岩石学実験2013年	 

熱伝導と接触変成作用
Heat Conduction and Contact Metamorphism
Simon Wallis, Nagoya University
ウォリス サイモン
INTRODUCTION 基本知識
The thermal structure of the earth is fundamental for understanding
many geological processes including the locations of volcanoes,
earthquakes and hydrothermal fields.
Heat transfer （熱輸送）can be divided into three different types:- conduction (熱伝導)- advection（熱移流）- radiation（熱輻射）In fluids and gasses advection and radiation are dominant. In rocks,
radiation is not an important heat transfer mechanism (or can betreated as part of conduction), but both advection and conduction
may be important. Conduction of heat energy (熱エネルギーの拡散) is
analogous to diffusion of matter (物質の拡散) and represents thetransfer of energy by movement and collisions of atoms, electrons andphonons. Advection is the process of transferring heat by movementof either hot or cold bodies with respect to one another.
One simple way to think of the difference between conduction andadvection is to consider a blob of ink in a river (Fig. 1). The
concentration of ink is analogous to temperature, and the distribution
across the river of the ink concentration (or temperature) at sometime and at some distance down stream is a function of how quickly
the ink spreads out (conduction) and how quickly the ink istransported down stream (advection).	  

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Figure 1. The difference between conduction and advection can be easily
thought of in terms of the transport of a spreading blob of ink in a flowing river.

One example of advection of heat is the inflow of cold lithosphericplates in subduction zones. Heat advection is also important insituation where there is movement of hot fluids in rocks, e.g. oceanicridges （海嶺）. Today we will focus on heat conduction and contact
metamorphism (接触変成作用）.

FOURIER’S LAW AND HEAT FLOW
フーリェ法則と熱流量
The conductive flow of heat through a substance is described by
Fourier's Law (フーリェ法則):	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	   Q = −kdT	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  (1)dz

Q is the heat flow (熱流量) in Watts per square meter (W m-2)(熱流束€heat flux-ともいう),k is the thermal conductivity (熱伝導率) (J s-1 m-1Kelvin-1), and dT/dz is the thermal gradient. The negative sign is there
because we define temperature and distance to increase in the same
direction and because Q flows from high T to low T. Fourier’s Law isan empirical law but has been well tested and until recently nosignificant exceptions were known. In recent years it has beensuggested that some modern materials exhibit different non-Fourier
conductive behaviour.
Typical values for k are 1.5–3.0 J s-1 m-1 Kelvin-1. The average heat	  2	  岩石学実験2013年	 
flow for continental areas is around 56 mW m-2 and for oceanic areas
78 mW m-2. The reason for this difference is the young age of theoceanic crust. At the earth’s surface, the amount of heat flowing fromwithin the earth is much less than the heat energy derived from thesun (~400 W m-2).

DERIVATION OF THE HEAT CONDUCTION EQUATION
熱伝導方程式の導出
The heat conduction equation is derived by considering two
relationships: one between heat flow, Q, and spatial gradients in T(Fourier’s Law) and the other considering the relationship between
heat flow, Q, and temporal gradients in T. By combining these two
relationships, we derive an equation that shows how T varies with both
space and time.
In this example we will consider a 1-dimensional situation—this isappropriate for narrow elongate intrusions such as dykes and sills. It isalso a very good approximation for the cooling of the outermost 100
km of the Earth. Let us consider a small volume of the material we areinterested in with thickness Δz (Fig. 2).

Figure. 2. To derive the heat conduction equation, we first consider a onedimensional region with thickness Δz, heat capacity c, density ρ and with adifference in the heat flow in (Qin) and the heat flow out (Qout).

The Law of conservation of energy tells us:	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  ∂T Qin − Qout=	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  (2).	  
∂tcρΔz€	  3	  岩石学実験2013年	 
Where c is the specific heat of the material (比熱) (J/K/kg) and ρ is thedensity. That is, if the heat flow out (Qout) of the area is less than theheat flow in (Qin) then the material is heating up at a rate (dT/dt > 0)
related to its specific heat and density. Different rocks will change
temperature by different amounts for the same input of heat energy.
Heat and temperature are not the same! Let us consider changes inunit time.	  
-The temperature rise for c joules is 1° for 1 kg of material.
-The temperature rise for Q joules is (J/c)° for 1 kg of material.
-The temperature rise for Q joules and ρΔz kg of material is ( T cρΔz )°.

