The basic two-dimensional boundary value problems of the fully coupled linear equilibrium theory of elasticity for solids with double porosity structure are reduced to the solvability of two types of a problem. The first one is the BVPs for the equations of classical elasticity of isotropic bodies, and the other is the BVPs for the equations of pore and fissure fluid pressures. The solutions of these equations are presented by means of elementary (harmonic, metaharmonic, and biharmonic) functions. On the basis of the gained results, we constructed an explicit solution of some basic BVPs for an ellipse in the form of absolutely uniformly convergent series.
1. Introduction
In a material with two degrees of porosity, there are two pore systems, the primary and the secondary. For example, in a fissured rock (i.e., a mass of porous blocks separated from one another with an interconnected and continuously distributed system of fissures), most of the porosity is provided by the pores of the blocks or primary porosity, while most of permeability is provided by the fissures or secondary porosity. When fluid flow and deformation processes occur simultaneously, three coupled partial differential equations can be derived [1, 2] to describe the relationships governing pressure in the primary and secondary pores (and therefore the mass exchange between them) and the displacement of the solid.
A theory of consolidation with double porosity structure was proposed by Wilson and Aifantis [1]. The physical and mathematical foundations of the theory of double porosity were considered in the papers [1–3], where analytical solutions of the relevant equations are also given. This theory unifies a model proposed by Biot for the consolidation of deformable single porosity media with a model proposed by Barenblatt for seepage in indeformable media with two degrees of porosity. The basic results and the historical information on the theory of porous media were summarized by De Boer [4]. However, Aifantis’ quasi-static theory ignored the cross-coupling effect between the volume change of the pores and fissures in the system. The cross-coupled terms were included in the equations of conservation of mass for the pore and fissure fluid and in Darcy’s law for solids with double porosity structure by several authors [5–8].
Porous media theories play an important role in many branches of engineering, including material science, petroleum industry, chemical engineering, and soil mechanics, as well as biomechanics. In recent years, many authors investigated the BVPs of the 2D and 3D theories of elasticity for materials with double porosity structure, publishing a large number of papers (some of these results can be seen in [9–17] and references therein). In those works, the explicit solutions on some BVPs in the form of series are given in a form useful for the engineering practice.
In the present paper, the basic two-dimensional boundary value problems of the fully coupled linear equilibrium theory of elasticity for solids with double porosity structure are reduced to the solvability of two types of a problem. The first one is similar to the BVPs for the equations of classical elasticity of isotropic bodies, while the second one is the BVPs for the equations of the pore and fissure fluid pressures. The solutions of these equations are presented by means of elementary (harmonic, metaharmonic, and biharmonic) functions. On the basis of the gained results, we constructed an explicit solution of some basic BVPs for an ellipse in the form of absolutely uniformly convergent series.
2. Basic Equations
Let x=x1,x2 be a point of the Euclidean 2D space R2. In what follows, we consider an ellipse with a double porosity structure that occupies the region Ω of R2.
The system of homogeneous equations of the linear equilibrium theory of elasticity for solids with double porosity structure can be written as follows [9]:(1)μΔu+λ+μgraddivu=gradβ1p1+β2p2(2)k1Δ-γp1+k12Δ+γp2=0,k21Δ+γp1+k2Δ-γp2=0,where u=u1x,u2xT is the displacement vector in a solid, p1x and p2x are the pore and fissure fluid pressures, respectively, β1 and β2 are the effective stress parameters, γ>0 is the internal transport coefficient and corresponds to the fluid transfer rate with respect to the intensity of flow between the pores and fissures, λ,μ,k1,k2 are all constitutive coefficients, kj=κj/μ′j=1,2, k12=κ12/μ′, k21=κ21/μ′, μ′ is the fluid viscosity, κ1 and κ2 are the macroscopic intrinsic permeabilities associated with matrix and fissure porosity, respectively, κ12 and κ21 are the cross-coupling permeabilities for fluid flow at the interface between the matrix and fissure phases, and Δ is the two-dimensional Laplace operator. We consider the vectors as column matrices, if necessary. Throughout this paper, it is assumed that β12+β22>0. Superscript “T” denotes transposition. We will suppose that(3)k1>0,k2>0,k1k2-k12k21>0,k0=k1+k2+k12+k21>0.
