1 Introduction

Lunar regolith is of great significance to the study of the nature of the Moon and its evolution^[1]. China's Chang'e-5 lunar probe will carry out lunar drilling activities in the near future. A coring drill with the help of an auger is one of the most effective tools to acquire the scientific information trapped inside the geologic formations in the sub-surface of the Moon^[2]. However, owing to the extreme environment, including low gravity, low temperature, and vacuum, drilling activities on the Moon are different from those on the Earth. In addition, because of the large distance between the Earth and the Moon, which results in a light-time communication delay, the drilling activities cannot be controlled in time, which means that a highly autonomous drilling system is necessary. Therefore, the drilling mechanism should be fully understood.

Many researchers have made efforts to explain the drilling mechanism by modeling and experimental verification. Mellor^[3] developed a mass-ramp model to predict the conveying properties in the helical groove of an auger for terrestrial drilling. Zacny and Cooper^[4] used a similar approach to analyze the removal of cuttings by an auger for drilling on Mars. Following Mellor's model, many researchers^[5-9] carried out theoretical analyses for the auger drill with various hypotheses to simplify the conveying procedure of drilling. With regard to the experimental studies, Zacny et al.^[10] summarized a series of data of drilling tests, and presented some empirical formulas for drilling power. Tang et al.^[11] used an industrial camera to record the flowing characteristics of removed cuttings and utilized an ultrasonic sensor in the hollow stem to monitor the core recovery. However, a dynamical model of the whole drilling process including conveyance and coring, which can also be verified by experiments, has not been proposed.

In this study, a drilling model involving the whole dynamics of the removal of cuttings and the collection of samples is established explicitly. Using the new model, the complicated relations among the loads on the drill, the coring amount, and the drilling parameters are first uncovered. Experiments are also performed to verify the model.

This paper is arranged as follows. Section 2 proposes a simplified drilling model including the conveyance and sampling mechanisms. Section 3 presents the experimental results, which are consistent with the simulation. Section 4 analyzes the whole drilling process that can be classified into three stages. Conclusions are drawn in Section 5.

2 Simplified model for drilling 2.1 Conveyance mechanism 2.1.1 Motion of soil chips in the conveying channel

Take an infinitesimal segment of the conveyed materials along the conveyance queue in the groove of the auger, as shown in Fig. 1. Let s be the length of the spiral along the groove, and w be the position of a segment in the queue relative to the helix. The position of the element ξ=s+w can be expressed by r=[x, y, z]^T in a fixed inertial frame, where

(1)

where R represents the radius of the auger, h is the screw pitch, and Here, u is the axial position of the starting point of the helix on the auger. The unit vectors such as t, n, and b representing the Frenet frames at s can be obtained by taking the partial derivatives of Eq. (1) with respect to s and letting w=0 as

(2)

where s=s/A, γ=R/A, and ς=h/(2πA).

Taking the derivatives with respect to time on both sides of Eq. (1) leads to the velocity and acceleration of the element, They are expressed as

(3)

where the dimensionless velocities, as well as the dimensionless accelerations, and are defined. Moreover, v_a is the dimensionless absolute velocity of the segment. Note that is the conveying speed.

2.1.2 Force analysis of the soil chips

According to the momentum conservation law focusing on the Euler field ΔV of the queue of cuttings with its boundary ΔS, one immediately has

(4)

where v represents the velocity of each position in V, and F and P are the volume force and stress, respectively. To avoid identifying a complicated constitutive relation of the granular material, many researchers have adopted several hypotheses to obtain some analytical predictions of the conveying dynamics^{[3-4, 7-9, 11]}. In this article, each field ΔV is taken as an element containing a rectangular cross-section Γ=ab_a with a width a and a height b_a ≤ b for a small piece of the cuttings queue, d s, where b is the height of the groove. It can be seen that represents the flux of the conveying channel. Its interactions with the stem and the borehole wall are subject to Coulomb friction. In addition, the density of the cuttings is constant. Then, the dynamic equation (4) results in

