On Bias Correction for Parameter Estimation Problems with Applications to Model Updating for Geosteering

Abstract

In the geosteering process, a local geomodel is continuously updated around the drill bit to guide the planning of the remaining well trajectory. It was shown that the ensemble-based data assimilation techniques were useful to perform the geomodel updating using directional resistivity data. In ensemble-based methods for parameter estimation, however, it is often assumed that the model error has zero mean and is uncorrelated in time. As a result, the model parameters will be over corrected when the model error has nonzero mean. There exist many studies on modeling and estimation of model error in state estimation problems using the ensemble-base methods. But very few studies have been on parameter estimation problems because it is often difficult to distinguish the effect of model error and the effect of poorly known model parameters. In this study, we investigate techniques to correct bias from model errors when the ensemble-base methods are used for parameter estimation, i.e. estimation of the geomodel for geosteering. The model parameters to be estimated include the depth of the top reservoir surface and reservoir thickness. We consider model error due to the bias in the resistivity model and due to small scale heterogeneities that are below model resolution. We show that when the model error contributes significantly to the data mismatch, explicit estimation of model errors is necessary to obtain a reasonable estimate of the reservoir boundaries. Typically a state augmentation approach is effective for joint model error and model parameter estimation. When the number of data is much larger than the number of model error parameters, a weighted least square approach is also reliable for model error estimation. The weighted least square approach does not require a prior probabilistic model for the model error parameters, but requires explicit knowledge of the data sensitivity to the model error parameters. The performance of both approaches depends on the selection of the model error parameters.