Fluid substitution in rocks saturated with viscoelastic fluids

Transcription

1 GEOPHYSICS, VOL. 7, NO. 2 MARCH-APRIL 2 ; P. E11 E122, 7 FIGS..119/ Fluid ubtitution in rock aturated with vicoelatic fluid Dina Makarynka 1, Bori Gurevich 2, Jyoti Behura 3, and Mike Batzle 4 ABSTRACT Heavy oil have high denitie and extremely high vicoitie, and they exhibit vicoelatic behavior. Traditional rock phyic baed on Gamann theory doe not apply to material aturated with vicoelatic fluid. We ue an effective-medium approach known a coherent potential approximation CPA a an alternative fluid-ubtitution cheme for rock aturated with vicoelatic fluid. Such rock are modeled a olid with elliptical fluid incluion when fluid concentration i mall and a upenion of olid particle in the fluid when the olid concentration i mall. Thi approach i conitent with concept of percolation and critical poroity, and it allow one to model andtone and unconolidated and.we model the vicoelatic propertie of a heavyoil-aturated rock ample uing CPA and a meaured frequencydependent complex hear modulu of the heavy oil. Comparion of modeled reult with meaured propertie of the heavy-oil rock reveal a large dicrepancy over a range of frequencie and temperature. We modify the CPA cheme to account for the effect of binary pore tructure by introducing a compliant poroity term. Thi dramatically improve the prediction. The predicted value of the effective hear modulu of the rock are in good agreement with laboratory data for the range of frequencie and temperature. Thi confirm that our cheme realitically etimate the frequency- and temperature-dependent propertie of heavy-oil rock and can be ued a an approximate fluid-ubtitution approach for rock aturated with vicoelatic fluid. INTRODUCTION Growing demand for hydrocarbon energy ource ha renewed interet in production from bituminou and heavy-oil reervoir. Heavy-oil reerve account for more than 6 trillion barrel-in-place worldwide triple the world reerve of conventional oil and ga Batzle et al., 26. However, the high vicoity of heavy oil make them difficult or impoible to produce uing conventional technology. A variety of method currently ued by the indutry aim at lowering the vicoity by injecting heat or chemical olvent. Thermal recovery method have been the mot efficient in heavy-oil production, with recovery rate a high a 8%. Injected heat alter the already-complex phyical propertie of the heavy-oil reervoir, uch a eimic velocity, denity, and attenuation. Tracking thee change with time-lape eimic technique can coniderably improve recovery efficiency. Appropriate rock-phyic modeling i important for quantitative interpretation of the change in eimic repone due to alteration in reervoir and fluid propertie during production. Seimic fluid-ubtitution modeling uing Gamann equation Gamann, 191 i routine in analyzing and interpreting eimic velocitie and amplitude Smith et al., 23. Given pore, frame, and fluid propertie of a rock, Gamann equation can be ued to predict the bulk modulu of a rock under quai-tatic condition. The correponding dynamic moduli can be obtained from Biot theory of poroelaticity Biot, 196, an extenion of Gamann theory to finite frequencie. One important reult of the Gamann/Biot theory i that the effective hear modulu i identical to the frame hear modulu. In other word, the exitence of the fluid ha no effect on the effective hear modulu of the rock. A known from laboratory meaurement Nur et al., 1984; Eatwood, 1993; Schmitt, 1999; Batzle et al., 26; Behura et al., 27; Han et al., 27a, heavy oil exhibit vicoelatic behavior uch that their moduli are frequency and temperature dependent. Oil that be- Manucript received by the Editor 18Augut 29; publihed online 9April 2. 1 Curtin Univerity of Technology, Department of Exploration Geophyic, Perth,Autralia. d.makarynka@curtin.edu.au. 2 Curtin Univerity of Technology, Department of Exploration Geophyic and CSIRO Diviion of Petroleum Reource, Perth, Autralia. b.gurevich@curtin.edu.au. 3 Formerly Colorado School of Mine, Department of Geophyic, Center for Wave Phenomena, Golden, Colorado, U.S.A.; preently BPAmerica, Exploration and Production Technology, Houton, Texa, U.S.A. jyoti.behura@bp.com. 4 Colorado School of Mine, Department of Geophyic, Center for RockAbue, Golden, Colorado, U.S.A. mbatzle@mine.edu. 2 Society of Exploration Geophyicit. All right reerved. E11

