ICE REPORT 18-2 February 218 A framework for arriving at bounds on effective moduli in heterogeneous poroelastic solids by aumik Dana and Mary F. Wheeler The Institute for Computational Engineering and ciences The University of Texas at Austin Austin, Texas 78712 Reference: aumik Dana and Mary F. Wheeler, "A framework for arriving at bounds on effective moduli in heterogeneous poroelastic solids," ICE REPORT 18-2, The Institute for Computational Engineering and ciences, The University of Texas at Austin, February 218.
A framework for arriving at bounds on effective moduli in heterogeneous poroelastic solids aumik Dana a, Mary F. Wheeler a a Center for ubsurface Modeling, Institute for Computational Engineering and ciences, UT Austin, Austin, TX 78712, UA Abstract We invoke the concepts of representative volume element (RE), statistical homogeneity and homogeneous boundary conditions to arrive at bounds on effective moduli for heterogeneous poroelastic solids. The homogeneous displacement boundary condition applicable to linear elasticity is replaced by a homogeneous displacement-pressure boundary condition to arrive at an upper bound (within the RE) while the homogeneous traction boundary condition applicable to linear elasticity is replaced by a homogeneous traction-flux boundary condition to arrive at a lower bound (within the RE). tatistical homogeneity is then invoked to argue that the bounds obtained over the RE are representative of the bounds obtained over the whole heterogeneous poroelastic solid. Keywords: RE, Homogeneous boundary conditions, tatistical homogeneity, Bounds, Effective moduli 1. Introduction We invoke the concepts of RE (see Hashin and htrikman [1], Hill [11], Hashin [8]) commonly used in the analysis of composite materials, as well as statistical homogeneity (see Beran [1], Kröner [13]) to analyse porous solids. This approach, albeit slighty less accurate than the theory of two-scale homogenization, offers avenues for arriving at bounds on effective poroelastic moduli. The reader is refered to Fish [6] for a rendition of two-scale homogenization for the analysis of composite materials and Hollister and Kikuchi [12] for a comparison of two-scale homogenization and RE based analyses for porous composites. Email addresses: saumik@utexas.edu (aumik Dana), mfw@ices.utexas.edu (Mary F. Wheeler)
RE based methods decouple analysis of a composite material into analyses at the local and global levels. The local level analysis models the microstructural details to determine effective elastic properties by applying boundary conditions to the RE and solving the resultant boundary value problem. The composite structure is then replaced by an equivalent homogeneous material having the calculated effective properties. The global level analysis calculates the effective or average stress and strain within the equivalent homogeneous structure. The applied boundary conditions, however, cannot represent all the possible in-situ boundary conditions to which the RE is subjected within the composite. The accuracy of the RE approximation depends on how well the assumed boundary conditions reflect each of the myriad boundary conditions to which the RE is subjected in-situ. This is where the imposition of homogeneous boundary conditions plays a special role in arriving at bounds on effective properties. For the sake of convenience, given a bounded convex domain Ω R 3, we discard the differential in the integration of any scalar field χ over Ω as follows χ(x) χ(x) d ( x Ω) Ω Ω Likewise, given a bounded convex domain R 2, we discard the differential in the integration of any scalar field χ over as follows χ(x) χ(x) d ( x ) 