## Abstract

The acquisition of a permeability high enough to constitute an aquifer in crystalline rocks is a result of physico-chemical weathering through the transformation of minerals by the chemical processes of oxidation and hydration. The volumetric changes resulting from hydration induce stresses that fracture unweathered rocks. These reactions are exothermic, suggesting that the heat produced may generate a geothermal signature and even some hydrothermal effects. This paper develops a simplified model of thermal disturbance related to exothermic hydration to determine the relevance of this potential thermal effect. The fundamental thermal parameter is the rate of heat production, which is the product of the heat generation per unit volume multiplied by the velocity of the propagation of the front. When the front velocity is *c.* ≤1 mm a^{−1}, the temperature disturbance is negligible. The thermal effect only becomes significant if propagation of the front is extremely rapid (several centimetres per year). This observation led to an investigation of the instantaneous value of the front propagation velocity. This parameter was evaluated using a physical model coupling diffusion and chemical reactions with rock fracturing. Such extreme front velocities were only reached in exceptional circumstances; in most common situations, the weathering of crystalline rocks does not cause geothermal effects.

Unweathered, unfractured crystalline (plutonic and metamorphic) rocks have a low matrix porosity and low hydraulic conductivity. The formation of aquifers in such rocks (known as hard rock aquifers) is largely related to their weathering (Lachassagne *et al.* 2011). The typical sequence encountered in such rocks is illustrated by the weathering cross-section shown in the left-hand panel of Figure 1 (Lachassagne *et al.* 2014). Chemical weathering of some minerals induces swelling, such as the serpentinization of olivine in mafic rocks or the oxidation and hydration of biotite in granites. This volumetric expansion (Banfield & Eggleton 1990; Goodfellow *et al.* 2016) generates stresses which, in turn, induce fracturing of the rock (Wyns *et al.* 2004, 2015*a*; Royne *et al.* 2008). The resulting fracture network is known as a stratiform fractured layer (Lachassagne *et al.* in review). Its thickness can exceed 100 m (Dewandel *et al.* 2017). The fracture network is responsible for enhanced permeability, allowing interactions with the hydrosphere and, in particular, groundwater flow. It also favours the weathering process and the complete mechanical disintegration of rocks – for example, into a grussic saprolite for granites. The transformation from fresh rock into weathered rock (regolith = unconsolidated saprolite + underlying stratiform fractured layer) is a well-established fact, as illustrated in Figure 1. The rate of this transformation is generally evaluated on a steady-state assumption, which implies an order of magnitude consistent with the rate of denudation, at least as a long-term average (Dupré *et al.* 2003; Fletcher *et al.* 2006).

The transformation of unweathered crystalline rocks into regolith (saprolite and the stratiform fractured layer) is the result of several coupled phenomena, including chemical reactions and their mechanical consequences. The rate of these chemical reactions can be estimated directly by geochemical budgets using measurements of the concentrations of dissolved ions issuing from a given watershed (Dupré *et al.* 2003; Regard *et al.* 2016; Vasquez *et al.* 2016). However, their mechanical consequences are largely based on a scenario in which rock–fluid hydration reactions cause the volumetric expansion of minerals, resulting in an increase in the local stress field. This induces mechanical cracks in previously intact rock, which cause it to fracture and, in the case of granite, to eventually disaggregate into a grussic saprolite (Yakobson 1991; Buss *et al.* 2008; Rudge *et al.* 2010; Goodfellow *et al.* 2016). Many specific studies have modelled these coupled processes (weathering–fracturation) in ultramafic rocks such as peridotites (Kelemen & Matter 2008; Jamtveit *et al.* 2009; Rudge *et al.* 2010; Kelemen *et al.* 2011), but also, to a lesser extent, in granitic rocks (Fletcher *et al.* 2006; Goodfellow *et al.* 2016).

