To Be or not to Be ... Markovian?

Abstract:

For the modeling of flow and transport in the heterogeneous subsurface, random walk models that are based on the Markov hypothesis are frequently applied. However, the validity of the Markov hypothesis or more precisely the Chapman-Kolmogorov equation is verified only rarely. In this work, we present a new breakthrough method that resolves crucial difficulties that arise when verifying Markovianity. Our method is computationally efficient and robust in the sense that statistical errors are kept at low levels in spite of limited sample sizes. We demonstrate the capabilities of the new method in the context of macro dispersion in the heterogeneous subsurface. We find that depending on the heterogeneity level, Markovian models work well above a certain scale of interest for media with exponentially decaying and Gaussian log-conductivity correlation structures (see figure). Moreover, we find surprising similarities in the velocity dynamics of both structures, which is relevant when formulating predictive macro-dispersion models.