International Conference on Gas Hydrates (ICGH) (6th : 2008)

RELATING GAS HYDRATE SATURATION TO DEPTH OF SULFATE-METHANE TRANSITION Bhatnagar, Gaurav; Chapman, Walter G.; Hirasaki, George J.; Dickens, Gerald R.; Dugan, Brandon 2008-07-31

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   RELATING GAS HYDRATE SATURATION TO DEPTH OF SULFATE-METHANE TRANSITION   Gaurav Bhatnagar, Walter G. Chapman, George J. Hirasaki?   Department of Chemical & Biomolecular Engineering Rice University 6100 Main St., Houston, TX, 77005 USA  Gerald R. Dickens, Brandon Dugan Department of Earth Science Rice University 6100 Main St., Houston, TX, 77005 USA   ABSTRACT Gas hydrate  can precipitate in pore space  of  marine  sediment  when  gas concentrations exceed solubility  conditions  within  a  gas  hydrate  stability  zone  (GHSZ).  Here  we  present  analytical expressions that relate the top of the GHSZ and the amount of gas hydrate within the GHSZ to the depth of the sulfate-methane transition (SMT). The expressions are strictly valid for steady-state systems in which (1) all gas is methane, (2) all methane enters the GHSZ from the base, and (3) no  methane  escapes  the  top  through  seafloor  venting.  These  constraints  mean  that  anaerobic oxidation of methane (AOM) is the only sink of gas, allowing a direct coupling of SMT depth to net methane flux. We also show that a basic gas hydrate saturation profile can be determined from the SMT depth via analytical expressions if site-specific parameters such as sedimentation rate, methane  solubility  and  porosity  are  known.  We  evaluate  our  analytical  model  at  gas  hydrate bearing  sites  along  the  Cascadia  margin  where  methane  is  mostly  sourced  from  depth.  The analytical expressions provide a fast and convenient method to calculate gas hydrate saturation for a given geologic setting.  Keywords: gas hydrate, sulfate-methane transition, modeling, Cascadia Margin                                                      ?  Corresponding author: Phone: +1 713 348 5416 Fax +1 713 348 5478 E-mail: NOMENCLATURE jic     Mass fraction of component i in phase j ,lm eqbc Methane solubility at the base of GHSZ iD     Diffusivity of component i if     Normalized flux of component i iF     Mass flux of component i g      Function denoting integral of porosity term sL     SMT depth below seafloor tL     Depth of GHSZ below seafloor hL     Thickness of gas hydrate layer iM    Molecular weight of component i 1 2,Pe Pe  Peclet numbers for the two fluxes Q     Modified sum of Peclet numbers 1 2,r r   Fitting parameters for solubility curve hS      Gas hydrate saturation ,sedU Fluid flux due to sedimentation Proceedings of the 6th International Conference on Gas Hydrates (ICGH 2008), Vancouver, British Columbia, CANADA, July 6-10, 2008.  ,extU  Fluid flux due to external source ,totU  Net fluid flux sU      Sediment flux z        Depth below the seafloor j?      Density of phase j ?        Porosity ? ?       Porosity at great depths  Subscripts/superscripts/overscript: 0        Value at seafloor g        Gas phase h        Hydrate phase l ,  w   Water phase or component m       Methane component s        Sulfate component ~       Denotes normalized or dimensionless value  INTRODUCTION  Clathrate  hydrates  of  gas,  often  called  gas hydrates,  form  in  pore  space  of  marine  sediment along  continental  margins  [1,2].  Their  stability depends on temperature, pressure, salinity and gas composition.  In  general,  these  conditions  restrict gas hydrate occurrence to a finite region below the seafloor,  usually  referred  to  as  the  gas  hydrate stability  zone  (GHSZ)  [3].  However,  the  amount of  gas  hydrate  present  within  this  region  (?gas hydrate  saturation?)  can  vary  considerably,  both globally and locally, because it relates to dynamic inputs  and  outputs  of  gas,  principally  methane, over long (>105 yr) timescales [4,5].  The presence of gas hydrates in marine sediments implies  high  methane  concentrations  in  pore waters  at  shallow  sub-bottom  depths,  and  a significant  methane  flux  towards  the  seafloor (Figure  1).  This  upward  methane  flux  consumes dissolved sulfate, so that most, if not all, seafloor settings  with  gas  hydrate  exhibit  a  relatively shallow  and  sharp  sulfate-methane  transition (SMT),  a  depth  interval  where  pore  water  SO42- and CH4 concentrations approach zero [6,7,8]. For reasons  of  mass  balance,  the  depth  of  the  SMT should relate to the  uppermost  occurrence  of gas hydrate (Figure 1) [6].  We have developed a numerical model [9] that has been  revised  to  incorporate  a  dynamic  SMT  for systems  where  methane  is  supplied  from  depth [10].  