Acoustic and Electrical Properties of Tight Rocks: A Comparative Study Between Experiment and Theory

Abstract

The acoustic-electrical (AE) response of subsurface hydrocarbon reservoirs is highly affected by rock heterogeneity. In particular, the characterization of the microstructure of tight (low-permeability) rocks can be aided by a joint interpretation of AE data. To this purpose, we evaluate cores from a tight-oil reservoir to obtain the rock mineralogy and pore structure by X-ray diffraction and casting thin sections. Then, ultrasonic and resistivity experiments are performed under different confining pressures to analyze the effects of pores, microcracks and mineralogy on the AE properties. We have developed acoustic and electrical models based on effective-medium theories, and the Cole–Cole and triple-porosity equations, to simulate the response to total and soft (crack) porosities and clay content. The results show that these properties play a significant role. Then, a 3D rock-physical template is built and calibrated by using the core samples and well-log data. The template is applied to tight-oil reservoirs to estimate the rock properties, which are validated with log data. The good match between the predictions and these data indicates that the model can effectively explain the effects of the heterogeneous microstructure on the AE data.

Article Highlights

• Tight rock microstructure is analyzed with X-ray diffraction, thin sections and ultrasonic and electrical resistivity tests

• Rock acoustic-electrical properties are obtained by the effective-medium and triple-porosity theories

• Practical application is given based on an acoustic-electrical rock physics template

Appendices

Appendix 1: The geometrical factors (P, Q)

The coefficients P and Q for ellipsoidal inclusions are given in Berryman (1980) and Mavko et al. (2009),

$$\begin{array}{*{20}c} {P = \frac{1}{3}T_{1} ,} & {Q = \frac{1}{5}\left( {T_{2} - \frac{1}{3}T_{1} } \right)} \\ \end{array} ,$$

(13)

with the pertinent scalars T₁ and T₂ given by,

$$\begin{array}{*{20}c} {T_{1} = \frac{{3F_{1} }}{{F_{2} }}} & {T_{2} - \frac{1}{3}T_{1} = \frac{2}{{F_{3} }} + \frac{1}{{F_{4} }} + \frac{{F_{4} F_{5} + F_{6} F_{7} - F_{8} F_{9} }}{{F_{2} F_{4} }},} \\ \end{array}$$

(14)

where

$$F_{1} = 1 + G\left[ {\frac{3}{2}(g + \theta ) - J\left( {\frac{3}{2}g + \frac{5}{2}\theta - \frac{4}{3}} \right)} \right],$$

(15)

$$\begin{aligned} F_{2} = & 1 + G\left[ {1 + \frac{3}{2}(g + \theta ) - \frac{J}{2}\left( {\frac{3}{2}g + 5\theta } \right)} \right] + H\left( {3 - 4J} \right) \\ & + \frac{G}{2}\left( {G + 3H} \right)\left( {3 - 4J} \right)\left[ {g + \theta - J\left( {g - \theta + 2\theta^{2} } \right)} \right], \\ \end{aligned}$$

(16)

$$F_{3} = 1 + G\left[ {1 - (g + \frac{3}{2}\theta ) + J\left( {g + \theta } \right)} \right],$$

(17)

$$F_{4} = 1 + \frac{G}{4}\left[ {g + 3\theta - J\left( {g - \theta } \right)} \right],$$

(18)

$$F_{5} = G\left[ { - g + J\left( {g + \theta - \frac{4}{3}} \right) + H\theta \left( {3 - 4J} \right)} \right],$$

(19)

$$F_{6} = 1 + G\left[ {1 + g - J\left( {g + \theta } \right) + H\left( {1 - \theta } \right)\left( {3 - 4J} \right)} \right],$$

(20)

$$F_{7} = 2 + \frac{G}{4}\left[ {3g + 9\theta - J\left( {3g + 5\theta } \right)} \right] + H\theta \left( {3 - 4J} \right),$$

(21)

$$F_{8} = G\left[ {1 - 2J + \frac{g}{2}\left( {J - 1} \right) + \frac{\theta }{2}\left( {5J - 3} \right)} \right] + H\left( {1 - \theta } \right)\left( {3 - 4J} \right),$$

(22)

$$F_{9} = G\left[ {\left( {J - 1} \right)g - J\theta } \right] + H\theta \left( {3 - 4J} \right),$$

