Journal of Geophysical Research: Solid Earth

Reactive transport at stressed grain contact and creep compaction of quartz sand

Authors


Corresponding author: W. He, M-I SWACO, A Schlumberger Company, 5950 North Course Drive, Houston, TX 77072, USA. (whe4@slb.com)

Abstract

[1] A kinetic model is developed to investigate intergranular pressure solution. This model couples stress-induced dissolution at grain contacts, diffusion through grain boundaries, and precipitation in pore spaces. The rate-controlling processes are evaluated according to the dimensionless concentrations at grain contacts and in the pore fluid. Constrained by the experimental results of creep compaction of quartz sands, calculations suggest that the equilibrium concentration at stressed grain contacts (ceqb) does not exceed 1 order of magnitude higher than the hydrostatic equilibrium concentration (ceq) at the initial stages of creep compaction. However, ceqb decays rapidly with increasing compaction and becomes close to ceq after several percent strain. The diffusivity at grain contacts is 1–2 orders of magnitude lower than the diffusivity in pore fluid. The rate-controlling process is related to grain size and strain. An increase in grain size shifts the systems toward the diffusion-controlled regime, while an increase in strain shifts the systems from dissolution-controlled regime toward either diffusion-controlled regime (larger grains) or precipitation-controlled regime (smaller grains). Intergranular pressure solution appears to be very sensitive to pore-fluid chemistry. Even a slight supersaturation in the pore fluid could prevent the diffusion along grain contacts. This may explain why the strength of intergranular pressure solution varies widely in natural sandstones.

1 Introduction

[2] Intergranular pressure solution is regarded as an important mechanism of rock deformation in the upper crust and for the diagenesis of sedimentary rocks. It consists of stress-induced dissolution at grain contacts, diffusion along the grain boundaries, and precipitation on free surfaces [Weyl, 1959; Rutter, 1976, 1983; Raj, 1982; Tada et al., 1987; Spiers and Schutjens, 1990; Spiers and Brzesowsky, 1993; Dewers and Hajash, 1995; Hickman and Evans, 1995; Lehner, 1995; Kruzhanov and Stockhert, 1998; Renard et al., 1999; Revil, 2001; Gundersen et al., 2002; Niemeijer et al., 2002; Yasuhara et al., 2003, 2005; Revil et al., 2006]. The slowest process is the rate-controlling step for creep compaction.

[3] It is commonly accepted that solubility increases in response to increasing stress acting on the solid [Kamb, 1961; Paterson, 1973; Rutter, 1976; Robin, 1978; Lehner and Bataille, 1984; Elias and Hajash, 1992; Mullins, 1995; Fletcher and Merino, 2001; He et al., 2007; Croizé et al., 2010]. Several studies have modeled the dissolution and transport processes along stressed grain contacts [Rutter, 1976, 1983; Dewers and Ortoleva, 1990; Spiers and Schutjens, 1990; Ortoleva, 1994; Lehner, 1995; Shimizu, 1995; Renard et al., 1999; Revil, 2001; Yasuhara et al., 2003; Revil et al., 2006; Van Noort et al., 2008a; Van Noort and Spiers, 2009], which have contributed to our understanding of the solution-transfer creep compaction. However, the knowledge on the kinetic processes at stressed grain contacts is still very limited, mainly due to the difficulties of direct observations.

[4] Various studies have reached very different assumptions about the rate-controlling process [Weyl, 1959; Rutter, 1976, 1983; Raj, 1982; Gratier and Guiget, 1986; Tada et al., 1987; Mullis, 1991, 1993; De Meer and Spiers, 1995; De Meer et al., 1997; Revil, 1999; Niemeijer et al., 2002; Yasuhara et al., 2003; Zhang and Spiers, 2005; Gratier et al., 2009]. These differences may be related to the experimental conditions, material properties, or the modeling setting. Differences may also be due to the uncertainties in modeling parameters at grain-grain contacts, such as the magnitude and distribution of stress, the equilibrium concentrations, and diffusivity.

[5] Creep compaction rate of grain aggregates has been the focus of many theoretical and experimental studies [Raj, 1982; Rutter, 1983; Angevine and Turcotte, 1983; Spiers and Schutjens, 1990; Gratz, 1991; Schutjens, 1991; De Meer and Spiers, 1995; Dewers and Hajash, 1995; Lehner, 1995; Lemee and Gueguen, 1996; De Meer et al., 1997; Renard et al., 1999; Rutter and Wanten, 2000; He et al., 2002, 2003; Kay et al., 2006; Van Noort et al., 2008b]. These studies indicate that compaction rates depend on the physical and chemical conditions of the system and the material properties, such as stress, temperature, strain, grain size, and grain-boundary diffusivity. Nevertheless, both modeling and experimental studies have their limitations. Theoretical modeling is a simplification of a complex system which involves both chemical and mechanical processes. For experimental investigation, it is not always possible to create enough strain to understand the deformation mechanism and rate law. For example, it is unrealistic to achieve adequate strains to observe the transfer from loose sands to consolidated sandstones due to the slow compaction rate at diagenetic conditions [Dewers and Hajash, 1995; He, 2001; He et al., 2003]. An increase in temperature or stress can increase the compaction rate, but the deformation mechanisms may significantly change. However, these limitations can be minimized by combining theoretical modeling and experiments. Experimental observation can provide crucial constraints to theoretical modeling, while theoretical modeling can help understand the deformation behavior that experimental investigations have difficulty to observe and measure.

[6] This paper describes a model of dissolution and diffusion at a stressed grain contact and precipitation in a closed pore space to mimic pressure solution during creep compaction of quartz sand at diagenetic conditions. The model predicts the ability of one process to be the rate-limiting step by describing concentrations relative to two end-members—the equilibrium value in the pore space and the more poorly known equilibrium value at the stressed contacts. This approach avoids having to specify the solubility at the contact to determine the rate-controlling process. We show how the rate-controlling process evolves with strain along a path that is determined by grain surface roughness and a dimensionless parameter, which depends on the square of the grain size and inversely on the grain-boundary diffusivity.

[7] Due to the difficulty of direct experimental measurement, few published data on the equilibrium concentration (solubility) at stressed grain contacts are available. There are also large uncertainties on the reported values of diffusivity along grain contacts. We will evaluate the two parameters with experimental creep compaction rates and pore-fluid concentrations as modeling input. We then apply the dissolution-diffusion-precipitation model to the creep compaction of quartz sand at diagenetic conditions. Based on the modeling results, we analyze the factors influencing intergranular pressure solution in the diagenesis of sandstone.

