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AN EMPIRICALLY DERIVED KINETIC MODEL FOR ALBITIZATION OF

DETRITAL PLAGIOCLASE

RENEE J. PEREZ*

†

and JAMES R. BOLES

University of California, Santa Barbara, Geological Sciences Department Room 1006,

Webb Hall, Santa Barbara, California, 93106-9630

ABSTRACT. We propose an empirically derived model that estimates the extent of

reaction for albitization of plagioclase as a function of time, grain surface area, and

temperature from 0 to 200°C. We use a kinetic formulation independent of pressure,

and consistent with the Rate Law, which quantiﬁes the dependence of the reaction rate

on the initial concentration of the reacting material, and the Arrhenius equation. The

formulation is described by the function:

d[An]

dt

⫽ S

m

k[An

o

]

2

(1 ⫺ ⍀

)

where d[An]/dt is the rate change of the anorthite mole fraction with time, S

m

the

mineral surface area (cm

2

), k the rate constant 1/cm

2

s, [An

o

] is a constant representing

the initial anorthite mole fraction, and ⍀

a constant weighed average saturation index.

We derive the apparent activation energy (E

a

) and frequency factor (A), both present in

the rate constant k, by ﬁtting them to the extent of albitization measured in 11 samples

from the San Joaquin Basin. Subsequently, we test the model against two independent

albitization trends, one from the Texas Gulf Coast basin and one from the Denver

Basin of Colorado. Our results indicate that albitization in all three basins can be ﬁt by

an E

a

of 68 ⴞ 4 kJ/mole and A of (6.5 ⴞ 0.5) ⴛ 10

3

1/cm

2

Ma. The rate dependence on

temperature is consistent with experimental values for albite crystal growth and with

empirically derived precipitation rates of other diagenetic silicates such as illite and

quartz. The parameters and ﬁt suggest that albitization can be modeled as a surface

controlled reaction, primarily dependent on temperature.

introduction

Kinetics of Diagenetic Reactions

The aim of this paper is to present an empirical kinetic model for albitization of

detrital plagioclase. The model is derived directly from geologic data and is applicable

to at least three different sedimentary basins. Albitization, and most diagenetic

reactions in general, are modeled empirically because (1) the extrapolation of

laboratory measurements to geologic conditions generally requires very accurate and

precise data that are so far not attainable, (2) small errors in the temperatures and

heating rates during experimentation can become a major error in activation energies

and orders of magnitude error in frequency factors (Burnham and others, 1987), and

(3) changes in reaction mechanisms complicate the validation of kinetic parameters

for wide temperature ranges (Comer, 1992). An alternative approach to estimate rate

changes over large time scales (that is millions of years), is to model them using ﬁeld

observations and time-temperature burial data.

There are several examples of empirically derived reaction kinetics in geologic

systems. Some, for instance, are the kinetics of petroleum generation (Lopatin, 1971;

Waples, 1980), aromatization and isomerization of hydrocarbons (Mackenzie and

Mackenzie, 1983; Gallagher and Evans, 1991), and vitrinite reﬂectance (Burnham and

Sweeney, 1989). Similarly, empirically derived kinetics for illitization of smectite and

*Current afﬁliation: University of Calgary, Applied Geochemistry Group, Department of Geological

Sciences, Calgary, Alberta T2N 1N4 Canada

†

Corresponding author: Renee J. Perez, rene@earth.geo.ucalgary.ca

[American Journal of Science, Vol. 305, April, 2005,P.312–343]

312

epitaxial quartz overgrowth receive special and continual attention due to their

abundance in oil reservoirs (Walderhaug, 1994, 1996; Elliot and Mattissof, 1996). Our

literature review reveals, however, that albitization of feldspar, although present in all

rock settings, lacks a kinetic formulation useful for its prediction.

Albitization, A Brief Review

Albitization of plagioclase has been recognized since the turn of the twentieth

century (for example, Bailey and Grabham, 1909). Perhaps the best reviews can still be

found in Coombs (1954) and Deer and others (1963). In general, albitization refers to

the transformation of any An-Or-Ab solid solution into albite. The solid solution is

originally crystallized at high temperatures as igneous or metamorphic minerals,

whereas the albite precipitates at low temperatures. Albitization results in low albite

(Morad and others, 1990; Slaby, 1992), due to the high to low albite transition

temperature of 575 50°C (Smith, 1972; Moody and others, 1985). The newly formed

albite is porous (Boles, 1982), which, as most replacement reactions, implies a

dissolution/precipitation reaction mechanism such that the ion transport to the

reaction interface and removal away from the interface are not diffusion-transfer

limited (Putnis, 2002).

In sedimentary basins, albitization refers to a pseudomorphic replacement pro-

cess that may imply a coupled dissolution of detrital plagioclase and precipitation of

low albite at equal rates. The shape of the plagioclase parent crystal and internal

structural details are preserved in the daughter albite crystal. A priori, coupled

dissolution-precipitation reactions can not be considered to be controlled by the

dissolution step (Luttge and Metz, 1991, 1993), however, isotopic analysis and kinetic

experiments suggest that the major mechanism in feldspar replacement is dissolution

followed by precipitation, and the rate-controlling step is grain surface dissolution

(O’Neil and Taylor, 1967; Moody and others, 1985). Solid diffusion usually fails to

explain albitization, mainly due to the presence of the peristerite or miscibility gap,

which is evident in albitized grains by the presence of sharp reaction fronts separated

by zones of constant composition (Fox, ms, 1989). Furthermore, isotopic analysis

(O’Neil and Taylor, 1967; Stallard and Boles, 1989) and recent

18

O, H, D, and

Na/(NaK) mapping (Labotka and others, 2002) demonstrate that, during feldspar

replacement, the oxygen isotope distribution re-equilibrates in the product, which

indicates the breaking of Si-O and Al-O bonds, and the oxygen redistribution is

facilitated by cation exchange (O’Neil and Taylor, 1967; Labotka and others, 2002).

The preservation of internal details and oxygen re-equilibration strongly suggest that

the replacement mechanism involves ﬁne-scale solution and re-deposition in a structur-

ally organized ﬂuid ﬁlm at the interface between the exchanged minerals (O’Neil,

1977; Putnis, 2002).

Albitization is probably one of the most common alumino-silicate reactions in the

shallow crust of the Earth. It has been reported in deuterically altered granite and

granodiorite (for example, Bailey and Grabham, 1909; Hess, 1950; Deer and others,

1963; Fox, ms, 1989), alkali-carbonatite complexes (Bodart, 1980), spilitized lava

(Eskola and others, 1935; Rosenbauer and others, 1988), volcanic tuff (Coombs, 1954;

Boles and Coombs, 1977), and sedimentary basins (Tester and Atwater, 1934; Middle-

ton, 1972; Merino, 1975a; Land and Milliken, 1981; Boles, 1982; Gold, 1987; Boles and

Ramseyer, 1988; Morad and others, 1990). Albitization is usually described with Al

3

conservative reactions that reproduce the characteristic volume conservation observed

in thin sections. For example:

NaAlSi

3

O

8

䡠 CaAl

2

Si

2

O

8

H

4

SiO

4

Na

2NaAlSi

3

O

8

Al

3

Ca

2

4OH

(1)

313R. J. Perez and J. R. Boles 313

2SiO

2

⫹ 0.5H

2

O ⫹ H

⫹ Na

⫹ CaAl

2

Si

2

O

8

⫽ NaAlSi

3

O

8

⫹ 0.5Al

2

Si

2

O

5

OH

4

⫹ Ca

2

(2)

Merino (1975b) proposed equation (1), and Boles (1982) proposed equation (2).

Reaction 2 assumes that silica is provided by quartz, instead of being supplied by pore

ﬂuid in the form of Si(OH)

4

—for example Si(OH)

4

may be supplied from clay

diagenetic reactions (Boles and Franks, 1979).

