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Register a new userDeep-learning-based surrogate flow modeling and geological parameterization for data assimilation in 3D subsurface flow
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Data assimilation in subsurface flow 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 model for two-phase flow in 3D subsurface formations. This surrogate model, a 3D recurrent residual U-Net (referred to as recurrent R-U-Net), consists of 3D convolutional and recurrent (convLSTM) neural networks, designed to capture the spatial-temporal information associated with dynamic subsurface flow systems. A CNN-PCA procedure (convolutional neural network post-processing of principal component analysis) for parameterizing complex 3D geomodels is also described. This approach represents a simplified version of a recently developed supervised-learning-based CNN-PCA framework. The recurrent R-U-Net is trained on the simulated dynamic 3D saturation and pressure fields for a set of random `channelized geomodels (generated using 3D CNN-PCA). Detailed flow predictions demonstrate that the recurrent R-U-Net surrogate model provides accurate results for dynamic states and well responses for new geological realizations, along with accurate flow statistics for an ensemble of new geomodels. The 3D recurrent R-U-Net and CNN-PCA procedures are then used in combination for a challenging data assimilation problem involving a channelized system. Two different algorithms, namely rejection sampling and an ensemble-based method, are successfully applied. The overall methodology described in this paper may enable the assessment and refinement of data assimilation procedures for a range of realistic and challenging subsurface flow problems.
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A new low-dimensional parameterization based on principal component analysis (PCA) and convolutional neural networks (CNN) is developed to represent complex geological models. The CNN-PCA method is inspired by recent developments in computer vision using deep learning. CNN-PCA can be viewed as a generalization of an existing optimization-based PCA (O-PCA) method. Both CNN-PCA and O-PCA entail post-processing a PCA model to better honor complex geological features. In CNN-PCA, rather than use a histogram-based regularization as in O-PCA, a new regularization involving a set of metrics for multipoint statistics is introduced. The metrics are based on summary statistics of the nonlinear filter responses of geological models to a pre-trained deep CNN. In addition, in the CNN-PCA formulation presented here, a convolutional neural network is trained as an explicit transform function that can post-process PCA models quickly. CNN-PCA is shown to provide both unconditional and conditional realizations that honor the geological features present in reference SGeMS geostatistical realizations for a binary channelized system. Flow statistics obtained through simulation of random CNN-PCA models closely match results for random SGeMS models for a demanding case in which O-PCA models lead to significant discrepancies. Results for history matching are also presented. In this assessment CNN-PCA is applied with derivative-free optimization, and a subspace randomized maximum likelihood method is used to provide multiple posterior models. Data assimilation and significant uncertainty reduction are achieved for existing wells, and physically reasonable predictions are also obtained for new wells. Finally, the CNN-PCA method is extended to a more complex non-stationary bimodal deltaic fan system, and is shown to provide high-quality realizations for this challenging example.
Reduced Order Modeling (ROM) for engineering applications has been a major research focus in the past few decades due to the unprecedented physical insight into turbulence offered by high-fidelity CFD. The primary goal of a ROM is to model the key physics/features of a flow-field without computing the full Navier-Stokes (NS) equations. This is accomplished by projecting the high-dimensional dynamics to a low-dimensional subspace, typically utilizing dimensionality reduction techniques like Proper Orthogonal Decomposition (POD), coupled with Galerkin projection. In this work, we demonstrate a deep learning based approach to build a ROM using the POD basis of canonical DNS datasets, for turbulent flow control applications. We find that a type of Recurrent Neural Network, the Long Short Term Memory (LSTM) which has been primarily utilized for problems like speech modeling and language translation, shows attractive potential in modeling temporal dynamics of turbulence. Additionally, we introduce the Hurst Exponent as a tool to study LSTM behavior for non-stationary data, and uncover useful characteristics that may aid ROM development for a variety of applications.
We consider the use of probabilistic neural networks for fluid flow {surrogate modeling} and data recovery. This framework is constructed by assuming that the target variables are sampled from a Gaussian distribution conditioned on the inputs. Consequently, the overall formulation sets up a procedure to predict the hyperparameters of this distribution which are then used to compute an objective function given training data. We demonstrate that this framework has the ability to provide for prediction confidence intervals based on the assumption of a probabilistic posterior, given an appropriate model architecture and adequate training data. The applicability of the present framework to cases with noisy measurements and limited observations is also assessed. To demonstrate the capabilities of this framework, we consider canonical regression problems of fluid dynamics from the viewpoint of reduced-order modeling and spatial data recovery for four canonical data sets. The examples considered in this study arise from (1) the shallow water equations, (2) a two-dimensional cylinder flow, (3) the wake of NACA0012 airfoil with a Gurney flap, and (4) the NOAA sea surface temperature data set. The present results indicate that the probabilistic neural network not only produces a machine-learning-based fluid flow {surrogate} model but also systematically quantifies the uncertainty therein to assist with model interpretability.
Sensitivity analysis plays an important role in searching for constitutive parameters (e.g. permeability) subsurface flow simulations. The mathematics behind is to solve a dynamic constrained optimization problem. Traditional methods like finite difference and forward sensitivity analysis require computational cost that increases linearly with the number of parameters times number of cost functions. Discrete adjoint sensitivity analysis (SA) is gaining popularity due to its computational efficiency. This algorithm requires a forward run followed by a backward run who involves integrating adjoint equation backward in time. This was done by doing one forward solve and store the snapshot by checkpointing. Using the checkpoint data, the adjoint equation is numerically integrated. The computational cost of this algorithm only depends on the number of cost functions and does not depend on the number of parameters. The algorithm is highly powerful when the parameter space is large, and in our case of heterogeneous permeability the number of parameters is proportional to the number of grid cells. The aim of this project is to implement the discrete sensitivity analysis method in parallel to solve realistic subsurface problems. To achieve this goal, we propose to implement the algorithm in parallel using data structures such as TSAdjoint and TAO. This paper dealt with large-scale subsurface flow inversion problem with discrete adjoint method. This method can effectively reduce the computational cost in sensitivity analysis.
A new kinetic model for multiphase flow was presented under the framework of the discrete Boltzmann method (DBM). Significantly different from the previous DBM, a bottom-up approach was adopted in this model. The effects of molecular size and repulsion potential were described by the Enskog collision model; the attraction potential was obtained through the mean-field approximation method. The molecular interactions, which result in the non-ideal equation of state and surface tension, were directly introduced as an external force term. Several typical benchmark problems, including Couette flow, two-phase coexistence curve, the Laplace law, phase separation, and the collision of two droplets, were simulated to verify the model. Especially, for two types of droplet collisions, the strengths of two non-equilibrium effects, $bar{D}_2^*$ and $bar{D}_3^*$, defined through the second and third order non-conserved kinetic moments of $(f - f ^{eq})$, are comparatively investigated, where $f$ ($f^{eq}$) is the (equilibrium) distribution function. It is interesting to find that during the collision process, $bar{D}_2^*$ is always significantly larger than $bar{D}_3^*$, $bar{D}_2^*$ can be used to identify the different stages of the collision process and to distinguish different types of collisions. The modeling method can be directly extended to a higher-order model for the case where the non-equilibrium effect is strong, and the linear constitutive law of viscous stress is no longer valid.
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