2 Dissolution-Diffusion-Precipitation Model

[8] We consider a closed system consisting of an idealized cubic packing of homogeneous spherical grains undergoing isotropic volumetric strain. Each grain has diameter d and touches six other grains. The deformation along a single grain contact can be represented by Figure 1. The grain-contact radius, a, for small volume strains (ε) is given approximately by [He et al., 2002]

display math(1)
Figure 1.

Dissolution along a nominally flat contact of two identical spherical grains subject to a normal stress, σn.

[9] It is important to note that the strain in equation ((1)) is measured from an initial state of point contacts between spheres. In actual aggregates, the contacts will be finite in size, even before grain-contact dissolution occurs. For the convenience in developing the remainder of the model, we will use the relationship in equation ((1)); but bear in mind that real grain systems will start out at some non-zero strain, ε0.

[10] Since each grain undergoes the same evolution, we can then model the system by dividing the aggregate into regions, or “unit cells,” each consisting of a single contact and its surrounding fluid. Each “unit cell” is identical, so the volume strain within any cell is equivalent to the bulk volume strain of the aggregate. The enhanced stress at the contact is inversely proportional to the contact area and therefore varies with strain.

[11] We assume that grain contacts are filled with a thin film of fluid which transmits the stress between grains. Since grain solubility is enhanced by stress, we presume an equilibrium concentration in this film, ceqb, which may be significantly greater than the equilibrium concentration in the lower stress pore fluid, ceq. We do not need to specify the quantitative relationship between ceqb and stress in order to continue to develop the model, so we defer this discussion until section 5. For simplicity, we neglect variation in stress across the contact.

[12] We can write the kinetic equations describing the evolution of concentration along the grain contact, cb(r), and in the pore fluid, c, thus

display math(2)
display math(3)

where r represents the distance from the center of the circular grain contact; t is the time; Dgb is the diffusivity at the grain contact; ϕ is the porosity; and Rdiss, Rprec, and Rs are the kinetic rates of dissolution along the grain contact, precipitation in the pore space, and supply of dissolved materials to the pore space, respectively. The rate of dissolution and supply are taken to be positive and precipitation rate negative. For simplicity, rapid diffusivity in the pore fluid is assumed to remove any spatial gradients in c.

[13] The boundary conditions in (2) and (3) are symmetrical at the center of the contact (r = 0) and continuity of concentration at the edge of the contact (r = a):

display math(4a)
display math(4b)

[14] The dissolution and precipitation rates depend on the deviation of the concentration from the equilibrium value. The reaction rate (mol/cm3 sample/s) in the pore space is given by

display math(5)

where km is the dissolution rate constant for solid under hydrostatic conditions (mol/cm2 mineral surface/s) and Am is the specific reactive surface area (cm2 mineral surface/cm3 sample).

[15] We assume that dissolution at a grain contact follows a similar relationship to (5), but both the local equilibrium concentration at the contact and the dissolution rate constant are assumed to increase with stress. Rdiss (mol/cm3 fluid/s) is given by [Shimizu, 1995]

display math(6)

where Abs is the specific reactive surface area of a single grain contact (cm2 surface/cm3 fluid).

[16] Rs is the product of the flux of material diffusing out of a single grain contact, J (mol/s), and the density of contacts, nc (number of contacts/cm3 sample). Given a simple cubic pack of spherical grains undergoing isotropic compaction, the density of contacts is 3 times the density of grains, ng (grain/cm3 sample).

display math(7)

[17] The flux of solute from a grain contact is given by the diffusive flux at the edge of the contact multiplied by the surface area of the contact edge [Tada et al., 1987; Mullis, 1991; Lehner, 1995]:

display math(8)

where δ is the mean grain-contact thickness and inline image is the concentration gradient at the periphery of the contact. The resulting supply rate (mol/cm3 sample/s) is

display math(9)

[18] Substituting (5), (6), and (9) into (2) and (3) yields

display math(10)
display math(11)

[19] Because the contact radius (a), porosity (ϕ), and the grain-contact solubility (ceqb) are a function of strain, the kinetic rates on the right-hand sides of equations ((10)) and ((11)) are functions of time. However, for typical geologic strain rates, the changes in a, ϕ, and ceqb are very slow compared to the rate of change of the solute concentration at the grain contact and in pore fluid. Thus, solute concentration can be assumed to quickly reach a steady state for any given value of strain (i.e., ∂ cb/∂ t = 0 and ∂ c/∂ t = 0). With this approximation, equations ((10)) and ((11)) can be rewritten as

display math(12)
display math(13)

3 Dimensional Analysis

[20] Since, in the closed system, the concentrations at grain contacts and in pore fluids are always between the two equilibrium values, ceq and ceqb, we introduce two scaled concentrations that will vary from 0 (pore-fluid equilibrium) to 1 (grain-boundary equilibrium):

display math(14a)
display math(14b)

[21] The behavior of this system depends on these dimensionless concentrations. The rest of the analysis in this section will be independent of the actual values of ceq and ceqb.

[22] We use the contact radius, a, as a length scale

display math(15)

[23] Substituting equations (14) and ((15)) into (4), ((12)), and ((13)) leads to the non-dimensionalized system

display math(16)
display math(17)
display math(18a)
display math(18b)

where the two dimensionless parameter groups are

display math(19a)
display math(19b)

[24] The role of the two parameters, Θ and Ω, can be illustrated by rewriting them as ratios of characteristic time scales for different processes. Θ can be written as a ratio of the time scale for solute to diffuse out of the contact, to the time scale for grain-contact dissolution to add solute to the grain-boundary fluid:

display math(20)

[25] For large values of Θ, the time required for material to diffuse out of the contact becomes larger than the time required to dissolve the same amount of material. In this case, the concentrations at the contact will approach saturation at ceqb. When Θ is minute, diffusion will reduce the concentration along the grain boundary to a level very near the pore-fluid concentration.

[26] Ω can be written as the ratio of the time scale for precipitation to the time scale for the supply of dissolved material from the grain contacts:

display math(21)

[27] When Ω is very small, all the material removed from the grain contacts is rapidly precipitated in pore fluids. When Ω is large, precipitation is less effective, and the pore fluid could become supersaturated.

