GEOPHYSICS, VOL. 64, NO.4 (JULY-AUGUST 1999); P.1160-1171,16 FIGS.
Azimuth-dependent tuning of seismic waves reflected from fractured reservoirs
Michael A. Schoenberg *, Simon Dean, and Colin M. Sayers **
expressed in terms of the horizontal slowness, automatically accounting for the change of angle with azimuth for rays propagating through the layer and for the tuning effect which occurs for layers with thickness of the order of the wavelength. For low enough frequency (or equivalently, thin enough layers), approximate expressions for the reflection and transmission coefficient matrices and transmitted amplitudes are derived. These expressions demonstrate explicitly that all reflected pulses and all converted transmitted pulses have the same shape as the time derivative of the incident pulse, whereas for thicker layers, distinct reflections from the top and bottom of the layer are evident, particularly for small angles of incidence. When these reflections interfere, significant changes in pulse shape with azimuth are found which result from differences in the azimuthal variation of reflection coefficient from the top and bottom of the layer due to propagation effects in the layer.
ABSTRACT
Reservoirs with thickness less than the seismic wavelength can still contain significant amounts of hydrocarbons. Such layers exhibit a tuning effect which involves the interference of reflected waves from the top and bottom of the reservoir. Natural fractures in such reservoirs can play an important role in determining fluid flow, which makes the density and orientation of fractures of great interest. In the presence of one or more sets of aligned vertical fractures, the amplitude of reflected waves at nonzero offset varies with azimuth; hence, the tuning effect will vary with azimuth. For wavelengths much greater than typical fracture spacing, equivalent medium theory allows such a vertically fractured layer to be modeled as a monoclinic layer with a plane of mirror symmetry parallel to the layer. The variation in reflection and transmission coefficients with incidence and azimuthal angle for a thin vertically fractured layer can be
P-wave amplitude variation with offset (AVO) to characterize the fracture orientation. This possibility was first investigated by Mallick and Frazer (1991) for marine seismic data, who showed that the P-wave AVO response depends on the orientation of the shot line with respect to the vertical fractures and, therefore, that P-wave AVO can be used to determine fracture orientation. Pelissier et al. (1991) considered waves incident on boundaries between isotropic and transversely isotropic media, and showed that when the anisotropy axis is not normal to the interface, the scattering coefficients depend on azimuth. Lefeuvre and Desegaulx (1993) discussed how the directions characteristic of azimuthal anisotropy could be obtained by analyzing the variation of amplitude with azimuth and offset within bins selected from a 3-D surface seismic data set. Chang and Gardner (1993) investigated the effects of vertically aligned fractures on reflection amplitudes
INTRODUCTION
Because natural fractures in reservoirs and in the cap rock overlying the reservoir play an important role in determining fluid flow during production, the density and orientation of fractures is of great interest (Reiss, 1980; Nelson, 1985). Since oriented sets of fractures also lead to anisotropic (i.e., angle-dependent) seismic wave velocities, the use of seismic anisotropy to determine the orientation of fractures has received much attention. Although shear waves are considered to be more reliable indicators of fracture orientation than P-waves, considerable interest remains in the use of P-waves for determining fracture orientation because these form the basis of most commercial seismic surveys (Lefeuvre, 1994; Lynn et al., 1995a). Consequently, several authors have investigated the possibility of using the azimuthal variation in
Manuscript received by the Editor March 4, 1998; revised manuscript received December 28, 1998. *Schlumberger-Doll Research, Old Quarry Road, Ridgefield, Connecticut 06877-4108. E-mail:
[email protected]. Bullard Laboratories, University of Cambridge, Department of Earth Sciences, Madingley Road, Cambridge, CB3 OEZ England, United Kingdom. **Schlumberger Geco-Prakla, 1325 South Dairy Ashford, Houston, Texas 77077. E-mail:
[email protected]. © 1999 Society of Exploration Geophysicists. All rights reserved. 1160 Downloaded 25 Dec 2010 to 199.6.131.16. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/
Seismic Reflectivity from Fractured Reservoirs
using a physical model. Lynn et al. (1995a,b) related azimuthdependent P-wave AVO responses to open fracture orientation and relative fracture intensity (as deduced by commercial pay intervals) in upper Green River fractured gas reservoirs in Bluebell-Altamont field in northeastern Utah. Finally, Ruger (1997) and Riiger and Tsvankin (1995, 1997) gave an analytic expression for the azimuthal variation in the P-wave reflection coefficient for a vertically fractured medium (represented as a transversely isotropic medium with a horizontal axis of symmetry), which is valid for small anisotropy and material contrasts. Roger (1998) extended these results to orthorhombic media. In contrast to the case of a single interface, few authors have considered the case of a fractured layer. In a recent paper, Sayers and Rickett (1997) studied the reflection of P-waves from a 100-m-thick fractured layer as a function of the normal and tangential compliances Z N and Z T of the fractures. For ZN/Zr =1 and moderate offsets, the variation with offset of the reflection coefficient from the top of the fractured unit was found to be dominated by the contrast in Poisson's ratio between the gas sand and the overlying shale, the effect of fractures only becoming noticeable as the critical angle for the unfractured sandstone is approached. However, for reflections from the base of the fractured unit, the variation in reflection amplitude with azimuth is greater at conventional seismic offsets than for the reflection from the top. Azimuthal variations in the strength of the reflection from the top of the reservoir depend only on the variation in reflection coefficient, whereas the raypath is also a function of azimuth for reflections from the base of the fractured unit, leading to stronger, more visible, variations of AVO with azimuth. For thinner layers, the reflections from the top and bottom of the layers will interfere (tuning), and the purpose of the present paper is to examine the azimuthally-dependent tuning that occurs for thin vertically fractured reservoirs with thickness of the order of the wavelength. A correct treatment of this must include all mode conversions and multiples that may occur in the fractured layer, and frequency-dependent reflection coefficients are derived which include these effects. The approach presented automatically accounts for change of incidence angle with azimuth for rays propagating through the layer. The outline of the paper is as follows. In the next section, the seismic anisotropy resulting from one or more vertical fracture sets is discussed. An analysis of the scattered fields due to an incident plane wave on a thin fractured layer is then presented. The azimuthal tuning effect which results is then illustrated for the example of a single set of vertical fractures in an isotropic gas sand embedded in shale.
