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The rate of growth of sandstone-hosted calcite concretions
Geochimica PI Connochimiur Copyright 8 19’S Pergamon
0016-7037/9O/S3.OU
Aeta Vol. 54, pp. 3391-3399 Press U.S.A.
+ .oO
pit.F’rinted in
The rate of growth of sandstone-hosted calcite concretions MARK WILKINSON”* and MICHAELD. DAMPIER* ‘cement of Geology, Leicester University, L.&ester, UK ‘Department of Mathematics, Leicester University, Leicester, UK (Received January 8, 1990; accepted in revisedform September 25, 1990)
Abstract-Concretion growth within sandstones can be modelled as a three-stage process, comprising solute supply, solute transport, and surface reaction. Solute supply is not thought to be rate dete~ining. R. A. Bemer has proposed models for solute transport, which can be adapted to the complex calcitewater system. His model for concretion growth by diagenetic redistribution, which incorporates the replenishment of solute within the porewaters during growth by the dissolution of skeletal aragonite, is reassessed and a new solution presented. The model is also expanded to include the effects of species interconversion during transport. The relative importance of solute transport and surface-reaction can be assessed by calculating model growth times assuming each process to act in the absence of the other. In magnesium-free porewaters growth is transport controlled, in seawater it is surface-reaction controlled. In porefluids with intermediate magnesiumcalcium ratios, both processes must be modelled if growth times are to be accurately predicted, with the two processes being rate balanced at the surface of the concretion. The models can be easily modified to include the effects of mobile porefluids. The models have been applied to the concretions of the Valtos Sandstone Formation of Skye, UK. A 1 m diameter concretion is predicted to grow in 9.0 Ma in stationary porefluids, or 5.7 Ma in porefluids flowing at 1 m a-‘. INTRODUCI’ION MANY SANDSTONESHOST near-spherical or ellipsoidal concretions. Inside the concretions the porespace is almost totally occluded by a calcite cement, while, in contrast, the intervening ho&rock is usually poorly cemented (e.g., HUDSON and ANDREW& 1987). Individual concretions rarely exc&d 1.5 m in diameter; however, coalescence can result in the formation of laterally extensive cement sheets, which, when located within petroleum reservoirs, can lead to production problems ( KANTOROWICZet al., 1987 )_ Calcite concretions within sedimentary rocks have been the focus of increasing attention over the past 20 years. The majority of the work has concentrated upon shale-hosted concretions (e.g., ASTIN and SCOTCHMAN,1988; RAISWELL, 1988), but some attention has been paid to the sandstonehosted equivalents (e.g., JOHNSON, 1989; MCBRIDE, 1988; PIRRIE, 1987). There are several aspects of the formation and distribution of sandstone-hosted concretions that are at present poorly understood. This paper addresses one of the key problems: the determination of the concretion growth rate and, hence, the time required to form a concretion of a given size. Although this paper is primarily concerned with sandstone-hosted concretions forming under specific conditions, much of the theory can be applied to concretions forming under different conditions within a variety of host rocks. Concretion growth can be considered to be a three-stage process. Initially, solute must be introduced into the porewaters. For the purposes of this paper, this process will either be neglected (i.e., all solute will be assumed to be in solution * Present address: Department of Earth Sciences, Liverpool University, Brownlow Street, Liverpool, L69 3BX, UK. 3391
at time = 0) or shown to be rapid compared to the other processes, and, consequently, to be unimportant in determining concretion growth times. All current concretion growth models (e.g., BERNER,1968, 1980) assume an initially uniform ~t~bution of source material, be it initially in solution or in the solid phase. The second process comprises solute transport. Either porefluid flow, or diffusion, can act to move solute from the site of supply to the site of precipitation. BERNER ( 1968, 1980) has proposed several models for the transport sta8e of concretion growth. Finally, the solute precipitates as cement onto the concretion surface. This involves the dehydration of the solute ions and construction of the calcite lattice; these two processes are commonly grouped together and referred to as surface reaction. CALCIUM CARBONATE SUPPLY The supply of calcium carbonate to a growing concretion may involve transport of molecular species into the rock unit hosting the concretion. This would be the case for concretions growing within diffusive distance of the sediment-water interface, for example, within the sulphate-reduction zone in muds ( WIL~NSON, 1989a). In sandstones, the majority of the calcium carbonate is derived by the internal r~s~bution of detrital shell material (WILKINSON 1989b). The models presented within this paper all start with the porewaters in equilibrium with amgonite, a geologically common carbonate mineral. It can be shown, using published dissolution rate constants (e.g., WALTER, 1986), that agitated solutions will reach equilibrium with aragonite in only a few days. Even though equilibration times may be as long as two months in more quiescent porewaters ( MUCCI, 1983), dissolution is orders of
M. Wilkinson and M. D. Dampier
3392
magnitude more rapid than the other potentially rate-determining steps involved in concretion growth which are considered below. For the remainder of this paper it is assumed that solute supply is not rate determining.
where [C] is the concentration of species C, and D, is appropriate diffusion coefficient (Table 1). Rc, and R-, the net rates of the carbonate interconversion reactions and B, respectively), and R, is the net rate of the water sociation/dissociation reaction (C):
the are (A as-
SOLUTE TRANSPORT The processes driving transport from the solute source to the growing concretion are diffusion and porefluid flow. BERNER( 1968) has proposed two models for solute diffusion within spherically symmetric systems. The former, here referred to as the simple diffusion model, assumes the existence, prior to the initiation of concretion growth, of a fluid uniformly supersaturated in the cementing mineral. No allowance is made for the addition of solute to the porefluids once concretion growth has commenced. BERNER and MORSE ( 1974) applied this model to the calcite-water system and correctly predicted calcite dissolution times (dissolution and precipitation are similar mathematically). However, their calculations assume that each species diffuses without undergoing chemical reaction. In the calcite-water system, where carbon dioxide is present as carbonate, bicarbonate, and carbonic acid, this need not be the case. A more complete treatment of diffusion, as described below, hence requires consideration of the carbonate interconversion reactions ( BOUDREAU,1987; BOUDREAUand CANFIELD, 1988 ). The second model for difisional solute transport ( BERNER, 1980, pp. 108- 112) involves the dissolution of a source mineral during concretion growth and is known as diagenetic redistribution. Bemer’s mathematical treatment of this model is shown to be incorrect, and a revised version is presented below. This model must also be modified to account for species interconversion during diffusion. BERNER( 1968, 1980) also described the effects of porefluid flow upon concretion growth times. This is incorporated into both the simple diffusional model and the diagenetic redistribution model in a subsequent section. Simple Diffusion Model Incorporating Species Interconversion
(la) cl
(lb)
6r
+Rc, -R,
(1~)
(Id)
(C)
The R terms in Eqs. ( la-f) are not known but can be eliminated by algebraic manipulation ( BOUDREAU,1987) to yield two equations.: b[COzl + ~4HCO31 -
6t
6t
+ -4CO31
at
6[HCO31 4CO31 + &CO, .p + Dco, 6r
6r
6[Hl ------
4OHl
6t
6t
6iHCO31
Pa) 1)
2 .-6iCO31
St
6t
_ DOH.-d’oH1 6r x 6IHCO31
----iiT-
- DHCO,
2 * ho,
(2b)
Equations ( la), (2a), and (2b) can be simplified by assuming that, at any given time, the concentration gradients around a growing concretion are constant, i.e., that the system is in a steady state. This is an acceptable approximation as TABLE 1 - Symbols used in the text
D F Fd
FP
JX JP k Kcc Kcl.Kc2 Kw
r rL R : V
[Xl 0 n vd.vp
(10
(B)
Hz0 = H+ + OH-.
MP n PC02
.p4HCO31 HCO3
(A)
HCO; = CO:- + H+
G
The initial equations below describe the diffusion of the six components of the calcite-water system. They are analogues of those listed by BOUDREAU( 1987), modified for a spherically symmetric system:
_ R
C02(aq) + Hz0 = HCO; + H+
concentration of solute r=x (mol/cms) dffision coefficient (cm2/s) ratio of rates of concretion volume change for flowtng and static porewaters volume fraction of qrecipitated matexial wIthin the concretion volume fraction of precipitated material wlthln the host-rock flux of X (mol/cm2 .s) cement precipitation rate (mol/cm2 .s) surface reaction rate constant (mol/cm2.s) calcite solubllity constant bnol2/cmsl carbonate dissoctation reaction constants [mol2/cms) water dissociation constant (mol2/cms) mass of precipitated material within concretion (81 order of calcite preclpltation reaction psrtlal pressure of CO2 In porewaters prior to equilibration with arsgonlte. radial distance from concretion centie (cm) radius of locus of d&solution (cm) concretion radius (cm)
time k3)
porefluid flow velocity (cm.s- 1)
concretion
volume
(cm3)
concentration of species X (mol/cms) porosity supersaturation with respect to calcite intrinsic molar volume of source and cement (cms/mol)
3393
Growth of calcite concretions in sandstone the diffusion prohles adjust rapidly in comparison to concretion growth ( BOLJDREAU,pers. comm.; BERNER, 1980, p. 111) and produces three equations:
(34
The ionic fluxes at the concretion strained by two other equations: JG, = Jcs
= 0
(3b)
&.~-DOH.~-DHcol
Dca*[Ca] = Aca + JCa*R2/r Dtrco,.[HCO31+
2aACa + AH = AOH + AHC03 + 2*AC03.
(4a)
(4b)
x [HC03] - 2. Dco,.[CO3]
Aca = Dca.[Ca],
(sa)
+ D~co,*[HC031, (5b)
- D~co~*[HC031, - 2. Dee,-[C03],.