[28] Assuming a flat circular grain contact, the specific reactive surface area of a single grain contact, Abs, is given by 2/δ. The specific reactive surface area in pore spaces (Am) is the product of the reactive surface area of a single grain and the density of grains. At the beginning of compaction, Am is given by

display math(22)

where ϕ0 is the initial porosity and α is the ratio of the actual surface area of a grain to the surface area of a perfect sphere with the equivalent diameter d. The factor α is introduced to account for small-scale roughness and deviations from spherical grains. With increasing compaction, the reactive surface area decreases due to grain convergence and cementation [Angevine and Turcotte, 1983; Merino et al., 1983; Canals and Meunier, 1995]. However, for a small strain, the change in the reactive surface area is small. For simplicity, we assume that reactive surface area is constant during compaction.

[29] Using the above substitutions, we find that the two dimensionless parameters are functions of strain (ε) and a new parameter Γ:

display math(23a)
display math(23b)

where

display math(24)

[30] Both parameters increase with increasing strain: Θ is linearly dependent upon strain, due to the lengthening diffusion pathway; Ω increases as a result of the increase of the density of contact, but the change is small for small strains. Note that Θ and Ω have different dependencies on the dimensionless parameter, Γ, which varies directly with the square of grain size and inversely with the grain-boundary diffusivity.

4 Dimensionless Concentration and Rate-Controlling Processes

[31] Equations ((17)) and ((18b)) can be combined into a single boundary condition on ψb at the edge of the contact. Combining this with equations ((16)) and ((18a)) gives a system that can be solved for dimensionless concentration as a function of position along the contact:

display math(25)

where

display math(26)

and J0 and J1 are the zeroth- and first-order Bessel functions of the first kind.

[32] Figure 2 shows that there are three fundamentally different solutions: (A) high concentrations (ψb approaches 1) within the contact and low concentrations (ψ approaches 0) in the pore fluid, (B) high concentrations both along the contact and in the pore fluid, and (C) low concentrations both along the contact and in the pore fluid. In case A, the large concentration gradients at the contacts indicate that diffusion is the rate-controlling process. In case B, the large supersaturation of the pore fluid indicates that precipitation is the rate-controlling process. In case C, the small concentrations along the grain boundary mean that grain-boundary dissolution is the rate-controlling process. Therefore, the solutions can be succinctly classified based on the two extremes of the dimensionless concentration: the concentration at the contact center (r′ = 0), ψc,

display math(27)

and in the pore fluid (r′ = 1), ψ,

display math(28)
Figure 2.

The variation of the dimensionless concentration in the grain-boundary fluid (ψb) from the contact center (r′ = 0) to the pore space (r′ = 1), for three sets of parameter values that yield (A) a diffusion-controlled system (Θ = 10, Ω = 0.05), (B) precipitation-controlled system (Θ = 10, Ω = 3), and (C) a dissolution-controlled system (Θ = 0.2, Ω = 0.05).

[33] Figure 3 shows how ψc and ψ vary for a large range of values of Θ and Ω. For large values of Ω and small values of Θ (lower right), the concentration on the grain boundary is very close to that in the pore fluid, indicating that diffusion is very fast. Moving to the upper left of Figure 3, the contours diverge as diffusion becomes the rate-controlling process. We will divide up the Θ-Ω parameter space of the problem into three regions, in which the rate-controlling process is dissolution (Ds), diffusion (Df), or precipitation (Pr). We define cases where ψc ≤ 0.1 as dissolution controlled. If ψc > 0.1, either diffusion or precipitation is rate-controlling, depending on the value of ψ. If ψ < 0.5ψc, then there is a large concentration gradient at the contact, and we define this to be diffusion controlled. If ψ > 0.5ψc, then concentration in the pore fluid is relatively high, and we define this as precipitation controlled. The exact values used to draw the boundaries in Figure 3 are arbitrary but are useful in illustrating the regions of parameter space where different processes are important. Other values would merely move the fields slightly but not change their overall shape.

Figure 3.

Variation of the dimensionless concentrations at the center of a grain contact, ψb(0), (thick contour lines) and in the pore fluid, ψ, (thin contour lines) for a wide range of the parameters Θ and Ω. The contour labels apply to both sets of contours. In this and the subsequent figures, the red lines divide the three regions of parameter space where the rate of pressure solution is controlled primarily by a single process: Ds—dissolution controlled, Df—diffusion controlled, and Pr—precipitation controlled.

[34] A particular system will move through this parameter space with increasing strain (Figure 4). If we assume perfect spheres and extremely small contacts, then the pre-strain value of Θ will be near zero, and all systems will be in the dissolution-controlled region. If, however, we assume that all initial contacts are somewhat larger, equivalent to a non-zero initial strain ε0, it is possible that the system starts in one of the other dominant fields, depending on the other parameters. Θ will increase linearly with further strain, while Ω will remain relatively constant for small strains, and only begin to slowly increase when ε > 0.1. Therefore, systems will evolve on a path through parameter space toward either the diffusion-controlled region or the precipitation-controlled region.

Figure 4.

Variation of the dimensionless parameters Θ and Ω with increasing strain. Solutions are shown for five different values of the dimensionless combination, Γ, and a surface area factor, α = 1. The arrows show how the Γ = 102 curve would displace for α values of 10 and 0.1. The thin lines join points on each path with identical strain.

[35] The position of an evolution path will depend on the parameter Γ and on the surface area factor α. The curves on Figure 4 represent paths for different values of Γ with α = 1, while the arrows show how those paths would shift for an order of magnitude change in α. The situation for α < 1 depends on the availability of reactive surface area. If diffusion in the pore fluid is not fast enough to remove the concentration gradients, then precipitation will preferentially occur near the contacts; therefore, the surface area available for precipitation will be less than the entire grain surface in the pore. This effect can be modeled by α < 1.

[36] For large Γ (e.g., large grain size or small diffusivity), the system is diffusion controlled; for smaller values of Γ (e.g., smaller grains or larger diffusivity), the system remains dissolution controlled up to ε  = 10%, with precipitation becoming the rate-controlling process after that. Rougher grains (large α) shift the paths horizontally to the left (smaller Ω), moving the systems toward the diffusion-controlled region; conversely, smoother grains (small α) shift the paths horizontally to the right (larger Ω), moving the systems toward the precipitation-controlled region.

