Use of redundant data to reduce estimation errors in geochemical speciation

Use of redundant data to reduce estimation errors in geochemical speciation F. De Gasparia,b,∗, M.W. Saaltinka, J. Carrerab, L.J. Slootenb,c aGHS, Department of Geotechnical Engineering and Geosciences, Universitat Politecnica de Catalunya, UPC-BarcelonaTech, c/Jordi Girona 1-3, 08034 Barcelona, Spain bGHS, Institute of Enviromental Assessment and Water Research (IDÆA), CSIC, c/Jordi Girona 18, 08034 Barcelona, Spain cAlten Nederland, Rivium 1e straat 85, 2909 LE Capelle a/d IJssel, Nederland Abstract Speciation is the process of evaluating the concentrations of all the species in a chemical system from equilibrium conditions and measured data such as total concentrations of components, electrical conductivity, pH, redox potential, gas partial pressure. It is essential for analyzing geochemical data and defining the chemical composition of waters for geochemical modeling problems like evaluating the chemical composition of evaporating, diluting, mixing waters or reactive transport. We present an algorithm that reduces estimation errors in chemical speciation calculations by means of the use of redundant data. Redundant data are measurements and assumptions that exceed the minimum data set required, and therefore are not strictly necessary, to speciate a water sample. The proposed method was compared with the classical speciation algorithm on two synthetic examples. Our re- ∗Corresponding author Email addresses: [email protected] (F. De Gaspari), [email protected] (M.W. Saaltink), [email protected] (J. Carrera), [email protected] (L.J. Slooten) Preprint submitted to Applied Geochemistry November 17, 2014 NOTICE: this is the author’s version of a work that was accepted for publication in Applied Geochemistry. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Applied Geochemistry, [VOL 55, (April 2015)] DOI 10.1016/j.apgeochem.2014.12.005•¨ sults show that using redundant data improves speciation results reducing the estimation error between computations and measurements. Moreover, the larger the amount of redundant data, the better in terms of errors of the estimated concentrations. Keywords: Redundant data, Geochemical modeling, Speciation, Optimization problem 1. Introduction1 Geochemical modeling is important in Earth Sciences. In particular, it is2 required to assess problems ranging from weathering to the characterization3 of the chemical composition of water and processes that could influence its4 quality (Appelo and Postma, 2010; Bethke, 2008). Geochemical speciation5 is a key step of geochemical modelling that consists of evaluating concentra-6 tions of all the species in a chemical system from measured data (e.g., total7 concentrations of components, pH, alkalinity, gas partial pressures, electrical8 conductivity, redox potential) and equilibrium constraints. For this reason,9 it is often termed thermodynamic speciation.10 Speciation requires the solution of a non-linear system of equations and11 a lot of research has been focused on numerical issues that might arise when12 solving these equations. Several methods have been proposed to solve chem-13 ical equilibrium in a robust way in order to guarantee the convergence (Paz-14 Garc´ıa et al., 2013; Carrayrou et al., 2002; Brassard and Bodurtha, 2000)15 and many codes have also been released to deal with geochemical specia-16 tion calculations: GEMS3K (Kulik et al., 2013), Visual MINTEQ (Gustafs-17 son, 2011), CHEPROO (Bea et al., 2009), ORCHESTRA (Meeussen, 2003),18 2 MIN3P (Mayer et al., 2002), PHREEQC (Parkhurst et al., 1999) and its19 interactive version, PHREEQCi (Charlton et al., 1997), EQ3NR (Wolery,20 1983, 1992) and WATEQ4F (Ball and Nordstrom, 1991).21 Speciation calculations are subject to implicit sources of uncertainty which22 can derive from uncertainty in thermodynamic data, such as equilibrium23 constant values, or from errors in chemical analyses (i.e., analytical errors).24 These types of random errors can be referred to as ”aleatory uncertainty”.25 Misjudgment in the definition of the chemical system, such as failure to ac-26 count for some reactions or discarding