Devolatilization-induced pressure build-up: Implications for reaction front movement and breccia...

Devolatilization-induced pressure build-up: Implications forreaction front movement and breccia pipe formation

I . AARNES1, 2 , Y. PODLADCHIKOV3, H. SVENSEN1

1Roxar ASA, Lysaker, Norway; 2Physics of Geological Processes, University of Oslo, Blindern, Oslo, Norway; 3Institute of

Geophysics, University of Lausanne, Lausanne, Switzerland

ABSTRACT

Generation of fluids during metamorphism can significantly influence the fluid overpressure, and thus the fluid

flow in metamorphic terrains. There is currently a large focus on developing numerical reactive transport models,

and with it follows the need for analytical solutions to ensure correct numerical implementation. In this study, we

derive both analytical and numerical solutions to reaction-induced fluid overpressure, coupled to temperature and

fluid flow out of the reacting front. All equations are derived from basic principles of conservation of mass,

energy and momentum. We focus on contact metamorphism, where devolatilization reactions are particularly

important owing to high thermal fluxes allowing large volumes of fluids to be rapidly generated. The analytical

solutions reveal three key factors involved in the pressure build-up: (i) The efficiency of the devolatilizing reaction

front (pressure build-up) relative to fluid flow (pressure relaxation), (ii) the reaction temperature relative to the

available heat in the system and (iii) the feedback of overpressure on the reaction temperature as a function of

the Clapeyron slope. Finally, we apply the model to two geological case scenarios. In the first case, we investigate

the influence of fluid overpressure on the movement of the reaction front and show that it can slow down signif-

icantly and may even be terminated owing to increased effective reaction temperature. In the second case, the

model is applied to constrain the conditions for fracturing and inferred breccia pipe formation in organic-rich

shales owing to methane generation in the contact aureole.

Key words: analytical solutions, breccia pipes, contact metamorphism, devolatilization, fluid overpressure

Received 8 November 2011; accepted 24 May 2012

Corresponding author: Ingrid Aarnes, Physics of Geological Processes, University of Oslo, PO Box 1048 Blindern,

0316 Oslo, Norway.

Email: [email protected]. Tel: +47 22547842. Fax: +47 51818801

Geofluids (2012)

INTRODUCTION

Metamorphic devolatilization of sedimentary rocks is a key

process during progressive metamorphism. Usually, the

positive change in fluid volume upon devolatilization is lar-

ger than the corresponding decrease in solid volume, caus-

ing a net volume increase in the reaction. If the sudden

volume increase cannot be accommodated by the

host-rock, the fluid pressure will build up (Hanshaw &

Bredehoeft 1968; Thompson 1987; Walther 1990). Fluid

overpressure is present when the fluid pressure exceeds the

background hydrostatic pressure. Understanding the pro-

cesses related to the release of metamorphic carbon fluids

(CH4 and CO2) is important also in a broader perspective

of the global carbon cycle (e.g. Bickle 1996; Svensen et al.

2004; Aarnes et al. 2010).

There has been a recent focus on volume changing reac-

tions as a driving force to pressure build-up, both in the

case of isochoric conditions and in the presence of fluid flow

(e.g. Manning & Bird 1991; Corbet & Bethke 1992; Bred-

ehoeft et al. 1994; Dutrow & Norton 1995; Labotka et al.

2002; Jamtveit et al. 2004). In this study, we systematically

explore this process one step further, by deriving analytical

solutions for overpressure generation coupled to a thermally

driven reaction front, while at the same time, honouring

pressure relaxation by fluid flow. The analytical solutions

are cross-verified by similar numerical solutions. In the anal-

ysis, we identify the important parameters, which allow us

to estimate the magnitude of the fluid overpressure directly.

Hence, the analytical solutions presented in this study are

important tools that can be applied in their current state to

a wide range of geological phenomena.

Geofluids (2012) doi: 10.1111/j.1468-8123.2012.00368.x

� 2012 Blackwell Publishing Ltd

We present two case studies to illustrate some geological

applications: (i) the feedback of overpressure on the move-

ment of the reaction front; (ii) breccia pipe formation in

contact aureoles in organic-rich shales owing to the meth-

ane generation, as suggested by Svensen et al. (2007).

The model is solved for contact metamorphism specifi-

cally, but the results are also relevant for regional prograde

metamorphism, and other areas with high heat flow, such

as crustal shear zones and zones with magmatic activity

during crustal thickening (e.g. Leloup et al. 1999; Baxter

et al. 2002; Ague & Baxter 2007; Weinberg et al. 2009).

Contact metamorphism is of particular relevance owing

to the rapid heat transfer causing efficient metamorphic

devolatilization of the intruded sediments (e.g. Simoneit

et al. 1978; Jamtveit et al. 1992; Lasaga & Rye 1993;

Bishop & Abbott 1995; Cooper et al. 2007; Nabelek

2007; Aarnes et al. 2010). Overpressure during contact

metamorphism can be generated by several volume chang-

ing processes, such as boiling and thermal expansion of

pore fluids, buoyancy effects, release of magmatic fluids

and fluid generation within the contact aureole (e.g. Fur-

long et al. 1991; Hanson 1992, 1995; Nabelek & Labotka

1993; Osborne & Swarbrick 1997). There are several stud-

ies focused on the overpressures resulting from boiling and

thermal expansion of pore fluids, especially around large

plutons (Knapp & Knight 1977; Norton & Knight 1977;

Einsele et al. 1980; Delaney 1982; Manning & Bird 1991;

Cathles et al. 1997; Cui et al. 2001; Jamtveit et al. 2004).

However, we do not include this process as incorporation

of the equation of state for fluid, solid or pore compress-

ibility into our derivations is outside the scope of this

study.

Reactive transport models in porous media coupling

mass transport, heat, fluid flow and chemical reactions in

various geological systems are well documented in the liter-

ature (e.g. Lichtner 1985, 1988; Ortoleva et al. 1987;

Baumgartner & Ferry 1991; Steefel & Lasaga 1994; Kele-

men et al. 1995; Le Gallo et al. 1998; Xu & Pruess 2001;

Gaus et al. 2005). A key topic in these models is fluid–rock

interactions, with for example coupling of fluid flow and

solute concentrations, or coupling of solute concentrations

and temperature (Steefel & Maher 2009; and references

therein). There is, however, limited focus on the hydro-

logic driving forces for the fluid flow or on the pressure

feedback on the reactions. A standard simplification in the

derivations of the fluid flux is to consider conservation of

the fluid mass in a steady-state flow regime as a response

to thermal expansion only (Furlong et al. 1991; Ferry &

Gerdes 1998). The aim of this study is to make the first

step towards a more elaborate pressure model to be uti-

lized in fully coupled reactive transport modelling.

Several studies demonstrate the importance of devolatil-

ization reactions on fluid pressure and fluid flow through

the effective volume change (e.g. Connolly 1997; Balashov

& Yardley 1998; Nabelek 2009). A potentially important

implication of increased fluid pressure is the feedback on

the reaction front propagation for pressure-dependent

reactions (Colten-Bradley 1987; Osborne & Swarbrick

1997; Miller et al. 2003; Wang & Wong 2003). Dehydra-

tion of gypsum yields a relevant example of how the high

pore pressures inhibit the reaction until fluid flow relaxes

the overpressure (Wang & Wong 2003). The importance

of the pressure dependence of the reaction is among oth-

ers documented for the dehydration conditions in the

Southern Alps, New Zealand, where reducing the pressure

is the key for triggering dehydration reactions (Vry et al.

