Modeling the impact of nano-pores on mineralization in sedimentary rocks



[1] By limiting crystal size, nanometer-scale pores in geological media can control the effective solubility of mineral phases. Since mineralization is determined by solubility, this mechanism, termed pore-size controlled solubility (PCS), is potentially significant for the evolution of porosity and permeability during reactive transport. To demonstrate the potential impact in geological systems, we developed a new one-dimensional numerical model, using a moving boundary condition, to describe the mineralization in the rock matrix adjacent to a pressure solution (stylolite) interface. In the model, the porous domain initially possesses a polymodal pore size distribution, although this is allowed to change in response to mineral precipitation, and the evolution of porosity and pore size distributions was simulated both for systems with constant mineral solubility and for systems in which solubility was pore size controlled. Consistent with field observations, total porosity decreases near the stylolite interface in all the simulations. However, in systems with constant solubility, nanometer-scale pores close rapidly because of their high specific surface area; by contrast, when the PCS mechanism is included in the model, transient bimodal pore size distributions develop, with nano-pores remaining open throughout the simulation. Our simulations suggest that the combination of the PCS mechanism with kinetic models for mineral precipitation can account for the bimodal pore size distributions observed in sedimentary carbonate rocks. Furthermore, while the mechanism is unlikely to affect rocks such as sandstones, PCS can impact carbonate and clay-bearing sediments, which typically possess high levels of submicron porosity.

1. Introduction

[2] Porosity and permeability in geological formations are crucial parameters in any process that involves fluid flow and solute transport. As a result, diagenetic changes to the rock matrix are critical in many geological and industrial settings, including hydrocarbon recovery in subsurface reservoirs [Lucia, 2003], carbon sequestration in geological formations [e.g., Gaus et al., 2005; Xu et al., 2005], seismicity [Ague et al., 1998], and transport of contaminants in groundwater [e.g., Steefel and Lichtner, 1994, 1998].

[3] The pore-space within geological media is often composed of pores of different sizes ranging in scale from the macroscopic to the microscopic. Significantly, nanometer-scale pores can represent a substantial proportion of the rock porosity [e.g., Wang et al., 2003], strongly influencing mineralization by increasing the available reactive surface area. Furthermore, as surface area also affects permeability [Bear, 1972], the presence of nanoscale pores can strongly impact fluid flow through the rock.

[4] Very small pores may also have less conventional effects on system behavior, with some studies suggesting that pore size can modify mineral stability and solubility. In soils, ice segregation, a phenomenon whereby the migration of supercooled pore water through the porous matrix leads to the formation of ice layers, is related to pore size [Ozawa, 1997]; while ice crystallizes in pores larger than 1 μm, it fails to form in smaller pores. In a study of mineralized sandstones, Putnis and Mauthe [2001] found that halite cement had preferentially filled large pores, leaving many submicron-scale pores free of precipitate. More recently, Rijniers et al. [2005] used nuclear magnetic resonance to measure in situ solute concentrations in porous media; in that study they showed that the solubility of hydrated sodium carbonate in 10 nm pores was more than twice that of the bulk solubility.

[5] To illustrate the potential impact that pore size controlled solubility (PCS) can have on mineral precipitation and dissolution patterns, one can consider a solute being transported through a fluid-saturated monomineralic porous rock in which the pore space is dominated by small voids. If enough time is available, the solute concentration will eventually reach equilibrium, at which point the ion activity product will be equal to the value of the solubility product determined by the pore size. However, if the solute is then transported into a fracture opening or region containing larger pores, the fluid phase will be oversaturated with respect to the mineral phase and precipitation can take place. Conversely, when a solute moves from a zone populated by large pores into one containing small pores, the fluid phase becomes undersaturated and the matrix may begin to dissolve. As the range of pore sizes increases, system behavior could become increasingly complex, with precipitation dominating in large pores and dissolution occurring in smaller ones, or vice versa as was predicted in a numerical study of the PCS mechanism [Emmanuel and Berkowitz, 2007]. Furthermore, even when levels of disequilibrium induced by PCS are relatively small, mineralization can be rapid enough to reduce porosity by tens of percent on timescales of 103–105 years [Emmanuel and Berkowitz, 2007], and PCS could therefore have a significant impact on geological processes.

