Geometric variational approach to the dynamics of porous media filled with incompressible fluid
Tagir Farkhutdinov, François Gay-Balmaz, Vakhtang Putkaradze
GGeometric variational approach to the dynamics of porousmedia filled with incompressible fluid
Tagir Farkhutdinov ∗ and Fran¸cois Gay-Balmaz † and Vakhtang Putkaradze ‡ Abstract
We derive the equations of motion for the dynamics of a porous media filled with anincompressible fluid. We use a variational approach with a Lagrangian written as the sumof terms representing the kinetic and potential energy of the elastic matrix, and the kineticenergy of the fluid, coupled through the constraint of incompressibility. As an illustration ofthe method, the equations of motion for both the elastic matrix and the fluid are derived inthe spatial (Eulerian) frame. Such an approach is of relevance e.g. for biological problems,such as sponges in water, where the elastic porous media is highly flexible and the motionof the fluid has a ’primary’ role in the motion of the whole system. We then analyze thelinearized equations of motion describing the propagation of waves through the media. Inparticular, we derive the propagation of S -waves and P -waves in an isotropic media. Wealso analyze the stability criteria for the wave equations and show that they are equivalentto the physicality conditions of the elastic matrix. Finally, we show that the celebratedBiot’s equations for waves in porous media are obtained for certain values of parametersin our models. Contents S -waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 P -waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 ∗ Department of mathematical and statistical sciences, University of Alberta, Edmonton, AB, T6G 2G1Canada / ATCO SpaceLab, 5302 Forand St SW, Calgary, AB, T3E 8B4, Canada, [email protected] † CNRS - LMD, Ecole Normale Sup´erieure, 24 Rue Lhomond, 75005 Paris, [email protected] ‡ Department of mathematical and statistical sciences, University of Alberta, Edmonton, AB, T6G 2G1Canada / ATCO SpaceLab, 5302 Forand St SW, Calgary, AB, T3E 8B4, Canada, [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] J u l arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media The coupled dynamics of porous media filled with fluid, also known as poromechanics , hasbeen the subject of an active research for many decades. The foundational works in the areawere driven by applications to soil dynamics and geophysics, whereas in the latter years theapplications also included biomedical fields. The earlier developments were associated withthe works of K. von Terzaghi [1] and M. Biot [2, 3, 4] in the consolidation of porous media,and subsequent works by M. Biot which derived the time-dependent equations of motion forporomechanics, based on certain assumptions on the media, and considered the wave propaga-tion in both low and high wavenumber regime [5, 6, 7, 8]. There has been substantial amount ofnew work in the field of porous media, see [9, 10, 11, 12, 13, 14] and subsequent mathematicalanalysis of the models [15, 16, 17]. We refer the reader interested in the history of the field tothe review [18] for a more detailed exposition of the literature.While Biot’s equations, especially with respect to acoustic propagation in porous media,still remain highly influential today, subsequent investigations have revealed difficulties in theinterpretation of various terms through the general principles of mechanics, such as materialobjectivity, frequency-dependent permeability and changes of porosity in the model, as well asthe need to describe large deformations of the model [19]. The above-cited paper then proceedsin outlining a detailed derivation for the modern approach to saturated porous media equationswhich does not have the limitations of the Biot’s model. We shall also mention here two recentpapers [20, 21] where the equations for saturated porous media were further developed basedon the general thermodynamics principles of mechanics.By their very nature, variational methods involve fully nonlinear treatment of the inertialterms. The mainstream approach to the porous media has been to treat the dynamics asbeing friction-dominated by dropping the inertial terms from the equations. The equations wewill derive here, without the viscous terms, will be of infinite-dimensional Hamiltonian type.On the other hand, the friction-dominated approach gives equations of motion that are ofgradient flow type. The seminal book of Coussy [22] contains a lot of background informationand analysis. For more recent work, we will refer the reader to, for example, the studies ofmulti-component porous media flow [23], as well as the gradient approach to the thermo-poro-visco-elastic processes [24].Fluid-filled elastic porous media, by its very nature, is a highly complex object involvingboth the individual dynamics of fluid and media, and a highly nontrivial interactions betweenthem. The pores in the elastic matrix, and the fluid motion inside them, are micro-structuredelements that contribute to the macro-structured dynamics. Thus, the porous media mustinclude the interaction between the large scale dynamics and an accurate, and yet treatable,description of micro-structures. It has long been known in mechanics that variational principlesare ideally suited to treat complex, multi-component systems. Variational methods proceedformally by describing the Lagrangian of the system on an appropriate configuration manifold,and proceeding with variations to obtain the equations of motion in a systematic way. Theadvantage of the variational methods is their consistency, as opposed to the theories based onbalancing the conservation laws for a given point, or volume, of fluid. In a highly complex arkhutdinov, Gay-Balmaz, Putkaradze
Geometric variational approach to porous media
It seems to be also clear that it is a waste of effort to try to construct a true variational princi-ple as the Biot model contains a nonequilibrium variable, the increment of fluid contents whichrules out the existence of such a principle.
In spite of this difficulty, variational methods were actively applied to the field poromechanics.One of the earliest papers papers in the field was [25] where the kinetic energy of expansionwas incorporated into the Lagrangian to obtain the equations of motion. In that work, severalLagrange multipliers were introduced to enforce the continuity equation for both solid andfluid. The works [26, 27] use variational principles for explanation of the Darcy-Forchheimerlaw. Furthermore, [28, 29] derive the equations of porous media using additional terms in theLagrangian coming from the kinetic energy of the microscopic fluctuations. Of particular in-terest to us are the works on the Variational Macroscopic Theory of Porous Media (VMTPM)which was formulated in its present form in [30, 31, 32, 33, 34, 35, 36, 37, 38, 39], also sum-marized in a recent book [40]. In these works, the microscopic dynamics of capillary pores ismodelled by a second grade material, where the internal energy of the fluid depends on boththe deformation gradient of the elastic media, and the gradients of local fluid content. Thestudy of a pre-stressed system using variational principles and subsequent study of propagationof sound waves was undertaken in [41].One of the main assumptions of the VMTPM is the dependence of the internal energy ofthe fluid on the quantity measuring the micro-strain of the fluid, or, alternatively, the fluidcontent or local density of fluid, including, in some works, the gradients of that quantity. Thisassumption is physically relevant for compressible fluid, but, in our view, for an incompressiblefluid (which, undoubtedly, is a mathematical abstraction), such dependence is difficult tointerpret. For example, for geophysical applications, fluids are usually considered compressiblebecause of the large pressures involved. In contrast, for biological applications like the dynamicsof highly porous sponges in water, the compressibility effects can be neglected. For a trulyincompressible fluid, it is difficult to assign a physical meaning to the dependence of internalenergy of the fluid on the parameters of the porous media. We refer the reader to the theclassical Arnold’s description of incompressible fluid [42] as geodesic motion on the group ofvolume-preserving diffeomorphism in the three-dimensional space, in the absence of externalforces. In that theory the Lagrangian is simply the kinetic energy, as the potential energyof the fluid is absent, and the fluid pressure enters the equations from the incompressibilitycondition. The main result of the present paper is to extend this geometric description tothe motion of the fluid-filled porous media, for the case when the fluid inside the pores isincompressible, and, neglecting all thermal effects, without considering the internal energy offluid.Before we delve into detailed derivations, it is useful to have a discussion on the physics ofwhat is commonly considered the saturated porous media. In most, if not all, previous works,the saturated porous media is a combined object consisting of an (elastic) dense matrix, and anetwork of small connected pores filled with fluid. The fluid encounters substantial resistancewhen moving through the pores due to viscosity and the no-slip condition on the boundary. Insuch a formulation, it is easier to consider the motion of the porous matrix to be ’primary’, and arkhutdinov, Gay-Balmaz, Putkaradze
Geometric variational approach to porous media
In this Section we derive the equations of motion for a porous medium filled with an incom-pressible fluid by using a variational formulation deduced from Hamilton’s principle. We willfollow the description of both fluid and elastic matrix, individually, as outlined in the bookby Marsden and Hughes [44], where the reader can find the background and fill in technicaldetails of the description of each media.
We shall remark that the preferred description for the motion of an elastic body is achievedthrough the Lagrangian coordinates of the media as being the independent variables, andbalancing the forces in the spatial frame or the frame attached to the media. On the otherhand, the description of the fluid equation is traditionally done in the Eulerian (spatial) frame.The combined mixed fluid-material motion for porous media can thus be described in eitherframe. In order to connect with the earlier works by Biot and subsequent analysis of wavepropagation in the porous media, we compute the equations of motion in spatial coordinatesthroughout the paper.
Configuration of the elastic body and the fluid.
Suppose that at t = 0 the fluid andthe elastic body occupy completely a given volume B ⊂ R . By default, we are working witha three-dimensional system, although the equations of motion reduce trivially to the two- andone-dimensional cases. The motion of the elastic body (indexed by s ) and the fluid (indexedby f ) is defined by two time dependent maps Ψ and ϕ defined on B with values in R , withvariables denoted as x = Ψ ( t, X s ) and x = ϕ ( t, X f ). We assume that there is no fusionof either fluid or elastic body particles, so the map Ψ and ϕ are embeddings for all times t ,defining uniquely the mappings X s = Ψ − ( t, x ) and X f = ϕ − ( t, x ). We also assume that thefluid cannot escape the porous medium or create voids, so at all times t , the domains occupiedby the fluid B t,f = ϕ ( t, B ) and the elastic body B t,s = Ψ ( t, B ) coincide: B t,f = B t,s = B t .Finally, we shall assume for simplicity that the domain B t does not change with time, and will arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media B , hence both ϕ and Ψ are diffeomorphisms of B for all time t . An extension tothe case of a moving boundary is possible, although it will require appropriate modificationsin the variational principle. Velocities of the elastic body and the fluid.
