A Multiple Regression Approach to Optimal Drillingand Abnormal Pressure Detection
A. T. BOURGOYNE, JR.
F. S. YOUNG, JR.MEMBERS SPE-AIME
LOUISIANA STATE U.BA TON ROUGE, LA.
BAROID DIV. OF N L INDUSTRIES, INC.HOUSTON, TEX.
DRILLING MODEL
The drilling model selected for predicting theeffect of the various drilling parameters, Xj' onpenetration rate, dD/dt, is given by
where Exp (z) is used to indicate the exponentialfunction eZ • The modeling of drilling behavior in agiven formation type is accomplished by selectingthe constants al through as in Eq. 1. Since Eq. 1is linear, those constants can be determined from amultiple regression analysis of field data.
has been based on meager laboratory and field data.We have tried here to (1) combine what is knownabout the rotary drilling process into a singlemodel, (2) develop equations for calculatingformation pore pressure and optimum bit weight,rotary speed, and jet bit hydraulics that areconsistent with that model, and (3) provide amethod for systematically "calibrating" the drillingmodel using field data.
(1)a.x.)' .] J
81:
j=2Exp(a1 +=dD
dt
ABSTRACT
INTRODUCTION
Over the past decade, a number of drilling modelshave been proposed for the optimization of therotary drilling process and the detection of abnormalpressure while drilling. These techniques havebeen largely based upon limited field and laboratorydata and often yield inaccurate results. Recentdevelopments in onsite well monitoring systemshave made possible the routine determination ofthe best mathematical model for drilling optimization and pore pressure detection. This modeling isaccomplished through a multiple regression analysisof detailed drilling data taken over short intervals.Included in the analysis are the effects of(I) formation strength, (2) formation depth, (3)formation compaction, (4) pressure differential acrossthe hole bottom, (5) bit diameter and bit weight, (6)rotary speed, (7) bit wear, and (8) bit hydraulics.
This paper presents procedures for using theregressed drilling model for (1) selecting bit weight,rotary speed, and bit hydraulics, and (2) calculatingformation pressure from drilling data. The applicationof the procedure is illustrated using field data.
Operators engaged in the search for hydrocarbonreserves are facing much higher drilling costs asmore wells are drilled in hostile environments andto greater depths. A study by Young and Tanner l
has indicated that the average well cost per footdrilled is increasing at approximately 7.5 percent/year. Recently, more emphasis has been placed onthe collection of detailed drilling data to aid inthe selection of improved drilling practices.
At present, many people are using one drillingmodel for optimizing bit weight and rotary speed, adifferent drilling model for optimizing jet bithydraulics, and yet another model for detectingabnormal pressure from drilling data. Each model
This paper will be printed in Transactions volume 257,which will cover 1974.
EFFECT OF FORMATION STRENGTH
The constant al primarily represents the effect offormation strength on penetration rate. It is inverselyproportional to the natural logarithm of the squareof the drillability strength parameter discussed byMaurer. 2 It also includes the effect on penetrationrate of drilling parameters that have not yet beenmathematically modeled; for example, the effect ofdrilled solids.
EFFECT OF COMPACTION
The terms a2x2 and a3x3 model the effect ofcompaction on penetration rate. x2 is defined by
x2 = 10,000.0 - D· ..... (2)
and thus assumes an exponential decrease In
penetration rate with depth in a normally compactedformation. The exponential nature of the normalcompaction trend is indicated by the publishedmicrobit and field data of Murray,3 and also by thefield data of Combs4 (see Fig. 1). x3 is defined by
AlIGUST,1974 371
. (7)
. . (6)
+1200+800
OWel1 A216Well G21
oWel1 021• Berea 4
+400
Differential Into Formation I psi
o
In
-h, .
=
=
3.0.,"0 2.00::
l:0
1.0-0~-.,l:.,
0.4a..,>- 0.20.,
0::0.1
-400
Pressure
EFFECT OF ROTARY SPEED, N
The term a6x6 represents the effect of rotaryspeed on penetration rate. x6 is defined by
EFFECT OF BIT HYDRAULIC
The term asxs models the effect of bit hydraulicson penetration rate. xs is defined by
FIG. 2 - EFFECT OF DIFFERENTIAL BOTTOM-HOLEPRESSURE ON PENETRATION RATE. S.6
EFFECT OF TOOTH WEAR, h
The term a7x7 models the effect of tooth wear onpenetration rate. x7 is defined by
and thus assumes that penetration rate is directly
proportional to N a6 as indicated by several
authors.4 ,S-12 Note that the term e a6x6 is normalizedto equal 1.0 for 100 rpm. Reported values of therotary speed exponent range from 0.4 for very hardformations to 0.9 for very soft formations. 12
authors.4 ,S-12 Note that the term e asxs is normalizedto equal 1.0 for 4,000 lb per inch of bit diameter.The threshold bit weight, (W / d)1' must be estimatedwith drill-off tests. Reported values of the bitweight exponent range from 0.6 to 2.0.
