## Abstract

We explore theoretically the controls on dissolution of salt A, in an undersaturated brine of salts A and B. We show that, as the concentration of B increases, the dissolution rate of A decreases, for brine of given temperature. We also show that there is a sharper decrease in dissolution rate with increasing concentration, for concentrations of B above a critical value, where B limits the equilibrium concentration. We explore the implications of the predictions for dissolution of KCl or NaCl, by a mixed brine of NaCl and KCl, a common reaction that may arise in dissolution of evaporites. We predict that, with mixed-composition brine, KCl crystals dissolve more rapidly than NaCl crystals, unless the (far-field) brine is nearly saturated in KCl. We also predict that the dissolution rate of these salts is largely independent of fluid temperature and is controlled by compositional diffusion.

## 1. Introduction

There are natural and industrial processes, including salt dissolution during water flooding of porous rocks, melting caused by hot magma intrusion into the shallow crust of the Earth and solution mining of minerals, in which a soluble solid dissolves into a multi-component liquid solution.

In particular, in the context of evaporite formation and extraction, there is interest in the dissolution of one salt by an aqueous solution containing two or more solutes. The purpose of this paper is to quantify the dissolution rate as a function of the temperature and the concentration of each salt in the melt. We restrict attention to the situation in which diffusion is the dominant mechanism for heat and mass transport, building from the classical Stefan problem approach for equilibrium phase change (cf. Carslaw & Jaegar 1986; Woods 1992). Although in many natural situations, fluid convection may develop, there are regimes in which diffusion provides the dominant mass transport process. For example, as evaporites are formed, there are periods in which relatively fresh fluid may enter a lagoon (cf. Sonnenfeld 1985). This may then lead to dissolution of salt from the bed of the lagoon and formation of a stable stratification. Also, in the context of sea ice/ocean interaction, relatively warm, saline ocean water may impinge on the underside of sea ice. The ice may then melt into the ocean water, again forming a stable stratification (cf. van Andel 1994). In these cases, the phase change will be diffusion controlled. Even where there is a driving force for convection, our analysis provides a reference, with which to compare the effects of convection on the phase change. In §3, we build on the self-similar solutions for dissolution of a single species in brine developed by Woods (1992). We focus on the effect of a second salt in solution. We thereby identify (§§4 and 5) some of the controls exerted by fluid concentration and temperature on the dissolution rate.

## 2. The model

Figure 1 is a schematic of our model system, in which we indicate that the compositional boundary layers, adjacent to the dissolving solid, are much thinner than the thermal boundary layer, owing to the much smaller value of compositional diffusivity *O*(10^{−9} m^{2} s^{−1}) compared with the thermal diffusivity *O*(10^{−6} m^{2} s^{−1}).

In modelling equilibrium dissolution of one salt into a brine of two salts, we first describe conservation of the mass of each salt and conservation of heat in the liquid. We then model heat and salt conservation across the ablating interface. As solid dissolves into the brine, the concentration of the dissolving salt in the liquid increases; meanwhile, as the solid vacates space, the other salt can diffuse to fill that space only if there is a concentration gradient for it to descend, depressing the concentration of that salt near the interface. Both salts, therefore, diffuse in the liquid; this is important since the equilibrium condition at the interface depends on both salt concentrations.

### (a) Governing equations

We now develop a quantitative model for mass and heat conservation. If a unit volume of the brine contains a mass *m*_{W} of water, *m*_{A} of salt A and *m*_{B} of salt B, then we denote the concentrations of A and B by(2.1)

(2.2)Each salt may diffuse through the liquid. In general, the salts have different diffusion coefficients, which depend on the concentration of the other salt, but in the dilute limit, which, for simplicity, we assume applies herein, we take them to be constants *D*_{A} for salt A and *D*_{B} for salt B, leading to conservation equations(2.3)(2.4)where we have, in a Boussinesq approximation, neglected the effects of variation in the density of the fluid, *ρ*_{L}≡*m*_{A}+*m*_{B}+*m*_{W}. Once this is done, conservation of water (within the fluid) is automatically satisfied by conserving the two salts. *v* is a spatially uniform fluid speed, in the negative *x*-direction, whose origin will be made clear in the discussion of equation (2.9). Heat conduction through the liquid (*j*=L) and solid (*j*=S) involves the thermal diffusivity, *κ*_{j},(2.5)Usually, we consider the dissolving solid to be composed of salt A, but our model formulation can also handle either salt B or ice as the dissolving solid.

