model of seismic reflection from a permeable layer

. Analysis of compression wave propagation in a poroelastic medium predicts a peak of reﬂection from a high-permeability layer in the low-frequency end of the spectrum. An explicit formula expresses the resonant frequency through the elastic moduli of the solid skeleton, the permeability of the reservoir rock, the ﬂuid viscosity and compressibility, and the reservoir thickness. This result is obtained through a low-frequency asymptotic analysis of Biot’s model of poroelasticity. A review of the derivation of the main equations from the Hooke’s law, momentum and mass balance equations, and Darcy’s law suggests an alternative new physical interpretation of some coeﬃcients of the classical poroelasticity. The velocity of wave propagation, the attenuation factor, and the wave number, are expressed in the form of power series with respect to a small dimensionless parameter. The absolute value of this parameter is equal to the product of the kinematic reservoir ﬂuid mobility and the wave frequency. Retaining only the leading terms of the series leads to explicit and relatively simple expressions for the reﬂection and transmission coeﬃcients for a planar wave crossing an interface between two permeable media, as well as wave reﬂection from a thin highly-permeable layer (a lens). Practical applications of the obtained asymptotic formulae are seismic modeling, inversion, and attribute analysis.


Introduction
The classical theory of elasticity associates anomalously high reflection from a layer in a homogeneous medium with the tuning effect, which takes place when the thickness of the layer is equal to one-fourth of the wavelength [53]. However, field observation demonstrate anomalous seismic signal reflection from a thin fluid-saturated permeable layer in the low-frequency end of the spectrum, where the thickness of the reservoir is much smaller than a quarter of the wavelength. Moreover, frequency-dependent data analysis identifies the most productive spots in the reservoir [16,[25][26][27][28][29][30][31]52]. In this work, we explain this seemingly abnormal reflection by the interaction between the elastic wave and fluid flow.
The theoretical foundations of seismic wave propagation in a fluid-saturated porous medium have been established in the pioneer works by Frenkel, Gassmann, Biot, and others [7,8,19,[22][23][24]41]. Nikolaevskii [44,45] has developed a comprehensive theory of deformation, flow and heat transfer in porous media. Overviews of the Frenkel, Gassmann, and Biot theories and further development of poroelasticity are presented in [3,5,45,47,50]. Extensions accounting for local heterogeneities including double-porosity or layered media have been developed in [6,14,15,33,35,48,49]. The reflection and transmission coefficients predicted by Biot's theory for a wave crossing a planar interface have been calculated in [21,32]. The complexity of the expressions for the reflection and transmission coefficients derived from the exact Biot's solution makes unclear what is the relative impact of the rock and fluid properties on the magnitudes of the reflection and transmission coefficients. On the contrary, the simplified low-frequency asymptotic relationships obtained in this study yield approximate but relatively simple and practically useful expressions for the reflection and transmission coefficients.
The small parameter used in the asymptotic analysis below is equal to the product of the reservoir fluid mobility and density, and the frequency of the signal, multiplied by an imaginary unit. The velocities of the slow and fast Biot's waves, and the respective wave numbers and attenuation factors are expressed as power series with respect to this small parameter. The coefficients of the series are real-valued functions of the properties of the reservoir rock and fluid. Retaining only the leading terms of the series produce relatively simple mathematical expressions, which are valid in the low-frequency end of the spectrum including the seismic frequency band (10-100 Hz). Using these expressions, we study the reflection and transmission coefficients for a planar wave crossing a permeable interface at a normal incident angle. We further obtain that the fast-wave reflection coefficient from a thin permeable layer (a lens) attains a peak value. The corresponding peak frequency is expressed through the reservoir rock and the fluid properties. The magnitude of this frequency predicted by the obtained formulae is in agreement with the field observations [25,29]. The practical implications of the theory developed here are seismic modeling, inversion, and attribute analysis.
