ELSEVIER Journal of Petroleum Science and Engineermg 17 (1997) 113-121

JOURHA[ OF

P- r- UmM $amc EHGmEElUHG

The chemistry of scale prediction

Gordon Atkinson a Miroslaw Mecik b

a Department of Chemzstry and Biochemist~'. UniverstO' of Oklahoma, Norman, OK 73019, USA b Department of Phystcal Chemtstry, Technical Unicersity of Gdansk, 80-952 Gdansk, Poland

Received 10 September 1995: accepted 9 April 1996

Abstract

The complete thermodynamics of "scale"-forming minerals is discussed in detail. The effects of temperature, pressure, and non-ideal solutions are analyzed. The special case of CaCO 3 "scale" formation is considered in detail.

Keywords: scale; minerals: sulfates; calcite

1. Introduct ion

" S c a l e " formation is the precipitation of a solid mineral from a brine. Although there are many in- dustrial processes where scaling can be a problem, we have focussed our attention on those scales most common in oil and gas production. In this paper we address the chemistry of the most common scales - -

CaSO4.2H2Oc, ), CaSO41,1, SrSO4(~), BaSO4c,/. and CaCO3~s). Other scales occasionally encountered are FeCO 3, Fe20 3, SiO 2 and CaF 2. Cowan and Wein- tritt (1976) discuss this in an encyclopedic fashion.

All petroleum reservoirs contain connate brines. These are ancient seawaters modif ied chemically by millennia of interaction with the petroleum, a gas phase (if any) and the reservoir rock -c l ay matrix. The diversity and analysis of such brines is ad- dressed by Collins (1975). The fundamental chem- istry of br ine-minera l interaction is described in basic texts such as Garrels and Christ (1965). In the reservoir the brine is in chemical equilibrium with its surroundings at their temperature and pressure. It may or may not be saturated with respect to any

given mineral. But, as the brine is produced the equilibrium is disturbed by going to a lower tempera- ture and pressure. In the case of CaCO3~, ) the equi- l ibrium can also be disturbed by a degassing of CO2~g I from the brine a n d / o r petroleum. This can lead to precipitation of scale ranging from a minor annoyance to massive clogging of production tubing and above ground equipment. Scale precipitation can also be a serious problem during water f lood opera- tions where one is trying to replace diminishing gas pressure with injected water pressure. Here the reser- voir brine may be chemically incompatible with the injected brine. This is a common occurrence in offshore work where seawater is the common injec- tion fluid. This can be particularly insidious since formation damage may not be noticeable until long after flooding has begun. If the scaling occurs deep in the formation it may be inaccessible to any rea- sonable treatment process. Finally, scale formation can occur by reinjecting produced water as a means of disposal and by brine - COz~g ) interactions during a CO2(g ) flood.

Problems with previous models included but were

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114 G. Atkinson, M. Mecik / Journal of Petroleum Scwnce and Engineermg I7 (1997) 113-121

not limited to the following defects: 1. Ignoring the effect of pressure on scale thermody-

namics, 2. Adopting inadequate models for non-ideal behav-

ior of these concentrated brines, 3. Ignoring the non-ideality of CO2tg ~, 4. Considering the individual scales in separate pro-

grams. This last defect is particularly harmful since it is

beconaing more and more clear that simultaneous and/or sequential precipitation of the sulfate scales is quite common. And the formation of (Ba,Sr)(SO4) 2 solid solutions may be the norm as discussed by Yuan (1989).

2. General method and results

Our approach is described below using BaSO~s ) as an example for the sulfates. We have examined and expanded the data base available for the sulfates in a very critical and detailed way. This is described in a number of publications from this group. For this information see Raju and Atkinson (1985, 1989, 1990), and Howell et al. (1992). The only really good solubility data for all the sulfate systems is solubility in water and aqueous NaC1.

For such systems:

2+ 2 - 2 2 B a S O 4 ( s ) ~ B a ( a q ) + S O 4 ( a q ) , K~p = S T + ( 1 )

We c a l c u l a t e Ksp from the known thermodynamics of the system, not from just the solubilities. We then use t h e Ksp at the appropriate temperature and pres- sure, together with the critically analyzed experimen-

tal solubilities, S, to calculate what amounts to an "experimental activity coefficient" y+. We, then, fit the 3' ± values to a semi-empirical model.