(5~)
[Xl, represents the concentration of species Xa large distance from the concretion, which is equal to the initial concentration of that species for all rat time t = 0. Substituting Eqns. (5a-c) into (4a-c), respectively, yields Eqns. (6a-c), expressed for r = R , where [X lR is the concentration of species X at the concretion surface: = DC..([Cal,
- [Ca]R) R
(ha)
Jco,
Dco,~([CO2lm - [COZIR) + D~co,([HC031, - [HCO,]R)
+ Dcos ‘(lCO31m - [CO,],) R
(6b)
JH-JOH-JHCO,-2’Jco, DH*([HI,
=
-
[H]R)
-
DoH*([OHL
- DHCO~-([HCO,~, - bHCo31~) - 2. Dco~.([CO31, R
K, = [H].[OH].
(11)
(13)
The final constraint is provided by equilibrium between the porefluids at the growing concretion surface (r = R) and the precipitating calcite: K, = [Ca]R’[Co31R.
+ Dco,~[CO31,
=
(10)
(12)
(4c)
J, denotes the flux of species X at the concretion surface, and A denotes an integration constant. By allowing r to tend to a, the integration constants may be assessed:
A, = DH*[H]~ - Do~‘lOH1.z
(9)
= A,
- (JH - JO* - JHCQ - 2. Jco,)*R2/r.
JCQ+ JHCO~+
(8)
Equations (9) and ( 10) include six unknowns (the ion concentrations at r = R) and are mathematically indeterminate. However, by applying the partial equilibrium approximation-i.e., by assuming thermodynamic equilibrium to be maintained between those species involved in reactions (A)(C) at all points within the system ( BOUDREAU, 1987, and refs. therein)-three more constraints are added: Kc = ]HCO31.[Hl I [CO21
DH*[HI - DoH*[OHI - DHCO,
Ca
(7)
Equation (7) is the steady-state condition for the precipitation of stoichiometric calcite, and Eqn. ( 8 ) is the charge balance equation applied to the concretion surface. Substituting Eqns. (6a) and (6b) into (7) and Eqns. (6a) and (6~) into Eqn. (8) yields the following, where AX represents Dx.( [Xl,
(3c)
Dco,~[COzl
= AC + (Jco, + JHCQ + JHICOJ).R21r
J
+ JH = JHa, + JOH + 2. Jco,.
ACa = AC03 + AHC03 + AC02 = 0.
Integrating Eqns. (3a-c) twice, and using the condition r = R at the concretion surface to eliminate the first integration constant, yields
AC = Dco2.lCO2lm
JCO~
- [XIR)/R:
x dlHCO31 _ 2. Dco dlCO31 3*- dr dr
Dco;[CO31+
+ JHCO,+
and 2*Ja
x dlHCO31 + Dco dlCO31 .dr ’ dr
surface are also con-
- [OHIR) - lCO31R)
. (6~)
(14)
These equations may be solved, if the inital porefluid composition is known, to yield the composition of the porefluid at the surface of the growing concretion. This composition is found to be constant; i.e., it is not a function of time or of concretion radius. From this, growth times may be calculated; for example, from Eqn. ( 15) ( BERNER, 1968) R2 t = 2.u.
Dc..([Ca],
- [Cal,)
(15)
where u is the intrinsic molar volume of calcite. Diagenetic Redistribution Interconversion
Incorporating Species
Diagenetic redistribution involves the dissolution of a source material, its transport, and its precipitation ( BERNER, 1980, p. 108). For the case of concretion growth, a simple redistribution model was proposed by BERNER ( 1980), in which only solute transport is considered as a rate-detertnining process. A spherically symmetric concretion (Fig. 1) grows at the expense of an (initially) evenly distributed source material. If growth is assumed to be transport controlled, then dissolution will take place along a sphere located concentrically about the concretion.
M. Wilkinson and M. D. Dampier
3394
Let r = rL to determine B, where CL is the concentration the diffusing species at r = rL:
R2-J~
c=
R2* JR q5. rL* D *
-+c,-q5.r.D
of
(22)
Rearranging, - C)
-c$b’D*(G
= JR. R.( l/r - l/rL).
(23)
in the thtx at the concretion
surface
R
We are interested (r = R): J
with carbonate
SOL@
FIG. 1. Diagramatic representation of the diagenetic redistribution model of concretion growth (aher BERNER,1980, p. 109). The sohrtion of BERNER (1980)is, however, flawed, as it is a good app~ximation onfy for a limiting case and not for concretion growth in general. Specifically, the assumption is made that the solute gradient around the concretion, from the locus of dissolution (r = rL, C = C,) to the concretion surface (r = R, C = CR), is linear. This leads to an expression for the flux at the concretion surface given by Eqn. (S-44) (BERNER,1980,p. 110), J
R
-4.
=
D*(CL - CR) (TL--R) *
R
(17)
‘ rL’(cL
-
.
d”p -
--
_4.=.Rs
-
di diw, =
r=. I) A=R2
(181
* *dc
(191
dr D
dR .
Using JR. = - 4 9D - dC/dR (Fick’s First law), where JR is the flux at r = R , and 4 is the porosity of the host-sediment, dC -= dr
-R2’ JR $*r2.D’
(20)
(24)
(25-l
R
(26)
UP
Combining Eqns. ( 25 ) and ( 26 ) , dR -JR. v, -=--dt Fp ’
(27)
Substitute in Eqn. (24) and let L = (rL - R), i.e., the radial dimension of the zone of depletion: dR dt=
#*D-up*rL(C,R-L-F,
C,) .
(28)
BERNER (1980)usesJR = JL; however, this is only a good approximation for a limiting case. We use the following mass balance equation to derive the correct relationship between the two fluxes: ri. JL;
(29)
i.e., Jr. = JR. R2/&
.dc=A
J
4,r*R2-Fp,dR
~*v~*R~*J~ =4-?r. Integrating once, and using r = R for a boundary condition, defines the integration constant (A):
CR)
R.(rL-R)
Equation (24) correctly expresses the flux at the concretion surface and is used in place of Rqn. (5-44) of WERNER ( 1980). The concretion grows by addition ofthe surface flux ( BERNER, 1980); therefore, if MP is the mass of the precipitate within the concretion, v, the intrinsic molar volume of the precipitate, and Fp the volume fraction of the concretion occupied by the precipitate, the growth of the concretion is given by
(16)
A steady-state gradient in a spherically symmetric system is not, in general, linear but can be determined as follows. Let the radius of the concretion equal R and the radius of the locus of dissolution equal rL. Using a steady-state approximation, we may write Eqn. ( 17) from Fick’s Second law for spherical symmetry, where C is the concentration of any species, D the diffusion coefficient, and R i; r s rLi:
=-4-D
(30)
Hence, using an analogue of Eqn. (27) for the locus of dissolution,
where ud is the intrinsic molar volume of the source material, and Fd is the volume fraction of source material within the host-rock. Subtracting Rqn. (28) from ( 3 1) yields an expression for dL/dt:
Integrating once more,
t-211
dL -= dt
9+D*fCL-C,) L
(32)
3395
Growth of calcite concretions in sandstone Integrating this, and substituting the integration ( =0) found by putting t = 0, yields Eqn. (33):
constant
(33)
L=2.qb.D.(CL-CR). Substitute this into Eqn. (28): dR
4*D*vp*rL*(CL-CR) R - F,,
dt=
4i\
v,,,r, --R-F, (( 112 . t-“2 .
1 ’ 2.4.
D.(CL - C,)
\
1:
(34)
0.0
0.2
0.4
Integrate, with boundary conditions R = 0, t = 0, R=4.D.vp.rL.(CL-CR) R-F,
vd,R --(( Fd’r_,.
112 * t”2.
’ s.D.(C,
- C,)
JH
R4. F; De rt.(C,
2.vi.b.
- C,)
Vd’R ---
Vp’rL
Fd.rL
R. Fp
).
=
Vd,
Rf rL = (Fd/Fp)‘13.
(38)
Substituting Eqn. (38) into Eqn. (36), and approximating Fp = q5and vd = v, = v, yields the desired result: t =
r2*(1 - (FdFp)“3)
(39)
*
2.v.D.(CL-CR)
Predicted growth times are hence reduced by a factor of ( 1 - (Fd/Fp)‘13) relative to those for a system with no solute replenishment during growth. Note that if Fd/F,, is zero, i.e., there is no source material, the equation transforms to that of BERNER ( 1968). Figure 2 illustrates the difference between growth times predicted using Eqn. (S-56) of BERNER ( 1980, p. 112) and our Eqn. (39), for an arbitrary system where concretion growth in the absence of solute replenishment would take 10 Ma. Diagenetic redistribution can be modelled in the calcitewater system, allowing for species interconversion, when Eqns. (6a-c) become Jca
DCa’([~lm -
=
ICal,)
(40a)
R.( 1 - (Fd/F,)“3) +
Jco,
Dco,~([CO21,
-
JH~CO~ + JHCO,
=
-
[C021R)
iHCo31~) R.(
+ 41co,.([HC031,
+ Dco,~([CO3lm 1-
-
JOH
-
[co,],)
(&/F,)“‘) (40b)
JHCO,
&‘([Hl, -
(37) V,
-
(36)
The relationship between R and rL may be determined by considering the mass balance between total source dissolved and total cement precipitated. This neglects the small quantity of solute undergoing transport at any time:
Hence, if
1.0
(35)
Squaring and inverting, t=
0.8
FIG. 2. A comparison of concretion growth times calculated from the diagenetic redistribution models of WERNER (1980) and this paper. Growth time for a concretion in the absence of solute replenishment is arbitrarily set to 10.