(23)

with G, H and J given by

$$G = \frac{{\mu_{i} }}{{\mu_{m} }} - 1,$$

(24)

$$H = \frac{1}{3}\left( {\frac{{K_{i} }}{{K_{m} }} - \frac{{\mu_{i} }}{{\mu_{m} }}} \right),$$

(25)

$$J = \left[ {\frac{{\left( {1 - 2v_{m} } \right)}}{{2\left( {1 - v_{m} } \right)}}} \right],$$

(26)

where K_m, μ_m and v_m are the bulk and shear moduli and Poisson’s ratio of the host phase, respectively, K_i, and μ_i are the bulk and shear moduli of the phase i, and

$$\theta = \left\{ \begin{aligned} \frac{\alpha }{{\left( {\alpha^{2} - 1} \right)^{3/2} }}\left[ {\alpha \left( {\alpha^{2} - 1} \right)^{1/2} - \cosh^{ - 1} \alpha } \right] \hfill \\ \frac{\alpha }{{\left( {1 - \alpha^{2} } \right)^{3/2} }}\left[ {\cos^{ - 1} \alpha - \alpha \left( {1 - \alpha^{2} } \right)^{1/2} } \right] \hfill \\ \end{aligned} \right\},$$

(27)

for prolate (α > 1) and oblate (α < 1) spheroids, respectively, the α is aspect ratio, and

$$g = \frac{{\alpha^{2} }}{{1 - \alpha^{2} }}\left( {3\theta - 2} \right),$$

(28)

Appendix 2: The dispersion equation

The dispersion equation is given in Sun et al. (2016) and Zhang et al. (2017). The plane-wave analysis was performed by substituting a time harmonic kernel $e^{{i\left( {\omega t - k \cdot {\mathbf{x}}} \right)}}$ into Eq. (4). Then the complex wave number $k$ can be obtained as

$$\left| {\begin{array}{*{20}c} {a_{11} k^{2} + b_{11} } & {a_{12} k^{2} + b_{12} } & {a_{13} k^{2} + b_{13} } & {a_{14} k^{2} + b_{14} } \\ {a_{21} k^{2} + b_{21} } & {a_{22} k^{2} + b_{22} } & {a_{23} k^{2} + b_{23} } & {a_{24} k^{2} + b_{24} } \\ {a_{31} k^{2} + b_{31} } & {a_{32} k^{2} + b_{32} } & {a_{33} k^{2} + b_{33} } & {a_{34} k^{2} + b_{34} } \\ {a_{41} k^{2} + b_{41} } & {a_{42} k^{2} + b_{42} } & {a_{43} k^{2} + b_{43} } & {a_{44} k^{2} + b_{44} } \\ \end{array} } \right| = 0,$$

(29)

