Study of the Interaction Model of PDC Cutters with Rock

The existing PDC cutters with rock interaction models does not match the actual drilling condition, because WOB can’t be loaded on their doesn’t be considered. A drilling experiment was carried out in the experimental device in which cutters can be loaded WOB. The relationship of cutting area, cutting speed, back dip with the force of PDC cutter were considered. The multiple nonlinear regression analysis were done on the basis of the experimental data, a new cutter-rock interaction model was established. The cutter area is the p component influencing the force of cutters though the cluster analysis and principal component analysis. It is a logarithmic relationship between cutting velocity and the force of cutter, the theory of particle dynamics confirms this relationship.


INTRODUCTION
The interaction model of PDC cutters with rock is very important to forecast the ROP of the bit.Deyong and Wang (2005) and Ma and Wang (2006) established some PDC cutter rock interaction models on the cutting rock experiment using lathe, but their models have some problems.Their equipment can't WOB loaded on the bit.Their model don't consider the influence of the cutting velocity on the force of cutters.To solve these problems, a new drilling experiment was carried out, using a drilling machine which can loaded WOB on the bit, under different WOB, cutting velocity, rock and back rake angle of PDC cutter.

Equipment and methods of drilling experiment:
No. XY-2B drilling machine consists of electric motor, kelly bar, PDC bit, hydraulic system, sensors, data acquisition unit etc. in China University of Petroleum, shown in Fig. 1a.Hydraulic pressure l is 3, 4, 5, 6, 7 KN, respectively.The cutting velocity of PDC cutter is 0.05, 0.28, 0.66 and respectively.There are 15 kinds of PDC consists of three cutters, shown in Fig. 1b.The cutter's diameters are 13, 16, 19 mm and their back rake angle are 5, 10, 15, 20 and 25°, respectively.
Tangential force F c is in the direction of the cutter velocity.Normal force F n is in the normal to the rock surface, shown in Fig. 1c.The pressure sensors are used to measure the normal force of the PDC cutter.A The existing PDC cutters with rock interaction models does not match the actual drilling condition, because WOB can't be loaded on their equipment, the influence of the cutting velocity on the force of cutters doesn't be considered.A drilling experiment was carried out in the experimental device in which cutters can be loaded WOB.The relationship of cutting area, cutting speed, back dip angle of PDC cutter and rock performance with the force of PDC cutter were considered.The multiple nonlinear regression analysis were done on the basis of rock interaction model was established.The cutter area is the p component influencing the force of cutters though the cluster analysis and principal component analysis.It is a logarithmic relationship between cutting velocity and the force of cutter, the theory of particle dynamics confirms Back rake angle, force of cutter, interaction model, PDC bit, rock drill-ability The interaction model of PDC cutters with rock is very important to forecast the ROP of the bit.Deyong Ma and Wang (2006) established some PDC cutter rock interaction models on the cutting rock experiment using lathe, but their models have some problems.Their equipment can't WOB loaded on the bit.Their model don't consider the influence of the locity on the force of cutters.To solve these problems, a new drilling experiment was carried out, using a drilling machine which can loaded WOB on the bit, under different WOB, cutting velocity, rock and Wang Jiajun, Zou Deyong, Yang Guang, He Renqing and Chen Xiuping College of Petroleum Engineering in China University of Petroleum, Qingdao 266580,

MATERIALS AND METHODS
The existing PDC cutters with rock interaction models does not match the actual drilling condition, equipment, the influence of the cutting velocity on the force of cutters doesn't be considered.A drilling experiment was carried out in the experimental device in which cutters can be angle of PDC cutter and rock performance with the force of PDC cutter were considered.The multiple nonlinear regression analysis were done on the basis of rock interaction model was established.The cutter area is the principal component influencing the force of cutters though the cluster analysis and principal component analysis.It is a logarithmic relationship between cutting velocity and the force of cutter, the theory of particle dynamics confirms   1.

