## Abstract

The aim of this work is twofold. We show the construction of an objective relative acceleration for a two-component mixture and prove that its incorporation in the momentum source requires additional terms in partial stresses and in the energy. This may be interpreted as an influence of tortuosity, in the theory of saturated poroelastic materials, and a connection of tortuosity with fluctuations of the kinetic energy on a mesoscopic level of observation. The linearization of such a model yields Biot's equations, used in poroacoustics.

We demonstrate as well, that results for the propagation of acoustic waves in saturated poroelastic media, are qualitatively similar for Biot's model, and for the simple mixture model, in which both the tortuosity and the Biot's coupling between partial stresses are neglected. It is also indicated that the coupling constant of Biot's model obtained by means of the Gassmann relation may be too large, as it leads to very small differences in the speed of propagation of the P1-wave for small and large frequencies, which contradicts the data for soils.

## 1. Introduction

A celebrated property of Biot's model of two-component porous materials is related to equations of motion containing a contribution of the relative acceleration. Linear equations describing such a model in a chosen inertial frame of reference have the following form (e.g. Biot 1956; Tolstoy 1991):(1.1)where(1.2)and *e*^{S} denotes the macroscopical Almansi–Hamel *deformation tensor* of the skeleton, its trace, tr *e*^{S} is the *volume change* (small deformations) of the skeleton, *ϵ* is the *volume change* of the fluid, and this is related to the *increment of fluid content*, *ζ*, by the relation (1.2)_{3}. The current partial density of the fluid component is denoted by *ρ*^{F}. *ρ*_{0}^{S} and *ρ*_{0}^{F} are the constant initial partial mass densities connected to the true mass densities, *ρ*_{0}^{SR}, *ρ*_{0}^{FR}, in the following way:(1.3)where *n*_{0} is the initial porosity. The *macroscopic velocities* of both components are *v*^{S} and *v*^{F}, i.e. *v*^{F}−*v*^{S} is the seepage velocity. The material parameters *λ*^{S}, *μ*^{S}, *κ*, *Q*, *π* and *ρ*_{12}, are constant.

The literature on Biot's model is far from being unanimous in relation to the notation, and this creates considerable confusion. The above material parameters, which we use further in this work, are characteristic for the formulation of a two-component mixture. Usually, in soil mechanics, use is being made of the total bulk stress **T**=**T**^{S}+**T**^{F}, and the fluid partial stress is related solely to the pore pressure *p*. Namely, **T**^{F}=−*n*_{0}*p***1**.

For this reason the material parameters are introduced, for instance, in the following way (Stoll 1989):(1.4)On the other hand, in the standard reference book on linear acoustics of porous materials (Bourbie *et al*. 1987) the following form of the set (1.1) is used:(1.5)with the following relations among parameters,(1.6)Still another set of parameters is used by Allard (1993), where *ρ*_{12}=−*ρ*_{a}, *ρ*_{0}^{FR}=*ρ*_{0}, *ρ*_{0}^{S}=*ρ*_{1}, etc.

Let us return to the set (1.1). The parameter *ρ*_{12}, describing the contribution of the relative acceleration, is usually related to the *tortuosity* of the porous material. For example, in the works Berryman (1980) and Johnson *et al*. (1987), the following approximate relation between this parameter, the porosity *n*_{0}, and the tortuosity parameter *a*∈[1, ∞), is proposed:(1.7)

It is easy to show that the model, equation (1.1), is nonobjective. This means that the change of the reference frame to a noninertial system (a time dependent change of observer),(1.8)yields constitutive contributions in these equations following from the presence of the relative acceleration. These contributions appear additionally to the usual centrifugal, Coriolis, Euler and translational accelerations that are characteristic for the continuum mechanics in noninertial frames (Liu 2002).

This violation of objectivity is a concern because the solutions of practical problems must then depend on a chosen reference: we obtain entirely different solutions in, say, a laboratory reference, and in a moving reference on a turntable. This problem has been investigated in Wilmanski (2001).

The question arises of whether one could overcome this difficulty by assuming that the nonobjectivity follows from the linearization of some objective nonlinear equations. If this were the case, one would have to describe porous materials by Biot's equations solely in inertial reference systems, and a time-dependent change of reference would require solely an addition of classical acceleration terms and ignoring contributions from the relative acceleration. We investigate some aspects of this question in the present work.

