In many rocks, compressional velocity and bulk dens- ity have a positive relationship, so that as velocity increases so does density (Fig. 8.3).
Gardner et al. (1974) developed a series of (brine- saturated) lithology dependent relations of the form:
ρb¼ dV f
p, ð8:5Þ
whereρbis bulk density in g/cc, Vpis the P wave velocity in km/s, and d and f are constants. The various lithology dependent coefficients are shown inTable 8.1.
0
0
0.2
0.4
0.6
0.8
1
10
20
30
40
Porosity
K (GPa)
5
3
1
2, 4
Figure 8.2 Bounds on bulk modulus of a brine-filled sandstone, after Nuret al. (1998). In this case the constituents are quartz and brine. The curves shown are (1) Voigt average, (2) Reuss average, (3) upper Hashin–Shtrikman, (4) lower Hashin–Shtrikman and (5) modified Voigt bound.
30000 25000 20000 15000 12000 10000 8000 6000 7000 5000 3980 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Bulk density (g/cc)
Logarithm of bulk density (g/cc) Rock salt Limestone Dolomite Anhydrite Shale Sandstone Time-average (Sandstone) ρ=0.23V0.25 Velocity (ft/s) Logarithm of velocity (ft/s) 0.2 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 0.3 0.4 0.5
Figure 8.3 Velocity–density relationships in rocks of different lithology (re-drawn after Gardneret al.,1974.)
The Gardner relations may be used in transform- ing sonic or density logs for the purposes of replacing missing sections or in constraining the results of inversions for P and S reflectivity (e.g. White,2000). Owing to the lack of universal applicability it is advised that area-specific density– velocity relations are derived from the available data. In some cases it may be more appropriate to derive density from a transform based on shear velocity rather than com- pressional velocity (e.g. Potter and Stewart,1998).
8.2.2.2 Wyllie’s (‘time average’) equation
Wyllie et al. (1958) derived a relationship between velocity and porosity that fits data from well consoli- dated sandstones and limestones. The relation is essentially intuitive rather than based on physical principles. Wyllie’s equation relates the sonic interval transit time (i.e. the reciprocal of the seismic velocity, usually measured in μs/ft in sonic logs) to the weighted addition of interval transit times through the pore fluid and the mineral matrix. It is often referred to as the‘time-average’ equation):
t¼ ϕtf l+ð1 ϕÞt0, ð8:6Þ whereϕ is the porosity (as a fraction) and t, tfland t0 are the interval transit times in the rock, the fluid and the mineral matrix. Figure 8.4illustrates the form of Wyllie’s equation in the context of a selection of different (brine-bearing) sandstone data. It is evident that Wyllie’s equation falls at the top of the data- points, indicating that it is a model appropriate for stiff well consolidated sands. It is often observed to fit well with data from carbonate rocks (see chapter 7) where the mineralogy has a key role in establishing a relatively stiff rock frame.Figure 8.5illustrates some
carbonate data from Eberli et al. (2003). The best fit line drawn on the graph is close to a prediction using Wyllie’s equation. Intriguingly, high permeability car- bonates with moldic and intra-frame porosity plot above the trend and low permeability carbonates with microporosity and interparticle porosity plot below the trend (Eberli et al., 2003). In general, to quote Weger et al., (2009),‘carbonates with a large amount of microporosity, a complex pore structure (high specific surface), and small pores generally show low acoustic velocity at a given porosity. Samples with a simple pore structure (low specific surface) and large pores show high acoustic velocity for their porosity’.
When traditional porosity logs are not available, the Wyllie equation is sometimes used by petrophy- sicists as a way to calculate porosity:
ϕ ¼ t tð 0Þ= tf t0
ð8:7Þ Clearly the porosity estimate will be erroneous for rocks which fall below the Wyllie trend. An empirical correction to this porosity estimate that extends it to relatively unconsolidated rocks is to multiply it by 100/tsh, where tsh is the shale interval transit time at or near the depth of interest, inμs/ft.