2 E116 Makarynka et al. have like a Newtonian liquid at low frequencie can act a a nearly elatic olid at high frequencie. In between thee extreme cae, wave propagation in heavy oil i diperive and exhibit trong attenuation. Indeed, the heavier the oil, the more long-chain or highcarbon-number molecule preent, reulting in higher vicoity. Although vicoity i influenced by preure and ga content, it i primarily a function of oil gravity, compoition, and temperature. Depending on temperature condition, heavy oil can have a finite hear modulu even at eimic frequencie, making Gamann/Biot poroelatic contitutive equation inapplicable to heavy-oil rock. When a rock i aturated with heavy oil, it behavior alo become vicoelatic Behura et al., 27. A dramatic decreae of compreional velocitie in heavy-oil and with increaing temperature ha been oberved in laboratory experiment Toaya et al., 1987.Depite the fact that thi decreae wa in part caued by the abnormally high preure in the experimental etup, the main factor reponible for the velocity decreae i change in fluid propertie with temperature. Laboratory experiment provide evidence of the complex propertie of heavy-oil rock. In particular, Hanetal. 27a, 27b, 28 tudy variou effect on the compreional- and hear-wave velocitie of heavy-oil and, uch a rock texture, pore-fluid propertie, preure, and interaction between pore fluid and rock-frame propertie at different temperature. Hanetal. 27b alo tet Gamann prediction againt experimental data for a range of temperature. They indicate that a temperature decreae below 6 C, heavy oil become diperive and Gamann etimate no longer match the data. Vicoelatic behavior of the oil violate Gamann theory. Indeed, Gamann theory i baed on Pacal Law, which tate that, in the abence of body force, fluid preure i the ame throughout the pore pace. Thi law i not applicable to vicoelatic media. Biot theory i not applicable to heavy oil either becaue it ignore the vicoelatic phenomenon by neglecting the fluid hear tre in the microcopic pore-cale contitutive equation Pride et al., 1992; Gurevich, 22. Several approache to modeling effective elatic propertie of rock aturated with heavy oil have been propoed Marion and Nur, 1991; Eatwood, 1993; Tiklauri and Berenev, 23; Leurer and Dvorkin, 26; Ciz and Shapiro, 27; Da and Batzle, 28; Gurevich et al., 28. However, verification of thee model uing controlled laboratory experiment wa not poible. We propoe an alternative modeling cheme baed on equivalent-medium theory for rock aturated with vicoelatic fluid. Comparion with laboratory data allow u to refine the cheme uing the concept of dual poroity to obtain reaonable agreement between theory and experiment. Our approach can be ued for practical application to produce reliable etimate of effective frequency- and temperature-dependent propertie of heavy-oil rock. METHODOLOGY Effective-medium model Fluid aturation i an important parameter influencing the eimic propertie of rock. A mentioned, the effect of a Newtonian pore fluid on the elatic moduli of a rock can be predicted from the propertie of dry rock and fluid compreibility uing the Gamann/Biot theory. An alternative approach, which addree the problem of etimating elatic moduli from knowledge of rock contituent and microtructure, i baed on effective-medium