On the relative importance of global and squirt flow in cracked porous media

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1 On the relative importance o global and squirt low in cracked porous media Aamir Ali 1 and Morten Jakobsen 1, 1 Department o Earth Sciences, University o Bergen, Allegaten 41, 5007 Bergen, Norway. Centre or Integrated Petroleum Research, University o Bergen, Allegaten 41, 5007 Bergen, Norway. Summary A uniied theory o global and squirt low in cracked porous media was developed several years ago on the basis o a combination o the dynamic T-matrix approach to rock physics. The theory has been successully used to model ultrasonic velocity and attenuation anisotropy measurements in real rocks under pressure. At the same time, it was recently pointed out that this theory, which contain an established theory o interconnected cracks as a special case contains an error related to luid mass conservation. The error was recently corrected, and this paper represents an attempt to perorm a systematic study o the implications o uniied theory or the relative importance o global and squirt low in cracked porous media characterized by dierent microstructures and luid mobilities. Our numerical results suggest that squirt low dominates over global low and global low appears to be more important at higher requencies or more realistic models o microstructure. The attenuation peak o squirt low move towards lower requencies with the increasing luid viscosity i.e. changing saturating luid rom water to oil, while the global low attenuation peak move towards higher requencies with increasing luid viscosity. A previous observation o negative velocity dispersion in uniied theory still remain, even i we use the correct 1

2 eective wave number, when dealing with the phenomenon o wave-induced luid low in models o cracked porous media where global low eects dominates. The attenuation peak o the global low obtained using the correct wave number is always shited to the let as compared to the approximate solution. At seismic requencies global low eects are not so important and needs very high permeability and low viscosity to have an eect. Keywords: Wave induced luid low, Attenuation, Velocity dispersion, Cracked porous media 1 Introduction In seismic modelling, a cracked porous medium can sometimes be replaced by a long wavelength equivalent homogenous medium that can be both anisotropic and viscoelastic due to microstructural alignments and wave induced luid low, respectively. Wave induced luid low can occur in the orm o global or squirt low. Global low is caused by pressure gradients at the scale o the acoustic wavelength and in the direction o wave propagation, whereas squirt low is caused by the pressure gradients at the microscopic or mesoscopic scale and in directions that are potentially dierent rom that o the wave propagation. There have been several attempts to develop special (phenomenological or microstructural) theories o global low (Biot 196; Hudson et al. 1996), special (microstructural) theories o squirt low (Mavko and Nur 1975; O Connell and Budiansky 1977; Mavko and Jizba 1991; Mukerji and Mavko 1994; Dvorkin et al. 1995; Chapman 003), and uniied (phenomenological and microstructural) theories o global and squirt low (Dvorkin and Nur 1993; Hudson et al. 1996; Chapman et al. 00; Jakobsen et al. 003b; Jakobsen 004). The uniied theory o Jakobsen et al. (003b), which contains the theory o interconnected cracks developed by Hudson et al. (1996) as a special case, originally had an error related to luid mass conservation, but this error was recently corrected by Jakobsen and

3 Chapman (009). Unlike the uniied theory o Chapman et al. (00), the corrected version o the theory o Jakobsen et al. (003b) presented by Jakobsen and Chapman (009) can deal with eects o anisotropy as well as attenuation. In the uniied theories or global and squirt low presented by Hudson et al. (1996), * Jakobsen et al. (003b) and Jakobsen and Chapman (009), the eective stiness tensor C depends on the eective wave vector k and as well as on the angular requency ω. Since we are dealing with theories o the coupled physical process o wave-induced luid low, this is perhaps not surprising. In all previous studies, these non-local eects have been avoided simply by replacing the eective wave vector k by the unperturbed wave vector k associated with the waves in the solid reerence medium; that is, by using the approximation k k = ω /V, where ω is the angular requency, V is the speed o the wave mode under consideration in the solid matrix and k is the length o k. This approximation or eective wave vector k was considered to be one o the possible explanations or predictions o negative velocity dispersion in numerical experiments dealing with the phenomenon o waveinduced luid low in models o cracked porous media where global low eects dominates (Jakobsen and Chapman 009). An important aim o this study is to investigate the implications o uniied theory o Jakobsen and Chapman (009) or the relative importance o global and squirt low characterized by dierent microstructures and luid mobilities. A change in the viscosity may lead to a shit o the attenuation peak towards lower or higher requencies, depending on the mechanism o wave-induced luid low. The inluence o viscosity on attenuation peaks o the global and squirt low is dierent depending on the type o mechanism involved (Xi et al. 007). Also the experimental observations show that the high luid viscosity can act to shit the relaxation towards lower requencies (Winkler et al. 1985; Dvorkin et al. 1994). So it will 3