2.1. Temperature effect

We start with expressions of the standard form:

Cp, = A + B T + C / T 2 (2)

for the molar specific heat of each ion and each solid. Then, by standard thermodynamic integration techniques, we can obtain the free energy expression:

B T 2 C AG, ° = A T In T 2T 2T + Ih + IgT (3)

where I h and Ig are integration constants. A, B and C values for the solids are taken from the literature (Barin and Knacke, 1973, Barin et al., 1977) and those for the ions from Barner and Scheverman (1978). The I h and Ig values are evaluated with the NBS values for AG, ° (Parker et al., 1971) and Wag- man et al. (1982) and fine-tuned to reproduce the best 25°C K~p data. Then for an equilibrium such as BaSO 4 dissolution:

A A In T A B T Ac A I h Alg In Ksp - - - - + - - + - -

R 2 R 2 R T 2 R T R

(4)

where

A A = ABa2- + AsoJ - ABaso 4 (5)

and so forth.The equations for the sulfate minerals are summarized in Table 1.

Tab le 1

Ef fec t o f t empera tu re on the Ksp o f sulfates

Sol id A a A B A C A I h Alg (×10 -o)

BaSO4(s t 495 .534 - 1.9 1171 - 40.0731 - 200 ,488 3 ,740 12

S rSO4 , ) 641.541 - 1 .90146 - 42 .7605 - 251 ,748 4 ,102 24

CaSO4(s) 689.581 - 1 .094455 - 45 .0378 - 287 ,889 4 ,432 .90

C a S O 4 - 2H ~_O(s ) 763 .714 - 2 .04731 - 43 .2002 - 282 ,176 4 ,837.58

A A l n T ABT AC AI h AIg l n K = - - + - - + - -

R 2R 2RT 2 RT R

R = 8 .31435 J K -1 tool t. Opera t iona l range: 2 7 3 . 1 5 - 5 7 3 . 1 5 K.

G. Atkinson, M. Mecik / Journal o f Petroleum Science and Engineering 17 (1997) 113-121 115

2.2. Pressure effect

For a mineral dissolution such as:

2+ B a S O 4 ( s j ~ B a ( a q ) + SO2(~q) (6)

AV', the volume change, is always negative and A~', the compressibility change, is always negative. This is because the waters next to a divalent ion are under a high compressive field so their volume is de- creased and their compressibility is decreased com- pared to bulk water. Then thermodynamics allows us to derive the equation (to a very good approxima- tion):

K ( P ) AT/o . p A~,O . p 2

- - - - - + - ( 7 ) K(1) RT 2RT

where K(P), K(1)= Ksp at P bars pressure and 1 bar pressure, respectively; AV ° = (volume change) / (moles of salt dissolved); AK ° = (compressibility change)/(moles); and R=83 .15 cm -3 K 1 mol-l.Since both A ~ and AK ° are always negative the first term dominates at low pressures and is moderated by the second term at higher pressures. Note that the Ksp always increases with increased pressure.

The A~d°'s and AK°'s can be evaluated from the solution thermodynamics of the ions and the density

of the solid. In our calculational approach we correct Ksp to the fight temperature first and then correct to the fight pressure. This means that we need to know the Av°'s and A~'°'s at the appropriate temperature. This information is presented in Table 2.

2.3. Activity coefficients

Appropriate calculation of activity coefficients has presented a perennial problem to scale prediction. The most successful approach for brine systems has been that developed by K.S. Pitzer and coworkers and applied initially to brines by Harvie and Weare. The basic Pitzer equation for a system such as BaSO 4 in NaC1 is:

In y + = 4 f v + m M [2 BMS + 2 ECMs ]

+ mN[ BMC + BNS + ECMc + ECNs ] 2 t

+ mM[4BMs + 2CMs ]

+ mMmN[4B'Mc + 4B~s + 2CMc + 2CNs ] 2 t t

+ mN[aBNc + 2CNc ] + mN[SOMN + WSC ]

+ mN[20MN + Sqsc ]

+ m M m N [ ~JMNS -}- ~MSC ]

-~ T [ ~ J M N C 71- ~JNSC] ( 8 )

Table 2

Effect o f pressure on Ksp o f sulfates

a b c d

BaSO~s

AV ° - 5 7 . 1 1 2 6 . 4 3 . 10 - 2 - 1 7 . 9 8 - 10 - 4

A ~ , ~ ( × 1 0 3) 17.54 - 1 . 1 5 9 . 1 0 2 - - 1 7 . 7 7 . 10 4

SrSO4(s)

