Rock Physics for Fluid and Porosity Mapping in NE GoM Rock Physics for Fluid and Porosity Mapping in...
Transcript of Rock Physics for Fluid and Porosity Mapping in NE GoM Rock Physics for Fluid and Porosity Mapping in...
1
Rock Physics for Fluid and Porosity Mapping in NE GoM
JACK DVORKIN, Stanford University and Rock Solid Images
TIM FASNACHT, Anadarko Petroleum Corporation
RICHARD UDEN, MAGGIE SMITH, NAUM DERZHI, AND JOEL WALLS, Rock Solid Images
May 07, 2003
We apply a rock physics analysis to well log data from the North-East Gulf of Mexico
to establish an effective-medium transform between the acoustic and elastic impedance
on the one hand and lithology, pore fluid, and porosity on the other hand. These
transforms are upscaled and applied to acoustic and elastic impedance inversion volumes
to map lithology and porosity.
INTRODUCTION. The ultimate goal of the geoscientist is to determine the reservoir
properties (lithology and porosity) and conditions (pore fluid and pressure) from seismic
data. This goal can be achieved by applying rock physics transforms to a volume of
seismically-derived elastic properties. The basis for establishing rock physics transforms
are controlled experiments where the reservoir and elastic sediment properties are
measured on the same sample of earth at the same conditions. Such experimental data
come from well logs and cores.
The rock physics diagnostic is aimed at finding a deterministic effective-medium
model appropriate for the rock under examination, calibrating it to the data, and then
using it as the ultimate link (transform) between various rock properties. We select this
rational deterministic approach, as opposed to pure statistics, in order to (a) understand
and generalize trends seen in the data and use them outside of ranges available in the
data; and (b) determine the domains of applicability of the trends.
Below, the rock physics diagnostic is applied to well log data from the NE GoM and
then used to map fluid and quantify porosity from acoustic and elastic impedance
inversion volumes.
WELL LOG DATA. The interval under examination includes shales and several high-
2
porosity sand packages (Figure 1). The upper sands are water-saturated while the lower
sand contains hydrocarbons. The hydrocarbon-filled sand layer includes high-impedance
calcite streaks. The S-wave data are not available in the well.
One goal of this study is to quantify the porosity of the reservoir from seismically-
derived elastic parameters. This is why in Figure 1 we display an impedance-porosity
cross-plot together with the depth plots of gamma-ray (GR), saturation, porosity, and
impedance. The shale and wet sand form two separate branches in the impedance-
porosity frame. In the same range of the total porosity, the impedance in the shale is
smaller than in the wet sand. In both lithologies, evident are distinctive impedance-
porosity trends.
The impedance in the high-porosity hydrocarbon sand is smaller than in the wet sand
of comparable porosity mostly due to the effect of the pore fluid on the bulk modulus.
This is why the impedance-porosity trend present in the hydrocarbon reservoir is different
from that observed in the wet sand located above.
ROCK PHYSICS DIAGNOSTIC. In order to find an effective-medium model that will
provide a porosity-lithology-impedance transform, we first theoretically saturate, using
the
†
Vp -only fluid substitution, the entire interval under examination with the same pore
fluid, namely, the formation brine. This pore-fluid equalization (we also call this
procedure “common fluid denominator”) allows us to concentrate purely on the effects of
lithology and porosity on the elastic properties.
The impedance in the brine-substituted interval is plotted versus porosity in Figure 2,
top. It is color-coded by GR and the in-situ water saturation. The model-based
impedance-porosity curves from the uncemented sand/shale model are superimposed
upon the data points in the same figure. The sand data points are bound by the curves
drawn for pure quartz sand and sand with 25% clay content.
Model-based impedance-porosity curves are also drawn for rock with 75% and 100%
clay content. These curves serve at lower bounds for the shale data. Most of the shale
data lie above the 75% clay content curve which simply means that the shale is not pure
clay.
3
Figure 1. Well log curves (from left to right, GR, water saturation, the total porosity, and impedance)
and the impedance-porosity cross-plot. In the second, third, and fourth rows, we highlight (in red) the
shale, wet sand, and hydrocarbon sand, respectively. Depth is in fictitious units.
