Anisotropic dispersion and attenuation due to wave induced fluid flow: Quasi static finite element modeling in poroelastic solids

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1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2009jb006644, 2010 Anisotropic dispersion and attenuation due to wave induced fluid flow: Quasi static finite element modeling in poroelastic solids F. Wenzlau, 1,2 J. B. Altmann, 1 and T. M. Müller 1 Received 25 May 2009; revised 15 February 2010; accepted 26 February 2010; published 14 July [1] Heterogeneous porous media such as hydrocarbon reservoir rocks are effectively described as anisotropic viscoelastic solids. They show characteristic velocity dispersion and attenuation of seismic waves within a broad frequency band, and an explanation for this observation is the mechanism of wave induced pore fluid flow. Various theoretical models quantify dispersion and attenuation of normal incident compressional waves in finely layered porous media. Similar models of shear wave attenuation are not known, nor do general theories exist to predict wave induced fluid flow effects in media with a more complex distribution of medium heterogeneities. By using finite element simulations of poroelastic relaxation, the total frequency dependent complex stiffness tensor can be computed for a porous medium with arbitrary internal heterogeneity. From the stiffness tensor, velocity dispersion and frequency dependent attenuation are derived for compressional and shear waves as a function of the angle of incidence. We apply our approach to the case of layered media and to that of an ellipsoidal poroelastic inclusion. In the case of the ellipsoidal inclusion, compressional and shear wave modes show significant attenuation, and the characteristic frequency dependence of the effect is governed by the spatiotemporal scale of the pore fluid pressure relaxation. In our anisotropic examples, the angle dependence of the attenuation is stronger than that of the velocity dispersion. It becomes clear that the spatial attenuation patterns show specific characteristics of waveinduced fluid flow, implying that anisotropic attenuation measurements may contribute to the inversion of fluid transport properties in heterogeneous porous media. Citation: Wenzlau, F., J. B. Altmann, and T. M. Müller (2010), Anisotropic dispersion and attenuation due to wave induced fluid flow: Quasi static finite element modeling in poroelastic solids, J. Geophys. Res., 115,, doi: /2009jb Introduction [2] In generally anisotropic, dissipative media waves are attenuated differently in different directions. There has been a recent interest in the anisotropy of seismic attenuation as it may provide additional information about subsurface elastic properties. In particular, velocity and attenuation anisotropy have important implications for fracture characterization in exploration seismology [Liu et al., 1993, 2007]. Anisotropic dispersion in anisotropic media is also relevant for upscaling ultrasonic and sonic borehole measurements [Wong et al., 2008]. The observability of attenuation anisotropy is demonstrated by laboratory measurements of ultrasonic waves [Best et al., 2007; Chichinina et al., 2008; Zhu et al., 2007] as well as by VSP field measurements [Maultzsch et al., 2007]. 1 CSIRO Earth Science and Resource Engineering, Perth, Western Australia, Australia. 2 Now at Fraunhofer ITWM, Kaiserslautern, Germany. Copyright 2010 by the American Geophysical Union /10/2009JB [3] Wave attenuation in linear, isotropic viscoelastic media is studied in the classical works of Buchen [1971] and Borcherdt and Wennerberg [1985]. Variation of plane wave attenuation with the angle of incidence in layered, anelastic media has been analyzed by various authors [e.g., Winterstein, 1987; Krebes, 1983]. Recently, these formulations have been generalized to anisotropic viscoelasticity by Červený and Pšenčík [2005, 2008], Sharma [2008], or Borcherdt [2009], revealing the directionality of compressional and shear wave attenuation. Simplified formulas of attenuation anisotropy in the form of a so called Q matrix were introduced by Zhu and Tsvankin [2006] in analogy to the Thomsen velocity anisotropy parameters [Thomsen, 1986]. [4] Whereas anisotropic viscoelasticity is useful as a descriptive or phenomenological concept of wave propagation in real rocks [Carcione, 1992, 2001], it does not allow any inference about the underlying physical attenuation mechanism, and therefore cannot be directly used for rock property inversion. In order to interpret the observed velocity or attenuation anisotropy suitable rock physics models are required that link acoustic signatures to a 1of12

2 physical process. In other words, to get further insight from these laboratory observations it is important to analyze the attenuation anisotropy for a specific attenuation mechanism. [5] A very important seismic attenuation mechanism occurs in porous media that are heterogeneous on mesoscopic scales, i.e., on length scales larger than the pore size d but still smaller than the seismic wavelength l [Pride et al., 2004]. Denoting the scale of the heterogeneity by a, the relation of scales is expressed by d a : While grain sizes and pore diameters on the microscale d are typically several micrometers, and seismic wave length l is of the order of tens of meters, a is usually assumed to be on the centimeter scale. In a medium with mesoscopic heterogeneities, the propagating wave causes local fluid pressure differences across internal interfaces. This drives local fluid flow and governs the overall relaxation behavior with a characteristic frequency that depends explicitly