A~V ~ - 5 4 . 6 1 1 7 . 7 3 - 1 0 z - - 1 5 . 1 3 . 1 0 4

A~"~ ( × 10 - 3 ) 17.83 - - 1 . 1 5 9 . 10 2 - - 1 7 . 7 7 . 10 - 4

CaSO4ts ) AV ~ - 5 4 . 9 2 2 2 . 1 1 . 1 0 - 2 - 2 3 . 3 4 . 1 0 - 4

A ~ "° ( × 10 -3 ) 16.13 - -0 .944 • 10 - 2 -- 16 .52- 10 - 4

C a S O 4 • 2H201~ )

AY~ - 4 7 . 4 5 22.43 10 - 2 - 2 2 . 0 7 . 1 0 - 4

AR "° ( × 10 -3 ) 17.83 -- 1.543 • 1 0 - 2 - 16.01 - 1 0 - 4

- 3 . 7 2 2 - 10 0

17.06. . 10 6

- 3 . 7 2 2 . 1 0 ~

17.06 10 - 6

- 0 . 7 3 8 - 10 6

16.71 . 10 6

- 0 . 7 4 8 . 10 6

16 .84 . 10 - 6

ATe ~ = a + bt = ct 2 + dt 3

At~ ° ( × 10 -3 ) = a + bt + ct 2 + dt 3

AV ° units are c m 3 mol 1; A~,O units are cm 3 K t mol 1; t m degrees Celsius.

116 G. Atkmson, M Mecik / Journal of Petroleum Science and Engineering 17 (1997) 113-121

Tab le 3

T e m p e r a t u r e d e p e n d e n t P i t ze r c o e f f i c i e n t s

Salt Func t ion a b c d e

C a S O 4 /3(0) 0 . 20000 0 .002916 - 3 .872 X 10 - 6 0 0

/3 d ) 3 .19733 0 .001224 1.809 X 1 0 - 5 0 0

/3 (2) - - 54.24 0 .8283 - 0 .001782 0 0

CaC12 /3 co) 0 .3397 - 0 .1390 6 .286 x 1 0 - 5 5108 .0 46 .48

/3(1) 1.5050 - 0 . 0 1 6 4 7 2 .346 X 10 i 498 .24 0 .0

C + - 0 . 02679 0 .2462 - 1.184 x 1 0 - 4 - - 9676.3 - 84 .83

S r S O 4 13(°1 0 .20000 0 .002916 - 3 .872 X 1 0 - 6 0 0

/3 °~ 3 .1973 0 .00124 1.809 X 10 5 0 0

/3(2~ - 54.24 0 .8283 - 0 .001782 0 0

SrC12 /3 (°} 0 .2918 - 0 . 1 3 9 0 6 .826 x 10 5 5 ,108 .0 46 .48

13 (1; 1.5603 - 0 . 0 1 6 4 7 2 346 X 10 - 5 498 .24 0.0

C + - 0 . 00446 0 .2462 - 1 184 X 1 0 - 4 - 9 ,676.3 - 84.83

B a S O 4 /3 (m 0 .20 0 .002916 - 3 . 8 7 2 X 10 6 0 0

13 (1) 3 .1973 0 .00124 1.809 X 10 5 0 0

/3 t2~ - 5 4 . 2 4 0 .8283 - 0 . 0 0 1 7 8 2 0 0

BaC12 /3(0~ 0 .2628 - 0 . 1 3 9 0 6 .826 X 10 5 5 ,108 .0 46 .48

/3 d~ 1.4963 - 0 . 0 1 6 4 7 2 .346 × 10 5 - 4 9 8 . 2 0 0

C + - 0 . 0 1 9 3 8 0 .2462 - 1 184 × 1 0 - 4 - 9 . 6 7 6 . 0 - 8 4 . 8 3

The coefficients are obtained from experimental ac- tivity data on single electrolytes. This creates real problems because of the low solubility of the sulfate

scales. In addition, one must consider the "cross" electrolytes. That is, for BaSO 4 in NaC1 we also need the coefficients for BaCI~ and Na~SO 4. The

