AU621138B2

AU621138B2 - Apparatus for avoiding stuck drilling equipment and method of determining the probability of unsticking a stuck drill pipe

Info

Publication number: AU621138B2
Application number: AU39290/89A
Authority: AU
Inventors: W. Brent Hempkins; Roger H. Kingsborough; Wesley E. Lohec; Conroy J. Nini
Original assignee: Chevron Research and Technology Co
Current assignee: Chevron USA Inc
Priority date: 1988-08-03
Filing date: 1989-08-03
Publication date: 1992-03-05
Anticipated expiration: 2009-08-03
Also published as: ES2026747A6; CN1016204B; CN1040653A; DK378289D0; AU3929089A; DK378289A; EP0354716A1; NO893113L; NO893113D0

Description

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6 21138 COMMONWEALTH OF AUSTRALIA, PATENTS ACT 1952 COMPLETE SPECIFICATION 444 4 4 4 4 4 4 4 4,4 4 44, 4 4 4 044 o 4 44 4 4 44 4 NAME ADDRESS OF APPLICAN'T: Chevron Research C-ompany San-Francisco-C-alifornia United- States-of-America NAME(S) OF INVENTOR(S): W. Brent HEMPKJNS Roger H. KINGSBOROUGH Wesley E. LOHEC Conroy J. NIN a'd lr4C"Okji -j 4~ CL t- iis ADDRESS FOR SERVICE: DAVIES COLLISON Patent Attorneys 1 Little Collins Street, Melbourne, 3000.

COMPLETE SPECIFICATION FOR THE INVENTION ENTITLED: Apparatus for avoiding stuck drilling equipment and method probability of unsticking a stuck drill pipe The following statement is a full description of this invention, including performing it known to me/us:of determining the the best method of la 0000 o0 o o J o 0000 000 0 30 a 0 0 03 ooo°oi 00 00 0 3 oo 30 0 0 00 0 00 00 00 0 0 The present invention relates to apparatus for calculating the probability of drill pipe sticking during drilling of a well bore in a given geologic province. More specifically, it relates to apparatus for controlling or modifying drilling conditions in such a well bore to avoid sticking of the drill pipe either due to mechanical conditions affecting movement of a drill string in the well bore, such as high hole angle, oversize drill collars and the like, or due to differential sticking, as a result of excessive differential hydrostatic pressure by drilling fluid in the well bore on the drill pipe against a low-pressure earth formation surrounding the well bore. Other causes of drill string sticking may be further identified and evaluated by the same method, if desired. Further, it particularly relates to a method of and apparatus for determining the probability of freeing such a drill pipe should it become stuck.

It is a particular object of the present invention to control drilling of a well by statistically determining, as by calculating, or plotting, or both, the probability of a drill pipe sticking in a well bore and correcting well drilling conditions to avoid that result. Such probability is calculated from a multiplicity of independent and dependent variables which are physical quantities, or parameters, each representing a standard mechanical, 1 -0 .1 L i I'' -2chemical or hydraulic drilling condition normally measured or measurable in drilling a well. The same physical parameters in a multiplicity of wells are measured at depths where a drill string has become stuck and at a generally corresponding depth in a multiplicity of similar wells where the drill string has not stuck. The statistical probability of "o sticking is then calculated by a method of o.0o statistical analysis known as "multivariate analysis" i 0 10 from such similarly measured quantities at any single depth in any of such multiplicity of wells in a given 0d 0 geologic province where drill pipe sticking has 0 0 o0 occurred. "Geological province", as used herein, includes a geographical area of a geologic region in which a multiplicity of wells have been drilled and wherein similar consequences of earth formations, '0 0o such as shale-sand bodies or other lithologies of 0 d differing compositions, are normally encountered over a range of known well depths. Such measurements are a do 20 made in wells in which a drill pipe has become stuck in a significant number of instances, due to both mechanical and differential pressure or other conditions in the well bore, and in a similar significant number of wells where the drill pipe did not stick. The probability of avoiding sticking the drill pipe during drilling is increased by selectively controlling the values of such measured J quantities or variables in accordance with the coefficients determined for each variable by such analysis.

Monitoring and correcting the variable mechanical and hydraulic quantities of such data measured during drilling, in accordance with the i

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-3invention, is accomplished by multivariate analysis of a multiplicity of measured variables in each well of a multiplicity of wells. Such variables are measured in at least two classes or groups of wells, which may comprise wells in which the drill string has stuck and has not stuck, or stuck wells in which the drill string has been freed or not freed. The ooo.Oo analysis depends upon matrix algebra to generate a 0' well vector for each well to represent the selected 0 '10 variables of that well at a selected depth over the I given depth range. Each such algebraic -alue is then numerically recorded, or graphically plotted.

oH o Preferably the plot is within a multi-dimensional space with lower dimensional hyper-surfaces, such as hyperplanes which are selected to best separate distributions of the classes about the mathematical 0 o I mean or centroid of each of two or more classes of 0 0o wells from the mean or centroid of the other class or classes. The statistical probability of the selected multiplicity of related and unrelated (but measured and measurable) variables placing each well in the proper class permits a similar well vector generated 0. °from current or proposed drilling conditions in another given well to be determined. The current or proposed position of such a well vector is thus related to the mean or centroid of each group or class of wells. Control of drilling conditions in an individual well may then be evaluated, and if required, modified, by changing variables, such as drilling mud properties, hole angle, drill string composition, etc., to effect positively or negatively the numerical value of the well vector to change its plotted location relative to the spatial areas, or -4numerical means, representative of the respective classes of wells.

As used in the present specification and claims the numerical examples, including their calculation for recording or display, illustrate the general application of multivariate analysis by 0° apparatus and methods of this invention to indicate or measure the probability of sticking a drill pipe o by any cause or combination of causes. For 10 simplicity of explanation and understanding by those skilled in the arts of drilling, or statistics or both, the terms of the specific examples are generally defined as being linear and planar.

However, it will be understood that the following terms, as used in the detailed description and claims, are intended to be included within the 14, breadth of the following definitions: S"Multivariate analysis" means the application of methods for evaluating the simultaneous relationship of a large nunmer or multiplicity of variables or measurements (well parameters) for each of a multiplicity of objects (drilled or drillable well bores) by the use of data matrices to separate groups or classes of such objects and to properly classify another object into one or more groups or classes.

The term "plane" (including "mapping plane" or surface) is used to identify any surface or space definable whether two or multi- dimensional and either flat or curvilinear. The term also includes hyperplanes or surfaces in a space or sub-space that is useful for displaying or measuring distances between points of intersection, between such plane or surface and a well vector. More specifically, a plane is any space or surface such that the tangent to that surface is a constant, either scaler or some functional representation thereof.

'O-O The term "centroid" denotes a location in 0(09 multidimensional space which is a special volume or area containing the central tendency of well vectors 0 10 comprising a group of well vectors which are Qo sufficiently spaced from the central tendency of any other statistical group of well vectors.

The term "well vector" is used to indicate the vector solution of all measured or measurable o ii5 variables in any group or groups of wells analyzed by \o1 multivariate analysis or in any other well to be studied.

The term "mean" relative to well vectors and plane includes any measure of central tendency of 20 numerical values whether, arithmetic, geometric, harmonic or other mathematic forms, including modal tendencies.

"Arithmetic" means solutions involving manipulations of numbers or mathematical functions describable by numbers including both real and imaginary numbers which can be manipulated in any spatial context including both Euclidean and non- Euclidean spatial systems.

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1 Pt~ -6- "Coefficient" refers to any number or group of numbers which are observed or calculated and whose purpose is to describe or to be used to describe some condition of the arithmetic, mathematical or statistical condition or outcome of such observations or calculations.

0,o "Correlation" means any arithmetic S' expression involving the relation between one or more sets of observations. This includes all distance \1o0 relationships in any coordinate system. It includes the Pearson-product-moment correlation coefficient, 00. 0 which is convenient to calculate and has special 0 0 matrix properties as described herein.

"Centroid" is a way of expressing the central tendency of a multiplicity of measurements in 0o 0 any space (see mean).

O0 "Covariance" in general is an expression of the same relationships as correlation and may indeed be the same. Normally, covariance has dimensions, whereas correlation need not (see correlation).

"Discriminant" includes any devices, either mathematical or graphical, which can be used to separate groups of objects into conjunctive or disjointed "families" of objects with or without 25 probability estimates of group membership. This includes all linear and non-linear and all Euclidean or non-Euclidean spaces.

"Dispersion" is a measure of the distance of an object (well, etc.) from one or more measurements 1 i t -7of central tendency (see means or centroids). It may also refer collectively to a group of objects expressed as a scalar vector or matrix value.

"Standard deviation" or "Second Moment" are synonymous terms, as is Variance.

"Eigenvalue" is the characteristic or latent root of a vector or matrix. It may be real or imaginary.

0 0 0 o 04 S"Eigenvector" is a vector, real or o, :10 imaginary, associated with any eigenvalue. Generally o" o it is a vector with both magnitude and direction.

Two or more eigenvectors can determine a plane, generally in multidimensional space and, therefore, a hyperplane in that space or any sub-space thereof.

o a o °5 "Expert System" is any means for minimizing the behavioral skills or expertise of a human by storing knowledge of an expert or experts which allows operation of a process or procedure without such an expert being present to observe or measure 2 20 variable conditions and respond or act on such observations or measurements. The term is also known as knowledge based systems, or so-called artificial intelligence.

"Factor analysis" is an arithmetic, mathematic or geometric means of factoring a set of distance or dispersion measures into correlated or related subsets in which the objects (wells) have the highest relationship possible under the criterion for factoring. Examples of factoring include varimax, t -8minimax, quartimax, maximum or minimum entropy, as well as other methods.

The term "geometric" is generally, but not inclusive, of numerical descriptions of spatial representations of data or derived spaces, or hyperspaces, or sub-spaces thereof.

A "hyperplane" is any surface expressible by the numerical, arithmetic, geometric or other o0 0 0 operation which describes any space or sub-space, oaa 00 o" 10 including graphical means or projective geometry. In So general, the word plane will be used to describe a hyperplane.

The term "linear" has several definitions.

In one sense, linear may refer to a simple arithmetic computation using any ordinary mathematical method.

It may also imply a straight line or simple flat surfaces known generally as a plane; however, such lines or surfaces may be curved but defined by techniques generally known as linear transformations.

However, the term linear also implies in differential mathematics that derivatives of any relationships are also linear. For example, while logarithms, exponentials, etc. are not linear, under normal circumstances their derivatives may be used to linearize their transformations. Linear may also include non-linear spaces, variables or transformations thereof.

"Linear Program" (LP) means any optimization technique which is designed to minimize or maximize some objective function, often called a "cost" l. -9function, such that analysis results can be designed to yield solutions which optimize program results.

Herein it is meant that linear programming includes such non-linear approaches as quadratic or dynamic programming and mixed integer techniques as well as any other optimizing methods.

The term "map" refers to any mathematical, algebraic or projective means of portraying the O o o o* o spatial characteristics of any observed or derived «,4lO solutions in any space.

o o go o "Matrix" means any arrangement of observed or calculated numbers which can be arranged in any sort of array comprised of rows and columns in any sort of space, including linear or non-linear spaces.

o "5 Some "matrices" have special properties. We here o0o:. include the word matrix to pertain to such normally recognized spaces as Hilbert, Euclidean, linear, non- 0 0 o o linear, Positive, Definite, semi-Definite, etc., as may be applied to the problem.

00 00 "Multidimensional" as used herein includes the concept of one or more observed or collected variables and all results coming from manipulation of the variables and their derivative arithmetic or geometric operations, including any graphic representations.

"Mean" refers to any measure of the central tendency of a set of observations. Ordinary arithmetic operations can yield the simple arithmetic mean, median, mode, etc. Furthermore, there are harmonic, geometric, etc., means. As used herein,

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"mean" includes any measure of such a central tendency.

"Plane" (also see hyperplane) as used herein expresses any representation of a surface in any space or subset thereof.

.io "Projection" is any mathematical or graphical representation of observed or calculated data which shows the results as coincident with any 09 0 o surface or sub-surface in any space either by the 10 solution or an extension of the solution values.

BACKGROUND OF THE INVENTION Drilling deep wells, say over 6,000 ft. with conventional drilling fluids and without setting well 9o" casing to prevent drill pipe sticking, is a difficult and long-standing problem. Particularly in off-shore drilling, numerous deep wells are usually drilled from a single stationary platform within a work area generally less than 1/4 acre. Thus, the wells must .9 Fbe directionally drilled ("whip-stocked" or "jet deflected") at relatively high angles from vertical to reach substantial distances away from the single platform. In this way petroleum may be produced from formations covering substantial underground areas including multiple producing intervals.

In general, one of the most common methods of drilling such wells uses water-based drilling fluid to lubricate and flush rotary drill bit cuttings from the bore hole, but more particularly, it provides hydrostatic pressure or "head" in the I i pI^ -11well bore to control higher pressures that may be encountered in a petroleum-containing formation cut and traversed by the hole. Such hydrostatic head prevents "blow-out" or loss of gas or oil into the well during drilling. Further, the drilling fluid contains solid materials that form a mud cake on the wall of the well bore to seal any permeable formation traversed by the well during deeper drilling. Such water-based drilling fluids, including sea water, are .o :10 substantially cheaper than the alternative oil-based fluids from the standpoint of original cost, ao~ maintenance, formation evaluation, and protecting the *A 0 ocean environment. However, oil-based or chemicallybased fluids are useful in certain environments and accordingly are frequently used.

00' It has long been known that one of the 00l primary causes of drill string "sticking" is the effect of differential pressure between the hydroo 0 static head in the well bore and any porous, lowpressure earth formations through which the drill string passes. Under such conditions, the hydraulic 0'o; pressure difference between the well bore and the low pressure formation presses the drill pipe against the bore hole wall with sufficient force to prevent pipe movement. This occurs because the density or weight of the drilling fluid in the well bore creates a hydrostatic pressure against the pipe that is substantially greater than that in a porous earth formation traversed by the well bore. This is due to the filtrate (water in the drilling fluid) flowing into the low pressure formation from the well bore wall and thereby undesirably increasing the thickness of the "mud cake". This condition may occur in the :i 1 -12drill collar section of the drill string which is 88 o 8 8888 10 88 8 08 0 op O 88 8. 80'.

8 80 0 08 84 8 8 0 8 8 0888 J 80 *8 8 80 08 0 8 80 88 0 8 8 88 84 00 0 8 used to apply weight to the bit~ direcl..yabvth drill bit, but apparently more frequently, occurs at shallower depths where return mud flow around the smaller diameter drill string is less turbulent and hence relatively laminar. Thus, where the drill pipe lies close to one side of the well bore, as in slant holes, higher differential pressure across the drill pipe increases its adherence to the si.de of the well bore. In a worst case, this results in differentialpressure sticking of the drill string.

While sticking of the drill string can be avoided by setting casing or well pipe through the depth interval where such sticking may occur, the cost of setting pipe, and subseqaient increase in cost of formation evaluation through the pipe, is generally too high for normal drilling operations.

Correction of drill string sticking conditions often requires a decrease in the drilling fluid pressure in the well either by reducing the hydrostatic head of the drilling fluid or increasing solids content of the fluid to reduce filtrate loss, with subsequent buildup of a thicker filtrate, or mud cake, which increases pipe contact area.

Alternatively, sticking can sometimes be avoided by using smaller diameter drill pipe or fewer drill collars in the assembly above the drill bit. The problem of differential pipe sticking is frequently severe where a well encounters over-pressured formations either above or below low pressure formations cut by the bore hole. In such wells, the formation pressure exceeds the pressure normally -13expected due to hydrostatic head alone at such depths. Thus, in wells entering or passing through over- pressured formations, the counterbalancing hydrostatic pressure in the well cannot be reduced safely at either the shallower or deeper depths.

However, such increased pressures on the low pressure formations may substantially increase the risk of fracturing such a formation, with accompanying loss of liquid from the drilling fluid in the well into the fracture and increase in filter cake around the drill string. Preferably, lubricating fluid may be 0 "spotted" in the well at or near the drill string 0 sticking point to reduce friction between the drill 0 0: pipe and the filter cake or to displace drilling 00 o0 0°°.15 fluid in the annulus between the bore hole and the drill pipe. Alternatively, with water based fluid, oil, or oil based fluid may assist in breaking down the filter cake so that adhesion of the drill pipe to 0000 othe well bore is reduced enough to free it.

