Sparse geologic dictionaries for subsurface flow model calibration: Part II. Robustness to uncertainty

We introduced sparse geologic dictionaries for formulating and solving subsurface flow model calibration problems in Part I. A key assumption was the availability of a reliable prior model library for construction of a relevant sparse geologic dictionary that was used to constrain the inversion solution. In practice, however, the spatial connectivity in rock physical property distributions has to be inferred from uncertain and incomplete sources of information, including qualitative geologic interpretations, formation type and outcrop maps, as well as scattered measurements (e.g., well core, log data, and seismic maps). Thus, the geologic continuity model that is used for generating prior models is likely to carry significant uncertainty. In Part II, we investigate the performance of the proposed method under uncertain and incorrect prior continuity models. We show that, unlike the conventional reduced-order parameterization methods such as truncated SVD, diverse geologic dictionaries are robust against uncertainty in the structural prior model. This robustness is attributed to the selection property of the sparse reconstruction algorithm and the connectivity exhibited by the dictionary elements. Under highly uncertain prior models, the reconstruction problem is reduced to identifying and combining relevant elements from a large and diverse geologic dictionary. Diversity of the dictionary further enhances solution sparsity since a large number of dictionary elements will have negligible contribution to the solution. Consequently, the inversion provides considerable flexibility to accommodate significant levels of variability (uncertainty) in prior geologic models. As an extreme case, we also evaluate the performance of the proposed method when even the formation type (e.g., Gaussian versus non-Gaussian) is uncertain. In addition, we examine the sensitivity of the proposed method to noise in the flow data, where we observe solution underestimation effects due to l1-norm approximation of l0-norm regularization. We address this issues by replacing l1-norm with an adaptive lp-norm, where p decreases with iterations from an initial value p = 1 to a value of p = 0+ at final iterations. Using several numerical examples, we illustrate these important properties of the proposed inversion approach and compare its performance with that of the truncated singular vector parameterization.