Flow Regimes of Particle Settling in Suspensions and Non-Newtonian Fluids
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ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 22, 2014 Flow Regimes of Particle Settling in Suspensions and Non-Newtonian Fluids with Relevance to Cuttings Transport during Drilling Rune W. Time and A.H. Rabenjafimanantsoa University of Stavanger, Norway ABSTRACT some of these effects. As might be expected, Settling velocities of cuttings particles in the settling flow regimes and are quite drilling fluid (“mud”) is one of the most different for Newtonian versus non- important parameters for the efficiency of Newtonian liquids. Finally, we describe drilling long deviated wells. Numerical image analysis assisted determination of fall simulators have been developed worldwide speeds. to assist the optimization of cuttings transport. Models for particle settling velocities to enter these simulators are based INTRODUCTION on experimental conditions which not Cuttings transportation under real always reflect the situation they are used for drilling operations naturally involves a in industrial processes. whole spectrum of particle sizes. In this Effects which modify and disturb project both mono-dispersed and multiple settling velocities are drillstring movement, size slurries were tested. If the liquid flows liquid convection and turbulence, cuttings turbulent in itself it is a challenge to particles size and shape, particle-particle calculate frictional drag on particles, since interaction, and - most important - the the drag coefficient is based on the particle’s drilling mud rheology and composition. This Reynolds numBer. For multiphase flow this also includes oil-water dispersions and also is even more complex, since droplets’ free gas in the mud. interfacial deformation and the fluid flow is The experimental work of this paper has strongly interconnected. been developed to a large extent in Drilling companies report substantially connection with a Master thesis1 at the improved hole cleaning efficiency with oil- University of Stavanger. The motivation for water mud, and researchers2 also on invert the study was to determine the impact of oil-water mud using nano-particles for flow regimes on settling modes and falling emulsion stabilization. Furthermore, it has velocities. been suggested e.g. for underBalanced In the present paper the main focus is on drilling to utilize gas bubbles to modify the theoretical aspects and analysis methods flow pattern and create gas bubble assisted involved. More results from the laboratory lift of cuttings particles. Some preliminary tests will be described in the presentation for results of impact on cuttings settling from the NRS 2014 Conference. such multiphase flow situations will be Laboratory experiments using high addressed in this work. speed video recordings were used to reveal 115 THEORY Multiple particle (cloud) flow regimes Modelling of particle settling has been are often based on the particle volumetric carried out analytically since the 18th fraction as either dilute (< 0.01) or century for single particles in Newtonian dense ( > 0.01). Dense is further divided fluids, and later both experimentally and (Crowe9) into collision dominated (< 0.1) analytical-numerical for clouds of particles or contact dominated ( > 0.1). and for non-Newtonian liquids. For Newtonian fluids a particle starting Single particle settling in Newtonian fluids with an initial velocity v different from the The earliest models for particles falling surrounding fluid velocity u, will Be in fluids are dating back to Stokes3,4 (1851) 5 “relaxed” within a characteristic response for a pendulum and Basset (1889) for a time9 falling sphere, arriving at the well-known 2 dD equation for the drag force D under laminar V (3) creeping conditions, often referred to as 18C Stokes law; where d is the particle density, D the D6RU (1) diameter, and C is the surrounding fluid Frequently it is expressed using the drag viscosity. The governing equation of motion coefficient, CD, for the relaxation disregarding gravity is DCR(U)221 (2) dv 1 D 2 (u v) (4) 24 dt where C , U is the free stream fluid V D Re with the solution ; Ud vu(1et/ V ) (5) velocity, Re is the Reynolds The response time ranges from milliseconds number, with d Being the particle diameter for micron sized particles, to a second or and the kinematic viscosity. RemarkaBly more for particles in the cm range. This simple still, Stokes law has Been important relation is still associated with laminar flow, for 3 Nobel Prizes6, e.g. the famous Millikan so if the flow regime in the fluid itself is oil drop7 experiment to determine the turbulent the response time will normally electron charge. More information can Be decrease, while on the other hand the linked from the Stokes biography8. particle velocity fluctuations increase. In presence of gravity