The  present  analysis  differs  from  previous modeling  efforts  by  developing  a  complete analytical theory for relating gas hydrate saturation to  the  depth  of  the  SMT.  This  approach  for quantifying  gas  hydrate  abundance  is advantageous because it requires  only pore  water data from shallow piston cores.   Figure  1:  (a)  Schematic  representation  of  a  gas hydrate  system  showing  pore  water  sulfate  and methane concentrations go to zero at some shallow depth below the seafloor. Also shown are different system depths and parameters. (b) Close-up of the sulfate-methane transition (SMT) showing overlap of sulfate and methane profiles.  GAS  HYDRATE  SYSTEMS  AND  SULFATE DEPLETION  In  anoxic  marine  sediments,  depletion  of  pore water  sulfate  occurs through  two  main  reactions. First,  sulfate  gets  reduced  when  bacteria  utilize solid organic carbon molecules as a substrate [11]. The  other  important  reaction  is  anaerobic oxidation of methane (AOM), where communities of bacteria and archaea use dissolved methane as follows [12,13,14]:  24 4 3 2CH SO HCO HS H O? ? ?? ? ? ?         (1)  The presence of gas hydrates in shallow sediments implies  a  significant  methane  flux  towards  the seafloor,  which  can  make  the  second  route  for sulfate depletion significant [12,15,16,17]. In fact, in regions with even modest upward methane flux, such  as  Blake  Ridge,  sulfate  reduction  can  be dominated  by  AOM.  This  inference  can  be supported  through  careful  modeling  studies  or basic  observations [6,15,16,17]. For  example, the pore  water  sulfate  gradient  can  vary  by  large amounts  across  regions  with  methane  (e.g.,  a factor  of  16  in  the  Carolina  Rise-Blake  Ridge region)  despite  similar  sedimentation  rates  and total  organic  carbon  (TOC)  supply  [6,12].  We assume that  AOM is the  only sulfate sink in our model.  This  is  possible  because,  as  mentioned before, we focus on gas hydrate settings where the TOC  content  of  sediment  is  low,  and  where  all methane is supplied by deeper sources.  MATHEMATICAL MODEL  Model framework  We first derive a relationship between the depth of the  SMT  and  the  upward  methane  flux  using  a steady-state  mass  balance  equation  for  sulfate. This is followed by writing a two-phase methane balance for the system, which links the thickness of the gas hydrate layer and gas hydrate saturation to  the  methane  flux.  Finally,  by  relating  the methane flux to sulfate flux at the SMT, we show how the SMT depth is related to thickness of the gas hydrate layer and its saturation.  The  upward  methane  flux  to  shallow  sediment depends  on  the  net  fluid  flux  and  the  methane concentration  of  rising  pore  waters.  In  our modeling,  we  assume  that,  at  steady-state conditions,  gas  hydrate  extends  to  the  base  of GHSZ  because  of  continuous  sedimentation. Consequently, pore fluid methane concentration at this depth  equals the peak solubility  value  of the point of three-phase equilibrium [9].  Sulfate mass balance  Two  assumptions  are  made  in  formulating  the sulfate  mass  balance:  (1)  no  sulfate  depletion occurs within the sulfate reduction zone (SRZ) due to reduction by solid organic carbon; and, (2) both methane  and  sulfate  react  fast  enough  within  the SMT so that their concentrations drop to zero at a single  depth.  Geochemical  data  from  several  gas hydrate  settings  [6,7,8]  indicates  that  sulfate  and methane  can  co-occur  across  a  horizon  several meters  thick,  suggesting that  the  species  actually react  over  a  finite  depth  instead  of  a  sharp interface. However, we later normalize all depths by  the  depth  to  the  base  of  the  GHSZ,  which causes the finite SMT transition zone to approach a  relatively  sharp  interface  in  the  dimensionless form.  The steady-state sulfate mass balance is:  0ll sf f s f scU c Dz z? ??? ?? ?? ?? ?? ? , 0 sz L? ?      (2)  The vertical depth  z  is set to zero at the seafloor and  is  positive  downwards,  following  previous work [9]. The mass balance (equation (2)) implies that  the  mass  flux  of  sulfate,4SOF ,  remains constant within the SRZ, and can be rewritten as:  4ll sf f s f s SOcU c D Fz? ???? ?? , 0 sz L? ?         (3)  We recast this relation in dimensionless form. The vertical  depth  is  normalized  by  t   ( / t ). Consequently, the SMT depth  s  is also written in scaled  form  as  /s s t? ,  while  sulfate concentration  is  scaled  by 40SOc ,  its  value  in standard  seawater  (40/s s SOc c c?? ).  