Yaohui Wang

Piotr Bilinski

Francois Bremond

Antitza Dantcheva
email: antitza.dantcheva@inria.frbilinski@mimuw.edu.pl

G 3 AN: Disentangling Appearance and Motion for Video Generation

Creating realistic human videos entails the challenge of being able to simultaneously generate both appearance, as well as motion. To tackle this challenge, we introduce G 3 AN, a novel spatio-temporal generative model, which seeks to capture the distribution of high dimensional video data and to model appearance and motion in disentangled manner. The latter is achieved by decomposing appearance and motion in a three-stream Generator, where the main stream aims to model spatio-temporal consistency, whereas the two auxiliary streams augment the main stream with multi-scale appearance and motion features, respectively. An extensive quantitative and qualitative analysis shows that our model systematically and significantly outperforms state-of-the-art methods on the facial expression datasets MUG and UvA-NEMO, as well as the Weizmann and UCF101 datasets on human action. Additional analysis on the learned latent representations confirms the successful decomposition of appearance and motion. Source code and pre-trained models are publicly available 1 .

Introduction

Generative Adversarial Networks (GANs) [START_REF] Goodfellow | Generative adversarial nets[END_REF] have witnessed increasing attention due to their ability to model complex data distributions, which allows them to generate realistic images [START_REF] Brock | Large scale GAN training for high fidelity natural image synthesis[END_REF][START_REF] Karras | Progressive growing of gans for improved quality, stability, and variation[END_REF]18,[START_REF] Ledig | Photorealistic single image super-resolution using a generative adversarial network[END_REF][START_REF] Ma | Disentangled person image generation[END_REF][START_REF] Miyato | Spectral normalization for generative adversarial networks[END_REF][START_REF] Xu | Attngan: Finegrained text to image generation with attentional generative adversarial networks[END_REF]44], as well as to translate images [2, [START_REF] Isola | Image-to-Image Translation with Conditional Adversarial Networks[END_REF][START_REF] Dibeklioglu | Are you really smiling at me? spontaneous versus posed enjoyment smiles[END_REF][START_REF] Mahfuzur Rahman Siddiquee | Learning fixed points in generative adversarial networks: From image-to-image translation to disease detection and localization[END_REF]. While realistic video generation is the natural sequel, it is substantially more challenging w.r.t. complexity and computation, associated to the simultaneous modeling of appearance, as well as motion.

Specifically, in inferring and modeling the distribution of human videos, generative models face three main challenges: (a) generating uncertain motion, (b) retaining of human appearance throughout the generated video, as well as (c) modeling spatio-temporal consistency. Such challenges have been alleviated by conditioning the generation on potent priors such as input images [42], human keypoints [7] and optical flows [23]. This relates to learning to sample from conditional distributions, assuming access 1 https://wyhsirius.github.io/G3AN/ to the marginal distributions instead of learning joint distributions [START_REF] Pu | Jointgan: Multi-domain joint distribution learning with generative adversarial nets[END_REF].

Deviating from such approaches, in this work we focus on the highly intricate problem of video generation without prior knowledge w.r.t. either appearance or motion. Specifically, based on noise variables, we generate an appearance, e.g. human face and body, which we concurrently animate, by a facial expression or human action.

G 3 AN, our new generative model, is streamlined to learn a disentangled representation of the video generative factors appearance and motion, allowing for manipulation of both. A disentangled representation has been defined as one, where single latent units are sensitive to changes in single generative factors, while being relatively invariant to changes in other factors [4]. In this context, our G 3 AN is endowed with a three-stream Generator-architecture, where the main stream encodes spatio-temporal video representation, augmented by two auxiliary streams, representing the independent generative factors appearance and motion. A self-attention mechanism targeted towards high level feature maps ensures satisfactory video quality.

G 3 AN is hence able to generate realistic videos (tackling challenges (a) and (c)) by following a training distribution and without additional input, as well as is able to manipulate the appearance and motion disjointly, while placing emphasis on preserving appearance (challenge (b)).

In summary, our main technical contributions include the following.

• A novel generative model, G 3 AN, which seeks to learn disentangled representations of the generative factors appearance and motion from human video data. The representations allow for individual manipulation of both factors. • A novel three-stream generator, which takes into account the learning of individual appearance features (spatial stream), motion features (temporal stream) and smoothing generated videos (main stream) at the same time. • A novel factorized spatio-temporal self-attention (F-SA), which is considered as the first self-attention module applied to video generation, in order to model global spatio-temporal representations and improve the quality of generated videos.

• Extensive qualitative and quantitative evaluation, which demonstrates that G 3 AN systematically and significantly outperforms state-of-the-art baselines on a set of datasets.

Related Work

Despite the impressive progress of image generation, the extension to video generation is surprisingly challenging. While videos constitute sequences of temporally coherent images, video generation encompasses a majority of challenges that have to do with generation of plausible and realistic appearance, coherent and realistic motion, as well as spatio-temporal consistency. A further challenge, namely the generation of uncertain local or global motion, associated to future uncertainty, allows for multiple correct, equally probable next frames [START_REF] Walker | An uncertain future: Forecasting from static images using variational autoencoders[END_REF]. Finding suitable representation learning methods, which are able to address these challenges is critical. Existing methods include approaches based on Variational Autoencoders (VAEs) [START_REF] Diederik | Auto-encoding variational bayes[END_REF], auto-regressive models, as well as most prominently Generative Adversarial Networks (GANs) [START_REF] Goodfellow | Generative adversarial nets[END_REF].

While video generation tasks aim at generating realistic temporal dynamics, such tasks vary with the level of conditioning. We have video generation based on additional priors related to motion or appearance, as well as contrarily, video generation following merely the training distribution. We note that the latter is more challenging from a modeling perspective, due to lack of additional input concerning e.g. structure of the generated video. Therefore the majority of approaches to date include a conditioning of some kind.

Video generation with additional input. Due to challenges in modeling of high dimensional video data, additional information such as semantic maps [START_REF] Pan | Video generation from single semantic label map[END_REF][START_REF] Ting-Chun | Video-tovideo synthesis[END_REF], human keypoints [START_REF] Jang | Video Prediction with Appearance and Motion Conditions[END_REF]42,[START_REF] Walker | The pose knows: Video forecasting by generating pose futures[END_REF]7], 3D face mesh [START_REF] Zhao | Learning to forecast and refine residual motion for image-to-video generation[END_REF] and optical flow [23] can be instrumental as guidance for appearance and motion generation. This additional information is either pre-computed throughout the generated video [START_REF] Jang | Video Prediction with Appearance and Motion Conditions[END_REF][START_REF] Zhao | Learning to forecast and refine residual motion for image-to-video generation[END_REF]7] or predicted based on an initial input image [42]. The additional information guides conditional image translation, which though results in lack of modeling of spatio-temporal correlations.

Video generation from noise. Directly generating videos from noise requires the capturing and modeling of a dataset distribution. Existing works tend to reduce related complexity by decomposing either the output [START_REF] Vondrick | Generating videos with scene dynamics[END_REF] or latent representation [START_REF] Saito | Temporal generative adversarial nets with singular value clipping[END_REF]35]. VGAN [START_REF] Vondrick | Generating videos with scene dynamics[END_REF] was equipped with a two-stream spatio-temporal Generator, generating foreground and background separately. TGAN [START_REF] Saito | Temporal generative adversarial nets with singular value clipping[END_REF] decomposed the latent representation of each frame into a slow part and a fast part. Due to jointly modeling appearance and motion, generated results from VGAN and TGAN might comprise spatially unrealistic artefacts, see Figure 5. The closest work to ours is MoCoGAN [35], which decomposed the latent representation of each frame into motion and content, aiming at controlling both factors. However, there are two crucial differences between MoCoGAN and G 3 AN. Firstly, instead of only sampling two noise vectors for each video, MoCoGAN sampled a sequence of noise vectors as motion and a fixed noise as content. However, involving random noise for each frame to represent motion increases the learning difficulty, since the model has to map these noise vectors to a consecutive human movement in the generated videos. As a result, MoCoGAN gradually ignores the input noise and tends to produce a similar motion, as we illustrate in Figure 8. Secondly, MoCoGAN incorporated a simple image Generator aiming at generating each frame sequentially, after which content and motion features were jointly generated. This leads to incomplete disentanglement of motion and content. Deviating from that, we design a novel Generator architecture, able to entirely decompose appearance and motion in both, latent and feature spaces. We show that such design generates realistic videos of good quality and ensures factor disentanglement.

Disentangled representation learning. Learning disentangled representations of data has been beneficial in a large variety of tasks and domains [4]. Disentangling a number of factors in still images has been widely explored in recent works [START_REF] Chen | Infogan: Interpretable representation learning by information maximizing generative adversarial nets[END_REF][START_REF] Ma | Disentangled person image generation[END_REF][START_REF] Singh | Finegan: Unsupervised hierarchical disentanglement for fine-grained object generation and discovery[END_REF][START_REF] Lee | Diverse image-to-image translation via disentangled representations[END_REF]. In the context of video generation, an early approach for motion and appearance decomposition was incorporated in MoCoGAN. However, experiments, which we present later (see Figure 6), suggest that the results are not satisfactory.

Approach

In this work, we propose G 3 AN, a novel GAN architecture, aiming at generating videos in a disentangled manner from two noise vectors, z a ∈ Z A and z m ∈ Z M , which represent appearance and motion, respectively. G 3 AN consists of a three-stream Generator G, as well as a two-stream Discriminator D, as illustrated in Figure 1. While G aims at generating videos with the ability to modulate appearance and motion disjointly, D accounts for distinguishing generated samples from real data, in both, videos and frames, respectively.

Generator

Hierarchical Generator with G 3 -modules. We design G in a hierarchical structure of G 3 modules. Specifically, we have N levels of hierarchy, denoted as G 3 n=0...N -1 . The first G 3 module, G 3 0 accepts as input the two noise vectors z a and z m . The remaining modules G 3 n=1...N -1 , inherit the three feature maps F Sn-1 , F Vn-1 and F Tn-1 as their inputs from each previous G 3 n-1 module, see Figures 1 and2. Each G 3 n module consists of three parallel streams: a spatial stream G Sn , a temporal stream G Tn , as well as a video stream G Vn (Figures 1 and2). They are designed to generate three different types of features. The spatial stream G Sn , denoted by a blue line in Figures 1 and2, takes as input z a for n = 0 and F Sn-1 for n > 1, and generates 2D appearance features F Sn by upsampling input features with a Spatio-temporal fusion is the key-element to learn well disentangled features, the inputs of which are output feature maps F Sn , F Tn and F V n from the convolutional layers in each G 3 n module. The fusion contains three steps (see Figure 3). Firstly, spatial and temporal replications are applied on F Tn and F Sn respectively, in order to obtain two new feature maps F R Tn and F R Sn . Both new feature maps have the same spatio-temporal size as F V n . Next, F R Tn and F V n are combined through a position-wise addition, creating a new spatio-temporal embedding F V n . Finally, F R Sn is channel-wise concatenated with F V n , obtaining the final fused feature map F Vn . The feature maps F Sn , F Tn and F Vn represent inputs for the following G 3 n+1 module. Factorized spatio-temporal Self-Attention (F-SA). Despite Self-Attention (SA) has been successfully applied in image generation within SAGAN [START_REF] Zhang | Self-Attention Generative Adversarial Networks[END_REF], however it has not been explored in the context of spatio-temporal video generation yet. Here, we incorporate a spatio-temporal SA module, enabling G to utilize cues from all spatio-temporal feature positions and model relationships between widely sep- arated regions. However, computing correlation between each position with all the others in a 3D spatio-temporal feature map is very computationally expensive, particularly if it is applied on higher feature maps in G. Therefore, we propose a novel factorized spatio-temporal self-attention, namely F-SA, as shown in Figure 4. F-SA consists of a Temporal-wise SA (T-SA), followed by a Spatial-wise SA (S-SA). Such factorization reduces the computational complexity, allowing for application of the F-SA on larger feature maps.

