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Abstract

The present investigation deals with the flow through concentric annulus with the inner cylinder in rotation. This work has got its importance in the petroleum industries in relation to the wellbore drilling. In wellbore drilling, the issue of the hole-cleaning is very serious problem especially in case of the horizontal drilling process. The effect of the various parameters like slurry flow velocity, inner cylinder rotational speed, inlet solid concentration which affect hole cleaning was discussed. Their effect on the pressure drop, wall shear stress, mixture turbulence kinetic energy, and solid-phase velocity and slip velocity were analyzed, which are responsible for solid-phase distribution. Flow was considered to be steady, incompressible and two-phase slurry flow with water as carrier fluid and silica sand as the secondary phase. Eulerian approach was used for modeling the slurry flow. Silica sand was considered of spherical shape with particle size of 180 µm. ANSYS FLUENT software was used for modeling and solution. Plotting was done using Tecplot software and Microsoft Office.

Keywords

Water–sand slurry concentric annulus two-phase flow Eulerian approach Oil well drilling

Introduction

Flow through annulus is a widely encountered problem. Chemical process and petroleum industries, pipeline engineering, power plants, biomedical engineering applications, micro-scale fluid dynamics studies, food processing industries, geothermal flows, extrusion of molten plastics, etc. encounter many applicable situations of flow through such geometrical situation of concentric annulus. A large variety of fluids and industrial applications have been a major motivation for research in annular flow with varying degree of complexity. An extensive bibliographic list of work on annular flows has been presented by Escudier et al.^1,2

In the present work, emphasis is put on the application of the flow of the slurry formed by the combination of the drilling fluids, and the cuttings generated after drilling fluid have been passed through the drilling cutter. The hydraulics of this is very important part of study, as it has lot of bearing on the avoidance of the drilling defects Darley & Gray³, 1998 – which are of multiple varieties and needs to be avoided/reduced. Many drilling defects depend upon the quality of cleaning of the borehole being drilled.

The type of cuttings generated depends on the nature of the soil where the borehole is drilled. Usually, these are sandy in nature carrying the feature of various minerals. In present investigation, silica sand (the most common one) is considered as the solid-phase. After drilling fluid (presently water) passes through the drilling cutter zone, it carries the cuttings generated (sand is taken here for study). Slurry of the water and sand is formulated, which passes through the annular wellbore.

The above thought process was further strengthened and developed by the previous researches carried out in this area. These are discussed briefly below.

Oil & gas well drilling mud flow is usually operated in the turbulent regime. In the work of Nouri and Whitelaw⁴, three velocity components (axial, radial, and tangential) of a Newtonian and a weakly elastic shear-thinning non-Newtonian fluid have been measured in an annulus with an eccentricity of 0.5, a diameter ratio of 0.5, and an inner cylinder rotation of 300 r/min. The results show that the rotation had similar effects on the Newtonian and non-Newtonian fluids, with a more uniform axial flow across the annulus and the maximum tangential velocities in the narrowest gap in both cases. The turbulence intensities in the region of widest gap were uninfluenced by rotation, increased in the Newtonian fluid, and decreased in the non-Newtonian fluid in the region of the smallest gap.

Escudier et al.^1,2 reported experimental data for fully developed laminar flow of a shear-thinning liquid for both the concentric and an 80% eccentric annulus with and without centre body rotation. The working fluid was an aqueous solution of 0.1% xanthan gum and 0.1% carboxymethyl cellulose (CMC).

Kim and Hwang⁵ did an experimental investigation concerning the characteristics of vortex flow in a concentric annulus with a diameter ratio of 0.52, whose outer cylinder is stationary and inner one is rotating. Pressure losses and skin friction coefficients have been measured for fully developed flow of water and of 0.4% aqueous solution of sodium CMC, for the inner cylinder rotation speed of 0–600 r/min. Also, the visualization of vortex flows has been performed to observe the unstable waves.

Cruz and Pinho⁶ have obtained analytical solution of helical flow of fluids in concentric annuli due to inner cylinder rotation as well for Poiseuille flow in a channel skewed by the movement of one plate in span-wise direction, which constitutes a simpler solution for helical flow in the limit of very thin annuli. Expressions are derived for the radial variation of the axial and tangential velocities, as well as for the three shear stress and the two normal stresses using non-dimensional number as Reynolds number and Taylor number, etc.

Kelessidis and Bandelis⁷ have presented a critical review of the state-of-art modeling for efficient cutting transport during coiled-tube drilling and presented the critical parameters like pump rate, well dimension, fluid sizes, solid loading, and hole inclination, etc. affecting efficient cutting transport. They set up a laboratory system also.

Han et al.^8,9 investigated the hydraulic transport characteristics of a solid–liquid mixture flowing vertically upward where solid particles are carried by non-Newtonian fluids in a slim hole concentric annulus with rotating inner cylinder. Solid volumetric concentration and pressure drops were measured for the various parameters such as inclination of annulus, flow rate, and rotational speed of inner cylinder. Aqueous solution of sodium CMC (0.2–0.4%) and 5% bentonite solutions was taken for non Newtonian fluid one by one.

Frigaard and Ngwa¹⁰ have worked in predicting the rheological properties that are necessary to prevent the annular plug fluid from flowing under the action of buoyancy or indeed to predict how far the plug material may flow for given rheological properties for annular fluid flow in oil wellbore construction. Mathematically, these flows were modeled using a Hele-Shaw approximation of the narrow annulus. Zhiyuan and Baojiang¹¹ established the basic hydrodynamic models, including mass, momentum, and energy conservation equations for annular flow with gas hydrate phase transition during gas kick for deep water drilling. They investigated the behavior of annular multiphase flow with hydrate phase transition by analyzing the hydrate-forming region and the gas fraction in the fluid flowing in the annulus.

Gavrilov et al.¹² proposed a numerical algorithm for simulating steady and laminar flows of an incompressible fluid in annular channels with eccentricity and rotation of the inner cylinder.

Podryabinkin and Rudyak¹³ present results of numerical modeling for analysis of the moment and forces exerted on an eccentrically positioned rotating inner cylinder due to the annular flow between two cylinders with parallel axes. Laminar stationary fully developed flows of Newtonian and power law fluid flows are considered. An impact of annulus geometry, flow regime, and fluid characteristics is studied. The study indicates that the moment exerted on the inner cylinder increases monotonically with the eccentricity. Forces acting on the inner cylinder include pressure and viscous friction. The pressure forces provide a predominant contribution. When eccentricity does not exceed a certain critical value, the radial force pushes the inner cylinder to the channel wall. When eccentricity is large enough, the radial force reverses its sign, and the inner cylinder is pushed away from the outer wall. Circumferential component of the force has always the same direction and induces precession of the inner cylinder.

In these investigations, depending upon the type and nature of the flow, different solution methodology has been adopted. Usually, turbulent non-Newtonian flows are often encountered in the oil and gas industry. These fluids are used in the drilling of oil wells to transport the cuttings to the surface and to keep solids in suspension during stationary periods. In directional drilling, an eccentric annulus is often used; there is a tendency for the cuttings to accumulate in the narrowest gap where the velocity is lowest. The cutting generated during drilling process of the boreholes makes the non-Newtonian fluid as slurry comprising the supplied drilling fluids with the cutting particles generated. Although the cuttings transport is a directional process, still along with the axial annular flow involves lateral rotational effects because of the rotation of inner cylindrical portion of the annulus. Such combination of axial annulus flow along with the lateral rotation brings the flow in that zone to be in transition or in turbulence state. This tends to suppress such accumulation of cuttings. Thus, it is very much important to maintain turbulent flow situation.