Another aspect of weathering reactions is their exothermic nature, which may generate thermal anomalies and is the focus of the study reported here. This thermal impact seems to be important in the oceanic lithosphere, where the heat generated during the serpentinization of the oceanic lithosphere by the inflow of seawater into the rocks of the ocean floor is thought to be responsible for some of the anomalously high heat flow observed in the Indian Ocean (Verzhbitsky & Lobkovsky 1993; Delescluse & Chamot-Rooke 2008). Following Fyfe (1974), the serpentinization of olivine appears to be a highly exothermic process, releasing *c.* 290 kJ of heat per kilogram of olivine. It consumes large amounts of water and modifies the oceanic lithosphere (Emmanuel & Berkowitz 2006). The importance of this process in plate tectonics on a global scale was stressed by Kelemen & Hirth (2012).

In continental domains, the weathering of crystalline bedrock by meteoric waters is the result of the same type of coupled mechanisms and the potential thermal impact of exothermic reactions deserves a quantitative evaluation. Although no data is available for the heat produced in granitic rocks by the oxidation and hydration of biotite and feldspars, it seems likely that these reactions are also highly exothermic (Wyns *et al.* 2015*b*). Despite this fact, their thermal effect is generally neglected. In most textbooks (e.g. Kappelmeyer & Haenel 1974), chemical reactions are considered to be of minor importance as heat sources and only a few direct studies are available on the effects of weathering on the geothermal regime (Parry *et al.* 1980). Nevertheless, on a geological scale, the conjugate thermal effects of metamorphic dehydration reactions in the thermal evolution of the crust during metamorphic episodes has long been recognized (Walther & Orville 1982; Lyubetskaya & Ague 2009). Wyns *et al.* (2015*b*) suggested that the energy released by weathering may locally heat the crust or even contribute to hydrothermalism in crystalline rocks.

This study aimed to determine whether, in such a geological context, the chemical reactions of weathering can generate a thermal disturbance. It is based on a simple conducting model (thought to give an upper bound of the thermal effects) and on a conservative assumption about the enthalpy of the reactions. The amplitude of the thermal disturbance is computed and discussed as a function of various parameters and, in particular, as a function of the dynamics of the weathering processes. This dynamic aspect is important because, locally, weathering reactions produce heat only during a limited time. Therefore a key parameter is the velocity of the weathering front, although weathering only occurs when crystalline rocks are exposed to the ground surface. It is tentatively evaluated on the basis of a physico-chemical model of saprolite generation that couples mass transfer by diffusion in the fluid phase with mechanical rock fracturing. All the parameters used in this paper are given in Table 1.

## Assumptions and model

The weathering chemical reaction is exothermic and releases a heat *E* per unit volume (units: J m^{−3}). From Fyfe (1974), the serpentinization reaction 2H_{2}O + MgSiO_{2} + Mg_{2}SiO_{4} → Mg_{3}SiO_{2} (OH)_{4} generates *c.* 290 kJ kg^{−1}, which corresponds to an available volumetric heat *E* of *c.* 1 GJ m^{−3}. The value adopted throughout our paper is *E* = 0.5 × 10^{9} J m^{−3}, which is one-half of the value proposed for the serpentinization of peridotite. This value of *E* is arbitrary. However, as the equation is linear, the thermal effects of other values of *E* can easily be computed if more strongly justified values of *E* are found in the future. This value is roughly equivalent to the energy available during the combustion of 10 l of gasoline (1 TEP = 42 GJ). This seems huge. If the whole of this heat *E* was used locally for heating a cubic metre of rock with a heat capacity *ρC*, then the increase in temperature would reach Δ*T = E*/*ρC* = 0.5 × 10^{9}/2 × 10^{6} = 250°C. This is a theoretical and virtual upper bound of the temperature effect because dissipating mechanisms, such as diffusion and fluid advection, tend to disperse this thermal effect.

In contrast with the quasi-permanent heat generated by radioactivity in the Earth's crust, the weathering reaction only produces heat for a limited time Δ*t*. Its rate *A* (units: W m^{−3}) = *E/*Δ*t* is imposed by the kinetics of the chemical reaction. The continuing nature of the reaction requires that the chemical front moves on to weather new rock and propagates, mostly downwards (Lachassagne *et al.* 2011, 2014), with a velocity *V*, into the unweathered rock. Therefore a front needs to propagate (downwards) with a velocity *V* and a width *a*, both parameters being imposed by the chemical (weathering) and physical (fracturing) phenomena accompanying the transformation; this is illustrated in the right-hand panel of Figure 1, where the front is moving down linearily because the propagation velocity is assumed to be constant.

