The 3D companion sim treats the ground beyond the borehole wall as an infinite fixed-temperature reservoir and solves only a steady resistance stack. This 2D sim instead solves the actual transient radial diffusion equation in cylindrical coordinates for both the grout and native soil:
∂T/∂t = (1/r) ∂/∂r( r · α(r) · ∂T/∂r )
α = k / (ρ·c_p) [thermal diffusivity, m²/s]
It's solved with an explicit finite-volume scheme on ~115 concentric shells (grout: 0.016–0.075 m, soil: 0.075–5 m), with harmonic-mean conductances across the grout/soil interface so flux stays continuous. Heat enters at the pipe wall from the fluid side (film convection + pipe-wall conduction, Dittus–Boelter as in the 3D model) and diffuses outward; the far edge is held at the undisturbed ground temperature.
At steady state this reduces to exactly the same two log-resistance terms used in the 3D sim's resistance stack — R_grout = ln(r_bore/r_pipe)/(2πk_grout) and an equivalent R_soil = ln(r_max/r_bore)/(2πk_soil) — which is how this model was verified: run to steady state with a constant injected flux, the numerical profile matches the analytic ln(r) law to within a few percent (small, bounded finite-volume discretization bias, not a sign or instability error).
- Grout conductivity — sets the diffusivity of the inner shell; low conductivity slows the approach to steady state and raises the grout's own resistance.
- Soil conductivity — new lever this 2D model exposes that the 3D one can't: a low-conductivity soil (dry sand) both resists heat flow more and diffuses a "cold plume" (heating mode) or "hot plume" (cooling) more slowly, so it doesn't recover between cycles as fast as a high-conductivity saturated soil or rock.
- Time speed — simulated hours per real second; diffusion into soil is slow (days to weeks to fully equilibrate), so time is accelerated to make the plume's growth visible.
- Reset field — snaps the whole temperature field back to the undisturbed ground temperature and restarts the clock.
Watch the borehole-wall temperature (not the fixed far-field temperature) drift away from the undisturbed value under sustained one-directional operation — this thermal depletion, and the fact that recovery during idle periods is slower than depletion during use, is exactly why real ground-source systems are sized with duty-cycle and long-term ground-temperature drift in mind, something a pure steady-state resistance calculation can't show.