The original (not fluid-substituted) impedance is plotted versus porosity in Figure 2,
4
bottom. The uncemented sand/shale curve drawn for sand with 10% clay and for the in-
situ reservoir fluid is superimposed on the data. It can serve as an average impedance-
porosity transform in the reservoir sand. The corresponding best-fit equations are:
†
Ip =12.45 - 39.62f + 47.23f 2,
f = 0.741- 0.117Ip + 0.0051Ip2,
(1)
where the impedance (
†
Ip ) is in km/s g/cc and porosity (
†
f ) is in fraction.
Figure 2. Impedance versus total porosity color-coded by GR (left) and water saturation (right). Top:
Impedance calculated for the brine-saturated interval, including the reservoir sand. The model curves
come from the uncemented sand/shale model and are calculated for brine-saturated conditions. The
curves are (top to bottom) for zero, 25%, 75%, and 100% clay content, respectively. Bottom: The in-
situ impedance. The yellow curve is for 10% clay content and the in-situ reservoir fluid.
A comparison between the in-situ velocity and impedance data and the velocity and
5
impedance calculated using the uncemented sand/shale model with the in-situ clay
content, porosity, and saturation as input (Figure 3) indicates that the selected rock
physics transform is appropriate for the interval under examination.
Figure 3. Comparison between sonic data and modeled data. Well log curves (from left to right, GR,
water saturation, P-wave velocity, and impedance). In the velocity and impedance frames, the black
curves represent the well log data while the red curves are computed from the uncemented sand/shale
model using the clay content, porosity, and saturation curves as input. Depth is in fictitious units.
In Figure 4 we compare the porosity measured in the pay zone with that predicted
from the measured impedance using the transform from the second Equation (1). The
prediction is very close to the data.
Figure 4. Comparison between porosity data and prediction in the pay zone. In the porosity frame, the
black curve is measured porosity while the red curve is porosity calculated from the measured
impedance using the transform from Equation (1).
Based on these results, we adopt the uncemented sand/shale model as appropriate for
6
the well under examination. As we have already seen, this model can serve to predict
porosity from impedance. It can also be used to calculate the S-wave velocity and
Poisson’s Ratio (PR) which are not available at the well. Our calculations indicate that,
as expected, PR in the reservoir sand is low (about 0.13); PR in the wet sand is about
0.28; and PR in the shale is about 0.35.
UPSCALING. Elastic waves in the seismic frequency range usually sample large
portions of the subsurface whose dimensions are on the order of the wavelength. As a
result, the elastic structure of the subsurface extracted from seismic data can only
represent the elastic properties averaged over relatively large intervals. The details that
are apparent at the well log scale cannot be recovered from seismic data.
To illustrate this effect, consider synthetic seismic traces modeled at the well under
examination using a 50 Hz and 25 Hz Ricker wavelet and the velocity (
†
Vp and the model-
predicted
†
Vs) and bulk density at the well (Figure 5). This synthetic seismic modeling is
based on the convolution of the wavelet with the reflectivity series derived at the well.
The reflectivity
†
Rpp (q) at an angle of incidence
†
q is calculated according to Hilterman’s
approximation to the Zoeppritz equations:
†
Rpp (q) = Rpp (0)cos2 q + 2.25Dn sin2 q,Rpp (0) = D(0.5ln Ip ),
(2)
where
†
n is PR and
†
D indicates the difference between the lower and upper half-space at
the reflecting interface.
The pay zone is manifested by a positive reflection whose amplitude decreases with
offset. The positive normal reflection is due to the gradual impedance increase from the
shale above the pay zone, through the pay zone, and to the underlying hard shale. The
AVO effect is due to the small Poisson’s ratio characteristic of porous sand with
hydrocarbon.
