Quantifying uncertainty using level sets with stochastic motion

Jef Caers Liang Yang
In many geosciences applications, predicting requires building 3D models that are complex and cumbersome. As a result, often a single model, or some small variation of it, is built. One of the main bottlenecks lies in the creation of a high-resolution grid (million to billion cells) to provide enough detail and to constrain to data. Uncertainty is provided by generating many realizations, as sampled from some posterior distribution. Built into this common notion of modeling lies a fundamental inefficiency: building many accurate high-resolution models, each one of them being a poor approximation of actual reality (the truth). In this research, we provide a fundamentally new view on the same problem. Consider some geological geometry (ore, reservoir, fault) whose uncertainty needs to be quantified based on geological understanding rules and data. Instead of modeling these geometries and their uncertainty with many high-resolution models drawn by Monte Carlo, we will model their uncertainty by means of motion (velocity), representing uncertainty by means of an 3D velocity, as a stochastic process. To model 3D surfaces (e.g. faults, horizons, bodies) and their uncertainty, we will integrate this stochastic velocity into the level set equation. Level sets are an ideal way to represent mathematically complex surfaces without explicit grid-representations. This idea is applied to uncertainty quantification for groundwater systems, mineral resources and oil/gas reservoirs.