Quantifying Inflow and Infiltration (I/I) in Sewer Systems Using Temperature and Conductivity Sensors — Supplementary Information

Authors: Unknown Year: Unknown (data collection spans 2015, 2019, 2022–2023; journal identifier suggests 2024) Tags: inflow-infiltration, sewer-monitoring, water-quality-sensors, bayesian-calibration, hydraulic-simulation, groundwater-infiltration

TL;DR

This supplementary material documents the mathematical, computational, and empirical detail behind a method that uses in-sewer temperature and conductivity measurements to reconstruct base wastewater flow (BWF) and quantify inflow and infiltration (I/I) in real sewer catchments. It provides the simulation model equations, MAP calibration formulation, GWI threshold logic, and validation KGE scores that underpin the main paper's claims.

First pass — the five C's

Category. Supplementary information to a research prototype / applied modelling paper; contains mathematical derivations, simulation model specification, and empirical validation data.

Context. Urban drainage / sewer systems subfield. Builds directly on Zhang et al. (2018) conductivity-based I/I quantification; uses Nash (1959) instantaneous unit hydrograph for rainfall-to-inflow; Manning et al. (1891) for gravity pipe hydraulics; Li et al. (2023) for sewer pipe–soil heat transfer boundary conditions.

Correctness. Load-bearing assumptions: (1) residuals of the likelihood are normally distributed (Eq. S6); (2) α parameter follows a Laplace prior enabling sparsity (Eq. S7); (3) groundwater level variation from rainfall and from tidal/seawater effects can be linearly superposed (Eq. S11); (4) low-frequency atmospheric temperature linearly predicts groundwater, headspace air, and soil temperatures (Eq. S27). Assumptions (3) and (4) are plausible for shallow coastal systems but are stated without independent validation.

Contributions. - Boxplot-based threshold method (Qu4 = Qu3 + p·IQR) to identify valid calibration windows when permanent GWI keeps groundwater persistently above pipe invert. - MAP estimation via L-BFGS with mixed priors (Laplace on α, Gaussian on β and γ) formulated as negative log-posterior minimization. - Full physics-based sewer simulation coupling Saint-Venant hydraulics, advection-diffusion of temperature and conductivity, and a three-term groundwater discharge model (Eq. S12) for validation of I/I quantification. - Demonstration that average BWF (rather than a full BWF time series) still recovers total I/I reliably, broadening practical applicability.

Clarity. Equations are consistently numbered and symbols defined locally; however, as supplementary material it assumes familiarity with the main paper's notation and goals, making standalone reading difficult. Several figure captions are informative; Figure S2 and S6 schematics aid comprehension.

Second pass — content

Main thrust: A physics-informed simulation model is used to generate synthetic sewer temperature and conductivity data under two I/I scenarios (intermittent and permanent GWI), and a MAP-calibrated reconstruction algorithm recovers BWF with KGE > 0.85 in all tested cases, enabling mass-balance-based I/I quantification.

Supporting evidence: - BWF reconstruction on real data: Ta1 KGE = 0.9627 (train) / 0.9543 (test); Ta2 KGE = 0.9927 / 0.9620; Ca2 KGE = 0.9524 / 0.8716. - Simulation validation (Case1, intermittent GWI): temperature KGE = 0.8637, conductivity KGE = 0.9833. - Simulation validation (Case2, permanent GWI): temperature KGE = 0.8556, conductivity KGE = 0.9280. - Using average BWF instead of actual BWF time series reduces dynamic profile accuracy but total inflow and infiltration volumes remain reliably estimated (Figure S21; no numeric KGE reported for this sensitivity test). - Rainfall exclusion windows selected as w = 36 h (η = 0 mm), 48 h (η > 2 mm), 60 h (η > 5 mm).

Figures & tables: S10 (reconstruction Ta1 with KGE labels) and S17 (Case1/2 BWF comparisons) carry the validation argument; axes appear labeled in described figures. Error bars and confidence intervals are not shown on any figures. Statistical significance is not formally reported — KGE is the sole scalar metric. BIC results (Figure S9) select model structure but raw BIC values are not tabulated. Table S6 is the key quantitative summary.

Follow-up references: - Zhang et al. (2018) — foundational conductivity-based I/I quantification this work extends. - Li et al. (2023) — sewer pipe/soil heat transfer boundary condition modelling, directly reused here. - Nash (1959) — instantaneous unit hydrograph formalism used for rainfall-to-inflow and rainfall-to-groundwater modules.

Third pass — critique

Implicit assumptions: - Superposition of tidal and rainfall groundwater signals (Eq. S11) assumes independence and linearity; violated if tidal fluctuations modulate soil permeability or if rainfall occurs during tidal surge. - Linear low-pass filter model (Eq. S27) for groundwater/soil/air temperature from atmospheric data assumes stationarity of the thermal transfer function — could break during unusual weather or changes in soil moisture. - Groundwater conductivity treated as near-constant (based on 2011–2017 BOM data); any land-use change, drought, or contamination event would invalidate this. - Normal residuals assumed for likelihood (Eq. S6); in-sewer data with pump-induced hydraulic transients (acknowledged in S9) and sensor dropouts likely produce heavy-tailed or bimodal residuals.

Missing context or citations: - No comparison to RTK models (SWMM standard), despite the text claiming instantaneous UH "captures more dynamic information" — this claim is unquantified. - No engagement with other sensor-fusion or data-driven I/I approaches (e.g., machine learning surrogates) that would contextualize the MAP approach's advantage. - The p = 1.5 IQR multiplier for GWI threshold (S1) is borrowed from standard boxplot convention without justification for this specific hydrological application.

Possible experimental / analytical issues: - Validation is conducted on only three real datasets (Ta1, Ta2, Ca2) from a single catchment in coastal Queensland; generalization to inland, combined sewer, or larger catchments is undemonstrated. - Outlier removal by linear interpolation (S9, Figure S19) for pump transients is ad hoc; the fraction of data affected and the sensitivity of I/I estimates to this choice are not reported. - KGE is reported for BWF reconstruction but not directly for the final I/I flow estimates on real (non-simulated) data, where ground truth is unavailable — the core claim of I/I quantification accuracy rests entirely on synthetic simulation cases. - BIC model selection results (Figure S9) are displayed but not discussed quantitatively in the SI; it is unclear how sensitive results are to the chosen periodic structure (Nl values in Table S5). - No uncertainty quantification is propagated from MAP estimates through the I/I calculation to final flow estimates; MAP point estimates suppress posterior uncertainty.

Ideas for future work: - Replace MAP point estimation with full MCMC or variational inference to propagate parameter uncertainty into I/I quantification and report credible intervals. - Test the boxplot GWI threshold (S1) against physically derived thresholds or alternative anomaly-detection methods across diverse catchment types to assess robustness of the p = 1.5 choice. - Validate I/I estimates on a catchment with independent tracer or dye testing to provide ground truth beyond synthetic simulation. - Extend the tidal–rainfall groundwater superposition (Eq. S11) to a nonlinear interaction model and assess whether the simplification materially biases inferred infiltration volumes during storm–tide coincidence events.