US 3761701 A Abstract In drilling oil and gas wells, particularly by the rotary drilling technique, it is necessary periodically to pull the drill string from the well in order to replace a worn bit. The most economic drilling is that in which the type of bit and the drill rig operating conditions are matched to the characteristics of the rock being drilled so that the cost of drilling, between bit changes works out to the lowest possible value per foot. This invention concerns a rig floor drilling cost indicator which is basically an analog computer which graphically shows the fractional wear on the bit, the average cost of drilling per foot, and the incremental cost of drilling a foot of hole. These indications permit the driller or the tool pusher to determine with considerable accuracy both the arrangement for minimal cost drilling, and when to change the bit. Most of the equations solved by this computer are novel.
Claims available in Description (OCR text may contain errors) . i c, SR -25-73 United States Patent 1 1 3,761,701 Wilder et a1. Sept. 25, 1973 DRILLING COST INDICATOR OTHER PUBLICATIONS [76] Inventors; Lawrence B'Wilderclo Amoco Alterman Engineered Bit Logs Cut Drilling Cost, Production Company, PO. Box 591, world on, March 1969 38/43- Tulsa, Okla. 74102; Renic P. Vincent, deceased, late of Tulsa, Primary Examiner-FeliX D. Gruber Okla; Meta Luella Vincent, p l F l-l l administratrix, c/o Amoco Production Company, PO. Box 591, Tulsa, Okla. 74102 [57] ABSTRACT Filed? y 14, 1971 In drilling oil and gas wells, particularly by the rotary [21] Appl 162,693 drilling technique, it is necessary periodically to pull the drill string from the well in order to replace a worn Related Application D bit. The most economic drilling is that in which the type [63] Continuution-in-part of Ser. No. 874.562. Nov. 6. of bit and the drill rig operating conditions are matched 1969. to the characteristics of the rock being drilled so that the cost of drilling, between bit changes works out to [52] U.S. Cl 235/193, 72/151, 235/184, the lowest possible value per foot. This invention con- 235/197 cerns a rig floor drilling cost indicator which is basically [51] Int. Cl. G06g 7/00, 606g 7/48 an analog computer which graphically shows the frac- [58] Field of Search 235/193, 184, 194, tional wear on the bit, the average cost of drilling per 235/197; 346/30; 73/151, 151.5; 175/39 foot, and the incremental cost of drilling a foot of hole. These indications permit the driller or the tool pusher [56] References Cited to determine with considerable accuracy both the ar- UNITED STATES PATENTS rangement for minimal cost drilling, and when to 2 737 343 3/1956 Hinton 235/197 change the bit. Most of the equations solved by this 3,660,649 5/1972 Gilchrist et '31. 235/193 computer are novel 4 Claims, 8 Drawing Figures 3 if; l2 N SPRING l 36 I3 PATENTEnstrzsms sum 1 UF 8 T, DECEASED, IADMINISTRATRIX RENIC P. VINCEN BY META LUELLA VINCEN LAWRENCE B. WILDER Y INVENTORS BY f aLLC ATTORNEY PAIENTEDSEPZSIQB 3.761.701 ' sum 20F 8 TO SWIVEL 6 -9 t r-ll NORMALLY NORMALLY r' 23 OFF 2| ON llllllllllllIl-IIHH .IH'RI Ht-Ihlllllllllilllllll FIG.2 RENIC P. VICENT, DECEASED, BY META LUELLA VINCENT, ADMINISTRATRIX LAWRENCE B. WILDER INVENTORS ATTORNEY PATENTED SEPZSIB'IS SHEET 3 [IF 8 RENIC F? VINCENT, DECEASED, BY META LUELLA VINCENT, ADMINISTRATRIX LAWRENCE B. WILDER INVENTORS ATTORNEY PATENTEU8EP25'975 3.761301 SHEET BF 8 RENIC P. VINCENT, DECEASED, BY META LUELEA VINCENT, ADMINISTRATRIX LAWRENCE B. WI LDER INVENTORS By mzh wg, ATTORNEY v PATENTED 3.761.701 SHEET 5 BF 8 AMP. 2 a 4 T v AMP. CLOCK FIG. 5 RENIC F? VINCENT, DECEASED, BY META LUELLA VINCENT, ADMINISTRATRIX, LAWRENCE B WILDER INVENTORS ATTORNEY CLOSED ONLY WHEN DRILLING RENIC P. VINCENT, DECEASED, BY META LUELLA VINCENT, ADMINISTATRIX LAWRENCE B. WILDER INVENTORS ATTORNEY PATENTEU SEPZS I973 SHEET 7 OF 8 RE I PVINCENT, BY META LUELL NCENT, ADM X LAWRENC ILDER NTORS BYZQLAQ ATTORNEY PATENTEU 3.761 .701 I sum 80F 8 RENIC P. VINCENT, DECEASED, BY META LUELLA VINCENT, ADMINISTRATRIX LAWRENCE B. WILDER INVENTORS A TTORNE Y DRILLING COST INDICATOR BACKGROUND OF THE INVENTION 1. Field This invention pertains to an analog computer responsive to movement of the traveling block ofa rotary drill rig. With this single connection, it presents the driller in graphical form the fraction of the bit life used up in any particular time. He is also shown the incremental cost of drilling the last foot, and the total average cost per foot of drilling the current bit. The driller sets into the computer the average weight on the bit and the rpm at which the rotary table is revolving, as well as the anticipated bit life, the hourly cost of operating the drill rig, the purchase cost (or rental cost) of the bit, and the time required to make a round trip of the drill string to replace the bit. Preferably the entire computer is energized with alternating current at commercial frequency, for example, 60 hertz. 2. Description of the Prior Art Factors affecting the speed at which a well can be drilled and the economics of this drilling have been very widely discussed in the petroleum industry press. Factors that affect cost per foot, which is the usual measure of over-all effectiveness in drilling can be divided into two broad categories: l Those out of control of the operator; and (2) those under his control. The former includes formation characteristics, formation fluid content, formation fluid pressure, depth, location, etc. The latter category includes pipe and hole sizes, types of bit, weight on the bit, rotary speed, type of mud, hydraulics, and sometimes even rig selection. When it comes down to drilling any particular hole size required by a particular casing, the driller has under his control three main variables: Bit type, weight on bit, and rotary speed. It is known that each formation usually can be drilled best by one of four types of drilling action, such as scraping, as with drag bits, chipping, as with rolling cutter rock bits, crushing, as with tungsten carbide insert rock bits, and abrading, as with diamond bits. Except in very thick, uniform formations, the type of rock being drilled may change several times within the depth drillable by a single bit. Accordingly, the driller, or someone in supervision over him, selects the bit that experience indicates as most effective under the conditions expected. Once the bit has been selected and a trip has been made to place it on bottom, the major variables under the drillers control are the weight on the bit and the revolutions per minute at which the bit is rotated. Bit tooth wear increases more rapidly for each increment in