Best Rheological Model Drilling Fluids

3rd Annual European Rheology Conference, Hersonissos, 2006

Choosing the best rheological model for bentonite suspensions MAGLIONE Roberto and KELESSIDIS C. Vassilios 1

2

Consultant, Vercelli, Italy (contact: [email protected]) 2 Mineral Resources Engineering Department, Technical University of Crete, Chania, GREECE 1

(contact: [email protected])

INTRODUCTION - Water-bentonite suspensions are encountered in a variety of chemical, petroleum and waste treatment industries and occur in flow situations in complex geometries like pipe flow, concentric and eccentric annulus, and flow in rectangular ducts. Most of the time, flow of these suspensions are laminar and analytical solutions have been developed for a variety of rheological models that have been proposed in the past. Among these are the most commonly used Bingham plastic and power law models, and the Herschel-Bulkley, the Casson, and the Robertson-Stiff models. Rheological parameters are obtained with Couette viscometer data, normally using Newtonian shear rates in the viscometer. However, approaches have been suggested for using true, non-Newtonian shear rates1,2. MATERIAL AND METHODS - Four water-bentonite suspensions (S1, S7, S10, and S23), with different concentration of bentonite and densities ranging from 1,050 and 1,080 kg/m3, were tested in the lab with a rotational viscometer (Grace 3500). Using the reported viscometric data, Casson, Robertson-Stiff (RS) and Herschel-Bulkley (HB) rheological models were applied and the rheological parameters together with statistical coefficients such as the correlation coefficient (R2), the sum of square errors (SSE), and the root mean square error (RMSE) were computed. The appropriate rheological model was chosen from the best fit of rheograms to raw experimental data. Differences in the rheological behavior when using Newtonian and true shear rates in the viscometer gap as well as in the flow parameters for flow in pipes and annuli were then evaluated.

RESULTS

Pressure Gradient in Annuli (8 ½“–5”) 3,5 3,0 2,5 2,0 1,5 1,0 0,5 0,0

Shear Stress [Pa]

30,0

0

100

200

300

400

500

600

700

800

900

1000 1100

80 70 60 50 40 30 20 0

25,0

200

400

New tonian shear rate [1/s]

20,0

Ratio Ne-HB

15,0

Ratio Ne-Casson

5,0 0

100

200

300

400

500

600

700

800

Velocity Profile in Annuli (8 ½”–5”)

900 1000 1100

Herschel-Bulkley

Casson

Robertson-Stiff

Figure 1 - Experimentally derived and computed Newtonian shear rate-based rheograms for mud S1

velocity [m/s]

1,4

New tonian shear rate

1,2 1,0 0,8 0,6 0,4 0,2 0,0 0,06

Rheological model

τy [Pa] K, µp [Pa?sn] γo [1/s] N R2 SSE [Pa2] RMSE [Pa]

Robertson-Stiff 9,209 6,499 0,162 0,986 4,486 0,670

Herschel-Bulkley 8,475 3,401 0,256 0,988 3,877 0,623

Table 1 - Rheological parameters derived by Newtonian shear rate and statistical correlation coefficients for mud S1

Shear stress against shear rate calculated by the true shear rate expression is then compared to the experimentally data derived by the Newtonian shear rate (Fig. 2 and Table 2). Casson and RS true shear rates were computed by a non-linear regression method1. HB shear rate was calculated by using a numerical algorithm adapted in house2 from a hybrid procedure already proved to be effective in other fields of drilling fluids research3.