2010).

Evidences of significant overpressure generation during

contact metamorphism come from the development of

fracture networks (e.g. Nishiyama 1989; Manning & Bird

1991; Nabelek & Labotka 1993; Aarnes et al. 2011b), flu-

idization of the host sediment (Delaney 1982; Kokelaar

1982), formation of sandstone dykes (Walton & O’Sullivan

1950; Svensen et al. 2010) and hydrothermal vent com-

plexes and breccia pipes (Jamtveit et al. 2004; Svensen

et al. 2004, 2006, 2007; Planke et al. 2005). The latter

two are both vertical, circular channels extending from

within the aureoles to the paleosurface. The difference in

terminology is that hydrothermal vent complexes are

formed as a result of boiling and expansion of pore fluids

in shallow aureoles no deeper than approximately 1 km

where boiling of H2O can be expected (Jamtveit et al.

2004). Breccia pipes, on the other hand, origin in aureoles

several kilometres down where boiling of water is no

longer expected, and their documented existence provides

spectacular evidence for fluid pressure build-up in the

metamorphic aureoles (e.g. Svensen et al. 2007). However,

their formation mechanism is still not resolved, and we

therefore use the results of this study to investigate

whether the breccia pipes can be triggered by the reaction-

induced pressure build-up from methane release in heated

shales.

THEORETICAL BACKGROUND ANDMETHODS

Temperature of a cooling sill

Conductive heat transfer

The standard equations, for example, pressure and temper-

ature diffusion given in textbooks such as Carslaw & Jaeger

(1959), are derived under the assumption that these pro-

cesses operate alone. However, when dealing with coupled

processes, care must be taken to ensure that we utilize a

thermodynamically consistent set of equations that includes

all potential feedback processes. This is properly taken care

of in this study by deriving all equations from basic princi-

ples of conservation of energy, mass and momentum,

2 I. AARNES et al.


combined with a local thermodynamic equilibrium

approach. A detailed explanation of all the steps involved

in the derivation of the system of equations used in this

study can be found in Aarnes (2010). A list of all symbols

used in this study is presented in Table 1.

In our model, we represent the heat transfer between

the magma and host-rocks by a conductive heat equation

with latent heat of crystallization. The heat conduction

equation can be written

oT

ot¼ �T

o2T

oz2ð1Þ

where z is the vertical direction, t is time, T is tempera-

ture, �T ¼ �=�=Cp is the thermal diffusivity, Cp is the

specific heat capacity, q is density and k is the thermal

conductivity. We use this relation as a first-order approxi-

mation to the heat transfer, and hence do not consider

heat transport by fluid advection in this study. One key

parameter to determine the influence of advective heat is

the ratio of fluid available to carry the heat versus the

total rock volume to be heated. This was analysed in a

study of Podladchikov & Wickham (1994), who showed

that for a release of <10 wt% of fluid, heat advection is

negligible regardless of what numbers the other parame-

ters take. For our setup with the single release of no more

than approximately 10 wt.% of metamorphic fluids, the

diffusive heat transfer regime provides sufficient complex-

ity to effectively capture the bulk change in the tempera-

ture relevant for the processes involved. However, this

may be a too simplified assumption in other settings not

considered here, such as around large plutons or in aure-

oles with considerable amounts of heated pore fluids

flushing through the host-rocks (e.g. Bickle & McKenzie

1987). The model is further simplified by considering a

one-dimensional formulation, which captures the key

response in horizontally layered rocks, as well as giving

boundary conditions that allow us to solve the equations

analytically.

Latent heat of crystallization

Specific latent heat, Dh, is given in J ⁄ kg and is the amount

of energy released upon the phase transition from melt to

crystals in a cooling magma. We use two common meth-

ods to treat this reaction: (i) The sharp interface approach

used in the analytical solution, where we determine the

position of the reaction front and follow its progression

towards the centre of the intrusion with time; (ii) The dif-

fuse interface approach used in the numerical solution,

where the front is approximated over an temperature inter-

val between the liquidus (TL) and solidus (TS) tempera-

tures. The two methods are equivalent in the limit where

TS is similar to TL, which is the case, for example eutectic

crystallization. Although mafic intrusions are not eutectic,

the effect on cooling time between a linear heat capacity

model in comparison with more elaborated models is neg-

ligible for melts of mafic composition (Podladchikov &

Wickham 1994; Turcotte & Schubert 2002).

Table 1 A list of symbols used in this paper.

Symbols Descriptions Units

af Isobaric thermal expansion 1 ⁄ Kb Effective fluid compressibility 1 ⁄ Pa

Cp Heat capacity J ⁄ kg ⁄ K/ Porosity –

Dh Specific latent heat J ⁄ kg

jT Thermal diffusivity m2 ⁄ sjH Hydraulic diffusivity m2 ⁄ sL* Latent heat effect of crystallization J ⁄ kg ⁄ Kk Thermal conductivity J ⁄ s ⁄ m ⁄ Klf Fluid viscosity Pas

nT Nondimensional parameter in thermal regime –

nP Nondimensional parameter in pressure regime –

�mT Nondimensional rate of crystallization front –

�cT Nondimensional contact position –

�rT Nondimensional reaction front –

Pf Fluid pressure Pa

Pr Fluid pressure at the reaction front Pa

PV Pressure from volume change of reaction

(max overpressure)

Pa

DPf Fluid overpressure Pa

Phr Background hydrostatic pressure Pa

PC Contact pressure Pa

P* Normalized overpressure –

DPr Fluid overpressure at the reaction front Pa

P�r Normalized fluid overpressure at

the reaction front

–

dP ⁄ dT Clapeyron slope Pa ⁄ KQ total

f Source of total fluids generated at the front kg ⁄ m3

Qq,f Rate of fluid mass production per unit

volume of rock

kg ⁄ m3 ⁄ s

q Density kg ⁄ m3

Ri Nondimensional ratio of reaction

front vs. fluid flow

–

Rc Parameter describing pressure

dependence of reaction

–

Ste Stefan number -

t Time s

T Temperature K

TL Liquidus temperature of melt K

TS Solidus temperature of melt K

Thr Initial host-rock temperature K

Tm Initial melt temperature K

Tc Contact temperature between

intrusion and host-rock

K

Tr Reaction temperature isograd K

T* Normalized temperature –

Tc* Normalized contact temperature –

Tr0 Initial reaction temperature

(no pressure influence)

K

T effr Effective reaction temperature

(influenced by pressure)

K

DThp Parameter describing additional heating

for reaction to proceed

–

z Vertical direction m

zr Position of the reaction front m

Devolatilization-induced pressure buildup 3


The sharp interface approach is performed by treating

the latent heat release by a heat balance at the crystalliza-

tion front (e.g. Turcotte & Schubert 2002),

�Dhdzm

dt¼ � oT

oz

� �z¼zm

ð2Þ

This equation is used to derive the position of the reac-

tion front as a function of both heat conduction and the

latent heat release, where more latent heat results in a

slower crystallization front.