[37] The above analysis of rate-controlling processes reduces the question of rate-controlling process to a single dimensional group (Γ), a grain roughness parameter (α), and volumetric strain. It is independent of the stress magnitude across a grain contact. Thus, it avoids the effect of uncertainties in chemical potential and solubility which are related to stress.

5 Equilibrium Concentration (ceqb), Diffusivity (Dgb), and Prediction of Rate-Controlling Process

[38] To understand the kinetic processes at stressed grain contacts and make quantitative predictions about the rate-controlling process for actual sandstones from the solutions shown in section 4, we must estimate both equilibrium concentration (ceqb) and diffusivity (Dgb). These parameters are not easily measured directly by experimental methods. In this section, we will estimate ceqb and Dgb based on theoretical modeling with experimental measurements in compaction rates and pore-fluid concentrations as the modeling input.

5.1 Equilibrium Concentration (ceqb)

[39] It is commonly understood that ceqb increases with stress [Paterson, 1973]. But the understanding of the effect of stress on solubility is very limited. One approach for estimating ceqb is to relate the dynamic rates of grain-contact dissolution to the bulk compaction rate of a system if all compaction is due solely to grain-contact dissolution. Assuming that the grains are in a cubic packing arrangement undergoing isotropic compaction, we can write a kinematic equation relating the rate of material dissolution (moles/s), Ri, from both grains at a contact to the grain convergence rate, following He et al. [2002]:

display math(29)

where Vm is molar volume and inline image is the linear convergence rate of grains, which is approximately 1/3 the volumetric strain rate of the sample inline image.

[40] From the paths through phase space shown in Figure 4, we expect that dissolution is likely to be the rate-controlling process during the initial strains for very fine grain sizes of material. When dissolution is rate controlling, the concentration at the contact is near the concentration in the pore fluid; so from equation ((6)),

display math(30)

[41] Setting the product of Rdiss and the volume of the contact fluid (πa2δ) equal to Ri, gives an estimate of ceqb(1) as a function of sample compaction rate and pore-fluid concentration:

display math(31)

[42] We will examine experiments of creep compaction for two fine-grained pure (>99%) quartz aggregates: St. Peter quartz sand with grain sizes of 90–124 µm and novaculite with grain sizes of 10–60 µm [He, 2001; He et al., 2003]. The experimental systems have been described in detail by Hajash and Bloom [1991] and He et al. [2003]. The creep-compaction experiments were conducted at 150°C with an effective pressure of 34.5 MPa; first at dry conditions for 1 to 3 months and then at wet conditions for a few months. For St. Peter quartz sand, the experiment achieved ~1.7% of strains during ~2.6 months of creep compaction at wet condition. For novaculite, the experiments achieved ~3.6% of strains during ~3.5 months of creep compaction at wet conditions. The experimental compaction rates at several given strains are summarized in Table 1. During the initial loading of the system there was a strain (ε0) due to mechanical deformation (elastic deformation and micro-granulation, etc.). However, ε0 was not able to be measured exactly. We assume that the total strain before water injection was 1%.

Table 1. The Experimental Compaction Rate and Pore-Fluid Concentration and the Calculated ceqba
Strain (%)St. Peter Quartz Sand (90–124 µm)Novaculite (10–60 µm)
inline image (s−1)c/ceqceqb(1)/ceqceqb(2)/ceqinline image (s−1)c/ceqceqb(1)/ceqceqb(2)/ceq
  1. a

    The pore-fluid concentrations are constrained by experiments, and ceq is assumed to be 160 ppm [Dewers and Hajash, 1995; He et al., 2007].

11.0 × 10−81.032513.53.4 × 10−81.0527.87.5
1.55.4 × 10−91.021472.1 × 10−81.0417.54.5
2.71.2 × 10−91.013.936.8 × 10−91.036.42.5
4.6    8.7 × 10−101.021.71.5

[43] The understanding of pore-fluid chemistry in compacting systems has been greatly improved in recent years through direct experimental measurements and modeling calculations [Elias and Hajash, 1992; Dewers and Hajash, 1995; He et al., 2007]. However, in addition to stress-induced grain-contact dissolution, ultrafines and unstable surfaces which are created during compaction can also contribute to pore-fluid chemistry [He et al., 2007]. In order to better constrain the ceqb, we have to minimize the effects of ultrafines and unstable surfaces. Table 1 shows the summary of silica concentration of pore fluids in which the effects of ultrafines and unstable surfaces on pore-fluid chemistry were partially removed [Dewers and Hajash, 1995; He et al., 2007]. We calculated ceqb at several given strains by applying the data from Table 1 to equation ((31)) (Figure 5, black symbols). Since pore fluids are only slightly supersaturated (c/ceq ≤ 1.05), the compaction rate term has the dominant impact on the calculated results of ceqb.

Figure 5.

Calculated equilibrium concentration at stressed grain contact (ceqb normalized by ceq) with experimental compaction rates and pore-fluid concentrations (Table 1) as modeling input. The first 1% of strain is assumed to be caused by elastic and other mechanical deformations during initial loading. The black symbols are the calculated results from equation ((31)), and the open symbols are the calculated results from equation ((33)).

[44] Since mechanical processes such as grain rearrangement, micro-cracking, etc, likely contribute somewhat to compaction after the initial 1% strain [He, 2001; Chester et al., 2004; He et al., 2007], these estimates for ceqb represent the upper bound at a given strain.

[45] Under the same set of assumptions (dissolution rate controlled, all strain after mechanical compaction due to dissolution), we can independently estimate ceqb from pore-fluid concentrations. Assuming that the precipitation rate in pore space is approximately equal to the grain-contact dissolution rate, we have

display math(32)

[46] The grain-contact dissolution rate (mol/cm3 sample/s) on the right side of equation ((32)) is equal to the product of Rdiss (equation (30)), the volume of the contact fluid (πa2δ), and the density of contacts in the sample (equation (7)). Equation ((32)) can be rewritten as

display math(33)

where the superscript (2) in ceqb is used to distinguish the calculated values from those according to equation ((31)).

