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 T_f and background Geothermal Temperature Profile T_G(z).
| Fig. 1. Sample Temperature Flat Source Solution |
Outputs
T(t, z) | Temperature distribution alon transversal direction to |
Inputs
Equations
Heat flow equation for Semispace Linear Conduction:
| (1) | \frac{\partial T}{\partial t} = a^2 \Delta T = a^2\frac{\partial^2 T}{\partial z^2} |
Initial Conditions
| (2) | T(t=0, z) = T_G(z) |
Boundary conditions
| (3) | 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:
| (4) | 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 ( \zeta \sim 0) is given by expanding the integral:
| (5) | 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
| (6) | \zeta = \frac{z}{4 a t} |
The final solution for temperature above the flowing unit is represented by RHK pipe flow solution where TG is replaced with Tb from (5).
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 (5) in terms of temperature gain:
| (7) | 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:
| (8) | 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:
| (9) | 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:
| (10) | T_b(t, z) = T_G(z) + dT_{b,over}(t, z) + dT_{b,under}(t, z) |
Replacing the static value of T_G(z) in RHK model with dynamic value of T_b(t, z) one arrives to the final wellbore temperature model with account of heat exchange with surrounding rocks and cooling effects from flowing units (Semispace Linear Conduction).
See also
Physics / Fluid Dynamics / Linear Fluid Flow