Issue of the hole cleaning

If cleaning of the return passage, i.e. annular hole is not done properly by the completion fluids, then it may lead to the problems like pipe sticking, premature wear of drill bit, slow drilling, formation damages (like fracture, reduced permeability, etc.), excessive torque and drag, and trouble in logging and cementing, etc.

The various factors which are responsible for the efficiency of the hole cleaning are annular velocity of flow, hole inclination angle, drill string rotation, rate of penetration (ROP), drilling fluid property, and characteristics of cuttings.

These problems are crucially related to the fluid mechanics of transportation of the formation/completion fluids. For example, lack of turbulence level in the flow through annulus may cause improper carrying up of the cutting generated through the narrow zones of the eccentric cross-sections of annulus. This will lead to the hole cleaning problems. Cuttings transport during drilling has a major impact on the economics of the drilling process. Inefficient hole cleaning from the cuttings can lead to the numerous problems such as stuck pipe, reduced weight of the drill bit leading to the reduced ROP, transient hole blockage leading to lost circulation condition, extra pipe wear and extra cost because of the additives in the drilling fluid, and wasted time also.⁷

Similarly, presence of particles affects erosion life of the components involved as well as hydraulic performance. For example, many oil and gas producers are forced to address issue resulting from the production of formation sand along with oil and gas. The produced sand causes several problems such as potential damage to reservoir, sand accumulation in lines or equipment, sand separation issues, and erosion.¹⁴

Problem definition, boundary conditions, and numerical approach

Although multiple investigations have been conducted, the exploration into the hole cleaning parameters has been left somehow by the most of the researchers. Of course, Kelessidis and Bandelis⁷ and Han et al.,^8,9 etc. have explored this aspect experimentally. These works are very inspiring as well. A number of such aspects, which can affect hole cleaning, were depicted directly and indirectly in these investigations. But detailed comprehensive studies of the flow hydraulics of the slurry formed considering the concentric annulus with inner cylinder in rotation using CFD have been less taken care. Thus, the present work focuses on the computational study of the flow hydraulic aspects of the water–sand slurry through the annulus (annular wellbore) and studies the various flow parameters like pressure drop, solid-phase distribution, slip velocity, solid-phase velocity and mixture turbulent kinetic energy, which are linked closely with the cleaning of the borehole.

Model dimensions of the annular wellbore are considered, and the list of the input parameters considered is given in Table 1.

Table 1.

Model dimensions of the annular wellbore and list of the input parameters of carrier fluid characteristics, material and type of cuttings, and kinematic parameter.

Model geometrical parameter	Magnitude (mm)	Carrier fluid characteristics: Water was considered as carrier fluid. (a) Viscosity (µ) = 0.001003 kg/m-s. (b) Density (ρ) = 998.1 kg/m³.
Outer diameter of the annulus; D_o	80
Inner diameter of the annulus; D_i	60	Material and type of cuttings: (a) Sand for all types of particles with density = 2650 kg/m³. (b) Particle size = 180 µm (0.18 mm) spherical. (c) Cutting concentrations = 5%, 10% and 15%.
Length of the annulus; L_e	1500 (>50 D_H)
Hydraulic mean diameter of annulus: $D_{H} = D_{o} - D_{i}$	20	Kinematic parameters: (a) Annular flow velocity of the carrier phase and secondary phase = 3.0 m/s, 3.5 m/s, 4.0 m/s, 4.5 m/s and 5.0 m/s. (b) Rotational speed of inner cylinder of annulus = 50 r/min, 100 r/min, 150 r/min, 200 r/min and 250 r/min.
Outer cylinder is fixed. Centre of the inner cylinder is taken as reference center for measurement.

Flow is considered to be three-dimensional, incompressible, steady, and turbulent under the considered geometrical and kinematic values taken. Flow direction is along the positive z-axis, through the annulus. This slurry flow has been modeled using the Eulerian model approach using the ANSUS-FLUENT 12.0 software. As compared to the other modeling approaches, this approach is proven to be best suiting for dealing with the CFD prediction of slurry flow.^14–16 The numerical formulation is based on the finite volume method as implemented in ANSYS-FLUENT 12.¹⁷ SIMPLEC algorithm has been applied for dealing with pressure velocity coupling. Present study employs RNG $k - ɛ$ turbulence model, which is more preferred for the swirl kind of possibility that exists in the flow. Enhanced wall treatment option was used for proper modeling of the flow near the wall. QUICK scheme is considered for the discretization of momentum, k and ɛ terms. Pictorial view of the geometry and boundary conditions is being given in Figure 1. It is tabulated in Table 2 also.

Figure 1.

Pictorial view of the geometry and boundary condition for the annulus.

Table 2.

Boundary conditions.

Location	Boundary condition and mathematical description
Inlet (z = 0)	For carrier phase (primary phase) and solid phase (secondary phase) ${\vec{u}}_{S} = {\vec{u}}_{L} = Constant$ secondary phase concentration based on the volume fraction $C_{S} = C_{average} = Constant$ This is represented/accessed using the velocity inlet boundary condition in the ANSYS-Fluent software.
Outlet (z = finite)	For carrier phase (primary phase) Mass flow rate at inlet = Mass flow rate at outlet i.e. ${\overset{\cdot}{m}}_{inlet, L} = {\overset{\cdot}{m}}_{outlet, L} = Constant$ For solid-phase (secondary phase) Mass flow rate at inlet = Mass flow rate at outlet i.e. ${\overset{\cdot}{m}}_{inlet, S} = {\overset{\cdot}{m}}_{outlet, S} = constant$ This is represented/accessed using the outflow boundary condition in the ANSYS-Fluent Software. It can further be written as below $\frac{\partial p}{\partial x} = 0; \frac{\partial u_{S, L}}{\partial x} = 0; \frac{\partial v_{s, L}}{\partial x} = 0; \frac{\partial w_{s, L}}{\partial x} = 0$ This means zero pressure and velocity gradient boundary conditions are specified at the outlet by using the outflow boundary condition.
Inner wall of annulus	No slip boundary condition For the primary phase $u_{θ, L} = u_{θ, innerwall} = constant; u_{r, L} = 0; u_{z, L} = u_{z, innerwall} = 0$ For the solid-phase $u_{θ, S} = u_{θ, innerwall} = constant; u_{r, S} = 0; u_{z, L} = u_{z, innerwall} = 0$
Outer wall of annulus	No slip boundary condition For the primary phase $u_{θ, L} = u_{θ, outerwall} = 0; u_{r, L} = 0; u_{z, L} = u_{z, outerwall} = 0$ For the solid-phase $u_{θ, S} = u_{θ, outerwall} = 0; u_{r, S} = 0; u_{z, L} = u_{z, outerwall} = 0$

In the present work, the orthogonal gird has been used for meshing of the geometry considered for the flow system. This is depicted in Figure 2. For final simulation run, the distribution of the mesh considered was of distribution 50 × 150 × 120 (radial × azimuthal × longitudinal, respectively) comprising the total 774,000 number of cells in the computational domain.

