**1. Introduction: subject of rock physics, background, and brief history**

Rock physics is often called a "velocity-porosity" science. The idea behind this name is to predict the elastic-wave velocities in porous rock from its porosity or implement an inverse operation and interpret the velocity measured in a well or using seismic tomography or reflection techniques for the porosity of rock. It is important to mention that the elastic-wave velocities are related to the elastic moduli of rock as follows:

$$\mathbf{V}\_p = \sqrt{\frac{\mathbf{K} + \mathbf{4}/\mathbf{3G}}{\rho\_b}}; \mathbf{V}\_s = \sqrt{\frac{\mathbf{G}}{\rho\_b}}, \tag{1}$$

where *Vp* and *Vs* are the P- and S-wave velocities, respectively; *K* and *G* are the bulk and shear moduli, respectively; and *ρ<sup>b</sup>* is the bulk density. The latter quantity is related to the total porosity *ϕ* as

$$
\rho\_b = (\mathbf{1} - \phi)\rho\_s + \Phi\rho\_f,\tag{2}
$$

where *ρ<sup>s</sup>* is the density of the mineral matrix also called the solid component of the rock, while *ρ <sup>f</sup>* is the density of the pore fluid.

Important elastic constants used in rock physics are the bulk (*K*), shear (*G*), and compressional (*M*) moduli, as well as the P-wave (*Ip*) and S-wave (*Is*) impedances and Poisson's ratio (*ν*):

$$\mathcal{M} = \rho\_b V\_p^2; \quad \mathcal{G} = \rho\_b V\_s^2; \quad K = M - 4\sqrt{\mathcal{G}};$$

$$I\_p = \rho\_b V\_p; \quad I\_s = \rho\_b V\_s; \quad \nu = \frac{\mathbf{1}\left(V\_p / V\_s\right)^2 - 2}{2\left(V\_p / V\_s\right)^2 - \mathbf{1}}.\tag{3}$$

Most of natural rocks contain more than one mineral. In this situation, *ρ<sup>s</sup>* can be computed as the arithmetic average of the densities of the individual components:

$$
\rho\_s = \sum\_{i=1}^{N} f\_i \rho\_i,\tag{4}
$$

where *fi* is the volume fraction of the *i*-th mineral component in the mineral matrix and *ρ<sup>i</sup>* is its density. These individual densities can be found in handbooks, such as Mavko et al. [1]. They can vary between, e.g., 2.58 g/cc in clay and 4.93 g/cc in pyrite.

The same rule applies to the density of the pore fluid:

$$
\rho\_f = \mathbb{S}\_w \rho\_w + \mathbb{S}\_o \rho\_o + \mathbb{S}\_{\mathbb{S}} \rho\_{\mathbb{S}},\tag{5}
$$

where *Sw*, *So*, and *Sg* are the water, oil, and gas saturations in the pore space, respectively, and *ρw*, *ρo*, and *ρ<sup>g</sup>* are the densities of these pore fluid components. Of course, it is required that

$$\sum\_{i=1}^{N} f\_i = \mathbf{1} \tag{6}$$

and

$$\mathcal{S}\_w + \mathcal{S}\_o + \mathcal{S}\_\mathcal{g} = \mathbf{1}.\tag{7}$$

Because of the link between the elastic-wave velocities and elastic moduli as given by Eq. (1), it is often instructive to relate these elastic moduli to porosity. Such approach opens an avenue to using the so-called effective medium theories where the elastic moduli are theoretically related to porosity and the geometry of rock, referring to the spatial arrangement of pores and grains, as well as shapes of these pores and grains.

It has been discovered early that the velocity and elastic moduli not only depend on porosity, but also on the properties of the mineral frame. A rule of thumb is that at the same porosity, the softer the mineral frame, the smaller the elastic moduli of rock. For example, at the same porosity, rocks containing soft clays have velocities smaller than rocks dominated by stiffer quartz. Hence, rock physics is not only a "velocity-porosity" science but also a "velocity-porosity-mineralogy" science.