theorie e.g., Hahin and Shtrikman, 1963; Budianky, 196; Hill, 196; Mukerji et al., 199; Kazatchenko et al., 24. Thee theorie allow one to compute equivalent elatic propertie of a mixture of two or more elatic contituent. Unlike the Gamann/Biot theory, effective medium theorie require characterization of matrix and pore pace geometry. Furthermore, at ufficiently low frequencie, thee theorie cheme can be applied to mixture of vicoelatic contituent by conidering them a olid with frequency-dependent moduli. Thi i known a elatic-vicoelatic analogy or the correpondence principle Hahin, 197; Haddad, 199. Hence, effective-medium theorie can be applied to heavy-oil rock. To gauge the effect of vicoelaticity of the pore-filling material on the overall rock propertie, we ue Berryman 198, 1992 cheme baed on a elf-conitent theory. Thi cheme i a elf-conitent verion of the average T-matrix approximation of Küter and Tököz 1974 and i alo known a coherent potential approximation CPA. CPA i a veratile mixing law of the theory of compoite. It i known to provide reliable etimate of frame moduli of heterogeneou material Oguhwitz, 198. CPA ue the concept of elatic-wave cattering theory for the deformation of iotropic incluion and approximate the interaction of the incluion by replacing the background medium with an a-yet-unknown effective medium. Thi mean that the effect of many pore of a particular hape i given by olving the canonical problem of a ingle pore urrounded by a uniform medium with yet-unknown elatic propertie of the porou compoite. Milton 198 prove rigorouly that CPA i a realizable effective medium cheme, i.e., there exit a particular geometric arrangement of the contituent for which thi cheme i exact. A key property of CPA i that it i ymmetrical with repect to the contituent. Each contituent i treated equally in the cheme, meaning that no ingle contituent act a a hot to the other but that a more abundant contituent i the load-bearing one. Thu, a olidfluid mixture i modeled a a olid with fluid incluion of a particular hape when fluid concentration i mall and a a upenion of olid particle in the fluid when the olid concentration i mall. Thi property i conitent with the concept of percolation and critical poroity Nur et al., 199 and allow one to model ituation when heavy oil i a pore-filling fluid and when it i part of the rock matrix. CPA calculate the effective propertie of a porou rock uing known propertie of the olid matrix, the pore fluid, and the pore apect ratio. The implicit CPAformula for bulk K and hear G moduli of a two-component rock one olid phae, one fluid phae are K f K P f 1 K K P, G f G Q f 1 G G Q, where i poroity; K f and G f are bulk and hear moduli of the pore fill, repectively; K and G are bulk and hear moduli of the matrix grain material, repectively; and P and Q are invariant of the Wu tenor Wu, The component of thi tenor depend on the apect ratio of the pore and on the bulk and hear moduli of the pore fill, matrix material, and a-yet-unknown effective moduli K and G of the compoite. We ue the expreion of the Wu tenor for pheroidal incluion of arbitrary apect ratio to decribe the pore/grain geometry of typical andtone. Thee expreion are cumberome and can be found in Berryman 198 or Mavko et al Equation 1 and 2 are coupled and can be olved by iteration: 1 2 K n 1 K fp n f 1 K P n P n f 1 P n, 3

