Journal o f Hydrology, 54 (1981) 51--74
51
Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
THE ORIGIN AND EVOLUTION OF SALINE FORMATION WATER, LOWER CRETACEOUS CARBONATES, SOUTH-CENTRAL TEXAS, U.S.A.
LYNTON S. LAND and DENNIS R. PREZBINDOWSKI
Department of Geological Sciences, University of Texas at Austin, A ustin, TX 78 712 (U.S.A.) Amoco Production Company, Tulsa, OK 74102 (U.S.A.) (Accepted for publication April 16, 1981)
ABSTRACT Land, L.S. and Prezbindowski, D.R., 1981. The origin and evolution of saline formation water, Lower Cretaceous carbonates, south-central Texas, U.S.A. In: W. Back and R. L~tolle (Guest-Editors), Symposium on Geochemistry of Groundwater -- 26th International Geological Congress. J. Hydrol., 54: 51--74. Systematic chemical variation exists in formation water collected from a dip section through Lower Cretaceous rocks of south-central Texas. Chemical variation can be explained by an interactive water--rock diagenetic model. The cyclic Lower Cretaceous shelf carbonates of the Edwards Group dip into the Gulf of Mexico Coast "geosyncline", and can be considered, to a first approximation, as part of a complex aquifer contained by Paleozoic basement beneath, and by relatively impermeable Upper Cretaceous clay and chalk above. The hydrodynamic character of this carbonate system is strongly controlled by major fault systems. Major fault systems serve as pathways for vertical movement of basinal brines into the Lower Cretaceous section. Formation water movement in this sytem has strong upfault and updip components. The "parent" Na--Ca--C1 brine originates deep in the Gulf of Mexico basin, at temperatures between 200 and 250oc, by the reaction: halite + detrital plagioclase + quartz + water --> albite + brine Other dissolved components originate by reaction of the fluid with the sedimentary phases, K-feldspar, calcite, dolomite, anhydrite, celestite, barite and fluorite. Significant quantities of Pb, Zn and Fe have been mobilized as well. As the brine moves updip out of the overpressured deep Gulf of Mexico basin, and encounters limestones of the Stuart City Reef Trend (the buried platform margin), small amounts of galena precipitate in late fractures. Continuing to rise upfault and updip, the brine becomes progressively diluted. On encountering significant quantities of dolomite in the backreef facies, the Ca-rich brine causes dedolomitization. Although thermochemical consideration suggests that small amounts of several authigenic phases should precipitate, most have yet to be found. Minor amounts of several kinds of calcite spar are present, however. As the brine evolves by dilution and by cooling, no systematic changes in any cation/C1 ratio occur, except for regular updip gain in Mg as a result of progressive dedolomitization. The formation water, highly diluted by meteoric water, eventually discharges along faults as hot mineral water.
0022-1694/81/0000--0000/$02.50 © 1981 Elsevier Scientific Publishing Company
52 INTRODUCTION The origin of saline formation water in sedimentary basins has been problematical since it was first recognized that basinal fluids typically contain dissolved solids in concentrations considerably in excess of seawater. Vast differences in major-ion ratios quickly dispelled early assumptions that basinal fluids were connate and represented buried seawater (Chave, 1960). Since then, different mechanisms have been advocated to account for the composition of subsurface water, and indeed, different mechanisms probably operate in basins with different lithologies and different burial histories. In some cases saline formation water m a y evolve in near isochemical rock--water systems during burial, as increasing temperature and pressure induce reactions which transfer c o m p o n e n t s from the solid to the dissolved state. At the other end of the spectrum, fluid bearing no resemblance to the interstitial burial water may be imported from another part of the basin, or even from outside the basin, for example, by meteoric recharge, and modified by rock--water interaction. The primary goal of our study was to d o c u m e n t the burial diagenetic history of the Lower Cretaceous Edwards Group in south-central Texas with emphasis on the most deeply buried rocks. Our basic questions were: H o w has burial affected the rocks? Does rock chemistry or mineralogy record early or late (burial) diagenetic events? What percentage of cementation (or any other diagenetic reaction) is due to burial? In attempting to answer these questions we systematically sampled formation water from 43 producing oil and gas wells from eleven fields in five Texas counties. Thirteen additional samples were obtained from updip water wells. The present paper reports our water analyses, and a model which, we believe, best accounts for our observations. We do not pretend to fully understand this gigantic aquifer, much of which is inaccessible, and preface our remarks with the recommendation that more data are badly needed. S t u d y area
In Early Cretaceous time a carbonate platform existed over most of what is n o w Texas, fringed by a " r e e f " complex (bank edge) which separated extensive areas of shallow-water carbonate sedimentation from the ancestral Gulf of Mexico. Between 300 and 1000 m of Lower Cretaceous shallow-water limestone, dolomite and evaporites accumulated over Paleozoic and Lower Mesozoic rocks, and was subsequently covered by clay and chalk deposited by Late Cretaceous transgressions. Burial of the platform occurred as thick clastic deltaic complexes prograded over it throughout much of the Tertiary. Today, the bank edge (the Stuart City " r e e f " ) is buried to depths of between ~ 3 . 7 and 4.7 km in the study area. Fig. 1 is a map of the study area showing the major zones of faulting and the oil and gas fields from which samples have been obtained. Because of the distribution of producing oil and
53 BALCONES FAULT ZONE
/A,,
LULING FAULT Z O N E
BEXAR
TROUGH
~
~
Antonio, / I
Cree~
~
/
BAD WATER LINE
.,/ .
WILSON
.~ ~
ATASCOSA TROUGH
/
Kened~ Jourdonton :horlotte
/
/I
Muil
// //
f
Foshin
i ~l,lonleolo
6
Son Miguel Creek
~nee Oil ~ Fields
_IVE MC MULLEN
OAK
Gos
o ~ ~
-~
MILES
Fig. 1. Map of the study area showing the position of the Stuart City Reef Trend and the producing fields which were sampled. Lower Cretaceous peritidal sediments are exposed in outcrop today northwest of the Balcones Fault zone, whereas the Stuart City Trend (the bank edge) is buried today to depths of between 3700 m (southwest) and 4700 m (northeast) in the study area.
gas fields, collection of samples from a complete dip section and at all depths in the section is not possible. As a result, our samples are concentrated at five distinct depth intervals, i.e. ~ 4 2 0 0 m (the Stuart City shelf edge), 3 2 0 0 m (the Karnes Trough), 2 1 0 0 m (the Atascosa Trough), 7 8 0 m (the Luling Fault zone), and shallower depths (water wells). All areas of production are associated with fault trends, and all samples come from the uppermost zones of the thick "aquifer".
