Assimilating Hydraulic Conductivity Data Using Multiscale Training Images

Abstract:

Hydraulic conductivity is the most important and at the same time the most uncertain parameter in numerical simulations. One problem commonly faced is that the data are usually collected at scales that are different than the elements used in numerical models. The question of how to calculate the permeabilities at the numerical element scale is driven by the hydraulic conductivity values that increase with larger scales, and also by the fact that the permeability is not an additive variable. Therefore, a common challenge occurs when different hydraulic conductivity measurements (often corresponding to different spatial scales, e.g. well tests, slug tests and permeameter tests) coexist in a field study and have to be integrated simultaneously. In the context of multiGaussian models, analytical formulations are available that can estimate equivalent hydraulic conductivity. However, when multiGaussian assumptions are not adopted, such as in the case of multiple-point statistics simulations, such analytical scale relationships are not valid. 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 multi-scale training image to incorporate simultaneously conditioning data at several spatial scales. 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 to ensure both local conditioning at the coarsest scales and physically-driven scale relationships.