The menu on Rock Strength currently include ther following options.

  • Compute Dynamic Modulus
  • Static Youngs Modulus Correlations
  • Compute Static Bulk Modulus and Static Shear Modulus.
  • Biot Coefficient Correlations
  • Friction Angle Correlations
  • UCS Correlations
  • Tensile Strength

Each of these options is discussed in more detail in the sections below. 

 

A Comment on Porosity. Some of the equations shown below use porosity $\phi$ as an input curve. The literature from which these equations are sourced are often not specific about what sort of porosity this is, and it could either be a total porosity $\phi_t$ or an effective porosity $\phi_e$. The current version of the "Porosity and Grain Density computation" (see section 7.5) computes an Apparent Total Porosity curve called PHIA which should be able to provide a good estimate of Total Porosity in water filled formations.. The porosity section will get refined in a later version.


9.7.1  Compute Dynamic Modulus

The Shear Modulus $G_{dyn}$, Bulk Modulus $K_{dyn}$, Young's Modulus $E_{dyn}$ and Poisson's Ratio $\nu_{dyn}$ are functioned directly from the DT compressional, DT shear and Bulk Density curves as follows.

With the $\rho_b$ in units of gm/cm3 and slowness in units of us/ft, the resulting modulus will be in units of GPa.

$$\begin{align} G_{dyn} &= {{304.8 \cdot 304.8  \cdot \rho_b } \over {\Delta T_{shear}}^2} \tag{1} \\ \\  K_{dyn} &= {{304.8 \cdot 304.8 \cdot \rho_b}\over {\Delta T_{comp}}^2} - {{4  G_{dyn}} \over 3} \tag{2} \\ \\ E_{dyn} &= {{9 G_{dyn}   K_{dyn}}\over {G_{dyn} + 3   K_{dyn}}} \tag{3} \\ \\ {\nu}_{dyn} &= {{3 K_{dyn} - 2  G_{dyn}}\over {6  K_{dyn} + 2   G_{dyn}}} \tag{4} \end{align}$$

The dynamic Poisson's Ratio ($\nu_{dyn}$) computed by this method will be identical to the Poisson's Ratio computed directly from the compressional and shear DT curves (see section 7.6), and provides a confirmation of the computational process above.

A plot of the four computed curves is automatically produced.


9.7.2  Static Youngs Modulus Correlations

A number of correlations are provided to estimate the static Youngs Modulus from other curves. The correlations below are found in literature and give a wide range of responses, as can be seen from the graphs below, so it's important to find a correlation that is suitable for the dataset being processed. Ideally, a user defined correlation should be used where the user defined parameters are detemined through empirical analysis that are applicable to the dataset.

Unless otherwise noted, the $E_{dyn}$ and $E_{sta}$ in the equations below are in units of GPa. Within the software, unit conversion from GPa is done automatically, should it be required.

  • Morales 1993. See reference 2. The $E_{dyn}$ and $E_{sta}$ in these equations, as written below, have units of psi. $$\begin{align} \log E_{sta} &= 2.137 + 0.6612 \cdot \log E_{dyn} &&\text{for $\phi$ 0.10 to 0.15} \tag{5}\\ \log E_{sta} &= 1.829 + 0.6920 \cdot \log E_{dyn}  &&\text{for $\phi$ 0.15 to 0.25} \tag{6}\\ \log E_{sta} &= -0.4575 + 0.9402 \cdot \log E_{dyn} &&\text{for $\phi$ > 0.25 } \tag{7} \end{align}$$
  • Modified Morales 1997. This equation is a modification of the Morales method, by incorporating $\phi$ as a function variable, rather than having 3 seperate equations. The $\phi$ type is unknown, but suspect it may be $\phi_{eff}$. $$\begin{align} E_{sta} = E_{dyn}  \left( -2.21 \cdot \phi + 0.963 \right) &&\text{} \tag{8} \end{align}$$
  • Lacy 1997. See reference 4 and reference-1 section 2.5.5 for more details on these equations. The $E_{dyn}$ and $E_{sta}$ in these equations, as written below, have units of Mpsi. $$\begin{align}  E_{sta} &=0.018\cdot  {E_{dyn}}^2 + 0.422 \cdot  E_{dyn} &&\text{for general} \tag{9} \\  E_{sta} &=0.0293\cdot  {E_{dyn}}^2 + 0.4533 \cdot  E_{dyn} &&\text{for sandstone} \tag{10}\\  E_{sta} &=0.0428\cdot  {E_{dyn}}^2 + 0.233 \cdot  E_{dyn} &&\text{for shale} \tag{11} \end{align} $$
  • Plumb-Bradford 1998. See reference 3. $$ E_{sta} =0.0018 \cdot  {E_{dyn}} ^ {2.7} \tag{12} $$
  • Wang 1999. See references 5 and 7. $$\begin{align} E_{sta} &= 0.4145 \cdot E_{dyn} + 1.050 &&\text{for < 15 GPa (soft rock)} \tag{13} \\  E_{sta} &= 1.153 \cdot E_{dyn} - 15.2 &&\text{for > 15 GPa (hard rock)} \tag{14} \end{align} $$
  • Canady 2010. See reference 6. $$ E_{sta} = \ln \left( E_{dyn} + 1 \right) \left( E_{dyn} - 2 \right) / 4.5\tag{15} $$
  • User defined correlations. Four user-defined correlations are provided, that allow the values of $a0$ and $b0$ in the following equations to be selected by the user. The $\phi$ used in some of these equations can be either $\phi_t$ or $\phi_e$ depending on the user specifications. $$\begin{align} E_{sta} &= {a0} \cdot E_{dyn} + {b0} &&\text{defaults: a0=0.74, b0=5.568} \tag{16}\\  E_{sta} &= {a0} \cdot {E_{dyn}} ^ {b0} &&\text{defaults: a0=0.5036, b0=1.0} \tag{17}\\  E_{sta} &= {a0} \cdot \exp \left( {b0} \cdot \phi \right) &&\text{defaults: a0=50.77, b0=-17.8} \tag{18}\\ E_{sta} &= {a0} \cdot \left( 304.8 / \Delta T_{comp}  \right) ^ {b0}  &&\text{defaults: a0=0.076, b0=3.23} \tag{19} \end{align}$$

 

If the plot of the dynamic elastic modulus is present, then upon completion of the static Youngs's modulus computation using one of the available methods, that curve will be added to the plot.

