Inversion of the seismic amplitude into impedance is attractive because impedance is a layer property while amplitude is a property of the interface between layers. Impedance is directly measured during controlled experiments in the lab and in the well, together with porosity, mineralogy, pressure and saturation. As a result, one can first establish rock physics transforms from the rock's elastic properties into its bulk properties and conditions and then apply them to the seismic impedance to describe the subsurface.
The scale of data acquisition is very important in this workflow. Remember that while a rock physics transform, such as that between velocity and porosity, is established at the lab or log scale of only inches and feet, we aspire to apply it to seismic data on the scale of tens and hundreds of feet. Direct and unconditional use of such rock physics transforms, applied to seismic impedance volumes without accounting for scale effects, will produce erroneous results.
Log-derived rock physics transforms
Consider log curves from an offshore well with several sand layers (Figure 1). The total porosity in the sand is greater than in the surrounding shale. As a result, we observe a substantial negative impedance difference between the shale and gas sand while the impedance difference between the wet sand and shale is essentially nonexistent.
Figure 2 displays the impedance-porosity cross-plots for these data, color-coded by gamma-ray. The sand data points appear blue, while the shale is yellow-red. The gas-sand impedance-porosity trend is parallel and lies below the wet-sand trend (left frame). After theoretically substituting the in-situ pore fluid with the formation brine throughout the well, we observe that the impedance in what was originally the gas sand becomes the same as in the wet sand (right frame).
The curves shown in the cross-plot come from the uncemented (soft) sand/shale rock physics model designed to relate the velocity to porosity and clay content in soft clastic sediment. In the left frame of Figure 2, the upper two curves are for wet sand with zero and 10% clay content, while the lower two curves are for gas sand with the same clay content. In the right frame where the data points are for wet rock, each of the five model curves is produced for a fixed clay content ranging from zero for the upper curve to 100% for the lower curve with 20% clay increment, and for 100% brine saturation. The model curves accurately describe the trends observed in the data: The sand data lie between the zero clay and 20% clay curves, while the shale data lie between the 20% clay and 100% clay curves.
This model constitutes a site-specific log-scale rock physics transform between the total porosity, mineralogy and impedance. Can it be directly applied to the seismic impedance?
Seismic impedance
To address this question, we construct
a one-dimensional earth model with three gas sand layers of progressively increasing thickness (Figure 3). The porosity, clay content and water saturation in the sand are constant, 0.4, 0.05 and 0.2, respectively. The shale background also has constant porosity 0.35 and clay content 0.8 and is fully water-saturated. The log-scale impedance in the section is calculated from porosity, mineralogy and fluid according to the soft sand/shale model.
A simple way of calculating the seismic-scale impedance is via the Backus upscaling, which uses a running harmonic average of the elastic modulus. This upscaled (seismic) impedance profile is shown in Figure 3 in magenta. The seismic impedance is the same as the log-scale impedance in the thick sand layer located at the bottom, while it is noticeably different in the thin layer at the top. If this seismic impedance is used with the log-derived impedance-porosity transform, the predicted porosity in the sand will be about 0.3 instead of 0.4 (Figure 3, right-hand frame). This example illustrates the dichotomy due to scale in geophysical interpretation: A log-scale relation should not be blindly applied to seismic-scale data; porosity at a point cannot be always correctly mapped from seismic impedance. The reason is that an elastic property at a point cannot be accurately recovered from an experiment that employs large wavelengths. Downscaling is simply impossible without additional assumptions about the structure of the subsurface.
Instead of making such assumptions (which are often baseless away from well control), let us ask ourselves: Is there a scale-independent volumetric reservoir property and, if there is, is there a scale-independent seismic attribute to map this property?
Reservoir property/attribute
One scale-independent reservoir property is the total pore volume, which is often expressed as porosity times thickness. Generally, it is the integral of porosity with respect to depth taken within the reservoir: We call this property the accumulated porosity and plot it versus depth in Figure 4 for the synthetic earth model.
As a scale-independent seismic attribute that could be related to this property, we propose using the integral of the anomaly of the inverse impedance, where the anomaly is defined as the difference between the values in the reservoir and background. This is a cumulative seismic attribute (CATT).
The CATT curves calculated for the log-scale impedance and seismic-scale impedance are shown in Figure 4. This attribute appears almost scale-independent. A cross-plot of the cumulative porosity and CATT shown in Figure 4 indicates that total pore volume can be uniquely determined form CATT no matter at which scale the latter attribute is measured. The reason for this scale independence is that the upscaling of the elastic moduli is done by means of the harmonic average. Therefore, the anomaly of the inverse modulus will be exactly scale-independent, while the anomaly of the inverse impedance is approximately scale independent.
CATTS from seismic
Consider next a normal-incidence synthetic trace obtained by the convolution of a Ricker wavelet with the reflectivity series in the earth model examined here (Figure 5). The uncalibrated seismic impedance can be calculated as the exponent of the integral of the trace. By calibrating it with the log-scale impedance at the gas sand in the bottom of the interval, we obtain the calibrated seismic impedance trace. Finally, by integrating the anomaly of the inverse of this seismic impedance, we obtain the seismic CATT (Figure 5).
This seismic-derived CATT is very close to the log-scale CATT. Moreover, a cross-plot of the former and the accumulated porosity in Figure 5 indicates that this scale-independent reservoir property can be recovered from a seismically derived CATT.
These results indicate that porosity
in thin sub-resolution layers cannot be correctly mapped by directly applying rock physics impedance-porosity transforms to seismic impedance volumes simply because the upscaled seismic impedance often differs from
the actual fine-scale values. Instead, we propose mapping the product of porosity and thickness or, more precisely, the
total pore volume of the reservoir. This cumulative measure of porosity can be related, by means of rock physics, to
a new class of seismic attributes introduced here.
These are cumulative attributes (CATTS) which are calculated, for example, by an integration of a seismic impedance anomaly along the seismic trace. While the seismic impedance (acoustic and elastic alike) can be,
simply speaking, estimated by integrating the trace, CATTS are estimated by integrating the trace repeatedly. This
new class of seismic attributes potentially can be used in many geological environments for the purpose of mapping cumulative (or integrated)
rock properties from seismic.
We envision that CATTS can be constructed in different ways for different situations, for the acoustic and elastic impedance alike, in order to map different cumulative reservoir properties. The concept of this new class of seismic attributes is a solution to the problem of downscaling inherent in mapping reservoir properties from seismic data. The principle offered here is to map cumulative reservoir properties through cumulative seismic attributes.
Conclusion
Deriving reservoir properties from surface seismic data is a difficult task. Two major factors work against us. The first is non-uniqueness - a range of possible reservoir models will fit our seismic observations. The second challenge relates to overcoming scale effects. This paper suggests one solution to this problem. There is an increasing trend to use seismic data, along with traditional engineering data, to help quantify reservoir properties even to the point of using such data for economic predictions. For such methods to take hold, we will need to consistently and robustly address non-uniqueness and scale effects in our inversion workflows.

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