3 Fluid ubtitution in heavy-oil rock E117 G n 1 G fq n f 1 G Q n Q n f 1 Q n. 4 Equation 3 and 4 can be ued to etimate the effective elatic moduli of a aturated rock. The iteration proce require a firt gue of K 1 and G 1, which can be calculated uing the Voigt-Reu-Hill average. The cheme can alo be ued to calculate the dry-frame propertie K dry, G dry by etting the bulk and hear moduli of the aturating fluid to zero rock with empty pore. Complex hear modulu of heavy oil To apply CPA to calculate the effective propertie of a rock aturated with heavy oil, we need to know it complex temperature- and frequency-dependent hear modulu. For vicoelatic material, the hear modulu G i complex and repreent the ability of a material to tore energy elatically and to diipate it a a vicou fluid: G G ig, where G, in vicoelaticity theory, i a torage modulu and G i a lo modulu Ferry, 198. The frequency dependency of the complex hear modulu of heavy oil can be approximated by Cole-Cole empirical diperion equation Cole-Cole, 1941; Gurevich et al., 28, which relate complex hear modulu with hear moduli at the low- and high-frequency limit, angular frequency, and relaxation time. The relaxation time depend on fluid vicoity, which i a function of temperature. According to Hanetal. 28, the temperature dependency of heavy oil can be decribed by three main tage, depending on the oil liquid point: liquid tate, quai-olid tate, and olid tate. Thi dependency can alo be approximated by empirical relationhip e.g., Begg and Robinon, 197. The dependency of propertie of linear vicoelatic material on frequency and temperature can be implified by uing the temperature-frequency uperpoition principle, which tate that the diperion curve for different temperature are the ame if the frequency i normalized by ome temperature-dependent parameter William et al., 19. Thi mean, for intance, that the Cole-Cole diperion equation can be ued for all temperature uing the Begg and Robinon 197 vicoity-temperature relationhip. To perform fluid ubtitution for heavy-oil rock, frequency- and temperature-dependent complex modulu mut be ued in equation 3 and 4 intead of G f. Comparion with known olution for dry and liquid-aturated rock Before applying CPA to heavy-oil rock, we tet it applicability to the limiting cae of Newtonian liquid/olid mixture. Figure 1 how CPA prediction for dry and water-aturated rock with a quartz matrix for a range of poroitie and apect ratio. The predicted bulk moduli are compared with the empirical model of Krief et al. 199 and the Hahin-Shtrikman HS bound Hahin and Shtrikman, The Krief et al. etimate of K dry correpond more cloely to the CPA etimate with.2, which are conitent with value of apect ratio for and-related pore reported in literature e.g., Xu and White, 199. A expected, all CPA etimate lie within HS bound. In Figure 2, we compare the difference between CPA etimate of aturated and dry moduli K K dry with Gamann prediction for quartz/water mixture for realitic crack denitie. crack denity i related to poroity and apect ratio of the pheroidal crack by 3 /4. Thi comparion i important becaue any mixture model for rock aturated with a vicoelatic ubtance mut be conitent with Gamann equation when the pore-filling ubtance i a Newtonian fluid. At the ame time, model baed on iolated ellipoidal pore are not exactly Gamann conitent becaue fluid preure in the pore varie with their orientation Thomen, 198; Mavko and Jizba, A can be een from Figure 2, the CPA reult are in the bet agreement with Gamann prediction for pherical pore and how the maximum deviation from Gamann etimate for relatively flat pore. For apect ratio.2.4 and typical poroitie of up to.2, we oberve atifactory agreement with Gamann prediction. The dicrepancy i le than 1 GPa. We thu conclude that CPA i approximately Gamann conitent within typical meaurement error. a) K dry (GPa) b) K (GPa) Poroity (fraction) Poroity (fraction) α Kr HS α Kr HS Figure 1. Bulk moduli of dry a and aturated b quartz andtone veru poroity. CPA etimate for different apect ratio olid line, Hahin-Shtrikman bound ymbol and Krief et al. model dahed line. Comparion of CPA etimate of a dry bulk modulu of a rock with quartz matrix and b bulk modulu of a water/ quartz mixture. Hahin-Shtrikman bound HS and Krief et al. model Kr for different apect ratio and poroitie.

