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<ul><li><p>60 Journal of Canadian Petroleum Technology</p><p>Cutting Efficiency of a Single PDC Cutter on Hard Rock</p><p> G. Hareland </p><p>University of Calgary</p><p>W. Yan new Mexico Institute of Mining and Technology </p><p>r. nYGaard* University of Calgary</p><p>J.l. WISe Sandia national laboratories</p><p>*currently with Missouri University of Science and Technology</p><p>Peer reviewed PaPer (review and Publication Process can be found on our website)</p><p>IntroductionPDC bits have gained popularity in drilling for petroleum due </p><p>to its long bit life together with its ability to maintain a high rate of penetration (ROP). The shearing action induced by fixed cut-ters has shown to be more efficient for penetrating rock than the crushing effect of the teeth or inserts on the rolling cones of a roller bit(1-4). However, PDC bits have traditionally had limitations when encountering hard formations(5). Therefore, PDC bits are not yet preferred for hard formations encountered with mining, petroleum or geothermal energy. Enhanced geothermal energy, i.e., geo-thermal energy from considerable (+3 km) depths, is seen as one </p><p>AbstractPolycrystalline diamond compact (PDC) bits have gained </p><p>wide popularity in the petroleum industry for drilling soft and moderately firm formations. However, in hard formation ap-plications, the PDC bit still has limitations, even though recent developments in PDC cutter designs and materials steadily im-proves PDC bit performance. The limitations of PDC bits for drilling hard formations is an important technical obstacle that must be overcome before using the PDC bit to develop competi-tively priced electricity from enhanced geothermal systems, as well as deep continental gas fields. Enhanced geothermal energy is a very promising source for generating electrical energy and, therefore, there is an urgent need to further enhance PDC bit per-formance in hard formations. </p><p>In this paper, the cutting efficiency of the PDC bit has been analyzed based on the development of an analytical single PDC cutter force model. The cutting efficiency of a single PDC cutter is defined as the ratio of the volume removed by a cutter over the force required to remove that volume of rock. The cutting efficiency is found to be a function of the back rake angle, the depth of cut and the rock property, such as the angle of internal friction.</p><p>The highest cutting efficiency is found to occur at specific back rake angles of the cutter based on the material properties of the rock. The cutting efficiency directly relates to the internal angle of friction of the rock being cut.</p><p>The results of this analysis can be integrated to study PDC bit performance. It can also provide a guideline to the application and design of PDC bits for specific rocks.</p><p>of the promising energy sources in the U.S.(6) Deep continental gas developments also suffer from a low rate of penetration and high bit wear for wells drilled. </p><p>The objective of this paper is to develop an analytical model to study the cutting efficiency of PDC bits in hard rock formations which can improve the future design of PDC bits and, thereby, im-prove the cost efficiency of wells in hard formations.</p><p>PDC BitMost PDC bits are composed of a hard matrix body which is </p><p>milled out of a solid block of steel. The matrix body is milled out in such a shape that the bits contain blades where the actual PDC cutters are glued or braced on, and open areas where the cut-tings and mud flow can escape to the annulus (Figure 1). Figure 1 </p><p>FIGURE 1: Example of an 81/2 inch diameter drag PDC bit(3).</p></li><li><p>June 2009, Volume 48, No. 6 61</p><p>shows a typical 8 in diameter PDC bit from one of the leading PDC bit manufacturers(3). In Figure 1, the PDC cutters are placed on the gold coloured blades. The flow pathways for mud and cut-tings are coloured blue. Each cutter is fixed on the blade and rock is removed when the cutters are dragged in a circular path when the bit is rotated at the bottom of the hole (Figure 2). Figure 2 shows a sketch of the circular path for a single cutter while rotating the bit. In Figure 3, a sketch in the vertical plane of a single cutter is shown. The free surface is the bottom of the wellbore. The cutter penetrates the rock based on the point load on each cutter origi-nated from the applied weight on the bit (WOB). The cutter is tilted with an angle, , with respect to the rock. The angle, , is the back rake of the cutter. How efficient the PDC bit is in removing rock is dependant on several factors. Increased rock hardness reduces ROP. Increased WOB and revolutions per minute (RPM) will in-crease the ROP if cuttings are removed efficiently(4). The number of cutters, their back rake and other design features of the bit will also affect ROP(4). And lastly, the PDC cutter material and cutter material design affects ROP throughout the bit cutter life(7). For instance, cutters made up from fine (10 m) diamond grain size provides higher ROP and less abrasion than those from coarse (70 m) diamond grains(7). Further, rounding the cutter edges and in-creasing the sintering pressures under cutter production make them more thermally stable and gives dramatically improved bit perfor-mance when drilling in hard formations(1, 7). </p><p>Of all the different areas mentioned above, to improve PDC bit performance in hard rock applications, the question we would ad-dress here is how the orientation of the cutter can improve the PDC bit performance.