vp.rL -’ R-F, )
2
0.6 Fd/FP
-
-2.Jm, [HIR)
D~co,*([HCOslm -
= R.(
-
DoH*([OHL -
-
[OHIR)
-
ic031R)
[HC031~)
2~&o,~(tCO31m
1 - (F,j/FP)“3)
(4Oc) When Eqns. (40a-c) are substituted into Eqns. (7) and (8), Eqns. (9) and ( 10) are produced as before, and may be used with Eqns. ( 1l- 14) to calculate the porefluid composition at the concretion surface. This composition is independent of the effects of solute replenishment to the porefluid during growth. Concretion growth times due to diagenetic redistribution can hence be calculated from Eqn. (39) as applied to the calcium ion. Figure 3 shows predicted concretion growth times for various geologically reasonable values of Fd/F,. These are calculated assuming that the porewaters had a partial pressure of CO2 of 0.0 1 prior to closed-system equilibration with aragonite (see below for a discussion of the significance of this value). For Fd/F, = 0.1, a 1 m diameter concretion is predicted to grow in 5.1 Ma. Porefluid Flow Although porefluid flow rates within the subsurface are poorly constrained, it may be generally assumed that, in sandstones within the subsurface, the porefluids will not be stationary. The relative importance of diffusion and flow may be compared using the Peclet Number, Pe ( BERNER, 1980). RAISWELL (1988) used this to demonstrate that, within shales, diffusion is the dominant transport mechanism. For sandstones with externally impressed flow, flow will be the dominant transport mechanism over distances greater than porespace dimensions ( BERNER, 1980,pp. 117- 118 ) . The effects of porefluid flow on growth rates may be modelled using the factor F of NIELSON (1961))the ratio of the rate of change of concretion volume in flowing fluids to that in static fluids. For the case of concretion growth controlled solely by transport processes, fluid flow does not influence the solute concentrations at the growing concretion surface,
M. Wilkinson and M. D. Dampier
3396
5 s
$ s
Jp is derived from Eqn. (5) of WALTER ( 1986), which on substitution yields Eqn. (45):
80 t = R/v. 60 40 20 0
s
!
0
I
I
I
1
2
3
’
I
I
4
5
Time/Ma FIG. 3. Concretion growth times calculated for the diagenetic redistribution model. Curves for various values of Fd/F, are shown. Note that Fd/F, = 0 corresponds to the simple diffisional growth model. PCs = 0.0 1.The calculations for this hgure do not incorporate any degree of surface-reaction control.
and the calculated calcium concentrations may be substituted into Eqn. (41) (Eqn. 8 of BERNER, 1968) to derive growth times directly, where U is the porewater flow velocity: (R - Dca/(0.715~U))~(l t=
1.715-U.
+ R*U/Dca)0.7’5 + &./(0.715*U). u.([Ca], - [Cal,)
(41)
Equation (4 1) is only applicable to the simple diffusional model. Solute fluxes for the diagenetic redistribution model can be calculated using Eqn. (42), as, once again, the solute concentrations at the concretion surface are independent of porefluid flow velocity: [(R - &J(O.715~U))~( 1 + R.u/Dca)o.715 +DcJ(o.715~u)]~(l -(Fd/Fp)“3) t= 1.715-U. u*([Ca], - [Cal,)
’
(42)
The use of F in the above involves a degree of approximation, as F is a function of the solute distribution around the growing cement body and will differ in the calcite-water system from that found in the simple two ion system modelled by NIELSON (1961). SURFACE REACTION demonstrated that the rate of concretion growth within shales is determined by the rate of surface reaction. For concretions growing within sandstones the controlling process is less apparent. The relative importance of the two processes may be compared by calculating model growth times for both transport and surface reaction control. The slower of the processes will be rate determining. Model growth times assuming surface reaction to be the sole ratedetermining process may be calculated by fixing the porefluids at the concretion surface in equilibrium with the source mineral. The rate of change of concretion volume is then given by Eqn. (43): RAISWELL
($l*dV=)4
( 1988)
.?r. R=. v.c$. Jp. dt = 4-r.
R=.c$.dR.
(43)
Symbols are as above and as in Table 1. The growth relationship follows directly: t = R/(v.
Jp).
(44)
k.(Q
- 1)“.
(45)
Omega is the degree of supersaturation of the solution, k a rate constant, and n is the order of the precipitation reaction. A more rigorous approach would use the Plummer-WigleyParkhurst equation described by Reddy et al. ( 198 1) . Figure 4 plots growth times (for a 1 m diameter concretion, from a porefluid in equilibrium with aragonite) calculated from Eqn. (45). The supersaturation of the porefluid (B) is calculated using equilibrium constants for 25°C tabulated in Drever ( 1982 ) . Rate constants and reaction orders are taken from REDDY et al. ( 198 1) and MUCCI and MORSE ( 1983 ) (see Table 2). These have been experimentally determined for a range of solution compositions and show calcite precipitation rates to be highly sensitive to the magnesium-calcium ratio of the porefluid. Hence, model growth times are shown for three values of Mg/Ca, including Mg/Ca = 5, the value for seawater. Other ions have a similar, though lesser, effect upon precipitation rates (e.g., sulphate and phosphate; WALTER, 1986 ) and are neglected. Figure 4 also plots growth times calculated for a 1 m diameter concretion growing by diagenetic redistribution (as above, with Fd/F, = 0.1). The assumption is made that the porewaters have entered the system with no solute except for a variable quantity of carbon dioxide (plotted as the y axis). They equilibrate with aragonite before concretion growth commences, with the fluid composition calculated by the methods described by DREVER ( 1982) for a closed system. The value of diffusion coefficients are discussed below. From Fig. 4 it is apparent that, in magnesium-free porewaters, concretion growth rate will be controlled by the rate of solute diffusion, but in Mg-rich porewaters (e.g., those derived from seawater) it will be controlled by the rate of surface reaction. However, in porewaters with an intermediate Mg-Ca ratio, the two processes take place at similar rates,
100
2
Ibtg/ca=5
10 1
0.011 0.00
/
,
0.02
I
Diffusion
,
0.04 PCO,
only
,
0.06
. Mg/Ca=O , *
0.08
,
0.10
/atm
FIG. 4. Growth times for a I m diameter concretion growing in waters initially in equilibrium with aragonite vs. the partial pressure of carbon dioxide ( Pco2)in the porewatersprior to reacting with the aragonite. Curves are shown for growth controlled by diffusion only (Fd/Fp = 0.1) and for growth controlled by surface reaction only. In the later case, three curves indicate the influence of porefluid magnesium-calcium ratios (assumed constant duringgrowth) on predicted growth times.
Growth of calcite concretions in sandstone
3397
TABLE2 - Rateconstantsand ordersof reactbn for the caldte predpitatlon reaction at 25OC Rate
con&ant Order Remarks 2.60 CaHCCbsolution Reddy et al. 1661 2.71 seawater, Mg/Ca=l Mucd & Morse. 1963 3.24 seawater, Mg/Ca=5 Mucct &Morse. 1963
1.56e-11 2.76e-13 676e-15
and neither will dominate. The calculation of accurate concretion growth times hence requires that both processes be modelled. If the rates of surface reaction and diffusion are similar, the porefluids at the concretion surface will not be in equilibrium with either the source mineral or the cementing mineral (calcite). They will be undersaturated with respect to the source but supersaturated with respect to the precipitate. This degree of supersaturation must be assessed in order to calculate growth times. The poretluid composition at the surface of the concretion can be calculated numerically as for the dilfusive model, except that Eqn. (46) replaces the calcite equilibrium condition previously used: & = k*(Q - 1)“.
(46)
The degree of surface solute supersaturation is a function of concretion radius. This is illustrated by Fig. 5, which was calculated for conditions discussed below. The growth of small concretions is surface-reaction dominated; hence, the degree of surface supersaturation is high. As growth proceeds, so diffision slows relative to surface reaction, and the degree of supersaturation at the concretion surface falls. In Mg-free porewaters dithrsion becomes the dominant rate-determining process before the concretion reaches 1 cm in diameter, while in seawater, surface reaction will be rate determining throughout growth. The calcium flux around the concretion, for a given radius, can be determined from the differential form of Eqn. (46) (Fig. 6), and hence growth times may be calculated ( Fig. 7 ) . APPLICATION OF THE MODELS
,001
.Ol
.l
1
10
100
1000
Diameter/cm FIG. 6. Calculated calcium fluxes around a growing concretion for various poretluid magnesium-calcium ratios and conditions as per Fig. 5.
Jurassic Valtos Sandstone Formation of the Inner Hebrides, Scotland, UK. These are located within prograding, upwardscoarsening, mudstone-sandstone deltaic sediments which built into a lagoon of low or fluctuating salinity (HUDSON and HARRIS, 1979). The only common macrofossil is the aragonitioshelled bivalve Neomiodon , which by mass balance and isotopic modelling has been shown to contribute around 90% of the concretionary carbonate (WILKINSON 1989b). The sands around the concretions, which are only lightly cemented, are now devoid of fossils (HUDSONand ANDREW& 1987). The concretions have been convincingly demonstrated to be of burial diagenetic origin by HUDSON and ANDREWS ( 1987 ), and to have grown within porefluids of meteoric origin (see also, WILKINSON, 1989b), at temperatures of 2535°C (HUDSON and ANDREW& 1987, Fig. 4). The cements are low magnesium ( MgCOs = 1.2 mol% average; WILKINSON, 1989b), ferroan calcite, as now are the Neomiodon shells preserved within them (WILKINSON,1989b). The concretions can be classified into three geometrical types, of which only the spherically symmetric ones are easily modelled.
The concretion growth models presented within this paper were initially developed for calcite concretions within the 100
E
< !I
1.5 --
a
1.4-
25
1.3:
20
Ek 3 12*I 1.1 1.0 1 .OOl
80
$ e, 60 $ 5 40
4 .Ol
.l
1
10
100
1000
Diameter /cm FIG. 5. Degree of supersaturation with respect to calcite at the concretion surface (St) vs. concretion diameter for various solution compositions calculated assuming balanced diffusion and surface rcaction rates. Pcq = 0.01; F,/Fd = 0.1.
