**4. Theoretical velocity-porosity models**

There are two kinds of elastic moduli versus porosity effective medium models: (a) inclusion models and (b) grain-based models. The first kind models build a rock from the zero-porosity endpoint by placing inclusions into the solid matrix [1]. These models are perhaps relevant to some carbonate rocks where the pores appear as inclusions in calcite or dolomite matrix. The second kind assumes that the rock is formed by solid grains which comprise an uncemented grain pack at the highporosity endpoint (also called the critical porosity) and, as the porosity is reduced, the original pack is altered either by grain contact cement or by smaller grains deposited in the pore space between the original larger grains, or a combination of these two processes.

As an example of the **inclusion models**, consider the differential effective medium model (DEM), where spheroidal pores are placed inside the solid matrix. A spheroid is an ellipsoid with two large diameters equal to each other and the third diameter smaller or equal to these two. The ratio of the small to large diameter is called the aspect ratio *α*≤ 1. If the spheroid is a sphere, *α* = 1. The inputs are the bulk and shear moduli of the mineral matrix and those of the inclusions.

## *Rock Physics: Recent History and Advances DOI: http://dx.doi.org/10.5772/intechopen.92161*

**Figure 11** (top) shows how the bulk and shear moduli depend on the total porosity for pure calcite rock with the bulk and shear moduli of the mineral 76.8 and 32.0 GPa, respectively, and its density 2.71 g/cc. The pores are empty, meaning the bulk and shear moduli of the inclusions are zero. In the same figure (bottom), we plot the respective *Vp* and *Vs*. The aspect ratio is different for each of the curves shown. It is 1.00 for the upper curves and gradually decreases to 0.50, 0.20, 0.10, and 0.01 for the curves below. The smaller the aspect ratio, the smaller the elastic moduli and velocities at a fixed porosity.

**Figure 12** is the same as **Figure 11** except that we use a single aspect ratio 0.10 and compare the results for empty inclusions with those for water-filled inclusions where the bulk modulus is 2.25 GPa and density is 1.00 g/cc.

We observe that *both* the bulk and shear moduli increase for pores filled with water as compared to empty pores. So do *Vp* and *Vs*. This means that DEM is not consistent with Gassmann's fluid substitution theory [9] which predicts that the shear modulus is pore-fluid-independent and *Vs* reduces upon saturation due to increasing bulk density.

Notice that DEM curves connect two endpoints, one at zero porosity where the elastic moduli of rock are those of the mineral matrix and the other at 100% porosity where the elastic moduli are those of the inclusions (fluid in the pores). About three decades ago, Nur observed that most natural rocks simply do not exist in the entire zero to 100% porosity range. The maximum geologically plausible porosity for clastic rocks (sands and sandstones) is about 0.40. It may be higher in

### **Figure 11.**

*Elastic moduli (top) and velocities (bottom) versus porosity computed using DEM model for a pure calcite rock. The aspect ratio corresponding to the top curves is 1.00 and for the bottom curve 0.01. The aspect ratio gradually decreases to 0.50, 0.20, and 0.10 for the curves in between (top to bottom).*

### **Figure 12.**

*Same as Figure 11 but for a single aspect ratio 0.10 and for empty pores (black) and pores filled with water (blue).*

carbonates, such as chalks, that can have porosity up to 0.50. This porosity can be even higher for foam-like formations, such as volcanic rock (pumice) or artificially manufactured glass foam. This maximum porosity is called the **critical porosity**. This concept was formalized in Nur et al. [15].

One implication of the critical porosity concept is that the high-porosity endpoint should be at the critical porosity rather than at 100% porosity. It gave rise to the so-called modified elastic bounds. The simplest example is based on the upper elastic bound (also called the Voigt bound) for a composite made of two elastic components ("1" and "2") with the compressional and shear moduli *M*1, *G*<sup>1</sup> and *M*2, *G*2, respectively.

Assume that *M*<sup>2</sup> = *G*<sup>2</sup> = 0. Then the respective moduli of this composite (*M* and *G*) at porosity *ϕ* cannot exceed

$$M = (\mathbf{1} - \phi)M\_1; G = (\mathbf{1} - \phi)G\_1. \tag{29}$$

These two curves are plotted in **Figure 13**. In the same figure, we plot Han's [13] data for low-clay-content samples at 50 MPa confining pressure. These data fall way below the upper bound curves for pure quartz with *M*<sup>1</sup> = 96.6 GPa and *G*<sup>1</sup> = 45.0 GPa.

The modified bounds use the same equations, but with porosity scaled by the critical porosity *ϕc*:

$$\mathbf{M} = (\mathbf{1} - \phi/\phi\_c)\mathbf{M}\_1; \mathbf{G} = (\mathbf{1} - \phi/\phi\_c)\mathbf{G}\_1; \phi \le \phi\_c,\tag{30}$$

giving modified curves that are much closer to the data (**Figure 13**).
