Block-Upscaling Using Multi-Rate Mass Transfer

Real aquifers are heterogeneous over an evolving range of scales. Often anomalous transport behavior is observed e.g. the tailing of breakthrough curves or the scale effect of dispersivity. This behavior cannot be represented by the ADE-type equations and transport problems in hydrogeology are usually too complex to be solved at the pore scale. Thus new upscaled transport equations are needed which allow to model large scale heterogeneities on the numerical grid and small scale heterogeneities within the transport equation. Multi-Rate Mass Transfer (MRMT) models are known to successfully represent anomalous transport. The resulting transport equation is characterized by a memory function in time. The main limitation of MRMT models resides in the fact that this memory term is determined by some best-fit procedures against measured breakthrough curves and is not easily related to measurable quantities of clear physical meaning. Our work aims at relating the memory function of MRMT models to the underlying heterogeneity structure. Particularly, we investigate how the memory function changes with varying block size. The approach is based on the following steps. First, we model a plume on a single realization of a heterogeneous transmissivity field with a very fine grid resolution and solving the ADE with constant local-scale dispersivity. We assume this model to reproduce a perfectly known reality where the ADE is valid (i.e., at the pore scale). Breakthrough curves are observed at the outlet of the domain and reproduced by a homogeneous MRMT model. Then the original heterogeneous domain is divided into blocks of identical size and the same procedure is applied to the individual blocks in order to obtain breakthrough curves and corresponding MRMT representations. This is repeated for various block sizes. The resulting set of memory functions are then compared and related to block size and underlying heterogeneity. We show for short time scales that heterogeneity causes tailing of breakthrough curves and MRMT models are able to reproduce much better the original field than the upscaled ADE. We then propose an empirical relationship of the MRMT memory functions to (1) measurable geostatistical properties and (2) the block size of the numerical model.