Transient pressure analysis of polymer flooding fractured wells with oil-water two-phase flow

The oil-water two-phase flow pressure-transient analysis model for polymer flooding fractured well is established by considering the comprehensive effects of polymer shear thinning,shear thickening,convection,diffusion,adsorption retention,inaccessible pore volume and effective permeability reduction.The finite volume difference and Newton iteration methods are applied to solve the model,and the effects of fracture conductivity coefficient,injected polymer mass concentration,initial polymer mass concentration and water saturation on the well-test type curves of polymer flooding fractured wells are discussed.The results show that with the increase of fracture conductivity coefficient,the pressure conduction becomes faster and the pressure drop becomes smaller,so the pressure curve of transitional flow goes downward,the duration of bilinear flow becomes shorter,and the linear flow appears earlier and lasts longer.As the injected polymer mass concentration increases,the effective water phase viscosity increases,and the pressure loss increases,so the pressure and pressure derivative curves go upward,and the bilinear flow segment becomes shorter.As the initial polymer mass concentration increases,the effective water phase viscosity increases,so the pressure curve after the wellbore storage segment moves upward as a whole.As the water saturation increases,the relative permeability of water increases,the relative permeability of oil decreases,the total oil-water two-phase mobility becomes larger,and the pressure loss is reduced,so the pressure curve after the wellbore storage segment moves downward as a whole.The reliability and practicability of this new model are verified by the comparison of the results from simplified model and commercial well test software,and the actual well test data.

A USM-Θ two-phase turbulence model for simulating dense gas-particle flows

A second-order moment two-phase turbulence model for simulating dense gas-particle flows （USM-Θ model）, combining the unified second-order moment twophase turbulence model for dilute gas-particle flows with the kinetic theory of particle collision, is proposed. The interaction between gas and particle turbulence is simulated using the transport equation of two-phase velocity correlation with a two-time-scale dissipation closure. The proposed model is applied to simulate dense gas-particle flows in a horizontal channel and a downer. Simulation results and their comparison with experimental results show that the model accounting for both anisotropic particle turbulence and particle-particle collision is obviously better than models accounting for only particle turbulence or only particle-particle collision. The USM-Θ model is also better than the k-ε-kp-Θ model and the k-ε-kp-εp-Θ model in that the first model can simulate the redistribution of anisotropic particle Reynolds stress components due to inter-particle collision, whereas the second and third models cannot.

Coupled Model of Two-phase Debris Flow,Sediment Transport and Morphological Evolution

The volume fraction of the solid and liquid phase of debris flows, which evolves simultaneously across terrains, largely determines the dynamic property of debris flows. The entrainment process significantly influences the amplitude of the volume fraction. In this paper, we present a depth-averaged two-phase debris-flow model describing the simultaneous evolution of the phase velocity and depth, the solid and fluid volume fractions and the bed morphological evolution. The model employs the Mohr–Coulomb plasticity for the solid stress, and the fluid stress is modeled as a Newtonian viscous stress. The interfacial momentum transfer includes viscous drag and buoyancy. A new extended entrainment rate formula that satisfies the boundary momentum jump condition （Iverson and Ouyang, 2015） is presented. In this formula, the basal traction stress is a function of the solid volume fraction and can take advantage of both the Coulomb and velocity-dependent friction models. A finite volume method using Roe’s Riemann approximation is suggested to solve the equations. Three computational cases are conducted and compared with experiments or previous results. The results show that the current computational model and framework are robust and suitable for capturing the characteristics of debris flows.

REVIEW ON MATHEMATICAL ANALYSIS OF SOME TWO-PHASE FLOW MODELS

The two-phase flow models are commonly used in industrial applications, such as nuclear, power, chemical-process, oil-and-gas, cryogenics, bio-medical, micro-technology and so on. This is a survey paper on the study of compressible nonconservative two-fluid model, drift-flux model and viscous liquid-gas two-phase flow model. We give the research developments of these three two-phase flow models, respectively. In the last part, we give some open problems about the above models.

IMPROVED SUBGRID SCALE MODEL FOR DENSE TURBULENT SOLID-LIQUID TWO-PHASE FLOWS

The dense solid-phase governing equations for two-phase flows are obtained by using the kinetic theory of gas molecules.Assuming that the solid-phase velocity distributions obey the Maxwell equations,the collision term for particles under dense two-phase flow conditions is also derived. In comparison with the governing equations of a dilute two-phase flow,the solid-particle's governing equations are developed for a dense turbulent solid-liquid flow by adopting some relevant terms from the dilute two-phase governing equations.Based on Cauchy-Helmholtz theorem and Smagorinsky model, a second-order dynamic sub-grid-scale(SGS)model,in which the sub-grid-scale stress is a function of both the strain-rate tensor and the rotation-rate tensor,is proposed to model the two-phase governing equations by applying dimension analyses.Applying the SIMPLEC algorithm and staggering grid system to the two-phase discretized governing equations and employing the slip boundary conditions on the walls,the velocity and pressure fields,and the volumetric concentration are calculated.The simulation results are in a fairly good agreement with experimental data in two operating cases in a conduit with a rectangular cross-section and these comparisons imply that these models are practical.