The fluid velocity u f and elastic solid ve-locity u s , measured relative to the fixed coordinate system, i.e. , in the Eulerian representation,are given as usual by u f ( t, x ) = ∂ t ϕ (cid:0) t, ϕ − ( t, x ) (cid:1) , u s ( t, x ) = ∂ t Ψ (cid:0) t, Ψ − ( t, x ) (cid:1) , (1)for all x ∈ B . Note that since ϕ and Ψ keep the boundary ∂ B invariant, the vector fields u f and u s are tangent to the boundary, i.e. , u f · n = 0 , u s · n = 0 , (2)where n is the unit normal vector field to the boundary. One can alternatively impose that ϕ and Ψ (or only Ψ ) keeps the boundary ∂ B pointwise fixed. In this case, one gets no-slipboundary conditions u f | ∂ B = 0 , u s | ∂ B = 0 , (or only u s | ∂ B = 0) . (3) Elastic deformations of the dry media.
In order to incorporate the description of theelastic deformations of the media in the potential energy, we consider the deformation gradientof Ψ denoted F ( t, X s ) = ∇ Ψ ( t, X s ) . (4)In the spatial frame, we consider the Finger deformation tensor b ( t, x ) defined by b ( t, x ) = F F T ( t, X s ) , (5)where x = Ψ ( t, X s ), see the paragraph below for the intrinsic geometric definition of b . Incoordinates, we have F iA = ∂ Ψ i ∂X As , b ij = ∂ Ψ i ∂X As ∂ Ψ j ∂X As with the summation over A is assumed.In general the deformation of an elastic media without fluid leads to b (cid:54) = Id (the unittensor). The potential energy V of deformation of the dry media thus depends on b . However, inour case there is another part that leads to the elastic potential energy, namely, the microscopicdeformations of the pores that we shall describe below. Internal deformation of the pores and constraint.
Let us now consider the volumeoccupied by the fluid in a given spatial domain. We assume that the fluid fills the porescompletely, so the volume occupied by the fluid in any given spatial domain is equal to the netvolume of pores in that volume. Let us take the infinitesimal Eulerian volume d x and definethe pore volume fraction g ( t, x ), so that the volume of fluid is given by g ( t, x )d x . There aretwo aspects to take into account to obtain the available volume to the fluid, namely, the localconcentration of pores c ( t, x ) and the infinitesimal pore volume v ( t, x ).If, for example, the pores are “frozen” in the material, they simply move as material moves.Then, the change of the local concentrations of pores c ( t, x ) due to deformations is given by c ( t, Ψ ( t, X s )) | det F ( t, X s ) | = c ( X s ) , (6) arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media c ( X s ) is the initial concentration of pores in the Lagrangian point X s . Using thedefinition (5) of the Finger tensor b gives det b ( t, x ) = | det F ( t, X s ) | , hence we can rewrite theprevious relation as c ( t, x ) (cid:112) det b ( t, x ) = c ( X s ) . In the case of an initially uniform porous media, i.e. , c = const, this formula shows that theconcentration c ( t, x ) is a function of the value b ( t, x ) of the Finger deformation tensor c ( b ) = c √ det b . (7)Note that from (6), the concentration of pores satisfies ∂ t c + div( c u s ) = 0 . On the other hand, the pores themselves can expand and contract, which one can un-derstand as modeling the pores through infinitesimally small elastic volumes filled with fluid.When the pores expand, they generate stress in the material; however, the stress averagedover any volume that is much larger than the size of the pores, is going to vanish. We thusintroduce an additional dependence of the elastic part of the media on the infinitesimal volumedenoted V ( t, X s ) in the Lagrangian description. Its Eulerian version is v ( t, x ) with v ( t, Ψ ( t, X s )) = V ( t, X s ) . (8)These two considerations lead to the following constraint on the total volume of pores,which is more easily written in the spatial description: g ( t, x ) = c ( b ( t, x )) v ( t, x ) . (9) Conservation law for the fluid.
In what follows, we will consider an incompressible fluid,as that case has not been studied in the literature in sufficient details. The density of the fluiditself is denoted as ρ f = const. We can thus discuss the conservation of the volume of fluidrather than the mass. Let us now look at the volume of fluid g ( t, x )d x from a different pointof view. The fluid must fill all the available volume completely, and it must have come fromthe initial point X f = ϕ − ( t, x ). If the initial volume fraction at that point was g ( X f )d X f ,then at a point t in time we have g ( t, x ) = g (cid:0) ϕ − ( t, x ) (cid:1) J ϕ − ( t, x ) , J ϕ − := det (cid:0) ∇ ϕ − (cid:1) . (10)Differentiating (10), we obtain the conservation law for g ( t, x ) written as ∂ t g + div( g u f ) = 0 . (11)The mass of the fluid in the given volume is ρ f g d x . Note that the incompressibility conditionof the fluid does not mean that div u f = 0. That statement is only true for the case where noelastic matrix is present, i.e. , for pure fluid. In the porous media case, a given spatial volumecontains both fluid and elastic parts. The conservation of volume available to the fluid is thusgiven by (11). Conservation law for the elastic body.
The mass density of the elastic body, denoted ρ s , satisfies an equation analogous to (10), namely, ρ s ( t, x ) = ρ s, (cid:0) Ψ − ( t, x ) (cid:1) J Ψ − ( t, x ) , (12)where ρ s, ( X s ) is the mass density in the reference configuration. The corresponding differen-tiated form is ∂ t ρ s + div( ρ s u s ) = 0 . (13) arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media Intrinsic geometric formulation.
To understand the transport equation of the Finger de-formation tensor, it is advantageous to reformulate geometrically its definition. We assumethat the reference configuration B is endowed with a reference Riemannian metric G , locally de-noted G = G AB dX As dX Bs and we consider its inverse G − . It is a symmetric two-contravarianttensor locally denoted G − = G AB ∂∂X As ∂∂X Bs with G AB G BC = δ AC . Then, the Finger deforma-tion tensor is the symmetric two-contravariant tensor obtained by pushing forward G − by theelastic configuration Ψ , namely b = Ψ ∗ G − . (14)For a domain in three-dimensional Euclidian space, the Riemannian metric is simply an iden-tity, and is often not included in the considerations. However, the differential-geometric con-siderations here are important, e.g. for evolution of porous shells, which we do not considerhere. The geometric description presented here is explained in details in [44]. Using localcoordinates, one notes that when G is the Euclidean metric, (14) reduces to (5). Using (14)and (1), we get the transport equation for b as ∂ t b + £ u s b = 0 , where £ u s denotes the Lie derivative of a two-contravariant tensor, given in coordinates by( £ u s b ) ij = ∂b ij ∂x k u ks − b kj ∂u is ∂x k − b ik ∂u js ∂x k . (15)Let us now formulate (7) intrinsically i.e. , without the use of the local coordinates. Given aRiemanian metric γ on the spatial domain, the Jacobian J Ψ of Ψ is defined by Ψ ∗ µ γ = J Ψ µ G ,where µ γ = √ det γ d x and µ G = √ det G d X s are the Riemannian volume forms. From this,one expresses intrinsically the Jacobian of Ψ in terms of the Finger deformation tensor as J Ψ ◦ Ψ − = µ γ µ b − , where the Riemannian metric b − is the inverse of b .Since equation (6) can be written intrinsically as ( c ◦ Ψ ) J Ψ = c , we get c = c ◦ Ψ − J Ψ ◦ Ψ − = ( c ◦ Ψ − ) µ b − µ γ . If c = const, we get the expression c ( b ) = c µ b − µ γ which is the intrinsic version of (7). Summary of the variables in the Lagrangian and Eulerian descriptions.
From thediscussion above, the independent variables in the Lagrangian descriptions are the two embed-dings and the infinitesimal volume, i.e. , Ψ ( t, X s ) , ϕ ( t, X f ) , V ( t, X s ) . (16)In the Eulerian description the variables are u f ( t, x ) , u s ( t, x ) , v ( t, x ) , g ( t, x ) , ρ s ( t, x ) , b ( t, x ) , (17)defined from the Lagrangian variables in (1), (8), (10), (12), (14), respectively. arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media Lagrangian.
For classical elastic bodies, the potential energy in the spatial description de-pends on the Finger deformation tensor b , i.e. , V = V ( b ). If the pores are present and theirvolumes change due to their expansion or contraction, the potential energy changes even whenthere are no net deformation of the porous media. Thus, the potential energy of elastic porousmaterial must depend on both variables b and v , and we write V = V ( b, v ).The Lagrangian of the porous medium is the sum of the kinetic energies of the fluid andelastic body minus the potential energy of the elastic deformations: (cid:96) ( u f , u s , ρ s , b, g, v ) = ˆ B (cid:20) ρ f g | u f | + 12 ρ s | u s | − V ( b, v ) (cid:21) d x . (18)Note that the expression (18) explicitly separates the contribution from the fluid and theelastic body in simple physically understandable terms. The interaction between the fluid andthe media comes from the critical action principle involving the incompressibility of the fluid.We shall derive the equations of motion for an arbitrary (sufficiently smooth) expression for (cid:96) ( u s , u f , ρ s , b, g, v ), and will use the physical Lagrangian (18) for all computations in the paper. Variational principle and incompressibility constraint.
