Thermal Storage and Transport Properties of Rocks, I: Heat Capacity and Latent Heat

Synonyms

Specific heat capacity; Thermal capacity; Volumetric heat capacity

Definition

Specific heat capacity c. Physical property defining the amount of sensible heat that can be stored in or extracted from a unit mass of rock per unit temperature increase or decrease, respectively. Isobaric and isochoric specific heat capacities are defined at constant pressure and volume, respectively; dimension: J kg⁻¹ K⁻¹.

Thermal capacity (also: volumetric heat capacity) ρ c. The product of isobaric specific heat capacity and density. Physical property defining the amount of sensible heat that can be stored in or extracted from a unit volume of rock per unit temperature increase or decrease, respectively; dimension: J m⁻³ K⁻¹.

Thermal storage properties

The thermal regime of the Earth is defined by its heat sources and sinks, the heat storage and transport processes, and their corresponding physical properties. The storage properties are discussed below. The transport properties, thermal conductivity and thermal diffusivity, are dealt with in this volume in the companion chapter “ Thermal Storage and Transport Properties of Rocks, II: Thermal conductivity and diffusivity .”

Heat can be stored as sensible heat and enthalpy of transformation. In the Earth, sensible heat is defined by the heat capacity of rocks, and the enthalpy of transformation by their latent heat of fusion.

Heat is transmitted within the Earth mainly by diffusion (in the steady state: conduction), advection, and radiation. Generally, heat diffusion dominates heat radiation within the lithosphere of the Earth, at temperatures below about 1 000 K. For sufficiently large flow rates, convection-driven heat advection, however, can be equally or even more efficient than diffusion. The heat diffusion–advection equation for a porous medium is:

$${\frac{{\partial (\rho {{\text {c}}}_{\text{P}}{\text{T}})}}{{\partial {t}}} = \nabla \cdot \left( {\underbrace {{\mathbf{\lambda}}\nabla {\text{T}}}_{{{\text{q}}_{\text {dif}}}} - \underbrace {{{(\rho {c_{\text{P}}})}_{\text{f}}}{\text{T}}{\mathbf{v}}}_{{{\text {q}}_{\text {adv}}}}} \right) + \underbrace {\text{A}}_{\text{generation}},} $$

(1)

where T is temperature in K, t time in s, ρ density in kg m⁻³, c isobaric specific heat capacity in J kg⁻¹ K⁻¹, λ thermal conductivity in W m⁻¹ K⁻¹, v specific fluid discharge (volume flow rate per unit cross section) in m s⁻¹, A (radiogenic) heat generation rate in W m⁻³; subscript “f” refers to fluid properties. Diffusive bulk storage and transport properties for heat in rocks governing this equation are thermal capacity ρ c_P, also referred to as volumetric heat capacity, and thermal conductivity λ. Advection is governed by the thermal capacity and specific discharge of the flowing fluid, (ρ c_P)_f and v, respectively.

Heat advection does not require very large flows for becoming as or more efficient than heat diffusion. This is often the case in sedimentary basins (see, e.g., Clauser et al. (2002) for a literature review). But fluid-driven heat advection may be also important in crystalline rocks and on a crustal scale (e.g., Clauser, 1992). The non-dimensional Péclet and Nusselt numbers, Pe and Nu, are ratios of specific heat flows q_adv and q_dif (Equation 1) indicating the efficiency of (fluid flow driven) advective heat transport versus heat conduction for given flow geometries. For instance, assuming flow of magnitude v over a distance L and a temperature difference T₁–T₀ one obtains:

$$\begin{array}{ll} {\text{Pe}} = \frac{{{{\text{q}}_{\text {adv }}}}}{{{{\text{q}}_{\text {dif}}}}} = \frac{{{{(\rho {\text{c}})}_{\text{f}}}{\text{v}}({{\text{T}}_1} - {{\text{T}}_0})}}{{\lambda \,{{{{{{\text{T}}_1} - {{\text{T}}_0}}}} \left/ {\text{L}} \right.}}} = \frac{{{{(\rho {\text{c}})}_{\text{f}}}{\text {vL}}}}{\lambda }; \\ {\text{Nu}} = \frac{{{{\text{q}}_{\text {adv}}}\;{ + }\;{{\text{q}}_{\text {dif}}}}}{{{{\text{q}}_{\text {dif}}}}} = \frac{{{{\text{q}}_{\text {adv}}}}}{{{{\text{q}}_{\text {dif}}}}} + 1 = {\text{Pe}} + 1.\end{array} $$

(2)

Thus advection or diffusion (in the steady state: conduction) dominate for Pe > 1 or Pe < 1, respectively (in terms of the Nusselt number for Nu > 2 or Nu < 2, respectively). At temperatures above 1 000 K, heat is propagated increasingly as radiated electromagnetic waves, and heat radiation begins to dominate diffusion (see Thermal Storage and Transport Properties of Rocks, II: Thermal Conductivity and Diffusivity , this volume).

