Low-porosity and low-permeability reservoirs characterization using low-frequency seismic attribute

Abstract

The low-frequency seismic reflection is important for the characterization of hydrocarbon reservoirs. Previous research has proved that the frequency-dependent component of the low-frequency reflection coefficient is approximately proportional to the reservoir fluid mobility. A low-frequency seismic attribute is defined to extract an approximate measurement of reservoir fluid mobility from seismic reflection data. Based on the high-resolution time–frequency decomposition technology, we apply this seismic attribute to detect hydrocarbons in shale and igneous reservoirs with low porosity and low permeability. The application results illustrate that the low-frequency seismic attribute can not only indicate the spatial distribution of oil and gas, which can help design the optimal landing point and trajectory of directional drilling, but also be approximately proportional to the production of oil and gas, which can be used for resource evaluation during the early exploration stage. We also give a brief discussion on the working mechanism of the low-frequency seismic attribute. The present work may provide some references for the subsequent exploration and research of reservoirs with low porosity and low permeability.

Appendices

Appendix 1

Review of harmonic solution to Eq. (1)

Substituting Eq. (2) into (1) yields

$$\left\{ \begin{gathered} \omega^{2} \rho_{b} \left( {U^{F} E^{F} + U^{S} E^{S} } \right) - i\omega \rho_{f} \left( {W_{o}^{F} E^{F} + W_{o}^{S} E^{S} } \right){ = }\beta^{ - 1} \left[ {\left( {k^{F} } \right)^{2} U^{F} E^{F} + \left( {k^{S} } \right)^{2} U^{S} E^{S} } \right] - i\left( {k^{F} P^{F} E^{F} + k^{S} P^{S} E^{S} } \right), \hfill \\ \omega \left( {k^{F} U^{F} E^{F} + k^{S} U^{S} E^{S} } \right) + i\omega \phi \beta_{f} \left( {P^{F} E^{F} + P^{S} E^{S} } \right) = i\left( {k^{F} W_{o}^{F} E^{F} + k^{S} W_{o}^{S} E^{S} } \right), \hfill \\ \left( {1 + i\omega \tau } \right)\left( {W_{o}^{F} E^{F} + W_{o}^{S} E^{S} } \right){ = }i\kappa \eta^{ - 1} \left( {k^{F} P^{F} E^{F} + k^{S} P^{S} E^{S} } \right) + \omega^{2} \rho_{f} \kappa \eta^{ - 1} \left( {U^{F} E^{F} + U^{S} E^{S} } \right), \hfill \\ \end{gathered} \right.$$

(12)

where $E^{F} = \exp \left[ {i\left( {\omega t - k^{F} x} \right)} \right]$ and $E^{S} = \exp \left[ {i\left( {\omega t - k^{S} x} \right)} \right]$. All the unknown parameters (k, U, P, W_o) in Eq. (1), corresponding to the fast and slow P-waves, satisfy the following equation

$$\left\{ \begin{gathered} \left( {\omega^{2} \rho_{b} - k^{2} \beta^{ - 1} } \right)U + ikP - i\omega \rho_{f} W_{o} = 0, \hfill \\ k\omega U + i\omega \phi \beta_{f} P - ikW = 0, \hfill \\ \omega^{2} \rho_{f} \kappa \eta^{ - 1} U + ik\kappa \eta^{ - 1} P - \left( {1 + i\tau \omega } \right)W_{o} = 0. \hfill \\ \end{gathered} \right.$$

(13)

Simplify Eq. (13) to (14) by eliminating P.

$$\left\{ {\begin{array}{*{20}c} { - v^{2} \left( {\beta \rho_{f} } \right)\left( {\phi \beta_{f} \beta^{ - 1} } \right)\left( {\rho_{b} \rho_{f}^{ - 1} - 1} \right) + \phi \beta_{f} \beta^{ - 1} + 1 = \chi ,} \\ {v^{ - 2} \left( {\beta \rho_{f} } \right)^{ - 1} - \left( {\rho_{b} \rho_{f}^{ - 1} - 1} \right) = \left( {\tau \rho_{f}^{ - 1} \kappa^{ - 1} \eta - i\varepsilon^{ - 1} - 1} \right),} \\ \end{array} } \right.$$

