Reza Sohrabi
email: reza.sohrabi@unine.ch

Benjamin Malvoisin

Adriano Mazzini

Stephen A Miller

Multi-GPU based 3D numerical modeling of fluid migration and clay dehydration influence on

Keywords: High-Performance Computing (HPC), Graphics Processing Unit (GPU), 3D numerical modeling, Lusi, Hydrothermal system Highlights

Introduction

3D numerical modeling is a powerful tool for investigating processes occurring in the Earth's interior such as hydrothermal circulation or geothermal exploitation. The dynamics of such complex systems involve coupled and nonlinear flow, heat transport and reaction. Quantitative analysis of these systems is challenging and relies on numerical solutions of coupled partial differential equations (PDEs). Considerable effort during the last decades focused on various algorithms for solving PDEs on multi-CPU platforms, but these algorithms have made limited use of evolving hardware changes such as GPU-computing. 3D numerical modeling of hydrothermal systems presents a major challenge due to the physical processes involved (Ingebristen et al., 2010;[START_REF] Jeanne | A 3D hydrogeological and geomechanical model of an Enhanced Geothermal System at The Geysers, California[END_REF] over both short and long-terms [START_REF] White | Thermal waters of volcanic origin[END_REF][START_REF] Bowen | Geothermal resources[END_REF][START_REF] Ingebritsen | Groundwater in geologic processes[END_REF][START_REF] Lowenstern | Monitoring a supervolcano in repose: Heat and volatile flux at the Yellowstone Caldera[END_REF][START_REF] Hurwitz | The fascinating and complex dynamics of geyser eruptions[END_REF]. Scientific models have been developed for heat and mass transport through porous media with progressively more complex physics [START_REF] White | TOUGHREACT Version 2.0: A simulator for subsurface reactive transport under non-isothermal multiphase flow conditions[END_REF][START_REF] Steefel | Reactive transport modelling: an essential tool and a new research approach for the earth sciences[END_REF][START_REF] Bangerth | Deal. II-a general purpose object-oriented finite element library[END_REF][START_REF] Zhang | Recent magmatism drives hydrocarbon generation in north-east Java, indonesia[END_REF][START_REF] Gaston | MOOSE: a parallel computational framework for coupled systems of nonlinear equations[END_REF][START_REF] Flemisch | DuMux: DUNE for multi-{phase,component,scale, physics,…} flow and transport in porous media[END_REF][START_REF] Kolditz | OpenGeoSys: an open source initiative for numerical simulation of thermo-hydro-mechanical/chemical (THM/C) processes in porous media[END_REF][START_REF] Hammond | Evaluating the performance of parallel subsurface simulators: an illustrative example with PFLOTRAN[END_REF][START_REF] Su | Parallelization of MIN3PTHCm: a highperformance computational framework for subsurface flow and reactive transport simulation[END_REF]. The need for predictive tools led to the development of reactive models considering reaction kinetics and reaction induced changes in porosity [START_REF] Steefel | Reactive transport modelling: an essential tool and a new research approach for the earth sciences[END_REF][START_REF] Lichtner | Incorporating solid solutions in reactive transport equations using a kinetic discrete-composition approach[END_REF][START_REF] Lichtner | Upscaling pore-scale reactive transport equations using a multiscale continuum formulation[END_REF]Xu et al., 2008). However, none of these have used multi-GPU technology.

The Lusi eruption (Java, Indonesia) provides an unprecedented opportunity to test large scale numerical modeling for a hydrothermal system at its infancy. This system, erupting since the 29 th of May 2006 in East Java, elicited debate in science and politics [START_REF] Van Noorden | Mud volcano floods Java[END_REF]. During the early phases, a sequence of aligned clastic eruptions appeared through fractures that followed the same orientation as the Watukosek fault system, a NE-SW tectonic structure in NE Java (Figure 1). This structure can be observed from satellite images and exhibits fractures (including antithetic lineaments), in the field and on seismic data [START_REF] Mazzini | Strike-slip faulting as a trigger mechanism for overpressure release through piercement structures. Implications for the Lusi mud volcano, Indonesia[END_REF][START_REF] Moscariello | Genesis and evolution of the Watukosek fault system in the Lusi area (East Java)[END_REF][START_REF] Obermann | Seismicity at Lusi and the adjacent volcanic complex, Java, Indonesia[END_REF][START_REF] Sciarra | Radon and carbon gas anomalies along the Watukosek fault system and Lusi mud eruption, Indonesia[END_REF][START_REF] Mazzini | Relevant methane emission to the atmosphere from a geological gas manifestation[END_REF]. Fluids quickly flooded large parts of the Sidoarjo Regency urban area, permanently displacing 60,000 people [START_REF] Richards | Report into the past, present, and future social impacts of Lumpur Sidoarjo[END_REF].

Fifteen years later, Lusi continues to erupt boiling mud breccia and displays perpetual geyser-like behavior [START_REF] Karyono | Lusi, a clastic-dominated geysering system in Indonesia recently explored by surface and subsurface observations[END_REF] with flow rates of ~ 80,000 m 3 /day and, with significant earthquake-triggered increases, up to ~120,000 m 3 /day [START_REF] Miller | More than ten years of Lusi: A review of facts, coincidences, and past and future studies[END_REF][START_REF] Mazzini | Relevant methane emission to the atmosphere from a geological gas manifestation[END_REF]. This system displays complex interactions between hydrothermal fluids and carbonate-and clay-rich formations in an active tectonic setting neighboring a volcanic complex.

Numerical modeling helps decipher the mechanisms involved in Lusi hydrothermal system formation and evolution. Mud flow is generally studied by considering a mud reservoir linked to the surface by a mud filled conduit. [START_REF] Zoporowski | Modelling eruption cycles and decay of mud volcanoes[END_REF] reproduced the oscillatory behavior of the eruption with such a geometry. [START_REF] Rudolph | A prediction of the longevity of the Lusi mud eruption, Indonesia[END_REF] considered the effect of overpressure in the mud reservoir on flow through the conduit, and eruption longevity. [START_REF] Davies | Probabilistic longevity estimate for the LUSI mud volcano, East Java[END_REF] also estimated the evolution of the eruption rate with time but with a probabilistic approach. Collignon et al. (2018) determined the influence of conduit radius and clast presence on flow velocity in the conduit. Numerical modeling was also used at Lusi to study other processes than mud flow. [START_REF] Istadi | Modeling study of growth and potential geohazard for LUSI mud volcano: East Java, Indonesia[END_REF] studied the surface area affected by the eruption by combining topographic data with measured mud flow while considering processes such as faulting and subsidence. [START_REF] Svensen | Modelling of gas generation following emplacement of an igneous sill below LUSI, east Java, Indonesia[END_REF] used a one-dimensional thermal model to predict the influence of magma emplacement at depth on the emission of CO 2 . [START_REF] Mazzini | Strike-slip faulting as a trigger mechanism for overpressure release through piercement structures. Implications for the Lusi mud volcano, Indonesia[END_REF] studied the link between strike-slip faulting and fluidization with several rheological models. [START_REF] Tanikawa | Fluid transport properties and estimation of overpressure at the Lusi mud volcano, East Java Basin[END_REF] modelled the generation of overpressure during diagenesis. All these models shed light on processes involved in the eruption.

However, three-dimensional models were not developed, preventing the complex geological structure observed at Lusi [START_REF] Mazzini | Triggering and dynamic evolution of the LUSI mud volcano, Indonesia[END_REF] from being reproduced. Moreover, the contribution of heated fluids generated by magmatic intrusion emplacement and clay dehydration is still enigmatic.

This study investigates the evolution of the Lusi system by exploring hydrothermal fluid circulation and coupled hydro-thermo-mechanical interactions and its potential use for geothermal resources.

We report on a series of numerical simulations using multi-GPU computing to model physical processes involved in this eruptive clastic system. We show that evolution of the Lusi hydrothermal eruption is likely connected to the generation of high fluid pressures by dehydration reactions as proposed by [START_REF] Mazzini | Triggering and dynamic evolution of the LUSI mud volcano, Indonesia[END_REF][START_REF] Mazzini | A new hydrothermal scenario for the 2006 Lusi eruption, Indonesia[END_REF][START_REF] Mazzini | The geochemistry and origin of the hydrothermal water erupted at Lusi, Indonesia[END_REF] and petrographic data analyses [START_REF] Malvoisin | Deep hydrothermal activity driving the Lusi mud eruption[END_REF]Zaputlyaeva et al., 2020). These authors revealed that the clasts originating from the deepest sedimentary formation (i.e. Ngimbang Formation shale source rock) intersected by the Lusi conduit have been exposed to temperatures higher than 350°C. However, the Ngimbang Formation sediments have never been included in any numerical simulation representing the Lusi system. Our modeling shows that dehydration can generate a substantial overpressure and the fluids necessary to support eruption at Lusi. Our model represents the first attempt to model the couplings between reaction, deformation and fluid flow at high-resolution. Here we present a novel approach using models developed to simulate more than 7 billion numerical grid cells in a relatively short computational time (less than a week).

We explore this hypothesis with two different scenarios where in magmatic intrusion and related hydrothermal fluid migration at depth (500°C and 1,000°C beneath Lusi) induces dehydration within the sediment package and, ultimately, we quantify this contribution to the overall fluid outflow. Finally, we investigate the influence of simple reactions and temperature changes on porosity and fluid pressure to determine the evolution of fluid flow. The investigation of this complex system requires the development of long-term computational multi-physics approaches at high-resolution in three dimensions.