00 0 It is also known that a drill string may °0 stick in a drilling well because of mechanical problems between the drill string and the well bore itself. Such a condition can sometimes occur by what 0°00 is known as the "keyseat effect". That is, a keyseat is created when the drill string collar or a pipe joint erodes a circular slot near the size of the outside diameter of the collar or tool joint in one fi side of the larger circular bore hole, as originally cut by the drill bit. Such a slot can create greatly increased friction or drag between the drill string and the earth formation and result in seizure of the drill collars when an attempt is made to pull the string out of the hole and the collars become wedged

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-14"" in the keyseat. Such problems can also be created by excessive weight on the drill string so that the drill string buckles in the lower section and particularly where the bore hole is at a high angle, t say in excess of 60* from vertical, or the well bore includes more than one change of direction, such as an S-curve or forms one or more "dog-legs" between the drilling platform and the drill bit. It is also known that in mechanical sticking of a drill string, earth formations around the well may be sufficiently unstable so that the side wall collapses into the well bore and thereby sticks the pipe.

oi O 0O:o It is estimated that the cost to the o« petroleum industry for stuck drill pipe in drilling wells is on the order of one-hundred to five-hundred million dollars per year. Such an estimate is based on the fact that from 30% to 50% of all wells so stick the drill pipe and, in some offshore areas the 0' opercentage is even higher. The cost to rectify each occurrence can be on the order of $500,000 or more.

0 60' The extent of each pipe sticking problem generally depends upon the amount of time the operator is willing to "wash over" the stuck section of the drill 00 «4 pipe (after unthreading and removal of the unstuck portion), or to "fish" by otherwise manipulating the drill string. Correction may also include spotting or completely replacing the drilling fluid with a lubricating drilling fluid. Failure to free a stuck drill string results either in abandoning the well bore or side-tracking the bore hole above the stuck point. This may include loss of the drill bit, collars and stuck lengths of pipe in the bore hole.

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More recently, increased use of downhole drill logging and monitoring tools such as those known as Measure While Drilling (MWD) systems, have introduced new financial risks to drilling if the drill pipe stickp. Such a system is normally incorporated as a section of the drill pipe or collars at the lower end of the drill string.

Preferably, it is assembled into the drill string either directly adjacent the drill bit, or as near o0 10 the bit as possible.

o o Because each of these tool systems cost about $250,000, loss of the lower portion of the fo'> drill string, including such a tool in a stuck well, Q is now more costly than ever. Even where the tool can be insured, the premiums for such insurance necessarily are costly if there is little control over the probability of sticking the drill pipe.

o6*O"o The problem of sticking pipe has been described in numerous publications in the literature, particularly as it relates to differential sticking of the drill string in the well bore. As explained in the literature, drilling fluid (returning from the o4 drill bit to the earth's surface) in the annulus around the drill string may lose enough liquid into a porous formation penetrated by the bore hole so that no drilling fluid circulates around one side of the drill string. Because drilling fluid is normally thixotropic, the fluid gels when it is not flowing.

This creates sticking of the drill pipe. As noted above, such sticking occurs generally where the drilling fluid (and particularly water-based drilling fluid) contains too few solids or fluid loss control -16agents allow the thickness of the mud filter cake to increase between the drill string and the side of the well bore due to excessive liquid loss. Such literature is primarily directed to methods of avoiding differential sticking by assuring that the drilling fluid is tailored to match the earth formation characteristics penetrated by the well bore.

In drilling deep wells, where intimate 0 knowledge of the formations is not available, and 0 particularly where low pressure formations are S° ,encountered, workers in the art have found it o.p difficult to predict and take corrective or preventive action prior to such drill pipe sticking.

Further, while stuck pipe problems can be avoided by deeper casing of the bore hole around the drill string, such casing is expensive and in general o*O* undesirable because it limits formation evaluation 0 with conventional well logging tools. Increased 20 drilling costs are also a primary reason that Oo oil-based drilling fluid is not as desirable as water-based fluids, unless essential to the drilling operation. For example, many formation evaluation, or well logging, tools such as resistivity and selfpotential measuring devices depend upon the use of water-based drilling fluids because such fluids are electrically conductive through the earth formation, rather than insulative, as in the case of oil-based drilling fluids. While the formation evaluation tools, such as radioactive tools, can be used in cased wells, such services are generally more costly and have interpretive problems which further increase t drilling costs. Additionally, the cost of preventing -17pollution of water by oil based fluids can be exorbitant due to the high cost of transportation of spent fluids and cuttings to land-based toxic waste dumps, as compared to conventional drilling systems.

Accordingly, it is highly desirable, if at all possible, to drill with conventional water-based drilling fluids while still avoiding drill pipe sticking.

oa An equally serious problem, after a well has a 10 stuck the drill string, is whether, and for how long, o° 0 the operator should "fish" or take remedial steps 0 o including oil spotting, drill string jarring or the o 0" 0 like, to try to recover or unstick a drill string in the current well. Sidetracking and abandoning a 0 0 portion of the drill string is another costly decision which must be evaluated with little or no help from currently known methods apart from the 0"0o intuitive judgment of the driller.

0o 0 Examples of patents that disclose methods 00,o20 and apparatus to avoid cr remedy stuck pipe include the following: 0' Patent 4,428,441 Dellinger proposes the use of non-circular or square tool joints or drill 1 collars, particularly in the drill string directly 25 above the drill bit. Such shape assures that circulation is maintained around the drill pipe and 4 reduces the sealing area between the pipe and the side wall where the differential pressure may act.

However, such tools are expensive and not commonly available. Further, they may tend to aggravate the keyseat problem in relatively soft formations since t <i -18the square edges of such collars may tend to cut the side wall in high angle holes.

Patent 4,298,078 Lawrence proposes using a special drill section directly above the drill bit to permit jarring the drill bit if the pipe tends to stick. Additionally, valves in the tool may be actuated to release drilling fluid around the drill string to assist in preventing or relieving stuck drill string condition.

o 0 10 Patent 4,427,080 Steiger is directed to binding a porous layer on the outside of the drill string. Such a coating is stated to prevent K differential pressure sticking of the pipe by increasing liquid flow around the drill string.

Patent 4,423,791 Moses discloses avoiding differential sticking by use of glass beads in the drilling fluid to inhibit formation of a seal by the 9 av filter cake between the drill string and the well bore adjacent to a low pressure zone.

While it has been proposed heretofore to 4statistically study the probability of relieving differential sticking of a drill pipe, such statistical analyses typically have been directed to the problem of estimating minimum soaking time and A^ 25 maximum fishing time that may be economically devoted to unsticking the stuck drill pipe. Such a procedure is disclosed in an article published at the Offshore Technology Conference of 1984 entitled "Economic and Statistical Analysis of Time Limitation for Spotting Fluid in Fishing Operations" by P.S. Keller et al Sii -19- "Stickiness Factor A New Way of Looking at Stuck Pipe", IADC/SPE paper 11383, 1983 Drilling Conference, pages 225-231 by T.E. Love is directed to a statistical study of "stickiness factor" for evaluating the probability of freeing stuck pipe by use of an empirical formula that evaluates several significant variables in drilling a well, namely, the length of open hole, mud weight, drilling fluid loss, and length of the bottom hole assembly. The empirical formula was developed from wells in which drill pipe had become stuck and those in which drill pipe had not stuck by cross-correlation of 14 primary :0o parameters measured in connection with drilling wells oo* in a given area of the Gulf of Mexico. The primary purpose of the formula is to determine the chance of freeing stuck pipe and in guiding the well by controlling only the chosen variables used in the empirical formula.

S020 In the prior art, no suggestion is made to D b" 20 use statistical analysis of differentially stuck wells alone, or along with mechanically stuck wells as disclosed herein, to determine the probabilities of sticking the drill pipe or that a stuck drill pipe can be freed by modifying only certain measured well variables. Further, the prior art does not contemplate modification of a plurality of variables over a range of realizable values for each variable at any given depth in the well sufficiently to alter well drilling conditions that will, in fact, divert the drill string from a high probability of sticking to a non-stuck probability.

Studies have also been reported by M.

Stewart (Speech to Society of Petroleum Engineers, New Orleans Chapter, New Orleans, LA, 1984) on the problem of setting casing at particular depths based upon statistical studies of differentially stuck pipe, particularly in the Gulf Coast, in wells that enccunter over-pressured formations so as to avoid both inadequate bore hole hydrostatic head on such formations and possible fracturing of lower pressure formations by excessive pressure.

o" o Further, in previously known methods of o estimating the probability of freeing a drill pipe Poo., after it had become stuck differentially, none of the 15 prior art suggests the evaluation of a multiplicity S" of measurable variables by multivariate analysis to determine whether "fishing" or other remedial actions should be initiated or continued. As indicated in i!ao the above-noted article by T. E. Love, predictions based on cross-correlation of a few variables, including time to spot oil-based drilling fluid in aoo past occurrences of stuck drill pipe, have been relied upon to determine whether or not a drill pipe can be freed. Such results are not sufficiently s5 reliable to warrant general use, because they do not consider responses from other correlated variables.

As particularly distinguished from such prior proposals, we have found that by multivariate analyses of a significant multiplicity of stuck wells, including one group of such stuck wells wherein the drill pipe was freed and another such group of stuck wells the drill string was not freed, wells which had a high probability that they could 1 i /i i 11 BBBBB BBBB BBBB^ 1 -21not be freed can be clearly distinguished from wells having a high probability that they can be freed.

BRIEF SUMMARY OF THE INVENTION In one aspect, the invention provides apparatus for statistically determining the probability that a drill string in a well bore will stick while drilling in a geological province wherein the drill strings have stuck in one group (N 1 of multiplicity of well bores and the drill strings have not stuck in well bores of another group (N 2 of said multiplicity of well bores (Nr) which comprises; means for storing measurable values of a multiplicity of drilling variables at a selectable depth in each well bore of the two groups (N 1

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2 of well bores wherein the drill string stuck and wherein the drill string did not stick, multivariate analysis means for receiving the stored values for each well bore and for determining a non-dimensional coefficient for each of said drilling variables representing its relative contribution to a single well vector at a ,r selected depth for each well bore of said multiplicity of well bores, vector generating means for summing the products of each coefficient multiplied by its corresponding measurable value for a sirgle well bore to form a single vector for a selected well bore, centroid generating means for generating at least a group centroid of the vectors of said single well bores stored in each of said two groups, *til and means for selectively recording at least said group centroids and another single well bore vector for the measurable variables at a selected depth in another well bore to indicate the proximity of said other well bare -1 vector to said group centroids as a measure of the probability of sticking the drill string in said other single well bore at said selected depth.

The invention also provides apparat.:s for evaluating the effect of modifying a plurality of measurable variables used in drilling a well bore to decrease the probability of sticking the drill string within an earth formation traversed by said well bore which comprises: automatic data processing means, including first means forI ji '910910,cmsdat.125,39290Chcv,13 i 1 t N 1 7 22- 00 00 0 0 00 0 00 0 00 00 00 0 0 storing and forming at least one multivariate analysis matrix of a multiplicity of measurable well drilling variables in each of a multiplicity of well bores including at least two classes of well bores drilled in a similar geologic province, said two classes including one multiplicity of well bores wherein the drill pipe stuck and another multiplicity of well bores wherein the drill pipe did not stick, said multivariate matrix,- for each class including for each well a substantially identical multiplicity of drilling condition variables measured at a selected depth in each well over a given depth interval, means for computing a distance function coefficient for each variable of at icast the total discriminant matrix for all wells in said at least two classes of well bores, second means for storing the values of said coefficients, means for selectively calculating a single valued vector representing each well bore in said total matrix and any other well bore drilled in said similar geological province, each of said well vectors being the sum of each of its measured variables in said first storage means scaled by its respective distance function coefficient in said second storage means, means for recording at least the centroids of said well vectors for at least said two classes of wells and at least two well vectors of said other well at said selected depth, one of said at least two well vectors having a plurality of said measurable variables modified in an amount and to an extent to move said one well vector relative to the other well vector and means for indicating the change of distance of at least one of said two well vectors relative to said centroids.

The invention further provides apparatus for decreasing the probability of sticking the drill pipe during drilling of a well in a given geological province which comprises: means for recording each of a multiplicity of mechanical and fluid paramieters used in drilling a multiplicity of well bores in the geological province, means for recording a set of said multiplicity of parameters for each of )190csa.2532Ohv1 23a multiplicity of wells, one group of said multiplicity of wells having been drilled wihout sticking the drill pipe and another group of said multiplicity of wells havigr stuck the drill pipe, multivariate analysis means for determining a coefficient for each of said multiplicity of parameters in a total group of said multiplicity of wells, each coefficient being representative of the relative impo-tance of one of said parameters to form a vector for each well, means for identifying at least one centroid for each of said groups of wells from said vectors, and means for determining for a single set of a corresponding multiplicity of said parameters at a given depth in a single well bore the distance of its corresponding well vector to at least one of said centroids said distance indicating the probability that said single well bore is a member of at least one of said groups and the probability that a drill string will stick in said single well bore while drilling using said single set o r parameters at said depth, The invention still further provides a method of determining the o probability that a stuck drill pipe in a well bore can be freed which comprises: recording the same multiplicity of well drilling parameters M in each of a multiplicity of well bores N wherein a drill string has stuck, said multiplicity of well bores including one group of wells wherein the drill string was freed and another group of wells wherein the drill string was not freed, by multivariate statistical analysis, computing a coefficient for each of clm said multiplicity of well drilling parameters in said groups of wells, computing the centroid of well vectors for each of said groups of wells, each of said well vectors being determined in accordance with each of said multiplicity of parameters measured in a well scaled by its corresponding coefficient, said centroids differentiating said well vectors of said groups of wells wherein the drill string was not freed from wells wherein said drill string was freed, and computing the well vector for another well bore in which the drill string has stuck from said coefficients and the values of the measured parameters M.

determining the distance of said well vector for said other well to said 910910sdat.125,3290ChevS -24centroids to indicate the probability that the drill string in said other well bore is substantially closer to the cenroid of wells wherein the drill string was freed than the centroid of wells wherein the drill string was not freed.

The invention also provides automatic apparatus for simulating drilling of a well bore with reduced probability of sticking the drill string based upon a data base of a multiplicity of drilling parameters M in a multiplicity of wells N previously drilled over a depth wherein the drill string has become stuck in well bores in a given geological province, said data base including means for storing a multiplicity of measured values of a drilling fluid system and drill string parameters for each of said multiplicity of drilled well bores, said multiplicity of drilled wells including at least one group of well bores in which the drill string stuck and another group of well bores in which the drill string did not stick over a range of depths similar to the depths of said one group, said apparatus including means for forming a total matrix from said multiplicity of measured variables for each of said multiplicity of wells, means for computing the roots of said total matrix, means for computing from said roots the coefficient for each of said o 0: multiplicity of measured parameters in said total matrix, means for computing well vectors for selected wells of said two groups of wells from said coefficients and the corresponding measured parameter in each of said selected wells, and the distance between each well vector and the boo centroids of the well vectors of at least said two groups of wells, means for repetitively computing another well vector from said coefficients and a preselected sequence of values, for a plurality of said parameters to simulate drilling of another well, in said geological province and means for selectively indicating changes in distance of each of said other well vectors by changes in said values relative to said centroids as a measure of the effect that said preselected sequence of values would improve the probability of not sticking the drill pipe in a well bore at a selected depth.

The invention further provides apparatus for controlling the drilling of a well bore to reduce the probability of sticking the drill pipe while drilling said 910910.cmsdat.125,39290chv,16 i i

U

well bore with a rotary bit and drilling fluid, comprising: means for storing a multiplicity, M, of measurable well drilling parameters measured in a multiplicity, N, of well bores drilled under comparable drilling conditions in at least two different groups of well bores, said stored parameters having been contemporaneously measured at a selected depth in each well bore and said at least two groups of well bores include one group in which a drill string stuck and another group in which the well bore was drilled without sticking the drill string through similar depth intervals of wells in which the drill strings stuck, means for entering said stored parameters into a separate matrix formed for each of said two groups of N wells, with each of said measured parameters being an element xjI in a common group array (row or column), and such group matrix for each of said N wells selected as a member of its respective group; where, in each of said following matrices and equations, j indexes any well in any group; i indexes any variable in any of said wells; and N is the number of wells in each group and M is the number and type of parameter or variable in each group; o means for computing for each group a Mean (average) Vector, X, of each parameter and forming therefrom a corresponding group Standard Variance Vector, Si: o m wherein said Mean Vector Xi is

N

SXi l/N x i 25 o where j N (wells) and i n (variables) 910910,cmsdat.125,39290Chev,17 9 -26and said Variance Vector S i is:

N

Si (xji X) 2 jii and the Standard Deviation Vector s i of each element of said group is: si -fsi means for converting each of said vectors s, into a Pearson product-moment correlation matrix R=rik wherein the value between any two parameters, say xj, and Xjk is defined as the group variance-covariance matrix, Cik

N

Cik 1 (xji Xi) (Xjk k) 1 N-1 Si, k 1,2,3 M and the group correlation matrix elements, rik Cik/SiSk express the linear dependence or relationship, of said pair of x's, (say i 1, k 2) and so that each of said coefficients rik is expressed in a square, symmetrical group matrix R where the i's and k's refer to each parameter in the total group population, additional means for converting said vectors s, into a within group correlation matrix, similarly defined so that the j's refer only to the members of its group and the X's and si's refer only to the mean and standard deviations, respectively, of the parameters of that group, means for similarly forming a weighted average of the two within group correlation matrices to form a total matrix RT which is symmetric, square, positive and semi-definite, means for solving for the roots of a matrix Q, wherein Q is the product of the inverse of the within group correlations matrix and the between group correlation matrix (total group correlation matrix minus within r 4; correlation matrix) such that the relations are: t f 910910,cmsdat.12539290Chev,18 ^x~Lcyc I -27- T=B +W where T total correlation matrix B between group correlation matrix W within group correlation matrix and Q W 1

B

and the solution is: t wherein the roots are the eigenvalues g and associated eigenvectors,) g, I is the identity matrix, and g is the number of roots which exist, means for multiplying each original measured parameter of a well in the original matrix in said storage means by its corresponding eigenvector coefficient' g, and means for separately summing the products for each array of measured variables for each well, S(h) recording means for storing the sums of said products as the well o vector for each well in each group as a representation of the probability that each of said wells in each group is correctly assigned to its proper group and to identify the coordinates of the centroid of each of said two groups of wells; means for similarly multiplying each eigenvector coefficient with each measured parameter in another well drilled within the geological province and within said depth range of said multiplicity of wells and means for summing the resultant products as a well vector of said other well bore; means for recording the coordinates of said other well vector; and means for comparison of said coordinates of said other well vector with said coordinates of said centroids of said two groups to indicate the probability of sticking the drill pipe by using the drilling parameters measured in said other well bore for continued drilling of said other well bore.