Heat capacity

Heat can be stored and delivered as sensible heat or as latent heat required or liberated by phase changes. This and the next two paragraphs are concerned with sensible heat. Following this, latent heat will be discussed.

Sensible heat capacity C is defined as the ratio of heat ΔQ required to raise the temperature of a mass M of rock by ΔT. For each molecule of mass m, this temperature increase requires an energy of (f/2) k ΔT, where f is the number of degrees of freedom of the molecule and k = 1.380 650 4(24) × 10⁻²³ J K⁻¹ is Boltzmann’s constant. For a body of mass M, a temperature increase of ΔT requires an energy of ΔQ = (M/m) (f/2) k ΔT. Thus the heat capacity of the body at constant volume is:

$$ {{\text{C}}_{\text{V}}} = \frac{{\Delta {\text{Q}}}}{{\Delta{\text{T}}}} = \frac{\text{f}}{2}{\text{k}}\frac{\text{M}}{\text{m}}= \frac{\text{f}}{2}{\text{k}}{{\text{N}}_{\text{A}}} =\frac{\text{f}}{2}{\text{R}}\;\left( {{\text{J}}{{\text{K}}^{ -{1}}}} \right), $$

(3)

where Avogadro’s number N_A = 6.022 141 79(30) × 10²³ mol⁻¹ equals the number of molecules or atoms in an amount of substance of 1 mol and R = k N_A = 8 314 472(15) J mol⁻¹ K⁻¹ is the molar Gas constant (numerical values for N_A, R, k, and all other physical constants used in this chapter are from CODATA, 2006). For solids, f= 6, corresponding to the three degrees of freedom of potential and kinetic lattice vibration energy in each space direction. Accordingly, the heat capacity of one mole of substance, the molar heat capacity at constant volume is constant:

$$ {{\text{C}}_{{\text{V,mol}}}} = 3{\text{k}}{{\text{N}}_{\text{A}}} = 3{\text{R}} = 24.94\;{\text{ (J mo}}{{\text{l}}^{ - 1}}{{\text{K}}^{ - 1}}) $$

(4)

where N_A = 6.022 141 79(30) × 10²³ mol⁻¹ is Avogadro’s number, the number of molecules in one mole of substance, and R = k N_A the molar gas constant. Isobaric heat capacity C_P,mol is larger than isochoric heat capacity C_V,mol because additional work is required for volume expansion. Both are related by:

$$ {{\text{C}}_{{\text{P,mol}}}} = {{\text{C}}_{{\text{V,mol}}}} +{\text{R = }}\frac{{{\text{f + 2}}}}{{2}}\,{\text{R (J}}\,{\text{mo}}{{\text{l}}^{ - {1}}}\,{{\text{K}}^{ - {1}}}{)}{.}$$

(5)

With C_V,mol from Equation 4 and assuming, as above, f = 6 this yields:

$$\begin{array}{lll} {{\text{C}}_{{\text{P,mol}}}} & = 3{\text{R}}+ {\text{R}} = 4{\text{R}} \\ & = 33.26\;{\text{(J)}\,{\text{mo}}{{\text{l}}^{ - {1}}}\,{{\text{K}}^{ -{1}}}},{\text{ or:}} \\ & {{\text{C}}_{{\text{P, mol}}}} -{{\text{C}}_{{\text{V,mol}}}} = {\text{R}}.\end{array}$$

(6)

Equation 4, the Dulong–Petit law, is satisfied well for heavy elements. In contrast, molar heat capacities of lighter elements remain below this limiting value, the lower the temperature the smaller C_V,_mol.