1.1. The argument on bounds on effective moduli using displacement boundary conditions vis-á-vis traction boundary conditions 1 Consider the case in a linear elasticity boundary value problem where the in-situ boundary conditions differ from the applied boundary conditions, but produce the same average RE strain. In this case the average stiffness predicted by the RE analysis must be greater than the actual stiffness by the principle of minimum strain energy 2. The in-situ boundary conditions would minimize the energy while the assumed boundary conditions 1 The language used in this module is similar to the language used in Hollister and Kikuchi [12] due of its ease of explanation 2 The reader is refered to Fung [7] for a rendition of the classical extremum principles in elasticity, including the principle of minimum strain energy and the principle of minimum complementary energy 2
would be admissible and by definition produce greater energy. The average stress within the RE under assumed boundary conditions must be higher to produce a higher energy. The same argument holds for applied tractions boundary conditions with the principle of minimum complementary energy. In this situation, the applied traction boundary condition will produce a higher complementary energy than an in-situ traction condition for the same average stress giving a higher compliance and therefore a lower stiffness. Thus, RE analyses under applied displacements give an upper bound on apparent stiffness while applied tractions give a lower bound. 1.2. Homogeneous boundary conditions R 1 R 2 12 2 1 Figure 1: Two phase body. 1 2 12 12 Consider a two phase body with phases, as shown in Figure 1, occupying regions R 1 and R 2 with displacement fields u (1) i (x) and u (2) i (x) respectively. The volume of the two phase body is, the phase volumes are 1 and 2, the bounding surface is and the interface is 12. 1.2.1. Expression for average strain The strains ɛ ij (see (2.8)) are defined as ɛ ij = 1 2 (u i,j(x) + u j,i (x)) (1.1) 3
The volume average ɛ ij of ɛ ij is given by ɛ ij = 1 ɛ ij (x) (1.2) ubstituting (1.1) in (1.2), we get ɛ ij = 1 2 (u i,j (x) + u j,i (x)) which, in lieu of the divergence theorem, can be written as ɛ ij = 1 [ 2 1 (u (1) i n j + u (1) j n i ) + 2 (u ] (2) i n j + u (2) j n i ) (1.3) where 1 and 2 are the bounding surfaces of R 1 and R 2. Now each of 1 and 2 are composed of part of the external surface and the interface 12. Therefore, (1.3) may be rewritten as ɛ ij = 1 [ 2 = 1 2 (1) (u i n j + u j n i ) + (ui n j + u (1) j n i ) + (u 12 12 (2) i n j + u (2) ] j n i ) } {{ } (1.4) (u i n j + u j n i ) (1.5) where the interface integral terms in (1.4) cancel out as n i (and n j ) is always the outward normal. 1.2.2. Expression for average stress Let the stress field inside the body be σ ij (x) and the body force per unit volume be F i (x). The body is assumed to be in quasi-static equilibrium, so that at every point (see (2.6)), σ ij,j + F i = On the external surface, the tractions are prescribed t i () = σ ij n j 4
The average stress is defined by σ ij = 1 σ ij (1.6) We substitute the following (σ ik x j ),k = σ ik,k x j + σ ik δ jk = F i x j + σ ij into (1.6), to get σ ij = 1 ((σ ik x j ),k + F i x j ) = 1 = 1 [ (1) x j σ ik n(1) k + = 1 1 [ x j σ ik n k + 12 2 [ (2) x j σik n(2) k + t (1) i {}}{ + x j σ (1) ik n(1) k σ ik x j n k + 12 F i x j ] t (2) i {}}{ x j σ (2) ik n(2) k } {{ } F i x j ] + F i x j ] (1.7) (1.8) where the second equality in (1.7) with the surface integral follows from the divergence theorem and the integrands of the interface surface integrals in (1.8) cancel one another at each interface point and thus the two