Inverse laplace transform to fit soil water retention curve and estimate the pore size distribution

Soil Water Retention Curve (SWRC) provides crucial information for understanding soil moisture retention, essential for agriculture, hydrology, engineering and environmental science applications. Many SWRC fitting models in the literature are based on empirical equations without a direct physical meaning. However, SWRC data is physically related to the soil’s porous structure and its interactions with the wetting fluid. Hence, the curve’s behavior reflects the porous complexity. Non-physical model equations might even be able to fit the data to be used in several applications; however, the search for physically fitting models representing the SWRC data as a smooth continuous distribution function can reflect new insights and information about this heterogeneous porous media. In this regard, the well-established physically-based Kosugi model is based on the assumption of lognormal pore size distributions. However, a general approach for any modality and distribution shape could be interesting. This paper proposes applying the mathematical method known as “Inverse Laplace Transform” (ILT) to fit the Soil Water Retention Curve using a weighted superposition of exponential decays. This multi-exponential approach involves working with two physically related parameters, the amplitude and its respective characteristic matric potential, which are physically interpreted as the amount of pores that empty at that suction head. The ILT-EXP method proposed was implemented in Python software to fit the curves, and it is now available in an online web app. The evaluation of the ILT-EXP model to fit SWRC data is discussed, presenting its potential to estimate soil pore size distribution of multimodal samples. One advantage of ILT-EXP over other multimodal models is that it does not need to know how many modal components are present in the SWRC data, being automatically determined by the method. Finally, a statistical fitting comparison of 439 SWRC data, with six other classical models is discussed. The results indicate that fitting with the ILT-EXP model demonstrates strong potential, making it a powerful method for handling multimodal curves. This approach represents a novel and robust method for estimating a smooth, continuous soil pore size distribution.