“ ANALYTICAL SEISMIC WAVE ATTENUATION AND VELOCITY DISPERSIONIN LAYERED DOUBLE-POROSITY MEDIA „

Due to differences of rock properties such as porosity, permeability and compressibility between different regions in the porous media, pressure gradients are induced between those different regions and lead to local fluid flow. When seismic wave propagates in the porous media, the local fluid flow process is a main cause of wave attenuation and velocity dispersion. The local fluid flow mechanism of the layered porous model has been studied by many authors in the numerical approaches without analytical wave equations and solution for this kind of rock physics models. In this study, we first establish a layered double-porosity model saturated with a single fluid and derive the wave equations. According to the derived novel wave equations, then we calculate the phase velocity and quality factor in the layered double-porosity media based on plane wave analysis. The results demonstrate that there are three kinds of wave modes named as the fast P-wave and two slow P-wave in layered porous media when P-wave propagates through the model perpendicularly. Finally, we study the effects of local fluid flow on the mesoscopic loss mechanism by analyzing the attenuation and the velocity dispersion of seismic waves in the low frequency range. dispersion for White model by solving the Biot’s equations. Gurevich et al. [1997] considered a finely layered porous rock and calculated the P wave quality factor for this rock. Wang et al. [2013] studied the properties of wave attenuation and velocity dispersion with the variation of gas saturation and porosity based on the White model. Berryman and Wang [2000] developed phenomenological equations and obtained the coefficients in these equations to study oil and gas reservoirs contained storage porosity and fracture/crack porosity. Chen et al. [2016] thought that wave-induced fluid flow will be affected by the multi-phase fluid, and through the analyses of seismic wave attenuation and reflection behavior to verify. Pride et al. [2004] proposed a unified theory to analyze three P wave attenuation models for sedimentary rocks. The fluid flow due to heterogeneity is at different scales in each model. Sun et al. [2015] compared several analytical models to study the wave attenuation and dispersion due to wave-induced fluid flow and to analyze velocity-saturation relations. In order to study the P-wave dissipation process in a double-porosity media, Sun et al. [2016] proposed a triple-layer patchy model saturated with two immiscible fluids. Chapman et al. [2017] believed that the frequency scaling behavior of seismic attenuation is affected by the spatial distribution of the fluid, and they demonstrated this theory from the experimental investigation of the Berea sandstone saturated with two fluid phases.Local fluid flow (LFF) process induces wave attenuation and dispersion in the seismic frequency band when wave propagates in the porous media. To study the LFF process in the double-porosity medium, Ba et al. [2011] proposed a double-porosity model and derived the equations of motion for this model. However, the fluid kinetic energy inside the spherical pockets is neglected in this paper when they calculated the kinetic energy of the LFF process and the dissipation function. In order to solve this problem, Ba et al. [2014] derived the equations of motion by rewriting the kinetic energy function including the LFF velocities inside and outside the inclusions and the dissipation function. To predict the attenuation and dispersion in double-porosity media, Zheng et al. [2017] presented a new form of wave equations and solved these equations by using an analytical method. Ba et al. [2016] proposed a new doubleporosity model to explain strong velocity dispersion in tight siltstone with clay-filled pores obtained from the experiment. To study the effects of fabric and saturation inhomogeneities on wave attenuation and velocity dispersion, Ba et al. [2017] proposed a double double-porosity model and derived the governing equations for this model based on Hamilton’s principle. There are one fast P wave and four slow Biot waves in the media when the compressional waves propagate in the double double-porosity model. Based on the T matrix approach, Agersborg et al. [2009] modeled the effective viscoelastic properties of cracked porous media and explain the effects of LFF in double-porosity media. Song et al. [2016] proposed an analytical solution of effective dynamic shear modulus and analyzed the relation between wave-induced fluid flow and shear modulus by calculating the S wave attenuation and velocity dispersion. White et al. [1975] proposed a layered porous model which is saturated with both water and gas alternately. Pressure gradients between two layers induce LFF, namely, inter-layer flow which have been studied by many authors. Brajanovski et al. [2005] studied the influences of fractures on rock properties by modelling fractures as thin porous layers in porous media. Based on the propagator matrix approach, they obtained the effective P-wave modulus of the fractured porous material from the dispersion equation. Gelinsky et al. [1998] analyzed the effects of medium heterogeneity on P wave propagating in the layered medium and obtained the expressions of the phase velocity and the attenuation coefficient of P wave. Krzikalla and Müller [2011] performed elastic up-scaling of thinly layered rocks by using the established Backus averaging technique. Norris [1993] proposed a theory for the P wave attenuation and velocity dispersion in porous layered media saturated with fluid. In this work, we consider a layered double-porosity model to describe the wave-induced local fluid flow in porous layers, and analyze the mesoscopic loss mechanism due to LFF. In section 2, we first establish a layered double-porosity model containing two layers saturated with a single-fluid. Then we review the Hamilton’s principle and derive the expressions of potential energy, kinetic energy and dissipation function including LFF term by extending the Biot-Rayleigh theory [Biot, 1956, 1962; Rayleigh, 1917; Ba et al., 2011]. Substituting the expressions of potential energy, kinetic energy and dissipation function into Lagrange’s equations, we obtain the equations of motion for layered double-porosity model. In section 3, we perform the numerical simulations and analyze the effects of some rock parameters such as viscosity, permeability and thickness of layers on wave attenuation and velocity dispersion. ZHAO ET AL.