The invention further provides apparatus for predicting the effectiveness of changing a well drilling variable to decrease the probability that a drill pipe 910910nsdat.12539290Chev,19 i_ L -28 a a 0 a.

0 Jad a a a 4I* DAi r, will stick in a well bore being drilled with a rotary drill string comprising, data file means for storing each individually measured value of a multiplicity of controllable well drilling variables or parameters at a selected depth in each of a multiplicity of well bores drilled in a geological province wherein a drill string has stuck in a significant group of said multiplicity of well bores and another significant group of said well bores was drilled to a selected depth without the drill string sticking, means for converting each of said stored values in said data file means into an element xji of an array X forming a total group matrix, means for converting each of said stored values in said data file means in said group of well bores in which the drill string stuck into a similar element of an array forming a stuck well group matrix, means for converting each of said stored values in said data file means in said group of well bores in which the drill string did not stick into a similar element of an array forming a non-stuck well group matrix, means for calculating at least two correlation matrices from said total group matrix and at least one of said matrices for said stuck well group and said not-stuck well group, said two correlation matrices including a total group correlation matrix, T, a between group correlation matrix, B and a 20 within group correlation matrix, W, means for solving the eigenvalues and associated eigenvector coefficients for each measured variable of a matrix Q, wherein Q is derived from T B W to give Q W' 1 B -wherein W 1 is the inverse of matrix W, means for selectively calculating a single well vector from the sums of 25 the products of the recorded value of each variable in a selected well bore multiplied by its eigenvector coefficient, said sum defining said single well vector for said selected well bore of said total group matrix, and any other well bore drilled in said geological province, means for recording each well vector in said stuck well group matrix and said not-stuck well group matrix to establish a group centroid for each of said group matrices, and S means for indicating the relative distance of the well vector of another 910910lOnsdat.12539290Chev,20 29well bore to the centroids of said well vectors of said stuck well group and said non-stuck well group whereby a change in value of a selected variable of said multiplicity of measurable variables in said other well bore may be calculated by said well vector calculating means to measure the effectiveness of said change in said selected drilling variable to move the well vector of said other well bore toward said non-stuck well group centroid and thereby decrease the probability of sticking the drill string in said other well bore using said change in value of said selected variable.

Embodiments of the present invention are particularly directed to a method of, and apparatus for, evaluating the probability of correctly classifying the current or expected status of a well being drilled, or to be drilled in a known geologic province (as discussed above) without precise knowledge of the formations to be encountered. Then, based upon the evaluated probabilities, any selected one or more of a multiplicity of variable conditions or quantities that affect drilling fluid physical and chemical properties, drill string O configuration, and bore hole characteristics in earth formations traversed by o* a the well bore may be controlled to correct or maintain drilling conditions in a 0 a well to minimize the probability of sticking the drill string. Moreover, if a drill a' string becomes stuck, the probability of the cause for such sticking may be S° 20 identified so that efforts to divert or relieve the drill string may be optimally such sticking.

In embodiments of the invention statistical analysis of the probability of a drill string sticking in a well bore is predicted not only for differential pressure problems, as primarily addressed by prior workers in the field, but also for other causes of sticking that may be substantially unrelated to differential pressure, but A' 910910,cmsdat.12539290Chv,21 44 3o which have been found to be equally important in avoiding drill string sticking. In particular, by statistical analysis of at least two types or groups of wells, namely those in which sticking has occurred and those wells that were drilled without the drill string sticking, we have found that the method and apparatus of the present invention makes possible significant improvement in directing current well drilling, or in planning future well drilling, either in a current well or a well to be drilled. Thus, the present invention may be used to control drilling *00a 0 either continuously or periodically over given depths o 0O of a well, and if the drill string sticks, to evaluate not only the probability of how it became 0°1O stuck, but also to determine whether to begin or continue fishing or other remedial action, or to abandon further efforts to relieve the drill string.

0o o0 In theAmethodsof the present invention, statistical control of drilling is carried out by collecting and recording measurements in an adequate °o number of types of wells where the drill string has, or has not, stuck in a given geological province. A data base, or file, is first collected by a storage I means and assembled to form data matrices containing a multiplicity of values for each well, based on drill string and bore hole parameters at a selected well depth in each well. Such data matrices are then assembled in groups or classes of wells which include a multiplicity of wells in which the drill string has become stuck and another multiplicity of wells that have been drilled, without becoming stuck, through a depth interval comparable to wells in the stuck class. Also, a data matrix is similarly formed of

L-,

I~

'4 31 all wells and their measured values. In a preferred form, a probability "map" is created by plotting or recording a vector, referred to herein as a "well vector", based upon manipulation of the data matrices for each variable in each well of the matrices, or another well in the same or a similar geological province.

Under the most general, but not overall inclusive circumstances, each measured variable in 000 10 each group of wells, is an element, xji, (column or S row) of an array. The size or order of each such Sarray, or matrix, is equal to the selected number of variables M recorded in each well and the complemen- 0 00o tary column or row of the selected number N of wells.

From each such matrix, a variance-covariance (or other equivalent measure of similarity) matrix for each such variable relative to other variables in all other wells of its class, or group, is developed.

o* 0 From these matrices, the Pearson-product-moment correlation coefficient, or any similar measure of o o° similarity/dissimilarity, matrix for each class of wells is developed. Then, by procedures known as multivariate analysis, the eigenvalues and o0 eigenvectors (or other characteristic functions) of these similarity/disimilarity matrices are resolved.

In most cases, the eigenvalue and eigenvectors then Si determine the relative contribution or coefficient of each of the multiplicity of measured variables in the pertinent matrices. Such coefficients provide well vectors falling into substantially distinct groups that statistically group about a central cluster or tendency, called the mean or centroid, for wells in each group. Such groups are spatially separable and -11 i j 3) 2 recordable for graphic display, or analysis, of any well vector relative to all wells sampled in a given geological province. The coordinates of a well vector in any hyperspace are established by multiplying each measured variable by the coefficient value for that variable and summing such values to locate the intersection of the data vector in that hyperspace or sub-space.

In a common ternary linear method of o" carrying out the invention, multivariate discriminant 000 analysis of the data matrices includes recording the data and graphically finding a mathematical o r. hyperplane which optimally separates two groups.

15 Where three groups are evaluated, the third group is separated by another hyperplane so that two hyperplanes separate the three groups. In this case the three groups may then be plotted as projections "oa onto a surface representing the two hyperplanes.

o 20 Each well vector representing one of the wells in the O 00 complete set of wells may thus be projected onto the oo plotting surface as a point. From these points the intergroup distances between a well in any group and the centroid of each group may be calculated. The o~ E25 grand centroid of all such values may also be determined, and, if desired, recorded for mapping or i plotting on the plotting surface. Based upon the calculated probability that each well is correctly classified in its proper group, the probabilities of correctness may be contoured in any sub-space. Where the probabilities are nearly equal that a well belongs to either of two groups, the vector intersection point will normally fall near the intersection of the hyperplanes or plotting surfaces and the area 33 between the centroids for each group will be approximately equal. Accordingly, the further the well vector point is removed from such an intersection toward one of the group centroids, the greater the probability that the well is correctly classified. As will be understood by those skilled in the art of statistics, the highest probability that a well is correctly classified is where the normalized or scaled distance of a well vector to the total within-group or grand well cenfroid is a *oio maximum and the distance of the well vector away from °o all other groups centroids is a maximum. Such values 0 o J: or distances may be expressed either numerically or

S

c graphically.

From the probability "map" it is then possible to plot the progress of a drilling well based on the same measured multiplicity of variables.

As before, the coordinates of a well vector on the 00 map are established by multiplying each measured variable by the coefficient value for that variable and summing such values to locate the intersection of the well vector on the map surface or surfaces at its current drilling depth. Control of selected ones of the measured variables within realizable limits may then be modified to change well drilling conditions sufficiently to move the coordinates of the well vector projection toward or near the "never stuck" probability centroid.

For example, in the three group case, where the analysis of the multiplicity of measured variables generates a well vector which correlates current well drilling with mechanical sticking of the 4; C 1 34 drill string, such conditions have been found to heavily depend upon angle of the bore hole to vertical, bore.hole diameter, size of drill collars, and total depth of the bore hole, as well as frictional forces (drag) and torque on the drill string, but they also relate to drilling fluid hydraulic and chemical properties. Where such well vector projection lies at a point that primarily corresponds to a high probability of differentially sticking the drill pipe, such vector may heavily depend upon drilling fluid characteristics, such as density, viscosity, gel strength, water loss, and 2 flow rate; but it may also relate to depth and angle of deflection of the bore hole. Other measured drill 15 system variables that may cause sticking problems are also desirably evaluated by the present method, such as true vertical depth, drill fluid pH, and drilling gas. In each instance, of course, such measured aO 4 variables are adjusted only within the allowable 0 i range of their usable (physically feasible) values.

.ro Because the multiple measured parameters at any depth in each well adequately and clearly delineate the probability that during drilling of any

I

1 awell within the sampled depth interval will fall into the correct category of any two or more groups, such i a well to be drilled, or being drilled, may be controlled to "steer" its drilling conditions away from any sticking hazard and toward the probability of not sticking the drill string.

In accordance with a preferred form of apparatus for carrying out the invention, data storage or file means are provided for storing each

QU

910910,cmsdat.l2539290Chev,13 individually measured value of a multiplicity of controllable well drilling variables or parameters in each of a multiplicity of individual well bores in a geological province wherein drill strings have been stuck. In the case of stuck wells, the selected depth for such measurements is preferably the depth nn :where the drill pipe actually stuck, but may include the history of conditions prior to sticking, 0 including depth. For wells in the non-stuck well matrix, one depth is randomly selected within the range of the depths where the drill string stuck in 00 0 other matrix wells. Each measured value stored in the data file means then forms input to means for converting such stored values into elements of matrix arrays. These arrays corresponE', for example, to all wells in the total group, a stuck well group, and a t not-stuck well group.

o, In general, and for simplicity, a linear or other algebraic and statistical proposition is defined for two groups of wells or more. It is :00: "particularly convenient to analyze only two or three groups in the multidimensional space of this problem, but greater numbers of groups may be analyzed without departing from the scope of the present disclosure.

The data array matrices are then analyzed in calculating means adapted to compute at least two multivariate variance-covariance and/or correlation matrices, one of which is the total group matrix (T) and the other is either the between-group correlation matrix or the pooled within group correlation matrix so that T B W. The eigenvector coefficient solutions of these matrices is then computed by discriminant analysis means so that such fi^l, mutvraevrac-oai eado orlto maries one ofwihi h oalgopmti T ii

'A

If

'A

00Q 10

(C

4 coefficients maximize the determinant of the ratio of the distances of each group of well vector betweengroup centroids to the distances of well vectors within a group centroid, for all wells in the stuck and not-stuck groups. In a preferred form of apparatus, the three matrices are formed by means for calculating the mean corrected sum-of-squares of the cross-products for the "scores" of all stored values in each matrix. This includes means for calculating a standard mean deviation of each data element in each well to generate a standard variance-covariance matrix for the total group and each class of wells, and a standard correlation matrix, called a Pearson product-moment correlation coefficient matrix. The latter is obtained by means for cross multiplication of the corresponding measured variables and addition of their cross products for all possible pairs of wells in each matrix and scaling by the appropriate standard deviations. From 20 such matrices %he eigenvalue and corresponding eigenvector coefficients are determined from the between well group matrix the within group well matrix and the total group well matrix by means for solving B=T-W. For convenience, it is desired to solve the equation: lBI Q=IWI, expressed as Q=W-IB, in matrix algebra notation, such that Q is a maximum.

Solution of the determinantal matrix Q is then: 1QAI~u where A an eigenvalue 3 the eigenvector associated with a given A, and I the identity matrix (representing Ww 1 W 1 the inverse of W

I

37 From such eigenvalue and eigenvector coefficients, a well vector for each well in the total group may then be generated. Such vector is generated by means for calculating the sums of the products of the value of each recorded parameter multiplied by each eigenvector coefficient. A 0 multiplicity of well vectors from the multiplicity of •wells is stored by recording means and, if desired, indicated by plotting means to display a probability 0"Oo10 map applicable to the entire geological province.

Points plotted on the map may be displayed as points about the centroids of points which represent wells that are known to have stuck either by differential pressure, or because of mechanical 15 problems and wells where the drill string did not 0/ 0stick.

o.o Each of the three groups may be similarly separated by a technique known in statistics as "multivariate analysis". In such technique, the three groups are separated by intersecting mathematical hyperplanes. Each well vector from multidimensional space is then resolved into a set of coefficients representable as a point on a mapping surface. This permits vector projections from multidimensional space to be separated to the maximum extent and the vectors plotted. By contouring the probability of each well as represented by its vector coefficients on the mapping surface, it is thereby possible to separate probablistically wells that became differentially stuck from those in which the drill string became mechanically stuck, and both are separated from the "non-stuck" drill string i h** 3^ vectors. Then, in a well being currently drilled and from individual measurements of the same variables at any level in the well bore, the coefficients for each such variable may be used to calculate the predicted well vector, which is the sum of each coefficient multiplied by its corresponding, currently measured variable value. These sums yield the vector 9r00 coordinates of the well vector being controlled for o display on the mapping surface. The probability that .o 10 the present position of the drilling well vector will or will not result in the drill string becoming stuck may then be evaluated by its distance from the centroids of the three groups. From such calculated position, the controllable variables mud weight, solids, drill collar size, etc.,) in the oftft drilling well may be evaluated and modified to move the probability of the drilling well toward the coordinates of the map that represent a desired high 0 6 o probability that the well is in the "not stuck" region. Such a procedure makes possible analysis and directional control of the drilling well to preclude r the probability of sticking the drill pipe in a drilling well.

In accordance with another significant aspect of the present invention, multivariate analysis may be similarly used to evaluate the probability that a stuck drill string can be freed.

For such analysis a plurality of wells in which the drill string has stuck forms the total group in which a multiplicity of drilling parameters are recorded for each well at a given depth. The given depth, as above-discussed, may be either at the depth where the i drill string became stuck or at another depth near I. i' 1 1 91o9JOcmsdat-12,392WhevII7 39 -U1the sticking level. Wells which were then subsequently freed and those which were not freed are then grouped together for the multivariate analysis.

Accordingly, group matrices are formed and calculated to determine the variance-covariance, or similar measure of similarity of, distance of each well vector to the group centroids or means. However, in such analysis, the eigenvector coefficients that minimize the ratio of the distance of a well l 10 vector to its within group centroid to the d distance between group centroids, are scalar. Based on a significant number of wells which became stuck, the wells in which the drill string could not be freed cluster about their centroid and are clearly distinguishable by such scalar values, from the centroid of wells where the drill string was freed.

Desirably, the multiplicity of measurable values may include time values, or other variables based on when a drill pipe becomes stuck, such as time elapsed between sticking and "spotting" oil in the stuck well.

The method of determining the probability of releasing or freeing a stuck drill string may be carried out either independent of well drilling control or other forecasting aspects of the present invention. As a subsequent analysis method to normal drilling operations, the apparatus or method aspects of predicting whether the drill string can be freed may use substantially the same multiplicity of well variables.

Further objects and advantages of the present invention will become apparent from the MI4 910910ocpnsdat.1253929OChev,18 Ao following detailed description of the preferred embodiments of the present invention, including the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS Fig. 1 is a perspective cross-sectional ao" elevation view representing a plurality of wells 0"o drilled from a single off-shore platform and india o cates several types of deep, highly deflected, wells to which the well drilling method of the present invention is particularly applicable to improve the So probability of avoiding sticking the drill pipe in the well bore.

