DLVO-Based Estimates of Adsorbed Water Film Thicknesses in

Article pubs.acs.org/Langmuir

DLVO-Based Estimates of Adsorbed Water Film Thicknesses in Geologic CO2 Reservoirs Tetsu K. Tokunaga* Earth Sciences Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, ms 70-108B Berkeley, California 94720, United States ABSTRACT: When supercritical carbon dioxide (scCO2) is injected into deep subsurface reservoirs, much of the affected volume consists of pores containing both water and scCO2, with water films remaining as the mineral-wetting phase. Although water films can affect multiphase flow and mediate reactions at mineral surfaces, little is known about how film thicknesses depend on system properties. Here, the thicknesses of water films were estimated on the basis of considerations of capillary pressure needed for the entry of CO2 and disjoining pressures in films resulting from van der Waals and electric double-layer interactions. Depth-dependent CO2 and water properties were used to estimate Hamaker constants for water films on silica and smectite surfaces under CO2 confinement. Dispersion interactions were combined with approximate solutions to the electric double layer film thickness−pressure relationship in a Derjaguin−Landau−Verwey− Overbeek (DLVO) analysis, with CO2 as the confining fluid. Under conditions of elevated pressure, temperature, and salinity commonly associated with CO2 sequestration, adsorbed water films in reservoir rock surfaces are typically predicted to be less than 10 nm in thickness. Decreased surface charge of silica under the acidic pH of CO2-equilibrated water and elevated salinity is predicted to compress the electric double layer substantially, such that the dispersion contribution to the film thickness is dominant. Relative to silica, smectite surfaces are predicted to support thicker water films under CO2 confinement because of greater electrostatic and dispersion stabilization.

1. INTRODUCTION Geologic sequestration of carbon dioxide in deep reservoirs may become an important component in efforts to mitigate climate change, with saline reservoirs likely to provide the greatest storage capacity.1,2 Supercritical carbon dioxide (scCO2) injected into deep reservoirs for geologic carbon sequestration typically resides in pores containing the native water (brine) at varying saturations. Experimental evidence indicates that in the presence of scCO2 the aqueous phase will often remain the wetting fluid.3−5 Thus, large-scale reservoir simulations of CO2 storage commonly model the immiscible water−CO2 equilibrium and flow behavior using relationships developed for porous media variably saturated with water and air.6,7 Pore-scale models of immiscible water−CO2 behavior also assume that mineral surfaces remain water wet.8 However, the wetting behavior of water−scCO2 is currently not well understood3,9,10 and contributes to the uncertainty in the predictions of CO2 mobility and reservoir storage capacities.11 Although it is commonly assumed that water is retained by capillarity and adsorption, an understanding of how the thicknesses of adsorbed water films depends on system properties is not known. The presence of water films remaining on mineral surfaces of reservoir pore walls under geologic CO2 sequestration may have important implications for the residual trapping of CO2, and for long-term geochemical reactions.12,13 The wetting of mineral surfaces by water and the stability of water films are © 2012 American Chemical Society

complicated by heterogeneous surface chemistry and microtopography. Natural rock and mineral surfaces exhibit scaledependent roughness ranging from nanometers characteristic of basic crystallographic units14 up to micrometers and millimeters for grain boundaries and fractured surfaces.15 For mineral− water−air systems, such rough microtopography has been decomposed into combinations of capillary channels and flat surface segments.16,17 In such treatments, the wetted volume retained in channels is calculated on the basis of the relation of interfacial curvature to capillary pressure, and the adsorbed water film thickness is calculated with an adsorption isotherm or a disjoining pressure model. For geologic media variably saturated with water (brine) and scCO2, there is currently relatively little information on either capillary or adsorption influences on the behavior of these fluid phases. For example, models of scCO2 flow through pores with adsorbed water films have been developed18 but do not account for the dependence of the film thickness on the disjoining pressure. More recently, water films were experimentally measured on forsterite surfaces in contact with scCO2 saturated to various levels with water.13 Although that study confirms the stability of water films, it does not quantitatively address the dependence on the energy of the film phase, especially at the relatively low disjoining pressures Received: November 13, 2011 Revised: May 3, 2012 Published: May 7, 2012 8001