The  net  fluid flux  ( ,f tot )  can  be  written  as  the  sum  of  two components:  ,f sed   due  to  sedimentation-compaction  and  ,f ext   due  to  upward  external flow  (Appendix  A1).  This  enables  definition  of two Peclet numbers that compare each fluid flux to methane diffusion, as follows:  ,1f sed tmU LPeD? , ,2f ext tmU LPeD?         (4)  The  sulfate  balance  (equation  (3))  can  now  be rewritten in dimensionless form as (Appendix A1):  441 201 1( )11ll s ssmSO tf SO mD cPe Pe cD zF Lc D? ??? ?? ??? ? ?? ? ?? ? ?? ? ? ?? ?? ??? ?? ???    ,0 s?         (5)  where  ? ?   is  the  reduced  porosity ? ? ? ?? ?   and  ?   is  ? ? /? ? .  The porosity  model,  assuming  hydrostatic  pore pressure  and  equilibrium  compaction,  and  details of  non-dimensionalization  are  given  in  Appendix A1.  To  simplify  the  notation  we  define  the following groups:  1 21 ( )Pe Pe Q?? ? ? ?? ?? ?                                     (6) 444011SO tSOf SO mF L fc D?? ? ?? ??? ?                               (7)  where  Q denotes the modified net fluid flux and 4SOf   is  a  modified  sulfate  flux.  Using  these definitions, equation (5) can be written as:  41 ll ss s SOcQc D fz???? ? ?? ?? ? ?? ?? ????          (8)  where  s? = /s m . The first boundary condition (B.C.)  is  applied  at  the  seafloor  where  the normalized sulfate concentration is equal to unity, while the second is applied at the base of the SRZ (i.e.,  the  SMT),  where  normalized  methane  and sulfate concentrations are zero:  B.C.:  1ls ?   at     0,   and        (9) B.C.:  0ls ?   at     s?                              (10)  With these B.C.s, equation (8) can be integrated to give the steady-state sulfate profile [18]:  1 exp ( ) ( )( )1 exp (0) ( )sssssQ g z g LDc zQ g g LD? ?? ?? ?? ?? ?? ?? ? ?? ?? ?? ?? ????? ???,       0 s?    (11) where  ( )   is  a  function  obtained  from  the integral of the porosity term and is given as:  2 ln( (1 ) (1 ) )( ) 1zz eg z ? ? ? ? ??? ? ? ?????       (12)  The  following  expression  for  sulfate  flux  (4SOf ) can  also  be  derived  as  a  function  of  s?   and  Q using equations (8) and (11):  41 exp (0) ( )SOssQfQ g g LD? ? ?? ?? ?? ?? ???      (13)  Relationship between sulfate and methane flux  We  now  rewrite  the  sulfate  flux  in  terms  of  the methane flux from depth. At the base of the SRZ, the molar fluxes of methane and sulfate are equal, due to the 1:1 stoichiometry of the AOM reaction (equation (1)) [6,12]. Thus, the sulfate  mass  flux (4SOF )  can  be  written  in  terms  of  the  methane mass flux (4CHF ) from below as follows:  44 44SOSO CHCHMF FM? ? ,      at     sz L???           (14)  Substituting equation (14) into equation (7) yields:  44 44 401 11SOtSO CHf SO m CHMLf Fc D M?? ?? ?? ??? ?        (15)  To  simplify  the  notation,  we  introduce  a dimensionless methane flux 4CHf :  4 4,1 11tCH CHf m eqb mLf Fc D?? ??? ??? ?        (16)  where  ,lm eqb  is the methane solubility at the base of  GHSZ.  Using  this  notation,  equation  (16)  is used to express the dimensionless methane flux in terms of the dimensionless sulfate flux:  44SOCHffm?? , where 44 4,0lSO m eqbCH SOM cmM c?       (17)  To summarize, using equations (13) and (17), we obtain  the  following  expression  between  s?   and 4CHf : 44/1 exp (0) ( )SOCHssf Q mfm Q g g LD? ?? ?? ?? ?? ?? ?? ?? ???          (18)  Methane mass balance  We  now  perform  similar  mass  balances  on methane and water, and apply them to two distinct spatial domains. The first domain extends from the SMT to the top of gas hydrate, whereas the second domain extends from the top of gas hydrate to the base of the GHSZ (Figure 1).   The two-phase (aqueous and hydrate) steady-state methane mass balance, valid from the SMT to the base of the GHSZ, is:  1 0(1 )l hf f m h m hlmh f mUU c S cz cS Dz? ? ??? ?? ??? ???? ? ??? ?? ?? ??? ?,  s t           (19)  The methane flux invariance can be restated as:  4(1 )1ll h mf f m h m h h f mCHU cU c S c S DzF? ? ? ? ?? ?? ??    ,    s t     (20)  To  non-dimensionalize  this  equation,  we  utilize the following scalings:  ,ssf sedUUU?? , ,ll mm lm eqbccc?? ,  ,hh mm lm eqbccc?? , hhf?????          (21)  Using the  water  mass balance, the  methane  mass balance  (equation  (20))  is  rewritten  in  the following form (see Appendix A2 for derivation):  ? ?? ? 41 1 111 1l h h lm h h m w mlmh CHPeUQc S c c ccS fz? ?? ?? ????? ?? ?? ?? ? ? ? ?? ? ?? ? ?? ? ?? ?? ?? ? ??? ??    ,  1s z? ?       (22)  Dissolved  methane  zone  (SMT  to  top  of  gas hydrate)  The methane mass balance (equation (22)) is first applied to the region extending from the SMT to the  top  of  the  hydrate  layer  (Figure  1).  In normalized form, the thickness of the gas hydrate layer becomes  /h h t? , while depth to the top of  hydrate  is  (1 h? ).  