Fabio F Real

Anas Batou

Thiago G Ritto
email: tritto@mecanica.ufrj.br

Christophe Desceliers

Keywords: bit-rock stochastic interaction model, drillstring dynamics, hysteretic friction, experimental identification

This paper aims at constructing a stochastic model for the hysteretic behaviour of the nonlinear bit-rock interaction of a drill string under torsional vibrations using field data. The proposed model takes into account the fluctuations of the stick-slip oscillations and the hysteretic effect due to the nonlinear bit-rock interaction observed during the drilling process. These fluctuations are modelled by introducing a stochastic process associated with the variations of the bit-rock interaction during the drilling. The parameters of the stochastic model are calibrated with field data. Finally, independent trajectories of the proposed stochastic model are generated and used to simulate the stochastic response of the drill string torsional dynamics in presence of random bit-rock interaction. The stochastic model is then used to estimate statistics on the stability of the drill string.

Introduction

Drill string is a slender structure used for exploitation of oil reserves. A top drive rotates the system at the top, which transmits the torque to the bit that drills the rock. The design and analysis of drill strings are usually performed using a computational model. Several drill string models have been proposed in the literature. The complete dynamics of a drill string includes axial, torsional, and lateral vibrations [START_REF] Tucker | On the effective control of torsional vibrations in drilling systems[END_REF]; [START_REF] Ritto | Nonlinear dynamics of a drill-string with uncertain model of the bitrock interaction[END_REF]. In addition, in the development of directional drilling, advanced features taking into account the horizontal dynamics analysis might be necessary [START_REF] Ritto | Drill-string horizontal dynamics with uncertainty on the frictional force[END_REF]; [START_REF] Cunha | Computational modeling of the nonlinear stochastic dynamics of horizontal drillstrings[END_REF]. Torsional control is also an issue [START_REF] Patil | A comparative review of modelling and controlling torsional vibrations and experimentation using laboratory setups[END_REF]; [START_REF] Ritto | Active control of stick-slip torsional vibrations in drill-strings[END_REF].

The present paper focuses on the analysis of the torsional vibration of the drill string, which is nonlinear due to the bitrock interaction. Field observations [START_REF] Pavone | Application of high sampling rate downhole measurements for analysis and cure of stick/slip in drilling[END_REF]; [START_REF] Ritto | Validation of a drill string dynamical model and torsional stability[END_REF]; [START_REF] Real | Hysteretic Bit/Rock Interaction Model to Analyze the Torsional Dynamics of a Drill String[END_REF] show that the torque on bit varies nonlinearly with the bit speed. Furthermore, these observations show hysteretic cycles which fluctuate during the drilling process [START_REF] Real | Hysteretic Bit/Rock Interaction Model to Analyze the Torsional Dynamics of a Drill String[END_REF]. Torsional vibration is a problem in drill string dynamics, especially when its length reaches thousands of meters. In some conditions, the lateral and axial vibrations are small, and the torsional vibration is the main concern [START_REF] Ritto | Validation of a drill string dynamical model and torsional stability[END_REF]; it might lead to stick-slip oscillations [START_REF] Kyllingstad | Study of slip/stick motion of the bit[END_REF]; [START_REF] Leine | Stickslip vibrations induced by alternate friction models[END_REF]; [START_REF] Tucker | On the effective control of torsional vibrations in drilling systems[END_REF]; [START_REF] Richard | Self-excited stick-slip vibrations of drill bits[END_REF]; [START_REF] Khulief | Vibration analysis of drillstrings with self-excited stick-slip oscillations[END_REF]; [START_REF] Kreuzer | Controlling torsional vibrations of drill strings via decomposition of traveling waves[END_REF]; [START_REF] Patil | A comparative review of modelling and controlling torsional vibrations and experimentation using laboratory setups[END_REF]; [START_REF] Hong | Identification and control of stick-slip vibrations using Kalman estimator in oil-well drill strings[END_REF]. In this severe conditions, the bit sticks (zero speed) then slips (high speed), and that might cause, for instance, measurement equipment failure, low rate of penetration, bit damage, and fatigue [START_REF] Wu | Identifying the Root Cause of Drilling Vibration and Stick-Slip Enables Fit-for-Purpose[END_REF].

In [START_REF] Real | Hysteretic Bit/Rock Interaction Model to Analyze the Torsional Dynamics of a Drill String[END_REF], the authors have highlighted large fluctuations of the nonlinear bit-rock interaction law during the drilling. These fluctuations which are mainly due to the random soil mechanical properties during the drilling, and have to be taken into account in order to study the stability of the drill string with a good robustness. Objectives of the present paper are (1) to characterize and construct a probabilistic model of these fluctuations and (2) then to estimate the probability of instability of the drill string including the stochastic non-linear bit-rock interaction model. In the literature, only few papers are concerned with the probabilistic modelling of the bit-rock interaction. In [START_REF] Spanos | Nonlinear stochastic drill-string vibrations[END_REF], a stationary random process is considered to model lateral forces at the bit. In [START_REF] Ritto | Stochastic drill-string with uncertainty on the imposed speed and on the bitrock parameters[END_REF]), uncertainties are considered in the bit-rock interaction parameters. In [START_REF] Ritto | Nonlinear dynamics of a drill-string with uncertain model of the bitrock interaction[END_REF], an adaptation of the nonparametric probabilistic approach Soize C ( 2000) is proposed to model globally the uncertainties in the bit-rock interaction model. For the two later referred papers, the probabilistic models are not time-dependent, i.e., the bit-rock interaction model is random but does not vary during the drilling. Constructing such a stochastic computational models including the stochastic fluctuations of the bit-rock interaction forces would be helpful for robust optimization of the drill string [START_REF] Ritto | Robust optimization of the rate of penetration of a drill-string using a stochastic nonlinear dynamical model[END_REF].

In the present work, the nominal model of the drill string is constructed as two torsional beams which are discretized by means of the finite element method Jansen JD (1993); [START_REF] Khulief | Finite element dynamic analysis of drillstrings[END_REF]; Ritto TG and Sampaio R ( 2012)), and a reduced-order model is constructed using the elastic modes. The non-linear bit-rock interaction forces are modelled as a point torque applied at the end of the bottom hole assembly (BHA). This simple model enables the efficient construction of the stability maps of the drill string. Then the stochastic non-linear bit-rock interaction model is constructed by introducing a multiplicative stationary Gaussian stochastic process. Field data are used to calibrate the power-spectral density function of this stochastic process. Once calibrated, independent realizations are generated and statistics on the drill string dynamical behaviour are estimated using the Monte Carlo simulation method. The results we obtain show that taking into account the stochastic fluctuations of the nonlinear bit-rock interaction enable an estimation of the probability of failure with respect to the stability of the drill string and offers robust method to design the drill string and its working condition (minimum top drive velocity for instance).

