GB2205421A - Computer-controlled model for determining internal friction angle, porosity, and fracture probability - Google Patents

Description

220-4"', 1 COMPUTER-CONTROLLED MODEL FOR DETERMINING INTERNAL FRICTION

ANGLE, POROSITY, AND-FRACTURE PROBABILITY

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to a computer-based system for calculating the internal friction angle, porosity, and the fracture probability of a formation in a borehole at the drill bit level. The invention can perform "on line" while the borehole is being drilled, or "off line" after the borehole is completed.

2. The Prior Art

For many years, the oil drilling industry has sought new techniques to, extract information with regard to the fluid flow properties (e.g., porosity and permeability) of subsurface formations, particularly while the drill bit is still in operation. Such information not only helps predict the probability of commercial hydrocarbon deposits in a formation but, if developed soon enough, can predict blowout conditions before critical situations occur.

Thus, "real time" information about the fluid flow properties of strata cut by a borehole can help drill operators predict dangerous conditions which jeopardize life, the environment, and costly drilling equipment. Additionally, the information enables operators to drill more efficiently, and more accurately determine hydrocarbon producing possibilities of the well.

One method for determining the fluid-producing potential of a subsurface formation analyzes the nuclear magnetic resonance (NMR) properties of small samples of the formation carried to the surface by the drilling mud during the drilling operation. U.S. Patent No. 4,389,612 to Brown discloses an NMR system which measures the spinlattice relaxation time of hydrogen protons of interstitial fluids within a formation sample. From these measurements, the flow properties of the sample, including porosity, permeability, and recoverable fluid content can be determined.

The drawback of this system is that it cannot determine porosity at the bottom of the borehole on a "real time" basis. Instead, the NMR system must wait for the drilling fluid to carry strata samples ("cuttings") up from the bottom for the borehole before nuclear magnetic resonance testing can take place. Specif ically, the time necessary to process a strata sample and perform the NMR analysis takes place in intervals of 25 to 30 minutes. The analysis will be provided for a particular depth increment which will vary with the rate of penetration (e.g., at 100 ft/hr, the sample depth increment would be between 42 and 50 feet). For better depth resolution, it is necessary to retain more closely spaced samples of analysis after drilling is completed, or at some other location away from the well site. In either case, further delay is introduced and there are practical (time and budget) limitations on the number of samples which can be analyzed.

Thus, the information acquired by the NMR system at the surface does not include fluid property information at the bottom of the hole on a continuous "real time" basis, which facilitates accurate prediction of commercial hydrocarbon content, and control of blowout conditions during the drilling operation.

Further disadvantages of the NMR system occur when drilling into fractured formations. In such formations, the porosity may be on too large a scale to be adequately represented in samples as small as the drill cuttings acquired by the NMR system. Also, when drilling fractured formations, it is common for a large portion of the circulating drilling fluid and its contained drill cuttings to be lost into the formation fractures. In both situations, the value of the NMR analysis applied to the drill cuttings is severely reduced. In such situations, to obtain samples of a suitable size or from zones of lost circulation, it may be necessary to collect sidewall cores after drilling is completed. This operation would be time consuming and expensive to carry through. Although the NMR technique increases the efficiency of drilling operations to offset high operating costs, the system is inherently subject to the disadvantage of having to evaluate potentially inaccurate information which is being acquired at a relatively slow rate.

1 4 - SU'.V,ARY OF THE INVENTION This invention provides apparatus and methods for determining at least three characteristics of strata cut by a borehole on a "real time" basis, namely, the internal friction angle, porosity, and fracture probability. The internal friction angle is a property of the formation defining the relationship between shear and normal stresses to a failure plane in the formation. It is determined by calculating the shear and normal stress created by the drill bit in the formation, leading to failure or fracture of the formation. The tangent of the internal friction angle is termed the coefficient of friction of the formation. From the coefficient of friction angle, the invention determines porosity, which enables drill operators to determine storage capacity of the formation. From the porosity determinations, the invention determines fracture probability to indicate production capability of the formation.

To calculate the internal friction angle, the invention simulates apparent and theoretical model formulae of a chosen physical characteristic (e.g., fracture volume, penetration, depth, etc.) of a drill bit operating within a borehole. The apparent model formula is an accurate mathematical formula of the chosen physical characteristic, and the parameters of the formula are all known constant values or varying values measured by sensors. The theoretical -model formula is a mathematical representation of the chosen physical characteristic, and it has one unknown parameter, the internal friction angle. All the other parameters in the theoretical formula are either known constants or varying values measured by sensors. Because all the parameters for the apparent model formula (Ma) are known values, an actual value for Ma can be determined. The theoretical model formula (Mt), which is a function of the internal friction angle 0, is set equal to Ma, and the internal friction angle 0 is itera tively determined so that the absolute value Of Ma-Mt(O) is less than or equal to e, where e is a specified toler ance acceptable for field work. The value calculated for the internal friction angle 0 is then used to calculate porosity 1 from the relationship given by 6 (-KtanO) 1-6 - Ae W The parameters A and K are calibration transform coefficients correlated from values of "average" porosities determined for several nearby wells by techniques, which may be conventional, such as NMR or core analysis. The transform coefficients make the determination of porosity more accurate because they take into account the cohesive strength of the formation which the mathematical model formulae do not.

The porosity is one of the most important lithologic determinations of the formation. Porosity determines the storage capacity of the formation for oil and gas, and fracture probability can be calculated from it to predict the probability of oil/gas production from the formation.

Fracture probability is essentially a combination of cohesive index and porosity. The cohesive index depends on the amount of shear stress required to cause shear failure in the formation and is therefore related to the degree of consolidation of the formation. The lower the cohesive index, the lower the degree of consolidation and the greater the probability of oil production in a formation. Also, the higher the porosity, the greater the probability of having an economic quantity of oil in the formation. By combining the two indicators to calculate fracture probability, a more accurate indicator of the fluid flow properties.of the formation is derived. Cohesive index is determined by empirical comparison of plots of void ratio, 6/1-6, versus the coefficient of friction, tano, for a noncohesive formation with that of the calibrated transform. Then, to determine a percentage of fracture probability, the cohesive index is scaled to a number between 0 and 1 (called cohesive rating CR), and the porosity is scaled to a number between 0 and 1 (called porosity rating IR). Fracture probability, pr(frac), is then calculated from the relationship given by pr(frac) = CR x 6R.

