Article pubs.acs.org/EF
Dynamic Models in Multiphase Flow Ole Jorgen Nydal* Department of Energy and Process Engineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway ABSTRACT: Transport of oil and gas mixtures in pipelines involves flow dynamics on a wide range of time and length scales. Liquid slugs and waves occur at scales extending from diameters (“hydrodynamic slug flow”) and up to riser lengths (“severe slugging”). The approach in one-dimensional dynamic multiphase flow models for analysis of flow dynamics depends upon the scale to be resolved. Types of models are discussed, and a hybrid two-fluid model and a slug tracking model are in particular described. A two-fluid model is applied on a stationary grid in the gas−liquid stratified flow region until a slug is formed (or initiated), when a slug tracking method with a moving grid takes over. The performance of the model is demonstrated in relation to three types of cases with different time and length scales: Two- and three-phase severe slugging, hydrodynamic slugging after a bend, and a pigging case to simulate the rapid release of a hydrate plug. The computations are compared to experimental laboratory data.
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INTRODUCTION Transporting reservoir fluids in subsea pipelines from wells to processing units involves multiphase flows of complex gas−oil− water−solid mixtures. The term “flow assurance” is used as a broad term for a large range of challenges related to the safe design and operation of such transportation systems. Deposits (salts, gas hydrates, and wax particles) can block the pipe; corrosion can destroy the pipe; and flow transients can render the pipeline difficult to operate. The flow assurance issues are of major concern, because there are often no backups to the failure of a subsea transportation system. Assessment of flow assurance challenges requires the knowledge of the local thermohydraulic conditions along the pipeline: pressure, wall and fluid temperatures, fluid composition, flow regimes, liquid accumulation, and phase velocities. Flow models, which aim at predicting these local flow conditions, are therefore a basic requirement for flow assurance management. This is the background for the continuing efforts on improving flow simulators for multiphase transportation systems. The following is a discussion on some current modeling challenges and illustration of some computational examples on dynamic flows at different time and length scales.
regimes, phase fractions, and velocities. For a given undulating pipeline system, one design question is how these flow variables vary with the pipe diameter. The reservoir conditions limit the transport capacity of the system, and the design analysis is then often a balance between reducing the pressure loss at the cost of increasing the liquid accumulation. Temperature Profiles. Heat losses from the pipe to the surroundings leads to a cooling of the pipeline and its contents, from the wellhead and along the seabed. This can bring the fluid system into undesired pressure and temperature ranges, where solids precipitate and gas hydrates form, with the subsequent risk of wall deposits and plugging of the pipe. The modeling problem then also includes the outer wall heattransfer problem. The cooling of pipelines during shutdown operations is of particular importance, because the fluid-related effects at low temperatures can lead to startup problems afterward. Flow Dynamics. Whether a flow condition should be termed dynamic or steady state is a matter of scales. At sufficiently small length and time scales, multiphase flows are all dynamic, with fluctuations in phase fractions, pressures, and velocities. Small-Scale Slug Flow. Slug flow is a dynamic flow with alternating flows of gas and liquid plugs. In a laboratory setup with straight pipes, slug lengths vary typically between 10 and 100 numbers of diameters. The modest scale of this flow dynamics is such that it normally does not pose severe operational problems for pipeline operations. The propagation of large breaking roll waves is also a dynamic flow phenomenon, on a similar scale as liquid slugs. Because such waves contribute to the liquid transport, they are also subject to the modeling concerns. Large-Scale Dynamics. The large-scale flow dynamics are clearly of operational concern. These are flow fluctuations on
