ࡱ>  nhq HYbjbjt+t+ AA`Pzm]   6JJJ$nnnn>*\n+a* uaaaaaaa!bdTaJ"aA[JJ* A[A[A[JJannJJJJaA[BA[^`|JJa fnn8"aModelling of Terrain Induced Slug Flow  Diploma Thesis by Jon Kolsum Reutz Autumn 2000 Department of Chemical Engeneering NTNU Statement I hereby declare that this work is done in compliance with the exam-ragulations at NTNU. Trondheim, 14/12-2000Jon Kolsum Reutz Abstract Slug flow is a major problem in oil production. Some of the main problems are overfilled separators and tripping of compressors. ABB Industri AS have been working with slug control at well Brage A-27, in order to avoid such difficulties. The work in this thesis is done in cooperation with ABB. The object of this thesis has been modelling of terrain induced slug at Brage A-27. A simple model would be a good basis for development of a structure for slug control. However, the modelling was a very complex problem, and the work has been fully concentrated towards this issue. Both modelling in Simulink and Matlab has been tried. Simulation in Simulink was not successful. The problems were large simulation time and a very incomplete implementation of conservation of momentum. In Matlab better results were achieved. After a lot of modifications and different approaches had been tried, a model that generated slug flow was obtained. The model has not the same geometry as Brage A-27, since it was almost impossible to start simulations with the original geometry and in a slugging flow regime. The model shows a behaviour that was expected and typical for a system in a slugging regime. It should therefore be suitable for examination and implementation of slug control. However, numerical troubles were difficulties through the entire work, and many hours work was used to cope with these problems. As for control of the system, some previous work is shown, and controlled and manipulated variables are suggested.  TOC \o "1-3" 1 Introduction  PAGEREF _Toc501438653 \h 6 2 Slug Flow Theory  PAGEREF _Toc501438654 \h 7 2.1 Riser Induced Slug Flow  PAGEREF _Toc501438655 \h 7 2.2 Terrain Induced Slug Flow  PAGEREF _Toc501438656 \h 8 2.3 Hydrodynamic Slug Flow  PAGEREF _Toc501438657 \h 9 2.4 Slug Control  PAGEREF _Toc501438658 \h 10 3 Case description  PAGEREF _Toc501438659 \h 12 4 Modelling of Slug Flow  PAGEREF _Toc501438660 \h 17 4.1 Basic theory for two-fluid modelling  PAGEREF _Toc501438661 \h 17 Conservation of mass  PAGEREF _Toc501438662 \h 17 Conservation of momentum  PAGEREF _Toc501438663 \h 17 Conservation of energy  PAGEREF _Toc501438664 \h 18 Pressure calculations  PAGEREF _Toc501438665 \h 18 Discretization  PAGEREF _Toc501438666 \h 19 Algebraic relations  PAGEREF _Toc501438667 \h 21 4.2 Well Conditioned Model  PAGEREF _Toc501438668 \h 24 5 Results  PAGEREF _Toc501438669 \h 25 5.1 The Simulink Models  PAGEREF _Toc501438670 \h 25 Assumptions  PAGEREF _Toc501438671 \h 25 Equations and relations  PAGEREF _Toc501438672 \h 25 Control volumes  PAGEREF _Toc501438673 \h 27 State in the control volumes  PAGEREF _Toc501438674 \h 28 5.2 The Matlab Models  PAGEREF _Toc501438675 \h 28 Assumptions  PAGEREF _Toc501438676 \h 28 Equations and relations  PAGEREF _Toc501438677 \h 29 Control volumes  PAGEREF _Toc501438678 \h 30 State in the control volumes  PAGEREF _Toc501438679 \h 31 Mathematical solver routines  PAGEREF _Toc501438680 \h 39 6 Discussion  PAGEREF _Toc501438681 \h 40 The Simulink Models  PAGEREF _Toc501438682 \h 40 The Matlab Models  PAGEREF _Toc501438683 \h 41 Modelling structure  PAGEREF _Toc501438684 \h 41 Changes in inflow  PAGEREF _Toc501438685 \h 41 Geometry and initial conditions  PAGEREF _Toc501438686 \h 42 Shear stress at the phase interface  PAGEREF _Toc501438687 \h 42 Discretization  PAGEREF _Toc501438688 \h 43 Mathematical solving routines  PAGEREF _Toc501438689 \h 43 Suggestions to further Work  PAGEREF _Toc501438690 \h 44 7 Conclusions  PAGEREF _Toc501438691 \h 45 8 Notation  PAGEREF _Toc501438692 \h 46 9 References  PAGEREF _Toc501438693 \h 48 Appendix 1: Perimeters  PAGEREF _Toc501438694 \h 49 Appendix 2: Flow Sheet in Simulink  PAGEREF _Toc501438695 \h 50 Appendix 3: Matlabcode for the first Matlab model  PAGEREF _Toc501438697 \h 51 Appendix 4: Matlabcode for the final Matlab model  PAGEREF _Toc501438698 \h 59 Appendix 5: Additional Simulations  PAGEREF _Toc501438699 \h 64  1 Introduction This thesis has been written at Department of Chemical Engineering, NTNU, autumn 2000. ABB Industri AS, Oslo, gave the subject of the thesis. Professor Sigurd Skogestad has been supervisor, and Ph.D. Morten Dalsmo has been contact person at ABB. Ph.D. students Espen Storkaas and Vidar Alstad have been of great help. In this period of time, ABB have been working with control of terrain induced slugging at Norsk Hydros well Brage A-27, at the Fensfjord formation. Terrain induced slugging and other forms of slugs are major problems in oil production offshore. Slug flow is a cyclic phenomenon where gas and liquid flows vary from zero to substantial values. When the slug unit comes out of the pipeline the flow rates will change from zero to maximum value during a short period. At the outlet both phases are sent into separators and then the gas goes through compressors. Large variations in flow rates as a result of slug flow may overfill the separators and trip the compressors. This is of course a problem, and is often solved by choking at the outlet. Choking reduces the production and is not an ideal way to handle slugs. The well is already modelled in OLGA. The model represents the slug behaviour in a good manner, and shows that it is possible to apply slug control on the well. The subject of this thesis has been to implement an even simpler model in Matlab. The intention was to deduce a model based control structure on basis of the Matlab model. The control structure would then have been tested in OLGA. If good results were achieved the control structure could have been used on the actual process. However, the modelling of slug flow proved to be very difficult. The work done in this thesis has therefore been concentrated on constructing a valid slug model. Already early in the work it was clear that there would not be any time to deduce a control structure and simulations in closed loop. 2 Slug Flow Theory There are many different kinds of slug flow, and the reasons why they arise vary. However the behaviour and results of slug flow are similar. Slugging may be a problem in risers as well as in wells. In risers, the main problem is that the gas and liquid (water and oil is often considered to be one phase, called liquid) flow rates at the outlet have large variations. It is common that the liquid flow rises from zero to a substantial value, and is followed by a similar change in the gas rate. The magnitude of the changes will of course differ between processes. These fluctuations in rates lead to two major difficulties. At first, when the liquid rate rises, the first separator in the separator train is most likely completely filled. This will effect the separation in the first separator and the rest of the separator train. When the liquid rate starts to drop the gas rate will increase with a steep gradient. Large variations in the gas rates may cause the compressors to trip. In wells, problems arise as results of fluctuations in gas and liquid flow, but the slug is not necessarily transported through the riser and to the outlet. This depends on whether the well is a satellite well or connected to a manifold. A satellite well is not connected to other wells before it reaches the platform, and slugs are therefore transported all the way. If the well goes through a manifold, slugs may be buffered down and are not necessarily noticeable at the outlet. However, if there are few wells connected to the manifold, the buffer effect may be small, and slugs can be transported through the riser. Slugging in one well may also cause trouble for the manifold and the wells connected to it. If slugging increases the wellhead pressure, it may result in decreased or stopped production from the other wells. 2.1 Riser Induced Slug Flow Riser induced slug flow is very much alike the terrain induced slug flow, and is illustrated in figures 2.1.1 to 2.1.4  Figure 2.1.1: Slug formation Figure 2.1.2: Slug movement into the separator  Figure 2.1.3: Blow-out Figure 2.1.4: Liquid fallback The figures show the four phases it is normal to divide induced slug flow into: Slug formation. Liquid is accumulated and the slug unit is generated. Slug movement. The movement starts when the liquid height increases beyond the height of the riser. Blow-out. The slug unit accelerates to a high velocity when gas reaches the bottom of the riser and expands. Liquid fallback. Remaining liquid falls back to the bottom of the riser, and the cyclic behaviour starts again. The cyclic process is described in detail in the next chapter. It should also be mentioned that the figures illustrate severe slugging. Severe slugging is the term that is used when the slug unit is longer than the height of the riser. There is also another variation of riser induced slug flow, other than severe slugging. If the gas flow is not totally blocked by the liquid plug, some of the gas will slip through and drag some liquid through the pipeline. The riser will in this case never be completely filled with liquid, and the slug unit is shorter than the riser is. This kind of riser induced slug flow does not have the same cyclic behaviour as severe slugging, and is documented in Taitel et.al. and Jansen et.al. (1996). 