TWO-PHASE FLOW AND DISPLACEMENT IN ECCENTRIC ANNULI: A CFD STUDY N.A. Caruso, Q.D. Nguyen and H. Zhang School of Chemical Engineering, University of Adelaide, Adelaide, Australia, 5005
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TWO-PHASE FLOW AND DISPLACEMENT IN ECCENTRIC ANNULI: A CFD STUDY N.A. Caruso, Q.D. Nguyen and H. Zhang School of Chemical Engineering, University of Adelaide, Adelaide, Australia, 5005 ABSTRACT Cementing operations are critical in the petroleum industry to isolate formations. Safety, environment impact and the economic success of a project depend on a successful cementing campaign. Cementing involves pumping spacer fluid to displace drilling mud, then cement in order to create the bond between the casing and formation. It is very rare that the casing is concentric in the borehole; therefore, it is almost always offset, usually to an unknown extent. It is for these reasons that a good knowledge of one fluid displacing another is needed, so that cementing operations may be run with greater certainty. The aim of this work is to build a functional 3-dimensional CFD model to capture the effects of non-newtonian fluid displacements in eccentric annuli. Effects such as eccentricity of the well bore and rotation of the inner cylinder and fluid rheology may have large influences on the displacement efficiencies and flow phenomena. The model was run using ANSYS CFX 13.0 with fluid flow occurring under laminar conditions with negligible mixing between phases. It was found that the model can predict the effects of eccentricity and rotational flows on displacement efficiency and frictional pressure losses. The simulations confirm the experimental observations that rotation of the inner cylinder, even at low speeds, can significantly improve the displacement performance in highly eccentric annuli. Furthermore, for viscoplastic fluids, mild rotational flows lead to decreases in friction losses, which are strongly dependent of the annular eccentricity. INTRODUCTION An abundance of research, both theoretical and experimental, currently exists on single phase flows through eccentric annuli for Newtonian and non-newtonian fluids. Recently, research focus has begun to look at the process of displacing one fluid by another, especially considering the needs of the oil and gas industry in cementing operations. Cementing operations are critical to the success of any field development. Although the displacement of fluids in a two fluid system has been gaining attention in recent years, there is still not a great deal of published research, and much of it is still at a descriptive stage. Concerning flows through eccentric annuli, there are still many uncertainties that are faced. Some of the most important physical parameters and uncertainties include eccentricity (offset of inner cylinder), rotation of the inner cylinder as well as surface tensional parameters. The scope of this study is to build a computational fluid dynamic (CFD) model using ANSYS CFX software that is able to simulate the flow and displacement of two non- Newtonian fluids in eccentric annuli. The model is developed for laminar, transient flows with no mixing between fluids in vertical eccentric annuli. Comparison with experimental data from flow visualisation and displacement efficiency measurements has been used to validate the CFD model. This paper begins by explaining the model equations and theory as well as the simulation parameters. It then describes results from simulations completed to date concerning displacement interfaces, velocity profiles, displacement efficiency, and frictional pressure losses associated with two-fluid displacement and flow. NUMERICAL METHOD Displacement Model The basic equations used to model flow for fluids that behave as a continuous substance are the equations of continuity and linear momentum. In the homogeneous fluid model, the transport equations are solved for the bulk fluid rather than for each individual phase (Bird et al., 2007). The homogeneous transport equations are formed by the summation of the transport equations for each phase, and consist of the continuity and momentum equations as follows. ρ + ρv = 0 t T ( ρv) + ( ρvv + η( v + ( v ) )) + p = S t where S is a momentum source term, ρ is the fluid density, η is the fluid viscosity, p is pressure and v is the velocity vector. The transport equations were solved using the finite volume method available in the ANSYS CFX software. Fluid properties, such as viscosity and density, were calculated as volume-weighted averages in each element. The displacement was modelled by considering the displacing and displaced fluids to be immiscible. The effect of interfacial tension usually associated with sharp immiscible interfaces was considered by using the Continuum Surface Force (CSF) model developed by Brackbill et al. (1992). The CSF approach treats surface tension force not as a surface force but as an extra body force that can be included in the momentum equation. The discontinuities or sharp interfaces where fluid changes from one phase to another are thus replaced by a continuous transition region within which the surface tension force is acting. Flow Geometry and Simulation Parameters The annular flow geometry used is a vertical annulus with dimensions as shown in Table 1. For the CFD simulations, a tetrahedral mesh was constructed for the flow domain using the ANSYS DesignModeler and its meshing routine. An example of the mesh used containing elements is illustrated in Figure 1. Further, two different mesh refinements were employed to study the effect of mesh size on the simulated profiles. Fig. 1: Typical fine mesh used for simulation. 