Quasistatic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 211

how the Darcy permeability law could be obtained from a model for the flow of a
viscous fluid through a rigid porous solid. These ideas, plus the example of the
compressed, fluid-saturated sponge described in Sect. 8.1 , convey the physical
meaning of Darcy's law in three different models.

8.5 Matrix Material and Pore Fluid
Incompressibility Constraints

The two incompressibility constituent-specific constraints for poroelasticity are that
matrix material and the pore fluid is incompressible. These two incompressibility
constraints are kinematic in nature, requiring that both materials experience no
volume change at any stress level. Constraints of this type introduce an indetermi-
nate pressure in both the fluid and the matrix material that must be equal in both
materials at any location from the requirement of local force equilibrium. Thus the
two assumptions are compatible.
The requirement that the fluid be incompressible is implemented by requiring
that the reciprocal of the bulk modulus of the fluid tends to zero as the instantaneous
density tends to the initial density,

1
K f

lim
r f ! r fo

! 0, or by imposing the condition that

the fluid density r fo are equal. Recall

that, in the case of a compressible fluid above, the value of the fluid content

r f be a constant; thus r f and its initial value

can
change if the fluid density changes or if the porosity changes. However, in the case
of the incompressible fluid since

z

r f ¼ r f o , the fluid content z ( 8.16 ) can change only

if the porosity changes,

z ¼ f f o : (8.28)

r f ¼ r fo ,

A similar change occurs for the Darcy's law ( 8.17 ) when

K T q ¼ f v ¼ð 1 =mÞ K r p ð x ; t Þ; K ¼ : (8.29)

The requirement that the matrix material be incompressible involves a slightly
longer development. Hooke's law for the matrix material is written as

E m T m

¼ S d

(8.30)

and the incompressibility constraint for the matrix material is the requirement that
the dilatational strain,

E m T m ¼ T m

U

E kk ¼ U S d S d U tr E m ¼ ¼ (8.31)

vanish for all possible stress states T m , thus

E m

U

) U S d ¼ S d U

¼ 0 ¼ 0 : (8.32)