To calculate internal friction angle, porosity, and fracture probability, the system of this invention need not wait for information to come up through the borehole in the form of rock samples, or even by mud pulse telemetry (i.e., mud pulses generated at the bottom of the borehole sent to sensors at the top of the borehole). Instead, sensors at the drilling rig detect all the values needed for the calculations on a "real time" basis. Thus, the system determines formation properties at the bottom of the borehole, while drilling is occurring and can make accurate predictions regarding hydrocarbon productivity and blowout possibilities on a "real time" basis. Thus, the invention increases the efficiency of drilling operations.

Moreover, the system need not operate on a "real time,, basis. For example, the information necessary to perform the model formula calculations can be stored in a memory device, and accessed later to perform the calculations. Furthermore, as more accurate model formulae for existing and new drill bits become available, the system can be easily modified and enhanced to determine more accurately and efficiently the internal friction angle and porosity.

7 - BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a drilling rig equipped for detecting sensory inputs and simultaneously determining internal friction angle, porosity, and fracture probability at the bottom of a borehole in accordance with this invention; FIG. 2 is a diagram showing the operation of a multiplexer for accessing data from the sensory monitoring system; FIG. 3 is a schematic block diagram of the information accessed by the multiplexer and sent by parallel buses to data processing equipment; FIG. 4 is a schematic block diagram of a system for determining "real time" and non-real time porosity and internal friction angle of a formation; FIG. 5 is a schematic block diagram of the drilling model in accordance with this invention; FIG. 6 is a flow chart of a software algorithm for iteratively calculating the internal friction angle; FIG. 7 is a plot of the void ratio versus coefficient of friction (tangent of the internal friction angle, tanO); FIG. 8 is a plot of porosity, 5, versus depth, for the NMR technique and for the uncalibrated drilling model; FIG. 9 is a plot of the void ratio, 5/1-M, versus coefficient of friction, tanO, for the uncalibrated and calibrated drilling model; FIG. 10 is a table of average porosity, average cohesive index, and average fracture probability for three different zones of a formation drilled by a test well; and FIG. 11 is a cross sectional view of a Fracture Crater wherpu is the included angle of the crushed rock- /wedge;_and the curved portion of the cross section (AB) is modeled by polar equation r=r eetan. 0 DESCRIPTION OF SPECIFIC EMBODIMENTS

Referring to FIG. 1, a well 10 is drilled into an earth formation with a rotary drilling rig 12, which includes the usual derrick 14, derrick floor 16, drilling line 18, hook 20, swivel 22, kelly joint 24, kelly height chamber 25, rotary table 26, and drill string 28 that includes conventional drill pipe 30 secured to the lower end of the kelly joint 24 and to the upper end of a section of drill collars 32 which carry a drill bit 34. Drilling liquid (or mud, as it is commonly called in the field) circulates from a mud pit 36 through a mud pump 38, a mud supply line 41 (a flexible rotarl hose), and into the swivel 22. The drilling mud flows down through the kelly joint, drill string and drill collars, and through jets (not shown) in the lower face of the drill bit. The drilling mud flows back up through the annular space between the outer diameter of the dr-11 string and the well bore to the surface, where it is returned to the mud pit through a mud return line 42. Also shown is the usual shaker screen 43 for separating formation cuttings from the drilling mud before the mud returns to the mud pit.

A sensory monitoring system (SMS) 48 monitors all of the varying parameters of the drilling operation and converts all the information into a computer-readable digital format. Five of the many monitored parameters are (1) mud pressure, (2) kelly height, (3) hook load, (4) weight on bit, and (5) rotary speed of the drill bit.

Sensor 50 measures kelly height. Sensor 50 consists of a kelly height chamber 25, service box 57, and a flexible connecting hose 59 carried by the rotary hose. Basically, a column of water is maintained in a U-shaped leg 60 of hose 59 connecting the kelly height chamber to the service box 57. As the kelly-height raises and lowers, the hydrostatic pressure of the water in hose 59 var-es. A pressure transducer 51 in the service box 57 detects the variation and produces a corresponding analog signal, which is sent to the SMS 48 where --- is stored and converted to digital format.

The weight-on-bit and the hook load are determined from an analog.signal from a sensor 58 mounted on the deadline anchor 59 of the draw works 61. :he weight supported by the hook 20 or hook load (HL) is the total weight of the drill string (WOS) minus the weight on the bit (WOB), i.e., HL=WOSWOB. The sensor 58 converts the weight (amount of pull) on the deadline anchor 59 to an analog signal proportional to the hook load, and sends a signal to the SMS where it is converted to a digital hook 20.

signal, which varies with the weight on tAre Another sensor 62 detects the rotary speed of the rotary table 26. The sensor 62 consists cf- a magnet (not shown) and a reed switch (not shown). The magnet is attached to the rotary table, and, as the rotary table turns, the magnet swings past and closes the reed switch with each rotation. Each closing sends a pulse the magnet to the sensor 62. By tracking the time that on the rotary table 26 takes to rotate 360', the sensor 62 calculates the velocity of the rotary table. This information is transmitted to the SMS 48 via electrical conductor 64 (FIG. 2) where it is stored and converted into digital format.

Referring to FIG. 4, block 48 represents the sensory monitoring system described above. The sensors on the rig measure the varying drilling parameters necessary for the Drilling Model program 74 described below. The SMS 48 converts analog signals from the sensor to digital signals (binary coded decimal).

A multiplexer 66 polls each of the sensors in turn and receives the sensor readings (FIG. 2). The multiplexer converts the binary coded decimal signal into an ASCII code, and a unique ASCII address is assigned signifying each sensory signal. The ASCII signals and addresses for all of the sensors are placed on a parallel bus 65 which transmits the signals to data processing equipment 68 (FIGS. 3 and 4). Here, a control program 70 directs the data to a table in the memory of the data processing equipment 68. Computations are performed upon the data, and new data is added to the data table (e;g., rate of formation penetration by the drill bit is computed from time and kelly height, average mud densities computed from mud density readings over a specified time interval, etc.). At a particular time interval specified by the control program 70, the data processing equipment 68 transfers a copy of the data table to a new memory storage 72. Here, the data awaits access by the Drilling Model program 74.

The Drilling Model is a software program 74 which may operate "on line", i.e., while the drill bit is in operation, or "off line", i.e., after the drill bit operation is completed. In either case, when the Drilling Model program performs its calculations, the system acts as if it is operating under a near "real time" basis. By having the data processor equipment 68 assemble and calculate necessary data parameters, and then by storing the data in storage 72, there is no need for the Drilling Model 74 to keep pace with the "real time" data generated by the sensors.