We review the equations of poroelasticty by relating them to Darcy's law, Hooke's law, and momentum and mass balance equations. Although the obtained equations are essentially the same the original Biot's equations [7,8], our approach provides new physical interpretations of some poroelasticity coefficients. For instance, we demonstrate that the Biot-Willis coefficient α [10] is related to the relative surface area of the grains, which is not exposed to the fluid. Gassmann's results [23] appear as the zero-frequency limit of the asymptotic solution presented in this study.
We assume that the deformations are small and the macroscopic stressstrain relationship for the skeleton are adequately described by the Hooke's law [43]. We have to modify Darcy's law to account for dynamic and nonequilibrium effects in fluid flow. We demonstrate that this modification of Darcy's law is equivalent to a linearization of the dynamic permeability discussed in [15,17,39] for a periodic oscillatory flow. The dynamic component of the modified Darcy's law does not enter the leading terms of the asymptotic expressions. Thus, we conclude that the classical Darcy's law formulation [2,18,37,38] is sufficient for the low-frequency analysis.
The present work, to a certain degree, is an extension of the work [51]. Besides a slightly different choice of the small parameter, we abandon the assumption of grain stiffness employed in [51]. More importantly, in the study of reflection and transmission coefficients for a permeable interface.
The paper is organized as follows. In Section 2, we briefly overview the derivation of the equations of poroelasticity from the basic theoretical principles of flow and deformation in porous media. In Section 3, we obtain an asymptotic harmonic-wave solution valid in the low-frequency range. In Section 4 we obtain explicit expressions for the reflection and transmission coefficients for a planar compression wave crossing a permeable interface. The resonant frequency of a fast wave reflection from a permeable layer is studied in Section 5. Finally, Section 6 summarizes the findings and formulates the conclusions.

The Model
Throughout this work, the porous medium is assumed to be homogeneous and isotropic and the fluid is newtonian. The linear dimensions of an elementary volume of the medium are small relative to the linear dimensions of the entire medium, but large relative to the size of individual pores or grains constituting the solid skeleton. The total stress in bulk medium is the resultant of three components: the elastic stress in the solid skeleton, the fluid pressure, and viscous friction in the fluid flow relative to the solid skeleton. In a linear approximation, these components are decoupled, so they can be considered separately and then summed up.
Linear elasticity of drained skeleton. The macroscopic stress in the drained skeleton σ s is a bulk-area average of the the grain-to-grain contact forces, Figure 1. Let u = (u x , u y , u z ) denote the vector of macroscopic displacement of the solid skeleton and Ξ = {u ij } denote the macroscopic strain tensor: The Hooke's law for an isotropic and uniform medium says: where K and µ are the bulk and shear moduli of the drained skeleton, and I is an identity tensor [43]. Note that the macroscopic skeleton moduli are different from the bulk and shear moduli of the grain material. Figure 1. Force balance at an individual grain in a clean rock. The force F g at a plane cross-section orthogonal to a unit normal e z equals the sum of contact forces F i shown as solid arrows, and fluid pressure acting at the part of the grain surface in located the positive with respect to e z half-space, shown as dashed arrows.
