## ABSTRACT

The bending portion of a multiple-stabilizer bottom hole assembly can be considered as a string of several simply-supported beams loaded by axial compressive forces together with transverse ones. By the beam-column theory, a system of "equations of three moments" can be obtained. The essential aim to design a bottom hole assembly with special requirements is to produce a desirable reactive side force on the bit from the formation. While there are numerous parameters that would influence the value of this force, the effect of each of them can be evaluated from the position it occupies in those equations. This paper is particularly focused on the analysis of pendulum assemblies with two stabilizers and affirms their obvious advantages over those with only one stabilizer or no stabilizer.

## INTRODUCTION

When the angle of inclination or a borehole drilled in crooked regions is confined within very small value, simple pendulum assembly without any stabilizer may be used, but the weight on the bit should be confined to very small values, too. Many investigators had tried various procedures to promote the effectiveness of the pendulum by using heavy metal collars,^{(1)} or by installing one or more stabilizers at proper places on the drill collar string.^{(2,3)} When the borehole is inclined and the bit weight is applied, the lower portion of a multi-stabilized bottom hole assembly will be bended in a special manner. Indeed, quite a \ill\ methods to solve this problem,^{(4,5,6,7,8)} had been published, still, it is presented here another one by applying the beam-column theory. An assembly is, then, cut at tne stabilized junctures into several simply-supported beams submitted to axial compression in company with lateral loads.

The parameters which would influence the force distribution and deflecting configuration of a multiple stabilizer bottom hole assembly include: (see fig. 2)

D

_{B}= diameter or bit, cm.P

_{B}= weight on bit, kg.α = angle of inclination, deg.

D

_{ci}= outside diameter of i-th collar, cm.W

_{i}= effective unit weight of collar, kg/cm.γ = specific gravity of drilling fluid, kg/cm

^{3}.I

_{i}= moment of inertia, cm^{4}.E = modulus of elasticity, kg/cm

^{2}.EI

_{i}= flexural rigidity, kg/cm^{2}.D

_{sj}= diameter of j-th stabilizer, cm.l

_{i}= length of i-th drill collar section, cm. (in general, i = j+1)e

_{j}= $12DB-Dsj$ = radial clearance of stabilizer, cm.

It will be seen later in this paper, the various roles played by all of the above parameters can be exposed by examining the positions they seat in a system or "equations of three moments" derived by the beam-column theory. To do a specific drilling job successfully, a good designed bottom hole assembly should create an appropriate side force on the bit, so that, a resultant force including a component from the formation will be formed to drive the bit to the proper direction.

## SIDE FORCE ON BIT

Fig. 1 show how a side force R_{B} is created at the bit on the lowest section l_{1} of a multiple stabilizer bottom hole assembly of pendulum type. This side force is shown directing upward because it is a reactive force of a simply-supported beam l_{1} with the bit as the lower end. The transverse loads acting on the beam include a bending moment M_{1} at the upper end S_{1}, besides the uniformed distributed load q_{1}

By the static equilibrium condition, the magnitude of R_{B} may be computed by the following equation:

Where P is the average value of the axial compressive load and may be taken as:

Equation (1) may be regarded as a criterion to appraise the effectiveness of different pendulum designs. (see table 1) But, right now, the internal bending moment M_{1} at stabilizer S_{1} is still an unknown quantity. It is the purpose of this paper to derive the formulas to calculate M_{1} and other moments at the supporting stabilizers first, and then to see how to choose those optional parameters to render R_{B} as large as possible.

## ASSEMBLY WITH TWO STABILIZERS

As shown in Fig. 2, a pendulum assembly with two stabilizers is taken as a sample to illustrate how the beam-column theory can be utilized to solve the bottom hole assembly problems. A pendulum assembly is characterized by its long heavy collar section l_{1} above the bit. After cutting off at the stabilizers S_{1} and S_{2}, the bending portion of the assembly would be divided into three spans, l_{1}l_{2}l_{3}, with two internal bending moments M_{1}, M_{2} appeared as end couples. Each span may be considered as a simply-supported beam. Two lengths of them, l_{1} and l_{2}, can be arbitrarily selected and therefore are taken as know quantities in the derivation, the third one l_{3}, its upper end being the point of tangency T is, however, an unknown, which, following and M_{1} and M_{2}, represents the third unknown to be determined.

