A COMPARISON OF SIMULATION APPROACHES FOR SOLID LIQUID FLOWS

Transcription

1 3 rd Brazilian Conference on Boiling, Condensation and Multiphase flow Curitiba, 7 9 May 2012 A COMPARISON OF SIMULATION APPROACHES FOR SOLID LIQUID FLOWS J. A. A. Oliveira Jr.*, C.E. Fontes*, J. Z. Souza*, A. T. A. Waldmann, A. L. Martins *ESSS, Rodovia SC 401, Km 01, 600, Parq.Tec Alfa, 5º andar - Sl. 401, Bairro João Paulo PETROBRAS/CENPES/PDGP/IRF, Av. Horácio de Macedo, 950 Cid. Universitária - RJ ABSTRACT Fluid losses are still today one of the most challenging problems in well construction. Difficulties in drilling and unsuccessful cementing jobs are frequent events, especially while dealing with fractured reservoirs. Most strategies to control losses are empirical and in some situations, detrimental effects cannot be avoided. This article deals with unique modeling efforts to understand the dynamics of bridging fractured zones. The main tool adopted to address the problem was the Computational Fluid Dynamics (CFD). The study includes two different modeling strategies: Discrete Element Simulation and Granular Eulerian CFD approach. The first method solves the particle trajectory equations individually, considering collision and cohesion effects. Despite of the reliability of the approach, computation effort is huge and limits the number of particles in the system. The Eulerian approach, on the other hand treats statistically the particulate system, generating a probabilistic field of occurring one or the other phase at given space and time. This approach obviously generates minor computation costs. The main goal was to study particle deposition inside fractures due to losses through the external walls of an axial annular flow. ANSYS FLUENT and EDEM were the adopted simulation tools. INTRODUCTION During the process of drilling and well completion, a phenomenon that may occur is the drilling fluid loss to the reservoirs. This phenomena is intensified when fractured zones are present and the effect is even greater when these fractures are connected to a network of natural fissures. It is known that the increase of fluid loss can cause additional difficulties in operation, besides increasing the drilling cost. Today, several lines of research are being developed to understand this phenomenon, and especially, how to control it or eliminate it. One of these lines of research is the use of numerical simulation. In this way, the present work show simulations concerning liquid-solid flow field behavior in the dynamics of bridging fractured zones. This paper presents some efforts made jointly by Petrobras and ESSS to modeling this phenomenon and the process used today for its control. SIMULATION APPROACHES Different approaches in simulation have been performed to investigate the liquid-solid flow field behavior; and new approaches, based on Eulerian models and Granular Kinetic Theory, are being tested and implemented in ANSYS FLUENT software. The initial approach studied was the use of Discrete Element Simulation (DEM) coupled with Computational Fluid Dynamics (CFD) simulations. The main advantage of this method is the high level of detail obtained about the solid material behavior, since each particle is accompanied throughout the flow. The main disadvantage of this method is, for the case of the present study, its high computational effort. Nevertheless, data obtained using CFD-DEM approach can be used to generate results to compare and validate models with another multiphase approach. Another multiphase model used is the Eulerian-granular approach. This has been successfully used in the simulation of gas-solid flows and it has been recently extended for used in liquid-solid flows. The advantage of this method is the reduced computational cost. However, the disadvantage is the dependence of constitutive relationships required to model additional terms of the solid phase. In order to have a better knowledge of these method and theirs particularities, a simple comparison between these two approaches will be presented. CFD-DEM Simulation The first approach used by the group in the development of this project was the application of a coupling between CFD and DEM techniques. CFD is a well-known technique for simulation of fluid flow problems involving various additional physics (heat transfer, chemical reactions, flow in porous media, etc.). The software used, ANSYS FLUENT, applies the finite volumes method [1] for solution of conservation of mass and momentum [2]. In the case of its coupling with DEM, the form of the equations used is that for multiphase flows as two-fluid model [3]. DEM is a technique used exclusively to handle cases involving particulate materials. In this method, each particle is followed using Lagrangian view along its way. The interaction with other particles and solid walls is accounted considering the shape of the particle and physical properties of materials, using, for example, a nonlinear contact model such as Hertz-Mindlin [4]. In addition, DEM model is able to account the effects of field forces (gravity, magnetic force, etc.). A coupling between CFD software (ANSYS FLUENT ) and DEM software (DEM Solutions EDEM ) allowed the use of both tools simultaneously. The CFD software is used to solve the continuous phase (fluid) and DEM software to solve the motion of dispersed phase (particles). The coupling is