In our G 3 AN, we apply F-SA on the output of the G 3 3 in the G V stream, which achieve best video quality. We report related evaluation results of applying F-SA at various hierarchy-layers of the G 3 AN in Section 4.

Discriminator

Towards improving both video and frame quality, similar to MoCoGAN, we use a two-stream Discriminator architecture, containing a video stream D V and an image stream D I . During training, D V accepts a full video as input, whereas D I takes randomly sampled frames from videos.

Training

Given our two-stream Discriminator architecture, G 3 AN simultaneously optimizes D V and D I . Both losses use the GAN loss function proposed in DCGAN [START_REF] Radford | Unsupervised representation learning with deep convolutional generative adversarial networks[END_REF]. The objective functions of G 3 AN can be expressed as

G * = min G max D I ,D V L(G, D I , D V ), (1)
L(G, D I , D V ) = L I (G, D I ) + L V (G, D V ), (2)
where L I denotes the loss function related to D I , L V represents the loss function related to D V .

L I = E x ∼p data [log(D I (x ))] + E za∼pz a ,zm∼pz m [log(1 -D I (G(z a , z m ) ))], (3)
L V = E x∼p data [log(D V (x))] + E za∼pz a ,zm∼pz m [log(1 -D V (G(z a , z m )))], (4)
G attempts to generate videos from z a and z m , while D I and D V aim to distinguish between generated samples and real samples. (•) characterizes that T frames are being sampled from real and generated videos.

Experiments 4.1. Implementation details

The entire network is implemented using PyTorch. We employ ADAM optimizer [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF] with β 1 =0.5 and β 2 =0.999. Learning rate is set to 2e -4 for both G and D. Dimensions of latent representations constitute 128 for z a and 10 for z m . We set N = 5 in order to generate videos of 16 frames with spatial scale 64 × 64. We randomly sample T = 1 frame from each video as input of D I . More implementation details can be found in SM.

Datasets

We evaluate our method on following four datasets. Facial expression datasets. The MUG Facial Expression dataset [START_REF] Aifanti | The mug facial expression database[END_REF] contains 1254 videos of 86 subjects, performing 6 facial expressions, namely happy, sad, surprise, anger, disgust and fear. The UvA-NEMO Smile dataset [9] comprises 1240 video sequences of 400 smiling individuals, with 1 or 2 videos per subject. We crop faces in each frame based on detected landmarks using [START_REF] Bulat | How far are we from solving the 2d & 3d face alignment problem? (and a dataset of 230,000 3d facial landmarks)[END_REF] 2 .

Action recognition datasets. The Weizmann Action dataset [START_REF] Gorelick | Actions as space-time shapes[END_REF] consists of videos of 9 subjects, performing 10 actions such as wave and bend. We augment it by horizontally flipping the existing videos. The UCF101 dataset [START_REF] Soomro | UCF101: A Dataset of 101 Human Action Classes From Videos in The Wild[END_REF] contains 13,320 videos of 101 human action classes. Similarly to TGAN [START_REF] Saito | Temporal generative adversarial nets with singular value clipping[END_REF], we scale each frame to 85 × 64 and crop the central 64 × 64 regions.

In all our experiments, we sample video frames with a random time step ranging between 1 and 4 for data augmentation.

Experimental Results

We test our method both quantitatively and qualitatively, providing results of four experiments. Specifically, firstly we evaluate and compare videos generated from G 3 AN, VGAN, TGAN and MoCoGAN, quantitatively and qualitatively on all four datasets. Next, we test conditional and unconditional video generation, where we aim to demonstrate the effectiveness of the proposed decomposition method.

Then, we manipulate the latent representation, providing insight into each dimension of the two representations. We proceed to add appearance vectors and study the latent representation. Finally, we conduct an ablation study, verifying the effectiveness of our proposed architecture.

Quantitative Evaluation

We compare G 3 AN with three state-of-the-art methods, namely VGAN, TGAN, as well as MoCoGAN. We report two evaluation metrics on the above four datasets. In particular, we use the extension of two most commonly used metrics in image generation, the Inception Score (IS) [START_REF] Salimans | Improved techniques for training GANs[END_REF] and Fréchet Inception Distance (FID) [START_REF] Heusel | Gans trained by a two time-scale update rule converge to a local nash equilibrium[END_REF], in video level by using a pre-trained 3D CNN [12] as feature extractor, similar to Wang et al. [START_REF] Ting-Chun | Video-tovideo synthesis[END_REF].

The video FID grasps both visual quality and temporal consistency of generated videos. It is calculated as µ -µ 2 + T r(Σ + Σ -2 Σ Σ), where µ and Σ represent the mean and covariance matrix, computed from real feature vectors, respectively, and µ, and Σ are computed from generated data. Lower FID scores indicate a superior quality of generated videos.

The video IS captures the quality and diversity of generated videos.

It is calculated as

exp(E x∼pg D KL (p(y|x) p(y))),
where p(y|x) and p(y) denote conditional class distribution and marginal class distribution, respectively. A higher IS indicates better model performance.

We report FID on MUG, UVA-Nemo and Weizmann datasets, and both FID and IS on UCF101. Since IS can only be reported, when GAN and feature extractor are trained on the same dataset, we do not report it on other datasets.

Comparison results among different methods are presented in Table 1. Our method consistently achieves the lowest FID on all datasets, suggesting that videos generated by G 3 AN entail both, best temporal consistency and visual quality. At the same time, the obtained highest IS on UCF101 indicates that our method is able to provide the most diverse samples among all compared methods. Such evaluation results show that proposed decomposition method allows for controlling the generated samples, and additionally facilitates the spatio-temporal learning of generating better quality videos. Generated samples are illustrated in Figure 5. In addition, we conduct a subjective analysis, where we asked 27 human raters to pairwise compare videos of pertaining to the same expression/action, displayed side by side. Raters selected one video per video-pair. We randomized the order of displayed pairs. We had an equal amount of pairs for each studied case (e.g. G 3 AN / Real videos). The posed question was "Which video clip is more realistic?". We report the mean user preference in Table 2.

MUG

We observe that human raters express a strong preference for the proposed framework G 3 AN over MoCoGAN (84.26% vs. 15.74%), TGAN (87.31% vs. 12.69%) and VGAN (90.24% vs. 9.76%), which is consistent with the above listed quantitative results. Further, we compare real videos from all datasets with the generated video sequences from our method. The human raters ranked 25.71% of videos from our G 3 AN as more realistic than real videos, which we find highly encouraging.

Methods

Rater preference (%)

G

Qualitative Evaluation

We conduct an unconditional generation experiment utilizing the Uva-NEMO dataset, where we fix z a and proceed to randomly vary motion, z m . Associated generated samples from G 3 AN and MoCoGAN are shown in Figure 6. While we observe the varying motion in the video sequences generated by G 3 AN, the appearance remains coherent. Hence, our model is able to successfully preserve facial appearance, while altering the motion. Therefore, this suggests that our three-stream design allows for manipulation of appearance and motion separately. On the contrary, video sequences generated by MoCoGAN experience constant motion, despite of altering z m . Further, we leverage on labels of the MUG and Weizmann datasets, in order to analyze conditional video generation. Towards this, we here concatenate a one-hot category vector and motion noise z m , feeding it into G T . We note that the inputs of G S and G V remain the same as in the setting of unconditional generation. Related results show that when varying motion-categories, while having a fixed appearance, G 3 AN correctly generates an identical facial appearance, with appropriate category-based motion (facial expressions and body actions), see Figure 7. Further, we note that appearance is very well preserved in different videos and is not affected by category-alterations. In addition, in the same conditional setting, we note that when varying the noise z m , G 3 AN is able to generate the same category-motion in different ways. This indicates that z m enables an intra-class diversity.

In videos generated by MoCoGAN, we observe a correctly generated motion according to given categories, however we note that the category also affects the appearance. In other words, MoCoGAN lacks a complete disentanglement of appearance and motion in the latent representation, see Figure 8. This might be due to a simple motion and content decomposition in the latent space, which after a set of convolutions can be totally ignored in deeper layers. It is notable that G 3 AN effectively prevents such cases, ensured by our decomposition that occurs in both, latent and feature spaces.

Latent representation manipulation. While there is currently no general method for quantifying the degree of learnt disentanglement [START_REF] Higgins | beta-vae: Learning basic visual concepts with a constrained variational framework[END_REF], we proceed to illustrate the ability of our model to learn latent representations by ma- For both datasets, each line is generated with random zm. We observe that same category (smile and one hand waving) is performed in a different manner, which indicates that our method is able to produce intra-class generation. See SM for more samples.

nipulating each dimension in the appearance representation. We show that by changing values of different dimensions in the appearance representation, we are able to cause a modification of specific appearance factors, see Figure 9. Interestingly such factors can be related to semantics, e.g. facial view point in Figure 9a, mustache in Figure 9b, and color of pants in Figure 9c. We note that motion is not affected by altering the appearance representation. Similarly, when altering values of different dimensions in the motion representation, we observe that factors such as starting position, motion intensity and moving trajectory are being affected, see Figure 10. Such observations show that our method learns to interpolate between different data points in motion-and appearance-latent spaces, respectively.

Addition of appearance representations. We here add appearance vectors, aiming to analyze the resulting latent representations. Towards this, we generate two videos V a and V b by randomly sampling two sets of noises, (z a0 , z m0 ) and (z a1 , z m1 ). Next, we add z a0 and z a1 , obtaining a new appearance z a2 . When combining (z a2 , z m0 ) and (z a2 , z m1 ), we observe in the two new resulting videos a sum- mary appearance pertaining to z a0 and z a1 , with identical motion as z m0 and z m1 , see Figure 11.

