Capillary effects on hydrate stability in marine sediments

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2010jb008143, 2011 Capillary effects on hydrate stability in marine sediments Xiaoli Liu 1 and Peter B. Flemings 2 Received 3 December 2010; revised 4 April 2011; accepted 15 April 2011; published 12 July [1] We study the three phase (Liquid + Gas + Hydrate) stability of the methane hydrate system in marine sediments by considering the capillary effects on both hydrate and free gas phases. The capillary pressure, a measure of the pressure difference across a curved phase interface, exerts a key control on the methane solubility in Liquid + Hydrate (L + H) and Liquid + Gas (L + G) systems. By calculating the L + H and L + G solubilities as a function of water depth (pressure) and pore size (interface curvature), we show how the solubility requirements for forming both gas hydrate and free gas can be met in a three phase zone. The top of the three phase zone shifts upward in sediments as the water depth increases and the mean pore size decreases. The thickness of the three phase zone increases as the distribution of pore sizes widens. The top of the three phase zone can overlie or underlie the bulk three phase equilibrium depth. At Blake Ridge, we predict that the three phase zone is 27.7 m thick and that the top of the three phase zone lies 13 m above the predicted bulk equilibrium depth. This reconciles the observation of the bottom simulating reflector (BSR) at Blake Ridge that is shallower than the predicted bulk equilibrium depth. In contrast, at Hydrate Ridge where water depth is shallower, we predict that the three phase zone is 20.4 m thick and that the top of the three phase zone lies 0.7 m below the predicted bulk equilibrium depth. Our model, which predicts an upward shift in the top of free gas occurrence with increasing water depth (pressure), is compatible with worldwide observations that the BSR is systematically shifted upward relative to the bulk equilibrium depth as water depth (pressure) is increased. Citation: Liu, X., and P. B. Flemings (2011), Capillary effects on hydrate stability in marine sediments, J. Geophys. Res., 116,, doi: /2010jb Introduction [2] Gas hydrate is an ice like compound that contains methane and/or other low molecular weight gas in the water molecule lattice. Whether hydrate can exist as a stable phase depends on pressure (P), temperature (T), gas abundance, water activity, and the pore size of host materials [Sloan and Koh, 2008]. Gas hydrates are generally distributed in marine sediments along continental margins, where temperature is low, pressure is high, and adequate supplies of methane are available. Marine sediments that host methane hydrates range from coarse grained sands to fine grained shales and thus have a wide range of pore sizes [Clennell et al., 1999]. Gas hydrates in confined pores are subject to capillary effects, which can disturb the P T condition for hydrate stability. Stability conditions of natural gas hydrates must be understood so that the total amount of gas in gas hydrate fields can be estimated, the temperature at the base of the observed hydrate stability zone can be predicted, and the feasibility of extracting natural gas from a gas hydrate field can be determined. 1 ExxonMobil Upstream Research Company, Houston, Texas, USA. 2 Jackson School of Geosciences, University of Texas at Austin, Austin, Texas, USA. Copyright 2011 by the American Geophysical Union /11/2010JB [3] The capillary effect on hydrate stability has been measured in synthetic [Handa and Stupin, 1992; Uchida et al., 1999; Østergaard et al., 2002; Anderson et al., 2003, 2009] and natural samples [Uchida et al., 2004]. Various authors have presented theoretical models to quantify the effect of pore size on hydrate stability on the basis of statistical thermodynamics [Clennell et al., 1999; Henry et al., 1999; Smith et al., 2002; Sun and Duan, 2007]. Fluid flow models were also developed [Malinverno, 2010; Daigle and Dugan, 2011] to quantify capillary effects on hydrate distribution in layered porous media. These experimental and modeling studies suggested that capillary effects greatly inhibit hydrate growth in narrow pores by depressing water activity. Water activity is a measure of the energy state of water in the system and the depressed water activity increases the aqueous CH 4 solubility required to form hydrate in pores. Consequently, the capillary effect on hydrate stability is equivalent to that of an inhibitor (e.g., salt). [4] Almost all of these studies accounted for the capillary effects of the hydrate phase in pores, but they neglected the capillary effects of the gas phase. The excess energy required to create the curved gas water interface in pores demands a supersaturation of the gas component in solution [Clennell et al., 1999; Henry et al., 1999]. This effect is termed capillary supersaturation. Capillary supersaturation inhibits the formation of free gas phase in pores, which can counteract the capillary inhibition of hydrate in pores. 1of24

2 Table 1a. Physical Parameters in Simulations Variable Physical Meaning Units Reference s gw Gas water interfacial tension J m 2 Clennell et al. [1999], Henry et al. [1999] s hw Hydrate water interfacial tension J m 2 Clennell et al. [1999], Henry et al. [1999] gw Gas water contact angle 180 Clennell et al. [1999], Henry et al. [1999] hw Hydrate water contact angle 180 Clennell et al. [1999], Henry et al. [1999] R Gas constant J mol 1 K 1 Henry et al. [1999] V b Molar volume of water in hydrate lattice 22.6 cm 3 mol 1 Henry et al. [1999] [5] Only a few studies [Clennell et al., 1999; Henry et al., 1999] considered the capillary effects of both the gas hydrate and free gas phases. In contrast to previous studies, Henry et al. [1999] suggested that capillary effects within sediment with a constant pore size would promote rather than inhibit three phase stability; this model predicts that the top of the free gas zone is deeper than would be predicted if capillary effects are not considered. However, the opposite behavior is observed at Blake Ridge: the bottom simulating reflector (BSR), which marks the top occurrence of free gas, is shallower than the depth of bulk three phase stability [Ruppel, 1997]. The BSR