岩石物理

SCIENCE CHINAEarth Sciences© Science China Press and Springer-Verlag Berlin Heidelberg 2011 *Corresponding author (email: tangxiam@)• RESEARCH PAPER •September 2011 Vol.54 No.9: 1441–1452doi: 10.1007/s11430-011-4245-7A unified theory for elastic wave propagation through porousmedia containing cracks—An extension of Biot’sporoelastic wave theoryTANG XiaoMing *College of Geosciences and Technology, China University of Petroleum (East), Qingdao 266555, ChinaReceived November 25, 2010; accepted March 3, 2011; published online July 25, 2011Rocks in earth’s crust usually contain both pores and cracks. This phenomenon significantly affects the propagation of elastic waves in earth. This study describes a unified elastic wave theory for porous rock media containing cracks. The new theory extends the classic Biot’s poroelastic wave theory to include the effects of cracks. The effect of cracks on rock’s elastic prop-erty is introduced using a crack-dependent dry bulk modulus. Another important frequency-dependent effect is the “squirt flow” phenomenon in the cracked porous rock. The analytical results of the new theory demonstrate not only reduction of elas-tic moduli due to cracks but also significant elastic wave attenuation and dispersion due to squirt flow. The theory shows that the effects of cracks are controlled by two most important parameters of a cracked solid: crack density and aspect ratio. An appealing feature of the new theory is its maintenance of the main characteristics of Biot’s theory, predicting the characteristics of Biot’s slow wave and the effects of permeability on elastic wave propagation. As an application example, the theory cor-rectly simulates the change of elastic wave velocity with gas saturation in a field data set. Compared to Biot theory, the new theory has a broader application scope in the measurement of rock properties of earth’s shallow crust using seismic/acoustic waves.poroelasticity, wave propagation, cracked medium, rockphysicsCitation: Tang X M. A unified theory for elastic wave propagation through porous media containing cracks—An extension of Biot’s poroelastic wave theory.Sci China Earth Sci, 2011, 54: 1441–1452, doi: 10.1007/s11430-011-4245-7The formulation of the theory for elastic wave propagation in porous media has long been an important topic of geo-physics. A fundamental phenomenon is the universal pre- sence of pores and cracks in rocks that consist of the earth’s shallow crust (Figure 1). Compared to pores, cracks usually have a much smaller aspect ratio, i.e., crack thickness is much smaller than its lateral size. The presence of both pores and cracks in rocks makes it quite complicated to de-scribe the elastic properties of the rock media. Neither the poroelastic wave mechanics represented by Biot theory nor the cracked medium elasticity theory based on O’Connell and Budiansky work can adequately model the elastic wave response of such media. The quest for a unified elastic wave theory for the porous rock containing cracks is an important subject in this branch of geophysics.In the 1950s and 1960s, researchers like Eshelby [1] and Walsh [2] began to study the effects of cracks on the elastic properties of media containing cracks. Afterward, O’Connell and Budiansky [3–7] published a series of papers on this subject. They pointed out an additional effect of flu-id-saturated cracked media, i.e., a crack with its thickness being much smaller than its lateral dimension can deform easily under external forces. The expulsion of the fluid from cracks produces a local fluid flow called “squirt flow”, re-1442Tang X M. Sci China Earth SciSeptember (2011) Vol.54 No.9Figure 1Pores and cracks seen from a thin section of a sandstone rock. sulting in the viscoelastic behavior of the cracked medium. On the other hand, Biot [8–10] formulated his well known poroelastic wave theory for porous media. Assuming homogeneous distribution of pores in a medium, the theory mainly considers viscous coupling of pore fluid with solid matrix and analyzes the dynamic response of the fluid- sat-urated porous medium to external wave excitations. The main contributions of Biot theory are the prediction of a slow wave in porous media and the relationship between medium permeability and elastic wave propagation. For this reason, Biot theory is widely used in applications related to elastic wave exploration of earth’s shallow crust (e.g., acoustic logging, see Tang and Cheng [11], and seismic data interpretation, etc.). However, the theory has been shown to be inadequate in describing the behavior of real rocks in various seismic/acoustic exploration application problems. For example, Dvorkin and Nur [12] point out that the theory significantly under-predicts the elastic wave dis-persion and attenuation measured in real rocks. The main