SCIENCE CHINA Physics, Mechanics & Astronomy

SCIENCE CHINA Physics, Mechanics & Astronomy Article June 014 Vol.57 No.6: 1068 1077 doi: 10.1007/s11433-014-5435-z Wave dispersion and attenuation in viscoelastic isotropic media containing multiphase low and its application YANG Lei 1,, YANG DingHui 1, & NIE JianXin 3 1 Department o Mathematical Sciences, Tsinghua University, Beijing 100084, China; Computational Geophysics Laboratory, Tsinghua University, Beijing 100084, China; 3 State Key Laboratory o Explosion Science and Technology, Beijing Institute o Technology, Beijing 100081, China Received June 4, 01; accepted January 1, 014; published online April 17, 014 In this paper, we introduce the complex modulus to express the viscoelasticity o a medium. According to the correspondence principle, the Biot-Squirt (BISQ) equations in the steady-state case are presented or the space-requency domain described by solid displacements and luid pressure in a homogeneous viscoelastic medium. The eective bulk modulus o a multiphase low is computed by the Voigt ormula, and the characteristic squirt-low length is revised or the gas-included case. We then build a viscoelastic BISQ model containing a multiphase low. Through using this model, wave dispersion and attenuation are studied in a medium with low porosity and low permeability. Furthermore, this model is applied to observed interwell seismic data. Analysis o these data reveals that the viscoelastic parameter tan is not a constant. Thus, we present a linear requency-dependent unction in the interwell seismic requency range to express tan. This improves the it between the observed data and theoretical results. multiphase low, viscoelastic, BISQ model, wave dispersion, attenuation PACS number(s): 91.30.Cd, 91.60.Ba, 91.60.Lj, 91.60.Np Citation: Yang L, Yang D H, Nie J X. Wave dispersion and attenuation in viscoelastic isotropic media containing multiphase low and its application. Sci China-Phys Mech Astron, 014, 57: 10681077, doi: 10.1007/s11433-014-5435-z Corresponding author (email: dhyang@math.tsinghua.edu.cn) In lithology, predictions rom seismically derived properties, such as velocity, impedance, and velocity ratio, are o great importance in seismic exploration and reservoir characterization [1 3]. The inluence o porous luids on seismic waves is also critical [4 7]. Since Gassmann [8] derived the ormula o elastic modulus or a solid saturated by luids, many studies have been conducted on porous media. Using the Lagrange equations and Newton s law, Biot [9,10] derived the porous elastic wave equations, and predicted that dilatational waves o the second kind (the slow P-wave) existed in the system. Plona [11] and Bouzidi et al. [1] experimentally observed the slow P-wave predicted by Biot s theory, and Berryman [13] conirmed Biot s idea theoretically. However, many researchers have ound that large velocity dispersion and strong attenuation o waves cannot be explained by Biot s theory. Mavko has shown that this phenomenon is mainly caused by the squirt-low mechanism o porous luids [14,15]. Traditionally, the description o the squirt-low mechanism ocused on micro-characteristics, e.g., an individual porous geometry, which made it diicult to extend the squirt-low mechanism. Hence, Dvorkin et al. [16,17] presented a consistent theory (BISQ) that combined the concepts o Biot with the squirt-low mechanism. This explained the phenomena o the waves large velocity dispersion and strong attenuation. On the basis o the constitutive relations in the space-requency domain [18] and the isotropic description o the solid\luid coupling eect, Parra [19,0] extended the BISQ theory to include transversely Science China Press and Springer-Verlag Berlin Heidelberg 014 phys.scichina.com link.springer.com

Yang L, et al. Sci China-Phys Mech Astron June (014) Vol. 57 No. 6 1069 isotropic poroelastic media. Based on Biot s theory and the assumption o anisotropy in the solid/luid coupling eect, Yang [1,] generalized Dvorkin s heuristic model to high-dimensional anisotropic cases, and presented a general BISQ model that dealt with the Biot mechanism and the squirt-low mechanism uniormly. Marketos [3] studied the application o the BISQ model to clay squirt lows in reservoir sandstone, and Cui et al. [4] studied the eatures o the slow P-wave in a porous medium using the BISQ model. Nie et al. [5] presented a revised unsaturated BISQ model. They studied the wave velocity dispersion and attenuation in partially saturated sandstone [5] and conducted a reservoir parameter inversion [6]. Based on the generalized BISQ equations, Yang et al. [7,8] perormed a wave-ield simulation using two types o lux-corrected transport dierence method, and Wang et al. [9] perormed a numerical simulation o a 3-D two-phase anisotropic medium with a staggered-grid higher-order inite dierence method. Viscoelasticity is another important actor that inluences wave dispersion and attenuation. Cheng et al. [30] introduced Christensen s viscoelastic parameter and presented