Importance

The question of which reaction best represents albitization, however, remains a

subject of great debate but the implications are clear: overall consumption of Na

,

Si

4

,H

, and release of Ca

2

and Al

3

. In other words, albitization reﬂects mass

transfer (Merino, 1975b; Boles, 1982; Aagaard and others, 1990; Morad and others,

1990) and causes chemical changes in basinal pore ﬂuids (Fisher and Boles, 1990;

Davisson and Criss, 1996). Furthermore, albitization inﬂuences the porosity and

permeability of ﬂuid reservoirs, especially when CO

2

supply is constant. Kaolinite and

calcite precipitation (as products of reactions 1 and 2) are commonly reported in the

San Joaquin Basin as a consequence of the albitization of plagioclase (Merino, 1975b;

Boles, 1984; Hayes and Boles, 1992; Wilson and others, 2000). Clearly, albitization is

one of the most important and common reactions in all rock settings, and yet little is

known about its transformation rate and predictability.

The Thermodynamic Problem

In spite of the reaction’s universality, the temperature range of the so called

“albitization zone” is not the same everywhere. Among some possible causes are the

grain reactivity, degree of fracturing, initial composition, structural states (Ramseyer

and others, 1992), and grain provenance (Boles and Ramseyer, 1988). Our literature

review reveals, however, that the appearance of the albitization zone may be associated

with differences in sediment ages and burial-thermal histories of the basins (Gold,

1987; Pittman, 1988; Ramseyer and others, 1992). This premise leads us to postulate

that time, temperature, and heating rates may strongly inﬂuence the replacement zone

as well, and that albitization is kinetically controlled by a critical activation energy.

One unanswered question is—Why are ﬂuid compositions, at depths and tempera-

tures less than the albitization zone, commonly in the albite dickite stability ﬁeld?

(Merino, 1975b; Boles, 1982; Morad and others, 1990; Helgeson and others, 1993).

Pore ﬂuid compositions demonstrate that this is the case in several sedimentary basins

(ﬁg. 1). The equilibrium boundaries in ﬁgure 1 are based on reaction 2 assuming that

the silica is provided by quartz. The estimated error on the equilibrium boundary,

using errors assigned to the Gibbs free energy values at 25° C given by Robie and

Waldbaum (1968), and using standard error estimate formulas is 1.0 pH unit (Boles,

1982). If silica is assumed to be provided by pore ﬂuids, for example supplied by clay

reactions (Boles and Franks, 1979) in the form of H

4

SiO

4

in equilibration with quartz

at the appropriate temperatures, the equilibrium boundaries would be 0.2 to 0.3 pH

units higher at ﬁxed calcium and sodium activity ratio (log([a

Ca

2

]/[a

Na

])) than those

shown in the plot (Boles, 1982). If the aqueous silica concentrations were higher than

required for equilibration with quartz, the albite-dickite stability ﬁeld would expand

even more, and pore ﬂuid data would plot even further in the albite-dickite stability

ﬁeld. Additionally, reservoir ﬂuids from the San Joaquin Basin are saturated with

respect to albite, allowing low albite precipitation in open pore spaces at temperatures

as low as 43°C, yet albitization (as a replacement) doesn’t start until the temperatures

are higher than 83°C (Boles and Ramseyer, 1988).

It is clear that ﬂuid chemistry plays an important role in albitization (Baccar and

others, 1993), but time and temperature may be the main controlling factors. In other

314 R. J. Perez and J. R. Boles—An empirically derived

words, the albitization zone is thermodynamically constrained by the pore ﬂuid

chemistry in basins. The reaction onset, however, is kinetically controlled and pro-

ceeds at an appreciable rate after a critical activation energy is reached, presumably at

temperatures higher than 70°C.

relevant kinetic experimentation and modeling

There is a long tradition of dissolution and precipitation experiments and

modeling involving feldspars; Helgeson and others (1984), Hellmann (1994), and

Blum and Stillings (1995) give excellent reviews. Most experiments on albite precipita-

tion have been performed at green schist and lower amphibolite P-T-X conditions

(250-500°C and up to 1 kbar), generally in 0.1 M NaCl and 0.05 M Na

2

SiO

3

solutions

(Matthews, 1980; Moody and others, 1985). Alkali metal halides, such as NaCl and

CaCl, have a dramatic effect on reaction kinetics (Winkler and Luttge, 1999), and it

may be reasonable to consider that this is the case of subsurface environments.

However, these temperature conditions are not observed in sedimentary basins and

these experiments do not truly reproduce the nature of a pseudomorphic replacement

(Merino and others, 1993).

Previous modeling of plagioclase albitization includes Baccar and others (1993)

and Wilson and others (2000). The former study focused on the minimum time of

albitization from 60 to 150° C in a closed system. It allowed a critical saturation before

nucleation and growth, and assumed an initial ﬂuid volume, ﬂuid composition, and

grain size distribution. The rate constants were derived using speciﬁc dissolution and

precipitation kinetic constants and thermodynamic equilibrium constants. Baccar and

others (1993) concluded that the albitization rate decreases with the extent of the

reaction (which increases with temperature) and that plagioclase is not stable at high

Pco

2

conditions. These results give insight into dissolution-precipitation processes,

Fig. 1. Equilibrium boundary of albitization according to reaction 2 (see text), after Boles (1982). The

error in the equilibrium constant calculated from free energy values given by Robie and Waldbaum (1968) is

1 pH unit. We assume quartz provides silica. The plot shows that reservoir ﬂuids are in the albite stability

ﬁeld and yet albitization was not observed.

315kinetic model for albitization of detrital plagioclase

such as the effect of Pco

2

, but not into replacement mechanisms. The preservation of

plagioclase internal details such as twin planes and cleavages, requires, among still

many unknown parameters, at least: a) plagioclase to dissolve and albite to grow at

equal and slow volumetric rates (Merino and others, 1993), b) a stress-controlled

coupled dissolution/precipitation mechanism (Merino and Dewers, 1998), c) a struc-

turally self-organized ﬂuid ﬁlm at the mineral interface (O’Neil and Taylor, 1967;

Reyhani and others, 1999), and d) differences between parent and daughter crystal

solubility (Putnis, 2002).

The latter study, (Wilson and others, 2000), considered albitization as a process

independent of ﬂuid composition. The model used an activation energy of 65 kJ/mol

based on plagioclase dissolution experiments at low temperatures (Blum and Stillings,

1995) and ﬁt the reaction order and the frequency factor to a single compositional

data (Boles and Ramseyer, 1988). This empirical approach was useful to simulate the

chemical evolution of the pore ﬂuids in the San Joaquin Basin. It also proved to be

successful by yielding results consistent with geologic observations. The model, how-

ever, lacked a chemical afﬁnity term, validation in other settings, and was also

independent of a surface area variable. After reviewing these two studies, we concluded

that an albitization kinetic model could be better formulated by including a surface

area variable, a saturation index term, and by testing it widely against data from several

basins.

hypothesis

We postulate that the albitization rate is a function of time, temperature, ﬂuid

composition, and surface area. Our rate equation follows a kinetic formulation

consistent with the Rate Law and the Arrhenius equation (Lasaga, 1984). Rate laws

quantify the dependence of reaction rates on the concentration of the starting

material. They take the general form:

dC

dt

⫽ kC

n

(3)

where dC/dt is proportional to the change in concentration of C raised to the power of

n. The Arrhenius equation quantiﬁes changes in the rate constant k with temperature

and has the form:

k ⫽ Ae

E

a

/Rt

(4)

where A is the frequency factor, E

a

the activation energy, R the universal gas constant,

and T temperature. In our work, the kinetic parameters (E

a

and A) present within the

reaction rate k are ﬁtted using solely molar compositional trends with depth, which are

implicit functions of time and temperature. In reality, plagioclase composition, pore

ﬂuid pH, and temperature are the main controlling factors of the rate constant.

However, different plagioclase compositions occur together in samples with identical

pore ﬂuid and temperature history, strongly suggesting that the chemical composition

of the plagioclase has a stronger effect on the rate constant than pore ﬂuid (Ramseyer

and others, 1992). Furthermore, based on the following observations:

(1) pore water from marine basins have a limited initial variation in pH values and

chemical composition (Fisher and Boles, 1990); and

(2) oil ﬁeld brines are almost always in equilibrium or saturated with respect to

albite and unsaturated with respect to anorthite (Helgeson and others, 1993),

We assumed that (1) albitization is a dissolution reaction, which should have an E

a

between 40 and 80 kJ/mol (Lasaga, 1984), and (2) during albitization the sum of the

anorthite, orthoclase, and albite mole fraction is one at all times.

