We derive a contrast expansion formalism for the effective conductivity of heterogeneous materials (media) with local properties described by arbitrary continuous random fields, significantly generalizing the widely used two-phase (binary-field) models. The theory produces a rapidly convergent Neumann series that, upon Gaussian closure via a Hermite expansion, yields closed-form first-, second-, and third-order approximations, which achieve percent-level accuracy at first order for isotropic media. For anisotropic media, second-order approximations achieve sub-$2\%$ accuracy across a wide range of local property contrasts and correlations. Our formalism provides mathematically rigorous structure-property closures, with significant implications for the discovery and design of graded and architected materials with tailored transport properties.

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• Electrical conductivity
• Random & disordered media
• Effective medium approach
• Green's function methods