In this work we study reactive flows through porous media. We suppose dominant Peclet's number, dominant Damkohler's number and general linear reactions at the pore boundaries. Our goal is to obtain the dispersion tensor and the upscaled model. We introduce the multiple scale expansions with drift for the problem and use this technique to upscale the reactive flow equations. Our result is illustrated with numerical simulations for the dispersion tensor. (C) 2009 Elsevier Ltd. All rights reserved.