INTRODUCTION
When seismic waves propagate in underground porous media saturated with fluid, pressure gradients are induced between different regions and lead to fluid flow relative to the solid. This kind of fluid flow is known as wave-induced fluid flow, which can lead to attenuation and velocity dispersion of seismic waves. According to the heterogeneities of the different scales, we can divide attenuation theories due to wave-induced fluid flow into macro-scale, microscale and meso-scale attenuation theory [Chapman et al., 2002;Müller et al., 2010]. Biot [1956Biot [ , 1962 described quantitatively the wave attenuation and dispersion induced by macroscopic fluid flow. Nur [1975, 1979] proposed a squirt-flow mechanism, which is a typical micro-scale attenuation theory. This theory emphasizes that because of the heterogeneity of the individual pore scale, the fluid in the pores is squeezed out the pores when seismic wave propagates in porous media. Squirt-flow mechanism predicts wave attenuation much higher than that predicted by Biot model. Dvorkin and Nur [1993] presented a uniform model named as BISQ model, which combined Biot theory and squirt-flow mechanism. This model, however, cannot describe seismic wave attenuation in the seismic frequency range. White [1975] introduced a partially saturated porous model to describe the meso-scale attenuation and dispersion in the low frequency band. The mesoscale heterogeneity is much higher than individual pore scale and smaller than wavelength scale. Odé [1979a, 1979b] analyzed the attenuation and velocity dispersion for White model by solving the Biot's equations. Gurevich et al. [1997] considered a finely layered porous rock and calculated the P wave quality factor for this rock. Wang et al. [2013] studied the properties of wave attenuation and velocity dispersion with the variation of gas saturation and porosity based on the White model. Berryman and Wang [2000] developed phenomenological equations and obtained the coefficients in these equations to study oil and gas reservoirs contained storage porosity and fracture/crack porosity. Chen et al. [2016] thought that wave-induced fluid flow will be affected by the multi-phase fluid, and through the analyses of seismic wave attenuation and reflection behavior to verify. Pride et al. [2004] proposed a unified theory to analyze three P wave attenuation models for sedimentary rocks. The fluid flow due to heterogeneity is at different scales in each model. Sun et al. [2015] compared several analytical models to study the wave attenuation and dispersion due to wave-induced fluid flow and to analyze velocity-saturation relations. In order to study the P-wave dissipation process in a double-porosity media, Sun et al. [2016] proposed a triple-layer patchy model saturated with two immiscible fluids. Chapman et al. [2017] believed that the frequency scaling behavior of seismic attenuation is affected by the spatial distribution of the fluid, and they demonstrated this theory from the experimental investigation of the Berea sandstone saturated with two fluid phases.Local fluid flow (LFF) process induces wave attenuation and dispersion in the seismic frequency band when wave propagates in the porous media. To study the LFF process in the double-porosity medium, Ba et al. [2011] proposed a double-porosity model and derived the equations of motion for this model. However, the fluid kinetic energy inside the spherical pockets is neglected in this paper when they calculated the kinetic energy of the LFF process and the dissipation function. In order to solve this problem, Ba et al. [2014] derived the equations of motion by rewriting the kinetic energy function including the LFF velocities inside and outside the inclusions and the dissipation function. To predict the attenuation and dispersion in double-porosity media, Zheng et al. [2017] presented a new form of wave equations and solved these equations by using an analytical method. Ba et al. [2016] proposed a new doubleporosity model to explain strong velocity dispersion in tight siltstone with clay-filled pores obtained from the experiment. To study the effects of fabric and saturation inhomogeneities on wave attenuation and velocity dispersion, Ba et al. [2017] proposed a double double-porosity model and derived the governing equations for this model based on Hamilton's principle. There are one fast P wave and four slow Biot waves in the media when the compressional waves propagate in the double double-porosity model. Based on the T matrix approach, Agersborg et al. [2009] modeled the effective viscoelastic properties of cracked porous media and explain the effects of LFF in double-porosity media. Song et al. [2016] proposed an analytical solution of effective dynamic shear modulus and analyzed the relation between wave-induced fluid flow and shear modulus by calculating the S wave attenuation and velocity dispersion. White et al. [1975] proposed a layered porous model which is saturated with both water and gas alternately. Pressure gradients between two layers induce LFF, namely, inter-layer flow which have been studied by many authors. Brajanovski et al. [2005] studied the influences of fractures on rock properties by modelling fractures as thin porous layers in porous media. Based on the propagator matrix approach, they obtained the effective P-wave modulus of the fractured porous material from the dispersion equation. Gelinsky et al. [1998] analyzed the effects of medium heterogeneity on P wave propagating in the layered medium and obtained the expressions of the phase velocity and the attenuation coefficient of P wave. Krzikalla and Müller [2011] performed elastic up-scaling of thinly layered rocks by using the established Backus averaging technique. Norris [1993] proposed a theory for the P wave attenuation and velocity dispersion in porous layered media saturated with fluid. In this work, we consider a layered double-porosity model to describe the wave-induced local fluid flow in porous layers, and analyze the mesoscopic loss mechanism due to LFF. In section 2, we first establish a layered double-porosity model containing two layers saturated with a single-fluid. Then we review the Hamilton's principle and derive the expressions of potential energy, kinetic energy and dissipation function including LFF term by extending the Biot-Rayleigh theory [Biot, 1956[Biot, , 1962Rayleigh, 1917;Ba et al., 2011]. Substituting the expressions of potential energy, kinetic energy and dissipation function into Lagrange's equations, we obtain the equations of motion for layered double-porosity model. In section 3, we perform the numerical simulations and analyze the effects of some rock parameters such as viscosity, permeability and thickness of layers on wave attenuation and velocity dispersion.

