Impact of diagenesis and pore aspects on the petrophysical and elastic properties of carbonate rocks from southern Lebanon

Abstract

Carbonate rocks form under widely variable depositional settings and are susceptible to complicated diagenetic processes which influence their petrophysical and elastic properties. Understanding the controlling factors on these rock parameters is crucial for the interpretation of sonic logs and seismic reflection profiles. Herein, we collect a large number of carbonate rock samples exposed at southern Lebanon to examine the lithofacies, pore types, diagenetic processes, and their impact on the petrophysical and elastic properties. Collected samples belong to the Upper Cretaceous, Eocene, and Miocene limestone beds and are dominated by bioclastic packstone/wackestone from carbonate shelf deposits which are more susceptible to early diagenesis. Measured porosity is generally moderate to high and varies between 10 and 40%, with a pore system dominated by intraparticle, moldic, and vuggy pores. A small amount of porosity is represented by fractures and interparticle pores. Permeability is very low due to the dominance of isolated intraparticle porosity and small micropores. The measured density and the seismic wave velocities are low due to the moderate/high porosity and the presence of a significant (~ 17%) non-carbonate matrix. Many diagenetic features, such as micritization, cementation, compaction, and dissolution, impacted porosity and permeability differently (dependent on the pore throat size) and led also to the widely variable but generally low seismic wave velocities. The wide scatter observed in the porosity-velocity data cannot be explained solely by the microfacies, pore types, or mineralogy. Instead, we used effective medium theories to explain the variability of seismic velocities with porosity and the petrographic characteristics of the studied rocks. Modeling results show that, in addition to porosity, composition, rock texture, pore types, and the pore aspect ratios have significant impacts on the elastic properties of the studied samples which could explain the observed variations of seismic wave velocities at a given porosity. These findings are crucial for a better characterization of both onshore and offshore carbonate rocks which may host hydrocarbon, groundwater, and geothermal energy resources.

Appendix A

The differential effective medium (DEM) theory and the self-consistent approximation (SCA) are used to calculate the effective bulk modulus and Poisson’s ratio of heterogeneous rocks.

The coefficient P and Q used in Equations (1, 2, 3, 4) to calculate the effective young’s modulus and Poisson’s ratio depend on the inclusion shape and the inclusion aspect ratio. The equation used to calculate P and Q for a spherical, penny crack and ellipsoidal inclusions are detailed in A.1, A.2, and A.3, respectively (Mavko et al. 2009).

• A.1 For a spherical inclusion

  $$P=\frac{{K}_{m}+\frac{4}{3}{\mu }_{m}}{{K}_{i}+\frac{4}{3}{\mu }_{m}}$$
  $$Q=\frac{{\mu }_{m}+{\zeta }_{m}}{{\mu }_{i}+{\zeta }_{m}}$$

where $\zeta =\frac{\mu }{6}\frac{(9K+8\mu )}{(K+2\mu )}$.

• A.2 For a penny crack inclusion

  $$P=\frac{{K}_{m}+\frac{4}{3}{\mu }_{i}}{{K}_{i}+\frac{4}{3}{\mu }_{i}+\pi \alpha {\beta }_{m}}$$
  $$Q=\frac{1}{5}\left[1+\frac{8{\mu }_{m}}{4{\mu }_{i}+\pi \alpha ({\mu }_{m}+2{\beta }_{m})}+2\frac{{K}_{i}+\frac{2}{3}({\mu }_{i}+{\mu }_{m})}{{K}_{i}+\frac{4}{3}{\mu }_{i}+\pi \alpha {\beta }_{m}}\right]$$

  where $\beta =\mu \frac{\left(3K+\mu \right)}{(3K+4\mu )}$ and $\alpha$ is the crack aspect ratio.

• A.3 For an ellipsoidal inclusion (Berryman 1980)

• $$P=\frac{1}{3}{T}_{iijj}$$
  $$Q=\frac{1}{5}({T}_{ijij}-\frac{1}{3}{T}_{iijj})$$

  where

  $${T}_{iijj}=3\frac{{F}_{1}}{{F}_{2}}$$
  $${T}_{ijij}-\frac{1}{3}{T}_{iijj}=\frac{2}{{F}_{3}}+\frac{1}{{F}_{4}}+\frac{{F}_{4}{F}_{5}+{F}_{6}{F}_{7}-{F}_{8}{F}_{9}}{{F}_{2}{F}_{4}}$$

  where

$${F}_{1}=1+A\left[\frac{3}{2}\left(\kern0.1500emf+\theta \right)-R\left(\frac{3}{2}f+\frac{5}{2}\theta -\frac{4}{3}\right)\right]$$

$$\begin{aligned}F_2=1&+A\left[1+\frac32\left(f+\theta\right)-\frac12R\left(3f+5\theta\right)\right]+B\left(3-4R\right)\\&+\frac12A(A+3B)(3-4R)\left[f+\theta-R(f-\theta+2\theta^2)\right]\end{aligned}$$

$${F}_{3}=1+A\left[1-\left(f+\frac{3}{2}\theta \right)+R\left(f+\theta \right)\right]$$

$${F}_{4}=1+\frac{1}{4}A\left[\kern0.1500emf+3\theta -R\left(\kern0.1500emf-\theta \right)\right]$$

$${F}_{5}=A\left[-f+R\left(f+\theta -\frac{4}{3}\right)\right]+B\theta (3-4R)$$

$${F}_{6}=1+A\left[1+f-R\left(f+\theta \right)\right]+B(1-\theta )(3-4R)$$

$${F}_{7}=2+\frac{1}{4}A\left[3f+9\theta -R\left(3f+5\theta \right)\right]+B\theta (3-4R)$$

$$F_8=A\left[1-2R+\frac12f\left(R-1\right)+\frac12\theta\left(5R-3\right)\right]+B(1-\theta)(3-4R)$$

$${F}_{9}=A\left[\left(R-1\right)f-R\theta \right]+B\theta (3-4R)$$

$$A=\frac{{\mu }_{i}}{{\mu }_{m}}-1$$

$$B=\frac{1}{3}\left(\frac{{K}_{i}}{{K}_{m}}-\frac{{\mu }_{i}}{{\mu }_{m}}\right)$$

$$R=\frac{(1-2{\nu }_{m})}{2(1-{\nu }_{m})}$$

$$f=\frac{{\alpha }^{2}}{1-{\alpha }^{2}}(2\theta -2)$$

$$\theta=\left\{\begin{array}{c}\frac\alpha{\left(\alpha^2-1\right)^{3/2}}\left[\alpha\left(\alpha^2-1\right)^{1/2}-\cosh^{-1}\alpha\right],for\;prolate\;spheroid\;i.e.,\alpha>1\\\frac\alpha{\left(1-\alpha^2\right)^{3/2}}\left[\cosh^{-1}\alpha-\alpha\left(1-\alpha^2\right)^{1/2}\right],for\;oblate\;spheroid\;i.e.,\alpha<1\end{array}\right.$$