A Data-Driven Method for Estimating Sewer Inflow and Infiltration Based on Temperature and Conductivity Monitoring

Authors: Jingyu Ge, Jiuling Li, Ruihong Qiu, Tao Shi, Chenming Zhang, Zi Huang, Zhiguo Yuan Year: 2024 Tags: sewer-inflow-infiltration, temperature-conductivity-sensing, prophet-time-series, mass-energy-balance, base-wastewater-flow-reconstruction, urban-drainage

TL;DR

Replaces costly, fouling-prone flow meters for sewer I/I quantification by using temperature and conductivity sensors: a Prophet-model algorithm reconstructs the base wastewater flow (BWF) signal, then mass/energy balances resolve inflow and infiltration ratios separately. Reconstruction KGE reaches 0.88–0.99 on real catchment data; simulated I/I quantification KGE reaches 0.76–0.97.

First pass — the five C's

Category. Research prototype — new methodology combining time-series modelling with first-principles mass/energy balance, demonstrated on real sensor data (reconstruction step) and simulation (quantification step).

Context. Sewer I/I quantification subfield. Builds on: Zhang et al. 2018a (conductivity-based unit hydrograph calibration, direct predecessor); Aumond & Joannis 2008 (dilution model assuming constant conductivity, shown to be limited); Taylor & Letham 2018 (Prophet time-series decomposition model, adopted as the reconstruction backbone); Guo et al. 2022 (experimental ranking of turbidity, T, and conductivity as I/I tracers, motivates the sensor choice).

Correctness. Core load-bearing assumptions: (1) conductivity obeys conservation like a dissolved-mass tracer; (2) surface water and groundwater are distinguishable from each other and from BWF in both T and conductivity; (3) permanent groundwater infiltration (GWI) can be folded into BWF without distortion; (4) BWF flow rate magnitude is independently known. Assumptions 1 and 2 are physically reasonable but site-dependent; 3 and 4 are acknowledged limitations, not fully resolved.

Contributions. - Prophet-based algorithm that reconstructs BWF temperature and conductivity from in-sewer time series, explicitly handling trends, multi-period Fourier terms, and special events. - Closed-form analytical solution (Eq. 7) for simultaneous inflow and infiltration ratios (α, β) from three paired mass/energy balance equations requiring no geophysical coefficients. - Demonstration that BWF reconstruction generalises across two real Australian catchments with average testing KGE 0.8842–0.9939. - Simulation-based validation of the full I/I pipeline on a real small sewer network under two groundwater scenarios, with total-I/I KGE of 0.9730 (Case 1) and 0.8894 (Case 2).

Clarity. Well-structured and readable; mathematical notation is consistent. Supplementary material carries critical model details (parameter values, simulation inputs, BIC tests) that are not self-contained in the main text, reducing reproducibility from the main paper alone.

Second pass — content

Main thrust: Fit a Prophet time-series model to in-sewer T and conductivity during confirmed dry-weather periods to reconstruct the BWF baseline; then plug measured deviations and independently monitored surface-water/groundwater T and conductivity into mass/energy balance equations to yield inflow and infiltration as fractions of BWF flow—no flow meter needed.

Supporting evidence: - BWF reconstruction: average testing KGE across 10 random splits = 0.9332 (T_a1), 0.9630 (T_a2), 0.8842 (C_a2); all training KGE ≥ 0.94 (Table 2). - Case 1 I/I quantification (intermittent GWI): inflow KGE = 0.8873, infiltration KGE = 0.8622, total I/I KGE = 0.9730 (Table 3). - Case 2 (permanent GWI absorbed into BWF): inflow KGE = 0.8357, infiltration KGE = 0.7628, total I/I KGE = 0.8894 (Table 3); accuracy drop attributed to increased BWF profile complexity. - Simulated network: 0.549 km rising main + 2.098 km gravity pipes; 11 inflow locations, 10 infiltration points; sensor noise = 1% of mean (per HACH specs); 5-minute sampling; 3-month simulation windows. - Separation of inflow vs. infiltration fails (produces swapped attribution) when ΔT₂/ΔC₂ ≈ ΔT₃/ΔC₃, i.e., when surface-water and groundwater T–conductivity signatures converge; authors identify this analytically via Eq. 9.