4 be very interesting to investigate the eects o viscosity in the context o relative importance o global and squirt low. A urther aim is to investigate i this velocity dispersion will remain negative in these models, i we use the eective wave vector k rather than the unperturbed wave vector k; that is, i we implement the global low part o the theory in a proper manner. We use an iterative method or solving the nonlinear equations associated with the uniied theory o global and squirt low in cracked porous media, where the eective stiness tensor depends on the requency ω and eective wave vector k. A quadratic equation representing microstructural and phenomenological theories o wave-induced luid low in isotropic media to the irst order in porosity and crack density is also presented. We also present and apply a simple model or the eects o viscosity on the relaxation time constant or squirt low associated with a particular pore shape/orientation. Uniied theory o global and squirt low in cracked porous media We consider a model in which a solid contains inclusions or cavities characterized by dierent shapes, orientations and spatial distributions labeled by n = 1,..., N. The eective stiness tensor * C is given by Jakobsen et al. (003a, b), where and ( 0) C = C + C : ( I + C : C ), (1) C = v t, () C 1 N r= 1 N N ( s) ( s) = v t : G : t v. (3) r= 1s= 1 ( rs) d Here C represents the elastic properties o the solid matrix, : denotes the double scalar product (see Auld 1990), I 4 is the (symmetric) identity or ourth-rank tensors and (rs) G d is given by the strain Green s unction integrated over an ellipsoid having the same aspect ratio 4

5 ( s r) as p ( x x ), which in turn gives the probability density or inding an inclusion o type s at x, given that there is an inclusion o type r at point x (Jakobsen et al. 003a, b). The t- matrix or an inclusion o type r ully saturated with a homogenous luid is given by (Jakobsen and Chapman 009) t d d = t + t : S : ( I ) : C, (4) and 1 t d = C : ( I 4 + G : C ). (5) Here, 1 S = ( C ) is the compliance tensor o the solid matrix material; I is the identity tensor or second-rank tensors; is the dyadic tensor product (see Jakobsen et al. 003a, b), (r) G is a ourth-rank tensor given by the strain Green s unction (or a material with properties given by C ) integrated over a characteristic spheroid having the same shape as cavities o type r (Jakobsen and Chapman 009) and (r) is a second-rank tensor (luid polarization tensor) that relates the luid pressure to the applied stress. The luid polarization tensor (r) is given by Jakobsen and Chapman (009) under the assumption that the cavities are o the same scale-size and the squirt low relaxation constant τ is independent o the shape and orientation. The analysis o Chapman et al. (00) suggests that τ depends on the scale-size o the cavity, suggesting that the theory can easily be extended to model the cracked/ractured porous media under the assumption o inclusions or cavities o type r having dierent sizes. Ater letting the τ having an index r dependent on scale-size o the cavities the -tensor has the orm = ~ Θ r φ I : K d 1 + iωγ τ 1 + iωγ iωτ τ κ I : K d, (6) where, 5

6 1 d γ = + κ I : ( K S ) : I, (7) 1 K d = ( I 4 + G C ) : S, (8) 1 Θ ~ φ γ ik ki k jκ = κ, (9) r 1 + iωγ τ η ω(1 Δ ) and K ( ) ki k jκ r Δ = τ. (10) φη Here K are the components o the eective permeability tensor o the cracked/ractured porous media and k i represents the component o the eective wave number vector, where i, j =1,,3, κ is the bulk modulus or the luid, η is the viscosity o the luid, φ is the total porosity and ω is the angular requency o the wave, respectively. For the case o anisotropic media, the real-valued phase velocities and attenuation actors can be obtained by inserting the viscoelastic eective stiness tensor C into the Christoel equation (see Appendix-A), which can be solved by using eigenvalue/eigenvector method (Jakobsen et al. 003b; Carcione 007). The phase velocity is the reciprocal o the slowness and is given in the component orm by (Carcione 1995, 007) 1 1 Vp = Re Iˆ. () V The quality actor Q is deined as the ratio o the peak strain energy to the average loss energy density (Auld 1990), and is deined by (Carcione 1995, 007) Re( V ) Q =. (1) Im( V ) 6