The diffuse interface approach, also known as the effec-

tive heat capacity method, is frequently introduced for

numerical solutions (e.g. Lewis et al. 1996). We can treat

the latent heat release as an effective heat capacity increase

in the interval between TS and TL,

Cpeff ¼ Cpð1þ L�Þ for TS<T<TL

Cpeff ¼ Cp for T � TS and T � TL

ð3Þ

where the latent heat effect is approximated by

L� ¼ DhCp TL�TSð Þ.

Fluid pressure during prograde metamorphism

Fluid pressure evolution

The simplified hydraulic diffusion equation can be written as

oPf

ot¼ �H

o2Pf

oz2; ð4Þ

where Pf is the fluid pressure, �H ¼ k=ð’��f Þ is the

hydraulic diffusivity coefficient, k is permeability, lf is fluid

viscosity, / is porosity and b is the effective fluid compress-

ibility.

Compressibility of the porous matrix adds to the total

storage, and we make a rough approximation to this effect

by using an effective fluid compressibility. More sophisti-

cated models incorporating the effect of matrix deforma-

tion will require full coupling of the fluid flow equation to

the system of equations describing the matrix deformation

and total stress history and is not within the scope of this

study. The response of matrix to a localized fluid pressure

build-up is investigated in Rozhko et al. (2007) for a non-

reactive case.

The effect of gravitational flow owing to buoyancy is

incorporated implicitly through solving for pressures in

excess of the background hydrostatic pressures.

Reaction-induced pressure

The key unknown parameter to be solved for in this study

is the fluid pressure at the reaction front, Pr, as a result of

the fluids released from the host-rock. It is coupled to

temperature through the devolatilizing reaction front, zr,

which we assume for simplicity is coinciding with a

reaction temperature isograd, Tr (Fig. 1). The position of

this reaction front is controlled by the efficiency of heat

diffusion into the host-rock. If the generated fluids cannot

move away from the reaction front, the overpressure is

controlled by the effective volume change in the reaction,

which can be readily calculated. We refer to this process as

the isochoric pressure build-up. However, if the fluids are

flowing away from the reaction front, the overpressure at

the front is reduced by this distribution of the pressure fur-

ther out into the host-rock (Fig. 1).

By analogy with the conservation of energy for latent

heat (Eq. 2), the reaction-induced pressure source can be

incorporated into the analytical solution using the sharp

interface approach, by setting up a mass balance relation at

P

z

PrPhr

∆Pr

T

z

TmThr

Tr

TS

TL

Tc

Melt

Crystals

Reacted host-rock

Reaction front ( zr )

Crystallization front ( zm )

Melt + crystals

Ho

st-r

ock

Intr

usi

on

Temperature profileat a given time

Pressure profileat a given time

P

Fig. 1. A schematic drawing of temperature and

pressure profiles at a certain time after sill

emplacement. There are two moving fronts; the

crystallization front (zm) and the reaction front

(zr). The crystallization front (zm) marks the inter-

face between melt and crystals and moves

towards the centre of the intrusion with time.

Latent heat is released upon crystallization in the

zone where melt and crystals exist. The reaction

front follows the reaction temperature isograd

(Tr) and marks the interface between the devola-

tilization reactions and the un-reacted host-rock.

The release of fluids at the reaction front builds

up overpressure Pr relative to background hydro-

static pressure Phr.

4 I. AARNES et al.


the reaction front. Conservation of mass requires that the

fluid production rate at the reaction front equals the fluid

expulsion rate away from the reaction front,

1

�f

� 1

�s

!Q total

f

dzr

dt¼ � k

�f

oDPf

oz

� �z¼zr

; ð5Þ

where Q totalf is the source term for the total amount of fluids

generated at the front (kgfluid ⁄ m3rock) and DPf = Pf - Phr is

the fluid overpressure, that is, the fluid pressure relative to

the background hydrostatic pressure (Phr) prior to any dev-

olatilization reaction. The left-hand side of Eq. (5) describes

the fluid production rate at the reaction front, and the right-

hand side of Eq. (5) describes the expulsion rate out of the

front following Darcian flux down the pressure gradient.

This balance assumes a steady-state fluid flux away from the

reaction front and further into the host-rock.

In the numerical solution of pressure, the devolatiliza-

tion reaction is more conveniently implemented as a sepa-

rate source term in Eq. (4),

oPf

ot¼ �H

o2Pf

oz2þ 1

�f

� 1

�s

!Q�;f

��; ð6Þ

where 1 ⁄ qf - 1 ⁄ qs is the net volume change in the reaction,

qf is the fluid density (kgfluid ⁄ m3fluid), qs is the solid density

(kgrock ⁄ m3rock) and Qq,f is the rate of fluid mass production

per unit volume of rock (kgfluid ⁄ m3rock ⁄ s) (e.g. Aarnes

2010). The source term (Qq,f) encompasses the combined

fluid products from all devolatilization reactions occurring

at the reaction front. Eq. (6) shows that the fluid pressure

evolves through time owing to the imbalance between the

reduction in pressure through diffusive flux and the

increase in pressure through the source term. Although we

focus on devolatilization, the source term can describe any

volume changing process such as melting or expansion of

pore fluids. For thermal expansion in an isochoric system,

the source term becomesf

��dTdt , where af is isobaric thermal

expansion of the fluid (e.g. Domenico & Palciauskas

1979). This process is not considered in this study as the

volume change from thermal expansion of existing pore

fluids is negligible compared to the volume change in a

devolatilization reaction going from solid to fluid.

FORMULATION OF ANALYTICALSOLUTIONS

Temperature

The analytical solutions presented in this paper are simpli-

fied representations of the complex nature and are focused

on reaction-induced pressure build-up with drainage, and

feedback mechanisms on the metamorphic reactions. We

need the specification of boundary conditions as well as a

few basic assumptions about our system to arrive at these

solutions. The relatively simple, yet nontrivial structure of

the final equations allows us to acquire a good understand-

ing of the key parameters involved.

We start by presenting the well-known analytical solu-

tion for temperature in a contact metamorphic setting, as

we believe that understanding this approach is the key to

recognize the similar procedure applied for the hydraulic

regime, as well as the coupling between the two.

The analytical solution to the thermal equation (e.g.

Turcotte & Schubert 2002) can be written

T �ð�T Þ ¼erfcð�T Þ

1þ erf ð�mT Þ; ð7Þ

where

�T ¼ z=2ffiffiffiffiffiffiffiffi�T tp

ð8Þ

is a nondimensional parameter combining time and space, �mT

is related to the progression of the crystallization front and

T � ¼ ðT � ThrÞ=ðTm � ThrÞ ð9Þ

is the normalized temperature. Thr is the initial host-rock

temperature and Tm is the initial melt temperature. Eq. (7)

is the classical solution to the linear second-order differen-

tial equations in a semi-infinite half-space (e.g. Carslaw &

Jaeger 1959; Crank 1979; Turcotte & Schubert 2002;

Philpotts & Ague 2009). The error function has the prop-

erties that erf ð0Þ ¼ 0 and erf ð1Þ ¼ 1; and erfcð�Þ ¼1� erf ð�Þ. The boundary conditions are the following:

T �ð�mT Þ ¼ T �m ¼ 1 at the crystallization front, and

T �ð�1T Þ ¼ T �hr ¼ 0 infinitely far into the host-rock.