Condition (9) represents ascalar constraint for every point of an infinite-dimensional system. Formally, such constraintcan be treated in terms of Lagrange multipliers. The application of the method of Lagrangemultipliers for an infinite-dimensional system is quite challenging, see recent review papers[45, 46]. In terms of classical fluid flow, in the framework of Euler equations, the variationaltheory introducing incompressibility constraint has been developed by V. I. Arnold [42], withthe Lagrange multiplier for incompressibility related to the physical pressure in the fluid. Wewill follow in the footsteps of Arnold’s method and introduce a Lagrange multiplier for theincompressibility condition (9). By analogy with Arnold, we will also treat this Lagrange mul-tiplier as related to pressure, as it has the same dimensions, and denote it p . Since (9) refers tothe fluid content, the Lagrange multiplier p relates to the fluid pressure. This will be furtherjustified by the equations of motion (32) below, connecting pressure with the derivatives ofthe potential energy with respect of the pores’ volume. Note that p may be different fromthe actual physical pressure in the fluid depending on the implementation of the model. Fromthe Lagrangian (18) and the constraint (9), we define the action functional in the Euleriandescription as S = ˆ T (cid:20) (cid:96) ( u f , u s , ρ s , b, g, v ) − ˆ B p (cid:0) g − c ( b ) v (cid:1) d x (cid:21) d t . (19)The equations of motion are obtained by computing the critical points of S with respect toconstrained variations of the Eulerian variables induced by free variations of the Lagrangianvariables. Indeed, it is in the Lagrangian description that the variational principle is justified,as being given by the Hamilton principle with constraint. One also notes that the constraint(9) is holonomic when expressed in terms of the Lagrangian variables (16) via the relations(8), (10), (14). This justifies that this constraint can be incorporated via the introduction of aLagrange multiplier. The constrained variations of the Eulerian variables induced by the freevariations δ Ψ , δ ϕ vanishing at t = 0 , T are computed by using the relations (1), (10), (12), arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media δ u f = ∂ t η f + u f · ∇ η f − η f · ∇ u f δ u s = ∂ t η s + u s · ∇ η s − η s · ∇ u s δg = − div( g η f ) δρ s = − div( ρ s η s ) δb = − £ η s b , (20)where η f and η s are defined η f = δ ϕ ◦ ϕ − , η s = δ Ψ ◦ Ψ − (21)and the variations δv and δp are arbitrary. In the case of the boundary conditions (2) it followsfrom (21) that η f and η s are arbitrary time dependent vector fields vanishing at t = 0 , T andtangent to the boundary ∂ B : η s · n = 0 , η f · n = 0 . (22)In the case of no-slip boundary conditions (3), we have η f | ∂ B = 0 , η s | ∂ B = 0 , (or only η s | ∂ B = 0) . (23) Incorporation of external and friction forces.
Frictions forces, or any other forces,acting on the fluid F f and the media F s can be incorporated into the variational formulationby using the Lagrange-d’Alembert principle for external forces. This principle reads δS + ˆ B (cid:0) F f · η f + F s · η s (cid:1) d x d t = 0 , (24)where S is defined in (19) and the variations are given by (20). Such friction forces are usuallypostulated from general physical considerations. If these forces are due exclusively to friction,the forces acting on the fluid and media at any given point must be equal and opposite, i.e. F f = − F s , in the Eulerian treatment we consider here. For example, for porous media it iscommon to posit the friction law F f = − F s = K ( u s − u f ) , (25)with K being a positive definite matrix potentially dependent on material parameters andvariables representing the media. In particular, the matrix K depends on the local porosity,composition of the porous media, deformation and other variables. The general functional formof dependence of K on the variables should be of the form K = K ( b, g ). For example, whendeformations of porous media are neglected, i.e. , assuming and isotropic and a non-movingporous matrix with b = Id, Kozeny-Carman equation is often used, which in our notationis written in the form K = κg / (1 − g ) , with κ being a constant, see [47] for discussion.In general, the derivation of the dependence of tensor K on variables g and b from the firstprinciples is difficult, and should presumably be obtained from experimental observations. Ingeneral, the anisotropy of K is related to the geometry of the pores. The shape of the poresand their distribution in space will dictate the numerical values of K for each given point inspace, and the deformation of the pores’ geometry will determine the functional dependence K = K ( b, g ). For the purpose of this paper, we will implicitly assume the dependence on flowvariables without specifying them explicitly in the formulas. For computations in Section 4dedicated to the description of propagation of linear disturbances about the steady state, such arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media K on variables is not important. If there are other external forces acting on thesystem, then, in general, F f + F s (cid:54) = 0. This situation can happen, for example, if either thefluid or the media is either electrostatically charged or laden with magnetic particles, and issubjected to the electric or magnetic field. The equations that we derive in the general settingare valid for arbitrary external forces F f and F s . For explicit computations, we assume theexpression (25). General form of the equations of motion.
In order to derive the equations of motion,we take the variations in the Lagrange-d’Alembert principle (24) as δS + ˆ B (cid:0) F f · η f + F s · η s (cid:1) d x d t = ˆ B (cid:20) δ(cid:96)δ u f · δ u f + δ(cid:96)δ u s · δ u s + δ(cid:96)δρ s δρ s + (cid:18) δ(cid:96)δb + pv ∂c∂b (cid:19) : δb
12 + (cid:18) δ(cid:96)δg − p (cid:19) δg + (cid:18) δ(cid:96)δv + pc ( b ) (cid:19) δv + (cid:0) g − c ( b ) v (cid:1) δp
12 + F f · η f + F s · η s (cid:21) d x d t = 0 . (26)The symbol “ : ” denotes the contraction of tensors on both indices. Substituting the ex-pressions for variations (20), integrating by parts to isolate the quantities η f and η s , anddropping the boundary terms leads to the expressions for the balance of the linear momentumfor the fluid and porous medium, respectively, written in the Eulerian frame. This calculationis tedious yet straightforward for most terms and we omit it here. The main difficulty is thecalculation of the terms related to the evolution of the tensor b , which we now show in somedetails.Denoting by Π the 2-covariant symmetric tensor field δ(cid:96)δb + pv ∂c∂b , we compute the fourthterm on the right hand side of (26) by using (15) as follows: ˆ B Π : δb = − ˆ B (Π : £ η b )d x = − ˆ B Π ij (cid:18) ∂b ij ∂x k η k − b kj ∂η i ∂x k − b ik ∂η j ∂x k (cid:19) d x = − ˆ B (cid:18) Π ij ∂b ij ∂x k η k + η i ∂∂x k (cid:16) Π ij b kj (cid:17) + η j ∂∂x k (cid:16) Π ij b ik (cid:17)(cid:19) d x + ˆ B ∂∂x k (cid:16) Π ij b kj η i + Π ij b ik η j (cid:17) d x = − ˆ B (cid:18) Π ij ∂b ij ∂x k + ∂∂x i (cid:0) Π kj b ij (cid:1) + ∂∂x j (cid:0) Π ik b ij (cid:1)(cid:19) η k d x + 2 ˆ ∂ B Π ij b kj η i n k d s = − ˆ B (cid:18) Π ij ∂b ij ∂x k + 2 ∂∂x i (cid:0) Π kj b ij (cid:1)(cid:19) η k d x + 2 ˆ ∂ B (Π ij b kj n k ) η i d s , (27)where in three-dimensional case, n i are the components of the normal vector n . For compact- For a general metric G , the rigorous statement is that n i is the one form associated to the normal vectorfield n via the Riemannian metric. arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media diamond operator (Π (cid:5) b ) k = − Π ij ∂b ij ∂x k − ∂∂x i (cid:0) Π kj b ij (cid:1) (28)whose coordinate-free form readsΠ (cid:5) b = − Π : ∇ b − · b ) . (29)The result of (27) thus reads ˆ B Π : δb = ˆ B (Π (cid:5) b ) · η d x + 2 ˆ ∂ B [(Π · b ) · n ] · η d s. (30)The equations of motion also naturally involve the expression of the Lie derivative of amomentum density, whose global and local expressions are £ u m = u · ∇ m + ∇ u T · m + m div u ( £ u m ) i = ∂ j m i u j + m j ∂ i u j + m i ∂ j u j . (31)With these notations, the Lagrange-d’Alembert principle (26) yields the system of equations ∂ t δ(cid:96)δ u f + £ u f δ(cid:96)δ u f = g ∇ (cid:18) δ(cid:96)δg − p (cid:19) + F f ∂ t δ(cid:96)δ u s + £ u s δ(cid:96)δ u s = ρ s ∇ δ(cid:96)δρ s + (cid:18) δ(cid:96)δb + pv ∂c∂b (cid:19) (cid:5) b + F s δ(cid:96)δv = − pc ( b ) , g = c ( b ) v∂ t g + div( g u f ) = 0 , ∂ t ρ s + div( ρ s u s ) = 0 , ∂ t b + £ u s b = 0 . (32)When the boundary conditions (3) are used, no additional boundary condition arise fromthe variational principle. In the case of the free slip boundary condition (2), the variationalprinciple yields the condition[ σ p · n ] · η = 0 , for all η parallel to ∂ B , (33)where σ p := − (cid:18) δ(cid:96)δb + pv ∂c∂b (cid:19) · b. (34)This is shown by using (30). Physically, the condition (33) states that the force t = σ · n exerted at the boundary must be normal to the boundary (free slip).The first equation arises from the term proportional to η f in the application of theLagrange-d’Alembert principle. The second condition and the boundary condition (33) arisefrom the term proportional to η s and via the use of (30). The third and fourth equations arisefrom the variations δv and δp . The last three equations follow from the definitions (10), (12),(14), respectively. In the derivation of (32), we have used the fact that on the boundary ∂ B , η s and η f satisfy the boundary condition (22). Remark 2.1 (Discussion of the form of the Lagrangian) . Equations (32) allow for an arbitraryform of the dependence of the Lagrangian on the variables. The derivatives of the Lagrangianwith respect to the variables entering (32) should be considered to be variational derivatives. arkhutdinov, Gay-Balmaz, Putkaradze
Geometric variational approach to porous media ρ s and its spatial derivatives ∇ ρ s , e.g. (cid:96) = ˆ B (cid:96) ( ρ s , ∇ ρ s , u s , . . . )d x then δ(cid:96)δρ s = ∂(cid:96) ∂ρ s − div ∂(cid:96) ∂ ∇ ρ s , and similarly with other variables such as u s , ρ f , v etc . Thus, equations (32) are capable ofincorporating very general physical models of the porous media. However, it is important tonote that in our model, we do not assume that the energy of the fluid depends on any kindof strain measure of the solid or the fluid. The pressure p in (32) is obtained purely from theaction principle with the action (19). In that sense, our paper follows the framework of fluiddescription due to Arnold [42]. Specific form of the equations.