Below the Debye temperature, Θ_D, heat capacity varies with temperature. Θ_D tends to zero as T³ as absolute temperature approaches zero. Θ_D falls in the range 85 K and 450 K for most substances and 200 K and 1 000 K for most minerals (Stacey and Davis, 2008; Table 1). Therefore, heat capacity in the Earth can be well explained by classical Debye theory, in particular in the mantle and except for a thin crustal layer near the Earth’s surface. There are, however, exceptions such as beryllium (Θ_D = 1 440 K) and diamond (Θ_D ≈ 1 800 K). These are caused by the so-called freezing of vibrational or rotational degrees of freedom, which cannot absorb heat any more at low temperature. Therefore heat capacity tends to zero close to absolute zero.

Thermal Storage and Transport Properties of Rocks, I: Heat Capacity and Latent Heat, Table 1 Debye temperature Θ_D and mass number A of selected elements (Kittel, 2004)

Isobaric and isochoric specific heat capacity

Specific heat capacity c of a substance is defined as heat capacity C related to unit mass:

$$ {\text{c}} = \frac{{\Delta {\text{Q}}}}{{{\text{M}}\,\Delta{\text{T}}}} = \frac{\text{f}}{{2}}\,\frac{\text{k}}{\text{m}} =\frac{\text{f}}{{2}}\,\frac{\text{k}}{{{{\text{A}}_{\text{r}}}\,{{\text{m}}_{\text{u}}}}}\left( {{\text{J k}}{{\text{g}}^{ - {1}}}{{\text{K}}^{ - {1}}}}\right)\,, $$

(7)

where m_u = 1.660 538 782(83) × 10⁻²⁷ kg is the atomic mass constant, defined as ¹/₁₂ of the atomic mass of the carbon isotope ¹²C, and A_r the atomic mass of a substance relative to m_u. Isobaric specific heat capacity c_P is larger than isochoric specific heat capacity c_V because additional work is required for volume expansion. Their ratio, the adiabatic exponent, is:

$$ {{\text{c}}_{\text{P}}}/ {{\text{c}}_{\text{V}}} = \left( {{\text{f}} + {2}} \right)/ {\text{f}}\cdot $$

(8)

Alternatively, isobaric specific heat capacity c_P can be expressed by enthalpy H(T,P) = E + P V, a state function of temperature and pressure, where E, P, and V are internal energy, pressure, and volume, respectively. In a closed system, the change in internal energy dE is the sum of the change in heat dQ and the work dW delivered: dE = dQ + dW. If we only consider volume expansion work, dW = −P dV, the change in enthalpy dH becomes:

$$\begin{array}{ll} {\text{dH}}({\text{T,\,P}}) & = {\text{dE}} + {\text{pdV}} + {\text{VdP}} = {\text{dQ}} + {\text{VdP}} \\ &= {\left( {\frac{{\partial {\text{H}}}}{\text{T}}} \right)_{\text{P}}}{\text{dT}} + {\left( {\frac{{\partial {\text{H}}}}{\text{P}}} \right)_{\text{T}}}{\text{dP}}\cdot \end{array}$$

(9)

Comparing coefficients, we obtain:

$$ \frac{\text{dQ}}{{\text{dT}}} = {\left( {\frac{{\partial {\text{H}}}}{{\partial {\text{T}}}}} \right)_{\text{P}}} \doteq {{\text{c}}_{\text{P}}}. $$

(10)

Thus, Equation 10 defines isobaric specific heat capacity c_P as the first derivative of enthalpy with respect to temperature. Comparison of Equations 7 and 10 shows that both expressions are equivalent for dQ = ΔQ/M, and the isobaric enthalpy change ΔH is equal to the specific heat content ΔQ/M.

Isobaric and isochoric specific heat capacity are related to compressibility β = ΔV/(VΔP) and its inverse, incompressibility or bulk modulus K = VΔP/ΔV, by c_P/c_V = β_T/β_S = K_S/K_T (e.g., Stacey and Davis, 2008), where subscripts T and S refer to isothermal and adiabatic conditions, respectively. Inserting the thermodynamic relation β_T = β_S + α² T/(ρ c_P) (e.g., Birch, 1966) between isothermal and adiabatic compressibility yields the relative difference between isobaric and isochoric specific heat capacity:

$$ {{\text{c}}_{\text{P}}}/{{\text{c}}_{\text{v}}} = {1} + \alpha \gamma {\text{T}}, $$

(11)

where α = ΔV/(VΔT) is the volume expansion coefficient,

$$ \gamma = \frac{{\alpha {{\text{K}}_{\text{S}}}}}{{\rho {{\text{c}}_{\text{P}}}}} = \frac{{\alpha {{\text{K}}_{\text{T}}}}}{{\rho {{\text{c}}_{\text{V}}}}}, $$

(12)

the dimensionless Grüneisen parameter, and ρ density. Inserting the expressions for α and K into Equation 12 yields:

$$ \gamma = \frac{1}{{\rho {{\text{c}}_{\text{P}}}}}\frac{{\Delta {\text{V}}}}{{{\text{V}}\,\Delta {\text{T}}}}\frac{{{\text{V}}\Delta {\text{P}}}}{{\Delta {\text{V}}}} = \frac{{\Delta {\text{P}}}}{{\rho {{\text{c}}_{\text{P}}}\Delta {\text{T}}}}. $$

(13)

Thus the Grüneisen parameter γ is the relative pressure change in a material heated at constant volume.

For solids, i.e., f = 6, the absolute difference between isobaric and isochoric specific heat capacity follows from Equations 11 and 8:

$$ {{\text{c}}_{\text{P}}} - {{\text{c}}_{\text{V}}} = \frac{{{{\text{K}}_{\text{T}}}{\alpha^2}{\text{T}}}}{\rho } = \frac{{3{{\text{K}}_{\text{S}}}{\alpha^2}{\text{T}}}}{{4\rho }}. $$

(14)

For crustal rocks (γ = 0.5; α = 20 μK⁻¹; T < 10³ K; ρ = 2 600 kg m⁻³; K_S < 75 GPa (Dziewonski and Anderson, 1981; Stacey and Davis, 2008)), the difference between isobaric and isochoric specific heat capacity is less than 9 J kg⁻¹ K⁻¹ or 1% according to Equations 14 and 11, respectively. Thus, the distinction between isobaric and isochoric specific heat capacity is negligible for crustal rocks at temperatures below 1 000 K. However, it need be made for mantle rocks. From here on, “specific heat capacity” will always refer to isobaric specific heat capacity, denoted simply by the letter c without the subscript “P.”

This classical treatment of heat capacity is sufficient for temperatures above the Debye temperature. In the Earth, temperature exceeds the Debye temperature everywhere except in the crust (Stacey and Davis, 2008). Therefore, in experiments at room temperature and atmospheric pressure, we observe deviations from the values predicted by Equations 3–14, which are based on the classical Dulong–Petit theory. The lower the temperature, lighter the element, and stronger the lattice bonding become, the larger are these deviations. Clearly, interpretation of heat capacity below the Debye temperature is beyond classical mechanics and requires quantum mechanical treatment. This is, however, beyond the scope of this text and interested readers are referred to standard physics textbooks (e.g., Tipler and Mosca, 2007). Therefore heat capacity at crustal temperatures should not be calculated from Equations 3 and 14 but rather be measured or calculated from appropriate, quantum mechanical equations.

Čermák and Rybach (1982) compiled data on isobaric specific heat capacity for different rock-forming minerals and different igneous, metamorphic, volcanic, and sedimentary rocks as well as the corresponding variations with temperature.

Measuring techniques

Specific heat capacity c can be measured directly or derived as the isobaric derivative of enthalpy H with respect to temperature. Specific heat capacity of rocks varies with temperature, pressure, porosity, and saturants. Accordingly, in situ values may deviate from laboratory data according to temperature, pressure, and type and content of pore fluid.

Numerous steady-state and transient calorimetric methods are available for measuring specific heat capacity. The most popular are mixing or drop calorimeters and heat flux differential scanning (DSC) calorimeters. The first method yields an absolute value; the second one is a comparative method. All of these methods and their details are discussed in the literature (e.g., Hemminger and Cammenga, 1989; Brown, 2001; Haines, 2002) to which interested readers are referred. The isobaric enthalpy change (or specific heat content) ΔH of solids may be determined by the method of mixtures using a Bunsen-type calorimeter, in which the unknown isobaric enthalpy change of a sample relative to a base temperature, e.g., 25 °C, is compared to the corresponding known isobaric enthalpy change of platinum (Kelley, 1960; Somerton, 1992).