The paper is organized as follows. Section 2 describes the nominal drill string computational model, including the nonlinear bit-rock interaction. Section 3 presents the proposed probabilistic model for the bit-rock interaction, which is modelled as a nonstationary stochastic process to take into account its stochastic fluctuations during the drilling. The numerical results are presented in Section 4, where deterministic and stochastic dynamical response are analyzed. The last section brings the concluding remarks. Figure 1 presents a scheme of the torsional system considered in the analysis. A vertical wellbore, with only torsional vibrations, is taken into account. The system is composed of drill pipes (about 5,000 meters) and by the bottom hole assembly (BHA, about 400 meters). In the present paper lateral and axial vibrations are not modelled. The extension of the approach presented in this paper including lateral and axial vibrations is straightforward.

A constant angular speed Ω is imposed at the top and a reaction torque acts at the bottom due to the bit-rock interaction. In this paper, the torsional dynamics of the drill string is analysed by solving the torsional beam equation:

ρI p ∂ 2 θ(x, t) ∂t 2 -GI p ∂ 2 θ(x, t) ∂x 2 = T (x, t), (1)
where θ(x, t) is the angular rotation about the longitudinal xaxis related to time (t), T (x, t) is the torque per unit length, I p is the cross sectional polar area moment of inertia, and ρ and G are the mass density and shear modulus of the material of the column. The boundary conditions related to the imposed angular speed at the top are given by

θ(0, t) = Ωt θ(0, t) = Ω , (2)
and the initial conditions are

θ(x, 0) = 0, θ(x, 0) = Ω , (3)
As proposed in [START_REF] Real | Hysteretic Bit/Rock Interaction Model to Analyze the Torsional Dynamics of a Drill String[END_REF], the rotational displacements about a rotating frame is considered. Let θ rel (x, t) be the relative torsional degree of freedom in the rotating frame associated to the top sectional area (at x = 0). We introduce the absolute rotational displacement as

θ(x, t) = Ωt + θ rel (x, t). ( 4
)
The system is discretized by means of the finite element model, where linear shape functions are applied. Let u(t) be the vector of θ rel (x, t) nodal values related to the drill string mesh. Note that, in the rotating frame, the angular displacement is fixed at the top because there is no relative displacement between the top drive and the first node, at the top of the drill string. Adding a proportional damping to the system, the vector u(t) is solution of the matrix equation

[M ]ü(t) + [D] u(t) + [K]u(t) = T( u(t), ü(t))), (5)
where [M ] is the mass matrix, [D] is the damping matrix, [K] is the stiffness matrix, and T( u(t), ü(t)) is the torque vector. According to Eqs. ( 3) and ( 4), the initial conditions in the rotating frame read

u(0) = 0, u(0) = -Ω1. ( 6
)
where 1 = [I n×1 ] a vector with all entries equal to one. All the components of the torque vector are zero except the one corresponding to the drill bit node (corresponding to x = L). For this node, the nonlinear torque applied to the bit is denoted by T bit ( θbit (t), θbit (t)) and will be described in the next section.

The normal modes of the conservative homogeneous system are used to construct a reduced-order model. The m first eigenvalues 0 < λ 1 ≤ λ 2 ≤ . . . ≤ λ m associated with elastic modes {ϕ 1 , ϕ 2 , . . . , ϕ m } are solutions of the generalized eigenvalue problem

[K] ϕ = λ[M ] ϕ . ( 7
)
The reduced-order model is obtained by projecting the full computational model on the subspace spanned by the m first elastic modes calculated using Eq. ( 7). Let [Φ] be n × m matrix whose columns are the m first elastic modes. We can then introduce the following approximation

u(t) = [Φ] q, (8)
in which q is the vector of the m generalized coordinates which are solution of the reduced matrix equation

[ M ]q(t) + [ D] q(t) + [ K]q(t) = T( q(t), q(t)), (9)
with the initial conditions

q(0) = 0, q(0) = -Ω [ M ] -1 [Φ] T [M ]1. (10)
In these equations,

[ M ] = [Φ] T [M ] [Φ], [ D] = [Φ] T [D] [Φ] and [ K] = [Φ] T [K] [Φ]
are m × m mass, damping and stiffness reduced-order matrices, and T( q(t), q(t)) =

[Φ] T T([Φ] q(t), Φ]q(t)) is the vector of the reduced-order generalized torque. The set of equations ( 8), ( 9) and ( 10) can be solved using commonly used integration schemes, such as the Euler scheme or the Runge-Kutta, for instance.

Probabilistic model for the bit-rock interaction with hysteretic effects

Figure 2 shows the field data [START_REF] Ritto | Validation of a drill string dynamical model and torsional stability[END_REF]; [START_REF] Real | Hysteretic Bit/Rock Interaction Model to Analyze the Torsional Dynamics of a Drill String[END_REF] that will support the proposed model presented in this section. The downhole information used in this paper was acquired using a downhole mechanics measurement unit capable of providing both real-time measurement through mud telemetry and continuously recorded high-frequency data throughout the run. The sub, installed at the BHA above the bit, contains a suite of 19 sensors sampled at 10,000 Hz and downsampled and filtered prior to recording at 50 Hz.

Following is a list of 50-Hz data recorded in this sub: triaxial accelerations; gyro rpm; magnetometer rpm; axial loading; torque; bending moment [START_REF] Shi | Integrated Efforts to Understand and Solve Challenges in 26-in Salt Drilling, Gulf of Mexico[END_REF]. The field data shown in Fig. 2 was measured at the BHA, very close to the drill bit. A window average smoothing was used to filter measurements noise and obtain these stickslip cycles. A deterministic bit-rock interaction model that takes into account the observed hysteresis (non-reversibility) effect was proposed in [START_REF] Real | Hysteretic Bit/Rock Interaction Model to Analyze the Torsional Dynamics of a Drill String[END_REF]. Hysteretic (non-reversible) behaviour is more evident when the drill bit is slipping (high torsional oscillations). This macroscopic phenomenon is related to a frictional memory due a friction force's delay, where the bandwidth of the loop increases according to the drill string acceleration Hess DP and Soom A (1990); [START_REF] Olsson | Friction models and friction compensation[END_REF]; [START_REF] Wojewoda | Hysteretic effects of dry friction: modelling and experimental studies[END_REF]. The deterministic hysteretic bit-rock interaction model considered in the present paper [START_REF] Real | Hysteretic Bit/Rock Interaction Model to Analyze the Torsional Dynamics of a Drill String[END_REF] reads

T bit ( θbit , θbit ) = b0 tanh(b1 θbit ) + b2 θb 4 bit 1 + b3 θb 5 bit (1 + H( θbit θbit )), , (11)
H( θbit , θbit ) = β 1 tanh(β 2 θbit ) , (12)
where β 1 and β 2 are two positive parameters. Note that the hysteretic cycle is limited (1 ± β 1 ), and if H equals to zero there is no hysteresis.

