbiased treatment. Numerical experiments are also carried out to demonstrate the strength of the improved IMPES scheme.

A continuum model of multi-phase reactive transport in igneous systems

SUMMARY Multiphase reactive transport processes are ubiquitous in igneous systems. A challenging aspect of modelling igneous phenomena is that they range from solid-dominated porous to liquid-dominated suspension flows and therefore entail a wide spectrum of rheological conditions, flow speeds and length scales. Most previous models have been restricted to the two-phase limits of porous melt transport in deforming, partially molten rock and crystal settling in convecting magma bodies. The goal of this paper is to develop a framework that can capture igneous system from source to surface at all phase proportions including not only rock and melt but also an exsolved volatile phase. Here, we derive an n-phase reactive transport model building on the concepts of Mixture Theory, along with principles of Rational Thermodynamics and procedures of Non-equilibrium Thermodynamics. Our model operates at the macroscopic system scale and requires constitutive relations for fluxes within and transfers between phases, which are the processes that together give rise to reactive transport phenomena. We introduce a phase- and process-wise symmetrical formulation for fluxes and transfers of entropy, mass, momentum and volume, and propose phenomenological coefficient closures that determine how fluxes and transfers respond to mechanical and thermodynamic forces. Finally, we demonstrate that the known limits of two-phase porous and suspension flow emerge as special cases of our general model and discuss some ramifications for modelling pertinent two- and three-phase flow problems in igneous systems.

Solution of Three-Phase Flow Problems in Three-Dimension Reservoir with Nonlinear Filtration Law

The work is devoted to the methods of solving three-phase fluid flow problems in the reservoirs. It is assumed that in domain with high velocities Darcy’s law is violated and a nonlinear filtration law is used. The methods for solving three-phase flow problems in porous media with a nonlinear filtration law based on the decomposition methods are proposed. The proposed methods are implemented on the heterogeneous computing systems.

A numerical model for multiphase liquid–vapor–gas flows with interfaces and cavitation

We are interested in multiphase flows involving the liquid and vapor phases of one species and a third inert gaseous phase. We describe these flows by a hyperbolic single-velocity multiphase flow model composed of the phasic mass and total energy equations, the volume fraction equations, and the mixture momentum equation. The model includes stiff mechanical and thermal relaxation source terms for all the phases, and chemical relaxation terms to describe mass transfer between the liquid and vapor phases of the species that may undergo transition. First, we present an analysis of the characteristic wave speeds associated to the hierarchy of relaxed multiphase models corresponding to different levels of activation of infinitely fast relaxation processes, showing that sub-characteristic conditions hold. We then propose a mixture-energy-consistent finite volume method for the numerical solution of the multiphase model system. The homogeneous portion of the equations is solved numerically via a second-order wave propagation scheme based on robust HLLC-type Riemann solvers. Stiff relaxation source terms are treated by efficient numerical procedures that exploit algebraic equilibrium conditions for the relaxed states. We present numerical results for several three-phase flow problems, including two-dimensional simulations of liquid–vapor–gas flows with interfaces and cavitation phenomena.

Three-phase fluid displacements in a porous medium

Oil in a reservoir is usually found together with water and gas. Often a mixture of water and gas is used to displace such oil. In this work, we present the Riemann solution for such three-phase flow problem. This solution encodes the dependence of recovery on the injected proportion, the proportion initially present, and the viscosity of the several fluids. We use the wave curve method to determine the Riemann solution for initial and injection data in the above-mentioned class. We verify the [Formula: see text]-stability of the Riemann solution with variation of data. We do not establish uniqueness of the Riemann solution, but we believe that it is valid.

Newton Power Flow Methods for Unbalanced Three-Phase Distribution Networks

Two mismatch functions (power or current) and three coordinates (polar, Cartesian and complex form) result in six versions of the Newton–Raphson method for the solution of power flow problems. In this paper, five new versions of the Newton power flow method developed for single-phase problems in our previous paper are extended to three-phase power flow problems. Mathematical models of the load, load connection, transformer, and distributed generation (DG) are presented. A three-phase power flow formulation is described for both power and current mismatch functions. Extended versions of the Newton power flow method are compared with the backward-forward sweep-based algorithm. Furthermore, the convergence behavior for different loading conditions, R / X ratios, and load models, is investigated by numerical experiments on balanced and unbalanced distribution networks. On the basis of these experiments, we conclude that two versions using the current mismatch function in polar and Cartesian coordinates perform the best for both balanced and unbalanced distribution networks.

