- Email: [email protected]
Experimental studies of rock abrasiveness using a fractal approach
International Journal of Rock Mechanics & Mining Sciences 54 (2012) 37–42
Contents lists available at SciVerse ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Technical Note
Experimental studies of rock abrasiveness using a fractal approach Hui Zhang a,n, Deli Gao a, Dongtao Liu b, Boyun Guo c a b c
MOE Key Lab of Petroleum Engineering, China University of Petroleum, Beijing, China Supervision and Technology Company, CNOOC, Shen Zhen, Guang Dong, China Department of Petroleum Engineering, University of Louisiana at Lafayette, Lafayette, Louisiana
a r t i c l e i n f o Article history: Received 1 March 2011 Received in revised form 3 May 2012 Accepted 18 May 2012 Available online 9 June 2012
1. Introduction Currently, fractal studies on rock mechanics have developed into three basic fields. The first one is based on the primary assumption that the natural structure of rock could be regarded as the fractal structure. This is the mathematic foundation and frame of rock mechanics, including reacquainting and re-establishing the physics quantities and laws of physics and mechanics in fractal space [1]. The second direction is the deeper studies of rock mechanics fractal phenomenon, fractal nature and fractal mechanism. It emphasizes and quantifies the descriptions of some complicated physical and mechanical behaviors of fractal mechanism [2]. The third direction is the industrial application of rock mechanics fractal study. Appling this result can solve the practical engineering problems, and obtain quantitative prediction of complicated rock mechanics problems accurately [3]. At the present time, however, studies on rock abrasiveness seldom employ the method of fractal approach. The capacity of rock abrading the cutting edge material called abrasiveness. The existing methods of determining rock nonabrasive are basically as following: drilling and grinding method, grinding method, micro bit drilling method and friction abrasion method. These methods are all indoor-experimental methods with many deficiencies. Firstly, it is difficult to reflect the characteristic of the underground rock in the environment of high temperature and high pressure. Secondly, it is hard to reflect the variation of rock properties in heterogeneous stratum. Thirdly, it is limited by the core material, especially in the areas where lithology varies greatly, since the measured data may lack
n Correspondence to: MOE Key Lab of Petroleum Engineering, China University of Petroleum, No. 18 Fuxue Road, Changping District, Beijing 102249, China. Tel.: þ86 10 89733702. E-mail address: [email protected] (H. Zhang).
1365-1609/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijrmms.2012.05.022
of contrast. Fourthly, it is difficult to establish a continuous section of rock drillability. Lastly, it is a waste of manpower and financial resources. This paper focuses on the microscopic structure of rock. Based on the relationship between the abrasiveness and the internal friction angle of rock [4], combining the rock mechanics parameters in the test with the applying of the fractal geometry theory, this paper mainly discusses the quantitative relationship between the fractal characteristics (rock porosity fractal features and granularity fractal characteristics) and the internal friction angle of rock. Thereby, it can provide an important theoretical foundation for evaluating the abrasiveness of rock during drilling by debris (sand-like) in fabricating yard. This work serves as practical reference for drilling bit selection and parameter optimization.
2. Rock porosity and rock granularity fractal dimension calculation method 2.1. Pore fractal dimension calculation method According to the principle of fractal geometry [5], if the rock porosity distribution accords with fractal structure N(r), the number of rock pores which radius are greater than r, and r follows a power function relationship [6]: Z rmax NðrÞ ¼ rðrÞdr ¼ ArD1 ð1Þ r
where r max is the maximum pore radius of rock, r(r) is the density function of rock porosity distribution, A is a coefficient of proportionality, and D1 is the pore fractal dimension. The expression of r(r) could be obtained by differentiating the above equation with respect to r, then, according to r(r), the accumulated volume V(r) which pore radius less than y is
38
H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 37–42
deduced as follows: Z r ABD1 3D1 3D1 rðrÞUBr3 dr ¼ ðr r min Þ VðrÞ ¼ 3D1 r min
ð2Þ
where r min is the minimum pore radius, B is a constant relating to pore structure (B ¼1 when the pore is a cube; B¼ 4p/3 when the pore is a sphere), Similarly, the bulk volume of rock pore Va could be obtained. According to V(r) and Va, we can obtain the accumulated volume fraction S, whose pore radius is less than r: 3D1 r S¼ ð3Þ r max As we know, the resistance of oil–gas migration in arbitrary magnitudes of rock throats follows the equation of capillary pressure: 2scos y pc ðrÞ ¼ r
ð4Þ
where pc is the capillary pressure corresponding to pore radius r, s is the surface tension of liquid, with s ¼ 480 N/m for mercury, y is the contact angle of liquid, with y ¼1401 for mercury. According to (3) and (4), S could be expressed as pc D1 3 ð5Þ S¼ pmin where pmin is the capillary pressure corresponding to the maximum pore radius r max , (the threshold capillary pressure). Taking natural logarithm on both sides of formula (5) yields lnS ¼ ðD1 3Þlnpc þ ð3D1 Þlnpmin
ð6Þ
Formula (6) means that if the rock porosity owns the characteristics of fractal structure, a linear correlation should be noticed between the capillary pressure of wetting properties: lnS and lnpc . According to the regression analysis of correlation coefficient can explain pore fractal structure. Assume the slope of a linear regression is a, and combined with formula (6), the pore fractal dimension formula D1 could be shown as D1 ¼ 3 þa
ð7Þ
accumulated quality of the particles whose diameters are smaller than R, and MT be the total quality of all the particles. Particle size distribution meet Weibull distribution model: a MðRÞ R ¼ 1exp ð9Þ MT s where s is related to the average diameter, usually take the maximum diameter particles, and a is an undetermined constant, called the Weibull Modulus. If R/s 51, we expand the above equation in series and ignore all terms after the second one; this leads to a MðRÞ R ¼ ð10Þ MT s According to (8) and (10), D2 1
R
R3 Ra1 ¼ Ra4
ð11Þ
and then, the rock granularity fractal dimension D2 could be shown as follows: D2 ¼ 3a
ð12Þ
Taking natural logarithm on both sides of formula (10) gives MðRÞ R ¼ aln ð13Þ ln MT s Formula (13) means that if there are fractal distribution characteristics in the grain size of rock, ln[M(R)/MT] and ln(R/s) should be linearly correlated. Linear regression analysis is given of the linear gradient Weibull Modulus a, and then from formula (12), granularity fractal dimension D2 could be obtained. 3. Experiments on fractal characteristics and the correlation Experimental core: 20 pieces of rock cores are selected from Tarim Oilfield, Changqing Oilfield and Zhongyuan Oilfield for this fractal characteristics test. The fundamental information of cores and their internal friction angle are shown in Table 1. Note that the rock’s internal friction angle is obtained from triaxial rock mechanics tests. 3.1. Rock porosity fractal characteristics test
2.2. The rock granularity fractal dimension calculation method Set rock grain diameter as R, through sieve analysis [7], we can get the number N(R) of grain diameter which is greater than R [8] if the number of particles satisfied: Z 1 NðRÞ ¼ rðRÞdR ¼ kRD2 ð8Þ R
where D2 is the grain size fractal dimension, and r(R) is the density function of particle size distribution. Let M(R) be the
The relationship between capillary pressure of rock and saturation of wetting phase (or non-wetting phase) is called capillary pressure curve. There are mainly three methods to determine the capillary pressure curve so far: Semi-permeable partition method, Mercury intrusion method and Centrifuging. The advantages of Mercury intrusion method: it consumes less time in measurement; there are no strict requirements on the shape and size of rock. Mercury intrusion method is used in the experiment of this paper.