is interpreted to record the top of free gas occurrence, not necessarily the base of gas hydrate occurrence [MacKay et al., 1994; Holbrook et al., 1996]. [6] We first describe the capillary effects on three phase stability in a single pore size. In this case, the change in three phase stability is determined by the relative increases of two equilibrium solubilities as a function of the chosen pore size and water depth. Then we develop an equilibrium model to predict hydrate and free gas stability conditions given a pore size distribution. The measured pore size distributions at Blake Ridge and Hydrate Ridge and synthetic curves are used as examples. We show that the occupancy of gas hydrate and free gas in pores of different sizes allows a three phase zone to develop in sediments. The thickness of the three phase zone depends on the water depth, the amount of gas available, the pore size distribution, and the phase preference to occupy large pores. [7] Unlike Henry et al. [1999], we calculate the phase fractions (S g and S h ) both within and outside the three phase zone and highlight the critical effect of water depth on capillary inhibition of hydrate stability. We predict that capillary inhibition of hydrate stability in sediments, as evidenced by the difference between the BSR depth and the theoretical base of the GHSZ (Gas Hydrate Stability Zone), is more significant in deep water than in shallow water. This prediction is consistent with observations of the BSR depths compiled around the world. 2. Equilibrium Calculation in a Given Pore Size 2.1. Thermodynamic Calculation [8] CH 4 can exist in marine sediments as hydrate, free gas or dissolved in water. When immiscible phases coexist in porous media, the capillary effect is present [Clennell et al., 1999]. We assume the mineral surface is hydrophilic: consequently, the pore water is the wetting phase, and the hydrate and gas are the nonwetting phases. As a result, the grain surface is always coated with a water film, whereas the hydrate or gas forms a convex surface to the liquid. The pressure difference across the curved interface is capillary pressure, which is a function of surface curvature, interfacial energy and wettability. Hydrate and gas, as nonwetting phases, have higher phase pressures than does ambient water at capillary equilibrium. [9] If a spherical pore with radius r is filled by hydrate or gas, the interface mean curvature () within this pore is 2/r. The surface energy per unit area over this interface is given by the interfacial energy (s). Wettability, the ability of a phase to maintain contact with the grain surface, is expressed as the contact angle subtended between the interface and the grain surface (). If a pore is saturated with either gas or hydrate, the gas water (P cgw ) or hydrate water (P chw ) capillary pressure in a pore of radius r is P cgw ¼ 2 gw cos gw ; ð1þ r P chw ¼ 2 hw cos hw ; ð2þ r where s gw and s hw are gas water and hydrate water interfacial energy, gw and hw are gas water and hydrate water contact angle. These equations indicate a mechanical balance in which the pressure difference across an interface is supported by the surface energy of this interface. In this model, we assume the hydrate and gas phases are completely nonwetting, i.e., gw = hw = 180. The gas water surface has a larger interfacial energy (s gw = J/m 2 ) than does the hydrate water surface (s hw = J/m 2 )[Clennell et al., 1999; Henry et al., 1999] (Table 1a). [10] Distribution of hydrate and free gas within the sediment column depends on two P T dependent equilibrium solubility curves [Liu and Flemings, 2006]: (1) the liquidhydrate (L + H) methane solubility curve, in which gas hydrate is at equilibrium with dissolved gas in water, and free gas is absent; and (2) the liquid gas (L + G) methane solubility curve in which free gas is at equilibrium with dissolved gas in water, and hydrate is absent. Gas pressure (P g = P w + P cgw ) is used in the model of Duan et al. [1992] to predict the L + G equilibrium. The model of Henry et al. [1999] that includes the capillary term is used to predict the L + H equilibrium. These calculations are slightly different from those presented by Liu and Flemings [2006], in which the L + G and L + H equilibria are predicted from water pressure alone. [11] Hydrates in pores have greater chemical potentials and higher solubilities than in bulk. The increase in L + H solubility due to the capillary effect is described by the Gibbs Thomson equation [Clennell et al., 1999; Henry et al., 1999]: DC LþH C LþH0 ¼ D RT ¼ 2 hw cos hw nv r RT ; ð3þ 2of24

3 Figure 1. The L(liquid) + G(gas) and L + H(hydrate) solubilities (black lines) in bulk condition and in fine pores (100, 50, 20 and 10 nm) at (a) Hydrate Ridge and (b) Blake Ridge. Both L + G and L + H solubilities in pores are increased relative to the bulk condition. The bulk three phase stability (horizontal dash line) is determined by the intersection of bulk L + G and L + H solubilities. As pore size decreases, the three phase stability in fine pores (black dots) shifts downward, increasing the thickness of hydrate stability. The gray triangles represent the three phase stability for hydrate in 10 nm pores and free gas in 50 nm pores, which lies above the bulk condition. Mbsl, meters below sea level; mbsf, meters below seafloor. where R is the gas constant, T is the temperature in Kelvin, Dm is the chemical potential increase for hydrate changing from the bulk to the confined condition, n is the hydrate stoichiometry factor, V b is the molar volume of water in the hydrate lattice, C L+H is the CH 4 solubility in equilibrium with hydrate, and subscript 0 denotes the bulk condition. Equation (3) indicates that the capillary force due to the curved hydrate liquid interface increases the supersaturation (DC L+H ) with decreased pore size. [12] Supersaturation is also necessary to form gas bubbles in confined pores. According to Henry s law, gas solubility in liquid is directly proportional to gas pressure (P g = P w + P cgw ). Thus, the increase in L + G solubility due to capillary force is approximated as DC LþG C LþG0 P cgw P w ¼ 2 gw cos gw rp w ; ð4þ where C L+G is CH 4 solubility in equilibrium with gas. Equation (4) is only an approximation; methane gas can deviate from this ideal behavior as pressure increases. At shallow depth, gas water capillary pressure (P cgw ) represents a large fraction of ambient water pressure (P w ). For example, at 1 km below sea level, P cgw in 50 nm pores (2.88 MPa) represents 28.8% of the ambient pressure (P w ); thus the L + G solubility in 50 nm pores increases approximately by 28.8% relative to the bulk condition (Figure 1a) Equilibrium Solubilities in a Given Pore Size [13] We calculate L + H and L + G solubilities in several pore sizes, using the P T salinity conditions representative of Hydrate Ridge and Blake Ridge (Table 1b). Hydrate Ridge has a shallower water depth and a steeper thermal gradient ( 790 m and 59 C/km, respectively) [Tréhu, 2006] than Blake Ridge ( 2780 m and 36.9 C/km, respectively) [Henry et al., 1999]. Increasing temperature 3of24