</p><p>Cutting Efficiency</p><p>A convenient way to improve PDC bit performance is to re-late the cutting efficiency to the volume of rock removed by the cutter and the force spent on that removing action. A term, spe-cific volume, is introduced here to describe the cutting efficiency. It is defined as:</p><p>Specific Volume =Volume of rock removed in oone major chip</p><p>Maximum force required to remmove that volume of rock </p><p>Volume of Rock Removed By the Cutter in One Chip</p><p>The first requirement needed for analyzing the cutting effi-ciency is the volume of rock removed by the cutter in one major </p><p>cycle (thus, the major cutting chip). The cutting shape is defined by two curves (in two dimensional view, see Figure 4):</p><p>y k x x y</p><p>y k x</p><p>a a</p><p>b b</p><p>= ( ) +=</p><p>0</p><p>2</p><p>0</p><p>.............................................................................. (1)</p><p>where ya is the failure surface of the rock assumed to follow a parabolic representation and yb represents the rock surface of the wellbore. The axes in Figure 4 are oriented such that the y-axis is parallel to the cutter face and the x-axis is normal to the cutter face. </p><p>The way the rock is fragmented into cutting chips by the PDC cutter, and thereby controls the shape of the curve ya, is a two stage phase process: crack initiation and crack propagation, shown in Figure 4. The position of the crack initiation is determined pri-marily by the magnitude of stress which will be infinite on the cutter tip if we assume a linear elastic homogenous rock mate-rial and that the PDC cutter is a rigid body. Linear elasticity is an acceptable assumption for hard rocks(8, 9). Since the compressive strength of the rock is orders of magnitude higher than the tensile strength, the crack initiation is caused by tension. The direction of the initial crack at the cutter tip is 0 (crack initial angle). Once the crack is initiated, it will continue to propagate as a shear frac-ture until the chip is formed. The crack angle () will change since the failure stress state is changing from tensile to maximum shear failure. The coefficients in Equation (1) for ya are derived in Yan</p><p>(9) and are given as: </p><p>FIGURE 2: Sketch showing the cutting path, cutting direction and side force on the cutter when bit rotates.</p><p>FIGURE 3: Sketch showing the PDC cutter and parameters acting on it.</p><p>x</p><p>y</p><p>( )x y0 0,</p><p>( )x y1 1,</p><p>ya</p><p>yb</p><p>dx</p><p>y yb a</p><p>FIGURE 4: Sketch showing the chip dimension.</p></li><li><p>62 Journal of Canadian Petroleum Technology</p><p>xh</p><p>yh</p><p>kh</p><p>c</p><p>k</p><p>c</p><p>c</p><p>ac</p><p>b</p><p>0</p><p>0</p><p>0</p><p>2</p><p>20 25</p><p>1</p><p>=</p><p>= </p><p>= +( )=</p><p>tan</p><p>.sin</p><p>tan</p><p>......................................................................... (2)</p><p>The failure curve, ya, initiation point (x0, y0) is a function of the depth of cut, hc, the back rake angle, , and the crack initiation angle. The shape of the failure surface, ya, is made more realisti-cally by introducing a function for the constant, ka, in the parabolic Equation (1). The failure surface, ka, is a function of the depth of the cut, hc, fitting constant, c, and crack initiation angle, 0. The free surface, yb, is only a function of the back rake angle, , and the length of the cutting chip along the x-axis.</p><p>The shaded area in Figure 4 which represents a cross sectional area of the cuttings at an arbitrary distance from the cutter is given as:</p><p>ds y y dxb a= ( ) ..................................................................................... (3)</p><p>The whole cross sectional area of the cutter can be obtained by integrating Equation (3) from x0 to x1.</p><p>S y y dxb ax</p><p>x</p><p>= ( )0</p><p>1</p><p> ................................................................................... (4)</p><p>Carrying out the integration, notice that:</p><p>xe</p><p>hc12</p><p>01 25 21= ( ).</p><p>tansin</p><p>................................................................. (5)</p><p>and rearrange to get the area of the cutter shown in Figure 4 as:</p><p>S c h c hc c= +1 22</p><p> ........................................................................................ (6)</p><p>where</p><p>c ca b</p><p>1 00 0 00 0833 1</p><p>2= +( ) </p><p>. sin</p><p>sin</p><p>tan</p><p>22</p><p>2 0 0 0 0 0</p><p>214</p><p>1 1c a b b= + ( ) </p><p>tansin sin</p><p> ................................ (7)</p><p>and</p><p>a e</p><p>b e</p><p>02</p><p>02</p><p>1 25 1</p><p>1 25</p><p>= </p><p>=</p><p>.</p><p>.