Table 1 Basic information of core samples. No.
Rock number
Lithology
Internal friction angle (1)
No.
Rock number
Lithology
Internal friction angle (1)
1 2 3 4 5 6 7 8 9 10
1-122-127 4-1-12 4-1-17 No.1 1-4 5-7-13 9-12-35 2-14-153 2-43-153 6-60-180
Sandstone Sandy mudstone Gray gas bearing fine sandstone Sandstone Grayish brown sandy shale Sandstone Sandstone Grayish brown oil-patched fine sandstone Grayish brown oil-patched fine sandstone Grayish brown oil-patched fine sandstone
32.81 37.39 37.22 36.39 38.78 36.18 37.37 37.35 36.68 36.76
11 12 13 14 15 16 17 18 19 20
6-5-180 09A09 2A 2B 6B6 6A6 09B09 1-1-86 s2 s1
Grayish brown oil-patched fine sandstone Sandstone Sandstone Sandstone Sandstone Sandstone Sandstone Sandy mudstone Sandy mudstone Muddy unequal-granular sandstone
37.57 30.31 34.75 34.52 37.45 35.22 34.79 38.97 40.82 37.06
H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 37–42
Fig. 1 is capillary pressure curve of rock sample No. 1. According to the calculation method of pore fractal dimension that mentioned earlier in this paper. Least square is used in the linear regression
0.0
Rock core㧦NO.1 Regression equation㧦y=-0.68213x+1.24035 Correlation coefficient㧦-0.99098
ln(S)
-0.5
-1.0
-1.5
2
3 ln(Pc)
4
39
analysis of the experimental data of rock sample No. 1. We can get the regression equation: y¼ 0.68213xþ1.24035 (Correlation coefficient¼0.991), which displays a highly matching result. The Residual Analysis result is shown in Fig. 2. The variation range of residual varies in reasonable bounds, indicating a reliable model. Combined with formula (7), we can obtain the pore dimension of rock sample No. 1: D1 ¼3þ( 0.68213)¼ 2.31787. Taking the similar linear regression and residual analysis to the remaining nineteen rock samples as that of rock sample No. 1. The pore fractal dimension of these 19 cores could be obtained and shown in Table 2, where the rock samples’ pore structures reveal obvious fractal characteristics and the pore size distribution follows the self-similar power–law distribution. Matching each rock sample’s internal friction angle in Table 1 and pore fractal dimension data in Table 2 with the corresponding rock sample number respectively, the regression curve obtained by least square regression method is shown in Fig. 3, and its residual analysis is shown in Fig. 4. Both the regression equation and the correlation coefficient are shown in Table 5. Based on the correlation coefficient (0.19733) and the residual of pore fractal dimension showed in Fig. 4, rock pore fractal dimension and the internal friction angle indicates a poor correlativity. Therefore, there is a poor relationship between the rock pore fractal characteristics and the rock abrasiveness, and rock pore fractal characteristics cannot be used to determine the level of rock abrasiveness.
Fig. 1. Capillary pressure curve of rock No. 1.
Regression equation: y=2.49049x+29.7693
40
Correlation coefficient: Rock internal friction angle /°
residual
0.1
0.0
0.19733
35
-0.1 30 2
3 ln pc
4
2.2
Fig. 2. Residual analysis about S and Pc of rock No. 1.
2.4
2.6 2.8 Pore fractal dimension
3.0
Fig. 3. Rock internal friction angle vs. Pore fractal dimension.
Table 2 Porosity fractal dimension of rock sample. No.
Straight line slope
Pore fractal dimension
Correlation coefficient
No.
