AI-driven triple-optimization based on ant colony optimization for minimizing deformation energy in three-dimensional analysis of bottom-hole assemblies

doi:10.1007/s10483-026-3399-9

Abstract

Abstract:

An accurate analysis of the three-dimensional (3D) deformed configuration of a bottom-hole assembly (BHA) is critical for predicting and controlling well paths in directional drilling. Among the various numerical approaches, the weighted residual method exhibits superior performance owing to its semi-analytical nature, enabling high accuracy in handling diverse boundary conditions. In previous studies, the weighted residual method has been coupled with a dual optimization process to determine the 3D deformation and tangency point of the BHA. However, its applicability is limited by the conventional treatment of contact interactions between the BHA and wellbore wall. Specifically, stabilizer-wellbore contacts are regarded as predefined boundary conditions, rather than solving these contact positions as unknown variables consistent with actual downhole conditions. This limitation reduces modeling fidelity in complex downhole environments. To address this limitation, this study enhances the optimization-based weighted residual method by introducing ant colony optimization (ACO) to solve the 3D contact problem. In the proposed framework, the bending energy of the deformed BHA is conceptualized as “food”, while the contact positions and orientations of stabilizers are assigned a measure of “taste”. Through this metaphor, the ACO algorithm employs artificial “ants” to explore the optimal stabilizer locations and orientations that minimize the BHA bending energy, thereby refining the computed 3D deformation. The simulation results demonstrate that integrating ACO into the previously established dual optimization framework enables the effective determination of the contact configuration between the BHA and wellbore wall. As a result, the overall accuracy of the 3D BHA deformation analysis is significantly improved. In one representative case study, the bending energy of the BHA is reduced by 55.6% compared with that obtained from the original dual-optimization method.

Key words: weighted residual method, directional drilling, bottom-hole assembly (BHA), artificial intelligence (AI)-driven triple optimization, ant colony optimization (ACO)

2010 MSC Number:

O232

Qinfeng DI, Dakun LUO, Heyuan YANG, Tianxin LI, Wenchang WANG, Feng CHEN, H. ZHANG. AI-driven triple-optimization based on ant colony optimization for minimizing deformation energy in three-dimensional analysis of bottom-hole assemblies. Applied Mathematics and Mechanics (English Edition), 2026, 47(6): 1417-1432.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3399-9

https://www.amm.shu.edu.cn/EN/Y2026/V47/I6/1417

Figures/Tables 11

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Table 1

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Table 2

Calculation results of the neighborhood enumeration method"

$e s1$ /mm	$δ s1$ /(°)	$e s2$ /mm	$δ s2$ /(°)	Deformation energy/J
1.35	109	1.85	132	637.624 361 0
1.35	109	1.85	133	637.632 619 4
1.35	109	1.85	134	637.637 574 4
1.35	110	1.85	132	637.727 590 6
1.35	110	1.85	133	637.735 849 0
1.35	110	1.85	134	637.741 079 2
1.35	111	1.85	132	637.599 310 7
1.35	111	1.85	133	637.847 061 6
1.35	111	1.85	134	637.851 741 3
1.35	109	1.95	132	637.504 064 2
1.35	109	1.95	133	637.504 890 1
1.35	109	1.95	134	637.513 423 7
1.35	110	1.95	132	637.606 467 9
1.35	110	1.95	133	637.607 569 0
1.35	110	1.95	134	637.616 653 2
1.35	111	1.95	132	637.718 231 1
1.35	111	1.95	133	637.719 332 2
1.35	111	1.95	134	637.728 141 1
1.45	109	1.85	132	635.807 521 2
1.45	109	1.85	133	635.817 431 2
1.45	109	1.85	134	635.816 054 8
1.45	110	1.85	132	635.918 458 5
1.45	110	1.85	133	635.928 643 8
1.45	110	1.85	134	635.927 267 4
1.45	111	1.85	132	636.037 378 9
1.45	111	1.85	133	636.047 564 3
1.45	111	1.85	134	636.047 289 0
1.45	109	1.95	132	635.800 363 9
1.45	109	1.95	133	635.801 465 0
1.45	109	1.95	134	635.808 622 3
1.45	110	1.95	132	635.690 527 7
1.45	110	1.95	133	635.689 701 9
1.45	110	1.95	134	635.697 960 2
1.45	111	1.95	132	635.919 834 9
1.45	111	1.95	133	635.920 936 0
1.45	111	1.95	134	635.928 643 8

Table 2

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Variable	Definition	Unit
U_i	Displacement of ith segment of BHA in x-direction	m
V_i	Displacement of ith segment of BHA in y-direction	m
l	Curvilinear coordinate along axis of BHA	m
L_i	Length of ith segment	m
E_i	Modulus of elasticity of ith segment	Pa
I_i	Inertia moment of cross-section for ith segment	m⁴
M_ti	Torque applied to ith segment	N · m
q_wi	Distributed weight per unit length of ith segment	N/m
α_i	Inclination angle at ith segment location	rad
B_i	Axial force along z-direction acting on lower end of ith segment	N
F_xi(F_yi)	Internal force of ith segment in x(y)-direction	N
N_j	Contact force between jth stabilizer and wellbore wall	N
f_sw	Friction coefficient between stabilizer and wellbore wall	—
f_a(f_t)	Equivalent axial (tangential) friction coefficient at stabilizer-wellbore interface	—
v	Rate of penetration of drill bit	m/h
Ω	Self-rotation angular velocity of BHA	r/min
D_oj	Outer diameter of jth stabilizer	m