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Theoretical model for the formation and accumulation of marine gas hydrates in compacting sediments Kingdon, Kevin Andrew 1998

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A Theoretical Model for the Formation and Accumulation of Marine G c Hydrates in Compacting Sediments By Kevin Andrew Kingdon Honours B.Sc. (Space and Communication Sciences), York University  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS  FOR T H E D E G R E E OF  M A S T E R OF  SCIENCE  in T H E FACULTY OF G R A D U A T E STUDIES D E P A R T M E N T OF E A R T H A N D O C E A N  SCIENCES  We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH COLUMBIA  June 1998 © Kevin A . Kingdon, 1998  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Earth and Ocean Sciences The University of British Columbia 129-2219 Main Mall Vancouver, BC, Canada V6T 1Z4  Date:  June.  XS  7  \99<Z  11  Abstract Marine gas hydrates are commonly found along continental margins where a unique combination of stable pressure and temperature conditions and an adequate gas supply exist to generate the ice-like substance. The vast quantities of methane contained in the hydrate structure could significantly affect global climate if released. The abundant amount of gas contained within these structures could also represent a potential future energy resource, although the cumbersome recovery of gas trapped in the solid hydrate structure currently designate this process as economically unfeasible. The remote locations of gas hydrates has hampered studies of the natural setting and consequently many aspects of the physical processes that control hydrate formation in the marine environment remain unresolved. A quantitative model for the formation and accumulation of gas hydrates in compacting marine sediments is presented. Conservation principles are used to develop a mathematical model which is described by a set of governing equations that represent the physical processes.  The numerical method of lines is employed to obtain time varying solutions  for hydrate volume fraction, temperature, gas and salt. Jump boundary conditions are imposed at the base of the hydrate stability zone where there can be a discontinuity in hydrate volume fraction. The widely varying geographical and geological distribution of hydrates indicates that the amount of organic material and its deposition are the dominant factors controlling hydrate growth within hydrate stability zones. A variety of these settings can be achieved through the model presented here by considering different sedimentation rates and organic carbon contents as inputs to the calculations. The necessary levels of these quantities required for hydrate formation is investigated by examining a range of sedimentation rates and total organic carbon contents.  Typical values of continental margin organic carbon contents and sedimentation rates are sufficient for hydrate evolution in both active and passive margins. In both cases, the gas is supplied by in situ biogenic production. Specific settings are also modelled by inputing representative values for sedimentation rates and organic carbon contents into the computations. The formation rates for hydrate regions are believed to correspond to the length of time required to develop current hydrate volume fractions from biogenic gas production. The possibility of attaining steady state conditions, observed in results for typical continental margins, indicates that hydrates may have been in existence for longer time periods. Hydrates are absent in deep sea regions because of insufficient organic material and slow sedimentation rates. Increasing the total organic content over typical continental margin values magnifies the hydrate volume fraction since biogenic production augments the amount of gas available for hydrate formation by the extra available organic carbon. When the sedimentation rate is increased, the amount of hydrate produced is not increased significantly, but the time scales on which the hydrate develops are altered because of changes in the burial rate of organic material, and correspondingly, the rate at which gas is produced through the decay of buried organic material. Global distributions of total organic carbon are analyzed and results obtained from model calculations are used to predict where the conditions for hydrate formation are favourable.  IV  Contents Abstract  "  List of Figures  i  x  Acknowledgement  x  1  1  2  Introduction 1.1  History and Definitions  1  1.2  The Hydrate Environment  3  1.3  Observables  3  1.3.1  Seismic  4  1.3.2  SaHnity/Chlorinity  5  1.3.3  Other Measurements  6  1.4  Gas Supply  6  1.5  Structure of Thesis  9  Development of the Governing Equations  10  2.1  The Physical Environment  10  2.2  Definitions of Environment Variables  12  2.3  Conservation of Mass  13  CONTENTS 2.4  Conservation of Energy  14  2.5  Conservation of Gas  18  2.6  Conservation of Salt  19  2.7  Kinetic Effects and Hydrate Conservation  20  2.8  Governing Equations  21  2.8.1  Dimensionless Forms  22  2.8.2  Deriving Expressions for Fluid and Sediment Velocities  24  2.9 3  v  Summary  Numerical Solution  25 27  3.1  The Numerical Methods of Lines  27  3.2  Jump Boundary Conditions  29  3.3  Initial and Boundary Conditions  32  3.3.1  Gas Conditions  32  3.3.2  Salt Conditions  32  3.3.3  Temperature Conditions  33  3.3.4  Hydrate Conditions  33  Approximating Spatial Derivatives  34  3.4.1  36  3.4  Centered Differences and Upwinding  3.5  Choice of integrator and Stiffness  37  3.6  A Representative Example  38  CONTENTS 3.7  4  vi  Summary  43  Results 4.1  Sedimentation Rate and Biogenic Production  43  4.1.1  Obtaining Estimates of Biogenic Production Rates from TOC . . .  44  4.1.2  Organic Carbon Requirements for Hydrate Formation  46  4.1.3  Hydrate Volume Fraction Profiles for Ranges of TOC and Sedimentation Rate  4.1.4  4.1.5  47  Gas Concentration Profiles for Ranges of TOC and Sedimentation Rate  49  Salt Concentration Profiles for Ranges of TOC and Sedimentation Rate  5  41  51  4.2  Inferring Hydrate Volume Fractions from Salinity Profiles  54  4.3  Summary  57  Global Implications  58  5.1 Some Real World Scenarios to Model  58  5.2  Continental Margins  59  5.2.1  Implications for Active Continental Margins  65  5.2.2  Blake Ridge  67  5.3 Deep Sea  72  5.4  Global Hydrate Occurrences  73  5.5  Summary  76  CONTENTS 6  Conclusions  References A  Nomenclature and Notation  B  Resume - Abstract in French  C  Zusammenfassung - Abstract in German  viii  List of Figures 1.1  The physical environment for the pore fluid migration model of gas supply, after Hyndman and Davis, (1992)  8  2.1  A n illustration of B S R migration  11  3.1  The numerical method of lines applied to the hydrate environment  28  3.2  Fluxes across the depth of discontinuity in hydrate volume fraction  31  3.3  The effect of gas concentration boundary conditions on hydrate volume fraction  34  3.4  Alleviating numerical stabilities in the calculation of spatial derivatives. . .  37  3.5  Representative example: t = 125 ky  39  3.6  Representative example: t = 334 ky  40  3.7  Representative example: t = 700 ky  41  4.1  Effects of increasing total organic carbon content on the hydrate volume fraction  48  4.2  Effects of increasing total organic carbon content on the gas concentration  50  4.3  Effects of increasing total organic carbon content on the salt concentration  52  4.4  Investigating the hydrate volume fraction inferred from salinity measurements 56  5.1  Time evolution of fields for typical continental margin parameters  61  5.2  Gas fluxes out of the top and bottom of the computational domain  63  5.3  Time evolution of fields for Blake Ridge with TOC=1.5% and dS/dt = 22 cm/ky  66  LIST OF FIGURES  ix  5.4  Geographic location of the Blake Ridge, after Holbrook et al., (1996).  ...  5.5  Time evolution of fields for typical continental margin sedimentation rate, high TOC  5.6  67  69  Time evolution of fields for typical continental margin TOC, high sedimentation rate  70  5.7  Gas concentration profile for deep sea regions  72  5.8  Worldwide locations of known and inferred gas hydrate deposits, from Kvenvolden, (1994)  5.9  74  Organic carbon distribution in the sediments of world oceans and seas, from Premuzic et al, (1982)  75  X  Acknowledgements My graduate career was very fortunate to benefit from the supervision of Bruce Buffett. Any ideas which appear clever in this thesis were no doubt inspired by him. Every time that I entered his office in a state of frustration, I invariably left the room with a much more positive attitude; in part, because he would stimulate plenty of ideas to attempt to alleviate the problem, but also because of his uncanny ability to inspire the confidence required to undertake daunting tasks. There are not enough superlatives to describe my appreciation for his guidance and his friendship. There were plenty of others who created a salubrious environment within the geophysics group. Charly Bank was generous enough to take a stranger from Ontario into his apartment and has remained a friendly presence ever since that first day of my arrival in Vancouver. Gwenn Flowers has been a truly great friend, a colleague who can identify with some of the more inauspicious occurrences of the graduate student experience, and who's early work in gas hydrates was particularly inspirational. Mathieu Dumberry and Mike Caputi were the other two members of the triumvirate incoming class. It seems that the first people you get to know when starting in a new place are the other new people. I couldn't have asked for two better people than Mike and Mathieu. I thank Camille Li for providing the other half of the highest potential procrastinating office combination, it is scary to think what would have happened were we actually in the same office. Colin Farquharson always provided me with a soccer ball and an excuse to temporarily abandon my computer. Leonard Pasion and Sean Walker kept me connected with the important things in life such as television and professional wrestling. Stephane Rondenay was the ideal office mate and shared my visions for rock-stardom. A special thanks to Barry Zelt, the hardest working man in cribbage. Friends outside of Vancouver have unknowingly sent some of the most timely emails, often providing laughs with impeccable timing. Finally, I would like to thank my family for their unwaivering support even when they were not exactly sure what I was doing or why it was taking so long.  XI  You can take the boy out of Sarnia but you can't take the Sarnia out of the boy. [J. Finn, personal communication, 1998].  1  Chapter 1 Introduction The ubiquitous actors that seem to be something to everyone are gas hydrates: curious, ice-like solids made up of gas molecules—mostly methane—caged in a crystal lattice of water [Appenzeller, 1991]  1.1  History and Definitions  Gas hydrates are composed of a mixture of water and gas wherein the host water molecule traps the guest gas molecule in an ice-like structure. There are no chemical bonds between the guest and host molecules. Such chemical compounds are referred to as inclusion compounds. Inclusion compounds are further divided into many categories, one of which is a clathrate. The term clathrate refers to a structure where guest molecules occupy the cavities of the lattice host molecules. When the host molecule is water, the description clathrate hydrate is used. For a more complete review of clathrate structures, see Sloan [1990]. For brevity, we refer to these clathrate hydrates simply as hydrates, or since the guest molecule is methane, as a gas hydrate. Because the large volumes of methane gas estimated to be contained within these layers [Kvenvolden, 1988; Gornitz and Fung, 1994], the potential as a future energy resource has garnered significant interest [MacDonald, 1990; Monastersky, 1996]. There have been successful extractions of natural gas from the Messoyakha hydrate reservoirs in the former Soviet Union [Makogon, 1981] but a technique has yet to be designed which allows for safe and economical recovery of methane.  INTRODUCTION  2  The role of gas hydrates in global climate change has also been discussed because methane is a strong greenhouse gas [Englezos and Hatzikiriakos, 1994]. Releases of large quantities of methane from the hydrate structure have been linked with major glaciation cycles and associated perturbations in the global carbon cycle [Nisbet, 1990a,b; Loehle, 1993; Dickens et al, 1997a]. In a period of glacial maximum, the sea level is lowered as water from the oceans is incorporated into the expanding ice sheets. This lowering of the sea level will alter the pressure conditions and could result in significant hydrate dissociation. If large amounts of methane are consequently released, increasing the global temperature and therefore producing a negative feedback to glaciation, it has been suggested that hydrates could play an important role in restricting the extent of glaciation limits [Paull et al, 1991]. Hydrates are also familiar as a nuisance to the gas industry. It was first reported by Hammerschmidt [1934] that these hydrates could form in and plug high pressure pipelines. Hydrate formation has been curtailed by the addition of methanol and other inhibitors to the fluids which flow through the pipelines. The prohibitive cost of supplying these inhibitors to often remote locations motivates current research towards a better understanding of hydrate formation [Savidge, 1995]. The danger of exploration drilling penetrating a hydrate layer, which causes dissociation and release of flammable gas, has led to the development of special techniques for drilling in hydrate infested regions [Goodman, 1981]. Recently there have been observations made by Charles Fisher of two meter long tube worms living on and in mounds of gas hydrates, apparent evidence of a new marine ecosystem [Mlot, 1997; Holden, 1997]. Even the mysterious Bermuda triangle has been attributed to gas hydrates . The presence of hydrate in the pore space tends to act as a cementing agent [Stoll, 1974] and if this hydrate were removed by some process, submarine landslides [Rothwell et al, 1998; Paull et al, 1996b] could occur resulting in large releases of gas [Kvenvolden, 1993]. If there was consequently an ebullition of methane gas, it was  INTRODUCTION  3  suggested by Mclver [1982] that ships could become negatively buoyant and the engines of planes could be choked causing these otherwise inexplicable disappearances. It is evident from the substantial interest in gas hydrates and its consequences by such diverse research groups that an understanding of hydrate formation and accumulation is essential for a determination of the role which hydrates play in these and other processes. A model which captures the fundamental understanding of hydrate growth and development as well as the associated time scales is essential for interpreting the importance of hydrates in the many phenomena which have been attributed to hydrate occurrence.  1.2  The Hydrate Environment  The location of gas hydrates are determined by pressure and temperature conditions and by the supply of gas. Evidence has been found for hydrate existence on land in permafrost regions [Collett, 1992] and in marine sediments below the seafloor [Brown et al, 1996; Holbrook et al, 1996; Yuan et al, 1996; Miller et al, 1991]. This thesis will only consider the marine environment. Sediments of nearly the entire ocean floor are within the temperature and pressure conditions for stability yet marine gas hydrates are predominantly found along continental margins. It is these regions which uniquely produce the required amount of gas to permit hydrate formation.  1.3  O bser vables  Hydrate occurrences are confined to regions which make in situ measurements technically and economically prohibitive. Fortunately, remote sensing techniques can be employed to detect hydrate layers from the sea surface. Also, projects such as the Ocean Drilling Program (ODP) are able to bring drill cores to the surface where geochemical and geo-  INTRODUCTION  4  physical testing can be undertaken to elicit a better understanding of the in situ hydrate environment.  1.3.1  Seismic  The primary detection technique for marine gas hydrates has been seismic reflection. An excellent example of implementing these seismic methods to delineate hydrate occurrences is given by Shipley [1979]. In seismic sections, a reflection was observed that paralleled the ocean floor and hence was given the name bottom simulating reflector or BSR. Hyndman and Spence [1992] used various seismic techniques to study the nature of the BSR, the distribution and concentration of hydrate in the sediments, and to conclude that there is no seismically detectable free gas. A method for estimating the amount of in situ gas hydrates by Lee et al. [1993] uses amplitude information above the BSR in seismic data. The BSR exhibits a negative polarity which implies an impedance decrease when traversing from materials above the BSR to those below. The impedance contrast can be explained by at least two scenarios. A small amount of fast velocity hydrated sediments overlying a zone with a significant amount of slow velocity free gas would produce a large impedance contrast [Minshull and White, 1989; Miller et al, 1991]. An alternative explanation, favoured by Hyndman and Davis [1992], involves a top layer with much more hydrate above the BSR and no free gas required below the BSR. The difference between the very fast, highly hydrated sediments above and the normal sediments below the BSR is sufficient to generate the required impedance contrast. The plausibilities of these different scenarios can be explored with our numerical simulations. We can observe when and where free gas exists and address the question of what is responsible for the observed impedance contrast.  