## Abstract

The topic of the previous work of Albers and Wilmanski was the study of monochromatic surface waves at the boundary between a porous medium and a vacuum. This article is an extension of this research to the propagation of surface waves on the interface between a porous halfspace and a fluid halfspace. Results for phase and group velocities and attenuations are shown in dependence on both the frequency and the surface permeability. In contrast to classical papers on surface waves where only the limits of the frequency *ω*→0, *ω*→∞ and the limits of the surface permeability (fully sealed and fully open boundary) were studied, we investigate the problem in the full range of both parameters. For the analysis we use the ‘simple mixture model’ which is a simplification of the classical Biot model for poroelastic media. The construction of a solution is shown and the dispersion relation solved numerically. There exist three surface waves for this boundary: a leaky Rayleigh wave and both a true and a leaky Stoneley wave. The true Stoneley wave exists only in a limited range of the surface permeability.

## 1. Introduction

This paper is a continuation of the work of Albers & Wilmanski (2005*a*) on the numerical analysis of surface waves in poroelastic media. In the previous paper, the frequency dependence of monochromatic surface waves at the boundary of a poroelastic halfspace and vacuum was shown. Whereas at a plane boundary of a homogeneous linear elastic material solely a true Rayleigh wave appears, at the boundary of the porous medium with vacuum this wave becomes leaky and additionally a second surface wave, a true Stoneley wave, emerges.

A true surface wave propagates along the direction parallel to the surface and decays exponentially with depth, while leaky surface waves are attenuated in the surface direction and radiate energy into bulk or other surface waves. It is customary in the literature on waves in porous materials to call leaky surface waves pseudosurface waves (e.g. Feng & Johnson (1983), Chao *et al*. (2004)), i.e. a *leaky surface wave is identical with a pseudosurface wave*. In this article, we prefer to call these waves *leaky surface waves* (as e.g. done in Schroeder & Scott (2001)).

In the present article the boundary between a porous medium and an ideal fluid is investigated. This means that we have one additional component compared to the boundary porous medium/vacuum. Thus, besides the three bulk waves in the porous medium (fast longitudinal wave *P*1, slow longitudinal wave *P*2 and transversal wave *S*) there exists also a *P*-wave in the fluid. These four waves combine into three surface waves: a leaky Rayleigh wave and both a true and a leaky Stoneley wave. Their acoustic properties (phase and group velocities, attenuations) will be shown in this paper in dependence on two quantities: the frequency *ω* and the surface permeability parameter *α*.

In contrast to earlier works in this field, we investigate the acoustic properties of surface waves in the whole range of both parameters. Only in the work of Gubaidullin *et al*. (2004) were these properties analysed for all frequencies but only for the two limit values of the surface permeability *α* (1/*T* in their notation). In spite of the use of the full Biot model and another analytical method of investigation results presented in their work coincide with our results for the limit values of *α*.

The investigation of the full range of frequencies reveals that the transition from low frequencies to high frequencies is not monotonous or even smooth. As intermediate frequencies are essential for practical purposes such an investigation becomes of paramount importance. Practical ranges may vary from 100 MHz in the electronics industry (e.g. the testing of surface coatings by nanomaterials) to 1–100 Hz for field testing of soils and rocks in geophysics. Intermediate frequencies appear as well, e.g. in civil engineering and medicine.

Apart from the frequency range, properties of the boundary described by the surface permeability are also investigated in the present work. The variation of this second parameter, *α*, which controls the intensity of the in- and outflow of the fluid from the porous medium, brought to light that the true Stoneley wave exists only for very small values of this parameter, i.e. for a boundary which is almost sealed. Attenuations of both leaky waves show an interesting behaviour in dependence on the frequency: for two frequencies there appear resonance effects. They seem to be not only theoretically but also experimentally observed in Chao *et al*. (2004) and may be related to characteristic frequencies of the solid and the fluid, respectively.

## 2. Theoretical background

### (a) Biot's model

The most popular and widespread model for the theoretical description of linear processes in fluid-saturated poroelastic media is the *Biot model* (see the 1941 and 1956 papers of M. A. Biot which can be found in Tolstoy (1992)). It is based on the following momentum balances(2.1)where **v**^{F}, **v**^{S} are the macroscopic (average) partial velocities of the fluid and of the skeleton, respectively, *e*^{S} denotes the symmetric Almansi–Hamel tensor of small deformations of the skeleton, *ε* is the volume change of the fluid. They fulfil the following relations(2.2)where *ζ* denotes the change of fluid contents commonly used as a field in Biot's model instead of *ε*. If we introduce partial displacement fields **u**^{S}≡* u*,

**u**^{F}≡

*then(2.3)The quantities , denote constant (initial) mass densities, are the initial true mass densities,*

**U***n*

_{0}is the initial porosity. The material parameters

*λ*

^{S},

*μ*

^{S},

*κ*,

*Q*,

*π*,

*ρ*

_{12}are assumed to be constant. The parameter

*Q*describes the coupling of partial stresses in the Biot's model, while

*ρ*

_{12}is attributed to the tortuosity

*a*. The permeability coefficient

*π*corresponds to the classical Darcy coefficient and

*κ*is the compressibility of the fluid.