6
Time/Ma FIG. 7. predicted concretion growth times for balanced surface reaction and diffusion. Curves are shown for three porefluid magnesium-calcium ratios. Also shown, for comparison, is the predicted growth curve for the simple diffusion growth model, which lies close to that calculated for Mg-free porewaters. Conditions are as for Fig. 5.
3398
M. Wilkinson and M. D. Dampier
For the purpose of this paper it is assumed that meteoric porewaters enter the sandstone bearing nothing but carbon dioxide derived from either the atmosphere or the soil. Once within the sand the water attained equilibrium with the aragonite, by closed-system reaction, hence becoming supersaturated with respect to calcite. This supersaturation drives concretion growth. The concentration of carbon dioxide (in its various ionic forms) within the meteoric water has a marked influence upon predicted concretion growth times, as illustrated in Fig. 4. If the porewaters entering an aquifer are in equilibrium with the present-day atmosphere, Pco, will be 10-3.5 atmospheres. In passing through the soil profile, this value becomes enhanced by a factor of IO-100 (HOLLANDet al., 1986). As the COz content of the Jurassic atmosphere is unresolved. the best estimate of initial Pco, in the waters in which the Valtos Sandstone Formation concretions formed seems to be 10m2 atmospheres. This is the value used by DREVER (1982). The activity coefficients of all ions are taken as one, and a general diffusion coefficient of low5 cm* s-’ is adopted, except that the hydronium ion diffusion coefficient is taken to be 2.5 times this value ( BERNERand MORSE, 1974). Accurate values at infinite dilution in free solutions are given in LI and GREGORY ( 1974). For calcium, carbonate, and bicarbonate, these have a mean value of 9.8 cm2 s-’ and a standard deviation of 1.1 cm 2 s-’ . All the growth estimates presented in this paper are hence considered to contain at least a 10% possible error. Allowance should also be made for sediment tortuosity. However, this effect will be small, as host-sediment porosity was around 40% at the time of growth and is neglected. Using the simple diffusion-controlled growth model, which neglects solute replenishment and surface-reaction effects, a growth time of 9.5 Ma is predicted for a 1 m diameter concretion. Carbonate, bicarbonate, and calcium are predicted to diffise towards the concretion, while carbonic acid diffuses away. This is in accordance with the calcite-precipitating reactions, which consume bicarbonate and carbonate, and liberate carbonic acid (e.g., BERNERand MORSE, 1974 ). It can be shown ~tro~phi~ly that the aragonitic shell material within the Valtos Sandstone Formation was undergoing dissolution at the time of concretion growth ( WILKINSON, 1989b). Hence, the diagenetic redistribution model is more appropriate than the simple model used above. The concretions comprise approximately 40% cement, and the host-rock contained an estimated average of 4% aragonite prior to dissolution (WILKINSON, 1989b). FJF, is hence equal to 0.1, reducing predicted growth times by a factor of 0.54 relative to those of the simple diffusion model (Fig. 3). A I m diameter concretion will hence form in 5.1 Ma. However, the Mg/Ca ratio in the porefluids from which the Valtos Sandstone Formation concretions grew has been estimated, using published distribution coefficients, as between 0.2 and 1.0 (WILKINSON, I989b). Figure 1 indicates that, under these conditions, surface reaction and diffusion operate at comparable rates, and both must be modelled to predict realistic growth times. In the absence of published rate data for 0 -z Mg/Ca < 1, a ratio of 1.0 has been used. The best estimate of the time to grow a 1 m diameter concretion in stationary porewaters is 9.0 Ma (Fig. 7).
100 E 80 4 3 3 .60 8
40 20 0+------S
0
2
I
1
4
6
Time/Ma FIG. 8. Concretion growth times for static and flowingporewaters. Calculated for FJ F, = 0.1, an initiJ partial pressure of CO2 in the porefluids of 0.0 1,and a constant m~n~ium-c~cium ratio of I.
The effects of porewater flow upon growth time are shown in Fig. 8 for FdjF, = 0.1 and MgfCa = 1.0. If the average porewater velocity is 1 m a-‘, the predicted growth time for a 1 m diameter concretion is reducted to 5.7 Ma. These times are comparable with those estimated by MC BRIDE( 1988 ) . CONCLUSIONS
Concretion growth times are a function of the rates of solute supply, solute transport, and surface reaction. Of these, solute supply (aragonite dissolution) is the most rapid and has not been modelled. Solute transport can be modelled either as a diffusive process or as diffusion enhanced by porefluid flow. Although solute transport and surface reaction invariably act together, they can be modelled in isolation, and growth times can be calculated assuming either of these processes to be the sole mte~ete~ining process. The rate of concretion growth within shales has been shown to be determined by the rate of surface-reaction processes ( RAISWELL, 1987), while in sandstones either process may dominate depending upon the prevailing geological conditions. In Mg-free porewaters, diffusive transport will be the slower, and hence dominating, process. In porewaters with Mg/Ca = 5 (e.g., seawater), surface reaction will control growth rates due to the inhibiting effect of magnesium upon calcite precipitation. For porewaters of intermediate magnesium-calcium ratio, diffusive transport and surface reaction take place at comparable rates, and both processes must be modelled to produce realistic concretion growth time estimates. For geoiogicalty reasonable conditions, as calculated for the Valtos Sandstone Formation, Scotland, a 1 m diameter concretion will take 9.0 Ma to form. This growth time is reduced to 5.7 Ma if the porewaters are moving at I m a-‘. These times are comparable with those estimated by MCBRIDE (1988). Acknowledgments-One of the authors (MW) is in receipt of a research grant from Shell Exploration and Production UK, for which he is grateful. Discussions with B. P. Boudreau and J. D. Hudson contributed greatly to the content of the paper, as did the reviews of R. A. Bemer, A. Morse, and J. D. Rimstidt. M. J. Norry and R. Raisweh also contributed. Thanks to D. Whitehead for reading the rnanu~~pt and J. Lynch for assistance with diagram preparation. Edimrial hand&:
J i. Drever
Growth of calcite concretions in sandstone REFERENCES ASTIN T. R. and SCOTCHMANI. C. ( 1988) The diagenetic history of
some septarian concretions from the Kimmeridge Clay, England.
Sedimentology 35, 349-368. BERNERR. A. ( 1968) Rate of concretion growth. Geochim. Cos-
mochim. Acta 35471-483. BERNERR. A. ( 1980) Early Diagenesis, a Theoretical Approach. Princeton University Press. BERNERR. A. and MORSEJ. W. ( 1974) Dissolution kinetics of calcium carbonate in sea water. IV. Theory of calcite dissolution.
Amer. J. Sci. 274, 108-134. B0UDRnAUB. P. ( 1987) A steady state diigenetic model for dissolved carbonate species and pH in the porewaters of oxic and anoxic sediments. Geochim. Cosmochim. Acfa 51, 1985-1996. BOUDREAUB. P. and CANFIELDD. E. ( 1988) A movisional diaaenetie model for pH in anoxic porewaters: Appli&tion to the FOAM Site. J. Mar. Research 46,429-45X DREVERJ. 1.( 1982) The GeochemistryofNatural Waters. PrenticeHall. HOLLANDH. D., LAZARB., and MCCAFFREYM. ( 1986) The evolution of the atmosphere and oceans. Nature 320,27-33. HUDSON J. D. and ANDREWSJ. E. ( 1987) The diagenesis of the Great Estuarine Group, Middle Jurassic, Inner Hebrides, scottand In Diagenesis of Sedimentary Sequerzes (ed. J. D. MARSHALL); Geoi. Sot Spec. Pubf. 36, pp. 259-276. Blackwell Scientific Publications. JOHNSONM. R. ( 1989) P~eogeo~phic significance of oriented calcareous concretions in the Triassic Katberg Formation, South Africa. J. Sediment. Petrol. 59, 1008-1010. KANTOROWICZJ. D., BRYANTI. D., and DAWANSJ. M. ( 1987) Controls on the geometry and distribution of carbonate cements in Jurassic sandstones: Bridport Sands, southern England and Viking Group, Troll Field, Norway. In Diagenesis of Sedimentary
3399
Sequences (ed. J. D. MARSHALL);Geol. Sot. Spec. Publ. 36, pp. 103-l 18. Blackwell Scientific Publications. LI Y. and GREGORYS. ( 1974) Diffusion of ions in sea water and in deep sea sediments. Geochim. Cosmochim. Acta 38,703-7 14. MCBRIDEE. F. ( 1988) Contrasting diagenetic histories of concretions and host-reek, Lion Mountain Sandstone (Cambrian), Texas. Geol. Sot. Amer. 100, 1803-1810. MUCCIA. ( 1983) The soiubility of calcite and aragonite in seawater at various salinities, temperatures, and one atmosphere total pressure. Amer. J. Sci. 283,780-799. Muccr A. and MORSEJ. W. ( 1983) The in~~mtion of Mg2+ and Sr*+ into calcite ovetgrowthsz Infhtences of growth rate and solution composition. Geochim. ~osm~him. Acta 4?,217-233. NIELSONA. E. ( 1961) Diffusion controlled growths of a moving sphere. The kinetics of crystal growth in potassium perchlorate mecinitation. J. Phvs. Chem. 74. 309-320. PI~RIE-D. ( 1987) Or&ted c&are&s concretions from James Ross Island, Antarctica. British Antarctic Surv. Bull. 754 I-50. RAISWELLR. ( 1988) Evidence for surface reaction-controlled growth of carbonate concretions in shales. Sedimentology 35, 57 l-575. REDDYM. M., PLUMMERL. N. and BUSENBERGE. ( 1981) Crystal growth of calcite from CaHCOr solution at constant PC02 and 25 ‘C: A test of the calcite dissolution model. Geochim.Cosmochim. Acta45, 1281-1291. WALTERL. M. ( 1986) Relative efficiency of carbonate dissolution and precipitation during diagenesis: A progress report on the role of solution geochemistry. In Roles of Organic Matter in Sediment Diagenes~s(ed. D. L. GAUTIER); SEPb4 Spec. Publ. 38, pp. II 1. Society of Economic Paleontologists and Mineralogists. WILKINSONM. (1989a) Discussion: Evidence for surface reaction controled growth of carbonate concretions in shales. Sedimentology
36,951-953. WILKINSONM. ( 1989b) Sandstone-hosted concretionary cements of the Hebrides, Scotland. Ph.D. thesis, Leicester University, UK.
