Plane-wave reflection coefficients for gas sands at nonnormal angles of incidence. CVJV,) - 2 o = Z[(V,/v,)Z - 1)

Transcription

1 GEOPHYSICS, VOL. 49, NO. IO (OCTOBER 1984); P , 20 FIGS., 1 TABLE Plane-wave reflection coefficients for gas sands at nonnormal angles of incidence W. J. Ostrander* ABSTRACT The P-wave reflection coefficient at an interface separating two media is known to vary with angle of incidence. The manner in which it varies is strongly affected by the relative values of Poisson s ratio in the two media. For moderate angles of incidence, the relative change in reflection coefficient is particularly significant when Poisson s ratio differs greatly between the two media. Theory and laboratory measurements indicate that high-porosity gas sands tend to exhibit abnormally low Poisson s ratios. Embedding these low-velocity gas sands into sediments having normal Poisson s ratios should result in an increase in reflected P-wave energy with angle of incidence. This phenomenon has been observed on conventional seismic data recorded over known gas sands. INTRODUCTION During the past decade, the use of bright spot type analysis in petroleum exploration has become increasingly common. Oil companies, both large and small, are making use of the fact that high-intensity seismic reflections may be indicators of hydrocarbon accumulations, particularly gas. Bright spot exploration has significantly increased the recent success ratio for wildcat gas wells. Nonetheless, problems still do exist. Many seismic amplitude anomalies are not caused by gas accumulations, or they are caused by gas accumulations which are subcommercial. The latter problem is difficult to resolve. However, amplitude anomalies caused by nongaseous, abnormally high- or low-velocity layers may have distinguishing characteristics. This paper proposes a method which potentially may distinguish between gas-related amplitude anomalies and nongas related anomalies. Notable observations contained herein are (I) the somewhat surprising effects of Poisson s ratio on P-wave reflection coefficients and (2) the existence of these effects in seismic amplitude anomalies related to gas accumulations. Poisson s ratio BACKGROUND Poisson s ratio, sometimes denoted by the Greek letter small sigma (o), is a somewhat neglected elastic constant. It is related to other elastic constants by a simple set of equations. In particular, Poisson s ratio for an isotropic elastic material is simply related to the P-wave (V,) and S-wave (V,) velocities of the material by CVJV,) - 2 o = Z[(V,/v,)Z - 1) This equation indicates that Poisson s ratios may be determined dynamically using field or laboratory measurements of both VP and I$ Poisson s ratio also has a physical definition. If one takes a cylindrical rod of an isotropic elastic material and applies a small axial compressional force to the ends, the rod will change shape. The length of the rod will decrease slightly, while the radius of the rod will increase slightly. Poisson s ratio is defined as the ratio of the relative change in radius to the relative change in length. Common isotropic materials have Poisson s ratios between 0.0 and 0.5. Incompressible materials such as liquids will have Poisson s ratios of 0.5, while spongy materials might have ratios closer to zero. Reflection coefficients In 1940, a classic article was published by Muskat and Meres showing the variations in plane-wave reflection and transmission coefficients as a function of angle of incidence. Since then, several additional articles on the subject have appeared in the literature, including those by Koefoed (1955, 1962) and Tooley et al. (1965). Using the simplified Zoeppritz equations given by Koefoed (1962), one can show that four independent variables exist at a single reflecting/refracting interface between two isotropic media: (1) P-wave velocity ratio between the two bounding media; (2) density ratio between the two bounding media; (3) Poisson s ratio in the upper medium; and (4) Poisson s ratio (1) Presented at 52nd Annual International SEC, Meeting in Dallas, Texas, on October 21, Manuscript received by the Editor February 1% 1983; revised manuscript received April 30, *Chevron U.S.A., Inc., 2003 Diamond Boulevard, Concord, CA d m 1984 Society of Exploration Geophysicists. All rights reserved. 1637

