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Abstract: Abstract The 3D reconstruction of porous media plays a key role in many engineering applications. There are two main methods for the reconstruction of porous media: physical experimental methods and numerical reconstruction methods. The former are usually expensive and restricted by the limited size of experimental samples, while the latter are relatively cost-effective but still suffer from a lengthy processing time and unsatisfactory performance. With the vigorous development of deep learning in recent years, applying deep learning methods to 3D reconstruction of porous media has become an important direction. Variational auto-encoder (VAE) is one of the typical deep learning methods with a strong ability of extracting features from training images (TIs), but it has the problem of posterior collapse, meaning the generated data from the decoder are not related to its input data, i.e. the latent space Z. This paper proposes a VAE model (called SE-FBN-VAE) based on squeeze-and-excitation network (SENet) and fixed batch normalization (FBN) for the reconstruction of porous media. SENet is a simple and efficient channel attention mechanism, which improves the sensitivity of the model to channel characteristics. The application of SENet to VAE can further improve its ability of extracting features from TIs. Batch normalization (BN) is a common method for data normalization in neural networks, which reduces the convergence time of the network. In this paper, BN is slightly modified to solve the problem of posterior collapse of VAE. Compared with some other numerical methods, the effectiveness and practicability of the proposed method are demonstrated. PubDate: 2022-06-23
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Abstract: Abstract Recent developments in predicting microemulsion phase behavior for use in chemical flooding are based on the hydrophilic-lipophilic deviation (HLD) and net-average curvature (NAC) equation-of-state (EoS). The most advanced version of the HLD-NAC EoS assumes that the three-phase micelle characteristic length is constant as parameters like salinity and temperature vary. In this paper, we relax this assumption to improve the accuracy and thermodynamic consistency of these flash calculations. We introduce a variable characteristic length in the three-phase region based on experimental data that is monotonic with salinity or other formulation variables, such as temperature and pressure. The characteristic length at the boundary of the three-phase region is then used for flash calculations in the two-phase lobes for Winsor type I/II. The functional form of the characteristic length is made consistent with the Gibbs phase rule. The improved EoS can capture asymmetric phase behavior data around the optimum, whereas current HLD-NAC based models cannot. The variable characteristic length formulation also resolves the thermodynamic inconsistency of existing phase behavior models that give multiple solutions for the optimum. We show from experimental data and theory that the inverse of the characteristic length varies linearly with formulation variables. This important result means that it is easy to predict the characteristic length in the three-phase region, which also improves the estimation of surrounding two-phase lobes. The results show that the optimum solubilization ratio can change significantly by a factor of two when variable characteristic length is included as temperature and pressure change. This can in turn greatly impact the interfacial tension (IFT) at optimum. This improved physical understanding of microemulsion phase behavior should aid in the design of surfactant blends and improve recovery predictions in a chemical flooding simulator. PubDate: 2022-06-14
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Abstract: Abstract Upscaling methods such as the dual porosity/dual permeability (DPDP) model provide a robust means for numerical simulation of fractured reservoirs. In order to close the DPDP model, one needs to provide the upscaled fracture permeabilities and the parameters of the matrix-fracture mass transfer for every fractured coarse block in the domain. Obtaining these model closures from fine-scale discrete fracture-matrix (DFM) simulations is a lengthy and computationally expensive process. We alleviate these difficulties by pixelating the fracture geometries and predicting the upscaled parameters using a convolutional neural network (CNN), trained on precomputed fine-scale results. We demonstrate that once a trained CNN is available, it can provide the DPDP model closures for a wide range of modeling parameters, not only those for which the training dataset has been obtained. The performance of the DPDP model with both reference and predicted closures is compared to the reference DFM simulations of two-phase flows using a synthetic and a realistic fracture geometries. While the both DPDP solutions underestimate the matrix-fracture transfer rate, they agree well with each other and demonstrate a significant speedup as compared to the reference fine-scale solution. PubDate: 2022-06-10