others, can also lead to errors in speci-27 ation. These can be defined as ”epistemic uncertainty”. They arise from an28 incomplete or inadequate characterization of the system (Gupta et al., 2012),29 such as assuming the neutrality of a solution when it is not electrically bal-30 anced, or imposing equilibrium with phases that are not. The effect of errors31 propagation in geochemical calculations has been extensively studied. In32 particular, the effect of aleatory errors has been investigated by Weber et al.33 (2006); Denison and Garnier-Laplace (2005); ¨ Odegaard-Jensen et al. (2004);34 Nitzsche et al. (2000); Cabaniss (1999, 1997); Criscenti et al. (1996); Merino35 (1979), while Smith et al. (1999) examined the connection between aleatory36 and epistemic errors. Although the origin and propagation effects of both37 types of errors are different, they can be treated in the same way through38 probability density functions, e.g., by means of mean and standard deviation39 values.40 All these studies use a fixed number of data to solve the speciation. Geo-41 chemical speciation, in fact, requires a fixed minimum number of data, includ-42 ing equilibrium assumptions, equal to the number of independent variables43 3 of the system (i.e., number of species). For example, a carbonate system is44 characterized by four degrees of freedom (see Section 2.1). Therefore, four45 data (e.g., total concentrations of inorganic carbon and calcium, pH) or hy-46 potheses (e.g., water activity equal to 1) are needed. However, extra data47 might be available (e.g., alkalinity, electrical conductivity or redox potential)48 or extra assumptions about the system might be made (e.g., equilibrium with49 calcite or CO2(g)in equilibrium with the atmosphere). Chemical analyses of50 waters, for example, often provide extra data and also the analytical errors51 associated to each of them.52 We term these extra data as redundant and we claim that speciation53 calculations can benefit from their use, while aknowledging analytical errors.54 The aim of this paper is to present an algorithm to include redundant data55 in speciation calculations and to prove that their use can improve the results56 by reducing estimation errors. We also claim that increasing the number of57 redundant data helps decreasing the estimation errors even further.58 2. Methodology59 We start by analyzing a speciation example to clarify the differences be-60 tween the traditional and the proposed method. This allows us to formalize61 the problem statement and to propose a solution algorithm.62 2.1. Speciation of a carbonate system63 We consider the problem of calculating the concentrations of dissolved64 species in a carbonate system. This system has received extensive attention65 from the scientific community, e.g. to study seawater intrusion in carbonate66 coastal aquifers (Werner et al., 2013; Bear, 1999; Back et al., 1979, amongst67 4 many others), including geochemical processes occurring in the mixing zone68 between freshwater and saltwater (Sanz et al., 2011; De Simoni et al., 2007;69 Rezaei et al., 2005), and to analyze the feasibility of CO2sequestration in70 deep aquifers (Saaltink et al., 2013; Duan and Li, 2008; Xu et al., 2006).71 The most simple chemical system consists of 9 species (Ns= 9) and the72 following 5 equilibrium reactions (Nre = 5):73 OH−+H+H2Olog K1= 13.995 CO2− 3+H+HCO− 3log K2= 10.328 CO2(aq)+H2OHCO− 3+H+log K3=−6.344 CO2(g)+H2OHCO− 3+H+log K4=−7.813 CaCO3(s)+H+Ca2+ +HCO− 3log K5= 1.848 (1) The number of degrees of freedom of this system is Ns−Nre = 4. This74 means that 4 data or assumptions are needed to solve the speciation prob-75 lem. Speciation codes normally use this criterion. Optionally species with76 constant activity can be decoupled and eliminated, e.g., water if the system77 is sufficiently diluted (aH2O= 1) or proton if the pH is fixed (aH+= 10−pH ),78 to reduce the number of unknowns. Numerous methods have been pro-79 posed to eliminate constant activity species in reactive transport calculations80 (Kr¨autle, 2011; De Simoni et al., 2005; Kr¨autle and Knabner, 2005; Molins81 et al., 2004; Saaltink et al., 1998). Regardless of the decision