The preferred embodiment of the Drilling Model 74 is a software program which accepts the data in storage 72 by a bus 73. However, the step/or data storage 72 may be circumvented when processing of the drilling model needs to access the data on a "real time" basis. FIG. 5 shows a schematic block diagram of the Drilling Model. After the data is input at 75 the bit type is determined at 76. The bit type algorithm 76 determines whether the bit s a Rollercone or a Polycrystalline Diamond Compact (PDC)/Diamond bit. Rollercone bits are either milltooth or insert kinds of bits, and the PDC/Diamond bits are usually core or full-hole bits. The software algorithm simulating the drill bit in operation is selected by the drill bit type algorithm 76. The purpose of these formulae is to determine formation fracture volume, internal friction angle, and porosity as the bit operates in the borehole.

In the preferred embodiment, the Rollercone bits are modeled by two formulae, the apparent fracture volume of a bit tooth formula 78 and the theoretical fracture volume of a bit tooth formula 80. The apparent fracture volume formula 78 mathematically represents the volume of rock fractured by a bit tooth impact in the borehole. The formula assumes that all of the rock fractured by a tooth impact is removed by mud flow beneath the bit. That is, 100% of the fractured rock is hydraulically cleaned by the mud flow from the borehole. In addition, all of the parameters in this Rollercone formula are known values which were previously held in the data storage 72 or input directly from data processing equipment 68.

The apparent volume fracture per impact of a Rollercone bit is derived in the following way. For a tricone Rollercone bit rotating on the bottom of a borehole, the first parameter to consider for modeling is the number of revolutions that a single cone of the bit makes. Assuming Db to be the diameter of the bit in inches, and Dc to be the diameter of a cone in inches, then Db divided by Dc is the number of revolutions that a cone makes for a single revolution of the drill bit. If the number of teeth is n, then there are:

3n (Db/Dc) (tooth impacts per revolution of the bit).

If Va represents the apparent volume (ins3) of the fornation fractured per bit tooth impact, and N the revolutions of the bit per minute, then the volume of formation fractured per minute is:

3n (Db/DC) Va14f ins3/min.

If we assume that a bit advances at penetration rate R (ft/hr), the volume formation fracture per minute is:

(ITDb 2/4)(12R/60), ins3/min.

Therefore, the apparent fracture volume per tooth impact is:

3 Va _ IT DcDbR/(60Nn), ins Dc, Db, n are all constant values. R and N may vary continuously as they are measured by sensoring monitoring system 48, and input to the drilling model algorithm 74 at 75. By having the value of all of the parameters, the algorithm calculates an actual value for the apparent fracture volume per tooth impact.

The second formula representing the Rollercone bit is a theoretical fracture volume per tooth impact. This model is designed so that the only unknown is the internal friction angle 0. In the preferred embodiment, the theoretical fracture volume (ins3) is represented by the following formulas:

bp2COSO (2sin20-1) + (1+2sin20)e Q3n/2)-0)tano Vt 2_ _s -i-n:

where Wtan(6-OM-sino) P = (6 m+( 6m-2 6p)sinO)E)M 14 The derivations for these formulae are discussed in Appendices A and B, and both of the above fcrmulae have been empirically tested for their accuracy.

In the above formula, Vt equals the theoretical volume of rock fractured per bit tooth impact, which is represented in ins3, b equals tooth width which is in inches, P equals the tooth penetration depth which is also in inches, 6 equals tooth-included angle, W equals the weight-on-bit in pounds, - 6m equals mud hydrostatic pressure in pounds per square inch (psi), 6p equals pore pressure in psi, and M equals the number of teeth in contact with the formation at any one time. All the values for these parameters have been suppl-Jed from data storage 72, which is down-loaded by line 73 or 71 to the drilling model 74 of FIG. 3. All parameters, except the internal friction angle 0 are known values. These parameters are either constant or varying values updated continuously by the sensor inputs, which are eventually input to the drilling model 74.

The internal friction angle is found by setting the apparent fracture volume equal to the theoretical fracture volume, and then iteratively selecting values for the internal friction angle to reduce the difference between the theoretical and apparent fracture volumes to an insignificant amount. This is expressed mathematically in the following way. If the theoretical fracture volume is Vt (0), i.e., a function of the internal friction angle 0, and if the apparent fracture volume is Vai then there is some internal friction angle 0. so that:

-e <va-vt (0) <e, where e is a predetermined tolerance.

The above equation is solved by iteratively testing for different values of 0. Specifically, it is solved by using the interval halving algorithm shown in FIG. 6.

More particularly, the interval halving algorithm is initialized by first choosing a minimum value for 0 and a maximum value for 0 at 112. The test value for the internal friction angle, 0 test, is then calculated at 118 by:

0 test(118) olow + 0 max, 2 and then the theoretical volume is calculated at 120 using 0 test (118) If the difference between the theoretical fracture volume and the apparent fracture volume does not satisfy the tolerance equation -e <va-vt(O) <e, then the test value for the internal friction angle is placed in the relevant boundary value, either 0 low or 0 max That is, if the difference Vt - Va calculated at 122 is greater than e set at 126, then the 0 boundary min is set at 130 equal to 0 test(130) On the other hand, if the difference between Vt - Va is less than -e set at 128, then the boundary 0 max is set at 132 equal to 0 test(132)' In time, as new 0 test values are tested, the value for Vt(O)test gets closer to the value for Va once the internal halving routine determines a value for 0 test which satisfies the tolerance equation at 124, then an acceptable value for 0 has been determined.

In the preferred embodiment, PDC/Diamond bits are also simulated by two formulae, the apparent blade penetration depth of a drill bit at 84 (FIG. 5) and the theoretical blade penetration of a drill bit at 86. The apparent - 16 blade penetration model 84 mathematically represents the blade penetration of a drill bit or core bit in operation at the bottom of a borehole. All the parameters in the PDC/Dianond bit apparent penetration formulae are known values which were previously stored in the data storage 72 or inputted directly from the data processing equipment. Although PDC blanks or diamonds are arranged in particular patterns and varied density on the working surface of a bit, the net effect is that of a blade removing a volume of formation on each bit revolution. The apparent blade penetration depth of a PDC/ Ciamond bit is derived by the following formula. The cutting rate of a bit is expressed as the surface area of the bottom of the borehole times the rate of advance of the bit:

d. (v) (D) 2 d (h) - = n - dt 2 dt (1) where V is the volume (ins.3) of formation cut, D is the bit diameter of a drill bit or the annular cutting structure diameter of a core bit, ins, and h is the depth in the formation of the bit, in ins. The parameter d(h)/dt can also be expressed as a function of the, rate of penetration R, so that:

d(h) = R/5, (ins/min.) dt By placing this expression in equation (1) above, the volume of formation cut per minute is:

d (V) 7r D2 R (2) (3) dt 20 on the other hand, for a bit which advances a vertica-l distance Y inches in one revolution, the following formula represents the volume (ins3/rev) removed per revolution:

d (V) rev 7r D2 y 4 (4) If the bit makes N revolutions per minute, then the volume (ins3/min) cut per minute is given by:

d (V) dt ir D2 y N 4 (5) By equating equations (5) and (3), Y, the bit advance per revolution is:

Y = -E, ins/rev. 5N where Y is the bit advance per revolution (ins/rev), R is the rate of penetration (ft/hr), and N is the bit rotary speed (revs/min).

