This study presents a method for obtaining risk-qualified estimates of various statistics that characterize the migration of a nonreactive tracer in heterogeneous porous media. The steps required to get these estimates are as follows: (1) the solute body is assumed to be composed of a large number of imaginary solute particles; (2) a Lagrangian description of the migration of the solute particles in space is adopted; (3) the Lagrangian description is transformed into an Eulerian one by a quasi-linear method, whereby the translation of the particles in space is simulated through successive, spatially correlated displacements; (4) the plume's various statistics are obtained by averaging over the ensemble of simulated particles, and, lastly, (5) simultaneous simulation of the motion of two particles in space entails the spatial moments of the plume and their variances, and in addition, the concentration is obtained qualified by its variance. The model is compared with numerical simulations and is found to compare quite favorably even for log conductivity variances as large as 2.6. This study shows that (1) after a certain travel time, the concentration variance becomes largest next to the plume center and gets smaller with distance from the plume center; (2) the concentration coefficient of variation grows with distance from the plume center and prediction of the low concentration on the periphery of the plume is subject to the largest uncertainty; (3) the concentration coefficient of variation grows with time while the concentration variance gets smaller with time; and (4) interpretation of field data depends on the interpolation method and in particular on the size of the averaging domain.