Golmshtok-2014-The impact of faulting on the s(5)

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 50

component of the conductive heat flow at the base ofthe layer is constant and equal to q0.

Both before and after the onset of faulting, the layeris fed by water, which percolates upwards from theunderlying strata and seeps through the lower bound??ary of the layer within a wide zone near the escarpmentin the basement topography at a constant verticalvelocity .v0

Before faulting, porosity φs(z) within this laterallyhomogeneous layer is only depth dependent, so is theeffective permeability κ(e)s(z) of the sedimentary rocks,which is a function of porosity. Far from the channel,the porosity and permeability of sediments remain thesame as before faulting.

We assume that after the termination of faulting,which resulted in the intense fracturing of the sedi??mentary material, the porosity on the axis of the chan??nel φc(z) decreases with depth much slower than φs(z).

The permeability κ(e)

c(z) on the axis also increases to aconsiderable degree compared to depth.

κ(e)s(z) at the sameSuppose that in the process of the channel forma??tion, the degree of fracturing and the mechanicalproperties of the rock gradually change both in time(during the interval Θ) and in space (within the inter??val r0?Δr0≤r≤r0+Δr0) from the values on thechannel axis to the values in the regions remote fromthe channel (i.e., the initial values characterizing themedium before the onset of fluid conduit formation).Then, the expression for the porosity is

φ=φ(r,z,t)=φs(z)+[φc(z)–φs(z)]F(r,t),

(25)

where F(r,t) is a smooth nonnegative function, with0≤F(r,t)≤1. This function is zero if t ≤ 0 and/orr≥r0+Δr0 and it is equal to unity if t≥Θ simulta??neously with r≤r0?Δr0. The permeability and otherporosity??dependent physical properties are expressedin a similar way.

For F(r,t), we use the following function

F(r,t)=[1–(r–r0,Δr0)(t–τ,τ).We assume that the depth variations in the porosityof the sedimentary unit follow Athy’s law (Athy, 1930),which is characteristic of terrigenous sediments, and

write φs(z)=φ0exp??sz is the porosity of the rocks at the bottom(k)and φc(z)=φ0exp??czwhere φ(k

),of the lake. The depth dependences for both poros??

0ities only differ by the compaction parameters ??and k

??k

sc.With the onset of channel formation, the channel isfilled with additional amounts of water, which riseswithin the zone above the edge of the impermeable

No. 4 2014

536GOLMSHTOK

ancient scarp in the basement. Considering this addi??tional water inflow, we write the expression for the ver??tical velocity of water flowing through the base of thesedimentary layer (z = –H):

vb(r,t)=v0+vaF(r,t),

where va is the additional velocity of water inflowalong the channel axis.

At any given moment of time t > 0, in any verticalcolumn extending throughout the entire layer, the lat??ter is subdivided into two areas. In the first (lower)area, the pores of the sediment are only saturated withwater. This area is bounded from above by the surfacez=z0ph, corresponding to the base of the zone of gashydrate stability before the beginning of their dissoci??ation, when pressure on it was equal to p0ph and equi??

0librium temperature was Tph.

The second area accommodates the reactions ofmethane hydrate dissociation, which occur when theconditions of methane hydrate stability in the mediumare violated due to the formation of the permeablechannel, causing changes in the thermophysical prop??erties of the sediments and additional heat and masstransfer due to the water filtration flow. We assume thathydrate saturation of the pore space is constant andequal to δh everywhere between the phase boundaryand the bottom of the lake.

The expression for the effective permeabilities towater and gas in the common form for the both areas is

whereas for gas at z≥z0ph (the permeability of gas isequal to zero at z≤z0ph), we obtain

(e)

κs,g

3(e)

(s)[φssg]κ0??????????????????????????????????????(1

(e)2

[1–φs]3(e)

(c)[φcsg]κ0??????????????????????????????????????(1

(e)2

[1–φc]

+3sw),

(e)κc,g

+3sw).

Here, variables sw and sg are determined byformulas(16)–(17), and the terms φ(se)sw, φ(se)sg, and

e)(e)φ(csw, φcsg are determined by (15) and (18) with thecorresponding substitutions of φ for φs and φe.