In §2, we introduce the notion of objective relative acceleration. We follow here the same line as Drew *et al*. (1979), who considered a problem of the suspension of bubbles in a fluid. In contrast to that work, we apply a Lagrangian description (e.g. Wilmanski 1998). Then we show that indeed a nonlinear poroelastic two-component model yields Biot's model by linearization.

In the third section, we provide some thermodynamical arguments to show that a nonlinear objective model with a contribution of relative accelerations is thermodynamically admissible if we add some nonlinear contributions to the partial stresses and to the free energy. They reflect in the simplest manner the existence of fluctuations of the microstructural kinetic energy caused by the heterogeneity of momentum in the representative elementary volume. The existence of such fluctuations as a result of the tortuosity of porous materials was indicated by Coussy (1991). Kinematic considerations concerning a structure of such fluctuations are presented in the article of Kosinski *et al*. (2002). Further quotations concerning this issue can be also found in this work. However, the constitutive part of a model based on such considerations has not been presented. There exist some attempts to derive Biot's model with the contribution of relative acceleration by means of Hamilton's principle, based on the fluctuating kinetic energy. As the true variational principle does not hold for dissipative systems, the dissipation through fluctuation and diffusion is accounted for by pseudo-potential and pseudo-variational principles. This does not seem to be the right way to handle irreversible processes. For this reason, we rely rather on the nonequilibrium thermodynamics in our considerations.

For completeness, in §4, we show the conditions for propagation of acoustic waves (hyperbolicity) and, in §5, differences in the behaviour of bulk monochromatic waves in porous materials within the linear Biot model and a simplified model (the so-called ‘simple mixture model’) where the coupling through relative accelerations is left out.

Let us mention in passing that the lack of relative accelerations in the model does not mean that the influence of tortuosity is neglected. Certainly, the permeability of the material described by the parameter *π* in our notation contains an influence of the morphology of the porous materials and this includes an influence of tortuosity.

## 2. Objective relative acceleration

We consider a two-component continuum consisting of a solid skeleton and of a fluid. The motion of the skeleton is assumed to be described by the following twice continuously differentiable function:(2.1)where ℬ denotes the reference configuration of the skeleton and *T* is the time-interval. The velocity, acceleration and the deformation gradient of the skeleton are defined by the relations(2.2)Certainly, the value **F**^{S}=1 corresponds to the reference configuration for, say, *t*=*t*_{0} in which *f*^{S}(**X**, *t*_{0})=**X**.

The motion of the fluid is described by the transformation of the Eulerian velocity field *v*^{F}=*v*^{F}(* x*,

*t*) defined on the space of current configurations, , of the skeleton. We have(2.3)The acceleration of the fluid is then given by(2.4)where is the so-called Lagrangian velocity of the fluid with respect to the skeleton.

We proceed to determine the transformation rules for the above quantities specified by the relation (1.8). The relations (2.2) and the time differentiation of the relation (1.8) yield the following quantities in the new reference system:(2.5)where the dot denotes the time derivative.

We assume that the transformation rule for the velocity field of the fluid component has the same form as it does for the skeleton(2.6)Consequently,(2.7)Bearing these relations in mind, we can now easily derive the transformation of the acceleration of the fluid. We obtain immediately(2.8)where the definition of the Lagrangian velocity has been used.

Owing to the presence of contributions dependent solely on the choice of the frame, we say that velocities , and accelerations , are nonobjective. Consequently, their difference is also nonobjective. We have(2.9)For this reason, the difference of accelerations cannot be used as a constitutive variable in a construction of the macroscopic model of a two-component system.

In the paper Drew *et al*. (1979), a method has been proposed to overcome these difficulties in the Eulerian description of suspensions. We shall use a similar way in the Lagrangian description. If we take the gradient of the transformation relations for velocities, we obtain(2.10)Consequently, we can write(2.11)where is arbitrary.

Substitution of this relation in equation (2.9) yields(2.12)It means that the quantity(2.13)is objective, i.e.(2.14)We call this quantity an *objective relative acceleration.* As an objective variable, it can be incorporated into the set of constitutive variables. Obviously, there exists a class of such accelerations specified by the constitutive coefficient, .

It is easy to see that a linear momentum source, , in an isotropic material, would contain a term , as required by the relations (1.1) of Biot's model. The open question is whether the second law of thermodynamics admits this type of contribution in a fully nonlinear model. We address this problem in §3.