4 E118 Makarynka et al. FLUID-SUBSTITUTION SCHEME USING CPA If the effective pore apect ratio i known, the effective elatic propertie of a rock can be computed by iterating equation 3 and 4. Here, we aume that the rock tructure can be approximated by ellipoid with the ame effective apect ratio. Of coure, pore geometry of real rock i more complicated and conit of pore of different ize and hape. However, ellipoid often are ued to capture ome eential propertie of uburface void, providing imple parameterization of the enormou complexity of the real pore pace. For a given rock ample, the effective pore apect ratio can be etimated from dry moduli in Gamann fluid ubtitution, K dry i the key input parameter a well. Given K dry, the effective apect ratio can be etimated by the invere CPA cheme equation 3 and 4 with fluid moduli et to zero. The apect ratio can alo be derived uing the invere CPA cheme if we know the bulk modulu K of the rock aturated with a fluid with known bulk modulu e.g., water.alternatively, apect ratio can be etimated from hear moduli. Once i etimated, we can calculate the effective moduli of a aturated rock for any pore-filling material uing the forward CPA cheme. The CPA fluid-ubtitution workflow involve three tep. Firt, invere CPA i ued to etimate the effective uing known dry or aturated bulk or hear modulu. Second, we obtain the frequency- and temperature-dependent complex hear modulu of the pore fill G f from laboratory meaurement. If uch meaurement are not available in the range of temperature and frequencie required, they can be interpolated uing known empirical relationhip Cole-Cole, 1941; Begg and Robinon, 197 and the frequencytemperature uperpoition principle William et al., 19; Ferry, 198. Finally, we calculate the effective elatic moduli of a rock aturated with vicoelatic fluid uing the forward CPA cheme equation 3 and 4 with the complex hear modulu of pore fill G f. K K (GPa) dry ε < Poroity (fraction) α Figure 2. Difference between water-aturated and dry moduli of a rock with quartz matrix a a function of poroity and apect ratio for crack denitie.: CPA dahed line veru Gamann etimate olid line. COMPARISON WITH LABORATORY MEASUREMENTS Uvalde heavy-oil rock We apply our CPA fluid-ubtitution cheme to Uvalde heavy-oil rock from Texa, U.S.A. The laboratory meaurement conducted on thi rock have been reported in the literature Batzle et al., 26; Behura et al., 27; Da and Batzle, 28. The Uvalde rock i a carbonate aturated with extremely vicou heavy oil with anapi denity of five 1.12 g/cm 3. It ha poroity of approximately 2%, and permeability of md. A hear rheometer wa ued to meaure the complex hear modulu of the rock at temperature ranging from 3 C to 3 C and frequencie ranging from.1 to 8 Hz. Then the oil wa extracted and it complex hear modulu wa meaured in the ame frequency range and temperature ranging from 3 C to 2 C. The meaured torage modulu of the Uvalde rock and the extracted oil for the range of frequencie and temperature ranging from 3 C to 2 C are hown in Figure 3. The miing data point have been interpolated. The modulu of the heavy-oil-aturated rock how a moderate dependency on frequency and a trong influence by temperature. The torage modulu G of the rock increae with frequency for temperature le than C. At higher temperature, the behavior of the torage modulu G i weakly dependent on frequency. The trend oberved for the rock hear propertie look imilar to thoe for the extracted oil. The Uvalde oil how noticeable diperion for temperature le than 18 C. At low temperature, the extracted heavy oil upport a hear wave; but with increaing temperature, it torage modulu decreae rapidly, imilar to what i oberved for the rock hear modulu. A in the cae with the rock, at higher temperature the torage modulu of the oil G f i nearly independent of frequency. In general, G f increae with increaing frequency for all temperature. Therefore, we can conclude that the mechanical repone of a heavy-oil rock with changing temperature and frequency i trongly influenced by the behavior of the pore-filling oil. Modeling with a ingle apect ratio Given that the meaurement of the hear propertie of the Uvalde oil and Uvalde rock are available, we ue the