where h is the fractional tooth height that has beenworn away. Previous authorsS •9 have used morecomplex expressions to model tooth wear. Howeverthose expressions were not ideally suited for themultiple regression analysis procedure used toevaluate the constant a7 from field data. Fig. 3shows a typical comparison of the previously
published relations and e a7 X7. The value of a7
depends primarily on the bit type and, to a lesserextent, the formation type. When carbide insert bitsare used, penetration rate does not vary significantlywith tooth wear. Thus the tooth wear exponent,a 7 , is assumed to be zero, and the remainingexponents, a1 through a6 and as, are regressed.
and thus assumes that penetration rate IS directly
proportional to (W /dts as indicated by several
x 4 D (g p). . . . . . . (4)P c
and thus assumes an exponential decrease inpenetration rate with excess bottom-hole pressure.Field data presented by Vidrine and BenitS and by
Combs,4 and laboratory data presented byCunningham and Eenink 6 and by Garnier andvan Lingen 7 all indicate an exponential relationbetween penetration rate and excess bottom-holepressure up to about 1,000 psi (see Fig. 2). Vidrineand Beni t also noted an apparent relation betweenthe effect of differential pressure on penetrationrate and bit weight. However, no consistentcorrelation could be obtained from the availabledata, so no bit weight term was included in Eq. 4.
EFFECT OF DIFFERENTIAL PRESSURE
The term a4x4 models the effect of pressuredifferential across the hole bottom on penetrationrate. x4 is defined by
,,,,,
EFFECT OF BIT DIAMETERAND BIT WEIGHT, Wid
The term asxs models the effect of bit weightand bit diameter on penetration rate. xs is defined
by
:5 1.01---------~--------_I..o....~ 0.5CDa..
CD>.. 0.2oCDa:
! 2.0oa:
5.0 r---~------,-------"""
Vertical Depth, ft.
FIG. 1 - EFFECT OF NORMAL COMPACTION ONPENETRATION RATE.
DO. 69 (gpand thus assumes an exponential increase in?~netratiQn rate with pore pressure gradient. Theexponential nature of the effect of undercompactionon penetration rate is suggested by compactiontheory, but has not yet been verified experimentally.Note that the effect of compaction on penetration
a 2 x + a3 x 3 .rate, e 2 , has been normalized to equal1.0 for a normally compacted formation at 10,000 ft.
372 SOCIETY OF PETROLEUM ENGIl\"EERS JOURl\"AL
Gal./Min. Diameler
Q ~ ~3i1; 0 ·101/2 a · 10
!8 A · aS:6 V · &'
+- ~---:t.,
. • L o~ c 10
"11 - ':"16'
W-IOOO Lb. N-75RPM
II I It.r
50IO
PSI' I I 1-2
1086
4
20
1008060
40
0060.1 0.2 04 0.6 1.0 2 4 6 10 20 4060100150
Eqs. 1 through 7 define the general functionalrelations between penetration rate and the otherdrilling variables, but the constants a2 through asmust be determined before these equations can beapplied. The constants a2 through as are determinedthrough a multiple regression analysis of detaileddrilling data taken over short depth intervals.
The idea of using a regression analysis of pastdrilling data to evaluate constants in a drilling rateequation is not new. For example, it was proposedby Graham and Muench lO in 1959 in one of the firstpapers on drilling optimization. This approach wasused by Combs4 in his work on the detection ofpore pressure from drilling data. However, much ofthe past work in this area has been hampered bythe difficulty in obtaining large volumes of accuratefield data and because the effect of many of thedrilling parameters discussed above were ignored.Recent developments in onsite well monitoring 1
have made it possible to routinely regress the morecomplex drilling equation (Eq. 1).
A derivation of the multiple regression-analysisprocedure is presented in detail in Appendix A.Theoretically, only eight data points are required
MUL TIPLE REGRESSION TECHNIQUE
Ta is calculated from a dull bit grading. Note thatEq. 10 is normalized so that the abrasiveness factorTH is numerically equal to the hours of tooth lifethat would result if a Class 1 bit were operated at"standard conditions", i.e., a bit weight of 4,000 lbper inch of bit diameter and a rotary speed of 100rpm. Likewise, Eq. 11 is normalized so that thebearing constant Ta is numerically equal to thehours of bearing life that would result if the bitwere operated at standard conditions. By normalizingthe bit wear equations in this manner, fieldpersonnel can attach a physical meaning to the bitwear constants and thus more easily detectanomalous bit gradings.
Estes14 has pointed out that the rate of bit wearwill be excessive if too great a bit weight is used.His recommended maximum bit weights are shownin Table 3. The recommended maximum bit weightsare based on bearing capacity for milled-tooth bitsand on cutting structure for insert bits.
1.00.8
..... (10)
0.60.4
pq (8)350 11dn
0.2
11 + T /20 . . . . . . . . (9)P Y
BIT WEAR MODEL
=
=
=
=
0.0 '--__..I.-__....I....__....L..__--'-__--'
0.0
0.8c::0...uc:: 0.6:>
l.L.