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Abstract: Abstract Kriging is a standard method for conditioning surfaces to observations. Kriging works for vertical wells, but may produce surfaces that cross horizontal wells between surface observations. We establish an approach that also works for horizontal wells, where surfaces are modeled as a set of correlated Gaussian random fields. The constraints imposed by the horizontal wells makes the conditional surfaces non-Gaussian. We present a method for exact conditional simulation and an approximation for prediction and prediction uncertainty. Thousands of constraints can be handled efficiently without numerical instabilities. The approach is illustrated with synthetic and real examples that show how the constraints influence the surfaces and reduce uncertainty. PubDate: 2022-06-02
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Abstract: Abstract A three dimensional parallel implementation of Multiscale Mixed Methods based on non-overlapping domain decomposition techniques is proposed for multi-core computers and its computational performance is assessed by means of numerical experiments. As a prototypical method, from which many others can be derived, the Multiscale Robin Coupled Method is chosen and its implementation explained in detail. Numerical results for problems ranging from millions up to more than 2 billion computational cells in highly heterogeneous anisotropic rock formations based on the SPE10 benchmark are shown. The proposed implementation relies on direct solvers for both local problems and the interface coupling system. We find good weak and strong scalalability as compared against a state-of-the-art global fine grid solver based on Algebraic Multigrid preconditioning in single and two-phase flow problems. PubDate: 2022-06-01
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Abstract: Abstract In this work we present an efficient implementation of Eulerian TVD methods. We apply parallelization strategies based entirely on GPU for the solution of the 2D transport equation in heterogeneous porous media. Additionally, a parallel strategy is proposed for the generation of exponentially correlated lognormally distributed permeability fields in GPU. The programs are developed using C++/CUDA. The implemented methods are used to solve advective dominant problems, in a context of Monte Carlo type simulations to numerically determine the longitudinal and transversal macrodispersion coefficients averaging over 100 simulations for permeability fields for a large range of variances. The following types of transport are considered for testing: pure advection, advection-diffusion and advection-dispersion. The performance in terms of the computation time of explicit and implicit methods are compared. We show that the implemented algorithms allow to efficiently solve problems in computational domains of up to 134.5 million cells in a single GPU. PubDate: 2022-06-01
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Abstract: Abstract Traditional stochastic algorithms for characterizing fracture networks are purely based on statistical inferences from outcrop images, and therefore the models produced, may not be physically realistic because they may not be consistent with the process of propagation and termination of fractures. These processes are better represented in geomechanical models of the fracturing process. However, full-physics numerical models are computationally inefficient for modeling fractures at a reservoir scale while accounting for material heterogeneities. More importantly, geomechanical simulations yield deterministic results, thus failing to represent the inherent uncertainties due to input properties and paleo stress conditions. In order to facilitate geomechanical characterization, in this research, a number of small-scale, high fidelity, finite discrete element method (FDEM) based forward models are executed and the relationship between prevailing stress conditions and the fracture propagation direction is inferred using Machine Learning (ML) approaches. We develop a ML based fracture network modeling approach that is orders of magnitude faster, efficiently scalable and may extend the capabilities of statistical fracture modeling approaches by accounting for the physical process of fracture propagation and uncertainties associated with geomechanical parameters. The application and effectiveness of this ML based modeling approach is demonstrated using a synthetic case and a case study from Teapot Dome, Wyoming based on the fracture characteristics inferred from the FMI logs near well 67-1-x-10 in the Tensleep Formation reported by Schwartz (2006). PubDate: 2022-06-01
