Motivation

The Temperature Flat Source Solution @model is one of the fundamental solutions of temperature diffusion equations modelling the temperature conduction in linear direction (see Fig. 1).

This temperature profile is very common in subsurface studies, particularly in modelling the temperature above and below the lateral reservoir flow with a temperature

Outputs

Inputs

Equations

\frac{\partial T}{\partial t} = a^2 \Delta T = a^2\frac{\partial^2 T}{\partial z^2}

T(t=0, z) = T_G(z)

T(t, z=0) = T_f = {\rm const}, \quad T(t, z \rightarrow \infty) = T_G(z)

Solution

T(t,z) = T_f + (T_G(z) - T_f) \cdot \frac{2}{\sqrt{\pi}} \int_0^{z/\sqrt{4at}} e^{-\xi^2} d\xi

Approximations

T(t,z) = T_f + (T_G(z) - T_f) \cdot \Bigg[  1- \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta}  + \frac{3}{4 \zeta^3} \bigg) \Bigg]

See also

Heat flow equation for Semispace Linear Conduction:

\frac{\partial T}{\partial t} = a^2 \Delta T = a^2\frac{\partial^2 T}{\partial z^2}

Initial Conditions

T(t=0, z) = T_G(z)

Boundary conditions

T(t, z=0) = T_f = {\rm const}, \quad T(t, z \rightarrow \infty) = T_G(z)

The exact solution is given by following formula:

T(t,z) = T_f + (T_G(z) - T_f) \cdot \frac{2}{\sqrt{\pi}} \int_0^{z/\sqrt{4at}} e^{-\xi^2} d\xi

A fair approximation at late times (

T(t,z) = T_f + (T_G(z) - T_f) \cdot \Bigg[  1- \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta}  + \frac{3}{4 \zeta^3} \bigg) \Bigg]

where

\zeta = \frac{z}{4 a t}

The final solution for temperature  above the flowing unit is represented by RHK pipe flow solution where T_G is replaced with T_b from 

For the intervals between two injection units the one needs to account for the SLC contribution from upper flowing unit and from lower flowing unit which can be done using the superposition.

First, let's rewrite 

dT(t, z) = T(t,z) - T_G(z)= -  (T_G(z) - T_f) \cdot    \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta}  + \frac{3}{4 \zeta^3} \bigg) 

Now one can write down the temperature disturbance from the overlying flowing unit A1:

dT_{b,over}(t, z) = T_{b,up}(t,z) - T_G(z)= -  (T_G(z) - T_{f, A1}) \cdot    \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta}  + \frac{3}{4 \zeta^3} \bigg) 

and from the underlying flowing unit A2:

dT_{b,under}(t, z) = T_{b,up}(t,z) - T_G(z)= -  (T_G(z) - T_{f, A2}) \cdot    \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta}  + \frac{3}{4 \zeta^3} \bigg) 

The background temperature disturbance between the flowing units will be:

T_b(t, z) = T_G(z) + dT_{b,over}(t, z) + dT_{b,under}(t, z)

Replacing the static value of 

See Also

Geology / Geothermal Temperature Field / Geothermal Temperature Profile

Physics / Fluid Dynamics / Linear Fluid Flow 

[ Temperature Flat Source Solution @model ] [ Geothermal Temperature Profile @model ]

Reference