2 Table 1: Geometric and simulation parameters for the CFD model. Fluid I/II System Outer cylinder ID (m) Inner cylinder OD (m) Length (m) Eccentricities (%) 0, 25, 50, 90 Axial flow rate (L/min) 10 Inner cylinder rotational speeds (rpm) 0, 10 Time of Simulation (s) 140 Flow regime Laminar Properties of the fluids used are summarised in Table 2. The displaced fluid was denoted as Fluid I and the displacing fluid as Fluid II in the simulation. Both fluids were modelled as Bingham plastic fluids described by the following constitutive equation: τ y η = + µ p & γ Where & γ is the shear rate, τ y is the yield stress and µ p is the plastic viscosity. Densities of both fluids were set at 1000 kg/m 3. Table 2: Properties of the fluids used for the simulation Fluid τ y (Pa) µ p (Pa.s) ρ (kg/m 3 ) Fluid I Fluid II The inlet flow rate was set at 10 L/min, and the outlet pressure was set to 0 Pa, averaged across the outlet. Simulation time was set to be 140 s such that at least three annular volumes of displacing fluid were pumped into the annulus. No slip boundary conditions were applied to both the inner and outer walls of the cylinder. The initial conditions were set with a static pressure of 0 Pa, and the annulus full of displaced fluid. The solver parameters were set up in CFX-Pre for the transient flow using a second order backward Euler scheme and the high resolution advection scheme was used (ANSYS CFX-Solver Theory Guide, 2010). Residual targets for each step in time were set to 10-4, which were met in the simulation, and double precision was employed. The CFD model was validated by comparing the simulation results with displacement interface profiles obtained by flow visualisation and displacement efficiency measurements from the work by Deawwanich et al. ( 2008). RESULTS AND DISCUSSION Displacement Interface Profile Examples of the simulated interfacial profiles for axial displacement of one fluid by another in an annulus of 50% eccentricity are shown in Figure 2. Also shown for comparison are experimentally observed profiles reported by Deawwanich et al. (2008) for the same fluid system, geometry and operating conditions. In both cases, the displacing fluid was pumped in at a constant flow rate of 10 L/min, and there was no inner cylinder rotation. It can be seen that the moving interface between the displacing 3 fluid and the fluid displaced is not flat but is inclined toward the wide part of the annulus (left side of the column). This behaviour is partly caused by the fluids preferring to flow in the wider parts of the annular channel, where resistance to flow is lower, while fluids on the narrower part either flow slowly or not at all. Consequently, the displacing fluid can be seen to channel through the wide annular region, by-passing the displaced fluid on the narrowest side. The shape of the interface obtained from simulations is quite similar to the experimental observations, especially with the presence of the channelling front in the wide part of the annulus. Due to the optical resolution of experimental images, it is difficult to distinguish the base and tip of the front; yet what appears to be the base of the phase front agrees well with the shape generated by the simulations. t =10s t = 20s t = 30s Fig. 2: Comparison between experimental and simulated displacement interfaces at 3 different elapsed times from start-up. Annular eccentricity = 0.5; narrow side at right. The effect of annular eccentricity on the interfacial profile is illustrated in Fig. 3. The snapshots were taken from the simulations at 20 seconds after flow started, at the same flow rates of the displacing fluid. First, with the concentric annulus the fluid-fluid interface appears to be flat, indicating a piston-like displacement. As the degree of eccentricity in increased, the interface becomes steeper toward the widest part of the annulus with an increase in the peak velocity. The fluid on the narrow side of the eccentric annuli, however, exhibits little movement and would stop flowing at high eccentricities. 