surface integrals cancel each other. Also, since σ ij (x) = σ ji (x), it follows that σ ij = σ ji. Thus (1.8) can be symmetrized in the form σ ij = 1 [ ] (x j t i + x i t j ) + (x j F i + x i F j ) 2 (1.9) 1.2.3. Homogeneous displacement boundary conditions A homogeneous displacement boundary condition is refered to the case of boundary prescribed displacements (over the entire boundary) corresponding to uniform strain (over the entire boundary) as follows u i () = ɛ ijx j (1.1) ubstituting (1.1) in (1.5), we get ɛ ij = 1 2 (ɛ ik x kn j + ɛ jk x kn i ) = 1 2 (ɛ ik x k,j + ɛ jk x k,i) (1.11) 5
= 1 2 (ɛ ik δ kj + ɛ jk δ ki) = 1 2 (ɛ ij + ɛ ji) = ɛ ij (1.12) where the second equality in (1.11) follows from the divergence theorem. (1.12) clearly shows that the RE average strain in case of homogeneous displacement boundary condition is equal to the uniform boundary strain that the boundary condition corresponds to. 1.2.4. Homogeneous traction boundary conditions A homogeneous traction boundary condition is refered to the case of boundary prescribed tractions (over the entire boundary) corresponding to uniform stress (over the entire boundary) as follows t i () = σ ijn j (1.13) ubstituting (1.13) in (1.9), with zero body forces, we get σ ij = 1 (x j σik 2 n k + x i σjk n k) = 1 (x j,k σik 2 + x i,kσjk ) (1.14) = 1 (δ jk σik 2 + δ ikσjk ) = 1 (σji + σ 2 ij) = σ ij (1.15) where the second equality in (1.14) follows from the divergence theorem. (1.15) clearly shows that the RE average stress in case of homogeneous traction boundary condition is equal to the uniform boundary stress that the boundary condition corresponds to. 2. Model equations 2.1. Flow model Let the boundary Ω = Γ f D Γf N where Γf D is the Dirichlet boundary and Γf N is the Neumann boundary. The fluid mass conservation equation (2.1) in the presence of deformable and anisotropic porous medium with the Darcy law (2.2) and linear pressure dependence of density (2.3) with boundary conditions (2.4) and initial conditions (2.5) is ζ t + z = q (2.1) 6
z = K µ ( p ρ g) = κ( p ρ g) (2.2) ρ = ρ (1 + c (p p )) (2.3) p = g on Γ f D (, T ], z n = on Γf N (, T ] (2.4) p(x, ) = p (x), ρ(x, ) = ρ (x), φ(x, ) = φ (x) ( x Ω) (2.5) where p : Ω (, T ] R is the fluid pressure, z : Ω (, T ] R 3 is the fluid flux, ζ is the fluid content, n is the unit outward normal on Γ f N, q is the source or sink term, K is the uniformly symmetric positive definite absolute permeability tensor, µ is the fluid viscosity, ρ is a reference density, φ is the porosity, κ = K µ is a measure of the hydraulic conductivity of the pore fluid, c is the fluid compressibility and T > is the time interval. 2.2. Poromechanics model Let the boundary Ω = Γ p D Γp N where Γp D is the Dirichlet boundary and Γp N is the Neumann boundary. Linear momentum balance for the anisotropic porous solid in the quasistatic limit of interest (2.6) with small strain assumption (2.8) with boundary conditions (2.9) and initial condition (2.1) is σ + f = (2.6) f = ρφg + ρ r (1 φ)g (2.7) ɛ(u) = 1 2 ( u + ( u)t ) (2.8) u n 1 = on Γ p D [, T ], σt n 2 = t on Γ p N [, T ] (2.9) u(x, ) = x Ω (2.1) where u : Ω [, T ] R 3 is the solid displacement, ρ r is the rock density, f is the body force per unit volume, n 1 is the unit outward normal to Γ p D, n 2 is the unit outward normal to Γ p N, t is the traction specified on Γp N, ɛ is the strain tensor, σ is the Cauchy stress tensor given by the generalized Hooke s law (see Coussy [5]) σ = σ + Mɛ α(p p ) (2.11) 7