316 R. J. Perez and J. R. Boles—An empirically derived

The Saturation Index Hypothesis

The dissolution rate of a given plagioclase composition depends on the rate

constant k, and the saturation index . In our model the rate constant k is the ﬁtted

parameter and we will deal with it in more detail later. The saturation index ,ata

ﬁxed temperature and water composition, is commonly deﬁned as:

⫽

IAP

K

eq

(5)

where IAP is the ionic activity product of the solution, and K

eq

the equilibrium

constant. From the log

10

positive numbers represent supersaturation and negative

undersaturation (for example Bethke, 1996). Based on ﬁeld observations that suggest

that anorthite dissolution is the albitization rate-limiting step we assume that reaction 2

is adequate to represent albitization throughout geological time in sedimentary basins.

Again, reaction 2 may be written as:

2quartz ⫹ 0.5water ⫹ H

⫹ Na

⫹ anorthite ⫽ albite ⫹ 0.5dickite ⫹ Ca

2

(6)

in which the IAP is deﬁned by:

IAP ⫽

a

Ca

䡠 a

dickite

1/2

䡠 a

albite

a

anorthite

䡠 a

Na

䡠 a

H

䡠 a

H

2

O

1/2

䡠 a

quartz

2

(7)

where the a’s are the activities of both, the solids and aqueous species. In this work, the

solid phases were considered pure with unit activity, except anorthite. The anorthite

content of plagioclase being albitized is 0.35 mole fraction, and its activity is 0.45

(Saxena and Ribbe, 1972). The equilibrium constant K

eq

is calculated from the Gibbs

free energy of the reaction. The free energy change at any T and P and solution

composition Q, in its simplest form, is given by:

r

G

T,P

⫽

r

H

°

298

⫹

冕

298

T

r

Cpdt ⫺ T

冉

r

S

°

298

冕

298

T

r

Cp

T

dt

冊

⫹

冕

1

P

r

Vdp ⫹ RTlnQ (8)

where

r

G

T,P

is the Gibbs free energy

r

H

°

298

enthalpy,

r

S

°

298

entropy,

r

Cp heat

capacity, and

r

V the molar volume change for the reaction, all in calories/mole (table

1). The subscript 298 refers to the reference temperature 298.15 Kelvin. At equilib-

rium, Q becomes K

eq

, and the free energy change of the reaction is zero so that:

lnK

eq

⫽

冋

r

H

°

298

冕

298

T

r

Cpdt ⫺ T

冉

r

S

°

298

⫹

冕

298

T

r

Cp

T

dt

冊

⫹

冕

1

P

r

Vdp

册

RT ⫹ ln0.45 (9)

As shown above, the equilibrium constant for any reaction is ultimately a function of P,

T, ﬂuid composition, and solids composition. To calculate K

eq

for reaction 2 we used

thermodynamic data from two different, but consistent, sources. Free energies of

formation, enthalpies of formation, and third law entropies for all solid phases were

obtained from Robie and Waldbaum (1968). We used the database reported by Robie

and Waldbaum (1968) instead of the more recent and standard database SUPCRT92

(Johnson and others, 1992) for the following reasons:

317kinetic model for albitization of detrital plagioclase

Table 1

Formation water compositions associated with partial or complete albitization. Data compiled from several sources

Concentrations in mg/liter

*Reported as Si

TGCB Gulf Coast Basin

SJB San Joaquin Basin

NI No Information

References 1) Boles (1982); 2) Fisher and Boles (1990).

Temperature in degrees Celsius

Depth in meters

318 R. J. Perez and J. R. Boles—An empirically derived

a) Consistency with previous studies. Boles (1982) determined the albitization

stability ﬁeld and errors in the free energy of formation and equilibrium constant

of reaction 2 using the Robie and Waldbaum (1968) database.

b) Contrary to Robie and Waldbaum’s database, SUPCRT92 lacks standard enthalpy

and Gibbs free energy data for Dickite, which is part of the products in the

reaction under study.

d) Robie and Waldbaum’s database contains thermodynamic data for Dmisteinber-

gite, an hexagonal anorthite polymorph used, as we will explain later, as a proxy

to evaluate the inﬂuence of the atomic arrangement and crystalline structure on

the transformation kinetics.

On the other hand, the standard Gibbs free energy, enthalpy and entropy of

formation of anorthite, low albite, and quartz present in Robie and Waldbaum (1968)

differ by less than 1.4 percent from those present in SUPCRT92, so, in spite of the

other differences, we felt comfortable using heat capacity data from SUPCRT92 as

summarized by Anderson and Crear (1993).

In order to keep our model simple and ease the calculation of the equilibrium

constant we made several simpliﬁcations. First, justiﬁed by the narrow range of

temperatures being considered here, we assumed that the heat capacity for ions was

constant. We also assumed that our model is pressure independent, and the net solid

volume change is close to zero (Boles, 1982) thereby assuming P 1 bar in equation

(9) and neglecting the pressure correction term. Consequently, lnK

eq

is left as a

function of temperature and can be represented by the following polynomial equa-

tion:

log

10

K

eq

⫽ p

1

T

3

⫹ p

2

T

2

⫹ p

3

T ⫹ p

4

(10)

where p

1

1.7513 10

7

, p

2

2.7925 10

4

, p

3

0.1685, and p

4

37.993 (ﬁg.

2). The error in the equilibrium constant at 298.15 K, calculated from standard error

estimation formulas is 0.98. It is likely that this error may expand at higher

temperatures. The thermodynamic values used are summarized in table 2. As we will

discuss further in the text, we plan to qualitatively examine the effect of the anorthite

crystalline structure on albitization, in other words, on reaction 2. Thus, ﬁgure 2 also

illustrates the log

10

K

eq

for reaction 2 assuming that an hexagonal anorthite (an

anorthite polymorph) detrital grain is being albitized, in which case we use different

standard free energy and entropy values (table 2). Heat capacity data for hexagonal

anorthite are, to our knowledge, not available, so we retained the same heat capacity

equations. The equilibrium constant for reaction 2 assuming hexagonal anorthite can

also be ﬁtted to equation (10) with different coefﬁcients: p

1

1.4412 10

7

, p

2

2.2995 10

4

, p

3

0.13938, and p

4

30.421. The error in log

10

K

eq

is larger than

0.98, but no further calculations were performed.

Speciation and ion activities were, on the other hand, calculated with

SOLMINEQ.88 (Kharaka and Barnes, 1973; Perkins and others, 1990). We used

reservoir ﬂuid compositions from the San Joaquin and Gulf Coast Basins, at depth

intervals where albitization is present and thermodynamically active (see our table 1;

data from Boles, 1982; Fisher and Boles, 1990). Based on the water chemistry data we

assumed that the oil ﬁeld waters are in equilibrium with quartz, so that the activity of

the solid quartz activity is one. This assumption is justiﬁed by the presence of

authigenic quartz in intervals above the albitization zone, also by ﬂuid chemistry data

(Wilson and others, 2000). The SOLMINEQ.88 code is only partially consistent with

the origins of the thermodynamic database presented in table 2. However, the

analyzed Na

and Ca

2

concentrations in the reservoir waters differ by less than 10

percent from those concentrations calculated by the code, clearly indicating that the

319kinetic model for albitization of detrital plagioclase

complexation of these ions in minerals is negligible. Consequently, any thermody-

namic calculation of Na

and Ca

2

activities would be consistent with the thermody-

namic database used in our paper. The resulting activities, ionic activity products, and

equilibrium constants for a variety of ﬂuid compositions and temperatures used in our

model are shown in table 3.

The former table shows the variation of the saturation index from San Joaquin

and Gulf Coast Basin ﬂuids that are currently at temperatures between 100 and 130° C,

and where albitization is actively occurring (Boles, 1982). However, considering the

waters’ ages are considerably different, in general several millions of years of differ-

ence, and that they have gone through different time-temperature paths, it could be

considered that the variation of is relatively small. Thus, given 1) the narrow range of

Fig. 2. Plot showing the equilibrium constant for reaction 2 assuming pure anorthite and its hexagonal

polymorph (see text). The log

10

K is calculated with equation (9) as a function of temperature. Thermody-

namic data is from Robie and Waldbaum (1968) and Johnson and others (1992). The error in log

10

K

(anorthite) at 298.15 K and 1 bar, calculated from standard error estimation formulas and from thermochemi-

cal errors in the former database is 0.98. It is likely that this error may expand at higher temperatures.