Ablation Study

We here seek to study the effectiveness of proposed G 3 AN architecture, as well as the effectiveness related to each component in the proposed Generator. Towards this, we firstly generate videos by removing G S and G T , respectively, in order to verify their ability of controlling motion and appearance. We observe that when removing G T , the model is able generate different subjects, however for each person the facial movement is constant, see Figure 12 (top). Similarly, when G S is removed, changing motion will af-fect subject's identity, whereas the appearance vector loses its efficacy, see Figure 12 (middle). When removing both, G T and G S , appearance and motion are entangled and they affect each other, see Figure 12 (bottom). This demonstrates the effective disentanglement brought to the fore by the streams G S and G T . We proceed to demonstrate the contribution of G S , G T and F-SA in the Generator w.r.t. video quality. In this context, we remove each component individually and report results on the four datasets in Table 3. The results show that after removing all three components, video quality is the poorest, which proves that all of them contribute to the final results. Individually, G S plays the most pertinent role, as removing it, decreases FID most profoundly for all datasets. This indicates that generating appearance features separately can be instrumental for good quality videos. Moreover, our results confirm the necessity of F-SA in our approach. Transposed Convolutions. Then, we compare the proposed factorized transposed spatio-temporal (1+2)D convolution, standard transposed 3D convolution, and transposed (2+1)D convolution, when used in G V w.r.t. video quality. We carefully set the number of kernels, allowing for the three networks to have nearly same training parameters. We report the results of the quantitative evaluation in Table 4. Both convolution types, (1+2)D and (2+1)D outperform standard 3D kernels w.r.t. generated video quality. (1+2)D is slightly better than (2+1)D, and the reason might be that the (1+2)D kernel uses more 1 × 1 kernels to refine tempo-ral information, which we believe to be important in video generation tasks. Where to insert self-attention? Finally, we proceed to explore at which level of the Generator, F-SA is the most effective. We summarize performance rates in Table 5. Inserting F-SA after the G 3 3 module provides the best results, which indicates that middle level feature maps contribute predominantly to video quality. As shown in GAN Dissection [START_REF] Bau | Gan dissection: Visualizing and understanding generative adversarial networks[END_REF], mid-level features represent semantic information, e.g., object parts while high-level features represent local pixel patterns, e.g., edges, light and colors and low-level features do not contain clear semantic information, which could be the reason, why F-SA achieves the best result in G 3 3 module.

Architecture MUG UvA Weizmann UCF101 FID ↓ FID ↓ FID ↓ FID ↓ IS ↑ w/o G S ,G

MUG

MUG

Conclusions

We have presented the novel video generative architecture G 3 AN, which leverages among others on (i) a threestream Generator that models appearance and motion in disentangled manner, as well as (ii) a novel spatio-temporal fusion method. We have performed an extensive evaluation of our approach on four datasets, outperforming quantitatively and qualitatively the state-of-the-art video generation methods VGAN, TGAN and MoCoGAN. Further, we have shown the ability of G 3 AN to disentangle appearance and motion, and hence manipulate them individually.

Future work involves the design of GAN models for video generation of high resolution videos.

Figure 1 :

 1 Figure 1: Overview of our G 3 AN architecture. G 3 AN consists of a three-stream Generator and a two-stream Discriminator. The Generator contains five stacked G 3 modules, a factorized self-attention (F-SA) module, and takes as input two random noise vectors, za and zm, aiming at representing appearance and motion, respectively. Details of architecture can be found in Supplementary Material (SM).

Figure 2 :

 2 Figure 2: G 3 module architecture.Factorized transposed spatio-temporal convolution has been proposed by Wang et al. in[START_REF] Wang | ImaGINator: Conditional Spatio-Temporal GAN for Video Generation[END_REF]. It explicitly factorizes transposed 3D convolution into two separate and successive operations, M transposed 1D temporal convolution followed by a 2D separate spatial convolution, which is

Figure 3 :

 3 Figure 3: Spatio-temporal fusion.

Figure 4 :

 4 Figure 4: Factorized spatio-temporal Self-Attention (F-SA).

Figure 5 :

 5 Figure 5: Comparison with the state-of-the-art on MUG (top-left), Weizmann (top-right), UvA-NEMO (bottom-left) and UCF101 (bottom-right). More samples are presented in SM.

Figure 6 :

 6 Figure 6: Unconditional video generation of G 3 AN and MoCo-GAN on Uva-Nemo. For each model, we fix za, while testing two zm instances (top and bottom lines). See SM for more samples.

  (a) MUG: Happiness (b) Weizmann: One hand waving

Figure 7 :

 7 Figure 7: Conditional video generation on MUG and Weizmann.For both datasets, each line is generated with random zm. We observe that same category (smile and one hand waving) is performed in a different manner, which indicates that our method is able to produce intra-class generation. See SM for more samples.

Figure 8 :

 8 Figure 8: Comparison between G 3 AN and MoCoGAN. Given fixed za and zm, as well as two condition-labels smile and surprise, G 3 AN and MoCoGAN generate correct facial expressions. However, while G 3 AN preserves the appearance between rows, MoCoGAN alters the subject's appearance.

  (a) Manipulation of third dimension on UvA-NEMO (b) Manipulation of third dimension on MUG (c) Manipulation of second dimension on Weizmann

Figure 9 :

 9 Figure 9: Latent appearance representation manipulation. For each dataset, each row shares the same motion representation, whereas from top to bottom values in one dimension of appearance representation are increased. See SM for more samples.

  (a) Manipulation of sixth dimension of MUG (b) Manipulation of second dimension on Weizmann

Figure 10 :

 10 Figure 10: Latent motion representation manipulation. For each dataset, each row shares the same appearance representation, whereas from top to bottom values in one dimension of the motion representation are increased. See SM for more results.

Figure 11 :

 11 Figure 11: Addition of appearance representations. We add the appearance vectors of two samples (top rows of (a) and (b)), and obtain the sum-appearance in each bottom row. We inject motion pertained to each top appearance of (a) and (b) and are able to show same motion within lines of (a) and (b).

1 Figure 12 :

 112 Figure 12: Ablation study. Generated videos obtained by removing GT (top row), removing GS (middle), and both (bottom row).

Table 1 :

 1 Comparison

		UvA Weizmann	UCF101
		FID ↓ FID ↓	FID ↓	FID ↓ IS ↑
	VGAN	160.76 235.01	158.04	115.06 2.94
	TGAN	97.07 216.41	99.85	110.58 2.74
	MoCoGAN 87.11 197.32	92.18	104.14 3.06
	G 3 AN	67.12 119.22	86.01	91.21 3.62

with the state-of-the-art on four datasets w.r.t. FID and IS.

Table 2 :

 2 Mean user preference of human raters comparing videos generated by the respective algorithms, originated from all datasets.

	3 AN / MoCoGAN	84.26 / 15.74
	G 3 AN / TGAN	87.31 / 12.69
	G 3 AN / VGAN	90.24 / 9.76
	G 3 AN / Real videos	25.71 / 74.29

Table 3 :

 3 Contribution of main components in G.

	T ,F-SA 117.10 164.04 252.97 127.09 2.78
	w/o G S ,G T	113.44 159.54 176.73 120.17 3.16
	w/o G S	109.87 129.84 141.06 117.19 3.05
	w/o F-SA	85.11 128.14	97.54	98.37 3.44
	w/o G T	82.07 121.87	94.64	96.47 3.16
	G 3 AN	67.12 119.22	86.01	91.21 3.62

Table 4 :

 4 UvA Weizmann UCF101 FID ↓ FID ↓ Comparison of various convolution types in G.

			FID ↓	FID ↓ IS ↑
	3D	93.51 149.98	154.21	117.61 2.88
	(2+1)D 73.08 141.35	95.01	98.70 3.36
	(1+2)D 69.42 140.42	87.04	96.79 3.07

Table 5 :

 5 Comparison of inserting F-SA at different hierarchical levels of G 3 AN.

		UvA	Weizmann	UCF101
		FID ↓ FID ↓	FID ↓	FID ↓ IS ↑
	G 3 0 G 3 1 G 3 2 G 3 3	83.01 188.60 72.54 178.64 69.02 160.12 67.12 119.22	96.38 99.66 97.53 86.01	100.37 3.09 126.12 2.74 112.36 3.03 91.21 3.62

https://github.com/1adrianb/face-alignment

Acknowledgment. This work is supported by the French Government (National Research Agency, ANR), under Grant ANR-17-CE39-0002.


Andrey Myagkiy
email: andrey.myagkiy@univ-lorraine.fr

Laurent Truche

Michel Cathelineau

Fabrice Golfier

Revealing the conditions of Ni mineralization in the laterite profiles of New Caledonia: insights from reactive geochemical transport modelling

Keywords: Laterite, Transport Modelling, Weathering, Kinetics, Water-rock interactions

New Caledonia is one of the world's largest nickel laterite deposits that form from intense chemical and mechanical weathering of a peridotite bedrock. As a result of such a weathering process a subsequent downward migration of Si, Mg and Ni takes place, which eventually leads to redistribution of the elements in depth and over time depending on their mobility. Being released from ultramafic parent rock to groundwater, the mobility of nickel is to a great extent controlled by sorption, substitution and dissolution/precipitation processes. Therefore, the final profile of nickel enrichment is the result of the superposition of these possible fates of nickel. The way how Ni is redistributed in between them represents and defines its mineralization in laterite. Knowledge of these processes along with factors, controlling them appears to be a key to detailed understanding of laterite formation. In this study a numerical model, which solves the reaction-transport differential equations, is used to simulate the formation of laterite profile from ultramafic bedrock with particular emphasis on modelled Ni enrichment curve, its comparison with in situ observations, and detailed understanding of trace elements mobility. Since nickel deposits in New Caledonia is characterized by oxide and hydrous Mg silicate ores, three different concurrent fates of Ni deposition in a profile were taken into account in the modelling: i) Ni in a goethite crystal lattice, ii) Ni sorbed on weak and strong goethite sorption sites, and iii) Ni precipitated with silicates (garnierite). Simulations were performed using PHREEQC associated with llnl.dat thermodynamic database that has been edited in order to account garnierite minerals used in the calculations. The work outline is represented by: i) long term (10 Ma) simulation of nickel laterite formation and evolution, ii) analysis of mobility of the elements and understanding its controlling factors, iii) comparison of modelled and in situ Ni enrichment profile and analysis of nickel distribution in between different retention processes, iv) modelling and in depth understanding of these retention processes. The modelling reveals that the vertical progression of the pH front controls thickening of iron-rich zone, explains the vertical mobility of the elements and governs the Ni enrichment. Adsorption itself plays an important role in lateritization process retarding Ni mobility, but i) becomes significant in a narrow range of pH (slightly alkaline) due to competition of Mg and Ni for sorption sites and ii) does not explain such a high nickel content in limonite nowadays, suggesting that Ni is held in goethite mostly by stronger ties i.e. substituted for Fe in the crystal lattice of iron oxyde. 1-D modelling appears to be a powerful tool in understanding the general behavior of trace elements upon the formation of laterite and at the same time reveals that locally Ni mineralizations should be explained by more complex processes, such as lateral transfers, convective flows and preferential pathways.