dx.doi.org/10.1021/la2044587 | Langmuir 2012, 28, 8001−8009

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associated with scCO2 entry into reservoir pores. Because of its low solubility in scCO2, how water redistributes within complex networks of pores by invading scCO2 is ultimately governed by its phase continuity, which is strongly influenced by the presence and thickness of water films. Over longer timeframes, reservoir mineral dissolution and carbonate precipitation reactions important to the permanent trapping of C depend on the extensiveness of hydrated mineral surfaces.13,19,20 For these reasons, fluid-phase distributions within pores control transport and reactions that determine the effectiveness of geologic C sequestration. Therefore, a better understanding of how the film thickness depends on the disjoining (capillary) pressure is clearly necessary for developing mechanistic, predictive models of geologic C sequestration. This study considers the problem of adsorbed water film thicknesses on flat mineral surfaces, in the presence of CO2. Predictions of how adsorbed water film thicknesses depend on surface electrostatic potentials, film solution ionic strength, and disjoining pressure are obtained with approximate solutions to the Poisson−Boltzmann equation. The electric double layer contribution to film stabilization is combined with the dispersion (van der Waals) contribution to relate film thicknesses to disjoining pressures using the Derjaguin− Landau−Verwey−Overbeek (DLVO) approach. However, a prerequisite for the occurrence of adsorbed aqueous films confined by scCO2 in porous media is the entry of scCO2 into previously water-saturated pores. Once scCO2 has entered a pore space, hysteresis in capillary saturation21 allows it to remain in pores over a wider range of pressures. Before suitable capillary and DLVO relations applicable to geologic CO2 sequestration can be estimated, the depth dependences of several underlying parameters first need to be determined.

Figure 1. CO2 phase diagram25 and typical variations of P and T with depth y below the earth’s surface.2,22−24

3. CAPILLARY ENTRY CRITERIA CO2 must enter originally water-filled pores before water films can become confined between solid surfaces and CO2. The condition for the immiscible displacement of water is given by the Young−Laplace equation, 2γ cos θ ψc = (1a) r which relates the capillary pressure (CO2 phase pressure minus the aqueous phase pressure) ψc required for CO2 entry into a characteristic pore radius r to the H2O−CO2 interfacial tension γ and the potentially hysteretic contact angle θ. In terms of the harmonic mean interfacial radius of curvature R = r/cos θ, the Young−Laplace equation (not restricted to cylindrical pores) is 2γ ψc = (1b) R Thus, adsorbed water films coexisting with the immiscible CO2 phase can occur when ψc > 2γ/R, although pore size distributions and resultant hysteresis in saturation−ψc relations complicate the application of this simple relation to porous media. The two fluid phases are taken to be in mutual equilibrium such that no net dissolution of one phase into the other occurs. Such solubility equilibria are expected to be more applicable at greater distances from CO2 injection wells. Under these conditions, water retained by capillarity in pendular rings and depressions within mineral surface microtopography coexists with adsorbed water films (Figure 2). The scaling relation shows that in finer-pore-size (finer-grain-size) media, larger ψc values are required to allow adsorbed films to exist. Conversely, lower fluid−fluid interfacial tensions permit the existence of films under lower-magnitude ψc. The ratio of gravity to interfacial tension forces (the Bond number31,32) ΔρgR2/γ, where Δρ is then density difference between water and scCO2, is only ca. 10−3 for even moderately large R of 100 μm. Thus, the gravitational distortion of interfacial curvature within reservoir pores is negligible. Note that γ values for CO2−water at depths of ≥0.67 km are less than half the value for air−water at surface temperature and pressure (Table 1), indicating that ψc required for CO2 entry into pores of a given size is much lower than air-entry values.3 Decreases in wettability (increased θ) lead to even lower ψc

2. DEPTH DEPENDENCE OF BASIC PROPERTIES Unlike most other systems in which capillary and DLVO calculations are commonly applied, significant increases in pressure P and temperature T encountered with depth below the earth’s surface cause variations in several basic properties affecting equilibrium. The importance of P and T variations is evident upon considering their depth profiles given by typical geothermal gradients ranging from 20 to 30 K km−1 2,22,23 and a hydrostatic pressure gradient of 10.5 MPa km−1.24 Allowed P− T combinations associated with these gradients, given surface conditions of 0.101 MPa and 288 to 298 K, are shown in Figure 1, superimposed on the CO2 phase diagram. This figure illustrates the fact that only a small subset of possible P−T combinations is relevant to subsurface fluid conditions. Wide variations in CO2 properties can be anticipated from the trace of the P−T depth on its phase diagram. Highly depthdependent properties include the H2O−CO2 interfacial tension, which affects the capillary pressure and pore drainage; the density of CO2, which affects capillary and DLVO interactions; the dielectric constants of H2O and CO2, which affect electrostatic and van der Waals interactions; and the Hamaker constants, which are central to calculating van der Waals interactions. Approximate depth dependencies of properties affecting capillarity and the adsorption of water in equilibrium with CO2 are presented in Table 1 for selected depths. The variation of these parameters with depth will be discussed in following sections. 8002