Sampling and analysis Formation water was obtained from producing wells as close to the wellhead as possible, either from the first separator or from the well-head itself. Only the most prolific water-producing wells in each field were sampled. In the case of gas wells, considering gas production, and temperature and pressure changes during production, less than 5% of the water produced is calculated to be distillate.
54 At each well, three samples were taken whenever possible. A 500-ml sample, acidified with HNOa, was taken for chemical analysis and 125-ml samples were taken for 5180 and 5D analyses (untreated) and for d13C analysis (added to 5 ml of saturated SrC12 and 10 ml of 5 N NaOH). In addition, as much information as possible was obtained on production rates of oil/gas and water, well manipulation (fracturing, acidizing, etc.), well treatment, scaling problems, and previous water analyses. Well scale was obtained wherever possible. Although innumerable possibilities exist for errors in sampling of this sort, and it is very difficult, if n o t impossible, to determine in situ formation conditions, the data presented and discussed (Table I) represent an internally consistent data set. In the course of our sampling we typically obtained erratic results from gas wells producing less than ~ 1 . 5 9 m 3 (10 bbl) of water per day, or wells from which gas aspirated during collection. Water from these wells usually had enriched 5D-values and reduced total dissolved solids (TDS) content. In the case of three samples where several wells connected to a single separator, or wells in which multiple completions were indicated, one or more components (especially K) had aberrant concentrations relative to nearby samples. We present these data in Table II, b u t have n o t included them in our discusion. Wells which were n o t producing water more-or-less continuously, or which had been extensively manipulated (acidized or treated) were n o t sampled. No water-flooding is used in this area, and we have no reason to suspect that the analyses reported in Table I represent anything other than the ion ratios of in situ water. Because of the care taken during sampling, we believe the ionic strengths are representative as well.
Analytical results Fig. 2A--D presents the results of the cation analyses plotted vs. chloride, using a different symbol for each depth interval sampled. Sulfate is very dilute in these waters, less than 300 mg/l, and was n o t routinely analyzed. Alkalinity is also very low, and determination of bicarbonate is complicated by the presence of organic alkalinity (Carothers and Kharaka, 1980) and loss of CO2 as the water is produced. We made no a t t e m p t to determine in situ carbonate saturation states. Regarding the major cations, and chloride, with the exception of Mg (discussed below), the relative ionic proportions of water deeper than ~ 2 km are approximately constant. We observe no statistically significant differences in Ca/C1 or Na/C1 ratios among any samples from this area. The Na/C1 ratio is neither that of halite nor that of seawater (Fig. 2). Sr and K do n o t correlate as well with chloride, in part due to more severe analytical difficulties. There is a suggestion, especially in the 3200and 2100-m depth intervals, that s o m e shallower water contains slightly less Sr and K relative to chloride than does deeper water and that a correlation with depth m a y exist.
Production depth (ft.*) Temperature (°c)
13,076--13,253 13,033--13,288 13,065--13,261 13,384--13,452
13,624--14,002 13,446--13,922 13,804--13,898 13,716--13,971
161 163 174 168
157 162 160 157
13,440--13,666 13,575--13,645 13,424--13,587
10,935--10,960 10,904--10,918 10,906--10,920 10,953--10,980
10,817--10,835 10,524--10,822 10,600--10,664 10,712--10,772 10,628--10,712
123 125 121 129 118
114 132 137 133
176 170 171
L.M. G u b b e i s No. 1
10,149--10,182
112
Exxon, San Miguel Field, McMuUen County:
E m m a T a r t t No. 1 W.T. H u r t No. 3 R W.T. H u r t No. 2U W.T. Hurt No. 1U E.S. K o e h l e r No. 1U
Gulf Oil, Fashing Field, A tascosa County:
Y. Cisneros No. 1 A. Dugie No. 2 Krnse No. 1 C. Kainer No. 2
Sswll Oil, Person Field, Karnes County:
R u h m a n n No. 1 R u c k m a n B No. 1 8chulz No. 1
M.G.F. Oil Co., Monteola Field, Bee County:
A. G o r d o n No. 2 A. O i s o n No. 1 J . A . L e p p a r d No. 1 S.E. T u r n e r No. 2
Shell Oil, Pawnee Field, Bee County:
A. Wernli No. 1 J. P o i s o n No. 1 E. R o ] f N o . 1 C. S t r a w n No. 1
General Crude, Kenedy Field, Karnes County:
Well
C h e m i c a l analyses o f oil-field w a t e r s , s o u t h - c e n t r a l T e x a s
TABLE I
1.059
1.094 1.083 1.127 1.158 1.096
1.165 1.147 1.133 1.165
1.141 1.131 1.153
1.075 1.071 1.049 1.052
1.213 1.052 1.061 1.213
Density ( g / c m 3)
--
80 -246 386 180
296 -48 260
----
370 ~800 240 366
118 -229 156
HCO 3 (rag/l)
20.1
35.9 32.0 51.0 63.5 38.0
63.0 56.5 52.0 64.0
55.2 37.5 57.2
30.5 26.0 17.6 22.2
74.0 19.3 21.5 77.5
Na (g/l)
2,050
2,800 1,400 2,900 4,400 1,400
4,200 3,400 2,900 4,400
3,370 2,800 3,470
1,400 2,220 610 510
4,800 340 890 5,600
K (rag/l)
5.63
10.8 8.9 13.9 18.9 10.3
20.1 17.6 14.9 20.5
15.6 26.2 17.7
7.39 9.08 6.93 3.98
24.5 6.77 8.53 28.6
Ca (g/l)
556
830 760 1,220 1,330 900
1,670 1,580 1,380 1,710
630 520 660
340 330 201 220
830 360 370 800
Mg (rr "/1)
1,120
1,650 2,170 2,950 3,250 2,400
2,380 2,240 1,880 1,310
2,740 1,800 3,210
1,610 1,240 615 1,110
4,390 868 1,090 4,980
Sr (rag/l)
47.0
86.6 71.5 110.6 140.0 83.2
145.0 125.0 114.3 146.0
123.6 111.3 132.9
64.4 59.1 41.0 44.7
173.0 43.5 51.5 189.0
C1 (g/l)
4.75
5.85 5.8 5.8 6.05 6.6
6.2 -5.1 5.9
4.7 6.2 4.55
6.25 5.45 5.75 6.7
5.45 -4.95 5.45
ph
+13.1
+6.7 +8.5 +12.3 +12.9 +13.0
+10.4 +8.5 +10.7 +10.3
+14.4 +29.6 +18.0
+13.9 +9.5 +13.6 +14.7
+12.8 +3.4 +11.4 +9.7
61SO (°/00 )
--
+10 --10 --15 --15 --14
--9 --7 --14 --21
--26 --18 --6
--22 --20 --21 --15
--17 --27 --21 --17
5D (%)
--
------
881 ----
580 470 620
-241 -177
701 --858
Br (ppm)
ol ol
Production depth (ft.*) (°C)
7,386--7,392 -7,350--7,396 7,380--7,384
7,568--7,573 7,599--7,606 7,455--7,451
6,996--7,000 6,440-6,572
6,921--6,929 6,917--6,927
---
81 74
--83
89 99 116 89
--50
2,613--2,616
2,657--2,659 --
1
22
52 49
2,548--2,562 2,633--2,635
No. 31 1
* 1 ft. = 0 . 3 0 4 8 m .