(5)

where the positive conveyance velocity is assumed to guarantee negative frictional directions at the interface between the groove and the conveyance queue of cuttings. In Eq. (5), μ₁ is the coefficient of friction at the surface of the groove, and μ₂ is the one between the cuttings and the borehole wall. The dimensionless variables and parameters above are set as follows:

2.1.3 Relations of stresses to elements of the soil

Unlike solid materials, the correlation within the components of the stress at a segment of the conveyance queue is dependent on the moving boundary. The bottom component of the stress on the flute of the groove is related closely to the lateral stress at the borehole. The lateral friction increases the bottom stress, and in turn strengthens the lateral friction itself. To figure out how these components of stress at the conveyance queue correlate, Janssen's model is introduced here. In Fig. 1(b), a segment in the conveying channel is considered to be quasi-statical, and the horizontal normal stress is proportional to the vertical normal stress, which is expressed as

(6)

where K is Janssen's constant coefficient.

In the vertical direction b, we can write the force balance for a slice db of the material in a segment of the granules (see Fig. 1(b)) as

(7)

where α is the helical angle, and β denotes the angle between the friction τ and the tangential direction of the conveying groove. Here, we assume that all the materials in the segment have the same velocity. Thus, the direction of the friction is invariant on the height b at a segment. sin β=v_b/v_a and cos β=v_t/v_a, where v_a is the absolute velocity of the segment, and v_t and v_b are its two components in the tangential direction t and the binormal direction b.

Integrate the differential equation (7) from a bottom boundary σ_{b, 2} to an upper boundary σ_{b, 1} with the lateral average components of the stress defined as

(8)

to obtain correlations among the components of the stress in the conveyance queue, which satisfies

(9)

where

in which

2.1.4 Governing equations of conveyance

Because a stable drilling process is required in actual engineering problems, we set the accelerations in the dynamic equations (5) to be zero as Then, they become

(10)

The variables in the equations are time-independent for a stable conveying process. To simplify the model, we assume that the velocities of the conveyed segments are consistent and independent of the position s at the queue.

By substituting the stress characteristics in Eq. (9) into the governing equations (10), a differential equation with respect to dp and the dimensionless conveying velocity is given as

(11)

where

According to the conveyance in the groove of the auger, two cases need to be analyzed. One represents an unsaturated state where b_a < b, and the other is a saturated state where b_a=b. First, when the auger's groove is not fully saturated or marginally saturated, namely, b_a ≤ b, the upper component of stress on each segment is zero, σ_{b, 1}=0. Put σ_{b, 1}=0 into Eq. (9) to obtain σ_{n, 1}=p=ν₂σ_{b, 2}. Then, we substitute them into the second and the third equations of (10) to have a constant p(s), namely, dp=0. Thus, the differential equation (11) degenerates to the algebraic equations of the pressure p and the conveying velocity ,

(12)

According to the relations σ_{n, 1}=p=ν₂σ_{b, 2} with a constant p obtained above, we can quickly draw a conclusion that the stress distribution in the conveying channel is almost constant.

In the second case, the conveying channel is full, and the redundant soil accumulation leads to high pressure at the bit and the inlet of the groove. As a consistent conveying velocity along the groove independent of the position s has been assumed, the differential equation (11) can be integrated along the auger as

(13)

where C₁ is a constant, determined by an initial condition, S denotes the dimensionless length of the conveyance, S=S/A=2πH/h, and H represents the current length of the drill stem under the ground.

We set the boundary condition on the surface as p(S)=p^*. By putting it into Eq. (13), the constant C₁ is obtained as Thus, by Eq. (13), the pressure at the bit p(0) is related to the outside pressure p^* as follows:

(14)

As Eqs. (12) and (14) reveal the relation between the conveying capacity and the pressure at the bottom, in the following we discuss how the pressure at the bottom forms and affects the conveying state.

