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

All **grain-based theories** exploit the critical porosity concept. We start with the **contact-cement** theory where it is assumed that the grains are not subjected to any confining stress at *ϕc*, and, as a result, the elastic moduli are zero, and porosity reduction is due to cement rims enveloping the grains (**Figure 14**). Such contact cement acts to rapidly increase the elastic moduli of the grain pack due to the dramatically expanding contact areas between the grains as porosity decreases, as explained in Dvorkin et al. [4], where the theoretical equations are given as well. This model is only valid in the very high-porosity range.

The **soft-sand** model assumes that at the critical porosity and the elastic properties of the grain pack are given by the Hertz-Mindlin [16] contact theory. This theory assumes that the grain pack is made of identical spherical grains whose elastic properties are those of the mineral (solid) matrix as given by Eq. (25). Combined with the mean field approximation that assumes that all grains are subject to identical local stresses and have the same average number of contacts per

### **Figure 13.**

*Upper and modified upper elastic bounds for the compressional (left) and shear (right) moduli versus porosity. The critical porosity is 0.36. Data are from Han's [13] sandstone dataset for the clay content below 7% and with the elastic-wave velocities measured on dry samples at 50 MPa confining pressure.*

### **Figure 14.**

*Schematic modes of porosity reduction. From top to bottom: Contact-cement and stiff-sand model; soft-sand model; and constant-cement model (adopted from Dvorkin et al. [4]).*

grain *n* (also called the coordination number), the respective dry rock bulk (*K*HM) and shear (*G*HM) moduli are

$$K\_{HM} = \left[\frac{n^2(\mathbf{1} - \boldsymbol{\phi}\_c)^2 G\_s^2}{18\pi^2(\mathbf{1} - \boldsymbol{\nu}\_s)^2} P\right]^{\frac{1}{\frac{1}{2}}}, \quad G\_{HM} = \frac{5 - 4\boldsymbol{\nu}\_s}{5(2 - \boldsymbol{\nu}\_s)} \left[\frac{3n^2(\mathbf{1} - \boldsymbol{\phi}\_c)^2 G\_s^2}{2\pi^2(\mathbf{1} - \boldsymbol{\nu}\_s)^2} P\right]^{\frac{1}{\frac{1}{2}}},\tag{31}$$

where *P* is the differential pressure (Eq. (28)) and *Gs* and *ν<sup>s</sup>* are the shear modulus and Poisson's ratio of the solid matrix, respectively. This model implies that porosity reduction is not due to contact-cement deposition but instead due to smaller particles deposited away from grain contacts (**Figure 14**).

The coordination number *n* in an identical grain pack at the critical porosity is about 6.

It is assumed in Eq. (31) that the grains have infinite friction (no slip) at their contacts. If we allow only the fraction *f* of these contacts to have infinite friction while the rest of the contacts are frictionless and can slip, the equation for *K*HM does not change but *G*HM becomes now

$$G\_{HM} = \frac{2 + \mathfrak{F} - \nu(\mathbf{1} + \mathfrak{F})}{\mathfrak{F}(2 - \nu)} \left[ \frac{3n^2(\mathbf{1} - \phi\_c)^2 G^2}{2\pi^2 (\mathbf{1} - \nu)^2} P \right]^{\frac{1}{5}}.\tag{32}$$

This parameter *f* is called the shear stiffness correction factor.

Finally, to obtain the dry rock bulk (*K*Soft) and shear (*G*Soft) moduli at any porosity *ϕ* < *ϕc*, we use the modified (critical porosity scaled) lower Hashin-Shtrikman bound (e.g., [4]):

$$\begin{aligned} K\_{\text{Sof}} &= \left[ \frac{\phi/\phi\_c}{K\_{\text{HM}} + \frac{4}{3} G\_{\text{HM}}} + \frac{1 - \phi/\phi\_c}{K + \frac{4}{3} G\_{\text{HM}}} \right]^{-1} - \frac{4}{3} G\_{\text{HM}}, \\ G\_{\text{Sof}} &= \left[ \frac{\phi/\phi\_c}{G\_{\text{HM}} + z\_{\text{HM}}} + \frac{1 - \phi/\phi\_c}{G + z\_{\text{HM}}} \right]^{-1} - z\_{\text{HM}}, \quad z\_{\text{HM}} = \frac{G\_{\text{HM}}}{6} \left( \frac{9K\_{\text{HM}} + 8G\_{\text{HM}}}{K\_{\text{HM}} + 2G\_{\text{HM}}} \right). \end{aligned} \tag{33}$$

It is important to emphasize that the critical porosity endpoints here do not necessarily have to be given by the Hertz-Mindlin contact theory. Alternatively, these values can be selected from experimental data. What is most important in this model is the usage of the "soft" connection between the two porosity endpoints.

