EP0293767A2

EP0293767A2 - Computer-controlled model for determining internal friction angle, porosity, and fracture probability

Info

Publication number: EP0293767A2
Application number: EP19880108441
Authority: EP
Inventors: John Duncan Macpherson; Alun Howard Whittaker
Original assignee: Exploration Logging Inc
Current assignee: Baker Hughes Oilfield Operations LLC
Priority date: 1987-06-03
Filing date: 1988-05-26
Publication date: 1988-12-07
Also published as: NO882403D0; GB8811936D0; GB2205421A

Abstract

An apparatus and method for determining three characteristics of strata cut by a borehole on a "real time" basis, namely, the internal friction angle φ , porosity Φ, and fracture probability pr(frac). To calculate the internal friction angle φ, the invention simulates apparent and theoretical model formulae of a chosen physical characteristic 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 φ. The theoretical model formula is set equal to the apparent model formula, and the internal friction angle ₀ is interatively determined. The internal friction angle φ is then used to calculate porosity from the relationship given by

The parameters A and K are calibration transform coefficients calibrated from values of "average" porosity as determined by several nearby wells. The facture probability of the formation is calculated from the relationship given by

where CR is the cohesive rating and ΦR is the porosity rating.

Description

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. Specifically, 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.

SUMMARY 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 (M_a) are known values, an actual value for M_a can be determined. The theoretical model formula (M_t), which is a function of the internal friction angle 0, is set equal to M_a, and the internal friction angle 0 is iteratively determined so that the absolute value of M_a-M_t(0) is less than or equal to ±e, where e is a specified tolerance acceptable for field work. The value calculated for the internal friction angle 0 is then used to calculate porosity from the relationship given by
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, Φ/1-Φ, versus the coefficient of friction, tan0, 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 c^pR). Fracture probability, pr(frac), is then calculated from the relationship given by
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.

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, tan0);
FIG. 8 is a plot of porosity, cp, versus depth, for the NMR technique and for the uncalibrated drilling model;
FIG. 9 is a plot of the void ratio, Φ/1-Φ, versus coefficient of friction, tan0, 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 where a 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_oe^θtanφ.

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 rotary 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 drill 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 varies. 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 it 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. The 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=WOS-WOB. 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 signal, which varies with the weight on the hook 20.
Another sensor 62 detects the rotary speed of the rotary table 26. The sensor 62 consists of 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 to the sensor 62. By tracking the time that the magnet 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 is 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 * (DbiDc) (tooth impacts per revolution of the bit).

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

3n * (D_b/D_c)*Va*N, ins^3/min.

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

(_πD_b ²/4)(12R/60), ins^3/min.

Therefore, the apparent fracture volume per tooth impact is:

V_a ⁼ π *D_c*D_b*R/(60*N*n), ins³.

D_c, D_b, 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:
where
The derivations for these formulae are discussed in Appendices A and B, and both of the above formulae have been empirically tested for their accuracy.
In the above formula, V_t equals the theoretical volume of rock fractured per bit tooth impact, which is represented in ins , 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, 6_m equals mud hydrostatic pressure in pounds per square inch (psi), ó_P 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 supplied 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 V_a, then there is some internal friction angle 0, so that:
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, 0test, is then calculated at 118 by:
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
then the test value for the internal friction angle is placed in the relevant boundary value, either Ø_lowor 0 _max. That is, if the difference V_t- V_a calculated at 122 is greater than e set at 126, then the Ø_min boundary is set at 130 equal to Ø_test(130). On the other hand, if the difference between V_t- V_a 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 0test values are tested, the value for V_t(Ø)_test gets closer to the value for V_a. 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, PDCiDiamond 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 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/Diamond 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/Diamond 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:
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:
By placing this expression in equation (1) above, the volume of formation cut per minute is:
On the other hand, for a bit which advances a vertical distance Y inches in one revolution, the following formula represents the volume (ins3/rev) removed per revolution:
If the bit makes N revolutions per minute, then the volume (ins³/min) cut per minute is given by:
By equating equations (5) and (3), Y, the bit advance per revolution is:
ins/rev. where Y is the bit advance per revolution (ins/rev), R is the rate of penetration (ftlhr), 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:
in_/rev, 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 theoretical blade penetration depth is represented by the following formula:
A derivation of the above formula can be found in Appendix C. In the above formula, P_t is the theoretical blade penetration depth, which is represented in ins/rev, W is the weight on the bit, β is the blade backrake angle, X is the total number of cutters on the bit, X_R 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, ó_M is the mud hydrostatic pressure, C is the formation cohesive strength, and 6p 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 updated continuously 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 reduce the difference between the theoretical and apparent blade penetration depths to an insignificant amount. Specifically, this may be expressed mathematically in the following way. The theoretical blade penetration depth is represented by P_t, and it is expressed as a function, P_t(Ø), of the internal friction angle 0. Then, if the apparent blade penetration depth is P_a, there is some interval friction angle 0, so that:
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, V_a is replaced with P_a, and V_t(Ø) is replaced with P_t(Ø).
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, Ø_test, is then calculated at 118 by
and then the theoretical penetration depth is calculated at t20 using 7_test(118). If the difference between the theoretical penetration depth and the apparent penetration depth do not satisfy the tolerance equation
then a new value for 0test is determined by the interval halving algorithm previously described. Once the interval halving routine has determined a value for 0test 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:
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:
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
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 techniques^R1, 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 R1 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 placed into :aula Ae^-KtanØ, where A and K are still uncalibrated. An uncalibrated porosity is calculated from the expression Ae^-KtanØ. 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 is 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 "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 NMR 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,
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
where tanØα_M = internal friction angle calculated by drilling model assuming C is to be zero; tan0F = actual internal friction angle of the formation;
C = cohesive strength of the formation; and Ò_N - 6p = normal stress across a failure plane in the formation.
Referring to FIG. 9, a plot of the void ratio, Φ/1-Φ, versus the coefficient of friction, tan0, for the case when the cohesive strength of the formation is zero (β(o)), and a similar plot by calibrated drilling model (β-(c)), are shown. As shown in the graph, values for the cohesive index, "C/(O_N- Op)" are empirically determined on a "real time" basis by empirically evaluating the horizontal difference between the zero cohesive strength plot, ₆(o), and the calibrated plot, f3(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:
where C_R is a continuously changing cohesive index rating, scaled to a scaling ratio (Si) between 0 and 1 by the following formula:
where C_max is a maximum value for Cmdex. As the drilling model produces calibrated plots of void ratio, Φ/1-Φ, versus the coefficient of friction, tan0, 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. The scaling ratio has a maximum value of 1 for formations with no cohesive strength, and a minimum value of zero for formations where Cin_dex > C_max. The scaling ratio (Si) determines a rating for cohesive index so that the lower the cohesive index. the greater the rating becomes.
Φ_R is a continuously changing porosity rating of a formation scaled to a scaling value (Sz) between 0 and 1 by the following formula:
where Φ_d is equal to the porosity determined by the calibrated drilling model (FIG. 9), Φ_ave is a statistically weighted average value for porosity determined by the drilling model, and Φ_max is the largest statistically significant value for porosity determined by the drilling model. Φ_ave and Φ_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. The rating for porosity has a maximum of one for the highest porosity rating, and a minimum of zero for formations where Φ_R < 0.
C_R and Φ_R 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.
The invention has been described in an exemplary and preferred embodiment, but is not limited thereto. Those skilled in the art will recognize that a number of additional modifications and improvements can be made to the invention without departure from the essential spirit and scope. Therefore, the invention is not limited by the above disclosure, but only by the claims.