Secondly we can say that €	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	   Qout = Qin + Δz€

∂Q.	  	  	  	  	  	  	  	  	  	  	  	  (3)∂zWe can now rewrite the first equation to relate the spatial variation intemperature
with the rate of temperature change.
€	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  	  ∂T=∂tQin − Qin − Δz∂Q∂zcρΔz∂Q= ∂z cρ−€We can now€ include the substitute for Q using Fourier's Law,

∂T=∂t(∂ k ∂T ∂z∂z)cρk ∂ 2T=cρ ∂z 2

€k ∂ 2T.
=cρ ∂z 2€	  Now we let k/cρ = κ where κ (m2/s) is the thermal diffusivity and so€	  4	  

岩石学実験2013年	 

∂T∂ 2T=κ 2∂t∂z(4).

This is the fundamental equation of heat conduction in one dimension.

€QUALITATIVE UNDERSTANDING OF HEAT CONDUCTION
熱伝導の定性的な理解
Heat conduction is controlled by a second order differential equation
relating temperature, time and position. Changes in temperature byconduction only occur if there are curves in the heat profile. If there isa linear gradient then no change in temperature takes place. Thegreater the curvature is, the greater the rate of change oftemperature. This means that if sharp changes are introduced in thethermal structure, for instance by intruding a hot magma into a cold
country rock, then the temperature at the boundaries will changemuch faster than elsewhere. Positive curvature causes a rise in T, whilenegative curvature causes a decrease in T—protrusions of the thermalstructure get flattened out and depressions get filled in.
A second important point is that if two conducting solid objects arebrought together, the temperature at the boundary is the average ofthe two temperatures. This means that if a 1000°C magma is intruded
into rock with a temperature of 200°C, then the initial temperature at
the boundary will be (1000+200)/2 = 600°C.

APPLICATION TO MAGMATIC INTRUSIONS
貫入岩体への適応例
We can now use our understanding of heat conduction to predictchanges in temperature around an igneous intrusion intruded into cold
country rock.	  

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Figure. 3 The change in the temperature around an intrusion with time.
Time, t’, temperature, T’, and distance, Z’, are expressed as
dimensionless quantities with definitions as follows:

T − T0κtTʹ′ =	  (T is
2 (t is time and kappa is the thermal diffusivity) ,Ts − T0hthe temperature, Ts is the original temperature of the intrusion and T0 is theoriginal temperature of the country rock and the temperature at infinity),Zʹ′ = z /h (z is the distance and h is the half width of the intrusion)€tʹ′ =€€A common boundary condition is to assume initial uniform temperatureextending to infinity (Fig. 3). This means that after an infinite amountof time the thermal anomaly created by the dyke will disappear. If weassume the intrusion was instantaneous then the initial thermal profile
will be a square wave. The temperature along the contact will be theaverage of the intrusion and country rock. So we know the initialconditions and final conditions. The heat conduction equation here
tells us what happens in between.
We know from our result above that the most rapid changes intemperature will occur where the gradients are steepest, i.e. near thecontacts, which are the corners in the thermal structure shown infigure 3. These corners will become rounded and the heat will moveoutwards.
Within the body we will see a gradual cooling. Outside we will see avery different behaviour. First the temperature will rise and then fall.
Also the peak of temperature will not occur at the same time in allplaces. We can see that even for a simple situation there will be acomplex variation in the relationships between temperature-time	  6	  岩石学実験2013年	 
history of rocks and their position.

PARAMETERS OF THERMAL CONDUCTION IN ROCKS
岩石における熱伝導のパラメター
Something very fortunate is that although c and k vary considerably κ
only varies very little for most rock types. It is usually considered tobe 10-6 m2 s-1. However, it is about half this for calcite-rich rocks.
Changing κ	  affects the time scale for heating but does not affect thepeak temperature attained.
The most common problem is to determine how the temperature
varies with time and space for a given intrusion geometry, magma
temperature and pre-intrusion temperature. Different magmas vary intemperature by several 100s of degrees. Intrusion sizes can vary frommm to 10s kilometers.
We will look in more detail at the modeling during the practical. To getsome idea, the time t taken for a body with half width h to reachapproximately half its original temperature in the core of the intrusion
is related to its size and κ by the equation:h2(Table 1).
t=κh (m eters)	  1	  10	  100	  1000	  10000	  €T(seconds)	  t(years)	  106 	  108	  1010	  0.0317	  3.17317	  1012	  1014	  31700	  3.17x106	  Table 1. Cooling time scales for intrusions of different sizes.

