Predicting the occurrence, distribution, and evolution

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 104, NO. B3, PAGES , MARCH 10, 1999 Predicting the occurrence, distribution, and evolution of methane gas hydrate in porous marine sediments Wenyue Xu and Carolyn Ruppel School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta Abstract. Using a new analytical formulation, we solve the coupled momentum, mass, and energy equations that govern the evolution and accumulation of methane gas hydrate in marine sediments and derive expressions for the locations of the top and bottom of the hydrate stability zone, the top and bottom of the zone of actual hydrate occurrence, the timescale for hydrate accumulation in sediments, and the rate of accumulation as a function of depth in diffusive and advective end-member systems. The major results emerging from the analysis are as follows: (1) The base of the zone in which gas hydrate actually occurs in marine sediments will not usually coincide with the base of methane hydrate stability but rather will lie at a more shallow depth than the base of the stability zone. Similarly, there are clear physical explanations for the disparity between the top of the gas hydrate stability zone (usually at the seafloor) and the top of the actual zone of gas hydrate occurrence. (2) If the bottom simulating reflector (BSR) marks the top of the free gas zone, then the BSR should occur substantially deeper than the base of the stability zone in some settings. (3) The presence of methane within the pressure-temperature stability field for methane gas hydrate is not sufficient to ensure the occurrence of gas hydrate, which can only form if the mass fraction of methane dissolved in liquid exceeds methane solubility in seawater and if the methane flux exceeds a critical value corresponding to the rate of diffusive methane transport. These critical flux rates can be combined with geophysical or geochemical observations to constrain the minimum rate of methane production by biogenic or thermogenic processes. (4) For most values of the diffusion-dispersion coefficienthe diffusive end-member gas hydrate system is characterized by a thin layer of gas hydrate located near the base of the stability zone. Advective end-member systems have thicker layers of gas hydrate and, for high fluid flux rates, greater concentrations near the base of the layer than shallower in the sediment column. On the basis of these results and the very high methane flux rates required to create even minimal gas hydrate zones in some diffusive endmember systems, we infer that all natural gas hydrate systems, even those in relatively low flux environments like passive margins, are probably advection dominated. 1. Introduction In recent years the observational data set on the in situ conditions associated with gas hydrate occurrences has expanded rapidly, primarily due to the success of Ocean Drilling Program (ODP) Leg 164 [Paull et al., 1996], which drilled through the gas hydrate-bearing zone and the bottom simulating reflector (BSR) on the Blake Ridge sediment drift deposit. A key outcome of such field-based programs has been a clearer understanding of in situ porosities [Paull et al., 1996], advective flux rates (constrained by pore water chemistry [Egeberg and Dickens, 1999] and thermal data [Ruppel, 1997]), gas hydrate concentrations [Holbrook et al., 1996; Dickens et al., 1997], hydrate distribution patterns [Collett, 1998; Egeberg and Dickens, 1999], and other factors that can only be well parameterized by direct sampling of marine sediments in hydrate-bearing zones. Despite the rapid recent improvement in the quality and quantity of observational data, fundamental questions remain Copyright 1999 by American Geophysical Union. Paper number 1998JB /99/1998JB about gas hydrate formation and stability in marine sediments. Most importantly, few studies have analyzed the physical parameters controlling the evolution of the methane hydrate zone or quantified the role of various processes and variables in governing the rate of gas hydrate formation and the distribution of gas hydrate in marine sediments. In this study we address this gap by developing an analytical model for the two-component (methane gas and water), three-phase (gas hydrate, free gas, and water containing various amounts of dissolved methane) system subject to coupled mass, momentum, and energy conservation. We first describe the model and necessary assumptions and then simplify the coupled equations to derive expressions for the location of interfaces bounding the actual zone of methane gas hydrate occurrence and the zone of methane gas hydrate stability. Finally, we examine the timescales of gas hydrate accumulation in diffusion- and advection-dominated systems and develop expressions to describe the distribution of gas hydrate in homogeneous marine sediments. To maximize the potential applicability of our results, we show how the analytical model can be combined with simple observables (e.g., water depth at the seafloor, thickness of the methane hydrate stability zone (MHSZ), BSR depth, heat

2 5082 XU AND RUPPEL: GAS HYDRATE FORMATION IN MARINE SEDIMENTS Temperature (øc) :.'.?.:::.':::.i.;"..;.i.::".?;.::"?.?.::".'.'...'.?.::".?:..: ' '.i;:1:.'::)).?)}.?..:.'.).i.:.!.;.'?.?.?.::.'::.'.i?.;"5).:.' :5.).::.'.?);".':;.).:.? :-.'.??.'5)::::::.)J'::::.)J: ¾.)? occurrence zone bottom of hydrate "'"' - : Z bottom occurrence of hydrate zone (BGHZ),..... o. 0;. ;. 1'N O M 1, Ms 1 stability gone (BGHS) _ ov, ;o;o oo _;oo;oo_;oo_ oo; Vo vo o o o o ovo o o o Oo vo o o o o {o o;o oo oooo ;oo;ooooo o ; o o ovo ovo o o;o;o ooo oo ;O oo;oo;c _ t o p o f free g as zone x500 oooo, oo o!!! o %%%0 o ooooooo ooooooooo oooo oo oooo: oooo ; oo oo ooioooo:ooo : o-o-o-o-o-o-o-oo o o o;; o, oo O ooooo ooo ooo ooooo _:oo o.oooo!oo!!: : oooooooooooooooooi ooooooooo ooo!oo o o o o o o 5: o O o o o oo!ooioo!: 'Oo, oo oo;o_o _oo;oog Ooo;g FREE G A S ;g!' ee % eo o;o O; o oo g o%o g;g oo?oo%%oo%ooo oooo o ;oooo o o oogoooooooooo ::o o-o-o-o o-o- o o o: oo ii o-o-o-o-o-o-o-o-o- - oo o;o o oo; o-oo o i i}i-o-o-o-, o-oo o o : o o o-o;o O oooooo0o oooo: oo oo oo oo oo o oo I,ooo 0OoOOOOOoO :oooooo :oo : ooo oo oo oo O.o o. o I : ooi:!oo : Oo Mass fraction of methane (kg/kg) Figure 1. The relationship among the actual zone of gas hydrate occurrence (MHZ), the zone of methane gas hydrate stability (MHSZ), and the free gas zone for a particular geotherm and an assumed seafloor pressure corresponding to 2500 m water depth. The location of the base of the MHSZ lies at the depth at which the geotherm crosses the stability boundary. The base of the MHZ does not coincide with the base of the MHSZ in this case because upward flux of methane (qm) does not exceed the critical value for the system. The free gas zone only exists at depths for which the mass fraction of methane M exceeds methane solubility (M.t) below the base of the MHSZ. The scenario illustrated here, in which the free gas zone lies far below the base of the MHSZ and is separated from the MHSZ by an intervening layer of sediment lacking both gas hydrate and free gas, may explain observations Site 994 of ODP Leg 164. flux, etc.) to directly and quantitatively constrain fluid advection rates through gas hydrate systems and the rate of methane production due to biogenic or thermogenic processes. The framework described here therefore provides a fundamental link among the rates and parameters that describe the physical, chemical, and biological processes occurring in marine methane hydrate provinces. 2. Background and Previous Research Natural gas hydrates are an icelike, crystalline form of (mostly) methane plus water stable under a range of pressure and temperature conditions that characterize the uppermost tens to hundreds of meters in continental margin marine sediments, terrigeneous deposits in deep freshwater lakes (e.g., Lake Baikal), and permafrost regions. Figure 1 shows the methane hydrate stability field as a function of depth (pressure) and temperature. To the left of the stability curve, water ideally coexists only with methane hydrate, while water and free gas are stable to the right of the curves. The phase boundary itself marks the coexistence of all three phases: gas hydrate, free gas, and water. Factors such as salinity [de Roo et al., 1983], gas composition [Sloan, 1990], capillarity [Clennell et al., 1995], and possibly sediment mineralogy [Cha et al., 1988] can affect the position of the stability boundary. The three-phase nature of the gas hydrate system presents challenges to the application of typical, immiscible, two- phase flow through porous media approaches [e.g., Bear, 1979]. Previous researchers have therefore adopted various simplifications to study the gas hydrate system, and most analytical treatments have focused on solution of the Stefan problem, in which a cooling or warming front created by the instantaneous juxtaposition of media at different initial temperatures passes through the porous sediments containing gas hydrate or its components. For example, Yousif et al. [1991] devise a numerical model for the dissociation of gas hydrate in a one-dimensional, three-phase system subjecto isothermal depressurization and particular initial and boundary conditions, while Selim and Sloan [1990] develop an analytical model that assumes immobility of water released during hydrate dissociation and temperature distributions that may be unrealistic for most natural sediment systems. The approach most similar to the one adopted here is that of Rempel and Buffett [1997], who write a system of coupled conservation equations to track the formation of gas hydrate