The situation becomes more complex if we consider the effects of the pore fluid on the elastic moduli (and velocities) of a porous composite. It is intuitively clear that the less compressible the pore fluid (water versus gas), the stiffer the entire rock, meaning that its bulk modulus is higher. Now we are talking about "velocityporosity-mineralogy-fluid."

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

The science of rock physics also includes understanding and quantification of other rock properties, such as hydraulic permeability and electrical resistivity, and their relation to other attributes, namely, porosity, rock texture, and mineralogy.

Generally, contemporary rock physics treats natural rock as a holistic object whose various properties (attributes) are extracted from experiments simulating processes, such as elastic-wave propagation, fluid and electrical transport, nuclear magnetic resonance (NMR), and breakage. We seek a theoretical understanding of interrelations between such attributes and their mathematical quantification. Such relations are also called rock physics models (RPM) or transforms. Needless to say that such quantification has to be "as simple as possible but not simpler."

Finally, the newest branch of rock physics is digital rock physics (DRP) whose mandate is to "image and compute," image rock at the pore scale and digitally simulate various processes within the digital image. For example, simulations of viscous fluid flow yield permeability, simulations of electrical charge transport yield resistivity, and simulations of deformation under stress yield the elastic moduli. Let us now review some of historic developments in rock physics.

Arguably, the first rock physics velocity-porosity transform was introduced by Wyllie et al. [2]. It simply states that the total P-wave traveltime through rock with porosity *ϕ* is the sum of the travel times through the mineral and fluid parts of the rock. This is why it is called the time-average equation. In terms of the P-wave velocities, this formulation is

$$\frac{1}{V\_p} = \frac{1-\phi}{V\_{\text{ps}}} + \frac{\phi}{V\_{\text{pf}}},\tag{8}$$

where *Vp* is the P-wave velocity, *V*ps is the velocity in the mineral phase, and *V*pf is that in the fluid phase. Examples for 100% quartz and 100% dolomite rock are shown in **Figure 1**. Also shown is an example for rock with mixed 50% quartz and 50% dolomite mineralogy. At the same porosity, *Vp* is highest in stiffer dolomite, lowest in softer quartz, and falls in between for the mixed mineralogy. The pore fluid was water with *V*pf = 1500 m/s.

Equation (8) is purely empirical in spite of its physically meaningful form. Indeed, in real rock, the mineral and fluid parts are not arranged in layers to enable a simple summation of the respective traveltimes. Still, this equation gives a reasonably accurate approximation for *Vp* in "fast" sediments as discussed in Mavko et al. [1]. Also note that it can only work for rock with liquid since in vacuum dry rock, *V*pf = 0. Yet, as have been shown by seismic experiments on the moon, *Vp* in such sediment is finite.

### **Figure 1.**

Vp *versus porosity according to the Wyllie et al. [2] and Raymer et al. [3] transforms for quartz, dolomite, and mixed mineralogy. Legend in the middle refers to all plots.*

Equation (8) has dominated petrophysical interpretation of velocity for porosity for a long time. It gave rise to the so-called sonic porosity computed from wireline velocity data as

$$\phi = \frac{V\_p^{-1} - V\_{\rm ps}^{-1}}{V\_{\rm pf}^{-1} - V\_{\rm ps}^{-1}}.\tag{9}$$

The next historic equation was introduced by Raymer et al. [3]:

$$V\_p = \left(\mathbf{1} - \boldsymbol{\phi}\right)^2 V\_{p^\sf s} + \boldsymbol{\phi} V\_{p^\sf f}. \tag{10}$$

As Eq. (8), it is purely empirical, derived from wireline data. Still, it is very meaningful as it can be applied to rock with any fluid inside, even where *V*pf = 0. As shown in Dvorkin et al. [4], it is more accurate than the Wyllie et al. [2] time average if applied to "fast" consolidated sediments. Velocity-porosity examples according to this equation are also shown in **Figure 1**.

We conclude this section by presenting equations relating the electrical resistivity to porosity and absolute hydraulic permeability to porosity.