This  region ( 1s h? ? ) does not contain any hydrate, so that  equation  (22)  can  be  simplified  by  setting 0h ? :  41 ll mm CHcQc fz???? ? ?? ?? ? ?? ?? ??? ,  1s hL z L? ? ?? ??  (23)  Methane  concentration  is  zero  at  the  SMT  ( s? ), and is equal to the solubility curve at the top of the hydrate layer. Hence, the two boundary conditions for this equation are:  B.C.(1):  0lm ?  at     s?           (24) B.C.(2):  ? ?, 1hlm m sol Lc c ?? ?? ?  at     1 hz L? ???       (25)  where  , , ,lm sol m sol m eqb   is  the normalized  methane  solubility  (mass  fraction)  in pore  water  in  equilibrium  with  gas  hydrate  as  a function of the scaled depth ( z ). Analogous to the solution of the sulfate mass balance, equation (23) can  be  integrated  with  the  above  boundary conditions  to  give  the  following  expressions  for methane flux and concentration:  ? ?? ?4, 1|1 exp (1 ) ( )hm sol LCHh sQcfQ g L g L??? ?? ???? ?       (26)  ? ?? ?, 11 exp ( ) ( )( ) |(1 )1 exp( )hslm m sol LhsQ g z g Lc z cg LQg L?? ?? ?? ? ?? ?? ?? ? ?? ?? ?? ?? ?? ????? ? ??                      ,  1s h? ?     (27)  Gas hydrate zone (top of hydrate to base of GHSZ)  The  methane  mass  balance  equation  (22)  is  now applied to the region extending from the top of the gas  hydrate  layer  to  the  base  of  the  GHSZ (1 1h z? ? ).  In  this  region,  the  pore  water methane  concentration  is  constrained  by  the solubility  curve,  which  causes  gas  hydrate saturation  ( h )  to  be  the  primary  dependent variable,  which  gives  the  following  relation  in terms of gas hydrate saturation:  ? ? ? ?41,, ,1 1( )11( ) 1 ( )sm sol h hh hm w m sol h m solCHPeUQc z Sc c c z S c zf? ?? ?? ????? ??? ?? ? ? ? ?? ? ?? ? ?? ?? ??? ?? ???? ? ?? ?                        , 1 1h z? ?     (28)  Several  previous  simulation  results  have  shown that gas hydrate saturation monotonously increases from zero at the top of the gas hydrate layer to a maximum  value  at  the  base  of  the  GHSZ [9,10,18,19,20]. We use this observation to impose the  constraint  that  gas  hydrate  saturation  goes  to zero as the top of the hydrate layer is approached through this spatial domain. This condition can be written mathematically as:  0h ?  as   ? ?h ??       (29)  Substituting the above  condition in  equation (28) gives:  4, ,1( ) ( )m sol m sol CHQc z c z f???? ? ?? ?? ?? ??? ?? ?,               as  ? ?h ??       (30)  We now have three equations (18, 26, and 30) in terms of four unknowns ( s? , 4CHf ,  Q and  h? ). Hence,  by  using  s?   as  an  input,  the  other  three unknowns can be calculated.  Coupled equations for  s?  and  h?   In this section, we obtain two non-linear coupled equations in terms  of the three  variables  s? ,  h?  and  Q.  First,  we  eliminate 4CHf   between equations  (18)  and  (26),  which  amounts  to equating the sulfate flux to the methane flux from depth at the SMT:  ? ?? ?, 1|/1 exp (1 ) ( )1 exp (0) ( )hm sol Lh sssQcQ mQ Q g L g Lg g LD??? ? ? ?? ? ?? ?? ?? ?? ?? ?? ???? ???          (31)  Secondly,  we  equate  methane  flux  in  the  region containing dissolved methane to the methane flux in the region containing gas hydrate. This helps to eliminate 4CHf   between  equations  (26)  and (30), yielding:  ? ?? ?? ? ? ?, 1, ,1 1|1 exp (1 ) ( )1| |hh hm sol Lh sm sol m solL LQcQ g L g LQc c????? ??? ?? ? ?? ?? ? ??? ?? ??? ??? ??? ?      (32)  Once  s?  is known for a particular site, equations (31) and (32) can be solved iteratively (e.g., using a Newton-Raphson  or bisection algorithm) to get hL?  and  Q.  Gas hydrate saturation profile  A  major  advantage  of  our  formulation  is  that  it gives an analytical expression for the gas hydrate saturation  profile  (below  the  top  of  the  hydrate layer) through equation (28). This equation can be rearranged  to  give  the  saturation  profile  as  a function of scaled depth  z , as follows:  ? ?? ?4 ,,1,,( ) ( )11 ( )1( )CH m solm solhh hh m w m solm solf Qc z c zS z PeUc c c zc z???? ??? ?? ??? ? ??? ?? ??? ?? ?? ?? ? ??? ??? ??? ?? ?? ?? ? ? ??? ?? ? ? ??? ?                   , 1 1h z? ?       (33)  RESULTS  We  first  summarize  the  overall  calculation procedure to obtain the results: a.  Given  ? ?,m sol z   and  other  site-specific parameters  ( m ,  s? ,  ?   and  ? ),  solve  coupled equations (31) and (32) to obtain  Q and  h? . b. Using these values, calculate 4CHf  , from any of the three expressions, (18), (26) or (30). c. Substitute into equations (11) and (27) to get the sulfate  and  methane  concentration  profiles, respectively. d. By specifying the parameters  h ,  1 ,  hm  and hwc , equation (33) gives the gas  hydrate saturation profile  ? ?h z  within the GHSZ.  