LAYERED DOUBLE-POROSITY MODEL
Based on the White model, we first consider a layered double-porosity model containing two layers shown in Figure 1. Figure 1a shows a layered medium which contains of alternating layers saturated with two immiscible fluids. We choose an elementary volume in the laminated medium, as shown by the black box in the Figure 1a. Figure 1b is the diagram of the layered double-porosity model. It is interesting to note the layered double-porosity model and the elementary volume we choose are not exactly the same. Each layer in the layered double-porosity model has a different porosity but is saturated with the same fluid. Assume that φ 10 and φ 20 are the porosity in upper and lower layers, respectively. The absolute porosities of two layers can be written as φ m = V m φ m0 , m = 1, 2, where V m , m = 1, 2 are the volume fraction of upper and lower layers relative to the model, respectively. The total porosity of the model is φ = φ 1 + φ 2 . Moreover, we denote the thickness of upper layer by X 1 and the total thickness of the model by X 2 , the length and the width of the model by l 1 and l 2 , respectively.
To analyze the propagation mechanism of the seismic waves, we consider the following assumptions: (1) each layer in the layered double-porosity model is isotropic and homogeneous; (2) there is no fluid flow across the top and the bottom of the model and the boundary conditions between the two layers are open; (3) the wavelength of the seismic wave is much longer than the total thickness of the model. Macro-flow and LFF are induced when P wave propagates through model perpendicularly, as shown by black arrow and white arrow in Figure 1, respectively. We denote the dynamic thickness of fluid in upper layer by X, which is a function of time.
We denote by x i , i = 1, 2, 3 the spatial variables and by t the time variable, the displacement vector of the solid by u i and the average fluid displacement vector in two layers by U i (m) , m = 1, 2, respectively. Furthermore, we define the strain components of the displacement vector of the solid and fluid: (1) and introduce the bulk strain of the solid and fluid as (2)

HAMILTON'S PRINCIPLE
In order to study the attenuation and velocity dispersion of seismic waves in layered porous media, we derive the equations of motion by using the Hamilton's principle. The Lagrange energy density can be expressed as where T and W are the kinetic energy and potential energy in the model, respectively. The Lagrange's equations can be written as (4) where u i is the generalized coordinates, denoting the displacement vectors u i , U i (1) and U i (2) , respectively; D is the dissipation function. The point above a variable represents the derivative of the time. The governing equation can be expressed as (5) where the δ is the bulk strain increment of fluid due to LFF. In order to obtain the equations of motion for layered double-porosity model, we need to derive the expressions of potential energy, kinetic energy and dissipation function and substitute these expressions into Lagrange's equations and governing equation.

EXPRESSION OF POTENTIAL ENERGY FUNCTION
The potential energy function of layered doubleporosity media has five independent variables, which can be expressed as W = W (I 1 ,I 2 ,I 3 , ξ 1 ,ξ 2 ) where I 1 , I 2 and I 1 are the first, second and third invariants of solid strain; ξ 1 and ξ 2 denote the fluid volumetric strain in upper and lower layers, respectively. Based on Ba et al., [2011], the strain energy function in the layered double-porosity media can be written as: (7) where A, N, Q 1 , Q 2 , R 1 , R 2 are the six elastic coefficients (see Appendix A). Here we used Taylor expansion and ignored the higher order terms beyond the second order in the derivation. According to the mass conservation of fluid, we can get φ 1 (-φ 2 δ) + φ 2 (φ 1 δ) = 0. The effective variation of fluid volume in upper layer is φ 2 δ ≈ 1 -V 1 /V, where V 1 and V are the fluid static and dynamic volume in upper layer whose the thicknesses are X 1 and X, respectively. Then, the bulk strain increment of fluid δ can be expressed as:

EXPRESSIONS OF KINETIC ENERGY AND DISSI-PATION FUNCTION
The kinetic energy function for layered doubleporosity model can be written as: with ρ s and ρ f denoting the solid and fluid mass densities, respectively; υ Expanding the second item on the right of the Equation (9) gives (11) Let us analyze the terms (12) The relative velocity of the global fluid flow υ (m) global, i can be expressed as (13) Substituting the Equation (13) into Equation (12) gives (14) Assume that the angle between the propagating direction of the P wave and the normal direction of the interface between two layers is θ, we have υ .
Since the propagating direction of the P wave is perpendicular to the interface between two layers, the velocity of inter-layer flow and the absolute velocity of fluid flow are equal, which can be written as υ  (11), (13) and (15) into Equation (9) gives (16) where (17) is the kinetic energy function of the LFF process and ρ 00 , ρ 0m , ρ mm are five density coefficients (see Appendix A).
The dissipation function in this case is denote the dissipative coefficients in upper and lower layers; η and κ m denote the viscosity and the permeability of upper and lower layers, respectively; D L is the dissipation function of the LFF process.

KINETIC ENERGY AND DISSIPATION FUNCTION OF THE LFF PROCESS
In this section, we derive the expression of the kinetic energy and dissipation function of the LFF process by using the generalization of Rayleigh's theory. First, we calculate the fluid velocity in upper and lower layers, respectively. The one-dimensional continuity equation of fluid in upper layer is (20) where ẋ upper denote the fluid particle velocity in upper layer. After the calculation of the Equation (20), we have (21) where Ẋ denote the fluid particle velocity at the interface between two layers. Generalizing Rayleigh's theory to the layered double-porosity model, we have (22) where ẋ lower denote the fluid particle velocity in lower layer.
Second, we derive the kinetic energy function of the LFF process in upper and lower layers, respectively. The kinetic energy function of the LFF process in upper layer is (23) The fluid kinetic energy of the LFF process in lower layer is (24) Thus the kinetic energy of the LFF process can be described as the sum of the fluid kinetic energy of upper and lower layers, i.e.

EQUATIONS OF MOTION AND PLANE WAVE ANALYSIS
Substituting the expressions of potential energy, kinetic energy and dissipation function into Equations (4) and (5), and assuming the seismic wave propagates along the x 1 direction, we derive the following equations of motion: We indicate the unknowns of Equations (31a) ~ (31d) as the form of plane wave solutions (32) where ω denote the angular frequency, and k 1 = k 10 -iα is the complex wave number, with k 10 and α denoting the real wave number and attenuation coefficient. Substituting Equation (32) into Equation (31d) gives the expression of δ. Then substituting the expression of δ and Equation (32) into Equations (31a) ~ (31c) gives three solutions of P wave number (see Appendix B).
The complex velocity and the phase velocity can be expressed as a and the attenuation factor and the quality factor can be written as (34) where f = ω/2π is frequency.