The thickness of the active zone (here the regolith) is *a = V*Δ*t*, so that the average rate of heat generation becomes *A = EV*/*a*. More usefully, the instantaneous rate of heat generation per unit surface (called rate of heat release), *Aa*, is defined by:

The quantity *Aa* has units of W m^{−2}, i.e. of a flux of heat (as the geothermal heat flow). It characterizes the thermal effect of the reaction at least when thermal equilibrium is realized. A similar expression has been proposed for characterizing the thermal effects of the serpentization of the oceanic lithosphere (Verzhbitsky & Lobkovsky 1993).

Assuming that heat propagates purely by conduction, the thermal effect (temperature perturbation) due to this release of heat can be computed as a function of the relevant thermal parameters: the thermal conductivity and heat capacity. The calculation assuming only conduction provides an upper bound of the expected increase in temperature. In fact, if fluid advection or convection was to occur due to active groundwater flow, the resulting effect would tend to dissipate the increase in temperature more efficiently. The rock is assumed to be one-dimensional (as a function of depth *z*), with homogeneous properties (mineralogy, petrofabrics) and the front velocity *V* is constant and vertical. This assumption of homogeneous properties is reasonable for granites, where the minerals are randomly distributed. The one-dimensional assumption is also consistent with the observation that, in a geodynamic context favouring the ‘accumulation’ of a thick weathering profile, characterized by a flat or gently sloping surface of the Earth, the weathering profile is parallel to the topographic surface (Lachassagne *et al.* 2011, 2014). The layer where heat is generated, i.e. the depth range where minerals are submitted to chemical weathering, is then assimilated to a heat-producing sheet of thickness *a*. In fact, *a* is very small with respect to the thickness of the weathering profile, which reaches several tens of metres (Lachassagne *et al.* 2011, 2014) and, of course, with respect to the underlying fresh rock. Analytical solutions are given for the temperature disturbance due to such a moving heat source as a function of rate of release of heat, the velocity of the heat source, the depth and the duration.

### First assumption: steady state with front velocity V = erosion rate

The steady-state assumption is that, with respect to the rock, the velocity of the front is equal to the erosion rate, or at least to the chemical erosion rate (Regard *et al.* 2016). This means that an existing weathering profile is the ‘climax’ one and its thickness is stable. Long-term regolith production rates in granitic rocks are of the order of a few m myr^{−1} (myr = million years; Riebe *et al.* 2004). A similar order of magnitude is measured for the rates of denudation (5–100 m myr^{−1}) (e.g. Regard *et al.* 2016; Vasquez *et al.* 2016). According to this assumption, the velocity of the front *V* is equal to the denudation rate or, at least, to the chemical denudation rate, which is *c.* 60% of the total denudation rate (Regard *et al.* 2016). This leads to a chemical denudation rate of 0.3–30 m myr^{−1}, the rest being linked to physical processes.

As illustrated in Figure 2a, the temperature equation is solved in a frame moving downwards with velocity *V* with origin at the front. In this frame, the front remains immobile at the vertical coordinate *z* = 0, whereas the ground surface remains at *z = −h* as a result of erosion.

If *λ* is the thermal conductivity of the medium and *ρC* is its heat capacity, then *κ* = *λ*/*ρC* is the diffusivity. For crustal rocks, typical values of these thermal parameters are of the order of 2 W m^{−1} K^{−1} for *λ*, 2 × 10^{6} J m^{−3} K^{−1} for *ρC* and 10^{−6} m^{2} s^{−1} for *κ* (Kappelmeyer & Haenel 1974). The conductive equation for the temperature disturbance *T* with respect to the background temperature due to a heat release of *Aa = EV* (assumed to be equivalent to a plane source at *z* = 0) satisfies the heat equation (equation 2) above and below the level *z* = 0:

Equation (2) holds both for *−**h* < *z* < 0 and *z* > 0. At the level *z* = 0, the heat source acts as a contribution to the energy balance and is assimilated to a boundary condition. For *z* = 0, the solution *T*(*z*) is continuous, but its derivative is discontinuous due to the presence of the sheet:

The steady-state assumption requires *∂T/∂t* = 0 and boundary conditions are imposed at *z =* −*h* and *z =* +∞.
*z = −h* imposes a null disturbance
*z* = +∞.

