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 .

GLOSSARY > Temperature Flat Source Solution @model > image2019-11-26_17-32-3.png

Fig. 1. Sample Temperature Flat Source Solution

Outputs

Temperature distribution

Inputs

	Time lapse after the temperature step from up to
	Spatial coordinate along the transversal direction to constant temperature plane
	Boundary temperature at
	Thermal diffusivity of the surroundings

Equations

Driving equation

Initial conditions

Boundary conditions

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

T(t=0, z) = 0

T(t, z=0) = T_f = {\rm const}

T(t, z \rightarrow \infty) = 0

Solution

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]

where

Error function

Motivation

Outputs

See also