[6] A potentially rewarding way to assess the effect of PCS is to compare mineralization patterns with reactive transport models. A major challenge for many systems, however, is that the source of the precipitated mineral phase is often unclear, leading to poorly constrained boundary conditions and, subsequently, ambiguous results. One geological system that can offer at least a partially constrained diagenetic scenario is the mineralized rock matrix adjacent to stylolites, suture-like features in rock that result from pressure solution (stress-induced dissolution at grain to grain contacts). Such features are common in sedimentary rocks, forming during diagenesis or in response to tectonically controlled forces. In many systems, the stylolite represents an interface at which rock is being dissolved and is typically perpendicular to the direction of main compressive stress; moreover, some fraction of the dissolved mineral phase often recrystallizes in the porous matrix [e.g., Mapstone, 1975; Wong and Oldershaw, 1981; Oelkers et al., 1996; Walderhaug and Bjørkum, 2003].

[7] The primary aim of the present study is to simulate the impact that PCS can have on the evolution of porosity and pore size distributions during mineralization in the rock matrix adjacent to stylolites. The results of the study lead to predictions regarding porosity patterns and pore size distributions in stylolite-bearing systems which can be tested against field measurements. Furthermore, we show that the combination of the PCS mechanism with standard kinetic equations for mineral precipitation can lead to bimodal pore size distributions, which could account for similar diagenetic features observed in sedimentary carbonate rocks.

2. Theory of Pore-Size Controlled Solubility

[8] It is well established that crystal size can play an important role in mineral solubility, with crystals typically dissolving more readily as their size decreases. This effect is related to the change in interfacial energy of the growing crystal and the effective solubility, Sd, is often given by [Adamson, 1990; Scherer, 2004]

equation image

where S0 is the bulk solubility, νm is the molar volume of the mineral, R is the gas constant, T is temperature, and γ is the interfacial energy. In addition, ζ represents the crystal curvature, defined as the rate of change of volume with respect to surface area; thus for a spherical crystal possessing a radius of curvature rc (Figure 1a), ζ = 2/rc and equation (1) becomes

equation image

While equation (2) predicts that smaller crystals will be more soluble, this effect is typically only significant in microscopic crystals.

Figure 1.

Schematic diagram showing the geometries for crystal growth explored in (a) the study by Emmanuel and Berkowitz [2007] and (b) the present work. In Figure 1a, growth occurs along the axis of cylindrical pores and the curvature of the growing crystals remains fixed. In Figure 1b, crystal growth is perpendicular to the walls of a square cuboid-shaped pore; as the pore is filled, the radius of curvature of the crystal-fluid interface decreases, leading to an increase in solubility. In both cases, the contact angle is defined by the angle θ and the characteristic size of the pore is defined by the dimension d, which, in the case of cylindrical pores, is also equal to the pore radius.

[9] Clearly, this argument also applies to the solubility of minerals growing in voids within a porous medium. However, unlike crystals growing freely in solution, the maximal radius of curvature for crystals confined within rigid pores is, in most cases, no longer equivalent to the radius of the crystal but is instead a function of the pore size; for example, for a crystal growing in a cylindrical pore (see Figure 1a), the radius of curvature is related to the radius of the pore, d, by rc = d/cosθ, where θ is the contact angle between the substrate and the crystal surface. Substituting this relationship into equation (2) yields [Scherer, 2004]

equation image

and for typical parameter values relatively high levels of supersaturation can be achieved in submicron-scale pores (Figure 2). Although the contact angle, θ, determines whether or not Sd is greater than the bulk solubility, crystals growing in voids typically possess convex rather than concave surfaces [Scherer, 2004], in which case θ < π/2; thus, for many common mineral systems Sd will always exceed S0. Moreover, similar relationships can be developed for different pore geometries, and in all cases the pore dimensions effectively limit the maximum value of rc; moreover, for pores in which multiple crystals are growing, the pore size still determines the minimum value of solubility as smaller crystals will be even more soluble. Thus, as a general rule, the smaller the pore size, the higher the effective solubility.

Figure 2.

Contour plot of normalized effective saturation (Sd/S0) as a function of pore size, d, and interfacial energy, γ, for a crystal growing in a cylindrical pore. Calculated from equation (3) using values in Table 1 with T = 373 K. For goethite, calcite, and gypsum, γ = 1.6, 0.094, and 0.026 J m−2, respectively [Stumm and Morgan, 1996].

