Patent Publication Number: US-2021190755-A1

Title: Method, apparatus and system for estimation of rock mechanical properties

Description:
BACKGROUND 
     An on-going challenge in the oil and gas industry is to obtain reliable mechanical properties of formations having anisotropic material compositions. Such formations can have a plurality of millimeter-scale layers laminated together with each layer composed of different types of materials (e.g., clay, silica, or organic matter) and thus each having different mechanical properties. Consequently, different core plug specimens can have mechanical properties that can differ by several orders of magnitude. For example, different core plug specimens obtained from a whole core with a spacing of tens of centimeters between core plug specimens may have significant variations in mechanical properties, since different laminations of material can be present in the different plug specimens. Thus, classical rock mechanical property analysis obtained from such whole core operations cannot be reliably used for formations with anisotropic mechanical properties. 
    
    
     
       BRIEF DESCRIPTION 
         FIG. 1A  presents a cross-sectional front view of an example apparatus embodiment of the disclosure; 
         FIG. 1B  presents a cross-sectional side view of the example apparatus shown in  FIG. 1A ; 
         FIG. 1C  presents a cross-sectional bottom view of the example apparatus shown in  FIG. 1A ; 
         FIG. 2  presents a perspective view sketch of an example system embodiment of the disclosure; 
         FIG. 2A  presents a perspective view sketch of an application fixture prior to placing the disc-shaped sample in the applicant portion to contact the shaped surface of the end cap of the application fixture; 
         FIG. 3  presents a flow diagram of selected steps of an example method embodiment of the disclosure; 
         FIG. 4  presents a perspective view of an example design of simulated force application fixtures to apply compressive stress to a simulated disc-shaped sample in accordance to the principles of the disclosure; 
         FIG. 5  presents a perspective view of example simulated axial displacement distributions of a simulated disc-shaped sample subjected to a simulated compressive stresses from simulated force application fixtures in accordance to the principles of the disclosure; and 
         FIG. 6  presents a plot of Young&#39;s modulus (E) versus compressive strength for disc-shaped samples of rock formations tested using a conventional system and method (squares), and tested using a prototype apparatus, system and method of the disclosure (triangles). 
     
    
    
     DETAILED DESCRIPTION 
     As part of the present invention, we recognized that prior testing methods often require the acquisition of multiple core specimens from formations with anisotropic material compositions to obtain just limited numbers of mechanical properties. In contrast, the approach of the present disclosure can obtain multiple different mechanical properties from a single core specimen. Since minimal specimen material is required (e.g., a single specimen), the effect of material composition heterogeneity in the formation of the measured mechanical properties can be minimized Consequently, the approach of the present disclosure provides a simple and economical way to more precisely obtain mechanical properties of any mechanically competent formation, including formations with anisotropic material compositions. 
     As further disclosed below, the apparatus, system and method of the disclosure facilitate compression testing of a single core plug specimen in multiple directions to detect rock formation anisotropic mechanical properties. In particular, multiple disc-shaped sample of the single core plug specimen can be sequentially compression tested at different angles, relative to an in situ force experienced by formation from which the core plug specimen was obtained, to evaluate the extent of heterogeneity in the mechanical properties. 
     In the drawings and descriptions to follow, like parts are typically marked throughout the specification and drawings with the same reference numerals, respectively. The drawn figures are not necessarily to scale. Certain features of this disclosure may be shown exaggerated in scale or in somewhat schematic form and some details of conventional elements may not be shown in the interest of clarity and conciseness. Specific embodiments are described in detail and are shown in the drawings, with the understanding that they serve as examples and that they do not limit the disclosure to only the illustrated embodiments. Moreover, it is fully recognized that the different teachings of the embodiments discussed, infra, may be employed separately or in any suitable combination to produce desired results. 
