Estimating Organic Enrichment in Shale Gas Reservoirs Using Elastic Impedance Inversion Based on an Organic Matter−Matrix Decoupling Method

Abstract

The accumulation of organic matter is the basis for gas generation and significantly affects the ultimate gas production in shale reservoirs. Estimation of organic enrichment using seismic data is essential for shale gas characterization. The commonly used correlations between elastic properties and organic matter content for a particular area are locally applicable but may not be workable for other zones. Herein, a general physics-based approach is proposed to predict organic enrichment in shales. An organic matter-matrix decoupling amplitude variation versus offset (AVO) formula is constructed to straightforwardly quantify seismic signatures of organic matter via an introduced organic matter-related factor (M_c). Then, the elastic impedance (EI) function is established from the decoupling AVO formula to compute M_c. The proposed EI inversion method is suitable for capturing organic enrichment, particularly in the case of inadequate petrophysics information for reliable evaluation of M_c using log data as a constraint in the inversion. The developed AVO formula and EI function regard the organic matter as solid pore-fillings, presenting a more reasonable model for organic shales. Numerical tests show that M_c exhibits enhanced sensitivity to organic matter content with respect to the regularly used elastic properties. The real data applications indicate that the estimated M_c agrees well with the gas production in horizontal development wells, suggesting that M_c is a good indicator of favorable gas areas. The proposed approach may have broader potential applications and can be extended to detect other fluids and solid-saturated hydrocarbon reservoirs such as shale oil, heavy oil, and gas hydrates.

Appendices

Appendix 1

Establishment of the Organic Matter–Matrix Decoupling AVO Formula

According to Russell et al. (2011), P- and S-wave velocities, V_P and V_S, can be expressed using Eqs. (1) and (2) as:

$$V_{P} = \sqrt {{{\left( {K_{{{\text{dry}}}} + \frac{4}{3}\mu_{{{\text{sat}}}} + M_{k} } \right)} \mathord{\left/ {\vphantom {{\left( {K_{{{\text{dry}}}} + \frac{4}{3}\mu_{{{\text{sat}}}} + M_{k} } \right)} \rho }} \right. \kern-0pt} \rho }} ,$$

(16)

$$V_{S} = \sqrt {{{\left( {\mu_{{{\text{dry}}}} + M_{\mu } } \right)} \mathord{\left/ {\vphantom {{\left( {\mu_{{{\text{dry}}}} + M_{\mu } } \right)} \rho }} \right. \kern-0pt} \rho }} ,$$

(17)

where ρ is the bulk density of shale.

Then, the two organic matter-related parameters in Eqs. (1) and (2) have the following forms:

$$M_{k} = K_{{{\text{sat}}}} - K_{{{\text{dry}}}} = \rho V_{P}^{2} - \frac{4}{3}\rho V_{S}^{2} - \left( {\gamma_{{{\text{dry}}}}^{{2}} - \frac{4}{3}} \right)\mu_{{{\text{dry}}}} ,$$

(18)

$$M_{\mu } = \mu_{{{\text{sat}}}} - \mu_{{{\text{dry}}}} = \rho V_{S}^{2} - \mu_{{{\text{dry}}}} ,$$

(19)

where γ_dry represents the dry frame’s V_P/V_S ratio.

Aki and Richards (2002) proposed a linearized approximation of the PP-wave reflectivity as follows:

$$R_{{{\text{PP}}}} \left( \theta \right) = \left( {1 + \tan^{2} \theta } \right)\frac{{\Delta V_{P} }}{{2\overline{V}_{P} }} + \left( {\frac{{ - 8\sin^{2} \theta }}{{\gamma_{{{\text{sat}}}}^{2} }}} \right)\frac{{\Delta V_{S} }}{{2\overline{V}_{S} }} + \left( {1 - \frac{{4\sin^{2} \theta }}{{\gamma_{{{\text{sat}}}}^{2} }}} \right)\frac{\Delta \rho }{{2\overline{\rho }}},$$

(20)

where θ is the average of the incidence and refraction angles; $\overline{V}_{{\text{P}}}$, $\overline{V}_{{\text{S}}}$, and $\overline{\rho }$ are the averaged velocities and bulk density across the boundary, and the bar “–” will be omitted in the following expressions for simplicity; ∆ denotes the difference in according parameters across the interface; γ_sat represents the averaged V_P/V_S ratio of rocks.