The problem of the frequency dependence for intermediate frequencies has for a long time been ignored in the literature, mostly due to numerical difficulties within Biot's model. One of the first attempts, e.g. to investigate surface waves in two-component porous materials stems from Deresiewicz (1962). In this paper, he studied the boundary porous medium/vacuum using the Biot equations. He numerically calculated Rayleigh wave velocities and attenuations. But he wrote:Because of its complexity, the secular (i.e. dispersion; B.A.) equation does not lend itself to analytical study for intermediate values of the frequency. Accordingly, a numerical study was undertaken, of the variation of velocity and dissipation per cycle with frequency, for a material whose elastic and dynamical coefficients were available, with several curious results.

One of these ‘curious’ results is a minimum of the phase velocity in the region of small frequencies which we also found for the Rayleigh wave (Albers & Wilmanski 2005*a*). It is impressive how much the author already knew about the Rayleigh wave in the 1960s. However, it is strange that he did not get the Stoneley solution from the dispersion equation.

Feng & Johnson (1983) use Biot's theory to search numerically primarily the velocities of various surface waves. They distinguish between the true slow surface wave and the leaky surface waves and calculate their velocities at an interface between a fluid half-space and a half space of a fluid-saturated porous medium, as we do as well in the sequel. However, they rely on a somewhat different approach to obtain results for the surface waves. In contrast to our approach, Feng and Johnson focus only on the high-frequency range and they limit their attention to the open-pore and to the sealed pore situation (i.e. in our notation: *α*=∞ and *α*=0, respectively, and in their notation: *T*=0 and *T*=∞). In dependence on the stiffness of the skeleton (in their notation: longitudinal modulus of the skeleton frame , where *K*_{b}: bulk modulus of porous drained solid, *N*: shear modulus of the drained porous solid) and e.g. on the coefficient of added mass (‘tortuosity’, in their notation: *α*) they have investigated the existence of the surface modes. Their numerical results for material parameters for water as the fluid (in both *z*<0 and *z*>0 regions) and fused glass beads as the porous medium agree with our results. Otherwise the ranges of investigation of Feng and Johnson and ours do not coincide. In order to compare the results we have calculated the stiffness of the sandstone of our example in their notation. For the first set of parameters , for the other one . Feng and Johnson investigate a range . They found that for an open-pore surface situation, the true surface wave exists for a limited range of material parameters and changes continuously into a slightly leaky Stoneley wave as its velocity crosses over the slowest bulk wave velocity. For the sealed pore situation there exist simultaneously a true surface wave (for all values of material parameters) and a leaky Stoneley wave.

### (b) Simple mixture model

In our analysis we rely on a simpler model than that of Biot. We neglect two effects:

the added mass effect reflected in Biot's model by off-diagonal contributions to the matrix of partial mass densities (the parameter

*ρ*_{12}),the static coupling effect between partial stresses (the parameter

*Q*).

The first one is neglected because it yields a non-objectivity of Biot's equations and, additionally, an unphysical dependence of the attenuation on the tortuosity (see Wilmanski 2001, 2005*a*).

The second contribution, the coupling of partial stresses *Q*, is neglected because it yields only *quantitative* corrections without changing the *qualitative* behaviour of the system, at least in the range of a relatively high stiffness of the skeleton in comparison with the fluid. This has been analysed for bulk waves in Albers & Wilmanski (2005*b*).

Bearing these remarks in mind it seems to be appropriate to rely on the simplified model (‘simple mixture model’ in which *Q*=0, *ρ*_{12}=0). It has the advantage of reducing essentially technical difficulties. We present here the linear form of the ‘simple mixture’ model for a two-component poroelastic saturated medium (for details see Wilmanski 1999).