3 XU AND RUPPEL: GAS HYDRATE FORMATION IN MARINE SEDIMENTS 5083 in the two-phase system (water and gas hydrate). The key differences between this study and Rempel and Buffett's [1997] work are some aspects of our analytical formulation, our emphasis on the role of fluid advection and methane supply rates, and our adoption of initial and boundary conditions and problem parameters relevant for the natural marine system. In addition, this study, in contrast to some previous work, yields an integrated picture of the entire marine methane hydrate system, including both the methane hydrate zone and the free gas zone. (sedimentation rate), the mass fluxes q of liquid (subscript ), gas (subscript g) and hydrate (subscript h) phases are given by e =-krekpe (Vp+ pe,)+4 Sep, e, (1) Og =-krgkp- --g (Vp+pg )+CpSgpg g, (2) #g Oh : (PShPh s, (3) 3. Model Formulation where p is pressure, g represents gravitational acceleration, S stands for volume saturation of individual phases, Pi and /f2 i are We model the gas hydrate system as consisting of two densities and dynamic viscosity of different phases, respectively, components and three phases. Water and methane gas and krg and krl denote relative permeability functions for the gas constitute the system components, while gas hydrate and liquid, which are dependent on the saturations S. (subscript h), free gas (g), and water plus dissolved methane Conservation of mass for the system is given by (l) representhe three phases. Consideration of a fully threephase model is justified on several grounds. First, although simple thermodynamiconsiderations dictate that only two ((P ) q- V.( t e + t g + Oh )-- Qm, (4) phases should be present for P-T conditions not coincident where t denotes time, p is bulk density, and Qm represents the with the stability boundary, in dynamic natural systems, rate of methane gas production in the sediments. With Qm as particularly those undergoing rapid phase changes in response the source term, conservation of mass for methane gas is to forcing, we postulate that three phases may coexist in a written as: region of finite thickness. Second, observational data such as those obtained on ODP Leg 164 [Paull et al., 1996] suggest that three phases probably coexist within the hydrate stability zone. Finally, by considering the three-phase system, the 3(½PM ) +V.( em e + gmg + hmh) = V.(CPKmSeVMe)+Qm, (5) natural evolution of the physical system can be tracked where M denotes mass fraction of methane in individual without adopting the numerous simplying assumptions of earlier workers [Rempel and Bufferr, 1997]. In other words, if phases and Km represents the product of the dispersiononly two phases are present at a given location in the system, diffusion coefficient Dm and the representative fluid density of the mathematical formulation we adopt here will naturally the system, which can nominally be taken as 1000 kg m -3 for evolve to reflect the existence of only two phases. the sake of converting between Km and Din. The presence of Several assumptions are made in the formulation of the only one liquid' phase diffusion-dispersion term on the right equations for the three phase system. First, we assume that side of (5) is justified because Mg is identically one, the Darcy's law can be applied to the system. Darcy's law is diffusivity of hydrate is negligible, and the gradient of Mh is widely used to describe a broad range of porous media small compared to the gradient of Me. phenomena, and this assumption is unlikely to introduce Conservation of energy is cast in terms of enthalpy H and appreciable errors in low flow rate systems like those temperature T and written as considered here. Second, the variations in potential and kinetic energy are considered negligible compared to the - [cpph + (1- c) )PsCsr] + V. ( eh e + gh g + hsh ) variation in thermal energy. Third, capillary effects are = V- (AVT) + Qe, (6) neglected despite their possible role in affecting the hydrate stability boundary [e.g., Clennell et al., 1995]. Capillary where subscript s refers to the sediment matrix, C denotes effects can be easily incorporated into later numerical models specific heat capacity, Qe represents a heat source term related of these system equations but may obscure some of the basic to methane production, and A is effective thermal physics if considered at this point. Fourth, the kinetics of the conductivity, which is written as a linear combination of the phase transition between hydrate and water plus methane is thermal conductivities for different components of the system: ignored, following the justification given by Rempel and Buffett [1997]. Fifth, to make the problem tractable, we A = (1- ½)A,,, +½(SeA e + SnAg q- ShAh). (7) assume that liquid, gas, and methane hydrate are in thermal This method of combining thermal conductivities assumes and chemical equilibrium and that diffusion of methane gas in parallel flow of heat through both the fluids and solids gas hydrate and free gas phases is negligible. Finally, [Farouki, 1986] and yields an upper limit for effective bulk although incorporation of a conservation equation for salinity thermal conductivity. is straightforward, the first-order approach taken here focuses Finally, the balance among the phases can be incorporated only on a system with fixed salinity (35%o NaC1). by writing bulk density p, bulk enthalpy ph, and bulk mass The fundamental equations governing the two-component, fraction of methane pm as linear combinations of the three-phase system in porous sediments are conservation of properties of the individual phases: momentum, fluid mass, energy, and methane gas. Assuming that Darcy's law describes the conservation of momentum for multiphase fluid flow through oceanic sediments of porosity ½ and permeability k that are moving downward at a velocity u.,. p = Se.p + Sgp g + Sht9h, ph = SepeH e + Sgt9 ghg + Sht9hH h, (8) (9)

4 5084 XU AND RUPPEL: GAS HYDRATE FORMATION IN MARINE SEDIMENTS pm = SfpfMf + SgpgMg + ShPhM h, (10) Table 1. Typical Parameters Used in Calculations. where the saturations S are subject to the constitutive equation: I=Se+Sg+S h. (11) The saturations are determined from equations (8-11): where Se= Ae Sg - Sh _ h (12) ZX, = t9gph[(hg - H)(M h - M)-(H h - H)(Mg - M)] n)(m,, A,, = ptpg[(ht - H)(Mg - M)- (Hg - H)(Mt - M)] A = A e +Ag +A h. 4. Results (13) The mathematical framework described above can be used to derive fundamental relationships that quantitatively describe the position of the zone of actual gas hydrate occurrence within the hydrate stability field, the distribution of gas hydrate in end-member marine systems, and the time for formation of the gas hydrate zone. We here use observational data to constrain values for many of the physical parameters given in the notation section and demonstrate the significance of analytical expressions for evaluating the behavior of natural hydrate systems. Throughouthis paper, we apply the formulation to a standard case described by the physical parameters given in Table 1. Some values in Table 1 have been chosen to crudely represent those that might apply to marine systems like the Blake Ridge. For example, thermal conductivity of-1 W m -1 K 'l, sediment porosity of 0.5, and bottom water temperature of 3øC are consistent with observations made on ODP Leg 164 [Paull et al., 1996]. Other numerical values given for the standard system are chosen to represent the middle of the range of reasonable variations (10-6 to m 2 S -1) for such parameters as diffusion-dispersion coefficient D m [e.g., Krooss and Leythaeuser, 1988]. Finally, some of the values given in Table 1 simply reflect the adoption of certain assumptions (e.g., no sedimentation and no energy or methane source terms) in modeling this system. For practical purposes, the model is completely insensitive to reasonable variations in parameters such as M0, mass fraction of methane in the bottom water. Actual values for Mo are on the order of 104 kg kg -1 or smaller under most conditions [Ward, 1992]. Since the solubility of methane in seawater M..t is on the order of l0-3 kg kg 'l, almost any Mo value that would be realistic for a generalized natural system would effectively provide a seafloor boundary condition of zero for methane