Improved stiff string torque and drag prediction using a computationally efficient contact algorithm

Figures & data

Table 1. Bond graph elements

Variables	Symbol	Constitutive law	Causality constraints
Flow		$\mathit{f}=\mathit{f}(\mathit{t})$	Fixed flow out
Effort		$\mathit{e}=\mathit{e}(\mathit{t})$	Fixed effort out
Inertia		$\mathit{f}=\frac{1}{I}\int \mathit{e}\mathit{dt}$	Preferred integral
		$\mathit{e}=\mathit{I}\frac{\mathit{df}}{\mathit{dt}}$
Capacitor		$\mathit{e}=\frac{1}{C}\int \mathit{f}\mathit{dt}$	Preferred integral
		$\mathit{f}=\mathit{C}\frac{\mathit{de}}{\mathit{dt}}$
Resistor		$\mathit{e}=\mathit{R}\mathit{f}$	None
		$\mathit{f}=\frac{1}{R}\mathit{e}$
Transformer		$\begin{array}{rl}& e_2=\mathit{n}{f}_1\\ & f_1=\mathit{n}{f}_2\end{array}$	Flow-in effort-out or Effort-in flow-out
Gyrator		$\begin{array}{rl}& e_2=\mathit{n}{f}_1\\ & f_1=\mathit{n}{f}_2\end{array}$	Effort-in effort-out or flow-in flow-out
1-junction		$\begin{array}{rl}& e_2=e_1-e_3\\ & f_1=f_2=f_3\end{array}$	One flow input
0-junction		$\begin{array}{rl}& f_2=f_1-f_3\\ & e_1=e_2=e_3\end{array}$	One flow input

Figure 1. Discetized drill string segments are connected together with axial, torsional, and bending springs (all are included simultaneously).

Figure 2. BG formulation of 3D rigid drill string MBD segment (Rideout et al. 20131984).

Figure 3. Test cylinder and the test point used in the 3D contact algorithm.

Figure 4. When drill string endpoint is located at the top or bottom of the wellbore cylinder.

Figure 5. Arbitrary vector going through test point (T) and point in cylinder axis (S).

Figure 6. BG model of the drill string with the contact algorithm.

Figure 7. Summary of contact calculation algorithm.

Table 2. Sample well profile calculation in cartesian coordinates.

SurveyDepth (m)	Incli. (α) (deg)	Azim. ($\mathit{\epsilon }$) (deg)	$\mathrm{\Delta }$TVD $(D_{\mathit{Mi}})$ (m)	$\mathrm{\Delta }$North $(\mathrm{\Delta }{L}_n)$ (m)	$\mathrm{\Delta }$East $(\mathrm{\Delta }{M}_n)$ (m)	TVD (Dn) (m)	North (Ln) (m)	East (Mn) (m)
0	0	0	0.00	0.00	0.00	0.00	0.00	0.00
1000	0	0	1,000.00	0.00	0.00	1,000.00	0.00	0.00
1050	5	0	49.95	2.18	0.00	1,049.95	2.18	0.00
1100	10	0	49.57	6.53	0.00	1,099.53	8.71	0.00
1150	15	0	48.82	10.82	0.00	1,148.34	19.53	0.00
1200	20	0	47.69	15.04	0.00	1,196.03	34.56	0.00
1250	25	0	46.20	19.13	0.00	1,242.23	53.70	0.00
1300	30	0	44.36	23.09	0.00	1,286.58	76.79	0.00
1350	35	0	42.18	26.86	0.00	1,328.76	103.65	0.00
1400	40	0	39.68	30.44	0.00	1,368.44	134.09	0.00
1450	45	0	36.88	33.78	0.00	1,405.31	167.87	0.00
1500	45	5	35.37	35.32	1.54	1,440.68	203.19	1.54
1550	45	10	35.37	35.05	4.61	1,476.05	238.24	6.16
1600	45	15	35.37	34.52	7.65	1,511.42	272.76	13.81
1650	45	20	35.37	33.72	10.63	1,546.79	306.48	24.44
1700	45	25	35.37	32.66	13.53	1,582.16	339.14	37.97
1750	45	30	35.37	31.36	16.33	1,617.53	370.50	54.30
1800	45	30	35.37	30.62	17.68	1,652.90	401.12	71.97

Figure 8. The gap between wellbore cylinders and overlap.

Figure 9. Comparision of contact force between FEA and FCTD models.

Figure 10. Comparision of horizontal displacement between FEA and FCTD models.

Figure 11. Deviated well profile used for validation.

Figure 12. Off-bottom torque comparison between measured vs. simulated.

Figure 13. Comparision of torque predictions between FCTD andMSC Adams® contact alogirthms.

Rideout D, Arvani F, Butt S, Fallahi E. 2013. Three-dimensional multi-body bond graph model for vibration control of long shafts-application to oilwell drilling. In: Proceeding. Integrated Modeling and Analysis in Applied Control and Automation; September 2013; Athens, Greece. CAL-TEK SRL. p. 25–27.

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