320 R. J. Perez and J. R. Boles—An empirically derived

compositions of albitized detrital plagioclase grains, 2) the relatively small variations

for waters of considerably different ages (millions of years) and different T-t paths, and

3) the occurrence of different plagioclase compositions in samples with identical pore

ﬂuid and temperature history, we hypothesize that a weighted average integration of a

saturation index as a function of temperature, may be useful to simplify the variations

of water chemistry of the studied basins.

All our assumptions may seem, a priori, very simplistic and far from realistic but they are

constrained by ﬁeld data and petrographic observations, and they are not unreasonable. First,

normally all replacement reactions are considered isovolumetric (Merino and others,

1993; Nahon and Merino, 1997; Merino and Dewers, 1998). Second, it might be

justiﬁed that during the pseudomorphic replacement, the coupled dissolution-

precipitation of plagioclase and albite, establishes locally a small-scale steady state or at

least a “quasi” steady state composition of the pore ﬂuid, leading to a local saturation

that could be assumed to be about constant over long time intervals. This idea would

be particularly justiﬁed if the Al

3

and Ca

2

leave the local system at a constant rate

and ﬂuid transport would not be too fast, as in the case in deep crustal settings away

from fractures and faults, where permeabilities could be in the range of tenths of

milidarcies and the average ﬂow velocity is, at times, less than 1 cm/Ma (Garven, 1995).

Table 2

Thermodynamic values used to calculate the equilibrium constant for reaction 2

*Delta of the reaction assuming anorthite

**Delta of the reaction assuming hexagonal anorthite

G° Standard Gibbs Free Energy of Formation (Calories/mole)

S° Standard Entropy “Third Law” of phase (Calories/K mole)

H° Standard Enthalpy of Formation (Calories/mole)

a Calories/mole

b Calories/mole K

2

c Calories/mole K

5

The sum considers the stoichiometry of the reaction

Heat capacity of pure phases and thermodynamic data for aqueous ions from SUPCRT92 and

summarized by Anderson and Crear (1993)

321kinetic model for albitization of detrital plagioclase

Table 3

Input data for saturation index calculation. Speciation performed with SOLMINEQ.88

IAP Ionic activity product Keq Equilibrium constant

322 R. J. Perez and J. R. Boles—An empirically derived

general kinetic models and equations

Lasaga (1984, 1995, and 1998) explains all theoretical and fundamental ap-

proaches to describe dissolution and precipitation rates of minerals. The general

kinetic formulation for mineral/ﬂuids interactions is equation (11):

Rate ⫽ S

m

Ae

E

a

/RT

a

OH

n

䡠 g I

写

i

a

i

n

i

f

r

G (11)

where the Rate is a function of the reactive surface area S

m

(cm

2

), the frequency factor

A (1/s), the apparent overall activation energy E

a

(kJ/mol) the temperature T (K), the

universal gas constant R (J/K mol), the OH

activity raised to the power of na

OH

n

(dimensionless), the ionic strength of the solution g(I )(I has units of molality times

the ionic charge squared ), the activity products a

i

n

of the rate-determining specie i

(dimensionless) raised to the power of n, and a general function of the free energy of

the reaction f(

r

G) that can take many forms, where

r

G has units of J/mol. Another

widely used empirical kinetic model, written as an adaptation of equation (11) by

Aagaard and Helgeson (1982) is:

Rate ⫽ S

m

k

m

a

i

m

1 ⫺ (12)

where S

m

is the interfacial area (cm

2

) between the mineral and the aqueous solution,

k

m

is the rate constant (1/s), a

i

n

is the activity of the rate-determining aqueous species i

in solution. On the other hand, the kinetics of pseudomorphic replacement have been

formulated with two simultaneous reactions that occur at equal volumetric rates,

driven by a force of crystallization generated by supersaturation (Maliva and Seiver,

1988; Dewers and Ortoleva, 1990; Merino and others, 1993; Nahon and Merino, 1997).

In this case, the sum of the rates equals zero as in equation (13)

R

a

⫹ R

b

⫽ 0 (13)

where R

i

is the linear dissolution rate of mineral i in cm/s represented by equation

(14):

R

i

⫽ k

i

1 ⫺ (14)

where k

i

(cm/s) and

i

have the usual meanings. Equations (11) through (14) depend

on the aqueous ionic species present during the reaction and are founded on a

well-developed physical-chemical theoretical basis, and only laboratory-based measure-

ments, with good control of ﬂuid composition (in either closed or open systems) can

be most easily used for a rigorous interpretation of kinetic rate laws and reaction

mechanisms. However, the prediction and modeling of the ﬂuid saturation path in

sedimentary basins is far from attainable. The variation and evolution of chemical and

ionic species through geologic time is even more difﬁcult to predict or model with

precision. Furthermore, invalid extrapolations of experimental kinetic parameters to

geological settings and our poor understanding of reaction mechanisms (and of many

other variables) limit the applicability of rigorous and experimental approaches to

ﬁeld examples.

Based on the reasons explained previously, most empirical models of diagenetic

reactions take a different and simple approach; they reduce and ignore a vast amount

of variables, speciﬁcally ﬂuid compositions by avoiding or leaving constant the satura-

tion index or chemical afﬁnity parameter in the rate expression. The models lack a

physical-chemical basis (Oelkers and others, 2000), however, they can successfully

predict dissolution and precipitation over long time scales, which is the ultimate goal

of these studies.

323kinetic model for albitization of detrital plagioclase

The smectite to illite conversion during shale diagenesis provides one example of

empirically derived kinetics. Several models for this particular reaction have been

proposed and evaluated (Elliot and Matisoff, 1996). The most recent model (Huang

and others, 1993) computes the rate with:

dS

dt

⫽ k K

S

2

(15)

where dS/dt is the change of the smectite fraction with respect to time t, k is the rate

constant, [K

] is a ﬁxed and constant potassium concentration (in ppm), and S is the

dimensionless fraction of smectite in the illite/smectite raised to the power of 2.

Another example is quartz overgrowth precipitation. Commonly, models for

quartz precipitation couple diffusion, kinetic, and thermodynamic aspects of the

reaction (Oelkers, 1996; Oelkers and others, 2000). In contrast, a simpler mathematic/

kinetic model based on time-temperature and surface area (Walderhaug, 1994 and

1996) continues to be successfully tested in basins around the world (for example

Awwiller and Summa, 1997; Lander and Walderhaug, 1999; Perez and others, 1999a,

1999b). The quartz precipitation model uses a logarithmic rate function dependent on

temperature (eq 16):

Rate ⫽ a10

b 䡠Tt

(16)

where a (moles/cm

2

) and b (1/°C) are pre-exponential and exponential constants and

T(t) is temperature expressed as a function of time. The precipitated quartz volume Vq

(cm

3

) is calculated as a sum of integrals within sequences of time steps i (eq 17):

Vq ⫽

M

a

冘

i1

n

冉

A

i

冕

t

i

t

i1

10

b 䡠Tt

dt

冊

(17)

where M is quartz molar volume (g/mole), is quartz density (g/cm

3

), A

i

is surface

area (cm

2

), and t is time (sec). As we explained previously, these empirical models

depend mainly on the variations of the heating rate through geologic time, obtained

through basin modeling (Siever, 1983), and lack an afﬁnity term.

albitization kinetic model

Based on previous formulations in the San Joaquin Basin (Wilson and others,

2000), we modeled albitization with a kinetic/mathematical formulation expressed in

equation (18) by:

d An

dt

⫽ S

m

kAn

o

2

1 ⫺

(18)

where d[An]/dt is the change of the anorthite mole fraction with respect to time t, k is

the rate constant in 1/cm

2

s, [An

o

]

2

is the initial anorthite mole fraction squared, and

the weighted average saturation index. In equation (18) S

m

is the area of the

mineral-solution interface (Lasaga, 1984) and is expressed in equation (19) in the

following form:

S

m

⫽

A

m

V

m

䡠 v (19)

where, A

m

is the mineral surface area (cm

2

), V

m

is the mineral volume assuming

spherically-shaped grains (cm

3

), and v is the unit volume of the sandstone, which

324 R. J. Perez and J. R. Boles—An empirically derived

includes porosity and is equal to 1 cm

3

. Simplifying, the surface area interface results in

equation (20):

S

m

⫽

6

d

v (20)

where d is the grain diameter (cm). In all analyzed samples the grain size ranges from

0.012 0.001 to 0.086 0.018 cm, varying from one sample to another, however,

average grain diameter is 0.031 0.002 cm (table 4). These values are obtained by

measuring the mean grain diameter in 40 grains per thin section, using a standard

polarized light petrographic microscope.

