Geochimlca CI Cosmoclt~n~rcu.4c1u Vol. 56. pp. 4315-4320 0016.7037/92/$5.00 + .oo Copyright 0 1992 Pergamon Press Ltd. Printed in U.S.A.

LETTER

The question of uniqueness in geochemical modeling

CRAIG M. BETHKE Depa~m~nt of , University of Illinois, Urbana, IL 6 1801, USA

(Received June 24, 1992; accepfcd in revised,form Septambcr 29. 1992)

Abstract-Contrary to prevailing opinion, geochemical models that predict the distribution of species in aqueous fluids do not always give unique results. To constrain ~lculations of this type, geochemical modelers commonly set the activities of certain dissolved species or of certain gases and many times assume equilibrium between the fluid and one or more . In such cases, there may be distinct geochemical systems that satisfy equally well the conditions posed by the modeler. The modeling software may locate any of the roots to the governing equations, depending on the point at which iteration begins. Modeling algorithms may favor discovery of one of the roots, but not necessarily the root that is most geochemically meaningful or most appropriate to the problem at hand.

INTRODUCTION constrains the fluid’s composition. The reasons include a lack of analytical data for certain components, suspected error in branches of the field have come GEOCHEMISTS INNEARLY ALL the analysis, a need to adjust the model results to compensate to depend on calculated models of the equilib~um states of for error in the therm~ynamic data or the computation of multicomponent fluids to interpret the chemistries of waters activity coefficients, or the presence of certain elements in in nature and in the laboratory, and to trace the reaction too small to detect with the analytical methods processes by which these waters evolve (e.g., et NORDSTROM in use. He might, for example, assume that the fluids silica al., 1979; PLUMMER et al., 1983). In calculating the equilib- content reflects equilib~um with quartz or tridymite and that rium dist~bution of species in a water, the geochemist gen- aluminum is controlled by equiiib~um with erally assumes that his result is unique (i.e., that there is a a clay ; his model, therefore, is posed in terms of a single mathematical that honors the constraints he combination of mass balance and mass action constraints. has placed on the problem). to the multicomponent In this letter, I show that when the muIticomponent equi- equilibrium problem are, except in highly nonideal cases, librium problem is posed by ~mbining mass balance and unique when the calculation is entirely constrained by the mass action constraints, as is common practice in geochemical lluid’s bulk composition. In this case, the modeler sets the modeling, the calculated answers are not always unique. mole number of each thermodynamic component in the fluid. There are sometimes multiple, physically meaningful roots Such constraints on the fluid’s bulk composition are known that, in the mathematical sense, satisfy the problem equally as mass balance constraints. well. Rest&s in such cases depend on the starting point chosen In practice, geochemical modelers define their calculations for iterating to a solution, which is selected arbitrarily by the by mixing mass action constraints with mass balance con- modeling software. Further, there is no reason to believe that straints. Mass action constraints include measurements of numerical methods in use today tend to locate the most geo- the activities of individual species, such as pH and pe, and chemically reasonable root to a nonunique problem. species concentrations, such as dissolved oxygen content. Each a~umption that a mineral co-exists at equilib~um with UNIQUENESS IN CHEMICAL MODELING the water also constitutes a mass action constraint, as does each assumption of the of a gas. That a set of equations has a unique root can sometimes A modeler might begin to define a problem with an analysis be proved mathematically. Such a proof is valuable because of a fluid’s composition. The analysis provides a measure of methods of solving the equation set can be designed without the total concentrations of components such as Na+, K”, fear that the computation process wili influence the calculated Ca++ , Cl-, and SO;-. The total sodium concentration ZNa+, result. When iterative methods, such as those applied in geo- for example, includes the molality of the element present in chemical modeling, are used to solve an equation set with a the free ion Na+ as well as in complexes such as NaSO;, unique root, for example, there need be no concern that NaHCO,, NaCl, and NaOH. Lacking a measurement of the choice of the starting point for the iteration affects the answer. total concentration pYH+of the H+ component (i.e., TOTH Some equation sets have multiple roots, each of which in the notation of MOREL, 1983), the modeler might set pH may or may not be physically meaningful. Once a root has to a value measured with an electrode, or fix pH by setting been located, it is necessary to consider whether the answer the fluid’s CO* fugacity. For various reasons, the modeler is meaningful and whether a preferred root might be found may assume that equilibrium with one or more minerals with further effort. When locating roots by iteration, the mo- 4315 4316 C. M. Bethke deler may need to repeat the iteration from a number of EXAMPLES OF NONUNIQUE SOLUTIONS starting points that span the range of meaningful values; even In the following sections I give three examples, chosen for then there is no assurance that all possible solutions, or even their simplicity and familiarity to geochemists, in which two the most meaningful one, will have been found. fluids of differing composition satisfy the constraints of Chemical modeling developed from efforts to calculate the . In each example I set a combination thrust of rocket fuels (see historical sketches by ZELEZNIK of mass balance and mass action constraints and use a com- and GORDON, 1968; VAN ZEGGEREN and STOREY, 1970; puter model to converge by iteration to a fluid chemistry that SMITH and MISSEN, 1982). By calculating the equilibrium honors those constraints. Depending on the values chosen volume of a combusted fuel from its bulk composition, theo- to start the iteration, the program converges to one of two reticians could determine thrust directly, saving the expense fluid chemistries that equally well satisfy the definition of of testing fuels individually. Mathematicians (e.g., BRINKLEY, chemical equilibrium. 