Introduction
In deep ice-core drilling, when approaching the bedrock where ice temperature is close to the melting point the mechanical drilling of ice becomes very difficult and unstable, reducing the penetration obtained by single drilling to a few tens of centimetres. This so-called ‘warm-ice problem’ is the most crucial unresolved issue in deep ice-core drilling. It is considered to result from the fact that near the melting point of ice the refreezing of ice chips, melted partially during cutting, blocks the pathway of chips into the chips chamber (Fig. 1a), or refrozen chips on the cutters (Fig. 1b) hinder subsequent cutting.
Fig. 1. (a) Refrozen chips blocking the pathway of chips into chips chamber. (b) Refrozen chips on the cutter.
One possible way of overcoming this problem is to prevent chips from refreezing using an ethanol–water solution (EWS), which is already being used in North Greenland Icecore Project (NorthGRIP) and European Project for Ice Coring in Antarctica (EPICA) drilling (Reference AugustinAugustin and others, 2007a, Reference Augustin, Panichi and Frascatib). The drawbacks of this method are that ethanol might intrude into the grain boundaries of ice cores and influence the properties of ice, and that the extraction of ice cores from the core barrel at the surface becomes very difficult because the ice melted by EWS refreezes between the ice core and core barrel during the drill’s ascent to the surface where the temperature is usually very low.
Another way of approaching the problem is to prevent ice chips from melting by reducing the cutting heat, which would improve the ice-coring performance near the melting point. In metal-cutting technology, this cutting heat has been well studied both theoretically and experimentally. In this paper, we apply the metal-cutting theory to ice and examine the heat generated during coring, taking into account the mechanical and thermal properties of the ice and cutters.
Model
Notation
Forces during cutting
We consider the forces between the ice and each cutter and ice deformation when the cutters of the ice-core drill are cutting ice. Figure 2 shows the force balance between the
Fig. 2. Force balance between ice and cutter.
ice and a cutter. With respect to the cutter, the friction force on the rake plane F T, the normal force on the rake plane N T, the shear force on the shear plane F S and the normal force on the shear plane N S can be expressed as
where F V is the normal force on the ice plane, F H is the tangential force, α is the rake angle of the cutter and is the shear angle. Here, F V, F H and the resultant cutting force R can be expressed as
where τS is the shear stress on the shear plane, t 1 is the cutting depth, b is the cutting width, β is the friction angle and ω is the angle between the shear plane and R. Here we assume that the shear plane coincides with the maximum shear stress plane, which implies ω = 45˚ .
The friction coefficient is defined as
Heat generated by cutting
The total heat generated per unit time by cutting is given by the sum of the heat generated by the shear deformation of ice Q S and the friction heat on the rake plane of the cutter Q f as follows:
where V S is the shear velocity of ice and V C is the chip velocity. Here V is the cutting velocity, as shown in Figure 2, and can be expressed as
where D b is the core barrel diameter and B r is the barrel rotation speed.
Assuming that a ratio (f 1) of the heat generated at the shear zone flows into the chip and a ratio (f 2) of the friction heat generated at the rake plane of the cutter flows into the chip, the ratio of the heat that flows into the chip (R c), cutter (R t) and ice body (R i) to the total heat generated can be expressed as
The temperature increase in the chip due to shear deformation T S can be expressed as
where C i is the specific heat of ice and p i is the density of ice. With regard to the chip, Equation (17) yields the mean temperature rise of the shear plane in terms of the unknown f 1. The same temperature rise is given in the work-piece (ice body) in the vicinity of the shear plane. Reference Loewen and ShawLoewen and Shaw (1954) developed a solution for this workpiece temperature rise using Jaeger’s solution (Reference JaegerJaeger, 1942) for the mean temperature at the interface of a perfectly insulated slider moving across a conducting surface with velocity V as heat is continuously and uniformly supplied to the interface; they obtained the following expression for f 1:
where γ S is the shear strain at the shear plane and K i is the thermal diffusivity of the work-piece material.
As for the temperature rise at the rake plane of the cutter, the friction between the chip and the cutter can be regarded
as a heat source moving in relation to the chip, and at the same time stationary in relation to the cutter. The following solutions to the temperature increase in the chip because of friction T f were developed by Loewen and Shaw (1954):
These solutions are well known in metal cutting and continue to be used as a good approximation. Using these equations, we calculated the total temperature increase in the chips due to the heat generated because of cutting:
Results and Discussion
Friction coefficient and yield shear stress S of ice In order to calculate the heat generated by cutting as described above, we need to determine the values of and S. The friction coefficient between ice and steel (cutter) depends on the temperature and the relative velocity between them. Reference Evans, Nye and CheesemanEvans and others (1976) studied the variation in the friction coefficient for various temperatures and velocities and found that the friction coefficient between ice and steel was proportional to the difference between the ambient temperature and the melting temperature and proportional to –1/2 power of the velocity. Based on these experimental results, we derived the following equation which yields the friction coefficient between the ice and the cutter:
where T m (˚C) is the melting temperature of ice, T (˚C) is the ambient temperature and V C (m s–1) is the chip velocity.