Calculated heat capacity

When no direct measurements can be performed, the isobaric enthalpy change and specific heat capacity of rocks can be calculated according to Kopp’s law, Equation 15, as the arithmetic mean of the individual mineralogical and fluid contributions weighted by the volume fractions n_i of the N individual phases relative to total rock volume:

$$ \Delta {\text{H}} = \sum\limits_{{\text{i}} = 1}^{\text{N}} {{{\text{n}}_{\text{i}}}\,\Delta {{\text{H}}_{\text{i}}};\quad {\text{c}} = \sum\limits_{{\text{i}} = 1}^{\text{N}} {{{\text{n}}_{\text{i}}}\,{{\text{c}}_{\text{i}}}}; \quad } 1 = \sum\limits_{{\text{i}} = 1}^{\text{N}} {{{\text{n}}_{\text{i}}}}. $$

(15)

Based on data for various minerals (e.g., Kelley, 1960; Berman and Brown, 1985; Somerton, 1992; Waples and Waples, 2004), the isobaric enthalpy change ΔH or specific heat capacity c can be computed from Equation 15 for any rock consisting of an arbitrary number of minerals with given volume fractions.

Temperature dependence

Derived from measured variation of isobaric enthalpy change ΔH with temperature of various oxides, Kelley (1960) suggested a second-order polynomial for fitting ΔH from which c_P = (∂H/∂T)_P can be easily calculated. Somerton (1992) and Clauser (2006) report ΔH and c values of various rock-forming oxides and pore fluids. An alternative approach is fitting heat capacity measured at different temperatures directly to polynomials of various degrees (e.g., Maier and Kelley, 1932; Berman and Brown, 1985; Fei and Saxena, 1987; Holland and Powell, 1996; Robertson and Hemingway, 1995). Waples and Waples (2004) provide a discussion of the various approaches. The polynomial proposed by Berman and Brown (1985),

$${{{\text{C}}_{{\text{P,mol}}}} = {{\text{k}}_0} + {{\text{k}}_1}\,{{\text{T}}^{ - {0}{.5}}} + {{\text{k}}_2}\,{{\text{T}}^{ - {2}}} + {{\text{k}}_3}\,{{\text{T}}^{ - {3}}}{\text{ (T in K),}} }$$

(16)

works over a large temperature range and yields no values incompatible with the Dulong–Petit law for high temperatures. Table 2 lists values for the coefficients k₀–k₃ in Equation 16 determined from fits of heat capacity of selected minerals measured at different temperatures.

Thermal Storage and Transport Properties of Rocks, I: Heat Capacity and Latent Heat, Table 2 Coefficients for calculating isobaric molar heat capacity C_P,mol (J mol⁻¹ K⁻¹) (From Equation 16, Berman, 1988)

As an alternative, Waples and Waples (2004) propose a statistical approach describing the general temperature dependence of all rocks and minerals, which can be rescaled easily for individual rocks and minerals. To this end, measured specific heat capacity c_P was normalized by the corresponding normalizing value c_P,n at 200 °C (473.15 K), a temperature at or near which data was available. The resulting polynomial regressions yielded much better coefficients of determination R² for data measured on nonporous rock (R² = 0.93) than for those measured on minerals (R² = 0.62) while the trends were similar. The regression on the combined data for minerals and nonporous rocks yields an expression for the normalized specific heat capacity of a mineral or nonporous rock at arbitrary temperature T with a coefficient of determination R² = 0.65:

$$\begin{array}{ll}{{\text{c}}_{{\text{P,n}}}}({\text{T}})&= 0.716 + 1.72 \times {10^{ - 3}}\,{\text{T}} - 2.13 \\& \times {10^{ - 6}}\,{{\text{T}}^2} + 8.95 \times {10^{ - 10}}\,{{\text{T}}^3}\text, (T\;in\; ^\circ {\text{C}}) \cdot\end{array}$$

(17)

Equation 17 can be rescaled for any mineral or nonporous rock at any temperature T₂ provided a value C_P(T₁) measured at temperature T₁ is available, for instance from any of the compilations of Berman and Brown (1985), Fei and Saxena (1987), Berman (1988), Holland and Powell (1996), or Robertson and Hemingway (1995):

$$ {{\text{c}}_{\text{P}}}({{\text{T}}_2}) = {{\text{c}}_{\text{P}}}({{\text{T}}_1})\,\frac{{{{\text{c}}_{{\text{P,n}}}}({{\text{T}}_2})}}{{{{\text{c}}_{{\text{P,n}}}}({{\text{T}}_1})}}. $$

(18)

Additionally, Waples and Waples (2004) consider the variation of specific heat capacity with lithology, where interested readers find a specific discussion regarding coals of different carbon content or maturity.

