Efficient Inverse Modeling Framework for Energy Transition Applications Using Operator-Based Linearization and Adjoint Gradients
Authors: Tian Xiaoming , Delft TU , Voskov Denis , Delft TU 
Published: March 2023

Abstract
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.