Straight line slope
Pore fractal dimension
Correlation coefficient
1 2 3 4 5 6 7 8 9 10
0.68213 0.01713 0.29652 0.15576 0.25307 0.26119 0.23639 0.40855 0.71609 0.49095
2.31787 2.98287 2.70348 2.84424 2.74693 2.73881 2.76361 2.59145 2.28391 2.50905
0.99098 0.96326 0.98891 0.99016 0.93725 0.99432 0.99205 0.99532 0.94967 0.99559
11 12 13 14 15 16 17 18 19 20
0.44951 0.29477 0.16547 0.22049 0.37083 0.28619 0.33738 0.17787 0.1905 0.58443
2.55049 2.70523 2.83453 2.77951 2.62917 2.71381 2.66262 2.82213 2.8095 2.41557
0.99184 0.97769 0.99297 0.98933 0.99624 0.98625 0.99787 0.98487 0.99598 0.99403
40
H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 37–42
3.2. Rock particle granularity fractal characteristics tests
0.0
Rock core㧦NO.1 Regression equation㧦y=0.80569x+0.78034 Correlation coefficient㧦0.96769
-0.5 ln(M(R)/MT)
Rock grain size analysis test in Table 1 is carried out after the sample is crushed; the data is shown in Table 3. The fractal curve of test rock granularity distribution could be obtained by linear regression analysis based on formula (13). Then, combined the fractal curve of particle granularity fractal dimension with formula (12), the value of particle granularity fractal dimension of each rock sample and the correlation coefficient could be achieved. Fig. 5 is the Granularity fractal dimension of the experimental data of rock sample No. 1. Least square is used in the linear regression analysis of the experimental data of rock sample No. 1. The regression equation is: y ¼0.80569xþ0.78034, (correlation coefficient: 0.96769), which displays a highly matching result. And the Residual Analysis result is shown in Fig. 6. The variation range of residual varies in reasonable bounds, indicating a reliable
-1.0 -1.5 -2.0 -2.5 -4.0
-3.5
-3.0
-2.5 -2.0 ln(R/Rmax)
-1.5
-1.0
Fig. 5. Granularity fractal dimension of rock sample No. 1.
5 0.4
0.2 Residual
Residual
0
0.0
-5 -0.2
2.2
2.4
2.6 2.8 Pore fractal dimension
-4
3.0
-3
-2
-1
ln(R/Rmax)
Fig. 4. Residual of pore fractal dimension.
Fig. 6. Residual of granularity fractal dimension.
Table 3 Analytical data of particle size (R-particle diameter/mm). Sample no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Sample total weight (g)
20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 19.3 20
Content of various size grade (g) R¼ 1000
R¼ 710
R¼ 500
R¼ 355
R¼ 250
R¼ 180
R¼ 125
R ¼90
R¼ 75
R ¼63
R ¼45
R ¼28
R ¼10
R ¼1
0 0 0 0 0 0 0.01 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0.01 0.02 0 0.04 0.01 0.01 0 0.01 0 0.01 0.02 0.01 0.01 0.03 0.01 0.01 0
0.01 0 0 0.27 0.04 0.01 0.23 0.02 0.01 0.02 0.02 0.01 0.12 1.96 0.21 0.47 0.2 0.02 0.01 0
0.01 0.01 0 4.61 0.16 0.12 2.29 0.03 0.03 0.01 0.01 0.11 3.49 6.67 4.41 5.2 3.66 0.06 0.03 0.02
0.06 0.15 0.06 5.03 0.39 4.18 5.04 0.29 0.56 0.08 0.28 2.01 6.04 3.78 4.91 4.72 6.21 0.42 0.22 0.09
1.08 0.66 0.58 2.68 0.89 5.94 3.46 2.81 3.87 1.25 2.92 5.62 2.79 1.99 2.55 2.44 2.84 1.29 0.65 0.31
6.27 1.3 2.24 1.58 0.99 2.62 1.92 4.73 4.16 4.84 4.7 3.21 1.56 1.16 1.56 1.48 1.55 1.64 0.95 0.82
5.05 4.12 6.81 1.62 2.93 2.25 1.94 4.05 3.45 5.38 4.1 2.71 1.59 1.18 1.63 1.54 1.57 2.98 2.84 5.99
0.59 1.64 1.01 0.27 0.53 0.36 0.36 0.57 0.53 0.79 0.57 0.41 0.28 0.23 0.25 0.36 0.29 1.42 0.92 2.01
1.47 2.31 1.65 0.48 1.34 0.66 0.78 1.32 1.35 1.39 1.05 0.83 0.49 0.32 0.47 0.56 0.49 1.37 1.94 2.72
1.67 2.92 1.89 0.75 3.43 1.04 0.99 1.59 1.53 1.54 1.36 1.42 0.77 0.51 0.65 0.67 0.81 2.28 2.91 2.64
1.49 2.24 1.25 0.51 3.22 0.71 0.75 1.23 1.24 0.97 0.83 0.93 0.45 0.23 0.26 0.5 0.46 2.38 2.88 1.43
0.02 0.02 0.02 0.01 0.03 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.02 0.02 0.03 0.03 0.02
2.28 4.63 4.49 2.18 6.03 2.1 2.17 3.33 3.24 3.71 4.13 2.72 2.4 1.94 3.08 2.03 1.87 6.1 5.91 3.95
H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 37–42
41
Table 4 Rock granularity fractal dimension. No.
Straight line slope
Granularity fractal dimension
Correlation coefficient
No.
Straight line slope
Granularity fractal dimension
Correlation coefficient
1 2 3 4 5 6 7 8 9 10
0.80569 0.56007 0.58362 0.63865 0.42666 0.69873 0.58305 0.63491 0.63436 0.6198
2.19431 2.43993 2.41638 2.36135 2.57334 2.30127 2.41695 2.36509 2.36564 2.3802
0.96769 0.98214 0.95699 0.96864 0.98743 0.96619 0.96793 0.96626 0.97135 0.94943
11 12 13 14 15 16 17 18 19 20
0.54171 0.75816 0.71193 0.80499 0.66114 0.74387 0.77329 0.36844 0.35351 0.59257
2.45829 2.24184 2.28807 2.19501 2.33886 2.25613 2.22671 2.63156 2.64649 2.40743
0.94945 0.9784 0.97718 0.98389 0.97855 0.98296 0.97839 0.97193 0.95836 0.95592
Regression equation : y=14.86488x+1.1118 Correlation coefficient㧦 0.85721
2
Residual
Rock internal friction angle /°
40
35
0
-2
-4 30 2.2 2.2
2.4 2.6 granularity fractal dimension
2.4 Rock granularity fractal dimension
2.8
2.6
Fig. 8. Residual of granularity fractal dimension.
Fig. 7. Rock internal friction angle vs. Granularity fractal dimension.