2.2 Terrain Induced Slug Flow The mechanisms of terrain induced slug have been well documented. Taitel et.al., Zeng et.al. 1994 and De Henau and Raithby (1995) all describes the mechanisms thoroughly. Havre et.al. and Per Fuchs (1997) have also done studies of this cyclic process. Figures 2.1.1 to 2.1.4 illustrate the principles behind terrain induced slug, all though it actually illustrates severe slugging in a riser. Terrain induced slug flow is a result of the geometry in the well. When the angle in the pipeline increases from a horizontal or negative state to a positive value, liquid build-up will occur. Oil and water is accumulated in the lower parts of the pipeline. After some time the amount of liquid will cover the entire inner section of the pipeline. The liquid acts like a plug, which prevents gas flow. Small amounts of the gas will manage to bubble through the liquid plug, but the main part of the gas accumulates upstream the liquid. As a result of the gas accumulation there will be an increase in pressure. At a certain time the pressure has increased enough to move the generated liquid plug. The pressure is at this point larger than the hydrostatic force from the liquid. According to Havre et.al., the acceleration of mass is exponentially unstable, and the trajectory goes through an exponentially unstable manifold (zone) into a stable manifold. This causes a limit cycle, which is known as terrain induced slug flow. It is worth noting that: -The pipe stays within the unstable manifold for a very short period of time -The different parts as the pipeline may be in the unstable manifold at different time as the slug moves along the pipeline. This means that there can be several unstable modes in the process, but the acceleration of mass is at least one unstable modus and there are not necessary more. The transportation of the liquid plug will either die out or continue all the way through the pipeline and to the manifold. If the slug is transported through an area where the pipelines angle is negative (falling pipe), the liquid plug may disappear. In a downhill there will not be any accumulated liquid, and the slug unit may loose liquid until the flow is stratified. In many cases this does not happen, and the slug unit is transported to the outlet, and the already mentioned problems will arise. When the transportation of the accumulated liquid plug starts the pressure will gradually drop. The flow in to the well will then enlarge, because the pressure drop between the well and the reservoir increases. Liquid will again start to accumulate in the lower regions until a new liquid plug is generated and transported by high pressure. Terrain induced slug flow is a cyclic process, and the slugs are often generated with a constant frequency. The frequency depends on the actual process, but can be as large as several hours. 2.3 Hydrodynamic Slug Flow As mentioned, terrain induced slug and riser induced slug are much alike. Hydrodynamic slug flow does not have many similarities with the other two kinds of slug flow. Bendiksen et.al (1996) state that hydrodynamic slug flow is an unstable flow in straight pipelines, and it is not periodic in either time or space. The slug units experience alternation in velocity, length and hold-up as they flow through the pipeline. The hydrodynamic slug unit is formed by an unstable mechanism in stratified flow. Haandrikaman et.al. exemplifies the unstable mechanism with an operational change or by an instability of the gas/liquid interface. The pipelines cross section is not necessary blocked by liquid, and a region with higher liquid hold-up may pass through the pipeline like a wave. Transient slugs generated by instability of the gas/liquid interface can occur in separated flow conditions, and may be stable and increase in length. Whether the slug continues to grow or dies out depends on the state in the pipeline Figure 2.3.1 is taken from Taitel et.al. and represents the slug profile in a good manner. The slug profile is not unique for hydrodynamic slug flow, but represents all the other variants as well.  Figure 2.3.1: Slug profile The slug body is subdivided into two main sections: The liquid zone of length ls, and the film zone of length lf. The film zone consists of liquid film and Taylor-bubble, and the liquid zone often contains gas bubbles. 2.4 Slug Control Slugging can be avoided by choking and increased gas lift. This is postulated by Jansen et.al., among others. Unfortunately, this is not an ideal way to handle slug flow. The amount of gas available for gas lift is usually limited, and it would be expansive to use too much gas. Unstable slug flow may be forced into a stable area by choking the valve. Choking will of course lead to decreased production. Normally, it is desirable to produce at a maximum within the systems defined operation limits. Choking is therefore not a too good option, although it is used. The idea behind slug control is to operate with a high choke opening, in an unstable area, without generation of slug flow. This will ensure a high production rate, without tripping compressors and overfilling separators. By measuring the pressure in different parts of the pipeline it is possible to detect slug generation. The liquid build up is prevented by adjusting the choke opening. If there is a down hole choke, this can be used for adjustment as well. By controlling the valve opening a high production rate without slug flow is achieved. Figure 2.4.1 is a rough sketch showing how ABBs slug controller can be implemented.  Figure 2.4.1: Slug Control The picture is from ABB Industri, EOC. 3 Case description Brage A-27 is a satellite well at the Fensfjord formation. The pipeline transports oil, water and gas to the platform. A three-phase model of the well is already implemented in OLGA. The OLGA model is based on measurements done at the separator on Brage. At a well, oil, gas and water flow into the pipeline at several points. In OLGA the model has two feed points. The first is at the end at the bottom, and the second is located 1482 meters (horizontal) away. The geometry of Brage A-27 as it is modelled in OLGA is shown in figure 3.1.  Figure 3.1: Geometry of Brage A-27 in the OLGA model The pipeline is parted into 22 sections in the OLGA model. The inner diameter is 0.1214 m from the first feed point to the second and 0.1594 m from the second feed point and to the wellhead. Variables describing the case are given in table 3.1. Table 3.1: Different parameters at Brage A-27 Pressure at outlet, [bara]8.3Pressure at inlet, [bara]75Pressure in reservoir, [bara]85Temperature at inlet, [C]90Temperature at outlet, [C]60Temperature in reservoir, [C]100Inflow gas, [kg/s]2.28Inflow liquid,[kg/s]4.9Molar weight gas, [kg/mole]55.55.10-3Molar weight liquid, [kg/mole]205.19.10-3Roughness in pipeline, [m]4.5.10-4 These values are gathered from Dalsmo (1999) and Dalsmo and Jansen (1999). The inflows are based on a test period when the well was operated under slugging conditions, depicted in figure 3.2 and 3.3. Slugging at Brage A-27 is a result of both hydrodynamic and terrain induced slug. It is assumed that terrain induced slugging is the main problem at Brage A-27. Trend plots from the OLGA model are depicted in figure 3.2 and 3.3. Figure 3.2 shows the variation of gas mass rate at the well head, and figure 3.3 shows the variation of liquid mass rate at the same point.  EMBED Word.Picture.8  Figure 3.2: Variations in gas flow rate at the outlet  Figure 3.3: Variations in liquid flow rate at the outlet Both figures illustrate the large changes in the rates, and that the process oscillates with a frequency at approximately 0.6 slugs/hour. This leads to problems with separators and compressors. The OLGA model depicts the slug behaviour at Brage A-27 in a satisfying manner, and has been used to investigate the possibilities of implementing slug control at Brage A-27. Figure 3.4 shows the variations in the downhole pressure. The downhole pressure is used as set point when slug control has been tried at Brage A-27.  Figure 3.4: Downhole pressure variations Both testing of slug control in the OLGA model and offshore show very promising results. Figure 3.5 shows simulation of slug control in OLGA.  Figure 3.5: Slug control simulated in OLGA The simulation is run for 30 hours, where the first 15 hours are in closed loop and the 15 last are in open loop. A PI-controller with gain, Kp = -1.6.10-6 and integration time, (I = 1000 sec was used. The well did not slug as long as the pressure was held approximately constant. 4 Modelling of Slug Flow When modelling slug flow, it is common to assume two phase flow instead of three phases. Oil and water is the considered as one phase called liquid. In this thesis it is made different assumptions for different approaches, but it is always considered to be two phase flow. 4.1 Basic theory for two-fluid modelling The theory behind two-fluid models is described in Bendiksen et.al (1990). Bendiksen et.al (1990), De Henau and Raitby (1995), Geankoplis (1993) and Bird et.al. (1960) have also been used to confirm and supply the theories and equations. Conservation of mass All these models include equations describing the conservation of mass, momentum and energy. The conservation of mass is expressed by the following two equations  EMBED Equation.3  (4.1.1)  EMBED Equation.3  (4.1.2) which are given in Bendiksen et.al. (1990). Equation 4.1.1 is mass conservation for gas and 4.1.2 is mass conservation for the liquid phase. A [m] is the cross section in the pipe, v [m/s] is velocity, f [-] is volume fraction, ( [kg/m3] is density, x [m] is length, (G is mass transfer between the phases, G is a possible mass source and (e and (d are entrainment and deposition rates. Subscripts G and L indicates gas and liquid. There is also a conservation equation for liquid droplets, but it is assumed to be no liquid in the gas phase and vice versa, so that equation is therefore never used in this thesis. The assumption of no mixing of the phases holds through the entire thesis. Conservation of momentum The equations for conservation of momentum given in Bendiksen et.al. (1990) are as follows:  EMBED Equation.3  (4.1.3)  EMBED Equation.3  (4.1.4) The symbols denote the same for the momentum equations as for the mass equations. In addition, subscript i denotes interface, FD is the droplet drag, g [kg/m s2] is the gravity force, S [m] is wetted perimeter, ( [-] is the angle of inclination and vR [m/s] is the difference between gas and liquid velocity. In addition to the terms that already are neglected in the mass equations, fl is considered not to be dependent of space so the last term cancels out. The equations for conservation of motion are not always written like this. In De Henau and Raithby (1995), the terms expressing interface friction and friction against the wall are formulated quite different. By using Bird et.al. (1960) it is quite easy to show that the terms are equal. Conservation of energy The conservation of energy is given by equation 4.1.5:  EMBED Equation.3  (4.1.5)  EMBED Equation.3  E is the internal energy per unit mass, h is elevation, HS is the enthalpy from mass sources and U is the heat from pipe walls. Pressure calculations There are several ways to calculate the pressure. The most obvious is to use ideal gas law, equation 4.1.6.  