Figure 2.

Representation of the meshing style adopted for the concentric annulus.

The governing equations in ANSYS FLUENT 12.0 are solved using the segregated solver. In segregated solver, equations are solved sequentially unlike the coupled solver where the equations are solved in a coupled manner. In the segregated solution method, each governing equation is linearized implicitly with respect to the dependent variable in that governing equation. This results in a system of linear equations with one equation for each cell in the domain. SIMPLEC algorithm has been used for pressure–velocity coupling. A residual convergence of 10 ⁻ ³ has been obtained for the governing variables viz. continuity, u- and v- momentum equation of solid- and liquid-phase, k and ɛ, and volume fraction. The under-relaxation parameters were kept slightly lower than the default values (Tables 3 and 4).

Table 3.

Initial under-relaxation parameters.

S.N.	Parameter	Value	S.N.	Parameter	Value
1	Pressure	0.3	6	Turbulent kinetic energy	0.8
2	Density	1	7	Specific dissipation rate	0.8
3	Body forces	1	8	Turbulence viscosity	1
4	Momentum	0.7	9	Slip velocity	0.1
5	Volume fraction	0.5

Table 4.

Final under-relaxation parameters.

S.N.	Parameter	Value	S.N.	Parameter	Value
1	Pressure	0.2	6	Turbulent kinetic energy	0.5
2	Density	1	7	Specific dissipation rate	0.5
3	Body forces	1	8	Turbulence viscosity	0.8
4	Momentum	0.5	9	Slip velocity	0.1
5	Volume fraction	0.4

Mathematical and numerical formulation

Governing equations

In the continuum-mechanical or Eulerian approach, the volume-averaged, steady continuity, and momentum equations are presented for each phase. Let ${\vec{u}}_{s}$ , ${\vec{u}}_{L}$ , $C_{s}, C_{L}$ , $ρ_{s}$ and $ρ_{L}$ denote the velocity, volumetric concentration, density of the solid-phase (suffix ‘s’), and carrier phase (suffix ‘L’), respectively. Sum of solid and liquid concentrations is equal to unity. With these definitions, using intrinsic average quantities, the steady continuity equations for solid- and liquid-phases are written as where k is the representative index for solid- and liquid-phases. Similarly, the steady momentum equation for solid- and liquid-phases is $\nabla . (C_{k} ρ_{k} {\vec{v}}_{k} {\vec{v}}_{k}) = - C_{k} \nabla_{p} + \nabla . (C_{k} τ_{k}) + C_{k} ρ_{k} \vec{g} + \vec{R} + C_{k} ρ_{k} ({\vec{F}}_{k} + {\vec{F}}_{lift, k} + {\vec{F}}_{vm, k})$ (2) where p is the pressure, $τ_{k}$ is the stress tensor, $\vec{g}$ is the acceleration due to gravity, and $\vec{R}$ is the interaction force between the phases. The pressure is assumed to be same for all the phases. The pressure at the interface between the phases is assumed to be the average pressure.

In the last term in right-hand side of equation 2, ${\vec{F}}_{k}$ is the body force, ${\vec{F}}_{lift, k}$ is a lift force, and ${\vec{F}}_{vm, k}$ is the virtual mass force per unit mass of phase k. The lift force acting on solid-phase S in a carrier phase L is computed as ${\vec{F}}_{lift, s} = - 0.5 ρ_{L} α_{s} ({\vec{v}}_{L} - {\vec{v}}_{S}) \times (\nabla \times {\vec{v}}_{L})$ (3)

The lift force is added to the right side of the momentum equations of solid and carrier phases with an opposite sign in the latter, i.e. ${\vec{F}}_{lift, L} = - {\vec{F}}_{lift, s}$ .

Virtual mass effect occurs when the solid phase accelerates relative to the carrier phases L. The inertia of the carrier-phase mass encountered by the accelerating particles exerts a virtual mass force on particles. This is given as ${\vec{F}}_{vm} = 0.5 α_{s} ρ_{L} (\frac{d_{L} {\vec{v}}_{L} - d_{S} {\vec{v}}_{s}}{dt})$ (4) where $\frac{d_{k}}{dt}$ denotes the phase material derivative of the form of $\frac{d_{k} (Ø)}{dt} = \frac{d (Ø)}{dt} + ({\vec{v}}_{k .} \nabla) Ø .$ It should be noted that ${\vec{F}}_{vm, L} = - {\vec{F}}_{vm, S}$ .

The interactive force predominantly consists of interactive drag between the solid-phase (S) and the carrier-phase (L). The interaction force ${\vec{R}}_{SL}$ (suffix ‘SL’ indicates solid–liquid interaction) is subjected to the condition that ${\vec{R}}_{SL} =$ ${\vec{R}}_{LS}$ . This interaction force between solid- and liquid-phases can be further simplified as ${\vec{R}}_{SL} = K_{SL}$ ( ${\vec{v}}_{s} - {\vec{v}}_{L}$ ) where $K_{SL}$ is the inter phase momentum exchange coefficient. The fluid–solid exchange coefficient ( $K_{SL}$ ) from the model of Wen and Yu (1966)²¹ available is one of them $K_{SL} = \frac{3}{4} C_{D} \frac{α_{s} α_{L} ρ_{L} [{\vec{v}}_{s} - {\vec{v}}_{L}]}{d_{S}} α_{L}^{- 2.65}$ (5) $C_{p} = \frac{24}{α_{L} {Re}_{S}} [1 + 0.5 (α_{L} {Re}_{S})^{0.687}]$ (6) where ${Re}_{S} = \frac{ρ_{L} d_{S} [{\vec{v}}_{s} - {\vec{v}}_{L}]}{μ_{L}}$ (7)

In FLUENT computations, the viscosities are obtained by making an analogy between the random motion arising from the particle-particle collision and the thermal motion of molecules in gas according to the kinetic theory of gases. The solid stress tensor contains shear viscosity $μ_{S}$ , arising from the particle momentum exchange due to translation and collision. The shear viscosity of the solid is obtained as the assumption of collision kinetic parts as per the following expressions $μ_{S} = μ_{S, col} + μ_{S, kin}$ (8) $μ_{S, col} = \frac{4}{5} C_{S} ρ y_{S} d_{S} g_{o} (1 + e_{ss}) (\frac{T_{S}}{π})^{1 / 2}$ (9)

And $μ_{S, kin} = \frac{10 ρ_{S} d_{S} \sqrt{T_{S} π}}{96 C_{S} (1 + e_{SS}) g_{o}} [1 + \frac{4}{5} g_{o} C_{S} (1 + e_{SS})]^{2}$ (10)

