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Numerical model of laser spallation drilling of inhomogeneous rock⁎
Proceedings the Conference 2017 Johannesburg, Africa, December Africa, 7-8, 2017 Proceedings of of South the Control Control Conference Africa, 2017 Johannesburg, Africa, December 7-8, 2017 Proceedings of South the Control Conference Africa, 2017 Johannesburg, South Africa, December 7-8, 2017 Available online at www.sciencedirect.com Johannesburg, South Africa, December 7-8, 2017
ScienceDirect IFAC PapersOnLine 50-2 (2017) 43–46 Numerical model of laser spallation drilling Numerical model of laser spallation drilling inhomogeneous rock drilling Numericalofmodel of laser spallation of rock of inhomogeneous inhomogeneous rock ∗∗ ∗
Rehema A. Ndeda ∗ Sebusang E Sebusang ∗∗ ∗ Sebusang ∗∗ ∗∗∗ Rehema A. E Rehema A. Ndeda Ndeda E Sebusang Sebusang Rapelang Marumo Erich O. Ogur ∗∗∗∗ ∗ Sebusang ∗∗ ∗∗∗ ∗∗∗∗ Rehema A. Ndeda E Sebusang ∗∗∗ Erich O. ∗∗∗∗ Rapelang Marumo Ogur Rapelang MarumoSebusang Erich O. Ogur ∗∗∗ ∗∗∗∗ Rapelang Marumo Erich O. Ogur ∗ University of Botswana, Private Bag, UB, 0061, Gaborone (e-mail: ∗ ∗ University of Botswana, Private Bag, UB, 0061, Gaborone (e-mail: Private Bag, UB, 0061, Gaborone (e-mail: [email protected]). ∗ University of Botswana, Private ∗∗University of Botswana, [email protected]). [email protected]). University of Botswana, Private Bag, Bag, UB, UB, 0061, 0061, Gaborone Gaborone (e-mail: (e-mail: ∗∗ [email protected]). ∗∗ University of Botswana, Private Bag, UB, 0061, University of Botswana, Private Bag, UB, 0061, Gaborone (e-mail: (e-mail: [email protected]).Gaborone ∗∗ Private Bag, UB, 0061, Gaborone (e-mail: ∗∗∗University of Botswana, [email protected]). [email protected]). University of Botswana, Private Bag, UB, 0061, Gaborone (e-mail: ∗∗∗ [email protected]). ∗∗∗ University of Botswana, Private Bag, UB, 0061, Gaborone (e-mail: University of Botswana, Private Bag, UB, 0061, Gaborone (e-mail: [email protected]). ∗∗∗ Private Bag, UB, 0061, Gaborone (e-mail: ∗∗∗∗University of Botswana, [email protected]). [email protected]). Technical University of Kenya, P. O. Box 52428 00200, Nairobi, ∗∗∗∗ [email protected]). ∗∗∗∗ Technical University of Kenya, P. O. Box 52428 00200, Nairobi, Technical University of Kenya, P. O. Box 52428 00200, Nairobi, Kenya (e-mail: [email protected]) ∗∗∗∗ Technical University of Kenya, P. O. Box 52428 - 00200, Nairobi, Kenya [email protected]) Kenya (e-mail: (e-mail: [email protected]) Kenya (e-mail: [email protected]) Abstract: Spallation of rock during laser drilling is dependent mainly on temperature on Abstract: of laser is mainly temperature on Abstract: Spallation of rock rock during during laser drilling drilling is dependent dependent mainly on temperature on the surface, Spallation material properties and presence of discontinuities within theon rock. A numerical Abstract: Spallation of rock during laser drilling is dependent mainly on temperature on the surface, material properties and presence of discontinuities within the rock. A numerical the surface, material properties and presence of discontinuities rock. A numerical model of laser spallation of heterogeneous granitic rock using thewithin Finitethe Element(FE) method the surface, material properties and presence of discontinuities within the rock. A numerical model of laser spallation of heterogeneous granitic rock using the Finite Element(FE) method model of laserThe spallation of heterogeneous granitic rock using the Finite Element(FE) method is presented. developed model accounts for the presence of microcracks within the rock. model of laser spallation of heterogeneous granitic rock using the Finite Element(FE) method is The developed model accounts for presence of microcracks within the rock. is presented. presented. The developed model accounts for the the presence of of microcracks within the rock. Convective boundary conditions are also considered and the effect propagation of these cracks is presented. developed model accounts theand presence of of microcracks within theresults rock. Convective boundary conditions are considered the propagation of cracks Convective boundary conditions are also also considered and the effect effect of propagation of these these cracks on spallation isThe investigated. Temperature and for stress profiles generated are discussed. The Convective boundary conditions are also considered and the effect of propagation of these cracks on spallation is investigated. Temperature and stress profiles generated are discussed. The results on spallation investigated. on Temperature stress profiles generated discussed. results indicate stressis development the surfaceand of the rock and its effect onare eventual rockThe fracture. on spallation investigated. on Temperature stress profiles generated discussed. results indicate stress the of rock and effect eventual rock fracture. indicate stressis development development on the surface surfaceand of the the rock and its its effect on onare eventual rockThe fracture. indicate stress(International development on the surface of theControl) rock and its effect on eventual © 2017, IFAC Federation of Automatic Hosting by Elsevier Ltd. Allrock rightsfracture. reserved. Keywords: modelling, laser, spallation, finite element, stress. Keywords: modelling, modelling, laser, laser, spallation, spallation, finite finite element, element, stress. stress. Keywords: Keywords: modelling, laser, spallation, finite element, stress. 1. INTRODUCTION perature distribution within the rock, sharp temperature 1. INTRODUCTION INTRODUCTION perature distribution within the sharp temperature 1. perature between distribution within area the rock, rock, sharp temperature gradient the heated and the adjacent regions 1. INTRODUCTION perature distribution within the rock, sharp temperature gradient between the heated area and the adjacent regions gradientby between thethermal heated conductivity area and the adjacent regions the low of most rocks is The study of rock fracture due to thermal means is neces- created gradient between the heated area and the adjacent regions created by the low thermal conductivity of most rocks is The study of rock fracture due to thermal means is necescreated by the low thermal of most rocks is mechanism causingconductivity spallation. This mechanism The study of rock fractureapplications due to thermal means methods is neces- another sitated by the numerous of thermal created by the lowapplicable thermal conductivity of most rocks is another mechanism causing spallation. This mechanism The study of rock fracture due to thermal means is necessitated by the numerous applications of thermal methods another mechanism causing spallation. This mechanism also seems to be for relatively homogeneous sitated by the numerous applications of thermal in underground excavation such as blasting and methods tapping mechanism causing et spallation. mechanism also to for sitated by the resources. numerous applications of thermal in geothermal underground excavation such as as blasting blasting and methods tapping another also seems seems to be bebyapplicable applicable for relatively homogeneous rocks as studied Hartlieb al.relatively (2012).Thishomogeneous in underground excavation such and tapping of In addition, recent developments also seems to be applicable for relatively homogeneous rocks as studied by Hartlieb et al. (2012). in underground excavation such as blasting and tapping of geothermal resources. In addition, recent developments as studied by Hartlieb et al. (2012). of geothermal resources. addition, recent seek to introduce more In thermal means of developments rock drilling, rocks Several been conducted in laser spallation. rocks as studies