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 $\begin{array}{l}T_f\end{array}$ .

Fig. 1. Sample Temperature Flat Source Solution

Outputs

$\begin{array}{l}T(t, z)\end{array}$

Temperature distribution

Inputs

$\begin{array}{l}t\end{array}$	Time lapse after the temperature step from $\begin{array}{l}T(z=0) =0\end{array}$ up to $\begin{array}{l}T(z=0) =T_f\end{array}$
$\begin{array}{l}z\end{array}$	Spatial coordinate along the transversal direction to constant temperature $\begin{array}{l}T(z)= T_f\end{array}$ plane $\begin{array}{l}z=0\end{array}$
$\begin{array}{l}T_f\end{array}$	Boundary temperature at $\begin{array}{l}z=0\end{array}$
$\begin{array}{l}a\end{array}$	Thermal diffusivity of the surroundings

Equations

Driving equation

Initial conditions

Boundary conditions

(1)	$\begin{array}{l}\displaystyle \frac{\partial T}{\partial t} = a^2 \Delta T = a^2\frac{\partial^2 T}{\partial z^2}\end{array}$

(2)	$\begin{array}{l}\displaystyle T(t=0, z) = 0\end{array}$

(3)	$\begin{array}{l}\displaystyle T(t, z=0) = T_f = {\rm const}\end{array}$

(4)	$\begin{array}{l}\displaystyle T(t, z \rightarrow \infty) = 0\end{array}$

Solution

(5)	$\begin{array}{l}\displaystyle T(t,z) = T_f \cdot \left[ 1- \mbox{erf} \left( \frac{z}{\sqrt{4at}} \right) \right] = T_f \cdot \left[ 1- \frac{2}{\sqrt{\pi}} \int_0^{z/\sqrt{4at}} e^{-\xi^2} d\xi \right]\end{array}$

where

$\begin{array}{l}\mbox{erf}(x)\end{array}$

Error function

Page tree

Motivation

Outputs

See also