mass fraction. One notable exception will be at sites at which gas hydrate occurs at the seafloor. At these locations, Mo must locally exceed M,.t, which requires Parameter g To Us Value 9.81 ms -2 3øC 0 m s '1 ½ 0.5 Zo 2000 m Dm 1.3x10 '9 kg m 4 s ' Mo 0 kg kg 'l qt' 10 's to 6.4x10 -s kg m -2 s -1(~0.3 mm yr -1 to 2 mm yr - ) qc mw m '2 qh 0 kg m -2 s ' (hydrate flux rate) Qm 0 kg m -3 s - Qe 0 W m '3 k!x10-4 m W m' K ' Pt 1024 kg m '3 C e 4.18 kj kg -1 K '1 gt 8.87x10-4 kg m ' s ' to gain fundamental physical insights into all natural gas hydrate systems in marine sediments through the application of a simple model and the exploration of a limited, but realistic, parameter space. While different numerical results would be obtained for different values of constants, the fundamental relationships discussed here should be applicable to a broad range of gas hydrate settings Hydrate Stability Zone Versus Actual Zone of Hydrate Occurrence - We first introduce terminology that distinguishes the methane hydrate stability zone (MHSZ) from the actual (observed) methane hydrate zone (MHZ) of accumulation. The location of MHSZ is governed by pressure and temperature. While pressure and temperature profiles also largely control the location of the zone of actual hydrate occurrence (MHZ), the precise positions of the bottom and top boundaries of the MHZ depend on the availability of methane in excess of its solubility in seawater (Figure 1). We examine the initial formation and evolution of onedimensional (vertical) systems in which Qm=O, Qe=O, and U.=0 (no sedimentation). For this analysis, parameters, k,, P e, t e, re., Ph, Mr, and D m are assumed constant as given in Table 1. C e is specific heat capacity of liquid water and relates enthalpy to temperature through the relation H e = CeT. Boundary conditions are constant flux of total mass q methane qm and energy q at a large depth below the base of the MHSZ and constant pressure (P0), temperature (To) and mass fraction of methane (M0) at the seafloor Steady state approximations. In most cases, up to two phases (hydrate and liquid) may be present above the base of the MHSZ, but only liquid is mobile. Owing to the relatively slow rate of gas hydrate formation and dissociation in most natural systems, we assume that the pressure and temperature variations associated with its formation and accumulation are to first order negligible compared to total mass and heat transport in the system. This assumption renders the system steady state under certain boundary conditions of high fluid flux q and methane flux qm. Since we do not explicitly consider systems in which gas hydrate conditions. For simplicity, we also assume that the amount of occurs at the seafloor, we choose a small value of Mo. hydrate in pore space is sufficien tly small that the liquid phase Despite these justifications for the choice of the parameters occupies almost the entire available porosity. With these for the standard system, we note that the goal of this paper is assumptions the governing equations for the one-dimensional

5 XU AND RUPPEL: GAS HYDRATE FORMATION IN MARINE SEDIMENTS 5085 system reduce from (4) and (6) to qf =-kpe # (dp,dz +peg 1 dt (14) qe - q f CeT - 2,, (15) dz where qt' and q are constant fluxes of total mass and energy, respectively, and z is the spatial coordinate that originates at the seafloor and points upward. In general, the bulk mass fraction of methane cannot be viewed as steady state during the formation and accumulation of hydrates. However, owing to the absence of free gas and the immobility of gas hydrate, all methane transport occurs within the liquid phase, and (5) can be reduced to describe steady state methane transport: dme qm - qf Me -C)KmSe, (16) dz where S e = 1, based on our previous assumptions, and qm represents depth-dependent methane flux, which is not a constant within the MHZ. Below the base of the MHZ, qm is a constant given by the corresponding boundary condition. Above the top of the MHZ, qm(z) reduces to a constant qm,, which corresponds to the methane gas flux at the top boundary of the MHZ. The methane mass fraction in liquid is steady state but does vary spatially within the MHZ Location of MHSZ boundaries. In static hydrate systems the theoretical positions of the top and bottom boundaries of the MHSZ can be determined by combining estimates of hydrostatic pressure and geothermal gradient with hydrate stability constraints. However, in dynamic systems, the positions of the MHSZ boundaries depend on the flux of variou system components and must be written as functions of the mass and energy transport. For the steady state system, which is not the same as a static system, integration of (14) and (15) yields expressions for pressure p as a function of depth z and fluid flux qt: P- Po - -[, kp e + Peg z, (17) and for temperature T as a function of depth z, energy flux q, and fluid flux q : (qe-qj'ceto -qfcet z - - ln q 0 (18a) qfce z X(T - To) qf=o. (18b) qe The variables P0 and To in (17) and (18) denote seafloor pressure and temperature, respectively. Combining (17) and (18) yields upward lqf kpe 11! qfce 2' Iqe-qfCeTol ' P-Po- +Peg In. qr 0 (19a) P- Po - Peg2' (T- To) qe q; =0. (19b) When (19) is coupled with empirical methane hydrate stability curves (e.g., B. Tohidi et al., On the mechanism of gas hydrate formation in subsea sediments, submitted to Journal of the American Chemical Society, 1997, hereinafter referred to as Tohidi et al., submitted manuscript), the pressure and temperature at both the bottom (subscript sb) and top (subscript st) of the MHSZ can be determined. In practice, for typical values of mass flux and other parameters, the pressure perturbation associated with the first term in parentheses in (17) is almost always negligible. Thus mass flux does not appreciably alter the estimate of hydrostatic pressure at a given depth within the sediment column. On the other hand, (18) and Figure 2 clearly demonstrate the strong dependence of temperature on q and qt. Because heat flux values determined from deep in situ measurements of temperature and assumed thermal conductivity values implicitly include any nonzero mass flux component (the advective component), the expression given in (18) can be combined with other constraints on the approximate depth of the MHSZ to bracket the rate of mass flux q through natural marine hydrate systems. The MHSZ merely describes a zone in which gas hydrate is stable if it is present. When they adopt an empirical stability curve like those given by Dickens and Quinby-Hunt [1994] or Tohidi et al. (submitted manuscript) and do not consider the role of fluid flux in modifying the pressure and temperature of the system, researchers implicitly assume that hydrate systems are static. In real settings, the stability of the dynamic hydrate system depends on pressure and temperature parameters that are modified weakly (17) and sometimes strongly (18), respectively, by the flux of components. Thus, while various empirical relationships can always be used to determine the stability conditions for static systems, imprecise knowledge of the actual pressure or temperature conditions in sediments can lead to incorrect predictions about the position of the stability zone. In particular, extrapolation of nearsurface thermal measurements to great depths within the sedimentary column [e.g., Yamano et al., 1982] and failure to consider the role that even minor advective fluxes play in modulating temperature regimes can lead to significant disparities between the actual and predicted depths to the base of the gas hydrate stability zone. It is important that this potential disparity be clearly distinguished from the disparity between the base of the stability zone (MHSZ) and the base of the actual zone of hydrate occurrence (MHZ), which is discussed below. Figure 2 shows the methane hydrate stability curve and the position of the base of the MHSZ for a range of water depths (zo) and several constant mass (qt) and constant energy (i.e., heat; q0 flux rates. The upward expulsion of liquid through the system (qs) causes the geotherm to become nonuniform (nonlinear), lowering the temperature with respect to that of the static system (qs=0) at any depth in the sediment column and deepening the base of the MHSZ for any assumed water depth. The relative magnitude of MHSZ deepening is greatest when flux rates are high, the assumed water depth is large, or the total energy flux is small. For example, for a flux rate of 30 mw m -2 in sediments beneath 3000 m of water, an liquid mass flux rate of 10-8 kg m -2 s - (equivalent to an upward fluid velocity of---3x10-4 m yr - ) results in deepening of the base of the MHSZ by -20 m, from 558 to 578 m below seafloor (mbsf). Comparisons between the endmember cases shown in Figure 2 indicate that the degree of MHSZ deepening increases with increasing water depth (point A versus C) and with decreasing total energy (heat) flux (point B versus C).