2 responsible for the exchange of information between the phases, especially informing CFD software the volume fraction of dispersed phase in each cell and transferring from one side to the other the information from forces (drag, especially) and their reactions. A more detailed description of the equation used in this coupling can be found in the documentation of EDEM [5]. Using this approach, various simulations were carried out in a very extensive study investigating the effect of geometry characteristics of the particle and the boundary condition imposed. Figure 1 shows a case where one can observe the carrying of particles of the main flow (in annular space between the column and external wall) into the fractures. Some of the difficulties encountered in this approach were: The elevated computational effort of simulations using CFD-DEM coupling; the lack of known boundary conditions and data for validation and; the lack of fluid-particles interaction models (drag models) for flows with packing of particles close to its maximum value. The high computational effort becomes evident when it counts the number of particles in the interior of the computational domain. Given the huge extension of geometry and the small size of the particles, it becomes impractical to reproduce the full 3D simulation process. Studies in small areas (such as the slice shown in Fig. 1) or on a laboratory scale are possible. This will assist in future developments of the project. About the uncertainty of boundary conditions, it is due to difficulty of obtain precise data during drilling process. phase considered as an interpenetrating medium in an Eulerian frame. Exchange terms accounts for quantities transfer among the phases. In the present work just the mass and momentum balance equations were solved for each phase. The same mass balance equation over an infinitesimal control volume [7] is valid for both phases (fluid and solid) and is show in Eq. (1), the indexes q and p stand for the phase indexes, with q representing the phase being balanced. The momentum balance equations for each phase are similar but present differences on the internal stresses modeling. For the fluid phase the momentum balance equation is given by Eq. (2). In this equation, f stands for the fluid phase index and p for any additional phase. (1) (2) The internal stresses are represented as usual using the fluid viscosity and the fluid flow shear rate. In this Eulerian granular approach, the solids phase is treated as a pseudo-fluid which transport equations have special terms to deal with additional stresses due to the internal particles interaction. The solids phase momentum balance equation is given by Eq (3), details about the derivation of this equation is available in the literature [6]. Figure 1: Example of simulation performed using CFD-DEM approach in the fracture case. Moreover, despite being a physically consistent approach on dealing with the particle-particle interaction, this modeling lacks on drag correlations for dense packing and has no models for additional fluid-particle interaction forces. It is also necessary to validate the simulation for use in engineering projects. Thus, the group develops an experimental test rig to perform controlled experiments of the deposition of particulate material into the reservoir fractures. This step of project is discussed in a later section. CFD Simulation with Granular Kinetic Theory As the cost of a complete CFD + DEM coupled simulation is prohibitive to deal with the cases of interest, the development of the simulation methodology was directed toward the use of CFD simulations using multiphase Eulerian approach (also known as multi-fluid approach). This method uses balance equations for mass and momentum (and for other additional quantities, such as energy, when necessary) for each This equation is very similar to the momentum balance equation for a fluid phase. The differences are the presence of a solids pressure gradient term ( p s ) and a different definition of internal stresses including solid shear and bulk viscosities ( ), the definition of these stresses are given in Eq. (4). The additional forces term is used to account for fluid-particle interactions other than drag. These so-called non-drag forces, such as virtual mass, lift and turbulent dispersion, are sometimes more important than the drag itself. The presence of this term is another advantage of the usage of a CFD Eulerian modeling. (3)