Mottaghy et al. (2005) used a second-order polynomial in temperature to fit the variation of isobaric specific heat capacity with temperature measured on a suite of meta-sedimentary, volcanic, magmatic, and metamorphic rocks:

$$\begin{array}{ll} {{\text{c}}_{\text{P}}}({\text{T}}) = {{\text{A}}_0} + {{\text{A}}_1}{{\text{T}}^1} \\ \quad\qquad \quad+ {{\text{A}}_2}{{\text{T}}^2}({{{\text{c}}_{\text{p}}}{\text{in J k}}{{\text{g}}^{ - 1}}{ }{{\text{K}}^{ - 1}}{, 1}\,^\circ {\text{C}} \leq {\text{T}} \leq 100\,^\circ {\text{C}}})\cdot \end{array}$$

(19)

The average values for the coefficients A₀–A₂ determined from a regression of c_P(T) measured over a temperature range of 1 °C–100 °C on 26 samples from seven boreholes are: Ā₀ = 0.074717725 J kg⁻¹ K⁻¹; Ā₁ = 1.862585346 J kg⁻¹ K⁻²; Ā₂ = −2510.632231 J kg⁻¹ K⁻³.

Based on a composition of 30 % quartz, 60 % feldspar (albite), and 10 % phyllosilicates (5% phlogopite, and 5 % annite), Whittington et al. (2009) suggest average “bulk crustal” molar specific heat capacity equations based on end-member mineral data for two temperature ranges, separated by the transition at 846 K (~573 °C) between α- and β-quartz:

$$ \begin{array}{lll} {{\text{C}}_{{\text{P,mol}}}}({\text{T}}) = \\ \begin{cases} 199.50 + 0.0857{\text{T}} - 5.0 \times {{10}^6} {{\text{T}}^2}{;} & {\text{T}} \leq 846{\text{K}}\\229.32 + 0.0323{\text{T}} - 47.9 \times {{10}^{ - 6}}{{\text{T}}^2}{;} & {\text{T}} 846{\text{K}} \\\end{cases}\\ \qquad\qquad\qquad\qquad\qquad\qquad({{{\text{C}}_{{\text{P,mol}}}}{\text{ in J mo}}{{\text{l}}^{ -1}}{{\text{K}}^{ - 1}}})\cdot \end{array}$$

(20)

Assuming an average molar mass of 0.22178 kg mol⁻¹, this yields the variation of isobaric specific heat capacity c_P with temperature shown in Figure 1.

Thermal Storage and Transport Properties of Rocks, I: Heat Capacity and Latent Heat, Figure 1

Variation of specific heat capacity c_P, the phonon components of thermal diffusivity and thermal conductivity κ_p and λ_p, respectively, and thermal capacity ρ c_P = λ_p/κ_p with temperature in an average crust according to Equation 20.

Volumetric heat capacity: thermal capacity

When heat capacity is related to unit volume rather than to unit mass or unit amount of substance, it is referred to as volumetric heat capacity or thermal capacity. It can be calculated as the product of specific heat capacity c and density ρ or as the ratio of thermal conductivity λ and thermal diffusivity κ by

$$ \rho {\text{c}} = \lambda / \kappa. $$

(21)

Again, Kopp’s law yields the rock’s bulk thermal capacity (ρ c)_b as:

$$ {(\rho \,{\text{c}})_{\text{b}}} = (1 - \phi )\,{(\rho \,{\text{c}})_{\text{s}}} + \phi \sum\limits_{{\text{i}} = 1}^{\text{N}} {{{\text{S}}_{\text{i}}}\,{{(\rho {\text{c}})}_{\text{i}}}}, $$

(22)

where φ is porosity, (ρ c)_s thermal capacity of the rock skeleton, S_i fractional saturation, and (ρ c)_i thermal capacity of the i^th fluid phase in the pore space. The skeleton thermal capacity itself may be calculated again from Kopp’s law for a given mineral assemblage and the corresponding volume fractions of the solid phase from Equation 15. Because of the low density of air and gas – about three orders of magnitude lower than that of water and rock – the contribution of the gas phase to thermal capacity can often be ignored. In this case, N = 2 for the fluid phases water and oil or N = 1 for water only. Expressions for the density of various fluids are reported in Clauser (2006). Data on density of various minerals and rocks are listed, e.g., in Wohlenberg (1982a,b) or Olhoeft and Johnson (1989). Using Equations 17 and 18, Waples and Waples (2004) analyzed a substantial collection of density and specific heat capacity data from various authors and transformed specific heat capacity and thermal capacity to a uniform reference temperature of 20 °C (Table 3).

