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

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 = 4DL 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 =Pe1 is the ratio of the rate of conductive versus advective heat transfer in the fluid, and gamma = 2DLBi 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.