Based on Russell et al. (2011), Eq. (20) can be rearranged as the following forms:

$$R_{{{\text{PP}}}} \left( \theta \right) = \frac{{\frac{1}{2}\Delta \rho V_{{\text{P}}}^{2} + \frac{1}{2}\rho V_{{\text{P}}} \Delta V_{{\text{P}}} \sec^{2} \theta - 2\left( {\Delta \rho V_{{\text{S}}}^{2} + 2\rho V_{{\text{S}}} \Delta V_{{\text{S}}} } \right)\sin^{2} \theta }}{{\rho V_{{\text{P}}}^{2} }},$$

(21)

where the expression γ_sat = V_P/V_S is considered.

Treating γ_dry as a constant and applying the chain rule of the multivariable calculus to Eqs. (18) and (19) give:

$$\Delta M_{k} = \frac{{\partial M_{k} }}{{\partial V_{{\text{P}}} }}\Delta V_{P} + \frac{{\partial M_{k} }}{{\partial V_{{\text{S}}} }}\Delta V_{S} + \frac{{\partial M_{k} }}{\partial \rho }\Delta \rho + \frac{{\partial M_{k} }}{{\partial \mu_{{{\text{dry}}}} }}\Delta \mu_{{{\text{dry}}}} ,$$

(22)

$$\Delta M_{\mu } = \frac{{\partial M_{\mu } }}{{\partial V_{{\text{S}}} }}\Delta V_{S} + \frac{{\partial M_{\mu } }}{\partial \rho }\Delta \rho + \frac{{\partial M_{\mu } }}{{\partial \mu_{{{\text{dry}}}} }}\Delta \mu_{{{\text{dry}}}} .$$

(23)

Substituting Eqs. (18) and (19) into Eqs. (22) and (23) produces:

$$\Delta M_{k} = 2\rho V_{{\text{P}}} \Delta V_{{\text{P}}} + V_{{\text{P}}}^{2} \Delta \rho - \frac{4}{3}\left( {\Delta \rho V_{{\text{S}}}^{2} + 2\rho V_{{\text{S}}} \Delta V_{{\text{S}}} } \right) - \left( {\gamma_{{{\text{dry}}}}^{2} - \frac{4}{3}} \right)\Delta \mu_{{{\text{dry}}}} ,$$

(24)

$$\Delta M_{\mu } = \Delta \rho V_{{\text{S}}}^{2} + 2\rho V_{{\text{S}}} \Delta V_{{\text{S}}} - \Delta \mu_{{{\text{dry}}}} ,$$

(25)

Equations (24) and (25) can be further rearranged as follows:

$$\Delta \rho V_{{\text{S}}}^{2} + 2\rho V_{{\text{S}}} \Delta V_{{\text{S}}} = \Delta M_{\mu } + \Delta \mu_{{{\text{dry}}}} ,$$

(26)

$$\rho V_{{\text{p}}} \Delta V_{{\text{P}}} = \frac{1}{2}\left( {\Delta M_{k} - V_{{\text{P}}}^{2} \Delta \rho + \frac{4}{3}\Delta M_{\mu } + \gamma_{{{\text{dry}}}}^{2} \Delta \mu_{{{\text{dry}}}} } \right).$$

(27)