     Unless otherwise specified, any use of any form of the terms such as “press,” “connect,” “engage,” “couple,” “attach,” or any other term describing an interaction between elements is not meant to limit the interaction to direct interaction between the elements but include indirect interaction between the elements described, as well. In the following discussion and in the claims, the terms “including” and “comprising” are used in an open-ended fashion, and thus should be interpreted to mean “including, but not limited to.” Further, any references to “first,” “second,” etc. do not specify a preferred order of method or importance, unless otherwise specifically stated but are intended to designate separate elements. The various characteristics mentioned above, as well as other features and characteristics described in more detail below, will be readily apparent to those skilled in the art with the aid of this disclosure upon reading the following detailed description of the embodiments, and by referring to the accompanying drawings. 
     One embodiment of the disclosure is an apparatus for measuring mechanical properties of a rock formation (e.g., e.g., shale, sandstone or other rock formations for oil and gas production as familiar to those skilled in the pertinent arts).  FIGS. 1A, 1B and 1C  present a cross-sectional front, side and bottom views, respectively, of an example apparatus  100  embodiment of the disclosure. 
     With continuing reference to  FIGS. 1A-1C  throughout, embodiments of the apparatus  100  can comprise force application fixtures  105  configured to apply opposing compressive forces  107   a ,  107   b  to a disc-shaped sample  110  of a core plug specimen from the rock formation. Non-limiting examples of types of core plug specimens include rotary sidewall, regular, whole, outcrop or other core plugs familiar to those skilled in the pertinent arts. Each of the force application fixtures  105  can include an applicant portion  115  having an end cap  120 . The end cap  120  can have a shaped surface  125  that is configured to conform with a portion  130  of a non-planar side  132  of the disc-shaped sample  110  such that substantially an entire volume of the disc-shaped sample (e.g., at least about 90 percent of the total volume, and in some embodiments at least about 99 percent of the total volume) experiences a compressive stress when the opposing compressive forces  107   a ,  107   b  are applied to the portions  130  of the non-planar side  132  of the sample  110 . 
     Embodiments of the disc-shaped sample  110  can be substantially cylindrically shaped, e.g., a shape substantially corresponding to the shape of a core plug specimen from which the sample is derived from. For instance, some embodiment of the disc-shaped sample  110  can have a cylindrical shape, e.g., as defined by planar base ends  135  having substantially circular shaped perimeters where the diameter  140  of the base ends do not vary by more than about ±20 percent, and in some embodiments, not more than about ±10 percent, and in some embodiments not more than about ±1 percent, around the entire circumference of the base end  135 . 
     In some embodiments, the disc-shaped sample  110  can be further processed to have one or more flat faces on the side  132 , but, still retain a substantially cylindrically shape, with the base ends  135  having the substantially circular shaped perimeters. As a non-limiting example, the sample  110  can be processed by further cutting or grinding such that the base ends  135  have hexagonal, octagonal or decagonal shaped perimeters with the side  132  having 6, 8 or 12 flat faces, respectively distributed around the circumference of the base end  135 . 
     For any embodiments of the disc-shaped samples  110 , to facilitate the application of the compressive stress to the entire volume of the disc-shaped sample  110 , the shape of the end cap surfaces  125  can be configured to have a shape that is a negative or inverse of the shape of the portion  130  of the non-planar side  132  of the disc-shape sample  110  so as to wrap around and conformaly contact the portion  130  of the sample&#39;s side  132 . 
     For some embodiments, the surfaces  125  of the end caps  120  are configured to extend along the non-planar side  132  of the disc-shaped sample  110  (e.g., along the circular shaped side of a substantially cylindrically shaped sample), such that the end-to-end distance  145  of each of the end caps  120  is a value in a range from about 0.7 to about 0.9 times a diameter  140  of the disc-shaped sample  110 . 
     In some embodiments, having an end-to-end distance  145  equal to about 0.8 times the diameter  140 , represents a balance between avoiding the risk of the end caps  120  touching and damaging each other during stress testing versus avoiding the introduction of non-compressive stresses during testing. For instance, for some end cap designs where the end-to-end distance  145  is less than about 0.7 times the diameter  140  the application of the opposing compressive forces  107 ,  107   b , may undesirably produce a mixed-mode of mechanical failure comprising both compressive and shear stresses. For instance, for some end cap designs where the end-to-end distance  145  is greater than about 0.9 times the diameter  140  the application of the opposing compressive forces  107 ,  107   b , may undesirably result in the two end caps  120  contacting each other during the mechanical failure testing of the sample, resulting in damage to the end cap  120  or other parts of the apparatus  100 . 
