ISRM Suggested Methods for Determining Thermal Properties of Rocks from Laboratory Tests at Atmospheric Pressure

Appendix (to 4.2)

If a point heat source is located at (x′, y′, z′), the differential equation of heat conduction,

$$ \frac{{\partial^{2} T}}{{\partial x^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{{\partial^{2} T}}{{\partial z^{2} }} = \frac{1}{a}\frac{\partial T}{\partial t} $$

(28)

is satisfied by (Carslaw and Jaeger 1959, Sect. 10.2, p. 256):

$$ T = \frac{Q}{{8(\pi a {\text{t}})^{{\frac{3}{2}}} }}\exp \left[ { - \frac{{\left( {x - x^{{\prime }} } \right)^{2} + \left( {y - y^{{\prime }} } \right)^{2} + \left( {z - z^{{\prime }} } \right)^{2} }}{{4a{\text{t}}}}} \right] $$

(29)

where a is the thermal diffusivity (m²/s), Q is the heat strength (K m³), more precisely Q is the temperature by which an amount of heat released would raise a unit volume of the medium. As t → 0 the expression (29) tends to zero at all points except $\left( {x^{{\prime }} ,y^{{\prime }} ,z^{{\prime }} } \right)$, where it becomes infinite. It is easy to check that the total quantity of heat in the infinite region is equal to QC where C is the volumetric heat capacity.

Consider heat emitted at the origin for times t > 0 at the rate q in heat units per unit time (i.e. q in J/s hence W) and an infinite medium moving uniformly past the origin with velocity v parallel to the axis of x. The temperature T can be calculated at a fixed point (x, y, z) at time t. In the infinitesimal time interval dt′ at time t′, qdt′ heat units were emitted at the origin at time t′ (i.e. qdt′ = QC); also the point of the infinite medium, which at time t is at (x, y, z), was at time t′ at [x − v(t − t′), y, z]. Thus the temperature T at time t at point (x, y, z) due to the heat qdt′ emitted at t′ is, by (29),

$$ T\left( {x,y,z,t} \right) = \frac{{q{\text{d}}t^{{\prime }} }}{{8C\left[ {\pi a\left( {t - t^{{\prime }} } \right)} \right]^{{\frac{3}{2}}} }}\exp \left\{ { - \frac{{\left[ {x - v\left( {t - t^{{\prime }} } \right)} \right]^{2} + y^{2} + z^{2} }}{{4a\left( {t - t^{{\prime }} } \right)}}} \right\} $$

(30)

and the temperature T at time t due to heat emitted at the origin is

$$ \begin{aligned} T\left( {x,y,z,t} \right) & = \frac{q}{{8C(\pi a)^{{\frac{3}{2}}} }}\int\limits_{0}^{t} { - \frac{{\exp \left\{ { - \frac{{\left[ {x - v\left( {t - t^{{\prime }} } \right)} \right]^{2} + y^{2} + z^{2} }}{{4a\left( {t - t^{{\prime }} } \right)}}} \right\}}}{{\left( {t - t^{{\prime }} } \right)^{{\frac{3}{2}}} }}} {\text{d}}t^{{\prime }} \\ & = \frac{q}{{2R\lambda \pi^{{\frac{3}{2}}} }}\exp \left( {\frac{vx}{2a}} \right)\int\limits_{{\frac{R}{{2\sqrt {\text{at}} }}}}^{\infty } {\exp \left( { - \xi^{2} - \frac{{v^{2} R^{2} }}{{16a^{2} \xi^{2} }}} \right)} {\text{d}}\xi \\ \end{aligned} $$

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where $R = \sqrt {x^{2} + y^{2} + z^{2} }.$

This is a solution for the amount of heat for finite time t. If t → ∞, a steady thermal regime is established, and the temperature T at (x, y, z) is given by (Carslaw and Jaeger 1959, Sect. 10.7, p. 267)

$$ T\left( {x,y,z} \right) = \frac{q}{4\pi \lambda R}\exp \left[ { - \frac{{v\left( {R - x} \right)}}{2a}} \right] $$

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If the point heat source is located on a surface of a half-infinite medium, the temperature at (x, y, z) is given by:

$$ T\left( {x,y,z} \right) = \frac{q}{2\pi \lambda R}\exp \left[ { - \frac{{v\left( {R - x} \right)}}{2a}} \right].$$

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