The temperature at the contact between the sill and the

host-rock (Tc) and the temperature at which the reaction

occurs (Tr) are of key interest in our model. They are found

from the substitution of nT with �cT ¼ 0 (nondimensional

contact) and �rT (nondimensional front), respectively, giving

Tc ¼ Thr þðTm � ThrÞ1þ erf ð�m

T Þð10Þ

Tr ¼ Thr þðTm � ThrÞerfcð�r

T Þ1þ erf ð�m

T Þð11Þ

The reaction fronts

The parameter �mT describes how fast the crystallization

front reaches a given position. The location of this crystalli-

zation boundary in the dimensional coordinate system zm

is found by substituting for z in Eq. (8),

zm ¼ �2�mT

ffiffiffiffiffiffiffiffi�T tp

: ð12Þ

This relation shows that �mT can be interpreted as a con-

stant controlling the progression of the crystallization front

in the dimensional coordinate system. The implicit equa-

tion for calculating the constant �mT is



Ste ¼ Dh

Cp Tm � Thrð Þ ¼exp ��m2

T

� �ffiffiffip

�mT 1þ erf �m

T

� �� ð13Þ

where the left-hand side is the Stefan number (Ste) (e.g.

Turcotte & Schubert 2002). The equation defines a

unique correspondence between Ste and �mT . The unknown

�mT can be found by specifying the values in the left-hand

side of Eq. (13) and solve the equation iteratively. The

nondimensional contact temperature T �c is dependent on

the Stefan number, where more latent heat released upon

crystallization corresponds to a higher contact temperature.

The spatial position of the devolatilizing reaction front

zr is controlled by the temperature evolution,

zr ¼ 2�rT

ffiffiffiffiffiffiffiffi�T tp

ð14Þ

where the constant �rT is the position of the reaction front

(i.e. the isograd of the reaction temperature) in the nondi-

mensional coordinate system. �rT can be interpreted as a fac-

tor controlling the progression of the devolatilizing reaction

front in the aureole; for a large value of �rT , the reaction

front will be located further away from the contact at a cer-

tain time compared to a smaller value. It thus plays a similar

role as the �mT does for determining the movement of the

crystallization front. The way to find the proper expression

for �rT is nontrivial and will be discussed in a later section.

Pressure

The pressure equation is solved analytically analogous to

the temperature equation by three main steps: (i) transfor-

mation of the partial differential equation (Eq. 4) through

dimensional analysis, (ii) specification of the boundary con-

ditions and solving the equation with the aid of, for exam-

ple, the technical computing software Maple and (iii)

treating the pressure source at the reaction front (Eq. 5).

As boundary conditions, we set the overpressure infi-

nitely far from the reaction front to zero, DPf ð�1P Þ ¼ 0,

and the fluid overpressure at the contact to an unknown

constant, DPf ð0Þ ¼ PC . The analytical solution for over-

pressure then becomes

DPf �Pð Þ ¼ erfc �Pð ÞPC ; ð15Þ

where the nondimensional parameter nP is defined as

�P ¼ z=2ffiffiffiffiffiffiffiffiffi�H tp

: ð16Þ

In the dimensional coordinate system, Eq. (15) can be

written as

DPf z; tð Þ ¼ erfczffiffiffiffiffiffiffiffiffi�H tp� �

PC : ð17Þ

In this equation, we can see the similarity with the ana-

lytical solution of temperature. However, it is not possible

to solve this equation without the proper expression for

the contact pressure PC, which will be discussed in the fol-

lowing section.

The contact pressure

The expression for PC can be found from solving the con-

servation of mass relation (Eq. 5) at the reaction front.

Rearranging Eq. (5) with proper substitution and solving

the expression for PC yields:

PC ¼1

�f

� 1

�s

!Q total

f

’�

ffiffiffip

�rT

ffiffiffiffiffiffiffi�T

�H

r� �exp �r2

T

�T

�H

� �: ð18Þ

Time cancels out of Eq. (18), which implies that the

contact pressure is not a function of time in our analysis.

This is consistent with our assumption of a constant PC as

the boundary condition of the pressure equation. This

does not imply that contact pressure cannot vary, but that

it varies considerably less than the fluid pressures at other

locations with our model setup.

Because we have obtained an expression for the pressure

at the contact, we can substitute this into Eq. (17) to

obtain the final analytical solution for the overpressure,

DPf �Pð Þ ¼ erfc �Pð Þ1

�f

� 1

�s

!Q total

f

’�

ffiffiffip

�rT

ffiffiffiffiffiffiffi�T

�H

r� �

� exp �r2T

�T

�H

� �ð19Þ

We can more easily identify the physical meaning of this

solution by assigning a nondimensional parameter Ri to

the group:

Ri ¼ �rT

� �2�T

�H; ð20Þ

which describes the competition between the progression

of the reaction front and the efficiency of the hydraulic dif-

fusion. A large value for Ri is equivalent to a high reaction

intensity, that is, when the reaction front is faster than the

fluid flow. Furthermore, we recognize that the ratio of

constants in Eq. (19) defines the overpressure generated at

constant volume, PV, that is, at isochoric conditions,

PV ¼1

�f

� 1

�s

!Q total

f

’�: ð21Þ

Isochoric condition implies no release of overpressure by

fluid flow. PV defines the maximum overpressure that can

be generated. The source term in the pressure equation

used for the numerical solution (Eq. 6) can be recovered

by taking the rate of Eq. (21), to illustrate that the two

solutions are consistent.

6 I. AARNES et al.


Introducing the new terms, the analytical solution to

pressure (Eq. 19) reduces to

P� ¼ erfc �Pð ÞffiffiffiffiffiffiffiffiffiRip

exp Rið Þ ð22Þ

where P* = DPf ⁄ PV is the nondimensional overpressure.

This is the analytical solution to be implemented in models

incorporating overpressure generated by devolatilization in

a permeable aureole.

Resolving overpressure at the reaction front

We explore the solution in Eq. (22) further to obtain an

explicit solution to the overpressure at the reaction front,

DPr, which is the maximum overpressure generated by the

reaction for a given hydraulic diffusivity. This is performed

by substituting z = zr into the Eq. (22), which gives the

nondimensional overpressure at the reaction front P�r

P�r ¼DPr

PV¼

ffiffiffiffiffiffiffiffiffiRip

exp Rið ÞerfcffiffiffiffiffiffiRip� �

ð23Þ

This equation quantifies the overpressure instantly gen-

erated at the front upon fluid production minus the over-

pressure relaxed through fluids flowing away from the

front. The solution to Eq. (23) is plotted in Fig. 2 for dif-

ferent theoretical values of Ri.

For large values of Ri (> 1), the fluids are stagnant rela-

tive to the reaction front. As a result, the overpressure

approaches isochoric conditions, and Eq. (23) reduces to

P* � 1. This relation can be found from Taylor expansion

of Eq. (23) around Ri = ¥. In the opposite case, when Ri

is small (<<1), Eq. (23) reduces to

P� �ffiffiffiffiffiffiffiffiffiRip

ð24Þ

found from series expansion of Eq. (23) around Ri = 0.

This expression provides a simple approximation to the

overpressure generated at the front with drainage of fluids

out of the front, revealing that the parameters in Ri are

the key parameters controlling the overpressure.