7 3 Exact analytical solution o wave-induced luid-low in isotropic models with randomly oriented ellipsoids In this section, we derive exact analytical solution or a model o randomly oriented ellipsoids to the irst order in porosity or crack density. We start rom the relation o eective stiness to the irst order in porosity or crack density C * = C + t, (13) where is the porosity and t is the averaged t-matrix or a single communicating cavity. Using equation (55) rom Jakobsen et al. (003b) or the averaged t-matrix with modiied term Θ ~ rom Jakobsen and Chapman (009) in order to account or global low in a consistent manner and substituting in equation (13), we get C * = C + t d ~ ΘZ + iωτκ X iωγτ (14) Here, ω is the angular requency, κ is the bulk modulus or luid and τ is the squirt low relaxation time. The relations or Z, and X are given and discussed in detail by Jakobsen et al (003b). We now have C ~ = ΘZ + iωτκ d + C 1 + i ωγτ * X where, (15) d C = C + t d. (16) The relation or velocity in terms o stiness or an isotropic medium is given by * C V =, or ρ C * ρv =. (17) Equation (15) can be written as, C * ~ + = ΘZ iωτκ X d + C 1 + iωγτ. (18) 7

8 The relation or modiied term Θ ~ is given by (Jakobsen and Chapman, 009) 1 Θ ~ ( ) ( ) = ( ) = ( 1 ) N r r γ iκ Kkik j κ, (19) r r + iωγ τ η ω Δ where κ τ Δ = Kkik j. φη Substituting Δ in relation or Θ ~ (equation (19)) and simpliying by using the relation or the wave-number as ollowing * ω ω ω ω ρ V = k = k = k =. (1) k V * C C ρ Ater simpliication and re-arranging we are let with a simple quadratic equation in C * written as C * + κ τk ( η γ + iωγ τη ) κ η Z ω ρz 3 + C iη ωγτκ Z * + iω γτ κ K ( κ K ω ργτ iκ K iωτκ X ρz γη ) + ic ω τκ X ωρ C d d κ K K ρ = 0. η γ ic ωρ C d d κ K ωγ τη ω ργτ () Using equation (17) and representing equation () in terms o velocity V, we have ( ) A + V B + C = 0, V (3) where A = ρη γ + iρωγ C = ic B = κ K d ω τκ X iη ωγτκ Z φk. κ K ω C τη, d iωτκ X γη, d ω ργτ iκ K ωρ C η γ ic d κ K ω γτ + κ τk ω Z ωγ τη κ η Z 3 + iω γτ κ K Z (4) Equation (4) is a quadratic equation which can be solved by a quadratic ormula giving two solutions. The irst solution is or the ast P-wave mode and the second solution is 8

9 slow P-wave mode (Biot 1956). We have only used the solution or the ast P-wave mode in our numerical experiments. The analysis or the slow P-wave mode is beyond the scope o this study and will be addressed in another paper. 4 Numerical Experiments In this section, we report rom numerical experiments dealing with the eect related with dierent microstructures and luid mobilities on the relative importance o global and squirt low in cracked/ractured porous media. The exact analytical solution derived in section 3 is limited to isotropic models o solids with randomly oriented ellipsoids to the irst order in porosity or crack density. We have also used the iterative method or solving the nonlinear equations associated with the uniied theory o global and squirt low in cracked porous media, where the eective stiness tensor depends on the requency ω and eective wave vector k. The starting point o the iterations is being the wave-number associated with the waves in the solid reerence medium. The iterative solution is important to implement the global low part correctly, when investigating the relative importance o global and squirt low in complex microstructural models and to the higher order in porosity and crack density. For the background elastic properties, we take κ = 37 GPa, μ = 44 GPa, and ρ =.5 g/cm 3 to simulate the properties o quartz. The characteristic time scale constant or squirt low relaxation time or micro-porosity (pores and micro-cracks) with water as a saturating luid was taken to be τ = 10 5 s. We have applied a simple model or the eects w o viscosity on the relaxation time constant or squirt low associated with a particular pore shape/orientation (also see Chapman 001). The squirt low relaxation time or other luids (oil or gas) can be calculated using the relation: τ w τ = η, (5) η w 9

10 where τ and η are the relaxation time and viscosity o the corresponding luid (can be oil o gas), while η w is the viscosity o water. The viscosity o water, oil and gas is 3 10 Pa s, Pa s and 5 10 Pa s, respectively (see Pointer et al. 000). 4.1 Isotropic models Spherical pores To investigate the relative importance o global and squirt low or a simple model o spherical pores, we have divided the analysis in three numerical examples. The comparison or each example is presented in the orm o T-matrix estimates o velocity and attenuation spectra o the plane wave propagation or dierent luid mobilities. In the irst example (Fig. 1), we compare the exact, iterative and approximate solutions (the approximation or the wave-number k k = ω /V ) to the irst-order in porosity. The T-matrix estimates are obtained using the irst-order T-matrix approach. The second example (Fig. ) is same as example 1, but or higher porosity. The T- matrix estimates obtained in this example may not be strictly valid at higher porosities, but the purpose is to investigate the perormance o exact analytical solution. The third example (Fig. 3) deals with the comparison o only iterative and approximate solutions or higher porosity. The T-matrix estimates are obtained using the higher-order T-matrix approach with a spherically symmetric correlation unction. For a model o spherical pores, there is no dependence on the characteristic time or squirt low, because no local pressure gradients exist or such kind o a model. We only observe global low in all the three examples (Figs. 1, and 3) characterized by negative dispersion at higher requencies, when the porous rock model is considered to be ully saturated with water or oil. No global low is observed in the case when the porous rock model is considered to be ully saturated with gas (Figs. 1, and 3). The attenuation peak or 10

