Golmshtok-2014-The impact of faulting on the s(2)

530GOLMSHTOK

where coefficient κ0 depends on the structure of thesediment and has the dimension of area.

Following Leibenzon (1947), we express the gaspermeability as

κg

(e)

of the skeleton material, water, and methane, respec??tively.

2.2. Constitutive Equations

With allowance for methane hydrate dissociationand with the Henry’s law in the form suggested byLeibenzon (1947), the water mass conservation equa??tion is

??(ρwφesw)Dpg?δh

??????????????????????????????????????–cHφesw????????????+??(ρwvw)=–γρhφ????????????,(7)

dt?t?tand the mass conservation equation for methane gas is ?(ρgφesg)Dp????????????????????????????????????+cHφesw????????????g

?tdt

(8)

??δh

+??(ρgvg)=–(1–γ)ρhφ????????????,

?tHere, pw and pg is pressure in the gas and liquid phases;

(φesg)=κ0????????????????????????????????(1+3sw).2

(1–φe)

3

(3)

For calculating the effective thermal conductivity,we use the “effective medium method” (Allen, P. andAllen, J., 2005), which assumes that the medium isfilled with a multicomponent material. Component j(j=1,...,n) has thermal conductivity λj and occupiesrelative volume δVj (Σnj=1δVj = 1). The constituents ofeach component are randomly distributed in the rockvolume. In our case, the medium contains a mineralmatrix, pore water, free methane gas, and methanehydrate, i.e., n = 4, and the effective thermal conduc??tivity λe is obtained by solving the following quarticequation:

??h?1=3?1+φesw+φesg+φδ(4)?.

λe?2λe+λsk2λe+λw2λe+λg2λe+λh?Here, λsk = 2.5 W/(m K), λw = 0.6 W/(m K), λg=0.037 W/(m K) , and λh = 0.4 W/(m K) are the thermalconductivities of the skeleton material, water, meth??ane gas (Sultan et al., 2004), and methane hydrate(Sultan et al., 2004; Sloan and Koh, 2007), respec??tively.

The heat capacity of the material at constant pres??sure is a derivative of enthalpy with respect to temper??ature (Landau and Lifshits, 1940). Because the phasechange from methane hydrate to methane gas isaccompanied by a jump in enthalpy, the effective vol??umetric heat capacity of sediment should be used inthe same way as it was done by Bonacia et al. (1973) inthe one??dimensional problem of water freezing:

e=ρC+Lρhφδhδ(T–Tph),

(5)

Dpg?p

????????????=??????????g+vw??pg,dt?t

(9)

cH is Henry’s constant; vw is the pore water filtration

velocity determined according to Darcy’s law as

κw

vw=–????????????(?pw+ρwgez),

μw

(e)

(10)

μw ≈ 0.001 Pa ? s is the dynamic viscosity of water; g isthe acceleration of gravity; ez is the unit vector alongthe z axis; vg is the gas filtration velocity; ρg is the den??sity of methane gas described by the real gas equationof state as

pg

ρg=????????????????????,

ZRgT

(11)

where δ(x) is the Dirac delta??function; L = 430 kJ/kgis the latent heat of crystallization (dissociation) ofmethane hydrate (Sultan et al., 2004; Sloan and Koh,2007); ρC is the volumetric heat capacity of the sedi??ment without allowance for the jump in enthalpy dur??ing hydrate dissociation and ignoring the gas contribu??tion due to its relatively small density and saturation,which is equal to

??hρhCh+φeswρwCw;ρC=(1?φ)ρskCsk+φδ

(6)

ρsk = 2670 kg/m3 is the density of the mineral matrix

material; ρw = 1000 kg/m3 is the density of fresh wateror, for example, ρw = 1032.5 kg/m3 is the density ofsaline water at the salinity of 35‰; ρh = 913 kg/m3 isthe methane hydrate density; and Csk ≈ 1000, Cw =4187, and Ch = 2080 are the heat capacities in J/(kg K)

and Rg = 518.35 J/(kg ? K) is the gas constant for meth??ane; Z=Z(p,T) is the compressibility factor for meth??ane, which ranges within 0.73 to 0.87 with pressureand temperature in sediments ranging within 7 to25MPa and 275 to 300 K, respectively (Chekalyuk,1965). Here, Z is assumed to be 0.8, i.e., to its averagevalue.

