Measurements of Seismic Anisotropy in Synthetic Rocks with Controlled Crack Geometry and Different Crack Densities

Abstract

Seismic anisotropy can help to extract azimuthal information for predicting crack alignment, but the accurate evaluation of cracked reservoir requires knowledge of degree of crack development, which is achieved through determining the crack density from seismic or VSP data. In this research we study the dependence of seismic anisotropy on crack density, using synthetic rocks with controlled crack geometries. A set of four synthetic rocks containing different crack densities is used in laboratory measurements. The crack thickness is 0.06 mm and the crack diameter is 3 mm in all the cracked rocks, while the crack densities are 0.00, 0.0243, 0.0486, and 0.0729. P and S wave velocities are measured by an ultrasonic investigation system at 0.5 MHz while the rocks are saturated with water. The measurements show the impact of crack density on the P and S wave velocities. Our results are compared to the theoretical prediction of Chapman (J App Geophys 54:191–202, 2003) and Hudson (Geophys J R Astron Soc 64:133–150, 1981). The comparison shows that measured velocities and theoretical results are in good quantitative agreement in all three cracked rocks, although Chapman’s model fits the experimental results better. The measured anisotropy of the P and S wave in the four synthetic rocks shows that seismic anisotropy is directly proportional to increasing crack density, as predicted by several theoretical models. The laboratory measurements indicate that it would be effective to use seismic anisotropy to determine the crack density and estimate the intensity of crack density in seismology and seismic exploration.

Appendix

In the case of a material containing aligned or randomly orientated cracks, the overall elastic properties derived by Hudson contain the second order in the concentration. Both the single scattering formulae, which are correct to the first order in crack density $\varepsilon_{\text{c}}$, and crack–crack interactions, which are correct to the second order in crack density, are accounted. The model considered a plane wave propagating through the medium with cracks, but the pores in the background matrix are neglected.

The stiffness tensor of the medium contain cracks in the Hudson model is given as

$$ C_{ij}^{{}} = C_{ij}^{0} + C_{ij}^{1} + C_{ij}^{2} . $$

(4)

in which $C_{ij}^{{}}$ is the total stiffness tensor, $C_{ij}^{0}$ is the background, $C_{ij}^{1}$, and $C_{ij}^{2}$ are the first- and second-order effects of the cracks:

$$ C_{ij}^{0} = \left[ {\begin{array}{*{20}c} {\lambda + 2\mu } & \lambda & \lambda & 0 & 0 & 0 \\ \lambda & {\lambda + 2\mu } & \lambda & 0 & 0 & 0 \\ \lambda & \lambda & {\lambda + 2\mu } & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu \\ \end{array} } \right], $$

(5)

$$ C_{ij}^{1} = - \frac{{\varepsilon_{c} }}{\mu }\left[ {\begin{array}{*{20}c} {\lambda^{2} \overline{U}_{33} } & {\lambda^{2} \overline{U}_{33} } & {\lambda (\lambda + 2\mu )\overline{U}_{33} } & 0 & 0 & 0 \\ {\lambda^{2} \overline{U}_{33} } & {\lambda^{2} \overline{U}_{33} } & {\lambda (\lambda + 2\mu )\overline{U}_{33} } & 0 & 0 & 0 \\ {\lambda (\lambda + 2\mu )\overline{U}_{33} } & {\lambda (\lambda + 2\mu )\overline{U}_{33} } & {\lambda (\lambda + 2\mu )\overline{U}_{33} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {\mu^{2} \overline{U}_{11} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {\mu^{2} \overline{U}_{11} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(6)

$$ C_{ij}^{2} = \frac{{\varepsilon_{c}^{2} }}{15}\left[ {\begin{array}{*{20}c} {\frac{{\lambda^{2} q}}{\lambda + 2\mu }\overline{U}_{33}^{2} } & {\frac{{\lambda^{2} q}}{\lambda + 2\mu }\overline{U}_{33}^{2} } & {\lambda q\overline{U}_{33}^{2} } & 0 & 0 & 0 \\ {\frac{{\lambda^{2} q}}{\lambda + 2\mu }\overline{U}_{33}^{2} } & {\frac{{\lambda^{2} q}}{\lambda + 2\mu }\overline{U}_{33}^{2} } & {\lambda q\overline{U}_{33}^{2} } & 0 & 0 & 0 \\ {\lambda q\overline{U}_{33}^{2} } & {\lambda q\overline{U}_{33}^{2} } & {(\lambda + 2\mu )q\overline{U}_{33}^{2} } & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{2\mu (3\lambda + 8\mu )}{\lambda + 2\mu }\overline{U}_{11}^{2} } & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{2\mu (3\lambda + 8\mu )}{\lambda + 2\mu }\overline{U}_{11}^{2} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]. $$

