A METHOD FOR MODELING DIRECTIONAL BEHAVIOUR
OF BOTTOMHOLE ASSEMBLIES WITH DOWNHOLE
MOTORS
Igor KRULJAC1, Boris KAVEDŽIJA 2 , Ivanka JÜTTNER3
Abstract
Well path control is one of the most important and very complex drilling phase while
drilling deviated or horizontal well. Many authors have put tremendous amount of effort
to developed sophisticated computer programs that would enable engineers to
understand various mechanisms and variables that affect BHA's behaviour. These
include: weight-on-bit, rotary speed, penetration rate, hole inclination and direction,
geology and bits. Vast majority of developed programs are based upon finite element
methods due to their suitability for computer use.
Programs based on finite element methods calculate BHA position and the forces
exerted by the BHA on the wellbore. Bit side force and the bit tilt angle with respect to
the wellbore centerline are two key parameters for predicting directional response of
BHA. One simplification that is being made in all current models is that contacts of BHA
with wellbore are being made in stabilizers.
Analytical method for calculating side forces as well as contact points is being
proposed in this paper. Proposed method is based upon solving statically single or
multiple undefined beam depending on number of contact points between drill string and
wellbore wall. The proposed method provides the capability to evaluate and to adjust the
actual well profile in an interactive, real time mode thus enabling wellpath steering.
1. Introduction
Directional drilling is the ability of directing the bit along some predetermined trajectory
toward a target. Number of techniques has arisen in time to achieve this goal like
whipstocks, jetting bit, mud motors with bent subs just to name a few. Also, as a method
for directing the bit, certain BHAs have been used. Usually those BHAs have a tendency
to build, drop or hold the angle, but how big those tendencies are usually unknown.
Various authors have addressed issue of determining those tendencies creating computer
programs. Most of them were based upon finite element methods [1], [3], [10]
Detailed post-analyses [1] have shown that in many cases incorrect decisions have been
made by drilling personnel due to insufficient knowledge about BHA behaviour and
variables that affect it, thus resulting in premature pulling out and unnecessary change of
BHA configuration. Mistakes could be avoided if the analysis of BHA behaviour were
1
Igor Kruljac, B.Sc. in Petroleum Engineering; Faculty of Mining, Geology & Petroleum Engineering,
Pierottijeva 6,10000 Zagreb, Croatia; tel: ++385 1 4605 149, fax: ++385 1 4836 074; E-mail [email protected]
2
Prof.dr.sc. Boris Kavedžija; Faculty of Mining, Geology & Petroleum Engineering, Pierottijeva 6, 10000
Zagreb, Croatia; tel: ++385 1 4605 463, fax: ++385 1 4836 074; E-mail [email protected]
3
Prof.dr.sc. Ivanka Jüttner; Faculty of Mining, Geology & Petroleum Engineering, Pierottijeva 6,10000
Zagreb, Croatia; tel: ++385 1 4605 453, fax: ++385 1 4836 074; E-mail [email protected]
done in real time and thus not only save money but actually steer the wellpath varying
some technical parameters such as WOB for instance.
Programs based on finite element methods calculate BHA position and the forces exerted
by the BHA on the wellbore. Bit side force and the bit tilt angle with respect to the
wellbore centreline are two key parameters for predicting directional response of BHA.
One simplification that is being made in all current models is that all contacts of BHA
with wellbore are being made in stabilisers and that is not always true.
Proposed method enables engineers to calculate correct points of contact between BHAs
and wellbore and thus enabling them to predict BHA response on various parameters
such as WOB, hole curvature, stiffness of mud motors and drill collars and diameters of
drilling tools used.
2. Considered assembly of the lower part of drilling string
While analysing possible position of drilling string in the hole, special elaboration of
lateral forces that occur in contact points of bent drilling string with wall should take
place. Analysing possible positions of drill string in wellbore (position of contact points
between drill string and wellbore), bending intensity that are product of complex
loadings that are being put on drill string during drilling should not be neglected as well
as reaction forces that occur in contact points [5].