     As illustrated in  FIGS. 1A-1C , embodiments of the end caps  120  of the applicant portions  115  of the force application fixtures  105  can have identically shaped surfaces  125 . However, in other embodiments, the end caps  120  could have differently shaped surfaces  125 , provided that the shaped surfaces  125  still conform to the portion  130  of a non-planar side  132  of the sample so that substantially the entire volume of the sample experiences the compressive stress when the opposing compressive forces  107   a ,  107   b  are applied. 
     As further illustrated in  FIGS. 1 a -1 c   , some embodiments of the force application fixtures  105  can each include a connection portion  150 , the connection portions  150  configured to attach the force application fixtures  105  to actuators  155  of a load frame device  160 . For instance, the connection portion  150  can include a threaded opening  165  to allow attachment to one of two opposing actuators  155  of a hydraulic load frame device  160 . One skilled in the pertinent arts would appreciate how the connection portion  150  could be adapted to have other connection mechanisms to the actuators  155 . 
     As also illustrated in  FIGS. 1A-1C , some embodiments of the force application fixtures  105  can each include notches  170  to hold in place end pins  172 , the end pins  172  being part of sensors of an extensometer device configured to measure axial displacements of the disc-shaped samples  110  during the application of the opposing compressive forces  107   a ,  107   b . For instance, each force application fixture  105  can include two notches, e.g., on opposite sides of the fixture  105 , shaped so that the end pins  172  are held in place during mechanical testing of the sample  110 . 
     As further illustrated in  FIG. 1B , some embodiments of the end caps  120  have a width  180  that is substantially equal to or greater than a thickness  185  of the disc-shaped sample  110 , the width  180  being in a dimension that is perpendicular to the dimension of the end-to-end distance. In some embodiments of the end caps  120 , the width  180  can be substantially equal to the thickness  185  of the disc-shaped sample  110  (e.g., equal to the thickness within about ±10 percent, or within about ±1 percent, in some embodiments). Configuring the end caps  102  to have a width  180  that is substantially equal to or greater than the sample&#39;s thickness  185  can facilitate applying the compressive stress to the entire volume of the disc-shaped sample  110 . For instance, for embodiments where the end caps  102  have a width  180  that is less than the sample&#39;s thickness  185 , there may be an increased risk of introducing non-compressive stresses into the sample (e.g., bending or shearing stresses) when the opposing compressive forces  107   a ,  107   b  are applied to the disc-shaped sample  110 . 
     In some embodiments, the end-to-end distance  145  of each of the end caps  120  is a value in a range from about 1.4 to about 2.5 times (and in some embodiments, about 1.4 to about 1.8 times) the width  180  of the end caps  120 , the width of the end caps being in the dimension that is perpendicular to a dimension of the end-to-end distance  145 . Having the end-to-end distance  145  equal to about 1.4 to about 1.8 times the width  180  of the end caps  120  can facilitate applying the compressive stress to the entire volume of the disc-shaped sample  110 , e.g., when the sample  110  has a diameter  140  to thickness  185  ratio of about 2:1. 
     For instance, consider samples  110  having thicknesses  185  of 0.5, 1 or 2 inches and diameters of 1, 2 or 4 inches, respectively. The width  180  of the end caps  120  can be configured to have a width  180  of at least about 0.5, 1 or 2 inches, respectively, and the surface  125  of end caps  120  can be configured such that the end-to-end distance  145  equals a value in the range from about 1.4 to about 2.5 times the width  180  (e.g., end-to-end distances  145  ranging from about 0.7 to 1.25 inches, 1.4 to 2.5 inches or 2.8 to 5.0 inches, respectively for widths  180  of 0.5, 1 or 2 inches, respectively). 