Note that BVPs for system (2) which contain p1x and p2x can be investigated separately. Then if supposing pjx as known, we can study BVPs for system (1) with respect to ux. By combining the obtained results, we arrive at explicit solutions of BVPs for systems (1)-(2).
Obviously, from system (2), we have the following equations for p1x and p2x:(4)ΔΔ+λ12pj=0,j=1,2,where(5)λ1=iγk0k1k2-k12k21,k0=k1+k2+k12+k21.
It is easy to see that the solutions of system (2) have the form(6)p1=ϕ+m1ϕ1,p2=ϕ+ϕ1,where(7)m1=-k2+k12k1+k21,Δϕ=0,Δ+λ12ϕ1=0.
First, we assume pj to be known. Then, for ux, we get the following nonhomogeneous equation:(8)μΔu+λ+μgraddivu=gradβ1+β2ϕ+β1m1+β2ϕ1.
It is well known that the general solution of (8) has the form(9)u=v+v0,where v is a general solution of the equation(10)μΔv+λ+μgraddivv=0and v0 is a particular solution of the nonhomogeneous equation(11)v0=1λ+2μgradβ1+β2φ0-β1m1+β2λ12ϕ1;here,(12)Δϕ=0,Δφ0=ϕ,Δ+λ12ϕ1=0.
3. Basic Equations in the Elliptic Coordinate System
Let us rewrite equalities (6), (9), and (10)–(12) in elliptic coordinates system ξ,η0≤ξ<∞,0≤η<2π (see Appendix A). We obtain the following.
Formula (9) takes the form(13)u~=v~+v~0.
Formula (11) takes the form(14)v~0=1λ+2μ1h∂Ψ∂ξe→ξ+∂Ψ∂ηe→η,where(15)Ψ≔β1+β2φ-0-β1m1+β2λ12ϕ1¯,h=sinh2ξ+sin2η,φ0¯=φ0¯ξ,η,ϕ1¯=ϕ1¯ξ,η.
Equations (12) take the form(16)Δϕ¯ξ,η=0,1h2∂2ϕ1¯∂ξ2+∂2ϕ1¯∂η2+λ12ϕ1¯=0,(17)1h2∂2φ-0∂ξ2+∂2φ-0∂η2=ϕ¯.
Expressions (6) can be rewritten in the following form:(18)p-1=ϕ¯+m1ϕ¯1,p-2=ϕ¯+ϕ¯1.
Equation (10) in the elliptic coordinates system takes the form [18–21](19)∂D∂ξ-∂K∂η=0,∂u-∂ξ+∂v-∂η=κ-2κμ′h02D,∂D∂η+∂K∂ξ=0,∂v-∂ξ-∂u-∂η=1μ′h02K,where κ=41-ν,μ=E/21-ν,h0:=cosh2ξ-cos2η,u-=2hv~1/c2,v-=2hv~2/c2, (in what follows, we assume that c=1), v~1 and v~2 are the components of displacement vector v~v~1,v~2 along the normal and the tangent to the curve ξ=const, κ-2/κμD is divergence of displacement vector v~, 1/μK is a rotor of displacement vector v~, ν is Poisson’s ratio, and E is Young’s modulus.
Hooke’s law can be written as follows:(20)h02μσξξ=h02μD-2∂v-∂η-2h02sinh2ξu--sin2ηv-,h02μσηη=h02μD-2∂u-∂ξ+2h02sinh2ξu--sin2ηv-,h02μτξη=h02μK+2∂u-∂η-2h02sin2ηu-+sinh2ξv-.
4. Boundary Value Problems
For systems (1) and (2), we pose the following BVPs.
Find in domain Ω=0≤ξ<ξ1,0≤η<2π a regular solution Uu,p1,p2∈C2Ω to systems (1) and (2) satisfying one of the following boundary conditions (see Figure 1):(21)for ξ=ξ1:au-=f11,v-=f12,p-1=f13,p-2=f14,orbh02μσξξ=g11,h02μτξη=g12,p-1=f13,p-2=f14,orcu-=f11,v-=f12,∂p-1∂η=f23,∂p-2∂η=f24,ordh02μσξξ=g11,h02μτξη=g12,∂p-1∂η=f23,∂p-2∂η=f24.Here fjk and gjk are given functions.
