An Extension of Gardner's Model for the Hydraulic Conductivity Curve.

Gardner’s equation relating the hydraulic conductivity (K) to suction (s) is largely used in soil hydraulics. However, in general Gardner’s model  is not valid beyond a narrow suction range. Here we extended  the relative hydraulic conductivity (Kr) depletion, calculated by Gardner’s model as an inverse exponential of s, into a log(Kr) depletion, calculated as an inverse exponential of log(s). We called this arrangement as the Gardner’s dual model (GD). The transition suction (so) beyond which the classical Gardner’s model is no more valid corresponds to an inflection point commonly observed on the log(Kr) vs. log(s) curve. From Gardner’s model: log[Kr(so)]=-log(e)so/λ, where e is the Neper constant and λ the Gardner’s macroscopic capillary length. Imposing that Kr(s) be continuous and smooth at so, we propose that for s>so:

log[Kr(s)] = log[Kr(so)] –(so β/λ){1 – exp[-log(s/so)/β]},

where β, called the conductive depletion coefficient, is  positive and dimensionless. So, the GD model for Kr has three parameters (so, β, λ).  We selected 153 K(s) datasets from the UNSODA database (including saturated K) and compared the RMSE statistics, in log(K) residues, calculated by the GD formulation with the RMSE calculated by the Mualen-van Genuchten (MVG) model with two fitting parameters (Ko, a matching point at saturation; L, the pore connectivity parameter).We notice that both models have two degrees of freedom for matching the log(K) vs. s data. The mean RMSE’s for the GD and MVG models were 0.38 and 0.47, respectively. In general the GD model worked better than MVG also among the textural classes. The mean so varied from 24 cm (clayey soils) to 61 cm (sandy soils). So, beyond its analytical simplicity, the proposed model has large potential to describe the hydraulic conductive curve.