vs predictors revisited.pdf

R o c k p h y s i c s

288 The Leading Edge March 2014

SPECIAL SECTION: R o c k p h y s i c s

VS predictors revisited

AbstractVarious shear-velocity (VS) predictors differ from one

another, and this difference affects a synthetic AVO gather produced at a well or for a synthetic earth model. The differ-ence between gathers produced by using different predictors might be well within the noise level of field data and will not necessarily affect site-specific seismic-based hydrocarbon indicators.

IntroductionVS (shear-velocity) predictors are many, and so are

the choices the geoscientist faces when deriving VS from VP in the well where shear-wave data are poor or absent. As a result, we often encounter a multiple-choice situ-ation in which the 100% correct answer (the ground truth) is not available. To that end, our objective is to investigate by how much various predictors differ from one another and, most important, how that difference affects one of the ultimate goals of VS prediction — pro-ducing a synthetic AVO gather catalogue to serve as a field guide for interpreting the observed seismic anom-aly for rock properties and conditions.

Our modeling here concerns clastic rock, in which we assume that the only two mineral components pres-ent in the matrix are quartz and clay. For the properties of the pore fluids, we assume brine salinity of 150,000 ppm; oil API gravity 30; gas grav-ity 0.65; and gas-to-oil ratio (GOR) 160 (maximum GOR for these in-puts, according to Batzle and Wang, 1992). By using these inputs in the Batzle and Wang (1992) equations and by assuming that the pore pres-sure is 30 MPa and temperature is 75°C, we obtain the elastic proper-ties and densities listed in Table 1. The table also lists the properties of the mineral components of the rock matrix (from Mavko et al., 2009). Modeling will be conducted for sand with 95% quartz and 5% clay and for shale with 10% quartz and 90% clay.

Predictors in wet sand and shaleWe start with producing the

elastic properties of wet sand and shale using the soft-sand model (Mavko et al., 2009) in the poros-ity range of zero to 40%. We assume that the differential pressure is 30 MPa, critical porosity is 40%, and

JACK DVORKIN and GARY MAVKO, Stanford University

the coordination number (the average number of contacts per grain) is six. The resulting VS is plotted versus compressional velocity (VP) in Figure 1. In this figure, we also plot the re-spective Poisson’s ratio (v) versus VP.

Next, we apply various VS predictors to the wet-soft-sand VP and compute the respective VS and v. The first predictor is by Greenberg and Castagna (1992):

Component (g/c3) K(GPa) G(GPa) VP(km/s) VS (km/s)

Brine 1.094 3.245 0.000 1.722 0.000Oil 0.724 0.725 0.000 1.001 0.000Gas 0.205 0.073 0.000 0.597 0.00060% oil + 40% brine

0.872 1.051 0.000 1.098 0.000

80% gas + 20% brine

0.383 0.090 0.000 0.485 0.000

Quartz 2.650 36.60 45.00 6.038 4.121Clay 2.580 21.00 7.000 3.429 1.647

Table 1. Density ( ), bulk modulus (K), shear modulus (G), and P- and S-wave velocity of the pore fluid and mineral components used in modeling.

Figure 1. (left) VS versus VP and (right) Poisson’s ratio versus VP for (a) sand and (b) shale whose elastic properties were computed using the soft-sand model, as explained in the text. The legend in top left plot relates to all four plots. The mudrock line is marked in (b). GC indicates the Greenberg-Castagna VS predictor.

Dow

nloa

ded

10/2

1/15

to 1

24.1

95.4

.82.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

March 2014 The Leading Edge 289


(1)

where L is the number of pure-mineral lithologic constitu-ents; fi are the volume fractions of these constituents in the whole mineral phase; aij are empirical coefficients; Ni is the order of polynomial for constituent i; VP is the measured P-wave velocity; and VS is the predicted S-wave velocity. The velocity is in kilometers per second. The coefficients aij are given in Table 2. The results are plotted in Figure 1 as dot-ted curves.

The next predictor is by Vernik et al. (2002) for wet sand,

(2)

and for shale,

(3)

It is followed by the Krief et al. (1990) equation

(4)

where VP f is the velocity in the pore flu-id and VP s and VSs are the P- and S-wave velocities, respectively, in the mineral matrix.

We also use the Williams (1990) re-lations for wet sand,

(5)

and for shale,

(6)

where VP and VS are in kilometers per second, as they are in all the preceding equations.

To the shale curves (Figure 1b), we also add the famous mudrock equation by Castagna et al. (1985),

(7)

where, as before, velocity is in kilome-ters per second.