Table 5 Regression relationships of rock internal friction angle vs. Pore fractal dimension and Granularity fractal dimension (x-fractal dimension, y-rock internal friction angle). Rock structure parameters
Regression equation
Correlation coefficient
Pore fractal dimension Granularity fractal dimension
y¼2.49049x þ29.76931 0.19733 y¼14.86488xþ 1.1118 0.85721
model. Combined with formula (12), the granularity dimension of rock sample No. 1 can be obtained: D2 ¼ 3 0.80569¼2.19431. Taking the similar linear regression and residual analysis for the rest nineteen rock samples as that of rock sample No. 1, and the granularity fractal dimension of these nineteen cores could be obtained and showed in Table 4. Therefore, the rock samples’ granularity size shows obvious fractal characteristics; the particle granularity distribution follows the self similar power–law distribution. In order to study the relationship between particle granularity fractal characteristics and the rock abrasiveness, it is necessary to match each rock sample’s internal friction Angle in Table 1 and particle granularity fractal dimension data in Table 4 with the corresponding rock sample number respectively. The regression curve is shown in Fig. 7, and its regression equation is y¼14.86488
Granularity fractal dimension
2.7
Fitting equation:
y=3.0409x-0.0572
Correlation coefficient: -0.7925 2.4
2.1
0
50
100 150 Mean particle size /um
200
250
Fig. 9. Granularity fractal dimension vs. Mean particle size/mm.
R/s þ1.1118, with a correlation coefficient of 0.85721; see Table 5. The residual analysis is shown in Fig. 8. According to Table 5, rock granularity fractal dimension and the internal friction angle shows a strong correlativity. The higher rock granularity fractal dimension value owns the larger internal friction angle. In brief, the rock abrasiveness becomes larger while the rock internal friction angle is increasing. Therefore, and rock
42
H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 37–42
can serve as guidance for on-site drilling tool selection, rock breaking improvement and can enhance rock-breaking efficiency. Additionally, drill cuttings in this experiment show some advantages of adequate source, lower cost etc. Our main conclusions are as follows:
Granularity fractal dimension
2.7 Fitting equation: y=-2.60178x+2.60924 Correlation coefficient: -0.83941
2.4
2.1 0.00
0.05
0.10
0.15
Porosity Fig. 10. Granularity fractal dimension vs. Porosity.
Table 6 Regression relationships of mean particle size fractal dimension vs. granularity and porosity (x-rock structure parameters, y-granularity fractal dimension). Rock structure parameters Average particle size Porosity
Regression equation 0.0572
y¼ 3.0409x y¼ 2.60178x þ2.60924
Correlation coefficient 0.7925 0.83941
(1) The evaluation method of rock abrasiveness, fractal approach, used in this article is not limited by the size of the rock debris. Therefore, it is very suitable for the PDC bit in drilling intervals with smaller cuttings. (2) Rock pore structure and Rock grain size distribution show typical fractal characteristics, which could fit the self-similar rule statistically. (3) Pore fractal dimension has little relationship with rock abrasiveness, while rock granularity fractal dimension is on intimate terms with rock abrasiveness. It could serve as a characteristic quantity to reflect the level of rock abrasiveness. Moreover, it indicates that the rock abrasiveness is closely related to the rock microstructure. (4) This work established a quantitative relationship model between the rock granularity fractal dimension and the internal friction angle of rock through experimental results: y¼14.86488x þ1.1118. (5) Evaluation test of the rock abrasiveness with drill cuttings in field owns its advantages of adequate source, lower cost and full interval analysis.
Acknowledgments granularity fractal characteristics can be used to determine the level of rock abrasiveness. 3.3. Experimental results analysis Test results show that Rock granularity fractal dimension is closely related to the rock abrasiveness. In order to better explain this phenomenon, it is necessary for us to discuss the relationship among the rock granularity fractal dimension, the particle size of rock and the size of rock porosity. The scatterplot chart as shown in Fig. 9 and Fig. 10, and its regression equation and the correlation coefficient are shown in Table 6. According to the table above, the smaller the mean particle size of rock, the higher the rock grain size fractal dimension. The smaller of the rock porosity, the higher of the rock granularity fractal dimension. Therefore, we can explain the relationship between the grain size fractal dimension and the rock abrasiveness: particle size and distribution of rock is closely related to the mechanical properties. The higher the rock grain size fractal dimension is, the smaller porosity and more density the rock has, accordingly, the more abrasive ability the rock can provide, and vice versa.
4. Conclusion In this paper, we make a series of experiments on rock abrasiveness by the fractal approach. The experiments results
This work is financially supported by National Natural Science Foundation Project (Grant no. 51174220), PetroChina Innovation Foundation (Grant no. 2011–5006–0310), the State Key Laboratory of Petroleum Resource and Prospecting (Grant no. PRPDX2008-08), National Oil and Gas Major Project (Grant no. 2011ZX05009), National 973 Project (Grant no. 2010CB226703).
References [1] He W, Zhong FX. The fractal nature of pore structure and application. Gas Industry 2000;20:67–70 in Chinese. [2] Bagde MN, Raina AK. Rock mass characterization by fractal dimension. Eng Geol 2002;63:141–55. [3] Hlebszevitsch JC, Breda E. A fractal model for the vertical distribution of sands in the Perales Oil Fields in the Basin of Golfo San Jorge. In: Proceedings of the SPE Latin American and Caribbean Petroleum Engineering Conference, Buenos Aires, 2001. Ppaer SPE 69541. [4] Xie HP. Fractal rock mechanics. Beijing: Science Press; 1996 in Chinese. [5] Li KW. Theoretical development of the Brooks-Corey capillary pressure model from fractal modeling of porous media. In: Proceedings of the SPE/DOE 14th Symposium on Improved Oil Recovery, Tulsa, Oklahoma, 2004. Paper SPE 89429. [6] Wei H, Zhang YS. Rock fractal characteristics of microscopic pore distribution. Rock Mech. Eng 2000;19:318–20 in Chinese. [7] Zhang WD, Ge HK. Unconsolidated sand reservoir distribution of particle size. Oil Drilling Tech 2003;31:20–2 in Chinese. [8] Billi A, Storti F. Fractal distribution of particle size in carbonate cataclastic rocks from the core of a regional strike-slip fault zone. Tectonophysics 2004;384:115–28.
