Integrating Multiple Scales of Hydraulic Conductivity Measurements in Training Image-Based Stochastic Models

It has been widely demonstrated that the hydraulic conductivity of an aquifer increases with a larger portion of the aquifer tested. This poses a challenge when different hydraulic conductivity measurements coexist in a field study and have to be integrated simultaneously (e.g. core analysis, slug tests and well tests). While the scaling of hydraulic conductivity can be analytically derived in multiGaussian media, there is no general methodology to simultaneously integrate hydraulic conductivity measurements taken at different scales in highly heterogeneous media. Here we address this issue in the context of multiple-point statistics simulations (MPS). In MPS, the spatial continuity is based on a training image (TI) that contains the variability, connectivity, and structural properties of the medium. The key principle of our methodology is to consider the different scales of hydraulic conductivity as joint variables which are simulated together. Based on a TI that represents the fine-scale spatial variability, we use a classical upscaling method to obtain a series of upscaled TIs that correspond to the different scales at which measurements are available. In our case, the renormalization method is used for this upscaling step, but any upscaling method could be employed. Considered together, the different scales obtained are considered a single multi-scale representation of the initial TI, in a similar fashion as the multiscale pyramids used in image processing. We then use recent MPS simulation methods that allow dealing with multivariate TIs to generate conditional realizations of the different scales together. One characteristic of these realizations is that the possible non-linear relationships between the different simulated scales are statistically similar to the relationships observed in the multiscale TI. Therefore these relationships are considered a reasonable approximation of the renormalization results that were used on the TI. Another characteristic of these realizations is that they can be directly conditioned to local data, and the data can then be considered at any of the scales considered. The realizations obtained satisfy the conditioning data exactly across all scales, but it comes at the expense of an approximate representation of the physical scale relationships. In order to mitigate this approximation, we apply a kriging-based correction to the finest scale that ensures local conditioning at the coarsest scales. The method is tested on a series of synthetic and field-based examples where it gives good results and shows potential for the integration of different measurement methods in hydrogeology.