Some comparisons of these correlations are shown below with $E_{sta}$ plotted as a function of $E_{dyn}$. 

The Morales, Lacy and Wang correlations are shown individually above. The Morales correlation includes 3 curves depending on the $\phi$, while the Modifed Morales is a function of both $E_{dyn}$ and  $\phi$ and generates a cluster of points that follow a trend curve.

The composite plot below additionally shows the Plumb-Bradford (Red curve) and Canady (orange curve)

   


9.7.3  Compute Static Bulk Modulus and Static Shear Modulus.

There are no good correlations between the static and dynamic Poisson's Ratio, so it is just assumed that they are the same. That is, use $multiplier = 1.0$ in the equation (20) below.

$$\nu_{sta} = \nu_{dyn} \times  {multiplier} \tag{20} $$

The static Bulk Modulus and static Shear Modulus are estimated from the static Youngs Modulus and the static Poissons Ratio as follows.

$$\begin{align}  G_{sta} &= {E_{sta} \over {2  \left( 1+ \nu_{sta} \right) }} \tag{21} \\  \\ K_{sta} &= {E_{sta} \over {3\left( 1- 2 \nu_{sta} \right) }}  \tag{22} \end{align}$$


9.7.4  Biot Coefficient Correlations

Refer to the comment regarding porosity $\phi$ at the top of this page.


9.7.5  Friction Angle Correlations

Refer to the comment regarding porosity $\phi$ at the top of this page.


9.7.6  UCS Correlations

A number of correlations are provided to estimate the UCS from other curves. The correlations provided include:

  • Plumb Sandstone-1998. See reference 3. $$ UCS = 2.280 + 4.1089 \cdot E_{sta} \tag{23}$$
  • Chang Sandstone-2006. See reference 8. $$ UCS = 46.2 \cdot \exp \left( 0.027 \cdot E_{sta} \right) \tag{24} $$
  • Vernick Sandstone-1993. See reference 9. The $\phi$ type is unknown, it could be either $\phi_t$ or $\phi_e$. $$ UCS = 254  {\left( 1 - 2.7  \phi \right)}^2 \tag{25}$$
  • User defined correlations. Six user-defined functions are provided. The values of $a0$ and $b0$ in the following equations are to be selected by the user. The $\phi$ used in some of these equations can be either $\phi_t$ or $\phi_e$ depending on the user specifications. $$\begin{align}  UCS &= {a0} \cdot {E_{sta}} ^ {b0} &&\text{defaults: a0=7.97, b0=0.91} \tag{26} \\ UCS &= {a0} \cdot {\phi} ^ {b0} &&\text{defaults: a0=2.922, b0=-0.96} \tag{27} \\  UCS &= {a0} \cdot \exp \left( {b0} \cdot E_{sta}\right) &&\text{defaults: a0=46.2, b0=0.027} \tag{28} \\  UCS &= {a0} \cdot \exp \left( {b0} \cdot \phi\right) &&\text{defaults: a0=277, b0=-10} \tag{29} \\  UCS &= {a0} \cdot \left( 1 - {b0} \cdot \phi \right) ^ 2 &&\text{defaults: a0=276, b0=3.0} \tag{30} \\ UCS &= {a0} \cdot  \left( 304.8 / \Delta T_{comp}  \right) ^ {b0} &&\text{defaults: a0=0.77, b0=2.93} \tag{31} \end{align}$$

The first three correlations are provided for example. The user-defined options will allow the use of the large number of possible correlations that are available in the literature.

If the plot of the dynamic elastic modulus is present, then upon completion of the UCS computation using one of the available methods, that curve will be added to the plot.


9.7.7  Tensile Strength

 Computes the tensile strength from the UCS.$$\begin{align} TSTR = factor \times UCS &&\text{default value of factor = 0.1} \tag{} \end{align}$$


9.7.8  References

  1. Zhang, J.J., JJ, Applied Petroleum Geomechanics.
  2. Morales, Marcinew. 1993 Fracturing of high-permeability formations: mechanical properties correlations. SPE 26561
  3. Bradford, Fuller, Thompson, Walsgrove 1988. Benefits of Assessing the Solids Production Risk in a North Sea Reservoir Using Elastoplastic Modeling. SPE 47360
  4. Lacy, 1997. Dynamic rock mechanics testing optimized for fracture design. SPE 38716.
  5. Nur, Wang. 1999. Seismic and Acoustic Velocities in Reservoir Rocks: Recent Developments, vol 10. Society of Exploration Geophysics.
  6. Canady. 2010. Method for full-range young’s modulus correction. SPE 143604
  7. Brotons, et-al, 2016. Improved correlation between the static and dynamic elastic modulus of different types of rocks.
  8. Chang, Zoback, Khaksar. 2006. Empircal Relations between rock strength and physical properties in sedimentary rocks. Journal of Petroleum Science & Engineering.
  9. Vernik, Bruno, Bovberg. 1993. Empirical relations between compressive strength and porosity of siliclastic rocks. Int. J. Rock Mech. Min. Sci. & Geomech.