Fig. 2 is a perspective elevation view of a 1'5 portion of a well bore illustrating one type of problem involved in mechanically sticking a drill string, namely, a small diameter keyseat formed by the drill pipe in the side of the well bore.

Fig. 3 is a perspective elevation view of a 0 portion of a well bore illustrating a drill string sticking against a low pressure formation due to differential pressure.

Fig. 4 is a cross-sectional view through the drill string and well bore in the direction of the arrows 4-4 in Fig. 2, indicating a drill pipe in a keyseat.

Fig. 5 is a bar graph of survey angles of well deviations from vertical in a significant number of wells drilled in a given geological province which

"I

L

-3a pressure problems.

Fig. 6 is bar graph of measured depth ranges of wells in the sample of Fig. 5 plotted against the percent of total occurrences of sticking, as between mechanical and differential pressure, and those that Bo"° did not stick.

4oo 6o Oo o Fig. 7 is a bar graph similar to Figs. 5 and 6 showing hole-size range plotted against percent of 10 total occurrences of mechanical and differential

S

pressure sticking.

Fig. 8 is a stuck pipe probability "map" in which the vector of each well is plotted as a point o intersection of its vector from multidimensional space with a two-dimensional surface. Such surface is a projection onto the two hyperplanes which o separate the three spatial vector groups representing the three classes of wells, which were stuck (1) r. mechanically or by differential pressure and (3) those that were not stuck.

Fig. 9 is a stuck pipe probability map in which the probability of each well being correctly classified into its correct group is contoured.

Fig. 10 is a plot of the progress of a single well, which was analyzed by the sampled variables at regular depth intervals before it became stuck differentially. The plot indicates the course of the well as it proceeded from a probability of not being stuck, through the probability of being either 1 i' i i' 910910 sdat.125290CDev O i, mechanically or differentially stuck, to a high probability at the end condition that the drill string would, and in fact did, become differentially stuck.

Fig. 11 is a triangular graph of well *vectors shown in Figure 9.

So a j Fig. 12 is a plot of well vectors generated by a simplified, artificial example of four o. measurable variables in three wells in each of three o ,'10 different groups or classes of wells, and the centroids of each group as calculated by a computer program.

t Fig. 13 is a graph illustrating use of the *present invention to determine the probability of freeing a stuck well by using multivariate analysis to identify whether such a well is in a group of wells in which a stuck drill string was freed or in another group which was not freed.

Fig. 14 is a schematic diagram of the essential elements of a preferred form of system for carrying out the invention for determining and displaying the probability of sticking the drill pipe in a well bore.

Fig. 15 is a more detailed schematic diagram of the system of Fig. 14 and illustrates optional forms of apparatus, including Linear Programming for ranges of variables to control well drilling and to evaluate the probability of freeing a drill pipe in a i stuck well.

243 DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT OF THE PRESENT INVENTION Fig. 1 indicates in elevation and partially in perspective, a fixed off-shore drilling platform 10 of the type normally used to develop a major portion of one or more underwater producing forma- S tions. The well drilling control system of the o O^Oo present invention is particularly applicable to such o o drilling because a plurality, say 10 to 30 wells such i as 11, 12, 13, and 14 and 15 are drilled from a single platform 10 at high deflection angles to vertical to develop an underwater petroleum reservoir 16 extending over several thousand feet laterally from the platform. As indicated, the wells 11 to 15 are selectively drilled at differing angles and may include one or more "dog legs" 18 (different angles So'Q' to vertical). They may even take S-curve configurations, as in well 11 or 14, in drilling to a desired depth.

Such configurations may either be planned because of geological conditions or occur inadvertently during o. drilling.

a o It has long been known that high angle wells have a tendency to stick the drill pipe. This is particularly true at depths in excess of 12,000 feet but may occur at virtually any depth. It has generally been assumed that such sticking is due to differential pressures between the well bore and an earth formation acting on the drill pipe; such differential pressure being due to higher pressure in the well bore than in a formation traversed by the well bore. In some geological provinces, including offshore wells in the Gulf of Mexico, high pressures i are frequently encountered at relatively shallow hI -34depths; that is, the pressure in such a formation exceeds the normal vertical gradient of hydrostatic or geostatic head expected at that depth. (Normal well pressure is essentially the pressure of fluid in a well bore at a given depth.) To control over-pressured formations, the well pressure, as applied by the density of the drilling fluid, or mud, o in the hole, must exceed pressure in the formation.

°o 0 However, at greater depths in the well, formation 0o °i0: pressures may be nearer to normal for such depth.

Accordingly, to maintain adequate well pressure .ao. opposite the upper high-pressure formation, hydrostatic pressure on the lower formations may be excessive. Such excessive well pressure may fracture the formation, with resulting loss of drilling fluid o* to the fornation and consequent blow-out danger.

o 0 In drilling wells with excessive bore hole o, pressure through lower pressure, permeable formations 0 a using water-bsed drilling fluid, water may flow into the formation. Such flow is through the well bore S'°o mud or filter cake 20 around well bore 21, which normally is a thin layer of gelled solids that seal off the permeable formation 23. This flow may cause excessive precipitation of solids on the filter cake.

The condition is indicated at 22 in Figs. 2 and 3.

Continuing flow of liquid into the formation increases the thickness of the filter cake and increases the con'act area of the drill pipe 17 so that the drill pipe seals or sticks against the wall of well bore 21. An increase in the filter cake thickness additionally tends to make restoring drilling fluid circulation between the drill pipe and the well bore difficult.

S!

i -rH-* 1 r. .=re r Further, the thixotropic drilling fluid returning to the surface from the drill bit and flowing over the remaining area of the bore hole 21 may become relatively laminar so that the fluid tends to set up or gel. As is well known in the drilling art, the precise cause of such differential sticking is frequently difficult to determine. Hence, correcting such a condition is, in general, by trial and error.

o °o 10 Further, the prospect for correcting a "stuck condition may determine how much non-drilling rig time the operator can afford to use in "fishing" or other operations, as opposed to the cost of abandoning that portion of the well bore. Such abandonment frequently requires sidetracking the hole I above the last pipe section that is not stuck. This 0oo may require explosively cutting or unthreading the drill pipe above the stuck point and perhaps setting 0o 0 a plug in the bore hole, with consequent loss of equipment including drill collars and bit, or other devices. The well is then redrilled to the same 00 depth, and deeper if possible. Accordingly, knowing the probability of avoiding sticking or unsticking a differentially stuck drill string, as well as knowing the probability that the drill string is mechanically stuck, rather than differentially stuck are of high Seconomic value. This is particularly true where rig cost is on the order of thousands of dollars per hour, as in offshore drilling.

Figs. 2 and 4 illustrate a portion of a drill pipe 17 above the drill collars 25 and drill i l v7 1 J A_~i tI bit 27. As shown, substantially all of the drill pipe 17 is smaller in diameter than bore hole 21, as originally cut by drill bit 27. Generally, the drill pipe proper is more flexible than the bottom hole assembly, including drill collars 25 and drill bit i I GO a a 0 Qso 600 0 0040 o o0 0 0 0 0 0 0 00 00 0 0 0 0 0 0000 a a 0 O o o 0e 0 0 00 or a 00 a 0 00 20 04« 00 0 0 0 o 0 C7. uuord ungly at nign angles, the arill pipe may tend to sag against one side of the well bore wall.

The drill string in such a condition may mechanically cut the side of the well bore as at 29 in Fig. 2 and 4 to form what is known as a "keyseat". Under such conditions, the diameter of drill pipe 17, or joints between pipe sections are smaller than the drill collar sections or drill bit. When the pipe is then moved up or down (as in a "round trip" of the drill string to change bits), the pipe or joints may cause the pipe to mechanically stick in the bore hole.

Other mechanical problems may result from collapse of a low pressure or unstable formation into the well bore. While it has been known that a drill string may become stuck both by differential pressure conditions and mechanical problems, it has been commonly assumed that the greatest danger is in differential sticking and prior practice has generally been to assume that any stuck well is differentially stuck.

We have found from our statistical study of numerous cases of pipe sticking that such an assumption is not necessarily true. As a result, conventional methods of attempting to avoid sticking a drill pipe, or to unstick the pipe, may not be appropriate to the statistically most likely or probable cause of either mechanical, or differential 1-2Z L u C~.x

O

D

a

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-7 sticking, or both. Accordingly, a method of minimizing the probability of sticking a drill pipe in a drilling well was a long felt need in well drilling.

Our study included well drilling variables measured in the same geological province in several hundred wells, some of which were known to have stuck due to differential pressure. Other wells in the o- 10 province were known, or suspected, to have stuck due "o to mechanical problems. However, in the same "0 'o geological province a significant number of wells 040 were drilled where the drill string did not stick.

All were drilled over a significant geological area in the Gulf of Mexico. In general, the wells sampled in such geological province involved wells drilled deeper than 6,000 feet in a basin having generally similar, or common, geological structure. Such wells were drilled basically through sand and shale strata forming traps for petroleum reservoirs, such as those around salt domes or terminated by faults.

In general, according to the methodology of our study, a stuck pipe case was classified as differential if the following occurred: Full circulation was maintained after sticking.

The pipe became stuck while sitting motionless in the well.

46 A There was a known high differential pressure between the wellbore and a permeable formation.

There was no sign of borehole instability.

aO" Basically, all other occurrences of drill a *a pipe sticking which did not fit the description of 0b o differential sticking were classified as mechanical.

The cases in this category included sticking due to a 10 borehole failure or collapse, keyseating, and undergauge boreholes.

As will be explained more fully below, on the order of 20 taken from some 100 well variables o a OPa'

S

s selected drilling variables in each oa a well were measured. The values of such quantities o oo were recorded at selected depths in each well of a multiplicity of wells in each of three classes mechanically stuck, differentially stuck, or non- 0 t" stuck). The relative number of wells in each of the three classes is indicated in Figs. 5, 6 and 7.

Fig. 5 shows in bar graph form the percent of wells in the sampled number where pipe became stuck mechanically or differentially over a range of from 0O to 75' deviation from vertical. Fig. 6 indicates, in a bar graph, the distribution of the three classes of wells forming the data matrices plotted as a function of depths of the wells. Fig. 7 is a bar graph of the hole size range of wells in the sample of stuck wells.

As a guide to selection of variables for measurement, statistical techniques known as

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-4L 4 AA 94ar o o 00a 0 06 correlation, principal component and factor analysis can be used. The goal of the exploratory analysis was to use such analysis to determine whether interrelationships were noted. It was then necessary to determine whether the interrelationships were different for each classification group: mechanically stuck, differentially stuck, or non-stuck. These analyses were conducted on the three groups separately. The results indicated that there were distinct factor patterns in each group. In addition, on the basis of these interrelationships and factors, several variables were highlighted as having potential use as key variables in the subsequent multivariate analysis techniques.

Figs. 8, 9 and 10 are probability plots of well vectors (eigenvector projections) onto hyperplanes or maps of each well in each of the three classes of wells. These plots or maps were developed by multivariate analyses of measured variables in each of the three classes. These maps indicate that the three classes of wells can be readily distinguished with sufficiently high probability so that, by measuring the same multiplicity of measured variables at any depth in a well being drilled, the drilling conditions in the well may be plotted to control the well while it is being drilled. Such control may be either by preplanning the drilling program or by implementing corrective action during drilling. A typical well during drilling is plotted, as in Fig. 10, to show its progress, relative to the three conditions, on such a two-dimensional map, based on projections of the well vector groups to the best separating hyperplanes.

a 20 6 a

I

u c a 4 control, and as shown in Figs. 8, 9, and 10, is by statistical analysis using a method known as multivariate discriminant analysis. Discriminant analysis is used to determine whether groups with different predetermined attributes mechanically stuck, differentially stuck, and non- S. stuck wells) can actually be distinguished as such, on the basis of data from each group. A discriminant analysis involves the solution of a set of equations a 00 that are derived from matrices describing the o0 "between-group distances" and the "within-group 0° distances" of all wells in a particular group about their joint means (centroids) and about the grand joint mean, or grand centroid. It is necessary to solve an eigenvalue problem so that the determinantal 000 ratio of the within-group distance to the betweenio group distance is a minimum. This solution yields a set of eigenvectors or coefficients that, when o 0 normalized, can be multirlied by the original data and summed to yield a discriminant value or discriminant score. In a given geological province, 0 o .d a significant number of wells, each of the three types of wells, is used to form statistically reliable samples. Numerical matrices are then developed for each group using the same multiplicity of variables. It will be apparent to those skilled in the applicable arts that similar probability maps can be developed for other geological provinces from such a multiplicity of significantly different measured drilling variables, selected in accordance with the desires of the well driller and the statistics of the controlling variables.

i'

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Fig. 8 is illustrative of separation of three groups by two intersecting hyperplanes, as indicated by three lines intersecting at the center of the plot. These lines are the best separating boundaries, as determined by such hyperplanes, here shown as linear.

Fig. 9 is similar to Fig. 8 and illustrates so-called "iso-probability" contour lines in each of 0o00 oao q the three groups indicating the probability that each °o 1 0 well vector is correctly plotted within the assigned o r group. The well plotted in Fig. 10 is on the same 0,3 0 0o vector coefficient map as the wells plotted in 0 Figs. 8 and 9.

Fig. 11 illustrates in a triangular graph 15 an alternative method of plotting the probability data of the wells shown in Fig. 9 for each of the three classes of wells. As indicated, the nearer o'o" each well is to the probability apex (100%) in each class, the greater the probability that it is correctly classified. Thus, corrective action can be o undertaken through modification of the most significant contributing variables. In Fig. 11, a total of 82 wells were plotted, 36 wells stuck the drill pipe differentially, 33 did not stick the drill pipe, and 13 stuck the drill string mechanically. Of these wells, 59 were plottable within 90% probability that they were correctly predicted; that is, within the shaded triangular areas at the three corners of Fig. 11; the remaining well vectors were then plotted as indicated by the individual well vector plots for r A i 0.001 0 0 0 00 0 0 G 00f 0 00 0 040 0 0 0 0 0000 0 4 0 ol a 0 00 0 0 009 0 04 O 0 0 a0 mechanically stuck, 10 differentially stuck, and 8 not-stuck, wells.

Alternatively, as explained in connection with apparatus in accordance with Figs. 14 and the distances between the group and grand centroids and the distance of a given well vector to each of the centroids may be recorded and indicated as distance values alone, with plotting or mapping a graphic alternative only.

Further in accordance with the present invention, multivariate analysis of a multiplicity of well parameters in a multiplicity of wells may be similarly used to evaluate the probability that a drill string that became stuck by differential pressure can be freed. Fig. 13 illustrates in bar graph or histogram form such analysis of 43 wells in which the drill string had stuck. In all, 23 wells of the total group were freed and 20 were not freed.

The multiplicity of drilling parameters recorded for each well at a given depth may be the same as those used to determine probability of the drill string sticking as described in detail above. Preferably, the analysis additionally includes a variable that reflects the time interval from drill string sticking to "spotting" of oil in the well to break the filter cake between the bore hole and a porous formation.

As indicated above, such oil is normally added to the drilling fluid when the drill string becomes stuck to lubricate the bore hole or to break the filter cake.

By statistical analysis of the total group of freed and not freed wells, or as determined by previous tracking of the well for probability of sticking, the 06 4 Ilk- 1 1-X'r 53 00ooo0 oo o 0000 10 0 e0 a 0o 0 00 0 o 0 00 ho 0 well vectors for each of the freed and not freed wells are plotted. Wells which were subsequently freed and those which were not freed are seen to cluster together by such multivariate analysis.

Because only two group matrices are formed, determination of the variance-covariance of well vectors for each well relative to the centroids of the two groups need be only scalar (one dimensional) values. The eigenvector coefficients which minimize the ratio of the distances of well vectors to their within group centroid to the distance between the group centroids thus has only magnitude. Based on such values, it will be seen that differentially stuck wells that could not be freed closely cluster about their centroid of well vectors of about 0.66 on the scale of Fig. 13. This centroid is clearly distinguishable from well vectors where the drill string was freed. While the centroid of such freed well vectors is not as well defined, a centroid of 0.54 may be chosen as the average value of the probability that a drill string can be freed. As noted in regard to determining probability of not sticking the drill pipe based on such well vectors, the smaller the ratio of the distance of a stuck well vector from one group centroid to its distance to the other group centroid, the greater the probability that a well is properly classified as to whether it can, or cannot, be freed. The graph of Fig. 13 indicates that wells that could not be freed are particularly clustered about their well vector centroid and are best separated from the freed well vector centroid by a value of about 0.60. This gives a well operator strong indications, in early efforts to free the drill string, ar to the amount of time 0 00) 00 0 0 00 oo 0o 0 o 1 i 1. i~

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Ir 2*:4 57h _V6_ g m 0 a4 0 o Au o 0 0 00 0 o 00 0 00Q 0 0 00 0 0 0 a e 00 0 0 0 and cost that should be incurred before abandoning or sidetracking the well. Because each fishing job in a deep well may cost on the order of $500,000 or more (primarily based on lost time in drilling), such probability determination can save substantial drilling expense. The individual historical cost items of stuck wells may also be used as additional variables in matrix analysis of these wells.