     Based upon the present disclosure, one skilled in the pertinent arts would appreciate how the end-to-end distance  145  and the width  180  of the end caps  120  could be adjusted depending upon the particular size and shape of the sample  110  to be tested, e.g., so as to ensure that substantially an entire volume of the disc-shaped sample experiences a compressive stress during the application of the opposing compressive forces  107   a ,  107   b . For example, if a cylindrically shaped sample  110  has a diameter  140  to thickness  185  ratio of about 4:1 or 1:1 then the end-to-end distance  145  and/or width  180  of the end caps  120  may be adjusted, e.g., as guided by simulations and experiments such further disclosed below, to ensure that substantially the whole volume of such samples  110  experience the compressive stress. 
     Another embodiment of the disclosure is a system for measuring mechanical properties of a rock formation.  FIG. 2  presents a perspective view of a sketch of an example embodiment of the system  200  of the disclosure.  FIG. 2A  presents a perspective view sketch of an application fixture  105  prior to placing the disc-shaped sample  110  in the applicant portion  115  to contact the shaped surface  125  of the end cap  120  of the application fixture  105 . The system  200  can include any of the embodiments of the apparatus  100  discussed in the context of  FIGS. 1A-1C . 
     With continuing reference to  FIGS. 1A-2A  throughout, the system  200  can include the force application fixtures  105  (e.g., two opposing fixtures) of the apparatus  100 , a load frame device  160 , and an extensometer device  210 . 
     The load frame device  160  can include actuators  155  (e.g., two opposing actuators) configured to apply opposing compressive forces  107   a ,  107   b  to a disc-shaped sample  110  of a core plug specimen. 
     The extensometer device  210  can include sensors  215 ,  217  (e.g., linear variable differential transformer sensors) to measure axial and lateral displacements (e.g., typically in in vertical and horizontal directions, respectively, when the device is oriented as shown in  FIG. 2 ) of the disc-shaped sample when the opposing compressive forces  107   a ,  107   b  are applied to the disc-shaped sample. For instance, some embodiments of sensors  215  can include end pins  172  configured to be coupled to the force application fixtures (e.g., physically resting inside of notches  170  in the side of the connection portion  150  of the fixture  105 ). The end pins  172  can be connected to arms  220  of the axial displacement sensor  215 , the arms  220  configured to record axial displacements of the sample  110  during compression stress testing. For instance, some embodiments of lateral displacement sensor  217  include one or more sensor arms  222  (e.g., a pair of arms  222  coupled to opposite base ends of the sample) configured to hold the base ends  135  of the disc-shaped sample  110  the lateral displacement sensors arms  222  are configured to measure the lateral displacement of the disc-shaped sample  110  when the opposing compressive forces  107   a ,  107   b  are applied. For instance, when the disc-shaped sample  110  has a thickness  185  of about 0.5 inches, then the lateral displacement sensor arms  222  can be configured to hold an about 0.5 inch thick sample. One skilled in the pertinent art would understand how the axial and lateral displacements of the disc-shaped sample  110  could be recorded by the sensors  215 ,  217  in the form of voltages, which are then transmitted (e.g., via data transmission lines  225 ,  227 ) to a recording and control module  230  of the extensometer device  210  for data processing. 
     As discussed above in the context of  FIGS. 1A-1C , the force application fixtures  105  are couplable to the actuators  155 , e.g., through a connection portion  150  of the fixture. As also discussed in the context of  FIGS. 1 a -1 c   , in some embodiments, each of the force application fixtures  105  includes an applicant portion  115 , having an end cap  120 . The end cap  120  has a shaped surface (e.g., surface  125 ) that is configured to conform with a portion  130  of a non-planar side  132  of the disc-shaped sample  110  such that substantially an entire volume of the disc-shaped sample experiences a compressive stress when the opposing compressive forces  107   a ,  107   b  are applied to the portions of the non-planar side  132 . 
     Another embodiment of the disclosure is a method of measuring mechanical properties of a rock formation.  FIG. 3  presents a flow diagram of selected steps of an example method  300  of the disclosure. The method  300  can be implemented by any of the embodiments of the apparatus  100  and the system  200  such as disclosed in the context of  FIGS. 1A-2 . 