The rate constant k follows the Arrhenius equation (21):

k ⫽ A exp

冋

E

a

RTt

册

(21)

where A and E

a

are apparent pre-exponential and exponential parameters in 1/cm

2

s

and kJ/mole respectively, R the universal gas constant (J/K mole), and T(t)is

temperature in K expressed as a function of time t.

The initial anorthite molar fraction value [An

o

] is obtained through ﬁeld measure-

ments. It is determined on detrital plagioclase grains with an ARL microprobe

analyzer, model SMX-SM equipped with three wavelength-dispersive spectrometers

and a Tracor Northern TN2000 energy dispersive system, by simultaneously analyzing

for sodium, potassium, and calcium, aluminum, and silicon. Each grain is analyzed on

the rim and in the core, and 50 grains are usually analyzed per thin section from

different depth intervals and temperatures (Boles, 1982; Boles and Ramseyer, 1988;

Pitmman, 1988).

Based on the reasons explained above we used a mean saturation index term

(

) obtained with a weighted average integration. The method to obtain

is as

follows: a) ﬁrst we calculate several numerical values of (deﬁned in equation 5) using

waters compositions from different reservoirs of different ages, depths and burial

temperatures where albitization is active (table 3); b) observing a slight temperature

dependence of these values, we ﬁt them to a function of temperature–which is

implicitly a function of time; c) we weight averaged the function with equation (22), so

that:

⫽

冕

T

TdT

冕

dT

(22)

The integration along the (T ) function (solution to equation 22), neglecting the

extreme values and within the temperature of interest, results in a constant log

10

2.632. This positive number suggests that pore ﬂuid compositions are in the albite

dickite stability ﬁeld, which is consistent with ﬁeld observations. In short, to start the

albitization, pore ﬂuids must be supersaturated with respect to albite, that is, in the

albite stability ﬁeld according to reaction 2.

It is well known that, in laboratory experiments, dissolution and precipitation

reactions are usually not independent of ﬂuid composition, and such assumption

would only be valid if the reaction occurs far from equilibrium. Thus, a constant

may

be somewhat problematic. However, based on (1) the small range of water composi-

325kinetic model for albitization of detrital plagioclase

Table 4

Summary of data sets used as input in our model. Data include unpublished well

names, and samples’ temperature, age, and grain size.

*Gulf Coast Basin data from Boles (1982), the San Joaquin Basin data from Boles and Ramseyer (1988) and the Denver Basin data from Pittman (1988)

**Present day depth

***Present day temperature

Ab stands for ablite, Or for orthoclase and An for anorthite

s.d. stands for standard deviation of spot analysis per sample [100 spots for Boles (1982) and Boles and Ramseyer (1988) and 20–27 spots for Pitmann (1988)]

N.I. No information

The grain size units are in millimeters

326 R. J. Perez and J. R. Boles—An empirically derived

tions present in the San Jaoquin and Gulf Coast Basin where albitization is active

(Boles, 1982), (2) a narrow albitization temperature window, (3) the relatively small

range of albitized plagioclase compositions (ﬁg. 2), we propose that the heating rate

may be the most important parameter for the actual kinetics of the replacement.

Particularly because experiments suggest that this is the case when the reaction

products grow on the reactant (Luttge and others, 1998), as in the case of replacement

reactions.

In addition, at depths greater than 3 to 4 km, where albitization occurs, the rock

permeabilities are on the order of the tenths of milidarcies and the rate of ﬂuid

transport is very small; and it is reasonable to assume that some minerals can establish

locally a steady state or at least a “quasi” steady state composition of the pore ﬂuid,

leading to a local saturation that could be assumed to be about constant over long time

intervals.

Numerical Solution

The solution of equation (18) is reached by the integration of equation (23):

冕

dAn ⫽

冕

t

0

t

m

Ae

E

a

/R 䡠Tt

6

d

v An

o

2

1 ⫺

dt (23)

An analytical integration of the right hand side of equation (23) from time t

0

to t

m

is

not trivial due to the time dependence of the temperature, and some proposed

rational approximations (for example Burnham and others, 1987) introduce errors at

high temperatures (Comer, 1992). Alternatively, a numerical solution can be reached

if T(t) is divided in increments, and a mean value

T

i

is calculated within the (i) (i

1) time steps. In the general case, the mean temperature

T

i

may be calculated

through a weighted average integration expressed in equation (24) as:

T

i

⫽

冕

t

i

t

i1

Ttdt

冕

t

i

t

i1

dt

(24)

If the increments are small and considered linear, T(t) takes the form of equation (25)

Tt ⫽

冘

i1

n

m

i

t ⫹ d

i

(25)

where the slope m is the temperature change rate (in °C/s), t is time, and d is the initial

temperature (in °C), and

T

i

is reduced to an average between two time steps, that is

(T

i

T

i1

)/2. If a time-temperature curve, represented by the function TT(t), is

divided in small linear increments from (T

i

,t

i

)to(T

i1

,t

i1

), the slope m

i

between

each interval would be:

m

i

⫽

T

i1

⫺ T

i

t

i1

⫺ t

i

(26)

and the initial temperature d

i

is:

d

i

⫽ T

i

⫺

冉

T

i1

⫺ T

i

t

i1

⫺ t

i

冊

t

i1

(27)

327kinetic model for albitization of detrital plagioclase

Our model uses variable times steps, between 0.3 and 0.7 million years, depending

upon the sample age and temperature history. The model results are independent of

the time step length, although intervals of 0.5 million years are preferred because they

give best results. The substitution of T(t)by

T

i

results in equation (28):

冕

d An ⫽

冕

t

i

t

i1

Ae

E

a

/R 䡠

T

6

d

v An

o

2

1 ⫺

dt (28)

Although equations (23) and (28) may seem similar, the substitution simpliﬁes the

integration, because

T is now a constant. Finally, the integration is numerically

expressed with equation (29):

An

i1

⫽

冘

i0

n

冉

An

o

i

⫺ Ae

E

a

/R

T

i

6

d

vAn

o

i

2

1 ⫺

t

i1

⫺ t

i

冊

(29)

where i is the time step.* During albitization of detrital plagioclase in the studied

basins, the anorthite mole fraction decreases from An

35

to An

0

, and the albite mole

fraction increases simultaneously from Ab

65

to Ab

99

(table 4). By deﬁnition the sum of

the mole fractions is unity, that is:

An ⫹ Ab ⫹ Or ⫽ 1 (30)

where An, Ab, and Or are the anorthite, albite, and orthoclase mole fractions.

Therefore, the simultaneous gain of the albite component, as a result of albitization, is

computed with equation (31)

Ab

i1

⫽ 1 ⫺

冘

i1

n

冉

An

o

i

⫺ Ae

E

a

/R

T

i

6

d

v An

o

i

2

1 ⫺

t

i1

⫺ t

i

冊

⫺ Or

o

(31)

where [Ab] is the ﬁnal albite mole fraction per time step i,[Or

o

] is the initial orthoclase

mole fraction that could be set constant to 0.05 0.01, or simply ignored.

The only unknown parameters in equation 29 are E

a

and A. When only two

parameters are being optimized, both least squares ﬁt and direct search techniques are

efﬁcient methods (Gallagher and Evans, 1991). We approached the problem with a

direct grid search, using an iteration method, which breaks the time-temperature

domain into a series of discrete blocks, where equations 23 through 31 are calculated.

We began our search of A by ﬁxing E

a

from 60 to 70 kJ/mol following Walther and

Wood (1984) and Lasaga (1984). We decided to search with small E

a

and wide A

increments. Our decision was based on the limited E

a

range (60 E

a

80 kJ/mol)

reported for a great number of dissolution reactions of alumino silicates (Lasaga,

1984) and speciﬁcally for feldspar hydrolysis reactions at neutral pH within the

temperature of interest (Hellman, 1994; Blum and Stillings, 1995).