4 Table 1b. Blake Ridge and Hydrate Ridge P T Salinity Conditions Blake Ridge, ODP Site 997 Hydrate Ridge, ODP Site 1250 Surface temperature ( C) Geothermal gradient ( C/km) Water depth (m) Hydrostatic gradient (MPa/km) Pore water salinity (wt.%) Reference Ruppel [1997], Henry et al. [1999] Tréhu [2006] and pressure with depth have opposite effects on the equilibrium CH 4 solubilities [Henry et al., 1999]: (1) L + H solubility increases greatly with increasing T and decreases slightly with increasing P; (2) L + G solubility decreases with increasing T and increases with increasing P. The net effect is that the L + H solubility increases with depth, while the L + G solubility decreases with depth. Their intersection marks the triple point or the depth of three phase stability, where liquid, gas, and hydrate coexist (Figure 1). The depth of bulk three phase stability is predicted at 119 mbsf (meters below seafloor) at Hydrate Ridge (Figure 1a) and 481 mbsf at Blake Ridge (Figure 1b). [14] Decreasing the pore size shifts the depth of threephase equilibrium downward (Figure 1). This effect is much more pronounced at Hydrate Ridge than at Blake Ridge: for example, for a 10 nm pore at Hydrate Ridge, the depth of the triple point is m beneath the bulk equilibrium depth (Figure 1a), whereas at Blake Ridge it is 34.5 m below the bulk equilibrium depth (Figure 1b). The conclusion that the three phase equilibrium depth is deeper in small pores than in bulk condition is in contrast to previous studies [Handa and Stupin, 1992; Uchida et al., 1999; Østergaard et al., 2002; Anderson et al., 2003;Uchida et al., 2004;Anderson et al., 2009]. The reason is that these authors only considered the capillary effects on hydrate and not the capillary effects on gas. We describe the mechanism for this behavior below. [15] For a given pore size, if the increase of CH 4 solubility for the liquid gas system (DC L+G ) is larger than the increase of CH 4 solubility for the liquid hydrate system (DC L+H ), the depth of the three phase equilibrium shifts downward. Figure 2. The ratio of DC LþG DC LþH at (a) Hydrate Ridge and (b) Blake Ridge. DC L+G and DC L+H are the increases of the L + G and L + H solubilities in confined pores relative to bulk condition. The depths of bulk three phase equilibrium (dash lines) are marked. DC L+G is greater than DC L+H, suggesting that capillary effects increase the depth of three phase stability condition in a given pore size. 4of24

5 Figure 3. Three phase stability conditions in bulk condition (black solid line) and in fine pores (100, 50 and 20 nm) by considering capillary effects on hydrate only (Figure 3a), gas only (Figure 3b), and both hydrate and gas (Figure 3c) phases. The P T profiles at Hydrate Ridge and Blake Ridge (dash lines) are plotted for reference. (a) Case A represents the maximum capillary inhibition of hydrate stability. The three phase stability conditions in fine pores are displaced to higher P or lower T, relative to bulk condition. (b) Case B represents the maximum capillary promotion of hydrate stability. (c) Case C is a mixture of the above two endmembers. The three phase stability conditions in fine pores are shifted to lower P or higher T. DC L+H (equation (3)) and DC L+G (equation (4)) are calculated as a function of pore size and depth, and their ratio is greatest at the seafloor and decreases with depth (Figure 2). The calculated DC L+G is always greater than DC L+H ( DC LþG DC LþH > 1). Thus, for a given pore size, the depth of three phase equilibrium (i.e., the intersection of L+G and L+H solubility curves) always shifts downward relative to the bulk condition. [16] The values for DC LþG DC LþH at the bulk equilibrium depth for Hydrate Ridge lie between 3.23 and 4.2 (Figure 2a), whereas at Blake Ridge they lie between 1.38 and 1.44 (Figure 2b). This drives the larger downward shift of the three phase stability condition at Hydrate Ridge than at Blake Ridge, for a given pore size (Figure 1). [17] The capillary effects on hydrate stability in the twophase (L + H) and three phase (L + G + H) fields are different. In the two phase field, the concentration of CH 4 dissolved in liquid must exceed the L + H solubility to precipitate hydrate. The capillary force in confined pores results in capillary supersaturation (equation (3)), increasing the CH 4 solubility required to form hydrate. The net result is to inhibit hydrate formation in the two phase field, where hydrate is formed from dissolved CH 4. By contrast, where three phases coexist, capillary effects result in a deeper depth of three phase equilibrium for a given pore size (Figures 1 and 2) Three Phase Stability in a Given Pore Size [18] To understand how capillary pressure impacts the distribution of hydrate and gas in the subsurface, we compute the three phase equilibrium curve by considering: (1) capillary effects on hydrate only; (2) capillary effects on gas only; (3) capillary effects on both hydrate and free gas (Figure 3). [19] In Figure 3a, we consider the capillary effects only on the hydrate (equation (3)); thus hydrate is present in pore, while free gas is present in bulk. This scenario is commonly assumed in hydrate equilibria measurements [Østergaard et al., 2002; Anderson et al., 2003, 2009]. In this case, three phase stability is determined by comparing the L + H solubility in pores with the L + G solubility in bulk condition (Figure 1). As pore size decreases, the three phase