In the original published article, one of the affiliation details for corresponding author "Denis Voskov" were incorrectly given as "Department of Energy Resources Engineering, Stanford University, CA, USA" but should have been "Department of Energy Science and Engineering, Stanford University, CA, USA". It is updated in this correction. The original article has been corrected.

In this study, we conduct a comprehensive history matching study for the FluidFlower benchmark model. This benchmark was prepared and organized by the University of Bergen, the University of Stuttgart, and Massachusetts Institute of Technology, for promoting understanding of the complex physics of geological carbon storage (GCS) through in-house experiments and numerical simulations. This paper synthesizes the experiences of history matching the benchmark data encountered by the Delft-DARTS and CSIRO participants. History matching is first performed based on a low-dimensional-zonated structured model using a simple Poisson-like solver. The permeability of six facies and two faults is inferred in this stage to match the digitized concentration data. The history matching is then further enhanced to richer spatial and physical models to capture the spatial variation of permeability and buoyancy effects, using an unstructured grid. Efficient adjoint methods are used to evaluate the gradient used in the optimization of data misfits or equivalent Bayesian log-likelihoods. With efficient optimization methods available for both maximum a posteriori model inference and Randomized Maximum Likelihood methods for model uncertainty, we perform history matching using both binary and continuous concentration observations. The results show that the tracer plumes in the enriched model match the experimental plumes more accurately compared with the results from the parsimonious-zonated model. The history matching results based on the concentration observations provide more similar plume shapes compared with the case based on the binary observations. The permeability difference between the model before and after history matching reveals that the tracer plume zone and the high permeable zone are the regions of high sensitivity in terms of data misfit between the model response and observations. Surprisingly, CO ₂ concentration plume forecasts based on these history-matched models were not especially sensitive to the improvements observed in the enhanced model.

In this study, we present an efficient and flexible adjoint-based framework for history matching and forecasting geothermal energy extraction at a large scale. In this framework, we applied the Principal Component Analysis to reduce the parameter space for representing the complex geological model. The adjoint method is implemented for gradient calculation to speed up the history-matching iteration process. Operator-based linearization (OBL) used in this framework makes the calculation of the physical state and its derivatives very efficient and facilitates the matrix assembly in the adjoint method. This study primarily focuses on history matching based on combined observation of well production and in-situ electromagnetic measurements to predict the temperature front. However, different types of misfit terms can be added to the objective function based on practical considerations. For example, our history-matching case studies include model misfit terms applied for regularization purposes. The measurement data is extracted from the true model, and realistic measurement errors are considered. Also, in this work, we propose an optimal weighting strategy for the terms of the objective function to balance their sensitivity with respect to the model control variables. The high efficiency of the framework is demonstrated for the geothermal doublet model implemented at the heterogeneous reservoir with multiple realizations. The framework allows for generating posterior Randomized Maximum Likelihood (RML) estimates of the entire ensemble of the realizations with a reasonable computational cost. Results show that the framework can achieve reliable history-matching results based on the doublets production data and the reservoir electromagnetic measurement.

Successful deployment of geological carbon storage (GCS) requires an extensive use of reservoir simulators for screening, ranking and optimization of storage sites. However, the time scales of GCS are such that no sufficient long-term data is available yet to validate the simulators against. As a consequence, there is currently no solid basis for assessing the quality with which the dynamics of large-scale GCS operations can be forecasted. To meet this knowledge gap, we have conducted a major GCS validation benchmark study. To achieve reasonable time scales, a laboratory-size geological storage formation was constructed (the “FluidFlower”), forming the basis for both the experimental and computational work. A validation experiment consisting of repeated GCS operations was conducted in the FluidFlower, providing what we define as the true physical dynamics for this system. Nine different research groups from around the world provided forecasts, both individually and collaboratively, based on a detailed physical and petrophysical characterization of the FluidFlower sands. The major contribution of this paper is a report and discussion of the results of the validation benchmark study, complemented by a description of the benchmarking process and the participating computational models. The forecasts from the participating groups are compared to each other and to the experimental data by means of various indicative qualitative and quantitative measures. By this, we provide a detailed assessment of the capabilities of reservoir simulators and their users to capture both the injection and post-injection dynamics of the GCS operations.

