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# Generalized and Fundamental Solutions of Motion Equations of Two-Component Biot’s Medium

By Lyudmila Alexeyeva and Yergali Kurmanov

Submitted: November 25th 2019Reviewed: March 10th 2020Published: April 22nd 2020

DOI: 10.5772/intechopen.92064

## Abstract

Here processes of wave propagation in a two-component Biot’s medium are considered, which are generated by arbitrary forces actions. By using Fourier transformation of generalized functions, a fundamental solution, Green tensor, of motion equations of this medium has been constructed in a non-stationary case and in the case of stationary harmonic oscillation. These tensors describe the processes of wave propagation (in spaces of dimensions 1, 2, 3) under an action of power sources concentrated at coordinates origin, which are described by a singular delta-function. Based on them, generalized solutions of these equations are constructed under the action of various sources of periodic and non-stationary perturbations, which are described by both regular and singular generalized functions. For regular acting forces, integral representations of solutions are given that can be used to calculate the stress-strain state of a porous water-saturated medium.

### Keywords

• Biot’s medium
• solid and liquid components
• Green tensor
• Fourier transformation
• regularization

## 1. Introduction

Various mathematical models of deformable solid mechanics are used to study the seismic processes of earth’s crust. The processes of wave propagation are most studied in elastic media. But these models do not take into account many real properties of an ambient array. These are, for example, the presence of groundwater, which affects the magnitude and distribution of stresses. Models, which take into account the water saturation of earth’s crust structures, presence of gas bubbles, etc., are multicomponent media. A variety of multicomponent media, complexity of processes associated with their deformation, lead to a large difference in methods of analysis and modeling used in solving such problems.

Porous medium saturated with liquid or gas, from the point of view of continuum mechanics, is essentially a two-phase continuous medium, one phase of which is particles of liquid (gas) and other solid particles are its elastic skeleton. There are various mathematical models of such media, developed by various authors. The most famous of them are the models of Biot, Nikolaevsky, and Horoshun [1, 2, 3, 4, 5]. However, the class of solved tasks to them is very limited and mainly associated with the construction and study of particular solutions of these equations based on methods of full and partial separation of variables and theory of special functions in the works of Rakhmatullin, Saatov, Filippov, Artykov [6, 7], Erzhanov, Ataliev, Alexeyeva, Shershnev [8, 9], etc. In this regard, it is important to develop effective methods of solution of boundary value problems for such media with the use of modern mathematical methods.

Periodic on time processes are very widespread in practice. By this cause, here we consider also processes of wave propagation in Biot’s medium, posed by the periodic forces of different types. Based on the Fourier transformation of generalized functions, we constructed fundamental solutions of oscillation equations of Biot’s medium. It is Green tensor, which describes the process of propagation of harmonic waves at a fixed frequency in the space–time of dimension N = 1, 2, 3, under the action concentrated at the coordinates origin. By using this tensor, we construct generalized solutions of these equations for arbitrary sources of periodic disturbances, which can be described as both regular and singular distributions. They can be used to calculate the stress-strain state of a porous water-saturated medium by seismic wave propagation.

## 2. The parameters and motion equations of a two-component Biot’s medium

The equations of motion of a homogeneous isotropic two-component Biot’s medium are described by the following system of second-order hyperbolic equations [1, 2, 3]:

xtRN×0.

Here N is the dimension of the space. At a plane deformation N = 2, the total spatial deformation corresponds to N = 3, at N = 1 the equations describe the dynamics of a porous liquid-saturated rod.

We denote us=usjxtejis a displacement vector of an elastic skeleton, uf=ufjxtejis a displacement vector of a liquid, and ejj=1Nare basic orts of Lagrangian Cartesian coordinate system (everywhere by repeating indices, there is summation from 1 to N).

Constants ρ11,ρ12,ρ22have the dimension of mass density, and they are associated with densities of masses of particles, composing a skeleton ρsand a fluid ρf, by relationships:

ρ11=1mρsρ12,ρ22=mρfρ12,

where m is a porosity of the medium. The constant of attached density ρ12is related to a dispersion of deviation of micro-velocities of fluid particles in pores from average velocity of fluid flow and depends on pores geometry. Elastic constants λ,μare Lama’s parameters of an isotropic elastic skeleton, and Q, R characterize an interaction of a skeleton with a liquid on the basis of.