Introduction

Nickel laterite ores account for over 60% of global Ni supply [START_REF] Butt | Nickel laterite ore deposits: Weathered serpentinites[END_REF], thus, making it an important exploration target in anticipation of the future demand of Ni. One of the biggest nickeliferous laterite reserves in the world is located in New Caledonia, 1300 km east of Australia in the southwest Pacific Ocean. Representing around 30% of the

A C C E P T E D

M A N U S C R I P T

ACCEPTED MANUSCRIPT

global resources in nickel, these laterites are actively worked by various mining companies (SLN-ERAMET, KNS, SMSP etc.).They occur at the top of the obducted ultramafic massif with an average thickness of 30-40 m, resulting from the weathering of the Peridotite Nappe. The initial nickel content of the parent rock, that consists of olivine and various form of serpentine (antigorite, chrysotile, lizardite, polygonal serpentine), can contain up to 0.4 wt% of Ni [START_REF] Ulrich | Péridotites et serpentinites du complexe ophiolitique de la Nouvelle-Calédonie[END_REF][START_REF] Herzberg | Nickel and helium evidence for melt above the core-mantle boundary[END_REF][START_REF] Golightly | Nickeliferous laterite deposits[END_REF]. The weathering of the olivine-rich ultramafic rocks and their serpentinized equivalents by meteoric water induces the selective leaching of Fe, Mg, Si and Ni and leads to the formation of a lateritic profile together with an exceptional Ni enrichment [START_REF] Trescases | L'élvolution gélochimique supergèlne des roches ultrabasiques en zone tropicale: Formation des gisements nickélifères de Nouvelle-Calédonie: Mémoires ORSTOM[END_REF][START_REF] Golightly | Progress in understading the evolution of nickel laterites[END_REF]. The limonite level, formed at the top of the profile (Fig. 1), is mainly composed of iron oxi-hydroxides and followed by the saprolitic level where hydrous Mg silicate de-posits form, especially in the mid to lower saprolite.

Ni concentration increases constantly throughout the limonite horizon, but the highest Ni grade (locally 2w% up to 5 wt%) is found in the saprolite horizon which represents about one third of the total Ni laterite resources (Fig. 1). In this horizon, Ni is concentrated in nickeloan varieties of serpentine, talc-like, chlorite and sepiolite, some of which are poorly defined and known informally as "garnierite" [START_REF] Butt | Nickel laterite ore deposits: Weathered serpentinites[END_REF]. Transition between the saprolite and limonite horizons is marked by a sharp increase in MgO and SiO 2 content and, consequently, by drastic change in mineralogy (Fig. 1). The so-called garnierite is mostly located along fractures [START_REF] Trescases | L'élvolution gélochimique supergèlne des roches ultrabasiques en zone tropicale: Formation des gisements nickélifères de Nouvelle-Calédonie: Mémoires ORSTOM[END_REF]. Under the saprolite level, a poorly altered peridotite is situated. The basis of the peridotite nappe hosts numerous pure magnesite veins, sometimes associated with amorphous silica [START_REF] Quesnel | Formation of silica and magnesite veins in the Massif of Peridotite of Koniambo: Geometric and stable isotopes data. 13th SGA Biennial Meeting[END_REF].

Figure 1: A schematic log of the New Caledonian Peridotite Nappe correlated with approximate elemental composition of different zones.

Turquoise line represents nickel distribution in a profile. Modified after [START_REF] Ulrich | Weathering effects on the mineralogical and geochemical composition of the New Caledonia ophiolite[END_REF][START_REF] Troly | Nickel deposits in New Caledonia: some factors influencing their formation[END_REF][START_REF] Guilbert | The geology of ore deposits[END_REF] The enrichment of nickel in the weathering profile is governed by many interplaying factors that include mineralogical composition of parent rock, climate, chemistry and rates of chemical weathering, tec-tonics, and drainage [START_REF] Lelong | Pedogenesis, chemical weathering and processes of formation of some supergene ore deposits[END_REF][START_REF] Golightly | Nickeliferous laterite deposits[END_REF][START_REF] Golightly | Progress in understading the evolution of nickel laterites[END_REF][START_REF] Gleeson | Nickel laterites: A review[END_REF][START_REF] Freyssinet | Ore-forming processes related to lateritic weathering[END_REF][START_REF] Butt | Nickel laterite ore deposits: Weathered serpentinites[END_REF]. After being leached from rockforming minerals nickel is thought to be redeposited be-

A C C E P T E D M A N U S C R I P T

ACCEPTED MANUSCRIPT low, mostly associated with i) Ni-rich ferruginous oxide ore where it is sorbed and/or substituted for Fe in the crystal lattice [START_REF] Cornell | Simultaneous incorporation of Mn, Ni and Co in the goethite (α -FeOOH) structure[END_REF][START_REF] Fischer | Observations and modelling of the reactions of 10 metals with goethite: adsorption and diffusion processes[END_REF][START_REF] Singh | Incorporation of Cr, Mn and Ni into goethite (α -FeOOH): Mechanism from extended X-ray absorption fine structure spectroscopy[END_REF][START_REF] Carvalho-E-Silva | Incorporation of Ni into natural goethite: An investigation by X-ray absorption spectroscopy[END_REF][START_REF] Dublet | Ni speciation in a New Caledonian lateritic regolith: A quantitative X-ray absorption spectroscopy investigation[END_REF], ii) a wide variety of neo-formed silicate minerals such as talc-like and sepiolite-like minerals [START_REF] Cathelineau | Raman spectra of Ni-Mg kerolite: effect of Ni-Mg substitution on O-H stretching vibrations[END_REF][START_REF] Villanova-De-Benavent | Garnierites and garnierites: Textures, mineralogy and geochemistry of garnierites in the Falcondo Ni-laterite deposit, Dominican Republic[END_REF][START_REF] Wells | Mineralogy and crystal chemistry of garnierites in the Goro lateritic nickel deposit, New Caledonia[END_REF], and iii) partly weathered primary lizardite, in which Mg to different extent has been exchanged by Ni [START_REF] Golightly | Nickeliferous laterite deposits[END_REF][START_REF] Golightly | Progress in understading the evolution of nickel laterites[END_REF]. Three different mechanisms have been suggested in the literature in order to explain the association between nickel and goethite: i) isomorphous substitution of Ni for Fe into the goethite structure, ii) association of Ni with poorly crystalline phase, and iii) adsorption of Ni to goethite surface. [START_REF] Singh | Incorporation of Cr, Mn and Ni into goethite (α -FeOOH): Mechanism from extended X-ray absorption fine structure spectroscopy[END_REF] and [START_REF] Carvalho-E-Silva | Incorporation of Ni into natural goethite: An investigation by X-ray absorption spectroscopy[END_REF] found and confirmed using EXAFS that Ni 2+ is associated with goethite via isomorphous substitution for Fe 3+ . Using sequential leaching studies of Indian laterite samples [START_REF] Swamy | Physico-chemical characterization and sulphatization roasting of low-grade nickeliferous laterites[END_REF] reported that the quantity of chemisorbed Ni was negligible and most of the nickel was bound in the crystal lattice with some associated with amorphous goethite. Actually such an association is quite complex and literature suggests that there is a continuum going from one stage to other [START_REF] Davis | A model for trace metal sorption processes at the calcite surface: Adsorption of Cd and subsequent solid solution formation[END_REF][START_REF] Stipp | Cd uptake by calcite, solid-state diffusion, and the formation of solidsolution: Interface processes observed with nearsurface sensitive techniques (XPS, LEED and AES)[END_REF]. Therefore, an initial fast adsorption step is followed by a slow step of incorporation of the adsorbed species into the crystal structure with formation of solid solution. [START_REF] Schultz | Adsorption and desorption of metals on ferrihydrite: Reversibility of reaction and sorption properties of the regenerated solid[END_REF] experimentally documented the irreversible adsorption of metals on ferrihydrite surfaces. Adsorption/desorption of 10 -5 M Ni 2+ on 10 -3 M ferrihydrite revealed that after the overnight exposure only about 25 % of the Ni 2+ could be desorbed and the remainder stayed bound in the ferrihydrite structure. Similar hysteresis were found by [START_REF] Mckenzie | The adsorption of lead and other heavy metals on oxides of Mn and Fe[END_REF], [START_REF] Brümmer | Reaction kinetics of the adsorption and desorption of nickel, zinc and cadmium by goethite. I. Adsorption and diffusion of metals[END_REF], [START_REF] Barrow | Reaction kinetics of the adsorption and desorption of nickel, zinc and cadmium by goethite. 2. Modeling the extent and rate of reaction[END_REF][START_REF] Gerth | Retention of Ni, Zn and Cd by Si-associated goethite[END_REF]. Further recrystallization of the ferrihydrite leads to a release of the nickel over time. Direct evidence was established for a negative correlation between the bulk Ni content and the crystallinity of goethite as a function of depth in the New Caledonian Ni laterites [START_REF] Dublet | Ni speciation in a New Caledonian lateritic regolith: A quantitative X-ray absorption spectroscopy investigation[END_REF][START_REF] Dublet | Goethite aging explains Ni depletion in upper units of ultramafic lateritic ores from New Caledonia[END_REF].

There are a number of scientific articles dealing with modelling of regolith formation from bedrock. For example Soler andLasaga (1996, 1998) have been working on an advection-dispersion-reaction model of bauxite formation, [START_REF] Fletcher | A spheroidal weathering model coupling porewater chemistry to soil thicknesses during steadystate denudation[END_REF] and [START_REF] Lebedeva | A reactive diffusion model describing transformation of bedrock to saprolite[END_REF] on granite bedrock (albite + FeO-bearing phase + quartz) transformation to saprolite (goethite + kaolinite + quartz), [START_REF] Navarre-Sitchler | A reactivetransport model for weathering rind formation on basalt[END_REF] on a reactive transport model for weathering rind formation on basalt etc. Nevertheless, the mobility and transport of trace elements in ultramafic bedrock was not yet modelled with the only exception of a recent study by [START_REF] Domènech | Reactive transport model of the formation of oxide-type nilaterite profiles (punta gorda, moa bay, cuba)[END_REF]. Their work provides valuable insights into the formation of the different laterite horizons in the profile from partially serpentinized peridotite. However, the analysis seems to miss important elements that suggest Ni should be included into the goethite structure and cannot be explained by adsorption alone. In addition, the formation of garnierite is not considered and Ni enrichment is shown only in oxide part of deposits. Thus, the processes of Ni retention in a profile and their controlling parameters are still poorly understood.