INTRODUCTION  5  The B S R is observed as a single symmetrical pulse likely representing the sharp impedance contrast expected at the base of the hydrate layer. If this is indeed the case, there is no seismic evidence of the top of the hydrate layer which implies a gradual increase in hydrate with depth to a maximum at the BSR. Hydrate does not exist below the B S R since the geothermal gradient causes the temperature to surpass the maximum temperature at which hydrates remain stable. This absence of a hydrate layer upper boundary in seismic sections is problematic in assigning a thickness to the layer or in estimating the total amount of hydrate contained within a given volume.  1.3.2  Salinity/Chlorinity  Evidence of gas hydrate existence has been confirmed through analysis of chlorinity measurements. As hydrate forms, salt ions are excluded from the crystal structure in an analogous fashion to the exclusion of salts as sea water freezes. The hydrate structure then consists of freshwater and the excluded salts are free to diffuse away over time. When a sample of hydrate is brought to the surface, the hydrate has often melted, releasing the freshwater contained in the hydrate structure and therefore the chlorinity measurement appears freshened relative to the expected value of seawater at that depth. The amount of freshening has been related to the amount of hydrate which was contained in the sample [Hesse and Harrison, 1981; Hesse, 1990] . Chlorinity is preferred over salinity in geochemical studies because it is less affected by other geochemical processes [Kastner, 1995], but both concern the exclusion of salts associated with hydrate formation.  INTRODUCTION 1.3.3  6  Other Measurements  There has been a significant volume of work undertaken concerning the behaviour of gas hydrates in controlled laboratory situations such as experiments on hydrate filled sediments [Dvorkin and Nur, 1994], tests of phase equilibrium [Dickens and Quinby-Hunt, 1997], detecting the phase change interface associated with hydrate formation using temperature measurements [Rempel, 1994] and observations of peculiarities associated with hydrate production [Stern et al, 1996]. Insights gained here can certainly provide information relevant to modelling the marine setting. On a recent ODP leg, a specialized apparatus was successfully deployed which allowed the sampling of in situ hydrates and the capability of keeping the hydrate in its stable form as it was brought through rapidly varying pressure and temperature conditions to the surface [Dickens et al, 1997b]. The ability to sample in situ marine gas hydrates will undoubtedly lead to insights and constraints which can be incorporated into models. Studies of the dissociation of hydrates in porous media [Tsypkin, 1991; Selim and Sloan, 1990; Yousif et al, 1991] have been motivated by the desire for a better understanding of methods for extracting gas from hydrate reservoirs.  1.4  Gas Supply  The question of how gas is supplied to grow hydrate is a point which is still debated. Geochemical analysis shows that methane accounts for almost 99% of the gas. Gases which are comprised of such high percentages of methane are formed in two distinct manners. One method is the biogenic production resulting from decaying organic matter. The other formation mechanism occurs at depths greater than 6 km and at temperatures above 200° C where thermogenic methane is produced in abundance due to the thermal cracking of existing hydrocarbons [Kvenvolden, 1983]. There has been evidence of thermogenic  INTRODUCTION  7  methane in gas hydrate regions [Brooks et al, 1985], but the biogenic methane scenario is favoured for two reasons. Gas hydrates are found in shallow marine sediments where temperatures are well below thermogenic values. Even if the assumption is made that the gas was generated at depth and subsequently migrated into the gas hydrate stability zone, a comparison of carbon isotopic compositions of methane and carbon dioxide (the immediate precursor to methane in biogenic production [Whiticar et al, 1986]) supports the hypothesis that the methane was a result of microbial reduction of carbon dioxide by decaying organic materials [Kvenvolden, 1993]. While the fact that the methane is biogenic in origin is generally agreed upon, the mechanism responsible for supplying methane to the hydrate stability zone is still debated  [Paull et al, 1994; Bangs et al, 1995; Borowski et al, 1996; Brown et al, 1996]. Hydrates have been found both on active margins where subduction is occuring [Hyndman and Spence, 1992] and on passive margins such as continental rises [Paull et al, 1995]. Not surprisingly, these two very different physical environments motivate two competing theories for the gas supply [Kvenvolden, 1985]. Hyndman and Davis [1992] use the upward fluid expulsion that commonly occurs in subduction zone accretionary wedges due to tectonic thickening to explain gas supply. This setting is shown in figure 1.1. In their model the methane is produced below the hydrate stability zone, but not at thermogenic depths. The gas is then removed from the rising pore fluids and incorporated into the hydrate structure. Their argument against in situ biogenic production is based on the large amount of hydrate inferred from seismic and chlorinity measurements. There is simply not enough carbon available through biogenic production and therefore a method of transporting gas into the hydrate stability field is required. The method proposed by Hyndman and Davis [1992] involves the flux of gas from below, transported by upward migrating fluids. These fluids are also credited with removing the  INTRODUCTION  8  excess chloride that one would expect to find in a hydrate producing environment. But where is all of this gas originating? It is almost certainly not from depths since significant thermogenic production is not expected.  Methane production  Figure 1.1: The physical environment for the pore fluid migration model of gas supply. Dissolved methane is carried upwards by moving pore fluids and removed from solution as the fluids enter the hydrate stability zone. After [Hyndman and Davis, 1992]. The fact that there is known to be fluid expulsion in these subduction zone accretionary wedges [Davis et al, 1990] is important for gas transport and therefore the development of hydrate layers in these environments. However, fluid expulsion cannot be accepted as an all encompassing explanation of gas supply to hydrate zones because there have been discoveries of significant hydrate occurence in passive margins where the fluid migration is not likely to play such an important role in supplying gas. In these regions, it has been proposed that the gas is supplied by decaying organic matter [Martens and Berner, 1974]. A n immediate question arises for this scenario: Is there enough available carbon to produce the amount of gas required for hydrate formation? Paull et al, [1994] suggest that such a scenario could work but that the concentration of methane required to achieve saturation was a concern. A new phase diagram [Zatsepina  INTRODUCTION  9  and Buffett, 1997] would seem to help but perhaps not solve this problem. Assuming that there is sufficient carbon reserves, the next logical question is can in situ biogenic production generate enough methane to allow hydrate formation? Since hydrate layers do indeed exist in these passive margins, in situ production must play a significant role. There is, however, a process which could enhance biogenic production. The effects of compaction due to ongoing sedimentation could be important.  The  continual burial of organic material provides a steady supply of carbon. Burial can also concentrate methane deeper into the hydrate stability region. Compaction of the sediments results in a reduction in porosity with depth because of the increasing load of overlying sediments. This reduction of porosity at depth causes a flux of pore fluid upwards as pore space is reduced, producing a gas transport mechanism similar to the upward migrating fluids proposed by Hyndman and Davis [1992] for the subducting zone environment.  1.5  Structure of Thesis  The second chapter presents some initial definitions and concepts used to describe the physical characteristics of the hydrate environment. Based on these quantities, a set of equations which govern the formation and accumulation of hydrates in compacting marine sediments are developed. The third chapter explains why a numerical approximation to these governing equations is required and how it is implemented. Chapter four contains the results of the numerical simulations for a range of marine parameter values and a discussion of the implications of these results. The fifth chapter addresses global hydrate issues and includes specific examples of typical marine settings. The final chapter summarizes the results and makes suggestions for future work.  10  Chapter 2 Development of the Governing Equations We could use up two Eternities learning all that is to be learned about our own world...Mathematics alone would occupy me eight million years. [Mark Twain, Notebook # 22, Spring 1883 - Sept. 1884]  2.1  The Physical Environment  In this chapter I present a model for the formation and accumulation of gas hydrates in compacting sediments. The effects of compaction are an important issue when considering different scenarios of gas supply. This ongoing compaction plays a significant role in transporting the gas produced by in situ biogenic production into the region where pressure and temperature conditions allow for stability of gas hydrates.  Sedimentation and  the associated compaction are also significant in explaining the observed peak in hydrate volume fraction that occurs at the base of the hydrate zone, which is often associated with a reflector that parallels the seafloor (e.g. the BSR). Figure 2.1 illustrates the recycling of gas that occurs at the base of the hydrate stability zone. The B S R is believed to coincide with a depth below which the temperature is too high for hydrates to be stable. The geothermal gradient below the seafloor is known and the depth of the B S R can be predicted. As sedimentation continues, the depth of the B S R remains fixed relative to the seafloor. A portion of the previously stable gas hydrate layer lies below the current stability depth predicted by the geothermal gradient as sediments accumulate above. Consequently, the  DEVELOPMENT  11  OF THE GOVERNING EQUATIONS  hydrate will dissociate, releasing methane that migrates upwards into the current stable zone and reforms hydrate.  h y d r a t e dissociation as a result o f BSR migration  |  Temperature  Temperature Figure 2.1: A n illustration of BSR migration. In (a), the hydrate stability zone is limited by the geothermal gradient. In (b), ongoing sedimentation causes the seafloor to rise relative to a fixed frame. The seafloor, BSR, and geothermal gradient of the initial state (a) are indicated by the dashed lines in (b). This shifting of the BSR implies that some of the hydrates from (a) now he outside the hydrate stability zone (shaded regions) for (b). Hydrate dissociation results and free gas is then transported up across the new B S R where it can reform hydrate. It is now our goal to develop a mathematical model which describes the formation and accumulation of gas hydrates in a marine environment. This model is an extension to the work of Rempel and Buffett [1997] who developed expressions for hydrate formation and accumulation in a uniform porous media. Thermodynamically stable conditions for hydrates exist beneath the entire ocean bottom, however the vast majority of hydrate reservoirs have been discovered along continental margins. This occurs because these regions uniquely produce an adequate supply of free gas. In situ biogenic production occurs in the organically rich sediments of continental margins. Sedimentation and compaction then bring this gas into the hydrate stability zone. The frame of reference used to model hydrate accumulation is fixed to the moving  DEVELOPMENT  OF THE G O V E R N I N G  EQUATIONS  12  seafloor as shown in figure 2.1. The top of the computational domain is the seafloor and the base is some fixed distance below the seafloor with the B S R occuring near the middle of the domain. Ongoing sedimentation means that the top of the computation domain continually moves upwards relative to the sediments. The porosity profile assumed is an exponential decrease with depth, similar to that suggested by Hyndman and Davis [1992]. which is constant in the moving frame of reference. Consequently, the porosity distribution with depth in the moving frame is constant over time. Even though sedimentation results in compaction effects that decrease the pore space, the pore volume at a prescribed depth below the moving sedimentation front is constant so the porosity depth profile remains unchanged over time. In order to model compaction effects accurately, it is necessary to consider both the fluid  flux,  • dS across the surface of the control volume and the flux of sediments, v, • dS  (which may or may not contain hydrate) across the surface of the same control volume, where V / and v are the velocities of the fluid and sediments respectively. The conservation s  equations are derived in the frame of the moving sedimentation front so that the seafloor always corresponds to z = 0.  2.2  Definitions of Environment Variables  Consider an infinitesimal volume V in the seafloor comprised of fluid, sediment, and hydrate with total mass M. The contributions of the fluid, sediments and hydrate to the total mass are denoted by the subscripts / , s and h respectively. The density p is defined by _ 9  M _ M + M + M  ~ V ~  f  s  h  V + V, + v f  h  1  ' '  DEVELOPMENT  OF THE GOVERNING EQUATIONS  13  which may be expressed in terms of the constituent volumes (fluid, sediment, and hydrate) by p  =  pfVf + p,V,+ phVh  ^ ^ 2  2  Since the hydrates and the fluid occupy only the pore spaces of V and the sediments will occupy the remaining volume, we can use the definition of porosity <f> to write:  Now define the hydrate volume fraction h as the fraction of pore space which the hydrate occupies. By letting  V V +V  (2.4)  h  f  h  it follows that Vh  V.  Vf  y = h<f>  V ^  1  - *  ^  =  (2-5)  These definitions can then be used to rewrite equation (2.2) as p = p (f>(l - h) + p,(l - <f>) + p h<j>. f  2.3  (2.6)  h  Conservation of Mass  Any change in the mass contained within the control volume is due to a net mass flux across the boundaries of the control volume. The total mass within the control volume is obtained by summing the individual masses of the hydrate, fluid, and sediments as represented by <ph + <t>(l - h) Ph  Pf  + (1 - <t>) AdV. P  (2.7)  DEVELOPMENT  OF THE GOVERNING EQUATIONS  14  If we equate the rate of change of mass with the mass fluxes across the boundaries of an arbitrary volume, we obtain 8  4>hp + (j>(l h  h)p  f  + (1 -  <p)p.  dt  = -  V • [v <t>hp ] t  h  V • [v <p(l f  -  h)p ] f  (2-8)  J  -  V • [v,(l  - <f>)Ps]  where the sohd hydrate is assumed to move with the sediments. Since <f> does not vary with time in the frame moving with the interface and  and  f, h  P  P  p  s  are also assumed to be  constant, (2.8) simplifies to  <l>(p ~Pf)^ h  = -PHV  • [vs<f>h] -  pV  • [v^(l  f  -  h)}  -  P  s  V • [v.(l  -  <p)].  (2.9)  A similar analysis for the conservation of sediment mass only yields V • [v,(l -</>)]= 0  (2.10)  so that the conservation of total mass given in (2.9) may be reduced to give an expression for hydrate and fluid components:  <t>{Ph -Pf)-Qi  =  • [vs<t>h]  -  PfV  •  M ( l-  h)].  (2.11)  These expressions will be utilized extensively in the derivation of the remaining governing equations.  2.4  Conservation of Energy  A change in the energy inside of the control volume may be expressed in terms of enthalpy changes due to heat fluxes or changes in phase. Heat fluxes can be due to advection of fluid or sohd (hydrate or sediments) as well as conductive transport through the sediment matrix. There will also be a release of latent heat associated with the hydrate formation.  DEVELOPMENT  OF THE GOVERNING EQUATIONS  15  The total heat content within the control volume, V is given by  X  \p <p(\ - h)H + (l f  f  Pt  - 4>)H + p <t>hH ] dV S  h  (2.12)  h  where Hf, H , and Hh are the specific enthalpies of the fluid, sediments, and hydrate a  respectively. This total heat content inside the control volume is altered by heat fluxes across the surface. The contribution due to the advection of fluid into the control volume is described by -  J  <p(l -  Pf  -dS = - ^V-  h)H Wf  [<£(1 -  Pf  f  dV  h)H Vf] f  (2.13)  where the divergence theorem has been used to convert the surface integral into a volume integral. In our subsequent discussion, it is useful to expand the volume integral and write the fluid transport of heat as  - J <p(l - h)H v • dS = Pf  }  f  P f  J  H V • {<t>{\ - h)v } f  + 0 ( 1 - h)\  f  • VH  f  f  dV.  (2.14) A heat flux across the surface of the control volume is also generated by a flux of sediments and any hydrate contained within the pore space due to the compaction. I make the assumption here that the hydrate is transported along with the sediments at the same velocity, v . The heat flux due to the transport of the sediments and hydrate into the s  control volume is then given by - / \p <f>hH + p (l - <f>)H ]v • dS = - j V - \{p 4hH Js Jv 1  h  h  t  J  t  t  h  1  = - /  Jv  +  h  (l-<p)H.}v ]dV  Pt  J  t  PhH V • (<f>hv ) + p d)h\ • VH h  s  h  s  h  L  + (l-<p)v Ps  s  VH^dV (2.15)  where I have made use of the conservation of solid ( V • [(1 — <f>)v ] = 0). Changes in heat 3  content due to conduction depend on the thermal conductivity, which can vary with the  DEVELOPMENT  OF THE GOVERNING EQUATIONS  16  volume fractions of solid, fluid, and hydrate. A n effective thermal conductivity, K (h), is e  defined as the volume average of the individual constituent thermal conductivities. The conductive heat flux into the control volume can then be expressed as  J K {h)VT • dS = J V • [K (h)VT]dV. e  (2.16)  e  Since the change in total heat is attributed to (2.14), (2.15), and (2.16) and since the control volume, V is arbitrary, the integrands can be equated to yield — [ (j>{l - h)H + p (l - <p)H + p h<j>H \ = -pfH V P}  f  3  s  h  h  - (j>(l - h)w • VH - p H V Pf  f  -p.(l-  f  h  • [ 0 ( 1 - h)v ]  f  f  • [<f>hv ] - <f>hv • VH  h  3  Ph  3  (2-17)  h  0 ) v , • VH + V • [K (h)VT]. S  e  Expanding the left hand side terms, equation (2.17) can be written as pm  -  1  )~of  h  +M  1  ~  = - p HfV  ^~dT ~ ^{pfHf  p h  ^'~QT  -  )~oi  phEh  • [ 0 ( 1 - h)v ] - <p(l - h)w • VH - p H V  f  f  Pf  f  f  - p cf>hv • VH - (l - 0 ) v -VH + Vh  3  h  Pa  s  S  h  • [d>hv ]  h  3  (2-18)  [K {h)VT]. e  Some simplifications to (2.18) are obtained using the expression for conservation of mass given in (2.11). Multiplying (2.11) by Hh and rearranging gives <f>p H — = <t>PfH -^ - H V h  h  h  Ph  h  • [v <j>h] - H V t  P}  •  h  [ 0(l V /  - h)]  (2.19)  which, when substituted into (2.18), gives <Ki - h) ^  - 0 ) ^ + p H -^  +  9  Pf  - (H Pf  f  d  h  = 4> (H Pf  f  h  ) §  - H )V • [ 0 ( 1 - h)v ] - <p(l - h)v • VH h  f  Pf  f  - <j>hv • VH - p.(l - 0 ) v , -VH + VPh  H  3  h  3  f  (2-20)  [K (h)VT]. e  It is now useful to express the changes in enthalpy in terms of temperature which is a more readily measurable quantity. The enthalpies of the fluid, sediment, and hydrate are related  DEVELOPMENT  OF THE GOVERNING EQUATIONS  17  to temperatures by dH = C dT  (2.21)  p  where separate values of C are assumed for sediment, fluid, and hydrate. Thermodynamic p  equilibrium between the sediments, fluid, and hydrate over a small volume is assumed so that dT is common to all components. The difference between the enthalpy of the fluid and the hydrate is the latent heat of formation of the hydrate so we define the latent heat,  Lh as L = Hh  H.  f  (2.22)  h  Writing (2.20) in terms of temperature gives r) F r  C (h) — = - C}<p(l - h)v • V T - C£<phv. • V T - p.C;(l - <f>)v • V T  P  p  Pf  f  Ph  s  + V • [K (h)VT] + <f>p L -^ ~ PfLhV • W - h)v ] e  f  h  f  (2.23) where pC (h) = [pf<f>(l — h)C^ + p (l — 4>)C + phhfiCp] is the average heat capacity of p  s  p  the sediment matrix. The first three terms on the right-hand side of (2.23) represent changes in temperature due to transport of fluid, hydrate and sediment respectively. The next term represents the diffusion of heat through both the bulk sediment and the pore fluid. The penultimate term represents a release of latent heat as hydrate forms, while the final term is a 'squishing' term resulting from fluid being expelled due to compaction. The amplitude of the final term depends on the enthalpy difference, L = (Hf — Hh)-, because n  hydrate is carried into the volume during the compaction of the sediments as the fluid is expelled.  18  DEVELOPMENT OF THE GOVERNING EQUATIONS  2.5  Conservation of Gas  Compaction of the sediments will also influence the gas concentration in the control volume. For the idealized case of uniform porosity sediments, gas is transported only by pore fluid and by molecular diffusion. However, if sediments are allowed to compact, the solid hydrate occupies sediment pore space which can carry gas into the control volume. Gas is also supplied to the system through decaying organic material via biogenic production which is denoted by the term c . The mass of gas in the control volume is either dissolved in g  the fluid or contained in the hydrate structure. Therefore, a change in the mass of gas is expressed by  d f — J  dV  p <p(l - h)c + p (f)hc f  h  a  (2.24)  h  where c is the mass fraction of gas in the pore fluid and c is the mass fraction of gas in g  n  the hydrates. As was done earlier in the development of the energy equation, the flux of gas into the control volume due to advecting fluid can be expressed as a volume integral using the divergence theorem: - J p <p(l - h)c v f  g  f  • dS = -  P  J  J [ c V • ( 0 ( 1 - h)w ) + <p{\ - h)w • Vc )] dV. f  3  f  fl  (2.25)  The mass of gas within the control volume is also changed as hydrate is fluxed into the control volume due to the compaction associated with ongoing sedimentation as described by  L  I  p (f)hc v • dS = -p I h  h  s  h  CfcV •  {4>hv ) s  +  <f>hv •dV. Vc 3  h  (2.26)  The diffusion of gas through the pore fluid into the control volume is represented by J^ (l-h)D Vc -dS Pf(p  g  g  = J V • [ 4>(l - h)D V Pf  g  dV.  (2.27)  DEVELOPMENT  OF THE GOVERNING EQUATIONS  19  Equating the rate of change of gas with fluxes out of an arbitrary volume gives after simplification (using substitution from conservation of water and gas and assuming that  Ch is a constant) (1 - h)^f  = (c g  )^  Ch  - ±(c - c )V • [4>(1 g  h  fc) ] V/  (2.28)  + (1 - h)v • Vc + f  g  • [<f>(l - h)D Vc ] + c . g  g  g  The expression in (2.28) describes the rate of change of gas due to the various causes identified above.  The first term on the right-hand side represents the changes due to  incorporation of gas into the hydrate structure. The second term accounts for squishing effects of compaction. The next two terms describe changes in gas concentration due to advection and diffusion respectively. The final term is a source term that represents biogenic production of gas from decaying organic matter in sediments.  2.6  Conservation of Salt  The concentration of salt in the pore fluid is important in modelling the marine hydrate environment. As hydrate forms, salt is excluded and salinity increases.  Salinity has a  modest influence on the phase equilibrium [Zatsepina and Buffett, 1998], but it is more important for estimating hydrate volume fractions from in situ drill samples. When a sample is retrieved from the seafloor and observed at the surface, the hydrate dissociates because of the changes in pressure and temperature conditions that occur as the sample is brought to the surface. Freshening occurs as pure water in the hydrate structure is released back into the pore fluid. With a knowledge of the starting salinity (i.e. prior to drilling) and assuming that any decreases in salinity must be the result of hydrate dissociation, hydrate volume fractions can be estimated. The development of an equation describing the rate of change of salt is analogous  DEVELOPMENT  OF THE GOVERNING EQUATIONS  20  to that for the gas equation with a few exceptions. There is no salt contained in the hydrate structure or transported with the sediments so the only transport mechanisms are an advective flux due to transport by pore fluids and diffusion of salt through the fluid. The time evolution of salt is then governed by ( 1 - h)^  = c^ s  • [ 0 ( 1 - h)v ] - ( 1 - h)w • Vc +  -  f  f  3  •[0(1 -  h)D.Vc,]. (2.29)  2.7  Kinetic Effects and Hydrate Conservation  The three conservation equations described in the preceeding sections determine the time rate of change of gas concentration, c , temperature, T, and salt concentration, c . A g  3  fourth unknown in these three equations is the hydrate volume fraction, h. Consequently, a fourth equation is necessary for a complete mathematical description of the problem. One possibility is to impose thermodynamic equilibrium on the variables. Specifically, the equilibrium temperature T is related to the gas concentration at a prescribed pressure eq  (e.g. T = T(c ,P)). Equivalently, the equilibrium gas concentration may be expressed eq  g  as a function of temperature and pressure. An alternative approach is to relate the rate of change of hydrate volume fraction to some measure of thermodynamic disequilibrium. Here I will choose afirstorder disequilibrium model where ^ is proportional to the deviations of gas concentration from the equilibrium value c , provided that the temperature lies eq  within the hydrate stability zone. The constant of proportionality K is chosen to describe the reaction rate. Large values of K ensure small deviations from equilibrium and in the limit as K —» oo, thermodynamic equilibrium is achieved. The amount of hydrate within the control volume changes according to thisfirst-ordermodel, but it also be altered as hydrate is carried into the control volume due to the compacting sediments. Superposition  DEVELOPMENT  OF THE GOVERNING EQUATIONS  21  of these two effects gives fc=-^-(4>v h)+K(c -c ). s  g  (2.30)  eq  The first-order disequilibrium model described by the second term in (2.30) is a conventional empirical relationship for which there are experimental estimates of K [Uchida, 1996].  2.8  Governing Equations  Equation (2.30) for the growth of hydrate completes the set of equations, but it can also be used to simplify the conservation equations for energy (2.23), gas (2.28), and salt (2.29). I develop the simplified equation for conservation of energy by isolating and focusing on the two terms in the energy balance that depend on Lh = (Hf — Hh) . Manipulations for the remaining equations are analogous and the results are listed below. Combining (2.11) for the conservation of gas and fluid components with (2.30), gives ^  = ^7-[v <Kl-h)]  + K(c -c )  f  g  S 1I1C6 ~Z  where K = r J  r  (2.31)  e9  1, it is assumed in subsequent developments that K ~ K.  J  The two terms dependent on Lh in the energy equation (2.23) can now be rewritten as  dh 0 ( 1 - h)v + 4>p L — = p L (f>K(c - c ). 1  -pfL V H  •  f  f  h  h  h  g  (2-32)  eq  The substitution of this term into equation (2.23) consequently leads to a new, more compact expression for the energy equation  c (h) ?±  P  p  =  -V .VT S  p <phc  h  h  v  + .c;(i-0) P  Cg  -pjCl^i-h)v C ). eq  f  •vr  (2.33)  DEVELOPMENT  OF THE GOVERNING EQUATIONS  22  When a similar process is applied to both the gas and salt equations, the respective governing equations are given by  p (l - h)<& = - (l - h)<pv • Vc + V • [ 0 ( 1 - h)D Vc ] f  Pf  f  g  Pf  g  g  (2.34)  d t  - Ph<f>(Ch - Cg) K(Cg ~ C ) + Cg e q  dc {\ - h)4>-^ = -p (l  Pf  f  - h)<f>w • V c . + V • [ 0 ( 1 - h)D.Vc,] + p (j>c K(c - c ). f  Pf  h  s  g  (2.35) It should be noted that no additional approximations or assumptions were required to obtain this latest version of the governing equations.  2.8.1  Dimensionless Forms  The formation and accumulation of gas hydrate is governed by the equations (2.33) to (2.35) and (2.30). These equations may be solved in three dimensions, although hydrates frequently occur in regions where fluid velocities as well as gradients in temperature, gas concentration, and salt concentration are mainly vertical. This motivates approximations to the governing equations where the variables are dependent only on the depth below the seafloor. The computations are simplified significantly, but the essential features of the problem are retained. In specialized regions of enhanced flow such as in fracture zones or where gradients are no longer primarily vertical, a 1-D depth dependent solution may not be appropriate. It is convenient to write equations in dimensionless form for a number of reasons. We are free to define meaningful length scales and reference values. Numerical solutions can then be scaled up or down without the need for further computations. Dynamical similarity also allows for a convenient comparison mechanism between numerical marine scale scenarios and laboratory simulations. Obviously the marine scale cannot be implemented  eq  DEVELOPMENT  OF THE GOVERNING  EQUATIONS  23  in a laboratory as there are not many labs equipped to contain 2-3 km depths of ocean water! However, an apparatus can be constructed which is geometrically similar (i.e. a scaled down version of the marine environment) and may be useful in eliciting information concerning physical processes in the marine environment. The 1-D equations are written in dimensionless form by defining dimensionless temperature, gas and salt concentrations as: P  m  Po  ~  ± — —. AT„  Cg  _  Cg  0  ^c = —Ac 9 0  :  C/j  _  Cg  0  c = —Ac h  g  c. ~ — s  g  Cg  C  s  o  Ac  s  (2.36) where T , c , and c Q  go  so  are reference values and A T , Ac , and Ac are typical variations over g  the region of interest. For example, if we choose c  go  s  = 0 and take Ac to be the maximum g  solubility of methane in seawater then a dimensionless value of c = 1 corresponds to the gas g  concentration in a saturated solution. Thus, values of c > 1 would imply supersaturation g  and the presence of gas bubbles. Time is converted into a dimensionless quantity by use of the thermal diffusion timescale U = -Ifrr  (2.37)  where Kd(h) is evaluated at h — 0 and z is a typical length scale (say, the depth from the d  seafloor to the BSR). We can then define a corresponding velocity scale by v = £ = ^ d  which allows us to define the following non dimensional variables:  (2.38)  DEVELOPMENT 2.8.2  24  OF THE GOVERNING EQUATIONS  D e r i v i n g E x p r e s s i o n s for F l u i d a n d S e d i m e n t Velocities  Estimates of the sediment and fluid velocities are obtained from the equations for the conservation of mass when the sedimentation rate is prescribed. The approach adopted in this study follows that of Hutchison [1985]. Recall from § 2.3 that we can write the conservation of mass in 1-D as d_  d_ <p(l-h(z))v (z) dz:  + dz  f  (2.40)  0  4>hv  s  assuming that pf £s p and that dh/dt is small. Also the 1-D expression for the conservation n  of solid is given by d_ (1 - <p)v dz  a  (2.41)  0.  Integrating these expressions with respect to z then gives (2.42)  <f>(z) [1 - h(z)]v (z) + 4>{z)h(z)v {z) = d ( « ) f  a  (2.43)  I - <f>(z)}v (z) = C (t). a  2  where G\(t) and C ( i ) are integration constants that depend only on time. At the seafloor, 2  where z = 0, the sediment velocity is equal to the sedimentation rate (i.e. v = ds/dt), so s  we write (2.43) as  i-^(o)|^ = c(t).  (2.44)  a  According to (2.43), the sediment velocity, v becomes a  v {z) a  [1 - 0(0)] ds l-<f>(z)] dt  (2.45)  We now proceed to find an expression for the fluid velocity, Vf. At some depth, say z — L, compaction will have reached the stage where the fluid is largely expelled and any  DEVELOPMENT  OF THE GOVERNING EQUATIONS  25  residual fluid moves with the sediment velocity so we can assume that Vf Pa v . There is a  no hydrate at the depth z = L when L is large enough because the temperature implied by the geothermal gradient lies well outside of the hydrate stability field. Using h — 0 at z = L, we can use (2.42) to evaluate C\(i): <p(L)v (L) = d ( « ) .  (2-46)  f  Since Vf f« v at z = L, we can substitute for Vf(L) in (2.46) using the expression for v (z) a  a  in (2.45) to obtain  *  L  )  [i - 4>(L)] di =  G l { t )  -  {  2  A  7  Equating the expressions for C\(t) as given in (2.42) and (2.47) and solving for the fluid velocity as a function of depth gives  HL)  [1 - 0(0)]  h(z)  1 ds  [l-«&(*)]J ' dt  _<P{Z)[\ -<HD]  We now have obtained expressions for the velocities of the sediments and the fluid as functions of depth. Both Vf(z) and v (z) are functions of the sedimentation rate, which a  is a prescribed parameter in the calculations. Consequently Vf(z) and v (z) are known a  functions of depth once h(z) is determined.  2.9  Summary  We conclude this chapter by listing the 1-D governing equations in dimensionless form. ^  =^ M ( l - h ) ]  dt 4>oz  pC (h) P  dT  + K(c --c ) g  (2.49)  eq  dT dT c;{i-4>)\^-PfCifti-h)v  = -v \ t>c + h  s  Ph(  p  ph<l>L  Ps  .  d / ~  -^r ^-Ce )+Q=[K (h) K  h  q  e  li  r  dT  (2.50)  )  DEVELOPMENT  OF THE GOVERNING Ph  v  ' dt  Pf  (c - c )K(c - c ) + h  26  EQUATIONS  g  g  eq  1 d -^QZ 0 ( 1  + Co  (2.51)  dc  s  0 ( 1 - h)D. ~dl  (2.52)  Numerical solutions to equations (2.49)-(2.52) are presented in the following chapter.  27  Chapter 3 Numerical Solution / have tried to avoid long numerical computations, thereby following Riemann's postulate that proofs should be given through ideas and not voluminous computations. [David Hubert, Report on Number Theory, 1897].  The mathematical model developed in the previous chapter describes the formation and accumulation of gas hydrates. The equations for temperature, gas, salt, and hydrate are coupled and non-linear. Analytical solutions are precluded without major simplifications so we rely on numerical methods to obtain solutions. (hereafter referred to as NML)  The numerical method of lines  is well suited for problems with one spatial dimension.  