Fluid pressure. In order to evaluate the fluid pressure contribution to the total stress, consider forces acting in the planar cross-section, which Figure 1 displays as the horizontal axis. We assume that the cross-section intersects sufficiently many grains. The fluid pressure portion of the total stress includes two components: the pore pressure and the portion of the stress in the skeleton which is the reaction on the pore pressure. Inside the pores, the fluid pressure contribution amounts to pφ, where p denotes the pressure of the fluid and φ is the porosity of the medium. In an individual grain, the fluid pressure acts through the grain surface. In general, only a portion of this surface is exposed to the fluid, whereas the remaining part is excluded from the fluid-solid interaction by the contacts with the neighbor grains. The force F g in Figure 1 equilibrates the sum of contact forces, F i , and the integral of the pressure over the portion of the grain surface which is in contact with the pore fluid. Summing up over all grains in the cross-section, the total action of the fluid pressure inside the skeleton is characterized by (1 − φ)α φ p, where α φ is a dimensionless geometric factor accounting for the portion of the average portion of the grain surface, which is excluded from a contact with the fluid. Clearly, 0 ≤ α φ ≤ 1. After adding pφ, the pore pressure contribution in the pores, one obtains: where α = φ + α φ (1 − φ), and σ p is the entire pore pressure contribution to the total stress. Note that φ ≤ α ≤ 1. The two extreme values of α are α = 1 and α = φ. The first one describes a medium where the average grain-to-grain contact area is negligibly small, for example in unconsolidated sand. The other extreme case can be represented by a medium where the skeleton is a bundle of infinite cylindrical columns.

AN ASYMPTOTIC MODEL OF SEISMIC REFLECTION FROM A PERMEABLE LAYER 5
The argument above is not new: our approach is similar to that developed in [40].
Fluid flow. If the skeleton moves with a constant acceleration, then Darcy's law for the flow relative to the skeleton can be written in the form: where W denotes the Dracy velocity of the fluid relative to the skeleton, η and ̺ f denote fluid viscosity and density, and κ is the absolute permeability of the medium. The left-hand side expresses the viscous drag force acting between the fluid and the solid skeleton, whereas the second term on the right-hand side accounts for the body force acting on the fluid in a non-inertial reference frame associated with the skeleton. Equation (4) is an expression of force balance. In an oscillating system, the fluid flows relative to the skeleton with an acceleration. Therefore the body force on the right hand side of Equation (4) must include an additional term ̺ f 1 φ ∂W ∂t .
In addition, the steady flow paths in the tortuous pore channels establish not instantaneously, the drag force on the left-hand side depends not only on the instantaneous Darcy velocity, but also on its variation. In a linear approximation, we write where τ * is a parameter having the dimensionality of time. Thus, finally, one obtains: where parameter τ = τ * + κ̺ f ηφ accounts both for the non-equilibrium effects of non-steady fluid flow and fluid inertia. We will call Equation (6) dynamic Darcy's law. If the motion of the skeleton can be neglected, Equation (6) reduces to the model of filtration with relaxation [1]. If, in addition, the flow is steady, Equation (6) yields the classical Darcy's law (gravity is neglected throughout this work). Different Darcy's law modifications accounting for dynamic and nonsteady effects have been developed in the past and are discussed below, after deriving the asymptotic low-frequency solution. Equation (6) with τ * = 0 has been obtained by Frenkel [22] and Nikolaevskii [44,45]. We have found the form of Equation (6) most convenient for the low-frequency asymptotic analysis below. This analysis shows that the parameter τ enters only the higher-order terms. On other words, one can assume τ = 0 in the low-frequency end of the spectrum.
Flow-imposed shear stress. A shear wave preserves the volume and porosity, and does not affect the pore pressure. The viscose friction between the fluid flowing in the pores and the skeleton develops a distributed drag force proportional to the Darcy velocity of the fluid relative to the skeleton. The Darcy velocity is orthogonal to the wave propagation, see Figure 2. Interaction of the porous layer between planes with coordinates x and x+dx with the rest of the formation includes the shear stress in the skeleton, and the drag force by the fluid flowing relative to the skeleton on both sides of the layer. Since the Darcy velocities of the fluid at the two sides of the layer are different, the drag forces are different as well. Therefore, in addition to the shear stress in the skeleton, there is a component coming from the Darcy velocity variation in direction orthogonal to the wave propagation. We denote this stress by σ f .
