Transient heat transfer in a horizontal well in hot dry rock – Model, solution, and response surfaces for practical inputs

Highlights: • Geothermal energy system analysis of heat transfer (HT) to a horizontal well. • FEA solution of equations governing TH in the rock mass and working fluid. • Identified range of the 3 dimensionless parameters for physically meaningful inputs. • Developed Response Surface Models (ROMs) for fluid Temperature vs time (T-t). • Demonstrate that ROMs accurately & rapidly evaluate T-t histories. This article presents a solution to the geothermal problem of transient heat production from hot dry rocks using a horizontal well. Dimensionless forms of the governing equations are derived, including conduction in the rock, convection between the wellbore rock and the fluid, and advection and conduction in the fluid along the well. Ten model material and geometric parameters are reduced to three dimensionless parameters: $\alpha =4\frac{L}{D}\mathrm{St}$ is the ratio of the rate of heat storage in the fluid to the rate the heat convection to the fluid from the rock, $\beta =\frac{1}{\mathrm{Pe}}$ is the ratio of the rate of conductive versus advective heat transfer in the fluid, and $\gamma =2\frac{L}{D}\mathrm{Bi}$ is the ratio of heat convection to the rock from the fluid to the rate of heat depletion in the rock. An axisymmetric finite element method (FEM) program is developed and yields the solution for the temperature of the rock mass and fluid over time. For physically meaningful inputs, analysis indicates that the effect of $\beta$ is negligible, that combinations of very large $\alpha$ and very low $\gamma$ (and vice versa) do not occur, and that all other things being equal, increasing $\alpha$ or decreasing $\gamma$ leads to higher fluid outlet temperatures. System response at different times and dimensionless temperature are captured in contour plots for values of $\alpha$ and $\gamma$ spanning practical injection rates, well geometries and fluid and rock properties. Power law response surface models are fitted using model outputs at discrete intervals and provide a means to accurately and rapidly compute the temperature-time histories of practical geothermal systems.