Ellipse with double porosity.
Note that when stress components h02/μσξξ and h02/μτξη are given on boundary ξ=ξ1, then the BVP is solvable if the principal vector and the principal moment of external stresses are equal to zero.
Clearly, BVPs for (2) can be investigated separately (Problem B). Then, by admitting p1x and p2x as known, we can study BVPs for system (1) with respect to ux (Problem A). By combining the obtained results, we arrive at the explicit solutions of BVPs for systems (1)-(2) and (21).
Problem A.
First we state the BVPs for the semiellipse, when the conditions of uninterrupted continuation of solutions (i.e., the conditions of symmetry or antisymmetry) are given on η=0, η=π, and ξ=0 [22]. Thus, we obtain the solutions of BVPs for the ellipse.
Now let us formulate the boundary value problems for semiellipse as follows: find in domain Ω1=0≤ξ<ξ1,0≤η<π a regular solution v~u-,v-∈C2Ω of (19) satisfying the following boundary conditions (see Figure 2):(22)forη=0:av-=0,τξη=0⟺v-=0,∂u-∂η=0orbu-=0,σηη=0⟺u-=0,∂v-∂η=0,(23)forη=π:av-=0,τξη=0⟺v-=0,∂u-∂η=0orbu-=0,σηη=0⟺u-=0,∂v-∂η=0,(24)forξ=0:au=0,τξη=0⟺u=0,∂v∂ξ=0orbv=0,σξξ=0⟺v=0,∂u∂ξ=0,(25)forξ=ξ1:au-=f11,v-=f12orbh02μσξξ=g11,h02μτξη=g12.
Semiellipse.
The solutions of (19) and (22)–(25) are represented by means of two harmonic functions φ1ξ,η and φ2ξ,η. Consider(26)u-=sinh2ξ1cothξ∂φ1∂η+∂φ2∂ξ+κ-1φ2sinhξcosη-cosh2ξ1tanhξ∂φ1∂ξ-∂φ2∂η-κ-1φ1coshξsinη,v-=-cosh2ξ1tanhξ∂φ1∂η+∂φ2∂ξ+κ-1φ2coshξsinη-sinh2ξ1cothξ∂φ1∂ξ-∂φ2∂η-κ-1φ1sinhξcosη,D=κμcosh2ξ-cos2η∂φ1∂ξ-∂φ2∂ηcoshξsinη+∂φ1∂η+∂φ2∂ξsinhξcosη,K=κμcosh2ξ-cos2η∂φ1∂ξ-∂φ2∂ηsinhξcosη-∂φ1∂η+∂φ2∂ξcoshξsinη.
The components of stress vector (17) can be written as(27)h02μσξξ=2sinh2ξ1∂∂η∂φ1∂ξ-∂φ2∂ηcoshξ-κ-2∂φ1∂η-κ∂φ2∂ξsinhξcosη+2cosh2ξ1∂∂η∂φ1∂η+∂φ2∂ξsinhξ+κ∂φ1∂ξ+κ-2∂φ2∂ηcoshξsinη-4sinhξ1+ξsinhξ1-ξcosh2ξ-cos2η∂φ1∂ξ-∂φ2∂ηcoshξsinη+∂φ1∂η+∂φ2∂ξsinhξcosη,h02μσηη=-2sinh2ξ1∂∂η∂φ1∂ξ-∂φ2∂ηcoshξ-κ∂φ1∂η-κ-2∂φ2∂ξsinhξcosη-2cosh2ξ1∂∂η∂φ1∂η+∂φ2∂ξsinhξ+κ-2∂φ1∂ξ+κ∂φ2∂ηcoshξsinη+4sinhξ1+ξsinhξ1-ξcosh2ξ-cos2η∂φ1∂ξ-∂φ2∂ηcoshξsinη+∂φ1∂η+∂φ2∂ξsinhξcosη,h02μτξη=-2cosh2ξ1∂∂η∂φ1∂ξ-∂φ2∂ηsinhξ-κ-2∂φ1∂η-κ∂φ2∂ξcoshξsinη+2sinh2ξ1∂∂η∂φ1∂η+∂φ2∂ξcoshξ+κ∂φ1∂ξ+κ-2∂φ2∂ηsinhξcosη+4sinhξ1+ξsinhξ1-ξcosh2ξ-cos2η∂φ1∂ξ-∂φ2∂ηsinhξcosη-∂φ1∂η+∂φ2∂ξcoshξsinη.