An improved version of the Wyllie equation was published by Raymer et al. (1980):
V ¼ 1 ϕð Þ2V0+ϕVfl, ð8:8Þ
Table 8.1. Coefficients for Gardner’s relations. Note that, based on practical experience, the coeficients initially published for limestone have been amended fromd ¼ 1.50, f ¼ 0.225 to d ¼ 1.55, f ¼ 0.3 (H. Morris, personal communication).
Lithology d f Ss/sh avg 1.741 0.25 Shale 1.75 0.265 Sandstone 1.66 0.261 Limestone 1.55 0.3 Dolomite 1.74 0.252 Anhydrite 2.19 0.16 Porosity Vp (m/s) Wyllie Reuss 0 2000 2500 3000 3500 4000 4500 5000 5500 6000 0.1 0.2 0.3 0.4
Figure 8.4 Crossplot of porosity vs compressional velocity for various sandstone datasets: purple points– data from Han et al. (1986), dark blue points– Tertiary sandstone dataset from North Sea, dashed red line– Reuss mix of water and quartz, red and yellow lines– trends from Oseberg high porosity data (Dvorkin and Nur, 1996), blue lines– trends from Troll high porosity data (Dvorkin and Nur,1996) with upper line 20MPa effective pressure and lower line 5MPa effective pressure, red points– selected unconsolidated sand data.
where ϕ is the fractional porosity and V, V0 and Vfl are the velocities of the rock, the mineral matrix and the fluid.
The Raymer–Hunt–Gardner model is a flexible model that can be readily calibrated to measured data. An example of the application of the model in describing dispersed shale in fluvial sandstones is shown in Fig. 8.6. Of course the application of a model in this way requires accurate estimates of min- eralogy fractions and porosity.
Following the work of Raymer et al. (1980), Raiga- Clemenceau et al. (1988) provided an improvement in the prediction of porosity from sonic by including lithology-specific dependence: ϕ ¼ 1 t0 t 1 x , ð8:9Þ
where t0¼ matrix slowness, t ¼ sonic slowness and x is a constant dependent on lithology.
Common values for matrix slowness and the fit- ting coefficient‘x’ are shown inTable 8.2.
Interestingly, this sonic transform is based on the formulation:
t
t0¼ 1 ϕð Þ
x, ð8:10Þ
which is of a similar form to the relationship of resistivity and porosity (Archie,1942):
R Rw
¼ aϕm¼ F ð8:11Þ
where
R¼ true formation resistivity,
m¼ cementation component (seeTable 8.3for typical values),
a¼ constant, sometimes referred to as the tortuosity factor,
F¼ formation factor.
The relationship of sonic and resistivity measure- ments is discussed inSection 8.2.2.5.
Spikes and Dvorkin (2005) have noted that the Wyllie time average equation is not consistent with
7000 6000 5000 4000 3000 2000 1000 0 10 20 30 40 50 60 Porosity (%) V p m/s
Predominant pore type Microporosity
Interparticle/crystalline prorosity Density cemented
Moldic porosity Intraframe porosity
Figure 8.5 Crossplot of velocity (at 8 MPa effective pressure) versus porosity of various pore types of (brine filled) carbonates with an exponential best fit curve through the data for reference (re-drawn after Eberliet al.,2003).
0 10 20 30 4 12 10 8 6 14 Sand Shale Vcl 0 1 Total Porosity (%) Acoustic Impedance km/s.g/cc Shaley Sand Sandy Shale Raymer-Hunt Model
Figure 8.6 Use of the Raymer–Hunt empirical model to describe changes in porosity and clay content (Dvorkinet al.,2004).
the Gassmann (1951) fluid substitution equation. Therefore, the time average equation should only be used with brine as the fluid. Substituting to another fluid would require the use of the Gassmann formu- lation. The Raymer–Hunt–Gardner equation on the other hand is broadly consistent with Gassmann, sug- gesting that any pore fluid may be used with this model (Spikes and Dvorkin,2005).