Layout 6 ANNALS OF GEOPHYSICS,60, 6, S0664, 2017; doi: 10.4401/ag-7450 Computation of wave attenuation and dispersion, by using quasi-static finite difference modeling method in frequency domain Qazi Adnan Ahmad1,*, Guochen Wu2, Wu Jianlu3 1 China University of Petroleum (East China), School of Geoscienc, Qingdao, Shandong, China 2 China University of Petroleum (East China), College of Geo Resources and Information, Department of Geophysics Dong Ying, Shan Dong, China 3 China University of Petroleum Huadong, Dongying, Shandong, China Article history Received May 21, 2017; accepted October 9, 2017. Subject classification: Exploration geophysics; Seismic methods; Seismology; Waves and wave analysis; measurements and monitoring; Mathematical Geophysics. S0664 ABSTRACT In seismology, seismic numerical modeling is regarded as a useful tool to interpret seismic responses. The presence of sub- surface heterogeneities at various scales can lead to attenua- tion and dispersion during seismic wave propagation. In ongoing global research, the study of wave attenuation and velocity dispersion due to wave induced fluid flow (WIFF) at mesoscopic scale become the subject of great interest. Al- though, seismic modeling technique is efficient in estimating wave attenuation and velocity dispersion due to wave induced fluid flow (WIFF) at mesoscopic scale. It is possible to further improve the efficiency to accurately predict wave attenuation and velocity dispersion at mesoscopic scale. To achieve this goal, a quasi-static finite difference modeling method in fre- quency domain is implemented to estimate frequency depen- dent P-wave modulus of mesoscopic heterogeneous porous media. The estimated complex and frequency dependent P- wave modulus will assist to estimate frequency dependent wave attenuation and velocity dispersion within a saturated porous media exhibiting mesoscopic heterogeneities. The pro- posed quasi-static finite difference modeling method is further validated with theoretically predicted high and low-frequency limits and also with the analytical solution of White’s 1-D model which is for rock saturated with two immiscible fluids creating heterogeneity at mesoscopic scale. Furthermore, the proposed method is further extended to rock saturated with three phase fluids exhibiting heterogeneity at mesoscopic scale. Subsequently, seismic wave attenuation (inverse quality factor Q-1) and the effects on P-wave velocity in 1-D models with dif- ferent patch size under same gas saturation were also com- puted. Our proposed quasi-static method is simple to be im- plemented by the computing scheme of parallelization and have a potential to extend it for two-dimensional case com- paratively in a flexible way. 1. Introduction The main challenge we are facing in oil and gas ex- ploration is the estimation of effects on wave character- istics due to the presence of multiscale subsurface heterogeneities. During wave propagation, the hetero- geneous nature of earth subsurface results into pressure gradient at different spatial scale. Consequently, the re- sulted pressured gradient will accelerate the fluid to flow (at different scales) from high pressured zone to com- paratively low-pressure zone. The wave induced fluid flow (WIFF) at wavelength scale is named as macro- scopic, whereas, at pore scale, it is termed as micro- scopic, on the other hand, at a scale much smaller than wavelength but larger than the pores scale it is named as mesoscopic fluid flow [Muller et al. 2010]. The influences of fluids on seismic responses re- main a key topic among researchers. Biot’s [M. A. Biot 1956, M. A. Biot 1956] classical theory of poroelasticity led the investigation regarding wave attenuation and ve- locity dispersion in a saturated porous medium. In his theory, macroscopic scale heterogeneities due to pres- ence of single phase fluid were outlined as predicted cause of wave attenuation and velocity dispersion. Wave propagation through such macroscopic heterogeneous media creates pressure gradient at wavelength scale, equilibration of this pressure gradient results into a loss in wave energy. Two compressional and one shear wave was predicted in Biot’s theory, experimental results also validate the presence of slow P-wave [Plona 1980, Kelder et al. 1997]. Meanwhile, another promising feature in Biot’s theory is the low-frequency limiting velocities are identical with Gassmann predicted velocities. Although Biot’s research opens new horizons in the field of geo- physical prospecting, however, the outcomes of Biot’s theory deals significantly at higher frequencies subse- quently, it exceedingly under estimate