A new finite volume approach for transport models and related applications with balancing source terms

We develop a new finite volume scheme for numerically solving transport models associated with hyperbolic problems and balance laws. The numerical scheme is obtained via a Lagrangian–Eulerian approach that retains the fundamental principle of conservation of the governing equations as it is linked to the classical finite volume framework. As features of the novel algorithm we highlight: the new scheme is locally conservative in balancing the flux and source term gradients and preserves a component-wise structure at a discrete level for systems of equations. The novel approach is applied to several nontrivial examples to evidence that we are calculating the correct qualitatively good solutions with accurate resolution of small perturbations around the stationary solution. We discuss applications of the new method to classical and nonclassical nonlinear hyperbolic conservation and balance laws such as the classical inviscid Burgers equation, two-phase and three-phase flow problems in porous media as well as numerical experiments for nonlinear shallow water equations with friction terms. In addition, we consider the case of the source term which is discontinuous as a function of space x. We also extend the Lagrangian–Eulerian framework to the two-dimensional scalar conservation law, along with pertinent numerical experiments to show the performance of the new method.

Lattice Boltzmann modeling of three-phase incompressible flows.

In this paper, based on multicomponent phase-field theory we intend to develop an efficient lattice Boltzmann (LB) model for simulating three-phase incompressible flows. In this model, two LB equations are used to capture the interfaces among three different fluids, and another LB equation is adopted to solve the flow field, where a new distribution function for the forcing term is delicately designed. Different from previous multiphase LB models, the interfacial force is not used in the computation of fluid velocity, which is more reasonable from the perspective of the multiscale analysis. As a result, the computation of fluid velocity can be much simpler. Through the Chapman-Enskog analysis, it is shown that the present model can recover exactly the physical formulations for the three-phase system. Numerical simulations of extensive examples including two circular interfaces, ternary spinodal decomposition, spreading of a liquid lens, and Kelvin-Helmholtz instability are conducted to test the model. It is found that the present model can capture accurate interfaces among three different fluids, which is attributed to its algebraical and dynamical consistency properties with the two-component model. Furthermore, the numerical results of three-phase flows agree well with the theoretical results or some available data, which demonstrates that the present LB model is a reliable and efficient method for simulating three-phase flow problems.

Hydrodynamic Simulation and Optimization of an Oil Skimmer

Oil spills can cause severe environmental damage. In situ burning or chemical dispersant methods can be used in many situations; however, these methods can be highly toxic and fail in slightly rough seas. Oil recovery techniques have also been developed to recover oil using skimmer equipment installed in ships. The challenges arise when a vessel is operated in heavy sea and current conditions. An oil skimmer has recently been developed by Extreme Spill Technology (EST) Inc. for automated oil recovery using a vacuum device installed in a vessel. Initial tests have shown that the prototype vessel is efficient in oil recovery. This paper presents the numerical and experimental studies of the hydrodynamic performance of the vacuum tower installed in the oil skimmer developed by EST. While the principle of the vacuum mechanism for oil skimming is simple, the hydrodynamic aspects of the recovery process is very complicated since it involves multiphase and multiscale moving interfaces, including oil, water, atmospheric air, and attenuate compressible air on the top part of the vacuum tower, and moving interface of oil slick, oil droplets, and air bubbles of different scales. The recovery process was simplified into a three-phase flow problem involving oil, water, and air and was simulated by using a computational fluid dynamics (CFD) method. The volume of fluid (VOF) method was employed to capture the moving surfaces between the fluid phases. Model tests were carried out to simulate the oil recovery process and for validation studies. Numerical results were compared with the experimental data. Studies were also extended to optimize the geometry of the tower for maximum oil recovery.

