  We present a model for upscaling the time-dependent effective retardation
factor in hierarchical porous media with multimodal reactive mineral facies.
The model extends the approach by Deng et al. (2013) in which they expanded a
Lagrangian-based stochastic theory presented by Rajaram (1997) in order to
describe the scaling effect of retardation factor. They used a first-order
linear approximation in deriving their model to make the derivation tractable.
Importantly, the linear approximation is known to be valid only to variances of
0.2. In this article we show that the model can be derived with a higher-order
approximation, which allows for representing variances from 0.2 to 1.0. We
present the derivation, and use the resulting model to recalculate the
time-dependent effective retardation for the scenario examined by Deng et al.
(2013).

Fluid Dynamics