For granular flows, a solid pressure is calculated independently and is used for the pressure gradient term, $\nabla ρ_{S}$ in the solid-phase momentum equation. A Maxwellian velocity distribution is used for the particles; a granular temperature is hence introduced into the model and is appeared in the expression for the solid pressure and viscosities. The solid pressure is composed of kinetic term and second term due to particle collisions, so that $P_{S} = C_{S} ρ_{S} T_{S} + 2 ρ_{S} (1 + e_{SS}) C_{S}^{2} g_{o} T_{S}$ (11) where $e_{SS}$ is the coefficient of restitution for particle collisions, $g_{o}$ is the radial distribution function, and $T_{S}$ is granular temperature. The granular temperature (T_S) is proportional to the kinetic energy of the fluctuating particle motion. The function $g_{o}$ is a distribution function that governs the transition from “compressible” condition with $α < α_{S, \max}$ , where the spacing between the solid particles can continue to decrease, to the “incompressible” condition with $α = α_{S, \max}$ , where no further decrease in the spacing can take place. The radial distribution function is defined as $g_{o} = [1 - (\frac{α_{S}}{α_{S, \max}})^{1 / 2}]^{- 1}$ (12)

Standard wall functions are applied liquid- and solid-phase along the annulus walls in FLUENT implementation as $U = \frac{1}{k} \ln (Ey) and y = \frac{ρ C_{μ}^{1 / 4} k_{p}^{1 / 2} y_{p}}{μ}$ (13) where k_p is the turbulence kinetic energy at point P, and y_p is distance from point P to the wall.

Mathematical modeling of turbulence

Primarily, there are three different turbulence models which are implemented for multiphase $k - ɛ$ turbulence modeling. These turbulence model options are:

Mixture turbulence model;

Dispersed turbulence model;

Turbulence model for each phase.

Most simple and general amongst three options is the mixture turbulence model. It uses mixture velocities and mixture properties, and is assumed to sufficiently capture the important features of turbulent flow. It represents the first extension of the single phase $k - ɛ$ model, and it is applicable when the phases separate, for stratified or nearly stratified multiphase flows, and when the density ratio between the phases is close to 1. In these cases, use of the mixture properties and mixture velocity are sufficient to capture the important feature of the turbulent flow. The present steady primarily concerns itself with the mixture k $- ɛ$ turbulence research work. The governing equations for mixture based RNG k $- ɛ$ turbulence model are given as follows $\frac{\partial}{\partial t} (ρ_{m} k) + \nabla . (ρ_{m} {\vec{v}}_{m} ɛ) = \nabla . (\frac{μ_{l, m}}{σ_{k}} \nabla k) + G_{k, m} - ρ_{m} ɛ$ (16) and $\frac{\partial}{\partial t} (ρ_{m} ɛ) + (ρ_{m} {\vec{v}}_{m} ɛ) = \nabla . (\frac{μ_{l, m}}{σ_{ɛ}} \nabla ɛ) + \frac{ɛ}{k} (C_{1 ɛ,} G_{k, m} - C_{2 ɛ} ρ_{m} ɛ)$ (17)

The mixture density and velocity, $ρ_{m}$ and ${\vec{v}}_{m}$ , are computed respectively, by the following relationship $ρ_{m} = \sum_{i = 1}^{N} α_{i} ρ_{i} and {\vec{v}}_{m} = \frac{\sum_{i = 1}^{N} α_{i} ρ_{i} {\vec{v}}_{i}}{\sum_{i = 1}^{N} α_{i} ρ_{i}}$

This takes the following form for two phase solid–liquid flow case $ρ_{m} = α_{s} ρ_{s} + α_{L} ρ_{L} and {\vec{v}}_{m} = \frac{α_{s} ρ_{s} {\vec{v}}_{S} + α_{L} ρ_{L} {\vec{v}}_{L}}{α_{s} ρ_{s} + α_{L} ρ_{L}}$ (18)

Turbulent viscosity, $μ_{t, m} = ρ_{m} C_{μ} \frac{k^{2}}{ɛ}$ and the production of turbulence kinetic energy, $G_{k, m}$ are computed by the following formula $G_{k, m} = μ_{t, m} (\nabla {\vec{v}}_{m} + (\nabla {\vec{v}}_{m})^{T}) : \nabla {\vec{v}}_{m}$ (19)

The constants in the above equations are same as in the single-phase standard $k - ɛ$ turbulence model, and $σ_{k} = 1$ , $σ_{ɛ} = 1.3$ , $C_{1 ɛ} = 1.44$ and $C_{2 ɛ} = 1.92$ are empirical constants.

The description above for the mixture-based turbulence model is presented based on the standard $k - ɛ$ model. RNG $k - ɛ$ model was derived using a rigorous statistical technique (called renormalization group theory). It is similar in form to the standard $k - ɛ$ model, with the addition of following refinements:

RNG model has an additional term in its $ɛ -$ equation that significantly improves the accuracy of the rapidly strained flows.

The effect of swirl on turbulence is included in the RNG model, enhancing the accuracy for the swirling flows.

The RNG theory provides an analytical formula for turbulent Prandtl number while the standard $k - ɛ$ model uses user specified constant values.

While the standard $k - ɛ$ model is a high Reynolds number model, the RNG theory provides an analytically derived formula for the effective viscosity that accounts for low Reynolds number effects. Effective use of this feature depends, however, on an appropriate treatment of the near-wall region.

Validation and mesh refinement study

First, the predicted results of CFD Software ANSYS FLUENT 12.0 were compared for single-phase concentric annular turbulent flow situations with the analytical results. Then, the results of water – sand slurry prediction in flow through pipe were compared with the previously published experimental work.

Validation with the analytical results

For flow of water through a concentric annulus with small annular gap, three-dimensional CFD results were validated against the turbulent flow analytical results (Figure 3). CFD modeling was done for slim-hole concentric annulus (outer diameter 44 mm and inner diameter 30 mm) for a Newtonian fluid (water is considered here) with density and viscosity of 998.9 kg/m³ and 0.001003 kg/ms. SIMPLE algorithm was taken for resolving pressure –velocity coupling. Standard $k - ɛ$ model with enhanced wall treatment and pressure gradient corrections were considered to deal with turbulent situation of the flow considered.

Figure 3.

Validation for present prediction with analytical results for turbulent flow through concentric annulus.

Validation with the previous experimental work of slurry pipeline

Kaushal et al.¹⁶ conducted experiments on a horizontal pipeline loop of 54.9 mm diameter. The experiment was performed by using the particles of the glass beads in water as carrier fluid. Mean particle diameter was 125 µm for flow velocity up to 5 m/s. The case was modeled using CFD software ANSYS FLUENT 12.0 for 30% inlet concentration. These results show (Figure 4) that the CFD predictions are in good agreement with the experimental results. The maximum concentration values (which is 15%) considered for study in the present work is well below the 30% concentration used in the validation.

Figure 4.

Comparison of the pressure gradient for CFD prediction and experimental work¹³ of slurry flow through pipe for inlet slurry concentration of 30%.

Further, it is worthwhile to validate the experimental results related to the solid concentration variations and solid velocity variation along the diameter of the pipe for slurry flow. This will complete the full validation scenario. For this purpose, two different cases have been undertaken these are as below.

Slurry – A: $u_{s, L} = 3.78 m / s$ , $C_{s} = 9.18 %$ , $D = 51.5 mm$ , $d_{p} = 165 μ m$ ;

Slurry – B: $u_{s, L} = 3.2 m / s$ , $C_{s} = 11.93 %$ , $D = 50.7 mm$ , $d_{p} = 520 μ m$ .