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ON THE FLOW PROBLEM A pipeline carries oil−water−gas mixtures: what are the potential flow problems that we need a model to predict? Flow Details. It would, of course, be of great value to have information on the local flow conditions inside the pipe: local phase fractions on the pipe cross-section, wall wetting condition, local deposition rates to the wall, etc. The sort of details computational fluid dynamics (CFD) provides for single-phase flows would be very useful for multiphase systems. This is, however, beyond the reach of current simulators, both regarding models and computation times. Flow Capacity of Pipelines. Under steady flow conditions, we need to know the state of the flow along the pipe, in terms of flow variables, which are volume-averaged over a pipe segment. These would typically be pressure, temperature, flow © 2012 American Chemical Society
Special Issue: Upstream Engineering and Flow Assurance (UEFA) Received: February 16, 2012 Revised: April 17, 2012 Published: April 27, 2012 4117
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multiphase mixtures is going to be a long-term challenge, including also numerical methods for free surface flows. In discussions on the future path for the development of pipeline models, it has been questioned whether 3D simulators will be the computational multiphase pipeline tools of the future. Simulation technology, including models, numerical methods, and computer hardware, is progressing, and 3D simulations may be more feasible in the future. The current status suggests, however, that one-dimensional (1D) models will prevail as design models for long pipeline simulations also in the long-term future. An intermediate concept could be an alternative, in which the 3D model is integrated in space along the spanwise direction of the pipe cross-section. This gives a two-dimensional (2D) computational scheme for multiphase flow, where the circular geometry of the pipe is taken into account in the averaging process.9 The following are some comments on challenges and experiences with computational methods for 1D models for dynamic multiphase pipeline simulations. Challenges in 1D Models. In a 1D model, all variables are only functions of x (the coordinate along the pipe) and time. The variables (liquid fraction, pressure, velocities, and temperatures) are then averaged properties over the pipe cross-section. The scale discussion then applies only to discretization in time (time resolution of the model) and in length (pipe segment lengths). A discussion on improvements of flow simulators then includes model formulation, closure relations, and numerical methods.
length scales of pipelines and risers and time scales on the order of transport times through the system. A pipeline−riser system can give unstable flows, with regular liquid buildup and subsequent violent blow out into the receiving facility (“severe slugging”, “riser slugging”, and “terrain slugging”). This is a compressibility problem, where trapped gas upstream of a blockage is compressed and subsequently provides the pressure for the accelerated blowout. This flow problem was a strong motivation for starting the development of the dynamic flow simulator OLGA1 in the early 1980s. Flow correlations for the estimation of steady-state pressure drops and liquid fractions were available, although largely based on small-scale experimental data, but dynamic models for the prediction of flow instabilities were at the time in demand. Nuclear codes were available,2,3 and other dynamic flow simulators have later been developed for oil−gas applications as well.4−8 Expansion effects in long risers or wells are also a class of flow instabilities that can lead to operational limitations. Even small amounts of accumulated gas upstream of a long vertical section can cause flow oscillations as it enters the vertical pipes, expands, and causes acceleration and flushing of the system. Pipelines are often operated dynamically. A network of wells feeding a common transport pipeline normally experiences changes as the production from individual wells varies in time. Shutdown and restart cases, planned production changes, and pigging procedures are other operations that require knowledge of the dynamic flow effects. A pig is the name for a device that is inserted into a pipeline, to remove liquid accumulation or wall deposits or for inspection purposes. The pig is driven by the flow itself and pushes accumulated fluids in front of it by the pressure buildup behind the pig. Surge waves are a class of flow transients that can be experienced in gas pipelines. They are long-wavelength waves passing through the pipeline after production changes. Even with small liquid fractions in the system, these waves can give large liquid transients into the receiving facilities, depending upon the length of the pipeline.