Reaction front controlled by temperature

The parameters within Ri need to be specified to find the

exact fluid overpressure. The thermal and hydraulic diffu-

sivities are assumed to be known model parameters found

from, for example, experiments. However, �rT at the reac-

tion front is still an unknown constant that needs to be

calculated. When the reactions are only controlled by tem-

perature, we can find �rT from,

Tr � Thr

Tc � Thr¼ erfc �r

T

� �: ð25Þ

Remember that Tc is a function of �mT and hence the

latent heat release in the intrusion is incorporated in this

equation. Eq. (25) can be solved iteratively.

Figure 3 shows one solution to the expected overpres-

sure as a function of permeability and reaction tempera-

ture, by evaluating the results in Eq. (23) and Eq. (25) for

realistic physical parameters. The fluid flow is primarily

controlled by the permeability of the host-rock in front of

the reaction interface, and higher permeability implies

more flow and thus reduction in the overpressure. The val-

ues for the fixed parameters are given in the caption of

Fig. 3. This figure translates the result from Fig. 2 into

∆Pr/P

V

Ri = (ξT)2κT/κHr

Isochoricoverpressure

Overpressure reducedby fluid flow

∆Pr/PV~√�Ri ∆Pr/PV~1

100

10–1

10–2

10–1 100 101 10210–210–310–4

Fig. 2. The solution to pressure at the front (Eq. 23) as a function of the

parameter Ri (Eq. 20). When the reaction front driven by heat conduction

is much larger than the fluid flux (Ri>1), the overpressure approaches the

volume change in the reaction (PV), that is, isochoric pressure build-up.

Fluid flux out of the dehydration front reduces the overpressure. The reduc-

tion is proportional to a factorffiffiffiffiffiffiffiffiffiRip

.

400

500

100

200

300

10–19 10–1510–16 10–14 10–1310–20 10–1710–18

108

104

107

106

105

Fluid overpressure (∆Pr ) [P

a]

Isochoricoverpressure∆Pr ~PV

Overpressurereduced by flow

Permeability [m2]

Rea

ctio

n te

mpe

ratu

re (T

r –T hr

) [°C

]

Tr ~Thr

Tr ~TC

Fig. 3. Fluid overpressure ( DPr) as a function of both the reaction temper-

ature and the degree of fluid flow, that is, mainly permeability. A low reac-

tion temperature will promote efficient movement of the reaction front,

and fluids will be liberated faster than they are able to flow out, hence

increasing the pressure. We have fixed the other parameters to the follow-

ing values: Tc)Thr = 550�C and Ste = 0.5, corresponding to a

Tm-Thr = 1100�C; jT = 10)6 m2 ⁄ s, which is the typical heat diffusivity for

most rocks (Delaney 1982); lf/b = 10)12corresponding to a H2O or CH4

fluid (Ague et al. 1998; Wangen 2001), and an isochoric pressure of PV of

108 Pa (Aarnes et al. 2008). The value for PV can be calculated from

Eq. (21) for the effective volume change in the reaction of interest. Note

that at relatively high permeabilities (10)16 to 10)15 m2), the generated

overpressure can be significant (>10 MPa).



realistic geological examples and shows that overpressures

between 1 and 10 MPa are possible even at permeabilities

of 10)13 to 10)16 m2. For rocks with smaller permeability

than 10)17 m2, the overpressure approximates PV of the

reaction. For reaction temperatures higher than about

400�C above the background host-rock temperature, isoch-

oric overpressure is only expected for very tight rocks, such

as unfractured shales or siliceous carbonates (<10)18 m2).

Comparison of numerical and analytical solutions

We have compared the analytical solutions to equivalent

numerical solutions to cross-verify the solutions. The

numerical solution is performed by explicit finite difference

method in 1D with a resolution of 1 · 1001. For the

treatment of the latent heat in the numerical solution, we

use the diffuse interface approach, with an effective heat

capacity in the thermal diffusivity coefficient (Eqs. 1

and 3). The source term of the pressure is calculated

explicitly by using Eq. (6). The comparisons are made in

the dimensional coordinate system, which makes the pre-

sented solution one out of several possible contact meta-

morphic systems. As an initial setup, we use T ð�d < z < 0Þ¼ Tm � Thr for the intrusion, and T 0 < z <1ð Þ ¼Thr � Thr ¼ 0 for the host-rock in both the analytical and

the numerical solutions. The initial condition for overpres-

sure is zero throughout the domain, that is, the fluid pres-

sure equals hydrostatic pressure.

The comparison shows that the sharp interface approach to

latent heat gives the same temperature profile as the diffuse

interface approach (Fig. 4A). Because of the different ways of

treating the latent heat, the two solutions differ slightly in the

crystallization interval. When solving the analytical expres-

sion, we include a condition specifying that �T <� �mT ¼ ��m

T ,

to extend the validity range of the analytical solution from

the crystallization front to the intrusion centre.

The analytical solution to the fluid pressure in the aure-

ole (Eq. 22) with mass balance at the front (Eq. 5) coin-

cides perfectly with the numerical solution to Eq. (6)

(Fig. 4B). Hence, we demonstrate the equivalence of the

sharp interface approach and a separate source term for the

fluid pressure. The validity range of the analytical solution

is extended from the intrusive contact to the reaction front

by Pf ð0 < z < zrÞ ¼ DPr . The extent of the reaction front,

�rT , is found from solving Eq. (25) for the thermal values

given in the caption of Fig. 4.

GEOLOGICAL APPLICATIONS

Reaction closure by fluid overpressure

A pressure-sensitive reaction front

In this section, we will use the calculated overpressure at

the reaction front to evaluate the feedback on the reaction

front progress. When the fluid pressure increases as a result

of dehydration, the equilibrium conditions for the reaction

are simultaneously shifted along the reaction curve in the

P-T space. We have illustrated this with tree hypothetical

Clapeyron slopes: (i) temperature controlled, (ii) interme-

diate and (iii) pressure controlled (Fig. 5). When the reac-

tion initiates, the temperature is equal to reaction

temperature (Tr) and pressure is equal to the background

host-rock pressure (Phr). As the fluids are liberated, the

fluid pressure increases by DPr. This elevated pressure cre-

ates a new equilibrium temperature condition for the reac-

tion depending on the Clapeyron slope. The new effective

reaction temperature requires more heating before the

reaction can continue, thus slowing down the progress of

the reaction front. If the reaction is predominantly pres-

sure-sensitive (line 3 in Fig. 6), this may even terminate

the reaction front progress.

Evaluating the pressure sensitivity

To evaluate the effect of the reaction closure quantitatively,

we correct Eq. (25) for the influence of the overpressure

and Clapeyron slope of the reaction by replacing the

–50 –25 0 25 50 75 1000

20

40

60

80

100

z [m] P

f–Phr

[MP

a]

Numericalanalytical

intrusion

Reaction front z

r

host-rock

Crystallization front z

m

0

200

400

600

800

1000

1200

T–T hr

[ºC

]

Temperature and pressure after 10 years

Numericalanalytical

PV

ΔPr = Pr–Phr

Tm-Thr

Tr–Thr

(A)

(B)

Fig. 4. Model result comparing the numerical and analytical solutions to

temperature (Eqs. 1 and 8) and pressure (Eqs. 3 and 16) for a sill with a

half-thickness of 50 m after 10 years. (A) Temperature relative to host-rock

temperature, with a latent heat of 320 kJ ⁄ kg; Tm)Thr = 1100;

Tr)Thr = 300; and jT = 10)6m2 ⁄ s. At the crystallization front, it is possible

to identify a small difference in the geometry of the two approximations

treating latent heat, while the overall result of the temperature profiles is

identical. (B) Overpressure relative to background host-rock pressure, with

a PV of 108 Pa (100 MPa) and jH = 10)6 m2 ⁄ s. The pressure profiles from

the numerical and analytical solutions are identical.

