ELSEVIER
Journal of Applied Geophysics 36 (1997) 1-19
Fractured rock mass characterization parameters and seismic properties- Analytical studies Fred Kofi Boadu * Department of Civil and Environmental engineering, Duke Universi~, Durham, NC 27708-0287, USA
Received 22 August 1996; accepted 5 February 1997
Abstract Analytical studies have been camed out based on a modified displacement discontinuity fracture model to characterize the relations between seismic properties and fractured rock mass parameters. Models of a fractured medium are developed to represent vertically aligned fractures embedded in an intact rock material. Seismic properties, including velocities and quality factors (Q), are computed from seismic waves transmitted through the medium. The geometrical properties of the fractures, that is, the fracture length, spacing, and aperture, are assumed to exhibit fractal or Weibull behavior. The discontinuity index ld, fracture density parameter C, and the RQD, which provide a descriptive measure of either the hydraulic properties or the strength of the fractured medium, are computed for simulated distributions of the geometric fracture properties. These parameters are analyzed and then related to the seismic properties. The investigations suggest that consistent low seismic velocity and Q values associated with a fracture zone are indicative of a permeable or hydraulically transmissive zone. Lower rock strengths are also associated with lower seismic velocity and Q values. The results show that reasonable inferences characterizing the hydraulic and strength properties of fractured rock mass may be derived from measured seismic properties. Keywords: fractures; seismic properties; hydraulic transmissitivity
1. I n t r o d u c t i o n
Characterization of a fractured rock mass utilizing information from seismic waves is attracting great interest in applications such as waste isolation in fractured terrain, fractured hydrocarbon reservoir characterization, assessing the integrity of foundations and underground caverns, and for creation and use of artificial fractures in geothermal developments, to mention a few. The mechanical properties of a fractured medium that dictate its seismic param* Fax: + 1-919-6605219; e-mail:
[email protected]
eters, such as compressional and shear wave velocities and quality factors (inverse of attenuation), are influenced by the geometric and material properties of the fractures. These properties include the fracture length, aperture, spacing between the fractures (fracture density), type of infilling material (viscosity), and the areal fraction of opposing fracture faces in contact. These same fracture properties also strongly influence the hydraulic transport properties of the medium. To make seismic methods useful in nondestructive evaluation of the subsurface, there is the need to establish realistic relationships between the characteristic physical properties of a fractured rock
0926-9851/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PH S 0 9 2 6 - 9 8 5 1 ( 9 7 ) 0 0 0 0 8 - 6
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F.K. Boadu/Journal of Applied Geophysics 36 (1997) 1 19
mass such as its fracture density, mean fracture length, and average fracture spacing and other quantifiable properties than can be measured remotely (e.g., seismic velocity and attenuation). Seismic refraction work (Crampin et al., 1980; Idziak, 1988) and crosshole seismic studies (Worthington, 1984; Wong et al., 1987) based on transmission of seismic wave energy across a fracture zone have been used to demonstrate correlations between decreases in compressional (Vp) and shear (V~) wave velocities and increases in fracture frequency or density. In addition to velocity versus fracture frequency relations, increases in attenuation have been observed in response to increased fracturing in crosshole seismic studies (Wong et al., 1983). Field results (O'Donoghue and O'Flaherty, 1977; Aki et al., 1982) have shown that other factors besides fracture frequency, such as the in-situ stress state, fracture length, presence of fluid and related pressure, and the presence of fracture filling material also affect the velocity and attenuation of seismic waves in rock masses. These findings show that the seismic velocity and attenuation effects are directly related to the intrinsic properties of the fractures, including their geometric and material properties. For seismic measurements to be useful in imaging and assessing the mechanical and hydraulic transport properties of a fractured medium, understanding how the geometric and material properties influence a propagating seismic waveform is important. A single fracture, for example, has been shown in some applications to dominate the hydraulic properties (Brown, 1989), and mechanical properties (Goodman, 1976) of a fractured rock mass. Study of this idea requires a fracture model which has the potential for relating the fracture parameters of the fractured medium to the seismic properties. Recently, Boadu (1994) and Boadu and Long (1996) proposed a fracture model (modified displacement discontinuity, MDD) for seismic propagation in fractured media which takes into account the geometric and material properties of the fractures. Experimental data were compared with MDD model results and the fit was reasonable. In this paper, parameters that characterize the strength and hydraulic state of a fractured medium are computed and related to its seismic properties. The fractured rock mass parameters described by the discontinuity index, the fracture density parameter, the
linear fracture density, and the rock quality designation (RQD) are computed from the distribution of geometric fracture properties. Relations between the fractured rock parameters and seismic properties are demonstrated. First, the modified displacement model (MDD) is briefly discussed; full details of the model description are given in Boadu and Long (1996). Second, analytical transmission calculations are used to illustrate the modifications of seismic waveforms when propagating through a series of vertically aligned fractures with varying lengths and spacings. Finally, seismic parameters are estimated from waves propagating through a fractured medium having a known distribution of fracture properties. Relationships between the seismic properties and the fractured rock parameters are then investigated and analyzed.
