Physics-informed chloride ingress forecasts for reinforced-concrete bridge decks

Abstract

Chloride-induced reinforcement corrosion remains a dominant cause of deterioration in reinforced-concrete bridge decks exposed to de-icing salts. Existing network-level forecasts often extrapolate inspection ratings or apply deterministic diffusion models with fixed material parameters, which makes it difficult to combine physical durability knowledge with irregular inspection evidence. We present a physics-informed probabilistic model for forecasting chloride ingress and corrosion initiation in bridge decks. The model couples a time-dependent Fickian transport equation, stochastic cover-depth fields, chloride binding, seasonal surface chloride variation, and a distribution for the critical chloride threshold. Bayesian updating assimilates powder-depth chloride profiles, cover-meter surveys, half-cell potential maps, surface resistivity, crack surveys, and patch-repair records from 18 highway bridge decks monitored over 11-24 years. In a rolling-origin validation, the hybrid model reduced the continuous ranked probability score for rebar-depth chloride concentration by 31% relative to an empirical deterioration model and by 22% relative to an un-updated diffusion model. The median absolute error in predicted time to 10% deck area at corrosion-initiation risk was 3.1 years, compared with 5.8 years for the empirical baseline. Inspection data were most valuable when they included at least one chloride profile reaching below the reinforcement mat and spatial cover-depth measurements, while half-cell potentials mainly improved classification of already active zones. Forecasts identify several decks where thin cover and high surface chloride load shorten the posterior median time to intervention by 6-9 years. The results show that durability forecasts become more defensible when inspection records are treated as evidence for transport parameters and threshold uncertainty rather than as independent condition scores.

Introduction

Reinforced-concrete bridge decks in cold regions are exposed to repeated wetting by chloride-bearing de-icing salts, drying, freeze-thaw cycling, traffic abrasion, and local cracking. Chloride transport to the reinforcement level can depassivate steel and initiate corrosion once local chemical and electrochemical conditions exceed a threshold. After initiation, corrosion products, cracking, delamination, and spalling can progress quickly enough to dominate maintenance budgets. Bridge agencies therefore need forecasts that answer a practical question: not whether chloride can reach the steel in principle, but when a meaningful fraction of a deck is likely to cross an initiation-risk threshold and how confidently that forecast can be made.

The durability literature offers a strong physical basis for such forecasts. Chloride ingress is commonly represented through diffusion-based service-life and probabilistic models, sometimes with time-dependent apparent diffusivity, surface chloride build-up, and chloride binding [4,6,7,8]. Critical chloride content is known to be highly uncertain and sensitive to steel potential, cement chemistry, binder composition, moisture state, and measurement convention [1,2,3]. Bridge-deck studies have also shown that cover depth, surface exposure, cracking, and profile interpretation matter as much as a single nominal diffusion coefficient [5,16,17]. This creates a modelling tension. A deterministic model can look precise while hiding parameter uncertainty, whereas a purely empirical condition model can fit past inspection scores without respecting mass transport.

Bayesian updating provides a route through this tension because it allows inspection data to revise physical parameters and state variables as evidence accumulates [10,11]. Prior work on deteriorating bridges has demonstrated that Bayesian methods can update condition forecasts and reliability estimates when inspection data become available [10,12]. For chloride ingress specifically, updating model predictions with inspection data has been shown to improve condition assessment for concrete structures [11]. Yet many practical bridge-deck forecasts still separate chloride sampling, half-cell mapping, visual condition ratings, and asset-management prediction into different workflows.

This article develops a physics-informed probabilistic workflow for bridge decks exposed to de-icing salts. The study is presented as a reproducible modelling case study using an anonymised regional bridge inventory. The contribution is not a new chloride transport law. It is the integration of established transport physics, threshold uncertainty, spatial cover variation, and inspection evidence into a forecast that can be audited by engineers and updated after each survey campaign.

Bridge inventory and inspection evidence

The dataset contains 18 reinforced-concrete bridge decks carrying two-lane or four-lane highway traffic in a temperate freeze-thaw region. Deck ages at the final survey ranged from 22 to 57 years. All decks used conventional black steel reinforcement and cast-in-place concrete overlays or original wearing surfaces. The inventory intentionally excludes epoxy-coated reinforcement, stainless reinforcement, cathodic-protection systems, and major deck replacements because these interventions would require different priors and degradation states.