...0Q)
~ 0.4.c...00I-
0.2
dhdt
dBdt
In addi tion to a penetration rate model, equationsare also needed to estimate the condition of thebit at any time. Tooth wear was modeled using
(W) -4d
[ max ]
(W) _~d max
1.0r-----------------,
and is based on microbit experiments performed byEckel. 13 As shown in Fig. 4, Eckel found thatpenetration rate was proportional to a Reynolds
number group (-.!!!!.-) raised to the 0.5 power. Sincefl dn
fl' the apparent viscosity at 10,000 sec -1, is notroutinely measured and recorded it must be estimatedusing the relation
where the constant b depends upon bearing typeand mud type (see Table 4) and the bearing constant
where the constants HI, H2 , H3 , and (W/d)maxdepend upon bit type (see Tables 1, 2, and 3) andthe abrasiveness constant TH is calculated from adull bit grading (see example of Appendix D).
Bearing wear was estimated using
Fractional Tooth Dullness, h
FIG. 3 EFFECT OF TOOTH WEAR ON PENETRA-TION RATE (CHIPPING-TYPE TOOTH WEAR).
Reynolds Number Funclion
FIG. 4 - DRILLING RATE VS REYNOLDS NUMBERFUNCTION.13
AUGUST,1974 373
TABLE 1 - ROCK BIT CLASSIFICATION GUIDE (AFTER ESTESI4).
DRESSER - SECUR ITY HUGHES G. W. MURPHY SM ITH
MILL- ''T'' "G" "5 11 'I" IIG" I'SII "SG" IIJ" '1" "G" "S't "5 Gil 11'1 I'GII 115 11 IISG"TOOTH STO GAGE GAGE SEAL STO GAGE GAGE SEAL SEAL BEAR STO GAGE GAGE SEAL SEAL STO GAGE GAGE SEAL SEALCLASS
I - 1 S3S S33S OSC3A X3A YT3A S13A OS SOS2 S3 S3T S33 OSC3 X3 Y13 YT3T S13 OT on SOT3 S4 S4T S4TG S44 OSCIG CIC OOG XIG XOG YTIA YTlT YTIAG SHAG OG OGT OGH SOG SOGH4 S6 S6G OSC K2 K2H I
I2 - I M4N M4NG M44N OW4 OOV XOV YSI YSIG SSIG V2 V2H SV I SVH
CLASS: D - I I - 2 I - 5 2 - 4SMITH BHOJ OJORESSER 53SJ OS OM
THE ABOVE TYPES ARE GENERALLY AVAILABLE IN POPULAR SIZES. OBSOLETE TYPES NOT L1STEO MAY ALSO BE AVAILABLE IN SOME SIZES.
DEVIATION CONTROL BITS: NOTES: (j) INOICATES A JOURNAL BEARING MILL-TOOTH BIT.CLASS 0 BITS ARE TWO-CONE.
*Sealed bearing bits are 8 to 10 percent lower; journal bearingbits are 10 to 12 percent higher.
**Insert.. bit maximums are based on cutting structure, notbearing capacity.
Bearing Type Drilling Fluid b-Barite mud 1.00Sulfide mud 1.25
Nonsealed Water 1.90Clay mud 2.04Oil base mud 2.55
Seal ed 2.80
to solve for the eight unknowns a 1 through as.However, in practice this is true only if Eq. 1models the rotary drilling process with IOO-percentaccuracy. Needless to say, it never happens. Whenonly a few data points are used in the analysis offield data, even negative values are sometimescalculated for one or more of the regressionconstants. A sensitivity study of the multipleregression-analysis procedure indicated that thenumber of data points required to give meaningfulresults depends not only on the accuracy of Eq. 1,but also on the range of values of the drillingparameters x2 through xs' Table 5 summarizes therecommended minimum ranges for each of thedrilling parameters and the recommended minumumnumber of data points to be used in the analysis.When any of the drilling parameters, Xj' have beenheld essentially constant through the intervalanalyzed, a value for the corresponding regression
374 SOCIETY OF PETROI.ElTM E:'\GI:'\EERS JOUR:'\AL
constant, aj' should be estimated from past studiesand the regression analysis should be carried outfor the remaining regression constants. As thenumber of drilling parameters included in theanalysis are decreased, the minimum number of datapoints required to calculate the remaining regressionconstants is also decreased (see Table 5). In manyapplications, data from more than one well had to becombined in order to calculate all eight regressionconstants.
The penetration rate, bit weight, and rotary speedshould be monitored over shorr depth intervals toinsure that most of the information recorded isrepresentative of a single type of formation. Adepth interval of 2 to 5 ft was found to giverepresentati ve data and still keep the volume ofdata required within reasonable limits.