6 5086 XU AND RUPPEL: GAS HYDRATE FORMATION IN MARINE SEDIMENTS Location of MHZ boundaries. The locations of 300 Temperature (øc) depth is given by the solubility of methane gas in seawater (M,. ), which can be written as a function of pressure and temperature [Zatsepina and Buffett, 1997] or, through application of (17) and (18), as a function of depth alone. the base and top of MHZ can be determined by combining steady state profiles of the mass fraction of dissolved methane in seawater with system pressure and temperature parameters. Within the MHZ the dissolved methane mass fraction at any Above the top of the MHZ, methane transport is described by (16), with constant qm(z) equal to qmt at the top of the MHZ. Integrating (16) from the top of MHZ (z=zt,) to the seafloor, where methane mass fraction in seawater is Mo, yields for different qt' values (PKm q f I qmt qmt-qfmø - q f Msl(Zlt ) 1 Zlt '- - ln. qs e0 (20a) ½Km ztt =- [M,.t(ztt)- Mo] q =0. (20b) qmt Figure 2. The role of fluid flux in controlling the depth to the base of the MHSZ in dynamic gas hydrate systems. The light curves representhe approximate phase boundary between gas hydrate and liquid to the left and free gas and liquid to the right for the indicated water depth at the seafloor. The curves show geotherms calculated with upward fluid flux (qs) values of 0 (solid curve), 0.5 mm yr - (dashed curve), and 1 mm yr - (dotted curve). Since these geotherms are calculated assuming a constant q value, adding a fluid flux component effectively lowers the local geothermal gradient. For a given heat flux value the difference between the depth to the base of the MHSZ with and without fluid flux increases with increasing water depth, as shown by a comparison of results at locfftions A and C in the diagram. For a given water depth the difference between the depth to the base of the MHSZ with and without fluid flux increases with decreasing energy flux (comparison of results at B and C). In (20), qmt '- q I dm"'l ) (21). f Msl(Zlt )- gm dg z=ztt ' Knowing methane solubility M,.t as a function of z (or p and T), the position of the top of the MHZ (ztt) can be calculated from (20) for the steady state, two-phase system. Note that the position of the top of the MHZ is not dependent on the total methane flux qm. On the basis of gas hydrate stability conditions the MHSZ typically extends above the seafloor into the water column. However, the low mass fraction of methane gas that remains in the liquid both at the seafloor and at shallow depths in the sedimentary column (where the mass fraction of methane is less than its solubility) means that the top of the MHZ should almost always occur deeper than the seafloor, even in the absence of the sulfate reduction zone. As shown in Figure 1, the base of the MHZ (ztb) may often be shallower than the base of MHSZ (z >z.,.b) due to the dependence of zt on the rate of methane supply. The location zt, of the base of MHZ can be determined iteratively using qm = q f Msl(Zlb )-(PKm dz (dmsll(22) Applying (21) and (22) together yields the thickness of the MHZ, which is shown in Figures 3 and 4 to depend on a combination of the rates of fluid flux qt energy flux q, methane flux qm, and the water depth at the seafloor. For a given rate of fluid flux, increasing methane flux results in a thicker MHZ up to the critical point denoted by the attainment of constant MHZ thickness in Figure 3. This critical point occurs when the base of the MHZ instantaneously coincides with the base of the MHSZ and the top of the free gas zone. Beyond the critical point, further increases in methane flux rate will not produce a thicker MHZ. Note that only the bottom of the MHZ can change position with varying qm. For given qt' and q values the top of the MHZ is effectively fixed by the methane mass fraction in seawater (Mo). Increasing Mo within some limit generally results in the top of the MHZ lying closer to the seafloor. In Figure 3 the illustrated link between energy (heat) flux and the thickness of the MHZ is mostly indirect and is related to the fact that gas hydrate is stable to greater depths in the sedimentary column for low heat flux and low geothermal gradients. However, given that total energy flux (background conductive heat flux combined with an advective fluid flux

7 . XU AND RUPPEL: GAS HYDRATE FORMATION IN MARINE SEDIMENTS oo- 6O0 o Methane flux ( rn kg s-m... 3 :i::i:::::: : :.:.iii:: ::::::::3:,E :,:: rn : :::::::..::... :: ;:::::::: ::::.m. : :: :. top free gas:%: : ' I ' I overprint) is the value actually measured with standard marine geophysical methods, the parameterization in terms of energy flux provides an obvious connection to geophysical observables in gas hydrate systems. For constant rates of fluid and methane flux an increase in energy flux produces a thinner MHZ. The dependence of MHZ thickness on fluid flux rates is less straightforward, as shown in Figure 4. Increasing the fluid flux rate increases the rate of methane flux required to form an MHZ. Once the MHZ reaches a critical thickness (denoted by the constant thickness portions of the curves in :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Figure 4), further increases in the methane flux rate do not affect the thickness of the MHZ but may cause more free gas to accumulate below the MHZ. Increasing the fluid flux rate for a fixed total energy flux value lowers the geotherm (Figure 2), making gas hydrate stable to greater depths and lowering the solubility at a given depth. Both effects tend to produce a thicker MHZ for higher fluid flux rates. The results shown in Figures 3 and 4 indicate that combining heat flow observations, measurement of water depth, and inferences about advective flux rates (from analysis of pore waters) should provide a means for predicting the thickness of the MHZ, an important exercise for estimation of resource potential. The formulation developed here can also be used to predict the depth to the top of the free gas zone, shown as the dotted curves in Figure 3b. For methane flux rates lower than the critical value the top of the free gas zone is not coincident with the base of the MHZ/MHSZ, and the MHZ is separated from the free gas zone by a layer of sediment containing neither gas hydrate nor free gas, as shown in Figure 1. The transition between the scenario shown in Figure 1 and the situation in which the top of the free gas zone and the base of ====================== ' : *: o...%. ':: ii:: ::... ' '": %. <q:% ::::::... '' ' 4 '...,. Figure 3. The top and bottom of the actual zone of gas hydrate occurrence (MHZ) as a function of methane flux rates at constant water depths of (a) 1000 m, (b) 2000 m, and (c) 3000 m at the seafloor, assuming a constant fluid flux rate of 10-8 kg s- m '2(-0.3 mm yr- ). Solid curveshow the top (shallow curve) and bottom (deep_curve) of the MHZ for a measured heat flux of 40 mw m-. 2 Dark shaded and light shaded curves correspond to MHZ boundaries for a heat flux of 50 and 30 mw m -2, respectively. The dotted curves in Figure 3b show the depth of the top of the free gas zone as a function of qm. To the left of the intersection of the dotted curve with the solid curve denoting the base of the MHZ, the top of the free gas zone is separated from the MHZ by sediments that contain neither free gas nor gas hydrate, the scenario illustrated in Figure m, 40 mw m ' 0.5 mm/yr' ß t mm/yr I - \ \ \ 00- \ ' \ - 2 I 3 I 4 I,. 6 I 7 I 8 I 9 I 10 Methane flux ( kg s-- m -2) Figure 4. Depth to top and bottom of the MHZ as a function of methane flux rates at 2000 m water depth and 40 mw m -2 energy flux. The MHZ thickness becomes constant once the base of the MHZ reaches the base of the MHSZ. The critical methane flux rate for the existence of gas hydrate in the sediments is marked by an MHZ of 0 thickness at the left side of the diagram (see also Figure 5b).

