We discuss positron-annihilation lifetimes for a set of illustrative bulk materials within the framework of the weighted-density-approximation (WDA). The WDA can correctly describe electronpositron correlations in strongly inhomogeneous systems, such as surfaces, where the applicability of (semi-) local approximations is limited. We analyze the WDA in detail and show that the electrons which cannot screen external charges efficiently, such as the core electrons, cannot be treated accurately via the pair-correlation of the homogeneous electron gas. We discuss how this problem can be addressed by reducing the screening in the homogeneous electron gas by adding terms depending on the gradient of the electron density. Further improvements are obtained when core electrons are treated within the LDA and the valence electron using the WDA. Finally, we discuss a semi-empirical WDA-based approach in which a sum-rule is imposed to reproduce the experimental lifetimes.