Three-phase power flow analysis in sequence component frame using Hybrid Particle Swarm Optimization

Power flow is an important tool in power system studies. Usually, a power system is assumed to be balanced and the power flow analysis is carried out for the same. Practical power system networks are highly unbalanced and it is required to obtain load flow (power flow) solution for such unbalanced networks. The unbalancing in power system is unavoidable due to the untransposed transmission lines and imbalance in loading. Hence, a power flow analysis dealing with three-phase configuration of power systems is necessary for proper planning and stability studies. The paper presents an algorithm for solving the three-phase load flow problem in symmetrical component frame using Hybrid Particle Swarm Optimization (HPSO) technique. A sequence coupled transformer model and a sequence line model are used. The objective function is formulated using the sequence models of three-phase power system and is solved using Hybrid PSO. The proposed method eliminates the formation of Jacobian matrix and its inversion and thus reduces the complexity of the problem. Ill-conditioned power system problems can also be solved by this method. Without any modification, this algorithm can also be applied to obtain load flow solution of the balanced networks. The validity of the proposed method is verified by a numerical example.

Stochastic analysis of transient three-phase flow in heterogeneous porous media

In this manuscript, we extend the stochastic analysis of transient two-phase flow (Chen et al., Water Resour Res 42:W03425, 2006) to three-phase flow, i.e., water, air, and NAPL. We use the van Genuchten model and the Parker and Lenhard three-phase model to describe the relationships between phase saturation, phase relative permeability, and capillary pressure. The log-transformations of intrinsic permeability Y(x) = ln k(x), soil pore size distribution parameter β ow (x) = ln α ow (x) between water and NAPL, and β ao (x) = ln α ao (x) between air and NAPL, and van Genuchten fitting parameter $$\bar{n}({\bf x}) = \ln {\left[{n{\left({\bf x} \right)} - 1} \right]},$$ are treated as stochastic variables that are normally distributed with a separable exponential covariance model. The Karhunen–Loeve expansion and perturbation method (KLME) is used to solve the resulting equations. We evaluate the stochastic model using two-dimensional examples of three-phase flow with NAPL leakage. We also conduct Monte Carlo (MC) simulations to verify the stochastic model. A comparison of results from MC and KLME indicates the validity of the proposed KLME application in three-phase flow. The computational efficiency of the KLME approach over MC methods is at least an order of magnitude for three-phase flow problems. This verified stochastic model is then used to investigate the sensitivity of fluid saturation variances to the input variances.

A Comparison of Techniques for Coupling Porous Flow and Geomechanics

Summary This paper compares three techniques for coupling multiphase porous flow and geomechanics. Sample simulations are presented to highlight the similarities and differences in the techniques. One technique uses an explicit algorithm to couple porous flow and displacements in which flow calculations are performed every timestep and displacements are calculated only during selected timesteps. A second technique uses an iteratively coupled algorithm in which flow calculations and displacement calculations are performed sequentially for the nonlinear iterations during each timestep. The third technique uses a fully coupled approach in which the program's linear solver must solve simultaneously for fluid flow variables and displacement variables. The techniques for coupling porous flow with displacements are described and comparison problems are presented for single-phase and three-phase flow problems involving poroelastic deformations. All problems in this paper are described in detail, so the results presented here may be used for comparison with other geomechanical/porous-flow simulators.

Simultaneous Estimation of Absolute and Relative Permeability by Automatic History Matching of Three-Phase Flow Production Data