cause of this discrepancy resides in the assumption of ho-mogeneity of the pore distribution. This assumption over-looks the “squirt flow” effect resulting from local fluid movement at the pore scale.Various theories have been developed to describe and model the heterogeneous pore distribution and the associat-ed local fluid flow. White [13], Johnson [14], and Carcione et al. [15] model the local flow using a heterogeneous pore-fluid distribution. Hudson [16–18] and Kuster and Toksöz [19] use elastic wave scattering theory to describe the effects of heterogeneity within a rock. Mavko and Nur [20] and Dvorkin and Nur [12] develop the “squirt flow” theories to describe the local fluid flow. The latter theory is often called the BISQ theory. A typical category of theoret-ical work is the theory of double-porosity models, as de-veloped by Pride and Berryman [21, 22], Berryman and Wang [23], and Berryman [24]. In a double-porosity me-dium, the local flow results from the pressure difference between the two pore systems of different structures. Ba [25, 26] uses Biot theory to analyze the elastic wave propagation effects of the double-porosity system. The double-porosity model needs to consider the elastic interaction between the two pore systems and the local flow in the model is gov-erned by parameters such as characteristic relaxation fre-quency and fluid transport coefficient, etc. The present study will discuss a more specific “double-porosity” system: the porous medium containing cracks. This is also a subject of significant practical importance. In this medium, the elastic interaction between cracks and pores is handled by a pore-crack consistency approach, whereas the associated local flow is described by the squirt flow between cracks and pores. Under this approach, the characteristic relaxation frequency and fluid transport parameters are described by two important parameters of a cracked medium: crack den-sity and aspect ratio.The physical picture of “squirting” flow in a cracked po-rous medium is quite straightforward to perceive. Compared to pores, the flat- or narrow-shaped cracks can easily con-tract and expand under external wave excitation, squeezing the fluid into neighboring pores and generating squirt flow. For years, researchers in this field have been seeking a uni-fied elastic wave theory in order to describe the elastic property and wave propagation characteristics of the cracked porous medium. Although the work of O’Connell and Budiansky [3–7] (hereafter referred as OB theory) re-lates the medium’s elastic property to cracks and predicts the viscoelastic property of the medium caused by squirt flow, their theory does not contain the effects of permeabil-ity and the related slow wave propagation. Thomsen [27] formulates a theory to relate OB theory to Biot theory and calls it the Biot-consistent theory. The theory describes the effects of fluid in the pore-crack system under the frame-work of Biot theory. Although the theory discusses the low-frequency scenario of the pore-crack system, corre-sponding to the low-frequency limit of the Biot theory, the formulation of the theory can be extended to the high-frequency regime of the Biot theory to obtain an elas-tic wave theory for the full frequency range. In the exten-sion of the theoretical formulation to the high-frequency regime, the effects of squirt flow between cracks and pores become prominent. As pointed by Mavko and Jizba [28], the squirt flow occurring at the pore-crack scale is strongly frequency-dependent. At low frequencies, the fluid ex-change between pores and cracks has sufficient time to complete. At high frequencies, this exchange may not have enough time to take place, resulting in “locking” the fluid in the narrow space of cracks. Dvorkin and Nur [12] (i.e., the BISQ model, hereafter referred as DN theory) propose a unified theory to relate Biot theory and squirt flow effects in order to model the significant elastic wave attenuation and dispersion measured in real rocks. However, the DN theory, as will be pointed out later, does not include the two most important parameters of the cracked medium: crack density and aspect ratio. More importantly, the theory does not cor-Tang X M. Sci China Earth Sci September (2011) Vol.54 No.91443rectly model the propagation characteristic of Biot’s slow wave, which is the fundamental property of Biot theory. In this regard, DN theory cannot be regarded as the unified elastic wave theory for the porous medium containing cracks.This study presents a detailed analysis of the squirt flow effect between cracks and pores and relates this effect to the important parameters of a cracked medium. The incorpora-tion of the cracked medium parameters into Biot formula-tion results in a unified elastic wave theory for the cracked porous medium. The