where σ is the in situ stress, M is the fourth order anisotropic elasticity tensor and α is the Biot tensor. The symmetry of the stress and strain tensor shows that M ijkl = M jikl = M ijlk = M klij (i, j, k, l = 1, 2, 3) α ij = α ji (i, j = 1, 2, 3) The inverse of the generalized Hooke s law (2.11) is given by (see Cheng [4]) ɛ = C(σ σ ) + 1 3 CB(p p ) (2.12) where C is the fourth order anisotropic compliance tensor, C(> ) is a generalized Hooke s law constant and B is a generalization of the kempton pore pressure coefficient B (see kempton [14]) for anisotropic poroelasticity. The symmetry of the stress and strain tensor shows that C ijkl = C jikl = C ijlk = C klij (i, j, k, l = 1, 2, 3) B ij = B ji (i, j = 1, 2, 3) 2.3. Fluid content The fluid content ζ is given by (see Cheng [4]) ζ = Cp + 1 3 CB : (σ σ ) 1 M (p p ) + α : ɛ (2.13) where M(> ) is a generalization of the Biot modulus (see Biot and Willis [3]) for anisotropic poroelasticity. 2.4. The poroelastic tensorial constitutive equation and the strain energy density The generalized Hooke s law (2.11) and its inverse (2.12) written in indicial notation as σ ij = σ ij + M ijkl ɛ kl α ij (p p ) ɛ ij = C ijkl (σ ij σ ij ) + 1 3 CB ij(p p ) are rewritten (respectively) in the contracted notation as σ β = σ β + M βθ ɛ γ α β (p p ) 8
ɛ β = C βθ (σ β σ β ) + 1 3 CB β(p p ) where the transformation is accomplished by replacing the subscripts ij (or kl) by β (or θ) using the following rules ij (or kl) β (or θ) ij (or kl) β (or θ) 11 1 23 (or 32) 4 22 2 31 (or 13) 5 33 3 12 (or 21) 6 and the fourth order tensors M 3 3 3 3 and C 3 3 3 3 are represented as second order tensors M 6 6 and C 6 6 respectively. The constitutive equation for the poromechanics model, under the assumption of zero in-situ stress (σ (x) = x Ω) and zero in-situ pore pressure (p (x) = x Ω), is then given by σ 6 1 = M 6 6 ɛ 6 1 α 6 1 p 1 1 (2.14) On the other hand, the constitutive equation for the flow model, under the assumption of zero in-situ stress (σ (x) = x Ω) and zero in-situ pore pressure (p (x) = x Ω), is then given by ζ 1 1 = α 6 1 : ɛ 6 1 + 1 M p 1 1 (2.15) In lieu of (2.14) and (2.15), we can write κ 7 1 { }} {{ }} {{ }} { σ 6 1 = M 6 6 α 6 1 ɛ 6 1 ζ 1 1 α T 6 1 A 7 7 1 M γ 7 1 p 1 1 (2.16) Biot [2] defined the strain energy density of porous elastic medium as U (υ) = 1 (σ : ɛ + pζ) (2.17) 2 where υ (u, p) is the poroelastic field. In lieu of (2.16) and (2.17), the strain energy density of porous elastic medium is given as U (υ) = 1 2 κt γ (2.18) 9
3. Procedural framework to arrive at bounds on effective poroelastic moduli We now invoke the principles derived in ection 1 for linear elastic solids to arrive at bounds on effective moduli for heterogeneous linear poroelastic solids. The homogeneous displacement boundary condition applicable to linear elasticity is replaced by a homogeneous displacement-pressure boundary condition. The homogeneous traction boundary condition applicable to linear elasticity is replaced by a homogeneous traction-flux boundary condition. 3.1. Homogeneous displacement-pressure boundary condition Let the poroelastic body with no body forces be subjected to the homogeneous boundary condition u i ( Ω) = ɛ ij x j p( Ω) = p (3.1) The γ vector in (2.16) can be separated into 6 vectors in each of which there occurs only one non-vanishing strain ɛ ij and one vector in which there occurs only one non-vanishing pressure p as follows γ 1 γ 2 γ 3 γ4 γ5 γ 6 γ 7 γ 3 = + + + + + + γ 1 γ 2 γ 4 γ 5 γ 6 γ 7 (3.2) In lieu of (3.2), the poroelastic boundary field (u, p) can be decomposed as u 1 γ1 x 1 γ4 x 3 γ6 x 2 u 2 γ2 = + x 2 γ5 + + + x 3 γ6 + x 1 + u 3 γ3 x 3 γ4 x 1 γ5 x 2 p γ 7 (3.3) Because of the superposition principle of the linear theory of elasticity (and hence by natural extension poroelasticity), the poroelastic υ (u, p) field which is produced by (3.1) is 1