a viscoelastic BISQ model in the EDA medium. They also studied the velocity dispersion and attenuation o waves based on their viscoelastic BISQ model or the EDA medium case. Nie et al. [31,3] investigated the eect o the clay contents on the ast P- wave velocity dispersion and attenuation. However, both Cheng et al. [30] and Nie et al. [31,3] neglected wave dispersion and attenuation in seismic requency ranges, especially below 100 Hz. Nie et al. [31,3] did not consider the eect o viscoelasticity on the S-wave. In contrast, the luid in reservoir media is usually a multiphase mixture including oil, water, and gas. However, previous viscoelastic BISQ models did not consider the eects o multiphase low. Thereore, it is necessary to consider the eects o multiphase low on waves. Hence, in this paper, we present a viscoelastic isotropic BISQ model containing multiphase low. We use the Voigt ormula to compute the eective bulk modulus according to eective medium theory, and introduce the complex modulus according to viscoelastic theory. This allows us to revise the characteristic squirt-low length. On the basis o the BISQ model given in this paper, we analyze the velocity dispersion and attenuation o waves under conditions o low porosity and permeability, and apply the model to observed interwell seismic data. This conirms the validity and accuracy o the model presented in this study. 1 Viscoelastic isotropic BISQ model containing multiphase low 1.1 Wave equations o viscoelastic isotropic BISQ model Yang [1,] presented a generalized BISQ model that deals with the Biot mechanism and the squirt mechanism simultaneously. Based on his work, and according to the correspondence principle, the steady-state BISQ equations in the space-requency domain described by solid displacements and luid pressure in homogenous viscoelastic media are presented as ollows: xux x y z uy uz xy xz P 0, x ux x y xu x y z uz P 0, y z y ux uy x z y z xu x y z P 0, z u u x y u z FS x y z h P P 0, x y z where and are the viscoelastic Lamé constants o the solid rame, whose relationship to the elastic Lamé constants will be discussed in the next section, u x, u y, and u z are the displacements o the solid rame in the x-, y-, and z-directions, respectively, P is the luid pressure, is the porosity, is the poroelastic coeicient, F is the Biot low coeicient, and S is the characteristic squirt-low coeicient. The relevant expressions are as ollows [1,,5]: 1 1 1 1 F,, K Q Q Ks 1 a c c i,, J1 R a c 1, i, RJ0 R F a K S h FS i, 1, Ks y z (1)

1070 Yang L, et al. Sci China-Phys Mech Astron June (014) Vol. 57 No. 6 where K is the eective bulk modulus o the luids, K is the solid bulk modulus, K s is the bulk modulus o the solid phase, is the eective density o the luid, a is the additional coupling density, is the permeability, and R is the characteristic squirt-low length. 1. Viscoelastic parameters and multiphase low parameters The parameters o a viscoelastic medium are always time dependent [33], which causes some diiculties in their selection. Dierent researchers propose dierent solutions. For instance, Cheng [30] introduced Christensen s viscoelastic parameters into the BISQ model, and corrected the relevant parameters: tan i, 1 ln, 0 tan G G const. G G G G G0 in which G is the complex modulus in the viscoelastic medium, G 0 is the modulus o the elastic medium, tan is the phase o the complex modulus, and 0 is the reerence requency. G, the real part o the complex modulus G, corresponds to the stored energy over one period, and G, the imaginary part o the complex modulus G, corresponds to the dissipative energy over one period. As tan is deined as the ratio o the real and imaginary parts o the complex modulus, larger values o tan imply a stronger attenuation. By analyzing the experimental results, we discover that tan is not a constant, but a unction o angular requency. Thus, the viscoelastic Lamé coeicients should be corrected as ollows: tan g( ) tan i, 0 1 ln, tan, 0 () tan i, 0 1 ln, tan, 0 where g( ) is an unknown unction o angular requency. Ater analyzing the observed interwell seismic data, we ind that the viscoelasticity is relatively large at low requencies and relatively small at high requencies. This relationship can be approximated by a heuristic linear unction o the angular requency in the interwell seismic requency range, that is tan tan1 g 1 tan 1, 1 where tan and tan 1 correspond to the viscoelasticity (3) at angular requencies o and 1, respectively. Because gas can be treated as a special kind o luid, the eective density o a luid composed o oil, water, and gas can be computed by s s s. (4) w w o o g g The eective bulk modulus o this mixture can be expressed by Voigt s ormula [34]: where s w, K swk, w soko sgkg (5) s o, and s g denote the saturation o the water, oil, and gas, respectively. From eqs. (4) and (5), it can be deduced that when s g vanishes, the porous luid degenerates to a double-phase low, and when s g and s w both vanish, the porous luid degenerates to a single-phase low. 