[10] As mentioned previously, the PCS mechanism can lead to dissolution in small pores and precipitation in larger ones, an effect that is similar to Ostwald ripening during which large crystals grow at the expense of smaller, more soluble crystals. In fact, as both effects are ultimately driven by crystal size, the concurrent mineral dissolution and crystallization that results from PCS can be considered a form of Ostwald ripening in which the porous medium acts as an additional constraint on the distribution of crystal sizes.

3. Simulation of Stylolite-Bearing Systems

3.1. Governing Equations

[11] There is much evidence to suggest that stylolites represent an important source of cement for their surrounding rocks. Reduced porosity has been measured in proximity to stylolite interfaces [Figure 3; Harms and Choquette, 1965; Dunnington, 1967; Tada and Siever, 1989; Oelkers et al., 1996], while preferential cementation close to stylolites has also been observed in a range of rock types, including chalk [Mapstone, 1975], limestone [Wong and Oldershaw, 1981; Ricken, 1986] and sandstone [Walderhaug and Bjørkum, 2003]. Thus it is reasonable to conclude that mineral dissolution at stylolites is accompanied by diffusion into the rock matrix and subsequent recrystallization [Bathurst, 1971; Merino et al., 1983; Tada and Siever, 1989].

Figure 3.

Measured porosity in stylolite-bearing sandstone from the Barents Sea. Data are from the study by Walderhaug and Bjørkum [2003]. The dashed line represents a regression line for the data. The reduction of porosity close to the stylolites strongly suggests that dissolution at the pressure solution interface was followed by diffusion into the rock matrix and recrystallization.

[12] While mineralization associated with stylolite formation and pressure solution has been modeled previously in both one-dimensional and two-dimensional domains [e.g., Angevine and Turcotte, 1983; Merino et al., 1983; Oelkers et al., 1996; Wangen, 1998; Gundersen et al., 2002], the potential influence of nanoscale pores on the evolution of porosity has not yet been explored. Here we simulate mineral dissolution at the interface between a stylolite and a porous rock matrix by considering a one-dimensional porous domain undergoing dissolution at a constant rate, v, at one of its boundaries (Figure 4); as a result of the dissolution, the domain continually shrinks in size, thus representing a moving boundary problem. We note here that although mineral dissolution in stylolite-bearing systems is ultimately driven by an externally applied stress, modeling a stress condition at the boundary introduces a significant level of additional complexity and uncertainty. By contrast, a constant velocity boundary is both physically reasonable and readily implemented. In addition, we have chosen to restrict mineral dissolution to the stylolite interface; while intergranular pressure solution can be found in stylolite-bearing rocks [Tada and Siever, 1989], stylolites are often the dominant form of pressure solution [e.g., Walderhaug and Bjørkum, 2003].

Figure 4.

Schematic diagram of the model domain. The simulations treat the rock matrix as a one-dimensional domain with a dissolution front that moves at a velocity v, so that the size of the domain at any time is equal to h. The solute flux at the stylolite boundary is determined by the rate at which the rock matrix is dissolved (Ji = v(1 − ϕ)/νm). The left-hand boundary is stationary, and a zero flux condition (∂Ci/∂x = 0) is imposed.

[13] To treat the problem, it is convenient to consider solute transport within a monomineralic porous domain; for simplicity, we examine a system in which the mineral phase undergoes congruent dissolution and precipitation and has a chemical formula AB, where A and B are different ionic species. Assuming that no advection takes place and that diffusion and mineral precipitation dominate mass transfer, a standard equation can be written for each reacting species, i,

equation image

where ϕ is porosity, Ci is the solute concentration, and De is the effective diffusion coefficient, defined as De = Dw/τ; Dw and τ indicate the free water diffusion coefficient and tortuosity factor respectively. Although tortuosity is a function of porosity [Boudreau, 1996], τ is expected to remain in the range ∼3–6 throughout our simulations; as a result, De should not vary significantly and it is treated as constant in our model.