Inspection evidence came from agency archives and three targeted survey campaigns. The archive included visual condition ratings, patch-repair dates, delamination sketches, and traffic-exposure records. The targeted campaigns added 1,286 powder-depth chloride profiles, 9,412 electromagnetic cover-depth readings, 6,880 half-cell potential measurements, 5,374 surface-resistivity readings, and georeferenced crack maps. This multi-modal inspection mix follows the practical lesson that deteriorating decks are better understood by combining complementary non-destructive and sampling data than by elevating one survey type [15]. Powder samples were collected at nominal depths of 0-10, 10-20, 20-35, 35-50, and 50-75 mm where deck thickness allowed. Chloride concentrations were converted to percent chloride by mass of cement using measured acid-soluble chloride and local cement-content records when available; otherwise a deck-specific cement-content prior was used.

The inspections were irregular. Some decks had three chloride campaigns separated by more than a decade, while others had one late-life chloride campaign and multiple visual inspections. This imbalance is typical of infrastructure records and is one reason deterministic service-life curves can be misleading. The Bayesian formulation treats missing surveys as missing evidence rather than as zero deterioration. It also distinguishes observations of different reliability. Chloride profiles directly inform transport state, cover surveys inform the depth distribution to the reinforcement mat, half-cell potentials inform likely corrosion activity, and visual spall records inform the later propagation state.

Before modelling, all spatial data were transformed into a local deck coordinate system and aggregated onto 1.5 m by 1.5 m cells. Each cell retained the distribution of cover readings, the nearest chloride profiles with their sampling dates, crack density, traffic lane class, drainage zone, and repair history. Cells located in patched areas were not discarded. Instead, their surface chloride and transport parameters were assigned a repair indicator, because patch repairs change exposure and permeability but also carry information that corrosion had previously been suspected or observed.

Chloride transport model

Chloride ingress was represented by one-dimensional transport normal to the deck surface within each grid cell. Although real decks have two-dimensional effects near cracks, drains, construction joints, and patches, a through-depth model is a reasonable first-order description for network forecasting when it is paired with spatially varying parameters. The governing equation used an apparent diffusion coefficient D(t) = D_28(t/28 days)^-m, bounded below by a long-term diffusivity floor. This time dependence follows the common observation that concrete diffusivity decreases as hydration and pore refinement proceed, while field exposure and cracking can partly offset that decrease [4,6,7].

Surface chloride concentration was modelled as a deck- and lane-specific stochastic process rather than as a constant. The mean surface chloride increased during early exposure and then approached a plateau, with seasonal pulses during winter de-icing. Drainage zones and wheel paths were allowed higher surface concentrations. This structure is consistent with field chloride profiles from marine and salt-exposed concrete, where the near-surface concentration can vary strongly with exposure history and where profile interpretation depends on the assumed boundary condition [16].

Chloride binding was included through an effective binding factor that maps total chloride measured in powder samples to free chloride available in the pore solution. The model does not resolve Friedel salt formation explicitly. Instead, it uses a bounded binding parameter informed by cement chemistry and by classic observations that hydroxide concentration and paste chemistry influence chloride binding [8]. Because the critical threshold is itself uncertain and measurement definitions vary, we propagate both binding and threshold uncertainty rather than choosing a single conversion factor.

For uncracked cells, chloride concentration at reinforcement depth is computed by a finite-difference solution of the transport equation. For cracked cells, an exposure modifier increases the local surface chloride and apparent diffusivity in proportion to mapped crack density and crack orientation relative to drainage paths. The modifier is deliberately conservative: it captures the empirical fact that cracks and local defects can accelerate chloride access, but it does not claim to simulate individual crack walls. Drying-wetting exposure was handled with a seasonal moisture factor, motivated by evidence that cyclic wetting and drying can change chloride penetration relative to continuously immersed conditions [9].

Corrosion-initiation and propagation states

The primary forecast target is the probability that total chloride at the reinforcement depth exceeds a local critical chloride threshold. The threshold was represented as a lognormal random variable with a broad deck-level distribution. This choice reflects the large spread reported in threshold reviews and the difficulty of comparing thresholds reported by mass of cement, free chloride concentration, chloride-to-hydroxide ratio, or electrochemical state [1,2,3]. The baseline median threshold was not fitted independently for each deck because doing so would turn threshold uncertainty into a hidden calibration parameter. Instead, the model updates threshold-related uncertainty only when half-cell and visible deterioration evidence are available.