Substituting Eqs. (26) and (27) into Eq. (21), we have:

$$\begin{gathered} R_{{{\text{PP}}}} \left( \theta \right) = \left( {\frac{1}{4}\sec^{2} \theta } \right)\frac{{\Delta M_{k} }}{{\rho V_{{\text{P}}}^{2} }} + \left( {\frac{1}{3}\sec^{2} \theta - 2\sin^{2} \theta } \right)\frac{{\Delta M_{\mu } }}{{\rho V_{{\text{P}}}^{2} }} \\ + \left( {\frac{1}{4}\gamma_{{{\text{dry}}}}^{2} \sec^{2} \theta - 2\sin^{2} \theta } \right)\frac{{\Delta \mu_{{{\text{dry}}}} }}{{\rho V_{{\text{P}}}^{2} }} + \left( {\frac{1}{2} - \frac{1}{4}\sec^{2} \theta } \right)\frac{\Delta \rho }{\rho }. \\ \end{gathered}$$

(28)

Using µ_dry = µ_sat –M_µ according to Eq. (19) and setting µ = µ_sat by neglecting the subscript “sat” for simplicity, we further rearrange Eq. (28)as:

$$\begin{gathered} R_{{{\text{PP}}}} \left( \theta \right) = \left( {\frac{1}{4}\sec^{2} \theta } \right)\frac{{\Delta M_{k} }}{{\rho V_{{\text{P}}}^{2} }} + \left[ {\left( {\frac{1}{3} - \frac{{\gamma_{dry}^{2} }}{4}} \right)\sec^{2} \theta } \right]\frac{{\Delta M_{\mu } }}{{\rho V_{{\text{P}}}^{2} }} \\ + \left( {\frac{{\gamma_{{{\text{dry}}}}^{2} }}{4}\sec^{2} \theta - 2\sin^{2} \theta } \right)\frac{\Delta \mu }{{\rho V_{{\text{P}}}^{2} }} + \left( {\frac{1}{2} - \frac{1}{4}\sec^{2} \theta } \right)\frac{\Delta \rho }{\rho }. \\ \end{gathered}$$

(29)

Dividing Eqs. (18) and (19) by ρV_P² on both sides gives:

$$\frac{{M_{k} }}{{\rho V_{{\text{P}}}^{2} }} = 1 - \frac{{4V_{{\text{S}}}^{2} }}{{3V_{{\text{P}}}^{2} }} - \left( {\gamma_{{{\text{dry}}}}^{2} - \frac{4}{3}} \right)\frac{{\mu_{{{\text{dry}}}} }}{{\rho V_{{\text{P}}}^{2} }} = 1 - \frac{4}{{3\gamma_{{{\text{sat}}}}^{2} }} - \left( {\gamma_{{{\text{dry}}}}^{2} - \frac{4}{3}} \right)\frac{{\mu_{{{\text{dry}}}} }}{{\rho V_{{\text{P}}}^{2} }},$$

(30)

$$\frac{{M_{\mu } }}{{\rho V_{{\text{P}}}^{2} }} = \frac{{V_{{\text{S}}}^{2} }}{{V_{{\text{P}}}^{2} }} - \frac{{\mu_{{{\text{dry}}}} }}{{\rho V_{{\text{P}}}^{2} }} = \frac{1}{{\gamma_{{{\text{sat}}}}^{2} }} - \frac{{\mu_{{{\text{dry}}}} }}{{\rho V_{{\text{P}}}^{2} }}.$$

(31)

Substituting Eq. (31) into Eq. (30) produces:

$$\frac{{M_{k} }}{{\rho V_{{\text{P}}}^{2} }} = 1 - \frac{{4M_{\mu } }}{{3\rho V_{{\text{P}}}^{2} }} - \gamma_{{{\text{dry}}}}^{2} \frac{{\mu_{{{\text{dry}}}} }}{{\rho V_{{\text{P}}}^{2} }},$$

(32)

Equation (32) can be expressed considering µ_dry = µ_sat–M_µ = ρV_s² – M_µ:

$$\frac{{M_{k} }}{{\rho V_{{\text{P}}}^{2} }} = 1 - \frac{{4M_{\mu } }}{{3\rho V_{P}^{2} }} - \gamma_{{{\text{dry}}}}^{2} \frac{{\rho V_{S}^{2} }}{{\rho V_{P}^{2} }} + \gamma_{{{\text{dry}}}}^{2} \frac{{M_{\mu } }}{{\rho V_{{\text{P}}}^{2} }} = 1 - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{{\gamma_{{{\text{sat}}}}^{2} }} + \left( {\gamma_{{{\text{dry}}}}^{2} - \frac{4}{3}} \right)\frac{{M_{\mu } }}{{\rho V_{P}^{2} }}.$$