Dix a n d McKean T h o m a s Dix No. C. K n o b l o c h No. A n d e r s o n No. 1 T h o m a s Dix No.
Gulf Oil, Darst Creek Field, Guadalupe County:
E.J. Pruitt No. 8 E.J. Pruitt No. 7
Exxon, Horn Field, A tascosa C o u n t :
J.T. U p p r i g h t No. 1 E.J. Pruitt No. 36
Exxon, Charlotte Field, A tascosa County:
D u r e n a n d Richter No. 5 D u r e n a n d Richter No. 2 C o w a r d a n d Couch No. 1
Exxon, lmogene Field, A tascosa County:
S.P.J.S.T. L o d g e No. 1 S.P.J.S.T. Lodge No. 2 J. S a n d e e n No.'s 1 a n d 2 A.N. M o u r s a n d No. 2
1.019 1.020 1.019 1.018 1.019
1.059 1.053
1.077 1.090
1.153 1.142 1.132
1.134 1.149 1.139 1.132
Na
730 650 514 316 --
---
6.99 6.63 6.71 6.50 7.17
25.1 23.1
32.5 33.0
49.0
--
---
63.3 56.5
53.0 56.0 53.8 51.5
(g/l)
---
334 739 >800 >800
(g/cm 3 ) (mg/l)
HC03
Density
Temper-
ature
Exxon, Jourdanton Field, A taacnsa County:
Well
T A B L E I (continued) K
Ca
260 260 250 250 260
281 253
605 875
3,190 2,340 1,175
2,300 2,950 2,475 1,885
1.77 1.67 1.71 1.58 1.63
3.53 2.26
6.33 7.50
17.5 16.6 15.0
15.8 17.5 16.3 15.7
(rag/l) (g/l)
580 550 550 540 540
890 647
1,360 1,526
2,270 2,210 2,060
2,150 2,350 2,250 2,350
(mg/l)
Mg
CI
53 53 51 49 69
490 320
390 355
2,150 2,300 2,350
768 925 890 1,750
14.4 13.9 14.2 13.5 14.8
47.6 42.6
63.8 68.7
134.0 125.0 83.8
119.0 128.0 117.0 114.0
(rag/l) (g/l)
Sr
6.5 6.7 6.7 6.5 --
5.85 6.6
5.55 5.75
5.25 5.1 5.2
5.9 4.95 5.3 5.6
pH
--2.5 3.0 --2.3 --1.7 --3.6
+5.2 +5.2
+6.4 +6.5
+10.8 +10.6 +10.4
+9.5 +12.5 +9.2 +10.1
~ZSo (°/0o )
--27 --26 22 --17 --23
--24 --20
--16 --22
--14 18 --20
--18 --18 --13 --15
6D (%)
85 -----
---
-441
----
732 ----
Br (ppm)
f~n
Production depth (ft.*) Temperature (°C)
15,655----15,814
-13,690--13,778
--
13,440--13,696
10,890-10,916 10,930-10,960 10,925--10,938
7,399--7,403
78
134 130 132
* 1 ft. = 0 . 3 0 4 8 m .
Dix and McKean No. 11
2,533--2,548
49
Gulf Oil, Darst Creek Field, Guadalupe County:
O.H. Pfeil No. 2
Exxon, Jourdanton Field, A tasco~a County:
L. Urbanezyk No. 1
T. Y a n t s N o . ' s I a n d 2 C. Wishert No. 1
Shell Oil, Person Field, Karnes County:
Boone No. 1
--
148
166 --
194
M.G.F. Oil Co., Monteola Field, Bee County:
Hay No. 1
Shell Oil, Buchel Field, Bee County."
E.P. B e n h a m No. 1 S.E. T u r n e r No. 1
Shell Oil, Pawnee Field, Bee County:
McDoweU No. 1
General Crude, Kenedy Field, Karnes County:
Well
1.022
1.063
1.079 1.155 1.121
1.076
1.098
.
1.044
1.013
Density (g/cm3)
.
620
256
180 ---
--
450
.
291
350
HCO3 (mg/1)
.
22.4
22.2 56.5 47.0
27.2
31.4
.
18.7
.
Na (g/l)
.
540
860
.
1,050
250 3,400 3,700
1,940
.
.
K (mg/l)
.
.
.
.
7.23
17.2 21.8 13.9
8.6
14.1
3.32
Ca (g/l)
.
.
1,030
520 1,460 960
445
800
.
174
.
Mg (mg/1)
.
.
860
480
810 2,230 1,970
1,330
4,000
.
