Numerical Solutions of the Classical and Modified Buckley-Leverett Equations Applied to Two-Phase Fluid Flow

Climate change is a reality. The burning of fossil fuels from oil, natural gas and coal is responsible for much of the pollution and the increase in the planet’s average temperature, which has raised discussions on the subject, given the emergencies related to climate. An energy transition to clean and renewable sources is necessary and urgent, but it will not be quick. In this sense, increasing the efficiency of oil extraction from existing sources is crucial, to avoid waste and the drilling of new wells. The purpose of this work was to add diffusive and dispersive terms to the Buckley-Leverett equation in order to incorporate extra phenomena in the temporal evolution between the water-oil and oil-water transitions in the pipeline. For this, the modified Buckley-Leverett equation was discretized via essentially weighted non-oscillatory schemes, coupled with a three-stage Runge-Kutta and a fourth-order centered finite difference methods. Then, computational simulations were performed and the results showed that new features emerge in the transitions, when compared to classical simulations. For instance, the dispersive term inhibits the diffusive term, adding oscillations, which indicates that the absorption of the fluid by the porous medium occurs in a non-homogeneous manner. Therefore, based on research such as this, decisions can be made regarding the replacement of the porous medium or the insertion of new components to delay the replacement.

Pressure transient characteristics of non-uniform conductivity fractured wells in viscoelasticity polymer flooding based on oil-water two-phase flow

Polymer flooding in fractured wells has been extensively applied in oilfields to enhance oil recovery.In contrast to water,polymer solution exhibits non-Newtonian and nonlinear behavior such as effects of shear thinning and shear thickening,polymer convection,diffusion,adsorption retention,inaccessible pore volume and reduced effective permeability.Meanwhile,the flux density and fracture conductivity along the hydraulic fracture are generally non-uniform due to the effects of pressure distribution,formation damage,and proppant breakage.In this paper,we present an oil-water two-phase flow model that captures these complex non-Newtonian and nonlinear behavior,and non-uniform fracture characteristics in fractured polymer flooding.The hydraulic fracture is firstly divided into two parts:high-conductivity fracture near the wellbore and low-conductivity fracture in the far-wellbore section.A hybrid grid system,including perpendicular bisection(PEBI)and Cartesian grid,is applied to discrete the partial differential flow equations,and the local grid refinement method is applied in the near-wellbore region to accurately calculate the pressure distribution and shear rate of polymer solution.The combination of polymer behavior characterizations and numerical flow simulations are applied,resulting in the calculation for the distribution of water saturation,polymer concentration and reservoir pressure.Compared with the polymer flooding well with uniform fracture conductivity,this non-uniform fracture conductivity model exhibits the larger pressure difference,and the shorter bilinear flow period due to the decrease of fracture flow ability in the far-wellbore section.The field case of the fall-off test demonstrates that the proposed method characterizes fracture characteristics more accurately,and yields fracture half-lengths that better match engineering reality,enabling a quantitative segmented characterization of the near-wellbore section with high fracture conductivity and the far-wellbore section with low fracture conductivity.The novelty of this paper is the analysis of pressure performances caused by the fracture dynamics and polymer rheology,as well as an analysis method that derives formation and fracture parameters based on the pressure and its derivative curves.

Evaluation of frictional pressure drop correlations for air-water and air-oil two-phase flow in pipeline-riser system

Accurate prediction of the frictional pressure drop is important for the design and operation of subsea oil and gas transporting system considering the length of the pipeline. The applicability of the correlations to pipeline-riser flow needs evaluation since the flow condition in pipeline-riser is quite different from the original data where they were derived from. In the present study, a comprehensive evaluation of 24prevailing correlation in predicting frictional pressure drop is carried out based on experimentally measured data of air-water and air-oil two-phase flows in pipeline-riser. Experiments are performed in a system having different configuration of pipeline-riser with the inclination of the downcomer varied from-2°to-5°to investigated the effect of the elbow on the frictional pressure drop in the riser. The inlet gas velocity ranges from 0.03 to 6.2 m/s, and liquid velocity varies from 0.02 to 1.3 m/s. A total of885 experimental data points including 782 on air-water flows and 103 on air-oil flows are obtained and used to access the prediction ability of the correlations. Comparison of the predicted results with the measured data indicate that a majority of the investigated correlations under-predict the pressure drop on severe slugging. The result of this study highlights the requirement of new method considering the effect of pipe layout on the frictional pressure drop.

