Résumé
We perform Stokes flow simulations on digitized images of a Berea sandstone sample obtained through micro-tomography imaging and segmentation processes. We obtain accurate information on the transport using a streamline reconstruction algorithm which uses the velocity field obtained from the flow simulation as input data. This technique is based on the method proposed by Pollock (Groundwater, 1988) but employs a quadratic interpolation near the rock mesh cells of the domain similarly to Mostaghimi et al. (SPE, 2012). This allows an accurate resolution of the velocity field near the solid interface which plays an important role on the transport characteristics, such as the probability density of first arrival times and the growth of the mean squared displacement, among others, which exhibit non-Fickian behavior. We analyze Lagrangian and Eulerian velocity statistics and their relation, and then focus on the ergodicity and the stationarity properties of the transport. We analyze the temporal evolution of Lagrangian velocity statistics for different injection conditions, and find quick convergence to a limiting velocity distribution, indicating the transport to be near-stationary. The equivalence between velocity samplings within and across streamlines, as well as the independency of the statistics on the number of sampled streamlines, lead as to conclude that the transport may be modeled as ergodic. These characteristics then allow us to upscale the 3-dimensional simulations using a 1-dimensional Continuous Time Random Walk model. This model, parametrized by the velocity results and the characteristic correlation length obtained from the above-mentioned simulations, is able to efficiently reproduce the results and to predict larger scale behaviors.