8 5088 XU AND RUPPEL: GAS HYDRATE FORMATION IN MARINE SEDIMENTS Base MHZ - Base MHSZ - Top of free gas b. For existence of MHZ I I I. /'"" '" mmlyr ' I ' I ' I Water depth (m) 0.5 mm/¾r 30 rn Win-2... O. 5 rn rn/yr 0.3 rn rn/yr I ' [ I Water depth (m) Figure 5. (a) Critical methane flux q,, for the base of the zone of methane hydrate occurrence (MHZ) to coincide with the base of the stability zone (MHSZ) and the top of the free gas zone, as calculated for a range of fluid flux rates, energy flux, and water depths at the seafloor. The critical flux represents the flux necessary for the bulk mass fraction of methane M to exceed methane solubility M,.z(z.,.o) at the base of the MHSZ in the region beneath the MHSZ. Only the curve corresponding to 40 mw m -2 is p21otted for most of the fluid flux rates. The relationship between the 40 mw m -2 and 50 mw m' curves at low (0.3 mm yr - ) and high (2 mm yr - ) fluid flux rates is explained in the text. (b) The critical methane flux qm for the existence of methane hydrate in sediments, assuming constant energy flux rates of 30, 40, and 50 mw m and flmd flux rates of q/of 0.3 to 2 mm yr -. As water depth increases, the critical methane flux for gas hydrate to form in sediments decreases. Only the 40 mw m -2 curve is plotted from most values of fluid flux. the MHZ/MHSZ become coincident occurs over a very flux, is strongly dependent on the total rate of fluid flux, and narrow range of methane flux values for given seafloor depth has intermediate dependence on the water depth. Most and energy flux. This relationship can be exploited to infer methane flux rates at locations like the Blake Ridge, where free gas is observed at -90 m below the MHSZ at Site 994 but notably, a twofold increase in the rate of fluid flux through the system results in near doubling of the critical methane flux value. Although our analytical formulation only applies to the immediately below the MHZ/MHSZ at adjacent Site 995 interpretation of gas hydrates in systems with homogeneous [Holbrook et al., 1996]. sediments, Figure 5a clearly demonstrates that sediments characterized by high advection rates require a high rate of 4.2. Critical Rates of Methane Supply Gas hydrate extending to the base of the stability zone. A common misconception is that the base of the actual zone of hydrate occurrence in marine sediments (base of methane supply if the gas hydrate is to extend as deep as the base of the MHSZ. This result has important implications for assessment of the resource potential of gas hydrate deposits. In Figure 5a the critical methane flux rate for energy flux MHZ) should coincide with the base of the MHSZ. In fact, of 50 mw m -2 is greater than that for energy flux of 30 mw the base of the MHSZ is only equivalent to the base of the MHZ if methane mass flux qm exceeds a critical value that produces a bulk mass fraction of methane M in excess of m -2 at all water depths when fluid flow occurs at 0.3 mm yr '. However, at a fluid flow rate of 2 mm yr ' the critical methane flux rate for energy flux of 50 mw m -2 is less than that for 30 methane solubility M,q(z.,.,) in the region beneath the MHSZ. mw m -2. This result reflects a fundamental transition Figure 5a shows the results of calculations to determine this critical methane flux value as a function of water depth, between a more diffusion-dominated system (qt.=0.3 mm yr - ) and a more advection-dominated system (qt =2 mm yr- ). various values of q and energy flux qe. The critical methane Increasing fluid flux rates for a constant energy flux lowers flux rate necessary for the base of the MHZ to coincide with the bottom of the MHSZ is almost independent of the energy the geotherm, increases the thickness of the stability zone, and affects the methane solubility curve that controls the position

9 XU AND RUPPEL: GAS HYDRATE FORMATION IN MARINE SEDIMENTS 5089 of MHZ boundaries. For higher rates of fluid flux the advective effect related to lowering the geotherm and increasing the thickness of the stability zone dominates. For lower rates of fluid flux the diffusive effect associated with changes in the methane solubility curve is more important Critical conditions for occurrence of gas hydrate. For the MHZ to exist at all in marine sediments requires that the depth to the base of the MHZ (calculated from (22)) exceed the depth to the top of the MHZ (from (20) and (21)). A critical methane flux value qm necessary for the existence of the MHZ is shown in Figure 5b as a function of water depth at the seafloor for various qs values and a range of energy flux rates qe. Owing to decreased methane solubility at higher pressures [Zatsepina and Buffett, 1997], the critical methane flux for the formation of an MHZ decreases as water depth increases, meaning that formation of an MHZ is slightly more favored at great water depths than in shallower waters very close to continental margins. However, the critical methane flux rate for the occurrence of the MHZ is most strongly dependent on the rate of fluid flux. Doubling the rate of fluid flux results in a significant, although not doubled, increase in the critical methane flux rate. Figure 5b also shows the much weaker dependence of the critical methane flux rate on variations in total energy flux. Results that constrain the critical methane flux rate for the If the BSR marks the top of a free gas zone, then the presence of free gas is a requirement for the existence of the impedance contrast that defines the BSR. The analytical model indicates that free gas may exist in a zone below the base of the MHSZ (e.g., Figures 1 and 3) when the saturation of methane in the liquid phase exceeds methane solubility, an interpretation with two important implications: First, since the free gas zone will, in some cases, be separated from the overlying MHZ and MHSZ by an intervening sediment layer that contains neither hydrate nor free gas (Figure 1), the BSR should sometimes occur below the MHZ and MHSZ, at the top of the free gas zone. An example of this situation may be Site 994 of Ocean Drilling Program Leg 164, where a reflector with negative impedance contrast lies at 550 mbsf [Holbrook et al., 1996], far below the base of the MHZ (-450 mbsf) as defined on the basis of chloride anomalies [Paull et al., 1996]. Second, in cases where physical and chemical conditions do not favor the development of a free gas zone, perhaps no BSR may develop. The nearly ubiquitous presence of BSRs in regions where hydrate-bearing sediments are believed to occur implies that "normal" conditions for the occurrence of gas hydrate systems may involve (1) the flux of fluids containing methane in excess of solubility at some depth below the MHSZ, which will produce a free gas zone and an associated BSR, and/or (2) an older episode of either high methane flux or lower temperature that has trapped free gas layers (and produced BSRs) in systems not characterized by conditions conducive to the formation of contemporary free gas zones. If the impedance contrast associated with the BSR is produced by the seismic velocity contrast between hydratebearing sediments above and hydrate-free sediments below [Dvorkin and Nut, 1993], then the BSR should lie at the base of the MHZ and wholly within the MHSZ under many existence of the MHZ (Figure 5b) or the coincidence of the base of the MHZ and the base of the MHSZ (Figure 5a) have potential applicability for estimating the minimum rates of methane production by biogenic or thermogenic processes. For example, if gas hydrate is detected in a region characterized by a known heat flux and a mass flux rate constrained using pore water analyses, then Figure 5b can provide an estimate of the critical methane flux rate, which can be linked to the minimum rates and extent of methanogenic processes in the sediments. circumstances, as illustrated in Figure 1. Thus the BSR would be associated with P-T conditions that place it within the 4.3. Interpretation of the Bottom Simulating Reflector The presence of a bottom simulating reflector (BSR), a seismic reflector with a negative impedance contrast, has long hydrate stability field, not at the boundary between the stability fields for hydrate plus liquid and gas plus liquid. On the Blake Ridge, Ruppel [1997] shows that high-quality in situ temperature measurements to within 50 m of the BSR at been considered diagnostic of