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

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Acknowledgments

Authors thank Prof. R. Ulusay for helpful discussions and advice during the manuscript preparation and revisions. Authors express also their thanks to reviewers A. Musson, R. Schellschmidt, Ömer Aydan, to the members of the ISRM Commission on Testing Methods E. Quadros, S. Kramadibrata, J. Muralha, to the ISRM Board members S. Read, D. Stead and M. He for their important scientific and technical recommendations and corrections during reviewing that helped to improve the manuscript.

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Correspondence to Y. Popov.

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Please send any written comments on this Suggested method to Prof. Resat Ulusay, President of the ISRM Commission on Testing Methods, Hacettepe University, Department of Geological Engineering, 06800 Beytepe, Ankara, Turkey. E-mail: resat@hacettepe.edu.tr

Appendix (to 4.2)

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

$$ \begin{aligned} T\left( {x,y,z,t} \right) & = \frac{q}{{8C(\pi a)^{{\frac{3}{2}}} }}\int\limits_{0}^{t} { - \frac{{\exp \left\{ { - \frac{{\left[ {x - v\left( {t - t^{{\prime }} } \right)} \right]^{2} + y^{2} + z^{2} }}{{4a\left( {t - t^{{\prime }} } \right)}}} \right\}}}{{\left( {t - t^{{\prime }} } \right)^{{\frac{3}{2}}} }}} {\text{d}}t^{{\prime }} \\ & = \frac{q}{{2R\lambda \pi^{{\frac{3}{2}}} }}\exp \left( {\frac{vx}{2a}} \right)\int\limits_{{\frac{R}{{2\sqrt {\text{at}} }}}}^{\infty } {\exp \left( { - \xi^{2} - \frac{{v^{2} R^{2} }}{{16a^{2} \xi^{2} }}} \right)} {\text{d}}\xi \\ \end{aligned} $$
(31)

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)

$$ T\left( {x,y,z} \right) = \frac{q}{4\pi \lambda R}\exp \left[ { - \frac{{v\left( {R - x} \right)}}{2a}} \right] $$
(32)

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

$$ T\left( {x,y,z} \right) = \frac{q}{2\pi \lambda R}\exp \left[ { - \frac{{v\left( {R - x} \right)}}{2a}} \right].$$
(33)

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Popov, Y., Beardsmore, G., Clauser, C. et al. ISRM Suggested Methods for Determining Thermal Properties of Rocks from Laboratory Tests at Atmospheric Pressure. Rock Mech Rock Eng 49, 4179–4207 (2016). https://doi.org/10.1007/s00603-016-1070-5

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