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

MODELING NATURAL GAS HYDRATE EMPLACEMENT: A MIXED FINITE-ELEMENT FINITE-DIFFERENCE SIMULATOR Schnurle, Philippe; Liu, Char-Shine; Wang, Yunshuen Jul 31, 2008

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Proceedings of the 6th International Conference on Gas Hydrates (ICGH 2008), Vancouver, British Columbia, CANADA, July 6-10, 2008.  MODELING NATURAL GAS HYDRATE EMPLACEMENT: A MIXED FINITE-ELEMENT FINITE-DIFFERENCE SIMULATOR Philippe Schnurle ∗ Institute of Oceanography National Taiwan University PO BOX 23-13, Taipei 106 TAIWAN Char-Shine Liu Institute of Oceanography National Taiwan University PO BOX 23-13, Taipei, TAIWAN Yunshuen Wang Central Geological Survey, MOEA P. O. Box, 968, Taipei, TAIWAN ABSTRACT Gas hydrates are ice-like crystalline solids composed of a hydrogen bonded water lattice entrapping low-molecular weighted gas molecules commonly of methane. These form under conditions of relative high pressure and low temperature, when the gas concentration exceeds those which can be held in solution, both in marine and on-land permafrost sediments. Simulating the mechanisms leading to natural gas hydrate emplacement in geological environments requires the modeling of the temperature, the pressure, the chemical reactions, and the convective/diffusive flow of the reactive species. In this study, we take into account the distribution of dissolved methane, methane gas, methane hydrate, and seawater, while ice and water vapor are neglected. The starting equations are those of the conservation of the transport of momentum (Darcy’s law), energy (heat balance of the passive sediments and active reactive species), and mass. These constitutive equations are then integrated into a 2-dimentional finite element in space, finite-difference in time scheme. In this study, we are able to examine the formation and distribution of methane hydrate and free gas in a simple geologic framework, with respect to geothermal gradient, dewatering and fluid flow, the methane in-situ production and basal flux. The temperature and pressure fields are mildly affected by the hydrate emplacement. The most critical parameter in the model appears to be the methane (L+G) and hydrate (L+G+H) solubility: the decrease in methane solubility beneath the base of the hydrate stability zone (BHSZ) critically impacts on the presence of free gas at the base of the BHSZ (thus the presence of a BSR), while the sharp decrease of hydrate solubility above the BHSZ up to the sea bottom critically impact on the amount of methane available for hydrate emplacement and methane seep into the water column. Keywords: gas hydrates, methane solubility, finite-elements, simulation  ∗  Corresponding author: Phone: +8862 2736 4029 Fax +8862 2362 6092 E-mail: schnurle@oc.ntu.edu.tw  NOMENCLATURE ci Specific heat [J/kg/K] Dij Bulk diffusivity [m2] ei Compressibilty [Pa-1] Fi Generation term [kg/ m2/s] g Gravitational acceleration [9.81 m/s2] Hi Enthalpie [kg] i Index of reactive specie [w,g,h,d] j Index of phase [a,g,s] ki Permeability [m2] L Latent heat [430 KJ/kg] Mi Mass fraction [kg] Pi Partial pressure [Pa] T Temperature [K] Ui Darcy flux [Kg/m2/s] ∇Gradient (∂/∂x, ∂/∂y) λi Thermal conductivity [W/K/m] ηi Viscosity [kg/m/s] Φ Porosity ρi Density [kg/m3] INTRODUCTION In the last decade, conceptual models for the kinetics of gas hydrate formation and dissociation in porous media (e.g. [1,2]), as well as for the transport of fluid and gas leading to natural gas hydrate emplacements (e.g. [3,4,5,6]) have considerably matured. Gas hydrate form under conditions of relative high pressure and low temperature, when the gas concentration exceeds those which can be held in solution, both in marine and on-land permafrost sediments. Thus, simulating the mechanisms leading to natural gas hydrate emplacement in geological environments requires the modeling of the temperature, the pressure, the chemical reactions, and the convective/diffusive flow of the reactive species. Two chemical components (i=methane, water), and 3 phases (j=aqueous, gas and solid) are considered, resulting in 6 unknown mass fractions Mij. In