Claims

1. A computer-implemented drilling model for determining 0, the internal friction angle, of a formation in a borehole drilled by a drill bit comprising the steps of:

a) modeling V_a, the apparent fracture volume of a bit tooth by a formula where each parameter in the formula is a known constant or varying value;

b) sensing the varying values for the V_a formula with sensory equipment;

c) modeling V_t, the theoretical fracture volume fracture of a bit tooth by a formula where Ø, the internal friction angle, is the only unknown parameter of the formula, and any other parameter is a constant or varying value which can be sensed by a sensor;

d) sensing the varying values for V_t formula with sensory equipment; and

e) determining 0, the internal friction angle, of a bit by setting the V_a, apparent fracture volume equal to V_t, the theoretical volume fracture,

whereby the internal friction angle is iteratively computed by the computer, testing values of 0, so that the absolute value of V_a- V_t is less than or equal to e, where e is a predetermined tolerance.

2. The method of determining 0, internal friction angle, of claim 1 further comprising the step of computing porosity, Φ, from the relationship given by

where the constants A and K are calibration coefficients which take into account the cohesive strength of the formation.

3. The method of determining Φ, porosity, of claim 2 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, R,, for the nearby wells;

c) plotting porosity versus depth, R₂, calculated by the uncalibrated drilling model; and

d) correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, substantially traces Ri, whereby the cohesive strength of formation is taken into account by the drilling model.

4. The method of determining Φ, porosity, of claim 2 further comprising the step of calculating A and K, calibration transform coefficients, which account for the 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 the nuclear magnetic resonance properties of formation samples from nearby wells;

b) plotting the formation porosity versus depth, Ri, for the nearby wells;

d) correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, substantially traces Rt, whereby the cohesive strength of formation is taken into account by the drilling model.

5. The method of determining A and K, calibration transform coefficients, of claim 3 or 4 further comprising the step of calculating cohesive index, C_index, comprising the steps of:

a) plotting void ratio, Φ/1-Φ, versus the coefficient of friction, tan0, β(o), calculated by the drilling model when cohesive strength of formation is zero;

b) plotting void ratio, Φ/1-Φ, versus the coefficient of friction, tan0, β(c), calculated by the calibrated drilling model; and

c) calculating C_index by empirically evaluating the difference between the zero cohesive strength plot, β(o), and the calibrated plot, β(c), for a particular value of the void ratio, Φ/1-Φ.

6. The method of determining C_index, cohesive index, of claim 5 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

where C_max is the largest statistically significant value for the C_index;

b) determining the porosity rating, Φ_R, of the formation from the relationship given by

where Φ_d is equal to the calibrated drilling model porosity, Φ_ave is equal to the statistically weighted average porosity, and Φ_max 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) ⁼ C_R * Φ_R.

7. The method of determining Φ, porosity, of claim 2 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 samples from the borehole as it is drilled;

b) plotting porosity versus depth, R₂, calculated by the uncalibrated drilling model; and

c) correlating R₂ to R, by empirically determining values for A and K so that R₂, on average, substantially traces R,, whereby the cohesive strength of formation is taken into account by the drilling model.

8. The method of determining Φ, porosity, of claim 2 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, Ri, by testing the nuclear magnetic resonance properties of samples from the borehole as it is being drilled;

c) correlating R₂ to R, by empirically determining values for A and K so that R₂, on average, substantailly traces Ri, whereby the cohesive strength of formation is taken into account by the drilling model.

9. The method of determining A and K, calibration transform coefficients, of claim 7 or 8 further comprising the step of calculating cohesive index, C_index, comprising the steps of:

a) plotting void ratio, Φ/1-Φ, versus the coefficient of friction, tanØ, β(o), calculated by the drilling model when cohesive strength of formation is zero;

c) calculating C_index by empirically evaluating the horizontal difference between the zero cohesive strength plot, β(o), and the calibrated plot, β(c), for a particular value of the void ratio, Φ/1-Φ.