### 2.1 Biot’s law for stresses

σij=λkusk+Qkufkδij+μiusj+jusiσ=mp=Rkufk+QkuskE2

Here σijxtare a stress tensor in a skeleton, and pxtis a pressure in a fluid component. External mass forces acting on a skeleton Fs=Fjsxtejand on a liquid component Ff=Fjfxtej.

Further we use the next notations for partial derivatives: k=xk,uj,k=kuj, Δ=kkis Laplace operator.

There are three sound speeds in this medium:

c12=α1+α124α2α32α2,c22=α1α124α2α32α2,c32=ρ22μα2E3

where the next constants were introduced as:

α1=λ+2μρ22+Rρ112Qρ12,α2=ρ11ρ22ρ122,α3=λ+2μRQ2.

The first two speeds с1,c2с1>c2describe the velocity of propagation of two types of dilatational waves. The second slower dilatation wave is called repackaging wave. A third velocity c3corresponds to shear waves and at ρ12=0coincides with velocity of shear wave propagation in an elastic skeleton (c3<c1).

We introduce also two velocities of propagation of dilatational waves in corresponding elastic body and in an ideal compressible fluid:

cs=λ+2μρ11,сf=Rρ22

## 3. Problems of periodic oscillations of Biot’s medium

Construction of motion equation solutions by periodic oscillations is very important for practice since existing power sources of disturbances are often periodic in time and therefore can be decomposed into a finite or infinite Fourier series in the form:

Fsxt=nFnsxeiωnt,Ffxt=nFnfxeiωntE4

where periods of oscillation of each harmonic Tn=2π/ωnare multiple to the general period of oscillation T. Therefore, it is enough to consider the case of stationary oscillations, when the acting forces are periodic on time with an oscillation frequency ω:

Fsxt=Fsxeiωt,Ffxt=FfxeiωtE5

The solution of Eq. (1) can be represented in the similar form:

usxt=usxeiωt,ufx=ufxeiωtE6

where complex amplitudes of displacements usx, ufxmust be determined. If the solution has been known for any frequency ω, then we get similar decomposition for displacements of a medium:

usxt=nusnxeiωnt,ufxt=nufnxeiωntE7

which give us the solution of problem for forces (4).

We get equations for complex amplitudes by stationary oscillations, substituting (6) into the system (1):

To construct the solutions of this system for different forces, we define Green tensor of it.

## 4. Green tensor of Biot’s equations by stationary oscillations

Let us construct Umjxωeiωtjm=12Nfundamental solutions of the system (1) for the forces in the form:

Fxt=FsFf=δkjekδk+Njekδxeiωt,E9
k=1,,N,j=1,,2N.

Here δkj=δjkis the Kronecker symbol, and δxis the singular delta-function. They describe a motion of Biot’s medium at an action of sources of stationary oscillations, concentrated in the point x = 0. The upper index of this tensor (…[k]) fixes the current concentrated force and its direction. The lower index corresponds to component of movement of a skeleton and a fluid, respectively, k=1,,Nandk=N+1,,2N.

Their complex amplitudes Umjxωjm=12Nsatisfy the next system of equation:

λ+μUj,jik+μUi,jjk+ω2ρ11Uik+QUj,jik+Nω2ρ12Uik+N+δxδjk=0QUj,jik+ρ12ω2Uik,tt+RUj,jik+N+ρ22ω2Uik+N+δxδj+Nk=0j=1,,2N,k=1,,2N.E10

Since fundamental solutions are not unique, we’ll construct such, which tend to zero at infinity:

Uijxω0atxE11

and satisfy the radiation condition of type of Sommerfeld radiation conditions [10].

Matrix of such fundamental equations is named Green tensor of Eq. (8).

## 5. Fourier transform of fundamental solutions

To construct Umjxω, we use Fourier transformation, which for regular functions has the form:

Fφx=φ¯ξ=RNφxeiξxdx1dxNF1φ¯ξ=φx=12πNRNφ¯ξeiξxdξ1.dξN

where ξ=ξ1ξNare Fourier variables.

Let us apply Fourier transformation to Eq. (10) and use property of Fourier transform of derivatives [10]:

xjiξjE12

Then we get the system of 2 N linear algebraic equations for Fourier components of this tensor:

λ+μξjξjU¯jkμξ2U¯jkQξjξjU¯j+Nk+ρ11ω2U¯jk+ρ12ω2U¯j+Nk+δjk=0QξjξjU¯jkRξjξlU¯j+Nk+ρ12ω2U¯jk+ρ22ω2U¯j+Nk+δj+Nk=0j=1,,N,k=N+1,,2NE13

By using gradient divergence method, this system has been solved by us. For this the next basic function were introduced:

f0kξω=1сk2ξ2ω2,fjkξω=fj1kξω,j=1,2;E14

and the next theorem has been proved [11, 12].

