Compositional Flow in Fractured Porous Media: Mathematical Background and Basic Physics - Selected Topics of Computational and Experimental Fluid Mechanics - page 13

isotropic solid are the same in all directions. Although anisotropy is important in
some regions of the Earth's interior, isotropy has proven to be a reasonable first-
order approximation for most parts. Therefore, if we assume isotropy, C ijkl will be
invariant with respect to rotation and the number of independent parameters reduces
to two parameters only so that

C ijkl = ʻʵ kk ʴ ij + 2 μʵ ij , (37)

ʴ ij is the Kronecker delta, and ʻ where are called the Lamé parameters of

the material, which characterize the rigidity of the solid. These two parameters com-
pletely describe the linear stress-strain relation within an isotropic solid.

and μ

is termed
the shear modulus and is a measure of the resistance of the material to shearing, while
the other parameter, ʻ

μ

, does not have a simple physical explanation. In the case of
pure shear, the elastic constant that comes into play is G , while for isotropic

compression the strain tensor is proportional to the identity tensor. The elastic con-
stant relating the pressure to the decrease of volume is K

μ =

= ʻ + 2 μ , where K is

called the compressibility modulus.
Mathematically, a heterogeneous material is represented by a random medium

M , i.e., a family of media M(ˉ) whose members differ slightly from the homoge-

neous (reference) medium M 0 , where ˉ is a point in the sample space ʩ (space of

events). In probability theory of an experiment

or observation. To close the statistical description we must provide the mathemati-
cal model with a probability distribution over the members

ʩ consists of all possible outcomes ˉ

. However, most
heterogeneous materials are nonequilibrium, quenched disordered media and so we
have no access to a true statistical ensemble. Thus, theory and experiment can only
be connected through some kind of ergodicity. The equivalence between theoretical
and observational averaging in heterogeneous media is an open problem and will not
be addressed here. Therefore, our description of a heterogeneous system as a random
medium is based on a heuristic model for the local spatial variations.
Rigorous methods are mostly limited to the study of simple models. In particu-
lar, we model the heterogeneous medium by introducing spatial fluctuations of the
material density and Lamé coefficients, i.e.,

M(ˉ)

ˁ R ( x ) = ˁ R , 0 + ʵˁ R , 1 ( x ),

ʻ(

) = ʻ 0 + ʵʻ 1 ( x x ), (38)

μ( ) = μ 0 + ʵμ 1 ( ),

x x

where

ˁ R , 0 = ˁ R ( x ) ,

2
3 G

ʻ 0 = ʻ(

) = − ,

x K

μ 0 = μ( x ) = G ,