An integrated workflow is presented to model the seismic response of a fractured carbonate reservoir. Carbonate rocks are known to have a more complicated, heterogeneous pore structure than sandstone rocks. The workflow uses P-wave and porosity log data to calibrate a multi-porosity model using the extended Xu-White model (Xu and Payne, 2009). The model is then up-scaled to investigate the low frequency seismic response through the inclusion of aligned meso-scale fracture sets. Analysis is shown for log data measured in a fractured limestone reservoir. The rock physics model was used to predict the P-wave and P-to-S wave reflectivity from the top of the limestone reservoir. The results demonstrate that rock physics is able to play an important role in modelling and characterizing the seismic response of carbonate reservoirs.

BACKGROUND

Approximately 50% of the world’s known hydrocarbon reserves are contained in carbonate reservoirs. Central to the seismic characterization of carbonate reservoirs is an understanding of the effect of porosity on the elastic moduli. Carbonate rocks are known to exhibit heterogeneous pore space structures that are both complex and highly variable. Dissolution of the carbonate rock can result in moldic pores and vugs that behave very differently under the passage of a seismic wave when compared with conventional inter-granular pores observed in sandstone reservoirs. Fractures also commonly play an important role in production from carbonate reservoirs. Areas of high fracture density can act as high permeability channels within an otherwise low permeability carbonate reservoir.

Xu and White (1995) developed a multi-porosity model for shaley sandstones by combining the Kuster-Toksöz (1974) and differential effective medium theories with Gassmann’s equation. Xu and Payne (2009) demonstrated that this model can be extended to predict velocities in carbonate rocks. The extended Xu-White model (referred to as the Xu-Payne model) divides the total pore volume into four components: (i) clay-related pores, (ii) inter-particle pores, (iii) microcracks and (iv) stiff pores. Each pore component is incrementally added to the rock matrix to calculate the elastic properties of the resultant effective medium. One versatile feature of the Xu-Payne model is that individual pore components can be included into the model such that they are either isolated, or, in perfect fluid connectivity with the remaining pore space.

The pore space model is calibrated using a cross-plot of P-wave velocity against porosity (Kumar & Han, 2005). Thus, a multi-porosity model can be interpreted for a carbonate reservoir from sonic and porosity log data (Figure 1). Sonic logs measure the effective rock properties in a region close to the borehole wall with a probing wavelength approximately 10-30cm. Measurements at this scale are not able to fully characterise meso-scale fracture networks that are important to reservoir production. The multi-porosity model can be up-scaled to include the low-frequency seismic response of a fracture network using the effective medium theory described by Schoenberg and Sayers (1995). Parameterization of the fracture properties can come from a mixture of FMI, VSP and laboratory data.

METHOD

A multi-porosity model is calibrated for a particular reservoir using petrophysical log data. The clay-related pore space is accounted for using the shale volume curve. Pores in clay are expected to be water saturated, bound and of low aspect ratio (0.02-0.05). The remaining porosity can be divided into contributions from typically three representative pore space components, each with a characteristic pore aspect ratio, see Table 1. This step is performed by inverting the measured log data on a sample by sample basis. The generated multi-porosity model is one valid model that explains the log data. When core samples are available, the relative contribution of each pore component should be compared to expectations from petrographic analysis in each carbonate interval.

The Xu-Payne approach permits two types of fluid substitution to be performed in the rock’s pore space. Fluids contained in an isolated pore space component cannot flow to equalise pressure gradients created by the passage of a seismic wave. The response due to these pores is modelled by inserting fluid saturated inclusions with the differential effective medium. Saturation changes are considered by modelling inclusions with different pore fluids. In contrast, the pore space components that are assumed to be in perfect fluid connectivity are modelled by inserting dry inclusions and fluid substitution to the desired saturation is performed as a final step using Gassmann’s equation.

Schoenberg and Sayers (1995) showed that the elastic moduli of a fractured rock can be obtained as the sum of the compliance tensor of the unfractured rock and the compliance tensor for each set of aligned fractures. The compliance tensor of the unfractured rock is calculated for the calibrated multi-porosity model in the previous step. A single set of rotationally symmetric fractures is fully described by four parameters: fracture density, strike azimuth, fracture normal and tangential compliance. FMI log data are used to interpret the average strike of an aligned fracture set. FMI data collected in horizontal, or, deviated wells can also be useful in estimating the density of vertical fractures. The range of published values for fracture compliance has been compiled by Worthington (2007). Lubbe et al. (2007) suggested that a normal-tangential compliance ratio of 0.5 is a representative average value for modelling. Alternatively, anisotropic Thomsen parameters that characterise the fractured rock can be estimated from VSP surveys.

