In this master thesis several novel DEIM formulations are proposed that enable the construction of non-linearly stable hyper-reduced order models (hROMs) of the incompressible Navier-Stokes equations. The hROMs have the same mass, momentum and energy conservation properties as the previously proposed ROM, but they do not suffer of prohibitively expensive computational scaling when the number of POD modes is increased. The first of the proposed methods is the least-squares discrete empirical interpolation method (LSDEIM), which is based on a constrained minimization. The second method is the Sherman-Morrisson discrete empirical interpolation method (SMDEIM), which applies a rank-one correction to the conventional DEIM to conserve energy. The third method is the decoupled least-squares discrete empirical interpolation method (DLSDEIM), which is a generalization of the LSDEIM that allows increasing the size of the measurement space. All methods result in structure-preserving DEIM formulations that have an equivalent computational scaling as the conventional DEIM, but provide provably stable, structure-preserving hROMs. Furthermore, the use of the principal interval decomposition (PID) in the construction of the reduced and DEIM spaces is considered to beat the Kolmogorov barrier.

Due to the energy transition, an increasing number of oil pipelines will be decommissioned in the foreseeable future. As a part of the decommissioning process, the oil needs to be flushed from the pipelines. Some models to predict the required water velocity to flush oil from pipelines are available, but these are limited in accuracy, and data to validate these models are lacking. The aim of this MSc research project is to gain a better understanding of the oil flushing process in pipelines using water, and to provide validation data for the existing models. Thereto lab experiments were carried out in the flow loop of TNO in Rijswijk. The length of the pipe is almost 6 m and the diameter is 56 mm. The experiments are performed by filling the experimental pipe section with oil and subsequently flushing it using four pipe volumes of water. The first set of experiments, for a pipe inclination of 0° to - 5°, was performed with superficial flushing velocities ranging from 0.05 m/s to 1.5 m/s, for two different types of oil. Data were collected using a mass flow meter system and a video capturing system. These data were subsequently analysed using the software tool MATLAB. The first set of experiments was also modelled in the multi-phase flow modelling tool OLGA. A second set of experiments was performed to find the transition point between stratified and mixed flow, which is relevant for the flushing efficiency, by varying the inclination in a range of 0° to -90° relative to the horizontal plane, using a superficial water velocity of 0.1 m/s. In addition to the mass flow measurements, data were collected with a video system during all sets of experiments, and subsequently processed to determine the phase distribution throughout the pipe during, and after the flushing process. It was established that the flushing efficiency is proportional to the superficial water velocity. For both types of oil, for inclinations between 0° and -5°, a superficial water velocity of 0.35 m/s suffices to flush all oil when flushing four pipe volumes of water. For all considered inclinations and pipe geometries and for both oil types, the bulk (≥ 95%) of the oil is removed from the pipe for a flushing velocity of 0.25 m/s. In a horizontal orientation, the lighter and less viscous Fuchs Renolin DTA7 oil is flushed more efficiently from the pipe compared to the Mobil Velocite No.10 oil. This effect diminishes for downward inclined orientations and is reversed for an inclination of -5°. In OLGA it is possible to easily vary for example the geometry of the pipeline and show the effect on the two-phase flow. Discrepancies were found, however, when the flow was evaluated around the 0° inclination angle. Small deviations from this angle of inclination gave significantly different results. For the 0° to 90° range of inclinations, and a superficial velocity of 0.1 m/s, the transition zone between stratified flow and mixed flow was established to be in a range of 7.5° to 20° relative to the horizontal plane. Either decreasing or increasing the inclination angle from this point is beneficial to the flushing process. Apart from the increased understanding of the flushing process summarised above, the data collected on the flushing process under various sets of conditions can be used to validate models for predicting oil flushing processes.

Currently, the paper producing industry is a significant energy consumer and contributes to the excessive greenhouse emissions. In order to become carbon neutral by 2050, it is evident that the paper industry must become more sustainable as well. Most of the energy is consumed in the dewatering process. Therefore, optimizing this process is key for efficient operation and reducing the carbon footprint. Since the late 1950's, there have been several models proposed to describe the dryer section of a paper machine. However, the majority did not capture internal transport phenomena resulting in inadequate models for heavier paper grades. In this work, a comprehensive physical-numerical model is developed that describes the internal transport phenomena in multicylinder paper drying. The model solves a set of two differential equations describing the moisture content and temperature in the thickness direction of the sheet. The model can be solved by imposing time-varying boundary- and initial conditions. The model includes unfelted and double-felted cylinders. Correlations for multiple thermodynamic properties are proposed and evaluated. Heat and mass transfer coefficients at the open surface are determined using the Chilton-Colburn analogy. Furthermore, the sorptive characteristics are fully accounted for. In order to find the most applicable thermodynamic properties, the model is validated against data gathered in a field survey, carried out as part of this project. In this field survey, a paper grade of 203 [g/m^2] running at 403 [m/s] was investigated. Temperature measurements were performed in order to determine the conditions per cylinder. A final moisture content and two temperature profiles were used as test criteria to validate the model. Furthermore, the survey gave direct insight on the state of the dryer section: Cylinder 23 was found to be in the flooded state and the warm-up time was too long. The model was found to slightly overestimate the final moisture content: the calculated final wet-basis moisture content at the end of the pre-dryer section was 0.235 [kg/kg], which is 0.022 [kg/kg] above the measured final moisture content. The calculated temperature profiles closely followed measured values. The effect of changing three operating conditions was evaluated. This involved fixing the relative humidities in the pockets, felting the first 10 cylinders and varying the machine speeds. By fixing the relative humidities in the pockets, the final moisture content increased by almost 20%. However, a 60% reduction in volumetric flow rate over the fans could be achieved. Felting the first 10 cylinders results in a 50% reduction in the warm-up length and a 8.9% decrease in the final moisture content. Lastly, it was found that for an increased machine speed, the final moisture content increases as well, due to reduced heat transfer and drying time. This Master Thesis Project was carried out by the author at the company Kadant Johnson in the Netherlands.

