Self-similar models proposed by Villermaux and Schweich (1992, J. Phys. II France 2, 1023-1043) are modified and applied to simulate convection and nonlinear adsorption of a solute in a disordered porous medium. We explore the dispersive behaviour of three self-similar patterns. Conservation equations are integrated numerically for the Langmuir adsorption isotherm, and an analytical procedure is developed for the rectangular isotherm. Breakthrough curves rapidly approach an asymptote with a nonzero breadth as the isotherm becomes increasingly concave-downward, in contrast to results of conventional dispersion models. The networks are simple to use, and we discuss how they can be applied in practice.