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Table 1. Approximate Values of Properties Affecting Capillarity and the Adsorption of Water Films at Selected Depths depth, km T, K P, MPa γ, mN m−1a ρH2O, Mg m−3

0 293 0.10 72 0.998

0.67 313 7.1 31 0.993

1.00 323 10.6 29 0.989

1.34 333 14.1 27 0.984

2.00 353 21.1 26 0.975

ε H2 Ob

80.25

73.39

69.93

66.62

60.89

n H2 Oc

1.3333b

1.332

1.331

1.330

−20

3.71 × 10 scCO2 0.567

3.65 × 10−20 scCO2 0.638

1.117

1.26

1.327

1.372

1.041

1.106

1.133

1.147

6.2 × 10−22 −8.7 × 10−21 −4.3 × 10−20 3.0 1.0 0.4

4.0 × 10−21 −6.8 × 10−21 −3.3 × 10−20 3.0 0.9 0.4

6.2 × 10−21 −6.0 × 10−21 −2.9 × 10−20 3.0 0.9 0.4

7.5 × 10−21 −5.6 × 10−21 −2.7 × 10−20 2.9 0.9 0.4

3.73 × 10 air 0.00120d

3.73 × 10 CO2 gas 0.206

εCO2

1.00022d

nCO2

1.00028

d

A22, J A132(SiO2), J A132(smec), J κ−1 (n∞= 10 mol m−3), nm κ−1 (n∞= 100 mol m−3), nm κ−1 (n∞= 500 mol m−3), nm

∼0 −1.0 × 10−20 −4.9 × 10−20 3.1 1.0 0.4

−20

1.326

3.72 × 10 scCO2 0.455

A33, J CO2 (or air) phase ρCO2, Mg m−3

−20

−20

Values for interfacial tension γ were estimated from Chalbaud et al.26 and Bachu and Bennion.27 bDielectric constants for water eH2O were calculated from Bradley and Pitzer.28 cRefractive index nH2O values for water at 589 nm are from Harvey et al.29 dProperties for water-saturated air are from Lide.30 Hamaker constants A33 of H2O and A22 of CO2 at several depths were calculated by using relative permittivities eCO2 and refractive indices nCO2 in the Lifshitz equation (eq 5). Hamaker constants for van der Waals interactions with water between minerals and CO2, A132, were calculated for SiO2−water−CO2 and smectite−water−CO2 using A11 = 6 × 10−20 J for SiO2, A11 = 2 × 10−19 J for smectite, and the combining relation (eq 4), as described in the text. Electric double layer (Debye) lengths k−1 were calculated with eq 7. a

The dispersion or van der Waals pressure component on flat substrates varies with the reciprocal of the third power of film thickness, f, through

values for CO2 entry. Recent measurements of the water wettability of silica in the presence of scCO2 indicate that θ does increase with increased pressure.9,10,33,34 Therefore, ψc values for CO2 entry into specific pore sizes may decrease substantially with depth. Adjustment of the entry criterion for the wettability alteration through cos θ from the Young− Laplace equation provides a potentially useful approximate correction, although the precise θ scaling of overall capillary pressure−saturation relations is limited to simple tube geometry.35,36 Under conditions of pore drainage to a given equilibrium state, the chemical potential of capillary water and adsorbed water films is identical. Thus, ψc in pendular rings is in equilibrium with the disjoining pressure of water films, Π, the surface-area-normalized differential change in the Gibbs free energy per change in the film thickness (Figure 2).

