Pore-size controlled solubility (PCS) is incorporated into continuum equations for fluid transport and porosity evolution. The physical properties of a porous domain, in particular pore-size, can modify the effective solubility of minerals, allowing highly supersaturated fluids to exist within submicron-scale pores of rocks; when fluid flows from small pores into larger ones, or vice versa, precipitation or dissolution may occur. Using numerical simulations, we demonstrate that the PCS mechanism can account for the filling of large pore spaces during transport though a heterogeneous rock matrix. Furthermore, depending on flow and initial conditions, the steady state porosity patterns that develop may be heterogeneous. The mechanism is expected to be of significance during diagenesis and fracture mineralization.
 The precipitation and dissolution of minerals in porous media occurs in numerous geological and industrial settings. Such reactions can significantly change the porosity, permeability, and other physical characteristics of the porous matrix, and thus often have important implications in fields as diverse as oil recovery and pollutant transport in groundwater.
 Minerals crystallize or dissolve when they are in contact with a fluid which is either supersaturated or undersaturated, respectively, with respect to the reacting minerals. Dissolution reactions increase porosity, and hence permeability, and can act to enhance further dissolution. By contrast, precipitation reactions tend to clog the pore space, restricting flow and mass transfer, thereby limiting additional precipitation. In laboratory experiments and numerical simulations of mineral precipitation from supersaturated fluids, the inlet rapidly becomes clogged, preventing the transport and precipitation of the dissolved species farther downstream [e.g., Dijk and Berkowitz, 1998; Hilgers and Urai, 2002]. Nevertheless, in nature, pervasive mineralization and porosity reduction of rocks is commonplace, suggesting that such models do not adequately represent geological systems.
 The apparent inconsistency between these studies and field observations can, in part, be resolved by considering the path by which a fluid becomes supersaturated. Traditionally, three main mechanisms have been thought to be responsible for causing supersaturation: (1) pressure and temperature changes during flow through the porous medium, (2) matrix dissolution resulting in supersaturation with respect to a different mineral (e.g., dissolution of calcite and precipitation of gypsum [Singurindy and Berkowitz, 2003]), and (3) mixing-induced supersaturation [e.g., Emmanuel and Berkowitz, 2005]. While these pathways are important in many systems, other mechanisms could help to account for a wider range of mineralization observed in porous rocks.
 One additional process that is likely to play a central role in geological formations is pore-size controlled solubility (PCS), which occurs as a result of the surface tension associated with crystals growing in confined pores. This surface tension gives rise to an excess pressure within the crystal, analogous to the capillary pressure at a liquid-vapor interface within a pore [Rijniers et al., 2005]. By considering the chemical potential at the pore scale, the excess pressure (also termed the crystallization pressure) can be related both to the pore size and to the effective solubility of the mineral [Scherer, 1999; Flatt, 2002; Rijniers et al., 2005]. Although a number of factors, such as interfacial energy and pore and crystal geometry, ultimately determine the magnitude of the effect, fluids in small pores often reach high degrees of supersaturation without mineral precipitation occurring. Furthermore, it is often the case that the smaller the pore, the higher the degree of supersaturation.
 Clearly, the PCS mechanism has important implications for systems with spatially heterogeneous pore-size distributions; in a mono-mineralic system, fluid flowing through a region dominated by small pores will achieve a high degree of supersaturation; upon reaching a fracture or zone with large pores, the fluid will no longer be in equilibrium with the porous matrix and mineral precipitation will occur. Conversely, fluid flowing from a high porosity zone into a low porosity region will be relatively undersaturated, and dissolution of the matrix will dominate. In systems with a range of pore sizes, the system is much more complex as, locally, precipitation may dominate in large pores, while dissolution may occur in smaller ones, or vice versa.
 The theoretical relationship between pore size and solubility has recently been demonstrated experimentally by in situ measurements of solute concentrations using NMR [Rijniers et al., 2005]. In that study, the solubility of hydrated Na2CO3 in 10 nm pores was found to be more than twice that of the bulk solubility, showing that PCS is a feasible mechanism for achieving high supersaturations in porous media.
 In geological systems, observations suggest that the porous matrix can strongly influence saturation and crystallization. In soils, ice segregation, a phenomenon whereby the migration of supercooled pore water through the porous matrix leads to the formation of discrete layers of ice, is related to pore size [Ozawa, 1997]; while ice does not form in pores smaller than 1 μm, it can crystallize in larger pores. In rock formations, the seemingly trivial observation that minerals tend to precipitate in fractures and openings in rocks poses a number of difficulties for conventional mechanisms of porosity reduction. At an analogue site for nuclear waste repositories at Maqarin, Jordan, an examination of mineralized fractures suggested that the fractures sealed before the rock matrix [Steefel and Lichtner, 1998]; critically, such an observation requires the rate of precipitation in the fractures to be far higher than in the rock matrix, which can be achieved with the PCS mechanism. Furthermore, in a study of halite mineralization in sandstone, Putnis and Mauthe  observed that the cement was confined to large pores, a finding that was interpreted as evidence of pore-size controlled solubility.
