**2. Effect of pore fluid on elastic properties**

Laboratory experiments measuring the elastic-wave velocities in rock often show that the presence of the fluid in the pores strongly affects the elastic properties (**Figure 4**). Such dramatic results, especially for *Vp*, are in part due to the fact that such experiments are commonly conducted at very high frequencies, on the order of 1 MHz. In this frequency range, the fluid in the pores is "unrelaxed" and acts to strongly reinforce the soft mineral frame, thus increasing the bulk modulus (e.g., [1]).

Arguably, the most important contribution to rock physics is Gassmann's fluid substitution theory [9]. This theory allows us to compute the bulk modulus of porous rock filled with Fluid A if this modulus is known (measured) in the same rock but filled with Fluid B. These derivations were conducted under the assumption that the wave-induced pore pressure oscillations equilibrate within the sample over the wave period, meaning that Gassmann's is a low-frequency theory. Hence, it is applicable at the wireline and seismic frequency ranges. It helps predict the seismic response of rock filled with any hypothetical fluid if it is measured in situ where the pore fluid is known. For example, if the elastic properties of rock are measured in situ in rock 100% filled with water, we can predict these properties in the same rock but filled with oil or gas.

### **Figure 4.**

Vp *(left) and* Vs *(right) of high-porosity unconsolidated sand versus hydrostatic confining pressure. The pore pressure is constant 0.1 MPa. Squares are data obtained in ultrasonic pulse transmission experiments on the water-saturated sample. Circles are for the room-dry sample (after Zimmer [8]).*

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

Gassmann's theory provides the bulk modulus in fluid-saturated rock (*K*Sat) as a function of the dry rock bulk modulus (*K*Dry), the bulk modulus of the solid phase (*Ks*), that of the pore fluid (*K <sup>f</sup>* ), and total porosity (*ϕ*). It assumes that the shear modulus is fluid-independent

$$K\_{\rm Sat} = K\_s \frac{\phi K\_{D\gamma} - (1 + \phi) K\_f K\_{D\gamma\gamma} / K\_s + K\_f}{(1 - \phi) K\_f + \phi K\_s - K\_f K\_{D\gamma\gamma} / K\_s}, \quad G\_{\rm Sat} = G\_{\rm Dry}. \tag{17}$$

The latter equation can be rearranged as follows:

$$K\_{D\gamma} = K\_s \frac{\mathbf{1} - (\mathbf{1} - \phi)\mathbf{K}\_{\text{Sat}}/\mathbf{K}\_s - \phi\mathbf{K}\_{\text{Sat}}/\mathbf{K}\_f}{\mathbf{1} + \phi - \phi\mathbf{K}\_s/\mathbf{K}\_f - \mathbf{K}\_{\text{Sat}}/\mathbf{K}\_s}, \quad \mathbf{G}\_{D\gamma} = \mathbf{G}\_{\text{Sat}}.\tag{18}$$

Equations (17) and (18) provide us with a fluid substitution recipe as follows. Assume that we know the bulk modulus *K*SatA of rock saturated with Fluid A whose bulk modulus is *K*fA and density is *ρ*fA. Then from Eq. (17), we obtain:

$$K\_{D\gamma} = K\_s \frac{1 - (1 - \phi) K\_{\text{Sat}A} / K\_s - \phi K\_{\text{Sat}A} / K\_{f\text{A}}}{1 + \phi - \phi K\_s / K\_{f\text{A}} - K\_{\text{Sat}A} / K\_s} \,. \tag{19}$$

The bulk modulus *K*SatB of the same rock saturated with Fluid B is (Eq. (17)):

$$K\_{\rm SatB} = K\_s \frac{\phi K\_{\rm Dry} - (1 + \phi) K\_{\rm fB} K\_{\rm Dry} / K\_s + K\_{\rm fB}}{(1 - \phi) K\_{\rm fB} + \phi K\_s - K\_{\rm fB} K\_{\rm Dry} / K\_s},\tag{20}$$

where *K*fB is the bulk modulus of Fluid B.

Of course, the shear modulus of the rock remains the same, no matter what fluid it is saturated with.

It is important to remember that the bulk density *ρ<sup>b</sup>* of the rock is also a function of the pore fluid. It depends on the porosity and density of the fluid (*ρfA* or *ρfB*):

$$
\rho\_{\rm bB} = \rho\_{\rm bA} - \phi \rho\_{\rm fA} + \phi \rho\_{\rm fB}, \tag{21}
$$

where *ρbA* and *ρbB* are the bulk densities of the rock with the two pore fluids, respectively.