Π (f ) =

vdw

vdW

el

6πf 3

(3a)

where A132 is the Hamaker constant for the interactions among (1) the solid substrate, (3) the water film, and (2) CO2. Thus, the film thickness resulting from van der Waals interactions is ⎛ −6π Π vdW ⎞1/3 f=⎜ ⎟ ⎝ A132 ⎠

(3b)

Although eqs 3a and 3b were developed for films on flat, solid surfaces, the influence of interfacial curvature on smoothly curved substrates in granular porous media is minor.39 Thus, for simplicity, the influences of interfacial curvature are not included here. The combining relation can be used to estimate the nonretarded A132 from the Aii of component materials through40

4. DLVO DESCRIPTION OF AQUEOUS FILMS EQUILIBRATED WITH CO2 For simplicity, the condition ψc > 2γ/R is taken as the criterion allowing the adsorbed water films to coexist with capillary water. When this condition is met, the DLVO framework can be applied to describe the dependence of the adsorbed film thickness f on the disjoining pressure Π, taken as the sum of pressure contributions from the van der Waals dispersion, ΠvdW(f), and electrostatic components, Πel(f),37,38 as Π(f ) = Π (f ) + Π(f )

−A132

A132 ≈ ( A11 −

A33 )( A 22 −

A33 )

(4)

Hamaker constants of many common minerals (A11) at room temperature and pressure are available in the literature.41,42 Measurements of A11 over wide ranges of P−T values for solids are limited but indicate negligible changes in conditions associated with CO2 sequestration.43,44 Suitable values for CO2 (A22) and water (A33) were not found for the elevated pressures and temperatures associated with geologic sequestration. Therefore, we used the Lifshitz equation40

(2)

Here, the dispersion and electrostatic components will be evaluated under P−T conditions characteristic of geologic CO2 sequestration, and then their predicted combined effects on film thickness will be considered.

⎛ εi − 1 ⎞2 3hνe (ni 2 − 1)2 3 Aii = kBT ⎜ ⎟ + 4 16 2 (ni 2 + 1)3/2 ⎝ εi + 1 ⎠ 8003

(5)


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The depth-dependent A132 values for SiO2 are presented in Table 1. For smectite, a value of A33 = 2 × 10−19 J48 was used with the combining relation to obtain its A132 values reported in Table 1. From the combining relation (eq 4), stable films are predicted when A33 values are between the values of A11 and A22.49 Calculated A132 values indeed indicate that van der Waals interactions in these mineral−water−CO2 systems support stable adsorbed films. The A132 values range from −5.6 × 10−21 to −4.9 × 10−20, with more negative values indicative of more stable (thicker) water films. Electrostatic interactions between excess charge at interfaces and ions in the aqueous phase provide the other strong influence on the adsorbed film thickness. A conceptual model of the distributions of water and electrostatic potentials in various locations within a porous medium is shown in Figure 2. Wide ranges in electric double layer conditions can coexist in extremely close proximity on the grain scale, depending on the degree of aqueous phase saturation, which in turn depends on pore size distributions and the capillary potential (assumed to be in local equilibrium with the disjoining potential). In this study, we are primarily concerned with electric double layers in water films bounded by solid−water and water−CO2 interfaces, under conditions suitable for DLVO analyses. Confined between the solid−water and CO2−water interfaces, the distribution of ions within the aqueous phase is affected by charged interfaces and by the aqueous film thickness. The main features of the electrical double layer at flat interfaces are described by the 1D Poisson−Boltzmann equation,40,50 expressed for the variation in electrostatic potential ψe in symmetric electrolyte solutions along the x direction, which is normal to the planar solid−water interface

Figure 2. Conceptual model of the pore-scale water distribution after CO2 injection, showing (a) the local equilibrium of capillary and adsorbed water, (b) water and electric double layers between widely separated solid surfaces, (c) water and electric double layers between narrowly separated solid surfaces, and (d) an adsorbed water film and half-space approximation of its electric double layer.

d2ψe dx

2

=

⎛ −ezψe ⎞⎤ ezn∞ ⎡ ⎛ ezψe ⎞ ⎢exp⎜ ⎟ − exp⎜ ⎟⎥ εoε3 ⎢⎣ ⎝ kBT ⎠ ⎝ kBT ⎠⎥⎦

(6a)