Methodology and model set-up

The local stratigraphy has been constrained from borehole data complemented by regional studies, seismic data interpretations, and analysis of erupted clasts [START_REF] Mazzini | Triggering and dynamic evolution of the LUSI mud volcano, Indonesia[END_REF][START_REF] Malvoisin | Deep hydrothermal activity driving the Lusi mud eruption[END_REF][START_REF] Moscariello | Genesis and evolution of the Watukosek fault system in the Lusi area (East Java)[END_REF][START_REF] Samankassou | Origin and age of carbonate clasts from the Lusi eruption, Java, Indonesia[END_REF]Zaputlyaeva et al., 2020). The identified units and formations have been used by [START_REF] Sohrabi | Numerical modeling of the Lusi hydrothermal system: Initial results and future challenges[END_REF] to propose the first 3D geological model for the Lusi eruption: Altered sand, shale and clay from the Pucangan Formation (blue) in Figure 2a, Bluish Grey clay from the Upper Kalibeng Formation (purple), volcanoclastics from the Upper Kalibeng Formation (red), carbonates from the Kujung-(Prupuh)/Tuban Formations (orange), and mudstones from the Ngimgbang Formation (yellow). The model also includes two major faults or shear zones (red) crossing Lusi, the Watukosek strike slip system and the Siring antithetic faults system (Figure 2a). This 3D heterogeneous geological numerical model is then implemented in a newly developed numerical High-Performance Computing (HPC) tool for subsurface fluid dynamics [START_REF] Sohrabi | GEYSER: 3D thermo-hydrodynamic reactive transport numerical simulator including porosity and permeability evolution using GPU clusters[END_REF]. To our knowledge, there is no other numerical tool that can couple physical processes such as clay dehydration, considering porosity and permeability evolution through time, in three-dimensions at such high-resolution (more than 7 billion numerical cells).

Lusi is a sediment-hosted hydrothermal system connected at depth with a neighboring magmatic complex located ~10 km to the SW [START_REF] Miller | More than ten years of Lusi: A review of facts, coincidences, and past and future studies[END_REF]. Geochemical studies show that Lusi vents a mixture of gases that indicate formation temperatures of up to 400 °C [START_REF] Mazzini | A new hydrothermal scenario for the 2006 Lusi eruption, Indonesia[END_REF]. Measurements reveal that Lusi represents one of the largest natural gas systems on Earth, with CH 4 emissions up to 0.1 Tg/year and CO 2 release comparable to large volcanic systems [START_REF] Mazzini | Relevant methane emission to the atmosphere from a geological gas manifestation[END_REF]. High helium isotopic ratios, with mantle-derived signature, have been measured from the gas collected at Lusi (R/Ra = 7) and are those measured at the fumaroles of the neighboring volcanic complex (R/Ra = 7.3) [START_REF] Mazzini | A new hydrothermal scenario for the 2006 Lusi eruption, Indonesia[END_REF][START_REF] Inguaggiato | The Arjuno-Welirang Volcanic Complex and the connected Lusi system: geochemical evidences[END_REF][START_REF] Sciarra | Radon and carbon gas anomalies along the Watukosek fault system and Lusi mud eruption, Indonesia[END_REF]. Regional tomography studies reveal that the Lusi conduit is connected at 4.5 km depth with the volcanic complex through a magmatic intrusion/hydrothermal fluids migration propagating towards the NE of Java Island [START_REF] Fallahi | The plumbing system feeding the Lusi eruption revealed by ambient noise tomography[END_REF]. In agreement with these evidences, studies completed on clast erupted from the Lusi crater, reveal that they had been exposed to temperatures >350 °C, inconsistent with the local geothermal gradient extrapolated to the apparent source depth [START_REF] Malvoisin | Deep hydrothermal activity driving the Lusi mud eruption[END_REF]Zaputlyaeva et al., 2020).

Accordingly, for our modeling, we consider an additional heat source supplied by magmatic intrusion and related hydrothermal fluid migration at depth supplementing the initial conductive geothermal gradient of 42°C/km measured prior to the eruption [START_REF] Mazzini | Triggering and dynamic evolution of the LUSI mud volcano, Indonesia[END_REF]. The model does not include magma flow and dynamics, only the added heat source, assumed to be at a depth of 5.5 km. We investigate two scenarios, one with fluid temperature at 500°C, consistent with a linear interpolation of the thermometry data obtained on clasts erupted at Lusi [START_REF] Malvoisin | Deep hydrothermal activity driving the Lusi mud eruption[END_REF]Zaputlyaeva et al., 2020), and the second with fluid temperature of 1,000°C, that considers a nonlinear temperature evolution with depth. The surface temperature is set to 27°C [START_REF] Svensen | Modelling of gas generation following emplacement of an igneous sill below LUSI, east Java, Indonesia[END_REF]. For rock properties, we used the parameters found in [START_REF] Svensen | Modelling of gas generation following emplacement of an igneous sill below LUSI, east Java, Indonesia[END_REF] for the lithology, combined with analyses from [START_REF] Mazzini | A new hydrothermal scenario for the 2006 Lusi eruption, Indonesia[END_REF] and [START_REF] Tanikawa | Fluid transport properties and estimation of overpressure at the Lusi mud volcano, East Java Basin[END_REF]. We assume that the flow properties of supercritical CO 2 are similar to those of water because supercritical CO 2 is ten times more compressible than water, but it is also about ten times less viscous [START_REF] Miller | Aftershocks driven by a high-pressure CO 2 source at depth[END_REF].

This allows modeling fluid flow by considering only one fluid phase, which is a valid assumption for most of the modeling domain except near the surface where gas forms. This may have consequences on temperature estimates at shallow depth because the latent heat of vaporization is not incorporated in the heat conservation equation. Because we consider a one-phase H 2 O-CO 2 mixture, mud transport is not simulated. Clay dehydration and mass and heat transport in porous media are linked to porosity and permeability changes, which may affect outflow at the surface.

Dehydration reactions involve large and sometimes rapid volume changes related to free water release from the clay mineralogical structure. They occur not only during diagenesis but also as a result of pressure and temperature increase during metamorphism [START_REF] Vidal | Thermodynamic modelling of clay dehydration, stability and compositional evolution with temperature, pressure and H2O activity[END_REF]. The reaction of the two main clay layers (Upper Kalibeng Formation and Ngimgbang Formation) (Figure 2b) may thus change the thermo-hydrodynamic properties of the sediment-hosted Lusi system.

Physical model

The aim of our work is to model the impact of clay dehydration on the Lusi fluid flow. Modeling the complex couplings between reaction, fluid flow and deformation is usually performed with preexisting codes (e.g. Xu et al., 2011). However, the achievable resolution using such models is low (approximately several thousands of grid cells; [START_REF] Izadi | The effects of thermal stress and fluid pressure on induced seismicity during stimulation to production within fractured reservoirs[END_REF][START_REF] Shabani | TOUGHREACT-CO2 Bio-A new module to simulate geological carbon storage under biotic conditions (Part 2): The bio-geochemical reactive transport of CO2-CH4-H2-H2S gas mixtures[END_REF], precluding the inclusion of complex geological structures. Fluid migration and flow at Lusi are apparently controlled by the Watukosek Fault system [START_REF] Mazzini | Strike-slip faulting as a trigger mechanism for overpressure release through piercement structures. Implications for the Lusi mud volcano, Indonesia[END_REF]) so that it is essential to use at least moderately high-resolution. To attain such definition, the physical model is simplified in an effort to highlight processes playing a first-order role on fluid flow at Lusi (dehydration, heating, permeability evolution). The approach presented here does not use separate modules to implement reaction, fluid flow and deformation, and instead fully coupled fluid pressure and porosity variations are calculated in single equations [START_REF] Malvoisin | Coupling changes in densities and porosity to fluid pressure variations in reactive porous fluid flow: Local thermodynamic equilibrium[END_REF]. We use a continuum porous media approach and solve the governing nonlinear equations with the Finite Difference (FD) method. Clay dehydration reactions can play a significant role in the evolution of porosity and permeability and may also contribute to over-pressure development. Production of fluid through dehydration reactions establishes pore pressure gradients that then must diffuse to the surroundings. [START_REF] Mazzini | Triggering and dynamic evolution of the LUSI mud volcano, Indonesia[END_REF][START_REF] Mazzini | The geochemistry and origin of the hydrothermal water erupted at Lusi, Indonesia[END_REF] measured lower salinity in the fluids expelled at Lusi than in seawater, and attributed this dilution to clay dehydration based on enrichment in B, Li and 18 O and a depletion in 2 H in the expelled fluids. Analyses on side well cores from the BJP1 borehole, located in the vicinity of Lusi, revealed that illite-smectite transformation (a dehydration reaction) occurs in the sediments of the Bluish Grey clays from the Upper Kalibeng Formation [START_REF] Mazzini | Triggering and dynamic evolution of the LUSI mud volcano, Indonesia[END_REF].

We model the temperature-dependent clay dehydration by fitting the water content/temperature curve found in [START_REF] Vidal | Thermodynamic modelling of clay dehydration, stability and compositional evolution with temperature, pressure and H2O activity[END_REF] for the illite-smectite transformation. We use the 3D geological model proposed by [START_REF] Sohrabi | Numerical modeling of the Lusi hydrothermal system: Initial results and future challenges[END_REF] to generate a distribution of porosity and permeability, and the 3D multi-GPU simulator developed by [START_REF] Sohrabi | GEYSER: 3D thermo-hydrodynamic reactive transport numerical simulator including porosity and permeability evolution using GPU clusters[END_REF] for clay dehydration.

Mass conservation

We compute fluid pressure (Pf), temperature (T), porosity ( , permeability (k), and fluid generation from clay dehydration as follows.