Abstract This paper focuses on the simultaneous estimation of the absolute permeability field and relative permeability curves from three-phase flow production data. Irreducible water saturation, critical gas saturation, and residual oil saturations are assumed to be known. The two-phase relative permeability curves for an oil-gas system and the two-phase relative permeability curves for an oil-water system are represented by power law models. The three-phase oil relative permeability curve is calculated from the two sets of two-phase curves using Stone's Model II. The adjoint method is applied to three-phase flow problems to calculate the sensitivity of production data to the absolute permeability field and the parameters defining the relative permeability functions. Using the calculated sensitivity coefficients, absolute permeability, and relative permeability fields are estimated by automatic history matching of production data. Introduction The main objective of this paper is to consider the feasibility of estimating absolute permeability fields and parameters that define relative permeability functions by automatic history matching of production data obtained under multiphase flow conditions. While the topic is not new, to the best of our knowledge, no paper in the petroleum engineering literature has considered this problemunder three-phase flow conditions. It appears that Archer and Wong(1) were the first authors to consider the estimation of relative permeability curves by applying a reservoir simulator to history match laboratory core flood data. They estimated only parameters that define the shape of relative permeability curves for simple empirical relative permeability models and adjusted relative permeabilities by a trial and error method during the history matching. Sigmund and McCaffery(2) were the first to apply nonlinear regression to the problem of history matching laboratory core flood data. They used power law expressions to model relative permeability curves and estimated only the two exponential parameters in these formulas. Kerig and Watson(3) considered a similar problem. They calculated predicted data from the Buckley- Leverett model, used cubic splines to parameterize relative permeability curves and compared relative permeability estimates obtained with such a representation to those obtained using a power law functional form. They showed that, in general, power law models do not contain enough degrees of freedom to represent the truth well, whereas cubic splines with a small number of knots appear to be sufficiently flexible to yield more accurate estimates of true relative permeability curves. In their results, they assume absolute permeability is known. Lee and Seinfeld(4) considered the simultaneous estimation of the absolute permeability field and relative permeabilities for a two-dimensional, two-phase flow oil-water system. They assumed power law relative permeability curves and assumed that the end point values of relative permeabilities were known. Thus, only the two exponents in the power law relative permeability functions were estimated. They modelled the two-dimensional isotropic heterogeneousermeability field using bi-cubic B-splines. In the specific xamples considered, they matched pressure and water cut data at wells producing from an oil reservoir under waterflood. Tikhonov(5) regularization was used to stabilize the nonlinear least squares problem.

History Matching of Three-Phase Flow Production Data

Summary Adjoint equations for 3D, three-phase flow problems are developed and implemented to calculate the sensitivity of production data to permeability fields and well skin factors. Typically, the development of adjoint equations is rather tedious, but the formulation given here makes the development and numerical solution of adjoint equations straightforward. The procedure for calculating sensitivity coefficients is coupled with a fully implicit finite-difference simulator to develop an automatic history-matching procedure. Even if one does not wish to fully automate history matching, sensitivity coefficients are useful for understanding the physics that control 3D multiphase flow and the influence of local reservoir properties on production data. Procedures to estimate the value of particular types of data [pressure, producing gas/oil ratio (GOR), and water/oil ratio (WOR)] for reducing the uncertainty in estimates of reservoir properties are applied and discussed.

Sensitivity Coefficients for Three-Phase Flow History Matching

Abstract Changing the value of the porosity or the horizontal or vertical permeability in any grid cell in a reservoir simulator by a small amount often results in a small change in the value of a property predicted by the simulator. The ratio of change in prediction to change in reservoir property is called the sensitivity coefficient. In this paper, we describe the use of the adjoint system of equations to compute the sensitivity of wellbore pressure, water-oil ratio, and gas-oil ratio to changes in gridblock permeability and porosity. Unlike some other methods of computing sensitivity coefficients, this method is applicable for problems with large numbers of model parameters and for problems in which cross-flow and compressibility are significant. Although the adjoint system has previously been used to compute the gradient of an objective function for three-phase flow data with respect to model parameters, the sensitivity coefficients are significantly more powerful as they enable the use of Newton-like methods with quadratic convergence, i.e., estimation of the covariance of model estimates based on the inverse of the Hessian, and they provide insight into the information content of data. We use several three-phase flow examples with solution-gas drive, gas injection, and gravity segregation to illustrate these ideas. Introduction In this work, we develop a computationally efficient adjoint procedure for computing sensitivity coefficients for general threedimensional, three-phase flow problems. Although originally introduced for linear problems by Chen et al.(1) and Chavent et al.,(2) the adjoint method has also been applied to multiphase flow history matching problems; see, for example, Wasserman et al.,(3) Lee and Seinfeld(4), Yang and Watson(5), and Makhlouf et al.(6) These authors applied the adjoint method to compute the gradient of the sum of the squares of the production data mismatch terms and minimized the data mismatch function using steepest descent or conjugate gradient algorithms. The steepest descent and conjugate gradient methods tend to converge slowly, however, so Newton-like methods are advantageous when the number of data is small. For example, Wu et al.(7) applied the adjoint method to calculate sensitivity coefficients of wellbore pressure and WOR with respect to model parameters in two-dimensional, two-phase (oil and water) problems and used a Gauss-Newton algorithm to minimize the data mismatch function. Approximately ten iterations were required to obtain convergence for a problem with 1,250 model parameters. A more common choice for computing sensitivity coefficients for multiphase flow problems is the sensitivity equation(8) or gradient simulator method(9). Tan and Kalogerakis(10) used the sensitivity equations to compute the sensitivity of production data to a relatively few model parameters. Zonation was used in their work as a method for reducing the number of parameters, and correlation between parameters was used to determine which parameters should be combined. They discussed the importance of computation of sensitivities for estimation of covariance of model parameters, and for obtaining the most rapid convergence possible with the Gauss-Newton method. Tan(11) used the sensitivity equation approach to compute sensitivities for a three-dimensional, three-phase flow problem.