Theorem 1.Components of Fourier transform of fundamental solutions have the form:

forj=1,N¯,k=1,N¯,
U¯jk=iξjiξkβ1f21+β2f22+β3f23++1α2ρ12δj+Nkρ22δjkf03,U¯j+Nk=iξjiξkγ1f21+γ2f22+γ3f23μα2δj+Nkξ2f231α2ρ11δj+Nk+ρ12δjkf03;
forj=1,,Nk=N+1,,2N
U¯jk=iξjiξkNη1f21+η2f22+η3f23++1α2ρ12δj+Nkρ22δjkf03
U¯j+Nk=iξjiξkNς1f21+ς2f22+ς3f23μα2δj+Nkξ2f231α2ρ11δj+Nk+ρ12δjkf03

where the next constants have been introduced as:

D1=1α2υ12,υlm=cl2cm2,q1=Qρ12λ+μρ12,q2=ρ11RQρ12,d1=λ+μρ22Qρ12,d2=Qρ22Rρ12,d3j=ρ12сj2Qj=12βj=1j+1D1сj2α2υ3jd1bsj+d2d3j,β3=с32α2υ31υ32d1bs3+d2d33;γj=1j+1D1сj2α2υ3jq1bfj+q2d3j,γ3=D1с32υ12α2υ31υ32q1bf3+q2d33;ηj=1j+1D1сj2α2υ3jdjd3j+d2bjs,η3=с32υ12α2υ31υ32d1d33+d2b3s;ςj=1j+1D1сj2α2υ3jq1d3j+q2b4js,ς3=с32υ12α2υ31υ32q1d33+q2b3sbfj=ρ22υfj,bsj=ρ11υjs.

This form is very convenient for constructing originals of Green tensor.

## 6. Stationary Green tensor construction: radiation conditions

In this case let us construct the originals of the basic function but only over ξby constant frequency:

Φ0mxω=Fξ1f0mξω

which, in accordance with its definition (14), satisfies the equation:

cm2ξ2ω2f0m=1E15

Using property (12) for derivatives from here, we get Helmholtz equation for fundamental solution (accurate within a factor cm2):

Δ+km2Φ0m+cm2δx=0,km=ωcmE16

Fundamental solutions of Helmholtz equation, which satisfy Sommerfeld conditions of radiation:

atr
Ф'0mrikmФ0mr=Or1,N=3,Ф'0mrikmФ0mr=Or1/2,N=2.

are well known [10]. They are unique. Using them, we obtain:

forN=3
Ф0m=14πrc2eikmr,km=ωcm;
forN=2
Ф0m=i4c2H01kmr,

where Hj1kmris the cylindrical Hankel function of the first kind:

forN=1
Ф0m=sinkmx2kmcm2.

These functions (subject to factor eiωt) describe harmonic waves which move from the point x = 0 to infinity and decay at infinity.

The last property is true only for N = 2,3. In the case N = 1, all fundamental solutions of Eq. (16):

d2dx2+km2Φ0m+cm2δx=0,

do not decay at infinity.

From Theorem 1, the next theorem follows.

Theorem 2.The components of Green tensor of Biot’s equations at stationary oscillations with frequency ω, which satisfy the radiation conditions, have the form:

forj=1,N¯,k=1,N¯,
Ujkxω=ω2m=13βm2Φ0mxjxk+1α2ρ12δj+Nkρ22δjkΦ03,Uj+Nkxω=ω2m=13γm2Φ0mxjxk++μδj+Nkα2ω2c32δx+k32Φ0mρ11δj+Nk+ρ12δjkα2Φ03;
forj=1,,Nk=N+1,,2N
Ujkxω=ω2m=13ηm2Φ0mxjxk+1α2ρ12δj+Nkρ22δjkΦ03Uj+Nkxω=ω2m=13ςm2Φ0mxjxk++μα2ω2c32δx+k32Φ0mδj+Nk1α2ρ11δj+Nk+ρ12δjkΦ03

where

forN=1
d2Ф0mdx2=12cm2kmkm2sinkmx2kmδx;
forN=2
2Ф0mxjxk=i4cm20.5km2H0kmrH2kmrr,jr,k+kH11kmrr,jk;
forN=3
Ф0mxjxk=14πrcm2eikrr,jr,kikm1r2+1r2+r,jkikm1r;
km=ωcm,r=x,r,j=xjr,r,ij=1rδijxixjr2..