The validation results are shown on Figures 5 and 6. Figure 5 shows variation of solid concentration along the pipe diameter for slurry – A and Slurry – B, whereas Figure 6 shows variation of solid velocity along the pipe diameter for slurry – A and Slurry – B. The above two cases are different from the case of Figure 5 where pressure gradient validations were performed. Purposely, we have taken different cases to bring more generalization in the experimental validation of the present CFD prediction for slurry pipelines. There is a good agreement between the CFD predictions using fluent and experimental data. In Figure 6, the average error is around 3% and maximum error is around 5.6% in comparison with the experimental data. For Figure 6, the average error is around 4% and the maximum error is around 5.7%. The cases were modeled using ANSYS-FLUENT 12.0 by Eulerian multiphase approach. Water was considered as the carrier phase and sand was considered as secondary phase with the above given parameters for slurry A and slurry B.

Figure 5.

Variation of solid concentration along the pipe diameter for slurry – A and Slurry – B.

Figure 6.

Variation of solid velocity along the pipe diameter for slurry – A and slurry – B.

Mesh refinement study

Different mesh sizes (with node distribution) as well as annulus dimension, flow parameters, and slurry characteristics considered for the mesh independent studies are shown in Table 5 below.

Table 5.

Different mesh sizes, annulus dimensions, flow parameters, and slurry characteristics used for the mesh refinement study.

Model geometrical parameter		Value		Slurry characteristics		Value
Wellbore diameter (outer diameter of the annulus), D_o		80 mm (0.08 m)		Carrier fluid		Water
Drill pipe diameter (inner diameter of the annulus), D_i		60 mm (0.06 m)		Particle material type (sand)		2650 kg/m³
Length of the annulus, L_e		2.5 m (>100 D_H)		Particle size		180 µm
Hydraulic mean diameter of annulus, $D_{H} = D_{o} - D_{i}$ =		20 mm		Particle concentration		20% by volume
Concentric case is being considered only.				Annular velocity and drill pipe r/min
Node distribution (radial × azimuthal × longitudinal)	A (15 × 90 × 150)	B (30 × 110 × 100)	C (40 × 120 × 100)	D (50 × 130 × 100)	E (50 × 150 × 120)	F (50 × 150 × 200)
Total no of cells	202,500	312,000	432,000	580,000	774,000	1,290,000

Based on the details of Table 5, the CFD computation was conducted for using the mixture-based Eulerian multiphase approach. Turbulence was modeled using the RNG k-ɛ model, as this is useful where swirl situation arises in the flow (FLUENT theory guide). Pressure gradients were obtained for the different mesh. These are tabulated in Table 6. Apart from this, the various meshes were compared also on the basis on the solid-phase velocity distribution (Figure 7) and solid-phase concentration (Figure 8).

Figure 7.

Variation of the solid velocity along plane P1 for various mesh sizes.

Figure 8.

Variation of the solid concentration along the plane P1 for various mesh sizes.

Table 6.

Mesh refinement test based on the pressure gradient.

Node distribution (radial × azimuthal × longitudinal)	A (15 × 90 × 150)	B (30 × 110 × 100)	C (40 × 120 × 100)	D (50 × 130 × 100)	E (50 × 150 × 120)	F (50 × 150 × 200)
Total number of cells	202,500	312,000	432,000	580,000	774,000	1,290,000
Pressure drop per unit length (kPa/m)	30.483	28.369	27.627	27.011	27.000	26.992

Slurry – A:

u_{s, L} = 3.78 m / s

C_{s} = 9.18 %

D = 51.5 mm

d_{p} = 165 μ m

Slurry – B:

u_{s, L} = 3.2 m / s

C_{s} = 11.93 %

D = 50.7 mm

d_{p} = 520 μ m

From Figures 7 and 8, it is obvious that the mesh types A (202,500), B (312,000) and C (432,000) are highly deviated as compared to the other mesh sizes. Considering the pressure gradient variation (Table 6) and variation of the solid velocity (Figure 7), and solid concentration (Figure.8), mesh size of type E of node distribution 40 × 120 × 100 (radial, azimuthal, and longitudinal nodes, respectively), having the total 774,000 number of cells, was chosen finally for the purpose of the analysis for further study. Mesh type F may lead to very long computational time.

Apart from the finalization of mesh size, some additional turbulence models other than, the RNG k-ɛ model, were also tested. These are as follows.

One equation Spalart–Allmaras model (along with mixture multiphase model),

SST k-ω model (along with mixture multiphase model),

Standard k-ɛ model (along with Eulerian multiphase model),

Realizable k-ɛ model (along with Eulerian multiphase model),

The results obtained by using mixture-based multiphase models are not able to have similar accurate results as compared to the Eulerian based models. Thus, SST k-ω model also could not yield accurate results. Eulerian implementation of the SST k-ω model is not available in FLUENT. Also, the SST 4 – equation model is computationally more costly as well as this is also not available with the Eulerian implementation. Amongst the remaining two models, i.e. standard k−ɛ and realizable k−ɛ model, results of the realizable k−ɛ and RNG k−ɛ model were close. The comparison was sought for the velocity distribution as well as of the concentration profile distribution. Considering the physics of the problem undertaken in the present investigation, the RNG k-ɛ model (along with Eulerian multiphase model) has been found to be sufficiently suited for the present analysis. This opinion was further strengthened by the other researchers such as Najafia et al.¹⁸, Gupta and Kumar¹⁹, Escue and Cui²⁰, investigations, who have also adopted the RNG k−ɛ model approach for the swirl flows in their investigations. Apart from this ANSYS-FLUENT also recommends RNG k−ɛ model for swirling flows.

Results and discussions

Cleaning of the inclined and horizontal holes presents two major concerns which are not present in vertical hole cleaning. One concern is the existence and thickness of a bed of cuttings on the lower side of the hole. The other is the sliding of a bed of cuttings down-hole. A bed of cuttings is said to be stable if it does not have a tendency to slide and unstable if it does. Presence of the cuttings bed in the hole in the course of the drilling places the tripping of the drill pipe in jeopardy. This is further worsened by the cases when drilling fluid circulation is halted due to some reasons. Thicker beds are more likely to form in more deviated holes and unstable beds are more likely in the less deviated sections of the hole.

Most cuttings bed problems occur with an oil base and normally pressured zones. Usually, water-based mud aid in the disintegration of the cuttings, and thus may support the subsequent removal. Further cuttings from the over-pressured zones are more buoyant (cuttings less dense and the mud is more dense) and have less cohesiveness than cuttings from normally pressured zones and therefore are less likely to form as thick bed if any at all.

It has been found that thinner cutting beds are formed and maintained by the presence of the turbulence in the drilling fluid flow, rotation of the drill string, and flat velocity profile at the circulation rate of the circulating drilling fluids. These findings have been supported by laboratory as well as filed test data.

The most basic requirement for thinning of the cuttings bed formation is that the solid particles present (silica sand particles considered here) in the drilling fluid (considered water here) should not get opportunity to get settled down at the bottom of the horizontal wellbore.

