Formation Thermal History, Shut-In and Restart Calculator

Calculation sequence

1. Select one axial segment with an initially uniform wellbore and formation temperature. Supply constant formation conductivity and volumetric heat capacity, the total wellbore heat capacity per length, and an effective interface conductance per length. The heat input is a net source after any axial effects have been assessed outside this model; it is not an inlet fluid temperature or circulating mass rate.
2. Solve 40 and 80 logarithmically spaced cylindrical finite volumes coupled to wellbore storage. Apply heat for the first duration, zero input during shut-in, then the same heat rate on restart. A matrix exponential advances each segment without smearing the switching times. Check total stored wellbore-plus-rock energy against the integrated source at every returned time.
3. Require outer radius beyond eight diffusion lengths over the entire history and at least five well radii, keeping the insulated boundary away from the thermal disturbance. Return only if wellbore temperature refinement stays below 0.1 K and temperatures stay within 200-600 K. The shut-in response retains prior rock heating rather than resetting the formation or applying a steady-state heat-loss factor.
4. Calculations remain local in a dedicated WebAssembly worker. The existing self-hosted scientific engine is about 33 MB raw on first use; the envelope additionally loads the pinned CoolProp engine and eccentric flow loads a 178 KB FEM package. Nothing is downloaded until calculation. No composition, heat history or measurements are sent to a calculation server. Save retains canonical input values; file backup can be shared with colleagues, while CSV/PDF retain samples, assumptions and sources. Browser storage alone can be lost.

Worked check

With rw=0.1 m, k=2.5 W/(m.K), formation heat capacity 2.5 MJ/(m3.K), wellbore capacity 50,000 J/(m.K), H=50 W/(m.K) and +100 W/m for 24 h / shut-in 24 h / restart 24 h, the default wellbore warms from 300 K to about 311.732495 K, cools to 302.084492 K, then reaches 312.975423 K. These are synthetic verification inputs.

Synthetic examples for reproducibility, not measured field results.

Assumptions and limitations

• One local axial segment, uniform initial state and constant properties. No axial advection, geothermal gradient, full circulating well profile, casing-layer resolution or phase change.
• Prescribed net heat rate: positive heats, negative cools, and shut-in means zero external heat input, not zero wellbore-to-rock heat transfer. Steel/fluid inventory and effective conductance must be supplied consistently.
• Insulated distant finite domain guarded by a diffusion-length test. This is not an arbitrary-duration infinite-reservoir solution. Energy/grid checks do not establish field accuracy or replace a calibrated transient well model.

Sources and maintenance

Published by PetroCalcHub Team. Implementation and source checks: 2026-10-22. This does not claim independent expert review.

• You, Rahnema and McMillan: unsteady wellbore heat transmission (2016)
• SciPy: matrix exponential algorithm

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