1.3 Characteristic squirt-low length o the gas-included case Marko et al. [15] showed that waves undergo large velocity dispersion and strong attenuation not only in the saturated reservoir medium, but also in the unsaturated case under the action o the squirt-low mechanism. Consequently, Dvorkin et al. [17] assumed that the squirt-low length o unsaturated rock decreased rom its initial value in a ully saturated rock, and corrected the characteristic squirt-low length as R R0 S, where S is saturation. To describe the squirt-low mechanism or a reservoir medium with multiphase low, we urther correct Dvorkin s ormula as: R R S S (6) 0 o w, where R 0 is the characteristic squirt-low length without gas and water, and R is the characteristic squirt-low length in the presence o gas and water. In physical terms, R 0 is the average length required to produce a squirt-low eect identical to the cumulative eect o local low in pores o various shapes and sizes, and is intimately related to the pore-space geometry o a given rock [16]. Dvorkin [16] also assumed this parameter to be a undamental property o the rock, which can thus be determined experimentally. 1.4 Formulae o the phase velocity and attenuation o waves To study the wave velocity dispersion and attenuation, we irst solve the viscoelastic BISQ equation (1) by the Fourier method [,30]. The basic concept o the Fourier method is that the solutions are harmonic. In detail, we irst suppose that solutions to eq. (1) are harmonic. Then substituting these harmonic solutions into eq. (1), we obtain a system o linear equations with respect to the solid displacements and luid pressure. To obtain non-zero solutions o the solid

Yang L, et al. Sci China-Phys Mech Astron June (014) Vol. 57 No. 6 1071 displacements and luid pressure, we let the determinant o the coeicient matrix vanish. As a result, we obtain a dispersion equation and urther obtain wave-number solutions by solving the dispersion equation. Finally, we obtain the phase velocity and attenuation o the ast P-wave, slow P-wave, and S-wave using the deinitions [16] o v k and 1 Q Im( k) Re( k), where k is the wave number. These computational ormulae are given as ollows, which are similar to the equations presented by Nie [5], except or slight dierences in the ormula or the S-wave velocity and attenuation. where V p1, Re Vs Re 1 Im b c,, bc Q Re bc p1, 1 1 Im x,, Q Re x s x x b, FS 4 x c b. FS Numerical simulation In this section, we use numerical simulations to investigate the wave velocity dispersion and attenuation in a viscoelastic medium containing multiphase low. The computational parameters are shown in Table 1. Here, η is the viscosity o the luid, is the porosity, R 0 is the characteristic squirtlow length without water and gas, ρ w is the water density, ρ o is the oil density, ρ g is the gas density, ρ s is the mineral density or the solid phase density, ρ a is the additional density, K w, K o, and K s are the bulk moduli o water, oil, and minerals, respectively, and λ 0 and μ 0 are the elastic Lamé coeicients..1 Eects o viscoelasticity on the velocity dispersion and attenuation o waves To study the eects o viscoelasticity on wave velocity dispersion and attenuation, we select the ollowing parameter values: the gas saturation is 0%, the oil and water saturations are both 40%, the reerence requency is 300 Hz, and the viscoelastic parameter tan is varied between 0.01, 0.1, and 0.. Figure 1 shows both the phase velocity dispersion and attenuation curves o three dierent waves (ast P-wave, slow P-wave, and S-wave) with respect to the requency and viscoelastic parameter. The numerical results reveal that the viscoelastic parameter has remarkable inluence on the (7) (8) Table 1 Computational parameters Parameters Units Values Parameters Units Values cp 1 K w GPa 1 0.1 K o GPa 1.5 R 0 mm 1 K g GPa 0.000131 w kg/m 3 1000 K s GPa 38 o kg/m 3 880 0 GPa 10 g kg/m 3 1.19 0 GPa 8 s kg/m 3 650 a kg/m 3 40 velocity dispersion and attenuation o the ast P-wave and the S-wave, and almost no eect on the slow P-wave. In particular, Figure 1(a) shows that the phase velocity o the ast P-wave in a viscoelastic medium is lower than that in the elastic medium at low requencies, and is larger than that in the elastic medium at high requencies. This implies that the viscoelasticity enlarges the phase velocity dispersion o the ast P-wave. As the viscoelasticity strengthens, the phase velocities o the ast P-wave become much lower in the low-requency range and much higher at high requencies. In other words, the dispersion increases with enhanced viscoelasticity. Our results is consistent with Batzle s experiments [35]. Figure 1(b) shows that the attenuation o the ast P-wave in a viscoelastic medium strengthens compared with that in the elastic medium. As the viscoelasticity increases, the attenuation o the ast P-wave is clearly enhanced, and appears to be requency-dependent. The enhancement is more evident in the seismic requency range (<100 Hz), especially when tan reaches 0., where