2. Modified displacement discontinuity model (MDD) The theory of MDD was recently developed by Boadu (1994) and Boadu and Long (1996) and only a brief summary is given here. Although other elegant fracture models do exist, the use of the MDD model provides the necessary information needed for the investigations and analyses in this paper. The MDD fracture model incorporates the relevant fracture parameters: fracture length, fractional contact area, viscosity of the infilling fluid, and, in a slightly modified form, the aperture. Many investigators have modeled wave propagation in fractured rock mass. The displacement discontinuity theory (Schoenberg, 1980) has been used to model wave propagation in fractured rock mass (Schoenberg, 1980; Schoenberg and Douma, 1988). Dispersive interface waves have been shown to propagate along a fracture (Gu et al., 1996a). All these methods assume infinite fracture length. Fracture models based on boundary integral methods (Gu et al., 1996b) have also been developed. However, such methods are computationally burdensome. Other investigators have used effective medium theories to model wave propagation in fractured rock mass (Hudson, 1981; Gibson and Ben-Menahem, 1991). Effective medium theories, however, do not account for all the wave propagation effects, specifically
F.K. Boadu / Journal of Applied Geophysics 36 (1997) 1-19
wave attenuation. The reader is referred to Boadu (1994) and Boadu and Long (1996) for extensive literature review on fracture models and elastic wave propagation in fractured rock mass. The MDD model is a suggested alternative to the fracture models cited above which incorporates the fracture parameters of interest. In the MDD fracture model, we consider two half spaces bounded by rough surfaces that are in contact with one another in a horizontal plane defined by the direction vector r normal to the surface. Regarding the boundary between the two half spaces as a non-welded contact, the boundary conditions for an incident elastic wave can be described as: ~'m(r) =
t~rm( r) - - =
6t
m=P,S
lim
Ar-~0
[ t~em(r + Ar) Ar---~0 6t
the viscosity of the pore-filling material, the fractional area of opposing fracture surfaces in contact (for partially closed fractures), and the frequency of the seismic wave. For an incident longitudinal wave at an angle 0 to the plane of the fracture, the complex reflection and transmission coefficients are (Boadu, 1994; Boadu and Long, 1996)
~ ~p + (1 - A p p )
rpp = -App
¢~pp~p -~- l 1
1
tpp =App ~pp~p + 1 + (1 - A p p ) ¢~ps ~s -F 1 '
rps = - A p s ¢~pp~p -~- 1 +
Tm( r + Ar) Zm
6~s ¢s Aps ~bps~s + 1 '
1
/PS = -Aps (1)
where Zp and z s are the respective normal and tangential stresses; Ep and e s are normal and tangential strain, respectively, and r is the radius vector determining the plane of the fracture in space. The magnitude of the 'jump' in the discontinuity of the displacement or velocity is determined by the fracture impedance Z,, (Zp is the longitudinal impedance and Z s is the transverse impedance). The impedance of the fracture is obtained by exploiting the well-established analogies between mechanical and electrical quantities (Anderson, 1985). Formal relations between equations for acoustical wave motion and electric transmission lines allow the fracture to be treated as a transmission line for passage of seismic waves. The corresponding equations of motion are then solved for the local transmission and reflection coefficients for an incident P or S-wave. When there is no elastic link (non-welded contact), the transmission of the seismic waves takes place in the form of a blow with some inertia resulting in the delay of the transmission process. The ratio of the impedance of the intact rock to that of the fracture is termed the inhomogeneity factor ~,~. This is an important parameter which significantly influences the seismic wave response. It is a function of the acoustic or elastic impedance of the intact rock, the fracture length and its aperture,
~ps ~s ¢~PS~S q- 1 '
~ Cp
"rm(r+Ar),
lim
3
¢~pp~p -k 1
1
+ Aps
¢~PS~S -I- 1
(2)
where App, Aps, t~pp and ~bps are all functions of the angle of incidence and Poisson's ratio of the intact material. For SH waves, the expressions for the reflection and transmission coefficients are ~:s cos 0
FSH
1 + ~s cos 0 ' 1
tSH
1 + ~S COS 0"
(3)
The geometric characteristics of the fracture have profound frequency dependent effects on the reflection and transmission coefficients. The dimensionless quantity ~p,s is defined by the ratio of the impedance of the intact medium to the fracture impedance (Schoenberg, 1980). This ratio is here termed the inhomogeneity factor ~p,~ and is given by Wp,~
~p,s = 2Zp, s
(4)
where, for example, Wp,s is the P or S-wave impedance and Zp,s fracture impedance. The expression for the fracture impedance is fully derived in Boadu and Long (1996). For realistic situations where the fracture opening is negligible compared with the length of the fracture and where seismic frequencies are employed, the influence of the fracture opening
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F.K. Boadu / Journal of Applied Geophysics 36 (1997) 1-19
on the dynamic characteristics of the propagating wave is minimal. At higher frequencies however, the effect may be influential (Gibson and Ben-Menahem, 1991). The expression for the fracture impedance for an incident compressional wave can be obtained as (Boadu and Long, 1996) Er Zcp yp( 1 - r) l Zp = 2 i w'---h+ 1 coth 2 '
(5)
where E is the Young's modulus of the intact rock, r is the fractional area of the fracture surface in contact, l is the length of the open fracture, h is the region of influence (similar to the first Fresnel zone), Z c and y are, respectively, the characteristic impedance and the propagation constant derived in Appendix A. The subscript p denotes an incident compressional wave. Similar expressions can be written for incident S-waves as derived in Boadu and Long (! 996).
3. Statistical distribution and fractai characteristics of fracture properties Natural fracturing processes occur at several different scales although observations suggest that the frequency distributions of fracture length, aperture, and spacing are scale invariant. For example, the distributions of fracture spacings as obtained by Boadu and Long (1994a) in a field environment (exposed fractured rock in Georgia and South Carolina) were determined to exhibit fractal characteristics. In the same study the authors also noted that the fracture spacings fit the Weibull distribution to a reasonable degree. The general definition of a fractal takes the form (Barnsley, 1993) N~( r) = Cr -D
(6)
where NF(r) is the number of fractures with linear dimension greater than r, C is a constant and the exponent, D, is known as the fractal dimension. The Weibull distribution provides a flexible characterization of several classic distributions including the exponential. The cumulative distribution function (CDF) for a Weibuil model is given by N(r) A/T
exp[ =
1 --
~]a
(7)
where N ( r ) is the cumulative number of fracture fragments with a size or length less than r, and Nv is the total number of fracture fragments. The parameters or and A are known as shape and scale parameters, respectively (Bury, 1975). A particular appealing property of the Weibull distribution is the simplicity of obtaining expressions for other parameters such as the mode, mean, median, and the variance (Bury, 1975). Velde et al. (1993) and Kojima et al. (1989) examined the distribution of fracture lengths and concluded that they exhibit f¥actal behavior. Barton and Hsieh (1989) and Brown (1989) analyzed the distributions of fracture apertures from a wide spectrum of rock types and found them to exhibit fractal behavior. Finally, the distribution of fractional contact area in single natural fractures in quartz monzonite was measured and found to be fractal (Sahimi, 1993). Other researchers have established that fracture spacing and aperture also bear other distributions, for example, lognormal distribution (La Pointe, 1988) and exponential distribution (Hudson and Priest, 1983). In general, differences in the distribution of fracture properties are due to the differences in the mechanical or tectonic processes creating the fractures. Boadu and Long (1994b) have suggested that the distributions of the fracture properties depend on the physical generating mechanism. That is, a fracture property (e.g., the fracture spacing) having an exponential distribution may be the result of a fracturing process caused by a single dominant operating stress. On the other hand, fractal-distributed fracture parameters may be the result of a repetitive fracturing process. It is understandable that any realistic modeling involving fractured rock mass must take into account such distributions. While the average spatial orientation of the fractures in the subsurface may be consistent on a regional basis, the lengths and spacings between individual fractures may vary considerably, even between neighboring fractures. Fracture lengths have been shown to influence the transmission of seismic waves significantly (Boadu and Long, 1996). In the numerical calculations performed in this work, fractal fracture lengths were generated with varying fractal dimensions randomly chosen between 0.1 and 0.9. Fractal fracture apertures were also generated and used in the model. Different methods of generating fractals have been presented in Barnsley (1993)
F.K. Boadu / Journal of Applied Geophysics 36 (1997) 1-19
and are not discussed further here. Fracture spacings were generated from a Weibull distribution according to a procedure described in Boadu and Long (1994b). The basic aim here is to compute the seismic response of a geologic section containing a set of fractures with properties of varying distributions. Such numerical experiments are expected to provide the theoretical basis necessary for field or laboratory scale interpretations regarding the relations between fracture parameters and seismic properties.
