ments would provide the theoretical base neces- sary for field scale interpretation of relations
between seismic and hydraulic parameters.
2. Theoretical formulations
For two half spaces bounded by rough sur- faces that are partly in contact with one another,
the boundary conditions for an incident wave are
t r s lim t
r q D r
Ž . Ž
.
m m
D r ™0
d e
r d
e r q D r
Ž . Ž
.
m m
s lim
D r ™0
dt dt
t r q D r
Ž .
m
q ; m s P,S
1
Ž .
Z
m
where t and t are the respective normal and
p s
tangential stresses, e and e are normal and
p s
tangential strain, r is the radius vector determin- ing the plane of the fracture in space and Z
is
m
the fracture impedance. The magnitude of the ‘jump’ in the discontinuity of the displacement
is determined by the fracture impedance Z ,
m
where Z is the longitudinal wave impedance
p
and Z is the transverse wave impedance.
s
The impedance of the fracture is obtained by exploiting the well-established analogies be-
tween mechanical
and electrical
quantities Ž
. Anderson, 1985 . The existence of formal rela-
tions between equations for acoustical wave motion and electric transmission lines allows us
to treat the fracture as a transmission line for passage of seismic waves. The corresponding
equations of motion have been developed and solved for the transmission and reflection coef-
Ž ficients for an incident P-wave Boadu, 1997a,b;
. Boadu and Long, 1996 . The ratio of the
impedance of intact rock and that of the fracture is termed the inhomogeneity factor j . This is
m
an important parameter which greatly affects the seismic wave response. It is a function of the
acoustic impedance of intact rock, fracture length and opening, viscosity of infilling mate-
rial, and fraction of fracture surface area in contact, and the frequency of the seismic wave
Ž
. Boadu, 1997a,b; Boadu and Long, 1996 .
For an incident longitudinal wave at an angle u to the plane of the fracture, the complex
reflection and transmission coefficients have been found to be:
f j f j
pp p
ps s
R s yA q 1 y A
Ž .
pp pp
pp
f j q 1 f j q 1
pp p
ps s
1 1
T s A q 1 y A
Ž .
pp pp
pp
f j q 1 f j q 1
pp p
ps s
f j f j
pp p
ps s
R s yA y A
ps ps
ps
f j q 1 f j q 1
pp p
ps s
1 1
T s yA q A
2
Ž .
ps ps
ps
f j q 1 f j q 1
pp p
ps s
where A , A , f and f
are all functions
pp ps
pp ps
of the angle of incidence and the Poisson’s ratio Ž
of the intact material Boadu, 1997a,b; Boadu .
and Long, 1996 . For SH-waves, the expres- sions for the reflection and transmission coeffi-
cients are
j cos u
s
R s y
SH
1 q j cos u
s
1 T
s .
3
Ž .
SH
1 q j cos u
s
The geometric characteristics of the fracture affect the reflection and transmission coeffi-
cients which are also frequency dependent. The amplitude and phase of a waveform propagating
across a single fracture have been shown to
Ž change significantly
Boadu, 1997a,b; Boadu .
and Long, 1996 . From the spectrum of the transmitted waves, the lower frequencies un-
dergo the least change. In the reflected waves however, the high frequencies dominate.
Ž .
The method described in Boadu 1997a will be used in this study to obtain synthetic seismo-
Ž . grams from the global reflection coefficient R
for a stack of layers including a fractured medium with fractally distributed fracture pa-
rameters. Synthetic seismograms that include reflections from the fractured layer are then
computed using reflectivity methods. The ap- proach involves integration of reflection and
transmission
coefficients for
unfractured Ž
. Ž
. welded
and fractured unwelded
interfaces into a recursive scheme, from which the hy-
bridized global reflection and transmission coef- ficients are computed. This methodology can
also account for the absorptive properties of the intact rock. Any realistic modeling of fluid
transport in fractured terrain or estimation of its strength properties has to take into account such
Ž distribution
properties Boadu
and Long,
. 1994a,b . In the numerical experiments de-
scribed in this paper, fractal fracture lengths, spacings and apertures with varying fractal di-
mensions were generated using the method de-
Ž .
scribed in Boadu 1997a; b . Fractures, when present in geologic medium,
can significantly affect its flow characteristics. As illustrated in Fig. 1, for inclined fractures at
an angle a with respect to the hydraulic gradi-
n
Ž .
ent e.g., Fracture 1 , with finite length L ,
f
width W and thickness b , the discharge Q
f n
fd
across a section containing a suite of fractures is Ž
. Hossain, 1992 :
Q s q P W P cos
2
a
fd fd
f
s c P f L rL P W P cos
2
a . 4
Ž .
Ž .
f f
d f
Here, c is the hydraulic discharge through a
f
continuous fracture per unit width, under unit hydraulic gradient. For practical purposes, the
Ž .
function f L rL is taken as L rL , or as
f d
f d
Ž L
f
yL
d
.
Ž .
e Witherspoon, 1986 where L
is the
f
finite length and L is related to the thickness
d
of the section h by L s hrcos a. The sum
d
contributions from R fractures divided by the section area gives the effective permeability K :
a R
K s 1rA Q
Ž .
Ý
a fd n
ns 1 R
3
s 1rV c P A P cos a
5
Ž .
Ž .
Ž .
Ý
f n f n
n ns 1
where V is the volume of the section and A is
f n
the area of the nth fracture. Experimental and Ž .
field measurements have substantiated Eqs. 4 Ž .
Ž and 5 for known fracture systems Hossain,
. 1992 . The total permeability is the sum of the
Ž .
Fig. 1. Matrix block with fractures of varying lengths, orientations and thicknesses after Boadu, 1997a . In this study, intact rock has permeability of 10
y12
mrs.
intact rock permeability and the fracture perme- ability K .
a
Similarly, the fracture porosity F can be
f
computed from the fracture parameters as:
R
L P t
Ý
f n f n
ns 1
F s 6
Ž .
f
L P h
d
where t is the thickness of the nth fracture.
f n
Conventionally, Fourier transform techniques have been employed to ascertain the average
properties of signals with very limited or no information about the local variations in their
properties. Additional information about the sig- nals which retain local information can be ob-
Ž tained by performing time–frequency complex
. Ž
. waveform analysis of the signals Cohen, 1995 .
Such analyses provide new insights about the medium through which the waves have propa-
gated. In this paper, time–frequency analysis of
Ž . Ž
. Ž .
Fig. 2. a Synthetic seismogram vertical incidence for a stack of geologic layers with no fractured layer, b instantaneous Ž .
Ž . frequency, c instantaneous bandwidth and d instantaneous amplitude. The model parameters are listed in Table 1.
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