The transportation of high viscosity oil is an ongoing research topic for quite some years because of its importance in industrial applications and the increasing energy requirements of the world population. The water lubricated oil core with a core-annular flow configuration is a viable option for the transport of high viscosity oil. It was observed that there are significant differences between the predictions from the RANS simulations and the experimental data for turbulent oil-water core-annular flow obtained in the flow loop in the lab at TU Delft. Thus, this study is performed to understand the difference between the RANS simulations and the experimental data. 1D and 2D OpenFOAM simulations for turbulent vertical core-annular flow are performed using the Launder-Sharma low-Re k-ε turbulence model. The pressure drop and the oil holdup are imposed while the total flow rate and the water-cut is obtained as the output from the simulations. The results from the OpenFOAM simulations are compared with the DNS data available in literature to understand the impact of the turbulence characteristics and the interfacial waves. Both 1D and 2D simulations were carried out. The 1D model does not include the interface waves. The 2D axisymmetric model is able to capture the travelling interface waves. The 1D OpenFOAM simulations are carried out for five cases with different oil holdups and the results are compared with the asymptotic wall laws. It is concluded that the turbulence is sufficiently resolved by the 1D simulations. The 2D simulations are also performed for five different oil holdups. The 2D simulations are first validated by performing grid independence tests. The dependence of the simulation results on the streamwise domain length, presence of gravity and oil viscosity is also analysed. It was found that the streamwise domain length and the oil viscosity used does not have a significant influence on the prediction of the results. The comparison of the RANS simulations with the DNS data shows that the difference depends on the turbulence levels inside the simulation domain. The difference for the total flow rate between the 2D OpenFOAMsimulations and the DNS data is found to be 10% for fully turbulent core-annular flow while the difference increases to 29% for core-annular flow with no turbulence. The friction factor for the 2D Open- FOAM simulations is found to be about 17% lower for fully turbulent flow while it is 39% lower than the DNS predictions for the simulation with no turbulence. The difference for the holdup ratio (when comparing 2D OpenFOAM simulations with respect to DNS) decreases first from 15% higher for the low oil holdup cases to 8% higher for simulation with oil holdup fraction 0.71 and then increases to 14% higher for the highest oil holdup case. It is also observed from the comparison of the mean streamwise velocities and the mean shear stresses that the turbulence level in the simulation domain impact the extent of the difference. The agreement in the wave characteristics between the 2D OpenFOAM simulations and the DNS data depends on the mixture of wavelengths present in the simulation domain. The RANS simulations predict a specific dominant wavelength while the DNS data gives a mixture of wavelengths for lower oil holdups. The dominance of a specific wavelength increases with increasing oil holdup for the DNS predictions.

In the oil and gas industry, it is crucial that the transport of hydrocarbons through long pipelines occurs in the most efficient and safest way. These pipelines can cover up to several tens to hundreds of kilometres. Hydrocarbons are not the only fluids present within the pipelines, often they are filled with a mixture of oil, gas, water and even sand particles. The flow phenomena associated with this multiphase mixture complicate the transport operations of the industry. The operations covered in the present study are: flushing, batch transport and hydrate inhibition.% in which is focussed on the physical challenges on.

Simulation techniques exist to reproduce the multiphase flow phenomena. However, three-dimensional Computational Fluid Dynamics (CFD) models require very much computational power and the one-dimensional two-fluid models, such as OLGA and COMPAS, are not capable of capturing all involved flow phenomena. The goal of the present study is to develop a one-dimensional model of high accuracy to reproduce miscible fluid-fluid displacements in long pipelines, in which the focus is on the displacement within the aqueous phase of a multiphase mixture. The model must be able to reproduce slip velocities and gravity currents within this aqueous phase without solving the momentum equations. Furthermore, the model must account for turbulent dispersion, mass transfer, pipeline inclination and differences in thermodynamic properties of the fluids.

Two virtual regions are created in the model in which both fluids can be present. The upper region is associated with the less dense fluid and the lower region is related to the denser fluid. The mass of the regions is tracked by a mass conservation equation. To account for the changes in the concentrations of the fluids in the regions two advection-diffusion equations are solved. These three equations are coupled by exchange terms. The velocity variables are solved by a slip relation between the regions and by a velocity equation derived from the incompressibility condition. All equations are cross-sectionally averaged to ensure the one-dimensionality of the model. To reproduce the appropriate slip interface between the regions several slip relation functions are derived. An empirical relation for the turbulent axial dispersion is implemented in the advection-diffusion equations of the two regions.

The one-dimensional miscible fluid-fluid displacement model is discretised by employing a finite volume method on a staggered grid, by using the Crank-Nicolson time integration and different schemes for the advective terms. The diffusion terms are discretised by the central approximation method. The flow variables are solved by a coupled approach using Picard's method to linearise the nonlinear terms. Moreover, the possibilities of applying Newton's method for linearisation have also been studied. The model is validated by comparing the flow solution of a single-phase advection-diffusion equation to the analytical solution of a turbulent diffusion equation. Even the flow solutions obtained on relatively coarse grids turn out to be of high accuracy.

By simulating miscible fluid-fluid displacement cases it is shown that the model is able to reproduce slip velocities and gravity currents under different circumstances. A linear empirical relation for the volumetric transfer terms is derived by matching the numerical simulation results to experimental results and to CFD results of a specific miscible fluid-fluid displacement case. The CFD results were also obtained as a part of this study; thereto the Fluent CFD package was applied. The CFD results were also used to obtain the exchange terms in the one-dimensional flow model.

The constant impingement of sand particles on the wetted surfaces of a centrifugal dredge pump causes the performance of the pump to deteriorate. Therefore, components or complete pumps have to be replaced after some time. The use of numerical models to estimate the erosion wear in centrifugal dredge pumps can give insight into the principal erosion zones and the magnitude of the erosion in those zones. This information can be used to increase the erosion resistance of pump components. In addition, instructions for the maintenance of the pump can be composed based on the expected erosion wear. In that way, the efficiency of dredge processes can be increased considerably. This Master's Thesis project is carried out by the author during a 12-month period at Damen Dredging Equipment in Nijkerk, The Netherlands. The focus of the project is on developing, validating and demonstrating a numerical model capable of estimating the erosion wear due to slurry flow on the impeller blades of a centrifugal dredge pump by using Computational Fluid Dynamics (CFD). Within the CFD framework, it is found that the appropriate model for calculating the flow of the slurry (water with sand particles) is the Eulerian-Lagrangian method. The numerical computations are performed using the commercial software ANSYS Fluent. For the turbulence of the water, the k-ω SST turbulence model is used. The collisions of the particles with the wall are modeled using the Grant-Tabakoff model, whereas the linear soft sphere model is used for the inter-particle collisions. Using the aforementioned combination of models several studies are conducted, starting with a validation study using a submerged impinging jet benchmark. From that study, it is found that the fluid and particle velocity fields can be calculated with reasonable accuracy. For the same type of problem, the numerical erosion profile is compared to experimental values. It appeared that there is a considerable dependency of the erosion profile on the material hardness. In addition, the result of the four-way coupled method is influenced to a large extent by the values of the collision model parameters. By comparing the resulting erosion profile from the two-way coupled model with the experimental erosion, a good qualitative agreement is found. However, the magnitude of the erosion is overpredicted by this model. In addition, a verification and validation study was performed for the impeller. For the latter, new experiments were conducted in a facility that is available within the company Damen Dredging Equipment. By measuring a number of points before and after a 56 hour experiment, the resulting erosion could be determined. The four-way coupled computation for the impeller could not be fully completed within this project. However, the two-way coupled numerical model showed good correspondence with the experimental results. The currently available model is capable of predicting the maximum erosion zones on the impeller blades. This information could for example be used to improve the design of the impeller or to get a first order estimation of the lifetime of an impeller. The applicability of the numerical model that is developed during the current project can be extended by for instance including the pump volute.