Tab le 4

M i x i n g cor rec t ion p a r a m e t e r s

P a r a m e t e r A n h y d r i t e G y p s u m S r S O 4 B a S O 4

B 1 - 3 . 7 3 6 6 4 . 1 0 - 3 - 82 .6504 5 . 8 5 1 2 0 . 1 0 - 4

B 2 4 .02891 . 1 0 - 3 109 .860 - 5 .04085 • 1 0 - 4

B 3 - 1 . 2 7 5 8 9 . 1 0 3 - 3 6 . 2 7 0 0 1 05712 • 10 4

B 4 31 .4958 0 .548561 - 5 .43328 • 10 2

B 5 - 33 .5633 - 0 . 7 4 1 5 5 4 4 .72342 ' 10 2

B 6 10.6681 0 . 2 5 6 6 3 8 0 - 1 .00413 10 - 2

B 7 - 0 . 0 8 7 0 3 6 9 - 9 . 3 0 8 2 2 " 10 - 4 1 .67755

B 8 0 .0913125 1 . 2 5 7 6 9 . 10 3 - 1 .47354

B9 - 0 .0290883 - 4 . 3 8 3 2 6 . 1 0 - 4 0 .318368

Blo 7 . 8 9 5 8 1 . 10 5 0 0 - 1 . 7 2 1 5 5 . 10 - 3

B u - 8 . 1 1 5 0 0 . 1 0 5 0 .0 1 . 5 2 9 4 2 . 1 0 3

B12 2 .59349 • 1 0 - 5 0 0 - 3 . 3 5 8 7 4 . 1 0 - 4

N u m b e r o f da ta po in t s 93 50 106

T r a n g e 298 573 2 7 3 - 3 8 3 2 9 8 - 3 7 3

I r a n g e 0 . 1 - 4 . 6 0 . 1 - 5 . 2 0 . 1 - 4 . 3

- 2 . 5 6 1 1 7 0 . 10 5

2 . 9 4 6 6 8 . 10 5

- - 8 . 6 8 8 6 2 . 10 - 4

1 . 6 6 6 3 3 . 1 0 - 3

- - 1 . 8 9 5 8 8 - 10 - 3

5 .61661 • 10 - 2

3 .55903

4 .02156

- 1.19878

2 . 5 0 5 3 2 - 10 3

- 2 . 8 1 7 8 2 . 10 3

8 4 8 4 2 4 • 10 - 4

108

298 523

0 . 1 - 4 . 0

G. Atkinson. M. Mecik / Journal of Petroleum Science and Engmeering 17 (1997) 113-121 117

sources for all the coefficients are given in Atkinson et al. (1991). Because of a lack of data some sets must be modeled.

Each Pitzer coefficient's temperature variation can be represented by equations of the form:

~(" = a + b ( T - TR) - c(T 2 - TR)

1 T

All these parameters are summarized in Table 3. Unfortunately, we do not know the Pitzer mixing

terms as a function of temperature. So we have taken our "experimental" activity coefficients, the Pitzer formalism, and an empirical extension to solve this problem:

In y += n r; + raM[ B 1 + B 1/2 + B 3 / ) ]

+ + (B4 + B511/2 + B61)T

+ (B7 + BsI 1/2 + B91)T 2

q - ( B l o q - B , l I 1 / 2 q - B I 2 I ) T 3 ( 1 0 )

This horrendous collection of parameters was fitted by NLLSQ means and is only justified by goodness of fit and the large number of experimental points available. The parameters are given in Table 4 with a brief description of the data base and its limits.

In addition to the above one can make a correc- tion for the effect of pressure on activity coefficients. However, the only system where this effect is known well is NaC1-H20. We have examined this and found that the pressure correction only becomes substantial ( ~ 10%) at high NaC1 concentrations and high pressure ( ~ 100 atm).

Table 5 The MgSO 4 effect

2+ ~ [MgSO~] Mg(aq) + SO~q; ~-~

log K h = 158.40-62.1601og T - 4810.6

T + 0.046298T

Range: 0-200°C (T in K)

K(A P I A~ "~ X P A~.~ × p e I n - - F - -

KA(1 ) RT 2RT

Arff ° = 1 0 . 1 7 - 7 . 8 1 × 1 0 - 2 ( T - 2 7 3 . 1 5 ) + 6 . 8 8 X 1 0 4 ( T - 273.15) 2 +0.54X 10 6(T-273.15) 3

AR'°X 10 -3 = --8.271 +0.136x 10- 2(T -273 15)+8.468 X 10-4(T -273.15) 2 -- 8.706X 10 -6 X(T -273.15) 3

AV ° in cm3/mol; AR "° in cm 3 mol- 1 bar- 1; T m K: P in bars: R = 83.15 cm 3 bar K 1 tool- ~; range. 0-150°C

the effect of pressure on the equilibrium reaction. Increasing pressure pushes the equilibrium back to the left.

Table 5 summarizes the temperature and pressure effects. The temperature effect is based on the work of Marshall (1967). The pressure effect is based on the work of Millero and Masterton (1974). The activity coefficients used are based on the work of Wu et al. (1969) on MgSO 4 in NaC1 solutions. Since this work was only at 25°C, we used the temperature dependence of the CaSO4-NaCI system described above.

