Quasistatic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 210

Problems
8.3.1. Prove that U

T

tr T .
8.3.2. Show that the isotropic elastic compliance tensor S , twice contracted with U ,
the six-dimensional vector representation of the three-dimensional unit ten-
sor 1 , U

¼

U , is given by ( 8.19 ).
8.3.3. Show that, in the case of isotropy, A is given by

S

aU , where

a is the isotropic

( K d /K m )]. effective stress coefficient given by a ¼ [1

8.3.4. In the case of isotropy show that U S d C d S d U is equal to K Reff =ð

2 ,

K Reff Þ

S d S d

and thus A A is given by A A

1
K d

K m and

1 1

K f

¼ L ¼

K Reff þ fð

K d
ðK m

1
K m

Þ

2 .

Þ

8.4 Darcy's Law

The constitutive equations of poroelasticity developed thus far are the strain-stress-
pore pressure relations ( 8.1 ) and the fluid content-stress-pore pressure relation
( 8.17 ). The other constitutive relation of poroelasticity is Darcy's law, which relates
the fluid mass flow rate, r f v , to the gradient ( r p ) of the pore pressure p ,

H T q ¼ðfr f =r fo Þ v ¼ H ð p Þr p ð x ; t Þ; H ð p Þ¼ ð p Þ; ð : Þ

4 36D repeated

where the symmetry in H has been shown to hold for material symmetries greater
than monoclinic (Chap. 4 ) . In this equation r f is the fluid density and r fo is a
reference value of the fluid density. It will be assumed that H is independent of pore
fluid pressure and that H ¼ K / m , where K is the intrinsic Darcy's law permeability

tensor, and is the fluid viscosity. The intrinsic permeability tensor K has units of

length squared and is a function of the porous structure only, not the fluid in the
pores; thus Darcy's law takes the form

m

K T q ¼ðfr f =r fo Þ v ¼ð 1 =mÞ K r p ð x ; t Þ; K ¼ ; (8.26)

where it has been shown that the symmetry in K holds for material symmetries
greater than monoclinic and where the volume flux q has the dimension of velocity
because it is the volume flow rate per unit area. In the case of isotropy, Darcy's law
is written in the form

q ¼ðfr f =r fo Þ v ¼ð k =mÞr p ð x ; t Þ: (8.27)

Recall from Sect. 1.8 the lumped parameter model for Darcy's law and consider
that idea again in the present context. Also, recall from Sect. 7.5 that it was shown