Onshore and offshore multiphase pipelines provide a safe and effective solution for the transport of fluids from wells to production plants in the oil and gas industry. For the control of the accumulated amount of liquid present in the pipeline (the so-called liquid holdup) and for the regular maintenance of the pipeline, pigs (Pipeline Inspection Gauges) are conventionally used. To reduce the pig-generated volume (which is the amount of extra liquid that is removed from the pipeline due to the pig traverse), research in the past decades showed that a bypass pig can be used as a more attractive alternative. With the bypass pig, the size of the slugcatcher (which temporarily stores the pig-generated volume at the end of the pipeline) can be reduced. To reduce the risk of stalling of the bypass pig in the pipeline and the risk of exceeding the available liquid storage volume of the slugcatcher, undersized ball pigs can be deployed prior to launching the bypass pig. The undersized ball pig will be smaller than the pipe internal diameter (e.g., 0.9 ⋅ D) and will not experience significant friction with the pipe wall. Furthermore, the pig-generated volume will be smaller than what is found with the bypass pig as the ball pig will leak some liquid volume; this means that not all liquid is driven out of the pipe by the pig, but some liquid is moved to and left behind the pig. Although ball pigging is already used in some field operations, its use can be extended if the fluid physics are better understood and reliable models for preparing such operations are available.
In this Master Thesis Project, small scale experiments were carried out to investigate the undersized ball pig behaviour in a 52 mm diameter, 60 m long multiphase flow loop located in the Process & Energy Laboratory at the campus of Delft University of Technology. Air and water are used as the working fluids. The performed experiments were used to get a better understanding of the parameters that determine the flow around an undersized ball pig. The pig velocity and the liquid leakage were of particular interest. Two pressure sensors, a liquid hold-up sensor, three cameras, and a weighing scale were used as the measurement system. Three buoyant pigs were created using a 3Dprinter. The three ball pigs varied in size: 42 mm, 45 mm, and 48 mm. In addition, a flexible ball pig of 50 mm was used. The pigs were released in a stratified flow through a pig launcher. The presence of (wavy) stratified flow in this flow loop maintained up to a certain upper limit for the flowrates (u_SG = 4 m/s, u_SL = 0.0628 m/s). Higher liquid flow velocities would rapidly create intermittent slugs. 
The pigging observations showed that it is recommended for industrial applications to create buoyant pigs (e.g., 3Dprinted balls) when using a diameter ratio (the ratio between the pig diameter and the inner pipe diameter) of approximately 0.9. This reduces the risk of stalling of the undersized ball pig. From the local video recordings, it can be observed that undersized ball pigs accumulate liquid while creating one or multiple liquid slugs. The created liquid slug will propagate and separate from the pig, whereafter a new liquid accumulation at the front of the pig is created. The local pig velocity is found to be oscillatory in time, due to liquid slug accumulation and liquid slug propagation. It is found that the normalized pig velocity (being the pig velocity divided by the mixture velocity) increases with increasing superficial gas velocity according to three different regions. In the first region, almost no increase in the normalized pig velocity can be observed. In the second region, the normalized pig velocity increases significantly. In the third region, the normalized pig velocity reaches a stable level. The transition from the first to the second region was found to have a local Reynolds number of about 2000-3000 for the gas. The normalized pig velocity in the second region and in the third region can be approximated with a power law correlation for the various pigs. 
It is found that the liquid leakage increases with increasing superficial gas velocity and decreases with increasing superficial liquid velocity. The liquid leakage versus the superficial gas velocity increases linearly from approximately u_SG = 2 m/s onward. A larger diameter ratio results in a lower liquid leakage. The difference in liquid leakage between the various pigs decreases for large superficial gas velocities (u_SG ≈ 4 m/s). For low superficial gas velocities (u_SG ≈ 0.5 m/s), a liquid leakage below zero was found. Undersized ball pigging becomes ineffective for these low velocities because the liquid flow will overtake the pig instead of leaking past the pig. The experimental leakage results have been approximated with a power law fit. At last, an analytical leakage model, provided by the OLGA commercial package for dynamic multiphase pipeline simulations, is used to compare the experimental liquid leakage. It is found that the OLGA model significantly underpredicts the values of the experimentally found liquid leakage. Improving this model is desirable so that it can be used in multiphase pipe simulations in the industry. 
The results in this study show that undersized ball pigging is associated with some distinct observed phenomena. These observations contribute to the understanding of the fundamental fluid flow structures around an undersized ball pig. The influence of the diameter ratio and the fluid flow velocities on the pig velocity and on the liquid leakage were determined. The measured influence of these parameters will help in the estimation of the pig velocity and the estimation of the leaked volume when industrial pipeline conditions are known. 
It is recommended for future research to focus on improved models to determine the pig velocity and the liquid leakage such that the use of undersized ball pigs can be reliably extended. Additional pigging experiments with local video recordings can be helpful.