EMBED Equation.3  (4.1.6) R [J/ K mole] is the gas constant, T [K] is temperature, V [m3] is volume, m [kg] is mass, M [kg/moles] is molar weight and P [Pa] is pressure. However ideal gas law is recommended for low-pressure regions. A correctional factor, Z, can be introduced at higher pressures. Smith et.al. (1996) have probably the best approaches when modelling the pressure. At higher pressures either the virial equations or the van der Waals equation should be used. The van der Waals equation is given as  EMBED Equation.3  (4.1.7) where a and b are functions of critical temperature and pressure. Out of the two virial equations the following is recommended:  EMBED Equation.3  (4.1.8) B is a virial coefficient, found experimentally. Discretization When a model contains partial differential equations (PDE-system) it is common to keep time as a continuos variable while the rest of the variables are discretized. Instead of a PDE-problem an ODE-problem arises. The ODE-problem is consists of more equations and is larger than the PDE-problem. However, it is easier to solve. Two discretization methods are described in this chapter. The line method The line method is described in Stren and Hertzberg (1995). Spacial variables are discretized by initialising a grid vector with discretization points. A state vector is also defined, containing the systems state at each discretization point. If there are more than one spacial variable, matrixes will be initialised instead of vectors. The states are now continuos in time but discreet variables in space. From the state vector and grid vector an estimate of the spacial derivatives are found from equation 4.1.9.  EMBED Equation.3  (4.1.9) The line method is used for discretization of the mass balance. Discretization of the mass balance The equations describing conservation of mass are based on equations 4.1.1 and 4.1.2, and that the simplifications related to the ( and G-terms hold. The equations are discretized by the line method, and can be written as  EMBED Equation.3  (4.1.10)  EMBED Equation.3  (4.1.11) The cross section and density is considered constant for each sampling time. 4.1.10 and 4.1.11 are on a volume basis, and by multiplying with the volume of the control volume, A(x, the differential equations for conservation of mass are  EMBED Equation.3  (4.1.12)  EMBED Equation.3  (4.1.13) where W [kg/s] is mass rate. The equations are not dependent of each other. The phases are de coupled, and there is no mass transfer between the phases as mentioned earlier. The staggered grid method The staggered grid method is described in Banerjee (1999). The method is a way to calculate the momentum balance at different descretization points than for the mass balance, and is compulsory for two-phase systems. By calculating at different points the possibility of unstable behaviour at steady state is avoided. It can be showed, either by von Neumann analysis or by linearisation, that discretization with the line method gives unstable modes. Some of the systems poles will be in the right half plane. The staggered grid method is necessary when discretizing the momentum-balance, and is given in equation 4.1.14.  EMBED Equation.3  (4.1.14) G is the differential variable, and is equal to the density multiplied with superficial velocity ((u). Superscript n denotes values at the time step n and j is the discretization point. F is the total friction loss. Equation 4.1.14 holds for both phases. It is not shown the discretization of the conservation of momentum in this chapter, because the modelling of momentum conservation has been done by different approaches. Figure 4.1. describes the staggered grid method.  Figure 4.1: Discretization by the staggered grid method The figure shows how the pipeline is discretized into control volumes, and where the conservation equations are applied. Algebraic relations The cross section in the pipeline is given by equation 4.1.15, and the volumes by equation 4.1.16 to 4.1.18.  EMBED Equation.3  (4.1.15)  EMBED Equation.3  (4.1.16)  EMBED Equation.3  (4.1.17)  EMBED Equation.3  (4.1.18) L [m] is length of the control volume and r [m] is the radius in the control volume, Vt [m3] is the total volume, Vl [m3] is liquid volume and Vg [m3] is gas volume. Volume fractions, fl and fg [-], are calculated from equations 4.1.19 and 4.1.20.  EMBED Equation.3  (4.1.19)  EMBED Equation.3  (4.1.20) The gas density,(g [kg/m3], is expressed from equation 4.1.21  EMBED Equation.3  (4.1.21) where mg is mass of gas. The middle density, (m [kg/m3], in the pipeline is a function of density and volume fraction of both liquid and gas.  EMBED Equation.3  (4.1.22) Mass fractions of liquid and gas, xl [-] and xg [-], are found by equations 4.1.23 and 4.1.24:  EMBED Equation.3  (4.1.23)  EMBED Equation.3  (4.1.24) When calculating the conservation of momentum, ((u)ph is the differential variable. Subscript ph denotes phase, which can be gas (g) or liquid (l), ( [kg/m3] is density and u [m/s] is superficial velocity. The relation between the differential variable and the mass flow is described in equation 4.1.25.  EMBED Equation.3  (4.1.25) Equation 4.1.26 and 4.1.27 gives the superficial and actual velocity from ((u)ph.  EMBED Equation.3  (4.1.26)  EMBED Equation.3  (4.1.27) Superficial velocity is denoted u and actual velocity is denoted v. A is total cross section and Aph is the cross section of the phase. Some geometric relations are requested as well. In order to calculate different interface perimeters, a table with Al and corresponding ( [rad] values is generated. ( is an angle that indicates how much liquid and gas there is in the pipeline, and geometric calculations are done on basis of (. When the pipeline contains only gas ( is zero, and when it is filled with liquid ( is pi. Geometric illustration is given in Appendix 1. The perimeters are calculated from equations 4.1.28 to 4.1.30, given in Fuchs (1997).  EMBED Equation.3  (4.1.28)  EMBED Equation.3  (4.1.29)  EMBED Equation.3  (4.1.30) S [m] is the perimeter between the phases (i), gas and wall (g) and liquid and wall (l), r [m] is the radius in the pipeline. On basis of the perimeters, the hydraulic diameters can be found from 4.1.31 and 4.1.32, as described in Fuchs (1997).  EMBED Equation.3  (4.1.31)  EMBED Equation.3  (4.1.32) Dh,l [m] is hydraulic diameter for the liquid phase, and Dh,g is hydraulic diameter for the gas phase. The Reynolds numbers for the flow are calculated from equation 4.1.33, found in Geankoplis (1993).  EMBED Equation.3  (4.1.33) Subtext ph denotes phase as before and ( [kg/ms] is viscosity for the phases. In order to calculate the shear stress between the phases and at the wall, the friction factors must be determined. They are found by equations 4.1.34 to 4.1.38 from Fuchs (1997).  EMBED Equation.3  (4.1.34)  EMBED Equation.3  (4.1.35)  EMBED Equation.3  (4.1.36)  EMBED Equation.3  (4.1.37)  EMBED Equation.3  (4.1.38) ( [-] is friction factor and ( [m] is the roughness. The friction factor will be different in turbulent and annular flow. The friction factor with the lowest value will be used in further calculations. The shear stress is determined on basis of the absolute velocity and friction factor:  EMBED Equation.3  (4.1.39)  EMBED Equation.3  (4.1.40)  EMBED Equation.3  (4.1.41) ( [N/m2] is shear stress, and the equations represent the shear stress respectively for the phase interface, the interface between liquid and wall and the interface between gas and wall. These equations are also from Fuchs (1997). 4.2 Well Conditioned Model When modelling, it is essential to follow some basic rules to ensure a good model. Moe and Hertzberg (1994) explain most of these guidelines. When a model is put together it is critical that it is well conditioned. In a well conditioned model has each differential equation a differential variable assigned to it, and each algebraic equation has an algebraic variable assigned to it. This is assured by carefully examining all the equations of motion, internal equations and rate equations. Designing control volumes is also very important, especially considering simulation of the model. If there are many variables in each control volume, and many control volumes, it may take extremely long time to run simulations. On the other hand, it is essential to get enough information. Areas with large transients should therefore be discretized finer than other sections. Different ways to discretize is described in Stren and Hertzberg (1995) and Banerjee (1999). 5 Results 5.1 The Simulink Models In 1999 Eik (1999) did similar work in his diploma thesis. Although he did not succeed completely, it was natural to recapitulate some of the work and figure out whether it was a good approach. Assumptions A brief overview over the simplifications and assumptions done in the Simulink models: Ideal gas law is valid There are only two phases, liquid (oil and water) and gas No interaction between the phases (no mass transfer) The momentum balance is applied over the interface between two control volumes No momentum is conserved over the interface Incompressible liquid Constant temperature in the pipeline (conservation of energy is unnecessary) Pressure loss due to friction is added into a total friction coefficient The mass flow and density is constant over a control volume Equations and relations Two models in Simulink were tested, model S1 and model S2. Both models are based on the general principles in chapter 4. The difference between the models is how the gas and liquid rates were calculated. Both methods are based on the momentum balance given in Geankoplis (1993):  EMBED Equation.3  (5.1.1) ( [-] is inclination angle and ( [N/m] is shear stress. Subscript m denotes middle values and is used when the equation represents combined conservation of momentum. If combined conservation is not assumed, the conservation of momentum is calculated from two equations identical to 5.1.1, but subscript m is altered to g (gas) and l (liquid). It can be argued that this balance is valid for the interface between two control volumes, and therefore it will not be any accumulation of momentum, no hydrostatic pressure and the shear stress will be included in a total resistance coefficient. This argument is not especially good, but it results in valve characteristics that are easy to implement in the model. The assumptions made concerning conservation of momentum is typical for non-continuos systems. They indicate that the pressure is non-continuos over the control volume interfaces. For a system with two phase flow, this is not likely. Model S1 For model S1 it is assumed that there is one combined momentum balance for both phases. For a combined momentum balance the valve characteristic is given by equation 5.1.2:  EMBED Equation.3  (5.1.2) Subscript m is for middle values, n is control volume, and out and in refers to values at the interface between control volumes. To estimate the pressure at the interfaces the hydrostatic pressure has to be added or subtracted from the middle pressure in the control volume. The hydrostatic pressure is added for Pin, while this will be at a lower position than the middle, and it is opposite for Pout. Equation 5.1.2 can be transformed into equation 5.1.3 by combining 4.1.22 with constant mass flow and multiplying with middle density and cross section.  