3 (4) The solids pressure term accounts for the resistance of the solid to packing, tending to infinite as the solid phase approaches the maximum packing limit. The solid viscosities represent the losses due to friction (of shear and volumetric compression/expansion). The local value of the solids pressure and of each solid viscosities is dependent of the local volume fraction and of a local granular temperature. The definition of this granular temperature is the basis of the Granular Kinetic Theory [6]. This theory uses a similarity between the suspended granular movement with the molecular thermal movement. This granular temperature represents the fluctuating velocity of the particles (around the average value solved by these equations). A number of different correlations are available to calculate the solids pressure, solid viscosities, radial distribution function and other properties needed for each term. The choice of which correlations should be used in a given case is still dependent of the phenomena contemplated and the ranges of physical properties of both phases and operating conditions. Fluid-Solid flow regimes One important point is the correct modeling of the solid phase flow patterns. In a simplistic point of view we can define three different flow patterns in fluid-solid flow [8]. Figure 2 shows these three regimes. media with elastic structural properties, the interaction regime of particles at this fixed bed is called elastic regime. All the initial development on the Granular Kinetic Theory was focused on the modeling of suspended flows, such as the gas-solid flows inside fluidized beds. As a consequence, the correlations for the different solid stress terms were developed and validated (according to the case) to deal only with this particular regime. As the usage of this approach increased, a strong interest on modeling the other regimes appeared and at the moment different groups are investing on the development of such models. Using ANSYS FLUENT, models are available to deal with the suspended and frictional regimes. The correct capture of the different regime is important when dealing with dense fluid-particulate flow with deposition, as is the case of the formation fissures blocking by injection of particulate material. The correct representation of these phenomena will be vital on the evaluation of fissure blockage due to particle-fluid-walls interaction. MODELING APPROACHES COMPARISON TEST CASE A simple test case was developed to compare the results of the CFD + DEM coupling with two simulations using the Eulerian approach, one with models just for the collisional regime and other with models for the collisional and frictional regime. This simple case could be executed using a twodimensional approach when solved with the CFD Eulerian method. The solution with the CFD + DEM coupling was performed using a small slice of the channel in the third dimension (just a few particle diameters deep). Test case setup Figure 2: Fluid-solid (particulate) flow patterns: a) Suspended particles on a flowing fluid; b) Suspended particles and a moving bed of particles at the bottom of the pipe and; c) Suspended particles, a moving particle bed and a fixed bed of particles at the bottom of the pipe. In the suspended regime, the particles can interact (according to the concentration) only via collisions. The time of these collisions, when compared with the travel time between collisions, is quite small. This fact simplifies the development of models to deal with solid stresses. This regime is also called the collisional regime. As the average flow velocity decreases, a moving bed of particles is formed at the bottom of the pipe. These particles are in constant interaction by friction. The hypothesis of instantaneous collisions is no longer valid. Just a few models are available to deal with this regime. The regime of particles interaction inside the moving bed is called the frictional regime. With a further decrease in the average flow velocity more particles accumulate at the bottom of the pipe and a lower layer of fixed particles is formed. So, as defined in many references in the area (Peker, et al., 2007), a three-layer flow occurs. The particles in the fixed bed behave like a porous Figure 3 shows the geometry used in this modeling comparison test case. This geometry uses dimensions consistent with the annular space between the drilling column and the wellbore in a horizontal well. The case was planned using data from gravel packing operations were it is expected to the particles to settle forming the three layers (the suspended upper layer, the moving bed and the lower fixed bed of particles). Figure 3: Geometry of the test case used for modeling approaches comparison. Figure 4 shows the general description of boundary conditions used in this test case. The detail of the particles release plane is necessary when setting up the CFD + DEM simulation. For the CFD granular setup the particles are injected with the fluid stream at the velocity inlet.