on the scale of the heterogeneity and on rock permeability. Often, this local flow mechanism is referred to as wave induced fluid flow [e.g., Müller and Gurevich, 2005]. Since multiscale heterogeneities are present in porous rocks, seismic attenuation due to wave induced flow affects a large frequency range and plays a major role at seismic frequencies. This is particularly important in fractured porous rocks. [6] The assumption of scale separation as introduced in equation (1) allows to apply Biot equations of poroelasticity [Biot, 1962] with varying coefficients to describe the effects of mesoscopic flow fields on the seismic velocities and attenuation. Applications of Biot s theory include waveinduced flow in fractured rocks [Berryman, 2007; Gurevich et al., 2009] and rock with patchy saturation [Toms et al., 2007]. In the seismic frequency range, wave induced flow can be modeled using the quasi static or low frequency version of Biot s theory. This is possible since inertial effects of fluid flow in the pore space can be neglected and the pressure equilibration process is governed by fluid pressure diffusion [Pride, 2005]. Since the seismic wavelength is assumed to be much larger than the mesoscopic heterogeneities the effective material behavior of the poroelastic composite can be described using viscoelasticity [Masson and Pride, 2007; Rubino et al., 2009; Wenzlau and Müller, 2009]. In the most general case the effective material behavior requires the concept of anisotropic viscoelasticity which has not yet been analyzed for attenuation and dispersion resulting from wave induced flow. [7] For simple shapes of heterogeneities such as spheres and layers or particular random arrangements of heterogeneities analytical solutions for attenuation and dispersion of compressional waves are available [White et al., 1975; Norris, 1993; Müller and Gurevich, 2005]. However, for arbitrary internal medium geometry attenuation and dispersion are unknown. One possibility to estimate dispersion and attenuation is to perform numerical wave simulations using the spectral ratio method or frequency shift method [e.g., Helle et al., 2003; Carcione and Picotti, 2006; Rubino et al., 2007]. In these simulations, wave signals are recorded along the propagation path, then dispersion and attenuation are calculated by measuring traveltime and the log amplitude decay. A different approach was suggested by Masson ð1þ and Pride [2007] who simulate quasistatic relaxation based on a finite differences solver for the dynamic Biot equations. Using their approach, the model size is considerably reduced to only one representative elementary volume (REV). Also, finite element simulations are used to model attenuation and dispersion [Rubino et al., 2007, 2009]. [8] In this work, the ideas of Masson and Pride [2007] and Rubino et al. [2009] are further developed and generalized. We propose a new and efficient strategy for estimating effective elastic moduli from time domain, quasi static relaxation experiments using a finite element solver for poroelastic rheology. By applying the Fourier transform we obtain complex, frequency dependent elastic stiffness tensor components, which are then used to derive velocity dispersion and attenuation for both, compressional and shear waves. To validate the approach we compute compressional wave attenuation and dispersion for two simple composites and compare it with known, exact solutions. For the case of a 3 D model containing an ellipsoidal inclusion we extract 5 components of the viscoelastic stiffness tensor thus demonstrating the applicability of the approach in effectively anisotropic media. We compute the angle and frequency dependence of attenuation and dispersion for all wave modes. 2. Effective Viscoelastic Rheology for Porous Media [9] At seismic frequencies, the effects of wave induced fluid flow are completely described by the quasistatic Biot theory, i.e., by two coupled equations for the stress equilibrium within the porous matrix and the Darcy flow of pore fluid. Writing spatial derivatives in the i direction as i and using Einstein repeated indices convention they are given by 0 j ij ðx k ; tþ; ð2þ _w iðx k ; tþ i px ð k ; tþ; ð3þ where t ij and p are the components of the stress tensor and the fluid pressure, respectively. In the Darcy equation (3), _w i are the components of the filtration velocity, defined as the relative fluid volume flux per unit area. The hydraulic parameters are permeability and fluid viscosity h. Equations (2) and (3) are coupled through the poroelastic stress strain relations, since both the stress and the strain depend on the displacement fields of the matrix and the fluid, u i and w i [Biot, 1962] ij ¼ 2" ij þ k u k ij þ k w k ij ; p ¼ M@ k u k M@ k w k ; with the strain " ij =(u i,j + u j,i )/2 and d ij being the Kronecker symbol. In these poroelastic constitutive relations, l u and m are the undrained Lamé parameters, a is the Biot coefficient and M is the so called pore space modulus. We provide a definition of the poroelastic parameters in Appendix C. [10] The calculation of effective elastic properties of a porous medium involves an upscaling process in which the medium properties on the macroscale are calculated from the properties on the mesoscale. In order to emphasize this ð4þ ð5þ 2of12

