Mathematical Model Of Two-phase Flow Research Articles

Determining relative permeability and capillary pressure from mixed-wet core floods

The main goal of this paper is a laboratory method development for determining the capillary pressure and relative permeability from coreflooding test specifically for mixed-wet rocks. Previously, the steady-state transient test (SSTT) was designed to determine both relative permeability (Kr) and capillary pressure (Pc) from a single steady-state coreflood by using the transient data of routine steady-state tests. On the contrary to the previous works, where SSTT was developed for completely hydrophobic or hydrophilic outcrop cores, this paper determines Kr and Pc for mixed-wet cores. Depending on the capillary pressure equation applied, the mathematical model for two-phase flow in mixed-wet rocks contains one or two extra constants, so some extra measurements must be taken during SSTT, to determine Pc and Kr curves. We change the experimental protocol by establishing a uniform initial saturation Swi by evaporation method. We also include bump floods at two high rates for the alternative determination of saturation and relative permeability for residual oil. A new inverse solver determines the set of model constants that employs van Genuchten's Pc formula and a vaporization procedure for initial core saturation. The number of resulting model coefficients is equal to seven and matches the dimension of the measured data array for mixed-wet cores.The developed SSTT for mixed-wet rocks is applied and validated for both high-salinity (HS) and low-salinity (LS) waterfloods. Strong agreement between laboratory steady-state, transient, and bump-flood data and those obtained by mathematical modelling – R2 is above 0.99 for steady-state data and above 0.9 for transient data - validates the proposed SSTT procedure and inverse solver for mixed-wet cores and HS and LS waterfloods. Our findings indicate that low salinity has a greater effect in expanding the range of positive capillary pressure compared to high salinity. This suggests an increase in water-wettability as the salinity decreases.

Quantitative characterization of imbibition in fractured porous media based on fractal theory

In low-permeability reservoirs, such as shale and tight sandstone, imbibition is an important mechanism for enhancing oil recovery. After hydraulic fracturing treatment, these reservoirs create a network of fracture pathways for fluid flow. Therefore, understanding the imbibition mechanisms in fractured porous media and quantitatively characterizing oil–water distribution are crucial for the development of low-permeability reservoirs. In this study, a mathematical model of two-phase flow in porous media with branching fractures was established. The phase-field method was employed to track the oil–water interface, and quantitative characterization of imbibition was conducted based on fractal theory, and the effects of wetting phase injection rate, the number of disconnected fractures, fracture spacing, and fracture morphology on imbibition in branched fracture porous media were discussed. The research findings indicate that in branched fracture porous media, both co-current and countercurrent imbibition processes occur simultaneously, and there exists a diffusion interface layer with a certain thickness at the oil–water interface. The hydraulic pressure generated by the wetting phase injection rate provides the driving force for imbibition oil recovery, but it also affects the contact time between the wetting and non-wetting phases. The presence of disconnected fractures hinders the propagation of hydraulic pressure, reducing the effectiveness of imbibition. The imbibition displacement zone is limited and occurs only within a certain range near the fractures. As the number of branching fractures increases, the channels for the wetting phase to enter matrix pores are enhanced, resulting in higher efficiency of imbibition displacement of the oil phase. The results of this research can provide guidance for the design of fracturing programs and recovery prediction in low-permeability reservoirs.

Investigation of the Influence of Inner Geometries of the Receiver Tube of a Parabolic Trough Collector System Used for Water Heating

Solar collectors are devices that help to harvest solar thermal radiation and convert into other forms of energy. Because of its low fabrication cost and simple design, the Parabolic Trough Collector (PTC) is one of the finest alternatives for medium-temperature (i.e., in 80°𝐶𝐶-250°𝐶𝐶 range) needs among all types of concentrated solar collectors. The absorber tube or the receiver tube is one of the main critical components of a PTC. Thermal radiation of the sun falling onto a parabolic reflector is reflected to its focal line where the receiver tube is placed. Therefore, the performance of a PTC system significantly depends on Receiver Geometries (RGs). A locally developed PTC system with a new and accurate tracking system was designed and fabricated in the current study and steam generation inside the receiver was considered in the performance analysis. The experimental analysis was performed using seven different RGs. The experiments were carried out by measuring generated steam every 10 minutes for all the RGs, and simultaneously, solar insolation data was taken by a domestic solar Photovoltaic (PV) system with a capacity of 5 kW. The steam generation data was taken only when the power output of the reference solar PV system was ≥ 4kW. Having tested seven different RGs, the researchers were able to improve the steam generation to a maximum of 10.0% compared to the reference geometry using direct experimental data. This was 8.1% in accordance with the statistical approach via multiple linear regression model. A three-dimensional mathematical model for two-phase flow was formulated for a closed domain. Restricting this model into twodimensional geometry, the steam generation process was simulated as a TPF problem using ANSYS Fluent software. The volume fraction profiles clearly show the phase transition, which leads to evaporation and condensation inside the tube.