Let us rewrite conditions (25) when ξ=ξ1 in the following equivalent form:(28)(a)2h02coshξ1sinηu-+sinhξ1cosηv-=-12sinh2ξ1∂φ1∂ξ-∂φ2∂η+κ-1φ1,2h02sinhξ1cosηu--coshξ1sinηv-=12sinh2ξ1∂φ1∂η+∂φ2∂ξ+κ-1φ2 orb2μcoshξ1sinησξξ+sinhξ1cosητξη=sinh2ξ1∂∂η∂φ1∂η+∂φ2∂ξ+κ∂φ1∂ξ+κ-2∂φ2∂η,2μsinhξ1cosησξξ-coshξ1sinητξη=sinh2ξ1∂∂η∂φ1∂ξ-∂φ2∂η-κ-2∂φ1∂η+κ∂φ2∂ξ.
In (28), functions φ1 and φ2 are harmonic functions, which, by using the method of separation of variables [23, 24], can be presented as follows:(29)φi=∑n=1∞φin,i=1,2,where(30)φ1n=B1nsinhnξcoshnξ1sinnη,φ2n=B2ncoshnξcoshnξ1cosnη orφ1n=B1ncoshnξcoshnξ1cosnη,φ2n=B2nsinhnξcoshnξ1sinnη.When n=0, then φ10=B10+a02ξ+a03η+a04ξη and φ20=B20+b02ξ+b03η+b04ξη, where B10,a02,…,b04 are constants. From the condition of limitation of gradφi0=1/h∂φi0/∂ξ+∂φi0/∂α in the focuses of ellipse and condition of periodicity of functions φi0 along η coordinate, there ensues the following: a0j=0,b0j=0,j=2,3,4. Therefore, φ10=0 and φ20=A20 or φ10=A10, and φ20=0. When on boundary ξ=ξ0 the values of the stress vector are given, then the rigid displacements of the ellipse are defined by the equalities. φ10=0 and φ20=A20 and φ10=A10, and φ20=0.
By substituting (22)–(24) and (29) into (28), for determining unknown B1n and B2n, we get the following infinite system of algebraic equations, whose main matrix has a block-diagonal form (see Figure 3).
Diagram of the main matrix of the algebraic equation system.
The dimension of all matrices Dii=1,2,… is 2×2 and detDi≠0. When i→∞, detDi→M, where M≠0.
Thus, the system is uniquely solvable. Let us substitute the obtained values, B1n and B2n, into φ1 and φ2. As a result, we obtain the vector v~u-,v-.
Problem B.
Find regular solutions p1 and p2 to (2) in domain Ω=0≤ξ<ξ1,0≤η<2π satisfying one of the following boundary conditions (see Figure 1):(31)whenξ=ξ1:ap1=f13,p2=f14orb∂p1∂η=f23,∂p2∂η=f24.
The solution of (2) is (18), where function ϕ1¯ is a solution of (16). The solution of (16) by applying the method of separation of variables is reduced to solutions of the Mathieu differential equation or modified Mathieu differential equations (see Appendix B) [25]. These solutions are known as Mathieu functions. Thus, ϕ1¯ξ,η=Rξ·Φ2η, where Φ2η is the solution of Mathieu differential equation d2Φ2ηη/dη2+a-2qcos2ηΦ2ηη=0 (see Appendix C), and R is the solution of modified Mathieu differential equations d2Rξ/dξ2-a-2qcosh2ξRξ=0 (see Appendix D), where a≔c+1/2λ12,q≔1/4λ12,c=const is an arbitrary constant.
