Because of the complex coupling relationship between water flow equation and heat flow equation, it is quite difficult to solve it strictly by analytical method, and generally only numerical method can be used. Only by ignoring many quadratic processes and interactions can it be approximated as a simple convection-dispersion problem and an analytical solution can be obtained.

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.