wave attenuation and velocity dispersion within low-frequency limits [Vo- gelaar et al. 2007]. Accordingly, extrinsic attenuation (scattering) dominates in Biot’s theory, when the wave- lengths are of the order of grain scale (microscopic scale). Therefore, some new techniques are required to overcome these existing challenges. In order to compute wave attenuation and velocity dispersion at grain (microscopic) scale Mavko and Nur [Mavko et al. 1975, Mavko et al. 1979, Dvorkin et al. 1995] presented the concept of fluid flow from soft crack to stiff pores, that eventually causes wave attenuation and velocity dispersion. Several analytical solutions are also put forward to estimate wave attenuation and ve- locity dispersion due to WIFF at microscopic scale [Mavko et al. 1975, Mavko et al. 1979, Budiansky et al. 1976, Palmer et al. 1980, Dvorkin et al. 1995, Chapman et al. 2002, Pride et al. 2003, Gurevich et al. 2009]. Apart from the ability to precisely predict wave attenuation and velocity dispersion at sonic frequency frequencies, in- vestigations reveal that the microscopic scale theories lack to predict wave attenuation and velocity dispersion within seismic frequency band. This inadequacy moti- vates the researchers in opening new avenues to estimate the influences of WIFF at mesoscopic scale. The study of wave propagation through saturated porous media demands to estimate inevitable wave at- tenuation and velocity dispersion within seismic fre- quency band. Wave propagation through saturated porous rock exhibiting heterogeneity at mesoscopic scale (which is larger than microscale but much smaller than the macro scale) causes significant attenuation and ve- locity dispersion within a seismic frequency band [Muller et al. 2010]. Pressure gradient arises at mesoscopic scale, due to presence of heterogeneity within solid (i.e. be- tween complaint pores/cracks and stiff pores) and fluids (i.e. due to compressibility difference between saturating fluids) at mesoscopic scale. This heterogeneity give rise to fluid flow at mesoscopic scale that eventually leads to wave attenuation and velocity dispersion [Pride 2004]. Within a porous rock presence of two or more im- miscible fluids at mesoscopic scale arises patchy satura- tion. Exploring patchy saturation in a spherical manner, White [White 1975] conducted a significant work and de- clared that mesoscopic fluid flow plays an influential role in wave attenuation and velocity dispersion. White’s con- cept was further extended to patchy saturation in layered form [White et al. 1975]. In subsequent studies, Dutta and Ode [Dutta et al. 1979] further investigated gas-water patchy saturation in terms of Biot’s poroelastic theory [M. A. Biot 1956]. Johnson [Johnson 2001] analyzed con- sequences of arbitrary shape patchy saturation on wave characteristics and figured out the low and high-fre- quency limits for p-wave velocity. Additionally, with the growing importance of mesoscopic scale theory, up- coming researchers analyzed wave attenuation and velocity dispersion due to the presence of three-dimen- sional heterogeneities at mesoscopic scale [Müller et al. 2005, Toms et al. 2007]. In recent years, there has been a growing interest in analyzing the effects of mesoscopic heterogeneity on wave characteristics. During seismic and acoustic propa- gation through saturated porous rock with mesoscopic heterogeneity, significant wave attenuation and velocity dispersion can be produced due to WIFF [Muller et al. 2010] within seismic frequency band. In order to analyze the effects of meso-scopic fluid flow on wave character- istics, seismic modeling is proved as an effective tool which is frequently applied in recent studies concerning estimation of wave attenuation and velocity dispersion [Masson et al. 2006, Rubino et al. 2011, Rubino et al. 2012, Quintal et al. 2011, Milani et al. 2015]. Masson and Pride [Masson et al. 2007] proposed an efficient method to es- timate wave attenuation and velocity dispersion due to wave induced fluid flow (WIFF) at mesoscopic scale. Ac- cording to their proposed quasi-static creep test, they used finite difference method in the time domain for the solution of Biot’s quasi-static equations for wave propa- gation in poroelastic media. Furthermore, Rubino [Ru- bino et al. 2009] and Quintal [Quintal et al. 2011] have put forward two different quasi-static numerical strategies, in which they used the finite element method in frequency and time domain, for the computation of seismic atten- uation due to wave-induced fluid flow. Exploiting the quasi-static finite element method in frequency domain, Carcione [Carcione et