(7)

in which $q = 15\left( {\frac{\lambda }{\mu }} \right)^{2} + 28\left( {\frac{\lambda }{\mu }} \right) + 28$. In the case of water saturation,

$$ \overline{U}_{11} = \frac{16}{3}\frac{\lambda + 2\mu }{3\lambda + 4\mu },\,\quad \overline{U}_{33} = 0. $$

(8)

The Chapman’s model considers frequency-dependent seismic anisotropy in fractured rocks through knowledge of rock porosity, permeability, fracture density, and pore fluid properties. The model is based on fluid interactions at two scales: randomly aligned microcracks and aligned mesoscale fractures.

The stiffness tensor given by Chapman (2003) is

$$ C_{ijkl} (\omega ) = C_{ijkl}^{0} - \phi_{p} C_{ijkl}^{1} - \varepsilon_{c} C_{ijkl}^{2} - \varepsilon_{f} C_{ijkl}^{3} , $$

(9)

where $C_{ijkl}^{0}$ is the isotropic background matrix of porous rock, $C_{ijkl}^{1}$, $C_{ijkl}^{2}$, and $C_{ijkl}^{3}$ are the contributions from pores, grain size cracks, and meso-scale fractures, respectively. $\phi_{\text{p}}$ is the porosity, $\varepsilon_{\text{c}}$ is the crack density, and $\varepsilon_{\text{f}}$ is the fracture density.

$$ \varepsilon_{\text{c}} = \frac{{N_{\text{c}} a_{\text{c}}^{3} }}{V};\quad \varepsilon_{f} = \frac{{N_{\text{f}} a_{\text{f}}^{3} }}{V}, $$

(10)

where $N_{\text{c}}$ and $N_{\text{f}}$ are the number of cracks and fractures in volume V, $a_{\text{c}}$ and $a_{\text{f}}$ are the radium of cracks and fractures, respectively. The model parameters are the functions of the elastic tensor (isotropic matrix $\lambda \;{\text{and}}\;\mu$), fracture parameters, fluid properties, frequency, relaxation time $\tau_{\text{m}}$ of micro-scale pores and cracks, $\tau_{\text{f}}$ of meso-scale fractures.

In fact, fluid flow in the model takes place at two scales, micro-scale squirt flow in pores and cracks and meso-scale flow in large fractures. The grain size local flow is related with the squirt flow relaxation time $\tau_{\text{m}}$; the flow at fracture scale is related with the larger relaxation time $\tau_{\text{f}}$ which depends on the fracture size. In the Chapman model, the relaxation time corresponding to the fractures,$\tau_{\text{f}}$, is related to the fracture scale and micro-scale relaxation time $\tau_{\text{m}}$ as

$$ \tau_{\text{f}} = \frac{{a_{\text{f}} }}{\varsigma }\tau_{\text{m}} , $$

(11)

in which $a_{\text{f}}$ is the fracture scale $\varsigma$ is the grain size scale. and $\tau_{\text{m}}$ is given by

$$ \tau_{\text{m}} = \frac{{c_{\text{v}} \eta (1 + K_{\text{c}} )}}{{\sigma_{\text{c}} \kappa \varsigma c_{1} }}, $$

(12)

where $\eta$ is the fluid viscosity, $\kappa$ is the permeability, ${\text{c}}_{\text{v}}$ is the volume of the individual cracks, ${\rm K}_{\text{c}}$ is the inverse of the crack space compressibility, $\sigma_{\text{c}} = \pi \mu r/[2(1 - \nu )]$ is the critical stress in which r is the aspect ratio of the cracks, $\nu$ is the Poisson’s ratio of the isotropic rock matrix, and $c_{1}$ is the number of connections to other elements of the pore space.

