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### TITLE : LAMINAR FLOW OF NON-NEWTONIAN FLUIDS IN ECCENTRIC ANNULI

BY : CHAM Development Team - G.Garnier

FOR : Validation of Non-Newtonian Models

DATE : 1997 PHOENICS Version : 3.0

### PURPOSE OF THE CALCULATIONS:

• PHOENICS calculations are performed for the laminar flow of viscous non-Newtonian fluids in eccentric annuli.

• The main objective is to validate the PHOENICS implementation of the Herschel-Bulkley (HB) rheological model against analytical and experimental data.

• The problem has practical relevance in the flow of drilling muds in the annular space between the drillstring and the rock. Moreover, during normal drilling, it is known that the drillstring is not centered. Hence, numerical predictions can help the drilling engineer to adapt fluid rheology and flow rate to the specific well geometry, so that cutting transport is efficient, while annular pressures remain below the formation fracturation pressure.

• The HB model relates the shear stress tau to the strain rate Gamma via the consistency index K, the power law index n and the yield stress tauy (fig. 1) , i.e. tau= tauy + K*(Gamma**n).

• The HB model simplifies to the power-law model when tauy=0. For values of n<1, the fluid is pseudoplastic (shear thinning), and for values of n>1, it is dilitant (shear thickening).

• The motion only commences when the yield stress is exceeded.

• The HB model reduces to the Bingham plastic model when n=1.
• When tauy=0 and n=1, the fluid is Newtonian.

• Calculations are made for fully-developed laminar flow of both power-law and Herschel-Bulkley fluids so as to calculate the velocity profile, the frictional resistance, and hence pressure drop, by use of the single-slab solver.

• We define the dimensionless geometric parameter: the eccentricity, e, which is equal to zero for a concentric annulus and is equal to one for fully eccentric annulus:

```     e= delta / (Rin - Rout)
```
wherein delta is the distance between the two centers, Rin the inner radius, Rout the outer radius.

• The grid is a BFC grid, as shown in figure 2.
1. HERSCHEL-BULKLEY FLUID: FLOW CURVE
2. ECCENTRIC ANNULI:GRID

• The first step is to validate the PHOENICS implementation on flow of Power-Law fluid in eccentric annuli. For this case, Haciislamoglu and Langlinais [1] have developped a correlation for the pressure drop :

```     (dP/dZ)ecc=(dP/dZ)conc * R
```
wherein

• (dP/dZ)conc is the pressure drop for the concentric annulus flow. It can be calculated by the Reed coreelation, using the effective diameter [2].

• R is given by the equation which is valid for eccentricities from 0 to 0.95, pipe diameter ratios of 0.3 t 0.9 and Power- Law index of 0.4 to 1

```      R= 1 - .0072(r**.8454)(e/n) -1.5(r**.1852)(e*e)(n**.5)
+ .96(e**3)(n**.5)(r**.2527)

wherein

r = Din/Dout
```
This correlation accuracy is within +/- 5 percent. The PHOENICS predicted pressure drops are compared with this correlation in the range of parameters this one is valid. The differences are within the same accuracy.

### 3. POWER-LAW FLUID :

The second step is to validate PHOENICS implementation on flow of Herschel-Bulkley fluid in eccentric annuli. There is no existing correlation. We have hence validated the computations with Haciislamoglu and Langlinais results [1] (finite difference coding).

The flow properties are:

```  _ n = .7
_ K = .25 kg/ (m sec**n)
_ tauy = 2.394 N/m**2
_ flow rate Q = 1.2618e-2 m**3/sec
_ Din = 25.4 cm
_ Dout = 12.7 cm
```

We first compare the PHOENICS pressure drops. We can notice the well- known effect: For a constant flow rate, frictionnal pressure losses are decreasing with increasing eccentricity. The results show good agreement with Haccislamoglu and Langlinais pressure losses.

### 4. HERSCHEL-BULKLEY FLUID :

Then, we check the velocity profiles by comparing two extremum velocities: the maximum velocity of the largest and the narrowest part of the annulus. The results are in good agreement (fig. 5) with Haccislamoglu and Langlinais works.

We can notice the effect of increasing eccentricity: The velocity in the narrowing part of the annulus is reduced, while the fluid is rushing into the widening part (fig. 6). For high eccentricities (greater than .5), there is a high velocity plug (flat velocity profile) in the wide part of the annulus, while a no-flow (stagnant) region is created in the narrow part.

A few velocity profiles are given for differents eccentricities (fig. 7,8,9,10,11)

### References

[1] M. Haciislamoglu, J. Langlinais, 1990, "Non-Newtonian flow in eccentric annuli", Journal of Energy Ressources pp 163,169

[2] T.D. Reed, Conoco Inc, A.A. Pilehvari, U. of Tulsa, 1993, "A new model for laminar, transitionnal, and turbulent flow of Drilling muds", Society of Petroleum Engineers, SPE 25456, pp 39, 52

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