Frequency dependent attenuation and dispersion in patchysaturated

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1 Frequency dependent attenuation and diperion in patchyaturated porou rock Huixing Zhang, ritopher A. Innanen. Key Lab of Submarine Geocience and Propecting Technique, MOE, Ocean Univerity of China;. Department of Geocience, Univerity of Calgary Summary From eimic wave equation in modified patchy-aturated model etablihed on the bai of White model, we derived the formula of reciprocal quality factor and velocitie of the two kind of P-wave and analyzed the eimic attenuation and velocity diperion of the fat P-wave in modified patchyaturated rock within eimic band. Owing to the importance of poroity, permeability and fluid aturation, we alo tudied and analyzed the effect of thee three factor on eimic attenuation and velocity diperion of fat P-wave in patchy-aturated rock within the eimic band. Introduction Attenuation and diperion often occur a eimic wave propagate in underground media, epecially in oil and ga reervoir (Chapman et al., 006; Quintal et al., 0; Yan et al., 04). A good undertanding of eimic attenuation and velocity diperion i of great importance to eimic interpretation and inverion and alo i helpful to infer the fluid type and property. However, the phyical mechanim reponible for uch high attenuation and diperion in the eimic band are currently not fully undertood. In eimic exploration, we are mot intereted in the wave-induced fluid flow mechanim. At preent, numerou model for eimic attenuation and velocity diperion from wave-induced flow have been developed with varying degree of rigor and complexity (Müller et al., 00). Thee model can be categorized roughly into three group: the Biot model (Biot, 956), the BISQ model and the patchy-aturated model. The predicted attenuation baed on the Biot model can only be applied for the ultraonic band and wa much maller than that meaured in the eimic frequency range(-00hz) (Berryman, 988). Dvorkin and Nur (993) propoed the Biot-Squirt (BISQ) model that combined both mechanim baed on one-dimenional iotropic pore. Unfortunately, the eimic attenuation predicted by BISQ model only applie to the ultraonic frequency range (Dvorkin et al., 994; Dvorkin et al., 995). Furthermore, it cannot decribe the relative relation of the eimic amplitude between P-wave of the firt and econd kind (Bordakov, 999). Wave induced fluid flow can alo be caued by preure gradient between area of the rock which are much larger than the typical pore ize but much maller than the eimic wavelength. Thi kind of fluid flow i called meocopic flow. Meocale flow wa modeled by White (975) and White et al. (975). Since the White model, which i alo known a the patchy-aturated model, wa etablihed, many cientit tudied eimic attenuation and velocity diperion in the patchy aturated model (Dutta and Ode, 979a, 979b; Lopatnikov and Gurevich, 988; Yin et al., 99; Gurevich and Makarynka, 0; Quintal, 0; Tiato and Quintal, 04; Hu et al., 04; Yao et al., 05; Pimienta et al., 05; Spencer and Shine, 06). Pride et al. (004) illutrated that the microcale quirt flow mechanim decribe attenuation in the eimic frequency range inufficiently, wherea the meocale flow model can account for the attenuation in the low frequency range. Johnon (00) modified the White model and theoretically analyzed the wave attenuation characteritic in patchy-aturated rock. However, Johnon didn t propoe a new eimic wave equation for patchy-aturated porou rock. On the bai of White model, we (Zhang and He, 05) propoed a modified patchy-aturated model and etablihed the correponding eimic wave equation for partially aturated porou rock. GeoConvention 07

