PhD Thesis (2022-2025)

Title:  Transdimensional inversion of flow data in geomodeling

Supervisors: Guillaume Caumon (GeoRessources, Université de Lorraine), Mustapha Zakari (GeoRessources, CNRS), Thomas Bodin (ENS Lyon, CNRS)

Inverse problem theory is a very powerful way to reduce subsurface uncertainty by updating earth model parameters to reflect some new information. Although the theory is very general, its application has mainly been considered globally. A notable exception is the use of adjoint techniques to efficiently identify where model changes will have an impact on a particular physical process. These methods have gained significant popularity in subsurface flow problems (e.g., Ackerer et al., 2014) and in seismology (Fichtner et al., 2006). However, these approaches mainly consider continuous model parameters. In geomodeling, however, some model components are discrete at the scale of concern (e.g., minerals, facies, fractures, layers), hence call for indicator (binary) variables or object-based parameterizations. Then, the number of parameters itself becomes an unknown of the inverse problem. To address this issue, random vector parameterization in conjunction with point processes have been proposed (Cherpeau et al., 2012). Transdimensional inversion (Bodin et al., 2009) has also been considered, and provides a rigorous way to deal with a variable number of model parameters. However, transdimensional methods so far have been applied to relatively simple model parameterizations such as Voronoi diagrams.
The first objective of this PhD is to consider more suitable geological parameterizations in transdimensional inversion, such as layer boundaries defined by implicit functions, faults and fractures (Wellmann & Caumon, 2018). For example, one could allow for subdividing or merging geological layers in a stratigraphic domain in a particular transdimensional iteration. Introducing or removing faults could also be considered. To measure the effect of this type of transition, we will consider the physical problem of fluid flow in layered porous media.
The second objective is to assess whether and how to best exploit the local aspect of model updating in transdimensional methods to reduce the computational burden. For this, a significant focus will be to consider the effect of topological changes in the geomodel by considering flow-based upscaling methods (Durlofsky, 2005; Nœtinger et al., 2005). In these approaches, machine learning may be considered to accelerate the exploration of the relationships between model space and data space.
This PhD project is connected to the ongoing PhD thesis of Capucine Legentil (2019-2022), who focuses on the computational geometry challenges of local tetrahedral mesh updating. However, the methodology will first be developed on two-dimensional (2D) models to facilitate prototyping and to minimize the computational costs.
Once the transdimensional approach has been demonstrated on 2D models, three-dimensional applications will be considered, considering for instance sub-seismic faults in the Volve hydrocarbon reservoir (Norwegian North Sea) or the regional geothermal system of Basse-Terre (Guadeloupe).

About
I am a third year PhD student graduate of Numerical Geology Master’s degree at Nancy school of Geology (Nancy, France), and Petroleum Engineering Master’s degree at the University of Pau and the Adour countries (Pau, France). I also had a professional experience as BRGM intern and employee in an SME about airborne geophysics acquisition and processing, and as TotalEnergies intern about flow simulation on unstructured meshes. I developed 3 years ago a C++ SKUA-GOCAD plugin for geosteering and model updating using a Bayesian framework. Currently, I am working for my PhD thesis on inverse problems and transdimensional Markov chain Monte Carlo applied on history matching. The purpose is to help reservoir engineers to get optimal petrophysical models that can be used as input of a flow simulation for hydrocarbon exploitation or CO2 storage.