[14] In this approach, the porous medium is treated as a continuum in which the pore space consists of voids with a distribution of sizes; each discrete pore size, defined by the dimension d (see Figure 1), is assigned a partial porosity, ϕd, which is related to the total porosity, ϕ, by the following expression

equation image

In addition, Qi in equation (4) is a kinetic term related to the total rate at which mineralization occurs; here we define this as the sum of the partial rate terms (Qid which is defined in equation (11)), associated with pore size d, such that

equation image

[15] If the initial size of the domain is H, the position, h, of the moving boundary at time t is simply given by h = Htv. Assuming that mass is conserved at the stylolite boundary (i.e., all the dissolved mass diffuses back into the matrix), the boundary conditions for equation (4) are

equation image


equation image

where Ji is the solute flux (i.e., moles of dissolved species per unit area per time). Such a condition also places an upper limit on the amount of dissolution that can occur: defining the size of the domain at equilibrium (he), a mass balance equation for the amount of rock required to fill all the voids in the dissolving domain can be written such that

equation image

Thus, for a domain with initially uniform porosity, he = H(1 − ϕ∣t=0). Furthermore, by choosing a static zero flux condition for the boundary at x = 0, it is implicitly assumed that the system is symmetrical about that point, so that the initial spacing between two stylolite interfaces is 2H and the combined velocity at which they approach each other is 2v.

[16] In addition to influencing solute transport, mineralization also alters the porosity according to the expression

equation image

for a porous medium in which the PCS mechanism is negligible Qi has the form [Stumm and Morgan, 1996]

equation image

where s is the specific surface area (surface area per unit of bulk volume), K is a kinetic rate coefficient, Sp is the pore ion activity product, and β is an empirical reaction order. For the case of precipitation (equation image < 1), the sign of Qi is positive. In addition, in the system examined here, QA = QB, and if the concentrations of the species are equal and the solutes are ideal, Sp = CA2 and only one transport equation is required.

[17] Here the partial terms, Qid (equation (6)), are given by [Emmanuel and Berkowitz, 2007]

equation image

where sd is the partial specific surface area related to the total specific surface area by

equation image

Note that in equation (11), the constant solubility (S0) of equation (10) has been replaced by a pore-size dependent solubility (Sd), and it is this feature that enables different levels of saturation to be reached in pores of different size but identical solute concentrations.

[18] It is worth emphasizing at this point that in adopting a continuum approach we assume a volume-averaged concentration that is the same for each pore. However, numerical simulations have demonstrated that solute concentrations and mineral precipitation may be heterogeneous, even at the pore scale, if the effect of diffusive transport is dominated by that of the chemical reactions [Kang et al., 2003; Tartakovsky et al., 2007]. In the simulations carried out in the present paper, the reaction rate is relatively slow, occurring over millions of years, and diffusion is likely to smooth out most of the pore-scale variation. Moreover, Kang et al. [2006] showed that volume-averaging techniques are likely to be valid in systems close to equilibrium, as is the case in the current study, and a continuum approach is likely to represent a valid approximation of the system being considered.

3.2. Non-dimensional Equations

[19] To facilitate the parameterization of the problem and the numerical solution of the equations, dimensionless quantities can be obtained by scaling the distance with the initial domain size (equation image = x/H), concentrations by the bulk equilibrium value (equation imagei = Ci/C0, where C0 = equation image), and time with the characteristic period for the complete dissolution of the domain (equation image = t/t0, where t0 = H/v); from this we obtain a dimensionless transport equation

equation image

where Pe is the Peclet number, defined as Pe = vH/De, and equation imagei is the dimensionless rate term. In addition, the position of the dissolution front is expressed as equation image = 1 − equation image and the boundary conditions for the equation are as follows:

equation image


equation image

[20] The nondimensional equation for the partial porosity is given by

equation image


equation image

Here the nondimensional specific surface area is equation imaged = sdH while the dimensionless rate coefficient is defined as κ = K/vC0. Note that in this formulation, κ is a measure of the rate of mineral precipitation relative to the rate of stylolite dissolution. Furthermore, while κ can be set for a given simulation, equation imaged can vary both spatially and temporally and it is therefore necessary to specify the length scale, H, to solve the equations.