Also in Figure 5 shown are the pseudo-impedance (
†
Ipp ) and pseudo-PR (
†
n p ) curves
which were calculated from the synthetic traces by using Hilterman’s AVO
approximation – the pseudo-impedance is simply the cumulative sum of the normal-
incidence trace while pseudo-PR is a linear combination of the cumulative sum of the far-
7
offset trace and the pseudo-impedance to produce Poisson’s ratio reflectivity:
†
Ipp = T0(t )dt0
t
Ú , n p = [cos-2 q Tq (t)dt0
t
Ú - Ipp ]/(2.25tan2 q), (3)
where
†
T0 is the normal-incidence trace;
†
Tq is the trace at angle
†
q ; and
†
t is the two-way
travel time.
Figure 5. Synthetic seismic at the well. From left to right: Synthetic gather (black) and stack (red);
impedance and PR at the well; and pseudo-impedance and pseudo-PR calculated from the synthetic
traces. Travel time is in fictitious units.
The pseudo-impedance and pseudo-PR curves presented in Figure 5 capture the
8
character of the impedance and PR at the well but, at the same time, omit the details and,
effectively, smooth the well log curves.
Because of this smoothing effect, it is not clear whether the impedance-porosity
transform expressed in Equation (1) can be useful at the seismic scale, how it has to be
corrected (upscaled), and, finally, how to interpret porosity estimates that are based on
the seismic-scale impedance values.
To investigate the effects of upscaling on the elastic properties and also on rock
physics transforms, we use the Backus average which estimates the seismic-scale
response by harmonically averaging of the elastic moduli at the well. The bulk density is
upscaled by means of arithmetic averaging. Then the upscaled impedance and PR are
calculated from the Backus-upscaled elastic moduli and arithmetically-upscaled density
according to the appropriate theory-of-elasticity equations. The upscaled elastic curves
as well as the arithmetically-averaged porosity curve are shown in Figure 6. This display
confirms what we already learned from the synthetic seismic traces: the log-scale details
of the interval will be omitted in seismic data.
Figure 6. Original (black) and upscaled (bold cyan) porosity, impedance, and PR curves at the well.
Depth is in fictitious units.
UPSCALED IMPEDANCE-POROSITY TRANSFORM. A question to address is whether
the transform from the impedance to porosity derived from well log data --
†
f = 0.741- 0.117Ip + 0.0051Ip2 -- is valid at the seismic scale. To investigate this
problem, let us plot the porosity versus impedance at the log scale and seismic scale
(Figure 7). The upscaled trend generally follows the original porosity-impedance trend
but slightly overestimates the porosity at a given impedance value. The best-linear-fit
9
transform between the upscaled impedance and averaged porosity is
†
fu = 0.205 - 0.037(Ipu - 6.5), (4)
where the impedance is in km/s g/cc and porosity is in fraction, and the subscript
†
u refers
to the upscaled values.
This equation is site-specific and should not be generalized because the results of
upscaling may be strongly influenced by the size and shape of the reservoir and
surrounding strata. In general, the validity of log-scale rock physics transforms at the
seismic scale improves as the reservoir becomes thicker and the variations of the rock
properties with depth around the reservoir become smoother.
Figure 7. Log-scale porosity versus log-scale impedance (black); log-scale porosity versus upscaled
impedance (blue); and upscaled porosity versus upscaled impedance (red). The yellow line is from
Equation (4).
INFLUENCE OF HIGH-IMPEDANCE LAYERS ON REFLECTION. Let us investigate is
whether the high-impedance layers that bound the reservoir affect the seismic reflection
and, eventually, porosity estimates from the seismic impedance. To address this
question, we alter the elastic curves at the well by reducing the impedance in the three
high-impedance layers to the surrounding impedance values (Figure 8). Then we
calculate the Backus-upscaled impedance curves.
10
Figure 8. The impedance curve at the reservoir. Black is for the measured impedance; red is for the
impedance artificially reduced within the high-impedance layers; cyan is for the upscaled original
impedance curve; and yellow is for the upscaled reduced impedance curve. Depth is in fictitious units.
