3-D Rock-Physics Templates for the Seismic Prediction of Pore Microstructure in Ultra-Deep Carbonate Reservoirs

Abstract

Ultra-deep carbonate reservoirs have low porosity, a complex pore space with microcracks of varying aspect ratio and dissolved pores, that affect the seismic and transport properties. We propose a rock-physics model based on penny-shaped inclusions in the framework of the double-porosity theory to estimate rock features, such as the crack porosity and aspect ratio, and stiff porosity. Based on this model, a 3-D rock-physics template is built, calibrated at the ultrasonic and seismic frequency bands, from attenuation, P-wave impedance and V_P/V_S ratio to quantitatively evaluate the effect of those features. Attenuation is estimated by using the spectral-ratio and improved frequency-shift methods. The template is applied to ultra-deep carbonates of the S work area of the Tarim Basin (China). The predictions agree with the well-log data and field production reports. In general, the higher the crack aspect ratio, the higher the storage and transport capacity of the reservoir. Therefore, these crack features can be used as indicators of these reservoir properties.

Appendices

Appendix A: Factors P and Q

Following [20, 43], we have

$$P_{s} = \frac{{(1 - v_{s} )}}{{6(1 - 2v_{s} )}} \times \frac{{4(1 + v_{s} ) + 2\alpha_{{{\text{stiff}}}}^{2} (7 - 2v_{s} ) - \left[ {3(1 + 4v_{s} ) + 12\alpha_{{{\text{stiff}}}}^{2} (2 - v_{s} )} \right]g}}{{2\alpha_{{{\text{stiff}}}}^{2} + (1 - 4\alpha_{{{\text{stiff}}}}^{2} )g + (\alpha_{{{\text{stiff}}}}^{2} - 1)(1 + v_{s} )g^{2} }},$$

(A1)

$$\begin{aligned} Q_{s} = & \frac{{4(\alpha_{{{\text{stiff}}}}^{2} - 1)(1 - v_{s} )}}{{15\left\{ {9(v_{s} - 1) + 2\alpha_{{{\text{stiff}}}}^{2} (3 - 4v_{s} ) + \left[ {(7 - 8v_{s} ) - 4\alpha_{stiff}^{2} (1 - 2v_{s} )} \right]g} \right\}}} \\ & \; \times \left\{ \begin{gathered} \frac{{8(1 - v_{s} ) + 2\alpha_{{{\text{stiff}}}}^{2} (3 + 4v_{s} ) + \left[ {(8v_{s} - 1) - 4\alpha^{2} (5 + 2v_{s} )} \right]g + 6(\alpha_{{{\text{stiff}}}}^{2} - 1)(1 + v_{s} )g^{2} }}{{2\alpha_{{{\text{stiff}}}}^{2} + (1 - 4\alpha_{{{\text{stiff}}}}^{2} )g + (\alpha_{{{\text{stiff}}}}^{2} - 1)(1 + v_{s} )g^{2} }} \hfill \\ - 3\left[ {\frac{{8(v_{s} - 1) + 2\alpha_{{{\text{stiff}}}}^{2} (5 - 4v_{s} ) + \left[ {3(1 - 2v_{s} ) + 6\alpha_{{{\text{stiff}}}}^{2} (v_{s} - 1)} \right]g}}{{ - 2\alpha_{{{\text{stiff}}}}^{2} + \left[ {(2 - v_{s} ) + \alpha_{{{\text{stiff}}}}^{2} (1 + v_{s} )} \right]g}}} \right] \hfill \\ \end{gathered} \right\} \\ \end{aligned}$$

(A2)

where

$$g = \left\{ \begin{gathered} \frac{{\alpha_{{{\text{stiff}}}}^{2} }}{{(1 - \alpha_{{{\text{stiff}}}}^{2} )^{3/2} }}\left( {\arccos \, \alpha_{{{\text{stiff}}}}^{2} - \alpha_{{{\text{stiff}}}}^{2} \sqrt {1 - \alpha_{{{\text{stiff}}}}^{2} } } \right)\;\;(\alpha_{{{\text{stiff}}}}^{2} < 1) \hfill \\ \frac{{\alpha_{{{\text{stiff}}}}^{2} }}{{(1 - \alpha_{{{\text{stiff}}}}^{2} )^{3/2} }}\left( {\alpha_{{{\text{stiff}}}}^{2} \sqrt {1 - \alpha_{{{\text{stiff}}}}^{2} } - \arccos h \, \alpha_{{{\text{stiff}}}}^{2} } \right)\;\;(\alpha_{{{\text{stiff}}}}^{2} > 1) \hfill \\ \end{gathered} \right.$$

(A3)

where $v_{s} = (3K_{s} - 2\mu_{s} )/(6K_{s} + 2\mu_{s} )$ is the Poisson ratio.