stability curve progressively shifts to higher pressure and lower temperature (Figure 3a), decreasing the P T area of hydrate stability. The decrease in the P T area of hydrate stability corresponds to a decrease in the depth of threephase stability. The shift in three phase stability is more significant for high P T values, where the increase in L + H solubility due to the capillary pressure is greater. [20] In Figure 3b, we consider capillary effects only on the gas phase (equation (4)): hydrate is present in bulk, while free gas is present in pores. The three phase stability is determined by comparing the L + G solubility in pores with the L + H solubility in bulk condition (Figure 1). As pore size decreases, the three phase equilibrium curve progressively shifts to lower P and higher T (Figure 3b), increasing the P T region of hydrate stability. This effect deepens the depth of three phase stability. There is a greater increase in L + G solubility at low P T, since P cgw represents a larger fraction of P w at shallow depth (equation (4)). Thus the downward shift in three phase stability is larger for low P T values. [21] In Figure 3c, we consider capillary effects on both hydrate and gas: thus hydrate and gas are present in separate pores (i.e., the pores contain either hydrate or gas). The shift in three phase stability is more complicated in this case, 5of24

6 Figure 4. Conceptual model of hydrate formation within porous media at two phase stability. Porous media is considered as a matrix of various spherical pores (r 1 > r 2 > r 3 ), connected by pore throats. (a) Initially the pore network is filled with water (blank area). (b) As CH 4 is added to the system, hydrate (green) appears first in the largest pores r 1. A thin water film separates pore hydrate from the grain surface. (c) With increased gas supply, hydrate progressively grows into the smaller pores r 2. At this point, CH 4 concentration in solution is determined by the relatively smaller pore size r 2. (d) Eventually, all of the pores are filled with hydrate. Hydrates in r 1 and r 2 are in contact with a supersaturated solution that is determined by the smallest pore r 3. Gas bubble formation in porous media is envisioned similar to the above process. because hydrate and free gas have opposite effects on threephase stability. DC LþG DC LþH is greater than 1 (Figure 2), suggesting that a greater CH 4 supersaturation is required to form gas than hydrate for a given pore size. As a result, the three phase stability curve shifts to lower P and higher T (Figure 3c), increasing the P T region of hydrate stability. The magnitude of the shift is smaller than that in Figure 3b, because capillary inhibition on hydrate phase partially counteracts the capillary effect on the gas phase in pores. 3. Conceptual Model of Hydrate Formation in a Discrete Pore Size Distribution [22] We now examine hydrate formation in sediments with a discrete pore size distribution at two phase (L + H) 6of24 stability (Figure 4). Porous sediment is envisioned as a matrix of spherical pores of various sizes, connected by narrow pore throats. As an example, three distinct pore radii (r 1 > r 2 > r 3 ) are randomly distributed in the pore network. There is no hydrate initially and the network is liquid filled (Figure 4a). CH 4 is gradually added to the system. The smallest supersaturation of methane is required to form hydrate in the largest pore. Once the dissolved CH 4 concentration exceeds this value, hydrate first appears in the largest pores (r 1 ) and fills them (Figure 4b). If hydrates form in smaller pores, the larger supersaturation required for sustaining hydrates in smaller pores would lead to their dissolution and coalescence in larger pores.

7 Figure 5. Coexistence of hydrate (H), gas (G) and liquid water (L) in porous media. (a) Hydrate and gas are present in pores of different sizes to achieve three phase equilibrium. At three phase equilibrium, the L + G solubility in pore r g is equal to the L + H solubility in pore r h. (b) Coexistence of H + G+Lin sediment is envisioned analogous to a meteorological problem [Defay et al., 1966], where snow crystal and raindroplet are immersed in humid vapor. [23] The hydrate filling pores are not necessarily next to each other (Figure 4b), because: (1) hydrate is precipitated from methane dissolved in liquid and (2) liquid, as a wetting phase, is well connected. There is a thin water film present between the hydrate and the confining wall (Figure 4b). This solution film is a diffusion path to transfer dissolved methane between the hydrate and the surrounding solution. Thus, a uniform CH 4 concentration is achieved in solution and hydrates in r 1 are in equilibrium with the surrounding solution. At this point, hydrate impinges on the pore wall to arrest its further growth; hydrate has a higher phase pressure than the ambient liquid. [24] Further hydrate growth requires progressively greater supersaturation from an increased CH 4 supply (Figure 4c). As more CH 4 enters the system, hydrate will form in the next largest pore r 2. At this point, the concentration of CH 4 in solution is determined by pore r 2. The solution is in equilibrium with hydrate in r 2 but supersaturated with respect to hydrate in r 1. Thus, CH 4 may diffuse to the surface of hydrate in r 1 along the water film, and hydrate can grow against the confining wall. Hydrate progressively grows into pore r 3 as more CH 4 is supplied to the system (Figure 4d). At this point, the concentration of CH 4 in solution is determined by the smallest pore r 3. The CH 4 solubility required to precipitate hydrate in the system increases greatly as hydrate grows into smaller pores. 