The production of heavy oil from reservoirs requires much more effort than the production of lighter oils. This is due to the fact that this heavy oil has a significantly higher viscosity; namely, typically a factor 10 to 10000 more viscous than water. This higher viscosity makes the transport through pipes less economically viable due to the enormous pressure drop in the pipe. An interesting way to reduce this pressure drop is by creating core annular flow, which is a two phase flow where the less viscous fluid forms an annulus around the viscous fluid. This significantly reduces the pressure drop in comparison to single phase oil flow. This research focuses on determining the water hold-up in lab experiments of core-annular flow in a horizontal pipe. With the water hold-up and the water cut also the so-called hold-up ratio is known, which is a measure for the oil-water slip. The experiments were conducted for a constant oil flow rate of 0.35 l/s in horizontal pipe with 21 mm diameter. Different water cuts (10 to 20%) and oil/water viscosity ratios (600, 3000) were measured. To determine the water hold-up, the area of the pipe which is occupied by water (or oil) is required. In order to do this the flow is visualised with a high speed camera. Due to the difference in refractive index of the different media that the light has to travel through to eventually reach the lens of the camera, some optical distortion is encountered, which has to be corrected for. The correction procedure was done through a ray tracing analysis which produced a calibration curve that could deduce the actual position of the oil-water interface from the recorded movie. From the temporal and spatial movement of the oil-water interface the waves on this surface were analysed: wave length, wave frequency, wave speed. This was done through applying an autocorrelation function to the interface data. Besides the flow visualisation (to determine the water hold-up and wave characteristics), a pressure transducer was used to measure the pressure drop over a one metre long section of the pipe. The flow visualisation proved that for low water cuts (10%) the hold-up ratio is significantly higher than that for the higher water cuts (15% and 20%). This means that there is relatively more water accumulation and therefore more oil-water slip when the water cut is reduced. The waves on the interface become longer and the frequency becomes lower with an increasing water cut resulting in an almost constant wave speed. To see the effect of a lower viscosity the temperature of the oil was increased from 20 C to 40 C (which decreases the oil-water dynamic viscosity ratio from 3000 to 600). This did not have a noticeable effect on the water hold-up. However, the wave length decreased and the frequency increased which still resulted in a similar wave speed. The results from the experiment have been compared to CFD simulations carried out by PhD candidate Haoyu Li. Interestingly the CFD simulation gives a pressure drop which is roughly 30% lower than the values measured during the experiment. The measured water hold-up fraction is 0.257, whereas the CFD simulation gives 0.255; however, the oil/water interface determined by the simulation is not in agreement with the experimentally recorded interface. The wave frequency given by the CFD simulation is half that which is recorded during the experiment, while the wave length given by the CFD simulation is twice that which is recorded during the experiment. These differences between the CFD simulation and the experiment still lead to an almost equal wave speed.

In the oil and gas industry the occurrence of slug flow in flowlines and risers can cause operational problems. Such flowline-riser systems are used to transport the oil and/or gas from the location of the wells to a production platform, where the fluids might be separated into single phases. Slug flow conditions imposefluctuations in the production rate, which may lead to the flooding of the separators, trips of compressors or pumps, and to increased loads on the supports of the pipeline and piping sections at the production platform.Therefore, measures have to be taken to mitigate the slugging, which can be a reduction of the production rate, making adjustments to the pipeline system, or adding surfactants to the flow. For vertical flow in production wells, the use of surfactants is a proven technology. Here the creation of a foam through adding asurfactant can increase the production life time, as the accumulation of liquid, which typically occurs when the reservoir pressure has decreased at the end of field life, is prevented. Far less is known yet about the effect of using surfactants for the mitigation of hydrodynamic slugs in a nearly horizontal flowline or the mitigation of severe slugs in risers. From a previous Master’s Thesis project by Pronk [1], in which lab experiments were carried out in the same flowline-riser facility as used in the present study, it was concluded that growing slugs could be suppressed by adding a surfactant to create foam. The surfactants also influence the characteristics of the severe slugging cycle, but they are not able to fully suppress it. Re-analysis of the measurement data by Pronk hasturned out that instead of growing slugs, severe sluggingwas measured and therefore that severe slugging can be suppressed using surfactants and no conclusions can be drawn on growing slugs or hydrodynamic slugsfrom her research. There is limited literature on the effect of adding a surfactant to hydrodynamic slug flow in horizontal flowlines, but there are some indications that hydrodynamic slugs can be suppressed, though requiring a higher concentration of the surfactant than vertical flow. Suppressing slug flow using surfactants should eliminate the fluctuations in the flow and can lead to a lower pressure drop. For this research experiments have been carried out in the Severe Slugging Loop at the Shell Technology Centre Amsterdam. This flow loop consists of a horizontal flowline of 50 m, a U-turn and a 15 m section. From there onward the pipe starts declining at an angle of -2.54° over a length of 35 m. Thereafter there is a short horizontal section of 5 m before arriving at the base of the riser, which has a height of 16.8 m and an internal diameter of 32 mm. The other sections have an internal diameter of 50.8 mm. For this research the configuration was used where air and water are connected through a Y-sprout at the start of the flow line. The air is supplied as pressurized air of 6 barg. The water is supplied by a pump, and it is separated from the airat a section at the top of the riser and recirculated through the system. The used surfactant is the household detergent Dreft. At several points along the flowline and riser the flow conditions have been measured using pressure indicators and at two locations along the flowline the pressure difference over 3 m has been measured using differential pressure indicators. Through the inspection windowat the end of the flowline the flowis recorded by using a GoPro camera. Furthermore, over a distance of 40mthe acoustic energy of the flow has been measured by a Distributed Acoustic Sensing (DAS) system. The DASmeasurements have been used to determine the flow behaviour of the flow. In the case of slug flow, the velocity of the slugs was deduced from the DAS data. Adding a 2000 ppm concentration of the surfactant Dreft Original has resulted in the full suppression of hydrodynamic slug flow. This has been verified using the pressure data, the video data and the DAS data in combination with the velocity tracking tool. The waves that were present without surfactant are decreased to a much smaller wave height when the surfactant is added; therefore, the flow pattern is changed from slug flow to wavy stratified flow. The severe slugging cycle can also be suppressed using 2000 ppm concentration of the surfactant Dreft Original. At the same flow conditions, the gas is able to lift more liquid through the riser. The measurements by the DAS system were quantified using a velocity tracking algorithm. The measurement data can now be used to accurately measure the slug velocity and to determine whether slug flow or stratified flow is present in the system. This measurement method is ready for scale-up. When the concentration of surfactant is sufficient to suppress slug flow, the flow can be described from visual observations as a threephase flow, containing gas, foam and liquid. The foam absorbs the stresses imposed by the other phases, because of its non-Newtonian fluid properties. The waves are no longer able to grow until they reach the top of the pipe and are diminished into slow moving dampened waves of the foam layer.