Integrated numerical simulation of hydraulic fracturing and production in shale gas well considering gas-water two-phase flow

Based on the displacement discontinuity method and the discrete fracture unified pipe network model,a sequential iterative numerical method was used to build a fracturing-production integrated numerical model of shale gas well considering the two-phase flow of gas and water.The model accounts for the influence of natural fractures and matrix properties on the fracturing process and directly applies post-fracturing formation pressure and water saturation distribution to subsequent well shut-in and production simulation,allowing for a more accurate fracturing-production integrated simulation.The results show that the reservoir physical properties have great impacts on fracture propagation,and the reasonable prediction of formation pressure and reservoir fluid distribution after the fracturing is critical to accurately predict the gas and fluid production of the shale gas wells.Compared with the conventional method,the proposed model can more accurately simulate the water and gas production by considering the impact of fracturing on both matrix pressure and water saturation.The established model is applied to the integrated fracturing-production simulation of practical horizontal shale gas wells.The simulation results are in good agreement with the practical production data,thus verifying the accuracy of the model.

Arbitrary High-Order Fully-Decoupled Numerical Schemes for Phase-Field Models of Two-Phase Incompressible Flows

Due to the coupling between the hydrodynamic equation and the phase-field equation in two-phase incompressible flows,it is desirable to develop efficient and high-order accurate numerical schemes that can decouple these two equations.One popular and efficient strategy is to add an explicit stabilizing term to the convective velocity in the phase-field equation to decouple them.The resulting schemes are only first-order accurate in time,and it seems extremely difficult to generalize the idea of stabilization to the second-order or higher version.In this paper,we employ the spectral deferred correction method to improve the temporal accuracy,based on the first-order decoupled and energy-stable scheme constructed by the stabilization idea.The novelty lies in how the decoupling and linear implicit properties are maintained to improve the efficiency.Within the framework of the spatially discretized local discontinuous Galerkin method,the resulting numerical schemes are fully decoupled,efficient,and high-order accurate in both time and space.Numerical experiments are performed to validate the high-order accuracy and efficiency of the methods for solving phase-field models of two-phase incompressible flows.

Numerical Simulation of Oil-Water Two-Phase Flow in Low Permeability Tight Reservoirs Based on Weighted Least Squares Meshless Method

In response to the complex characteristics of actual low-permeability tight reservoirs,this study develops a meshless-based numerical simulation method for oil-water two-phase flow in these reservoirs,considering complex boundary shapes.Utilizing radial basis function point interpolation,the method approximates shape functions for unknown functions within the nodal influence domain.The shape functions constructed by the aforementioned meshless interpolation method haveδ-function properties,which facilitate the handling of essential aspects like the controlled bottom-hole flow pressure in horizontal wells.Moreover,the meshless method offers greater flexibility and freedom compared to grid cell discretization,making it simpler to discretize complex geometries.A variational principle for the flow control equation group is introduced using a weighted least squares meshless method,and the pressure distribution is solved implicitly.Example results demonstrate that the computational outcomes of the meshless point cloud model,which has a relatively small degree of freedom,are in close agreement with those of the Discrete Fracture Model(DFM)employing refined grid partitioning,with pressure calculation accuracy exceeding 98.2%.Compared to high-resolution grid-based computational methods,the meshless method can achieve a better balance between computational efficiency and accuracy.Additionally,the impact of fracture half-length on the productivity of horizontal wells is discussed.The results indicate that increasing the fracture half-length is an effective strategy for enhancing production from the perspective of cumulative oil production.

Identifying the enhancement mechanism of Al/MoO_(3) reactive multilayered films on the ignition ability of semiconductor bridge using a one-dimensional gas-solid two-phase flow model

Energetic Semiconductor bridge(ESCB)based on reactive multilayered films(RMFs)has a promising application in the miniature and intelligence of initiator and pyrotechnics device.Understanding the ignition enhancement mechanism of RMFs on semiconductor bridge(SCB)during the ignition process is crucial for the engineering and practical application of advanced initiator and pyrotechnics devices.In this study,a one-dimensional(1D)gas-solid two-phase flow ignition model was established to study the ignition process of ESCB to charge particles based on the reactivity of Al/MoO_(3) RMFs.In order to fully consider the coupled exothermic between the RMFs and the SCB plasma during the ignition process,the heat release of chemical reaction in RMFs was used as an internal heat source in this model.It is found that the exothermal reaction in RMFs improved the ignition performance of SCB.In the process of plasma rapid condensation with heat release,the product of RMFs enhanced the heat transfer process between the gas phase and the solid charge particle,which accelerated the expansion of hot plasma,and heated the solid charge particle as well as gas phase region with low temperature.In addition,it made up for pressure loss in the gas phase.During the plasma dissipation process,the exothermal chemical reaction in RMFs acted as the main heating source to heat the charge particle,making the surface temperature of the charge particle,gas pressure,and gas temperature rise continuously.This result may yield significant advantages in providing a universal ignition model for miniaturized ignition devices.