4. Equilibrium Calculation in a Broad Pore Size Distribution 4.1. Model Assumption and Description [25] We present an equilibrium model to calculate the phase saturations in sediments where capillary effects are present. We assume the following: (1) Porous sediment can be described as a matrix of spherical pores connected by very narrow pore throats (Figure 4). (2) The pores are completely water wet; the solid surface is everywhere covered by a thin water film (Figure 5a), which prevents hydrate and gas from contacting the solid surface directly. The film is sufficiently thin for the curvature of the nonwetting phase interface to be approximated by the dimensions of the pore space. (3) There is no capillary hysteresis between hydrate growth and dissociation. (4) Because we are studying hydrate formation over geological timescales, we assume this is an equilibrium process; there are no kinetics or mass transfer limitations. (5) Gas and hydrate, if any, are present in separate pores to minimize surface energy (Figure 5a). They are in contact only through the liquid phase. This allows the approximation of the interface curvature by pore radius in equations (1) and (2). (6) CH 4 is the only hydrateforming gas. (7) Pore pressure is hydrostatic. (8) The lithology is uniform, meaning the same pore size distribution is present at all depths. (9) The same M CH4 (total CH 4 concentration) is available at all depths without any need for fluid transport and CH 4 transport is not considered. [26] M CH4 (total CH 4 concentration) is an input to the model and we estimate it from pressurized core sampler (PCS) measurements. We report the total CH 4 concentration in g CH 4 /dm 3 pore vol. (dm is decimeter) as Dickens et al. [1997] and Henry et al. [1999] did. A total CH 4 concentration of 40 g CH 4 /dm 3 pore vol. is assumed in this paper, which is similar to that measured with PCS at Blake Ridge during ODP (Ocean Drilling Program) Leg 164 [Dickens et al., 1997]. This total concentration is enough to fill pores with hydrate or free gas. To differentiate M CH4 (total CH 4 concentration) from dissolved CH 4 concentration, we also term M CH4 as CH 4 quantity in the following discussion. [27] This model includes the following features. (1) The model accounts for the capillary effects on both gas hydrate and free gas in pores. Capillary force can result in some pores being gas saturated while other pores are hydrate saturated or water saturated. (2) The partitioning of components among the phases is calculated based on the assumption of thermodynamic and capillary equilibrium. At three phase equilibrium, the L + H solubility calculated from the smallest hydrate filling pore is equal to the L + G solubility calculated from the smallest gas filling pore. [28] The assumption that hydrate and free gas are present in separate pores (Figure 5a) is supported by theory [Defay et al., 1966; Clennell et al., 1999] and experiment [Tohidi et al., 2001]. If three phases coexist in the same pore, water forms a continuous film on the pore wall and both hydrate and gas are convex to water. In this system, there are three interfaces: gas water, hydrate water, and hydrate gas. Capillary force balance across these interfaces requires s hg = s gw + s hw [Defay et al., 1966; Clennell et al., 1999], where s hg is the interfacial energy between hydrate and gas. As a result, gas hydrate and free gas would occupy different pores to avoid the highest energy interface between them. The micromodel experiment of hydrate formation [Tohidi 7of24

8 et al., 2001] also illustrated that in systems with free gas, the hydrates form in different pores than those with gas present. The balancing of capillary forces (i.e., pore size and wettability) controls the distribution of gas, water and hydrate within pores. [29] The coexistence of hydrate and free gas in porous sediments is envisioned as analogous to the coexistence of snow crystal and liquid drop immersed in humid vapor (Figure 5b) [Defay et al., 1966]. Both solid and liquid are convex to the vapor. The triple point where three phases coexist depends on the relative sizes of solid crystal and liquid drop. Similarly, the shift of three phase (L + G + H) stability in porous sediments relative to the bulk condition depends on the pore sizes that hydrate and gas phases occupy. [30] We showed that with a single pore size, the interface where three phase stability is present is shifted downward relative to the bulk condition (Figure 1). However, with multiple pore sizes, the depth of the top occurrence of free gas can shift upward. For example, the gray triangle in Figure 1 represents the CH 4 solubility required to precipitate gas hydrate in 10 nm pores or nucleate free gas in 50 nm pores. It corresponds to the three phase stability condition for gas in 50 nm pores and hydrate in 10 nm pores, which is located above the bulk condition. This suggests that the presence of gas in larger pores than hydrate may shift the top of free gas occurrence above the bulk condition. [31] The mass conservation of methane is described as S w w C CH4 þ S h h f CH4 þ S g g ¼ M CH4 ; S w þ S h þ S g ¼ 1; with supplemental constraints r g ¼ 2 gw cos gw =P cgw S g ; S h ; ð7þ r h ¼ 2 hw cos hw =P chw S g ; S h ; ð8þ where C CH4 is the CH 4 concentration in solution, f CH4 is the mass fraction of CH 4 in hydrate, r w, r h and r g are the water, hydrate and gas densities, S w, S h and S g are the water, hydrate and gas pore saturations, M CH4 is the total pore space CH 4 concentration, and r h and r g are the smallest hydrate and gas filling pores. When three phases coexist, there are two nonwetting phases (hydrate and gas). The capillary pressure curves P cgw and P chw (equations (7) and (8)) are described as a function of both the phase saturations (S g and S h ) and the distribution of hydrate and gas in pores. Two equilibrium solubilities C L+G and C L+H are computed as a function of pressure (P), temperature (T), salinity (X), and pore size (r). The smallest hydrate and gas filling pores (r h and r g ) determine the L + H and L + G solubilities in a broad distribution of pore size (equations (3) and (4)). r w is calculated as a function of P w, T and X, while r g as a function of P g and T. Enough CH 4 is present to form hydrate or gas. If C L+G < C L+H, only liquid and gas are present, S h =0, and C CH4 = C L+G. In this case, equations (5) and (6) collapse to S w w C LþG þ S g g ¼ M CH4 ; S w þ S g ¼ 1: ð5þ ð6þ ð9þ ð10þ At this point, we have two unknowns (S w and S g ) and two equations. Similarly, if C L+G > C L+H, only liquid and hydrate are present, S g =0,andC CH4 = C L+H. Where three phases coexist, we have the additional constraint that C LþG r g ¼ CLþH ðr h Þ: ð11þ Now there are three unknowns (S g, S h and S w ) and three equations (equations. 