In [START_REF] Real | Hysteretic Bit/Rock Interaction Model to Analyze the Torsional Dynamics of a Drill String[END_REF], the parameters of this bitrock interaction model have been calibrated such that it fits with the mean field data cycle. Figure 3 compares the field data with the deterministic model. It can be seen that, in average, the bit-rock interaction model, Eq.( 11), presents a reasonable good fit comparing to the field data mean cycle. Nevertheless, it is noticed a considerable time fluctuation of the field data cycles. This fluctuation is mainly due to the variation of the soil mechanical properties during the drilling (the measurement noise has already been filtered using a window averaging). The probabilistic modelling of this fluctuation is the objective of this paper.

If we take a close look at the field data, we notice that the torque on bit varies from about 7 to 10.5 kNm, when the bit speed is close to 1 rad/s. For higher speeds, 20 rad/s, the torque on bit varies from about 4 to 5.8 kNm. The distance between the higher and lower values is very different, depending on the bit speed: 3.5 and 1.8 kNm. But, for both speeds the torque on bit is varying plus or minus 20%. Therefore, the present paper proposes to take into account the stochastic fluctuations of the bit-rock interaction model by including a multiplicative stochastic process to the deterministic model:

T bit ( θbit (t), θbit (t)) = T bit ( θbit (t), θbit (t))(1 + η(t)), (13)
where η(t) is a centred stochastic process η(t) which can be rewritten as

η(t) = T bit ( θbit (t), θbit (t)) T bit ( θbit (t), θbit (t)) -1. ( 14
)
The experimental stochastic process η exp (t) can be computed using Eq. ( 14) and the field data related to the torque on bit. This stochastic process is shown in Fig. 4. Unfortunately, there is not enough field data to completely characterize this stochastic process. We will assume here that η(t) is a centred stationary Gaussian stochastic process. This assumption will be verified in future work with more experimental data. The power spectral density (PSD) is estimated using the periodogram method [START_REF] Priestley | Spectral Analysis and Time Series[END_REF]. Figure 5 shows the estimated field data PSD. Regarding Figure 5, the PSD is constant until a critical frequency and then decreases linearly (in log-log scale). This type of PSD is often encountered when addressing turbulent forces [START_REF] Batou | Experimental identification of turbulent fluid forces applied to fuel assemblies using an uncertain model and fretting-wear estimation[END_REF]. Then the proposed PSD model S(f ) is written as where f 0 , A 0 , a and b are the parameters of the model. These parameters are calibrated using the experimental PSD such that f 0 = 0.27, A 0 = -7.6, a = -3.13 and b = -11.67 (with appropriate units). Figure 6 compares the calibrated PSD model with field data PSD, where a reasonable agreement is observed. With the PSD S(f ) in hands one can generate independent realizations of the stochastic process η(t) using a classical generator of Gaussian process [START_REF] Benaroya | Probability Models in Engineering and Science[END_REF]. Figure 7 shows two independent trajectories of η(t), which again give reasonable agreement with the observed process η exp (t) shown in Figure 4.

log(S(f )) = A 0 for f < f 0 , log(S(f )) = a log(f ) + b for f ≥ f 0 , (

Simulation of the stochastic drill string dynamics

Analysis of one realization of the stochastic drill string dynamical response

The previous section was concerned with the construction of a stochastic bit-rock interaction model including hysteretic cycles fluctuations. A stationary stochastic process was introduced for the computation of the torque on bit, Eq.( 13).

The stochastic bit-rock interaction model is then added to the torsional drill string model, Eq.( 9), and the stochastic non-linear dynamical response of the drill string is computed and analysed. Note that Eq.( 9) becomes random because of the random bit-rock interaction, Eq.( 13). The general integration scheme for one realization of the stochastic bitrock interaction model is presented in Algorithm 1. Table 1 contains the parameters of the drill string used for the simulation. The mass and stiffness matrices are constructed using 100 finite elements, after convergence check. The generalized damping matrix is diagonal with damping ratios equal to 0.005 for the first mode, 0.03 for the second and third modes, and 0.005 for all the other modes. The first five natural frequencies computed for the system are: 0.13, 0.42, 0.74, 1.07, 1.41 Hz.

The non-linear equation ( 9) is solved using a modified Euler scheme with a time step 0.512 ms. For one realization of the stochastic bit-rock interaction model, Figure 8 shows the stochastic response of the drill string in the stationary regime. In comparison with Figure 2, it can be observed the same stochastic behaviour but with a slightly larger value of the maximum angular velocity. This difference due to the Finite Element model used here which can not reproduce all the complexity of a real drill string dynamics. But the objective of the paper concerns the construction Algorithm 1: Simulation of the drill string dynamics.

INITIALIZATION:

Generate a realization of stochastic process η(t);

q 0 = 0; q0 = -Ω [ M ] -1 [Φ] T [M ]1; LOOP: for k = 1, . . . , (n t ) do
Update the angle and angular speed (depending on the integration scheme):

( qi-1 , qi-1 ) → ( qi , qi ) ; ui = [Φ] qi ; θbit,i = ubit,i + Ω ; θbit,i = übit,i ; Calculate the torque on bit: of the stochastic bit rock interaction model which should not depend on the computational model of the drill string. Figure 9 compares the steady-state of random bit response using the proposed stochastic bit-rock interaction model with the corresponding time range response obtained using the deterministic model described by Eq. ( 11). We are interested in range While a 4-cycles periodic regime is reached for the deterministic case, as expected, no periodicity is observed in the stochastic case.

T bit,i ( θbit,i , θbit,i ) = T bit ( θbit,i , θbit,i )(1 + η i ) ;
For the deterministic case, the 4-cycles periodic sequence for the duration of the stick and slip phases are respectively (1.34, 2.53, 2.48) s and (5.4, 6.28, 6.41) s . For the stochastic case, these durations are random and their associated probability distribution (obtained statistically using 750 cycles) are represented on Fig. 10. We can see is these figures a large variability of the stick and slip durations with modes closed to values corresponding to the values calculated with a deterministic bit-rock interaction model.