(14)

where

$$v = \frac{\omega }{k}, \, \chi = \frac{{iW_{o} }}{\omega U}, \, \varepsilon { = }\rho_{f} \frac{\kappa }{\eta }\omega .$$

(15)

Introduce some new variables as follows

$$\lambda = \tau \rho_{f}^{ - 1} \kappa^{ - 1} \eta , \, v_{f}^{2} = \left( {\beta \rho_{f} } \right)^{ - 1} , \, \gamma_{\rho } = \left( {\rho_{b} \rho_{f}^{ - 1} - 1} \right), \, \gamma_{v} = \phi \beta_{f} \beta^{ - 1} , \, V = v^{2} v_{f}^{ - 2} ,$$

(16)

and turn Eq. (14) into (17).

$$\left\{ \begin{gathered} - V\gamma_{v}^{{}} \gamma_{\rho } + \gamma_{v}^{{}} + 1 = \chi , \hfill \\ V^{ - 1} - \gamma_{\rho } = \left( {\lambda - i\varepsilon^{ - 1} - 1} \right)\chi . \hfill \\ \end{gathered} \right.$$

(17)

Eliminate χ from Eq. (17) and acquire

$$\left[ { - i\varepsilon \gamma_{v} \gamma_{\rho } \left( {\lambda - 1} \right) - \gamma_{v} \gamma_{\rho } } \right]V^{2} + \left\{ {i\varepsilon \left[ {\left( {\lambda - 1} \right)\left( {\gamma_{v} + 1} \right) + \gamma_{\rho } } \right] + \left( {\gamma_{v} + 1} \right)} \right\}V - i\varepsilon = 0.$$

(18)

The expressions of exact solutions to Eq. (18) are quite cumbersome. It is advisable to seek asymptotic solutions. Due to ε → 0 in low-frequency domain, Silin et al. (2004) approximated the solutions as the power series with respect to iε, i.e.,

$$V = V_{0} + i\varepsilon V_{1} - \varepsilon^{2} V_{2} + ...$$

(19)

Substitute Eq. (19) into (18), abandon the higher-order terms of iε and attain

$$V^{F} = \frac{{\gamma_{v} + 1}}{{\gamma_{v} \gamma_{\rho } }} + i\varepsilon \frac{1}{{\gamma_{v} \left( {\gamma_{v} + 1} \right)}}, \, V^{S} = i\varepsilon \frac{1}{{\gamma_{v} + 1}}.$$

(20)

Substitute Eq. (20) into (17) to estimate χ.

$$\chi^{F} = - i\varepsilon \frac{{\gamma_{\rho } }}{{\gamma_{v} + 1}}, \, \chi^{S} = \gamma_{v} + 1 - i\varepsilon \frac{{\gamma_{v} \gamma_{\rho } }}{{\gamma_{v} + 1}}.$$

(21)

Finally, v, k, P and W_o can be estimated under the assumption that U is known, shown in Eqs. (3) and (4).

Appendix 2

Review of low-frequency reflection coefficient in porous media

Assuming that slow P-wave does not propagate and fast P-wave is normal incidence, we obtain Eq. (22) by virtue of Eqs. (4) and (6).

$$\left\{ {\begin{array}{*{20}l} {\left( {1 + R} \right)U_{1} = U_{2}^{F} + U_{2}^{S} ,} \hfill \\ {\frac{{k_{1} }}{{\beta_{1} }}\left( {1 - R} \right)U_{1} = \frac{{k_{2}^{F} }}{{\beta_{2} }}U_{2}^{F} + \frac{{k_{2}^{S} }}{{\beta_{2} }}U_{2}^{S} ,} \hfill \\ {k_{1} \frac{{1 - \chi_{1} }}{{\phi_{1} \beta_{{f_{1} }} }}U_{1} = k_{2}^{F} \frac{{1 - \chi^{F} }}{{\phi_{2} \beta_{{f_{2} }} }}U_{2}^{F} + k_{2}^{S} \frac{{1 - \chi^{S} }}{{\phi_{2} \beta_{{f_{2} }} }}U_{2}^{S} ,} \hfill \\ {\chi_{1} U_{1} = \chi_{2}^{F} U_{2}^{F} + \chi_{2}^{S} U_{2}^{S} .} \hfill \\ \end{array} } \right.$$

