Static and Dynamic Moduli of Malm Carbonate: A Poroelastic Correlation

Abstract

The static and poroelastic moduli of a porous rock, e.g., the drained bulk modulus, can be derived from stress–strain curves in rock mechanical tests, and the dynamic moduli, e.g., dynamic Poisson’s ratio, can be determined by acoustic velocity and bulk density measurements. As static and dynamic elastic moduli are different, a correlation is often required to populate geomechanical models. A novel poroelastic approach is introduced to correlate static and dynamic bulk moduli of outcrop analogues samples, representative of Upper-Malm reservoir rock in the Molasse basin, southwestern Germany. Drained and unjacketed poroelastic experiments were performed at two different temperature levels (30 and 60 $^\circ$C). For correlating the static and dynamic elastic moduli, a drained acoustic velocity ratio is introduced, corresponding to the drained Poisson’s ratio in poroelasticity. The strength of poroelastic coupling, i.e., the product of Biot and Skempton coefficients here, was the key parameter. The value of this parameter decreased with increasing effective pressure by about 56 $~\%$ from 0.51 at 3 MPa to 0.22 at 73 MPa. In contrast, the maximum change in P- and S-wave velocities was only 3 % in this pressure range. This correlation approach can be used in characterizing underground reservoirs, and can be employed to relate seismicity and geomechanics (seismo-mechanics).

Appendices

Appendixes

A Gassmann equation

Gassmann’s equation relates the drained to the undrained bulk moduli. Gassmann’s equations can be written in terms of the strength of the poroelastic coupling $\alpha B$ (Berryman 1999):

$$\begin{aligned} K_\mathrm{u}=\frac{K_\mathrm{d}}{1-\alpha B} \end{aligned}$$

(17)

$$\begin{aligned} \mu _\mathrm{d}=\mu _\mathrm{u}. \end{aligned}$$

(18)

This relation (Eq. 17) shows how the stiffness of a loaded rock is related to the strength of the poroelastic coupling. In principle, the drained and undrained shear moduli, i.e., $\mu _\mathrm{d}$ and $\mu _\mathrm{u}$, are identical (Berryman 1999). At low frequencies, the Gassmann equation relates the drained and undrained elastic moduli (Le Ravalec and Guéguen 1996).

B Picking the arrival-times of ultrasonic waves

A review of time-picking techniques is given by Sarout et al. (2009). The Akaike Information Criterion (AIC) was used for picking arrival times of ultrasonic waves. The AIC picker is a statistical function for which the global minimum defines the onset time of the signal, as described by Akaike (1974), Akazawa (2004), and Kurz et al. (2005):

$$\begin{aligned} \mathrm{AIC}(J)= J.\log (\mathrm{var}(R(1,J)))+(N-J-1).\log (\mathrm{var}(R(1+J,N))). \end{aligned}$$

(19)

This formulation applies two sliding ranges (windows) to the signal, where J is a counter range through the signal and N is the total number of data samples. $\log$ and var denote the logarithm and variance functions, and R(a, b) determines the interval range of the recorded voltage.

C Transient poroelasticity

The coupling of fluid mass diffusion with volumetric deformation results in a coupled pore pressure diffusion, i.e., the pore pressure diffusion is coupled with the rate of change of the volumetric strain. The diffusivity coefficient D can be written in terms of drained and undrained elastic moduli, the mobility ratio $\frac{k}{\mu _\mathrm{f}}$, and poroelastic coefficients to be (Guéguen and Boutéca 2004),

$$\begin{aligned} D=\frac{BK_\mathrm{u}}{\alpha } \frac{k}{\mu _\mathrm{f}}\frac{(K+\frac{4\mu }{3})}{(K_\mathrm{u}+\frac{4\mu }{3})}. \end{aligned}$$

(20)