1947; WHITE et al., 1958; BOYNTON, 1960) applied consid- I used the computer program React, developed at the Uni- erable effort in analyzing the multicomponent equilibrium versity of Illinois ( BETHKE, 1992), to perform the calcula- problem, including the uniqueness of its roots. WARGA tions. React, like most programs of this class (WOLERY, ( 1963) considered the problem of a thermodynamically ideal 1983), locates roots to the governing equations using Newton- solution of known bulk composition. He proved that the free Raphson iteration, The program converged in each case to energy surface representing the sum of the free energies of within one part in 10 lo, the approximate limit of numerical individual species, when traced along trajectories satisfying precision that can be carried on a 64 bit computer. The cal- mass balance, is concave upward. The surface, hence, has a culations, based on thermodynamic data from the dataset single minimum point and therefore a unique equilibrium compiled at Lawrence Livermore Laboratories ( DELANYand state. VAN ZEGGEREN and STOREY ( 1970, pp. 31-32) cite LUNDEEN,1989), employ the extended Debye-Htickel equa- several other studies that offer mathematical proofs of tions ( HELGESON,1969) to compute, as a function of ionic uniqueness. strength, I, activity coefficients for aqueous species. As I Modeling techniques were introduced to demonstrate, the results are true roots of the multicomponent through the hand calculations of GARRELSand THOMPSON equilibrium problem that could be located by any computer ( 1962) and GARREL~and MACKENZIE( 1967), and computer model; they are not specific to the React code, the choice of programs written by HELGESON( 1968) and HELGESONet thermodynamic data, or the method for calculating activity al. ( 1969, 1970). A considerable number of numerical meth- coefficients. ods have since been developed to solve the multicomponent equilibrium problem for geochemical systems (e.g., KARPOV Example l-Calcium Content Set by Equilibrium with and KAZ'MIN, 1972; KHARAKA and BARNES,1973; TRUES- Fluorite DELLand JONES, 1974; CRERAR, 1975; WOLERY and WAL- TERS, 1975; WESTALLet al., 1976; BALLet al., 1979; WOLERY, We first consider a fluid of known F concentration whose 1979; PARKHURSTet al., 1980; REED, 1982; LEHMANNand Ca content is set by assuming equilibrium with fluorite FABRIOL, 1989). Despite the attention devoted to these geo- ( CaF2). Two roots to the problem arise because of the spe- chemical models, the literature contains virtually no discus- ciation of F. In dilute solutions, the free ion F- dominates sion of the question of uniqueness, beyond citing the early among F species. The reaction proofs. The uniqueness proofs, however, are limited in two regards: CaF2 @ Ca++ + 2F- (1) they ( 1) consider thermodynamically ideal solutions and (2) fluorite assume that, as in the rocket fuel problem, the calculation is requires that the Ca content vary inversely with F concen- posed in terms of mass balance constraints. In materials sci- ence, CARAM and SCRIVEN (1976) and OTHMER (1976) tration. By reaction 1, increasing F concentration leads to less calcic solutions, and vice versa. showed that nonideal solid solutions can have local minima in free energy in addition to the global minimum; the local In Ca-rich fluids, especially at elevated , the minima represent metastable equilibria. Metastable equilibria CaF* ion pair can be more concentrated than the F- ion. satisfy the governing equations of a chemical model, so mod- The CaF+ activity exceeds that of F- at 200°C whenever the els applied to these solid solutions would have nonunique activity of Ca ++ is greater than about 10m3; at 300°C the roots. In geochemistry, activity relations in aqueous systems ion pair is favored at Ca+’ activities as small as 1O-4.8. Where are perhaps more linear than those in some solid solutions, CaF+ dominates, the reaction and there have been no reports of metastable equilibria re- CaFz + Ca++ F? 2CaF+ sulting from nonideality. Nonetheless, this question bears (2) fluorite further consideration, especially with respect to brines, which are complex thermodynamically (e.g., PITZER, 1987 ). controls fluorite solubility. Reaction 2, in contrast to reaction The second limitation of the uniqueness proofs, however, 1, requires that fluids become proportionally richer in Ca as is critical. Geochemical models differ from chemical models their F contents increase. applied in other fields in that they are commonly posed in Fluids ofthe same F content but two distinct Car+ activities terms of mass action as well as mass balance constraints. In can, by these reactions, exist at equilibrium with fluorite. A geochemical modeling, the system’s bulk composition is sel- plot of fluorite stability (Fig. 1) shows that the mineral is dom known completely; contrary to prevailing opinion, the soluble at low and high Ca activities. At 200°C setting the uniqueness proofs do not apply in such cases. activity of dissolved F to 10-3.5 allows two equilibrium ac- Uniqueness of geochemical models 4317