As it is done in the Biot model, in the simple mixture model, a process is described by the *macroscopic* fields *ρ*^{F}(*x*, *t*), the partial mass density of the fluid, **v**^{F}(*x*, *t*), **v**^{S}(*x*, *t*), the velocities of the fluid and of the skeleton, respectively, *e*^{S}(*x*, *t*), the symmetric tensor of small deformations of the skeleton and *n*, the porosity. The following set of linear equations is satisfied by those fields:(2.4)We have already introduced , which are the constant reference values of partial mass densities and porosity, respectively. There also appear constant material parameters *κ*, *λ*^{S}, *μ*^{S}, *β*, *π*, *τ*_{n}, *δ*, *Φ*. In particular, *κ* denotes the macroscopic compressibility of the fluid component, *λ*^{S} and *μ*^{S} are macroscopic elastic constants of the skeleton, *β* is the coupling constant between the components, *π* denotes the bulk permeability coefficient, *τ*_{n} describes the relaxation time of porosity and *δ*, *Φ* are equilibrium and non-equilibrium changes of porosity, respectively. As we have already done in the work Albers & Wilmanski (2005*a*) for the analysis of surface waves, the coupling parameter *β* is assumed to be zero. Then the problem of evolution of porosity described by equation (2.4)_{5} can be solved separately from the rest of the problem. We investigate the approximation of a very large relaxation time of porosity (*τ*_{n}→∞) compared to the inverse of characteristic frequencies. This is justified in applications to soils where viscous effects related to porosity do not seem to appear. Then the non-equilibrium porosity *n* is determined by volume changes of the skeleton and of the fluid: . Consequently, they do not give any independent contribution to the wave problem.

### (c) Boundary conditions

For the determination of surface waves in saturated poroelastic media conditions for *z*=0 are needed. We are going to consider the interface between a saturated porous material and an ideal fluid. Boundary conditions for such an interface were formulated by Deresiewicz & Skalak (1963). Here we have slightly changed their notation. The quantities outside of the porous medium are denoted by a sub- or superscript ‘+’ sign. Then the boundary conditions have the following form:(2.5)Here , are *x*-, and *z*-components of the displacement **u**^{S}, respectively, and , denote *z*-components of the displacements **u**^{F} and **u**^{F+}, respectively.

The total stress tensor * T* is the sum of the partial stress tensors of the skeleton, , and of the fluid, . The stress tensor of the fluid outside the porous medium is given as . and are reference values of the pore pressure and the pressure in the fluid outside.

*ρ*

^{F+}denotes the partial mass density of the fluid in the +-region and is its constant reference value.

*κ*

^{+}describes the compressibility of the fluid outside the porous medium.

Simultaneously, , , , , are squares of the front velocities of the bulk waves in the porous material: *P*1 (fast wave), *S* (shear wave), *P*2 (slow wave, also called Biot's wave), and of the *P*-wave in the fluid, respectively. In the case of Biot's model there would be an additive contribution in the numerator of *c*_{P1} of the coupling parameter *Q* which is of the order of a few percent of *λ*^{S} (see Albers & Wilmanski (2005*b*) for a detailed analysis).

Two of the boundary conditions, (2.5)_{1} and (2.5)_{2}, describe the continuity of the full traction, , on the boundary; (2.5)_{3} reflects the continuity of the fluid mass flux, and condition (2.5)_{4} specifies the mass transport through the surface. The difference of the pore pressures on both sides of the boundary determines the in- and outflow through the boundary. *α* denotes a surface permeability coefficient, which corresponds to 1/*T* in the works Feng & Johnson (1983), Gubaidullin *et al*. (2004) and *p*^{F+} is the external pressure. Condition (2.5)_{4} relies on the assumption that the pore pressure *p* and the partial pressure *p*^{F} satisfy the relation at least in a small vicinity of the surface.

Some words are appropriate to explain the notion of *surface permeability*. The parameter represents surface effects. The flow of the fluid is ‘straight’ both outside the porous medium and inside the channels of the porous medium. However, at the entrance to the porous medium the flow is disturbed: the fluid has to find its way through the voids between the solid particles. In principle we consider a flow which is perpendicular to the boundary but in a small boundary layer it deviates from this direction. A detailed structure of the boundary determines the resistance against the inflow and, consequently, the surface permeability *α*. The value *α*=0 corresponds to a sealed pore situation, for *α*=∞ the boundary is completely open.

## 3. Construction of the solution

### (a) Introduction

Investigating the interface between a porous medium and a liquid we have to consider additional equations for the liquid outside of the porous medium. We distinguish this part of the system by the sign ‘+’.