studied have by Hartlieb et al. (2012). of geothermal resources. In addition, recent developments seek to introduce introduce more thermal means ofcommon. rock drilling, drilling, seek to more thermal means of rock with laser assisted drilling being the most Rock Several studies have been conducted in spallation. Several studies have been conducted in laser laser (2012) spallation. Experimental studies conducted by Ghassemi and seek to introduce more thermal means rock drilling, with laser assisteddue drilling being the most common. Rock Several with laser assisted drilling being the most Rock is heterogeneous to its constitution ofofcommon. different grain studies have been conducted in laser spallation. Experimental studies conducted by Ghassemi (2012) and Experimental studies conducted by Ghassemi (2012) and Kobayashi et al. (2009) have been useful in determining with laser assisted drilling being the most common. Rock is heterogeneous heterogeneous due tosizes its constitution constitution of different different grain Experimental is its of grain types, grain sizes,due poreto and microcracks. Application studies conducted by Ghassemi (2012) and Kobayashi et al. (2009) have been useful in determining Kobayashi et al. (2009) have been useful in determining the onset of cracking in rock exposed to laser radiais heterogeneous due to its constitution of different grain types, grain sizes, pore sizes and microcracks. Application types, grainloads sizes, on pore sizesinitially and microcracks. Application of thermal loads causes cracking, with Kobayashi et al. (2009) have been useful in determining the onset of cracking in rock exposed to laser radiathe onset of the cracking in rockmade exposed to laser radiaAmong observations were the variation of types, grain sizes, pore sizes and microcracks. Application of thermal thermal loads on loads initially causes cracking, cracking, with tion. of loads on loads initially causes with further heating causing melting. the onset of the cracking in rock exposed toaslaser radiation. Among observations made were the variation of tion. Among the observations made were the variation crack density with increase in temperature well as the of thermal loadscausing on loads initially causes cracking, with tion. Among the observations made were the variation of further heating causing melting. further heating melting. of crack density with increase in temperature as well as the crack density withthe increase temperature as well as the within grains in and grain boundaries. These The useheating of laserscausing has been lauded due to the reduced wear variations further melting. crack density with increase in temperature as well as the variations within the grains and grain boundaries. These The use of lasers has been lauded due to the reduced wear variations within the grains and grain boundaries. These The use of lasers has been lauded due as to well the reduced because their non-contact nature, as theirwear ver- works have enabled the determination of several threshold the grains and grain These works enabled the determination of several threshold The useinof lasers has been of lauded due as to well the reduced because their of non-contact nature, as well as(Soleymani theirwear ver- variations works have havewithin enabled the determination ofboundaries. several threshold temperatures which have formed a suitable baseline for because ofterms their non-contact nature, as their versatility means material removal works have enabled the determination of several threshold temperatures which have formed a suitable baseline because of their non-contact nature, as well as their versatility in terms terms of means means ofused material removal (Soleymani temperatures which The haveneed formed a suitable modelling baseline for for attempts. for numerical is satility in of material removal et al., 2013). Lasers can beof to either cause(Soleymani melting of modeling which have formed a suitable modelling baseline for modeling numerical is satility inorterms of means material removal (Soleymani et al., al., 2013). Lasers canwhich beofused used to either either causestresses meltingare of temperatures modeling attempts. attempts. The need for numerical modelling is underpinned by theThe factneed thatfor experimental studies are et 2013). Lasers can be to cause melting of the rock spallation, means thermal modeling attempts. The need for numerical modelling is underpinned by the fact that experimental studies are et al., 2013). Lasers can be used to either cause melting of the rock or spallation, which means thermal stresses are underpinned by the fact that experimental studies are not able to fully describe the thermal stress evolution for the rockinorthe spallation, which thermal The stresses induced rock in order to means cause fracture. focusare of underpinned by the fact that experimental studies are not able to fully describe the thermal stress evolution for the rockin orthe spallation, which thermal The stresses are induced in the rock in order order to means cause fracture. The focus of microcrack not able to development fully describeinthe stress evolution for thethermal rock due to the expansion induced rock in to cause fracture. focus of research on laser spallation has been primarily been on the ablerock. to development fully describein stress evolution for microcrack the rock to the expansion induced rock in order toparameters cause fracture. The of not research in onthe laser spallation has been primarily primarily beenfocus on and the microcrack development inthe thethermal rock due due to computational the expansion of the In addition, improvement in research on laser spallation has been been on the mechanism of spallation, the of spallation microcrack development in the rock due to the expansion of the rock. In addition, improvement in computational research on laser spallation has been primarily been on the mechanism of spallation, the parameters of spallation and of the rock. In addition, improvement in computational as well as numerical methodology enables the mechanism the parameters of spallation and capacity the behaviorofofspallation, the different rock types during spallation the rock. In addition, improvement in computational capacity well as numerical methodology enables mechanism ofof thehas parameters of spallation and of the behavior behavior ofspallation, the different rock types during spallation capacity ascomplex well as phenomena. numerical methodology enables the the analysis ofas the the different rock types during spallation (Olaleye, 2010). Spallation been attributed to several capacity ascomplex well as phenomena. numerical methodology enables the analysis of the behavior of the different rock types during spallation (Olaleye, 2010). Spallation has been attributed to several analysis of complex phenomena. (Olaleye, 2010). Spallation has been attributed to several mechanisms. The presence of different minerals with dif- Numerical methods have sufficient popularity in the study analysis of complex phenomena. (Olaleye, 2010). Spallation has been attributed to several mechanisms. The presencecauses of different different minerals with difmechanisms. The presence of minerals with ferent rates of expansion increased strain at difthe of Numerical methods have sufficient popularity in the the study Numerical have sufficient popularity in thermal methods effects on rock on both the macroandstudy mimechanisms. The presence of different minerals with different rates of expansion causes increased strain at the ferent ratesofofthe expansion at the boundaries minerals,causes hence increased promotingstrain cracking. In Numerical methods have sufficient popularity in the study of thermal effects on rock on