As input, we used (1) compositional data that revealed increasing albitization with

depth and temperature (table 2), and (2) time-temperature curves obtained through

1-D burial-thermal modeling of the basins. We performed the basin modeling using

Genesis

tm

(Zhiyong, 2003).

*Note that [An

o

]

i

2

is a constant per time step.

328 R. J. Perez and J. R. Boles—An empirically derived

model fit and tests

San Joaquin Basin.—We began our search in the San Joaquin Basin. Callaway

(1971) provided an excellent review of its geologic framework. In summary, the basin

contains more than 7 km of Tertiary Miocene sediment, and underwent down warping

by compaction during its burial history. In the center of the basin, where we focused

our study, the present depths represent maximum burial. For the albitization kinetic

ﬁt, we selected from the published compositional data (Boles and Ramseyer, 1988) a

total of 11 samples from nine different wells (table 4). The selection was based on the

availability of thin sections and well data. We constructed time-temperature histories

for each sample and determined grain sizes by averaging the diameter of 40 plagioclase

grains per sample.

We extracted input parameters from the literature for basin modeling pur-

poses, including stratigraphic information from ﬁeld-scale structural cross sections

(California Oil and Gas Division, 1985), formation ages (COSUNA, 1984 and Wood

and Boles, 1991). The latter paper also provided formation thickness and subsi-

dence rates. We assumed a constant basal heat ﬂow of 50 mW/m

2

, based on present

day thermal data (Lachenbruch and Sass, 1980) and geothermal studies in the San

Joaquin Basin (Dumitru, 1988; Wilson and others, 2000) and we used thermal

conductivity values compiled in Genesis

tm

by Zhiyong (2003). It is important to

emphasize that heat ﬂuxes in the San Joaquin Basin were likely lower than 50

mW/m

2

during active subduction, and that thermal gradients have recovered since

then (Dumitru, 1988). Figure 3 depicts the resulting burial histories. The kinetic

modeling search (ﬁg. 4) indicates that the albitization trend can be reasonably

simulated using an apparent activation energy of 68 4 kJ/mole and apparent

frequency factors of (6.5 0.5) 10

3

1/cm

2

Ma. The error bars in the data points

(ﬁg. 4) represent the standard deviation, computed as the square root of the

variance, which is in turn calculated as the average squared deviation from the

Fig. 3. Several time-temperature paths of albitized samples from the San Joaquin Basin used in the

kinetic model.

329kinetic model for albitization of detrital plagioclase

mean. The errors in E

a

and A represents the range of values in E

a

and A required to

enclose the variation in the compositional data.

Tests Results

Once we calculated the set of unknown parameters in the San Joaquin Basin, we

proceeded to test them against plagioclase compositional trends from the Texas Gulf

Coast (Boles, 1982) and Denver Basin of Colorado (Pittman, 1988).

Texas Gulf Coast Basin.—Galloway and others (1994) provided a synthesis of the

geologic, structural, and depositional history of the onshore Tertiary deposits present

in the southern part of the Texas Gulf Coast Basin. The stratigraphic column in the

southern part of the Texas Gulf Coast consists of 4 to 6 km of sedimentary rocks, buried

in a simple down-warped fashion without major regional uplifts (Murray, 1960).

Present-day depths and temperatures in the basin represent, in many cases, maximum

values (Boles, 1982). We used the kinetic model and parameters derived from the San

Joaquin Basin ﬁt to simulate the plagioclase compositional trend reported for the

Oligocene Frio Formation of south Texas (Boles, 1982). The data comprise eight

samples distributed over eight wells from six different counties. Grain sizes were

determined by measuring the mean diameter of 50 plagioclase grains in thin sections

(table 4).

We constructed 1-D burial-thermal curves for each sample using Genesis

tm

(Zhiyong, 2003). From Dodge and Posey’s (1981) set of subsurface structural cross

sections, we obtained well locations, formation tops, gamma-rays/spontaneous poten-

tial logs, bottom hole temperatures, and biostratigraphic information. We used the

biostratigraphy information from the COSUNA (1984) chart and the U.S. Department

of Interior Biostratigraphic Chart for the Gulf of Mexico (OCS) Region (2002) to

derive formation ages. For the time-temperature reconstruction, the present day heat

Fig. 4. Initial model ﬁt to albitization in the San Joaquin Basin. Measured and calculated data are

plotted as functions of temperature. The ■ symbol with vertical lines represents data from Boles and

Ramseyer (1988) with standard deviation computed as the square root of the variance. Symbols E, 〫, ‚, and

▫ represent sequences of simulations performed for different samples. The data ﬁt an E

a

of 68 4 kJ/mol

and an A (6.5 0.5) 10

3

1/cm

2

Ma. The E

a

and A errors represent the required deviation mean value to

enclose the standard deviation of the compositional data.

330 R. J. Perez and J. R. Boles—An empirically derived

ﬂux (53 mW/m

2

) was set constant from the middle Oligocene to the present. The heat

ﬂux was calculated by the code using formation temperatures and thermal conductiv-

ity values compiled by Zhiyong (2003). We calibrated our time-temperature histories

with temperature measurements taken along the depth of the wells (Dodge and Posey,

1981). Results are plotted in ﬁgure 5.

Using the previous results, we modeled albitization and computed the loss of the

anorthite component (ﬁg. 6A). Additionally, due to the fact that the orthoclase

fraction in a detrital plagioclase remains constant and is usually less than 5 percent, an

albitized grain can be conveniently studied as a binary system. In a binary system, when

the compositional data is normalized to 100 percent, the variance of each component

is the same, and consequently, the standard deviations are equal. Thus, we modeled

the gain of the albite component using the standard deviation from the anorthite mole

fraction (ﬁg. 6B). Ten waters analysis from ﬂuid reservoirs that have gone through

different time-temperature trajectories, have signiﬁcantly different ages (millions or

years), and where albitization is active, the waters have relatively similar saturation

index values compared to subsurface waters from the San Joaquin Basin (table 3).

Thus for modeling purposes, we used the same weight averaged log

10

2.632 value

as in the San Joaquin Basin. The albite gain or anorthite loss during albitization in

Tertiary Gulf Coast Basin can be simulated assuming reaction 2 with an apparent

activation energy of 68 4 kJ/mole and apparent frequency factors of (6.5 0.5)

10

3

1/cm

2

Ma. Again, errors of E

a

and A represent the values required to enclose the

variations in the compositional data. The error bars of the data (ﬁg. 6B) represent the

standard deviation calculated as the square root of the variance.

Denver Basin.—We tested the kinetic model against data from the Cretaceous

Terry sandstone in the Spindel ﬁeld, Denver Basin of Colorado. Pittman (1988 and

1989) presented a geologic synthesis of the basin and plagioclase albitization data. The

history of the basin is rather complex. The heat ﬂux varied through time resulting in

Fig. 5. Time-temperature history of albitized samples from the Gulf Coast Basin, derived from

basin-burial modeling.

331kinetic model for albitization of detrital plagioclase

Fig. 6(A) Plot illustrating simulated and measured anorthite mole fractions versus temperature in the

Gulf Coast Basin. Measured and calculated data are plotted as a function of temperature. The ■ symbol

represents data extracted from Boles (1982). Vertical lines represent standard deviation of 100 spot analyses

per sample, computed as the square root of the variance. All other symbols represent sequences of

simulations calculated for ﬁne-grained sandstones deposited between 28.6 and 32.1 Ma (see legend). Fit

corresponds to E

a

68 4 kJ/mole and A (6.5 0.5) 10

3

1/cm

2

Ma. The errors represent the range of

values in E

a

and A required to enclose the variation in the composition data.

Fig. 6(B) Calculated and measured albite mole fractions as a function of temperature in Gulf Coast

Basin. The ■ symbol represents data from Boles (1982). The vertical lines represent standard deviations

(square root of variance) of anorthite component. Vertical lines represent standard deviation of 100 spot

analyses per sample.

332 R. J. Perez and J. R. Boles—An empirically derived

changing thermal gradients and heating rates, and the sedimentary section was

uplifted 800 m during Early Paleocene and Miocene time. Burial and thermal

histories were calibrated against R

o

proﬁles, K/Ar data, and multiple temperature logs

(Pittman, 1988).