THE STUDY ON THREE-DIMENSIONAL MATHEMATICAL MODEL OF RIVER BED EROSION FOR WATER-SEDIMENT TWO-PHASE FLOW

Based on the tensor analysis of water-sediment two-phase how, the basic model equations for clear water flow and sediment-laden flow are deduced in the general curve coordinates for natural water variable-density turbulent how. Furthermore, corresponding boundary conditions are also presented in connection with the composition and movement of non-uniform bed material. The theoretical results are applied to the calculation of the float open caisson in the construction period and good results are obtained.

An improved semi-empirical friction model for gas-liquid two-phase flow in horizontal and near horizontal pipes

Pressure drop and liquid hold-up are two very important fluid flow parameters in design and control of multiphase flow pipelines.Friction factors play an important role in the accurate calculation of pressure drop.Various empirical and semi-empirical closure relations exist in the literature to calculate the liquid-wall,gas-wall and interfacial friction in two-phase pipe flow.However most of them are empirical correlations found under special experimental conditions.In this paper by modification of a friction model available in the literature,an improved semiempirical model is proposed.The proposed model is incorporated in the two-fluid correlations under equilibrium conditions and solved.Pressure gradient and velocity profiles are validated against experimental data.Using the improved model,the pressure gradient deviation from experiments diminishes by about 3%;the no-slip condition at the interface is satisfied and the velocity profile is predicted in better agreement with the experimental data.

Three-dimensional CFD simulation of inlet structure flow in pumping station based on Eulerian solid- liquid two-phase flow model

Sediment deposition in the pumping station has a huge negative impact on unit operation.The three-dimensional CFD method has been used to simulate inlet structure flow in pumping station based on the Eulerian solid- liquid two-phase flow model. The numerical results of the preliminary scheme show that sediment deposition occurs in the forebay of pumping station because of poor flow pattern therein. In order to improve hydraulic configuration in the forebay,one modified measure reconstructs water diversion weir shape,and another measure sets a water retaining sill in the approach channel. The simulation results of the modified scheme prove that back flow in the forebay has been eliminated and the sediment deposition region has also been reduced.

SECOND-ORDER MOMENT MODEL FOR DENSE TWO-PHASE TURBULENT FLOW OF BINGHAM FLUID WITH PARTICLES

The USM-θ model of Bingham fluid for dense two-phase turbulent flow was developed, which combines the second-order moment model for two-phase turbulence with the particle kinetic theory for the inter-particle collision. In this model, phases interaction and the extra term of Bingham fluid yield stress are taken into account. An algorithm for USM-θ model in dense two-phase flow was proposed, in which the influence of particle volume fraction is accounted for. This model was used to simulate turbulent flow of Bingham fluid single-phase and dense liquid-particle two-phase in pipe. It is shown USM-θ model has better prediction result than the five-equation model, in which the particle-particle collision is modeled by the particle kinetic theory, while the turbulence of both phase is simulated by the two-equation turbulence model. The USM-θ model was then used to simulate the dense two-phase turbulent up flow of Bingham fluid with particles. With the increasing of the yield stress, the velocities of Bingham and particle decrease near the pipe centre. Comparing the two-phase flow of Bingham-particle with that of liquid-particle, it is found the source term of yield stress has significant effect on flow.

Heat transfer model of two-phase flow across tube bundle in submerged combustion vaporizer

In order to optimize the design of the submerged combustion vaporizer(SCV), an experimental apparatus was set up to investigate the heat transfer character outside the tube bundle in SCV. Several experiments were conducted using water and CO_2 as the heat transfer media in the tubes, respectively. The results indicated that hot air flux, the initial liquid level height and the tube pitch ratio had great influence on the heat transfer coefficient outside the tube bundle(ho). Finally, the air flux associated factor β and height associated factor γ were introduced to propose a new hocorrelation. After verified by experiments using cold water, high pressure CO_2 and liquid N_2 as heat transfer media, respectively, it was found that the biggest deviation between the predicted and the experimental values was less than 25%.