methane hydrate-bearing ODP Site 997 predict temperatures at the BSR to be too cold provinces. The origin of the BSR is the subject of some controversy in the literature: Some observations imply that the BSR marks the base of hydrate bearing sediments (base of the MHZ [Stoll and Bryan, 1979; Dvorkin and Nut, 1993]), while other results supporthe interpretation that the BSR lies at the top of the free gas zone [Minshull et al., 1994; Holbrook et al., 1996]. The analytical model developed here provides a means for estimating the depth to both the top of the free gas zone and the base of the MHZ in marine hydrate provinces (Figure 3) and may thus help to constrain the nature of the BSR in real settings. Explicitly, the depth of the base of the MHZ is calculated iteratively from (22). The position of the top of the free gas zone can be determined using an expression similar to (20) and assuming that the methane gas solubility for the region below the base of the MHSZ can be approximated as a constant M,. =M,.dz,.,) [Zatsepina and Bufferr, 1997]. This discussion necessarily focuses narrowly on addressing only the physical processes that may explain the presence of the BSR with respect to the free gas zone or gas hydrate zone in a steady state system, not on the observation that the BSR mimics seafloor morphology in most settings. by 0.5 to 1.3øC if the BSR represents the base of hydrate stability zone and if currently available stability curves are valid. If the physical significance of the BSR is as a marker between hydrate-bearing sediments above and sediments lacking hydrate below, then the BSR could often occur within the MHSZ according to our calculations. This phenomenon may explain the occurrence of BSRs at overly shallow (colder than predicted temperatures or lower than predicted pressures for gas hydrate dissociation) depths within some settings. For the Blake Ridge, where vertical seismic profiling (VSP) data indicate the presence of a thick free gas zone lying just below the BSR [Holbrook et al., 1996], close mutual proximity of the base of the MHZ, the base of the MHSZ, and the top of the free gas zone can perhaps not be resolved with current data. Thus the BSR may lie wholly within the MHSZ, at the base of the MHZ and at appropriate, not overly cold, temperatures. The data compilation of Booth et al. [1996] also includes many BSR P-T estimates that plot entirely within the methane hydrate stability field. Although these estimates are generally not obtained from high-quality thermal measurements deep in the sediments, at least some of these results may imply the occurrence of BSRs within the

10 5090 XU AND RUPPEL: GAS HYDRATE FORMATION IN MARINE SEDIMENTS MHSZ, a depth consistent with the interpretation that the BSR timescale necessary for some gas hydrate to be present may lie at the base of the MHZ under some conditions. everywhere within a 200-m-thick MHZ in a diffusion In most settings, researchers interpret the BSR as dominated system is shown in Figure 6a for a range of Dm and coinciding simultaneously both with the top of the free gas K,,values. Diffusion-dispersion coefficients in natural zone and the base of the MHZ/MHSZ, as would be the case systems may vary widely between 10-6 and m 2 s -l beyond the critical point in Figure 3. In our simplification of [Krooss and Leythaeuser, 1988]. While the standard case the governing equations we have assumed a sedimentation (Table 1) for this study adopts a Dm value in the central part of rate of zero. However, it is clear that rapid sedimentation this range, Figure 6a clearly reveals the strong dependence of may cause gas hydrate near the base of the MHZ to encounter temperatures too high for continued stability of gas hydrate the diffusive timescale for formation of a layer of gas hydrate on the Dm value. For D,, less than, m 2 S -1 the [e.g., Minshull et al., 1994; Pecher et al., 1998]. If the characteristic time to form a 200-m-thick MHZ is of the order amount of methane released by dissociation exceeds methane solubility, then a free gas zone could indeed develop immediately below the MHSZ/MHZ, even if the methane flux of tens to hundreds of millions of years. For larger D,, values, Figure 6b illustrates that the diffusive timescale is similar to the advective timescale, which is given by rate is not sufficiently rapid to produce free gas at this depth in the absence of sedimentation. :: (24) q.! 4.4. Timescales for Gas Hydrate Accumulation in Natural Systems The steady state approach can also be used to predict the timescales for the formation of gas hydrate in natural systems in the absence of sedimentation or erosion, sea level change, and bottom water temperature variations (due to climate change and certain oceanographic phenomena). The timescale and distribution of gas hydrate formation will differ depending on the location of the primary methane supply. For methane originating (due to biogenic processes) within the hydrate stability zone, the timescale for gas hydrate accumulation can be estimated if Qm is known. When methane (either thermogenic or biogenic) enters the MHSZ from below, the timescale for the accumulation of gas hydrate in the sediments depends primarily on the relationship of the mass fraction of methane M beneath the bottom of the MHSZ to methane solubility at the base of the MHSZ Msl(Zsb ). In the following analysis, we focus on systems in which methane is supplied from below the MHZ and assume that Sh is small, meaning that changes in porosity, permeability k, and thermal conductivity, accompanying formation of gas hydrate can be ignored. Accounting for variations in these physical parameters will require numerical, instead of analytical, modeling of the system. Three relationships between the mass fraction of methane M and the methane solubility at the base of the MHSZ are possible in natural systems: 1. For M<M,.i(z.,.o), the base of the MHZ is shallower than the base of the MHSZ, as illustrated in Figure 1, and a free gas zone is either completely absent or exists well below the base of the MHSZ. 2. For M=M.,.i(zsb), the base of the MHZ coincides with the base of the MHSZ. Free gas exists just below the MHSZ. 3. For M>M,.t(z.,.b), the base of the MHZ coincides with the base of the MHSZ, and large amounts of free gas are present beneath the MHSZ. For cases 1 and 2 the diffusive timescale, which represents the amount of time for some gas hydrate to be present everywhere within the MHZ due to diffusivedominated methane transport, is given by : --, (23) Km where AZl = Zlt- Zlb denotes the thickness of the MHZ. The As shown in Figure 6b, a low flux rate (-0.2 mm yr - ) advective system would produce a 200-m-thick MHZ in -0.5 m.y., the same time required to form this MHZ in a diffusiondominated system with D,,of---1.3x10-9 m 2 s '. Similarly, a diffusion-dominated system would require a Dm value of-10 -s m 2 s - to generate the same 200-m-thick MHZ that an advective-dominated system with flux of 1 mm yr - could produce in -0.1 m.y. The dependence of (23) and (24) on only assumed constants (dispersion-diffusion coefficient Dm or fluid flux rate q ) and observed thickness of the MHZ means that this formulation can be applied to natural systems to loosely constrain the minimum ages of gas hydrate deposits. For case 3 (M>M.,. (z.,h)) the timescale governing gas hydrate formation in the diffusion-dominated system is the same as given in (23) above. The major difference between case 3 and cases 1 and 2 is therefore in the nature of the advection-dominated system. As methane (free gas) is transported upward across the base of the MHSZ for case 3, the rapid phase transition from free gas to gas hydrate releases latent heat, and the volume change associated with the phase transition produces a change in the pressure of the system. If heat transport and pressure diffusion are efficient enough to quickly dissipate the latent heat and pressure change associated with the phase change, then gas hydrate will accumulate rapidly near the bottom of the MHZ/MHSZ, eventually clogging the void space in a thin layer of sediment just above the base of the MHSZ. The cessation of advective methane transport across the clogged layer means that further accumulation of gas hydrate within the MHZ must be diffusion dominated, unless the thin clogged layer can be breached and the free gas advection restored. Similar inferences have been drawn from the theoretical analysis of the water-ice transition in permeable media [Griffiths and Nilson, 1992] Accumulation and Distribution of Gas Hydrate in Natural Systems The expressions given in (23) and (24) provide only a rough measure of the timescale for minimal accumulation of hydrate throughout the MHZ. A more useful parameter is the time necessary for methane hydrate to fill porosity ½ to any average saturation Sh:

11 XU AND RUPPEL: GAS HYDRATE FORMATION IN MARINE SEDIMENTS 5091 Diffusive Timescale Advective vs. Diffusive Timescales 500 Diffusion-dispersion coefficient Dm( m S 10-7 Diffusion-dispersion coefficient D m 10-0 ] ]0-7 ] I I ß 400. ', I I K m (kg m I I 0.0 ' I, I I Flux rate (mm yr - ) Figure 6. The timescales for some gas hydrate to be present everywhere within a 200-m-thick MHZ for (a) diffusion and (b) advection-dominated systems, based on the timescales given in equations (23) and (24). (a) For values of D,,, less than m 2 S-I, the timescale for the formation of a 200-m-thick MHZ is much greater than 1 m.y. In practice, such a thick MHZ could rarely develop in diffusive end-member systems. (b) For Dm values less than m 2 S '1, advection is generally more efficient than diffusion at producin gas hydrate everywhere within a 200-m-thick zone. The diffusion curve plotted here is an enlarged view of the results in Figure 6a and shows the characteristic time as a function of Din, which is read off the top axis. The advection curve shows the characteristic time as a function of fluid flux rate expressed in units of mm yr -1 and read off the bottom axis. See text for further discussion. ( S h - T'= ghk-- ) ß (25) 3Sh/3t denotes the methane hydrate accumulation rate for the whole MHZ, which is found from (5): Sh t,tp ( p h M h _ p (26) 1, M s l )(firm d2 d z Msl 2 q.t dmsl d z 1 ' The appendix details the steps required to determine the first and second derivatives of solubility for application of this expression in the evaluation of real hydrate systems. The expression given in (26), which is similar to that obtained by Rempel and Buffett [1997] for advection-dominated methane transport, can be used to predict the distribution of methane hydrate in natural end-member systems, subject to simple assumptions. Equation (26) shows that the accumulation rate is independent of methane flux qm. However, the magnitude of qm determines whether gas hydrate accumulation will occur at all, an effect that can be seen for the fully diffusiveadvective systems in Figures 3, 4, and 5. Figure 7 shows the rate of gas hydrate accumulation as a function of depth in end-member diffusion (q =0 in 26) and advection (Km=Dm=O in 26) systems characterized by the sam energy flux (40 mw m -2) and homogeneous physical properties (e.g., thermal conductivity, sediment porosity, sediment permeability). The results predict important differences between the distribution of gas hydrate in diffusion- and advection-dominated gas hydrate systems. Diffusion-dominated hydrate accumulation with Dm less than -lx10-8 m 2 s -1 is characterized by the formation of gas hydrate distributed uniformly at great depths within the MHSZ. Even for very high Dm values, diffusion-dominated systems will still produce only a thin gas hydrate layer. Thus, although Figure 6b demonstrates that the time needed to form a 200-mthick MHZ is much less than 1 m.y. for high Dm values, Figure 7 shows that diffusion-dominated systems normally cannot produce such a thick MHZ. Note that diffusion-endmember systems characterized by large Dm only produce a nonzero thickness MHZ when the rate of methane supply is sufficiently rapid, in this case 10-8 kg m -2 s -1, which is several orders of magnitude larger than the methane flux values adopted for the calculations shown in Figures 3, 4, and 5. Using a more realistic value of qm ( 10-1ø kg m -2 s -1) produces the same accumulation rate and the same thickness MHZ as for the higher qm value but only for Dm of m 2 s -1 and smaller.

12 5092 XU AND RUPPEL: GAS HYDRATE FORMATION IN MARINE SEDIMENTS Diffusion- vs. Advection-Dominated Systems Diffusive End-member I \ dx e 500 I I I I I I I Accumulation Rate (10- s - ) s s o 10 Accumulation Rate (!0 -I4 s -1) Figure 7. (a) Accumulation rate (r?svot) as a function of depth for end-member diffusion and advection-dominated gas hydrate systems, assuming constant total energy flux q of 40 mw m -2. For the diffusion-dominated end-member (solid curves), fluid flux qt is taken as zero. The group of curves labeled 1 shows accumulation rates for the diffusive end-member for D,,, values of to l0 -ll m 2 S -1. Curve 2 corresponds to Dm of 1.3x10-8 m 2 S -1 and curve 3 represents the results accumulation for D,, rate of 1.3x10 of v x m 10 2 s -15' -I s -1. The A results very high for Dm rate of 1.3x10 methane - m flux 2 s -I qm lie off (10 the -8 kg graph, m -2 s- a[)an is adopted here because smaller values of q,,, do not produce an MHZ for the high Dm diffusive end-member case. For the advection-dominated end-member (dotted curves and overlying dashes on curves a-e), the diffusion-dispersion coefficient (D,1,) is set to 0. Curves a-e correspond to fluid flux rates of 0.5, 0.75, 1.5, 3, and 6 mm yr-, respectively. Dotted curves show the accumulation rate in the presence of high methane supply rates (qm=10-8 kg m -2 s-l). Overlying crosses on curves a-d indicate the accumulation rates as a function of depth for lower, and more realistic, methane supply rates (qm = 10 kg m -2 s-j). See text for further discussion. (b) Enlarged view of the gas hydrate accumulation rate plotted in linear-log space for the diffusive end-member cases. This plot shows the same results as those in Figure 7a, and the labels on the curves denote Dm values. The results in Figure 7 also indicate that end-member Blake Ridge, which is proximal to a passive margin and advection systems produce a thicker layer of gas hydrate than which lacks large fault systems that focus flow over diffusion-dominated systems. For the system modeled here, significant length scales, the low average concentration of gas values of q.r greater than -1.5 mm yr - lead to higher hydrate within the relatively homogeneous sediments concentrations of gas hydrate near the base of the MHZ than [Holbrook et al., 1996; Dickens et al., 1997] may imply either shallower in the sedimentary column. Thus both the thickness low advection rates or diffusion-dominated flow. However, and degree of uniformity of gas hydrate deposits may be used as rough indicators of the degree to which advection dominates in natural systems. The results on the distribution of gas hydrates in advective and diffusive systems have important implications for the the thickness of the gas hydrate zone (-240 m [Paull et al., 1996]) probably implies advection-dominatedeposition since even a very high methane supply rate would only produce a relatively thin gas hydrate zone for the diffusive end-member case. Thus the Blake Ridge and other passive interpretation of natural gas hydrate provinces. Most margin settings may represent not the diffusion-dominated critically, regardless of tectonic setting (e.g., passive margin, active margin), Figure 7a implies that nearly every province where significant gas hydrate deposits occur is likely to be dominated by advective transport of methane-laden fluid. It is often assumed that passive margin sediments might be end-member, but rather the low flux advective end-member. Active margins, which are usually characterized by higher fluid flux rates and higher energy fluxes (heat flow), are more difficult to evaluate using this simple model. As shown earlier for a combined diffusive-advective system (Figure 3), loosely associated with diffusion-dominated mass transport, the thickness of the MHZ depends on a combination of fluid while the hydrologic regime on active margins may produce flux, methane flux, energy flux, and water depth. Thus MHZ advection-dominated gas hydrate systems. In settings like the thickness alone may not be entirely diagnostic of the nature of