Contents lists available at SciVerse ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Technical Note
Experimental studies of rock abrasiveness using a fractal approach Hui Zhang a,n, Deli Gao a, Dongtao Liu b, Boyun Guo c a b c
MOE Key Lab of Petroleum Engineering, China University of Petroleum, Beijing, China Supervision and Technology Company, CNOOC, Shen Zhen, Guang Dong, China Department of Petroleum Engineering, University of Louisiana at Lafayette, Lafayette, Louisiana
a r t i c l e i n f o Article history: Received 1 March 2011 Received in revised form 3 May 2012 Accepted 18 May 2012 Available online 9 June 2012
1. Introduction Currently, fractal studies on rock mechanics have developed into three basic fields. The first one is based on the primary assumption that the natural structure of rock could be regarded as the fractal structure. This is the mathematic foundation and frame of rock mechanics, including reacquainting and re-establishing the physics quantities and laws of physics and mechanics in fractal space [1]. The second direction is the deeper studies of rock mechanics fractal phenomenon, fractal nature and fractal mechanism. It emphasizes and quantifies the descriptions of some complicated physical and mechanical behaviors of fractal mechanism [2]. The third direction is the industrial application of rock mechanics fractal study. Appling this result can solve the practical engineering problems, and obtain quantitative prediction of complicated rock mechanics problems accurately [3]. At the present time, however, studies on rock abrasiveness seldom employ the method of fractal approach. The capacity of rock abrading the cutting edge material called abrasiveness. The existing methods of determining rock nonabrasive are basically as following: drilling and grinding method, grinding method, micro bit drilling method and friction abrasion method. These methods are all indoor-experimental methods with many deficiencies. Firstly, it is difficult to reflect the characteristic of the underground rock in the environment of high temperature and high pressure. Secondly, it is hard to reflect the variation of rock properties in heterogeneous stratum. Thirdly, it is limited by the core material, especially in the areas where lithology varies greatly, since the measured data may lack
n Correspondence to: MOE Key Lab of Petroleum Engineering, China University of Petroleum, No. 18 Fuxue Road, Changping District, Beijing 102249, China. Tel.: þ86 10 89733702. E-mail address: [email protected] (H. Zhang).
1365-1609/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijrmms.2012.05.022
of contrast. Fourthly, it is difficult to establish a continuous section of rock drillability. Lastly, it is a waste of manpower and financial resources. This paper focuses on the microscopic structure of rock. Based on the relationship between the abrasiveness and the internal friction angle of rock [4], combining the rock mechanics parameters in the test with the applying of the fractal geometry theory, this paper mainly discusses the quantitative relationship between the fractal characteristics (rock porosity fractal features and granularity fractal characteristics) and the internal friction angle of rock. Thereby, it can provide an important theoretical foundation for evaluating the abrasiveness of rock during drilling by debris (sand-like) in fabricating yard. This work serves as practical reference for drilling bit selection and parameter optimization.
2. Rock porosity and rock granularity fractal dimension calculation method 2.1. Pore fractal dimension calculation method According to the principle of fractal geometry [5], if the rock porosity distribution accords with fractal structure N(r), the number of rock pores which radius are greater than r, and r follows a power function relationship [6]: Z rmax NðrÞ ¼ rðrÞdr ¼ ArD1 ð1Þ r
where r max is the maximum pore radius of rock, r(r) is the density function of rock porosity distribution, A is a coefficient of proportionality, and D1 is the pore fractal dimension. The expression of r(r) could be obtained by differentiating the above equation with respect to r, then, according to r(r), the accumulated volume V(r) which pore radius less than y is
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H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 37–42
deduced as follows: Z r ABD1 3D1 3D1 rðrÞUBr3 dr ¼ ðr r min Þ VðrÞ ¼ 3D1 r min
ð2Þ
where r min is the minimum pore radius, B is a constant relating to pore structure (B ¼1 when the pore is a cube; B¼ 4p/3 when the pore is a sphere), Similarly, the bulk volume of rock pore Va could be obtained. According to V(r) and Va, we can obtain the accumulated volume fraction S, whose pore radius is less than r: 3D1 r S¼ ð3Þ r max As we know, the resistance of oil–gas migration in arbitrary magnitudes of rock throats follows the equation of capillary pressure: 2scos y pc ðrÞ ¼ r
ð4Þ
where pc is the capillary pressure corresponding to pore radius r, s is the surface tension of liquid, with s ¼ 480 N/m for mercury, y is the contact angle of liquid, with y ¼1401 for mercury. According to (3) and (4), S could be expressed as pc D1 3 ð5Þ S¼ pmin where pmin is the capillary pressure corresponding to the maximum pore radius r max , (the threshold capillary pressure). Taking natural logarithm on both sides of formula (5) yields lnS ¼ ðD1 3Þlnpc þ ð3D1 Þlnpmin
ð6Þ
Formula (6) means that if the rock porosity owns the characteristics of fractal structure, a linear correlation should be noticed between the capillary pressure of wetting properties: lnS and lnpc . According to the regression analysis of correlation coefficient can explain pore fractal structure. Assume the slope of a linear regression is a, and combined with formula (6), the pore fractal dimension formula D1 could be shown as D1 ¼ 3 þa
ð7Þ
accumulated quality of the particles whose diameters are smaller than R, and MT be the total quality of all the particles. Particle size distribution meet Weibull distribution model: a MðRÞ R ¼ 1exp ð9Þ MT s where s is related to the average diameter, usually take the maximum diameter particles, and a is an undetermined constant, called the Weibull Modulus. If R/s 51, we expand the above equation in series and ignore all terms after the second one; this leads to a MðRÞ R ¼ ð10Þ MT s According to (8) and (10), D2 1
R
R3 Ra1 ¼ Ra4
ð11Þ
and then, the rock granularity fractal dimension D2 could be shown as follows: D2 ¼ 3a
ð12Þ
Taking natural logarithm on both sides of formula (10) gives MðRÞ R ¼ aln ð13Þ ln MT s Formula (13) means that if there are fractal distribution characteristics in the grain size of rock, ln[M(R)/MT] and ln(R/s) should be linearly correlated. Linear regression analysis is given of the linear gradient Weibull Modulus a, and then from formula (12), granularity fractal dimension D2 could be obtained. 3. Experiments on fractal characteristics and the correlation Experimental core: 20 pieces of rock cores are selected from Tarim Oilfield, Changqing Oilfield and Zhongyuan Oilfield for this fractal characteristics test. The fundamental information of cores and their internal friction angle are shown in Table 1. Note that the rock’s internal friction angle is obtained from triaxial rock mechanics tests. 3.1. Rock porosity fractal characteristics test
2.2. The rock granularity fractal dimension calculation method Set rock grain diameter as R, through sieve analysis [7], we can get the number N(R) of grain diameter which is greater than R [8] if the number of particles satisfied: Z 1 NðRÞ ¼ rðRÞdR ¼ kRD2 ð8Þ R
where D2 is the grain size fractal dimension, and r(R) is the density function of particle size distribution. Let M(R) be the
The relationship between capillary pressure of rock and saturation of wetting phase (or non-wetting phase) is called capillary pressure curve. There are mainly three methods to determine the capillary pressure curve so far: Semi-permeable partition method, Mercury intrusion method and Centrifuging. The advantages of Mercury intrusion method: it consumes less time in measurement; there are no strict requirements on the shape and size of rock. Mercury intrusion method is used in the experiment of this paper.
