Advances in Water Resources 96 (2016) 180–189 Contents lists available at ScienceDirect Advances in Water Resources journal homepage: www.elsevier.com/locate/advwatres Upscaling of solute transport in disordered porous media by wavelet transformations Mahsa Moslehi a , Felipe P.J. de Barros a,∗ , Fatemeh Ebrahimi b , Muhammad Sahimi c a Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, California 90089 2531, USA b Department of Physics, Faculty of Sciences, University of Birjand, Birjand 97178 51367, Iran c Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, California 90089 1211, USA a r t i c l e i n f o Article history: Received 29 March 2016 Revised 21 July 2016 Accepted 21 July 2016 Available online 22 July 2016 Keywords: Flow and transport Heterogeneous porous media Upscaling wavelet transform Computational efficiency a b s t r a c t Modeling flow and solute transport in large-scale (e.g. on the order of 10 3 m heterogeneous porous me- dia involves substantial computational burden. A common approach to alleviate the problem is to utilize an upscaling method that generates models that require less intensive computations. The method must also preserve the important properties of the spatial distribution of the hydraulic conductivity (K) field. We use an upscaling method based on the wavelet transformations (WTs) that coarsens the computa- tional grid based on the spatial distribution of K. The technique is applied to a porous formation with broadly distributed and correlated K values, and the governing equation for solute transport in the for- mation is solved numerically. The WT upscaling preserves the resolution of the initial highly-resolved computational grid in the high K zones, as well as that of the zones with sharp contrasts between the neighboring K, whereas the low-K zones are averaged out. To demonstrate the accuracy of the method, we simulate fluid flow and nonreactive solute transport in both the high-resolution and upscaled grids, and compare the concentration profiles and the breakthrough times. The results indicate that the WT upscaling of a K field generates non-uniform upscaled grids with a number of grid blocks that on aver- age is about two percent of the number of the blocks in the original high-resolution computational grids, while the concentration profiles, the breakthrough times and the second moment of the concentration distribution, computed for both models, are virtually identical. A systematic parametric study is also car- ried out in order to investigate the sensitivity of the method to the broadness of the K field, the nature of the correlations in the field (positive versus negative), and the size of the computational grid. As the broadness of the K field and the size of the computational domain increase, better agreement between the results for the high-resolution and upscaled models is obtained. © 2016 Elsevier Ltd. All rights reserved. 1. Introduction It is of fundamental and practical importance to incorporate the spatial heterogeneity of porous geological formations in the mod- els of flow and transport in such media (Dagan et al., 1989; Ru- bin, 2003; Sahimi, 2011). At the field scale (e.g. orders of 10 2 − 10 3 m), subsurface properties, such as the permeability, vary many or- ders of magnitude across multiple length scales (e.g. from 1 m to 10 3 m or larger) (Dagan et al., 1989; Rubin, 2003; Sahimi, 2011). It is well recognized that the spatial fluctuations of the permeabil- ity field, i.e. many orders of magnitude difference between per- meability values, have a significant role in the spreading rates of ∗ Corresponding author. E-mail addresses: moslehi@usc.edu (M. Moslehi), fbarros@usc.edu (F.P.J. de Barros), f_ebrahimi@birjand.ac.ir (F. Ebrahimi), moe@usc.edu (M. Sahimi). solute plumes, as well as estimates of their early or late arrival times. Thus, neglecting the effect of subsurface heterogeneity, and in particular the spatial distribution of the permeability, in numer- ical simulation leads to erroneous prediction of solute transport, which will have severe consequences for health risk assessment (de Barros and Rubin, 2008; Maxwell et al., 1999), the likelihood of extreme events (de Barros and Fiori, 2014; Dentz and Tartakovsky, 2010; Henri et al., 2015), and for reactive mixing (Dentz et al., 2011; Luo et al., 2008). In general, to obtain accurate predictions for solute mixing at the field scale and calculate the properties that characterize the process, such as the distribution of travel times and the disper- sion coefficients, numerical simulation of flow and transport in large-scale porous media requires a computational grid with high enough resolution to represent the variability of the hydrogeologi- cal properties (Ababou et al., 1989; de Dreuzy et al., 2007). Simula- tion with such high-resolution computational grids entails solving http://dx.doi.org/10.1016/j.advwatres.2016.07.013 0309-1708/© 2016 Elsevier Ltd. All rights reserved.