Designing -Well Paths To Reduce Drag and Torque M.C. Sheppard, * SPE, Schlumberger Cambridge Research C. Wick, Anadrill-Schlumberger T. Burgess, SPE, Anadrill-Schlumberger

Summary. A deviated well with an undersection trajectory (Le., a trajectory lying below the conventional tangent section and constantly building to target) can exhibit lower drag and torque than a conventional well geometry in certain circumstances. The influence of well geometry on drag and torque is discussed, making use of the results of the theoretical model. Although an undersection well may have a reduced overall drag, the side forces in the drill collars will be increased. This can lead to a greater danger of sticking in the bottornhole assembly (BHA). Furthermore, because of the enhanced side forces near the bit, an undersection well may also exhibit greater torque. The relative merits of constant-build and catenary trajectories are discussed.

Introduction Drag and torque loss are phenomena particularly associated with deviated wells. Drag is experienced as a load in addition to the rotat­ing string weight when tripping out of the hole. Torque is lost from the rotating drillstring during drilling so that the power available at the bit for rock destruction is considerably lower than that applied at the rotary table. Problems of drag and torque loss are normally found to occur together and can be particularly marked in long­reach wells.

There are a variety of sources of drag and torque loss, including differential sticking, key seating, hole instabilities, poor hole clean­ing, and the frictional interaction associated with the side forces along the drillstring. The side-force profile is essentially determined by well geometry, so we can broadly separate drag problems into those caused by poor hole condition or inappropriate mud weight, and those associated with the well path.

In this paper, we discuss the extent to which the drag and torque loss associated with the frictional interaction of the drillstring with the wellbore can be reduced by a suitable choice of well path. It has been suggested I that a catenary well path will reduce torque and drag. We will analyze this proposal and investigate the efficacy of less exotic solutions.

Friction Model for Drag and Torque Loss To estimate the drag and torque loss associated with different well geometries, a "frictional model" will be used that is based on the assumption that the drag force on the drill string at any location is proportional to the side force acting there. The coefficient of propor­tionality, K (not necessarily constant), thus appears in the model as a sliding-friction coefficient. The resultant frictional force acts against the motion of the drillstring and leads to drag when trip­ping out and torque loss when rotating.

The tension and drag profiles for a well are derived in Appendix A. The well trajectory may be parameterized by the distance s along the well path from the bit (so that s=O is the bottom of the well).

It is convenient to work in terms of an effective tension, ais), which is the sum of the true tension, a(s), and the product of the mud pressure acting at s and the cross-sectional area of the pipe; i.e.,

ae(s) =a(s) +p(s)A(s). . ............................. (1)

The tension profile may then be derived from the a e profile, which is given by

aae [[ a() ]2 -=Wb cos ()(s)±K ae-+Wb sin ()(s) as as

+ rae : sin O(S)rrh, ........................ (2)

• Now with Anadrill·Schlumberger.

Copyright 1987 Society of Petroleum Engineers

344

where W b is the buoyed weight per unit length of the pipel collar. The + K term applies in cases of tripping out, while - K applies in cases of tripping in. In cases of a rotating drillstring, the tension profile is derived from Eq. 2 with K=O. The + sign in front of Wb takes account of the parameter s running from the bit up the drillstring.

10hancsik et at. 2 propose an expression in terms of the tension only and take no account of the mud pressure. Explicit inclusion of the mud pressure avoids confusion over the location of the neu­tral point (effective tension=O) and the influence of buoyancy forces on curved pipe sections. Tension and effective tension agree at the surface.

From the profile derived from Eq. 2, the drag profile, F(s), is given by

...................... (3)

In cases of a rotating drillstring, the drag, F(s), is considered to act at an appropriate radius (usually the tool-joint radius in the drill­pipe, the collar radius in the drill collar, or the stabilizer radius in the BHA), giving rise to a local torque loss. The total torque loss is the sum of these contributions.

Drag Profiles for Planned and Drilled Wells. Analytical solu­tions ofEq. 2 for planar wells (i.e., a{3las=O) are described in Ap­pendix B. In general, however, Eq. 2 will be solved by an iterative method.