new theory not only maintains the main characteristics of the original theories, but also signif-icantly enhances the capability of modeling and predicting elastic wave propagation in such a medium. Thus the new theory has a broader application scope in the measurement of rock property in earth’s shallow crust using elas-tic/acoustic waves.1 Generalization of Biot theory to porous me-dia containing cracksBefore carrying out our theoretical analyses, let us firstsummarize the main results of Biot theory for use in the later theoretical development. In Biot’s theory, three types of wave can propagate in a porous medium. These are fast compressional, shear, and slow compressional waves.The fast compressional wave and shear wave are mainly related to the solid matrix, although they are also affected by pore fluid. The main contribution of Biot theory is the prediction of the presence of the slow compressional wave. The slow wave is mainly borne in the pore fluid, although it is also affected by the motion of the porous matrix. Using the notations of Tang and Cheng [11], we write out the equations for calculating the wavenumber of the three Biot waves:p p s k k k ±== (1) In the subscripts for the wavenumbers p and s represent compressional and shear waves, respectively, and the sym-bols + and − represent fast and slow compressional waves,respectively. Other symbols in the wavenumber expressionsare given below:0200021,2(4/3/),()(),.βμαβααρραρθω±⎡=⎣=−++==−+=d p s f s s f b b c b K k c b b b b (2) In eq. (2), ω is angular frequency, K and K d are respec-tively the saturated and dry bulk moduli of the medium, and μ is the shear modulus. The remaining three parameters are given below:(1),1,().s f d s f s K K K K ρρϕρϕαβϕαϕ=−+=−=+− (3) In eq. (3), ϕ is porosity, ρf and K f are density and bulk modulus of pore fluid, respectively, and ρs and K s are re-spectively density and bulk modulus of the solid grain. The following two parameters are related to pore fluid flow:22ˆ,(),f f i ρρρωθθκωηω=+= (4)where a frequency-dependent permeability parameter, called dynamic permeability, as derived by Johnson et al. [29], is given by1/200(),1/()/()2κκωτκρωηϕτκρωηϕ=⎡⎤−−⎢⎥⎣⎦f f i iin which κ0 is the static Darcy permeability, τ is tortuosity,and η is fluid viscosity. The simple functional form of the dynamic permeability can easily characterize the low- and high-frequency regimes of the Biot theory. At low frequen-cies, 0()κωκ→ and the Biot slow compressional wave motion is characterized by a viscous fluid flow. Above the Biot’s critical frequency 0/,c f ωηϕτρκ= ()κω→/f i ηϕτρω the Biot slow wave is characterized as a prop-agational wave.The relative motion between pore fluid and porous ma-trix due to permeability will cause certain attenuation anddispersion for the fast and slow compressional waves andthe shear wave. The dispersion and attenuation are respec-tively calculated using:{}{}{}1Re ,2Im Re ,v k Q k k ω−== (5) where Q is quality factor, Re{k }and Im{k }denote taking thereal and imaginary part of the wavenumber k , respectively,k can be the wavenumber of any wave in eq. (1), fast or slow compressional, or shear wave, and v is the phase ve-locity of the corresponding wave. For the fast compression-al and shear waves, the attenuation and dispersion caused by pore fluid flow is usually quite small compared to thosemeasured in real rocks (see Dvorkin and Nur [12]).The slow compressional wave can exhibit significant at-tenuation and dispersion. In the low-frequency condition,the slow wave motion is actually the diffusive viscous fluid1444Tang X M. Sci China Earth Sci September (2011) Vol.54 No.9flow described by Darcy’s law, whereas the fast compres-sional and slow waves are governed by Gassmann’s theory. The wave velocities are given below:p s v v +== (6) where the bulk modulus of the saturated porous medium isgiven by the Gassmann equation [27]:2(),d s f K K K K ααϕϕ⎡⎤=+−+⎣⎦ (7) whereas the shear modulus μ is the same for dry and satu-rated conditions. Because Biot theory reduces to Gassmanntheory in the low-frequency limit, the theory is sometimes called Biot-Gassmann theory. It follows from the above equations that the elastic property of the Biot theory is gov-erned by five fundamental parameters: K s , ϕ, K f , μ, and K d . Here the dry modulus K d is introduced as an independentparameter into the Biot theory. Because of this property, thevalue of K d can be properly adjusted to suit various applica-tions of the Biot theory. For example, Brie et al. [30] adjust this parameter to effectively model the change of fast com-pressional wave velocity with pore fluid saturation. For the same reason, Thomsen [27] introduces the effects of cracks into his Biot-consistent theory by using a crack-dependent K d to calculate the fluid-saturated bulk modulus of eq. (7) for the