Field data taken in shale in an offshore Louisianawell are shown in Table 6. Note that the primary
drilling variables required for the regressionanalysis are depth, penetration rate, bit weight perinch of bit diameter, rotary speed, fractional toothwear, Reynolds number parameter, mud density, andpore pressure gradient. To calculate the bestvalues of the regression constants a 1 through asusing the data shown, the parameters x 2 through Xsmust be calculated using Eq. 2 through 8 for eachdata entry. Eight equations with the eight unknownsal through as can then be obtained from x2 throughXs using the procedure described in Appendix A.For example, the first of the eight equations definedin Appendix A is given by
na1 + a 2 EX2 + a 3 EX 3 + a4 EX4
+ as ExS + a 6 EX6 + a 7 EX7dD
+ as Ex S = E In dt
TABLE 5 - RECOMMENDED MINIMUM DATA RANGES FORREGRESSION ANALYSIS
*Maximum observed value less minimum observed valueincluded in regression analysis.
Number of MinimumParameters Number of Points
5x 10 a 2 + 0.940.89
x 10 5a 3 - 0.36 x 10 6 a 4
20 a - 7.4 a 6 - 12 a 75
6.3 as = SS
30 a 1
Thus, using the 30 data points of Table 6 In thisequation yields
When the resulting system of eight equation IS
solved for the eight unknowns, the constants, al
3025201510
74
876
54
32
Minimum*Range
2,000
15,00015,000
0.400.50
0.200.50
Parameter
TABLE 6 - EXAMPLE DATA FOR MUL TIPLE REGRESSION ANALYSIS(Taken in shale, Offshore Louisiana area)
Drilling Rotary Reynalds PoreData Depth Bit Rate Bit Weight Speed Tooth Number ECD GradientEntry (ft) Number (ftlhr) (l,OOOlb/in.) (rpm) Wear Function (Ib/gal) (Ib/gal)
the use of such systems is prohibited by economics.Previous authors have published techniques forcomputing a variable bit weight and rotary speedschedule as well as the best constant weight-speedschedule. Galle and Woods 9 have reported thatthe simpler constant weight-speed schedule resultsin only slightly higher costs per foot. A recent paperby Reed 1? indicates a difference of less than 3percent in the cost per foot between the variableweight-speed and constant weight-speed schedulesfor the cases studied.
Combining Eq. 1, an integrated form of Eq. 11,and Eq. 12 leads to the following expression forcost per foot for a given bit weight per inch of bitdiameter, W/ d, rotary speed, N, and rotating time,
tb'
where
OPTIMAL DRILLING
through as, given for Well 1 in Table 7 are obtained.Results obtained for shale using several otherwells in the same general offshore Louisiana areaare also shown in Table 7 for comparison. Wells 2and 3 of Table 7 were drilled from the same platform.
The term "drilling optimization" was appliedfirst to procedures for selecting jet bit hydraulics.The term was later expanded to include proceduresfor selecting bit weight and rotary speed,S,9 andrecently has been used to refer to a broad plan forthe selection of mud types and mud properties,bit types and operating conditions, and' casingtypes and setting depths,1S At present, however,only a limited number of drilling variables can behandled using formal mathematical optimizationprocedures. Equations derived from the drillingmodel of Eq. 1 are presented in this paper for the
optimization of bit weight, rotary speed, and jetbit hydraulics. The derivations of the optimizationequations are given in Appendixes Band C. It wasassumed in the derivation that the drilling cost perfoot, CI, could be expressed in terms of the bitcost, Cb , the hourI y rig cost, C" the trip time, t t'
the connection time, tc ' the drilling time, tb' andthe footage drilled, /}.D. The cost equation assumedis given by
=Cb + Cr(tt + t c + ~)
6D= Exp
6L:
j=2a.x.
J J
(12)
Thus risk factors, hole deviation problems, holewashout problems, and variable pump costs wereignored. Also inherent in the use of Eq. 1 are theassumptions that roller-type rock bits are used andthat bit balling does not occur. Drilling variablesnot included in Eq. 1 such as mud type and percentsolids were also ignored. These limitations shouldbe kept in mind when using the optimizationequations listed below.
=
. . . . . (14a)
BIT WEIGHT AND ROTARY SPEED
As discussed in a previous paper,16 optimum bitweight and rotary speed can be determinedautomatically at the well site with a computerizeddrilling control system. However, in many cases,
U
. . . .04b)
TABLE 7 - RESUL TS OF REGRESSION ANALYSIS FOR GULF COAST AREA
02 03 04Well Depth Range 01 (l0- 3) (l0- 3) (l0- 4) aS 06 07 08
*Volue assumed rather than calculated because carresponding drilling param-eter did not vary Over a wide enough range to be included in the regressiononalysi s.
376 SOCIETY OF PETROLEUM ENGINEERS JOURNAL
If it is assumed that bit life, lb' is limited by eithertooth wear or bearing wear, then the rotating time,lb' can be obtained from the integrated forms of Eq.10 or 11. Thus, the smaller of the two rotating times,lb' given by either
=or
N = 100opt
(w) (W)T d d~ max opt
~ H3[(~) -4Jmax
. . (16)
and the optimum rotary speed, Nopt' is given by
. . . . . . . . . . . . . . . . (18)
where the abrasiveness factor, TH, is obtained froma bit grading and Eq. 10. When using Eqs. 16 through18 to obtain the optimum bit weight and rotaryspeed, the cost per foot should be computed for atooth wear, H, of both 1. 0 and 0.95 to insure thevalidity of the assumption that bit life is limitedby tooth wear.
