Mathematical Geosciences

, Volume 45, Issue 1, pp 87–101

# Interpretation of Na–Cl–Br Systematics in Sedimentary Basin Brines: Comparison of Concentration, Element Ratio, and Isometric Log-ratio Approaches

Article

DOI: 10.1007/s11004-012-9436-z

Engle, M.A. & Rowan, E.L. Math Geosci (2013) 45: 87. doi:10.1007/s11004-012-9436-z

## Abstract

Mathematicians and geochemists have long realized that compositional data intrinsically exhibit a structure prone to spurious and induced correlations. This paper demonstrates, using the Na–Cl–Br system, that these mathematical problems are exacerbated in the study of sedimentary basin brines by such processes as the evaporation or dissolution of salts owing to their high salinities. Using two published datasets of Na–Cl–Br data for fluids from the Appalachian Basin, it is shown that log concentration and Na/Br versus Cl/Br methods for displaying solute chemistry may lead to misinterpretation of mixing trends between meteoric waters (for example shallow drinking water aquifers) and basinal brines, partially due to spurious mathematical relationships. An alternative approach, based on the isometric log-ratio transformation of molar concentration data, is developed and presented as an alternative method, free from potential numerical problems of the traditional methods. The utility, intuitiveness, and potential for mathematical problems of the three methods are compared and contrasted. Because the Na–Cl–Br system is a useful tool for sourcing solutes and investigating the evolution of basinal brines, results from this research may impact such critical topics as evaluating sources of brine contamination in the environment (possibly related to oil and gas production), evaluating the behavior of fluids in the reservoir during hydraulic fracturing, and tracking movement of fluids as a result of geologic CO2 sequestration.

### Keywords

Brine Isometric log-ratio Compositional data analysis Appalachian Basin Produced Water

## 1 Introduction

Sodium (Na) and chloride (Cl) are typically two of the most abundant constituents in formation waters from sedimentary basins, and understanding their source provides basic information regarding the nature and evolution of the brines under investigation. It has long been recognized that the systematics of Na–Cl-bromide (Br) have the potential to provide this information (Rittenhouse 1967; Carpenter 1978). Bromide is virtually excluded from the halite lattice (Zherebtsova and Volkova 1966; McCaffrey et al. 1987); thus seawater (or paleoseawater) evaporation past halite saturation will result in a brine enriched in Br relative to Na and Cl, and with lower Na/Br and Cl/Br ratios than occur in seawater. Conversely, dissolution of halite by fresh water or seawater will produce brine that is heavily depleted in Br relative to Na and Cl, that is, with higher Na/Br and Cl/Br ratios than in seawater. These relationships have most commonly been investigated by plotting raw concentrations of Na and Cl versus Br or by plotting concentration ratios of Na/Br versus Cl/Br. Both of these methods can be shown mathematically to induce spurious correlations (shown below) and have the potential to generate results that may be misleading. The phrase spurious correlation was introduced to the mathematics literature by Pearson (1897), and remains commonly used to indicate a correlation induced by improper treatment of the data, as opposed to a natural relationship between the variables. As will be discussed, spurious correlations that go unrecognized may be misinterpreted as resulting from a natural reaction or process. A newer approach using isometric log-ratios, a transformation developed by Egozcue et al. (2003), can be applied to compositional geochemical data and avoids these statistical and mathematical problems. In this paper, we discuss some of these mathematical issues related to brine compositions, and compare interpretation of Na–Cl–Br systematics for two sets of brine data from the eastern United States portion of the Appalachian Basin, using concentration data, concentration ratios, and isometric log-ratio approaches.

Given the current focus on fluids being generated from sedimentary basin brines and public concern regarding potential contamination of drinking water as a result of CO2 sequestration and hydrocarbon extraction, including flowback fluids from hydraulic fracturing, the presentation of potentially misleading interpretation of brine geochemistry is of major concern. The importance of proper treatment of geochemical data from brines does not apply only to Na, Cl, and Br, but to all constituents carried in these fluids. For these reasons, the application of multivariate log-ratio techniques to examine the controls and composition of brines is an important area of research. However, much of the literature on compositional data analysis has been viewed as arcane or unapproachable by many geoscientists. Topics such as robust statistical methods and dealing with missing values and non-detects were only recently developed or are still being researched (Filzmoser et al. 2009a, 2009b; Hron et al. 2010; Palarea-Albaladejo et al. 2011). While this may have delayed the development of compositional data analysis in the field of geochemistry, hopefully the benefits of such approaches will be increasingly recognized.