Finally, we can compute the elastic-wave velocities, as well as other seismic attributes, once we know the elastic moduli:

$$\mathbf{V}\_{pB} = \sqrt{\frac{\mathbf{K}\_{\text{SatB}} + 4\sqrt{\mathbf{G}\_{D\eta}}}{\rho\_{bB}}}; \quad \mathbf{V}\_{sB} = \sqrt{\frac{\mathbf{G}\_{D\eta}}{\rho\_{bB}}},\tag{22}$$

and

$$I\_{pB} = \rho\_{bB} V\_{pB}; \quad \nu\_B = \frac{\mathbf{1} \left(V\_{pB} / V\_{sB}\right)^2 - 2}{\left(V\_{pB} / V\_{sB}\right)^2 - \mathbf{1}},\tag{23}$$

where *I*pB and *ν<sup>B</sup>* are the P-wave impedance and Poisson's ratio of the rock filled with Fluid B, respectively. Although the shear modulus *G* is pore-fluidindependent, *Vs* is since the bulk density varies with varying fluid.

Let us refer to a later important development in theoretical fluid substitution. It stemmed from the fact that Gassmann's theory [9] requires the knowledge of the

### *Geophysics and Ocean Waves Studies*

bulk modulus that can only be computed using Eq. (1) if both *Vp* and *Vs* (and the bulk density *ρb*) are known. In practice, the shear wave velocity may not be available. To address this issue, Mavko et al. [10] derived an approximate (but quite accurate) *Vp*—only fluid substitution theory that uses the compressional modulus *<sup>M</sup>* <sup>¼</sup> *<sup>ρ</sup>bV*<sup>2</sup> *<sup>p</sup>* instead of the bulk modulus *K*. The functional form in this theory is the same as that in Gassmann's:

$$\begin{split} \mathcal{M}\_{\text{Sat}} &\approx \mathcal{M}\_{s} \frac{\phi \mathcal{M}\_{\text{Dry}} - (\mathbf{1} + \phi) \mathcal{K}\_{f} \mathcal{M}\_{\text{Dry}} / \mathcal{M}\_{s} + \mathcal{K}\_{f}}{(\mathbf{1} - \phi) \mathcal{K}\_{f} + \phi \mathcal{M}\_{s} - \mathcal{K}\_{f} \mathcal{M}\_{\text{Dry}} / \mathcal{M}\_{s}}, \\ \mathcal{M}\_{\text{Dry}} &\approx \mathcal{M}\_{s} \frac{\mathbf{1} - (\mathbf{1} - \phi) \mathcal{M}\_{\text{Sat}} / \mathcal{M}\_{s} - \phi \mathcal{M}\_{\text{Sat}} / \mathcal{K}\_{f}}{\mathbf{1} + \phi - \phi \mathcal{M}\_{s} / \mathcal{K}\_{f} - \mathcal{M}\_{\text{Sat}} / \mathcal{M}\_{s}}. \end{split} \tag{24}$$

**Figure 5** shows an example of the results of fluid substitution (pure water) on the elastic properties of high-porosity sand measured in the laboratory [11] at roomdry conditions. Clearly, the pore fluid has a dramatic effect on Poisson's ratio. Such plots are basis for in situ fluid identification from seismic data.

Let us finally describe the details required in practical fluid substitution, specifically the computation of *Ks*, *ρs*, *K <sup>f</sup>* , and *ρ <sup>f</sup>* .

The elastic moduli of the multi-mineral rock matrix *Ks* and *Gs* can be obtained using Hill's average (e.g., [1]) as

$$K\_{\mathfrak{s}} = \frac{K\_V + K\_R}{2}, G\_{\mathfrak{s}} = \frac{G\_V + G\_R}{2}, \tag{25}$$

where

$$\begin{aligned} K\_V &= \sum\_{i=1}^N f\_i \mathbf{K}\_i, \mathbf{G}\_V = \sum\_{i=1}^N f\_i \mathbf{G}\_i, \\\ K\_R^{-1} &= \sum\_{i=1}^N f\_i \mathbf{K}\_i^{-1}, \mathbf{G}\_R^{-1} = \sum\_{i=1}^N f\_i \mathbf{G}\_i^{-1}, \end{aligned} \tag{26}$$

where *N* is the number of the mineral components, *fi* is the volume fraction of *i* th mineral, and *Ki* and *Gi* are the bulk and shear moduli of the *i* th component. The pure-mineral elastic moduli, as well as their densities, can be found in various sources, including Mavko et al. [1].

The bulk modulus of the pore fluid is

### **Figure 5.**

*Sand experimental data and fluid substitution. Left. The bulk and shear moduli versus confining pressure as measured (dry) and water-substituted using Gassmann's theory [9]. Middle.* Vp *and* Vs *versus confining pressure as measured (dry) and water-substituted. Right. The P-wave impedance versus Poisson's ratio as measured (dry) and water-substituted, color-coded by the confining pressure.*

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

$$\frac{1}{K\_f} = \frac{\mathcal{S}\_w}{K\_w} + \frac{\mathcal{S}\_o}{K\_o} + \frac{\mathcal{S}\_\mathcal{g}}{K\_\mathcal{g}},\tag{27}$$

where *Kw*, *Ko*, and *Kg* are the bulk moduli of water, oil, and gas, respectively. To estimate these moduli, as well as the densities used in Eq. (5), we refer to [12].