Table 1 Basic information of core samples. No.
Rock number
Lithology
Internal friction angle (1)
No.
Rock number
Lithology
Internal friction angle (1)
1 2 3 4 5 6 7 8 9 10
1-122-127 4-1-12 4-1-17 No.1 1-4 5-7-13 9-12-35 2-14-153 2-43-153 6-60-180
Sandstone Sandy mudstone Gray gas bearing fine sandstone Sandstone Grayish brown sandy shale Sandstone Sandstone Grayish brown oil-patched fine sandstone Grayish brown oil-patched fine sandstone Grayish brown oil-patched fine sandstone
32.81 37.39 37.22 36.39 38.78 36.18 37.37 37.35 36.68 36.76
11 12 13 14 15 16 17 18 19 20
6-5-180 09A09 2A 2B 6B6 6A6 09B09 1-1-86 s2 s1
Grayish brown oil-patched fine sandstone Sandstone Sandstone Sandstone Sandstone Sandstone Sandstone Sandy mudstone Sandy mudstone Muddy unequal-granular sandstone
37.57 30.31 34.75 34.52 37.45 35.22 34.79 38.97 40.82 37.06
H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 37–42
Fig. 1 is capillary pressure curve of rock sample No. 1. According to the calculation method of pore fractal dimension that mentioned earlier in this paper. Least square is used in the linear regression
0.0
Rock core㧦NO.1 Regression equation㧦y=-0.68213x+1.24035 Correlation coefficient㧦-0.99098
ln(S)
-0.5
-1.0
-1.5
2
3 ln(Pc)
4
39
analysis of the experimental data of rock sample No. 1. We can get the regression equation: y¼ 0.68213xþ1.24035 (Correlation coefficient¼0.991), which displays a highly matching result. The Residual Analysis result is shown in Fig. 2. The variation range of residual varies in reasonable bounds, indicating a reliable model. Combined with formula (7), we can obtain the pore dimension of rock sample No. 1: D1 ¼3þ( 0.68213)¼ 2.31787. Taking the similar linear regression and residual analysis to the remaining nineteen rock samples as that of rock sample No. 1. The pore fractal dimension of these 19 cores could be obtained and shown in Table 2, where the rock samples’ pore structures reveal obvious fractal characteristics and the pore size distribution follows the self-similar power–law distribution. Matching each rock sample’s internal friction angle in Table 1 and pore fractal dimension data in Table 2 with the corresponding rock sample number respectively, the regression curve obtained by least square regression method is shown in Fig. 3, and its residual analysis is shown in Fig. 4. Both the regression equation and the correlation coefficient are shown in Table 5. Based on the correlation coefficient (0.19733) and the residual of pore fractal dimension showed in Fig. 4, rock pore fractal dimension and the internal friction angle indicates a poor correlativity. Therefore, there is a poor relationship between the rock pore fractal characteristics and the rock abrasiveness, and rock pore fractal characteristics cannot be used to determine the level of rock abrasiveness.
Fig. 1. Capillary pressure curve of rock No. 1.
Regression equation: y=2.49049x+29.7693
40
Correlation coefficient: Rock internal friction angle /°
residual
0.1
0.0
0.19733
35
-0.1 30 2
3 ln pc
4
2.2
Fig. 2. Residual analysis about S and Pc of rock No. 1.
2.4
2.6 2.8 Pore fractal dimension
3.0
Fig. 3. Rock internal friction angle vs. Pore fractal dimension.
Table 2 Porosity fractal dimension of rock sample. No.
Straight line slope
Pore fractal dimension
Correlation coefficient
No.
