Neural network backpropagation (NNBP) and adaptive neuro-fuzzy inference system (ANFIS) applied to predict incremental gas rate following hydraulic fracturing — across 11 input parameters, 20 actual datasets, validated against December 2011 field actuals.
The study selected 11 parameters as inputs, representing three domains of hydraulic fracturing performance. All data was calibrated by nodal analysis to a common datum (VLP production system) to make datasets from different wells comparable.
Feedforward backpropagation network — 11 inputs, 1 hidden layer with 1 unit, 1 output. Activation functions: logsig and purelin. Dataset split: 12 training, 7 no-CBL, 1 December 2010 testing.
Sugeno FIS with subtractive clustering (cluster radius determines rule number = number of datasets). Hybrid optimization: backpropagation + least squares. Validation dataset: 4% of total data.
Both methods achieved R² approaching 1 in training. The decisive difference appeared in testing scenarios — when training data was progressively reduced and testing data increased:
| Method | 1 test point | 2 test points | 3 test points |
|---|---|---|---|
| NNBP | 0.990 | 0.988 | 0.935 |
| ANFIS | 0.999 | 0.989 | 0.990 |
ANFIS combines fuzzy inference system (FIS) with backpropagation neural networks — this combination provides better generalization with fewer training points than standard NNBP.
Sensitivity analysis using the trained networks (running without each input parameter and calculating delta R²), combined with simple linear regression for confirmation, identified four parameters with average influence exceeding 10%:
A reduced 4-parameter ANFIS model (using only these four drivers) achieved R² = 0.97537 (validation) and R² = 0.99992 (testing) — demonstrating that four parameters can explain the majority of hydraulic fracturing performance variance.
The final ANFIS network was deployed as an Excel macro application for gas rate prediction with additional proppant mass sensitivity analysis. The model was tested against the most recent actual data from a job performed in December 2011 — data the model had never seen during training.
The application predicted 0.400 Mmcfd; the actual job result was 0.470 Mmcfd. R² = 0.96985 — excellent match with actual field data within operating tolerance.