5, 6 and 11).The equations are solved in the following manner when three phases coexist. We reduce the unknown S w by substituting equation (6) (S w =1 S g S h ) into equation (5). First, an initial guess of hydrate saturation (S h ) is made in equation (5). Then, a gas saturation (S g )is calculated from equation (5). r g and r h are then calculated from equations (7) and (8). C L+H and C L+G are then calculated through equations (3) and (4). If these two values are not equal (equation (11)), then a new value for hydrate saturation is assumed in equation (5) and the procedure is repeated until equation (11) is satisfied. The solution is dependent on the capillary behavior. Specifically the solution depends on how gas and hydrate fill different pore sizes and how r g and r h are determined from P cgw and P chw (equations (7) and (8)). We discuss this below Calculation With Simple Pore Size Distributions Three Phase Equilibrium With Discrete Versus Continuous Pore Size Distributions [32] We first perform the above calculation with two simple (discrete versus continuous) pore size distributions (PSD). The P T salinity conditions at Hydrate Ridge are used in both cases (Table 1b): 790 m water depth, 10 MPa/km hydrostatic gradient, 4 C surface temperature, 59 C/km geothermal gradient, and 3.5 wt.% salinity. The discrete versus continuous PSD models have significantly different effects on the three phase stability (Figure 6). [33] In the discrete PSD case, we assume a pore size distribution and then convert it to a capillary pressure curve. The discrete model is composed of three distinct pore sizes (100, 50 and 20 nm) (Figure 6c). Each pore size represents a number of pores and these three pore sizes represent 30%, 40% and 30% of the total pore volume, respectively. This pore size distribution is converted to the gas water and hydrate water capillary pressures (equations (1) and (2)), using the corresponding interfacial energy and wetting angle (Table 1a). The capillary pressure curve has three steps (Figure 6b), where a range of nonwetting phase saturations corresponds to the same capillary pressure or pore size. In the discrete PSD case, three phase stability may exist at the depths that represent the intersections of the L + H and L + G solubilities in all possible combinations of the three pore sizes (marked by the arrows in Figure 6a), depending upon the quantity of CH 4 available. At a depth different than these intersections (horizontal dashed line, Figure 6a), there is no combination of pore sizes where the L + H and L + G solubilities are equal. [34] The three phase stability may appear at these discrete intersections (arrows in Figure 6a), only when certain CH 4 quantities are reached. As we show below (section 4.2.2), only gas and liquid coexist initially with increasing CH 4 quantity at the depth of three phase equilibrium in 50 nm pores (horizontal solid line, Figure 6a). Three phase equilibrium appears only when a certain CH 4 quantity is reached. Different 8of24

9 Figure 6. Effects of the discrete (Figures 6a 6c) versus continuous (Figures 6d 6f) pore size distributions (PSD) on three phase stability. (a) In the discrete PSD case, the three phase equilibrium can only exist at the intersections of the L + H and L + G solubilities in the combination of these three pore sizes (marked by the arrows). The horizontal solid line marks the depth of three phase equilibrium in 50 nm pores. (b) The capillary pressure curve in the three pore size case has three steps. (c) The three pore sizes represent 30%, 40% and 30% of total pore volume. (d) In the continuous PSD case, the three phase equilibrium can exist in the gray area bounded by the L + H and L + G solubilities in pore sizes 100 nm and 20 nm, depending on the quantity of CH 4 available. Illustrated are 5 combinations of pore sizes that can achieve three phase stability at the given depth (horizontal solid line). The markers from 8A to 8F correspond to Figure 8. The horizontal solid line marks the depth of three phase equilibrium in 50 nm pores. (e) The capillary pressure linearly increases with the nonwetting phase saturation (1 S w ). (f) Pore sizes are continuously distributed between 100 nm and 20 nm. CH 4 quantities are required for three phase stability to appear at these discrete intersections. [35] We next consider the impact of a continuous distribution of pore sizes (Figures 6d 6f). The gas water capillary pressure linearly increases with the nonwetting phase saturation from 1.44 MPa to 7.2 MPa (Figure 6e). The pore size continuously varies from 100 nm to 20 nm (Figure 6f). Each pore size in this range represents an infinitesimal fraction of the total pore volume: the y axis of Figure 6f represents the pore volume dv between pore sizes r and r + dr. In this case, the three phase system may exist within the area bounded by the L + H and L + G solubility curves in pore sizes of 100 nm and 20 nm (gray area in Figure 6d), depending on the amount of CH 4 available. Because pore sizes are continuously distributed between 100 nm and 20 nm, there are many combinations of pore sizes that achieve three phase equilibrium within this area. At the depth of three phase equilibrium in 50 nm pores (947.8 mbsl meters below sea 9of24

10 Figure 7. Evolution of phase saturations in the case of three discrete pore sizes (100, 50 and 20 nm). We consider a parcel of sediment at the depth that corresponds to the three phase equilibrium in r 2 =50nm pores (horizontal solid line in Figure 6a). (a) CH 4 is gradually added to the system. (b) Initially, only gas (red) is present in the r 1 pores until (c) gas fills all of the r 1 pores. At 36 g CH 4 /dm 3 pore vol., all CH 4 cannot exist in (d) free gas plus dissolved phase or (e) gas hydrate (green) plus dissolved phase. (f) The increased CH 4 quantity results in the conversion of a portion of free gas to gas hydrate, which competes with free gas for the r 1 pores. level, horizontal line in Figure 6d), five of these combinations are selected for illustration (see the discussion in section 4.2.3): r h = 100 nm, r g = 59.3 nm; r h = 75 nm, r g = 55.9 nm; r h = r g = 50 nm; r h = 35 nm, r g = 43.9 nm; r h = 20 nm, r g = 33.1 nm. In the first two cases, hydrate is present in larger pores than gas. In the last two cases, gas is present in larger pores than hydrate. [36] In summary, a pore size distribution