4. Numerical transmission computations One-dimensional numerical experiments based on transmission of waves in a simulated fractured medium were performed as described below, using a model having a distribution of vertically aligned fractures or cracks with different lengths, apertures, and spacings. A homogeneous and isotropic geologic section (e.g., sandstone) is assumed, having imbedded fractures or cracks of different lengths and spacings. Fig. 1 illustrates the model showing fractures of different length (L) and spacing (h) imbedded in a material assumed to be homogeneous and isotropic. The material between adjacent fractures is assumed to be intact rock with thicknesses corresponding to specified spacings. The input seismic signal, a plane wave source wavelet, is transmitted from one end of the geologic section containing the fractures to a receiver at the end of the section. When the propagating wavefront encounters a fracture, it undergoes a transmission and reflection process. Converted waves (P to SV) are not considered since the source waves are assumed to propagate in a direction normal to the fracture plane (analogous to normal-inci-
I
t_. 1-
5
dence wave propagation in reflection seismology). Varying orientations of the fractures can be accounted for in the MDD model (Boadu and Long, 1996). The generalized global transmission coefficient T~n for a unit input signal arriving at the receiving end of the section is expressed as t l ( n - 1)t(n - 1)n exp( iqb{._ 1)) Tin = 1 + r l ( . _ l ) r { . _ , ) . e x p ( - - Z i d P ( . _ l ) ) " -
(8)
In the above equation, t.{._ 1) and r.(._ l) are, respectively, the local transmission and reflection coefficients across the nth and ( n - 1)th section, ~b. = wh./v, where co is the angular frequency of the propagating waveform, and h. and v. are, respectively, the thickness and velocity of the nth intact material. For example, the global transmission coefficient after the wave has traveled through the second fracture or first spacing (Fig. 1) from the indicated source is T13 expressed as tl2 t23 exp( iq02 ) T13 = 1 + r12r23 exp(2 - iq~2) ' -
(9)
A recursive approach (e.g., Fokkema and Ziolkowski, 1987) is used to propagate the local transmission coefficients at each fracture given in Eqs. (2) and (3) to a global form at the receiver end. This scheme is similar to the one developed in Boadu (1996) for the global reflection coefficients. These methods are essentially the same as the propagator matrix method (Aki and Richards, 1980) which has been shown by Hovem (1995) to contain all the relevant physics entailed in wave propagation. The local reflection and transmission coefficients as a result of a wave striking a fracture boundary are
Fractured Zone
_1 --I
Source
!
i-- 1 Receiver J I
[ hn-I
I !
Fig. 1. A model illustration of wave propagation (transmission) in a fractured medium. Fractures are modeled as discontinuities of varying lengths ( L ) and spacings (h) embedded in an otherwise homogenous isotropic medium. Analysis is based on plane wave source representation.
6
F.K. Boadu/Journal of Applied Geophysics 36 (1997) 1-19
given by Eqs. (2) and (3) for incident P and SH waves, respectively. The seismic wave response in the frequency domain can be obtained by multiplication of the global transmission response (Eq. (7)) by the Fourier transform of the input signal. The resulting synthetic seismogram is given by the inverse transform of the resulting product. The input signal is a Ricker wavelet which, in the frequency domain, is expressed as
0.8 ~
r',
0.6
r~
0.4
~ = L r r
0.2
}0
.,
(10)
- - Incident wave
lr
--Rece,ve0 ....
03,
!
]1
0.6p
I
/ dq
0.4 t
t
0.2"
-0.2 -04 -06
0
0.002
0.004
0.006
0.008 0.01 Time (sec)
0012
0.014
0.016
0018
Fig. 2. An illustration of incident and received waveforms (P-wave) for Model A as described in the text.
/ i
-0.4
~ I i I I '1
-0.8 -1
where K = 27r 2f2, fM denotes the peak frequency of the propagating waveform, and t~ is an arbitrary time delay. As already mentioned, reflection and transmission processes at the fractures exhibit dispersive behavior. Consequently, the expressions for the local transmission and reflection coefficients given in Eq. (2) are frequency dependent. These coefficients are functions of the fracture length, fracture opening, viscosity of the material filling the fracture opening, incident angle of the wave, the velocities (P and S) of the intact rock in which the fractures are imbedded, and the frequency of the propagating waveform. Fig. 2 shows an incident waveform and the received waveform after propagating through a set of fractures. The peak frequency of the incident waveform is 900 Hz and the maximum frequency is 2700 Hz. The model (Model A) consists of four vertically
Received wave
< -0.2
-06
F(w) = K--~3-~to2e ('°2/2")-i'°',
L--
i 0.002
t, i , i 0,004 0.006
i i 0,008 0.01 Time (see)
i 0.012
0.014
i 0.016
i 0.018
Fig. 3. An illustration of incident and received waveforms (P-wave) for Model B as described in the text.