In this work, we describe our decisions made to perform the FluidFlower simulation study and discuss various aspects of the benchmark that are different from our usual subsurface simulation practice. We will discuss the impact of various modeling choices on the outcomes of the simulation models, such as gridding, discretization, and solver strategies, and the lessons learned, taking into account the different conditions of the FluidFlower study compared to conditions commonly dealt with in subsurface simulation. We will start with a brief description of the DARTS framework utilized for compositional simulation, the thermodynamic and physical modeling related to the atmospheric CO2 -brine system, and the modeling workflow used in our benchmark submission. Additionally, we describe a custom nonlinear solver developed for the atmospheric benchmark conditions to improve convergence including the linear solver strategy since our default two-stage preconditioner does not perform effectively. To make meaningful comparisons between each of the modeling choices, we define a baseline model which is a simplified version of our setup in the main FluidFlower benchmark. The baseline model is then used to study the effect of Cartesian and unstructured meshes and a two-point flux approximation compared with a multi-point flux approximation for capturing the physics at play. We conclude our work with lessons learned and future recommendations.

In this paper, we present an efficient inverse modeling framework for energy transition applications. The key feature of this framework is a combination of adjoint gradients and Operator-based Linearization (OBL) technique to achieve high efficiency in inverse modeling based on forward simulations. This framework allows conducting the history matching of practical industrial applications using the gradient descent method with considerable model control variables in a reasonable time. Generally, the inverse modeling of industrial applications involves large amounts of gradient calculations in algorithms based on gradient descent. In this study, we analytically compute the gradient using the adjoint gradient method as an alternative to the widely used numerical gradient method where many time-consuming forward simulation runs are needed. In the adjoint gradient approach, the objective function is linearly combined with the governing equation by introducing a Lagrange multiplier. That allows for finding the analytical gradient in a backward manner. The developed adjoint gradient method takes full advantage of the OBL efficiency and flexibility when assembling the Jacobian and some relevant derivatives. We demonstrate the applications of the proposed inverse modeling framework to different energy transition applications, including petroleum production, extraction of geothermal energy, and CO₂ storage. We demonstrate various treatments of objective function definitions, well controls, and measurement errors for these industrial applications. For petroleum production, the proposed framework is tested on the multiphase multi-component flow problem, which is illustrated by an example of data-driven Discrete Well Affinity model. For this application, only production data is considered. The geothermal problem involves an additional energy balance equation and various property calculations for water and steam. In this application, together with the production data, additional electromagnetic monitoring is used in the history matching process. The results show that electromagnetic monitoring significantly improves the inversion process. We conclude the description of our framework with an application relevant to CO₂ sequestration process. The CO₂ storage modeling is complicated due to the complex physical phenomena to be considered. In this application, tracer data are used as an additional observation, which allows considering uncertainties in the dynamics of CO₂. In this study, the adjoint gradient method is specially designed and customized for OBL infrastructure of the Delft Advanced Research Terra Simulator (DARTS). This allows us to design the general-purpose inversion module with efficient gradient computation, while most existing simulation platforms lack this capability. Based on the multiphysics simulation engine in DARTS, the various observation information can be combined in the proposed framework. This allows us to solve the general-purpose inverse modeling problems for most energy transition applications.

In this paper, we describe adjoint gradient formulation for the Operator-Based Linearization modeling approach. Adjoint gradients are implemented in Delft Advanced Research Terra Simulator (DARTS) framework and applied for history matching using a proxy methodology. Due to the application of adjoint gradients, the computational efficiency of the discrete well affinity (DiWA) proxy model for production forecast is significantly improved. That allows us to derive several important extensions. The proxy methodology is further extended and validated for 3D three-phase black-oil problems. The results show that the gradient-based regression can provide good history matching and reconstruct a true petrophysical characterization when the initial guess is generated based on highly reliable geological information. For cases with a limited or not sufficient geological characterization, an efficient stochastic application of DiWA proxy model is proposed. This approach consists of massive sampling procedures for collecting different realizations based on high-fidelity statistics with filtering. These realizations are generated stochastically because they are not conditioned to any production information but the basic geological statistics of the reservoir. The trained DiWA proxy model demonstrates a small deviation between the model response and the observation data. When applying the refined DiWA model for the training, the error between the model response and observation data can be further reduced. The forecast based on the trained model has slightly larger variability but the deviation is still reasonable. The enhanced DiWA methodology presents an efficient and robust technique for creating an ensemble of stochastic proxy models that can be used in production forecast, flow diagnostic, and optimization.

A physics-based data-driven model is proposed for forecasting of subsurface energy production. The model fully relies on production data and does not require any in-depth knowledge of reservoir geology or governing physics. In the proposed approach, we use the Delft Advanced Reservoir Terra Simulator (DARTS) as a workhorse for data-driven simulation. DARTS uses an operator-based linearization technique that exploits an abstract interpretation of physics benefiting computational performance. The physics-based data-driven model is trained to fit data increasing the fidelity of the model forecast and reflecting significant changes in reservoir dynamics or physics over its history. The model is examined and validated for both synthetic and real field production data. We demonstrate that the developed approach is capable of providing accurate and reliable production forecast on a daily basis, even if the exact geological information is not available.