it is much stronger than in the logging requency range. From Figures 1(c) and 1(d), it can be seen that both the dispersion and attenuation curves o the slow P-wave predicted by the viscoelastic BISQ model are almost superposed on those predicted by the elastic model. This indicates that viscoelasticity has very little impact on the phase velocity dispersion and attenuation o the slow P-wave. Figure 1(e) shows that the phase velocity o the S-wave has no dispersion in the elastic medium, because the dispersion curve is constant or all requencies. However, in the viscoelastic medium, the S-wave undergoes dispersion, and this intensiies as the viscoelasticity increases. The relationship between the phase velocity o the S-wave and the requency is linear in the logarithmic coordinates. This implies that the dispersion o the S-wave is exponentially strengthened. This is consistent with Batzle experiments [35]. In Figure 1(), the attenuation curves o the S-wave in the viscoelastic medium are constant, which implies that the attenuation o the S-wave has the same characteristics at dierent requency ranges. We can also see that an increase in viscoelasticity enhances the attenuation.

107 Yang L, et al. Sci China-Phys Mech Astron June (014) Vol. 57 No. 6 Figure 1 Eects o viscoelasticity on phase velocity dispersion and wave attenuation. (a) The phase velocity dispersion curves o the ast P-wave, (b) the attenuation curves o the ast P-wave, (c) the phase velocity dispersion curves o the slow P-wave, (d) the attenuation curves o the slow P-wave, (e) the phase velocity dispersion curves o the S-wave, () the attenuation curves o the S-wave.. Eects o permeability on the velocity dispersion and attenuation o waves Permeability is one o the most important actors to aect the dispersion and attenuation o waves. To investigate the eects o permeability on the dispersion and attenuation o waves, we select the ollowing parameter values: the gas saturation is 0%, both the oil and water saturations are 40%, the viscoelastic parameter tan is 0.05, the reerence requency is 300 Hz, and the permeability is varied between 0.01 md, 0.1 md, and 1 md. The other parameter values are as in Table 1. Figure (a) shows that, in an isotropic medium containing three-phase low, the phase velocity dispersion curves o the ast P-wave predicted by the viscoelastic and elastic BISQ models have similar characteristics. The velocities predicted by the viscoelastic BISQ model are lower at low requencies, and higher at high requencies, than those predicted by the elastic model. The viscoelastic model also exhibits dispersion at both low and high requency ranges, in contrast to the constant velocity o the elastic case. Figure (b) shows that the attenuation curves o the ast P-wave predicted by the viscoelastic BISQ model have a similar shape to those o the elastic model, albeit with a much higher degree o attenuation. As the permeability decreases, the peak value o the attenuation strengthens, and the dominant requency (the requency corresponding to the peak value o the attenuation) shits to the lower end o the range. This is because when permeability decreases, pore connectivity o the solid rame becomes poor. Thus, the pore luids are isolated or semi-isolated. Hence, the pore luids are dashpots embedded in the solid rame, and the propagating waves are viscous. The viscoelastic mechanism becomes the dominant actor aecting the wave s high dispersion and strong attenuation. Thereore, the results predicted by the viscoelastic model are larger than those predicted by the elastic model or the same value o permeability. Comparing Figures (a) and (b), we ind that the dispersion and attenuation o the ast P-wave are consistent: requencies at which large phase velocity dispersion occurs correspond to strong attenuation. Figure (c) shows that the shit in dominant requency is linear with respect to permeability. Figure 3(a) shows the phase velocity dispersion curves o the slow P-wave predicted by the viscoelastic BISQ model. These are all superposed on the curves given by the elastic BISQ model or the same value o permeability. The phase velocity dispersion mainly occurs in the high requency range. As permeability decreases, the velocities o the slow P-wave decrease sharply, and the dispersion at the high requency range weakens. Figure 3(b) also shows that the

Yang L, et al. Sci China-Phys Mech Astron June (014) Vol. 57 No. 6 1073 Figure Eects o permeability on the phase velocity dispersion and attenuation o the ast P-wave. (a) The phase velocity dispersion curves o the ast P-wave, (b) the attenuation curves o the ast P-wave, (c) the curve o dominant requency o ast P-wave attenuation versus permeability. Figure 3 Eects o the permeability on phase velocity dispersion and attenuation o the slow P-wave and the S-wave. (a) The phase velocity dispersion curves o the slow P-wave, (b) the attenuation curves o the slow P-wave, (c) the phase velocity dispersion curves o the S-wave, (d) the attenuation curves o the S-wave.

1074 Yang L, et al. Sci China-Phys Mech Astron June (014) Vol. 57 No. 6 attenuation curves predicted by the viscoelastic BISQ model are superposed on the curves given by the elastic BISQ model or the same value o permeability. This implies that the eect o the viscoelastic mechanism on the slow P-wave is less. This conclusion is the same as that drawn by Nie et al. [3]. Meanwhile, Figures 3(a) and (b) show that the phase velocity and attenuation o the slow P-wave increase with increasing permeability. In act, Biot [9,10] concluded that the slow P-wave was mainly aected by luid motion, and the presence o the slow P-wave was related to the luids in porous media. In other words, the slow P-wave is related to the permeability. Thus, the velocity o the slow P-wave increases with permeability. Overall, the attenuation o the slow P-wave is very strong. This is why the slow P-wave is diicult to observe, especially at the seismic requency range. By Comparing Figures and 3, it is easy to see that the attenuation o the two P-waves has dierent characteristics: the attenuation o the slow P-wave is relatively weak at requencies where the dispersion is large, whereas the attenuation o the ast P-wave is relatively strong at requencies with high dispersion. Figures 3(c) and 3(d) show that the dispersion and attenuation curves o the S-wave are dierent or the viscoelastic and elastic models, but that the value o the permeability has almost no eect on these characteristics o the S-wave. This is because the squirt low eect on the S-wave can be ignored []. Thus, the permeability, relevant to the squirt-low mechanism, has almost no eect on the dispersion and attenuation o the S-wave..3 Eects o gas content on the velocity dispersion and attenuation o waves For the purpose o investigating the eects o the gas content on the velocity dispersion and attenuation o waves, we select the ollowing parameter values: the oil saturation is 40%, the permeability is 0.1 md, the viscoelastic parameter tan is 0.01, the reerence requency is 300 Hz, and the gas saturation varies between 0%, 30%, and 60%. The other parameters are the same as in Table 1. Figure 4 illustrates the eect o gas content on the velocity dispersion and attenuation o dierent kinds o waves. Figure 4(a) shows that, in a medium with low permeability and porosity, the phase velocity o the ast P-wave increases at low requencies and decreases at higher requencies as the gas content increases. In other words, the velocity dispersion o the ast P-wave decreases with the increase in gas. Figure 4(b) shows the eect o the gas content on the attenuation o the ast P-wave. As the gas level increases, the attenuation exhibits a weakening trend at the seismic Figure 4 Eect o the gas content on phase velocity dispersion and attenuation on the waves. (a) The phase velocity dispersion curves o the ast P-wave, (b) the attenuation curves o the ast P-wave, (c) the phase velocity dispersion curves o the slow P-wave, (d) the attenuation curves o the slow P-wave, (e) the phase velocity dispersion curves o the S-wave, () the attenuation curves o the S-wave.

Yang L, et al. Sci China-Phys Mech Astron June (014) Vol. 57 No. 6 1075 requency range, interwell seismic requency range, and acoustic logging well requency range. The corresponding physical explanation is that increasing the gas content weakens the squirt-low mechanism, and thus decreases both the dispersion and attenuation o the ast P-wave. A comparison o Figures 4(a) and 4(b) reveals a correlation between the dispersion and attenuation o the ast P-wave, that is, large velocity dispersion occurs at requencies with strong attenuation. The eect o the gas content on the slow P-wave is small in the viscoelastic isotropic medium containing multiphase low. Figure 4(c) shows that the velocity dispersion o the slow P-wave is mainly concentrated in the high requency range. As the gas content increases, the velocities decrease slightly, and the velocity dispersion also exhibits a downward trend. Figure 4(d) shows that the strongest attenuation o the slow P-wave appears in the seismic requency range (<100 Hz), and the attenuation curves o this wave with dierent gas contents are almost superposed, which implies that the gas content has little impact on the attenuation o the slow P-wave. Figure 4(e) shows that the S-wave velocities increase a little with the increase in gas, but the dispersion trend remains unchanged. Figure 4() shows superposed attenuation curves, which imply that gas content has no eect on the attenuation o the S-wave. The S-wave dispersion may be related to the eective density and viscoelastic parameter tan. The viscoelastic parameter tan remains unchanged, so the dispersion trends o S wave remain the same. However, increasing the gas level in porous media decreases the eective density. As a result, the S-wave velocity increases. 3 Application to observed interwell seismic data Interwell seismic technology and logging well technology are usually used to detect reservoir structure and parameters. To veriy the validity o the model presented in this paper, we apply it to observed interwell seismic data rom a limestone area located in Kankakee. To obtain the velocity dispersion, two interwell seismic traces were selected. The source and the two receivers were in the same vertical plane at a depth o 196 m. The resulting phase velocity dispersion curves, obtained using the spectra ratio method, are plotted in Figure 5 together with the theoretical phase velocity dispersion curves. The reservoir parameters used in the theoretical computation are shown in Table [0]. These parameters were obtained rom high-resolution reverse VSP and interwell seismic experiments. As complementary data, logging well data were also used to determine the parameters [36]. The permeability and squirt-low length in Table are the averages o those used by Parra [0]. Ater some analysis, we select the ollowing interpolation unction to express the viscoelastic parameter tan: Table Reservoir parameters Parameters Units Value Parameters Units Value cp 1800 K w GPa 1 0.05 K o GPa 1.5 R 0 mm 3 K g GPa 0.0001 w kg/m 3 1000 K s GPa 75 o kg/m 3 880 0 GPa 1 g kg/m 3 1.19 0 GPa 18 s kg/m 3 400 κ md 050 a kg/m 3 40 0.48 0.68 g g 00 0.68. 4000 00 In the example o application to real data, we reasonably choose the values o 0.48, 0.68, and g(ω) or the viscoelastic parameter tan, which is dependent on the strong attenuation o the investigated area. In act, Parra [36] ound that both logging well data and interwell seismic data revealed very strong attenuation o waves in Kankakee limestone. This implies that the medium viscoelasticity is very high. Thereore, Parra [36] suggested the use o a requency-dependent viscosity model to explain this phenomenon. However, Parra [36] did not provide a viscoelastic model. Additionally, rock core inormation indicated that the porous matrix rock was saturated by oil, water, and gas, and Kankakee limestone was surrounded by a background o shale [36]. Shale is a ine-grained, classic sedimentary rock composed o a mix o lakes o mud and tiny ragments o other minerals. Because the mud o the background shale weakens the elasticity o the solid rame [3], there is greater attenuation and thus enhanced eects o viscosity on waves. Figure 5 depicts the it between the observed data and the theoretical curves predicted by the elastic BISQ model, Nie s viscoelastic BISQ model, Cheng s viscoelastic BISQ model, and our proposed model. We can see that the theoretical velocity dispersion curve o the ast P-wave predicted by the elastic model is lat, and does not it the observed velocities. Nie s model its the observed data at relatively high requencies o above 000 Hz, but provides a poor it at requencies below this. I we apply our viscoelastic BISQ model containing multiphase low to the data, the it between the theoretical velocities and the observed velocities improves greatly, and the dispersion trend o the observed data can be sketched. However, the dispersion characteristics o the P-wave velocity vary slightly at dierent requency ranges. The theoretical P-wave velocities corresponding to tan = 0.68 it the observed velocities very well at low requencies (<000 Hz), and are much larger than the observed data at high requencies (>000 Hz). In contrast, the theoretical P-wave velocities corresponding to tan = 0.48 it the observed velocities very well at high requencies (>000 Hz), and are much larger than the observed data at (9)