The Casson Dusty Nanofluid: Significance of Darcy–Forchheimer Law, Magnetic Field, and Non-Fourier Heat Flux Model Subject to Stretch Surface

This work aims to offer a mathematical model for two-phase flow that investigates the interaction of Casson nanofluid and dust particles across a stretching surface. MHD Darcy–Forchheimer porous medium and Fourier’s law through Cattaneo–Christove thermal flux are also considered. The governing equations for the two phases model are partial differential equations later transmuted into ordinary ones via similarity transforms. The Runge–Kutta method with the shooting tool is utilized numerically to solve the boundary layer equations computed in MATLAB to obtain numerical results for various pertinent parameters. The numerical outcomes of momentum, temperature, and concentration distribution are visible for both phases. The results of the skin friction, heat transfer coefficients, and the Sherwood number are also visible in the graphs. Furthermore, by comparing the current findings to the existing literature, the validity of the results is confirmed and found to be in good agreement. The fluid velocity is reduced against increasing strength of Casson fluid parameter, enhanced the fluid phase and dust phase fluid temperature. The temperature declines against the growing values of the relaxation time parameter in both phases. Dusty fluids are used in various engineering and manufacturing sectors, including petroleum transportation, car smoke emissions, power plant pipes, and caustic granules in mining.

A nonlinear mathematical model for two-phase flow in nanoporous media

A nonlinear mathematical model for two-phase flow in nanoporous media

The coupled model of transient non-equilibrium interphase mass transfer rate of sliding bubble and two-phase flow in variable gradient drilling

The aim of this research is to investigate the influence of key parameters of variable gradient drilling by injecting hollow-glass spheres (Referred to as HGS, it is a low-density and micron-sized hollow glass sphere) on the interphase mass transfer rate of the surface boundary layer of the sliding bubble in the annulus and the flow law of the two-phase flow in the wellbore after the gas intrusion occurred. Firstly, the key parameters of variable-gradient drilling, such as HGS volume fraction, HGS density, the depth of the filter separator, were coupled with the model of non-equilibrium interphase mass transfer rate and the mathematical model of two-phase flow to establish a new two-phase flow model of transient non-equilibrium interphase mass transfer for variable-gradient drilling. Secondly, the existing classic model and measured data were used for comparative analysis with the model in this paper, and then the accuracy of the model in this paper was verified. Thirdly, based on the on-site data, the numerical calculations and sensitivity analysis were carried out using the model in this paper. Finally, a comparative analysis of the interphase mass transfer rate was performed under equilibrium and non-equilibrium conditions. In addition, the influence on non-equilibrium interphase mass transfer rate of surface boundary layer of the bubble, bottom hole pressure, overflow rate and overflow rate exerted by different separator positions, HGS volume fraction, HGS density, and pumping rate was studied. The results indicated that the interphase mass transfer rate under the non-equilibrium condition was lower than that under the equilibrium condition when the both were at the same well depth or at the same circulation time. The interphase mass transfer rate enhanced with the increase of pumping rate, the depth of filter separator and HGS density, but decreased with the increase of HGS volume fraction. The bottom-hole pressure was negatively correlated with the depth of the filter separator and the volume fraction of HGS, on the contrary, it was positively correlated with the HGS density. Similarly, the gas void fraction, overflow rate and overflow were positively correlated with the depth of filter separator and HGS volume fraction, but negatively correlated with the HGS density. This research can provide a certain theoretical reference for variable gradient drilling.