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MODEL FORMULATIONS The term “two-fluid model” is used here for the model scheme where each region in a multiphase mixture has its separate momentum equation. A region can be a continuous singlephase region or a dispersed region (dispersed droplets or bubbles). The “mixture model” denotes the case where a mixture momentum equation is applied. A full model includes mass balances for each field and energy equations. The velocities required in the mass balances are given from separate momentum equations or mixture momentum equations supplied with slip relations. For a two-field flow model, the set of equations for field k (e.g., a dispersed or a continuous phase) would then typically be as follows: Mass balances:
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ON FLOW MODELS A computational model for multiphase flow is a set of mass, momentum, and energy conservation equations for the different flow fields. The equations are discretized and solved numerically, with space and time integration of the equation system for given boundary and initial conditions. The momentum and energy equations require closure relations: models for internal momentum and heat transfer in the fluids, between the phases and with the boundaries. The formulation of these closure relations depends upon the time and length scales, which are targeted for the model formulation. Three-Dimensional (3D) Models. For single-phase CFD models, the mass and momentum equations can be solved directly on a very fine grid [“direct numerical simulations” (DNS)]. Because DNS computational times are often prohibitively large, models are formulated on a larger scale. The step up in scale means that new closure models must be supplied to take into account the averaged flow behavior at the larger scale. For single-phase flow this means turbulence models, which describe the momentum transfer because of turbulent fluctuations. For multiphase flow, this closure problem is much more extensive, because the larger scale now also involves multiphase mixtures, with interface transfer terms depending upon the local flow regime. Needless to say, the modeling of turbulence in dispersed and separated
∂αkρk ∂t
+
∂Wk =Ψ ∂x
For field k, Wk = αkρkUk is the momentum per unit volume, ρk is the density of the phase in the field, αk is the cross-sectional phase fraction, Uk is the cross-section averaged velocity, and Ψ includes all of the mass-transfer terms (phase change and mixing terms). The two-fluid model applies two momentum equations: ∂Wk ∂WkUk ∂p + = −αk − Fk ± Fi − Gk + Ok ∂t ∂x ∂x
Acceleration of the field k is caused by the action of pressure forces (∂p/∂x), friction (wall Fk and interface Fi), and gravity forces Gk. Ok includes various other momentum-transfer terms, such as level-gradient terms, phase-change and droplet4118
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approximated as steady flows. Slug flow and large-amplitude wavy flows, however, involve characteristic time and length scales and the propagation of gas−liquid fronts and possibly liquid−liquid fronts (oil−water). Two strategies can then be considered. One is to resolve all scales (slugs and roll waves), and the other is to resolve only the large-scale dynamics (severe slugs) and treat the small scale as averaged flow.
exchange terms, and possibly correction terms, to render the set of equations well-posed. The mixture model (or “drift flux model”) applies a mixture momentum equation: ∂p ∂Wm ∂WmUm + =− − Fm − Gm ∂t ∂x ∂x
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The mixture momentum now changes according to the pressure forces, the wall friction of the mixture (Fm), and the gravity of the mixture (Gm). The mixture momentum equation is supplied with a slip relation, which is normally some sort of algebraic relation relating the phase velocities to each other (and to the mixture velocity through the volumetric flow conservation). There are different forms for slip relations (or drift flux relations). A simple gas−liquid relation is
AVOIDING SMALL-SCALE PHENOMENA (AVERAGED WAVE/SLUG FLOW) Representing the pipeline with a very fine numerical grid can give unacceptable high computation times. Applying a large grid means that slugs and waves must be treated as averaged flows and not resolved individually. Wavy flow can then be regarded as separated flow, with tuned friction closure relations. A “unit cell” slug flow model is often used for averaged slug flow. A unit cell slug flow model is based on a characteristic slug−bubble sequence and a combination of a separated flow model in the bubble region and a mixture model in the slug region. The models are coupled with mass balances and supplied with relations for slug propagation (bubble velocity relations), gas entrainment rates, and slug length estimates (or slug frequency). Because the grid now is designed to be larger than the slug unit scale, flow regime transition criteria need to be implemented. OLGA applies a criterion based on continuity in phase fractions across the transitions. This is numerically favorable and also consistent with high-pressure transition criteria, but it also leads to constraints in the unit cell model. The flow model applied in the bubble region should be equal to the model applied for stratified flow (droplets and friction closures). The slip and friction models in the slug region and the slug void fraction model should also be equivalent to the models applied for bubble flow. The continuity-based stratified−slug transition criterion does not allow for simulation of hysteresis effects at low-pressure transitions (unstable stratified flow transition criteria). One example is the generation of initial very long liquid slugs at the point of unstable stratified flow, because of excess liquid buildup beyond the holdup values where slug flow can be sustained and would also have been initiated by the continuity criterion.