Fig. 3 shows the acting forces and angular deflections at the ends of three beams, or rather beam-columns, putting horizontally. In elementary mechanics of materials, we usually apply the method of "Three Moments Equations" to solve a continuous beam problem. Now, each of these beams in Fig 3 is loaded axially besides the uniformly distributed forces q_{i} = w_{i}sinα: (i = 1, 2, 3.) the same technique can, still, be utilized to establish the corresponding equations by satisfying the following conditions:

- (3)$Condition\u2009\u2009of\u2009\u2009continuity\u2009\u2009at\u2009\u2009S1\theta 1\'=\u2212\theta 1"$
- (4)$Condition\u2009\u2009of\u2009\u2009continuity\u2009\u2009at\u2009\u2009S2\theta 2\'=\u2212\theta 2"$
- (5)$Zero\u2009\u2009slope\u2009\u2009at\u2009\u2009point\u2009\u2009of\u2009\u2009tangency\u2009\u2009T\u2009\u2009\theta \tau =0$

The formulas expressing the above angles of rotation can be found from Timoshenkos book^{(9)} as:

It should be noted, here, that three trancendental functions. X_{i}, V_{i}, W_{i}, representing the effects of axial compressive forces on the transverse bending deflections, do appear in the above five equations and are expressed by the following formulas:

Where

and P_{i} = average axial compressive force on i-th section. Substituting eqs. (6), (7), (8), (9), (10), into eqs. (3), (4), (5), and simplifying, then the following important equations to determine the unknown values of M_{1}, M_{2}, and M_{3} are obtained:

(Assume E_{1} = E_{2} = E_{3} = E = 2.1 × 10^{6} kg/cm^{2})

In solving the above simultaneous equations, as the unknown length l_{3} is entangled in the trancendental functions X_{i}, V_{i}, and W_{i}, a reasonable value of l_{3} is assume at the beginning, and then use the iteration method to get the final answers.

Now, it is the time to examine eqs. (15), (16), (17) to fix up the related parameters in order to design a most effective pendulun assembly with two stabillizers.

In general, at first, a long and heavy enough section, is chosen for l_{1}, then the problem is, with known values of l_{1}, W_{1}, I_{1}, α, P_{B}, D_{B}, how to choose those parameters such as l_{1}, D_{c2}, e_{1}, e_{2}, etc, so as to make the value of R_{B} in E_{q}(1) as large as possible. E_{q}(1) indicates that a large positive value of M_{1}, is profitable. Careful examination will reveal that for a larger M_{1}, the following choices are preferable:

smaller value of l

_{2}thinner section of L

_{2}larger value of e

_{1}e

_{2}as small as possible.

Among these preferences, the third one is harmful, as it will produce a larger negative part in E_{q}(1) ( i.e $Pel1$. If both a large positive M_{1} and a small e_{1} are required, assemblies with three stabilizers may be used.

Two illustrative examples are given below, shotting that there are two options, one is featured with large value of e_{1} the other with very slender section of l_{2}. The solutions give the main numerical results only.

Example 1:

solution: The main numerical results are obtained as:

P

_{1}= 15000 − $12$ 1.84 × 1800 = 13344kge

_{1}= 1st stabilizer radial clearance = $12$(24.4−21.4)=1.5cme

_{2}= 2nd stabilizer radial clearance = $12$(24.4−24.4)=0e

_{3}= radial clearance atT = $12$ (24.4-17.8) = 3.3cm

M

_{1}= 21100kg-cmM

_{2}= -66400kg-cml

_{3}= 2300cmR

_{B}= $12$ x 1.84 x sin3° − $211001800$ − $13344x1.51800$=87.3kg

Example 2: Given l_{1} = 1850cm,

W_{1}=W_{2}=1.34kg/cm (7" collar in mud with γ=1.2),

EI_{1}=EI_{2}=10 x 10^{9}kg/cm^{2},

l_{2}=1000cm,

W_{2}=0.225kg/cm (5" pipe),

EI_{2}=1.25 x 10^{9}kg-cm^{2},

D_{B}=21.5cm(8$12$" bit),

D_{c1}D_{c3}=17.8cm,

D_{c2}=12.7cm,

D_{s1}=D_{s2}=21.3cm,

α = 2°

P_{B} = 12000 kg

Solution:The main numerical results are obtained as:

P

_{1}= 12000 − $12$ 1.34 × 1850 = 10760 kge

_{1}= e_{2}= 0.1 cme

_{3}= 1.85 cmM

_{1}= 1200 kg - cmM

_{2}= - 3000kg-cml

_{3}= 1650 cmR

_{B}= $12$ x 1.34 x sin2° 1850−$12001850$ − $10760x0.11850$=\ill\

## PENDULUM ASSEMBLY WITHOUT ANY STABILIZER

Without any stabilizer, it is the simplest bottom hole assembly which is naturally welcomed by all drillers and under certain conditions is reliable too. But in crooked area this will result an unfavorable conditon as only very light weight can be exerted on the bit.

