Stress Dependency of Permeability Represented by an Elastic Cylindrical Pore-Shell Model: Comment on Zhu et al. (Transp Porous Med (2018) 122:235–252)

Abstract

Stress dependency of permeability of porous rocks is described by means of a theoretical elastic cylindrical pore-shell model. This model is developed based on a bundle of elastic capillary tubes representation of the preferential flow paths formed in heterogeneous porous rocks. The radial displacement caused in tubes by the pore fluid pressure applied over the surface of the elastic cylindrical flow tubes is expressed by a Lamé-type equation. The radial displacement is incorporated into the Kozeny–Carman relationship to determine the variation of the permeability of porous rocks by variation of the pore fluid pressure. The solution of this equation yields a semi-analytical equation which provides accurate correlations of the stress dependency of the permeability data of porous rocks. The errors associated with the previous formulation of this problem by Zhu et al. (Transp Porous Med 122:235–252, 2018) are explained in view of the present formulation.

Appendix A: Definition of Coefficients and Calculation of Parameters

Definition of the coefficients A, B, C, and D and the calculation of parameters E,$\upsilon$, c_p, and p_o are described in the following. Substituting Eq. (11) into Eq. (10) yields:

$$ \begin{aligned} \frac{{r_{0} }}{{r_{1} }} & = \frac{{\left[ {\left( {\frac{1 - \upsilon }{{E\alpha \phi_{\text{o}} c_{p} }}} \right)\alpha \phi_{\text{o}} \left[ {1 - c_{p} \left( {p_{\text{o}} - p_{1} } \right)} \right] + \left( {\frac{1 + \upsilon }{{E\alpha \phi_{\text{o}} c_{p} }}} \right)} \right]\left[ {p_{\text{o}} - \left( {p_{\text{o}} - p_{1} } \right)} \right] + \left( {\frac{{1 - \alpha \phi_{\text{o}} }}{{\alpha \phi_{\text{o}} c_{p} }}} \right) + \left( {p_{\text{o}} - p_{1} } \right)}}{{\left[ {\left( {\frac{{1 - \alpha \phi_{\text{o}} }}{{\alpha \phi_{\text{o}} c_{p} }}} \right) + \left( {p_{\text{o}} - p_{1} } \right)} \right]}} \\ & = \frac{{\left( {\frac{1 - \upsilon }{E}} \right)\left( {p_{\text{o}} - p_{1} } \right)^{2} + \left[ {1 - \frac{1 - \upsilon }{{Ec_{p} }} - \frac{1 + \upsilon }{{E\alpha \phi_{\text{o}} c_{p} }} - \left( {\frac{1 - \upsilon }{E}} \right)p_{\text{o}} } \right]\left( {p_{\text{o}} - p_{1} } \right) + \frac{{1 - \alpha \phi_{o} }}{{\alpha \phi_{\text{o}} c_{p} }} + \left( {1 - \upsilon + \frac{1 + \upsilon }{{\alpha \phi_{\text{o}} }}} \right)\frac{{p_{\text{o}} }}{{Ec_{p} }}}}{{\left( {\frac{{1 - \alpha \phi_{o} }}{{\alpha \phi_{o} c_{p} }}} \right) + \left( {p_{o} - p_{1} } \right)}} \\ & = \frac{{A\left( {p_{\text{o}} - p_{1} } \right)^{2} + B\left( {p_{\text{o}} - p_{1} } \right) + C}}{{D + \left( {p_{\text{o}} - p_{1} } \right)}} \\ \end{aligned} $$

(A.1)

Comparison of coefficients in Eq.(A.1) gives the following relationships:

$$ D = \frac{{1 - \alpha \phi_{\text{o}} }}{{\alpha \phi_{\text{o}} c_{p} }},\,{\text{or}}\,{\text{solving}}\,{\text{for}}\,c_{p} = \frac{{1 - \alpha \phi_{\text{o}} }}{{\alpha \phi_{\text{o}} D}} $$

(A.2)

$$ A = \frac{1 - \upsilon }{E},\,{\text{solve}}\,{\text{for}}\,\frac{1 + \upsilon }{E} = A + \frac{2\upsilon }{E},\,{\text{or}}\,{\text{for}}\,E = \frac{1 - \upsilon }{A}\,{\text{after}}\,{\text{calculating}}\,\upsilon \,{\text{as}}\,{\text{follows}} $$

