Stress-Dependent Porosity and Permeability of Porous Rocks Represented by a Mechanistic Elastic Cylindrical Pore-Shell Model

Abstract

The stress dependency of the porosity and permeability of porous rocks is described theoretically by representing the preferential flow paths in heterogeneous porous rocks by a bundle of tortuous cylindrical elastic tubes. A Lamé-type equation is applied to relate the radial displacement of the internal wall of the cylindrical elastic tubes and the porosity to the variation of the pore fluid pressure. The variation of the permeability of porous rocks by effective stress is determined by incorporating the radial displacement of the internal wall of the cylindrical elastic tubes into the Kozeny–Carman relationship. The fully analytical solutions of the mechanistic elastic pore-shell model developed by combining the Lamé and Kozeny–Carman equations are shown to lead to very accurate correlations of the stress dependency of both the porosity and the permeability of porous rocks.

Appendix: Semi-analytical Formulation of Stress-Dependent Porosity and Permeability of Porous Rocks

The stress dependence of porosity was approximated by the following empirical exponential decay equation by Zhu et al. (2018) and later by Civan (2019a). The linear dependence on the effective stress was expressed by a truncated Taylor series expansion and then using Eq. (10) as:

$$ \phi = \phi_{\text{o}} \exp \left[ { - c_{\text{p}} \left( {p_{\text{o}} - p} \right)} \right] \cong \phi_{\text{o}} \left[ {1 - c_{\text{p}} \left( {p_{\text{o}} - p} \right)} \right] = \phi_{\text{o}} \left[ {1 + c_{\text{p}} \left( {\frac{{\sigma_{\text{o}} - \sigma }}{\rm X}} \right)} \right] $$

(A-1)

Consequently, by applying Eq. (A-1), Civan (2019a) derived the following semi-analytic equation for stress-dependent permeability:

$$ \frac{{D^{{\prime }} + \sigma }}{{K^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} }} = a^{{\prime }} \sigma^{2} + b^{{\prime }} \sigma + c^{{\prime }} $$

(A-2)

where a′, b′, c′, and D′ are some parameters.

Here, it is shown that Eq. (A-1) can be derived also by neglecting several terms in Eq. (14) and applying Eqs. (2) and (3) as the following.

$$ \begin{aligned} \sigma & = p_{\text{o}} {\rm X} + \sigma_{\text{o}} - \frac{{E{\rm X}}}{1 - \upsilon }\left[ {\frac{{r_{0} r_{2}^{2} - r_{0} r_{1}^{2} - r_{1} r_{2}^{2} + r_{1}^{3} }}{{r_{1}^{3} + \left( {\frac{1 + \upsilon }{1 - \upsilon }} \right)r_{2}^{2} r_{1} }}} \right] \\ & \cong p_{\text{o}} {\rm X} + \sigma_{\text{o}} - \frac{{E{\rm X}}}{1 - \upsilon }\left[ {\frac{{r_{1}^{3} }}{{\left( {\frac{1 + \upsilon }{1 - \upsilon }} \right)r_{2}^{2} r_{1} }}} \right] = p_{\text{o}} {\rm X} + \sigma_{\text{o}} - \frac{{E{\rm X}\alpha^{2} }}{{\left( {1 + \upsilon } \right)r_{2}^{2} }}\phi \\ \end{aligned} $$

(A-3)

Solving Eq. (A-3) for porosity $\phi$ yields the above given Eq. (A-1) as:

$$ \begin{aligned} & \phi = \frac{{p_{\text{o}} \left( {1 + \upsilon } \right)r_{2}^{2} }}{{E\alpha^{2} }}\left[ {1 + \frac{1}{{p_{\text{o}} }}\left( {\frac{{\sigma_{o} - \sigma }}{\rm X}} \right)} \right] = \phi_{\text{o}} \left[ {1 + c_{\text{p}} \left( {\frac{{\sigma_{\text{o}} - \sigma }}{\rm X}} \right)} \right] \\ & {\text{where }} \\ & \phi_{\text{o}} = \frac{{p_{\text{o}} \left( {1 + \upsilon } \right)r_{2}^{2} }}{{E\alpha^{2} }},\quad c_{\text{p}} = \frac{1}{{p_{\text{o}} }} \\ \end{aligned} $$

(A-4)

This exercise reveals that Eq. (A-1) involves a significant simplification compared to the full mechanistic model developed in this paper.