Normalized methane solubility curve  An important parameter in our formulation is the methane  solubility  curve  within  the  GHSZ.  For sake  of  demonstration  and  simplicity,  we approximate the solubility  curve  ? ?,m sol z  by an exponential function similar to the form proposed in  [21]. We  start  with  this  simple  two  parameter solubility function:  ? ? 2, 1 r zm sol z re ?? ?                              (34)  This  solubility  curve  is  scaled  by  methane solubility  at  the  base  of  the  GHSZ,  so  that  its normalized  value is  equal to unity at  1. This constraint  yields  the  following  relationship between  1  and  2 :  21 1rre ?   ?       21rr e??       (35)  which allows us to reduce equation (34) to a single parameter equation:  ? ? ? ?2 1, r zm sol z e? ? ?? ?         (36)  This  simple  equation  (36),  with  a  single  fitting parameter,  2 ,  yields  very  good  fits  to  solubility curves  (Figure  2)  obtained  through  rigorous thermodynamic  models  (e.g.,  [9]),  for  two different  seafloor  depths,  seawater  salinity, seafloor  temperature  of  3?C,  and  a  geothermal gradient of 0.04?C/m.   Figure  2:  Comparison  of  normalized  methane solubility  curves,  computed  from  rigorous thermodynamic  models  versus  those  obtained from equation (36). Two different seafloor depths are  considered,  with  the  corresponding  fitting parameters,  2 , listed in the inset.  Effect of   s?  on the gas hydrate system  We now explain the system in terms of the input parameter,  s? . The  following constant parameter values are assumed for all results shown later: ?  = 6/9,  ?  = 9 (which correspond to  0  = 0.7, ? ?  = 0.1),  hm  = 0.134,  h  = 0.9, 4CHM = 16, 4SOM  = 96, seawater sulfate concentration equals 28 mM, and  s?  = 0.64 [22].  Figure  3:  Effect  of  net  fluid  flux  and  variable SMT  depths  on  steady-state  sulfate  and  methane concentration  profiles.  Specifying  s?   uniquely constrains  the  sulfate  and  methane  concentration profiles, as well as the top of the gas hydrate layer. The  methane  solubility  curve  corresponds  to seafloor depth  of 1000m, seafloor temperature  of 3?C, and geotherm of 0.04?C/m (Figure 2).   Figure  4:  Effect  of  net  fluid  flux  and  variable SMT depths on steady-state gas hydrate saturation profiles. Shallow SMT depths indicate higher net methane  flux  from  depth  and  higher  gas  hydrate saturation within the GHSZ.  1  equals 0.1 for all three cases. Numerical simulation results (crosses) from  the  model  of  Bhatnagar  et  al.  [10]  match well  with  the  analytical  saturation  profiles (curves).  Figure  3  shows  steady-state  sulfate  and  methane concentration profiles, obtained through equations (11)  and  (27),  for  three  different  scaled  SMT depths.  The  solubility  curve  corresponding  to seafloor depth of 1000 mbsl ( 2  = 0.625) is used. Due to co-consumption of sulfate and methane at the SMT, shorter  s?  indicates higher methane flux from below, thereby leading to a shallower top of the gas hydrate layer (Figure 3).  Gas  hydrate  saturation  profiles  as  a  function  of scaled SMT depths show higher saturations within the  GHSZ  with  decreasing  s? ,  again  due  to  net increase in methane flux (Figure 4). Increase in the thickness of the hydrate layer with decreasing  s?  is  also  evident  from  the  saturation  profiles.  We further compare steady-state gas hydrate saturation profiles obtained from simulation results (crosses) of  Bhatnagar  et  al.  [10],  which  reveal  good agreement  between  the  theory  developed  in  this paper and the numerical formulation. The profiles in  Figures  3  and  4  clearly  highlight  that  each distinct value of  s?  results in a unique profile for dissolved  sulfate,  methane  and  gas  hydrate saturation.  Application to Cascadia Margin sites  The  Cascadia  Margin  is  an  accretionary  margin characterized by pervasive upward fluid flow with localized  gas  venting  [7,8].  Results  from  Ocean Drilling  Program  (ODP)  Leg  204  and  Integrated Ocean  Drilling  Program  (IODP)  Expedition  311 have  given  great  insight  into  the  complex  and heterogeneous  gas  hydrate  distribution  at  several sites  drilled  along  this  margin  [7,8,23].  Sites  in this  region  are  characterized  by  relatively  high fluid fluxes and low average total organic carbon (TOC) content [8,24], which is indicative of a gas hydrate setting  where  fluids from  depth  form the dominant methane source. This makes sites along Cascadia  Margin  a  good  location  to  test  our model.  We  use  SMT  depths  and  other  data  for three  Cascadia  Margin  sites  (Table  1)  to  predict gas  hydrate  saturations,  average  saturation  and depth to the first occurrence of gas hydrate below the  seafloor  (Table  2).  