CA2036021A1 - Intersection solution method for drill bit design - Google Patents

Abstract

INTERSECTION SOLUTION METHOD
FOR DRILL BIT DESIGN

ABSTRACT OF THE DISCLOSURE
A process for distributing cutters along the profile of a drill bit and a drill bit produced by such a process is provided by which at least during one stage of bit design the cutters are distributed in a manner providing that each cutter traces a path of predetermined area.
Two methods for determining that area are presented. The cutter distribution process provides for accurate cutter placement by taking into account each cutter's radial distance from the center line, respective placement along the profile, and intersection with other cutters. The placement of each cutter element along a conventional bit profile is determined by defining the bit profile in terms of a continuous line of alternately facing acute isosceles triangles the legs of which are equal to the cutter radius. The development of this geometry is achieved by the mathematical solution of the intersections of circles with lines and of circles with circles.

Description

sAcXGRouN~ OF ~rHE INVE~TIO~
This invention relates to the design of drill bits and, more particularly, it concerns the distribution of earth boring cutting elements along the profile of a drill bit.
Conventional earth boring drill bits and particularly those co~monly known as drag bit~ have cutting ~ur~ace made up Or a ~umher of polycrystallin~
diamond compact (PDC) cut ers. Each of the PDC cutters is normally mounted on a tungsten carbide stud or cylinder which is received within a corresponding aperture o~ the drill bit during bit ~abrication. PDC
cutter element~ such as ST~AT~PAX cutting element5 from Gen~ral Electric Company are readily commercially available.
~ he drill bit de~ign and manufacturing industry has made a significant ef~ort to distribute the individual : ~utters about the drill bit to provids the most efficient operation. In particular, a variety of methods and technique~ have been developed so as to produce a cutter digtribution which provides ~or uni~orm cUtter wear in order to maximiz~ the service liPe of the drill bit.
In the past, bit designers spaced cutters at uniform i~crement~ along radii extending from the central axis of th~ drill bit. SUch a de~ign ha~ not been found to be completsly ef~ective since it doe~ not take into account factors, such a~, differiny bit pro~ile , unequal cutter wear due to unequal volum~ cutting, and increased cutter wear relate~ to higher cutter vQlocitiQs.
An attempt to empirically determine an optimum cutter dl~tribution is described in a technical paper ~6~

entitled "Optimization of Radial Distribution of STRATAPAX Cutters in Rock Drilling Bits" by J. D. Barr~
This paper wa5 presented to the Eneryy Sources Technology Conference at New Orleans in February, 1980. In this paper, a power law model is assumed to relate cutter wear rate to cutter velocity and area of cut and, thus, the distribution of cutters i5 made according to the formula:
1/S=K R wher~ E=b/c. In this fo~mula, S is the radial spacing between cutters, K is an empirical constant, R
is the distance of the cutting edge from the central axis of the bit, E is a spacing exponent, b is a velocity exponent, and c is an area exponent. An upper limit for ~he value of S was selected at about 1/3 or 1/4 of the cutter diameter to insure adequate redundancy in th~
event of loss of a particular cutter or even two adjacent cutters. A disadvantage o~ the design m~thod disclosed in this paper is a r~quirement for empirical wear measurements from uged bits in the particular ~,aterial to be drilled. Also, experiments undertaken by the author of the paper and summarized therein resulted in considerable scatter ~rom theoretical wear patterns.
Another approach to the distributien of cutters on drill bits iq described in a publication released by Sandia Laboratories entitl~d ~Stratapax Computer Program"
and authored by Richard F~ Ashmore et al. The publication i3 identified by a number S~ND77-1994 and was printad in April, 1978. Thi~ publication describes a complex computer progra~ which calculates the volume of ~aterial cut by each cutter on a drill bit. A number of variables such a~ the location of the cutt~r in radial distance ~rom thQ central axis o~ bit, the angular locat~on o~ tho individual cutters ~rom an arbitrary base line in a plane p~rpendicular to the central axis of the bit and tho po~ition o~ th~ indlvidual cutters along the length o~ th~ bit are input into the computer progra~.
Then, th~ progr~m optir~ize~ the po~itloning o~ a number of cutter~ by trial and error iteration in an attempt ~o 2~360~.