11 global low move towards relatively high requencies when viscosity is increased or this particular model i.e. saturating luid o the porous matrix is changed rom water to oil (Figs. 1, and 3). An increase in the matrix permeability rom 1 to 10 Darcy shits the global low attenuation peak to relatively lower requencies (Figs. 1, and 3). In the irst example, the exact, iterative and approximate solutions clearly ollow each other. Very small dierences exist or the case o approximation with respect to the exact and iterative solution (Fig. 1). For the second example, observable dierences exist in the case o approximation with respect to the exact and iterative solution (Fig. ). The global low attenuation peak o the approximate solution is shited to right as compared to the exact and iterative solution (Fig. ). Similarly in the third example, the global low attenuation peak o the approximate solution is shited to the right as compared to the iterative solution (Fig. 3). Randomly oriented micro-cracks The investigation o relative importance o global and squirt low or a model o randomly oriented micro-cracks is also similarly divided in three numerical examples as or the case o spherical pores (Figs. 4, 5 and 6). For a model consisting o randomly oriented micro-cracks, we observe squirt low (positive dispersion) at lower requencies due to dierent orientations o randomly oriented micro-cracks along with global low (negative dispersion) at higher requencies, when the porous rock model is considered to be ully saturated with water or oil (Figs. 4, 5 and 6). The squirt and global low parts change their positions i.e. we observe global low at lower requencies and squirt low at higher requencies, when porous matrix is considered to be ully saturated with gas (Figs. 4, 5 and 6). The magnitude o global low part dominates over the squirt low part or this particular model (Figs. 4, 5 and 6). The attenuation peaks o global and squirt low move towards relatively high and low requencies when viscosity is increased or this particular model i.e. saturating luid o the porous matrix is changed rom water to oil

12 (Figs. 4, 5 and 6). The attenuation peak or the global low part shits toward relatively low requencies when matrix permeability is increased rom 1 Darcy to 10 Darcy (Figs. 4, 5 and 6). The exact, iterative and approximate solutions clearly ollow each other in the irst numerical example. Very small dierences exist or the case o approximation with respect to the exact and iterative solution only or the global low part (Fig. 4). For the second example, observable dierences exist in the case o approximation with respect to the exact and iterative solution only or the global low part (Fig. 5). The global low attenuation peak or the case o approximate solution is shited to right as compared to the exact and iterative solution (Fig. 5). Similarly in the third example, the global low attenuation peak o the approximate solution is shited to the right as compared to the iterative solution (Fig. 6). Spherical pores and randomly oriented micro-cracks Fig. 7 shows the result o a comparison o the iterative and approximate solutions or a model o spherical pores and randomly oriented cracks in the orm o T-matrix estimates (obtained using higher-order T-matrix approach with a spherically symmetric correlation unction) o velocity and attenuation spectra o plane wave propagation to higher-order in porosity and crack density or dierent luid mobilities. We observe both squirt and global low with the dominance o squirt low part or the case when the porous rock model is considered to be ully saturated with water or oil. The squirt low or the case o the porous rock model saturated with water or oil is observed at relatively lower requencies depending on the squirt low relaxation time, while global low is observed at relatively higher requencies. Only squirt low is observed at relatively higher requencies when the porous rock model is considered to be ully saturated with gas. The squirt and global low attenuation peaks move towards relatively low and high requencies, when viscosity is increased or this 1