and

$$\begin{aligned} a_{11} = A + 2N + \left( {Q_{1} \phi_{2} - Q_{2} \phi_{1} } \right)M_{0}^{{\left( {12} \right)}} + \left( {Q_{2} \phi_{3} - Q_{3} \phi_{2} } \right)M_{0}^{{\left( {23} \right)}} , \hfill \\ a_{12} = Q_{1} + \left( {Q_{1} \phi_{2} - Q_{2} \phi_{1} } \right)M_{1}^{{\left( {12} \right)}} + \left( {Q_{2} \phi_{3} - Q_{3} \phi_{2} } \right)M_{1}^{{\left( {23} \right)}} , \hfill \\ a_{13} = Q_{2} + \left( {Q_{1} \phi_{2} - Q_{2} \phi_{1} } \right)M_{2}^{{\left( {12} \right)}} + \left( {Q_{2} \phi_{3} - Q_{3} \phi_{2} } \right)M_{2}^{{\left( {23} \right)}} , \hfill \\ a_{14} = Q_{3} + \left( {Q_{1} \phi_{2} - Q_{2} \phi_{1} } \right)M_{3}^{{\left( {12} \right)}} + \left( {Q_{2} \phi_{3} - Q_{3} \phi_{2} } \right)M_{3}^{{\left( {23} \right)}} , \hfill \\ a_{21} = Q_{1} + \phi_{2} R_{1} M_{0}^{{\left( {12} \right)}} ,\quad a_{22} = R_{1} + \phi_{2} R_{1} M_{1}^{{\left( {12} \right)}} , \hfill \\ a_{23} = \phi_{2} R_{1} M_{2}^{{\left( {12} \right)}} ,\quad a_{24} = \phi_{2} R_{1} M_{3}^{{\left( {12} \right)}} , \hfill \\ a_{31} = Q_{2} - R_{2} \left( {\phi_{1} M_{0}^{{\left( {12} \right)}} - \phi_{3} M_{0}^{{\left( {23} \right)}} } \right),\quad a_{32} = - R_{2} \left( {\phi_{1} M_{1}^{{\left( {12} \right)}} - \phi_{3} M_{1}^{{\left( {23} \right)}} } \right), \hfill \\ a_{33} = R_{2} \left( {1 - \phi_{1} M_{2}^{{\left( {12} \right)}} + \phi_{3} M_{2}^{{\left( {23} \right)}} } \right),\quad a_{34} = R_{2} \left( { - \phi_{1} M_{3}^{{\left( {12} \right)}} + \phi_{3} M_{3}^{{\left( {23} \right)}} } \right), \hfill \\ a_{41} = Q_{3} - \phi_{2} R_{3} M_{0}^{{\left( {23} \right)}} ,\quad a_{42} = - \phi_{2} R_{3} M_{1}^{{\left( {23} \right)}} , \hfill \\ a_{43} = - \phi_{2} R_{3} M_{2}^{{\left( {23} \right)}} ,\quad a_{44} = R_{3} \left( {1 - \phi_{2} M_{3}^{{\left( {23} \right)}} } \right), \hfill \\ b_{11} = - \rho_{00} \omega^{2} + i\omega \left( {b_{1} + b_{2} + b_{3} } \right),\quad b_{12} = - \rho_{01} \omega^{2} - i\omega b_{1} , \hfill \\ b_{13} = - \rho_{02} \omega^{2} - i\omega b_{2} ,\quad b_{14} = - \rho_{03} \omega^{2} - i\omega b_{3} , \hfill \\ b_{21} = - \rho_{01} \omega^{2} - i\omega b_{1} ,\quad b_{22} = - \rho_{11} \omega^{2} + i\omega b_{1} ,b_{23} = b_{24} = 0, \hfill \\ b_{31} = - \rho_{02} \omega^{2} - i\omega b_{2} ,\quad b_{33} = - \rho_{22} \omega^{2} + i\omega b_{2} ,b_{32} = b_{34} = 0, \hfill \\ b_{41} = - \rho_{03} \omega^{2} - i\omega b_{3} ,\quad b_{44} = - \rho_{33} \omega^{2} + i\omega b_{3} ,b_{42} = b_{43} = 0, \hfill \\ \end{aligned}$$

(30)

where the Biot dissipation coefficients (Biot 1962 and Sun et al. 2016) and the permeabilities of the three phases (Vaughan et al. 1986 and Mavko et al. 2009),

$$ \begin{array}{*{20}l} {b_{1} = \phi_{1} \phi_{10} \frac{{\eta_{f}^{{}} }}{{\kappa_{1} }},} \hfill & {b_{2} = \phi_{2} \phi_{20} \frac{{\eta_{f}^{{}} }}{{\kappa_{2} }},} \hfill & {b_{3} = \phi_{3} \phi_{30} \frac{{\eta_{f}^{{}} }}{{\kappa_{3} }},} \hfill \\ {\kappa_{1} = {{DR}}_{12}^{2} \phi_{1}^{3} ,} \hfill & {\kappa_{2} = \frac{{\kappa_{0} \phi_{2}^{3} }}{{\left( {1 - \phi_{2} } \right)^{2} }},} \hfill & {\kappa_{3} = \frac{{\kappa_{0} \phi_{3}^{3} }}{{\left( {1 - \phi_{3}^{2} } \right)}},} \hfill \\ \end{array} $$

(31)

where D = 50 and κ₀ = 75.54 mdarcy, and

$$\begin{aligned} S_{12} = & \frac{{ - \phi_{1} \phi_{2}^{2} R_{12}^{2} \omega \left( {\rho_{f} \omega \left( {1/5 + \phi_{10} /\phi_{20} } \right) + i\left( {\eta /\left( {5\kappa_{1} } \right) + \eta /\kappa_{2} } \right)\phi_{10} } \right)}}{3} - \phi_{2}^{2} R_{1} - \phi_{1}^{2} R_{2} , \\ S_{23} = & \frac{{ - \phi_{3} \phi_{2}^{2} R_{23}^{2} \omega \left( {\rho_{f} \omega \left( {1/5 + \phi_{30} /\phi_{20} } \right) + i\left( {\eta /\left( {5\kappa_{3} } \right) + \eta /\kappa_{2} } \right)\phi_{30} } \right)}}{3} - \phi_{3}^{2} R_{2} - \phi_{2}^{2} R_{3} , \\ M_{0}^{{\left( {12} \right)}} = & \frac{{\left( {Q_{1} \phi_{2} - Q_{2} \phi_{1} } \right)/S_{12} + \phi_{1} \phi_{3} R_{2} \left( {Q_{2} \phi_{3} - Q_{3} \phi_{2} } \right)/\left( {S_{12} S_{23} } \right)}}{{1 + \left( {\phi_{1} \phi_{3} R_{2} } \right)^{2} /\left( {S_{12} S_{23} } \right)}}, \\ M_{1}^{{\left( {12} \right)}} = & \frac{{\phi_{2} R_{1} /S_{12} }}{{1 + \left( {\phi_{1} \phi_{3} R_{2} } \right)^{2} /\left( {S_{12} S_{23} } \right)}}, \\ M_{2}^{{\left( {12} \right)}} = & \frac{{ - \phi_{1} R_{2} /S_{12} + \phi_{1} \phi_{3}^{2} R_{2}^{2} /\left( {S_{12} S_{23} } \right)}}{{1 + \left( {\phi_{1} \phi_{3} R_{2} } \right)^{2} /\left( {S_{12} S_{23} } \right)}},\quad M_{3}^{{\left( {12} \right)}} = \frac{{ - \phi_{1} \phi_{2} \phi_{3} R_{2} R_{3} /\left( {S_{12} S_{23} } \right)}}{{1 + \left( {\phi_{1} \phi_{3} R_{2} } \right)^{2} /\left( {S_{12} S_{23} } \right)}}, \\ M_{0}^{{\left( {23} \right)}} = & \left( { - M_{0}^{{\left( {12} \right)}} \phi_{1} \phi_{3} R_{2} + Q_{2} \phi_{3} - Q_{3} \phi_{2} } \right)/S_{23} ,\quad M_{1}^{{\left( {23} \right)}} = - M_{1}^{{\left( {12} \right)}} \phi_{1} \phi_{3} R_{2} /S_{23} , \\ M_{2}^{{\left( {23} \right)}} = & \left( { - M_{2}^{{\left( {12} \right)}} \phi_{1} \phi_{3} R_{2} + \phi_{3} R_{2} } \right)/S_{23} ,\quad M_{3}^{{\left( {23} \right)}} = \left( { - M_{3}^{{\left( {12} \right)}} \phi_{1} \phi_{3} R_{2} - \phi_{2} R_{3} } \right)/S_{23} . \\ \end{aligned}$$

(32)

The stiffness and density coefficients are

$$\begin{aligned} A = \left( {1 - \phi } \right)K_{s} - \frac{2}{3}N - \frac{{K_{s} }}{{K_{f} }}\left( {Q_{1} + Q_{2} + Q_{3} } \right),\quad N = \mu_{b} , \hfill \\ Q_{1} = \frac{{\phi_{1} \beta_{1} K_{s} }}{{\beta_{1} + \gamma }},\quad Q_{2} = \frac{{\phi_{2} K_{s} }}{1 + \gamma },\quad Q_{3} = \frac{{\phi_{3} K_{s} }}{{\beta_{1} \gamma + 1}}, \hfill \\ R_{1} = \frac{{\phi_{1} K_{f} }}{{{{\beta_{1} } \mathord{\left/ {\vphantom {{\beta_{1} } \gamma }} \right. \kern-\nulldelimiterspace} \gamma } + 1}},\quad R_{2} = \frac{{\phi_{2} K_{f} }}{{1 + {1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-\nulldelimiterspace} \gamma }}},\quad R_{3} = \frac{{\phi_{3} K_{f} }}{{{1 \mathord{\left/ {\vphantom {1 {\beta_{1} \gamma }}} \right. \kern-\nulldelimiterspace} {\beta_{1} \gamma }} + 1}}, \hfill \\ \gamma = \frac{{K_{s} }}{{K_{f} }}\frac{{\phi_{1} \beta_{1} + \phi_{2} + {{\phi_{3} } \mathord{\left/ {\vphantom {{\phi_{3} } {\beta_{2} }}} \right. \kern-\nulldelimiterspace} {\beta_{2} }}}}{{\left( {1 - \phi } \right) - {{K_{b} } \mathord{\left/ {\vphantom {{K_{b} } {K_{s} }}} \right. \kern-\nulldelimiterspace} {K_{s} }}}}, \hfill \\ \end{aligned}$$