0016-7037/9O/S3.OU
Aeta Vol. 54, pp. 3391-3399 Press U.S.A.
+ .oO
pit.F’rinted in
The rate of growth of sandstone-hosted calcite concretions MARK WILKINSON”* and MICHAELD. DAMPIER* ‘cement of Geology, Leicester University, L.&ester, UK ‘Department of Mathematics, Leicester University, Leicester, UK (Received January 8, 1990; accepted in revisedform September 25, 1990)
Abstract-Concretion growth within sandstones can be modelled as a three-stage process, comprising solute supply, solute transport, and surface reaction. Solute supply is not thought to be rate dete~ining. R. A. Bemer has proposed models for solute transport, which can be adapted to the complex calcitewater system. His model for concretion growth by diagenetic redistribution, which incorporates the replenishment of solute within the porewaters during growth by the dissolution of skeletal aragonite, is reassessed and a new solution presented. The model is also expanded to include the effects of species interconversion during transport. The relative importance of solute transport and surface-reaction can be assessed by calculating model growth times assuming each process to act in the absence of the other. In magnesium-free porewaters growth is transport controlled, in seawater it is surface-reaction controlled. In porefluids with intermediate magnesiumcalcium ratios, both processes must be modelled if growth times are to be accurately predicted, with the two processes being rate balanced at the surface of the concretion. The models can be easily modified to include the effects of mobile porefluids. The models have been applied to the concretions of the Valtos Sandstone Formation of Skye, UK. A 1 m diameter concretion is predicted to grow in 9.0 Ma in stationary porefluids, or 5.7 Ma in porefluids flowing at 1 m a-‘. INTRODUCI’ION MANY SANDSTONESHOST near-spherical or ellipsoidal concretions. Inside the concretions the porespace is almost totally occluded by a calcite cement, while, in contrast, the intervening ho&rock is usually poorly cemented (e.g., HUDSON and ANDREW& 1987). Individual concretions rarely exc&d 1.5 m in diameter; however, coalescence can result in the formation of laterally extensive cement sheets, which, when located within petroleum reservoirs, can lead to production problems ( KANTOROWICZet al., 1987 )_ Calcite concretions within sedimentary rocks have been the focus of increasing attention over the past 20 years. The majority of the work has concentrated upon shale-hosted concretions (e.g., ASTIN and SCOTCHMAN,1988; RAISWELL, 1988), but some attention has been paid to the sandstonehosted equivalents (e.g., JOHNSON, 1989; MCBRIDE, 1988; PIRRIE, 1987). There are several aspects of the formation and distribution of sandstone-hosted concretions that are at present poorly understood. This paper addresses one of the key problems: the determination of the concretion growth rate and, hence, the time required to form a concretion of a given size. Although this paper is primarily concerned with sandstone-hosted concretions forming under specific conditions, much of the theory can be applied to concretions forming under different conditions within a variety of host rocks. Concretion growth can be considered to be a three-stage process. Initially, solute must be introduced into the porewaters. For the purposes of this paper, this process will either be neglected (i.e., all solute will be assumed to be in solution * Present address: Department of Earth Sciences, Liverpool University, Brownlow Street, Liverpool, L69 3BX, UK. 3391
at time = 0) or shown to be rapid compared to the other processes, and, consequently, to be unimportant in determining concretion growth times. All current concretion growth models (e.g., BERNER,1968, 1980) assume an initially uniform ~t~bution of source material, be it initially in solution or in the solid phase. The second process comprises solute transport. Either porefluid flow, or diffusion, can act to move solute from the site of supply to the site of precipitation. BERNER ( 1968, 1980) has proposed several models for the transport sta8e of concretion growth. Finally, the solute precipitates as cement onto the concretion surface. This involves the dehydration of the solute ions and construction of the calcite lattice; these two processes are commonly grouped together and referred to as surface reaction. CALCIUM CARBONATE SUPPLY The supply of calcium carbonate to a growing concretion may involve transport of molecular species into the rock unit hosting the concretion. This would be the case for concretions growing within diffusive distance of the sediment-water interface, for example, within the sulphate-reduction zone in muds ( WIL~NSON, 1989a). In sandstones, the majority of the calcium carbonate is derived by the internal r~s~bution of detrital shell material (WILKINSON 1989b). The models presented within this paper all start with the porewaters in equilibrium with amgonite, a geologically common carbonate mineral. It can be shown, using published dissolution rate constants (e.g., WALTER, 1986), that agitated solutions will reach equilibrium with aragonite in only a few days. Even though equilibration times may be as long as two months in more quiescent porewaters ( MUCCI, 1983), dissolution is orders of
M. Wilkinson and M. D. Dampier
3392
magnitude more rapid than the other potentially rate-determining steps involved in concretion growth which are considered below. For the remainder of this paper it is assumed that solute supply is not rate determining.
where [C] is the concentration of species C, and D, is appropriate diffusion coefficient (Table 1). Rc, and R-, the net rates of the carbonate interconversion reactions and B, respectively), and R, is the net rate of the water sociation/dissociation reaction (C):
the are (A as-
SOLUTE TRANSPORT The processes driving transport from the solute source to the growing concretion are diffusion and porefluid flow. BERNER( 1968) has proposed two models for solute diffusion within spherically symmetric systems. The former, here referred to as the simple diffusion model, assumes the existence, prior to the initiation of concretion growth, of a fluid uniformly supersaturated in the cementing mineral. No allowance is made for the addition of solute to the porefluids once concretion growth has commenced. BERNER and MORSE ( 1974) applied this model to the calcite-water system and correctly predicted calcite dissolution times (dissolution and precipitation are similar mathematically). However, their calculations assume that each species diffuses without undergoing chemical reaction. In the calcite-water system, where carbon dioxide is present as carbonate, bicarbonate, and carbonic acid, this need not be the case. A more complete treatment of diffusion, as described below, hence requires consideration of the carbonate interconversion reactions ( BOUDREAU,1987; BOUDREAUand CANFIELD, 1988 ). The second model for difisional solute transport ( BERNER, 1980, pp. 108- 112) involves the dissolution of a source mineral during concretion growth and is known as diagenetic redistribution. Bemer’s mathematical treatment of this model is shown to be incorrect, and a revised version is presented below. This model must also be modified to account for species interconversion during diffusion. BERNER( 1968, 1980) also described the effects of porefluid flow upon concretion growth times. This is incorporated into both the simple diffusional model and the diagenetic redistribution model in a subsequent section. Simple Diffusion Model Incorporating Species Interconversion
(la) cl
(lb)
6r
+Rc, -R,
(1~)
(Id)
(C)
The R terms in Eqs. ( la-f) are not known but can be eliminated by algebraic manipulation ( BOUDREAU,1987) to yield two equations.: b[COzl + ~4HCO31 -
6t
6t
+ -4CO31
at
6[HCO31 4CO31 + &CO, .p + Dco, 6r
6r
6[Hl ------
4OHl
6t
6t
6iHCO31
Pa) 1)
2 .-6iCO31
St
6t
_ DOH.-d’oH1 6r x 6IHCO31
----iiT-
- DHCO,
2 * ho,
(2b)
Equations ( la), (2a), and (2b) can be simplified by assuming that, at any given time, the concentration gradients around a growing concretion are constant, i.e., that the system is in a steady state. This is an acceptable approximation as TABLE 1 - Symbols used in the text
D F Fd
FP
JX JP k Kcc Kcl.Kc2 Kw
r rL R : V
[Xl 0 n vd.vp
(10
(B)
Hz0 = H+ + OH-.
MP n PC02
.p4HCO31 HCO3
(A)
HCO; = CO:- + H+
G
The initial equations below describe the diffusion of the six components of the calcite-water system. They are analogues of those listed by BOUDREAU( 1987), modified for a spherically symmetric system:
_ R
C02(aq) + Hz0 = HCO; + H+
concentration of solute r=x (mol/cms) dffision coefficient (cm2/s) ratio of rates of concretion volume change for flowtng and static porewaters volume fraction of qrecipitated matexial wIthin the concretion volume fraction of precipitated material wlthln the host-rock flux of X (mol/cm2 .s) cement precipitation rate (mol/cm2 .s) surface reaction rate constant (mol/cm2.s) calcite solubllity constant bnol2/cmsl carbonate dissoctation reaction constants [mol2/cms) water dissociation constant (mol2/cms) mass of precipitated material within concretion (81 order of calcite preclpltation reaction psrtlal pressure of CO2 In porewaters prior to equilibration with arsgonlte. radial distance from concretion centie (cm) radius of locus of d&solution (cm) concretion radius (cm)
time k3)
porefluid flow velocity (cm.s- 1)
concretion
volume
(cm3)
concentration of species X (mol/cms) porosity supersaturation with respect to calcite intrinsic molar volume of source and cement (cms/mol)
3393
Growth of calcite concretions in sandstone the diffusion prohles adjust rapidly in comparison to concretion growth ( BOLJDREAU,pers. comm.; BERNER, 1980, p. 111) and produces three equations:
(34
The ionic fluxes at the concretion strained by two other equations: JG, = Jcs
= 0
(3b)
&.~-DOH.~-DHcol
Dca*[Ca] = Aca + JCa*R2/r Dtrco,.[HCO31+
2aACa + AH = AOH + AHC03 + 2*AC03.