[47] This indicates that the increase in equilibrium concentration due to stress is inversely proportional to strain, but directly related to the supersaturation of the pore fluid (which also varies with strain). Equation ((33)) is developed assuming that grain-contact dissolution is the only silica source added to pore fluid. But as discussed above, ultrafines and unstable surfaces can also contribute to pore-fluid chemistry and the experimental pore-fluid concentrations represent the upper limit of the contribution from grain-contact dissolution. Therefore, the calculated ceqb based on the experimental pore-fluid concentration may represent the upper limit of the actual values. The estimates of ceqb from the values of Table 1 and equation ((33)) are shown as open circles in Figure 5. These values are consistently lower than those obtained from equation ((31)) based on compaction rate and pore-fluid concentration. One possible reason for this difference is that the ultrafines and unstable surfaces, which could lead to higher supersaturation, had been partially removed or dissolved by pore-fluid flush while the contribution of mechanical deformation to compaction rate is much less affected by it. The effects of mechanical processes on compaction rates and pore-fluid concentration are strongest during the initial stages of compaction and decreases with increasing compaction [Chester et al., 2004; He et al., 2007]; therefore, the calculated ceqb values would more closely reflect the actual values at higher strains because of the smaller effects of mechanical processes.

[48] Experiments also show that pore-fluid concentrations become approximately saturated at low strains if the effect of ultrafines and unstable surfaces is reduced by flushing distilled water through the samples or by long-term fluid flow [He, 2001; He et al., 2007]. This indicates that grain-contact dissolution is not able to create significant supersaturation after a low strain. For example, the experiment with St. Peter quartz sand of grain sizes of 250–350 µm indicated that pore-fluid concentration was approximately saturated when compaction achieved ~1.3% of strain at wet condition [He, 2001]. Calculations from equation ((33)) indicate that ceqb does not exceed 2–3 times of hydrostatic equilibrium concentration (ceq) or even approximately equal to ceq by ~1.3% of strain achieved at wet conditions.

[49] The lower limit of ceqb can be constrained by pore-fluid concentration. To initiate the diffusion of solute from grain contact to pore fluid, ceqb must be greater than the pore-fluid concentration. The pore-fluid concentration shown in Table 1 indicates that ceqb is greater than 1.01ceq–1.05ceq at the indicated strains. The lower limit of ceqb can be further constrained by the pore-fluid concentration if the effects of ultrafines and unstable surfaces on pore-fluid chemistry are largely still present. Table 2 shows pore-fluid concentrations at some specific strains during creep compaction of quartz sand in nominally closed systems (no fluid samples were removed except for one fluid sample from SP41). These compaction experiments were conducted at 150°C with an effective pressure of 34.5 MPa (confining pressure ≈ 46.2 MPa, pore pressure ≈ 11.7 MPa) [He, 2001], similar to those discussed previously. Studies of post-test samples indicate that intergranular pressure solution was one of the important mechanisms of creep compaction. Therefore, ceqb should be greater than 1.23ceq at ~1.5% of strain and greater than 1.04ceq at ~4.4% of strain.

Table 2. Silica Concentration of Pore Fluids During Creep Compaction of Quartz Sand in Nominally Closed Systems
Exp. #Grain Size, µmElapsed Time (h)Strain (%)SiO2 (ppm)Saturation State
  1. Elapsed time is calculated from the injection of distilled water to the systems. Error for the measurement of silica concentration is ±5 ppm. ceq is assumed to be160 ppm [Dewers and Hajash, 1995; He et al., 2007].

SP38250–35030201.41971.23
SP41124–18016601.61961.23
SP41124–18026801.81841.15
SP4090–12433713.11821.14
NOVA510–6025204.41661.04

[50] Compaction experiments of quartz sand by Schutjens [1991] and Niemeijer et al. [2002] suggest that intergranular pressure solution is the dominant compaction mechanism at higher temperatures. Silt quartz was used as the test material in Niemeijer et al. [2002], and dissolution was regarded as the rate-controlling process. Very fine grained sands were used as the test material in Schutjens [1991], and we assume that dissolution is also the rate-controlling process. The Niemeijer et al. data and equation ((31)) yield ceqb values less than 1 order of magnitude over ceq even when the effective pressure is up to 150 MPa (Figure 6a). This data also indicate that ceqb increases by up to a factor of 3.5 as effective pressure increases from 50 MPa to 150MPa. When the effective pressure is 15.0MPa, ceqb is approximately equal to ceq (Figure 6b). In all the cases, ceqb decreases with increasing compaction (strain).

Figure 6.

Calculated equilibrium concentration at stressed grain contact (ceqb normalized by ceq) with experimental compaction rates as modeling input. Temperature, effective pressure, and grain size are indicated: (a) experimental results from Niemeijer et al. [2002] as modeling input and (b) experimental results from Schutjens [1991] as modeling input. The first 2%–7% of strains during initial compaction could be mainly contributed by the mechanical processes such as grain coordination and rearrangement. For comparison purpose, we eliminate these strains in the figures.

[51] Based on the above calculations, we find that (1) ceqb does not exceed 1 order of magnitude over ceq at the initial stages of compaction and (2) ceqb decays with increasing strain and reaches close to ceq as creep compaction achieves several percent of strain.

[52] A third way to estimate ceqb as a function of strain is from thermodynamic considerations. This allows us to evaluate the evolution of ceqb at higher strains that experiments have not been able to achieve. Assuming that the activity coefficient is close to unity, ceqb can be written as [Paterson, 1973; Lehner, 1995; Shimizu, 1995; Kruzhanov and Stockhert, 1998; Revil, 2001]

display math(34)

where Δu is the difference in chemical potential between the stressed grain contacts and free faces of grains, R is gas constant, and T is temperature. Δu is dependent on the normal stress at the contact [Paterson, 1973; Lehner, 1995; Mullins, 1995; Shimizu, 1995; Fletcher and Merino, 2001]:

display math(35)

[53] Δu is also affected by the strain energy difference between the stressed grain contact and free surfaces [Gal and Nur, 1998; Revil, 2001; Sahal et al., 2004], but this effect is small compared to σnVm. Equations ((34)) and ((35)) indicate that ceqb increases exponentially with stress at the contact, which decreases with increasing strain due to the gradual growth of the grain-contact area. For a random packing of identical spheres, the intergranular force F can be related to the macroscopically applied effective pressure (Pe) [Dvorkin and Yin, 1995; Fjar et al., 2008]:

display math(36)

where Nc is the average number of contacts per sphere. The mean normal stress at grain contact, inline image, can be written as

display math(37)

where λ is a numerical factor which is approximately 1 under the assumption of cubic packing undergoing isotropic compaction. Note that strain (ε) includes the combined contribution of deformation during initial loading and creep by intergranular pressure solution. Equation ((37)) indicates that inline image is independent of grain size, as suggested by Renard et al. [1999].