3 Figure 1. Elementary volume of a heterogeneous porous material containing mesoscale heterogeneities. change of scales between the fields on the mesoscale and on the macroscale, we use capital letters for the upscaled quantities. In the case of stress and strain, both scales are related through a spatial average over one REV as Z T ij ðþ¼ t 1 V REV ij ðx k ; tþdv; ð6þ E ij ðþ¼ t 1 Z " ij ðx k ; tþdv: ð7þ V REV with the REV sample volume V = R dv. [11] On the macroscale, the stress strain relation of a linear viscoelastic solid is expressed by T ij ðþ¼ t _R ijkl ðþ*e t kl ðþ; t such that the stress field T ij is calculated from a convolution of the strain E kl and the so called tensor of relaxation rate _R ijkl [Carcione, 2001]. Adapting the notation of Aki and Richards [1980], this quantity is conveniently written as a sum of an instantaneous part and a viscoelastic part as _R ijkl ðþ¼ t : Cijkl u ðþþ_y t ijkl ðþ t ; ð9þ u where C ijkl is the unrelaxed stiffness tensor, d(t) the delta distribution and _Y ijkl the viscoelastic part of the relaxation rate tensor. Applying the Fourier transform to equation (8), one obtains the stress strain relation in the frequency domain as ~T ij ð! Þ ¼ ~C ijkl ð! Þ~E kl ð! Þ; ð8þ ð10þ with the circular frequency w, and C ~ ijkl denoting the effective stiffness tensor. ~C ijkl is a complex, frequency dependent quantity that is related to the relaxation rate tensor by Fourier transform (denoted by F) as : ~C ijkl ð! Þ ¼ F _R ijkl ðþ t ¼ Cijkl u 1 þf _Y ijkl ðþ t : ð11þ [12] Note that in equations (9) and (11), as well as later on in equation (17), the usual summation convention does not apply, but all operations are carried out elementwise. This is indicated by the ¼ : symbol. 3. Numerical Simulation of Poroelastic Relaxation [13] In order to obtain the effective stiffness tensor of a given porous sample, we perform numerical simulations in the time domain using the finite element solver Abaqus for poroelastic rheology. First, the term relaxation experiment and the transformation from the time domain to the frequency domain are explained. Then we give the necessary boundary conditions that allow us to assess every single component of the effective stiffness tensor for the case of vertical transverse isotropy. Finally, rock physics formulas are given to compute dispersion and attenuation of compressional and shear waves from the stiffness tensor. [14] Consider a heterogeneous, fluid saturated, porous rock sample with the height L that is confined laterally, sealed hydraulically and represents one REV of a porous 0 rock, as shown in Figure 1. Initially, a global mean strain E kl is imposed instantaneously on the sample, resulting into a heterogeneous fluid pressure distribution and an initial mean stress state Tij 0 ¼ T ij ðt ¼ 0Þ ð12þ within the sample. The following pressure diffusion process governs the relaxation of the whole sample, such that while maintaining the overall strain, the mean stress decreases until the sample reaches the relaxed state T 1 ij ¼ lim T ij ðþ: t t!1 ð13þ This so called relaxation experiment allows the effective viscoelastic behavior of the heterogeneous porous medium to be characterized. [15] Inserting a Heaviside step function as strain input signal E ij ðþ¼e t ij 0 Ht ðþ; ð14þ 3of12

4 Figure 2. Schematic view of a relaxation experiment. (a) A step change in strain E is imposed in the form of a Heaviside function H(t), such that (b) the stress response is identical to the relaxation curve R(t). (c) The relaxation rate is the sum of an instantaneous part C u d(t) and a viscoelastic part C u _Y(t). (d) From the relaxation rate, the complex stiffness is obtained by Fourier transform, as indicated by the symbol F. into equation (8) and applying the theorem stiffness tensor ~C ijkl is then obtained by time differentiation and Fourier transform as indicated in equation (11). The procedure is depicted schematically in Figure 2. The input strain signal is shown in Figure 2a, at positive times it is has the constant value E 0, which is used for normalization. The stress response, divided by the input strain yields the relaxation curve R, given in Figure 2, varying between the unrelaxed limit C u and the relaxed limit C r, calculated similar to equation (18), but with the relaxed stress state T ij. Time differentiation of the relaxation curve yields the relaxation rate (Figure 2c) which is transformed into the frequency domain to calculate the complex stiffness, shown in Figure 2d. As the relaxation function, the real part of the stiffness varies between the relaxed and the unrelaxed limits, which correspond to low and high frequency material behavior, respectively. [16] The stiffness tensor ~C ijkl of an anisotropic medium without symmetry has 21 independent components which cannot be computed from a single 3 D relaxation experiment. In the most general case, it is necessary to apply six basic deformation states and measure in each experiment all components of the stress tensor in order to fully describe the material behavior [Barbero, 2008]. Here, we define a basic deformation state such the strain tensor has only one single nonzero component. [17] The situation simplifies in the case of a medium with vertical transverse isotropy (VTI) which we are analyzing in the following. A viscoelastic VTI medium has only 5 independent stiffness tensor components and these are found by using three basic relaxation experiments, two uniaxial strain tests and one shear test, as depicted in Figure 3. [18] This means that in order to obtain the effectively VTI properties of a poroelastic medium, one has to solve equations (2) and (3) within the computational domain W under the following boundary conditions: 9 u 3 ¼ E 0 L on G T ; >= u i n i ¼ 0 on G n G T ; ð19þ w i n i ¼ 0 on G; >; E 33 _fðþ*h t ðþ¼f t ðþ; t ð15þ one finds that the stress response is directly proportional to the relaxation tensor R ijkl, such that E 11 ð20þ T ij ðþ¼r t ijkl ðþe t kl 0 : ð16þ 9 u 1 ¼ E 0 L on G L ; >= u i n i ¼ 0 