Since the net effect of the individual cutting elements of a diamond or PDC bit is that of a blade, then it can be considered that this "blade" removes a volume of formation for each revolution of the bit. Therefore, the bit advanced per revolution represents the depth of penetration of a blade per revolution of the bit, and rewriting the last equation gives:

p - R, in/rev, 5N where P equals the depth of penetration of the blade, in/rev. This equation represents the apparent penetration depth of a blade where the parameters R and N are varying values which are continuously updated by the sensoring monitoring system 48 and inputted to the drilling model algorithm 74 at 75. Thus, the algorithm can calculate an actual value for the apparent blade penetration depth per revolution.

The second formula representing the PDC/Diamond bit is a theoretical blade penetration depth in inches per revolution. This model simulates the PDC/Diamond bit as it operates in.the borehole. The model is designed so that the only unknown of the formula is the internal friction angle 0. In the preferred embodiment, the theoret-4cal blade penetration depth is represented by the following formula:

Pt. = 2 W (X - XR) Cot (0 + 8) (1 - sinO) 1 S/rev. X D [6M (1 + sinO) + 2 (c cos(O - 6p) sinO)] A derivation of the above formula can be found in Appendix C. In the above formula, Pt is the theoretical blade penetration depth, which is represented in ins/rev, W is the weight on the bit, 8 is the blade backrake angle, X is the total number of cutters on the bit, XR is the number of redundant cutters 'on the bit, D is the diameter of a drilling bit or the annular cutting structure diameter of the bit, 6M is the mud hydrostatic pressure, C is the formation cohesive strength, and 6 p is the formation pore pressure.

All the values of these parameters are supplied from data storage 72 which is downloaded by line 73 to the drilling model 74 of FIG. 4. All the parameters, except the internal friction angle, 0, are known values which are either constant or varying values which are updatedcontinuously by the sensor inputs at 48.

The internal friction angle is solved by setting the apparent blade penetration depth equal to the theoreti- cal blade penetration depth and then iteratively selecting values for the internal friction angle, which red uce the difference between the theoretical and apparent blade penetration depths to an insignificant amount. Specifical ly, this may be expressed mathematically in the following way. The theoretical blade penetration depth is represented by Pt, and it is expressed as a function, Pt(O), of the internal friction angle 0. Then, if the apparent blade penetration depth is Paf there is some interval friction angle 0, so that:

-e < Pa - Pt(O) < e, where e is a predetermined tolerance.

As with the Rollercone bit calculations, the above equation is solved by iteratively testing different values for 0. Specifically, it is solved by using the internal halving algorithm shown in FIG. 6; however, va is replaced with Pa, and Vt(O) is replaced with Pt(O).

More particularly, the interval halving algorithm in FIG. 6 is initialized by first choosing a minimum value for 0 and a maximum value for 0 at 112. The test value for the internal friction angle, 0 test' is then calculated at 118 by 0 = 0 min + Omax, test (118) 2 and then the theoretical penetration depth is calculated at 120 using 0 test(118)4 If the difference between the theoretical penetration depth and the apparent penetration depth do not satisfy the tolerance equation -e <Pt(O) - Pa <e, - 20 then a new value for 0 test is determined by the interval halving algorithm previously described. Once the interval halving routine hasdetermined a value for 0 test which satisfies the tolerance equation, then an acceptable value for 0 has been determined.

After the internal friction angle is calculated for either a Rollercone bit, or a PDC/Diamond bit, the next step is to calculate the porosity of the formation. The porosity of formation is calculated for both kinds of bits at module 90 (FIG. 5), which solves for porosity from the relationship:

as = Ae (-K tanO) 1 - ds 0 where the parameters A and K are calibration transform coefficients. The transform coefficients make the determination of porosity more accurate because they take into account the cohesive strength of the formation which the uncalibrated drilling model calculations did not take into account. From a large number of Sonic Interval Transit Time logs values of A and K have been obtained which allow calculation of a porosity which gives good general agreement with porosity determined by the Sonic log using the relationship:

ts = 0.6393 (-2.2 tanO) - as 0 This calculated porosity term is the uncalibrated drilling porosity. More accurate values or calibrated drilling porosities may be determined if the values of the transform coefficients are determined from nearby wells or from the current well using NMR methods or core analysis which will be discussed in more detail in the following paragraphs.

The following is a detailed description of two scenarios for determining the transform coefficients A and K for

3 - Ae (-K tanO) l-(l 1 at a particular well site. In the first scenario, assume that there are several nearby wells where the porosity has been determined by a nuclear magnetic resonance technique, as disclosed in U.S. Patent No. 4,389,612 to Brown, which is incorporated herein by reference. Plots of the porosity versus depth are then made. The plots for several nearby wells are collected and combined to get an "average" plot, R1 for porosity versus depth for the entire area (FIG. 8). A similar plot of porosity versus depth R2 determined by the uncalibrated drilling model is also plotted. As shown by the graph in FIG. 8, the "average" plot produced by the NMR techniquesRl, may not correspond exactly to the plot produced by the uncalibrated drilling model R2.

In the preferred embodiment, calibration of the drilling model is performed by a software algorithm. More particularly, information from the nearby wells is loaded and stored within data storage 72 (FIG. 4) of the computer system. Then, when an uncalibrated plot of porosity versus depth is calculated by the drilling model, a correlating algorithm empirically calculates the values for A and K, so that the drilling model curve, R2 "on average", follows the NMR curve Rl through calibrated points within a specified depth interval (FIG. 8). More particularly, for several points within a depth interval of the formation, the drilling model iteratively determines values for the internal friction angle, 0. These values of 0 are -KtanO placed into the formula Ae /, where A and K are still uncalibrated. An uncalibrated -porosity is calculated -KtanO from the expression Ae The drilling model porosity determination is then compared against the "average" NMR porosity for the same depth interval (FIG. 8). A and K are then empirically determined so that the drilling model produces values for porosity, equal to the NMR porosity, for the same depth interval.