to eliminate82 them, we refer generically to these methods as the traditional speciation83 methods, as they should all yield the same results.84 Being the degrees of freedom for system (1) equal to 4, the concentrations85 5 of all species can be calculated from four known data: total concentration of86 calcium, alkalinity, activity of water and pH for example87                Catot : [Ca2+]−x1= 0 Alkalinity : [HCO− 3] + 2[CO2− 3]+[OH−]−[H+]−x2= 0 Water Activity : aH2O−x3= 0 pH :−log aH++x4= 0 (2) where [ ] represents molal concentration (mol/kgw). x1,x2and x4are88 actual measurements representing Catot,Alkalinity and pH, while x3is the89 value of water activity fixed to 1. We term these kind of equations ”data90 equations”. These must be solved together with the mass action laws deriving91 from system (1)92 fMAL =Selog a−log k= 0 (3) where ais a vector containing the activities of the Nsspecies, Seis a ma-93 trix (Nre×Ns) with the stoichiometric coefficients of the equilibrium reactions94 and kis a vector (Nre) of equilibrium constants.95 Generalizing the traditional speciation method we can say that N1=96 Ns−Nre data equations need to be solved together with N2=Nre mass97 action laws, fMAL:98    g(c)−x= 0 fMAL(c) = 0 (4) where cis the vector of concentrations of the Nsspecies, xa vector of N1 99 data and g(c) defines operations to be applied to cin order to compute what100 6 is measured (e.g., linear combinations of species concentrations to obtain101 measured components, or −log(γH+·[H+]) to obtain pH, where γH+is the102 proton activity coefficient). Typically data equations contain balances of103 total concentrations, electrical charge, alkalinity, total dissolved inorganic104 carbon (TIC), pH values, redox potential or electrical conductivity.105 The traditional algorithm to speciate consists of five steps: (1) dividing106 the species in two sets of N1=Ns−Nre primary and N2=Nre secondary107 species (Steefel and Yabusaki, 2000) with concentrations c1and c2, respec-108 tively; (2) guess an initial value of primary concentrations; (3) use fMAL to109 calculate c2=f(c1); (4) use data xto solve g(c1,c2)−x= 0 for c1, (5)110 repeat steps (3) and (4) until convergence.111 This work is focused on cases in which the number of available data is112 larger than N1. In this case, the resulting data equations cannot be solved113 exactly. Instead, they need to aknowledge measurement errors.114 For example, if measurements of total dissolved inorganic carbon (TIC)115 and pressure of gas (PCO2(g)) were available and we wanted to apply zero116 charge balance and equilibrium with calcite as well, the data equations could117 be rewritten as118 7                                        Catot : [Ca2+]−x1=ε1 Alkalinity : [HCO− 3] + 2[CO2− 3]+[OH−]−[H+]−x2=ε2 Water Activity : aH2O−x3=ε3 pH :−log aH++x4=ε4 TIC : [CO2(aq)]+[HCO− 3]+[CO2− 3]−x5=ε5 PCO2(g): log aCO2(g)−x6=ε6 Charge Balance : [H+] + 2[Ca2+]−[OH−]−[HCO− 3]−2[CO2− 3] = ε7 Calcite Eq. : log aCa2+ + log aHCO− 3 −log aH+−log K5=ε8 (5) where x5is the measured TIC,x6is log(PCO2(g)) and x7and x8are119 equal to zero because of the zero charge balance and equilibrium constraints120 (x7corresponds to the saturation index of calcite, null at equilibrium). εi,121 i= 1, ..., 8, represent measurement errors that need to be taken in account122 since the system to be solved has become overdetermined (i.e., the number123 of data is larger than N1). The data set (5) presents 4 redundant data.124 The algorithm to solve data equations (5) together with mass action laws125 (3) to speciate is explained in the following section.126 2.2. Speciation with redundant data: Problem statement127 If redundant informations are used to solve a speciation problem, system128 (4) can be re-defined as follows129    g(c)−x=ε fMAL(c) = 0 (6) 8 The differences of system (6) from the traditional speciation problem130 defined in (4) are the dimension of gand x(dim(g) = dim(x) = Nd>131 N1) and errors in measurements εwhich are included. εcan incorporate132 analytical errors in data, such as in data 1 to 4 in system (5), and uncertainty133 about the correct model, such as charge balance and equilibrium with calcite134 