2.2 Sampling mechanism

In the drilling process, in addition to the soil (cuttings) removed by the auger, part of the in situ soil enters the core tube. As shown in the sketch in Fig. 2(a), with the penetration of the drill, the flexible tube, which is dragged by a rigid rope in the hollow tube, stretches to wrap around regolith samples. The upward speed of the tube relative to the stem is obviously the feeding speed Here, we define a coring speed representing the speed of the soil entering the inlet of the bit. Then, in terms of the radius of the inlet being the same as the tube, the fill level of the flexible tube, the so-called coring rate c, can be expressed as

If the coring speed does not exceed the penetration speed namely, c ≤ 1, the weight of the samples is supported not only by the pressure at the boundary of Region Ⅰ and Region Ⅱ, but also by the friction upwards from the flexible tube, when the flexible tube is dragged upwards. Otherwise, the pressure between Region Ⅰ and Region Ⅱ is the sum of the weight of the samples and the friction from the tube, because the friction turns downwards. In this way, the force balances of a slice dz of the material can be written as follows:

In Region Ⅰ,

(15)

In Region Ⅱ,

(16)

Here, r is the radius of the core tube.

We assume that the stress distribution in the core tube is subject to Janssen's model, satisfying σ_r=K_sσ_z, where σ_r is the lateral stress, and K_s is a constant. In addition, the friction in Region Ⅰ, τ₁=μ_b σ_{r, 1}, and the friction in Region Ⅱ, τ₂=μ_f σ_{r, 2}, are considered in this paper. Here, μ_f is the dynamic friction factor between the soil and the flexible tube, and μ_b is the dynamic friction factor between the soil and the bit. Therefore,

(17)

By substituting Eq. (17) into Eqs. (15) and (16) and integrating these differential equations with the stress boundary σ_{z, 2}(h_s)=0, where h_s is the height of samples in the centering tube, the stress distributions are then given as follows:

In Region Ⅰ,

(18)

In Region Ⅱ,

(19)

Here, h_b is height of the bit.

Then, the pressure at the bottom of the bit can be expressed as

(20)

However, if the coring speed exceeds the penetration speed the friction between the flexible tube and the samples changes. The pressure at the bottom can be written as

(21)

The relation between the pressure at the bottom (bit) and the height of the coring is shown in Fig. 3. Two cases are encountered. If the coring speed is less than the penetration speed the pressure at the bottom finally reaches a constant. Janssen's effect makes the pressure at the boundary of Region Ⅰ and Region Ⅱ tend to be a constant with an increase in the coring height. The friction in Region Ⅱ offsets most of the weight of the samples in the tube. Then, the constant pressure results in the pressure at the bottom approaching a constant in turn. This case corresponds to the solid curve in Fig. 3. However, the pressure rises abruptly and exponentially if the coring speed exceeds the penetration speed Once the friction in Region Ⅱ changes the direction, Janssen's effect makes the friction increase exponentially along the height of the samples. Thus, the pressure at the bottom increases correspondingly.

Because the soil surrounding the drill bit is in a flowing state, we assume that the pressure in the soil is the same at any position in this region for simplification. Namely, the pressure at the inlet of the conveying channel equals the pressure at the inlet of the core tube. Thus, the coring model in this subsection can be combined with the conveyance model in Subsection 2.1. Once the pressure at the bit is determined by the coring speed with the coring model (see Eq. (20)), the conveying velocity in the auger groove can be calculated by the conveyance model (see Eq. (12) or (14)), and the conveying velocity in turn affects the coring speed, owing to the mass conservation law.

2.3 Loads on the bit

To correctly reflect the drilling force and torque, the interaction between the soil and the bit is needed. In this study, we employ an empirical relation for the cutting force proposed by Mellor^[3]. The total axial thrust on the bit F_b is expressed as

(22)

where p_b is the pressure on the bit, the factor B is a constant, and n is the revolution per unit time.

The total torque on the bit T_b is expressed as

(23)

where τ_b=μ_bp_b is the tangential stress.