These  sites  include  ODP Site 889 and IODP Sites U1325 and U1326.  Site  889  (ODP  Leg  146)  has  been  previously modeled  as  a  gas  hydrate  system  dominated  by deeper methane sources [9,20]. Davie and Buffett [20] fit the pore water chloride profile at Site 889 using  a  coupled  numerical  model  with  methane supply  from  depth.  Their  results  indicate  peak hydrate  saturation  close  to  2%  at  the  base  of GHSZ  and  average  saturation  <1%  within  the GHSZ [20]. This result agrees favorably with our simulation  that  shows  peak  saturation  of  about 2.7% at the base of GHSZ (Figure 5) and average saturation of 0.6% across the entire GHSZ (Table 2).  Hyndman  et  al.  [25]  calculated  gas  hydrate saturation  between  25-30%  of  pore  space  in  the 100 m interval above the base of GHSZ at Site 889 using  resistivity  log  data.  However,  subsequent calculations  using  a  different  set  of  Archie parameters have revised this estimate to 5-10% in that  100  m  interval  [26].  Although  several parameter uncertainties confront such geochemical and  geophysical  estimates  [8,27],  average saturation  predicted  using  our  SMT  based  model concurs with the lower estimates at Site 889.    Figure  5:  Steady-state  gas  hydrate  saturation profiles  computed  from  scaled  SMT  depths  at Cascadia  Margin  Sites  889,  U1325  and  U1326. Scaled  SMT  depth  is  highest  for  Site  889  and lowest  for  Site  U1326,  implying  higher  methane flux  and  greater  gas  hydrate  saturation  at  Site U1326  and  relatively  low  methane  flux  and hydrate saturation at Site 889.   For the IODP Expedition 311 sites, drilled along the  northern  Cascadia  Margin,  we  compare  our predictions  with  average  saturations  computed from  chloride  anomalies  and  resistivity  log  data (Table  2).  Average  saturation  is  calculated  from chloride  data  by  assuming  a  background  in  situ chloride  profile  and  attributing  the  relative  pore water freshening to gas hydrate dissociation (e.g., [27]).  Average  saturation  is  obtained  from resistivity  data  using  the  Archie  equation  and parameters  given  in  [8].  Average  saturation  over the  GHSZ  at  Site  U1325  is  estimated  from resistivity data and chloride anomalies to be 3.7% and  5.3%,  respectively.  Corresponding  estimates from resistivity and chlorinity for Site U1326 are 6.7%  and  5.5%,  respectively.  These  values compare  favorably  with  3.1%  and  6.6%  average saturation  from  our  SMT  based  model  at  Sites U1325 and U1326, respectively (Table 2).  In  general,  we  get  good  first  order  agreement between  average  gas  hydrate  saturations  derived using resistivity logs/chloride anomalies and those predicted  using  our  model,  although  our  model consistently  predicts  lower  average  saturation  at all  three  sites  along  Cascadia  Margin.  For estimates  from  resistivity  logs,  a  possible explanation for the deviation is that interpretations of  resistivity  logs  depend  on  knowledge  of formation  water  resistivity  and  three  empirical constants, which are hard to constrain in clay-rich sediments.  Moreover,  most  studies  employing transport  models  (e.g.,  [9,10,20,28])  predict  gas hydrate to  first occur  well below the seafloor. In contrast,  log-based  results  often  predict  gas hydrate  starting  immediately  below  the  seafloor. This will cause saturations from transport models to be lower than those predicted  using resistivity log  data.  Similarly,  estimation  of  hydrate saturation from  chloride  data is quite sensitive to the choice of baseline curves. Apart from the small deviations between  model and  chloride/resistivity log  predictions,  our  model  gives  a  good  average estimate of gas hydrate saturation.  CONCLUSIONS  We  have  developed  analytical  expressions  to estimate gas hydrate saturation from scaled depth of  the  sulfate-methane  transition  (SMT)  for  gas hydrate  systems  dominated  by  deep-methane sources. This scaled SMT depth is the ratio of the dimensional depth of the SMT below the seafloor to  the  depth  of  the  gas  hydrate  stability  zone (GHSZ)  below  the  seafloor.  Using  simple  one-dimensional  mass  balances  for  sulfate  and methane,  we show that  net methane  flux in such deep-source  systems  uniquely  determines  the scaled SMT depth, the thickness of the gas hydrate layer and gas hydrate saturation within the GHSZ. Steady-state  results  show  that  as  the  SMT becomes  shallower,  methane  flux  and, consequently,  gas  hydrate  saturation  increases. Average  saturations  over  the  GHSZ  at  three Cascadia  Margin  locations,  calculated  from  our method, are 0.6%, 2.3% and 5.5%  for Sites 889, U1325  and  U1326,  respectively.  