Sr (mg/l)
Chemical analyses o f oil-field w a t e r with o n e or m o r e c o m p o n e n t s having a b e r r a n t c o n c e n t r a t i o n s , s o u t h - c e n t r a l T e x a s
T A B L E II
52.7
66.5 133.0 106.0
64.0
81.2
37.2
CI (g/l)
6.6
5.9
6.1 4.2 5.7
5.0
5.7
6.55
5.55
pH
--2.9
+3.2
+7.8 +10.0 +6.4
+18.0
+13.1
+14.2
+11.8
+0.9
51SO (%)
--25
--
--4 --8 +12
--25
--26
--21
--15
+2
5D (°/co)
--
344
265 ---
--
--
--
--
--
Br (ppm)
58
The relation between Mg and chloride is m o s t informative. Because the correlation between Ca and chloride is excellent (Fig. 2B), we present the data as a scatter plot of Mg against Ca (Fig. 3}. It must be stressed again that our "depth intervals" are imposed by sampling limitations, and samples are unavailable at intermediate depths in areas where no oil or gas production occurs. Clearly, Fig. 3 shows that shallower waters are progressively and systematically enriched in Mg relative to Ca (or chloride). Each "depth interval" has a characteristic Mg/Ca (or Mg/C1) ratio, and that ratio varies systematically in the dip section. Fig. 4 presents a scatter plot of 8D vs. depth. All water is depleted in deuterium relative to SMOW, and there is a tendency for samples to be
(~)
~-
•
•
Q~
=a
e °
e-
e l~
tuNa+
2 Im
•
e ~o e e
1
0
°~.
o
i
2
3
~
~
mOV 0.8
® 0 0
0.6'
ala~
raCe2+0.4 ,
• ~aao ll
I I
0.2' •o
•
I
eA
0 0
i
2
3
mCl-
4
5
6
59 0.16
© 0.12
%,
em
rnK÷
0.08
0.04
•w ~A
0
3 mci-
0
4
6
0.08
® 0.06
msr2+ 0.04
0.02' m
•e
•e
0
o
i
2
~
,~
~
rnci-
Fig. 2. Scatter plot of molality of: (A) sodium; (B) calcium; (C) potassium: and (D) strontium, respectively, vs. molality of chloride. Circles depict samples from ~ 4 2 0 0 m (Stuart City Trend), squares depict samples from ~ 3 2 0 0 m (Karnes Trough), and triangles depict samples from ~ 2 1 0 0 m (Atascosa Trough). Note that the deepest samples are the most saline, but relatively low salinity samples can be found at any depth. The Na/C1 ratio of the brine is constant, and less than that of seawater. Note also that the Ca/Cl ratio is essentially constant.
somewhat enriched with increasing depth. Within each of the three deeper zones, there is a suggestion that shallower wells are slightly depleted relative to deeper wells, but no parallel relationship with chlorinity exists. The 5180 of the water is buffered by the Edwards limestone, as originally explained by
60
0.8
• /20
"C
./. 0.6 •
mca2+
~ 0
eC
0.4 i
i
0.2
0
o
o.b6
0~03
o.b9
o.i2
rnMg 2.
Fig. 3. Scatter plot of molality of calcium vs. molality of magnesium. Symbols as in Fig. 2. Note that unlike previous data, each depth interval has a consistent and unique Mg/Ca (or Mg/C1) ratio. The lines are best fits to the data points, and using the slopes of the lines and the data of Rosenberg and Holland (1964), an equilibrium temperature for calcite-dolomite has been calculated. In all cases the calculated temperatures are much hotter than the reservoirs today or at any time in their past. For the three depth intervals, reservoir temperatures today are ~ 1 6 5 ° C (circles), 126°C (squares) and 89°C (triangles). Therefore all water is undersaturated with dolomite.
~30
I-
•
•
Ill
-20
•
6D
P
(%0)
•
•
el
-10
0
1
2 ~ DEPTH (kin)
4
5
Fig. 4. Scatter plot of 6D vs. depth. Note that all samples are depleted relative to SMOW, but most are slightly enriched relative to local meteoric water today (6D = - - 2 1 % 0 ) . Note also that shallower water in each depth interval (fault trend) is in cases depleted relative to deeper samples in the same fault trend.
61 Clayton et al. (1966), and the 51SO of the water in almost all cases is in equilibrium with the limestone at the reservoir temperature (Prezbindowski, 1981). No useful information relative to the origin of the water can be obtained from oxygen-isotope data except to be sure that the water has been in contact with the rocks long enough to have equilibrated with them. Discussion
In trying to account for the data presented above, we considered three possible "models": Model 1. Meteoric water is recharging the outcrop of the aquifer, moving downdip, and evolving in chemical composition as rock--water interactions take place at increasing depths and temperatures. Model 2. Connate water (alternatively, Cretaceous seawater, a Cretaceous sabkha brine, or Cretaceous meteoric water) has evolved to the present formation water compositon by closed-system burial diagenesis, or diagenesis in a system permitting loss of material by compaction, b u t n o t addition. Model 3. Brine is formed in the deep Gulf of Mexico by reaction of deep basinal water with Jurassic evaporites, is injected into the aquifer at depth, and moves updip.