(4a)
(4b)
x [HC03] - 2. Dco,.[CO3]
Aca = Dca.[Ca],
(sa)
+ D~co,*[HC031, (5b)
- D~co~*[HC031, - 2. Dee,-[C03],.
(5~)
[Xl, represents the concentration of species Xa large distance from the concretion, which is equal to the initial concentration of that species for all rat time t = 0. Substituting Eqns. (5a-c) into (4a-c), respectively, yields Eqns. (6a-c), expressed for r = R , where [X lR is the concentration of species X at the concretion surface: = DC..([Cal,
- [Ca]R) R
(ha)
Jco,
Dco,~([CO2lm - [COZIR) + D~co,([HC031, - [HCO,]R)
+ Dcos ‘(lCO31m - [CO,],) R
(6b)
JH-JOH-JHCO,-2’Jco, DH*([HI,
=
-
[H]R)
-
DoH*([OHL
- DHCO~-([HCO,~, - bHCo31~) - 2. Dco~.([CO31, R
K, = [H].[OH].
(11)
(13)
The final constraint is provided by equilibrium between the porefluids at the growing concretion surface (r = R) and the precipitating calcite: K, = [Ca]R’[Co31R.
+ Dco,~[CO31,
=
(10)
(12)
(4c)
J, denotes the flux of species X at the concretion surface, and A denotes an integration constant. By allowing r to tend to a, the integration constants may be assessed:
A, = DH*[H]~ - Do~‘lOH1.z
(9)
= A,
- (JH - JO* - JHCQ - 2. Jco,)*R2/r.
JCQ+ JHCO~+
(8)
Equations (9) and ( 10) include six unknowns (the ion concentrations at r = R) and are mathematically indeterminate. However, by applying the partial equilibrium approximation-i.e., by assuming thermodynamic equilibrium to be maintained between those species involved in reactions (A)(C) at all points within the system ( BOUDREAU, 1987, and refs. therein)-three more constraints are added: Kc = ]HCO31.[Hl I [CO21
DH*[HI - DoH*[OHI - DHCO,
Ca
(7)
Equation (7) is the steady-state condition for the precipitation of stoichiometric calcite, and Eqn. ( 8 ) is the charge balance equation applied to the concretion surface. Substituting Eqns. (6a) and (6b) into (7) and Eqns. (6a) and (6~) into Eqn. (8) yields the following, where AX represents Dx.( [Xl,
(3c)
Dco,~[COzl
= AC + (Jco, + JHCQ + JHICOJ).R21r
J
+ JH = JHa, + JOH + 2. Jco,.
ACa = AC03 + AHC03 + AC02 = 0.
Integrating Eqns. (3a-c) twice, and using the condition r = R at the concretion surface to eliminate the first integration constant, yields
AC = Dco2.lCO2lm
JCO~
- [XIR)/R:
x dlHCO31 _ 2. Dco dlCO31 3*- dr dr
Dco;[CO31+
+ JHCO,+
and 2*Ja
x dlHCO31 + Dco dlCO31 .dr ’ dr
surface are also con-
- [OHIR) - lCO31R)
. (6~)
(14)
These equations may be solved, if the inital porefluid composition is known, to yield the composition of the porefluid at the surface of the growing concretion. This composition is found to be constant; i.e., it is not a function of time or of concretion radius. From this, growth times may be calculated; for example, from Eqn. ( 15) ( BERNER, 1968) R2 t = 2.u.
Dc..([Ca],
- [Cal,)
(15)
where u is the intrinsic molar volume of calcite. Diagenetic Redistribution Interconversion
Incorporating Species
Diagenetic redistribution involves the dissolution of a source material, its transport, and its precipitation ( BERNER, 1980, p. 108). For the case of concretion growth, a simple redistribution model was proposed by BERNER ( 1980), in which only solute transport is considered as a rate-detertnining process. A spherically symmetric concretion (Fig. 1) grows at the expense of an (initially) evenly distributed source material. If growth is assumed to be transport controlled, then dissolution will take place along a sphere located concentrically about the concretion.
M. Wilkinson and M. D. Dampier
3394
Let r = rL to determine B, where CL is the concentration the diffusing species at r = rL:
R2-J~
c=
R2* JR q5. rL* D *
-+c,-q5.r.D
of
(22)
Rearranging, - C)
-c$b’D*(G
= JR. R.( l/r - l/rL).
(23)
in the thtx at the concretion
surface
R
We are interested (r = R): J
with carbonate
SOL@
FIG. 1. Diagramatic representation of the diagenetic redistribution model of concretion growth (aher BERNER,1980, p. 109). The sohrtion of BERNER (1980)is, however, flawed, as it is a good app~ximation onfy for a limiting case and not for concretion growth in general. Specifically, the assumption is made that the solute gradient around the concretion, from the locus of dissolution (r = rL, C = C,) to the concretion surface (r = R, C = CR), is linear. This leads to an expression for the flux at the concretion surface given by Eqn. (S-44) (BERNER,1980,p. 110), J
R
-4.
=
D*(CL - CR) (TL--R) *
R
(17)
‘ rL’(cL
-
.
d”p -
--
_4.=.Rs
-
di diw, =
r=. I) A=R2
(181
* *dc
(191
dr D
dR .
Using JR. = - 4 9D - dC/dR (Fick’s First law), where JR is the flux at r = R , and 4 is the porosity of the host-sediment, dC -= dr
-R2’ JR $*r2.D’
(20)
(24)
(25-l
R
(26)
UP
Combining Eqns. ( 25 ) and ( 26 ) , dR -JR. v, -=--dt Fp ’
(27)
Substitute in Eqn. (24) and let L = (rL - R), i.e., the radial dimension of the zone of depletion: dR dt=
#*D-up*rL(C,R-L-F,
C,) .
(28)
BERNER (1980)usesJR = JL; however, this is only a good approximation for a limiting case. We use the following mass balance equation to derive the correct relationship between the two fluxes: ri. JL;
(29)
i.e., Jr. = JR. R2/&
.dc=A
J
4,r*R2-Fp,dR
~*v~*R~*J~ =4-?r. Integrating once, and using r = R for a boundary condition, defines the integration constant (A):
CR)
R.(rL-R)
Equation (24) correctly expresses the flux at the concretion surface and is used in place of Rqn. (5-44) of WERNER ( 1980). The concretion grows by addition ofthe surface flux ( BERNER, 1980); therefore, if MP is the mass of the precipitate within the concretion, v, the intrinsic molar volume of the precipitate, and Fp the volume fraction of the concretion occupied by the precipitate, the growth of the concretion is given by
(16)
A steady-state gradient in a spherically symmetric system is not, in general, linear but can be determined as follows. Let the radius of the concretion equal R and the radius of the locus of dissolution equal rL. Using a steady-state approximation, we may write Eqn. ( 17) from Fick’s Second law for spherical symmetry, where C is the concentration of any species, D the diffusion coefficient, and R i; r s rLi:
=-4-D
(30)
Hence, using an analogue of Eqn. (27) for the locus of dissolution,
where ud is the intrinsic molar volume of the source material, and Fd is the volume fraction of source material within the host-rock. Subtracting Rqn. (28) from ( 3 1) yields an expression for dL/dt:
Integrating once more,
t-211
dL -= dt
9+D*fCL-C,) L
(32)
3395
Growth of calcite concretions in sandstone Integrating this, and substituting the integration ( =0) found by putting t = 0, yields Eqn. (33):
constant
(33)
L=2.qb.D.(CL-CR). Substitute this into Eqn. (28): dR
4*D*vp*rL*(CL-CR) R - F,,
dt=
4i\
v,,,r, --R-F, (( 112 . t-“2 .
1 ’ 2.4.
D.(CL - C,)
\
1:
(34)
0.0
0.2
0.4
Integrate, with boundary conditions R = 0, t = 0, R=4.D.vp.rL.(CL-CR) R-F,
vd,R --(( Fd’r_,.
112 * t”2.
’ s.D.(C,
- C,)
JH
R4. F; De rt.(C,
2.vi.b.
- C,)
Vd’R ---
Vp’rL
Fd.rL
R. Fp
).
=
Vd,
Rf rL = (Fd/Fp)‘13.