Optimizing Two-Phase Flow Heat Transfer:DCS Hybrid Modeling and Automation in Coal-Fired Power Plant Boilers

In response to escalating challenges in energy conservation and emission reduction,this study delves into the complexities of heat transfer in two-phase flows and adjustments to combustion processes within coal-fired boilers.Utilizing a fusion of hybrid modeling and automation technologies,we develop soft measurement models for key combustion parameters,such as the net calorific value of coal,flue gas oxygen content,and fly ash carbon content,within theDistributedControl System(DCS).Validated with performance test data,thesemodels exhibit controlled root mean square error(RMSE)and maximum absolute error(MAXE)values,both within the range of 0.203.Integrated into their respective automatic control systems,thesemodels optimize two-phase flow heat transfer,finetune combustion conditions,and mitigate incomplete combustion.Furthermore,this paper conducts an in-depth exploration of the generationmechanismof nitrogen oxides(NOx)and low oxygen emission reduction technology in coal-fired boilers,demonstrating a substantial reduction in furnace exit NOx generation by 30%to 40%and the power supply coal consumption decreased by 1.62 g/(kW h).The research outcomes highlight the model’s rapid responsiveness,enabling prompt reflection of transient variations in various economic indicator parameters.This provides a more effective means for real-time monitoring of crucial variables in coal-fired boilers and facilitates timely combustion adjustments,underscoring notable achievements in boiler combustion.The research not only provides valuable and practical insights into the intricacies of two-phase flow heat transfer and heat exchange but also establishes a pioneering methodology for tackling industry challenges.

Simulation and Analysis on the Two-Phase Flow Fields in a Rotating-Stream-Tray Absorber by Using Computational Fluid Dynamics

The flow field of gas and liquid in a φ150mm rotating-stream-tray (RST) scrubber is simulated by using computational fluid dynamic (CFD) method. The sismulation is based on the two-equation RNG κ-ε turbulence model, Eulerian multiphase model, and a real-shape 3D model with a huge number of meshes. The simulation results include detailed information about velocity, pressure, volume fraction and so on. Some features of the flow field are obtained: liquid is atomized in a thin annular zone; a high velocity air zone prevents water drops at the bottom from flying towards the wall; the pressure varies sharply at the end of blades and so on. The results will be helpful for structure optimization and engineering design.

Three-dimensional Computational Fluid Dynamics Modeling of Two-phase Flow in a Structured Packing Column

Characterizing the complex two-phase hydrodynamics in structured packed columns requires a power- ful modeling tool. The traditional two-dimensional model exhibits limitations when one attempts to model the de- tailed two-phase flow inside the columns. The present paper presents a three-dimensional computational fluid dy- namics （CFD） model to simulate the two-phase flow in a representative unit of the column. The unit consists of an CFD calculations on column packed with Flexipak 1Y were implemented within the volume of fluid （VOF） mathe- matical framework. The CFD model was validated by comparing the calculated thickness of liquid film with the available experimental data. Special attention was given to quantitative analysis of the effects of gravity on the hy- drodynamics. Fluctuations in the liquid mass flow rate and the calculated pressure drop loss were found to be quali- tatively in agreement with the experimental observations.

Fluid−Structure Interaction of Two-Phase Flow Passing Through 90° Pipe Bend Under Slug Pattern Conditions

Numerical simulations of evolution characteristics of slug flow across a 90°pipe bend have been carried out to study the fluid−structure interaction response induced by internal slug flow.The two-phase flow patterns and turbulence were modelled by using the volume of fluid(VOF)model and the Realizable k−εturbulence model respectively.Firstly,validation of the CFD model was carried out and the desirable results were obtained.The different flow patterns and the time-average mean void fraction was coincident with the reported experimental data.Simulations of different cases of slug flow have been carried out to show the effects of superficial gas and liquid velocity on the evolution characteristics of slug flow.Then,a one-way coupled fluid-structure interaction framework was established to investigate the slug flow interaction with a 90°pipe bend under various superficial liquid and gas velocities.It was found that the maximum total deformation and equivalent stress increased with the increasing superficial gas velocity,while decreased with the increasing superficial liquid velocity.In addition,the total deformation and equivalent stress has obvious periodic fluctuation.Furthermore,the distribution position of maximum deformation and stress was related to the evolution of slug flow.With the increasing superficial gas velocity,the maximum total deformation was mainly located at the 90°pipe bend.But as the superficial liquid velocity increases,the maximum total deformation was mainly located in the horizontal pipe section.Consequently,the slug flow with higher superficial gas velocity will induce more serious cyclical impact on the 90°pipe bend.