 Studies examining the PCS mechanism have typically focussed on effects at the pore scale [e.g., Putnis and Mauthe, 2001; Wiltschko and Morse, 2001]; however, no studies have yet explicitly incorporated the PCS effect into the continuum equations describing flow, solute transport, and porosity evolution in porous media. Here, we use a 1D numerical model to examine the effect of pore size on precipitation and dissolution in a porous medium. We describe how the PCS mechanism can be integrated into conservation equations, and we explore some of the dynamic and steady-state porosity patterns that may develop. Implications of the model for future experimental and theoretical research are also discussed.
2. Theoretical Development
 Solute transport and the evolution of porosity in a porous domain can be described by a series of coupled equations. Here, we use a form of the classical, local scale advection-diffusion equation to account for the transport of a reacting species i,
where ϕ is the total porosity, Ci is the solute concentration, De is the effective diffusion coefficient, q is the specific fluid flux, and Qi is a sink/source term derived from empirical equations which accounts for the effects of precipitation and dissolution. The effective diffusion coefficient is defined here as De = ϕDw/τ, where Dw is the free water diffusion coefficient and τ is the tortuosity.
 To derive an expression for the sink/source term Qi, the effect of pore size on solubility must first be considered. To simplify the problem, we consider the pore space in the domain to be comprised of interconnected cylindrical pores (Figure 1). Furthermore, crystal growth and dissolution are restricted to the axial direction of the cylindrical pore. This idealized representation of the porous matrix enables a relatively straightforward expression to be used for the ratio between the effective solubility, Se, and the bulk solubility, S0, of a mineral [Rijniers et al., 2005]
where νsc is the molar volume of the salt in the crystal, m is the number of ions per unit salt, R is the gas constant, T is absolute temperature, γ is the surface tension or interfacial energy, θ is the contact angle between the crystal-liquid interface and the pore wall, and r is the radius of the pore (Figure 2). As θ is usually larger than 90°, the term (−) is typically positive. Thus, for many situations Se > S0, and the fluid in the pore may be supersaturated with respect to a bulk solution; in this case, the smaller the pore radius, the higher the effective solubility. By substituting typical values, such as those in Table 1, into equation (2), significant degrees of supersaturation can be reached for pore sizes at the submicron level. In addition to being pore-size dependent, supersaturation is also strongly related to interfacial energy (Figure 3) which can vary significantly between minerals; for relatively insoluble goethite γ = 1.6 J m2, while for more soluble gypsum the value is only 0.026 J m2 [Stumm and Morgan, 1996]. Importantly, even for minerals with relatively low interfacial energies, low degrees of supersaturation are sufficient to drive mineralization on geological time scales.
 As the porous medium is characterized by cylindrical pores of radius rj, pore length Lj, and the number of pores per unit volume, nj, the total porosity is given by
In addition, the specific surface area, j, corresponding to each pore size is given by
and is related to the total specific surface area by = Σjj.
 For a mineral of the form AB, where A and B are different ionic species, the molar rate of mineral precipitation or dissolution per bulk unit volume is also equal to the magnitude of Qi (equation (1)). In a porous medium with a single pore size, the rate equation is often given as a function of the distance from equilibrium, and has the form [Emmanuel and Berkowitz, 2005]
where is the total specific surface area, Kr is a kinetic rate coefficient, Sp is the pore ion activity product (CACB), and β is an empirical reaction order. The sign of Qi is positive if dissolution occurs ( < 1) and negative for the case of precipitation ( > 1). Here, a similar function is used to describe the rate of dissolution/precipitation associated with each pore size, j,
such that Qi = ΣjQij.
 As crystallization occurs only along the longitudinal axis of the pores, the rate of change in the pore length is given by the expression
where Lj ≥ 0. Moreover, the rate of change in porosity can be related to the Qij term using the expression
 In principle, this set of equations can be used to describe complex porous systems over the whole range of pore sizes. However, considering porous media with only two characteristic pore radii can demonstrate some of the interesting features of such systems, and this is explored in the next section.
3. Numerical Simulations
 The objective of the simulations presented here is to demonstrate some of the large scale effects that may result from pore-size controlled saturation. For simplicity, we consider a porous domain, comprised of a single mineral type, with regions of both unimodal and bimodal distributions of pore radii. The 1D domain is 1 m in length, and small-pore porosity is present throughout, while in the central region of the domain, large-pore porosity is also present; such a domain could represent a single flow conduit, such as a partially filled fracture zone or a high porosity band, bounded by impermeable media. While a 1D domain cannot capture all of the features that might be encountered in multi-dimensional systems, such as flow divergence resulting from changes to the porous medium, the simulated effects on porosity shown here should be valid for a wide range of scenarios. Parameter values for the model are given in Table 1 and the caption for Figure 4.