Straight line slope
Pore fractal dimension
Correlation coefficient
1 2 3 4 5 6 7 8 9 10
0.68213 0.01713 0.29652 0.15576 0.25307 0.26119 0.23639 0.40855 0.71609 0.49095
2.31787 2.98287 2.70348 2.84424 2.74693 2.73881 2.76361 2.59145 2.28391 2.50905
0.99098 0.96326 0.98891 0.99016 0.93725 0.99432 0.99205 0.99532 0.94967 0.99559
11 12 13 14 15 16 17 18 19 20
0.44951 0.29477 0.16547 0.22049 0.37083 0.28619 0.33738 0.17787 0.1905 0.58443
2.55049 2.70523 2.83453 2.77951 2.62917 2.71381 2.66262 2.82213 2.8095 2.41557
0.99184 0.97769 0.99297 0.98933 0.99624 0.98625 0.99787 0.98487 0.99598 0.99403
40
H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 37–42
3.2. Rock particle granularity fractal characteristics tests
0.0
Rock core㧦NO.1 Regression equation㧦y=0.80569x+0.78034 Correlation coefficient㧦0.96769
-0.5 ln(M(R)/MT)
Rock grain size analysis test in Table 1 is carried out after the sample is crushed; the data is shown in Table 3. The fractal curve of test rock granularity distribution could be obtained by linear regression analysis based on formula (13). Then, combined the fractal curve of particle granularity fractal dimension with formula (12), the value of particle granularity fractal dimension of each rock sample and the correlation coefficient could be achieved. Fig. 5 is the Granularity fractal dimension of the experimental data of rock sample No. 1. Least square is used in the linear regression analysis of the experimental data of rock sample No. 1. The regression equation is: y ¼0.80569xþ0.78034, (correlation coefficient: 0.96769), which displays a highly matching result. And the Residual Analysis result is shown in Fig. 6. The variation range of residual varies in reasonable bounds, indicating a reliable
-1.0 -1.5 -2.0 -2.5 -4.0
-3.5
-3.0
-2.5 -2.0 ln(R/Rmax)
-1.5
-1.0
Fig. 5. Granularity fractal dimension of rock sample No. 1.
5 0.4
0.2 Residual
Residual
0
0.0
-5 -0.2
2.2
2.4
2.6 2.8 Pore fractal dimension
-4
3.0
-3
-2
-1
ln(R/Rmax)
Fig. 4. Residual of pore fractal dimension.
Fig. 6. Residual of granularity fractal dimension.
Table 3 Analytical data of particle size (R-particle diameter/mm). Sample no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Sample total weight (g)
20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 19.3 20
Content of various size grade (g) R¼ 1000
R¼ 710
R¼ 500
R¼ 355
R¼ 250
R¼ 180
R¼ 125
R ¼90
R¼ 75
R ¼63
R ¼45
R ¼28
R ¼10
R ¼1
0 0 0 0 0 0 0.01 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0.01 0.02 0 0.04 0.01 0.01 0 0.01 0 0.01 0.02 0.01 0.01 0.03 0.01 0.01 0
0.01 0 0 0.27 0.04 0.01 0.23 0.02 0.01 0.02 0.02 0.01 0.12 1.96 0.21 0.47 0.2 0.02 0.01 0
0.01 0.01 0 4.61 0.16 0.12 2.29 0.03 0.03 0.01 0.01 0.11 3.49 6.67 4.41 5.2 3.66 0.06 0.03 0.02
0.06 0.15 0.06 5.03 0.39 4.18 5.04 0.29 0.56 0.08 0.28 2.01 6.04 3.78 4.91 4.72 6.21 0.42 0.22 0.09
1.08 0.66 0.58 2.68 0.89 5.94 3.46 2.81 3.87 1.25 2.92 5.62 2.79 1.99 2.55 2.44 2.84 1.29 0.65 0.31
6.27 1.3 2.24 1.58 0.99 2.62 1.92 4.73 4.16 4.84 4.7 3.21 1.56 1.16 1.56 1.48 1.55 1.64 0.95 0.82
5.05 4.12 6.81 1.62 2.93 2.25 1.94 4.05 3.45 5.38 4.1 2.71 1.59 1.18 1.63 1.54 1.57 2.98 2.84 5.99
0.59 1.64 1.01 0.27 0.53 0.36 0.36 0.57 0.53 0.79 0.57 0.41 0.28 0.23 0.25 0.36 0.29 1.42 0.92 2.01
1.47 2.31 1.65 0.48 1.34 0.66 0.78 1.32 1.35 1.39 1.05 0.83 0.49 0.32 0.47 0.56 0.49 1.37 1.94 2.72
1.67 2.92 1.89 0.75 3.43 1.04 0.99 1.59 1.53 1.54 1.36 1.42 0.77 0.51 0.65 0.67 0.81 2.28 2.91 2.64
1.49 2.24 1.25 0.51 3.22 0.71 0.75 1.23 1.24 0.97 0.83 0.93 0.45 0.23 0.26 0.5 0.46 2.38 2.88 1.43
0.02 0.02 0.02 0.01 0.03 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.02 0.02 0.03 0.03 0.02
2.28 4.63 4.49 2.18 6.03 2.1 2.17 3.33 3.24 3.71 4.13 2.72 2.4 1.94 3.08 2.03 1.87 6.1 5.91 3.95
H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 37–42
41
Table 4 Rock granularity fractal dimension. No.
Straight line slope
Granularity fractal dimension
Correlation coefficient
No.
Straight line slope
Granularity fractal dimension
Correlation coefficient
1 2 3 4 5 6 7 8 9 10
0.80569 0.56007 0.58362 0.63865 0.42666 0.69873 0.58305 0.63491 0.63436 0.6198
2.19431 2.43993 2.41638 2.36135 2.57334 2.30127 2.41695 2.36509 2.36564 2.3802
0.96769 0.98214 0.95699 0.96864 0.98743 0.96619 0.96793 0.96626 0.97135 0.94943
11 12 13 14 15 16 17 18 19 20
0.54171 0.75816 0.71193 0.80499 0.66114 0.74387 0.77329 0.36844 0.35351 0.59257
2.45829 2.24184 2.28807 2.19501 2.33886 2.25613 2.22671 2.63156 2.64649 2.40743
0.94945 0.9784 0.97718 0.98389 0.97855 0.98296 0.97839 0.97193 0.95836 0.95592
Regression equation : y=14.86488x+1.1118 Correlation coefficient㧦 0.85721
2
Residual
Rock internal friction angle /°
40
35
0
-2
-4 30 2.2 2.2
2.4 2.6 granularity fractal dimension
2.4 Rock granularity fractal dimension
2.8
2.6
Fig. 8. Residual of granularity fractal dimension.
Fig. 7. Rock internal friction angle vs. Granularity fractal dimension.