In the present work, we used a reactive transport modelling in order to study the development of the secondary nickel ores upon vertical progression of the alteration front. We analyse the effect of key parameters governing i) sorption processes on a goethite surface , ii) precipitation/dissolution of secondary minerals, and iii) kinetic dissolution of serpentinized olivine due to meteoric water flow (Fig. 2). Moreover, the effect of pimelite/kerolite (Ni-bearing phases) being present as a solid solution and incorporation of Ni into the goethite structure were also studied. The role of sorption processes on goethite in Ni redistribution in space during the mineralization, as well as the detailed understanding of their mechanisms have particular emphasis in this paper. Direct comparison of Ni distribution obtained after the modelling with its typical distribution in the profiles provides us in depth understanding of the weathering process in laterite deposits, reveals the validity of the model and gives new insights into the other processes that might have also governed the mobility of elements and formation of these increasingly important Ni deposits.

Materials and methods

Modelling the hydrodynamic system

A reactive multicomponent 1-D transport model of supergene enrichment of lateritic Ni deposits has been simulated by assuming the weathering of a onedimensional vertically oriented column of serpentinized olivine due to steady state meteoric water flow. The code used for the simulations is PHREEQC (V 3.1.4) [START_REF] Parkhurst | Description of input and examples for PHREEQC version 3-A computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF]. The 1-D column is defined by a series of 40 cells of 0.5 m. in length. The velocity of water in each cell is determined by the length of the cell, porosity and amount of annual rain precipitation. Solute concentrations at some point on a flowline may change by i) advection of concentration gradi-

A C C E P T E D

M A N U S C R I P T

ACCEPTED MANUSCRIPT

ents, ii) reactions with the solid material, and iii) dispersion and diffusion. The Advection-Reaction-Dispersion equation that describes these changes along the flowline and implemented on Phreeqc is the following:

∂c ∂t x = -ν ∂c ∂t t - ∂q ∂t x + D L ∂ 2 c ∂x 2 t ( 1
)
where c is the solute concentration (mol.l -1 ), v is pore water flow velocity (m.s -1 ), D L is hydrodynamic dispersion coefficient (m 2 .s -1 ) and q is the concentration in the solid (mol.l -1 of pore water). In our calculations we neglect dispersion and set D L to 0. According to in situ observations, indeed, transport of the chemical components is controlled by advection and high reaction rates. Neglecting dispersion will cause sharper reaction front and higher concentration peaks. In other words, spreading of the concentration front should be slightly underestimated.

Flux boundary conditions (Cauchy) were defined for the first and last cell. A slightly acidic tropical rainwater with pH=5.6 due to its equilibrium with atmospheric CO 2 represents an incoming solution and is injected in the first cell (Fig. 3). Regarding separately the values of the experimental rates of kinetic dissolution of the parent rock constituents (olivine, serpentine, enstatite), presented in the literature (e.g. [START_REF] Brantley | Chemical weathering rates of pyroxenes and amphiboles[END_REF]; [START_REF] Pokrovsky | Kinetics and mechanism of forsterite dissolution at 25 • C and pH from 1 to 12[END_REF]; [START_REF] Wilson | Weathering of the primary rockforming minerals: Processes, products and rates[END_REF]; [START_REF] Thom | Chrysotile dissolution rates: Implications for carbon sequestration[END_REF]), one may note that olivine has a highest rate of the dissolution process among them. The rates of enstatite and serpentine are extremely sluggish compared to those of olivine, being one to a few orders of magnitude lower depending on pH. This makes olivine the main initial mineral in a system that provides nickel and other elements (Mg, Si and Fe). In this way, the numerical modelling was performed assuming olivine in each cell of the 1-D column at initial state (Fig. 2).

The New Caledonia is situated in seasonally humid wet savannas, thus, characterized by summer rainfall of 900-1800 mm and a 2-5 months winter dry season [START_REF] Butt | Nickel laterite ore deposits: Weathered serpentinites[END_REF]. Nevertheless, these values are related to the current climatic circumstances and most likely were not the same during all 10 Ma of laterite formation. According to calculations of [START_REF] Thorne | Climate change and the formation of nickel laterite deposits[END_REF] Ni laterites develop where rainfall exceeds 1000 mm/y and mean monthly temperatures range between 22-31 • C (summer) and 15-27 • C (winter). For our simulations the value of 2000 mm/y was chosen as acceptable.

Porosity of the ultramafic bedrock was chosen to be about 2% but is highly dependent on the fractures density [START_REF] Join | Système hydrogéologique d'un massif minier ultrabasique de nouvelle-calédonie[END_REF][START_REF] Jeanpert | Analyse préliminaire des donées hydrogéologiques du massif du Koniambo[END_REF]. Due to code limitations, the porosity remains constant throughout the calculations. There are some other codes with improved hydrodynamic part (PhreeqcRM, Crunchflow, etc) that handle porosity changes but from a 1-D modeling point of view such an assumption has a minor influence on the results of simulation. Indeed, when magnesium and silica are leached from limonite zone a total mass loss of up to 70-80% appears there. At the same time, the density ahead of the pH front, where silicates form, will still be much closer to the initial value than in the goethite zone. Since 1-D column does not allow any kind of lateral fluid movements, the fluid flow rate is constant along the column. In real profiles, the goethite zone density is so low that it collapses under the weight of the profile to a near constant value [START_REF] Golightly | Nickeliferous laterite deposits[END_REF]. The total computing time for 1-D transport calculations is defined as 250,000 pore volumes of filling solution that are moved through the column (10 7 shifts of solution per 40 column cells). The real geological time in years could be estimated from residence time of solution in a cell multiplied by number of solution shifts. Since the residence time per cell was defined as 100,000 seconds, it will correspond to a summary computing time of about 10 Ma. We take this value as a connection of our results to real geological time but it should be noted that in reality the residence time will greatly change (decrease) due to porosity changes, formation of fractures and lateral dissipations of fluid. Therefore, this value of 10 Ma is an upper limit and certainly overestimated. Further planned investigations will require implementation of this code into a 2D/3D fully coupled hydrological model to resolve this drawback.

Modelling the geochemical system

Olivine composition in the model was selected according to [START_REF] Trescases | L'élvolution gélochimique supergèlne des roches ultrabasiques en zone tropicale: Formation des gisements nickélifères de Nouvelle-Calédonie: Mémoires ORSTOM[END_REF] as following: Mg 1.82 Fe 0.17 Ni 0.01 Al 0.006 SiO 4 . Olivine dissolution is assumed to be kinetically controlled whereas the precipitation of secondary weathering products is considered to occur according to local equilibrium. [START_REF] Pokrovsky | Kinetics and mechanism of forsterite dissolution at 25 • C and pH from 1 to 12[END_REF] provided in their work the following steady state specific dissolution rate for olivine at 25 • C in CO 2 -free solutions:

log(r) = -0.5pH -6.64
(2)

Which is consistent with an Arrhenius equation of the form of:

r = Aa n H + exp(-E a /RT ) (3)
where r signifies the specific dissolution rate of olivine in (mol.m -2 .s -1 ), a n H + is the activity of protons (mol.l -1 ), A refers to a pre-exponential factor, E a designates an activation energy, R represents the gas constant, and T denotes absolute temperature.

Generally, the effect of atmospheric CO 2 on olivine dissolution rate is very weak in acid to neutral solutions but may be important in alkaline conditions. Results of [START_REF] Wogelius | Olivine dissolution at 25 • C -effects of pH, CO 2 , and organicacids[END_REF] show a decrease of the dissolution rate at elevated pH and low CO 2 pressure while experiments conducted under similar conditions by [START_REF] Pokrovsky | Kinetics and mechanism of forsterite dissolution at 25 • C and pH from 1 to 12[END_REF] and [START_REF] Golubev | Experimental determination of the effect of dissolved CO 2 on the dissolution kinetics of Mg and Ca silicates at 25[END_REF] did not account such a decrease in comparison with their CO 2 -free experiments. Current modelling was performed according to the equation ( 2) and without taking into account the effect of presence of CO 2 .

The overall olivine reaction rate in the model is as follows:

R d = r A 0 V m m 0 n (4)
where R d signifies the overall reaction rate (mol.l -1 .s -1 ), r is the specific dissolution rate (mol.m -2 .s -1 ) extracted from equation ( 2), A 0 refers to

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT
the initial surface area of the solid (m 2 ), V designates the volume of solution (l), m 0 represents the initial moles of solid, and m is the moles of solid at given time.

The factor (m/m 0 ) n accounts for changes in surface area during dissolution of olivine. For a monodisperse population of uniformly dissolving spheres or cubes, n = 2/3, since m is proportional to the volume, or r 3 (here r is the radius of the sphere or the side of the cube), while the surface area is proportional to r 2 .

The rate expression is embedded in the PHREEQC third order Runge-Kutta-Fehlberg algorithm [START_REF] Parkhurst | Description of input and examples for PHREEQC version 3-A computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF]. Equilibrium phase composition is calculated before a kinetic calculation is initiated and after a kinetic reaction increment. Calculations were done at 25 • C with the code PHREEQC associated with the llnl.dat thermodynamic database [START_REF] Johnson | Database from thermo.com.v8.r6.230 prepared at lawrence livermore national laboratory[END_REF], that has been edited in order to account for garnierite minerals. The reactions, involved in the precipitation or dissolution of Mg and Ni end-member phyllosilicates are given in Table 1.

Basically, Mg-Ni-phyllosilicates are parts of solidsolution series extending from Mg to Ni end-members (e.g. [START_REF] Springer | Compositional and structural variations in garnierites[END_REF][START_REF] Springer | Falcondoite, nickel analogue of sepiolite[END_REF]; [START_REF] Reddy | Characterisation of Ni silicate-bearing minerals by UV-vis-NIR spectroscopy. Effect of Ni substitution in hydrous Ni-Mg silicates[END_REF]) and effect of them being present as an ideal solid solutions instead of separate minerals will be discussed. The process of substitution of Ni for Mg in the lattice of serpentine is not considered here due to the lack of thermodynamic data needed to perform such kind of modeling. [START_REF] Trescases | L'élvolution gélochimique supergèlne des roches ultrabasiques en zone tropicale: Formation des gisements nickélifères de Nouvelle-Calédonie: Mémoires ORSTOM[END_REF] demonstrated that many ground waters from the base of the nickel laterite profiles in New Caledonia are saturated or supersaturated with respect to talc-like mineral but rarely saturated in serpentine. For the moment serpentine has been seldom reported to form under earth-surface conditions and, thus, suppressed from equilibrium phases that were allowed for precipitation. Instead, in nature the precipitation rates favor the formation of metastable sepiolite and kerolite in Al-poor environments [START_REF] Stoessell | 25 • C and 1 atm dissolution experiments of sepiolite and kerolite[END_REF]. Thus, for transport calculations at 25 • C the list of secondary weathering products is following: Ferrihydrite, Gibbsite, Saponite-Mg, Kerolite, Pimelite, Sepiolite, Falcondoite, Quartz. Ferrihydrite is known to be an initial precursor phase in the formation of goethite but is difficult to be identified using standard characterisation techniques (such as PXRD) due to its poorly crystalline nature. According to [START_REF] Cornell | The effect of nickel on the conversion of amorphous iron(iii) hydroxide into more crystalline iron oxides in alkaline media[END_REF] the introduction of nickel to the system increases the stability of the ferrihydrite phase, inhibiting its transformation to goethite and also suggesting that ferrihydrite could be still present in nickeliferous laterites. Goethite and hematite form later on from ferrihydrite and the ratio and cristallinity of these products are strongly influenced by pH. [START_REF] Schwertmann | Effect of pH on the formation of goethite and hematite from ferrihydrite[END_REF] showed that at 24 • C maximum hematite was formed in between pH 7 and 8, while maximum goethite at pH below 4 and above 12. This way, relying on a diagram of pH change with depth, one can suppose what will be the final products in each part of profile.