From the foregoing detailed description of the method aspects of the present invention, it will be seen that a system can be implemented as shown in Figs. 14 and 15. As shown schematically, well data accumulated in unit 50 flows into automatic data processing means 52 to generate the required sequence of events to analyze the multiplicity of measured variables in a multiplicity of wells to generate well vectors useful to control selected well variables for continued drilling of a well or to generate displays of probabilities if drilling is continued using selected values of such variables.

Fig. 14 particularly illustrates the general system which includes means for accumulating well data, as in Well Data Storage file or means 50. The basic well data comprises a multiplicity of values or parameters of measured drilling variables M for each of a multiplicity of wells N. The data may be taken from previously drilled wells, or on a cumulative basis over a period of time, as by frequent, say daily, entry of values from individual wells.

Entries of data may either be key-stroked into well data, or file storage, 50 or introduced as a stream of data via modem, tape, disc, or computer from the 0000 o S 0 00 0° 20 i s

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crLJS r i _Aq- "tour" sheets or daily logs of each well. Stored data in well data file 50 is then fed to Matrix Converter 55 of data processing means 52. The major elements of processing means 52 are Multivariate Analysis means 56 and Well Vector Generator means 58.

Matrix Converter 55 establishes a grand or total matrix of all values of the measured well parameters for each well, and at least two additional matrices formed by classes, or sub-groups, of the total group of wells. Multivariate Analysis means 56 then resolves each matrix into a set of coefficients o representing each variable of the matrix. The coefficients are stored in Well Vector Generator 58 o i to generate an individual well vector for each well in the total matrix from the stored variable values in Well Data Storage means 50. The same variables for any other well or group of wells or sets of such fo2"oA variables from simulated or actual well drilling a aconditions may be similarly computed by feeding such sets of data from Test Well Unit 100 to Well Vector fom d Generator 58 through input line 57. As discussed above, each well variable is multiplied by its corresponding coefficient and the products are SA summed.

The centroids of each group of thll vectors and the total group are computed and their relative distances determined and their values stored in Well Vector and Centroid Recorder 60. Such recorded values preferably include the relative distances between individual well vectors, the distance of individual well vectors to their respective within group centroid, as well as their relative distance to other group centroids and the grand centroid. These j sAet or L i'r! i i- ;r recorded values are then available for display and analysis by Display unit 62. Unit 62 may either present the recorded values as a map, as shown in Figs. 8 to 13, or as numerical values alone.

As indicated more fully in Fig. Multivariate Analysis means 56 includes Matrix Converter 55 which is supplied with a multiplicity of variables M from each of a multiplicity of wells N o°o for each group of wells or the total wells from Data o 3.a 10 Storage means 54. Converter 55 then selectively "o forms a matrix for each desired group or class of o0O wells using the same multiplicity of well drilling variables for each well in its respective group. As indicated schematically, each matrix formed in Matrix Converter 55 may be stored in registers or compilers indicated as Total Well Matrix register 63, Non-stuck ooo o *Well Matrix register 64, Stuck Well Matrix :o register 65, Mechanically Stuck Well Matrix register 66, Differentially Stuck Well Matrix °oC, 20 register 67, Freed Well Matrix register 68, and Non- Freed Well Matrix register 69. It will be understood, of course, that the matrix in each of registers 63 to 69 is selectively connectable to the multivariate analysis computation apparatus for serial processing through Standard Mean Vector Computer 70, Standard Variance Vector Computer 72 and Variance-Covariance Matrix Computer 76. In this arrangement, a matrix stored in one of the registers 63 through 69 is fed to Standard Mean Vector Computer 70 for computation of the values of Xi. The output of computer 70 is supplied to the Standard Variance Computer 72 for calculation of S i and then to Variance-Covariance Matrix Computer 76.

6P;7 V r I 57 3- Preferably, such computation includes formation of a Pearson-product correlation matrix, or another similarity matrix for the computation of Cik.

Although each of the matrices stored in the Non-Stuck Well register 64 and Stuck Well register may be separately computed and compiled to form values stored in the Total Group Correlation Matrix, T, unit 78, preferably the total group correlation 0 matrix is separately computed from total well data 10 stored in register 63. Such total well data may be no formed by any desired groups of well data, and need .o not contain all groups where fewer groups, such as 0 o 0 where Freed and Not-Freed Well Computers are to be O separately evaluated. The within group correlations are then calculated from the output of Variance- Covariance Matrix Computer 76 in Within Group Correlation Matrix Computer 80 to determine the value o0 of W. Similarly, Between Group Correlation Matrix is calculated by unit 82, wherein the within group So0",20 correlation matrices and the total group correlation matrix are suitably combined to produce B T-W.

0 For computation of Total Matrix Product Q W- 1 B, in unit 86 the inputs of matrix B stored in unit 82 and W-1 in Inverse Matrix Computer 88 are required. Solution of the total matrix product formed in computer 86 is obtained by computation of the eigenvalues Ag which exist the minimum of number of groups minus one or the number of variables). Similarly, eigenvectors 1 g are calculated for the same M number of variables. The values obtained in unit 84 are then stored in Eigenvector and Eigenvalue Register 90 for subsequent L :I produced from the original well data in each group of wells as supplied from Data Storage unit 54 through line 57. Data from any subsequent well, indicated as being stored in Test Well Data unit 100, will include the individual M variables for that well. Thus, data from wells in the same or a similar geological province may be evaluated, whether actual o0 measurements or simulated data. Well Vector o Generator 58 multiplies and sums the total of each eigenvector times its corresponding variable delivered from either data storage 54 or test well o* o data in unit 100. Individual well vectors are stored in Well Vector Register 92 of Recorder 60. These may be directly displayed as by Display unit 62 or used to compute group centroids or a total group centroid o 4 in Group Centroid Generator and Register 94.

d Preferably as indicated in connection with 0o..20 the general description of the invention, a grand mean, or grand centroid, and group centroids for each selected groups of wells, will be recorded as plotting data for display on a surface. The data will include relative distances between group centroids, and their distance to the grand centroid.

Similarly, distance from any selected well vector to any group centroid may be calculated and stored in Distance I:mdicator 96 for read-out or display as numbers. Display of the complete suite of well vectors, or any selected summations or parts, may then be plotted as by Centroid, Well Vector and Probability Plotter 98.

ii 0'

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0900 000090 a a« o 09 o a 0 0 0 09 00 o 0 Where only two groups are to be evaluated as explained above, a simple distance indication is adequate to distinguish a well selected from data storage unit 54, or a test well, relative to the centroids of the two groups. As indicated, such display may be suitable for a system such as that described in connection with Fig. 13 for distinguishing between freed wells and not freed wells after a well has become stuck. Where multiple groups are to be displayed as in Figs. 8 to 12, plotting of the centroids and an individual well is preferred, and, preferably probability contours, as indicated in Fig. 8, may be added to the display prepared by the plotter. However, it will be understood that, where the centroids are well known for a group of wells, say as between wells that are stuck mechanically, differentially and not-stuck, a single distance indicator for a test well relative to any of the three groups may be quite sufficient for display. The distance may be expressed either as a ratio of numbers, or as a number indicative of relative distances, without actual display of the data on a map, or other graphic plot.

The arrangement of Fig. 15 also indicates means for optimizing variable values that can be selected within given ranges of values to correct or avoid the probability of sticking a drill pipe in a well under given conditions in the test well. Data stored in unit 100 may be diverted to Linear Program (LP) Variable Ranges unit 102 for successively supplying physically feasible values for each selectable variable to Well Vector Generator 59. The resultant values are then recorded in Register 92 and 0 00 0 00 00 0 0 00 0 C i i r v r ~L ^iil 1 supplied to LP Optimizer 104 along with centroid data from Centroid Generator and Register 94.

Optimizer 104 is set to terminate operation of unit 102, as by switch means 106, when a selected value, or a selected value within a range of values, optimally moves the predicted well vector away from the stuck well centroid or centroids and as near the non-stuck well centroid as economically feasible.

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EXAMPLES

To illustrate development of the abovediscussed method, a condensed outline of the specific steps including the mathematical basis is set forth.

Such steps include calculating the probabilities of sticking the drill pipe and avoiding sticking while drilling a well bore with water-based or other drilling fluids. A simplified illustration of use of Ssuch steps to control drilling is then given in a O 10 specific numerical example. The steps are as Sfollows: o Prior to drilling a well bore, measuring a multiplicity, M, of well drilling variables in a multiplicity, N, of wells drilled under comparable drilling conditions in three different groups of a 0" wells, the measured variables being at a given depth in each well bore and the three groups being where a drill string has either become mechanically stuck during drilling or (ii) become stuck by differential pressure between the well bore and a permeable earth formation traversed by said well bore, or (iii) has drilled through depth inter- Svals of wells selected in or (ii) without sticking; forming each of said three groups of N wells in step into a separate matrix, X, in which each of the M measured variables is an element, xji, in a group array (row or column), and the f~ I Ils Uj r

I

Lo2 complementary group array (row or column) is one of the N wells selected as a member of its respective group. As used in the following matrices and equations, j indices indicate any well in a group, i indices indicate any variable in any well; and N is the number of wells in each group. The number of wells in each group need not necessarily be the same number, but the variables must be the same number and type in each group; forming a Standard Mean (average) Vector, Xi, whose components are the mean of each measured variable in a given group. The mean vector Xi is then used to form a corresponding group Standard Variance Vector, Si: Mathematically, the ith component, xji of the Mean Vector Xi is: 0 10 0o a 0 00 0 o o o o o o O o0 o 00o 00 0 20 r S 9 Xi 1/N xji

L~~I

where i M (variables) and j N (wells in group) and the ith component, Si, of said Variance Vector S is: i Si 1/(N-1) 1C (X Xi) 2 j=1 where i is defined, as above and the Standard Deviation Vector, s i of each variable of said group is: Si Si g-_ 1 i i i forming the Pearson Product-Moment Correlation or other similarity matrix R, wherein the covariance between any two variables, say xji and xjk, is defined as Cik.

In this formulation, xji is defined as the value of the ith variable for the jth well and xjk is defined as the value of the kth variable for the jth well.

Thus, Cik is the covariance between the ith and kth (or any two) variables for all wells in a group.

o 4' Algebraically, 04 o N 00

N

Sik (xji Xi) (xjk Xk) 04 Cik 0 *15 N-l j= 1 00 j, k The elements of Within Group Correlation Matrix are rik Cik/sisk.

0040 00 0°o That is, each of the coefficients rik is expressed in 00 a square, symmetrical group matrix R where the subscript i's and k's identify each pair of variables in the group. The entries in the matrix express the dependence, or relationship, of pairs of variables in the group of wells, and thus may be defined as the Pearson Product moment correlation.

then forming a weighted average of the three Within Group Correlation Matrices that defines a pooled matrix RT which is symmetric, square, positive, and semi-definite, L Y;" r ri- r~ solving the total matrix Q for Q I I is a minimum wherein the solution is given as the product of the inverse of the Within Group Correlations Matrix with the Between Group Correlations Matrix where Matrix B equals the Total Correlation Matrix, T, minus the Within Correlation Matrix, W. Algebraically, the relations are: T= B+ W 004 where T =Total Correlation Matrix B Between Group Correlation Matrix :4,W =Within Group Correlation Matrix

W-

1 -the inverse of W so thatQ W- B and solving aieve r IQ Iis. th 0dniymtiadgi h whrbei eleen gii the oignal (atent froot) of step bari by ind corrgponinhegt associinatcefcin asigevetfrom is thendety maix and g is*h pouthe (g futilyn each origina measured vralst rvd "iscores" for the discriminant functions, i'

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115 a w 10 0 0 a 04 0 0o 0 0 09 0 0 0 00 0 0 0 0 0 P 0a a oa 2 *0420 plotting or recording the discriminant scores for each well as a well vector representation of the probability of a well being correctly located in its assigned class; determining the centroids or means of each group of well vectors; and then correlating another well vector to the centroids of the well vector groups by multiplying and summing the products of 4 g and each corresponding variable of such other well which is drilled or to be drilled within said geological province and said depth range to determine the probability of sticking the drill string by its relative distance from the stuck well and not-stuck well centroids.

To illustrate use of the method of the present invention, a simplified artificial example based on a linear approach is calculated as follows: A total of M 4 variables are measured in each well in each of three groups or classes of wells. It will be apparent that in actual practice this example is not statistically valid. However, the same procedure will apply to a larger number of measured variables, say 20 or more in each well, and that a larger number of wells, say 40 to 1200 wells or more, will be included in each matrix.

Selection of the wells for identification in each of the three groups, as noted above, is made on the basis of the values measured for the set of variables at a selected depth in each well. The 14 4 i0

POOD

o o a o a o 0 P 00 0 o tt 0 o p p a O values in the set for stuck drill strings are preferably the last values measured at substantially the depth at which each drill string became stuck, mechanically or differentially. However, values of the variables measured in a stuck well just before the drill string became stuck may also be used. In practice, a single set of variables for each nonstuck well is selected at a randomly chosen depth within a typical range of depths of the differentially and mechanically stuck wells.

Matrices are then assembled for the "i" variables and wells in each of the three groups as follows: PoPP 0, P FIRST OF 3 GROUPS: 3 WELLS AND 4 VARIABLES VARIABLES, i=1 i=2 i=3 i=

WELLS,

j=l X 11 9750 13.7 4750 70 j=2 9500 14.5 5000 60 j=3 10000 13.1 4500 Xji= 50 where the variable in columns i 1 to i 4 are example, i=l is Total Depth (feet) i=2 is Mud Weight (lbs/gal) i=3 is Drill Weight on bottom (pounds) i=4 is Hole Angle to Vertical (degrees) .0 .0 .0 for :4 In the example, the column mean Xi for each variable is determined as:

N

Xi 1/N xji; j=l j a

F

r h r 9 2- 67 where: i 1,2,3 and 4 (VARIABLES) j 1,2 and 3 (WELLS) For example in the first column: Xi 1/3(9750 9500 10000) 9750 Similarly for each of the other columns, the means are calculated as: MEANS OF FIRST GROUP Xi= 1 Xi=2 Xi3 Xi=4 9750.00 13.77 4750.00 60.00 0 0 o: 10 In the above example the variance for each S column is calculated as follows: 00 0 0 0 For the first column of the data, the variance S i is calculated as: S: S 1 (9750-9750)2 (9500-9750)2 (10000- N-1 a 0 9750)2] 62,500 g o (as used in the following tables, 62,500 in computer "floating point" notation is 0I 8 0.625 X 105 and expressed as 0.625E+05 The standard deviation is the square root of the variance, which gives si=250.00 in the present example. Similarly, standard deviations for the other four columns of variables in this example are: si =s1 si s2 si s3 si s4 250.00 0.702 250.00 10.00 t /O 't\ 7j In order between the varia is then formed.

calculated as:

N

Cik 1 N-I j=l to express linear relationships bles, the VARIANCE/COVARIANCE MATRIX The entries in the matrix can be (xji-Xi) (xjk-Xk) co 0 aa~a 000* 00 a *0* 20 0 00 0 a 0,0 a 0 0 -0 0 0 a 60 00 0 0 0 where the index j refers to the wells and indices i,k represent variables.

noted that when the values of i and k or i=k, the covariance is the same as in the group It will be are the same, the variance.

The VARIANCE-COVARIANCE MATRIX for the first group of wells is then: Variables II 1 4 1 0.625E+05 2 -0.175E+03 3 -0.625E+05 4 -0.125E+04 2 175E+03 0. 493E+00 0. 175E+03 0. 300E+01 3 625E+05 0. 175E+03 0. 625E+05 0. 125E+04 4 125E+04 0. 300E+01 0. 125E+04 0.100E+03 When the diagonal entries are divided by the variance of that variable and when the off-diagonal elements are divided by the product of the standard deviations of the variables represented by that row-column intersection, the covariance for the variables at the row one intersection with column two is divided by the standard deviations of variable 1 and variable the resulting matrix is the Within Group Correlation Matrix. That is, the entries for this matrix R 1 are rik Cik/sisk

I

49 The WITHIN GROUP CORRELATION MATRIX, R 1 for the first group of wells in the example is then: Variables 1 2 3 4 0.100E+01 -0.996E+00 -0.100E+01 -0.500E+00 -0.996E+00 0.100E+01 0.997E+00 0.427E+00 -0.100E+01 0.997E+00 0.100E+01 0.500E+00 -0.500E+00 0.427E+00 0.500E+00 0.100E+01 By definition, this matrix is symmetrical about the diagonal, the intersection of row 1 with 0 column 2 is the same as the intersection of row 2 c-£lwith column 1. (That is, the correlation of the ith 0 fto kth variable is the same as the correlation of the kth to ith variable.) The Within Group Correlation Xoo*, Matrix has the special properties that it is square, positive, and semidefinite all its characteristic roots are non-negative).