We now use the Lagrangian function (cid:96) defined in (18)and compute the derivatives δ(cid:96)δ u f = ρ f g u f , δ(cid:96)δ u s = ρ s u s , δ(cid:96)δρ s = 12 | u s | ,δ(cid:96)δb = − ∂V∂b , δ(cid:96)δg = 12 ρ f | u f | , δ(cid:96)δv = − ∂V∂v . (35)For the Lagrangian in (18), using (29) and the third and fourth equations in (32), the diamondterm in (32) simplifies as (cid:18) − ∂V∂b + pv ∂c∂b (cid:19) (cid:5) b = − (cid:18) pv ∂c∂b − ∂V∂b (cid:19) : ∇ b − (cid:20)(cid:18) pv ∂c∂b − ∂V∂b (cid:19) · b (cid:21) = g ∇ p + ∇ (cid:18) V − ∂V∂v v (cid:19) − (cid:20)(cid:18) pv ∂c∂b − ∂V∂b (cid:19) · b (cid:21) . Then, the equations of motions (32) become ρ f ( ∂ t u f + u f · ∇ u f ) = −∇ p + 1 g F f ρ s ( ∂ t u s + u s · ∇ u s ) = g ∇ p + ∇ (cid:18) V − ∂V∂v v (cid:19) − (cid:20)(cid:18) pv ∂c∂b − ∂V∂b (cid:19) · b (cid:21) + F s ∂V∂v = pc ( b ) , g = c ( b ) v∂ t g + div( g u f ) = 0 , ∂ t ρ s + div( ρ s u s ) = 0 , ∂ t b + £ u s b = 0 . (36)together with the boundary condition (33) in which the stress tensor σ p in (34) reads σ p = − (cid:18) pv ∂c∂b − ∂V∂b (cid:19) · b , ( σ p ) ik = − (cid:18) pv ∂c∂b kj − ∂V∂b kj (cid:19) b ij . (37)The divergence term in the media momentum equation (second equation above) is theanalogue of the divergence of the stress tensor for an ordinary elastic media: This term,however, contains the contribution from both the potential energy and the fluid pressure.These equations define the coupled motion of an incompressible fluid and porous media.We are not aware of these equations having been derived before. arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media Remark 2.2 (Equations of motion with external equilibrium pressure) . If the media is sub-jected to a uniform external pressure p , then the equations of motion are derived by changingthe Lagrangian to (cid:96) p → (cid:96) + ( p − p )( g − c ( b ) v ). In that case, equations (32), and, similarly, (36)are altered by simply substituting p − p instead of p . In what follows, we shall put p = 0. We are now going to proceed to prove that our model yields strict dissipation of mechanicalenergy in the presence of friction forces. This is important in order to demonstrate that ourderivation is physically consistent. Fortunately, variational methods are guaranteed to provideenergy conservation for the absence of friction, and when the friction forces are introducedcorrectly, also guaranteed to provide energy dissipation. Let us consider the energy densityassociated to the Lagrangian (cid:96) given by e = u f · δ(cid:96)δ u f + u s · δ(cid:96)δ u s + ˙ v δ(cid:96)δ ˙ v − L , (38)where L denotes the integrand of (cid:96) . Note that in our case (cid:96) does not depend on ˙ v hence thethird term vanishes. For the general system (32), and its explicit form (36), to be physicallyconsistent, we need to prove that in the absence of forces F s and F f , the total energy E = ´ B e d x is conserved. When these forces are caused by friction, we must necessarily have˙ E ≤ u · £ u m = u · (cid:16) u · ∇ m + ∇ u T · m + m div u (cid:17) = div (cid:0) u ( m · u ) (cid:1) , (39)which easily follows from its coordinates expression in (31). Then, using equation (39) andsystem (32), we compute ∂ t e = u f · ∂∂t δ(cid:96)δ u f + u s · ∂∂t δ(cid:96)δ u s − δ(cid:96)δρ s ∂ t ρ s − δ(cid:96)δb : ∂ t b − δ(cid:96)δg ∂ t g − δ(cid:96)δv ∂ t v = − div (cid:20) u f (cid:18) u f · δ(cid:96)δ u f (cid:19) + u s (cid:18) u s · δ(cid:96)δ u s (cid:19) − (cid:18) δ(cid:96)δg − p (cid:19) g u f − δ(cid:96)δρ s ρ s u s + 2 u s · (cid:18) δ(cid:96)δb + pv ∂c∂b (cid:19) · b (cid:21) + (cid:18) δ(cid:96)δg − p (cid:19) ∂ t g + δ(cid:96)δρ s ∂ t ρ s + (cid:18) δ(cid:96)δb + pv ∂c∂b (cid:19) ∂ t b − δ(cid:96)δρ s ∂ t ρ s − δ(cid:96)δb : ∂ t b − δ(cid:96)δg ∂ t g − δ(cid:96)δv ∂ t v + u s · F s + u f · F f = − div J − p∂ t g + pv ∂c∂b : ∂ t b − δ(cid:96)δv ∂ t v + u s · F s + u f · F f , (40)where we denoted by J the vector field in the brackets inside the div operator. The last termin these brackets has the local expression (cid:18) u s · (cid:18) δ(cid:96)δb + pv ∂c∂b (cid:19) · b (cid:19) k = 2 u is (cid:18) δ(cid:96)δb ij + pv ∂c∂b ij (cid:19) b jk = − σ p · u s . The sum of the second, third, and fourth terms in last line of (40) cancel thanks to the thirdand fourth equations in (32). We thus get the energy balance ∂ t e + div J = u s · F s + u f · F f . arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media E = ˆ B ( u s · F s + u f · F f ) d x − ˆ ∂ B J · n d s . (41)From the boundary conditions (2) and (33) we have u s · n = 0, u f · n = 0, and [ σ p · n ] · u s = 0 onthe boundary ∂ B , so that J · n = at the boundary. In the case of the boundary conditions (3),we have J | ∂ B = 0. In the absence of external forces, when F f and F s are caused exclusivelyby the friction between the porous media and the fluid, we have F f = − F s . Since in that case˙ E ≤
0, we must necessarily have˙ E = ˆ B F s · ( u s − u f ) d x ≤ . (42)If one assumes (25) for the friction, i.e. , F s = K ( u s − u f ), then K must be a positive operator, i.e. , K v · v ≥
0, for all v ∈ R and for any point x ∈ B . Let us start with connecting to the case considered frequently in the literature, namely, thecase of a compressible fluid moving inside a matrix made out of elastic compressible material.In this case, the fluid pressure is no longer a Lagrange multiplier, but has to be found fromthe identities regarding the internal energy of the fluid as a function of its density. We referthe reader to [48] for background in classical thermodynamics. If the volume fraction occupiedby the fluid is φ , the volume fraction of the elastic matrix is then 1 − φ . In the generalthermodynamic description, the specific internal energy of the material e is a function of itsdensity ρ and specific entropy S , with the pressure being given as p = ρ ∂e∂ρ . This formulais correct whether the thermodynamics effects are considered, i.e. S is varying, or ignored, i.e. S =const. If the effective density of the fluid is ρ f , and its volume fraction is φ , then themicroscopic density of the fluid is ¯ ρ f = ρ f /φ , so the internal energy of the fluid is a functionof ¯ ρ f , i.e. , e f = e f ( ¯ ρ f ). Similarly, the microscopic density of the solid is ¯ ρ s = ρ s / (1 − φ ). Itis natural to assume that the internal energy of the elastic solid depends on both ¯ ρ s and theFinger deformation tensor b , e s = e s ( ¯ ρ s , b ). Thus, the physically relevant Lagrangian takes theform (cid:96) ( u f , u s , ρ f , ρ s , b, φ ) = ˆ B (cid:20) ρ f | u f | + 12 ρ s | u s | − ρ f e f (cid:18) ρ f φ (cid:19) − ρ s e s (cid:18) ρ s − φ , b (cid:19)(cid:21) d x. (43)Proceeding as in the derivation of (32), we obtain the following system, written in terms of ageneral Lagrangian: ∂ t δ(cid:96)δ u f + £ u f δ(cid:96)δ u f = ρ f ∇ δ(cid:96)δρ f ∂ t δ(cid:96)δ u s + £ u s δ(cid:96)δ u s = ρ s ∇ δ(cid:96)δρ s − δ(cid:96)δb : ∇ b − δ(cid:96)δb · b∂ t ρ f + div( ρ f u f ) = 0 , ∂ t ρ s + div( ρ s u s ) = 0 , ∂ t b + £ u s b = 0 δ(cid:96)δφ = 0 (44) arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media ρ f ( ∂ t u f + u f · ∇ u f ) = − ρ f ∇ (cid:18) e f + ¯ ρ f ∂e f ∂ ¯ ρ f (cid:19) = − φ ∇ (cid:18) ¯ ρ f ∂e f ∂ ¯ ρ f (cid:19) ρ s ( ∂ t u s + u s · ∇ u s ) = − ρ s ∇ (cid:18) e s + ¯ ρ s ∂e s ∂ ¯ ρ s (cid:19) − ρ s ∂e s ∂b : ∇ b + 2 div ∂e s ∂b · b = − (1 − φ ) ∇ (cid:18) ¯ ρ s ∂e s ∂ ¯ ρ s (cid:19) + 2 div ∂e s ∂b · b∂ t ρ f + div( ρ f u f ) = 0 , ∂ t ρ s + div( ρ s u s ) = 0 , ∂ t b + £ u s b = 0¯ ρ f ∂e f ∂ ¯ ρ f = ¯ ρ s ∂e s ∂ ¯ ρ s =: p where ¯ ρ f := ρ f − φ , ¯ ρ s = ρ s φ . (45)The last equation, coming from the variation in δφ , states the equality of pressure in bothelastic and fluid part of the system. We can transform the system to the following form: ρ f ( ∂ t u f + u f · ∇ u f ) = − φ ∇ pρ s ( ∂ t u s + u s · ∇ u s ) = − (1 − φ ) ∇ p + div σ el ∂ t ρ f + div( ρ f u f ) = 0 , ∂ t ρ s + div( ρ s u s ) = 0 , ∂ t b + £ u s b = 0 (46)where p := ¯ ρ f ∂e f ∂ ¯ ρ f = ¯ ρ s ∂e s ∂ ¯ ρ s , σ el := 2 ∂e s ∂b · b Equations similar to (46) appear, for example in [20], with additional thermodynamical effects.These thermodynamics effects can be incorporated in our model as well if we allow the energiesof the fluid and solid part in the Lagrangian (43) to depend on the entropies of fluid S f andsolid S s , such as e f = e f ( ¯ ρ f , S f ) and e s = e f ( ¯ ρ s , S s ), with additional equations for advectionof the entropy and heat exchange between