4 Figure 4: General description of boundary conditions for the test case. The operating conditions and material properties applied are presented in Tab. 1. These properties and operating conditions are also consistent with data from a gravel packing operation. Again, these values were used to force the presence of the three-layer flow inside the channel. Table 1: Physical properties and operating condition used for the test case. Data Value Units Particle diameter m Particle density ³ kg/m³ Particle injection velocity m/s Fluid density kg/m³ Fluid viscosity Pa s Fluid injection velocity m/s Simulation results All the simulations were performed for a time around 20s (physical time). The clock time spent on each simulation was drastically different. Both simulations using the CFD granular approach took one day using a workstation. The simulation using CFD + DEM coupling took three weeks on the same equipment. This huge difference is due to the constantly increasing time spent by the DEM solver to run, increasing as the number of particles increased with time, and also due to the small time step that had to be used on the coupled simulation. Time steps used in DEM simulations are always very small, to precisely capture particle-particle interaction, when coupled with CFD, to keep consistency between the simulations, the CFD solver is also forced to use smaller time steps than a standard CFD solution for the same phenomena. Figure 5 shows the comparison of the solid phase volume fraction field for all the solutions. Some important conclusions can be derived from its analysis. The first one comes from a comparison between the two CFD granular solutions. It is very clear that the lack of models to deal with the frictional (and also with the elastic) regimes of particle interaction has a strong influence on the result quality. Despite the fact that both simulations presented volume fractions limited to the values defined at the simulation setup, showing that the solids pressure models were consistently used, the simulation performed without the frictional regime modeling showed an abnormal flow of the particulate material, resembling a more fluid behavior than actually expected. The simulation using the models available to deal with the frictional regime has shown much better agreement with the expected behavior. When comparing the simulation using the Eulerian granular with frictional regime modeling with the CFD + DEM coupling simulation we observe that both present a clear limiting particle bed height with a reasonable agreement between their height values. The main difference between these results is the angle of the particle bed at the end of the deposition zone. Here it is interesting also to compare the solid phase velocity profiles for a better understanding of this characteristic. Figure 6 compares the axial velocity profiles for the solid phase among all the results. It is clear that the result using the CFD approach without frictional regime model is far from agreement with the other results. It is also clear that none of the simulations were able to capture the fixed bed formation, including the CFD + DEM coupling. It could be expected that the CFD + DEM simulation would be able to capture this regime since this approach is supposed to be the most precise for particleparticle interaction. One can attribute the failure on capturing this fixed bed to the fact, already mentioned, that the models available at the coupling for drag forces are very simplistic, lacking special treatment for the dense packing case. Figure 7 compares the solid phase volume fraction profiles among all results. It can be seen that the three approaches are similar in the intermediate packing region, with the curves showing the same derivatives. One can conclude that the CFD granular approach is, then, successful in reproducing these regions. The simulation using the CFD granular approach with modeling for the frictional regime agreed very well with the CFD + DEM simulation, and it was expected not to capture the fixed bed since no model was applied to deal with elastic particle interaction. If well-adjusted and validated, the CFD approach using correct granular models can be used to model this kind of phenomena with satisfactory results. Figure 5: Comparison of solid phase volume fraction fields for the solutions of the test case using the different approaches. Figure 6: Solid phase axial velocity profiles measured at 0.5m from the inlet.