The procedure for the construction of the solutions which is used in this work has already been applied in earlier works (e.g. Albers & Wilmanski 2005*a*; Edelman & Wilmanski 2002; Wilmanski 2002 or Wilmanski & Albers 2003). While in Edelman & Wilmanski (2002) an asymptotic analysis of the high-frequency properties of surface waves in function of the wavelength 1/*k* (*k*—wave-number) within the simple mixture model was carried out, here and in the other papers, we consider monochromatic waves with a given *real frequency ω*. This is necessary for the investigation of a far-field boundary value problem with a harmonic surface source of waves.

The additional equations for the fluid in the exterior of the porous material read(3.1)Here *ρ*^{F+} denotes the partial mass density of the fluid in the +-region and is its constant reference value. *κ*^{+} describes the compressibility of the fluid.

*Comment*. The identification of compressibility coefficients in porous materials can be done in many ways and they do not necessarily give the same results. To make this issue clear, we present here two approaches.

One of them is commonly used in micro–macro transitions for granular materials. Such a transition is described by relations *ρ*^{F}=*nρ*^{F+}, *p*^{F}=*np*^{F+}, , , and, under the condition of constant porosity *n*=*n*_{0}, we have *κ*=*κ*^{+}. However, a simple wave analysis shows that this relation cannot hold for dynamical processes as the speeds of waves carried by the fluid component are, respectively, , , and these are, of course, different. The situation does not improve essentially if we introduce Biot's coupling. This means that linear models used in the description of waves cannot be based on the above quoted simple micro–macro relations. They possess their own macroscopic status and their effective material parameters account for such microscopic effects as scattering of waves. Different macroscopic compressibilities for the same material inside and outside of the porous medium are caused by the channels of the porous medium. While a wave can take a free ‘path’ in the fluid outside, inside the porous medium waves are slower due to reflection on the boundaries of the channels. This is, certainly, not present in simple micro–macro transitions.

We assume the fluid to be water. A wave in water travels with the velocity of around *c*_{+}≈1500 m s^{−1} and the corresponding compressibility *κ*^{+}=2.25×10^{6} m^{2} s^{−2}. Inside the porous medium the velocity of the *P*2-wave is assumed to be *c*_{P2}=1000 m s^{−1} to which corresponds a compressibility *κ*=*c*^{2}_{P2}=1×10^{6} m^{2} s^{−2}.

### (b) Compatibility with field equations

We introduce the displacement vectors **u**^{S}, **u**^{F} and **u**^{F+} for the skeleton, for the fluid inside the porous medium and for the fluid outside the porous medium, respectively. Both latter quantities do not have any direct physical bearing. They are introduced only for the technical symmetry of considerations. According to Helmholtz's theorem we have then(3.2)The following solutions for monochromatic waves in the *x*-direction are considered for the two-dimensional case:(3.3)Here, *A*^{S}, *A*^{F}, *B*^{S}, *B*^{F}, *A*^{F+}, and are complex amplitudes of the corresponding fields. As is customary i denotes the imaginary number.

Relations (3.3) are substituted in the field equations of the simple mixture model (equation (2.4) without balance of porosity and equation (3.1)). This leads to the following compatibility conditions:(3.4)

### (c) Dimensionless notation

A dimensionless notation has the advantage to connect characteristics of the surface waves to those of the better known bulk waves (e.g. with the velocity of the *P*1-wave *c*_{P1}). To this aim some dimensionless quantities are defined as follows:(3.5)Here *τ* is an arbitrary reference time. It may be chosen as which would lead to *π*′=1 or it may be identical with the relaxation time of porosity *τ*_{n}. Further, we make an arbitrary choice of this normalization parameter.

### (d) Ansatz

For simplicity we further omit the prime in equation (3.5). Substitution of these quantities in equations (3.4) yields(3.6)The coefficients *A*^{F}, *A*^{S}, *B*^{S} and *A*^{F+} are for homogeneous materials independent of *z*. This leads, in contrast to heterogeneous media where they depend on *z*, to a differential eigenvalue problem which can be easily solved. Consequently, we seek solutions in the form(3.7)In these relations the exponents *γ*_{1}, *γ*_{2}, *ζ* must possess negative real parts and *γ*^{+} must possess a positive real part to describe a surface wave. These properties result from Sommerfeld conditions at infinity (*z*→∞).

Substitution in equation (3.6) yields relations for the exponents: *ζ*_{1,2}/*k* follow from(3.8)*γ*_{1,2,3,4}/*k* result as solutions of the relation(3.9)and as solutions of(3.10)Simultaneously, we obtain relations for the eigenvectors(3.11)where(3.12)with(3.13)However, we still have four unknown constants *B*_{s}, *A*_{f}^{2}, *A*_{s}^{1}, *A*_{f}^{+} which have to be determined by the boundary conditions.