We have accepted model 3 as best accounting for the observed facts, and present our discussion accordingly. Fig. 3 presents, we believe, a compelling argument for this model. Because this is a carbonate aquifer, the Mg/Ca ratio of the water is certainly related to the calcite--dolomite system. Clay constitutes only a few percent of the rock section. Dolomite is abundant in outcrop of the Edwards Formation, and decreases in abundance into the subsurface, being limited in distribution in cores deeper than 1 km. Water shallower than ~ 1 km approaches equilibrium with calcite plus dolomite using the Kdolomite-ValUeSof Langmuir (1971) or Hsu (1963) (Pearson and Rettman, 1976). A different situation exists with respect to the deeper water, however, using the relation: log(mca2÷/mMg2÷) = - - 1 0 0 0 T -1 + 2.98
where T is in kelvins, derived from experimental data in 2 M divalent cationchloride (MC12 ) solutions at elevated temperatures (Rosenberg and Holland, 1964). We assume, as did Rosenberg and Holland (1964), that 7ca2÷ --~ 7Mg2÷. Since the ionic strength of Eosenberg and Holland's experimental solutions (6) is similar to or greater than ionic strengths observed in the Edwards Formation (1.3--5.5), Rosenberg and Holland's molality ratios should be comparable to our molality ratios irrespective of the values of the activity coefficients. Using the 1 M data of Rosenberg et al. (1967) (ionic strength = 3) would additionally favor the following argument, namely that all the water deeper than ~ 1 km is grossly undersaturated with respect to dolomite, and the
62 water becomes increasingly undersaturated with increasing depth. Consider the three possible models in light of these data: Model 1. It is impossible for water to move downdip, lose Mg relative to Ca, becoming progressively m o r e undersaturated with dolomite. This model can therefore be ruled out, unless the Mg is being controlled by reactions other than those involving calcite--dolomite. We are unaware of any other Mg-bearing phase of volumetric significance in this carbonate aquifer. Model 2. If water, no matter what its original salinity, was in near equilibrium with calcite + dolomite or was oversaturated with dolomite by virtue of a high Mg/Ca ratio at shallow depth (like Cretaceous seawater or a sabkha brine), it could never evolve to its present composition with increasing temperature (burial) by dolomitization because the water is everywhere undersaturated with dolomite today. It is very unlikely that the very low Mg/Ca ratio of the brine could be primary because neither Cretaceous seawater nor any reasonable Cretaceous sabkha brine are likely to have had such a composition. Model 3. Because most of the dolomite in these rocks is associated with the shallower (bankward) parts of the carbonate platform (Rogers, 1967), a Ca-rich water moving updip would encounter progressively more and more dolomitic country rock as it moved updip, and could thus dedolomitize the rock and increase the Mg/Ca ratio of the water. The amount of Mg gained by the water would be limited by the a m o u n t of dolomite originally present in the rocks and by the channelization of the flow. Because dolomite is absent in the platform margin, the water could remain undersaturated with dolomite until significant quantities of dolomite were encountered updip. Fig. 5 is a photomicrograph of a sample from the J o u r d a n t o n field in the Atascosa Trough. Blocky spar calcite lines vuggy pores in the dolomitic limestone, and the poikilotopic spar encircles many of the dolomite crystals, some of which are extremely corroded. There is a distinct transition zone between the calcite lining the pores and the porous dolomite groundmass making up the bulk of the rock. This transition zone is characterized by an increase in corroded and replaced dolomite crystals as one moves toward the pore space. The average 5XSO of the poikilotopic spar calcite (dedolomite) is--5.3°/00 (PDB) as compared to 51SO of --2.1%0 for the dolomite and tiXSo o f - - 3 . 8 °/00 for nearby bulk limestone. These data are consistent with a subsurface origin for the spar, whereas the isotopic composition of the bulk limestone and dolomite is due to early, pre-burial meteoric diagenesis (Prezbindowski, 1981). Both petrography and isotopic data indicate that dedolomitization is the latest diagenetic process to significantly alter these rocks. We suggest that dedolomitization is being caused by the present-day formation water. A second line of argument concerns the TDS c o n t e n t of the water, which averages 3 m in chloride. Why does the concentration of dissolved solids in the water increase with increasing depth [a c o m m o n p h e n o m e n o n in sedimentary basins (Rittenhouse et al., 1969)] ? Edwards water is of the Na--Ca--C1 type
63
Fig. 5. Photomicrograph of dedolomite (poikilotopic spar calcite) from the Pheil No. 3 well, Atascosa County, 2253 m, white light. Width of field is 940prn. Note that many of the dolomite rhombs are badly corroded where they have been unreplaced by the spar. (again, very typical of sedimentary basins), dominated by NaC1, which accounts for ~ 70% of the TDS. The only reasonable source for the dissolved Na and chloride in solutions this concentrated is halite. Although other sources of chloride have been proposed, accounting for the vast a m o u n t of chloride in the large volume of water in this aquifer from any other source except halite is very unlikely. Processes such as ultrafiltration (clay-membrane filtration, e.g., Hanshaw and Coplen, 1973; Kharaka and Smalley, 1976), suffer overwhelming objections in cases such as this (model 2). The rocks we are studying are carbonates, generally containing less than 1% clay, and nowhere more than 10% clay. Such clay as is present in this section is concentrated in the basal part of the section and increases in abundance updip. The closest adjacent shales are I km above these deposits, separated from them by impermeable chalk. In addition, the Edwards and all overlying units are normally pressured at the present time. No evidence of earlier overpressuring exists and the rocks may have never been overpressured since they and the overlying clastics have never been more deeply buried than they are today. If membrane filtration is important then the membranes are far removed from the water and a driving mechanism (overpressure) has apparently always been absent. If membrane filtration is involved, why are essentially constant ionic proportions always observed in the water (except for Mg)? One would expect systematic variation in, say, Ca/Na with increasing salinity if filtration were operative, as certain species (Na +) passed through the shale membrane
64 more easily than other species (Ca 2+) (White, 1965). Yet the Ca/Na ratio of the water is constant (Fig. 2A and B). Although 5D does seem to increase slightly with increasing depth, no correlation with salinity exists and the observed variation is far less than predicted experimentally (Coplen and Hanshaw, 1973). In addition, all water is depleted in deuterium relative to SMOW [or the Cretaceous ocean (Lowenstam, 1961)]. If the brine represents the residue after filtration of seawater, meteoric water, or a sabkha brine, it should be enriched in deuterium, not depleted. For these reasons, membrane filtration by shales is believed to be of no consequence in this geological system. Could meteoric water, moving downdip, evolve into the Edwards brine (model i ) ? Although meteoric water plays an important role in diluting upward-moving brine, it cannot evolve into a brine by interaction with the available country rock. In other areas where carbonate coastal plains of similar lithology are known to be recharged by meteoric water, the tremendous increase in TDS as we observe in the Edwards is not found. For example, in the Florida aquifer, water at 500 m depth contains only ~ 8 5 0 mg/1 TDS and the dominant anion is sulfate (Back and Hanshaw, 1970), compared to 4000 mg/l in the Cl-dominated Edwards. In addition, the junction