achieve equal volume cuts for each cutter. This relatively complex program 3U f~ers from several disadvantages. For example, a system operator must first select a pattern of cutter distribution to initialize the computer program. This initial selection of cutter position effects the usefulnsss of tne iteration technique.
A simplified approach to cutter placement in which a plurality of cutters are pl~ced in a pattern approaching an ideal equal volume cutting arrangement whil~ ~inimizing the mathematical steps necessary to calculate the desired positions is described in U.S.
Patent No. 4,~75,606 issued to Morgan L. Crow on October 9, 1984. This simpli~ied approa~h positions the cutters so that the annular area between radially adjacent cutters as projected onto a com~on plane perpendicular to the central axis o~ the drill bit is a constant. Certain groups o~ cutter~ are positioned to prevent a central core in the material being drilled and to provide a ; 20 desired ker overlap. Although this approach produces drill bit~ having reasonabl~ wear characteristics,~ it does not take into account variations in the bit profiIe, di~rence~ in wear du~ to dif~erent cutter velocities, and the ~ctual intersection 9 f the cutters with each other.
In light o~ the foregoing, there i~ a need ~or an improved proce5s ~or accurately distributing cutters along the profile of a drill bit.

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S~MMARY OF THE ~NvENTIoN
In accordance with the pres~nt invention, a process for distributing cutter~ along the pro~ile of a drill bit and a drill bit produced by such a process is provided.
The cutter distribution process o~ th~ present invention proYides for accurate cutter placement by taking into accou~t each cut~er's radial distance from the bit center line, respective placement along the pro~ile, and intersection with other cutters.
In the practice of the present inve~tion, the placement o~ each cutter element along a conventional bit profile is dete~mined by defining th2 bit profile in terms of a continuous line o~ alternately ~acing acute isosceles triangles the leg~ of which ars equal to the cu~er radiu3. The combination o~ isosceles triangles is developed by solvin~ ~or a s~ries of circle-line int~rsections and circle-circls intersections. This mathe~atical solution o~ intersections to represent a bit profile in accordance with the present invention applies ~o conventional bit proiles having a plurality of radially distributed cutter elements, bu~ does not apply to center cutter3 and cases of extraordinary spread. ~ :
Among the ob~ects o~ the present invention are, there~ore, th~ provision o~ a process which provides a simpl~ and yet effective solution for tha distribution of Gutterg along the profil~ o~ a drill bit. Another ob~ect of th~ present invention i~ to provide such a process by which each cutter is distributed 50 as to ~ trace a path of predetermined area taki~g into account ; 30 their radial distances from the center line, their respectiv~ placement along the profile, and their intersection with each other. Yet another object of the present invention is to provide a drill bit having a cutter distri~ution which provides tor uni~orm cutter wear and th~reby maximizes the servic~ life o~ the drill bit. Yet still another ob~ect o~ the presen~ invention is the provi~ion o~ a procesa ~or cutter distribution 2~

which provides a more accurate cutter placement than is possible using conventional cutter placement techniques.
Other obj~cts and further acope of applicability o th~
present invention will become apparent ~rom the detailed description to follow taken in conjunction with the acco~panying drawings in which like parts are designated by like re~erence numerals.

~ ~ 3~30 ~J~i DESCRI~ION 0~ 2 DRAWINGS
Fig. 1 is a perspective view of an exemplary drill bit designed in accordance with the present invention;
Fig. 2 is a pro~ile representation of the cutters on an exe~plary drill bit rotated to a single radius from the central axis with variables identified as they appear in the equations in this application; and, Fig. 3 is a chart illustrating two methods of PDC
compaction.