(38)
Substituting Eqn. (38) into Eqn. (36), and approximating Fp = q5and vd = v, = v, yields the desired result: t =
r2*(1 - (FdFp)“3)
(39)
*
2.v.D.(CL-CR)
Predicted growth times are hence reduced by a factor of ( 1 - (Fd/Fp)‘13) relative to those for a system with no solute replenishment during growth. Note that if Fd/F,, is zero, i.e., there is no source material, the equation transforms to that of BERNER ( 1968). Figure 2 illustrates the difference between growth times predicted using Eqn. (S-56) of BERNER ( 1980, p. 112) and our Eqn. (39), for an arbitrary system where concretion growth in the absence of solute replenishment would take 10 Ma. Diagenetic redistribution can be modelled in the calcitewater system, allowing for species interconversion, when Eqns. (6a-c) become Jca
DCa’([~lm -
=
ICal,)
(40a)
R.( 1 - (Fd/F,)“3) +
Jco,
Dco,~([CO21,
-
JH~CO~ + JHCO,
=
-
[C021R)
iHCo31~) R.(
+ 41co,.([HC031,
+ Dco,~([CO3lm 1-
-
JOH
-
[co,],)
(&/F,)“‘) (40b)
JHCO,
&‘([Hl, -
(37) V,
-
(36)
The relationship between R and rL may be determined by considering the mass balance between total source dissolved and total cement precipitated. This neglects the small quantity of solute undergoing transport at any time:
Hence, if
1.0
(35)
Squaring and inverting, t=
0.8
FIG. 2. A comparison of concretion growth times calculated from the diagenetic redistribution models of WERNER (1980) and this paper. Growth time for a concretion in the absence of solute replenishment is arbitrarily set to 10.
vp.rL -’ R-F, )
2
0.6 Fd/FP
-
-2.Jm, [HIR)
D~co,*([HCOslm -
= R.(
-
DoH*([OHL -
-
[OHIR)
-
ic031R)
[HC031~)
2~&o,~(tCO31m
1 - (F,j/FP)“3)
(4Oc) When Eqns. (40a-c) are substituted into Eqns. (7) and (8), Eqns. (9) and ( 10) are produced as before, and may be used with Eqns. ( 1l- 14) to calculate the porefluid composition at the concretion surface. This composition is independent of the effects of solute replenishment to the porefluid during growth. Concretion growth times due to diagenetic redistribution can hence be calculated from Eqn. (39) as applied to the calcium ion. Figure 3 shows predicted concretion growth times for various geologically reasonable values of Fd/F,. These are calculated assuming that the porewaters had a partial pressure of CO2 of 0.0 1 prior to closed-system equilibration with aragonite (see below for a discussion of the significance of this value). For Fd/F, = 0.1, a 1 m diameter concretion is predicted to grow in 5.1 Ma. Porefluid Flow Although porefluid flow rates within the subsurface are poorly constrained, it may be generally assumed that, in sandstones within the subsurface, the porefluids will not be stationary. The relative importance of diffusion and flow may be compared using the Peclet Number, Pe ( BERNER, 1980). RAISWELL (1988) used this to demonstrate that, within shales, diffusion is the dominant transport mechanism. For sandstones with externally impressed flow, flow will be the dominant transport mechanism over distances greater than porespace dimensions ( BERNER, 1980,pp. 117- 118 ) . The effects of porefluid flow on growth rates may be modelled using the factor F of NIELSON (1961))the ratio of the rate of change of concretion volume in flowing fluids to that in static fluids. For the case of concretion growth controlled solely by transport processes, fluid flow does not influence the solute concentrations at the growing concretion surface,
M. Wilkinson and M. D. Dampier
3396
5 s
$ s
Jp is derived from Eqn. (5) of WALTER ( 1986), which on substitution yields Eqn. (45):
80 t = R/v. 60 40 20 0
s
!
0
I
I
I
1
2
3
’
I
I
4
5
Time/Ma FIG. 3. Concretion growth times calculated for the diagenetic redistribution model. Curves for various values of Fd/F, are shown. Note that Fd/F, = 0 corresponds to the simple diffisional growth model. PCs = 0.0 1.The calculations for this hgure do not incorporate any degree of surface-reaction control.
and the calculated calcium concentrations may be substituted into Eqn. (41) (Eqn. 8 of BERNER, 1968) to derive growth times directly, where U is the porewater flow velocity: (R - Dca/(0.715~U))~(l t=
1.715-U.
+ R*U/Dca)0.7’5 + &./(0.715*U). u.([Ca], - [Cal,)
(41)
Equation (4 1) is only applicable to the simple diffusional model. Solute fluxes for the diagenetic redistribution model can be calculated using Eqn. (42), as, once again, the solute concentrations at the concretion surface are independent of porefluid flow velocity: [(R - &J(O.715~U))~( 1 + R.u/Dca)o.715 +DcJ(o.715~u)]~(l -(Fd/Fp)“3) t= 1.715-U. u*([Ca], - [Cal,)
’
(42)
The use of F in the above involves a degree of approximation, as F is a function of the solute distribution around the growing cement body and will differ in the calcite-water system from that found in the simple two ion system modelled by NIELSON (1961). SURFACE REACTION demonstrated that the rate of concretion growth within shales is determined by the rate of surface reaction. For concretions growing within sandstones the controlling process is less apparent. The relative importance of the two processes may be compared by calculating model growth times for both transport and surface reaction control. The slower of the processes will be rate determining. Model growth times assuming surface reaction to be the sole ratedetermining process may be calculated by fixing the porefluids at the concretion surface in equilibrium with the source mineral. The rate of change of concretion volume is then given by Eqn. (43): RAISWELL
($l*dV=)4
( 1988)
.?r. R=. v.c$. Jp. dt = 4-r.
R=.c$.dR.
(43)
Symbols are as above and as in Table 1. The growth relationship follows directly: t = R/(v.
Jp).
(44)
k.(Q
- 1)“.
(45)
Omega is the degree of supersaturation of the solution, k a rate constant, and n is the order of the precipitation reaction. A more rigorous approach would use the Plummer-WigleyParkhurst equation described by Reddy et al. ( 198 1) . Figure 4 plots growth times (for a 1 m diameter concretion, from a porefluid in equilibrium with aragonite) calculated from Eqn. (45). The supersaturation of the porefluid (B) is calculated using equilibrium constants for 25°C tabulated in Drever ( 1982 ) . Rate constants and reaction orders are taken from REDDY et al. ( 198 1) and MUCCI and MORSE ( 1983 ) (see Table 2). These have been experimentally determined for a range of solution compositions and show calcite precipitation rates to be highly sensitive to the magnesium-calcium ratio of the porefluid. Hence, model growth times are shown for three values of Mg/Ca, including Mg/Ca = 5, the value for seawater. Other ions have a similar, though lesser, effect upon precipitation rates (e.g., sulphate and phosphate; WALTER, 1986 ) and are neglected. Figure 4 also plots growth times calculated for a 1 m diameter concretion growing by diagenetic redistribution (as above, with Fd/F, = 0.1). The assumption is made that the porewaters have entered the system with no solute except for a variable quantity of carbon dioxide (plotted as the y axis). They equilibrate with aragonite before concretion growth commences, with the fluid composition calculated by the methods described by DREVER ( 1982) for a closed system. The value of diffusion coefficients are discussed below. From Fig. 4 it is apparent that, in magnesium-free porewaters, concretion growth rate will be controlled by the rate of solute diffusion, but in Mg-rich porewaters (e.g., those derived from seawater) it will be controlled by the rate of surface reaction. However, in porewaters with an intermediate Mg-Ca ratio, the two processes take place at similar rates,
100
2
Ibtg/ca=5
10 1
0.011 0.00
/
,
0.02
I
Diffusion
,
0.04 PCO,
only
,
0.06
. Mg/Ca=O , *
0.08
,
0.10
/atm
FIG. 4. Growth times for a I m diameter concretion growing in waters initially in equilibrium with aragonite vs. the partial pressure of carbon dioxide ( Pco2)in the porewatersprior to reacting with the aragonite. Curves are shown for growth controlled by diffusion only (Fd/Fp = 0.1) and for growth controlled by surface reaction only. In the later case, three curves indicate the influence of porefluid magnesium-calcium ratios (assumed constant duringgrowth) on predicted growth times.
Growth of calcite concretions in sandstone
3397
TABLE2 - Rateconstantsand ordersof reactbn for the caldte predpitatlon reaction at 25OC Rate
con&ant Order Remarks 2.60 CaHCCbsolution Reddy et al. 1661 2.71 seawater, Mg/Ca=l Mucd & Morse. 1963 3.24 seawater, Mg/Ca=5 Mucct &Morse. 1963
1.56e-11 2.76e-13 676e-15
and neither will dominate. The calculation of accurate concretion growth times hence requires that both processes be modelled. If the rates of surface reaction and diffusion are similar, the porefluids at the concretion surface will not be in equilibrium with either the source mineral or the cementing mineral (calcite). They will be undersaturated with respect to the source but supersaturated with respect to the precipitate. This degree of supersaturation must be assessed in order to calculate growth times. The poretluid composition at the surface of the concretion can be calculated numerically as for the dilfusive model, except that Eqn. (46) replaces the calcite equilibrium condition previously used: & = k*(Q - 1)“.
(46)
The degree of surface solute supersaturation is a function of concretion radius. This is illustrated by Fig. 5, which was calculated for conditions discussed below. The growth of small concretions is surface-reaction dominated; hence, the degree of surface supersaturation is high. As growth proceeds, so diffision slows relative to surface reaction, and the degree of supersaturation at the concretion surface falls. In Mg-free porewaters dithrsion becomes the dominant rate-determining process before the concretion reaches 1 cm in diameter, while in seawater, surface reaction will be rate determining throughout growth. The calcium flux around the concretion, for a given radius, can be determined from the differential form of Eqn. (46) (Fig. 6), and hence growth times may be calculated ( Fig. 7 ) . APPLICATION OF THE MODELS
,001
.Ol
.l
1
10
100
1000
Diameter/cm FIG. 6. Calculated calcium fluxes around a growing concretion for various poretluid magnesium-calcium ratios and conditions as per Fig. 5.
Jurassic Valtos Sandstone Formation of the Inner Hebrides, Scotland, UK. These are located within prograding, upwardscoarsening, mudstone-sandstone deltaic sediments which built into a lagoon of low or fluctuating salinity (HUDSON and HARRIS, 1979). The only common macrofossil is the aragonitioshelled bivalve Neomiodon , which by mass balance and isotopic modelling has been shown to contribute around 90% of the concretionary carbonate (WILKINSON 1989b). The sands around the concretions, which are only lightly cemented, are now devoid of fossils (HUDSONand ANDREW& 1987). The concretions have been convincingly demonstrated to be of burial diagenetic origin by HUDSON and ANDREWS ( 1987 ), and to have grown within porefluids of meteoric origin (see also, WILKINSON, 1989b), at temperatures of 2535°C (HUDSON and ANDREW& 1987, Fig. 4). The cements are low magnesium ( MgCOs = 1.2 mol% average; WILKINSON, 1989b), ferroan calcite, as now are the Neomiodon shells preserved within them (WILKINSON,1989b). The concretions can be classified into three geometrical types, of which only the spherically symmetric ones are easily modelled.
