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LETTER the Question of Uniqueness in Geochemical Modeling

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LETTER the Question of Uniqueness in Geochemical Modeling 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 Geology, 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 fugacities of certain gases and many times assume equilibrium between the fluid and one or more minerals. 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 concentrations 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 concentration is controlled by equiiib~um with erally assumes that his result is unique (i.e., that there is a a clay mineral; his model, therefore, is posed in terms of a single mathematical solution that honors the constraints he combination of mass balance and mass action constraints. has placed on the problem). Solutions 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 fugacity 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 chemical equilibrium. 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 geochemistry 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 temperature, the minima represent metastable equilibria. Metastable equilibria CaF* ion pair can be more concentrated than the F- ion.
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