1161
with the fracture normal, the horizontal tangential direction, and the vertical), that is, its compliance is fully specified by a normal compliance Z N , a horizontal tangential compliance Z H , and a vertical tangential compliance Z v (Schoenberg and Sayers, 1995). The homogeneous anisotropic medium equivalent (in the long wavelength limit) to this fractured medium is up-down symmetric (i.e., monoclinic with a horizontal mirror plane of symmetry). Orthorhombic media with a horizontal plane of mirror symmetry, trigonal media with a vertical threefold symmetry axis, and transversely isotropic (with horizontal symmetry axis) media are special cases of azimuthally dependent media with up-down symmetry. The thin layer's 6 x 6 stiffness matrix Cp (the subscript £ referring to the thin layer) is the inverse of its compliance matrix, S e , that is, C k = Sx 1 . The compliance matrix is more suitable for modeling fractured media since the introduction of fractures can be modeled by the addition of a fracture compliance matrix to the compliance matrix of the unfractured host medium, as shown by Schoenberg and Sayers (1995). A general monoclinic medium has 13 elastic constants; its stiffness or compliance matrix is of the form 0 0 • 0 0 • 0 0 • 0 0 0 • • 0 ' 0 0 0 • • 0 •
0 0 •
where the dots denote nonzero elements. It will be seen below that the introduction of vertical fractures does not affect the (1, 3), (2, 3), (3, 3), and (3, 6) elements of the compliance matrix. Thus for the case considered here, where the background host medium (denoted by subscript b) is assumed to be azimuthally isotropic (S b13 = Se23 and Sb36 = 0), the compliance matrix of the vertically fractured medium is also subject to these constraints, that is, S^13 = Sf23 and Se36 = 0. However, the vanishing of SP36 and the equality of S f13 and Sf23 does not imply the corresponding relations hold for the stiffness matrix. Following Nichols et al. (1989) and Schoenberg and Sayers (1995), the compliance matrix for a medium with n sets of aligned fractures (assuming that the possible effects of any fracture intersections can be neglected) can be written n
MODELING VERTICALLY FRACTURED ROCK Consider a thin fractured layer between two isotropic or transversely isotropic halfspaces. The thin layer is assumed to consist of one or more sets of aligned vertical fractures in a background host medium which, in its unfractured state, is assumed to be azimuthally isotropic (i.e., it is either isotropic or transversely isotropic with a vertical axis of symmetry). The fractured thin layer, then, is azimuthally anisotropic by virtue of the presence of the vertical fractures. The excess compliance attributable to each of the sets of aligned vertical fractures is assumed to be diagonal (in a coordinate system aligned
Sc = S6 ^ S f l
,
(1)
q=1
where S f ) is the fracture compliance matrix of the qth set of aligned fractures and S b is the compliance matrix of the unfractured host medium. Under the assumption that each of the sets of aligned fractures is vertical, and that the qth fracture set, q = 1, ... , n, is specified by normal fracture compliance ZNq ^, horizontal tangential fracture compliance ZHq ^, vertical tangential fracture compliance Z(9) , and strike 0 (q ) measured from the horizontal x2-direction away from the x 1 -axis (so the normal to the qth fracture set is 0 q from the horizontal xl (
)
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1162
Schoenberg et al.
direction towards the x2-axis), S (9) is given by
0
0 ZN4) 0 0 0 0
s 12(q) s22(q) 0
0
0
S(q) 26
0
0
0
0
0
0
0
0
0
0
`S44 )
'S45 )
0
0
0
0
s45 )
s55 )
0
S 16(9)
(4)
S26
0
0
0
S66
S11=N g )
H
+
(2) 0
(q)
Z cos20
(4)
cos20(q)
(q)
S (q) = ZNq) — ZH) (1 — cos 49 ( q ) 8
),
W _ ZNq) + ZHq) —Z Nq) _ Z(y) (q), cos 49
66
2
2
(q) + ZNq) — ZHq) S (q) — sin 20 Z sin 4B(q) 16
N
2
4
(q) (9) S (q) — sin20 (q) — ZN — ZK sin40 ( q ) , 26 2 Z N 4 S44 ) = Z ( q )1 — c 20(s)
O + cos Sss) — Z1
S (4) — Z (q) Sin 20 45 — V
20(q)
(q)
2
This can be derived by carrying out the matrix multiplication given by Nichols et al. (1989) or directly using 4th-rank tensor notation. Note that when Z (q) = ZHq) , the 49 q dependence of the excess fracture compliance vanishes, S vanishes, and S is independent of 9 q)• For 0 ( q) = 0 (strike along the x 2 -direction) and for 0 (q ) =n/2 (strike along the x 1 -direction), the S q are given by (
)
(
(
0
0
0
0
0 0 0 0 0
0
0
0
0 0 0 0 0
0
0
0 0
0 0 0 Z q 0 (
0
0
0
0
0
0
0
respectively.