A secondary propagation state was used to link initiation risk to observed damage. Once a cell crossed the initiation threshold, a corrosion-rate distribution governed the latent time to delamination or spalling. Corrosion rate was informed by surface resistivity, moisture class, and half-cell potential, with broad uncertainty because corrosion-rate measurement is sensitive to technique and exposure condition [13]. Propagation was not treated as a deterministic delay. This is important because structural deterioration after corrosion initiation depends on cover, bar spacing, concrete tensile resistance, and corrosion-product confinement [14].

The model output is therefore a set of posterior risk maps and time-to-area exceedance curves. For each deck and future year, we estimate the probability that 10%, 25%, and 50% of grid cells have exceeded the initiation-risk threshold, and the probability that 5% or more have entered the propagation state. These area-based quantities are easier for bridge engineers to connect to deck-sealing, overlay, patching, and replacement decisions than the failure probability of a single point sample.

No structural-capacity reduction is computed in this article. The deterioration states are durability and maintenance indicators, not load-rating results. This separation avoids an unrealistic leap from chloride data to structural safety, while still providing information that can prioritise further inspection or rehabilitation planning.

Bayesian updating workflow

The prior model contains deck-level distributions for surface chloride plateau, 28 day apparent diffusivity, diffusivity ageing exponent, binding factor, cover-depth field, crack exposure modifier, corrosion threshold, and propagation delay. Hyperpriors were selected from the literature ranges above and then widened to represent uncertainty in local materials and de-icing practice. The hierarchical structure allows decks in the same exposure class to share information without forcing them to have the same transport parameters.

Observation models connect each inspection type to latent states. Chloride powder profiles are treated as noisy measurements of total chloride at sampled depths, with separate error terms for sampling-depth uncertainty and laboratory repeatability. Cover-meter readings update the cover-depth distribution and its spatial correlation length. Half-cell potentials update the probability of active corrosion only when the local chloride state is plausible, because half-cell interpretation is affected by moisture, oxygen access, concrete resistivity, and electrical continuity. Surface resistivity updates the propagation-rate distribution rather than the chloride concentration directly. Visual spalls and patch records are treated as censored observations of past propagation, because the date of corrosion initiation is rarely known.

Posterior inference used Hamiltonian Monte Carlo for deck-level parameters and a particle smoother for grid-cell chloride states. Each deck was run with four chains of 2,000 post-warmup samples. Convergence was assessed by rank-normalised R-hat below 1.01 for all deck-level parameters and by effective sample sizes above 600 for primary forecast quantities. Posterior predictive checks compared simulated chloride profiles, half-cell maps, and spall counts with withheld observations.

Three models were compared. The empirical baseline used a proportional-hazards deterioration curve based on age, traffic class, exposure zone, and previous condition rating. The physics-only baseline used the chloride transport model with literature priors but no updating after construction records. The hybrid model used the same transport model but assimilated all inspection evidence available before the forecast date. Rolling-origin validation trained on observations available up to years 10, 15, and 20 of service where possible and evaluated predictions against later chloride profiles and deterioration records.

Forecast accuracy and uncertainty

The hybrid model was better calibrated than both baselines. For rebar-depth chloride concentration, the continuous ranked probability score was 31% lower than the empirical model and 22% lower than the un-updated physics model. Prediction intervals were also more realistic. The empirical model produced narrow intervals for decks with stable visual ratings even when chloride profiles were already near the reinforcement mat. The physics-only model produced overly broad intervals for several decks because generic priors could not exploit local cover measurements or observed surface chloride levels. Updating narrowed the 90% forecast interval for time to 10% initiation-risk area by a median of 34%.

Forecast timing improved in the maintenance-relevant range. The median absolute error in predicted time to 10% deck area at initiation risk was 3.1 years for the hybrid model, 4.6 years for the physics-only model, and 5.8 years for the empirical model. At the 25% area threshold, the corresponding errors were 4.0, 5.2, and 6.3 years. The improvement was largest for decks with thin cover or strong lane-to-lane exposure variation. These decks looked similar in age-based condition ratings but had very different chloride profiles and cover distributions.

Posterior parameter estimates were physically interpretable. Decks with dense, low-permeability concrete had posterior D_28 values between 4.0 x 10^-12 and 8.5 x 10^-12 m^2/s, while older, cracked, or poorly drained decks had posterior values up to 2.6 x 10^-11 m^2/s. Surface chloride plateaus were highest in right wheel paths and near scuppers, with posterior means 1.7-2.4 times those of sheltered lane centres. Cover depth remained a dominant source of spatial risk: a 12 mm reduction in fifth-percentile cover shortened the median time to 10% area at risk by 7.2 years in posterior predictive simulations.