⎛ ezψ ⎞ 2ezn∞ sinh⎜ e ⎟ εoε3 ⎝ kBT ⎠

(6b)

or

with i = 2 for CO2 to calculate A22 for CO2 on the basis of its relative permittivity ε2 and refractive index n2, the Planck constant h, and the primary electronic absorption frequency in the ultraviolet region νe (ca. 3 × 1015 s−1). The same procedure was followed with i = 3 to obtain the Hamaker constant for H2O. To use eq 5 to estimate A22 for CO2, suitable values of εCO2 and nCO2 must be selected, and these depend on pressure and temperature. Using approximate temperature and pressure profiles described previously, literature values for CO 2 densities,25 εCO2,45 and nCO246 were obtained for 0.67, 1.00, 1.34, and 2.00 km depths (Table 1). It should be noted that the value obtained for 0.67 km depth corresponds to CO2 as a dense gas, whereas the other depths are associated with scCO2 (Figure 1). By applying these values of εCO2 and nCO2 to eq 5, we calculated highly depth-dependent values of the Hamaker constant for CO2 presented in Table 1. A similar procedure was applied to calculate the much less depth-sensitive Hamaker constants for H2O, A33, using sources listed in Table 1. Silica and smectite were selected as representative substrates for calculations because of their very common occurrence and because their A11 values cover most of the range of van der Waals interactions for films on common minerals.39 Using an intermediate value of 6 × 10−20 J from two sources42,47 for the A11 of SiO2, we calculated A132 for water films confined between SiO2 and fluid phases consisting of air, CO2(g), and scCO2.

d2ψe dx

2

=

where e is the electron charge, z is the ion valence, n∞ is the ion concentration (number density) in the bulk solution, εo is the vacuum permittivity, ε3 is the dielectric constant of water, kB is the Boltzmann constant, and T is the Kelvin temperature. The electric double layer decay length or Debye length κ−1 is given by κ −1 =

εoε3kBT 2e 2z 2n∞

(7)

and typically ranges from ∼1 nm up to a few tens of nanometers in pore water, varying with the inverse square root of ionic strength. As noted earlier, T and ε3 values are depthdependent. Values of κ−1 for selected monovalent ion concentrations and depths are listed in Table 1, showing only minor decreases in κ−1 with depth. In terms of κ−1 and the dimensionless electrostatic potential Ye = zeψe/kBT relating the importance of ψe to kBT/e, the 1D Poisson−Boltzmann equation can be expressed as d2 Ye = κ 2 sinh Ye dx 2 8004

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potentials are large enough to allow pore drainage39 and will not be considered here.

Although the nonlinear Poisson−Boltzmann equation can be solved numerically for given sets of parameters, some analytical approximations reveal system behavior and have been shown to be reasonably accurate for many purposes. The double-layer compression approximation (CA) developed by Gregory51 is one convenient model that performs well for surfaces having low- to intermediate-magnitude electrostatic potentials (typical of many geologic mineral surfaces) over a wide range of surface separations and is suitable for use over moderately wide ranges of ionic strength. The electrostatic contribution to the disjoining pressure, Πel, in a symmetric electrolyte solution between parallel charged plates is related to f in the compression approximation through

5. RESULTS AND DISCUSSION 5.1. Dispersion Influences. In the limit of negligible interfacial surface charge at either mineral−water and water− CO2 interfaces, dispersion interactions are still predicted to provide stabilizing influences for water films because A11(mineral) > A33(water) > A22(CO2), leading to negative values of A132. As noted previously, the Hamaker constant for dispersion interactions A132 depends not only on the specific mineral substrate but also on the depth, primarily because of the large increases in CO2 density and A22(CO2) with increased P. The disjoining pressure-dependent thicknesses of water films on SiO2 and smectite, with air versus CO2 as the confining fluid, were calculated with eq 3b using the values of A132 in Table 1. The predicted ΠvdW-dependent water film thicknesses at the earth's surface (0 km depth, air) and at 1 and 2 km depths (scCO2) are shown in Figure 3. The higher A132 values

⎧ 1/2 ⎪ ⎡ ⎛ κf ⎞ ⎤ Π(f ) = n∞kBT ⎨2⎢1 + 0.25(Ye,1 + Ye,2)2 csc h2⎜ ⎟⎥ ⎝ 2 ⎠⎦ el ⎪ ⎣ ⎩ −

(Ye,1 − Ye,2)2 exp( −κf ) 1 + 0.25(Ye,1 + Ye,2)2 csc h2

κf 2

()