Bending intensity of drilling string depends on many factors some of which, while
drilling with downhole drilling motor, should be pointed out: average deviation angle of
wellbore at considered section of the well, curvature radius of wellbore, weight and
stiffness of drilling string, buoyancy (density) of mud, weight on bit, BHA and clearance
between elements of drilling string and wellbore wall.
In elaboration of mentioned dependence and in order to reduce number of theoretical
stabilizing systems, some simplified assumptions have been made4 :
1. diameter of drilling hole is equal to the diameter of the bit [10]
2. wellbore walls are absolutely hard
This means that lateral actions of the bit does not widen the wellbore but either deflects
wellbore or stays inside current wellbore path. Second assumption simply means that
there is no cutting in at contact points between BHA and wellbore walls.
3. Methods for determining bending moments and lateral forces in contact points
of drilling string with the wall
Bottom part of drilling string is considered as continuous beam on solid supports that
aren’t in a straight line, burdened with concentrated longitudinal force and uniform
(even) transversal force. Looking at the problem in this way, then the beam is statically
single or multiply undefined.
Let’s consider the ways of calculating required parameters on example of triple
undefined beam (Fig. 1).
It can be assumed that bending moment in point A (bit) is equal to zero (MA = 0) and
bending moment in point F (in drill collar area) presents a limiting moment, that is, it is
proportional to stiffness of drill collars and deviation of the hole (MF = Mg = - kZ), since
drill collars here do not touch the wall in one point, they are rather laying at considerable
4
The term stabilizing system is assumed to be a possible position of lower part of
drilling string in the hole characterised by geometrical relation of drilling string elements
an the wall.
length of the wall from point F onward [4]. In this case the following values are
unknown:
e = distance of limiting contact point from upper stabilizer,
MC = bending moment in contact point of stabilizer C and the wall,
ME = bending moment in contact point upper stabilizer and the wall,
while the values:
g = distance between lower stabilizer from the bit,
h = distance between stabilizers (h = LT – g)
are known.
Fig 1
Elaborating parts of beam shown on Fig 1, bending angle of drilling string in point C can
be presented with following equation [11]:
Q ⋅ g 3 ⋅ sin α
M ⋅g
(1)
θCA =
⋅ χ(u g ) + C ⋅ Ψ (u g )
24 ⋅ K
3⋅ K
Values χ(ug) and Ψ(ug) stated in equation 1 present Berry’s functions that have general
form:
3 ⋅ (tgu − u )
3  1
1 
(2)
;
χ(u ) =
; Ψ (u ) =
⋅
−
u3
2 ⋅ u  2 ⋅ u tg 2u 
while in this case:
g
Pdl − Q ⋅ cos(α )
g
2
(3)
ug = ⋅
2
K
This bending angle of the drilling string, assuming that diameter of drilling hole is equal
to diameter of drilling bit (Ddh = Ddl), can be expressed with this equation:
A − C Z⋅g
(4).
θCA =
−
2⋅g
2
Equalling right sides of equations (1) and (2):
A − C Z ⋅ g Q ⋅ g 3 ⋅ sin α
M ⋅g
(5).
−
=
⋅ χ(u g ) + C ⋅ Ψ (u g )
2⋅g
2
24 ⋅ K
3⋅ K
Resulting equation is equation with one unknown (MC). Using identical procedure
bending angle of drilling string in point E could be expressed as:
3
(6).
(θEC = ) E − C − Z ⋅ h = Q ⋅ h ⋅ sin α ⋅ χ(u h ) + M C ⋅ h ⋅ Ψ (u h ) + M E ⋅ h ⋅ Φ (u h )
2⋅h
2
24 ⋅ K
3⋅ K
6⋅K
that is
3
(7).