If we assume that the transform coefficients A and K change linearly over the interval of calibration points, a constant value-or linearly changing value for A and K can be chosen so that cohesive strength of the formation:s taken into account for the entire borehole. As additional information becomes available from other nearby wells, the transform coefficients are updated to ensure the highest accuracy of the drilling model for the particular borehole being drilled.

In the second scenario, assume that there are no nearby wells, and that there are no other formations anywhere like the site presently being drilled. Under this scenario, accurate "real time" calculations of internal friction angle, porosity, and fracture probability cannot be determined by the drilling model alone because the information necessary to perform the calibration algorithm, as discussed aboVe, is not available on a "real time" basis. Instead, a portion of the well is completely drilled, and information calculated by the drilling model is maintained in the data storage 72 (FIG. 4) of the computer system. In real time, uncalibrated drilling porosity is calculated by the drilling model in online mode. Some time later, when samples become available, the NMR technique discussed above is used to determine porosity versus depth for the particular well being drilled. The calibration algorithm performed for the first scenario above is then performed in this scenario in order to calibrate the drilling model to account for cohesive strength. Although the drilling model in this scenario cannot be used to determine fluid properties of the formation on a 11-real time" basis, information obtained can be used to better evaluate the fluid flow properties of the formation on an "off line" basis. It is envisioned that once the N11AR techniques, or other methods capable of determining porosity which are not affected by formation cohesive strength, can determine porosity versus depth on a "real time" basis, the second scenario for calculating fracture probability will also perform on a "real time" basis.

Information obtained under either scenario can be used to calculate the cohesive index, CI, CI C- 1 (6N-6p) of the formation, which accounts for the uncalibrated drilling porosity's inaccuracy. In the preferred embodiment, this calculation is also performed by a computer software algorithm.

It can be shown by the Navier-Coulomb criterion of failure that the relationship between the coefficient of friction as calculated by the drilling model and the cohesive strength of the formation, is C tanO DM _ tanOF + (6N-6p) where tanO DM.0 internal friction angle calculated by drilling model assuming C is to be zero; tanOF = actual internal f riction angle of the f ormation; C = cohesive strength of the formation; and 61; 6 p = normal stress across a failure plane in the formation.

Referring to FIG. 9, a plot of the void ratio, tS/11, versus the coefficient of friction, tanO, for the case when the cohesive strength of the formation is zero (3(o)), and a similar plot by calibrated drilling model (8(c)), are shown. As shown in the graph, values for the cohesive index, 11C1(6N - 6p)11 are empirically determined on a "real time" basis by empirically evaluating the horizontal difference between the zero cohesive strength plot, B(o); and the calibrated plot,"B(c), for particular values of the void ratio. The cohesive index accurately represents the amount of shear stress required to cause shear failure in the formation at the drill bit level. Thus cohesive index is an indicator of the degree of consolidation of a formation. When the value for cohesive index, C index' is low, the probability for oil in the formation increases.

With the cohesive index and porosity of the formation calculated, the fracture probability of the formation at the drill bit level is determined at module 91 (FIG. 5) from:

pr(frac) = CR d5R, where CR is a continuously changing cohesive index rating, scaled to a scaling ratio (S,) between 0 and 1 by the following formula:

CR -4 C max C - C index max J1 where Cmax is a maximum value for C index. As the drilling model produces calibrated plots of void ratio, 1/1-6, versus the coefficient of friction, tanO, values for cohesive index are determined, and the frequency for each value of cohesive index is plotted on a histogram. The largest value for C index which is statistically significant on the histogram is the value chosen for C max T1.--- scaling ratio has a maximum value of 1 for formations with no cohesive strength, and a minimum value of zero for formations where C index > C max' The scaling ratio (Sl) determines a rating for cohesive index so that the lower the cohesive index, the greater the rating becomes.

MR is a continuously changing porosity rating of a formation scaled to a scaling value (S2) between 0 and 1 by the following formula:

5d 5ave 45R max 5 ave i where Md is equal to the porosity determined by the calibrated drilling model (FIG. 9) 1 Mave is a statistically weighted average value for porosity determined by the drilling model, and M max is the largest statistically significant value for porosity determined by the drilling model. I ave and 5 max are empirically determined from a histogram of porosities continuously updated by the drilling model. The scaling ratio (S2) determines a rating for porosity so that the larger the porosity, the greater the rating becomes. 7 The rating for porosity has a maximum of one for the highest porosity rating, and a minimum of zero f or f ormations where MR < 0 - CR and 5R are multiplied together to obtain a probability which is the most reliable determination of the oil production capability of the formation. The greater the fracture probability calculated at a particular depth of the formation, the greater the likelihood that the formation can produce oil at a commercial rate.

Once the actual values for porosity and fracture probability have been determined by the drilling model, they are sent to the peripherals 100, described below. Referring to FIG. 5, the peripherals are divided into three different kinds, namely, CRT terminal 92, a memory storage 94, a printer and/or plotter 96. The porosity determined from the drilling model is plotted against depth of the drill bit in the borehole on printer 96, or the calculated porosity for each drill depth and internal friction angle are permanently stored in memory 94. The porosity can be also plotted on the CRT terminal 92 for an observer's "real time" review. obviously, any combination of information with regard to the porosity and the internal friction angle can either be displayed on a terminal, stored in memory, or printed by a plotter.

Experiments demonstrating the drilling model calibrated with the NMR technique were performed in an experimental well. We used the drilling model of this invention to calculate drilling porosity, cohesive index, and fracture probability for three different zones penetrated by the well. The analysis of the three zones is shown in FIG. 10. The drilling model determined that zone A would not be productive, zone B, having the highest fracture probability, would provide commercial production, and zone C, having the highest porosity would be a wet well, i.e., producing water instead of oil. This study proved the drilling model, in combination with the NMR technique, to be an accurate formation evaluator. In particular, the fracture probability determination proved to be a reliable indicator of oil producing capability. This test confirmed that in the oil bearing zone, if the calculated fracture probability is high, meaning that the cohesive index is low, and that the calculated porosity is high, then the likelihood for commercial oil producing capability is high.