Integrating Eq. 2 and applying Eq. 3 allows the construction of the entire drag profile for a particular well. This (1) allows esti­mation of hook load during tripping out and torque loss and (2) provides insight into distribution of drag in a particular geom­etry. The construction depends, however, on a reasonable estima­tion of K(s).

For a well already drilled, an effective value of K can be deter­mined by comparing the hook load during tripping or the torque losses during drilling with the solutions ofEqs. 2 and 3 expressed as functions of K. This approach leads to the evaluation of a global friction factor for the entire well. Experiments 2 have consistently given rise to values of K in the range of 0.2 to 2.4 with a prepon­derance of values around 0.3.

With measurement-while-drilling technology, our experience has been unique in having direct access to the downhole weight on bit (WOB) and the downhole torque. These measurements can be used to analyze the transfer of torque and weight from the surface to

SPE Drilling Engineering, December 1987

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the bit on a foot-by-foot basis during drilling_ The resultant fric­tion log can be used to diagnose drilling and wellbore problems in real time.

An example of such an analysis is presented in Fig. 1. In this example, the average hole inclination was 22° [0.38 rad] over the length of the section. The build/drop rates were <0.2°/100 ft [0.115xlO-3 rad/m] and the turn rates were <0.5°/100 ft [0.286 x 10-3 rad/m]. Fig. I, a through c, displays the downhole WOB, the difference between surface and downhole WOB, and the torque losses between the rotary table and the bit. Fig. Id shows the friction factor determined by the downhole WOB and the torque losses with Eq. 3.

The friction factor will vary for a variety of reasons_ In particu­lar, the global coefficient reflects differences in the local value in

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Fig. 2-Planned well trajectory.

cased- and openhole sections. The friction factor is also subject to the effects of the stiffness of the BHA and the action of the stabilizers in different formations.

To compare expected drag and torque loss for planned wells (with a view to choosing an appropriate geometry), a guess at K(s) must be made. In the absence of data from offset wells, a value of 0.3 will generally be suitable on the basis of currently available data. The results of such a comparison will certainly be qualitatively reasonable and sufficient to choose a particular well plan, as long as the source of the drag is frictional, as previously described.

Figs. 2 and 3 display a planned build and tangent well with turn and the associated drag profile (when tripping out), torque loss pro­file, and a e profile. The well details and friction coefficient are given with the figures.

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Fig. 3-Expected drag torque and tension profiles for planned well.

SPE Drilling Engineering, December 1987 345

Fig. 4-General undersection well trajectory.

Drag Reduction and Well Geometry Planar Wells. The majority of drilled wells have some (often small) degree of turn, and the numerical solutions of Eq. 2 are required to determine the drag and torque loss profiles. On the other hand, most planned wells are considered to lie in a plane given by {3={3const. Analytic solutions (see Appendix B) to Eq. 2 exist for such wells:

..................................... (4)

where

F(s) = KWb sin O(s) + Wb cos O(s) ..................... (5)

and a(s) is the curvature profile, (-8018s), for the well. The influence of turn on these solutions is frequently small, so

that an analysis of the planned planar wells will be fruitful. Analy­sis of wells with turn is qualitatively similar to what will be de­scribed here. We will consider planar directional wells of the following generic type.

1. The well will have a vertical section from the surface to the kickoff point (KOP) at a depth D (see Fig. 4).

2. The well is subsequently caused to build at some constant rate until an inclination angle OJ is obtained. This point will be referred to as the end of the initial-build section. In general, the initial-build section is required by the directional driller to clear the near-surface steel work (particularly important on dense platforms) and to de­fine an initial direction, (3eonst. Thus, in practice, OJ will be non­zero, although the methods to be described can usually be applied with OJ=O.

3. The final section of the well follows a trajectory with some curvature profile a(s), where, as before, s parameterizes the trajec­tory and s=O denotes the target location.

Given the necessity of an initial-build section, we wish to inves­tigate the influence of a(s) in the lower section of the well on the

346

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til .Q

<Xl .::t. C\J 0 0

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Fig. 5-Hook load vs. 9, for catenary and constant-build trajectory .

expected drag and torque. From Eq. 3, the expression determining the drag in a planar well is given by

K[ue(s)a(s)-Wb sin O(s)] . .......................... (6)

Reducing the magnitude of this term reduces drag. We shall consider two classes of well trajectories that reduce the

side-force term: catenary trajectories and constant-build trajectories.