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D~TAILED DESCRIPTION OF THE PREFERRED EMBODIMENT$
In Fig. 1 o~ the drawings, an exemplary drill bit of the presant invention is generally designated by the ~eference numeral 10 and shown to include a bit body 1 having a central axis of rotation 14. A plurality o~
individual PDC cutt~rs 16 are mounted on respective wings 18 in a distribution that will be described hereinafter.
The spaces 20 between th~ wing5 18 permit drilling mud and cuttings from the working face to pass up along the gage of the bit and eventually to the surf~ce ~or disposal in a conventional manner.
The distribution of the cutters 16 on the drill bit 10 is critical to the e~fective and efficient operation of the drill bit. If too much stress or wear is encountered on individual cutters, the drill bit can be rendered ine~ ective pre~aturely. HencP, the design goal is to achieve unifor~ wear on each cutter so as to maximize the service life and e~fectiveness of the drill bit.
Many conventional cutter distribution techniques do not take bit pro~ile into account when positioning cutters. The result is that cutters do not cut equal paths or wear equally. A check o~ one bit showed errors greater than 13% in single revolution areas o~ coverage.
Moreover, there is no way to automatically detect or correct error9. The cutter placement proces~ o~ the present invention takes a more rigorous approach to accurate cutter placement which includes the evaluation o~ actual inter~ection~ of cut~ers along the rsal profile o~ th~ bit.
In accordance with the present invention and as shown in Fig. 2 o~ the drawinq, cutter placement along a bit profile can be determined by solving aquations for intersections Or circles with lines and circles with circles. Thi~ i9 possible with the ~i~ualixation ot the pro~ being repre~en~ed by a con'cinuou~ series of ':

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isosceles triangles whereas the legs of the isosceles triangles are equal to the radii of the cu~ters.
In accordance with the present invention and with reference to Fig. 2 of the drawings, durinq at least one stage of a bit design process cutter distribution i5 determined as follows:

I. Determine the geometry of the profile about an origin. Based in part on the desired cutter exposure.
BFP - Bit Face Profile II. Determine the expressions f or the loci of points through the center of the cutters and the in~lection point at which changes in ~eometry take place.
CLP = Cutter Centerline Profile I~I. Determine the amount o~ center cutter o~fset to be included, i~ any.
CC0 = Center Cutter Offset (examples in this text will usa 0) -IV~ The bslow listed variables are recognized and the total area to be cut is determined:
R - Hole Radius H s Hole Ar~a N = Number o~ cutters n J CuttQr Nu~ber (N to 1, descending) r ~ Cutter Radiu3 - Area assigned to cutter number n Amax = Maximum Area for any cuttvr Smax 2 Maximum Separation of cutter intersection~
~ = 3.1~16 ~
V. The li~it o~ allowable separation of cutter int2rsections i~ det~rmined. Thi~ distance is normally relatQd to th~ Cuter ~adiu~ (r3.
e.g. Smax ~ r , ' ~360~:~

VI. Determin~ the area to be covered by each cu-ter preferably by a method of mathematical compaction. This process designates the area per cutter in such a way as to compensate for S increases in wear as a result of the higher velocities cutter experience towards the outer perimeter o~ the bit. The compaction oP the designated cutters take place beginning ~ith the maximum separation, along the length to be compacted. Two methods are shown in Fig . 3 o f th~ drawings and will be demonstrated here. The first method involves the use of the sums of powers of integers. The second method uses a linear compaction wi~h adjustable slope.
i. Method 1:
This method determin~s cutter area assignments by use of the 'Sums of the Powers o~ the Integers' with the equation ~or the first power demonstrated. This method should bc applied within the limits of maximum cutter separation as described above. ~h~ power used determines the rate of compaction.
p = N(N~1)/2 ~n = (I~/P~ ~ }I
ii. Method 2:
Thi method determines cutter area assignment~ by u5e of an assigned number of cutters deter~ining a slope of compaction.
Alternatively the slope o~ compaction can be standardized and the numb9r of cutters determ~ned therein. The slope determines the rate of compaction. By having adju~table coDpaction raten, corrections can be mad~ for di~erent drillinq conditions.