equal to the sum of the seven fields which are produced by the application of each of (3.3), separately, on the boundary. uppose that γ 1 = 1 and let the resulting poroelastic υ (u, p) field be denoted by υ (γ 1) {v (γ 1) 1, v (γ 1) 2, v (γ 1) 3, v (γ 1) 4 } T. Then, when γ 1 1, the field is by linearity γ 1 υ(γ 1). imilar considerations for each of γ 2, γ 3, γ 4, γ 5, γ 6 and γ 7 on the right hand side of (3.3) and superposition show that the poroelastic υ (u, p) field in the body due to (3.1) on the boundary can be written in the form υ = γ k υ(γ k) (3.4) where k is summed. The γ field at any point is then given by γ(x) = γ k γ( υ (γ k) ) (3.5) Finally, the κ at any point is given in view of (2.16) and (3.5) as κ(x) = γ k A(x)γ( υ (γ k) ) (3.6) where A(x) are the space dependent poroelastic moduli of the heterogeneous body and k is summed. Let (3.6) be volume averaged over a RE. The result is written in the form κ = A γ k where the upper bound on effective poroelastic moduli is obtained as A 1 A(x)γ ( υ (γ k) ) (upper bound) 3.2. Homogeneous traction-flux boundary condition Let the poroelastic body with no body forces be subjected to the homogeneous boundary condition t i ( Ω) = σ ij n j ζ( Ω) = ζ (3.7) The κ vector in (2.16) can be separated into 6 vectors in each of which there occurs only one non-vanishing stress σ ij and one vector in which there occurs only one non-vanishing 11
fluid content ζ as follows κ 1 κ 2 κ 3 κ 4 κ 5 κ 6 κ 7 κ 3 = + + + + + + κ 1 κ 2 κ 4 κ 5 κ 6 In lieu of (3.2), the poroelastic boundary field (t, ζ) can be decomposed as t 1 κ 1 n 1 κ 4 n 3 κ 6 n 2 t 2 κ 2 = + n 2 κ 5 + + + n 3 κ 6 + n 1 + t 3 κ 3 n 3 κ 4 n 1 κ 5 n 2 ζ κ 7 κ 7 (3.8) (3.9) Because of the superposition principle of the linear theory of elasticity (and hence by natural extension poroelasticity), the poroelastic υ (t, ζ) field which is produced by (3.1) is equal to the sum of the seven fields which are produced by the application of each of (3.3), separately, on the boundary. uppose that κ 1 = 1 and let the resulting poroelastic υ (t, ζ) field be denoted by υ (κ1) {v (κ 1) 1, v (κ 1) 2, v (κ 1) 3, v (κ 1) 4 } T. Then, when κ 1 1, the field is by linearity κ 1 υ(κ1). imilar considerations for each of κ 2, κ 3, κ 4, κ 5, κ 6 and κ 7 on the right hand side of (3.3) and superposition show that the poroelastic υ (t, ζ) field in the body due to (3.1) on the boundary can be written in the form υ = κ k υ(κ k) (3.1) where k is summed. The κ field at any point is then given by κ(x) = κ k κ( υ (κ k) ) (3.11) Finally, the γ at any point is given in view of (2.16) and (3.11) as γ(x) = κ k A 1 (x)κ ( υ (κ k) ) (3.12) 12
where A(x) are the space dependent poroelastic moduli of the heterogeneous body and k is summed. Let (3.12) be volume averaged over a RE. The result is written in the form γ = A 1 κ k where the effective poroelastic compliance is obtained as A 1 1 A 1 (x)κ ( υ (κ k) ) which is inverted to obtain the lower bound on the effective poroelastic moduli ( 1 A A 1 (x)κ ( υ (κ k) )) 1 (lower bound) 3.3. Invoking statistical homogeneity The essence of statistical homogeneity is explained in Appendix A. The crux of the argument is that the stress and strain fields in a very large heterogeneous body that is statistically homogeneous, subject to homogeneous boundary conditions, is statistically homogeneous. The argument is that if the field is statistically homogeneous, then the volume average taken over the RE approaches the whole body average, wherever the RE may be located. Appendix A. tatistical homogeneity 3 Consider a collection of N fibrous cylindrical specimens, each of which is refered to a system of axes, the origin of