(33)

$$\begin{aligned} \begin{array}{*{20}c} {\rho_{11} = \frac{1}{2}\phi_{1} \rho_{f} \left( {1 + \frac{1}{{\phi_{10} }}} \right),} & {\rho_{22} = \frac{1}{2}\phi_{2} \rho_{f} \left( {1 + \frac{1}{{\phi_{20} }}} \right),} & {\rho_{33} = \frac{1}{2}\phi_{3} \rho_{f} \left( {1 + \frac{1}{{\phi_{30} }}} \right),} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\rho_{01} = \phi_{1} \rho_{f} - \rho_{11} ,} & {\rho_{02} = \phi_{2} \rho_{f} - \rho_{22} ,} & {\rho_{03} = \phi_{3} \rho_{f} - \rho_{33} ,} \\ \end{array} \hfill \\ \rho_{00} = v_{1} (1 - \phi_{10} )\rho_{s1} + v_{2} (1 - \phi_{20} )\rho_{s2} + v_{3} (1 - \phi_{30} )\rho_{sh} - \rho_{01} - \rho_{02} - \rho_{03} , \hfill \\ \end{aligned}$$

(34)

where K_s, K_b and K_f are the bulk moduli of the mineral mixture, skeleton and fluid, respectively, ρ_s1, ρ_s2 and ρ_sh are the mineral densities corresponding to the three phases, ρ_f is the fluid density, and

$$\begin{aligned} \beta_{1} = \frac{{Q_{1} R_{2} }}{{Q_{2} R_{1} }} = \frac{{\phi_{20} }}{{\phi_{10} }}\left[ {\frac{{{1 \mathord{\left/ {\vphantom {1 {K_{s1} }}} \right. \kern-\nulldelimiterspace} {K_{s1} }} - {{\left( {1 - \phi_{10} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \phi_{10} } \right)} {K_{b1} }}} \right. \kern-\nulldelimiterspace} {K_{b1} }}}}{{{1 \mathord{\left/ {\vphantom {1 {K_{s2} }}} \right. \kern-\nulldelimiterspace} {K_{s2} }} - {{\left( {1 - \phi_{20} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \phi_{20} } \right)} {K_{b2} }}} \right. \kern-\nulldelimiterspace} {K_{b2} }}}}} \right], \hfill \\ \beta_{2} = \frac{{Q_{2} R_{3} }}{{Q_{3} R_{2} }} = \frac{{\phi_{30} }}{{\phi_{20} }}\left[ {\frac{{{1 \mathord{\left/ {\vphantom {1 {K_{s2} }}} \right. \kern-\nulldelimiterspace} {K_{s2} }} - {{\left( {1 - \phi_{20} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \phi_{20} } \right)} {K_{b2} }}} \right. \kern-\nulldelimiterspace} {K_{b2} }}}}{{{1 \mathord{\left/ {\vphantom {1 {K_{sh} }}} \right. \kern-\nulldelimiterspace} {K_{sh} }} - {{\left( {1 - \phi_{30} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \phi_{30} } \right)} {K_{b3} }}} \right. \kern-\nulldelimiterspace} {K_{b3} }}}}} \right]. \hfill \\ \end{aligned}$$

(35)

where K_b1, K_b2 and K_b3 are the skeleton bulk moduli of the crack inclusions, host and clay inclusions, respectively, and K_s1, K_s2 and K_sh are the mineral bulk moduli corresponding to the three phases.