     With continuing reference to  FIGS. 1-3  throughout, the embodiments of the method  300  can include providing a core plug specimen from the rock formation (step  310 ). One skilled in the pertinent art would be familiar with the operation of conventional coring tools, such as rotary coring tools, to obtain such a specimen. For instance, in some embodiments, providing the core plug specimen from the rock formation can include using downhole core drilling equipment to cut a cylindrically-shaped core plug specimen from axially diagonally or laterally oriented side wall portions of a well bore and catching and storage equipment to capture and transport the core plug sample to the surface for mechanical testing as disclosed herein. 
     Embodiments of the method  300  can also include dividing the core plug specimen into disc-shaped samples (step  315 ). One skilled in the pertinent art would be familiar with the operation of conventional cutting or grinding tools to divide the specimen into, e.g., same-dimensioned disc-shaped samples  110 . For instance, in some embodiments, the dividing of the core plug specimen into same-dimensioned disc-shaped samples, as part of step  315 , can include separating the specimen into the disc-shaped samples such that each of the disc-shaped samples have a thickness to diameter ratio value that is in a range from about 1:1 to 1:4, and in some embodiments, about 1:2. For instance, in some embodiments, core plug specimen having a diameter of about 1 to 2 inches (2.5 to 5 cm) and length of about 1 to 4 inches (2.5 to 10 cm) can be cut into multiple 0.5 inch or 1 inch thick disc-shaped samples and the end faces of each of the sample can be grinded to obtain the desired thickness to diameter ratio. 
     Embodiments of the method  300  can further include successively placing each one of the disc-shaped samples into to an apparatus (step  320 ), e.g., one sample  110  at a time into apparatus  100 , such that force application fixtures  105  of the apparatus can apply opposing compressive forces  107   a ,  107   b  to the disc-shaped samples  110  such that substantially an entire volume of the disc-shaped samples  110  is configured to experience a compressive stress. 
     Embodiments of the method  300  can further include successively applying the opposing compressive forces to each of the disc-shaped samples  110  (step  325 ). For each of the disc-shaped samples  110 , the opposing compressive forces  107   a ,  107   b  are applied in directions that are at different angles relative to a direction of in situ force experienced by the core plug specimen. For instance, the direction of in situ force experienced by the core plug specimen can be substantially perpendicular to the average plane of the substantially parallel laminated layers (e.g., average plane  240  of the layers  245 ,  FIG. 2 ) of the disc-shaped samples  110  and therefore each of different the samples  110  can be rotated such that directions of the opposing compressive forces  107   a ,  107   b  are applied at different angles (e.g., angle  250 ,  FIG. 2 ) relative to the average plane  240  of the laminated layers  245  of the samples  110 . 
     For example, in some embodiments, the opposing compressive forces  107   a ,  107   b  can be successively applied, as part of step  325 , to three different disc-shaped samples  110  at different angles  250  equal to about 0, 45 and 90 degrees, respectively. However, in other embodiments, additional samples  110  could be compression tested at different angles (e.g., intermediate angles of 22.5 and 67.5 degrees, respectively), to provide more detailed information about anisotropy of the mechanical properties present in the specimen from which the samples  110  are derived. 
     Embodiments of the method  300  can further include successively measuring axial and lateral displacements of the disc shaped samples during the application of the opposing compressive forces until failure of the disc-shaped sample is detected (step  330 ). The axial displacement can be measured along a first axis (e.g., axial axis  255 ,  FIG. 2 ), the first axis being parallel to the directions of the opposing compressive forces  107   a ,  107   b . The lateral displacement can be measured along a second axis (e.g., lateral axis  260 ,  FIG. 2 ), the second axis being perpendicular to the directions of the opposing compressive forces  107   a ,  107   b.    
     Some embodiments of the method  300  can further include determining a measured compressive strength of each of the disc shaped samples from the measured axial and lateral displacements obtained for each of the disc shaped samples (step  335 ). One skilled in the pertinent arts would be familiar with the equations used to determine compressive strength from such displacement measurements. For example determinations of unconfined compressive strength can be performed by compression of cylindrical spacemen and detecting force of its mechanical failure with resulted permanent deformation. 