Total mass conservation in a porous and reacting medium is:

(1)

where , , are the porosity (volume fraction of fluid), density and velocity, respectively, and subscripts 's' and 'f ' denote the properties of the solid and the fluid. Mass of nonvolatile species is conserved by [START_REF] Malvoisin | Coupling changes in densities and porosity to fluid pressure variations in reactive porous fluid flow: Local thermodynamic equilibrium[END_REF]:

(2)

where the amount of fluid into the solid X s is calculated at the equilibrium with a fit of the data of [START_REF] Vidal | Thermodynamic modelling of clay dehydration, stability and compositional evolution with temperature, pressure and H2O activity[END_REF], which gives X s as a function of temperature in the smectite/illite system.

Conservation of momentum

Conservation of fluid momentum is expressed by Darcy's law:

(3)

where the permeability (k) is related to the porosity through the Kozeny-Carman relationship with an exponent 3 and a constant background permeability [START_REF] Kozeny | Uber kapillare leitung der wasser in boden[END_REF]Carmen, 1937;Carmen, 1956), is the fluid viscosity, is the gravitational constant, and is the fluid pressure.

The fluid pressure considering the effects of clay dehydration on volume change (rock density constant without viscous deformation) can be expressed as [START_REF] Malvoisin | Coupling changes in densities and porosity to fluid pressure variations in reactive porous fluid flow: Local thermodynamic equilibrium[END_REF]:

(4)

where the partial time derivatives are replaced by material derivatives ( ) and and are the fluid and pore (crack) compressibility. The effective compressibility is described by:

(5)

The hydro-mechanical porosity change due to fluid production by metamorphism can be expressed by [START_REF] Malvoisin | Coupling changes in densities and porosity to fluid pressure variations in reactive porous fluid flow: Local thermodynamic equilibrium[END_REF]:

(6)

Energy conservation

Energy heat conservation over an elemental volume of medium (solid and fluid phase) at thermal equilibrium so that T = T s = T f [START_REF] Nield | Convection in porous media[END_REF] is:

(7)
where c is the specific heat of the phase, is the thermal conductivity and q is the heat production per unit volume. Equations 1 to 7 results in a highly nonlinear system of differential equation solved here by using the Finite Deference (FD) method.

The general coupled parameters used for simulations are shown in

Results

We generated a numerical grid for multi-GPU computing from the 3D geological model (Figure 3).

We performed 3D simulations in space and time to quantify the effects of clay dehydration in the Upper Kalibeng (purple) and the Ngimgbang (yellow) Formations. We numerically investigated two scenarios with different heated fluid temperatures (500°C and 1,000°C). Time zero was chosen as the onset of the fluid eruption in 2006. Figures 4 and5 show the calculated parameters through the shear zones/faults as pressure (P f ), temperature (T), porosity ( ), permeability (k) and the quantity of fluid (X s ).

We simulated up to 200 years of the system evolution. However, we present only the first 50 years of simulation because fluids produced by dehydration are expected to mostly exhausted by this time. The temperature distributions for the two considered scenarios are different (Figure 4a and Figure 5a). We observe over pressure in the Upper Kalibeng clay Formation of up to 5-10 MPa (Figure 4b and Figure 5b). Through the 50 years of simulation, we perceive that the Ngimgbang and the Upper Kalibeng Formations liberate ~ 17 % and ~ 40 % of their stored water, respectively. This amount of water release modifies the porosity and the permeability of the shear zones/faults (Figure 4c, d and Figure 5c,d). Figure 4e and Figure 5e show that clay layers had already been largely dehydrated by diagenesis before Lusi started to erupt (e.g. illization, [START_REF] Mazzini | Triggering and dynamic evolution of the LUSI mud volcano, Indonesia[END_REF].

Figure 6 and Figure 7 show the evolution of the temperature through the entire Lusi system. These calculated temperatures do not represent the fluid temperature at shallow depth, because two important shallow physical processes are not considered in this work. The latent heat of vaporization, as already discussed, is not implemented neither is the meteoritic and groundwater recharge in the system. Both these processes are extremely efficient in significantly dropping the fluid temperatures of the active system at shallow depths. Figure 8 and Figure 9 show the porosity through the two shear zones/faults (Watukosek and Siring faults) during 50 years of activity. We observe that the temperature of fluid source (500°C or 1,000°C) does not strongly modify the overall temperature distribution in the system over this time scale. The system comes to equilibrium after a certain time. The calculated porosity evolution shows that a 500°C fluid source generates less change in the rock properties of the Ngimgbang Formation than in the Upper Kalibeng Formation.

We computed the amount of fluid released from the two different clay layers (Upper Kalibeng and Ngimgbang Formation) after Lusi began to erupt (Figure 10). The Upper Kalibeng Formation releases four times (~ 4x) more water than the Ngimgbang Formation. Simulated results within the first decade overestimate by a factor of ~ 2 to 3 the measured outflow of fluids at the Lusi site [START_REF] Mazzini | Triggering and dynamic evolution of the LUSI mud volcano, Indonesia[END_REF][START_REF] Mazzini | A new hydrothermal scenario for the 2006 Lusi eruption, Indonesia[END_REF]. The simulated volume of water released is 240,000 m 3 /day for a fluid source at 1,000°C and 230,000 m 3 /day for fluids at 500°C during the first decade.

Simulated outflow caused by clay dehydration was around 60,000 m 3 /day after 25 years and 343 m 3 /day after 50 years for a fluid source at 1,000°C. Using instead 500°C, simulated water release was around 51,000 m 3 /day after 25 years and 184 m 3 /day after 50 years. After 25 years, the temperature in the conduit (Watukosek and Siring faults intersection) is constant at depths greater than ~ 1,500 m (Figure 4a and Figure 5a). As a result, fluid production in the two clay layers ceases (Figure 4e and Figure 5e).

Discussion

Hydrothermal system of Lusi: evolution and lifetime

We investigated the influence of a high-temperature intrusion/hydrothermal fluids migration at depth on clay dehydration, using a high-resolution 3D numerical model. We integrate previous field data measurements and calculations [START_REF] Mazzini | Triggering and dynamic evolution of the LUSI mud volcano, Indonesia[END_REF][START_REF] Mazzini | A new hydrothermal scenario for the 2006 Lusi eruption, Indonesia[END_REF][START_REF] Mazzini | 10 years of Lusi eruption: Lessons learned from multidisciplinary studies (LUSI LAB)[END_REF].

Results show that dehydration from two clay layers will continue to drive the Lusi system. The model proposed here invokes clay dehydration after ascent of hot fluids from a deep source. This process was suggested by geochemical measurements [START_REF] Mazzini | Triggering and dynamic evolution of the LUSI mud volcano, Indonesia[END_REF][START_REF] Mazzini | A new hydrothermal scenario for the 2006 Lusi eruption, Indonesia[END_REF][START_REF] Malvoisin | Deep hydrothermal activity driving the Lusi mud eruption[END_REF] and we demonstrate here that it can explain aqueous fluid ascent and eruption at Lusi. Flow dynamics controlled by hot fluid ascent may partly reproduce a geyser behaviour, even though the Lusi pulsations are not predicted here [START_REF] Mazzini | Triggering and dynamic evolution of the LUSI mud volcano, Indonesia[END_REF][START_REF] Zoporowski | Modelling eruption cycles and decay of mud volcanoes[END_REF][START_REF] Karyono | Lusi, a clastic-dominated geysering system in Indonesia recently explored by surface and subsurface observations[END_REF][START_REF] Lupi | Enhanced hydrothermal processes at the new-born Lusi eruptive system, Indonesia[END_REF]. Geysering is "almost" always associated with multiphase flow. The fact that it does not occur in these simulations probably reflects, at least in part, the single-phase approximation. The Lusi system is of potential resource interest on the long term, regardless the initial temperature of the fluid source. The simulations show fluid flow mainly controlled by the highly permeable pathways provided by fault/shear zones, and these could be targeted for geothermal energy.

Model simulations show that dehydration of the underlying clay layers around ~ 1,500 m (Upper Kalibeng Formation) and ~ 5,000 m (Ngimgbang Formation) could release additional fluid in the future. We investigated two scenarios, and both overestimate by a factor of 2-3x fluid production when compared with on-site measurements. However, these measurements only considered outflow from the main vents [START_REF] Mazzini | Relevant methane emission to the atmosphere from a geological gas manifestation[END_REF] and do not include the large quantities of water released from other active vents.

Our model considers one fluid phase and does not implement the latent heat of vaporization. It is thus unable to model vaporization and its effect on temperature. After magma emplacement at depth, the temperature progressively increases towards the surface up to values above 100°C after 10 years of simulated eruption (Figure 5). The lack of heat extraction due to vaporization also induces temperature overestimation at depth after 10 years of simulated eruption. This effect is more pronounced near the surface, that is in the Upper Kalibeng Formation rather than in the Ngimgbang Formation. The main objective of our model is to investigate the effect of clay dehydration on fluid dynamics at Lusi. Dehydration is complete in the Ngimgbang Formation in the first 10 years of the eruption. Two thirds of the maximum dehydration and of the porosity increase also occur in the Upper Kalibeng Formation during this period (Figure 5c and Figure 5e).

Neglecting vaporization in our model thus only affects one third of the predicted dehydration after 10 years of eruption in the Upper Kalibeng Formation. Vaporization will induce cooler temperature in this formation than predicted with the model. This will slow down its dehydration. The duration of the eruption of 50 years predicted here is thus a minimum. The amount of liberated fluid could also be overestimated by a maximum of one third since vaporization could also reduce the maximum temperature reached in the Upper Kalibeng Formation during the eruption.