aligned fractures of average length 0.8 m imbedded in a section of length 2 m. These fractures are a subset of a series of fractures having fractal-distributed fracture lengths (fractal dimension 0.5). The fracture spacings follow a Weibull distribution. The fractures are assumed to be filled with water of viscosity 0.001 N s / m 2. The compressional wave and shear wave velocities of the intact material are 5000 and 3200 m / s , respectively. As shown in Fig. 2, the transmitted wave experiences considerable amplitude reduction and time delay relative to the incident waveform. The shape of the transmitted wave is also distorted relative to the incident wave. As the waves interact with the fractures, they are slowed down. The constructive and destructive interference in the propagating wave components results in amplitude reduction and changes in the waveform shape. Fig. 3 illustrates the waveform after propagating through Model B. This model (same intact rock velocities as in Model A) consists of seven vertically aligned fractures of mean length 0.8 m from the same series of fractures specified in Model A. The amplitudes of the transmitted waveform are further diminished in comparison to those in Model A. The amplitude spectra of the received waveforms in Models A and B are shown in Figs. 4 and 5, respectively. The spectral amplitudes for the transmitted waveforms are diminished with respect to the incident waveform for both models. As inferred from
F.K. Boadu / Journal of Applied Geophysics 36 (1997) 1-19 1
/
7
\
O.i
0.9 /
~
/
0.8
-
/--
~
L
,%%w:2 °
-
- Incident w a v e
Received w a v e
0.8
0,7
0.4
0.6
0.2
'
i
t
< -0.2
0,4
,
x \
0.3
-0.4
0.2
-0,8
0.1
-0,~
\ /
i
0
800
1000
1500
2000
-1 2500
j
L p q i
0,002
0.004
0.006
Frequency (Hz)
0,008
0.01
0.012
0.014
0.016
0.018
T i m e (sec)
Fig. 4. Amplitude spectra of incident and received waveforms (P-wave) for Model A as described in the text.
Fig. 6. Incident and received waveforms (SH-wave) for Model A as described in the text.
the time-domain waveforms, the amplitudes of the waveform associated with Model B are significantly reduced relative to those of Model A. Besides the relative amplitude reduction, there is a shift in the peak amplitude toward lower frequencies. This observation demonstrates the selective frequency absorption characteristics of waves propagating through fractured media. In the spectrum of the transmitted waves as opposed to reflected waves, the lower frequencies undergo the least change while the high frequencies dominate in the spectrum of the reflected waves (Pyrak-Nolte et al., 1990; Boadu and Long, 1996). The fractures behave as a filter to attenuate the high frequencies in the propagating waveform.
Fig. 6 shows the incident and received SH-signal after transmission through the fractures for Model A. The peak frequency of the incident waveform is 800 Hz and the m a x i m u m frequency is 2400 Hz. Though one can observe decrements in signal amplitude, it is relatively stronger and less attenuated than the Pwave for the same model. The transmitted SH-wave associated with Model B is shown in Fig. 7. The relative amplitude reduction is smaller in comparison with that experienced by the P-wave. The amplitude spectra for Models A and B are shown in Figs. 8 and 9, respectively. In both cases, the shift of their peak frequencies toward lower frequencies and the attenu-
1 / o, O.E
/
0.7
i
~ /
\
incident w a v e Received wave
'
J
0.8
Iii
0.6
f i
\
- Incident w a v e
Received wave
~ i P
\
t
--
t
0.4
i O,e
--
0.2
'
I
i
--0.2 /
\\
0.3
~
--0,4
\ 0.2
~
I
\\
--0.6
I
;i q I qt
--0.81 500
1000
1500
2000
2500
Frequency (Hz)
Fig. 5. Amplitude spectra of incident and received waveforms (P-wave) for Model B as described in the text.
-10
0.002
0.004
0.008
0.008
001
0.012
0014
0.018
0.018
Time (sec)
Fig. 7. Incident and received waveforms (SH-wave) for Model B as described in the text.
8
F.K. Boadu/Journal of Applied Geophysics 36 (1997) 1-19
/
O,9F
/
X
[~ L
,
I
~Cicdeieerdwaveve
0.8t 0.7~
iii[ i
o4l
l
/
0.3
/
J
o.2~
o.1 ~
.\ ',\
\ \ \
velocity and attenuation, a simple geometry consisting of multiple parallel fractures is considered. A fracture system may consist of R different sets of fractures of different orientations. If the jth set has a linear fracture density (number of fractures per meter) of F/ in the direction of wave propagation, and any interactions between intersecting fractures are neglected to a first approximation, then the attenuation or absorption coefficient c~ and the velocity V of the fracture system can be characterized as (see Boadu and Long, 1996 for derivations)
/
500
o
1000 1500 Frequency(Hz)
2000
R
ce = ozo +
V0
V= 1+
5. Estimation of seismic parameters Natural rock formations are usually sectioned by a system of fractures characteristic to the particular rock type. To estimate the seismic parameters of
~
0.9
Incidentwave Receivedwave
0.8 0,7 0.6
'
\\
~0.5
'\\,\ \
0.4
i
03
/
\\
0.2:- / i' / 0.1 ~ 0
\, \
\ \ ,.\ .\
500
1000 1500 Frequency(Hz)
,, \\
\ . 2000
Fig. 9. Amplitude spectra of incident and received waveforms (SH-wave) for Model B as described in the text.
(11)
j=l
Fig. 8. Amplitude spectra of incident and received wavelbrms (SH-wave) for Model A as described in the text.
ation of high frequency signal amplitudes is moderate compared with the P-waves for comparable frequencies. The SH-waves interact with the fractures in a manner different from the P-waves and, therefore, the waveform features after transmission are different.
E E,(-lnltppI)
VoS,~=,Fj(-arg(tvv)/w )
.