Due to the calculation of the elastic constants following the interaction energy approach of Eshelby (1957), the original form of the Chapman model is limited to low porosity. To make the Chapman model more applicable to real data, a slightly modified version was described by Chapman et al. (2003) through introducing the $\lambda^{0}$ and $\mu^{0}$ which were calculated from the measured $V_{\text{p}}^{0}$ and $V_{\text{s}}^{0}$ and density of the rock. Additionally, the model requires a $C_{ijkl}^{0} (\varLambda ,{\rm M})$ term to be defined in a way that the fracture and pore corrections to velocities are applied at a specific frequency ($w_{0}$). Thus:

$$ \varLambda = \lambda^{0} + \varPhi_{{{\text{c,}}\;{\text{p}}}} (\lambda^{0} ,\mu^{0} ,fw_{0} );\quad {\rm M} = \mu^{0} + \varPhi_{{{\text{c,}}\;{\text{p}}}} (\lambda^{0} ,\mu^{0} ,w_{0} ), $$

(13)

where $\varPhi_{{{\text{c,}}\;{\text{p}}}}$ is an elastic correction term that is proportional to $\varepsilon_{\text{c}}$ and $\varepsilon_{\text{f}}$ and with,

$$ \lambda^{0} = \rho (V_{\text{p}}^{0} )^{2} - 2\mu^{0} ;\quad \mu^{0} = \rho (V_{s}^{0} )^{2} . $$

(14)

Then equation is written as follows:

$$ C_{ijkl} (\omega ) = C_{ijkl}^{0} (\varLambda ,{\rm M},w) - \phi_{\text{p}} C_{ijkl}^{1} (\lambda^{0} ,\mu^{0} ,w) - \varepsilon_{c} C_{ijkl}^{2} (\lambda^{0} ,\mu^{0} ,w) - \varepsilon_{f} C_{ijkl}^{3} (\lambda^{0} ,\mu^{0} ,w). $$

(15)

In this form, the correction for pores, microcracks, and fractures which describe the frequency dependence and anisotropy of the rock can be calculated with physical properties obtained from measured velocities. In the case of high porosity, the model is simplified by setting the mircocrack density as zero. Therefore,

$$ C_{ijkl} (\omega ) = C_{ijkl}^{0} (\varLambda ,{\rm M},w) - \phi_{p} C_{ijkl}^{1} (\lambda^{0} ,\mu^{0} ,w) - \varepsilon_{f} C_{ijkl}^{3} (\lambda^{0} ,\mu^{0} ,w). $$

(16)

Based on the theoretical model, experimental data measured from synthetic samples are compared with theoretical results. The input parameters for theoretical calculation are the properties of the background matrix measured from the blank sample and fracture density. To model the anisotropy caused by fractures, the background anisotropy should be taken into account. A modified version of the Chapman model was generated by Chapman et al. (2003) to account for background layering anisotropy in the blank rock. Tillotson et al. (2011) used this simplified equation to model samples with background anisotropy: by replacing $C_{ijkl}^{0} (\varLambda ,{\rm M},w)$ with $\left[ {C_{ijkl}^{\text{background}} + \varPhi_{{{\text{c,}}\;{\text{p}}}} C_{ijkl}^{1} (\lambda^{0} ,\mu^{0} ,w,\;{\text{water}})} \right]$; thus

$$ C_{ijkl} (\omega ) = \left[ {C_{ijkl}^{\text{background}} + \varPhi_{{{\text{c,}}\;{\text{p}}}} C_{ijkl}^{1} (\lambda^{0} ,\mu^{0} ,w, \;{\text{water}})} \right] - \phi_{\text{p}} C_{ijkl}^{1} (\lambda^{0} ,\mu^{0} ,w) - \varepsilon_{\text{f}} C_{ijkl}^{3} (\lambda^{0} ,\mu^{0} ,w), $$

(17)

in which the background stiffness $C_{ijkl}^{\text{background}}$ is formed by the measured velocities $(V_{\text{p}} \; (0{^\circ })$, $V_{\text{p}} \; (45^\circ )$, $V_{\text{p}} \; (90^\circ )$, $V_{\text{sh}} \; (0{^\circ })$ and $V_{\text{sh}} \; (90{^\circ }))$ and density of the background rock.