As shown in Fig 4 there exists only one unknown quantity which is the length l between the bit B and the point of \ill\ T and can be easily determined by the following equation:

Where e = radial clearance of collar = $12DB-Dc$

The reactive side force R_{B}, in the present case, then consists of two items as:

From the above relations, it can be seen that as the value of P_{B} is small, X is nearly equal to one, l is large and consequently R_{B} will be fairly large. So this is a good deviation correction tool. But when P_{B} is large, X will grow to a value much greater than one, and l will become short and at the same time, the negative part of eq (19) grows larger at even greater rate and consequently, R_{B} will be reduces to a very small value. Such a situation will be illustrated apparently in Example 4.

Example 3

Solution: The main numerical results are obtained as:

Example 4:

Solution: Table 1 gives the main numerical results.

## PENDUIUM ASSEMBLY WITH ONE STABLIZER

Woods and Lubinski pointed out in their paper (2) that more weight may be carried on the bit by using one stabilizer properly placed on the assembly. As shown in Fig 5, the pendulum length l_{1}. can be fixed to value much longer than 1 in Fig 4. Besides, the negative item in Eq (19) may be materially eliminated. Some profits are really obtained, but, on the other hand, a negative M_{1} of large value is induced at S_{1} which will nullify the profits to a certain extent.

It is very easy, now, to derive the required equations to determine the unknown quantities M_{1} and l_{3} as:

Example 5: Given D_{B} = 24.4cm, D_{c1} = 20.3cm,

D_{c2} = 17.8cm, D_{s1} = 24.4cm,

W_{1} = 1.84 kg/cm (γ = 1.2),

W_{2} = 1.34 kg/cm l_{1} = 1800cm

α=3° P_{B}=15000

Solution: The main numerical results are obtained as:

The above calculated value of R_{B} is even smaller than that of Example 3. It indicates that the pre-selected I_{1} is \ill\ short, and also explains the harmful effect of M_{1},

Example 6: Given D_{B} = 21.5cm, D_{c1}=D_{c2}=17.8cm

W_{1} = W_{1} = 1.34 kg/cm (γ = 1.2),

D_{s1} = 21.3 cm, l_{1} = 1850cm,

α=2° P_{B}=12000kg

Solution: The main numerical results are obtained as:

## COMPARISON OF SIDE FORCES

It is stated that the magnitude of reactive side force at the bit represents the effectiveness to prevent crookedness of different designs of pendulum assemblies. The numerical results obtained in the above six illustrative examples are compilled in Table 2. The influencing parameters have not been optimized, and there are still rooms for improvement, however, the advantages of those assemblies with two stabilizers can be observed clearly. It should be pointed out here once more that there exist two options of combination one with small D_{s1} and stiff l_{2} and the other with large D_{s2} and flexible l_{2}.

## FIELD VERIFICATIONS

Field testings had been performed on eight wells with depths ranging from 1200m to 1800m in crooked regions of different classes of severity. All of the experimental results show very close agreement with the theoretical predictions and may be explained by the following typical examples:

Example 1. (A) Well number: No. Tong 13

Exp. Results:

Weight on bit: 16000kg to 18000kg.

(B) Offset well for reference. No. Tong 10 used 7" collar pendulum without any stabilizer. Max. Weight on bit, 12000kg reached 15° at depth 1300m

Example 2. (A) Well number: No, Jing 113

Exp. results:

Weight on bit 12000kg to 14000kg

(B) offset wells: average of several wells. $812$ bit use 7" pendulum (no stabilizer). Weight on bit, \ill\ to 14000kg. α_{1} doubled as much at same depths.

Example 3. (A) well number: Ko. Wangdong 12-7

Exp. results

(B) offset well: No. Wangdong 9-7. Use 8" pendulum without any stabilizer. Wt on lit, 3000kg about the same as (A).

## CONCLUSIONS

According to beam-column theory, a multiple stability bottom hole assembly can be treated similarly as a continuous beam as in elementary mechanies of materials. A systems of equations of three moments can be obtained to find simple and accurate solution.

The problem of designing an effective bottom hole assembly in general is essentially to produce a desirable side force on the bit by proper selections of related influencing parameters. As the role played by such parameters and be judged by the positions they occupied in those equation of three moments, the designer can manage his job straightforward to get the optimum results without much troubles.

Particular attention is focused on the analysis of pendulum assemblies with two stabilizer. Theoretical analysis shows that with certain special arrangement, they are much more effective to resist crookedness than those with only one stabilizer or no stabilizer. The theoretical prediction:, confirmed by field testings. Pendulum assemblies may be further improved by using of three stabilizers.

The theory proposed in this paper is also applicable to assemblies in building-up, maintaining, or dropping the angle of inclination.