### (b) Phase equilibrium and saturation

We assume that the concentration of the liquid in contact with the solid interface is given by the phase-equilibrium value. This has the form shown in figure 2, which corresponds to the KCl–NaCl system. Each contour corresponds to the equilibrium concentrations for a given temperature. If the brine contains a second salt, then the equilibrium concentration of the dissolving phase, for a given temperature, decreases. Furthermore, as the concentration of the second salt increases beyond a critical point (the kink in each contour of figure 2*b*), there is a rapid decrease in the equilibrium concentration of the dissolving salt. For convenience, we use a simplified form for this equilibrium surface, given by(2.6)where *a*_{i}, *b*_{i} and *c*_{i} are parameters obtained by fitting to the empirical data reproduced in figure 2*a*. The best-fit values are tabulated in the left-hand part of table 1. This model approximates the equilibrium surface by three planes, intersecting at linear boundaries, which divide the concentration space into three regimes: the high A concentration regime (equilibrium with solid A), where *a*_{A}, *b*_{A} and *c*_{A} are relevant, the ‘A-dominated branch’; the high B concentration regime (equilibrium with solid B), where *a*_{B}, *b*_{B} and *c*_{B} are relevant, the ‘B-dominated branch’; and the low concentration regime (equilibrium with solid ice), where *a*_{I}, *b*_{I} and *c*_{I} are relevant, the ‘ice-dominated branch’. Our description (equation (2.6)) could be seen as a linearized, local approximation to the high-order polynomials that it is traditional (cf. Hall *et al*. 1988; Sterner *et al*. 1988) to fit to the equilibrium data.

Figure 3 illustrates the importance of the equilibrium surface in figure 2: the condition of the fluid at the dissolving interface is constrained to lie on that surface.

### (c) Interfacial boundary conditions

At the interface, we require conservation of A, B and heat. If the interface has position *h*(*t*) at time *t*, then heat conservation across the interface requires(2.7)where the subscript *k* indexes the solid composition: *k*=A for salt A; *k*=B for salt B; or *k*=I for ice. In our numerical and graphical examples, and in our qualitative conclusions, we examine only solid salts, *k*∈{A, B}, but our algebraic expressions are also valid for ice, *k*=I. *l*_{k} is the ratio of the specific latent heat of dissolution of material *k* to its specific heat capacity. Also, the temperature is continuous across the interface(2.8)Conservation of A across the interface requires(2.9)where *δ*_{kl} is the Kronecker delta: *δ*_{kl}=1 where *k*=*l*; *δ*_{kl}=0 where *k*≠*l*; and *v* is the (spatially uniform) fluid speed, away from the interface, required to allow for volume expansion on phase change, associated with the differing densities *ρ*_{L} of the liquid and *ρ*_{S} of the solid (cf. Chiareli & Worster 1995). Similarly, conservation of B across the interface requires(2.10)Since the solid and liquid have different densities, we also account for the conservation of total mass(2.11)We can substitute from equation (2.11) into equations (2.9) and (2.10), to give(2.12)(2.13)where *r*_{LS} is the density ratio *ρ*_{L}/*ρ*_{S}.