NUMERICAL SIMULATIONS
In order to verify the correctness of the method in this work, we calculate the phase velocities and quality factors of the three P waves respectively, and compare these results with those of White et al. (1975). The rock parameters involved in the calculation process are shown in Table 1, where K s , μ s and ρ s are the bulk modulus, shear modulus and mass density of solid grains; K f and ρ f are the bulk modulus and mass density of the pore fluid; c 1 and c 2 are the consolidation parameters of upper and lower layers, and c s are the shear consolidation parameter.
The dry rock bulk moduli of upper and lower layers can be expressed as  Figure 2a, the phase velocities increase with frequency in the low frequency range. LFF process induces the velocity dispersion. Moreover, the curves of phase velocity move toward low frequencies with increasing of the fluid viscosity. As is shown in Figure 2b, the curves of dissipation factor have two peaks, one of which induced by LFF process is in the low frequency band, the other caused by Biot's friction is in the high frequency band. In addition, the low frequency peaks move toward the low frequencies with the increase of the fluid viscosity. The black solid lines in Figure 2 are the results calculated by White et al. [1975]. As is shown in Figure 2, the two results predicted by the layered double-porosity model and the layered White model are similar in the low frequency band, which can verify the correctness of the method in this work (Since the layered White model focuses on the attenuation and velocity dispersion of seismic waves in the low frequency range, the black solid lines have no attenuation peak and dispersion in the high frequency range).
Then we analyze the effects of the thickness of upper layer on the phase velocity and the dissipation factor. Here we use two ways to change the thickness of upper layer: (1) only the thickness of upper layer is changed and the thickness of the model is a constant; (2) the thicknesses of upper layer and the model are changed simultaneously to ensure they satisfied X 2 = 2X 1 . Figure 3 displays the phase velocity ( Figure 3a) and the dissipation factor (Figure 3b) of the fast P wave obtained using the first way, where the model saturates  the ambient water. The velocity dispersion and the attenuation become obvious with decreasing of the thickness of upper layer or increasing of the thickness of lower layer, which illustrate high-porosity high-permeability thin layer can induce high velocity dispersion and strong attenuation. The results obtained by using the second way are shown in Figure 3c and 3d. The curves of phase velocity in the low frequency band and the LFF peaks move toward low frequencies with the increase of the thickness of upper layer. Figure 4 displays the phase velocity (Figure 4a and 4c) and the dissipation factor (Figure 4b and 4d) of the fast P wave for different permeability, where the thickness of upper layer X 1 = 0.1 m and the model saturates the ambient water. As is shown in Figure 4, the curves of phase velocity and dissipation factor do not change significantly with decreasing of the permeability of the upper layer. Furthermore, the curves of phase velocity and the LFF peaks move toward high frequencies with the increase of the permeability of the lower layer. These results indicate that the effective permeability of the model should be determined by the layer with lower permeability when the permeability of two layers is different. Therefore the effective permeability of the model does not change very much when the permeability of the layer with higher permeability changes, so that the influence on attenuation and velocity dispersion is not significant. However, the effective permeability of the model will increase with the increase of permeability of the layer with lower permeability, which leads to the curves of phase velocity and the LFF peaks move toward high frequencies.
In this study, we obtain two slow P waves (P2 and P3), whose phase velocity and dissipation factor for different fluids are shown in Figure 5. Figure 6 displays the phase velocity (Figure 6a and 6c) and the dissipation factor (Figure 6b and 6d) of two slow P waves by using the two ways of changing thickness. In the low frequency range, the P2 and P3 waves are dominated by the diffusive mode, and the internal friction between ZHAO ET AL.  to zero at low frequencies, and decrease with frequency. In addition, the dissipation factor of P3 wave increase with frequency in the low frequency range and decrease with frequency in the high frequency band.

CONCLUSIONS
We presented a layered double-porosity model and derived the equations of motion to study the effects of rock properties on seismic wave attenuation and velocity dispersion in isotropic poro-elastic media. Based on the White model, the present model consists of two layers saturated with a single-fluid. According to the Hamilton's principle and the extension of Biot-Rayleigh theory, we derived the equations of motion by calculating the expressions of potential energy, kinetic energy and dissipation function and obtained three P wave modes. In the numerical simulation, we calculated the phase velocity and quality factor for these three P waves. The fast P wave attenuation and velocity dispersion are induced by LFF process in the low frequency range. The curves of phase velocity and the low frequency peaks move toward seismic frequency band with the increase of viscosity or the decrease of permeability. The internal friction force leads to the attenuation and velocity dispersion of two slow P waves. This study fully indicates that local fluid flow is a main cause of the mesoscopic loss mechanism in the layered double-porosity medium. However, this study doesn't consider the patchy saturation and the situation of oblique incident of seismic wave. These questions will be analyzed in a future work.