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Abstract: Abstract Ensemble Kalman filters are based on a Gaussian assumption, which can limit their performance in some non-Gaussian settings. This paper reviews two nonlinear, non-Gaussian extensions of the Ensemble Kalman Filter: Gaussian anamorphosis (GA) methods and two-step updates, of which the rank histogram filter (RHF) is a prototypical example. GA-EnKF methods apply univariate transforms to the state and observation variables to make their distribution more Gaussian before applying an EnKF. The two-step methods use a scalar Bayesian update for the first step, followed by linear regression for the second step. The connection of the two-step framework to the full Bayesian problem is made, which opens the door to more advanced two-step methods in the full Bayesian setting. A new method for the first part of the two-step framework is proposed, with a similar form to the RHF but a different motivation, called the ‘improved RHF’ (iRHF). A suite of experiments with the Lorenz-‘96 model demonstrate situations where the GA-EnKF methods are similar to EnKF, and where they outperform EnKF. The experiments also strongly support the accuracy of the RHF and iRHF filters for nonlinear and non-Gaussian observations; these methods uniformly beat the EnKF and GA-EnKF methods in the experiments reported here. The new iRHF method is only more accurate than RHF at small ensemble sizes in the experiments reported here. PubDate: 2022-06-01
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Abstract: Abstract This work investigates an ensemble-based workflow to simultaneously handle generic, nonlinear equality and inequality constraints in reservoir data assimilation problems. The proposed workflow is built upon a recently proposed umbrella algorithm, called the generalized iterative ensemble smoother (GIES), and inherits the benefits of ensemble-based data assimilation algorithms in geoscience applications. Unlike the traditional ensemble assimilation algorithms, the proposed workflow admits cost functions beyond the form of nonlinear-least-squares, and has the potential to develop an infinite number of constrained assimilation algorithms. In the proposed workflow, we treat data assimilation with constraints as a constrained optimization problem. Instead of relying on a general-purpose numerical optimization algorithm to solve the constrained optimization problem, we derive an (approximate) closed form to iteratively update model variables, but without the need to explicitly linearize the constraint systems. The established model update formula bears similarities to that of an iterative ensemble smoother (IES). Therefore, in terms of theoretical analysis, it becomes relatively easy to transit from an ordinary IES to the proposed constrained assimilation algorithms, and in terms of practical implementation, it is also relatively straightforward to implement the proposed workflow for users who are familiar with the IES, or other conventional ensemble data assimilation algorithms like the ensemble Kalman filter (EnKF). Apart from the aforementioned features, we also develop efficient methods to handle two noticed issues that would be of practical importance for ensemble-based constrained assimilation algorithms. These issues include localization in the presence of constraints, and the (possible) high dimensionality induced by the constraint systems. We use one 2D and one 3D case studies to demonstrate the performance of the proposed workflow. In particular, the 3D example contains experiment settings close to those of real field case studies. In both case studies, the proposed workflow achieves better data assimilation performance in comparison to the choice of using an original IES algorithm. As such, the proposed workflow has the potential to further improve the efficacy of ensemble-based data assimilation in practical reservoir data assimilation problems. PubDate: 2022-06-01
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Abstract: Abstract The accuracy and the limits of validity of the discontinuous pressure model, which describes fluid flow inside a fracture using a subgrid scale approach, is assessed by comparing simulation results with those from direct simulation using Stokes flow. While the subgrid scale approach assumes a unidirectional flow, the Stokes model includes both velocity components. This is at the cost of meshing the interior of the fracture, which is here achieved through a spline-based mesh generation scheme. This scheme explicitly couples the spline representing the discontinuity to the fracture mesh and thereby alleviates the (re)meshing requirements for the interior of the fracture. The subgrid model and the direct simulation of Stokes flow approaches are compared by simulating a typical case containing a pressurised fracture, highlighting the advantages of using a subgrid model for the range in which its assumptions are valid, and showing its capabilities to accurately include the influence of the fracture on the porous material even outside this range. PubDate: 2022-06-01
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Abstract: Abstract In heavy oil displacement by fluid injection, severe instability can occur due to the adverse mobility ratio, gravity segregation or compositional effects. However, when estimating relative permeability, most analytical methods assume a stable front in the displacement, which may be highly erroneous when used with unstable displacements. Quite often in such cases, history matching using a high-resolution model is preferred, however, it is computationally inefficient or impractical in some cases. This work describes a relatively fast methodology for estimating relative permeability from displacement with instability and compositional effect. It involves defining a set of 2-dimensional (2D) coarse-grid models and the tuning of a three-parameter correlation. By this approach, an attempt is made to resolve the fine-scale information without direct solution of the global fine-scale problem. The results of the coarse-grid history matching corrected by the proposed approach in this study showed that the improved methodology is three times faster, and required less than half the memory of a high-resolution 2D model. PubDate: 2022-05-31
