Automated Geologically-Consistent History Matching of Facies Distribution and Reservoir Properties in Inter-Well Space by Adjoint Methods
Authors: Zakirov E. S. , Lyubimova O. V. , Indrupsky I. M. , Anikeev D. P. , Arkhipova E. Yu. 
Published: October 2017

Abstract
Comprehensive geological analysis at the stage of 3D geological modeling is necessary to obtain a high-quality flow model. The next important step is history matching to dynamic well data without distortion of the initial geological basis of the model. It is necessary to preserve static data on facies indexes, porosity and other parameters in the well gridcells, adjusting their distributions in the inter-well space, while maintaining the geological concepts of the object. The purpose of the study presented in this paper was the development of automated computationally efficient algorithms for solving this problem.
Earlier, based on the computationally efficient adjoint methods, we developed algorithms and their implementation in the in-house SimMatch® simulator for geologically consistent history matching with identification of parameters of the variograms and porosity-to-permeability relations for a given distribution of facies in the inter-well space. In this study, we make transition from the discrete representation of facies at wells to the continuous values in the inter-well space reflecting the fractional contribution of the facies in the formation of model cell properties. Considering implicit parametric dependencies on static data at wells and variogram parameters for properties and facies, a computationally efficient algorithm was developed for consistent adjustment of the distributions of facies and reservoir properties during history matching of the 3D model.
The algorithms developed are implemented within the frameworks of the forward and inverse problems. In the forward problem, a distribution of a continuous facies parameter is constructed, taking into account well data and the variogram for the facies. In current implementation, the continuous "facie" value is interpreted as being transitional (weighted) between the adjacent integer values, which is typical for geological environments with sequential change of facies usually modeled by methods such as the truncated Gaussian simulation. Further, for each facie, distribution of the reservoir parameters (porosity, permeability) is independently constructed, taking into account the variogram for porosity and the porosity-to-permeability relation for this facie. The resulting property value in each cell is determined by weighing by the portions of the "pure" facies.
Within the framework of the inverse problem, the parameters of anisotropic variograms for the facies and for the reservoir properties within each facie, as well as the coefficients in the porosity-to-permeability relation for each facie, serve as the control parameters. For efficient implementation of the automated gradient procedure for adjustment control parameters, the adjoint problem is solved at each iteration, and the object function gradient with respect to the control parameters is calculated taking into account the implicit dependencies of the reservoir properties in the model cells on the variogram parameters for the facies and reservoir properties. The results of approbation of the approach on a realistic example of the 3D reservoir model and on a 3D model of a real deposit section are presented.
During the study, effecient algorithms for consistent adjustment of the facies and reservoir properties distributions in a 3D model were for the first time constructed on the basis of adjoint methods and implemented in the in-house simulator. The advantage of the approach is the significant reduction in computational costs (number of simulator runs) for solution of the inverse problem in comparison with alternative methods of automated history matching, while preserving the consistency of the facies and reservoir properties distributions with the original principles of the geological model construction.