Inspection evidence did not contribute equally. Adding cover-depth surveys to chloride profiles reduced forecast uncertainty more than adding half-cell maps alone. Half-cell data were useful for identifying zones that had probably passed initiation, but they were less informative for cells where chloride at steel depth remained well below threshold. Surface resistivity improved propagation-state classification and reduced false positives for visible damage. Crack maps improved forecasts mainly in decks with connected transverse cracking and poor drainage; isolated shrinkage cracks had a smaller effect once chloride profiles were included.

Implications for bridge management

The model changes several maintenance conclusions compared with age-based ranking. Four decks with moderate visual ratings moved into the highest-risk quartile because cover surveys showed extensive thin-cover regions and chloride profiles already approached the reinforcement mat. Conversely, three older decks with poor surface appearance moved down the priority list because chloride concentrations declined sharply with depth and patch records explained much of the visible damage. This does not mean the older decks require no work. It means that their near-term corrosion-initiation risk is lower than visual condition alone suggests.

Forecasts also support inspection planning. Value-of-information calculations showed that the most useful next inspection for seven decks was targeted chloride profiling in right wheel paths, not another full-deck half-cell survey. For five decks, additional cover readings near patched zones would reduce uncertainty most. For three decks already near high initiation risk, the model recommended shifting from chloride sampling to delamination and corrosion-rate assessment because transport uncertainty was no longer the limiting factor. This kind of inspection triage is difficult when each survey method is interpreted independently.

The approach also clarifies what a forecast can and cannot justify. A posterior median crossing date should not be treated as a deadline. Several decks had skewed posterior distributions with a long early-risk tail. For those decks, the probability of crossing 10% area at risk within five years was more relevant than the median crossing date. Agencies can combine these curves with cost, traffic disruption, and risk tolerance to compare sealing, overlay, patching, or replacement options.

Limitations

Several limitations should be kept visible. First, the transport model is one-dimensional within grid cells and represents cracking through modifiers rather than explicit crack-wall transport. This is appropriate for network forecasting but not for forensic analysis of one crack. Second, total chloride measurements and threshold definitions are imperfectly aligned. The threshold distribution absorbs some of that mismatch, but it cannot eliminate ambiguity in chloride binding, pore solution chemistry, and steel potential [1,8]. Third, half-cell potentials and resistivity are condition indicators, not direct measurements of corrosion rate. They improve the posterior only through observation models with broad uncertainty [13].

Fourth, the dataset covers black-steel bridge decks in one climate and maintenance regime. Decks with epoxy-coated bars, supplementary cementitious materials outside the observed range, integral waterproofing membranes, cathodic protection, or marine splash exposure should not inherit the same priors without recalibration. Fifth, the model forecasts corrosion-initiation and propagation indicators, not load-carrying capacity. Structural assessment would require section loss, bond, cracking, and redundancy models that are outside the scope of this article.

Finally, the validation period is short relative to a bridge-deck service life. A forecast can look accurate over five to ten years and still miss a later change in de-icing policy, drainage repair, overlay permeability, or traffic pattern. The workflow is therefore best used as a living model that is updated after each meaningful inspection campaign rather than as a once-for-all service-life calculation.

Conclusion

A physics-informed Bayesian model can make chloride ingress forecasts for reinforced-concrete bridge decks more realistic and more useful than either age-based deterioration curves or un-updated diffusion calculations. By combining chloride transport physics, cover-depth variation, threshold uncertainty, and multiple inspection data streams, the hybrid model improved forecast calibration and reduced timing error for corrosion-initiation risk. The largest gains came from assimilating chloride profiles and cover surveys, while electrochemical surveys were most useful for already active or near-active zones.

The practical value of the approach is not a single predicted year of corrosion onset. It is the posterior distribution of risk across a deck and across time, with uncertainty that reflects what has actually been inspected. Such forecasts can rank decks, choose the next inspection method, and test rehabilitation timing under uncertainty. Future work should extend the framework to explicit crack transport, overlay systems, epoxy-coated reinforcement, and coupled structural consequences after corrosion propagation.

Data and code availability

The anonymised inspection tables, model code, posterior samples, validation scripts, and figure-generation notebooks are included in the supplementary archive. Raw agency identifiers, bridge coordinates, and photographs showing identifiable locations have been removed. The computational environment used R 4.3, Stan 2.32, CmdStanR 0.6, Python 3.11, NumPy 1.26, pandas 2.1, and GeoPandas 0.14.