assumptions in system (5).135 When solving speciation problems, it is common to use data equations136 which are either linear combinations of concentrations (e.g., TIC, alkalinity)137 or linear combinations of log-activities (e.g., equilibrium with minerals or138 gases). Moreover, the errors (ε) of both types of data equations can have a139 normal or log-normal distribution. Therefore, the expressions of g(c) must140 be defined and calculated accordingly to the types of data equations (see141 Appendix A for details).142 System (6) is overdetermined, therefore a non-linear least square fitting143 is required to minimize ε, as described below.144 2.3. Algorithm145 We want to find the solution of (6) that minimizes the sum Sof the146 weighted squares of the difference between measured and calculated data,147 defined as148 S=εtV−1ε(7) where Vis the covariance matrix (Nd×Nd) of measurement errors. With-149 out loss of generality, we will assume errors to be not correlated, so that V150 is a diagonal matrix, containing the variance of each i-th measurement, σ2 e,i.151 9 the σevalues of all data one at a time and calculated MSElog and V ar∗as273 function of the standard deviation of measurement errors of every constraint.274 We chose two values of σe: the first larger than the one used to generate the275 perturbed measurements (σe= 0.35 > σg), to simulate a higher uncertainty276 associated to the data, and the second smaller (σe= 0.09 < σg), to simulate277 more certain data values.278 3. Results279 3.1. Gypsum example280 Figure 3 shows the results of the traditional speciation method, i.e. using281 only log[Ca2+] data. It can be noticed that the five points moved on the282 equilibrium line, since the equilibrium with gypsum was imposed as certain283 condition, along a line parallel to the y-axis which represents the imposed284 calcium concentration data. In this case MSElog=0.28 and V ar∗=0.154.285 Note that V ar∗coincides with its expected value, σ2 e, once ln is converted to286 log10. Afterwards, the proposed method was tested, i.e., using both log[Ca2+]287 and log[SO2− 4] data. The results are shown in figure 4. It can be observed288 that while some of the points moved further from the exact solution with289 respect to the classical speciation results (white and black triangles), the290 others moved closer to the exact solution. However, for all the points the291 proposed algorithm minimizes the distance between measured and calculated292 data. The calculated mean squared error in this case was 0.23, smaller than293 0.28 obtained with the traditional method. Moreover, V ar∗=0.077, which294 coincides again with its expected value, σ2 e/2.295 The same methodology was employed to compare the two methods for296 16 the 1500 measurement points of figure 2 and the resulting mean squared error297 decreased from 0.029 for the traditional method to 0.016 for the proposed298 methods. The value of V ar∗also decreased by half, confirming its expected299 value: from 0.156 to 0.078 for traditional and proposed methods, respectively.300 3.2. Carbonate example301 The results of the comparison between the two methods in terms of302 MSElog are shown in figure 5. It can be noticed that the value of the mean303 squared error for the traditional method (solution 1) is barely larger than304 0.05, while it is smaller for the solutions using redundant data (solutions 2,305 3 and 4). Moreover, increasing the number of redundant data used in the306 speciation contributes to reduce more the MSElog value: it decreases from307 0.04 using only one redundant data (solution 2) to 0.016 using 3 redundant308 data (solution 4).309 Figure 6 shows the effect of changing the standard deviation associated310 to measurements (σe). Obviously its value does not affect solution 1, which311 is the result of a traditional speciation calculation. Neither does the value312 of σefor aCO2(g)have an effect on solution 2 (figure 6e) because this solution313 does not use aCO2(g)data. Nor does the σefor the assumption of equilibrium314 with calcite have an effect on solutions 2 and 3 (figure 6f), for the same315 reason. In general one can observe that the use of an incorrect