
Citation: | Hicham Ferroudji, Ahmed Hadjadj, Titus N Ofei, Ahmed Haddad. Effects of the Inner Pipe Rotation and Rheological Parameters on the Axial and Tangential Velocity Profiles and Pressure Drop of Yield Power-Law Fluid in Eccentric Annulus[J]. Petroleum Drilling Techniques, 2020, 48(4): 37-42. DOI: 10.11911/syztjs.2020066 |
Drilling fluid mostly behaves as non-Newtonian fluid and it can be modelled by the Herschel-Bulkley model, which is also called yield power-law (YPL). This model provides accurate results for a wide range of shear rate. In the present paper, a numerical study of the Herschel-Bulkley fluid through the eccentric annulus (E=0.5) was performed for the laminar flow regime using finite volume method (FVM). Effect of the inner pipe rotation and rheology parameters (yield stress τ0, consistency index K and behavior index n) on the axial and tangential velocity profiles and pressure drop gradient were studied. Results showed that increasing the inner pipe rotation from 100 to 400 rpm induces an increase of 120 % of the maximum axial velocity. Low value of the behavior index (n=0.2) causes the appearance of the secondary flow in the wide region of the annulus. The variation of the inner pipe rotation and rheological parameters of the Herschel-Bulkley fluid have a negligible effect on the tangential velocity profile in the wide region of the eccentric annulus. Furthermore, It was observed that the increase of the inner pipe rotation from 0 rpm to 400 rpm causes a decrease of 10% of pressure drop gradient of yield power-law fluid for all eccentric annulus (E=0.2, E=0.4, E=0.6 and E=0.8).
The flow of non-Newtonian flow through annular section has received much attention of researchers to provide optimized solutions for industry challenges. Mud displacement in the annular section between drilling string and casing is one of its industry applications. The drilling fluid is used to carry out different functions, such as maintaining borehole stability, transporting drilled cuttings to the surface and cooling the bit[1], and preventing costly problems when proper hydraulics design of the well is needed during the process. Among models that correctly predict the rheological behavior of drilling muds, Herschel-Bulkley, also known as yield power-law, fits for a large range of shear rates[2]. C. Nouar et al[3], carried out a numerical and experimental study of Herschel-Bulkley fluid flow through annulus with rotation of the inner pipe, they found that rotation of the inner pipe causes a decrease of the axial velocity gradient
In the present paper, the influence of rheological parameters (yield stress τ0, consistency index K and behavior index n) and inner pipe rotation on the distribution of the axial and tangential velocity profile of Herschel-Bulkley fluid in the wide and narrow regions of eccentric annulus are studied. Effects of these parameters on pressure drop gradient are evaluated utilizing a CFD commercial code ANSYS Fluent 17.0 as well.
To study the impact of the inner pipe rotation and rheological parameters (yield stress τ0, consistency index K and behavior index n) on the axial and tangential velocity profiles of Herschel-Bulkley fluid in annular section, the flow is assumed to be fully developed, incompressible, steady and isothermal in laminar regime.
The continuity equation is expressed as:
∇⋅v=0 | (1) |
where v is flow rate.
The momentum equation governing the flow in annulus is expressed in terms of τ in cylindrical coordinates
ρ(∂vr∂t+vr∂vr∂r+vθr∂vr∂θ+vz∂vr∂z−v2θr)=−∂p∂r−[1r∂(rτrr)∂r+1r∂τθr∂θ+∂τzr∂z−τθθr]+ρgr | (2) |
ρ(∂vθ∂t+vr∂vθ∂r+vθr∂vθ∂θ+vz∂vθ∂z+vrvθr)=−1r∂p∂θ−[1r2∂(r2τrθ)∂r+1r∂τθθ∂θ+∂τzθ∂z+τθr−τrθr]+ρgθ | (3) |
ρ(∂vz∂t+vr∂vz∂r+vθr∂vz∂θ+vz∂vz∂z)=−∂p∂θ−[1r∂(rτrz)∂r+1r∂τθz∂θ+∂τzz∂z]+ρgz | (4) |
where
The rheological behavior of the non-Newtonian fluid is modeled as:
τ=τ0+K(˙γ)n | (5) |
where
In the present study, Herschel-Bulkley fluid flows through eccentric annulus (E=0.5) to simulate mud pattern during the drilling of horizontal well, in which the eccentricity is defined as:
E=2eDo−Di | (6) |
where E is eccentricity of the inner cylinder, e is the distance between the centers of the inner and outer pipes(m), Do is diameter of the outer cylinder(m), Di is diameter of the inner cylinder(m).