+ Z^ 8 ZH cos4B (q ) ,
12
0
THE SCATTERED FIELDS DUE TO AN INCIDENT PLANE WAVE ON THE THIN FRACTURED LAYER
2
9 _ ()
0
0 0 0 0 0 Zy9)
X(2) — 3Z Nq) + Z Hq) — Z^q) 8
0
0 0 0 Z(9) 0 0
(q) _ (q) + ZN g ZH COS 4B (R) ,
0
0
s 16 ("
3Z (g) + Z (g) Z (g)
0
0
0
S 12 (q)
with
0
0
0
S(9) _
ZN9)
0
0
S(4) 11
S
0
)
0 0 0 0 0 ZHR
)
)
The solution for transmission and reflection coefficients at an interface between two general anisotropic halfspaces requires, in general, the solution of six equations in six unknowns, and the coefficients of the unknowns involve the solutions of a sixthorder polynomial for six vertical slownesses for each medium (Fryer and Frazer, 1984). However, the problem is greatly simplified if each medium is at least monoclinic with a plane of mirror symmetry parallel to the interface, a condition that we call "up-down symmetry." In this case, because of the mirror symmetry, the eigenvalues (vertical slownesses) for each medium are a solution of a bicubic equation, the eigenvectors (polarization vectors) always appear as mirror symmetric pairs, and the elements of the 6 x 6 coefficient matrix for each medium can be fully expressed in two 3 x 3 coefficient matrices. The formalism of Schoenberg and Protazio (1992) was designed to take advantage of this simplified situation. Explicit expressions for the reflection and transmission coefficient matrices at an interface between anisotropic media were derived in terms of the two 3 x 3 coefficient matrices, or "impedance matrices," designated X and Y (one pair for each medium in the problem). The matrices X and Y relate the field variables (i.e., components of velocity and traction at any depth) to the amplitudes of the possible down-going and (with a change of sign) up-going waves that can exist in the medium. The formalism is easily extended to the case (considered here) of a vertically fractured layer, whose elastic behavior is long wavelength equivalent to that of an up-down symmetric anisotropic medium sandwiched between two half-spaces. The calculation of the transmissivity and reflectivity of such an anisotropic layer is essentially reduced to finding these required frequency-independent impedance matrices. For transient plane waves, this calculation has to be performed but once, for the horizontal slowness of interest. With these matrices in hand, the calculation of transmission and reflection coefficient matrices is reduced to a few lines of computer code. Finding impedance matrices requires calculating the three eigenvalues (vertical slownesses squared) for downward propagating waves taken, in general, in order of smallest to largest (denoted with subscripts P, S, and T), and the three corresponding eigenvectors (associated polarizations) of the Christoffel equations. For isotropic and VTI media, which always have an in-plane (vertical plane) polarized shear (SV) wave and a cross-plane (horizontal) polarized shear (SH) wave, we will let S and T be synonymous with SV and SH, respectively. From the assumption of up-down symmetry, an upward wave vertical slowness is merely the negative of the corresponding downward wave vertical slowness, and the upward polarization vectors are merely the mirror images of the downward ones. The formalism takes care of this automatically.
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Seismic Reflectivity from Fractured Reservoirs
For an isotropic medium of density p, compressional speed a, and shear speed ,B, the 3 x 3 impedance matrices (needed when there is conversion between in-plane and cross-plane waves through the presence of another azimuthally anisotropic medium) are given by,
X
=
a l
F^ 1 35/th
a'^2
2^3S/^h
-pa(1 - 2^ 2
^h)
-
1163
where, A - X - 'XrC(coh)X, 1 X - iX 'XeS(coh)Y, i Y -
B=Y-'YEC(c)h)Y-'Y-iY1Y - S(wh E e ) Xe1X. The dependence on wh (i.e., on frequency x layer thickness) is conveyed only through the diagonal cosine and sine matrices,
2/h 1/^h 0
2 p, 3 h^3s
C(wh) =
Y=
(3)
cos w3 h
0
0
0
cos w^Q3s h
0
0
0
Cos w^j 3 T h
sin wi^ 3 p h 0 0
0
0 0 sin w^j3T h
-2 pa^ 2 41 3 p -p1(1 - 2,2 h) / h Pi 2 ^2 3T 1 ^h
-2pap 2 ^2 3p - P2( 1 - 2fl 2 ^h)/th - P 2 ^1^3T/ h a^3p
S(coh) =
0
where h = 1 + 2 is the magnitude of the horizontal slowness, and 3p , X35 , and i 3T are the vertical slownesses given by 3p = q-2 - h,
S = 3T =
sin wi e3s h
0
It is easy to see that for the nonevanescent case (for which X, Y, XE , and Y E are real matrices), the only effect of a change in sign of wh is a change in sign of the imaginary parts of T and R. Note that since lim wn - o A = lim h _ o B = I (the identity matrix), then, as expected, T -* I and R 0 as wh 0. When the medium parameters inside the layer approach those of the medium outside, A -* B -- C(wh) - iS(wh); again, as expected, R 0 and T - C(wh) + i S(wh), which accounts for the phase difference over thickness h in the now homogeneous medium.
2 - h.
This is the explicit representation of the matrices that relate amplitudes to components of velocity and stress. Note that XI = - pa,43s while jYl = p 2 a,B 3 4 3T 3p , so X and/or Y are singular only at critical angles. This is the case for anisotropic media also. Impedance matrices for a general monoclinic medium, derived by Schoenberg and Protazio (1992), are given by
e p1
e st
eT1
e p2
e s2
e T2
X = - (C13ep, +c36ep2)^l
— (C13es, + C36es2)^1
- (C13eT1 +C36eT2 )^1
- (C23ep2 + C36ep1)^2
— (C23es2 + C36es,)^2
- (C23 eT2 + C36 eT,) ^2
-C33 e p3 3p
—C33es3 3S
-C33 eT3 ^3T
(4)
Y
=
— (C55 1 + C45^2)ep3
—(C55^1 + C45 2)es3
— (C55ep1 +C45ep2)^3p
— (C55es, + C45eS2) 3S
—(C45^1 + C44 2)ep3
—(C45^1 + C44^2)eS3
—
(C45ep1 +C44ep2)^3p
e p3
— (C45e51 + C44eS2) 3S
es3
—(C55^1 + C45^2)eT3 —
(C55eT, +C45eT2 )^3 T
—(C45^1 + C44^2)eT3 —
(C45eT, +C44eT2 )^3 T eT3
where ep, e s , and e T are the polarization vectors associated When wh times the largest vertical slowness in the problem with the downgoing waves. is small enough so that squares and higher powers of vertical slowness times wh may be neglected, we can write, Consider the case of a vertically fractured layer (whose properties are denoted by subscript f) sandwiched between idenC(wh) I, tical halfspaces with impedance matrices X and Y. It is convenient to refer the incident wavefield to the top of the layer 0 0 (6) ^Z3p and the transmitted wavefield to the bottom of the layer. The transmission and reflection matrices, specified by T(^ 1 , ^z , wh) S(wh) wh 0 ^^ 3s 0 = wh 3 , and R(^ 1 , ^z , wh), respectively, are 0 0 t3 T
T = 2(A + B) ', -
(5) R = z (A - B)T = (A - B)(A + B) ', -
where i; e3 is the diagonal matrix of the vertical slownesses in the layer. In this low frequency-thin layer approximation, A : I-iwhX - 'X ' ^ E3 Yf 1 Y and BSI-icwhY-1Y'^13Xe'X,
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1164
Schoenberg et al.