The heat conductivity and heat capacity in themedium are described, as before, by Eqs. (4), (5), and(6), in which porosity φ is determined by (25).In order to model the impact of the developingchannel, we assume the following values for theparameters: r0 = 875 m; Δr0=r0 τ = 2500 years;

??s= φ0=0.65; 4.7 × 10–4 m–1 is the quantity deter??k

mined when constructing the velocity model for sedi??ments in the Central Baikal Basin (Golmshtok, 2000);??c=k??s v0 = 1 mm/yr; va = 5 cm/yr (seek

Section4.2 for the details concerning the choice ofthis value); ρw = 1000 kg/m3; Hw = 1400 m; δh = 0.1 ishydrate saturation of Baikal sediments according tothe deep water drilling (Kuz’min et al., 1998); Tw =276.45 K is the lake bottom temperature in the deep??

water part of the Central Baikal Basin; and κ(0 = 5 ×

c)(s)10–13 m2, κ(0 = 1000κ0.

κe

(w)(g)

=κs,w+[κc,w–κs,w]F(r,t),

(e)

(e)(e)(e)

κe=κs,g+[κc,g–κs,g]F(r,t),

e)??h)φs and φ(??where, denoting φ(se)=(1?δc=(1?δh)φc,

according to (2) and (3) we have for water

In this study, we disregard the effects of sedimenta??tion. In this case, the steady??state pressure pst(z) andtemperature Tst(z), in the sediment layer before fault??ing with given p0, z0ph,Tw, and vb are described by thefollowing formulas:

(c)κs,w

?(s)φ30s

????????????,z<z?κ0????????????????????ph,2

?(1–φs)=?

3e)?(s)[φ(s]0sw????????????????????????,z≥z?κ0??????????????ph,(e)2?[1–φs]

?(c)φ30c

????????????,z<z?κ0????????????????????ph,2

?(1–φc)=?

3e)?(c)[φ(

0csw]??,z≥z?κ0????????????????????????????????ph,(e)2?[1–φc]

pst(z)=ρwg(hw–z)

(26)

?zph0?0dzdz

????????????????????+??????????????????????????,z<z?μwv0??????ph,(–)(+)κ(z)0κ(z)?z+?zph

?0?0dz????????????????????,z≥z?μwv0??????ph,(+)

κ(z)?z

(c)κc,w

∫∫

(27)

∫

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 50 No. 4 2014

THE IMPACT OF FAULTING ON THE STABILITY CONDITIONS OF GAS HYDRATES537

?T0ph+(T0ph–T)1??????–??????e??????1??(??????z????)e00,z<zph,T?w??1st(z)=?–e0

?(28)

?01–??????e??????2????(????z????),z≥z0?Tw+(Tph–Tw)????????1–eph.0

Here,

eexp?1(z)=?dz?

?–ρwCwv0∫

??z

λ??????(??–????)

????????????(z)??,e?0

2(z)=exp??–ρwCwv0∫

????????dz?(???????????????????,z

λ+)

(z)?e0

0=e2(zph);

κ(?)(z) coincides with κ(e)0

s,w in (26) at z<zph, whereas

κ(+)(z) is the same function κ(e)0

s,w, but at z≥zph and withthe replacement φ(e)=(1?δ0sh)φs, sw=sw; the heat

conductivity λ(?)

(z) is defined from (4) at φe=sw=1, sg=0, and δ

??φs,

h=0, and λ(+)(z) is found from thesame equation but at φδ=s0

e=(1?h)φs and sww,

s0g=sg, and δ

??h=δh.The steady??state vertical conductive heat flowthrough the base of the layer for a given value of z0ph isdetermined from the formula for of the parameters of the medium cited above, it mea??Tst(z). With the valuessures q0 = 23.7 mW/m2.

Prior to estimating the impacts of faulting on thestability of gas hydrates in the considered segment ofthe axial zone of the Central Baikal Basin, we analyzethis problem without considering the phase transfor??mation of methane hydrate to methane gas. In otherwords, we numerically solve the system of equations(19a) and (24) with L = 0 with the initial conditions forpressure and temperature (27) and (28), and boundaryconditions on the surfaces of the layer z=?H and z = 0,as well as on the axis r=0, where ?pw?r=0 and?T?r = 0 due to the axial symmetry of the problem.In our calculations, we replace the radially infinitepart of the sediment layer outside the channel by abounded region with the internal and external radiir=r0 and r=R, respectively. The value of R is speci??fied in such a way that within this distance from theaxis, the radial components water filtration velocity