weight on the bit until the bit can fail due to compressional tooth failure.'lf the bit is toothed, there is little reason to apply weight beyond the point where the teeth penetrate their entire height into the formation. At the other end of the scale, there is a minimum weight below which a bit will not drill because the unit stress on the rock has not reached rock failure strength. Most investigators state that drilling rate is directly proportional to the bit weight multiplied by a constant that is a fucntion of the formation, of the type of bit, and the drilling fluid. (See Engineering Design of Drilling Operations," Jack H. Edwards, Drilling and Production Practice, AIME 1964, page 40; Analytical Determination of Optimum Bit Weight and Rotary Speed Combinations, J. W. Graham and N. L. Muench, Society of Petroleum Engineers, Preprint Paper 1349-G, page 4.) In 1958 the American Association of Oilwell Drilling Contractors sponsored a series of field tests to determine the effects of weight and speed on drilling rate. The results were published in the Petroleum Engineer for Jan. 1958, pages B40 to B52. In this reference on page B43 E. M. Galle and H. B. Woods show drilling rate proportional to weight on bit raised to the 1.2 power. Galle and Woods in a later article, Best Constant Weight and Rotary Speed for Rotary Rock Bits, 1963 AIME Drilling and Production Practice, pages 48-55, showed that the rotary speed-drilling rate relationship was different from soft and from hard formations. They used an exponent of one when the average or hard formation was being drilled, and six-tenths for very soft formations. Other authors have given various values for the exponent, generally less than one, or at maximum not greater than 1.1. A number of patents have considered the matter of drilling parameters and their effects on drilling economics. The single reference that appears closest to the work disclosed herein is the Dellinger et al. US. Pat. No. 3,364,494 in which the footage drilled by the bit is automatically plotted against the drilling time on a mechanical graph-drawing machine, there being a suitable offset for the trip time and an equivalent time (in terms of rig costs) for the bit cost. The plot is drawn on graph paper bearing a special set of lines in terms of costs per foot. The driller can, by looking at the graph, determine the condition under which most economic average drilling is obtained. Arps in US Pat. No. 3,345,867, teaches the measure of wear on roller type rock bits by determining the relative rotational speed of the .drilling string to that of the rotating cones. The rotating cone speed is determined from the vibration of the drill string as the teeth on the roller impact the formation. Other more crude types of drilling rate indicators are shown in earlier patents, such as Nichols US. Pat. No. 2,287,819, using a rotary chart on which the feet drilled are marked off against the time of drilling; the Mizell Reissue US Pat. No. 21,297, which shows production ofa somewhat similar chart, and the Pearson US Pat. No. 2,935,871, concerning a drilling rate plotter. In general these arrangements are useful but lack precision. In many cases they do not even attempt to consider the most valuable economic factors required to drill a well under optimum conditions. For example, no one has been able to plot in the past on a strip chart the fraction representing the bit life lost in drilling, say, a foot of formation in comparison with the total estimated life of the bit. Similarly, one has not been able to log directly on such a strip chart the incremental cost involved in drilling the last foot. The charts purporting to measure average driling cost, such as, for example, the Dellinger et al. system, do not, in fact, directly or linearly record the average drilling cost, but require extrapolative reading in terms of a series of cost indicator lines. On the other hand, the computer which we describe herein does permit graphing these important variables directly on a strip chart. The results obtained could perhaps be compared to those which can be obtained by other means, for example, by digital data manipulation in a general-purpose programmed computer. Such units are described in the World Oil of Jan. I968 on page 88, and World Oil for Feb. l, 1968, page 40-]. Such equipment is more delicate and much more expensive, thus can be justified only in the cases of very expensive wells. Finally, it is known that other types of analog computers are available, for example one described in Drilling for Jan. 1969 on page 42, which monitors rotary speeds and bit weights, multiplies these two quantities together and compares it against the optimum product already set into the machine, then operates a brake control unit to modify the bit weight and correct it to the desired condition. Such equipment is again useful, but it will not produce the important economic parameters available when using our analog computer. SUMMARY OF THE INVENTION Basically the driller adjusts certain dials which control potentiometers or equivalent electrical apparatus setting up electrical signals related to the weight on the bit, the average bit rpm, the rig cost per unit time, the bit cost, and the like, and which read out directly in the form of graphs on a strip chart. The fraction of the bit life expended by drilling up to the present, the incremental drilling cost for drilling the last unit length in the well, and the average cost per foot, including appropriate trip time and bit cost, for the total formation section that has been drilled by the current bit are then plotted. The only attachment between the computer and the drill rig is a taut-line arrangement attached to the traveling block or the swivel and passing around a cathead or drum at the computer so that it rotates in direct relation to the downward motion of the drilling bit through the formations. The graphs are always available to the driller so that he can make appropriate corrections in rotary speed or weight on the bit in order to secure further optimization of drilling costs and to know when it is desirable to change bits. BRIEF DESCRIPTION OF THE DRAWINGS The attached drawings form a part of this disclosure and are to be read in conjunction therewith. In these FIGS, FIG. 1 represents in very schematic form a drill rig with attachment from the swivel or traveling block to the drilling cost indicator. FIG. 2 shows the arrangement for sending out periodic electrical signals showing each time that a stated unit length of drilling has been accomplished (for example, each foot). FIG. 3 shows the basic analog computer arrangement for measuring the