The exact solution is:
*h* < *z* ≤ 0,
*z* ** ≥** 0.

For *z* of the same order as *h*, it is easy to see that *Vz/κ* is *c.* 10^{−6} *z* ≪ 1 and equations (7) & (8) are approximated by:
*u*, exp(*u*) is *c.* 1 + *u*.

The vertical profile *T*(*z*) is shown in Figure 2a. Between the ground surface and the level *z* = 0, the temperature disturbance increases almost linearly and remains constant for *z* > 0. The temperature disturbance is assimilated to that resulting from a heat-producing sheet located at *z* = 0. Typical figures used for the parameters are: *E* = 5 × 10^{8} J m^{−3}, *λ* = 2 W m^{−1} K^{−1}, *V* = 3 × 10^{−13} m s^{−1} (equivalent to 10 m myr^{−1}), *EV* = 1.5 × 10^{−4} W m^{−2}. For *h =* 100 m (weathering profiles of *c.* ≥100 m are common; Lachassagne *et al.* 2014; Dewandel *et al.* 2017), the resulting temperature disturbance occurs at and below *h* and is of the order of 0.01°C, i.e. practically undetectable. Such a small temperature disturbance is robust with respect to the various parameters. It must be remembered that the assumption of pure conduction (neither advection nor convection) gives an upper bound of the actual disturbance.

As a steady state imposes the velocity of the front *V*, the rate of heat release is so slow that most of the heat is dissipated at the Earth's surface, where a zero temperature disturbance is imposed. At the steady state, the heat released (*EV = aA c.* 10^{−4} W m^{−2}) is *c.* 1000 times smaller than the background geothermal heat flow (of the order of 0.1 W m^{−2}). However, we now explore the possibility of a transient thermal phenomenon associated with a sudden increase in the front velocity *V*.

### Alternative assumption: transient state and front velocity V ≫ erosion rate

The same equation with *∂T/∂t* ≠ 0 is solved for a transient beginning at *t* = 0 with *T*(*z*, *t*) = 0. The solution is derived from that of the moving source of heat (Carslaw & Jaeger 1959) and is developed in Appendix A. The condition of imposed surface temperature is ignored (assumption valid for *t* < *h ^{2}*/

*κ*, where

*h*is the initial depth of the front) and the solution is:

This solution is illustrated on Figure 2b in non-dimensional units. The normalized temperature profile *TρC*/*E* is displayed as a function of normalized depth *ζ = zV/κ* for several normalized times: *τ = tV*^{2}/*κ*.

The disturbance increases in amplitude and width as function of *t*. At *z* = 0 it is written, for small *t*:

Assuming that from *t* = 0, the velocity *V* reaches 0.2 × 10^{−8} m s^{−1} = 0.06 m a^{−1}. Then *EV* is *c.* 1 W m^{−2} (about ten times the geothermal flux) and the temperature disturbance *T* reaches 1.5°C for *t* = 1 year, 5°C for 10 years and 15°C for *t* = 100 years (which still obeys *τ* < *h ^{2}/κ*). It is difficult to envisage longer periods of time in this theoretical exercise for such an extreme velocity. Using these parameters, the thermal effect is noticeable because, due to the limited duration of the heat pulse (from a geological standpoint), the generated heat is applied for a limited range of time. It is clear, however, that such a velocity

*V*cannot be maintained for a long time (over a geological timescale of several thousand years). Because neither advection nor convection is taken into account, this again leads to the upper bound of the temperature disturbance. Nevertheless, it appears that in the case when the exothermic weathering reaction (hydration) occurs as sudden bursts, the generated heat may result in a significant increase in the local temperature at specific times.