yielding
^n —
T ti I+ i 2
(X -'
M)=
Xe^e 3 Ye t Y + Y -1 Ye^e 3 Xe 1X ) (
7
27r
[ao +a 1 cosw n t+a 2 cos2w n t], Its
0,
otherwise;
)
R -i 2 (X -1 Xe^e 3 Ye 1 Y - Y -1 Ye^e 3 Xe 1 X)-
(
8
)
wn is its nominal radial frequency, and S2„ = wn h/,B bV is the di-
Thus, these approximate expressions for R and T — I are linearly proportional to iwh/2. Since multiplication by iw in the frequency domain is equivalent to differentiation in the time domain, all reflected pulses and all converted transmitted pulses have the shape of the time derivative of the incident pulse, provided that the medium above and below the layer is the same. This agrees the results of earlier work for normal incidence of a plane wave on a thin isotropic layer (Widess, 1973; Koefoed and de Voogd, 1980). At any horizontal slowness, the solution, even for P to P reflection, depends on wh^P3Q , Q = P, S, T being small enough so that the approximations of equation (6) are valid for all the vertical slownesses. Thus it makes sense to form a dimensionless frequency as wh divided by the smallest vertical shear speed. But as a reasonable approximation (with an eye to keeping the normalization slowness independent of the fracturing), we use the vertical shear slowness of the unfractured host medium (assumed to be at least transversely isotropic), Pb /C5 5b = $bv , to define a dimensionless frequency:
mensionless nominal frequency. Normalizing the pulse so that its total area is unity forces a o = 1, and for the pulse to be continuous at its start and end points [i.e., for f (fn/w) = 0], a 2 =a 1 — 1. The pulse's peak value is a, w n /n . As wn -* oc (T -+ 0), the pulse approaches a delta function. The Fourier transform of a time limited cosine signal is n
_
The results of this approximation [equation (6)] are reasonable for Q up to about unity. THE BLACKMAN-HARRIS INCIDENT PULSE AND CORRESPONDING REFLECTED PULSES The tuning effect of a layer involves the interference of the reflections from the top and bottom of a layer. When the layer is azimuthally anisotropic, this tuning affect will depend on azimuth. This effect on P to P, P to SV, and P to SH reflected waves from the layer contains information about layer thickness, the dominant fracture orientation in the layer, and perhaps also the ratio of Z N /Z T . The presence of significant gas in the fractures results in Z N /Z T approaching values close to unity (Schoenberg and Sayers, 1995); a lower ratio of Z N /Z T results when the fractures are liquid saturated (liquids having relatively high bulk modulus) or from the additional presence of cement or clay within the fractures. The tuning effect may be best appreciated by calculating and displaying the pulse shape of the reflected waves both as a function of angle and of azimuth for a given simple incident plane P-wave pulse. Note that when the upper half-space is isotropic, the horizontal slowness components in terms of incidence angle and azimuth are given by a
cos 1p,
Ei°t COS riCO n t dt
rl w n (-1)n sin rrw/[O n (w/Cwn)Z 0)/w n (0)/w) 2 — n 2
so the spectrum of the three-term cosine window, or what is usually called a three-term Blackman-Harris window (Harris, 1978), of equation (8) is given by F (w) =
SlnJrw/w,1
1 — aiw
2 /c^n
+ ( a1 — 1)w 2 /co n
7rw/CC n C^2/2_1
w2/w2_4
nw/w n (w 2 /wn - 1) (60 2 /wn - 4) ,
NbV
sing
J
sin 7rw/wn (-4 + 3a1)w 2 /wn + 4 (9)
wh
S2=— R
=
27r
pn/o'n
sing 2 =
a
COS C^J,
where 9 is the incidence angle, q5 is the azimuth measured from the 1-direction, and a is the upper medium's P-wave speed. To concentrate on azimuthal dependence, because that is the new phenomenon of interest, traces identified with a particular reflected wave will be shown as a function of azimuth with angle 0 held constant for three values: 9 = 15°, 30°, and 45°. Assume the incident P-wave pulse is a three-term cosine window of total width T = 2n/w , which may be written as
a sync function multiplied by a rational function in w/w n whose
simple poles at ±1 and ±2 are canceled by zeros of that sync function. Thus F (w) is analytic for all real w. This spectrum has zeros at w/wn = ±n, n > 3, and at any root of the numerator of the rational function. For the main lobe to span —3 < w/w„ < 3, the roots of the numerator, denoted by wo /w,,, must be outside this range, that is, NO Z 4
cwn 4-3a1
> 9,
or a i > 32/27 1.185. The conventional three-term Blackman window uses a 1 = 25/21 1.1905 for which wo /wn = 3.055, forcing F (w) to vanish just outside the main lobe. We used this value for a l even though the lowest maximum absolute value of F (w) for all w, w/w n I > 3, is achieved when coo /w„ I = 3.47 (a value that has been found numerically) corresponding to an optimal value of a 1 = 1.2226 and a maximum side lobe peak —64.18 db lower than the main lobe peak. Both the conventional and optimal values of ai agree with those stated by Harris (1978), taking into consideration that Harris constrained the value of the pulse at t = 0 to be unity instead of our constraint that the integral of the pulse be unity. However, for all values of al between 32/27 and 4/3, the peak values of the side-lobes are small enough relative to the main lobe peak that a good approximation to the reflected pulses is obtained by calculating the inverse transform of F(w)R QP (w) (Q = P, SV, SH) by integrating over the main lobe, which requires calculation of the reflection coefficients only over the range —3 < w/m„ < 3. To find the reflected field at any point in the upper medium for an incident plane P-wave pulse u,, f (t) where u,,,, is an arbitrary P-wave amplitude of dimension length, (so that 2aiu/T is the particle velocity at the center of the pulse), it is necessary to find the inverse transform of that field in the
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Seismic Reflectivity from Fractured Reservoirs
Fourier domain. As X and Y transformed downward wave amplitudes into velocity and stress components (in particular, X transformed downward wave amplitudes into vi, v2, and a3, while Y transformed them into a5, a4 and v3), these components may be written as i
Vi V2 = UincX( 1, 2)F(w) a3 RPP(^1, 2; (o)exp —iww^3p(, 2)X3
X
Rsp(, 2; a)) exp —iUJ^3 S ( 1, 2)x3 RTP(^1, 2;CD) eXp —iw 3 T ( 1, 2)x3
x exp i w(i;lxl + 2x2),
(10) Q5 a4 = — UincY(41, S4 2)F(w) V3
RPP(^1, 2;w) exp —iw 3 p ( i, 2)X3 X
RSP(^1, 2; w) exp —iw^3 s (1, 2)x3 RTP(1, 2; w) exp
—
a=3.3, 3 = 1.7, p=2.35, a b =4.2, 1b = 2.7, Pb = 2.49,
x exp i w(1x1 + 2x2).