standard316 deviation can worsen the solution with respect to the one obtained with the317 correct standard deviations. However, the quality of the solution in terms318 of MSElog improves using redundant data with respect to the traditional319 speciation, despite a wrong choice about σevalue. Decreasing the uncertainty320 relative to the equilibrium with calcite assumption (figure 6f) improves the321 17 solution even with respect to the one obtained with correct σevalues. This322 is because the correct value of σgfor this datum is 0, not 0.17 (see Table 2).323 The effect of σerelative to alkalinity and TIC (figure 6c, d) are very similar324 as their values are very close, due to the fact that in this pH range the325 concentration of HCO− 3is predominant with respect to carbonate species326 or OH−concentrations. Finally, it seems that changing the uncertainty327 relative to Catot (figure 6a) does not affect the solution. Nevertheless, when328 a large number of redundant data is used, such as in solution 4, the standard329 deviation seems to have a minor effect on the estimation error, MSElog.330 The effect of an incorrect value of σeon V ar∗was also analyzed. Only the331 results for aH+and alkalinity are reported in figure 7 as the most representa-332 tive. It can be noticed that the error variance can be big for the traditional333 speciation method (solution 1), while it slightly decreases when redundant334 data are used (solutions 2, 3 and 4). Moreover, the more redundant data335 are used, the more the variance of estimation error decreases, converging to336 a value corresponding to the true standard deviation of measurement errors337 (σe=0.17).338 4. Conclusions339 We proposed a speciation algorithm that uses redundant data and aknowl-340 edges measurement errors, on the assumption that redundant data will reduce341 estimation errors in geochemical calculations.342 We compared the proposed method with the traditional speciation method343 in terms of logaritmic mean squared error, MSElog and mean value of es-344 timation error variance, V ar∗. We tested both algorithms by means of two345 18 synthetic examples. Both MSElog and V ar∗using redundant data are con-346 sistently smaller than in the traditional method.347 The effect of measurement errors was examined in a carbonate system348 example. The algorithm is sensitive to the variance of measurement errors.349 Also, a wrong value of the standard deviation can worsen the results with350 respect to the ones obtained with the correct standard deviation. However,351 the effect of its value depends on the type of data associated to it. A wrong352 error associated to measurements can still improve the results in terms of353 mean squared error and variance of estimation error with respect to a tradi-354 tional speciation method, especially when a large number of redundant data355 are used.356 Therefore we argue that the proposed method can improve the quality of357 the speciation results, reducing estimation errors.358 Appendix A. Error definitions359 The errors εallowed in the proposed method can be additive or mul-360 tiplicative. Additive errors should lead to gaussian distributions, whereas361 multiplicative to lognormal distributions. Depending on the type of error,362 the function g(c) and the data xmust be defined accordingly: aritmetic or363 logaritmic for additive and multiplicative errors, respectively.364 Data used in speciation calculations are typically combinations of concen-365 trations or activity values. The former, which we name balance equations,366 are linear combinations of concentrations representing total concentrations,367 alkalinity, charge balance or TIC values. The latter are usually employed to368 fix pH values or to impose equilibrium with minerals or gases.369 19 Distinguishing between these two types of data equations we can define:370 ε=Bc −x(19) for the balance equations and371 εi= Ns ∏ n=1 anLin −xi(20) for each i−th activity combination, respectively. Bis a matrix of di-372 mension (Nb×Ns) and Nbis the number of balance equations. Bcontains373 different coefficients depending on the type of balance equation: ionic charge374 for charge