## Discussion

### Maximum temperature

The heat input associated with weathering is potentially able to give a temperature excess *E/ρC* = 250°C if the weathered rock was perfectly isolated, which is a purely virtual scenario. In fact, for the time at which it has been generated, the heat is advected and diffused (due to conduction), so that the potential maximum thermal disturbance *E/ρC* is multiplied by a dimensionless coefficient *g* < 1, accounting for conductive losses.

In the steady-state assumption proposed here, the rate of chemical weathering (rate of regolith production) would be compensated by the rate of erosive denudation. The velocity *V* of the heat front is then equal to the rate of denudation, so that the geometry remains stable, as shown on Figure 2a. On a long timescale, this assumption is consistent with geochemical studies where the analysis of the long-term weathering rate is based on a large-scale budget of the ions exported by surface and groundwater (Riebe *et al.* 2004). Using this assumption the maximum temperature disturbance is reached at and below the depth *h* of the heat front (Fig. 2a) and is written as:

The coefficient *g = Vh/κ* is then small. For a *V* of a few metres per million years and *h* of *c.* 100 m, *g* is typically of the order of 10^{−5}–10^{−4}, so the thermal disturbance is vanishingly small (<0.01°C).

In contrast with the previous assumption, the alternative considers the possibility that the chemical reaction proceeds by sudden bursts during which the instantaneous front velocity *V* becomes very large for a while. Assuming that such a burst lasts for a time *t*, assumed to be smaller than the characteristic diffusion time *h ^{2}*/

*κ*, the temperature disturbance is at a maximum at the front and is written as:

The coefficient *g* is written as *Vt*, the distance travelled by the front at time *t*, and *√*(*πκt*), the thickness of the thermal disturbance generated by the heat source. Although <1, the coefficient *g* increases with time as *√t* and may reach appreciable values. For a front velocity *V* of the order of 6 cm a^{−1} (60 km myr^{−1}), *g* is *c.* 0.6 × 10^{−2} for *t* = 1 yr and *c.* 0.6 × 10^{−1} for *t* = 100 yr, therefore an appreciable transient temperature increase of up to a few tens of degrees may be obtained.

The overall result is that the thermal effects of chemical weathering on the temperature of rocks are generally negligible, except in very singular situations where the front velocity reaches very large values. If such a circumstance actually occurs, the assumed front velocity *V* (of the order of several centimetres per year) would be several hundred times larger than the long-term erosion rate (an order of magnitude of extreme erosion or exhumation rates is *c.* 1 km myr^{−1} or 1 mm a^{−1}; Wölfler *et al.* 2016). Of course, such an extreme velocity, say *V* = 6 cm a^{−1}, could not be maintained over a long period of time because, in 1000 years, the front would travel *c.* 60 m. It is therefore important to argue the possibility that such extreme *V* values could occur, at least occasionally.

Such occurrences of transient acceleration of the weathering front velocity would be limited to short periods of time to remain consistent with the long-term erosion rate. Long-term chemical rates based on the budget of the ions exported by surface and groundwater can hardly be used to infer this instantaneous velocity of the weathering front. This is because the geochemical data are integrating large-scale and long-term trends; moreover, the timescales over which they apply are not calibrated and are spatially variable (Green *et al.* 2006). Nevertheless, we present here some arguments in favour of this possibility of a large velocity. These arguments are based on a physico-chemical model describing the dynamics of a weathering front limited by diffusion and crack propagation.

### Front velocity for granite weathering according to a reaction-induced cracking model

The propagation of the weathering front can be modelled as a physico-chemical phenomenon accounting for fluid transport, mass transfer and mechanical fracturing of the previously intact rocks. Such a model of reaction-induced cracking has already been developed by Rudge *et al.* (2010) for application to the serpentinization and carbonatation of peridotites. The output of their model is the velocity of the front and the typical length of cracks opened in the rock in an assumed steady-state regime. Because this available model has been completely solved analytically for a wide range of physical and chemical parameters, it can be tentatively applied to the weathering of granite.