between saline and potable water (the "Bad Water" line just downdip from the Balcones Fault zone corresponding to 1000mg/1 TDS, Fig. 1) is a relatively distinct feature, not a gradation over many kilometers. Because there is no halite associated with the country rocks, chloride supplied by saline water moving updip, n o t meteoric water moving downdip, is the only possible explanation for the rapid basinward increase in chloride. Isochemical evolution during burial to yield the water we observe in the rocks today is also unlikely. It has been adequately demonstrated that during and soon after deposition, the Cretaceous sediment deposited on the Comanche shelf was massively altered by meteoric diagenesis (Mueller, 1975; various papers in B e b o u t and Loucks, 1977; Prezbindowski, 1981). The interstitial water in which the sediment was initially deposited (either seawater or, rarely, a sabkha brine) had been replaced by many pore volumes of meteoric water prior t o the beginning of burial in the Late Cretaceous. If the TDS c o n t e n t of the average Edwards water t o d a y was produced by dissolution of synsedimentary halite by connate meteoric water or even connate marine water (assuming the synsedimentary evaporites s o m e h o w escaped Cretaceous meteoric diagenesis), then an average Edwards water today containing ~ 2 g N a ÷ / 1 would need to have dissolved ~ 2 . 3 cm 3 of halite per liter of water. Assuming an average post-diagenetic porosity of ~ 1 0 % , then 0.6% of the original rock section would need to have been halite. Bedded halite has never been described or postulated from Edwards carbonates. Only very minor amounts of gypsum are still present, which, together with solution--collapse features, are most c o m m o n l y associated with restricted areas of the platform and exposed today far updip and in outcrop. No evaporites whatsoever are associated with the buried platform margin (the Stuart City
65 Reef Trend), and anhydrite or former evaporites (solution--collapse features) are rare in rocks which are buried today to depths in excess of 2500 m, and are localized in the fault troughs. Thus the origin of the Edwards brine from dissolution of synsedimentary evaporites is unlikely. There simply were insufficient volumes of evaporites in the rock section, and what little evaporites were deposited were mostly removed in Cretaceous time prior to burial (Rose, 1972). Carpenter (1978) proposed that a brine from Mississippi, similar to that described here, was a modified sabkha brine, or a true connate fluid buried along with its cogenetic evaporites. Aside from the arguments already presented, t h a t very little gypsum or anhydrite and no bedded halite are associated with Edwards deposits, other arguments oppose such an origin for the Edwards brine. Volumetrically, such a connate brine would have been insignificant, even if evaporites had been extensive (Garret, 1970; Kinsman, 1971). A connate sabkha brine should have been enriched in deuterium relative to SMOW, y e t all the Edwards brine we analyzed is depleted. Carpenter's model proposes modifications of a connate sabkha brine by mineral--water reactions (principally dolomitization) to yield the observed formation water. We have already pointed out t h a t the Mg/Ca ratio of the Edwards brine disputes the role of dolomitization in the formation of the brine as the water c a n n o t continue to lose Mg beyond saturation with respect to dolomite as temperature increases with burial. Carpenter uses the C1/Br of the water as another line of evidence, arguing t h a t the C1/Br of formation water resembles seawater-derived brine from which considerable halite has precipitated. Conventional wisdom has it that a brine derived from the dissolution of halite will have the C1/Br of the halite. Such a solution would contain less Br relative to C1 than seawater or a sabkha brine, because of the value of the distribution coefficient: D = (Br/C1)crystal/(Br/C1)solution which is ~ 0 . 0 3 5 at 25°C (Holser, 1979b). However, conventional wisdom is apparently flawed in this case. If the partitioning of bromide between halite and brine were reversible and represented a true homogeneous equilibrium, then a solution formed by dissolving halite initially precipitated from seawater should have a C1/Br exactly t h a t of seawater, n o t that of the salt. At equilibrium, if the distribution coefficient were a true reversible equilibrium constant, then the solution formed by dissolving halite would be considerably enriched in bromide relative to the salt. In order to investigate the behavior of Br during halite dissolution, we reacted Jurassic halite from a Texas salt dome with distilled water on a shaker table at 25°C. The C1/Br ratio for the solution after only 18hr. of reaction (13,500) was lower than t h a t of either the halite or the solution immediately after reaction (both 17,000). After nearly eight months the C1/Br of the solution had fallen to an apparent steady state (8000), but did not approach
66 the value to be expected of an homogeneous reversible equilibrium with D = 0.035 (600). Similar behaviour of halite on dissolution has been noted by Kuhn (1968) and a similar mechanism has been suggested to account for very low Br concentrations in halite (Holser et al., 1972). Holser (1979b) summarizes some of the kinetic problems that have plagued C1/Br studies, and which m a y explain our observations. Therefore the molar C1/Br ratio of the Edwards brine (160--250, Table I) might be due to simple dissolution of Jurassic halite since: (1) in situ basinal Jurassic halite should contain more bromide than the displaced and recrystallized basin--margin material used in our experiment; and (2) the distribution coefficient of Br in halite increases with increasing temperature (Holser, 1979b). We find arguments based on C1/Br ratios unsatisfactory at this stage of our knowledge, and recommend that our experimental observations be tested on a more rigorous basis, and at higher temperatures such as exist during burial. A more satisfactory explanation for the TDS content of Edwards brine is that a highly saline fluid originates deep in the Gulf of Mexico by reaction with Jurassic evaporites, moves up-fault and up-dip, and is diluted and undergoes changes in composition by rock--water interaction as it moves.
Origin of "parent" Edwards brine Many explanations have been proposed for the origin of Na--Ca--C1 formarion water. In addition to mechanisms which we have rejected (e.g., membrane filtration), Carpenter (1978) proposed that such a brine results from the generalized reaction: halite + carnallite + water + limestone -+ Na--Ca--C1 brine + dolomite His model is clearly inapplicable in this case unless the water is produced at a temperature of at least 320°C (see Fig. 3), the temperature in equilibrium with calcite, dolomite and the Mg/Ca of the deepest brine. At a reasonable average Gulf of Mexico coast geothermal gradient of ~ 3 0 ° C / k m , this would require ~ 1 0 km of burial, whereas the Cretaceous section we are studying has never been more deeply buried than it is today ( ~ 5 k m ) . In addition, carnallite is a rare evaporite mineral, and Carpenter's stoichiometry requires 10% of the evaporite section to be carnallite, a figure hardly in accord with the abundance of carnallite in cored evaporites (Holser, 1979a). No subsurface dolomitization of these rocks has been detected, and in fact the opposite reaction is taking place (Fig. 5). As a general explanation for Na--Ca--C1 brines, Carpenter's (1978) hypothesis appears to us to be unsuitable. We propose another reaction, namely: detrital plagioclase + halite + water -+ Na--Ca--C1 brine + albite for the following reasons. First, albitization is a known process in the Gulf of Mexico coast subsurface (Garbarini and Carpenter, 1978; Boles, 1979;