fraction of the bit life expended in drilling a single unit length in the formation. FIG. 4 shows the apparatus in the computer for summing up and graphically recording the total fraction of the bit life expended up to the present. FIG. 5 shows the equipment involved in plotting the incremental cost, that is the cost to drill a single additional foot under the drilling parameters set in the machine. FIG. 6 shows the equipment used in preparing a chart of the average cost per foot involved in all drilling with the current bit. FIG. 7 shows in isometric view one form of the drilling cost indicator. FIG. 8 represents one form of chart produced by the drilling cost indicator. DESCRIPTION OF THE PREFERRED EMBODIMENT When drilling rock formations using a rotary drill rig, the important variables immediately available for control by the driller are the weight on the bit W and the rotational speed of the drill bit R. Increase of the weight on the bit increases the drilling rate, as does increase in the revolutions per minute of the drill bit. On the other hand, increases in weight or bit rpm also decrease the sharpness of the bit whether it be any of the four ordinary types of bit, and thus has a tendency to decrease drilling rate. The most effective combination of these two variables produces the greatest over-all effectiveness and least cost per foot drilled. Generally it has been found difficult to predict in advance the part of the total bit life which has been expended by drilling under a particular set of circumstances, and it has not been possible to determine the most economic conditions for the drilling to proceed. This is particularly true when drilling in a region where there has not been much prior drilling, but is also frequently found true even though many wells have been drilled in a particular oil or gas field. We have found it was possible to combine the drilling variables into a set of equations which can be formulated into the design of an analog computer enabling the driller to read from strip charts produced by the drilling cost indicator the fraction of the anticipated bit life expended in drilling with a particular bit up the the present, the incremental drilling cost, and the current average drilling cost per foot for the current bit. These strip charts permit the driller to determine when he can improve drilling performance by increasing or decreas' ing the weight on the bit or the bit rpm, and the point at which it is advisable to change the bit. The computer constructed in accordance with these equations is an analog device which is connected mechanically to the rotary drill rig only at the rotary swivel, or equivalent, so that the location of the bit as drilling proceeds is automatically fed into the computer. The driller adjusts six static parameters which constitute inputs into the computer. These variables are the cost of the bit, the cost of the rig per hour, the cost of making the round trip of the drilling string to replace the bit (or its equivalent, the time to accomplish the bit change and replace the new bit at the formation), the bit life factor (explained below), the rotary speed and the weight on the bit. These last two parameters may be adjusted or up-dated by the operator at any time during the life of the bit when the driller changes these variables. Another variable input is real time, which is generated by clocks within the computer. The computer is furnished with electric current, preferably alternating current of the normal power frequency, for example, 60 hertz in the United States. This electric energy actuates the computer to produce the strip chart outputs. Ordinarily the strip charts are driven past the recording pens in direct relation to the depth at which drilling is occurring, but the strip charts may also be driven past the pens at a constant time rate. The first equation involved is the fraction n of the anticipated bit life N which is expended in drilling a unit distance in the ground, for example, one foot. Here we use the formula where R is the rotary speed, i.e., revolutions per unit time at which the bit is rotated, 01 is an empirical constant set into the computer, W is the weight on the bit, B is a second empirical constant set into the computer, and t t is the time expended in drilling the k" unit distance (foot). The anticipated bit life N is defined as the life of the bit from the time that the bit in its new condition first commences drilling until it is so dull that it no longer drills economically, and, accordingly, should be replaced. It follows that k=K N: where K is the total footage drilled with a bit up to the point where it is completely dull and must be replaced. The fraction of the anticipated bit life N expended in drilling K feet of formation, called Z, is given by This is one of the variables which is printed out on a strip chart by the analog computer and thus informs the driller when the bit is approaching the end of its useful life. The second useful equation solved by this analog computer is the incremental drilling cost, that is, the cost of drilling only the last unit length in the well, for example, the k' foot. This quantity, called Y, is given y where C is equal to the bit cost, C is equal to the total cost required to replace the bit (and is thus equal to the product of the rig cost per unit time times the total time involved in making a round trip of the drill string in replacing the bit), and C is the rig cost per unit time. From inspection of this formula (4) it is apparent that Y is dependent upon the fraction of the anticipated bit life expended in drilling this k" foot, and the time required to drill it. Finally, the third quantity charted by the drilling cost indicator is the average drilling cost per unit distance in the ground (ordinarily per foot) for the current accumulated footage drilled by the bit in use. This quantity X is given by X=(C1+ C2 1 k o)l/ where C is the rig cost per unit time, I, is the total time required to replace a bit, and 1,, is the time that the current bit commenced drilling. Several values which have been mentioned above need further explanation. The exponents a and B are quantities which are determined by analysis of drilling records in the region of interest. a can be determined by operating a drill rig with a new bit at constant weight on the bit while varying the rotary speed and drilling through a homogeneous formation. Values for this quantity which, of course, vary as the formations change, have been determined by numerous authors, see for example Effect of Rotary Speed on Drilling Rate by P. L. Moore, Oil & Gas Journal, vol. 58, No. 33, Aug. 15, 