(22)

Simplify Eq. (22) by eliminating χ₁ to the following expression

$$\left( {\begin{array}{*{20}c} { - 1} & 1 & 1 \\ {\sqrt \varepsilon } & {X\sqrt \varepsilon } & {X\sqrt \varepsilon } \\ 0 & {\phi_{1} \phi_{2}^{ - 1} X\sqrt \varepsilon } & {Y + Z\sqrt \varepsilon } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} R \\ {U_{2}^{F} U_{1}^{ - 1} } \\ {U_{2}^{S} U_{1}^{ - 1} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} 1 \\ {\sqrt \varepsilon } \\ {\sqrt \varepsilon } \\ \end{array} } \right),$$

(23)

where $X = \frac{{\beta_{1} }}{{\beta_{2} }}\frac{{v_{1} }}{{v_{f} }}\sqrt {\frac{{\gamma_{v} \gamma_{\rho } }}{{\gamma_{v} + 1}}} \sqrt \varepsilon$, $Y = \frac{i - 1}{{\sqrt 2 }}\frac{{\phi_{1} \beta_{{f_{1} }} }}{{\phi_{2} \beta_{{f_{2} }} }}\frac{{v_{1} }}{{v_{f} }}\gamma_{v} \sqrt {\gamma_{v} + 1}$, $Z = \left( {\gamma_{v} + 1} \right)$, and v₁ = ω/k₁ is the velocity of incident P-wave, which can be viewed as a constant. By solving Eq. (23), we can attain the asymptotic expression for the low-frequency reflection coefficients of incident P-wave

$$R = \frac{{a - b - \left( {d + e} \right)\sqrt \varepsilon }}{c + b - e\sqrt \varepsilon },$$

(24)

where

$$\left\{ \begin{gathered} a = \frac{{v_{1} }}{{v_{f} }}\gamma_{v} \sqrt {\gamma_{v} + 1} \left( {\frac{{\beta_{1} }}{{\beta_{2} }} + \frac{{\phi_{1} \beta_{{f_{1} }} }}{{\phi_{2} \beta_{{f_{2} }} }}} \right), \, b = \frac{{\beta_{1} }}{{\beta_{2} }}\frac{{\phi_{1} \beta_{{f_{1} }} }}{{\phi_{2} \beta_{{f_{2} }} }}\frac{{v_{1}^{2} }}{{v_{f}^{2} }}\left( {\gamma_{v} + 1} \right)\sqrt {\gamma_{v} \gamma_{\rho } } , \, c = \frac{{\phi_{1} \beta_{{f_{1} }} }}{{\phi_{2} \beta_{{f_{2} }} }}\frac{{v_{1} }}{{v_{f} }}\gamma_{v} \sqrt {\gamma_{v} + 1} , \hfill \\ d = \frac{1 + i}{{\sqrt 2 }}\frac{{v_{1} }}{{v_{f} }}\frac{{\beta_{1} }}{{\beta_{2} }}\sqrt {\frac{{\gamma_{v} \gamma_{\rho } }}{{\gamma_{v} + 1}}} , \, e = \frac{1 - i}{{\sqrt 2 }}\left( {\gamma_{v} + 1} \right) - \frac{1 - i}{{\sqrt 2 }}\frac{{v_{1} }}{{v_{f} }}\sqrt {\frac{{\gamma_{v} \gamma_{\rho } }}{{\gamma_{v} + 1}}} \left[ {\frac{{\beta_{1} }}{{\beta_{2} }}\left( {\gamma_{v} + 1} \right) + \frac{{\phi_{1} \beta_{{f_{1} }} }}{{\phi_{2} \beta_{{f_{2} }} }}} \right]. \hfill \\ \end{gathered} \right.$$

(25)

Multiplying simultaneously the numerator and the denominator of Eq. (24) by (c + b + e√ε), and then ignoring the terms of order ε yields

$$R = \frac{a - b}{{c + b}} + \frac{{\left( {a - b} \right)e - \left( {d + e} \right)\left( {c + b} \right)}}{{\left( {c + b} \right)^{2} }}\sqrt \varepsilon ,$$

(26)

which can also be written as Eq. (7).