[54] From equations ((34)), ((35)), and ((37)), we have

display math(38)

[55] In an actual compacting system, the distribution of contact forces is a strong function of the packing structure [Chan and Ngan, 2005, 2006] and the factor λ varies depending on the actual packing pattern of grains. The constraints on ceqb suggest that λ is 0.03–0.13. According to equation ((38)), ceqb can be related to ceqb1 by

display math(39)

ceqb can be evaluated if its value, ceqb1, is known at a specific strain, ε1. We evaluate ceqb as a function of strain given three different ceqb1 assumptions (10ceq, 5ceq, and 2ceq at ε = 0.01) that are consistent with the above constraints. The calculations suggest that ceqb decreases rapidly with increasing strain when ε < 3% and then gradually approaches to the equilibrium value under hydrostatic pressure, ceq (Figure 7). When ε > 10%, ceqb becomes very close to ceq.

Figure 7.

Equilibrium concentration at stressed grain contact (ceqb normalized by ceq) as a function of strain (ε) for three different assumptions, ceqb = 10ceq, ceqb = 5ceq, and ceqb = 2ceq, at the initial creep compaction (ε1 = 0.01). The first 1% of strain is assumed to be caused by elastic and other mechanical deformations during initial loading.

5.2 Diffusivity (Dgb) at Stressed Grain Contacts

[56] Similar to ceqb, Dgb is not easily measured by experimental methods. There are large uncertainties on the reported values of Dgb. Estimates for Dgb vary from values close to that of bulk pore fluid [Horn et al., 1989; Revil, 2001; Dysthe et al., 2002a; Alcantar et al., 2003], to 1 order of magnitude smaller [Kruzhanov and Stockhert, 1998; Gratier et al., 2009], to 5–7 orders of magnitude smaller [Rutter, 1976; Tada et al., 1987; Farver and Yund, 1991]. Effect of stress on diffusivity is noticed in the experimental studies of Nogami and Tomozawa [1984], which found that the diffusivity of water in silica glass decreases with increasing compressive stress and increasing hydrostatic pressure.

[57] Here we will use the experimental measurements of compaction rate and fluid chemistry to constrain the diffusivity at stressed grain contacts. Assuming that the dissolution rate at a contact (Ri; equation (29)) is approximately equal to the flux of material diffusing out of a grain contact (J; equation (8)), we have

display math(40)

where cb(0) is the silica concentration at the center of grain contact. δDgb is the effective grain-boundary diffusivity. The concentration difference between grain contacts and pore fluid, (cb(0) − c), is difficult to obtain by direct measurement, but its value can be constrained as 0 ≤ (cb(0) − c)  (ceqb − ceq). (cb(0) − c) increases as the diffusivity decreases and becomes close to (ceqb − ceq) if diffusion is the rate-controlling process. We will first use the compaction rates and fluid chemistry of St. Peter quartz sand as the modeling input. Calculations suggest that δDgb is 10−18–10−19 m3/s. If the grain-boundary thickness (δ) is in the order of 10−8–10−9 m [Tada et al., 1987; Gratz, 1991; Renard and Ortoleva, 1997], then Dgb is 10−9–10−10 m2/s, which is at the same magnitude or 1 order of magnitude smaller than the diffusivity in the pore fluid.

[58] Note that flat grain contacts are assumed in the modeling; however, observations of the deformed St. Peter quartz sand show that micro-fracturing develops at grain contacts, with some fractures connecting to pore fluid [He et al., 2003, 2007; Chester et al., 2004, 2007]. Experiments of natural quartzites suggest that micro-fracturing at grain contacts may lead to significant increase in deformation rate by several orders of magnitude [Gratz, 1991; Den Brok, 1998]. Experimental investigations also show that pressure solution may strongly depend on the roughness of the grain contact [Dysthe et al., 2002b, 2003; Gratier et al., 2009; Van Noort et al., 2011]. These findings suggest that the calculated diffusivity at grain contact could be significantly escalated due to the enhanced deformation rates by micro-fracturing along grain contacts.

[59] Micro-structural studies using optical microscopy and SEM indicate that novaculite with silt-sized grains has much smoother grain surfaces and less tendency for micro-cracking to develop at stressed grain contacts [Chester et al., 2004, 2007; He et al., 2007]. Evolution of fluid chemistry in quartz compaction systems also suggests that micro-cracking is strongest during the initial stage of compaction and its effect decays with increasing compaction [He et al., 2007]. This relationship between micro-fracturing and strain is consistent with the experimental studies of Dysthe et al. [2003], which indicate that the roughness evolves through time by dissolution and its effect on compaction rate decays with increasing compaction. The experiments on novaculite lasted for 4–6 months and achieved strains of ~4.5% [He, 2001]. Here we assume that at the end of the compaction experiments, the effect of micro-cracking on novaculite compaction is minimal. Calculation by using the experimental results of novaculite as the modeling constraints suggests that δDgb is 10−19–10−20 m3/s. For δ = 10−8–10−9 m, Dgb is 10−10–10−11 m2/s, which is 1 to 2 orders of magnitude smaller than the diffusivity in pore fluid. We will use these δDgb estimations in the following calculation for creep compaction of quartz sand.

5.3 Rate-Controlling Process of Creep Compaction of Quartz Sand

[60] We can re-draw Figure 4 to show the evolution of rate-controlling process for creep compaction of quartz sand with a range of grain sizes. System temperature is assumed to be 150°C. In order to reflect the effects of diffusivity on rate-controlling processes, calculations are conducted on both δDgb = 10−19 m3/s and δDgb = 10−20 m3/s. The grain properties are taken from characteristics of St. Peter quartz sand [He, 2001; He et al., 2003]. Initial porosity is ~35%. For an average grain size of 300 µm, the surface area from BET measurements is ~900 cm2/cm3. The roughness parameter, α, derived from equation ((22)) has a value of ~7. The initial surface areas for other grain sizes of material can be derived from equation ((22)).