Pressure Drop of Liquid–Solid Two-Phase Flow in the Vertical Tube Bundle of a Cold-Model Circulating Fluidized Bed Evaporator

A cold-model vertical multi-tube circulating fluidized bed evaporator was designed and built to conduct a visualization study on the pressure drop of a liquid–solid two-phase flow and the corresponding particle distribution.Water and polyformaldehyde particle(POM)were used as the liquid and solid phases,respectively.The effects of operating parameters such as the amount of added particles,circulating flow rate,and particle size were systematically investigated.The results showed that the addition of the particles increased the pressure drop in the vertical tube bundle.The maximum pressure drop ratios were 18.65%,21.15%,18.00%,and 21.15%within the experimental range of the amount of added particles for POM1,POM2,POM3,and POM4,respectively.The pressure drop ratio basically decreased with the increase in the circulating flow rate but fluctuated with the increase in the amount of added particles and particle size.The difference in pressure drop ratio decreased with the increase in the circulating flow rate.As the amount of added particles increased,the difference in pressure drop ratio fluctuated at low circulating flow rate but basically decreased at high circulating flow rate.The pressure drop in the vertical tube bundle accounted for about 70%of the overall pressure drop in the up-flow heating chamber and was the main component of the overall pressure within the experimental range.Three-dimensional phase diagrams were established to display the variation ranges of the pressure drop and pressure drop ratio in the vertical tube bundle corresponding to the operating parameters.The research results can provide some reference for the application of the fluidized bed heat transfer technology in the industry.

An experimental study of drag reduction by nanofluids in slug two-phase flow of air and water through horizontal pipes

This study investigates the effect of injecting nanofluids containing nano-SiO_2 as drag reducing agents(DRA) at different concentrations on the pressure drop of air-water flow through horizontal pipe.The test fluid used in this study was air-water with nano-SiO_2 particles at 0.1%-1%mass concentration.The test sections of the experimental set-up were five pipes of the same length of 9 m with ID from 0.0127m-0.03175m(0.5 to 1.25 in).Airwater flow was run in slug flow regime under different volumetric flow rates.The results of drag reduction(η%)indicated that the addition of DRA could be efficient up to some dosage.Drag reduction performed much better for smaller pipe diameters than it did for larger ones.For various nanosilica concentrations,the maximum drag reduction was about 66.8%for 0.75%mass concentration of nanosilica.

MATHEMATICAL MODEL OF TWO-PHASE FLUID NONLINEAR FLOW IN LOW-PERMEABILITY POROUS MEDIA WITH APPLICATIONS

A mathematical model of two-phase fluid nonlinear flow in the direction of normal of ellipse through low-permeability porous media was established according to a nonlinear flow law expressed in a continuous function with three parameters, a mass conservation law and a concept of turbulent ellipses. A solution to the model was obtained by using a finite difference method and an extrapolation method. Formulas of calculating development index not only before but also after water breaks through an oil well in the condition of two-phase fluid nonlinear flow in the media were derived. An example was discussed. Water saturation distribution was presented. The moving law of drainage front was found. Laws of change of pressure difference with time were recognized. Results show that there is much difference of water saturation distribution between nonlinear flow and linear flow; that drainage front by water moves faster, water breaks through sooner and the index gets worse because of the nonlinear flow; and that dimensionless pressure difference gets larger at the same dimensionless time and difficulty of oil development becomes bigger by the nonlinear flow. Thus, it is necessary that influence of nonlinear flow on development indexes of the oil fields be taken into account. The results provide water-flooding development of the oilfields with scientific basis.

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.

Linearization and analytic solution of fluid dynamics of cell two-phase flow

Linearized equations of fluid dynamics of cell two phase flow for one dimensional case are proposed. Based on the equations, an analytic solution is derived, in which the frequency of wave is observed. The frequency formula consists of all important parameters of the fluid dynamics. In our observation, the group velocity and phase velocity of the motion of wave propagation are explicitly exhibited as well.