this study, I take into account the distribution of dissolved methane (Md=Mma), methane gas (Mg=Mmg), methane hydrate (Mh=Mms), and water (Sw=Mwa), while water vapor (Mv=Mwg), and ice (Mi=Mws) are neglected. Through time, the concentrations of these species evolve due to transport in the porous medium: the fluid flow (Uf), and gas flow (Ug) are responsible for the convection of water, dissolved methane, and free gas; The hydrate phase is fixed within the sediment matrix. Chemical species are also subject to diffusion. Thus the starting equations are those  of the conservation of the transport of momentum (Darcy’s law), energy (heat balance of the passive sediments and active reactive species), and mass: Ui = -ki /ηi (∇Pi + ρi g); where i=w,g (1) [Φ∑ρi∂Hi/∂t + (1-Φ)ρscs∂T/∂t + ∑Ui∇Hi] = ∇ (k∇T) , where i=w, g, h, d (2) [∂ΦMi/∂t + ∑Uj∇Mij] =∇ (Φ∑Dij∇Mij) + Ri , where j= w, g and i=w, g, h, d (3) The boundary conditions include some surfaces on which values of the problem unknowns are specified (e.g. Ui=e), for instance points of known temperature or species concentration. Some other surfaces may have constraints on the gradients of these variables across the surface (e.g. n.∇Ui=f), for instance points of known fluid or gas flux. In thus study, the constitutive equations are rearranged and integrated into a mixed finite element in space, finite-difference in time scheme, programmed with FreeFem++ (developed at the Laboratoire Jacques-Louis Lions in Paris). Finiteelement modeling of phase transitions in multiphase fluid flowing though porous media has received considerable attention (e.g. [7,8,9]). Finite-elements benefit from high level programming language and advanced numerical solvers. Although the coupling between thermodynamics, kinetics, and transport processes from the microscopic to the geologic scale presents a numerical challenge, this framework appears sufficient to examine the formation and distribution of methane hydrate and free gas in a simple geologic settings, with respect to geothermal gradient, dewatering and fluid advection rate, the methane in-situ production and basal. 1-D MODEL In sediments, diffusive transport of ions and molecules is primarily controlled by their concentration gradient according to Fick’s diffusion equation; Hence, in 1D a homogeneous porosity medium filled with constant density fluid, the methane vertical flux is approximated by: Ud = -Φ Ddg ∇zMdg ; (4) The bulk diffusivity is in turn dependant on the porosity and tortuosity of the sediment [10]. We can first assume an equal methane diffusion coefficient in both the gas and dissolved phase (with no diffusion of the hydrate phase): Ddg = Dgg = Dm / [1+n(1-Φ)], (5)  with n=3 for muddy sediment [11,12]. Our 2D model further considers binary diffusion coefficients in the liquid and gaseous phase following Pruess and Moridis [13]. Methane solubility at high pressure conditions have received considerable attention, through experimental [14], and modeling studies [15,16,17]. Figure 1 presents methane solubility versus pressure and temperature, showing that considerable amounts of dissolved methane should be held in the pore fluid of sediment before methane is made available to hydrate precipitation.  BGHSZ  L+G  gradient just at the BGHSZ reaching 10 and 14 mg/m, for (4 and 10 K/100m, respectively). Therefore in conditions of forming hydrate (i.e. methane in excess of the solubility), the vertical gradient of the solubility, through diffusion, induces considerable upward transport of dissolved methane in the stability zone and downward transport below. Thus at the BGHSZ for ∂ΦMi/∂t=0 (i.e. Fick’s law), the local solubility gradient requires a methane influx no less than 1.7 to 2.4 10-12 kg/m2/s or 3 to 4.8 mol/m2/kyr (for 4 and 10 K/100m, respectively), assuming Φ=50% and Dm =8.7 10-10 m2/s.  L+G+H  In order to exsolve methane and to precipitate hydrate, this diffusive flux need to be refurbished with methane by in-situ biogenic generation, advection of free/dissolved methane from below, porosity loss, or a combination of these mechanisms.  