Stochastic stability analysis

Now the analysis is extended to quantify statistics on the stability threshold of the system, as the imposed speed at the top varies. The torsional stability of a drill string can be quantified through the stick-slip severity factor, defined by Ritto TG et al. ( 2017)

γ SS (Ω) = θmax bit (Ω) -θmin bit (Ω) 2 Ω , (16)
where θmax bit (Ω) and θmin bit (Ω) are the minimum and maximum bit velocity in the steady-state regime. In case of pure slipping, γ SS = 0. If there are stick-slip oscillations then γ SS > 0.

First, the deterministic system is analyzed. The stick-slip severity factor in the range B = [6; 27] rad/s is plotted in Fig. 11. As expected, the stick-slip severity factor decreases when imposed rotation at the top increases. If the speed at the top is lower than 16 rad/s, γ SS is greater than 0.5. For the stochastic bit-rock interaction model, the stickslip severity factor becomes random and its statistics are estimated using the Monte Carlo simulation method with n s = 500 samplings. For each Monte Carlo loop, an independent realization of the stochastic bit-rock interaction model is generated, a realization of the stochastic angular velocity is calculated and the corresponding stick-slip severity factor is determined on the stationary regime. Figure 13 shows statistics on the random stick-slip severity factor. The convergence with respect to the number of samplings n s is analysed by introducing the convergence function

CV (n s ) = 1 n s ns i=1 B γ SSi (ω) 2 dω, (17)
where γ SSi corresponds to the i th calculated realization of the stick-slip severity factor, and B is the integration domain.

The convergence function which is plotted in Figure 12. We can see in this figure that a good convergence is achieved using 500 realizations. The statistical envelope shown in Fig. 13, due to the stochastic bit-rock interaction, yields large fluctuation in the random stick-slip severity factor. This means that bit-rock interaction variability has a direct impact on the drill string stability and should therefore be taken into account for a robust analysis of the drill string dynamics. The results show that when the top speed is about 20.5 rad/s the system has 5% probability of having the value of γ SS greater than 0.5. The probability of instability increases as the speed decreases. This result brings much more information comparing with the deterministic result, where, for the same threshold of γ SS < 0.5, the limit speed of 16 rad/s was obtained.

Concluding Remarks

In the present paper, a new probabilistic model for the bitrock interaction model is proposed. This model includes a multiplicative stochastic process to take into account fluctuations of the torque on bit during the drilling. The stochastic model was calibrated with field data, and it takes into account hysteretic cycles and their stochastic fluctuations. The proposed bit-rock probabilistic model can be constructed independently from the computational model of the column.

The deterministic and stochastic torsional dynamics of a drill string are analyzed and a reasonable agreement between model predictions and field data is observed. The statistics of the stick and slip duration were also analyzed. A considerable impact of the proposed stochastic model on the torsional stability of the system was observed. For this end, the stick-slip severity factor is computed and statistical envelopes are plotted for varying imposed speed at the top. A robust estimation of the minimum angular speed is estimated using these plots.

In future works, additional experimental data will be collected in order to validate our assumptions on the probability distribution of the stochastic process η(t) introduced in the stochastic non-linear bit rock interaction model. Also, the PSD is likely to depend on the imposed angular speed. The future experimental results will enable the characterization of this angular speed dependency.

Figure 1 .

 1 Figure 1. General scheme of a drill string.

Figure 2 .

 2 Figure 2. Field data stick-slip cycles: measured torque T exp bit

  for θbit > 0, in which b 0 , b 1 , b 2 , b 3 , b 4 , b 5 are positive parameters to be fitted such that 0 < b 4 < b 5 . The bit-rock interaction model Eq. (11) Real FF et al. (2018) contains a hysteretic function H, which is a function of the bit angular acceleration (to separate forward and backward phases) and which is defined by

  The optimal parameters are b 0 = -3478, b 1 = 938, b 2 = 2.56, b 3 = 0.38, b 4 = 0.78, and b 5 = 1.1, β 1 = 14%, β 2 = 10.6.

Figure 3 .

 3 Figure 3. Bit-rock interaction: field data (black), mean field data cycle (green) and bit-rock interaction model with hysteresis (red).

Figure 4 .

 4 Figure 4. Stochastic process η exp (t) obtained experimentally with field data.

Figure 5 .

 5 Figure 5. Field data PSD.

Figure 6 .

 6 Figure 6. Comparison between the calibrated PSD and the field data PSD.

Figure 7 .

 7 Figure 7. Two independent realizations of the stochastic process η(t).

Figure 8 .

 8 Figure 8. One realization of the simulated bit-rock interaction.

Figure 9 .

 9 Figure9. Steady-state of bit angular velocity: deterministic bit-rock interaction model (top, 4-cycles periodic regime is observed) and stochastic bit-rock interaction model (bottom, no periodicity is observed) -5,250 s of total simulation time.

Figure 10 .

 10 Figure 10. Random stick and slip phases duration: probability density functions (black line) and corresponding 4-cycles periodic values calculated with a deterministic bit-rock interaction model (red vertical lines).

Figure 11 .

 11 Figure 11. Stick-slip severity factor for a deterministic bit-rock interaction. Red dashed line indicates the stability threshold of γSS.

Figure 12 .

 12 Figure 12. Convergence function ns → CV (ns).

Figure 13 .

 13 Figure13. Random stick-slip severity factor for a stochastic bit-rock interaction. Solid lines: mean response and 90% confidence region. Dashed line: deterministic case (black), stability threshold (red).

Table 1 .

 1 Calculate the torque vector T i ; Calculate the reduced-order torque vector Ti = [Φ] T T i Drill string characteristics.

	DP	BHA

Prepared using sagej.cls

Acknowledgements

The authors are grateful to the financial support of the Brazilian agencies CAPES, CNPq, and FAPERJ, as well the "Agence Nationale de la Recherche" (Contract HiMoDe, ANR-12-JS09-0014).