Catenary Well Plans. Th.e class of catenary curves includes the case where the effective tension in a uniform-density string satisfies

Wb sin O(s) ue(s) = ................................ (7)

a(s)

A general catenary curve will give rise to a reduction in the side force, although the side force is rendered zero only if a particular tension is applied to the string. It is worth noting that for positive curvature a(s), the condition ofEq. 7 requires a nonzero, positive value of the effective tension everywhere. This condition is violated at least at the bit during tripping and for much of the BHA during drilling so that it is not possible to have zero side forces every­where in the drillstring.

A significant problem with a catenary well path is the directional control required to drill even an approximate catenary curve. This leads us to consider the drag reduction efficiency of more readily achievable trajectories.

Constant-Build Trajectories. In practice, the building character­istics of a well trajectory are achieved by the strategic placement of stabilizers in the BHA. Usually, a given BHA at constant WOB has a tendency to build at a fairly constant rate,3 although, of course, varying downhole conditions will influence this tendency. This observation suggests that it would be of interest to explore the drag-reduction properties of a constant-build-rate trajectory rather than a catenary. Such a well would have a plan as shown in Fig. 4, with the final section of the well trajectory having a cur­vature a(s), given by a(s) =aconst.

Although the drag term, F(s), given by Eq. 3 remains nonzero over most or all of the trajectory in this case, the magnitude of F(s) can readily be reduced by an appropriate choice of a const ' and a reduction in drag is achieved. The following analysis shows that the theoretical reduction in drag and torque achievable with the constant-build trajectory is usually close to the improvement as­sociated with the catenary trajectory.

Drag Properties of Constant-Build vs. Catenary Trajectories. The total hook load during tripping out from full depth and the ro-

SPE Drilling Engineering, December 1987

tary torque were calculated for a typical model well to exhibit the possible reduction in drag and torque obtained by implementing curved trajectories. Qualitatively, the results presented for this case are similar to those obtained for a variety of model wells. In par­ticular, the similarity of the drag reduction obtained in the case of a catenary trajectory and a constant-build trajectory is typical.

Here we will consider the drag and torque properties of an ideal­ized well plan based on Well 3 described in Ref. 2. The fixed points on the model trajectory are as follows. The well is considered to be drilled vertically to a KOP at a depth of 2,400 ft [732 m]. The inclination angle then builds at a rate of 5 0 11 00 ft [2.86 x 10 - 3

rad/m] to some angle (ji, as depicted in Fig. 4. The target location is considered to be at a vertical depth of 9,000 ft [2744 m] and displaced horizontally from the rig location by 6,000 ft [1829 m]. Drilled as a conventional build-tangent well, this would correspond to a 44.5 0 [0.78-rad] well.

The model drillstring was configured with 372 ft [113 m] of 6V2-in. [16.5-cm] drill collar and 840 ft [256 m] of 5-in. [12.7-cm] heavyweight pipe with 5-in. [12.7-cm] drillpipe to the surface. A mud weight of 9.8 Ibm/gal [1174 kg/m3] was used. In this exam­ple, a value of 0.4 was postulated for the coefficient of friction, K, to simulate severe conditions. Torque-loss calculations were made with WOB assumed at 38,000 Ibf [169 kN].

Fig. 5 shows the hook load during tripping out from full depth as a function of the angle (ji (the angle attained at the end of the initial-build section). Note that as (ji is reduced, the curvature, a(s), in the section downstring from the initial-build section increases, as does the inclination achieved at the target. The upper curve gives the hook load in the case of a constant-curvature trajectory, while the lower curve shows the same quantity in the case of a catenary well.

A substantial reduction in hook load is evident. For the conven­tional trajectory with (ji=44.5° [0.78 rad], a tripping hook load of 320,000 Ibf [1423 kN] would be experienced. Reducing (ji to 300 [0.52 rad] leads in both constant-curvature and catenary­trajectory cases to a drag reduction of about 55,000 Ibf [245 kN]. Note that for (ji greater than 30 0 [0.52 rad], the two trajectories are equally effective.