1960, p. How to Calculate Bit Weight and Rotary Speed for Lowest-Cost Drilling by E. M. Galle and H. B. Woods, Oil & Gas Journal, vol. 58, No. 46, Nov. 14, 1960, pp. 167-176; and A Method of Utilizing Existing Information to Optimize Drilling Procedures by J. W. Langston, a paper presented at the Annual Fall Meeting of the SPE of AIME, Oct. 3-6, 1965, Denver, Colorado. Published values, plus a recent analysis of bit records, indicate ranges for a from about 0.1 to 1.1, averaging around 0.33fLacking better information, the value ofa may be set at this figure of +0.33. Similarly, the value of B can be determined by drilling with a fresh bit through a homogeneous formation keeping a constant rpm on the bit and varying the weight applied to the bit. Values of B have been found by other investigators to lie between +1 and +2 (for example the Galle and Woods and Langstron references mentioned above, and Computerized Drilling Control by F. S. Young, SPE Paper 2241, Oct. 29, 1968, Houston, Texas), but our correlations indicate for modern bits ,8 should lie between the values of about +0.14 and about +0.74. A value of +0.34 may be employed in the absence of more complete knowledge on a particular formation. The value for the total anticipated bit life N may be determined by use of the computer itself, drilling through a homogeneous formation and noting the value of Z while also noting the value of X. Where the value ofX has decreased to a minimum and starts to increase, the bit is sufficiently dulled so that it is desirable to change bits. The corresponding value for Z plotted by the computer at this point is multiplied by the value for N which was set into the computer during this drilling and becomes the value of N to be used the next time when drilling through this same formation. Equations (3) to (5) are, of course, too complicated for use on the rig floor. Accordingly, this analog computer was developed for determining and plotting the various quantities desired, specifically Z, Y, and X as defined above. If desired the driliing cost indicator may also plot the values of W and R, particularly of use when these values are changed by the driller while using a single bit. The apparatus is enclosed in a box, one form of which is shown in FIG. 7. The computer 1 is mounted on the floor of the rig (see FIG. 1) or at least close to the rig framework 2 so that a line 3 may be attached to the swivel 4 or the traveling block immediately above it. The line 3 is wrapped about a drum or cathead 5 and is maintained in taut condition by having the end not connected to the swivel run over a pulley 6 attached to the rig framework 2 and down to a weight 7. The drum 5 is spool shaped with the major part of the spool of a fixed diameter, preferably such that the circumference is one unit distance, for example. one foot, 'l'hus, (sec FIG. 2) each time that the swivel moves down one foot due to drilling of one foot by the bit, the drum 5 rotates through one complete revolution. The shaft of the drum 5 is mounted in the frame of the computer box (not shown) on bearings 8. A source of electric energy, preferably an alternating current generator 9 is used with the drilling cost indicator. It may be incorporated into the indicator itself, but ordinarily one uses a commercial source of alternating current or the rig generator. In either case it is assumed that its output voltage, present across terminals 10, is essentially constant in amplitude. Two other sets of output terminals 11 and 12 are used in connection with two switches 20 and 21 to produce useful signals employed in the drilling cost indicator. These switches 20 and 21 are of a type frequently called micro switches, that is, a physically small switch mounted on the frame of the computer box (not shown) and equipped with insulated mechanical contacts which actuate the switch upon external motion of a cam or the like. Switch 20 is a normally open switch, which closes when the mechanical contact is depressed. Switch 21 is a normally closed switch which opens upon actuation of the mechanical contact, Thus from the wiring diagram shown in FIG. 2 it is apparent that there will be electric energy across terminals 12, except when the contact on switch 21 is actuated. Across terminals 11 there will be electric energy only when the contact of switch 20 is actuated. The micro switches are actuated from a cam wheel 22 containing two separate but closely spaced actuators or cams. These can be in the form of steel pegs or equivalent projecting from the cam wheel 22. As shown in FIG. 2, lowering of the swivel rotates cam wheel 22 in the direction of the arrow shown on the cam wheel circumference. Accordingly, the lefthand peg on cam wheel 22 actuates switch 20 once per revolution just slightly ahead of the righthand peg which actuates switch 21. As the result of this, electric energy is momentarily applied across terminals 11 once per revolution just slightly before electric energy is interrupted across contacts 12. A magnetic clutch 23 connects the shaft from the drum 5 to that of the cam wheel 22. It is energized from contacts through switch 16. This switch is closed by the driller whenever a new bit has been placed on the bottom and drilling is to commence. Similarly, this switch is opened by the driller whenever drilling ceases and a round trip is to be made to replace the bit. The shaft carrying cam wheel 22 also carries a small gear 24 which mates with a large gear 25, the gear ratio being preferably 1:30 to 1:50, for example 1:40. The shaft of gear 25 supported by bearing 8 from the computer box frame (not shown) drives a variable resistor 26, preferably of the helically wound potentiometer type, sometimes called a Helipot, or the like. Thus the resistance between terminals 27 is always directly proportional to the total rotations of the drum or cathead 5 while drilling is progressing, and, accordingly, is a measure of the total footage drilled by the current bit. Equation 1 defined the fraction of the anticipated bit life expended in drilling a foot of formation (or in general, a unit distance). The drilling cost indicator makes the computation of n automatically in a fashion shown in FIG. 3. Potentiometer 28 (Preferably of the helically wound type) is wound with wire which may be of either varying cross section or varying resistivity, or both, such that the total resistance varies from the zero position at some constant to the a power. Such potentiometers are commercial items widely available for a large choice of a and over a range of a sufficiently wide to include the range previously discussed, that is from about +0.15 to about +0.45. Thus when one sets the