Quantification of uncertainty in production injection forecasting is an important aspect of reservoir simulation studies. Conventional approaches include intrusive Galerkin-based methods (e.g., generalized polynomial chaos (gPC) and stochastic collocation (SC) methods) and non-intrusive Monte Carlo (MC) based methods. Nevertheless, the quantification is conducted in reformulations of the underlying stochastic PDEs with fixed well controls. If one wants to take various well control plans into account, expensive computations need to be repeated for each well design independently. In this project, we take advantages of the equation-free spirit of convolutional neural network (CNN) to overcome this challenge and thus achieve the flexibility of efficient uncertainty quantification with various well controls. We are interested in the development of surrogate models for uncertainty quantification and propagation in reservoir simulations using a deep convolutional encoder-decoder network as an analogue to the image-to-image regression tasks in computer science. First, a U-Net architecture is applied to replace conventional expensive deterministic PDE solver. Then we adopt the idea from shape-guided image generation using variational U-Net and design a new variational U-Net architecture for "control-guided" reservoir simulation. Backward propagation is learned in the network to extract the hidden physical quantities and then predict the future production by the learned forward propagation using the hidden variable with various well controls. Comparisons in computational efficiency are made between our proposed CNN approach and conventional MC approach. Significant improvements in computational speed with reasonable accuracy loss are observed in the numerical tests.
In CS231N class, we have learned two different approaches to image generation in the context of deep learning: Variational Auto-Encoder (VAE) @cite_16 and Generative Adversarial Networks (GAN) @cite_8 . In @cite_0 , a conditional U-Net for shape-guided image generation, conditioned on the output of a VAE for appearance is presented. The separation between shape and appearance is carefully modelled and thus an explicit representation of the appearance, which can be combined with new poses, is obtained. Motivated by this work, we established the analogue between shape-guided image and control-guided saturation map. Similarly, the hidden appearance in image is analogous to the underlying permeability map. Under this framework, the variational U-Net is trained to learn the underlying Darcy's law. We take advantages of the nature of CNN, allowing the freedom of saturation map generation under various well controls. Efficient uncertainty quantification tasks can be conducted then to provide valuable evaluations of potential injection plans.
@cite_0 @cite_16 @cite_8
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Deep generative models have demonstrated great performance in image synthesis. However, results deteriorate in case of spatial deformations, since they generate images of objects directly, rather than modeling the intricate interplay of their inherent shape and appearance. We present a conditional U-Net [30] for shape-guided image generation, conditioned on the output of a variational autoencoder for appearance. The approach is trained end-to-end on images, without requiring samples of the same object with varying pose or appearance. Experiments show that the model enables conditional image generation and transfer. Therefore, either shape or appearance can be retained from a query image, while freely altering the other. Moreover, appearance can be sampled due to its stochastic latent representation, while preserving shape. In quantitative and qualitative experiments on COCO [20], DeepFashion [21, 23], shoes [43], Market-1501 [47] and handbags [49] the approach demonstrates significant improvements over the state-of-the-art. We propose a new framework for estimating generative models via an adversarial process, in which we simultaneously train two models: a generative model G that captures the data distribution, and a discriminative model D that estimates the probability that a sample came from the training data rather than G. The training procedure for G is to maximize the probability of D making a mistake. This framework corresponds to a minimax two-player game. In the space of arbitrary functions G and D, a unique solution exists, with G recovering the training data distribution and D equal to ½ everywhere. In the case where G and D are defined by multilayer perceptrons, the entire system can be trained with backpropagation. There is no need for any Markov chains or unrolled approximate inference networks during either training or generation of samples. Experiments demonstrate the potential of the framework through qualitative and quantitative evaluation of the generated samples.