Complex values of *ζ*_{1,2}, *γ*_{1,2,3,4} and result from the dissipation caused by the relative motion, i.e. the influence of the permeability *π*. As a consequence, solutions decay in *z*-direction but, simultaneously, they vibrate.

### (e) Insertion into boundary conditions

Using the constitutive relations(3.14)and assuming that(3.15)which means that the initial external pressure is equal to the initial pore pressure, we insert equations (3.2)–(3.4) into the boundary conditions (2.5) and obtain in dimensionless form (again omitting the primes)(3.16)where we have used the relation if .

### (f) Dispersion relation

Inserting the ansatz for the solutions (3.7) into these boundary conditions we obtain the following problem for the four unknown constants *B*_{s}, *A*_{s}^{1}, *A*_{f}^{2}, *A*_{f}^{+}:(3.17)where(3.18)(3.19)This homogeneous set yields the *dispersion relation:* determining the *ω*−*k* relation.

### (g) High- and low-frequency approximations

*High frequencies*. For high frequencies (physical dimensions) we have *δ*_{s}=*δ*_{f}=0 and the exponents and . The dispersion relation follows in the form(3.20)where(3.21)This is the classical Rayleigh dispersion relation. We consider in the following two cases of the dispersion relation (3.20).

*α*=0 (impermeable boundary; i.e. sealed porous medium in contact with an external fluid)We get from equation (3.20)(3.22)The case

*α*=0 does not correspond to the case of the boundary porous medium/vacuum because the fluid outside of the porous medium yields a pressure on the boundary. However, if additionally to*α*=0 also*r*^{+}=0 we have the same conditions and equation (3.22) must be identical to the Rayleigh dispersion relation for the boundary porous medium/vacuum (e.g. Wilmanski & Albers 2003)(3.23)This is, indeed, true if we cancel*γ*^{+}/*k*on both sides after setting*r*^{+}equal to zero.*α*→∞Here, the other two terms of equation (3.20) remain and we obtain after division by

*α*(3.24)

Both equations (3.22) and (3.24) remind equation (3.23) but they are both modified by the influence of the fluid outside.

*Low frequencies*. In contrast to the impermeable boundary with vacuum, analytical calculations in this case become very complicated. Therefore, we investigate this case solely on a numerical example.

## 4. Numerical results

### (a) Procedure and parameters

The problem has been solved for the complex wave number, *k*, using the two computing packages Maple 7 and Maple V Release 5.1. In principle, it is possible to use the existing equation solvers although, for the calculations with complex variables, one has to perform a very careful manual bookkeeping and, in addition, they need a very extensive main memory. This makes the numerical evaluation far from being trivial. It has been observed that the used packages calculate only *one of the solutions* for *k* for any choice of sign combinations of exponents *γ*_{1}, *γ*_{2}, *γ*^{+} and *ζ* changing between branches of solution by the variation of exponents without any apparent reason. Interestingly, the programs for some sign combination of exponents produce two solutions which are close to each other but, for most frequencies, not the same.

Inspection of the above solutions shows that they contain roots of quantities which are either already complex or may change the sign. As is well known, this means that there exist many Riemann surfaces. It has already been seen in the case of classical Rayleigh waves (Schroeder & Scott 2001; Wilmanski 2005*b*) that in order to obtain a true surface wave one has to choose a proper Riemann plane, otherwise one obtains leaky surface waves. This is, of course, also visible in the numerical procedure which yields solutions on both Riemann surfaces which are related to the true surface wave as well as to solutions for leaky waves. An additional problem arises due to the fact that complex solutions in the present case follow not only from the choice of the Riemann surface but also from the attenuation through dissipation. Consequently, a numerical analysis has to be done with a particular care.

From the complex results for *k* we are able to determine the normalized velocities of the surface waves , and the corresponding normalized attenuations Im *k*_{i}.

The calculations have been performed for the following data which correspond to water saturated sandstones(4.1)While some of the results for the boundary porous medium/vacuum have been shown for the varying bulk permeability parameter, *π*, this coefficient is constant here, namely *π*=10^{7} kg m^{−3} s^{−1}.1 However, instead of this we show for this boundary the influence of the surface permeability, *α*. We demonstrate that two surface waves, both of them leaky, exist in the whole range of frequencies for each choice of *α*. They correspond to the classical Rayleigh wave, and to a Stoneley wave which is produced due to presence of the fluid outside the porous medium, respectively. The latter can be supported by the fact that it also appears for the sealed pore situation if there is a fluid outside. Moreover we obtain a third type of wave which also appeared on the boundary porous medium/vacuum, namely a true Stoneley wave. However, this wave exists only for small values of *α*.