The surface complexation model of [START_REF] Dzombak | Surface Complexation Modeling:Hydrous Ferric Oxide[END_REF], that takes into account binding of metals and protons on both strong and weak sites of ferrihydrite, which develops a charge depending on the ions sorbed, has been used in order to simulate adsorption process. The adsorption constants required to parameterize the model for Ni adsorption at the surface of ferrihydrite were obtained by fitting experimental data [START_REF] Dzombak | Surface Complexation Modeling:Hydrous Ferric Oxide[END_REF] and are given in Table 2.

Two sites are defined for a diffuse-double-layer: i) Hfo s, which is strong binding site, and ii) Hfo w, which is weak binding site. [START_REF] Dzombak | Surface Complexation Modeling:Hydrous Ferric Oxide[END_REF] used 0.2 mol weak sites/mol ferrihydrite and 0.005 mol (2.5% of weak sites) strong sites/mol ferrihydrite with a surface area 5.33×10 4 (m 2 .mol -1 ) and a weight of 89 g Hfo/mol Fe. To be consistent with their model, the relative number of strong and weak sites is kept constant as the total number of sites varies. In reactive 1-D transport modeling presented in this paper oxi-hydroxide surfaces change in proportion as the ferrihydrite dissolves or precipitates.

Table 2: Thermodynamic data for a diffuse-double-layer surface "Hfo"(Hfo stands for hydrous ferric oxide, i.e. ferrihydrite), derived from [START_REF] Dzombak | Surface Complexation Modeling:Hydrous Ferric Oxide[END_REF].

Reaction

Log k

Hfo sOH + H + = Hfo sOH + 2 7.18 Hfo sOH = Hfo sOH -+ H + -8.82 Hfo sOH + Ni 2+ = Hfo sONi + + H + 0.37 Hfo wOH + H + = Hfo wOH + 2 7.18 Hfo wOH = Hfo wOH -+ H + -8.82 Hfo wOH + Ni 2+ = Hfo wONi + + H + -2.5
It is worth noting that the llnl.dat database contains thermodynamic data for sorption of many trace metals. Once surface Ni complexes, Hfo s and Hfo w, are introduced in the model, the effect of other cations, such as Mg 2+ , Fe 2+ , Ca 2+ can be accounted for. Cations competition for sorption at the surface of ferrihydrite will, thus, influence the mobility of leaching metals.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT
In order to analyse this competition a transport modelling through one-cell with flux-open boundaries was performed. The input of one-cell model is identical to the main one with 40 cells. This model has been used in order to reduce calculation time and gain insight into the aforementioned competition of metals for sorption sites and precipitation of Ni-goethite solid solution with 2wt% of Ni. It should be noted that in a results and discussion section we always use the name goethite in spite of the fact simulations are done with ferrihydrite constants (same chemical formula and different solubility constant value). This latter choice is applied to avoid any kind of confusions since ferrihydrite is being more geochemical rather than geological term.

Results and discussion

1-D Reactive transport model

Figure 4 shows the evolution of the paragenetic sequence and the development of the secondary nickel ores as a function of time. The results represent mass fractions of secondary weathering products formed after 50,000 PV, 150,000 PV and 250,000 pore volumes (PV) of filling solution that are moved through the column and corresponds to 2 Ma, 6 Ma and 10 Ma, respectively. The reactive transport model reproduces both the formation of the laterite profile in time with thickening of iron-rich zone and change of the Ni content with depth. Here, nickel supplied in the system by olivine dissolution was supposed to be redistributed in between solution, adsorption on goethite and dissolution/precipitation of Ni-bearing minerals. In the following chapter of detailed understanding of Ni behaviour during weathering is displayed. As can be seen from Figure 4, the obtained chemical profile clearly distinguishes the ferruginous/aluminous residual part where Si, Mg, and Ni are leached, from the zone where silicates start to precipitate. The effect of silicate weathering on the water chemistry is primarily the addition of cations (Ni 2+ , Mg 2+ , Fe 2+ , Fe 3+ ; Al 3+ ) and silica. pH variation is balanced on one hand by silicate weathering reactions which consume H + and on the other hand by leaching of the elements out of the cell and renewing with low pH meteoric water which tend to decrease the pH value. Migration of the pH front, subdividing slightly acidic at the surface to alkaline pH at depth, appears to have the main influence in controlling the precipitation/dissolution of neo-formed silicate minerals, and, thereby, regulating the transport of elements in the profile. The limonite zone is represented by goethite and minor precipitations of gibbsite. Beneath, after the pH front, the secondary weathering products are represented by falcondoite (Ni bearing phase), and massive precipitations of sepiolite and Saponite-Mg. Considering the results after 2 Ma, one can observe that part of olivine mass is not yet dissolved, providing pH up to 11 at the bottom of the column, and that pimelite (Ni talclike) precipitation (indicated by a narrow black line on Fig. 4) occurs at depth, after the pH front. Pimelite precipitation is no more observed after 4 Ma. Indeed, after 4 Ma, olivine disappears and the maximum pH value decreases from 11 to 9.5 after the pH front.

It worth noting that in New Caledonia the silicates, precipitated in fractures or in a form of concentric zoning of Ni-bearing silicates [START_REF] Cathelineau | Nickel dispersion and enrichment at the bottom of the regolith: formation of pimelite target-like ores in rock block joints (Koniambo Ni deposit, New Caledonia)[END_REF], are mostly represented by talc-like minerals. This is in contrast to sepiolite-like phases (falcondoite, sepiolite), obtained in current numerical modelling which is more common for Falcondo Ni-laterite deposits observed in Dominican Republic (Villanova-de-Benavent et al., 2014). In order to find out why sepiolite-like precipitated instead of talc-like minerals let us turn to the literature. According to [START_REF] Stoessell | 25 • C and 1 atm dissolution experiments of sepiolite and kerolite[END_REF] the difference in solubility between kerolite and sepiolite is small enough that precipitation kinetics is probably decisive in their formation. The metastability of kerolite relative to sepiolite, determined in the study of [START_REF] Stoessell | 25 • C and 1 atm dissolution experiments of sepiolite and kerolite[END_REF], was suggested by [START_REF] Jones | Clay mineral diagenesis in lacustrine sediments. Studies in Diagenesis[END_REF]. He reported kerolite as a precipitate at Stewart springs (California, USA) from waters with pH above 12, but with aqueous silica concentrations below (25 • C and 1 bar) quartz saturation. [START_REF] Jones | Clay mineral diagenesis in lacustrine sediments. Studies in Diagenesis[END_REF] suggested that with increasing pH and Mg content in water and decreasing silica content, kerolite should precipitate at the expense of sepiolite. Same conclusion might be reached for pimelite (Ni talc-like) and falcondoite (Ni sepiolite) precipitation since they are the Ni counterparts of kerolite and sepiolite, respectively. It is consistent with the numerical results (Fig. 4) where, as it was mentioned above, pimelite (narrow black line, 2 Ma column) precipitated in the deeper part of profile where high pH of about 11 was maintained by dissolution of still presenting olivine. In this way it could be concluded that for precipitation of talc-like minerals at 25 • C in Al-poor environments it is necessary and sufficient to have an ample amount of olivine a system.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT
It should be noted that in real profiles olivine would persist for longer time due to the different character of the porosity distribution. In fact, the extreme instability of olivine results in its replacement by silicate phases in situ with the cations in the solutions diffusing outwards to joints, where the water is flowing downwards. Because of this the olivine in the bulk material does not disappear before 6 million years as in the model. Nevertheless, an accurate representation of the process will not be achieved until 2D modelling is attempted.

Returning to the results of the model the entire dissolution of olivine in the column after 6 Ma and 10 Ma (Fig. 4) led to a change in pore water pH with values of 5.6 close to the surface to about 9.5 at the bottom. Present day water analyses at Koniambo (northwest of New Caledonia) indicated similar pH condition: from 5.5 near the surface to about 10 at depth, either in the laterite or in the serpentinized peridotite [START_REF] Jeanpert | Analyse préliminaire des donées hydrogéologiques du massif du Koniambo[END_REF]. Concerning the mass fraction of Ni bearing sepiolite (falcondoite), its accumulation and downward movement with pH front migration can be observed. The increase of Ni content with time and depth is shown in Figure 5.

It should be noted that the Ni concentration in wt% here is related to initial mass of olivine in a cell, serving as reference in order to be able comparing it through time. Normalizing to the current mass of minerals presented in a cell adds a complexity in following comparison of Ni enrichment since this mass is changing due to the leaching. Despite the fact that in the Figure 5 at 250,000 PV (10 Ma) maximum concentration of nickel (4.8 wt.% ) seems to be lower than in previous step (6 wt.% after 200,000 PV, or 8 Ma), it appears that the thickness of the Ni rich zone is also higher after 250,000 PV than after lower PV circulation. Therefore, the precipitation of Ni in the two closest cells after 250,000 PV (10 Ma) (Fig. 4) with Ni content of 3.06 wt.% and 4.80 wt.% at 9.25 m and 9.75 m of depth respectively, yields an overall equivalent Ni content of 7.89 wt.%. This value is much higher comparing to typical New Caledonian profile presented in Figure 1 which exhibits up to 2-3 wt.% of Ni distributed throughout limonite and saprolite zone. The cause of it might be related to precipitation of silicates at local equilibrium in our model, implying that simultaneous precipitation occurs once favorable conditions are reached in a cell and re-sulting in a narrow zone of Ni precipitation. Also, in nature, nickel, released due to peridotite dissolution, may be partly lost by lateral movement in solution.