Claims (44)

CLAIMS:

1. A method of determining 0, the internal friction angle of a formation in a borehole drilled by a drill bit, the method comprising a) modelling an apparent model formula (M a of a drill bit operating within a borehole; b) modelling a theoretical model formula (M t). oL a drill bit operating in a borehole, where the only unknown parameter of the theoretical model formula is 0, the internal friction angle of formation, all other parameters having values which are known or can be sensed by sensors; c) sensing the varying values for both the actual and theoretical models; and d) determining 0 by setting the value of the apparent model formula equal to the value of the theoretical model formula whereby 4 is iteratively computed, setting values of 0 so that the absolute value of M a - M t is less than or equal to e where e is a predetermined tolerance.

2. A method as claimed in claim 1, in which the model formulae are the apparent and theoretical fracture volumes of a bit tooth.

A method as claimed in claim 1, in which the model formulae are the apparent and theoretical blade penetration depths.

4. A method as claimed in any preceding claim, including the step of computing porosity, 6, from the relationship given by 1 = Ae (-K tan), 1-ffi where the constants A and K are calibration coefficients which take into account the cohesive strength of the formation. 1.

- 28 A method as claimed in claim 4, further comprising the step of calculating A and K, calibration transform coefficients, which account for cohesive strength of formation for the borehole being drilled, comprising the steps of:

a) determining the porosity of the formation versus depth, R1, by analyzing formation samples from nearby wells; b) plotting the formation porosity versus depth, R1, for the nearby wells; C) plotting porosity versus depth, R 21 calculated by the uncalibrated drilling model; and d) correlating R 2 to R 1 by empirically determining values for A and K so that R 21 on average, substantially traces R1, whereby the cohesive strength of formation is taken into account by the drilling model.

6. A method as claimed in claim 4, further comprising the step of calculating A and K, calibration transform coefficients, which account for the cohesive strength of formation for the borehole, the porosity of the formation versus depth, R1, is determined by analyzing the nuclear magnetic resonance properties of formation samples from nearby wells.

7. A method as claimed in claim 4, further comprising the step of calculating A and K, calibration transform coefficients, which account for cohesive strength of formation for the borehole being drilled, comprising the steps of:

a) measuring and plotting porosity versus depth, R1, by testing samples from the borehole as it is drilled; b) plotting porosity versus depth, R 21 calculated by the uncalibrated drilling model; and 29 - c) correlating R 2 to R 1 by empirically determining values for A and K so that R 21 on average, substantially traces R,, whereby the cohesive strength of formation is taken into account by the drilling model.

8. A method as claimed in claim 4, further comprising the step of calculating A and K, calibration transform coefficients, which account for cohesive strength of formation for the borehole being drilled, comprising the steps of:

a) measuring and plotting porosity versus depth, R,, by testing the nuclear magnetic resonance properties of samples from the borehole as it is being drilled; b) plotting porosity versus depth, R 2' calculated by the uncalibrated drilling model; and c) correlating R 2 to R 1 by empirically determining values for A and K so that R 2' on average, substantially traces R,, whereby the cohesive strength of formation is taken into account by the drilling model.

A method as claimed in any one of claims 6 to 8, further comprising the step of calculating cohesive index, C index' comprising the ste PS of:

a) plotting void ratio, 1/1-1, versus the coefficient of friction, tan, j6(o), calculated by the drilling model when cohesive strength of formation is zero; b) plotting void ratio, V1-1, versus the coefficient of friction, tan, P(c), calculated by the calibrated drilling model; and c) calculating Cindex by empirically evaluating the horizontal difference between the zero cohesive strength plot, 0(o), and the calibrated - plot, P(c), for a particular value of the void ratio C5/ 1 -5.

10. A method as claimed in claim 9, further comprising the step of calculating the fracture probability, pr(frac), of the formation, comprising the steps of:

a) determining the cohesive index rating, c R' of the formation from the relationship given by C C max C index max where C is the largest statistically C index; max value for the b) determining the porosity rating, 6., of the formation from the relationship given by I d ave R max ave where 6 d is equal to the calibrated drilling model porosity, 6 ave is equal to the statistically weighted average porosity, and &max is equal to the statistic ally significant maximum porosity of the formation; and C) determining fracture probability, pr(frac), from the relationship given by:

significant Pr(frac) = C R (I R'

11. An apparatus for implementing a drilling model, for iteratively determining 0, the -internal friction angle, of a formation in a borehole drilled by a bit, comprising:

a) means for modelling an apparent model formula (M a) of a drill bit operating within a borehole; b) means for modelling a theoretical model formula (M t) of a drill bit operating within a borehole, where the ' only unknown parameter of the theoretical model formula (M t) is f, the internal friction angle, of formation, all other variables are known constant values or varying values which can be sensed by sensors; C) means for sensing the known varying values for both the apparent and theoretical model formula of a drill bit in operation; and d) means for iteratively computing the internal friction angle, from the relationship so -hat the absolute value of M M is less than or I a- t equal to e, where M a'- the actual model of the drill bit, is a calculated value, M t (+) the theoretical model of the drill bit, is a function of, the internal friction angle, and e is a predetermined tolerance.

12. Apparatus as claimed in claim 11, in which the model formulae are the apparent and theoretical fracture volumes of a bit tooth.

13. Apparatus as claimed in claim 11, in which the model formulae are the apparent and theoretical blade penetration depths.

14. Apparatus as claimed in any of claims 11 to 13, further comprising the means for computing porosity 6 from the relationship given by = Ae (-K tan) 7 i - where the constants A and K are calibration coefficients which take into account the cohesive strength of the formation.

15. Apparatus as claimed in claim 14, further comprising the means for calculating A and K, calibration transform coefficients, which account for cohesive strength of formation for the borehole being drilled, comprising:

a) means for determining the porosity of the formation versus depth, R1, analyzing formation samples from nearby wells; b) means for plotting the fcrmation porosity versus depth, R,, for the nearby wells; C) means for plotting poros---,y versus depth, R 21 calculated by the uncalibrated drilling model; and d) means for correlating R2 to R 1 by empirically determining values for A and K so that R 21 on average, substantially traces R1, where the cohesive strength of formation is taken into account by the drilling model.

16. Apparatus as claimed in claim 15, in which the means for determining the porosity of the for.pation versus depth, R1, analyzes the nuclear magnetic resonance properties of formation samples from nearby wells.

17. Apparatus as claimed in claim 14, further comprising the means for calculating A and K, calibration transform coefficients, which account for cohesive strength of formation for the borehole being drilled, comprising:

a) means for measuring and plotting porosity versus depth, R1, by testing samples from the borehole as it is drilled; b) means for plotting porosity versus depth, R 21 calculated by the uncalibrated drilling model; and c) means for correlating R 2 to R 1 by empirically determining values for A and K so that R 2' on average, substantially traces R1, where the cohesive strength of formation is taken into account by the drilling model.