2.5. CaIculational strategy for sulfates

2.4. The magnesium effect

MgSOa~s) is usually not discussed in scale predic- tion because of its high solubility. This can create serious errors in brines where the Mg 2+ concentra- tion is high. Mg 2+ forms ion pairs with the SO42- ion:

2+ ~- [MgSO4 °] (11) Mglaq} + SO~(aq )

Any SO42- tied up in an ion pair is not available to form a precipitate with another bivalent ion. This process is endothermic and, so becomes more impor- tant at elevated temperature. One must also consider

In our program one must choose between anhy- drite and gypsum since the thermodynamic transition temperature between the two seems to be routinely violated in real life. As an example, let us assume that anhydrite has been selected. The following equa- tions are then established (ignoring charges):

[Ba][SO4]TB = KB [BaSO4~s) ]

[SF] [SO4 ] . f2 = K c [SrSO4ts)]

[Ca] [SO 4 ]yA 2 = K A [CaSO4(,, ]

Kp[Mg][SO4]Yp 2 = [MgSO4 °]

(11a)

118 G. Atkinson, M. Meclk / Journal of Petroleum Sctence and Engmeering 17 (1997) 113-121

The appropriate K ' s for the input temperature and pressure are then calculated. The appropriate activity coefficients for the input temperature, pressure, and brine composition are calculated. The mass balance equations [Mg]x, [Ca] T, [Sr] v, [Ba] T and [SOa] v are set up from the input analytical concentrations. The program then solves the nine equations simultane- ously and calculates the amount of each solid formed and reports it. For those of a mathematical bent the nine simultaneous equations boil down to the solving of a cubic equation. And, in fact, only one of the three possible roots makes any physical sense.

2.6. The CaCO 3 problem

The CaCO3(s) equilibrium problem needs to be approached in a different way than the sulfate prob- lem for two reasons. First, H2CO3(aq ) is a far weaker acid t h a n H2SO4(aq ). The [CO 2- ] is very low under

the pH conditions common to the oil field. Second, the gas phase must be considered explicitly if a gas phase is present. In the presence of a gas phase the equilibrium is most usefully written as:

2+ CaCO3{s) + CO2(g ) q- H20 ~ Ca(aq) + 2 H C O ~ a q )

(12)

and in the absence of a gas phase as

e+ CaCO3{s) + CO2(aq ) + H 2 0 ~- Ca t , q ) + 2 H C O ~ , q )

(13)

the temperature dependence of these equilibria has been given previously in Atkinson and Mecik (1994), and is repeated in Table 6 together with the equilib- rium that connects reactions (12) and (13).

The main complication here is the occurrence of CO~(g) in reaction (12). The molar volume of CO2(~) varies greatly with both temperature and pressure.

Table 6 Temperature-dependent constants for In K

In K A A I n T ABT AC AI h Alg - - + - - + - -

R 2R 2RT 2 RT R

2+ CaCO3(~) + H20(l ) + CO2(g ) ~ Ca(aq~ + 2HCO31aq ), K 1 (1')

Solid A A A B AC A I h A lg (× 10 0)

Calcite - 239,623 0.18866 9.0767 83,810.8 - 1,540.62

Range: 0 to 300°C. Source: previous papers.

2+ +2HCO3{aq)" K~ (5') CaCO3(sl + HzOI1 ~ + CO2{aq ) ~- Cafaql

Solid A A A B A C A I h ~ lg ( X l 0 -6 )

Calcite 282.476 - 0.7958 - 14.5318 - 102,360 0 1,772.44

Range: 0 to 200°C. Source: previous papers.

CO2(g ~ ~ CO2(aq),K H

AA ~B AC ~lh Alg (× 10 -6)

-80 .384 0.18166 6.1255 31,661.2 -495 .94

Range: 0-250°C, temperature in K. Source: Plummer and Busenberg (1982)

G. Atkinson. M. Mecik / Journal of Petroleum Sctence and En gineermg 17 (1997) 113-121 119

"'CO2(aq)" in reaction (13) is really our short-hand notation for the sum of dissolved CO, in the unreac- tive form and that which has reacted to form HzCO 3. We use the data of Read (1975) to evaluate the partial molar volume of and partial molal compress- ibility of "CO2taq ) ' ' . The values for the ions, water, and CaCO3(~) were evaluated as in the sulfate case. The results are summarized in Table 7 for the case where no gas phase is present.

