This paper presents an alternative numerical tool to model the one—and two—dimensional contaminant transport through saturated porous media using a meshfree method called Element Free Galerkin Method (EFGM). In EFGM, an approximate solution is constructed entirely in terms of a set of nodes and no characterization of the interrelationship of the nodes is needed. The advection-dispersion equation with sorption is considered to illustrate the applicability of EFGM. The unknown concentration values are approximated by using moving least-square approximants, which are constructed using a weight function, a linear basis function and a set of non-constant coefficients. The weak form of the governing equation is formulated and Lagrange multiplier method is used for enforcing the essential boundary conditions. Three numerical examples are presented and the results are compared with those obtained from the analytical and finite element results. © 2008 Taylor & Francis Group, LLC.