2 1636 Ostrander 0.4 t q=0,=0.3 u;, _-- qt0.4 ctpo.1 q= 0.3 s,= t 1, ;;. 0 F v Y -0.1 k c* VPZZ R.!L DR VPI P, _ r Rz09D&0 9./# FIG. I. Plot of P-wave reflection coefficient versus angle of incidence for constant Poisson s ratios of 0.2 and 0.3. in the lower medium. These four quantities govern plane-wave reflection and transmission at a seismic interface. Since Muskat and Meres (1940) had very little information on values of Poisson s ratios for sedimentary rocks, they used a constant value of 0.25 in all their calculations, i.e., Poisson s ratio was the same for both media. Results similar to theirs are shown in Figure 1 for various velocity and density ratios and constant Poisson s ratios of 0.2 and 0.3. One would conclude from these results that angle of incidence has only minor effects on P-wave reflection coefficients over propagation angles commonly used in reflection seismology. This is a basic principle upon which conventional common-depth-point (CDP) reflection seismology relies. The work of Koefoed (1955) is of particular interest since his calculations involved a change in Poisson s ratio across the reflecting interface. He found that by having substantially different Poisson s ratios for the two bounding media, large changes in P-wave reflection coefl$cients versus angle of incidence could result. Koifoed showed that under certain circumstances? reflection coefficients could increase substantially with increasing angle of incidence. This increase occurs well within the critical angle where high-amplitude, wide-angle reflections are known to occur. Figures 2 and 3 illustrate an extension of Koefoed s initial computations. Figure 2 shows P-wave reflection coefficients from an interface, with the incident medium having a higher Poisson s ratio than the underlying medium. The solid curves represent a contrast in Poisson s ratio of 0.4 to 0.1, while the dashed curves represent a contrast of 0.3 to 0.1. One may AH kl t Ly ; 0. 1 : 0. o- 0 F v!y -0. k! oz ,l L vpz, R pz, DR VPI PI FIG. 2. Plot of P-wave reflection coefficient versus angle of incidence for a reduction in Poisson s ratios across an interface. FIG. 3. Plot of P-wave reflection coefficient versus angle of incidence for an increase in Poisson s ratios across an interface.

3 Reflection Coefficients for Gas Sands 1639 Table 1. List of measured Poisson s ratios for various sedimentary rock types* POD10 ET AL. (1968) GREEN AIVEA SHALE HAMILTON (1976) SHALLOW MARINE SEUIMENTS GREGORY (1976) CONSOLlllATEO SEDIMENTS BRINE SATURATED GAS SATURATED DOMENICO (1976) SYNTHETIC SANDSTONE BRINE SATURATED 0.41 GAS SATURATED 0.10 DOMENICO (1977) OTTAWA SANDSTONE BRINE SATURATE GAS SATURATED 0.10 conclude from these curves that if Poisson s ratio decreases going into the underlying medium, the reflection coefficient decreases algebraically with increasing angle of incidence. This means positive reflection coefficients may reverse polarity and negative reflection coefficients increase in magnitude (absolute value) with increasing angle of incidence. Figure 3 shows the opposite situation to that shown in Figure 2. Here Poisson s ratio increases going from the incident medium into the underlying medium. In this case, the reflection coefficients increase algebraically with increasing angle of incidence. Negative reflection coefficients may reverse polarity, and positive reflection coefficients increase in magnitude with increasing angle of incidence. The foregoing three illustrations point to a strong need for more information on Poisson s ratio for the various rock types encountered in seismic exploration. This is particularly important when one considers the long offsets commonly in use today and the resulting large angles of incidence. It will become evident later that this phenomenon has an important effect on bright spot analysis. For additional computations of reflection and transmission coefficients, the reader should refer to Koefoed (i962) and Tooiey et ai. (i-965j. MEASUREMENTS OF POISSON S RATIO In the Handbook of Physical Constants, Birch (1942) lists Poisson s ratios for various materials, including many rock types. However, little significance can be placed on these values because of the methods and environments of measurement. As will become obvious later, any air or gas in cracks or pore spaces can severely alter measurement of Poisson s ratio. Until recently, other published measurements for sedimentary rocks were quite limited. Many comprehensive measurements of Poisson s ratios for sedimentary rocks were reported in the literature during the 1970s. Hamilton (1976) presented a review of measurements made for shallow marine sediments including both sands and shales. His results showed that shallow, unconsolidated marine sediments to depths of ft had Poisson s ratios between 0.45 and Gregory (1976) gave results including fluid saturation effects for many consolidated sedimentary rocks. His samples included