Multi-Level Inversion Based on Mesh Decoupling

Abstract

Understanding the permeability of the subsurface is a crucial step to simulate fluid flow in the subsurface. A parameter estimation problem for the flow equations can be solved to find the permeability. The robust identification of material parameters remains a significant challenge. In classical approaches, the non-linear least-squares problem is formulated as a non-linear optimization problem in which the partial differential equation governing the permeability field acts as a constraint. These approaches lead to large scale problems and are, therefore, computationally challenging. This thesis proposes a new approach based on mesh decoupling of state and design variables. This approach allows treating the design variables on various scales of resolution without comprising the accuracy of the state and adjoint solver. The method is implemented on a one-dimensional and two-dimensional Poisson problem taken from the literature. The first numerical results show that the multilevel approach is able to accelerate the initial stages of the search procedure significantly. Without their guidance, supervision, insights, and help, I would not be able to do this work. I thank them for believing in me and willing to accept the challenge of combining pure mathematical research with petroleum engineering aspects to explore new ideas that can benefit both aspects of science. During the work, both have let me the flexibility to test my thoughts, some time to fail, all to be a better student and researcher for later stages. This approach has taught me a lot, and I thank them for that.I thank Dr. Ir. Femke Vossepoel for her interest in my work and for being a committee member for my thesis defense. During my studies, the courses and interactions with professor Vossepoel have encouraged me to find interest in geostatistics and different interpolation methods.I want to thank the TU Delft university staff for the different courses, classes, presentations, and fieldwork. All that prepared me to be a better scientific researcher, petroleum engineer, and a co-worker.Last but not least, I would like to thank my dear family and especially my dear mother for the constant support in the challenging times and for encouraging me to go abroad and pursue an MSc. v Contents List of Figures ix List of Tables xi 7 ConclusionBibliography