Subsurface
Clairvoyance.

HEND is a physics-informed machine learning model that predicts subsurface hydraulic properties from electrical signals.

Global Stats
R²0.72Coefficient of determinationThe fraction of the variance in measured log permeability that the model explains. 1 is a perfect fit; 0 is no better than predicting the mean.
RMSE0.483log m²Root mean squared errorTypical size of the prediction error in log permeability (log m²). Squaring before averaging weights large misses more heavily.
MAE0.356log m²Mean absolute errorThe average absolute difference between predicted and measured log permeability. Less sensitive to outliers than RMSE.

This model is produced from real data using HEND.Click on layers for details

Subsurface Characterisation From Electrical Conductivity

HEND characterises unconsolidated subsurface materials from non-invasive electrical resistivity tomography (ERT) measurements.

Predicted properties
  • Permeabilityk
  • Hydraulic conductivityK
  • Effective pore areaAe
  • Solid-phase electrical conductivityσs
  • Porosities (total and effective)φ, φe
  • Archie cementation exponentm
  • and further petrophysical parameters

Validation On Aquifer Fields

Ground TruthGround truthThe true permeability field of the synthetic aquifer, known by construction. HEND is scored against it.Predicted By HEND (Q50)Predicted by HENDPermeability predicted by HEND from the electrical conductivity data. The quantile shown is Q50 (the median) on the log scale and Q62.5 on the linear scale.Residual ErrorResidual errorThe difference between the prediction and the ground truth at each location. The closer to zero, the better the match.

HEND performance on a synthetic channelised aquifer (1 × 1 km). HEND recovers the geometry and connectivity of the high-permeability channels. Residuals are largest along channel margins, where permeability changes most sharply. The prediction shown is the median (Q50).

HEND performance on a synthetic aquifer recharge case (1 × 1 km). HEND reproduces the location and extent of the high-permeability zones in the lower left of the domain, as well as the fine-scale heterogeneity of the surrounding material. Residuals are small and largely unstructured, with mild deviations around the margins of the high-permeability zones. The prediction shown is the median (Q50).

R²0.719Coefficient of determinationThe fraction of the variance in true log permeability that the prediction explains. 1 is a perfect fit.
RMSE0.483log₁₀ m²Root mean squared errorTypical size of the prediction error in log permeability. Large misses are weighted more heavily.
MAE0.356log₁₀ m²Mean absolute errorThe average absolute difference between predicted and true log permeability.
ME−0.047log₁₀ m²Mean error (bias)The average signed difference between predicted and true log permeability. Near zero means no systematic over- or under-prediction.

A Physics-Informed Machine-Learning Pedotransfer Function

HEND is a physics-informed machine-learning pedotransfer function (ML-PTF) built on a new closed-form permeability formulation that links permeability to pore-scale electrical properties. A ladder of three artificial neural networks predicts m, then σs, then Gc, each feeding the next. An Extra-Trees quantile regression forest, tuned with Optuna’s TPE sampler, then predicts permeability at nine quantiles.

Training Data
6,288
laboratory measurements, global dataset
Model
3 ANNs + ExT-QRF
neural-network ladder and quantile forest
Primary Output
Q05–Q95
nine permeability quantiles

HEND model pipeline. Electrical and soil data pass through three chained neural networks that predict m, σs and Gc. These feed an ExT-QRF model, tuned with TPE in Optuna, which predicts permeability at nine quantiles from Q05 to Q95, with Q50 as the median. The equation k = (Ae/C)Gc then gives the remaining term Ae, and the result is drawn as slice profiles through the ground.

Applications In Hydrogeology, Geotechnics And Soil Physics

HEND estimates Ks from co-located electrical and soil measurements, so it can extend sparse direct tests across a surveyed area.

Aquifer Characterisation

Spatially distributed Ks from ERT or EMI surveys to parameterise groundwater flow models between boreholes and pumping tests.

Geotechnical Site Investigation

Preliminary permeability for seepage, dewatering and consolidation analyses, ahead of in-situ falling-head or packer tests.

Infiltration And The Vadose Zone

Field-scale infiltration capacity as an input to vadose-zone models and to irrigation scheduling.

Solute Transport

Advective velocity estimates, v = Ks·i / ne, to compare travel times and select monitoring locations.

Peer-Reviewed Research

HEND combines a new permeability theory with machine learning to give stable predictions across a global dataset. It is publicly available for non-commercial use.

Publications

  1. Theory

    Rafeek, O., Ab Razak, M. S., Zulkafli, Z., & Nik Daud, N. N. (2026). An analytical pore-scale model for predicting permeability from electrical conductivity. Geosciences, 16(11), 442.

    DOI10.3390/geosciences16110442
  2. HEND

    Reference to be added upon acceptance of the manuscript.

Authors

  • Omar Ali Rafeek
  • Mohd Shahrizal Ab Razak
  • Zed Zulkafli
  • Nik Norsyahariati Nik Daud

Department of Civil Engineering, Universiti Putra Malaysia

Enquiries

water@scients.co