Estimating anisotropic permeability from attenuation anisotropy using 3C-2D data

Transcription

1 Estimating anisotopic pemeability om attenuation anisotopy Estimating anisotopic pemeability om attenuation anisotopy using C-D data Nicolas W. Matin and R. James Bown BSTRCT It has been expeimentally poed that the attenuation, Q, athe than elocity, is stongly coelated with pemeability (Klimentos and McCann, 990; kba et al., 99). Thus it should be possible to elate attenuation anisotopy with anisotopic pemeability in patially o completely satuated ocks (Gelinsky et al., 994). mathematical appoach is pesented o estimating anisotopic pemeability om C-D data, using the naow elationship between seismic attenuation and pemeability as pedicted by Biot s laws o isotopic satuated poous media. This appoach could be applied to media with anisotopic pemeability due to tansese isotopy poduced by a stack o hoizontal layes o azimuthal anisotopy caused by etical actues. INTRODUCTION Recently, Gibson and Toksöz (990) pedicted how the pemeability would ay with diection in actued ocks based on seismic elocity anisotopy. Futhe, the expeimental data o Han (987) and Klimentos and McCann (990) obtained on sandstone samples show that the attenuation coeicient is moe stongly elated to clay content than elocity. On the othe hand, Klimentos and McCann show a stong systematic elation between clay content and pemeability and they conclude that attenuation is the key acto in detemining pemeability. dditionally, the wok o Gelinsky and Shapio (994a, b) shows that, o seismic equencies, the attenuation anisotopy is popotional to the pemeability anisotopy. In thei study, anisotopy is consideed twoold, consisting ist o a weak anisotopy o the elastic constants and second o a much stonge anisotopy o the system's pemeability. They showed that the absolute alue o the qsv-wae attenuation is lage than that o the q waes o all equencies in a mateial with anisotopic pemeability and a homogeneous and isotopic ame. Then it is easible to obtain aluable inomation about the eseoi anisotopy om the study o the behaio o the seismic attenuation with diection using C-D seismic data. It is expected that by estimating the attenuation o, SV and waes one can obtain a bette image o the spatial distibution o the pemeability in the eseoi. CREWES Reseach Repot Volume 7 (995) 7-

2 Matin and Bown THEORY In geneal tems, a eseoi can pesent at least two dieent types o anisotopy: (i) tansese isotopy, caused by ine hoizontal layeing (Fig. a), and (ii) azimuthal anisotopy due to paallel (o nealy paallel) etical aligned cacks o actues, o due to unequal hoizontal stesses (Fig. b). Fo both anisotopies the elastic popeties ay depending on the diection o measuement. In the ist case, waes geneally tael aste hoizontally, along layes, than etically acoss layes. Fo mateials showing azimuthal anisotopy, waes taeling along the actue diection ¾ but within the competent ock ¾ geneally tael aste than waes cossing the actues. VERTICL XIS OF SYMMETRY Y X Y HORIZONTL XIS OF SYMMETRY X Z Z (a) (b) Fig.. Simple geometies assumed o elastic anisotopy.(a) In layeed ock, elastic popeties ae uniom within hoizontal layes, but may ay etically and om laye to laye.(b) In etically cacked ocks, elastic popeties ae uniom in etical planes pa-allel to the cacks, but may ay in the diection acoss the cacks. In both cases, howee i thee exists any anisotopy caused by hoizontal ine layeing o actues, it means that, besides the anisotopic static pooelastic stiness tenso, the mateial will show a dynamic eect o anisotopic pemeability as well. In act, o mateials showing tansese isotopy, the pemeability ¾ the ease with luids low though ock ¾ measued paallel to the layes o poous sedimentay ocks can be geate than the etically measued pemeability ( k > k ). In the othe h case, the etical actues acts as baies to the luid low and the pemeability measued pependicula to the actue planes is smalle than the pemeability o the ock matix measued paallel to the actues ( k < k ). h 7- CREWES Reseach Repot Volume 7 (995)

