Geology Reference
In-Depth Information
Figure 8.15
Schematic illustration of
the key assumptions in Gassmann's
equation.
Solid phase:
homogenous
and isotropic
Pore space totally
interconnecting
Fluid is frictionless
(ie low viscosity)
Shear modulus
unaffected by
pore fluid
No coupling
between solid
and fluid phases
'Low' frequency
model
is simply the arithmetic average of the various solid
and fluid components of the rock, weighted according
to their volume fractions. The effect of fluids on
velocity is more complicated.
The usual starting point for modelling fluid sub-
stitution effects at seismic frequencies is Gassmann
Some key assumptions in the Gassmann model
(Wang and Nur,
1992
;
Fig. 8.15
) are that:
the solid is homogeneous and isotropic,
all the pore space is in communication,
wave-induced pressure changes throughout the
pore space have time to equilibrate during a
seismic period (the low-frequency assumption),
s
equation (Gassmann,
1951
; Geertsma and Smit,
1961
). It describes rocks in terms of the bulk moduli
of a two-phase medium (fluid and mineral matrix).
The reader is referred to
Chapter 2
for an introduc-
tion to elastic moduli. Gassmann
'
the fluid that fills the pore space is frictionless
(i.e. low viscosity),
no coupling between solid and fluid phases.
s equation is applic-
able to rocks with intergranular porosity and more or
less uniform grain size. It can be written as:
K
sat
'
Given these constraints Gassmann
'
s model
in the
strictest sense is likely to apply only to
sand-
stones with moderate to high porosity at low (seismic)
frequency (
Figs. 8.16
and
8.17
). In this scenario there
is sufficient time for fluid movement between pores
during the passage of the wave. At high frequency (i.e.
in the laboratory) this process is limited leading
to increased pore stiffness and higher velocity. To
account for this dispersion effect, Gassmann needs
to be extended using additional models, such as those
proposed by Biot (
1962
) and Mavko et al.(
1998
). It is
worth noting that laboratory measurements on dry
rocks are independent of frequency and can be used
directly in Gassmann relations. Sonic frequencies are
typically around the transition from low to high fre-
quency as defined by Biot (
1956
) and it is generally
assumed that Gassmann can be applied to porous
rocks at these frequencies. Tight sands and shaley
sands are scenarios in which it
'
clean
'
K
d
K
fl
K
sat
¼
K
d
+
K
0
K
0
K
0
K
fl
ð
8
:
20
Þ
ϕ
μ
sat
¼ μ
d
,
where K
sat
is the bulk modulus of the fluid-saturated
rock, K
0
is the bulk modulus of the matrix material,
K
d
is the bulk modulus of the dry rock frame, K
fl
is
the bulk modulus of the pore fluid, and
is the
ϕ
(fractional) porosity,
μ
sat
is the shear modulus of the
fluid-saturated rock and
μ
d
is the shear modulus of
the dry rock frame. The second part of
Eqs. (8.20)
simply states that the shear modulus is not affected by
pore fill, effectively because shear waves do not travel
through fluids. The equations introduce the concept
of the moduli of the dry rock frame, which is the rock
frame with all fluid (liquid or gas) removed. It is
possible to measure the dry rock modulus in the
laboratory, but in most Gassmann applications using
wireline log data this modulus is inverted from the
other inputs. As will be discussed, the dry rock modu-
lus is an important parameter for quality control of
Gassmann results.
is possible that
Gassmann
s assumptions are violated. This will be
discussed in
Section 8.5.
It is generally held that Gassmann is appropriate
for carbonate rocks with relatively homogenous
pore systems (e.g. Wang and Nur,
1992
;Wang
'
160