Let us assume that harmonic function ϕ¯ from (18) is represented in the form of series:(32)ϕ¯=∑n=0∞Cncoshnξcosnη+∑n=1∞Dnsinhnξsinnη.Taking into account the boundary conditions (31), to determine unknown coefficients Cn and Dn, we obtain the system of infinite algebraic equations, whose main matrix has a block-diagonal form (Figure 3). By solving this system, we find ϕ¯ξ,η.
For determining φ-0 from (17), let us consider the particular cases: keeping in mind the homogeneous boundary conditions, we get Cn=0 or Dn=0. Let us assume that Dn=0.
Substituting (32) in (17), we obtain the following equation:(33)∂2φ-0∂ξ2+∂2φ-0∂η2=∑k=0∞Ckcoshkξcoskηcosh2ξ-cos2η=∑n=0∞Encoshnξcosnη.The solution of (33) is sought in the following form:(34)φ-0=φ-0g+φ-0p,where φ-0g is a harmonic function and it can be obtained in the same way as ϕ¯. Thus, we have(35)Δφ-0g=0,φ-0g=∑n=0∞Tncoshnξcosnη.φ-0p is the particular solution of the following equation:(36)∂2φ-0p∂ξ2+∂2φ-0p∂η2=∑n=0∞Encoshnξcosnη.The solution of (36) is sought in the form(37)φ-0p=∑n=0∞Qncoshnξcosnη.Substituting (37) in (36), we obtain the relations between Qn and En and hence it follows that(38)φ-0=∑n=0∞Tncoshnξcosnη+∑n=0∞Qncoshnξcosnηor(39)φ-0=∑n=0∞Mncoshnξcosnη,where Mn=Tn+Qn.
Quite similarly, we obtain the solution, when Cn=0. By combining the obtained results, we obtain an explicit solution of (17).
Remark 1.
When we have the nonhomogeneous boundary conditions, then the coefficients in (32) are different from zero, Cn≠0,Dn≠0, and function φ-0 will be obtained by above-mentioned method for each series, separately.
By substituting φ-0 and ϕ¯1 in (14), we obtain function v~0. Finally, from (13), we obtain displacement u~ in Ω at arbitrary point. Stress components are defined from (20).
5. Conclusions
The main results of this work can be formulated as follows:
The system of equations of the linear equilibrium theory of elasticity for solids with double porosity structure is written in terms of elliptic coordinates.
The problems are reduced to the solvability of two types of problem. The first one is similar to the BVPs for the equations of classical elasticity of isotropic bodies, while the second one is the BVPs for the equations of the porous and fissure fluid pressures.
Analytical (exact) solutions are obtained for 2D BVPs for the ellipse with double porosity structure.
By using the above-mentioned method, the following is possible:
It is possible to construct explicitly the solutions of basic BVPs for systems (1) and (2) for simple cases of 2D domains (circle, plane with circular hole.) in the form of absolutely and uniformly convergent series that are useful in the engineering practice.
It is possible to obtain numerical solutions of the boundary value problems.
It is possible to construct explicitly the solutions of basic BVPs of the systems of equations in the modern linear theories of elasticity, thermoelasticity, and poroelasticity for materials with microstructures and for elastic materials with double porosity for a circle, and so forth.
In practice, such BVPs are quite common in many areas of science. The potential users of the obtained results will be the scientists and engineers working on the problems of solid mechanics, micromechanics and nanomechanics, mechanics of materials, engineering mechanics, engineering medicine, biomechanics, engineering geology, geomechanics, hydroengineering, applied and computing mechanics, and applied mathematics.
AppendixA. Some Basic Formulas in Elliptic Coordinates
If x,y-∞<x<+∞,-∞<y<+∞ are Cartesian coordinates and ξ,η are elliptic coordinates 0≤ξ<∞,0≤η<2π, then x=ccoshξcosη, y=csinhξsinη, where c is scale factor (Figure 4) [26].
Elliptic coordinate system.
The coordinate curves are ellipses and hyperbolas:(A.1)x2c2cosh2ξ0+y2c2sinh2ξ0=1,ξ0=const≠0,x2c2cos2η0-y2c2sin2η0=1,η0=const≠0,π2,π,32π.h≔hξ=hη=c/2cosh2ξ-cos2η=csinh2ξ+sin2η are metric coefficients (Lamé coefficients).