Appendix B: Factors P ^*i and Q ^*i

Following [60, 61], we have

$$P^{*i} = \frac{{K_{s} + \frac{3}{4}\mu_{i} }}{{K_{i} + \frac{3}{4}\mu_{i} + \pi \alpha \beta_{s} }}$$

(B1)

$$Q^{*i} = \frac{1}{5}\left[ {1 + \frac{{8\mu_{s} }}{{4\mu_{i} + \pi \alpha \left( {\mu_{s} + 2\beta_{s} } \right)}} + 2\frac{{K_{i} + \frac{2}{3}\left( {\mu_{i} + \mu_{s} } \right)}}{{K_{i} + \frac{4}{3}\mu_{i} + \pi \alpha \beta_{s} }}} \right]$$

(B2)

where $\beta_{s} = \mu_{s} \frac{{\left( {3K_{s} + \mu_{s} } \right)}}{{\left( {3K_{s} + 4\mu_{s} } \right)}}$.

Appendix C: Wave Propagation Equation of Penny-Shaped Inclusion Model

Following [38], the wave propagation equations based on penny-shaped inclusion are

$$\begin{aligned} 2G\nabla e_{ij} + & \lambda_{c} \nabla e - \alpha_{1} M_{1} \nabla (\xi^{(1)} - \phi_{1} \phi_{2} \varsigma ) - \alpha_{2} M_{2} \nabla (\xi^{(2)} + \phi_{1} \phi_{2} \varsigma ) \\ & = \rho_{0} \ddot{u} + \rho_{f} \ddot{w}_{i}^{(1)} + \rho_{f} \ddot{w}_{i}^{(2)} \\ \end{aligned}$$

(C1)

$$\scriptsize \alpha_{1} M_{1} \nabla e - M_{1} \nabla (\xi^{(1)} - \phi_{1} \phi_{2} \varsigma ) = \rho_{f} \ddot{u} + m_{1} \ddot{w}_{i}^{\left( 1 \right)} + \frac{\eta }{{k_{1} }}\frac{{\phi_{10} }}{{\phi_{1} }}\dot{w}^{(1)}$$

(C2)

$$\alpha_{2} M_{2} \nabla e - M_{2} \nabla (\xi^{(2)} + \phi_{1} \phi_{2} \varsigma ) = \rho_{f} \ddot{u} + m_{2} \ddot{w}_{i}^{\left( 2 \right)} + \frac{\eta }{{k_{1} }}\frac{{\phi_{10} }}{{\phi_{1} }}\dot{w}^{(2)}$$

(C3)

$$\begin{aligned} &\left( {\frac{3}{8} + \frac{{\phi_{20} }}{{2\phi_{10} }}\ln \frac{{L + R_{0} }}{{R_{0} }}} \right)\phi_{1}^{2} \phi_{2} \rho_{f} R_{0}^{2} \ddot{\varsigma }\\&\quad + \left( {\frac{3\eta }{{8\kappa_{2} }} + \frac{\eta }{{2\kappa_{1} }}\ln \frac{{L + R_{0} }}{{R_{0} }}} \right)\phi_{20} \phi_{1}^{2} \phi_{2} R_{0}^{2} \dot{\varsigma } \\ &\quad = \phi_{1} \phi_{2} \left( {\alpha_{1} M_{1} - \alpha_{1} M_{2} } \right)e + \phi_{1} \phi_{2} \left( {M_{2} \xi^{(2)} - M_{1} \xi^{(1)} } \right) \\ &\quad\quad+ \phi_{1}^{2} \phi_{2}^{2} \left( {M_{1} + M_{2} } \right)\varsigma \\ \end{aligned}$$