THREE DIMENSIONAL NUMERICAL MODELING OF MULTIPHASE FLOW AND TRANSPORT

The coupled multiphase flow and transport problem is solved by the finite element method, using linear three-dimensional tetraeder elements. The three-phase flow problem is solved simultaneously for the fluid pressures. The strong nonlinearities are treated with a Newton–Raphson iteration scheme with relaxation. Due to the kinetic of interphase mass transfer and reactions, the equations for water, gas, and solid phase concentrations are strongly coupled. Therefore, both transport equations for water and gas are solved simultaneously, while the equation for the solid phase is solved directly. The numerical model was tested with analytical, quasi-analytical and other numerical models. After the verification first applications were performed.

On the selection of primary variables in numerical formulation for modeling multiphase flow in porous media

Selecting the proper primary variables is a critical step in efficiently modeling the highly nonlinear problem of multiphase subsurface flow in a heterogeneous porous-fractured media. Current simulation and ground modeling techniques consist of (1) spatial discretization of mass and/or heat conservation equations using finite difference or finite element methods; (2) fully implicit time discretization; (3) solving the nonlinear, discrete algebraic equations using a Newton iterative scheme. Previous modeling efforts indicate that the choice of primary variables for a Newton iteration not only impacts computational performance of a numerical code, but may also determine the feasibility of a numerical modeling study in many field applications. This paper presents an analysis and general recommendations for selecting primary variables in simulating multiphase, subsurface flow for one-active phase (Richards' equation), two-phase (gas and liquid) and three-phase (gas, water and nonaqueous phase liquid or NAPL) conditions. In many cases, a dynamic variable switching or variable substitution scheme may have to be used in order to achieve optimal numerical performance and robustness. The selection of primary variables depends in general on the sensitivity of the system of equations to the variables selected at given phase and flow conditions. We will present a series of numerical tests and large-scale field simulation examples, including modeling one (active)-phase, two-phase and three-phase flow problems in multi-dimensional, porous-fractured subsurface systems.

Coarse scale simulation of horizontal wells in heterogeneous reservoirs

An upscaling approach appropriate for the generation of coarse scale simulation models of horizontal wells is developed and applied. The method entails the solution of a general, single-phase, well-driven flow problem over a limited fine scale domain. Coarse scale simulation parameters (transmissibilities and well indexes) are computed from this solution using either an averaging procedure or, to achieve higher degrees of accuracy, an optimization procedure. The method is applied to single- and three-phase flow problems in several different heterogeneous permeability fields. Global coarse scale simulation results using the near-well upscaling procedure are shown to be in consistently close agreement with the reference fine grid simulation results. Results using standard coarse scale models (with no near-well upscaling) are shown to be in considerable error in many cases. The overall upscaling method can be generalized to more complex, deviated and multilateral well configurations.