The concretion growth models presented within this paper were initially developed for calcite concretions within the 100
E
< !I
1.5 --
a
1.4-
25
1.3:
20
Ek 3 12*I 1.1 1.0 1 .OOl
80
$ e, 60 $ 5 40
4 .Ol
.l
1
10
100
1000
Diameter /cm FIG. 5. Degree of supersaturation with respect to calcite at the concretion surface (St) vs. concretion diameter for various solution compositions calculated assuming balanced diffusion and surface rcaction rates. Pcq = 0.01; F,/Fd = 0.1.
6
Time/Ma FIG. 7. predicted concretion growth times for balanced surface reaction and diffusion. Curves are shown for three porefluid magnesium-calcium ratios. Also shown, for comparison, is the predicted growth curve for the simple diffusion growth model, which lies close to that calculated for Mg-free porewaters. Conditions are as for Fig. 5.
3398
M. Wilkinson and M. D. Dampier
For the purpose of this paper it is assumed that meteoric porewaters enter the sandstone bearing nothing but carbon dioxide derived from either the atmosphere or the soil. Once within the sand the water attained equilibrium with the aragonite, by closed-system reaction, hence becoming supersaturated with respect to calcite. This supersaturation drives concretion growth. The concentration of carbon dioxide (in its various ionic forms) within the meteoric water has a marked influence upon predicted concretion growth times, as illustrated in Fig. 4. If the porewaters entering an aquifer are in equilibrium with the present-day atmosphere, Pco, will be 10-3.5 atmospheres. In passing through the soil profile, this value becomes enhanced by a factor of IO-100 (HOLLANDet al., 1986). As the COz content of the Jurassic atmosphere is unresolved. the best estimate of initial Pco, in the waters in which the Valtos Sandstone Formation concretions formed seems to be 10m2 atmospheres. This is the value used by DREVER (1982). The activity coefficients of all ions are taken as one, and a general diffusion coefficient of low5 cm* s-’ is adopted, except that the hydronium ion diffusion coefficient is taken to be 2.5 times this value ( BERNERand MORSE, 1974). Accurate values at infinite dilution in free solutions are given in LI and GREGORY ( 1974). For calcium, carbonate, and bicarbonate, these have a mean value of 9.8 cm2 s-’ and a standard deviation of 1.1 cm 2 s-’ . All the growth estimates presented in this paper are hence considered to contain at least a 10% possible error. Allowance should also be made for sediment tortuosity. However, this effect will be small, as host-sediment porosity was around 40% at the time of growth and is neglected. Using the simple diffusion-controlled growth model, which neglects solute replenishment and surface-reaction effects, a growth time of 9.5 Ma is predicted for a 1 m diameter concretion. Carbonate, bicarbonate, and calcium are predicted to diffise towards the concretion, while carbonic acid diffuses away. This is in accordance with the calcite-precipitating reactions, which consume bicarbonate and carbonate, and liberate carbonic acid (e.g., BERNERand MORSE, 1974 ). It can be shown ~tro~phi~ly that the aragonitic shell material within the Valtos Sandstone Formation was undergoing dissolution at the time of concretion growth ( WILKINSON, 1989b). Hence, the diagenetic redistribution model is more appropriate than the simple model used above. The concretions comprise approximately 40% cement, and the host-rock contained an estimated average of 4% aragonite prior to dissolution (WILKINSON, 1989b). FJF, is hence equal to 0.1, reducing predicted growth times by a factor of 0.54 relative to those of the simple diffusion model (Fig. 3). A I m diameter concretion will hence form in 5.1 Ma. However, the Mg/Ca ratio in the porefluids from which the Valtos Sandstone Formation concretions grew has been estimated, using published distribution coefficients, as between 0.2 and 1.0 (WILKINSON, I989b). Figure 1 indicates that, under these conditions, surface reaction and diffusion operate at comparable rates, and both must be modelled to predict realistic growth times. In the absence of published rate data for 0 -z Mg/Ca < 1, a ratio of 1.0 has been used. The best estimate of the time to grow a 1 m diameter concretion in stationary porewaters is 9.0 Ma (Fig. 7).
100 E 80 4 3 3 .60 8
40 20 0+------S
0
2
I
1
4
6
Time/Ma FIG. 8. Concretion growth times for static and flowingporewaters. Calculated for FJ F, = 0.1, an initiJ partial pressure of CO2 in the porefluids of 0.0 1,and a constant m~n~ium-c~cium ratio of I.
The effects of porewater flow upon growth time are shown in Fig. 8 for FdjF, = 0.1 and MgfCa = 1.0. If the average porewater velocity is 1 m a-‘, the predicted growth time for a 1 m diameter concretion is reducted to 5.7 Ma. These times are comparable with those estimated by MC BRIDE( 1988 ) . CONCLUSIONS
Concretion growth times are a function of the rates of solute supply, solute transport, and surface reaction. Of these, solute supply (aragonite dissolution) is the most rapid and has not been modelled. Solute transport can be modelled either as a diffusive process or as diffusion enhanced by porefluid flow. Although solute transport and surface reaction invariably act together, they can be modelled in isolation, and growth times can be calculated assuming either of these processes to be the sole mte~ete~ining process. The rate of concretion growth within shales has been shown to be determined by the rate of surface-reaction processes ( RAISWELL, 1987), while in sandstones either process may dominate depending upon the prevailing geological conditions. In Mg-free porewaters, diffusive transport will be the slower, and hence dominating, process. In porewaters with Mg/Ca = 5 (e.g., seawater), surface reaction will control growth rates due to the inhibiting effect of magnesium upon calcite precipitation. For porewaters of intermediate magnesium-calcium ratio, diffusive transport and surface reaction take place at comparable rates, and both processes must be modelled to produce realistic concretion growth time estimates. For geoiogicalty reasonable conditions, as calculated for the Valtos Sandstone Formation, Scotland, a 1 m diameter concretion will take 9.0 Ma to form. This growth time is reduced to 5.7 Ma if the porewaters are moving at I m a-‘. These times are comparable with those estimated by MCBRIDE (1988). Acknowledgments-One of the authors (MW) is in receipt of a research grant from Shell Exploration and Production UK, for which he is grateful. Discussions with B. P. Boudreau and J. D. Hudson contributed greatly to the content of the paper, as did the reviews of R. A. Bemer, A. Morse, and J. D. Rimstidt. M. J. Norry and R. Raisweh also contributed. Thanks to D. Whitehead for reading the rnanu~~pt and J. Lynch for assistance with diagram preparation. Edimrial hand&:
J i. Drever
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some septarian concretions from the Kimmeridge Clay, England.
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mochim. Acta 35471-483. BERNERR. A. ( 1980) Early Diagenesis, a Theoretical Approach. Princeton University Press. BERNERR. A. and MORSEJ. W. ( 1974) Dissolution kinetics of calcium carbonate in sea water. IV. Theory of calcite dissolution.
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3399
Sequences (ed. J. D. MARSHALL);Geol. Sot. Spec. Publ. 36, pp. 103-l 18. Blackwell Scientific Publications. LI Y. and GREGORYS. ( 1974) Diffusion of ions in sea water and in deep sea sediments. Geochim. Cosmochim. Acta 38,703-7 14. MCBRIDEE. F. ( 1988) Contrasting diagenetic histories of concretions and host-reek, Lion Mountain Sandstone (Cambrian), Texas. Geol. Sot. Amer. 100, 1803-1810. MUCCIA. ( 1983) The soiubility of calcite and aragonite in seawater at various salinities, temperatures, and one atmosphere total pressure. Amer. J. Sci. 283,780-799. Muccr A. and MORSEJ. W. ( 1983) The in~~mtion of Mg2+ and Sr*+ into calcite ovetgrowthsz Infhtences of growth rate and solution composition. Geochim. ~osm~him. Acta 4?,217-233. NIELSONA. E. ( 1961) Diffusion controlled growths of a moving sphere. The kinetics of crystal growth in potassium perchlorate mecinitation. J. Phvs. Chem. 74. 309-320. PI~RIE-D. ( 1987) Or&ted c&are&s concretions from James Ross Island, Antarctica. British Antarctic Surv. Bull. 754 I-50. RAISWELLR. ( 1988) Evidence for surface reaction-controlled growth of carbonate concretions in shales. Sedimentology 35, 57 l-575. REDDYM. M., PLUMMERL. N. and BUSENBERGE. ( 1981) Crystal growth of calcite from CaHCOr solution at constant PC02 and 25 ‘C: A test of the calcite dissolution model. Geochim.Cosmochim. Acta45, 1281-1291. WALTERL. M. ( 1986) Relative efficiency of carbonate dissolution and precipitation during diagenesis: A progress report on the role of solution geochemistry. In Roles of Organic Matter in Sediment Diagenes~s(ed. D. L. GAUTIER); SEPb4 Spec. Publ. 38, pp. II 1. Society of Economic Paleontologists and Mineralogists. WILKINSONM. (1989a) Discussion: Evidence for surface reaction controled growth of carbonate concretions in shales. Sedimentology
36,951-953. WILKINSONM. ( 1989b) Sandstone-hosted concretionary cements of the Hebrides, Scotland. Ph.D. thesis, Leicester University, UK.