Spatial derivatives are calculated using both centered differences and upwind differences. Special attention must be given to the spatial derivatives at the base of the hydrate zone where the hydrate volume fraction may be discontinuous and therefore jump conditions on the solution must be applied to ensure that the conservation equations are obeyed. These conditions are completely analagous to conditions that arise in moving-boundary problems and, in fact, the base of the hydrate zone may be viewed as a moving boundary in the reference frame of the sediments which are continually buried.  3.1  The Numerical Methods of Lines  Perhaps the biggest advantage of the NML  is its conceptual simplicity. Excellent back-  ground material and applications of this method can be found in [Schiesser, 1991; Madsen  NUMERICAL  SOLUTION  28  and Sincovec, 1974]. By discretizing the spatial domain into N points, each partial differential equation is transformed into N ordinary differential equations dependant only on time. These /V coupled equations can then be integrated forward in time using standard solvers. The calculation of spatial derivatives and the choice of integrator are important decisions and will be discussed later but the ideas are straightforward and can be illustrated here using the hydrate model as an example. z=0  ^  *  seafloor  T  «  sc  z=0.5  *  -at-  *-  B S R  ,= 1.0  t = 0  t = dt  t — tout  i n t e g r a t e s p a t i a l l y d i s c r e t i z e d s o l u t i o n f o r w a r d in t i m e  Figure 3.1: The numerical method of lines applied to the hydrate environment. Stars represent discretized points in the spatial domain at which h, T, c , and c are to be solved. g  3  The computational depth domain of the marine environment is discretized into N points, where 7Y is chosen to be an odd number to ensure that the midpoint (i.e. the B S R as predicted by three phase equilibrium) occurs at a grid point. At each of the N points, approximations of the spatial derivatives are obtained so that expressions for the time rates of change of hydrate volume fraction, temperature, gas, and salt are evaluated and subsequently integrated forward in time (see figure 3.1).  NUMERICAL 3.2  SOLUTION  29  Jump Boundary Conditions  The geothermal gradient determines the depth below which hydrates are no longer stable. This situation was previously described in § 2.1. The base of the hydrate stability zone (hereafter referred to as HSZ) is chosen to coincide with the middle of the computational domain. The position of the HSZ occurs at the same depth relative to the sedimenting seafloor because the temperature of the seafloor is fixed and hydrate formation does not significantly perturb the temperature in the sediments. The depth of hydrate occurrence can also be limited by gas supply. This is generally the case at early times before in situ biogenic production has generated enough gas to exceed the solubility values at all depths down to the base of the HSZ. Numerical calculations show that hydrate is initially formed at shallow depths (see the representative example presented in § 3.6). The biogenic production will, in time, release enough gas to exceed the solubility at all depths within the HSZ provided there is sufficient organic carbon. Sedimentation transports hydrate down below the seafloor. The dissociation and migration of gas back into the HSZ eventually focusses hydrate at the base of the stability zone. As a result, the highest hydrate concentrations occur at the base of the hydrate stability zone but the formation is slow enough that the latent heat release barely alters the geothermal gradient.  Immediately below the HSZ,  the hydrate concentration is constrained to be zero since the temperature exceeds the stability conditions. This change in hydrate concentration provides a plausible explanation of seismic observations in some environments. The B S R appears so prominently in seismic data because of a large impedance contrast. One explanation is the change from a highly hydrated region, with fast velocity, to one which overlies a region void of hydrate [Hyndman and Davis, 1992] and possibly containing free gas [Minshull and White, 1989; Miller et al., 1991]. Although this discontinuity is a reasonable explanation of the seismic observations, allowances are required in developing a numerical implementation which includes such a  NUMERICAL  SOLUTION  30  discontinuity. The hydrate volume fraction is modelled as a piecewise continuous function with a jump discontinuity at the base of the HSZ (at z = 0.5 in the computational domain). This discontinuity in the hydrate volume fraction at the B S R introduces discontinuities in the gradients of temperature, gas and salt concentrations in order to satisfy the conservation equations. Attempting to resolve these sharp gradients with a fine grid significantly hinders the computational efficiency. Instead, spatial derivatives are calculated independently on either side of the discontinuity. Jump conditions are imposed at the discontinuity to connect the solutions on either side. The procedure for obtaining the jump boundary condition is outlined here for the case of the gas concentration field. The remaining jump conditions for temperature and salt concentration are simply listed since their derivations are analogous. Consider a thin region which encompasses the discontinuity. Note that for the remainder of this section, the ~ notation for dimensionless quantities is dropped for clarity of presentation.  Figure 3.2  illustrates the fluxes of mass, gas, energy and salt above and  below the discontinuity in hydrate volume fraction.  Conservation of mass requires the  equality of the mass flux in and out of the region:  • v+<l>h p + v +  h  +  0(l - h )  = vripf  +  Pf  f  (3.1)  which may be written in a more convenient form as ^[  - _ + ( l - h + ) ] =v h+  (3.2)  +  V f  Vf  where the superscripts  s  +  Ph  and ~ are used to represent quantities evaluated above and below  the hydrate discontinuity respectively. The hydrate h~ on the lower side of the discontinuity vanishes. A similar treatment of gas conservation gives  h c v+ +  Ph  h  + (1 - h+)p c+vl - p (l - h )D Vc+ +  f  f  g  = p c-vf  - pD V f  g  c  (3.3)  NUMERICAL  31  SOLUTION  Conservation of Gas  Conservation of Mass  <j>hp c v, + (l-h)p <f)c v h  v,(\-4>)p. + v.cj>h,p + v j>(l-h)p h  f(  Pf  hydrate—  hydrate —  no hydrate -  no hydrate -  v,(l-<j>)p, + v (l>p f  h  }  g  f  - {l-h)(f>D Vc  f  g  Pf<t>c Vf -  f  <j)D Vc  9  Conservation of Salt  g  P!  g  g  Conservation of Energy p,H,(l-<j>)v,+ p H <f>hv, h  {l-h)4>p c,v,-  (l-h)(t>p D,Vc  f  +  f  hydrate  hydrate —  no hydrate —  no hydrate -  <j>pjc,v - D Vc,(j>p f  s  f  p.H.{l-4>)y.+  f  h  p H,<l>(l-h)v -KVT f  PiH ^r -KVT t  t  Figure 3.2: Fluxes across the depth of discontinuity in hydrate volume fraction. Cases shown for conservation of mass, gas, energy and salt. The gas concentration must be continuous since any jump would be immediately erased by diffusion. Using (3.2), we obtain  p,[(l - h)D Vc } _ = h (c +  g  +  g  Ph  h  - c )v+. g  (3.4)  which implies that the imbalance in the diffusion of dissolved gas into and out of the volume is equal to the rate at which gas is released by dissociation of hydrate. The hydrate is carried across the interface with velocity v which is continuous (e.g. v^" = v~). The jump s  conditions for temperature and salt are given by  p [(1 - h)D.Vc.] _ = - h c vt +  +  f  "«vr]+ =  Ph  - <pL h v +  Ph  h  s  a  (3.5) (3.6)  NUMERICAL  SOLUTION  32  The method by which these jump conditions are implemented in the numerical solution procedure is described in § 3.4.  3.3  Initial and Boundary Conditions  In addition to the jump conditions at the discontinuity in hydrate volume fraction, a numerical solution of the governing equations requires initial conditions for each variable and boundary conditions at the ends of the computational domain.  3.3.1  Gas Conditions  It is assumed that there is initially no gas in the system, i.e. that c = 0 at all points. As g  the solution progresses forward in time, gas accumulates due to biogenic production. Once the gas concentration exceeds the solubility in the HSZ, the excess gas triggers hydrate production. Two boundary conditions are also needed for the gas concentration as a result of the second order spatial derivatives in (2.51). I choose to set c = 0 at the seafloor g  (5 = 0). Seawater in the open ocean has low methane concentrations and if this value is chosen for the origin c , the dimensionless gas concentration is zero (see § 2.8.1). The go  second boundary condition is the no flux boundary condition imposed at the base of the computational domain (z = 1.0) so that there is no diffusive flux of gas out of the base of the computational domain.  3.3.2  Salt Conditions  The background salinity of seawater is assumed for the initial condition on c . s  Since  the salinity of seawater is chosen as the characteristic scale for c , the initial condition s  in dimensionless variables is c = 1.0. s  Two boundary conditions are also necessary to  NUMERICAL  SOLUTION  33  solve the salt equation, (2.52). The salinity at the seafloor is set equal to the background salinity of seawater (c = 1.0 at z = 0) and a no flux condition is imposed at the base of a  the computational domain  3.3.3  = 0 at z = 1.0).  Temperature Conditions  The geothermal gradient below the seafloor determines the temperature.  Initially the  temperature increases linearly from a dimensionless value of zero at the seafloor to a value of T = 1.0 at the base of the computational domain. Since the temperature profile remains essentially invariant, the boundary conditions can be obtained from the initial conditions  (f = 0 at z = 0; f = 1.0 at z = 1.0).  3.3.4  Hydrate Conditions  It is assumed that hydrate is not present at the initial time. As the solutions evolve in time, sufficient gas develops to allow hydrate formation. No boundary conditions are required for the hydrate, but by imposing the gas concentration to be zero at the seafloor, the gas concentration is always constrained to be less then the solubility value and therefore it is impossible for hydrate to form in the upper portion of the sediments (see figure 3.3). The dashed curve represents the solubility limit of the gas. By choosing the boundary condition of c = 0 at 5 = 0, for the gas, the c solution is constrained to go to zero at the seafloor g  g  and hence for values above z ~ 0.15 in figure 3.3, the gas concentration lies below the solubility values which implies that hydrate cannot form in this uppermost region.  NUMERICAL  34  SOLUTION  Figure 3.3: The gas concentration lies below solubility values (indicated by the dashed curve) in the upper regions of (a) and consequently in (b), hydrate cannot form in the upper portions of the computational domain. 3.4  Approximating Spatial Derivatives  The accuracy of solutions obtained via NML is dependent, in part, on the accuracy of the spatial derivative approximations. A centered difference approximation is chosen for calculating the spatial derivatives of the temperature, gas concentration, and salt concentration, but upwind differences are used for the hydrate equation. It is necessary to calculate these spatial derivatives in two distinct regions: an upper region and a lower region on either side of the discontinuity in hydrate volume fraction. Differences cannot be calculated across the discontinuity in hydrate volume fraction because the derivatives are not defined here. Instead, the jump conditions are used to relate the derivatives on either side of the discontinuity. Special attention is required for obtaining values of the fields at the depth of the discontinuity. Numerical instabilities arise when attempting to resolve the sharp gradients in the fields at this depth. Jump conditions are implemented to determine c , c , and T g  3  NUMERICAL  SOLUTION  35  at the discontinuity. In effect, the values of c , c , and T at the discontinuity are adjusted g  3  at each step to ensure that the conditions on the gradients i n (3.4), (3.5) and (3.6) are honoured as the solution evolves. As an example, the expression for the gas concentration at the discontinuity is developed here based on a discretized approximation of the jump boundary condition in (3.4). In the discrete approximations that follow, n is assumed to be a point corresponding to the depth of the discontinuity, (n + 1) is the point immediately above the discontinuity, within the HSZ and the point (n — 1) lies immediately below the discontinuity within the region where hydrate is unstable. The ~ notation used to denote dimensionless quantities is again dropped here for the purpose of clarity. Writing the gas jump condition in terms of values at grid points gives  {  '  1  ^z  h  dT  ]  7 t D  =  { a  h  ~  a  )  '  (  }  where: dc  c" - c "  n +1 n  9  _  1  .9  9  dz  dd -  + 1  9  dz  c " - - c"  1  1  _  dz  g\  9_  __3  dz  and c™ is the value of the gas concentration at the depth of the hydrate volume discontinuity, the quantity which is to be solved for. Straightforward algebra leads to the expression c = n  g  "—r^ 2 - h [1 - ^-h v^dz  •  1  n  (3-9)  n  The jump boundary conditions for temperature and salt concentration are given in (3.10) and (3.11). i  _LI  Ph<t>h v™KLhdz n  rpn—l _|_ rpn-\-i. r_ ' rt  ( i - h )c: n  c. =  2 - h (l n  +i  +cr  1  £-h v^dz} n  8  kAT  11  (3.10)  (3.11)  NUMERICAL  SOLUTION  36  At each timestep in the numerical implementation, the midpoints are calculated using (3.9) - (3.11) based on the values of c , c and T at that particular time step. Spatial g  s  derivatives are then approximated for the non-midpoints and the solution is integrated forward in time another step.  This process is repeated until the desired final time is  achieved. Note that when there is no hydrate present at the base of the HSZ (i.e. h = 0), n  the value of the gas concentration field at z{n) determined from (3.9) would simply be an average of the point above (eg. c™ ) and the point below (c™ *). Deviations from the +1  mean when h  n+1  is not zero introduce discontinuities into the gradients in c , c and T to g  s  enforce (3.9) - (3.11).  3.4.1  Centered Differences and Upwinding  Equation (2.49) for the hydrate volume fraction differs from the other equations in that there is no diffusive term. Attempting to use a centered difference scheme for approximating the derivatives in hydrate volume fraction causes the solution to exhibit non-physical oscillations indicating a numerical instability. This is a common effect in first order hyperbolic PDEs [Schiesser, 1991]. In this problem, hydrate is transported with velocity v down the sediment column due to the ongoing sedimentation. Therefore the hydrate s  volume fraction above the point where the derivative is to be approximated may be more influential than the point below and so it may be beneficial to consider upwind points in the calculation of  The result of calculating the spatial derivatives of h using a two  point upwind approximation are compared with calculations using centered differences in figure 3.4. The calculations using centered differences develop numerical instabilities which are absent in the calculation using upwind differences. On the other hand, the upwinding method has the drawback of introducing some measure of numerical diffusion into the solution, but, if necessary, these effects can be reduced by implementing a higher order  NUMERICAL  SOLUTION  37  upwind approximation with additional points.  Figure 3.4: Numerical instabilities that occur when (a) centered differences are employed to solve for the hydrate volume fraction. These instabilities are not evident when (b) upwind differences are used.  3.5  Choice of integrator and Stiffness  Stiffness arises in the solution of differential equations where the dependent variable can be expressed in terms of two or more different scales in the independent variable [Derrick and Grossman, 1987]. Small time steps are required for initial transitory regions that exhibit rapid oscillations but larger time steps are computationally desirable once these initial oscillations have subsided and the solution is smoother. Fortunately there are a number of integrators available that are specifically designed for stiff systems of ODE's, I used the routines of Shampine and Reichelt [1997] which are part of the Matlab package. Choosing a finer grid (i.e. dividing the depth into an increasing number of discrete points) will also increase the stiffness of the problem because increasing the resolution tends to increase the spread in eigenvalues [Schiesser, 1991].  