     Some embodiments of the method  300  can further include determining a measured Young&#39;s Modulus for each of the disc shaped samples (step  340 ) using the measured compressive strength of each of the disc shaped samples (e.g., measured in step  335 ). One skilled in the pertinent arts would be familiar with the equations (e.g., E=σ/ε, where σ is compressional stress and ε is axial strain) used to determine Young&#39;s Modulus from such measured compressive strength values. 
     Some embodiments of the method  300  can further include determining a measured cohesive strength (step  345 ), a measured friction angle (step  350 ), or a measured Poisson&#39;s ratio (step  355 ) of each of the disc shaped samples, using the measured compressive strength (e.g., determined in step  335 ) of each of the disc shaped samples and a tensile strength measured for a portion of the core plug specimen (step  360 ). One skilled in the pertinent arts would be familiar with how to determine tensile strength from core plug specimens, e.g., using ASTM standard D3967. For example a disc shape specimen can be compressed from two opposed non planar sides, using force applicators that are substantially flat or slightly curved to assure minimum area of contact between applicators and the specimen. The diameter to thickness ratio is typically in a range about 2 to 2.5. The compressive force is continuously increased until failure of the specimen. The Tensile strength (σ t ) can be calculated by formula: 
       σ t =2 F /(π dL )
 
     where, F equals the compressive force at failure, d equals the specimen&#39;s diameter and L equals the specimen&#39;s thickness. 
     One skilled in the pertinent arts would be familiar with how to determine Poisson&#39;s ratio from such compressive strength and tensile strength measurements. For example Poisson&#39;s ratio (ν) can be determined from the equation: 
       ν=−ε z /ε x ,
 
     where ε z  equals axial strain during compression and ε x  equals strain in the direction perpendicular to the direction of compression. 
     One skilled in the pertinent arts would be familiar with how to determine Cohesive strength and Friction angle can be also determined using Mohr-Coulomb failure envelope. ASTM D7012, Standard Test Methods for Compressive Strength and Elastic Moduli of Intact Rock Core Specimens under Varying States of Stress and Temperatures. 
     Some embodiments of the method  300  can further include determining an equivalent unconfined compression strength of the disc shaped samples (step  370 ). Determining the equivalent unconfined compression strength in accordance with step  370  can include determining a conventional slope and conventional intercept from plots of conventionally determined unconfined compression strength versus conventionally determined Young&#39;s Modulus (step  372 ), e.g., by linear regression analysis. The unconfined compression strength and the conventionally determined Young&#39;s Modulus can be determined from one or more different core specimens obtained from the rock formation, or, in some embodiments from other rock formation including other types of rock formations than the rock formation from which the core plug specimen was obtained. One skilled in the pertinent arts would be familiar with how to determine the unconfined compression strength conventionally determined Young&#39;s Modulus. For example Young&#39;s modulus (E) can be determined using equation: 
         E=C   1 σ/ε,
 
     where C 1  equals a predetermined correlation coefficient, σ equals compressional stress, and equals axial strain. The compressive strength σ c  equals C 2 σ f , where C 2  is another predetermined correlation coefficient and σ f  equals measured stress at failure. 
     Determining the equivalent unconfined compression strength (step  370 ) can include determining a measured slope and measured intercept (step  374 ) from plots of the measured compressive strength (e.g., as determined in step  335 ) versus a measured Young&#39;s Modulus (e.g., as determined in step  340 ) for at least one of the disc shaped samples  110 , e.g., by linear regression analysis. 
     Determining the equivalent unconfined compression strength (step  370 ) can include determining a slope calibration factor (step  376 ) (e.g., as ratios of the measured slope to the conventional slope) and an intercept calibration factor (e.g., as ratios of the measured intercept to the conventional intercept). 
     The slope calibration factor and the intercept calibration factor can then be used to convert the measured compressive strength to the equivalent unconfined compression strength in accordance with step  370 . 
     Based on the present disclosure one skilled in the pertinent arts would understand how analogous steps could be performed to determine an equivalent unconfined cohesive strength, friction angle, or Poisson&#39;s ratio for each of the samples  110 . 