It is important to note that in the well described, it is impossible to achieve a correctly tensioned catenary unless the BHA sticks. In such circumstances, the entire length of the drillpipe between the end of the initial-build section and the top of the heavyweight pipe would lift off at a given hook load from the bore wall and allow an efficient transmission of the pull to the BHA. Even in this case, however, the constant-build trajectory is almost as effective as the catenary trajectory.

Finally, Fig. 6 shows the rotary torque as a function of (}i for the constant-build case. For the conventional well, the torque lost between the surface and the bit is 22,500 Ibf-ft [30.5 kN' m] (reflect-

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ing the severe conditions}, while the constant-build trajectory with (}i - 300 [0.52 rad] will reduce this loss by 4,500 Ibf-ft [6100 N·m]. The catenary would give a similar reduction.

Detailed Anal,sis of Constant-Build Trajectories The redistribution in the side forces becomes a crucial issue if hole problems or differential sticking at the BHA are expected.

Drag Reduction. Given the comparable efficiency of the catenary and the constant-build trajectories in reducing drag, attentionwill now be focused on the more easily attainable trajectory-i.e., the. constant -build case.

It has been shown that the total drag in a drilling system may be reduced by implementing an undersection, constant-curvature trajectory. It is important, however, to consider in detail the way in which the side forces are redistributed in a curved trajectory.

Figs. 7 and 8 display the drag profile, the torque-loss profile, and the (J e profile for the well described earlier as functions of pipe depth. Fig. 7 shows these quantities for a tangent-build trajectory case (conventional deviated well), while Fig. 8 deals with a constant­build trajectory case determined by the condition (}i =30 0 [0.52 rad] (angle attained at the end of the initial-build section).

.....

It has already been determined that the total drag reduction achieved by setting (}i =30 0 [0.52 rad] would be about 55,000 Ibf [245 kN]. It is immediately clear from a comparison of the two

.... ......

1000 2000 3000 4000 5000 6000 7000 8000 9000 - 10000 11000 12000

pipe depth

Fig. 7-Drag torque and tension for conventional well plan.

SPE Drilling Engineering. December 1987 347

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Fig. 8-Drag torque and tension for 9, =30 0 undersection well.

figures, however, that the drag in the BRA is in fact greater in the case of the undersection well than in the conventional case. This is a direct consequence of the greater side forces that act at the BRA in the undersection case. Similarly, a comparison of the torque loss for the two cases leads to the same conclusion.

The cause of the increased side forces at the BRA is the increased target angle associated with the undersection well. In the case of the conventional well, this angle is 44.5 0 [0.78 rad], while for the 8; =30 0 [0.52-rad] undersection well, the target angle is increased to 58° [1.01 rad]. This gives rise to an increased side force essen­tially through the Wb sin 8 term occurring in Eq. 3. This increase in target angle is a general property of the undersection wells de­scribed here, and for the case of the catenary wells, the enhancement is greater than in the case of the constant-build wells.

Away from the drill collar and heavyweight sections, the drag in the undersection well is significantly reduced compared with the conventional well. This reduction is marked in the initial-build sec­tion, which is also reduced. The reduction of the drag in higher sections of the well outweighs the increased drag in the lower sec­tions so that an overall drag reduction is obtained.

It becomes clear then that the overall drag reduction should be viewed with caution. The suggestion to implement an undersection well must be based on a careful analysis of the nature of the drag problem.

In cases where contingent problems are anticipated-e.g., hole instability, differential sticking, or poor hole cleaning-it is likely to be important to keep the side forces at the BRA as low as possible. In such circumstances, the conventional well plan would prove the appropriate choice.

It cannot be overemphasized that the techniques described here must be used as a planning aid in conjunction with a good under­standing of the likely hole conditions. If no information is available (from offset wells), then the techniques must be used with extreme caution.

