We report on the parametric study of composite-vortex patterns formed by displaced singly-ringed Laguerre-Gauss beams. We find that rich structures of vortices appear and disappear as the phase, topological charge and displacement of the component beams is varied. The net topological charge depends on various factors. When the beams are collinear or nearly collinear the net charge is the largest topological charge of the two component beams. When they are displaced by about one or more beam widths the net charge is the sum of the topological charges of the component beams plus the charge of vortices created by the shear phases in the region in between the two beams. The shear charges depend on the parameters of the problem. The experimental measurements are consistent with the expectations, although the measured location of the vortices is not necessarily in agreement with the predictions. © 2009 SPIE.