We analyze flow in heterogeneous media composed of multiple materials whose hydraulic properties and geometries are uncertain. Our analysis relies on the composite media theory of Winter and Tartakovsky <2000, 2002>, which allows one to derive and solve moment equations even when the medium is highly heterogeneous. We use numerical solutions of Darcy flows through a representative composite medium to investigate the robustness of perturbation approximations in porous medium with total log conductivity variances as high as 20. We also investigate the relative importance of the two sources of uncertainty in composite media, material properties, and geometry. In our examples the uncertain geometry by itself captures the main features of the mean head estimated by the full composite model even when the within-material conductivities are deterministic. However, neglecting randomness within materials leads to head variance estimates that are qualitatively and quantitatively wrong. We compare the composite media approach to approximations that replace statistically inhomogeneous conductivity fields with pseudohomogeneous random fields with deterministic trends. We demonstrate that models with a deterministic trend can be expected to give a poor estimate of the statistics of head.