a seismic wave after propagating through a set of fractures with known distribution of fracture
parameters is used to characterize the fractured medium.
3. Numerical experiments
Fig. 2 shows the three seismic attributes, the instantaneous frequency, bandwidth and ampli-
tude computed from the wave train for simple Ž
model in which no layer contains fractures see .
Table 1 . The attributes are very distinct with no interference. The instantaneous frequency and
bandwidth values are fairly constant and equal to the peak frequency of the propagating signal.
Fig. 3 illustrates the seismic attributes for the situation where one of the layers is fractured
Ž
. Table 1 . Each attribute plot exhibits a distinct
pattern in the demarcated time zone containing the fractures. On the instantaneous amplitude
Ž .
plot, four peaks time zone 0.6–0.7 s corre- sponding to maxima in the reflection strengths
are recognized. These maxima correspond to reflections from fractures with dominant lengths.
The relatively longer fractures give relatively stronger reflection amplitudes than the shorter
ones. Composite reflections as anticipated from a heterogenous medium such as a fractured zone
are seen on the plot.
The variations in the character of the instan- taneous frequency correlates with that of the
reflection strength as seen in Fig. 3. The instan- taneous frequency is sensitive to changes in
acoustic impedance. Values are constant and equal to the dominant frequency for fracture-free
zones of maximum transmissitivity and lower for waves that have propagated through a frac-
tured zone. The smooth variations result from interaction of waveforms which have reflected
off various fractures with different lengths Ž
. Boadu, 1997a . The wave energy experiences
frequency discrimination and as a result the reflected energy at certain frequencies is prefer-
entially attenuated. Progressive reduction in in- stantaneous bandwidth in the fracture zone is
also visible. Integrating information from the various seismic attributes from fractured zones
may be useful in the detection, location and characterization of fractured zones in rock
masses.
To estimate seismic velocities, we will use Ž
. the model developed by Boadu 1997b for a
multiple set of parallel fractures forming a sys- tem of fractures. The absorption coefficient a
and the velocity c of a fracture system can be
Ž .
obtained Boadu, 1997b
R
a s a q D yln T
7
Ž .
Ž .
Ý
k pp , k
ks1
c
m
c s 8
Ž .
R
yarg T
pp, k
1 q c D
Ý
m k
ž
v
ks1
where T is the P–P transmission coefficient
pp, k
for the k th system of fractures. The fracture
Table 1 Layer parameters for model used in experiment. An illustration of the parameters of the individual layers used in the
computation of the synthetic seismograms. Intact rock in layer 3 has permeability of 10
y12
mrs
3
Ž . Ž
. Ž
. Ž
. Layer
Thickness m V
mrs V mrs
Density kgrm Q
p s
1 10
1000 600
2600 100
2 10
2500 1500
2650 150
Ž .
3 Fractured 30
3000 1800
2660 200
4 10
3500 1980
2660 250
5 `
4900 2700
2650 300
Ž . Ž
. Fig. 3. a Synthetic seismogram vertical incidence for a stack of geologic layers with one of the layers containing
Ž . Ž .
fractures. The time zone known to contain composite reflections from fractures is shown, b instantaneous frequency, c Ž .
instantaneous bandwidth and d instantaneous amplitude.
frequency of the k th system of fractures in the direction of propagation is D , and c
and a
k m
denote respectively the velocity and absorption coefficient of the intact rock. This model allows
for input of any distribution of fracture parame- ters.
The transit times in the fracture zone for different models are estimated for a series of
controlled numerical experiments where the properties of the fractures and the medium are
known. For each experiment, the fractured layer is characterized by varying fracture parameters,
that is, fracture lengths, apertures and spacing. These parameters are fractally distributed with
fractal dimension randomly varying within the range 0.1 to 0.9, the fracture frequency ranges
from 0.006 to 0.5 and the fracture lengths range from a minimum of 0.1 m to 4 m. These ranges
of values are reasonable representations of frac- ture parameters for tight sedimentary and ig-
Ž .
neous rocks Chernyshev and Dearman, 1991 . All the fractures are horizontally oriented. We
should note, however, that both the seismic and Ž
Ž . the hydraulic models expressed above Eqs. 4
Ž .. and 5
can accommodate fractures of any ori- entation.
For a given model, the ratio of the transit time in the fractured to that without fractures is
computed and correlated against the fracture Ž
. porosity and permeability Boadu, 1997b . This
ratio was computed for both P- and SH-waves. Fig. 4 shows the relation between fracture per-
Ž .
Ž
p
. meability K
and P-wave velocity ratio V .
f r
The results show a good correlation between the velocity ratio and fracture permeability. Veloc-
ity ratios decrease with an increase in fracture permeability. An equation expressing this rela-
tion in least squares sense is,
log K s y1.49 y 10.5V
p
9
Ž .
f r
with a root-mean square error of 0.018. Simi- larly, a correlation can be established between
fracture porosity and the velocity ratio. Fig. 5 shows a plot of the logarithm of fracture poros-
ity in percent vs. velocity ratio. As shown in the figure, low velocity ratios correlate with high
fractures porosity with a relation of the form log F s y6.87 q 3.72V
p
10
Ž .
f r
with RMS value of 0.06. Both fracture permeability and porosity were
also related to the SH-wave velocity ratio. Fig. 6 illustrates the correlation between the fracture
permeability and SH-wave velocity ratio. Al- though there is some degree of scatter, in gen-
eral, the velocity ratio decreases with an in- crease in fracture permeability. The equation
describing the relation is established as
log K s y0.56 y 10.96V
s
11
Ž .
f r
with an RMS error of 0.042. The scatter may be due to the fact that SH-waves are relatively
insensitive to fractures in the transmission pro- Ž
cess compared with P-waves Boadu and Long, .
Ž .
1996 . As explained by Boadu and Long 1996 , there are cut off frequencies at which a given
fracture length does not influence the velocity reduction. These cut off frequencies are lower
Fig. 4. Relationship between logarithm of fracture permeability and the velocity ratio for P-waves. The least-squares regression provides an equation describing the relationship.
Fig. 5. Relationship between logarithm of fracture porosity in percent and the velocity ratio for P-waves. The least-squares regression provides an equation describing the relationship.
for SH-waves than P-waves and as such, the contribution from most of the fractures with
lengths greater than the characteristic length may not be effective in the delay process. The
relation between fracture porosity vs. velocity ratio for SH-waves is illustrated in Fig. 7 and
Fig. 6. Relationship between logarithm of fracture permeability and the velocity ratio for SH-waves. The least-squares regression provides an equation describing the relationship.
Fig. 7. Relationship between logarithm of fracture porosity in percent and the velocity ratio for SH-waves. The least-squares regression provides an equation describing the relationship.
shows some scatter compared with that of P- wave case. Velocity ratio decreases with an
increase in fracture porosity. The equation de- scribing this relation is given as
log F s y2.57 y 1.70V
s
12
Ž .
f r
with the RMS error estimated to be 0.1585.
4. Conclusions