## 3. Thermodynamical admissibility

A nonlinear poroelastic two-component model requires the formulation of field equations for the following fields(3.1)where *ρ*^{F} is the partial mass density of the fluid per unit volume in the reference configuration of the skeleton, i.e. in the current configuration it is given by the relation *ρ*_{t}^{F}=ρ^{F}*J*^{S}^{−}^{1}, *J*^{S}≔det **F**^{S}. *T* is the absolute temperature of the medium common for both components, and *n* is the current porosity. Other symbols have the same meaning as before.

The partial mass density of the skeleton in the reference configuration, *ρ*^{S}, does not appear among the fields because it is constant in a homogeneous material without mass exchange between components.

These fields are assumed to fulfil the following set of balance equations (e.g. Wilmanski 2004)1(3.2)(3.3)(3.4)(3.5)(3.6)(3.7)where **P**^{S} and **P**^{F} denote the first Piola–Kirchhoff partial stress tensors, is the momentum source, *ϵ* is the specific internal energy per unit mass of the mixture, **Q** is the heat flux vector, *n*_{E} describes the so-called equilibrium porosity, **J** is the porosity flux, and is the porosity source.

The porosity balance equation (3.7) yields the model essentially beyond the frame of Biot's model owing to the contribution of the relaxation source, . It was introduced some years ago (Wilmanski 1998) and analysed in numerous papers. For instance, the applicability in the theory of abrasion has been discussed by Kirchner (2002).

It can be shown (Wilmanski 2003) that changes of porosity predicted by the linearized porosity balance equation are identical with those following from Biot's model and the Gassmann relations, provided the relaxation time of porosity goes to infinity (i.e. ). However, it should be mentioned that many other approaches to the problem of evolution of volume fractions, porosity, etc. appear in the literature. One of the most popular forms of such an evolution equation follows from the so-called principle of equilibrated pressures introduced by Goodman, Cowin, Nunziato, Passman and others (see, for example, Kirchner (2002) for references and discussion). Even though in some applications such an approach may by advantageous to the porosity balance, we do not discuss it any further in this work.

In order to obtain field equations from the above balance equations, we have to specify constitutive relations for these quantities, i.e.(3.8)which must be functions of constitutive variables. In this work, the set of constitutive variables is chosen as follows:(3.9)Once the function(3.10)is given, we obtain a closed system of differential equations for fields .

It has been shown (Wilmanski 2002, 2004*b*) that the existence of coupling between partial stresses requires a constitutive dependence on some gradients of fields. Analysis has been performed for the model with a dependence on the porosity gradient. It was shown that within a linearized model, one obtains the Biot coupling described by the constant *Q*.

We shall not include this point in the thermodynamical analysis of this work. It may be shown that the existence of such a dependence yields possibilities of additional couplings, but it has no influence on the thermodynamical admissibility of a dependence on the relative acceleration. Simultaneously, the analysis is much simpler without these additional gradient constitutive variables. Consequently, the constitutive variable **N**:=Grad *n* does not appear in the list (3.9).

We say that constitutive relations (3.10) satisfy the *second law of thermodynamics* if the following entropy inequality,(3.11)is satisfied by all solutions of field equations. In this inequality, *η* is the specific entropy and **H** is its flux.

This requirement is equivalent to the following inequality which must hold for *all fields:*(3.12)where(3.13)are Lagrange multipliers dependent on constitutive variables, .

The exploitation of the second law of thermodynamics in the general case is technically impossible. Therefore, we make simplifying assumptions sufficient for the second law to be satisfied and yielding explicit limitations on constitutive relations. They are as follows:

The material is isotropic. Consequently, scalar constitutive functions, for instance, depend on vector and tensor variables solely through invariants. This assumption will be applied in some steps of our proofs. Some relations are simpler in a general form and then we do not introduce this limitation.

The dependence on the relative velocity, , is at most quadratic. This assumption is related to the structure of the nonlinear contribution to the objective relative acceleration. We motivate its form further.

The dependence on the temperature gradient

**G**is linear. We could skip this assumption at the cost of some technicalities, but experience with the thermodynamical construction of poroelastic models shows that it does not yield any undesired results.The dependence on the deviation of porosity,

*n*, from its equilibrium value*n*_{E}, Δ_{n}≡*n*−*n*_{E}, is quadratic.The dependence on the relative acceleration,

**a**_{r}, is linear.Higher order combinations of variables

**G**, , Δ_{n},**a**_{r}can be neglected.

Further, as we shall see, these assumptions limit thermodynamical considerations to a vicinity of the thermodynamical equilibrium similar to that appearing in the classical Onsager thermodynamics.