The Laplacian operator in the elliptic coordinates has the form(A.2)Δf=1h2∂2f∂ξ2+∂2f∂η2.
The operator grad has the form (A.3)∇f≔gradf=1h∂f∂ξe→ξ+∂f∂ηe→η,where f=fξ,η.
The Helmholtz equation can be written as(A.4)1h2∂2f∂ξ2+∂2f∂η2+λ12f=0,where f=fξ,η,and biharmonic equation has the form (A.5)ΔΔf=Δf2=1h2∂2f∂ξ2+∂2f∂η22=1h4∂4f∂ξ4+2∂4f∂ξ2∂η2+∂4f∂η4=0.
B. Solution of Helmholtz Equation in the Elliptic Coordinates
Helmholtz equation in the elliptic coordinateshas the following form:(B.1)1sinh2u+sin2v∂2F∂u2+∂2F∂v2+k2F=0.Let us solve this equation by using the method of separation of variables.
Let function F be sought in the form(B.2)Fu,v=Uu·Vv,and then Helmholtz equation takes the following form:(B.3)1sinh2u+sin2vVd2Udu2+Ud2Vdv2+k2UV=0.From here, we obtain(B.4)1sinh2u+sin2v1Ud2Udu2+1Vd2Vdv2+k2=0.Let us rewrite the last equation in the following form:(B.5)1U∂2Udu2+k2sinh2u+1V∂2Vdv2+k2sin2v=0.From here, we get(B.6)1Ud2Udu2+k2sinh2u=c,c+1Vd2Vdv2+k2sin2v=0.Thus,(B.7)d2Udu2-c-k2sinh2uU=0,d2Vdv2+c+k2sin2vV=0.Take into account the following identities:(B.8)sinh2u=12cosh2u-1,sin2v=121-cos2v.We obtain(B.9)d2Udu2-c-12k2cosh2u-1U=0,d2Vdv2+c+12k21-cos2vV=0.From here, we get(B.10)∂2U∂u2-c+12k2-12k2cosh2uU=0,d2Vdv2+c+12k2-12k2cos2vV=0.Let us introduce the notations a≔c+1/2k2 and q≔1/4k2; then we have d2U/du2-a-2qcosh2uU=0; this is the modification differential equation of Mathieu. Consider d2V/dv2+a-2qcos2vV=0; this is the differential equation of Mathieu.
C. The Solutions of Differential Equation of Mathieu
Let us assume that c=0; then a:=λ1/22 and q≔1/4λ12. The differential equation of Mathieu takes the form(C.1)∂2Φ2η∂η2+a-2qcos2ηΦ2η=0.The solutions of this equation have the following forms [25]:
When q>0,(C.2)ce2mq,η=∑n=0∞A2n2mqcos2nη,se2m+2q,η=∑n=0∞B2n+22m+2qsin2n+2η,ce2m+1q,η=∑n=0∞A2n+12m+1qcos2n+1η,se2m+1q,η=∑n=0∞B2n+12m+1qsin2n+1η.
When q<0,(C.3)ce2m-q,η=-1m∑n=0∞-1nA2n2mqcos2nη,se2m+2-q,η=-1m∑n=0∞-1nB2n+22m+2qsin2n+2η,ce2m+1-q,η=-1m∑n=0∞-1nA2n+12m+1qcos2n+1η,se2m+1-q,η=-1m∑n=0∞-1nB2n+12m+1qsin2n+1η,
where, for ce2mq,η,(C.4)aA0-qA2=0,a-4A2-qA4+2A0=0,a-4n2A2n-qA2n+2+A2n-2=0,n≥2;for ce2m+1q,η,(C.5)a-1-qA1-qA3=0,a-2n+12A2n+1-qA2n+3+A2n-1=0,n≥1;for se2m+1q,η,(C.6)a-1+qB1-qB3=0,a-2n+12B2n+1-qB2n+3+B2n-1=0,n≥1;and, for se2m+2q,η,(C.7)a-4B2-qB4=0,a-4n2B2n-qB2n+2+B2n-2=0,n≥2(here A2n2m≡A2n,…,B2n+22m+2≡B2n+2).
