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Orange: produced brine. Dotted: first breakthrough (shortest streamline). The white dot is the current year.
The science
Flow. Water injected at Q (kg/s) into a confined aquifer of thickness H and withdrawn at a well a distance L away moves as a 2D source-sink pair (Darcy flow, u = -(k/μ) grad p). The pore velocity is v = Qv/(2 pi H φ) (r1/|r1|² - r2/|r2|²). Integrating the pressure gradient along the line between the wells gives the pump pressure Δp = μ Qv / (2 pi k H) * 2 ln(L/rw), and the pump power is Qv Δp / ηp.
Thermal front. Rock and water are assumed in local thermal equilibrium, so heat is carried at the retarded speed q ρwcw / (ρc)agg, where (ρc)agg = φ ρwcw + (1-φ) ρrcr. The fluid needs tf = pi φ H L² / (3 Qv) along the straight line between the wells, so the cold front reaches the producer after tth = pi L² H (ρc)agg / (3 Qv ρwcw) (Gringarten, Witherspoon and Ohnishi, 1975). Doubling the spacing multiplies it by four.
Streamtubes. The sim traces 90 equal-flux streamlines with a 4th-order Runge-Kutta integrator. Each carries 1/90 of the flow and delivers cold water at its own retarded arrival time, so the produced temperature is T = Tres - (Tres - Tinj) f(t), with f the fraction of the flow already broken through. Streamlines that would need more than about a thousand years never break through.
Electric power. Pth = m cw (Tprod - Tinj). An organic Rankine cycle is taken at 50 % of the Carnot limit, η = 0.5 (1 - Tc/Th) with a 25 C condenser, 4 % parasitic load, minus the pump power.
Honest simplifications. A 2D homogeneous isotropic aquifer of constant thickness; sharp thermal front (no dispersion or conduction from the caprock and basement, which in reality prolongs the life and smooths the decline); constant water properties (viscosity at the mean of reservoir and injection temperature); no wellbore friction, thermosiphon or scaling; no regional flow; no fractures or short-circuits. Rock constants (2650 kg/m³, 850 J/kg/K), the 50 % Carnot fraction, pump efficiency 0.6 and the geothermal gradient are typical illustrative values, not site data. The 3D block is vertically exaggerated and the wells are drawn thicker than real ones.