Bearing these assumptions in mind, we can write the following representations of constitutive functions:

partial stresses,(3.14)

internal energy and entropy,(3.15)

fluxes of energy, entropy, porosity,(3.16)where all coefficients are functions of variables ;

momentum source,(3.17)with coefficients dependent again on variables .

The notation of some coefficients in the above relations corresponds to that which is customary in the literature.

The contributions with the coefficients *ϵ*_{d} and *η*_{d} to the energy and entropy are motivated by fluctuations of the microstructural kinetic energy caused by the tortuosity. We do not introduce any additional microstructural variable describing changes of tortuosity. For this reason, a macroscopic influence of tortuosity can be solely reflected by the seepage velocity, which in our model corresponds to the Lagrangian velocity, . The classical kinetic energy in this model is given by . Consequently, the correction may be considered as an *added mass effect* resulting from tortuosity.

As we see further, the dependence of partial stresses on this velocity, introduced in the simplest form by equation (3.14), is then required by the consistency of the model with the second law of thermodynamics. In other words, we show further that coefficients *σ*^{S}, *σ*^{F} in the stress relations (3.14) and the coefficient *ϵ*_{d} in the energy relation (3.15) are connected (see formula (3.57)).

The exploitation of the second law of thermodynamics (3.12) is standard (e.g. Müller 1985). We apply the chain rule of differentiation to constitutive laws. Here we skip the rather cumbersome technical details.

Linearity of the second law of thermodynamics with respect to time derivativesyields(3.18)(3.19)(3.20)(3.21)(3.22)(3.23)These identities still contain linear contributions of Grad **F**^{S}, Δ_{n} and quadratic contributions of the latter, as well as quadratic contributions of Lagrangian velocity. As they should hold for arbitrary fields, coefficients of these contributions must vanish separately. After easy analysis we obtain(3.24)(3.25)The second law of thermodynamics is also linear with respect to the following spatial derivatives:(3.26)We have listed them in the order of the further analysis and, simultaneously, skipped the derivative Grad **a**_{r} because it does not contribute to the second law owing to the relations (3.24). The linearity with respect to quantities (3.26) yields a set of identities and leaves a residual inequality that is essentially nonlinear. It defines the *dissipation* in the system and has the following form:(3.27)Hence, the state of thermodynamical equilibrium defined by appears if(3.28)i.e. the temperature gradient, relative motion (diffusion) and the relaxation of porosity cause the deviation from the equilibrium.

Clearly, the fourth assumption yields the linearity of and Λ^{n} with respect to Δ_{n}. In addition, the above inequality yields homogeneity of these functions, i.e. we can write(3.29)where *τ* and *λ*^{n} can be solely functions of variables . Consequently, we also obtain(3.30)It is worth mentioning that owing to equation (3.24) the relative acceleration does not contribute to the dissipation. This property of the model follows from the fact that the model does not possess any independent field of tortuosity that could relax to the thermodynamical equilibrium.

Now, we return to the coefficients of spatial derivatives of fields. The vanishing coefficient of yields the following results:(3.31)(3.32)(3.33)(3.34)where(3.35)are the main invariants of the Cauchy–Green deformation tensor **C**^{S}.

The coefficient of yields(3.36)(3.37)(3.38)Consequently, bearing equations (3.32) and (3.37) in mind,(3.39)Next, we consider the coefficient of Grad *ρ*^{F}. We have(3.40)(3.41)Similarly, the coefficient of Grad **F**^{S} yields(3.42)(3.43)where the components of tensors **Ξ**^{S} and **Ξ**^{F} in Cartesian coordinates are given by the relations(3.44)Under our assumptions, the contribution of Grad Δ_{n} does not yield any restrictions.

Finally, the last condition follows from the vanishing coefficient of Grad **G**, and it has the form(3.45)Inspection of the results (3.41) and (3.42) for thermal coefficients yields(3.46)where the standard argument (e.g. Wilmanski 2004) has been used.

It is not quite clear what limitations on partial stress tensors are imposed by conditions (3.40) and (3.42). Derivatives of partial stresses, with respect to the mass density, *ρ*^{F}, as well as with respect to the deformation gradient, **F**^{S}, seem to restrict elastic properties of the system in equilibrium. This does not seem very plausible. Hence, we assume that the coefficient vanishes, i.e.(3.47)Then the multipliers of momentum equations vanish as well. As the consequence of equations (3.23), (3.25), (3.33) and (3.39), we obtain(3.48)It is convenient to introduce the following notation:(3.49)Then, for Lagrange multipliers we have(3.50)and the relation (3.23) implies the following classical formula for the internal energy:(3.51)Simultaneously, the relations (3.40) and (3.42) yield(3.52)(3.53)These relations yield the following integrability condition for the multiplier :(3.54)Consequently, integration of equation (3.50)_{1} leads to the following additive splitting of the free energy, *ψ*:(3.55)The above separation property is characteristic for the so-called *simple mixtures*.