both the macroand miof thermal effects on rock on both the macroand microscale levels. The finite element method has been used in ferent rates expansion causes increased strain at the boundaries ofofthe the minerals, hence of promoting cracking. In of boundaries of minerals, hence promoting cracking. In addition, variation of the location the different minerals thermal effects on rock on both the macroand microscale levels. The finite element method has been used in croscale levels. The finite element method has been used in the investigation of rock cracking due to thermal exposure. boundaries of the minerals, hence promoting cracking. In addition, variation of the the locationvariation of the the different different minerals croscale addition, of location of minerals within thevariation rock, caused a distinct in the behavior levels. The finite element method has been used in the investigation of rock cracking due to thermal exposure. the investigation of rock cracking to element thermal exposure. Agha et al. (2004) developed a due finite model to addition, variation of the location of the different minerals within the rock, caused a distinct variation in the behavior within the rock, distinct variation theal., behavior of the rock masscaused (Brkica et al., 2015; Walshin et 2012). the investigation ofand rock cracking due to element thermal exposure. Agha et al. (2004) developed aa penetration finite model to Agha et al. (2004) developed finite element model to predict the rate depth of during laser within thenecessitated rock, a et distinct variation theal., behavior of the thehas rock masscaused (Brkic et al., 2015; 2015; Walshin et et al., 2012). of rock mass (Brkic al., Walsh 2012). This the inclusion of heterogeneity in any Agha al.without (2004) developed finite element model to predict rate and depth during laser predictetthe the rate and depth of ofa penetration penetration during laser consideration of melting. This model of the rock mass (Brkic et al., 2015; Walsh et al., 2012). This has necessitated the inclusion of heterogeneity in any machining This necessitated the inclusion of heterogeneity any studyhas attempts into thermal spallation. In terms ofin tempredict the rate and depth of penetration during laser machining without consideration of melting. This model machining without consideration melting. This model inhomogeneities within theofrock. A more inclusive This necessitated the inclusion of heterogeneity any ignored studyhas attempts into thermal thermal spallation. In terms terms of ofintemtemstudy attempts into spallation. In machining withoutbyconsideration melting. This model ignored inhomogeneities within theof(2012) rock. Afocused more inclusive ignored inhomogeneities within the rock. A more inclusive model developed Walsh et al. on the study attempts into thermal spallation. In terms of temFunding was provided by the IEEE Control Systems Society Outignored inhomogeneities within the (2012) rock. Alevel more inclusive model developed by Walsh et al. focused on the model developed by Walsh et al. (2012) focused on the behavior of granitic rock on a grain scale during hyby Control Society Out Funding reach Fund was and provided the INTRA-ACP (METEGA) Scholarship Program. Funding was provided by the the IEEE IEEE Control Systems Systems Society Outmodel developed byrock Walsh al. (2012) focused on the behavior of granitic on aaetgrain scale level during hy behavior of granitic rock on grain scale level during hyreach Fund and the INTRA-ACP (METEGA) Scholarship Program. Funding was provided by the IEEE Control Systems Society Outreach Fund and the INTRA-ACP (METEGA) Scholarship Program. behavior of granitic rock on a grain scale level during hyreach Fund and the INTRA-ACP (METEGA) Scholarship Program.
Copyright © 2017 IFAC 43 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 43 Copyright © under 2017 IFAC IFAC 43 Control. Peer review responsibility of International Federation of Automatic Copyright © 2017 IFAC 43 10.1016/j.ifacol.2017.12.008
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analyzed numerically with constant temperature applied to one boundary. The temperature profile, T(r) for the annular sample along the radial distance, r can be theoretically calculated from (5). Ta In( rb ) + Tb In( ar ) T (r) = (5) In( ab ) Figure 1 shows the comparison between the theoretical solution obtained from Equation 5 and the solution obtained from ANSYS. There is good agreement between these results, validating the accuracy of the ANSYS Model.
drothermal spallation. The model determined that stress concentration was affected by properties of the rock surface such as thermal conductivity and grain distribution within the material. The research, however, used a fluid jet as the source of thermal energy, which differs distinctly from the coherent and directional nature of laser energy. Variation of sources of energy, varies the behavior of the rock, with some sources capable of irradiating wider areas while other sources such as laser only illuminate a smaller spot. Research into the effects of the laser on rock has primarily focused on analyzing laser-rock interaction with a view to understanding the specific energy and rate of penetration into the rock(Brkic et al., 2015; Adeniji, 2014). There have been few numerical studies into the influence of laser on rock damage. This study seeks to develop a numerical model for thermal stress development in granite rock when exposed to laser power. Heterogeneity on the rock surface is proposed through statistical distribution of grains with different mechanical properties throughout the granite rock. The temperature and stress distributions within the rock is discussed and the effect on spallation is determined. The accuracy of the model is validated by comparison to theoretical solutions.
Fig. 1. Comparison between temperature profiles obtained from theoretical and numerical calculations.
2. GOVERNING EQUATIONS Governing equations of the model will be classified into two: heat transfer within the rock and rock deformation due to the stress generated. Laser beam of intensity, I, is directed on the rock and is assumed to have a Gaussian distribution. Heat conduction through the rock is described by (1). ∂T k∇2 T = ρc (1) ∂t Material properties are represented by thermal conductivity, k, density, ρ and specific heat, c. Heat loss due to radiation is assumed to be negligible and hence ignored. The relationship between the stress and strain components in the rock due to the temperature effects are defined by σij = λkkδij + 2Gij − β∆T δij (2) The change in temperature, ∆T is the difference between the current temperature and the initial temperature. The thermal stress coefficient, β is denoted by β = (3λ + 2G)α (3) where α is the thermal expansion coefficient. λ and G are the Lame constant and shear modulus respectively. The strain tensor, ij can also be defined in terms of the displacement vector, ui 1 ij = (ui,j + uj,i ) (4) 2
According to Timoshenko and Gere (1972), radial stress at each point on the body can be calculated from Equation 6. The verification of the thermal stress results are realized in Figure 2. b2 − 1 Eα(Ta − Tb ) In( rb ) 2 − rb2 (6) σr = − 2(1 − ν) In( ab ) a2 − 1
Fig. 2. Verification of thermal stress model 3.2 Creation of the physical model
3. NUMERICAL MODEL
A three dimensional geometry is created in ANSYS with radius of 15mm and thickness of 10mm. In order to capture heterogeneity of the rock, two minerals are considered, that is, quartz and feldspar. The properties of the minerals vary according to Table 1 as recorded by Yu et al. (2015). The laser heat flux is loaded on one boundary of the workpiece and the thermal load used as input to the stress model.