13 XU AND RUPPEL: GAS HYDRATE FORMATION IN MARINE SEDIMENTS 5093 the flux system. However, when the diffusive and advective effects are separated, as they are in Figure 7, it is clear that the thickness of the gas hydrate zone directly reflects the dominance of the advective component and that large rates of gas hydrate accumulation over a relatively thick zone can only be consistent with advective deposition. As an example, the northern Cascadia margin is characterized by high concentrations of gas hydrate in an -125-m-thick layer [Hyndman et al., 1999]. While the thickness of the gas hydrate layer may not be strictly diagnostic of the dominant flux mechanism, the high inferred concentrations of gas hydrate (25-30% of pore space) imply that this margin represents a higher flux advective setting. 5. Conclusions BSR in a gas hydrate province might imply the absence of free gas, an observation with important implications for the mass fraction of methane present in the system below the base of the MHSZ. 4. If the BSR represents the base of the methane hydratebearing layer (MHZ), then the BSR may occur wholly within the MHSZ since the base of the MHZ will lie within the MHSZ for methane flux rates that are less than a critical value (conclusion 2 above). The BSR could therefore occur at pressure and temperature conditions lower than those at the base of the MHSZ. 5. Failure to account for the flux of fluid components in the analysis of dynamic, but steady state, gas hydrate systems can lead to poor predictions about the depth to the base of the MHSZ. Thus extrapolation of near-surface heat flow data to the depth of the BSR and the adoption of the assumption that the BSR represents the base of the MHSZ may lead to an inaccurate parameterization of a given gas hydrate province. 6. If the thickness of the MHZ can be constrained using geochemical measures (e.g., chloride anomalies [Hesse and Harrison, 1981]), downhole logs, or geophysical methods, The results of this steady state, coupled conservation then the approximate timescale necessary for some gas formulation have important implications for the interpretation hydrate to be present everywhere within the MHZ can be and analysis of natural marine hydrate systems. determined for end-member diffusion- and advection- 1. Disregarding the fact that gas hydrate cannot occur dominated systems, particularly if advective fluid flux rates within the sulfate reduction zone, there are clear physical can be bracketed using pore water geochemistry or heat flow explanations for the observation that the MHSZ typically data. Such a calculation provides constraints on the age of a extends above the seafloor, although gas hydrate is commonly gas hydrate deposit. Diffusion-dominated systems absent from the upper tens to hundreds of meters of sediment. characterized by large diffusion-dispersion coefficients Gas hydrate can only accumulate at shallow depths in the (>-10-9 m 2 S '1) could theoretically produce an MHZ of the sedimentary section if the mass fraction of methane remains same thickness over the same timescale as advective systems in excess of local methane solubility as the gas-laden liquid with slow fluid flux rates. However, because diffusive endascends through the shallow sediments. The position of the member systems (no advective fluid flux) only form gas top of the MHZ is largely controlled by the boundary hydrate in thin zones, even in the presence of very large condition at the seafloor, namely, the mass fraction of methane flux, it is unlikely that thick gas hydrate deposits methane in bottom water. For gas hydrate to be stable at the ever occur in diffusive end-member systems. seafloor requires a locally high mass fraction of methane in 7. Several characteristics of the distribution of gas hydrate bottom water and high methane and fluid flux rates. in homogeneous natural systems can be used to distinguish 2. The analysis reveals an obvious explanation for the whether the system is to first-order diffusion or advection disparity in the depths to the base of the MHZ (actual gas dominated. The diffusive end-member system is characterized hydrate occurrence), the base of the MHSZ, and the top of the by a thin layer of gas hydrate that generally accumulates at a free gas zone. For a given total mass flux qf these interfaces slow rate and in a relatively uniform fashion deep within the will only coincide when the rate of methane mass flux below MHSZ. The advective end-member system produces a thick the MHSZ exceeds a critical value qm. In natural gas hydrate zone of gas hydrate that usually accumulates rapidly, even for provinces in which the base of the MHZ and the base of the relatively low fluid flux rates. For the standard system MHSZ nearly coincide, this critical qm value provides an modelled here fluid flux rates faster than -1.5 mm yr -1 tend to important constraint on the minimum rate of methane produce higher concentrations of gas hydrate near the base of production as a result of either biogenic or thermogenic the MHZ than at shallower depths. The observation of high processes. For systems in which microbial activity below the concentrations or large thicknesses of gas hydrates in a base of the stability zone is the primary source of methane, natural system may imply advection-dominatedeposition, our analytical framework provides a first-order link between although high concentrations of gas hydrate can also be critical rates of microbial methanogenesis and the direct produced by lengthy accumulation of gas hydrates in physical and chemical observations that constrain the diffusion-dominated systems with high diffusion-dispersion thickness of the MHZ. coefficients. In practice, observations at both passive and 3. The nearly constant methane solubility below the active margins probably are most consistent with the MHSZ means that for certain methane flux rates a free gas interpretation of these systems as advection dominated. zone could develop far below the base of the MHSZ and be Passive margins likely represent not the diffusion-dominated separated from the base of the MHZ by a zone of sediment end-member for gas hydrate systems, but rather the low flux containing neither free gas nor gas hydrate. In such cases, advective end-member that can be directly contrasted with the any seismic impedance contrast associated with the top of the active margin high flux advective end-member. free gas zone (e.g., the BSR?) should occur tens to hundreds of meters below the MHZ. In this scenario the absence of a Appendix Derivation of the equation for the accumulation rate of gas hydrate (26) requires repeated application of the chain rule to determine the derivatives of solubility:

14 5094 XU AND RUPPEL: GAS HYDRATE FORMATION IN MARINE SEDIMENTS dmsl dz d 2 Ms! dz 2 (A1) Watkins for constructive reviews. C. R. thanks B. Buffett, B. Clennell, B. Tohidi, J. Dickens, and T. Collett for advance copies of their work; P. Wallace, I. Pecher, B. Clennell, G. Dickens, W. S. Holbrook, and W. Wood for discussions and comments; and J. Nimblett for technical assistance. This research was partially supported by the following grants to C.R.: JOI-USSSP F and F for ODP Leg 164 and ODP Leg 170 postcruise research and NSF OCE The pressure and temperature gradients and their derivatives are calculated from (17) and (18) as Notation d2p =0 dz 2 dt qe - q f CeT = - (A2) dz 2. d2t q f Cr dt dz 2 2, dz' g gravitational acceleration (m s-2). p pressure (Pa). P0 pressure at seafloor (Pa). z depth below seafloor, positive up (m)., t depth to bottom and top of MHZ (m). t.,., t.,. depth to bottom and top of the MHSZ (m). T temperature (øc). To temperature at seafloor (øc). u sedimentation rate (m s- ). sediment porosity (dimensionless) Dm diffusion-dispersion coefficient (m 2 s- ). M methane mass fraction (kg kg- ). M0 methane mass fraction in seawater at seafloor (kg kg- ). M.t solubility of methane gas (kg kg- ). q total mass flux rate (kg m -2 s- ). qm methane flux rate (kg m '2 s- ). q energy flux rate (W m-2). Qm rate of methane production in sediments (kg m -3 s-i). Qe rate of heat production in sediments (W m-3). k bulk permeability (m2). H bulk enthalpy (J kg- ). X bulk thermal conductivity (W m-lk- ). p bulk density (kg m-3). Variables Related to Phases in the Three-Phase System subscript denoting liquid phase. g subscript denoting methane free gas phase. h subscript denoting gas hydrate phase. Pi density of phase i (kg m'3). kri relative permeability function for phase i (dimensioness). C, specific heat capacity of phase i (J kg - K-I). gi dynamic viscosity of phase i in fluid (kg m -1 s-l). 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