Inverse problems associated with reservoir characterization are typically under-determined and often have difficulties associated with stability and convergence of the solution. A common approach to address this issue is through the introduction of prior constraints, regularization or reparameterization to reduce the number of estimated parameters.
We propose a dual scale approach to production data integration that relies on a combination of coarse-scale and fine-scale inversions while preserving the essential features of the geologic model. To begin with, we sequentially coarsen the fine-scale geological model by grouping layers in such a way that the heterogeneity measure of an appropriately defined ‘static’ property is minimized within the layers and maximized between the layers. Our coarsening algorithm results in a non-uniform coarsening of the geologic model with minimal loss of heterogeneity and the ‘optimal’ number of layers is determined based on a bias-variance trade-off criterion. The coarse-scale model is then updated using production data via a generalized travel time inversion. The coarse-scale inversion proceeds much faster compared to a direct fine-scale inversion because of the significantly reduced parameter space. Furthermore, the iterative minimization is much more effective because at the larger scales there are fewer local minima and those tend to be farther apart. At the end of the coarse-scale inversion, a fine-scale inversion may be carried out, if needed. This constitutes the outer iteration in the overall algorithm. The fine-scale inversion is carried out only if the data misfit is deemed to be unsatisfactory.
We demonstrate our approach using both synthetic and field examples. The field example involves waterflood history matching of a structurally complex and faulted offshore turbiditic oil reservoir. Permeability and fault transmissibilities are the main uncertainties. The geologic model consists of more than 800,000 cells and 10 years of production data from 8 producing wells. Using our dual scale approach, we are able to obtain a satisfactory history match with a finite-difference model in less than a day in a PC. Compared to a manual history matching, the dual scale approach is shown to better preserve the geological features and the pay/non-pay juxtapositions in the original geologic model.