suspect that the iteration favors discovery of the roots that are most physically meaningful.

Example 2-pH Set by Mineral Equilibrium

A common difficulty in constructing a geochemical model (e.g., REED and SPYCHER, 1984) is determining the in situ pH of a hot fluid, perhaps one sampled from a hydrothermal experiment or a deep well. Direct measurement by electrode is difficult or impossible, so the modeler must estimate the pH on the basis of other information. One alternative is to assume that pH is fixed by equilibrium with a mineral present in the subsurface or experiment. Since the solubilities of alu- minum hydroxide and aluminosilicate minerals vary strongly with pH, these phases are commonly employed in such cal- culations. This choice is perilous, however, because many aluminum minerals, including the clays and micas, are am- y300°c 200°C photeric and hence equally soluble at high and low values 4.5 I I I I * I -7 -6 -5 -4 -3 -2 -1 of pH. log a Ca++ Consider as an example a fluid extracted from a hydro- thermal experiment in which boehmite ( AlOOH) reacts with FIG. 1. Reactions controlling the equilibrium solubility of F in a fluid at 200°C. Under acidic conditions, AlOH++ and as a function of Ca ion activity at 200 (solid lines) Al( OH): predominate among aluminum species (Fig. 2). and 300°C (dashed lines). Vertical axis shows log activity of the predominant F-bearing species in aqueous solution. Fluorite is soluble On the basis of an analysis of the fluids aluminum content, at a specific activity (horizontal line) either as F- at small Ca++ pH is set by the reaction activity (point A) or as CaF+ at high Ca++ activity (point B). AlOOH + 2H + 9 Hz0 + AlOH ++ boehmite

tivities ofCa++: 10m4.3and 1O-‘.6 (points A and B); the cor- or responding activities at 300°C are lO-‘j’ and 10m3-‘ . Geochemical modeling software can discover either of the AIOOH + H+ S Al(OH);. solutions, depending on the starting point for iteration. Table boehmite I shows dual solutions reached for a hypothetical fluid having The hydroxy species Al(OH), dominates under alkaline a pH of 5 and known Na and F concentrations. The Cl con- conditions, where the reaction centration is set by ionic charge balance, as is common prac- tice in geochemical modeling. At each temperature, the it-

eration finds a solution at low Ca content in which F- is the -3 7 dominant F-bearing species, and another at high Ca concen- 200°C tration in which most F exists in the CaF+ species. In my tests, Newton-Raphson iteration tended to converge toward the solution at low Ca content unless I carefully ma- -4 nipulated the starting point for iteration. At 3OO”C, for ex- ample, the easily found solution has a Ca content of about 10e4 molal. This concentration is improbably small for a natural fluid of this salinity, but the iteration was unlikely to find the more reasonable answer in which the Ca concentra- tion was about 10-l molal. There is, therefore, no reason to

Table1. Non-unique solutions calculated for fluids at 200°C and 300°C in -6 equilibrium with fluorite.