8 I. AARNES et al.


effective reaction temperature with the initial reaction tem-

perature Tr0. From geometrical constraints, we find that

T effr ¼ Tr0 þ ðdT =dPÞDPr . Substitution of T eff

r into Eq.

(25) gives a pressure-dependent movement of the reaction

front,

dT

dP

DPr

Tc � Thr

� �þ Tr0 � Thr

Tc � Thr

� �¼ erfc �r

T

� �: ð26Þ

Although the Clapeyron slope is commonly presented as

dP ⁄ dT, we use the inverse to be consistent with the equa-

tions.

From this equation, we are able to identify the key

dependencies of �rT : (i) the reaction front temperature, (ii)

fluid flow out of the front and (iii) the Clapeyron slope.

To make a simple 2D representation of these three depen-

dencies, we have to make �rT as an explicit function of

Clapeyron slope and the influence of overpressure by com-

bining parameters and utilize approximate relations for DPr

and �rT . The first approximation P�r �

ffiffiffip

�rT

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�T =�H

pis

valid for the pressure build-up in the fluid flow regime

(Fig. 2). The second approximation erfcð�rT Þ � 1 is valid

for small values of �rT (<0.1). Employing these simplifica-

tions gives the expression

�rT �

dPV

dT

Tc � Tr

DPr

ffiffiffiffiffiffiffiffiffiffi1

�H

�T

r: ð27Þ

This expression can be simplified by defining two dis-

tinct parameters

�rT �

1

DThpRcð28Þ

where the pressure dependence is quantified by

Rc ¼ dT

dPV

DPr

Tc � Thrð Þ

ffiffiffiffiffiffiffiffiffiffi�T

�H

rð29Þ

and the additional heating of the host-rock required for

the reaction to proceed is quantified by

DThp ¼Tc � Thr

Tc � Tr: ð30Þ

Figure 6 shows the approximated value for �rT as a func-

tion of Rc and DThp. Red colours represent efficient reac-

tion fronts, and blue colours represent reaction closure.

We identify two domains, one where the reaction is tem-

perature controlled (vertical lines) and one where the reac-

tion is influenced by both temperature and pressure

(inclined lines). In the temperature domain (Rc < 1), the

reaction front progress is only dependent on the degree of

heating required for the host-rock to reach the effective

reaction temperature. A large value of DThp is equivalent of

a large degree of host-rock heating before the reaction

temperature is reached ðTr � TcÞ; hence, the reaction front

is moving very slowly.

solid

Aso

lid B

+ F

luid

solid

Aso

lid B

+ F

luid

solid A

solid B + Fluid

Pr

Phr

P

Tr Tc T

1 23

Treff~Tr

Treff~(Tc + Tr)/2

Treff~Tc

ΔPr

Fig. 5. Three schematic end-member reaction lines drawn in a P-T dia-

gram for a temperature-controlled reaction (1), an intermediate (2) and

a pressure-controlled reaction (3). When the temperature has hit the

reaction curve at Tr and a background pressure of Phr, three possibilities

exist for the reaction progress: (i) The reaction curve is temperature con-

trolled, and an increase in pressure (DPr) will simply move the reaction

condition along the curve up-pressure without changing Tr. (ii) The reac-

tion curve depends both on temperature and pressure, and an increase

in pressure will shift the reaction conditions to higher temperatures. (iii)

The reaction curve is mainly pressure dependent, and a little change in

pressure will require a large change in the temperature at which reaction

takes place.

log10(ΔThp)

log 10

(Rc)

log10

(ξΤ)r

Degree of heating penalty

Tr ~Thr

Deg

ree

of fl

uid

over

pres

sure

Reaction closure by fluid overpressureFast

Closed

Rate of reaction front

Tr ~Tc

Rc < 1 1

2

3Rc >> 1

Fig. 6. The solution of Eq. (28) for �rT as a function of the nondimensional

parameter Rc (Eq. 29) and degree of thermal overheating DThp (Eq. 30).

When Rc is larger than 1, then the reaction proceeds according to the tem-

perature conditions. However, if Rc is much larger than 1, then the over-

pressure generated owing to the reaction will work to inhibit the

movement of the reaction front, and hence closing the reaction. The black

lines (1, 2 and 3) represent theoretical scenarios for reactions with different

Clapeyron slopes, that is, the curves 1, 2 and 3 from Figure 5. Steep slopes

(curve 1) have limited the influence of the generated overpressure and are

thermally controlled, while reactions with gentle slopes (curve 3) will be

strongly influenced by the reaction-induced overpressure and will require

more heating for the reaction to proceed. We have fixed the contact tem-

perature to Tc ) Thr = 680�C as an intermediate choice, and changes in this

value have negligible influence on the diagram.



In the pressure-controlled domain, the absolute value of

the overpressure reduced by flow and variations in the Cla-

peyron slopes (dT ⁄ dP) will influence the reaction front.

Larger dT ⁄ dP of the reactions slows down the movement

of the reaction front by requiring a higher effective reac-

tion temperature. In the case of gentle Clapeyron slopes,

the overpressures created can be large enough to terminate

the reaction owing to the very high effective reaction tem-

peratures required. When this is the case, the reaction front

can only move further into the aureole if the overpressure

is reduced by drainage, for example by fracturing of the

host-rock (i.e. increasing the permeability). This is equiva-

lent of moving downwards in the diagram to the condi-

tions where a more efficient reaction front is expected.

Geological examples

Figure 7 shows the pressure dependency on the reaction

front (Eq. 27) using plausible physical parameters of per-

meability and Clapeyron slope, keeping the other parame-

ters fixed. This clearly shows that for a reaction such as

dehydration of smectite to illite with a relatively shallow

Clapeyron slope (approximately 10)5�C ⁄ Pa), the reaction

front can be 2–3 orders of magnitude slower. Indeed, it

has been shown that this reaction can terminate when the

pressure increases (Colten-Bradley 1987). A dT ⁄ dP of

approximately 10)5�C ⁄ Pa and an overpressure of 10 MPa

is equivalent of a approximately 100�C increase in the

effective reaction temperature (Eq. 26).

The decarbonation reaction of calcite + quartz to pro-

duce wollastonite and CO2 can have a dT ⁄ dP of approxi-

mately 4 · 10)6�C ⁄ Pa, depending on the CO2 content in

the fluid (Philpotts & Ague 2009). In this scenario, the

movement of the reaction front can be retarded by at least

one order of magnitude.

Breccia pipe formation and venting

We apply the analytic solution of pressure at the reaction

front (Eq. 23) to investigate whether the pressure gener-

ated from the methane generation in shales is large enough

to fracture the rocks and thus cause breccia pipe formation,

with subsequent venting of the aureole fluids to the atmo-

sphere (Fig. 8A). Figure 8B shows a schematic figure of a

breccia pipe originating in a shale aureole. The pipes are

associated with sill intrusions at depths greater than the

boiling point of pure H2O (approximately 1 km), which

implies another driving force than expansion of pore fluids

as suggested for hydrothermal vent complexes (Jamtveit

et al. 2004). Figure 8A shows the conditions where breccia

pipes can potentially form as a function of depth and per-

meability, using a tensile strength of 10 MPa above the

lithostatic pressure as a conservative fracturing condition.