## 2 Mathematically Spurious Relationships in Concentration Data for Brines

For decades geochemists have recognized that compositional data in their raw form (for example, ppm, %, mol L−1, mg kg−1) are prone to artifacts (Chayes 1960; Miesch 1969). The fundamental issue is that the raw concentrations of individual components in a sample are not free to vary independently, and a change in the abundance of one component must change the relative abundance of every other. The simplest generalization is a 2-component (binary) system: for instance, the mass proportions of <2 mm and ≥2 mm diameter fractions in a soil sample. The two components have a perfect negative and completely spurious correlation. As a result of a constant sum constraint (for example, 100 %), as one component increases the other must decrease by a corresponding amount. For more complex systems, Rollinson (1993) summarized the following numerical problems presented by raw concentration data: (1) negative biases may be induced in correlations, (2) the constant sum constraint forces correlation between components, and (3) subcompositions do not necessarily reflect variations present in the full data set (that is, subcompositional incoherence). These problems are more obvious in solid phase geochemistry, where the elements sum to an easily identified quantity (for example, 100 % or 1,000,000 mg kg−1). Bern (2009) showed that the enrichment of quartz during the weathering of soils generated a strong induced negative correlation between Si and many other elements, and generated artificially positive correlation between non-silicate mineral components. It has also been demonstrated that the statistical issues discussed by Rollinson (1993) apply to compositional data in aqueous geochemistry (Buccianti and Pawlowsky-Glahn 2005; Otero et al. 2005). In a comparison of compositional analysis techniques on water-quality data, Otero et al. (2005) demonstrated forced positive correlations between constituents as a result of the constant sum constraint and marked differences in interpretation.

The units of aqueous phase data (for example, mol L−1, mg kg−1) do not immediately indicate that the data are subject to the same issues as with solid-phase compositional data. The potential for artifacts lies in the fact that the denominator in the concentrations of solutes corresponds, except in the case of molality (discussed below), to the mass or volume of the entire solution, rather than just the solvent, that is, water. Take for instance data reported in a mass of solute per mass of solution (mass of solutes [mi] + mass of water [mwater]) basis, containing D number of solutes. The total mass of the system (mT) is given by Dividing both sides by mT puts a constant sum (k) constraint on the concentrations of the individual dissolved species plus water In the examples used here k sums to unity, but more generally in compositional data analysis, it represents a constant to which the parts sum. For instance, k is equal to 100 for data in percentages and 106 for data in parts per million. Equation (2) shows that if the mass of species i (mi) changes due to a gain or loss from the system, the concentration of i changes, because mT has changed as well. Because of the constant sum constraint, any change in mT will change the concentration of every species in the system, including water. A similar equation can be developed for data reported on a mass per volume basis, with the only difference being the replacement of mT with the product of the volume (VT) and density (ρT or mT/VT) of the solution If we assume for the moment that VT is constant, then Eq. (3) indicates that a change in mass of species i will impact the concentration of itself (because mi changed); the concentration, on a mass per volume basis, of every other species is unaffected due to the a corresponding change in ρT. The assumption that VT is constant lacks rigor, but introduces negligible error in dilute solutions (such as typical stream or lake water). However, VT and ρT can vary appreciably during the geochemical evolution of sedimentary basin formation waters whose solute content can reach 10–40 % by mass. This is most easily conceptualized by changes in the mass fraction of water in the solution through evaporation and dilution, processes that affect the concentration of every other constituent in the system. In a more generalized sense, with potential gain or loss on the order of 100 s of grams per liter, each individual constituent involved in the multitude of reactions during the evolution of a brine has the potential to markedly impact the concentrations data of all components in the system for purely mathematical reasons. In other words, the relationships shown in Eqs. (2) and (3) demonstrate that the concentrations of dissolved constituents in brines are not free to vary independently, but rather behave as a constrained system. One exception to this behavior is data reported in molality, where the denominator is the mass of the solvent itself. As such, addition or removal of solutes does not affect the mass of the solvent. However, as will be discussed in detail in Sect. 4, relationships between constituents with the same denominator (i.e., the mass of pure water, as in molality), are prone to spurious correlations (Pearson 1897). We argue then that the raw concentration data used to investigate geochemical processes in brines, such as Na–Cl–Br systematics, are not the most appropriate form of the data to use.

