How the wellbore model works

Each well is treated with Ramey's (1962) quasisteady energy balance, ṁcw dTf/ds = −(2πrcoU)(Tf − Te(z)), against a linear geotherm Te(z) = Tsurf + az with a = (Trock − Tsurf)/Lvert. The relaxation length is A = (ṁcw/2π) [ke + rcoU f(t)] / [rcoU ke], with the transient conduction function from the Hasan and Kabir correlations, f = 1.1281√tD(1 − 0.3√tD) for tD ≤ 1.5 and f = (0.4063 + ½ln tD)(1 + 0.6/tD) above it, where tD = αt/rwb² (Eqs. 4.6a and 4.6b). Injector (down): T(L) = Tsurf + aL − aA + (Tinj − Tsurf + aA)e−L/A. Producer (up): Twh = Tsurf + aA + (Tres,out − Tsurf − aL − aA)e−L/A. Because f(t) grows as the near-well rock equilibrates, A grows and both the injector gain and producer loss shrink with time — the opposite trend to reservoir drawdown.

Limitations worth knowing

The Gringarten solution assumes a constant inlet temperature, but the injector delivers a slowly changing one; the coupling here is quasistatic rather than a rigorous superposition. Friction dissipation (~1 °C per 5 MPa) and decompression cooling (~0.02 °C/MPa) are neglected, as are two-phase flow, non-constant fluid properties, and heat exchange in the lateral itself. Eqs. 4.6a and 4.6b remove the vanishing-radius error of the line source at early times, but Ramey's energy balance is still quasisteady and takes no account of the heat stored in the wellbore fluid itself, so the first days of any flow period remain approximate.

Note that the original Gringarten model (as implemented here) is known to be optimistic when the fracture length is not long compared to the height, because the Gringarten model assumes one-dimensional flow of water, whereas in reality the flow is two-dimensional so doesn't "touch" as much rock surface area. Barros-Galvis, Ehlig-Economides & Guevara, arXiv:2601.22316, "A Generalized Analytical Heat Transfer Model for EGS: Capturing Fracture Interactions and Correcting Classical Optimistic Predictions."