⎫ ⎪ − 2⎬ ⎪ ⎭

(9)

where Ye,1 and Ye,2 are dimensionless electrical potentials at opposing interfaces 1 and 2. This approximation is most satisfactory when |Ye,i| ≤ 2 . By representing the dimensionless electrostatic potential at the solid−water interface as Ye,1 and its value at the scCO2−water interface as Ye,2, Gregory’s CA model could be used to estimate the thicknesses of water films on solid surfaces confined by a scCO2−water interface. In the absence of measurements or estimates of electrostatic potentials at water− CO2 interfaces, most of the calculations to be presented were made with Ye,2 = 0. Adsorbed films bounded by uncharged fluid−fluid interfaces can also be treated as the half space between opposing surfaces of equal Ye,1, as was first done by Langmuir,52 albeit with additional constraints of high electrostatic potential at the solid−water interface and very low ionic strength. A subset of calculations were made with small values of Ye,2 to illustrate the impact of finite fluid−fluid interfacial potentials that have been measured in other fluid−fluid interfaces53,54 but not yet for H2O−CO2. An alternative relation between the disjoining potential and film thickness is provided by the linear superposition approximation (LSA, or the weak overlap approximation) for pairs of planar double layers.50 This approximation is suitable for cases where film thicknesses are large relative to the characteristic double-layer thickness (κf > 1) such that electrostatic potentials from each interface are approximately additive because interactions between opposing double layers are sufficiently small. The LSA-model-predicted pressure in a symmetric electrolyte film solution in equilibrium with its bulkphase concentration of n∞ and confined between parallel charged interfaces is given by ⎛ Ye,1 ⎞ ⎛ Ye,2 ⎞ Π(f ) = 64n∞kBT tanh⎜ ⎟ tanh⎜ ⎟ exp( −κf ) ⎝ 4 ⎠ ⎝ 4 ⎠ el

Figure 3. Calculated dispersion (van der Waals) contributions to adsorbed water film thicknesses on SiO2 and smectite under CO2 confinement. Approximate pore sizes associated with the capillary entry of CO2 are presented along the upper axis for 0 and 1 km depths.

for smectite relative to SiO2 result in predicted thicker f(ΠvdW) values for smectite. Note that increasing values of A22(CO2) with depth lead to thinner f(ΠvdW) values. The upper range of film thickness is generally not accessible in porous media because the capillary entry of CO2 is required to allow the coexistence of the three phases. This condition imposes poresize-dependent threshold ΠvdW values with magnitudes equal to ψc (eq 1). Characteristic pore sizes associated with ΠvdW values are shown on two scales along the upper x axis of Figure 3 for surface (0 km) and 1 km depths. Because the change in γ for the water−CO2 interface is relatively small in going from conditions at 1 to 2 km depth, the scale for the drainage pore diameter at 2 km is close to that for 1 km and is not shown. The offset in these upper axes reflect the lower-magnitude ψc associated with CO2 entry in deep reservoirs relative to air entry in surface soils and shallow aquifers of equivalent pore size. For reservoir pores with sizes in the 10 to 100 μm range, adsorbed films as thick as about 20 nm are possible. After the CO2 phase enters pores of lower-permeability strata including caprocks, the thicknesses of water films remaining on pore surfaces are predicted to be only a few nanometers at the most.

(10)

for dimensionless interfacial potentials Ye,1 and Ye,2. The LSApredicted pressures are also between the constant-surfacepotential- and constant-surface-charge-based predictions but are closer to the latter.51 The interfacial tension contribution to film pressure can be included by simply adding the grain scale capillary curvature term 4σ/λ to the left sides of eqs 9 and 10. However, this influence is relatively small when the capillary 8005


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5.2. Electrostatic Influences. The electrostatic influences on adsorbed water film thickness were examined with respect to influences of the depth below the earth’s surface, the aqueous solution ionic strength, and the electrostatic potential at the water−solid interface. For the case of the depth dependence at constant ionic strength and the mineral surface electrostatic potential, the main variations result from increased temperature and the decreased dielectric constant of water with depth (Table 1). Using the tabulated values of parameters for 0, 1, and 2 km depths with an ionic strength of 100 mol m−3 (0.1 M), ψe,1 = −25 mV, and ψe,2 = 0 mV, the CA-model-predicted film thickness dependences on Πel were calculated for several depths, as shown in Figure 4. Film thicknesses of less than 1 nm