Proof. By using originals of basic functions, property (12) of derivatives, we can obtain from formulas for Ujkin Theorem 1 the originals of all addends, besides that which contain factor ξ2. But using (16) we have:

ΔΦ=cm2δx+km2Φ0mξ2f0m=cm2+km2f0m

Then formulas of Theorem 2 follow from formulas of Theorem 1.

## 7. Generalized solutions by arbitrary periodic forces

Under the action of arbitrary mass forces with frequency ωin Biot’s medium, the solution for complex amplitudes has the form of a tensor functional convolution:

ujxt=UjkxωFkxeiωt,j,k=1,2N¯E17

Note that mass forces may be different from the space of generalized vector function, singular and regular. Since Green tensor is singular and contains delta-functions, this convolution is calculated on the rule of convolution in generalized function space. If a support of acting forces are bounded (contained in a ball of finite radius), then all convolutions exist. If supports are not bounded, then the existence conditions of convolutions in formula (17) requires some limitations on behavior of forces at infinity which depends on a type of mass forces and space dimension.

The obtained solutions allow us to study the dynamics of porous water- and gas-saturated media at the action of periodic sources of disturbances of a sufficiently arbitrary form. In particular, they are applicable in the case of actions of certain forces on surfaces, for example, cracks, in porous media that can be simulated by simple and double layers on the crack surface.

There is another interesting feature of the Green tensor of the Biot’s equations, which contains, as one of the terms, the delta-function that complicates the application of this tensor for solving boundary value problems based on analogues of Green formulas for elliptic systems of equations or the boundary element method. Here, when constructing the model, the viscosity of the liquid is not taken into account, which, apparently, leads to the presence of such terms, and it requires improvement of this model taking into account a viscosity.

## 8. Green tensor of Biot’s equations by non-stationary motion

To construct the non-stationary Green tensor, at first we also construct the originals of the basic functions in an initial space-time:

Φ0mxt=F1f0mξω=F1cm2ξ2ω21

They are originals of the classic wave equation:

2t2cm2ΔΦ0m=δtδxE18

Depending on the dimension of a space, solutions of this wave equation that satisfy the radiation conditions have the following form [10]:

Φ0mxt=14πcm2rδtrcm,N=3;E19
Φ0mxt=12πcmHctrcm2t2r2,N=2;E20
Ф0mxt=12Hcmtx,N=1.E21

Here Htis the Heaviside function, and singular function δtr/cmis the simple layer on the sound coner=cmt,r=x.

Using regularization of the general function ω1in the space of distribution [10]:

Htδx1iω+i0

and the properties of Fourier transform of generalized functions convolution:

h=fgh¯=f¯×g¯

It is easy to show that the next lemma is true.

Lemma.The originals of the primitives of the basic functions satisfying the radiation conditions are representable in the following form:

for N = 3

Ф1mxt=Ф0mxtHtδx=Hcmtr4πc2r,Ф2mxt=Ф1mxtHtδx=cmtr+4πc3r;E22

for N = 2

Ф1mxt=12πc2lncmt+cm2t2r2r,Ф2mxt=12πc3cmtlncmt+cm2t2r2rcm2t2r2;E23

for N = 1

Ф1mxt=0,5cmtrHcmtr0.5cmtr+,Ф2mxt=12c2cmtr2Hcmtr12c2cmtr+2.E24

Using these functions and the properties of the Fourier transform, we obtain the components of the Green tensor from the formulas of Theorem 1. We formulate the result in the next theorem.