cracked porous medium. The cracked medium elasticity theory of OB [5, 6] pro-vides the relationship between elastic parameters and crack density and porosity (definitions given in the appendix) for penny-shaped cracks embedded in a porous solid. Using therelationship in the Biot-consistent theory [27], K d can beexpressed as0(1)2,3(12)B d B K νμν+=− (8)where μ0 is dry shear modulus for the cracked porous solid.01,1(45)2,15(1)(1)(5)32,45(2)ps B B B B B B B B B B b b B ϕμμεννννν⎛⎞=−−⎜⎟−⎝⎠−=−−−=− where ε is crack density, ϕp is porosity of the pore spacewithout cracks (definition given in eq. (A-12) of the appen-dix), μs is shear modulus of solid grain, and v B is Poisson’s ratio of the cracked porous solid under dry condition, which must be solved in connection with the formulation of Biot theory (see Thomsen [27]). By substitution of K d of eq. (8), as found using the approach of the Biot-consistent theory, into the Biot equations (1) through (4), one can calculate theelastic property of the porous solid in the presence of cracks. However, to extend the calculation into the frequency re-gime using the Biot equations, one must consider another effect. This is the “squirt flow” effect induced by the elasticwave propagation through the cracked porous medium. Thefollowing discussions will be focused on this subject. Let us first review the result of DN theory [12], which will be compared with our result to be obtained. The squirt flow model of the DN theory results in modifying the β parameter in eq. (3) to become11102()1,()J J ξββξξ−−⎡⎤→−⎢⎥⎣⎦ (9) whereξ= DN theory introduces a parameter R , called squirt flow length. Other parameters in the above formula are defined ineqs. (3) and (4); J n (n =0, 1) is the first kind Bessel function of order n .Let us now consider the fluid-saturated porous rock model with cracks (Figure 2). This model is an abstraction of the real rock shown in Figure 1. Assume that the cracks are randomly oriented in the 3D space. Apply an external pressure P to the model. When P is incremented by a small amount δP (which may result from the disturbance of apassing elastic wave), fluid volume V f of the pores and cracks in rock will be compressed, giving rise to a change in fluid pressure δP f . The two pressure increments are relatedby (see Chen [31])Figure 2 Model for analyzing squirt flow between crack and pore. A crack is modeled as fluid-filled penny-shaped space in the cylindrical co-ordinates (r , z ). An external pressure change δP induces pore pressure change δP p and crack pressure change δP c . The pressure-induced crack expansion and contraction generate squirt flow.Tang X M. Sci China Earth Sci September (2011) Vol.54 No.9144511,11d f d sK KP P K K δδ−=− (10)where K is the bulk modulus of the fluid-saturated cracked porous rock to be determined. The change in the fluid vol-ume can be decomposed into two contributions: one due to pores and the other due to cracks: .f p c V V V Δ=Δ+Δ (11)We now analyze each term in the above equation. Let usfirst consider the term on the left hand side. In the deriva-tion of Gassmann equation (e.g., Thomsen [27]), there is noassumption about the shape of the fluid-saturated part of the rock. The fluid volume may shape like round pores, narrow cracks, or consist of both cracks and pores. Following the derivation of Gassmann equation (e.g., Chen [31]), this term is related to the rock porosity and dry bulk modulus, and the grain modulus, as111(),fd sf f f s V K K P P P V K δδδϕΔ−=−+(12)where the dry modulus K d can be calculated using eq. (8). The use of this equation accommodates the effects of cracks on the dry modulus. The first term on the right side of eq. (11) is the compression of pore fluid volume, which can beexpressed in terms of the pore fluid pressure change and the fluid bulk modulus:.p ppfV P V K δΔ=(13) The second term on the right side of eq. (11) is related to squirt flow due to cracks. Modeling the fluid motion along the narrow space inside a crack as the flow of an incom-pressible viscous fluid, one can easily deduce that the vol-ume compression of the crack fluid by the external pressure change δP should equal to the amount of fluid squeezed out from the crack into the neighboring pores. By denoting the crack fluid volume change by q and substituting eqs. (12) and (13) into eq. (11), the latter equation, after division by V f δP f , becomes11111.p p d s f s f f f f f P V K K P q P K K P V P V δδϕδδδ⎛⎞−−+=+⎜⎟⎜⎟⎝⎠(14) According to the model of Figure 2, one can see that crackvolume is much smaller than the pore volume, such that the total fluid volume is basically the pore fluid volume, i.e., V f ~V p . For the same reason, the crack fluid volume squeezed into pores is also small, to the extent not to signif-icantly change the pore pressure, such that δP f ~δP p . (Note this condition holds exactly at low frequencies. Using the analysis result of the appendix (eq. (A-11)), it can be shown that c p P P δδ→ in the low-frequency limit, and thus bothδP c and δP p should be equal to δP f ). Based on the above analysis, the first term at the right side of eq. (14) reduces to the compressibility of fluid, 1/K f . The analysis for the se-cond term, which relates to the squirt flow due to cracks, is quite lengthy and involved and is therefore described indetail in the appendix. With the result from the appendix and the relationship between δP and δP f , as given in eq.