The water retention curve (WRC) and the hydraulic conductivity function (HCF) are key ingredients in most analytical and numerical models for flow and transport in unsaturated porous media. Despite their formal derivation for a representative elementary volume (REV) of soil complex pore spaces, these two hydraulic functions are rooted in pore-scale capillarity and viscous flows that, in turn, are invoked to provide interpretation of measurements and processes, such as linking WRC with the more difficult to measure HCF. Numerous conceptual and parametric models were proposed for the representation of processes within soil pore spaces and inferences concerning the two hydraulic functions (WRC and HCF) from surrogate variables. We review some of the primary models and highlight their physical basis, assumptions, advantages, and limitations. The first part focuses on the representation and modeling of WRC, including recent advances such as capillarity in angular pores and film adsorption and present empirical models based on easy to measure surrogate properties (pedotransfer functions). In the second part, we review the HCF and focus on widely used models that use WRC information to predict the saturated and unsaturated hydraulic conductivity. In the third part, we briefly review issues related to parameter equivalence between models, hysteresis in WRC, and effects of structural changes on hydraulic functions. Recent technological advances and monitoring networks offer opportunities for extensive hydrological information of high quality. The increase in measurement capabilities highlights the urgent need for building a hierarchy of parameters and model structures suitable for different modeling objectives and predictions across spatial scales. Additionally, the commonly assumed links between WRC and HCF must be reevaluated and involve more direct measurements of HCF. The modeling of flow and transport through structured and special porous media may require special functions and reflecting modifications in the governing equations. Finally, the impact of dynamics and transient processes at fluid interfaces on flow regimes and hydraulic properties necessitate different modeling and representation strategies beyond the present REV-based framework.

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