Sparse geologic dictionaries for subsurface flow model calibration: Part I. Inversion formulation

Inference of heterogeneous rock properties from low-resolution dynamic flow measurements often leads to underdetermined nonlinear inverse problems that can have many non-unique solutions. The problem is usually regularized by reducing the number of unknown parameters and/or incorporating direct or indirect prior information. In subsurface flow modeling, structural connectivity in hydraulic properties plays a critical role in determining local and global flow and displacement processes. When reliable prior information about the structural connectivity of a formation is available it can be used to discourage implausible inversion solutions. In this two-part paper, we introduce a geologically-inspired conceptual framework, geologic dictionaries, for reconstructing complex subsurface physical properties from the flow data. We evaluate the performance of our method under both reliable and highly uncertain prior knowledge and measurements. In Part I, we present inversion with sparse geologic dictionaries, learned from prior models, for estimation of complex heterogeneous subsurface hydraulic properties. We show how learning methods can be adapted to build, from a prior training library, specialized sparse geologic dictionaries that contain relevant structural elements (words) for constructing the solution of subsurface flow inverse problems. The key property of the constructed geologic dictionaries that we invoke during flow data integration is its sparsity; that is, we require that only a small subset of geologic dictionary elements be sufficient for accurately approximating any prior model realizations in the training library, and hence the model calibration solution. Using the sparsity property of the geologic dictionary, we formulate and solve the nonlinear model calibration as a feature estimation problem. To construct a solution, we adopt an iteratively reweighted least-squares (IRLS) algorithm to identify the important dictionary elements by minimizing a sparsity-regularized data misfit objective function. We illustrate the flexibility and effectiveness of the proposed method by applying it to a series of numerical experiments in multiphase flow systems and compare it with parameterization by truncated singular vectors. (C) 2011 Elsevier Ltd. All rights reserved.