Numerical results for velocities are normalized by the *P*1-velocity. Imaginary parts of the wavenumber *k* determine the damping of waves. It is normalized by the product with the *P*1-velocity and the relaxation time (see equation (3.5)). This means, for our parameters, that the values presented in the figures are 400 times smaller ((2500×10^{−6})^{−1}) than in real physical units (m^{−1}).

### (b) Dependence of phase velocities and attenuations on the frequency

To have an impression of the results of the simpler problem of the boundary between a porous medium and a vacuum and in order to compare them to the results on the present problem figure 1 is included. It shows the normalized velocities and attenuations of both the leaky Rayleigh wave and the true Stoneley wave which appear at this boundary.

Figure 2 shows both the phase velocities and the attenuations of all three surface waves appearing at the interface between a porous halfspace and a fluid halfspace. On the left-hand side the velocities are plotted and figures on the right-hand side show the attenuations. Both quantities are given for a wide range of frequencies between 1 Hz and 100 MHz. The different curves correspond to various values of the surface permeability parameter *α*. As explained above, *α=0* means that the surface is completely impermeable while *α*=∞ corresponds to an open pore situation. Both the frequency and the attenuations are shown in logarithmic scale while the velocity is presented in normal scale.

#### (i) Phase velocities

The velocity of the Rayleigh wave is smaller than the velocity of the bulk shear wave, *c*_{S}, whose normalized value is . Curves for different values of *α* each have a low and a high-frequency limit which is unequal to zero. While for small frequencies the velocity is the same independently of *α*, the high-frequency limit decreases with increasing *α*. For the open pore situation the difference between high and low-frequency limits is approximately one half of the difference for a close boundary. In between there is a steep increase in the Rayleigh velocity. The range of frequencies for this increase is smaller for small values of *α* (10–100 kHz) than for large values of *α* (1–500 kHz). Moreover, for the latter, there appears a small plateau in this zone. This may be an indication of the change of the Riemann surface which is, however, much better pronounced in the attenuations.

The velocity of the leaky Stoneley wave behaves similar to the Rayleigh wave. Also this wave possesses high and low-frequency limits unequal to zero and the steep increase in between appears in the same frequency region. However, in contrast to the Rayleigh wave, for this wave the high-frequency limit is larger for bigger values of *α* than for smaller ones. The frequency behaviour of this wave is—at least for small *α*—not monotonous. A maximum value appears in the region of order 100 kHz. Interestingly, the velocity of this wave is smaller than that of the Rayleigh wave although it is driven by the fluid outside of the porous medium whose longitudinal bulk wave is faster than that of the fluid inside the porous medium (*c*_{f}^{+}=0.6, *c*_{f}=0.4). This result has been obtained also by Feng & Johnson (1983) within Biot's model. As we will see in §4*c* the true Stoneley wave exists only for small values of *α*, and, therefore, we show its behaviour only for two values of *α*. For these the velocities do not differ substantially. They start form zero at *ω*=0 and increase until around 100 kHz where they reach a high-frequency limit which is a little bit smaller than the velocity of the *P*2 wave, .

#### (ii) Attenuations

Let us start with the attenuation of the true Stoneley wave. This has the same appearance as that obtained for the boundary porous medium/vacuum (see figure 1). We show a log–log-plot of this attenuation. It starts from zero as *ω*=0 and reaches a horizontal asymptote at around 100 kHz. The only amazing point is that in the mapped region of frequencies, starting from 1 Hz the value for *α*=10^{−4} s m^{−1} is much smaller than for *α*=0 even though the difference in velocities is small. It is surprising because, for the other attenuations, the value for *α*=0 is smaller than all other values for different *α*.