The upscaling results indicate that the influence of the high-impedance layers on the
upscaled impedance values is small. To further support this conclusion, we generate
synthetic traces using the impedance curves at the well with and without the high-
impedance layers (Figure 9). The reflections appear to be virtually unaffected by the
high-impedance layers.
This apparently counter-intuitive result is due to the fact that the averaging of the
elastic properties in the subsurface is harmonic and as such gives larger weights to the
softer parts of the interval.
POROSITY FROM UPSCALED IMPEDANCE. Next we apply Equation (4) to the
upscaled impedance curve to assess the accuracy of porosity estimates from the seismic-
scale impedance. The estimated porosity is compared to the log-scale porosity and also
to the arithmetically-averaged porosity in Figure 10. The porosity curve derived from the
upscaled impedance is essentially the same as the arithmetically averaged porosity curve.
Both curves give a reasonably accurate average estimate to the log-scale porosity.
11
Figure 9. Synthetic traces with (top) and without (bottom) high-impedance layers in the reservoir
zone. The display is the same as in Figure 5.
12
Figure 10. Left: Water saturation curve in the reservoir zone. Right: the log-scale porosity curve
(black); arithmetically upscaled porosity curve (red); and porosity derived from the upscaled
impedance (yellow). Depth is in fictitious units.
POROSITY FROM SEISMIC IMPEDANCE INVERSION. Next we use acoustic and elastic
impedance inversion sections obtained from real seismic data to identify the reservoir and
estimate the total porosity within the reservoir. In the section displayed in Figure 11, the
reservoir is identified by a low-value-PR strip, as is expected from the above rock physics
analysis and pseudo-impedance and pseudo-PR inversion curves shown in Figures 5 and
9. The minimum-value strip in the PR inversion section does not precisely coincide in
space with the maximum-value strip in the impedance-inversion section. This is
expected from the impedance structure of the well log profile where the maximum of the
impedance is located slightly below the reservoir while the minimum of the PR is located
precisely at the reservoir.
The reservoir in the inversion section is identified as the low-PR strip. The porosity
in the reservoir is directly calculated from the impedance inversion using Equation (4).
The calculated porosity section is displayed in Figure 11. The same approach as used in
the vertical section is employed to create a seismic porosity cube from 3D acoustic and
elastic impedance inversion (Figure 12). The dipping sand layer filled with hydrocarbon
is apparent in the volume. The curved section of the sand by a constant-time horizon
visible in the 3D display is an apparent indication of a three-way closure.
13
Figure 11. Left: PR inversion from real seismic. Middle: Acoustic impedance inversion from real
seismic. Right: Seismic porosity derived from the impedance inversion in the reservoir zone. Depth
and cross-line distance are in fictitious units. The vertical bars indicate the location of the well. The
well log curves of saturation (left); GR (middle); and porosity (right) are superimposed upon the
vertical sections.
Figure 12. Mapping fluid and porosity from 3D acoustic and elastic impedance inversion. From left
to right: PR inversion; acoustic impedance inversion; and total porosity produced from the acoustic
impedance according to Equation (4).
CUMULATIVE POROSITY. An important desired reservoir characterization parameter
is the cumulative porosity (
†
Cf ) also known as the total pore volume in the reservoir, or
net-to-gross. This quantity is often expressed as the product of the porosity (
†
f ) and
reservoir thickness (
†
h):
†
Cf = fh . Strictly speaking, it is the integral of the porosity with
respect to depth, calculated within the reservoir:
†
Cf = f(z)dz,zT
zB
Ú (5)
14
where the integration is between the top (
†
zT ) and bottom (
†
zB ) depths of the reservoir.
Moreover, only the accumulated porosity within the pay zone is of immediate
interest. To calculate the cumulative pay-zone porosity (
†
CfP ), Equation (5) should be
modified as:
†
CfP = f(z) Sw <1dz,zT
zB
Ú (6)
where condition
†
Sw <1 simply means that during the integration, porosity outside of the
pay zone is not counted.
The cumulative pay-zone porosity is plotted versus depth in Figure 13 for the original
(log-scale) curve as well as for the upscaled porosity. The two curves are very close to
each other which means that the seismically derived
†
CfP and the log-scale
†
CfP are
essentially the same.