13 particular model i.e. saturating luid o the porous rock model is changed rom water to oil. The global low attenuation peak shits toward relatively low requencies when matrix permeability is increased rom 1 to 10 Darcy. Very small dierences exist or the global low part (Fig.7), and the global low attenuation peak is shited to the right or the case o approximate solution as compared to the iterative solution. 4. Anisotropic models Aligned micro-cracks Fig. 8 shows the result o a comparison o the iterative and approximate solutions or a model o aligned micro-cracks in the orm o T-matrix estimates (obtained using higher-order T-matrix approach with correlation unction equal to the aspect ratio o micro-cracks) o velocity and attenuation spectra o plane wave propagation to higher order in porosity and crack density or dierent luid mobilities. Like in the case o spherical pores, there is no dependence on the characteristic time or squirt low in this case, because no local pressure gradients exist or such kind o a model. We observe global low characterized by negative dispersion at higher requencies, when the porous rock model is considered to be ully saturated with water, oil or gas. The magnitude o global low is smaller, when the porous rock model is considered to be ully saturated with gas. The global low attenuation peak move towards relatively high requencies when viscosity is increased or this particular model i.e. saturating luid o the porous rock model is changed rom water to oil. An increase in matrix permeability rom 1 to 10 Darcy shits the global low attenuation peak towards relatively lower requencies. Observable dierences exist in the approximate solution with respect to iterative solution and the global low attenuation peak is shited towards the right or the approximate solution as compared to the iterative solution (Fig. 8). 13

14 Pores and aligned micro-cracks Fig. 9 shows the result o a comparison o the iterative and approximate solutions or a model o pores and aligned micro-cracks in the orm o T-matrix estimates (obtained using higher-order T-matrix approach with correlation unction equal to the aspect ratio o microcracks) o velocity and attenuation spectra o plane wave propagation to higher-order in porosity and crack density or dierent luid mobilities. We observe squirt and global low at low and high requencies with the dominance o squirt low part the case when the porous rock model is considered to be ully saturated with water or oil (Fig. 9). We only observe squirt low at higher requencies when the porous rock model is considered to be ully saturated with gas. The squirt and global low attenuation peaks move towards relatively low and high requencies, when viscosity is increased or this particular model i.e. saturating luid o the porous rock model is changed rom water to oil. The global low attenuation peak shits toward relatively low requencies when matrix permeability is increased rom 1 to 10 Darcy. Very small dierences exist or the global low part (Fig. 9) or the case o approximate solution as compared to the iterative solution. Spherical pores, randomly oriented micro-cracks and aligned meso-ractures We inally consider a more realistic model consisting o pores, randomly oriented cracks and one set o aligned meso-ractures. The orientation o meso-ractures is 45 o. For this case, the elastic background properties are κ = 70 GPa, μ = 9 GPa, and ρ =. 71g/cm 3 to simulate the properties o calcite. For aligned meso-ractures the characteristic time scale constant with water as the saturating luid was calculated depending on the size/length o the ractures using the lowing relation (Chapman 003, Agersborg et al. 007) r τ m τ =. (6) ξ 14

15 Here, τ is the squirt low relaxation time or the meso-ractures, r is the radius o 6 ractures, ξ is the size o the grains ( m assumed in this study) and τ m is the squirt low relaxation time or micro-porosity. The squirt low relaxation times corresponding to dierent luids (oil or gas) can be computed using equation (5). The characteristic time scale constant or squirt low relaxation time or micro-porosity (pores and randomly oriented cracks) with water as a saturating luid is taken to be in this study is 0.5 m. ( τ m ) w = 10 7 s. The radius o ractures In order to obtain the iterative solution or this particular case, eective permeability K along with the correct use o wave-number (eective wave-number k ) must be used, because two types o permeabilities exist in this model i.e. matrix permeability (due to pores and micro-cracks) and racture permeability (due to aligned meso-ractures). This is also explained by the act that the terms or eective permeability components ( K ) and eective wave number components ( k ) always come together in Jakobsen and Chapman (009) i k j theory indicating a summation over the indices i, j =1,, 3 (see equation (9)). Their contraction can be written as by expanding the summation over indices and collecting the identical terms K kik j Kk1 + K1k1k + K13k1k3 + K k + + K 3kk = K k. (7) To obtain the independent eective permeability components or this particular model, we used the eective permeability model o Jakobsen (007) (see appendix-b). The eective wave vector k is given by where k is the length o k = k l (8) k and l in spherical coordinates is given by l1 sinθ sinϕ l = = l sinθ cosϕ. (9) l3 cosθ 15

16 Then the components o eective wave vector k are given by k k k 1 3 = k = k = k l1 l l3. (30) Fig. 10 shows the result o comparison o iterative and approximate solutions or a model o pores, randomly oriented cracks and meso-ractures in the orm o T-matrix estimates (obtained using higher-order T-matrix approach with correlation unction equal to the aspect ratio o meso-ractures) o velocity and attenuation spectra o plane wave propagation to higher order in porosity and racture density or dierent mobilities. Approximate solution or this case was obtained by using approximation or wave-number as well as the matrix permeability instead o eective permeability. Due to the introduction o one set o aligned meso-ractures, we observe squirt low (positive dispersion) associated with aligned set o meso-ractures as well as due to microporosity (pores and micro-cracks). The squirt low peak associated with the aligned mesoscopic racture set is observed at seismic requencies, while the squirt low peak associated with micro-porosity (pores and randomly oriented cracks) is observed at higher requencies, respectively, when the porous rock model is considered to be ully saturated with water or oil. We only observe squirt low associated with meso-ractures and micro-porosity at higher requencies, when the porous rock model is considered to be ully saturated with gas. Global low characterized by negative dispersion is also observed at relatively higher requencies, with water or oil as the saturating luids in the porous rock model. The attenuation peak or squirt and global low move towards relatively low and high requencies when viscosity is increased or this particular model i.e. saturating luid o the porous rock model is changed rom water to oil. The attenuation peak o the global low shits toward relatively lower requencies when matrix permeability is increased rom 1 to 10 Darcy. 16