To predict the flow regime of yield power-law fluid in the annulus, Reynolds number is calculated using the relationship presented by K. Madlener et al[11]:
ReYPL=ρDnhu2−nτ08(Dhu)n+K(3m+14m)n8n−1 | (7) |
where ReYPL is the Reynolds number of yield power-law fluid, u is the bulk flow velocity(m/s), m is the local shear rate(s–1), Dh is the hydraulic diameter(m) which are calculated as:
Dh=Do−Di | (8) |
m=nK(8uDh)nτ0+K(8uDh)n | (9) |
To prevent the entrance effects, the length of the cylinders is selected to be longer than hydrodynamic entry length, which is given as[12]:
Lh,laminar=0.05(Do−Di)Re | (10) |
where Lh, laminar is length of the hydrodynamic entry.
The rheological parameters of Herschel-Bulkley fluid and geometry characteristics are:
The domain of the fluid flow is meshed into 800 000 elements (20 radial divisions, 80 circumferential divisions and 500 axial divisions), where the number of elements is selected to ensure the results independence of mesh model used as well as keeping the number of elements as low as possible to save computational time, as shown in the Fig. 1 and Fig. 2. The commercial code ANSYS Fluent 17.0 based on finite volume method is used to solve the differential equations, where the flow equations are integrated over each control volume. The solution is assumed to be converged, when the convergence criteria 10–4 is reached.
Results obtained from the numerical model are compared to those of experimental work of C. Nouar et al[3] are shown in Fig. 3. The Simulation and experimental results are in good agreement in which the mean error are of 7.7% and 6.4% for dimensionless axial and tangential velocity, respectively.
Comparison of the experimental and numerical results reveals the ability of numerical simulation to provide accurate results.
Fig. 4 exhibits that the increase of the inner pipe rotation from 100 to 400 r/min causes an increase of 120 % of the maximum axial velocity in the narrow region of the annulus, which could improve the well cleaning process by the transportation of cuttings from the lower side of the horizontal annulus preventing the formation of cuttings bed. However, the increase of the inner pipe rotation, induces a reduction of the velocity gradient (∂u/∂r) near the outer pipe in the wide region of the annulus. Similar to the conclusion reported by C. Nouar et al[3].
Fig. 5 shows that the increase of yield stress from 8 to 32 Pa causes a decrease of 6.1 % and 3.6 % of the maximum axial velocity in the wide and narrow regions, respectively.
When the consistency index diminishes from 16 to 4 Pa·sn, the axial velocity profile becomes more flat in the wide region of the annulus, however, a negligible effect is observed for the narrow region, as exhibited in the Fig. 6.
As the behavior index n decreases from 0.8 to 0.2, the axial velocity profile becomes flat in the wide region of the eccentric annulus, as shown in the Fig. 7. While a negligible effect is observed in the narrow region except for n=0.8, where the negative values of the axial velocity profile could be attributed the rotation of the inner pipe which probably increase pressure drop in the annulus.
For all range of the inner pipe rotation, the tangential velocity decreases dramatically near the inner pipe rotation in the wide region of the annulus. As the fluid moved away from the inner pipe, this decrease becomes gradual except for ω=100 r/min, the decrease becomes more gradual in the center of annulus. After that, the tangential velocity begins to decease sharply again until the outer pipe, as presented in the Fig. 4.