The calculation of the effect of pressure in the presence of the gas phase is quite complicated since ArC ~ must be written, according to reaction (12), as:

Ag"° = ~Ca g+ q- 2~HCO; -- ~CO.,~, -- ~HeO,,, -- ~C.CO:(~,

(14)

and AWco,,g ' varies enormously with temperature and pressure. All the partial molar volumes, except 2XWco,,g ' were described as cubic polynomials, while C O 2 ~ was treated in a different way. Thus we rewrite Eq. (14) as:

A~[/"O = A I ~ , 1 - - A~co21g, (15)

where Ar¢~,I denotes the sum of partial molal vol- umes for liquids and solids in reaction (12) which can be calculated as:

AV,'~,I = A~c,.,+ + 2 ~ c o , - V H R O , I ) - A ~ c a c o ....

(16)

On the other hand, the CO21g ) molar volumes cannot be described well using simple polynomial formulas.

Table 7

Effect of pressure on C a C O 3 dissolution. " n o gas p h a s e "

2+ + 2 H C O ~ a q l , K~ (5") CaCO3(s) +H20(1 ) +CO2(aq) ~ Cafaql

In = ( P - l ) + 2RT ( - 1 ) (20 ' )

AV ° = - - 6 1 . 9 0 + 2 0 . 2 3 1 • 10 et - 2 4 4 5 . 1 0 4t2 - - 0 . 6 0 3 . 1 0 - 6 t 3

(umts o f Awl/° are cm 3 m u l - 1, t in °C)

A h "~ × 1 0 3 = 1 3 . 5 1 7 + 3 2 2 2 . 1 0 2 t - 1 5 . 1 3 8 . 1 0 4 t 2 +

1 2 . 9 1 9 . 1 0 - 6 t 3

(units o f Zk~ "° are cm 3 b a r - i t o o l - i, t m °C)

Range: temperature, 0 - 2 2 5 ° C ; pressure, 1 a tm to 2000 bar

Table 8

Molar volumes o f C O , g : as a funct ion o f temperature and pres-

sure

Coeff icient P > Pd P < Pd

for Eq. (24)

B l 12,330.9 - 1,140.04

B 2 - 128.17 7.0075

B 3 0 .43099 - 0 015736

B 4 - 0 . 0 0 0 4 6 8 4 8 1 .2388 .10 5

B 5 - 38.995 - 23.989

B 6 0.44731 0 .16158 B 7 - 0 . 0 0 1 6 1 2 1 - 3 . 6 0 8 4 . 1 0 4

B 8 1 .8501-10 6 2 . 6 7 6 9 - 1 0 7

B 9 - 50.00 82.418

BI0 47 .806 82.428 B H - 0 . 1 2 2 4 5 2 . 0 5 6 8 . 1 0 3

B12 8 .00"10 - 7 - 1 .9861 .10 - 6

where Pa = 3 2 . 9 4 + 1.25t, P is in bars and t m °C. Range:

temperature, 2 5 - 1 5 0 ° C ; pressure, 0 - 2 0 0 bar

In this case we interpolated the existing literature data using the following equation:

~ C O 2 ( g , = ( B 1 + B 2 t + B 3 t 2 + B 4 t 3 )

+(B5 + B6t + By t2 + Bst3)p

+(Bg+Btot+BI,t2+B:2t3)/P (17)

At present we have reinterpolated the CO 2 molar volumes reported by Angus et al. (1976) in a wider range of pressures. To get more accurate results, we divided the temperature-pressure range by the straight line Pa = 32.94 + 1.25t ( P is in bars, t in °C). The resulting two sets of coefficients (below or above the dividing line) are presented in Table 8. It is important to point out that the existing literature data used in this interpolation cover the range of both temperatures and pressures. That allows us to get the CO, molar volume at each temperature and pressure immediately, so, in this case the compress- ibilities are no longer necessary. Thus, the compress- ibility change of reaction (13) can be calculated as A/~,: (for solids and liquids only):

Ah'~,l = ~ c _,++ 2 ~ c o ; - ~ o , , - ~caco: (18)

and the compressibility term in the pressure correc- tion of the solubility product contains the solids and

120 G. Atkinson, M. Mecik / Journal of Petroleum Science and Engineering 17 (1997) 113-121

Table 9

Effect of pressure on CaCO 3 dissolution, "'gas phase"

2+ C a C O 3 ( s I + H 2 O o ) + C O a l g ~ ~-~ Ca(aq)+2HCO31aq ), K 1 (1")

[ ll~l ] RT 2RT

AV" = - 2 6 . 6 9 + 0.146365t - 15.7085- 10 4 t - 1 .0566 .10-6 t3

(units of A.K "~ are cm 3 m o l - 1, t in °C)