18. Apparatus as claimed in claim 14, further comprising the step of calculating A and K, calibration transform coefficients, which account for cohesive strength of formation for the borehole being drilled, comprising the steps of:

a) means for measuring and plotting the porosity versus depth, R,, by testing the nuclear magnetic resonance properties of samples from the borehole as it is being drilled; b means for plotting porosity versus depth, R 21 calculated by the uncalibrated drilling model; and c) means for correlating R 2 to R 1 by empirically determining values for A and K so that p 21 on average, substantially traces R,, whereby the cohesive strength of formation is taken into account by the drilling model.

19. Apparatus as claimed in any of claims 16 to 18, further comprising means for calculating cohesive index, C index' comprising:

a) means for plotting void ratio, 1/1-1, versus the coefficient of friction, tan, 8(o), calculated by the drilling model when cohesive strength of formation is zero; b) means for plotting void ratio, 1/1-h, versus the coefficient of friction, tan, p(c), calculated by the calibrated drilling model; and c) means for calculating C by empiric index ally evaluating the horizontal difference between the zero cohesive strength plot, p(o), and the calibrated plot, P (c), for a particular value of the void ratio, 6/1_1.

20. Apparatus as claimed in claim 19, further comprising the means for calculating the fracture probability, pr(frac), of the formation, comprising:

a) means for determinig the cohesive index rating, C of the formation from the relationship R' 1 given by 34 - C C R - max C c index, max is the largest statistically significant where C max value for the C index; rating, 1 R' given by b) means for determining the porosity of the formation from the relationship d ave R - 6 1 max ave where 6 d I is equal to the calibra':ed drilling model porosity, 6 ave is equal to the statistically weighted average porosity, and 6 max is equal to the statistically significant maximum porosity of the formation; and c) means for determining fracture probability, pr(frac), from the relationship given by: pr(frac) = C R 6 R

21. A computer-implemented drilling model method for determining porosity, 6, 0- a formation in a borehole comprising the steps of:

a) determining 0, the internal friction angle, of the formation; and b) using to compute porosity, 6, from the relationship given by 1 ffi -j = Ae (-K tan) 9 where the constants A and K are calibration coefficients which take into account the cohesive strength of the formation.

22. A method as claimed in claim 21, in which is determined by the steps of:

a) modelling V ay the apparent fracture volume by a single tooth of the bit, by 1 - 35 v Tr D c D b R a -70 N n_ where D c is the diameter of the bit cone, D b is the diameter of the bit, R is the rate of penetration of the bit, N is the rotary speed of the bit, and n is the number of teeth per bit cone; b) using the V a just computed, determining the internal friction angle, from the relationship v a to Vt, theoretical fracture volume, given by:

v bp 2 cos C(2sin 20_1) + (1+2sin 2)e((3Tr/2)-0)tanfl t 2sin345 where b is the width of each tooth of the Rollercone bit, and where p is determined by Otan(94)(1-sinO) P - (dm+(cIM-2aP)sinP)bPI where W is the weight on the bit, G is the tooth included angle, (f m is the mud pressure, C'p is the pore pressure, and M is the number of teeth on forma tion at any one time, whereby the internal friction angle is iteratively computed by the computer, testing values of 4, so that the a bsolute value of V. - Vt is less than or equal to e, where e is a predetermined tolerance.

23. A method as claimed in claim 21, in which 0 is determined by the steps of:

a) modelling P a' the apparent blade penetration depth given by p - R a 5N where R is the rate of penetration, and N is the bit rotary speed; b) using P a to determine the internal friction angle, from the relationship P a to Pt, theoretical blade penetration depth, given by 2W(X-X R) cot(o+p) (1-sin)n t XD [cr ( 1+sino) + 2(Ccos(f -o) sinf) 3 m P where W is the weight on the bit; A is the blade backrake angle; X is the total number of cutters on the blade; X R is the number of "redundant" cutters; D is the diameter of the drilled hole; d m is the mud hydrostatic pressure; C is the formation cohesive strength; and P is the formation pore pressure, whereby 0, the internal friction angle, is iteratively computed by the computer by testing values sothat the absolute value of P a - P t is less than or equal to e, where e is a predetermined tolerance.

24. A method as claimed in claim 22 or claim 23, further comprising the steps of sensing R, the rate of penetration, and N, the rotary bit speed.

25. A method as claimed in claim 22 or claim 23, further comprising the step of calculating A and K, calibration transform coefficients, which account for cohesive strength of formation for the borehole being drilled, comprising the steps of:

a) determining the porosity of the formation versus depth, R1, by analyzing formation samples from nearby wells; b) plotting the formation porosity versus depth, R1, for the nearby wells; C) plotting porosity versus depth, R 21 calculated by the uncalibrated drilling model; and d) correlating R 2 to R 1 by empirically determining values for A and K so that R 21 on average, traces R1, whereby the cohesive strength of formation is taken into account by the drilling model.

26. A method as claimed in claim 25, in which the porosity of the formation versus depth, R,, is determined by analyzing the nuclear magnetic resonance properties of formation samples from nearby wells.

27. A method as claimed in claim 22 or claim 23, further comprising the step of calculating A and K, calibration transform coefficients, which account for cohesive strength of formation for the borehole being drilled, comprising the steps of:

a) measuring and plotting porosity versus depth, R1, by testihg samples from the borehole as it is drilled; b) plotting porosity versus depth, R21 calculated by the uncalibrated drilling model; and c) correlating R 2 to R, by empirically determining values for A and K so that R 2, on average, substantially traces R,, whereby the cohesive strength of formation is taken into account by the drilling model.

28. A method as claimed in claim 27, in which the porosity is measured by testing the nuclear magnetic resonance properties of samples from the borehole as it is being drilled.

29. A method as claimed in any of claims 25 to 28, further comprising the step of calculating cohesive index, C index' comprising the steps of:

a) plotting void ratio, 6/1-6, versus the coefficient of friction, tano, j8(o), calculated by the drilling model when cohesive strength of formation is zero; b) plotting void ratio, (1/1-6, versus the coefficient of friction, tan, p(c), calculated by the calibrated drilling model; and c) calculating C index by empirically evaluating the horizontal difference between the zero cohesive strength Plot, 9(o), and the calibrated plot, B(c), for a particular value of the void ratio 6/1_(3.