(12)
Here, V0 and c% denote, respectively, the velocity and absorption coefficient of the intact rock section and tpp is the transmission coefficient associated with an incident and received P wave. From the above equations, knowledge of tve (from seismic measurements)and V0 and c% (from published work or measurements) can define the fracture density F. As mentioned previously, the transmission coefficients tpp are frequency dependent functions of the fracture length, aperture, fractional contact area, viscosity of infilling material, and the incident angle of the propagating wave. Similar expressions are derived for S-wave attenuation and velocity in fractured systems by replacing tpp with tsH of Eq. (3) in Eqs. (11) and (12), respectively. Since the velocity and attenuation are computed from the transmission coefficients, it is expected that the velocity and attenuation will also be frequency dependent and functions of the aforementioned fracture properties, including the linear fracture density. In estimating the velocities and quality factors, their values are computed at two frequency intervals near the peak of the transmitted wave and averaged to obtain representative values. The peak frequencies are obtained from the power spectra of the transmitted waveforms. This approach is justified since most of the energy in the propagating waves is borne by the transmitted waves (Pyrak-Nolte et al., 1990; Boadu and Long, 1996). According to O'Connell and Budiansky (1978), a spatially attenuated plane wave of angular frequency
F.K. Boadu / Journal of Applied Geophysics 36 (1997) 1-19
~o(= 27r f ) traveling in the x-direction can be written in the form
U( x, t) =e-~Xe/'°(' (~/v))
(13)
where V is the velocity of the plane wave and c~ is the attenuation coefficient. For highly dissipative materials, the equation relating Q to c~ must include a second-order term, since the energy stored depends on the derivative of the complex modulus with respect to frequency as well as the magnitude of the modulus (O'Connell and Budiansky, 1978). The quality factor Q is thus given by
Q( o ) = =
o#( ~V) - ( ~V)/o~ 2 ~Q(U) (27rf)/(~v)
- (aV)/(Zqrf) 2
(14)
The above equation is applicable to both P and S waves and is used to compute the quality factors for the medium in the neighborhood of the peak frequency. As the wave propagates, reflections from individual fractures interfere with the transmitted waveform resulting in a slightly out-of-phase contribution to the total waveform. An accumulation of such reflections reduces the phase coherency, causing an apparent attenuation effect.
6. Fracture properties Fracture density quantifies an important geological condition of a fractured rock mass which may be used to characterize its mechanical and hydraulic state. The ratio of the seismic velocity in the fractured rock to that of the intact section defines the degree of fracturing. The crack density e, which is used in most of the existing crack models to establish relations with seismic velocity (Crampin et al., 1980; Leary and Henyey, 1985), depends on the number of fractures per unit volume and the geometry of the cracks. The definition of crack density for N ellipsoidal cracks or discs of radius a in a volume V is
~= Na3/V.
(15)
The various parameters defining the crack density
9
are not directly measurable and may be difficult to estimate in the field or under laboratory conditions. A more useful parameter is perhaps the linear fracture density (number of fractures per unit length), usually measured normal to the average strike of the fractures along a scanline under field conditions. Its value is obtained by counting the number of fractures intersecting a unit length of the scanline. In this paper, the effects of linear fracture density F, fracture density parameter C, rock quality designation (RQD), and discontinuity index I d on the seismic parameters are analyzed using the MDD model. The fracture density parameter has been shown to have a strong correlation with the transmissivities of fractured geothermal reservoirs (Watanabe and Takahashi, 1995) and has been suggested to be used in the prediction of the hydraulic properties of reservoirs. The fracture density parameter C is defined as
F C=
(1 --In train)
(16)
where 0 i is the orientation of the ith set of fractures (0 = 0 ° for vertical fractures assuming vertical flow), (.) denotes average, F is the linear fracture density defined as the number of fractures per unit length, and rmin is the smallest fracture length. In geological engineering practice, rock mass quality is quantified by measurements along scanlines or drill cores. One of the earliest of such measures of rock competence is the RQD devised by Deere (1963) and was based on the number and spacings between fractures. RQD is simply defined as the sum of lengths of rock pieces (intact lengths) or fracture spacings greater than 10 cm (4 in.) expressed as a percentage of the total length of the scanline. Strong and massive rocks which have their intact lengths greater than 10 cm have an RQD of 100%, while for an intensely fractured rock with most fractures spacing less than 10 cm, the RQD may approach zero. The discontinuity index I a is used as an indicator of whether or not a fractured rock mass is permeable (Wei et al., 1995). This index is defined based on the permeability threshold of jointed rock mass for a representative volume using the percolation theory. For an average fracture length L in a given distribu-
10
F.K. Boadu / Journal o["Applied Geophysics 36 (1997) 1-19
tion, the discontinuity index is defined as (Wei et al., 1995) average fracture length I~ = = FL. (17) average fracture spacing The average fracture spacing is by definition the inverse of the linear fracture density. Fracture length tends to have a stronger influence on permeability than fracture density (Long and Witherspoon, 1985). Thus, fractured rock masses with shorter and higher fracture density will have a lower permeability that those with longer fractures and lower density. Hence fractured rock mass will tend to be permeable if I d > 1 and, thus, the permeability increases with 1j. The four properties of the fractured rock mass just described are termed fractured rock mass parameters. Each of the four properties of the fractured medium bears a quantifiable character that can be used to characterize the mechanical or hydraulic conditions of the medium. The fracture apertures are not explicitly used in the computations of these four properties, though the MDD can be formulated to account for the aperture. Next, these representative parameters will be related to the seismic velocity and quality factor of the fractured rock formation.
7. Relations b e t w e e n fractured r o c k m a s s parameters and seismic velocities
One of the cardinal aims of this paper is to investigate, through numerical experiments, the possibility of the existence of relations between seismic velocities and fractured rock mass parameters. This section illustrates and discusses such relations. The model (Model C) considered is one in which intact rock velocity is assumed to be 4950 m / s for P waves, 2920 m / s for S waves, and density is 2500 k g / m 3, typical of Georgia granites (Lama and Vutukuri, 1978). The fracture openings are assumed to be water-filled. Fractal fracture length distributions were randomly generated following the procedure by Boadu and Long (1994b)with varying fractal dimensions (0.1-0.9). The minimum and maximum fracture lengths for all simulations were 0.05 and 0.85 m, respectively. The fracture spacing distributions follow the Weibull distribution as described in Boadu and Long (1994b) with the mean varying randomly between 0.2 and 1 m. The fractured zone spans about 2 m. The distance from the edges of the model
boundaries and the fractured zone is 0.5 m (see Fig. 1). In all the simulations, the fracture openings were assumed to be 4 /xm, though they do not affects the dynamics of the model as explained earlier. Fig. 10(a-d) show the variation of P-wave velocity as functions of the fractured rock mass parameters. Except for one outlier point ( ~ 3000 m / s ) in Fig. 10(a), the seismic velocity associated with the fracture model decreases with increasing discontinuity index. This realization implies that, for a fractured medium, a decrease in seismic velocity is, in general, the result of an increase in fracture permeability. After the seismic velocity of the fractured rock mass has dropped to roughly 25% of its intact rock value ( ~ 1000 m/s), it becomes insensitive to further increases in permeability. This information may be useful in practical applications when mapping permeable zones in fractured terrain. Thus, zones having velocity Vp < 1000 m / s in such a geologic formation may be identified as permeable zones.