Detailed results for water–sand slurry flow through the concentric annulus with inner cylinder in rotation are presented here. First, the results are illustrated for a particular value of the operational and geometric parameters. Afterwards, the effect of inner cylinder rotation, inlet bulk velocity, and inlet solid concentration on the solid (cuttings) concentration variation has been considered. Then, the effect of turbulence kinetic energy (TKE) and slip velocity was corroborated on the bed formation and solid particles distribution pattern within the borehole.

Overall cross-sectional variation of the cuttings (solid) concentration at particular section (Figure 9 to 15)

Figures 9(a) to (d) and 10(a) to (d) show the contour plot of the variation of the solid concentration at a particular cross-section. The cross-section considered is taken at axial distance of z = 1.2 m. Total length of the annulus taken is L = 1.25 m. This satisfies the criterion of the fully developed flow. Since the flow which is occurring is within the turbulent regime, thus pipe length L should be greater than the entry length L_e (≈50 D_h i.e. 1 m). Thus, the cross-sectional plane considered is within the fully developed region.

Figure 9.

Effect of variation of the linear speed on cross-sectional variation of the solid concentration at outlet for 250 r/min inner cylinder rotation and 15% inlet solid concentration. (a) For 3 m/s bulk velocity, (b) for 3.5 m/s bulk velocity, (c) 4.5 m/s bulk velocity and (d) 5 m/s bulk velocity.

Figure 10.

Effect of variation of the inner cylinder rotational speed on cross-sectional variation of the solid concentration at outlet for 3 m/s bulk velocity and 15% inlet solid concentration. (a) For 50 r/min inner cylinder rotation, (b) for 100 r/min inner cylinder rotation, (c) for 200 r/min inner cylinder rotation and (d) For 250 r/min inner cylinder rotation.

Figure 9(a) to (d) illustrates the variation of the solid concentration for a constant inner cylinder rotational speed (anticlock wise 250 r/min) and 15% inlet solid concentration for four different bulk annular flow velocity of the slurry, i.e. 3 m/s, 3.5 m/s, 4.5 m/s and 5 m/s. For the justified comparison among all the diagrams, the scale is taken of the equal range. As it is clear from the diagrams, the zone of maximum concentration of the solid-phase is towards bottom side and shifted in anticlock wise manner (i.e. along the direction of rotation of the inner cylinder) due to the effect of the rotation of the inner cylinder. The highest amount of the concentration settling at the bottom is for the least bulk slurry velocity, i.e. 3 m/s (Figure 9(a)). When the bulk slurry velocity increases to 3.5 m/s (Figure 9b) then to 4.5 m/s (Figure 9(c)), the settling of the solid particles is lesser. Figure 9(d) shows more mixing of the solid particles within the cross-section, for 5 m/s. Thus by increasing the bulk slurry velocity, better transportation of the solid particles can be achieved. From the point of view of the cutting transportation, it is better to increase the velocity of the slurry flow. At lower flow velocity, there is tendency of formation of the bed. Higher the flow velocity higher will be the throw of the solid particles in the upwards manner by the effect of the rotation. Thus, the flow distribution can move towards the homogeneous nature of the distribution of the solid particles.

Along the same lines, the effect of inner pipe rotation is obvious from Figure 10(a) to (d) for a particular flow velocity 5 m/s and 15% inlet solid concentration. As the rotational speed increases, the zone of the maximum solid particle concentration lying towards lower side of the annulus shifts/rises up towards upper part following the direction of the rotation of the inner cylinder. The combination of the higher bulk slurry velocity and higher rotational speed of the inner cylinder give better distribution of the cutting in the cross-section.

To get more clear quantified values of the cross-sectional distribution of the cuttings within the annulus, further, Figures 12 to 15 are presenting the bar chart for solid concentration distribution at four specified locations namely Sector 1, Sector 2, Sector 3 and Sector 4 (refer Figure 11) of a particular cross-section (outlet cross-section z = 1.25). The values of the solid-phase concentration amount in the bar chart for various sectors of a particular cross-section present the area weighed average of solid-phase concentration within the zone of that particular sector.

Figure 11.

Location of the quadrants 1, 2, 3 and 4 and location of the radial planes P1, P2, P3 and P4.

Figure 12.

Cross-sectional solid-phase concentration distribution for 5 m/s inlet bulk velocity and 15% inlet solid concentration.

Figure 12 is the bar chart representation of the area weighed average of solid-phase concentration in sector-wise manner at particular cross-section for 5 m/s inlet bulk slurry velocity and 15% inlet solid concentration for 50, 100, 200, and 250 r/min (anti-clockwise). Sectors 3 and 4 have the maximum and comparable contribution, whereas Sectors 1 and 2 are lagging behind them in solid concentration transportation. Increase in the rotational speed of the inner cylinder causes reduction in the share of the solid-phase concentration for sectors 3 and 4 and causes increment in the case of sectors 1 and 2. The same trend is obvious from the bar chart presented in Figure 13, which shows the area weighed average of solid-phase concentration in sector-wise manner at particular cross-section for 3 m/s inlet bulk slurry velocity and 15% inlet solid concentration for 50, 100, 200, and 250 r/min (anti-clockwise). By comparing Figures 12 and 13, it is obvious that higher slurry velocity causes more uniformity in the solid-phase concentration, within the flow field.

Figure 13.

Cross-sectional solid-phase concentration distribution for 3 m/s inlet bulk velocity and 15% inlet solid concentration.

Increase in the contribution of the averaged solid cuttings concentration amount by the sector 1 and sector 2 and reduction in the sector 3 and sector 4 contributions is better indicator of the homogeneity achieving target within the cross-section. Of course, the complete homogeneity/uniformity is not possible within the cross-section for many reasons. Still, the parameters which are assisting towards that target will naturally contribute towards the reduction in the bed thickness. This means cleaning of the borehole will be better in that case.

Figure 14 is illustrating the bar chart of the sector-wise variation of the area weighted average of solid-phase concentration distribution for 250 r/min and 4 m/s slurry velocity for 5%, 10%, and 15% inlet concentration. For a particular combination of the slurry velocity and inner cylinder rotational speed, the effect of the inlet solid concentration is same. Solid concentration at different section is found as per the following trend, Cs₄ > Cs₃ > Cs₂ > Cs₁.

Figure 14.

Cross-sectional solid-phase concentration distribution for 250 r/min at 4 m/s inlet bulk velocity.

Figure 15.

Cross-sectional solid-phase concentration distribution for constant Rossby number 12.732 and 15% inlet solid concentration.

Figure 15 is the sector-wise variation of the area weighted solid-phase concentration distribution for constant Rossby number and 15% inlet concentration. It is clear from the previous discussion that the trend of solid-phase variation is constant for the various inlet solid concentrations, whereas the trend changes with the variation in the slurry velocity and the inner cylinder rotational speeds. Rossby number (V/ωR_in) represents the combined effect flow velocity and wall rotation, i.e. amount of the swirl present in the domain considered. Since the Rossby number considered here is 12.732 clearly stating the inertial forces in the transverse direction is more as compared to the tangential inertia contributed by the rotation of the cylinder. Thus, the effect of the bulk flow velocity is higher as compared to the speed of the rotation of the inner cylinder even when the ratio of the r/min/bulk velocity is same (i.e. 250/5 = 200/4 = 150/3 = 50).