 In the simulations, fluid enters the left side of the domain at a constant velocity, and is assumed to have already reached chemical equilibrium with the small-pore matrix. Precipitation or dissolution reactions can occur, and as a result the cylindrical pores are allowed to change in size in the axial direction only (i.e., Ls and Ll can lengthen or shorten), although the pore radius and number of pores per unit volume are kept constant.
 The coupled equations determining solute transport and porosity evolution outlined in Section 2 are solved numerically using a finite element method utilizing the FEMLAB® software package. The scheme employs a mesh consisting of 3840 Lagrange quadratic elements together with a time dependent linear system solver. It is important to stress that while the values of the solubility parameters and reaction rate coefficients of the mineral phase that we use are based on those of calcite in a saline fluid, the simulations demonstrate some of the general features that might be encountered during reactive transport in porous rocks.
 In Figures 5a–5b, the evolution of total porosity (ϕ = ϕs + ϕl) is shown for two simulations with different fluid fluxes: in the first, q = 3 × 10−6 m s−1 (∼0.26 m d−1), while in the second q = 10−6 m s−1 (∼0.09 m d−1). At the higher velocity (Figure 5a), porosity decreases uniformly in the central region, until a steady-state uniform porosity pattern is reached throughout the domain. In contrast, in the second simulation (Figure 5b) in which q is lower, porosity decreases non-uniformly in the central region, and by the end of the simulation the pattern is still heterogeneous, although an overall reduction in porosity is observed. Between the inlet and the left boundary of the bimodal region, porosity is constant, while in the central bimodal region overall porosity increases from left to right, reaching a maximum at the right boundary between bimodal and unimodal domains. Farther downstream in the unimodal region, porosity slowly decreases, although values are actually higher relative to the initial condition.
 By examining the components comprising the total porosity, ϕs and ϕl, some insight can be gained into how such patterns form (Figures 5c–5d). In the second simulation, ϕs increases in both the bimodal region and the downstream unimodal region, while ϕl is reduced uniformly. Upon entering the large-pore zone, the fluid is oversaturated in the large pores and the mineral begins to precipitate; at the same time, the precipitation also acts to reduce the degree of saturation of the fluid in the small pores, causing dissolution. As the fluid enters the unimodal region again, it is still undersaturated and the solid matrix continues to dissolve. Overall, the system only stabilizes once the large pores in the central region are filled, and precipitation and dissolution reactions cease entirely. The reason such patterns are not observed during the higher fluid flux simulation is that the rapid transport acts to homogenize the concentration throughout the domain; the removal of dissolved material due to precipitation is balanced by the influx of new material, thereby keeping the fluid in the small pores close to equilibrium.
4. Concluding Remarks
 In this paper, we describe how pore-sized controlled solubility can be incorporated into continuum equations for fluid transport and porosity evolution. Furthermore, using numerical simulations, we demonstrate that the PCS mechanism can account for the filling of large pore spaces during flow though a heterogeneous rock matrix, and the formation of heterogeneous porosity patterns. In addition, the time scale of the simulations (500 ky), indicates that the PCS mechanism is relevant primarily over geological time-scales.
 Clearly, real systems are far more complex than the model presented here, and the main limitations can be summarized as follows: (1) in rocks, pore-radii are not characterised by single values but are distributed over a wide range of sizes; (2) pore-size distribution will evolve continually as long as precipitation and dissolution reactions occur; (3) ideal cylinders of uniform size and spatial distribution do not accurately describe the pore geometry of actual geological media; (4) natural systems often involve more than one mineral; (5) reaction kinetics in geological media could be more complex than the empirical rate laws used in the simulations; (6) additional mechanisms related to pore size, such as nucleation inhibition [e.g., Putnis et al., 1995], are not considered in the model.
 Despite these limitations, the model demonstrates the significance that the PCS mechanism has for geological processes. Perhaps the most unexpected result of the simulations is that, depending on the flow conditions, the steady state porosity patterns can be either homogeneous or heterogeneous. Thus, in addition to representing an important mechanism for the infilling of fractures and high porosity regions, PCS may also lead to the increase of porosity by dissolution, and the mechanism can therefore account for the formation of heterogeneous porosity patterns in rocks which also show evidence of mineralization. While the theoretical and experimental basis of pore-size controlled solubility is sound, evidence from field observations and petrographic studies is still required to assess the impact of PCS in geological settings.
 The Israel Science Foundation is thanked for financial support (contract 598/04). We also thank two anonymous reviewers for their constructive comments.