3.3. Moving Boundary Transformation

[21] To treat the moving boundary problem, we transform the coordinate system to a domain of fixed size employing a similar approach to that used by Gundabala et al. [2006] to model film drying. The transformations that achieve this are simply

equation image


equation image

using the chain rule, the differentials in the new coordinates are found to be

equation image


equation image

Substituting equation (18b) into the nondimensional transport equation (equation (13)) and expanding the left hand side yields

equation image

which then reduces to

equation image

In transformed coordinates, the equation for the evolution of partial porosity (equation (15)) simplifies to

equation image

[22] For a given initial domain size (H) and set of boundary conditions, the two nondimensional parameters are expected to have an important effect on the evolution of porosity in the system: the Peclet number Pe effectively controls both the rate at which the stylolite interface recedes and the flux of material into the porous domain, while κ, the nondimensional rate coefficient determines how rapidly mineralization occurs. Furthermore, some constraints can be placed on the expected magnitude of these parameters in geological systems. Assuming a domain size of centimeters to meters with dissolution occurring over tens of thousands to millions of years, it follows that v is in the range 10−16–10−11 m s−1, and a typical value for De of ∼10−9 m2 s−1 yields 10−9 < Pe < 10−2. An estimate for κ can be obtained from the rate coefficient, K, of the precipitation reaction; however K is highly dependent on the mineral system and can be strongly affected by the presence of inhibitors. Experiments for calcite suggest that K is probably in the range 10−11–10−7 mol m−2 s−1 [e.g., Zhong and Mucci, 1993]; taking S0 = 0.4603 mol2 m−6 for calcite yields C0 = 0.679 mol m−3 and a corresponding range for κ of 100–109. In this study, our analysis will present the results of simulations with Pe values in the lower end of the estimated range and κ values in the middle end, which ensures that the product κPe remains much smaller than unity; as κPe is a measure of the effect of reaction relative to the effect of diffusion, such a condition implies that the continuum assumption is valid [Kang et al., 2003; Tartakovsky et al., 2007].

[23] Previous studies have employed similar transport equations and reaction terms to those adopted in the current model to predict the evolution of total porosity [e.g., Angevine and Turcotte, 1983; Merino et al., 1983; Oelkers et al., 1996; Wangen, 1998; Gundersen et al., 2002], and at very low levels of dissolution (i.e., short times), constant domain models should produce overall similar results to the moving boundary approach we present here; however, at higher levels of dissolution the moving boundary model is expected to more accurately reproduce the behavior of real systems. Moreover, the changes in pore size distribution which are explored in the current model have not previously been considered.

3.4. Model Representation of the Porous Medium

[24] To simulate the impact of nano-pores on the evolution of porosity it is necessary to define a conceptual model of the porous domain. In a preliminary numerical study of the PCS mechanism, Emmanuel and Berkowitz [2007] simulated the growth of crystals along the axis of cylindrical pores (Figure 1a). However, while partial porosities were allowed to change, the pore radii, and hence the effective solubilities, remained fixed; in addition, the initial pore size distribution was entirely bimodal. In the present paper we explore a more realistic model in which the pore size distribution is initially polymodal and allowed to evolve during the simulations. While some minerals grow evenly on the surface of a porous substrate, many minerals will grow as individual crystals within the pore; in our model crystals grow perpendicular to the axis of square cuboid-shaped pores (i.e., tubular pores with a square cross section; see Figure 1b). Importantly, when this is the case, the solubility will depend on the curvature of the crystal, which is limited by the pore size, rather than on the actual curvature of the pore itself. In addition to being analytically tractable and possessing fewer parameters, the model also allows the radius of curvature to evolve throughout the simulations, thereby reproducing the kind of behavior that can be expected in natural systems.

[25] Here we choose to represent the pore space as a bundle of interconnected tubes, such that the partial porosity for pore size d can be defined as

equation image

where Ad is the cross-sectional area of a pore of size d and Ld is the total length of d-sized pores per unit volume. Using basic geometrical relationships, it can be shown that for square cuboid-shaped pores, the radius of curvature of the growing crystal is related to the dimension d by

equation image

where ω = (π/2 − θ)/2; the cross-sectional area of the individual pores is defined as

equation image

while the reactive specific surface area is

equation image

From equations (22) and (24), the specific surface area can also be given as

equation image

and it can be seen that when all other parameters are constant, sd increases with decreasing pore size. Although alternative models of the pore space may more accurately represent the complexity of pore geometry and its subsequent evolution during mineralization [e.g., Moctezuma-Berthier et al., 2002], this basic representation is suitable for demonstrating the overall evolution of porosity and pore size distributions.

3.5. Numerical Simulations

[26] The coupled equations determining solute transport and porosity evolution outlined in sections 3.4 and 3.5 are solved numerically using the COMSOL Multiphysics® software package based on finite elements. The scheme employs a mesh consisting of 3840 Lagrange quadratic elements together with a time-dependent linear system solver; higher mesh densities did not have a significant affect on the results.

