Numerical Analysis of Seismoelectric Conversion in Stratified Low-Permeability Porous Rocks

Abstract

A new set of complex dynamic localized response coefficients (LRC-NEW), related to the fast P waves, slow P waves, S waves, and electromagnetic (EM) waves, respectively, is derived to overcome the numerical inaccuracy occurring while computing the seismoelectric (SE) wave fields in stratified low-permeability porous media based on the reflectivity method. In addition, the dynamic electrokinetic coupling coefficient is modified to make it applicable for the electrical double layer of arbitrary thickness. After a thorough validation, we find that the EM waves simulated by using the known LRC expressions differ significantly with the analytical solutions when the static permeability is below 10^–15 m², while those simulated by using LRC-NEW achieve excellent fits with the analytical solutions even when the static permeability is as low as 10^–20 m². A sensitivity study demonstrates that an optimum static permeability exists where the amplitude of evanescent SE responses reaches its maximum. The normalized root-mean-square amplitude of the evanescent SE signals in low-permeability rocks such as shales is more sensitive to permeability than seismic waves, even when the interdependence of porosity and static permeability is considered. Sensitivity studies also manifest that evanescent SE responses are sensitive to salinity and water saturation even in low-permeability shales. Notably, the amplitude of the evanescent SE conversions increases monotonically with water saturation, which is much simpler than the relationship between the seismic amplitude and water saturation. Therefore, the SE method has the potential to characterize the permeability, salinity, and water saturation of low-permeability reservoirs.

Appendices

Appendix A: The Frequency-Dependent Coefficients Derived By Pride (1994)

The complex and frequency-dependent electrokinetic coupling coefficient L derived by Pride (1994) can be written as

$$L\left( \omega \right) = - \frac{\phi }{{\alpha_{\infty } }}\frac{{\varepsilon_{w} \zeta }}{{\eta_{w} }}\left( {1 - 2\frac{{\tilde{d}}}{\Lambda }} \right)\left[ {1 - i\frac{m}{4}\frac{\omega }{{\omega_{t} }}\left( {1 - 2\frac{{\tilde{d}}}{\Lambda }} \right)^{2} \left( {1 - i^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} \frac{{\tilde{d}}}{\delta }} \right)^{2} } \right]^{{ - \frac{1}{2}}} ,$$

(A1)

where $\phi$ is the porosity, $\alpha_{\infty }$ is the tortuosity, $\omega_{t}$ is the transition frequency, $\delta$ is the viscous skin depth and $\Lambda$ is the second fundamental porous material geometry term (Pride, 1994).

The frequency-dependent effective electrical permittivity $\tilde{\varepsilon }$ (Pride, 1994; Pride & Haartsen, 1996) can be expressed as

$$\tilde{\varepsilon }\left( \omega \right) = \varepsilon + \frac{i}{\omega }\sigma \left( \omega \right) - \tilde{\rho }\left( \omega \right)L^{2} \left( \omega \right),$$

(A2)

where

$$\sigma \left( \omega \right) = \frac{{\phi \sigma_{f} }}{{\alpha_{\infty } }}\left\{ {1 + \frac{2}{{\sigma_{f} \Lambda }}\left[ {C_{em} + C_{os} \left( \omega \right)} \right]} \right\},$$

(A3)

$$\tilde{\rho }\left( \omega \right) = \frac{i}{\omega }\frac{\eta }{\kappa \left( \omega \right)},$$

(A4)

where $\varepsilon$ is the permittivity, $\sigma_{f}$ is the pure electrolyte, $C_{em}$ and $C_{os} \left( \omega \right)$ are the electromigration conductance and electroosmotic conductance, The detailed expressions of $\sigma_{f}$, $C_{em}$, and $C_{os} \left( \omega \right)$ can be found in Pride (1994) (their Eqs. 72, 194, and 206).

The slowness of S wave and EM wave derived by Pride and Haartsen (1996) can be written as

$$s_{s,em}^{2} { = }\frac{1}{2}\left\{ {\frac{{\rho_{t} }}{G} + \mu \tilde{\varepsilon }\left( {1 + \frac{{\tilde{\rho }L^{2} }}{{\tilde{\varepsilon }}}} \right) \pm \sqrt {\left[ {\frac{{\rho_{t} }}{G} - \mu \tilde{\varepsilon }\left( {1 + \frac{{\tilde{\rho }L^{2} }}{{\tilde{\varepsilon }}}} \right)} \right]^{2} - 4\mu \frac{{\rho_{f}^{2} L^{2} }}{G}} } \right\},$$

(A5)

where

$$\rho_{t} \,{ = }\,\rho - \frac{{\rho_{f}^{2} }}{{\tilde{\rho }}}.$$

(A6)

The slowness of Pf wave and Ps wave (Pride & Haartsen, 1996) are defined as

$$s_{pf,ps}^{2}\, { = }\,\frac{1}{2}\left\{ {\gamma \mp \sqrt {\gamma^{2} - 4\frac{{\tilde{\rho }\rho }}{{MH - C{}^{2}}}\left( {\frac{{\rho_{t} }}{\rho } + \frac{{\tilde{\rho }L^{2} }}{{\tilde{\varepsilon }}}} \right)} } \right\},$$

(A7)

where

$$\gamma = \frac{{\rho M + \tilde{\rho }H\left( {1 + {{\tilde{\rho }L^{2} } \mathord{\left/ {\vphantom {{\tilde{\rho }L^{2} } {\tilde{\varepsilon }}}} \right. \kern-0pt} {\tilde{\varepsilon }}}} \right) - 2\rho_{f} C}}{{MH - C^{2} }}.$$

(A8)