An alternative "stiff" connection between the aforementioned endpoints is given by the modified upper Hashin-Shtrikman bound as

$$\begin{aligned} K\_{\text{Stiff}} &= \left[ \frac{\phi/\phi\_c}{K\_{HM} + \frac{4}{3}G\_s} + \frac{1 - \phi/\phi\_c}{K + \frac{4}{3}G\_s} \right]^{-1} - \frac{4}{3}G\_s, \\\ G\_{\text{Stiff}} &= \left[ \frac{\phi/\phi\_c}{G\_{HM} + z} + \frac{1 - \phi/\phi\_c}{G\_s + z} \right]^{-1} - z, \quad z = \frac{G\_s}{6} \left( \frac{9K\_s + 8G\_s}{K\_s + 2G\_s} \right), \end{aligned} \tag{34}$$

where, once again, *Gs* and *Ks* are the shear and bulk moduli of the solid matrix, respectively.

This stiff connection, also called the stiff-sand model, can serve to connect the contact-cement curve with the zero-porosity endpoint.

Yet another model belonging to this family is the constant-cement model. It assumes that the grains have initial contact cementation with further porosity

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

reduction due to the placement of small particles away from grain contacts (**Figure 14**). The functional form of this model is the same as in the soft-sand model (Eq. (33)) but with artificially high coordination number.

Examples of velocity-porosity curves according to the aforementioned grainbased theories are shown in **Figure 15**, where we assumed that both the grain and cement materials are pure quartz; *n* for the soft-sand model is 6, while it is 20 for the constant-cement model; and the differential pressure is 20 MPa. The shear stiffness correction factor is 1.

**Figure 16** shows an example of using the constant-cement model to describe the elastic behavior of unconventional gas shale, while **Figure 17** is an example of applying the stiff-sand model to carbonate reservoirs. The parameters of the models are provided in the captions. These two examples show that the grain-based theories given here are appropriate not only for clastic sediments but also in very different lithological settings.

**Figure 18** shows laboratory data obtained at 30 MPa confining pressure on dry high-porosity, almost pure-quartz sand samples from the North Sea. In this classic example, the higher-velocity dataset is contact-cemented turbidite sand, while the

**Figure 15.**

*Velocity-porosity curves according to the soft-sand, stiff-sand, contact-cement, and constant-cement models as explained in the text.*

### **Figure 16.**

Vp *(left) and* Vs *(right) versus porosity for gas shale from wireline data adjusted for 100% water saturation. The color code is the sum of the clay and kerogen volume fractions (red for high and blue for low). The model curves are computed to bound the data. These curves are from the constant-cement model with the coordination number 12, differential pressure 26 MPa, critical porosity 0.40, and shear stiffness correction factor 1 (adopted from Dvorkin et al. [14]).*

### **Figure 17.**

*Velocity- (left) and impedance-porosity (middle) plots showing chalk (gray) and lower-porosity carbonate (black) data points from wireline data adjusted for 100% water saturation. Graph on the left is the impedance versus Poison's ratio plot, also for 100% water saturation conditions. The curves are from the stiff-sand model with the coordination number 6, differential pressure 30 MPa, critical porosity 0.40, and shear stiffness correction factor 1. The two model curves are for the two slightly different properties of the pure calcite end member (adopted from Dvorkin and Alabbad [17]).*

### **Figure 18.**

Vp *(left) and* Vs *(right) versus porosity for two high-porosity sand datasets as explained in the text. The model curves marked in the plots are computed for 30 MPa differential pressure, critical porosity 0.40, coordination number 7, shear stiffness correction factor 1, and dry rock.*

lower-velocity dataset is friable and virtually uncemented sand. The former data can be matched by the contact-cement curves transitioning into the stiff-sand trajectories. The latter data are matched by the soft-sand curves.