(θEF = ) d z − E − Z ⋅ e = q ⋅ e ⋅ sin α ⋅ χ(u e ) + M E ⋅ e ⋅ Ψ (u e ) + M F ⋅ e ⋅ Φ(u e )
2⋅e
2
24 ⋅ k
3⋅ k
6⋅k
while in this case Barry's function take form of:
3  1
1 ;
(8)
Φ(u ) = ⋅ 
−

u  sin 2u 2 ⋅ u 
g+h
Pdl − Q ⋅ 
 ⋅ cos α
h
 2 
(9)
uh = ⋅
2
K
e
Pdl − Q ⋅ (g + h ) ⋅ cos α − q ⋅ ⋅ cos α
e
2
(10)
ue = ⋅
2
k
Q = Q'⋅U ;
q = q′ ⋅ U
Berry’s functions χ(u); Ψ(u) and Φ(u) characterise influence of axial force on buckling
line of elaborated beam. Presented equations (5), (6) and (7) present system of three
equations with three unknowns out of which unknowns MC, ME and e can be calculated.
After calculating all unknowns, determining of reactionary forces of the wall at contact
points can take place. For elaborated case (Fig 1) these reactions will be:
Q ⋅ g ⋅ sin α M C
(11).
RA =
+
2
g
Q ⋅ (g + h ) ⋅ sin α M C M E − M C
(12).
RC =
−
+
2
g
h
Q ⋅ h ⋅ sin α + q ⋅ e ⋅ sin α M C − M E M F − M E
(13).
RE =
+
+
2
h
e
In the way presented, after forming adequate system of equations, all unknowns for all
theoretically possible cases of positioning bottom part of drilling string in the bore hole
can be determined.
In the calculations above, calculation of triple undefined beam was presented i.e. BHA
has only three contacts with borehole wall besides contact of bit with the bottom.
Unfortunately this is not the only possible distribution of contact points. The BHA can
be in contact with walls in one, two three and even more points (Fig 2) and not all
contacts must be made in stabilizers. Some contacts may occur in drill collar due to its
buckling under longitudinal load exerted on the bit. Until now usual practice is to
assume that all contacts are being made in stabilizers, and thus, having three points one
can construct a circle and try to predict the toolface direction i.e. direction of the
resultant force vector. As mentioned, distribution of contact points could be very
different [5].
Fig 2.
Equations 5, 6 and 7 present transcendental functions and thus they can be solved only
using one of numerical solution techniques derived from Taylor series function
approximation [2] such as Newton-Raphson, Secant or False-Position method. NewtonRaphson method has been chosen due to its simplicity and its fast convergence toward
numerical solution.
4. Computer program development
Development of BHA and wellpath prediction/analyzing programs for a long time has
been a privilege of companies that could spend great financial resources for research and
development and required great programming skills. Today, advancements in computer
hardware as well as software have created suitable environment to run programs on
smaller, more affordable computers and thus enabling wider population of scientists to
develop and test software.
All programming has been done using Mathematica 4.1. by Wolfram Research Inc.
Mathematica is a fully integrated environment for technical computing, due to its
simplicity and powerful built in calculating algorithms thus enables one to concentrate
more on creating and testing a mathematical models rather than waste time to develop
and optimize algorithms for various kinds of numeric and symbolic calculations.. In
engineering, Mathematica has become a standard tool for both development and
production of various products. One of the unique features of Mathematica is possibility
of performing algebraic as well as numeric computations.
4.1 Input data preparation
Each part of BHA is represented with several properties such as: length, diameter,
weight , stiffness as well as position number i.e. number that represents the position of
certain part of BHA counting from the bit. All the data is divided in separate list matrices
d= A, T, C, T, Ef, t, t
such as:
representing list of diameters of various parts of BHA assemblies. In the same manner
lists for length (l), weight (q) and stifness (K) are being made. Out of list of lengths
cummulative list of lengths (lt) is being made representing distance from the bit to
certain point in BHA.
For instance for situation presented on Fig 1 input data would take the following form:
d= A,C, El, F ;l= 0,g,h,e ;q= 0,qT,qT,qTs;
K= 0,kT,kT,kTs;
and list of cumulative lengths would be:
lt= 0, g, g+ h, e+g+ h
Other input data needed for calculation is weight on bit (WOB).
4.2 Barry's function module
As mentioned above Barry's functions characterize influence of axial force on buckling
line of elaborated beam. The argument of Barry's function u depends on WOB and
position of contact point and distance to previous contact point (Equations 3, 9, 10).