Water liberation through dehydration calculated here depends in part on the uncertainties associated with the 3D geological model. Further, the physics applied here do not consider the effect of reaction on heat budget (Eqs. 1 to 7), the influence of boiling, or possible cooling by meteoric water. This explains why the results found in Figure 4a and Figure 5a greatly overestimate shallow temperature in the shear zones/faults zone at later simulation times. The overly high simulated temperatures may also explain why the extent of dehydration in the Upper Kalibeng Formation is 15% higher in the model (82% of dehydration) than in the natural samples (65% illitization; [START_REF] Mazzini | Triggering and dynamic evolution of the LUSI mud volcano, Indonesia[END_REF]. Groundwater recharge from the sediment-hosted system and fluid flow in the shallow surface linked to the Arjuno-Welirang volcanic complex should also be considered in future simulations, to encompass the hydrogeological complexity of the region. This is the first 3D computational model of the Lusi's mud eruption that estimates the contribution of clay dehydration to fluid generation at depth (Figure 11). However, it considers single-phase flow of an aqueous fluid. This approximation is justified at depth greater than ~ 1,800 m, since large part of the mud is believed to originate in the Upper Kalibeng Formation [START_REF] Mazzini | Triggering and dynamic evolution of the LUSI mud volcano, Indonesia[END_REF]. As mud is more viscous than pure aqueous fluid, this may lead to overestimation of fluid velocity in the top part of our model. [START_REF] Zoporowski | Modelling eruption cycles and decay of mud volcanoes[END_REF], [START_REF] Rudolph | A prediction of the longevity of the Lusi mud eruption, Indonesia[END_REF] and [START_REF] Collignon | Modelling fluid flow in clastic eruptions: Application to the Lusi mud eruption[END_REF] investigated mud flow between the mud source and the surface with single-phase, two-dimensional models considering mud as a viscous fluid. However, the impact of interplay between the various phases is still poorly constrained. This hampers our ability to accurately predict flow dynamics, especially near the surface where mud and aqueous fluid coexist and gas can form. Developing a multi-phase numerical model of Lusi should thus be targeted for future numerical investigation.

Further, even though our model provides more complete treatment of the geological complexity at Lusi, it does not solve for stress evolution. We consider the influence of the Watukosek Fault System on permeability but do not model fluidization and subsidence associated with faulting, which is known to play a key role on fluid dynamics at Lusi [START_REF] Istadi | Modeling study of growth and potential geohazard for LUSI mud volcano: East Java, Indonesia[END_REF][START_REF] Mazzini | Strike-slip faulting as a trigger mechanism for overpressure release through piercement structures. Implications for the Lusi mud volcano, Indonesia[END_REF][START_REF] Osorio Riffo | Tectonic insight and 3-D modelling of the Lusi (Java, Indonesia) mud edifice through gravity analyses[END_REF]. Finally, other processes may deserve to be considered for in future largescale numerical modeling; including thermal maturation of organic matter in contact with a magmatic intrusion at depth [START_REF] Svensen | Modelling of gas generation following emplacement of an igneous sill below LUSI, east Java, Indonesia[END_REF], and focusing of seismic waves, which may favor fluidization [START_REF] Lupi | Lusi mud eruption triggered by geometric focusing of seismic waves[END_REF][START_REF] Lupi | Correction: Corrigendum: Lusi mud eruption triggered by geometric focusing of seismic waves[END_REF].

If, in the future, we see a notable increase of the geothermal gradient across the entire sedimenthosted hydrothermal system of East Java, we will have to recalculate and re-estimate the initial conditions invoked in our simulations. In addition, if the fluid outflow at the Lusi site does not decrease on a decadal scale, we might infer that the fluid discharging from the Lusi vent is recharged by other sources.

Implications for potential geothermal resources

Hydrothermal systems can be found in many places and used as geothermal energy resources, as in Italy (e.g. Lardarello), Iceland (e.g. Blue Lagoon), and the U.S.A. (e.g. The Geysers). Each of these systems has a specific hydrogeological environment and its own complexity. At Lusi dehydration may be an important element. The Salton Sea (California, U.S.A.), a sediment-hosted hydrothermal system [START_REF] Svensen | Processes controlling water and hydrocarbon composition in seeps from the Salton Sea geothermal system, California, USA[END_REF][START_REF] Mazzini | Fluid origin, gas fluxes and plumbing system in the sediment-hosted Salton Sea Geothermal System (California, USA)[END_REF], may bear some resemblance to Lusi [START_REF] Mazzini | 10 years of Lusi eruption: Lessons learned from multidisciplinary studies (LUSI LAB)[END_REF]. However, Salton Sea system has been geologically active for a long while compared to Lusi. There, deep heated fluids fill the system and structural clay dehydration causes porosity evolution at a different scale. Our results show that for 50 years following the Lusi inception in 2006, the dynamics and thermodynamics will be strongly dependent on this process, which induces irreversible physical changes to the rock via water release.

Our high-resolution 3D numerical simulations track the controlled rock property changes (dehydration) over 50 years. The models suggest that the clay dehydration should decrease after 25 years and may stop after 50 years. The complex volcanic structure in the vicinity of Lusi is a potential source of geothermal energy for the surrounding population as reflected by the simulation distribution of temperature in time and space (Figure 6 and Figure 7). The simulations suggest evolving property profiles which may control hydro-thermo-mechanical structure over time.

Conclusions

We present 3D numerical simulations using multi-GPU computing of fluid thermo-hydrodynamics with porosity and permeability evolution driven by hydrothermal fluid intrusion at 5.5 km depth below the active Lusi eruption. Starting from 2006, the results demonstrate that Lusi decreasing less outflow due to decreasing clay dehydration (in the Upper Kalibeng Formation and Ngimgbang Formation) over the next decade. Our simulations suggest that the contribution of dehydration to water release may essentially stop after 25 years of activity and completely stop after 50 years.

Clay mineral illitization, already ongoing before the eruption, continues mainly in the Upper Kalibeng Formation but also in the Ngimgbang Formation under conditions of increasing temperature and pressure. This process is irreversible and may be complete after about 50 years.

The simulations are consistent with the degree of illitization measured on site. Heating of the Upper Kalibeng Formation could produce hydro-fractures which will enhance a preferential pathway for fluid to the surface. Porosity and permeability evolution in time can have a nonnegligible effect.

The Lusi activity may still persist after 50 years because of the groundwater recharge of the entire hydrothermal system linked to the Arjuno-Welirang volcanic complex.

The initial temperature of the source fluids (500°C or 1,000°C) does not have a significant influence on the entire system. Temperatures are better reproduced during the first decade of Lusi activity with a hypothetic source at 500°C. The 10 MPa calculated overpressure calculated and the heated fluids coming from 5.5 km are primary controls on the clay dehydration rates.

3D numerical modeling can inform decision making on natural or engineered systems. High-Performance Computing (HPC) is a valuable technique to model physical processes in a reasonable computational time. Geothermal reservoir valorization is an important topic today. Decisions have to be made regarding use of renewable energy. Deeper boreholes and complementary coupled numerical physical analyses are needed to evaluate this potentially large geothermal energy supply.

Figure 1 :

 1 Figure 1: a) Inset map of Indonesia and topographic map showing the Arjuno-Welirang volcanic complex to the south with some of the surrounding hydrothermal springs. Lusi and other mud volcanoes are located to the north of the island inside the sedimentary basin. b) Aerial image showing erupting craters in the central part of the inaccessible hydrothermal pond.

Figure 2 :

 2 Figure 2: Conceptual model for fluid pressure increase controlled by dehydration in a deep hydrothermal system or geothermal reservoir. Under conditions of increasing temperature and pressure, clay dehydration reactions begin. a) 3D conceptual model considering five main stratigraphic units and two main shear zone/faults at Lusi site. b) 3D model set-up for simulations, considering deep heated fluids filling the entire Lusi hydrothermal system.

Figure 3 :

 3 Figure 3: 3D structural numerical grid with initial parameters permeability (k) and porosity ( ) distribution at high-resolution (384x384x384) for multi-GPU calculation. This numerical grid is based on the 3D geological model of Lusi presented in Sohrabi et al. (2018).

Figure 4 :

 4 Figure 4: Evolution of simulated a) temperature, b) pressure, c) porosity, d) permeability and e) water liberation through the Lusi shear zones/faults after fluid intrusion at 500°C after: 0 years (red), 5 years (green), 10 years (blue), 25 years (magenta) and 50 years (dashed cyan).

Figure 5 :

 5 Figure 5: Evolution of simulated a) temperature, b) pressure, c) porosity, d) permeability and e)water liberation through the Lusi shear zones/faults after fluid intrusion at 1,000°C after: 0 years (red), 5 years (green), 10 years (blue), 25 years (magenta) and 50 years (dashed cyan).

Figure 6 :

 6 Figure 6: Evolution of temperature from an initial geothermal gradient of 42°C/km throughout the system and fluid intrusion at 500°C with input parameters presented in the 3D geological model (Figure 3). Times are since Lusi erupted.

Figure 7 :

 7 Figure 7: Evolution of temperature from an initial geothermal gradient of 42°C/km throughout the system and fluid intrusion at 1,000°C with input parameters presented in the 3D geological model (Figure 3). Times are since Lusi erupted.

Figure 8 :

 8 Figure 8: Evolution of porosity after fluid intrusion at 500°C with input parameters presented in the 3D geological model (Figure 3). The two shear zones/faults are shown with a focus on the two clay layers (Upper Kalibeng Formation and Ngimgbang Formation). Times are since Lusi erupted.

Figure 9 :

 9 Figure 9: Evolution of porosity after fluid intrusion at 1,000°C with input parameters presented in the 3D geological model (Figure 3). The two shear zones/faults are shown with a focus on the two clay layers (Upper Kalibeng Formation and Ngimgbang Formation). Times are since Lusi erupted.