A Pressure-Drop Model for Oil-Gas Two-Phase Flow in Horizontal Pipes

The accurate prediction of the pressure distribution of highly viscous fluids in wellbores and pipelines is of great significance for heavy oil production and transportation.The flow behavior of high-viscosity fluids is quite different with respect to that of low-viscosity fluids.Currently,the performances of existing pressure-drop models seem to be relatively limited when they are applied to high-viscosity fluids.In this study,a gas-liquid two-phase flow experiment has been carried out using a 60 mm ID horizontal pipe with air and white oil.The experimental results indicate that viscosity exerts a significant influence on the liquid holdup and pressure drop.At the same gas and liquid volume,both the liquid holdup and pressure drop increase with an increase in the viscosity.Combining two existing models,a modified pressure drop method is developed,which is applicable to horizontal pipes for different viscosities and does not depend on the flow pattern.This new method displays a high accuracy in predicting the new experimental data presented here and other published data in literature.

Two-Phase Flow of Blood with Magnetic Dusty Particles in Cylindrical Region: A Caputo Fabrizio Fractional Model

The present study is focused on the unsteady two-phase flow of blood in a cylindrical region.Blood is taken as a counter-example of Brinkman type fluid containing magnetic(dust)particles.The oscillating pressure gradient has been considered because for blood flow it is necessary to investigate in the form of a diastolic and systolic pressure.The transverse magnetic field has been applied externally to the cylindrical tube to study its impact on both fluids as well as particles.The system of derived governing equations based on Navier Stoke’s,Maxwell and heat equations has been generalized using the well-known Caputo–Fabrizio(C–F)fractional derivative.The considered fractional model has been solved analytically using the joint Laplace and Hankel(L&H)transformations.The effect of various physical parameters such as fractional parameter,Gr,M and
γ on blood and magnetic particles has been shown graphically using the Mathcad software.The fluid behaviour is thinner in fractional order as compared to the classical one.

An improved large eddy simulation of two-phase flows in a pump impeller

An improved large eddy simulation using a dynamic second-order sub-grid-scale （SGS） stress model has been developed to model the governing equations of dense turbulent particle-liquid two-phase flows in a rotating coordinate system, and continuity is conserved by a mass-weighted method to solve the filtered governing equations. In the cur- rent second-order SGS model, the SGS stress is a function of both the resolved strain-rate and rotation-rate tensors, and the model parameters are obtained from the dimensional consistency and the invariants of the strain-rate and the rotation-rate tensors. In the numerical calculation, the finite volume method is used to discretize the governing equations with a staggered grid system. The SIMPLEC algorithm is applied for the solution of the discretized governing equations. Body- fitted coordinates are used to simulate the two-phase flows in complex geometries. Finally the second-order dynamic SGS model is successfully applied to simulate the dense turbu-lent particle-liquid two-phase flows in a centrifugal impeller. The predicted pressure and velocity distributions are in good agreement with experimental results.

Numerical simulation of the effect of void fraction and inlet velocity on two-phase turbulence in bubble-liquid flows

There are contradicted opinions on whether bubbles enhance or reduce the liquid turbulence. In this paper, the effect of void fraction and inlet velocity on the bubble-liquid two-phase turbulence of the multiple bubble-liquid jets in a two-dimensional channel is studied by using the two-phase second-order moment turbulence model. The results confirm the phenomena observed in experiments and reported in references that at a low void fraction and low inlet velocities the bubbles enhance the liquid turbulence, whereas at a high void fraction and high inlet velocities the bubbles reduce the liquid turbulence.

Numerical simulation of predicting and reducing solid particle erosion of solid-liquid two-phase flow in a choke

Chokes are one of the most important components of downhole flow-control equipment. The particle erosion mathematical model, which considers particle-particle interaction, was established and used to simulate solid particle movement as well as particle erosion characteristics of the solid-liquid two-phase flow in a choke. The corresponding erosion reduction approach by setting ribs on the inner wall of the choke was advanced. This mathematical model includes three parts： the flow field simulation of the continuous carrier fluid by an Eulerian approach, the particle interaction simulation using the discrete particle hard sphere model by a Lagrangian approach and calculation of erosion rate using semiempirical correlations. The results show that particles accumulated in a narrow region from inlet to outlet of the choke and the dominating factor affecting particle motion is the fluid drag force. As a result, the optimization of rib geometrical parameters indicates that good anti-erosion performance can be achieved by four ribs, each of them with a height （H） of 3 mm and a width （B） of 5 mm equaling the interval between ribs （L）.