Mathematically spurious relationships in raw compositional data exist because the constant sum constraint limits them to a hyperplane (multi-dimensional surface) known as the simplex (Aitchison 1986), and does not allow them to vary freely throughout positive real space $$(\mathbb{R}^{\mathrm{D}}_{+})$$. Although the simplex is contained within $$\mathbb{R}^{\mathrm{D}}_{+}$$, it is defined by a geometry which is not Euclidean (Pawlowsky-Glahn and Egozcue 2001). Evidence of non-Euclidean behavior includes the ability to calculate confidence intervals or uncertainties around concentration data which exceed the limits of the simplex (for example, <0 or >100 %) (Buccianti and Magli 2011). For proper geometrical treatment of compositional data, one approach is to apply members of the family of log-ratio transformations (Aitchison 1986; Egozcue et al. 2003). Despite these concerns, the use of raw concentration data is still popular in the study and interpretation of sedimentary basin brines.

## 3 Na, Cl, and Br Concentration Plots

Many previous workers have suggested that plotting concentrations of Na and Cl versus Br, in comparison to seawater evaporation paths, can be used to help understand the origin of basin brines (Carpenter 1978; Hanor 1994; Nativ 1996; Kharaka and Hanor 2007). For plotting purposes, we have modeled the behavior of Na, Cl, and Br in these systems using two basic scenarios: (1) evaporation of seawater and (2) dissolution of seawater by Br-free halite (Figs. 1 and 2). Modeling results from seawater evaporation were compared with empirical data from McCaffrey et al. (1987) and show good agreement (Figs. 1 and 2).

To adequately account for the effect of high ionic strength on activity coefficients, Na and Cl concentrations were modeled using the virial activity model of Harvie–Møller–Weare (Harvie et al. 1984). Bromide is not a component in the Harvie–Møller–Weare model, so it was assumed to behave conservatively. Potassium (K) was used as a surrogate for its behavior, so modeling was terminated once K-bearing minerals began to precipitate. Volume changes as a result of solute and water addition and removal can be modeled successfully using these Pitzer-type activity models (Monnin 1989). For instance, modeling results suggest a decrease in Br molarity (down 12.5 % at the point of halite saturation) as a result of an increase in VT during halite dissolution by seawater in Figs. 1a and 2a.

The Na–Cl–Br reactions involved in basin brines derived originally by the evaporation of seawater are well understood. As seawater evaporates, the saturation indices of a variety of minerals are surpassed leading to a process of cascading mineral precipitation at different degrees of evaporation (DE), which is the factor by which solutes have been concentrated by evaporation relative to unevaporated bulk seawater. The data of McCaffrey et al. (1987) indicate the degrees of seawater evaporation associated with formation of a series of minerals: calcite (DE = 1.8), gypsum (DE = 3.8), halite (DE = 10.6), magnesium-bearing salts (DE ∼ 70), and K-bearing salts (DE > 90) (Fig. 1a). Only reactions involving Na, Br, and Cl, namely precipitation of halite, sylvite and other Na, Cl, and Br bearing minerals, obviously impact the relationship between the concentrations of these elements. However, because these data are presented in the raw form, the concentrations of these constituents are linked to all other components of the system.

In cases of evaporite dissolution as a dominant source for solutes, large quantities of Na and Cl as well as other constituents contained in the salts are added to the waters, with relatively little Br (Zherebtsova and Volkova 1966; McCaffrey et al. 1987). Similar to a seawater evaporation scenario, input and output of large quantities of constituents have potential to obscure the intrinsic relationships between the components of the system, as described in Sect. 2.

To better understand the potential impacts of using differing mathematical treatments of data to investigate Na–Cl–Br geochemistry in sedimentary basin brines, two datasets in the Appalachian Basin are examined. We are in no way attempting to second-guess the findings from the original studies, but rather found the datasets particularly illustrative for the purposes of this paper. As our first example, Cl and Br concentration data for brine and groundwater samples in and around the Oak Ridge National Laboratory in eastern Tennessee examined by Nativ (1996), hereafter referred to as the Nativ dataset, are plotted (Fig. 1a). Plots of concentrations of both Na versus Br (not shown, but provided in the original reference) and Cl versus Br (Fig. 1a) fall off of the modeled seawater evaporation trend. The low Cl:Br ratio was not fully explained in the original paper but the plot between Cl and Br concentrations shows a near linear trend from data points with a TDS <5000 mg L−1 to those exceeding 200,000 mg L−1, suggesting a mixing pathway from freshwater to seawater evaporated by a DE of 40–45. Note that the points corresponding to a TDS of 5001–35,000 mg L−1 appear to be a mixture of roughly 10 % evaporated seawater and 90 % fresh water. A similar trend is observed for the Na and Br data.