## 3. Self-similar solutions

### (a) The form of the solutions

Guided by the results of Woods (1992), who presented self-similar solutions for a single salt, we have developed analogous solutions for the ternary system. We express solutions in terms of the error function(3.1)where the normalized Gaussian(3.2)The solutions depend on *x* and *t* only through the dimensionless similarity variable(3.3)By analogy to this variable, we define a time-independent boundary position in *η* space (and dissolution rate constant)(3.4)

#### (i) The fluid region

We consider first the fluid region of the system, to the left of the boundary (figure 1). Here, the solutions are(3.5)(3.6)and(3.7)The governing equations suggest and are key dimensionless numbers. Δ*C*_{A}, Δ*C*_{B} and Δ*T*_{L} remain to be determined. The variable *ξ*=*η*+(1/*r*_{LS}−1)*λ* represents a translation of the similarity variable *η*, adjusted to allow for advection of salt and heat by the fluid velocity−*v*. These solutions satisfy equations (2.3) and (2.4), and the fluid version of equation (2.5), as well as the far-field conditions (figure 1).

#### (ii) The solid region

Next, we turn to the solid region, to the right of the boundary (figure 1). Here, only temperature is relevant and the solution is(3.8)The governing equations suggest that is a key dimensionless number. Δ*T*_{S} remains to be determined. This solution satisfies the solid version of equation (2.5), as well as the far-field conditions (figure 1).

### (b) Constraints on the solutions from boundary conditions

#### (iii) Boundary between a fluid region and a solid region

Substituting equation (3.5) into equation (2.12) gives(3.9)The concentration gradient, driving diffusion of salt A away from the boundary, is controlled by the competition between a term representing supply of salt A from the dissolving solid and a term representing the uptake of salt A by the newly liquid volume. By a similar analysis based on equations (2.13) and (3.6),(3.10)Substituting equation (3.7) into equation (2.7) gives(3.11)while substituting equation (3.7) into equation (2.8) gives(3.12)The heat fluxes, driven by the temperature gradients on either side of the boundary, differ by the uptake of latent heat in dissolution.

### (c) Constraints on the solutions from the equilibrium conditions

#### (iv) Boundary between an all-fluid region and an all-solid region

At the boundary, equation (2.6) applies, i.e.(3.13)

Substituting Δ*T*_{S} from equation (3.12) into equation (3.13),(3.14)Equation (3.14) pertains to the *i*-dominated branch of equation (2.6); to choose whether *i* is ‘A’, ‘B’ or ‘ice’, given *C*_{A0}, *C*_{B0} and *λ*, we calculate the concentrations *C*_{A0}+Δ*C*_{A}*G*(*λ*/*r*_{LS}) and *C*_{B0}+Δ*C*_{B}*G*(*Rλ*/*r*_{LS}) at the boundary and choose the *i* value in whose concentration-space domain the boundary concentrations fall. Equation (3.14), which we term the ‘dispersion relation’, is an implicit determination of *λ* and is the key result of this paper.

## 4. Discussion

We have solved the dispersion relation for a range of cases to illustrate how the dissolution behaviour depends on the concentration of each salt in the brine.

First, we consider the solid to be pure KCl, and we explore (§4*a*) the effect of changing the NaCl concentration in the fluid on the dissolution rate. We contrast this to the rate of dissolution of solid NaCl in the same fluid conditions (§4*b*) to illustrate the impact of the presence of two salts in solution. We also show (§§4*a* and 4*b*) that variations in the temperature of the fluid have a much smaller effect. This results from the equilibrium constraint at the interface, combined with diffusion of solute. Similar results are found in all problems considered in this work. We then (§4*c*) generalize the results, allowing the far-field fluid to contain both NaCl and KCl. We explore how the dissolution rates of NaCl and KCl vary in this two-salt brine. To these ends, we choose parameters suitable for a system where salt A is KCl and salt B is NaCl (for clarity about the substances involved, we will henceforth use subscripts *K* and *N*, not *A* and *B*). The parameter values can be found in the right-hand part of table 1.