2 Thi paper i a equel to our previou work (Zhang and He, 05). In thi paper, we obtain the reciprocal quality factor Q - through olving the eimic wave equation to tudy the frequency dependent attenuation of the patchy-aturated model. Moreover, we tudy the effect of poroity, permeability and fluid aturation on eimic attenuation and velocity diperion in patchy-aturated rock. Theory and/or Method The modified patchy-aturated model we etablihed before i hown in FIG.. Thi model wa propoed according to the White model which uggeted that ome region were fully aturated with water and other were fully aturated with ga. FIG. a how the rock keleton and two kind of fluid where the dot repreent for fluid and dahe repreent for fluid. For convenience, we rearrange the model to be a in FIG. b, which conit of two concentric phere with radii R a and R b. The volume of the inner phere, which i aturated with fluid (ga pocket in the White model), i the total pace of pore filled with fluid in FIG. a. The outer pace repreent the rock keleton and the pore filled with fluid. Beide White aumption (White, 975), we alo aumed that there i no movement between fluid and the keleton but there i relative movement between fluid and the keleton. On the bai of the above, the dilatational wave equation in modified patchy-aturated model were etablihed a follow: with ρ= ( φρ ) + φsρ f + φsρ f and S+ S =. θ ε ρ + ρ f = H θ + γd ε, () t t θ ε η ε ρf + m = γd θ + D ε t t t, () In the above equation, ρ i the total ma of the patchy-aturated porou rock per unit volume, and i the ma denity of olid grain; S, ρ and η (for later ue) are the aturation, denity and vicoity of fluid, repectively; S, ρ f and η are the aturation, denity and vicoity coefficient of fluid, repectively; θ and ε are the volume train of the olid ( frame rock containing fluid ) and that of fluid relative to olid, repectively; φ and are the poroity and permeability of the rock, repectively; H i the plane-wave modulu of partially-aturated rock, H = K + 43 µ and µ are bulk and hear moduli of the patchy-aturated rock, repectively; γ and D are elatic coefficient which are related to parameter φ, S, S f and K (Zhang and He, 05). K f and K are the bulk moduli of fluid, fluid and olid grain, repectively; m i a parameter with the dimenion of denity and can be computed through m= ρf φs,where i a tructure contant depending on the pore tructure and orientation; t i time. The bulk modulu K of the patchy-aturated rock can be computed by the following formula Johnon(00) etablihed: KBGH KBGW K( ω) = KBGH. (3) ζ + ζ iωτ / ζ (a) (b) FIG.. Modified patchy-aturated model for porou media ρ GeoConvention 07

3 For the expreion of K BGH BGW angular frequency of the eimic wave., ζ and τ ee our previou work (Zhang and He, 05). ω i Solving wave equation () and (), we can obtain the reciprocal quality factor and velocitie: with v, ω ω = =, Q k Re( k ),, v α α Im( k ),,,,, = = =, π f k, Re( k, ) η 4 4 η 3 A= 4γ D DH, B= Dρω 4γ Dρ f ω + mω H + i ωh, C = ρ ω mρω i ρω, f (4) k, ± = A B B 4AC, k = Re( k ), α = Im( k ),,,,, where k, and α, repreent the wavenumber and attenuation coefficient of two kind of P-wave, repectively; v, and Q, repreent the velocity and the reciprocal quality factor of two kind of P-wave, repectively. The ymbol Re and Im repreent real part and imaginary part of a complex number, repectively. Example FIG. how that the reciprocal quality factor and velocity of the fat P-wave change with frequency when the rock ha different poroity, permeability and ga aturation. The rock parameter we ued are in Table. The parameter we varied are poroity φ, permeability and aturation S and S. Table. Elatic parameter of the patchy-aturated model frame parameter value fluid parameter value φ 0.3 S 0.5 K Pa S 0.5 µ Pa ρ 650 K Pa 3 K f m kg m 3 ρ 000 ρ 78 Pa R a 7.937cm f η 3 R b 0cm kg m 3 kg m Pa η Pa From FIG. a, we can ee that the fat P-wave exhibit large attenuation in patchy-aturated rock within the eimic frequency band and the attenuation become higher with increaing frequency and poroity, becaue the amount of fluid increae with increaing poroity at the ame degree of aturation. From FIG. b, we can ee that the peak of the reciprocal quality factor move to high frequency with increaing GeoConvention 07 3