10. The method of determining C_index, Cohesive index, of claim 9 further comprising the step of calculating the fracture probability, pr(frac), of the formation, comprising the steps of:

where C_max is the largest statistically significant value for the C._ndex;

d) determining fracture probability, pr(frac), from the relationship given by:

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

a) means for modeling V_a, the apparent fracture volume, by calculating V_a from a formula representing the apparent volume fracture of the bit,
where each parameter in the formula is a known value, and each known value is either a constant or a varying value;

b) means for sensing the varying values of the V_a formula with sensory equipment;

c) means for modeling V_t, the theoretical fracture volume, by calculating V_t from a formula representing the bit,

where 0, internal friction angle, is the only unknown variable of the formula, any other variable is a constant value or a varying value which can be sensed by a sensor; and

d) means for sensing the known varying variables of said V_t formula with sensory equipment: and

e) means for determining 0, internal friction angle, of the bit by setting the V_a actual volume fracture equal to the theoretical volume fracture,

whereby the internal friction angle is iteratively computed by the computer, testing values of Ø, so that the absolute value of V_a- V_t is less than or equal to e, where e is a predetermined tolerance.

12. The apparatus for determining 0, internal friction angle, in claim 11 further comprising the means for computing porosity from the relationship given by

_- 13. The apparatus for determining Φ, porosity, of claim 12 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, Ri, analyzing formation samples from nearby wells;

b) means for plotting the formation porosity versus depth, R_i, for the nearby wells;

c) means for plotting porosity versus depth, R₂, calculated by the uncalibrated drilling model; and

d) means for correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, substantailly traces Ri, where the cohesive strength of formation is taken into account by the drilling model.

14. The apparatus of determining Φ, porosity of claim 12 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 determining the porosity of the formation versus depth, Ri, by analyzing the nuclear magnetic resonance properties of formation samples from nearby wells;;

d) means for correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, substantially traces Ri, whereby the cohesive strength of formation be taken into account by the drilling model.

15. The method of determining A and K, calibration transform coefficients, of claim 13 or 14 further comprising the means for calculating cohesive index, C_index, comprising:

a) means for plotting void ratio, Φ/1-Φ, versus the coefficient of friction, tan0, β(o), calculated by the drilling model when cohesive strength of formation is zero;

b) means for plotting void ratio, Φ/1-Φ, versus the coefficient of friction, tan0, β(c), calculated by the calibrated drilling model; and

c) means for calculating C by empirically evaluating the horizontal difference between the zero cohesive strength plot, β(o), and the calibrated plot, β(c), for a particular value of the void ratio, Φ/1-Φ.

16. The apparatus for determining Cindex, cohesive index, of claim 15 further comprising the means for calculating the fracture probability, pr(frac), of the formation, comprising:

a) means for determining the cohesive index rating, C_R, of the formation from the relationship given by

where C_max is the largest statistically significant value for the C_index;

b) means for determining the porosity rating, Φ_R, of the formation from the relationship given by

where Φ_d is equal to the calibrated drilling model porosity, Φ_ave is equal to the statistically weighted average porosity, and is equal to the maximum porosity of the formation; and

c) means for determining fracture, pr(frac), from the relationship given by: pr(frac) ⁼ CR Φ_R.

17. The apparatus for determining Φ, porosity, of claim 12 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, Ri, by testing testing samples from the borehole as it is drilled;

b) means for plotting porosity versus depth, R_z, calculated by the uncalibrated drilling model; and

c) means for correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, substantially traces Ri, where the cohesive strength of formation is taken into account by the drilling model.

18. The apparatus of determining Φ, porosity of claim 12 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, Ri, by testing the nuclear magnetic resonance properties of samples from the borehole as it is being drilled;

b) means for plotting porosity versus depth, R₂, calculated by the uncalibrated drilling model; and

c) means for correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, substantially traces R₁, whereby the cohesive strength of formation is taken into account by the drilling model.

19. The apparatus for determining A and K, calibration transform coefficients, of claim 17 or 18 further comprising the means for calculating cohesive index, C_index, comprising:

c) means for calculating C_index by empirically evaluating the horizontal difference between the zero cohesive strength plot, β(o), and the calibrated plot, 8(c), for a particular value of the void ratio, Φ/1-Φ.

20. The apparatus for determining C_index, cohesive index, of claim 19 further comprising the means for calculating the fracture probability, pr(frac), of the formation, comprising:

where C_max is the largest statistically significant value for the C_index;

c) means for determining fracture probability, pr(frac), from the relationship given by:

21. A computer-implemented drilling model iterative method for determining Ø, the internal friction angle, of a formation in a borehole drilled by a drill bit comprising the steps of:

a) modeling P_a, the apparent blade penetration depth given by a formula where each parameter in the formula is a known constant or a varying value which can be sensed by a sensor:

b) sensing the varying values for the P_a formula with sensory equipment;

c) modeling P_t, the theoretical blade penetration depth given by a formula where 0, the internal friction angle, is the only unknown variable of the formula, and any other variable of the formula is a constant value or a varying value which can be sensed by a sensor;

d) sensing the varying value for the P_t formula with sensory equipment; and

e) determining 0, the internal friction angle, of a blade of the bit by setting the apparent blade penetration depth equal to P_t, the theoretical blade penetration depth,

whereby 0, the internal friction angle, is iteratively computed by the computer, testing values of 0, so that the absolute value of V_a- V_t is less than or equal to e, where e is a predetermined tolerance.