[61] The resulting variations in Θ and Ω with increasing strain for different grain sizes of quartz sands are shown in Figure 8. For very fine to very coarse quartz sand, the rate of intergranular pressure solution is controlled by diffusion. For very fine sand to silt, dissolution at the contacts is the rate-controlling process at the early stages of compaction, with diffusion or precipitation becoming the rate-controlling process at a strain determined by grain sizes. The transition strains from dissolution controlling to diffusion controlling are <15% for very fine sand and up to 30% for silt. For very fine to medium silt, the precipitation rate in the pore space becomes a factor, but precipitation in the pores is never the rate-controlling mechanism at strains <30%. These modeling results suggest that for most sandstones, creep compaction may be rate controlled by diffusion from an early stage of compaction. But for siltstone or very fine grained sandstone, dissolution could be the rate-controlling process, as suggested by the experimental studies of Niemeijer et al. [2002] and Van Noort et al. [2008b].

Figure 8.

Variation of the dimensionless parameters Θ and Ω with increasing strain for different grain sizes of quartz sand. The surface area factor, α, is taken to be 7. The thin lines join points on each path with identical strain.

[62] Note that an increase in grain size shifts all systems toward the diffusion-controlled regime, while an increase in δDgb shifts all systems away from diffusion control. In addition, a smoother grain surface (smaller α) will also shift the curves horizontally to the right (away from diffusion control).

6 Creep-Compaction Rate of Quartz Sand at Diagenetic Conditions and the Controlling Factors

[63] Natural quartz sandstones are produced by complex coupled chemical and physical processes operating over geologic time. Intergranular pressure solution is regarded as an important mechanism of compaction during diagenesis of quartz sandstones [Houseknecht, 1987, 1988], although other processes (e.g., grain sliding, cataclasis, and fluid-assisted cracking) may operate concurrently or even dominate depending on the conditions [Dickinson and Milliken, 1995]. Here we will assume that intergranular pressure solution is the dominant compaction mechanism of quartz sandstone and will use the determinations of ceqb and Dgb from section 5 in our model to make quantitative prediction about the evolution of creep-compaction rate as a function of strain at diagenetic conditions.

[64] The rate laws of creep compaction can be obtained from the above theoretical modeling. If dissolution is the rate-controlling process (c ≈ ceq), equation ((31)) suggests that

display math(41)

[65] If diffusion is the rate-controlling process (cb(0) ≈ ceqb and c ≈ ceq), creep-compaction rate can be derived from equation ((40)):

display math(42)

[66] Like many previous models [Raj, 1982; Angevine and Turcotte, 1983; Rutter, 1983; Shimizu, 1995; De Meer et al., 1997], the above models indicate that creep-compaction rate varies as 1/d3 if grain-boundary diffusion is the rate-controlling process and as 1/d if dissolution at grain contacts is the rate-controlling process.

[67] Regardless of the rate-controlling mechanism, creep-compaction rate inline image can be derived by assuming that the grain-contact dissolution rate is equal to the supply (diffusion) rate of the material from grain contact to pore fluid. Rewriting equation ((40)) gives

display math(43)

where inline image evolves as a function of strain and can be obtained by numerically solving equations ((2)) and ((3)). Equations ((41)), ((42)), and ((43)) all indicate that strain rate inline image as a function of strain (ε) depends on the magnitude and evolution of ceqb.

[68] We will calculate inline image as a function of strain according to equation ((43)). In order to evaluate the effectiveness of the modeling, we will first simulate two different experimental results of Schutjens [1991]. The first set of experiments was conducted at 350°C and intergranular pressure solution was regarded as the dominant compaction mechanism. The second set of experiments, using the same material, was conducted at 300°C, and both intergranular pressure solution and micro-cracking contributed to the compaction.

[69] There is good correlation between the experiments and modeling results for the experiment at 350°C, indicating that the model developed here can simulate compaction rates when intergranular pressure solution is the dominant compaction mechanism. For the experiment at 300°C, the compaction rates decay much faster than the model prediction (Figure 9). This discrepancy could be related to the compaction mechanisms due to a faster decaying in compaction rate for micro-cracking [Schutjens, 1991; He, 2001]. The difference in decaying between experimental and modeling compaction rates could be used as an indication of compaction mechanisms.

Figure 9.

Modeling creep-compaction rates as a function of strain (line) and the experimental results of creep compaction of quartz sand (markers). The experiments were conducted at 300°C and 350°C, respectively with an effective pressure of 15 MPa [Schutjens, 1991].

[70] Figure 10 shows a similar attempt to model the experimental compaction rates at diagenetic conditions [He, 2001; He et al., 2003]. For the small strains that could be achieved in experiments, it is possible to fit that data with either a power law or an exponential decay. But creep-compaction rates as a function of strain represented by equation ((43)) obey power laws. The fast decaying of compaction rate in the early stage of compaction (low strains) is mainly related to the fast decrease in ceqb. According to the modeling results, the timescale to achieve 20% of strain for St. Peter quartz sand with grain sizes of 250–350 µm is ~260 years and the time required to achieve 20% strain for grain sizes of 90–124 µm is ~50 years.

Figure 10.

Modeling creep-compaction rates as a function of strain (line) and the experimental results of creep compaction of quartz sand (markers) for three different grain sizes: 90–124 µm, 124–180 µm, and 250–350 µm. The experiments were conducted at 150°C with an effective pressure of 34.5 MPa [He, 2001; He et al., 2003].

[71] However, mechanical processes such as grain rearrangement and micro-cracking also contributed to the achieved compaction in the experiments [Chester et al., 2004, 2007]. Therefore, the calculated results shown in Figure 10 represent the upper limit of creep-compaction contribution from intergranular pressure solution. Though it is difficult to accurately evaluate the intergranular pressure solution, we can constrain it based on the understanding of ceqb. Assuming ceqb is 2ceq during the initial compaction, which is close to its lower limit of ceqb, the timescale to achieve 20% of strain via pressure solution for St. Peter quartz sand with grain sizes of 250–350 µm is still less than 1000 years. These calculated time scales are small compared to the time scale for burial, suggesting that intergranular pressure solution could be widely developed in quartz sandstone.

[72] Observations of natural sandstone indicate that the diagenetic role of intergranular pressure solution varies widely. Some sandstones show well-developed pressure-solution structure, while others lack any visual evidence. As discussed above, the concentration gradient between stressed grain contact and hydrostatic pore fluid is the driving force for diffusion of material. In order to better understand the controlling factors of intergranular pressure solution, we will examine the fluid chemistry evolution at grain contacts and in pore fluid in compacting sandstone according to the dissolution-diffusion-precipitation model.