Following program listing represents function for calculating longitudinal force needed
for calculating Barry's function argument.
F1 i1_
, i2_ := ModuleP, t, g , P = 0;
t=If
If i1< i2, i1, i2 ;
g= If i1< i2, i2, i1 ;
Fori= 1, iŁ t, i++, P = P+ q i l i ;
l i
Fori= t+ 1, i Ł g, i++, P = P+ q i
2
P
;
This programm takes as arguments positions of contact points between which the
longitudinal force is being calculated. It would be more clear if we now take a look at a
function for calculating Barry's function argument and compare it with equations 3, 9,
10:
u i1_
, i2_
, ku_ :=
Abslt i1 - lt i2
2
P- F1 i1, i2 Cosa
ku
Function u takes as arguments boundary points between the desired contact points as
well as stiffness in interval of interest.
Barry's function program listing is now very simple and is as follows:
c u_ := 3
F
Tanu - u
u3
Y u_ :=
3 1
1
2u 2u Tan2u
u_ :=
3
1
1
u Sin2u 2u
4.3 Bending angle
From equations 4, 5, 6 and 7 it is obvious that to develop equations for bending moments
it is needed to calculate that angle from two perspectives, one of which is geometric
conditions in wellbore. To simplify this, introduction of one simple argument that can
take value of –1 or 1 can resolve the problem of stating the bending angle with single
function:
2fi- d i1 - d i2 - ZAbslt i1 - lt i2
AnGi1_
,s1_
,i2_
,fi_:= - s1 2Abs
lt i1 - lt i2
2
The AnG function takes four arguments and uses one global variable Z that represents
wellbore curvature. Argument i1 is the position of point in which the angle is calculated,
i2 represents the direction of angle, i.e. if we whish to calculate angle θCA for situation on
Fig 1 argument i1 would have value of 2 (representing the point of first contact above
bit) and i2 will have the value of 1 (representing bit). In the situation of Fig 1 argument
of s1 would have value of –1. In general, argument s1 will have value of –1 if the contact
point i1 is bellow contact point i2, and 1 if it is above contact point i2. Argument fi
presents diameter of concern, depending on distribution of contact points. For instance,
if the contact points are on the opposite sides of wellbore, fi will have value of diameter
of wellbore. If the contact points are on the same side, fi will have the value of greater
diameter between diameter at point i1 and i2.
A procedure explained under chapter 4.2 presents subroutines for a module that
calculates bending angle depending on bending moments (Equation 1 and right side of
equations 6 and 7). The Mkut module has a three input arguments; i1 the point for which
the angle is being calculated, i2 the direction of angle and ku the stiffness in the region
limited by points i1 and i2.
Mkuti1_
,i2_
,ku_:= Moduleub, p , p= Abslt i1 - lt i2 ;ub= u i1, i2,ku;
q i1 Sina p3
M i2 p
M i1 p
c ub +
F ub +
Y ub
24ku
6ku
3ku
As can be seen, module first calls the subroutine u to calculate Barry's function argument
and then as a result gives a bending moment.
4.4 Resulting equation
Module Eq is a final module that incorporates all of the above modules and as a result
gives a equations needed for calculating bending moments in contact points (Equations
5, 6, 7)
Eqi1_Integer
, s1_Integer
, i2_Integer
, ku_
, fi_ := AnGi1, s1, i2, fi == Mkuti1, i2, ku
To test this module we can use a sample data for Fig 1 presented above. For bending
angle θCA we must input:
Eq 2, - 1, 1, K 2 , d 1
And result will give:
A- C
1
qTAbsg 3 Sina c u 2, 1, kT
MbAbsg Y u 2, 1, kT
ZAbsg ==
+
2Absg 2
24kT
3kT
Which is exactly the same as equation 5.