Figure 10 :

 10 Figure 10: Release water due to the dehydration of the two main clay layers over 50 years after Lusi started to erupt in 2006. The red line shows results for the fluid intrusion at 1,000 °C and the blue line results for fluid intrusion at 500 °C. The green circles are outflow data measured at the Lusi vent.

Figure 11 :

 11 Figure 11: Conceptual model based on the numerical simulations performed in this work. Heated fluids associated with magmatic intrusion at depth induces clay dehydration feeding the Lusi mud eruption.

  Figure 1

Table 1 .

 1 Parameter values chosenfor the fully coupled 3D numerical model (Table1) are inputs measured and calculated from previous studies.

Table 1 :

 1 Parameters definition and values used.

	Symbol	Definition	Parameters	Units
		Fluid pressure	-	Pa
		Density of fluid	-*	kg•m -3
		Density of solid	2650	kg•m -3
		Velocity of fluid	-	m•s -1
		Compressibility of fluid	1e -10	Pa -1
		Compressibility of solid	1e -9	Pa -1
		Effective Compressibility	-	Pa -1
		Porosity	-	-
	k	Permeability	-	m 2
		Mass fraction of fluid in solid	-	wt %
	T	Temperature	-	°C
		Fluid specific heat	4190	J/kg•K
		Solid specific heat	1000	J/kg•K
		Fluid thermal conductivity	0.6	W/ m•K
		Solid thermal conductivity	2.1	W/ m•K
		Fluid heat production	-	J/ m 3 / s
		Solid heat production	-	J/ m 3 / s
	g	Gravity acceleration	9.81	m 2 s -1

* calculated with the Equation of State (EOS) from[START_REF] Sun | New equations for density, entropy, heat capacity, and potential temperature of a saline thermal fluid[END_REF].

Acknowledgements

We appreciate helpful discussions with G. Jansen, Y.Y. Podladchikov and B. Valley. We thank the Editor and anonymous reviewers for very constructive suggestions.

Funding

This work was funded by a grant from the Swiss National Science Foundation (SNSF) Project n°200021-16005/1 and by the European Research Council under the European Union's Seventh Framework Programme Grant agreement No. 308126 (LUSI LAB project) and the Research Council of Norway through its Centers of Excellence funding scheme, Project number 223272 (CEED) and the HOTMUD Project number 288299

Data availability

The relevant data to reproduce the numerical results are provided in the manuscript. The remaining data used in this study are available from cited references. All data and materials that support the findings of this study can be made available, in some form, to any researcher for purposes of reproducing or extending the analysis upon reasonable request to the corresponding author R.S.

Code availability

The numerical code used in this study is available from cited reference (with restrictions) upon reasonable request.

Competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Flore Villaret
email: flore.villaret@edf.fr

Xavier Boulnat

Pascal Aubry

Julien Zollinger

Damien Fabrègue

Yann De Carlan

Modelling of delta ferrite to austenite phase transformation kinetics in martensitic steels: application to rapid cooling in additive manufacturing

Keywords: Steel, Martensite, Phase transformation kinetics, Electron BackScatter Diffraction (EBSD), Metal Additive manufacturing

In this paper, the high temperature transformation kinetics of delta ferrite to austenite (δ → γ) phase transformation is modeled by thermodynamic and diffusion calculations. It appears that, in martensitic steels, the δ → γ transformation is very fast (a few milliseconds) as soon as the first austenite nuclei appears. Classically the austenitic phase will thus systematically be observed in the material during conventional elaboration processes. However, in powder metallurgy and additive manufacturing, it is possible to obtain sufficiently high quenching rates (up to 10 6 °C/s) so that the γ phase does not have time to appear. The calculations presented here allow to rationalize the understanding of the microstructures of powders and different additive manufacturing materials.

They enable to understand why ferrite or martensite is sometimes obtained in the final microstructure. From the calculations made, an original CCT (Continuous Cooling Transformations) diagram starting from the δ phase is proposed. This understanding is one more step toward the control of microstructure and properties of additively manufactured martensitic steels.

INTRODUCTION

High-chromium martensitic steels are widely used in energy applications, thanks to their good mechanical and corrosion properties at high temperature and their resistance to irradiation swelling [START_REF] Pandey | Effect of normalizing temperature on microstructural stability and mechanical properties of creep strength enhanced ferritic P91 steel[END_REF][START_REF] Shankar | Low cycle fatigue behavior and microstructural evolution of modified 9Cr-1Mo ferritic steel[END_REF][START_REF] Zinkle | Materials challenges in nuclear energy[END_REF][START_REF] Cabet | Ferritic-martensitic steels for fission and fusion applications[END_REF]. These steels are called martensitic because of their ability to keep a martensitic microstructure under low cooling rates, which are induced in conventional manufacturing steps, such as forging.

The extreme temperature gradients and the very fast cooling rates associated with additive manufacturing processes can lead to out-of equilibrium microstructures. As reported by Herzog et al. [START_REF] Herzog | Additive manufacturing of metals[END_REF], many studies have been done on austenitic stainless steels such as 316L or 304L, but only few studies deal with martensitic steels, focusing on maraging and tools steels.

As a matter of fact, steels which exhibit a martensitic microstructure with the conventional elaboration processes can exhibit a ferritic or ferritic/martensitic microstructure after additive manufacturing. Such unusual microstructures were already reported in literature. For example, in precipitation hardened steels such as the 17-4PH, Sun et al. [START_REF] Sun | Effect of laser scan length on the microstructure of additively manufactured 17-4PH stainless steel thin-walled parts[END_REF] noticed that changing the laser beam path length and wall thickness in Laser Beam Powder Bed Fusion (PBF-LB) allowed to switch from a ferritic to a martensitic microstructure. In the same way, Adeyemi et al. [START_REF] Adeyemi | Influence of laser power on microstructure of laser metal deposited 17-4 ph stainless steel[END_REF] observed that increasing laser power leads to more ferrite in Laser Beam Direct Energy Deposition (DED-LB). Vunnam et al. [START_REF] Vunnam | Effect of powder chemical composition on the as-built microstructure of 17-4 PH stainless steel processed by selective laser melting[END_REF] has shown that, with the same PBF-LB building parameters, a small change in the powder chemical composition can also change the microstructure.

In reduced activation 9Cr steels, Liu et al. [START_REF] Liu | Effect of scanning strategy on microstructure and mechanical properties of selective laser melted reduced activation ferritic/martensitic steel[END_REF] and Jiang et al. [START_REF] Jiang | Strong and ductile reduced activation ferritic/martensitic steel additively manufactured by selective laser melting[END_REF] observed that it is possible to control the amount of ferrite by changing the scanning strategy in PBF-LB. Zhong et al. [START_REF] Zhong | Microstructures and mechanical properties of a modified 9Cr ferritic-martensitic steel in the as-built condition after additive manufacturing[END_REF] chose instead to specifically design an alloy composition for DED-LB process to obtain the martensitic microstructure. This is also possible with the good processing conditions, as demonstrated for example in DED-LB by Xia et al. [START_REF] Xia | Microstructure evolution and mechanical properties of reduced activation steel manufactured through laser directed energy deposition[END_REF].

Alnajjar et al. [START_REF] Alnajjar | Evidence of austenite by-passing in a stainless steel obtained from laser melting additive manufacturing[END_REF], as others authors, explain the formation of this microstructure by the possibility of partially or entirely "skipping" the austenitic domain due to the high cooling rate in additive manufacturing but no model could be proposed to account for the diversity of the microstructures obtained.

In this article, the growth kinetics of an austenite nucleus as a function of the cooling rate is computed by thermodynamic and diffusion calculation. This demonstrates to what extent it is possible to avoidor notthe formation of austenite during cooling and allow to understand the microstructures in martensitic steels obtained by additive manufacturing.

A Fe-9Cr-1Mo martensitic steel is used as a demonstration alloy. In this example, the variation of the cooling rate will be obtained by comparing different sizes of powder particles and PBF-LB and DED-LB builds. Indeed, the cooling rate is one or two magnitude order higher in PBF-LB [START_REF] Ma | A comparison on metallurgical behaviors of 316L stainless steel by selective laser melting and laser cladding deposition[END_REF].

EXPERIMENTAL TECHNIQUES

Samples were built using a Fe-9Cr-1Mo pre-alloyed steel powder. Chemical composition and powder size distribution used in this work are given in Table 1.

This powder was gas atomised, so it has spherical particles, a microstructural analysis of the powder is presented later, in 4.2. DED-LB and PBF-LB samples were built using parameters listed in The microstructure was observed on a ZEISS Sigma HD Field Emission Gun Scanning Electron Microscope (FEG-SEM) with a Bruker Electron BackScatter Diffraction (EBSD) camera with a voltage of 20 kV. In PBF-LB samples, observations were conducted in the plan parallel to the Building Direction (BD).

Scanning strategy

Single tracks Island scan + 67° rotation between each layers

Samples geometry

Walls 20x30 mm Cube 10x10x10 mm

In DED-LB samples, observations were conducted in the plan parallel to the Building Direction and the Scanning Direction.

The step size used for acquisition of all EBSD maps is 0.2 µm/pixel. Inverse Pole Figures (IPF) are projected along the building direction. A threshold angle of 10° was used to define grain boundaries. EBSD Band Contrast was used to distinguish ferrite from martensite by computing the mean band contrast by grains with the method described in [START_REF] Vunnam | Effect of powder chemical composition on the as-built microstructure of 17-4 PH stainless steel processed by selective laser melting[END_REF][START_REF] Nowell | Differentiating ferrite and martensite in steel microstructures using electron backscatter diffraction[END_REF][START_REF] Mujica | Microstructure and mechanical properties of laser-welded joints of TWIP and TRIP steels[END_REF][START_REF] Wilson | Application of orientation imaging microscopy to study phase transformations in steels[END_REF]. An adapted threshold was then used to separate ferrite grains from martensite.