The former transform relates the resistivity *Rt* of rock fully saturated with conductive fluid (brine) with resistivity *Rw* as

$$F = \frac{R\_{\rm f}}{R\_w} = \frac{1}{\phi^m},\tag{11}$$

where *F* is called the formation factor and *m* is the cementation exponent. In many sandstones *m* is approximately 2; however it may be much larger in carbonates [1]. **Figure 2** shows experimental data for Fontainebleau sandstone [5] with Eq. (11) curves for *m* = 1.5, 2.0, and 2.5 superimposed.

At partial brine saturation, *Sw* < 1, the resistivity of rock *R*tS not only depends on porosity but also on saturation *Sw* as

$$\frac{R\_{t\mathcal{S}}}{R\_w} = \frac{1}{\phi^m \mathcal{S}\_w^n} = \frac{F}{\mathcal{S}\_w^n},\tag{12}$$

### **Figure 2.**

*Left:* F *versus porosity according to Eq. (11) for* m *= 1.5, 2.0, and 2.5 with Fontainebleau experimental data shown as symbols. Right:* RtS*/*Rw *ratio versus water saturation for* ϕ *= 0.2 and* m *=* n *= 2.0 (Eq. (12)).*

**Figure 3.** *Permeability versus porosity plots as explained in the text.*

where *n* is the saturation exponent. This exponent is much more elusive than *m* since laboratory experiments measuring resistivity at partial saturation are scarce. Generally, *n* should be larger than 1.0 and approach 2.0. An example of *R*tS*=Rw* versus *Sw* is shown in **Figure 2** for porosity 0.2, *m* = 2.0, and *n* = 2.0.

Both Eqs. (11) and (12) were discovered by Archie in 1942 [6] and remain the cornerstone of resistivity interpretation for hydrocarbon saturation in the wellbore. Various modifications of these equations dealing with resistivity interpretation in sediments containing clays and shales are discussed in Mavko et al. [1].

The historic absolute permeability prediction equation is called the Kozeny-Carman [7] formula. It is based on an extremely idealized representation of pores as a set of parallel pipes inclined to the direction of pore pressure gradient at an angle *α*. The tortuosity *τ* of these pores is defined as

$$
\pi = \mathbf{1}/\cos a \ge \mathbf{1}.\tag{13}
$$

The permeability *k* is also a function of the specific surface area *S* defined as the ratio of surface of the pore space *S*Pore to the total volume *V* of the rock sample:

$$\mathcal{S} = \mathcal{S}\_{\text{Pore}} / V. \tag{14}$$

A variable alternative to *S* is the grain size (or grain diameter) *d*. The Kozeny-Carman equation reads [1]

$$k = \frac{1}{2} \frac{\phi^3}{S^2 \tau^2} = \frac{1}{72} d^2 \frac{\phi^3}{(1 - \phi)^2 \tau^2}. \tag{15}$$

A modified version of this equation is based on the assumption that *k* becomes zero not at zero porosity but at a finite and very small porosity value *ϕ<sup>p</sup>* called the percolation porosity:

$$k = \frac{1}{72} d^2 \frac{\left(\phi - \phi\_p\right)^3}{\left[1 - \left(\phi - \phi\_p\right)\right]^2 \tau^2}. \tag{16}$$

It follows from Eq. (15) that the unit of absolute permeability is length squared. However, traditionally, the permeability unit is Darcy (D) or milli-Darcy (mD). One D is 10�<sup>13</sup> m<sup>2</sup> , while one mD = 10�<sup>15</sup> m<sup>2</sup> .

**Figure 3** shows experimental permeability data for Fountainebleau sandstone and two North Sea sand sets with an Eq. (16) curve superimposed for *d* = 0.25 mm, *τ* = 2.5, and *ϕ<sup>p</sup>* = 0.02. This theoretical curve matches the Fountainebleau data, while the permeability from the other two datasets falls below this curves. The reason is the varying grain size as discussed in Mavko et al. [1].