Ug = CUm + uo where C represents the phase distribution effects and uo represents the averaged local slip effects. The principles in the OLGA model are that five fields are tracked for the case of the separated gas−liquid flow regimes: three layers (gas, water, and oil) and two droplet fields (oil and water droplets in gas). The required five velocity fields are determined from three momentum equations (one for each layer) and two slip relations (gas and water droplets and gas and oil droplets). One energy equation is applied under the assumption of thermal equilibrium, with equal temperature in all phases. A similar concept is applied in LedaFlow, except for LedaFlow using two energy equations and mass balances for all of the possible dispersed fields (in total, nine mass balances).8 The flow regimes can change along the pipe and in time during dynamic flow simulations. A two-fluid model is typically suitable for separated flows, where the slip can vary very much. A mixture model is more intuitive for mixed flows, where the slip is quite limited because of the strong phase interactions (bubble flows or droplet flows). Two computational approaches can then be considered. One is to switch between solving a two-fluid model and a mixture model, depending upon the regime identification in each numerical grid cell at each time step. The other is to retain solving the same set of equations for all regimes (two fluid or mixture) and only change the closure models depending upon the regime. OLGA applies the latter method, solving the same two-fluid scheme for all regimes. For mixed flows, where a mixture model is desired, the friction models in the two-fluid scheme are tuned to yield the same result as the mixture model.10 Single-phase liquid flow is numerically solved as a compressible flow problem.
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RESOLVE ALL SCALES (CAPTURING AND TRACKING METHODS) Models can be designed to resolve the time and space evolution of individual slugs and waves. Averaged slug or wavy flow models should then not be necessary to formulate. This could be a general model concept, where the small-scale dynamics, including the slug and wave initiation from unstable stratified flows, can be captured by refining the numerical grid. Two strategies can be followed or a mixture of the two: front capturing with a fixed grid or slug tracking with a moving grid. Front Capturing. The idea with capturing methods is to solve the set of conservation equations on a sufficiently small numerical grid such that fronts are resolved with the desired accuracy.11,12 The positive features of this approach are that one model formulation can be applied for all flow conditions and flow regime changes should evolve directly from the model and need not be implemented as separate transition models. Implementing flow regime transition models often gives challenges related to both the formulation of the transition
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CLOSURES The closure requirements for the two-fluid models are mainly the three friction terms (Fk, Fg, and Fi). For the mixture model, a mixture friction (Fm) and a slip relation is required. These closure relations are often empirically suggested on the basis of experimental data, and the formulation of the closures will depend upon the scales for which the models are designed. The models are solved on a numerical grid, and the grid size and time step must be consistent with the assumed model scale and will then determine the resolution of the model. The multiphase flow regimes in pipe flows span configurations from smooth stratified flows, gravity waves, large breaking waves, droplet flows, slug flows, and bubbly flows. Bubbly flows and small-amplitude stratified wavy flows can be 4119
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This is a multiscale method, with different grid sizes for different variables. The gridding can schematically be as indicated in Figure 3. For the case of two liquids (three-
criteria and the implementation of different models across the transitions. Challenges remain on deriving universal models, illposedness, numerical diffusion, and transition to single-phase flow. In practical implementations, some sort of front identification is normally also needed for capturing schemes, to choose appropriate closure models for the different regimes on each side of the front (Figure 1). The penalty of capturing schemes is that a fine grid does lead to long computational times, but parallelization methods and hardware capabilities are also steadily progressing.
Figure 3. Illustration of a hybrid scheme with moving grid.
phase flow), a grid in the slug is needed for the liquid mass balances, to track the oil and water phase fractions from an oil− water slip model. The implementation is object-oriented, meaning that the models in the grid sections for the compressible and incompressible regions can be modified in terms of new classes, without the need for conditional tests in the main program. Implementation of a pig is an example where the pig will be an inherited liquid section and the behavior (friction and bypass) can be specified separately through virtual functions. Figure 4 illustrates the scheme, where also the fixed grid for the wall energy equation is indicated.
Figure 1. Illustration of a fine numerical grid for the front capturing scheme.
Front Tracking. Front tracking schemes apply moving grid points at the fronts. The positive aspects of this approach are that numerical diffusion is avoided, front propagation behavior can be implemented directly, and the computational times are more reasonable, because the number of grid points can be orders of magnitude less than for capturing schemes. Front tracking can be implemented with a moving grid scheme or with a fixed grid, applying subgrid tracking models that modify the fluxes and the closures for the fixed grid model (Figure 2). OLGA applies the latter scheme.