3 Experimental verification 3.1 Testbed for drilling and coring

The drilling and coring system studied here is depicted in Fig. 4(a). The bit and stem have the same radius of R=15.5 mm. The pitch of the auger is set as h=12 mm. The helical groove has a flute with a radial width of a=0.85 mm and an axial length of b=10.5 mm. The barrel is 2.5 m long and 0.52 m in diameter. The drill string is driven by servo motors with a constant rotation speed and penetration speed 1 m deep into the long barrel of the simulant. As the bit cuts and stirs the in situ simulant into a fluid state, part of the simulant is discharged by the helical groove of the rotating auger. The other simulant enters the inlet at the bottom of the bit, and a coring tube in the hollow stem is used to form the coring sample. The coring tube in the hollow stem is dragged by a fixed rigid rope while the string is drilling into the soil. Thus, the coring tube has the same speed as the penetration speed relative to the drilling stem though the coring tube does not actually move. In the drilling tests, the loads on the bit are obtained over time by the force sensors, and the coring amount is also weighted up.

3.2 Lunar soil simulant

The lunar simulant is manufactured of basaltic pozzolana, according to the composition of lunar samples brought back by the Apollo missions. The grain size of the returned soil sample ranges from about 40 µm to 800 µm^[12]. Thus, in our experiments, the simulant particle size ranged from 0.1 mm to 1 mm, and the bulk density was ρ=1.74 g/cm³. The moisture content of the simulant was less than 0.5%, and the moisture content of the air in the lab was also carefully controlled. The simulant was uniform and relatively incompressible after a vibrating process. After these treatments, the soil became dense and cohesive (see Fig. 4(b)), and the mechanical properties of the simulant were close to the real lunar regolith.

3.3 Comparison between the experimental and computational results

To validate the drilling model, a series of drilling tests were conducted with different drilling parameters (penetration and rotation speeds). Figures 5(a)-5(d) and 6(a)-6(d) show comparisons between the experimental and computational drilling drags/torques varying with the penetration, for different penetration speeds = 40 mm/min, 50 mm/min, 60 mm/min, and 80 mm/min, under the same rotation speed n=120 r/min. All the parameters for simulation are listed in Table 1.

Both the experiment and simulation indicate that the drag force/torque increases with the penetration depth at the beginning, and reaches a constant quickly. The drag force/torque is mainly attributed to the pressure at the bit and is influenced by the conveyance mechanism. For specific drilling parameters, the loads on the drill are determined by the conveying state, but are independent of the penetration depth.

Other penetration rates such as = 20 mm/min, 30 mm/min, 50 mm/min, and 100 mm/min are added into the experiment when n=120 r/min. Figure 7 plots the logarithm of the drag force/torque varying with the penetration speed . It is clear from the figure that the drag force/torque changes very slowly in the experiment when the penetration speed is less than a threshold, and increases exponentially when the penetration speed is high. The simulation by the dynamic model indicates that the drag force/torque remains constant if < 53 mm/min, and increases exponentially with the increasing penetration speed when > 53 mm/min.

Figure 8 plots the coring rate in the tube varying with the revolution per penetration defined as . It is interesting to find that the coring rate is linear with the revolution per penetration in a proper interval approximately equal to [2.2, 4.3] of the revolution per penetration. The interval corresponds to [28, 53] of the penetration rate , when n=120 r/min. From Fig. 7 we can see that the drag force/torque is constant in this interval. This is not accidental. The two values of the revolution per penetration, 2.2 and 4.3, are critical points, which are discussed in the following section. The two critical values determine the effective working interval and the optimum drilling parameters.

4 Model usage and three drilling stages

Based on the previous experimental and computational results, as shown in Fig. 9, the whole drilling process can be classified into three stages with respect to the ratio between the rotation and the penetration speed , (Ⅰ) zero-coring stage; (Ⅱ) linear-coring stage; (Ⅲ) saturated-coring stage.