17 Very small dierences exist in the case o approximate solution as compared to iterative solution only or the global low part. The global low attenuation peak in the case o approximate solution is shited to right as compared to the iterative solution (Fig. 10). 5 Conclusions We have investigated the relative importance o global and squirt low in cracked/ractured porous media using the uniied theory (Jakobsen and Chapman, 009) or dierent microstructures and luid mobilities. The magnitude o squirt low dominates over global low and global low appears to be important at higher requencies or more realistic microstructures (models like pores and randomly oriented cracks or pores, randomly oriented cracks and aligned mesoscopic ractures). The attenuation peak o squirt low move towards relatively low requencies with the increase o viscosity i.e. changing saturating luid rom water to oil, while the global low attenuation peak move towards relatively high requencies with the increase o viscosity. The attenuation peak o the global low obtained using the approximate wave number is always shited to the right as compared to the solution with correct wave number (exact analytical or iterative solution). The observations o negative velocity dispersion in Jakobsen and Chapman (009) theory still remain, even i we use the correct eective wave number, when dealing with the phenomenon o wave-induced luid low in models o cracked /ractured porous media where global low eects dominates. We may also conclude rom these numerical experiments that at seismic requencies global low eects are not so important and needs very high permeability and low viscosity to have an eect. 17

18 Acknowledgements PhD student Aamir Ali would like to thank Higher Education Commission (HEC), Pakistan, and Department o Earth Sciences, University o Bergen, Norway, or providing the necessary unding to complete this work. Reerences Agersborg R., Jakobsen M., Ruud B.O., and Johansen T.A., 007. Eects o pore luid pressure on the seismic response o a ractured carbonate reservoir, Studia Geophysica et Geodaetica 51, Auld, B.A., Acoustic Fields and Waves in Solids, Krieger Publishing, Malabar, FL. Biot, M.A., Theory o propagation o elastic waves in a uid saturated porous solid. I. Low requency range and II. Higher-requency range. J. Acoust. Soc. Am., 8, Biot, M. A., 196. Mechanics o deormation and acoustic propagation in porous media, J. appl. Phys., 33, Carcione, J.M., Constitutive model and wave equations or linear, viscoelastic, anisotropic media, Geophysics, 60, Carcione, J.M., 007. Wave ields in real media: Wave propagation in anisotropic, anelastic, porous and electromagnetic media: Elsevier Science Publishing Company, Inc. Chapman, M., 001. Does luid viscosity inluence seismic velocity? 63rd Annual International EAGE Meeting, Amsterdam, The Netherlands, - 15 June. Chapman, M., V. Zatsepin, S., and Crampin, S., 00. Derivation o a microstructural poroelastic model, Geophys. J. Int., 151, Chapman, M., 003. Frequency dependent anisotropy due to meso-scale ractures in the presence o equant porosity, Geophys. Prosp., 51,

19 Dvorkin, J., and Nur, A., Dynamic peorelasicity: a uniied model with the squirt and Biot mechanism, Geophsyics, 58, Dvorkin, J., Nolen-Hoeksema, R., and Nur, A., The squirt low mechanism: Macroscopic description, Geophsyics, 59, Dvorkin, J., Mavko, G. and Nur, A., Squirt low in ully saturated rocks, Geophysics, 60, Hudson, J.A., Liu, E., and Crampin, S., The mechanical properties o materials with interconnected cracks and pores, Geophys. J. Int., 14, Jakobsen, M., Hudson, J.A., and Johansen, T.A., 003a. T-matrix approach to shale acoustics. Geophys. J. Int., 154, Jakobsen, M., Johansen, T.A., and McCann, C., 003b. The acoustic signature o luid low in complex porous media, J. Appl. Geophys., 54, Jakobsen, M., 004. The interacting inclusion model o wave-induced luid low, Geophys. J. Int., 158, Jakobsen, M., 007. Eective hydraulic properties o ractured reservoirs and composite porous media, J. Seism. Expl., 16, Jakobsen, M., and Chapman, M., 009. Uniied theory o global and squirt low in cracked porous media. Geophysics, 74, WA65-WA76. Mavko, G., and Jizba, D., Estimating grain-scale luid eects on velocity dispersion in rocks, Geophysics, 56, Mavko, G., and Nur, A., Melt squirt in the asthenosphere, J. Geophys. Res., 80, Mukerji, T., and Mavko, G., 1994, Pore luid eects on seismic velocity in anisotropic rocks, Geophysics, 59,