In addition, integration in equation (3.52)_{1} yields(3.56)where we have accounted for relations (3.45) and (3.46).

Inspection of relations (3.34), (3.38) and (3.48) leads immediately to the following identification of constitutive coefficients, coupled to the relative acceleration:(3.57)Simultaneously, relation (3.31) taken together with equations (3.49), (3.55) and (3.56) for partial stresses **P**_{0}^{S} and relation (3.36) for partial stresses **P**_{0}^{F} yields(3.58)Hence, as mentioned in §1, the partial stresses do not possess a coupling term characteristic for Biot's model, and this fallacy of the model can be removed by additional constitutive variables.

For practical purposes, it is convenient to transform equations of the model to *Eulerian coordinates*. We write them in an arbitrary, *noninertial* reference system. The set of balance equations (3.2) then has the form shown below.

mass balance for the fluid component:(3.59)

momentum balance for the skeleton:

momentum balance for the fluid:

energy balance:

porosity balance:(3.63)

The external forces *ρ*_{t}^{S}*b*^{S} and *ρ*_{t}^{F}*b*_{t}^{F}, called *apparent body forces*, contributing to momentum balance equations, have the following structure:(3.64)where *ρ*_{t}^{S}*b*_{b}^{S}, *ρ*_{t}^{F}*b*_{b}^{F} are true (e.g. gravitational) body forces, and *ρ*_{t}^{S}*i*^{S}, *ρ*_{t}^{F}*i*^{F} are called *inertial body forces*. In order of appearance in the above relations, they consist of the inertial force of relative translation, Coriolis force, Euler force and centrifugal force. They depend on the matrix of angular velocity, **Ω**, of the noninertial system with respect to an inertial one. Certainly, the inertial body forces vanish in an inertial reference system. It should be mentioned that the partial accelerations appearing in the above partial momentum balance equations, combined with apparent body forces, are objective, that is, invariant with respect to the transformation (1.8).

The remaining notation used above is as follows:(3.65)however, the Cauchy stress tensors **T**^{S} and **T**^{F} are given by the following constitutive relations:(3.66)(3.67)with the free energy given by(3.68)The energy, *ϵ*, and the energy flux vector, ** q**, are given by(3.69)and the porosity flux has the form(3.70)It is easy to see that the linearization of the above set for isothermal processes without the source of porosity leads to Biot's equations (equations (1.1)) without the coupling constant,

*Q*.

There remains the question of the practical estimation of additional parameters, *ρ*_{12}^{0} and . The added mass coefficient, *ρ*_{12}^{0}, has been extensively studied in literature concerning Biot's model. The parameter is new. There seem to exist various possibilities for its estimation connected to the fact that it appears in contributions which may be time independent. As an example, let us consider a stationary isothermal process in which, in a chosen inertial frame of reference, the skeleton does not move (i.e. *v*^{S}=0). Such a flow of the fluid through a porous material is described by the mass balance and by the momentum balance for the fluid. For simplicity we neglect changes of porosity. Then we have(3.71)The correction of the permeability coefficient, *π*, driven by volume changes of the fluid div *v*^{F} seems to be very small. However, the correction of mass density appearing on the left-hand side of this equation may be essential and measurable. For instance, in an irrotational flow (rot *v*^{F}=0) we have approximately(3.72)where *p*=*p*^{F}/*n* is the pore pressure and *p*_{0} its constant reference value. If the pressure increment is of the order of, say, 10 kPa, the velocity of the fluid must be of the order of 1 m s^{−1} to make both contributions of the similar order. Practically measurable would be the influence of for much smaller velocities that seem to be plausible, at least for rocks.

## 4. Propagation of fronts of acoustic waves in Biot's model

As already mentioned, the linearization of the above-presented model yields the contribution of the difference of accelerations in Biot's model, written in a chosen inertial frame of reference. Consequently, one can ask whether such a contribution, and the contribution of the coupling of stresses, reflected by the material parameter, *Q*, essentially influence the results for the propagation of acoustic waves in porous materials. There even exist claims in the literature that the added mass effect is necessary for the existence of the so-called ‘Biot wave’.