slider of the potentiometer at a distance L from the be ginning. the resistance R b L". where b is a constant. This potentiometer 28 is mounted in the computer with an appropriate dial which can be manually adjusted, for example to set on the dial of the revolutions per minute (R of the drill bit. If the voltage output of the generator 9 is e. the voltage e, is equal to b, e R", where b, is a constant. Accordingly. this potentiometer provides means for generating a voltage in a circuit directly proportional to R". This voltage is applied across a second potentiometer 29. The resistance of this potentiometer is chosen in a like manner to that of 28 to vary according to some constant to the B power. It is also preferably of the helical potentiometer type, and is furnished with a dial permitting one to manually adjust the slider to any desired value. In this case the numerical value set on the dial is the weight on the bit W. Accordingly e is directly proportional to the total voltage across the potentiometer. X IV". Thus e; is equal to I): RW. where b: is a constant. The voltage e is in turn applied across a third potentiomter 30 which is arranged with a linear resistance per unit length, and which is preferably again a helical potentiometer. Unlike potentiometers 28 and 29, potentiometer 30 is adjusted to be driven by a clock not a manually adjustable dial. This clock (31) is an electric synchronous motor energized from terminals 12 and actuates a shaft 32 which is connected to the shaft of potentiometer 30 through a magnetic clutch 33, also actuated from terminals 12. Mechanical spring 34, of suffcient strength to return the shaft to zero position whenever clock is not driving it, is also connected to the shaft of potentiometer 30. This arrangement of clock, clutch, and spring (sometimes called a resettable relay) acts in the following manner: As soon as terminals 12 are energized, i.e., at the start of drilling each foot, clock, (31) commences to rotate synchronously in accordance with the constant power frequency applied and clutch 33 applies the rotation ofthe clock output shaft 32 to move the slider of potentiometer 30 linearly in accordance with time during the time contacts 12 are energized. The maximum position of the slider of potentiometer 30 is directly proportional to the total time elapsed in drilling one foot of formation, therefore proportional to (1 t Accordingly, the output voltage across terminals 13 from the three concatinated potentiometers 28, 29 and 30 is directly proportional to R"W (na Thus, there is generated across terminals 13 at the end of drilling each foot an electric voltage directly proportional to the quantity n One of the lines 13 is connected in series with an adjustable impedance, resistor 35, preferably a helical linear-resistance variable resistor furnished with a dial, adjustable for manual set ting. The value of this resistance is set proportional to the anticipated bit life N. This circuit is closed through a recording current meter 36, which may be either a separate unit or more preferably, one channel of a multi-channel strip chart recorder. It records the current passing through the meter when a separate, recording potential is applied across two terminals. In this case the recording voltage is that applied across terminals 13, that is, the output voltage generated between the slider and base of potentiometer 30. Accordingly, the recording ammeter 36 and the adjustable impedance or resistor 35 make up a high impedance recording voltmeter for measuring the voltage generated between base and slider of potentiometer 30. As a result, the strip chart on recorder 36 records the maximum value of the voltage across terminals 13 divided by the series impedance of the recording meter 36, impedance 35, and the resistance looking back into the concatinated potentiometers. The resistance of each potentiometer preferably is of the order of 100 ohms, that of the current meter 36 is ordinarily ohms or less, and the maximum value of the series impedance 35 is in the range of 100,000 to 10,000 ohms. The adjustable impedance 35 has a magnitude which is large compared to that of the other impedances in the circuit. Accordingly, the value of the current through the recording current meter 36, printed on the chart is i= [b4 MW" (t t ,)]/N, where b, is a constant. The recorder uses one of the customary chart drives which may be energized from the alternating current source 9, but preferably the chart is driven by a flexible shaft turned by the shaft of drum 5, just as well log charts are driven at the present time. Thus the strip chart will be moved in accordance with depth of the drilling bit, and the periodic recording upon it is the value of n,,, appropriately plotted at the k" foot. The chart may also be driven by a clock so that n is plotted as a function of the elapsed time. A convenient, but not necessary, arrangement is shown in FIG. 3 for additionally recording on strip charts as functions of depth both the rotary speed of the drill rig and the weight on the bit. It was stated that the value of the rotary speed was manually set into the drilling cost indicator by adjustment of the dial on potentiometer 28. This dial can be mechanically coupled through well-known commercial means to drive the slider of a second potentiometer 37 mechanically mounted for simultaneous shaft movement. The dashed line 38 represents this arrangement. The resistance of potentiometer 37 is made linear and the voltage across terminals 10 is applied to the potentiometer. Thus the voltage e developed from one end to the slider of the potentiometer 37 is directly proportional to the revolutions per minute of the bit, or R. A recording strip chart voltmeter 39 is then connected to record the value of the voltage e upon actuation of a recording voltage to terminals 11 which cause the stylus of this recorder to contact the strip chart. Preferably this strip chart is mechanically arranged to be driven from the same source as that of the recorder 36 either with reference to footage drilled or to the elapsed time (in fact, this is true of all strip charts, so no further mention will be made of this fact). Similarly, if it is desired to record the weight on the bit W, the slider of potentiometer 29 is mechanically connected (dashed line 40) to the slider of a third potentiometer 41 which again is a helical potentiometer with linear resistance, the terminals of which are connected to terminals 10. A recording voltmeter 42 is connected to measure the slider voltage e which is directly proportional to W. Equaltion (3) is one of the important drilling variables which is to be graphically