NUMERICAL  3.6  SOLUTION  38  A Representative Example  Typical solutions for the four fields are presented here for three distinct time periods: before hydrate formation has occurred (figure 3.5), after hydrate has formed in the sediment but before it is transported to the base of the HSZ (figure 3.6), and finally a scenario where hydrate production has reached the base of the HSZ and the focussing of hydrate at the B S R becomes evident (figure 3.7). Since the main goal of this section is to demonstrate how the onset of hydrate formation modifies the various fields, we consider a fast sedimentation rate of 22 cm/ky and total organic carbon (TOC) of 1% (the procedure for converting T O C into a biogenic production rate is outlined in § 4.1.1). Solutions are only presented up to the time when hydrate reaches the base of the HSZ. A more detailed investigation of these parameters is presented in the next two chapters. Hydrate does not form until the gas reaches solubility values within the HSZ. In figure 3.5, the gas concentration due to in situ biogenic production has not yet achieved the solubility values indicated by the dashed curve, and therefore hydrate formation has not yet occured at t = 125 ky. The salt concentration remains constant at the value associated with the background salinity of seawater (c = 1.0) prior'to hydrate formation. s  At the later time of 334 ky, shown in figure 3.6, hydrate has begun to form. The plots of hydrate volume fraction and gas concentration show that hydrate occurs at depths where the gas concentration has reached solubility values. As hydrate is formed, the local salinity is increased as excluded salts are introduced to the pore fluids. The salt concentration increases, reaching a maximum value at a depth corresponding approximately to the peak in hydrate volume fraction. Diffusion transports salt to the seafloor and deeper into the sediments.  The no flux condition at the base of the computational domain causes the  salinity to slowly increase below the hydrate zone. Changes in temperature are not evident  NUMERICAL  39  SOLUTION  Hydrate volume fraction 0  o  0.02  0.04  0.06  0.08  0.1  0.5  Temperature 1  1.5  0.2  S 0.4 Q  0.6 0.8 1  Gas concentration 0.2  0.4  0.6  O.I  Salt concentration 0.5  1  1.5  Figure 3.5: At an early time of « 125 ky, the onset of hydrate formation has not yet occurred. Sedimentation rate of 22 cm/ky and T O C = 1% were chosen. in the results because thermal diffusion is approximately three orders of magnitude more efficient then compositional diffusion and, unlike gas and salt, the temperature can also be conducted through the sediments. Consequently, temperature variations from the linear profile described by the geothermal gradient are not evident. Even latent heat release due to hydrate formation does not cause a perceptible perturbation in the temperature field. The presence of salt, excluded during hydrate formation, will effect the stability temperature of hydrates and hence the B S R location but the influences of salt are expected to be small and are not considered here. Hydrate has reached the base of the HSZ in the solutions at t = 700 ky in figure 3.7. The profile of the hydrate volume fraction agrees qualitatively with the inferences made  NUMERICAL  SOLUTION Hydrate volume fraction  40 Temperature  Figure 3.6: After ~ 334 ky, hydrate formation has now occurred. Hydrate formation corresponds to an increase in the salt concentration in the hydrated region as salts are excluded during hydrate formation. Gas concentration must also have reached solubility values in the hydrate producing region. from seismic observations. A gradual increase in hydrate volume fraction is observed with distance above the B S R , but an abrupt decrease below this depth occurs as hydrate is no longer stable. Since hydrate cannot form below the HSZ, gas concentrations above solubility in this region indicate the presence of free gas. The gas concentration plot shows evidence for the existence of free gas below the hydrate layer because the gas concentration exceeds solubility below the BSR. This plot is somewhat misleading though, because the gas concentration profile exceeds the solubility values by a larger margin as depth below the B S R increases. We don't however expect the free gas to increase with depth; it should rather be concentrated at a layer immediately below the BSR. This is probably due to the  NUMERICAL  SOLUTION  41  Hydrate volume fraction  Temperature  Figure 3.7: Hydrate has reached the base of the HSZ after fa 700 ky. Free gas is now apparent below the base of the HSZ.  dc no-flux boundary condition (e.g. -^S- at the base of the computational domain. The salt concentration also increases because more salt has been excluded due to increased hydrate formation.  3.7  Summary  The numerical implementation developed in this chapter for the mathematical model presented in the second chapter is now ready to be tested. Chapter four outlines the procedure used for obtaining the biogenic production rate required by the governing equations from an estimate of total organic carbon.  Results are presented for different sedimentation  NUMERICAL  SOLUTION  42  rates and a range of total organic carbon contents to gauge typical requirements of these quantities that permit hydrate formation in the marine environment.  43  Chapter 4 Results Nature never tends to act in a simple way. [Bernoulli].  4.1  Sedimentation Rate and Biogenic Production  The two parameters which are the most poorly constrained in the governing equations are the sedimentation rate and the biogenic production rate. Hydrate occurrences are observed mainly on continental margins which are known to have fast sedimentation rates and high levels of biologic activity. The sedimentation rates in these areas vary with location but estimates are available. The biogenic production rate is not so well constrained.  It is  dependant on the available organic carbon content in the ocean and the efficiency of the conversion of this carbon into methane. The abundant biological activity in the surface water ensures an adequate supply of organic material in the sediments which is a factor that makes continental margins viable for hydrate formation. Biological activity in the sediments is probably a function of the carbon supply. In the model calculations, the production rate is explicitly related to the carbon supply. The amount of total organic carbon in the sediments (hereafter referred to as T O C ) limits the amount of methane that can be produced and therefore the biogenic production of methane cannot exceed the limits imposed by the T O C estimates. This constraint on the biogenic production rate is described in the next section. Applying this method, a range of T O C values required for hydrate production in marine sediments can be found.  RESULTS 4.1.1  44  Obtaining Estimates of Biogenic Production Rates from T O C  The T O C is the carbon content in marine sediments (e.g. useable carbon). The rate c  g  at which gas is added to the system due to decaying organic matter is clearly limited by the T O C , but there is no straightforward relationship. This section describes the method used to obtain a biogenic production rate for a given T O C value. The assumption is made that the available amount of methane is produced at a rate such that a parcel of organic material is fully depleted when it reaches the base of the computational domain. Results from the Blake Ridge show that an extremely well developed and distinct B S R occurs when T O C values are roughly 1.5% [Paull et al., 1996a]. These values are high when compared with those obtained previously by Paull et al. [1994], who claimed that continental rise settings rarely have T O C exceeding 1%. Not all of the organic carbon is available to microorganisms and it is commonly assumed that only half of the T O C is metabolizable, although lower fractions have also been suggested. A representative case is presented here for obtaining a biogenic production rate for T 0 C = 1 % . Values of T O C define the mass of carbon as a fraction of the mass of sediments (dry weight). If it is assumed that half of the T O C is metabolizable, the mass of carbon per cubic meter is given by = 0.005(1 - 4>)ps  M  c  (4.1) = 7.8 k g / m  3  where an average porosity of <f> ~ 0.4 has been assumed and p is the density of the 3  sediments. Following Paull et al. [1994], methanogenesis is assumed to occur in two stages as described by (4.2). If CH 0 is the typical carbon, then the reactions 2  2(7 # 0 — • CH + C0 4  2  2  C0 + 4H —> GH + 2H 0 2  2  4  2  (4.2)  RESULTS  45  produce one mole of methane for each mole of organic carbon. The molar mass of the typical carbon CH 0  is 30 g/mol and combining this with the value of Mc from (4.1),  2  gives 260 moles of CH 0 in the cubic meter sample. Assuming the 1:1 conversion of CH 0 2  2  into CH4, the mass of methane contained in the sample can be calculated as M  = (260 mol/m )(0.016 kg/mol) 3  CHi  (4.3) = 4.16 k g / m  3  The gas concentration c is defined as the ratio of the mass of gas to the combined mass of g  gas and fluid. The maximum gas concentration that can develop due to decaying organic matter, assuming that no diffusion occurs is given by 3  M  4.16 kg/m  CHi  "  M +M i  9  CHi  404.16 k g / m  f uid  3  = 0.0103.  (4.4)  Scaling this value of c™ with the representative scale Ac = 0.003 gives the dimensionless ax  g  9  value -rnax 9  =  ^max _g  Ac,9  =  3 43^  (4.5)  The biogenic production rate is chosen such that all of the organic material is consumed over the computational domain. This is simply a mathematical convenience. Since the base of the hydrate is located in the middle of the computational domain, half of the methane is produced in the hydrate zone and half is produced below. The biogenic production rate c is determined by the expression g  ^jnax 1,  = IT  <«)  where At is the time required for sediments to traverse the computational domain at a prescribed sedimentation rate. The biogenic production rate is chosen such that a parcel of organic material will have converted all of its available decayable organic matter by the  46  RESULTS  time that it exits the computational domain. Assuming a dimensionless sedimentation rate of  the time to traverse the dimensionless distance z = 1 is given by  (4.7) When expression (4.7) is combined with (4.6), the biogenic production rate is given by (4.8) where the sedimentation rate is prescribed and c™ can be readily calculated for a given ax  T O C content using equation (4.5).  4.1.2  Organic Carbon Requirements for Hydrate Formation  A n estimate of the T O C required for hydrate production is investigated by experimenting with different rates of biogenic production. The goal is to show the amount of organic carbon that is essential for hydrate formation at various sedimentation rates. Combinations of sedimentation and production rates are sought which most closely match the regions of observed and inferred hydrate estimates. By obtaining the combinations of parameters that allow hydrate formation, model results can be tested and applied to actual hydrate settings. Sedimentation rates and T O C values from specific sites can be used as inputs to the model, allowing the hydrate volume fractions predicted by the model to be compared with observations from the sites.  Once required parameter ranges are determined for  hydrate formation, worldwide hydrate occurrences can be estimated.  47  RESULTS  Hydrate volume fraction TOC(%) 0.25 0.5 1.0 1.5  H  - 22cm/ky 0 0.043 0.186 0.395  H  = llcm/ky 0 0 0.118 0.411  Table 1: Hydrate volume fractions as a function of T O C and sedimentation rate. Results were obtained after approximately 12 million years. 4.1.3  Hydrate Volume Fraction Profiles for Ranges of T O C and Sedimentation Rate  Numerical results produced by my model for a sedimentation rate of 22 cm/ky [Borowski et ai, 1996] are compared with observed hydrate occurrences on the Blake Ridge. This sedimentation rate is thought to be typical for the Blake Ridge region, but smaller values have also been reported [Kvenvolden, 1985]. Consequently, a more conservative estimate of 11 cm/ky is also used in the computations. Table 1 illustrates the hydrate volume fractions for the two different sedimentation rates over a range of T O C values. Two important and visceral conclusions are evident in the plots of hydrate volume fraction shown in figure 4.1. Raising the T O C increases the hydrate volume fraction because more organic carbon produces more methane through biogenic production. As more methane is added to the system, the gas concentration will eventually exceed the solubility value in the hydrate stability zone and any additional methane introduced will allow hydrate formation to commence. It is also apparent in figure 4.1 that a faster sedimentation rate corresponds to a larger hydrate volume fraction for a given value of T O C . For a T O C of 0.5%, the slower sedimentation rate does not permit any hydrate to form whereas the faster sedimentation rate yields enough methane for hydrate production. At 1.0% T O C , the faster sedimentation  48  RESULTS  Hydrate volume fraction 0.1  02  0.3  0.4  0.5  Figure 4.1: Effects of increasing total organic carbon content on the hydrate volume fraction. T O C values are 0.5% in (a), 1% in (b) and 1.5% for (c). Two sedimentation rates are shown for each T O C value. A sedimentation rate of 11 cm/ky is represented by the sohd curve and the 22 cm/ky sedimentation rate is illustrated with a dashed line. Time is « 12 M y for all three plots.  RESULTS  49  rate produces a larger hydrate volume fraction at the base of the HSZ. The dissociation and upward migration of methane cause hydrate to build up at the base of the HSZ (as described in § 2.1). A faster sedimentation rate supplies a greater amount of organic material that will subsequently be converted into methane, and therefore the solubility limit is achieved earlier. This fact is also evident in equation (4.8) which shows that a higher sedimentation rate corresponds to a higher biogenic production rate c in the gas equation, (2.51). When g  the T O C is increased further to 1.5%, it appears the sedimentation rate no longer plays a significant role in determining the extent of the hydrate volume fraction since the results are very similar for the two different sedimentation rates. Perhaps this indicates a region where T O C values are a more important factor than sedimentation rate. Since the T O C is the same for the two curves of figure (4.1c), similar hydrate volume fraction profiles indicate that variations in the sedimentation rate are inconsequential for high organic carbon contents.  4.1.4  Gas Concentration Profiles for Ranges of T O C and Sedimentation Rate  It is instructive to compare the preceeding plots of hydrate volume fraction with the corresponding plots of the gas concentration in figure 4.2. When hydrate formation occurs, the gas concentration must be at or above the equilibrium (e.g. solubility) gas concentration. Below the hydrate zone, a gas concentration that lies above the equilibrium value indicates that the solubility has been exceeded and free gas is present. This is particularly important near the base of the B S R where the presence of free gas is thought to create a low velocity zone which plays a significant role in the impedance contrast responsible for producing the B S R [Minshull and White, 1989; Miller et al, 1991]. The gas concentration in figure 4.2a confirms why the hydrate volume fraction was zero in figure 4.1a. The gas concentration had not yet reached the solubility value and so hydrate could not begin to  50  RESULTS Gas concentration 0 K  \  0.5  1  '  '  1_Jj '  2 '  2.5 '  3 I  Figure 4.2: Effects of increasing total organic carbon content on the gas concentration. T O C values are 0.5% in (a), 1% in (b) and 1.5% for (c). Two sedimentation rates are shown for each T O C value. A sedimentation rate of 11 cm/ky is represented by the solid curve and the 22 cm/ky sedimentation rate is illustrated with a dashed line. Time is « 12 My for all three plots.  RESULTS  51  form. However, for the faster 22 cm/ky sedimentation rate, the 0.5% T O C is sufficient to allow for hydrate formation and there is even some free gas appearing below the BSR. For a value of T O C = l % , both sedimentation rates allow the gas concentration to reach the equilibrium value and therefore permit hydrate production. Free gas exists for both sedimentation rates however a higher concentration is obtained with the faster sedimentation rate (i.e. the gas concentration exceeds the solubility curve by a larger amount). Environment conditions under which free gas does and does not exist can be definitively determined from the numerical calculations, but the amount of free gas that exists and the thickness of the free gas layer are difficult to quantify with model results. This is due to the no flux boundary condition at the base of the HSZ. The no flux boundary condition means that gas cannot diffuse out of the bottom boundary. Consequently the gas accumulates below the HSZ in figure 4.2 and whenever it exceeds the solubility, the excess gas is present in the form of bubbles. This accumulation of gas could be alleviated by shutting off biogenic production below the HSZ or by moving the bottom boundary further below the base of the HSZ. On the other hand, the distribution of free gas below hydrate layers is not well understood and for the purposes of this study, the ability to detect the onset of free gas is satisfactory. When the T O C is increased to 1.5%, the concentration of free gas below the base of the HSZ increases for both sedimentation rates relative to figures 4.2a and 4.2b as a consequence of more methane being produced from the extra organic carbon introduced for a higher T O C value.  4.1.5  Salt Concentration Profiles for Ranges of T O C and Sedimentation Rate  Salinity has been used as a proxy measurement of hydrate volume fraction because as hydrate forms, salt is excluded. Over time, this salt diffuses away so that when a sample from depth is brought to the surface and the hydrate is allowed to dissociate, there appears  52  RESULTS  Salt concentration .0.99  1  1.01  1.02  1.03  1.04  1.05  1.06  1.07  99  1  1.01  1.02  1.03  1.04  1.05  1.06  1.07  ,0 99  1  1.01  1.02  1.03  1.04  1.05  1.06  1.07  a  CX  Q  o 0.1 0.2 0.3 0.4  CX 0.5  Q  0.6 0.7 0.8 0.9 1  CX  Q  Figure 4.3: Effects of increasing total organic carbon content on the salt concentration. T O C values are 0.5% in (a), 1% in (b) and 1.5% for (c). Two sedimentation rates are shown for each T O C value. A sedimentation rate of 11 cm/ky is represented by the solid curve and the 22 cm/ky sedimentation rate is illustrated with a dashed line. Time is & 12 My for all three plots.  