4 (a) (b) (c) (d) Fig. 3: Simulated interfacial profiles at 20 sec as a function of eccentricity. (a) concentric 0%, (b) 25%, (c) 50%, (d) 70% and (e) 90%. (e) Effect of rotating the inner cylinder during axial displacement on the interfacial profile is illustrated in Figure 4 where the simulations are compared with experiments for flow in an annulus of 50% eccentricity. When there is purely axial flow, the displacement interface exhibits the characteristic fingering profile toward the wide side of the annulus. With rotation of the inner cylinder at 10 rpm, the displacing fluid finger is dispersed and moved toward the narrow side of the annulus caused by a tangential velocity superimposed on the axial velocity forming helical flow through the annulus. As the displacement interface followed a helical flow path, it forces the displacing fluid to move to the narrow annular region and promotes flow of the displaced fluid. The results show that the effect of inner cylinder rotation became more significant at higher degrees of cylinder eccentricity, where it reduced the imbalance in flow velocities between the wide and narrow sides of the annulus. Cylinder rotation affects the displacement both through the additional tangential flow that drives more fluid to the narrowest part of the annulus, and by convective mixing of fluids at the interface. (a) (b) (c) (d) Fig. 4: Effect of cylinder rotation on displacement profile: (a) simulations, no rotation; (b) simulations, 10 rpm; (c) experimental, no rotation; (d) experimental, 10 rpm. 50% eccentric annulus. Flow rate = 10 L/min. Elapsed time = 20 sec. 5 Velocity Profiles Figure 5 presents examples of the simulated axial velocity profiles at the outlet, as a function of annular eccentricity and cylinder rotation. The velocity contours for the purely axial flow show a peak of the axial velocity located in the centre of the wide gap, and a decrease in velocity toward the narrow gap. The velocity is essentially zero in the narrow half of the annular cross section. As eccentricity increases, the axial velocity peak becomes narrower and moves more into the wide gap, leaving a larger region stagnant. When rotation is applies to the inner cylinder, the velocity profile become distorted. In the examples shown, the peak axial velocity is swept into the narrowing gap in the direction of rotation. Also in the presence of rotational flow, the peak axial velocity is not as high as in purely axial flow, and there is a more even distribution of axial velocity resulting in more flow through the entire the annulus. This can also be seen from the displacement fronts for the rotating case (Figure 4b), where the interface is flatter and more uniform. (a) (b) Fig.5: Effect of rotation and eccentricity on axial velocity profile at outlet of annulus. (a) 50% eccentricity, no rotation; (b) 50% eccentricity, rotation at 10 rpm; (c) 70% eccentricity, no rotation; (d) 70% eccentricity, 10 rpm. Flow rate = 10 L/min. (c) (d) Displacement Efficiency Effectiveness of the displacement process is usually determined by the displacement efficiency, which can be defined as the fraction of the total annular volume occupied by the displacing fluid at any given time (Tehrani et al., 1992). Displacement efficiency was computed from the simulations simply as the volume fraction of the displacing fluid present in the flow domain at any instant. The results obtained are expressed as a function of the number of annular volumes of the displacing fluid pumped. The latter can be considered as a dimensionless time of displacement (Tehrani et al., 1992). Examples of displacement efficiencies calculated from simulations are shown in Figure 6 for displacements in 50% eccentric annuli. Also shown for comparison are experimental displacement efficiency data taken from Deawwanich et al. (2008) for the same fluid system and flow geometry under the same flow rate. 6 1 1 Displacement Efficiency Displacement Efficiency Experimental Data 0.2 Experimental Data Simulation Simulation Annular Volumes Pumped (a) Annular Volumes Pumped Fig. 6: Simulated and experimental displacement efficiencies for 50% eccentric annuli. (a) axial flow only; (b) axial flow with cylinder rotation at 10 rpm. Flow rate: 10 L/min. (b) The displacement efficiency results obtained from the simulations appear to match experimental values reasonably well, especially in the early and final stages of the displacement. It is evident that the experimental results tend to deviate from the initial straight line earlier than the simulations. This may be due to a more elongated displacement front observed in experiments, which would result in an earlier breakthrough - when the displacing fluid first appears at the exit, compared to the simulations. The reason can also be applied to the fact that the simulated displacement efficiencies deviate more sharply after breakthrough, while the change in the experimental data is more gradual. The simulation results also confirm the experimental observations that rotation of the inner cylinder improves the displacement in eccentric annuli. As illustrated in Figure 6(b), even a slow rotation at 10 rpm manages to increase the displacement significantly, leading to possible complete removal of the displaced fluid. As shown earlier with the simulated displacement interface profiles and the velocity profiles, rotational flow coupled with axial flow