Thermal Storage and Transport Properties of Rocks, I: Heat Capacity and Latent Heat, Table 3 Typical values or ranges for density ρ, isobaric specific heat capacity c_P, and thermal capacity ρ c_P of selected rocks at 20°C (Waples and Waples, 2004; Petrunin et al., 2004)

The mean thermal capacity of “impervious” rocks was found at 2.300(46) MJ m⁻³ K⁻¹ by Roy et al. (1981). This is acceptably close to the mean of 2.460(65) MJ m⁻³ K⁻¹ found by Waples and Waples (2004) for inorganic minerals.

Based on density, specific heat capacity, and thermal conductivity measured at room temperature, Mottaghy et al. (2005) determined thermal capacity as inverse slope of a regression of diffusivity on thermal conductivity according to Equation 21. All values fell well within ±20 % of the average of 2.3 MJ m⁻³ K⁻¹ recommended by Beck (1988).

Mottaghy et al. (2008) determined average values for thermal capacity according to Equation 21 for metamorphic and magmatic crystalline rocks as the inverse slope of a linear regression of values of thermal diffusivity versus thermal conductivity measured at temperatures in the range 20 °C–300 °C:

$$ \kappa ({\text{T}}) = \frac{{\lambda ({\text{T}})}}{{{\text{m}} + {\text{nT}}}},{\text{ (T in}} ^\circ {\text{C),}} $$

(23)

Regression of data measured on seven samples collected along a profile crossing the Eastern Alps from north to south and on nine samples from the northern rim of the Fennoscandian Shield near the Kola ultra-deep borehole SG-3 yielded m = 20 66(70) kJ m⁻³ K⁻¹, n = 2.2(4) kJ m⁻³ K⁻², R² = 0.97 and m = 2404(91) kJ m⁻³ K⁻¹, n = 3.6(5) kJ m⁻³ K⁻², R² = 0.92, respectively. This yields a range of thermal capacity for the Alpine and Fennoscandian data of about 2.1 MJ m⁻¹ K⁻¹–2.7 MJ m⁻¹ K⁻¹ and 2.4 MJ m⁻¹ K⁻¹–3.5 MJ m⁻¹ K⁻¹, respectively, in the temperature range 20 °C–300 °C.

The product of an average density of 2 700 kg m⁻³ for the crust and isobaric specific heat capacity calculated for an average molar mass of 0.22178 kg mol⁻¹ according to Equation 20 (Whittington et al., 2009) yields the variation of thermal capacity ρ c_P (Equation 21) with temperature shown in Figure 1. Its increase with temperature by about a factor of 1.7 in a temperature interval of 1 000 K demonstrates that the effect of temperature is stronger for the phonon component of thermal diffusivity κ than for phonon thermal conductivity λ_p due to the increase of specific heat capacity. Assuming a constant density throughout the crust implies that the increase and decrease in density due to the increase in pressure and temperature, respectively, partly cancel each other and that these changes are small compared to those of specific heat capacity and thermal diffusivity.

Latent heat

Solidification of magma and melting of rocks as well as freezing and thawing of water in soils or rocks liberates or consumes heat, respectively. The like applies to mineral phase changes such as those associated with the seismic discontinuities at 410 km, 520 km, and 660 km in the transition zone from 400 km–600 km between the upper and lower mantle. These mineral phases are chemically identical but differ with respect to crystal structure and therefore elastic properties. This is why this transition is seen in the seismic wave field. Phase transitions require a certain pressure P and temperature T, but also a specific relation between these two state variables expressed by the so-called Clapeyron slope dP/dT = ΔS(P,T)/ ΔV(P,T) (the inverse of the Clausius–Clapeyron equation), where S and V are entropy and volume, respectively. This means that the depth where a certain phase transition occurs varies with the ambient temperature in the crust. Positive and negative values for the Clapeyron slope are associated with exothermic and endothermic reactions, respectively.