Effect of Annular Gas-Liquid Two-Phase Flow on Dynamic Characteristics of Drill String

Natural gas hydrate (NGH) is a kind of new type green energy source with giant reserves which has been thought of highly by energy explorers in the world. However, NGH breaks down to produce some natural gas that enters the annulus and flows together with the drilling fluid. The gas-liquid two-phase flow can have an impact on the work of the drill string. Therefore, it is important to study gas-liquid two-phase flow in the annulus on the dynamic characteristics of the drill string. In this article, taking a single drill string as the research object, a fluid-structure coupled finite element mathematical model of two-phase flow in the annulus and drill string is established based on computational fluid dynamics and computational structural dynamics theory. The finite element numerical simulation method is used to analyze the influence of drilling fluid and natural gas in the annulus on the dynamic characteristics of the drill string. The simulation analysis shows the following: (1) The motion of drilling fluid or natural gas in the annulus will reduce the natural frequency of the drill string, and the drilling fluid has a greater impact on the natural frequency of the drill string. (2) When single-phase drilling fluid flows in the annulus, the displacement peak in different directions, maximum equivalent stress, and strain of the drill string increase with the increase of the drilling fluid flow velocity or pressure, and the drilling fluid pressure has a more significant effect. (3) When the gas-liquid two-phase fluid flows in the annulus, the displacement peak, maximum equivalent stress, velocity amplitude, and acceleration amplitude of the drill string all increase with the natural gas flow velocity and natural gas content increase, and the natural gas flow velocity has a more significant effect.

The Effect of Interlayer on Water Cut Rise in a Bottom Water Reservoir

A bottom water reservoir is universal in all reservoirs in the world. In the exploitation process of the bottom water reservoir, because the reservoir is in complete contact with bottom water, the natural energy supply of the bottom water reservoir is sufficient. At the same time, in the production process, the pressure drop generated by the formation is mainly concentrated near the wellbore, which leads to a short period of waterless oil production, fast water breakthrough, and fast water cut rise, which seriously affects the overall recovery factor and increases the risk of oil filed exploitation. Since the distribution of an interlayer enhances the heterogeneity of the reservoir, it has a prominent impact on the flow of oil and water in the bottom water reservoir. However, people know little about the impact of the interlayer on the bottom water reservoir. In this paper, the mathematical model of two-phase flow in the bottom water reservoir with the interlayer is proposed, and the numerical solution of the model is obtained by a finite element method. The influence of the distribution of the interlayer on the water cut rising law of the bottom water reservoir is analyzed. The results show that the bottom water is blocked by the interlayer and its own gravity and the rising height of the bottom water is limited, so that the interlayer has a significant inhibitory effect on the coning of the bottom water. The closer the interlayer is to the oil-water interface, the faster the water cut will rise; the narrower the interlayer is, the faster the bottom water ridge will advance; when the width of the interlayer is the same as that of the reservoir, the water cone will not appear; the higher the permeability of the interlayer is, the faster the upper body of the water cut will be; when the permeability of the reservoir and the interlayer is over 1000, the influence of the interlayer on the water cut rise is not obvious; the thickness of the interlayer on the water cut rise law has little effect. Finally, through the case analysis, it is proposed that the new horizontal wells should be arranged in the position where the interlayer develops. The findings of this study can help for the better understanding of control laws of the interlayer on the bottom water and providing a crucial theoretical basis for the development of the bottom water reservoir.

Dynamics and combustion of single aluminium agglomerate in solid propellant environment

Aluminized composite propellant used in solid rocket motors contain a lot of aluminium particles because high combustion energy is generated and propulsion efficiency increases by burning aluminium particles. The combustion of aluminum occurs in a significant portion of the combustion chamber and produces aluminum oxide smokes and residues that are carried into the flowfield. Agglomerates have non-spherical shape, and consist of aluminium droplet and oxide particle (oxide cap) attached to the droplet. Unlike the liquid droplet ignition, the solid oxide film blocks the liquid aluminum from the penetration of the oxidizer hence prevents the particle from its ignition. Development of robust models of aluminum particle dynamics is essential in the design of advanced propulsion systems. The mathematical model of two-phase flow around a single aluminum droplet with oxide cap is developed. The model solves the continuity, momentum, energy and species continuity equations simultaneously to obtain the species and temperature profiles and the burning time of droplet. The results of numerical simulations are compared with predictions from semi-empirical correlations and computational data.

