History Matching With Learned Sparse Dictionaries

Abstract

This article, written by Senior Technology Editor Dennis Denney, contains highlights of paper SPE 133654, ’History Matching With Learned Sparse Dictionaries,’ by Mohammad-Reza M. Khaninezhad, Behnam Jafarpour, SPE, and Lianlin Li, Texas A&M University, prepared for the 2010 SPE Annual Technical Conference and Exhibition, Florence, Italy, 19-22 September. The paper has not been peer reviewed. History matching of heterogeneous hydrocarbon reservoirs is carried out to construct predictive reservoir models. Estimating individual-gridblock heterogeneous-reservoir properties from limited static data and spatially averaged dynamic-production measurements inherently leads to an ill-posed inverse problem that can have several nonunique solutions. To achieve solution sparsity, an iteratively reweighted-sparsity regularized least-squares algorithm is used to combine and weigh a small subset of relevant model elements selectively from a learned dictionary that explains the observed data. Introduction History matching of hydrocarbon-reservoir models is required to integrate production data that constrain uncertain previous descriptions of model parameters (i.e., spatially distributed petrophysical properties). The main goal of history matching is to construct updated models capable of predicting future performance of the reservoir more accurately. Reconstructing petrophysical properties is a challenging inverse problem because of data scarcity and because these properties are heterogeneous. When spatial discretization of the forward-simulation model is used to describe unknown spatial parameters in history matching, the problem typically becomes ill posed because the number of unknown parameters overwhelmingly exceeds available observations. Consequently, several nonunique solutions can exist that respect available observations but exhibit different spatial continuity and flow predictions. Several techniques can regularize ill-posed history-matching problems. Usually, regularization involves incorporating additional prior assumptions to eliminate implausible solutions and improve the stability of solution algorithms. Reparameterization Reparameterization of the original spatial problem is a regularization approach for ill-posed inverse problems in which a reduced number of parameters is used to approximate the unknown-petrophysical-property maps. Reparameterization is facilitated when reliable a priori information about structural attributes of the solution is available. Mathematical transformations provide an effective reparameterization approach when the properties of the unknown parameters are exploited better in the transform domain. For example, principal-component analysis (PCA) is effective in reparameterizing correlated parameters by elimination of their correlation (redundancy). Other popular transform-domain reparameterization methods that have compression properties include the discrete-cosine transform (DCT) and the discrete-wavelet transform (DWT).