The weathering of granite has been the object of detailed studies (Buss *et al.* 2008; Goodfellow *et al.* 2016), emphasizing that the oxidation of biotite is a key initial reaction. Nevertheless, the exothermic reactions associated with the chemical front are probably more complex. In the model of this study, they are assumed to occur simultaneously. The weathering of biotite is described by a single general reaction (see the reference to Asselborn 2013):

The basic processes used are illustrated in Figure 3, in a framework moving along with the unknown velocity *V* of the weathering front. Above the front, lying at *z* = 0, the rock is transformed into regolith, where the hydraulic conductivity is increased and (ground)water is more easily renewed than below (initial content *w*_{0} in mol m^{−3}). Below the front the water can only diffuse into the intact rock and reacts with biotite (after the key initial reaction of iron oxidation) to generate some dilating reactant (kaolinite). As the reaction proceeds, the biotite content *b* decreases from its initial value in the intact rock (initial content *b*_{0}) because the chemical reaction begins as soon as water diffuses slowly into the intact rock. The volumetric increase in the reacted products induces fractures in the intact rock, thereby propagating the front deeper into the intact rock. The progression of the front is limited by the slow diffusion of fluid in the intact rock, by the kinetics of the reaction and by the generation of cracks advancing in the intact rock (see Fig. 3). The equations describing this scenario – including diffusive transport, chemical reactions and the propagation of cracks – are described and solved in Rudge *et al.* (2010). Based on this simplified one-dimensional model, and assuming that a steady state is realized, the equations are solved for the front velocity *V* and typical crack length *L*.

The application of the solution of Rudge *et al.* (2010) requires the definition of several parameters. Some of them are well constrained from mechanical tests, such as the mechanical parameters of unweathered granitic rocks (Young's modulus *Y*, Poisson's ratio *ν* and the fracture toughness *K*_{c}, which characterizes the ability of the rock to resist the growth of a pre-existing crack). Other parameters are estimated, such as the diffusion coefficient *D* of the active fluid and the time constant of the hydration reaction *µ* deduced from the biotite dissolution constant measured in the laboratory (Acker & Bricker 1992; Kalinowski & Schweda 1996). The composition of intact rocks (porosity *ϕ* = 0.005 and the volumetric percentage of biotite *α* = 0.05) apply to standard monzonitic granite (Navarre-Sitchler & Brantley 2015). Reaction (16), with its stoichiometric numbers, produces a volumetric variation *δv*/*v* = 0.2 due to the transformation of biotite into hydrated kaolinite. This value of 0.2 is highly consistent with the results of Banfield & Eggleton (1990); it gives confidence to the chosen parameter set. The application of these parameters to the reaction-induced cracking model of Rudge *et al.* (2010) is developed in Appendix B.

Using these parameters, this reaction-induced cracking model predicts a front velocity *V* of the order of 56 mm a^{−1} and a typical crack opening length *L* of 6 mm. Such a high velocity is consistent with that previously assumed in the alternative thermal model developed here for the generation of appreciable thermal anomalies. Such high velocities of the reaction front of several kilometres per million years of peridotite serpentinization have already been proposed to explain heat flow anomalies observed on the oceanic lithosphere (Delescluse & Chamot-Rooke 2008).

In the case of the near-surface weathering of granitic rocks, it is clear that such a high velocity, even if it is reached, cannot be maintained over a long time, particularly in geodynamically stable regions such as those where the accumulation of weathering profiles prevail. Nevertheless, it is possibile that the front is advancing by rapid jumps and stops between jumps because the chemical reaction lacks reactants – for example, the reaction front may be short of water due to local clogging of the fractures in the fractured layer (and also the alteration of their diffusion properties). This possibility cannot be formally rejected, although its occurrence should be considered as exceptional in time and space. Nevertheless, in some instances, it could serve as an alternative explanation to the classical interpretation of deeply rooted fractures to explain some thermal anomalies in stable crystalline areas (this hypothesis being cited, for instance, by Roy & Le Guern 2015).

## Conclusions

The weathering of crystalline rocks is responsible for the creation of porosity and permeability. It may also contribute to the release of heat because chemical reactions such as oxidation and hydration are exothermic. This study aimed to discuss the possible thermal impact of the chemical weathering of crystalline rocks, with the suggestion that this heat may even contribute to hydrothermalism, in particular in granitic rocks.