sandstones, limestones, and chalks ranging in porosity from 4 to 41 percent. The work of Domenico (1976, 1977) is of special interest because it applies to many of our shallow gas fields which have related seismic amplitude anomalies. In both a synthetic high-porosity glass bead and a highporosity Ottawa sandstone mixture, Domenico found marked changes in Poisson s ratios between brine and gas saturations. In these unconsolidated 38 percent porosity specimens, the replacement of brine with gas reduced Poisson s ratio from 0.4 to 0.1. A summary of the foregoing results is shown in Table 1. Several conclusions can be drawn from measurements of Poisson s ratios for sedimentary rocks. First, unconsolidated, shallow, brine-saturated sediments tend to have very high Poisson s ratios of 0.40 and greater. Second, Poisson s ratios tend to decrease as porosity decreases and sediments become more consolidated. Third, high-porosity brine-saturated sandstones tend to have high Poisson s ratios of 0.30 to And fourth, gas-saturated high-porosity sandstones tend to have abnormally low Poisson s ratios on the order of The above conclusions result from the fact that the shear modulus u of a rock does not change when the fluid saturant is changed. However, the bulk modulus k does change significantly (Gassmann, 1951). The bulk modulus of a fluid-saturated rock is a function of the bulk moduli of the fluid, the grains, and the dry rock framework. The bulk modulus of a brine-saturated rock is greater than that of gas-saturated rock because brine is significantly stiffer than gas. This results in the P-wave velocity (VP) of the brine-saturated rock being considerably higher than that of a gas-saturated rock from equation (2). The S-wave velocity (V,) defined in equation (3) is only affected by a small change in the density p. Since density is reduced by a gas saturant, the S-wave velocity is slightly increased with gas saturation. Equations (2) and (3) show the relationships among these parameters. and 0.P, v,=p 1.2 Analysis of Gassmann s equation shows that the weaker the framework modulus, the greater the differences between brine and gas saturations. This explains the dramatic differences observed in poorly consolidated rocks such as those analyzed by Domenico. Depth of burial and differential pressure also influence the elastic behavior of rocks. Differential pressure is the difference between the overburden pressure and the fluid pressure and is generally a monotonic function of depth. As differential pressure increases, the bulk and shear moduli of a rock increase, resulting in greater P- and S-wave velocities. Sediment consolidation and increased differential pressures tend to decrease fluid saturation effects with increased depth of burial. Theoretical results also support the large reduction in Poisson s ratio as gas replaces brine as the saturant in high-porosity sandstones. Using the equations of Gassmann (1951), one can compute theoretical P-wave and S-wave velocities as a function

4 1640 Ostrander 7K % GAS SATURATION (a) 0.0 J ? a GAS SATURATION Cc) 4.5K ? a GAS % GAS SATURATION FIG. 4. Plots of (a) P-wave velocity. (b) S-wave velocity, (c) Poisson s ratio, and (d) V,/V, ratio as a function of gas saturation. of percent gas saturation. Poisson s ratios can then be computed using equation (1). Theoretical results for a 35 percent porosity sandstone buried at ft are shown in Figures 4a. 4b, 4c, and 4d. In Figure 4c, one sees that the major change in Poisson s ratio occurs with less than 10 percent gas saturation. From 10 to 100 percent gas saturation, Poisson s ratio changes very little around an average value of These characteristic gas saturation curves have been supported by laboratory measurements (Domenico, 1976,1977; Gregory, 1976). GAS SAND MODEL Using the foregoing review of physical parameters, one can now devise a hypothetical gas sand model. This model can be used to analyze plane-wave reflection coefficients as a function of angle of incidence. Calculations can be made for the reflections originating from both the top and base of the gas sand. Figure 5 shows a three-layer gas sand model with parameters which might be typical for a shallow, young geologic section. Here, a gas sand with a Poisson s ratio of 0.1 is embedded in shale having a Poisson s ratio of 0.4. There is a 20 percent velocity reduction going into the sand, from k/s to ft;s, and a 10 percent density reduction from 2.40 g/cm3 to 2.16 g/cm3. These parameters result in normal-incidence reflection (a) -_---_-2_ SHALE -.-z-1 $3 =I 0, r-_rt_z_7z_ - - -_ - _ & , -0.4 FIG. 5. Three-layer hypothetical gas sand model.