200°C 300°C Solution 1 Solution 2 Solution 1 Solution 2 PH 5.0 5.0 5.0 5.0 -7 I I C Ca++ (molal) .0026 1.0 .00013 .090 2 3 4 5 6 7 8 BF- .80x10“ .80x10m3 .80xW3 .80x10-’ PH free F- .34x10-’ .021x10-3 .73x10-3 .027~10-~ free CaF+ .050x 1o-3 .75x10-3 .027x10m3 .77x10-3 FIG. 2. Equilibrium activities of Al-bearing species in aqueous B cl- * 1.0 3.0 1.0 1.2 solution as a function of pH at 200°C in the presence of boehmite C Na+ 1.0 1.0 1.0 1.0 (solid lines) and kaolinite plus quartz (dashed lines). Aluminum is I (molal) .91 2.7 1.0 1.1 soluble at a specific activity (horizontal line) either under acidic con- ditions as species Al(OH):, AlOH++, or Al+++ (e.g.. point A) or *Set by charge balance under alkaline conditions as AI( (point B). 4318 C. M. Bethke

AlOOH + 2HzO -i; Al(OH); •l- H’ boehmite controls pH. Hematite To calculate a model of the fluid under experimental con- ditions, we use the fluid’s composition as might be determined by chemical analysis and assume that the experimental pH was constrained by equilibrium with the boehmite. In the calculation, we take a 0.1 molal KC1 solution containing small amounts of dissolved silica and carbonate. The ~uminum G-55 concentration is 10 pmolal, and pH is set by equilibrium 0 with boehmite. The calculation results (Table 2) show that the model may converge to one root at low pH, where AlOH++ and Al(OH): predominate, or to another at high pH, where most Al is present as Al(OH);. Similar results could be obtained, for example, by assuming that equilibrium with kaolinite and quartz fixes pH and silica activity in a ______~~~ fluid of known Al content, or that muscovite and quartz fix b I * , these values when the Al and K contents are given. 2 3 4 6 7 8

Example 3--O* Fugacity Set by Sulfide Equilibrium FIG. 3. -pH diagram drawn at 100°C for the Fe-S-H20 sys- tem, showing speciation of S (dashed line) and the stability fields of An additional problem encountered in geochemical mod- Fe minerals (solid lines). Calculation assumes that S and Fe species eling is constraining the redox state of a natural water. Direct are present at activities of 1O-‘3and 10e4, respectively. Broken line information is not always available, so the modeler may elect at bottom of diagram is the stability limit of water at I atm . to set the oxidation state by equilibrium with a mineral be- At pH 4 (vertical line), there are two oxidation states (points A and lieved to be in contact with the fluid. Consider as an example B) in equilibrium with pyrite under these conditions. a reduced fluid of known Fe and S content assumed to be in equilib~um with pyrite. Pyrite sulfur, which is in the S’- oxidation state, may dissolve by oxidation to sulfate ( S6+) for instance, by setting fo, in terms of the fugacity of a gas of inte~ediate oxidation state, such as Nz or SO*. Fe& + Hz0 + 7/202(g) Ft Fe’+ + 2SO;- + 2H+ pyrite DISCUSSION or by reduction to HzS( S*- ) Calculated models of the chemistries of natural waters, as is broadly recognized, may differ from reality for a number Fe!& + Hz0 + 2H+ F! Fe+’ f 2HzS + ‘/2O2(g). of reasons. A model may be based on an inaccurate chemical pyrite analysis or erroneous or incomplete thermodynamic data. The methods for determining activity coefficients for aqueous As such, there are two redox states that satisfy the assumption species may not be sufficiently accurate, or the software may of ~uilib~um with pyrite (Fig. 3). not converge to within a smail enough tolerance to compute Table 3 shows the results calculated for a hypothetic~ 1 the concentrations of minor species. One or more reactions molal NaCI solution with Fe and S in IO mmolal concentra- in the real system may not approach equilibrium for kinetic tion. We set pH to 4 and assume that equilibrium with pyrite reasons, contradicting the assumptions implicit in computing controls oxygen fugacity. Depending on the starting values the model. In a thermodynamically nonideal system, as noted for the iteration, the model converges either to a fluid dom- above, it is even possible for the modeling software to locate inated by sulfate with an fc, of 10W50or an H,S-dominated a local rather than global minimum in free energy, and hence fluid whose fo, is 1O- 67. Analogous results can be obtained,