D. The Solutions of the Modification Differential Equation of Mathieu
The modification differential equation of Mathieu has the form(D.1)∂2Rξ∂η2-a-2qcosh2ξRξ=0.The solutions of this equation are as follows [25]:
If q>0,(D.2)Ce2mq,ξ=∑k=0∞A2kJ2k2qsinhξ,Ce2m+1q,ξ=∑k=0∞-1k+1A2k+1J2k+12qcoshξ,Se2m+1q,ξ=∑k=0∞B2k+1J2k+12qsinhξ,Se2m+2q,ξ=tanhξ∑k=0∞-1k2kB2kJ2k2qcoshξ.
If q<0, then the argument of Bessel function is 2-q=i2q.
E. Physical Motivation of Double Porosity Model
The double porosity model has received a lot of attention from mathematicians and from engineers. In such a model, there are two pore systems with different permeability. There is first a set of isolated porous blocks of low permeability (sometimes called matrix), surrounded by network of high permeability connected porous medium (usually called fractures network). Pores are pervasive in most of the igneous, metamorphic, and sedimentary rocks in the earth’s crust. In fact, porosity found in the earth may have many shapes and sizes, but two types of porosity are more important. One is the matrix porosity, and the other is fracture or crack porosity which may occupy very little volume, but fluid flow occurs primarily through the fracture network.
In physical terms, the theory of poroelasticity postulates that when a porous material is subjected to stress, the resulting matrix deformation leads to volumetric changes in the pores. The pores are filled with fluid. The presence of the fluid results in the flow of the pore fluid between regions of higher and lower pore pressure.
The physical process of coupled deformation is governed by the following equations:
The equations of motion [27, 28]:(E.1)tlj,j=ρu¨l-Fl,l,j=1,2,3,
where tlj is the components of total stress tensor, ρ>0 is the reference mass density, and F is the body force per unit mass. We assume that subscripts preceded by a comma denote partial differentiation with respect to the corresponding Cartesian coordinate, repeated indices are summed over the range (1,2,3), and the dot denotes differentiation with respect to t (here t denotes the time variable; t≥0).
The equations of fluid mass conservation [5]:(E.2)divv1+ζ˙1+β1e˙rr+νp1-p2=0,divv2+ζ˙2+β2e˙rr-νp1-p2=0,
where v(1) and v(2) are the fluid flux vectors for pores and fissures, respectively.
elj is the components of strain tensor:(E.3)elj=12ul,j+uj,l,l,j=1,2,3.β1 and β2 are the effective stress parameters and γ is the internal transport coefficients and corresponds to a fluid transfer rate respecting the intensity of the flow between pores and fissures, γ≥0, and ς1 and ς2 are the increments of fluid in pores and fissures, respectively, and are defined by(E.4)ς1=α1p1+α12p2,ς2=α21p1+α2p2.α1 and α2 measure the compressibility of pore and fissure systems, respectively, and α12 and α21 are the cross-coupling compressibility for fluid flow at the interface between the two pore systems at a microscopic level.
The constitutive equations (extending Terzaghi’s effective stress concept to double porosity) [1, 5]:(E.5)tlj=2μelj+λerrδlj-β1p2+β1p2δlj,l,j=1,2,3.
λ and μ are the Lame constants, and δlj is Kronecker’s delta.
Darcy’s law for material with double porosity [7, 8]:(E.6)v1=-1μ′κ1gradp1+κ12gradp2-ρ1s1,v2=-1μ′κ21gradp1+κ2gradp2-ρ2s2,
where μ′ is the fluid viscosity, κ1 and κ2 are the macroscopic intrinsic permeabilities associated with the matrix and fissure porosity, respectively, and κ12 and κ21 are the cross-coupling compressibilities for fluid flow at the interface between the matrix and fissure phases; ρ1,s(1) and ρ2,s(2) are the densities of fluid and the external forces for the pore and fissure phases, respectively.
Substituting (E.3)–(E.6) into (E.1) and (E.2), assuming that u=u1,u2, pj=pjx1,x2, uj=ujx1,x2, ρ=0, ρ1=0, sj=0 and Fj=0, we obtain the system of homogeneous equations of motion in the coupled linear theory of elasticity for solids with double porosity structure (see (1) and (2)).