Conflict of interest

The authors declare no conflict of interest in preparing this article


Marvin Gennesson
email: marvin.gennesson@ascometal.com

Julien Zollinger
email: julien.zollinger@univ-lorraine.fr

Dominique Daloz
email: dominique.daloz@univ-lorraine.fr

Bernard Rouat
email: bernard.rouat@univ-lorraine.fr

Joëlle Demurger
email: joelle.demurger@ascometal.com

Hervé Combeau
email: herve.combeau@univ-lorraine.fr

Three Dimensional Methodology to Characterize Large Dendritic Equiaxed Grains in Industrial Steel Ingots

Keywords: industrial ingot, steel, dendritic grain size

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Steel manufacturing requires more homogeneous and cleaner products in terms of inclusions, chemical segregations and porosity. Chemical segregations at the product scale cannot be reduced once the product is fully solidified and are responsible for the differences of mechanical properties in rolled products. Those macrosegregations arise because of the relative motion between the sedimenting solid grains and the thermosolutal convected liquid phase. Consequently, the number, shape and size of the grains are expected to have a great impact on the relative motion and thus on the final macrosegregation profile. Good characterization methods to measure grain properties are of primary importance. Moreover, improvements in casting processes need multi-scale and multi-physics numerical modeling. To be realistic, these models need an extensive description of the grain properties at the beginning of solidification [START_REF] Kumar | Prediction of equiaxed grain structure and macrosegregation in an industrial steel ingot: Comparison with experiment[END_REF][START_REF] Combeau | Prediction of macrosegregation in steel ingots: Influence of the motion and the morphology of equiaxed grains[END_REF][START_REF] Combeau | Predictive Capabilities of Multiphysics and Multiscale Models in Modeling Solidification of Steel Ingots and DC Casting of Aluminum[END_REF][START_REF] Heyvaert | Modeling of the Coupling of Microstructure and Macrosegregation in a Direct Chill Cast Al-Cu Billet[END_REF][START_REF] Bousquet-Melou | Average momentum equation for interdendritic flow in a solidifying columnar mushy zone[END_REF][START_REF] Ni | A volume-averaged two-phase model for transport phenomena during solidification[END_REF][START_REF] Goyeau | Numerical calculation of the permeability in a dendritic mushy zone[END_REF][START_REF] Beckermann | Modelling of macrosegregation: Applications and future needs[END_REF][START_REF] Gu | Simulation of convection and macrosegregation in a large steel ingot[END_REF][START_REF] Ahmadein | Analysis of macrosegregation formation and columnar-to-equiaxed transition during solidification of Al-4 wt.% Cu ingot using a 5-phase model[END_REF]. Those numbers are not accessible for industrial castings, and postmortem techniques need to be developed.

Conventional metallography uses 2D measurements for equiaxed dendritic grain size measurements [START_REF] Watt | Solidification mapping of a nickel alloy 718 laboratory VAR ingot[END_REF][START_REF] Ivanoff | Microstructure Characterization of Nickel Alloy 718 with Automated Optical Image Processing[END_REF][START_REF] Zlatičanin | Characterization of microstructure and properties of AlCuMg alloys[END_REF][START_REF] Wang | Method for Automatic Quantification of Dendrite Arm Spacing in Dendritic Microstructures[END_REF][START_REF] Wang | Method for Automatic Quantification of Dendrite Arm Spacing in Dendritic Microstructures[END_REF]. More recently, computer-aided methods that take binary images as the input can be found in the literature. On these binary images, combined segmentation/erosion methods to isolate individual dendrites have also been applied to equiaxed dendritic structures [START_REF] Watt | Solidification mapping of a nickel alloy 718 laboratory VAR ingot[END_REF][START_REF] Ivanoff | Microstructure Characterization of Nickel Alloy 718 with Automated Optical Image Processing[END_REF], as well as intercept techniques, [START_REF] Zlatičanin | Characterization of microstructure and properties of AlCuMg alloys[END_REF][START_REF] Wang | Method for Automatic Quantification of Dendrite Arm Spacing in Dendritic Microstructures[END_REF][START_REF] Wang | Method for Automatic Quantification of Dendrite Arm Spacing in Dendritic Microstructures[END_REF]. They only measure primary and secondary dendrite arm spacings. Moreover, these image processing techniques give only qualitative and non-comparative results as they are very sensitive to the segmentation and filtering steps, which remove very small objects. Finally, the major drawback of these techniques is that for non-convex 3D objects, no stereological relationship exists to estimate the 3D properties of the grain distribution. It is then necessary to move to 3D in order to characterize equiaxed dendrites.

Advanced techniques like computed X-ray tomography [START_REF] Fuloria | Microtomographic characterization of columnar Al-Cu dendrites for fluid flow and flow stress determination[END_REF][START_REF] Tolnai | In situ synchrotron tomographic investigation of the solidification of an AlMg4,7Si8 alloy[END_REF][START_REF] Yang | Characterisation of the 3-D dendrite morphology of magnesium alloys using synchrotron X-ray tomography and 3-D phase-field modelling[END_REF][START_REF] Wang | Growth orientations and morphologies of α-Mg dendrites in Mg-Zn alloys[END_REF] or serial cutting/polishing have been used to assess the real 3D grain size of dendritic structures [START_REF] Alkemper | Three-dimensional characterization of dendritic microstructures[END_REF][START_REF] Madison | Characterization of three-dimensional dendritic structures in nickel-base single crystals for investigation of defect formation[END_REF]. The large size of equiaxed grain structures, which is in the order of magnitude of 1 cm, in addition to the low chemical contrast as encountered in heavy steel ingots makes using X-ray computed tomography quite impossible. The same problem exists with commercially available serial polishing microscopes for steel. For such large structures, conventional cutting remains the best solution. An example was given by Laren and Fredriksson in 1972 to identify the shape of one columnar grain (five slices) and one equiaxed grain (three slices) [START_REF] Laren | Relations between ingot size and microsegregations[END_REF]. However, the small number of cutting planes and number of grains prevented drawing a general conclusion about dendritic equiaxed grains.

The current study in this paper proposes a macro serial cutting method to characterize 3D centimetric equiaxed grains and to estimate the 3D volumetric equiaxed grain density for industrial steel ingots. The method used along with its advantages and its drawbacks are presented with one example of an industrial ingot. It is depicted that the grain density, the characteristic size, the shape and the orientation of the 3D equiaxed grain distribution can be determined with this method.

Materials and Methods

Material and Microscopy

The steel grade used in this study is 100Cr6 (ASTM 52100). 100Cr6 is a through hardening steel used for bearing steel. It is industrially cast as ingots, and its nominal composition is given in Table 1. A 6.2 ton 100Cr6 bottom poured ingot was cast in an Asco Industries plant at Fos-sur-Mer, France, for use in this study. as seen in Figure 1a. As a part of conventional metallographic characterization, one central slice was cut along the main axis of the ingot and divided into thirty sections. Polishing followed by macroetching in a warm 15 wt % HCl aqueous solution was performed on each section to reveal the solidification structure. It must be noted that electron back-scattering diffraction cannot be used for characterization of low-alloyed steel primary structures. 100Cr6 fully solidifies in γ-austenite. At lower temperatures, primary austenite undergoes solid state phase transformations, and all the primary crystallographic structure is transformed. In the as-cast state, 100Cr6's microstructure is mostly fully pearlitic. Dendrites exist at room temperature only as chemical segregations that can be revealed with an appropriate reagent.