Shifting from coal and oil towards sustainable energy sources is currently an urgent and global challenge. Compared to coal and oil, natural gas is a cleaner fossil fuel and may serve as a transition fuel. To meet future demands for gas there is a desire to optimize the production from existing reservoirs with their production facilities. In case of wet natural gas wells this gas production can be hampered by the presence of liquids, namely water and condensate, thus creating a multiphase flow in the production tubing. The maximum gas well capacity is limited by the ever reducing upward gas flow velocity due to the subsurface reservoir pressure that is reducing over time. The reduced gas velocity results in the inability to transport the liquid upward from the downhole location to the surface. The accumulated liquids will then block the well bore (so-called liquification), making further gas production impossible. One solution it to decrease the cross sectional flow area by inserting an inner pipe in the existing tubing, thus creating an annulus. This results in a higher gas velocity. The multiphase flow characteristics in the annulus tubing are not yet fully understood, such as the pressure drop, the liquid hold-up and the different flow regimes. Therefore in the present study lab experiments were carried out to further investigate those characteristics to enable optimisation of predictive flow modeling. The main focus is on the behaviour of the liquid film along both pipe walls. In this study a 12 m mid-scale vertical annulus with a 124 mm diameter outer pipe and a 100 mm diameter inner pipe was designed and constructed. The experiments were conducted at atmospheric conditions using an air-water mixture as the working fluids. Multiple combinations of gas and liquid throughputs were studied. The liquid at the inlet could either be injected on the inner pipe, the outer pipe or equally divided on both pipes. The position of the inner pipe with respect to the outer pipe was adjustable to create different eccentricities. To measure the local film heights flush mounted conductance sensors were designed, built and installed in the half circumference of both pipes. The efficacy of the sensors was thoroughly studied and the sensors were subjected to multiple test cases. The concentric annulus experiments showed that the pressure drop does not depend on the method of liquid injection. However, the liquid hold-up fraction at lower liquid throughputs was observed to be lower for the single wall liquid injection as compared to injection on both pipes. The redistribution of the liquid film for the single liquid injection depends on the liquid throughput. Larger liquid throughputs gave thicker liquid films containing a wavier gas-liquid interface from which droplets are atomized and subsequently migrated to the other pipe wall. Furthermore, it was found that the critical superficial gas velocity at which liquid loading occurs is 14 m/s, which is neither dependent on the method of liquid injection nor on the liquid throughput. The designed sensors were able to provide a good indication of the film heights. Single phase experiments, using air as the working fluid, showed that the pressure drop is reduced by 30$\%$ in the fully eccentric case in comparison to the concentric case. The eccentric annulus configuration with air-water flow showed that the liquid films along both pipe walls may merge. This so-called liquid bridging mainly occurs at higher eccentricities and causes flow reversal in the narrow gap. This results in a much higher liquid hold-up and pressure drop. Eccentricity induces redistribution of the liquid film from the inner pipe to the outer pipe at higher superficial gas velocities. It was found that eccentricity promotes an unequal film height along the circumference of each pipe: the film height decreases when going from the narrow gap towards the wide gap.

The climate change agreements require large investments in renewable energies. However, with the increasing demand for energy, immediately stopping the production and use of fossil fuels is not realistic. Natural gas is twice as clean as coal and hence could be considered as a transition fuel, which can help to find a balance between the requirements of fighting climate change and producing more and cleaner energy.The high-pressure, underground reservoirs supply natural gas which is usually wetted with liquid hydrocarbons and water. Natural gas is lifted through a production tubing to the surface, with a relatively high velocity and the drag between the gas and liquid will also bring the oil (or condensate) and water to the surface. However, over time the gas supply from the reservoirs will deplete, which gives a decreasing gas velocity, until liquid starts to fall. This is known as liquid loading, which can finally lead to the killing of the gas production. One of the solutions to operate such aging reservoirs is to reduce the cross-sectional area of the production tubing through making use of an annulus (which actually is the space in between an outer pipe and an inner pipe).Liquid loading is noticed to occur when the flow regime changes from annular flow to churn flow. A good understanding of this multiphase flow behaviour in an annulus is desired. Therefore, this Master Thesis project was devoted to obtaining new experimental data and to develop a model for upward gas-liquid flow in a vertical annulus. Experiments with air and water were carried out in the small-scale flow loop of TNO in Delft. The air and water throughputs were varied, and both concentric and eccentric pipe-in-pipe configurations were measured. Both the pressure drop and liquid holdup were measured. The model is able to predict the pressure gradient and holdup for a range of superficial gas and liquid velocities, using different eccentricities in the annulus configuration. The model predicted the pressure gradient within 10% of the experimental values at high superficial liquid velocities (>0.5 cm/s). However, for a superficial liquid velocity of 0.5 cm/s, the model was overpredicting the pressure drop by 100%, which could be attributed to the partial dry-out on the tube walls. To determine the superficial gas velocity, for a given pressure drop and holdup, a modified version of the Wallis correlation was used for the interfacial friction factor. The gas velocity was predicted within 5 to 10% at low holdups (< 0.075). For higher holdups the gas velocity was overpredicted by about 40%. The primary modifications made to the model for the eccentric cases are based on applying a number of grid cells in circular direction allowing for the inclusion of a closure for the variation of the circular film thickness both at the inner and at the outer pipe wall. A verification of the closure of the amplitude or of the closure of the film thickness variation is required to confirm the dependency between this parameter, the eccentricity and the holdup. However, with the applied closure of the amplitude the pressure gradient with respect to the holdup was predicted within 6% accuracy for the eccentric cases.

The transport of highly-viscous oil in the core-annular flow regime in a horizontal pipe is investigated with a numerical simulation study. In this flow type the viscous oil is lubricated by a water annulus along the pipe wall. The Launder-Sharma low-Reynolds number k-ε turbulence model is used with the Volume-of-Fluid solver in interFOAM. Three types of simulations were carried out: 3D multiphase flow in a pipe section, 2D multiphase flow in an axi-symmetric pipe section (wedge-shaped section), in which gravity is ignored, and 2D single phase water flow in the annulus, using an imposed wavy boundary. This enabled to study the effect of the viscous oil core on the annulus behaviour, such as the turbulence structures and possible dispersion of oil into the water annulus. In particular the rationale of a 'solid-core' assumption is discussed through the comparison between simulations for the annulus that use the 'interface-bounded' flow and 'wall-bounded' flow.

A main finding of this study is that the use of the Compressive Volume of Fluid (CVOF) method in the multiphase simulations gives spurious dispersion of oil in the water annulus. This is the primary cause of the over-prediction of the pressure gradient simulations carried out in a previous study. Therefore, the Coupled Level-Set Volume of Fluid (CLSVOF) interface capturing method has been implemented in the 3D multiphase model. The new predictions show a good agreement with the experimental data at 20, 30, 40℃ (corresponding to a descrease in the oil viscosity). The oil dispersion in the annulus region close to the wall is diminished by using the new method while the sharpness of the interface is not improved. The CLSVOF interface capturing method is concluded to be an efficient and reliable interface capturing method for the numerical prediction of core-annular flow.