20 O Connell, R.J., and Budiansky, B., Viscoelastic properties o luid-saturated cracked solids, J. Geophys. Res., 8, Pointer, T., Liu, E., and Hudson, J.A., 000. Seismic wave propagation in cracked porous media, Geophys. J. Int., 14, Shahraini, A., Ali, A., and Jakobsen, M., 010. Characterization o ractured reservoirs using a consistent stiness-permeability model: ocus on the eects o racture aperture, Geophys. Prospect., 59, Van Gol-Racht, T.D., 198. Fundamentals o Fractured Reservoirs Engineering, Elsevier. ISBN Winkler, K.W., Dispersion analysis o velocity and attenuation in Berea sandstone, J. Geophys. Res., 90, Xi, D., Liu, X., and Zhang, G., 007. The Frequency (or Time) Temperature Equivalence o Relaxation in Saturated Rocks, Pure appl. Geophys., 164,

21 Appendix-A Christoel equation or viscoelastic media From Hooke s law the stress-strain relation can be written as using the matrix notation (Carcione, 007) σ = C e (A-1) where σ is the stress tensor, e is the strain tensor and C is the stiness tensor. The equation o motion in the absence o body orces can be written as (Carcione, 007) where tt Γ u = ρ u, (A-) T Γ = C, (A-3) where u is the displacement vector and the symmetric gradient operator using the matrix representation is given by (Auld, 1990; Carcione, 007) 1 = (A-4) 0 The strain displacement relation is given by (Carcione, 007) e = T u. (A-5) A general plane wave solution or the displacement vector o the body waves is given by (Carcione, 007) [( k )] u = u exp i ω t, (A-6) 0 x where u 0 represents a constant complex vector, ω is the angular requency and k is the wave-number vector. The particle velocity is given by time derivative o equation (A-6) given by v = t u = iωu. (A-7) In the absence o body orces ( = 0), we consider plane waves propagating along the direction given by (Carcione, 007) 1

22 kˆ = l +, (A-8) 1eˆ 1 + leˆ l3eˆ 3 where l 1, l 1 and l 1 are the direction cosines. The wave-number vector k can be written as (Carcione, 007) k = k, k, k ) = k( l, l, l ) = k ˆ, (A-9) ( k where k is the magnitude o the wave-number vector. The spatial dierential operator in equation (A-4) can be replaced by l l3 l ik 0 l 0 l3 0 l 1 ikl. (A-10) 0 0 l3 l l1 0 Now substituting the time derivative as t i ω and using equation (A-10) in equation (A-), we get (Carcione, 007) where k Γ u = ρω u, (A-) T Γ = L C L, (A-1) is the Christoel matrix. The dispersion relation is given by (Carcione, 007) where det( Γ ρv I3 ) = 0. (A-13) V ω =, (A-14) k is the complex velocity. The equation (A-13) is known as the Christoel equation. Using equation (A-14), the components o the slowness and attenuation vectors can be expressed in terms o the complex velocity as (given by Carcione, 1995) and 1 s = Re kˆ, V (A-15) 1 α = ω Im kˆ. V (A-16)

23 Appendix-B The eective permeability tensor The eective permeability tensor * K o the ractured porous reservoir model, assuming that the distribution o ractures is same or all racture amilies is given by (Jakobsen 007) where Here, * K = K + K ( I + g d K ), (B-1) K = v τ. (B-) 1 r K is background or matrix permeability, I is the (Kronecker-delta) identity or second rank tensors, (r) v is the volume concentration or ractures o type r and g d is a tensor given by the strain Green s unction integrated over an ellipsoid determining the symmetry o the correlation unction or the spatial distribution o ractures (see Jakobsen 007, Shahraini et al. 010). The Here, (r) τ or a single racture o type r is given by (Jakobsen 007) [ I g ( K )] 1 τ = ( K K ) K. (B-3) (r) g is a second-rank tensor given by the pressure gradient Green s unction integrated over a characteristic spheroid having the same shape as inclusions o type r (see Jakobsen 007, Shahraini et al. 010), and (r) K is a second-rank tensor o permeability coeicients or ractures o type r, which can be estimated using the cubic law given by (Van Gol-Racht 198) K ( a ) = I. (B-4) 1 3