RESULTS  53  to be a freshening of salinity values. For the case of the slower sedimentation rate with 0.5% T O C , we saw in figure 4.1a that no hydrate forms and consequently no salt is excluded. The salt concentration in this case (figure 4.3a, solid line) does not deviate from the initial background salinity of seawater (c = 1.0). A 1% T O C value produced hydrates 3  for both sedimentation rates yet larger hydrate volume fractions were produced with the fast sedimentation rate. The salt concentration profiles of figure 4.1b correlate well with the hydrate results as a faster sedimentation rate produces higher salt concentrations, indicating that more salt has been excluded, evidence of hydrate production. The salt concentration results for TOC=1.5% seem somewhat problematic. The hydrate volume fraction plots of figure 4.1c showed that for a T O C value of 1.5%, the hydrate profiles were very similar. If this is the case, similar amounts of salt should have been excluded and the salinities of figure 4.3c should be complimentary. Both salinity profiles are freshened below the HSZ, although not to values in excess of seawater salinity. This discrepancy is probably due to difficulties in resolving the gradients in c for the large hydrate volume fractions s  associated with high T O C and fast sedimentation rates. The absence of freshening above seawater values in both salinity profiles for 1.5% T O C is a result of the large amount of salt excluded during the significant hydrate production. Over time, this salt diffuses into the region below the HSZ but cannot diffuse out of the computational domain as a result of the no flux boundary condition and therefore salt concentration values are raised above seawater salinity. The process of inferring hydrate volume fractions from salinity measurements assumes that the majority of salts excluded in the hydrate formation have diffused away and a background salinity of seawater is assumed as the reference state for freshening.  The  results shown in figure 4.3 indicate that this assumption is not entirely true. In the next section, a comparison is presented between the hydrate volume fraction results achieved  RESULTS  54  when assuming a background salinity of seawater with those obtained using the salinity calculated from numerical solutions of the governing equations.  4.2  Inferring Hydrate Volume Fractions from Salinity Profiles  Measurements of chlorinity profiles as proxy measurements of hydrate profiles [Hesse and Harrison., 1981; Hesse, 1990] have been used in many hydrate occurrences. The numerical salinity profile that is obtained from my model can be used in a similar manner. Comparisons of the inferred h with the actual numerical h can be used to test the validity of salinity as a proxy measurement. Suppose that the hydrate volume fraction which is computed in the model is brought to the surface. All of this hydrate dissociates just as would be the case for an in situ sample brought to the surface. The dissociation of hydrate due to the pressure changes releases the freshwater stored within the hydrate structure, freshening the pore fluid. An expression can be derived for the salt concentration after freshening by integrating the salt equation, (2.52). It is assumed that a closed sample is brought to the surface so that there is no advection or diffusion effects. In this case, the salt equation can be written as (4.9)  where: c is the assumed background salinity c is the freshened salinity after all hydrate is dissociated h is hydrate volume fraction profile which we wish to obtain. So  3f  a  Integrating (4.9) and solving for the reduced salinity (e.g. freshening), gives the expression cs  f  —  c, (l c  h) a  (4.10)  RESULTS  55  In geochemical analyses, the value of c  3f  is measured once a sample has been brought to  the surface and the value of c has been assumed. A value for h is obtained by assuming So  0  that all of the freshening observed in c is due to the dissociation of hydrate. The results Sf  of my model can be used to simulate the geochemical analysis method and compare the inferred hydrate with the actual amount computed from the model. The measured value for the freshened salinity in this test is obtained using the hydrate volume fraction from the model and assuming that the background salinity is the output salt concentration from the model. Equation (4.10) can be used to arrive at the 'observed' profile for the freshened salinity (shown in figure 4.4a). This freshened salinity profile, c  Sf  is the starting point in  the geochemical analysis as there is no a priori knowledge of the hydrate volume fraction, h (the quantity to be estimated by the geochemical analysis) nor the background salt 0  concentration, c . So  The amount of freshening, and hence the hydrate volume fraction inferred from that freshening, is determined by the assumed background reference salinity. It is typically assumed that the background salinity is just a constant value corresponding to the salinity of seawater [Hesse and Harrison, 1981]. However, model predictions indicate that the salinity profile deviates from a uniform background value. The salinity increases with depth over the domain of the HSZ due to hydrate formation. The hydrate volume fraction predicted using a constant background salinity of seawater is shown in figure 4.4b (dashed line) and compared with the hydrate volume fraction from the model calculations. The hydrate volume fraction inferred from salinity measurements shows some discrepancies with the model output, particularly in the shallower part of the hydrate zone. The depth of first hydrate occurrence and hence the thickness of the hydrate layer is underestimated.  The maximum amount of hydrate occuring at the base of the HSZ is also  underestimated but this effect is small. The most serious consequence is that estimates of  56  RESULTS  Salt concentration 0.8  0.85  0.9  0.95  1  1.05  a  Hydrate volume fraction 0  0.05  0.1  0.15  0.2  0.25  Figure 4.4: Investigating the hydrate volume fraction inferred from salinity measurements. In (a) the salinity output from the model is represented by the sohd curve while the dashed line is the freshened salinity once all of the hydrate output from the model (the sohd curve in (b)) has been dissociated. The hydrate volume fraction inferred using a background salinity of seawater is shown by the dashed line in (b). Results are for TOC=l%, dSjdt = 22 cm/ky, and t 12 My. hydrate volumes from salinity measurements tend to be too low. This discrepancy would probably not be discovered through seismic analysis because the gradual increase in hydrate volume fraction with depth is not sufficient to produce a reflection. Consequently the top of the hydrate zone is difficult to detect by seismic methods.  RESULTS  4.3  57  Summary  Preliminary applications of the model illustrate that the two main factors which influence the model's output are the sedimentation rate and the TOC. As more organic material is made available, more methane is produced. If a large enough amount of methane is supplied to the HSZ, solubility is exceeded and hydrate formation occurs. Faster sedimentation rates transport hydrate to the base of the HSZ and focus hydrate in that region. In the following chapter, these ideas will be applied to typical marine settings so that model outputs can be compared with results and theories for these regions.  58  Chapter 5 Global Implications Submarine gas hydrate, a combination of water and methane, may be the motherlode of hydrocarbon deposits—a pool of organic carbon much larger than global oil reserves, and capable of interacting with the atmosphere to influence climate cycles.  [MacDonald, 1997].  5.1  Some Real World Scenarios to Model  With the knowledge obtained from the results in the previous chapter, I apply my numerical model to specific settings with known hydrate occurrences. Using typical sedimentation rates and organic carbon contents on continental margins, I gauge the amount of hydrate formation and investigate why one particular area may be more apt to develop hydrate than another area. One such location is the Blake, Ridge, a passive continental margin where hydrate existence has been confirmed both by seismic observations of the BSR [Holbrook et al, 1996; Katzman et al, 1996] and through in situ samples [Dickens et al, 1997b]. Detailed studies of this area are very valuable for testing my model with an actual geologic setting. Data published in the literature is used both to constrain the model and to serve as a reference for comparing the model predictions. I also use typical sedimentation rates and TOC contents from active margins to demonstrate that biogenic gas production associated with sedimentation is capable of producing hydrates in settings where hydrates have previously been attributed to removal of methane from rising pore fluids [Hyndman  GLOBAL IMPLICATIONS  59  and Davis, 1992]. The model results also explain why there is little evidence of hydrate occurrence in the deep sea. Since hydrate stability conditions are predominant in the deep sea, the absence of hydrate in these areas is likely a consequence of inadequate gas supply. Hyndman and Davis, [1992] attribute the absence of hydrate in these regions to the lack of a significant gas transport mechanism such as the upward migrating fluids associated with an active margin that is required to concentrate gas within the HSZ. I will use the model to show that hydrate would not be expected in these regions because sedimentation rates in the deep sea are small and the supply of organic matter in these deep sea environments is low (see figure 5.9). Presumably, the lack of organic carbon in the deep ocean basins is due to the lower level of biologic activity as compared with continental margins. The model results indicate that typical sedimentation rates and TOC values in the deep sea will not produce enough gas to permit hydrate formation. The supporting calculations are described in § 5.3. There have also been attempts to estimate worldwide distribution of hydrates [Gornitz and Fung, 1994; Smith and Judge, 1994; Kvenvolden, 1994]. The model results obtained here can be used to generate global estimates of hydrate occurrences. With a knowledge of required sedimentation rate and TOC for hydrate production, global ranges of these parameters can be analyzed to observe locations where my model predicts that hydrate should exist.  5.2  Continental Margins  The results of Chapter 4 demonstrated that relatively fast sedimentation rates and high organic carbon contents are favourable for hydrate production. Typical values for these  GLOBAL IMPLICATIONS  60  quantities at continental margins are 10 cm/ky for sedimentation rate and organic content of 1% [S. Calvert, personal communication, 1998]. Model calculations for these typical continental margin parameters are shown in figure 5.1. After 1.2 My, hydrate has begun to form within the HSZ. The hydrate is confined to the middle region of the HSZ and no free gas is evident in the gas concentration profile. The salt concentration exhibits a salinity increase in the region of hydrate production with a constant value that is slightly above seawater salinity in the bottom half of the computational domain. In twice this time, 2.4 My, hydrate has reached the bottom of the HSZ but the recycling of gas below the HSZ (see § 2.1) has not yet begun. Once the pore fluid reaches solubility values, the gas released through hydrate dissociation will exist as free gas. The gas concentration profile at 2.4 M y in figure 5.1 shows that free gas does not yet exist immediately below the HSZ (although it does appear deeper as a result of the no flux boundary condition imposed in the computations that allows gas to slowly accumulate below the HSZ). This absence of free gas immediately below the HSZ bolsters the implication of the hydrate volume fraction profile which suggests that hydrate has reached the base of the HSZ but not yet begun to focus in that region as a result of gas recycling below the HSZ. The increase of salinity within the HSZ with time corresponds to higher hydrate volume fractions and therefore higher salt concentrations as more water is removed by hydrate formation. Below the HSZ, the profile is freshened as salt concentration values lie below the background salinity of seawater indicating that hydrate has been forced out of the HSZ and consequently the hydrate structure dissociates, releasing fresh water into the surrounding pore fluids. The build up of hydrate at the base of the HSZ is evident in the profiles after 4.8 M y has  GLOBAL  61  IMPLICATIONS  Salt concentration  0.7 0.8 0.9 -  Figure 5.1: Time evolution of hydrate volume fraction, gas concentration and salt concentration for a typical continental margin sedimentation rate (10 cm/ky) and organic carbon content (1%). Results are shown for t=1.2, 2.4, 4.8 and 12 My.  GLOBAL IMPLICATIONS  62  elapsed. Because the hydrate structure holds such large amounts of methane [Hyndman, 1992], this increasing hydrate volume fraction should quickly lead to the gas saturation of pore fluids and the presence of free gas at the base of the HSZ. The transition from large hydrate volume fraction to a region of low velocity free gas can explain the presence of an impedance contrast which produces the B S R in seismic data. In figure 5.1, the hydrate volume fraction peaks at the base of the HSZ, but abruptly drops to zero below this depth, providing an ample supply of fast velocity hydrated sediments. The gas concentration plots exceed solubility values below the HSZ, indicating the presence of low velocity free gas. The salt concentration still exhibits freshening below the HSZ and the effect is even more pronounced at this later time as more hydrate is dissociated below the base of the HSZ. The final time in the calculations is 12 My. The peak volume fraction of hydrate increases by only 2% but is still focussed at the base of the HSZ. Both the amount of free gas inferred from gas concentration plots and the freshening below the HSZ observed in the salt concentration plots have also increased slightly. It appears that changes in the solutions of the various fields have diminished, indicating that a steady state is developing. Biogenic production should have produced at least twice as much gas than existed at the previous time of 4.8 My, yet the gas concentration profile does not exhibit such a significant increase nor is the hydrate volume fraction significantly altered. There is slightly more free gas at 12 M y than at 4.8 M y yet it is insufficient to explain the amount anticipated from a constant biogenic production over this time. What happened to the gas generated through biogenic production over this 7.2 M y period? The amount of gas produced over this time period cannot be completely accounted for by a combination of hydrate, gas dissolved in pore fluids, and free gas which implies that gas is being removed from the system. A comparison between the amount of gas added to and lost from the system should indicate whether a steady state is possible. Because the production rate is known (and dependant  GLOBAL  63  IMPLICATIONS  on sedimentation rate and T O C ) , the amount of gas added to the system as a result of in situ biogenic production is readily calculated for a known sedimentation rate and organic carbon content. Gas is lost from the system because of a diffusive flux out of the top and an advective flux of gas out of the bottom of the computational domain due to transport by pore fluids. PfCpDgS/Cg Z=  0  Z=  .5  Z=  Pf4>c  &  1  9f v  Figure 5.2: Gas fluxes out of the top and bottom of the computational domain. Biogenic production occurs throughout the computational domain. Gas concentration profile illustrated is after 12 M y and used typical continental margin parameter values. If the system is indeed approaching steady state, the gas introduced to the system via biogenic production should be approximately balanced by the gas lost due to fluxes out of the top and bottom of the computational domain. The estimate of the gas lost in (5.1) is slightly lower than the gas produced, according to (5.2). The gas that is produced in excess of the amount lost is incorporated into the hydrate structure and into free gas below the HSZ because both the hydrate volume fraction and gas concentration increase slightly from 4.8 M y to 12 My.  