neutrality, coefficients defining alkalinity or TIC, or component375 matrix elements for total concentration. Lis a matrix of dimension (Na×Ns)376 containing the coefficients of the activity for every species involved in the377 combination. Nais the number of activity conditions imposed.378 If we want to use a log-distribution instead of a normal distribution of379 errors, one should use:380 ε= ln(Bc)−ln x(21) for the balance equations and381 ε=Lln a−ln x(22) for the activity combinations, respectively.382 20 Appendix B. Jacobian calculation383 The jacobian, containining the derivatives of εwith respect to the state384 variables ln c1at every step of the iterative method, can be calculated as385 ∂εi ∂ln c1,j =∂εi ∂c1,j ·c1,j = B1,ij ·c1,j + N2 ∑ l=1 B2,il ·∂c2,l ∂c1,j ·c1,j i= 1, . . . , Nb j= 1, . . . , N1 (23) from the definition (19) whereas by means of definition (21) results386 ∂εi ∂ln c1,j =∂ln zi ∂ln c1,j =1 zi ·∂zi ∂ln c1,j =1 zi ·(B1,ij ·c1,j + N2 ∑ l=1 B2,il ·∂c2,l ∂c1,j ·c1,j) i= 1, . . . , Nb j= 1, . . . , N1 (24) being387 zi=(N1 ∑ m=1 B1,im ·c1,m + N2 ∑ l=1 B2,il ·∂c2,l ∂c1,j ·c1,j)(25) Matrices B1and B2are the parts of matrix Brelative to primary and388 secondary species, respectively, and the derivatives of secondary concentra-389 tions with respect to primary concentrations can be calculated considering390 21 that at every step of the iterative method the total derivative of fMAL with391 respect to primary species concentrations is null:392 dfMAL dln c1 =∂fMAL ∂ln c1 +∂fMAL ∂ln c2 ∂ln c2 ∂ln c1 = 0 (26) Those derivative can be calculated by means of the following linear system393 ∂fMAL ∂ln c2 ∂ln c2 ∂ln c1 =−∂fMAL ∂ln c1 (27) The conversion to ∂c2/∂c1is straightforward, recalling that dln x/dx =394 1/x:395 ∂c2,i ∂c1,j =c2,i c1,j ∂ln c2,i ∂ln c1,j (28) The derivatives of (22), remembering the definition of activity (a=γ·c)396 and that γ=f(c), read397 ∂εi ∂ln c1,j = L1,ij + N1 ∑ m=1 L1,im ∂ln γ1,mj ∂ln c1,j + + N2 ∑ l=1 L2,il (∂ln γ2,lj ∂ln c1,j +∂ln c2,lj ∂ln c1,j ) i= 1, . . . , Na j= 1, . . . , N1 (29) Matrices L1and L2are the parts of matrix Lrelative to primary and398 secondary species, respectively.399 The derivatives of (20) with respect to the state variables can be calcu-400 lated from401 22 ∂εi ∂ln c1,j = Ns ∏ n=1 anLin · ·[L1,ij + N1 ∑ m=1 L1,im ∂ln γ1,mj ∂ln c1,j + N2 ∑ l=1 L2,il (∂ln γ2,lj ∂ln c1,j +∂ln c2,lj ∂ln c1,j )] i= 1, . . . , Na j= 1, . . . , N1 (30) References402 Appelo, C.A.J., Postma, D., 2010. 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Advances in Water Resources 34, 137–151.459 25 1 2 3 4 0.01 0.02 0.03 0.04 0.05 0.06 M S E lo g solution ID Figure 5: MSElog for traditional speciation method (solution 1) and proposed method (solutions 2, 3 and 4) obtained with σevalues of table 2. 32 1234 0.01 0.02 0.03 0.04 0.05 0.06 e = 0.09 e = 0.35 e = 0.17 M SE lo g solution ID (a) 1234 0.01 0.02 0.03 0.04 0.05 0.06 M SE lo g solution ID (b) 1234 0.01 0.02 0.03 0.04 0.05 0.06 M SE lo g solution ID (c) 1234 0.01 0.02 0.03 0.04 0.05 0.06 M SE lo g solution ID (d) 1234 0.01 0.02 0.03 0.04 0.05 0.06 M SE lo g solution ID (e) 1234 0.01 0.02 0.03 0.04 0.05 0.06 solution ID (f) M SE lo g Figure 6: MSElog obtained changing values of σeof each data: (a) Catot; (b) aH+; (c) alkalinity; (d) TIC; (e) aCO2(g); (f) calcite equilibrium. 33 1 2 3 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 2 3 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 V ar * (a) V ar * sol ution ID e = 0.09 e = 0.17 e = 0.35 (b) sol ution ID Figure 7: V ar∗obtained for different values of σeof (a) aH+and (b) alkalinity. 34 Table 1: Exact solution for carbonate example. Species Values c∗Ca2+ 4.92 ·10−4 [mol/l]H+5.51 ·10−9 HCO− 39.63 ·10−4 CO2(aq)1.07 ·10−5 CO2− 39.78 ·10−6 OH−2.01 ·10−6 a∗CO2(g)3.16 ·10−4 [bar] SI Calcite 0.0 (Sat. Index) 35 Table 2: Mean values and standard deviations adopted to generate 1500 realizations of data for the carbonate example. Data µ σgσeεequation Catot 4.92 ·10−40.17 0.17 (21) aH+5.29 ·10−90.17 0.17 (22) Alkalinity 9.85 ·10−40.17 0.17 (21) TIC 9.84 ·10−40.17 0.17 (21) aCO2(g)3.16 ·10−40.17 0.17 (22) Calcite Eq.0 0 0.17 (22) Units are in mol/l except for aCO2(g)which is expressed in bar. 36