the two phases. We shall postpone this discussionof thermal effect for our follow-up work in order not to distract from the main message of thepaper. However, within the framework of this paper, it is worth noting that the internal energiesof the fluid and solid are completely separated: the internal energy of the fluid depends onlyon the internal variables of the fluid, and, correspondingly, the internal energy of the elasticmatrix depends only on the internal variables of the elastic material. The interaction betweenthe terms comes from equality of pressure and follows from the equations of motion; it doesnot have to be assumed a priori . Thus, we believe, our approach is consistent with the classicalLagrangian approach of dealing with the systems with several interacting parts. Let us now connect this description of compressible fluid and solid to the case of incompressiblefluid and compressible solid. We shall keep the same variables as in the derivation of (45) tokeep the notation consistent, and then show how to connect the resulting equations with(32). The difference between the cases of compressible and incompressible fluids comes to twofundamental restrictions:1. Since the microscopic density of fluid ¯ ρ f , also denoted ρ f earlier, is constant, the internalenergy of the fluid do not depend on ¯ ρ f . arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media φ = ( φ ◦ ϕ − ) J ϕ − , equivalent to (10). We remindthe reader that ϕ − ( x , t ) is the inverse of the Lagrangian mapping for fluid particles, alsoknown as the back-to-labels map. Physically, this law states that all the fluid in a givenmicroscopic volume of porous media has appeared from its initial source at t = 0.Note that the incompressibility condition presented above is similar to the conservation ofmass in [25] (Eq. (10) taken for the case of fluid only). In spite of this similarity, thereis an important difference to keep in mind: in [25], the conservation law is written for both compressible fluid and solid parts. In our case, no additional conservation laws are necessary inthe case of compressible fluid and solid, so there is only one incompressibility condition for fluidfor the incompressible fluid case, and none for the compressible fluid case. The conservationlaw for the compressible part in our theory is satisfied automatically, and no extra Lagrangemultipliers are necessary. The action functional (19), incorporating the constraint with theLagrange multiplier p , rewritten in the new variables, becomes S p = ˆ T (cid:20) (cid:96) ( u f , u s , ρ f , ρ s , b, φ ) + ˆ B p (cid:0) φ − ( φ ◦ ϕ − ) J ϕ − (cid:1) d x (cid:21) d t . (47)While the method works for an arbitrary Lagrangian, the physically relevant form of theLagrangian to consider is given by (cid:96) ( u f , u s , ρ f , ρ s , b, φ ) = ˆ B (cid:20) ρ f | u f | + 12 ρ s | u s | − ρ s e s (cid:18) ρ s − φ , b (cid:19)(cid:21) d x. (48)Note that compared to the previous form for compressible fluid case (43), the term ρ f e f ( ¯ ρ f )is now absent from (48). Using the identity δ (cid:2) ( φ ◦ ϕ − ) J ϕ − (cid:3) = − div (cid:0) ( φ ◦ ϕ − ) J ϕ − η f (cid:1) , (49)we get the following set of equations written for a general Lagrangian (cid:96) : ∂ t δ(cid:96)δ u f + £ u f δ(cid:96)δ u f = ρ f ∇ δ(cid:96)δρ f − φ ∇ p∂ t δ(cid:96)δ u s + £ u s δ(cid:96)δ u s = ρ s ∇ δ(cid:96)δρ s − δ(cid:96)δb : ∇ b − δ(cid:96)δb · b∂ t ρ f + div( ρ f u f ) = 0 , ∂ t ρ s + div( ρ s u s ) = 0 , ∂ t b + £ u s b = 0 φ = ( φ ◦ ϕ − f ) J ϕ − f , δ(cid:96)δφ + p = 0 . (50)In the case of the physically relevant Lagrangian (48), we obtain ρ f ( ∂ t u f + u f · ∇ u f ) = − φ ∇ pρ s ( ∂ t u s + u s · ∇ u s ) = − ρ s ∇ (cid:18) e s + ¯ ρ s ∂e s ∂ ¯ ρ s (cid:19) − ρ s ∂e s ∂b : ∇ b + 2 div ∂e s ∂b · b = − (1 − φ ) ∇ (cid:18) ¯ ρ s ∂e s ∂ ¯ ρ s (cid:19) + 2 div ∂e s ∂b · b∂ t ρ f + div( ρ f u f ) = 0 , ∂ t ρ s + div( ρ s u s ) = 0 , ∂ t b + £ u s b = 0 ∂ t φ + div( φ u f ) = 0 , ¯ ρ s ∂e s ∂ ¯ ρ s = p. (51) arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media ∂ t ρ f + div( ρ f u f ) = 0 and ∂ t φ + div( φ u f ) = 0 imply that ρ f = ¯ ρ f φ with ¯ ρ f aconstant. Note that the last equation of (51), states that the thermodynamic pressure inthe solid, defined through the derivatives of the internal energy function e s , is equal to theLagrange multiplier p . Thus, physically, the Lagrange multiplier p is equal to the pressureinside the solid, so it also acquires the physical meaning of the pressure in the fluid. However,that physical meaning is elucidated only after the equations of motion (51) are derived andcannot be inferred a priori .A quick calculation shows that the system (51) is equivalent to the equations (32) derivedearlier, under the change of variables g = φ, c ( b ) = ρ s , v = 1 ρ s − ρ s . (52)That equivalence is proved by assuming the internal energy of the solid in the form V ( b, v ) = ρ s ( b ) e s ( ¯ ρ s , b ) , ¯ ρ s := ρ s ( b )1 − ρ s ( b ) v . (53)Substitution of that expression for the internal energy of the solid into (32) gives (51). Webelieve that such calculation is useful since it connects our earlier derivation (32) with theinformation on the compressible case, and also elucidates the nature of the variable φ . It isuseful to recall the quote from [19] mentioned in the Introduction, where the nature of thisvariable was suggested to preclude the existence of a variational principle. Our theory presentedhere shows that the variable describing the fluid content has to be considered carefully in thevariational principle (47), or, equivalently, in (19) earlier, as a constraint through the geometricvariational formulation presented here. The understanding of the role of this variable, webelieve, is the key to the derivation of the variational principle for porous media, and wasperhaps the source of difficulty in explaining the incompressible fluid case in previous works.The physical meaning of v becomes clear from the last formula of (52). Indeed, choose m s to be a given mass of elastic solid, then m s ρ s is the volume of occupied by the porous elasticsolid, and m s ¯ ρ s is the volume occupied by the (imaginary) elastic solid without any porosity.Thus, the quantity m s (cid:16) ρ s − ρ s (cid:17) is the volume occupied by the fluid per unit mass of the solid,and therefore the quantity v = ρ s − ρ s is the physical meaning of specific volume of the fluid’scontent, measured per unit mass of the elastic solid. We linearize equations (36) about the equilibrium state( u f , u s , ρ s , b, g, v, p ) = ( , , ρ s , b , g , v , p ) , (54)where each component on the right-hand side of (54) with a subscript 0 is a constant. Theequilibrium condition reads ∂V∂v (cid:12)(cid:12)(cid:12)(cid:12) = p c . (55)where F | denotes the value of a function F taken at the equilibrium (54). We consider thepotential V ( b, v ) to be general and assume, for simplicity, an unstressed state b = Id and arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media p = 0. Throughout this section, we shall assume friction forces of the form (25) with agiven constant general permeability tensor K . For simplicity of computations, we will eventu-ally further assume isotropic and uniform media, so the permeability tensor K will be takenproportional to a unity matrix. Notation.
In this chapter on linearization, we denote the value of a variable f evaluated atthe equilibrium with the index 0, i.e. , f . The spatiotemporal deviation from the equilibriumis then denoted as δf ( x , t ) (cid:39) f ( x , t ) − f ( x , t ), with δf assumed small. Note that this is thesame notation δ as for the variations used in the previous chapter. We hope that no confusionarises due to that clash of notation. Expression of the stress tensor.
The full stress tensor computed from (37) is σ p = σ el + c vpJ Id = σ el + cvp Id = σ el + gp Id , J = √ det b , where for c ( b ) = c /J , we used ∂c∂b = − c J b − and where σ el = 2 ∂V∂b · b is the elastic stress tensor associated to the potential V . The linearization of the full stresstensor is δσ p = δσ el + g δp Id , (56)where we recall that we chose p = 0 and that b = Id , so J | = 1. The linearization of theelastic stress tensor is written as δσ el = ∂σ el ∂b (cid:12)(cid:12)(cid:12)(cid:12) : δb + ∂σ el ∂v (cid:12)(cid:12)(cid:12)(cid:12) δv = 2 ∂ V∂b (cid:12)(cid:12)(cid:12)(cid:12) : δb + 2 ∂V∂b (cid:12)(cid:12)(cid:12)(cid:12) · δb + 2 ∂ V∂b∂v (cid:12)(cid:12)(cid:12)(cid:12) δv. (57) Linearization.