More generally, the shear stress σ f is a linear function of the following tensor: The coefficient ϕ is a function of the fluid viscosity η and the geometry of the pore space. Dimensional considerations suggest that ϕ = ϕ 0 η, where ϕ 0 is a dimensionless shape factor. Equation (7) has been written down by analogy with the classical fluid mechanics [42]. The stress σ f acts on saturated porous medium, whereas the flow is governed by Darcy's law. Therefore, this is different from the shear stress in the Brinkmann model [11]. We do not know any experimental data on stress σ f . The asymptotic analysis below shows that σ f enters only higher order terms and can be ignored in the low-frequency range.
Momentum balance in a planar wave. From the previous sections, the total stress, σ t , is the sum of all three stresses components defined in Equations (2), (3), and (7): σ t = σ s + σ p + σ f . Let us denote by ̺ g the density of the solid constituent,so that the bulk density is equal to The linearized momentum balance equation has the following from For a planar wave propagating in direction x, all the derivatives with respect to y and z vanish. A substitution of Equations (2), (3), and (7) reduces Equations (8) to Mass balance. The variation of the fluid mass in an elementary volume due to the deformation of the skeleton and fluid compression equals the total mass flux through the boundary of the volume. Assuming adiabatic linear fluid compressibility and retaining only the first-order terms, one obtains Here β f is the coefficient of adiabatic compressibility of the fluid: d̺ f ̺ f = β f dp. The linearized mass balance equation for the skeleton is: We assume that the compression of the grains is determined by the component of the skeleton stress coming from the volumetric strain (described by the term K∇ · u in Equation (2)) and the fluid pressure variations. In a linearized form, one obtains: Here K sg and K f g are the elastic moduli quantifying the compression of the grains by the volumetric strain of the skeleton and the fluid pressure variations, respectively. Thus, in terms of time derivatives, one obtains

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The time derivative of the porosity can be eliminated from Equations (10) and (11). After gathering similar terms, one obtains: where thee dimensionless coefficients γ K and γ β are defined by Some remarks. The skeleton is less stiff than the grain material: K ≤ K sg . Thus, φ ≤ γ K ≤ 1. At vanishing porosity, φ → 0, the skeleton strength approaches that of the grain material, K/K sg → 1. Therefore, if the transition to the zero porosity is smooth, which according to [13,34] holds true for a variety of heterogeneous media, then Equation (15) implies that γ β ≥ Kβ f φ. The compressibility of the fluid, β f , is independent of the porosity, whereas the bulk modulus of the skeleton, K, converges to that of the grain material as φ → 0. Thus, the product Kβ f does not vanish and Equation (16) imply that This estimate will be used below.
To compare the introduced coefficients to those of Biot and Willis [10], one can introduce a fluid displacement vector, w, as the integral of the Darcy velocity: Integration of Equation (14) with respect to t yields The divergence in the last term, ∇ · w, is equal to the fluid content parameter [4,10,20]. A comparison of the last equation to Equation [10, (30)] shows that where the subscript BW refers to the Biot-Willis coefficients in the notations of [10], and γ K and γ β are defined in Equation (15). Equation (32) below shows that in fact γ K = α, where α is the geometric factor in Equation (3) and (9).

Asymptotic harmonic wave solution
Let us seek a harmonic planar-wave solution to the obtained system of equations. That is, put Here k x denotes the x-component of the complex-valued wave vector: k = (k x , 0, 0). A substitution of Equation (21) into equations (9), (6), and (10) yields Like in the classical case, the compression and shear waves decouple and each component of the solution can be calculated separately.