al. 2012] and Rubino [Quintal et al. 2014] have discussed the effects of fracture connectivity and anisotropy on the seismic attenuation and dispersion. Lately, the effects on seismic characteristics of fractured magmatic geothermal reservoirs were numerically mod- eled [Grab et al. 2017] in a simple and efficient way. Despite the fact that, the application of seismic modeling to efficiently estimate wave attenuation and velocity dispersion due to wave induced fluid flow AHMAD ET AL. 2 3 ANALYSIS OF SEISMIC RESPONSES FROM MICRO STRUCTURES (WIFF) at mesoscopic scale has been improved in recent years. Nonetheless, it is possible to further improve the efficiency of the methods by put forwarding more flexi- ble methods for accurate prediction of wave attenuation and velocity dispersion at mesoscopic scale. To achieve this goal, recent work seeks to propose a simple and more effective method to compute P-wave modulus for a wide range of frequencies compared with prior to the finite difference in time domain methods. In this article, a new numerical technique is sug- gested in which finite difference in frequency domain is used to solve Biot’s [Biot 1941] quasi-static equations of consolidation based on the finite difference in time do- main approach, presented by Masson and pride [Masson et al. 2007]. Our suggested methodology eventually im- proves the efficiency to estimate seismic wave attenua- tion and dispersion in a 1D saturated medium exhibiting mesoscopic heterogeneity. We further predicted that seismic wave attenuation is intensively sensitive to vari- ation in patch size in a one-dimensional saturated porous medium. Both, the implementation and the computa- tional procedure for measuring the complex P-wave modulus of our suggested methodology is much easier. At the same time, our proposed method can be further extended to two-dimensional case, in a flexible way. The remainder of the paper is organized as follows: The outlines of relationships of 1D Biot’s poroelastic theory are given first. Afterwards, the presentation of poroelastic relations of Biot’s theory and the outlines of quasi-static finite difference in frequency domain are given, which will assist in computing wave attenuation and velocity dispersion in frequency domain. Then the implementation of proposed quasi-static numerical method in frequency domain and its comparison with analytical results of 1D White's model for rock saturated with two immiscible fluids is described. Then the nu- merically estimated results for different patch sizes hav- ing identical gas saturation are given. After that, the numerical measurements of 1D model saturated with three different kinds of fluids are presented. The end of the paper summarizes the results and draws the conclu- sions from our work. 2. Methodology 2.1 1D Biot’s poroelastic relationships We have implemented a quasi-static finite differ- ence modeling method in frequency domain for the estimation of frequency dependent complex P-wave modulus of a mesoscopic heterogeneous porous media. We have used the Biot’s [Maurice A. Biot 1941] quasi-static equations for linear consolidation in a sat- urated porous media in 1D frequency domain, which can be written as (1) (2) (3) (4) In above relations τzzand P defines total stress and fluid pressure, where Usand Uf represents the solid and fluid displacement. The bulk modulus and shear mod- ulus of the dry frame is indicated as k and m, where η, ω and k0 represents viscosity, angular frequency and permeability of the saturated porous medium, , α and M are given as (5) (6) In above equations, Ks and Kf represents the bulk modulus of the solid matrix and the fluid phase re- spectively φ represents the porosity of the media. 2.2 The outline of quasi-static finite difference in fre- quency domain The aim of this study is to compute complex and frequency dependent P-wave modulus for a certain fre- quency band in case of a saturated porous media ex- hibiting mesoscopic heterogeneity. To achieve this goal, many researchers have proposed various methods, in- cluding finite element method in time/frequency do- main and the finite difference method in time domain. To estimate wave attenuation and velocity dispersion due to WIFF at the mesoscopic scale, we have imple- mented traditional frequency domain finite difference method to compute complex and frequency dependent P-wave modulus. Then the boundary conditions are ad- justed in such a manner that the applied stress posi- tions, pressure, and solid/fluid displacement are remained fixed on the same grid point. In order to en- sure the undrained condition, we have applied the nor- mal stress