In §§ 5 and 6, we present an example of analysis of weak discontinuity (acoustic) waves for Biot's model, as well as the ‘simple mixture’ model in which both the coupling, *Q*, and the tortuosity coefficient (*a*−1) are assumed to be zero. Similarly to Biot's model, the latter model has already a rather extensive literature (for a review of the results, see Wilmanski (1999) and Wilmanski & Albers (2003)).

The main aim of this analysis is to show that the differences between these two models are solely quantitative. This has a particular bearing in applications to such complex problems as the propagation of surface waves, which play an important role in nondestructive testing of porous materials.

Let us repeat the set of equations of Biot's model, equations (1.1) and (1.2), with a small modification of the notation. For the fields *v*^{S}, *v*^{F}, *e*^{S} and *ϵ*, we have the field equations(4.1)where(4.2)We begin the analysis of this system by proving its hyperbolicity. To this aim, we consider the propagation of the front, , of the weak discontinuity wave, i.e. of a singular surface on which(4.3)where [[…]] denotes the jump of the quantity. On such a surface, the accelerations may be discontinuous, and we call their jumps the *amplitudes of discontinuity*,(4.4)Then the following compatibility conditions hold:(4.5)where *c* is the speed of propagation of the surface and ** n** its unit normal vector. The latter gives, of course, the direction of propagation of the wave.

Bearing (4.2) in mind, we obtain immediately(4.6)We evaluate the jump of field equations (4.1) on the surface . It follows immediately that(4.7)This is clearly an eigenvalue problem. We say that the system (4.1) is *hyperbolic* if the eigenvalues *c* are real and the corresponding eigenvectors (*a*^{S} and *a*^{F}) are linearly independent. We prove that this is indeed the case.

It is convenient to separate the transversal and longitudinal parts of the problem (4.7). The *transversal* part follows if we take the scalar product of the equations with a vector *n*_{⊥} perpendicular to ** n**. We obtain(4.8)Hence we have for the speed of the front(4.9)As

*ρ*

_{22}>0,

*μ*

^{S}>0, it follows the

*first condition for hyperbolicity*of the set (4.1):(4.10)This condition is obviously fulfilled because

*a*is not smaller than one.

The speed of propagation, equation (4.9), describes the shear wave. It is easy to see, that, in the particular case without the influence of tortuosity *a*=1, this relation reduces to the classical formula . In this case, according to (4.8)_{2}, the amplitude in the fluid *a*_{⊥}^{F} is zero, i.e. the shear wave is carried solely by the skeleton.

We proceed to the *longitudinal* part. To this aim, we take the scalar product of the relations (4.7) with the vector ** n**. It follows that(4.11)and the dispersion relation is as follows:(4.12)where(4.13)The eigenvalues of this problem have the form(4.14)where(4.15)It can be easily shown that under the condition (4.10)

*B*>0 for all

*a*≥1,

*Q*≥0. However,

*c*

^{2}, defined by (4.14), is positive only if the following additional condition on

*Q*is satisfied:(4.16)This is the second condition for hyperbolicity.

In the particular case where *a*=1 and *Q*=0, we have *c* equal to either *c*_{P1} or *c*_{P2}, which means that the set is unconditionally hyperbolic.

The two solutions for *c*^{2} define two longitudinal modes of propagation, P1 and P2. The P2 mode, called the Biot wave or the *slow wave* in the theory of porous materials, is also known as the *second sound,* and it appears in all two-component systems described by hyperbolic field equations. For instance, it is known in the theory of binary mixtures of fluids, in which it is applied to describe dynamical properties of liquid helium as discovered by Tisza (1938). For porous materials, it was discovered by Frenkel (1944).

## 5. Biot's model versus the simple mixture model on the example of monochromatic acoustic waves

The above analysis yields solely the propagation properties of the wavefront . We do not learn anything about, for instance, the attenuation of the waves. For this reason, we proceed to analyse monochromatic waves. As we see, the speeds of propagation obtained above follow in the limit of frequency *ω*→∞.

We seek solutions of equations (4.1) that have the form of the following monochromatic waves(5.1)where **V**^{S}, **V**^{F}, **E**^{S}, *E*^{F} are constant amplitudes, ** k** is the wave vector, and

*ω*is a real frequency. This should be understood in the following way: the wavenumber, is complex and the direction of propagation

**:=**

*n***/**

*k**k*real, which yields exp(i

**.**

*k***)=exp(−Im**

*x**k*(

**.**

*n***))exp[i(Re**

*x**k*(

**.**

*n***)−**

*x**ωt*)], i.e. contributions of the attenuation and of the progressive wave.