AUSTENITE GROWTH KINETICS MODELING DURING COOLING FROM DELTA PHASE

This section proposes a scenario to explain the microstructure differences in martensitic steels, based on the example of Fe-9Cr-1Mo powders, and steels built either by DED-LB or PBF-LB. The growth kinetics of an austenite nucleus during cooling, as a function of the cooling rate is calculated using diffusion equations.

Here, each transformation temperature is named as following (see also Figure Assuming that solidification takes place in the δ domain and that an austenite nucleus is instantly formed when T < Ae5 (these assumptions will be discussed in more details later), its growth during cooling is mainly controlled by diffusion [START_REF] Leone | The Ferrite to Austenite Transformation in Stainless Steels[END_REF][START_REF] Kobayashi | The formation and dissolution of residual δ ferrite in ASME Grade 91 steel plates[END_REF]. The idea here is to compute the size reached by an austenite grain after cooling for each process.

Since the cooling rates considered are very high, the model presented here is based on the assumption that austenite growth is controlled by the diffusion of interstitial elements such as carbon and nitrogen, while other elements do not have the time to diffuse [START_REF] Hemmer | A process model for the heat-affected zone microstructure evolution in duplex stainless steel weldments: Part I. The model[END_REF]. In the case of the Fe-9Cr-1Mo steel used as an example here, the nitrogen content is very low (4 ppm of nitrogen versus 1000 ppm of carbon), so only carbon diffusion will be considered.

The growth of an austenite grain during cooling can be calculated in the δ + γ and γ domains successively.

In the δ + γ domain, the thermodynamic equilibrium of carbon distribution between the two phases must be considered. The radius rδ+γ of a growing nucleus in this temperature range according to the cooling speed (v) can then be calculated with the following equation, adapted from [START_REF] Hertzman | An experimental and theoretical study of heataffected zone austenite reformation in three duplex stainless steels[END_REF]:

𝒓 𝜹+𝜸 = ∫ 𝑫(𝑻
Where D(T) is the carbon diffusion coefficient in γ phase, depending on the temperature T, and Ω is the supersaturation considering local thermodynamic equilibrium defined as:

𝛀 = 𝝎 𝑪 0 -𝝎 𝑪 𝜹/𝜸 𝝎 𝑪 𝜸/𝜹 -𝝎 𝑪 𝜹/𝜸 (2)
Where 𝜔 𝐶 0 is the nominal carbon composition of the steel, 𝜔 𝐶 𝛿/𝛾 is the carbon concentration of δ in equilibrium with γ and 𝜔 𝐶 𝛾/𝛿 is carbon concentration of γ in equilibrium with δ.

These carbon equilibrium contents of the δ and γ phases in the δ + γ domain, as well as the diffusion coefficients as a function of temperature are obtained from the data tabulated in the ThermoCalc software with the TCFE5 and MOB2 thermodynamic and mobility databases.

In the γ domain, the growth of the nucleus is only controlled by atomic diffusion, the final size rf of the new grain when Ae3 is reached is calculated according to the cooling speed using the following equation (adapted from [START_REF] Kobayashi | The formation and dissolution of residual δ ferrite in ASME Grade 91 steel plates[END_REF]):

𝒓 𝒇 = 𝒓 𝜹+𝜸 + ∫ 𝟐 (𝑫(𝑻). 𝒅𝑻 𝒗 ) 𝟎.𝟓 𝑨𝒆 𝟒 𝑨𝒆 𝟑 (3)
To carry out these calculations, the Fe-9Cr-1Mo equilibrium phase diagram is used. The calculated two-phase δ + γ domain therefore extends from Ae5 = 1420 °C to Ae4 = 1240 °C, and the single-phase γ domain from Ae4 = 1240 °C to Ae3 = 850 °C (Figure 1).

With this input data and using equations ( 1) and ( 3), it is possible to plot the size of an austenite grain when it reaches Ae3 as a function of the cooling rate (green curve in Figure 2). To test another hypothesis than a nucleus immediately formed at Ae5, other calculations have been made assuming that the austenite grain only forms at 1200 °C (red curve on Figure 2) or 1100 °C (blue curve on Figure 2). As expected, the size of the austenite grain is then much smaller when the temperature reaches Ae3. This model demonstrates the important effect of the cooling rate on the growth of an austenite grain: the faster the cooling rate, the less time the new austenite grain will have time to grow.

Two cooling rates representative of PBF-LB and DED-LB can be considered, according to the correlation set by Ma et al. [START_REF] Ma | A comparison on metallurgical behaviors of 316L stainless steel by selective laser melting and laser cladding deposition[END_REF], to illustrate the impact of the cooling rate on microstructures. In DED-LB (v = 10 4 °C/s), the size of an austenite grain can theoretically reach 61 μm. The time spent in the domains δ + γ and γ is therefore sufficient to allow the formation and growth of austenite grains, until the δ ferrite disappears completely.

In contrast, in PBF-LB (v = 10 6 °C/s) the time spent in the γ domain is 10 to 100 times shorter, leaving much less time for austenite grains to grow. With the cooling rate of 10 6 °C/s, an austenite grain only reaches a theoretical size of 6 μm if nucleation starts at Ae5 and can go down to 1.9 µm if the nucleation stage is delayed to lower temperature. This demonstrates that a very high cooling rate can lower the formation of austenite and then martensite in Fe-9Cr-1Mo.

The austenite formation and growth stop when the Ae3 temperature is reached.

From this temperature, the untransformed δ ferrite becomes stable again up to room temperature. Any austenite grains that may have formed are transformed into martensite when the Ms temperature is reached (around 400 °C).

From the calculations made, an original CCT (Continuous Cooling

Transformations) diagram starting from the δ phase is proposed (Figure 3).

For a fast PBF-LB cooling, austenite only has little time to form so the most of the δ ferrite is maintained up to room temperature. For a DED-LB cooling, or slower cooling, such as a conventional water quench, austenite replaces the δ ferrite and forms martensite at room temperature. Some phenomena are not considered in this model and may further reduce the size of the austenite grains previously calculated. For example, the initial formation of the austenite nucleus may be delayed (due to chemical heterogeneities for example): for nucleation at 1100 °C instead of 1420 °C, with a cooling rate of 10 6 °C/s, the size of the austenite grain when Ae3 is reached is only 1.9 μm (blue curve Figure 2). The possibility of forming such a segregation during solidification will be discussed in the next section. This model also does not consider capillarity, or precipitates formation (like M23C6), which could further slow the growth of austenite.

DISCUSSION AND ANALYSIS OF EXPERIMENTAL OBSERVATIONS

Segregation effects during the solidification

In paragraph 3, it is assumed that all the liquid is transformed into delta ferrite with the same composition to simplify the explanation of the model. In practical cases, it is known that fast solidification implies segregation [START_REF] Liu | Dislocation network in additive manufactured steel breaks strength-ductility trade-off[END_REF][START_REF] Godec | Quantitative multiscale correlative microstructure analysis of additive manufacturing of stainless steel 316L processed by selective laser melting[END_REF].

It is possible to consider the chemical segregation in the model, by using the phase diagram associated with the composition between and inside the solidification cell (it is the diagrams showed in Figure 5, with a correction considering the diffusion in solid delta phase as shown in Figure 6) instead of using the general Fe-9Cr-1Mo phase diagram (Figure 1). The purpose of this paragraph is not to show the application of the model in this case, but to show that segregation phenomena only really has an impact in PBF and can only limit the formation of austenite.

The very fast cooling undergone in additive manufacturing leads to an out-ofequilibrium solidification of the liquid. Under these conditions, the diffusion of the metallic elements in the solid is not fast enough to guarantee chemical homogeneity. On the other hand, the solid/liquid interface velocity is not fast enough to deviate from the local thermodynamic equilibrium [START_REF] Aziz | Continuous growth model for interface motion during alloy solidification[END_REF]. For most of the alloying elements used in steel, this implies that the solid formed at the end of solidification will be enriched compared to the solid formed at the beginning with partition coefficient k, defined as the ratio between the solid to the liquid composition, smaller than 1. The Gulliver-Scheil equations [START_REF] Scheil | Bemerkungen zur schichtkristallbildung[END_REF] can be used to account for this enrichment and to predict possible changes in the solidification sequence.

In the following, the solidification microstructure is assumed to be cellular rather than dendritic. The considered length scale for microsegregation is half the cellular spacing  (Figure 4(a)), it can be estimated using the geometrical model developed by Hunt [START_REF] Hunt | Cellular and primary dendrite spacings[END_REF], that has already been successfully applied to predict spacing in AM solidification microstructure [START_REF] Li | Solidification microstructure during selective laser melting of Ni based superalloy: Experiment and mesoscopic modelling[END_REF]. Calculations lead to values of  of 2 µm for DED-LB and 0.5 µm for PBF-LB. Experimentally, the measured solidification cell size is typically 1 to 5 μm in diameter, depending on the process and parameters used [START_REF] Ma | A comparison on metallurgical behaviors of 316L stainless steel by selective laser melting and laser cladding deposition[END_REF]. The small size of the solidification cells makes it difficult to experimentally observe compositional differences. The analysed surface in SEM-EDX is generally in the range of 0.1 pt/µm [START_REF] Moncel | Méthodes physiques d ' analyse en métallurgie[END_REF], which is too large to measure the chemical segregations induced by such solidification, STEM-EDX should be used for this purpose as done in [START_REF] Godec | Quantitative multiscale correlative microstructure analysis of additive manufacturing of stainless steel 316L processed by selective laser melting[END_REF] for example.

