Assuming that both water flow and heat flow are confined in the Z-axis direction, the Darcy velocity of groundwater is constant vz, the passive sink term and key parameters are regarded as constants, and the heat transfer equation can be abbreviated as
Groundwater motion equation
Where: (ρc)s represents the comprehensive specific heat capacity; Dzz is the thermal conductivity. When used to study the stable geothermal gradient, the following mathematical model can be established:
Groundwater motion equation
Where a = dzz/(ρ cfvz); T0 and Tb are boundary temperatures. Equation (6.11) belongs to the second order linear ordinary differential equation. According to Appendix 2, the general solution is as follows
Groundwater motion equation
Where: a and b are integer constants. Using the boundary conditions, we can get
Groundwater motion equation
Therefore, the relationship between geothermal gradient and depth is
Groundwater motion equation
Obviously, the geothermal gradient is influenced by the groundwater velocity vz. Geothermal flow q is also related to depth:
Groundwater motion equation
Substitute formula (6. 1 14) and formula (6. 1 16) into formula (6. 1 17), and get.
Groundwater motion equation
This shows that the geothermal flow value is constant.