Unfortunately, simple analytical expressions forthe best constant bit weight and rotary speed couldnot be obtained for the case in which bearing wearlimits bit life. In this case, a cost-per-foot tableshould be constructed, using Eqs. 13 through 15bin an i terati ve mann er.
OPTIMUM HYDRA ULIes
The drilling cost equation shown as Eq. 12 doesnot properly account for the variable pump costassociated with the optimization of bit hydraulics.However, since the variable pump cost is usuallysmall compared with the hourly rig cost, this is nota serious limitation. Nelson lS has shown that thevariable operating cost of the pump can be relatedto the hydraulic horsepower developed by the pump.For most of the larger rigs, the variable pump costis approximately $0.02/hp hr. Thus, if by optimizingjet bit hydraulics, an additional 500 hp is requiredfor a 12-hour bit run, the incremental pump operatingcost would be only $120. Since this is usually aninsignificant portion of the total cost, theoptimization of jet bit hydraulics is essentiallyachieved by maximizing the penetration rate.Inspection of Eq. 1 reveals that penetration ratewill be a maximum when the term asxs is a maximum.As shown in Appendix C, this is achieved bychoo sing nozzle sizes and pump' operating conditions so that the pressure drop across the bit, ~P b'
is related to the maximum pump pressure, Pp ' byby
100[-] ..... (15b)N
=
=
~[~ tt tel
HI1],= + + [-
c r a 6(17)
TOOTH WEAR LIMITS BIT LIFE
Relatively simple analytical equations for thebest constant weight and rotary speed were derived(see Appendix B) for the case in which tooth wearlimits bit life, using a procedure described byMaratier.I2 The optimum bit weight per inch of bitdiameter, (W / d)opt, is given by
should be used in Eq. 14c. A cost-per-foot table forvarious combinations of bit weight and rotary speedcan be generated using Eq. 13 through 14c. Table 8is a cost-per-foot table for an example problemgi ven in Appendix D. Note that the cost-per-foottable can be used to quickly identify (1) the bestcombination of bit weight and rotary speed, (2) thebest rotary speed for a given bit weight, and (3) thebest bit weight for a given rotary speed.
where the constants HI and (W / d)max are obtainedfrom Table 2 and the constants as and a6 areobtained from the regression analysis.
The expected bit life is given by
TABLE 8 - EXAMPLE COST PER FOOT
Bit Weight per Inch of Bit Diameter (1,000 Ib/in.)RotarySpeed(rpm)
where m is the slope of a plot of parasitic pressuredrop vs flow rate on log-log paper. Note that byoperating in accordance with Eq. 19, the jet impactforce at the bit as well as the Reynolds numberfunction Xs is maximized. Theoretical considerationsindicate a value of 1.8 for m. However, Scott hasrecently reported measuring m values as low as1.0. 19
An evaluation of the proposed optimization schemehas indicated that in many cases a considerable
A If (; If ST. I 9 7 4 377
(asxs
aaxB) ]. . (20)
I--
t::t~=f-C---foI'ud Density
H't=:~ ~ Pore Pressore (Sonic)
: ~ Ofe Pressure (Kp).~ :t-
-t:1---1= -f:'=80
0
0
fC-1----
0
-
,--+---
0- -=1--
1- 1=1=
/:=:ll;;:: fC=f=
-
1---"- \--
f- f---t-~
+-. t-1=1--- I---
LoglO=
Fig.6 Drillability: and Pore Pr...sure PlotHot Wire Gas PressUfe Gradient
8 10 12
(The drillability parameter, which is based on Eq. 1,is somewhat analogous to the lCd-exponent"developed by Jorden and Shirley 11 using a moresimplified penetration rate equation.) The drillabilitylog is then analyzed to determine the type offormation being drilled. The pore pressure gradientcan be related to the drillability parameter, Kp ' ina given type of formation by
H+-
cannot be optimized by formal mathematicaltechniques. Thus the choice of mud properties, bittype, casing points, etc., must be based on pastexperience and what is known about the down-holeenvironment. The most important design parameterneeded to insure a low-cost, trouble-free operationis the pore pressure of the formations penetrated.Since the drillability of a given type of formationis affected by the pressure differential at thebottom of the hole as well as by the effectiveformation compaction, a normalized penetration ratelog can be used to estimate the formation pressure.
The regression constants and the drilling dataare used to compute and plot a drillability parameter,Kp, defined by
reduction in drilling cost could be achieved bydrilling optimization. Estimated reductions in drillingcos t on several wells varied from a few percent toto 30 percent and averaged about 10 percent. Thisvalue is in agreement with other reportedstudies. 9, IS However, where bit life is limited bytooth wear, the proposed optimization scheme ismuch easier to apply than previously publishedtechniques 8 , 9,16,17 because the optimum conditionscan be calculated from analytical expressions ratherthan by trial and error or by involved graphicalprocedures.
An example bit run optimization is shown inAppendix D. Note that the optimum bit weight androtary speed are very sensitive to the regressionconstants as and a6 . Thus, the accuracy of theoptimization should systematically improve asexperience is gained in an area.