As the yield stress diminishes, the tangential velocity decreases more gradually in the center of annulus in the wide region, while a similar variation is observed near the outer and inner pipes, as showed in the Fig. 5.
It can be seen from Fig. 7 that tangential velocity profile in the center of annulus of the wide region decreases more gradually as the behavior index decreases from 0.8 to 0.2 where the tangential velocity profile presents negative values from center of the annulus until the outer pipe. The negative values of the tangential velocity explained by the presence of a secondary flow (also called counter rotating swirl) which rotates in the opposite direction of the inner pipe. The appearance of the secondary flow could decrease carrying capacity of the drilling mud, which affects the whole cleaning process.
In the narrow region of the annulus, the tangential velocity profile decreases dramatically as the fluid moves away from the inner pipe. Fig. 4 to Fig. 7 show that the variation of the inner pipe rotation and rheological parameters have a slight effect on the tangential velocity.
As can be seen in Fig. 8, the increase of the inner pipe rotation from 0 r/min to 400 r/min causes a decrease of pressure drop gradient of yield power-law fluid, this decrease is estimated around 10% for all eccentric annulus (E=0.2, 0.4, 0.6 and 0.8). Since shear thinning phenomenon tends to reduce the pressure drop of yield power-law fluid, inertial effects induced by rotation of the inner pipe are dominated by shear thinning phenomenon in the annulus. This trend is also depicted by R. M. Ahmed et al[7].
Fig. 9 depicts the effect of the yield stress of yield power-law fluid on pressure drop gradient, as shown in the figure, the increase of the yield stress from 8 Pa to 32 Pa induces an increment of 15% of pressure drop gradient for all eccentricities. This increase could be attributed to additional required stress on the fluid to initiate the flow which enhance inertial effects of yield power-law fluid in the annulus.
Fig.10 displays variation of pressure drop gradient of yield power-law fluid in the annulus when the flow consistency index increases from 4 Pa·sn to 16 Pa·sn. As seen in the figure, an increase of 150% of pressure drop gradient for all eccentricities caused by the increment of the flow consistency index. This considerable increase could be explained by the increase of the flow resistance in the annulus due to the increasing fluid viscosity.
Moreover, it can be stated that as the flow consistency index increases, the effect of the eccentricity of cylinders on pressure drop gradient becomes more pronounced.
Fig. 11 exhibits impact of the flow behavior index on pressure drop gradient of yield power-law fluid. As can be seen, an exponential increase of pressure drop gradient as the flow behavior index gets closer from Newtonian behavior where the fluid is less affected by shear thinning phenomenon, which makes inertial effects dominate the flow of yield power-law fluid in the annulus. It was also observed that for low values of the flow behavior index, the eccentricity of cylinders has a slight effect, however, as the flow behavior index gets greater, pressure drop gradient of yield power-law fluid decreases with the increase of the eccentricity.
For all rheological parameters and inner pipe rotation, pressure drop gradient of yield power-law fluid decreases with the increase of the eccentricity.
1) The increase of the inner pipe rotation from 100 to 400 r/min increases the axial velocity in the narrow region of the eccentric annulus, which could enhance the cleaning process in the lower part of the eccentric annulus. However, a slight effect is observed for the maximum axial velocity in the wide region of the annulus.
2) For the behavior index
3) Low value of the behavior index (
4) The variation of the inner pipe rotation and rheological parameters of the Herschel-Bulkley fluid have a negligible effect on the tangential velocity profile in the wide region for an annulus of
5) The increase of the inner pipe rotation from 0 rpm to 400 rpm causes a decrease of 10% of pressure drop gradient of yield power-law fluid for all eccentric annulus (E=0.2, 0.4, 0.6 and 0.8).
6) As the flow behavior index gets closer from the Newtonian behavior, the eccentricity of cylinders pressure drop gradient of yield power-law fluid decreases with the increase of the eccentricity.
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