Compression wave. The system of compression wave equations consists of only those equations (22), which involve the x-components of the skeleton displacement and the Darcy velocity: For low frequencies, formula ε = iλω defines a dimensionless small parameter, where the quantity λ = ̺ f κ η can be called kinematic reservoir fluid mobility. Let us define unknown dimensionless parameters: Here v 2 = ω 2 k 2 x and v 2 f = M ̺ f . Then, the system of equations (23) takes on the form: where We seek a solution to the system (25)- (27) in the form of power series in ε: where ". . . " denotes the higher order terms. At ε = 0, Equation (26) implies ζ 0 χ 0 = 0. One obtains two zero-order solutions: The superscripts S and F stand for the slow and fast waves. To determine the first-order terms, one has to solve the following system of equations: Note that ϑ is the only dimensionless parameter depending on the parameter τ in the dynamic Darcy's law, Equation (6). Since χ 0 ζ 0 = 0, the term involving ϑ vanishes both for the slow-wave and the fast-wave solutions. Consequently, the classical steady-state Darcy's law formulation is sufficient for the first-order asymptotic analysis. The solution to the system of equations (30) is:

AN ASYMPTOTIC MODEL OF SEISMIC REFLECTION FROM A PERMEABLE LAYER 11
Finally, Equations (29) and (31) can be rewritten in the form: and Hence, for the fast wave, where For the slow wave, In the last equation, the branch of the square root has been selected to guarantee that Im(k S x ) < 0, that is, (35) and (36) can be rearranged into the from: where Note that the estimate in Equation (17) velocities and attenuation factors for the fast and slow waves are By the definition of ε, the angular frequency ω is present both in the numerator and the denominator in the last equation. It is useful to rewrite it in an alternative from Clearly, V F ≫ V S = O |ε| and a F ≪ a S as ε → 0. Equation (16) also implies that, for φ → 0, the fast compressive wave velocity approaches the velocity of sound in a medium whose density is equal to the density of the grains and elastic moduli are those of the drained skeleton. Finally, the power series asymptotic expressions for the Darcy velocity and the fluid pressure have the following forms: Shear wave. The shear waves in the directions y and z are analogous to each other, so we consider in detail only the shear wave in the direction y. Since shear deformation includes zero volumetric strain, the mass balance equation (14) is an identity. Thus, only the following system of two equations needs be solved:

AN ASYMPTOTIC MODEL OF SEISMIC REFLECTION FROM A PERMEABLE LAYER 13
A convenient set of dimensionless variables is provided by the first and last equations (24), where the x-components are replaced with the respective y-components. The system of equations (49) transforms into the following: which is equivalent to Equations (25)-(26) if one puts ξ = 0 and replaces M with µ. The solution to the system (50) is: In the form of a power series in ε, this solution takes on the form: Note that the shear wave solution has no slow-wave component.
Returning to the physical quantities, one obtains where the superscript H denotes a shear wave. Thus, k H 0 = ̺ b /µ in the power-series expansion k H = ω k H 0 + k H 1 ε + O(|ε| 2 ) . For the attenuation factor, a H , and the velocity, V H , one obtains Further remarks Let us demonstrate that the equations above are consistent with the classical theory of poroelasticity.
Dynamic Darcy's law and flow-imposed shear stress. For an oscillatory fluid flow in a porous medium, Johnson et al. [39] have obtained a modification of Darcy's law, in which the coefficient of permeability depends on the frequency of the oscillations: At the zero frequency limit, the frequency-dependent coefficient of permeability must be equal to the classical Darcy permeability: κ(0) = κ. Therefore, one can write:

DMITRIY SILIN AND GENNADY GOLOSHUBIN
Using Taylor expansion, A(iω) = 1 + A ′ (0)iω + . . ., and truncating the higher-order terms, one obtains from Equation (55): The last equation is equivalent to Equation (6) for τ = A ′ (0). Equations (29), (31), and (52) imply that both, the dynamic term in the dynamic Darcy's law, Equation (6), and the flow-induced shear stress, affect only the power series terms of the order of O(|ε| 2 ) or higher. Thus, the classical steady-state formulation of Darcy's law is sufficient for the firstorder asymptotic approximation of the compression wave solution.