P0 on both sides of 1D synthetic sample and kept the fluid displacement as zero at both ends of the sample(Figure 1). By doing so, the solid displacement at both ends (Us top (ω) and Us bottom (ω)) of the sample can ∂τ zz ∂z = 0 zz = K + 4 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∂Us ∂z −αP ∂U f ∂z +a ∂Us ∂z + P M =0 iωU f + k0 η ∂P ∂z =0 τ m i = −1 α =1− K Ks M = Ks α −φ +φKs Kf( ) be measured easily. These measurements will assist in measuring the total strain (ω) in the synthetic sample. After estimating the strain and knowing the applied stress, the frequency dependent complex P-wave mod- ulus M (ω) can be easily computed by using equations (7-8). Eventually, seismic wave attenuation Q-1 (ω) and P-wave velocity Vp (ω) can be measured by using equa- tion [9-10) [Masson and Pride 2007]. (7) (8) (9) (10) Where L indicate the total length of the described model, is the effective density which can be obtained by using the formula, given below (11) Where ρs and ρf are the solid and fluid densities. Furthermore, the first-space derivative is obtained by applying second order space-operator of differencing approximation is given by (12) 3. Numerical Results 3.1 1D White’s model We have proposed a quasi-static finite difference numerical method to quantify seismic wave attenuation and dispersion in frequency domain. To verify our quasi- static finite difference method, we implemented our AHMAD ET AL. 4 ρ = (1−φ) ρs + φ f ∂F ∂z z= j*dz = (Fj+1 − Fj−1) / 2dz ( j=1,2,....N ) !ε ω( ) = Us top −Us bottom( ) L M ω( ) = P0 !εω ( ) Q−1 ω( ) = image M ω( )( ) real Mω ( )( ) Vp ω( ) = real Mω ( )( ) /ρ ρ Figure 2. White’s model. Figure 3. Numerical simulations of White’s model Stress (b) Fluid pressure (c) Solid displacement (d) Fluid displacement. (a) (b) (a) (b) (c) (d) Figure 1. The outline of quasi-static finite difference modeling in frequency domain. !ε 5 ANALYSIS OF SEISMIC RESPONSES FROM MICRO STRUCTURES proposed method for the calculation of seismic wave attenuation and velocity dispersion of the White’s 1D model (Figure 2a) exhibiting mesoscopic heterogeneity due to alternating layers of gas and water within a porous media. The statistical representative volume element (REV) enclosing a pair of alternating layers of saturated fluid, is given in Figure 2b. We implemented our proposed method on the described model having petrophysical properties given in table 1, where Dz = 0.1cm is the grid size and10cm as thickness of each layer. The results of total stress, pressure, solid and fluid displacement are given in Fig- ure 3. The comparison of our numerical results with the White’s [White et al. 1975] 1D layer model’s analytical results of seismic wave attenuation and dispersion is giv- en in Figure 4. The results validate the accuracy of our proposed method over a wide frequency range (10-2-105). 3.2 Numerical experiments with different patch sizes for the same gas saturation In order to compute seismic wave attenuation and velocity dispersion due to WIFF at mesoscopic scale, a novel technique is presented in this manuscript. The supposed caused of WIFF is due to the presence of alternating layers of gas and water within a porous media creating heterogeneity at mesoscopic scale. We further investigated the effect of variation in the patch size by keeping constant gas saturation. The detail of our 8 different cases about the varia- tion in patch size is given in Figure 5, the description of mesoscopic heterogeneity in first four cases indicate a gradual decrease in patch size where the random patch sizes are selected in other cases. The regions saturated with gas (brighter) and water (darker) are all divided into two parts in accordance with the size of 2 to 1 Figure 4. Seismic attenuation (a) and velocity dispersion (b) of White’s model. Skeleton Gas Oil Water Ks = 33.4 (GPa) Kf = 9.6x10 -3 (GPa) Kf = 1 (GPa) Kf = 2.2 (GPa) ρf = 2700 (kg/m3) ρf = 70 (kg/m 3) ρf = 700 (kg/m3) ρf = 1000 (kg/m3) k = 0.05 (D) η = 1.5x10-5 (Pa·s) η = 0.04 (Pa·s) η = 6x10-4 (Pa·s) ms = 1.4 (GPa) φ = 0.3 Figure 5. Seismic attenuation (a) and velocity dispersion (b) of rock samples for different cases. Table 1. Properties of the skeleton and fluids. (a) (b) (a) (b) respectively. The low-frequency limit of Gassmann- Wood and high-frequency limit of Gassmann-Hill [Müller et al. 2010] are also calculated by using the physical properties of the solid and the saturating flu- ids given in Table 1. Figure 6 shows the visual representation of fre- quency dependence of wave attenuation and velocity dispersion with a change in the size of the heterogene- ity. The results for the case of a decrease in patch