Learning generative models for geostatistical facies 1 simulation based on a single training image

7 Characterization of subsurface reservoirs often requires geological facies models to identify 8 areas with favorable rock properties. With the development of computing powers, deep 9 learning approaches, such as the generative adversarial networks (GANs), became widely used 10 for simulating complex geological models. However, training of the GANs typically requires 11 a large quantity of training data for updating neural parameters. This process is generally done 12 using traditional geostatistical methods based on multiple-point statistics or process-based 13 models to build the training data. In this study, we propose to train the GANs using one single 14 training image, a conceptual model from which the statistics of the geological patterns can be 15 extracted. The training image is first down-sampled to different scales, and the generator and 16 the discriminator are trained alternately for each scale. The training process is implemented 17 from the coarsest to the finest scale to learn the spatial statistics from the training image 18 progressively. We apply the proposed GANs to simulate the 2D Lena river delta and 3D 19 Descalvado aquifer analog model, in which complex geological patterns and structures from 20 the training image are successfully learned and reproduced by GANs. The gradual deformation 21 method is further applied to iteratively calibrate the random realizations by the generator to 22