(A.3)

$$ \begin{aligned} B & = 1 - \frac{1 - \upsilon }{{Ec_{p} }} - \frac{1 + \upsilon }{{E\alpha \phi_{\text{o}} c_{p} }} - \left( {\frac{1 - \upsilon }{E}} \right)p_{\text{o}} = 1 - \frac{1 + \upsilon }{{E\alpha \phi_{\text{o}} c_{p} }} - \left( {\frac{1 - \upsilon }{E}} \right)\left( {p_{\text{o}} + \frac{1}{{c_{p} }}} \right) \\ & = 1 - \frac{1 + \upsilon }{{E\alpha \phi_{\text{o}} c_{p} }} - A\left( {p_{\text{o}} + \frac{1}{{c_{p} }}} \right) = 1 - \left( {\frac{1 + \upsilon }{1 - \upsilon }} \right)\frac{A}{{\alpha \phi_{\text{o}} c_{p} }} - A\left( {p_{\text{o}} + \frac{1}{{c_{p} }}} \right) \\ \end{aligned} $$

(A.4)

Solution of the above equation gives:

$$ \upsilon = \frac{{\frac{1 - B}{A} - p_{\text{o}} - \frac{1}{{c_{\text{p}} }} - \frac{1}{{\alpha \phi_{\text{o}} c_{\text{p}} }}}}{{\frac{1 - B}{A} - p_{\text{o}} - \frac{1}{{c_{\text{p}} }} + \frac{1}{{\alpha \phi_{\text{o}} c_{\text{p}} }}}} $$

(A.5)

Alternatively, consider

$$ \begin{aligned} & C = \frac{{1 - \alpha \phi_{\text{o}} }}{{\alpha \phi_{\text{o}} c_{\text{p}} }} + \left( {1 - \upsilon + \frac{1 + \upsilon }{{\alpha \phi_{\text{o}} }}} \right)\frac{{p_{\text{o}} }}{{Ec_{p} }} = D + \left( {1 - \upsilon + \frac{1 + \upsilon }{{\alpha \phi _{\text{o}} }}} \right)\frac{{p_{\text{o}} }}{{Ec_{p} }} = D + \left( {\frac{1 + \upsilon }{{E\alpha \phi _{\text{o}} c_{p} }}} \right)p_{\text{o}} + \frac{{Ap_{\text{o}} }}{{c_{p} }} \\ & \quad = D + \left( {\frac{1 + \upsilon }{1 - \upsilon }} \right)\frac{{Ap_{\text{o}} }}{{\alpha \phi _{\text{o}} c_{p} }} + \frac{{Ap_{\text{o}} }}{{c_{p} }} = D + \left[ {\left( {\frac{1 + \upsilon }{1 - \upsilon }} \right)\frac{1}{{\alpha \phi _{\text{o}} }} + 1} \right]\frac{{Ap_{\text{o}} }}{{c_{p} }},\,{\text{or}}\,{\text{rearranging}} \\ & \left[ {\frac{{\left( {C - D} \right)c_{p} }}{{Ap_{\text{o}} }} - 1} \right]\alpha \phi _{\text{o}} - 1 = \left[ {\frac{{\left( {C - D} \right)c_{p} }}{{Ap_{\text{o}} }} - 1} \right]\alpha \phi _{\text{o}} \upsilon + \upsilon = \left\{ {\left[ {\frac{{\left( {C - D} \right)c_{p} }}{{Ap_{\text{o}} }} - 1} \right]\alpha \phi _{\text{o}} + 1} \right\}\upsilon \\ \end{aligned} $$

(A.6)

Then, the solution of the above equation gives:

$$ \upsilon = \frac{{\left[ {\frac{{\left( {C - D} \right)c_{p} }}{{Ap_{\text{o}} }} - 1} \right]\alpha \phi_{\text{o}} - 1}}{{\left[ {\frac{{\left( {C - D} \right)c_{p} }}{{Ap_{\text{o}} }} - 1} \right]\alpha \phi_{\text{o}} + 1}} $$

(A.7)

Further,

$$ B = 1 - \frac{1 + \upsilon }{{E\alpha \phi_{\text{o}} c_{p} }} - A\left( {p_{\text{o}} + \frac{1}{{c_{p} }}} \right),\,{\text{solving}}\,{\text{for}}\,\frac{1 + \upsilon }{{E\alpha \phi_{\text{o}} c_{p} }} = 1 - B - A\left( {p_{\text{o}} + \frac{1}{{c_{p} }}} \right) $$

(A.8)

Substitute Eq.(A.8) into Eq.(A.6) to obtain the following equation to solve for p_o if it is not already known:

$$ \begin{aligned} C & = D + \left( {\frac{1 + \upsilon }{{E\alpha \phi_{\text{o}} c_{p} }}} \right)p_{\text{o}} + \frac{{Ap_{\text{o}} }}{{c_{p} }} = D + \left[ {1 - B - A\left( {p_{\text{o}} + \frac{1}{{c_{p} }}} \right)} \right]p_{\text{o}} + \frac{{Ap_{\text{o}} }}{{c_{p} }} \\ & = D + \left( {1 - B - Ap_{\text{o}} } \right)p_{\text{o}} \\ \end{aligned} $$

(A.9)