sizes shows that the velocity dispersion and attenuation peaks shifted towards higher frequencies with the decrease in size of heterogeneity. From case 5 to 8 patch sizes are randomly selected, there is a little dif- ference in results of seismic wave attenuation and velocity dispersion. Infect the results of rock sample of the case 5 and case 6 have almost identical values of wave attenuation and velocity dispersion, also the rock samples having patch size as shown in case 7 and case 8 have the same behavior.It can be concluded from results that, the change in the spatial size of the saturating fluids hav- ing lesser compressibility values have fewer effects on seismic wave attenuation and velocity dispersion than the fluids having higher compressibility values. AHMAD ET AL. 6 Figure 6. Geometries of the partially saturated rock samples with different patchy sizes under the same condition of gas saturation, grey and black phases are saturated with gas and water respectively. Figure 7. Rock sample saturated with three kinds of fluid. Figure 8. Numerical simulations of the sample saturated with three kinds of fluid Stress (b) Fluid pressure (c) Solid displacement (d) Fluid displacement. (a) (c) (d) (b) 7 3.3 Application of quasi-static finite difference in frequency domain, on a 1D model, saturated with alternating layers of three different fluids We designed a new quasi-static finite difference technique for the estimation of seismic wave attenuation and velocity dispersion in frequency domain. Although several analytical/numerical studies have been carried out to estimate wave attenuation and velocity dispersion for rock saturated with two distinct fluid, little attention has been paid to predict wave attenuation and velocity dispersion for rock saturated with three different fluids. The problem with the analytical prediction of wave attenuation and velocity dispersion in rock saturated with three different fluids is that it is not easy to predict the complex P-wave modulus analytically. This part of the article demonstrates the feasibility of estimating the complex P-wave modulus and eventually the wave attenuation and velocity dispersion in rock saturated with three different fluids. We implemented the proposed quasi-static finite difference modeling in frequency domain to a 1D model saturated with alternating layers of three different kinds of fluids (Figure 7). Numerical simulations of the investigated sample are demonstrated in Figure 8 and the estimated results of wave attenuation and velocity dispersion are visualized in Figure 9. According to our findings, this is the first study in which rock saturated with three different sorts of fluids is investigated and interestingly, there are two peaks in the attenuation curve at seismic frequencies and also the results are stable over a wide frequency range. The attenuation peak appears at the low frequency is because of the reason that at the interface between gas and oil the complex P-wave modulus gets increased whereas the peak appears at high frequency is indicating the increase in complex P-wave modulus at the interface between water and gas. Velocity dispersion at the same frequencies where there is high modulus dispersion validates the concept of higher values of complex P-wave modulus at the interfaces between different fluids. Because of little difference between modulus of oil and water, attenuation at the interface between oil and water is disappeared in the attenuation curve. 3.4 Effects of changing the order of saturating fluids Presence of fluids within the subsurface rocks creates heterogeneity at a different scale which significantly influences the characteristics of propagating wave. Accurate computation of these influences on wave characteristics is an essential need of oil and gas exploration. We have proposed a quasi-static finite difference method in frequency domain to compute the influence of three different fluids within a porous rock. Figure 10 gives the pictorial view of numerical simulation of investigated sample with change in fluid order (i.e. 10a Gas, Oil, Water and 10b Gas, Water Oil) and the results of wave attenuation and velocity dispersion due to the change in the order of saturating fluids are demonstrated in Figure 11 (i.e. 11a wave attenuation and 11b Velocity dispersion). The black solid line shows the wave attenuation and velocity dispersion when the fluids are in order of Gas, Oil, and water while blue circle shows the order of Gas, Water, and Oil. It is interesting to notice that, there is a good match between Gas, Oil, Water and Gas, Water, Oil saturation. One possible reason for this