As our second example, we show Cl versus Br concentration data (Fig. 2a) for lower Pennsylvanian sandstones of the Pottsville Group in West Virginia from Price et al. (1937), hereafter referred to as the Price dataset. For these Appalachian Basin data, there is a trend line (Trend A in Fig. 2a) from evaporated seawater (DE ∼ 40) back to modern seawater for samples > 35,000 mg TDS L−1. This trend is typically interpreted as mixing between evaporated paleoseawater and unevaporated seawater (Dresel and Rose 2010). A second trend (Trend B in Fig. 2a) extends away from seawater to lower concentrations of both Br and Cl and could be interpreted as mixing unevaporated paleoseawater with fresh water. Additionally, a third group of points exhibit TDS concentrations > 35,000 mg L−1 plots to the left of the seawater evaporation pathway and appear to be highly depleted in Br. The cause for the depletion in Br for these samples is not readily evident in this type of plot, although a few of the points occur near the halite-dissolution pathway. In both of the example datasets (Nativ and Price), linear trends identified in the concentration plots should be interpreted with caution because constituents are prone to artificial correlations.

## 4 Comparison of Na/Br and Cl/Br Ratios

Visual interpretation of Na–Cl–Br systematics in basinal fluids was greatly improved upon by Walter et al. (1990) who plotted concentration ratios of Na/Br versus Cl/Br. Although individual element concentrations vary with concentration change in any other element, the element ratios are preserved. Thus the use of ratios avoids dealing with concentration artifacts from changes in total volume and total mass, such as the modeled decrease of Br molarity as a result of halite dissolution increasing the solute volume in Figs. 1a and 2a. The additive log-ratio (alr) of Aitchison (1986), one of the transformations mentioned in Sect. 2, has a similar structure to the ratios used in Figs. 1b and 2b, and, for a D-part composition, has the form such that the first D−1 components (for example, Na and Cl) are normalized by the Dth component (for example, Br). Assuming no major input of Na, Cl, or Br other than seawater or halite dissolution, Na and Cl derived from the evaporation of seawater should plot down and to the left of seawater while meteoric waters which have dissolved halite should plot up and to the right, all along a single linear trend. One minor disadvantage to this approach relative to using raw data is that these ratios only begin to change dramatically at DE > 10.6, so that evaporation to levels less than those for halite precipitation are not identifiable. Dilution of brines by meteoric water should not dramatically affect these ratios, and thus they represent the cumulative amount of evaporation or halite dissolution that has occurred during the evolution of the brine.

Despite the benefits of plotting element ratios rather than raw concentration data, several potential disadvantages exist with the ratio approach. First, because both axes include the same element in the divisor, Br, the two ratios are spuriously correlated. This issue with ratio correlation was pointed out more than 100 years ago (Pearson 1897) and has the potential to show a linear relationship between the ratios, suggesting a possible fluid evolution trend that may not truly exist. For this reason, statistical analysis of similarly structured alr-transformed data is avoided in correlation determination (Aitchison 1986). Second, the data still are not properly transformed into space with Euclidean geometry. Therefore, estimates of the degree of mixing and amounts of evaporation may be incorrect. Taking the natural log of the two axes, thus converting them to an alr form, does not solve the problem because alr-transformed data are defined by an oblique basis, and if mapped onto orthonormal axes, distances and angles may be deformed (Egozcue et al. 2003; Mateu-Figueras et al. 2011). The last potential issue with ratio plots is that fluids generated from seawater evaporation plot in the opposite direction from mean seawater composition, along a single linear path, from fluids enriched by halite dissolution. For this reason, mixtures of evaporated seawater with water that has dissolved halite can be difficult to recognize (Chi and Savard 1997).