67 Land and Milliken, 1981). Pettijohn (1963) pointed o u t that albite is the typical feldspar of deeply buried greywacke sequences, and could not have been detrital because pure albite-bearing source rocks are uncommon. But Pettijohn offered no explanation for this materially significant anomaly. Albitization is a plausible explanation. Second, interest in geothermal energy has stimulated research into various aqueous "geothermometers". One which has been used with success, based ultimately on feldspar equilibria, is the Ca/Na geothermometer. Using the data of Helgeson (1972, fig. 9), or the empirical formula of Fournier and Trusdell (1973), and the ion ratios of Edwards brine from Fig. 2., a temperature of ~ 2 0 0 ° C is calculated. This is a reasonable temperature for updip Jurassic salt in the Gulf of Mexico t o d a y (6000 m x 30°C/km). If we consider a rather simple case where the components of a brine are generated by successive addition of c o m m o n sedimentary minerals, halite plus water yields a brine of a fixed composition (roSa ÷, mcl ÷, pH, etc.) at constant temperature and pressure. Adding quartz and albite saturates the brine with silica and alumina. Adding the metastable phase intermediate plagioclase (the typical detrital c o m p o n e n t of sandstones) causes albitization and consequent replacement of Na in the brine with Ca. Predicting the Ca/Na ratio of the resultant brine accurately is difficult, and depends on the availability and reactivity of plagioclase and whether or n o t the brine remains in contact with halite + quartz + plagioclase. If a halite-saturated (or near-halite-saturated) brine flowed into a clastic section lacking halite as a phase, Na removed from the brine by albitization would cause undersaturation with respect to halite but Na could not be replaced by continued halite dissolution. At ~ 2 0 0 ° C a brine in equilibrium with albite and K-feldspar should have a Na/K activity ratio (or molality ratio) of ~ 1 6 (Helgeson, 1972, figs. 6 and 7), less than the value of 26 observed for Edwards brine. Edwards brine is therefore undersaturated with respect to K-feldspars. K-feldspars have been observed to be almost completely removed from Frio sandstones in Brazoria County, Texas, as shallow as 4 2 5 0 m (Land and Milliken, 1981), and we suggest one possible reason for the potassium composition of Edwards brine is that most of the K-feldspars may have been removed from the Jurassic and Lower Cretaceous sandstone aquifers before the Edwards rocks were as deeply buried as they are today. Thus as the Edwards brine forms, insufficient K-feldspars are present in the discharge conduits, and the brine remains undersaturated. We presume a similar situation to prevent the brine from equilibrating with dolomite. We have already pointed o u t that a brine temperature as high as 320°C is unlikely, and thus the most likely explanation for the Mg/Ca of the brine is that it forms in a dolomite-starved system. The two most saline brines (Wernli No. 1 and Strawn No. 1) contain only 96 and 8 4 p p m SiO2, respectively. If albitization had occurred at ~ 2 0 0 ° C , then the water should contain ~ 2 5 0 p p m SiO2 (Crerar and Anderson, 1971}.
68 But three facts suggest that Edwards brines t o d a y should contain less dissolved silica than when they initially formed: (1) the most saline brines we sampled are only ~ ] saturated with halite, and thus have probably been diluted since they formed; (2) cooling to 160°C, the present bottom-hole temperature of the wells, may have caused precipitation of about half the original silica in the deep aquifer; and (3) the solubility of silica in nearhalite-saturated brines m a y be lowered significantly because of the reduced activity of water. Clearly the SiO2 values obtained today do n o t preclude quartz-saturation at the time the brine originally formed. Therefore it is possible that a paucity of K-feldspar and dolomite in the pre-Edwards aquifers downdip from our study area is the reason for the undersaturation of the brine with respect to these phases. It should be noted that K-feldspar, quartz and dolomite are extremely rare, or absent entirely from the deep Edwards carbonates themselves. We suggest that the initial major-ionic composition of the brine is controlled by a tendency toward equilibrium with the c o m m o n sedimentary phases halite, quartz, plagioclase and K-feldspar. The other components of the brine are probably controlled by reaction with other c o m m o n sedimentary phases. Since 2mca~ + m~a÷ approaches mcF , Ca in the brine cannot be derived from dissolution of anhydrite. Dissolution of anhydrite would result in Ca being balanced by sulfate or bicarbonate (in the case of sulfate reduction). Since the concentrations of both sulfate and bicarbonate are very low, the cause of the high Ca concentration in the brine cannot be anhydrite dissolution. The sulfate concentration in the brine is controlled by the equilibration of anhydrite with a Ca-rich brine, and the bicarbonate by equilibration with calcite and CO2. The high Sr content, which exceeds Mg in the deepest brines, is probably controlled by equilibration with celestite in the evaporite section. Attempts to calculate actual brine compositions using existing algorithms (WATEQ - - Truesdell and Jones, 1973; SOLMNEQ Kharaka and Barnes, 1973; EQ3/6 -- Wolrey, 1979) have n o t been very successful at present writing, owing to lack of reliable thermochemical parameters in aqueous solutions near halite saturation and in excess of 150°C. Even so, values n o t t o o different from the Edwards brine ion ratios are obtained when halite, quartz, albite, anorthite, calcite, anhydrite, celestite and fluorite are " r e a c t e d " to saturation, and K-feldspar and dolomite held below saturation. Three other lines of evidence favor our hypothesis that Edwards brine evolves deep in the Gulf of Mexico basin and moves up-fault and up-dip to its present position. (1) The underlying Sligo carbonate section is known to be overpressured. Pressure gradients in the Sligo Formation range from ~ 1.88" 104 to 1.99" 104 Pa/m (0.83 to 0.88 psi/ft.), whereas a gradient of 1.15" 104 Pa/m (0.51 psi/ft.) was measured in the (Edwards) Wernli No. 1 well (Prezbindowski, 1981). The Strawn No. 1 well, which produced the most saline brine we sampled (Table 1), flowed at "~318 m 3 (2000 bbl) of water per day. Irrespective of -
-
69
800
600
ppm 400 Zn
200
I
0
1oo g Ci-/filer
Fig. 6. Scatter plot of Zn (ppm) vs. C1 (g/l) for five samples from the Stuart City Reef Trend. The most saline well, the Strawn No. 1, was sampled twice over an interval of 13 months, and both poin'ts have been plotted. No other samples had Zn contents greater than 1 ppm.
the measured bottom-hole pressure of the well, a 4084-m column of water having a density of 1.213 g/cm 3 exerts approximately the same pressure as a 4954-m column of fresh water having the same temperature change with depth. Considerably less than 870 m of relief exists between any possible area of updip recharge and the land surface above the Stuart City Trend, and therefore the formation water must have a strong updip c o m p o n e n t of movement. (2) We find minor amounts of galena associated with late fractures in the Strawn No. 1 well, the well which produced the most saline brine we collected. Dissolved Pb and Zn have been detected in concentrations greater than I ppm only in wells from the Stuart City Reef Trend and n o t from shallower zones. The most saline wells, the Strawn No. 1 and Wernli No. 1, contain the highest concentrations of heavy metals (Fig. 6), and the Strawn well additionally contains fluorite as scale. Although the data are admittedly sparse, the distribution of both the metallic phases and high dissolved metal values are consistent with an out-of-the-basin fluid movement. The source for the metals is unknown. Since the metals are associated with the bank margin, n o t shallow shelf deposits, concentration of the metals by algal mats (Davis, 1977) is n o t indicated. (3) Hydrocarbon typing suggests t h a t a significant fraction of the oil being produced from Edwards fields was sourced from Jurassic rocks (Prezbindowski, 1981). Again, out-of-the-basin transport is indicated.
