We propose a new concept for effective properties of composites with uncertain spatial arrangements of constitutive materials and within-material properties. Rather than replacing a heterogeneous property with a constant e ective parameter, we seek to preserve composite's internal macro structure. This general concept is used to derive an effective conductivity of composite porous media, when both material s geometry and conductivities within each material are uncertain. Our analysis uses a random domain decomposition to explicitly account for the separate effects of material and geometric uncertainty on ensemble moments of pressure and flux. We present a general expression for the effective (apparent) conductivity of such media and analyze it in detail for one- and two-dimensional steady flows in bounded random media composed of two materials with highly contrasting conductivities.