3.1 Verification of the Model In order to determine the suitability of the numerical model developed in ANSYS, comparison is made between both the temperature and thermal stress values and the theoretical values. A simple annular model with internal radius a of 5 mm and external radius, b of 30mm is 44
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Table 1. Granite Material Properties Property Density(g/cm3 ) Elastic Modulus(GPa) Poisson’s ratio Thermal expansion coefficient(e−6 /o C) Friction angle(o ) Uniaxial Tensile Strength (MPa)
Quartz 2.65 96 0.08 11 60 47
Feldspar 2.57 67 0.27 6 40 33
Fig. 4. Temperature distribution with depth within the material. Stress distribution in the rock is shown using the Von mises stress distribution in Figure 5. The region impinged by the laser experiences high stress of 361 MPa. According to studies conducted by Eberhardt (1998), damage begins when microcracking is initiated, and this occurs at 20 30% of the compressive strength of the material. This stress exceeds the compressive strength of the weaker mineral, feldspar, causing it to fracture. In addition, there is cracking within the quartz mineral since it exceeds the 30% threshold value. This agrees with the experimental observations of Brkic et al. (2015) and Chaki et al. (2008) who observed increased acoustic emission between the temperatures above 500o C denoting increased cracking. Further, this cracking was accompanied with the ejection of flakes or spalls off the surface. This can be attributed to the large temperature gradient between the heated region and those surrounding it.
The variation in thermal conductivity, k, within the rock with temperature between 20◦ and 600o C follows equation 7 as indicated by (Yu et al., 2015). T − 20 −1 (7) k = k0 − (k0 − 2.01) exp T + 130 where k0 is the thermal conductivity at ambient temperature. Once the geometry is created, the finite element mesh is generated and suitably refined in order to ensure accuracy as well as computational speed. The heat conduction equation is solved to determine the temperature distribution through the rock, which are subsequently used as load for the structural problem in order to determine thermal stress. 4. RESULTS AND DISCUSSION Since laser spallation is a static process, the laser beam is stationary on the surface of the workpiece. The power of the laser beam is high enough to initiate spallation but maintained just below the melting point of granite which is 1100◦ C. Figure 3 shows that the highest temperature on the center of the rock where the laser is focused is highest at 679◦ C and 931◦ C for laser heat flux of 0.75 MW/m2 and 1.0 MW/m2 respectively. According to experimental studies by Brkic et al. (2015), spallation begins at 500◦ C upto 700◦ C. Laser heat flux of 0.75MW/m2 is suitable for spallation since temperature is within the spallation range. A sharp decrease in temperature in the regions immediately surrounding the rock is observed, and is attributed to the poor thermal conductivity of granite. Because of the low thermal conductivity of granite, the temperature drop within the rock is very sharp, indicating the presence of a large temperature gradient. This is also a factor that aids spallation as indicated by Ndeda et al. (2017). This is also observed in Figure 4, which shows the temperature dissipation within the material.
Fig. 5. Stress distribution on the surface of the rock. In order to validate the results obtained from the numerical model, the results were compared to an analytical model used by Xu et al. (2004) according to Equation 8. √ y 2q κt √ (8) ierfc T = k 2 κt The temperature variation, T, with depth, y, is mainly k influenced by the thermal diffusivity of the rock, κ = ρc where k, ρ and c are the thermal conductivity, density and specific heat of the rock respectively. q represents the laser heat flux. Figure 6 shows the change in temperature with depth within the rock. 5. CONCLUSION A finite element model of laser cutting of granite using ANSYS finite element software is presented. The temperature and stress distribution results are analyzed. Results
Fig. 3. Temperature distribution within the rock after the heating period 45
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Ndeda, R., Sebusang, S.E., Marumo, R., and Ogur, E.O. (2017). On the role of laser pulses on spallation of granite. Lasers in Manufacturing and Materials Processing, 1–16. Olaleye, M. (2010). A review of light amplification by stimulated emission of radiation in oil and gas well drilling. Mining Science and Technology (China), 20(5), 752–757. Soleymani, M., Bakhtbidar, M., and Kazemzadeh, E. (2013). Experimental analysis of laser drilling impacts on rock properties. World Appl Sci J, 1(2), 106–114. Timoshenko, S. and Gere, J. (1972). Mechanics of materials. Van Nostrand Reinhold Co. Walsh, S.D., Lomov, I., Kanarska, Y., and Roberts, J.J. (2012). Simulation tools for modeling thermal spallation drilling on multiple scales. Technical report, Lawrence Livermore National Laboratory (LLNL), Livermore, CA. Xu, Z., Reed, C.B., Parker, R., and Graves, R. (2004). Laser spallation of rocks for oil well drilling. In Proceedings of the 23rd International Congress on Applications of Lasers and Electro-Optics, 1–6. Citeseer. You, M. (2015). Strength criterion for rocks under compressive-tensile stresses and its application. Journal of Rock Mechanics and Geotechnical Engineering, 7(4), 434–439. Yu, Q., Ranjith, P., Liu, H., Yang, T., Tang, S., Tang, C., and Yang, S. (2015). A mesostructure-based damage model for thermal cracking analysis and application in granite at elevated temperatures. Rock Mechanics and Rock Engineering, 48(6), 2263–2282.