The resulting time-temperature history of the Terry sandstone corresponds to a

single horizon at 1,325 m depth (ﬁg. 7). Not having compositional data at speciﬁcally

1,325 m depth, we chose the closest samples where data were available, corresponding

to those in the well Rademacher No. 1 at 1,283.2 and 1,286.9 m depth (see table 2, p.

752, Pittman, 1988). Albitized plagioclase in this setting was mainly reported in ﬁne to

very ﬁne-grained sandstone. Hence, we modeled albitization using three surface area

assumptions, calculated from three different sand grain sizes (1) ﬁne grain, with d

0.0177 cm, (2) very ﬁne grain with d 0.0088 cm, and (3) ﬁne to very ﬁne grain with

d 0.0125 cm.

Having no information regarding the oil ﬁeld water chemistry or saturation

indexes, we computed the reaction using the same log

10

2.632 used in the San

Joaquin and Gulf Coast Basin. Again, results indicate that the albitization data can be

consistently predicted with an apparent activation energy of 68 4 kJ/mole and

apparent frequency factors of (6.5 0.5) 10

3

1/cm

2

Ma (ﬁg. 8). The same ﬁtting

scheme performed with data from the San Joaquin Basin was performed indepen-

dently using data from the Louisiana Gulf Coast and Denver Basin, and the results

showed no skew.

model evaluation

Comparison with Other Diagenetic Reactions

We compared the albitization rate constant k with the rate constants of quartz

precipitation and smectite-to-illite models derived from ﬁeld data and burial-

temperature histories by different authors (ﬁg. 9). Although smectite-to-illite and

Fig. 7. Thermal history of the Terry Sandstone, Spindle ﬁeld, Denver Basin, after Pittman (1988).

Time-temperature proﬁle corresponds to a horizon at 1,325 meters depth and was used as input for

albitization model.

333kinetic model for albitization of detrital plagioclase

quartz precipitation are all very different reactions, their reaction rates were derived

with an empirical-numerical approach, similar to ours. Additionally, smectite-to-illite

and quartz overgrowth precipitation, together with albitization, constitute the main

diagenetic reactions in sedimentary basins: they are all strongly dependent on heating

rates and surface area or grain size. In our model, the rate constant k, is calculated with

equation (32), identical to equation (4), as shown below:

k ⫽ Ae

E

a

/RT

(32)

The k is plotted in ﬁgure 9. There k ranges from 1 10

21

1/cm

2

s at 25°C to

1 10

18

1/cm

2

s at 200°C (ﬁg. 9), assuming E

a

and A to be 68 kJ/mol and 6.5 10

3

1/cm

2

Ma and making the appropriate unit conversion to 1/cm

2

s.

Our rate constant values are higher than the quartz precipitation rate of Walder-

haug (1994) and smectite-illite transformation rates of Huang and others (1993), and

lower than Pytte and Reynolds’ (1988) and Velde and Vaseur’s (1992) smectite to illite

rates. We also compared our model to growth rates of albite (ﬁg. 9). Aagaard and

others (1990) estimated albite growth rates from albite dissolution rates (Knauss and

Wolery, 1986) and the principle of microscopic reversibility (Lasaga, 1984). According

to Aagaard and others (1990), dissolution would not normally be the limiting factor in

the case of K-feldspar albitization. This conclusion is reached by comparing the

relatively fast K-feldspar dissolution rate (Helgeson and others, 1984) with the rela-

tively slow albite precipitation rate—omitting nucleation.

Contrary to the K-feldspar case, in our model the rates of albitization (plagioclase

dissolution) are slower than the albite growth rates of all temperatures (ﬁg. 9), which is

Fig. 8. Plot of comparison between calculated and measured anorthite mole fractions in Denver Basin.

Measured and calculated data are plotted as a function of time. The ■ symbol represents two data points at

1,283.2 and 1,283.9 m depth (Pittman, 1988), and the error bar the standard deviation. Other symbols

represent sequences of simulations performed for very ﬁne-grained sandstones (‚), ﬁne-grained (

䉬

) and

very ﬁne to ﬁne sand ( ). Age of samples is 73.3 Ma. The data ﬁt an E

a

68 4 kJ/mole and A(6.5

0.5) 10

3

1/cm

2

Ma. The errors in E

a

and A represents the range of values in E

a

and A required to enclose

the variation in the composition data.

334 R. J. Perez and J. R. Boles—An empirically derived

fairly consistent with ﬁeld observations. In sedimentary basins, plagioclase dissolution

is the rate-limiting step of albitization based on the fact that low albite precipitates in

porous sandstones at temperatures lower than the temperature onset of albitization

(Ramseyer and others, 1992).

Sensitivity Studies and Model Limitations

We performed several sensitivity analyses to illustrate the model’s response to

variations in the input parameters. The common input to all simulations was an

arbitrarily chosen time-temperature proﬁle, somewhat similar to the thermal history of

the Gulf Coast and San Joaquin Basins (ﬁg. 10). From the simulation results (ﬁgs. 11

and 12), we are able to evaluate the effects of the initial composition and the surface

area on albitization. Additionally, based on the Denver and Gulf Coast Basin results, we

discuss the effect of heating rates on albitization.

Effects of initial composition.—The effect of the initial anorthite [An

0

] composition

on the rates of albitization is depicted in ﬁgure 11. From 0 to 65°C, the reaction rates

are relatively slow and all modeled initial compositions, An

30

Ab

65

,An

40

Ab

60

,An

60

Ab

30

and pure An

100

, remain constant. Between 75 and 125°C all compositions are partially

albitized. Finally, at temperatures over 125°C, the different compositional trends

converge to a single value, that is [An] 0 mole percent. It is clear that the

plagioclase’s detrital composition in terms of anorthite content affects the reaction,

but the effect is minimal at high temperatures. The wide standard deviations at low

temperature and small standard deviations at high temperature (ﬁgs. 4 and 6) of the

data (table 1), can both be partially explained by differences in the initial anorthite

fraction.

Fig. 9. Comparison of the albitization rate calculated in this paper assuming E

a

68 kJ/mole and A

6.5 10

3

1/cm

2

Ma)–with other rates of mineral reactions as a function of temperature. For illitization of

smectite (S/I) based on Pytte and Reynolds (1988), Velde and Vassseur (1992) and Huang and others

(1993); quartz precipitation based on Walderhaug’s (1994); albite growth rate based on Aagaard and others

(1990).

335kinetic model for albitization of detrital plagioclase

Another rate determining factor to examine, but well beyond the scope of this

paper, is the inﬂuence of the ordered (that is plutonic) versus disordered (that is

volcanic) atomic arrangement of the albitization kinetics. At present we know of no

Fig. 10. Time-temperature proﬁle representing a single burial cycle, similar to proﬁles from the San

Joaquin and Gulf Coast Basins. This curve was used as input for several sensitivity analyses illustrated in

ﬁgures 8 and 9.

Fig. 11. Sensitivity of albitization kinetic model to plagioclase initial composition, as a function of

temperature using proﬁle in ﬁgure 7. The initial plagioclase composition is function of An. Simulations show

that initial composition persists until 100°C, slows albitization at temperatures below 125°C, and

converges to a single composition trend over 135°C.

336 R. J. Perez and J. R. Boles—An empirically derived

thermodynamic data of ordering at constant anorthite content in plagioclase. Thus as

a proxy, we used thermodynamic data of an hexagonal polymorph named Dmisteinber-

gite. This polymorph is rare but has been reported on fractured surfaces in coal from

the Chelyabinsk coal basin, Southern Ural Mountains, Russia.

As mentioned the thermochemical data for hexagonal anorthite is taken from

Robie and Waldbaum (1968). Hence, we calculated the equilibrium constants of

reaction 2 assuming an hexagonal anorthite and graphed it in ﬁgure 2. We also

calculated

values for reaction 2 under the same assumption and using ﬂuid

compositions from the San Joaquin Basin and Gulf Coast Basin (table 3). The resulting

values of log

10

hexagonal

suggest that if Dmisteinbergite was present in the San Joaquin

Basin, reaction 2 would proceed far enough to the right until the saturation point,

either because the pore ﬂuids have been in contact with sufﬁcient amounts of the

mineral or have reacted with the mineral long enough. In short, pore ﬂuids are

saturated with respect to hexagonal anorthite, and we can only speculate that the

albitization of this polymorph would be faster, assuming the same rate constant.