The other groups of wells have the following statistics: SECOND OF 3 GROUPS: 3 WELLS AND 4 VARIABLES 00 VARIABLES i=1 i=2 i=3 i=4

WELLS

j= 1 5500.00 10.80 3700.00 21.00 j= 2 5000.00 10.40 3500.00 25.00 j= 3 6000.00 11.20 3250.00 30.00 Means of variables in the SECOND group are: Xi=1 Xi= 2 Xi 3 Xi= 3 5500.00 10.80 3483.33 25.33 L Lu Th te ruso elshv h olwn i. The components of the Standard Deviation Vector of this group are: si-l 500.00 si=2 0.400 sij3 225.46 VARIANCE-COVARIANCE MATRIX, for the variables measured for the second group of wells is: Variables--------> 1 2 3 *1 0.250E+06 0.200E+03 -0.625E+05 0.: 2 0.200E+03 0.160E+00 -0.500E+02 0.: 3 -0.625E+05 -0.500E+02 0.508E+05 4 0.125E+04 0.100E+01 -0.102E+04 0.: si= 4 4 4 1.25E+04 100E+01 102E+04 203E+02 at 0 t 0 aat4 fast o a I a. S at a o at a of ff90 o a a 00 4 15 a a a a The WITHIN GROUP CORRELATION second group of wells is: Variables-------> MATRIX, R 2 for the 2 3 4 1 0. 100E+0l 0. 100E+01 554E+00 0. 554E+00 2 0. 100E+01 0.l100E+01 -00 555E+00 0. 554E+00 3 554E+00 555E+00 0. 100E+01 100E+01 4 0. 554E+00 0. 554 E+00 100E+01 0. 100E+01 THIRD OF 3 GROUPS: 3 WELLS AND 4 VARIABLES i=l 1=2i= J= 1 7000.00 12.10 3875.00 J= 2 7250.00 12.00 4000.00 J= 3 8000.00 12.80 3950.00 MEANS OF VARIABLES IN THIRD GROUP: i=l i=2 i=3 7416.66 12.40 3941.66 i=4 35.00 48.00 40.00 i=4 41.00 17 <r STANDARD DEVIATIONS OF THE VARIABLES IN THIS GROUP

ARE:

i-1 520.45 1-2 0.44 1-3 62.86 1-4 6.56 VARIANCE -COVARIANCE for the third group Variables-------> MATRIX for the variab] of wells is: 44 £4

I

4 441£ 4 £4 44 #4 4 I £4 4 44 ~l5 #4 4 4£ 44 4 4 4 I 4 4 4 £4 £4 0 4 44 4 1 2 3 *1 0.271E+06 0.213E+03 0.1E5 2 0.213E+03 0.190E+00 0.625E-01 3 0.115E+05 0.625E-0l 0.395E+04 4 0.375E+03 -0.699E+00 0.400E+03 WITHIN GROUP CORRELATION MATRIX, R 3 Variables 1 0.l00E+0l 0.937E+00 0.350E+00 +2 0.937E+00 0.100E+01 0.228E-02 3 0.350E+00 0.228E-02 0.100E+01 4 0.110E+00 -0.245E+00 0.970E+00 Les measured 4 0. 375E+03 699E+00 0. 400E+03 0. 430E+02 0. 110E+00 245E+00 0.*970E+00 0.100OE+01 The three Within Group matrices R 1

R

2 and R 3 are weighted and summ~ed together to get the so-called Pooled Within Group matrix, for all wells in all the groups: (4 7 79k POOLED MATRIX Variables 1 0.117E+07 2 0.475E+03 3 -0.227E+06 4 0.750E+03 2 0. 47 5E+03 0. 169E+01 0. 250E+03 0. 660E+01 3 22 7E+-6 0. 250E+03 0. 23 5E+06 0. 127E+04 4 0. 750E+03 0. 660E+01 0. 127E+04 0. 327E+03 lat.9 a a Cola a aa 4 a a a Ca 0 ma a 101: a CCI a C C Ca a 44 a.

a a a a 4 a 404 a a a ma a a a a Ca a CI a a a a ma 44 I I I I The overall statistics for the nine wells in all groups combined are: MEAN, XT FOR TOTAL SAMPLE: i=i i=2 i=3 i=4 7555.55 12.29 4058.33 42.11 STANDARD DEVIATION VECTOR sT FOR TOTAL SAMPLE 1882.3816 1.3643 581.2178 16.3359 TOTAL CORRELATION MATRIX (RT) Variables-------> 2 3 4 1 0. 100E+01 0. 943E+00 0. 905E+00 0. 904E+00 2 0.94 3E+00 0. 100E+01 0. 927E+00 0. 902E+00 3 0. 905E+00 0. 927E+00 0. 100E+0l 0. 892E+00 4 0.904E+00 0. 902E+00 0. 892E+00 0. 100E+01

N

-I ~:i

I

73 The between group distances about the grand mean over all wells is calculated for the Between Group Matrix Variables-------> 04.44 4 4 *4.44 4.404 o 4.4.

44.

4.

4.4.

4. 4.

o 0 04.

40 0 04 0.~ 4 40$ 4.

4 0 ~l5 44 4.

4. '0 0 4.

4. '.4 4. 0 44 4.

0 ~4 2 3 4 1 o0.272E+08 0. 189E+05 0. 815E+07 0. 222E+06 2 o0. 189E+05 0. 132E+02 0. 56313+04 0. 154E+03 3 0. 815E+07 0. 563E+04 0. 247E+07 0. 665E+05 4 0. 2 22E+06 0. 154E+03 0. 665E+05 0. 181E+04 The eigenvalues of the total extracted: EIGENVALUE 1 73.461 EIGENVALUE 2 0.221 correlation matrix are 4~ O 4.

L

Checks may be made to establish the precision of the results (all checks should be the same value): (B l/9)t T (Bl1/2) where 'It" is the transpose of B 1 2 which is 1 SUM OF EIGENVALUES =73.5640259 TRACE OF$Bi/2 (Bl/ 1, 73.5639648 ROOTS OF W *)T 73.3556 0.2084 TRACE OF W- *T -73.5640

W

1 IT Trace thereof

I

74 where indicates the inverse of W and the percentage of the variation in the data explained by each eigenvalue should sum to 100t: PERCENTAGE OF TRACE WHICH EACH ROOT IS 99.72 0.28 The discriminant functions are calculated as: VECTORS OF W1 AS COLUMNS Eigenvectors or Discriminanit functions for the above example are: Variables Discriminant Functions------> o1 2 1 0.244E-02 0.139E-03 o4 0.274E-01 -0.809E-02 A simplified explanation of derivation of ::oo eigenvalues and discriminant functions or 1-Jo eigenvectors for a smaller, (unrelated) 2x2 matrix, 2,is given as follows: Take Matrix Qand solve the determinantal equation: Q Ai II~ o where I~ is the identity matrix, is the eigenvalue and -is the eigenvector.

Find~ the eigenvalues and eigenvectors ofQ where Q therefore: IQ -~I or 11 31 A L 9. 0 12~ 2 C,4 N kfj-qK.

or, (1 Ai 2 (2 Ai) 0. Thus, (1 Ai) (2 Ai) 6 -0 hence, 2. Th su a.

or Ai 2 3 4 0-- A 4 1 -1 e associated eigenvectors are found by bstitution: For A. 4 0090 o 04 00 0 04 0 \20 o o 0 o a 2 ad o o o o o o 1 A1 3 xl /i 13 0 or 2 2 X2 -3 3 x =0 2 -2 )(x2 Note coefficient matrix has rank 1 which necessarily implies there exists only one linear independent solution vector.

By inspection i is the vector.

b. For 2 -1.

0 0 0 0 0 00 «a oA 30 4 09< 0 C1 2 3 X 1 l2 2 2) 2) 3 (X 0 00 09« 4 0 0 0 Again there exists only one unique solution vector2 3 For the eigenvalues 4 and the eigenvectors are 1 1and 2 respectively.

1 2 Vj

A

I The eigenvectors define the discriminant functions when properly normalized.

10 0oo a 0 0 00 0 0 0 o0 0oo0 0 0 This example of a simplified computation of a 2x2 matrix example does not have the same properties of the correlation matrix, as one of the eigenvalues in this example is negative. However, such example was selected to illustrate the computation and, because even a simple matrix, as presented in the previous example of 3 groups, is somewhat too complex to be readily solved by a hand calculator. However, the 3x4 matrix of the example may be solved by a program such as those of SAS (Statistical Analysis System, SAS Institute, Raleigh, or BMDP4, Los Angeles, California. In such solution, after the eigenvectors are obtained, conventional statistical analysis can show the relative importance of each variable to the discriminant function as follows: SCALED VECTORS Variables Discriminant Functions 1 2 1 0.264E+01 0.150E+00 2 -0.130E+01 -0.130E+01 3 0.238E+01 0.996E+00 4 0.495E+00 -0.146E+00 From the table it will be noted that Variables 1 and 3 are the most important contributors to both Discriminant Functions (eigenvectors) 1 and 2.

Variable 2 makes some contribution to eigenvector 1 but relatively much more to eigenvector 2. Variable 4 contributes relatively little to either eignevector.

o «a oa os 0 0 0 00 0 o00 0 0 0b 4 t

NT

I

ao 4 1 t 4 A t ,If i t t i t 77 The statistical tests for significance are made using the Wilk's Lambda criterion and the F-ratio.

LAMBDA FOR TEST OF H 2 0.011

F

1 8 degrees of freedom of the numerator

F

2 6 degrees of freedom of the denominator.

FOR TEST OF H 2 with degrees of freedom (F1,F 2 F 6.36 Where H 2 is the null hypothesis that no relationships exist.

In the case of the example, the null hypothesis was rejected at the .01 probability level, there is only 1 chance in 100 that the calculated results could have arisen by chance.

Each well's discriminant "score" is calculated by multiplying the original values of each variable for the well by the corresponding discriminant coefficient pertaining to each variable and then summing the results for the four variables for each well in each group in the above example are: ORIGINAL VALUE OF VARIABLES TIMES EIGENVECTORS FIRST GROUP OF WELLS Wells 1 2 1 35.37 -3.14 2 34.91 -3.38 3 34.80 -2.86 j If L1 I_ L _i i~2 1 I- I:.L ORIGINAL VARIABLES TIMES EIGENVECTORS SECOND GROUP OF WELLS Wels 1 2 4 21.40 -2.60 5 19.70 -2.71 6 20.25 -3.92 ORIGINAL VARIABLES TIMES EIGENVECTOR THIRD GROUP OF WELLS Wells 1 2 7 25.00 -3.44 8 26.68 -3.15 9 27.24 -3.89 0944 This essentially completes the discriminant analysis.

"15 The results for each well in each of the three groups of wells may then be plotted as either a two dimensional plot, as in Figures 8-10, or in a triangular form, as in Figure 12 or in any other convenient numeric or graphic presentation.

The probabilities of correct classification can be calculated by conventional methods of statistical analysis using, for example, Chi-squared values, as follows: MEANS OF GROUPS IN TEST SPACE Variables Groups 9750.00 13.77 4750.00 60.00 S5500.00 10.80 3483.33 25.33 7416.65 12.30 3941.67 41.00 M M

I

ii Ii CENTROIDS OF Discriminant Groups GROUPS IN DISCRIMINANT SPACE, ROW-WISE Function 35.03 -3.13~ Joint 20.45 -3.08 discriminant 26.31 -3.49) means of the 3 Groups.

10 ,.ff t 9 9 9, 94 i 9 4 .4 *91 a 44 9 15 #9 44 9 9 DISPERSION IN

DISCRIMINANT

Wells 4

DISCRIMINANT

DI SCRIMINANT Wells

DISCRIMINANT

Wells SPACE FOR FUN CTIONS 0.090 -0.018 SPACE FOR

FUNCTIONS

0.75 0.18 GROUP 1 -0.018 0.072 GROUP 2 0.18 0.54 949* 4 4 0* 4 p4 4 9, 44 9, 4* 9 e £4 25 SPACE FOR GROUP 3

FUNCTIONS--------->

1.36 -0.16 -0.16 0.14 Using a Chi-squared approximation to a Bayesian statistic the probabilities are found.

J

1 2 3 1 2 3 1 1.334 322.918 76.613 1.000 0.000 0.000 2 1.334 307.021 64.589 1.000 0.000 0.000 3 1.331 295.166 74.808 1.000 0.000 0.000 4 2142.553 1.332 18.637 0.000 1.000 0.000 2722.738 1.333 32.018 0.000 1.000 0.000 6 2652.085 1.333 37.634 0.000 1.000 0.000 S o 10 7 1203.734 31.762 1.335 0.000 0.000 1.000 0 0 ,0 8 820.693 56.615 1.337 0.000 0.000 1.000 o 9 758.760 73.265 1.333 0.000 0.000 1.000 A* The results of such analysis of these groups, plotted in accordance with their eigenvectors, is shown in Fig. 12 wherein the nine wells are each plotted by their eigenvector coordinates. As indicated, the separation of the 0" three groups is perfect, since the problem for illustrative purposes is fabricated, but not 0° o20 necessarily unrealistic.

In an unknown, or poorly defined, geologic province, it may be desirable to initially determine whether groups of wells can be identified without dependable data on the causes of sticking the drill string. Multivariate analysis can also be used to determine the probability of so identifying such groups without specific information as to the cause of sticking or whether any of the total group of wells in fact stuck. In statistical terms, such evaluation is known as Q- mode analysis (as

©II

above.) In Q- mode, as compared to R- mode, where say 20 variables are evaluated, the variance covariance or Pearson-product correlation matrix is formed by summing the cross products of each well summed over the number of variables M. That is, Cjq, where j, q refers to q to N wells, is determined, rather than Cik, which refers to 1 to M variables, as above.

1 M Thus: Cj q M-1 r (Xij Xj) (X iq Xq) ~i=1 4 4 The covariance is determined for all wells across all Svariables and all other statistics may be calculated :15 according to the above for all matrix indices.

o Accordingly, if the number of wells is, say 1,200, the matrix would be 1,200 x 1,200 rather than x 20. The eigenvalue and eigenvector vector solution of such a matrix similarly generates oO° :20 variable vectors which will cluster about centroids 0 representative of variables which can then be inferred to represent different classes of stuck wells. If these classes exist, an R- mode analysis as taught herein can be guided as to probable causes 4C25 of drill string sticking.

6 t4 Best Mode From the foregoing example, it will be seen that for twenty or more measured variables at a given depth in each well and for 40 to 100 wells or more in each of the three classes, the calculations and graphic representations of each well are best performed by computer.

I 1 w ''l

'I

The calculations of each dimensionless matrix coefficient can be calculated with an (Hewlett Packard) hand-held computer or similar device for a few variables and wells. For large data sets, say 20 variables and 80 wells in each of three matrices, a program known as SAS, available from SAS Institute, Raleigh, NC, will perform statistical analysis as above described either on a large main frame computer or on a personal computer Such program is capable of performing all steps of 0ooo multivariate analysis, including matrix computation o O of principal components, factor, regression and ao discriminant analysis. Additionally, a text book by W.W. Cooley and P.R. Lohnes, "Multivariate Procedures for the Behavioral Sciences", John Wiley and Sons, New York,NY, 1962 presents FORTRAN code for statistical analysis. The graphic presentation of the three classes of wells and location of each well vector may be plotted using a program known as Lotus 1-2-3 available commercially from Lotus Development, oa Cambridge, MA. It can be used together with a PO program known as dBASE III, available from Ashton Tate, Culver City, CA to manage the data file.

Linear programs for calculating each individual well vector to plot and control a drilling well can be performed by a program known as OMNI, available from Haverly Systems, Inc., Danville, NJ. or by a Program MPSX, available from IBM Corp., White Plains, NY may also be used.

In a field application of the method of the present invention, the following commonly measured 5 ses ay2 arals n 0 el i ah ftre 83 well variables, or parameters, were used to establish the matrices.

measured well depth, true vertical well depth, open (uncdsed) hole length, rotary drill string drive torque rotary drill string drag, survey hole angle (from vertical), drilling fluid (mud) weight, drilling fluid plastic viscosity, drilling fluid yield point, 00°: (10) drilling fluid 10 second gel strength, °1 (11) drilling fluid 10 minute gel strength, 0 99 (12) API standard drilling fluid water loss 15 (filtrate), (13) drilling fluid pH, (14) drilling fluid chlorides content, bore hole size (diameter), (16) drilling fluid solids percent, 0 20 (17) drilling fluid water percent 0 t 0 (18) drilling fluid flow (pumping) rate, (19) drill collar outside diameter, and 00 vertical length of drill collar section of drill pipe.

ft i All variables are measured in accordance with API standards.

Various measures of gas content of drilling fluid, and gas type, have also been used with success.

NT

'i .1

I

In development of the well vector coefficients using multivariate analysis of the above-listed multiplicity of measured variables in a multiplicity of wells drilled in the subject geological province, the relative importance of the individual coefficients for each variable to redirect the probability vector of a drilling well between the groups can be calculated by stepwise discriminant analysis. For example, the important variables are as listed in Table 1.