As assumed at the beginning of section 3, the Scheil equation predicts that all the solidification occurs in δ ferrite (Figure 4(b)), despite the compositional differences that can be observed in the solid between the beginning and the end of solidification (Figure 4(c)). Such local differences in chemistry between the solid formed at the beginning and at the end of solidification (i.e. between the interior and the edge of the solidification cells) have already been observed in the literature, especially in 316L after PBF-LB [START_REF] Liu | Dislocation network in additive manufactured steel breaks strength-ductility trade-off[END_REF][START_REF] Godec | Quantitative multiscale correlative microstructure analysis of additive manufacturing of stainless steel 316L processed by selective laser melting[END_REF] or DED-LB [START_REF] Ziętala | The microstructure, mechanical properties and corrosion resistance of 316 L stainless steel fabricated using laser engineered net shaping[END_REF] and also in 300M low alloyed steel (40 NiSiCrMo 7) [START_REF] Liu | Effect of microstructure on the Charpy impact properties of directed energy deposition 300M steel[END_REF].

It would be possible to calculate the phase change temperatures during the cooling of each of the compositions of the solid formed during solidification. As the composition of the solid varies little, up to the last fractions of solid formed, we have chosen for simplicity, to represent only the two extreme compositions corresponding to the first and the last solid formed (Figure 5).

At the beginning of solidification, the composition of the solid is close to the nominal composition, the liquid solidifies into δ ferrite and can then be transformed into austenite at lower temperatures (Figure 5 (a)). At the end of solidification, the solid formed is enriched in alloying elements such as Cr, C, Mo and Mn (Figure 5 (b)). In this case, the liquid is still transformed into δ ferrite but these changes in chemical composition stabilize the ferrite and prevent the formation of austenite at lower temperatures (Figure 5 (b)). However, this more strongly ferrite-forming composition only affects a few percent of the solid formed at the end of solidification. These theoretical equilibrium calculations show that it is possible to form austenite on cooling in the majority of the solidified material, the driving force to form austenite decreases while the solid is enriched with solute elements.

However, since cooling in additive manufacturing is extremely rapid, it is necessary to consider the kinetic aspects of the δ→γ phase change to understand the ferritic/martensitic microstructures obtained in additive manufacturing, as it was proposed in the previous section.

Around 1360 °C, the δ phase solidification is complete and chemical segregations exist between the solid at the edge of the solidification cells, formed last, and the rest of the solid.

Between 1360 °C and 1250 °C, the δ ferrite is stable (Figure 5). It can be noticed that in this case Ae5 is lowered with respect to the calculation performed without segregation in section 3, were Ae5 = 1420 °C. In this temperature range, the segregations formed during solidification can be partially homogenized by diffusion. Using the well-known formula giving for a time interval dt and a temperature-dependent diffusion coefficient D(T), the characteristic diffusion length dL of a chemical element, it is possible to evaluate the effectiveness of homogenization in this temperature range:

𝒅𝑳 = √𝑫(𝑻). 𝒅𝒕 (4)
For a temperature increment dT and a given cooling speed v, the time spent at this temperature is expressed as:

𝒅𝒕 = 𝒅𝑻 𝒗 (5)
Combining ( 4) and ( 5), it is possible to compute for each temperature increment the diffusion length increment. Summing these increments allows to calculate the diffusion length in a temperature range for a given cooling rate:

𝑳 = ∫ 𝒅𝑳 = ∫ (𝑫(𝑻). 𝒅𝑻 𝒗 ) 𝟎.𝟓 (6)
As carbon is an interstitial element, its diffusion is much faster than that of substitutional metallic elements such as chromium or molybdenum. Here, only diffusion of carbon and chromium are considered. Chromium is chosen to represent the diffusion of the substitutional elements because it is the element that is present in the largest quantity in Fe-9Cr-1Mo. Together with carbon, it is mainly responsible for the control of the ferrite ↔ austenite transformation in this alloy.

20

For carbon, with a cooling rate of 10 6 °C/s (PBF-LB), the characteristic diffusion length is greater than the size of the solidification cells (LC, PBF-LB = 5.7 μm compared with PBF-LB/2 = 0.25 µm, and Lc,DED-LB = 57 µm compared with DED-LB/2 = 1 µm) (Figure 6 (a)). This shows that the diffusion of carbon in the δ domain is fast enough to ensure complete homogenization across the solidification cells, whether in DED-LB or PBF-LB.

For chromium, diffusion is slower. At a cooling rate of 10 6 °C/s (PBF-LB) the diffusion length is roughly the size of the solidification cells (LCr, PBF-LB = 0.21 μm to be compared with PBF-LB/2 = 0.25 µm ), which means that Cr segregations probably still remain marked at the edge of the solidification cells when entering the austenitic domain. However, at a velocity of 10 4 °C/s (DED-LB), the characteristic diffusion length is larger than the size of the solidification cells (LCr, DED-LB = 2.1 μm to be compared with DED-LB/2 = 1 µm), which means that the Cr segregations between the cells are attenuated during this cooling phase. At the end of solidification and cooling in the δ domain, most of the solid formed has a chemical composition that allows the formation of austenite. Only a few percent of the PBF-LB solid still has a ferrite-forming composition. It is usually admitted that the austenite nucleation is favored on defects, such as at the grain boundaries and intergranular solidification cells [START_REF] Hsieh | Effects of Cooling Time and Alloying Elements on the Microstructure of the Gleeble-Simulated Heat-Affected Zone of 22% Cr Duplex Stainless Steels[END_REF][START_REF] Gao | The isochronal δ → γ transformation of high Cr ferritic heat-resistant steel during cooling[END_REF]. In the case of the PBF-LB Fe-9Cr-1Mo, the chemical composition at these places favors the stabilization of the ferrite because of the Cr enrichment. Austenite nucleation should be more difficult because of this enrichment in PBF-LB, this may result in a delay in austenite nucleation and therefore in retention of δ ferrite at room temperature.

Analysis of powder microstructures

Powder microstructure was analyzed on polished cross section by EBSD (Figure 7). It is interesting to note that the microstructure is changing with the powder size: small particles are ferritic (Figure 7 Such microstructure changes in steel powders are not particularly common in the literature. However, in 2002, Pryds and Pedersen [START_REF] Pryds | Rapid solidification of martensitic stainless steel atomized droplets[END_REF] observed this phenomenon in a martensitic powder of X20 CrMoV 12-1 steel. To explain this result, the authors estimated the cooling rate of the powder particles as a function of their size. According to their calculations, a powder particle with a diameter around 10 μm is cooled at a speed of about 4.10 5 °C/s while a powder particle with a diameter around 50 μm is cooled at a speed of about 2.10 4 °C/s. This calculation shows that the size of the powder grain has a strong influence on the cooling rate.

Observations and these calculations agree with the model presented in section 3: in fast solidification conditions, the faster the Fe-9Cr-1Mo cools (i.e. the smaller the powder), the less martensite is formed. In their study, Pryds and Pedersen showed that austenite formed more easily in large diameter powders. This is because the higher number of grains and the slower cooling rate facilitate the germination of austenite in these powders. In small diameter powders, the lower grain count and faster cooling rate prevent austenite formation when cooling from liquid, which leads to a ferritic microstructure from the δ ferrite formed during solidification. This observation made in X20 CrMoV 12-1 steel can be extrapolated to the Fe-9Cr-1Mo steel studied here.

This demonstrates that for this chemical composition, the formation of the martensitic microstructure from the liquid phase is sensitive to the cooling rate between 10 4 °C/s and 10 5 °C/s.

However, it is important to keep in mind that powder particles are fully melted and mixed during the PBF-LB and DED-LB processes, meaning that their initial microstructure and eventual chemical segregations are erased during the building process. Prior austenite grains were determined using Kurdjumow-Sachs and Nishiyama-Wassermann relationships, with a method similar to the one used in ARPGE software [START_REF] Cayron | ARPGE: A computer program to automatically reconstruct the parent grains from electron backscatter diffraction data[END_REF]. This allows to plot prior austenite grains boundaries (in white on Since the real thermal cycle undergone by the material during the building is not precisely measured in this case, it is not possible to compare the grain sizes obtained here with the ones computed in section 3. Only the order of magnitude could be compared.

The increase in prior austenite grain size between the bottom and the top of the wall is explained by an increase of the temperature around the melt pool, as the height of the wall increases (cooling by conduction is reduced and the energy supplied by the laser accumulates) [START_REF] Xia | Microstructure evolution and mechanical properties of reduced activation steel manufactured through laser directed energy deposition[END_REF]. This increase in temperature is also accompanied by a reduction of the cooling rate. As the cooling rate decreases, the austenite nucleation is reduced. The temperature increase accelerates grain growth, leading to a higher austenite grain size at the top of the wall.

PBF-LB

In PBF-LB, the cooling rate during construction is higher than in DED-LB (approx.

10 6 °C/s in PBF-LB compared to 10 4 °C/s in DED-LB [START_REF] Ma | A comparison on metallurgical behaviors of 316L stainless steel by selective laser melting and laser cladding deposition[END_REF]). The increased cooling rate results in a more complex ferritic/martensitic microstructure (Figure 9(c)), which is unusual for bulk materials of equivalent composition.

In the middle of the wall, the ferrite forms coarse grains elongated in the building direction (BD), and characterized by a <001>//BD preferential orientation (red in Figure 9 (b)), which corresponds to the preferential solidification direction in cubic materials [START_REF] Niendorf | Highly Anisotropic Steel Processed by Selective Laser Melting[END_REF]. On this EBSD map, the size of the prior austenitic grains, determined in the same way as in Figure 8, is around 10 μm, which is much finer than in DED-LB.

The EBSD observation being a local observation, several large EBSD maps or optical micrograph are required to obtain a representative observation of the microstructure. Only the last layers are shown here in order to illustrate and validate the model presented. Detailed analysis of the last layer shows a majority of ferritic grains (Figure 9 (c)).