In wet sand, all the predictors pro-duce essentially identical results, barring Krief et al.’s (1990) equation, which pre-dicts Poisson’s ratio slightly smaller than predicted by the other relations. For wet

Figure 2. Same as Figure 1 but using the stiff-sand model.

Figure 3. Same as Figure 1a but for gas sand.

Lithology ai2 ai1 ai0

Sandstone 0 0.80416 –0.85588

Limestone –0.05508 1.01677 –1.03049

Dolomite 0 0.58321 –0.07775

Shale 0 0.76969 –0.86735

Table 2. Regression coefficients for the Greenberg-Castagna predictor. These coefficients are valid only if the velocity is in kilometers per second.

Dow

nloa

ded

10/2

1/15

to 1

24.1

95.4

.82.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/



shale, all predictors, including the mudrock equation, produce practically identical results.

Figure 2 shows the results of the same exercise but now using the stiff-sand model (Mavko et al., 2009) for both sand and shale. As before, Krief et al.’s (1990) equation produces Poisson’s ratio in wet sand smaller than all other predictors. In addition, unlike in the soft-sand case, the stiff-sand model-predicted Poisson’s ratio in wet sand is smaller than that predicted by the other equations (Greenberg and Castagna, Vernik, and Williams) and falls closer to that pre-dicted by Krief et al. (1990).

Predictors in gas sandTo compute the elastic proper-

ties of gas sand, we first use the soft-sand (or stiff-sand) model to compute the elastic properties of the sand at 100% brine saturation, including the wet-rock P-wave velocity VPwet and its bulk density bwet. Next, we apply a VS predictor to the wet-rock VPwet thus obtained. The result is VSwet from, e.g., the Greenberg-Castagna predic-tor. The respective shear modulus G is then computed as bwetVSwet. Finally, by assuming that the shear modulus does not depend on pore fluid, the gas-sand S-wave velocity is computed as

(8)

where bgas is the bulk density of the gas sand with 20% water saturation:

(9)

where w is the density of water and fg is the density of the composite pore fluid with 80% gas and 20% water (Table 1).

The corresponding VP in gas sand was computed by Gassmann’s fluid substitution performed on the wet-rock elas-tic properties obtained from the respective effective-medium

Figure 4. Same as Figure 2a but for gas sand.

Figure 5. Synthetic earth-model and seismic gathers at wet sand for VP computed using the soft-sand model and for VS predicted using the soft-sand model and the Greenberg-Castagna, Vernik, and Williams equations. (a) The VS predicted by the soft-sand-model. From left to right: clay content; porosity; water saturation; bulk density; VP and VS ; P-wave impedance; Poisson’s ratio; and synthetic gather. In the gather, the bounding trace on the left is for normal incidence, whereas the bounding trace on the right is for the incidence angle of about 45°. (b) From left to right, Greenberg-Castagna, Vernik, and Williams equations. Here, only Poisson’s ratio and respective gathers are shown because all other inputs are the same as used for the soft-sand-model gather generation.

Dow

nloa

ded

10/2

1/15

to 1

24.1

95.4

.82.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/



Figure 6. Same as Figure 5 but for sand with oil (40% water saturation).

model, namely the soft- or stiff-sand model. This process was used with all VS predictors except for that of Krief et al. (1990), in which VS in the sand with hydrocarbons was com-puted directly by using equation 4 with the respective fluid properties listed in Table 1.

Figure 3 shows the results using the soft-sand model. There is no discernable difference between the model curves in the VS-versus-VP plot. However, because Poisson’s ratio

is very sensitive to VS variations, the v-versus-VP plot better serves to elucidate the differences among various predictors used here. We observe that for VP > 3 km/s, all Poisson’s ratio values group around 0.10. For smaller VP, v can fall below 0.10 or become as high as 0.20 (the Vernik equation) or even 0.25 (the Williams equation).

For stiff gas sand (Figure 4), the stiff-sand model and Krief et al.’s (1990) equations both predict Poisson’s ratio at about

Dow

nloa

ded

10/2

1/15

to 1

24.1

95.4

.82.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/



Figure 8. Same as Figure 6 but for sand with gas (20% water saturation).

0.10. At the same time, the Greenberg-Castagna, Vernik, and Williams equations all predict higher v, approaching and even exceeding 0.20.

Effect on AVO in clastics: Synthetic examplesAn earth model used here is a thick blocky gas sand em-

bedded in shale. The clay content is, as before, 5% in sand and 90% in shale. The synthetic seismic traces were produced using a convolutional-model-based ray tracer with a 30-Hz Ricker wavelet.