These  values compare  favorably  with  averages  computed  from resistivity  log  and  chlorinity  data  for  all  sites. Hence,  our  analytical  formulation  provides  a simple and fast technique to constrain gas hydrate saturation in deep-source systems.  ACKNOWLEDGEMENTS  We acknowledge financial support form the Shell Center for Sustainability, the Kobayashi Graduate Fellowship,  and  the  Department  of  Energy  (DE-FC26-06NT42960).  REFERENCES [1]  Kvenvolden  KA.  Gas  hydrates:  Geological perspective  and  global  change.  Reviews  of Geophysics 1993; 31:173-187. [2] Sloan ED, Clathrate hydrates of natural gases, Marcel Dekker, New York, 1998. [3] Dickens GR. The potential volume of oceanic methane  hydrates  with  variable  external conditions. Organic Geochemistry 2001; 32:1179-1193. [4]  Dickens  GR.  Rethinking  the  global  carbon cycle  with  a  large,  dynamic  and  microbially mediated  gas  hydrate  capacitor.  Earth  and Planetary Science Letters 2003; 213:169-183. [5]  Buffett  BA,  Archer  D.  Global  inventory  of methane  clathrate:  Sensitivity  to  changes  in  the deep  ocean.  Earth  and  Planetary  Science  Letters 2004; 227:185-199. [6] Borowski WS, Paull CK, Ussler WIII. Global and local variations of interstitial sulfate gradients in  deep-water,  continental  margin  sediments: Sensitivity  to  underlying  methane  and  gas hydrates. Marine Geology 1999; 159:131-154. [7]  Tr?hu  AM,  Bohrmann  G,  Rack  FR,  Torres ME,  et  al.  (Eds),  Proceedings  of  the  Ocean Drilling Program, Initial Reports 2003, vol. 204, Ocean Drilling Program, College Station, TX. [8]  Riedel  M,  Collett  TS,  Malone  MJ,  and  the Expedition  311  Scientists  (Eds).  Proceedings  of the Integrated Ocean Drilling Program 2006, vol. 311,  Integrated  Ocean  Drilling  Program Management International Inc., Washington, DC. [9]  Bhatnagar  G,  Chapman  WG,  Dickens  GR, Dugan  B,  Hirasaki  GJ.  Generalization  of  gas hydrate  distribution  and  saturation  in  marine sediments  by  scaling  of  thermodynamic  and transport processes. American Journal of Science 2007; 307:861-900. [10]  Bhatnagar  G,  Chapman  WG,  Dickens  GR, Dugan B, Hirasaki GJ. Sulfate-methane transition as a proxy for average methane hydrate saturation in  marine  sediments.  Geophysical  Research Letters  2008;  35:L03611, doi:10.1029/2007GL032500. [11] Berner RA. Early Diagenesis: A Theoretical Approach, Princeton Univ. Press, Princeton, N.J., 1980. [12] Borowski WS, Paull CK, Ussler WIII. Marine pore-water  sulfate  profiles  indicate  in  situ methane  flux  from  underlying  gas  hydrate. Geology 1996; 24(7):655-658. [13]  Reeburgh  WS.  Methane  consumption  in Cariaco  trench  waters  and  sediments.  Earth  and Planetary Science Letters 1976; 28:337-344. [14]  Orphan  VJ,  House  CH,  Hinrichs  K-U, McKeegan KD, DeLong EF. Methane-consuming archaea revealed by directly coupled isotopic and phylogenetic analysis. Science 2001; 293:484-487. [15] Davie MK, Buffett BA. A steady state model for  marine  hydrate  formation:  Constraints  on methane  supply  from  pore  water  sulfate  profiles. Journal of Geophysical Research 2003; 108:2495, doi:10.1029/2002JB002300. [16]  Luff  R,  Wallman  K.  Fluid  flow,  methane fluxes,  carbonate  precipitation  and biogeochemical  turnover  in  gas  hydrate-bearing sediments  at  Hydrate  Ridge,  Cascadia  Margin: Numerical  modeling  and  mass  balances. Geochimica  et  Cosmochimica  Acta  2003; 67:3403-3421. [17] Snyder GT, Hiruta A, Matsumoto R, Dickens GR,  Tomaru  H,  Takeuchi  R,  Komatsubara  J, Ishida  Y,  Yu  H.  Pore  water  profiles  and authigenic  mineralization  in  shallow  marine sediments  above  the  methane-charged  system  on Umitaka Spur, Japan Sea. Deep-Sea Res. II 2007; 54:1216-1239. [18] Bhatnagar G. Accumulation of gas hydrates in marine sediments, Ph.D. Thesis, Rice University, Houston, TX, February 2008. [19]  Davie  MK,  Buffett  BA.  A  numerical  model for  the  formation  of  gas  hydrate  below  the seafloor.  Journal  of  Geophysical  Research  2001; 106:497-514. [20] Davie MK, Buffett BA. Sources of methane for  marine  gas  hydrate:  inferences  from  a comparison  of  observations  and  numerical models. Earth and Planetary Science Letters 2003; 206:51-63. [21]  Davie  MK,  Zatsepina  OY,  Buffett  BA, Methane  solubility  in  marine  hydrate environments.  Marine  Geology  2004;  203:177-184. [22]  Iversen  N,  J?rgensen  BB.  Diffusion coefficients  of  sulfate  and  methane  in  marine sediments:  Influence  of  porosity.  