The overall concentration of nickel of 7.89 wt.% at the last step of simulation (10 Ma) represents 97.2 % of the initial nickel content in the column and distributed as follow: i) 4.27 wt.% of Ni sorbed on ferrihydrite surface, ii) 95.7 wt.% precipitated with Ni-silicate (falcondoite) and iii) 1.69×10 -4 % in solution along the whole column. From Figure 5 one can observe the movement of transition zone from the limonite horizon to the saprolite, which is marked by a sharp increase in MgO content from 0 to up to 20 wt.%. Iron content meanwhile varies from values of about 65 wt.% before the reaction front down to 5 wt.% after it. Whilst the iron has been oxidized, the nickel remains bivalent and is somewhat mobile, more so than iron, less so than magnesium. This chemical distribution of elements is consistent with Figure 1 tact with saprolite, while in numerical modelling this zone comes out to be free of Ni. According to the modelling results, adsorbed nickel appears to be governed by pH front. Being initially retarded compared to fluid movement, adsorbed Ni has been eventually liberated into pore water at pH as low as 5.6 due to reaction front downward advancement upon silicates dissolution. Dependence of nickel sorption on pH along with competitive retardation is discussed in the following chapter, devoted to a detailed understanding of the weathering process. Yet, it could be concluded that adsorption can not explain such a high nickel content in limonite nowadays since the actual pH of soil waters can be in some instances as low as 4.5 due to dissolved organic acids, thus, suggesting that nickel is held here by stronger ties. The latter might be explained by isomorphous substitution of Ni for Fe in goethite and either significant process in soils related to specific binding of elements to the variable charge surfaces of organic matter. Adsorption on organic matter is similar to that on ferrihydrite since surface charge likewise depends on pH and solution composition, but the difference is that the sorption edge for organic matter is shifted to lower pH. It implies that at low pH from about 4 to 7 adsorption on organic matter will be dominating, but at higher pH the major role will be passed to adsorption on oxyhydroxides. Present study does not consider the issue of DOC due to its low concentration in well-drained and scarce in terms of soil terrains of New Caledonia.

Detailed understanding of the weathering process

Effect of pimelite/kerolite being present as solid solution

The numerical simulations presented above consider the Mg-Ni-phyllosilicates to be pure end-members although in reality they are being members of solidsolution series extending from Mg and Ni end-members (e.g. [START_REF] Springer | Compositional and structural variations in garnierites[END_REF][START_REF] Springer | Falcondoite, nickel analogue of sepiolite[END_REF]; [START_REF] Reddy | Characterisation of Ni silicate-bearing minerals by UV-vis-NIR spectroscopy. Effect of Ni substitution in hydrous Ni-Mg silicates[END_REF][START_REF] Villanova-De-Benavent | Garnierites and garnierites: Textures, mineralogy and geochemistry of garnierites in the Falcondo Ni-laterite deposit, Dominican Republic[END_REF]; [START_REF] Cathelineau | Raman spectra of Ni-Mg kerolite: effect of Ni-Mg substitution on O-H stretching vibrations[END_REF]). Usually solid solutions show deviations from ideal behavior and the activity coefficients, that account for non-ideality, become a function of the excess free energy of mixing. The free energy of mixing might be

A C C E P T E D

M A N U S C R I P T obtained from different non-ideal thermodynamic properties such as variations in fractionation factor, miscibility gaps etc. [START_REF] Glynn | Solid-solution solubilitites and thermodynamics: sulfates, carbonates and halides[END_REF]. Despite the fact that many researchers describe Mg-Ni-phyllosilicate members of solid solutions, the possibilities of modelling such a system remain poor due to the lack of thermodynamic data, thus, limiting simulations by assumption of ideality. In this way, for example, [START_REF] Galí | Ni-enrichment and stability of Al-free garnierite solid solutions: a thermodynamic approach[END_REF] have calculated Lippmann diagrams for the garnierite solid solutions. This subchapter aims at understanding the effect of e.g. pimelite and kerolite being present as a solid solution instead of separate mineral phases. Linear combination of reactions for kerolite and pimelite, presented in Table 1 results in Equation (5):

Mg 3 S i 4 O 10 (OH) 2 : H 2 O + 3Ni 2+ = Ni 3 S i 4 O 10 (OH) 2 : H 2 O + 3Mg 2+ (5)
At first, let us consider a situation where a solution is in equilibrium with kerolite and pimelite as separate minerals. The mass action expression for Equation (5) looks as follows:

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT K = [Ni 3 S i 4 O 10 (OH) 2 : H 2 O] Mg 3 S i 4 O 10 (OH) 2 : H 2 O × Mg 2+ 3 Ni 2+ 3 = K Kerolite K Pimelite = 10 14.33 (6)
Then, in Equation ( 6) the activities of two pure solid phases will be equal to one. Therefore, the ratio [Mg 2+ ] 3 [Ni 2+ ] 3 in solution is fixed to 10 14.33 and [Mg 2+ ] [Ni 2+ ] = 59795. The concentration of Ni 2+ is much smaller than of Mg 2+ because the solubility of pimelite is much smaller than of kerolite.

In the second situation Ni 2+ is incorporated in the kerolite structure as an ideal solid solution, which may be written as Ni λ Mg 3-λ S i 4 O 10 (OH) 2 : H 2 O, where λ represents the nickel mole fraction. In this way, activities of kerolite and pimelite in Equation ( 6) will no longer be equal to one as for separate minerals, but will depend on λ . Therefore, if Mg concentration remains the same as before, then the Ni concentration must now be lower. It can clearly be seen in Figure 6, that shows the effect of Ni substitution in pimelite/kerolite solid solution on the relative Ni concentration decrease in solution. Therefore, the common values of Ni mole fraction in New Caledonian pimelite/kerolite solid solutions, i.e around λ = 0.3-0.4, will lead to a decrease of the Ni concentration by about 20% from the case with pure phases on assumption that Mg concentration remains the same (Fig. 6). Thus, to conclude it should be said that the formation of solid solution lowers the aqueous concentration of the minor component, in our case nickel, but does not have a drastic effect on the global Ni mass balance. The latter means that despite the use of separate mineral phases our results of simulations may be considered as reliable.

Ni in solid solution with goethite

It has already been mentioned above that sorption alone cannot explain increase of Ni content throughout the limonite zone (Fig. 1). In our 1-D model all Ni, adsorbed on goethite, was released in solution and shifted at depth once the pH decrease below six after the front. Obviously, this latter phenomenon is not observed in the field, where Ni still persists in the limonite zone. According to [START_REF] Dublet | Ni speciation in a New Caledonian lateritic regolith: A quantitative X-ray absorption spectroscopy investigation[END_REF], nickel content up to 2wt.% is common for New Caledonian laterites, where

A C C E P T E D M A N U S C R I P T

ACCEPTED MANUSCRIPT

Ni 2+ is associated with goethite via isomorphous substitution for Fe 3+ . In order to account for this gap, here, we perform a sensitivity test that considers additionally to our previous transport model (Fig. 4) a precipitation of Ni-bearing goethite. In this way, reaction for precipitation of Ni-bearing goethite has been added in the thermodynamic database as follows:

Fe 0.98 Ni 0.03 OOH + 3H + = 0.98Fe 3+ + 0.03Ni 2+ + 2H 2 O (7)
The influence of such a small amount of Ni in structure of goethite on its equilibrium constant has been ne-glected. Indeed, this constant, according to Equation (7), will be defined as:

LogK = 0.98 × log Fe 3+ + 0.03 × log Ni 2+ + 3pH (8)
Considering that the Ni concentrations in the model are as low as 10 -9 to 10 -6 moles per liter, and taking into account Equation ( 8) one can see that the effect of Ni presence on the equilibrium constant will be minor. Incoming solution, olivine content and dissolution rate here are identical to those used in 1-D transport model previously presented and such a test case was done in order to get detailed understanding of the weathering processes in each cell. X-axis in Figure 7 represents pore volumes, which is a number that shows how many times filling solution moved through the cell. Therefore, this axis represents time of evolution of the bedrock until the formation of oxide-rich zone. This way the principal shape Ni enrichment curve in Figure 7(A) can be easily compared to Figure 1. Indeed, we can highlight three zones for Ni enrichment: i) a zone of plateau where Ni co-precipitated with goethite, followed at pH higher than six by ii) a peak and iii) a decrease up to the value of Ni concentration in parent rock. Of course, since we investigate a cell stand alone, it has not any income of the elements from the upper cells as it was in a column modelling, but the principal shape of Ni enrichment curve will remain the same with some differences in values. This shape, shown in Figure 7(A), is very close to that presented in Figure 1. Similarly as it is observed in the field, the zone of residual laterites (Fig. 7(A)) is mostly composed of iron

A C C E P T E D

M A N U S C R I P T

ACCEPTED MANUSCRIPT

oxy-hydroxides, and Ni, being more mobile, reaches its maximum concentrations at the base of this region. This area is marked by a sharp increase in pH and subsequently MgO content (Fig. 7(A)). The latter means that pH appears to be the most important chemical parameter that drives the enrichment of nickel in the weathering profile. Nevertheless, some differences can be spotted in between Figure 7(A) and Figure 1 chemical distributions. First and foremost, as one can see, oxide Ni deposits show a regular decrease of bulk nickel content towards the surface (Fig. 1) (e.g. [START_REF] Trescases | Weathering and geochemical behaviour of the elements of ultramafic rocks in New Caledonia[END_REF]; [START_REF] Wells | Mineralogy and crystal chemistry of garnierites in the Goro lateritic nickel deposit, New Caledonia[END_REF]; [START_REF] Dublet | Ni speciation in a New Caledonian lateritic regolith: A quantitative X-ray absorption spectroscopy investigation[END_REF]), while Figure 7(A) with the modelling results represents a stable plateau with value of 2 wt.% of Ni. [START_REF] Dublet | Ni speciation in a New Caledonian lateritic regolith: A quantitative X-ray absorption spectroscopy investigation[END_REF] have explained it by ageing of goethite in situ through successive dissolution and precipitation cycles during lateritization. Due to a strong mismatch with Fe 3+ both in terms of ionic radii and valence, nickel, substituted for Fe in goethite, is likely to be expelled during each dissolution/precipitation cycle which leads to purification and increase of crystallinity of goethite. In addition, such a decrease in Ni content might be related to the fact that some part of Ni in nature is also bonded in Mn-oxides (up to 10 %). As far as these Mn-oxides are more soluble than iron oxy-hydroxides, they have been leached from upper limonite zone and, therefore, a consequent drop in Ni grade appears.

Another difference lies in a form of a peak and distribution of Ni concentrations around it. Here, there is a number of possible processes that might have influenced the peak shape making it smoother and wider, namely: diffusion, dispersion, convective flows, kinetics of precipitation of silicates, etc. It is important to note that the shape of nickel enrichment curve represents itself a superposition of different processes of Ni retention that are all evolving in time due to chemical conditions change. In our simulations three different concurrent fates of Ni deposition in a profile were taken into account: i) Ni in a goethite crystal lattice, ii) Ni sorbed on weak and strong goethite sorption sites, and iii) Ni precipitated with silicates (garnierite). Therefore, the superposition that represents the Ni enrichment curve (red line) in Figure 7(A) stands for the sum of these three processes. The contribution of each process in the total Ni enrichment is shown in Figure 7(B).