Fig. 6. Change in temperature with depth within the rock indicate that the stresses due to increase in temperature exceed the strength of the weakest mineral within the rock, while causing cracking within the stronger mineral. These weaknesses eventually cause spallation. The model is validated using analytical results. While these studies underscore the usefulness of thermal stress in the fracture of rock and specifically in spallation, there is need for economic evaluation of the laser spallation process, indicating the cost implications. On-going research into the efficiency of laser energy as a power source and spall removal from drilling site will also enhance the possibility of industrial use of lasers. REFERENCES Adeniji, A.W. (2014). The applications of laser technology in downhole operations-a review. In IPTC 2014: International Petroleum Technology Conference. Agha, K.R., Belhaj, H.A., Mustafiz, S., Bjorndalen, N., and Islam, M.R. (2004). Numerical investigation of the prospects of high energy laser in drilling oil and gas wells. Petroleum science and technology, 22(9-10), 1173– 1186. Brkic, D., Kant, M., Meier, T., Schuler, M.J., and von Rohr, R. (2015). Influence of process parameters on thermal rock fracturing under ambient conditions. In Proceedings World Geothermal Congress 2015, Melbourne, Australia. Chaki, S., Takarli, M., and Agbodjan, W. (2008). Influence of thermal damage on physical properties of a granite rock: porosity, permeability and ultrasonic wave evolutions. Construction and Building Materials, 22(7), 1456–1461. Eberhardt, E. (1998). Brittle rock fracture and progressive damage in uniaxial compression. Ph.D. thesis, University of Saskatchewan Saskatoon. Ghassemi, A. (2012). A review of some rock mechanics issues in geothermal reservoir development. Geotechnical and Geological Engineering, 30(3), 647–664. Hartlieb, P., Leindl, M., Kuchar, F., Antretter, T., and Moser, P. (2012). Damage of basalt induced by microwave irradiation. Minerals Engineering, 31, 82–89. Kobayashi, T., Kubo, S., Ichikawa, M., Nakamura, M., et al. (2009). Drilling a 2-inch in diameter hole in granites submerged in water by co2 lasers. In SPE/IADC Drilling Conference and Exhibition. Society of Petroleum Engineers. 46
ScienceDirect IFAC PapersOnLine 50-2 (2017) 43–46 Numerical model of laser spallation drilling Numerical model of laser spallation drilling inhomogeneous rock drilling Numericalofmodel of laser spallation of rock of inhomogeneous inhomogeneous rock ∗∗ ∗
Rehema A. Ndeda ∗ Sebusang E Sebusang ∗∗ ∗ Sebusang ∗∗ ∗∗∗ Rehema A. E Rehema A. Ndeda Ndeda E Sebusang Sebusang Rapelang Marumo Erich O. Ogur ∗∗∗∗ ∗ Sebusang ∗∗ ∗∗∗ ∗∗∗∗ Rehema A. Ndeda E Sebusang ∗∗∗ Erich O. ∗∗∗∗ Rapelang Marumo Ogur Rapelang MarumoSebusang Erich O. Ogur ∗∗∗ ∗∗∗∗ Rapelang Marumo Erich O. Ogur ∗ University of Botswana, Private Bag, UB, 0061, Gaborone (e-mail: ∗ ∗ University of Botswana, Private Bag, UB, 0061, Gaborone (e-mail: Private Bag, UB, 0061, Gaborone (e-mail: [email protected]). ∗ University of Botswana, Private ∗∗University of Botswana, [email protected]). [email protected]). University of Botswana, Private Bag, Bag, UB, UB, 0061, 0061, Gaborone Gaborone (e-mail: (e-mail: ∗∗ [email protected]). ∗∗ University of Botswana, Private Bag, UB, 0061, University of Botswana, Private Bag, UB, 0061, Gaborone (e-mail: (e-mail: [email protected]).Gaborone ∗∗ Private Bag, UB, 0061, Gaborone (e-mail: ∗∗∗University of Botswana, [email protected]). [email protected]). University of Botswana, Private Bag, UB, 0061, Gaborone (e-mail: ∗∗∗ [email protected]). ∗∗∗ University of Botswana, Private Bag, UB, 0061, Gaborone (e-mail: University of Botswana, Private Bag, UB, 0061, Gaborone (e-mail: [email protected]). ∗∗∗ Private Bag, UB, 0061, Gaborone (e-mail: ∗∗∗∗University of Botswana, [email protected]). [email protected]). Technical University of Kenya, P. O. Box 52428 00200, Nairobi, ∗∗∗∗ [email protected]). ∗∗∗∗ Technical University of Kenya, P. O. Box 52428 00200, Nairobi, Technical University of Kenya, P. O. Box 52428 00200, Nairobi, Kenya (e-mail: [email protected]) ∗∗∗∗ Technical University of Kenya, P. O. Box 52428 - 00200, Nairobi, Kenya [email protected]) Kenya (e-mail: (e-mail: [email protected]) Kenya (e-mail: [email protected]) Abstract: Spallation of rock during laser drilling is dependent mainly on temperature on Abstract: of laser is mainly temperature on Abstract: Spallation of rock rock during during laser drilling drilling is dependent dependent mainly on temperature on the surface, Spallation material properties and presence of discontinuities within theon rock. A numerical Abstract: Spallation of rock during laser drilling is dependent mainly on temperature on the surface, material properties and presence of discontinuities within the rock. A numerical the surface, material properties and presence of discontinuities rock. A numerical model of laser spallation of heterogeneous granitic rock using thewithin Finitethe Element(FE) method the surface, material properties and presence of discontinuities within the rock. A numerical model of laser spallation of heterogeneous granitic rock using the Finite Element(FE) method model of laserThe spallation of heterogeneous granitic rock using the Finite Element(FE) method is presented. developed model accounts for the presence of microcracks within the rock. model of laser spallation of heterogeneous granitic rock using the Finite Element(FE) method is The developed model accounts for presence of microcracks within the rock. is presented. presented. The developed model accounts for the the presence of of microcracks within the rock. Convective boundary conditions are also considered and the effect propagation of these cracks is presented. developed model accounts theand presence of of microcracks within theresults rock. Convective boundary conditions are considered the propagation of cracks Convective boundary conditions are also also considered and the effect effect of propagation of these these cracks on spallation isThe investigated. Temperature and for stress profiles generated are discussed. The Convective boundary conditions are also considered and the effect of propagation of these cracks on spallation is investigated. Temperature and stress profiles generated are discussed. The results on spallation investigated. on Temperature stress profiles generated discussed. results indicate stressis development the surfaceand of the rock and its effect onare eventual rockThe fracture. on spallation investigated. on Temperature stress profiles generated discussed. results indicate stress the of rock and effect eventual rock fracture. indicate stressis development development on the surface surfaceand of the the rock and its its effect on onare eventual rockThe fracture. indicate stress(International development on the surface of theControl) rock and its effect on eventual © 2017, IFAC Federation of Automatic Hosting by Elsevier Ltd. Allrock rightsfracture. reserved. Keywords: modelling, laser, spallation, finite element, stress. Keywords: modelling, modelling, laser, laser, spallation, spallation, finite finite element, element, stress. stress. Keywords: Keywords: modelling, laser, spallation, finite element, stress. 