Radial variation of the cuttings (solid) concentration distribution at the outlet (Figures 16 to 18)

In order to get further clear picture of the bed formation possibility this is helpful to watch over the solid-phase distribution along the plane P4. Also radial distribution along other planes P1, P2, and P3 will be helpful to see that how the cuttings are spread throughout the cross-section by the effect of the flow variables.

Figure 16 shows the effect of the inlet solid concentration for 250 r/min rotation and 4 m/s slurry velocity; Figure 17 presents the effect of the cylinder rotation for 15% concentration at 3 m/s slurry velocity and Figure 18 presents the effect of constant ratio (50) of the inner cylinder r/min/bulk flow velocity (i.e. 250/5 = 200/4 = 150/3 = 50), leading to constant Rossby number 12.732 at 10% inlet solid concentration. Results have been displayed along the four different radial positions P1, P2, P3, and P4 (refer Figure 4.11) for a cross-section at axial location Z = 1.25 (i.e. outlet).

Figure 16.

Variation of solid concentration in volume fraction (Cs) along planes P1, P2, P3 and P4 for inner cylinder rotation of 250 r/min and inlet bulk flow velocity 4 m/s at various inlet solid concentrations.

Figure 17.

Variation of solid concentration in volume fraction (Cs) along planes P1, P2, P3 and P4 for inlet bulk flow velocity 3 m/s and inlet solid concentration of 15% for various inner cylinder rotational speeds.

Figure 18.

Variation of solid concentration in volume fraction (Cs) along planes P1, P2, P3 and P4 for constant Rossby number = 12.732 at 10% inlet solid concentration.

One interesting noticeable feature is that the maximum concentration is not occurring at the bottom of the horizontal well as obvious from the distribution pattern along the plane P4. This is the effect of the rotation of the inner cylinder. Figure 16 shows the clear peak concentration along all the planes P1, P2, P3, and P4. The maxima point shifts away from the inner cylinder in the case of P1, P2, and P3 planes, whereas for plane P4 it is closer to the inner cylinder, as the inlet solid concentration amount increases. Variation along P2 and P4 is affected by the gravity along with the flow velocity and r/min, but at planes P1 and P3, there is no effect of gravity. Thus, in that sense variation along planes P1 and P3 is very important. Because of rotation the solid concentration is thrown towards the outer diameter of the annulus in both the cases P1 and P3. The area under the curve is more for P1 and P3, which clearly indicate the increase in the solid-carrying capacity of flow when rotation is present.

The above discussed phenomenon can further clearly be observed by the results shown in Figure 17. This illustrates the effect of inner cylinder rotational speeds. Higher rotational speed throws out the solid particles away against the gravitational effects. From the variation along the plane P1, the distribution becomes more uniform by the effect of rotation. Higher rotational effects cause more mixing of the solid particles within the slurry flow field. By observing the area under the variation curve along the plane P4, area under the curve is smallest in case of the 250 r/min whereas highest in case of the 50 r/min rotation. It indicates the smaller chances of the settling of the solid or cuttings in case of the higher rotational speeds.

As the rotational speed increases, it causes reduction in the amount of the cutting available near the bottom of the horizontal wellbore. This is shown by the results of solid concentration distribution along the plane P4 (Figure 17). Even though for all the cases, concentration amount at the bottom is around 10%, the possibility of the settling is minimized by the inner cylinder rotation in the proportion to the rotational speed.

From Figure 18, which is drawn for the constant Rossby number, it is obvious to learn that the amount of the swirl is same in the slurry flow, the plots for r/min – slurry velocity combinations are not matching for any of the planes P1, P2, P3, or P4. This is also seen in Figure 17. For the same amount of the swirl present, the combination (i.e. 250 r/min and 5 m/s) which provides maximum turbulence to the leading effect on the removal of the cuttings. For slurry flow, the effect of the rotational speed and slurry flow velocity is completely different.

Correlating the solid-phase distribution with the mixture TKE and slip velocity variation at the outlet

There is a close correlation of the mixture TKE and slip velocity with the solid concentration distribution. Thus, by managing the extent of the mixture TKE and slip velocity, it is possible to control the bed thickness formation, which will thus help/assist in the cleaning of the borehole. The detailed discussion is presented below.

Figures 19 and 20 show the variation of mixture TKE and slip velocity variation, respectively, for the case 3 m/s and 15% inlet solid concentration for various inner cylinder rotational speed. Figures 21 and 22 present the variation of mixture TKE and slip velocity variation, respectively, for the case 5 m/s and 15% inlet solid concentration for various inner cylinder rotational speeds. Figures 23 and 24 show the variation of mixture TKE and slip velocity variation, respectively, for the case of 4 m/s and 250 r/min for various inlet solid concentrations (at 5%, 10% and 15%). Figure 25 & 26 present the variation of mixture TKE and slip velocity variation, respectively, for the case constant Rossby number 12.732 and 10% inlet solid concentration.

Figure 19.

Variation of the mixture TKE along planes P1, P2, P3 and P4 for slurry flow velocity 3 m/s and 15% inlet solid concentration.

Figure 20.

Variation of slip velocity along planes P1, P2, P3 and P4 for inlet slurry velocity 3 m/s and inlet solid concentration 15%.

Figure 21.

Variation of mixture TKE along the planse P1, P2, P3 and P4 for slurry flow velocity 5 m/s and 15% inlet solid concentration.

Figure 22.

Variation of slip velocity along planes P1, P2, P3 and P4 for inlet slurry velocity 5 m/s and inlet solid concentration 15%.

Figure 23.

Variation of mixture TKE along planes P1, P2, P3 and P4 for inner cylinder rotation 250 r/min, slurry flow velocity of 4 m/s at various inlet solid concentrations.

Figure 24.

Variation of the slip velocity along the planes P1, P2, P3 and P4 for inner cylinder rotational speed 250 r/min and inlet slurry velocity 4 m/s at various concentrations.

Mixture TKE is indicative of the extent of the turbulence presence within the flow of the liquid–solid mixture. Slip velocity has been defined in the present context as the difference between the slurry velocity and solid-phase particle velocity. This is considered as the indicator of how much the solid-phase particles slow down or accelerate as compared to the mixture velocity or overall slurry. This can be seen as the important parameters in deciding the possibility of carrying out the solid particles out from the flow domain. Zone or location of the higher value of the TKE represent the minimal slip velocity thereon, and hence the possibility of clustering of the cuttings in the form of the bed at the bottom could be avoided.

Zero value of the slip velocity refers to the idea that cuttings are being carried along with the flow. Positive value of the slip velocity is the indication of the particles lagging behind the main flow, whereas the negative values implies solid particles are moving faster than the main or bulk flow of the mixture. Thus, positive value of the slip velocity is an undesirable situation. This needs to be avoided.

By observing all the cases, the common feature amongst all the results of slip velocity is that it fluctuates to extremes near the wall, and at the inner walls the slip velocity values are the highest in most of the cases than its value at the outer wall of the borehole annulus.