Elastic moduli. The compressibility of the fluid and the moduli of drained skeleton can be determined in laboratory tests. The other two moduli introduces in Equation (12) also can be determined experimentally. For the first test, one can change the pore pressure by injecting or withdrawing fluid while maintaining the total stress constant. From the fluid compressibility and the variation of the fluid volume, one can evaluate the variation of the pore volume and, consequently, the variation of the total volume of the solid grains. Hence, from the known mass of the solid skeleton one obtains the variation of the average density of the grains. In an undrained uniaxial test, one also can measure the variation of the fluid pressure and the total volume of the saturated sample. Therefore, the variation of the average density of the grains can be calculated as well. Thus, knowing K from independent measurements, one obtains a system of two equations, which can be solved for the moduli K sg and K f g .
Biot's equations. Equations (32) and (20) show that the coefficient α introduced in Section (3) and the Biot-Willis coefficients alpha are the same. Further on, Equation (19) makes possible to eliminate the fluid pressure p from the first Equation (9) and Equation (6). Thus, using the notation (18), one obtains: The last system of equations is equivalent to Biot's equations [9] with the following mapping rule: The notations and the parameters on the left-hand sides of these relationships are from [9].
In some works, parameter m is associated with the formation resistivity factor [12]. In any case, parameter τ does not affect the zero and first-order terms in the asymptotic analysis above.
Nikolaevskii's model. Nikolaevskii [44,45] has developed a general model of poroelasticity based on mass, momentum, and energy balance. Besides purely mechanical deformation, his model accounts for the impact of heating on stress and deformation. For isothermal creeping flow, the model developed here is consistent with the Nikolaevskii model. The mass balance equations (10) and (11) where R is called the viscose resistance to the fluid flow [44]. In an isotropic medium, linearized Equations [44, (2.31), (2.65)] imply the following expression for R: Since the Darcy velocity and skeleton displacement are small, a substitution of Equation (62) into (61), after cancelling φ, gives: The latter equation is identical to Equation (6) if one puts and τ * = 0 τ = ̺ f κ ηφ .
Gassmann's model. The two elastic moduli, K sg and K f g , defined in Equation (12) relate the grain volumetric strain to the skeleton stress and fluid pressure, respectively. If the relationships hold true, then the expression K 1 + γ K 2 γ β yields Gassmann's bulk modulus [23,46]. Vanishing attenuation. For some combinations of parameters, the coefficient ζ F 1 defined in Equation (31) can vanish. In such a case, the first-order approximation of the the fast wave attenuation, Equation (42), is equal to zero. For example, ζ F 1 = 0 if γ K = √ γ β and γ ̺ = 2γ K simultaneously.

Normal reflection of a compression wave
Let two poroelastic media labelled by the superscripts 1 and 2 have a permeable plane interface at x = 0. A fast or slow incident wave generates four waves: fast and slow reflected waves, and fast and slow transmitted waves, Figure 3. Let us consider fast and slow incident waves separately.
Fast incident wave. A fast incident wave arriving from the half-space x < 0 generates four waves: the fast and slow reflected waves and the fast and slow transmitted waves. Thus, in the medium 1, the displacement can be characterized as whereas in medium 2, the skeleton displacement is Here R F F , R F S , T F F , and T F S are the respective reflection and transmission coefficients. The first letter in the superscript denotes the fast incident wave, whereas the second one denotes the fast or slow reflected or transmitted wave.