22. The method of determining 0, the internal friction angle, in claim 21 further comprising the step of computing porosity, Φ, from the relationship given by

23. The method of determining Φ, porosity, of claim 22 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, R_i, by analyzing formation samples from nearby wells;

b) plotting the formation porosity versus depth, Ri, for the nearby wells;

c) plotting porosity versus depth, R₂, calculated by the uncalibrated drilling model; and the method of determining Φ, porosity, of claim 22 further comprising the step of calculating A and K, calibration transform coefficients, which account for cohesive strength of formation in the borehole being drilled.

24. The method of determining Φ, porosity of claim 22 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 Ri, by analyzing the nuclear magnetic resonance properties of formation samples from nearby wells;

b) plotting the formation porosity versus depth, Ri, for the nearby wells;

d) correlating R₂ to R₁ by empirically determining the values for A and K so that R₂, on average, substantially traces R₁, whereby the cohesive strength of formation be taken into account by the drilling model.

25. The method of determining A and K, calibration transform coefficients, of claim 23 or 24 further comprising the step of calculating cohesive index, C_index, comprising the steps of:

b) plotting void ratio, Φ/1-Φ, versus the coefficient of friction, tan0, Φ(c), calculated by the calibrated drilling model; and

c) calculating C_index by empirically evaluating the horizontal difference between the zero cohesive strength plot, β(o), and the calibrated plot. β(c), for a particular value of the void ratio, Φ/1-Φ.

26. The method of determining C_index, cohesive index, of claim 25 further comprising the step of calculating the fracture probability, pr(frac), of the formation, comprising the steps of:

where C_max is the largest statistically significant value for the C_index;

c) determining fracture, pr(frac), from the relationship given by:

27. The method of determining Φ, porosity, of claim 22 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, Ri, by testing samples from the borehole as it is being drilled;

c) correlating R₂ to R, by empirically determining values for A and K so that R₂, on average, substantially traces Ri, where the cohesive strength of formation is taken into account by the drilling model.

28. The method of determining Φ, porosity, of claim 22 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 the nuclear magnetic resonance properties of samples from the borehole as it is being drilled;

c) correlating R₂ to R_i by empirically deter mining values for A and K so that R₂, on average, substantially traces Ri, whereby the cohesive strength of formation is taken into account by the drilling model.

29. The method of determining A and K, calibration transform coefficients, of claim 27 or 28 further comprising the step of calculating cohesive index, C_index, comprising the steps of:

30. The method of determining C_index, cohesive index, of claim 29 further comprising the step of calculating the fracture probability, pr(frac), of the formation, comprising the steps of:

where C_max is the largest statistically significant value for the C_index;

where Φ_d is equal to the calibrated drilling model porosity, Φ_ave is equal to the statistically weighted average porosity, and Φ_max is equal to the statistically significant maximum porosity of the formation: and

c) determining fracture, pr(frac), from the relationship given by:

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

a) means for modeling P_a, the apparent blade penetration depth, by calculating P_a from a formula representing the bit,

where each parameter in the formula is a known value or a varying value which can be sensed by a sensor;

b) means for sensing the varying values of the said Pa formula;

c) means for modeling P_t, the theoretical penetration depth, by calculating P_t from a formula representing the bit,

where 0, the internal friction angle, is the only unknown variable of the formula, and any other parameter is a constant value or a varying value which can be sensed by a sensor;

d) means for sensing the known varying values of said P_t formula; and

e) means for determining 0, the internal friction angle, of the bit by setting the P_a, the actual penetration depth equal to P_t, the theoretical penetration depth,

whereby the internal friction angle is iteratively computed by the computer, testing values of 0, so that the absolute value of P_a- P_t is less than or equal to e, where e is a predetermined tolerance.

32. The apparatus for determining 0, internal friction angle, in claim 31 further comprising the means for computing porosity, Φ, from the relationship given by

33. The apparatus for determining Φ, porosity, of claim 32 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:

d) means for correlating R₂ to R, by empirically determining values for A and K so that R₂, on average, substantially traces Ri, where the cohesive strength of formation is taken into account by the drilling model.

34. The apparatus of determining Φ, porosity of claim 32 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 determining the porosity of the formation versus depth, R₁, by analyzing the nuclear magnetic resonance properties of formation samples from nearby wells;

b) means for plotting the formation porosity versus depth, R₁, for the nearby wells;

35. The apparatus for determining A and K, calibration transform coefficients, of claim 33 or 34 further comprising the means for calculating cohesive index, C_index, comprising:

c) means for calculating C_index by empirically evaluating the horizontal difference between the zero cohesive strength plot, β(o), and the calibrated plot, β(c), for a particular value of the void ratio, Φ/1-Φ.

36. The apparatus for determining C_index, cohesive index, of claim 35 further comprising the means for calculating the fracture probability, pr(frac), of the formation, comprising:

where C_max is the largest statistically significant value for the C_index;

c) means for determining fracture, pr(frac), from the relationship given by:

37. The apparatus for determining Φ, porosity, of claim 32 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 plotting porosity versus depth, Ri, by testing the nuclear magnetic resonance properties of cuttings for the borehole as it is being drilled;

c) means for correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, traces R₁, where the cohesive strength of formation is taken into account by the drilling model.

38. The apparatus of determining Φ, porosity of claim 32 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 porosity versus depth, R₁, by testing the nuclear magnetic resonance properties of samples from the borehole as it is being drilled;

b) means for plotting porosity versus depth R₂, calculated by the uncalibrated drilling model; and

c) means for correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, substantially traces Ri, whereby the cohesive strength of formation is taken into account by the drilling model.

39. The apparatus for determining A and K, calibration transform coefficients,, of claim 37 or 38 further comprising the means for calculating cohesive index, C_index, comprising:

40. The apparatus for determining C_index, cohesive index, of claim 39 further comprising the means for calculating the fracture probability, pr(frac), of the formation, comprising:

where C_max is the largest statistically significant value for the C_index;

b) means for determining the porosity rating, D_R, of the formation from the relationship given by

where Φ_d is equal to the calibrated drilling model porosity Φ_ave is equal to the statistically weighted average porosity, and Φ_max is equal to the statistically significant maximum porosity of the formation; and

41. A computer-implemented drilling model iterative method for determining 0, the internal friction angle, of a formation in a borehole drilled by a drill bit, comprising the steps of:

a) modeling an apparent model formula (M_a) of a drill bit operating within a borehole;

b) modeling a theoretical model formula (M_t) of a drill bit operating within a borehole,

where the only unknown parameter of the theoretical model formula is 0, the internal friction angle, of formation, all other parameters are known constant values or varying values which can be sensed by sensors;

c) sensing the known varying values for both the actual and theoretical models of a drill bit in operation; and

d) iteratively computing 0, the internal friction angle, so that the absolute value of M_a- M_t(Ø) is less than or equal to e,

where M_a, the apparent model formula of the drill bit, is a calculated value, M_t(Ø), the theoretical model of the drill bit, is a function of 0, the internal friction angle, and e is a predetermined tolerance.

42. The method of determining 0, internal friction angle, in claim 41 further comprising the step of computing Φ, porosity from the relationship given by

where the constants A and K are calibration coefficients, which take into account the cohesive strength of the formation.

43. The method of determining Φ, porosity, of claim 42 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, Ri, by analyzing formation samples from nearby wells;

b) plotting the formation porosity versus depth, Ri, for the nearby wells;

d) correlating R₂ to R, by empirically determining values for A and K so that, R₂, on average, substantially traces Ri, whereby the cohesive strength of formation is taken into account by the drilling model.

44. The method of determining Φ, porosity of claim 42 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:

b) plotting the formation porosity versus depth, R₁, for the nearby wells;

d) correlating R₂, to R₁ by empirically determining the values for A and K so that. R₂, on average, substantially traces R₁, whereby the cohesive strength of formation be taken into account by the drilling model.

45. The method of determining A and K, calibration transform coefficients, of claim 43 or 44 further comprising the step of calculating cohesive index, C_index, comprising the steps of:

46. The method of determining C_index, cohesive index, of claim 45 further comprising the step of calculating the fracture probability, pr(frac), of the formation, comprising the steps of:

where C_max is the largest statistically significant value for the C_index;

c) determining fracture probability, pr(frac), from the relationship given by:

47. The method of determining Φ, porosity, of claim 42 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, Ri, by testing samples from the borehole as it is drilled;

48. The method of determining Φ, porosity, of claim 42 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, Rt, by testing the nuclear magnetic resonance properties of samples from the borehole as it is being drilled;

c) correlating R₂ to Ri by empirically determining values for A and K so that R₂, on average, substantially traces Ri, whereby the cohesive strength of formation is taken into account by the drilling model.

49. The method of determining A and K, calibration transform coefficients, of claim 47 or 48 further comprising the step of calculating cohesive index, C_index, comprising the steps of:

c) calculating Cin_dex by empirically evaluating the horizontal difference between the zero cohesive strength plot, β(o), and the calibrated plot, β(c), for a particular value of the void ratio, Φ/1-Φ.

50. The method of determining C_index, cohesive index, of claim 49 further comprising the step of calculating the fracture probability, pr(frac), of the formation, comprising the steps of:

where C_max is the largest statistically significant value for the Cin_dex;

c) determining fracture, pr(frac), from the relationship given by:

51. 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 modeling an apparent model formula (M_a) of a drill bit operating within a borehole;

b) means for modeling 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 0, 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 0, the internal friction angle, from the relationship so that the absolute value of M_a- M_t(Ø) is less than or equal to e,

where M_a, the actual model of the drill bit, is a calculated value, M_t(Ø) the theoretical model of the drill bit, is a function of 0, the internal friction angle, and e is a predetermined tolerance.

52. The apparatus for determining Ø, internal friction angle, in claim 51 further comprising the means for computing, Φ, porosity from the relationship given by

53. The apparatus for determining Φ, porosity, of claim 52 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:

54. The apparatus of determining Φ, porosity of claim 52 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 determining the porosity of the formation versus, R₁, by analyzing the nuclear magnetic resonance properties of formation samples from nearby wells;

d) means for correlating R₂ to R, by empirically determining values for A and K so that R₂, on average, substantially traces Ri, whereby the cohesive strength of formation is taken into account by the drilling model.

55. The apparatus for determining A and K, calibration transform coefficients, of claim 53 or 54 further comprising the means for calculating cohesive index, C_index, comprising:

56. The apparatus for determining C_index, cohesive index, of claim 55 further comprising the means for calculating the fracture probability, pr(frac), of the formation, comprising:

where C_max is the largest statistically significant value for the C_index;

57. The apparatus for determining Φ, porosity, of claim 52 further comprising the means for calculating A and K, calibration transform coefficients, which account for cohesive strength of formation for the borehole as it is being drilled, comprising:

a) means for measuring and plotting porosity versus depth, Ri, by testing testing samples from the borehole being drilled;

c) means for correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, subtantially traces R,, where the cohesive strength of formation is taken into account by the drilling model.

58. The apparatus for determining Φ, porosity of claim 52 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 porosity versus depth, Ri, by testing the nuclear magnetic resonance properties of samples from the borehole as it is being drilled;

c) means for correlating R₂ to R_i by empirically determining values for A and K so that R₂, on average, substantially traces R₁, whereby the cohesive strength of formation is taken into account by the driling model.