presented in strip chart form on the drilling cost indicator. The circuits shown in FIGS. 2 and 3 are arranged to give the individual values of n,,/ N. The circuit in FIG. 4 sums up the value of n N to produce a strip chart record of Z. In fact, this arrangement is so convenient that usually recorder 36 is omitted. The voltage across terminals 13 (FIG. 4) is applied in opposition to a voltage derived from a potentiometer 43 across the input to a servo-amplifier 44 driving an electric servo-motor 45. The total voltage applied across the outside terminals of the potentiometer 43 (preferably a helically wound potentiometer identical to potentiometer 30, i.e., with a linear resistance and an equal number of turns), are derived from terminals 10 by an isolating transformer 46 with a 1:1 ratio. The servo-motor 45 and its associated amplifier 44, accordingly, make up a conventionally known device for producing a voltage e equal to the voltage across terminals 13, i.e., so that the voltage applied to the input of servo amplifier 44 is zero. Accordingly, when the voltage across terminals 13 is zero (i.e., at the start of drilling each foot), the slider of potentiometer 43 is at the zero position and voltage a is Zero. As the voltage across terminals 13 increases, there is an equal and opposite growth to voltage 2 A second potentiometer 47 is physically identical to potentiometer 43 and is mounted to be driven from the same servo-motor shaft 48 through a magnetic clutch 49. This clutch is actuated from terminals 12. Potentiometer 47 is energized through isolating transformer 50 (identical to transformer 46) from terminals 10. With the arrangement shown in FIGS. 3 and 4, the first time the servo-system is actuated, the slider on potentiometer 47 will be actuated during the drilling of the first foot following insertion of a new bit, and will be moved along simultaneously with the slider of potentiometer 43 to a peak value representing the voltage directly proportional to n,. At this point, clutch 49 is de-energized while the voltage across terminals 13 goes to zero. The servomotor returns the slider on potentiometer 43 automatically to zero. As soon as cam wheel 22 has passed the position where its pegs are in contact with switches 20 and 21, i.e., when drilling on the second foot has started, voltage will gradually increase on terminals 13, clutch 49 will be re-engaged, and the slider on potentiometer 47 will gradually be moved up to a position representing the sum n, n Thus the voltage output e of potentiometer 47 represents at the end of each foot of drilling the summation of the maximum voltage across terminals 13 for all of the feet that have been drilled with the current bit and is therefore proportional to of the recorder 52 are deliberately kept quite low compared to that of impedance 51 which has been set to the value N. Accordingly, the current in the series circuit is the value of 2 as set out in equation (3). It should be added here that if no value is known for the anticipated bit life N, one simply adjusts impedance 51 to a convenient value and determines the maximum deflection on the strip chart from recorder 52. If this at the end of the use of that bit is not the value 1, but the value Z the driller now knows the correct value for N is the old value divided by Z. Thus, after a use ofa particular bit in a particular formation, the value of N is known. The next equation solved by this analog computer is that set out as (4), i.e., the cost of drilling only the k" foot. (Exactly the same arrangement may be employed for indicating the cost of drilling a different interval than 1 foot, for example, the last 2 feet, 3 feet, etc.) This arrangement is shown in FIG. 5. Here three identical isolation transformers 53, 54, and 55 energized re spectively from terminals 13, 13 and 10, supply voltage to identical helical potentiometers 56, 57, and 58, each of which has a dial for manual adjustment, calibrated in dollars. That for potentiometer 56 represents bit cost. That for 57 represents trip cost, i.e., the product of the hourly rig cost multiplied by the time in hours required to replace a bit at the current drilling depth. That for potentiometer 58 represents rig cost per hour. When the values appropriate to the current drilling condition have been set in by the driller, the voltage e is directly proportional to bit cost X n The voltage e,; is similarly directly proportional (with the same constant of proportionality) to the product of trip cost X n The series impedance (an adjustable resistor, preferably a helical potentiometer) 59 is identical to impedance 51 and, though not shown in FIGS. 4 and 5, is mounted for simultaneous revolution of the two potentiometer shafts, so that when one dials in the proper value of N for impedance 51, it is simultaneously set on impedance 59. The impedance of 59 is preferably 100 times that of potentiometers 56 and 57, at least. Resistance 60 completes the series circuit and has a resistance low in comparison with that of impedance 59, for example, less than one one-hundredth of this impedance. Accordingly, the input voltage across amplifier (61) is directly proportional to the current flowing through it which is therefore directly proportional to n N (C, C The setting of the slider on potentiometer 58 was directly proportional to the rig cost. Therefore, to the same proportionality as the quantities immediately above, the voltage e represents rig cost. This voltage is applied both to terminals (used in conjunction with the circuit of 56), and to the outside terminals of another helical potentiometer 62 of linear resistance. The slider of potentiometer 62 is shown in FIG. 5 to be driven by the resettable relay shown in FIG. 3. These are the clock, (31), the magnetic clutch 33, and the resetting spring 34. Accordingly, the slider of potentiometer 62 moves from its zero voltage position exactly as did the slider of potentiometer 30 to produce an indication directly proportional to the incremental drilling time, that is, the time (1,,- t The voltage e is directly proportional to the product of rig cost times incremental drilling time. This voltage is applied to the input of amplifier (63) which is identical to amplifier (61 The outputs of these two amplifiers are connected in series and the sum of the two voltages from these two amplifiers is fed to a recording strip chart voltmeter 64. Accordingly, the voltage recorded when terminals 11 of recorder 64 are energized will be directly proportional to the quantity Y of equation (4) and the strip chart thus records the incremental drilling cost for each foot or equivalent unit of distance, plotted against