adds a thermodynamic degree of freedom to the system. Hydrate and gas bubbles can occupy pores of different sizes to establish a zone of three phase equilibrium that extends over a depth interval. At three phase equilibrium, the L + G solubility predicted by free gas in pores of radius r g must equal the L + H solubility predict by hydrate in pores of radius r h. There are many such combinations of pore radius in sediments with a broad, continuous pore size distribution to achieve threephase equilibrium (Figure 6d) Saturations With a Discrete Pore Size Distribution at Three Phase Equilibrium [37] We calculate the phase saturations when three pore sizes are present (r 1 = 100 nm, r 2 = 50 nm, and r 3 = 20 nm). We consider a volume of sediment at the depth of threephase equilibrium in r 2 pores (horizontal solid line, Figure 6a) and gradually add CH 4 to it. The appearance of gas hydrate and free gas in pores is determined by the CH 4 concentration r1 (Figure 7). C L+G is the CH 4 solubility required to form gas in r 1 pores. Since the assumed depth (horizontal solid line, Figure 6a) is beneath the depth of three phase stability in the largest pore size r 1,CH 4 is initially stored in the dissolved phase with gas first appearing in r 1 pores only when the r1 dissolved CH 4 concentration reaches C L+G (Figure 7a). As the total CH 4 quantity continues increasing, the gas saturation increases correspondingly, reaching 20.5% at 20 g CH 4 /dm 3 pore vol. (Figure 7b). Aqueous CH 4 concentration is sustained at C L+G with increasing CH 4 quantity, because all gas r1 remains in r 1 pores at this stage. Initially there is no hydrate in the system. The gas saturation continues to increase until it fills all pores of radius r 1 (30% of the entire pore volume) at 28.4 g CH 4 /dm 3 pore vol. (Figure 7c). [38] A more complicated situation arises as we continue to increase the CH 4 quantity. There are three possible scenarios for the increased CH 4 quantity of 36 g/dm 3 pore volume (Figures 7d 7f). First, we assume that at this CH 4 quantity, all CH 4 would exist as free gas plus dissolved CH 4 (Figure 7d). Then, the gas saturation is calculated to be 33.3%, which fills all r 1 pores and some r 2 pores. In this case, the CH 4 concentration in solution must be increased to C r2 L+G, which is the CH 4 solubility required to form gas in r 2 pores. This CH 4 solubility (C r2 L+G ) is greater than that required to form hydrate in r 1 pores (C r1 L+H ) (Figure 6a), which would drive gas to form hydrate in r 1 pores. Therefore, it is not stable for all CH 4 to be in free gas phase plus dissolved in the water. Second, we consider another extreme case: all CH 4 is present in hydrate and in the dissolved phase (Figure 7e). In this case, the hydrate saturation would be 28.3%. As a result, all hydrate remains in r 1 pores. However, at this depth, the 10 of 24

11 Figure 8. Evolution of phase saturations in the case of continuous pore size distribution (PSD). We consider a parcel of sediment at the depth that corresponds to the three phase equilibrium in 50 nm pores (horizontal solid line in Figure 6d). CH 4 is gradually added to the system. (a) The first gas appears in the largest pores (100 nm). (b) Initially, only gas (red) is present in the system and gas successively fills smaller pores as the CH 4 quantity increases. (c) The first hydrate appears in the largest pores (100 nm). (d) More hydrate (green) appears in the system as the CH 4 quantity increases. Gas and hydrate equally occupy the pores larger than r h, and gas fills the pores between r h and r g. (e) Hydrate and gas equally occupy the pores larger than 50 nm. (f) The system has more hydrate than gas. Gas and hydrate equally occupy the pores larger than r g, and hydrate fills the pores between r g and r h. CH 4 solubility required to form hydrate in r 1 pores (C r1 L+G )is larger than that required to form gas in r 1 pores (C r1 L+G ) (Figure 6a), which would drive hydrate dissolution. Therefore, the second scenario is not stable either. Since the above two scenarios are not stable, the system must exist at threephase equilibrium for the given CH 4 quantity. A portion of free gas preexisting in the system is converted to hydrate for satisfying both mass conservation (equation (5)) and phase equilibrium (equation (11)) conditions. Both hydrate and gas must fill down to r 2 (r g = r h = r 2 ) in order to achieve threephase equilibrium at this depth (Figure 7f). [39] A question remains about how hydrate and gas are distributed in r 1 pores, when the dissolved CH 4 concentration is controlled by the smaller r 2 pores (Figure 7f ). At this point, CH 4 concentration is uniform in solution (C CH4 = r2 C L+G = C r2 L+H ), which is larger than both solubilities required to form hydrate (C r1 L+H ) and gas (C r1 L+G )inr 1 pores (Figure 6a). This concentration allows both hydrate and free gas to be stable in r 1 pores. We assume equilibrium behavior. This assumption suggests that any dissolved CH 4 concentration above the solubilities required for hydrate or gas formation in r 1 pores can precipitate hydrate or gas. Thus we assume that hydrate and gas have equal chances to precipitate in r 1 pores (i.e., they compete for the r 1 pores), if a supersaturation is sustained by the smaller r 2 pores. We will discuss the implications of this assumption in section Saturations With a Continuous PSD at Three Phase Equilibrium [40] We next illustrate how phase saturations evolve as total CH 4 is increased for a continuous PSD. We consider the same depth as the previous example (horizontal solid line, Figure 6d). r h and r g are the smallest hydrate and gasfilling pores (Figure 8). The first gas appears in the largest pore when the dissolved CH 4 concentration equals the solubility of gas formation in 100 nm pore (Figure 8a). In this case, pore size continuously varies between 100 nm and 20 nm, and each pore size in this range represents an infinitesimal fraction of pore volume (Figure 6f). As the CH 4 quantity increases, free gas progressively penetrates into smaller pores (Figure 8b) and the CH 4 concentration in liquid increases correspondingly. In Figure 8c, free gas fills pores as small as 59.3 nm. The CH 4 solubility required to form gas in 59.3 nm pores equals that required to form