Pipeline networks are extensively used in the oil and gas industry to transport fluids under multiphase flow conditions from the wells to the production platforms or plants. Over time, the flow rate of the production fluid decreases as the reservoir pressure decreases. When the flow rate decreases to below a certain value, unstable flow with liquid slugs occurs and the transportation of the fluids becomes more difficult. This is because of the high amount of liquid accumulation (holdup) in the pipeline. To extend the operation life of the production system, it is important to maintain the continuous production at low flow rates. Regular pigging, which refers to transporting a “pig” (Pipeline Inspection Gauge) through the pipeline, has been a conventional way of managing the liquid holdup in the oil and gas industry. Also "by-pass" pigs have been actively used to manage the liquid holdup since they generate smaller slug volumes compared to the traditional non-bypass pigs. However, since the pipeline can be as long as a hundred (or more) kilometers, the volume of the pig-generated slug at low flow rates can be very large. Even this reduced slug volume produced by using a by-pass pig may exceed the capacity of the downstream separator or slug catcher. Moreover, by-pass pigs with a large by-pass area have a high chance to get stuck in the pipeline at low flow rates. To enable pigging operation at those conditions, the use of an undersized ball pig was proposed. Here the ball pig is transversed through the pipeline prior to the by-pass pigging operation. A few test runs with undersized ball pigs were carried out in the F14 multiphase pipeline in Sarawak. To improve its performance, it is of much interest to study the detailed flow around an undersized ball pig. Although in actual operation the ball pig will be used under multiphase conditions (i.e. gas and liquid flow), in this study, as a starting point, single-phase flow was considered. This study investigated the detailed flow around an undersized ball pig in a horizontal pipe with the help of Ansys Fluent version 18.2. First, benchmark data from experiments are used to find the best numerical model which can be used to simulate the flow around a sphere. The comparison between experimental data and numerical results shows that the 2D laminar simulation can accurately capture the flow when Re < 300. The 2D simulation with the SST k-ω turbulence model (low-Reynolds number correction) gives an accurate prediction of the drag coefficient when 300 < Re < 5E5. The selected numerical models are then used to study the flow around an undersized ball pig in a pipe. For each set of conditions, this requires to carry out a number of simulations at different ball pig velocities, and find the velocity that gives zero drag force on the ball pig. The undersized ball pig is first moved to the centre, and 2D simulations were performed. The force analysis around the undersized ball pig at low Reynolds number shows that at equilibrium state, the undersized ball pig experiences a positive pressure force and a negative viscous shear force. Moreover, the computed profile of the normalized terminal velocity of the pig with a diameter ratio of 0.5 at various Reynolds numbers shows that the normalized terminal pig velocity profile experiences three stages when the Reynolds number is increased. In the first stage where Re < 365, the motion of the pig is dominated by the viscous shear force and the normalized terminal velocity of the pig is constant. When Re increases, the flow enters the second stage where the inertia force starts to affect the motion of the pig and the normalized terminal velocity of the pig decreases dramatically. When Re is further increased to above 5340, the motion of the pig is completely affected by the inertia force and the normalized terminal velocity of the pig stays in a stable range when Re increases. After this detailed study of the flow around an undersized ball pig with a fixed diameter ratio, simulations of the flow around undersized ball pigs with various diameter ratios are studied to find the influence of the pig diameter on the terminal velocity of the pig. The results show that the obtained profiles of the normalized pig terminal velocity against Re at various diameter ratios have a similar shape as when the diameter ratio was 0.5; the terminal pig velocity decreases when the pig diameter increases at a given Re. A further data regression analysis shows that the normalized pig terminal velocity in the region when Re < 300 and Re > 10000 has a second-order polynomial relationship with the diameter ratio. After the simulations for the flow around an undersized ball pig that moves along the axis of the pipe, the pig is moved to the bottom of the pipe and the influence of this position change is studied at a diameter ratio of 0.9. Due to this position change, a 3D simulation is required. First a 3D simulation for the previous configuration with the pig at the centre is performed. This simulation shows that the flow is actually unsteady and asymmetric. The terminal pig velocity at a given Reynolds number obtained from the 3D simulation is slightly higher than the one obtained from the 2D simulation. As the 2D simulation cannot capture the asymmetry of the flow, it provides less accurate results. The 3D simulation results of the flow past an undersized ball pig moving along the axis of the pipe is then compared with the one moving along the bottom of the pipe. The comparison shows that the ball pig terminal velocity is decreased when the pig is moved from the centre to the bottom of the pipe.

Steady state multiphase flow models are used for the design and operation of pipelines that are used in the oil and gas industry. The predictions obtained with these models contain uncertainties, which arise from model assumptions and simplifications, and from uncertainties in the input parameters. In this thesis we investigate the effect of uncertainties in the input parameters (such as the wall roughness or the superficial velocities) on two main quantities of interest in steady state multiphase flow in pipelines: the liquid holdup and the pressure drop.
The approach that we take is to describe the uncertain input parameters by probability density functions (PDFs), propagate these through the steady state models, and obtain a PDF for the quantities of interest. We use two methodologies from the field of Uncertainty Quantification (UQ) to perform the propagation step: Monte Carlo sampling, the current standard in the literature, and Polynomial Chaos Expansion (PCE), our proposed approach. Furthermore, we use Sobol indices to perform a sensitivity analysis that ranks the input parameters depending on their contribution to the output.
Our proposed UQ methodology is applied on two commonly used multiphase flow models in the oil and gas industry: a 0-D model, the Shell Flow Correlations (SFC), and a 1-D model, PIPESIM.
First, application to the SFC reveals that PCE is much more efficient than Monte Carlo sampling, and an improvement of several orders of magnitude is achieved in terms of the number of samples required for a given accuracy, while the evaluation of a single sample requires the same computation time for the two methods. Furthermore, with UQ the flow pattern maps commonly used in industry can now be displayed in a probabilistic way. This allows the quantification of the probability that a flow regime (e.g. slug, stratified) occurs under given conditions. These are significant improvements compared to existing work that handles uncertainties in multiphase flow models.
Second, application of PCE to a multiphase pipeline, known as Goldeneye, modelled in PIPESIM revealed that several uncertain variables, namely the wall roughness, the ambient temperature and the outlet pressure, play a role in determining the liquid holdup and the pressure drop. This is in contrast to what was assumed in an earlier benchmarking study. Furthermore, our probabilistic approach allows us to make predictions under uncertainty. For instance, we can now predict that the liquid holdup of the Goldeneye pipeline will have a 65% probability to be lower than the value of 1496 m3, the value obtained when using a deterministic setting.