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Abstract: Abstract Well placement optimization is commonly performed using population-based global stochastic search algorithms. These optimizations are computationally expensive due to the large number of multiphase flow simulations that must be conducted. In this work, we present an optimization framework in which these simulations are performed with low-fidelity (LF) models. These LF models are constructed from the underlying high-fidelity (HF) geomodel using a global transmissibility upscaling procedure. Tree-based machine-learning methods, specifically random forest and light gradient boosting machine, are applied to estimate the error in objective function value (in this case net present value, NPV) associated with the LF models. In the offline (preprocessing) step, preliminary optimizations are performed using LF models, and a clustering procedure is applied to select a representative set of 100–150 well configurations to use for training. HF simulation is then performed for these configurations, and the tree-based models are trained using an appropriate set of features. In the online (runtime) step, optimization with LF models, with the machine-learning correction, is conducted. Differential evolution is used for all optimizations. Results are presented for two example cases involving the placement of vertical wells in 3D bimodal channelized geomodels. We compare the performance of our procedure to optimization using HF models. In the first case, 25 optimization runs are performed with both approaches. Our method provides an overall speedup factor of 46 relative to optimization using HF models, with the best-case NPV within 1% of the HF result. In the second case fewer HF optimization runs are conducted (consistent with actual practice), and the overall speedup factor with our approach is about 8. In this case, the best-case NPV from our procedure exceeds the HF result by 3.8%. PubDate: 2022-05-25
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Abstract: Abstract Thermal-hydro-mechanical (THM) coupled fracture propagation is common in underground engineering. Rock damage, as an inherent property of rock, significantly affects fracture propagation, but how it influences the THM coupled fracturing remains stubbornly unclear. A pore-scale THM coupling model is developed to study this problem, which combines the lattice Boltzmann method (LBM), the discrete element method (DEM), and rock damage development theory together for the first time. This model can more accurately calculate the exchanged THM information at the fluid-solid boundary and fluid conductivity dependent on fracture and rock damage. Based on the developed model, the synergistic effect of injected temperature difference (fluid temperature below rock temperature) and rock damage (characterized by the parameter “critical fracture energy”, abbreviated as “CFE”) on fracture propagation of shale are investigated particularly. It is found that: (1) the generation of branched cracks is closely related to the temperature response frontier, and the fracture process zone of single bond failure increases in higher CFE. (2) through the analysis of micro failure events, hydraulic fracturing is more pronounced in the low CFE, while thermal fracturing displays the opposite trend. The fluid conductivity of fractured rock increases with a higher injected temperature difference due to the more penetrated cracks and wider fracture aperture. However, this enhancement weakens when rock damage is significant. (3) in the multiple-layered rock with various CFEs, branched cracks propagating to adjacent layers are more difficult to form when the injection hole stays in the layer with significant rock damage than without rock damage. PubDate: 2022-05-16
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Abstract: Abstract Numerical simulation of surfactant flooding using conventional reservoir simulation models can lead to unreliable forecasts and bad decisions due to the appearance of numerical effects. The simulations give approximate solutions to systems of nonlinear partial differential equations describing the physical behavior of surfactant flooding by combining multiphase flow in porous media with surfactant transport. The approximations are made by discretization of time and space which can lead to spurious pulses or deviations in the model outcome. In this work, the black oil model was simulated using the decoupled implicit method for various conditions of reservoir scale models to investigate behavior in comparison with the analytical solution obtained from fractional flow theory. We investigated changes to cell size and time step as well as the properties of the surfactant