(33)

Equation (33) can be further rearranged as follows:

$$\frac{1}{{\rho V_{{\text{P}}}^{2} }} = \frac{{1 - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{{\gamma_{{{\text{sat}}}}^{2} }}}}{{M_{k} + \frac{4}{3}M_{\mu } - \gamma_{{{\text{dry}}}}^{2} M_{\mu } }}.$$

(34)

Substituting Eq. (34) into Eq. (29) produces:

$$\begin{gathered} R_{{{\text{PP}}}} \left( \theta \right) = \left[ {\left( {1 - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{{\gamma_{{{\text{sat}}}}^{2} }}} \right)\frac{1}{4}\sec^{2} \theta } \right]\frac{{\Delta M_{k} }}{{M_{k} + \frac{4}{3}M_{\mu } - \gamma_{{{\text{dry}}}}^{2} M_{\mu } }} + \left[ {\left( {\frac{1}{3} - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{4}} \right)\left( {1 - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{{\gamma_{{{\text{sat}}}}^{2} }}} \right)\sec^{2} \theta } \right]\frac{{\Delta M_{\mu } }}{{M_{k} + \frac{4}{3}M_{\mu } - \gamma_{{{\text{dry}}}}^{2} M_{\mu } }} \\ + \left( {\frac{{\gamma_{{{\text{dry}}}}^{2} }}{4}\sec^{2} \theta - 2\sin^{2} \theta } \right)\frac{\Delta \mu }{{\rho V_{{\text{P}}}^{2} }} + \left[ {\frac{1}{2} - \frac{1}{4}\sec^{2} \theta } \right]\frac{\Delta \rho }{\rho }. \\ \end{gathered}$$

(35)

Equation (35) is further rearranged using ρV_P² = γ_sat²ρV_S² = γ_sat²μ:

$$\begin{gathered} R_{{{\text{PP}}}} \left( \theta \right) = \left[ {\left( {1 - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{{\gamma_{{{\text{sat}}}}^{2} }}} \right)\frac{1}{4}\sec^{2} \theta } \right]\frac{{\Delta M_{k} }}{{M_{k} + \frac{4}{3}M_{\mu } - \gamma_{{{\text{dry}}}}^{2} M_{\mu } }} + \left[ {\left( {\frac{1}{3} - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{4}} \right)\left( {1 - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{{\gamma_{{{\text{sat}}}}^{2} }}} \right)\sec^{2} \theta } \right]\frac{{\Delta M_{\mu } }}{{M_{k} + \frac{4}{3}M_{\mu } - \gamma_{{{\text{dry}}}}^{2} M_{\mu } }} \\ + \left( {\frac{{\gamma_{{{\text{dry}}}}^{2} }}{{4\gamma_{{{\text{sat}}}}^{2} }}\sec^{2} \theta - \frac{2}{{\gamma_{{{\text{sat}}}}^{2} }}\sin^{2} \theta } \right)\frac{\Delta \mu }{\mu } + \left[ {\frac{1}{2} - \frac{1}{4}\sec^{2} \theta } \right]\frac{\Delta \rho }{\rho }. \\ \end{gathered}$$

(36)