The mass and momentum balance imply that the skeleton displacement, the Darcy velocity of the fluid, the total stress, and the fluid pressure must be continuous at the interface. Using notations (24), one obtains the following system of boundary conditions To obtain asymptotic expressions for the reflection and transmission coefficients, we rewrite Equations (67)-(70) in an approximate form, retaining only the leading zero-order and the next after zero-order terms. For the first two equations, these terms are the constant ones and the ones which are linear in ε. For the last two equations, these terms are the constant ones and the ones proportional to |ε|. One obtains: Here By virtue of Equations (33), Thus, the slow-wave reflection and transmission terms vanish in Equation (73). Taking into account Equation (39), Equation (73) reduces to where Z i , i = 1, 2, are the modified acoustic impedances:

DMITRIY SILIN AND GENNADY GOLOSHUBIN
We seek asymptotic expressions for the transmission and reflection coefficients in the form We limit our calculations to the two leading terms only. Putting ε = 0 yields: Equations (72) and (74) imply The determinant of the linear system of equations (82)-(83) is obviously positive: D > 0. Hence, For the next terms of the fast wave reflection and transmission coefficients, one obtains Finally, Slow incident wave. A slow incident wave also generates four waves: the fast and slow reflected waves and the fast and slow transmitted waves. In the Medium 1, for a slow incident wave, one obtains whereas in the Medium 2, Here R SS , R SF , T SF , and T SS are the respective reflection and transmission coefficients. Equation (67)-(70) transform into After dropping the higher-order terms, one obtains As in the previous subsection, we seek asymptotic expressions for the transmission and reflection coefficients in the form (79) limiting our analysis by the two leading terms only. Putting ε = 0 yields: Hence, using Equations (96) and (98), we obtain

Reflection from a permeable layer
Consider reflection of a fast incident wave from a permeable layer of thickness H. Let this layer, labelled by the superscript 2 , be sandwiched between two media whose properties will be labelled by the superscript 1 , Figure 4.
Both at the top and at the bottom of the layer, an incident wave generates two pairs of transmitted and reflected slow and fast waves. We consider two signals generated by reflection of an incident fast wave from the layer. In the first case, the signal is transmitted into the layer as a slow wave, reflected from the bottom of the layer as a fast wave, and further transmitted back into the upper medium 1 as a fast wave. In the second case, the signal is transmitted into the layer as a fast wave, reflected from the bottom as a slow wave, and transmitted into the upper medium as a fast wave. Figure 4 shows schematically the path of the signal for each of the two configurations. In both cases, the slow wave constitutes only one of the four segments of the whole path. Since the fast wave reflections from the top and bottom of the layer cancel each other, the considered paths generate reflected signals neglecting multiple reflections.
Both paths are similar to each other, so we consider in detail only the first one. Let U 0 be the amplitude of the incident fast wave. Then, by virtue of Equations (80) and (85), the amplitude of the transmitted slow wave inside the layer is equal to U reflected from the bottom of the layer is equal to The exponential function in the last equation comes from the slow-wave attenuation. The fast-wave attenuation factor, a F , is small of higher order relative to a S , see Equation (42). Note that the indices 1 and 2 must be permutated in the left Equation (102) for a correct evaluation of the reflection coefficient from the bottom R SF 0 . Finally, for the amplitude of the signal reflected from the layer, one obtains the following expression through the amplitude of the original incident wave: For a correct evaluation of T F F 0 , the transmission coefficient for the signal reflected from the bottom and crossing the top interface, the indices 1 and 2 must be permutated in Equation (81). Using Equation (44), the ε-dependent factors in the product on the right-hand side of Equation (104) can be gathered in the form: This function is the permeability-based reflection factor. It attains a maximum value of The peak frequency is For example, if M = 10 +10 Pa, γ β + γ 2 K ≈ 2.5, κ = 1 Darcy, η = 10 −3 Pa-s, and H = 0.5 m, then ν max ≈ 8 Hz, Figure 5. This estimate is in a agreement with field observations of the low-frequency gas shadow [16].
Similarly, for the second path, the reflected signal amplitude is The transmission coefficients T F F 0 in Equations (103) and (104) correspond to different direction of wave propagation through the top interface and, in general, are different.
The travel times for both pathes are the same (approximately 30 ms for the numerical parameters mentioned above), as well as the peak frequencies.
The phase shifts due to the travel times also are the same. Therefore, in the superposition of the reflected signals, the amplitudes (104) and (108) sum up.