Substitution of this solution in field equations yields the following compatibility conditions:(5.2)As usual, the problem of the existence of such waves reduces to the eigenvalue problem with the eigenvector [**V**^{S}, **V**^{F}]. As before, we split the problem into two parts: in the direction *k*_{⊥} perpendicular to ** k** (transversal modes) and in the direction of the wave vector

**(longitudinal modes).**

*k*For transversal modes (monochromatic shear waves) we have(5.3)The dispersion relation can be written in this case in the following form:(5.4)i.e.(5.5)Consequently, neither the phase speed, *ω*/Re *k*, nor the attenuation, Im *k*, of monochromatic shear waves is dependent on the coupling coefficient, *Q*.

In the two limits of frequencies, we then have the following solutions:(5.6)The first result checks with the results of the classical one component model commonly used in soil mechanics. The speed in the second one is identical with that of formula (4.9). Hence, the propagation of the front of shear waves is identical with the propagation of monochromatic waves of infinite frequency. Let us note that the attenuation in this limit is finite.

We demonstrate further properties of these monochromatic waves on a numerical example.

For longitudinal modes we obtain the dispersion relation(5.7)or, after easy manipulations,(5.8)Let us check again two limits of frequencies: *ω*→0, and *ω*→∞.

In the first case we obtain(5.9)Obviously, we obtain two real solutions of this equation:(5.10)These are the squares of the speeds of propagation of two longitudinal modes in the limit of zero frequency. Clearly, the second mode P2-wave does not propagate in this limit. Both limits are independent of tortuosity. The result (5.10) checks with the relation for the speed of longitudinal waves used in the classical one component model of soil mechanics, provided *Q*=0.

In the second case we have(5.11)This coincides with relation (4.12). Consequently, the limit *ω*→∞ indeed gives the properties of the front of acoustic longitudinal waves in the system.

Simultaneously, we obtain the following attenuation in the limit of infinite frequencies:(5.12)Hence, both limits of attenuation for the P1-wave and P2-wave are finite.

We proceed to the presentation of a numerical result in the whole range of frequencies, *ω*∈[0, ∞). We use the following numerical data:(5.13)

Speeds *c*_{P1}, *c*_{P2} and *c*_{S}, the mass density *ρ*_{0}^{S} (i.e. for the porosity *n*_{0}=0.4) and the fraction *r*=*ρ*_{0}^{F}/*ρ*_{0}^{S} possess values typical of many granular materials under a confining pressure of a few atmospheres and saturated by water. In units standard for soil mechanics, the permeability, *π*, corresponds to approximately 0.1 Darcy. The coupling coefficient, *Q*, has been estimated by means of the Gassmann relation (e.g. Wilmanski 2004*a*). The tortuosity coefficient *a*=1.75 follows from Berryman's formula (1.7)_{2}.

Transversal waves that described relation (5.5) are characterized by the following distribution of speeds and attenuation in function of frequency (figure 1). The solid lines correspond to the solution of Biot's model and the dashed lines to the solution of the simple mixture model.

It is clear that the qualitative behaviour of the speed of propagation is the same in both models. It is a few per cent smaller in Biot's model than that in the simple mixture model in the range of high frequencies. A large quantitative difference between these models appears for the attenuation. In the range of higher frequencies, it is much smaller in Biot's model, that is, tortuosity decreases the dissipation of shear waves.

The latter property is illustrated in figure 2, where we plot the attenuation of the front of shear waves, i.e. , as a function in the tortuosity coefficient, *a*. This behaviour of attenuation indicates that damping of waves created by the tortuosity, which is connected in the macroscopic model to the relative velocity of components, is not related to the scattering of waves on the microstructure. It is related instead to the decrease in the macroscopic diffusion velocity in comparison with the difference of velocities on the microscopic level, owing to the curvature of channels and volume averaging. Fluctuations are related solely to this averaging and not to temporal deviation from time averages (lack of ergodicity).

We proceed to longitudinal waves. The solid lines on the following figures correspond again to Biot's model, the dashed lines to the simple mixture model. In order to separately show the influence of the tortuosity, *a*, and of the coupling, *Q*, we also plot the solutions with *a*=1 (dash-dotted lines) and the solutions with *Q*=0 (dash-double-dotted lines).

