An Iteratively Reweighted Algorithm for Sparse Reconstruction of Subsurface Flow Properties from Nonlinear Dynamic Data

A challenging problem in predicting fluid flow displacement patterns in subsurface environment is the identification of spatially variable flow-related rock properties such as permeability and porosity. Characterization of subsurface properties usually involves solving a highly underdetermined nonlinear inverse problem where a limited number of measurements are used to reconstruct a large number of unknown parameters. To alleviate the non-uniqueness of the solution, prior information is integrated into the solution. Regularization of ill-posed inverse problems is commonly performed by imparting structural prior assumptions, such as smoothness, on the solution. Since many geologic formations exhibit natural continuity/correlation at various scales, decorrelating their spatial description can lead to a more compact or sparse representation in an appropriate compression transform domain such as wavelets or Fourier domain. The sparsity of flow-related subsurface properties in such incoherent bases has inspired the development of regularization techniques that attempt to solve a better-posed inverse problem in these domains. In this paper, we present a practical algorithm based on sparsity regularization to effectively solve nonlinear dynamic inverse problems that are encountered in subsurface model calibration. We use an iteratively reweighted algorithm that is widely used to solve linear inverse problems with sparsity constraint (known as compressed sensing) to estimate permeability fields from nonlinear dynamic flow data. To this end, we minimize a data misfit cost function that is augmented with an additive regularization term promoting sparse solutions in a Fourier-related discrete cosine transform domain. This regularization approach introduces a new weighting parameter that is in general unknown a priori, but controls the effectiveness of the resulting solutions. Determination of the regularization parameter can only be achieved through considerable numerical experimentation and/or a priori knowledge of the reconstruction solution. To circumvent this problem, we include the sparsity promoting constraint as a multiplicative regularization term which eliminates the need for a regularization parameter. We evaluate the performance of the iteratively reweighted approach with multiplicative sparsity regularization using a set of waterflooding experiments in an oil reservoir where we use nonlinear dynamic flow data to infer the spatial distribution of rock permeabilities, While, the examples of this paper are derived from the subsurface flow and transport application, the proposed methodology also can be used in solving nonlinear inverse problems with sparsity constraints in other imaging applications such as geophysical, medical imaging, electromagnetic and acoustic inverse problems.