with the subscript b denoting the unfractured background medium of the layer. These values give $ 2 /a 2 = 0.265 and, equivalently, a dynamic Poisson's ratio of v = 0.319 for the shale, and $ /ab = 0.413 and a dynamic Poisson's ratio of Vb = 0.148 for the unfractured sandstone. In the absence of fractures, the PP critical angle is 51.8°. These medium properties are those used by Sayers and Rickett (1997) in their model 1 simulations. Figure 1 shows the reflection coefficients Rpp and R s p for PP and PSV reflections as a function of P-wave incidence angle
Defining
1
°O /
1
3^n
rQp(t) = fF(w)RQp( 1, 2; w)e 2n
F(w)RQP(^I, 2; w)e
1165
anisotropic layers (when the azimuthal anisotropy is caused by vertical fracturing), results will be given for the case of an isotropic gas sand containing a single set of rotationally symmetric vertical fractures (Z y = Z H - Zr). The fractured layer is then transversely isotropic with a horizontal axis of symmetry and is assumed to be encased in an isotropic shale with properties based on the classification scheme of Rutherford and Williams (1989). In this scheme, the AVO response of gas sands is classified according to the normal-incidence PP reflection coefficient R 0 . Class 1 gas sands have higher acoustic impedance for normal incidence than the encasing shale with relatively large positive values of R 0 . Class 2 gas sands have nearly the same acoustic impedance as the shale and are characterized by relatively small values of R 0 . Class 3 sands have lower impedance than the shale with relatively large negative values of R 0 . In this section, the effect of fractures on the AVO response of thin gas sands will be investigated using the class 1 example of Rutherford and Williams (1989) from the Pennsylvanian Hartshorn formation in the Arkoma Basin. This example was chosen because Sayers and Rickett (1997) showed that the effect of fractures is greatest when the acoustic impedence of the sandstone is greater than that of the shale. The shale and unfractured sandstone in this example have P- and S-wave velocities a and ,B (in km/s) and density p (in g/cm 3 ) given by
2)x3
iw3T(1,
imt
dlv
wc -
2r f 3mn Q = P, SV, SH,
dw ,
the reflected field in the time domain is given as Vi(t) v2(t)
= UincX(^1,
2)
Q3 (t) 0.2
rPP(t — 1X1 — 2X2 + 43 p(1 x
2)x3)
r SP(t — 1x1 — 2x2 + 3S(^1, 42)x3) 0.15
rTP(t — ^l xl — 2X2 + 3T(1 2)x3) ,
(11)
4-
0
a5 (t) a4(t)
_ UincY(^1, 2) —
V3(t)
C 0.1 0
U 04 0.05
[
x
rpp(t — lxl — 42x2 + i3p (^1, 2)x3) rSP(t — 1X1 — 2X2 + 3 S (^1, 2)x3) rTP(t —
0
1x1 — 2x2 + 3T(1 , 42)X3)
However, when plotting results for incident pulses, only the reflected pulse shapes will be plotted, that is, rpp(t), r sp(t), and r Tp(t). RESULTS
There are many types of layers that can be considered with many possible distributions of vertical fracture sets. In order to exhibit some of the possible tuning effects of azimuthally
-0.05
0
10
20
30
Angle of incidence (degrees)
40
50
FIG. 1. Reflection coefficients Rpp and R s p as a function of P-wave incidence angle for PP and PSV reflections for the case of a single interface with shale overlying an unfractured gas sand for the type 1 gas sand model of Rutherford and Williams (1989).
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1166
Schoe nberg et al.
for the case of a single interface with the shale overlying the unfractured gas sand. The mode-converted S-wave amplitude is seen to be significant for P-wave incidence angles greater than 5°. As expected for a gas sand overlain by shale, the PP reflection coefficient shows a strong variation with angle of incidence. Consider vertical fractures in this sandstone layer with fracture compliances Z N = Z T (the case of "scalar fractures") with the value of Z T chosen so that _
PbfbZT
= 0.2.
ST 1 + P b l b Z T
PP (8=15)
For a choice of axes with x 3 vertical and x l normal to the fractures, this results in the following elastic stiffnesses (in GPa) of the fractured sandstone: c 11 = 27.37, c 22 = c 33 = 43.43, C1 2 = C 13 = 4.75, c 23 = 7.12, c 44 = 18.15, c 55 = C66 = 14.52. The case Z N Z T is characteristic of either gas-filled fractures
or liquid-filled fractures in a very permeable host medium in which fluid can easily flow in and out of the fractures so that the liquid contributes little to the normal compliance of the fractures. For this example, Figures 2 to 10 show the variation of reflection coefficients with dimensionless frequency 0 for a P-wave incidence angle of 15°, 30°, and 45°, and azimuths 0°, 45°, and 90° measured with respect to the fracture normal direction. As
PP (9=15)
0.25
TO /0_1 C \
PD /0_i
i v.v>
0.9
19
0.08 0.2
0.8
18
0.07 0.7
v
0=90 0.15
0.6
17
0.06
0=90
0=45
l=0
C-i
0.1
I
0=0
0.4
0.05
0=45
0.05
0.5
0.3
0.03
0.2
0.02
0.1
P.