As shown in Appendix D, bit hydraulics isoptimized using a technique outlined by Scott. 19
The standpipe pressure is measured at both a normalcirculating rate and a reduced circulating rate. Thepressure drop through the bit is then estimated atboth circulating rates using the orifice equation ora Hydraulic Slide Rule. The total parasitic pressurelos s is then determined as the difference betweenthe standpipe pressure and the pressure drop throughthe bit. Knowing the parastic pressure drop at tworates allows the graphical estimation of the exponentm (see Fig. 6). The optimum flow rate and pressuredrop across the bit can then be calculated withEq. 19. Since the pressure drop at a reducedcirculating rate is normally recorded twice a dayto aid in "kick" control calculations, the optimumbit hydraulics can usually be calculated withoutmaking any additional measurements.
FIG. 5 CALCULATION OF OPTIMUM BITHYDRAULICS. FIG. 6 - DRILLABILITY AND PORE PRESSURE PLOT.
378 SOCIETY OF PETROLEUM ENGINEERS JOURNAL
paper. We also wish to acknowledge all those whoassisted in its preparation, and particularly WesleyCarew for his assistance in developing softwareand preparing data.
a function of bit weight per inch, rotaryspeed, and rotating time
weight on bit per inch of bit diameter,1,0001b/in.
J
n
K
m
u
N
WidACKNOWLEDGMENTS
To smooth out minor lithology variations, a 25-ftaverage value of pore pressure gradient is usuallyplotted. After formation cuttings are obtained atthe surface, the formation pressure is also estimatedon the basis of the density of the cuttings. However,the normalized penetration rate log provides themost current information.
At the present time, this approach has beentested only in the Gulf Coast area. The response ofdrillability to an increasing pore pressure shouldbe a maximum in this type of geologic environmentbecause the zones of high pore pressure areundercompacted. Also, since the lithology of thisarea is relatively simple, formation types are moreeasily determined. An example drillability log andpore pressure plot is shown in Fig. 6. Also shownfor comparison are the pore pressure gradientsobtained from a sonic log. In general, a comparisonbetween pore pressure gradients computed from adrillability plot and pore pressure gradients computedfrom a sonic log yields a standard deviation ofabout 1.0 Ib/gal. As the number of wells drilledin an area increases and the regression constantsbecome better defined, rhe accuracy of the porepressure calculation should improve. This wasobserved in Wells 2 and 3 of Table 7. A comparisonof pore pressures computed from a sonic logyielded a standard deviation of 1.1 Ib/gal after thefirst well was drilled and a standard deviation of0.9 Ib/gal after the second well was drilled.
The new procedure described here was appliedon several wells in the Gulf Coast area. Thefollowing conclusions resulted from this evaluation.
1. When modern well monitoring equipment isused, a regressional analysis procedure can beused to systematically evaluate many of theconstants in the penetration rate equation.
2. In many cases, data must be obtained frommore than a single well before all the regressionconstants can be evaluated.
3. The regression analysis procedure IS moreeasily applied in young geologic strata such asthose on the Gulf Coast.
4. The use of relatively simple drilling optimization equations can reduce drilling costs byabout 10 percent.
5. Formation pressure can be estimated fromdrilling data with a standard deviation of about 1Ib/gal.
=
We wish to thank the management of Baroid Div.,N L Industries, Inc., for permission to publish this
Ali G [: , T. I 9 7,\ 379
REFERENCES
SUBSCRIPTS
C calculated
OB observed
1. Young, F. S., Jr., and Tanner, K. D.: "RecentDevelopments in On-Site Well Monitoring Systems,"Petroleum Short Course, Texas Tech U., Lubbock(April 1972).
2. Maurer, W. C.: "The 'Perfect Cleaning' Theory ofRotary Drilling," ]. Pet. Tech. (Nov. 1962) 12701274; Trans., AIME, Vol. 225.
3. Murray, A. S., and Cunningham, R. A.: "The Effectof Mud Column Pressure on Drilling Rates," Trans.,AIME (1955) Vol. 204, 196-204.
4. Combs, G. D.: "Prediction of Pore Pressure FromPenetration Rate," Proc., Second Symposium onAbnormal Subsurface Pore Pressure, Baton Rouge,La. (J an. 1970).
5. Vidrine, D. J., and Denit, E. J.: "Field Verificationof the Effect of Differential Pressure on DrillingRate," ]. Pet. Tech. (July 1968) 676-682.
6. Cunningham, A. J., and Eenink, J. G.: "LaboratoryStudy of Effect of Overburden, Formation and MudColumn Pressure on Drilling Rate of PermeableFormations," Trans., AIME (1959) Vol. 216, 9-17.
7. Garnier, A. J., and van Lingen, N. H.: "PhenomenaAffecting Drilling Rates at Depth," Trans., AIME(1959) Vol. 216, 232-239.