Appendix B: Proof of the Mathematical Equivalence Between LRC-New and LRC-PH

Here, we will demonstrate that the LRC-NEW and LRC-PH are mathematically identical. By setting the determinant of Eq. (6) to zero, we can establish the following relationship:

$$G\tilde{\rho }\left( {s_{s,em}^{2} - \frac{\rho }{G}} \right)s_{s,em}^{2} - G\tilde{\rho }\left( {s_{s,em}^{2} - \frac{\rho }{G}} \right)\mu \tilde{\varepsilon } - \mu \left( {\frac{\eta L}{{\omega \kappa }}} \right)^{2} \rho + \mu \left( {\frac{\eta L}{{\omega \kappa }}} \right)^{2} Gs_{s,em}^{2} - \rho_{f}^{2} \mu \tilde{\varepsilon } + \rho_{f}^{2} s_{s,em}^{2} = 0,$$

(B1)

After rearranging Eq. (B1), it is possible to obtain

$$\frac{i}{\omega }\frac{\mu \eta GL}{\kappa }\frac{{\left( {s_{em}^{2} - {\rho \mathord{\left/ {\vphantom {\rho G}} \right. \kern-0pt} G}} \right)}}{{\left( {s_{em}^{2} - \mu \tilde{\varepsilon }} \right)}} = \frac{{\tilde{\rho }\left( {Gs_{em}^{2} - \rho } \right) + \rho_{f}^{2} }}{{\tilde{\rho }L}} = - \frac{{\rho_{f}^{{}} }}{{\beta_{em} }},$$

(B2)

and

$$\frac{i}{\omega }\frac{\eta G}{{\kappa \rho_{f}^{{}} }}\frac{{\left( {s_{s}^{2} - {\rho \mathord{\left/ {\vphantom {\rho G}} \right. \kern-0pt} G}} \right)}}{{\left( {s_{s}^{2} - \mu \tilde{\varepsilon }} \right)}} = - \frac{{\rho_{f}^{{}} }}{{s_{s}^{2} - \mu \tilde{\varepsilon }\left( {1 + {{\tilde{\rho }L^{2} } \mathord{\left/ {\vphantom {{\tilde{\rho }L^{2} } {\tilde{\varepsilon }}}} \right. \kern-0pt} {\tilde{\varepsilon }}}} \right)}} = - \beta_{s}.$$

(B3)

By substituting eqs (B2) and (B3) into Eqs. (9) and (10), respectively, we can deduce that

$$c_{em}^{NEW} = \frac{i\omega }{{\beta_{em} }} = \left( { - \frac{i\omega }{{\rho_{f}^{{}} }}} \right)*\left( {\frac{i}{\omega }\frac{\mu \eta GL}{\kappa }\frac{{s_{em}^{2} - {\rho \mathord{\left/ {\vphantom {\rho G}} \right. \kern-0pt} G}}}{{s_{em}^{2} - \mu \tilde{\varepsilon }}}} \right) = \frac{\mu \eta GL}{{\rho_{f}^{{}} \kappa }}\frac{{s_{em}^{2} - {\rho \mathord{\left/ {\vphantom {\rho G}} \right. \kern-0pt} G}}}{{s_{em}^{2} - \mu \tilde{\varepsilon }}} = c_{em}^{PH} .$$

(B4)

and

$$c_{s}^{NEW} = i\omega \mu L\beta_{s} = \left( { - i\omega \mu L} \right)*\left( {\frac{i}{\omega }\frac{\eta G}{{\rho_{f}^{{}} \kappa }}\frac{{s_{s}^{2} - {\rho \mathord{\left/ {\vphantom {\rho G}} \right. \kern-0pt} G}}}{{s_{s}^{2} - \mu \tilde{\varepsilon }}}} \right) = \frac{\mu \eta GL}{{\rho_{f}^{{}} \kappa }}\frac{{s_{s}^{2} - {\rho \mathord{\left/ {\vphantom {\rho G}} \right. \kern-0pt} G}}}{{s_{s}^{2} - \mu \tilde{\varepsilon }}} = c_{s}^{PH} .$$

(B5)

Similarly, if the determinant of the matrix in Eq. (12) vanishes, we can obtain

$$- \left( {Hs_{pf,ps}^{2} - \rho } \right)\left( {Ms_{pf,ps}^{2} - \tilde{\rho }} \right)\mu \tilde{\varepsilon } + \mu \left( {Hs_{pf,ps}^{2} - \rho } \right)\left( {\tilde{\rho }L} \right)^{2} + \left( {Cs_{pf,ps}^{2} - \rho_{f}^{{}} } \right)\mu \tilde{\varepsilon } = 0.$$

(B6)

By rearranging eq. (B6), it is possible to obtain the following relationship:

$$\frac{{Hs_{pf,ps}^{2} - \rho }}{{Cs_{pf,ps}^{2} - \rho_{f}^{{}} }} = \frac{{Cs_{pf,ps}^{2} - \rho_{f}^{{}} }}{{Ms_{pf,ps}^{2} - \tilde{\rho }\left( {1 + {{\tilde{\rho }L^{2} } \mathord{\left/ {\vphantom {{\tilde{\rho }L^{2} } {\tilde{\varepsilon }}}} \right. \kern-0pt} {\tilde{\varepsilon }}}} \right)}} = - \beta_{pf,ps}^{{}} .$$

(B7)

By substituting eq. (B7) into Eq. (13), we can deduce that

$$c_{pf,ps}^{NEW} = - \frac{\eta L}{{\kappa \tilde{\varepsilon }}}\beta_{pf,ps} = \frac{\eta L}{{\kappa \tilde{\varepsilon }}}\frac{{Hs_{pf,ps}^{2} - \rho }}{{Cs_{pf,ps}^{2} - \rho_{f}^{{}} }} = = c_{pf,ps}^{PH} .$$

(B8)