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УДК 539.422
DOI: 10.18303/2618-981 X-2018-3 -44-49
ПРИМЕНЕНИЕ НЕЛОКАЛЬНОЙ МОДЕЛИ НЕУПРУГИХ ДЕФОРМАЦИЙ К ДИНАМИЧЕСКОЙ ЗАДАЧЕ РАЗРУШЕНИЯ ГОРНОЙ ПОРОДЫ
Егор Владимирович Вторушин
Новосибирский технологический центр компании «Бейкер Хьюз», 630090, Россия, г. Новосибирск, ул. Кутателадзе, 4А, кандидат физико-математических наук, научный сотрудник, тел. (383)332-94-43, e-mail: [email protected]
Виталий Николаевич Доровский
Новосибирский технологический центр компании «Бейкер Хьюз», 630090, Россия, г. Новосибирск, ул. Кутателадзе, 4А, доктор физико-математических наук, советник по науке, тел. (383)332-94-43, e-mail: [email protected]
В лабораторных условиях, имитирующих трещиноватость породы при бурении, выявлены следующие закономерности: при увеличении скорости резания объемное разрушение уменьшает Размер трещиноватого слоя по гиперболическому закону, а сила сопротивления резанию линейно возрастает. Анализ решения двумерной задачи раскроя горных пород режущим долотом в рамках нелокальной упругопластической теории подтвердил экспериментальные результаты. Решение позволяет определить размер скальной стружки.
Ключевые слова: трещиноватость, вырубка, неупругие деформации.
NONLOCAL MODEL OF INELASTIC DEFORMATIONS APPLIED TO DYNAMIC PROBLEM OF ROCK CUTTING
Egor V. Vtorushin
Baker Hughes' Novosibirsk Technology Center, 4A, Kutateladze St., Novosibirsk, 630090, Russia, Ph. D., Researcher, phone: (383)332-94-43, e-mail: [email protected]
Vitaly N. Dorovsky
Baker Hughes' Novosibirsk Technology Center, 4A, Kutateladze St., Novosibirsk, 630090, Russia, D. Sc., Science Advisor, phone: (383)332-94-43, e-mail: [email protected]
The following consistent patterns have been revealed in the laboratory conditions imitating rock fracturing while drilling: as the cutting velocity increases, the bulk failure reduces the size of a fractured layer following the hyperbolic law, while the cutting resistance force increases linearly. Analysis of the solution of a 2D problem of rock cutting with a cutting-shearing bit within the framework of nonlocal elastoplastic theory has confirmed the experimental results. The solution allows one to determine the rock chip size.
Key words: rock fracturing, rock cutting, inelastic deformations.
The behavior of a drill string while drilling as well as the effects of the high-frequency torsional oscillations produced by cutting-shearing bit are determined by rock/cutter interaction. In this respect, laboratory experiments describing the interaction of a single cutter and fracturing rock become of crucial importance. A series of such experiments was carried out by K. Borisov [1], who experimentally established
the consistent patterns of rock/cutter interaction while cutting. In particular, he verified the physical reason of rock fracturing when the last is cut with a cutting-shearing cutter. Having analyzed Borisov's experimental results and G. Cherepanov's theoretical works [2] we have disproved the theory [3] that the rock fracturing is determined by the velocity characteristics of dry friction force relaxation. While analysis of drillstring high-frequency oscillations registered in the field has failed to confirm the theory [4], it has confirmed the validity of the experimental results obtained by K. Borisov, which has intensified our interest in his work, especially in terms of the nonlocal elasticity theory [4-6].
The main patterns of rock fracturing while cutting can be reduced to the three following assumptions: i) at constant cutting velocity and load on cutter, the cutting process enters in a bulk failure regime with a certain depth of cut (DOC); ii) DOC reduces with increasing cutting velocity according to the hyperbolic law (Fig. 1); iii) the cutting resistance force increases with the cutter's velocity (Fig. 2).
Fig. 1. The experimental hyperbolic dependence of DOC on cutting velocity
250
Pi 0
5.7 34 79 158 235
Cutter velocity V, cm/sec
Fig. 2. Cutting resistance force in bulk failure regime. Dynamic strengthening
while cutting
In order to explain the dependence of cutting force on cutting velocity as well as DOC reduction with cutting-velocity increase, the nonlocal theory of elastoplastic deformations was introduced. The main ideas behind the theory as well as analysis of the separated steady state can be found in [4]. The effects of the nonsteady solutions, their properties and corresponding equation system have been studied in [5]. Here, the theorization is reduced to including into the first law of thermodynamics dE0 = TdS + hikdgik /2 + ydR of the invariant R = dkdkgvv - didkgik from the second
derivative of the metric deformation tensor gik, which allows obtaining a class of spatial periodic solutions and determining the characteristic size of a rock chip. Zones with the maximum value of the parameter R (destruction) are identified as zones of rock failure, where E0, S, T denote the inner energy, volumetric unit entropy and temperature respectively. The time evolution [5] of the parameter y = P • svv + a • R conjugated with the destruction can be seen in Fig. 3. The parameters P, a determine the input of the destruction into the system's energy (see equation (1)), and svv denotes the trace of the deformation tensor sik. Dynamically, the process is followed by an abrupt increase in time of the first maximum value, its gradual reduction and a relatively fast growth of the second maximum, and the process covers all the other zones of the extreme value y*.