Table 3. Non-unique models calculated for a Table 2. Non-unique modek calculated for a fluid at 100°C in equilibrium with pyrite. fluid at 2OO’C in ~oili~om with bcehmite. Solution I Solution 2 Solution 1 Solution 2 PH 3.1 6.3 PH 4.0 4.0 -50 -67 Z K+ (molal) .I0 .lO h% fo, B Al+++ lox1o-6 10X10” Z Fe+’ (molal) ,010 ,010 free AlOH++ 4.3x10-6 1.5x10-‘* zs .OlO ,010 free Al(OH)f 3.0x10-6 1.8x10-9 zso~- .OlO 3.3x10-7 free Al( 5.5x10-9 9.2~10” XHzS(aq) .50x10-~ .OlO x cl- * .I0 .lO B SO, 3.0x10-3 3.oxio-3 Z Na+ (modal) 1.0 1.0 E HCOi 60x10-6 60~10~ Tel- * 1.0 1.0 I (mob@ .I0 .I0 I (molal) 1.0 1.0 *Set by charge balance *Set by charge balance Uniqueness of geochemical models 4319

find a metastable instead of stable equilib~um state. Such a geology Program: Amoco Production Co., Arco Gil and Gas Co., case, however, has yet to be reported. British Petroleum Res., Conoco, Inc., Du Pont Co., Exxon Production Res. Co., Illinois State Geological Survey, Japan National Oil Co., I consider in this letter a different source of error: the ex- Lawrence Livermore National Labs, Marathon Oil Co., Mobil Res, istence, even in thermodynamically ideal systems, of multiple and Dev. Co., Sandia National Labs, Texaco, Inc., and Union Oil or nonunique roots to the chemical equilibrium problem. It Co. of California. is clear from the examples above that nonunique roots arise Edit~)rial handling: G. Faure from the nature of the equilibrium problem because of the variety of dissolved species that can be present in aqueous REFERENCES solution. In each example, equilibrium with a mineral (flu- BALL J. W.. JENNEE. A., and NORDSTROMD. K. ( 1979) WATEQ2- orite, boehmite, or pyrite) set the activity of an aqueous spe- A computerized chemical model for trace and major element spe- cies(Ca++, H+ , or Oz). The controlling reaction can be writ- ciation and mineral ~quilib~a of natural waters. In Chemical ten in various ways, depending on which species predominate ~~(~dei~ng@Aqueous Systems (ed. E. A. JENNE); Amer. Chem. in solution. Dual roots occur when the reaction can be written Sot. Sv&osium Series 93, pp. 8 15-835. BETHKEC. M. ( 1992j The Geochemist’s Warkbench. A User’s Guide with the buffered species on either the reaction’s left or right to Rxn, A&. Tact, React, and Gtplot. Program, side. University of Illinois. The nonunique roots do not represent metastable ~uilib~a BOY~\;TONF. P. ( 1960) Chemical ~uilib~um in multicom~nent in a single fluid, nor do they reflect choice of modeling soft- polyphase systems. f. Chem. Phys. 32, 1880-l 88 1. ware, thermodynamic data, or the method for computing BRINKLEYS. R., JR. (1947) Calculation of the equilibrium com- position of systems of many components. J. Chem. Phys. l&107- activity coefficients. Instead, the dual roots represent the true 110. equilibrium states of two fluids of differing bulk composition CARAMH. S. and SCRIVEN L. E. ( 1976) Non-unique reaction equi- whose chemistries equally well satisfy the constmints imposed libria in non-ideal systems. Chem. Eng. Sci. 31, 163-168. on the calculation. CRERAR D. A. ( 1975) A method for computing multicomponent chemical equilibria based on equilibrium constants. Geochim. The solutions in the first example differ sharply in Ca con- Cosmochim. Acta 39, 1375-1384. tent and ionic strength. In the second example, one solution DELANYJ. M. and LUNDEENS. R. ( 1989) The LLNL thermochemical is acidic and one alkaline; in the third, S is present either as database. Lawrence