(10), one can use eq. (14) to obtain the bulk modulus for thefluid-saturated porous rock with cracks. Following the der-ivation of Gassmann eq. (e.g., Chen [31]), the bulk modulus can be written in a form analogous to eq. (7):2()//().d s f K K K K S ααϕϕω⎡⎤=+−++⎣⎦ (15)Compared with the Gassmann result in eq. (7), eq. (15) has an additional term S (ω) due to the contribution from squirt flow, whose functional form is given below:[]118(1)()π()()3114(1) 11().3νωεζζμνζμγ⎡⎤−−=−⎢⎥−⎣⎦−⎧⎫+−⎨⎬⎩⎭d s d fK K S f f K K K f (16) In eq. (16) the frequency variation factor f and its argu-ment ζ are respectively given by102()(),()J f J ζζζζ=ζ=(17) where γ is crack aspect ratio. Note K in eq. (16) is the same as the bulk modulus K in eq. (15). This is rather inconven-ient as eq. (15) becomes an implicit function for K . This inconvenience can be overcome by a first-order approxima-tion in which K is computed using eq. (7) without squirt flow effect. For the same approximation, μ is taken as the dry modulus of eq. (8) and the Poisson’s ratio ν can be computed either for the dry rock case or for the Biot- consistent case given in eq. (8). How K , μ, and ν are evalu-ated for eq. (16) will not significantly change the K value of eq. (15), because the squirt flow term in eq. (16) is con-trolled primarily by two most important fundamental pa-rameters of a cracked medium: crack density ε and aspect ratio γ, as will be discussed in more detail below.Our result shows that: 1) The magnitude of squirt flow effect is directly proportional to crack density. This effect vanishes in the absence of cracks corresponding to ε=0. 2) The S (ω) function due to squirt flow is a complex quantity; thus the bulk modulus K of eq. (15) is also a complex quan-tity. This means that the cracked porous medium system is a dissipative system, which will cause attenuation and disper-sion in an elastic wave passing through the medium. 3)1446Tang X M. Sci China Earth Sci September (2011) Vol.54 No.9Squirt flow varies with frequency following the variation ofthe frequency factor f . At low frequencies (0ω→), 1,f → meaning that the squirting effect attains its full strength because there is sufficient time to squeeze fluid out of carks. At high frequencies (ω→∞),0,f → mean-ing that fluid is locked in the crack because there is no suf-ficient time for the flow movement to take place. The tran-sition from the low- to high-frequency characteristics is controlled by the crack aspect ratio γ. In other words, γ de-termines the fluid relaxation time of the squirt flow.Based on the physical behavior of the squirting mecha-nism, i.e., low-frequency squirting and high-frequency locking of fluid in cracks, Mavko and Jizba [28] show that the dispersion of shear modulus can be derived from the dispersion of bulk modulus. A shear stress applied to the volume element in Figure 2 will produce a normal stress to the crack surface, which in turn will squeeze the crack to generate squirt flow. Assume that the cracks are randomly oriented in the 3D space. Averaging the orientation for the 3D space results in0011411,15K K μμ⎛⎞−=−⎜⎟⎝⎠(18) where the factor 4/15 is the result of the orientation averag-ing, μ and K are respectively the shear and bulk moduliwhen fluid is locked in the crack, which corresponds to ourhigh-frequency scenario; μ0 and K 0 are respectively theshear and bulk moduli when fluid is squeezed out or relaxedfrom the crack, which corresponds to our low-frequencyscenario. The value of μ0 and K 0 can be calculated usingGassmann equation. According to the result of eq. (A-11),c p P P δδ= at low frequencies, which is exactly the pres-sure equilibrium condition required by Gassmann’s theory [27, 31]. We can therefore calculate K 0 using eq. (7), and μ0 using the dry modulus of eq. (8). With this selection of K 0and μ0 and calculating the frequency dispersion of K usingeq. (15), the shear modulus calculated from eq. (18) natu-rally becomes a frequency-dependent modulus that contains the squirt flow effect.By substitution of K and μ from eqs. (15) and (18) into the Biot formulation represented by eq. (1) through eq. (4), and by calculating the dry bulk and shear moduli using eq. (8), we obtain a unified elastic wave theory for the cracked