EMBED Equation.3  (5.1.3) Wm,out,j is middle mass flow out of control volume j, and fm is a combined friction factor. The mass flow for each phase is calculated by multiplying the middle mass flow, Wm,out,j, with the respective mass fraction. This implies no slip, which is a very rough simplification. Model S2 For S2, the model based on momentum equations for each phase, both the equations for valve characteristics and mass flows will be similar to equations 5.1.2 and 5.1.3. Instead of two equations for middle values, there will be four equations two for each phase. The valve characteristics are in addition multiplied with the respective volume fractions. When modelling flow, it is also essential to take negative flow into account. The equations for negative flow will be the same as 5.1.2 and 5.1.3, but the upstream control volumes are changed. For calculation of regular flow between control volume j and j+1, control volume j will be upstream. When the flow is negative the control volume j+1 will be upstream. This will make a difference when assigning the correct values of the variables in 5.1.3. Control volumes The pipeline is discretized into seven control volumes. Characteristics of the control volume are given in table 5.1.1. Table 5.1.1: Control volume characteristics Control VolumeRadius, r [m]Inclination angle, ( [-]Horizontal length, x[m]10.06068-0.031815.420.060680.083666.330.07970.33470.940.07970.948938.550.07970.97750.860.07971.505547.370.07970.016.0 The different control volumes and the information flow between them are illustrated in Appendix 2. The diagram gives rough idea of how the model is put together, without going into details. I figure 5.1.1 a part of the diagram is shown.  Figure 5.1.1: Part of the Simulink flow diagram The figure shows that the flow information travels from inlet to outlet, while the pressure information goes the other way. Pressures and densities are sent from one control volume and back to the previous control volume. This is necessary in order to calculate mass flow on basis of pressure drop over the control volume interface. In figure 5.1.1 a subsystem block is illustrated. The subsystem arranges the outputs from the control volume so the variables are sent in the right direction. State in the control volumes To estimate the states in the control volumes is often the main challenge when modelling. The states depend on different algebraic relations, constants, conservation equations, the shape of control volumes and discretization methods. The states calculated during simulation are gas and liquid hold-up, gas and liquid velocity and pressure. The boundary conditions are pressure at the outlet and mass flow into the pipeline. These values are given in table 5.1.1. In the models the inflow is specified to be at a constant value. It is more realistic to let the inflow depend on the pressure difference between the pipeline and the reservoir, but constant flow is a reasonable assumption. The initial hold-up in the pipeline is calculated for a half-filled pipeline. Both were simulated using ode15s. There were two main problems when simulating the models. It was quite obvious that the simplification of conservation of moment were wrong. To simulate with valve characteristics instead was not a good solution. In addition, there was a large flow of information and it was difficult to iterate to consistent states. As a result the simulations had to be run for a tremendous long time. Therefore there are not given any graphical results for these simulations. As long as the simulations had to be run for too long time, and they did not seem to give good results, it was decided to try another approach. This was done in Matlab. 5.2 The Matlab Models The motivation for modelling in Matlab was to decrease the simulation time, and to implement better momentum balance. Use of the staggered grid method would also be complicated in Simulink. Except for the conservation of momentum, the modelling in Matlab is fairly similar to the modelling in Simulink. Assumptions A brief overview over the simplifications and assumptions done in the Matlab models: Ideal gas law is valid There are only two phases, liquid (oil and water) and gas No interaction between the phases (no mass transfer) Incompressible liquid Constant temperature in the pipeline (conservation of energy is unnecessary) The mass flow and density is constant over a control volume Constant crossection Equations and relations The difference between the models in Matlab and the models in Simulink, is mainly the modelling of momentum conservation. Pressure dependent flow at the inlet has also been tried in Matlab. The momentum balance In the Simulink models the conservation of momentum was simplified to rate equations, very similar to valve characteristics. In the Matlab models the modelling of conservation of momentum is much more complete. In Bendiksen et.al (1990) the conservation of momentum is given as in equation 4.1.3. In the Matlab models equation 5.2.1 and 5.2.2 are used.  EMBED Equation.3  (5.2.1)  EMBED Equation.3  (5.2.2) Both equations are simplified versions of equations given in Fuchs (1997) .The equations are discretized as described by the staggered grid method. This gives equation 5.2.3 and 5.2.4.  EMBED Equation.3  (5.2.3)  EMBED Equation.3  (5.2.4) The first term describes the change in velocity over the control volume. The second term characterise pressure drop over the control volume, while the third is the hydrostatic pressure. The last two terms represent the friction loss due to shear stress at the interface and against the wall. When the momentum balances were implemented, it was of course important to take negative flow into account. This was done by using downstream values instead of upstream values when negative flow was detected in a control volume. Boundary conditions The boundary conditions are the same in both Matlab models, as well as in the Simulink models. However, it has been tried to model the inflow by using valve characteristics. In this case the inflow is calculated from equation 5.2.5.  EMBED Equation.3  (5.2.5) K [(ms)-1] is valve constant and varies for gas and liquid, Preservoir [Pa] is the pressure in the reservoir and Pin [Pa] is the pressure at the inlet. Control volumes In the Matlab models, the control volumes were modelled in two different ways. Originally, the geometry in the model was supposed to be the same as at Brage A-27. This was modelled as in table 5.1.1. However, it was hard to start the simulations with a complex geometry and in a slugging flow regime. They would not converge. It was therefore decided to model a horizontal pipeline. The number of discretization points was increased when the new geometry was implemented. 30 control volumes of 100 meters describe the geometry. The pipeline was afterwards be bent in order to be more like a real case and to make it possible to generate slug flow. In the final model the last control volume is ascended 10 meters and the three before are risen 5 meters each. It adds up to 25 meters totally. This is equivalent with inclination angles at respectively 0.1 and 0.05 radians. It was also decided to remove the pipelines second feed point, since it would contribute with even more dynamics in the models. State in the control volumes Model structure 1 When the states in the control volumes were modelled, the first model mostly was based on for-loops. The model is given in Appendix 3. It was difficult for the model to converge, probably because of unsuitable initial conditions gathered from the OLGA model. The model was therefore simulated horizontally and at a zero-state: a consistent state with no initial flow, no pressure drop and no inflow of gas or liquid. The results from simulation were obviously very wrong. Also when little inflow at consistent conditions was added this happened. The instability in the gas flow is shown in figure 5.2.1, and this is of course not an expected behaviour, and it can not be correct either.  Figure 5.2.1: Gas flow at the outlet Although the values are relatively small, the tendency is quite clear. The process is unstable, and the variations grow larger as time goes. Figure 5.2.1 illustrates instability in both gas and liquid flow at the outlet. All the other variables, hold-ups and velocities in every other control volume, show the same behaviour. The reason for the unstable behaviour in this model seemed to start with small variations in the first control volume. These variations propagated through the pipeline, and were enlarged on the way. At one point it was assumed that this could be an effect of having constant flow into the pipeline. It was observed that the pressure increased at the start of the simulations, and it seemed strange if this should not lead to a reduced inflow. The inflow was therefore calculated from valve characteristics. Alas this did not stabilise the system, and probably brought more dynamics into the model. When none of the improvements or searches for errors gave positive results, the model was rewritten Model structure 2 The new model was implemented on a matrix basis. It is in theory identical with the previous, but the results are very different. The reason for this is probably either a wrong reference in the first model, or that the for-loops induce severe numerical trouble. The function that calculates the states in the pipeline is given in Appendix 4. Figure 5.2.2 shows the gas flow at the outlet when simulating the second Matlab model where the system is at a zero-state and minimal driving forces are added.  