The system (36) is linearized as follows: g ρ f ∂ t δ u f = − g ∇ δp + K ( δ u s − δ u f ) ρ s ∂ t δ u s = ∇ (cid:18) ∂V∂b (cid:12)(cid:12)(cid:12)(cid:12) : δb (cid:19) + div δσ p + K ( δ u f − δ u s ) ∂ V∂v (cid:12)(cid:12)(cid:12)(cid:12) δv + ∂ V∂v∂b (cid:12)(cid:12)(cid:12)(cid:12) : δb = c δp , δg = − c δb ) v + c δv∂ t δg + div( g δ u f ) = 0 , ∂ t δρ s + div( ρ s δ u s ) = 0 ,∂ t δb − δ u s = 0 , Def δ u s := (cid:16) ∇ δ u s + [ ∇ δ u s ] T (cid:17) . (58)To get the linearized balance of elastic momentum, we used the fact that the linearization ofthe term ∇ ( V − v ∂V∂v ) = ∇ ( V − pg ) for p = 0 in the second equation of (36) is computed as δ ∇ ( V − pg ) = ∇ (cid:16) ∂V∂v (cid:12)(cid:12)(cid:12) (cid:124) (cid:123)(cid:122) (cid:125) = p c =0 δv + ∂V∂b (cid:12)(cid:12)(cid:12) : δb (cid:17) − g ∇ δp . (59) arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media £ u s b at u s, = 0 and b = Id is − δ u s as a direct computation using (15) shows.For the linearized equations, we shall only need the coefficients of the linear and thequadratic expansions of the potential V ( b, v ) about the equilibrium. We thus define the coef-ficients: σ = ∂V∂b (cid:12)(cid:12)(cid:12)(cid:12) , ζ = v c ∂ V∂v (cid:12)(cid:12)(cid:12)(cid:12) , C = ∂ V∂b (cid:12)(cid:12)(cid:12)(cid:12) , D = ∂ V∂v∂b (cid:12)(cid:12)(cid:12)(cid:12) . (60)The coefficient ζ , from its definition, has the order of magnitude of the bulk modulus ofthe microscopic material itself, although it can depend on the pores concentration and theirarrangement in the matrix. Using this, the potential energy of the elastic deformation V ( b, v )about the equilibrium, up to the second order in deviations from equilibrium, and assuming V ( b , v ) = 0, is represented as V ( b, v ) (cid:39) σ : ( b − b ) + 12 ( b − b ) : C : ( b − b ) + c ζ v ( v − v ) + D : ( b − b )( v − v ) . (61)From (60) and (57), we have δσ p = δσ el + g δp Id = 2 C : δb + 2 σ · δb + 2 D δv + g δp Id . (62)The first term identifies the Hooke law connecting the linearized stress and linearized strain (cid:15) as follows σ := 2 ∂ V∂b (cid:12)(cid:12)(cid:12)(cid:12) : δb = 2 C : δb = 4 C : (cid:15) , (cid:15) := 12 δb (cid:39)
12 ( b − b ) , (63)where the definition of (cid:15) above is understood as a linearization of b about the equilibrium.We have intentionally denoted this linearized part of Finger tensor as (cid:15) since it happens to beexactly the standard linear strain used in elasticity, see (64) below.We shall now assume an isotropic and uniform material, which will be the case of study forthe remainder of the paper. Then, the tensor C in (63) has only two independent coefficients,and (63) becomes the familiar Hooke law for isotropic uniform materials, i.e. , σ ( (cid:15) ) = 4 C : (cid:15) = 2 G(cid:15) + ΛTr( (cid:15) ) Id , (64)where Λ and G are well known as Lam´e parameters for isotropic materials in continuummechanics . Furthermore, the tensors σ and D in (60) are proportional to the unit tensor,and it is convenient to express them as follows: σ = 12 g µ Id , D = 12 c ξ Id , with µ, ξ = const . (65)The constants µ and ξ , defined above, as well as coefficient ζ defined by (60), have the dimensionof Young’s modulus, i.e. , pressure. With all these assumptions (62) becomes δσ p = Λ2 Tr( δb ) Id + ( G + g µ ) δb + c ξδv Id + g δp Id . (66) Sometimes Lam´e coefficients are denoted λ and µ . We will avoid that notation since it clashes with thenotation used here, where λ denotes the growth rate of the disturbances as defined in (67), and µ characterizingthe residual stress according to (65). We hope no confusion arises from our use of notation for Lam´e coefficients. arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media Linear stability.
We now set δ u s = λ v e λt + i k · x , δ u f = λ u e λt + i k · x , δρ s = ρ s, e λt + i k · x , δb = b e λt + i k · x ,δg = g e λt + i k · x , δv = v e λt + i k · x , δp = p e λt + i k · x . (67)and equations (58) become g ρ f λ u = − g p i k + λ K ( v − u ) ρ s λ v = 12 (Λ + g µ )Tr( b ) i k + i ( G + g µ ) b · k + i k ( g p + c ξv ) + λ K ( u − v ) ζv v + ξ b ) = p , g = − c b ) v + c v ,g + g i ( u · k ) = 0 , ρ s, + ρ s i ( v · k ) = 0 ,b − i ( v ⊗ k + k ⊗ v ) = 0 . (68)By using the expression c ( b ) = c /J and the last equation in (68), we get c = − c Tr( b ) / − c i ( v · k ) so, from the constraint g = cv we have g = c v + c v = c ( v − iv ( v · k )) . (69)From the linearized continuity equation for g in (68) we get g = − ig ( u · k ), so combiningthis with (69) we obtain the expression of v as v = iv ( v · k − u · k ) , (70)and then, substituting this result into the third equation of (68), we deduce p = i (( ζ + ξ ) ( v · k ) − ζ ( u · k )) . (71)From (70) and (71), we get g p + c ξv = ig (( ζ + 2 ξ )( v · k ) − ( ζ + ξ )( u · k )) , (72)and the first two equations in (68) become g ρ f λ u = g k (( ζ + ξ ) ( v · k ) − ζ ( u · k )) + λ K ( v − u ) ρ s λ v = − (Λ + g µ )( v · k ) k − ( G + g µ )( v | k | + k ( v · k ))+ g k [( ζ + ξ )( u · k ) − ( ζ + 2 ξ )( v · k )] + λ K ( u − v ) . (73)Let us now compute the dispersion relation explicitly for the case when the dissipation inthe media is isotropic, so K = β Id for some β >
0. In that case, we obtain the dispersionrelation det S = 0 with the matrix S of the form S = (cid:20) λ ρ f g Id 00 λ ρ s (cid:21) + λβ (cid:20) Id − Id − Id Id (cid:21) + (cid:20) g ζ A − g ( ζ + ξ ) A − g ( ζ + ξ ) A g ( ζ + ξ ) A + B (cid:21) , (74)where A := k ⊗ k , B := (Λ + G + g (2 µ + ξ )) A + ( G + g µ ) | k | Id . (75) arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media Remark 4.1 (On formal equivalence of Lam´e coefficients) . One can notice that in (74), G , Λand µ only enter in combinations G + g µ and Λ + g µ . Therefore, the acoustic properties ofthe media with Lam´e coefficients G , Λ and with ∂V∂b (cid:12)(cid:12) = µg (cid:54) = 0 are the same as the acousticproperties of the media with the Lam´e coefficients replaced by the shifted values G → G + g µ , Λ → Λ + g µ, and with ∂V∂b (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (76)Since G > G + g µ > . (77)Or, more generally, two medias with G i , Λ i and ∂V∂b (cid:12)(cid:12) = µ i g , i = 1 , G + g µ = G + g µ and Λ + g µ = Λ + g µ . Nondimensionalization.