Experimental and numerical study of two-phase flow mixing in gas–liquid external-loop airlift reactor

The experimental and numerical study of two-phase gas–liquid flows in the external-loop airlift reactor is presented. The effect of riser-to-downcomer cross-sectional area ratio on the mixing hydrodynamic efficiency, gas holdup, and circulation liquid velocity is investigated. The transverse dimensions of vertical columns are 2.5 cm, 5 cm and 10 cm, the height is 2 m long. Low values of gas flow rate are tested (from 0.1 m3/h to 0.6 m3/h). The test rig with an open wide channel connecting the riser and the downcomer in the up, which provides the conditions for complete air release from water phase, is used. A numerical study of gas–liquid flow in the reactor's riser is also performed using the CFD solver Star-ccm+, which uses a three-dimensional mathematical model of two-phase flow based on the two-fluid Euler–Euler method. Both phases are calculated by solving the steady-state Reynolds-Averaged Navier–Stokes (RANS) conservation equations with the high Reynolds number k–ε turbulence model. The aerodynamic drag, shear-lift and added mass forces, and the turbulent dispersion of the flow are considered. The breakup-coalescence effect is also considered by solving the population balance equation. The numerical tool is used to examine the flow regime map of the experimentally measured flows. The flow regimes (bubble and bubble-to-slug) and transitions from the bubble to bubble-to-slug were identified experimentally and numerically using non-dimensional parameters. The formation of “bubble tracks” or chains of bubbles in the riser were experimentally observed at the flow regime which has been identified as bubble-to-slug. Experiments also have shown that the bubbles are carried away by the flow from the riser and the open top channel to the downcomer when the Reynolds number of the flow becomes bigger than 25,000. The liquid circulation rate remains constant with further increase in the gas flow rate. The gas phase holdup shows the lowest values as well as the liquid circulation rate has highest values when the ratio of column's cross-sectional dimensions is 1. Opposite, the highest values of gas holdup as well as the lowest liquid circulation rates is observed when the ratio is 0.25. The Wang correlation, for the drag coefficient, shows the best fit with the experimental data if the flow regime is bubble. The Schiller–Naumann correlation over-predicts values. A simple analytical model built on the balance of friction losses for a two-phase gas–liquid flow shows very good agreement with the experimental values of the gas phase holdup in the riser and the liquid circulation rate. No fitting parameters are used because of simple design of the experimental rig and low gas flow rate values.

Optimal slug size for enhanced recovery by low-salinity waterflooding due to fines migration

Abstract The objective of this study is modelling of low-salinity water (LSW) slug injection followed by continuous high-salinity chase drive. We discuss the case of fines-assisted low-salinity waterflood, where low salt concentration of the injected water results in mobilisation and migration of the in-situ reservoir fines and the consequent formation damage. This diverges the injected water flux into low-permeability zones and enhances sweep efficiency. We extend the mathematical model for two-phase flow with varying salinity and fines migration for the case of over and undersaturated fines, where the detaching drag torque, exerting the fine particle, exceeds or is below the attaching electrostatic torque, respectively. The laboratory study has been performed to determine how movable clays and the induced permeability damage is distributed between the layers. We distinguish between two competitive physics effects of “fines-assisted low-salinity”: (i) plugging of the high-permeability layer that results in the flux diversion into unswept zones yielding enhanced oil recovery, (ii) induced formation damage behind the waterfront in low-permeable zones that reduces the recovery. These opposite effects indicate the existence of an optimal low-salinity slug size that results in maximum recovery. The modelling shows that for all simulated cases there does exist an optimal low-salinity slug size, which is the main result of the work. It was found that the volume of optimal low-salinity slug has an order of magnitude of the pore volume of the high-permeability layer.