The S value for high-speed deformation such as ice-core drilling is not yet known, so we estimated it from ice-core drilling data at Dome Fuji, Antarctica. Using Equations (6)
and (13), the power consumed for cutting can be written as
Here W for cutting was estimated from the difference in the drill motor power used before and after cutting started. In this calculation, we assumed that the motor efficiency is 70%, although this value changes with torque. As listed in Table 1, with each dataset of α, B r, t 1, T and W, the τS value was calculated using Equations (8), (9), (12), (13), (23) and (24). The cutter width b was 20 mm and the core barrel diameter D b was 94 mm.
Table 1. Dataset of ice-core drilling at Dome Fuji to estimate the yield shear stress of ice for drilling
Figure 3 shows the τS vs T plot obtained using the Dome Fuji drilling data. It has been reported that during the high-speed deformation of ice (dγ/dt >10–4 s–1) under unconfined pressures, the yield strength of ice depends on neither the temperature nor the strain rate, and the maximum value of τS is 10MPa (Reference GoldGold, 1977). Reference TalalayTalalay (2003) reported experimental results on the cutting force of the Russian drill KEMS-112. According to his results, the yield strength of ice is close to the present estimation obtained from the Dome Fuji drilling data. Although the hydrostatic pressure might slightly increase the yield strength (Reference JonesJones, 1982) as compared to the unconfined test results, the present estimation from the drilling data shows the same order of yield strength and slight temperature dependence. Based on this result, we use the following equation to determine τS in the equations to calculate the cutting heat:
where τS is in MPa and T is in ˚C.
Each value of the parameter used in this calculation is listed in Table 2. The yield strength of ice depends on the grain size, crystal orientation, impurity content and so on. We do not take these effects into account in the present study because no detailed study has been performed taking into account these parameters in the high-speed deformation regime.
Table 2. Values of each parameter used in this study
Fig. 3. Yield stress of ice vs temperature T obtained from Dome Fuji drilling.
Chip temperature increase due to cutting
Figure 4 shows the values of T C for the chips (Fig. 4a, c and e) due to the heat generated by cutting, and the resulting chip temperature (Fig. 4b, d and f) calculated for various conditions of ambient (borehole) temperature T, barrel rotation speed B r, penetration pitch p t (=3t 1) and rake angle of the cutter α. In this figure, if we take into account the decrease in the pressure-melting point due to the hydrostatic pressure, the melting-point line should be T C = –2 to –3˚C for a depth of 3000 m. The value of α has more influence on the temperature increase of chips than B r and p t, and this increase in chip temperature becomes 2–4˚C under the condition that is commonly employed in polar deep ice drillings. This means that if the borehole temperature is approximately –5˚C at 3000m depth, the cutting chips should almost melt because of the cutting heat. This result is in very good agreement with the fact that the warm-ice problem is observed above a temperature of –5˚C (Augustin and others, 2007a, b).
Fig. 4. Temperature increase T c of chips (a, c, e) due to heat generation by cutting and the resulting chip temperature (b, d, f) calculated for various conditions with ambient (borehole) temperature T, barrel rotation speed B r, penetration pitch p t (=3t 1) and rake angle of cutter α.
On the other hand, the pressure-melting point decrease due to the additional hydrostatic pressure P s at the shear zone produced by R should also be taken into account in this problem involving the melting of chips. This amount of temperature decrease T p is given by
If p t is 4 mm, α is 40˚C and B r is 50 rpm, then T p is only 0.15˚C. (In this calculation, we used the value of dT/dp (0.074 KMPa–1) for pure water as the Clausius–Clapeyron constant. If we use the value for air-saturated water (0.098 KMPa–1), T p becomes 0.20˚C.) Therefore this contribution to chip melting is very small compared to the heat generated because of cutting.
Heat source and sink in cutting
As shown in Equation (10), the heat generated by cutting is the sum of the contribution from shear deformation of ice Q S and the contribution from friction on the rake plane of the cutter Q f. Figure 5 shows the values of Q S and Q f for various values of α at T = –5˚C, p t = 4 mm and B r = 50 rpm. Most of the heat is contributed by the shear deformation of ice, which can be reduced by increasing α. The heat generated by cutting is redistributed to the chips, cutter and ice body. The ratios of the heat flow to the chips and cutter depend on the ratio of the contact length of the chip on the rake plane of the cutter L to the cutter width b. Figure 6 shows the ratio of the heat flow into the chip R C, cutter R t and ice body R i for various values of L/b. The calculations were performed for two different thermal conductivities of the cutter (30 and 100Wm–1 K–1). This result shows that a larger contact length and larger thermal conductivity reduce the heat flow into the chips since more heat flows into the cutter. The T C values for the chips were calculated for various values of L and the thermal conductivity of the cutter κ t at T = –5˚C, α = 40˚, p t = 4 mm, B r = 50 rpm and b = 20mm (Fig. 7). If the cutter is made of a material with larger thermal conductivity such as ceramics (~100Wm–1 K–1) and a larger contact length (L/b~1), the temperature increase in the chips can be reduced by about 1˚C. However, if hard drilling in warm ice is caused by the refreezing of chips on the cutter, a cutter with low thermal conductivity and a small contact length can help prevent the chips from refreezing on the cutter, as the dissipation of heat from the chips onto the cutter is reduced.