RESULTS AND DISCUSSION
Cluster analysis: The similar variables were classified into a class using system clustering method, shown in Fig. 2. Cutting area is clustered early with cutting depth, contact arc and cutter diameters.Then, they are clustered with the force of cutters.Cutting area can comprehensively reflect the effect of cutter shape on the force of cutters, so we only consider cutting area in the interaction model of PDC cutters with rock.Rock drillability is clustered early with indentation hardness, compressing strength and tensile strength.Then, they are clustered with the force of cutters.Rock drill-ability can comprehensively reflect rock breaking efficiency, so we only consider rock drill-ability in the model.

Principal component analysis:
The main factors were found using principal component analysis, shown on the Table 2.The sum of tangential force's contribution rate and cutting area's contribution rate is over 76%.It is the same to normal force.So cutter area is the principal component influencing the force of cutters.

Regression analysis:
Relationship between cutting area and the force of cutter: When the back rack angle, cutting velocity and rock drill-ability is a constant, the force of cutters increases linearly with the cutting area shown in the Fig. 3.Because cutter area is the principal component influencing the force of cutters, the relationship between the force and cutting area can be expressed as a linearly function: Relationship between rock drill-ability and the force of cutter: When the back rack angle, cutting velocity and cutting area is a constant, the force of cutter Relationship between back rake angle and the force of cutter: When rock drill-ability, cutting velocity and cutting area is a constant, it is a quadratic relationship between back rake angle and the force of cutter, shown in the Fig. 5.When the back rake angle of cutter is over 10°, the normal force F n , tangential force F c increases with the back rake angle of cutter.

Relationship between cutting velocity and the force of cutter:
When rock drill-ability, back rake angle and cutting area is a constant, the normal force F n , tangential force F c increases with the cutting velocity.It is a logarithmic relationship between cutting velocity and the force of cutter, shown in the Fig. 6.Some authors Fairhurst (1955), Gray et al. (1962) and Roxborough and Pedroncelli (1982) ignored the effect of cutting velocity on the normal force and tangential force using forward rake cutter.Because their experiment's condition did not match drilling filed, their theory is not right.Other authors Dagrain and Tshibangu (2002) and Pelfrene et al. (2011) pointed out that the normal force and tangential force increases with the cutting velocity using backward rake cutter.In order to explain this logarithmic relationship, a finite element stress was done for dynamic cutting rock, using LS-DYNA finite element computer program, shown in Fig. 7. Massive cuttings will be cut down from the rock along the shear plane.The rock powder of broken rock zone can't move forward neither backward under the effect of squeezing, confined within the cutter and massive cuttings.
In our opinion, broken rock zone is a kind of granular materials.We invoked a dynamic mechanism studied in the field of granular materials.Hartley and Behringer (2003) have shown experimentally that increasing the shearing rate of a granular material, confined within the annular space of a rotor and a stator, increases the normal stress in the granular network.It is due to the increasing difficulty to reorganize the grain network as the shearing rate increases.According to this theory, the rock strength of broken rock zone depends on the cutting velocity.

CONCLUSION
• The cluster analysis and principal component analysis were done on the basis of the experimental data, cutter area is the principal component influencing the force of cutters.• It is found that our experimental data match the drilling filed data, using a drilling machine which can load WOB.• When rock drill-ability, back rake angle and cutting area is a constant, it is a logarithmic relationship between cutting velocity and the force of cutter.• When the back rack angle, cutting velocity and rock drill-ability is a constant, the force of cutters increases linearly with the cutting area.• When rock drill-ability, cutting velocity and cutting area is a constant, it is a quadratic relationship between back rake angle and the force of cutter.

NOMENCLATURE
Fig. 1: Equipment of text 2B drilling machine model of cutting rock

Fig. 2 :
Fig. 2: Cluster diagram increases linearly with rock drill-ability K d , shown in the Fig. 4.

Table 1 :
Rock parameter

Table 2 :
Contribution rate and eigenvalues of variables sensor is used to measure the tangential force of the PDC cutter.The averages of rock drill-ability, indentation hardness, compressing strength, tensile strength are shown in Table torque

Table 3 :
Experimental data and calculated value of the force of cutters