Macrostructures were imaged on a high resolution Expression 12000XL EPSON scanner (EPSON, Suwa, Japan). Of these etched structures, one sample has been chosen in the equiaxed zone and outlined in Figure 1. The sample is located 60 cm from the top of the ingot and 1.5 cm from the central axis of the ingot (drawn as the green dotted line). The distance from the edge of the sample to the

Serial Cutting

On the sample taken from the ingot in Figure 1a, 12 slices of 0.7 mm in thickness were cut by wire erosion, the diameter of which is 300 µm. Those slices are taken in the green part of Figure 1c with their normal vectors parallel to -→ z . The total analyzed thickness was 1 cm, which is the width of the green area along the ingot radial direction, -→ z . Marks were made before the cutting to ensure that each slice was properly tracked and the final slice had dimensions of 1.75 cm × 3.8 cm × 0.07 cm. Optical microscopy was used to control the slice widths, where no defect in parallelism was observed.

Manual Outlining of Individual Grains

All the slices were ground up to 4000 SiC paper, and the final polishing was performed with a 3 µm diamond suspension. The solidification structure was revealed by warm etching in a supersaturated picric acid aqueous solution (Bechet-Beaujard reagent). Micrographs were taken with a Axioplan 2/Axiocam MRc5 optical microscope (Carl Zeiss, Oberkochen, Germany). The motorized platform combined with Axiovision software (version 4.6) allowed macroscopic tile images to be created.

The freeware Fiji was used for image analysis [START_REF] Schindelin | Fiji: An open-source platform for biological-image analysis[END_REF]. Manual identification of each 2D grain was performed after histogram equalization. Orientations of the dendrite primary axis and secondary arms were used to outline one grain on the first slice. Once a grain was outlined on its first slice of appearance, primary axis and secondary axis of the same grain were once again used as indicators for outlining on the other slices. The process was repeated until the grain could not be seen on the other slices or until the last slice was reached. Other grains were then outlined with the same procedure. A total of 390 2D contours were drawn manually.

A custom Fiji macro was used to store all the data about each 3D grain as a Fiji region of interest. The macro also produced a colored image of each individual slice, where each grain remained the same color on each slice. The final output is presented in Figure 2.

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m)

3D Reconstruction

A program was written in Python 3 to reconstruct each grain in 3D as shown in Figure 3. After retrieving the data per grain from Fiji (as shown in Figure 3a) using the read-roi functions developed by Mary et al. [24], each 2D contour belonging to the same 3D grain needed to be discretized.

As the main goal is surface reconstruction, polar discretization was done on each contour with the same predefined array of angles. Henceforth, each angle is represented as a line, which is depicted in Figure 3b. If random points were taken on two successive contours, vertices could cross each other, and the surface would not be properly reconstructed. The chosen grain in Figure 3 is the white grain at the bottom of each slice in Figure 2a-i.

To ensure that the solution is unique for each angle, the discretization must be performed on a convex surface. The convex hull of the contour, i.e., the external blue contour in Figure 3b, was used. Each angle was then computed to the center of mass of the new convex contour. This point is the origin of all the blue lines, and each line represents one angle.

The intersection between the lines and the convex Hull contour are the points used for the reconstruction of the surface in 3D between two sets of contours. The final result for one grain is shown in Figure 3c. The number of 3D grains retrieved in the sample volume from the twelve slices is 171. This 3D reconstruction is only used for graphical purposes. (c) 3D reconstructed equiaxed grain. This the same grain as in (a). This grain is also colored in light grey at the bottom of each slice in Figure 2.

For the analysis, the use of grain area and grain volume is inconvenient in the international unit system. Image analysis studies often compare lengths by using equivalent circle and sphere diameters. Those equivalent diameters do not make any sense in the present case because the 2D dendrite morphology is very far from being circular.

The Feret diameter, which is defined as the longest distance between two points in a cloud of points, can be considered as a more physical length to describe the grains. For 2D objects, Feret diameters are found from the 2D point clouds created by the manually-drawn concave contours. An example of Feret estimation is shown in Figure 4a for the grain contour from Figure 3a. It can be seen that for concave contours, the Feret diameter may not lie in the drawn contour. For 3D objects, the point cloud for Feret calculation is the sum of all the 2D contours that belong to an individual grain.

In our dataset, it has been verified that grain area and grain volume are respectively proportional to the square and cube of the 2D and 3D Feret diameters. As a consequence, these two parameters are relevant to describe the geometry of equiaxed dendrites. They are also quite easy to handle, to compute elongation factors and also to define orientation angles. Feret diameters (2D or 3D) alone are sufficient to precisely describe a 3D object because they cannot describe if an object is isotropic, and other size parameters need to be introduced in other directions of the space.

Secondary and tertiary Feret diameters are defined similarly to 3D Feret diameter (also called 3D principal Ferret diameter): the secondary Feret diameter is defined as the longest distance found in a direction perpendicular to the principal Feret diameter, whereas the tertiary Feret diameter is the longest distance perpendicular to the principal Feret diameter and the secondary Feret diameters. To ensure that those other Feret diameters always numerically exist, the perpendicularity condition needs to be relaxed with a tolerance of ±5 • . This is because of the lack of data in between the different slices that creates non-existing z positions and angles in the 3D cloud point. Examples of the 3D principal, secondary and tertiary Feret diameters are given on Figure 4b for the 3D grain of Figure 3c. It has to be noted that with those definitions, nothing ensures that the three 3D Feret cross all at the same unique point.