Even though it is similar again, the quantitative differences are much more substantial for P1-waves (figure 3). This is primarily an influence of the coupling through partial stresses described by the parameter *Q*. The simple mixture model (*Q*=0, *a*=1), as well as Biot's model with *Q*=0, yields speeds of these waves, differing only by a few per cent (lower curves in figure 3*a*). The coupling, *Q*, shifts the curves to higher values and reduces the difference caused by the tortuosity. This result does not seem to be very realistic, because the real differences between low-frequency and high-frequency speeds measured in soils were in fact as large as is indicated by the simple mixture model. This may be an indication that the Gassmann relations give values that are much too large for the coupling parameter, *Q*, with respect to, these, indeed appearing in real granular materials.

Both the tortuosity, *a*, and the coupling, *Q*, reduce the attenuation quite considerably as indicated in figure 3*b*.

In spite of some claims in the literature, the tortuosity, *a*, does not influence the existence of the slow P2-wave (figure 4). Speeds of this wave are again qualitatively similar in Biot's model and in the simple mixture model. The maximum differences appear in the range of high frequencies and reach some 35%. The same concerns the attenuation even though quantitative differences are not so big (approximately 8%).

## 6. Conclusions

The analysis presented in this work yields the following conclusions:

(i) As demonstrated in the second section, it is possible to construct a relative acceleration for a two-component model of a poroelastic material in such a way that it transforms as an objective quantity. Additional contributions to the difference of partial accelerations are nonlinear and contain a single scalar constitutive parameter, .

As shown in §3, the linear dependence of the source of momentum on such a relative acceleration is thermodynamically admissible, provided the following conditions are fulfilled. The partial stress tensors must contain additional contributions quadratic in the relative velocity, with material coefficients determined by the combination of two parameters appearing in the contribution of the relative acceleration: the tortuosity coefficient, *a*, and the parameter The internal energy must contain an additional contribution of the kinetic energy of relative motion, with the constitutive coefficient dependent solely on the tortuosity, *a*. Such a model fulfils the second law of thermodynamics and the principle of material objectivity.

The linearization of the above described model yields Biot's contribution of relative accelerations. This is, of course, not objective anymore. Consequently, Biot's model can be used solely in inertial frames of reference. In noninertial frames, the transition from the nonlinear model yields apparent body forces but not additional terms, which would follow by the transformation of the system of Biot's equations.

(ii) We have demonstrated on the example of acoustic waves that tortuosity

*a*and the coupling parameter*Q*have a quantitative but not qualitative influence on results. We have compared results for Biot's model with these for the simple mixture model in which the tortuosity*a*=1 and the coupling parameter*Q*=0. We have proven that both models are hyperbolic provided the parameter*Q*satisfies a condition bounding this parameter from above. In particular, both models predict the existence of the P2-wave. Speeds and attenuations of monochromatic P1-, P2- and S-waves are qualitatively the same, but there are quantitative discrepancies, which we discuss below.(iii) Tortuosity introduced to the model through the relative acceleration yields dissipation solely owing to the modification of the relative motion. If we assume the permeability coefficient

*π*=0, the dissipation in isothermal processes without relaxation of porosity vanishes. This is owing to the fact that tortuosity, in contrast to porosity, is not introduced as a field described by its own field equation. This is an explanation of a rather unexpected behaviour of attenuation of monochromatic waves. Inspection of figures shown in this work makes it clear that the presence of tortuosity*a*≠1 yields a smaller attenuation, rather than bigger as it would be in the case of a dissipative field. This may be explained by the fact that tortuosity reduces the relative velocity,*v*^{F}−*v*^{S}, and, consequently, it reduces the contribution to dissipation,*π*(*v*^{F}−*v*^{S}).(*v*^{F}−*v*^{S}).(iv) We have demonstrated that a rather moderate value of the parameter

*Q*, suggested by the classical Gassmann relation for granular materials, leads to an unreasonable increment in the speeds of propagation and reduction of attenuation. In addition, the speed of propagation of monochromatic P1-waves becomes very flat as a function of frequency. This contradicts observations in soil mechanics and geotechnics, and it indicates that the Gassmann relation predicts excessively large values for this parameter. The situation would improve if we used the model proposed in Wilmanski (2002). This model contains a constitutive dependence on the porosity gradient, which yields a modification of Gassmann relations (Wilmanski 2003) and a considerable reduction of the parameter*Q*.

## Footnotes

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**F**^{ST}in the relation (3.5) denotes the transpose of**F**^{S}. Similar notation is used for other tensors.- Received April 26, 2004.
- Accepted September 22, 2004.

- © 2005 The Royal Society