Using the Na/Br versus Cl/Br ratio approach for the Nativ dataset, samples with >35,000 mg L−1 TDS plot along the seawater evaporation pathway, with higher salinity samples plotting further down the trend (Fig. 1b). The sample from well W&B is easily identified as an outlier on this plot, and may represent analytical error. Data from well W&B plots away from the other raw concentration data in Fig. 1a, but is located near the seawater evaporation pathway, making its delineation as an outlier less obvious than in the Na/Br versus Cl/Br ratio plot. In Fig. 1b, those samples with a TDS <5000 mg L−1 plot completely off the seawater evaporation path, suggesting that their ionic composition has had little if any input from the underlying brines. The narrow band in which Cl/Br ratios for these samples plot is within the range for precipitation samples collected at sites located > 100 km inland (∼ 170–270; Davis et al. 1998), suggesting that the data are representative of local meteoric water. These freshwater samples contain relatively high proportions of sulfate (SO4) and bicarbonate, relative to the deeper basin brines, suggesting that carbonate minerals (from the pervasive limestone and dolomite) and SO4-bearing minerals contribute significantly to the salinity of these fluids. Of most interest are the six samples exhibiting TDS values between 5001 and 35,000 mg L−1 (brackish water samples), which appear to represent roughly 10 % evaporated seawater and 90 % fresh water according to the concentration plots (Fig. 1a). Nativ (1996) interpreted their raw concentration data to be the result of mixing between fresh water and brine; one of these brackish samples contained high levels of tritium (83.7 T.U.), consistent with input of modern water. However, in the ratio plot of the data only three of these brackish water samples plot on or near the seawater evaporation trend. Those brackish water samples which plot near the evaporation trend exhibit molar calcium (Ca)/Na ratios ∼40–200 % greater and molar Cl/SO4 ratios 110–2200 % greater than those which plot off the trend, indicating that the two groups of samples are geochemically dissimilar. For those samples that do not plot near this trend, results from the concentration ratio method suggest that interpreted hydraulic connectivity between the surface water and the deep reservoirs containing the brines may have been over-estimated.

Data for brines from the Price dataset span the range of highly evaporated to heavily enriched by halite dissolution (Fig. 2b). The role of halite dissolution was not identifiable in the raw concentration plot (Fig. 2b), showing a limitation of this method. Those samples with TDS concentrations >35,000 mg L−1 that plot near the halite-dissolution pathway generally plot to the left of it, suggesting that they are either enriched in Cl/Br or depleted in Na/Br relative to the pathway. Many of the data for samples <35,000 mg L−1 TDS plot as either highly evaporated or having dissolved large quantities of halite. This suggests that either the brines were diluted several orders of magnitude by meteoric water or that some of the data are spuriously correlated and have no relationship with either end-member. Many of the data for samples containing 35,001–100,000 mg L−1 TDS plot near the modern seawater in Fig. 2b, but it is difficult to establish from the ratio approach whether these fluids are moderately evaporated or whether they represent mixtures of highly evaporated paleoseawater and halite-enriched waters.

## 5 A New Approach to Na–Cl–Br Systematics: Isometric Log-ratio Coordinates

In the years since the previous two methods were introduced, new mathematical techniques have been developed to work more effectively with compositional data, while avoiding the potential problems described in previous sections. The new methods, however, require a new way of thinking about and interpreting geochemical data. In our approach molar concentration data are converted to orthonormal coordinates, a basis of perpendicular axes of unit length, through the use of the isometric log-ratio (ilr) transformation (Egozcue et al. 2003). The particular form of coordinates employed here consists of groups of constituents, or parts. Selection of the order of the elements in the parts is arbitrary but one popular strategy to create meaningful coordinates is through the use of sequential binary partitions, wherein non-overlapping groups of parts, known as balances, are defined (Egozcue and Pawlowsky-Glahn 2005). In all cases, D−1 balances are generated from a system with D components. This lower degree of freedom exists for compositional data because the quantity or concentration of the final component, D, can be estimated as the difference between the sample total and the sum for the other D−1 components.