3 Estimating anisotopic pemeability om attenuation anisotopy s a esult o anisotopy, at a gien point in the medium the diection o the pessue-gadient ecto is, in geneal, dieent om that o the elocity ecto. In the geneal case o an anisotopic medium, thee will esult thee dieent low ates in each o the x, y, and z diections, wheeas in an isotopic medium the low is equal along all o these diections. Dacy s law o anisotopic media (Dullien, 979) should be witten as ollows: = k + k + k i µ i x i x i x (I=,, ), () o = µ () ( k ) whee,, and epesent the x, y, and z coodinates; k ij om the elements o a second-ode tenso, the alues o which depend on the oientation o the medium with espect to the coodinate system; is the elocity ecto, and is the pessue-gadient ecto. ssuming that anisotopic poous media ae othohombic, i.e., they hae thee mutually othogonal pincipal axes, the pemeability tenso K is symmetic ( k = k ) ij ji and otation o the coodinate system will poduce a diagonal matix when the thee coodinate axes ae aligned with the pincipal axes o the medium. Fo this paticula oientation o the medium, the pessue gadient and the elocity hae the same diection and, theeoe, in this case Dacy s law becomes: = k x i i µ i (I=,, ), () whee the thee dieent alues o k i ae still, in geneal, not equal. The theoy o Biot (956, 96) o wae popagation in an isotopic poous solid shows that thee ae two kinds o waes (ast and slow) and a single S wae. Both compessional waes hae dieent attenuations. The ast wae has itually constant phase elocity, although thee is a small tem aying as the squae o equency. The S wae also has constant elocity, as well as the same behaio o the attenuation with equency ( ). This theoy is alid only o a low-equency ange deined by the condition: < 05. ηφ πρ k whee and ρ ae the densities o the satuated ock and o the luid in the poe space, espectiely, is the poosity, h is the iscosity o the luid, k is the pemeability, and is the equency. The anisotopic pemeability entes the Biot equations though a dissipation tem. Because o the anisotopy, and S waes do not sepaate anymoe completely. n CREWES Reseach Repot Volume 7 (995) 7-

4 Matin and Bown wae appeas now. Then, the esults o Biot (White, 965) show that at low equencies, when we conside wae popagation in an anisotopic poous medium, the attenuation coeicient g o, SV and waes can be expessed as: γ π = σ ρ i k η γ SV = π s ρ k η SV (4) γ = π ρ k η whee is the phase elocity, g is the attenuation coeicient, and σ i is a dimensionless ock modulus. The, SV, and index o pemeability and phase elocity epesents the coesponding alues o both paametes associated with each wae mode. To analyze the seismic eects o global luid low in a system with anisotopic pemeability, we initially conside a simple model which in the static limit is homogeneous and isotopic. nisotopy o wae popagation is then a dynamic eect caused only by anisotopic pemeability. In this way we can extend Biot s esults o poous media to the case o anisotopic pemeability. Figue shows a sketch o an anisotopic poous solid. The axis etical to the Eath s suace is the z-axis and the angle q is deined between the slowness ecto, p, and the axis o symmety. I the pincipal axes o the pemeability tenso ae chosen as coodinate axes, the pemeability tenso becomes diagonal. Denoting the pincipal pemeabilities as k, k and the angles made by the popagation slowness ecto, p, with the pincipal axes as a, b, and g, we can expess the magnitude o the pemeability ecto k as: k = k cos α+ k cos β+ k cos γ, (5) s the pessue gadients poduced by the taeling wae and the phase elocity hae the same diection o othohombic anisotopic poous media, we hae: and k = k + k + cos θsin ξ sin θsin ξ k cos θ k = k + k + cos θcos ξ cos θsin ξ k sin θ (6) SV 7-4 CREWES Reseach Repot Volume 7 (995)