on G n G L ; >; w i n i ¼ 0 on G; Integrating equation (9) and inserting the result into equation (16) yields an expression for Y ijkl as with Y ijkl ðþ¼ t : T ij t ðþekl 0 1 Cijkl u Ht ðþ; ð17þ C u ijkl ¼ T 0 ij E0 kl 1 : ð18þ This means that the viscoelastic relaxation tensor Y ijkl (t) is directly calculated from the applied mean deformation state E 0 ij and the transient mean stress response T ij (t). The complex u 1 ¼ E 0 L on 9 G T ; >= u 1 ¼ 0 on G B ; u 2 ¼ u 3 ¼ w i n i ¼ 0 on G; >; E 13 ð21þ for uniaxial vertical compression, for uniaxial horizontal compression and for simple shear, respectively. Here, L is the sample edge length, G is the sample boundary, the indices T, B and L refer to the top (x 3 = L), bottom (x 3 =0) and left side (x 1 = 0) of the sample. The vector n i denotes the unit vector pointing in outward normal direction with respect to the boundary. 4of12

5 two independent components of the stiffness tensor. For example, the compressional and the shear wave relaxation functions are computed as R 3333 ðþ¼ t T 33ðÞ t ; ð29þ E 33 Figure 3. Three deformation states used to obtain all five components of the effective stiffness tensor (assuming effective transversal isotropy). (a) Vertical compression. (b) Horizontal compression. (c) Simple shear deformation. [19] In order to save computation time, two basic deformation states, uniaxial compression and simple shear, can be combined to form one deformation state that allows all but one component of the VTI stiffness tensor to be obtained by using just one numerical experiment. The result is u 3 ¼ u 1 ¼ E 0 L on G T 9 ; u 3 ¼ u 1 ¼ 0 on G B ; >= u 2 ¼ 0 on G F [ G K ; u i j G L ¼ u i j G R; w i n i ¼ 0 on G; >; E 33 þ E 13 ð22þ where the indices R, F and K refer to the right, the front and the back of the computational domain W, respectively. The five components of a VTI medium are then found using the relations R 1111 ðþe t 0 ¼ T 11 ðþ; t ð23þ R 1313 ðþ¼ t T 33ðÞ T t 11 ðþ t : ð30þ 2E 33 [21] It is well known that in anisotropic, viscoelastic solids, dispersion relations of compressional and shear waves are found by solving an eigenvalue problem of the Christoffel matrix where the eigenvalues correspond to complex velocities [e.g., Červený and Pšenčík, 2005]. In general, these velocities depend on the wave polarization, frequency, angle of incidence and on the degree of wave inhomogeneity, i.e., on the angle between the wave propagation direction and the direction of amplitude decrease. For homogeneous body waves, the directions of wave propagation and attenuation coincide, and Carcione [1992] gives simplified expressions for the complex velocities in this case. We provide the corresponding formulas in Appendix B. From the complex velocities ~V, dispersion and attenuation are derived as h i 1; V m ð!; Þ ¼ Re ~V 1 ð31þ Q 1 m m Im ~V 2 ð!; Þ ¼ m ; ð32þ Re ~V 2 m where the attenuation is given in the form of the inverse quality factor Q 1 and m ={P, SV, SH} for compressional, vertically polarized shear and horizontally polarized shear waves, respectively. R 1122 ðþe t 0 ¼ T 22 ðþ; t R 1133 ðþe t 0 ¼ T 33 ðþ; t R 1313 ðþe t 0 ¼ T 13 ðþ; t ð24þ ð25þ ð26þ 4. Results for Isotropic and Transverse Isotropic Media [22] In order to exemplify the applicability and accuracy of the proposed method, results are shown for a porous rock (consolidated sandstone (see Table 1)) with fluid pressure diffusivity D given by 2R 1212 ðþe t 0 ¼ T 11 ðþ T t 22 ðþ: t ð27þ D ¼ N : ð33þ The vertical P wave relaxation R 3333 requires an additional uniaxial compression experiment, as given by equation (19). Its value is computed as R 3333 ðþe t 0 ¼ T 33 ðþ: t ð28þ Note that in equation (22), we make use of a periodic boundary condition by tying together the displacements of the left and right edge of the model. [20] In the case of an effectively isotropic material, one uniaxial relaxation experiment is sufficient to calculate the Here, and h are the hydraulic permeability and the fluid viscosity, respectively. For a definition of the poroelastic parameter N, see Appendix C. [23] The synthetic sample is saturated with two fluids, 50% water and 50% gas. A periodically layered fluid distribution with spatial period L is assumed. For this case, an analytical solution is provided by White et al. [1975]. Results for the velocity dispersion and attenuation of a P wave are shown in Figure 4. We find that our numerically obtained estimates are in excellent agreement with the predicted values. The dispersion curve shows a smooth crossover from 5of12

6 Table 1. Material Properties Used in the Numerical Relaxation Experiments Background Soft Inclusion Gas Layer Matrix Grain bulk modulus K g (GPa) Matrix bulk modulus K d (GPa) Matrix shear modulus G d (GPa) Porosity Permeability (Darcy) Grain density r s (t/m 3 ) Fluid Bulk modulus K f (GPa) Density r f (t/m 3 ) Viscosity h (mpas) velocities around 3020 m/s at frequencies below the characteristic frequency! c ¼ 9D L 2 ð34þ to 3190 m/s at high frequencies above the characteristic frequency. Using 50 elements and 100 time steps, the standard deviation from the exact solution is less than 0.1%. In addition to this, the attenuation behavior is predicted with the same accuracy. [24] Finally, as a consequence of Hill s theorem, a partially saturated rock with isotropic matrix properties is always effectively isotropic, regardless of the distribution of the fluid phases inside the sample [Hill, 1963]. In the case of the previous example, this is easily checked by calculating the shear relaxation function according to equation (30). We confirmed that for the layered model depicted in Figure 5a, the shear modulus is indeed