situation could be, because of having little difference in compressibilities of water and oil, the pore pressure difference created by the passing wave in both cases is same. So the footprints of pore pressure equilibration at the interface between Gas/Oil and Gas/water occurs at the same frequencies. ANALYSIS OF SEISMIC RESPONSES FROM MICRO STRUCTURES Figure 9. Seismic attenuation (a) and velocity dispersion (b) of the sample saturated with three kinds of fluid. (a) (b) 3.5 Effects of changing the layer thickness of saturat- ing fluids For a visual representation of the dependence of layer thickness of three-phase fluids in a porous rock, the reader is referred to Figure 12. From the figure, it can be seen that with decreasing the layer thickness of saturating fluids, the phenomenon of pore pressure equilibration starts to occur at higher frequencies. The possible reason for the shift towards higher frequencies is that at a wavelength higher than layer thickness it’s AHMAD ET AL. 8 Figure 10. Numerical simulations of the sample saturated with three kinds of fluid in different order (a) Gas, Oil and Water (b) Gas, Water and Oil. (b) (a) 9 easy for the wave to pass through the layer, while it gets trapped (attenuate) at higher frequencies when the layer thickness becomes equal to the wavelength. The result of velocity dispersion shows the same trend of shifting towards higher frequencies while the upper and lower velocity limits remain the same. 3.6 Effects on wave attenuation and velocity dispersion due to change in thickness of layer’s having Gas saturation We started by investigating the effects of meso- scopic heterogeneity (due to the presence of fluids) on wave propagation. Several numerical experiments were carried out to compute the influence of varying fluids properties, on wave propagation. In results of Figure 13, we analyzed the influence of a change in thickness of a gas layer on wave characteristics, by keeping the total gas saturation constant. The analysis of numerical findings indicates that variation in thickness of gas layer at top and bottom significantly influences wave char- acteristics. The increase in thickness of gas layer at the top cause increase in attenuation at low frequencies due to presence of highly compressible fluid (gas), while ANALYSIS OF SEISMIC RESPONSES FROM MICRO STRUCTURES Figure 13. Seismic attenuation (a) and velocity dispersion (b) of the sample saturated with three kinds of fluid. (a) (b) Figure 12. Seismic attenuation (a) and velocity dispersion (b) of the sample saturated with three kinds of fluid. (a) (b) Figure 11. Seismic attenuation (a) and velocity dispersion (b) of the sample saturated with three kinds of fluid in different order. (a) (b) this attenuation peak gradually become flat with a de- crease in thickness of gas layer at the top. Maximum at- tenuation obtained when the thickness of gas layer becomes identical at top and bottom of the represen- tative element. 4. Conclusions This study is related to estimation of wave attenu- ation and velocity dispersion in fluid-saturated porous media. Our work, suggest a new technique that im- proves the ability to estimate complex P-wave modu- lus and eventually wave attenuation and velocity dispersion. In our proposed technique, by solving Biot’s quasi-static equations, a 1D finite difference scheme in frequency domain is used for the deduction of complex P-wave modulus of a mesoscopic heterogeneous porous media. In order to investigate the validity and accuracy of our proposed numerical scheme, we have compared our results with the analytical results of White’s 1D model. Our results are in good agreement with the analytical results of White's 1D model ex- hibiting mesoscopic heterogeneity due to saturation of two distinct fluids in a porous media. Moreover, we also investigated the effect of, variation in mesoscopic patches, on wave attenuation and velocity dispersion. The findings of our research are quite convincing, and thus the following conclusions can be drawn: The proposed scheme of quasi-static finite differ- ence in frequency domain, assist well in estimating more precise results of frequency dependent P-wave modulus over a wide frequency range (10-2-105). At- tenuation and velocity dispersion is much more sensi- tive to change in the patch size of fluid having higher compressibility values. Also, attenuation peak shift to- wards higher frequency with the decrease in patch size. Furthermore, variation in patch size shows negligible effect on high and low-frequency limit in case of iden- tical gas saturation. Acknowledgements. 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