A~,~ × 1 0 3 = 11 .883+2 .853 .10 2 t - 8 . 5 5 3 1 . 1 0 - 4 t 2 + 5 . 3 1 4 5 . 1 0 - 6t3

(units of A ~ "° are cm 3 bar - 2 mol J, t in °C)

Range: temperature, 0-150°C: pressure, 1 atm to 200 bar,

Wco ~,, from Table 8

liquids only. The final new formula used in our present calculations of the pressure effect in the "gas phase" approach is:

( P - 1)

AKL, + RT ( p 2 _ 1) (19)

while the numerical data for calculating A~, 1 and A/~,~ are presented in Table 9.

The equilibrium constant for reaction (6) can be written as:

[ C a 2+ ] [ H C O ; ]2 K 1 = X y3+ (20)

aH20 fCOziaq )

fco: is the fugacity of CO z. The values of the equilibrium constant, K 1, have been derived from the known thermodynamic properties of the species involved as described in our previous papers. For reaction (13), the equilibrium expression should be rewritten as:

[Ca2+][HCO;] 2 K, = × "/3_+ (21)

a H 2 0 aco2c~q)

where aco 2 is the activity of dissolved CO 2. This can be obtained from the equilibrium:

CO2{g ) ~ CO2(~q ) (22)

which is given by the Henry's law constant, K H.

aco 2 = KHfco 2 (23)

The fugacities of gaseous CO2 are calculated from the data presented by Majumdar and Roy (1956). The activity of water is calculated from the work of Silvester and Pitzer (1977) and Rogers and Pitzer (1982) using standard thermodynamic techniques. These data are all on aqueous NaC1 solutions which is consistent with the bulk of the solubility data on

C a C O 3 ( s / .

Finally, we need an expression for the activity coefficients of the C a ( H C O 3 ) 2. W e have taken a semi-empirical approach by calculating 7 3 (exp.) from:

Klaxa°°fc°" (24) y ± (exp.) = 4m 3

Here K 1, an2 o, fco2 are calculated as described above and m is the measured solubility of CaCO3(s~ in aqueous NaC1.

These solubility data are a very extensive set of data measured in this laboratory and presented at a symposium (1980). The data cover the ranges:

Temperature: CO 2 pressure: NaC1 concentrations:

25-150°C 0-200 bar 0-5 m (mol /kg H20)

Then we fit the 3' + (exp.) in the following fash- ion:

r = In y _+ (exp.) - In TDH (25)

Table 10

Act iwty coefficient parameter for Eq. (20)

Parameter " 'Gas phase . . . . No gas phase"

B 0 0.5865 + 0.0070 0.1595 + 0.0043

B 1 - 0.923 4- 0.018 - 0.7384 + 0.00112

B 2 0.909 + 0.012 0.8105 + 0.0076 B 3 - 0.0469 ___ 0.0012 - 0.04207 + 0.00071

B 4 4.742 _ 0.096 3.840 4- 0.059

B 5 - 4 . 1 2 3 +0 .090 - 3 .412+0.055 B 6 0.930 4- 0.022 0.776 4- 0.014 B 7 (1.0184-0.011). 10 -5 (7 3844-0.068). 10 -6

B s 0.0750 ___ 0.0011 0.10055 4- 0.00066

B 9 - 0.01318 4- 0.00035 - 0.00923 4- 0.00022

G. Atkmson. M. Mecik / Journal of Petroleum Science and En gmee ring 17 (I 997) 113-121 121

where TI~H is the standard form according to Pitzer:

f I 2 7=-] lnyon : 1 + 1 .2 f i + 1.2 ln(1 + 1.2VI ) (26)

A + is the Debye-Hiickel-Pitzer limiting slope. Y is then fitted to a polynomial of the form:

Y= B o + B~I 1/2 + B21 + B3I 2

T +( B41+ BsI3/~- + B612)In( ~ )

+ B v ( T 2 - T ~ ) + ( B 8 + B g l ) In P

T h e c o e f f i c i e n t s are g i v e n in T a b l e 10.

F o r any g i v e n b r i n e the i n p u t da t a r e q u i r e d are:

- - t e m p e r a t u r e

- - p r e s s u r e o f C O 2 (gas p h a s e p r e s e n t )

- - to ta l p r e s s u r e

- - C a 2 + c o n c e n t r a t i o n

- - H C O 3 c o n c e n t r a t i o n

- - concentration of other ions - - presence or absence of a gas phase - - pH Actually, the programs forces one to choose be-

tween CO 2 partial pressure, CO2caq , and pH as a control of the carbonic acid system. The program then calculates the remaining variables and the CaCO31s) if any.