15,0 10,0 5,0

0

100

200

300

400

500 600

700

800

900 1000 1100 1200

Herschel-Bulkley shear rate

Casson shear rate

Robertson-Stiff shear rate

New tonian shear rate

Figure 2 - Experimentally derived and computed true shear ratebased rheograms for mud S1 Rheological model

Casson τy [Pa] K, µp [Pa?sn] γo [1/s] n

R2 SSE [Pa2] RMSE [Pa]

13,670 0,0026 0,958 13,217 1,096

RobertsonStiff 8,754 6,090 0,162 0,986 4,489 0,670

HerschelBulkley 4,472 4,825 0,24 0,997 2,911 0,539

Table 2 - Rheological parameters derived by true shear rate and statistical correlation coefficients for mud S1

1200 1100 1000 900 800 500

1000

1500

2000

2500

3000

3500

4000

pump rate [L/min]

750 L/min (HB -Ne shear rate)

1500 L/min (HB -Ne shear rate)

750 L/min (HB -True shear rate)

1500 L/min (HB -True shear rate)

HB-Ne shear rate

HB-True shear rate

Figure 8 - Pressure drop gradient in laminar flow for HB model, derived by Newtonian and true shear rate for mud S1

Laminar to Transitional Critical Points 1,2

140

pipe

130

annulus

1,0

120 110

0,8

100 90

0,6

80 70

0,4

60 50 0

100

200

300

400

500

600

700

800

pump rate [L/min] HB-Ne shear rate

0,2 0,0

S10

HB-True shear rate

Figure 5 - Pressure drop gradient in laminar flow for HB model derived by Newtonian and true shear rate for mud S23

S23

S7

S1

Figure 9 -Ratio of critical flow rates, true to Newtonian shear rates

REFERENCES

300

250

1) Kelessidis VC, Maglione R, Modeling Rheological Behavior of Bentonite Suspensions as Casson and Robertson-Stiff Fluids Using Newtonian and True Shear Rates in Couette Viscometry, paper submitted to Powder Technology, 2006

200

150

100 200

400

600

800

1000

pump rate [L/min] HB-Ne shear rate

Shear Rate [1/s]

1300

0

0,11

150

0

0,0

0,10

Pressure Gradient in Pipes (4.27”)

pressure gradient [Pa/m]

20,0

0,09

Figure 4 - Velocity profile for HB model derived by Newtonian and true shear rate for mud S7

30,0 25,0

0,08

HB-True shear rate

1400

radius [m]

pressure gradient [Pa/m]

Casson 14,540 0,0026 0,955 14,090 1,132

0,07

1000

Figure 7 - Pressure drop gradient in laminar flow for HB model, derived by Newtonian and true shear rate for mud S10

pressure gradient [Pa/m]

1,6

Shear Rate [1/s]

800

Ratio Ne-RS

Figure 3 - Ratio of true over Newtonian shear rate for mud S1

0,0

600

pump rate [L/min] HB-Ne shear rate

10,0

Shear Stress [Pa]

pressure gradient [Pa/m]

Computed rheograms and statistical coefficients by Casson, RS, and HB models according to the Newtonian shear rate equation are reported and compared to the experimental one (Fig. 1 and Table 1).

ratio of true over Newtonian shear rate

4,0

2) Kelessidis VC, Maglione R, Shear Rates Corrections for Herschel-Bulkley Fluids in Couette geometry and Effects on Frictional Loss Estimation, paper submitted to SPE Journal, 2006 3) Maglione R, Robotti G, Romagnoli R, In-Situ Characterization of Drilling Mud, paper SPE 66285, SPE Journal 5 (4), December 2000

HB-True shear rate

Figure 6 - Pressure drop gradient in laminar flow for HB model, derived by Newtonian and true shear rate for mud S7

CONCLUSIONS

 Difference between the rheological behavior of a water-bentonite suspension computed either by applying the Newtonian or the true shear rate expressions to Casson, Robertson-Stiff, and Herschel Bulkley models, is observed.  The three-constant parameters Herschel–Bulkley rheological model, gives the best fit of the experimentally derived data computed either by the Newtonian or the true shear rate for all samples.  Robertson-Stiff rheological model gives comparable results to the Herschel-Bulkley model, even though the accuracy in the computation is slightly lower.  Computation with Casson model gives the lowest accuracy in the data fitting. For flow in pipe and annulus, the computed flow parameters, with the Herschel-Bulkley model, using Newtonian and true shear rates show small differences for the velocities profiles but up to 33% differences for the pressure gradient.  Variations in the predictions of the onset of transitional flow, both in pipes and annuli, is also observed when comparing results derived with Newtonian and true shear rates.