Hidden Information in Ill-posed Inverse Problems

Abstract

SUMMARYIt is well known that parameter updating of large-scale numerical reservoir flow models (a.k.a. 'computer assisted history matching') is an ill-posed inverse problem. Typically the number of uncertain parameters in a reservoir flow model is very large whereas the available information for estimating these parameters is limited. The classic solution to this problem is to regularize the unknowns, e.g. by penalizing deviations from a prior model. Attempts to estimate all uncertain parameters from production data without regularization typically lead to unrealistically high parameter values and therefore to updated parameter fields that have little or no geological realism. However, it has been suggested that the application of unregularized reservoir parameter estimation may still add value, because it, sometimes, gives an indication of the location of significant missing features in the model. We investigated under which conditions this perceived added value might occur. We conducted several twin experiments and applied unregularized parameter estimation to update uncertain parameters in a simple two-dimensional reservoir model that contained a major deficiency in the form of a missing high or low permeability feature. We found that in case of low-permeability barriers or high-permeability streaks it is indeed sometimes possible to localize the position of the model deficiency. To further analyze this behavior we conducted onedimensional experiments using a transfer function formalism to characterize the identifiability of the location and magnitude of model deficiencies (flow barriers).

Identifiability of Location and Magnitude of Flow Barriers in Slightly Compressible Flow

“…They argued, by use of numerical examples, that localized unrealistic-parameter values can be used as an indicator of model errors in the underlying reservoir model, a concept that they named "model maturation." In a follow-up study, Kahrobaei et al (2014) showed that the application of unregularized reservoir-parameter estimation may sometimes give an indication of the location of significant missing features in the model. In the present study we further analyze this phenomenon by addressing the identifiability of flow-relevant features.…”