[73] While the silica concentrations at grain contacts and in pore fluid evolve as a function of strain and grain size, Figure 11 shows that intergranular pressure solution alone does not cause significant supersaturation in pore fluid (thin contours), no matter the rate-controlling process. At small strains, either slow dissolution at the contact or slow diffusion out of the contact limits the pore fluid supersaturation. Even when precipitation is the rate-controlling process, this occurs when both the equilibrium concentration at grain contact (ceqb) and the pore-fluid concentration (c) approach ceq. For a wide range of grain sizes and strains, silica concentration at the center of grain contacts (cb(0)) is similar to the pore-fluid concentration (c) which is approximately equal to the equilibrium value (ceq).

Figure 11.

Silica concentrations at the center of grain contact in cb(0)/ceq (thick lines) and in the pore fluid in c/ceq (thin lines) as a function of grain size and strain at the assumptions of ceqb = 10ceq at ε = 0.2% with (a) δDgb = 10−19 m3/s and (b) δDgb = 10−20 m3/s.

[74] Note that the above fluid chemistry features at grain contacts and in pore fluid represent a simplified system of pure quartz sand with the assumption that no silica is transported into or out of the system. Nevertheless, it gives insight into how fluid chemistry in pore fluid affects intergranular pressure solution. The modeling results suggest that intergranular pressure solution is very sensitive to pore-fluid chemistry. Even a slight supersaturation in pore fluid would change the kinetics along grain contacts and could significantly slow and even prevent diffusion along grain contacts. Considering that the equilibrium concentration at grain contact (ceqb) is similar to the pore-fluid equilibrium concentration after several percent strains (Figure 7), we can conclude that an approximately saturated or undersaturated pore fluid is required to give rise to a well-developed pressure solution. In scenarios where long-term supersaturation in the pore fluid is present, cementation or quartz overgrowth could be the dominant mechanism for porosity reduction.

[75] Diagenesis in sedimentary basins is developed in open systems and the natural sandstone has much more complex mineralogy than the pure quartz sandstone. Fluid flow, mass transport, and fluid-rock interactions (dissolution and precipitation) in sedimentary basins could affect the development of intergranular pressure solution in sandstone. The factors not included in our model that could have the greatest effect on pressure solution are as follows:

  • [76] Fluid flow in sedimentary basins could strongly affect pore fluid chemistry, therefore also affecting intergranular pressure solution. Effects of pore-fluid flow on compaction are supported by experimental studies which indicate that creep-compaction rate increases as pore-fluid flow results in a decrease in pore-fluid concentration [He et al., 2003; Zhang et al., 2011]. In sedimentary basins, fluid flow along or against a temperature gradient could be a factor. Descending flow of cooler fluid leads to undersaturated pore fluid; therefore, long-term descending fluid flow could result in well-developed pressure solution. In contrast, ascending flow of hotter fluid leads to supersaturated fluids, which would inhibit pressure solution.

  • [77] Fluid-rock interactions in natural sandstone, such as the dissolution of feldspar and rock fragments, could create a pore fluid supersaturated in silica. Consequently, strong mineral alteration during diagenesis may suggest a poorly developed intergranular pressure solution.

  • [78] Pore-filling or grain-coating clays suppress quartz cementation and lead to higher pore-fluid concentrations [Tada et al., 1987; Ehrenberg, 1993; Oelkers et al., 1996]. Intergranular pressure solution may be less developed in sandstones with pore-filling or grain-coating clays, as suggested by several studies [Tada et al., 1987; Houseknecht, 1988; Dewers and Ortoleva, 1991; Ramm, 1992; Worden and Morad, 2000]. For the similar reason, grain-coating micro-crystalline quartz may also inhibit intergranular pressure solution [Worden and Morad, 2000; Maast et al., 2011].

  • [79] Quartz cementation or overgrowth could change the geometry of grain boundary. It could also partially or completely block the channels through which the dissolved material at grain contacts transports to pore fluid [Van Noort et al., 2008b].

[80] The above discussions suggest that extrapolation of experimental results to natural diagenesis must take account of factors such as fluid flow, rock compositions, fluid-rock interactions, and their evolutions during compaction. Consideration of these factors will help better constrain the modeling parameters while extrapolating the experimental data to natural rocks. The experimental results should not be over-extrapolated because the impact of compaction and cementation on the geometry of grain boundary during compaction is not well understood.

7 Conclusions

[81] The reactive transport at stressed grain contacts is characterized by a kinetic model which couples stress-induced dissolution at grain contacts, diffusion through grain boundaries, and precipitation in pore spaces. The rate-controlling processes are evaluated according to the dimensionless concentrations at grain contacts and in pore fluid. Constrained by the experimental results in creep-compaction rate and pore-fluid concentration during creep compaction of quartz sands, theoretical calculations are conducted to evaluate the equilibrium concentration (ceqb) and diffusivity (Dgb) at stressed grain contacts. Creep compaction of quartz sand at diagenetic condition is analyzed according to the dissolution-diffusion-precipitation model.

[82] Calculations suggest that ceqb does not exceed 1 order of magnitude higher than the hydrostatic equilibrium concentration (ceq) at the initial stages of creep compaction. However, ceqb decays rapidly with increasing compaction and becomes close to ceq after several percent of strain. The diffusivity at grain contacts (Dgb) is 1 to 2 orders of magnitude smaller than the diffusivity in pore fluid. The rate-controlling process is dependent on grain size and strain. An increase in grain size shifts the systems toward the diffusion-controlled regime, while an increase in strain shifts the systems from dissolution-controlled regime toward either diffusion-controlled regime (larger grains) or precipitation-controlled regime (smaller grains). Compaction rate decays according to power laws if intergranular pressure solution is the compaction mechanism. In sedimentary basins, fluid flow, mineral compositions, and fluid-rock interactions may affect the pore-fluid chemistry and therefore the development of intergranular pressure solution. This is because even a slight supersaturation in pore fluid could prevent the diffusion along grain contacts and retard the intergranular pressure solution. This may explain why the strength of intergranular pressure solution varies widely in natural sandstones.

Acknowledgments

[83] This work was partially funded from the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy, Grant No. DE-FG03-98ER14887. We sincerely thank Fred Chester, Judith Chester, Andreas Kronenberg, and Anthony Gangi for their numerous discussions and comments. We appreciate Katsman and the associate editor for their very thoughtful and valuable comments and suggestions.