Fig. 5. Comparison of Q s (heat by shear deformation) and Q f (heat by friction) for various α values at T = –5˚C, p t = 4 mm and B r = 50 rpm.
Fig. 6. Ratio of heat flow into the chip (R c), cutter (R t) and ice body (R i) for various values of L/b.
Fig. 7. T c of chips for various contact lengths L and thermal conductivities of the cutter κ t at T = –5˚C, α = 40˚, p t = 4 mm, B r = 50 rpm and b = 20mm.
Effect of fluid properties
Another important issue in the warm-ice problem is the drilling liquid. If the liquid density is smaller than that of the ice, then the cutting chips sink and accumulate at the bottom of the hole. This mixture of chips and liquid (called
‘slush’) produces friction heat between the hole wall and the drill head and raises the ambient temperature around the drill head. Now we examine the temperature increase in the slush, assuming that the space between the hole wall and the drill head is filled with slush from the bottom to height h, as shown in Figure 8. If the drill head rotates with a velocity V, the shear stress τc on a column plane is
where η is the viscosity of the slush and b is the width of the space, i.e. cutter width.
The friction heat generated per unit time by the rotation of this slush column is given by
The temperature-increasing rate of the slush is
where p sl is the density of the slush and C sl is the specific heat of the slush. These values are calculated assuming that 50% of the slush is made of ice chips and the remaining 50% of the drilling liquid (n-butyl acetate). Because the viscosity of this slush is unknown, we presume it to be of the order of 10 Pas on the analogy that the slush is made of water and snow (Reference Kobayashi, Izumi and KamiishiKobayashi and others, 1994). Figure 9 shows the temperature increase in the slush with time for various viscosities of the slush. This suggests that the friction heat because of the rotation of the slush column could raise the ambient temperature of the drill head by several degrees if the liquid density is less than the ice density and the chips remain at the bottom. This might be considered another bad characteristic of the warm-ice problem.
Fig. 8. Schematic diagram of the slush column and drill head at the bottom of the borehole.
Fig. 9. Temperature increase in the slush with time for various slush viscosities.
Conclusions
Based on the metal-cutting theory, we examined the heat generated during ice coring and the following conclusions were drawn:
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1. Most of the heat during cutting is generated by shear deformation at the shear plane of the ice; this heat raises the chip temperature by several degrees under the condition that is commonly used in polar deep ice drillings.
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2. The rake angle has more influence on the temperature increase in chips than the barrel rotation speed and penetration pitch.
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3. If the cutter is made of a material with larger thermal conductivity such as ceramics (~100Wm–1 K–1) and a larger contact length (L/b~1), the temperature increase in the chips can be reduced by about 1˚C.
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4. The friction heat because of the rotation of the slush column can raise the ambient temperature of the drill head by several degrees if the liquid density is less than the ice density and the chips remain at the bottom.
As shown in Equation (17), the increase in the chip temperature T C is proportional to the yield shear stress S. Because the value of τS estimated from the drilling data involves some uncertainty, T C might change by a few degrees. In order to precisely estimate T C, it is imperative to conduct laboratory tests to obtain the yield shear stress of ice for various temperatures, hydrostatic pressures, cutting speeds, grain sizes, crystal orientations, etc. The present analysis shows that a larger rake angle induces less heat at the shear zone. However, a rake angle greater than approximately 50˚ is unrealistic for drilling purposes. Therefore, chips cannot be prevented from melting in the vicinity of the melting point of ice. It is effective to rapidly dissipate the heat generated by shear into the tool section by having a large contact area, with the cutters made from a material having high thermal conductivity because the chips cannot melt immediately at the shear zone. At present, the only effective method of preventing the melted chips from refreezing and blocking the pathway into the chip chamber (which is the main cause of the warm-ice problem) seems to be to inject EWS at the bottom of the hole.
The density of the drilling liquid is another important factor for warm-ice drilling. As shown in Figure 9, if the density of the liquid is less than the density of ice, the cutting chips sink to the bottom and the friction heat generated by the drill head and the slush could raise the ambient temperature of the drill head by a few degrees. We should confirm this problem by laboratory experiments. n-Butyl acetate used at Dome Fuji has a larger density than ice even at 3000 m depth. This seems to have worked successfully in warm-ice drilling, and drilling progresses without difficulty down to 3000 m depth, where the hole temperature is about –3 to –4˚C.
Acknowledgements
We thank R. Greve, J. Schwander, F. Wilhelms and an anonymous reviewer for very helpful comments.