In contrast to the true surface wave the remaining leaky waves possess singularities in the attenuation for two intermediate frequencies. These frequencies seem to be related to characteristic frequencies and which have already appeared in the stability analysis of adsorption processes (Albers 2003). However, there exists an influence of the parameter *α*, responsible for dissipation, and, simultaneously, the location of the singularities changes with the variation of this coefficient. Consequently, as also indicated in some papers on Biot's model, the diffusion-driven resonances also appear in the surface waves. Their existence seems to be confirmed experimentally (for results of measurements see, in particular, Chao *et al*. (2004), Wisse *et al*. (2002)). We reproduce one figure from the paper Chao *et al*. (2004) (figure 3). It shows the measured damping coefficients of the pseudo-Stoneley wave (leaky Stoneley wave in our terminology) in a shock-induced borehole experiment. The formation is a Berea sandstone. The damping coefficient is given in a frequency range up to 50 kHz and we see that there also appear some well pronounced singularities. Little is known about their mathematical origin. However, the numerical analysis indicates that they appear due to the change of the Riemann surface. If one ignores the singularities and looks only at the connecting line between the maxima, the curves look similar to the curve for the boundary porous medium/vacuum. In any case, it is obvious that the curves show the leaky character as already observed for the Rayleigh wave for this boundary: for high frequencies the attenuation grows linearly and unbounded.

### (c) Dependence of phase velocities and attenuations on the surface permeability

In figure 4 we illustrate the behaviour of the three surface waves appearing at the boundary porous medium/fluid for the frequency limits *ω*→∞, *ω*→0 but in dependence on the surface permeability parameter *α*. In the first row of figure 4 we show normalized velocities and attenuations for the same material parameters as used in the last subsection. In the bottom row the same quantities are shown for another choice of material parameters: with unchanged mass densities and porosity and different velocities for two bulk waves (*c*_{P1}=3500 m s^{−1}, *c*_{S}=1750 m s^{−1}, *c*_{P2}, *c*_{P+} unchanged).

The upper left figure shows the phase velocities of the surface waves for the two limits of frequencies (*ω*→∞—solid lines, *ω*→0—dashed lines). Additional to the shown range of surface permeability parameters between 10^{−6} s m^{−1} and 10^{−1} s m^{−1}, we show on the left and right-hand side points corresponding to the limit cases *α*=0 and *α*=∞. It is obvious that the values for 10^{−6} and 10^{−1} already correspond well to the limit values. While both leaky surface waves exist in the whole range of surface permeabilities, the true surface wave appears only for small values of the surface permeability (approximately in the interval 0≤*α*≤10^{−3.9} s m^{−1}). Thus for a relatively open boundary (large *α*) no real surface wave exists.

It is obvious that for each wave the low-frequency value of the velocity (dashed lines) is independent of *α*. For the true surface wave this value is zero while it is bigger than zero for the leaky waves. Both the high-frequency limit of the leaky Rayleigh and of the leaky Stoneley wave change monotonously from the limit α=0 to the limit *α*=∞. However, the Rayleigh wave velocity is bigger for a dense boundary, while for the leaky Stoneley wave the limit for the open boundary is bigger. The velocity behaviour of the waves for the other choice of material parameters (*c*_{S}>*c*_{f+}) does not change substantially. Again, *c*_{R}<*c*_{S}, *c*_{St}<*c*_{f} and *c*_{leakySt}<*c*_{f+}. The latter is a little bit more obvious for this choice of material parameters than for the other choice. As shown above the last choice corresponds to a higher value of the modulus of the skeleton frame (5.53 instead of 3.95). For this stiffer medium the true Stoneley wave exists only for a still denser boundary, namely for approximately 0≤*α*≤10^{−4.8} s m^{−1}).

The figures on the right-hand side show the normalized attenuation of the three surface waves for a chosen frequency. As already mentioned, the true surface wave—the Stoneley wave—ceases to exist in the range of high surface permeabilities *α*. In the limit value, its attenuation becomes infinite. The remaining two leaky waves possess finite attenuation in the whole range of *α*.

### (d) Group velocities of the three surface waves

The figures for phase velocities of Rayleigh and Stoneley waves show that both of them depend on the frequency *ω*. In inhomogeneous media, waves of different frequency (or wavelength) in general propagate with different phase velocities. This phenomenon is known as *dispersion*. The dispersion in heterogeneous materials appears in a non-dissipative manner which is not the case in systems with diffusion. It is easy to see that dissipative waves considered in this article become non-dispersive in the non-dissipative limit *π*→0.