     Experiments 
     To further illustrate various features of the disclosure, an example embodiment of the apparatus of the disclosure and the example apparatus&#39;s use for measuring the mechanical properties of rock formation sample using embodiments of the system and method of the disclosure, are presented below. 
     Experiment 1: Simulation Experiments to Develop an Example Embodiment of the Apparatus 
     Computer simulation software (Abaqus FEA, Dassault Systémes Waltham, Mass.) was used to develop and test the ability of various model embodiments of the apparatus for their ability to created compression stress concentration for simulated cylindrically-shaped object, e.g., having a diameter to thickness ratio of 2:1 for some simulations. Simulated forces (e.g., from about 0 to 1E+7 Newtons) for various end cap shaped surface designs were evaluated for simulated displacement distributions and stress concentrations inside of a simulated object  410  to evaluate the end cap ability to produce compressive forces throughout defined volume elements  480  of the simulated object. 
       FIG. 4  presents a perspective view of an example simulated object  410  and example simulated end cap surfaces  424  of simulated force application fixtures  405  used to evaluate the distribution of compressive forces in a simulated object  410  in accordance to the principles of the disclosure. The object  405  was represented by about 500 volume elements  480  each of about 1 mm 3  in volume). 
       FIG. 5  presents a perspective view of a simulated object  410  showing a distribution plot of the numerical simulation results of calculated axial displacements for each of the simulated volume elements  480  ( FIG. 4 ) in axial dimension U and lateral dimension U 3 . The calculated axial displacement distribution of the elements  480  representing in the simulated object  410  are the result of a simulated application of opposing compressive forces  407 A  407 B in the axial dimension through simulated force application fixtures  404  such as presented in  FIG. 4 . 
     Based upon the present disclosure, one skilled in the pertinent arts would appreciate how analogous simulations could be conducted to develop end cap shapes appropriate for disc samples cut to have different shapes e.g., different thicknesses different diameters or top and bottom end faces with non-circular perimeters (hexagonal, octagonal or other shapes whose perimeters may have one or more planar portions). 
     Experiment 2: Evaluation of Prototype Force Application Fixtures 
     Prototype example force application fixtures were machined and experiments were performed to examine the correlation between mechanical properties of disc-shaped samples and data recorded using the prototype fixtures. A pair of prototype force application fixtures were machined to have shapes similar to the fixtures described in the context of  FIGS. 1A-2  for the testing of disc-shaped samples as described below. The end cap of the prototype force application fixtures was machined to have a shaped surface (e.g., surface  125   FIG. 1A ) conforming with a portion of a non-planar side of the disc-shaped samples such that the end-to-end distance of the end caps equaled 1.6 inches. A connection portion of the prototype force application fixtures was machined to include threaded openings so that the fixture could be attached to actuators of a hydraulically powered load frame device (Model Landmark®, MTS Systems Corporation, Minnesota). Notches were machined into sides of the prototype force application fixtures so that end pins of sensors of the extensometer device could be attached thereto to measure axial displacements of the disc shaped samples during the application of the opposing compressive forces. The axial and lateral displacements of the samples were measured using linear variable differential transformer sensors of an extensometer device (Model 632.90 MTS Systems Corporation, Minnesota). 
     Samples for use in with the prototype force application fixtures were formed by cutting core plug specimens obtained from different types of formations (e.g., specimen A for a shale formation, specimen B for a sand stone formation). The samples were trimmed as needed to form uniform sized disc-shape having a diameter of 2.0 inches (e.g., about 5 cm) and width of 0.50 inches (e.g., about 1.3 cm). The samples were then placed in between the prototype force application fixtures such that the end caps of the fixtures contacted the sides of the disc sample, and then the disc-shaped samples were subjected to compression until failure, e.g., opposing compressive forces ranging from about 2000 to about 14000 psi (about 13800 kPa to about 96500 kPa). 
       FIG. 6  presents plots of the Young&#39;s modulus (E) versus the compressive strength for disc-shaped samples of the rock formations specimens measured using the prototype force application fixtures, system and method (triangles), and, plots of conventionally measured Youngs modulus versus a conventional unconfined compressive strength (UCS) measured using a conventional apparatus and methods (squares). The conventional UCS measurements were conducted according to ATSM D7012 using the same load frame device but with conventional application fixtures. 