The green field represents conditions for reaction-induced

fluid pressures exceeding the tensile strength for a shale of

initially approximately 5 wt.% total organic carbon (TOC).

Also indicated on Fig. 8(A) are the venting conditions for

shales with initially approximately 10 wt.% TOC, where

breccia pipes can be expected for permeabilities lower than

approximately 10)16 m2 at 2–3 km depth. At 1 km, the

permeability can be as high as approximately 10)15 m2 and

still cause venting in this high TOC case. For shales with

initially approximately 1 wt.% TOC, permeabilities lower

than 10)18 m2 are required for venting to occur. Note that

the diagram in Fig. 8(A) assumes certain properties of the

system such as reaction temperature, gas compressibility

and tensile strength of the rocks given in the caption and

must be interpreted on the basis of these assumptions.

Hence, the pressure build-up from the metamorphic

fluid generation is feasible as a driving force for breccia

pipes in aureoles with permeabilities lower than approxi-

mately 10)17 m2 at 2–3 km depth. This situation corre-

sponds well with the paleodepth of the shale formations in

the Karoo Basin, where more than thousand breccia pipes

have been mapped (Svensen et al. 2006, 2007). Further-

more, low permeabilities are expected in unfractured shale

and crystallized igneous intrusions (10)23 to 10)18 m2)

(Brace 1980), both of which are associated with the brec-

cia pipes in the Karoo Basin (Aarnes et al. 2011b).

DISCUSSION

The model

Our model of significant overpressure generation during

prograde contact metamorphism is in contrast to other

10–20 10–16 10–1410–18

10–5

10–6

10–4

10–7

10–8

10–2

10–3

10–1

Permeability (m2)

Reaction front rate (ξ

T ) rCla

peyr

on s

lope

(°C

/Pa)

Overpressure control

on reaction front rate

Thermally controlledreaction front (ξT~1) r

Fig. 7. The rate of the reaction front (�rT ) is contoured as a function of per-

meability (fluid flow) and Clapeyron slope (overpressure), using the relation

derived in Eq. (27). The grey area represents conditions where the reaction

front progress is controlled by the reaction temperature only. The

other parameters are set to PV = 5 · 108 Pa, lf/b = 10)12 Pa)1,

jT = 10)6 m2s)1, Tr = 350�C, Thr = 100�C, Tm = 1200�C, Tc = 772�C, where

the latter is calculated from Eqs. 11a and 14. For reactions with Clapeyron

slopes of 10)5 �C ⁄ Pa, the front is moving up to two orders of magnitude

slower than what is expected for the thermally controlled front.

10 I. AARNES et al .


contact metamorphic models assuming no pressure varia-

tions, or minor fluid pressure gradients (Baumgartner &

Ferry 1991; Leger & Ferry 1993). The analytical solution

is an approximation to a more complex reality owing to,

for example, the required boundary conditions such as

constant pressure at the contact. The strength of the

model is that it is based on basic physical principles with

only a few assumptions about the system. The analytical

solution thus provides a good first step for further develop-

ment of such models, but can also be used in its present

form to explain a number of phenomena, among others

breccia pipe formation and reaction closure. The analytical

solutions are cross-checked with numerical modelling and

show that the solutions are not sensitive to the choice of

method used for treating the source terms (Fig. 4).

The model analyses the stresses generated from volume

changing processes after the emplacement. The stress field

can furthermore change in response to, for example, dis-

placement of sediments during sill emplacement (e.g. Pol-

lard 1973; Pollard & Johnson 1973) or crystallization (i.e.

negative volume change) within the sill (Aarnes et al.

2008; Svensen et al. 2010).

The pressure equation (Eq. 6) can easily be expanded

with several individual source terms, each defining a sepa-

rate process that affects the pressure, such as compaction

(e.g. Connolly 1997; Connolly & Podladchikov 1998),

thermal stresses or an array of different reactions involving

a net volume change. These processes are documented in

the literature and therefore not evaluated here.

To be able to implement the solution in a relatively sim-

ple way, we have considered that a major fluid release

occurs at one temperature, for example at 350�C. For the

metamorphic reaction front, the kinetics will play a role in

the fluid release. However, at a high temperatures such as

350�C, the conversion of organic matter to methane can

be approximated to instantaneous relative to the time of

the heat transfer. As an example, at laboratory conditions

of up to 365�C, 24 h is sufficient to convert all the organic

matter to gas by pyrolysis (e.g. Andresen et al. 1995).

Overstepping temperatures of dehydration reactions are

estimated from kinetic modelling to be only a few degrees,

with a maximum of 40�C (Walther & Wood 1984; Con-

nolly 1997). This is equivalent of shifting the effective

reaction temperature to that of the overstepping tempera-

ture for a discontinuous dehydration reaction and should

be considered to ensure a reaction rate that is fast relative

to the thermal pulse. In an aureole, the reaction tempera-

ture isograde comprises a variety of minerals and organic

matter that release fluids. To account for more reaction

fronts moving through the rock, more reaction terms can

be added to Eq. (6), similar to the one given. If only

sparse devolatilization reactions occur, this is equivalent of

having a smaller Q totalf source term.

We did not consider the effect of varying the heat diffu-

sivity, as this variation is within a few percent. The uncer-

tainty of this is negligible relative to the uncertainties in,

for example, permeability in our nondimensional parameter

Ri. Hence, we have assessed the various effects of varying

Pr/Plithostatic<1

Pr/Ptensile>1

P r =

P tens

ile

P r =

P litho

static

NO VENTINGNO VENTING

VENTINGVENTING

Dep

th (k

m)

Permeability (m2)

P v =

108 P

a

P V =

3×10

8 Pa

~1 w

t.% TO

C

~10

wt.%

TO

C

~5 w

t.% T

OC10–17 10–14 10–1310–1810–1910–20 10–1510–16

4.5

3.5

2.5

1.5

5

1

4

3

2

0.5

contact aureole

CH4

sill intrusion

organic–rich shale

sedimentaryrocks

(B)(A)

P v = 1

09 P

a

Fig. 8. (A) Depth of intrusion versus log of permeability by applying Eq. (23) to solve for overpressure generated as a result of organic cracking in contact

aureoles. For PV = 3 · 108 Pa corresponding to 5 wt.% TOC venting will occur when the permeability is lower than about 10)17 m2. In the region to the

right, the pressure will be reduced by fluid flow before any venting occurs. We also indicate the tensile strength curves for PV = 108 Pa and PV = 109 Pa. The

other parameters are set to Phydrostatic = qfgZ, Plithostatic = qhrgZ and Ptensile = qhrgZ + 10 MPa, where g = 9.81 m ⁄ s2, host-rock density is a linear function of

qhr = 2200–2600 kg ⁄ m3 for Z = 0.5–5 km, qf = 250 kg ⁄ m3 for 100 m (i.e. the generated CH4) and qf = 1000 kg ⁄ m3 (i.e. pore fluid H2O) for the rest of the

hydrostatic column. We use qhr = 2400 kg ⁄ m3 and /b = 10)9 Pa)1. (B) Schematic drawing of a vent complex rooted in the contact aureole. The pressure

build-up from methane generation in the low-permeable shales is larger than the tensile strength, resulting in fracturing and venting of the aureole fluids to

the atmosphere.



permeability and kept most other parameters constant for

simplicity. A systematic evaluation of the effect of different

thermal diffusivities on the reaction aureoles can be found

in Aarnes et al. (2011a).