the appropriate location of the current drilling bit. The third important quantity charted by the drilling cost indicator is the average drilling cost per foot over the k feet that have been drilled. This quantity was given by equation (5). The relatively simple circuit required is shown in FIG. 6. Three additional identical isolation transformers 65, 66, and 67 are respectively supplied from terminals l0, l5 and 15. These transformers in turn energize linear potentiometers, preferably of the helical type, 68, 69, and 70. Potentiometers 68 and 69 are arranged for manual adjustment by the driller. In fact, the shaft of potentiometer 68 is mechanically connected to the shaft of potentiometer 56, and, accordingly, is set to the bit cost. The slider of potentiometer 69 is set manually on an appropriate dial to the trip time. Accordingly, voltage e represents bit cost, 2 represents the product of rig cost times trip time, or trip cost. Linear potentiometer 70 has the slider actuated by a resetting relay similar to that shown in FIG. 3. Thus it is composed of a clock (71), a magnetic clutch 72, and a return spring 73, acting on shaft 74. Clock (71) and clutch 72 are energized from terminals 10 through a second independent pole of switch 16, the other pole of which was shown in FIG. 2. Clock runs as long as the driller has switch 16 closed. He closes this switch only during drilling, and, accordingly, the slider of potentiometer 72 moves linearly with respect to time during the entire period that drilling is going on. The gearing in clock differs from that in clock by a factor which can be appropriately 30 or 60, so that while the first resetting relay moves the slider of potentiometer 30 through a distance proportional to a distance of one foot, the slider for potentiometer 70 is moved onethirtieth or one-sixtieth of this distance. Since the setting of potentiometer 70 is proportional to the total drilling time, the voltage output e is directly proportional to the product of rig cost times total drilling time, i.e., C, (I t The three voltages e e and 2 are additively connected in series and applied across a large variable impedance consisting of potentiometer 26, which was also shown in FIG. 2. As there described, the resistance between terminals 27 is directly proportional to the total footage drilled by the current bit. The circuit is closed through a recording current meter 75 of low impedance. Since the impedances of the part of the potentiometers 68, 69, and 70 is this circuit are also low compared to that of the setting of potentiometer 26, the current through the circuit is essentially proportional to the sum of these three voltages divided by the impedance set in potentiometer 26. The current in the circuit is directly proportional to the quantity X set out in equation (5). This value is appropriately recorded since the actuating terminals of meter 75 go to terminals 11. It is to be noted that the slider of potentiometer 47 and that of potentiometer 26 should be returned to zero at the time the bit is changed. This can be accomplished very simply by providing the drilling cost indicator with a simple manual return, such as a crank 80, with appropriate gear 81 which rotates gear 25 back to zero position, where a mechanical stop (not shown) on gear 25 indicates that the zero position on potentiometer 26 has been reached. This same arrangement also returns potentiometer 47 to a zero setting. The drilling cost indicator may have a number of forms. One appropriate one is shown in FIG. 7. Here the case 82 is provided with a plurality of manual dials and appropriate counters for measuring the quantities set into the various potentiometers, such as bit cost, rig cost, trip time, trip cost, and expected bit life. The controls for setting in weight on the bit and rpm of the drilling string and bit preferably take a different physical form to distinguish these which may be changed during the course of drilling from the other quantities. Switch 16 is arranged conveniently as is return crank 80 for insuring that the driller will reset these during the trip. The strip charts themselves may be of any desired form. In FIG. 7 a multiple strip chart 83 has the three variables arranged from left to right in the order Z, X, and l for plotting the fraction of the total life of the bit expended, the average cost of drilling, and the incremental cost per foot. A specimen chart from such an indicator is shown in FIG. 8. In order to change the values of a and B it is simply necessary to obtain appropriate helical potentiometers with different values of these exponents, remove a section of the side of the drilling cost indicator, and install the new potentiometer where the old one was used. As has been mentioned earlier, lacking better information, potentiometer 28 will be used with an exponent a of 0.33, and the one involving B (29) will have the value of 0.34. In use it is to be noted that the average cost per foot for the first few feet of drilling is quite high (theoretically infinity at the start) and decreases rapidly as more footage is drilled. This is shown graphically by the middle chart in FIG. 8. At some time during the life of the bit the average cost will decrease to some minimum value, and remain about there. Drilling will continue with no appreciable change in this average if the formation drillable remains reasonably constant, until the bit wear becomes quite critical, which will cause the drilling rate to decrease, and increase the average cost of drilling. The bit can be pulled whenever this minimum has been reached, even if the bit is not worn out, without increasing the average cost per foot since the next bit can be expected to drill a similar interval in the same formation at the same average cost per foot. Printout of the equation for the incremental cost Y allows, in effect, for a continous drill-off test to be run while proceeding with normal drilling. For example, a driller may choose to increase both the weight on bit W and the rpm R. The next recorded cost for the increment may increase, decrease, or remain the same. If incremental drilling cost Y decreases (and this is more sensitive than the average drilling cost X), the driller will realize that he can further increase weight on the bit, or rpm, or both. In fact, this indication of the trend in the drilling cost is of very definite importance. Even if the value of N placed in the computer is quite different from the true value, this drilling cost indicator will still show when wrong combinations of rpm and weight on the bit are being employed. The fraction of bit life expended (Curve Z) furnishes the additional information