0.04
0.01
).1h
C 0
2
4 SZ
6
00
2
4
6
0
2
4
6
0
2
Q
4
6
0
FIG. 2. Amplitude and phase of the PP reflection coefficient FIG. 4. Amplitude and phase of the PT reflection coefficient R PP for the case of a layer of fractured gas sand within an RTP for the caseo a layer 1 gas sand awithin an isotropic shale for the type 1 gas-sand model for a P-wave isotropic shale for the type 1 gas-sand gas-sand model for a P-wave incidence angle of 15°. incidence angle of 15°. SP (0=15)
SP (0=15)
PP (0=30)
PP (O=30)
0.9
0.9
0.3 0.8 0=90
0.25 A
0.7
0.7
0.6
0.2
0=90
0.6 0.5
0.5 0.15
0=0
0.4
0.1
0.4 ^=90
@=0
0.3
0.3
0.2
0.2
0.1
0.1
0.05
00
2
4 0
6
0
0
2
46 Q
FIG. 3. Amplitude and phase of the PS reflection coefficient R s p for the case of a layer of fractured gas sand within an isotropic shale for the type 1 gas-sand model for a P-wave incidence angle of 15°.
-0
2
4
6
_0
2
0
4
6
Q
FIG. 5. Amplitude and phase of the PP reflection coefficient RPP for the case of a layer of fractured gas sand within an isotropic shale for the type 1 gas-sand model for a P-wave incidence angle of 30°.
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1167
Seismic Reflectivity from Fractured Reservoirs
expected, the variation in reflection coefficient with azimuth is small for small values of the P-wave incidence angle (Figures 2 and 3), but becomes significant as the incidence angle increases, particularly for values of the dimensionless frequency Q > 1 (Figures 5,6, 8, and 9). At small angles of incidence, the P-wave reflection coefficient is dominated by the contrast in the product of density and vertical P-wave velocity. At larger angles of incidence, the azimuthal variation in AVO is more significant. This is due in part to the critical angle for incidence in a plane perpendicular to the fractures being increased over that for incidence in a plane parallel to the fractures. Results for the reflected pulses at a fixed nominal frequency were calculated in three different frequency bands. The high-
frequency range is represented by Q = 5.818 corresponding to 50 Hz with a 50-m-thick layer of background shear speed of 2.7 km/s. The central range is represented by Q = 1.164 corresponding to 50 Hz with a 10-m-thick layer of the same background shear speed. The low-frequency range is represented by Q = 0.233 corresponding to 50 Hz with a 2-m-thick layer of the same background shear speed. Figures 11 to 16 show synthetic seismograms for a P-wave incidence angle of 30° and 45° calculated for the type 1 gas-sand model of Rutherford and Williams (1989) assuming S T = 0.2 for the two cases Z N = Z T . In the low-frequency regime (Figures 11 and 14), the reflected pulse is seen to resemble the derivative of the incident pulse, as expected. For a P-wave incidence angle of 30°
SP (0=30')
SP (0=30')
PP (9=45')
PP (0=45') 0.25
0.45 0.9 0.4
4=90
0.8
0.35
0.2
0.7
0.3
^=0 v 0.15
vro 0.5
0.25
0.2 0=0
0.2
0.05
0.1
0
0.7
0=45
I
¢=0
0=90 0.4
0.1
0.3
0.1
2
4
6
=90
0.5
0.4
0.15
0.8
0=90
0.6
v
0
0.9
0.3 0=45
0.2
0.05
0.1
2
4
6
0
2
4
°
6
0
FIG. 6. Amplitude and phase of the PS reflection coefficient R s p for the case of a layer of fractured gas sand within an isotropic shale for the type 1 gas-sand model for a P-wave incidence angle of 30 . 0
TP (9=30)
0
2
4
f
n
FIG. 8. Amplitude and phase of the PP reflection coefficient Rpp for the case of a layer of fractured gas sand within an isotropic shale for the type 1 gas-sand model for a P-wave incidence angle of 45°.
TP (0=30 )
SP (0=45')
SP (8=45')
0.45 0.4
0.65
4=0
0.6
0.35
0=45 °
0.55
0=45
0.3
0=0
v
K
=45
0.5
0.25
v
0=90 0.45
0.2 0.4
0=90 0.15 3.2
2
4
0
6
V0
0.3
0.05
3.1 0
0.35
0.1
2
4
6
0
FIG. 7. Amplitude and phase of the PT reflection coefficient R TP for the case of a layer of fractured gas sand within an isotropic shale for the type 1 gas-sand model for a P-wave incidence angle of 30°.
0.25
0
2
a
4
6
0.2
2
n
4
6
FIG. 9. Amplitude and phase of the PS reflection coefficient R s p for the case of a layer of fractured gas sand within an
isotropic shale for the type 1 gas-sand model for a P-wave incidence angle of 45°.
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1168
Schoenberg et al.
