Soil Water Retention Curve

Theory: SWRC AI

What the water retention curve is, how machine learning predicts it from soil properties and how the classic models are fitted to it.

Suction and water retention

In an unsaturated soil, water is held in the pores by capillarity and adsorption, at a pressure below that of the air. The difference is the matric suction, ψ = ua − uw: the drier the soil, the higher the suction needed to pull out the water that remains.

The soil-water retention curve (SWRC) relates the volumetric water content θ to suction ψ. It is the key property of unsaturated soils: hydraulic conductivity, shear strength and water storage all follow from it.

The suction at which a pore empties depends on its radius, by the Laplace equation — large pores drain first, at low suction:

\psi = \frac{2\, T \cos\alpha}{r}

T: surface tension of water (≈ 0.073 N/m at 20 °C) · α: contact angle · r: pore radius

Shape of the curve

Saturated zone: up to the air-entry value the soil stays saturated, with θ at the saturated content θs.
Desaturation zone: air enters the larger pores and θ drops quickly with suction.
Residual zone: only water adsorbed to the particles is left; θ approaches the residual content θr and changes little, up to about 10⁶ kPa, when the soil is dry.

Sands have large, uniform pores: a low air entry and a sharp drop. Clays hold water up to much higher suctions. Aggregated tropical soils, such as lateritic ones, have two pore families (between and within aggregates) and can show a bimodal curve with two steps.

The drying curve differs from the wetting curve (hysteresis). This app predicts the drying curve, which is what the lab tests in the training data measured.

Prediction by machine learning

Measuring the curve takes long tests (pressure plate, filter paper, tensiometers). Pedotransfer functions instead estimate it from simple soil properties. This app uses an Extra Trees (extremely randomized trees) model trained on measured drying curves.

Input	Accepted range	Why it matters
Sand, silt, clay, gravel	Ranges of the training data	Pore size distribution
Plasticity index	0–58%	Activity of the clay fraction
Porosity	35–61%	Void volume; gives θs
Suction ψ	0–10⁶ kPa	Point on the curve

Each tree splits the input space into regions at randomly chosen thresholds and predicts the mean measured water content in each region; the final prediction is the average over all trees. The 95% band comes from the model's validation error.

Outside the ranges of the training data the model would extrapolate without support, so the app rejects those values.

Fitted models

To use the curve in calculations and flow software, three classic models are fitted to the predicted curve by least squares, with all parameters non-negative. The saturated content θs is fixed at the porosity; θr and the other parameters are fitted, and R² measures the fit.

\theta = \theta_r + \frac{\theta_s - \theta_r}{\left[1 + (\alpha\psi)^n\right]^m}

van Genuchten (1980) · α: related to the inverse of the air entry · n: slope of the desaturation zone · m: asymmetry of the curve, fitted freely here

\theta = \theta_r + (\theta_s - \theta_r)\, e^{-\delta\psi}

Cavalcante & Zornberg (2017) · δ: desaturation rate; a single parameter that allows analytical solutions of unsaturated flow

\theta = \theta_r + (\theta_s - \theta_r) \left[\lambda e^{-\delta_1\psi} + (1 - \lambda)\, e^{-\delta_2\psi}\right]

Costa & Cavalcante (2021) · bimodal: δ₁ and δ₂ are the desaturation rates of each pore family and λ the weight of the first

Uses and limits

Infiltration and transient flow in unsaturated soils
Estimating the hydraulic conductivity function
Shear strength of unsaturated soils and slope stability under rainfall
Contaminant transport and cover barriers

The prediction is an estimate: it ignores hysteresis, structure, volume change on drying and mineralogy beyond what the plasticity index captures. For design, confirm it with tests on the soil at hand.

References

Fredlund, D. G.; Rahardjo, H. Soil Mechanics for Unsaturated Soils. New York: Wiley, 1993.
van Genuchten, M. Th. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal, 44(5), 892–898, 1980.
Cavalcante, A. L. B.; Zornberg, J. G. Efficient approach to solving transient unsaturated flow problems. I: Analytical solutions. International Journal of Geomechanics, 17(7), 2017.
Costa, M. B. A.; Cavalcante, A. L. B. Bimodal soil–water retention curve and k-function model using linear superposition. International Journal of Geomechanics, 21(7), 2021.
Geurts, P.; Ernst, D.; Wehenkel, L. Extremely randomized trees. Machine Learning, 63, 3–42, 2006.

Enter the soil's grain size, plasticity index and porosity to get its water retention curve predicted by machine learning, with the classic models fitted to it.

Retention curve

θ = θr + (θs − θr) / [1 + (αψ)ⁿ]ᵐ

Soil

Grain size

Sand%0–100%Silt%0–85%Clay%0–89%Gravel%0–5%

Total: 100%

Plasticity index%0–58%Porosity%35–61%

Model

Parameters

—

θs is taken as the porosity; the other parameters are fitted to the predicted curve.

Developed by Geofluxo with GEOAMB (Environmental Geotechnics Laboratory, UFBA). An Extra Trees model trained on measured drying curves predicts the volumetric water content from 0 to 10⁶ kPa of suction; accepted values are the ranges of the training data.