5 to implement drag correlations that gradually switch to a pressure drop correlation for a porous media. EXPERIMENTAL DESIGN Figure 7: Solid phase volume fraction profiles measured at 0.5m from the inlet. DEVELOPMENT OF NEW MODELS IN THE GRANULAR APPROACH As it was exposed, the granular approach fails to capture the fixed bed because no model is applied to consider elastic particle interaction. To deal with this phenomenon, some elements of kinetic granular theory must be changed. A comprehensive research is being done and new models are being implemented so that the correct approach for elastic regime can be considered. The definition of criteria to point out the change from frictional to elastic regimes is primordial. One possible approach is to use the soil mechanics yield limit criteria. In this model, the ratio between shear and normal stresses is an important parameter. If this ratio is smaller than the tangent of the internal friction angle, then the solid material supports the stresses in the elastic regime [10], [11]. As a first approximation one can use this as a criteria for regime change. As the ratio becomes greater than the tangent of the internal friction angle fluid-solid flow is in frictional regime. An important parameter to be considered is the compression direction of the solid, which is not necessarily normal to the wall or aligned with gravity. This direction changes along the flow, and the direction of the solid pressure gradient also is the compression direction of the solid. Figure 8 shows the particle volume fraction and the particle granular pressure in a fluid-solid flow. Due to the topics already discussed about the lack of welldefined boundary conditions in the fractures case and the need of experimental data to validate the simulations, the group invested in the design and construction of an experimental test rig, where tests for solid carrying to the interior of fractures could be performed under very controlled conditions. This test rig should represent the flow in the annular area between the drilling column and the external wall and must contain the fractures with a known format. This is necessary to make possible to reproduce the same conditions on the CFD simulations. The final design of the developed test rig is shown in Fig. 9. The rig consists of two rectangular channels (representing two spacing between a drilling column and the formation) and three different fractures with different thicknesses (the geometry of the fracture was kept with the same in all of three fractures). These fractures are arranged inclined to the main channel, so that tests can be performed with direction of the fractures aligned with the main flow or misaligned with it. Figure 10 shows the tortuous geometry of the fracture, prepared this way to better represent a natural fracture and promote the expected particulate material deposition inside the fracture. The format of the fracture was obtained from reservoir geological data in the literature [9]. Figure 9: Design of experimental bench tests to be used in the project. Figure 8: Particle volume fraction and particle granular pressure. Another important parameter to be changed is the drag law. In a fixed bed, since the solid phase is stationary, its behaviour is more similar to that of a porous media. Thus, it is necessary Figure 10: Fracture model used in experimental test bench.

6 CONCLUSIONS The simulation of fluid-solid flows is still an area of academic research but, as the results of the modeling comparison case have shown, with some experimental validation and some additional model implementation, it can be successfully applied to deal with cases of interest of the oil and gas industry. The simulation approach using the CFD + DEM coupling still lacks the implementation of better fluid-particle interaction models to be able to fully represent the phenomena. With this models implemented this approach can be used for extraction of properties needed for the CFD granular approach. The complete simulation of the process using only the CFD + DEM coupling is not possible at the moment due to the computational cost of such approach. The simulation with an Eulerian granular approach is the most promising one. Its computational cost is very reasonable and it is much simpler to implement new models on this methodology than on the CFD + DEM coupling. It is expected that an excellent agreement with experimental data can be reached and the methodology will be able to deal with an extense range of applications. Research is being done and new models are being implemented so that the Eulerian granular theory already present in ANSYS FLUENT is extended to represent the elastic regime. For all cases, experimental data will be needed to calibrate and validate models implemented. For this purpose the development and construction of the fracture flow test rig is very important. It will provide the engineers with detailed experimental results under controlled conditions, adequate to reproduction using the simulation methods. REFERENCES [1] C.R. Maliska, Transferência de Calor e Mecânica dos Fluidos Computacional, LTC Editora, 2ª Edição, [2] H. Schlichting, K. Gersten, Boundary-Layer Theory, Springer, 8 th Edition, [3] C.E. Brennen, Fundamentals of Multiphase Flow, Cambrige University Press, [4] A. di Renzo, F.P. di Maio, Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes, Chemical Engineering Science, 59, pp , [5] DEM-Solutions, EDEM-CFD Coupling Module for FLUENT User Guide, Edinburgh, [6] D. Gidaspow, Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions, Academic Press, [7] C. Crowe, M. Sommerfeld, Y. Tsuji, Multiphase Flows with Droplets and Particles, CRC Press LLC, [8] S. M. Peker, Ş. Ş. Helvaci, Solid-Liquid Two Phase Flow. Elsevier Science Ltd, [9] R. Nelson, Geologic Analysis of Naturally Fractured Reservoirs, 2 nd Edition, Gulf Professional Publishing, [10] T. W. Lambe and R. V. Whitman, Soil Mechanics, Wiley & Sons, [11] J. Atkinson, An introduction to the mechanics of soils and foundations, McGraw-Hill, 1993.