59. The apparatus for determining A and K, calibration transform coefficients, of claim 57 or 58 further comprising the means for calculating cohesive index, C_index, comprising:

60. The method of determining C_index, cohesive index, of claim 59 further comprising the means for calculating the fracture probability, pr(frac), of the formation, comprising the steps of:

where C_max is the largest statistically significant value for the C_index;

c) means for determining fracture, pr(frac), from the relationship given by:

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

a) determining 0, the internal friction angle, of the formation; and

b) using 0 to compute porosity, Φ, from the relationship given by

62. The method of determining Φ, porosity, of claim 61 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, R , by analyzing formation samples from nearby wells;

b) plotting the formation porosity versus depth, R_i, for the nearby wells;

c) plotting porosity versus depth, R₂, calculated by the uncalibrated drilling model.

63. The method of determining Φ, porosity of claim 61 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 R_i, by analyzing the nuclear magnetic resonance properties of formation samples from nearby wells;

b) plotting the formation porosity versus depth, R₁, for the nearby wells;

d) correlating R₂ to R₁ by empirically determining the values for A and K so that R₂, on average, substantially traces R₁, whereby the cohesive strength of formation is taken into account by the drilling model.

64. The method of determining A and K, calibration transform coefficients, of claim 62 or 63 further comprising the step of calculating cohesive index, Cindex, comprising the steps of:

65. The method of determining C_index, cohesive index, of claim 64 further comprising the step of calculating the fracture probability, pr(frac), of the formation, comprising the steps of:

where C_max is the largest statistically significant value for the C_index;

where Φ_d is equal to the calibrated drilling model porosity , Φ_ave is equal to the statistically weighted average porosity, and Φ_max is equal to the statistically significant maximum porosity of the formation; and

c) determining fracture probability, pr(frac), from the relationship given,by:

66. The method of determining Φ, porosity, of claim 61 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 samples from the borehole as it is being drilled;

c) correlating R₂ to Ri by empirically deter mining values for A and K so that R₂, on average, substantially traces R₁, where the cohesive strength of formation is taken into account by the drilling model.

67. The method of determining Φ, porosity, of claim 61 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:

c) correlating R₂ to R, by empirically determining values for A and K so that R₂, on average, substantially traces Ri, whereby the cohesive strength of formation is taken into account by the drilling model.

68. The method of determining A and K, calibration transform coefficients, of claim 66 to 67 further comprising the step of calculating cohesive index, C_index, comprising the steps of:

69. The method of determining C_index, cohesive index, of claim 68 further comprising the step of calculating the fracture probability, pr(frac), of the formation, comprising the steps of:

where C_max is the largest statistically significant value for the C_index;

where Φ_d is equal to the calibrated drilling model porosity, Φ_ave is equal to the statistically weighted average porosity, and Φ_max is equal to the statistically significant maximum porosity of the, formation; and

c) determining fracture probability, pr(frac), from the relationship given by:

70. An apparatus for implementing a drilling model for determining the porosity, Φ, of strata in a borehole comprising:

a) means for determining 0, the internal friction angle, of the compositional components within the strata; and

b) means for using 0, to determine porosity, Φ, from the relationship given by

where A and K are calibration coefficients which take into account the cohesive strength of the formation.

71. The apparatus for determining Φ, porosity, of claim 70 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;

c) means for plotting porosity versus depth, R₂, calculated by the uncalibrated drilling model.

72. The apparatus of determining o, porosity of claim 70 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 determining the porosity of the formation versus R₁, by analyzing the nuclear magnetic resonance properties of formation samples from nearby wells;

d) means for correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, substantially traces R , whereby the cohesive strength of formation is taken into account by the drilling model.

73. The apparatus for determining A and K, calibration transform coefficients, of claim 71 or 72 further comprising the means for calculating cohesive index, C_index, comprising:

c) means for calculating C_jndex by empirically evaluating the horizontal difference between the zero cohesive strength plot, β(o), and the calibrated plot, β(c), for a particular value of the void ratio, Φ/1-Φ.

74. The apparatus for determining Cindex, cohesive index, of claim 73 further comprising the means for calculating the fracture probability, pr(frac), of the formation, comprising:

where C_max is the largest statistically significant value for the C_index;

where 4_d is equal to the calibrated drilling model porosity, Φ_ave is equal to the statistically weighted average porosity, and Φ_max is equal to the statistically significant maximum porosity of the formation; and

75. The apparatus for determining Φ, porosity, of claim 70 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, Ri, by testing samples from the borehole being drilled;

c) means for correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, substantially traces R₁, where the cohesive strength of formation is taken into account by the drilling model.

76. The apparatus of determining Φ, porosity of claim 70 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 porosity versus depth, Ri, by testing nuclear magnetic resonance properties of samples from the borehole as it is being drilled;

c) means for correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, substantially traces R_i, whereby the cohesive strength of formation is taken into account by the drilling model.

77. The apparatus for determining A and K, calibration transform coefficients, of claim 75 or 76 further comprising the means for calculating cohesive index, C_index, comprising:

c) means for calculating C_index by empirically evaluating the horizontal difference between the zero cohesive strength plot, 8(o), and the calibrated plot, β(c), for a particular value of the void ratio, Φ/1-Φ.

78. The apparatus for determining C_index, cohesive index, of claim 77 further comprising the means for calculating the fracture probability, pr(frac), of the formation, comprising:

where C_max is the largest statistically significant value for the C_index;

79. A computer-implemented drilling model iterative method for determining porosity Φ of a formation in a borehole, drilled by a three-cone roller bit, comprising the steps of:

a) modeling V_a, the apparent fracture volume by a single tooth of the bit, by

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 0, the internal friction angle, from the relationship V_a to V_t, theoretical fracture volume, given by:

where b is the width of each tooth of the Rollercone bit, and where p is determined by

where W is the weight on the bit, o is the tooth included angle, 6_m is the mud pressure, 6p is the pore pressure, and M is the number of teeth on formation at any one time,

whereby the internal friction angle is iteratively computed by the computer testing, values of 0 , so that the absolute value of V_a- V_t is less than or equal to e, where e is a predetermined tolerance; and

c) using the 0, just determined, to compute porosity, Φ, from the relationship given by

80. The method of determining Φ, porosity in claim 79 further comprising the steps of sensing R, the rate of penetration, and N, the bit rotary speed.

