The Enskog repeated ring equations (ERRE) for tagged molecule motion are derived from the BBGKY hierarchy. The hierarchical deviation demonstrates the problem associated with using naive truncations for complex fluid systems. A moment-variational solution is suggested for the ERRE. The moment- variational method is applied to the ERRE for the two- and three-dimensional overlapping Lorentz gas (LG). Both self-consistent and non-self-consistent equations for the diffusion constantare solved. The results compare favorably with previous calculations and molecular dynamics (MD).