These ferritic grains are oriented in the direction of the thermal gradient of the last laser pass (arrows in Figure 9 (b)). A few martensite grains can be observed in this last layer, in the overlap zone between the melt pools (dotted lines in Figure 9). These observations show that martensite in PBF-LB materials is formed during the heating and subsequent cooling of the material induced by the construction of the next or adjacent layer. The ferrite comes directly from the solidification of the material. This observation is in good agreement with the model presented in section 3, which predicts the microstructure to be almost fully ferritic.

At the end of the first cooling cycle, the construction of the next layer causes a new heating/cooling cycle of the material and a repetition of all or part of the transformations seen previously. The areas close to the laser (above the blue line in Figure 10) will be melted and form the δ phase on cooling. The areas further away remain in the solid state (between the blue and red lines in Figure 10), and spend more time in the austenitic range (heating + cooling): they are partially austenitized and may form martensite on cooling. Below the red line in Figure 10, the time spent in the austenitic domain during the cycle is no longer sufficient to form austenite, and the δ ferrite does not transform. In the proposed scenario, the martensite observed in the rest of the PBF-LB sample is formed during the successive anneals caused by the construction of the upper layers. During these anneals, the time spent in the austenitic domain can be twice as long due to heating and cooling of the material, and the chemical segregation of the solidification is less marked, making nucleation easier. It is not showed in this paper but it is possible to use the formulas presented previously to predict the size of a nucleus formed during the second, third etc. thermal cycle, the only difference is that the time considered in the γ domain is different (longer at the second thermal cycle). If the time spent in the gamma domain is twice longer, the grain size will be √2 larger. This leads to the formation of large amounts of austenite and then martensite in the sample bulk.

Other examples from literature

With the cooling rate, the chemical composition also has a strong effect on as built microstructures. The chemical composition controls both the solidification sequence and the size of the austenitic domain, and thus the time spent in the austenitic domain during cooling.

In this way, Vuunam et al. [START_REF] Vunnam | Effect of powder chemical composition on the as-built microstructure of 17-4 PH stainless steel processed by selective laser melting[END_REF] showed that with constant manufacturing parameters, it is possible to increase the proportion of δ ferrite in 17-4PH

(precipitation hardened martensitic steel) by increasing the initial Creq / Nieq ratio in powder. These authors obtained a material with 90 % of ferrite for a Creq/Nieq ratio of 2.81, which is smaller to that of the Fe-9Cr-1Mo used here (Creq/Nieq = 3.0). In their experiment, the surface energy used for building is likely to be smaller (around 5 J/mm² versus 10 J/mm² here), which leads to a cooling rate that can be two to three times higher in their process, and therefore to a microstructure containing much more ferrite. In their experiment, the time spent in the austenitic domain is so small that even the reheating induced by the building of the next layers is not sufficient to allow the formation of austenite. By reducing the Creq/Nieq ratio, they succeeded to widen the austenitic domain and produce almost fully martensitic samples with the same building parameters.

In the same way, with a more ferrite-forming composition, the δ ferrite is maintained more easily. For example, Gao et al. [START_REF] Gao | The isochronal δ → γ transformation of high Cr ferritic heat-resistant steel during cooling[END_REF] obtained high fractions of δ ferrite in a Fe-9Cr-1W (around 30 %), for much lower cooling rates (10 2 °C/s) because their alloy contains half as much carbon as the one used here.

A slight variation in chemical composition can also alter the rapid solidification sequence and allow the appearance of austenite at the end of solidification, which then facilitates the disappearance of the ferrite δ. This is the case for example for the Fe-9Cr-1W of Sam et al. [START_REF] Sam | Delta ferrite in the weld metal of reduced activation ferritic martensitic steel[END_REF].

CONCLUSIONS

A scenario is proposed to explain differences observed in literature for the microstructure of martensitic steels after additive manufacturing. It is based on the calculation of the growth kinetics of austenite grains formed during cooling, as a function of the cooling rate. It explains why under certain conditions, martensite or ferrite is observed in the final microstructures. This model was successfully applied to the specific composition of a Fe-9Cr-1Mo steel, used in this work as an example. This approach could be used for other compositions of martensitic steels. Each step of the cooling process must be carefully checked, as the solidification path and the different equilibrium temperatures are very sensitive to the composition of the material.

This scenario shows that austenite nuclei may have more difficulty to form with PBF-LB due to the chemical segregation between the solidification cells. Then, the austenite grains do not have the time to grow. In contrast, in DED-LB, the time spent into the austenitic domain is longer, allowing austenite to grow and replace the δ ferrite formed at solidification.

It should be noted that these calculations show that the appearance of δ ferrite in PBF-LB for martensitic steels is very sensitive to the cooling rate, but also to the chemical composition. Indeed, the chemical composition controls both the solidification sequence and the size of the austenitic domain, and therefore the time spent in the austenitic domain during cooling.

This study highlights the important effect of the rapid solidification and cooling rate experienced in additive manufacturing on martensitic steels microstructures.

Finally, one other conclusion is that the time spent between Ae5 and Ae1, controls the final microstructure. This time can be tuned either by controlling the size of the domain, by changing the composition, or the cooling rate, with the building parameters.

This understanding is one more step toward the control of microstructure and properties of additively manufactured martensitic steels, by a wise choice of the process and its parameters in phase with the alloy chemical composition. Finally, this approach can also be applied to alloys, other than martensitic steels, with several high temperature phase changes.

ACKNOWLEDGEMENTS

Authors would like to thank Patrice Peyre and Thierry Malot (PIMM Laboratory, Arts et Métiers ParisTech) for their expertise and their help in the production of the DED-LB samples used for this study, Fernando Lomello (CEA-Saclay) for his help in the production of the PBF-LB samples and Elodie Rouesne (CEA-Saclay) for her help in EBSD acquisition and analysis.

  beginning of austenite formation (δ → δ + γ) -Ae4 for the end of austenite formation (δ + γ → γ) -Ae3 for the beginning of α ferrite formation (γ → α + γ) -Ae1 for the end of α ferrite formation (α + γ → α)

Figure 1 :

 1 Figure 1: Pseudo-binary diagram for Fe-xCr-1Mo

Figure 2 :

 2 Figure 2: Size of an austenite grain when Ae3 is reached, calculated with eq. (1) and eq. (3).

Figure 3 :

 3 Figure 3: Schematic CCT diagram of the Fe-9Cr-1Mo to report phases transformations in rapid cooling from the δ phase.

Figure 4 :

 4 Figure 4: (a) Scheme of the solidifying cellular structure with the considered characteristic volume, (b) Solidification sequence in Fe-9Cr-1Mo steel, (c) solid chemical composition evolution during solidification. Computed with ThermoCalc (TCFE5 database) according to Gulliver-Scheil equations.

Figure 5 :

 5 Figure 5: Compositions and equilibrium property diagrams of the solid composition at the beginning (a) and at the end (b) of solidification.

Figure 6 :

 6 Figure 6: Characteristic diffusion length between 1360 °C and 1250 °C (δ phase) for (a) carbon and (b) chromium. Diffusion coefficients in δ phase are extracted from MOB2 ThermoCalc database.

  (a)) and large particles are martensitic (Figure 7(c)). There is a critical size for which it is possible to observe ferriticmartensitic structures (Figure 7(b)).

Figure 7 :

 7 Figure 7: Different microstructures of the Fe-9Cr-1Mo powder (EBSD band contrast, IPF, and phase map): (a) a small ferritic particle, (b) a ferritic/martensitic particle and (c) a fully martensitic particle.

  microstructure at the top and in the middle of the wall was analysed by EBSD (Figure8). These EBSD maps indicate that DED-LB samples are fully martensitic. This martensitic microstructure agrees with the model presented in section 3.

Figure 8 (

 8 Figure 8 (c) and (f)), and to compute prior austenite grain size.

Figure 8 :

 8 Figure 8: EBSD analysis of DED-LB sample, (a), (b) and (c) show the microstructure in the top layer, and (d) (e) and (f) show the microstructure in the middle of the wall. (g) is the DED-LB sample as built. Both are fully martensitic.

Figure 9 :

 9 Figure 9: EBSD map of PBF-LB sample. The last melted layer is seen at the top of the map, melt pool tracks are visible on this last layer (dotted line). (a) Band contrast map, (b) IPF map projected along building direction (BD), arrows highlight the grain's direction change because of the thermal gradient and (c) phase map. (d) is a view of the as built samples.

Figure 10 :

 10 Figure 10: Scheme of liquid to δ, δ to γ, and γ to α transformation isotherms to explain the formation of martensite between the melts pool and below the last melt track.

Table 2 .

 2

	Fe	C	Cr	Ni	Mo	Mn	Si	N	D10 (µm) D50 (µm) D90 (µm)
	Bal. 0.10 9.3 0.22 1.04 0.48 0.28 0.004	10	30	65

Table 1 : Main alloying elements of the Fe-9Cr-1Mo powder (wt%).

 1

	Method	DED-LB	PBF-LB
	Setup	Optomec LENS 850R	Trumpf Truprint series 1000
	Laser power (P) (W)	400	100
	Scanning speed (v) (mm/s)	5	200
	Fe-9Cr-1Mo flow rate (g/min)	2	-
	Layer height (h) (µm)	200	20
	Shielding/carrying gas	Argon	Argon
	Spot diameter (Dspot) (µm)	1400	50
	Surface Energy density (P/v.Dspot) (J/mm²)	57	10
	Volume energy density (P/v.Dspot.h)(J/mm 3 )	285	500
	Cooling speed estimated from [14] (°C/s)	10 4	10 6

Table 2 : Building parameters.

 2