This study was aimed at improving the accuracy of the model predictions for the minimum fluid and inner wall temperatures for cold, low-pressure start-up of wells that produce oil or gas. Due to the large pressure drop over the well head choke, the so-called Joule-Thomson cooling will give a very low temperature of the
expanding gas jet. Low-temperatures can give brittle fracture of the material in the piping downstream of the choke. Models are needed to verify whether the material temperature remains above the lower-design temperature. For the model validation, Imperial College in London (on request by Shell) has carried out lab experiments with argon gas that expands through an orifice from 120 bara to 1 bara. Awaiting the results of the lab experiments, detailed simulations were carried out in the present study using the Fluent CFD programme.

The 3D, steady, compressible Reynolds-Averaged Navier-Stokes equations were solved with the SST k −ω model for the turbulence. The considered configuration is the same as in the lab. It consists of an upstream chamber with argon at 120 bara, that expands through a 5 mm long orifice with 1.55 mm diameter, into a square outlet section with 50 mm sides and 500 mm length. The inlet temperature is -17 oC and the outlet pressure is 1 bara. The supersonic flow leaving the orifice reaches a maximum Mach number of about 9, just before a shock to subsonic flow is found. The jet reaches very low temperatures due to isentropic expansion, and reaches the isenthalpic expansion temperature of 196 K (or -77 oC) downstream of the shock. The jet reaches the sides of the outlet at a distance of about 100 mm.

The maximum Mach number of about 9 predicted by Fluent is higher than the value of about 6 found in a previous simulation study that used the STAR-CCM+ CFD programme. To verify the Fluent results, the distributions of grid cells was varied and the number of grid cells was increased. Also, a MATLAB programme was written that solved the inviscid compressible equations (Euler equations) for an axisymmetric jet.

This confirmed the Fluent results. In addition to the 3D square outlet section, also 3D and 2D Fluent simulations were carried out for a cylindrical outlet (using a hydraulic diameter of 50 mm). The maximum Mach number and the jet structure (velocity, temperature) are not affected by the side walls. This is because the side walls are sufficiently far from the jet.

Furthermore, the temperature and the heat transfer at the walls of the outlet section were investigated. Thereto both adiabatic and non-adiabatic walls were considered. The ambient temperature is 20 oC. Thermal boundary layers are formed along the side walls, that are exposed to a temperature of 196 K (the isenthalpic expansion temperature) in the centre of the pipe, up to a distance of about 1.5 m, where the outer edge of the boundary layer reaches the centre of the pipe. Thereafter the centre line temperature increases due to heat inflow from the ambient.

The Interface Capturing method, which is a finite volume method as formulated by Queutey and Vissoneau for free surface immiscible, incompressible multiphase flows employs a collocated arrangement of unknowns
and achieves a discrete force balance for the case when the interface coincides with the faces of the control volumes in the computational domain. This constraint limits the applicability of the method. Furthermore,
the authors do not provide the exact formulation of the operators involved in the pressure velocity coupling. In the present research, a balanced-force numerical method is formulated, applicable for an interface that
neither has to coincide nor be aligned with the faces of the control volumes. The approach consists of the reconstruction of the values of the flow variables at the interface based on the interface jump conditions, with which the limit values of the normal derivatives at the interface are calculated. Furthermore, the construction of the operators of the discrete system is delineated to achieve a discrete force balance, by incorporating the reconstructed flow variables and employing a discretization which complies with the interface jump conditions. It is sufficient for a stationary discrete formulation to comply with the differential equation and the interface jump conditions. However, to apply this approach to solve unsteady flow problems the influence of the reformulated operators on the stability properties of the system should also be investigated.
The properties of the individual operators are analyzed as well as their behaviour when they are embedded in the complete solver algorithm. Results are shown for both steady and unsteady test cases and compared with numerical results obtained with OpenFOAM. The resulting framework avoids the occurrence of
spurious velocities as it discretely complies with the interface conditions.

Transportation of very viscous fluids through pipelines is a real challenge. With the depletion of light oils in reservoirs, it becomes economically favourable to harvest heavy crude oils to contribute to meeting the ever growing energy demand. A suitable candidate for the transportation of these very viscous oil is by means of core-annular flow. Core-annular flow is a flow regime of liquid-liquid two-phase flow, where a low viscous fluid in the annulus (water) is used to lubricate a high viscous fluid in the core (oil). The pressure drop is considerably reduced compared to single phase oil flow at the same oil flow rate.

In this study the influence of the oil viscosity on oil-water core-annular flow through a horizontal pipe is investigated experimentally. The fixed oil flow rate is set at 0.35 l/s at which the watercut is varied between 9% and 25% and the oil kinematic viscosity is altered by heating up the oil in a range from 3000 cSt at 20 °C to 400 cSt at 50 °C. Pressure drop measurements for stable core-annular flow are recorded with an electronic pressure transducer and are scaled with the calculated pressure drop of single phase oil flow.

Results for the scaled pressure drop at room temperature are compared to results by Ingen Housz et al. It is concluded that for increasing oil-water viscosity ratio, the scaled pressure drop decreases. At the highest considered viscosity ratio, the scaled pressure drop is almost independent of the watercut. Two models to predict the pressure drop (by Brauner and by Bannwart) are evaluated and deviations between the models and measurements are discussed. Visualisation by means of high-speed camera is applied, where a mirror is placed on top of the visualisation section to capture the front and top view simultaneously. For decreasing oil viscosity, the oil-water interface shows a more irregular wave pattern with shorter wave lengths.

Oil and gas production systems consist of flows in wells, in horizontal and inclined flowlines and in risers. Due to the reservoir composition and changes in pressure and temperature, the flow is often under multiphase conditions. With time, the reservoir matures and the reservoir pressure decreases. The gas rate drops below a critical gas rate, that is required to produce the liquid to the wellhead and the facilities. As a result, liquids build up down-hole in the well. This phenomenon is known as liquid loading, which may kill the well. One of the deliquification methods is to use surfactants, which are injected through a capillary at the bottom of the well. Recent field trials have shown that the injection of surfactants down-hole in a well can prevent liquid loading problems. The surfactants create a foam which forms relatively thick interfacial waves along the wall of the production tubing. The gas core gets more grip on the liquid film, making it easier to produce the liquid to the surface.

