Deep-learning-based surrogate modeling of flow and coupled flow-geomechanics for data assimilation in subsurface systems
Keywords: Energy Resources Engineering.

Data assimilation in subsurface systems is challenging due to the large number of flow simulations often required, and by the need to preserve geological realism in the calibrated (posterior) models. In this work we present a deep-learning-based surrogate-modeling framework, referred to as recurrent R-U-Net, for flow or coupled flow and geomechanics in subsurface formations. The recurrent R-U-Net consists of convolutional and recurrent (convLSTM) neural networks, designed to capture the spatial-temporal information associated with subsurface flow dynamics. The recurrent R-U-Net is trained on the simulated state fields for a set of O(1000) random geomodels. After training, the surrogate model provides very fast predictions of dynamic states (pressure, saturation and displacement when geomechanics is considered) and well responses for new geological realizations. The recurrent R-U-Net surrogate model is used for history matching in conjunction with CNN-PCA (convolutional neural network - principal component analysis), a recently developed deep-learning-based geological parameterization procedure. CNN-PCA preserves complex geological features by post-processing the PCA representation through use of a transform net. The recurrent R-U-Net surrogate model is first developed for 2D oil-water systems. Training samples comprise global pressure and saturation maps, at 10 time steps, generated by performing high-fidelity flow simulation for 1500 channelized geomodels defined on 80x80 grids. The training time for each of the (pressure and saturation) 2D recurrent R-U-Nets is 80 minutes on a Nvidia Tesla V100 GPU. After training, the recurrent R-U-Net provides predictions in 0.01 seconds, which represents a speedup of a factor of 1000 relative to high-fidelity simulation. The recurrent R-U-Net surrogate model is shown to be capable of predicting accurate dynamic 2D pressure and saturation fields and well rates for new geological realizations consistent with those used for training. Assessments demonstrating high surrogate-model accuracy are presented for an individual geological realization and for an ensemble of 500 test geomodels. The surrogate model is then used for history matching in a 2D channelized system. An optimization-based history matching procedure, randomized maximum likelihood with mesh adaptive direct search (MADS-RML), is applied. The overall approach provides substantial reduction in prediction uncertainty. High-fidelity numerical simulation results for the posterior geomodels (generated by the surrogate-based data assimilation procedure in combination with CNN-PCA) are shown to be in essential agreement with the recurrent R-U-Net predictions. Next, the 2D recurrent R-U-Net surrogate model is extended to handle 3D systems. This requires, in addition to the low-level implementation of some 3D network modules, the development of a 3D multiblock well model. The 3D recurrent R-U-Net is trained on 3D saturation and pressure fields for a set of 2500 channelized geomodels defined on 80x80x20 grids (a total of 128,000 cells). About 15 hours of training time are required for each of the 3D recurrent R-U-Nets on a Nvidia Tesla V100 GPU. The trained surrogate provides a single prediction in less than 0.04 seconds, while the high-fidelity simulations require about 7 minutes. Detailed flow predictions demonstrate that this recurrent R-U-Net surrogate model again provides accurate results for dynamic states and well responses for new geological realizations. Three different history matching procedures are assessed, with the 3D recurrent R-U-Net used for flow prediction and CNN-PCA applied for geomodel parameterization. The three methods are rejection sampling (RS), MADS-RML, and ensemble smoother with multiple data assimilation (ES-MDA). RS results provide the reference against which MADS-RML and ES-MDA posterior predictions are evaluated. We find that both MADS-RML and ES-MDA provide history matching results in general agreement with those from RS. MADS-RML is more accurate, however, and ES-MDA can display significant error in some quantities. Assessments of ES-MDA sensitivity to data error and the number of assimilation steps are also performed. Finally, we extend the 3D recurrent R-U-Net framework to treat coupled flow and geomechanics in CO2 storage settings. The problem domain for the high-fidelity solution includes the storage aquifer, a large surrounding region, overburden (which extends up to the Earth's surface), and bedrock. The full model is defined on a 60x60x37 grid, while the storage aquifer is represented by 40x40x12 blocks. The multi-Gaussian porosity and permeability fields in the storage aquifer are considered to be uncertain. Results from a set of 2000 high-fidelity full-order simulations provide the training data. An advantage of the 3D recurrent R-U-Net is that it can be trained to predict only the quantities of interest in particular domains. Here it is trained to provide pressure and saturation in the storage aquifer and vertical displacement at the Earth's surface. Five hours of training time are required for each of the three networks (corresponding to the three state variables) needed for this case. A single high-fidelity simulation takes about 0.8 hours of parallel computation (on 32 cores), while the trained surrogate provides predictions in less than 0.01 seconds. The storage aquifer states and the 2D surface displacement maps provided by the surrogate model display a high degree of accuracy, for both individual realizations and ensemble statistics. The 3D recurrent R-U-Net surrogate model is then applied with a rejection sampling procedure for history matching. The observations consist of a small number of surface displacement measurements. Significant uncertainty reduction in surface displacement and pressure buildup at the caprock is achieved. The history matching computations performed for this example would not be feasible using high-fidelity simulation