4 permeability. Furthermore, at low frequencie, the attenuation increae with decreaing permeability. The reaon i poibly that there i low fluid mobility when the permeability i low and thu vicoity of the fluid i relatively high and lead to higher attenuation. FIG. c how that the attenuation increae with decreaing ga aturation, which i conitent with the reult that Nie et al. (0) achieved through tudying wave attenuation in the BISQ model, and Deng et al. (0) obtained through tudying wave attenuation in a periodic layered patchy-aturated model. Kuteynikova et al. (04) alo concluded that the attenuation of the fat P-wave with water aturation of 90% i greater than that with water aturation of 83.6% through numerical modeling and laboratory meaurement of eimic attenuation in partially aturated rock. FIG. d how that velocity diperion of the fat P-wave become obviou with increaing poroity. When the rock poroity i 0%, there i very little velocity diperion, becaue the amount of fluid i too mall. Velocity diperion i obviou when the rock permeability i high, wherea it only appear in the low frequency band when the rock permeability i low (FIG. e). Therefore, low frequency i more important than high frequency for detecting oil and ga reervoir. FIG. f how that the velocity diperion increae with increaing water aturation, which alo agree with Deng et al. (0). poroity 30% poroity 0% poroity 0% permeability 00mD permeability 0mD permeability md ga aturation 50% ga aturation 40% ga aturation 30% Q Q Q Vp (m/) (a) poroity 30% poroity 0% poroity 0% Vp (m/) (b) permeability00md permeability0md permeabilitymd (d) (e) (f) FIG.. Attenuation and diperion of the fat P-wave with different poroity, permeability and ga aturation in patchy-aturated rock: (a) Reciprocal quality factor veru frequency with different poroity. (b) Reciprocal quality factor veru frequency with different permeability. (c) Reciprocal quality factor veru frequency with different ga aturation. (d) Velocity veru frequency with different poroity. (e) Velocity veru frequency with different permeability. (f) Velocity veru frequency with different ga aturation. Concluion We derived the reciprocal quality factor and velocity veru frequency from the eimic equation in a modified patchy-aturated model and tudied the eimic attenuation and velocity diperion of the fat P-wave in patchy-aturated rock. By tudying the effect of poroity, permeability and fluid aturation on eimic attenuation and velocity diperion, we obtained: Firt, eimic attenuation increae with increaing poroity and frequency and decreaing ga aturation within the eimic band. Attenuation peak move to high frequencie a rock permeability increae. Moreover, at low frequencie (below about 0Hz), attenuation increae with decreaing permeability. Second, velocity diperion become evere a the poroity goe up and the ga aturation goe down. It i obviou only at low frequencie when the permeability i low. Vp (m/) (c) ga aturation 50% ga aturation 40% ga aturation 30% GeoConvention 07 4

5 Acknowledgement We would like to thank the National Natural Science Foundation of China (No ), CREWES, and the China Scholarhip Council for upporting thi tudy. Reference Berryman, J. G., 988, Seimic wave attenuation in fluid-aturated porou media: Pure and Applied Geophyic, 8, Biot, M. A., 956, Theory of propagation of elatic wave in fluid-aturated porou olid. I. Low-frequency range: Journal of the Acoutical Society of America, 8, Bordakov, G. A., Ilyaov. K., Sekerzh-zenkovitch, S. Y., and Mikolaevki, E. Y., 999, Wave refraction with a porou plate in liquid----comparion of Biot and BISQ theorie: 6 t EAEG International Meeting, Expanded Abtract, -3. Chapman, M., Liu, E., and Li, X., 006, The influence of