Assimilating spatially dense data for subsurface applications—balancing information and degrees of freedom

Abstract

The degrees of freedom (DOF) in standard ensemble-based data assimilation is limited by the ensemble size. Successful assimilation of a data set with large information content (IC) therefore requires that the DOF is sufficiently large. A too small number of DOF with respect to the IC may result in ensemble collapse, or at least in unwarranted uncertainty reduction in the estimation results. In this situation, one has two options to restore a proper balance between the DOF and the IC: to increase the DOF or to decrease the IC. Spatially dense data sets typically have a large IC. Within subsurface applications, inverted time-lapse seismic data used for reservoir history matching is an example of a spatially dense data set. Such data are considered to have great potential due to their large IC, but they also contain errors that are challenging to characterize properly. The computational cost of running the forward simulations for reservoir history matching with any kind of data is large for field cases, such that a moderately large ensemble size is standard. Realization of the potential in seismic data for ensemble-based reservoir history matching is therefore not straightforward, not only because of the unknown character of the associated data errors, but also due to the imbalance between a large IC and a too small number of DOF. Distance-based localization is often applied to increase the DOF but is example specific and involves cumbersome implementation work. We consider methods to obtain a proper balance between the IC and the DOF when assimilating inverted seismic data for reservoir history matching. To decrease the IC, we consider three ways to reduce the influence of the data space; subspace pseudo inversion, data coarsening, and a novel way of performing front extraction. To increase the DOF, we consider coarse-scale simulation, which allows for an increase in the DOF by increasing the ensemble size without increasing the total computational cost. We also consider a combination of decreasing the IC and increasing the DOF by proposing a novel method consisting of a combination of data coarsening and coarse-scale simulation. The methods were compared on one small and one moderately large example with seismic bulk-velocity fields at four assimilation times as data. The size of the examples allows for calculation of a reference solution obtained with standard ensemble-based data assimilation methodology and an unrealistically large ensemble size. With the reference solution as the yardstick with which the quality of other methods are measured, we find that the novel method combining data coarsening and coarse-scale simulations gave the best results. With very restricted computational resources available, this was the only method that gave satisfactory results.