DRAG AND TORQUE CALCULATIONS FOR HORIZONTAL WELLS SIMPLIFIED FOR FIELD USE

Jiang Wu, Hans C. Juvkam-Wold
Texas A&M University
College Station, Tex.

Sliding drag and torque can increase dramatically in horizontal and extended-reach wells and may become the limiting factors in determining the horizontal length or extended-reach of a well.

Therefore, accurate calculations of drag and torque are necessary for drilling operations.

The proposed equations for calculating axial load, drag, and torque in the build section of horizontal or extended-reach wells do not require numerical segment approximation, so they are easy and accurate to use and especially suitable for use with a hand-held calculator in the field.

Drag and torque are the results of friction caused by moving pipe in the well bore; drag occurs while moving the pipe along the well bore, and torque occurs while rotating the pipe.

In drilling horizontal or extended-reach wells, excessive drag and torque may become troublesome both in the drilling operations and in the later completion operations.

Thus, estimating drag and torque is important, but the calculation of drag in the build section of the well bore is complicated by the effect of the axial load (tensile or compressive) on the lateral contact force which generates the sliding drag and in turn causes changes to the axial load itself.

The usual way to calculate approximate drag and torque values in the build section involves tedious numerical calculations: dividing the build section into many small pieces, assuming the axial load remains constant in those small pieces, calculating the friction drag for each of the pieces, and then summing these values to get the total drag over the entire build section. 1 This process is both time consuming and difficult for a field engineer.

The analysis of drag and torque is made easier by first establishing the differential equation for the axial load in the build section, then by obtaining the solutions for the explicit axial load, and then by determining the exact calculation formulas for drag and torque. 2

The dimensionless graphs and the proposed piecewise linear approximations are provided to illustrate the features of the drag and torque in the 90 build section of well bores and to simplify the calculation process for the engineer in the field.

PULLING OUT OF HOLE

Fig. 1 shows the pipe in a 90 build section of a well bore. It is considered that:

The well bore is drilled with a constant build-up rate.
The pipe (drillstring, casing, or tubing) contacts either the downside or the upside of the wall of the well bore and has a curvature almost the same as that of the well bore.
The shear force on the cross section of the pipe is negligible compared to the axial load in the force balance of a differential element of length deltas.

For such a differential element deltas of the pipe, the force equilibria in the x and y directions for pulling operations (ignoring inertial effects) are given by Equations 1 and 2, respectively. Note that when the differential element deltas contacts the upside of the well bore, the contact force N will be negative.

From Equations 1, 2, and 3, by taking the limit as delta alpha approaches zero we obtain Equations 4 and 5.

From Equation 6 it can be seen that the contact force N(alpha) is a function of a and may be positive or negative, depending upon the angle a and axial load F(alpha), when We and R are given. The angle alpha 1 is defined as the transition point, at which N(alpha) = 0, above which N(alpha) < 0, and below which N(alpha) > 0. Solving the differential Equations 4 and 5 results in Equations 7 and 8 in which alpha 0 and alpha 2 are the initial and final position angles of the build section. For the build section of horizontal wells, these values are 0 and pi/2, respectively.

Once the axial load is determined, the drag over the build section from alpha 0 to alpha 2 may be solved easily, as shown by Equations 11 and 12. The first step in the calculation process is to determine which equation should be used to calculate F(alpha 2) from F(alpha 0):

If all the pipe contacts the downside of the well bore along the build section (N(alpha) 0 for all alpha), then alpha 1 = alpha 2, and Equation 7 only will be used.
If all the pipe contacts the upside of the well bore (N(alpha)
When (alpha 2 alpha 1 alpha 0, both Equations 7 and 8 have to be used: first using Equation 7 to calculate F(alpha 1), which is the axial load at the transition point alpha 1, by trial and error and then using Equation 8 to calculate F(alpha 2).

Fig. 2 is a dimensionless plot of the transition point alpha 1 vs. the axial tension F(0) at the bottom of the 90 build section of horizontal wells for values of F(0) equal to or less than WeR for pulling operations. The transition point moves downward as the axial tension F(0) increases.

All the pipe will contact the upside of the well bore after F(0)/WeR exceeds 1.

Fig. 3 is a dimensionless plot of the axial tension at the top F(pi/2) vs. the axial tension at the bottom F(0), for pulling out of the 90 build section of a horizontal well.

Fig. 4 is a dimensionless plot of the total drag Fd vs the axial tension at the bottom F(0), for the same 90 build section for various friction coefficients mu ranging from 0.25 to 0.4. 3 The curved part of the plots for small values of F(0) (that is, F(0)/WeR < or equal to 1) results from the transition point being located within the build section (alpha 2 > alpha 1 alpha 0). Once all the pipe contacts the upside of the well bore (F(0)/WeR or equal to 1), a straight line relationship is established. By letting alpha 1 = 0 and alpha 2 = pi/2 in Equations 8 and 12, respectively, we obtain Equations 13 and 14.