fluid-ubtitution workflow decribed above. We firt predict the effective elatic propertie of the Uvalde rock aturated with the Uvalde oil uing CPA. Then we compare the predicted torage modulu of the rock with laboratory data. We perform our modeling for the full range of frequencie and temperature ranging from 3 C to 2 C. We take the phyical propertie of the grain material a thoe of calcite K 6 GPa, G 3 GPa, denity 2.71 g/cm 3. The dry hear modulu G dry of the Uvalde rock ha not been meaured, o we analyze the available value of the temperature- and frequency-dependent hear propertie of the oil and rock to obtain G dry.inour analyi, we aume that at high temperature and low frequencie, the pore fluid behave a a Newtonian fluid and ha no effect on the effective hear propertie of the rock. Indeed, careful analyi of laboratory data Figure 3 how that for temperature over 12 C, the hear modulu of the Uvalde rock i a real contant G 1.4 GPa. Thi mean the effective hear modulu G become identical to G dry. Next, we olve equation 3 and 4 to obtain uing the etimated G dry. Then we ue the forward CPA to compute the frequency- and temperature-dependent effective hear modulu of the Uvalde rock

5 Fluid ubtitution in heavy-oil rock E119 aturated with the Uvalde oil. In Figure 4a, we compare the calculated effective hear moduli of the rock ymbol with the laboratorymeaured moduli line. We can ee that CPA i unable to capture the trend of the frequency and temperature dependency of the rock hear propertie, epecially for low temperature. The etimated moduli are ignificantly lower than the meaured one. In Figure 4b, we compare the prediction of the effective torage modulu computed by the vicoelatic extenion of Gamann theory EG Ciz and Shapiro, 27 with laboratory data. The EG prediction ymbol deviate from the laboratory data line even more than the CPAetimate Figure 4a. Compliant poroity For additional inight into frequency- and temperature-dependent behavior of Uvalde rock, we compare torage moduli of the rock and oil a a function of temperature for different frequencie Figure. We oberve a ubtantial variation of the rock torage modulu G ymbol with temperature for different frequencie, contrary to the relatively moderate variation of the oil torage modulu G f line. For intance, G 8-Hz frequency varie from approximately 2 GPa up to 17 GPa, wherea G f varie from almot to 1.2 GPa. Thi ugget that a imple mixing law, uch a CPA with a ingle apect ratio, cannot account for thoe obervation. The effect of poroity and fluid on elatic propertie of rock i often modeled uing the binary tructure of the pore pace relatively tiff pore that occupy mot of the pore pace and relatively compliant oft pore preent at grain contact, reponible for the preure dependency of the elatic moduli. At high frequencie, fluid flow doe not have enough time to equilibrate pore-preure gradient between the two pore type, effectively iolating compliant pore from tiff pore and from each other. If a rock i aturated with vicoelatic fluid, thi effect may occur even at relatively low frequencie, a conidered in thi tudy. When heavy oil i in a near- a) 2 G' (GPa) 1 o T (C) b) 2 G' (GPa) 1 o T (C) Log frequency (Hz) Figure 3. Variation of a torage modulu G f of Uvalde heavy oil and b torage modulu G of Uvalde heavy-oil rock with frequency and temperature a meaured by Behura et al. 27. Figure 4. Comparion of frequency- and temperature-dependent torage modulu G predicted by a CPA with a ingle apect ratio ymbol and by b extended Gamann ymbol with laboratory meaurement conducted on a Uvalde heavy-oil rock ample line. T i temperature.