Relative permeability estimation of oil-water two-phase flow in shale reservoir

Oil-water two-phase flow is ubiquitous in shale strata due to the existence of connate water and the injection of fracturing fluid.In this work,we propose a relative permeability model based on a modified Hagen-Poiseuille(HP)equation and shale reconstruction algorithm.The proposed model can consider the nanoconfined effects(slip length and spatially varying viscosity),oil-water distribution,pore size distribution(PSD),total organic matter content(TOC),and micro-fracture.The results show that the increasing contact angles of organic matters(OM)and inorganic minerals(iOM)increase the relative permeability of both oil and water.As the viscosity ratio increases,the relative permeability of oil phase increases while that of water phase decreases,due to the different water-oil distribution.The effective permeability of both oil and water decreases with the increasing TOC.However,the relative permeability of water phase increases while that of oil phase decreases.The increasing number and decreasing deviation angle of micro-fracture increase the effective permeability of oil and water.However,microfracture has a minor effect on relative permeability.Our model can help understand oil-water twophase flow in shale reservoirs and provide parameter characterization for reservoir numerical simulation.

Determination of dynamic capillary effect on two-phase flow in porous media: A perspective from various methods

The relationship between capillary pressure and saturation plays a critical role in the characterization of two-phase flow and transport in aquifers and oil reservoirs. This relationship is usually determined under the static condition, where capillary pressure is the only function of saturation. However,considerable experiments have suggested that the dependence of capillary pressure on desaturation rate is under the dynamic condition. Thus, a more general description of capillary pressure that includes dynamic capillary effect has been approved widely. A comprehensive understanding of the dynamic capillary effect is needed for the investigation of the two-phase flow in porous media by various methods. In general, dynamic capillary effect in porous media can be studied through the laboratory experiment, pore-to macro-scale modeling, and artificial neural network. Here, main principle and research procedures of each method are reviewed in detail. Then, research progress, disadvantages and advantages are discussed, respectively. In addition, upscaling study from pore-to macro-scale are introduced, which explains the difference between laboratory experiment and pore-scale modeling. At last, several future perspectives and recommendations for optimal solution of dynamic capillary effect are presented.

AN ANALYTIC SOLUTION OF DENSE TWO-PHASE FLOW IN A VERTICAL PIPELINE

According to a mathematical model for dense two-phase flows presented in the previous pape[1],a dense two-phase flow in a vertical pipeline is analytically solved, and the analytic expressions of velocity of each continuous phase and dispersed phase are respectively derived. The results show that when the drag force between twophasesdepends linearly on their relative velocity, the relative velocity profile in the pipeline coincides with Darcy's law except for the thin layer region near the pipeline wall, and that the theoretical assumptions in the dense two-phase flow theory mentioned are reasonable.

A Thermal-Hydraulic Coolant Channel Module (CCM) for Single- and Two-Phase Flow

A theoretical “drift-flux based thermal-hydraulic mixture-fluid coolant channel model” is presented. It is the basis to a corresponding digital “Coolant Channel Module (CCM)”. This purpose derived “Separate-Region Mixture Fluid Approach” should yield an alternative platform to the currently dominant “Separate-Phase Models” where each phase is treated separately. Contrary to it, a direct procedure could be established with the objective to simulate in an as general as possible way the steady state and transient behaviour of characteristic parameters of single- and/or (now non-separated) two-phase fluids flowing within any type of heated or non-heated coolant channels. Their validity could be confirmed by a wide range of verification and validation runs, showing very satisfactory results. The resulting universally applicable code package CCM should provide a fundamental element for the simulation of thermal-hydraulic situations over a wide range of complex systems (such as different types of heat exchangers and steam generators as being applied in both conventional but also nuclear power stations, 1D and 3D nuclear reactor cores etc). Thereby the derived set of equations for different coolant channels (distinguished by their key numbers) as appearing in these systems can be combined with other ODE-s and non-linear algebraic relations from additional parts of such an overall model. And these can then to be solved by applying an appropriate integration routine. Within the solution procedure, however, mathematical discontinuities can arise. This due to the fact that along such a coolant channel transitions from single- to two-phase flow regimes and vice versa could take place. To circumvent these difficulties it will in the presented approach be proposed that the basic coolant channel (BC) is subdivided into a number of sub-channels (SC-s), each of them being occupied exclusively by only a single or a two-phase flow regime. After an appropriate nodalization of the BC (and thus its SC-s) and after applying a “modified finite volume method” together with other special activities the fundamental set of non-linear thermal-hydraulic partial differential equations together with corresponding constitutive relations can be solved for each SC separately. As a result of such a spatial discretization for each SC type (and thus the entire BC) the wanted set of non-linear ordinary differential equations of 1st order could be established. Obviously, special attention had to be given to the varying SC entrance or outlet positions, describing the movement of boiling boundaries or mixture levels along the channel. Including even the possibility of SC-s to disappear or be created anew during a transient.