fluid-enitive diperion and attenuation on AVO analyi: Geophyical Journal International, 67, Deng, J. X., Wang, S. X., and Du, W., 0, A tudy of the influence of meocopic pore fluid flow on the propagation propertie of compreional wave----a cae of periodic layered porou media: Chinee Journal of Geophyic (in Chinee), 55, Dutta, N. C., and Odé, H., 979a, Attenuation and diperion of compreional wave in fluid-filled porou rock with partial ga aturation (White model) Part I: Biot theory: Geophyic, 44, Dutta, N. C., and Odé, H., 979b, Attenuation and diperion of compreional wave in fluid-filled porou rock with partial ga aturation (White model) Part II: Reult: Geophyic, 44, Dvorkin, J., and Nur, A., 993, Dynamic poroelaticity: A unified model with the quirt and the Biot mechanim: Geophyic, 58, Dvorkin, J., Mavko, G., and Nur, A., 995, Squirt flow in fully aturated rock: Geophyic, 60, Dvorkin, J., Nolen-Hoekema, R., and Nur, A., 994, The Squirt-flow mechanim: Macrocopic decription: Geophyic, 59, Gurevich, B., and Makarynka, D., 0, Rigorou bound for eimic diperion and attenuation due to wave-induced fluid flow in porou rock: Geophyic, 77, L45-L5. Hu, Y. Q., Make, H. A., Valenza, J. J., and Johnon, D. L., 04, Frequency-dependent attenuation and elaticity in unconolidated earth material: Effect of damping: Geophyic,79, L4-L49. Johnon, D. L., 00, Theory of frequency dependent acoutic in patchy-aturated porou media: Journal of the Acoutical Society of America, 0, Kuteynikova, M., Tiato, N., Jänicke, R., and Quintal, B., 04, Numerical modeling and laboratory meaurement of eimic attenuation in partially aturated rock: Geophyic, 79, L3-L0. Lopatnikov, S. L., and Gurevich, B., 988, Tranformational mechanim of elatic wave attenuation in aturated porou media: Izvetija Academy of Science USSR, Phyic of the Solid Earth, 4, Müller, T. M., Gurevich, B., and Lebedev, M., 00, Seimic wave attenuation and diperion reulting from wave-induced flow in porou rock A review: Geophyic, 75, 75A47-75A64. Nie, J. X., Ba, J., Yang, D. H., et al., 0, BISQ model baed on a Kelvin-Voigt vicoelatic frame in a partially aturated porou medium: Applied Geophyic, 9, 3-. Pimienta, L., Fortin, J., and Guéguen, Y., 05, Experimental tudy of Young modulu diperion and attenuation in fully aturated andtone: Geophyic, 80, L57-L7. Pride, S. R., Berryman, J. G., and Harri, J. M., 004, Seimic attenuation due to wave-induced flow: Journal of Geophyical Reearch, 09, B00. Quintal, B., 0, Frequency-dependent attenuation a a potential indicator of oil aturation: Journal of Applied Geophyic, 8, 9-8. Quintal, B., Schmalholz, S. M., and Podladchikov, Y. Y., 0, Impact of fluid aturation on the reflection coefficient of a poroelatic layer: Geophyic, 76, N-N. Spencer, J. W., and Shine, J., 06, Seimic wave attenuation and modulu diperion in andtone: Geophyic, 8, D- D3. Tiato, N., and Quintal, B., 04, Laboratory meaurement of eimic attenuation in andtone: Strain veru fluid aturation effect: Geophyic, 79, WB9-WB4. White, J. E., 975, Computed eimic peed and attenuation in rock with partial ga aturation: Geophyic, 40, 4-3. White, J. E., Mikhaylova, N. G., and Lyakhovitkiy, F. M., 975, Low-frequency eimic wave in fluid aturated layered rock: Journal of the Acoutical Society of America, 57, Yan, F. Y., Han, D. H., Yao, Q. L., Zhao, L. X., 04, Prediction of eimic wave diperion and attenuation from ultraonic velocity meaurement: Geophyic, 79, WB-WB8. Yao, Q. L., Han, D. H., Yan, F. Y., and Zhao, L. X., 05, Modeling attenuation and diperion in porou heterogeneou rock with dynamic fluid modulu: Geophyic, 80, D83-D94. Yin, C. S., Batzle, M. L., and Smith, B. J., 99, Effect of partial liquid/ga aturation on extenional wave attenuation in Berea andtone: Geophyical Reearch Letter, 9, Zhang, H. X., and He, B. S., 05, Propagation and attenuation of P-wave in patchy-aturated porou media, Applied Geophyic,, GeoConvention 07 5