Reflection from a layered reservoir. Due to the factor of |ε|, the absolute value of the reflection coefficient is not large. However, frequently, a reservoir has a layered structure, where the permeability can differ between the layers Permeability-based reflection, ψ Figure 5. The permeability-based reflection factor ψ, Equation (105), attains a peak value at a low seismic frequency.
by orders of magnitude. Summation of the reflections from multiple layer enhances the peak reflection effect with a noticeable time delay relative to the first arrival. In the numerical example below, such a summation produces a noticeable effect on the reflection coefficient. For evaluation of the impedance contrasts and reflection coefficients for thick layered porous media, the conventional seismic amplitude analysis rely on Gassmann's model [23]. The relative easiness of assigning sensible values to the parameters in Gassmann's equation makes this model popular among exploration geophysicists [36]. In case of thin-layered heterogeneous reservoir, especially with significant variation of the porosity and permeability of the rock between the layers, the situation may be different. At the end of Section 3, we have demonstrated that Gassmann's equation can be obtained as the zero-frequency limit of the asymptotical solution obtained in the same section.
To evaluate the influence of the first-order terms on the reflection coefficients from a thin-layered reservoir, we consider a model of a 22 meter thick reservoir consisting of 22 one-meter thick layers of different porosities and permeabilities. Figure 6 (a) shows a plot of the computed fast-wave reflection coefficient, R, versus two way travel time from reservoir surface. The calculation accounts for the multiple reflections and the fast-slow and slowfast wave conversions at the interfaces, using Equations (80)-(81), (85)-(89), and (100)-(102). There is remarkable 10% difference between the result of application of asymptotic analysis and the result of calculation based solely on the zero-order terms (Gassmann's model), see Figure 6 (b). This difference is due the first-order terms in the expressions for asymptotic reflection coefficients and it is a product of the fluid motion relatively skeleton. The difference △R between R and the reflection coefficients evaluated from the zero-order approximation (Gassmann's model).

Summary and conclusions
A review of the derivation of Biot's equations of poroelasticity from the basic principles: momentum and mass balance equations, Hooke's law, and Darcy's law, suggests new physical interpretations for some coefficients of the classical poroelasticity. For example, the Biot-Willis coefficient α is related to the distribution of the surface of the grains between grain-to-grain and grain-to fluid contact.
Asymptotic analysis of a plane-wave solution at low frequencies leads to explicit relatively simple expressions for the velocity and attenuation of the fast and slow waves. The small parameter is a dimensionless quantity proportional to the product of the fluid mobility, density, and the frequency of the signal. The wave number, velocity, and attenuation factor are expressed as power series with respect to this parameter. The calculations yield that all the coefficients in the power series depend on the mechanical properties of the medium, but neither on the fluid mobility nor on the signal frequency.
The obtained asymptotic solutions lead to power-series expressions of the reflection and transmission coefficients for an elastic compression wave normally crossing a permeable planar interface between two media. It turns out that the leading frequency-dependent term is proportional to the square root of the frequency of the signal. The zero-order terms have been expressed through the acoustic impedances of the media, similarly to the classical theory.
Analysis of the reflection coefficient from a permeable layer (a lens) shows that the reflection of an incident fast wave including one slow-wave segment inside the layer is frequency-dependent and has a peak. The asymptotic relationships make possible an explicit evaluation of the maximum reflection coefficient and the peak-reflection frequency. For a realistic set of parameters, the maximum is attained at a low seismic frequency. Although the amplitude of the reflected signal is proportional to the absolute value of the small parameter and is small, the reflection from a number of such lenses can produce a noticeable effect. Such frequencies have been successfully used for imaging the most permeable areas of a hydrocarbon-bearing reservoir [25,29]. The results of asymptotic analysis have been applied for numerical evaluation of the reflection coefficient in a model of layered reservoir with variable permeability. Calculations with account for multiple reflections show a significant contribution of the frequency-dependent part of the asymptotic expansion. The practical applications of the theory developed here are seismic modeling, inversion, and attribute analysis.