81. The method of determining Φ, porosity, of claim 79 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:

b) plotting the formation porosity versus depth, R, for the nearby wells;

d) correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, traces R₁, whereby the cohesive strength of formation is taken into account by the drilling model.

82. The method of determining Φ, porosity of claim 79 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:

b) plotting the formation porosity versus depth, Ri, for the nearby wells;

d) correlating R₂ to R, by empirically determining the values for A and K so that R₂, on average, substantially traces R₁, whereby the cohesive strength of formation is taken into account by the drilling model.

83. The method of determining A and K, calibration transform coefficients, of claim 81 or 82 further comprising the step of calculating cohesive index, C_index, comprising the steps of:

b) plotting void ratio, Φ/1-Φ, versus the coefficient of friction, tan0, 8(c), calculated by the calibrated drilling model; and

84. The method of determining C_index, cohesive index, of claim 83 further comprising the step of calculating the fracture probability, pr(frac), of the formation, comprising the steps of:

where C_max is the largest statistically significant value for the C_index;

c) determining fracture probability, pr(frac), from the relationship given by:

85. The method of determining Φ, porosity, of claim 79 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_t, by testing samples from the borehole as it is drilled;

c) correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, substantially traces Ri, where the cohesive strength of formation is taken into account by the drilling model.

86. The method of determining Φ, porosity, of claim 79 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:

c) correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, substantially traces R_I, whereby the cohesive strength of formation is taken into account by the drilling model.

87. The method of determining A and K, calibration transform coefficients, of claim 86 further comprising the step of calculating cohesive index, C_index, comprising the steps of:

88. The method of determining C_index, cohesive index, of claim 87 further comprising the step of calculating the fracture probability, pr(frac), of the formation, comprising the steps of:

where C_max is the largest statistically significant value for the C_index;

c) determining fracture probability, pr(frac), from the relationship given by:

89. A computer-implemented drilling model iterative method for determining the porosity. Φ, of a formation in a borehole drilled by a diamond bit, comprising the steps of:

a) modeling P_a, the apparent blade penetration depth, given by

where R is the rate of penetration, and N is the bit rotary speed;

b) using P_a to determine 0, the internal friction angle, from the relationship P_a to P_t, theoretical blade penetration depth, given by

where W is the weight on the bit; β is the blade backrake angle; x is the total number of cutters on the blade; x_e is the number of "redundant" cutters; D is the diameter of the drilled hole; 6_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 so that the absolute value of P_a- P_I is less than or equal to e, where e is a predetermined tolerance;

where the constants A and K are calibrated coefficients which take into account the cohesive strength of the formation.

90. The method of determining porosity in claim 89 further comprising the step for sensing R, the rate of penetration, and N, the bit rotary speed.

91. The method of determining Φ, porosity, of claim 89 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) detemining the porosity of the formation versus depth, Ri, by analyzing formation samples from nearby wells;

b) plotting the formation porosity versus depth, Ri, from the nearby wells;

c) plotting porosity versus depth, R₂ calculated by the uncalibrated drilling model; and

92. The method of determining Φ, porosity of claim 89 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, R₁, by analyzing the nuclear magnetic resonance properties of formation samples from nearby wells;

b) plotting the formation porosity versus depth, Ri, for the nearby wells;

d) correlating R₂ to R, by empirically determining the values for A and K so that, on average, substantially traces R₁, whereby the cohesive strength of formation be taken into account by the drilling model.

93. The method of determining A and K, calibration transform coefficients, of claim 91 or 92 further comprising the step of calculating cohesive index, C_index, comprising the steps of:

a) plotting void ratio, Φ/1-Φ, versus the coefficient of friction, tan∅, β(o), calculated by the drilling model when cohesive strength of formation is zero;

c) calculating C_inciex by empirically evaluating the horizontal difference between the zero cohesive strength plot, β(o), and the calibrated plot, β(c), for a particular value of the void ratio, Φ/1-Φ.

94. The method of determining C_index, cohesive index, of claim 93 further comprising the step of calculating the fracture probability, pr(frac), of the formation, comprising the steps of:

where C_max is the largest statistically significant value for the C_index;

c) determining fracture, pr(frac), from the relationship given by:

95. The method of determining Φ, porosity, of claim 89 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, Ri, by testing sampler from the borehole as it is being drilled;

b) plotting porosity versus depth, R₂, calculated by the uncalibrated drilling model;

c) correlating R₂ to R, by empirically determining values for A and K so that R₂, on average, substantialy traces R_t, where the cohesive strength of formation is taken into account by the drilling model.

96. The method of determing Φ, porosity of claim 89 further comprising the step of calculating A and C, calibration transform coefficients which account for cohesive strength of formation for the borehole being drilled, comprising the steps of:

b) plotting porosity versus depth, Rz, calculated by the uncalibrated drilling model; and

c) correlating R₂ to Ri, by empirical determining values for A and C so that R₂, an average substantially traces R₁, whereby the cohesive strength of formation is taken into account by the drilling model.