porous medium. We use the following numerical examples to illustrate the important wave propagation characteristics of the new theory.2 Wave attenuation and dispersion characteris-tics of cracked porous mediumIn this section, we calculate the wave attenuation and dis-persion characteristics for porous rock containing cracks using the above described theory and method. The similari-ty and difference between the new theory and DN theory [12] will also be discussed. Table 1 lists the fundamental parameters of the new theory. Compared with Biot theory, the new theory has been added with two important parame-ters of a cracked medium: crack density and aspect ratio. Figures 3 through 5 show the calculation results for the fast compressional, shear, and slow compressonal waves,with the upper and lower figure corresponding to dispersion and attenuation, respectively. The calculated frequency range is from 1 to 107 Hz, plotted on logarithmic scale. The Figure 3(a) shows a clear dispersion effect (solid curve) for the fast compressional wave. For the parameter values used(see Table 1), the amount of dispersion is about 7%. Theshear wave shows a smaller dispersion (Figure 4(a)), mainlybecause of the factor of 4/15 of eq. (18). In fact, the disper-sion effect of both the fast compressional and shear wavescontains the contribution of two mechanisms: the squirtflow effect discussed by the current study, and the globalBiot flow. To demonstrate these two mechanisms, the at-tenuation (dashed curves) due to each mechanism is respec-tively plotted in Figures 3(b) and 4(b). In the calculation of the squirt flow effect using the new theory, medium perme-ability κ0 is set to zero, whereas in the calculation of the Biot flow effect, crack density ε is set to zero. ComparingTable 1 Values of fundamental parameters used in calculating the cracked porous medium elastic-wave theory for Figures 3–7Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 K s (GPa) 37.9 37.9 37.9 37.9 37.9 μs (GPa) 32.6 32.6 32.6 20.1 20.1 ρs (kg m −3) 2650 2650 2650 2650 2650 K f (GPa) 2.25 2.25 2.25 0.1–2.25 0.1–2.25 ρf (kg m −3)1000 1000 1000 1000 1000 ϕp 0.25 0.25 0.25 0.1–0.35 0.1–0.4 κ0 (mD) 0,1000 0, 1000 1000 1000 1000 η (cp) 1, 101, 102, 1031 1 1 1 τ 2.4 2.4 2.4 2.4 2.4 ε 0, 0.15 0, 0.15 0.15 0.15 0 γ0.0010.0010.0010.001–Tang X M. Sci China Earth Sci September (2011) Vol.54 No.9 1447Figure 3Dispersion (a) and attenuation (b) of fast compressional wave in a cracked porous rock. The attenuation (dispersion) includes the squirt- and Biot-mechanisms (dashed curves in (b)). Changing fluid viscosity (or crack aspect ratio) changes the location of fast-varying region in the dispersion curve (dashed curve family in (a)).the two attenuation curves with the total attenuation, one can easily conclude that the total attenuation is almost the sum of the two individual contributions. For the fast com-pressional wave, the attenuation due to Biot flow is much smaller than that due to the squirt flow, which shows that the Biot effect alone is generally inadequate to explain the substantial attenuation and dispersion observed in real rocks, and other effects, like the squirt flow effect under discussion, must be used. For shear wave, the squirt flow effect is about the same order as that of the Biot flow, due to the factor of 4/15 in eq. (18).Let us now discuss the propagation characteristics of slow compressional wave in the presence of squirt flow mechanism. The excitation and propagation of the slow wave in porous media is one of the key results of Biot theo-ry. We compare the slow wave characteristics of the new theory with those of the DN theory and elaborate why the latter theory cannot be regarded as the unified theory for the squirt flow at the local scale and Biot flow on the global scale.We consider the DN theory for the geometry of penny-Figure 4 Dispersion (a) and attenuation (b) of shear wave in a crackedporous rock. The attenuation (dispersion) includes the squirt- and Bi-ot-mechanisms (dashed curves in (b)).shaped cracks. For a single crack of thickness h, fluid-flow-related parameters such as permeability, porosity, etc.can be expressed as21,1,.12fhKκϕβ===The squirt flow length R in DN theory should now be thecrack radius. By using the above parameter expressions andthe definition of crack aspect ratio (eq. (A-12)), the fre-quency variation factor of DN theory in eq. (9) becomes thesame factor as in eq. (17):1112()1.()JJζββζζ−−⎡⎤→−⎢⎣⎦(19)Substituting the above modification of parameter β intothe Biot theory formulas, as given by eqs. (1)–(4), one cancalculate the attenuation and dispersion of the DN theoryfor the crack scenario.Figure 5 shows the comparison of dispersion and at-tenuation of slow compressional wave for the different the-ories. The new theory shows characteristics consistent。