time independent, implying dispersion free propagation of shear waves in partially saturated rocks (results not shown). [25] The situation changes if the matrix shear modulus is inhomogeneous, which in general results in an anisotropic effective behavior of the heterogeneous medium. As pointed out earlier, in this case, multiple relaxation experiments are necessary to determine all effective elastic moduli. [26] The special case of anisotropy is now considered where the geometry of the heterogeneity has a vertical symmetry axis, as e. g. in the case horizontal layering. Then, the corresponding effective medium possesses vertical transverse isotropy (VTI) and it is described by five independent elastic constants ~C 1111, ~C 1122, ~C 1133, ~C 3333 and ~C 1313 that are found as explained previously. [27] In order to exemplify the VTI case, two heterogeneous double porosity media will now be considered, both with an effectively anisotropic, viscoelastic material behavior. The first medium is a water saturated, 20% porosity consolidated sandstone with thin, soft embedded porous unconsolidated layers with 30% porosity. Material parameters for these two rock types are given in Table 1 and a picture of an REV is shown in Figure 5b. A second model is considered with the same material parameters, but with a soft 3 D ellipsoidal inclusion instead of the porous layer, see Figure 5c. The 3 D ellipsoid has an axis of rotational symmetry in vertical direction and an aspect ratio of 1:4. Both models are chosen such that the volume fraction of the inclusion is 2.83%. [28] For the layered case, exact theoretical solutions are available for the effective P wave modulus for normal incidence ~c 3333 (w) [Norris, 1993; Brajanovski et al., 2005]. In addition to that, the poroelastic Backus averaging gives the high and low frequency limits for all three wave modes (P, SV, SH). In the case of the ellipsoidal inclusion, a theoretical solution for the effective P wave modulus has been reported recently by Galvin and Gurevich [2006]. Their model includes fast to slow mode conversion but it is restricted to the case when the wave propagation direction is parallel to the vertical symmetry axis. In addition to that, their solution assumes a fluid filled inclusion and a weak shear modulus of the surrounding rock matrix, such that G d /K f < a/b. The advantage of numerical results if compared to the previously mentioned theoretical solutions is that it is possible to obtain the total complex, frequency dependent VTI stiffness tensor. Using numerical modeling, dispersion and attenuation of all three wave modes are therefore obtained for an arbitrary angle of incidence. [29] First of all, velocity and attenuation of all wave modes in the anisotropic models are a function of the angle of incidence. An overview of the degree of velocity anisotropy for the two double porosity media as compared to the partially saturated model is given in Figure 6. The curves show slowness surfaces of shear and compressional wave modes. For comparison, we also show the slowness surfaces for the effectively isotropic partially saturated model in Figure 6a. The strongest velocity anisotropy is observed for the 1 D layered double porosity model, with local minima of P wave slowness at zero and grazing angle of incidence. The SV wave slowness has a minimum at 45 Figure 4. (a) Velocity dispersion and (b) attenuation for the 1 D partially saturated rock sample shown in Figure 5a. Frequencies are normalized using the background diffusivity D and layer period L. 6of12

7 Figure 5. Three model geometries used in the relaxation experiments. (a) Water saturated consolidated sandstone model with a gas saturated layer of thickness h. It is used in the partial saturation benchmark test example. Double porosity models are (b) a water saturated consolidated sandstone containing a weak layer or (c) an ellipsoidal inclusion with rotational symmetry about the vertical axis. Total model dimension is L L L. and that of the SV wave at 0. The slowness anisotropy of the 3 D ellipsoidal model is qualitatively similar but less pronounced. The line thickness in the polar plots reflects the amount of dispersion, which is most clearly seen in the partially saturated model that is characterized by a strong frequency dependence of the P wave. [30] In addition to the phase velocity, the attenuation characteristics of the double porosity models are also anisotropic, as shown in the polar plots of Figure 7. While the P wave attenuation of the partially saturated model is independent of the propagation direction (Figure 7a), in the layered double porosity model (Figure 7b), only P and SV waves are attenuated, and the attenuation of the pure shear wave mode is zero. P wave attenuation has a maximum at 0 and it attains zero at approximately 55. SV wave attenuation is zero at = 0 and at = 90 and maximal at 45. [31] In the 3 D case, the P and SV waves behave qualitatively like in the 1 D case. However, there is neither for the P nor for the SV wave an angle of incidence where no attenuation occurs (Figure 7c). In addition to that, attenuation is also observed for the horizontally polarized SH wave, which is, however, always smaller than the attenuation of the P and SV waves. [32] A more detailed picture of P, SV and SH wave velocities is depicted in Figure 8 for the 1 D layered model and in Figure 9 for the 3 D ellipsoidal inclusion model. The amount of velocity dispersion is indicated by the grey shaded areas between the unrelaxed or high frequency limit and the relaxed or low frequency limit. The P wave velocities show local maxima at normal (0 ) and grazing (90 ) angle of incidence, with