Figure 4. Illustration of a hybrid scheme with pig implementation and fixed grid for wall temperatures.
The model in summary then consists of conservation equations, which are solved on open, deformable, and moving grids: three mass equations, two momentum equations on a sub-grid in the bubble region and an oil−water slip relation, one momentum equation for the slug region, with an oil−water slip model, two energy equations (gas and liquid), and one wall energy equation. The numerical method is based on semiimplicit time integration and upwind fluxes. The main drawbacks and limitations of the scheme are as follows: (1) The grid management requires a lot of attention, because grids of finite sizes come and go during the simulations (slugs and sub-grid sections can be created and removed dynamically). (2) The applied assumptions lead to limitations in flow scenarios, which can be simulated. Expanding bubbly flows is not possible under the current limitation of incompressible flow in the slug region. (3) Capturing the slug initiation for unstable stratified flows is possible numerically but difficult to make physically correct. Applying a large grid in the bubble region leads to challenges on formulation and implementation of sub-grid slug initiation models. Gravitydominated slug initiation in bends is less of a challenge. (4) Waves can be resolved if they are on the same scale as slugs. This requires an integral wave model similar to the slugs, and an attempt on that has also been made.16
Figure 2. Illustration of a numerical grid for the front tracking scheme.
The drawback of tracking methods is the elaborate grid management, to handle open and moving control volumes, which are dynamically destroyed and created as the flow evolves. Hybrid Concept. Some computational examples with a hybrid scheme are presented below. This is an extension of a simplified slug tracking scheme.13−15 A hybrid scheme includes both capturing (for evolution of liquid slugs) and tracking (for the subsequent slug dynamics). The basic ideas and experiences with the scheme are as follows: (1) The numerical grid is designed to represents the physical units of slugs and bubbles. This gives a non-diffusive scheme of slug and bubble tracking, not only local front tracking. (2) The computational domain is divided into compressible compressible (bubble) and incompressible (slug) regions. Each region can have a sub-grid, typically for segments with different inclinations. (3) Appropriate and numerical methods can be applied for the bubble region (a twofluid model or simplifications of it) and the slug region (incompressible flow). Because the liquid velocity is constant along the slug, an integral momentum equation can be applied for the whole slug length. We then avoid solving the momentum equation along a disretized incompressible unit, which would require very small time steps or an iterative procedure. (4) The local physics at fronts can be implemented directly. This relates to gas entrainment rates into liquid fronts and bubble propagation velocities. Implementation of a wake effect, in which the bubble velocity increases with a decreasing upstream slug length, is a useful mechanism for the prediction of the evolution of slug length statistical distributions.
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COMPUTATIONAL EXAMPLES The slug tracking scheme has been tested on several different types of flow dynamics where we also have laboratory experiments for comparisons: severe slugging, hydrodynamic slugging, roll waves, flushing, and pigging. The following gives some examples of dynamic cases at different time and length scales. Large-Scale Severe Slugging. Computations have earlier been compared to measured pressure amplitudes and slug 4120
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Figure 5. Screen shot of three-phase severe slugging simulation.17
Figure 6. Severe slugging experiments and simulation.18
frequencies for a large range of two-phase gas and liquid flow rates in a S-shaped riser configuration.16 The S shape adds to the complexity of the flow problem compared to a catenary, because there are two dips for liquid accumulation. Some experiments have later been made with three-phase flow in the S riser. Computationally, it is also a useful test case, because oil−water separation can occur during the slug buildup period. Separated oil and water flow will then enter the outlet separator during the slug blowout. Figure 5 shows a screen shot of the simulation results during the slug buildup. The slip model keeps the water back, and oil passes to the upper part of the liquid slug. The inlet pressure time recordings correspond well with the experiments.17 Only pressure was measured, because of limited instrumentation. Another case from the laboratory of Shell in Amsterdam proved to be useful, because it shows a coupling between small hydrodynamic slugs and large riser slugs. Figure 6 shows the pressure time recordings and a screen shot of the animation of the simulation results. A tank at the inlet provides the required upstream gas compressibility to obtain severe slugging. Liquid
