Closed-form expression for the multi-modal unsaturated conductivity function

In this note we derive a closed-form expression representing the hydraulic conductivity function for soils with a multi-modal pore size distribution. By combining the multi-modal representation of the retention function of Durner with the conductivity representation model of Mualem and following van Genuchten, we derive a simple analytical expression for the conductivity of soils with heterogeneous pore systems. Examples for the representation of bi-modal and tri-modal retention and corresponding conductivity curves demonstrate the applicability of the simple analytical expressions. It is concluded that the usefulness of the multi-modal representation of retention functions is increased by providing an analytical expression to directly calculate unsaturated hydraulic conductivities.