highest values at 90. A maximum in dispersion is found at 0. The maximum SV wave velocity is found at 45 in both models. [33] The frequency dependence of P and SH wave attenuation at normal and grazing incidence are shown in Figure 10. The first observation is that maximum attenuation throughout the whole frequency band is encountered for the P wave at vertical incidence ( = 0 ) for both models, but the values are higher for the layered case by a factor of 2.5 (note the different axes scaling in Figures 10a and 10b). Second, the depicted SH wave attenuation is zero for the layered case. This should be expected, since the dispersion for horizontally polarized shear waves is zero, see Figure 8c. Interestingly, SH wave attenuation is nonzero in the 3 D case of a model with the elliptical inclusion. Actually, the SH wave attenuation of vertically incident waves is comparable to that Figure 6. Slowness surfaces obtained from numerical simulations. The blue line corresponds to P waves, the red line corresponds to SV waves, and the black line corresponds to SH waves. Results are given for (a) the 1 D gas layer model, (b) the 1 D soft layer model, and (c) the 3 D model with a soft ellipsoidal inclusion. The vertical axis corresponds to normal incident waves while the horizontal axis corresponds to grazing incidence. The line thickness indicates the amount of velocity dispersion, which is strongest in the partially saturated case (Figure 6a). 7of12

8 Figure 7. The same as Figure 6 but for the maximum wave attenuation. (a) In the case of the partially saturated sample, the P wave (blue line) shows isotropic attenuation, and the shear waves propagate without attenuation. (b and c) In the double porosity models, the SV waves (red lines) are attenuated as well and attenuation shows anisotropic behavior. In the 3 D double porosity model (Figure 7c), attenuation occurs for the SH wave as well (black line). of vertically incident P waves, attaining a value of Q 1 SH = if compared to Q 1 P = Discussion [34] Patchy saturation and its influence on seismic attenuation has been analyzed extensively in the past, e.g., by White et al. [1975] and Norris [1992]. Toms et al. [2006] give a review of different models. Two important features of wave induced flow in porous media with partial saturation is the effective isotropy of the bulk medium and the absence of shear wave attenuation, which facilitates the theoretical analysis. Both features are also observed in the layered example shown in Figure 5a. The reason for the loss free shear wave propagation lies in the fact that shear waves do not change the pore volume and therefore do not excite the fluid pressure in the porous medium. [35] There is a qualitative difference between patchy saturated media and media with a heterogeneous rock matrix. In contrast to the patchy saturation case, the effective elastic behavior of the latter is in general anisotropic and an efficient coupling between shear and compressional waves occurs. One has to keep in mind that the fluid pressure is excited only if the strain field has locally dilatational components, as seen from the divergence terms in equation (5). In homogeneous porous media, the strain associated with shear wave motion is purely deviatoric, but this is no longer the case in heterogeneous double porosity media. In other words, in heterogeneous media, shear waves may perturb the fluid pressure distribution which entails pressure relaxation and therefore acoustic attenuation. The coupling can be zero as in the case of a horizontally polarized shear wave in layered media or very strong as in the case of a vertically polarized shear wave at 45 incidence angle. In fact, in the layered model, vertically polarized shear wave attenuation exceeds attenuation of compressional waves for angles of incidence between 25 and 80. [36] Our results are in agreement with the P wave attenuation measurement reported by Zhu et al. [2007, Figure 6], with the strongest attenuation occurring in the direction parallel to the symmetry axis. It remains to be investigated how the attenuation anisotropy manifests for damping mechanisms other than wave induced flow. Given that other attenuation mechanisms produce different directionality patterns, one might potentially infer from attenuation anisotropy measurements some information about the dominating loss mechanism. [37] Further insight into the behavior of heterogeneous porous solids is found by a more detailed comparison of the 1 D and the 3 D results. One fundamental observation is Figure 8. Phase velocity of compressional and shear waves as a function of angle of incidence obtained numerically for the 1 D layered double porosity model. The plots show (a) P wave velocities, as well as those of (b) vertically polarized and (c) horizontally polarized shear waves. The grey shaded areas between the limits of unrelaxed and the relaxed fluid pressure indicate the amount of dispersion. 8of12