1076 Yang L, et al. Sci China-Phys Mech Astron June (014) Vol. 57 No. 6 Figure 5 Comparison between the observed interwell velocities and the theoretical velocities. Stars denote the observed interwell seismic velocities. low requencies (<000 Hz). That is to say, the viscoelasticity is relatively strong at low requencies and relatively weak at high requencies. Cheng s result is consistent with the result with tan = 0.48 because both models consider the real and imaginary parts o the complex modulus to be requency-dependent. The physical mechanism may be explained as the low requency corresponding to a larger period, so the stress acting on the solid rame lasts longer during each period. Thus, the dissipation energy is relatively large. Ater analyzing the two theoretical dispersion curves corresponding to tan = 0.48 and tan = 0.68, we ound that we could not obtain a good it between theoretical and observed velocities over all requencies by considering tan as a constant. Thus, we concluded that the viscoelastic parameter tan should be a unction o the requency. Ater urther analysis, we chose the interpolation unction in eq. (9) to describe the viscoelastic parameter tan. Comparing the three theoretical velocity dispersion curves corresponding to tan = 0.48, 0.68, and g(ω) in Figure 5, we can see that the results corresponding to tan = g(ω), which are requency-dependent, can be used to approximate the observed data better than those corresponding to a constant tan rom low to high requency. On the basis o the above, we conclude that the viscoelasticity o the medium varies with requency in the interwell seismic requency range. Ater introducing a requency-dependent unction to express the viscoelastic parameter tan, the theoretical results it the observed data consistently well across all requencies. Thus, we believe this unction describes the viscoelasticity o the medium in the interwell seismic requency range. 4 Conclusion and discussion In this paper, we considered the eects o the Biot mechanism, the squirt mechanism, multiphase low, and viscoelasticity on waves. We developed and presented a viscoelastic isotropic BISQ model containing multiphase low according to the correspondence principle o viscoelastic theory and the eective medium theory. Based on our model, we studied the wave dispersion and attenuation in a viscoelastic medium with low permeability and low porosity. The validity and accuracy o the viscoelastic BISQ model containing multiphase low were conirmed by applying the model to observed interwell seismic data. Numerical results showed that the viscoelasticity o the medium aects the velocity dispersion and attenuation o the ast P-wave and the S-wave remarkably, but has almost no eect on the slow P-wave. Under conditions o low permeability and low porosity, the dominant attenuation requency o the ast P-wave shits linearly toward lower requencies as the permeability decreases. This plays a practical role in reservoir parameter prediction and exploration or gas and oil. An increase in gas content in porous media weakens the squirt-low mechanism, and impacts the dispersion and attenuation o the ast P-wave. In particular, as gas levels increase, the velocity o the P-wave was ound to increase in the low requency range and decrease at higher requencies, which weakens the dispersion o the ast P-wave. The attenuation o the ast P-wave decreases notably as gas content increases. Comparing the eects o viscoelasticity and gas content on the velocity dispersion and attenuation o the ast P-wave, we discovered a correlation between the dispersion and attenuation o the ast P-wave large dispersion occurred at requencies where strong attenuation appeared. Ater applying the model to the observed data, we discovered or the irst time that in the viscoelastic BISQ model the viscoelastic parameter tan is not a constant but a requency-dependent unction. In order to reveal the relationship between the viscoelastic parameter tan and requencies, we perormed quite a lot o analyses and presented a heuristic linear unction to approximate it. Numerical simulations showed when the linear requency-dependent unction was introduced, the theoretical results itted the observed interwell seismic data consistently well rom low requency to high requency. The great improvement showed the importance o requency-dependent viscoelasiticy, which has practical signiicance in deepening our knowledge o viscoelastic eects on waves. Though we have made great progress in the research, deiciency should be mentioned here and needs to be improved in the later research. The relationship between viscoelastic parameter tan and requency is considerably complex. The linear unction presented in this paper is heuristic, so it is just a kind o coarse approximation. So uture research work needs to ocus more accurately on unctions such as a piecewise linear unction or even a non-linear unction to approximate the requency-dependent relationship. Besides, in this paper, we did not supply experiment data to prove the squirt-low length or multiphase low due to the ex-

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