Livermore National Laboratory Report sulfate or H& In each case, the most approp~ate root could UCRL-2 1658. probably be selected on the basis of additional information GARREL~R. M. and MACKENZIEF. T. ( 1967) origin of the chemical compositions of some springs and lakes. In Equilibrium Concepts about the chemical system (for instance, which minerals be- in Natural Waters: Advances in Chemistry Series 67, 222-242. sides the buffer co-exist with the fluid, or whether H2S was Amer. Chem. Sot. detected). A geochemist relying on the widespread belief of GARRELSR. M. and THOMPSONM. E. ( 1962) A chemical model uniqueness, however, would not know to search further if for seawater at 25°C and one atmosphere total pressure. Amer. f. Sci. 268, 57-66. the model produced an inappropriate answer. HELGESONH. C. ( 1968) Evaluation of irreversible reactions in geo- It would be comforting to believe that the modeling soft- chemical processes involving minerals and aqueous solutions-I. ware tends to locate the most geochemically reasonable root Thermodynamic relations, Geochim. Cosmochim. Acta 32, 853- to a nonunique problem. As previously noted, however, the 877. model in the first example strongiy tended to find the root HELGES~NH. C. ( 1969) Therm~ynamics of hydrothermal systems at elevated and . Amer. J. Sci. 267, 729- with a low Ca content, even though for a natural water this 804. fluid seems unrealistically poor in Ca. In the second and third HELGESONH. C., GARRELSR. M., and MACKENZIEF. T. (1969) examples, my tests readily located either root, depending on Evaluation of irreversible reactions in geochemical processes in- where the iteration began. A modeler should not rely on ease volving minerals and aqueous solutions-II. Applications. Geo- chim. 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For this reason, a geochemist LEHMANNJ. and FABRIOLR. (1989) CEQCSY: un nouveau code should consider carefully the nature of the reactions with de calcul d’kquilibre des svst?mes muhit?hash. Commission of European Communities Report EUR 12299 FR. which he constrains a chemical model of a natural water; MORELF, M. M. f 1983) Principles o~,~q~at~~chemistry. J. Wiley. stability diagrams such as those in Figs. l-3 are useful for NORDSTROMD. K., PLUMMERL. N., WI~LEY T. M. L.. WOLERY this purpose. The modeler should write each buffering re- T. J., BALL J. W., JENNE E. A., BASSET R. L., CRERAR D. A., action as he believes it operates in nature or the laboratory FLORENCET. M., FRITZ B., HOFFMAN M., HOLDREN G. R., JR., LAFON G. M., MATTICO~ S. V.. MCDUFF R. E.. MOREL F.. and verify that this form of the reaction also controls his REDDY M. M., SPOSITOG., and THRAILKILLJ. (1979) A com- calculation results. parison of computerized chemical models for equilibrium calcu- lations in aqueous systems. In ChemiealModeiingqfAqueous Sys- Acknowledgments--I thank Paul Barton, Grant Garven, Blair Jones, tems (ed. E. A. JENNE); American Chemical Society Symposium Ming-Ku0 Lee, Mark Reed, and J. Donald Rimstidt for their Series 93. pp. 857-892. thoughtful reviews. This study was funded by NSF grants EAR 8% OTHMERH. G. f 1976) Nonuniqueness ofequilibria in closed reacting 52649 and EAR 86-01178. Computing facilities were provided systems. C&em. Eng. Sei. 31,993-1003. through the generosity of a consortium of affiliates to the Hydro- PARKH~JRSTD. L.. THORSTEN~ND. C., and PLUMMERL. N. ( 1980) 4320 C. M. Bethke

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