For angles θEC and θEF input would be:
Eq 3, 1, 2, K 3 , d 2
Eq 3, 1, 4, K 4 , d 3
and results would give:
-
C- El 1
ZAbsh ==
2Absh - 2
MbAbsh F u 3, 2, kT
qTAbsh 3 Sina c u 3, 2, kT
McAbsh Y u 3, 2, kT
+
+
24kT
6kT
3kT
and
-
El- F
1
ZAbse ==
2Abse
2
MdAbse F u 3, 4, kTs
qTAbse 3 Sina c u 3, 4, kTs
McAbse Y u 3, 4, kTs
+
+
6kTs
24kTs
3kTs
respectively.
It is obvious that the resulting equalities are same as equations 6 and 7.
Using this module it is easy to formulate needed equalities for any possible stabilizing
system [5] and then solve system of n equations for n unknowns where n represents the
number of contact points.
Solving such system would give us the value of bending moments in contact points as
well as distances between contact points. According to the results we can determine
whether the assumed stabilizing system is the correct one. If the bending moment in
some contact points turns out to be 0 then there is no contact in that points and we must
assume different stabilizing system and run the calculations again. Each change in
stabilizing system will require from us to rearrange input data. For instance, if we
assume that there is a contact point somewhere between the bit and first stabilizer, then
we mast increase the size of input lists d, l, q and K for one place representing the values
for new contact points, etc.
Once we have found the correct stabilizing system and got the resulting moment in
contact points, reaction forces i.e. side forces in contact points can be calculated.
5. Influence of change of weight on bit on bending moments and lateral forces
To examine influence of varying WOB on bending moments and reaction forces.
Following BHA was considered:
- drilling bit:
- downhole drilling motor
- stabilizers
- drill collar
diameter
Ddl = 0,1905 m (7 1/2’’)
diameter
DT = 0,164 m (6 1/2’’)
length
LT = 25,7 m
weight per meter Q’ = 128,785 kg/m
stiffness
K = 5,66 MNm2
diameter
C= E = 0,186 m (E = DT ⇒ none)
distance of first stabilizer from the bit
g = 13 m,
distance between stabilizers
h = 12,725 m
outer diameter dz = 0,152 m
inner diameter du = 0,0508 m
stiffness
k = 5.4 MNm2
weight per meter q’ = 64,1 kg/m
beside these, following parameters were accepted:
- bore diameter
Db = Ddl = 0,1905 m (7 1/2’’)
- average deviation angle
α = 0,3491 rad
- curvature of the hole axis
Z = 5,236 x 10-4 rad
- mud density
ρ = 1000 kg/m3
Weight on bit was gradualy changed between 49 to 196 kN. During the change of WOB
stabilization system was not changed (Fig 3).
G
F
f
E
e
D
d
c
C
b
B
a
A
Fig 3
Results of calculation are presented in Fig 4 and 5. From the presented data following
conclusions can be made:
- Bending moments and lateral forces in BHA do not increase together with increase of
WOB although such increase was anticipated. Possible explanation for this is that
increase of WOB doesn't increase influence of curvature and gravitational forces.
This was confirmed with calculation of bending angles of individual sections of
BHA. Those angles decrease from 10' to 1' due to increase of WOB.
- Stress due to bending moment and lateral forces have lowest values on BHA section
between bit and first stabilizer. Changes of those parameters are also negligible.
- Increase of WOB does not mean that the deviation tendency would also increase.
This behavior can be seen in Fig 4 as decrease in reaction forces while the WOB
increases from 98,1 to 196 kN. In that particular instance, 100 % increase of WOB
produced 10% decrease in directional force (reaction in point A decreased for 0,2
kN).
- In considered boundaries, both lateral forces and bending moments have the largest
values in points where the stabilizers have contact with wellbore.
- Greatest change in value is noticed at lower stabilizer (almost 150%).
3
8,00
RC
2
7,00
MD
1
RE
6,00
5,00
0
RF
M (kNm)
Rx (kN)
0
4,00
RD
50
100
150
200
250
MB
-1
MF
3,00
RB
-2
2,00
-3
ME
1,00
RA
MC
-4
0,00
0
50
100
150
WOB (kN)
Fig 4
200
250
-5
WOB (kN)
Fig 5
6. Conclusion
Elaborated method allows precise determination of the wellbore bending line of the
BHA, that is it can give accurate description of position of BHA in wellbore.
