Dynamic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 238

The next step in the development of the coupled wave equations is to substitute
the expression ( 9.9 ) for the solid stress tensor and the expression ( 9.10 ) for the pore
fluid pressure into the conservation of momentum for the solid phase ( 9.12 ) and the
conservation of momentum for the fluid phase ( 9.13 ), respectively. However
the analysis to this point does not include the force on the fluid phase set up by
the drag of the fluid moving over the solid surface. To account for this force the
flow-resistivity tensor R , defined as the inverse of the intrinsic permeability tensor K
(see ( 8.26 )), is introduced:

K 1

R ¼ : (9.14)

w represents the effect of the fluid-solid
interaction on the fluid phase. Formally it should be subtracted from the left hand
side of ( 9.13 ), but we add it to the right hand side, to the mass times acceleration
terms, thus

m R

The viscous resistive force

r p ¼ r f ð€ u þ J € w Þþm R _ w : (9.15)

Finally, accomplishing the substitutions mentioned above into ( 9.12 ) and ( 9.13 ),
but using ( 9.15 ) instead of ( 9.13 ), one obtains

2 u k 2 w k

Z ijkm @ M ij @

x j þ x j ¼ r€

u i þ r f € w i ; (9.16)

@ x m @ @ x k @

2 u k 2 w k

M km @ M @

x i þ x i ¼ fr f ð€

u i þ J ij € w j Þþm R ij w j : _ (9.17)

@ x m @ @ x k @

Equations ( 9.16 ) and ( 9.17 ) are two coupled wave equations for the solid
displacement field u and the displacement field w of the fluid relative to the solid.

Problems

9.2.1. Explain the differences between the quasistatic formulation and the dynamic
formulation of the theory of poroelasticity.
9.2.2. How does the composite elasticity tensor Z d , Z d

¼ C d ðA A

þ M Þ

, change

when the porosity of the porous medium vanishes?
9.2.3. Show that the coupled system of equations ( 9.16 ) and ( 9.17 ) reduce to the
wave equation for an anisotropic elastic continuum with no porosity.
9.2.4. Using the indicial notation substitute the expression ( 9.9 ) for the solid stress
tensor and the expression ( 9.10 ) for the pore fluid pressure into the conser-
vation of momentum for the solid phase ( 9.12 ) and the conservation of
momentum for the fluid phase ( 9.15 ), respectively, and derive the coupled
wave equations ( 9.16 ) and ( 9.17 ).