as to whether the bit still has considerably useful life or not. For example, assume that a sandstone formation is being drilled and there is evidence that the bit is becoming dulled, as indicated both on the X and Y curves. However, the value for Z shows the bit life is only half expended. A review of logs of offset wells shows that, for example, only an additional ten feet of sandstone remain to be drilled before a shale section is to be expected. The value for Z shows that drilling can be continued into the shale without fear of losing a cone from the bit. This would be more desirable than making a trip, since the new bit might be dulled to the point where it would not adequately drill the shale. This avoids an extra trip. It is realized that many changes could be made in the form of the computer without varying from the basic arrangement shown. The concatinated potentiometers could be replaced by variable ratio transformers, for example. Different arrangements to connect to the drill rig can be worked out. The example shown, however, is believed to illustrate the utility and simplicity of this particular type of drilling cost indicator. We claim: 1. Apparatus for the analog computation of drilling variables associated with a rotary drill rig comprising means (a) for generating a voltage in a circuit directly proportional to R, where R is the rate of rotation of the rotary table of said rig and a is an exponent relating drill bit life with rate of rotation of said bit, means (b) for generating from said voltage a second voltage varying directly as W, where W is the weight on the drill bit of said drill rig and B is an exponent relating drill bit life with weight on said bit, means (0) for generating from said second voltage a third voltage varying directly with the time I required to drill with said bit and said drill rig a unit distance in the ground, and an impedance (d) large compared with that of said means (c) connected in a closed circuit in series with the output of said means (0), in such a fashion that the total impedance in said circuit is substantially that of said impedance (d), whereby the current in said circuit varies as R W t. 2. Apparatus for the analog computation of drilling variables associated with a rotary drill rig comprising means (a) for generating a voltage in a circuit directly proportional to R", where R is the rate of rotation of the rotary table of said rig and 01 is an' exponent relating drill bit life with rate of rotation of said bit, means (b) for generating from said voltage a second voltage varying directly as W where W is the weight on the drill bit of said drill rig and B is an exponent relating drill bit life with weight on said bit, means (c) for generating from said second voltage a third voltage varying directly with the time t required to drill with said bit and said drill rig a unit distance in the ground, and a high impedance recording voltmeter, the impedance of which is large compared to that measured across the output of means (0) in the absence of said voltmeter, said recording voltmeter including a strip record and means for printing an indication of the voltage generated by means (c) on said strip record as drilling progresses. 3. Means for computing and recording drilling variables associated with a rotary drill rig comprising means (a) for successively generating a voltage directly proportional to [WWW where R is the rate of rotation of the rotary table of said rig, a is an exponent relating drill bit life with rate of rotation of said bit, W is the weight on the drill bit of said drill rig, B is an exponent relating drill bit life with weight on said bit, and t is the time to drill with said bit and said rig a uniform distance in the ground, for a succession of such uniform distances, means (b) responsive to said means (a) for generating a second potential directly proportional to the successive sum of a plurality of said voltages from means (a), an adjustable impedance (0) large compared with that of said means (b) connected in series with the output from said means (b), and a recording current meter connected in series with said impedance (0) and the output of said means (b) to complete a series circuit, said meter being adapted to record on a moving strip record separately the value of the current in said circuit for each of a plurality of times corresponding to each said successive sum. 4. Apparatus for the analog computation of drilling variables associated with a rotary drill rig comprising means (a) for generating a voltage in a circuit di recently proportional to R. where R is the rate of rotation of the rotary table of said rig and oz is an exponent relating drill bit life with rate of rotation of said bit, means (b) for generating from said voltage at the output from said means (a) a second voltage varying directly as W. where W is the weight on the drill bit of said drill rig and B is an exponent relating drill bit life with weight on said bit, means (c) connected to the output from means (b) for generating from said second voltage a third voltage varying directly with the time I required to drill with said rotary drill rig a unit distance in the ground, means (d) connected to the output of means (c) for generating from said third voltage a fourth voltage varying directly as the sum of the cost of said bit and the cost to place said bit at the bottom of the well being drilled by said rig, an adjustable impedance (e) in series with the output from said means (d) and large compared with the output impedance of said means ((1), a first amplifier (f) closing a circuit with the output of said means (d) and said impedance (e), for producing an output voltage directly proportional to the current in said circuit, means (g) for producing a fifth voltage directly proportional to the product of rig cost per unit time interval times the time required to drill with said bit and said drill rig said unit distance in the ground, a second amplifier (h) of substantially the same gain as that of said first amplifier (f), the input to which is connected to the output of said means (g) and the output of which is connected additively in series with that of said first amplifier (f), and a recording voltmeter connected in series with the outputs of said first amplifier (f) and said second amplifer (h), said voltmeter being adapted to record on a moving strip record the sum ofthe output voltage of said first amplifer (f) and said second amplifier (h) at a plurality of times corresponding to the successive driling of each of a plurality of integral multiples of said unit distance. 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