[27] The numerical model determines the evolution of porosity in a porous domain with an initial polymodal pore size distribution; 5 different pore sizes are defined (10−8, 10−7, 10−6, 10−5, and 10−4 m), each possessing a partial porosity of 5%. Additional parameter values used in the simulations are given in Table 1. We emphasize that while the physical values of the mineral phase are based on those of calcite in a saline fluid, the simulations demonstrate some of the general features that can be expected of mineralization in many stylolite-bearing systems. We also note here that the contact angle used in the simulations was somewhat arbitrary, and a different value could influence the kind of patterns observed.

Table 1. List of Symbols and Parameter Values
SymbolDefinitionSI UnitsValue
βEmpirical rate order 1
S0Bulk solubility productmol2 m−60.4603a
γInterfacial energyJ m−20.094b
νMolar volumem3 mol−13.69 × 10−5c
RGas constantJ mol−1 K−18.314
θPore wall–crystal contact angle π/4
HInitial domain sizem0.1

[28] In this study, simulations of systems with constant solubility (i.e., Sd = S0 in equation (11)) are compared with those for systems in which solubility is controlled by pore size. In addition, we present a parametric analysis of the impact of κ and Pe on model behavior.

4. Results and Discussion

4.1. Evolution of Total Porosity

[29] The constant solubility model may be applicable to some mineral systems and the simulations are useful as a test of overall system behavior. As expected, total porosity (ϕ) decreases as the stylolite interface is approached (Figure 5), and for many model parameters, porosity reduction is restricted to a narrow zone directly adjacent to the stylolite. Crucially, the width of this zone and its associated porosity gradient are strongly dependent on the value of κ, the dimensionless rate coefficient (Figures 5a–5c): at lower values, the porosity gradient is gradual, with the affected region spanning a relatively wide section of the domain, while at higher κ values the gradient is steeper and the affected region narrower.

Figure 5.

Simulated total porosity profiles adjacent to a stylolite interface for the constant solubility model. Dotted, dashed, and solid lines are for times equation image = 0, equation image = 0.025, and equation image = 0.05, respectively. A stationary boundary is imposed at x = 0, and the porous domain is steadily dissolved from the right hand side (i.e., the stylolite interface). In Figures 5a to 5c, Pe = 10−8, and the effect of the dimensionless kinetic constant, κ, is demonstrated: (a) κ = 103, (b) κ = 104, and (c) κ = 105. Other model parameters are indicated in Table 1. In Figures 5d to 5f, κ = 5 × 103 and Pe is varied: (d) Pe = 10−8, (d) Pe = 10−7, and (f) Pe = 10−6. The abrupt changes in the porosity gradient are due to the representation of the porous media by five distinct classes of pore size; as the stylolite interface is approached, each discontinuity in the porosity profile represents the transition into a region in which a particular class of pore has been totally filled.

[30] In addition, the Peclet number also plays an important role in determining the overall shape of the porosity profile (Figures 5d–5f). At low Pe values, the width of the porosity reduction zone is relatively high; as Pe increases, the zone narrows and the porosity gradient steepens.

[31] The effect of κ and Pe on the ϕ profiles is consistent with the physical nature of the model. At high rates of mineralization (i.e., high κ), the diffusing solute does not penetrate far into the system before equilibrium is reached, and the zone of porosity reduction is correspondingly narrow. Similarly, increasing values of Pe correspond to lower rates of solute transport into the porous domain, which again results in a smaller zone of porosity reduction.

[32] For the PCS simulations (Figure 6), the overall influence of κ and Pe was found to be similar to the constant solubility model. However, for identical parameter values the porosity profiles are noticeably different: the rate of porosity reduction at the stylolite boundary is lower and the width of the porosity reduction zone is greater.

Figure 6.

Simulated total porosity profiles adjacent to a stylolite interface for the PCS model. Dotted, dashed, and solid lines are for times equation image = 0, equation image = 0.025, and equation image = 0.05, respectively. (a) κ = 104, (b) κ = 105, and (c) κ = 106. Pe = 10−8, with boundary conditions and other model parameters as in Figure 5.

