Efficient History Matching Using a Multiscale Technique
Authors: Aanonsen Sigurd I. 
Published: February 2008

Summary
It is demonstrated that a method for multiscale history matching can be used to improve efficiency and/or quality of the solution when achieving a fine-scale match as compared to history-matching directly on the fine scale. Starting from a given fine-scale realization, coarser models are generated using a global upscaling technique in which the coarse models are "history matched" with respect to the solution at the fine scale. Conditioning to dynamic data is performed by history matching a coarse model, and this model is then successively refined using a combination of downscaling and history matching until a model that matches dynamic data is obtained at the finest scale. Bias in predicted data because of upscaling errors may be taken into account. The advantage of this procedure is that the large-scale corrections are obtained using fast models which—combined with proper downscaling procedures—provide a better initial model for the final adjustment on the fine scale. Coarse-scale history matching also provides a regularization of the fine-scale match, making the process less dependent on a correct prior model. With the proposed methodology, a series of models with varying degrees of complexity—all being consistent with both static and dynamic data—may be generated without additional cost. Effects of including a priori information and different initial downscaling techniques, such as sampling or block kriging with sequential Gaussian simulation (BKSGS), are investigated using two synthetic reservoir models.

Introduction
In Aanonsen and Eydinov (2006), a multiscale method was proposed for more-effective history matching of petrophysical properties. Starting from a given fine-scale realization, coarser models are generated by use of a global upscaling technique in which the coarse models are "history matched" with respect to the solution at the fine scale. Conditioning to dynamic data is accomplished by history matching a coarse model, and this model is then successively refined using a combination of downscaling and history matching until a model that matches dynamic data is obtained at the finest scale. It was demonstrated through some simple examples that this procedure may reduce the computational effort and/or improve the quality of the match as compared to history matching directly on the fine scale. In this paper, the technique is taken further, by taking into account bias in the coarse-scale solution. Also, effects of including a priori information and different initial downscaling techniques, such as sampling and BKSGS, are investigated.
In traditional multiscale estimation, a series of estimations is performed in which the resolution of the zonation is increased for each step in the sequence for examples, see Liu (1993); Chavent and Bissel (1998); and Yoon et al. (1999). This approach reduces both the computational effort and the overparameterization problem. The multiscale estimation was improved by Ben Ameur et al. (2002); Grimstad et al. (2003); and Grimstad and Mannseth (2004), who introduced a method for adaptive multiscale estimation in which the resolution is increased only in some regions of the reservoir of each stage in the sequence. Other types of multiscale approaches include updating a fine model on the basis of simulations performed on a coarse model (Mezghani and Roggero 2001).
Common for all of the above-mentioned multiscale techniques is that the computational grid is kept the same while the parameter resolution is changed. In the method used here, the forward problem is also computed on grids of varying resolution. The main challenge with the method is that it requires an efficient way for tuning the properties in every grid cell in the model. We have used commercial software in which the number of parameters (which, with this method, is equal to the number of active grid cells) is limited to 500. However, with software based on the adjoint method (Li et al. 2003; Gao and Reynolds 2006), this procedure may be applied to much larger cases—up to real field-model size.