(Figure 11) the PP reflection amplitude is seen to be greater at azimuth 0 = 90° than at q5 = 0 , whereas for an incidence angle of 45° (Figure 14), the reverse is true. This is consistent with the variation in reflection coefficient with azimuth shown in Fig- 0
TP (0=45)
Ti' (0=45)
C C c,
2
0
4
6
0
2
S2
4
6
S2
FIG. 10. Amplitude and phase of the PT reflection coefficient R TP for the case of a layer of fractured gas sand within an
isotropic shale for the type 1 gas-sand model for a P-wave incidence angle of 45°. ZN/Z T = 1, h = 2 m, to = 50 Hz (S2 = 0.233)
ures 5 and 8. The PS reflection amplitude varies only slightly with azimuth at a P-wave incidence angle of 30° (Figure 11), but shows a significant variation with azimuth at a P-wave incidence angle of 45° (Figure 14), in agreement with Figures 6 and 9. The PT reflection amplitude varies strongly with azimuth and vanishes for 0 =00 and 90°, corresponding to the incidence plane coinciding with a symmetry plane of the medium. In the high-frequency regime (Figures 13 and 16), separate reflections from the top and bottom of the layer are visible, particularly for small angles of incidence. For a P-wave incidence angle of 30°, the variation of the PP reflected pulse shape with azimuth is relatively small for Q = 0.233 and Q =1.164, but becomes significant for Q = 5.818 (Figure 13), where it is seen that the azimuthal variation in PP reflection coefficient is greater for the bottom of the layer than from the top, in agreement with the results of Sayers and Rickett (1997). Azimuthal variations in the strength of the reflection from the top of the reservoir depend only on the variation in reflection coefficient, whereas the raypath is also a function of azimuth for reflections from the base of the fractured unit, leading to stronger, more visible variations of AVO with azimuth. For an incidence angle of 45°, the PP reflected-pulse shape varies significantly with azimuth even for S2 = 0.233. The PS reflected pulse varies little with azimuth for a P-wave incidence angle of 30°, but varies significantly with azimuth for an incidence angle of 45°, particularly at higher values of Q. The PT reflected pulse varies strongly with azimuth and vanishes for P-wave incidence plane parallel
P-P Incidence angle = 30: Horizontal slowness = 0.1515 s/km
Z N /Z.r =1, h=10m, fo =5OHz (f2=1.164)
P-P Incidence angle = 30: Horizontal slowness = 0.1515 s/km
0 20
EE
E 40
e E H
d
60
-10
0
10
70 40 50 60 Azimuth (degrees) P-S Horizontal slowness = 0.1515 s/km
20
30
80
90
100
0 Azimuth (degrees) P-S Horizontal slowness = 0.1515 s/km -20 0 20
E N
E I-
40 60 80 -10
lo
80 -10
Azimuth (degrees) P-T Horizontal slowness = 0.1515 s/km
Azimuth (degrees) P-T Horizontal slowness = 0.1515 s/km
-20
-20
0
0
20
20
40
40
60
60
80 -10
0
10
20
30
40 50 60 Azimuth (degrees)
70
80
90
100
11. Synthetic seismograms for a P-wave incidence angle of 30° calculated the type 1 gas-sand model assuming S T = 0.2 and Z N = Zr for a 2-m-thick layer. FIG.
80 -10
0
10
20
30
40 60 50 Azimuth (degrees)
70
80
90
100
12. Synthetic seismograms for a P-wave incidence angle of 30° calculated the type 1 gas-sand model assuming S T = 0.2 and Z N = Z T for a 10-m-thick layer.
FIG.
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Seismic Reflectivity from Fractured Reservoirs
1169
derived in terms of the horizontal slowness. Our expressions
or perpendicular to the fractures. It achieves significant amplitude only at large values of Q (see Figures 7 and 10). Figure 14 shows a change in phase of the PP reflection with azimuth for a P-wave incidence angle of 45° resulting from the azimuthal variation in critical angle (Sayers and Rickett, 1997).
automatically account for the change of angle with azimuth for rays propagating through the layer together with the tuning effect which occurs for layers with thickness of the order of the wavelength. The nredictinne of the thenry were ilhictrntvrl fnr the rnca . fit ...
DISCUSSION AND CONCLUSIONS
b
......... ».,.. »..................»... » ..... b ....,
fractures, surrounded by isotropic shale. The normal and shear compliances of the fractures were assumed to be equal. For a thin gas sand, the reflected pulse resembles the derivative of the incident pulse, as predicted by the theory. For an incidence angle of 30°, the PP reflection amplitude is greater for incidence plane parallel to the fractures than perpendicular to the fractures, whereas for an incidence angle of 45°, the reverse is true. The PSV reflection amplitude varies little with azimuth at small angles of incidence, but shows a significant variation with azimuth at larger incidence angles as the critical angle of the unfractured gas sand is approached. This occurs because the critical angle is increased for incidence plane perpendicular to the fractures. For thicker layers, distinct reflections from the top and bottom of the layer become evident, particularly for small angles of incidence. For a P-wave incidence angle of 30°, the variation of the PP reflected pulse shape with azimuth is relatively
The solution for the transmission and reflection coefficients at an interface between two general anisotropic half-spaces requires, in general, the solution of six equations in six unknowns, and the coefficients of the unknowns involve the solutions of a sixth-order polynomial for six vertical slownesses for each medium. However, a thin but potentially hydrocarbon rich vertically fractured layer may be modeled as a monoclinic layer with a plane of mirror symmetry parallel to the layer. Because of the mirror symmetry, the solution for the reflected and transmitted fields due to an incident plane wave is greatly simplified since the elements of the 6 x 6 coefficient matrix can be fully expressed in two 3 x 3 coefficient matrices. The formalism of Schoenberg and Protazio (1992) was designed to take advantage of this simplified situation. Using this formalism, the variation in reflection and transmission coefficients with incidence and azimuthal angle for a thin vertically fractured layer was
ZN/ZT=1, h=2m, f o =50 Hz ()=0.233)
ZNIZ t = 1, h=50m, f0 =50 Hz (n=5.818)
P-P Incidence angle = 45: Horizontal slowness = 0.2143 s/km
P—P Incidence angle = 30: Horizontal slowness = 0.1515 s/km -20 0 20 E H
40 60 80 -10 v V
V
^ V
JV
Azimuth(degrees)
w V
VV
0
10
20
30
40
50
60
70
80
90
100
80
90
1f
Azimuth (degrees) P-S Horizontal slowness = 0.2143 s/km
,V VV
P-S Horizontal slowness = 0.1515 s/km
E
E
E I-
F
10 0^
60
Azimuth (degrees) P-T Horizontal slowness = 0.1515 s/km
0
0
10
20
30
40
50
80
70
Azimuth (degrees) P-T Horizontal slowness = 0.2143 s/km
-9n 0
N E
10
N
I-
10
E
mE
E E
i0
10 Azimuth (degrees)
FIG. 13. Synthetic seismograms for a P-wave incidence angle of 30° calculated the type 1 gas-sand model assuming S T = 0.2 and Z N = Z T for a 50-m-thick layer.
80 1 -10
I
0
I
10
20
liii
30
40
1
1
1
50
1
60
1
1
70
1
1
80
1
1
90
i
11
Azimuth (degrees)
FtG. 14. Synthetic seismograms for a P-wave incidence angle of 45° calculated the type 1 gas-sand model assuming S T = 0.2 and Z N = Z T for a 2-m-thick layer.
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1170
Schoenberg et al.