5 Reflection Coefficients for Gas Sands t u, k DC c. NO GAS IN SAND a; = NO GAS Frc;. 6. Plot of P-wave reflection coefficients versus angle of incidence for three-layer gas sand model. coefficients of and for the top and base of the gas sand, respectively. Changes in plane-wave reflection coefficient as a function of angle of incidence for several cases are shown in Figure 6. The two solid curves are those reflection coefficients resulting from the gas sand model parameters shown in Figure 5. The effect of transmission and refraction on the base of sand reflection have been taken into account. The horizontal axis is the angle of incidence referenced to the top of the sand. Because of refraction, a 40-degree incident angle at the top of the sand represents 01- RAY PATH ARC CENTERS - 0 T only 31 degrees incident angle at the base of the sand. The top of sand reflection coefficient changes from about to over 40 degrees while the base of sand reflection coefficient changes from about to Thus, the amplitude of the seismic waveform resulting from this complex reflection would increase approximately 70 percent over 40 degrees. The dotted curves in Figure 6 indicate what the reflection coefficients would be if Poisson s ratio in the sand were changed to 0.4. This would simulate the case of a low-velocity brinesaturated young sandstone embedded in shale. In this case, one sees only a slight decrease in the magnitude of the reflection coefficients as the angle of incidence increases. DATA ANALYSIS The obvious question at this point is how one can best observe and analyze changes in reflection coefficient with angle of incidence on today s conventionally recorded reflection seismic data. The answer lies in analyzing amplitudes on CDPgathered traces prior to stacking. In this way, one can observe changes in reflection amplitude versus shot-to-group offset. As shot-to-group offset increases, the angle of incidence increases monotonically. Angles of incidence There are several ways to estimate angles of incidence from the depth to a reflector and the shot-to-group offset. The first and most simple is the straight-ray approach where the angle of incidence Bi is given by where X is the shot-to-group ottset and Z is the reflector depth. If velocity increases with depth, which is most common, the angles computed from equation (4) will always be too small. In this situation, a better approach for estimating angles of incidence is illustrated in Figure 7. If the section interval velocity can be approximated in the form V, = V, + KZ, then all raypaths are arcs of circles whose centers are V,/K above the SUMMING IO INPUT PER OUTPUT.S.COP S. STACKING CHART DIAGRAM / OUTPUT + m _ / BOXES TRACES TRACE SHOTS FK. 7. Geometry for estimating angles of incidence for a velocity function of the form V, = V, + KZ. FE. 8. Trace-summing technique to increase S/N ratios.

6 1642 Oslrander q v

7 Reflection Coefficients for Gas Sands 1643 ground surface. Using the resulting geometry, one can derive the following relationship: 0, = tan ZX + V X/K Z2 + 2V,Z/K - X2/4 >. (5) An example calculation for K = , at a depth (Z) of ft, and an offset (X) of ft, gives an angle of incidence &Ii) of 34 degrees. Thus, one sees that angles of 30 degrees or more are not uncommon in today s CDP recording and are in many cases unmuted during CDP stacking. Trace summing-s/n improvement In viewing single CDP-gathered traces for amplitude variations, one major drawback occurs: poor signal-to-noise (S/N) ratios. As a means of signal enhancement, trace summing can prove most worthwhile. A method of partial trace summing is illustrated in Figure 8. Shown in the tigure is a stacking chart diagram on the right with an enlargement of the same in the upper left. The recording geometry is for 4%trace, single end, 24-fold CDP coverage with a near ottset of gl0 ft and a far offset of ft. Here CDP traces lie on a vertical line, common-offset traces on a horizontal line, and common-shot profile traces on the diagonals as shown. In the enlargement, individual traces are shown as small circles. To form a partial sum trace, all traces fall within boxes which are 5 CDPs by 4 offsets in dimension and which are summed together forming a single output trace. This limited summing will produce a