Introducing the factor N = M_μ/M_k can rearrange Eq. (36) to give the ultimate form of the organic matter-matrix decoupling formula:

$$\begin{gathered} R_{{{\text{PP}}}} \left( \theta \right) = \left[ {\frac{1}{4}\left( {1 - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{{\gamma_{{{\text{sat}}}}^{2} }}} \right)\frac{{\sec^{2} \theta }}{{1 + 4N/3 - \gamma_{{{\text{dry}}}}^{2} N}}} \right]\frac{{\Delta M_{k} }}{{M_{k} }} + \left[ {\left( {\frac{N}{3} - \frac{{\gamma_{{{\text{dry}}}}^{2} N}}{4}} \right)\left( {1 - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{{\gamma_{{{\text{sat}}}}^{2} }}} \right)\frac{{\sec^{2} \theta }}{{1 + 4N/3 - \gamma_{{{\text{dry}}}}^{2} N}}} \right]\frac{{\Delta M_{\mu } }}{{M_{\mu } }} \\ + \left( {\frac{{\gamma_{{{\text{dry}}}}^{2} }}{{4\gamma_{{{\text{sat}}}}^{2} }}\sec^{2} \theta - \frac{2}{{\gamma_{{{\text{sat}}}}^{2} }}\sin^{2} \theta } \right)\frac{\Delta \mu }{\mu } + \left[ {\frac{1}{2} - \frac{1}{4}\sec^{2} \theta } \right]\frac{\Delta \rho }{\rho } \\ \end{gathered}$$

(37)

For fluid saturation, we have M_µ = 0 due to µ_dry = µ_sat in Eq. (19). Therefore, the factor N = M_μ/M_k = 0 in this case, making Eq. (37) expressed as:

$$R_{{{\text{PP}}}} \left( \theta \right) = \left[ {\frac{1}{4}\left( {1 - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{{\gamma_{{{\text{sat}}}}^{2} }}} \right)\sec^{2} \theta } \right]\frac{{\Delta M_{k} }}{{M_{k} }} + \left( {\frac{{\gamma_{{{\text{dry}}}}^{2} }}{{4\gamma_{{{\text{sat}}}}^{2} }}\sec^{2} \theta - \frac{2}{{\gamma_{{{\text{sat}}}}^{2} }}\sin^{2} \theta } \right)\frac{\Delta \mu }{\mu } + \left[ {\frac{1}{2} - \frac{1}{4}\sec^{2} \theta } \right]\frac{\Delta \rho }{\rho },$$

(38)

Equation (38) has the same form of the formula proposed by Russell et al. (2011), with the term ΔM_k /M_k equivalent to the fluid term Δf /f in their paper.

Appendix 2

EI Inversion Based on the Organic Matter-Matrix Decoupling AVO Formula

According to the theory of Connolly (1999), the PP-wave reflection coefficient is expressed by EI:

$$R_{{{\text{PP}}}} \left( \theta \right) = \frac{{{\text{EI}}_{n + 1} \left( \theta \right) - {\text{EI}}_{n} \left( \theta \right)}}{{{\text{EI}}_{n + 1} \left( \theta \right) + {\text{EI}}_{n} \left( \theta \right)}} = \frac{1}{2}\frac{{\Delta {\text{EI}}\left( \theta \right)}}{{{\text{EI}}\left( \theta \right)}}.$$

(39)

Substituting Eq. (39) into Eq. (7) gives:

$$\frac{1}{2}\frac{{\Delta {\text{EI}}\left( \theta \right)}}{{{\text{EI}}\left( \theta \right)}} = \left[ {\frac{1}{4}\left( {1 - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{{\gamma_{{{\text{sat}}}}^{2} }}} \right)\frac{{\sec^{2} \theta }}{{1 + 4N/3 - \gamma_{{{\text{dry}}}}^{2} N}}} \right]\frac{{\Delta M_{{\text{c}}} }}{{M_{{\text{c}}} }} + \left( {\frac{{\gamma_{{{\text{dry}}}}^{2} }}{{4\gamma_{{{\text{sat}}}}^{2} }}\sec^{2} \theta - \frac{2}{{\gamma_{{{\text{sat}}}}^{2} }}\sin^{2} \theta } \right)\frac{\Delta \mu }{\mu } + \left( {\frac{1}{2} - \frac{1}{4}\sec^{2} \theta } \right)\frac{\Delta \rho }{\rho }$$

(40)