The development of the sequential binary partition (Table 1) and the corresponding ilr coordinates can be generated intuitively. For the first balance, Na and Cl are included in separate groups (+1 and −1, respectively) while Br is excluded (denoted by 0 in Table 1). This partition provides insight as to the net gain/loss of Na relative to Cl. For balance 2, Na and Cl are included in the first group (denoted by a +1 in Table 1) and Br is included in the second (denoted by −1). This partition can be interpreted to represent the exclusion of Br from mineral lattice during halite dissolution. The corresponding coordinates (z1 and z2) are calculated from the sequential binary partition via where ri are si are the number of parts coded with +1 and −1, respectively, xj and xl are the parts coded with and +1 and −1, respectively. The two resulting orthonormal coordinates, which serve as axes on which to plot the Na–Cl–Br compositional data of the samples, include
$$z_{1} = \frac{1}{\sqrt{2}} \ln\frac{[ \mathrm{Na} ]}{[ \mathrm{Cl}]}$$
(6)
and Disregarding the coefficients, these equations have structures similar to that of the Law of Mass Action (Buccianti 2011), with species concentrations raised to various powers in the denominator and numerator. Because the relative order of the variables in the sequential binary partition and subsequently the balances is somewhat subjective, we tested several other configurations and found them no more useful or intuitive than the version presented here. Using molar concentration data, as is done here, the first balance, z1, will have a value of zero when Na and Cl have a molar ratio of one (because the ln of 1 is 0). Therefore, the dissolution of halite pushes a fluid towards a z1 value of zero. However, the evaporation of seawater leads to an eventual depletion of Na relative to Cl because the Na/Cl molar ratio in seawater is <1, and the corresponding z1 values will decrease during evaporation. The second balance, z2, examines the ratio of Na and Cl to Br. As seawater evaporates, Br is progressively enriched in the residual fluid, while Na and Cl concentrations are limited by halite precipitation. Brines that formed by seawater evaporation will therefore have progressively smaller z2 values with increased evaporation. In a fluid whose salinity is derived from dissolution of halite, Br concentrations will be generally very low, and consequently, the corresponding z2 values will be large.
Table 1

Sequential binary partition and balances chosen for the Na–Cl–Br system

Balance

Na

Cl

Br

z1

+1

−1

0

z2

+1

+1

−1

To further illustrate the interpretation of data converted to orthonormal coordinates, data from Nativ and Price studies are plotted using isometric log ratios in Figs. 1c and 2c. The geochemical modeling pathway for seawater evaporation and corresponding data of McCaffrey et al. (1987), shown for reference, follow a curvilinear path away from modern seawater accounting for the gain in Br relative to Na and Cl (plotted on y-axis as z1), and the loss of Na relative to Cl (plotted on the x-axis as z2) that occurs during the evaporation process. Likewise, the calculated pathway showing the progressive dissolution of Br-free halite by seawater is also curvilinear, starting at seawater approaching a z1 of 0 with increasing values of z2. Using the virial activity model of Harvie–Møller–Weare (Harvie et al. 1984), halite dissolution by seawater is predicted to reach a maximum z2 value of 7.2, the point of halite saturation. Theoretically, even greater z2 values could be reached by progressive cycles of halite dissolution followed by dilution. Note that halite saturation points are reached on both pathways for seawater evaporation and halite dissolution.

The brine and groundwater data from the Nativ dataset fall into two general groups: (1) data that plot along the pathway for seawater evaporation (typically >35,000 mg L−1 TDS), and (2) data that plot off of the evaporation path (typically <5000 mg L−1 TDS). As was indicated on the molar ratio plot (Fig. 1b), there is no evidence for significant halite dissolution or mixing with waters that have reacted with halite. In Fig. 1c, as on the ratio plot (Fig. 1b), the point labeled W&B plots as an outlier, and many of the brackish to fresh water samples (that is, those with TDS <35,000 mg L−1) plot far above the seawater evaporation trend, suggesting that these samples contain little if any Na and Cl derived from the deeper brines, and are therefore only weakly hydrologically connected, if at all, to the deeper fluids.

In comparing the orthonormal coordinate mapping of the Price dataset (Fig. 2c) to the ratio concentration plot (Fig. 2b), both figures have data coinciding with portions of the seawater evaporation and halite-dissolution trends. However, in ilr coordinates, data from many of the samples containing <35,000 mg L−1 TDS plot away from either trend, suggesting that Na, Cl, and Br are not derived from either halite dissolution or seawater evaporation as suggested by Fig. 2b. This potential misinterpretation was likely a result of induced correlation by use of a common denominator (Br) in the ratios that are plotted.

Most of the samples from fluids with >100,000 mg L−1 TDS have Na/Br versus Cl/Br ratios that plot near modern seawater, or along the seawater evaporation trend (Fig. 2b). On the isometric log ratio plot, these samples plot near, but slightly below the seawater evaporation trend (Fig. 2c). Both plots, therefore, identify evaporated seawater as the primary source for Na and Cl in these samples, but the lower values for z1 relative to the seawater evaporation trend suggests that the Na/Cl ratio in these fluids has been reduced, possibly through albitization or formation of Na-rich clays (Hanor 1994; Kharaka and Hanor 2007). Notable too are the four data points for samples >100,000 mg L−1 TDS that plot to the right of the composition of modern seawater but markedly below the pathway for halite dissolution. Although location of these data in Fig. 2c suggests that they may represent halite dissolution followed by Na loss through albitization or uptake by clays, a more likely scenario is that these samples represent mixtures of evaporated seawater and water that dissolved halite. Mixing between two compositional end-members is described by where xi is the composition of the mixture, xA and xB represent the compositions of the mixture end-member A and B, respectively, and α (0,1) is the portion of end-member A in the mixture. The mixing trend follows a curvilinear pathway on the ilr plot. For example, using points A and B in Fig. 2c as hypothetical end members, the pathway representing their mixture plots on top of the continuum defined by the trends for seawater evaporation and seawater dissolution of halite. Translating point C back to the mixing trend, following the pathway accounting for ∼30 % Na loss from the original fluid due to albitization and formation of Na-rich clay minerals (shown as an arrow back to the halite-dissolution pathway), it would represent a mixture of approximately 92 % paleoseawater water that has dissolved halite and 8 % evaporated paleoseawater (Fig. 2c). This estimate falls in line with the predicted molar Na/Ca ratio of point C based on a simple mixing model (predicted = 5.05, actual = 5.48). However, more complicated mixing scenarios would likely require additional lines of evidence to sort out relative contributions (cf. Chi and Savard 1997).

## 6 Conclusions

Interpretation of Na–Cl–Br systematics using raw concentrations, ratios, and isometric log-ratio coordinates show similarities and also marked differences for the two example datasets (Table 2). The use of raw concentration plots has a long history in geochemistry, and for that reason, are typically very easy to interpret. However, in our examples, there were marked differences with interpretations in the raw concentrations and the ratio and isometric log-ratio plots. We contend that many of these differences in interpretations between the approaches result from spurious and induced correlations, as evidenced here, that exist because of improper application of mathematics given the geometry of each system. While requiring slightly more computation, the isometric log-ratio coordinates avoid these potential pitfalls and allow for graphical representation and interpretation of Na–Cl–Br systematics in sedimentary basin brine that is devoid of these mathematical inconsistencies. However, the plotting isometric-log ratio coordinates are also the least initially intuitive of the three methods and is the only method upon which mixing of end-members does not follow a linear pathway. The concentration ratio method, especially if converted to additive log-ratios by taking the logs, is nice balance between the other two methods, but the potential for spurious correlation due to the common denominator should elicit some caution in its use. Regardless of the mathematical rigor of the approach, false or misleading results can be generated from careless use of these methods as a result of ignoring external sources/sinks of the elements (for example, Br from drilling fluids or kerogen maturation), poor data QA/QC, improper sample collection, etc. In addition, multiple lines of evidence, as was done in the original papers for the two data sets examined here (Nativ 1996; Price et al. 1937), should be used for interpreting results. Reliance upon a single plot or technique should be cautioned against, as all methods have their downfalls. As such, care should be given to both the quality of the geochemical data generated and the methods used to interpret the results.
Table 2

Summary of differences between the three graphical approaches for the interpretation of Na–Cl–Br systematics in sedimentary basin brines

Method

Drawbacks

Concentration plot

• Requires little or no computation

• Interpretation of results straightforward

• Long history of use

• Concentrations may change due to loss or gain of other solutes or water

• Apparent mixing and reactions trends may be artifacts

Ratio plot

• Minimal computation

• Slightly less straightforward interpretation

• Ratios not affected by gain or loss of other solutes or water

• Common denominator may produce spurious correlation

• Axes are not truly orthogonal and plot may be distorted

• Mixtures may be difficult to identify

Isometric log-ratio plot

• Results free from spurious correlation

• Geometry of plot is fully Euclidean

• Exact formulation of balances can be modified as needed

• Requires most amount of computation

• Results initially less intuitive

• Mixing not linear

## Acknowledgements

This research was funded by the U.S. Geological Survey Energy Resources Program. The authors would also like to acknowledge Lin Ma (University of Texas at El Paso), Jennifer McIntosh (University of Arizona, U.S. Geological Survey), Ricardo Olea (U.S. Geological Survey), Josep Antoni Martín Fernández (Universitat de Girona), and two anonymous journal reviewers for providing critical comments and corrections to this paper.