## IBM Research - Ireland Internship Project: Cognitive inverse modelling and its application to hydraulic diffusivity inversion - overview

**Abstract:**

** **Extraction of fluids from porous media is critical for both petroleum resource management and supplying drinking water to a global population. Sparse sampling through wells and the heterogeneity of geologic formations make inverse estimation of the permeability field difficult and an under-determined problem. In this project we propose a cognitive strategy for estimating the heterogeneous diffusion coefficient – permeability field – of a 2D confined aquifer. The aquifer is modeled using a 2D linear Darcy equation within a relatively simple geometry. The permeability field is to be inferred from given measurements of water levels from a network of wells.

The key idea behind this project is to use state of the art spectral cluster algorithms to learn clusters in the input space, e.g. massive amounts of geo-physically plausible samples of permeability fields, from the data available in the output space, e.g. observed water heights at well locations. The expected outcome is:

- learning a number and centers of the clusters in the input space
- using the cluster central points as starting points for the subsequent inversion
- uncertainty quantification

Uncertainty quantification is a second important point in the project: clusters are similar in terms of the misfit function, hence we can expect that, after the inversion the members of the clusters will allow one to quantify the misfit variance associated to the optimized cluster centers. The student will work with IBM Research staff to develop/implement/test a cognitive inversion prototype for 2D Darcy flows. The work-flow of the project will be as follows:

- use existing conditional gaussian samplers to generate samples of the permeability field
- compute corresponding solutions of Darcy equation (existing simulator)
- creating the similarity matrix and learning the clusters (existing Matlab prototype)
- inversion (a combination of L-BFGS method with numerical gradient for the Darcy equation)
- sampling from the inverted permeability field (to be developed)

The final goal of the project is to develop/test the prototype and implement it as a C++ library of a general purpose.

**Ideal intern skills: **PhD student in Applied Mathematics, Control Theory, or Civil Engineering. Background in numerical analysis, Experience in C++ programming.