The P-wave velocity decreases with an increase in linear fracture density. The rate of decrease is quite pronounced for smaller fracture densities and is less for higher values. For example, a fracture density of two fractures per meter reduces the velocity to about 43% of its intact value, whereas a density of four fractures per meter reduces the velocity to about 32% and eight fractures per meter reduces the velocity to about 22% of the intact value. Velocity becomes insensitive to fracture density at increasingly higher values. A plausible explanation for this is that, as fracture density increases, the overall velocity becomes increasingly more sensitive to the fracture properties and less sensitive to the intact rock properties. One may thus infer that the ratio of the velocity in the fractured rock mass velocity to that of the intact rock could be used as an index for quantifying the degree of fracturing. Higher fracture density zones have been associated with high production zones in fractured hydrocarbon reservoirs. Using similar arguments for the discontinuity index, one may identify low velocity zones in fractured reservoirs for better production. Similar systematic relationships may also be developed for the other fractured rock mass parameters. In field application, one has to use higher frequencies to detect and characterize the fractures. The choice of frequency may depend on the attenuating characteristic of the intact
F.K. Boadu / Journal of Applied Geophysics 36 (1997) 1-19 3000
3000
2500
2500 X
2000
y~
2000
X
11
X 1500
X
>=
X XX
1000
1500
y~
X
1000
500
500
0
2
3
0
4
0
Discontinuity Index
2
4 6 8 Linear Fracture Density
2500
2500 + +
200C
2000
0
+
15oo 1000
+ +
++
0
>= 15oo
+
0
+
0 0
1000
0
0
0
500
50O 0 60
12
3000
3000
>=
10
70
80
90
100
110
RQD (%)
0
0
1
2
3
Fracture Density Parameter
Fig. 10. Variations of P-wave velocity with fractured rock mass parameters for Model C as described in the text: (a) discontinuity index la; (b) linear fracture density; (c) rock quality designation (RQD); (d) fracture density parameter.
rock material. This problem needs further investigation. It can be investigated numerically using the M D D model by making the velocities or, alternatively, the Young's modulus of the intact rock complex. As illustrated in Fig. 10(c), the compressional wave velocity decreases with a decrease in RQD. Thus, the P-wave velocity decreases as the rock mass competence measured by the RQD deteriorates. Several attempts have been made to relate fractured rock competence to the seismic velocities (e.g., Sjcgren et al., 1979). The difference in the present approach is that the fracture length, besides fracture spacing, is accounted for as one of the variables that significantly affect seismic velocities, making the relation more comprehensive. Since P-wave velocity in the fractured rock mass decreases with increasing fracture density and fracture density is a measure of the hydraulic transmissitivity, it is reasonable to infer that decreasing velocity implies increasing fracture transmissitivity. However, there is a limiting velocity ( ~ l l00 m / s ) below which the velocity becomes
insensitive to variations in fracture density (hydraulic transmissitivity). For delineating highly transmissive zones, this limiting velocity is an indicative characteristic of such fractured terrain. Note that this limiting velocity for hydraulic transmissitivity is slightly higher than that associated with the discontinuity index and the permeability of the fractured formations. The effects of fractured rock mass parameters on SH-wave velocity are shown in Fig. l l(a-d). The trends in the variations of each of these parameters with the SH-wave velocity are essentially similar to those of the P wave. The SH-velocity, in general, decreases with an increase in discontinuity index and fracture density, whereas it increases with an increase in RQD. However, there are distinct differences between the P-wave and S-wave relationships. For example, for a fracture density of two fractures per meter, the SH-velocity decreases to about 58% of its intact rock value, to about 48% for four fractures per meter, and to about 36% for eight fractures per meter. These decrements for SH waves are less than
F.K. Boadu / Journal of Applied Geophysics 36 (1997) 1-19
12 2500
2500
2000
2000 x x
1500
1500
× x
>-
x
x
1000 !
>x
x
1000
5OO
5OO
0 2
3
0
4
0
2
4
Discontinuity Index
2500
8
10
12
2500
2000
2000
+
0
+
o
+
1500
>-
+++
+
o
1500
+
o
>-
++
1000
o
o
1000
5OO
0 6O
6
Linear fracture Density
o
O
0
0
500
70
80 90 R Q D (%)
100
110
0 0
1
2
3
4
Fracture Density Parameter
Fig. 11. Variations of SH-wave velocity with fractured rock mass parameters for Model C as described in the text: (a) discontinuity index ld; (b) linear fracture density; (c) rock quality designation (RQD); and (d) fracture density parameter.
those for P-waves, indicating that P waves are more responsive to fractures than SH waves under the assumed transmission conditions. The higher decrements for P-waves compared with SH-waves are likely to be caused by the presence of water as the infilling material. Water has a significant effect on P waves by modifying the effective bulk modulus of the medium but no practical effect on SH waves since it does not support shear. The two velocities, when used in combination, may provide more comprehensive information about fractured rock masses. Variations in the ratio of compressional to shear wave velocity (Vp/VS) versus the fractured medium parameters are shown in Fig. 12(a-d). The Vp/V~ ratio decreases to about 70% of the intact rock value for a linear fracture density of two fractures per meter, 63% for a value of four fractures per meter, and 55% for a value of eight fractures per meter. These local variations of linear fracture density with Vp/V~ ratio are obviously much different from those observed in the values of either Vp or ~ . Thus, a combination of these three parameters; Vp, V~ and
Vp/~ could provide concise information for uniquely characterizing a fractured rock mass. This concept is planned to be used as a basis for a field-based study of fracture parameters in the future.