Figure 5.2.2: Gas flow at the outlet As can be seen from figure 5.2.2, the system is not at steady state before nearly 15000 seconds has passed. The reason for this is that tuning of the shear stress at the interface has been necessary to obtain reasonable values for velocity and hold-up, and when the pipeline is large with respect to the flow it takes a long time to reach steady state. Figure 5.2.2 shows simulation with a small change in the friction factor. It also shows that the variables are held within reasonable limits, and that the second Matlab model represents the system in a favourable manner. As figure 5.2.2 illustrates, the second and final Matlab model showed expected and promising results. In order to investigate the model further, and hopefully generate slug flow, the pipeline was bent upwards at the outlet. This was done by gradually increasing the inclination angle for control volumes near the outlet. At first an increase of 0.05 radians was tested. The simulation would not converge when run in ode15s. The Explicit Euler method was therefore tried. With Explicit Euler it took 48 seconds to simulate 1 second, and ode15s was therefore preferred as long as the increase in inclination angle was small. The principles behind the mathematical solver routines and the large real time are commented in the next chapter. Since the Explicit Euler was too slow ode15s was tried again, but with an increase of 0.001 radians. This worked well, but a hundred simulations were required to bend the pipeline 25 meters. During these simulations the model showed anticipated behaviour. Hold-up liquid grew and hold-up gas fell. The absolute velocities showed an opposite trend. After the pipeline was risen 25 meters it seemed that these modifications alone would not bring the system into a slugging area, unless it was risen several more meters. At this point there was not much time left for simulations, and it was decided to tune the friction between the gas and liquid phase. A reduction in the friction at the phase interface results in higher absolute gas rate and lower absolute liquid rate. In other words, there will be more liquid accumulation. At his point the friction factor coefficient had already been tuned down from 0.02 to 0.0008, and was further reduced to 0.00005. The hold-ups and flow rates started to oscillate. It seemed quite clear that the model managed to generate slug flow. Figure 5.2.3 and 5.2.4 show the variation in velocity, ((u)liquid and ((u)gas.  Figure 5.2.3: ((u)liquid at the outlet  Figure 5.2.4: ((u)gas at the outlet It can be seen from figure 5.2.3 and 5.2.4 that the system oscillates with a frequency equal to 6 slugs/1000 sec. The liquid velocity is at a maximum before the gas velocity. This behaviour is expected and characteristic for slug flow. In figures 5.2.5 and 5.2.6 the absolute velocity is portrayed.  Figure 5.2.5: Absolute liquid velocity, vl, at the outlet  Figure 5.2.6: Absolute gas velocity, vg, at the outlet These figures show the same as the two before, except that the variables are absolute. It is clear that the gas travels a lot faster than the liquid, which also is expected. This is among other a result of the tuned friction factor between the phases. Compared to results from other slug models, for instance the OLGA model, these oscillations are more like a sinus wave. There are three reasons for that: Use of the staggered grid method, rough discretization (large control volumes) and no valve at the outlet. The staggered grid method and the size of the control volumes will smooth out the slug. It is not possible to obtain an abrupt change in liquid hold-up at a certain point in the pipeline, as the model is now. A valve would also effect the slug when it reaches the outlet, and induce a steeper increase in velocity. Figures 5.2.7 and 5.2.8 show pressure variations at the inlet and the pressure profile through the pipeline.  Figure 5.2.7: Pressure variations at the inlet  Figure 5.2.8: Pressure profile The pressure variations at the inlet represent the variations in the horizontal part of the pipeline, since the systems dynamics is so fast. Figure 5.2.7 shows the build-up in pressure that leads to slug flow. If slug control had been implemented, this would be a possible controlled variable. By keeping the pressure at a constant value, slug flow can be avoided. Figure 5.2.8 shows that the pressure drop in the pipeline is mainly in the ascended parts of the pipeline. This is very typical. Figures 5.2.9 and 5.2.10 show variations in liquid volume fraction at the inlet and the profile through the pipeline.  Figure 5.2.9: Variations in liquid volume fraction at the outlet  Figure 5.2.10: Liquid volume fraction profile The figure depicting the liquid volume fraction trend, verifies that the liquid at the outlet shows slugging behaviour. However, the volume fraction never reaches 1, and the pipeline is therefore never completely filled with liquid. Some amount of gas will always slip through. Nevertheless, some gas is held back, and there is an accumulation and pressure increase. This leads to slug flow, although the slugs are not as large as possible. The liquid volume fraction profile is typical for a slugging system. There is a liquid build up in the non-horizontal part of the pipeline. The profile is relatively constant in time. This is because the slugs are small, and there are no real blow-outs that empty the pipeline. It was also done small steps in the gas and liquid flow rates at the inlet to see if this had any effect on the slug behaviour. The oscillations changed a bit, but the slug flow remains. Some of these results are given in Appendix 5. As long as the geometry and initial conditions in this model are so far from the ones in the OLGA model, it is no point to compare further. The model generates slug flow in a good way, and it is realistic that it can show similar behaviour as the OLGA model if the geometry is changed and the model further tuned. Mathematical solver routines Both these figures depict results from simulations run with Matlabs solver for stiff differential equations, ode15s. Ode15s is a solver that works very fast when the system is not far from steady state. When the pipeline was raised at the outlet, in order to improve the geometry, the system was brought rather far from the steady state. The ode-solver did not respond well to this, and would not converge. In order to run simulations, the model was simulated by using Explicit Euler. This method is described in Stren and Hertzberg (1995). When the Explicit Euler method is applied, it is required to determine the step size. The step size should be chosen one tenth of the systems smallest time constant. Time constants for the system is found from the eigenvalues. From the eigenvalues the systems poles are found, and all of them are in right half plane, which means that the model is stable. The smallest time constant was 0.003 second. The step size was chosen to be 0.005 seconds, which is larger than recommended, but with a smaller step size the simulations would take too long time. 0.005 seconds also gives a long simulation time, but some simulations with increased inclination angle at the outlet have been run. A time constant at 0.003 seconds indicates that the system is very sensitive. This has resulted in several numerical problems. 6 Discussion The object of this thesis was to model terrain induced slug flow at Brage A-27, and that a control structure could be deduced on basis of the model. It turned out to be rather difficult to create a good model, so the work is totally concentrated towards this point. The Simulink Models There are two main purposes of the Simulink models. First of all to understand most of the work done by Eik (1999). Second, it was a good way to understand the process on an early stage in the work. A model in Simulink is build such as the flow of information is very central. Information flow is an important issue, and it was very informative to build these models, regarding information flow. The boundary conditions are specified as constant pressure at the outlet and constant mass flow in the inlet. The mass flows are therefore calculated from the bottom and upward, while the pressure is calculated the other way. Flow of information is also the reason for the large simulation time. In case of negative flow it is necessary to get information about the states in the neighbour control volumes. Totally this adds up to quite a large amount of information. At the end it took 15 minutes to simulate 10 seconds. When the simulations are supposed to run several hours, this makes an impossible case. It is therefore not any use to plot graphical results. There is almost no information to get after so short simulations. It turned out to be most correct to model on a basis of two equations for conservation of momentum, in stead of one combined equation. This opens for negative flow of one phase while the other is positive. In slug flow this is important, as liquid often flows backward, but gas will rarely show such behaviour. However it was quite obvious that the equations for conservation of momentum had to be applied over the entire control volumes and not as a valve characteristics over the interfaces. It is not reasonable to assume that the velocities are constant over a control volume and change dramatic over the interfaces. The simplifications in the equations were also unnecessary rough. It was decided that the conservation of momentum should be implemented in a more complete manner, because these equations are very essential when modelling the slug flow. Implementation of the complete equations for conservation of momentum leads to an increase in differential variables. At this point it also seemed to be necessary to solve the model with the pressure as an algebraic variable. Simulink is not capable to solve DAE (differential algebraic equations) in a good manner. Combined with the fact that it took extremely long time to run simulations in Simulink, it was decided to do the rest of the modelling in Matlab. The Matlab Models In order to obtain a model which represent two phase flow in a desirable manner, several variations have been tested. The effects of these variations are described in the further chapters. The model that gave best results was the one implemented with matrix structure, and this model is used for simulation and generation of slug flow. Modelling structure There are mainly two Matlab models that have been tested, and the difference is the internal structure in the models. One is written with for-loops and the other is based on a matrix structure. Theoretically theses two models are alike. The equations, algebraic relations and initial and boundary conditions are the same. However, the models give very different results after ended