One can see that nickel mineralization begins with the formation of Ni-silicates at high pH, at the moment when ultramafic bedrock just starts to dissolve. At the same time the precipitation of Ni-goethite occurs, with subsequent adsorption of Ni on its surface. Due to the leaching of the elements out of the cell in course of olivine dissolution and refilling it by slightly acidic me-teoric water at each time step, the pH in a modelled cell tends to decrease. Thus, according to the Figure 7(B), at pH value above 9, Ni retention triggered by silicates precipitation prevails over the other processes. Then, the retention of nickel in goethite lattice becomes increasingly important with the maximum achieved at pH 6 and lower. In between these two important mechanisms, there is another one, related to adsorption of Ni on goethite. This process has its specific place during the weathering (Fig. 7(B)); it appears where pH is high enough to have sufficient charge of oxide surface to enhance sorption for binding of nickel. At the same time, at the values of pH of 8.5 or higher, it seems to be inhibited by some other process and subsequently replaced by precipitation of Ni-silicate (Fig. 7(B)). Therefore, adsorption is being rather a "transitional ore" since being released from silicates the adsorbed nickel further passes to the lattice of goethite or precipitates with it. As it has been noticed previously, the pH front in the nature most likely represents smoother curve due to diffusion, dispersion, convective flows etc., and the window of existence of such a "transitional ore" on the surface of oxides might be geometrically larger. The following subchapter is fully devoted to understanding of adsorption process of nickel alone in batch calculations, as well as in competitive environment of transport modelling.

Competition of elements for sorption sites

According to the results of the 1-D numerical transport modelling, presented in chapter (3.1), adsorption of nickel strongly depends on pH (Fig. 5 and 7(B)) and does not appear in limonite zone due to full dissolution of silicates and decrease of pH to the value of rainwater (i.e. 5.6). Indeed, the surfaces of oxides carry a charge that enhances sorption of metals and depends on pH and composition of the solution. Figure 8 represents a batch calculation of nickel distribution among the aqueous phase and strong/weak sorption sites of 0.09 g of goethite, while Figure 9 shows the shape of the sorption edge of Ni 2+ as a function of pH. These calculations are based on the Gouy-Chapman model from [START_REF] Dzombak | Surface Complexation Modeling:Hydrous Ferric Oxide[END_REF] and show the results of distribution for low (10 -8 M) and high (10 -4 M) nickel concentrations. Concentrations of Ni, representative in our modelling, generally lie in this interval, being lower than 10 -6 M.

As one can see from both This is related to the fact that adsorption of high Ni concentrations also occurs on weak sites, what can be seen from the Figure 8. Indeed, the distribution of Ni among the solution and sorption sites of goethite does also depend on the Ni concentration. At low concentrations of nickel (Fig. 8(A)) the strong binding sites outcompete the weak binding sites over the entire pH range. Therefore, at high pH, most of the nickel resides at the strong binding sites. In case of larger nickel concentrations (Fig. 8(B)), the strong binding sites prevail only at low pH.

Surface complexation theory appears to be a useful tool, helping to obtain a conceptual understanding of metal sorption behavior on mineral surfaces. While previous model (Fig. 8 and9) was oversimplified representing closed system, fixed goethite amount, and showing a sorption just for Ni, assumed to be the only metal in solution, the following one deals with competition of metals for sorption sites in a real case. The phenomenon is highlighted in Figure 10, where reactive transport modelling has been performed through one cell with fluxopen boundaries as in previous subchapter (3.2.2).

Once the goethite has been formed in the system, adsorption starts to regulate the mobility of metals and retarding their movement. As can be seen in Figure 10, Mg at some point starts occupying weak sites of goethite, attaining maximum at pH of about 9.2 where, as it was shown in Figure 9, nickel, being alone in the system, would have attained its maximum sorption. Thus, to conclude, adsorption of nickel on oxyhydroxide surfaces (goethite) should not be regarded for Ni alone due to the big influence of other cations releasing during the ultramafic bedrock alteration, in particular Mg. This competition of cations for the sorption sites is to large extent controlled by pH. Adsorption of Ni becomes important in a restricted range of pH around 8, quickly releasing at lower than neutral pH values to pore water and being minor due to competition with Mg at higher ones.

Conclusions

The formation of Ni laterite profile from ultramafic parent rock, at 25 • C, has been simulated by means of Phreeqc one-dimensional reaction-transport code. The main conclusions are listed below:

• The downward progression of the pH front controls thickening of iron-rich zone, explains the mobility of the elements and governs the Ni enrichment.

• Precipitation of talc-like silicates at the expense of sepiolite-like at 25 • C is favored by high pH due to dissolution of still persisting olivine.

• The use of separate silicate phases instead of their solid solutions in the simulations may be considered as reliable since it does not have a drastic effect on the global Ni mass balance.

• Modeling of Ni co-precipitation with goethite explains the presence of Ni in upper limonite zone, where adsorption cannot retain Ni anymore due to acidic pH. The decrease of bulk NiO content towards the surface might be explained by ageing of goethite in situ through successive dissolution and precipitation cycles during lateritization and also by dissolution of Mn oxides.

• Adsorption is being rather a "transitional ore" since after its release from silicates the adsorbed nickel further passes to the lattice of goethite or precipitates with it. The process is fully controlled by pH front and its shape.

• Nickel is more strongly adsorbed at high pH values than at low pH values up to some limit, when alone in the system.

• Adsorption of Ni becomes significant in a narrow range of pH due to the competition of Mg and Ni for sorption sites of goethite surface. Thus, it is of importance to take into account the competition between cations for sorption sites during the ultramafic bedrock alteration Presented in this paper reactive transport modelling of secondary nickel ore development is rather homogeneous (constant temperature, 1D flow) and aims at understanding the main chemical and hydrological influence on it. Further successive dissolution and precipitation cycles of the minerals during lateritization are very closely tied to inherited ores, tectonics, formation

A C C E P T E D

M A N U S C R I P T of preferential pathways, lateral movement, change of relief, higher temperatures, that might explain ore distribution heterogeneities observed on many Ni-laterite sites. Therefore, 2D/3D reactive transport simulations will be conducted to assess these aspects based on a coupling approach between geochemistry, hydrodynamics and heat and mass transfer. • Assessment of Ni distribution in between different fluid-rock interactions processes. Key chemical parameters that control the formation.

• The comparative role of adsorption and incorporation of Ni in goethite upon the formation of laterite.

Figure 3 :

 3 Figure 3: Bar chart of chemical composition of incoming solution (rainwater) at equilibrium with atmospheric CO 2 and subsequent pH of 5.6. In accordance with Trescases (1975).

Figure 2 :

 2 Figure 2: Interaction between a rock column and meteoric water: main variables taken into account.

Figure 4 :

 4 Figure 4: Mass fraction of secondary weathering products after 50,000 PV (2 Ma), 150,000 PV (6 Ma) and 250,000 PV (10 Ma). Yellow dotted line represents current pH front in the column.

Figure 5 :

 5 Figure 5: Migration of pH front along the profile with corresponding Ni, Mg and Fe concentrations in the secondary weathering products.

Figure 6 :

 6 Figure 6: Effect of Ni substitution in pimelite/kerolite solid solution (λ) on the relative Ni concentration decrease in solution (N).

Figure 7 :

 7 Figure 7: A represents the mobility of the elements in a 1-D reactive transport modelling through one cell; supergene enrichment of Ni and its retention by three different mechanisms: i) Sorption on goethite surface, ii) Retention in goethite structure, and iii) Precipitation with Ni-silicates. B shows the split (redistribution) of Ni between different retention mechanisms upon the formation of laterite.

Figure 7 (

 7 Figure 7(A) shows the chemical distribution of the elements in minerals versus pore volume (one million of PV represents one million years). Current reactive transport modelling has been performed through one cell with flux-open boundaries instead of 40 cells column.Incoming solution, olivine content and dissolution rate here are identical to those used in 1-D transport model previously presented and such a test case was done in order to get detailed understanding of the weathering processes in each cell. X-axis in Figure7represents pore volumes, which is a number that shows how many times filling solution moved through the cell. Therefore, this axis represents time of evolution of the bedrock until the formation of oxide-rich zone. This

  Figures 8 and 9, nickel is more strongly sorbed at high pH values than at low pH values. The shape of the sorption edge is shown in Figure 9 and actually depends on the Ni concentration. At low concentrations (e.g. 10 -8 M) the pH domain where Ni is sorbed to the surface of goethite increases from A C C E P T E D M A N U S C R I P TACCEPTED MANUSCRIPTpH 8 to 10.5 while at higher concentrations it reduces, general shape of the Ni sorption edge smooths, and the acid branch shifts to the higher pH.

Figure 9 :

 9 Figure 9: The sorption edge of (10 -8 M) and (10 -4 M) Ni 2+ at the surface of 0.09 g of goethite (ferrihydrite) as a function of pH in 0.1M NaCl solution.

Figure 10 :

 10 Figure 10: Competition of Mg and Ni for sorption sites of goethite, based on the 1-D transport through 1 cell as in a subchapter 3.2.2.

Figure 8 :

 8 Figure 8: Distribution of Ni among the aqueous phase and strong/weak surface sites of goethite (ferrihydrite) as a function of pH. Picture on the left (A) corresponds to low Ni concentration (10 -8 M), while B represents high Ni concentration (10 -4 M). Additional Y-axis represents charge balance of the solution. Solution was prepared with 0.1M NaCl, which permits to use HCl or NaOH as reactant to keep pH increasing, and 90 mg pre-equilibrated ferrihydrite.

•

  Development of 1D reactive transport model for studying Ni mineralization processes in the laterite profiles.

Table 1 :

 1 Dissolution reactions for Mg and Ni end-members of garnierite and Saponite-Mg, with the corresponding equilibrium constants at 25 • C and 1 bar.

	Mineral	Reaction	Log K
	Kerolite	Mg 3 Si 4 O 10 (OH) 2 : H 2 O + 6H + = 3Mg 2+ +4SiO 2 +5H 2 O	25.79 (Stoessell, 1988)
	Pimelite	Ni 3 Si 4 O 10 (OH) 2 : H 2 O + 6H + = 3Ni 2+ +4SiO 2 +5H 2 O	11.46 (Nriagu, 1975)
	Sepiolite	Mg 4 Si 6 O 15 (OH) 2 : 6H 2 O + 8H + = 4Mg 2+ +6SiO 2 +11H 2 O 30.44 (Stoessell, 1988)
	Falcondoite	Ni 4 Si 6 O 15 (OH) 2 : 6H 2 O + 8H + = 4Ni 2+ +6SiO 2 +11H 2 O	12.31 (Nriagu, 1975)
	Saponite-Mg	Mg 3.16 Al 0.33 Si 3.67 O 10 (OH) 2 + 7.32H + = 3.16Mg 2+ +0.33Al 3+ +3.67SiO 2 +4.66H 2 O	26.25 (llnl.dat)

Acknowledgments

This work has been supported by the French National Research Agency through the national program "Investissements d'avenir" with the reference ANR-10-LABX-21-01 / LABEX RESSOURCES21.