Geothermal Doublet Drawdown

What is modelled

Water injected at temperature T_i crosses a planar hot fracture of length L and height H in rock at temperature T_r, picking up heat conducted from both faces. As the rock next to the fracture cools, the produced temperature falls. The solution is the classic Lauwerier cold front.

Equations

• Reservoir temperature Tr = 15 + gradient × depth.
• Produced temperature T = Tr − (Tr − Ti) erfc(ξ), with ξ = λ L H N / (ṁ cw √(α t)), for N identical fractures sharing the flow.
• Thermal power ṁ cw (T − Ti); electric power through half the Carnot efficiency against a 25 C sink.
• Ten percent drawdown life (λ L H N / ṁ cw)² / (α ξ²) with erfc(ξ) = 0.1.

Checks run in the page

• Finite-difference fracture and rock model against the formula at 2, 8 and 20 years.
• Heat extracted from the rock against the heat carried out.
• The error function against numerical integration.
• The drawdown life gives exactly 10 % cooling.

Honest simplifications

The reservoir is N identical flat parallel fractures with uniform flow, the rock is homogeneous with fixed properties (conductivity 2.8 W/mK, volumetric heat capacity 2.5 MJ/m3K), the water travel time and fluid heat storage in the fracture are neglected, there is no flow short-circuiting, no thermal stress, no water loss and no pumping power. Real reservoirs have many fractures and the 3D page shows a whole caldera; this is the physics of a set of identical fractures only.

• Press Fast flow, short life and watch the cold front reach the production well within a few years.
• Press Large fracture and see the lifetime grow with the square of the heat transfer area.
• Press Hot caldera and read how a hotter reservoir lifts both the temperature and the electric efficiency.