70 CONCLUSIONS We present our conclusions with regard to the three " m o d e l s " we previously suggested as possible origins for the Edwards brine: (1) Arguments against brine formation by basinward recharging meteoric water (model 1), or modification of connate Cretaceous meteoric water or Cretaceous seawater (model 2): (a) No source of NaC1 is known from Edwards shelf carbonates. Very minor amounts of gypsum are known from outcrops of the inland shelf, or from the fault troughs (anhydrite). No evaporites are known to exist or have been postulated to have existed associated with the bank edge where the most saline brine is n o w found. (b) The "Bad Water" line is a sharp feature, not a gradual downdip increase in TDS. The "Bad Water" brine has preserved textures in the rocks which have been destroyed updip by the active meteoric regime today (Longman and Mench, 1978), indicating that the active meteoric regime has never extended further downdip than it does today. Additionally, the "Bad Water" contains abundant H2S , and occasionally hydrocarbons, facts inconsistent with a dominant basinward recharge, but consistent with sulfate reduction by upward-moving hydrocarbons on encountering the meteoric regime. (c) It is impossible for water to evolve to the Mg/Ca ratio of the brine by dolomitization, and all brine deeper than ~ 1 km is undersaturated with dolomite and is, in places, dedolomitizing. It is unlikely that any possible connate solution had such a low Mg/Ca ratio as observed in any brine below ~ 1 km. (d) Large volumes of water would be necessary to concentrate chloride from a dilute water by any mechanism, including membrane filtration. Such large volumes of water would have caused extensive isotopic reequilibration of the rocks, y e t the rocks largely retain their early diagenetic chemistry (Prezbindowski, 1981}. Therefore it is impossible to involve large volumes of isotopically depleted water in any postulated origin for the Edwards brine. (2) Arguments against brine formation by evolution of a connate Cretaceous brine (model 2): (a) No evaporites are known from the bank-edge part of the section where the brine is most saline t o d a y [see (1), (a) above]. (b) The Mg/Ca ratio of the brine is too low to have evolved to its present composition by dolomitization [see (1), (c) above]. (c) The volume of brine buried with the Cretaceous sediment, even if it had escaped the extensive Cretaceous meteoric diagenesis evidenced in the rocks, is insufficient to supply the volume of brine in the rocks today. (d) The a m o u n t of carnallite required relative to halite to yield the Ca content of the Edwards brine is much higher than observed in evaporite sections world-wide. Subsurface dolomitization, required to replace Mg
71
200"1
-
"
"
300 ~
•
o 22 H
Fig. 7. Schematic north--south cross-section through San Antonio and the Muil Field (see Fig. 1). Jurassic and Lower Cretaceous sediments (the " a q u i f e r " ) are shown by the limestone pattern since most are carbonates. They are overlain by Upper Cretaceous clay and chalk. The northern-most outcrops are associated with the Balcones Fault zone near San Antonio. The shallow part of the section is underlain by Paleozoic "b asem en t ", or by Triassic rocks further south. Jurassic salt is shown by solid h a t c h u r e s . D a r k e n e d areas at A, B and C depict the three producing zones, the Stuart City Reef Trend (the carbonate platform margin), the Karnes Trough and the Atascosa Trough, respectively. Note the faulting associated with them. The temperature axis is speculative below ~ 5 km.
derived from carnallite by Ca, is u n k n o w n in the study area and in fact the reverse reaction is occurring. (3) Arguments in favor of basinal origin of the Edwards brine (model 3): (a) Evaporites are abundant in the deep Gulf of Mexico, and albitization is a d o c u m e n t e d reaction of major significance. (b) The distribution of Mg in the brine is best explained by dedolomitization as the brine moves updip. Dedolomite has been found, and its isotopic chemistry is consistent with its formation in the present aqueous regime. (c) Galena and dissolved heavy metals are associated only with the most saline brines at the bank edge, indicating a basinward origin, and a model for their origin is consistent with at least some current theories (Banaszak, 1979). (d) Wells in the Stuart City bank edge flow, and the hydrodynamic drive is updip. (e) Hydrocarbons produced from the Edwards Formation have a strong Jurassic c o m p o n e n t indicating a downdip source. Fig. 7 presents a cross-section of the study area. We interpret the flow paths of the Edwards brine t o d a y to be up-fault and up-dip. We presume the samples from ~ 4 . 2 km to tap a slightly different source of brine than the shallower samples because of small but systematic differences in 8D (Fig. 6) and C1/Br ratio (Table I). It would be tempting to account for the distribution of hydrocarbons in this area by hydrodynamic trapping, but we are unconvinced that sufficient water m o v e m e n t c.an be taking place to accomplish migration. We face the same dilemma as faced by studies of sandstone diagenesis (e.g., Land and
72
Dutton, 1979), namely that some evidence (in this case the hydrocarbon distribution) favors extensive water movement, whereas other evidence (lack of extensive oxygen-isotopic modification of the country rock) does not. The volume of flow in this gigantic aquifer is, we believe, a major outstanding problem.
ACKNOWLEDGEMENTS
This research was supported by the National Science Foundation, NSF EAR-76 17774, and the Geology Foundation of the University of Texas. We sincerely thank the many oil and gas companies, and especially their production crews and supervisors, for their cooperation and patience in permitting and helping us to obtain water samples. Alden Carpenter provided the heavy metals analyses on samples collected for him at his request. Rick Schatzinger first detected the galena in the Strawn No. 1 core. Karl Hoops performed the chemical analyses and spent considerable time perfecting the bromide analytical method. We thank him for his careful, reliable analytical work. Charles Kreitler kindly pointed out editorial and logical flaws in an early manuscript.
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