TABLE 1 4 f 4 0, 4 IMPORTANCE OF VARIABLES IN ORDER OF SIGNIFICANCE AT 90% CONFIDENCE LEVEL OVERALL STUCK vs. NOT STUCK MECH vs. DIFF 15 SURVEY ANGLE SURVEY ANGLE HOLE SIZE HOLE SIZE TRUE VERTICAL DEPTH SURVEY ANGLE TRUE VERTICAL HOLE SIZE DRAG DEPTH OPEN HOLE MUD WEIGHT DRAG 10 MIN GEL WATER LOSS OPEN HOLE 10 SEC GEL CHLORIDES Ai FLOW RATE PERCENT WATER 10 SEC GEL SMUD WEIGHT PERCENT SOLIDS CHLORIDES PLASTIC VISCOSITY WATER LOSS TORQUE DRILL COLLAR O.D.* t (*Significant only at 89% confidence level.) SIn the list of variables the "Overall" column refers to movement of a well vector from one location to another on the plot or map. The "Stuck vs Not Stuck" column indicates the relative importance of modifying a measured variable to move from a Stuck well (Differential or Mechanical) area toward the Not-Stuck centroid. The "Mech. vs. Diff." column indicates the relative importance of each measured variable as between the position of a well vector in ?S11 the Mechanically stuck class rather than Differentially stuck class.

At a confidence level of 85% or less, it was found in one particular study that the drilling fluid variables, pH and yield point, and the variables, measured depth of the well bore and drill collar length were not independently significant, partly due to high correlations with other variables recorded at the 90% significance level, they were redundant.

a J oa- Based on the method of the present invention, a study was made of 35 wells not used in ^r the original data set. The purpose of the study was o o15 to determine the probability of correctly predicting sticking of drill pipe in a well bore drilled in the given geological province. A total of 49 predictions were made and in 41 cases the final outcome was Doo. correctly predicted as to its being properly classified into each of the three groups. Table 2 sets forth the results of such predictions at the o indicated depths in the 35 wells. Overall, such predictions were 82%-84% correct, depending upon what weight one gave to two wells which had multiple cases o s25 of sticking over large depth ranges.

o, I. 7 TABLE 2 SUMMARY OF FIELD RESULTS USING STUCK DRILL PIPE PROGRAM MAXIMUM ACTUAL PREDICTED WELL DEPTH CONDITION CONDITION MISCLASSIFIED 1 6570 NSTK NSTK 2 7556 NSTK NSTK 3 10986 NSTK DIFF X 4 7708 NSTK NSTK 5 5547 NSTK NSTK 6 4875 NSTK NSTK 7 9608 NSTK NSTK 8 12019 NSTK NSTK 01. 0o 9 :6536 NSTK NSTK 10 10998 NSTK NSTK o 11 5465 NSTK NSTK 12 15139 NSTK NSTK 13 9893 DIFF DIFF 11130 DIFF DIFF ST/#1 10701 DIFF DIFF 14 7238 DIFF DIFF ST/#1 10388 NSTK MECH X 9674 NSTK NSTK 16 7993 NSTK NSTK 17 12627 DIFF DIFF 12999 DIFF DIFF ST/#1 13823 DIFF DIFF 14036 DIFF DIFF 18 7673 DIFF NSTK X 19 14089 DIFF DIFF 15073 DIFF DIFF 10096 NSTK NSTK 21 8674 NSTK DIFF X 22 13409 NSTK DIFF X 23 5316 MECH MECH 6360 MECH MECH 8373 MECH MECH 12055 MECH MECH 12677 MECH MECH 24 17276 NSTK NSTK 9606 MECH MECH 26 9846 NSTK DIFF X 27 10125 NSTK NSTK 28 21045 NSTK NSTK 29 12560 NSTK NSTK 7520 DIFF DIFF 31 7510 DIFF DIF? 4 32 11849 MECH MECH 1 i il 13522 NSTK DIFF X 33 5421 NSTK NSTK 34 16506 DIFF DIFF 14691 NSTK DIFF X indicates that the well was sidetracked and redrilled from a level above the previous depth at which the drill pipe stuck to the next depth.) While the above description indicates that it is preferable to determine the probability of a .,10 drill string sticking using three or more groups of o0 wells, the method is clearly applicable to separate into such wells only two groups, such as wells wherein the drill string stuck in all of one group and did not stick in the other group. Such two ''15 groups may comprise all stuck wells, which are then used to identify those wells freed and those not freed. Alternatively, the analysis is applicable to distinguish only mechanical sticking from differential sticking. Corrective action for the measured variables, as each simultaneously contributes to the well vector at a particular depth, as related to the entire set of wells, is related to the individual coefficients for each variable.

It will be understood that the measured variables which make the greatest contribution to direct the well vector toward the non-stuck centroid and can most easily be modified in drilling the well may be evaluated before such variables are in fact changed.

Based on plots as shown in Figs. 9 and say on a daily basis, optimal values; of the variables to move the well vector into the non-stuck region may 'A!i 9 dS i 1 y\ s I be calculated using a linear program (LP) or other optimization methods. The LP, using reasonably constrained values for the given well, mud type, and hole conditions, calculates the extent of changes in the variables of the discriminant equation required to achieve the goal of collectively changing the variables to reach or approach the centroid of the not stuck wells.

Because an LP does not necessarily change 10 the variable in a manner consistent with common sense, it may be necessary to constrain the variables within the LP to maintain reasonable engineering values.

Two types of constraints may be used: functional constraints relating some of the variables, and boundary constraints to keep the variables within reasonable limits. For example, the functional constraints may be: An equation relating percent solids to mud weight in the drilling fluid.

Ten second gel values for drilling fluid cannot exceed ten minute gel values.

The sum of the drilling fluid content, solids percent and fluids percent, cannot exceed 100% (mass balance).

Boundary conditions or constraints are then set for the minimum and maximum value of each of the variables. Target location coordinates in the i non-stuck region are then also assigned and equated to the discriminant functions. The matrix is then solved to approach the target discriminant values as closely as possible without violating any of the constraints.

The LP optimization system may use, for example, Ashton Tate's dBase III for the input and output routines and Fortran for the LP matrix .ooo^ solution. Table 3 illustrates an example of the LP 0oo 10 input for three groups (two discriminant functions) t t and twenty variables. The Current Values (Column 2) o °o of the twenty variables (Column 1) are input along o o with target coordinates. Lower and Upper Limits for 00 the variables (Columns 3 4 respectively) are then assigned and an allowable range Down or Up (Columns and 6) of each variable if, indeed any change is possible, are set. As shown, in fact eight of the 000 1o twenty variables cannot be changed on any given day.

08° (It is also to be noted that limits can also be assigned to the target to allow some leniency in the 0oB 1 solution of the matrix.) From this input an LP matrix is created and 0o o0 S• solved by the simplex method, or any other comparable method. Occasionally a solution is not possible within the given boundary conditions and, thus, the constraints on the targets must be relaxed and the LP rerun. An example LP output is shown in Table 4.

The proposed LP Values (Column 1) are shown along with the actual Current Values (Column The proposed differences (Column 3) and the new values of the changed variable within the given limits are then shown in Columns 4 and 5. The X and Y target values b RA41 the values of the two eigenvector functions) are then plotted relative to the current X and Y coordinates of the drilling well.

Accordingly, the user is presented current eigenvector function values at any point in drilling a well bore. The user may select to change some, but perhaps not all, variables the well bore shallower). Upper and lower limits are then set on the variables that, practically, can be changed.

This then permits plotting the current location on 0 the probability plot and shows the "safe" position to o mo°: achieve the highest probability of not sticking the drill pipe. In the results shown in Table 4, it will be noted that among significant changes that could be made the operator can increase the mud weight Ibs/ft 3 decrease the drilling fluid water loss 2.3% I and decrease the chlorides content of the drilling 0 I fluid 2000 ppm. Other modifications such as drill m°mm collar diameter (increase) and length (decrease) are as indicated.

The well vector of the example well at the indicated depth (450 feet) was, in fact, within the O mechanically stuck well group, at a probability of about 90%. Corrections were calculated by the LP to move the well vector into the not-stuck well group, within a probability of about 99%, as may be seen by plotting the starting coordinates and the LP coordinates on the probability map of Fig. 9.

i_ 0a 4 0 0 o 0 4 4 -r L 4 0 0 j t LP DATA INPUT TABLE 3 WELL: EXAMPLE DATE: 06/12/85

VARIABLE

Measured Depth, feet True Vertical Depth,feet Casing Depth, feet Openhole Length, feet Torque Drag Survey Angle, degrees Mud Weight, lb/gal Plastic Visc.

Yield Point Sec. Gel Min. Gel Water Loss

CURRENT

VALUE

11000 10000 4500 6500 15000 50000 25.00 12.0 12 5 1 4 3.5 WELL VARIABLE

LOWER

LIMIT

11000 10000 4500 6500 15000 50000 25.00 11.5 8 3 0 2 1.0

REPORT

UPPER

LIMIT

11000 10000 4500 6500 15000 50000 25.00 12.5 16 11 4 10 4.5 STUCK DRILL PIPE OPTIMIZATION SYSTSH DOWN V 0 0 0 0 0 0 0 0 0 0 0 0 0.00 0.00 0.5 4 4 2 6 1 3 2 6 2.5 i 't, L i L c. I -i Y -1 apP p 0 *00 0 9 00 0 0 0-0 8000 0P~ 00 p 0 6* 0 00 04000 0 0 00 p *0 0 000w I 00 0 6 0 0 0 0 Opt 06 000 00 p p 00 000 coo LP DATA INPUT TABLE 3 (Continued)- WELL: EXAMPLE DATE: 06/12/85

CURRENT

VARIABLE VALUE pH 11.0 Chlorides, ppm 4000 Solids, percent 20 Water, percent 80 Hole Size, inches 12. 250 Mud Flow Rate, ft 3 /min 8.000 Drill Collar OD, inches 8.000 Drill Collar length, ft. 350 X Target -16.00 Y Target -9.00 X Coor -12.86 Y Coor -7.97 STUCK DRILL PIPE OPTIMIZATION SYSTEM WELL VARIABLE LO0WER

LIMIT

9.5 2000 12 75 12.250 7.500 7.*500 150 -16.50 -9.50

REPORT

UPPER

LIMIT

12.5 14000 18 85 12.250 9 .500 9.500 650 -15.50 -8.50

DOWN

1.5 2000 8 5 0.000 0.500 0.*500 200 0.50 0.50

UP

10000 0.000 1.500 1. 500 300 0.50 0.50 I WELL: EXAMPLE DATE: 06/12/85 SOLUTION IS OPTIMAL***

LP

VARIABLE VA LU-E Measured Depth feet 11000 True Vertical Depth,feet 10000 Casing Depth, feet 4500 Openhole Length, feet 6500 Torque 15000 Drag 50000 Survey Angle, degrees 25.00 Mud Weight, lb/gal 12.5 Plastic Visc. 16 Yield Point 5 Sec. Gel 1 Min. Gel 7 o 0 0 0~ 0 o C 00 o 0 0 0 -00 04 0 o 0 00 5~2~ TABLE 4 ooe 0 0 0 0 000000000 0 00 00 900 0 00 0 00 0 0 00 00 000 0 00 000 00 0 STUCK DRILL PIPE OPTIMIZATION SYSTEM LP OPTIMIZATION REPORT

CURRENT

VALUE

11000 10000 4500 6500 15000 50000 25.00 12.0 12 5 1 4

DIFFERENCE

0 0 0 0 0 0 0.00 0.5 4 6 3 3

LOWER

LIMIT

11000 10000 4500 6500 15000 50000 25.00 11.5 8 3 0 2

UPPER

LIMIT

11000 10000 4500 6500 15000 50000 25.00 12.5 16 11 4

I-

1 p WELL: EXAMPLE DATE: 06/12/85 SOLUTION IS OPTIMAL***

LP

VARIABLE VALUE Water Loss 1.2 PH 12.5 Chlorides, ppm 2000 Solids percent 16 Water percent 84 Hole Size inches 12.250 Mud Flow Rate ft 3 /min 525 Drill Collar OD inches 9.500 Drill Collar length ft. 150 X Target -15.75 Y Target -9.50 X Coor Y Coor TABL 4 Contnue LP4 OPTIMIZATON REPOR STUCK DRILL PIPE OPTIMIZATION SYSTEM

CURRENT

VALUE

3.5 12 .5 4000 20 80 12.250 450 8. 000 350 -16.00 -9.00 -12.86 -7.97

DIFFERENCE

-2.*3 1.5 -2000 -4 4 0.000 75 1.500 -200 0.25 -0.50

LOWER

LIMIT

1.0 9.5 2000 12 75 12.250 375 7.500 150 -16.50 9.50

UPPER

LIMIT

12.5 14000 18 12.250 525 9.500 650 -15.50 8.50

I

A

1 ;i cll 10 0000 0 0 V 04 0 00 0 0 0 4 00 0U~ 0 04 0r 4t 00 0..

0L 4 o 0t Various modifications and changes in the method of the present invention will become apparent to those skilled in the arts of statistical analysis and well drilling from the foregoing specification.

Such modifications may include planning an overall drilling program before the well is drilled, or even "spudded". In so using the method of the invention, the multiplicity of variables can be controlled on a periodic basis, say daily, to maintain the well vector within allowable limits. In this way, throughout drilling, the vector is kept as close as practical to the non-stuck centroid of wells drilled in the same or a similar geological province. Thus, the probability of not sticking the drill pipe in a directional well may be substantially improved.

Similarly, certain values of the multiplicity of measurable parameters used in multivariate discriminant analysis may be combined as dimensionless ratios either before, or in formation of the elements of the various matrices. In this connection, it is to be noted that each of the elements of the Pearson-product matrices are values of inherently mixed measurement units or dimensions and accordingly are now non-dimensional.

Additionally, in evaluating particular parameters for modification of a well vector to avoid sticking the drill string, other programs than linear programming may be employed, such as quadratic, dynamic, or non linear. Expert Systems may also be accommodated by the present method to improve further, but indirectly, the values of variables that require modification. For example, changes in the OOaO 0 00a 00 0 0 00 n 00 00 0 t I MV 4 l 1 8 04 64 6 '3 0404 '3 44 44 0 44 46) 4 0 @4 drilling fluid may be made by use of additives or change of weighting materials based on data stored embodying suites of skilled choices of an expert driller or geologist based on prior successes or failures in avoiding drill pipe sticking.

Additionally, it will be understood that geophysical and geological data may be used in a geologic province to further enhance the method of the present invention. For example, clay or shale data, including physical and chemical properties, may be used over known depth intervals, particularly where shale is encountered over significant depth intervals. Additionally, earth formation properties measured by electric, induction or nuclear well logging tools, including those measured while drilling, may be used as measured variables in the matrices and their contribution to the discriminant functions used to calculate well vectors representative of drilling conditions in the well bore.

The present invention is particularly well suited to utilize well logging data, geological data and drilling conditions detected using down hole measure-while-drilling (MWD) tool assemblies now frequently incorporated into a drill string. Such data is particularly desirable in wildcat or step-out wells where such data is not known in detail, or at all. Prompt evaluation of all measured variables, related and unrelated, in a currently drilling well is accordingly available for analysis by the present invention to determine the probability of sticking the drill string and the risk of losing both the well 4 It 1I

I

wi- 97 bore and the costly drill string section containing the detecting and measuring systems. Because these pipe sections are normally placed in the drill string, as near the bit as possible, loss of the lower end of the drill string in a stuck well represents a substantial investment and potential loss if this critical part of the drill string must be abandoned. Accordingly, this invention represents a substantial economic advantage in avoiding drill string sticking by improving control of well drilling 0 parameters or variables on a continuing basis.

0 0 ooee 0 All such modifications and changes coming within the spirit and scope of the following claims 0 "o15 are intended to be included therein.

00 0 0 O 00 0 0 0a0 o o #o 0 4 C I zI J i i

Claims

1. Apparatus for statistically determining the probability that a drill string in a well bore will stick while drilling in a geological province wherein the drill strings have stuck in one group (N 1 of multiplicity of well bores and the drill strings have not stuck in well bores of another group (N 2 of said multiplicity of well bores (NT) which comprises; means for storing measurable values of a multiplicity of drilling variables at a selectable depth in each well bore of the two groups (N 1 N 2 of well bores wherein the drill string stuck and wherein the drill string did not stick, multivariate analysis means for receiving the stored values for each well bore and for determining a non-dimensional coefficient for each of said drilling variables representing its relative contribution to a single well vector at a selected depth for each well bore of said multiplicity of well bores, *004 vector generating means for summing the products of each coefficient multiplied by its corresponding measurable value for a single well bore to form a single vector for a selected well bore, centroid generating means for generating at least a group centroid of ."the vectors of said single well bores stored in each of said two groups, and means for selectively recording at least said group centroids and another single well bore vector for the measurable variables at a selected t depth in another well bore to indicate the proximity of said other well bore vector to said group centroids as a measure of the probability of sticking the drill string in said other single well bore at said selected depth.