### (a) Dissolution of solid KCl by NaCl brine

In figure 4, we plot predictions for the dissolution rate of solid KCl (*k*=*K*), as a function of NaCl concentration in the far-field brine. Three curves are plotted corresponding to far-field brines at *T*_{f}=−10, 50 and 110°C. All of these temperatures are well below the melting point of the salt (into pure liquid salt). The *T*_{f} variation is deliberately extreme and is purely an input to produce model conclusions: the intention is that if we predict that, owing to the strong dependence of equilibrium temperature on fluid concentration, the dissolution rate is insensitive to this extreme temperature variation, then it will be even more insensitive to smaller, more geophysically realistic temperature variations. Exactly what is ‘geophysically realistic’ will depend on the details of the application. However, the existence of thermal contact between fluid and solid at the interface does not, in itself, prohibit large temperature contrasts between the far-field fluid and the far-field solid, where ‘far field’ means the distance from the interface is large compared with the (time-dependent) characteristic length for thermal conduction.

Addition of extra NaCl to the far-field brine reduces the dimensionless KCl dissolution rate significantly, from a maximum value of approximately 0.15 at *C*_{N0}=0. This decrease in dissolution rate becomes abruptly more rapid when *C*_{N0} passes a value which we term the *critical concentration* (point X, *C*_{N0}=0.26, *T*_{f}=50°C, *λ*=0.047). Eventually, at some higher *C*_{N0} (point Y, *C*_{N0}=0.28, *T*_{f}=50°C, *λ*=0), dissolution stops altogether. Later in this section, we will provide physical interpretations of the critical concentration and the stoppage-of-dissolution concentration. As the far-field brine temperature increases by 120 K, these two concentrations do not change perceptibly. The effect of changing the far-field brine temperature on the dissolution rate is smaller than that of changing the far-field NaCl concentration: even the enormous temperature shift between the highest and lowest contours changes the dissolution rate by only approximately 0.011 below the critical concentration and by only approximately 0.008 between the critical concentration and the stoppage-of-dissolution concentration.

To help interpret the variation of dissolution rate with concentration (figure 4), we now illustrate how the concentration and temperature of the liquid adjust from the far field to the interface, in three specific cases: A (*C*_{N0}=0, *T*_{f}=50°C, *λ*=0.14); B (*C*_{N0}=0.12, *T*_{f}=50°C, *λ*=0.10); and C (*C*_{N0}=0.27, *T*_{f}=50°C, *λ*=0.03), as marked in figure 4. To this end, in figure 5*a*, we plot predicted KCl–NaCl concentration profiles of the brine, for three far-field brine NaCl concentrations (points A, B and C in figure 4), along with the predicted locus, in KCl–NaCl concentration space, of interface brine conditions, as *C*_{N0} varies. Similarly, in figure 5*b*, we plot predicted KCl concentration–temperature profiles of the brine, for three far-field brine NaCl concentrations (points A, B and C in figure 4), along with the predicted locus, in KCl concentration–temperature space, of interface brine conditions, as *C*_{N0} varies.

Increasing the far-field NaCl concentration decreases the interface KCl concentration, and therefore decreases the KCl flux from the interface to the far-field brine, as indeed it must, to be associated with the reduced KCl dissolution rate we have described. In addition, as the far-field brine NaCl concentration increases, the interface temperature increases. This reduces the heat flux to the interface from both far fields. Again, this is necessary for the increase in far-field brine NaCl concentration to be associated with a decrease in dissolution rate, given a fixed latent heat. In figure 5*a*, one can see that the critical concentration (point X) is associated with the interface brine conditions switching from the KCl-dominated to the NaCl-dominated branch of equation (2.6). On the NaCl-dominated branch, the reduction of interface brine KCl concentration, as the far-field and interface brine NaCl concentrations increase, is quicker than on the KCl-dominated branch (*b*_{N}/*a*_{N}>*b*_{K}/*a*_{K}), as one would expect, given the associated rapid reduction of dissolution rate. The stoppage-of-dissolution concentration (point Y) is associated with the far-field brine being saturated in NaCl, and therefore unable to dissolve any KCl. Intriguingly (figure 5*a*), where there is NaCl in the far-field fluid, the NaCl concentration in the fluid varies with position (and therefore with time). This is necessary for dissolution, because, as the dissolution front passes by a given location, it replaces solid material, consisting entirely of KCl, with liquid, containing NaCl. Since the NaCl flux is zero in the solid, there has to be a non-zero flux and therefore an NaCl concentration gradient, in the fluid adjacent to the front, in order that the newly formed fluid element, produced by the dissolution, has a non-zero NaCl content. This transport also serves to reduce the magnitude of the NaCl concentration gradient by spreading out the constant far-field/dissolution-front concentration contrast (the latter concentration being fixed by phase equilibrium) over the system's increasing length-scale.