6 elatic tate, it cannot flow out of oft pore, tiffening the compliant grain contact. The preence of compliant pore may lead to additional diperion of compreional and hear wave in heavy-oil rock becaue of the dependency of the normal tiffne of thee contact on the torage modulu of the pore fill Mavko and Jizba, Poor performance of both CPA with a ingle apect ratio and EG may be explained by the preence of compliant pore in the rock matrix. Modeling with two apect ratio f E12 Makarynka et al. To account for the effect of double poroity, we introduce the compliant poroity term to equation 3 and 4: G' (GPa) 2 1 f (Hz) Temperature ( o C) Figure. Laboratory meaurement of torage moduli G of Uvalde heavy oil line and Uvalde heavy-oil rock line with ymbol for a range of frequencie and temperature. G' (GPa) Log frequency (Hz) o T (C) Figure 6. Comparion of frequency- and temperature-dependent torage modulu G predicted by CPA with two apect ratio ymbol with laboratory meaurement carried out on a Uvalde heavyoil rock ample line. K n 1 K f P n f c K f P n fc 1 K P n P n f c P n fc 1 P n, 6 G n 1 G f Q n f c G f Q n fc 1 G Q n Q n f c Q n fc 1 Q n, 7 where i tiff poroity that occupie mot of the pore pace, c i compliant poroity preent within the grain and at grain contact, and c. Now the problem become more complicated becaue equation 6 and 7 mut be inverted to etimate three unknown parameter: compliant poroity c, and apect ratio of compliant and tiff pore c and, repectively. For the fitting procedure, we ue dry hear modulu G dry and aturated hear modulu G meaured at C and two frequencie,.8 and 1.8 Hz. We aume the grain can be decribed by the apect ratio g.8 and keep it contant. The reult of the fitting or the invere CPA are c.1, 1, and c.4. Next, we ue the etimated parameter in the forward CPAcheme to predict the frequency- and temperature-dependent hear modulu G of the Uvalde rock. The reult are compared with meaurement in Figure 6. We oberve very good agreement between the CPA prediction ymbol and the laboratory data line. The only deviation larger than the meaurement error approximately 2.9 GPa; ee Behura et al., 27 i oberved at high frequencie for 3 C. We alo compute the hear quality factor Q inverely proportional to the attenuation coefficient from the predicted complex effective hear modulu of the rock. Figure 7b how Q predicted by CPA. The general trend of etimated attenuation i imilar to the meaured one Figure 7a. However, the meaured Q exhibit more ignificant variation with frequency. DISCUSSION The method decribed in thi paper a well a other method baed on effective-medium theorie are baed on analogy between elaticity and vicoelaticity. Thi analogy i limited to ufficiently low frequencie uch that the inertial term in the equation of vicoelaticity are mall and can be neglected Hahin, 197. Thi limitation i expected to be atified at eimic frequencie but may no longer be valid at higher frequencie where effect of quirt may become important Leurer and Dvorkin, 26. We have decribed the CPA fluid-ubtitution workflow auming a ingle apect ratio. One can ue a ingle apect ratio when the range of pore hape doe not vary ignificantly. Thi might be the cae when compliant grain-to-grain contact or intragranular poroity doe not ignificantly contribute to overall rock tiffne for intance, well-conolidated rock with welded grain or at high effective preure when compliant pore are cloed. However, in mot edimentary rock, compliant pore grain-to-grain contact with low apect ratio contribute to the total poroity and are reponible for the tre dependency of elatic propertie. When compliant pore are preent in the rock matrix, introducing a econd pore apect ratio into the cheme become important. Preent work aume that the poroity and geometry of the pore pace of the rock matrix are the ame for dry and oil-aturated rock, regardle of oil rheology. We alo aume that oil compoition doe not vary with temperature. Thee condition may be violated in field and laboratory tudie Behura et al., 27; Han et al., 27a. Note

7 Fluid ubtitution in heavy-oil rock E121 We have ued a elf-conitent mixing method known a coherent potential approximation a an alternative fluid-ubtitution cheme for rock aturated with vicoelatic fluid. Firt, CPA equation were inverted to obtain effective apect ratio of a rock from dry modulu. Then forward CPA with temperature- and frequency-dependent moduli of vicoelatic pore fill were ued to obtain aturated moduli. We teted CPA etimate of effective bulk moduli of a dry and a Newtonian fluid-aturated rock againt analytical olution. CPA prediction atify the Hahin-Shtrikman bound and are approximately conitent with Gamann equation and the Krief et al. model. A comparion with laboratory meaurement revealed that CPA with a ingle apect ratio i not adequate in predicting the frequency