Using the relationship $\frac{\Delta x}{x} \approx \Delta \ln x$ produces:

$$\Delta \ln \left[ {{\text{EI}}\left( \theta \right)} \right] = \left[ {\frac{1}{2}\left( {1 - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{{\gamma_{{{\text{sat}}}}^{2} }}} \right)\frac{{\sec^{2} \theta }}{{1 + 4N/3 - \gamma_{{{\text{dry}}}}^{2} N}}} \right]\Delta \ln M_{c} + \left( {\frac{{\gamma_{{{\text{dry}}}}^{2} }}{{2\gamma_{{{\text{sat}}}}^{2} }}\sec^{2} \theta - \frac{4}{{\gamma_{{{\text{sat}}}}^{2} }}\sin^{2} \theta } \right)\Delta \ln \mu + \left( {1 - \frac{1}{2}\sec^{2} \theta } \right)\Delta \ln \rho$$

(41)

By denoting the weighting coefficients in Eq. (41) as follows:

$$a\left( \theta \right) = \frac{1}{2}\left( {1 - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{{\gamma_{{{\text{sat}}}}^{2} }}} \right)\frac{{\sec^{2} \theta }}{{1 + 4N/3 - \gamma_{{{\text{dry}}}}^{2} N}}, \, \,\,\,b\left( \theta \right) = \frac{{\gamma_{{{\text{dry}}}}^{2} }}{{2\gamma_{{{\text{sat}}}}^{2} }}\sec^{2} \theta - \frac{4}{{\gamma_{{{\text{sat}}}}^{2} }}\sin^{2} \theta , \, \,\,\,c\left( \theta \right) = 1 - \frac{1}{2}\sec^{2} \theta ,$$

(42)

Equation (42) can be simplified in the following form:

$$\Delta \ln \left[ {{\text{EI}}\left( \theta \right)} \right] = a\left( \theta \right)\Delta \ln M_{{\text{c}}} + b\left( \theta \right)\Delta \ln \mu + c\left( \theta \right)\Delta \ln \rho$$

(43)

The EI-based formula is obtained by integrating and exponentiating both sides of Eq. (43):

$${\text{EI}}\left( \theta \right) = M_{{\text{c}}}^{a\left( \theta \right)} \mu^{b\left( \theta \right)} \rho^{c\left( \theta \right)}$$

(44)

According to Whitcombe et al. (2002), Eq. (44) should be properly scaled for directly comparing EI values at different incidence angles:

$${\text{EI}}\left( \theta \right) = A\left( {\frac{{M_{{\text{c}}} }}{{M_{{{\text{c}}0}} }}} \right)^{a\left( \theta \right)} \left( {\frac{\mu }{{\mu_{0} }}} \right)^{b\left( \theta \right)} \left( {\frac{\rho }{{\rho_{0} }}} \right)^{c\left( \theta \right)}$$

(45)

where M_c0, µ₀, and ρ₀ represent the averaged values of M_c, µ, and ρ calculated from log data for the target depth interval. The normalization factor A corresponds to EI(0°) in Eq. (44), which has the expression:

$$A = M_{{{\text{c}}0}}^{{\left( {1 - \frac{{\gamma_{{{\text{dry}}}}^{2} }}{{\gamma_{{{\text{sat}}}}^{2} }}} \right)\frac{1}{{2\left( {1 + 4N/3 - \gamma_{dry}^{2} N} \right)}}}} \mu_{0}^{{\frac{{\gamma_{{{\text{dry}}}}^{2} }}{{2\gamma_{{{\text{sat}}}}^{2} }}}} \rho_{0}^{\frac{1}{2}}$$

(46)

Taking the log function on both sides of Eq. (45) gives:

$$\ln \frac{{{\text{EI}}\left( \theta \right)}}{A} = a\left( \theta \right)\ln \frac{{M_{{\text{c}}} }}{{M_{{{\text{c}}0}} }} + b\left( \theta \right)\ln \frac{\mu }{{\mu_{0} }} + c\left( \theta \right)\ln \frac{\rho }{{\rho_{0} }}$$

(47)