Table 5 Regression relationships of rock internal friction angle vs. Pore fractal dimension and Granularity fractal dimension (x-fractal dimension, y-rock internal friction angle). Rock structure parameters
Regression equation
Correlation coefficient
Pore fractal dimension Granularity fractal dimension
y¼2.49049x þ29.76931 0.19733 y¼14.86488xþ 1.1118 0.85721
model. Combined with formula (12), the granularity dimension of rock sample No. 1 can be obtained: D2 ¼ 3 0.80569¼2.19431. Taking the similar linear regression and residual analysis for the rest nineteen rock samples as that of rock sample No. 1, and the granularity fractal dimension of these nineteen cores could be obtained and showed in Table 4. Therefore, the rock samples’ granularity size shows obvious fractal characteristics; the particle granularity distribution follows the self similar power–law distribution. In order to study the relationship between particle granularity fractal characteristics and the rock abrasiveness, it is necessary to match each rock sample’s internal friction Angle in Table 1 and particle granularity fractal dimension data in Table 4 with the corresponding rock sample number respectively. The regression curve is shown in Fig. 7, and its regression equation is y¼14.86488
Granularity fractal dimension
2.7
Fitting equation:
y=3.0409x-0.0572
Correlation coefficient: -0.7925 2.4
2.1
0
50
100 150 Mean particle size /um
200
250
Fig. 9. Granularity fractal dimension vs. Mean particle size/mm.
R/s þ1.1118, with a correlation coefficient of 0.85721; see Table 5. The residual analysis is shown in Fig. 8. According to Table 5, rock granularity fractal dimension and the internal friction angle shows a strong correlativity. The higher rock granularity fractal dimension value owns the larger internal friction angle. In brief, the rock abrasiveness becomes larger while the rock internal friction angle is increasing. Therefore, and rock
42
H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 37–42
can serve as guidance for on-site drilling tool selection, rock breaking improvement and can enhance rock-breaking efficiency. Additionally, drill cuttings in this experiment show some advantages of adequate source, lower cost etc. Our main conclusions are as follows:
Granularity fractal dimension
2.7 Fitting equation: y=-2.60178x+2.60924 Correlation coefficient: -0.83941
2.4
2.1 0.00
0.05
0.10
0.15
Porosity Fig. 10. Granularity fractal dimension vs. Porosity.
Table 6 Regression relationships of mean particle size fractal dimension vs. granularity and porosity (x-rock structure parameters, y-granularity fractal dimension). Rock structure parameters Average particle size Porosity
Regression equation 0.0572
y¼ 3.0409x y¼ 2.60178x þ2.60924
Correlation coefficient 0.7925 0.83941
(1) The evaluation method of rock abrasiveness, fractal approach, used in this article is not limited by the size of the rock debris. Therefore, it is very suitable for the PDC bit in drilling intervals with smaller cuttings. (2) Rock pore structure and Rock grain size distribution show typical fractal characteristics, which could fit the self-similar rule statistically. (3) Pore fractal dimension has little relationship with rock abrasiveness, while rock granularity fractal dimension is on intimate terms with rock abrasiveness. It could serve as a characteristic quantity to reflect the level of rock abrasiveness. Moreover, it indicates that the rock abrasiveness is closely related to the rock microstructure. (4) This work established a quantitative relationship model between the rock granularity fractal dimension and the internal friction angle of rock through experimental results: y¼14.86488x þ1.1118. (5) Evaluation test of the rock abrasiveness with drill cuttings in field owns its advantages of adequate source, lower cost and full interval analysis.
Acknowledgments granularity fractal characteristics can be used to determine the level of rock abrasiveness. 3.3. Experimental results analysis Test results show that Rock granularity fractal dimension is closely related to the rock abrasiveness. In order to better explain this phenomenon, it is necessary for us to discuss the relationship among the rock granularity fractal dimension, the particle size of rock and the size of rock porosity. The scatterplot chart as shown in Fig. 9 and Fig. 10, and its regression equation and the correlation coefficient are shown in Table 6. According to the table above, the smaller the mean particle size of rock, the higher the rock grain size fractal dimension. The smaller of the rock porosity, the higher of the rock granularity fractal dimension. Therefore, we can explain the relationship between the grain size fractal dimension and the rock abrasiveness: particle size and distribution of rock is closely related to the mechanical properties. The higher the rock grain size fractal dimension is, the smaller porosity and more density the rock has, accordingly, the more abrasive ability the rock can provide, and vice versa.
4. Conclusion In this paper, we make a series of experiments on rock abrasiveness by the fractal approach. The experiments results
This work is financially supported by National Natural Science Foundation Project (Grant no. 51174220), PetroChina Innovation Foundation (Grant no. 2011–5006–0310), the State Key Laboratory of Petroleum Resource and Prospecting (Grant no. PRPDX2008-08), National Oil and Gas Major Project (Grant no. 2011ZX05009), National 973 Project (Grant no. 2010CB226703).
References [1] He W, Zhong FX. The fractal nature of pore structure and application. Gas Industry 2000;20:67–70 in Chinese. [2] Bagde MN, Raina AK. Rock mass characterization by fractal dimension. Eng Geol 2002;63:141–55. [3] Hlebszevitsch JC, Breda E. A fractal model for the vertical distribution of sands in the Perales Oil Fields in the Basin of Golfo San Jorge. In: Proceedings of the SPE Latin American and Caribbean Petroleum Engineering Conference, Buenos Aires, 2001. Ppaer SPE 69541. [4] Xie HP. Fractal rock mechanics. Beijing: Science Press; 1996 in Chinese. [5] Li KW. Theoretical development of the Brooks-Corey capillary pressure model from fractal modeling of porous media. In: Proceedings of the SPE/DOE 14th Symposium on Improved Oil Recovery, Tulsa, Oklahoma, 2004. Paper SPE 89429. [6] Wei H, Zhang YS. Rock fractal characteristics of microscopic pore distribution. Rock Mech. Eng 2000;19:318–20 in Chinese. [7] Zhang WD, Ge HK. Unconsolidated sand reservoir distribution of particle size. Oil Drilling Tech 2003;31:20–2 in Chinese. [8] Billi A, Storti F. Fractal distribution of particle size in carbonate cataclastic rocks from the core of a regional strike-slip fault zone. Tectonophysics 2004;384:115–28.