9 Figure 9. The same as Figure 8 but for the 3 D ellipsoidal double porosity model. While the velocities are qualitatively similar to the 1 D case, the anisotropy and the dispersion are less pronounced in the 3 D case. that in the case of the layered medium, the frequencyattenuation curves (Figure 10a) are similar such that maximum attenuation occurs at the same frequency for horizontally and vertically propagating P waves. An explanation for this phenomenon can be made if one considers the possible paths of fluid pressure equilibration. In the 1 D case, equilibration can only occur vertically between internal layers within the model, independent of what deformation has been causing the fluid pressure disequilibrium in the unrelaxed state. Consequently, there exists only one characteristic time scale of fluid pressure relaxation, scaling with the layer period. [38] The situation is different in the case of a 3 D inclusion, which can be seen from the unrelaxed fluid pressure distributions corresponding to the different applied deformation states. In Figure 11, these distributions are given for uniaxial compression and pure shear deformation such as associated with P and SH waves under normal and grazing angles of incidence. The E 12 deformation state was obtained as the difference between the two horizontal compression states as p E 12 = p E 11 p E 22. [39] While in the case of uniaxial compression, fluid pressure gradients appear mainly across the interface between the inclusion and the surrounding matrix, in the case of shear deformation, double couple distributions of fluid pressure appear and fluid flow is induced mainly between different poles within the rock matrix. In addition to that, the spatial scale of fluid pressure distribution under horizontal shear is larger than in the case of vertical shear, entailing larger relaxation times for horizontally propagating SH waves and a shift of the characteristic frequency to lower frequencies. [40] In contrast to the scheme proposed by Masson and Pride [2007], the application of a fully implicit solver allows the free choice of the time step size, independent of numerical stability restrictions. Therefore, at the beginning of the experiment, a small time step is chosen and at larger simulation times, the time step is exponentially increased. The time step size is thus adapted optimally to the shape of the relaxation curve. Another advantage of the quasistatic finite element model if compared to a fully dynamic finite difference scheme, is that it excludes inertial effects and therefore, no resonance can occur in the model. As a consequence, a true step function can be applied without using a low frequency filter as in the work by Masson and Pride [2007], thus reducing the total simulation time. [41] Rubino et al. [2009] use a frequency domain finite element Biot solver to calculate effective viscoelastic properties of randomly heterogeneous porous rocks. They use time harmonic stress fields as boundary conditions. This setup allows the calculation of the bulk and shear moduli of a heterogeneous medium directly in the frequency domain, without calculating the relaxation tensor in the first place. Figure 10. Frequency dependent attenuation Q 1 derived from double porosity relaxation experiments, (a) for the layered model and (b) for the model with an ellipsoidal inclusion. P and SH waves are considered with normal (0 ) and grazing (90 ) angle of incidence. In contrast to the layered case, where shear wave attenuation is zero, it is nonzero for the 3 D geometry. 9of12

10 Figure 11. Unrelaxed fluid pressure distributions in the 3 D finite element model for different deformation states. Shown are profile cuts through the symmetry planes of the ellipsoidal inclusion, projected to the edges of modeling domain. Fluid pressure response due to uniaxial compression in (a) vertical and (b) horizontal directions are shown as well as pure shear within the (c) vertical and (d) horizontal planes. The fluid pressure distributions correspond to P 0, P 90, SH 0,andSH 90 waves. Rubino et al. [2009] demonstrate the accuracy of their approach by a comparison with the predictions of White s model. The new aspect of our simulations if compared to the previous works is the consideration of anisotropic behavior of double porosity media. This reveals for the first time the magnitude of SH and SV wave attenuation in addition to the compressional P wave mode. [42] The application of finite element modeling for the estimation of effective elastic properties is, however, still numerically expensive if 3 D geometries are considered. For the 3 D double porosity model with an ellipsoidal inclusion, an irregular mesh with tetrahedral elements was applied. The simulation of 60 time steps required to resolve the relaxation behavior takes 14 h on 10 nodes of the SGI Altix 350 located at the Geophysical Institute of Universität Karlsruhe. The model demands 8.9GB main memory. 6. Conclusions [43] We propose a methodology to efficiently compute macroscopic viscoelastic properties of a poroelastic solid containing arbitrary mesoscopic heterogeneities. Time domain finite element relaxation experiments yield all components of the relaxation rate tensor from which the complex, frequency dependent stiffness tensor is obtained by Fourier transform. Velocity dispersion and attenuation of compressional and shear wave modes, caused by induced fluid flow, are then calculated straightforwardly. A comparison of the simulation results for a 1 D layered model with the corresponding analytical solution reveals that velocity dispersion and attenuation are modeled with very high accuracy, i.e., with a standard deviation of less than 1%. [44] Examples of a layered double porosity medium and a 3 D medium with an ellipsoidal soft poroelastic inclusion have been analyzed. Both models show effectively vertical transverse isotropy (VTI). Three deformation states are applied to estimate the total VTI stiffness tensor. In the case of a 3 D ellipsoidal inclusion, we find that the amount of SV wave attenuation is comparable with that of P waves. In contrast to the 1 D case, the SH wave is dispersive and attenuating as well, although the attenuation is smaller than that of the P or SV waves. Each wave mode induces a characteristic spatial fluid pressure distribution governing the characteristic frequency of