Gradient-based optimization of flow through porous media: Version 3

These notes form part of the course material for the MSc course AES1490 "Advanced Reservoir Simulation" which has been taught at TU Delft over the past decade as part of the track "Petroleum Engineering and Geosciences" in the two-year MSc program "Applied Earth Sciences".
The notes cover the gradient-based optimization of subsurface flow. In particular they treat optimization methods in which the gradient information is obtained with the aid of the adjoint method, which is, in essence, an efficient numerical implementation of implicit differentiation in a multivariate setting.
Chapter 1 reviews the basic concepts of multivariate optimization and demonsrates the equivalence of the Lagrange multiplier method for constrained optimization and the use of implicit differentiation to obtain gradients in the presence of constraints.
Chapter 2 introduces the use of Lagrange multipliers and implicit differentiation for the optimization of large-scale numerical systems with the adjoint method. In particular it addresses the optimization of oil recovery from subsurface reservoirs represented as reservoir simulation models, i.e. space- and time-discretized numerical representations of the nonlinear partial differential equations that govern multi-phase flow through porous media. It also covers the use of robust adjoint-based optimization to cope with the inherent uncertainty in subsurface flow models and addresses some numerical implementation aspects.
Chapter 3 gives a brief overview of various further topics related to gradient-based optimization of subsurface flow, such as closed-loop reservoir management and hierarchical optimization of short-term and long term reservoir performance.