hydrate in 100 nm pores (marker 8C in Figure 6d), and thus the first hydrate appears in the largest pores (100 nm). [41] The system exists at three phase equilibrium as more CH 4 is added (Figure 8d). At this point, the dissolved CH 4 concentration (0.124 mol/kg water) exceeds the solubilities required to form hydrate in pores larger than 75 nm or form gas in pores larger than 55.9 nm (marker 8D in Figure 6d). Thus, hydrate and gas can coexist in pores larger than 75 nm. We assume that any dissolved CH 4 concentration above the solubility for hydrate or gas formation can precipitate hydrate or gas. Thus, hydrate and gas have equal chances to precipitate in the pores larger than 75 nm. Because hydrate fills 50% of the pore volume between 100 nm and 75 nm (Figure 8d), 2 S h is used as the nonwetting phase saturation on the capillary pressure curve (i.e., P chw (2 S h )inequation(8)) to determine the smallest hydrate filling pore (r h ) (Figure 6e). Free gas fills the pores between 55.9 nm and 75 nm (Figure 8d). 11 of 24

12 Figure 9. Pore size combinations used for determining the top and base of the three phase zone. The box represents total pore volume and the pore size increases to the right. Either hydrate (H) or gas (G) occupies the large pores, and liquid water (L) fills the small pores. r m is the largest pore size in sediment. r L+H h is the smallest hydrate filling pore size at L + H equilibrium. r L+G g is the smallest gas filling pore at L + G equilibrium. At the top of the three phase zone, gas exists only in the largest pore (r m ), while hydrate fills pores as small as r L+H h. At the base of the three phase zone, hydrate exists only in the largest pore (r m ), while gas fills pores as small as r L+G g. S g + S h is used as the nonwetting phase saturation on the capillary pressure curve (i.e., P cgw (S g + S h ) in equation (7)) to determine the smallest gas filling pore (r g ) (Figure 6e). At the CH 4 quantity of 29.8 g/dm 3 pore vol., the CH 4 concentration in solution (marker 8E in Figure 6d) is sufficient to form both hydrate and free gas in pores larger than 50 nm. As a result, hydrate and gas equally occupy the pores larger than 50 nm, i.e., S g = S h. S g + S h is used as the nonwetting phase saturation to calculate both the smallest hydrate and gas filling pores on the capillary pressure curve (r g = r h in Figure 8e). [42] More hydrate appears in the system as the CH 4 quantity increases (Figure 8f). At this point, CH 4 concentration in liquid (0.13 mol/kg water) exceeds the solubilities required to form hydrate in pores larger than 35 nm or form gas in pores larger than 43.9 nm (marker 8F in Figure 6d). Thus, hydrate and gas equally fill all pores larger than 43.9 nm, and hydrate fills the pores between 43.9 nm and 35 nm (Figure 8f). Thus, 2 S g and S g + S h are used as the nonwetting phase saturations on the capillary pressure curves (i.e., P cgw (2 S g ) in equation (7) and P chw (S g + S h )in equation (8)) to determine the smallest gas and hydratefilling pores, respectively. [43] Once the system reaches three phase stability, S h increases significantly with the increased CH 4 quantity, while S g does not change much (Figures 8d, 8e, and 8f). A decrease in r g (e.g., from 59.3 to 55.9 nm) requires a much larger decrease in r h (from 100 to 75 nm) to achieve threephase stability (Figure 6d). Thus, a great increase in S h is required to achieve three phase stability and the increased CH 4 quantity is largely partitioned into hydrate Determining the Thickness of the Three Phase Zone [44] When only two phases (Liquid + Hydrate or Liquid + Gas) are present, r h and r g depend on the phase saturations only. We use P cgw (S g ) in equation (7) and P chw (S h ) in equation (8) to compute r g and r h, respectively. If more CH 4 is available, S h or S g becomes larger and r h or r g becomes smaller correspondingly. We use r L+H h to denote the smallest hydrate filling pore if the system is at L + H equilibrium and r L+G g to denote the smallest gas filling pore if the system is at L + G equilibrium. [45] Hydrate and gas bubbles can occupy pores of different sizes to form a three phase zone over a depth interval. When the same CH 4 quantity is present at all depths, we can determine the top and base of the three phase zone using the L + G and L + H solubility curves in these three pore sizes (r m, r L+H h, r L+G g ) (Figure 9). r m is the maximum pore size present in sediment. When three phases coexist, r g and r h must be first calculated from the phase saturations and the capillary curves. At the top of the three phase zone, S g =0. We use P cgw (0) in equation (7) and P chw (S h ) in equation (8) to calculate r g and r h respectively. The gas and hydratefilling pores are r g = r m and r h = r L+H h. This combination suggests that at the top of the three phase zone, the first gas appears in the largest pore r m when hydrate fills down to the pore size r L+H h. At the base of the three phase zone, S h =0. We use P cgw (S g ) in equation (7) and P chw (0) in equation (8) to calculate r g and r h. Thus r g = r L+G g and r h = r m. The last hydrate disappears in the largest pore r m. [46] In summary, the top of the three phase zone is determined by the intersection of the L + G solubility curve in r m with the L + H solubility curve in r L+H h. The base of the three phase zone is determined by the intersection of the L + H solubility curve in r m with the L + G solubility curve in r L+G g. S h and r L+H h above the three phase zone change with depth even for the same CH 4 quantity at all depths, because the L + H CH 4 solubility changes with depth. Thus, the depth of the top of the three phase zone and its associated r L+H h are iteratively determined in our calculation. The same is also true for determining the base of the three phase zone and r L+G g. In practice, if S h above and S g below the threephase zone change only slightly with depth at the same CH 4 quantity, we can use S h in L + H system and S g in L + G system at the depths near the three phase zone to approximately determine r L+H h and r L+G g, respectively Pore Size Distributions at Blake Ridge and Hydrate Ridge [47] We next explore more realistic pore size distributions through analysis of porosimetry data from Hydrate 12 of 24