accumulates in the horizontal part at the same time as the riser is filled with liquid. During the blowout, the liquid in the horizontal section is transported toward the bend as hydrodynamic slugs and contributes to the initiation of a new severe slug in the bend. The computed riser slugging cycle does depend upon the slug initiation model in the horizontal part. The case was attempted with a fine grid and direct slug initiation from the two-fluid model (capturing) in the horizontal section. Using a larger grid and slug initiation from the transition criteria of unstable stratified flow (Kelvin-Helmholtz type of criterion) gave better correspondence with the experimental data than the capturing simulations.18 Small-Scale Hydrodynamic Slugging. Slug tracking simulation methods allow for implementation of a wake effect on the bubble propagation: a large bubble will accelerate as the upstream slug length is decreasing. This is a flow mechanism that can give the evolution of slug length statistical distributions from the tracking model. 4121
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Figure 7. Experiments and computations of slug length statistical distributions after a bend.19
Figure 8. Small-scale experiments and simulation of the plug propagating into a V-pipe geometry with and without initial liquid.14
forth as the liquid was accelerated and blown out of the system in a fraction of a second. The plug oscillations do not correspond completely with the computations; the friction properties of the plug was not measured; and there could also be uncertainties regarding gas leakage across the plug. The plug propagation was determined by digital analysis of high-speed videos, giving continuous values for the plug position and velocity during the blowout.
Slugs are initiated in the bend after accumulation of a predetermined critical amount. Because this is gravitydominated flow, the initiation mechanism is quite well-defined. The small initial slugs die after penetration by gas bubbles, and longer slugs survive for longer distances. The computed and measured distributions are shown in Figure 7 for three locations after the bend. It is in general difficult to predict hydrodynamic slugging well, but the correspondence for this case is quite good. The experiments were made with air−water in a small-scale setup (16 mm diameter and 2 m pipe lengths) using light diodes for slug detection.19 Case of a Rapid Pig Release. A small-scale setup was made to simulate the rapid release of a gas hydrate plug, after a sudden exposure to a higher upstream pressure.14 Single-sided pressure reduction to melt a hydrate plug can be a dangerous operation; there have been casualties with run-away plugs after release. Figure 8 shows the setup and results. A foam plug was inserted into the pipe, and a magnetic valve opened for the pressure in the tank at the inlet. For the case of an empty pipe, the plug was shot out at high velocities. For the bend case, the velocities were more moderate and the plug oscillated back and
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CONCLUSION
Dynamic multiphase flow models are important tools for design and operational support of multiphase pipelines. It is, however, difficult to arrive at one model concept that can cover the whole large span in time and length scales of the flow dynamics encountered in multiphase transport. Some model approaches are discussed, and the performance of a particular slug tracking model is demonstrated for three types of dynamic cases: severe slugging cases with large-scale two- and three-phase flow oscillations, a small-scale slug flow case, with the evolution of the slug length distributions after a bend, and a final pigging case, simulating a rapid blow out of a hydrate plug in a Vshaped pipe. 4122
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(16) Nydal, O. J.; Audibert, M.; Johansen, M. Experiments and modelling of gas−liquid flow in an S-shaped riser. Proceedings of the 10th British Hydrodynamics Research Group (BHRG) International Conference MULTIPHASE ’01; Cannes, France, June 13−15, 2001. (17) Kjeldby, T. K.; Henkes, R.; Nydal, O. J. A Lagrangian grid threephase slug tracking simulator. Proceedings of the 16th British Hydrodynamics Research Group (BHRG) International Conference on Multiphase Production Technology; Banff, Alberta, Canada, June 20−22, 2012. (18) Kjeldby, T. K.; Henkes, R.; Nydal, O. J. Slug initiation models in slug tracking simulation of sever slugging. Manuscript submitted for publication. (19) Renault, F.; Nydal, O. J. A simple slug capturing and slug tracking scheme for gas−liquid pipe flow. Application to slug length determination in a small scale loop. Manuscript to be published.
The slug tracking concept shows good comparisons to the presented experimental cases as well as a number of other dynamic flow cases. This type of model concept can potentially be a useful supplement to available general purpose models, for this particular class of slug flow problems.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
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