1. INTRODUCTION perature distribution within the rock, sharp temperature 1. INTRODUCTION INTRODUCTION perature distribution within the sharp temperature 1. perature between distribution within area the rock, rock, sharp temperature gradient the heated and the adjacent regions 1. INTRODUCTION perature distribution within the rock, sharp temperature gradient between the heated area and the adjacent regions gradientby between thethermal heated conductivity area and the adjacent regions the low of most rocks is The study of rock fracture due to thermal means is neces- created gradient between the heated area and the adjacent regions created by the low thermal conductivity of most rocks is The study of rock fracture due to thermal means is necescreated by the low thermal of most rocks is mechanism causingconductivity spallation. This mechanism The study of rock fractureapplications due to thermal means methods is neces- another sitated by the numerous of thermal created by the lowapplicable thermal conductivity of most rocks is another mechanism causing spallation. This mechanism The study of rock fracture due to thermal means is necessitated by the numerous applications of thermal methods another mechanism causing spallation. This mechanism also seems to be for relatively homogeneous sitated by the numerous applications of thermal in underground excavation such as blasting and methods tapping mechanism causing et spallation. mechanism also to for sitated by the resources. numerous applications of thermal in geothermal underground excavation such as as blasting blasting and methods tapping another also seems seems to be bebyapplicable applicable for relatively homogeneous rocks as studied Hartlieb al.relatively (2012).Thishomogeneous in underground excavation such and tapping of In addition, recent developments also seems to be applicable for relatively homogeneous rocks as studied by Hartlieb et al. (2012). in underground excavation such as blasting and tapping of geothermal resources. In addition, recent developments as studied by Hartlieb et al. (2012). of geothermal resources. addition, recent seek to introduce more In thermal means of developments rock drilling, rocks Several been conducted in laser spallation. rocks as studies studied have by Hartlieb et al. (2012). of geothermal resources. In addition, recent developments seek to introduce introduce more thermal means ofcommon. rock drilling, drilling, seek to more thermal means of rock with laser assisted drilling being the most Rock Several studies have been conducted in spallation. Several studies have been conducted in laser laser (2012) spallation. Experimental studies conducted by Ghassemi and seek to introduce more thermal means rock drilling, with laser assisteddue drilling being the most common. Rock Several with laser assisted drilling being the most Rock is heterogeneous to its constitution ofofcommon. different grain studies have been conducted in laser spallation. Experimental studies conducted by Ghassemi (2012) and Experimental studies conducted by Ghassemi (2012) and Kobayashi et al. (2009) have been useful in determining with laser assisted drilling being the most common. Rock is heterogeneous heterogeneous due tosizes its constitution constitution of different different grain Experimental is its of grain types, grain sizes,due poreto and microcracks. Application studies conducted by Ghassemi (2012) and Kobayashi et al. (2009) have been useful in determining Kobayashi et al. (2009) have been useful in determining the onset of cracking in rock exposed to laser radiais heterogeneous due to its constitution of different grain types, grain sizes, pore sizes and microcracks. Application types, grainloads sizes, on pore sizesinitially and microcracks. Application of thermal loads causes cracking, with Kobayashi et al. (2009) have been useful in determining the onset of cracking in rock exposed to laser radiathe onset of the cracking in rockmade exposed to laser radiaAmong observations were the variation of types, grain sizes, pore sizes and microcracks. Application of thermal thermal loads on loads initially causes cracking, cracking, with tion. of loads on loads initially causes with further heating causing melting. the onset of the cracking in rock exposed toaslaser radiation. Among observations made were the variation of tion. Among the observations made were the variation crack density with increase in temperature well as the of thermal loadscausing on loads initially causes cracking, with tion. Among the observations made were the variation of further heating causing melting. further heating melting. of crack density with increase in temperature as well as the crack density withthe increase temperature as well as the within grains in and grain boundaries. These The useheating of laserscausing has been lauded due to the reduced wear variations further melting. crack density with increase in temperature as well as the variations within the grains and grain boundaries. These The use of lasers has been lauded due to the reduced wear variations within the grains and grain boundaries. These The use of lasers has been lauded due as to well the reduced because their non-contact nature, as theirwear ver- works have enabled the determination of several threshold the grains and grain These works enabled the determination of several threshold The useinof lasers has been of lauded due as to well the reduced because their of non-contact nature, as well as(Soleymani theirwear ver- variations works have havewithin enabled the determination ofboundaries. several threshold temperatures which have formed a suitable baseline for because ofterms their non-contact nature, as their versatility means material removal works have enabled the determination of several threshold temperatures which have formed a suitable baseline because of their non-contact nature, as well as their versatility in terms terms of means means ofused material removal (Soleymani temperatures which The haveneed formed a suitable modelling baseline for for attempts. for numerical is satility in of material removal et al., 2013). Lasers can beof to either cause(Soleymani melting of modeling which have formed a suitable modelling baseline for modeling numerical is satility inorterms of means material removal (Soleymani et al., al., 2013). Lasers canwhich beofused used to either either causestresses meltingare of temperatures modeling attempts. attempts. The need for numerical modelling is underpinned by theThe factneed thatfor experimental studies are et 2013). Lasers can be to cause melting of the rock spallation, means thermal modeling attempts. The need for numerical modelling is underpinned by the fact that experimental studies are et al., 2013). Lasers can be used to either cause melting of the rock or spallation, which means thermal stresses are underpinned by the fact that experimental studies are not able to fully describe the thermal stress evolution for the rockinorthe spallation, which thermal The stresses induced rock in order to means cause fracture. focusare of underpinned by the fact that experimental studies are not able to fully describe the thermal stress evolution for the rockin orthe spallation, which thermal The stresses are induced in the rock in order order to means cause fracture. The focus of microcrack not able to development fully describeinthe stress evolution for thethermal rock due to the expansion induced rock in to cause fracture. focus of research on laser spallation has been primarily been on the ablerock. to development fully describein stress evolution for microcrack the rock to the expansion induced rock in order toparameters cause fracture. The of not research in onthe laser spallation has been primarily primarily beenfocus on and the microcrack development inthe thethermal rock due due to computational the expansion of the In addition, improvement in research on laser spallation has been been on the mechanism of spallation, the of spallation microcrack development in the rock due to the expansion of the rock. In addition, improvement in computational research on laser spallation has been primarily been on the mechanism of spallation, the parameters of spallation and of the rock. In addition, improvement in computational as well as numerical methodology enables the mechanism the parameters of spallation and capacity the behaviorofofspallation, the different rock types during spallation the rock. In addition, improvement in computational capacity well as numerical methodology enables mechanism ofof thehas parameters of spallation and of the behavior behavior ofspallation, the different rock types during spallation capacity ascomplex well as phenomena. numerical methodology enables the the analysis ofas the the different rock types during spallation (Olaleye, 2010). Spallation been attributed to several capacity ascomplex well as phenomena. numerical methodology enables the analysis of the behavior of the different rock types during spallation (Olaleye, 2010). Spallation has been attributed to several analysis of complex phenomena. (Olaleye, 2010). Spallation has been attributed to several mechanisms. The presence of different minerals with dif- Numerical methods have sufficient