lofold sum trace, and thus improve the S/N ratio by a factor of about 3. Repeating the procedure for groups of 4 offsets will produce 12 traces, all with improved S/N ratios. Displaying these 12 partially summed traces in increasing average shot-togroup offset gives a desirable product for analyzing amplitude information. Variations on this type of limited offset summing are easily implemented for different recording geometries. In the examples which follow, the reader will find that some CDP gathers are displayed as individual traces while others have been partially summed. The advantages of summing will become obvious. EXAMPLES Several examples will now be presented which illustrate apparent changes in reflection amplitude versus angle of incidence. All of the examples are in areas of well control, so the origin of the bright spot or amplitude anomaly is known. In two cases, the anomalies are caused by gas. while in the third, the seismic amplitude anomaly is caused by a high-velocity layer of basalt. Prior to drilling, all of these seismic anomalies were thought to be caused by gas-saturated sediments. The illustrations presented for each example are (1) a conventionally stacked section showing the given amplitude anomaly; and (2) CDP gathers at locations indicated by vertical dashed lines and the letters A, B, and C on the stacked sections. The processing flow, prior to the displays shown, employed standard techniques. These included spherical divergence correction, exponential gain, minimum phase-spiking deconvolution, statics, velocity analysis, normal moveout (NMO) removal, time-invariant band-pass filtering, and single long-gated trace equalization. No wavelet processing was done on these data, so no implications as to reflector polarity can be made. -ll-~,,. Line SV-1 SINGLE-FOLD CDP GATHERS SP 81 SP 80.., I I,,,.,;,!i.: pxij,yqj::~,;,i. ::: :: :;.i,... ~,!,,,,, ::I:.., 0 :.. I! ),,,!.,,:.,:::.,!I - 1.5,,:,j;r~~;:;.:::.iii: I::::, :: :::::,;: t.:;:,, r,,,ij:;~:j:,,,&,,,i...i*..*!,j..,,,,,,,,i :,,!:<!#I,....&...:.::.....L..I. ;:L...!:j::. I :,;:,;.,,I.,.*.,.,.;;;.; :I. ;;:,,.:,: ;::;..i!!:;::,,,;:,: :,,,, L):1:;.~,.,. ),,;,,;,, :, I,::.,!, I I:;),):, *,I.,,I.,), ~,;).I, :,,,,,,...,,,~:~..,..I *- 2.0,,,.,,.,,..,,)),... #..,./:,,I,,,,.:,,,,,.,.I)., I...,...L...~~..,l,,.,l..P*.~,I...I.. :.::.),I,. I...,, ~.~I...~.., rr....., lo-fold SUMMED CDP GATHERS FIG. 10. CDP gathers for location A on line SV-1. Shown in Figure 9 is a 24-fold CDP stacked seismic line over a large gas field in the Sacramento Valley. The sand reservoir occurs at a depth of about ft which corresponds to a seismic amplitude anomaly at about 1.75 s. The reservoir is a Cretaceous deep-sea fan deposit having a maximum net pay of 95 ft. The trap is both structural and stratigraphic with the reservoir being offset along a fault at about SP 95. The downthrown portion of the reservoir on the left is trapped against the fault, while the reservoir pinches out in the upthrown block at about SP 75. The velocity and density within the gas sand are substantially lower than the encasing shales, giving rise to strong seismic reflections at the top and base of the gas sand. A flat fluid contact reflection may be present at 1.8 s between SP 115 and SP 135. The discovery well is located at about SP 86 with the reservoir limits extending from about SP 75 to SP 130. CDP gathers from three locations, A, B, and C, are shown in Figures 10, 11, and 12, respectively. Both single-fold and lo-fold summed gathers are shown for locations A and B, while only the summed gathers are shown for location C. Shot-to-group of&et for all gathers increases to the left. These distances change on the summed gathers because the summing is done over four offsets. At the objective sand, the near-offset corresponds to about 5 degrees angle of incidence while the far-offset corresponds to about 35 degrees. A strong amplitude increase with increasing offset is apparent in the gathers at locations A and B shown in Figures 10 and 11. The IO-fold summing obviously improves S/N ratios, and an amplitude increase by a factor of about three is indicated from the near offset to far offset. The gathers at location C, shown in Figure 12, show no indication of amplitude increase with offset and in fact show a decrease. This possibly indicates