It is shown that such a potential thermal effect depends on the fundamental parameter *EV* (the rate of heat production), which is the product of heat generation *E* per unit volume by the velocity *V* of the weathering front. The solution of the conduction heat equation is used to evaluate the thermal anomaly assuming a (relatively large) heat generation of 5 × 10^{8} J m^{−3} for the reaction and various assumptions for the velocity. These assumptions give an upper bound for the thermal anomaly because neither water advection nor convection, which would dissipate heat, is taken into account in the computation. It is demonstrated that, on a long-term basis, the front velocity *V*, assumed to be coupled to the rate of denudation, is so low that the thermal effects of weathering are negligible.

The thermal effects may become significant only if extremely rapid front propagation *V* occurs (several centimetres per year). If such a burst occurs, its thermal effect is transient and increases with time as *√**t.* To discuss the possible occurrence of such chemical runaways, the front velocity *V* is evaluated using the available physico-chemical models coupling the transfer of mass by diffusion in the intact rock, the chemical reaction of some minerals and the occurrence of mechanical rock fracturing. An application of the model of Rudge *et al.* (2010) shows that such large front velocities (about 5 cm a^{−1}) could indeed be reached, although in exceptional circumstance in terms of both place and time. A thermal effect may then appear, but would be exceptional.

Thus, in most common situations, the weathering of crystalline rocks cannot cause a geothermal effect. Other more classical causes, such as deep fractures or an anomalous geothermal flux, must consequently be invoked. The very low to zero occurrence of geothermal anomalies or thermal springs in stable crystalline regions confirms the main results of this paper.

Further studies should focus on a better evaluation of the heat generation associated with weathering reactions and on the actual vertical distribution of weathering and temperature in crystalline areas where the weathering rate (or erosion conditions) can be constrained.

## Acknowledgements

The authors thank Robert Wyns and Laurent Guillou-Frottier who pointed out various aspects of the weathering of granitic rocks during the La Roche/Yon 2015 conference (www.cfh-aih.fr/index.php/colloques/78-colloque-socle-2015). Thanks are also due to Benoit Dewandel, Francis Lucazeau and Pierre Genthon for fruitful discussions. We also warmly thank the Corresponding Editor (U. Ofterdinger) and the two reviewers (one anonymous and B. Goodfellow) for their careful reviews, which helped to improve this paper.

## Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

## Appendix A: Transient temperature due to a moving heat-producing sheet

The solution of the heat equation is derived from the moving point source solution (Carslaw & Jaeger 1959) valid for a whole space medium, initially at 0°C temperature. Assume first a point heat source located at the origin and emitting, for *t*′ > 0, a heat source of intensity *q* per unit time. For time *t* > 0, the temperature at position *x, y, z* is given by the point source solution of the heat equation (2) valid for a medium moving along *z* with velocity *V*:
*x*′, *y*′, *z*′ = 0) with a surface source intensity *Q* per unit surface and unit time (units: W m^{−2}). In the present situation, *Q* will be identified with the rate of heat release *EV*. The temperature at *x*, *y*, *z* and time *t* is derived from an integration of equation (A1) with respect to *x*′, *y*′:
*z* > 0 so that *d*^{2}:
*z* < 0, the change of variable *t*→+∞, *θ*(*z*) tends towards *Q*/(*ρCV*) for *z* > 0 and towards *Q*/(*ρCV*)exp(*zV/κ*) for *z* < 0 as given by Carslaw & Jaeger (1959, p. 267). For *Q = EV*, this steady-state solution differs from that obtained in the main text because, in this development, no boundary condition *θ* = 0 was imposed at a level *z = −h* and therefore no heat sink exists. However it is easy to argue that, for small values of *t* such that *t < κ*/*h*^{2}, the influence of such a boundary (heat sink at ground surface) is negligible so that expressions (A3) and (A4) represent a fairly good approximation.