popularity in the study analysis of complex phenomena. (Olaleye, 2010). Spallation has been attributed to several mechanisms. The presencecauses of different different minerals with difmechanisms. The presence of minerals with ferent rates of expansion increased strain at difthe of Numerical methods have sufficient popularity in the the study Numerical have sufficient popularity in thermal methods effects on rock on both the macroandstudy mimechanisms. The presence of different minerals with different rates of expansion causes increased strain at the ferent ratesofofthe expansion at the boundaries minerals,causes hence increased promotingstrain cracking. In Numerical methods have sufficient popularity in the study of thermal effects on rock on both the macroand miof thermal effects on rock on both the macroand microscale levels. The finite element method has been used in ferent rates expansion causes increased strain at the boundaries ofofthe the minerals, hence of promoting cracking. In of boundaries of minerals, hence promoting cracking. In addition, variation of the location the different minerals thermal effects on rock on both the macroand microscale levels. The finite element method has been used in croscale levels. The finite element method has been used in the investigation of rock cracking due to thermal exposure. boundaries of the minerals, hence promoting cracking. In addition, variation of the the locationvariation of the the different different minerals croscale addition, of location of minerals within thevariation rock, caused a distinct in the behavior levels. The finite element method has been used in the investigation of rock cracking due to thermal exposure. the investigation of rock cracking to element thermal exposure. Agha et al. (2004) developed a due finite model to addition, variation of the location of the different minerals within the rock, caused a distinct variation in the behavior within the rock, distinct variation theal., behavior of the rock masscaused (Brkica et al., 2015; Walshin et 2012). the investigation ofand rock cracking due to element thermal exposure. Agha et al. (2004) developed aa penetration finite model to Agha et al. (2004) developed finite element model to predict the rate depth of during laser within thenecessitated rock, a et distinct variation theal., behavior of the thehas rock masscaused (Brkic et al., 2015; 2015; Walshin et et al., 2012). of rock mass (Brkic al., Walsh 2012). This the inclusion of heterogeneity in any Agha al.without (2004) developed finite element model to predict rate and depth during laser predictetthe the rate and depth of ofa penetration penetration during laser consideration of melting. This model of the rock mass (Brkic et al., 2015; Walsh et al., 2012). This has necessitated the inclusion of heterogeneity in any machining This necessitated the inclusion of heterogeneity any studyhas attempts into thermal spallation. In terms ofin tempredict the rate and depth of penetration during laser machining without consideration of melting. This model machining without consideration melting. This model inhomogeneities within theofrock. A more inclusive This necessitated the inclusion of heterogeneity any ignored studyhas attempts into thermal thermal spallation. In terms terms of ofintemtemstudy attempts into spallation. In machining withoutbyconsideration melting. This model ignored inhomogeneities within theof(2012) rock. Afocused more inclusive ignored inhomogeneities within the rock. A more inclusive model developed Walsh et al. on the study attempts into thermal spallation. In terms of temFunding was provided by the IEEE Control Systems Society Outignored inhomogeneities within the (2012) rock. Alevel more inclusive model developed by Walsh et al. focused on the model developed by Walsh et al. (2012) focused on the behavior of granitic rock on a grain scale during hyby Control Society Out Funding reach Fund was and provided the INTRA-ACP (METEGA) Scholarship Program. Funding was provided by the the IEEE IEEE Control Systems Systems Society Outmodel developed byrock Walsh al. (2012) focused on the behavior of granitic on aaetgrain scale level during hy behavior of granitic rock on grain scale level during hyreach Fund and the INTRA-ACP (METEGA) Scholarship Program. Funding was provided by the IEEE Control Systems Society Outreach Fund and the INTRA-ACP (METEGA) Scholarship Program. behavior of granitic rock on a grain scale level during hyreach Fund and the INTRA-ACP (METEGA) Scholarship Program.
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analyzed numerically with constant temperature applied to one boundary. The temperature profile, T(r) for the annular sample along the radial distance, r can be theoretically calculated from (5). Ta In( rb ) + Tb In( ar ) T (r) = (5) In( ab ) Figure 1 shows the comparison between the theoretical solution obtained from Equation 5 and the solution obtained from ANSYS. There is good agreement between these results, validating the accuracy of the ANSYS Model.
drothermal spallation. The model determined that stress concentration was affected by properties of the rock surface such as thermal conductivity and grain distribution within the material. The research, however, used a fluid jet as the source of thermal energy, which differs distinctly from the coherent and directional nature of laser energy. Variation of sources of energy, varies the behavior of the rock, with some sources capable of irradiating wider areas while other sources such as laser only illuminate a smaller spot. Research into the effects of the laser on rock has primarily focused on analyzing laser-rock interaction with a view to understanding the specific energy and rate of penetration into the rock(Brkic et al., 2015; Adeniji, 2014). There have been few numerical studies into the influence of laser on rock damage. This study seeks to develop a numerical model for thermal stress development in granite rock when exposed to laser power. Heterogeneity on the rock surface is proposed through statistical distribution of grains with different mechanical properties throughout the granite rock. The temperature and stress distributions within the rock is discussed and the effect on spallation is determined. The accuracy of the model is validated by comparison to theoretical solutions.
Fig. 1. Comparison between temperature profiles obtained from theoretical and numerical calculations.
2. GOVERNING EQUATIONS Governing equations of the model will be classified into two: heat transfer within the rock and rock deformation due to the stress generated. Laser beam of intensity, I, is directed on the rock and is assumed to have a Gaussian distribution. Heat conduction through the rock is described by (1). ∂T k∇2 T = ρc (1) ∂t Material properties are represented by thermal conductivity, k, density, ρ and specific heat, c. Heat loss due to radiation is assumed to be negligible and hence ignored. The relationship between the stress and strain components in the rock due to the temperature effects are defined by σij = λkkδij + 2Gij − β∆T δij (2) The change in temperature, ∆T is the difference between the current temperature and the initial temperature. The thermal stress coefficient, β is denoted by β = (3λ + 2G)α (3) where α is the thermal expansion coefficient. λ and G are the Lame constant and shear modulus respectively. The strain tensor, ij can also be defined in terms of the displacement vector, ui 1 ij = (ui,j + uj,i ) (4) 2
According to Timoshenko and Gere (1972), radial stress at each point on the body can be calculated from Equation 6. The verification of the thermal stress results are realized in Figure 2. b2 − 1 Eα(Ta − Tb ) In( rb ) 2 − rb2 (6) σr = − 2(1 − ν) In( ab ) a2 − 1
Fig. 2. Verification of thermal stress model 3.2 Creation of the physical model
3. NUMERICAL MODEL
A three dimensional geometry is created in ANSYS with radius of 15mm and thickness of 10mm. In order to capture heterogeneity of the rock, two minerals are considered, that is, quartz and feldspar. The properties of the minerals vary according to Table 1 as recorded by Yu et al. (2015). The laser heat flux is loaded on one boundary of the workpiece and the thermal load used as input to the stress model.
