**3. Variables influencing the elastic properties of rocks**

In addition to the pore fluid, there are two more important variables influencing the elastic properties of rocks, their mineralogy and the differential pressure *P*Diff (or stress) defined as the difference between the confining *P*Confining (the overburden) and pore pressure *P*Pore:

$$P\_{\text{Diff}} = P\_{\text{Conffining}} - P\_{\text{Pore}}.\tag{28}$$

Of course there are other influencing factors, such as rock texture (clastics versus carbonates versus unconventional shale), temperature, and diagenetic history. Here we only concentrate on the abovementioned two.

**Mineralogy.** As an example, let us examine the Han [13] laboratory dataset obtained on a large suite of sandstones with porosity ranging from zero to 30% and clay content between zero and 50%. **Figure 6** shows *Vp* and *Vs* versus porosity and color-coded by the clay content.

### **Figure 6.**

*Dry rock* Vp *(top) and* Vs *(bottom) versus porosity, color-coded by the clay content, at confining pressure 50 MPa (left) and 5 MPa (right) (after Han [13]).*

Obviously, the clay content plays a dramatic role acting to reduce both *Vp* and *Vs* at the same porosity. Also notice that the velocity-porosity-mineralogy trends are much more pronounced at 50 MPa. This is a commonly observed effect due to much clearer manifestations of key rock properties at high confining stress. The highporosity data point in **Figure 6** at porosity about 0.33 is for unconsolidated Ottawa sand sample. The effect of pressure on its velocities is very strong, similar to what we observe in **Figure 5** for a sand of different provenance.

Another striking example of velocity discrimination due to mineralogy comes from unconventional shale with data obtained by wireline logging in a vertical well (**Figure 7**). The data shown in this figure is for 100% wet rock, obtained by fluid substitution from in situ conditions. The velocity-porosity dataset forms an amorphous cloud (**Figure 7**, top) with both *Vp* and *Vs* varying by almost 1.5 km/s at the same porosity. However, as soon as we introduce a third variable, the sum of the clay and kerogen contents, we observe a clear velocity discrimination with the velocity decreasing as the fraction of this softest component of the solid matrix increasing (**Figure 7**, bottom).

The Raymer et al. [3] model also predicts a strong dependence of the velocity on mineralogy (**Figure 8**), as well as the pore fluid, the latter well pronounced at higher porosity.

**Stress.** The effect of the confining pressure on the velocity in sand can be clearly seen in **Figure 5** with *Vp* in dry rock increasing by about 2.0 km/s and *Vs* by 1.5 km/ s as *P*Diff varies from almost zero to 50 MPa. Two more examples are shown in

### **Figure 7.**

*100% wet rock* Vp *(left) and* Vs *(right) without accounting for mineralogy (top) and color-coded by the sum of clay and kerogen contents (bottom) (adopted from Dvorkin et al. [14]).*

### **Figure 8.**

Vp *versus porosity according to the Raymer et al. [3] model for dry rock (left) and 100% water-saturated rock (right). The mineralogy is quartz and clay. The upper curves is for zero clay content, while the bottom curve is for 100% clay. The in-between curves are for gradually increasing clay content with increment 10% (top to bottom).*

### **Figure 9.**

Vp *(left) and* Vs *(right) versus pressure for two dry sandstone samples from the North Sea (top) and Gulf of Mexico (bottom).*

**Figure 9**, one for a sandstone from the North Sea and the other from the Gulf of Mexico.

The velocity in carbonate rocks is often not as affected by stress as it is in clastic samples. The magnitude of this effect is often influenced by the presence of compliant cracks in the rock. Such cracks act to strongly affect the velocity at low

**Figure 10.**

*Same as Figure 9 but for two chalk samples from Ekofisk field in the North Sea. Velocities in sample A (porosity 0.38) is pressure dependent, while the velocities in sample B (porosity 0.31) hardly vary with pressure.*

pressure while they are open. As the pressure increases, these cracks close acting to increase the velocity (**Figure 10**, Sample A). In samples where the cracks are absent, the velocity hardly varies as a function of pressure (**Figure 10**, Sample B).

Notice that both historic velocity-porosity model by Wyllie et al. [2] and Raymer et al. [3] do not account for the dependence of the elastic-wave velocities on the confining stress. Both models are suitable for predicting the elastic properties at high, but not at low stress.

The velocity-stress dependence is important in understanding and predicting the seismic responses during hydrocarbon recovery, a process where the differential pressure may increase during production if the reservoir is depleted and the pore pressure is reduced, while the overburden remains constant. This differential pressure may decrease during enhanced oil recovery where water or gas are injected into the reservoir at high pressure, acting to reduce the difference between the overburden and pore pressure. Plots similar to that shown in **Figure 5** (right-hand frame) are useful in simultaneously assessing the effects of the pore fluid and differential pressure on the elastic attributes.