8 1644 Ostrander SINGLE-FOLD CDP GATHERS lo-fold SUMMED CDP GATHERS SP 101 SP 142 SP 141 SP 140 lo-fold SUMMED CDP GATHERS SP 102 SP 101 I,;,,,;;;-.6- I/ lt t111:; tttttttttttt Bbcbtttt Prr)Pttfffft,, I 1 b Wlll~ij tttttttt tt tt mbbuobiii ~~~rlrfttffr,/i,tbi/ - I.9 T9S FIG. 11. CDP gathers for location B on line SV-1 CDP GATHER LOCATIONS D A FIG. 12. CDP gathers for location C on line SV-1. an absence of gas in the vicinity of location C. This possibility is also supported by the presence of a gas-water contact in a well which would structurally project in at about SP 120. In this example and those which follow, no S-wave velocity data were available to model variations in reflection coefficient in shot-to-group offset. The P-wave sonic velocities within this gas reservoir may also be subject to error. The behavior of the observed seismic amplitude in shot-to-group offset is as expected from theory and laboratory measurements. However the magnitude of the change of amplitude may be influenced by cl B FIG. 13. Stacked seismic section for line GM- 1.

9 Reflection Coefficients for Gas Sands FOLD SUMMED CDP GATHERS 8-FOLD SUMMED CDP GATHERS FIG. 14. CDP gathers for location A on line GM-l CDP GATHER LOCATION u A FE. 15. CDP gathers for location B on line GM-l t I-c--l FIG. 16. Stacked seismic section for line FB-1.

10 1646 Ostrander lo-fold SUMMED CDP GATHERS FIG. 17. CDP gathers for location A on line FB-1. factors to be discussed later. Because one has great difficulty in separating out true reflection amplitudes, the interpreter typically must rely on relative changes, concentrating on anomalous behavior of the amplitude. Line GM-1 Figure 13 shows two gas-related seismic-amplitude anomalies on line GM-1 located in the Gulf of Mexico. The first of these anomalies is located on the left half of the seismic section at about 0.65 s. The second, deeper anomaly is toward the middle of the seismic section at about 1.35 s. Summed CDP gathers are displayed in Figure 14 for location A on the shallow anomaly and in Figure 15 for location B on the deeper anomaly. In both anomalies, it is quite apparent that reflection amphtude tends to increase with increasing offset. In the case of the shallower anomaly at location A, the effect of array attenuation and NM0 stretch on the fifth offset trace is obvious. Line FB-1 Shown in Figure 16 is a 24-fold CDP-stacked seismic line recorded in a virgin basin in Nevada. Several years, ago, a well was drilled on this line at SP 127 (location A) to a depth below 2.0 s. A seismic amplitude anomaly is indicated on the stacked seismic data at this location and at a time of about 1.6 s. Upon drilling, the amplitude anomaly was found to originate from a high-velocity basaltic interval of about 160 ft in thickness. As has happened elsewhere, the apparent bright spot is not due to the presence of gas in the sediments. The CDP gathers at the well location are shown in Figure 17. Here, there is a strong indication of a decrease in reflection amplitude with increasing offset or angle of incidence. This finding is consistent with a relatively uniform Poisson s ratio in the geologic section. Basalt is not expected to have an anomalous Poisson s ratio. FAR OFFSET Frc;. 18. Relationship between array attenuation, apparent reflector dip, and normal moveout. (1) Reflection coefficient (2) Array attenuation (3) Event tuning (4) Noise (5) Spherical spreading (6) Emergence angles (7) Reflector curvature (8) Spherical wavefronts (9) Transmission coefficients (10) Instrumentation/processing (11) Inelastic attenuation The first of these, the reflection coefficient, is the factor which one would like to observe. In actuality, one can only observe relative changes in reflection coefficient versus offset. If no other factors existed, one could simply observe the CDP gathers with a spherical divergence correction applied. However, because of the other offset-related amplitude factors listed above, simple observation of the reflection coefficient is not always feasible. Considered below are some of the other factors in more detail. Array attenuation Array attenuation arises because one generally does not have a point source and a point receiver. As the dip of the apparent reflector