[33] Previous models of mineralization in the rock matrix adjacent to stylolites [e.g., Wangen, 1998] have predicted overall similar reductions in porosity to those found in the constant solubility model, and such profiles are consistent with porosity measurements in stylolite-bearing rocks. For example, Oelkers et al. [1996] and Wang et al. [2003] examined porosity variations in stylolitic sandstones and found similar profiles (see Figure 3) to those produced in the simulations; in those studies the reduction of porosity was attributed primarily to reprecipitation of quartz in the matrix following dissolution near or at the stylolite. However, very little of the pore space in typical sandstones is submicron [Netto, 1993], and such systems are probably best modeled using a constant solubility approach. By contrast, in chalks and clays, in which nanoscale porosity is ubiquitous, very different types of porosity profiles could be recorded.

[34] An interesting feature of the present simulations that is not found in previous models is the relatively abrupt changes that occur in the porosity gradient. As discussed in section 4.2, such changes arise as a result of separating the porosity into distinct classes of pore sizes; discontinuities in the porosity gradient develop between regions in which a particular class of pore is completely filled and adjacent regions in which at least some of those pores remain open. Although the representation of a porous matrix by five classes of pore size may be an oversimplification for many systems, such a mechanism could account for the sometimes abrupt changes in total porosity recorded in the mineralized matrix near stylolites.

4.2. Evolution of Cumulative Porosity Patterns and Pore-Size Distributions

[35] An examination of changes in cumulative porosity (equation imageϕd) in the model domain sheds additional light on the governing dynamics of the two systems. In Figures 7a–7c, the cumulative porosity at different times for the constant solubility model is given as a function of pore size and spatial position in the evolving one-dimensional domain. At equation image = 0, the initial uniform polymodal pore size distribution can be observed; as the simulation proceeds the pores are filled hierarchically, with the smallest nano-sized voids being mineralized most rapidly. The order in which the pores are filled can be accounted for by considering the dependence of reaction kinetics on the partial specific surface area (sd), and the fact that sd is inversely proportional to pore size (equation (26)).

Figure 7.

Cumulative porosity (equation imageϕd) as a function of pore size (d) and dimensionless distance (x/H) at different times. In Figures 7a–7c, results for the constant solubility model are shown for (a) equation image = 0, (b) equation image = 0.05, and (c) equation image = 0.1. In Figures 7d to 7f, solutions for the PCS model are presented for (d) equation image = 0, (e) equation image = 0.05, and (f) equation image = 0.1. In both simulations, Pe = 10−8 and κ = 105, while other model parameters are the same as in Figure 5.

[36] As might be expected, the pattern of cumulative porosity for the PCS model is markedly different to that of the constant solubility model. In Figures 7d–7f, it can be seen that the intermediate pore sizes (10−7 < d < 10−5 m), rather than the smallest, are most rapidly filled. However, while the largest pores (d = 10−4 m) are slowly mineralized, the smallest nano-sized pores (d = 10−8 m) remain unfilled throughout the simulations.

[37] The differences between the two systems are clearly highlighted by examining the evolution of cumulative porosity and pore size distributions at the stylolite boundary. In the constant solubility model (Figures 8a–8b), the pore size distribution evolves from a polymodal to a unimodal distribution; importantly the mean pore size increases as mineralization proceeds. By contrast, in the PCS model a transient bimodal pore size distribution develops (Figures 8c–8d) with a steadily decreasing mean pore size; as the large pores continue to be filled, the system evolves slowly toward a unimodal distribution populated by nano-pores.

Figure 8.

(a) Cumulative porosity (equation imageϕd) and (b) partial porosity (ϕd) as a function of pore size at the stylolite interface at different times for the constant solubility model. Note the increase in mean pore size and the evolution toward a unimodal distribution. (c) Cumulative porosity and (d) partial porosity (ϕd) as a function of pore size at the stylolite interface for the PCS model. Note the development of a bimodal distribution. Model parameters are the same as in Figure 7.

[38] Importantly, the results from the PCS model also highlight the main difference between the current model and ones based solely on Ostwald ripening: while Ostwald ripening does lead to changes in crystal-sized distributions [Steefel and Van Cappellen, 1990], the mechanism on its own does not generate bimodal populations. In fact, in our model such a feature is a direct consequence of combining the PCS mechanism with kinetic rate laws.