Results

2D Identification of 3D Grains

As said in Section 1, common metallographic measurements are two dimensional. With our dataset, it is possible to retrieve this kind of measurement using only the 2D contours. If each slice were considered individually, the average 2D Feret diameters, < F 2D >, would vary through the slices, as shown in Figure 5. The represented error bars are 50 % of the standard deviation (0.5 • σ). The mean value, of all the grains on all slices, is 0.67 cm with a standard deviation of 0.15 cm. This mean value is not consistent with the variation observed through all the slices. On the first slice, the mean 2D Feret is 0.45 cm, whereas on the last seven slices, the mean 2D Feret length is close to 0.85 cm ± 0.1 cm. In other words, it means that grain size measurements can give results varying up to a factor of two just by changing the sampling position for the 2D measurement by 1 cm, i.e., measuring on the 0 cm slice depth or measuring on the 1 cm slice depth. It is the proof that gradients of structure sizes exist on very short scales in industrial ingots. Figure 6a shows the distribution in terms of 2D Feret diameter (F 2D ). The distribution follows a classical log-normal law that has been fitted with the blue line. The low probability of cutting 3D grains in a plan containing the principal Feret diameter is well described by a log-normal distribution law. The principal mode and the maximum Feret diameter are 0.72 cm and 2.2 cm, respectively. However, there is no information regarding the real shape of the grain. It is therefore impossible to understand what really is the grain size and also, for example, to infer if the grain distribution is uniform, bimodal or even more complex.

The volumetric grain number density can be at first approximated by dividing the surface grain number density by the mean 2D Feret diameter for each slice [START_REF] Underwood | Stereology, or the quantitative evaluation of microstructures[END_REF]. Although it is only valid for monodisperse and convex objects, it is a fast way of providing a lower boundary for the real 3D value. With this workflow, values from 0.5 × 10 7 -2 × 10 7 m -3 are retrieved. Those values are strongly under-evaluated because the calculation performed averages the values of the Feret diameter on each slice.

3D Grain Structures

The final 3D result is thereby presented in Figure 7. Each reconstructed grain envelope was given a random color. As for 2D measurements, it is also possible to observe the distribution of 3D Feret diameters as shown in Figure 6b. Consequently, the number of 3D Feret diameters is much lower than the number of 2D contours and equals the number of grains, i.e., 171 grains. The shape of the distribution is the same, but the characteristic values are slightly different. The principal mode and maximum values are respectively 0.8 cm and 2.75 cm. The ratio between 2D and 3D mean Feret diameters is 0.9. A short Monte Carlo study on 1000 random slices on monodisperse cubes and spheres gives values of 0.7 and 0.94, respectively, for the same ratio. The volumetric grain number density is 2.2 × 10 7 grains/m 3 .

x ( c m )

3D Grain Properties

In 3D, the Feret diameter is only a partial indicator of grain size. Further insight can be gained by finding the longest size in directions perpendicular to the principal Feret diameter. After calculations, three Feret diameters (principal, secondary and tertiary) are obtained in the direction perpendicular to each other with a decrease in lengths. For isotropic objects like spheres, the secondary Feret diameter equals the principal Feret diameter. For non-isotropic objects, they provide a quick way to determine if the objects are elongated. Elongation factors, represented by secondary or tertiary Feret diameters divided by the principal Feret diameter values, are plotted in Figure 8. From the two elongation factors, it can be seen that the dendritic grains are non-isotropic objects with complex shapes. The mean elongation factors are equivalent to a 5:3:2 ellipsoid. The most symmetrical case that exists is a 5:5:4 ellipsoid, and the most elongated one is equivalent to a 10:5:2 ellipsoid. There is no isotropic case in the analyzed volume. In comparison, the 2D elongation mean case is a 5:3 ellipse. The 2D value is close in comparison to the 3D one, but does not give any information regarding the 3D shape.

The orientation of dendritic grains is also of interest. The orientation of the principal Feret diameter is represented in Figure 9. To obtain a density plot, kernel density estimation was applied to the data with the use of the seaborn python library, [26]. This library is very useful because it allows one to directly see the density function for each variable along with the two-dimensional estimated density function. The colorscale values are directly proportional to the number of points by unit of area in the 2D space (θ x , θ z ). It can be seen that most of the principal Feret diameters are perpendicular to the slicing direction, -→ z , which is also the radial direction of the ingot. Moreover, the distribution of the principal Feret diameter orientation has two main characteristic orientations with respect to the x direction ( -→ x is oriented along --→ g , which is the opposite direction of gravity). Those characteristic values are located at the center of the darker spots in Figure 9 at 10 • and 65 • with respect to -→ x . Those characteristic orientations do not appear to be related in any way to the principal Feret diameters.

Discussion

The main drawback of the method is that all the grains are kept during the analysis. Usually in image analysis, objects that intercept picture sides are removed [START_REF] Coster | Précis D'analyse D'images[END_REF]. This is not the case in the present analysis because of the small size of each slice and also the large grain size. It is a source of errors in the estimation of grain size, orientation and shape factors. However, it can be corrected in the evaluation of the volumetric grain number density, which is the most important parameter in this study. This can be done by adding factors regarding the number of picture sides that the grain is crossing [START_REF] Coster | Précis D'analyse D'images[END_REF]. In the same way, some grains are only observed on one slice. The size of these grains is only estimated in 2D and thus under-evaluated. Last, but not the least, grains finer than the total distance between two observation planes are lost.

As previously stated, common practice for equiaxed grain size determination is 2D measurement, which is a fast way to obtain rudimentary results. In this case, the 2D grain size measurements allow us to establish a mean grain size of 0.67 cm. Previous measurements reported by Kumar et al. on the same kind of ingots gave experimental grain size values on the order of 0.1 cm via the ASTM E-112 intercepts technique [START_REF] Kumar | Experimental and Numerical Studies of the influence of hot top conditions on Macrosegregation in an industrial steel ingot[END_REF]. This kind of measurement gives values related to the secondary dendrite arm spacing and is by nature not adapted to dendritic equiaxed grain size measurements. Secondary arms that intercept the measurement line will be counted as the intercept and will decrease the mean intercepted length. The norm would only be applicable to the manually outlined grain contours that will be 2D convex contours. This case would be equivalent to the usual measurements of austenitic grain size.

In this study, 3D measurements show that grain size heterogeneities can arise in an industrial ingot on a scale less than 1 cm. On the first slice, the mean 2D Feret is 0.45 cm, whereas on the last seven slices, the mean 2D Feret length is close to 0.85 cm ± 0.1 cm. The 25 % of the largest grains fill 80 % of the analyzed volume. Equiaxed grains appear to be a polydisperse collection of objects, which was not possible to prove with two-dimension analysis alone. The ratio of 0.94 between 2D and 3D mean Feret diameters is a direct consequence of the polydispersity. Random cuttings of each 3D dendrite alone would not give results as close to unity because of the low shape isotropy (i.e., secondary and tertiary elongation factors far from one).