We start with modeling the reflections at soft wet sand. Figure 5 shows these synthetic results for 20% porosity shale and 30% porosity sand using the soft-sand model combined with the Greenberg-Castagna, Vernik, and Williams VS pre-dictors, as explained in the preceding section. Because Krief et al.’s (1990) model produces a relatively small Poisson’s ra-tio, similar to the soft-sand and stiff-sand models, now we will include in consideration only the three above-named pre-dictors because they produce larger v.

Figure 5 shows the results for the four VS variants (the first one according to the soft-sand model). The wet sand appears seismically transparent because of the very small difference between shale and sand impedances. The AVO effect is virtually nonexistent, no matter which VS predictor we use.

Our next exercise is for soft oil sand with 40% water saturation and the fluid properties listed in Table 1 (Figure 6). Figure 7 shows the respective AVO curves using the amplitudes picked at the trough marking the top of the oil sand.

All four VS predictors show amplitude reduction (increase in absolute values) versus the incidence angle at the top of the oil sand. As expected, this AVO effect becomes smaller as the oil-sand Poisson’s ratio becomes larger because of the

Figure 7. AVO curves picked at the troughs form the gathers shown in Figure 6 (marked as “Soft sand oil”) and in Figure 8 (marked as “Soft sand gas”). The upper curve in each set is for the Williams predictor (green), the next one for the Greenberg-Castagna predictor (dotted), the next for the Vernik predictor (red), and the lowest for the soft-sand model predictor (black).

difference between predictors. Still, the AVO effect is well pronounced for all four predictors.

Two questions to ask before selecting a single VS predictor to be used in a concrete case study are (1) whether the differ-ence in the AVO response is above the noise in seismic data and (2) whether the difference carries additional information pertaining to oil-reservoir detection and characterization.

Our next exercise is for soft gas sand with 20% water satu-ration and the fluid properties listed in Table 1 (Figure 8). The respective AVO curves extracted at the upper trough of

Dow

nloa

ded

10/2

1/15

to 1

24.1

95.4

.82.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/



each gather are also shown in Figure 7, along with the curves extracted for the oil sand. As in the latter case, the offset behavior differs, depending on which predictor we choose. At the same time, qualitatively, these AVO curves are very similar to one another. The main discriminator between oil and gas is the intercept rather than the gradient. Hence, we speculate that in the example presented here, the choice of the VS predictor is not of primary importance as far as hydrocar-bon identification is concerned.

Let us now examine the effect of consolidation and ce-mentation on the elastic properties of the interval by using the stiff-sand model for both shale and sand. In this example, we set the porosity at 10% in shale and 25% in sand. All other parameters and methods remain the same as in the soft-sand example. Figure 9 shows the synthetic gathers computed for stiff wet, oil, and gas sand. The AVO response changes from a weak Class I for wet sand to weak Class II for oil sand and weak Class III for gas sand.

The AVO curves extracted at the top of the sand interval for the oil and gas cases are plotted in Figure 10. As in the soft-sand case, the main discriminator for fluid detection ap-pears to be the AVO class. Still, for each class (II for oil and III for gas), the gradient varies depending on the VS predictor

Figure 9. Synthetic gathers computed for a (a) wet, (b) oil, and (c) gas stiff sand interval surrounded by shale. In each row, the gathers from left to right are for the stiff-sand model, Greenberg-Castagna predictor, Vernik predictor, and Williams predictor.

Dow

nloa

ded

10/2

1/15

to 1

24.1

95.4

.82.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/



used. Once again, before selecting a concrete predictor, the interpreter needs to find out whether the predictor-related differences in the synthetic seismic response are above the noise level and how crucial those differences are for hydro-carbon detection.

Real well exampleFigure 11 shows depth curves from an offshore gas well,

including gamma ray (GR), total porosity, water saturation, bulk density, and VP. The S-wave velocity data are not available, and hence, VS has to be predicted in order to gener-ate synthetic seismic gathers.

Rock-physics diagnostics conducted on this well data indicate that the velocity-porosity-clay behavior in the in-terval under examination can be explained quantitatively by the soft-sand model (Figure 12). To generate the diagnostics plot (Figure 12b), we first computed the wet-rock veloc-ity in the interval by using the VP -only fluid substitution (Mavko et al., 2009). The model curves superimposed on the wet-rock VP-versus-porosity crossplot are from the soft-sand model computed in the range of zero to 0.40 porosity and for clay content ranging from zero to 100%. The fluid used in the model had the formation brine bulk modulus and density.

The first VS predictor (Figure 11a) uses the soft-sand model. The three other predictors (Figure 11b) are the result of the Greenberg-Castagna, Vernik, and Williams equations. The respective VS and v at in situ conditions were obtained in the same way as in the preceding examples.

