An experimentally-verified approach for enhancing fluid drag force simulation in vertical oilwell drill strings

Figures & data

Figure 1. Procedure for RSM [34].

Table 1. Variables that affect the damping ratio.

Variable of control	Minimum	Maximum	Symbol	Units
Consistency index	0	2	$\kappa$	kg/m s
Power law (Flow behaviour) index	NA	NA	$n$	–
Yield stress	0	10	$\tau$	Pa
Velocity through pipe	0.1	1.5	$v$	m/s
Critical shear rate	NA	NA	${\dot{\gamma }}_c$	1/s
Minimum distance between wellbore and pipe	6.46	36.46	$d_{min}$	mm
Density of drilling fluid	1000	2000	$\rho$	kg/m3

Figure 2. Damping ratio ($\zeta$) vs. individual variables.

Table 2. Experiments and the respective results.

$Sim$	$d_{min}$	$\tau$	$\kappa$	$\rho$	$v$	$e$	$\zeta$	$F$
1	36.46	10	0	1750	1.5	0	14.02	12.91
2	21.46	5	1	1000	0.8	15	11.54	9.34
3	36.46	0	0	1000	0.1	0	4.45	0.00
4	6.46	10	2	1750	1.5	30	23.49	65.19
5	6.46	0	2	1750	0.1	30	17.30	11.50
6	6.46	10	0	1750	0.1	30	16.96	8.51
7	21.46	5	1	1750	0.8	15	11.86	9.03
8	21.46	5	1	1375	0.8	15	11.83	9.15
9	36.46	0	2	1000	1.5	0	7.93	162.14
10	6.46	10	2	1750	0.1	30	22.14	20.08
11	21.46	5	0	1375	0.8	15	9.34	2.76
12	6.46	0	2	1750	1.5	30	19.47	81.20
13	21.46	10	1	1375	0.8	15	13.90	5.14
14	36.46	10	2	1750	0.1	0	14.02	29.59
15	36.46	10	0	1000	0.1	0	8.95	12.82
16	36.46	10	0	1000	1.5	0	12.69	17.25
17	6.46	0	2	1000	0.1	30	17.34	11.52
18	36.46	0	0	1750	1.5	0	8.37	2.97
19	21.46	5	1	1375	0.8	15	11.83	9.22
20	6.46	10	2	1000	1.5	30	15.58	62.52
21	6.46	0	0	1750	1.5	30	8.37	2.97
22	21.46	5	1	1375	0.8	15	11.83	9.23
23	6.46	10	2	1000	0.1	30	23.20	20.08
24	6.46	0	2	1000	1.5	30	14.47	67.26
25	21.46	5	1	1375	0.8	15	11.83	9.23
26	36.46	0	2	1000	0.1	0	17.34	11.52
27	36.46	0	0	1750	0.1	0	8.33	0.02
28	36.46	5	1	1375	0.8	0	14.45	20.91
29	21.46	5	1	1375	0.1	15	14.07	7.65
30	21.46	5	1	1375	0.8	15	11.83	9.23
31	6.46	5	1	1375	0.8	30	14.45	20.93
32	36.46	0	2	1750	0.1	0	17.30	11.51
33	6.46	0	0	1000	1.5	30	7.26	1.70
34	6.46	10	0	1000	1.5	30	12.69	17.27
35	36.46	10	0	1750	0.1	0	16.96	8.52
36	21.46	5	1	1375	0.8	15	11.83	9.23
37	6.46	0	0	1750	0.1	30	8.33	0.02
38	36.46	10	2	1750	1.5	0	11.91	178.88
39	36.46	0	2	1750	1.5	0	9.51	175.48
40	21.46	5	1	1375	0.8	15	11.83	9.23
41	36.46	10	2	1000	1.5	0	7.59	165.65
42	21.46	5	2	1375	0.8	15	14.28	23.30
43	36.46	10	2	1000	0.1	0	13.57	28.84
44	21.46	5	1	1375	0.8	15	11.83	9.23
45	6.46	0	0	1000	0.1	30	7.25	0.01
46	6.46	10	0	1750	1.5	30	12.94	26.18
47	6.46	10	0	1000	0.1	30	17.74	8.53
48	36.46	0	0	1000	1.5	0	4.21	0.71
49	21.46	0	1	1375	0.8	15	11.54	9.37
50	21.46	5	1	1375	1.5	15	11.00	9.23

Figure 3. The approximation to a spring-mass-damper system.

Figure 4. Repeating unit of the lumped segment model.

Figure 5. $\zeta$- Predicted vs. actual.

Figure 6. Damping ratio model.

Table 3. ANOVA for reduced cubic model.

Source	Sum of Squares	df	Mean Square	F-value	p-value
Model	882.42	23	38.37	39.83	< 0.0001
A-d_min	1.137E-13	1	1.137E-13	1.180E-13	1.0000
B-Tau	142.48	1	142.48	147.92	< 0.0001
C-K	225.37	1	225.37	233.98	< 0.0001
D-Rho	41.36	1	41.36	42.94	< 0.0001
E-v	56.25	1	56.25	58.40	< 0.0001
AB	16.07	1	16.07	16.69	0.0004
AC	50.65	1	50.65	52.59	< 0.0001
AD	3.28	1	3.28	3.41	0.0764
AE	2.36	1	2.36	2.45	0.1294
BC	64.79	1	64.79	67.27	< 0.0001
BE	0.6626	1	0.6626	0.6880	0.4144
CD	0.0250	1	0.0250	0.0260	0.8732
CE	17.73	1	17.73	18.41	0.0002
DE	6.25	1	6.25	6.49	0.0171
A^2	7.76	1	7.76	8.05	0.0087
B^2	0.2610	1	0.2610	0.2709	0.6071
ABC	11.24	1	11.24	11.67	0.0021
ABE	19.23	1	19.23	19.96	0.0001
ACD	14.05	1	14.05	14.59	0.0007
ACE	24.40	1	24.40	25.33	< 0.0001
ADE	7.96	1	7.96	8.27	0.0079
CDE	19.26	1	19.26	20.00	0.0001
AB^2	8.35	1	8.35	8.67	0.0067
Residual	25.04	26	0.9632
Lack of Fit	25.04	19	1.32
Pure Error	0.0000	7	0.0000
Cor Total	907.46	49

Figure 7. Drag force: predicted vs. actual.

Figure 8. Axial drag model.

Figure 9. The 3D CAD model of the apparatus.

Figure 10. Experimental apparatus and equipment used.

Figure 11. Sample experimental data.

Figure 12. Damping ratio change in Newtonian fluid.

Figure 13. Viscometer test results of the non-Newtonian fluid.

Figure 14. Damping ratio change in non-Newtonian fluid.

Figure 15. 3D bond graph element.

Figure 16. Interface between elements.

Figure 17. BG element compliances.

Figure 18. Contact spring schematic [10].

Figure 19. Bending moment fluctuation.

Figure 20. Drill string’s apparent weight fluctuation.

Figure 21. Natural frequency vs. fluid velocity.

Figure 22. Radius ratio vs. Reynolds number.[40]

Figure 23. Velocity distribution with the changing eccentricity (Figures illustrate the change in maximum velocity with increasing eccentricity).

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