Equation 13 shows a straight line relationship between F(pi/2)/WeR and F(0)/WeR with a slope of F(0)/WeR with a slope of 3 mu pi/2. Equation 14 represents a straight line relationship between Fd/WeR and F(0)/WeR with a slope of (e mu pi/2 - 1).

Note also that the value of Fd in the curved part of the plot in Fig. 4 is less than mu WeR, which is the value of Fd calculated without considering the effect of axial load on the contact force N. The lowest Fd corresponds to alpha 1 approximately equal to 45 (referring to Fig. 2), meaning the transition point alpha 1 is now located at the middle of the 90 build section.

The piecewise linear approximation for the total drag in the 90 build section of the well bore and for pipe pulling operations is derived from the curves in Fig. 4 and shown by Equation 15.

RUNNING PIPE IN THE HOLE

Differential equations similar to Equations 4 and 5 may be established and analyzed in a similar manner for the cases in which pipe is tripped back into the hole.

The solutions to these equations result in Equations 16, 17, and 18, with A, B, alpha 0, alpha 1, and alpha 2 defined as before.

The subscript c for axial load F in these three equations indicates that the compressive load should be taken to be positive (Fc(alpha) - F(alpha) for convenient use in engineering calculations for cases involving running pipe in the hole.

The drag over the well bore section from 0-0 to (12 is now calculated by Equation 19.

Fig. 5 is a dimensionless plot of the axial compression at the top Fc(pi/2) vs. the axial compression at the bottom Fc(0), for running pipe in the 90 build section of a horizontal well.

Fig. 6 is a dimensionless plot of the total drag Fd vs. the axial compression at the bottom Fc(0), for running pipe in the same 90 build section.

The piecewise linear approximation for the total drag in the 90 build section of the well bore and for pipe running operations is derived from the curves in Fig. 6 to obtain Equation 20.

TORQUE IN THE BUILD SECTION

The torque generated in the build section is conceptually calculated by Equation 21 with the contact force N(alpha) still calculated by Equation 6 or Equation 18.

Because the friction is now generating the torque, it has almost no affect on the axial load.

The axial load equations for pulling-out-of-hole and running-in-the-hole are both reduced to the form in Equation 22.

There still may exist a transition point alpha 1, although its value is different from that in the drag calculations. The transition point alpha 1 is now simply determined by setting Equation 23 equal to 0 and solving.

The dimensionless plot of the transition point alpha 1 vs. the axial tension at the bottom of a 90 build section F(0)/WeR is presented in Fig. 2 as the top curve indicated by mu = 0.

For the case of a 90 build section with all the pipe contacting the downside of the well bore (when Fc(0) or equal to WeR), alpha 1 = alpha 2 = pi/2, Equation 24 becomes Equation 25 which indicates the straight line portion of the torque curve in Fig. 7 (when Fc(0) or equal to WeR). Fig. 7 is a dimensionless plot of the total friction torque T vs. the axial compression at the bottom Fc(0), for the 90 build section of a horizontal well.

The piecewise linear approximation for the total friction torque in the 90 build section of the well bore is derived from Fig. 7 to obtain Equation 26.

CONCLUSIONS

When the transition point is located within the build section, the calculations may become a little complicated because the trial and error approach first has to be used to determine the location of the transition point.
However, the calculations are straightforward for some problems in engineering, such as maximum horizontal length planning and the maximum hook load pull prediction, because all the pipe in the build section usually contacts only one side (either downside or upside) of the well bore under those circumstances.
The axial load has a great effect on the drag and torque calculations in the build section. When the axial load (tension or compression) becomes large enough to let the pipe contact only one side of the well bore, the drag and torque in the build section will increase proportionally with the increase in the axial load.
For the 90 build section, if F(0)/WeR 1 for pulling operations and Fc(pi/2)/WeR 0 for running or rotating operations, the drag in the build section will increase proportionally with the increase in the axial load F(0) or Fc(0) with a slope of (e mu pi/2 - 1).
The friction torque will increase proportionally with the increase in the axial load Fc(0) with a slope of (mu pi Dtj/4,000).

REFERENCES

Shuh, Frank J., "Horizontal well planning--build curve design," paper No. 890008, presented at the Centennial Symposium Petroleum Technology into the Second Century, held at New Mexico Institute of Mining & Technology, Socorro, N.M., Oct. 16-19, 1989.
Sheppard, M.C., Wick, C., and Burgess T., "Designing well paths to reduce drag and torque," SPE Drilling Engineering, December 1987, pp. 344-45.
Johancsik, C.A., Friesen, D.B., and Dawson, R., "Torque and drag in directional wells--prediction and measurement," Journal of Petroleum Technology, June 1984, pp. 987-92.
Carden, Richard S., "Equations determine maximum horizontal well bore length," OGJ, Dec. 26, 1988.