A monochromatic wave, as investigated in this section, is an idealization which is never strictly realized in nature. Most sources emit signals with a continuous spectrum over a limited frequency band. The group velocity *c*_{g} (for details in the case of real *k* see Aki & Richards 1980) for a given frequency *ω* is the velocity of transport of a wave package consisting of contributions from a band of frequencies around *ω*. Then, accounting for the fact that the real wavelength *k* and the phase velocity *c*_{ph} depend on the frequency (4.2)However, in our case the wavenumber is complex and a relation similar to (4.2) follows under some simplifying assumptions. We consider the wave consisting of a narrow band of frequencies near the middle frequency *ω*_{0}. The solution for the amplitude *A* can be described by a Fourier integral which accounts for all frequencies entering the band(4.3)Under the assumptions of small changes of the amplitude and small changes of damping *k*_{I} in the range , this is approximately(4.4)withFor the impermeable boundary (*α*=0) between a porous material and a fluid we present in figure 5 the group velocities of the surface waves. The group velocity of the leaky Stoneley wave behaves differently from both waves which also appeared for the boundary porous medium/vacuum. While they only possess a maximum for a certain frequency, the leaky Stoneley wave exhibits first a strong minimum and then a slight maximum. The mathematical reason is obvious: the growth in phase velocities of Rayleigh and Stoneley waves is almost monotonous but for the leaky Stoneley wave the phase velocity possesses a clear maximum (compare figure 2), and, as indicated in equation (4.4) the group velocity depends on the slope of the phase velocity.

It is clearly seen that the maximum group velocities exceed the corresponding limit values of the bulk waves. This indicates that surface waves cannot exist in the vicinity of the maximum point. Consequently, if we account for the resonance behaviour indicated in figure 2, the range of frequencies between two resonances is most likely forbidden for the propagation of surface waves.

## 5. Summary of results and conclusions

In the whole range of frequencies there exist two leaky surface waves: a leaky Rayleigh wave and a leaky Stoneley wave. In the range of very small values of the surface permeability parameter *α*, there exists a third true surface wave—a Stoneley wave. They possess the following attributes:

### (a) Leaky Rayleigh

The velocity of propagation of this wave lies in the interval determined by the limits

*ω*→0 and*ω*→∞. The high frequency limit is higher than the low-frequency limit. The velocity is always smaller than*c*_{S}, i.e. slower than the*S*-wave. As a function of*ω*it possesses at least one inflection point.For low frequencies the phase velocity for different values of the surface permeability

*α*remains almost constant. For high frequencies smaller values of*α*yield bigger velocities; for the open pore case the difference between high and low-frequency limits is approximately one half of the difference for a close boundary.The attenuation grows linearly and unbounded (the feature of a leaky wave), there appear singularities which depend on

*α*and seem to be related to the characteristic frequencies and .

### (b) Leaky Stoneley

The phase velocity of this wave behaves similarly to that of the leaky Rayleigh wave; however, the high-frequency limit is larger for bigger values of

*α*than for smaller ones; a maximum value appears in the region of order 100 kHz; the velocity of the leaky Stoneley wave is for each pair (*ω*,*α*) smaller than this of the leaky Rayleigh wave.Also the attenuation behaves similar to that of the leaky Rayleigh wave; however, the singularities are weaker dependent on

*α*.

### (c) True Stoneley

It exists only for small values of the surface permeability

*α*; for different values of*α*the velocity is nearly the same; it grows monotonically from the zero value for*ω*=0 to a finite limit which is slightly smaller (approximately 0.15%) than the velocity*c*_{P2}of the*P*2 -wave. The growth of the velocity of this wave in the range of low frequencies is much steeper than that of Rayleigh waves similarly to the growth of the*P*2-velocity.Both the velocity and attenuation of the true Stoneley wave approach zero as (which is not directly obvious due to the logarithmic scale of the figures).

The attenuation of the Stoneley wave grows monotonically to a finite limit for

*ω*→∞. It is slightly smaller than the attenuation of*P*2-waves.

The above described results show that the behaviour of true and leaky surface waves at intermediate frequencies is far from being trivial and cannot be ignored. Simultaneously, a rather sophisticated numerical analysis which yields this conclusion demonstrates that, at least at the present stage of evolution of numerical methods, the analysis of the full Biot model is technically almost impossible.

Let us add that neither finite element nor boundary element calculations can be applied to this analysis. The first one is not appropriate in the case of infinite problems (Sommerfeld conditions) and the second one requires the knowledge of the dynamical Green's function which is known only for some simplified problems (*π*≈0—no diffusion, and, consequently, no attenuation due to dissipation).

## Footnotes

This article is dedicated to my friend and teacher Prof. Krzysztof Wilmanski on the occasion of his 65th anniversary.

↵For instance, for water saturated sands

*K*∼10^{−2}÷10^{−3}m s^{−1},*p*∼10^{3}kg m^{−3}and*g*∼10 m s^{−2}⇒*π*∼10^{6}÷10^{7}kg m^{−3}s^{−1}(e.g. Bear 1988). In standard units of permeability this corresponds to approx. 1÷0.1 darcy.- Received February 18, 2005.
- Accepted September 12, 2005.

- © 2005 The Royal Society