CFD study of oil-in-water two-phase flow in horizontal and vertical pipes

The effectiveness of the classical Lagrangian method implemented in 3D CFD solver to simulate highly turbulent vertical and horizontal liquid-liquid dispersed flows where intensive turbulent migration of droplets make the flow behaviour complex is in the focus of the study. Dispersed oil-in-water two-phase pipe flow is analysed numerically using CFD solver ANSYS Fluent where a three-dimensional mathematical model of two-phase flow in pipes is developed. The Euler-Lagrange scheme is employed to resolve the interaction between the fluid (water) and droplet (oil) phases. The flow field of the continuous fluid phase is calculated by solving the Reynolds-averaged Navier-Stokes (RANS) conservation equations that are coupled with a high Reynolds number k-ε turbulence model and standard wall function. The Lagrangian method is used to resolve the motion of oil droplets. The drag, gravity, buoyancy and shear-lift forces are considered. The shear-lift force is of primary focus and is modelled and incorporated into the governing equations of the solver through the User Define Function (UDF) option as the external program. The two-way coupling procedure is employed. Here, droplet breakup and coalescence are not modelled. Mean droplet size is calculated using analytical methods. The model is validated with high level of accuracy against published experiments where the frictional pressure drop in vertical and horizontal dispersed oil-water flows is measured. The effect of mixture flow velocity and fluid phase superficial velocity on the droplet dynamics in vertical and horizontal pipes is investigated numerically. The transition from dispersed two-phase flow to stratified flow regime is analysed for the case of horizontal pipe flow. It has been found that the shear-lift force has much higher importance for the adequate flow representation as compared to the break-up/coalescence phenomena in the case of highly turbulent liquid-liquid dispersed pipe flows.

Analysis of local pressure gradient inversion and form of bubbles in Taylor flow in microchannels

Analysis of local pressure gradient inversion and pressure ‘jumps’ at gas-liquid Taylor flow in microchannels, which has been previously observed, but not explained by Kreutzer et al. (2005b) and Duran Martinez et al. (2015) is carried out. The new understanding originates from a mathematical model of two-phase flow developed before (Abiev (2008) for circular capillaries, Falconi et al. (2016) for micro channels with square cross-section). It is shown that the inverse flow of the fluid in the film around the bubble is caused by the inverse pressure gradient in the film.The key factors determining the Taylor bubble surface deformation near its ends are found. The deformation occurring at the nose and tail ellipsoidal surfaces and on the cylindrical surface near the tail of the bubble is caused by the balance of pressure in the liquid film along the bubble, the pressure level in the bubble and capillary pressure.It was shown that pressure difference in the liquid film along the bubble caused by peculiar properties of Taylor flow in micro channels is the prime cause of non-hemispherical form of the bubble ends, not vice versa. The obtained results are applicable to micro channels both with circular and square cross-section (in the latter case to the so-called axisymmetric form of the bubble or close to it).The wavy form of the bubble near its tail observed by Kreutzer et al. (2005b), Abadie et al. (2013), Hayashi et al. (2014), and Falconi et al. (2016) is shown herein to be governed by the balance between the capillary forces at the interface and wave-like pressure field near the ends of the bubble.The performed analysis provides a quite simple understanding of previously unexplored features of Taylor flow hydrodynamics (pressure oscillations near the ends of the bubble found by Kreutzer et al. (2005a, 2005b)) and allows a characterization of the shape of the bubble ends in general (concave or almost planar, convex, elongated convex). The results obtained will allow a more accurate modeling of developing gas-liquid slug flow and a calculation of the surface area of the bubbles in the description of mass transfer processes. Besides, the results of this paper are important for better understanding of bubbles’ formation process driving forces, thus giving a tool to engineers for better design of microreactors and micro heat exchangers. The bubble’s nose elongation results in a significant change in the liquid film thickness in this area, influencing thus mass transfer characteristics by the change in the gas-liquid interface and the velocity in the liquid film.The observed form of the bubbles could also be used in experimental praxis as a detector indicating pressure drop and two-phase flow rate in the Taylor flow.

A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media

In this work we consider a mathematical model for two-phase flow in porous media. The fluids are assumed immiscible and incompressible and the solid matrix non-deformable. The mathematical model for the two-phase flow is written in terms of the global pressure and a complementary pressure (obtained by using the Kirchhoff transformation) as primary unknowns. For the spatial discretization, finite volumes have been used (more precisely the multi-point flux approximation method) and in time the backward Euler method has been employed. We present here a new linearization scheme for the nonlinear system arising after the temporal and spatial discretization. We show that the scheme is linearly convergent. Numerical experiments are presented that sustain the theoretical results.