</p><p> ...................................................................................... (8)</p><p>To calculate the cutting chip volume, the two dimensional x, y cutting chip area, S, has to be multiplied with the width of the cut-tings in the horizontal, z-direction, given in Figure 5. The cutting chip volume is then given, approximately, by:</p><p>V w Se= .................................................................................................. (9)</p><p>where we is the equivalent width. The actual integration to obtain the entire three-dimensional surface over the shear plane is too complex to be integrated by any analytical method and some sim-plification is needed. Since the thickness of the chip is not large (small depth of cut), the shear surface can be treated as a plane </p><p>without losing too much precision. The plane is further projected onto the rock surface to get a half ellipse with the axis of a0 and b0, as shown in Figure 5. The equivalent width, we, can then be calcu-lated as the area of the ellipse, which is given as the numerator in Equation (10) divided by the half length (see Figure 5). </p><p>wa b</p><p>ae =</p><p>8</p><p>2</p><p>0 0</p><p>0</p><p> ....................................................................................... (10)</p><p>Since we is then only related to b0 and b0 is directly related to the depth of cut by(3):</p><p>b d h hc c c022= </p><p>.................................................................................. (11)</p><p>The equivalent width can then be calculated as:</p><p>w d h he c c c= 1 572.</p><p> ............................................................................ (12)</p><p>The only missing parameter is to calculate the crack initiation angle, 0. The crack initial direction is determined by the principal stresses at the crack tip. Since it is assumed that the crack is caused by tension at the cutter tip, the initial crack initiation angle can be calculated by: </p><p> 012</p><p>2=</p><p>arctan</p><p>xy</p><p>x y ....................................................................... (13)</p><p>Now we have the full solution to calculate the volume, V, for a cutting chip based on Equation (9). </p><p>Specific Volume</p><p>A factor, specific volume, is introduced to measure the cutting efficiency. The specific volume, V0, is defined as the volume of rock removed by the cutter divided by the resultant force on the cutter,</p><p>VVF0</p><p>= ................................................................................................. (14)</p><p>where, F is the resultant force (in pounds-force) of the horizontal and vertical forces acting on the cutter.</p><p>F P P Ph v s= + +2 2 2</p><p> .............................................................................. (15)</p><p>b0</p><p>a0</p><p>b we</p><p>FIGURE 5: Sketch showing the simplified shear plane for calculating we.</p></li><li><p>June 2009, Volume 48, No. 6 63</p><p>V is the volume (in cubic inches) of rock removed in one major chip and is given by Equation (9).</p><p>Equation (14) then provides a measure of the cutting efficiency in terms of specific volume. The highest efficiency occurs at the maximum specific volume. </p><p>Figure 6 and Figure 7 show the specific volume as a function of back rake angle and depth of cut (in inches), respectively. In Figure 6, the specific volume is calculated for different depths of cuts. All the curves show the similar shape with local maxima at 0 and 25 degrees. In Figure 6, the specific volume results are plotted as a function of depth of cut to identify any local maximum. As stated previously, the specific volume represents the cutting ef-ficiency and two important conclusions can be drawn from these plots. These are:</p><p>1. The specific volume reaches its local maximum at 0 and 25 back rake angle. This indicates that the best settings for back rake angle are either at 0 or at 25. Also, the effect of back rake angle on the specific volume becomes less significant as the depth of cut decreases. However, even when the depth of cut is less than 0.02 in, there is still a noticeable effect by the back rake angle, especially at 0. This implies that the back rake angle of a PDC cutter becomes more important at a larger depth of cut. </p><p>2. The specific volume versus depth of cut did not give any local maximum within the range of 0.01 ~ 0.08 in, except at 0.08 in. This indicates that no optimum depth of cut could be obtained. From the plot, it is clear that the larger the depth of </p><p>cut, the more efficient the cutting. However, as the depth of the cut increases, the incremental specific volume decreases. This suggests that a high rate of penetration for a PDC bit is appreciated in term of cutting efficiency only. In practice, the depth of cut cannot be too high since the blades where the cutters are mounted will come into contact and take up some of the weight of the bit and reduce the point load on the indi-vidual cutter. The limitation the blades impose on the cutter suggests that a depth of cut around 0.04 in is optimum.</p><p>Rock Properties in the Model</p><p>In the analysis above, the cutting chip is independent of any rock mechanical properties. To establish a failure criteria for the rock, the unconfined rock strength and internal friction angle has to be included. This was done by assuming the rock chip to be a linear elastic material with the shape of a wedge. There are two basic pa-rameters in the linear elastic derived force model(9): the half wedge angle, , and the ratio of the normal stress over shear stress on the failure curve, k0. These parameters were used in the model as con-stants. According to the laboratory observations, the half wedge angle takes the value of 25. The failure criteria k0 ta...</p></li></ul>