Fig. 3. Time evolution of the conjugated parameter y
_ _ i/2
The spatial distribution of the shear plastic deformations (<§ik <$ik ) can be seen in Fig. 4 (2D problem solution [7]). On the Fig. 3 the y parameter goes along the line connecting the cutter bottom in the direction of axis L while the cutter is moving with the constant velocity v.
The system below describes the dynamics of continuous medium in presence of the destruction field R and its conjugated parameter y in the frame of non-local model of inelastic deformations
Pv - dk°ik = 0, ¿jk ~(dkVj +d,vk)/2 = -~plk - cpjjbjk /3, aik = Xs jj Sik + 2№ik + pRSik, R/ 2 = "As jj + dtdk sik, ^jj= <^(ajj- 4Ay), = s[(aik - ajjSik / 3)+2(aAy - AySik/3)]^ (1)
2 2 Eq/ p = const + Tq(S / p - 50) + te.jj / 2p + ^sikI p + PRsjj I p + aR / 2p,
where p denotes the density and X, ^ - the elastic moduli, a^ - the stress tensor determined by the Murnaghan formulas, v - the elastic continuum's velocity of motion.
Fig. 5 demonstrates the solution of equation (1): y max to be the maximum value of the conjugated parameter y is calculated along axis L as a function of cutter motion for several sizes or rock layers cut while bulk failure.
Fig. 5. Maximum value y as a function of cutting velocity Now let us fix the value y = y*, as it is shown in Fig. 5. What we obtain are the hyperbolic dependences (Fig. 6) of DOC as a function of cutter's velocity for the fixed rock (y*).
Fig. 6. DOC while bulk failure as a function of cutter's velocity for different kinds of rock
The obtained solutions allow us to calculate the cutting force Fcut (v) (Fig. 7) that provides constant cutting velocity.
—DOFO,8mm O Borisov Lab data 0.75 mm
300
0
0 50 100 150 200 250 300
Cutter Velocity, cm/sec
Fig. 7. Cutting force/velocity Fcut (v) dependence. The red dots indicate the results obtained by K. Borisov and their non-linear character
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Thus, the two main results of the experimental dependences determining the main patterns of rock fracturing with a cutting-shearing cutter such as the dynamic rock strengthening Fcut (v) and the reduction of cut layer size h (v) find their explanations within the framework of the nonlocal theory of elastoplastic deformation in the assumption that rock bulk failure is followed by the maximum value of the conjugated parameter.
REFERENCES
1. Borisov K. I. A Scientific Approach to Evaluating Effectiveness of Dynamic Processes of Rock Failure While Drilling Wellbores Using Modern Shearing Tools: Dissertation for Doctor of Science in Engineering. Tomsk, 2012.
2. Cherepanov G. P. Theory of rock cutting// Strength of Materials. 1986. 18(8). P. 11031114.
3. Ford Brett J. The Genesis of Torsional Drillstring Vibrations// SPE, Drilling Engineering, September, 1992.
4. Zhang Z., Shen Y., Chen W., Bonstaff J. Shi. W., Tang K. Smith D. L., Arevalo Y. I. Continuous High Frequency Measurement Improves Understanding of High Frequency Torsional Oscillation in North America Land Drilling.
5. Guzev M. A., Paroshin A. A. NonEuclidean Model of the Zonal Disintegration of Rocks around an Underground Working // Journal of Applied Mechanics and Technical Physics. 2001. 42(1) P. 131-139.
6. Dorovsky V. N., Romensky E. I., Sinev A. V. Spatially non-local model of inelastic deformations: applications for rock failure problem // Geophysical Prospecting. 2015. 63(4) P. 11981212.
7. Vtorushin E. V., Dorovsky V. N. Nonlocal model of inelastic deformations applied to dynamic problem of rock fracturing // XIV International Exhibition and Scientific Congress "Interexpo GEO-Siberia 2017" Depths. Mining. Directions and technologies of quest, exploration and development of deposits. Economy. Geoecology. 2017 Vol. 3, P. 77-81.
8. Vtorushin E. V. Application of mixed finite elements to spatially non-local model of inelastic deformations// International Journal on Geomathematics. 2016. 7(2) P. 183-201.
© E. B. BmopywuH, B. H. ffopoecKUU, 2018
