attenuation peak due to Squirt flow mechanism shifts towards higher frequencies, whereas the
one that results from the Biot’s mechanism moves in the opposite direction. It is then to be
expected that for particular rock and fluid physi- cal parameters, these two peaks occur very far
apart in the frequency scale that they cannot be observed simultaneously or even individually in
an actual experiment due to limitations in mea- surement techniques. There is no doubt that
both the loss mechanisms described above coex- ist in saturated porous media loaded by a pass-
ing acoustic wave, their relative importance seems to be dependent on the nature of the
Ž .
porous medium whether consolidated or not and the frequency range of observation. Most of
the cases where Biot model has been recognised as an adequate mechanism to account for atten-
uation and velocity dispersion vs. frequency refer to unconsolidated sediments or highly per-
Ž meable rocks Bourbie et al., 1987; Yamamoto
´
. and Turgut, 1988; Yamamoto et al., 1994 ,
whereas reported cases where squirt flow is the most significant loss mechanism relate to exper-
iments carried out on consolidated sediments
Ž with relatively low permeability Mavko and
Nur, 1979; Jones, 1986; Murphy et al., 1986; .
Tutuncu and Sharma, 1992; Morig, 1993 .
¨
2. Theory
Ž .
Dvorkin and Nur 1993 , proposed a new
theory of dynamic poroelasticity, which simul- taneously
combines the
Biot’s mechanism
treated at macroscopic scale and the squirt flow Ž
mechanism treated at microscopic scale at indi- .
vidual pore level . This resulted in what is Ž
. known as the BiotrSquirt BISQ flow theory.
Model prediction of attenuation by the BISQ model seems to be more realistic for specific
case than that given by Biot’s mechanism alone. The novelty in this concept with respect to that
of Biot, comes mainly from the fact that in the BISQ model, the equation of fluid mass conser-
vation used to describe the fluid dynamic is two Ž
dimensional instead of one dimensional as in .
the initial Biot’s theory , that is, not only fluid flow parallel to wave propagation direction is
Ž .
considered Biot mechanism , but also fluid flow Ž
. in perpendicular direction
squirt flow . The principal difference between the BISQ model
and Biot’s model is felt through the expression of pore fluid pressure. In the Biot’s model this
Ž .
expression is given Dvorkin and Nur, 1993 by
EU g Eu
P s yF q
, 4
Ž .
ž
E x f E x
Ž whereas for the BISQ model, Dvorkin and Nur,
. 1993 it becomes:
2 J l R EU
g Eu
Ž .
1
P s yF 1 y q
, 5
Ž .
ž
l RJ l R E x
f E x
Ž .
where u and U are respectively solid and fluid displacement in the x-direction, g s a q f and
a is a poroelastic coefficient,
k
b
a s 1 y .
k
s
Ž . Ž .
The other parameters in Eqs. 4 and 5 are given as
y1
1 1
F s q
,
2
ž
r V f Q
f f
1 1
K
b
s 1 y f y
,
ž
Q K
K
s s
2
r v f q r rr
v
f s
f c
2
l R s R
q .
ž
F f
v R is the Squirt flow length, V the fluid acoustic
f
velocity, v the angular frequency, r the cou-
a
pling density and J , J represent Bessel func-
1
tions of first and second order, respectively. v
c
represents Biot’s characteristic frequency given by
v hf
c
s .
v k r v
f
The formulation of this theory like many others incorporating squirt flow mechanisms
ŽO’Connell and Budiansky, 1977; Murphy et .
al., 1986 continues to use parameters that de- pend on micro-scale geometry of porous medium
Ž architecture. Because local pore geometry e.g.,
. aspect ratio is one of the parameters that affects
relaxation frequency for local fluid absorption mechanism, we are confronted with the problem
of having different relaxation frequencies for rocks having a variety of pore geometry. More-
over, the BISQ model is based on one particular
Ž geometrical construction a cylinder with a con-
. stant pressure on its side while it is intended to
quantify in a wide range of rocks regardless of the particular geometry or spatial distribution of
the medium inhomogeneity that leads to local flow. Theoretical study by Gurevich and Lopat-
Ž .
Ž nikov 1995 shows that attenuation as well as
. dispersion is sensitive to the spatial distribution
of inhomogeneity, which may affect not only Ž
some parameters characteristic frequency or
. squirt flow length , but also the very form of the
attenuation and dispersion curves. To circum- vent this problem, we propose a new expression
for the pore fluid pressure, independent of the squirt flow length parameter R. This is achieved
by reconsidering the way in which the two-di- mensional fluid flow has been incorporated by
the BISQ model through the equation of fluid’s
Ž .
mass conservation. Dvorkin and Nur 1993 used the following equation:
f Er Ef
E
2
U y u
Ž .
f x
x
q q f
r Et
Et E xEt
f
E
2
z 1 Ez
q f q
s0, 6
Ž .
ž
ErEt r Er
where U is fluid displacement and u the solid
x x
displacement in the wave propagation direction Ž
. i.e. x-direction , z the fluid displacement in the
direction perpendicular to wave propagation di- Ž
. rection radial-direction observed on a repre-
sentative cylindrical volume of rock. Instead of considering a coordinate system where the fluid
flow in parallel and perpendicular directions to the wave propagation are independently treated,
we will use a coordinate system where the radial fluid displacement z is replaced by U . In
r
this case, the relative fluidrsolid displacement in the direction of P-wave propagation is ex-
pressed as
W s f U y u .
7
Ž .
Ž .
x x
x
Ž .