To help interpret the variation of dissolution rate with temperature (figure 4), we now illustrate how the concentration and temperature of the liquid adjust from the far field to the interface in three specific cases: D (*C*_{N0}=0.12, *T*_{f}=−10°C, *λ*=0.097); B and E (*C*_{N0}=0.12, *T*_{f}=110°C, *λ*=0.11), as marked in figure 4. To this end, in figure 6*a*, we plot predicted KCl concentration–temperature profiles of the brine, for three far-field brine temperatures (points D, B and E in figure 4), along with the predicted locus, in KCl concentration–temperature space, of interface brine conditions, as *T*_{f} varies. Similarly, in figure 6*b*, we plot predicted KCl–NaCl concentration profiles of the brine, for three far-field brine temperatures (points D, B and E in figure 4), along with the predicted locus, in KCl–NaCl concentration space, of the interface brine conditions, as *T*_{f} varies.

The KCl–NaCl concentration profiles for the three far-field brine temperatures are almost identical. The increase in interface KCl concentration, and therefore in KCl flux from the interface to the far-field brine, on increasing the far-field brine temperature, is tiny. This is consistent with the weak dependence of dissolution rate on far-field brine temperature. The decrease in interface NaCl concentration, and the attendant increase in NaCl flux from the far-field brine to the interface, on increasing the far-field brine temperature, is similarly small. However, the increase in interface temperature, as the far-field brine temperature increases, is substantial, albeit smaller than the increase in far-field brine temperature. Even though the change in concentration is small, a relatively large change in temperature is required, in order to maintain equilibrium, because the rate of change of the equilibrium temperature with concentration is so large (*a*_{K}=860 K, *b*_{K}=640 K). As far-field brine temperature increases, interface temperature also increases, so that part of the heat flux from the brine to the interface can be balanced by a heat flux from the interface to the solid, and vice versa: for the coldest far-field fluid temperature, there is more heat flux from the far-field solid to the interface than is needed to supply latent heat, and the need to transport away excess heat to the far-field fluid leads to an interface temperature higher than the far-field fluid temperature. The rate of dissolution is limited by the requirement to remain in equilibrium: therefore, as the temperature difference between the far-field brine and the far-field solid increases, a progressively smaller fraction of the heat flux to the interface is used to supply the latent heat of dissolution. This effect is shown in figure 7, where it is clear that, except in a narrow range of far-field brine temperatures, close to the far-field solid temperature, the heat flux *r*, used as latent heat of dissolution, is a tiny fraction of the heat flux *q*, delivered to the interface by the brine. This explanation of the minor importance of temperature, relative to concentration, relies on the liquidus being steep (large values of *a*_{k} and *b*_{k} compared with the available range of temperatures). For solids other than KCl (and NaCl), the liquidus may be shallower, and temperature may be more important; in particular, for solid ice, *a*_{I}=−44.0 K and *b*_{I}=−122.7 K. One might therefore expect the influence of temperature on dissolution rate, relative to that of NaCl concentration, to be approximately 5 times greater for solid ice than for solid KCl.