P-P Incidence angle = 45 : Horizontal slowness = 0.2143 s/km
E 20
E a E i=
40 60
-10
0
10
20
30
40 50 60 Azimuth (degrees)
70
80
90
100
P-S Horizontal slowness = 0.2143 s/km
-20 0
0
N
E 20
E a E H
F
40 60
^
r
r
^
^ •.r .r Azimuth (degrees)
r
:r
•^
80
i
V
^^ VV Azimuth (degrees) VV y P-S Horizontal slowness = 0.2143 s/km
V
^ v
VV
VV
VV
JV
VV
-10
v
P-T Horizontal slowness = 0.2143 s/km
vV
Azimuth (degrees)
VV
V
VV
P-T Horizontal slowness = 0.2143 s/km
-20
E
• ZN/Z1.=1, h=50m, fo =50 Hz (52=5.818)
ZN/ZT=1, h=10m, f 0 =50Hz (0=1.164) P-P Incidence angle = 45: Horizontal slowness = 0.2143 s/km
0
0
?0
E 20
a E
40 60
to Azimuth (degrees)
FIG. 15. Synthetic seismograms for a P-wave incidence angle of 45° calculated the type 1 gas-sand model assuming S T = 0.2 and Z N = Z T for a 10-m-thick layer. small for 0 - (wh/ 16bv ) < 1, but becomes significant at larger values of Q. This results from propagation effects in the layer which are automatically included in the present approach. The PSH reflected pulse varies strongly with azimuth and vanishes for P-wave incidence plane parallel or perpendicular to the fractures. Finally, it should be noted that distortions in the AVO signature may arise due to propagation phenomena in the overburden. For example, anisotropic focussing effects may significantly change the reflection response, even if the anisotropy is mild (Tsvankin, 1995). If the fractures are not confined to the reservoir layer, these effects will be azimuthally anisotropic. Horne and MacBeth (1997) reported amplitude variations of directly transmitted P-waves as a function of azimuth from walkaround vertical seismic profiles. Systematic variations were found which were consistent with the anisotropy directions obtained from shear-wave experiments. Thus an observed variation in reflection amplitude with azimuth may result from a combination of azimuthally dependent reflection and transmission effects. REFERENCES Chang, C. H., and Gardner, G. H. E, 1993, Effects of vertically aligned fractures on reflection amplitudes: An amplitude-versusoffset study: 63rd Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 769-771.
V
V
V
w
JV
Azimuth (degrees)
Vv
V
4V
w
FIG. 16. Synthetic seismograms for a P-wave incidence angle of 45° calculated the type 1 gas-sand model assuming S T = 0.2 and Z N = Z T for a 50-m-thick layer.
Fryer, G. J., and Frazer, L. N., 1984, Seismic waves in stratified anisotropic media: Geophys. J. Internat., 78, 691-710. Harris, F. J., 1978, On the use of windows for harmonic analysis with the discrete Fourier transform: Proc. IEEE, 66, 51-83. Horne, S., and MacBeth, C., 1997, AVA observations in walkaround VSPs: 67th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 290-293. Koefoed, 0., and de Voogd, N., 1980, The linear properties of thin layers, with application to synthetic seismograms over coal seams: Geophysics, 38,1254-1268. Lefeuvre, F, 1994, Fracture related anisotropy detection and analysis: "and if the P-waves were enough?:" 64th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 942-945. Lefeuvre, F., and Desegaulx, P., 1993, Azimuthal anisotropy analysis with P-waves—The AVO-AVAZ approach: 55th Mtg., Eur. Assoc. Expl. Geophys., Expanded Abstracts, CO31. Lynn, H. B., Bates, C. R., Layman, M., and Jones, M., 1995a, Natural fracture characterization using P-wave reflection seismic data, VSP, borehole imaging logs, and the in-situ stress field determination: Rocky Mountain Regional Symp., SPE 29595. Lynn, H. B., Simon, K. M., Layman, M., Schneider, R., Bates, C. R., and Jones, M., 1995b, Use of anisotropy in P-wave and S-wave data for fracture characterization in a naturally fractured gas reservoir: The Leading Edge, 14, 887-893. Mallick, S., and Frazer, L. N., 1991, Reflection/transmission coefficients and azimuthal anisotropy in marine seismic studies: Geophys. J. Internat., 105, 241-252. Nelson, R. A., 1985, Geologic analysis of naturally fractured reservoirs: Gulf Publ. Co. Nichols, D., Muir, F, and Schoenberg, M., 1989, Elastic properties of rocks with multiple sets of fractures: 63rd Ann. Internal. Mtg., Soc. Expl. Geophys., Extended Abstracts, 471-474.
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Seismic Reflectivity from Fractured Reservoirs
Pelissier, M. A., Thomas-Betts, A., and Vestergaard, P. D., 1991, Azimuthal variations in scattering amplitudes induced by transverse isotropy: Geophysics, 56,1584-1595. Reiss, L. H., 1980, The reservoir engineering aspects of fractured formations: Editions Technip. Ruger, A., 1997, P-wave reflection coefficients for transversely isotropic media with vertical and horizontal axis of symmetry: Geophysics, 62, 713-722. 1998, Variation of P-wave reflectivity with offset and azimuth in anisotropic media: Geophysics, 63, 935-947. Ruger, A., and Tsvankin, I., 1995, Azimuthal variation of AVO response for fractured reservoirs: 65th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1103-1106.
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1997, Using AVO for fracture detection: Analytic basis and practical solutions: The Leading Edge, 16, 1429-1434. Rutherford, S. R., and Williams, R. H., 1989, Amplitude-versus-offset variations in gas sands: Geophysics, 54, 680-688. Sayers, C. M., and Rickett, J. E., 1997, Azimuthal variation in AVO response for fractured gas sands: Geophys. Prosp., 45,165-182. Schoenberg, M., and Protazio, J., 1992, 'Zoeppritz' rationalized and generalized to anisotropy: J. Seis. Expl., 1, 125-144. Schoenberg, M., and Sayers, C. M., 1995, Seismic anisotropy of fractured rock: Geophysics, 60, 204-211. Tsvankin, I., 1995, Body-wave radiation patterns and AVO in transversely isotropic media: Geophysics, 60,1409-1425. Widess, M. B., 1973, How thin is a thin bed?: Geophysics, 38,1176-1180.
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