Effect of surface area.—The mineral/ﬂuid surface area interface is key in diagenetic

modeling (Oelkers, 1996). Our model is designed such that plagioclase dissolution

rate is inversely proportional to the grain diameter. A simple and direct inspection of

equation (18) shows that d[An]/dt S

n

so that [An]

i1

(6/d)v. Thus, given the same

initial composition and time-temperature path, smaller grains will albitize faster than

bigger grains. Figure 12 illustrates albitization trends plotted against temperature for

several grain diameters. Below 60°C the inﬂuence of the initial grain size on albitiza-

tion is small, but over 70°C the transformation rate increases as grain size decreases. As

a result plagioclase with different grain sizes may have different degrees of albitization

and/or albitization paths that persist over a wide temperature range. Perhaps the wide

deviations of the albitization trends may be explained by the standard deviations in

grain size distributions (table 4). However, the dissolution of minerals, and particularly

Fig. 12. Sensitivity of albitization to detrital grain size as a function of temperature. The initial

plagioclase composition is common to all simulations. Simulations indicate that grain size differences affect

the degree of albitization throughout the transformation. Effects of the grain size persist until high

temperatures are reached, but it minimizes at temperatures over 160°C.

337kinetic model for albitization of detrital plagioclase

feldspars, is certainly not uniform over all the surface. Leaching occurs at preferred

crystallographic sites, twin planes, microfractures, and dislocations.

An additional parameter not addressed in our study is the effect of permeability

on albitization. Fine-grained sandstone presumably has low initial permeability and

much of its original water could be lost through compaction/cementation by the time

of albitization (Boles and Coombs, 1977). Increasing albitization with increasing grain

size has been reported (Boles and Coombs, 1977), but these results may indirectly

reﬂect permeability effects rather than solely a surface area effect. Albitization requires

a local steady state composition of the pore ﬂuid, in which Ca

2

and Al

3

leave the

local system on a constant rate and ﬂuid transport is not rapid. Differences in

permeability and ﬂow rates would establish different saturation states of the ﬂuid over

short time intervals and would change from one depth to another dramatically. In

general, sandstone with contrasting permeability may have different anorthite content

through time and different degrees of albitization.

Effect of heating rates.—Heating rates can inﬂuence the dissolution, precipitation, and

even the nucleation rates of mineral reactions (Luttge and others, 1998). The effect of the

heating rate on albitization can be evaluated by comparing results from our simulations.

For instance the San Joaquin Basin consists of relatively young deposits, from 6.7 to 33 my,

but mostly less than 20 my old, and moderate to low thermal gradients between 25 and

27°C/km. At temperatures higher than 170°C, our modeling suggest that at low tempera-

ture albite has replaced almost all plagioclase, leaving the anorthite mole fraction below 5

percent. The Gulf Coast Tertiary Basin consists of sediments 30 to 35 my old and a

relatively high thermal gradient between 32 and 37°C/km. Albitization occurs there

between 90 and 130°C, and at temperatures higher than 130°C the plagioclase approaches

pure albite composition. In contrast, the Denver Basin was subject to periods of 50 my at

temperatures in the neighborhood of 90°C, and, in our model, plagioclase has not

reached pure albite composition as it does in the San Joaquin and Gulf Coast Basins. Based

on these three different sets of simulations, it is clear that albitization is primarily sensitive

to temperature rather than time.

The model dependence on temperature may also be evaluated by calculating

time-temperature end-member values for albitization. In other words, calculating

minimum temperatures at constant time (ﬁg. 13A) and minimum times at constant

temperature (ﬁg. 13B) required to albitize plagioclase. To perform these calculations,

we modeled albitization in ﬁne-grain sandstone, set the initial composition to

An

35

Ab

60

Or

5

, and used our average kinetic parameters: E

a

and A equal to 68 kJ/mole

and 6.5 10

3

1/cm

2

Ma.

For the speciﬁed conditions, sedimentary rocks as old as 500 my must be subject to

temperatures over 90°C in order to be albitized (ﬁg. 10A). The minimum temperature

decreases with time. Sandstone as old as 300 my must be subject to constant tempera-

tures over 95°C, and sandstone 200 my old at temperatures over 105°C. In contrast, at

temperatures over 170°C, albitization may occur in less than 0.6 my (ﬁg. 13B).

It is important to emphasize that the replacement has a strong dependence on

temperature and weak dependence on time (ﬁg. 13). Nevertheless, these results

suggest that albitization could occur slowly over long periods of time, and might occur

rapidly in rocks subject to rapid thermal events, for example, thermal alteration

associated with ﬂuid ﬂow along faults.

some model implications

With available compositional and petrologic data, our model may be used to put

limits on burial and thermal histories of basins. The albitization kinetics may be used as

a paleo-thermometer providing independent calibration for other thermal indicators,

such as vitrinite reﬂectance, TAI (Thermal Alteration Index, observed by spore

darkening), quartz precipitation overgrowth, and smectite-illite, in sedimentary basins.

338 R. J. Perez and J. R. Boles—An empirically derived

The knowledge of the degree of albitization, for a given time-temperature data set,

may be also useful for predicting the relative amounts of byproducts involved in

reaction (2). For instance, signiﬁcant Ca

2

enrichment in pore ﬂuids within the

albitized plagioclase zone has been reported (Fisher and Boles, 1990). The enrich-

ment is mainly caused by the uptake of Na

and release of Ca

2

implied in reaction

(2). The calcium release can now be quantitatively estimated in arkosic basins through

geologic time with the albitization kinetic model. In short, the model has the potential

Fig. 13(A). Minimum temperatures required to complete albitization at constant time assuming

ﬁne-grained sandstone, which we considered to be an average value for all three basins. (B) Minimum time

required to complete albitization at constant temperature. We assume ﬁne-grain size sandstone, E

a

68

kJ/mole and A 6.5 10

3

1/cm

2

Ma.

339kinetic model for albitization of detrital plagioclase

to lead to a better understanding of the evolution of basin ﬂuids in geological time

scales. The kinetic model may also be used by petroleum geologists for predicting the

precipitation of kaolinite or calcite (assuming a CO

2

supply) associated with albitiza-

tion, during burial in a given oil reservoir. Any estimation may be done by simply

integrating the kinetic equation through a time-temperature burial history of a given

siliciclastic formation, and applying basic rules of stoichiometry.

conclusions

Albitization of plagioclase trends present in sedimentary basins can be reasonably

reproduced with a kinetic equation that has a rate constant with a pre-exponential

parameter A (6.5 0.5) 10

3

1/cm

2

Ma, an activation energy E

a

68 4 kJ/mole

and an weighted average saturation index log

10

2.632. The only ﬁtted parameter

was A, the other parameters E

a

and omega were ﬁxed or calculated from real ﬁeld data

prior to the kinetic analysis. Our study demonstrates that time, surface area, initial

composition, and primarily temperature are the most important parameters control-

ling the replacement reaction. Our model was tested against data from three geologic

settings: Texas Gulf Coast Basin, Denver Basin of Colorado, and San Joaquin Basin with

different burial and thermal histories. These results evidence that the proposed

equation and its parameters can reasonably reproduce the extent of the albitization

within our knowledge of variables involved.

The kinetic parameters yield rates within the range of other diagenetic reactions

involving precipitation of silicates, speciﬁcally quartz overgrowth cementation, illite

precipitation, and with calculations of albite crystal growth rates. The initial composi-

tion of detrital plagioclase affects albitization during the initial stages. The grain size

affects albitization until the transformation is completed. Subtle differences in grain

size, initial composition, and to a less extent the degree of order of the albite

component may be enough to explain the wide standard deviations reported in the

literature that we reference. We conclude that the albitization zone depends more on

temperature and the heating rate, than time and ﬂuid chemistry. All variables,

however, are important and must be considered.

acknowledgments

We would like to acknowledge editorial work by Emma Perez and Dr. Arthur

Sylvester, and constructive comments by Drs. Thomas L. Dunn and Bradley Hacker.

Dr. Alicia Wilson provided important input with numerical methods and some burial

histories from the San Joaquin Basin. Comments and revisions by Dr. Rolland

Hellmann and the AJS anonymous reviewer greatly improved the content and theoreti-

cal basis of our study. The U.S. Department of Energy (DOE) funded our research

under grant no. 444033-22433.

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343kinetic model for albitization of detrital plagioclase