In the mass balance equations (7) and (8), porosityis taken out of the sign of time derivative in the right??hand sides of these equations because the sharpincrease in the mass fractions of pore water and gas perunit volume is exclusively caused by the decrease in thehydrate fraction at its dissociation, which is equallysharp.

The force of gravity for the gas component is usu??ally neglected; therefore, the equation of gas filtrationvelocity is

κg

vg=–?????????????pg,

μg

(e)

(12)

No. 4 2014

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 50

THE IMPACT OF FAULTING ON THE STABILITY CONDITIONS OF GAS HYDRATES531

where μg ≈ 10–5 Pa ? s is the dynamic viscosity of gas,which is assumed to be constant in our study.

Pressure pg in the gas phase, which is less wettingfor the mineral matrix, and pressure pw in the (wetting)water phase differ in the pore channels by the value ofcapillary pressure P(Barenblatt et al., 1984). In other words,

k, which depends on saturation swpg–pw=Pk(sw),

(13)

and the dependence of capillary pressure on water sat??uration in our case has the form Pk(sw) =Θ0J(sw where J(sw) is the Leverett

increase in water saturation and is determined experi??mentally for each sedimentary rock type; α is theinterphase surface tension in N/m; Θcharacteristic of wetting in the porous medium–liquid0 is the integralsystem (Barenblatt et al., 1984).

The porosity changes due to the change in the aver??age pressure =pwsw+pgsg = pw+Pk(sw)(1?sw)(Barenblatt et al., 1984). However, since the capillarypressure in the near??bottom sediments which arehighly saturated with water and typical for containinggas hydrates does not exceed a few kPa and hydrostaticpressure is about 10 MPa or higher, we assume thatporosity φ only depends on pdetermining the character of changes in porosity withw. In addition, whenpressure, we also take into account the fact that gassaturation in the effective pore space is relatively lowand therefore we assume that the entire free pore spaceis only filled with water.

Hereinafter, we assume that hydrate completelydissociates stepwise on the surface of the phase transi??tion and express the effective fraction of the pore??fill??

ing hydrate, ??sure in the pore water:

δ

h, as a function of temperature and pres??δ

??h=δh[1?σ(T?Tph)],(14)

where Tph=Tph(pw) is the equilibrium temperature

(phase transition temperature) at pw and σ(x) is theHeaviside unit step function.

Neglecting gas dissolution in water and water dis??placement, one can easily see that the fractions ofwater βwould be

w and gas βg in the pores of the sedimentβ(1–δ0

h)sw+γρ??hw=ρ??????δhσ(T–Tph),

w(15)

βδ0

?γρ??hg=(1–h)sg+?1–ρ????????

δhσ(T–Tph),

wwhere s00

pore space that is free of the solid phase (skeleton and

w and sg are the water and gas saturations of theIZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 50

hydrate) before the hydrate dissociation. In this case,s00

w+sg

= 1.With allowance for this, it is reasonable to cast thesaturations sw and sg and δw=φesw, δg=φesg in thefollowing form:

sw=w+ω, sg=g–ω=1–w–ω,

(16)

w=??βwβg

1??????–?????????δ??????, g=??h1??????–?????????δ??????,(17)

h

φesw=βwφ+φeω, φesg=βgφ–φeω,(18)where ω is the additional saturation due to the gas dis??solution in water and its displacement.

By substituting Eqs. (9)–(12), (14), and (16)–(18)into (7) and (8), we obtain after some transformations

ρw?Sw????????p????wt??–cHφesw????????p????gt+??(ρwvw)(19)

–cHφeswvw??pg=Qw–ρwφe?ω???????????t

,

ρ(1)?p(2)?pgSgas???????????gt+ρgSgas????w???????t

??+??(ρgvg)

(20)

+cHφeswvw??pg=Qgas+ρgφe?ω???????????t

,

where

?Sw=UwS–ω????????T????p??ph????W, S(1)sgas=?w???p????g+c????H??????s????w???φe,gρg?

(21)

S(2)

gas=U?????Tph???????p??????W,

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