Figures & tables: - Fig. 2 (frequency-amplitude bar charts): axes labeled, used to select Fourier periods; no uncertainty shown — purely deterministic selection criterion. - Fig. 3 (BWF reconstruction panels A–D): time axes labeled with dates, y-axes labeled with units (°C); no confidence intervals on reconstruction; visual correspondence is qualitative. - Fig. 4 (I/I quantification, Case 1, panels A-1/A-2/B-1/B-2/B-3): residuals plotted as green bars, units (m³/h) on flow panels; no statistical significance tests; error bars absent. Case 2 equivalent is in supplementary (Fig. S18), not in main paper. - Table 2: KGE mean and SD across 10 splits provided — this is the most statistically credible result in the paper. - Table 3: single KGE per scenario, no confidence interval, no sensitivity to simulation parameter choices.

Follow-up references: - Taylor & Letham 2018 ("Forecasting at Scale") — Prophet model architecture, required to understand reconstruction parameterisation. - Zhang et al. 2018a (J. Hydrol.) — conductivity-only IUH calibration approach, the closest prior method for comparison. - Figueroa et al. 2021 & 2023 (Water Res.) — in-sewer thermal-hydraulic model underpinning the simulation module's heat transfer equations. - Guo et al. 2022 (Water Sci. Technol.) — experimental basis for choosing T and conductivity over turbidity as I/I tracers.

Third pass — critique

Implicit assumptions: - Conductivity is treated as conserved like a dissolved mass (Eq. 6 third line), but conductivity is a non-linear function of ion composition and temperature; the approximation breaks down with mixing of waters having very different ionic compositions. This is not quantified. - BWF flow rate is assumed independently known (from dry-weather measurement or population estimates); errors in this quantity propagate directly into absolute inflow/infiltration estimates but are not analysed in the paper. - The Prophet model assumes BWF patterns are quasi-stationary over the reconstruction horizon; structural changes (e.g., new connections, industrial discharge changes) would invalidate the reconstruction without retraining. - Permanent GWI is defined as part of BWF and hence treated as zero infiltration — this is a definitional choice that obscures baseline leakage volumes and is noted only briefly. - Rainfall threshold parameters (η = 1, 3, 8 mm; w = 24, 48, 72 h) are stated but no sensitivity analysis is reported; the method's performance could be sensitive to these choices.

Missing context or citations: - No comparison against a flow-meter-based method on the same simulated network; KGE values for the proposed method cannot be contextualised without a baseline competitor. - Stable-isotope methods (Kracht et al. 2007, De Bondt et al. 2018) cited but not benchmarked against; their achievable accuracy at comparable cost is unaddressed. - No discussion of sensor drift or fouling for conductivity probes over multi-month deployments, despite this being a stated motivation for avoiding flow meters. - Uncertainty propagation from T/conductivity sensor noise (1% white noise added) through the algebraic Eq. 7 is not analytically characterised; the denominator of Eq. 7 can become near-zero, amplifying noise unpredictably.

Possible experimental / analytical issues: - The I/I quantification is validated entirely by simulation, not against real measured I/I volumes; the simulated network uses the same IUH and discharge model families that prior literature has criticised for parameter sensitivity — this is circular. - Only one small network (one Australian coastal city) is simulated; generalisability to larger, more complex networks with longer pipe travel times, multiple pumping stations, or different climates is undemonstrated. - The denominator of Eq. 7 approaches zero when surface-water and groundwater signatures are similar; no robust fallback or uncertainty flag is built into the algorithm beyond a qualitative observation in Section 4.3. - Ten random 70/30 splits are used for reconstruction validation, but splits are time-series data — random splitting can leak temporal structure, potentially inflating KGE on the test set. - Case 2 (permanent GWI) lumps GWI into BWF, so the reported infiltration KGE measures only rainfall-driven infiltration above the permanent baseline; the permanent component is never quantified or validated. - No ablation: it is unclear whether both temperature and conductivity are jointly necessary, or whether one alone would suffice; only the combined system is tested.

Ideas for future work: - Apply the method to real sewer networks with independent ground-truth I/I volumes (e.g., from dye-tracing or flow-difference campaigns) to replace the simulation-only validation. - Develop an automated calibration-sample selector directly from T/conductivity anomaly detection, removing dependence on external rainfall and borehole data. - Extend to multi-sensor spatial networks to localise I/I entry points, as sketched in Section 5.2, and assess the additional uncertainty introduced by propagating errors through multiple nodes. - Analyse error amplification in Eq. 7 as a function of the angle between the (ΔT₂, ΔC₂) and (ΔT₃, ΔC₃) vectors to define a site-suitability criterion before sensor deployment.