Three-dimensional analysis indicated that grains are elongated and not fully isotropic. The real shape of dendritic grains can be estimated by ellipsoids at the first order. This means that no good estimation, in terms of volumetric grain number density, can be made with only 2D measurements. For example, by neglecting the concave shape of equiaxed dendrites, it is possible to estimate the volumetric grain number density by dividing the mean 2D surface grain number density by the 2D mean Feret diameter, which is 0.7 × 10 7 m -3 . Though the order of magnitude is correct, the numerical value is four-times lower than the 3D value. It has to be noted that 2D and 3D Feret diameter calculations are performed on concave contours. The 2D and 3D Feret diameters as defined in this study may not be fully contained respectively in the surface or volume of interest. This underestimation accounts for one part of the difference between the volumetric density estimated from 2D measurement and the value estimated from the number of 3D grains.

This parameter is crucial to validate macrosegregation numerical models currently under development like finite volume or finite element models [START_REF] Heyvaert | Modeling of the Coupling of Microstructure and Macrosegregation in a Direct Chill Cast Al-Cu Billet[END_REF][START_REF] Gandin | Finite Element Multi-scale Modeling of Chemical Segregation in Steel Solidification Taking into Account the Transport of Equiaxed Grains[END_REF]. Numerical predictions by Kumar et al. for the ingot investigated in this study, consider three different cases with varying initial volumetric densities: 1 × 10 8 m -3 , 5 × 10 9 m -3 and 5 × 10 10 m -3 , respectively [START_REF] Kumar | Prediction of equiaxed grain structure and macrosegregation in an industrial steel ingot: Comparison with experiment[END_REF]. According to the authors, those initial values produce final castings with structures that are fully-dendritic, mixed globular/dendritic and fully-globular, respectively. The experimental value, 2.2 × 10 7 m -3 , is one order of magnitude less than the smallest value chosen for the simulation. The dendritic morphology as shown in Figure 2 clearly demonstrates that there is a good agreement between the experimental and the predicted morphology.

Finally, equiaxed grains are not randomly oriented. Their Feret diameters are always very close to the slicing The most probable angle is perpendicular to the radial direction (relatively to the ingot) and less than 15 • disoriented from the direction of gravity. Another characteristic angle exists, perpendicular to the slicing direction, at an angle of 65 degrees from the direction of gravity. These two characteristic angles probably come from the settling period in the liquid prior to packing and from the shape of the liquid pool. This kind of textural effect cannot be retrieved from 2D characterization.

Conclusions

A method for 3D characterization in industrial castings with large equiaxed dendritic structures has been proposed. Slicing of one sample was performed by wire erosion, and outlines of each grain were made precisely. This manual method allows the accurate determination of 2D properties along with 3D estimation of grain size. Equiaxed grains appear to be complex, non-isotropic and non-randomly-oriented objects. The volumetric grain number density as calculated is close to 2.2 × 10 7 grains/m 3 . This value is ten-times smaller than the currently used value in the literature so as to simulate this kind of ingot. Numerical calculations need to be performed to identify the underlying cause of the determined textural effect. As a part of future work, new measurements should be carried out at different heights on the central line of an industrial ingot to see the evolution in terms of equiaxed structures. The number of slices has also to be increased to improve the precision of the technique. Such data could really help to understand what is the origin of equiaxed grains and to provide better inputs for the numerical models.

Figure 1 .

 1 Figure 1. Description of the sample position in a 6.2 ton 100Cr6 ingot. The total height of the ingot is 267 cm. The red dotted line is the columnar-to-equiaxed transition. The green dashed line is the central axis of the ingot. (a) Macroetched central slice inside a 3D schematic view of the ingot. (b) Position of the sample for serial cutting. (c) Zoom on the red rectangle with the position of the serial cut metal in the green rectangle. The normal vector to the slices is directed along the Z axis.

Figure 2 .

 2 Figure 2. Solidification microstructure after FiJi processing. In the web version, each grain maintains the same color through all slices. (a) Z = 0 cm; (b) Z = 0.11 cm; (c) Z = 0.22 cm; (d) Z = 0.33 cm; (e) Z = 0.44 cm; (f) Z = 0.55 cm; (g) Z = 0.66 cm; (h) Z = 0.77 cm; (i) Z = 0.88 cm; (j) Z = 0.99 cm; (k) Z = 1.1 cm; (l) Z = 1.2 cm; (m) Associated coordinated system.

Figure 3 .

 3 Figure 3. Principle of 3D reconstruction with 2D outlining followed by polar discretization. (a) Same grain outlined on each slice. (b) Concave contours (red) resulting from the 2D outlining in one slice of (a) with its convex Hull contour (blue). Green lines shows the polar discretization of the blue contour.(c) 3D reconstructed equiaxed grain. This the same grain as in (a). This grain is also colored in light grey at the bottom of each slice in Figure2.

Figure 4 .

 4 Figure 4. Principle of Feret calculation. (a) 2D Feret for the contour of Figure 3b. The 2D Feret diameter is drawn as the black line. (b) 3D principal, secondary and tertiary Feret for the grain of Figure 3c. The 3D principal, secondary and tertiary Feret are drawn as the red, green and blue lines, respectively.

Figure 5 .

 5 Figure 5. Evolution of the mean 2D Feret diameter on all the slices.

Figure 6 .

 6 Figure 6. Histogram of measured Feret. (a) 2D Feret diameters. (b) 3D Feret diameters.

Figure 7 .

 7 Figure 7. 3D reconstructed envelope of the dendritic microstructure. Each color in the web version accounts for one grain.

Figure 8 .

 8 Figure 8. Comparisons of secondary and tertiary equiaxed dendritic grain elongation factors.

Figure 9 .

 9 Figure 9. Density map for 3D principal Feret orientations with respect to the x and z directions. Density function shape for θ x and θ z are respectively plotted parallel to the corresponding axis. The z direction is a radial direction of the ingot, and the x direction is opposite gravity.

Table 1 .

 1 Composition of 100Cr6 steel (wt %).

	Grade	C	Mn	Si	Cr	Cu	Ni
	100Cr6 0.98 0.32 0.22 1.4 0.12 0.14

Acknowledgments:

The authors would also like to thank Jacob Roman Kennedy and Manoj Joishi improving the use of English in the manuscript.

Funding: This work was supported by ASCO Industries and ANRTFrance under a CIFREPh.D. fellowship (Grant Number 2015-1286). This work was supported by the French State through the program "Investment in the future" operated by the National Research Agency (ANR) and referenced by ANR-11-LABX-0008-01 (LabEx DAMAS).

Conflicts of Interest:

The authors declare no conflict of interest.