Figure 5. Dependence of film thickness on the electrostatic component of the disjoining pressure at different levels of ionic strength in the bulk solution for conditions associated with a 1 km depth, ye,1 = −25 mV, and ye,2 = 0 mV.

aqueous films becoming more compressed as the ionic strength increases. The electrostatic potential at the solid−water interface, ψe,1, exerts a strong influence on film stability, with larger-magnitude values (positive or negative) supporting thicker adsorbed water films. Film thicknesses predicted by the CA when ψe,1 was varied from −30 to −3 mV are shown in Figure 6 for a

Figure 4. Dependence of the film thickness on the electrostatic component of the disjoining pressure for an ionic strength of 100 mol m−3, ye,1 = −25 mV, and ye,2 = 0 mV, calculated at several depths.

are below the resolution of the continuum approaches. Given the very similar results obtained with the CA and LSA (≤5% differences), predictions obtained with the LSA will not be shown. Note that increases in depth impart only very small decreases in film thickness, despite large changes in T and ε3. This limited depth variation results from opposing directions of change in these parameters that contribute to the double-layer thickness parameter κ−1 as a product (eq 7). Recall that κ−1 decreases very little with depth (Table 1). Thus, uncertainties in the geothermal gradient and depth-dependent ε3 are not expected to affect the electrostatic double layer pressure calculations significantly. In view of the predicted relatively minor depth dependence of the dispersion and electrostatic components of film thickness, the remainder of the calculations to be considered utilize P−T conditions at 1 km depth (Table 1). Keeping ψe,1 = −25 mV for the solid−water interface, the CA-based predicted influences of increased ionic strength from 10 to 500 mol m−3 are shown in Figure 5. As the ionic strength is increased, the adsorbed water film becomes thinner because the electric double layer charge balance is achieved within a shorter distance in more concentrated solutions of counterions. It should be noted that continuum models represent the electric double layer less accurately at ionic strengths greater than about 100 mol m−3. However, the 500 mol m−3 case is included in Figure 5 only to indicate the general qualitative trend of

Figure 6. Dependence of film thickness on the electrostatic component of the disjoining pressure at different solid−water surface electrostatic potentials for conditions associated with 1 km depth and an ionic strength of 100 mol m−3.

background ionic strength of 100 mol m−3. These results illustrate the importance of the surface potential, with highermagnitude ψe,1 stabilizing thicker films because of the higher concentration of counterions required within the film for charge balance. The relation between ψe,1 and the surface charge density, σ*, is given by the Grahame equation,55 which for 1:1 electrolyte solutions is ⎛ Ye,1 ⎞ σ * 8εoεrn∞kBT sinh⎜ ⎟ ⎝ 2 ⎠ 8006

(11)


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static contributions to film thickness were calculated using both CA (eq 9) and LSA (eq 10) methods and were found to be within 5% of each other. Therefore, only the CA-based DLVO calculations are presented in Figure 8. The upper x axis

On the basis of this relation, the ψe,1 values of −30, −10, and −3 mV cases presented in Figure 6 correspond to surface charge densities of 21, 7, and 2 mC m−2, respectively, (at an ionic strength of 100 mol m−3). These ranges of ψe,1 and σ* are representative of silica surfaces,56,57 with lower-magnitude values reached under acidic conditions (pH ∼3) around the silica point of zero charge (pzc).58 As the pH decreases toward the pzc, less charge balancing by counterions is required and the electrostatic stabilization of adsorbed aqueous films becomes weak. Because solutions equilibrated with scCO2 are at pH ∼3, silica, quartz, and other minerals with similarly low pzc's are predicted to have weakened water film stability under geologic CO2 sequestration. Although the electrostatic potential at the water−scCO2 interface, ψe,2, is currently unknown and is assumed to be negligible, the influences of small-magnitude values given the measured pH-dependent values of air−water54 and oil−water53 interfaces are worth considering. In both of these latter systems, ψe,2 is nearly zero at pH ca. 3 and increasingly negative at higher pH. With pH ∼3 expected for a large quantity of reservoir water in equilibrium with scCO2, small magnitudes of ψe,2 are reasonable to examine. In Figure 7, cases of ψe,2 = −3 and +3