GLOBAL  IMPLICATIONS  64  • Loss of gas from system: qtOp + qb0t = DgVCg  + CgVf  (5.1) = 2.28 x 1CT  3  ~c  • gas introduced via biogenic production (using method of § 4.1.1)  g  = 2.39  x  1(T  3  (5.2)  The steady state value for hydrate volume fraction occurs at approximately 12% for typical conditions at continental margins. This corresponds well with both hydrate estimates from seismic data [Singh et al., 1993] and in situ sampling [Dickens et al, 1997b]. Without the possibility of hydrates converging to a steady state solution, the age of these deposits would be limited by the time required to develop the observed volume fractions with a given biogenic production rate. However, if these deposits can achieve steady state values, it is feasible that hydrates can exist for much longer periods of time. The model results shown in figure 5.1 indicate that typical continental margins are capable of allowing hydrate formation and accumulation. The question then arises, why are hydrates not prevalent on all continental margins? There are many factors that can contribute to this disparity. The typical values of sedimentation rate and organic carbon content are likely to vary somewhat with different location environments. Recall from the results shown in figure 4.1a of § 4.1.2 that a T O C value of 0.5% was not adequate for hydrate production. Some continental margins may fall below the required threshold of organic material required to permit hydrate formation. There have been cases where low seismic velocities indicate the presence of free gas in the pore fluid and yet no discernable B S R is observed in seismic data [Holbrook et al, 1996]. Perhaps the biggest uncertainty is the efficiency and the processes by which the available organic carbon is converted to methane. We choose a value of 50% for the efficiency of the methanogenesis after Paull et  GLOBAL IMPLICATIONS  65  al. [1994] but lower estimates of 10% [Hyndman and Davis, 1992] and 2% [Claypool and Kaplan, 1974] have been suggested. A suite of results for varying efficiencies suggested  that for typical continental margins (TOC=l%, 2£ = 10 cm/ky) to prod uce hydrate, C Is  an efficiency of 40% is required. Special locations with faster sedimentation rates and higher organic carbon contents may allow for a lower efficiency. However, known values of sedimentation rate and organic carbon from the Blake Ridge indicate that the efficiency must exceed 10% in order to permit the development of hydrate. 5.2.1  Implications for Active Continental Margins  The values used to obtain the results of figure 5.1 were typical continental margin values, and hence are representative of both active and passive margins. The most apparent scenarios for in situ biogenic production are passive margins where the fluid expulsion, intrinsic to active margins, is not a factor. Rising pore fluids will likely play an important role for gas transport in active margins. The results presented above suggest thatfluidexpulsion may, however, not be the dominant cause of hydrate occurence in active environments. In the previous section, I demonstrated that biogenic production in a typical continental margin was sufficient to produce hydrates. Here, sedimentation and in situ biogenic production are found to be sufficient for producing hydrates in typical active continental margin settings. Thefluidexpulsion in active margins may, however, provide an increase in the amount of hydrate or allow hydrate formation in regions with lower than typical values of sedimentation rates by amplifying the amount of gas brought into the HSZ.  GLOBAL  66  IMPLICATIONS Hydrate volume fraction 0.1  0.2  03_  0.4  0.5  Gas concentration 2  3  Salt concentration 1.02  1.04  1.06  1.08  1.1  Figure 5.3: Time evolution of hydrate volume fraction, gas concentration and salt concentration for the Blake Ridge setting (sedimentation rate 22 cm/ky and organic carbon content 1.5%). Results are shown for t•= 0.24, 0.6, 1.2 and 12 My.  GLOBAL IMPLICATIONS 5.2.2  67  Blake Ridge  The Blake Ridge, geographic location shown in figure 5.4, is perhaps the most scrutinized study area of marine gas hydrates. Numerous seismic studies [Holbrook et al, 1996; Rowe  and Gettrust, 1993; Katzman and Holbrook, 1994; Korenaga et al. 1997] and geochemical investigations [Borowski et al., 1996; Paull et al., 1995] have been completed. It is also a location from which in situ hydrate samples have been recovered [Dickens et al., 1997]. 82°  80°  78°  76°  74° W  Figure 5.4: Geographic location of the Blake Ridge. B S R shown in the shaded region was used to infer hydrate occurence. Bathymetric contours are in meters. [After Holbrook et al, 1996]. The results of Chapter 4 indicated that fast sedimentation rates and high organic carbon contents are beneficial conditions for hydrate production within the HSZ. The structure of the Blake Ridge [Markl et al, 1970] is such that sedimentation rates as high as 22 cm/ky [Borowski et al, 1996] and T O C values of up to 1.5% [Paull et al, 1996a] have been reported. Results for this combination of sedimentation rate and organic carbon content are illustrated in figure 5.3. These rather substantial parameters probably represent an upper bound on hydrate production and may not prevail throughout the entire area.  GLOBAL IMPLICATIONS  68  On the other hand, these values are useful for examining the possible behaviours that can conceivably occur in nature. The hydrate volume fraction increases sharply at the base of the HSZ due to recycling of methane. This phenomena was previously observed for typical continental margins in figure 5.1. However, the hydrate reaches the base of the HSZ much faster and obtains concentrations much higher than the typical continental margin setting. A steady state does not appear to be attainable for these high parameter values. In fact, if the solution is allowed to proceed further in time, the hydrate volume fraction continues to grow until eventually /i ~ 1, indicating that the pore space has almost completely filled with hydrate. It appears that the T O C plays a more important role in achieving steady state conditions than the sedimentation rate. The T O C estimated on the Blake Ridge is 50% more than the typical continental margin and therefore should supply more gas to the HSZ for hydrate production. Doubling the sedimentation rate should also increase the gas supply, but the flux of gas transported by pore fluids out of the system at the base of the computational domain is also increased. The case of fully hydrated sediments (e.g. h « 1) implies a very substantial increase in both the gas contained within the hydrate structure, Ch and the free gas (the amount of gas outside of the HSZ in excess of solubility) as a result of the burgeoning T O C content supplying gas to the system faster than the diffusive and advective fluxes can remove gas. A n attempt to resolve the relative importance of these two quantities was investigated by individually increasing each parameter. Model outputs are shown for the two continental margin cases: a typical sedimentation rate with a high T O C (figure 5.5) and a typical T O C with a high sedimentation rate (figure 5.6). Comparing these two sets of plots illustrates that increasing the T O C of a typical continental margin by 50% produces hydrate volume fractions more than three times as large as those achieved for typical continental margins (figure 5.1) and twice as much as the results obtained by  GLOBAL  IMPLICATIONS  69  Figure 5.5: Time evolution of hydrate volume fraction, gas concentration and salt concentration for high TOC continental margin (sedimentation rate 10 cm/ky and organic carbon content 1.5%). Results are shown for t = 0.24, 0.6, 1.2 and 12 My.  GLOBAL  IMPLICATIONS  70  Figure 5.6: Time evolution of hydrate volume fraction, gas concentration and salt concentration for fast sedimenting continental margin (sedimentation rate 22 cm/ky and organic carbon content 1%). Results are shown for t = 0.24, 0.6, 1.2 and 12 My.  GLOBAL IMPLICATIONS  71  doubling the sedimentation rate over the typical continental margin value (figure 5.6). The gas and salt concentrations both increase without any indication of approaching a steady state solution in figure 5.5. The gas concentration increases steadily below the HSZ implying more free gas development with time. The salt concentration in the HSZ region reaches values substantially higher than those for the typical continental margin (see figure 5.1), a consequence of the increase in hydrate volume fraction.  Figure 5.6  illustrates the results for the fast sedimenting, typical T O C environment. Although the hydrate volume fraction and gas concentration both exhibit values larger than the those observed for the typical continental margin case of figure 5.1, the quantities are significantly smaller than those obtained from calculations using the high T O C , typical sedimentation rate parameters in figure 5.5. These comparisons indicate that T O C has a more important influence on the hydrate volume fraction. Changes in sedimentation rate will effect the efficiency with which organic material is brought into regions where it can be converted into methane and subsequently incorporated into the hydrate structure.  It is therefore  expected that changes in sedimentation rate will mainly effect the time scales on which hydrate is produced whereas variations in the T O C will alter the amount of methane that can be obtained through biogenic production and therefore the amount of hydrate that can form. The absence of freshening above seawater salinity values in the salt concentration profiles of both figure 5.5 and figure 5.6 may be related to sharp gradients at the base of the HSZ. Figure 4.3 suggests that freshening above seawater salinity does occur at earlier stages, when the gradients are not as problematic.  GLOBAL IMPLICATIONS  5.3  72  Deep Sea  The absence of hydrate in deep sea regions can be explained by examining the methane generation in marine sediments due to in situ biogenic production [Rice and Claypool, 1981]. Near the seafloor, available organic material is first consumed by aerobic bacteria, producing carbon dioxide which is dissolved in pore fluid or escapes into the atmosphere. If the sedimentation rate is slow, the organic content is low and oxygen is plentiful, it is possible for the organic material to be completely consumed by aerobic bacteria. In regions of rapid sedimentation, anaerobic conditions will occur at depths below the aerobic environment. Sulfates are reduced in the aerobic region which is important because significant methane production does not occur until sulfate concentrations approach zero [Canfield, 1991]. In the deeper anaerobic region, methane is the main product if the sulfates have been removed.  Gas concentration  Figure 5.7: Gas concentration for a sedimentation rate of 11 cm/ky and TOC=0.1% after 12 My. Equilibrium profile is indicated by the dashed curve.  The values assumed for the typical deep sea setting (sedimentation rate 11 cm/ky and T O C = 0.1%) are insufficient to allow hydrate formation to occur. The plot of gas  GLOBAL IMPLICATIONS  73  concentration in Figure 5.7 illustrates this point. I do not include a zone of sulfate reduction in my model and therefore some gas is present near the seafloor after 12 M y due to biogenic production. The production rate corresponding to these typical values is, however, so small that solubility has not yet been reached after 12 M y and therefore hydrate formation cannot commence.  5.4  Global Hydrate Occurrences  A worldwide location map of inferred and and known gas hydrate occurrences of volden  Kven-  [1994] is illustrated in figure 5.8. The marine gas hydrates are predominantly found  along continental margins. This is not surprising given the model calculations presented in this study: marine continental margins provide the unique combination of high organic carbon contents and fast sedimentation rates that produce prosperous conditions for hydrate formation. Can hydrates be expected on continental margins where experiments have not yet been undertaken to unequivocally confirm their existence? Can an areal extent of hydrate occurrence be estimated? Hydrates are found in diverse marine geologic setting such as the tectonically active Middle America Trench, the passive continental margin of the Blake Ridge, the rapidly sedimenting Gulf of Mexico and even in the Black Sea. This multifarious nature of occurrences implies that hydrate formation and accumulation can develop in sediments ranging from the coarse sands of the Alaskan North Slope to fine-grained hemipelagic muds and volcanic ash of the Middle America Trench  [MacDonald,  1990]. The fact that hydrates  have been observed over such a varying range of geologic settings suggests that geologic conditions are not the main factor. Instead, my model suggests that the amount of available organic matter and the sedimentation rate are the conspicuous factors that permit  GLOBAL  IMPLICATIONS  74  Figure 5.8: Worldwide locations of known and inferred gas hydrate deposits. Permafrost regions are indicated with squares, marine settings with circles and diamonds indicate regions where hydrate samples have been recovered, from [Kvenvolden, 1994]. hydrate growth. Estimates of the magnitude and spatial distribution of marine hydrates have been made by Gornitz and Fung [1994] for both the pore fluid expulsion and in situ biogenic production models. A measure of global organic content was calculated using data from the coastal zone colour scanner (CZCS). Marine chlorophyll concentrations measured by the CZCS are used as a proxy measurement of recently deposited carbon that is available for biogenic methane production. The results of Gornitz and Fung [1994] present an interesting comparison of the two gas supply theories and a prediction of global inventories of hydrate. However, the relationship between surface water productivity and hydrate occurrence is crude. Any area above 1% T O C is assumed to permit hydrate formation without any justification for why this particular choice of required carbon content was adopted. The results of my model indicate that sedimentation rates are also important, especially in  GLOBAL  75  IMPLICATIONS  areas where sedimentation rates may be high enough to allow hydrate formation with low T O C values (see figure 4.1a) or change the amount of hydrate with depth for a given T O C (see figure 4.1c). A more quantitative, realistic model of the hydrate formation process, such as the one presented in this thesis could substantially improve these global estimates. 120*  140'  ISO*  l«0*  160*  140*  120"  100*  80*  60"  40*  120"  140*  160"  180'  160"  140"  120"  100'  80'  60'  40'  °•!•;' mm. ••• t -i.oo %  ORGANIC  20*  O*  20*  40*  Vf  20*  40*  60*  60*  *00*  '2Q*  100  120  i .01 - 2.00 CARBON  Figure 5.9: Organic carbon distribution in the sediments of world oceans and seas, from [Premuzic et al., 1982]. Although detailed assessments of global inventories of hydrate are beyond the scope of this study, a few general comments are warranted. A knowledge of global organic content in marine sediments allows the results of my model to be extrapolated to link global estimates of T O C with hydrate occurrences. Consider the map of Premuzic et al. [1982] in figure 5.9 illustrating the distribution of organic carbon in the sediments of the worlds oceans and seas. The highest organic contents predominantly occur along the continental margins, corresponding very well with the observed and inferred hydrate locations from figure 5.8. I can now investigate global hydrate occurrences in a similar fashion to Gornitz and Fung  GLOBAL IMPLICATIONS  76  [1994] by observing which areas have sufficient organic carbon. We are not constrained to pick a single threshold value of T O C that will definitively allow for hydrate production because the information obtained from the model results indicates that there are other contributing factors, most notably, sedimentation rates. The organic carbon values in the deep sea are too low to permit hydrate formation based on my model's results. M y results indicate that the top two categories of T O C (i.e. 1% < T O C < 2%) should supply sufficient organic material for hydrate development. From figure 5.9, this range indicates that hydrates could be expected along the western coast of the American continents as well as in the surrounding areas of Japan, India, and much of the Asian and African Continental margins. The middle category in the organic carbon contents (0.5% < T O C < 1%) is likely to produce hydrates in some areas. For example, both the Blake Ridge and the Gulf of Mexico, which are known to contain gas hydrates have T O C values in this intermediate range which indicates that the fast sedimentation rates of these regions must contribute to permit hydrate formation. I suggest that in the intermediate range of T O C values, 0.5% < T O C < 1% local conditions such as sedimentation rates are the determining factor in hydrate production while the regions that fall within the higher range T O C are more certain to produce hydrates because of the abundance of available organic matter.  5.5  Summary  By applying the model to specific settings, it was found that values of sedimentation rate and organic carbon content for a typical continental margin are sufficient to produce hydrates through in situ biogenic production on both active and passive continental margins. A steady state was obtained for typical continental margins which could be important in determining long period trends of hydrate development. The effects of organic carbon content and sedimentation rates on the hydrate production were investigated separately and  GLOBAL IMPLICATIONS  77  it was found that increases in the sedimentation rate tended to effect the time scales on which the hydrate is produced while increasing the T O C generates more gas and therefore provides larger hydrate volume fractions. Results of the model are extrapolated to observe global trends of hydrate occurrences using a map of organic carbon contents of the oceans.  