Effect of copper nanoparticles on thermal behavior of two-phase argon-copper nanofluid flow in rough nanochannels with focusing on the interface properties and heat transfer using molecular dynamics simulation

A comparison between the efficacy of surface boundary structure and presence of nanoparticles on the condensation two-phase flow inside rough nanochannels has been accomplished by applying molecular dynamics procedure to evaluate the thermal conductivity and flow characteristics.Simulation is performed in a computational region with two copper walls containing rectangular rough elements under different saturated temperatures.The main properties of liquid–vapor interface including density and the number of liquid atoms,are obtained.It is observed that the density profile is more affected by nanoparticles than the roughness.Also,compared to the condensation of nanofluid in a smooth nanochannel,the rough wall causes a greater drop in the temperature at the early time steps and by development of liquid films,effects of the wall roughness reduce.At the first of the condensation process,adding nanoparticle causes that transferring argon particles to the liquid phase increases with a steeper slope.Furthermore,heat current autocorrelation function(HCACF)for nanofluid condensation flow over considered correlation time is analyzed and following that the thermal conductivity for different saturated conditions is calculated.It has been represented that at lower temperatures the roughness makes more significant influence on the heat transfer of two-phase flow,while at higher temperatures the importance of nanoparticles prevails.

Mechanism from particle compaction to fluidization of liquid–solid two-phase flow

A new model of particle yield stress including cohesive strength is proposed,which considers the friction and cohesive strength between particles.A calculation method for the fluidization process of liquid–solid two-phase flow in compact packing state is given,and the simulation and experimental studies of fluidization process are carried out by taking the sand–water two-phase flow in the jet dredging system as an example,and the calculation method is verified.

Introduction to investigations of the negative corona and EHD flow in gaseous two-phase fluids

Research interests have recently been directed towards electrical discharges in multi-phase environments.Natural electrical discharges,such as lightning and coronas,occur in the Earth＇s atmosphere,which is actually a mixture of gaseous phase（air） and suspended solid and liquid particulate matters（PMs）.An example of an anthropogenic gaseous multi-phase environment is the flow of flue gas through electrostatic precipitators（ESPs）,which are generally regarded as a mixture of a post-combustion gas with solid PM and microdroplets suspended in it.Electrical discharges in multi-phase environments,the knowledge of which is scarce,are becoming an attractive research subject,offering a wide variety of possible discharges and multi-phase environments to be studied.This paper is an introduction to electrical discharges in multi-phase environments.It is focused on DC negative coronas and accompanying electrohydrodynamic（EHD） flows in a gaseous two-phase fluid formed by air（a gaseous phase） and solid PM（a solid phase）,run under laboratory conditions.The introduction is based on a review of the relevant literature.Two cases will be considered：the first case is of a gaseous two-phase fluid,initially motionless in a closed chamber before being subjected to a negative corona（with the needle-toplate electrode arrangement）,which afterwards induces an EHD flow in the chamber,and the second,of a gaseous two-phase fluid flowing transversely with respect to the needle-to-plate electrode axis along a chamber with a corona discharge running between the electrodes.This review-based introductory paper should be of interest to theoretical researchers and modellers in the field of negative corona discharges in single-or two-phase fluids,and for engineers who work on designing EHD devices（such as ESPs,EHD pumps,and smoke detectors）.

Ultrasonic method for measuring water holdup of low velocity and high-water-cut oil-water two-phase flow

Oil reservoirs with low permeability and porosity that are in the middle and late exploitation periods in China＇s onshore oil fields are mostly in the high-water-cut production stage.This stage is associated with severely non-uniform local-velocity flow profiles and dispersed-phase concentration（of oil droplets） in oil-water two-phase flow,which makes it difficult to measure water holdup in oil wells.In this study,we use an ultrasonic method based on a transmission-type sensor in oil-water two-phase flow to measure water holdup in lowvelocity and high water-cut conditions.First,we optimize the excitation frequency of the ultrasonic sensor by calculating the sensitivity of the ultrasonic field using the finite element method for multiphysics coupling.Then we calculate the change trend of sound pressure level attenuation ratio with the increase in oil holdup to verify the feasibility of the employed diameter for the ultrasonic sensor.Based on the results,we then investigate the effects of oildroplet diameter and distribution on the ultrasonic field.To further understand the measurement characteristics of the ultrasonic sensor,we perform a flow loop test on vertical upward oilwater two-phase flow and measure the responses of the optimized ultrasonic sensor.The results show that the ultrasonic sensor yields poor resolution for a dispersed oil slug in water flow（D OS/W flow）,but the resolution is favorable for dispersed oil in water flow（D O/W flow） and very fine dispersed oil in water flow（VFD O/W flow）.This research demonstrates the potential application of a pulsed-transmission ultrasonic method for measuring the fraction of individual components in oil-water two-phase flow with a low mixture velocity and high water cut.