Figure 1. Methane hydrate and methane solubility in grams of methane per kilogram of pore fluid versus pressure and temperature. The methane hydrate phase transition (BGHSZ) separates the 2 phase L+G (water and gas) and the 3 phase L+G+H (water, gas and hydrate) behaviors. Diamonds mark the solubility for 3.5% Na [10]; Triangles mark the 3 phase solubility for 3.5% Na, upside-down triangles mark the 3 phase solubility for 1.9 % of Cl, and a dashed line marks the 2 phase solubility at atmospheric pressure [10]. As a matter of facts, in 1500 m of water depth and for a set of likely geothermal gradient (Figure 2): a) at sea-floor (277.5°K), 1 Kg of seawater holds up to 1.275 g of methane, b) the BGHSZ is located between 85 mbsf (4 K/100m) to 200 mbsf (10 K/100m), c) below the BGHSZ, the solubility drops with increasing pressure (and temperature to a lesser degree), providing opportunity for the presence of free methane, d) above the BGHZ, while the gaseous solubility increases, methane precipitation (from the dissolved phase) is increasingly possible, e) solubility drops by 60% across the hydrate stability zone, with the highest dissolution  Figure 2. Methane hydrate and methane solubility (in grams of methane per kilogram of pore fluid) in sediments located in 1500m water depth for a set of geothermal gradients from 4 to 10°/100m. Pressure is hydrostatic  2-D MODEL Having introduced the general processes involved in our simulation of methane hydrate emplacement, we can generalize these for a 2-D framework. Our case study consists of a 6250 m wide anticlinal structure, in 1000 m water depth. The initial porosity is 64% at the sea-floor and follows an exponential decay given as: Φ(z)= 0.3 + 0.34*exp(2 10-4 z). (5) The bottom water temperature is fixed at 277.5 K and a 40 mW/m2 basal heat flux is imposed. The initial pressure is hydrostatic and methane saturation is everywhere equal to the solubility resulting from the P,T equilibrium. As for the 1-D model, we compute the solubility gradient, the equivalent methane or pore-fluid fluxes (Figure 3).  Up to 1.4 kg/m3 of dissolved methane are needed at the toe and in the slope basin, with more favorable conditions for hydrate precipitation at the crest of the anticline (1.4 kg/m3). The negative gradient at the BGHSZ reaches -0.9 kg/m3 over several 10’s of meters in the slope basin, resulting in a zone of exsolution favorable to free gas accumulation. However, free gas is more likely observed at the crest of the anticline, where the solubility below the BGHZ is at its minimum. In the hydrate stability zone, a methane generation/flux of about 10-13 to 6 10-13 kg/m2/s is required to precipitate any hydrate. If the system is fed solely from its base by dissolved methane (1 kg/m3), a fluid flux of 10-14 kg/m2/s is required to keep full dissolution below the BGHSZ, and above 3 10-13 at the crest of the anticline and 6 10-13 kg/m2/s in the slope basin. 2-D SIMULATION In order for the total methane content to exceed solubility, to exhibit free gas at the BGHSZ, and for hydrate to form, fairly high methane generation is required: in our case study, we reset our initial methane content to 0.7 kg/m3. A basal fluid and methane flux of 10-12 and 3 10-13 kg/m2/s are imposed, respectively. Biogenic in-situ methane production is 10-14 kg/m3/s. Sediment compaction is not considered but a constant volumic fluid source of 10-14 kg/m3/s is introduced to simulate dewatering. The simulation is performed over 0.5 Ma at 1 kyr time steps. Material properties are given in Table 1. Figure 4 presents the methane budget during the simulation: the initial dissolved methane is for 4 tones; Full dissolution is reach after 375 kyr, then free gas saturation increases.  Figure 3. a) Methane solubility; b) Vertical solubility gradient; c) methane flux needed to counter-act diffusion resulting from the solubility gradient; d) Pore fluid (brine and dissolved methane) equivalent flux.  Figure 4. Methane budget.  Prior to 375 kyr, hydrate primarily precipitate from exsolved methane near the sea-floor, then from methane advection at the BGHSZ. Figure 5 shows the hydrate saturation reach after 0.5 Ma. The hydrate saturation reaches 3.8% at the BGHSZ near the structural highs, about 2% near the sea-floor at the toe of the anticline and in the slope basin, and is less than 1% in the remaining hydrate stability zone.  