LA PROMOCION Y SUS ESTRATEGIAS

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Figure 8.32 Dry rock parameters inverted from log data using Gasmmann’s equation; (a) porosity vs Kd, (b) porosity vsμ. Functions have

been derived (dashed orange curves) based on fitting the clean high-porosity data to the mineral point. For more discussion on the interpretation of the data scatter seeSection 8.5.

κ Pð Þ ¼e κ∞ 1+ΕκexpPePk μ Pð Þ ¼e μ_∞ 1+Ε_μexpPePμ , ð8:38Þ

where κ and μ are the bulk and shear modulus at effective pressure Pe, κ∞ and μ∞ are the asymptotic high-pressure values, and Pκand Pμare constants that characterise the rollover point beyond which the rock frame becomes relatively insensitive to pressure increase.

Vernik and Hamman (2009) published similar models for clean sandstones, parameterised in terms of dry velocities; Vpd¼ Vp0+bp Pe c:expðdPeÞ h i Vsd¼ Vs0+bs Pe c:expðdPeÞ h i , ð8:39Þ

where b, c and d are fitting parameters with physical meaning as follows. c correlates with microcrack density; when pressure is high enough to close the microcracks in the rock then c is zero and the equa- tions become linear, with slopes b that are inversely related to the level of consolidation and cementation. Note that d is related to the dominant aspect ratio of the microcracks.

In general it appears that rocks which are over- pressured and are relaxed through pressure draw- down will show relatively large dry rock changes consistent with laboratory measurements. However, if a normally pressured reservoir is drawn down then

the effect may well be less than that shown by labora- tory measurements (C. MacBeth, personal communi- cation). Drawdown in the overpressured situation results in the dry rock frame taking more of the weight of the overburden and thus the frame stiffens. In a normally pressured situation the effect is minim- ised as the rock frame already supports the overbur- den. In contrast to this, where pore pressures increase during hydrocarbon production (around water inject- ors), large changes in seismic amplitude have been observed (e.g. Sayers,2006).

Whilst laboratory data are important in assessing the dry rock sensitivity to effective pressure variations it should be noted that even when these data are available there is always some doubt as to the rele- vance of high-frequency measurements on small samples (micro-fabric measurement) to a field wide response under the influence of subsurface stress regimes. For example, the response in the low effect- ive pressure regime may be dominated by micro- cracks which may be natural or generated through pressure release of core material. The reader is referred to McCann and Southcott (1992) for a review of the issues in laboratory measurements.

8.2.6 Contact models

Contact models are based on mathematical principles of the interaction of granular materials and are applicable to sandstones. They are generally con- structed by determining high- and low-porosity dry rock end members which are then interpolated, often using modified Hashin–Shtrikman (1964) bounds.

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Eff Pressure (MPa) Nel A WOS Mio Gulf Rot A Rot B Nel B North Sea Cooper Rot C SNS C

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Figure 8.33 Dry rock parameters of sandstones and their sensitivity to effective pressure (trend data from MacBeth (2004).

Saturated elastic moduli are then calculated using Gassmann’s (1951) equation. These models effectively address the issues of effective pressure on unconsoli- dated grain packs and the effect of cement in stiffening the rock. When fit to log data it is possible to infer whether the sand is unconsolidated or cemented. Not only is this useful for seismic litho facies characterisation but it may also be useful in sanding assessment for reservoir production.

There are two commonly used contact models, the ‘friable sand’ model and the ‘cemented sand‘ model. These two models were derived by Dvorkin and Nur (1996) from a study of two sets of laboratory data on sandstones from the North Sea, one from the Oseberg Field and one from the Troll Field. Thin-section images showed slight quartz cementation of the grains in the Oseberg samples, whereas cementation was absent in the Troll samples.

With the ‘friable sand’ or ‘uncemented sand’ model, a high porosity dry rock end member is determined for a random pack of identical spherical grains at critical porosity. Hertz–Mindlin theory (Mindlin, 1949) is used. The lower Hashin–

Shtrikman bound then interpolates between this point and the mineral moduli at zero porosity. This interpolation represents the deterioration in poros- ity due to decrease in sorting. The model inputs are the solid phase bulk and shear moduli, critical porosity, effective pressure and the average number of contacts per grain (the ‘coordination number’). Empirical models relate the coordination number to porosity. The‘friable sand’ model can be effectively used to derive a constant clay model for sandy shales by substituting a much larger critical porosity (60%–70%) and a Reuss mix of clay and silt

components (assuming the silt is suspended in the clay) to derive the effective solid moduli (Avseth et al., 2005).

The ‘cemented sand’ or ‘contact cement’model assumes that the porosity reduces owing to the uni- form deposition of cement on the surface of the sand grains. Only a small amount of cement deposited at grain contacts is required to significantly increase the stiffness of the rock. The mathematical solution for this scenario was determined by Dvorkin et al. (1994). The model inputs are solid phase and cement phase bulk and shear moduli, and coordination number (representing the radius of contact of the cement layer). In the Oseberg samples clay cement gave slightly lower velocities than quartz cement. There is no effective pressure dependency built into the model.

Figure 8.34 illustrates examples of friable and cemented sand trends in data from the North Sea and Gulf of Mexico. The similarity of the trends in the two basins is striking. However, it should be remembered that there a large number of variables (i.e. sorting, shaliness, effective pressure, and amount of cement) that may give rise to similar values of velocity for a given porosity. The friable sand and cemented sand models are not necessarily mutually exclusive. For example, a combination of the friable and cemented sand models can potentially describe the situation where a sorting trend has a constant value of cement (i.e the constant cement model (Avseth, 2000; Avseth et al., 2005) (Fig. 8.35). It is recommended that the reader consults Avseth et al. (2005) for a thorough discussion of a variety of models describing specific sand/shale scenarios, including models to describe the dispersed clay and laminated shale effects described inChapter 5.

M (GPa) G (GPa) 10 20 30 2 4 6 8 10 0.1 0.2 0.3 0.4 Porosity 0.1 0.2 0.3 0.4 Porosity