The crystallization processes in the aluminum particles production technology

The physical and mathematical model of the crystallization process of liquid aluminum particles in the spray-jet of the ejection-type atomizer was proposed. The results of mathematical modeling of two-phase flow in the spray-jet and the crystallization process of fluid particles are given. The influence of the particle size, of the flow rate and the stagnation temperature gas in the ranges of industrial technology implemented for the production of powders aluminum of brands ASD, on the crystallization characteristics were investigated. The approximations of the characteristics of the crystallization process depending on the size of the aluminum particles on the basis of two approaches to the mathematical description of the process of crystallization of aluminum particles were obtained. The results allow to optimize the process parameters of ejection-type atomizer to produce aluminum particles with given morphology.

A BRIEF STUDY ABOUT TWO-PHASE FLOW (LIQUID + GAS) MATHEMATICAL MODELING

This work presents a brief study about mathematical modeling of two-phase flow (liquid + gas), approaching homogeneous phase flows and heterogeneous phase flows, in addition to mathematical modeling of pressure drop in flow restriction through abrupt expansions and abrupt contractions. Also presents a summary of flow patterns main types and a brief study about how the flow velocity influences these patterns.

Research on Numerical Simulation of Gas Storage Field in Aquifer Based on Fully Implicit Method

The mathematical model of Two-Phase flow of water and gas in aquifer is established based on the Darcy linear percolation law and law of conservation of mass of fluid mechanics. According to the fully implicit difference method, time and spatial discretization of the model lead to a set of nonlinear equations, which can be transferred into a group of linear equations. The linear equations can be solved using the Jacobi-iteration pretreatment because of its high stability and fast convergence speed of the solution positively. The distribution of the saturation and formation pressure with time in the process of gas injection is analyzed by using the mathematical model. The results show that the gas saturation increases with the gas injection rate, the permeability and reservoir thickness increasing whereas the formation pressure increases slowly under same cumulative gas injection rate. The formation pressure increases at first and then decreases to reach a steady value.

Mathematical modeling of two-phase flow and transport in an immobilized-cell photobioreactor

A one-dimensional two-phase flow and transport model is presented for a packed bed photobioreactor with transparent gel granules containing immobilized photosynthetic bacterial cells. The inherently coupled two-phase flow and mass transport, along with the biochemical reactions occurring in the photobioreactor are taken into account. The source term in the species conservation equation of the substrate is derived from a local transport model for a single gel granule. Model predictions of the glucose consumption efficiency and hydrogen production rate are in good agreement with experimental data. The results show that the photoinhibition of immobilized cells appears at incident light intensities higher than 6000 lux. It is the most suitable for photo-hydrogen production under neutral conditions and 30 °C of the influent substrate solution. Moreover, a high influent substrate solution flow rate results in a large hydrogen production rate due to the improved substrate transport from the bulk solution to gel granules.

Rapoport-leas two-phase flow model for anisotropic porous media

The Rapoport-Leas mathematical model of two-phase flow is generalized to include the case of anisotropic porous media. The formula for the capillary pressure, which specifies the relationship between the phase pressures, contains a scalar function of a vector argument. In order to determine the scalar function, the capillary pressure tensor and the tensor inverse to the tensor of characteristic linear dimensions are introduced. The capillary pressure is determined by the contraction of the second-rank tensors with a unit vector collinear to the phase pressure gradients, also assumed to be collinear. It is shown that the saturation function introduced for isotropic porous media (Leverett function) can be generalized to include anisotropic media and is now determined by a fourth-rank tensor. Generalized expressions for the Leverett and relative phase permeability functions are given for orthotropic and transversely isotropic media with account for the hysteresis of the phase permeabilities and capillary pressure.

Mathematical model of two-phase flow in accelerator channel

The problem of two-phase flow composed of energy-carrier phase (Newtonian liquid) and solid fine-dispersed phase (particles) in counter jet mill accelerator channel is considered. The mathematical model bases goes on the supposition that the phases interact with each other like independent substances by means of aerodynamics’ forces in conditions of adiabatic flow. The mathematical model in the form of system of differential equations of order 11 is represented. Derivations of equations by base physical principles for cross-section-averaged quantity are produced. The mathematical model can be used for estimation of any kinematic and thermodynamic flow characteristics for purposely parameters optimization problem solving and transfer functions determination, that take place in counter jet mill accelerator channel design.