Geochimica  et Cosmochimica Acta 1993; 57:571-578. [23]  Tr?hu  AM,  et  al.  Three-dimensional distribution  of  gas  hydrate  beneath  southern Hydrate Ridge: Constraints from ODP Leg 204 . Earth  and  Planetary  Science  Letters  2004; 222:845-862. [24] Westbrook  GK, Carson B,  Musgrave  RJ,  et al.  (Eds.),  Proceedings  of  the  Ocean  Drilling Program,  Initial  Reports  1994,  vol.  146  (Pt.  1), Ocean Drilling Program, College Statiom, TX. [25]  Hyndman,  RD,  Yuan  T,  Moran  K.  The concentration  of  deep  sea  gas  hydrates  from downhole electrical resistivity logs and laboratory data.  Earth  and  Planetary  Science  Letters  1999; 172(1):167-177. [26] Collett, TS. Quantitative well-log analysis of in-situ  natural  gas  hydrates,  Ph.D.  Thesis, Colorado School of Mines, Golden, CO, 2000. [27]  Egeberg  PK,  Dickens  GR.  Thermodynamic and  pore  water  halogen  constraints  on  hydrate distribution  at  ODP  Site  997  (Blake  Ridge). Chemical Geology 1999; 153:53-79. [28] Xu W, Ruppel C. Predicting the occurrence, distribution, and evolution of methane gas hydrate in  porous  marine  sediments.  Journal  of Geophysical Research 1999; 104:5081-5096.  APPENDIX  A1.  Non-dimensionalization  of  Sulfate  Mass Balance The  net  fluid  flux  in  the  system  ( ,f tot )  results from  the  combination  of  fluid  flux  due  to continuous  sedimentation  and  compaction  of sediments  ( ,f sed )  and  the  external  fluid  flux ( ,f ext ) [9,15,18]:  , , ,extU U U? ?          (A1)  In terms of Peclet numbers, equation (4), this sum can be written as:  , , ,1 2f tot t f sed t f ext tm m mU L U L U L Pe PeD D D? ? ? ?     (A2)  Multiplying equation (3) by  /t m  and dividing by 40SOc  gives:  ? ? 441 2 0lSOs s tsm f SO mFD c LPe Pe cD z c D? ??? ? ?????      (A3)  Porosity loss is modeled by relating it to effective stress  and  assuming  hydrostatic  pressure (equilibrium  compaction),  which  yields  the following  relationship  between  the  reduced porosity and normalized depth [9,18]:  ? ? z??? ?? ? ? ??        (A4)  where  ? ?  and ?  are reduced porosities defined in terms  of  the  maximum  ( 0 )  and  minimum  (? ? ) porosities achieved during compaction:  1? ?????????  ,  01? ????????           (A5)  Dividing equation (A3) by (1 ? ? ) we obtain the dimensionless  form  of  the  sulfate  mass  balance, which is equation (5) in the main text.  A2.  Non-dimensionalization  of  methane  mass balance The  steady-state  water  mass  balance  below  the SMT can be written as:  01l hsf f w h w hU S cz ? ? ??? ?? ? ?? ?? ?? ?, s t            (A6)  Equation (A6) can also be written in terms of the water flux (2H OF ) as:   ? ?2, ,1l hf f w h w h H Of sed f ext fUU c S c FU U? ? ???? ? ???     (A7)  Due  to  low  methane  solubility  in  water,  we assume  the  mass  fraction  of  water  in  aqueous phase to be unity. This gives us an expression for the water flux:  ? ?, , 1 hs hf f sed f ext h wfUU U U S c ??? ?? ? ? ?      (A8)  Substituting  this  expression  for  fluid  flux  into equation (20), we get:  ? ?? ? 4, , 111h lf sed f ext h w mflCHs h mh m h mf fUU U S c cFU cS c S Dz??? ??? ?? ? ?? ?? ? ?? ??? ?? ??? ? ?? ? (A9)   Similar  to  the  sulfate  mass  balance,  we  multiply the  above  equation  by  /t m   and  divide  by ,lm eqbc  to get the following dimensionless form:  ? ?? ? 41 11 2,1 11h l sh w h mlCHm th m h h lf m eqb mPeU PeUPe Pe S c cFc LS c Sz c D? ?? ?? ? ? ?? ?? ? ?? ?? ?? ??? ? ??? ??????                           (A10)  Equation  (A10)  can  be  divided  by  (1 ? ? )  to express  in  terms  of  the  reduced  porosity  and rearranged  to  get  the  dimensionless  methane balance, equation (22).             TABLES  Table 1: Site-specific parameters for Cascadia Margin sites   * Calculated from thermodynamic model [9] ** Calculated from fitting equation (36) to solubility curves obtained from thermodynamic model [9] a  ODP Leg 146 [24] b  IODP Expedition 311 [8] c   S?  was not available, hence assumed equal to nearest site U1325.  Table 2: Results for Cascadia Margin sites  Site  1   s?   t hL L?  (m)  h GHSZS  (calc.) h GHOZS  (calc.) h GHSZS  (res. log) a h GHSZS  (Cl-) b 889  0.068  0.058  113  0.6%  1.1%  -  <1% U1325  0.11  0.059  62  2.3%  3.1%  3.7%  5.3% U1326  0.11  0.060  36  5.5%  6.6%  6.7%  5.5%  a From Archie equation using LWD log data [8] b From fit to chloride data Site  S?  (cm/k.y.)  T0 (?C)  G (?C/m)  D0 (m)  sL  (m)  tL  (m)  ,lm eqb *  m  (eq. 17)  2r** (eq.36) 889a  25  3  0.054  1311  10  225  2.1?10-3  4.4  0.73 U1325b  38.3  3  0.06  2195  5  230  2.5?10-3  5.2  0.93 U1326b  38.3c  3  0.06  1828  2.5  126  2.3?10-3  4.8  0.86 


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