Pattern-based calibration of complex subsurface flow models against dynamic response data

Calibration of heterogeneous subsurface flow models usually leads to ill-posed nonlinear inverse problems, where too many unknown parameters are estimated from limited response measurements. When the underlying parameters form complex (non-Gaussian) structured spatial connectivity patterns, classical variogram-based geostatistical techniques cannot describe the underlying distributions. Modern pattern-based geostatistical methods that incorporate higher-order spatial statistics are more suitable for describing such complex spatial patterns. Moreover, when the unknown parameters are discrete (e.g., geologic facies distribution), conventional model calibration techniques that are designed for continuous parameters cannot be applied directly. In this paper, we introduce a novel pattern-based model calibration method to reconstruct spatially complex facies distributions from dynamic flow response data. To reproduce complex connectivity patterns during model calibration, we impose a geologic feasibility constraint that ensures the solution honors the expected higher-order spatial statistics. For model calibration, we adopt a regularized least-squares formulation, involving (i) data mismatch, (ii) pattern connectivity, and (iii) feasibility constraint terms. Using an alternating directions optimization algorithm, the regularized objective function is divided into a parameterized model calibration sub-problem, which is solved via gradient-based optimization. The resulting parameterized solution is then mapped onto the feasible set, using the k-nearest neighbors (k-NN) as a supervised machine learning approach, to honor the expected spatial statistics. The two steps of the model calibration formulation are repeated until the convergence criterion is met. Several numerical examples are used to evaluate the performance of the developed method.