3.1 Verification of the Model In order to determine the suitability of the numerical model developed in ANSYS, comparison is made between both the temperature and thermal stress values and the theoretical values. A simple annular model with internal radius a of 5 mm and external radius, b of 30mm is 44
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Table 1. Granite Material Properties Property Density(g/cm3 ) Elastic Modulus(GPa) Poisson’s ratio Thermal expansion coefficient(e−6 /o C) Friction angle(o ) Uniaxial Tensile Strength (MPa)
Quartz 2.65 96 0.08 11 60 47
Feldspar 2.57 67 0.27 6 40 33
Fig. 4. Temperature distribution with depth within the material. Stress distribution in the rock is shown using the Von mises stress distribution in Figure 5. The region impinged by the laser experiences high stress of 361 MPa. According to studies conducted by Eberhardt (1998), damage begins when microcracking is initiated, and this occurs at 20 30% of the compressive strength of the material. This stress exceeds the compressive strength of the weaker mineral, feldspar, causing it to fracture. In addition, there is cracking within the quartz mineral since it exceeds the 30% threshold value. This agrees with the experimental observations of Brkic et al. (2015) and Chaki et al. (2008) who observed increased acoustic emission between the temperatures above 500o C denoting increased cracking. Further, this cracking was accompanied with the ejection of flakes or spalls off the surface. This can be attributed to the large temperature gradient between the heated region and those surrounding it.
The variation in thermal conductivity, k, within the rock with temperature between 20◦ and 600o C follows equation 7 as indicated by (Yu et al., 2015). T − 20 −1 (7) k = k0 − (k0 − 2.01) exp T + 130 where k0 is the thermal conductivity at ambient temperature. Once the geometry is created, the finite element mesh is generated and suitably refined in order to ensure accuracy as well as computational speed. The heat conduction equation is solved to determine the temperature distribution through the rock, which are subsequently used as load for the structural problem in order to determine thermal stress. 4. RESULTS AND DISCUSSION Since laser spallation is a static process, the laser beam is stationary on the surface of the workpiece. The power of the laser beam is high enough to initiate spallation but maintained just below the melting point of granite which is 1100◦ C. Figure 3 shows that the highest temperature on the center of the rock where the laser is focused is highest at 679◦ C and 931◦ C for laser heat flux of 0.75 MW/m2 and 1.0 MW/m2 respectively. According to experimental studies by Brkic et al. (2015), spallation begins at 500◦ C upto 700◦ C. Laser heat flux of 0.75MW/m2 is suitable for spallation since temperature is within the spallation range. A sharp decrease in temperature in the regions immediately surrounding the rock is observed, and is attributed to the poor thermal conductivity of granite. Because of the low thermal conductivity of granite, the temperature drop within the rock is very sharp, indicating the presence of a large temperature gradient. This is also a factor that aids spallation as indicated by Ndeda et al. (2017). This is also observed in Figure 4, which shows the temperature dissipation within the material.
Fig. 5. Stress distribution on the surface of the rock. In order to validate the results obtained from the numerical model, the results were compared to an analytical model used by Xu et al. (2004) according to Equation 8. √ y 2q κt √ (8) ierfc T = k 2 κt The temperature variation, T, with depth, y, is mainly k influenced by the thermal diffusivity of the rock, κ = ρc where k, ρ and c are the thermal conductivity, density and specific heat of the rock respectively. q represents the laser heat flux. Figure 6 shows the change in temperature with depth within the rock. 5. CONCLUSION A finite element model of laser cutting of granite using ANSYS finite element software is presented. The temperature and stress distribution results are analyzed. Results
Fig. 3. Temperature distribution within the rock after the heating period 45
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Ndeda, R., Sebusang, S.E., Marumo, R., and Ogur, E.O. (2017). On the role of laser pulses on spallation of granite. Lasers in Manufacturing and Materials Processing, 1–16. Olaleye, M. (2010). A review of light amplification by stimulated emission of radiation in oil and gas well drilling. Mining Science and Technology (China), 20(5), 752–757. Soleymani, M., Bakhtbidar, M., and Kazemzadeh, E. (2013). Experimental analysis of laser drilling impacts on rock properties. World Appl Sci J, 1(2), 106–114. Timoshenko, S. and Gere, J. (1972). Mechanics of materials. Van Nostrand Reinhold Co. Walsh, S.D., Lomov, I., Kanarska, Y., and Roberts, J.J. (2012). Simulation tools for modeling thermal spallation drilling on multiple scales. Technical report, Lawrence Livermore National Laboratory (LLNL), Livermore, CA. Xu, Z., Reed, C.B., Parker, R., and Graves, R. (2004). Laser spallation of rocks for oil well drilling. In Proceedings of the 23rd International Congress on Applications of Lasers and Electro-Optics, 1–6. Citeseer. You, M. (2015). Strength criterion for rocks under compressive-tensile stresses and its application. Journal of Rock Mechanics and Geotechnical Engineering, 7(4), 434–439. Yu, Q., Ranjith, P., Liu, H., Yang, T., Tang, S., Tang, C., and Yang, S. (2015). A mesostructure-based damage model for thermal cracking analysis and application in granite at elevated temperatures. Rock Mechanics and Rock Engineering, 48(6), 2263–2282.
Fig. 6. Change in temperature with depth within the rock indicate that the stresses due to increase in temperature exceed the strength of the weakest mineral within the rock, while causing cracking within the stronger mineral. These weaknesses eventually cause spallation. The model is validated using analytical results. While these studies underscore the usefulness of thermal stress in the fracture of rock and specifically in spallation, there is need for economic evaluation of the laser spallation process, indicating the cost implications. On-going research into the efficiency of laser energy as a power source and spall removal from drilling site will also enhance the possibility of industrial use of lasers. REFERENCES Adeniji, A.W. (2014). The applications of laser technology in downhole operations-a review. In IPTC 2014: International Petroleum Technology Conference. Agha, K.R., Belhaj, H.A., Mustafiz, S., Bjorndalen, N., and Islam, M.R. (2004). Numerical investigation of the prospects of high energy laser in drilling oil and gas wells. Petroleum science and technology, 22(9-10), 1173– 1186. Brkic, D., Kant, M., Meier, T., Schuler, M.J., and von Rohr, R. (2015). Influence of process parameters on thermal rock fracturing under ambient conditions. In Proceedings World Geothermal Congress 2015, Melbourne, Australia. Chaki, S., Takarli, M., and Agbodjan, W. (2008). Influence of thermal damage on physical properties of a granite rock: porosity, permeability and ultrasonic wave evolutions. Construction and Building Materials, 22(7), 1456–1461. Eberhardt, E. (1998). Brittle rock fracture and progressive damage in uniaxial compression. Ph.D. thesis, University of Saskatchewan Saskatoon. Ghassemi, A. (2012). A review of some rock mechanics issues in geothermal reservoir development. Geotechnical and Geological Engineering, 30(3), 647–664. Hartlieb, P., Leindl, M., Kuchar, F., Antretter, T., and Moser, P. (2012). Damage of basalt induced by microwave irradiation. Minerals Engineering, 31, 82–89. Kobayashi, T., Kubo, S., Ichikawa, M., Nakamura, M., et al. (2009). Drilling a 2-inch in diameter hole in granites submerged in water by co2 lasers. In SPE/IADC Drilling Conference and Exhibition. Society of Petroleum Engineers. 46