Competing Interests
The authors declare that they have no competing interests.
WilsonR. K.AifantisE. C.On the theory of consolidation with double porosity19822091009103510.1016/0020-7225(82)90036-2ZBL0493.760942-s2.0-0020009001BeskosD. E.AifantisE. C.On the theory of consolidation with double porosity-II198624111697171610.1016/0020-7225(86)90076-5MR8702892-s2.0-0022882025KhaledM. Y.BeskosD. E.AifantisE. C.On the theory of consolidation with double porosity—III: a finite element formulation19848210112310.1002/nag.16100802022-s2.0-0021391937De BoerR.2000Berlin, GermanySpringer10.1007/978-3-642-59637-7KhaliliN.ValliappanS.Unified theory of flow and deformation in double porous media19961523213362-s2.0-0029777182KhaliliN.Coupling effects in double porosity media with deformable matrix200330222-s2.0-0942289039BerrymanJ. G.WangH. F.The elastic coefficients of double-porosity models for fluid transport in jointed rock19951001224611246272-s2.0-0029529019BerrymanJ. G.WangH. F.Elastic wave propagation and attenuation in a double-porosity dual-permeability medium2000371-2637810.1016/S1365-1609(99)00092-12-s2.0-0033883506SvanadzeM.De CiccoS.Fundamental solutions in the full coupled theory of elasticity for solids with double porosity2013655367390SvanadzeM.Fundamental solution in the theory of consolidation with double porosity200516123130TsagareliI.BitsadzeL.Explicit solution of one boundary value problem in the full coupled theory of elasticity for solids with double porosity201522651409141810.1007/s00707-014-1260-8MR33232552-s2.0-84938079404BasheleishviliM.BitsadzeL.Explicit solutions of the boundary value problems of the theory of consolidation with double porosity for the half-plane2012191414810.1515/gmj-2012-0002MR2901279ZBL1300.740182-s2.0-84860505040BitsadzeL.TsagareliI.The solution of the Dirichlet BVP in the fully coupled theory of elasticity for spherical layer with double porosity20165161457146310.1007/s11012-015-0312-zMR34996222-s2.0-84945589840KhaliliN.SelvaduraiA. P. S.A fully coupled constitutive model for thermo-hydro-mechanical analysis in elastic media with double porosity20033024752-s2.0-1642617812KhaliliN.SelvaduraiA. P. S.On the constitutive modelling of thermo-hydro-mechanical coupling in elastic media with double porosity2004255956410.1016/s1571-9960(04)80099-52-s2.0-77957059514StraughanB.Stability and uniqueness in double porosity elasticity2013651810.1016/j.ijengsci.2013.01.001MR30457342-s2.0-84874731487BitsadzeL.TsagareliI.Solutions of BVPs in the fully coupled theory of elasticity for the space with double porosity and spherical cavity20163982136214510.1002/mma.36292-s2.0-84948152434NowackiW.Translated from Polish B. E. Pobedri, Mir, Moscow, Russia, 1975 (Russian)UlitkoA. F.1979Kyiv, UkraineNaukova DumkaKhomasuridzeN. G.Elastic equilibrium of multi-layer plates on classical and moment theories2Proceedings of the All-Union Conference-Seminar in Tbilisi on the Theory of Numerical Method of Calculation of Plates and Shells1984346366KhomasuridzeN.ZirakashviliN.Some two dimensional elastic equilibrium problems of elliptic bodies1999493948KhomasuridzeN. G.The symmetry principle in continuum mechanics2007711202910.1016/j.jappmathmech.2007.03.008MR23319742-s2.0-34248166640BrownJ. W.ChurchillR. V.19935thNew York, NY, USAMcGraw-HillBitsadzeA. V.1980Mir PublishersMR587310JahnkeE.EmdeF.LeshF.Moscow, RussiaIzdat. Nauka1977 (Russian)BermantA. F.Moscow, RussiaFizmatgiz1958 (Russian)BiotM. A.Mechanics of deformation and acoustic propagation in porous media19623314821498MR0152238BiotM. A.Theory of finite deformations of pourous solids197221597620MR0297184