simulations. The first model, based one for-loops, gave rather unexpected and evidently wrong results during simulation. However, the second model, based on a matrix structure, returned sensible results. It is not quite clear why the two approaches gave so different behaviour. One possibility is that there are some wrong references in the for-loops. Nevertheless, this is not likely since the model has been worked through thoroughly searching especially for errors like this. The other possibility is that the for-loops induce some numerical problems. As long as the models are alike, except for the way the variables are arranged, numerical errors are the most likely explanation to the differences. To implement the model on a matrix basis was done in cooperation with Storkaas rather late in the period. This was the first successful improvement. A lot of other improvements were tried, without any success. Changes in inflow The reason for the unstable behaviour in the first model seemed to start with small variations in the first control volume. These variations propagated through the pipeline, and were enlarged on the way. At one point it was assumed that this could be an effect of having constant flow into the pipeline. It was observed that the pressure increased at the start of the simulations, and it seemed strange if this should not lead to reduced inflow. The inflow was therefore calculated from valve characteristics. Alas this did not stabilise the system, and probably brought more dynamics into the model. It was also tried to run simulations with only one phase. This was done merely to test how the first model responded, and if it could give any indication on what the problems were. One phase flow did not give any bright answers, so this approach was left since it is not important in slug modelling. In the geometry modelled in OLGA there are two inlets. To simplify the system the second feed point was removed. It is not easy to tell the exact effect of this, but it makes the system less complex. This is important while the system is very sensitive, and it is easier to converge and obtain steady state without the second inlet. Cancellation of the second feed point was done in both models. Geometry and initial conditions As mentioned in previous chapters, the geometry was altered so it described a horizontal pipeline. This is mainly because the flow regime is less complex, and it is easier to find consistent initial condition. It was impossible to run simulations in a slugging flow regime and geometry as in the OLGA model. Simulation went better when the pipeline gradually was exposed to increasing flow and steeper geometry. When initial conditions, as velocity, pressure drop and hold-up, were taken from Olga simulations, they were rarely consistent for this model. This resulted in large and fast varying gradients, and the system never stabilised, although different conditions were tried multiple times. Introductions of horizontal geometry and consistent initial conditions have been of great importance concerning convergence and stability of the model. Shear stress at the phase interface The conservation of momentum is important to describe the flow in the pipeline. Especially the term that gives the shear stress at the phase interface. The shear stress is calculated from algebraic relations. When these relations are used, it is important to notice that they are valid only for certain conditions. The equations for the shear stress given in the literature do therefore not necessarily describe the system in a good way. Different ways to express the shear stress are tried, and the chosen one has been tuned in order to give sought behaviour. The friction factor, (i, has been reduced with a factor 400. It is also worth noting the use of both superficial and absolute velocity when the conservation of momentum is calculated. The conservation equations are calculated with superficial velocity in the differential variable, and absolute velocity for the rest of the terms. This is according to Fuchs (1997). The Reynolds number is per definition calculated with superficial velocity. In this thesis absolute velocity is used, but it is multiplied with the volume fraction and it is therefore consistent use velocities and units. It is a known fact that the friction terms are dependent of the flow regime. The interface shear stress and the shear stress at the wall are generally larger for slug flow than for annular flow. In these models relations valid for annular flow calculate the shear stress. Some slug models check whether the system is in a slugging or annular flow regime, and calculates the shear stress on basis of the regime. At least two relations for the shear stress are therefore necessary. In this thesis it has been a goal to check whether only annular relation is sufficient, and what effect it has. In any event, this is an interesting approach since the system is modelled without any special implementations for slug. Many models only try to depict the precise slug behaviour. Instead it could be an idea to model a flow that could go into a slugging flow regime, but is controlled before the slugging is initiated. In other words the system is stabilised before the slugging starts instead of stabilising an already existing slug flow. Hopefully this is an approach that can lead to just that. Discretization In this thesis the staggered grid method has been used when the conservation of momentum has been discretized. Experience show that this is a good way to prevent an unstable solution when simulating at steady state. If this method was not used, it is possible that the pressure would oscillate around a certain value, even though there is no change in hold-up. The staggered grid method means that the mass balance and the momentum balance are not taken over the same control volume. The momentum balance is taken over control volumes that are skewed a half control volume length compared to the control volumes for the mass balance. To implement the staggered grid method as described in Banerjee (1999) is not a simple task. And at the boundaries special assumptions have to be made. The momentum balances are based on several upstream values. At the inlet and if there is negative flow near the outlet, these values have to be guessed. If there is need for values from a non-existing control volume, because it is supposed to be outside the pipeline, the values are approximated to be the same as in the nearest available control volume. This is not an entirely consistent assumption, but it works well enough. It is also worth noting that it is not needed to implement momentum balance over the first half control volume and mass balance over the last half control volume. The reason is that the inflow at the inlet is given as a boundary condition, and so is the pressure at the outlet. Mathematical solving routines Two mathematical methods have been used when simulating in Matlab. One is Matlabs own solver for stiff differential equations, ode15s. The other one is the Explicit Euler method. Ode15s can vary the step size, but Explicit Euler has constant step size. This means that ode15s takes small steps when the transients are large, and increases the step size when the system is near steady state. This is of course smart since the routine is both accurate and fast. Sometimes there are however very large transients, and the ode solver may have difficulties to converge. In these cases the Explicit Euler is very handy. Although the method is rather slow it will always make calculations, no matter how steep the gradients are. In this thesis both methods have been used. When simulating near steady state, ode15s has been preferred, and Explicit Euler is chosen when major changes in the states have been examined. Suggestions to further Work Although the final Matlab model is promising and shows fairly good results, there are still a lot of things to improve. Some of the most important are suggested in the following list: Geometry. The pipeline should be bent until the shape is the same as at Brage A-27. This includes adding the second feed point that is removed in this model. When the pipeline gets steeper the pressure near the inlet will increase, and the inflow gas can be increased as well. Implicit Euler. Implicit Euler is a mathematical solving method that may be useful for this model. The Explicit Euler works to slow, and ode15s does not handle large steps away from steady state. Implicit Euler works fast and converges for large changes. It might therefore be a good idea to try this in order to bend the pipeline several hundred meters upwards. Discretization. As already mentioned, the plots are smoothed out, and no abrupt changes appear. Finer discretization (smaller control volumes) is a possibility in order to obtain more detailed plots. Valve. A valve should be implemented at the outlet. This will also effect the shape of the velocity trend plots. In addition it is necessary in order to implement a controller on the system. Control. When the model represents Brage A-27 in a good manner, it is of course interesting to implement slug control. Pressure in the horizontal part is recommended as controlled variable, and the valve at the outlet should be used as manipulated variable. g-PROMS. When modelling in g-PROMS, it is supposedly easier to model PDE-systems than in Matlab. This is among others related to the need for discretization. g-PROMS could therefore be a better alternative than Matlab. 7 Conclusions The object of this thesis has been modelling of terrain induced slug flow at Brage A-27. A simple model would be a good basis for development of a structure for slug control. However, the modelling alone was a very complex problem, and this is where the work has been focused. As for the control of the system, some previous work is shown and controlled and manipulated variables are suggested. The modelling of terrain induced slug flow was first done in Simulink. The models implemented in Simulink were slow and did not represent the system in a good manner. A complicated flow of information resulted in slow simulations, and implementation of valve characteristics over control volume interfaces instead of conservation of momentum made the model unsuitable. It was decided to implement a new model in Matlab, and to apply the conservation of momentum in a much more complete way. First, a model based on for-loops was developed. This model gave rather unexpected results, and did not manage to generate slug flow. The model was evidently unstable, even at steady states where there supposedly was no flow at all. Several modifications were tried, but it did not help until the model was rewritten based on a matrix structure. Although the new model was in theory alike the old, it gave sensible and expected results. This is most likely a result of numerical problems in the first model. The new and final model was simulated as a horizontal pipeline at first. Gradually the control volumes near the outlet were bent upwards. The geometry is still far from the geometry at Brage A-27, but good enough to induce slug flow. Tuning of the interface friction has been important in order to make the model generate slug flow. The model generates terrain induced slugs with a frequency at 6 slugs per 1000 seconds, and shows typical behaviour for a system in a slugging flow regime. The model should therefore be suitable for examination and implementation of slug control. The main problem in this work has been numerical trouble. 8 Notation SymbolDescriptionUnitsACrossectionmaParameter in van der Waals Equation-BVirial Coefficient-bParameter in van der Waals Equation-DDiametermDhHydraulic DiametermEInternal EnergyJ/kgFTotal Friction lossm/s2FDDroplet dragkg m/s2fVolume Fraction-fmCombined Friction factor(kg/Pa)0.5 s-1GPossible Mass Sourcekg m/s2gGravity Forcem/s2HEnthalpyJ/kgHSEnthalpy from Mass sourceJhHeightmKValve Constant(ms)-1KPGain-LLengthmlLengthmMMolar Weightkg/molesmMasskgPPressurePaRGass ConstantJ/K.molerRadiusmReReynolds Number-SWetted PerimetermTTemperatureKUHeatJuSuperficial Velocitym/sVVolumem3vAbsolute Velocitym/svRDifference between absolute gas and liquid velocitym/sWMas Flow Ratekg/sxMass Fraction-(Inclination Anglerad(Roughnessm(eEntrainmentkg/ms(GMasstransferkg/ms(DDepositionkg/ms(Fiction Factor-(Viscositykg/ms(Internal Angle in Pipelinerad(Shear StressN/m2(IIntegral Times(Densitykgm3((u)Velocity used as Differential Variable when solving Conservation of Momentumkg(ms)-2SubtextDDroplets-GGas-gGas-gwGas-Wall-iInterface-jControl Volume Number-LLiquid-lLiquid-lwLiquid-Wall-mMidle-phPhasse-tTotal- 9 References Banerjee, S., Numerical Methods, Short Courses, Modelling and Computation of Multiphase Flows, Part 1, Zrich, 1999. Bendiksen, K.H., Malnes, D. and Nydal, O.J., On the Modelling of Slug Flow, Chem.Eng.Comm., Vols.141-142, pp. 71-102, 1996. Bendiksen, K.H., Malnes, D., Moe, R. And Nuland, S., The Dynamic Two-Fluid Model OLGA: Theory and Application, Journal of Petroleum Technology, SPE 19451, 1990. Bird, R.B., Stewart, W.E. and Lightfoot, E.N., Transport Phenomena, John Wiley & Sons, USA, 1960. Dalsmo, M. and Jansen, B., Modelling of Dynamics in Well A-27 at Brage, rapport for internal use in ABB Industri AS, 1999. Dalsmo, M., Input file from the OLGA model, unpublished, 1999. De Henau, V. And Raithby, G.D., A Transient 2-fliud Model for Simulation of Slug Flow in Pipelines. 1. Theory, , Int. J. Multiphase Flow, Vol. 21, No. 3, pp 335-349, 1995. Eik, K., Modellering og regulering av terrengindusert lsug ved tofasestrmning i rr, Diploma thesis at Department For Chemical Engineering, NTNU, 1999. Fuchs, P., Flerfase Rrstrmning, institute for refrigeration and air conditioning, NTNU, 1997. Geankoplis, C.J., Transport Processes and Unit Operations, 3. Ed., Prentice-Hall, USA, 1993. Havre,K., Stray,H. and Stornes,K.O., Stabilization of Terrain Induced Slug Flow in Multiphase Pipelines, submitted for publication in ABB review, year unknown. Haandrikman, G., Seelen, R., Henkes, R. And Vreengoor, L., Slug Control in Flowline/Riser Systems, Shell Research and Technology Centre Amsterdam, year unknown. Jansen, F.E., Shoham, O. and Taitel, Y., The Elimination of Severe Slugging Experiment and Modelling, Int. J. Multiphase Flow, Vol. 22, No. 6, pp 1055-1072, 1996. Moe, H.I. and Hertzberg, T., Modellering og Simulering av Dynamikk i Prosessanlegg, Compendium made for PROST, Institute for Chemical Engineering, NTNU, 1994. Smith, J.S., Van Ness, H.C. and Abbott, M.M., Introduction to Chemical Engineering Thermodynamics, 5.ed, McGraw-Hill, 1996. Stren, S. And Hertzberg, T., Numerisk Lsning av Partielle Differensialligninger, Compendium made for PROST, Institute for Chemical Engineering, NTNU, 1995. Taitel,Y and Barnea,D., Two-phase slug flow, Advances in heat transfer, vol. 20, year unknown. Zeng, G., Brill, J.P. and Taitel, Y., Slug Flow Behaviour in a Hilly Terrain Pipeline, Int. J. Multiphase Flow, Vol. 20, No. 1, pp 63-79, 1994 Appendix 1: Perimeters  Figure A1: Wetted perimeters Sl, Si and Sg are the wetted perimeters in the pipeline. D is the inner diameter, hl is liquid height and A is the the phases crossection. Appendix 2: Flow Sheet in Simulink  Figure A2: Flow sheet for the Simulink models Appendix 3: Matlabcode for the first Matlab model This function calculates the state in the control volumes, based on for-loops. %Function to calculate the state in the control volume %1.Revision: 18.09.00, by Jon Kolsum Reutz %2.Revision: 04.10.00, by Jon Kolsum Reutz %3.Revision: 09.10.00, by Jon Kolsum Reutz %4.Revision: 02.11.00, by Jon Kolsum Reutz %5.Revision: 09.11.00, by Jon Kolsum Reutz function sys = State_in_control_volume_3(t,x,u) global radius length angle next_value Table_1 Table_2 P %Assignation of defined inputs and outputs W_gas_inlet_1 = u(1); W_liquid_inlet_1 = u(2);, W_gas_inlet_2 = u(3);, W_liquid_inlet_2 = u(4);, P_outlet = u(5); %ASSIGNING VARIOUS CONSTANTS %Molar weight gas, Mg[kg/mol] Mg = 50.42*10^-3; %Molar weight liquid, Ml[kg/mol] Ml = 205.19*10^-3; %Constant density in the liquid phase, rho_l[kg/m3] rho_l= 834.41; %Gravity force, g[m/s2] g = 9.81; %Viscocity for gas and liquid, my[kg/ms] my_l = 1.4998*10^-3; my_g = 0.0117*10^-3; %Initialzing the matrix sys sys = []; %Gas constant for calculations with ideal gas law, R[J/K mol] R = 8.314; %Temperature, Tg[K] T = 348.15; %Correctionsfactor for unideal gas, Z[-] Z = 1; %Pressure in the reservoir, P_reservoir[Pa] P_reservior = 5.5*10^5; %Valve constants for flow from the reservoir into the pipe, valva_const[1/ms] valve_const_l = 4/3*10^-5; valve_const_g = 0.6/3*10^-5; %CALCULATING INITIAL CONDITIONS FOR EACH BLOCK for block_number = (1:7) %Inner radius in the pipeline, r[m] r = radius(block_number); if block_number == 7 r_next = 0; r_next_2 = 0; elseif block_number == 6 r_next_2 = 0; r_next = radius(block_number+1); else r_next = radius(block_number+1); r_next_2 = radius(block_number+2); end if block_number == 1 r_previous = 0; else r_previous = radius(block_number-1); end %Length of control volume, L[m] L = length(block_number); if block_number == 7 L_next = 0; L_next_2 = 0; elseif block_number == 6 L_next_2 = 0; L_next = length(block_number+1); else L_next = length(block_number+1); L_next_2 = length(block_number+2); end if block_number == 1 L_previous = 0; else L_previous = length(block_number-1); end %Elevation angle, alpha[rad] alpha = angle(block_number); %Assignations of inputs ml = x(block_number); mg = x(block_number+7); if block_number == 1 ml_previous = 0; mg_previous = 0; else ml_previous = x(block_number-1); mg_previous = x(block_number+6); end if block_number == 7 ml_next = 0; mg_next = 0; ml_next_2 = 0; mg_next_2 = 0; elseif block_number == 6 ml_next_2 = 0; mg_next_2 = 0; ml_next = x(block_number+1); mg_next = x(block_number+8); else ml_next_2 = x(block_number+2); mg_next_2 = x(block_number+9); ml_next = x(block_number+1); mg_next = x(block_number+8); end %Cross section in the pipe line, A[m2] A = pi*r^2; %Total volume in control volume(of lenght L), Vt[m3] Vt = pi*r^2*L; Vt_previous = pi*r_previous^2*L_previous; Vt_next = pi*r_next^2*L_next; Vt_next_2 = pi*r_next_2^2*L_next_2; %Liquid volume in control volume, Vl[m3] Vl = ml/rho_l; Vl_next = ml_next/rho_l; Vl_next_2 = ml_next_2/rho_l; Vl_previous = ml_previous/rho_l; %Gas volume in control volume, Vg[m3] Vg = Vt-Vl; Vg_next = Vt_next-Vl_next; Vg_next_2 = Vt_next_2-Vl_next_2; Vg_previous = Vt_previous-Vl_previous; %Volume fraction liquid, fl[-] fl = Vl/Vt; %Volume fraction gas, fg[-] fg = 1-fl; %Crossection covered by liquid, Al[m2] Al = fl*A; %Crossection covered by gas, Ag[m2] Ag = A-Al; %Assignation of inputs rho_u_l_out = x(block_number+14); rho_u_g_out = x(block_number+21); Wl_out = x(block_number+14)*Al; Wg_out = x(block_number+21)*Ag; P_in = P(block_number); if block_number == 1 % if (P_reservior-P_in)<0 % Wl_in = 0; % Wg_in = 0; % else % Wl_in = valve_const_l*(P_reservior-P_in); % Wg_in = valve_const_g*(P_reservior-P_in); % end Wg_in = W_gas_inlet_1; Wl_in = W_liquid_inlet_1; if Al>0 rho_u_l_in = Wl_in/Al; else rho_u_l_in = 0; end if Ag>0 rho_u_g_in = Wg_in/Ag; else rho_u_g_in = 0; end else rho_u_l_in = x(block_number+13); rho_u_g_in = x(block_number+20); Wl_in = x(block_number+13)*Al; Wg_in = x(block_number+20)*Ag; end if block_number == 7 P_out = P_outlet; rho_u_l_next = rho_u_l_out; rho_u_g_next = rho_u_g_out; rho_u_l_next_2 = rho_u_l_out; rho_u_g_next_2 = rho_u_g_out; elseif block_number == 6 rho_u_l_next_2 = rho_u_l_out; rho_u_g_next_2 = rho_u_g_out; P_out = P(block_number+1); rho_u_l_next = x(block_number+15); rho_u_g_next = x(block_number+22); else rho_u_l_next_2 = x(block_number+16); rho_u_g_next_2 = x(block_number+23); P_out = P(block_number+1); rho_u_l_next = x(block_number+15); rho_u_g_next = x(block_number+22); end %Taking the second feed point in account if block_number == 3 Wl_in = Wl_in + W_liquid_inlet_2; Wg_in = Wg_in + W_gas_inlet_2; if Al>0 rho_u_l_in = Wl_in/Al; else rho_u_l_in = 0; end if Ag>0 rho_u_g_in = Wg_in/Ag; else rho_u_g_in = 0; end end %Gas density, rho_g[kg/m3] if Vl>Vt Vl = Vt; Vg = 0; rho_g = 0; else rho_g = mg/Vg; end rho_g_out = rho_g; if block_number == 7 rho_g_next_2 = rho_g_out; rho_g_next = rho_g_out; elseif block_number == 6 if Vl_next>Vt_next Vl_next = Vt_next; Vg_next = 0; rho_g_next = 0; rho_g_next_2 = rho_g_next; else rho_g_next_2 = rho_g_next; rho_g_next = mg_next/Vg_next; end else if Vl_next_2>Vt_next_2 Vl_next_2 = Vt_next_2; Vg_next_2 = 0; rho_g_next_2 = 0; else rho_g_next_2 = mg_next_2/Vg_next_2; end if Vl_next>Vt_next Vl_next = Vt_next; Vg_next = 0; rho_g_next = 0; else rho_g_next = mg_next/Vg_next; end end if block_number == 1 rho_g_in = rho_g; else if Vl_previous>Vt_previous Vl_previous = Vt_previous; Vg_previous = 0; rho_g_in = 0; else rho_g_in = mg_previous/Vg_previous; end end %Gas and liquid velocity, vg and vl[m/s] if rho_g == 0 vg = 0; else vg = Wg_in/(Ag*rho_g); end if Al == 0 vl = 0; else vl = Wl_in/(Al*rho_l); end %Mass fraction liquid, xl[-] xl = ml/(mg+ml); %Mass fraction gas, xg[-] xg = 1-xl; %Calculating