which is at the same point in each specimen. The specimens have the same external geometry, however, their phase geometries, i.e. internal geometries may be quite different. In the theory of random processes, such a collection of specimens is called an ensemble and each specimen is a member of the ensemble. We consider the two points B 1, B 2 in each specimen member of the ensemble, which have the same position vectors x 1 and x 2 in each specimen. The probability that B 1 is in R 1 and B 2 is in R 1 is P (B 1 R 1, B 2 R 1 ) = P 11 (x 1, x 2 N 11 ) = lim N N 3 The language used in this module is similar to the language used in Hashin [9] due of its ease of explanation 13
imilarly, we have P (B 1 R 1, B 2 R 2 ) = P 12 (x 1, x 2 N 12 ) = lim N N P (B 1 R 2, B 2 R 1 ) = P 21 (x 1, x 2 N 21 ) = lim N N P (B 1 R 2, B 2 R 2 ) = P 22 (x 1, x 2 N 22 ) = lim N N Here N 11 is the number of times that both points fall simultaneously into phase 1, with analogous interpretations for N 12, N 21 and N 22. There are similar eight three point probabilities and in general 2 n, n point probabilities. uch n point probabilities may be written in the form P i1,i 2,..,i n (x 1, x 2,.., x n ) (A.1) where each of the subscripts i 1, i 2,..., i n assumes either one of the phase numbers 1, 2 and its position in the subscript sequence is attached to that corresponding to the position vector within the parenthesis. We now finally proceed to define the concept of statistical homogeneity. For this purpose, the system of n points entering into (A.1) may be considered as a rigid body which is described by the vector differences r 1 = x 1 x n, r 2 = x 2 x n,, x n 1 = x n 1 x n (A.2) uppose that any multipoint probability such as (A.1) depends only on the relative configuration of the points and not on their absolute position with respect to the coordinate system; then the ensemble is called statistically homogeneous. Mathematically this means P i1, P i2 i n (x 1, x 2,.., x n ) = P i1,i 2,..,i n (r 1, r 2,.., r n 1 ) (A.3) References [1] M. J. Beran. tatistical Continuum Theories, volume 9 of Monographs in tatistical Physics and Thermodynamics. Interscience Publishers Inc., 1st edition, 1968. [2] M. A. Biot. Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics, 33(4):1482 1498, 1962. 14
[3] M. A. Biot and D. G. Willis. The elastic coefficients of the theory of consolidation. Journal of Applied Mechanics, 24:594 61, 1957. [4] A. H. D. Cheng. Material coefficients of anisotropic poroelasticity. International Journal of Rock Mechanics and Mining ciences, 34:199 25, 1997. [5] O. Coussy. Poromechanics. Wiley, 2nd edition, 24. [6] J. Fish. Practical multiscaling. John Wiley and ons Inc, 1st edition, 214. [7] Y. C. Fung. Foundations of solid mechanics. International eries in Dynamics. Prentice Hall, 1965. [8] Z. Hashin. Analysis of composite materials a survey. Journal of Applied Mechanics, 5(3):481 55, 1983. [9] Z. Hashin. Theory of fiber reinforced materials. NAA, CR-1974. [1] Z. Hashin and. htrikman. On some variational principles in anisotropic and nonhomogeneous elasticity. Journal of the Mechanics and Physics of olids, 1(4):335 342, 1962. [11] R. Hill. Elastic properties of reinforced solids: ome theoretical principles. Journal of the Mechanics and Physics of olids, 11(5):357 372, 1963. [12]. J. Hollister and N. Kikuchi. A comparison of homogenization and standard mechanics analyses for periodic porous composites. Computational Mechanics, 1(2):73 95, 1992. [13] E. Kröner. tatistical Continuum Mechanics. CIM Courses and lectures 92. pringer, 1st edition, 1974. [14] A. W. kempton. The pore-pressure coefficients a and b. Géotechnique, 4(4):143 147, 1954. 15