results in a helical flow that forces the displacing fluid into the narrow side of the annulus, thereby reducing the flow imbalance due to inner cylinder offset. It is clear from both sets of results presented in Figure 6 that a discrepancy exits between simulated and experimental displacement efficiency results in the middle region. Experimentally, displacement efficiency was determined by measuring the composition of the mixed fluid phases, using the electrical conductivity method, at some location above the annular exit (Deawwanich et al., 2008). This method assumes that the fluid phases are perfectly mixed whose composition is directly proportional to the mixture conductivity. By contrast, displacement efficiency was determined directly from the computed displacing fluid phase occupying the simulation annular flow domain. However, it is possible that the mesh refinement used was not sufficient to capture the changes in phase compositions around the interfacial region where interphase mixing may take place. These and other effects, e.g. the nature of the displacement interface, will be investigated further in depth in future work. 7 Frictional Pressure Losses The CFD simulations also provided information on pressure distributions in the annulus that can be used to calculate frictional pressure losses as function of flow geometries, fluid rheological properties and flow conditions. In Figure 7 are shown the simulated axial pressure loss gradient profiles at a constant flow rate as a function of time (expressed as annular volumes pumped) for different degrees of eccentricity. The pressure loss gradient profiles have three distinct regions. Initially at the early stage of displacement, pressure losses are low and essentially constant due to the annulus primarily consisting of the displaced fluid. Next, with increasing quantities of the displacing fluid entering the annulus, pressure losses rises steeply with time due to a greater contribution of the more viscous displacing fluid to the overall frictional losses. After more than about one annular volume has been injected, pressure losses become stabilised and approach constant levels. The second stabilised region thus represents the condition under which the annulus is predominantly filled with the displacing fluid that controls the frictional energy loss dp/dz (Pa/m) dp/dz (Pa/m) 200 E=0% 200 E=25% E=50% E=70% E=90% Annular Volumes Pumped E=0% 100 E=25% E=50% E=70% E=90% Annular Volumes Pumped (a) Fig. 7: Simulated average pressure loss gradient profiles for different annular eccentricities. (a) No rotation; (b) Rotation at 10 rpm. Axial flow rate: 10 L/min. (b) Two significant observations can be made from the simulated results shown in Figure 7. Firstly, for the same axial flow rate, the frictional pressure losses decrease with an increase in annular eccentricity. This may be due to an increase in a wider section available for the fluid to flow with less resistance as the annulus becomes more eccentric. This phenomenon has been well established and documented for single phase flows in eccentric annuli (e.g. Haciislamoglu & Langlinais, 1990; Kelessidis et al., 2011; Sorgun, 2011). Secondly, for the same axial flow rate and eccentricity, the pressure loss decreases markedly when rotational flow is present. This drag reduction effect has been observed for single phase flows and can be attributed to the shearthinning nature of the fluid that exhibits a lower viscosity in response to an increase in the total shear rate produced by the helical flow in the annulus. It should be noted that at high rates of cylinder rotation, inertial effects may become significant compared to the shear-thinning effects in causing the pressure loss to increase with increasing rotational flows (Wan et al., 2000; Escudier et al., 2002). Furthermore, as shown by the simulations in Figure 7, the effects of eccentricity on pressure losses are less significant in helical flows than in purely axial flows. 8 CONCLUSIONS A functional 3-dimensional CFD model has been developed to capture the effects of non-newtonian fluid displacements in eccentric annuli. The model was run using ANSYS CFX 13.0 for laminar flow conditions with negligible mixing between phases. It was found that the model can predict the effects of eccentricity and rotational flows on displacement behaviour and efficiency and frictional pressure losses. The simulations confirm the experimental observations that rotation of the inner cylinder, even at low speeds, can significantly improve the displacement performance in highly eccentric annuli. Furthermore, for viscoplastic fluids, mild rotational flows lead to decreases in friction losses, which are strongly dependent of the annular eccentricity. Further work is underway to refine the model by investigating the combined effects between mesh parameters, interfacial properties and interphase mixing. REFERENCES ANSYS CFX-Solver Modeling Guide 2010, ANSYS Inc., Canonsburg. ANSYS CFX-Solver Th
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