It is convenient to define the dimensionless growth rates andwavenumbers by choosing the length scales L and time scales T : λ ∗ = T λ , k ∗ = L k . (78)and δ = ρ f g /ρ s is the ratio between the effective equilibrium density of the fluid and theequilibrium density of the elastic material.Let us define the following dimensionless matrices A ∗ = k ∗ ⊗ k ∗ , B ∗ = (cid:18) g (2 µ + ξ ) G (cid:19) A ∗ + (cid:16) g µG (cid:17) | k ∗ | Id . (79)Then, dividing S defined by (74) by ρ s , we obtain the following dimensionless dispersion matrixdefining the equation for nonlinear eigenvalues (growth rates) λ ∗ S ∗ = λ ∗ (cid:20) δ Id 00 Id (cid:21) + λ ∗ βTρ s (cid:20) Id − Id − Id Id (cid:21) + g T ρ s L (cid:20) ζ A ∗ − ( ζ + ξ ) A ∗ − ( ζ + ξ ) A ∗ ( ζ + ξ ) A ∗ (cid:21) + GT ρ s L (cid:20) B ∗ (cid:21) . (80)We are free to choose the time and length scales T and L , and we choose them in such a waythat the coefficients of λ ∗ (friction term) and the last term in (80) are equal to unity. Thiscorresponds to choosing T = ρ s β , L = T (cid:115) Gρ s . (81)Physically, T is the typical relaxation time in the porous media; L is the distance the elasticsound waves in the matrix filled with fluid propagate during that relaxation time. We thendefine the dimensionless quantities ζ ∗ = g T ρ s L ζ = g ζG , ξ ∗ = g T ρ s L ξ = g ξG , µ ∗ = g µG . (82)With these definitions, the nondimensionalized dispersion matrix takes the form: S ∗ = (cid:20) Id ( δλ ∗ + λ ∗ ) − Id λ ∗ − Id λ ∗ Id ( λ ∗ + λ ∗ ) (cid:21) + (cid:20) ζ ∗ A ∗ − ( ζ ∗ + ξ ∗ ) A ∗ − ( ζ ∗ + ξ ∗ ) A ∗ ( ζ ∗ + ξ ∗ ) A ∗ + B ∗ (cid:21) . (83)Equation det S ∗ = 0 defines a 12-th order polynomial in λ ∗ , and thus there are exactly 12roots λ ∗ = λ ∗ ( k ∗ ) in the complex plane. We now show that given k ∗ , all these roots can becomputed as S - and P -waves by considering subspaces parallel and orthogonal to a given k ∗ . arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media S -waves Let us consider the case in which ( u , v ) ⊥ k ∗ . Since A ∗ = k ∗ ⊗ k ∗ , we have A ∗ u = A ∗ v = .In other words, we only consider the displacements orthogonal to the wave vector k ∗ , whichis exactly the definition of an S -wave. We can set u ⊥ and v ⊥ to be parallel to a given vector ξ in the plane k ⊥∗ , i.e. , u ⊥ = u ξ and v ⊥ = v ξ . The eigenvalues have multiplicity 1 and arecomputed from the 2 × S ∗ ,s = (cid:20) ( δλ ∗ + λ ∗ ) − λ ∗ − λ ∗ ( λ ∗ + λ ∗ ) (cid:21) + | k ∗ | (cid:20) µ ∗ (cid:21) . (84)Since the space k ⊥∗ is two-dimensional, all the eigenvalues of det S ∗ = 0 given by (83) with u (cid:107) ξ and v (cid:107) ξ have multiplicity 2. The equation det S ∗ ,s = 0 given by (84) defines a fourth-orderpolynomial having 4 roots. Because of the multiplicity 2 of the S -waves, the total number ofroots for S -waves is 8.The condition det S ∗ ,s = 0 gives either λ ∗ = 0, or λ ∗ satisfying the following cubic equation: δλ ∗ + λ ∗ (1 + δ ) + λ ∗ k ∗ (1 + µ ∗ ) δ + k ∗ (1 + µ ∗ ) = 0 , k ∗ := (cid:107) k ∗ (cid:107) . (85)By Routh-Hurwitz’ criterion, the polynomial s + a s + a s + a is stable if a a > a . Thus,(85) is stable, i.e. , for any real k ∗ , Re λ ∗ <
0, as long as δ > δ is theratio of densities), and µ ∗ > −
1. Note that this is exactly the requirement (77) for consistencyof the media.Alternatively, instead of the dispersion relation λ ∗ = λ ∗ ( k ∗ ), it is common in the literatureto compute the attenuation of harmonic signals in porous media, in other words, k ∗ ( ω ∗ ) when λ ∗ = iω ∗ , with ω ∗ ∈ R being the frequency of forcing. In that case, from (85) we obtain k ∗ ( ω ∗ ) = ± ω ∗ (cid:115) δ + iδω ∗ (1 + µ ∗ )(1 + iδω ∗ ) . (86)As one can see, for δ > µ ∗ > −
1, Im k ∗ → ω ∗ →
0, so the attenuation oflow-frequency waves decreases with decreasing frequency, which is physically reasonable. Ifone considers propagation of waves for x >
0, one needs to choose the sign in the equation for k ∗ ( ω ∗ ) in such a way that Im k ∗ ( ω ∗ ) >
0, so the waves will be decaying as x → ∞ . P -waves Consider the case ( u , v ) (cid:107) k . In other words, we consider the disturbances parallel to thewave vector k , which is the definition of a P -wave. Then A ∗ u = ( k ∗ · u ) k ∗ = | k ∗ | u , and A ∗ v = ( k ∗ · v ) k ∗ = | k ∗ | v , and the dispersion relation det S ∗ = 0 takes the form det S ∗ ,p = 0for the 2 × S ∗ ,p = (cid:20) δλ ∗ + λ ∗ − λ ∗ − λ ∗ λ ∗ + λ ∗ (cid:21) + k ∗ (cid:20) ζ ∗ − ( ζ ∗ + ξ ∗ ) − ( ζ ∗ + ξ ∗ ) ζ ∗ + 2 ξ ∗ + Z (cid:21) , (87)where we defined for shortness k ∗ := (cid:107) k (cid:107) , Z := 2 + Λ G + 3 µ ∗ (88)and we used g ξG = ξ ∗ by (81) and (82). We rewrite this dispersion relation asdet (cid:20) δλ ∗ + λ ∗ + k ∗ ζ ∗ − λ ∗ − ( ζ ∗ + ξ ∗ ) k ∗ − λ ∗ − k ∗ ( ζ ∗ + ξ ∗ ) λ ∗ + λ ∗ + ( ζ ∗ + 2 ξ ∗ + Z ) k ∗ (cid:21) = 0 . (89) arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media λ ∗ , thus, for a given k ∗ there are 4roots corresponding to the P -waves. Combining with 8 roots for S -waves, we get the totalnumber of roots found being equal to 12, which is exactly the number of solutions for λ ∗ ( k ∗ )expected from (83). Thus, we have found all the roots of the equation (83). After computingthe determinant in (89) we get the following polynomial δλ ∗ + λ ∗ ( δ + 1)+ λ ∗ k ∗ ( ζ ∗ + δ ( Z + 2 ξ ∗ + ζ ∗ )) + λ ∗ k ∗ Z + k ∗ ( ζ ∗ Z − ξ ∗ ) = 0 . (90)For the stability of polynomial (90) we investigate the principal minors ∆ i , i = 1 , . . . δ + 1 Zk ∗ δ K K δ + 1 Zk ∗ δ K K (91)where we have defined K = ( ζ ∗ ( δ + 1) + δ ( Z + 2 ξ ∗ )) k ∗ , K = (cid:0) ζ ∗ Z − ξ ∗ (cid:1) k ∗ . (92)All ∆ i (with their exact forms given below) must be positive for stability. First, we notice thatthe conditions ∆ > > = δ + 1 > = ( δZ + ( δ + 1) ξ ∗ ) k ∗ > , (93)and are trivially satisfied.Next, we study the condition ∆ >
0. Since ∆ >
0, we can write, equivalently,∆ ∆ k ∗ = ζ ∗ Z − ξ ∗ > ⇔ ζ ∗ (cid:18) Λ G + 2 + 3 µ ∗ (cid:19) > ξ ∗ . (94)Finally, we compute the condition ∆ > k ∗ = δ Z + 2 δ ( δ + 1) ξ ∗ + ( δ + 1) ζ ∗ . (95)For stability of the steady state, we must have ζ ∗ >
0, otherwise v = v is not a stableequilibrium. Multiplying condition (95) by ζ ∗ , and adding/subtracting the term δ ξ ∗ , weobtain an equivalent formulation∆ k ∗ ζ ∗ = δ (cid:0) Zζ ∗ − ξ ∗ (cid:1) + ( δξ ∗ + ( δ + 1) ζ ∗ ) > . (96)which is satisfied as long as (94) is true. Since for physical reasons we necessarily have G > P -waves can be rewritten as ζ ∗ > G + Gµ ∗ ) + (cid:18) Λ + Gµ ∗ − G ξ ∗ ζ ∗ (cid:19) > . (97)Using the conditions (81) and (82), we can transform (97) to the following form which will beuseful for using the Sylvester criterion (106) below:2 ( G + g µ ) + (cid:18) Λ + g µ − g ξ ζ (cid:19) > . (98)We shall now show that the condition for the stability of the P -waves (94) is exactlyequivalent to the requirement for consistency of modified P -wave modulus in an isotropicmedium. arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media A digression: Linear stability of purely elastic media.
Let us now elucidate the phys-ical meaning of (94), which, as we show, is simply the condition on the stability of propagationof P -waves in an elastic media. Suppose a wave is propagating in an elastic media with Lam´ecoefficients (Λ , G ) in accordance with (64). The linearized equation for wave propagation is ρ s ∂ t δ u s = div σ ( (cid:15) ) ⇔ λ v = − G v | k | − (Λ + G )( k · v ) k , (99)where we assumed unstressed or relaxed elastic media i.e. σ = ∂V∂b (cid:12)(cid:12) = 0 so that µ = 0.For S -waves, v ⊥ k , and λ is purely imaginary if and only if G >
0. For P -waves, λ ispurely imaginary if 2 G + Λ >
0. The coefficient 2 G + Λ is also known as the P -wave modulusof the elastic media. As we shall see, the condition of positive P -wave modulus will play thecrucial part in the stability considerations.For further discussion, it is interesting to compute the general condition on the convexityof the potential energy in the purely elastic case. In this case In this case (61) reduces to V ( b ) (cid:39)
12 ( b − b ) : C : ( b − b ) (cid:39) G(cid:15) : (cid:15) + 12 Λ(Tr( (cid:15) )) = 2 G (cid:88) i>j (cid:15) ij + 12 X T Q X , X := ( (cid:15) , (cid:15) , (cid:15) ) , (100)and we have defined the quadratic form Q to be Q := G + Λ Λ ΛΛ 2 G + Λ ΛΛ Λ 2 G + Λ . (101)Assuming that the coefficients (cid:15) ij are independent numbers for a given deformations, thecondition on V to be positive definite is equivalent to the condition that the quadratic form Q is positive definite. By the Sylvester criterion, the quadratic form is positive definite if andonly if all the leading principal minors are positive, leading toa) 2 G + Λ > , b) 2 G + 2Λ > , c) 2 G + 3Λ > . (102)The first minor, i.e. , condition a) is exactly the stability of P -waves. The third condition c)is equivalent to the positivity of the bulk modulus of the material. The second condition b)follows from the first and the third conditions.As we shall see immediately below, the conditions for the well-posedness of the P -wavesand positive definite nature of the potential energy for the porous media follows closely thepurely elastic framework, with the appropriate corrections due to the dynamics of the pores v . Justification of (94) from the potential energy considerations.