5 Estimating anisotopic pemeability om attenuation anisotopy k = k + sin ξ k cos ξ whee x is the azimuthal angle o the plane containing the and SV waes with espect to the x axis. 0 X Y SV CQUISITION LINE ROGTION DIRECTION Z Fig.. opagation diection o, SV, and waes in an anisotopic poous media. The equations (4) and (6) pemit us elate the pincipal pemeability components k, k, and k with the attenuation coeicients γ, γ, and γ associated with SV the thee, SV, and wae modes. Combining these equations we obtain a linea system o equations with thee unknowns k, k as ollows: sin θcos ξ sin θsin ξ cos θ cos θcos ξ cos θsin ξ sin θ sin ξ cos ξ 0 k k k = (7) whee p η = γ π ρ ( ρ ρ σ ) i CREWES Reseach Repot Volume 7 (995) 7-5

6 Matin and Bown SV = γ η SV (8) π ρ ( ρ ρ) = γ η π ρ ( ρ ρ) The solutions k, k ae the pincipal pemeabilities o a mateial whose anisotopy o wae popagation is only due to anisotopic pemeability. But, as was elated beoe, cacks o layeing lead to an additional anisotopy o the elastic pooelastic stiness tenso. This static pooelastic anisotopy may be descibed by pooelastic -, SV-, and -wae phase elocities,, and and by the SV Thomsen (986) paametes (e, g, d), speciied by a omalism o luid-illed, aligned cacks (Schoenbeg and Douma, 988). This will diectly aect the phase elocities and become dominant o seismic equencies aecting the seismic attenuation as well. CONCLUSIONS mathematical backgound has been pesented o estimating the pincipal pemeabilities k, k by measuements o the seismic attenuation o, SV, and waes in C-D data. The estimated pemeabilities k, k o an homogeneous, isotopic, poous medium with a dynamic anisotopic pemeability, caused by actues o a stack o layes, depend on the seismic attenuation, phase elocity, q (the angle o wae popagation), and x (the azimuthal diection o the seismic line). Fo a moe detailed desciption o the pemeability behaio, it is necessay to conside the anisotopy o the static pooelastic stiness tenso as gien by the Thomsen paametes o luid-illed, aligned cacks. REFERENCES Biot, M.., 956, Theoy o popagation o elastic waes in a luid-satuated poous solid, low- and highe-equency ange: J. coust. Soc. m., 8, Biot, M.., 96, Mechanics o deomation and acoustic popagation in poous media: J. ppl. hys.,, Dullien, F.. L., 979, oous media: Fluid tanspot and poe stuctue: cademic ess, Inc., London. Gelinsky, S. and Shapio, S.., 994a, ooelastic elocity and attenuation in media with anisotopic pemeability: esented at the 64th nn. Intenat. Mtg., Soc. Expl. Geophys., Expanded bstacts, Los ngeles, 7., Gelinsky, S. and Shapio, S.., 994b, -wae elocity and attenuation in pooelastic tansese isotopic media: esented at the 56th Mtg., EEG, Vienna, ustia, pape CREWES Reseach Repot Volume 7 (995)

7 Estimating anisotopic pemeability om attenuation anisotopy Gibson, R.L.J. and Toksöz, M.N., 990, emeability estimation om elocity anisotopy in actued ock: J. Geophys. Res., 95, Han, D., 987, Eect o poosity and clay content on acoustic popeties o sandstones and consolidated sediments: h.d. thesis, Stanod Uniesity. Klimentos, T. and McCann, C., 990, Relationships among compessional wae attenuation, poosity, clay content, and pemeability in sandstones: Geophysics, 55, Schoenbeg, M. and Douma, J., 988, Elastic wae popagation in media with paallel actues and aligned cacks: Geophys. osp., 6, Thomsen, L., 986, Weak elastic anisotopy, Geophysics, 5, White, J.E., 965, Seismic waes: adiation, tansmission and attenuation: McGaw-Hill, Inc. CREWES Reseach Repot Volume 7 (995) 7-7