78  Chapter 6 Conclusions The ultimate goal of this work was the development of a theoretical model to describe the formation and accumulation of gas hydrates along passive continental margins. Continual sedimentation and compaction are key elements of the seafloor environment which are accounted for in the model. A quantitative description of the physical processes associated with hydrate growth is essential to understanding the origin of marine gas hydrates. The model was subsequently applied to the marine setting and competing ideas about hydrate formation were tested through model simulations. Governing equations were derived from the conservation of mass, energy, gas, and salt. Compaction effects were incorporated into the model through the choice of an exponentially decaying porosity distribution which remained fixed relative to a moving sedimentation front. The governing equations were derived in full vector form but since hydrate occurrences are often in regions where the gradients in temperature, gas, and salt are predominantly vertical, 1-D depth dependent versions of the governing equations were formulated to facilitate numerical solutions. The numerical method of lines was used to compute solutions. Special consideration was required to account for the discontinuity in hydrate volume fraction that occured at the base of the hydrate stability zone. A n estimate of biogenic gas production was based on values of the total organic carbon content in the sediments. The gas production enters the governing equations as a source term which represents the generation of gas through the decay of organic material. The two parameters which were the most poorly constrained were the biogenic production rate and the sedimentation rate. A range of values for these two parameters were investigated  CONCLUSIONS  79  and it was found that typical continental margin values were sufficient to allow for hydrate formation. Increasing the amount of available organic carbon corresponded to increases in the hydrate volume fraction predicted by the model. A n increased sedimentation rate concentrated hydrate deeper into the sediment column, eventually reaching the base of the hydrate stability zone. A n argument was made for the viability of in situ biogenic production along active margins. Typical sedimentation rates and organic carbon along active margins should be sufficient to form hydrate. The migrating fluids, which are indigenous to active margins, are likely to expedite the transport of gas and perhaps increase the amount of hydrate present or the speed of the accumulation process. Typical deep sea values for sedimentation rate and organic content do not produce adequate gas to initiate hydrate formation. The evolution of a hydrate containing region was observed through the analysis of time varying solutions. For typical continental margin values, growth in the hydrate volume fraction corresponded to an increase in the free gas below the HSZ as larger volumes of hydrate are forced out of the HSZ, dissociating and releasing gas from the hydrate structure when it is no longer stable. The salt concentration increases in hydrate producing regions, indicating the exclusion of salts. Below the HSZ, freshening is evident in the salinity profiles as the dissociation of hydrate releases fresh water. A steady state solution was investigated for the typical continental margin setting which could have important repercussions on the age of hydrate regions. A range of parameter values which were found to permit hydrate formation were extrapolated to address global occurrences. Global maps of organic carbon contents were considered to address the question of global occurrences. There appear to be two main regions where hydrates should arise, both occuring along continental margins. In the first  CONCLUSIONS  80  region, organic carbon contents are more than adequate to produce hydrate based on my model results. The second region involves areas which are borderline cases for having sufficient available carbon to permit hydrate formation. Because some of these areas have been identified as hydrate bearing regions, I postulate that the fast sedimentation rates characteristic to these regions make hydrate production viable. Ideas to expand on the current model include the addition of another equation to explicitly account for the free gas produced below the hydrate stability zone and the application of the model to a wider range of marine settings. Modelling results presented here can be extended and applied to better constrain predictions of global hydrate estimates. The insights gained through this work into the physical processes of the hydrate environment are essential for furthering knowledge of the many phenomena associated with marine gas hydrates.  81  References Appenzeller, T., Fire and ice under the deep-sea floor, Science, 252, 1790-1792, 1991. Bangs, N . L. B . , D. S. Sawyer, X . Golovchenko, The cause of the bottom-simulating reflection in the vicinity of the Chile triple junction, Proc. Ocean Drill. 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Cox, pp. 63-72, Butterworth, Boston, 1983. Lee, M . W . , D. R. Hutchinson, W . P. Dillon, J . J . Miller and W . F . Agena, Method of estimating the amount of in situ gas hydrates in deep marine sediments, Marine and Petroleum Geology, 10, 493-506, 1993. Loehle, C , Geologic methane as a source for post-glacial C O 2 increases: the hydrocarbon pump hypothesis, Geophys. Res. Lett., 20, 14, 1415-1418, 1993. MacDonald, G . J . , The future of methane as an energy resource, Annu. Rev. Energy, 15, 53-83, 1990. MacDonald, I. R., Bottom line for hydrocarbons, Science, 385, 389-390, 1997. Madsen, N . K and R. F . Sincovec, The numerical solution of nonlinear partial differential equations, Computational Methods in Nonlinear Mechanics, J . T. Oden et al., Eds., Texas Institute for Computational Mechanics, Austin, Tex., 371-380, 1974. Makogon, Yu. F., Hydrates of Natural Gas, translated from Russian by W . J . Cieslewicz, 237 pp., PennWell, Tulsa, Okla., 1981. Markl, R. G . , G. M . 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Walsh, The nature and distribution of organic matter in the surface sediments of world oceans and seas, Org. Geochem., 4, 63-77, 1982. Rempel, A . W . and B . A . Buffett, Formation and accumulation of gas hydrates in porous media, J. Geophys. Res., 102, B5, 10151-10164, 1997. Rempel, A . W . , Theoretical and experimental investigations into the formation and accumulation of gas hydrates, [Master's Thesis]: University of British Columbia, 122 p., 1994. Rice, D. D. and G . E . Claypool, Generation, accumulation, and resource potential of biogenic gas, Bull. Am. Assoc. Pet. Geol., 65, 5-25, 1981. Rothwell, R. G., J . Thomson and G. Kahler, Low-sea-level emplacement of a very large Late Pleistocene 'megaturbidite' in the western Mediterranean Sea, it Nature, 392 pg. 377-380, 1998. Rowe, M . M . and J . F . Gettrust, Fine structure of methane hydrate-bearing sediments on the Blake outer ridge as determined from deep-tow multichannel seismic data, J. Geophys. 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H . Abass, M . S. Selim, and E. D. Sloan, Experimental and theoretical investigation of methane-gas-hydrate dissociation in porous media, SPERE, (February 1991), 69-76. Yuan, T., R. D . Hyndman, G . D. Spence and B . Desmons, Seismic velocity increase and deep-sea gas hydrate concentration above a bottom-simulating reflector on the northern Cascadia continental slope, J. Geophys. Res., 101, B6, 13655-13671, 1996.  REFERENCES  86  Zatsepina, 0 . Ye. and B . A . Buffett, Conditions for the stability of gas hydrate in the seafloor, submitted to J. Geophys. Res., 1998. Zatsepina, 0 . Ye. and B. A . Buffett, Phase equilibrium of gas hydrate: Implications for the formation of hydrate in the deep sea floor, Geophys. Res. Lett., 24, no. 13, 1567-1570, 1997.  87  Appendix A Nomenclature and Notation The table below lists the symbols used throughout the text and the quantities which they represent. * bottom simulating reflector  BSR  equilibrium gas concentration gas concentration dissolved in pore fluid and in free gas  9  C  C  dimensionless biogenic gas production rate  9  reference gas concentration  9o  C  Ac  maximum solubility of methane is seawater  Ch  gas concentration in hydrate structure  C,  salt concentration dissolved in pore fluid  C,o  reference salt concentration  g  Ac  seawater salinity  s  C  specific heat capacity at constant pressure  CZCS  coastal zone colour scanner  p  Ci(t), C (t)  integration constants  D  chemical diffusion coefficient for gas  D  chemical diffusion coefficient for salt  h  hydrate volume fraction  H  enthalpy  HSZ  hydrate stability zone  K  reaction rate constant  2  9  s  t Symbols that appear with a  in the text represent dimensionless quantities  NOMENCLATURE  AND NOTATION  K (h)  effective thermal conductivity  Kd(h)  effective thermal diffusivity  L  depth well outside of hydrate stability zone  Lh  latent heat of formation  M  mass  NML  numerical method of lines  <p  porosity  p  density  e  pC (h)  average heat capacity of the sediment matrix  dS  sedimentation rate  p  ~dt  88  T  temperature  T  reference temperature  AT  typical temperature variation over the computational region  td  thermal diffusion timescale  V  volume  Vf  fluid velocity  v  s  sediment velocity  Vd  reference velocity  0  Subscripts and Superscripts /  fluid  h  hydrate  9  gas  s  salt  +  above the hydrate discontinuity below the hydrate discontinuity  89  Appendix B Resume — Abstract  in French translation by Stephane Rondenay  Les hydrates marins gazeux se retrouvent generalement le long des marges continentales, ou i l existe une combinaison unique de conditions de pression et de temperature stables ainsi qu'un apport adequat de gaz pour generer cette substance, qui est comparable a de la glace. Si elles etaient relachees, les vastes quantites de methane contenues dans les hydrates pourraient affecter le chmat global de facon significative. L'abondance de gaz present dans ces structures pourrait aussi representer une ressource energetique potentielle dans le futur, et ce malgre les complications reliees a sa recuperation, qui rendent presentement l'operation non viable au point de vue economique. Dans le passe, l'acces difficile aux hydrates gazeux a entrave l'etude du miHeu naturel dand lequel i l sont formes'. Consequemment, de nombreux aspects ayant attrait aux procedes controlant la formation des hydrates en milieux marin restent non resolus. Un modele quantitatif pour la formation et l'accumulation des hydrates gazeux dans les sediments marins en compaction est presente dans cette etude. Les pricipes de conservation sont utilises afin de developper un modele mathematique controle par une serie d'equations directrices qui representent les phenomenes physiques mis en jeux. L a methode numerique des Hgnes est employee afin d'obtenir les solutions pour la variation temporelle (i) de la fraction volumique des hydrates, (ii) de la temperature, (iii) du gaz et (iv) du sel. On impose un saut des conditions frontiere a la base de la zone de stabilite des hydrates, ou i l peut exister une discontinuity de leur fraction volumique. La grande variabilite de la distribution geographique et geologique des hydrates indique que la quantite de matiere organique et les conditions dans lesquels elles se deposent representent les facteurs controlant la croissance des hydrates a l'interieur des zones de stabilite. Le modele presente ici permet d'explorer une variete de conditions de formation, en considerant differents taux de sedimentations et differentes quantites de carbone organique comme donnees initiales. Les niveaux de ces parametres necessaires a la formation d'hydrates sont etudies en examinant un eventail de  RESUME  -  ABSTRACT  IN  FRENCH  90  taux de sedimentation et de quantites totales de carbone organique. Les valeurs typiques de quantite de carbone organique et de taux de sedimentation dans les marges continentales sont suffisantes pour expliquer revolution des hydrates a la fois dans les marges actives et passives. Dans ces deux cas, le gaz est genere par production biogenique  in  situ.  Des conditions specifiques sont aussi modelisees en introduisant  des valeurs representatives de taux de sedimentation et de contenu organique total dans les calculs. On considere que les taux de formation des zones d'hydrates correspondent au temps necessaire pour developper les volumes actuels de la fraction d'hydrates par production gazeuse biogenique. La possibilite qu'a le modele d'atteindre un etat stationnaire pour des conditions typiques aux marges continentales indique que les hydrates pourraient tres bien avoir existe pour de tres longues periodes de temps. Les hydrates sont inexistants dans les regions oceaniques profondes en raison de l'insuffisance de matieres organiques et des faibles taux de sedimentation. Une augmentation du contenu organique total par rapport aux valeurs typiques des marges continentales a pour effet d'accroitre le volume de la fraction d'hydrates, puisque la production biogenique augmente la quantite de gaz disponible pour la formation d'hydrates a partir du surplus de carbone organique disponible. L'augmentation du taux de sedimentation n'egendre pas d'accroissement significatif de la formation d'hydrates, mais modifie l'echelle temporelle de developpement des hydrates en raison de la variation du taux d'ensevelissement de la matiere organique a partir de laquelle le gaz est produit. Les distributions globales de carbone organique total sont analysees, et les resultats obtenus par modelisation sont utilises pour predire quelles sont les regions dotees des conditions ideales pour la formation des hydrates.  91  Appendix C Zusammenfassung  —  Abstract in German translation by Carl-Georg Bank  Marine  Gashydrate  findet  man  haufig entlang  Kontinentalrandern,  wo eine  auflergewohnliche Kombination herrscht von stabilen Druck- und Temperaturbedingungen sowie ausreichende Gaszufuhr, welche zur Bildung dieser eisartigen Strukturen fiihren. Eine Freisetzung der riesigen Menge an Methan aus diesen Hydratstrukturen konnte das globale Klima entscheidend beeinflussen. Obwohl zur Zeit die Gewinnung von Gas aus Hydraten unwirtschaftlich ist, konnte das Gas in der Zukunft eine nicht zu unterschatzende Energiequelle darstellen. Die Entlegenheit der Vorkommen behindert ihre Erforschung und deshalb sind viele Aspekte der physikalischen Prozesse, die die Bildung von Hydraten im marinen Umfeld kontr oilier en, unbekannt. Wir zeigen hier ein quantitatives Modell, welches die Bildung und Akkumulation von Gashydraten in kompaktierenden Meeressedimenten beschreibt. Mit Hilfe von Erhaltungsprinzipien entwickeln wir ein mathematisches Modell, welches die physikalischen Prozesse in einem Gleichungssystem verkorpert. Wir erhalten zeitlich variante Losungen fur den Volumenanteil an Hydrat, Temperatur, Gas und Salz tiber die numerische Methode der Linien. A n der Basis der Hydrat-Stabilitatszone fiihren wir eine Sprung-Randbedingung ein, weil dort eine Diskontinuitat im volumetrischen Hydratanteil herrscht. Die stark variierende geographische und geologische Verteilung von Hydraten deutet darauf hin, dafi Anteil und Ablagerung an organischem Material das Wachstum von Hydrat innerhalb der Hydratstabilitatszonen kontr oilier en. In unserem Modell kann eine Auswahl dieser Faktoren getroffen werden durch die Betrachtung unterschiedlicher Sedimentationsraten und unterschiedlicher Kohlenstoff-Inputs. Durch Vergleich einer Reihe von Sedimentationsraten und Anteilen an organischem Kohlenstoff werden die zur Hydrat bildung notwendigen Niveaus bestimmt. Typische an Kontinentalrandern vorzufindenden Anteile an organischem Kohlenstoff und Sedimentationsraten sind ausreichend zur Bildung von Hydraten sowohl an aktiven  ZUSAMMENFASSUNG - ABSTRACT IN GERMAN  92  als auch passiven Kontinentalrandern. In beiden Fallen wird das Gas durch in situ biogene Prozesse zugefiihrt.  Spezielle Umstande werden modelliert durch representative Werte  von Sedimentationsraten und organische Anteile in den Berechnungen. Wir nehmen an, dafl Hydrat-Formationsraten der Zeitdauer entsprechen, die notig ist um heutige Volumenanteile an Hydrat aus der Produktion von biogenem Gas zu bilden. Fur typische Kontinentalrander hat man zeitinvariante Zustande nachgewiesen. Dafi dies moglich ist, deutet auf eine lange Geschichte der Hydrate hin. Man findet keine Hydrate im tiefen Ozean, weil dort weder genug organisches Material noch ausreichende Sedimentationsraten zur Verfugung stehen. Vergrofiern wir den Anteil an organischem Material an Kontinentalrandern in unseren Modellen, erhalten wir einen grofieren volumetrischen Anteil an Hydrat, weil das zusatzlich zur Verfugung stehende organische Kohlenstoff durch biogenen Einflufl in Methan umgewandelt wird, der wiederum in Hydrate eingebaut wird. Steigern wir die Sedimentationsrate in unseren Modellen, vergrofiern wir nicht so sehr den Anteil an Hydrat, aber die Zeitspanne in welcher sich Hydrat bildet. Dies liegt an der Anderung der Sedimentation von organischem Material, die ihrerseits die Gasbildungsrate durch Verwesung desselben beeinflufit. Wir analysieren die globale Verteilung an organischem Kohlenstoff und bringen sie in Zusammenhang mit unseren Modellrechnungen um vorherzusagen, wo giinstige Bedingungen fur die Hydratbildung herrschen.  


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