It should be clear the sulfate program and the CaCO3~) operate independently. Therefore, a system that could precipitate both CaCO 3 and CaSO 4 would give overly pessimistic predictions.

A c k n o w l e d g e m e n t s

The authors acknowledge the support of the com- panies during this research:

Phase I: Akzo Chemie U.K., Aramco. ARCO, Conoco Exxon Chemical, Getty, Mobil, Oil Plus, Phillips, Shell, Sohio, and Texaco.

Phase II: Aramco, Kerr-McGee, Mobil, Petrobras, and Texaco.

Atkinson, G. and Mecik, M., 1994. CaCO 3 scale formation: How do we deal with the effects of pressure? Conf. Corrosion 94, Pap. No. 610, 12 pp.

Atkinson, G., Raju, K. and Howell, R.D., 1991. The thermody- namics of scale prediction. Soc. Pet. Eng., Richardson, TX, SPE Pap. 21021, 7 pp.

Barin, I. and Knacke, O., 1973. Thermochemical Properties of Inorganic Substances. Springer, Berhn, 120 pp.

Bann, I., Knacke, O. and Kubachewski, O., 1977. Thermochemi- cal Properties of Inorganic Substances - - A Supplement. Springer, Berlin, 60 pp.

Barner, H.E. and Scheverman, R.V., 1978. Handbook of Thermo- chemical Data for Compounds and Aqueous Species. Wiley, New York, NY, 105 pp.

Collins, A.G., 1975 Geochemistry of Oilfield Waters. Elsevier. Amsterdam, 496 pp.

Cowan, J.C. and Weintritt, D.J., 1976. Water-formed Scale De- posits. Gulf Publishing Co., Houston, TX, 596 pp.

Garrels, R.M. and Christ, C.L., 1965. Solutions, Minerals, and Equilibria. Freeman and Cooper, San Francisco, CA, 450 pp.

Howell, R.D., Raju, K. and Atkinson, G., 1992. Thermodynamics of scale mineral solubilities, 4. Experimental measurements of SrSO4(s) in H.O and aqueous NaC1 from 25 to 250°C and from 1 to 500 bar. J. Chem. Eng. Data, 37: 464-469.

Majumdar, A. and Roy, R., 1956. Fugaciues of CO 2. Geochim. Cosmochim. Acta, 10:311-317.

Marshall, W,L., 1967. Aqueous solutions at high temperature, XX. The dissociation constant and thermodynamic functions for MgSO 4 to 200°C. J. Phys. Chem., 71: 3584-3593.

Millero, F.J. and Masterton, W.L., 1974. Volume change for the formation of MgSO 4 in pairs at various temperatures. J. Phys. Chem., 78: 1287-1294.

Plummer, L.N. and Busenberg, E., 1982. Geochlm. Cosmochim. Acta, 46:1011.

Raju, K. and Atkinson, G., 1985. Thermodynamics of scale tmneral solubilities, 1. BaSQ(s) in H~O and aqueous NaCI. J. Chem. Eng. Data, 33: 490-495.

Raju, K. and Atkinson, G., 1989. Thermodynamics of scale mineral solubilities, 2. SrSO4(s) m aqueous NaCI. J. Chem. Eng. Data, 34: 361-364.

Raju, K. and Atkinson, G., 1990 Thermodynamics of scale mineral solubilities, 3. Calcium sulfate in aqueous NaC1. J. Chem. Eng. Data, 35: 361-367.

Rogers, P.S.Z. and Pitzer, K.S., 1982. Thermodynamics of aque- ous NaCI. J. Phys. Chem. Ref. Data, 11: 15-75.

Silvester, L.F. and Pitzer, K.S., 1977. Thermodynamics of elec- trolytes, 8. High-temperature properties, including enthalpy and heat capacity with applicaUon to sodium chloride. J. Phys. Chem., 81: 1822-1828.

Wu, Y.C., Rush, R.M. and Scatchard, G, 1969. Osmotic and activity coefficients for binary mixtures of NaC1, Na2SO 4. MgC12, and MgSO 4 in water at 25°C. J Phys. Chem., 73: 2047-2053.

R e f e r e n c e s

Angus, S., Armstrong, B. and de Reuck, K.M., 1976. Carbon Dioxide - - International Thermodynamic Tables of the Fluid State. Pergamon, London, 110 pp.