We did not include latent heat consumed by devolatiliza-

tion, as this only has a minor effect on the total reacted

aureole owing to the small volumes of dehydration products

relative to rock volumes (approximately 6 % smaller aureole)

(Aarnes et al. 2010). If the system of interest requires latent

heat of devolatilization, it is possible to add another source

term numerically. The treatment of such source term in the

analytical solutions is, however, highly nontrivial owing to

several feedback processes consistent with the fully coupled

system and is outside the scope of this paper.

For the reaction closure, we have assumed that the dehy-

dration reactions occur at local thermodynamic equilib-

rium. There is no conflict between this assumption and the

introduction of minor thermal overstepping of the equilib-

rium condition by simply using a higher initial reaction

temperature.

Overpressure reduction by fluid flow

Permeability is the key parameter controlling the efficiency

of fluid flow and has thus a major control on the magni-

tude of overpressure. It is unfortunately a relatively difficult

parameter to constrain for a full geological system, as it

can vary several orders of magnitudes (Ingebritsen & Man-

ning 2002). Unfractured shales, siliceous carbonates and

crystalline rocks are measured to have very low permeabili-

ties about 10)23 to 10)16 m2 (Brace 1980; Hanson 1995;

Cui et al. 2001), which makes significant overpressure gen-

eration in intruded shale systems likely. Laboratory experi-

ments show that dehydration of gypsum can generate

excess pore pressures on the scale of 150 MPa during

isochoric conditions (Wang & Wong 2003). This value

corresponds well with the results of this study.

One key result from our model is that devolatilizing sys-

tems with active fluid flow out of the reaction front can

still create significant overpressures, although reduced rela-

tive to the isochoric systems (Fig. 3). Even for relatively

high-permeable devolatilizing systems (approximately

10)15 m2), fluid overpressures of 10 MPa can be expected.

This is supported by dehydration experiments with the

hydrous mineral gypsum giving pore pressures up to

93 MPa when allowing for flow out of the dehydrating

sample (Olgaard et al. 1995; Wong et al. 1997).

Coupling of fluid flow to the system of equations

describing deformation of the rock matrix has not been

included in this study. Expansion and fracturing can

increase the storage capacity of the rock and could there-

fore act to lower the fluid overpressure. This can be

approximated in this study by using a relatively high effec-

tive compressibility for the geological examples.

Although we predict that these overpressures are gener-

ated instantly at the front, the hydraulic conductivity is of

major importance for the maintenance of the overpressure

(e.g. Bredehoeft & Hanshaw 1968; Ko et al. 1997). For

high permeabilities, the pressure build-up will only be tran-

sient and diffuse away after the devolatilization has ceased

(e.g. Wong et al. 1997).

Reaction closure by fluid overpressure

Most devolatilization reactions are mainly temperature-

dependent with typical Clapeyron slopes of approximately

10)7�C ⁄ Pa, at least at pressures above approximately 50 MPa

as calculated from thermodynamic phase equilibria (Connolly

2005, 2009) The influence of pressure on the effective reac-

tion temperature is thus on the order of 10–100�C for PV of

108–109 Pa. We can speculate that such overpressures could

occur at great depths (approximately 30–40 km) and thus be

relevant for the limited dehydration reactions at lithostatic

pressures of 800–1000 MPa, as suggested for the Southern

Alps, New Zealand (Vry et al. 2010).

There are some cases of negative dT ⁄ dP where the reac-

tion temperature decreases with increasing pressure at

higher pressures. An example of such reactions is dehydra-

tion of kaolinite at thermodynamic pressures above 109 Pa

(1 GPa) and 350�C (Chatterjee et al. 1984). This is

because the fluids at these pressures are so compressible

that they occupy less space as a fluid phase than in the

mineral phase. This gives a negative volume change in the

reaction and would result in an underpressure (e.g. Delany

& Helgeson 1978). In the case of lowering the pressure

on a negative Clapeyron slope, it would also require a shift

towards higher temperatures in the P-T diagram.

Overpressure and venting in natural systems

The magnitude of the overpressure that theoretically can

be generated by devolatilization reactions is significant

(>1 GPa). However, the rocks cannot always maintain such

pressures. Hydraulic fracturing will therefore put an upper

bound to the amount of overpressure generated (e.g. Wal-

ther & Orville 1982; Nishiyama 1989; Gueguen & Palci-

auskas 1994). The critical fluid pressure required to

fracture a rock matrix from a localized fluid overpressure

and the possible failure patterns is investigated by Rozhko

et al. (2007) for a nonreactive case.

Sill intrusions with thickness of 100 m intruding fluid-

producing rock types will ensure large-scale fluid generation

(Aarnes et al. 2011a,b). Furthermore, the intrusions can act

as a low-permeable seals themselves, thus increasing the

chance of venting and breccia pipe formation (Fig. 7). In

conclusion, we show that the new analytical solution can be

applied to infer that methane generation in organic-rich

contact aureoles can generate sufficient overpressure to



trigger large-scale fracturing of the host-rock and ultimately

breccia pipe formation, thus creating efficient pathways for

the aureole gases to reach the atmosphere. This is consistent

with the occurrence of several thousand breccia pipes origi-

nating in shale formations intruded by multiple sills in the

Karoo Basin.

CONCLUSIONS

In this study, we provide new analytical solutions to reac-

tive transport involving temperature, pressure, fluid flow

and metamorphic devolatilization reactions. The main con-

clusions about reaction-induced overpressure and the

implications for reaction fronts and aureole fracturing are

summarized:

(1) The overpressure is determined by the ratio of thermal

and hydraulic diffusion to the rate of the reaction

front.

(2) When fluid flow is more efficient than the advance of

the reaction front, the isochoric overpressure can be

reduced by several orders of magnitude, depending on

the effective permeability. Still, fluid overpressures of

tens of MPa can be generated.

(3) In the case of pressure-dependent reactions, the gener-

ated overpressure will shift the equilibrium conditions

towards correspondingly higher temperatures. The

effectively higher reaction temperatures will require

more heat to proceed and thus slow down the reaction

front and may even terminate it completely.

(4) The analytical solutions to overpressure are used to find

the conditions of reaction-induced fracturing, and thus

illustrate the feasibility of breccia pipe formation as a

response to methane release in shaly contact aureoles.

Low permeabilities (<10)17 m2) and high content of

organic material in the sediments (>5 wt%) are favour-

able conditions, consistent with the observations asso-

ciated with the breccia pipes.

To conclude, this model provides a theoretical basis that

can be applied to solve several geological processes related

to prograde metamorphism, and the analytical solutions

can be used with success for the verification of numerical

reactive transport models.

ACKNOWLEDGEMENTS

This study was supported by Grant 169457 ⁄ S30 from the

Norwegian Research Council. We would like to thank V.

M. Yarushina and N. S. C. Simon for important discus-

sions and help to improve the manuscript. Great thanks go

to all the participants in the PGP Wednesday Club for

more than 6 years of derivations and discussions on the

matter. Teng-Fong Wong and one anonymous reviewer

are thanked for valuable comments that helped strengthen

the manuscript.

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