becomes large, geophone arrays tend to reduce amplitudes of reflections. The same is true for shot arrays. This effect is greatest for shallow reflectors at long offsets and diminishes with greater depths of reflectors and shorter offsets. As illustrated in Figure 18, array attenuation is a result of NMO. For a flat-lying reflector, one can see that the apparent dip of the reflection across a geophone group array comes purely from NMO. This dip or slope is simply the derivative of the NM0 with respect to offset, dt,/dx. Using the NM0 equation one finds that OFFSET AMPLITUDE ANALYSIS dt At this point, the reader may wonder what type of amplitude balancing is desirable in order to analyze offset-dependent amplitude changes. One must then look at some of the major factors which affect the recorded amplitude of a reflection as a function of offset. Some of these factors are listed below. 7; = (T; + X2/Vf,,,s)1'2, (6) L = X(7 ;!& + XzVj,,,,)- *. dx In equations (6) and (7) 7; is the two-way event arrival time at shot-to-group offset X, To is the zero-offset two-way time and I$,,, is the root-mean-square (rms) velocity to the reflector. Using the above relationship and information about the (7)

11 Reflection Coefficients for Gas Sands 1647 VRMS OFFSET [FEET] 2K 4K 6K 8K 8800,/s s EFFECTIVE ARRAY LENGTH q 135 FT. FREQ.=28 cps FIG. 19. Plot of array attenuation versus two-way traveltime and shot-to-group offset. recording geometry, one can obtain an array attenuation plot similar to the one shown in Figure 19. The recording parameters corresponding to this figure are for an effective shot and group array length of 135 ft. The plot is for a frequency of 28 Hz and for the velocity function shown along the vertical axis. The contours are in decibels (db) and include the effects of both shot and group arrays. An overlay plot of this type is convenient in analyzing amplitudes on CDP gathers. Event tuning Event tuning is caused by differential traveltimes between two or more closely spaced reflections. This effect is illustrated in Figure 20 for a single gas sand with two reflectors, the top and base of the gas sand. The relationship between the two reflection arrival times and offset is shown by the two curves in the lower part of Figure 20. Because NM0 naturally decreases with increased record time the time difference between the top and base sand reflections will decrease with increasing offset, i.e., AT, < AT,. As with array attenuation, this effect is best analyzed through the NM0 equation given in (6). For small-event separations involved in tuning phenomenon, AT, is small and the ratio AT,/ATo becomes important. As this ratio deviates from one, differential tuning effects occur with changes in offset. Letting A7;/AT, approximate the derivative dt,/dt,, one has from (6) 2 = (1 + X2/T;V/,,,,)- 1:2, 0 Evaluation of equation (8), using realistic parameters for X, To, GAS I SAND ATo= T,- T* \ A& < ATo FIG. 20. Effect of differential traveltime on event tuning for thin gas sands. and Vrms, shows the derivative to have a minimum value of about 0.70 for extreme cases. This implies that interval time thicknesses for a thin interval such as a gas sand can decrease from near- to far-offset by about 30 percent. Differential tuning effects will therefore occur for two closely spaced reflections from near- to far-offset. Using the appropriate response curves, one can show that this differential tuning can cause amplitude changes of about 30 percent in the extreme cases. This effect can be either an increase or decrease in amplitude with offset depending upon bed thickness. For thin beds, such as many gas sands less than 50 ft thick, this effect results in a decrease in recorded amplitude with increasing offset. Noise Because noise can be strongly offset-dependent, its effect on analysis of offset amplitude can be significant. For example, vibrator data typically have noisy inside traces which after summing might be interpreted as lower reflection amplitude on the short-offset traces. Noise on the far-offset traces might have the opposite effect. This implies that any amplitude balancing which is data-dependent should be done on data which have the highest S/N ratio. Balancing of the noise energy will generally not balance low-energy signal.