[39] While we were unable to find reported profiles of pore size distributions in stylolite-bearing rocks that could be used to assess our model, there is nevertheless some field evidence to suggest that nano-pores are important in such systems. In a study by Carrio-Schaffhauser and Gaviglio [1990], porosity and pore size distributions were found to change significantly in rock adjacent to a normal fault at which both pressure solution and mineralization had occurred; porosity was found to decrease from 15% in the matrix to around 10% near the fault, with mean pore size falling from 0.1 to 0.03 μm. In the Upper Chalk formations of England (e.g., at Flamborough Head, Yorkshire), pressure solution at stylolite interfaces, and subsequent mineralization of the highly porous matrix, significantly reduces total overall porosity from around 45% to approximately 30% [Price et al., 1976]; moreover, pore size distributions (Figure 9) demonstrate that mineralized chalk has a lower median pore size (0.41 μm) than unmineralized chalk (0.65 μm). Notably, both types of chalk possess a significant level of pores in the <100 nm range, again suggesting that such small pores are not filled during the mineralization process. Detailed profiles of pore size distributions near stylolite interfaces in chalk formations should be a useful tool to assess the accuracy of the model presented here.

Figure 9.

Cumulative porosity relative to total porosity in mineralized and unmineralized chalk. The data were reconstructed from mercury intrusion measurements reported by Price et al. [1976] for the Upper Chalk of England. The light gray region is bounded by curves for two mineralized samples and represents the range of pore size distributions for mineralized chalk; the intermediate gray region is bounded by curves from two unmineralized samples showing the range for unmineralized rock; the darkest gray region represents the overlap between the two rock types. The average median pore size of the mineralized chalk is significantly lower than in unmineralized samples (0.41 μm and 0.65 μm, respectively); in addition, nanoscale pores (<100 nm) in both rock types are observed.

5. Concluding Remarks

[40] In our paper, we examine the influence of PCS on mineralization in natural systems. A moving boundary reactive transport model is presented which simulates the mineralization of a rock matrix adjacent to a stylolite interface; as the rock dissolves at the stylolite, the solute is allowed to diffuse back into the porous matrix where recrystallization occurs. The model demonstrates that in a system in which mineral solubility is independent of pore size (i.e., solubility is constant), nanometer-scale pores will close first because of their high specific surface area; by contrast, when the PCS mechanism is included in the model, transient bimodal pore size distributions can evolve, with nanometer-scale pores remaining unmineralized throughout the simulations. Such results suggest that measured profiles of pore size distributions in stylolite-bearing rocks could be used as a diagnostic feature to identify the influence of the PCS mechanism.

[41] Obviously, no model can completely capture all the processes associated with such a complex natural process as stylolitization, and a number of limitations associated with our simulations can be identified. More realistic pore geometry and reaction kinetics may lead to improvements in the model, while the inclusion of intergranular pressure solution in the matrix will make the model applicable to a much wider range of stylolite systems. Furthermore, the development of fully coupled mechano-chemical models in the future is likely to lead to an even better understanding of pressure solution and stylolitization.

[42] Despite such constraints, our model demonstrates the potential impact that the PCS mechanism may have, not only for mineralization in stylolite-bearing systems, but for other geological processes as well. Bimodal pore size distributions are often observed in sedimentary carbonates, strongly influencing hydrocarbon recovery [e.g., Pittman, 1971; Keith and Pittman, 1983]; petrographic evidence indicates that these porosity patterns form during diagenesis, and the evolution of bimodal distributions in our simulations suggest that PCS could account for such features. In addition, as the PCS mechanism can strongly influence the evolution of porosity in subsurface reservoirs, the mechanism has important implications for reactive transport during carbon sequestration; however, while PCS may be crucial in carbonate or clay-rich sediments, both of which can possess a significant proportion of nanoscale pores, the effect on typical sandstone formations is likely to be small due to their low levels of submicron porosity [Netto, 1993]. At present, new measurements of pore size distributions in a range of stylolite-bearing rocks are necessary to test the model predictions and to evaluate the impact of the PCS mechanism under geological conditions.


[43] Simon Emmanuel was generously supported by a Bateman Postdoctoral Fellowship from Yale University. Acknowledgment is made to the donors of the American Chemical Society Petroleum Research Fund for support of this research. We also thank Peter Eichhubl, Alexandre Tartakovsky, and an anonymous reviewer for their thoughtful comments, which improved the article.