We present an efficient adjoint model based on the deep‐learning surrogate to address high‐dimensional inverse modeling with an application to subsurface transport. The proposed method provides a completely code nonintrusive and computationally feasible way to approximate the model derivatives, which subsequently can be used to derive gradients for inverse modeling. This conceptual deep‐learning framework, that is, an architecture of deep convolutional neural network through combining autoencoder and autoregressive structure, efficiently produces an analogously analytical adjoint with the help of auto‐differentiation module in the popular deep‐learning packages. We intentionally retain training data at the specific time instances where the measurements are taken, the storage of the intermediate states and computation of their adjoint, therefore, are completely avoided. This proposed adjoint state method is tested on a synthetic two‐dimensional model for parameter estimations. The preliminary results reveal the feasibility of the proposed adjoint state method in term of computational efficiency and programming flexibility.

This paper presents a non-intrusive subdomain POD-TPWL (SD POD-TPWL) for reservoir history matching through integrating domain decomposition (DD), proper orthogonal decomposition (POD), radial basis function (RBF) interpolation, and the trajectory piecewise linearization (TPWL). It is an efficient approach for model reduction and linearization of general non-linear time-dependent dynamical systems without accessing to the legacy source code. In the subdomain POD-TPWL algorithm, firstly, a sequence of snapshots over the entire computational domain is saved and then partitioned into subdomains. From the local sequence of snapshots over each subdomain, a number of local basis vectors is formed using POD, and then the RBF interpolation is used to estimate the derivative matrices for each subdomain. Finally, those derivative matrices are substituted into a POD-TPWL algorithm to form a reduced-order linear model in each subdomain. This reduced-order linear model makes the implementation of the adjoint easy and results in an efficient adjoint-based parameter estimation procedure. Comparisons with the classic finite-difference-based history matching show that our proposed subdomain POD-TPWL approach is obtaining comparable results. The number of full-order model simulations required is roughly 2–3 times the number of uncertain parameters. Using different background parameter realizations, our approach efficiently generates an ensemble of calibrated models without additional full-order model simulations.

As an extension of our previous research on deep‐learning‐based adjoint‐state approach (Xiao, Deng, & Wang, 2021, https://doi.org/10.1029/2020wr027400), we present a two‐level hybrid neural network architecture to efficiently derive a model‐reduced adjoint‐based optimization workflow with a large‐scale inverse modeling example in geological carbon‐storage applications. The first level employs a nonlinear dimensionality reduction technique, that is, deep convolutional autoencoder, owing to its high scalability and the availability of high‐performance deep‐learning libraries for flexible implementation. Analogous to the reduced‐order linear model, the second level emulates a neural‐network‐powered linear transition unit based on the reduced subspace. This hybrid framework provides a readily available model‐reduced adjoint derivation with negligible computational cost and computer storage. Once the gradient is obtained in reduced space, it is correspondingly projected back to full space for the inverse modeling. The performance of the method is verified by estimating the space‐dependent stochastic permeability field using two geological carbon storage (GCS) process‐based multiphase flow models with an increasing complexity. The experimental results indicate that the surrogate model can accurately characterize the spatial‐temporal evolution of the CO₂ saturation fields using a relatively small number of training data. Although the proposed inversion workflow achieves comparable results similar to the conventional finite‐difference method, it is faster than high‐fidelity model simulations in several orders of magnitude; hence, it will be a beneficial simulation and modeling tool for engineers to manage the uncertainty of GCS processing.

Integrated optimization of well placement and hydraulic fracture parameters in naturally fractured shale gas reservoirs is of significance to enhance unconventional hydrocarbon energy resources in the oil and gas industry. However, the optimization task usually presents intensive computation-cost due to numerous high-fidelity model simulations, particularly for field-scale application. We present an efficient multi-objective optimization framework supported by a novel hierarchical surrogate-assisted evolutionary algorithm and multi-fidelity modeling technology. In the proposed framework, both the net present value (NPV) and cumulative gas production (CGP) are regarded as the bi-objective functions that need to be optimized. The hierarchical surrogate-assisted evolutionary algorithm employs a novel multi-fidelity particle-swarm optimization of a global–local hybridization searching strategy where the low-fidelity surrogate model is capable of exploring the populations globally, while the high-fidelity models update the current populations and thus generate the next generations locally. The multi-layer perception is chosen as a surrogate model in this study. The performance of our proposed hierarchical surrogate-assisted global optimization approach is verified to optimize the well placement and hydraulic fracture parameters on a hydraulically fractured shale gas reservoir. The proposed surrogate model can obtain both the NPV and CPG with satisfactory accuracy with only 500 training samples. The surrogate model significantly contributes to the convergent performance of multi-objective optimization algorithm.