Table 1: Aspect ratio of representative pore types used to construct the multi-porosity model (after Xu and Payne, 2009).

Pore component

Representative pore aspect ratio

Inter-particle pores

0.15

Microcracks

0.01

Stiff pores

0.80

EXAMPLE

Log data from a gas condensate, limestone reservoir are shown in Figure 2. The reservoir is observed to be clean, with a relatively low interpreted clay volume. Cross-plots of the P-wave velocity against porosity (Figure 1) indicate that a model containing both stiff pores and microcracks is required to explain the data in all three intervals. The P-wave velocity log was inverted to calculate the relative contribution of the two pore space components to the total porosity. In this example, the clay-related pores have been modelled as isolated but the remaining pore components are considered to be in fluid connectivity with each other. The calculated ratio of the two pore components and the modelled P-wave curve are shown in Figure 2.

Fracture information was interpreted from FMI image logs in the different intervals. A single set of sub-vertical (dip >50°) fractures, with an average fracture strike of 187°, were observed in the upper limestone interval. The fractures were not observed to continue upwards into the marl overburden and different fracture sets were interpreted in the lower limestone intervals. In this example, an excess fracture normal compliance of 6x10^{-11} Pa^{-1} was observed to generate a practical upper limit for modelling, generating a value of P-wave anisotropy (ε) of approximately 0.2, see Table 2.

The seismic response of the fractured reservoir was calculated in terms of the P and P-to-S AVO reflectivity (Vavryčuk and Pšenčík, 1998, Jílek, 2000). The AVO reflectivity patterns are shown for the top limestone horizon as a function of survey azimuth relative to the fracture strike direction (Figure 3). The P-wave reflectivity is only observed to differ from the isotropic reflectivity at very large angles of incidence. A much stronger anisotropic reflection pattern is observed in the PS data, with the PSh exhibiting the expected clover-leaf pattern.

Marl overburden

Upper limestone

Vp (km/s)

4.68

5.61

Vs (km/s)

2.86

2.91

ρ (g/cc)

2.65

2.71

ε

0.00

0.19

γ

0.00

0.14

δ

0.00

0.19

Figure 1:P-wave vs. porosity cross-plots for 3 limestone reservoir intervals. The solid lines show the reference trends for models containing only: 100% microcracks (dark red), 100% inter-granular pores (green) and 100% stiff pores (dark blue). The dashed lines show intermediate models containing contributions from 2 pore components. The upper interval (a) contains gas condensate and is modelled with a water saturation of 40%, the lower intervals are 100% water saturated..

Figure 2: Left-to-right: clay volume (brown = clay, blue = calcite), total porosity (coloured by saturation), pore space component (pink = stiff pores, blue = microcracks) and Vp logs (black = measured, red = calculated) from a carbonate reservoir. The upper limestone interval is between markers Limestone_A and Limestone_B. This interval contains a gas condensate and has an average water saturation of 40%. Calculationof the pore space component logs is described in the text

Figure 3: AVO reflectivity from the top limestone interface. Each subplot includes the AVO reflectivity curve along a certain survey azimuth (left) and the reflectivity as a function of survey azimuth for an incidence angle of 30° (right). Plots (a) and (b) show the AVO reflectivity for a survey azimuth parallel to fracture strike (modelled as 000°); plots (c) and (d) show the AVO reflectivity for a survey azimuth perpendicular to the fracture strike.The anisotropic P-wave reflectivity is shown in black, the anisotropic PSv reflectivity in blue, the anisotropic PSh reflectivity in pink, and the equivalent isotropic reflectivity in grey. Note the reflectivity scale is shown differently for the P-wave and P-to-S cases.

Conclusions

The seismic response of a carbonate reservoir can be studied through the construction of a multi-porosity rock physics model. The model is calibrated using log data from the reservoir and up-scaled to include the effects of meso-scale fracture sets. A particular feature of the model is the ability to include different pore space components as either isolated, or, in perfect fluid connectivity with the remaining pore space. The calibrated rock physics model can be used to study and interpret the offset and azimuth dependent seismic reflectivity of the reservoir.

An example is shown for a multi-porosity fractured limestone reservoir. The synthetic AVO indicates that it will not be possible to interpret any information about fracturing within the reservoir from the P-wave reflectivity alone. In this case, it would be necessary to analyze P-to-S converted wave reflections to study spatial variations in the fracturing.

Acknowledgements

The authors would like to thank their colleagues at Ikon Science who have supported this work and made it possible.

References

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