Let us now considerthe case of a general potential energy V ( b, v ) locally expressed about the equilibrium accordingto the quadratic expansion (61). Without loss of generality we consider the case with no linearterms, i.e. µ = 0, since the terms proportional to µ can be absorbed into G and Λ accordingto (76), with G → G + µ and Λ → Λ + µ . We have V ( b, v ) (cid:39)
12 ( b − b ) : C : ( b − b ) + c ζ v ( v − v ) + c ξ Tr( (cid:15) )( v − v ) (cid:39) G(cid:15) : (cid:15) + 12 Λ(Tr( (cid:15) )) + c ζ v ( v − v ) + c ξ Tr( (cid:15) )( v − v ) , (103) arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media V is a quadratic form of 7 variables: ( (cid:15) ij ) (6 elements from symmetry) and ( v − v ).However, the off-diagonal elements of tensor (cid:15) , namely ( (cid:15) , (cid:15) , (cid:15) ) enter only in terms ofsquares multiplied by G >
0. Thus, we rewrite (103) in the following form: V ( b, v ) (cid:39) G (cid:88) i>j (cid:15) ij + 12 X T · Q · X , X := ( v − v , (cid:15) , (cid:15) , (cid:15) ) T , (104)where we have defined a 4 × Q as Q := c ζv c ξ c ξ c ξc ξ G + Λ Λ Λ c ξ Λ 2 G + Λ Λ c ξ Λ Λ 2 G + Λ . (105)Assuming the independence of all components of the strain tensor (cid:15) ij , we see that V is aconvex, positive definite function if and only if the quadratic form Q is positive definite.The Sylvester criterion gives four stability conditions: ∆ = c ζv > = c ζv (cid:18) G + Λ − g ξ ζ (cid:19) > = 4 Gζc v (cid:18) G + Λ − g ξ ζ (cid:19) > = det Q = 4 c G ζv (cid:20) G + 3 (cid:18) Λ − g ξ ζ (cid:19)(cid:21) > , (106)where we recall that g = c v . The first condition of this system simply enforces the convexityof V with respect to the small changes in v about the equilibrium, and is thus very natural.To investigate the remaining three conditions, let us denote˜Λ = Λ − g ξ ζ . (107)We notice that the conditions for ∆ >
0, ∆ > > → ˜Λ. Thus, the new variable defined by (107) acquiresthe physical meaning of the effective value of the second Lam´e coefficient for the porous media.We remind the reader that the coefficients ζ and ξ encode the values of the second derivativesof V with respect to v and ( v, b ) respectively, and are thus appearing only in the description ofthe porous media. No corresponding values exist for the purely elastic media. It is thus evenmore surprising that the stability criteria for the porous media can be written in the form verysimilar to the elastic media through the combination of variable (107).Note also that the condition for the P -wave stability (97) for a general µ ∗ can now bewritten using the shift (76) as ∆ > i.e. , ∆ >
0, isequivalent to the requirement that the effective bulk modulus of a dry porous media is positive.
The dispersion relation S ( u , v ) T described by (74) can be mapped to a system of linear PDEs.Let us assume, for simplicity, an isotropic media and take K = β Id . We use the mapping of arkhutdinov, Gay-Balmaz, Putkaradze
Geometric variational approach to porous media k to differential operators in Fourier space as k ⊗ k → −∇ div and | k | → − ∆ to get ρ f g ∂ ∂t u + β ∂∂t ( u − v ) − g ζ ∇ div u + g ( ζ + ξ ) ∇ div v = ρ s ∂ ∂t v − β ∂∂t ( u − v ) + g ( ζ + ξ ) ∇ div u − ( g ( ζ + 2 ξ + 2 µ ) + Λ + G ) ∇ div v − ( G + g µ ) ∆ v = . (108)Note that the contribution from pressure in our system exactly cancel, which is reasonable,as the pressure fluctuations generated by the motion of porous media in an internal force andthus must vanish. The corresponding Biot’s system is given by ∂ ∂t ( ρ ( f )22 u + ρ v ) + β ∂∂t ( u − v ) − ∇ div ( R u + Q v ) = ,∂ ∂t ( ρ ( s )11 v + ρ u ) − β ∂∂t ( u − v ) − ∇ div ( Q u + P v ) + N ∇ × ∇ × v = (109)with N being shear modulus of the skeleton and the fluid/elastic body, assumed to be thesame. We shall note that Biot’s equations is not directly applicable to an incompressible fluid,since the expressions for the variables P, Q and R in (109) involve explicitly the bulk modulusof the fluid. However, if we proceed formally and use the equations from the literature andput K f = ∞ for an incompressible fluid, the expressions for P, Q and R in terms of the bulkmodulii of the porous skeleton K b and the elastic body itself K s , see e.g. , [49] are given by P = (1 − g ) K s + 43 N , Q = g K s , R = g K s − g − K b /K s . (110)Let us turn our attention to our theory described in (108), where we have set ρ = ρ = 0.The case of ρ (cid:54) = 0 an ρ (cid:54) = 0 can be easily incorporated by considering a more generalinertia matrix in the Lagrangian. There is also an exact correspondence between the frictionterms. Thus, we need to compare the coefficients of the spatial derivative terms. A directcomparison between Biot’s linearized system (109) and (108) gives R = g ζ by observing thecoefficients of the terms proportional to ∇ div u from the equations (109). From the termproportional to ∇ div v in the first equation of (109), we obtain Q = − g ( ξ + ζ ). Finally byusing ∇ × ∇ × v = ∇ div v − ∆ v we obtain the expressions of N and P . To summarize, theBiot’s coefficients ( P, Q, R, N ) are given by R = g ζ,Q = − g ( ξ + ζ ) ,N = G + g µ,P = (Λ + g µ ) + 2( G + g µ ) + g ( ζ + 2 ξ ) . (111)Note that the expression Λ + 2 G is also known as the P − wave modulus. In our case, this P -wave modulus is modified by a shift of Lam´e coefficients by g µ and additional terms ξ and ζ coming from the elasticity properties of the porous matrix. We investigate the non-dimensionalized dispersion relations (85) and (90) derived above inorder to explore the phase velocity, group velocity, and attenuation coefficients of wave prop- arkhutdinov, Gay-Balmaz, Putkaradze
Geometric variational approach to porous media λ = λ ( k ), we consider the responseof the system to a fixed frequency, as is common in the literature. Thus, we take λ = iω as a fixed parameter, and compute k = k ( ω ) from the dispersion relations. Then, the phasevelocity is given by v p = Re ω/k ( ω ). Once k ( ω ) is known, we compute the group velocity v g = Re dω/dk = Re ( dk/dω ) − by directly differentiating the dispersion relations as an im-plicit function and substituting ( ω, k = k ( ω )). We also present the attenuation coefficient forthe wave Im k ( ω ) and attenuation per cycle Im k ( ω ) / Re k ( ω ).According to (82), and the fact that ζ has the order of magnitude of the microscopic bulkmodulus, most materials will have ζ ∗ ∼ ξ ∗ ∼ Z ∼ δ tends to be large whereas for porous media made out of dense materialsconveying gas, δ is small. We thus explore both large and small values of δ in the simulations.In Figures 1-3 we present the results of computation of dispersion relation for the P -waves for aset of different parameters δ , Z , ζ ∗ and ξ ∗ . Only two roots of equation (90) are shown since theequation is a quadratic equation in k . The other roots correspond to the waves propagatingwith the same velocity and attenuation coefficient in the opposite direction. In Figures 4-6 wepresent the results of computation of dispersion relation for the S -waves for a set of differentparameter δ . The axes variables in the figures are dimensionless, rescaled according to thetime and length scales defined in (81). v e l o c i t y , V p h Phase velocity v e l o c i t y , V g r Group velocity k Attenuation Coefficient | k / k | Attenuation per Cycle
Figure 1: Velocities and attenuation coefficients for P -waves with δ = 3, Z = 3, ζ ∗ = 2 and ξ ∗ = 0. In this paper, we have derived the equations of motion for a porous media filled with anincompressible fluid. We have chosen to write all the equations in the Eulerian frame for boththe fluid and the porous media. Our equations are valid for arbitrary deformations, and, asfar as we are aware, are new. We have compared the linearized equations of motion to the arkhutdinov, Gay-Balmaz, Putkaradze
Geometric variational approach to porous media v e l o c i t y , V p h Phase velocity v e l o c i t y , V g r Group velocity k Attenuation Coefficient | k / k | Attenuation per Cycle
Figure 2: Velocities and attenuation coefficients for P -waves with δ = 1, Z = 3, ζ ∗ = 2 and ξ ∗ = 0. v e l o c i t y , V p h Phase velocity v e l o c i t y , V g r Group velocity k Attenuation Coefficient | k / k | Attenuation per Cycle
Figure 3: Velocities and attenuation coefficients for P -waves with δ = 1, Z = 1, ζ ∗ = 1 and ξ ∗ = 1. arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media v e l o c i t y , V p h Phase velocity v e l o c i t y , V g r Group velocity k Attenuation Coefficient | k / k | Attenuation per Cycle
Figure 4: Velocities and attenuation coefficients for S -waves with δ = 0 . µ ∗ = 0. v e l o c i t y , V p h Phase velocity v e l o c i t y , V g r Group velocity k Attenuation Coefficient | k / k | Attenuation per Cycle
Figure 5: Velocities and attenuation coefficients for S -waves with δ = 1 and µ ∗ = 0. arkhutdinov, Gay-Balmaz, Putkaradze Geometric variational approach to porous media v e l o c i t y , V p h Phase velocity v e l o c i t y , V g r Group velocity k Attenuation Coefficient | k / k | Attenuation per Cycle
Figure 6: Velocities and attenuation coefficients for S -waves with δ = 10 and µ ∗ = 0.Biot’s equations and found a correspondence between our equations and Biot’s equations, witha clear and physical interpretation of the parameters. We have also derived the stability of thelinearized system for the porous system which turned out to be equivalent to the requirementsthat the parameters of the dry porous media being physically consistent.For further studies, it will be interesting to consider the dynamics of active porous materialslike sea sponges. Recent work [50] has demonstrated interesting ’sneezing’ dynamics of afreshwater sponge, when the sponge contracts and expands to clear itself from surroundingpolluted water. Equations (36) can be readily modified to model such contraction, for example,by making the equilibrium value of v = v a prescribed function of time. We believe that asemi-analytic theory may be developed in that case under the assumption of radial or sphericalsymmetry, which is sufficiently close to the experimental case. This problem, as well as otherinteresting topics in active, fluid-filled porous media, will be considered in our upcoming work. acknowledgements. We are thankful for fruitful and productive discussions with Profs G. L.Brovko, D. D. Holm, A. Ibraguimov, T. S. Ratiu and D. V. Zenkov. VP is also thankful for thelively and informative discussion with the participants of G. L. Brovko’s seminar on ElasticityTheory at the Moscow State University (November 2018), where a preliminary version of thiswork was presented. FGB is partially supported by the ANR project GEOMFLUID 14-CE23-0002-01. TF and VP were partially supported by NSERC and University of Alberta.
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