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x~

H = N(yl) + (N/2) (Y2-y1) }~/N = (Yl) + (y2-y1)/2 Y1 = ( 2H/N) - YZ

I = Yl ~ m(X1) M = (Y2-Y1) / tX2-X1) let x1 = 1 x~ = N
Y2 - Amax Substituting:
M = Amax-[(2H/N)-Amax]/(n-1) With I and M solved for:
An = ~len ~ I
VII. Determine the Cutter Intersection Values ~CIV) fro~ the ar~as which have been calculated beginning with the gage cutter. This is done regardless o~ the method chosen to determine the area per cutter.
CIVn = Cutter Intersection Value CIY1 =
crvn = ~CIV~2 ~ An/3.1~1Sj~
VIII. Locate the gage cuttar center on the cutter centerline pro~ tangent to the gage.
Assuming th~ origin i3 on the bit centerline with y ~ O at the center of the ~a~e cutter, the c~n~er point is:
CCPn ~ Cutter Cent~point for cutter n CCP1 - (R-r,0) CC~E!n = (an~bn) Example: For an a.5~ bit with a Cutter radius of .262:
CCP~ ~ (4.25 -.262, 0) CCPt 3 ~3.988, 0) -%~

IX. Determine the Circle-Line Intersection This i~ the intersection between cutter number 1 (gage cutter) and the vertieal line CIV2.
CIPn = Cutter Intersection Point CIPn = ~,k) The Line is vertical therefore:
CI~l = 3~
The circl~ is the cutter ~ace:
~X-~n~2 + ~-bn~2 - ~2 The intPrs~ction of the circle and line is the set of point~ which solves the equatiuns for each simultaneously.
Substitute for X:
~ n)2 + tY-~n)2 ~ ~Z

This eli~inates X from th equation. Expanding this equation into its polynomial form:
2 Ybn ~ t ( ~:IVn~ _ an) 2 ~ ~,Z _ ~2] = O

This equation is quadratic in terms of Y. Let:
A = 1 B - -2b C :1~ [ (CIVnt~ -an)2 +bn2 _r2]

It follow~ that:
y ~ ~ ~ O

Then:
~ = t-B ~ ~2 _ 4AC)~/2A

There are two real solutions to the equation.
The solution for ~he larger value of Y is the only point oP interest.

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-Example contin~ed:
Let CIV2 = 4.16 ~o~ this example.
(X-3.988)2 ~ y2 = ~622 Substitute X = 4.16 y2 = .0390 Y = .1975 CIP2 = (4.16,.1975) ~his is the simplified solution for the gage cutter.
X. Designate a temporary ~esign Circle about the Cutter Intersection Point just located CIPn =
(j,k) center of the design circle (Figure 2).
The radius of the design circle is the same as that of the cutter. The center o~ the Design Circl~ is located at the apex o~ the isosceles triangle. The equation for the Design Circle '.
i5: :
(X_~2 ~ ~y_k)2 = r2 Example continued: The equation for the de~ign circle is;
(X-4.16)2 ~ (y_~1975)2 = (.262~2 XI. No~ solvs for the equation of the cutter Centerlin~ Profile. For a linear section of the profile, the equation describing that interval o the CLP is:
Y - HX+ I
Let e be the angle between the x~axis and the Cutt~r Centerline profile. Then:
M = Tan~
I = -M(CIVl ~ r) Y -- Ta~e~X) --~I(C~V1 -- r) Exa~ple cont~nued: The gage cutter lie~ on a bit pro~lle o~ 60 d~g. to th~ centerllne. Th~
equatlon de3cribing thi~ intQrval 0~ the CLP is ~ound a~ follows:

:
.. . . .
- .