Theorem 3.The components of Green tensor of motion equations of two-component Biot’s medium have the following forms:

Forj=1,N¯,k=1,N¯,
Ujkxt=m=13βm2Ф2mxjxk+1α2ρ12δj+Nkρ22δjkФ03x,Uj+Nkxt=m=13γm2Ф2mxjxk+μα2cm2δj+NkФ0mt+δx1α2ρ11δj+Nk+ρ12δjkФ03x
Forj=1,N¯,k=N+1,2N¯,
Ujkxt=m=13ηm2Ф2mxjxkN+1α2ρ12δj+Nkρ22δjkФ03x,Uj+Nkxt=m=13ςm2Ф2mxjxkN+μα2cm2δj+NkФ0mt+δx1α2ρ11δj+Nk+ρ12δjkФ03x

Here

for N = 1

d2Ф2mdx2=Hcmtxcm22cm2cmtx+δxE25

for N = 2

2Ф2mxjxk=Hcmtr2πcm3r32cm2t2r2cm2t2r2r,kr,jδjkcm2t2r2E26

for N = 3

2Ф2mxjxk=t4πcm2r2δcmtrr,kr,jtHcmtrrδjk3r,kr,j,E27
r,j=xj/r.

## 9. Generalized solutions of Biot’s equations by non-stationary forces

Using the properties of Green tensor, we obtain generalized solutions of non-stationary Biot’s equations under the action of arbitrary mass forces in the Biot’s medium, which satisfy the radiation condition at infinity. They have the form of tensor functional convolution:

ujxt=UjkxtFkxt,j,k=1,,2NE28

It’s taken according to the rules of convolution of generalized functions depending on the type of mass forces [10].

In order to get the classic solution, we must present formulas (28) in regular integral forms. For this, let us present matrix of Green tensor as sum of regular functions and singular functions, which contain delta-function:

Uxt=Uregxt+Usingtδx.

Then also write:

uxt=u1xt+u2xtE29

Here u1xtis representable by regular mass forces in the integral form:

u1xt=HtotRNUregxyτ×FsytτFfytτdy

The convolution with singular part is equal to:

u2xt=HtotUsingτ×FsxtτFfxtτ

In 3D space, there are convolutions with simple layers on sound cones (see (27)). To construct their integral presentation, use this rule:

αxtδcmtrFxt==Ht0tyx=cmταxyτFytyxcmdSy

Here the internal integral is taken over sphere with center in the point x, and its radius is equal to cmτ.

If components of acting forces Fxtare double differentiable vector function, it is convenient to use the property of differentiation of convolution [10]:

2Ф2mxjxkFxt=2xjxkФ2mFxt=Ф2mxt2Fxjxk

Substituting the formulas of Theorem 4 into (29), we obtain displacements and stresses of skeleton and liquid in Biot’s medium in spaces of dimension N = 1, 2, 3. Calculation of these convolutions by using these formulas essentially depends on the form of acting forces and gives possibility to construct regular presentation of generalized solution for wide class of acting forces, which are the classic solution of Biot’s equation.

## 10. Calculation of the stress state of Biot’s medium

Using Biot’s law (2), we can define the generalized stresses in skeleton and a pressure in a liquid:

σij=λlUlkFks+QlUlk+NFkfδij++μiUjkFks+jUik+NFkfσ=mp=RlUlk+NFkf+QlUlkFksE30

These formulas also can be written in integral form by using the same rules. But we can apply here the next lemma, which was proved in [11].

Lemma.Fourier transformations of the divergences of Green tensor have the next form:

byk=1,,N
FjUjk=D1iξkbf1f01ξωbf2f02ξωFjUj+Nk=D1iξkd31f01ξωd32f02ξωj=1,,N.
byk=N+1,,2N
FjUjk=iξkND1d31f01ξωd32f02(ξω)FjUj+Nk=iξkND1bs1f01ξωbs2f02(ξω)j=1,,N.

From this lemma, we can prove easily the next theorem.

Theorem 4.Divergences of elastic and liquid displacement of Green tensor have the next form:

fork=1,,N
jUjk=D1bf1kΦ01bf2kΦ02jUj+Nk=D1d31kΦ01d32kΦ02j=1,,N.
fork=N+1,,2N
jUjk=D1d31kNΦ01d32kNΦ02jUj+Nk=D1bs1kNΦ01bs2kNΦ02j=1,,N.

Substituting these formulas in (30), we define the stresses in the skeleton and the pressure in the liquid of Biot’s medium.

If we paste Φ02xωinstead of Φ02xtin formulas of this theorem, then formula (30) expresses complex amplitudes of stress tensor and pressure by periodic oscillations. It is used to determine stresses and pressure by solving the periodic problems (4).

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Lyudmila Alexeyeva and Yergali Kurmanov (April 22nd 2020). Generalized and Fundamental Solutions of Motion Equations of Two-Component Biot’s Medium [Online First], IntechOpen, DOI: 10.5772/intechopen.92064. Available from: