Abstract

This paper presents the development and validation of a nonlinear delay differential equation (DDE) model for borehole propagation in the inclination plane. Most importantly, built upon a quasi-linear model, the nonlinear approach incorporates information pertaining to the floating stabilizers and bit tilt saturation by formulating a linear complementarity problem. As a result, the outputs of the nonlinear model were in better agreement with the field data when compared with the quasi-linear model. The maximum modeling error of the nonlinear DDE is less than 1 deg over a drilled depth of 600 feet.

Issue Section:
Research Papers
Keywords:
nonlinear DDE model, quasi-linear DDE model, linear complementarity problem, floating stabilizers, bit tilt saturation angle, model validation, modeling
Topics:
Delay differential equations, Drilling, Modeling, Errors, Drill strings, Model validation

References

1.
Koesdwiady
,
A. B.
,
Elferik
,
S.
, and
Karray
,
F.
,
2016
, “
Adaptive Control for Directional Drilling Systems With Delay and Parameter Uncertainty
,”
Robot Intelligence Technology and Applications 4: Results From the 4th International Conference on Robot Intelligence Technology and Applications
Springer
, pp.
123
–
140.
Google Scholar
Crossref
Search ADS
 
2.
Lubinski
,
A.
, and
Woods
,
H.
,
1953
, “
Factors Affecting the Angle of Inclination and Dog-Legging in Rotary Bore Holes
,”
Drilling and Production Practice.
New York
. API-53-222. https://api.semanticscholar.org/CorpusID:129121011 Accessed February 22, 2023.
3.
Millheim
,
K.
,
Jordan
,
S.
, and
Ritter
,
C. J.
,
1978
, “
Bottom-Hole Assembly Analysis Using the Finite-Element Method
,”
J. Pet. Technol.
,
30
(
2
), pp.
265
–
274
.
Google Scholar
Crossref
Search ADS
 
4.
Rafie
,
S.
,
Ho
,
H. S.
, and
Chandra
,
U.
,
1986
, “
Applications of a BHA Analysis Program in Directional Drilling
,”
IADC/SPE Drilling Conference.
Google Scholar
Crossref
Search ADS
 
5.
Fischer
,
F. J.
,
1966
,
Analysis of Drilling String in Curved Bored Holes. SPE5071
.
6.
Neubert
,
M.
, and
Heisig
,
G.
,
1996
, “
Mathematical Description of the Directional Drilling Process and Simulation of Directional Control Algorithm
,”
Z. Angew. Math. Mech.
,
76
, pp.
361
–
362
.
Google Scholar
Crossref
Search ADS
 
8.
Cai
,
Z.
,
Lai
,
X.
,
Wu
,
M.
,
Chen
,
L.
, and
Lu
,
C.
,
2022
, “
Observer-Based Trajectory Control for Directional Drilling Process
,”
Asian J. Control.
,
24
(
1
), pp.
259
–
272
.
Google Scholar
Crossref
Search ADS
 
9.
Zhang
,
C.
,
Zou
,
W.
,
Cheng
,
N.
, and
Gao
,
J.
,
2018
, “
Trajectory Tracking Control for Rotary Steerable Systems Using Interval Type-2 Fuzzy Logic and Reinforcement Learning
,”
J. Franklin Inst.
,
355
(
2
), pp.
803
–
826
.
Google Scholar
Crossref
Search ADS
 
10.
Demirer
,
N.
,
Zalluhoglu
,
U.
,
Marck
,
J.
,
Darbe
,
R.
, and
Morari
,
M.
,
2019
, “
Autonomous Directional Drilling With Rotary Steerable Systems
,”
Proceedings of the American Control Conference
,
July
, pp.
5203
–
5208
.
Google Scholar
Crossref
Search ADS
 
11.
Marck
,
J.
,
Detournay
,
E.
,
Kuesters
,
A.
, and
Wingate
,
J.
,
2014
, “
Analysis of Spiraled-Borehole Data by Use of a Novel Directional-Drilling Model
,”
SPE Drill. Complet.
,
29
(
3
), pp.
267
–
278
.
Google Scholar
Crossref
Search ADS
 
12.
Shakib
,
M. F.
,
Detournay
,
E.
, and
van de Wouw
,
N.
,
2019
, “
Nonlinear Dynamic Modeling and Analysis of Borehole Propagation for Directional Drilling
,”
Int. J. Non. Linear Mech.
,
113
, pp.
178
–
201
.
Google Scholar
Crossref
Search ADS
 
13.
Perneder
,
L.
,
Marck
,
J.
, and
Detournay
,
E.
,
2017
, “
A Model of Planar Borehole Propagation
,”
SIAM J. Appl. Math.
,
77
, pp.
1089
–
1114
.
14.
Franca
,
L. F. P.
,
2010
, “
Drilling Action of Roller-Cone Bits: Modeling and Experimental Validation
,”
ASME J. Energy Resour. Technol.
,
132
(
4
), p.
043101
.
Google Scholar
Crossref
Search ADS
 
15.
Perneder
,
L.
, and
Detournay
,
E.
,
2013
, “
Steady-State Solutions of a Propagating Borehole
,”
Int. J. Solids Struct.
,
50
(
9
), pp.
1226
–
1240
.
Google Scholar
Crossref
Search ADS
 
16.
Kremers
,
N. A. H.
,
Detournay
,
E.
, and
Van De Wouw
,
N.
,
2016
, “
Model-Based Robust Control of Directional Drilling Systems
,”
IEEE Trans. Control Syst. Technol.
,
24
(
1
), pp.
226
–
239
.
Google Scholar
Crossref
Search ADS
 
17.
Ernst
,
S.
,
Pastusek
,
P.
, and
Lutes
,
P.
,
2007
,
Effects of RPM and ROP on PDC Bit Steerability
,
Paper Presented at the SPE/IADC Drilling Conference
, Amsterdam, The Netherlands.
18.
Pastusek
,
P.
,
Brackin
,
V.
, and
Lutes
,
P.
,
2005
,
A Fundamental Model for Prediction of Hole Curvature and Build Rates With Steerable Bottomhole Assemblies
,
Paper Presented at the SPE Annual Technical Conference and Exhibition
,
Dallas, TX
.
19.
Andreas Almqvist
,
2023
,
A Pivoting Algorithm Solving Linear Complementarity Problems—File Exchange—MATLAB Central
. https://www.mathworks.com/matlabcentral/fileexchange/41485-a-pivoting-algorithm-solving-linear-complementarity-problems, Accessed February 19, 2023.
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