2. Apparatus in accordance with claim I including means for selectively modifying a plurality of said multiplicity of values representing variables in said other well bore at a selected depth for generating at least one additional single well vector from the sum of the products of each of said values and its corresponding coefficient and means for visually displaying the distance of said single well vectors to said centroids as a measure of the changes in probability that the well bore values have been modified in an amount and to an extent to decrease the 9 .2 r~r- 910911Jpcsdat.125929Chv,1 i _II I i i i i~L_ IUI -99- A 4 44 44 4O 4 5444 4 4 4 4~ probability of sticking the drill string in said other well bore.

3. Apparatus for evaluating the effect of modifying a plurality of measurable variables used in drilling a well bore to decrease the probability of sticking the drill string within an earth formation traversed by said well bore which comprises: automatic data processing means, including first means for storing and forming at least one multivariate analysis matrix of a multiplicity of measurable well drilling variables in each of a multiplicity of well bores including at least two classes of well bores drilled in a similar geologic province, said two classes including one multiplicity of well bores wherein the drill pipe stuck and another multiplicity of well bores wherein the drill pipe did not stick, said multivariate matrix for each class including for each well a substantially identical multiplicity of drilling condition variables measured at a selected depth in each well over a given depth interval, means for computing a distance function coefficient for each variable of at least the total discriminant matrix for all wells in said at least two classes of well bores, second means for storing the values of said coefficients, means for selectively calculating a single valued vector representing each well bore in said total matrix and any other well bore drilled in said similar geological province, each of said well vectors being the sum of each of its measured variables in said first storage means scaled by its respective distance function coefficient in said second storage means, means for recording at least the centroids of said well vectors for at least said two classes of wells and at least two well vectors of said other well at said selected depth, one of said at least two well vectors having a plurality of said measurable variables modified in an amount and to an extent to move said one well vector relative to the other well vector and means for indicating the change of distance of at least one of said two well vectors relative to said centroids.

4. A system in accordance with claim I wherein said multivariate analysis LL9 910910,cmsdat.125,39290Chev, I :i ii I -100- means includes means for generating centroids of a multiplicity of well vectors for at least two groups of wells within said class of wells in which the drill string stuck and said indicating means includes means for displaying the distance of said other well vector to the centroids of said two groups of stuck wells.

A system in accordance with claim 4 wherein said means for generating said stuck well groups of well vectors includes means for generating at least one centroid for wells in which the drill string stuck mechanically and another centroid in which the drill string stuck differentially.

6. A system in accordance with claim 4 wherein said means for generating said two groups of well vectors additionally includes means for generating one centroid for a multiplicity of wells in which the drill string stuck and could not be freed and another centroid for a multiplicity of wells in which the drill string was stuck and then was freed.

7. Apparatus for decreasing the probability of sticking the drill pipe during drilling of a well in a given geological province which comprises: means for recording each of a multiplicity of mechanical and fluid a'e; parameters used in drilling a multiplicity of well bores in the geological S .province, means for recording a set of said multiplicity of parameters for each of S.l a multiplicity of wells, one group of said multiplicity of wells having been drilled without sticking the drill pipe and another group of said multiplicity of wells having stuck the drill pipe, multivariate analysis means for determining a coefficient for each of said multiplicity of parameters in a total group of said multiplicity of wells, each coefficient being representative of the relative importance of one of said parameters to form a vector for each well, means for identifying at least one centroid for each of said groups of wells from said vectors, and means for determining for a single set of a corresponding multiplicity of j said parameters at a given depth in a single well bore the distance of its corresponding well vector to at least one of said centroids said distance 4 Q910910,cmnsdat.125,39290Chev,3 ii -101- indicating the probability that said single well bore is a member of at least one of said groups and the probability that a drill string will stick in said single well bore while drilling using said single set of parameters at said depth.

8. Apparatus in accordance with claim 7 wherein said means for identifying the centrioids of said groups additionally includes means for identifying the centroids of at least two subgroups of said group of wells in which the drill string stuck.

9. Apparatus in accordance with claim 7 wherein said multivariate analysis means includes means for generating a multiplicity of well vectors for at least two groups of wells in which the drill string stuck and said recording means includes means for indicating the distances of said well vectors to said two group centroids.

Apparatus in accordance with claim 9 wherein said means for generating said groups of well vectors includes means for generating at least one centroid for wells in which the drill string stuck mechanically and another centroid in which the drill string stuck differentially.

11. Apparatus in accordance with claim 9 wherein said means for generating said groups of well vectors includes means for generating one centroid for wells in which the drill string stuck and could not be freed and another centroid for wells in which the drill string stuck and was then freed.

12. A method of determining the probability that a stuck drill pipe in a well bore can be freed which comprises: recording tile same multiplicity of well drilling parameters M in each of a multiplicity of well bores N wherein a drill string has stuck, said multiplicity of well bores including one group of wells wherein the drill string was freed and another group of wells wherein the drill string was not freed, by multivariate statistical analysis, computing a coefficient for each of said multiplicity of well drilling parameters in said groups of wells, computing the centroid of well vectors for each of said groups of wells, each of said well vectors being determined in accordance with each of said multiplicity of parameters measured in a well scaled by its corresponding coefficient, said centroids differentiating said well vectors of said groups of 910910,nsdat.125,39290Chev,4 r I r I t I~ II I c r. Is o 4 4 44$C -102- wells wherein the drill string was not freed from wells wherein said drill string was freed, and computing the well vector for another well bore in which the drill string has stuck from said coefficients and the values of the measured parameters M o determining the distance of said well vector for said other well to said centroids to indicate the probability that the drill string in said other well bore is substantially closer to the centroid of wells wherein the drill string was freed than the centroid of wells wherein the drill string was not freed.

13. The method of claim 12 wherein said multiplicity of measured parameters includes for each well a value representing time elapsed from the time that the drill string stuck to the time corrective fluid was placed, or spotted, in each of said two groups of wells and said other well.

14. Automatic apparatus for simulating drilling of a well bore with reduced probability of sticking the drill string based upon a data base of a multiplicity of drilling parameters M in a multiplicity of wells N previously drilled over a depth wherein the drill string has become stuck in well bores in a given geological province, said data base including means for storing a multiplicity of measured values of a drilling fluid system and drill string parameters for each of said multiplicity of drilled well bores, said multiplicity of drilled wells including at least one group of well bores in which the drill string stuck and another group of well bores in which the drill string did not sLick over a range of depths similar to the depths of said one group, said apparatus including means for forming a total matrix from said multiplicity of measured variables for each of said multiplicity of wells, means for computing the roots of said total matrix, means for computing from said roots the coefficient for each of said multiplicity of measured parameters in said total matrix, means for computing well vectors for selected wells of said two groups of wells from said coefficients and the corresponding measured parameter in each of said selected wells, and the distance between each well vector and the centroids of the well vectors of at least said two groups of wells, 1 i N 4 U. T~~ I 910910,nsdat.125,3929OhCv,5 1 -103- o~oo o 0 O 0 0 00 000 O c o0 oo «oo i 00 0 0 0 o 00 0 o 0000 0 e of oaa A 0 0 o 0101 96 00 a 0 0 00 0 «ac 0 0 0 o a o I means for repetitively computing another well vector from said coefficients and a preselected sequence of values, for a plurality of said parameters to simulate drilling of another well, in said geological province and means for selectively indicating changes in distance of each of said other well vectors by changes in said values relative to said centroids as a measure of the effect that said preselected sequence of values would improve the probability of not sticking the drill pipe in a well bore at a selected depth.

Apparatus in accordance with claim 14 wherein said other well vectors are simulated by means for preselecting ranges of values of a selectable plurality of said parameters to simulate modification of each of said plurality of parameters during drilling of a well at any selectable depth using values within said range.

16. Apparatus for controlling the drilling of a well bore to reduce the probability of sticking the drill pipe while drilling said well bore with a rotary bit and drilling fluid, comprising: means for storing a multiplicity, M, of measurable well drilling parameters measured in a multiplicity, N, of well bores drilled under comparable drilling condition: in at least two different groups of well bores, said stored parameters having oeen contemporaneously measured at a selected depth in each well bore and said at least two groups of well bores iiclude one group in which a drill string stuck and another group in which the well bore was drilled without sticking the drill string through similar depth intervals of wells in which the drill strings stuck, means for entering said stored parameters into a separate matrix formed for each of said two groups of N wells, with each of said measured parameters being an element xji in a common group array (row or column), and such group matrix for each of said N wells selected as a member of its respective group; where, in each of said following matrices and equations, j indexes any well in any group; i indexes any variable in any of said wells; and N is the number of wells in each group and M is the number and type of parameter or variable in each group; means for computing for each group a Mean (average) Vector, 910910,nsdat.125,39290Chev,6 w Oi 0 41 0 4 0 0 $4 -104- X, of each parameter and forming therefrom a corresponding group Standard Variance Vector, Si: wherein said Mean Vector Xi is N S 1N j-1 Xji where j N (wells) and i M (variables) 0000 0 0 0 0 0 0 o o iDo 0 0 0 0 0 0 00 0 0 0 0 0 00 4 Ut 01 0 I 0 a of 0a iorr 1 6 and said Variance Vector Si is: N Si L J-1 and the Standard Deviation Vector element of said group is: (xji Ri)2 s i of each means for converting each of said vectors si into a Pearson product-moment correlation matrix R=rik wherein the value between any two parameters, say xji and Xjk is defined as the group variance-covariance matrix, N Cik 1 (xj Ri) (xjk k) N-1 J=1 1, k 910910,cmsdat.12539290Chev,7 '-4 a -105- and the group correlation matrix elements, rik Cik/SiSk express the linear dependence or relationship, of said pair of x's, (say i 1, k 2) and so that each of said coefficients rik is expressed in a square, symmetrical group matrix R where the i's and k's refer to each parameter in the total group population, additional means for converting said vectors si into a within group correlation matrix, similarly defined so that the j's refer only to the members of its group and the Xi's and si's refer only to the mean and standard deviations, respectively, of the parameters of that group, means for similarly forming a weighted average of the two within group correlation matrices to form a total matrix Rr which is symmetric, square, positive and semi-definite, means for solving for the roots of a matrix Q, wherein Q is the product of the inverse of the within group correlations matrix and the between group correlation matrix (total group correlation matrix minus within correlation matrix) such that the relations are: T B W where T total correlation matrix .t B between group correlation matrix °4 W within group correlation matrix and Q W 1 B and the solution is: ao oo A wherein the roots are the eigenvalues 7g and associated eigenvectors, g, I is the identity matrix, and g is the number of roots which exist, means for multiplying each original measured parameter of a well in the original matrix in said storage means by its corresponding eigenvector coefficient g, and means. for separately summing the products for each array of measured variables for each well, o a recording means for storing the sums of said products as the well recording means for storing the sums of said 91091produ0ct.125,3929s as the well,8 t eniT m910910cmsdat.2539290Chev,8 me n for~- multilyin eac orgia me sue paa ee of a U 0040 00~4 a o a 0 0 oO 00 0 0 0 04*000 0040 *0 *000 0 4£ -106- vector for each well in each group as a representation of the probability that each of said wells in each group is correctly assigned to its proper group arnd to identify the coordinates of the centroid of each of said two groups of wells; means for similarly multiplying each eigenvector coefficient with each measured parameter in another well drilled within the geological province and within said depth range of said multiplicity of wells and means for summing the resultant products as a well vector of said other well bore; means for recording the coordinates of said other well vector; and means for comparison of said coordinates of said other well vector with said coordinates of said centroids of said two groups to indicate the probability of sticking the drill pipe by using the drilling parameters measured in said other well bore for continued drilling of said other well bore.

17. Apparatus in accordance with claim 16 wherein said means through include means for separately generating variance-covariance matrices and group correlation matrices for at least two groups of wells wherein the drill string stuck and wherein means includes means for forming a weighted average of the two within-group correlation matrices of said two groups to generate two separate centroids of stuck wells, and said recording means including means for comparing the recorded coordinate values of each well vector relative to the coordinates of said two centroids and the non-stuck well centroid.

18. Apparatus in accordance with claim 16 with the addition of means for storing limits for the values of each stored parameter of said multiplicity of measured parameters in said other well, and wherein means includes means for repetitively calculating the contribution of each of said parameters over a selected suite of values within said limits multiplied by its corresponding eigenvector coefficient and said comparison means includes means for indicating the effect of modifying at least one of said variables to alter the distance of said other well vector from the stuck group centroid toward the not stuck centroid.

19. Apparatus in accordance with claim 16, for additionally determining the A a 4 0 IIT 11 910910,cmsdat.125,39290Chcv,9 -107- probability that a drill string which may become stuck can be freed in a well bore where said drill string has become stuck which includes: means for additionally computing the eigenvector coefficients of another total matrix for at least two groups of wells formed as subgroups of wells in which the drill string stuck, one of said subgroups including wells in which the drill string was freed and the other subgroup including wells in which the drill string was not freed, means for computing the coordinates of at least the centroids of said stuck wells in which the drill strings were not freed, and were freed and means for computing the well vector coordinates for another well in which the drill string has stuck from the same multiplicity of measured parameters to form said other total matrix of well bores in which the drill string stuck, and means for comparing the coordinates of the well vector for said other stuck well with the coordinates of said centroids of said freed and not freed 0 wells to indicate the probability of freeing the drill pipe in said other stuck well. .0 0

20. Apparatus for predicting the effectiveness of changing a well drilling o 0* variable to decrease the probability that a drill pipe will stick in a well bore 0 being drilled with a rotary drill string comprising, data file means for storing each individually measured value of a 4 4 multiplicity of controllable well drilling variables or parameters at a selected I: depth in each of a multiplicity of well bores drilled in a geological province wherein a drill string has stuck in a significant group of said multiplicity of well bores and another significant group of said well bores was drilled to a selected depth without the drill string sticking, means for converting each of said stored values in said data file means into an element xj of an array X forming a total group matrix, means for converting each of said stored values in said data file means in said group of well bores in which the drill string stuck into a similar element of an array forming a stuck well group matrix, "i 9 means for converting each of said stored values in said data file means 910910,nisdat.125,3929OChev,10 -108- o C C C C C' o C C C *0CC Cf C C. 'CC. C 0*0* C,. C. 4* C C in said group of well bores in which the drill string did not stick into a similar element of an array forming a non-stuck well group matrix, means for calculating at least two correlation matrices from said total group matrix and at least one of said matrices for said stuck well group and said not-stuck well group, said two correlation matrices including a total group correlation matrix, T, a between group correlation matrix, B and a within group correlation matrix, W, means for solving the eigenvalues and associated eigenvector coefficients for each measured variable of a matrix Q, wherein Q is derived from T B W to give Q W1 B wherein XV' is the inverse of matrix W, means for selectively calculating a single well vector from the sums of the products of the recorded value of each variable in a selected well bore multiplied by its eigenvector coefficient, said sum defining said single well vector for said selected well bore of said total group matrix, and any other well bore drilled in said geological province, means for recording each well vector in said stuck well group matrix and said not-stuck well group matrix to establish a group centroid for each of said group matrices, and means for indicating the relative distance of the well vector of another well bore to the centroids of said well vectors of said stuck well group and said non-stuck well group whereby a change in value of a set'Cted variable of said multiplicity of measurable variables in said other well bore may be calculated by said well vector calculating means to measure the effectiveness of said change in said selected drilling variable to move the well vector of said other well bore toward said non-stuck well group centroid and thereby decrease the probability of sticking the drill string in said other well bore using said change in value of said selected variable.

21. Apparatus in accordance with claim 20 with the addition of further means for converting each of said stored values in said data file means in the group of well bores in which the drill string stuck into a similar element of an array to form at least one group matrix of mechanically stuck wells and at least another group matrix of differentially stuck wells, and wherein said recording 9109 10,cmsdat.125,39290Chev1I a s Y--c^cl~ -109- means includes means for separately recording the computed values of said mechanically stuck well group centroid and said differentially stuck well group centroid.

22. Apparatus in accordance with claim 20 for additionally determining the probability that a drill string can be freed in a well bore where the drill string has become stuck which includes: means for additionally computing the well vectors representing another total matrix for at least two groups of wells formed as subgroups of wells in which the drill string stuck, one of said groups including wells in which the drill string was freed and the other group including wells in which the drill string was not freed, means for computing at least the centroid of the well vectors of said stuck wells which had not been freed, means for computing the well vector for another well in which a drill 0*0 string has stuck from the same multiplicity of variables measured to form said other total matrix of well bores in which the drill string stuck, and means for comparing said well vector for said other stuck well with the centroids of at least said not freed wells to indicate the probability of freeing the drill pipe in said well. Dated this 10th day of September, 1991 Chevron Research Company by DAVIES COLLISON Patent Attorneys for the applicant(s) t 910910,crsdat.125,3929chev,12