30. A method as claimed in claim 29, further comprising the step of calculating the fracture probability, pr(frac), of the formation, comprising the steps of:

a) determining the cohesive index rating,.1 R' of the formation from the relationship given by Cmax - Cindex max where Cmax is the largest statistically significant value for the Cindex; b) determining the porosity rating, I., of the formation from the relationship given by &d &ave 6R = 6 max I ave where 6d is equal to the calibrated drilling model porosity, &ave is equal to the statistically weighted average porosity, and amax is equal to the statistically significant maximum porosity of the formation; and c) determining fracture probability, pr(frac), from the relationship given by:

pr(frac) = CR a R,

31. Apparatus ror Implementing a drilling model for determining the porosity, 6, o f strata in a borehole comprising:

a) means for determining the internal friction angle, of the compositional components within - 39 the strata; and b) means for usingo, to determine porosity, 1, from the relationship given by & Ae (-K tan) 1-6 - p where A and K are calibration coefficients which take into account the cohesive strength of the formation.

32. Apparatus as claimed in claim 31, in which the means for determining 0 comprises:

a) means for modelling V a' the apparent fracture volume, by a single tooth of the bit, by calculating v Tr ' D c D b R a __60 N n - where Dc is the diameter of the bit cone, Db is the diameter of the bit, R is the rate of penetration of the bit, N is the rotary speed of the bit, and n is the number of teeth per bit cone; b) means for using the V a to determine the internal friction angle, calculated from the relationship V to Vt, theoretical fracture volume a v bfp 2 cos C(2sin 20_1) + (1+2sin 2 0)e((31Y/2)-t)tang t 2sin3 ( where b is the width of each tooth of the Rollercone bit, and where p is determined by p W tan (9 4) ( 1 - sin) (cr +(olm-2cr)sinfl-bM m p where W is the weight on the bit, G is thetooth included angle, cr m is the mud pressure, (rp is the pore pressure, and M is thp number of teeth on forma tion at any one time, whereby the internal friction angle is iteratively computed by the computer, testing values of, so that the absolute value of Va - v t is less than or equal to e, where e is a predetermined tolerance.

33. Apparatus as claimed n claim 31, in which the means for determining comprises:

a) means for modelling P al the apparent blade penetration depth given by P - R 9 a 5N where R is the rate of penetration, and N is the bit rotary speed; b) means for using P a to determine 0, the internal friction angle, from the relationship P a to Pt, theoretical blade penetration depth, given by 2W(X-X R) cot(#+p) (1-sin) P t XD[(r ( 1+sini) + 2(CAcos(f sinf) 1 P m where W is the weight on the bit; A is the blade backrake angle; X is the total number of cutters on the blade; X R is the number of "redundant" cutters; D is the diameter of the drilled hole; cr m is the mud hydrostatic pressure; C is the formation cohesive strength; and P is the formation pore pressure, whereby, the internal friction angle, is iteratively computed by the apparatus by testing values so that the absolute value of P a - P t is less than or equal to e is a predetermined tolerance.

e, where

34. Apparatus as claimed in claim 32 or claim 33, further comprising the means for sensing R, the rate of penetration and N, the bit rotary speed.

35. Apparatus as claimed in claim 32 or claim 33, further comprising the means for calculating A and K, calibration transform coefficients, which account for cohesive strength of formation for the borehole being drilled, comprising:

a) means for determining the porosity of the formation versus depth, R1, analyzing formation samples from nearby wells; b) means for plotting the formation porosity versus depth, R1, for the nearby wells; C) means for plotting porosity versus depth, R21 calculated by the uncalibrated drilling model; and d) means for correlating R 2 to R 1 by empirically determining vlaues for A and K so that R 21 on average, substantially traces R1, where the cohesive strength of formation is taken into account by the drilling model.

36. Apparatus as claimed in claim 35, in which said means determines the porosity of the formation versus depth, R1, by analyzing the nuclear magnetic resonance properties of formation samples from nearby wells.

37. Apparatus as claimed in claim 32 or 33, further comprising means for calculating A and K, calibration transform coefficients which account for cohesive strength of formation for the borehole being drilled, comprising:

a) means for measuring and plotting porosity versus depth, R1, by testing samples from the borehole being drilled; b) means for plotting porosity versus depth, R 21 calculated by the uncalibrated drilling model; and c) means for correlating R 2 to R, by empirically determining values for A and K so that R 21 on average, traces R,, where the cohesive strength of formation is taken into account by the drilling model.

38. Apparatus as claimed in claim 37, 4n which said means measures the porosity by testing the nuclear magnetic resonance properties of samples from the borehole as it is being drilled.

39. Apparatus as claimed in any of claims 35 to 38, further comprising means for calculating cohesive index, C index' comprising:

a) means for plotting void ratio, 1/1-1, versus the coefficient of friction, tan, fl(o), calculated by the drilling model when cohesive strength of formation is zero; b) means for plotting void ratio, 1/1-6, versus the coefficient of friction, tan, A(c), calculated by the calibrated drilling model; and c) means for calculating C index by empirically evaluating the horizontal difference between the zero cohesive strength plot, P(o), and the calibrated plot, P (c), for a particular value of the void ratio, 6/1-6.

40. Apparatus as claimed in claim 39, further comprising the means for calculating the fracture probability, pr(frac), of the formation, comprising:

a) means for determinig the cohesive index rating, C R' of the formation from the relationship given by j 1 C C max C index max where C max is the largest statistically significant value for the C index; b means for determining the porosity rating, 6,, of the formation from the relationship given by d ave max &ave where & d' is equal - to the calibrated drilling model porosity, I ave is equal to the statistically weighted average porosity, and 6 max is equal to the statistically significant maximum porosity of the formation; and a) means for determining fracture probability, pr(frac), from the relationship given by:

pr(frac) = C R (I R'

41. A method of determining, the internal friction angle of a formation in a borehole drilled by a drilling bit, substantially as hereinbefore described with reference to the accompanying drawings.

An apparatus for implementing a drilling model for iteratively determining 07 the internal friction angle, of a formation in a borehole drilled by a bit substantially as hereinbefore described with reference to, and as illustrated in, the accompanying drawings.

43. A computer-implemented drilling model method for determining porosity, 5, of a formation in a borehole, substantially as hereinbefore described with reference to the accompanying drawings.

- 44

44. Apparatus for implementing a drilling model for determining the porosity 1, of strata in a borehole, substantially as hereinbefore descrbed with reference to, and as illustrated in, the accompanying drawings.

ir Published 1988 at The Patent Office, State House, 6671 gh Holborn, London WClR 4TP. Further copies maybe obtained from The Patent Office, Sales Branch, St Mary Cray, Orpington, Kent BR5 3RD. Printed by Multiplex techniques ltd, St Mary Cray, Kent. Con. 1/87.