## Appendix B: Material properties and front evolution following Rudge *et al.* (2010)

The intact rock is a granite with standard mechanical parameters: Young's modulus *Y* = 10^{11} Pa; fracture toughness *K*_{c} = 10^{6} Pa m^{1/2}; and Poisson's ratio *ν* = 0.2 (Meredith & Atkinson 1985). Its volumetric mineral composition indicates *α* = 5% biotite crystals; this corresponds to an initial molar volumetric concentration *b _{0}* = 300 mol m

^{−3}. The initial water content corresponds to a porosity

*ϕ*= 0.5%, i.e. a molar volumetric concentration

*w*

_{0}= 280 mol m

^{−3}. The diffusion coefficient of water is deduced from the self-diffusion coefficient of free water

*D*= 2 × 10

_{w}^{−9}m

^{2}s

^{−1}(Mills 1973). To account for the matrix effect in the porous medium the relevant diffusion coefficient is multiplied by the porosity

*D = D*

_{w}

*ϕ*.

The constant *µ* is the relative rate of biotite complete weathering (−2d*b/b* d*t*) at the front (the coefficient 2 coming from the stoichiometry of reaction 16). Its value is approached by the experimental data on the dissolution of biotite at room temperature and near-neutral pH (Acker & Bricker 1992; Kalinowski & Schweda 1996). These experiments give for 4 < pH < 8 a rate of biotite weathering of the order of *R* = 10^{−11} mol m^{−2} s^{−1}. For millimetre-scale biotite crystals, the specific surface is *c.* 1 m^{2} g^{−1} or *S* = 3 × 10^{6} m^{2} m^{−3}, which implies a volumetric dissolution rate d*b/*d*t = RS* = 3 × 10^{5} mol m^{−3} s^{−1} (the volumic mass of biotite is 3 × 10^{6} g m^{−3}). From these experiments using pure biotite, i.e. *c. b* = 6000 mol m^{−3}, the kinetic constant *μ* of the model (called *κ* in Rudge *et al.* 2010) is:
*b* moles of biotite into kaolinite is followed by an increase in the volume occupied by reactants (ratio *α* of the total volume) because they contain several layers of bound water: the assumed relative volumetric increase is *δv*/*v* = 20% (Banfield & Eggleton 1990). Following Rudge *et al.* (2010), this expansion results in an increase in stress. This stress, assumed to be linearly related to the level of transformation, is given by:
*L*. This means that, at the front, *K*_{c} *= σL*^{1/}^{2} where *L* is the length over which *σ* decays significantly. Below the front, in the unweathered rock, the fluid concentration is assumed to satisfy the diffusion equation and to react with the initial biotite at a rate characterized by *µ*. The solution of the conservation equations, once closed by the fracture criterion, gives the front velocity *V* and the length *L* over which a crack can propagate (Rudge *et al.* 2010).

Two non-dimensional numbers ruling the phenomenon are
*K _{c}*/

*Yβ*)

^{2}development and

With these parameter values, we obtain (Λ = 3.3 × 10^{−8}, Θ = 4 × 10^{−1}) lying in the asymptotic domain 2 of Rudge *et al.* (2010) in the (Λ,Θ) plane. In this domain, characterized by Λ ≪ 1 and Λ ≪ Θ^{5}, the propagation of cracks is already effective at the front of the reaction (as soon as a few per cent of the biotite of the intact rock is transformed). This asymptotic solution gives the front velocity *V* ≈ *D*^{3/5}(*βYμ*/*K _{c}*)

^{2/5}= 1.8 × 10

^{−9}m s

^{−1}(or 56 mm y

^{−1}) and the typical crack length

*L*≈ (

*K*/

_{c}D*βYμ*)

^{2/5}= 5 mm. These values can be modulated as a function of the leas well-known parameters, i.e.

*D*and

*µ*. In particular, the value of

*µ*(obtained through equation B1 from the rate

*R*in laboratory experiments) may be largely overestimated for natural situations (White & Brantley 2003). However, the dependance of

*V*and

*L*as a function of the later parameters is relatively weak (respectively as

*D*

^{3/5}

*µ*

^{2/5}or

*D*

^{2/5}

*µ*

^{−2/5}) so that the calculated values of

*V*remain largely >1 mm a

^{−1}.

- © 2018 The Author(s). Published by The Geological Society of London. All rights reserved