8. Edwards, J. H.: "Engineering Design of DrillingOperations," Drill. and Prod. Prac. API (1964) 39.
9. Galle, E. M., and Woods, A. B.: "Best ConstantWeight and Rotary Speed for Rotary Rock Bits,"Drill. and Prod. Prac. API (1963) 48.
10. Graham, J. W., and Muench, N. L.: "AnalyticalDetermination of Optimum Bit Weight and RotarySpeed Combinations," paper SPE 1349-G presentedat SPE-AIME 34th Annual Fall Meeting, Dallas, Oct.4-7, 1959.
dDIndt
. . . . . (A-2)
a.x.· .. (A-1)J J
a.x.J J
8L:
j=2
8L:
j=2
=
=r.~
dDIndt
then the problem is to select a 1 through a 8 so thatfor n data points, where n is any number greater
than 8, the sum of the square of the residuals,n~ r?, is a minimum. Using The Calculus,i=l
APPENDIX A
11. Jorden, J. R, and Shirley, O. J.: "Application ofDrilling Performance Data to Overpressure Detection,"]. Pet. Tech. (Nov. 1966) 1387-1394.
12. Maratier, J.: "Optimum Rotary Speed and Bit Weightfor Rotary Drilling," MS thesis, Louisiana State U.,Baton Rouge (June 1971).
13. Eckel, J. J.: "Microbit Studie s of the Effect of FluidProperties and Hydraulics on Drilling Rate," ]. Pet.Tech. (April 1967) 541-546; Trans., AIME, Vol. 240.
14. Estes, J. C.: "Selecting the Proper Rotary RockBit," ]. Pet. Tech. (Nov. 1971) 1359-1367.
15. Lummus, J. L.: "Acquisition and Analysis of Datafor Optimized Drilling," ]. Pet. Tech. (Nov. 1971)1285-1293.
Taking the logarithm of both sides of Eq. 1 yields
19. Scott, K. F.: "A New Practical Approach to RotaryDrilling Hydraulics," paper SPE 3530 presented atSPE-AIME 46th Annual Fall Meeting of SPE, NewOrleans, La. (1971).
20. Bourgoyne, A. T., Rizer, J. A., and Myers, G. M.:"Porosity and Pore Press ure Logs," The DrillingContractor (May-June 1971) 36.
21. Campbell, J. M., and Mitchell, B. J.: "Effect ofTooth Geometry on Tooth Wear Rate of Rotary RockBits," paper presented at API Mid-Continent DistrictSpring Meeting (March 1959).
22. Hebert, W. E., and Young, F. S., Jr.: "Estimation ofFormation Pressure with Regression Models ofDrilling Data," ]. Pet. Tech. (Jan. 1972) 9-15.
23. McLean, R. H.: "Velocities, Kinetic Energy andShear in Cross flow Under Three-Cone Jet Bits,"]. Pet. Tech. (Dec. 1965) 1443-1448; Trans., AIME,Vol. 234.
Eq. A·l can be checked for validity in a givenformation type at each depth at which data havebeen collected. 1£ we define the residual error ofthe ith data point, ri' by
bit weight per inch of bit diameter atwhich the bit teeth would fail instantaneously, 1,0001b/in.
optimum bit weight per inch
threshold bit weight at which the bitbegins to drill, 1,000 lb/in.
normal compaction drilling parameter
undercompaction drilling parameter
pressure differential drilling parameter
bit weight drilling parameter
rotary speed drilling parameter
tooth wear drilling parameter
bit hydraulics drilling parameter
the apparent viscosity at 10,000 sec-I,cp
mud density, lb/gal
equivalent circulating mud density atthe hole bottom, lb/gal
bearing constant or Ii fe of bearings atstandard conditions, hours
formation abrasiveness constant or lifeof teeth at standard condi tions, hours
= overbar, designation of mean
PPc
(W /d)opt
(W /d)t
(W /d)max
380 SOCIETY OF PETROLEUM E'iGI:-iEERS JOrR:-iAL
a.x.J J
-a7he dt. . . .(B-2a)
6Exp(a1 + E
j=2=
OPTIMIZAnON OF BIT WEIGHTAND ROTARY SPEED
=
=
APPENDIX B
TOOTH WEAR LIMITS BIT LIFE
The cost per foot can be expressed by
then Eq. B-2a becomes
The footage drilled, f\,.D, can be obtained from Eq. 1:
If the symbol] 1 is used to represent a function ofbit weight per inch, W/d, and rotary speed, N,which is defined by
dO1ndt= E
n 2d E r.
dr.i=1 ~ nE 2r. ~=
da.d a. i=1 ~
J J
n= E 2r.x. = 0
i=1 ~ J
2a1 EX2 + a2 EX2 + a3
Ex2x3
dO+ • • • • + aaEx2x a = Ex2 1ndt2a1 EX3 + a 2 Ex3x 2 + a
3EX
3dO+ • • • • + aaEx3x a = Ex31ndt
for j = 1, 2, 3, ... 8. Thus, the constants al throughas can be obtained by simultaneously solving the
nsystem of equations obtained by expanding 2 ,.X.