Torque Reduction. The problems of drag and torque loss, although closely linked, should really be considered separatcly. The reason for drawing this distinction is that the torque loss is more critically dependent on the redistribution of side forces than is the overall drag. Note, for example, that while the drag for the well described earlier decreases monotonically with decreasing 8;, the torque loss exhibits a minimum. Indeed there are circumstances in which the torque loss would be expected to increase for any value of 8; lower than its maximum value-i.e., the conventional well plan would provide the least torque requirement even though the overall drag might be reduced with an undersection well. In these circumstances, any attempt to drill an undersection well would exacerbate the torque problems.

348

Conclusions It has been shown that it is possible to reduce drag and torque by implementing certain undersection trajectories. Furthermore, this reduction can be achieved by the use of simple trajectories to target.

These trajectories, however, are invariably associated with higher side forces in the BRA. In the event that hole-stability problems or differential sticking are likely, then undersection trajectories will create greater problems than the conventional trajectories and should be avoided.

Nomenclature A(s) = pipe cross-sectional area at s, ft2 [m2]

D = depth, ft [m] F = general side force, Ibf [N]

F(s) = side force at s, lbf [N] g = gravitational acceleration, ft/sec2 [m/s2] K = sliding-friction coefficient

Pi = internal mud pressure, psi [kPa] Pm = mud pressure at (r.~), psi [kPa]

p(s) = mud pressure at s, psi [kPa] Po = mud pressure at Reference Point 0, psi [kPa] r.~ = polar coordinates on pipe element rl = outer pipe radius, ft [m] r2 = inner pipe radius, ft [m] s = pipe length from bit, ft [m]

~s = incremental pipe length ~T = difference between surface torque and downhole

torque, Ibf-ft [N'm] W = WOB, Ibf [N] ~W = difference between surface WOB and downhole

WOB, ·Ibf [N] Wb = buoyed weight of steel per unit length

W dh = downhole WOB, Ibf [N] a(s) = planar curvature at s

{J = azimuthal bearing angle ~(J = incremental change in bearing angle

8 = deviation angle ~8 = incremental change in deviation angle 8; = initial-build angle P = steel density, Ibm/ft3 [kg/m3]

Pm = mud density, Ibm/ft3 [kg/m 3]

u(s) = tension at s, Ibf [N] uAs) = effective tension at s, Ibf [N]

SPE Drilling Engineering, December 1987

Subscripts const = implication of constant value of associated variable

- = integration variable - = integration variable

References I. McClendon, R.T. and Anders, E.O.: "Directional Drilling Using

Catenary Method," paper SPE 13478 presented at the 1985 SPE/IADC Drilling Conference, New Orleans, March 6-8.

2. Johancsik, C.A., Friesen, D.B., and Dawson, R.: "Torque and Drag in Directional Wells-Prediction and Measurement," JPT (June 1984) 987-92.

3. Jogi, P.N., Burgess, T.M., and Bowling, J.P.: "Three-Dimensional Bottomhole Assembly Model Improves Directional Drilling," paper SPE 14768 presented at the 1986 IADC/SPE Drilling' Conference, Dallas, Feb. 10-12.

Appendix A-A Sliding Friction Model for Drag It is necessary to determine the tension profile, a(s) , for a drill­string with a general geometry. The parameter s is taken to denote the pipe distance from the bit. We assume that the interaction of the drillstring with the bore wall can be characterized by a coefficient of simple sliding friction, K. Thus if a side force, F, acts at some contact point along the drillstring, then the impedance to the mo­tion of the drill string is determined by a force of magnitude 1 KF I.

Consider an element of the drillstring with density p, cross­sectional area A(s), and mean incremental length As, as shown in Fig. A-I. A force F(s) acts orthogonally to the drillpipe element and with a magnitude determined by the local curvature of the string and, in the absence of buoyancy, the local tension, a(s):

[( AO )2 (A{3 )2J If,

F(s)=a(s) As +sin2 0 As As, .......... (A-I)

where A(s) is given by the product of the axial stress in the pipe and the pipe cross-sectional area.