Figure 8. Comparison of electrostatic-only and DLVO predictions of aqueous film thicknesses on SiO 2 and smectite under CO 2 confinement at 1 km depth.

indicates the nominal pore size associated with drainage at specific ψc = Π. Comparisons between electrostatic-only and DLVO (electrostatic plus dispersion) model calculations show that the dispersion contribution is predicted to be very important to film stability in the systems considered here. More broadly, the relative importance of electrostatic and dispersion interactions involves complex interactions of multiple variables (e.g., parameters in Table 1). Nevertheless, the dispersion component of the disjoining pressure is generally expected to be dominant in deep reservoir formations where the solution ionic strength is commonly high and surface potentials are low. Assuming that reservoir pore throat diameters are on the order of 10 μm, water films are expected to be less than about 10 nm in thickness upon initial CO2 entry. At higher capillary (disjoining) pressures, water film thicknesses are predicted to diminish to a few nanometers. Films can become even thinner when the confining scCO2 phase is undersaturated with respect to water, although at least monolayer hydration is likely to be maintained.13

Figure 7. Dependence of the electrostatic component of the film thickness on the disjoining pressure at different water−scCO2 interface electrostatic potentials (ye,2) for conditions associated with a 1 km depth, an ionic strength of 100 mol m−3, and an electrostatic potential of −10 mV at the solid−water interface (ye,1).

6. CONCLUSIONS This DLVO-based analysis provides estimates of water film thicknesses on SiO2 and smectite in equilibrium with CO2 as the confining fluid phase over ranges of conditions associated with geologic CO2 sequestration. The capillary entry criterion was included to constrain the range of disjoining pressures relevant to water films remaining in pores invaded by the CO2 phase. Using this constraint and suitable depth-dependent system properties, the DLVO models predict adsorbed film thicknesses that are typically less than about 10 nm. On the basis of Hamaker constants calculated in this study, van der Waals interactions provide the dominant film-stabilizing influence, especially for SiO2. In very dilute aqueous films on highly charges surfaces, the electrostatic contribution may dominate van der Waals interactions in stabilizing adsorbed films, although such conditions are uncommon in deep reservoirs. The residual water films provide hydrated environments in which reactions at reservoir mineral surfaces take

mV are compared with the previously shown neutral water− scCO2 interface at a 1 km depth for solid−water ψe,1 = −10 mV and I = 100 mol m−3. Negative values of ψe,2 interacting with negative ψe,1 stabilize thicker water films because of the requisite greater number of counterions needed for charge balance. In contrast, ψe,2 having a sign opposite that of ψe,1 promotes the collapse of the water film. 5.3. Combining van der Waals and Electrostatic Influences on Adsorbed Water Films. The influences of van der Waals dispersion and electrostatic forces are combined through eq 2 to obtain the overall DLVO dependence of adsorbed water film thicknesses on the disjoining pressure. The example calculations to be presented here are for hypothetical solid surfaces with ψe,1 = −5 and −50 mV for 1-km-depth P−T conditions, 100 mol m−3 ionic strength, and ψe,2 = 0 mV. These two ψe,1 values are assumed to approximate pH 3 conditions for SiO2 and smectite, respectively. As noted earlier, the electro8007


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place, probably with high sensitivity to film thickness. Improvements in the present results will become possible as more measurements of water, CO2, and surface properties become available, especially under their mutual equilibria over a range of P−T values and salinities. For films of only a few nanometers' thickness resulting from combinations of low surface electrostatic potential, high ionic strength, and/or high capillary pressure, the DLVO approach becomes very approximate. Such conditions characteristic of higher-salinity, lower-permeability brine reservoirs are being investigated with a variety of spectroscopic59,60 and computational methods.59,61 An improved understanding of CO2−H2O−mineral interactions being developed through these other approaches will likely yield information useful for refining DLVO analyses of films in CO2 reservoirs and caprocks. Thus, improvements in the DLVO description of water films under these extreme conditions may soon be achieved and may help to link our molecular-scale understanding with much larger pore-scale processes occurring in geologic CO2 sequestration.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS I thank the anonymous reviewers, especially reviewer 2 and Senior Editor Shaoyi Jiang, for their helpful comments. This material is based upon work supported as part of the Center for Nanoscale Control of Geologic CO2, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under award number DE-AC02-05CH11231.

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