Using U instead of z into Eq. 6 yields:
r
f Er Ef
E
2
U y u
Ž .
f x
x
q q f
r Et
Et E xEt
f
E
2
U 1 EU
r r
q f q
s 0. 8
Ž .
ž
ErEt r Er
From now on, we will consider that U and
x
U represent the axial and radial components of
r
the fluid displacement vector U, whereas u
x
and u represent the axial and radial compo-
r
nents of the solid displacement vector u, respec- tively.
Porosity differential is related to differentials of skeleton’s deformation and of the fluid pres-
Ž .
sure as Biot, 1941 : d P
df s a d e q ,
9
Ž .
Q where
Eu Eu
Eu
x y
z
e s q
q ,
10
Ž .
E x E y
E z represents the solid dilatation expressed in
cartesian coordinates, which is equivalent to:
Eu u
Eu
r r
x
e s q
q ,
11
Ž .
Er r
E x in axisymetrical cylindrical coordinates about
the x-axis. Ž
. Ž
. EfrEt is given by Dvorkin and Nur 1993
as Ef
E
2
u 1 EP
x
s a q
. 12
Ž .
Et E xEt
Q Et
Ž .
We observe that in Dvorkin and Nur 1993 , the differential of the skeleton’s deformation in
Ž . Eq. 9 is replaced by only the strain in the
direction of the wave propagation. We will as- sume the normal strains to be equivalent in all
Ž .
directions; hence, we can rewrite Eq. 12 using Ž
. Ž
. Eq. 11 to express EfrEt in the cylindrical
coordinate and obtain Ef
a Ee 1 EP
s q
Et 3 Et
Q Et a E
Eu u
Eu
r r
x
s q
q
ž
3 Et Er
r E x
1 EP q
. 13
Ž .
Q Et Ž
. Replacing EfrEt by this expression in Eq.
Ž . 8 and eliminating the fluid displacement by
Ž . using the relation 7 gives after some algebraic
Ž .
transformations see Appendix A : F
EW W
EW
r r
x
P s y q
q
ž
f E
r E x
r
Eu
Fg
Eu u
Eu
x 1
r r
x
q F y
q q
,
ž
E x f
Er r
E x 14
Ž .
where a
W s f U y u and g s
q f .
Ž .
r r
r 1
3 The expression of pore fluid pressure from
Ž .
Biot 1962 is given by:
EW W
EW
r r
x
P s yM q
q
ž
Er r
E x
Eu u
Eu
r r
x
y a M q
q ,
15
Ž .
ž
Er r
E x where the modulus M is given by
y1
f a y f
Ž .
M s q
. k
k
f s
It appears clearly that the effect of consider- ing radial fluid flow in the equation of fluid’s
mass conservation as proposed above, results mainly in the occurrence of an additional term
Ž .
namely F Eu rE x . It can be observed that the
x
two expressions between parenthesis in both Ž
. Ž
. Eqs. 14 and 15 represent divergence term,
hence, the new expression of pore fluid pres- sure, in what we will call as the reformulated
BISQ model is given by
F Eu
Fg
x 1
P s y
= P WqF
y
= P u,
16
Ž .
f E x
f
The notations = P W and = P u mean application of the divergence operator to the relative
fluidrsolid displacement and solid displace- ment, respectively.
Ž .
In the BISQ model Dvorkin and Nur, 1993 , only P-waves are considered. In a later paper,
Ž .
Dvorkin et al. 1995 devoted mainly to the
squirt flow mechanism, S-waves were included. Resulting expressions of the fast P-wave and the
S-wave velocities are still dependent on other parameters which are difficult to determine. We
will develop a 3-D extension of our model following exactly the method adopted by Parra
Ž
. 1997 in his extension of the BISQ model to a
transversally isotropic poroelastic medium. In so doing, the total pore fluid pressure for a 3-D
motion following the reformulated BISQ model for a poroelastic isotropic medium case, can be
written as
F P s y
= P W
t
f
Eu Fg
F
x 1
q F y
= P uy = P W
E x f
f
Eu Fg
F
y 1
q F y
= P u y = P W
E y f
f
Eu
Fg
z 1
q F y
= P u. 17
Ž .
E z f
This equation can be rewritten as F
P s y3 = P W
t
f
Eu Eu
Eu
x y
z
q F q
q
ž
E x E y
E z Fg
1
y 3 = P u.
18
Ž .
f We notice that the expression between paren-
thesis is a divergence term, therefore, the aver- age pore fluid pressure in 3-D wave motion for
an isotropic porous medium saturated with a fluid can be rewritten as
F F
Fg
1
P s y = P Wq
= P uy = P u,
19
Ž .
av
f 3
f where
F P s y
= P Wqa =P u , 20
Ž .
Ž .
av
f or simply
a q 2f a s
. 3
This expression of pore fluid pressure has been used in the modelling of acoustic wave
velocity and attenuation dispersion vs. fre- quency. Formation and saturant parameters used
Ž .
are those of Dvorkin and Nur 1993 as reported Ž
. by Parra 1997 . We avoid the lengthy mathe-
matical derivation as it has been clearly and extensively treated in the two papers cited above
and give directly expressions used to determine velocity and attenuation:
1 V s
, a s v Im
Y 21
Ž .
Ž .
p
Re Y
Ž .
where for our model,
2
r 1 y f q r f q vr u
Ž .
Ž .
a f
f
Y s ,
2
a M q F
f
y1
f r rr q f
hf
a f
u s y q j
. 22
Ž .
2
v r f
k r v
f f
Ž . Ž .
The notation Re and Im
indicate the real and complex part of a complex number.
3. Numerical results and discussion