8. Relations b e t w e e n of fractured r o c k m a s s par a m e t e r s and seismic quality factor The presence of fractures in a rock mass constitutes one of the major causes of scattering of body and surface seismic waves (Lerche and Petroy, 1986). This scattering process causes effective attenuation of the propagating waves. As already discussed, attenuation is usually expressed by the quality factor Q and is mainly caused by the mechanisms of scattering by heterogeneities (such as fractures) and by intrinsic absorption due to material anelasticity, The seismic quality factor due to scattering Q~c is frequency dependent (Lerche and Petroy, 1986), while the quality factor due material anelasticity Qi is
F.K. Boadu / Journal of Applied Geophysics 36 (1997) 1-19 1.6
1.6
1.4
1.4 ×
1.2
x x
x
1
>
1.2
x
x
13
x
x
>." >=
x
)E )E
1
0.8
0.8
0.6
0.6
)E
)E
i 1
2
3
4
5
2
4
Discontinuity Index
6
8
10
12
Linear fracture Density
1.6
1.6
1,4
1.4
0
+
1.2
+
+
+
++ ++
>
0.8
0.6
0.6 70
80
90
100
110
RQD (%)
0 0
0
0
0
1
0.8
60
0
1.2
+
0
1
2
0
0
0
3
Fracture Density Parameter
Fig. 12. Variations of the ratio of compressional wave velocity (Vp) to shear-wave velocity (V~) (Vp/V~ ratio) with fractured rock mass parameters for Model C as described in the text: (a) discontinuity index Id; (b) linear fracture density; (c) rock quality designation (RQD); and (d) fracture density parameter.
generally assumed to be frequency independent. The overall seismic quality factor QT, for a fractured medium is given by 1
1
1 + -
QT(/)
Oi
-
(18)
Qsc(f) "
The total quality factor was computed using this relationship assuming a non-attenuative intact rock material (Qi = ~ ) for both compressional and shear waves. The attenuation of the propagating waveform is due solely to the geometric and in-filling material properties of the fracture. Model C is used to investigate how QT varies with the characteristic parameters of the fractured medium. Fig. 13(a-d) show the variations of seismic quality factor for P-waves (Qp) versus the parameters of a fractured rock mass. As seen in Fig. 13(a), as a general trend, Qp decreases with an increase in the discontinuity index. A more attenuative medium (due to fractures) is likely to be more permeable. Quality
factor Qp also decreases with an increase in the number of fractures per meter (Fig. 13(b)). Although within certain ranges of the linear fracture density, Qp remains constant. The interference of the multiply reflected waves within the intact rock sections manifest with significance when the fractures are relatively longer and have smaller spacings. Different combinations of fracture lengths and spacings constituting similar or the same discontinuity index may produce different Q's. This effect may explain the localized constant Q values for varying linear fracture density values. Qp versus fracture density also exhibits a similar trend as shown in Fig. 13(d). Several useful interpretations can be derived from the result that transmissive fractured terrain will possess lower Q values. For example, since Qp decreases for lower values of rock competence parameter RQD (Fig. 13(c)), a highly attenuative fractured medium may be interpreted to be structurally incompetent and may require reinforcement. The
F.K. Boadu / Journal of Applied Geophysics 36 (1997) 1-19
14 6 5 4
X
X
C:~ 3
X
X
X XX X
2
X
1 0
1
2
3
4
0
~
0
2
4
Discontinuity Index
+ o=
+
+
60
++
70
+
d
+
+
80 90 RQD (%)
&
8
lO
,2
Linear Fracture Density
0
0 0
0
0
0
100
110
0
0
0
1 2 3 Fracture Density Parameter
Fig. 13. Variations of P-wavequality factor(Qp) with fractured rock mass parameters for Model C as described in the text: (a) discontinuity index ld; (b) linear fracture density; (c) rock quality designation (RQD); and (d) fracture density parameter.
quality factor Q~ for SH-waves and its dependence upon the various rock parameters is shown in Fig. 14(a-d). The trends for Q, are quite similar to those of Qv for P-waves. However, in general, the Q values are comparatively lower for P-waves than SH-waves. This observation has been reported by King et al. (1986) for crosshole measurements in columnar joints and is in accord with the present results. Fig. 15(a-d) show the variation of the ratio of P-wave to SH-wave quality factors (Qp/Qs) versus the fractured rock mass parameters. Although the trends are quite similar for of Qp and Qs, the shapes of the variations are different. In particular, the variations are more nonlinear for Qp/Qs compared with Qp and Q~. Another interesting observation is that the rates of change of the velocities are different from those of the quality factors. These characteristic differences may also be exploited for a more comprehensive characterization of fractured rocks. The combined use of Vp, V~, Vp/~, Qp, Q~ and Qp/Qs may provide a highly informative and unique tool for assessing and characterizing the mechanical and hydraulic conditions of fractured media. For
example, consistently lower values of the seismic parameters are good indicators of a highly fractured permeable or transmissive medium. Such a medium may, however, be weak in mechanical strength. Comparison of the spectra of reflected and transmitted waves after propagating through a fracture indicate that amplitude reduction is quite substantial in the reflected waves compared to the transmitted waves (Boadu and Long, 1996). This observation favors the use of transmission experiments, for example, cross-hole and refraction seismics for imaging fractures. Data in the frequency range 200 Hz to 3 kHz can be obtained with the crosshole geometry, which is a reasonable frequency range to locate and characterize fractures. An additional advantage of the crosshole geometry is that raypaths normal to the strike or azimuth of the fractures can be studied. This is the basis of the present work.
9. Summary and conclusions The modified displacement discontinuity (MDD) model has been used to investigate and analyze the
15
F.K. Boadu / Journal of Applied Geophysics 36 (1997) 1-19 6
5.5
~5 x
o"
d
x
5
)E )E
x x x
4.5
4
x
I
2
3
4.5 x
)E )E
x
4
4
0
Discontinuity Index
2
4
6
8
10
12
Linear fracture Density
6
+
5.5
5.5
+
d
5
o"
+
s
0
+
0
+
0
4+ +
4.5
0
4.5
0
4-
0
0
44 60
70
80
90
100
110
RQD (%) Fig.
14.
4
1
2
0
3
Fracture Density Parameter
Variations o f SH-wave quality factor ( Q s ) with fractured rock mass parameters for Model C as described in the text: ( a )
discontinuity index
ld; (b)
linear fracture density; (c) rock quality designation (RQD); and (d) fracture density parameter.