(C4)

where $e_{ij} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}\left( {\delta_{j} u_{i} + \delta_{i} u_{j} } \right)$ are the solid strain components, with i, j = 1, 2, 3, $e = \nabla \cdot {\mathbf{u}}$, $\xi^{(1)} = - \nabla \cdot {\mathbf{w}}^{(m)}$ is the fluid content increment (m = 1, 2 refer to the host and penny-shaped inclusions, respectively), and ${\mathbf{w}}^{(m)} = \phi_{m} \left( {{\mathbf{U}}_{m} - {\mathbf{u}}} \right)$, where ${\mathbf{U}} = \left( {U_{1} ,U_{2} ,U_{3} } \right)^{T}$ and ${\mathbf{u}} = \left( {u_{1} ,u_{2} ,u_{3} } \right)^{T}$ are the fluid and solid displacements, respectively, and the dot above a variable denotes a partial time derivative. The variation in fluid flow between the host medium and inclusion is denoted by $\varsigma$. $\phi_{10}$ and $\phi_{20}$ are the local porosities of the two skeletons (stiff pores and cracks), $\phi_{1} = \phi_{10} v_{1}$ and $\phi_{2} = \phi_{20} v_{2}$ are the absolute porosities of the two pore types, with $v_{1}$ and $v_{2}$ their respective volume ratio which satisfy $v_{1} + v_{2} = 1$, the total porosity is $\phi = \phi_{1} + \phi_{2}$, $\eta$ is the fluid viscosity, $\kappa_{1}$ and $\kappa_{2}$ are the host and inclusion permeabilities, respectively, $\rho_{0}$ and $\rho_{f}$ are the composite and pore-fluid densities, with $\rho_{0} = \left( {1 - \phi } \right)\rho_{s} + \phi \rho_{f}$ where $\rho_{s}$ is the grain density, $m_{1} = \frac{{\tau_{1} \rho_{f} }}{{\phi_{1} }}$ and $m_{2} = \frac{{\tau_{2} \rho_{f} }}{{\phi_{2} }}$ with $\tau_{1} = \frac{1}{2}\left( {1 + \frac{1}{{\phi_{1} }}} \right)$ and $\tau_{2} = \frac{1}{2}\left( {1 + \frac{1}{{\phi_{2} }}} \right)$ as the host medium and inclusion tortuosities, respectively. $G = \mu_{{{\text{SC}}}}^{*}$ is the bulk shear modulus of dry rock and the stiffnesses $\lambda_{c}$, $\alpha_{1}$, $\alpha_{2}$, $M_{1}$ and $M_{2}$ are given in Appendix D. The characteristic flow length is $L = \left( {\frac{{R_{0}^{2} }}{12}} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}$ and $R_{0}$ is the inclusion radius.

Appendix D: Stiffness Coefficients

The expressions of the stiffness coefficients are

$$\begin{aligned} \lambda_{c} & = \left( {1 - \phi } \right)K_{s} - \frac{2}{3}G + \left( {2 - \frac{{K_{s} }}{{K_{f} }}} \right)\left( {\phi_{1} \alpha_{1} M_{1} + \phi_{2} \alpha_{2} M_{2} } \right) \\ &\quad- \left( {1 - \frac{{K_{s} }}{{K_{f} }}} \right)\left( {\phi_{1}^{2} M_{1} + \phi_{2}^{2} M_{2} } \right) \end{aligned}$$

(D1)

$$\alpha_{1} = \frac{{\beta \phi_{1} K_{s} }}{{\gamma K_{f} }} + \phi_{1} ,\;\;\;\alpha_{2} = \frac{{\beta \phi_{2} K_{s} }}{{\gamma K_{f} }} + \phi_{2}$$

(D2)

$$M_{1} = \frac{{K_{f} }}{{\left( {\frac{\beta }{\gamma } + 1} \right)\phi_{1} }},\;\;M_{2} = \frac{{K_{f} }}{{\left( {\frac{1}{\gamma } + 1} \right)\phi_{2} }}$$

(D3)

$$\gamma = \frac{{K_{s} }}{{K_{f} }}\left( {\frac{{\beta \phi_{1} + \phi_{2} }}{{1 - \phi - \frac{{K_{b} }}{{K_{s} }}}}} \right)$$

(D4)

$$\beta = \frac{{\phi_{20} }}{{\phi_{10} }}\left[ {\frac{{1 - \left( {1 - \phi_{10} } \right)\frac{{K_{s} }}{{K_{b1} }}}}{{1 - \left( {1 - \phi_{20} } \right)\frac{{K_{s} }}{{K_{b2} }}}}} \right]$$

(D5)

where $K_{{{\text{SC}}}}^{*}$ = $K_{b}$ is the bulk modulus of the dry rock, $K_{i}$ = $K_{b1}$ and $K_{b2}$ are the dry-rock moduli of the host medium and inclusions.