Stage Ⅰ  No samples are collected. All the in situ soils stirred by the bit could be removed by the auger, and the conveying channel is unsaturated, b_a < b. The relation is incorporated with Eq. (12) to obtain the pressure p, the conveying velocity , and the height of the conveyance queue b_a. As the pressure along the conveyance queue remains constant, the pressure at the bit is the solution p. The bit pressure is small and less than a threshold pressure. Thus, the soil in Region Ⅱ cannot reach the entrance of the flexible tube, and no samples enter the coring tube. In Fig. 3, the threshold corresponds to the vertical dotted line marked by "height of the bit h_b". The simulation shows that the pressure at the bottom rises exponentially with the penetration per revolution in this stage.

Stage Ⅱ  The coring rate is linear with the revolution per penetration whereas the rotation speed is fixed as n=120 r/min (experiment). With the increment of penetration speed , more soils are stirred. With the help of the equation of continuity and Eq. (12), the height of the conveyance queue b_a increases accordingly. Then, the pressure at the bit increases with increasing b_a. Once the pressure exceeds the threshold determined by the height of the bit, the coring process begins. All the in situ soils stirred by the bit are removed by the auger and the core tube, which can be rewritten as

(24)

Equation (24) is combined with Eq. (12) to find the unknown variables. There are four variables, p, , b_a plus the coring speed , whereas only three equations are given, Eqs. (12) and (24). As such, the coring begins, and the height of coring increases. According to Fig. 8, the bottom pressure tends to p(0)= σ_z(∞) quickly after a short unstable adjustment and rebalance. Then, with the boundary equation p= σ_z(∞), the stable drilling procedure (Stage Ⅱ) could be calculated. Stage Ⅱ does not end until

Although Eq. (24) is obtained, we cannot say that we have proved the linear relation between the coring rate and the revolution per penetration because b_a and vary with . However, the experiments and the simulation indeed indicate such linearity. Further investigation needs to be carried out.

Stage Ⅲ  The coring rate has been saturated, whereas the drag force/torque at the bottom increases sharply. With the continuous increase in the penetration speed, more and more stirred soils need to be disposed of. If the coring speed exceeds , the friction between the samples and the flexible tube changes its direction downwards, which leads to extremely high pressure owing to the Janssen's effect, as shown by the dot-dashed line in Fig. 3. Because the high pressure resists coring, the coring speed cannot exceed . Let and combine the continuum equation (24) with the conveyance equations (12) to obtain the conveying height b_a, the pressure p, and the conveying velocity . In this case, the pressure increases exponentially with the conveying height b_a. When the conveying channel is fully filled, b_a=b, the exponential relation (14) and the coring model (20) are combined with the continuum equation (24) to find p(0), , and .

It seems to be clear that the drilling parameters at the boundary between Stage Ⅱ and Stage Ⅲ provide the optimum solution for a drilling mission, where the coring rate almost reaches the maximum, and the pressure at the bottom remains low.

5 Conclusions

A new drilling model that could predict the loads and coring amount has been proposed and validated experimentally in this paper. The conveying dynamics along the helical groove and the coring-sample mechanism in the core tube have been established based on the elementary analysis and Janssen's model to reflect the coupling effect of stress among different components. In addition, a simple cutting dynamic model for the interaction between the drill bit and the lunar regolith has also been introduced to reflect the cutting drag force/torque. Combining the three sub-models, a drilling model for a whole dynamic procedure has been established to uncover the complicated coupling relation among the conveying ability of auger, the coring rate in the core tube, and drilling parameters such as penetration and rotation speeds. In addition, a series of experiments with constant penetration and rotation speeds are conducted to verify the model. The comparisons between the simulation by the new model and the experimental results indicate three aspects of characteristics of the drilling dynamics, i.e., (ⅰ) the loads on the bit are almost independent of the penetration, when the penetration speed is lower than a threshold; (ⅱ) three obvious drilling stages with respect to the penetration speed are found, and the optimum drilling parameters are located at the boundary of Stage Ⅱ and Stage Ⅲ; (ⅲ) a linear relationship between the coring rate and the revolution per penetration is found in Stage Ⅱ. This state is also called the working stage because the samples begin to be collected and the drag force/torque remains low.