On the Optimal Time To Pull a Bit Under Conditions of Uncertainty
Authors: Vargo Louis G. 
Published: December 2007

Cost-effectiveness criteria are easier to state in theory than to apply in practice. The cumulative cost-per-foot (CCF) criterion for ending the use of a rock bit is an example. Ref. 1 simply advises, "When the cost per foot begins to increase, the bit should be pulled." This prescription then is illustrated by a numerical case in prescription then is illustrated by a numerical case in which the calculated CCF increased for the first time by less than 0.5% over a 10-ft depth increment. At this point there are two major sources of uncertainty in the point there are two major sources of uncertainty in the decision either to stop or to continue drilling:inaccuracies in the measurement and prediction of variable input quantities (footage, drilling time, and trip time) to the CCF equation andlack of precise knowledge of possible formation changes affecting future penetration possible formation changes affecting future penetration rates.
The first source suggests that a 0.5% increase may be more numerical than real; the second admits the chance that the CCF may "turn around" i.e., decrease to a new overall minimum after once increasing. A well-known drilling assistance program, COSTCAL, eliminates much of the first type of uncertainty. An incremental cost-per-foot (ICF) calculation is performed for each, say, 10-ft depth increase. performed for each, say, 10-ft depth increase. COSTCAL argues that bit and trip costs need not be prorated to the ICF's. Whether they are or not, the prorated to the ICF's. Whether they are or not, the principle of the program remains the sameas depth principle of the program remains the sameas depth increases, the ICF tends to overtake and eventually to exceed the CCF, and it becomes uneconomical to continue drilling with the used bit. We then would have a sharp criterion for bit pull-out when ICF first exceeds CCF if the second source of uncertainty were not present. The numerical example and discussion in Ref. 2 illustrate the resulting ambiguity by the statements: ". . . the bit should have been pulled at 11,210 feet" and "when ‘$‘ is to the right of ‘C' [ICF exceeds CCF] a change should be considered." The italics are added. The COSTCAL data are given in Table 1 for depths above and below the transition depth of 11,210 ft, at which the ICF first became greater than the CCF. These data show that pulling the bit at 11,210 ft is indeed premature; after an additional 40 ft of drilling, the CCF reached a new absolute minimum! What is needed at the depth at which ICF first exceeds CCF is a probabilistic test of the sequence of data points for the probabilistic test of the sequence of data points for the subsequent increments. The test should proceed from the question, "Now that the difference D = ICF-CCF first has become positive, what is the probability that there is a trend toward increasingly larger values of D?" If this probability is sufficiently large when combined with an probability is sufficiently large when combined with an inexact prediction of deeper strata changes and consequent penetration-rate effects, the driller could decide to penetration-rate effects, the driller could decide to pull the bit with full recognition of the complexities of pull the bit with full recognition of the complexities of the problem and with a nice combination of analysis and judgment. Fortunately, a very suitable test for the trend in D appears in the statistical literature. Called the Cox and Stuart Test for Trend, it tests any sequence of D's for an upward trend, beginning with the first D that is greater than zero. Denoting the number of D's generally by n, the number of comparisons by c, and the first D in a sequence by D1, the next D2, etc., we begin the test by grouping the D's into pairs (D1, Dc+1), (D2, Dc+2) . . . (Dn-c, Dn), where c = n/2 if n is even, and c = (n+1)/2 if n is odd. If Di less than Dc +i for any i, we replace the pair with a "+"; if Di greater than Dc +i, we write a "--" . In substance, we are comparing the first half of the sequence of n D's with the latter half. There are c distinct comparisons. The presence of more +'s for a given sequence length is stronger evidence for an upward trend in the D's. The Cox and Stuart Test for Trend is performed in this application in "real time," i.e., after several increments following our first indication that continued drilling is becoming uneconomical, we make the test. If the data show insufficient evidence to stop drilling, we drill another increment and reperform the test with the greater set of data. At no point are data discarded.
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