So we can see that 1m dykes may take only weeks to cool, but large
plutons may take millions of years. An example of cooling around apluton is given in figure 4. These data show that cooling has taken	  7	  岩石学実験2013年	 
several millions of years implying that the pluton must have a radius of
the order of 10km.

Figure 4. Cooling of the Quottoon Pluton in British Columbia (Canada) shownby radiometric dating. The dates record the time at which the rock reacheddifferent temperatures. It took several millions of years to cool significantly
implying the presence of a large pluton.

COMPLICATING FACTORS
その他の影響A more complete analysis of the temperature distribution around an
intrusion requires consideration of other factors. More complete
modeling requires numerical solutions to the conduction equation. Thisapproach can include other complications such as temperaturedependence of thermal conductivity (熱伝導の温度依存性), 2-d and 3-dgeometries of the intrusions, an initial thermal gradient in the region
with the intrusion (初期地温勾配), latent heat (潜熱), convection offluids near the intrusion (貫入岩体周囲の熱水循環) etc.
There are ways to incorporate some of these features in the 1-dmodeling we have looked at. We will look at two simple ways to treat
latent heat and advection.	  

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Latent heat (潜熱) is released as the liquid magma crystallizes. Theextra heat energy released maintains the temperature at a high value
until all the magma is crystallized. The contribution of latent heat canbe approximated by adding an amount equivalent to releasing all theenergy at the same time and treating it as an equivalent temperaturerise. For example some approximate values for basaltic magma are as
follows.
Latent heat of crystallization (潜熱) 4 x 105 J / kg
Specific heat (比熱) 103 J /Kg / °C
If the latent heat were instantaneously released and all used toincrease the temperature of the magma the rise would be(4 x 105)/103 = 400°C
This number can be added to the initial temperature for the magma.
This does not correspond to a physically meaningful process, but theresult does give a good approximation for the peak T around theintrusion. Numerical methods are required for accurate estimates of T
within and very close to the intrusion.
Advection of heat by the convection of hot fluids (熱水循環による熱移流) is quite common around intrusions. Convection transports heatmore effectively than conduction and results in a lower temperature
close to the intrusion and a higher temperature at a greater distance.
In many cases the zone of convection is restricted close to theintrusion. An example where both domains can be identified is theKakkonda (葛根田) hydrothermal area in NE Japan (Fig. 5), where ayoung and hot granite body has intruded to depths of about 3km
beneath the surface. The zone of convection can be modeled using ahigh value for κ, which has the same effect.	  

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Figure 5. Thermal structure in the Kakkonda hydrothermal region
showing the shallow regions with circulating water and small
differences in temperature compared to the deeper region with steep
thermal gradients due to conduction (after Ikeuchi et al., 1998).
Greater depth acts against free flow of fluid both due to pressure
closing gaps in the rock and the rapid deposition of minerals filling infractures.

REFERENCES	  Key References
Carslaw, H. S. & Jaeger, J. C. 2000. Conduction of Heat in Solids.
Oxford University Press, Oxford, pp. 510.	  

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- A comprehensive treatise on exact solutions to the heat conduction
equation for a range of geometries and boundary conditions. First
published in 1946 and still highly relevant. A classic textbook!
Spear, F. S. 1994. Metamorphic Phase Equilibria and PressureTemperature-Time Paths. Monograph of the Mineralogical Society ofAmerica, Ch. 3.
- A simplified approach to understanding the heat equation and some
applications to problems of importance in metamorphic geology.

Other References
Harrison, T. M., Armstrong, R. L., Naeser, C. W. & Harakal, J. E. 1979.
Geochronology and thermal history of the Coast Plutonic complex nearPrince Rupert, British Columbia. Canadian Journal of Earth Science, 16,400-410.
Ikeuchi, K., Doi, N., Sakagawa, Y., Kamenosono, H., Uchida, T. 1998.
High temperature measurement in well WD-1A and the thermal
structure of the Kakkonda geothermal system Japan. Geothermics 27,591-607.
Roetzel et al. 2003. Experiment and analysis for non-Fourier
conduction in materials with non-homogenous inner structure.
International Journal of Thermal Sciences, 42 541-552- Discussion of the reality of non-Fourier heat conduction.	 	 	  11