Due to the success of the application of surfactants in subsurface vertical wells, it is of interest to investigate whether surfactants can also help to overcome liquid management problems in flowline-riser systems on the surface. The objective of this work is therefore to find the effect of surfactants on two-phase (air-water) flows in flowlines and risers. Experiments were carried out in the Severe Slugging Loop at the Shell Research and Technology Centre Amsterdam. The flow loop consists of a 100 m horizontal and downward inclined flowline with an inner diameter of 0.051 m, followed by a 16.8 m riser with an inner diameter of 0.044 m. The working fluids are air and water, and operation is at atmospheric outlet pressure. Two configurations of the SSL were used. In configuration #1, both water and air are injected into the flowline. The mixture is transported through the flowline to the riser. In configuration #2, water is injected into the flowline. Air is injected directly into the riser base. The dish washing detergent Dreft is used as a surfactant to create foam. The concentration was systematically increased to find the effect on different types of slugging (in the flowline, in the riser and severe slugging at the riser base), the pressure gradient in the riser, the developing length in the riser and the liquid and foam holdup in the riser. Various measurement techniques were used: Differential Pressure Indicators (DPIs), Pressure Indicators (PIs), Distributed Acoustic Sensors (DAS), quick closing valves and flow visualization.

When operating the SSL in configuration #1 with a superficial gas and liquid velocity of u_sg= 1.21 m/s and u_sl= 0.4 m/s, slugging in the flowline is observed. The slugs are identified as growing slugs: the stratified flow builds up regularly due to a growing instability, forming a slug. The average length of the liquid body of the slug is 32 m, the average passing time of the liquid body of the slug is 51 s, and the average slug velocity is 0.66 m/s. Through DAS and pressure measurements one can see that the slugs are mitigated when the surfactant is added. The slugs completely disappear when an effective surfactant concentration of 1000 ppm is added to the air-water mixture.

When operating the SSL in configuration #1 with a superficial gas and liquid velocity of u_sg= 1.4 m/s and u_sl= 0.27 m/s, a severe slugging cycle is found. Without surfactants, the cycle has a period of 109 s in which the riser is filled entirely with the liquid body of the slug before is is pushed out by the air. Pressure drop measurements over the riser show that adding surfactants does not prevent the slugging cycle to occur. However, the creation of foam does increase the amount of gas in the riser, making the pressure build-ups more irregular. The slugging cycle reduces to 89 s when an effective surfactant concentration of 3000 ppm is added to the air-water mixture.

When operating the SSL in configuration #2 with a superficial gas and liquid velocity of u_sg= 0.37 m/s and u_sl= 0.27 m/s, slugging in the riser is observed. Differential pressure measurements were taken over a distance of 3 m at multiple locations: one at the riser base, and two at the top of the riser. The surfactant reduces the differential pressure along the riser. For a concentration of 3000 ppm, the differential pressure reduces by 5.1 % at the riser base and by 18.9 % at the top of the riser.

The pressure drop curve is used to analyze the flow behaviour for a range of superficial velocities. Two types of pressure drop curves were considered: 1) for a constant superficial liquid velocity (u_sl= 0.05 m/s), and 2) for constant gas-to-liquid ratio (GLR= 60 and GLR= 100). For the low gas flow rate region, i.e. the gravity dominated part of the curve, both methods show a decrease in pressure gradient for a surfactant concentration of 500 ppm or greater. In the high gas flow rate region, i.e. the friction dominated part of the curve, an increase in the effective concentration leads to an increase in the pressure gradient.

The developing length of the air-water mixture with and without surfactants is analyzed by means of differential pressure measurements. Measurements were taken over a distance of 3 m at multiple locations: one at the riser base, and two at the top of the riser. With an effective concentration of 3000 ppm, the differential pressure is slightly different for the two locations on top of the riser. This indicates that the flow is not fully developed at the top of the riser. The developing length is slightly increased with the addition of the surfactant.

The foam holdup is analyzed by measuring the height of the foam between two simultaneously closed quick closing valves on the riser. The experimental results are compared with the simulation results from the Shell Flow Correlations. The experimental results show a spread due to the transient behavior of the flow. Despite the spread, the experimental results follow the trend of the simulations. At low gas flow rates, surfactants decrease the foam holdup. At high gas flow rates, surfactants increase the foam holdup. However, the foam holdup decreases for increasing gas flow rates and eventually levels off to a constant holdup value.

It can be concluded from the small-scale experiments that surfactants: 1) mitigate (growing) slugs in flowlines, 2) do not remove the severe slugging cycle, 3) decrease the pressure gradient in the riser for small gas flow rates, 4) slightly increase the development length of the flow in the riser, and 5) decrease the foam holdup for low gas flow rates, and increase the foam holdup for high gas flow rates in the riser.
It is recommended to carry out the experiments on a larger scale, i.e. at higher temperatures, for larger pipe diameters, to better relate to existing flowline-riser production systems. It is also recommended to perform similar experiments to find the effect of the surfactant on other slug types, such as terrain slugs and hydrodynamic slugs, in the flowline.

Core-annular flow is an efficient flow regime for the transportation of viscous oils. The viscous oil in the core is surrounded and lubricated with an annulus of water. Water has a low viscosity and therefore reduces the pressure drop. Numerical simulations are performed for horizontal core-annular flow by using the Volume of fluid (VOF) method to solve the Reynolds-averaged Navier-Stokes equations (RANS) in OpenFOAM. Periodic boundary conditions are used for a small pipe section. With periodic boundary conditions the maximum wavelength at the oil-water interface is imposed together with the oil and water holdup fractions.

Numerical results are compared with recent experimental data. Oil viscosities between 3338 cSt at 20°C and 383 cSt at 50°C are solved with a fixed total flow rate. The 20°C simulation converged to the desired water cut of 20%, similar to the experiment. At this 20% water cut a comparable pressure
gradient is found with the experiment. At higher temperatures (i.e. lower viscosities) deviations from the desired water cut of 20% are obtained. The different water cut values lead to differences between the numerical and experimental pressure gradients.

Additional simulations are carried out for the oil viscosity of 718 cSt at 40°C. Instead of a fixed total flow rate these simulations are solved with a fixed pressure gradient. This solving method is found to be considerable faster. Different holdup fractions and pressure gradients are imposed which resulted in different water cut and flow rate values. Numerical results are interpolated for the experimental water cut values of 9%, 12% and 15% at similar flow rates. Differences of 98%, 24% and 37% are found for the water cut of 9%, 12% and 15% respectively. An interpolation is not possible for the experimental water cut of 20% as this experiment is outside the covered solution region of the numerical results. This difference is caused due to an incorrectly used domain length.

The study is finished with holdup estimations of the experiments. Flow visualizations from a high speed camera are used which are made during the experiments. Oil holdup fractions of 0.749 are found for a water cut of 20%. This holdup fraction corresponds exactly with the imposed holdup fraction for the different viscosity simulations which are solves with the fixed total flow rate. Only the 3338 cSt at 20°C converged, however, to the water cut of 20%.