the equation of Flow regimes – a brief classification motion becomes In this work the term “particle settling dv f (u v) (1 )g (6) flow regimes” refers to both single phase dt VP and multiphase fluids. For single phase C24 fluids there is also a marked difference in Where f is friction factor f D . particle settling dynamics between Re Newtonian and non-Newtonian fluids. For For complex fluids and particle systems it could be feasible to extend this to a multiphase systems there will also Be 10 substantial variation, depending on the “Langevin equation” approach ; dv 1 degree of mixing of the immiscible phases (u v) (1 )g F'(t) (7) (oil, water or polymer and gas) plus dt VP particles. Finally, for all the aBove cases where F’(t) describes the fluctuations caused single particles fall in a different way from by macroscopic collisions. This has been 28,29,30,31 clouds or assemBlies of particles . used in a published CFD model11. 116 Experiments were done to test the “… flow separation can be said to Begin at a accuracy of the relaxation equation for large Reynolds number of about 24. The point of particles, as described in the sections separation is at first close to the rear of the describing experiments and analysis. During sphere, and separation results in the formation fall the flow around larger particles undergo of a ring eddy attached to the rear surface of a transition from laminar to turbulent flow. the sphere. Flow within the eddy is at first While calculation of drag coefficient is quite regular and predictable, thus not fairly straightforward, it is more intricate to turbulent, but, as Re increases, the point of calculate free fall velocities since the separation moves to the side of the sphere, and unknown velocity is implicit both in the the ring eddy is drawn out in the downstream Reynolds numBer and in the drag direction and begins to oscillate and become 12 unstable. At Re values of several hundred, the coefficient. This can be overcome by using ring eddy is cyclically shed from behind the the Archimedes number sphere to drift downstream and decay as 3 2 4 g(P )d another (ring) forms. Also in this range of Re, Ar CD Re 2 (8) 3 turbulence begins to develop in the wake of the sphere. At first turbulence develops mainly Here , and are particle density, and P in the thin zone of strong shearing produced liquid density and viscosity respectively. by flow separation and then spreads out A range of different drag coefficients downstream, but as Re reaches values of a few have been developed. One such correlation thousand the entire wake is filled with a mass to cover a wide range of Reynolds numbers of turbulent eddies ..” is given by Morrison13, and data fitted to seven orders of magnitude. It is shown This “ring-wave” process has been inserted in Fig. 1 below and was used for visualised in another paper in these equation 6, in the Matlab program which proceedings15 using an immisciBle phase produced Fig. 8. (oil or air) clinging to the particles. Particle settling in Non-Newtonian liquids Particle settling in polymeric liquids is more complex than in the Newtonian case. Chhabra12 gives a good overview of studies of sphere motion in shear-thinning liquids. An alternative to Stokes drag law is given as F3dUYD (9) where the correction factor Y is given as CRe Y DPL , 24 (10) Vd2n n and RePL Figure 1. Data Correlation for Drag Coefficient m for Sphere (all Reynolds numBers)13. Y is then modelled in various ways, e.g like 12,16 Kawase 2 A fairly good and nicely illustrated ((3n 3)/ 2) 7n 4n 26 description of such a transition regime is Y3 (11) 5n(n 2) given in an MIT OpenCourse book14 as This analytical model is valid for Re < 20, follows; PL while Ceylan17 presented a correlation valid for Re<100. Matijaši and Glasnovi18 117 modelled drag coefficient up to 1000. Kelessidis and Mpandelis19 presented an explicit model for terminal velocity of pseudoplastic liquids beyond Re > 1000. Slurry transport in single phase fluids As described in the Theory section, the Behaviour of multiple particle flows can Be described in terms of the volumetric fraction of particles as either dilute or dense. However, this classification does not fully incorporate the dynamics of particles and fluid. A more complete description is Figure 2. Multiphase flow of air-oil-water and given in terms of the ratio of the particle sand particles forming a bed in an 8 cm i.d. diameter pipe. The flow has just stopped and the response time, V (Eq. 3) to the mean time interface layers become stratified. Maximum grain size is 1mm. between particle collisions. So, if V 1 C C The theory of colliding particles in non- the system is said to be dilute. And else it is Newtonian liquids is more complex, and dense. Contact flow is typical e.g. for simple correlations are not available. strongly sheared sand bed structures or Numerical simulations on simple systems fluidised beds.