Phase changes generally consume or deliver much more latent heat than can be stored or delivered as sensible heat: It requires a temperature increase of more than 500 K to equal by sensible heat the amount of latent heat required to melt 1 kg of granite, and still an increase of more than 80 K to equal by sensible heat the amount of latent heat required to melt 1 kg of sea ice (Table 4).

Thermal Storage and Transport Properties of Rocks, I: Heat Capacity and Latent Heat, Table 4 Comparison of isobaric specific heat capacity c_P and latent heat of melting L of granite, basalt, seawater, and freshwater (Stacey and Davis, 2008, supplemented)

The discontinuity at 410 km is generally associated with the transition in olivine from α-olivine to β-spinel, also named wadsleyite (e.g., Stacey and Davis, 2008). At expected pressure and temperature of 14 GPa and 1,600 K, respectively, corresponding values for the Clapeyron slope vary between 2.9 MPa K⁻¹ and 4.8 MPa K⁻¹ (Bina and Helffrich, 1994; Katsura et al., 2004; Stacey and Davis, 2008). The discontinuity at 520 km is associated with the transition from β-spinel (wadsleyite) into γ-spinel, also named ringwoodite (e.g., Stacey and Davis, 2008). At an expected pressure of 18 GPa, a temperature increase, such as by an ascending plume, would require a higher pressure for this transition according to inferred Clapeyron slopes of 4.0 MPa K⁻¹–5.3 MPa K⁻¹ (Helffrich, 2000; Deuss and Woodhouse, 2001). A second transition occurs between garnet and calcium-perovskite (CaSiO₃), where the iron in garnet goes into Ilmenite, and its CaSiO₃-component into calcium-perovskite. This reaction has a negative Clapeyron slope. The two slopes of different sign may shift the depth for two transitions into opposite directions, which is observed as a splitting of the 520 km discontinuity. The discontinuity at 660 km defines the transition into the lower mantle. It is caused by the transition of γ-spinel (ringwoodite) into magnesium-perovskite (MgSiO₃) and ferrous periclase (magnesiowüstite, (Fe,Mg)O). At an expected pressure of 23.5 MPa, this endothermic transition is associated with a Clapeyron slope of − 2.8 MPa K⁻¹ (Stacey and Davis, 2008).

The latent heat L that corresponds to these additional heat sources and sinks can be elegantly combined with the specific sensible heat capacities of the liquid and solid rock, c_l and c_s, respectively, into an effective bulk specific heat capacity c_eff. This effective specific heat capacity then accounts for the entire enthalpy change, including latent heat. In this approach, the latent heat effects are assumed to occur between the solidus and liquidus temperatures T₁ and T₂, respectively. The heat liberated by a solidifying (“freezing”) liquid phase is obtained by weighting by the volume fractions of liquid and solid phases, φ_l and φ_s, respectively. The enthalpy change of the rock volume then becomes dH_freezing = (φ_l c_l + φ_s c_s) dT + L dφ_l, and the effective heat capacity c_eff is:

$$ {\text{c}}_{\text{eff}}^{\text{freezing}} = \frac{\text{dH}}{{\text{dT}}} = {\phi_{\text{l}}}\,{{\text{c}}_{\text{l}}} + {\phi_{\text{s}}}\,{{\text{c}}_{\text{s}}} + {\text{L}}\;\frac{{{\text{d}}{\phi_l}}}{\text{dT}}. $$

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Conversely, when considering melting the solid phase, the enthalpy change of the rock volume is dH_melting = (φ_l c_l + φ_s c_s) dT + L dφ_s, and the effective heat capacity in this case c_eff is:

$$ {{\text{c}}_{\text{eff}}^{\text{melting}} = \frac{\text{dH}}{{\text{dT}}} = {\phi_{\text{l}}\,{{\text{c}}_{\text{l}} + {\phi_{\text{s}}}\,{{\text{c}}_{\text{s}}} + {\text{L}}\;\frac{{{\text{d}}{\phi_{\text{s}}}}}{\text{dT}}}}}. $$

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Cross-references

Geothermal Record of Climate Change

Heat Flow, Continental

Heat Flow, Seafloor: Methods and Observations

Thermal Storage and Transport Properties of Rocks, II: Thermal Conductivity and Diffusivity