The 3D slab view lumps microbially induced calcite precipitation (MICP) into a single number — a uniform dose that grows everywhere in the crack at the same rate. This 2D view solves the actual spatial problem instead: a longitudinal cross-section of the crack (length along the top, depth into the slab down the side), where dissolved calcium-lactate/urea substrate has to diffuse in from the exposed crack mouth before bacteria anywhere deeper can precipitate calcite from it.
∂C/∂t = D(x,y)·∇²C − R(C,T,moist) (substrate diffusion + consumption)
∂P/∂t = R(C,T,moist)·(1−P) (precipitate builds, saturates at 1)
R = Vmax · spore(·) · exp(−(T−30)²/2σ²) · moistureGate · C/(Km+C)
D(x,y) = D₀ · moistureGate · (1 − 0.85·P) (calcite clogs pores as it forms)
C(y=0) = moistureGate · lactate/(Km_lact+lactate) (fixed feed at the crack mouth)
D₀ = 8.64 mm²/day (≈10⁻¹⁰ m²/s), a realistic order-of-magnitude ionic diffusivity through tortuous, partially-saturated cement paste. Because substrate must diffuse down from the mouth, the near-surface region reaches full calcite saturation in days while the crack tip — a centimetre or so deeper — heals far more slowly, sometimes barely reaching partial saturation before you reset. That depth-limited healing front is real, well-documented behaviour in self-healing concrete literature, and it's the one thing a spatially-uniform 0D model (like the 3D view) cannot show at all.
- Spore density / lactate — spores set the reaction ceiling (Vmax scaling); lactate sets the surface boundary concentration via its own Michaelis–Menten saturation, exactly as it feeds the reaction in the 3D model.
- Temperature — same bell curve peaking at 30 °C, applied everywhere in the field.
- Moisture — gates both the boundary concentration (no film, no supply) and the reaction rate; below ~25% nothing happens anywhere in the crack.
- Clogging feedback — as calcite fills a cell, its local diffusivity drops, which is why the surface can seal completely while starving the interior of fresh substrate even further — a self-limiting effect of real MICP sealing.