Integrating multiple scales of hydraulic conductivity measurements in training image-based stochastic models

Hydraulic conductivity is one of the most critical and at the same time one of the most uncertain parameters in many groundwater models. One problem commonly faced is that the data are usually not collected at the same scale as the discretized elements used in a numerical model. Moreover, it is common that different types of hydraulic conductivity measurements, corresponding to different spatial scales, coexist in a studied domain, which have to be integrated simultaneously. Here we address this issue in the context of Image Quilting, one of the recently developed multiple-point geostatistics methods. Based on a training image that represents fine-scale spatial variability, we use the simplified renormalization upscaling method to obtain a series of upscaled training images that correspond to the different scales at which measurements are available. We then apply Image Quilting with such a multiscale training image to be able to incorporate simultaneously conditioning data at several spatial scales of heterogeneity. The realizations obtained satisfy the conditioning data exactly across all scales, but it can come at the expense of a small approximation in the representation of the physical scale relationships. In order to mitigate this approximation, we iteratively 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 examples where it gives good results and shows potential for the integration of different measurement methods in real-case hydrogeological models.