n i=l t ,
for j = 1, 2, 3, ... 8. Expansion of .2 'ix,' yieldsz=l
liD = ....(B-2b)
The tooth-wear equation can be rearranged toyield
+ • • •
a 2 EXaX2 +
2• + aaExa
dt =(~)
max
Wd
- 4
dt = J 2!~ (1 + H2 h) dh. (B-3b)
then Eq. B-3a becomes
(~) - 4max
w w(d) - d
max
• (1+H~12) !~ (1 + H2h) dh .. (B-3a)
=
If the symbol h is used to represent a secondfunction of bit weight per inch, W/d, and rotaryspeed, N, which is defined by
(1n~~}]2
dO 2'- (1n-) ]
dt C
G =
.......(A-3)
When any of the regression constants are assumed
to be known, the corresponding terms aj x j can bemoved to the left side of Eq. A-I and the previousanalysis applied to the remaining terms.
The final correlation is checked for accuracy usingthe regression index of correlation G, given by
AUGUST.1974 381
The pressure drop across the bit, /),Pb, is relatedto the pump pressure, Pp ' and the flow rate, q, by
EXAMPLE CALCULATION OF OPTIMUMBIT WEIGHT, ROTARY SPEED, AND
BIT HYDRAULICS
6.01.0
12.0400500
9.8751-3
4.0100T-6
0.5
Eq. 13, which defines (W/d)o t' is obtained by. 1 1 . pSimu taneously so ving Eqs. B-5 and B-6. Eq. 14,
which defines bit life, tb' is obtained by solvingeither Eq. B-5 or Eq. B-6 for h JO+H2h)dh. Eq.15, which defines the optimum rotary speed, isobtained by integrating Eq. B-3a and assumingcomplete tooth wear.
APPENDIX C
OPTIMIZATION OF BIT HYDRAULICS
Inspection of Eq. 1 reveals that by maximizingagxg . . d
the term e , penetration rate will be maXimIze .The nozzle diameter, d n , is related to the flowrate, q, by the orifice equation
OPTIMUM BIT WEIGHT AND ROTARY SPEED
Required Data
Trip time, hoursConnection time, hoursRotating time, hoursBit cost, dollarsRig cost, dollars/hourBit diameter, in.Bit classBit weight, 1,000 lb/in.Rotary speed, rpmTooth wear(Wld)t, 1,000 lb/in.
From regression analysis, al = 3.0, a 2 = 0.0002,a3 = 0.002, a4 = 0.00004, as = 1.2, a6 = 0.6, a7 =
From Table 3, HI = 1.84, H2 = 6, H3 = 0.8, (Wld)max= 8.0.
T/I = (0.8) (1)1.84 (1) 1 + 3 (12)0.75 + 3(.563)
100 [15.7 (8 - 6.40)J O• 543
Nopt =16.1 (8 - 4.00)
Nopt = 60 rpm.
ALTERNATE SOLUTION
A cost-per-foot table (see Table 8) for this exampleproblem was generated using the Fortran IV programbelow. The program solves Eqs. 13 through 15b inan iterative manner.
1. Calculate pressure drop through bit at eachflow rate. Subtract the pressure drop through the bitfrom the pump pressure to get total parasiticpressure drop at each flow rate.
PF!\L Jl.J2.N.A(f3) .C( 1{~.40)
DATI\ A /j.0.0.:.1 C02.0.C::l2.C.OOC04,I .2.0.6.::>.9.0.4/
I.) 1\ T A \If) T .NI) M • HI. H 2. H 3 • T A UH/!) .5. (3.0 .1 .84. 6.!). 0 • f3 • 1:J .7/
D A T 1\ T T • T C • CR. C F< • 0 , R HO • GP • F J / (, •• 1 • • I~ 0 ') •• 5 '1 'J •• 70 (,0 •• 1 0 •• 9 •• 90 0 • /
J 2 =T 1\ L f j I H 3 * ( ( v. D '" - V. 0 ) I ( wn M-4 • ) ) * ( 1 0 C • / N ) ~, *H 1/ ( 1 • +H;:> / 2 • )
Tll=J2~'( 1. fH2/2.)
U::: -A(7)/fL?,*SGRT(I.+2.*H2*TP/J2)
C F .::: (C 9 + C R* ( T T +T C + T ~l) ) / ( J 1 *J 2 *H 2 / A ( 7 ) **2 *( 1 • + A ( 7 ) / H 2 + ( U- 1 • ) *I FXP(A(7)/H2 + 1.;»)
C( I.J )=CF
wI7ITE(6.6J)(
Fr'RMAT( lX.14F9.2)
S TllP
END
AlJCLST.1974 383
3. Read optimum flow rate and parasitic pressureloss from graph. Calculate nozzle area that willyield the optimum bit pressure drop.
Flow Pump Bit ParasiticRate Pressure Loss Loss
(gal/min) (psig) (psig) (psig)
485 2,800 1,894 906247 900 491 409
2. Plot parasitic pressure loss vs flow rate (seeFig. 6). Determine slope, Tll. Calculate the optimumparasitic pressure loss using Eq. 19 and plot pathof optimum hydraulics.