Appendix E: Dispersion Equations

Substituting a plane-wave kernel into the differential equations (C1) - (C4), the complex wave number k can be obtained from

$$\left| {\begin{array}{*{20}l} {a_{11} k^{2} + b_{11} } \hfill &\quad {a_{12} k^{2} + b_{12} } \hfill &\quad {a_{13} k^{2} + b_{13} } \hfill \\ {a_{21} k^{2} + b_{21} } \hfill & \quad{a_{22} k^{2} + b_{22} } \hfill &\quad {a_{23} k^{2} + b_{23} } \hfill \\ {a_{31} k^{2} + b_{31} } \hfill &\quad {a_{32} k^{2} + b_{32} } \hfill &\quad {a_{33} k^{2} + b_{33} } \hfill \\ \end{array} } \right| = 0,$$

(E1)

where

$$\begin{array}{*{20}l} {a_{11} = \lambda_{c} + \frac{2}{3}G + \phi_{1} \phi_{2} \left( {\alpha_{1} M_{1} - \alpha_{2} M_{2} } \right)q_{1} ,} \hfill & {b_{11} = - \rho \omega^{2} } \hfill \\ {a_{12} = - \alpha_{1} M_{1} + \phi_{1} \phi_{2} \left( {\alpha_{1} M_{1} - \alpha_{2} M_{2} } \right)q_{2} ,} \hfill & {b_{12} = \rho_{f} \omega^{2} } \hfill \\ {a_{13} = - \alpha_{2} M_{2} + \phi_{1} \phi_{2} \left( {\alpha_{1} M_{1} - \alpha_{2} M_{2} } \right)q_{3} ,} \hfill & {b_{13} = \rho_{f} \omega^{2} } \hfill \\ {a_{21} = - \alpha_{1} M_{1} - \phi_{1} \phi_{2} M_{1} q_{1} ,} \hfill & {b_{21} = b_{12} } \hfill \\ {a_{22} = M_{1} - \phi_{1} \phi_{2} M_{1} q_{2} ,} \hfill & {b_{22} = - M_{1} \omega^{2} + i\omega b_{1} /\phi_{1}^{2} } \hfill \\ {a_{23} = - \phi_{1} \phi_{2} M_{1} q_{3} ,} \hfill & {b_{23} = 0} \hfill \\ {a_{31} = - \alpha_{2} M_{2} - \phi_{1} \phi_{2} M_{2} q_{1} ,} \hfill & {b_{31} = b_{13} } \hfill \\ {a_{32} = \phi_{1} \phi_{2} M_{2} q_{2} ,} \hfill & {b_{32} = 0} \hfill \\ {a_{33} = M_{2} + \phi_{1} \phi_{2} M_{2} q_{3} ,} \hfill & {} \hfill \\ \end{array}$$

(E2)

with

$$\begin{aligned} q_{1} = & \phi_{1} \phi_{2} \left( {\alpha_{1} M_{1} - \alpha_{2} M_{2} } \right)/Z \\ q_{2} = & - \phi_{1} \phi_{2} M_{1} /Z \\ q_{3} = & - \phi_{1} \phi_{2} M_{2} /Z \\ Z = & - \omega^{2} \left( {\frac{3}{8} + \frac{{\phi_{20} }}{{2\phi_{10} }}\ln \frac{{L + R_{0} }}{{R_{0} }}} \right)\phi_{1}^{2} \phi_{2} \rho_{f} R_{0}^{2}\\ & + i\omega \left( {\frac{{3\eta_{2} }}{{8\kappa_{2} }} + \frac{{\eta_{1} }}{{2\kappa_{1} }}\ln \frac{{L + R_{0} }}{{R_{0} }}} \right)\phi_{20} \phi_{1}^{2} \phi_{2} R_{0}^{2} \\ &- \phi_{1}^{2} \phi_{1}^{2} \left( {M_{1} + M_{2} } \right) \\ \end{aligned}$$

(E3)

The complex wave velocity is

$$v = \frac{\omega }{k}$$

(E4)

where k is the complex P-wavenumber. The P-wave phase velocity is

$$V_{p} = \left( {{\text{Re}} \left( \frac{1}{v} \right)} \right)^{ - 1}$$

(E5)

and the quality factor is

$$Q = \frac{{{\text{Re}} (k^{2} )}}{{{\text{Im}} (k^{2} )}}$$

(E6)

with $\omega = 2\pi f$, where $f$ is the frequency [54].