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4.25 -.262 - ~.988 Slope (M) = Tan 60 = -1.7321 Y-Intercept (I) - 6.9076 Y = -1.7321(X) ~ 6.9076 XII. Det2rmine the Circle-Line Intersection of the Design Circle and the Cutter Centerline Profile.
This will locate the center of the next cutter.

Equation of the circle (from step X.):
(X_j)2 + (y_k)2 = r2 Expanded:
~2 + y2 -2X~ -2Yk = r2 _j2 -kZ

Equation of the line (~rom step XI.):
Y = MX +I
Substituting:
X2 ~ (MS +x~2 ~2~ -2(N~ = r _~ Z _k2 Expanded to polynomial form:
X2 +~Z XZ + 2X~I +IZ 2Xj -2MXk -2Ik =
r2 ~j2 _~Z

Factoring out (~2 +1 (~ +l)X ~ (2MI -2~ ~2Mk)X +I2 _ 2Ik -r2 _j2 _k2 Thi3 equation i8 now in quadratic form and can be simplified for evaluation.
L4t A = t~
8 a ~2~ -2 1 ~2~]C) C = l~ 21s ~ ~ r2 ~ ~ Z ~ 1~2 It follow~ that:
AXZ ~ B~C ~ C ~ 9 .. ,. . :
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Then:
~ - r ~ 4AC)~/2A

The solution of the discriminant deter~ines the number of inte2sections. There will be t~o real solutions. The two solutions will represent the centers of two cutters n, and n~1. The solution with the lower value of X is the point of interest and this point will b2come CCP",,,1 = ( a,~,1, b,~ ~

Example continued:
Equation of the design circle is:
(X-4.1~0)2 + (y_~197~)2 - ~622 Equation of thQ cutter center line is:
Y = -1.7321(X) + 6.9076 Sub ituting:
~X -4.160)2 +(-1.7321(X) + 6.90~6 _,19.75)2 = (.262~2 ~2 -8.32X ~17.3056 +3.ooo2~2 -23. 2451X
+45.0254 - .0686 4.0002~ -31.5651X +62.262 ~ O

X - ~3105~51 ~ (996.356 - 996.241)~]/
.0004 2 8 , 3372 Th0 discri~inant is positive and therefore there ~S are two real solutions, meaning that the circle intersect~ th~ l~no tWiCQ.
X - 3.988 Thi~ i~ th~ center of th~ gage c~tter.

- 3.903 Thi i~ thQ center o~ thc n~xt cu~ter.

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, , ' XIII. As cutter positions progress inward from the gage, they may ~ow re~ch the curved portion of ths bit pro~ile. Th~ point at which th~ cu~ved portion of thQ pro~ile is tangent to the linear s portlon is referred to as the Inflection Point (Fig. 2). As the value of each c~tter cen~erpoint is determined, the abscissa must be compared to the Inflection Point. If the abscissa o~ thQ cutter centerp~int is less than that of the In~lection Point, then the solution of the centerpoint must ~e recalculated using the equations for the intersection of two circles as i~ described below. It is recognized that a given bit design may include multiple inflection points as a result of multiple changes in geometry and that the principl2s disclosed herein still apply~
XIV. The equation for the CLP in the curved i~terval, is ~or a circle with cen~er at point te,f) and radius p, as related to ths origin previo~sly described:
(X -e)2 + (y _f)2 = pz Therefore:
~2 ~y2 -2~ -2Yg =
2s Where:
C = p2 o~2 _~2 XV. The equation ~or the Design Circle around CIPn is:
(X~j)2 + (y-k)2 = r~
and:
~ ~y2 -2X~ ~2Y~ _ D
where:
D =
XVI. Subtracting the two circle equation~
2X(~ 2Y(k-f) ~ C - D
Y - ~ tt~ C ~ 2 .