12 1646 Ostrander Other factors Spherical spreading gives rise to a very predictable decay in seismic amplitude with time and offset. Newman (1973) presented correction factors which account for this amplitude decay if one has detailed information on subsurface velocities. However, in typical seismic data processing, a simple zero-offset correction is applied which may not correct nonzero offset traces properly. Emergence angles may become large for nonzero offset seismic traces. Nonvertical emergence angles will cause attentuation in seismic amplitude at a vertically responding geophone. One can show from Newman s work (1973) that the emergence angle is directly coupled to spherical spreading. For anomalously low or high surface velocities, these two factors can be quite large with offset. However, they are of opposite sign and somewhat cancel each other. Reflector curvature and nonplanar wavefronts at shallow depths can have effects on reflection coefficients as discussed by Krail and Brysk (1983). Transmission coefficients at nonvertical incidence angles are coupled with mode conversions and short path multiples and become difficult to analyze in general terms. Instrumentation, processing, and inelastic attenuation are also difficult to analyze in the shot-to-group offset domain. Surfaceconsistent processing can help reduce many of these distortions and also help with many noise problems. Amplitude balancing In all the examples presented here, the seismic data were simply trace-equalized over very long time gates. In using this method, relative changes in amplitude with offset are of prin- cipal significance. With a few exceptions, this method appears to work fairly well. Trace-equalizing data with a strong offsetdependent noise component has not been entirely satisfactory. Better techniques involving surface-consistent amplitude adjustment are currently under investigation and will undoubtedly improve our understanding of the phenomenon. CONCLUSIONS Two basic conclusions can be drawn from this paper: (1) Poisson s ratio has a strong influence on changes in reflection coefficient as a function of angle of incidence; and (2) analysis of seismic reflection amplitude versus shot-to-group offset can in many cases distinguish between gas-related amplitude anomalies and other types of amplitude anomalies. The methods of analysis presented here have proven to be useful in many of the world s gas provinces. However, the methods are not foolproof and experience has shown them to fail on occasion. Other factors which affect observed reflection amplitudes versus offset need to be considered. Amplitude balancing during processing is quite important. Additional information on Poisson s ratios for other rock types needs attention as well as the effects of depth of burial and sediment consoli- dation on Poisson s ratio. time and the drilling of additional wells will test the full utility of such methods in predicting the presence or absence of gas in the geologic section. ACKNOWLEDGMENTS The author wishes to thank Chevron, U.S.A. Inc. for permission to publish this paper. The author is also indebted to J. I. Foster, H. G. Lang, R. A. Seltzer, and D. D. Thompson for their advice, encouragement, and contributions to this project. REFERENCES Birch, F., ed, 1942, Handbook of physical.. constants: Geol. Sot. of Am., Special Paper 36. Domenico. S. N Effect of brine-gas mixture on velocity in an unconsolidated sand reservoir: Geophysics, 41,882%X94. ~ 1977, Elastic properties of unconsolidated porous sand reservoirs: Geophysics, 42, Gassmann, F , Elasticity of porous media: Vier. der Natur. Gesellschaft in Zurich, Heft I. Gregory, A. R., Fluid saturation effects on dynamic elastic properties of sedimentary rocks: Geophysics, 41, Hamilton, E. L., 1976, Shear-wave velocity versus depth in marine sediments: a review: Geophysics, 41, Koefoed, 0.: On the effect of Poisson s ratios of rock strata on the reflection coefficients of plane waves: Geophys. Prosp., 3, ~ 1962, Reflection and transmission coefficients for plane longitudinal incident waves: Geophys. Prosp., 10, Krail, P. M., and Brysk, H., 1983, Reflection of spherical seismic waves in elastic lavered media: geophysics Muskat, M.,-and Meres, M. W.: 1940, Reflection and transmission coefficients for plane waves in elastic media: Geophysics, 5, Newman, P., 1973, Divergence effects in a layered earth: Geophysics, 38, Podio, A. L., Gregory. A. R.. and Gray, K. E., 1968, Dynamic properties of dry and water-saturated Green River Shale under stress: J. Sot. of Petr. Eng Tooley, R. D., Spencer, T. W., and Sagoci, H. F., 1965, Reflection and transmission of plane compressional waves: Geophysics, 30, SS2-570.