effects of fractures on propagating seismic waves. In the author's previous work, computed velocities from the MDD model have been shown to compare well with experimental results. In the present paper, the MDD is used to simulate seismic wave propagation in fractured media. A fractured rock mass is modeled as a set of discontinuities embedded in an intact geological medium. The distribution of the geometric properties of the fractures; i.e., the fracture length, spacing, and aperture were assumed to exhibit fractal behavior or Weibull characteristics. Waves propagating through such fractured media are slowed down and attenuated. The time and frequency domain behavior of the waveform are shown to be dependent on the fracture density. The fractures cause a time delay in the propagating waveform and act as filter by attenuating the high frequency components in the spectrum of the waveform. In the process, the peak frequency in the received signal shifts toward the low-frequency end of the spectrum. Measures of rock strength (RQD) and the hydraulic properties (discontinuity index and fracture
density parameter) of a fractured rock mass and their relationship to seismic parameters (velocity and quality factors) have been investigated and analyzed. From the numerical studies presented, relationships between strength and hydraulic properties and the seismic parameters were inferred. Seismic velocities (both P and S) decrease with an increase in the discontinuity index Id. Similarly, the velocities decrease with an increase in the fracture density parameter. Low velocity zones may thus indicate high transmissitivity fractured zones. A fractured rock mass with high Id implies high permeability and, hence, lower-velocity fracture zones may have higher permeability. A fractured rock mass with lower strength or RQD will tend to possess lower seismic velocities. The variations in the relations between Vp/V~ ratio and the discontinuity index, fracture density parameter, and RQD are distinctly different from either Vp or V~ alone. This suggests that a combined use of Vp/V~ ratio, Vp, and VS may be employed to assess the engineering and hydraulic properties of a fractured rock mass.
F.K. Boadu / Journal of Applied Geophysics 36 (1997) 1-19
16
1
1
0.8
0.8 X
X X
0.6 O
X
X
0.6 x
x
O
0.4
0.4
0.2
0.2
2
3
0
4
0
Discontinuity Index
4
6
8
10
12
Linear Fracture Density
t
1
0.8
0.8 4-
+
+
0.6 ++
+
0
4-
0
0
++
0 0.4
0.2
0.2
70
0
0.6
0.4
0 60
2
80
90
100
110
RQD (%)
0
1
2
0
0
3
0
4
Fracture Density Parameter
Fig. 15. Variations of the ratio of compressional wave quality factor (Qp) to shear-wave quality factor (Qs) (Qp/Qs ratio) with fractured rock mass parametersfor Model C as described in the text: (a) discontinuity index ld; (b) linear fracture density; (c) rock quality designation (RQD); and (d) fracture density parameter. The seismic quality factor Q (P and S) decreases with an increase in I d and C, and also decreases with a decrease in RQD. In a similar observation, the ratio Qo/Q~ provides a different signature from either Qp or Q~ above in terms of its variation with I d, C, and RQD. Thus, the combined use of Qp/Qs, Qp and Q~ may be more useful than the individual parameters in retrieving relevant geologic and engineering information about a fractured rock mass. Although, in principle, attenuation information could be used in field measurements of a fractured rock mass, in practice, velocity information from traveltime measurements is more reliable and accurate than amplitude data. The results developed in this study are intended to be used as a basis for further research into the possibility of inferring strength and hydraulic properties of fractured rocks from seismic parameters.
American Chemical Society, for support of this research. Many thanks also go to the editor Dr. T.E. Owen and the reviewers for their suggestions and comments.
A p p e n d i x A. Derivation of fracture i m p e d a n c e Consider an aperture opening a between two parallel fracture planes subjected to an oscillatory wave of frequency w. If the opening is filled with a fluid of viscosity "q and density p, the frictional force P on the lower moving plane is given as (Landau and Lifshitz, 1959)
P = - Tlku cot ka
(A.1)
Acknowledgements Acknowledgement is made to the Donors of the Petroleum Research Fund, administered by the
where k = (1 + j ) / 6 , 6 = V;(2v/oJ), j = ~ 1 and u is the displacement and v is the kinematic viscos-
17
F.K. Boadu/ Journal of Applied Geophysics 36 (1997) 1-19
ity. This frictional force enters into the equation of motion, Eq. (A.9) of Boadu and Long (1996), as ( 6r + r )-Ev d- -Ydoyy
r-
h
-
u-~lkcot(ka)
dyu
~2 u
(A.2)
= p h d y 6t 2
and
d2V
~U r + / z h - - = 0. 6y
(A.3)
If we consider the stresses and the displacements to be harmonic in time and functions of y, then we can express them as: r = T ( y ) e i°~t,
u = U( y ) e i°''.
6y
E h
62u
u - r/k c o t ( k a ) u = ph
(A.5)
6t 2
and, from Eq. (A.4), we obtain ~U 6t
- -
=
"
i°gUe"°t;
~2R 6t 2
o)2Ue
E
6y
h
(A.11)
is given as
Ae-yy V = A e - ~ Y + Be yy,
I-
~[ E Yp =
U- ~/kcot(ka)U= -phto2U.
(A.7)
Also from Eqs. (A.3) and (A.4) we can obtain ~U T = - / z h Y6--"
(A.8)
BeVY
- -
z0
z0
(A.12)
where Z 0 is the characteristic impedance and y is the propagation constant and, in general, both are complex. In accordance with Eq. (A.10) and for an incident compressional wave, the propagation constant becomes
(A.6)
i°~t.
Substitution of Eq. (A.6) into Eq. (A.5) and further simplification yields 6T
--d y 2 -- TZv = 0
(A.4)
Eq. (A.2) simplifies to 6T
For small changes in stress and displacement, Eq. (A. 10) represents a second order differential equation with constant coefficients whose solution is well known. In analogy to transmission line circuits, the solution is also well known in terms of voltage V analogous to force T, and current I analogous to velocity given in terms of displacement U. Following Anderson (1985), the solution to the equation
r/k cot(ka)
tO092]
/xh
/x
#-~2
(A.13) "
The displacement is solved for by utilizing Eq. (A.8) from which the velocity is obtained by multiplying by ito. Thus, we have
1
ito
Zp
yp/xh
(A.14)
We can then write the expression for the characteristic impedance for incident P waves as
To solve for T, we differentiate Eq. (A.7) with respect to y and substitute from Eq. (A.8) to obtain
t~2T ~y2
zcp =
E
--T+ p.h 2
r/k cot(ka) /zh
T+
pro 2 /z
[ E
6yz
[ Ixh 2
,02
+ ph2~
T=0 (A.9)
•
(A.15) Similar expressions can be derived for the characteristic impedance for an incident S wave to obtain
which can be cast in the form 62T
---L-i- +
r/k cot(ka)
pro 2
Ixh
Ix
T=0. (A.10)
3(~ =
Eh 2
Eh
"
(A.16)
F.K. Boadu / Journal of Applied Geophysics 36 (1997) 1-19
18 and r/k cot(ka) to 2
Eh
+
ph2E) • (A.17)
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