Advanced Geologically-consistent History Matching and Uncertainty Evaluation

At the ECMOR-14 conference we proposed a method for automated geologically-consistent history matching of a 3D reservoir model and presented the results of its implementation as a numerical algorithm. In the approach considered, control parameters to be evaluated through dynamic data assimilation were geostatistical parameters of a 3D reservoir model. It was assumed that porosity within inter-well space was calculated by the kriging procedure based on measured values on wells. Permeability distribution was supposed to be calculated through its correlation dependence on porosity. Through the inverse problem solution, the most uncertain parameters of a geostatistical model were estimated, namely, parameters of an anisotropic semivariogram and the porosity-to-permeability dependence for each facies. The inverse problem solution algorithm was based on efficient methods of the optimal control theory (the adjoint methods). In the present study, the approach is further advanced by the transition from the deterministic kriging procedure to stochastic geostatistical methods such as sequential Gaussian simulation. The control parameters evaluated through the inverse problem solution – parameters of the anisotropic semivariogram and porosity-to-permeability dependence - are supplemented by the values in pilot points chosen with a special algorithm. For some common depositional environments, original algorithm has been also developed to adjust facies distribution within the inverse problem. Thus, firstly, it becomes possible to effectively identify heterogeneities of parameter distributions in the inter-well space. Secondly, efficient assessment of uncertainty in the parameters obtained by the inverse problem solution can be carried out. As an alternative implementation to the computationally-intensive approach based on the group analysis of an ensemble of history matched models, a simplified procedure has been implemented based on the linearization of the objective function at the optimal point. To obtain the covariance matrix, the sensitivity matrix is calculated by means of a special computationally-efficient procedure using the state variables variation problem - one of the subproblems of the adjoint-based algorithm for inverse problem solution. The paper provides a number of examples demonstrating performance of the proposed approaches and algorithms. Main contributions: - Geologically-consistent history matching based on adjoint methods has been extended to stochastic geostatistical formulations for evaluation of anisotropic semivariogram and porosity-to-permeability relation parameters for each facies and values in pilot points chosen with a special algorithm. - Original algorithm has been developed to constrain facies distribution to dynamic data. - Uncertainty assessment procedure for inverse problem solution has been implemented based on a subproblem of the adjoint algorithm for sensitivity and covariance matrix computation.