In figure 6*a*, the spatial variation of temperature, for *T*_{f}=50°C, is almost invisible, for two reasons. Firstly, it is small compared with the imposed variation of far-field fluid temperature, while the latter determines the range of the vertical axis. Secondly, the temperature variation happens over a much larger distance scale than the spatial concentration variation, so that almost all the temperature variation is compressed at the left-hand end of this graph, with concentration at its extreme low value. The same brine profile is presented in a different way, in figure 8, where we plot temperature as a function of dimensionless position, for *T*_{f}=50°C, which eliminates both causes of near-invisibility of the spatial temperature variation.

Interface temperature is not strongly constrained by phase equilibrium, because the phase-equilibrium relations are so steep that a tiny adjustment of concentration can maintain phase equilibrium through a huge range of temperatures. The leading-order contribution to interface temperature is a weighted average of the two far-field temperatures, just as if heat were being conducted from the hotter far field to the colder far field without any intervening phase-change front. The different thermal conductivities of NaCl and KCl lead to a small difference in the weightings. The dissolution introduces a second-order contribution to interface temperature through the need to drive heat flux into the interface to supply latent heat. This second-order contribution will be smaller than the leading-order contribution, because the latent heat of dissolution is small compared with the through-going heat flux (unless the two far-field temperatures are exceptionally close together; figure 7).

### (b) Dissolution of solid KCl or solid NaCl by NaCl brine

In figure 9, we compare predictions (figure 4) for the dissolution rate of solid KCl (*k*=*K*), as a function of NaCl concentration in the far-field brine, with predictions for the dissolution rate of solid NaCl (*k*=*N*), as a function of NaCl concentration in the far-field brine. For each solid, three curves are plotted corresponding to far-field brines at *T*_{f}=−10, 50 and 110°C.

The dissolution rate for solid NaCl decreases with increasing far-field NaCl concentration, reaching zero around the same concentration as for solid KCl. However, for solid NaCl, there is no critical concentration, i.e. no sudden change in the gradient of *λ* with respect to *C*_{N0}, owing to the lack of any KCl in the system. The effect of changing the far-field brine temperature on the dissolution rate is even smaller for solid NaCl than for solid KCl, with the dimensionless NaCl dissolution rate changing by only approximately 0.0014, over the range of accessible temperatures. For all far-field conditions with *C*_{K0}=0, the dissolution rate for solid KCl is quicker than that for solid NaCl. To understand this, first imagine that the dissolution rates for the two solids, with identical far-field conditions, were the same. To transport away the salt that is transferred from the solution into the brine, the NaCl concentration difference, between the interface and the far-field brine in the case with solid NaCl, must be about the same as the KCl concentration difference, between the interface and the far-field brine in the case with solid KCl. However, because *b*_{N}=4876 K is much larger than *a*_{K}=860 K, the interface temperature required for phase equilibrium will be higher in the case with solid NaCl (and a high NaCl concentration in the brine at the interface) than in the case with solid KCl (and a high KCl concentration in the brine at the interface). This raised interface temperature will reduce the temperature gradient and hence heat flux supplied from the hotter far field to the interface, and increase the temperature gradient and hence heat flux supplied from the interface to the colder far field, thereby reducing the difference in these fluxes which is available as latent heat for dissolution, by a greater margin than can be compensated by the smaller specific latent heat of NaCl: this is incompatible with the proposal that the dissolution rates for the two solids are the same, and we deduce that solid NaCl must dissolve more slowly than solid KCl.

To help interpret the variation of NaCl dissolution rate with concentration (figure 9), we now illustrate how the concentration and temperature of the liquid adjust from the far field to the interface in three specific cases: F (*C*_{N0}=0, *T*_{f}=50°C, *λ*=0.1096); G (*C*_{N0}=0.12, *T*_{f}=50°C, *λ*=0.0687); and H (*C*_{N0}=0.27, *T*_{f}=50°C, *λ*=0.0064), as marked in figure 9. To this end, in figure 10, we plot predicted NaCl concentration–temperature profiles of the brine for three far-field brine NaCl concentrations (points F, G and H in figure 9).