There is also a buoyancy contribution to the side force. The mag­nitude of this term is obtained by integrating the mud pressure acting on the free surface of the pipe. Relative to some Reference Point o where the mud pressure is Po, the mud pressure at (r,4», denoted Pm' is given by

Pm =Po+Pmgr sin 0 cos 4>, ....................... (A-2)

where Pm is the mud density and g the gravitational acceleration. Referring to Fig. A-2, the total contribution from the mud is given by

f(Po+2Pmgr] sin 0 cos 4»cos 4>r)d4>As o

- f(Po+2Pmgr2 sin 0 cos 4»cos 4>r2d4>As o

r2?r AO +2 J (rf-r})(l+cos 4»-po cos 4>d4> ........... (A-3)

o 2

in the vertical plane and

r2?r 2 2 A{3 2 J (r) -r2 )(1+ cos 4»-po cos 4> sin Od4> ........ (A-4) o 2

in the horizontal plane. Evaluating Eqs. A-3 and A-4 and combining with Eq. A-I yields

a total side force given by

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-5)

SPE Drilling Engineering, December 1987

where

ae=a(s)+p(s)A(s) ............................... (A-6)

and pes) is the local mud pressure previously denoted Po. In cases where the internal mud pressure Pi differs from the external pres­sure Po, ae is replaced by

a(s)+Po7frf -Pi7fr}.

However, this leads to no change in the final expression (Eq. A-9) obtained for the ae profile.

For a nonrotating drillstring tripping in or out of the hole, the frictional drag acts against the motion of the drillstring. Referring to Fig. A-I, the incremental change in the tension profile is given by

aa [[ ao J 2 - =A(s)pg cos O±K ae- +g(p-Pm)A(s)sin 0 as as

( a{3 )2J '/' aA(s)

+ ae-sin 0 -p(s)--, . .............. (A-7) as as

where + K implies tripping out and - K tripping in. The term p(s)[aA(s)/as] takes account of mud pressure acting at discontinuities in the pipe cross section. ,It is convenient to render this expression entirely in terms of ae .

We have

a a -p(s)A(s) = -Pmg cos OA(s)+p(s)-A(s), .......... (A-8) as as

a(S + 6S) buoyancy force

gpA(S)6S a(S)

Fig. A-1-Forces acting on an element of pipe.

Fig. A-2-Coordinate system for buoyancy calculation .

349

so from Eq. A-7,

±K[[ae :: +g(p-Pm)A(s)sin or +(ae : sin orJ 112 • ••••••••••••••••••••••••• (A-9)

The ae profile that gives the difference between the mud pressure and the compression in the pipe is thus expressed entirely in terms of the buoyed mass of the steel. Note that the value of s for which ae(s)=O gives the (drilling) neutral point of the system.

Appendix liS-Drag in Planar Wells A planar well is constrained by the condition

a{3 -=0 ........................................ (B-1) as

So, from Eq. A-9, the tension profile is obtained from solutions of

along with the condition

ae(s) =a(s) +p(s)A(s) , ............................ (B-3)

where Wb is the buoyed weight of steel per unit length of pipe.

350

Solutions to Eq. B-2 are found by obtaining the ae± solutions to

aae± (ao. ) --=Wb cos O±K ae -+Wb sm 0 , .......... (B-4) as ± as

and patching at points where aeCao/as) = Wb sin O. Thus solutions for ae+ apply in regions where ae(ao/as) > - Wb sin 0, and solu­tions for ae_ apply in regions where ae(ao/as) < - Wb sin O.

The solutions to Eq. B-4 take the form

a (s) =a e ± j' KaC\,)d, e ± eo ± 0

where

+e±j'KaCf)ds fSe±J' Ka(s)dsF (s)ds o J 0 ± '

o ........... (B-5)

F±(s)=+KWb sin O+Wb cos 0 .................... (B-6)

and a(s) is the curvature:

-a~ a(s)=- . ................................... (B-7)

as

The different planar wells are distinguished by the form of a(s).

51 Metric Conversion Factors degrees x 1.745 329 E-02

ft x 3.048* E-OI lbf x 4.448222 E+OO

lbf-ft x 1.355 818 E+OO

"Conversion factor is exact.

rad m N N'm

SPEDE

Original SPE manuscript received for review Oct. 5, 1986. Paper accepted for publication April 23. 1987. Revised manuscript received Sept. 4, 1987. Paper (SPE 15463) first present· ed at the 1986 SPE Annual Technical Conference and Exhibition held in New Orleans, Oct. 5-8.

SPE Drilling Engineering, December 1987