Proposed method can be used for very wide analysis of bending stress for all theoretical
stabilizing systems.
Knowing geological anisotrophy, proposed method can be used to predict direction in
which the drill bit will advance, and therfore, method can be used to steer the well.
Symbols:
A
= Ddl = drilling bit diameter (drilling bit diameter at point A)
C
= lower stabilizer diameter (stabilizer diameter at point C)
E
= upper stabilizer diameter (stabilizer diameter at point E)
K
= stiffness of downhole drilling motor
LT
= length of downhole drilling motor
M
= bending moment at considered point
= bending moment in contact point of stabilizer C and the wall,
MC
ME
= bending moment in contact point upper stabilizer and the wall
Mg
= limiting bending moment
Pdl
= weight on bit
Q
= weight per meter of downhole drilling motor in mud
Q’
= weight per meter of downhole drilling motor
U
= buoyancy factor
dZ
= diameter of drill collar
k
= stiffness of drill collar
q
q’
e
g
h
α
θ
= weight per meter of drill collar in mud
= weight per meter of drill collar
= distance of limiting contact point from upper stabilizer,
= distance between lower stabilizer from the bit,
= distance between stabilizers (h = LT – g)
= deviation angle
= bending angle at considered point
χ(u); Ψ(u) and Φ(u) are Berry’s functions
References:
[1]. Brett, J.F.; Gray, J.A.; Bell, R.K.; Dunbar, M. E.: "A Method of Modeling the
Directional Behavior of Bottomhole Assemblies Including Those With Bent Subs
and Downhole Motors", SPE 14767, IADC/SPE Drilling Conference, Dallas,
Texas, February 10-12, 1986.
[2]. Chapra, S.C.; Canale, R.P.: "Numerical Methods for Engineers: With
Programming and Software Applications", 3rd ed., McGraw-Hill, Boston, 1998.
[3]. Dahl, T.; Schmalhorst, B.: "A New Bottomhole Assebly Analysis Program for the
Prediction of th Borehole Path Based on Sophisticated Static Algorithm", SPE
21948, SPE/IADC Drilling Conference, Amsterdam, Netherlands, 11-14 March,
1991.
[4]. Gulizade, M.P.; Halimbekov B.M.; Kaufman, L.J.; Zelmanovič, G.M.:
"Issledovanje niza burilnoj kolonni pri turbinnom burenii naklonih skvažin s
krivim perevodnikom", Trudi AINH XXV, Baku, 1967.
[5]. Kavedžija, B.: "Parametry stabilizacji zestawu przewodu przy wierceniu otworow
kierunkowych turbowiertami", PhD disertation, AGH Krakow, Krakow, Poland
1978.
[6]. Konstantakopoulos, I.K.; Stamataki, S.K.: "A Computerized Method for the RealTime Well Path Monitoring and Placement in 3-D Space", SPE 37099,
International Conference on Horizontal Well Technology, Calgary, Alberta,
Canada, 18-20 November, 1996.
[7]. Larson, P.A.; Azar, J.J.: "Three-dimensional, Quasi-static, Drill Ahead BHA
Model for Wellbore Trajectory Prediction and Control", SPE 23530, May 3,
1991.
[8]. Lubinski, A.: "Maximum Permissible Dog-Legs in Rotary Boreholes, JPT
2/1961.
[9]. Millheim, K.K.; Apostal, M.C.: "The Effect of Bottom-Hole Assembly Dynamics
on the Trajectory of a Bit", SPE 9222, 55th Annual Fall Technical Conference
and Exhibition of the SPE of AIME, Dallas, Texas, September 21-24, 1980.
[10]. Millheim, K.K.; Gubler, F.H.; Zaremba, H.B.: "Evaluating and Plannig
Directional Wells Utilizing Post Analysis Techniques and a Three Dimensional
Bottom Hole Assembly Program"; SPE 8339, 54th Annual Fall Technical
Conference and Exhibition of SPE of AIME, Las Vegas, Nevada, September 2326, 1979.
[11]. Timoshenko S.P.: "Theory of Elastic Stability", New York, Toronto, London,
1961.