24 Figure 1. Spherical pores - First-order T-matrix estimates o velocity and attenuation spectra or dierent luid mobilities. The exact, iterative and approximate solutions are presented. The porosity o spherical pores is 10%. Panels (a) and (b) show results with permeability o 1 Darcy. Panels (c) and (d) show the same as panels (a) and (b) but with permeability o 10 Darcy (a) 4400 (c) Velocity (m/s) Da Velocity (m/s) Da Attenuation (1/Q) Da (b) Attenuation (1/Q) Da (d) Legend Water/Exact Water/Iterative Water/Approx. Oil/Exact Oil/Iterative Oil/Approx. Gas/Exact Gas/Iterative Gas/Approx Frequency (Hz) Frequency (Hz) Figure Same as Fig.1, but the porosity o spherical pores is 35%. 4

25 Figure 3. Spherical pores - Higher-order T-matrix estimates o velocity and attenuation spectra or dierent luid mobilities. The iterative and approximate solutions are presented. The porosity o spherical pores is 35%. Figure 4. Randomly oriented micro-cracks - First-order T-matrix estimates o velocity and attenuation spectra or dierent luid mobilities. The exact, iterative and approximate solutions are presented. The aspect ratio o randomly oriented cracks is 1/1000, while crack density is 0.1. Panels (a) and (b) show results with permeability o 1 Darcy. Panels (c) and (d) show the same as panels (a) and (b) but with permeability o 10 Darcy. 5

26 Figure 5. Same as Fig.4, but the crack density was 0.5. Figure 6. Randomly oriented micro-cracks - Higher-order T-matrix estimates o velocity and attenuation spectra or dierent luid mobilities. The iterative and approximate solutions are presented. The aspect ratio o randomly oriented cracks is 1/1000, while crack density is

27 4500 (a) 4500 (c) Velocity (m/s) Da Velocity (m/s) Da Attenuation (1/Q) (b) (d) 0.4 Attenuation (1/Q) 1 Da Da Legend Water/Iterative Water/Approx. Oil/Iterative Oil/Approx. Gas/Iterative Gas/Approx Frequency (Hz) Frequency (Hz) Figure 7. Pores and randomly oriented micro-cracks - Higher-order T-matrix estimates o velocity and attenuation spectra or dierent luid mobilities. The iterative and approximate solutions are presented. The aspect ratio o randomly oriented micro-cracks is 1/1000. The porosity and crack density is 35% and 0.4, respectively. Panels (a) and (b) show results with permeability o 1 Darcy. Panels (c) and (d) show the same as panels (a) and (b) but with permeability o 10 Darcy. Figure 8. Aligned micro-cracks - Higher-order T-matrix estimates o velocity and attenuation spectra or the plane wave propagation with polar and azimuthal angles o 30 o and 45 o or dierent luid mobilities. The iterative and approximate solutions are presented. The crack density o aligned micro-cracks is 0.3. Panels (a) and (b) show results with permeability o 1 Darcy. Panels (c) and (d) show the same as panels (a) and (b) but with permeability o 10 Darcy. 7

28 Figure 9. Pores and aligned micro-cracks - Higher-order T-matrix estimates o velocity and attenuation spectra or the plane wave propagation with polar and azimuthal angles o 30 o and 45 o or dierent luid mobilities. The aspect ratio o aligned micro-cracks is 1/1000. The porosity and crack density is 35% and 0.4, respectively. Panels (a) and (b) show results with permeability o 1 Darcy. Panels (c) and (d) show the same as panels (a) and (b), but or a permeability o 10 Darcy (a) 5800 (c) Velocity (m/s) Velocity (m/s) Da 10 Da Attenuation (1/Q) (b) 1 Da Attenuation (1/Q) (d) 10 Da Legend Water/Iterative Water/Approx. Oil/Iterative Oil/Approx. Gas/Iterative Gas/Approx Frequency (Hz) Frequency (Hz) Figure 10. Pores, randomly oriented micro-cracks and meso-ractures - Higher-order T- matrix estimates o velocity and attenuation spectra or the plane wave propagation with polar and azimuthal angles o 30 o and 45 o or dierent luid mobilities. The aspect ratio o randomly oriented micro-cracks and aligned meso-racture set is 1/1000. The porosity (spherical pores), crack density (randomly oriented cracks) and racture density (aligned set o mesoscopic ractures) is 10%, 0.01 and 0.4, respectively. Panels (a) and (b) show results with permeability o 1 Darcy. Panels (c) and (d) show the same as panels (a) and (b), but or a permeability o 10 Darcy. 8