and how it affects miscibility and flow. The main aim of this study was to understand pulse like behavior in the water bank, which we report for the first time, Our aim was to identify their cause and associated conditions. The pulses are induced by a sharp change in relative permeability as the interfacial tension changes. Pulses are diminished when adsorption is modeled, and ranged from 0.0002 kg/kg to 0.0005 kg/kg. The pulses are absent for high-resolution model of 5000 cells in x direction with a typical cell size as used in well-scale models. The growth or dampening of these pulses may vary from case to case but in this instance was a result of the combined impact of relative mobility, numerical dispersion, interfacial tension and miscibility. Oil recovery under the numerical problems reduced the performance of the flood, due to large amounts of pulses produced. Thus, it is important to improve existing models and use appropriate guidelines to stop oscillations and remove errors. PubDate: 2022-05-16
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Abstract: Abstract Artificial neural network-trained models were used to predict gas hydrate saturation distributions in permafrost-associated deposits in the Eileen Gas Hydrate Trend on the Alaska North Slope (ANS), USA and at the Mallik research site in the Beaufort-Mackenzie Basin, Northwest Territories, Canada. The database of Logging-While-Drilling (LWD) and wireline logs collected at five wells (Mount Elbert, Iġnik Sikumi, and Kuparuk 7–11–12 wells at ANS, plus 2L-38 and 5L-38 wells at the Mallik research site) includes more than 10,000 depth points, which were used for training, validation, and testing the machine learning (ML) models. Data used in training the ML models include the well logs of density, porosity, electrical resistivity, gamma radiation, and acoustic wave velocity measurements. Combinations of two or three out of these five well logs were found to reliably predict the gas hydrate saturation with accuracy varying between 80 and 90% when compared to the gas hydrate saturations derived from Nuclear Magnetic Resonance (NMR)-based technique. The ML models trained on data from three ANS wells achieved high fidelity predictions of gas hydrate saturation at the Mallik site. The results obtained in this study indicate that ML models trained on data from one geological basin can successfully predict key reservoir parameters for permafrost-associated gas hydrate accumulations within another basin. A generalized approach for selecting a well log combination that can improve model accuracy is discussed. Overall, the study outcome supports earlier work demonstrating that ML models trained on non-NMR well logs are a viable alternative to physics-driven methods for predicting gas hydrate saturations. PubDate: 2022-05-14
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Abstract: Abstract In order to study the efficiency of the various forms of trapping including mineral trapping scenarios for CO2 storage behavior in deep layers of porous media, highly nonlinear coupled diffusion-advection-reaction partial differential equations (PDEs) including kinetic and equilibrium reactions modeling the miscible multiphase multicomponent flow have to be solved. We apply the globally fully implicit PDE reduction method (PRM) developed 2007 by Kräutle and Knabner for one-phase flow, which was extended 2019 to the case of two-phase flow with a pure gas in the study of Brunner and Knabner. We extend the method to the case of an arbitrary number of gases in gaseous phase, because CO2 is not the only gas that threats the climate, and usually is accompanied by other climate killing gases. The application of the PRM leads to an equation system consisting of PDEs, ordinary differential equations, and algebraic equations. The Finite Element discretized / Finite Volume stabilized equations are separated into a local and a global system but nevertheless coupled by the resolution function and evaluated with the aid of a nested Newton solver, so our solver is fully global implicit. For the phase disappearance, we use persistent variables which lead to a semismooth formulation that is solved with a semismooth Newton method. We present scenarios of the injection of a mixture of various gases into deep layers, we investigate phase change effects in the context of various gases, and study the mineral trapping effects of the storage technique. The technical framework also applies to other fields such as nuclear waste storage or oil recovery. PubDate: 2022-04-18 DOI: 10.1007/s10596-022-10140-y
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Abstract: Abstract With the development of automatic measurement and data storage, vast quantities of data can be recorded and analyzed for heat transfer processes, which provides an opportunity to discover the transient heat transfer governing laws from the data. In this study, a machine learning-based sequential threshold ridge regression (STRidge) approach is applied to extract partial differential equations (PDEs) and tested on the heat conduction equation and conductive–convective heat transfer equation subjected to different boundary conditions, data volumes, and noise levels. Moreover, we studied the learning of governing equation of nonlinear transient heat transfer and used the improved STRidge with genetic algorithm to learn PDE with incomplete candidate library. The results showcase highly accurate identification of governing equations for heat transfer. And our results reveal the vast potential of the data-driven method in complex geothermal problems. PubDate: 2022-04-15 DOI: 10.1007/s10596-022-10145-7