97. The method of determining A and K, calibration transform coefficients, of claim 95 or 96 further comprising the step of calculating cohesive index, C_index, comprising the steps of:

98. The method of determining Ci_ndex, cohesive index, of claim 97 further comprising the step of calculating the fracture probability, pr(frac), of the formation, comprising the steps of:

where Cmax is the largest statistically significant value for the C_index;

c) determining fracture, pr(frac), from the relationship given by:

99. An apparatus for implementing a drilling model, for determining porosity Φ of strata in a borehole, drilled by a three-cone roller bit, comprising:

a) means for modeling V_a, the apparent fracture volume, by a single tooth of the bit, by calculating

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) means for using the V_a to determine the 0, internal friction angle, calculated from the relationship V_a, to V_t, theoretical fracture volume

where W is the weight on the bit, 0 -is the tooth included angle, Ó_m is the mud pressure, 6p is the pore pressure, and M is the number of teeth on formation at any one time,

whereby the internal friction angle is iteratively computed by the computer, testing values of 0, so that the absolute value of V_a- V_t is less than or equal to e, where e is a predetermined tolerance; and

c) means for using the 0to compute porosity Φ from the relationship given by

100. The apparatus for determining porosity in claim 99 further comprising the means for sensing R, the rate of penetration and N, the bit rotary speed.

101. The apparatus for determining Φ, porosity, of claim 99 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 detemining the porosity of the formation versus depth, Rt, analyzing formation samples from nearby wells;-

b) means for plotting the formation porosity versus depth, Ri, for the nearby wells;

d) means for correlating R₂ to R₁ by empir ically determining values for A and K so that R₂, on average, substantially traces R₁, where the cohesive strength of formation is taken into account by the drilling model.

102. The apparatus of determining Φ, porosity of claim 99 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:

d) means for correlating R₂ to R₁ by empirically determining values for A and K so that, R₂, on average, substantially traces Ri, whereby the cohesive strength of formation is taken into account by the drilling model.

103. The apparatus for determining A and K, calibration transform coefficients of claim 101 or 102 further comprising the means for calculating cohesive index, C_index, comprising:

104. The apparatus for determining C_index, cohesive index, of claim 103 further comprising the means for calculating the fracture probability, pr(frac), of the formation, comprising:

where C_max is the largest statistically significant value for the Cindex;

c) means for determining fracture, pr(frac), from the relationship given by:

105. The apparatus for determining Φ, porosity, of claim 99 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, Ri, by testing samples from the borehole as it is drilled;

c) means for correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, substantially traces R_i, where the cohesive strength of formation is taken into account by the drilling model.

106. The apparatus of determining Φ, porosity of claim 99 further comprising of 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 porosity versus depth, R₁, bu testing the nuclear magnetic resonance properties of samples from the borehole as it is being drilled.

c) means for correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, substantailly traces Ri, whereby the cohesive strength of formation is taken into account by the drilling model.

107. The apparatus for implementing a drilling model for determining A and K, calibration transform coefficients, of claim 105 to 106 further comprising the means for calculating cohesive index, Cin_dex, comprising:

108. The apparatus for determining C_index, cohesive index, of claim 107 further comprising the means for calculating the fracture probability, pr(frac), of the formation, comprising:

where C_max is the largest statistically significant value for the C_index;

109. An apparatus for determining the porosity Φ, of a formation in a borehole drilled by a diamond bit, comprising:

a) means for modeling P_a, the apparent blade penetration depth, given by

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 P_t, theoretical blade penetration depth, given by

where W is the weight on the bit, β is the blade backrake angle, X is the total number of cutters on the blade, x_e is the number of "redundant" cutters, D is the diameter of the drilled hole, 6_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 apparatus by testing values so that the absolute value of P_a- P_r is less than or equal to e, where e is a predetermined tolerance;

c) means for using the 0, just determined, to compute porosity, Φ, from the relationship given by

110. The apparatus for determining porosity in claim 109 further comprising the means for sensing R, the rate of penetration and N, the bit rotary speed.

111. The apparatus for determining Φ, porosity, of claim 109 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:

d) means for correlating R₂ to R, by empirically determining values for, A and K so that R₂, on average, substantially traces R,, where the cohesive strength of formation is taken into account by the drilling model.

112. The apparatus of determining Φ, porosity of claim 109 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:

d) means for correlating R₂ to R_i by empirically determining values for A and K so that R₂, on average, substantially traces Ri, whereby the cohesive strength of formation be taken into account by the drilling model.

113. The apparatus for determining A and K, calibration transform coefficients, of claim 101 or 112 further comprising the means for calculating cohesive index, C_index, comprising:

114. The apparatus for determining C_index, cohesive index, of claim 113 further comprising the means for calculating the fracture probability, pr(frac), of the formation, comprising:

where C_max is the largest statistically significant value for the C_index;.

where Φ_d is equal to the calibrated drilling model porosity, Φ_ave is equal to the statistically weighted average porosity, and Φ_max is equal to the statistically significant maximum porosity of the formation;

115. The apparatus for implementing a drilling model for determining Φ, porosity, of claim 109 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:

c) means for correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average, traces Ri, where the cohesive strength of formation is taken into account by the drilling model.

116. The apparatus of determining Φ, porosity of claim 109 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 porosity versus depth, R¹, by testing the nuclear magnetic resonance properties of samples from the borehole as it is being drilled;

b) means for plotting porosity versus depth R₂, calculated by the uncalibrated drilled model; and

c) means for correlating R₂ to R₁ by empirically determining values for A and K so that R₂, on average substantially traces Ri, whereby the cohesive strength of formation is taken into account by the drilling model.

117. The apparatus for determining A and K, calibration transform coefficients, of claim 115 or 116 further comprising the means for calculating cohesive index, C_index, comprising:

118. The apparatus for determining C_index, cohesive index, of claim 117 further comprising the means for calculating the fracture probability, pr(frac), of the formation, comprising:

where C_max is the largest statistically significant value for the C_index;