As far-field brine NaCl concentration increases, interface NaCl concentration remains stubbornly constant; therefore, the concentration difference between the interface and the far-field brine decreases, as does the NaCl flux from the interface to the far-field brine, as indeed it must, to be associated with a reduced NaCl dissolution rate. Simultaneously, the interface temperature increases noticeably, as permitted by the very steep (*b*_{N}=4876 K) liquidus, suppressing the temperature difference between both far fields and the interface, and therefore the heat flux from both far fields to the interface, which again is necessary for the reduced dissolution rate. The interface NaCl concentrations are unsurprisingly higher than those where the solid is KCl, which, given the steepness of the liquidus, means that the interface temperatures are a great deal higher than those where the solid is KCl; this latter fact makes for much lower heat fluxes than those where the solid is KCl, and is therefore consistent with dissolution rates being lower for solid NaCl than for solid KCl, even though KCl has the greater latent heat.

### (c) Dissolution of solid KCl or solid NaCl by a mixed brine

The effect on the dissolution rate, of including KCl in the far-field brine, is illustrated in figure 11.

Low to moderate far-field KCl concentrations suppress the dissolution rate, neither much distorting the form of its *C*_{N0}-dependence nor greatly affecting the differences between solid KCl and solid NaCl. However, for the highest far-field KCl concentration, *C*_{K0}=0.2, the dispersion curves have crossed over, so that NaCl dissolves more quickly than KCl.

## 5. Conclusions

We have developed and explored an equilibrium model for dissolution of a pure solid into a multi-component brine. We have focused on the KCl–NaCl–H_{2}O system and have developed a family of similarity solutions, which capture the dominant controls on the dissolution, associated with the temperatures of the solid and brine, with the identity (KCl or NaCl) of the solid and, most importantly, with the concentrations of NaCl and KCl in the far-field brine, relative to the equilibrium concentrations at the brine/solid interface.

We found that where the KCl-dominated branch of the equilibrium surface pertains (as it usually does for solid KCl), the fractional change in KCl dissolution rate with temperature, over the full (approx. 120 K) range of temperatures studied, is of order 11%, while the fractional change in KCl dissolution rate with concentration (of either salt), over the full (approx. 0.2) range of accessible concentrations, is of order 100%. This implies that concentration is an order of magnitude more important than temperature. Where the NaCl-dominated branch pertains (as it usually does for solid NaCl), the fractional change in NaCl dissolution rate with temperature, over the full (approx. 120 K) range of temperatures studied, is of order 2%, while the fractional change in NaCl dissolution rate with concentration (of either salt), over the full (approx. 0.28) range of accessible concentrations, is of order 200%. This implies that concentration is two orders of magnitude more important than temperature. To recap, for both solids, under this study's assumptions of equilibrium dissolution with diffusive transport, we predict the dissolution rates to be approximately independent of far-field brine temperature, in the range −10 to 110°C.

In addition to the predictions for the relative importance of concentration and temperature, the calculations show that, with the same far-field conditions, NaCl dissolves more slowly than KCl (unless the far-field brine is very close to saturation in KCl). This is especially interesting, since it implies that, with a mixed solid, dissolution (in the early time limit) will tend to produce a partially dissolved zone, in which solid NaCl coexists with a brine, containing both dissolved NaCl and dissolved KCl. The scenario with a mixed solid is the subject of further study.

This model can be applied directly to other salts, by finding appropriate parameter values for those salts to replace the KCl/NaCl values presented in table 1, and substituting those values into the algebraic equations presented herein. The possibility of this simple substitution is a key advantage of the analytical method we have adopted compared with numerical solution of the differential equations.

## Acknowledgments

We would like to thank Rio Tinto for financial support.

## Footnotes

- Received November 14, 2006.
- Accepted January 8, 2007.

- © 2007 The Royal Society