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Abstract: Abstract Estimation of porosity at a millimeter scale would be an order of magnitude finer resolution than traditional logging techniques. This enables proper description of reservoirs with thin layers and fine scale heterogeneities. To achieve this, we propose an end-to-end convolutional neural network (CNN) regression model that automatically predicts continuous porosity at a millimeter scale resolution using two-dimensional whole core CT scan images. More specifically, a CNN regression model is trained to learn from routine core analysis (RCA) porosity measurements. To characterize the performance of such approach, we compare the performance of this model with two linear regression models trained to learn the relationship between the average attenuation and standard deviation of the same two-dimensional images and RCA porosity. Our investigations reveal that the linear models are outperformed by the CNN, indicating the capability of the CNN model in extracting textures that are important for porosity estimations. We compare the predicted porosity results against the total porosity logs calculated from the density log. The obtained results show that the predicted porosity values using the proposed CNN method are well correlated with the core plug measurements and the porosity log. More importantly, the proposed approach can provide accurate millimeter scale porosity estimations, while the total porosity log is averaged over an interval and thus do not show such fine scale variations. Thus, the proposed method can be employed to calibrate the porosity logs, thereby reducing the uncertainties associated with indirect calculations of the porosity from such logs. PubDate: 2022-04-14 DOI: 10.1007/s10596-022-10143-9
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Abstract: Abstract We investigate reactive flow and transport in evolving porous media. Solute species that are transported within the fluid phase are taking part in mineral precipitation and dissolution reactions for two competing mineral phases. The evolution of the three phases is not known a-priori but depends on the concentration of the dissolved solute species. To model the coupled behavior, phase-field and level-set models are formulated. These formulations are compared in three increasingly challenging setups including significant mineral overgrowth. Simulation outcomes are examined with respect to mineral volumes and surface areas as well as derived effective quantities such as diffusion and permeability tensors. In doing so, we extend the results of current benchmarks for mineral dissolution/precipitation at the pore-scale to the multiphasic solid case. Both approaches are found to be able to simulate the evolution of the three-phase system, but the phase-field model is influenced by curvature-driven motion. PubDate: 2022-04-14 DOI: 10.1007/s10596-022-10142-w
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Abstract: Abstract Random reconstruction of three-dimensional (3D) digital rocks from two-dimensional (2D) slices is crucial for elucidating the microstructure of rocks and its effects on pore-scale flow in terms of numerical modeling, since massive samples are usually required to handle intrinsic uncertainties. Despite remarkable advances achieved by traditional process-based methods, statistical approaches and recently famous deep learning-based models, few works have focused on producing several kinds of rocks with one trained model and allowing the reconstructed samples to approximately satisfy certain given properties, such as porosity. To fill this gap, we propose a new framework with deep learning, named RockGPT, which is composed of VQ-VAE and conditional GPT, to synthesize 3D samples based on a single 2D slice from the perspective of video generation. The VQ-VAE is utilized to compress high-dimensional input video, i.e., the sequence of continuous rock slices, to discrete latent codes and reconstruct them. In order to obtain diverse reconstructions, the discrete latent codes are modeled using conditional GPT in an autoregressive manner, while incorporating conditional information from a given slice, rock type, and porosity. We conduct two experiments on five kinds of rocks, and the results demonstrate that RockGPT can produce different kinds of rocks with a single model, and the porosities of reconstructed samples can distribute around specified targets with a narrow range. In a broader sense, through leveraging the proposed conditioning scheme, RockGPT constitutes an effective way to build a general model to produce multiple kinds of rocks simultaneously that also satisfy user-defined properties. PubDate: 2022-04-11 DOI: 10.1007/s10596-022-10144-8
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