Figure 5. Methane hydrate saturation after 0.5 Ma. CONCLUSIVE REMARKS The model proposed in this study is still in its infancy. The finite-element approach appears well suited for this numerical challenge. In the near future, alternatives finite-element classes such as Raviart-Thomas elements, shall be tested for solving the Darcy flows. A better control on the physical properties and mechanical behavior of the sediments and chemical reactants is needed, in particular the hydrate permeability. Anaerobic methane oxidation and sulfate reduction could be introduced into the model in order to constrain the methane flux near the sea-floor. ρ c λ η e k  Water 1000 4180 0.58 8.9e-4 2.2e-5 e-15 krw  Gas 10 430 0.074 1.e-5 .001 e-15 krg  Hyd. 930 2080 0.62  Sedi. 2650 2200 3.0  Mixing None Linear Revil [18] None PV/RT van Genuchten  [13]  Table 1. Material properties. REFERENCES [1] Moridis GJ. Numerical studies of gas production from methane hydrates. SPE Journal 2003:8(4):359-370. [2] Sun X-F, Mohanty K-K. Kinetic simulation of methane hydrate formation and dissociation in porous media. Chem. Engrg Sci. 2006:61:3476 – 3495.  [3] Davie MK, Buffet B. Sources of methane for marine gas hydrate: inferences from a comparison of observations and numerical models. Earth Planet. Sci. Lett. 2003:206:51–63. [4] Guerrin K. Simulations of methane hydrate phenomena over geologic timescales: Part I. Effects of sediment compaction rates on methane hydrate and free gas accumulations. Earth Planet. Sci. Lett. 2003:206:65–81. [5] He L-J, Matsubayashi O, Lei X-L. Methane hydrate accumulation model for the Central Nankai accretionary prism. Marine Geology 2006:227:201–214. [6] Moridis GJ, Sloan ED. Gas production potential of disperse low-saturation hydrate accumulations in oceanic sediment. Energy Conservation and Management 2007:48:18341849. [7] Hughes TJR., Masud A., Wan J. A stabilized mixed discontinuous Galerkin method for Darcy flow. Comput. Methods Appl. Mech. Engrg. 2006:195:3347–3381. [8] Nakshatrala KB, Turner DZ, Hjelmstad KD, Masud A. A stabilized mixed finite element method for Darcy flow based on a multiscale decomposition of the solution. Comput. Methods Appl. Mech. Engrg. 2006:195:4036–4049. [9] Bernadi C, Pironneau O. Derivative with respect to discontinuities in the porosity. C. R. Acad. Sci. Paris 2002:335:661–666. [10] Shen L, Chen Z. Critical review of the impact of tortuosity on diffusion. Chemical Engineering Science 2007:62(14):3748-3755. [11] Iverson N, Jorgensen BB. Diffusion Coefficients of Sulfate and Methane in Marine Sediments: Influence of porosity. Geochimica et Cosmochimica Acta 1993;57:571:578 [12] Lin S, Hsieh W-C, Lim Y-C. Methane migration and its influence on sulfate reduction I the Good Weather Ridge, South China Sea, continental margin sediments. Terr. Atmos. Ocean. Sci. 2006:17(4):832-902. [13] Pruess K, Oldenburg C, Moridis G. TOUGH2 user’s guide, Version 2.0, Lawrence Berkeley. Laboratory Report LBL-43134, Berkeley, California, 1999:pp 203. [14] Chapoy A, Mohammadi AH, Richon D, Tohidi B. Gas solubility measurements and modeling for methane-water and methane-nbutane-water systems at low temperature conditions. Fluid Phase Equilibria 2004:220:113121.  [15] Tishchenko P, Hensen C, Wallmann K, Wong C-S. Calculation of the stability and solubility of methane hydrate in seawater. Chemical Geology 2006:219:37-52. [16] Duan Z, Mao S. A thermodynamic model for calculating methane solubility, density and gas phase composition of methane-bearing aqueous fluids from 273 to 523 K and from 1 to 2000 bar. Geochimica et Cosmochimica Acta 2006;70:3369:3386. [17] Sun R , Duan Z. An accurate model to predict thermodynamic stability of methane hydrate and methane solubility in marine environments. Geochemical Geology 2007;doi: 10.1016/j.chemgeo.2007.06.021. [18] Waite WF, Stern LA, Kirby SH, Winters WJ, Mason DH. Simultaneous determination of thermal diffusivity and specific heat in sI methane hydrate. Geophys. J. Int. 2007:169:767-774.  

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