the relaxation process. A general feature of our attenuation results for both the 1 D and the 3 D model is that maximum SV wave attenuation occurs at 45 and the local maxima of P wave attenuation occurring for normal and grazing incidence (see Figures 7b and 7c). We conclude that these patterns are characteristic for wave induced flow in VTI media. [45] The approach presented here can be easily applied to other types of internal medium heterogeneity. However, if the medium is not effectively VTI, up to six linear independent deformation states are required to estimate the full effective stiffness tensor. Appendix A: Integrals [46] In this paper, we use the Fourier transform according to the definitions F fðþ t g ð! Þ ¼ Z 1 1 F 1 ff ð! ÞgðÞ¼ t 1 2 ðþexp t ð {!tþdt; ða1þ Z 1 1 F ð! Þexp ð{!tþd!: ða2þ Since here, f is a causal function, integration (A1) is performed for t > 0 only. The convolution integral is defined as fðþ*g t ðþ¼ t Z 1 1 f ðþgt ð Þd: ða3þ Appendix B: Dispersion and Attenuation of Plane Time Harmonic Waves in VTI Media [47] Due to the vertical symmetry, a layered medium is VTI and thus characterized by five independent elastic moduli In compact notation, the stiffness tensor ~C ijkl has the form 0 1 ~C 1111 ~C 1122 ~C ~C 1122 ~C 1111 ~C ~C 1133 ~C 1133 ~C ðb1þ ~C B ~C A ~C of 12

11 where ~C 1212 is computed as (~C 1111 ~C 1122 )/2. In the isotropic case, one has furthermore ~C 1111 ¼ ~C 3333 ¼ ~P; ðb2þ ~C 1313 ¼ ~C 1212 ¼ ~G; ðb3þ they are given as [Carcione, 1992] 2~V 2 P ¼ p ~C 1111 sin 2 þ ~C 3333 cos 2 þ ~C 1313 þ ffiffiffiffi X ; ðb10þ 2~V 2 SV ¼ p ~C 1111 sin 2 þ ~C 3333 cos 2 þ ~C 1313 ffiffiffiffi X ; ðb11þ ~C 1122 ¼ ~C 1133 ¼ ~P 2~G: ðb4þ In the isotropic case, ~P and ~G are the P wave modulus and the shear wave modulus, respectively. [48] If the stiffness tensor of an effectively elastic or viscoelastic medium is known, In the general case, solving the equation of motion using a plane waves ansatz provides velocity dispersion and attenuation of three wave modes, i.e., one quasi compressional and two quasi shear modes. Following Červený and Pšenčík [2008], the calculation of the signatures involves solving the three systems for the polarization U k ðg ik ik ÞU k ¼ 0; i ¼ 1; 2; 3; ðb5þ ~V 2 SH ¼ ~C 1212 sin 2 þ ~C 1313 cos 2 : ðb12þ Please note that the structure of these equations is exactly the same as in the purely elastic case, where all elastic moduli and the velocities are real valued [Mavko et al., 1998]. Since in the viscoelastic case, the components of the stiffness tensor are complex, so are the velocities derived from equations (B10) (B12). Appendix C: Poroelastic Parameters [50] The poroelastic quantities a, M, N and the Lamé parameters l u and m are computed from the fundamental rock properties listed in Table 1 as where G ik the generalized Christoffel matrix given by G ik ¼ 1 ~ C ijkl p j p l : ðb6þ ¼ 1 K d K g ; ðc1þ The solution of this eigenvalue problem are three slowness vectors p i and polarization vectors U i corresponding to the quasi longitudinal P wave and horizontally and vertically polarized quasi shear waves SH and SV. The quantity r is averaged bulk density. From the slownesses and the polarizations, dispersion and attenuation in terms of the inverse quality factor are calculated as M ¼ þ 1 ; ðc2þ K g K f K d þ 4 3 N ¼ M G d K d þ 4 3 G d þ 2 M ; ðc3þ V ð!; ; DÞ ¼ 1 jre p i j ; ðb7þ u ¼ K d 2 3 G d þ 2 M; ðc4þ Q 1 ð!; ; DÞ ¼ 2S jim p j ; S i Re p i ðb8þ ¼ G d : ðc5þ where S i denotes the Poynting vector that can be obtained for each wave mode from the corresponding slowness and polarization. In general, the seismic signatures are frequency dependent, change with respect of the angle of incidence. In addition to that, the inhomogeneity of the wave, as controlled by the parameter D [Červený and Pšenčík, 2008], influences the dispersion and attenuation characteristics. [49] The situation is simplified when we restrict the attention to homogeneous waves, i.e., to the case when the real part and the imaginary part of the complex slowness vector are pointing in the same direction. Then, explicit expressions for V and Q are obtained for an effective viscoelastic VTI medium by first calculating three frequencydependent, complex velocities ~V P, ~V SV and ~V SH as a function of angle of incidence. Abbreviating X ¼ ~C 1111 ~C 1313 sin 2 ~C 3333 ~C 1313 cos 2 2 2sin þ ~C 1133 þ ~C ; ðb9þ [51] Acknowledgments. We thank the Deutsche Forschungsgemeinschaft and the sponsors of the PHASE consortium for supporting the research presented in this paper. References Aki, K., and P. Richards (1980), Quantitative Seismology: Theory and Methods, W. H. Freeman, San Francisco, Calif. Barbero, E. J. (2008), Finite Element Analysis of Composite Materials, CRC Press, Boca Raton, Fla. Berryman, J. G. (2007), Seismic waves in rocks with fluids and fractures, Geophys. J. Int., 171, Best, A., J. Sothcott, and C. McCann (2007), A laboratory study of seismic velocity and attenuation anisotropy in near surface sedimentary rocks, Geophys. Prospect., 55(5), Biot, M. A. (1962), Mechanics of deformation and acoustic propagation in porous media, J. Appl. Phys., 33(4), Borcherdt, R. D. (2009), Viscoelastic Waves in Layered Media, Cambridge Univ. Press, Cambridge, U. K. Borcherdt, R. D., and L. Wennerberg (1985), General P, type I S, and type II S waves in anelastic solids: Inhomogeneous wave fields in low loss solids, Bull. Seismol. Soc. Am., 75(6), Brajanovski, M., B. Gurevich, and M. Schoenberg (2005), A model for P wave attenuation and dispersion in a porous medium permeated by aligned fractures, Geophys. J. Int., 163, of 12

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