Also observing within the cross-section that positive value of the slip velocity near the bottom of the borehole is most undesirable situation. From Figures 19 and 20, it can be said that locations of the maximum mixture TKE is found near the walls which gives the fluctuation in the slip velocity from the maximum positive value (most undesirable) to the maximum negative (most desirable) value.

Figure 20 illustrates that for 3 m/s inlet velocity and 15% inlet solid concentration, the slip velocity (slurry velocity–solid-phase velocity) gets positive value just a little away from the wall. This is shows that particles are lagging behind the main stream of flow just near the walls. But immediately after this, there is sudden transition taking place in the slip velocity value. It goes to negative value, signifying that the solid particles are getting accelerated. This sudden transition zone of value of slip velocity is very interesting. This is obviously possible by the effect of presence of near-wall turbulence. From the results of the mixture TKE, these results can be correlated. The zone of the highest TKE is the zone of the fluctuation in the slip velocity. For the variation along the plane P4, it is observed that near the outer diameter of the borehole, slip velocity is very much decelerated as compared to the case with other planes. The smaller the rotational speed the higher the reduction in the solid velocity as compared to the slurry velocity.

But when the inlet slurry velocity increases to 5 m/s (Figure 21), then the effect of the swirl due to change in the rotational speed is swept away by the strong axial component of the higher inlet slurry velocity. Thus, level of the turbulence present in the flow domain is dependent on the comparative effect of the rotational speed vs. slurry flow velocity.

From Figure 22, it can be observed that at higher value of the inlet velocity, effect of the rotation is swept away by the axial effects, i.e. rotational speed has little effect, especially when operation is taking place at higher inlet solid concentration. But by this, it cannot negate the importance of the rotation of the inner cylinder and its impact on the slip velocity and other flow parameters considered. Along all the radial planes P1, P2, P3, and P4, it is observed that near to the outer cylinder, there is a possibility of the solid particles to accumulate due to presence of the stationary wall of the borehole annulus. Along the plane P4, the impact of the variation in the rotational speed is very minimal for 5 m/s and 15% inlet concentration.

Comparing the results of slip velocity as observed from Figures 20 and 22, it is clear that as inlet slurry flow velocity increases, the effect of the rotational speed is minimized due to strength of axial sweep.

Figure 23 depicts that the mixture TKE is depending upon the amount of the % inlet solid concentration in the borehole annulus. By comparing Figures 21 and 25, for the Rossby number 12.732, it is observed that the level of turbulence in the slurry, as represented by the mixture TKE depends additionally upon the amount of the % of the inlet solid concentration.

Figure 25.

Variation of mixture TKE along planes P1, P2, P3 and P4 for constant Rossby number = 12.732, at 10% inlet solid concentration.

By comparing Figures 23 and 24 for the case of the 10% inlet solid concentration, it is observed that apart from the Rossby number and the inlet % concentration, there are other factors, i.e. magnitude of the inner cylinder rotational speed and magnitude of the slurry inlet velocity, which affect the extent of turbulence in the slurry.

By observing all the cases, Figures 20 to 26, the common feature amongst all the results of slip velocity is that it fluctuates to extremes near the wall and at the inner walls the slip velocity values are highest, in most of the cases than its value at the outer wall of the borehole annulus.

Figure 26.

Variation of slip velocity along planes P1, P2, P3 and P4 for constant Rossby number 12.732 at 10% inlet solid concentration.

The common feature in all the cases considered is that the level of turbulence, as represented by the amount of the mixture TKE, is somewhere in the near mid-part of the annulus, whereas the maximum amount the turbulence present in the annulus is near to the inner and outer wall of the borehole annulus. This situation is very much desirable from the point of view of the cleaning of the annulus. The problem of the bed formation may be avoided by this.

Concluding remarks

Hole cleaning is very important concern for the horizontal wellbore. This was studied using the two-phase Eulerian, RNG k−ɛ-based CFD approach for the concentric annulus. Geometrical model considered was of the 20 mm diameteral thickness, signifying the situation of a slim-hole type oil wellbore drilling. Cleaning of the borehole can be judged by distribution of the cuttings concentration within the annulus cross-section. Thus, various parameters such as slurry bulk velocity, inner cylinder rotational speed, and inlet concentration have been discussed upon the concentration. Apart from this, the effect of the mixture TKE and slip velocity has been correlated with the solid concentration distribution. Few important findings based on the present research work are presented below:

Solid-carrying capacity, i.e. hole cleaning can be enhanced by increase in the bulk slurry velocity and by increase in the rotation of the inner cylinder. It causes the thorough mixing of the solid particles within the cross-section by overcoming the effect of the gravity which causes settling of the solid particles (Figures 9 and 10). Impact of the increase in the slurry velocity is more pronounced than increase in the rotational r/min (Figures 12 and 13). Increase in the concentration of the solid particles (cuttings) in quadrants 1 and 2 and reduction of it in quadrants 3 and 4 signify better situation for cuttings mixings and thus transportation of the cuttings. Impact of the rotational speed is more for the lower slurry bulk velocity values. Considering the quadrant 1, at 3 m/s velocity, four times increase in the rotational speed causes two times increases the solid concentration, but at 5 m/s velocity similar increase in the rotational speed causes only 1.25 times increase (Figures 12 to 15).

Further clear picture of the solid concentration distribution is available from Figures 16 to 18. Here, the possibility of the settling can be more visually observed by looking at the radial variations along the plane P4. This needs to be avoided. With the increase in the rotational speed, for a fixed inlet solid concentration, the location of the peak concentration (measured from the outer diameter) and the amount of the accumulation of the solid particles (measured by the area under the solid concentration distribution curve), refer Figure 17 for plane P4. Also the similar conclusion can be deduced from Figures 16 and 18.

Level of the turbulence as represented by the mixture TKE is the function of the inlet solid concentration, inlet slurry velocity, rotation of the inner cylinder and also on the relative magnitude of the swirl as presented by the Rossby number. Amount of the turbulence is least near the middle of the borehole annulus, whereas it is maximum at some distance from the inner and outer cylinder. At the walls obviously, it is zero then increases to maximum value before reaching to the lowest value near towards the middle part of the annulus, for all the cases shown in Figures 19, 21, 23 and 25. Near the wall, the settling possibility can be reduced by the higher slurry velocity and higher rotational speeds.

Slip velocity fluctuation is highly affected by the TKE. Positive value of the slip velocity signifies that the solid particles are lacking behind the main flow, and otherwise for the negative slip velocity. From comparison of the mixture TKE plots (Figures 19, 21, 23, and 25) with the respective slip velocity plots (Figures 20, 22, 24, and 26), it is very obvious that zones of the higher mixture TKE help the slurry to carry the solid particles with the flow.

Further, there is lot of scope for computational study of the slurry flow through eccentric annulus. This eccentric annulus presents a more close to the real situation for the horizontal well-drilling scenario. The case of the slurry flow through the eccentric annulus is also important from the research interests in Tribology and thermal devices-related applications as well.

Footnotes

Acknowledgements

The authors are thankful to the NIT,Raipur for all allowing us to use the facility of the ANSYS FLUENT software,Tecplot Software and institute Library.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) received no financial support for the research,authorship,and/or publication of this article.

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