theta for different Al if r == 0.06068 Table = Table_1; else Table = Table_2; end %Interpolation to find theta from the table, theta[rad] if Al>0 theta = interp1(Table(:,1),Table(:,2),Al); else theta = pi; end %Total perimeter in the pipeline, S[m] S = 2*pi*r; %Wetted perimeter for the interface gas/liquid, S_i[m] S_i = 2*r*sin(theta); %Wetted perimeter for the interface liquid/wall, S_l[m] S_l = theta/pi*S; %Wetted perimeter for the inteface gas/wall, S_g[m] S_g = S-S_l; %Hydraulic diameters, D_hl & D_hg[m] if Al == 0 D_hl = 0; else D_hl = 4*Al/S_l; end if Ag == 0 D_hl = 0; else D_hg = 4*Ag/(S_g+S_i); end %Defining Reynolds numbers, Re[-] Re_l = abs(vl*rho_l*D_hl/my_l); Re_g = abs(vg*rho_g*D_hg/my_g); %Friction factors, lambda[-] lambda_i = 0.02*(1+75*pi*2*r*fl/S_i); if D_hl == 0 lambda_l = 0; elseif Re_l == 0; lambda_l = 0; else lambda_l = 0.0055*(1+(2*10^4*0.00045/D_hl + 10^6/Re_l)^(1/3)); end if D_hg == 0; lambda_g = 0; elseif Re_g == 0; lambda_g = 0; else lambda_g = 0.0055*(1+(2*10^4*0.00045/D_hg + 10^6/Re_g)^(1/3)); end %Friction coeffisients for alternative interface friction if Re_g == 0 f_g_wall = 0; else f_g_wall = 0.046/Re_g^0.2; end f_interface = f_g_wall*(2+2.5*10^-5*Re_l/(2*r))*fg^2.5; %Hydrostatic pressure, P_hs[Pa] P_hs_gas = rho_g*fg*g*L*sin(alpha); P_hs_liquid = rho_l*fl*g*L*sin(alpha); %Delta P at phase interface, P_f_interface[N/m3] %P_f_interface = S_i/A*lambda_i/2*rho_g*vg^2*sign(vg); P_f_interface = f_interface*rho_g*vg^2/2*sign(vg); %Delta P at wall interface, P_g_wall & P_l_wall[N/m3] P_l_wall = S_l/A*lambda_l/8*rho_l*vl^2*sign(vl); P_g_wall = S_g/A*lambda_g/8*rho_g*vg^2*sign(vg); %Mass conservation equations dml_dt = Wl_in - Wl_out; dmg_dt = Wg_in - Wg_out; %Momentum conservation equations if rho_u_l_out < 0 drho_u_l_dt = (((rho_u_l_next)^2/rho_l-(rho_u_l_out)^2/rho_l)/L+(P_in-P_out)/L-P_hs_liquid/L-P_l_wall/fl+P_f_interface/fl); else drho_u_l_dt = (((rho_u_l_in)^2/rho_l-(rho_u_l_out)^2/rho_l)/L+(P_in-P_out)/L-P_hs_liquid/L-P_l_wall/fl+P_f_interface/fl); end if rho_u_g_out < 0 drho_u_g_dt = (((rho_u_g_next)^2/rho_g_next_2-(rho_u_g_out)^2/rho_g_next)/L+(P_in-P_out)/L-P_hs_gas/L-P_g_wall/fg-P_f_interface/fg); else drho_u_g_dt = (((rho_u_g_in)^2/rho_g_in-(rho_u_g_out)^2/rho_g_out)/L+(P_in-P_out)/L-P_hs_gas/L-P_g_wall/fg-P_f_interface/fg); end %Pressure calculation P(block_number) = (rho_g*R*T*Z/Mg); sys(block_number) = dml_dt; sys(block_number+7) = dmg_dt; sys(block_number+14) = drho_u_l_dt; sys(block_number+21) = drho_u_g_dt; end %sys = sys' t %pause Appendix 4: Matlabcode for the final Matlab model This function calculates the state in the control volumes, based on a matrix structure. %Function to calculate the state in the control volume %1.Revision: 18.09.00, by Jon Kolsum Reutz %2.Revision: 04.10.00, by Jon Kolsum Reutz %3.Revision: 09.10.00, by Jon Kolsum Reutz %4.Revision: 02.11.00, by Jon Kolsum Reutz %5.Revision: 09.11.00, by Jon Kolsum Reutz %6.Revisoin: 20.11.00, by Jon Kolsum Reutz function sys = State_in_control_volume_4(t,x,flag,u) global r L A alpha block_number Mg Ml rho_l g my_l my_g R T Z epsilon %Assignation of defined inputs and outputs W_gas_inlet_1 = u(1); W_liquid_inlet_1 = u(2); W_gas_inlet_2 = u(3); W_liquid_inlet_2 = u(4); P_outlet = u(5); %CALCULATING INITIAL CONDITIONS FOR EACH BLOCK %Assignations of inputs ml = x(1:block_number); mg = x(block_number+1:2*block_number); %Total volume in control volume(of lenght L), Vt[m3] Vt = A*L; %Liquid volume in control volume, Vl[m3] Vl = min(ml/rho_l,Vt*ones(block_number,1)); Vl = max(Vl,zeros(block_number,1)); %Gas volume in control volume, Vg[m3] Vg = Vt-Vl; %Vg(block_number) = Vt/2-Vl(block_number); %Volume fraction liquid, fl[-] fl = Vl/Vt; %fl(block_number) = Vl(block_number)/Vt*2; %Volume fraction gas, fg[-] fg = 1-fl; %Crossection covered by liquid, Al[m2] Al = fl*A; %Crossection covered by gas, Ag[m2] Ag = A-Al; %Rho_u in the inlets, rho_u[kg/sm2] rho_u_g_inlet = W_gas_inlet_1/A; rho_u_l_inlet = W_liquid_inlet_1/A; %Assignation of inputs rho_u_l = x(2*block_number+1:3*block_number); rho_u_g = x(3*block_number+1:4*block_number); Wl = rho_u_l*A; Wg = rho_u_g*A; %Gas density, rho_g[kg/m3] rho_g = mg./Vg; %Pressure calculation P = (R*T*Z*rho_g./Mg); %Superficial velicity, v[m/s] vl = rho_u_l/rho_l; vg = rho_u_g./rho_g; %Actual velicity, v[m/s] ul = vl*A./Al; ug = vg*A./Ag; %In case Vg is zero, the density is zero as well temp = isinf(rho_g); rho_g(find(temp)) = 0; %Calculating theta for different Al Table = Theta_and_Al_2(r); Table = Table'; for i = 1:block_number %Interpolation to find theta from the table, theta[rad] theta(i) = interp1(Table(:,1),Table(:,2),Al(i)); end %Total perimeter in the pipeline, S[m] S = 2*pi*r; %Wetted perimeter for the interface gas/liquid, S_i[m] S_i = 2*r*sin(theta'); %Wetted perimeter for the interface liquid/wall, S_l[m] S_l = S.*theta'/pi; %Wetted perimeter for the inteface gas/wall, S_g[m] S_g = S-S_l; %Hydraulic diameters, D_hl & D_hg[m] D_hl = 4*Al./S_l; D_hg = 4*Ag./(S_g+S_i); %Defining Reynolds numbers, Re[-] Re_l = abs(ul.*rho_l.*D_hl/my_l); Re_g = abs(ug.*rho_g.*D_hg/my_g); %Friction factors, lambda[-], taking both laminar and turbulent flow into account lambda_l = min(64./Re_l,0.0055*(1+(epsilon./D_hl*2*10^4 + 10^6./Re_l).^(1/3))); temp = isinf(lambda_l); lambda_l(find(temp)) = 0; lambda_g = min(64./Re_g,0.0055*(1+(2*10^4*epsilon./D_hg + 10^6./Re_g).^(1/3))); temp = isinf(lambda_g); lambda_g(find(temp)) = 0; lambda_i = 0.00005*(1+75*pi*2*r*fl./S_i); %Tuned down with from 0.02 coefficient to 0.00005 temp = isinf(lambda_i); lambda_i(find(temp)) = 0; %Shear stress at phase interface, P_f_interface[N/m2] tau_i = sign(ug).*lambda_i.*rho_g.*((ug-ul).^2)/2; %Shear stress at wall interface, tau[N/m2] tau_lw = sign(ul).*lambda_l./8.*rho_l.*ul.^2; tau_gw = sign(ug).*lambda_g./8.*rho_g.*ug.^2; %MASS COSERVATION EQUATIONS dmldt(1) = W_liquid_inlet_1 - Wl(1); dmldt(2:block_number) = Wl(1:block_number-1)-Wl(2:block_number); dmgdt(1) = W_gas_inlet_1 - Wg(1); dmgdt(2:block_number) = Wg(1:block_number-1)-Wg(2:block_number); %In order to find negative flow, imaginary parts of the square root of the flow is found and %compared to a zero-vector. A vector with values 1 and 0 is returned (1 for match, 0 for mismatch) dir_l=eq(imag(sqrt(rho_u_l)),zeros(length(rho_u_l),1)); dir_g=eq(imag(sqrt(rho_u_g)),zeros(length(rho_u_g),1)); %MOMENTUM CONSERVATION EQUATIONS %Hydrostatic pressure, P_hs[N/m3] P_hs_g = rho_g.*fg.*g.*sin(alpha); P_hs_l = rho_l*fl.*g.*sin(alpha); %ert=P_hs_l(block_number) %urt=P_hs_g(block_number) %Positive flow drhoul(1) = ((rho_u_l(1))^2/rho_l - rho_u_l_inlet^2/rho_l)/L; drhoul(2:block_number) = (rho_u_l(2:block_number).^2/rho_l - rho_u_l(1:block_number-1).^2/rho_l)/L; drhoug(1) = ((rho_u_g(1))^2/rho_g(1) - rho_u_g_inlet^2/rho_g(1))/L; drhoug(2:block_number) = ((rho_u_g(2:block_number).^2./rho_g(2:block_number)) - (rho_u_g(1:block_number-1).^2./rho_g(1:block_number-1)))/L; %Negative flow, at the outlet ther can not be nagative flow ndrhoul(1:block_number-1)=-(rho_u_l(1:block_number-1).^2/rho_l-rho_u_l(2:block_number).^2/rho_l)/L; ndrhoul(block_number)=-(rho_u_l(block_number).^2/rho_l-0.^2/rho_l)/L; ndrhoug(1:block_number-2)=-((rho_u_g(1:block_number-2).^2./rho_g(2:block_number-1))-(rho_u_g(2:block_number-1).^2./rho_g(3:block_number)))/L; ndrhoug(block_number-1)=-((rho_u_g(block_number-1).^2./rho_g(block_number))-(rho_u_g(block_number).^2./rho_g(block_number)))/L; ndrhoug(block_number)=-((rho_u_g(block_number).^2./rho_g(block_number))-0)/L; %Combining both negative and positive flow drhoul=dir_l.*drhoul'+abs(dir_l-1).*ndrhoul'; drhoug=dir_g.*drhoug'+abs(dir_g-1).*ndrhoug'; i_l=[1:block_number]'+abs(dir_l-1); i_g=[1:block_number]'+abs(dir_g-1); %The final equations for conservation of momentum drhouldt(1)=-drhoul(1)-0.5*(P_hs_l(1)+P_hs_l(2))-(P(2)-P(1))/L-S_l(i_l(1))/A*tau_lw(i_l(1))/fl(i_l(1))+S_i(i_l(1))/A*tau_i(i_l(1))/fl(i_l(1)); drhouldt(2:block_number-1)=-drhoul(2:block_number-1)-0.5*(P_hs_l(2:block_number-1)+P_hs_l(3:block_number))-(P(3:block_number)-P(2:block_number-1))/L-S_l(i_l(2:block_number-1))/A.*tau_lw(i_l(2:block_number-1))./(fl(i_l(2:block_number-1)))+S_i(i_l(2:block_number-1))/A.*tau_i(i_l(2:block_number-1))./(fl(i_l(2:block_number-1))); drhouldt(block_number)=-drhoul(block_number)-P_hs_l(block_number)-(P_outlet-P(block_number))/L-S_l(block_number)/A*tau_lw(block_number)/fl(block_number)+S_i(block_number)/A*tau_i(block_number)/fl(block_number); drhougdt(1)=-drhoug(1)-0.5*(P_hs_g(1)+P_hs_g(2))-(P(2)-P(1))/L-S_g(i_g(1))/A*tau_gw(i_g(1))/fg(i_g(1))-S_i(i_g(1))/A*tau_i(i_g(1))/fg(i_g(1)); drhougdt(2:block_number-1)=-drhoug(2:block_number-1)-0.5*(P_hs_g(2:block_number-1)+P_hs_g(3:block_number))-(P(3:block_number)-P(2:block_number-1))/L-S_g(i_g(2:block_number-1))/A.*tau_gw(i_g(2:block_number-1))./(fg(i_g(2:block_number-1)))-S_i(i_g(2:block_number-1))/A.*tau_i(i_g(2:block_number-1))./(fg(i_g(2:block_number-1))); drhougdt(block_number)=-drhoug(block_number)-P_hs_g(block_number)-(P_outlet-P(block_number))/L-S_g(block_number)/A*tau_gw(block_number)/fg(block_number)-S_i(block_number)/A*tau_i(block_number)/fg(block_number); t sys=[dmldt';dmgdt';drhouldt';drhougdt']; Appendix 5: Additional Simulations  Figure A5.1: ((u)g at the outlet after 15% increase in inflow gas  Figure A5.2: ((u)g at the outlet after 5% increase in inflow liquid  This was done at ABB Industri by the Author, Summer 2000  Simulink, version 3.0, is developed by The Math Works Inc., and is a simulation program for dynamic systemsbased on block diagrams. Simulink and Matlab are complete integrated programs.  Matlab, version 5.0, is developed by The Math Works Inc., and is a program for numerical calculations and visualistion of data. PAGE  PAGE  Abstract PAGE 3 Modelling of Terrain Induced Slug Flow Table of Contents PAGE  PAGE 4 Modelling of Terrain Induced Slug Flow Inrtoduction Slug Flow Theory Case Description PAGE 16 Modelling of Terrain Induced Slug Flow Modelling of Slug Flow PAGE 24 Modelling of Terrain Induced Slug Flow Results PAGE 39 Modelling of Terrain Induced Slug Flow Discussion PAGE 44 Modelling of Terrain Induced Slug Flow Conclusions PAGE 47 Modelling of Terrain Induced Slug Flow Notation References PAGE 48 Modelling of Terrain Induced Slug Flow Appendix 1: Perimeters PAGE 64 Modelling of Terrain Induced Slug Flow Appendix 2:Flow Sheet in Simulink Appendix 3:Matlabcode for the first Matlab model Appendix 4:Matlabcode for the final Matlab model Appendix 5:Additional Simulations  './=798 9 : H I X Y s t u v w   ! 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