and temperature dependency of the effective hear modulu of the Uvalde heavy-oil rock. We modified the cheme to account for the binary tructure of the pore pace. Inverted CPA wa ued to etimate three unknown parameter: compliant poroity and apect ratio of tiff and compliant pore from the laboratory-meaured effective dry and aturated hear moduli at a ingle temperature and at two frequencie. In thi cae, CPA reproduced frequency- and temperature-dependent behavior of the hear modulu and attenuation of Uvalde heavy-oil rock, proven by good agreement with laboratory data. Thi confirm that the propoed cheme provide realitic etimate of the propertie of heavy-oil rock and can be ued a an approximate fluid-ubtitution approach for rock aturated with vicoelatic fluid. ACKNOWLEDGMENTS We thank the ponor of the Curtin Reervoir Geophyic Conortium CRGC, CSIRO Earth Science and Reource Engineering, and Autralian Potgraduate Award APA for financial upport. We alo thank Kontantin Oypov, Oleg Dinariev, and Serge A. Shapiro for timulating dicuion. REFERENCES Figure 7. Variation of quality factor Q of Uvalde heavy-oil rock with frequency and temperature a a meaured by Behura et al. 27 and b a predicted by CPAwith two apect ratio. that we do not examine the preure and ga effect on moduli/velocitie, which might be important in the recovery proce. CONCLUSIONS Batzle, M., R. Hofmann, and D.-H. Han, 26, Heavy oil Seimic propertie: The Leading Edge, 2, Begg, H. D., and J. R. Robinon, 197, Etimation of the vicoity of crude oil ytem: Journal of Petroleum Technology, 27, Behura, J., M. Batzle, R. Hofmann, and J. Dorgan, 27, Heavy oil: Their hear tory: Geophyic, 72, no., E17 E183. Berryman, J. G., 198, Long-wavelength propagation in compoite elatic media II. Ellipoidal incluion: Journal of the Acoutical Society of America, 68, , 1992, Single-cattering approximation for coefficient in Biot equation of poroelaticity: Journal of the Acoutical Society of America, 91, Biot, M. A., 196, Theory of propagation of elatic wave in a fluid aturated porou olid. II. Higher frequency range: Journal of theacoutical Society ofamerica, 28, Budianky, B., 196, On the elatic moduli of ome heterogeneou material: Journal of the Mechanic Phyic of Solid, 13, Ciz, R., and S. A. Shapiro, 27, Generalization of Gamann equation for porou media aturated with a olid material: Geophyic, 72, no. 6, A7 A79. Cole, K. S., and R. H. Cole, 1941, Diperion and aborption in dielectric, I Alternating current characteritic: Journal of Chemical Phyic, 9, Da, A., and M. Batzle, 28, Modeling tudie of heavy oil in between olid and fluid propertie: The Leading Edge, 27, Eatwood, J., 1993, Temperature-dependent propagation of P-wave and S-wave in Cold Lake oil and: Comparion of theory and experiment: Geophyic, 8, Ferry, J. D., 198, Vicoelatic propertie of polymer: John Wiley & Son, Inc. Gamann, F., 191, Über die Elatizität poröer Medien: Vierteljahrchrift der Naturforchenden Geelchaft in Zürich, 96, Gurevich, B., 22, Effect of fluid vicoity on elatic wave attenuation in porou rock: Geophyic, 67, Gurevich, B., K. Oypov, R. Ciz, and D. Makarynka, 28, Modeling elatic wave velocitie and attenuation in rock aturated with heavy oil: Geophyic, 73, no. 4, E11 E122. Haddad, Y. M., 199, Vicoelaticity of engineering material: Springer. Han, D.-H., J. Liu, and M. Batzle, 28, Seimic propertie of heavy oil Meaured data: The Leading Edge, 27, Han, D.-H., Q. Yao, and H.-Z. Zhao, 27a, Complex propertie of heavy oil and: 77th Annual International Meeting, SEG, Expanded Abtract, Han, D.-H., H.-Z. Zhao, and Q. Yao, 27b, Velocity of heavy oil and: 77th Annual International Meeting, SEG, ExpandedAbtract, Hahin, Z., 197, Complex moduli of vicoelatic compoite I. General theory and application to particulate compoite: International Journal of Solid and Structure, 6, Hahin, Z., and S. Shtrikman, 1963, A variational approach to the theory of the elatic behaviour of multiphae material: Journal of the Mechanic and Phyic of Solid, 11, Hill, R., 196, A elf-conitent mechanic of compoite material: Journal of the Mechanic and Phyic of Solid, 13, Kazatchenko, E., M. Markov, and A. Mouatov, 24, Joint modelling of

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