The respective synthetic traces generated using a 30-Hz Ricker wavelet show Class III AVO response with the gradient

essentially the same for the first three predictors (soft sand, Greenberg-Castagna, and Vernik) and a less steep gradient for the Williams predictor (Figure 13). Once again, we speculate that in this case, any of the examined VS predictors could be used will little or no difference for predicting the expected AVO response and, by so doing, for establishing a site-specific hydrocarbon indicators.

Lesson and conclusionExercises presented here exemplify the problem-avoid-

ance approach: Instead of arguing which VS predictor is most appropriate at a given location, let us first clarify what

Figure 10. Same as Figure 7 but for stiff sand, as explained in the text.

Figure 11. Depth curves and synthetic gathers computed at an offshore gas well using four VS predictors (only VP is available in the original data). The display is the same as used in Figure 5 with (a) the soft-sand model results and (b) the Greenberg-Castagna, Vernik, and Williams results (left to right, respectively) (only Poisson’s ratio and the gather). In all graphs, the separation between horizontal gridlines is 10 m. The maximum angle of incidence in each gather is about 45°.

Dow

nloa

ded

10/2

1/15

to 1

24.1

95.4

.82.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/



we need this predictor for. If the goal is synthetic AVO generation for hy-drocarbon identification, the first step is to check the sensitivity of the re-sponse to a range of predictors. If this sensitivity is relatively small, mean-ing that all tested predictors provide qualitatively similar response and the differences between predictors could be well within the noise level, arguing about which predictor is best might be simply pointless.

The main lesson is that precision has to be commeasurable with the fi-nal objective: Good enough is good. By no means do we encourage the reader to blindly extend the results pre-sented here to all rock and fluid types. The geoscientist has robust quanti-tative tools to replicate these computations for site-specific conditions and then, based on the results, select appropriate models and predictors.

ReferencesBatzle, M., and Z. Wang, 1992, Seismic properties of pore flu-

ids: Geophysics, 57, no. 11, 1396–1408, http://dx.doi.org /10.1190/1.1443207.

Castagna, J. P., M. L. Batzle, and R. L. Eastwood, 1985, Relation-ships between compressional-wave and shear-wave velocities in clastic silicate rocks: Geophysics, 50, no. 4, 571–581, http://dx.doi.org/10.1190/1.1441933.

Greenberg, M. L., and J. P. Castagna, 1992, Shear-wave velocity es-timation in porous rocks: Theoretical formulation, preliminary verification and applications: Geophysical Prospecting, 40, no. 2, 195–209, http://dx.doi.org/10.1111/j.1365-2478.1992.tb00371.x.

Krief, M., J. Garat, J. Stellingwerff, and J. Ventre, 1990, A petrophysi-cal interpretation using the velocities of P and S waves (full-wave-form sonic): The Log Analyst, 31, 355–369.

Mavko, G., T. Mukerji, and J. Dvorkin, 2009, The rock physics hand-book: Tools for seismic analysis of porous media, 2nd ed.: Cam-bridge University Press.

Vernik, L., D. Fisher, and S. Bahret, 2002, Estimation of net-to-gross from P and S impedance in deepwater turbidites: The Leading Edge, 21, no. 4, 380–387, http://dx.doi.org/10.1190/1.1471602.

Williams, D. M., 1990, The acoustic log hydrocarbon indicator: 31st Annual Logging Symposium, Society of Professional Well Log Analysts, Paper W.

Acknowledgments: This work was supported by the Stanford Rock Physics and Borehole Geophysics industrial consortium.

Recommended reading. All models and equations used in this work are described and analyzed in Mavko, G., T. Mukerji,

Figure 12. Velocity versus porosity for the data shown in Figure 11. (a) In situ conditions. (b) Wet conditions. The data are color-coded by GR. The color bar in the plot in (a) pertains to both plots. (b) The model curves are from the soft-sand model computed for zero clay content (top curve) and 100% clay content (bottom curve), with the curves in between computed for the gradually increasing clay content with 20% increment.

Figure 13. AVO curves extracted from the synthetic gathers shown in Figure 12, at the trough corresponding to the top of the gas-sand interval. The lower group contains three curves because of the soft-sand model (black), Greenberg-Castagna (dotted, hidden behind the other two curves), and Vernik (red) predictors. The upper curve (green) is the result of the Williams predictor.

and J. Dvorkin, 2009, The Rock Physics Handbook: Tools for Seismic Analysis of Porous Media, 2nd ed.: Cambridge University Press.

Corresponding author: [email protected]

Dow

nloa

ded

10/2

1/15

to 1

24.1

95.4

.82.

Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

vs predictors revisited.pdf

Documents

sand vp

various vs predictors

left vs

respective vs

resulting vs

elastic properties of

dierent predictors

various predictors dier