     For the specimens obtained from shale rock formations, UCS measures were conducted on two different core samples oriented at 0 or at 90 degree angles (e.g., angle  250   FIG. 2 ) relative to the average plane of the substantially parallel laminated layers visible in such type A specimens (e.g., UCS-A0 and UCS-A90, respectively). Compressive strength measurements of the disc-shaped samples obtained from a same core sample, measured in accordance with the disclosure, were conducted with two different ones of the disc shaped samples oriented at 0 or 90 degrees relative to the average plane of the substantially parallel laminated layers visible in type B specimens (e.g., DAT-A0 and DAT-A90, respectively). For the specimens obtained from sand stone rock formations, due to the homogeneity of the specimens (e.g., no visible layers present) the UCS measurements of core samples (e.g., UCS-B) and the compressive strength measurements of the disc-shaped samples obtained from a sample core sample, measured in accordance with the disclosure, (e.g., DAT-B) were conducted with no particular orientation of the samples found to be important. 
       FIG. 6  also presents a linear regression analysis best-fit line (Line  1 ) of the combined UCS measurements obtained for the samples obtained from the shale and sand stone rock formations (squares) and a linear regression analysis best-fit line (Line  2 ) of the combined measurements obtained for the samples obtained from the shale and sand stone rock formations, measured in accordance with the disclosure. As illustrated regardless of the type of sample (e.g., sandstone or shale rock formations) the UCS measures follow a substantially same relationship of Young&#39;s modulus versus compression strength that can be modeled by line  1 . Likewise the measured Young&#39;s modulus and measured compressive strength measurements of the disc-shaped samples regardless of the type of sample, measured in accordance with the disclosure, followed a substantially same relationship of Young&#39;s modulus versus compression strength that can be modeled by line  2 . 
     As part of the present disclosure we recognized that the difference in the slope and intercept of line  1  and line  2  can be used to determine a slope calibration factor and an intercept calibration factor to convert the measured compressive strength, measured in accordance with the disclosure, into an equivalent UCS measurement measured using conventional procedures with conventional core samples. Moreover because different types of rock formations (e.g., sandstone and shale) followed substantially the same regression line, we expect that it may be possible, to just obtain such calibration factors just one time and then apply the resulting correlation factor generally to other specimens to determine equivalent UCS measurement, subject to the particular shape of sample. 
     Because the disc shape samples are all obtained from a same core specimen, we found that the variance in the measured compressive strength can be substantially smaller than, e.g., the results from conventional measurements made on different core specimens obtained from different locations in the rock formation. For instance, consider a shale formation where different core specimens obtained from locations that are only 10 inches apart from each other, and can have UCS values that vary by 30% or more, e.g., due to the different clay contents of the core specimens. The resulting average UCS from such different core specimens will have a variance of about ±30%. In contrast, the average measured compressive strength obtained from multiple disc shaped samples, divided from the same core specimen in accordance with the present disclosure, can have a variance of other a few percent. Moreover because the disc samples can be tested at different angles a fuller picture of the anisotropy in mechanical properties of the specimen can be obtained. 
     Consequently, it is possible to obtain more precise measurements for the core specimens than possible from conventional measurements made using two or more different core sample which of course will have to be from at least somewhat different locations of the rock formation. In turn, having more precise measurements allows more precise assessments of well-bore stability such as, better understands the ability of wellbores to tolerate stresses on the wellbore during various production stages of oil and gas extraction or better estimating how much fluid can be pumped into the wellbore and the weight of drilling mud can be used. 
     Additionally, we recognized, as part of the present disclosure, that relationships between conventionally measured tensile strength, cohesive strength, friction angle, or Poisson&#39;s ratio versus these properties, measured in accordance with the disclosure, could be determined and plotted similar to that presented in  FIG. 6  and a regression analysis of these relationships could be performed to determine calibration factors for these material properties, analogous to that presented above. 
     Those skilled in the art to which this application relates will appreciate that other and further additions, deletions, substitutions and modifications may be made to the described embodiments.