Figure 13. Cumulative porosity (left); cumulative impedance (middle); and cumulative inverse
impedance (right) versus depth. Black curves are for the log scale values while yellow curves are for
the upscaled values. The cumulative porosity is in fraction times meter; the cumulative impedance is
in km/s g/cc times meter; and the cumulative inverse impedance is in s/km cc/g times meter.
CUMULATIVE ATTRIBUTES. As we have already established, porosity can be related
to the impedance. Next question is which seismic attribute can be used to estimate the
cumulative porosity. One possibility is that such a cumulative attribute (CATT) is the
cumulative impedance in the pay zone (
†
CI p P ) which can be defined in the same manner
as
†
CfP :
15
†
CI p P = Ip (z)Sw <1
dz.zT
zB
Ú (7)
†
CI p P is plotted versus depth in Figure 13 (middle) for the original log-scale curve as
well as for the upscaled curve. Once again, the two curves are very close to each other
which means that the proposed integration operator will produce approximately the same
results at the seismic scale as at the log scale.
The next question is whether the proposed CATT can be uniquely translated into the
cumulative porosity within a seismic volume. To investigate this possibility, let us cross-
plot
†
CfP versus
†
CI p P at the log and seismic scale. The result shown in Figure 14 (left)
indicates that the cross-plot curves at both scales are close to each other which means that
the proposed CATT (
†
CI p P ) can be used to calculate the cumulative porosity from seismic.
Figure 14. Cumulative porosity versus cumulative impedance (left); and versus cumulative inverse
impedance (right). Black curves are for the log scale values while yellow curves are for the upscaled
values.
Another possible CATT may be the inverse cumulative impedance (
†
CI p-1P ):
†
CI p-1P = Ip
-1(z)Sw <1
dz.zT
zB
Ú (8)
This CATT is plotted versus depth in Figure 13 (right). The cumulative porosity is
cross-plotted versus
†
CI p-1P in Figure 14, right. In this cross-plot, the match between the
log-scale and seismic-scale values is even better than for the cumulative impedance on
16
the left. This is likely due to the fact that the elastic moduli upscale by means of
harmonic averaging and, in this particular case, such averaging rule may be
approximately valid for the impedance.
The approach proposed here essentially introduces a new class of seismic attributes
which are calculated from a seismic trace by repeated integration -- the seismic
impedance is due to a one-time integration of a trace which is approximately treated as
the reflectivity series and the cumulative impedance is due to one-time integration of the
seismic impedance or repeated integration of the trace.
CAVEAT -- DELINEATING PAY ZONE. An important condition for calculating
†
CfP
from a CATT was to integrate the latter within the pay zone. To meet this condition we
will have to employ an additional attribute, such as the seismic PR, to locate the pay
zone. Finding a threshold for this attribute to accurately delineate a pay is important for a
successful use of CATTS.
CONCLUSION. The rock physics diagnostic approach enables us to establish a site-
specific effective-medium model to transform the elastic rock properties into lithology,
fluid, and porosity. An effective-medium model allows us to systematically and
consistently perturb rock properties and conditions to predict the elastic response away
from a well. A model established for log-scale data has to be upscaled to the seismic
scale in order to be used with seismic inversion cubes. Such upscaling may call for
corrections in the model, especially so in a heterogeneous environment with large elastic
contrasts.
ACKNOWLEDGEMENT. Uwe Strecker helped with stratigraphic and geological
understanding of the example presented in this study.
SUGGESTED READING
Dvorkin, J., and Nur, A., 1996, Elasticity of High-Porosity Sandstones: Theory for Two North Sea
Datasets, Geophysics, 61, 1363-1370.
Mavko, G., Chan, C., and Mukerji, T., 1995, Fluid substitution: Estimating changes in Vp without
knowing Vs, Geophysics, 60, 1750-1755.
Backus, G.F., 1962, Long-wave elastic anisotropy produced by horizontal layering, JGR, 67, 4427-4441.