X ~ 3 This is the equation of a line which intersects either circle at the same two points, the solution 6et will solve for both equations:

Let M = (e -j)/(k-f) and I = (c D)/~k-~) Y = MX ~ I

Substituting thi~ value into the equation of the Design Circla X +(M~I)2 ~2~$ -2~ I) z D

Expanding into it~ polynomial form, x2 ~M~ x2 + 2XMI +I2 _ 2Xj -2MXk - 2Ik = D

(N~ 2~I -2~ ~2~k)X ~ I2 2Ik ~ D

XVII. Wherein this equation is quadratic in terms of X and the solution determines the number of intersections, let:
A ~ (~ + 1) B - 2(MI - j -kM~
C ~ 2k) - D

A~ ~ BX ~ C = O

~ BZ _ ~AC)]/2A
.
The positive value of the discriminant indicates two real roots exi~t and therefore two points o~ :
intersection. A zero value of the determinant would indicate that one real 301ution exists and ~:
~5 therefors circle~ are tangent.

Th~ larg~r valu~ o~ X correspond3 to the X valu~
oY th~ previou~ cutter center CCPn1, and the .~ . . .

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lower value will correspond to the next cu~ter center cCpn.

Substitute the lower x value into one o~ the two equations for a circle to solve for Y.

Other criteria sùch a~ force analysis (a determination of the direction and magnitude of the forces acting on each of the PDC cutters), back rake (the angle between the diamond face and the perpendicular to the hole face), side rake tthe angle between the cutter face and th2 radial line extending from the bit center ~o the face o ~he outter), cutter g~ometry since PDC
cutters are available in a variety of shapes and sizes, and cutter velocity during bit operation can be ~aken into account when selecting area to be cut. It will still be necessary to identify the intersections between cutters.
It appears impractical to design bits for a sing~lar rock hardness as most designs drill several rock types under a variety of operating conditions (weight-on-bit, RPM, Hydraulics, etc.) on any given application.
Moreover, most PDC bit~ have more than one geographical application ~or which ~hey are designed. The simplicity of having an adjustable rate o~ compaction allows designers to makz adjustments based on wear analysis of dull bits run in the area of interest.
Thus, it will be appreciated that as a result of the present invention, a highly effective process for dete~mining th~ distribution o~ cutter~ along the proPile of a drill bit and a drill bit produced in accordance with this proces is provided and by which the stated objective~ among others, are completely fulfilled. I~
i9 contemplated that modiSicat$ons and/or change~ may bs made in the $11u~trated smbodi~ent without departur~ from the invention. Accordingly, it is expre~sly intended tha~ the foregoing de cription and the accompanying ~33~

drawing~ are illustrative of a pre~erred embodiment only, not limiting, ~nd that the true spirit and scope o~ the present invention b~ determined by referenc~ to the appended claims.

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Claims (4)

1. In a process for determining the distribution of cutters along the profile of a drill bit, said process including the step of determining the placement of cutters along the profile of the bit so that each cutter traces a path of predetermined area, the improvement comprising:
determining the area to be covered by each cutter by a method of compaction, and determining the distribution of said cutters by representing the bit profile as a continuous line of sequentially inverted acute isosceles triangles, the legs of which are equal to the radius of the cutters, this being achieved by mathematical solution of the intersections of circles with lines and of circles with circles.

2. The process of claim 1, wherein the intersection of circles with lines and of circles with circles includes a circular representation of each cutter with vertical cutter intersection valve lines (CIV) representing intersection points of adjacent cutters.

3. The process of claim 2, wherein the center of a design circle having a radius equal to the radius of the cutters is located at the intersection point of a cutter intersection value line (CIVn+1) and a circle defining the cutter (n).

4. The process of claim 3, wherein the intersection of circles with lines and of circles with circles further includes the intersection of the design circle with a cutter centerline profile at two points of intersection with one defining the center of a first cutter (n) and the other defining the center of an adjacent cutter (n+1).