Rock Physics Modeling and Seismic AVAZ Responses of Fractured Reservoirs with Different Azimuths

Abstract

Fracture orientation and in particular the fracture strike play an important role in guiding the accurate estimation of the orientation of in situ stress fields. Fracture strike can be extracted based on the azimuthal seismic reflection responses. Thus, the characteristics of seismic reflection responses varying with fracture strike are critical for fracture strike estimation. In this work, a set of vertical fractures with different strikes is introduced into an isotropic host matrix based on the Schoenberg linear-slip theory, and the azimuthal seismic reflection response is modeled based on the reflection/transmission analysis at a plane interface. The effect of fracture parameters, i.e., fracture density, fracture aspect ratio, and fluid infill on the seismic reflection response, is analyzed. The influences of fracture strike on the stiffness matrix are simulated, and we further discuss the impacts on the P- and S-wave velocities. The influences of fracture strike on the reflection coefficient are depicted using the rose diagram method. The proposed workflow is applied to real seismic reflection data. The method is used to estimate the fracture strike with the seismic data in the Sichuan Basin work area by assuming that all fractures are vertically oriented, and the rose diagram is obtained close to the target layer. The results show that the fracture strike predicted by the method is in good agreement with the estimation based on the well-bore imaging data.

Appendix A: Calculation formulas of \(\xi\) and \(\eta\) in Eq. 6

$$t = \left[ {\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta } \right],$$

(A-1a)

$${\mathbf{t}}^{\prime} = \left[ { - \sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta } \right],$$

(A-1b)

$${\mathbf{p}} = \frac{1}{{\alpha_{{\text{b}}} }}\left[ {\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta } \right],$$

(A-1c)

$${\mathbf{p}}^{\prime} = \frac{1}{{\alpha_{{\text{b}}} }}\left[ { - \sin \theta \cos \varphi , - \sin \theta \sin \varphi ,\cos \theta } \right],$$

(A-1d)

$$\xi = {\mathbf{t}}_{i} {\mathbf{t}}_{i}^{^{\prime}} = \cos^{2} \theta - \sin^{2} \theta = \cos 2\theta ,$$

(A-2a)

$$\eta_{mn} = {\mathbf{t}}_{i}^{^{\prime}} {\mathbf{p}}_{j}^{^{\prime}} {\mathbf{t}}_{k} {\mathbf{p}}_{l} ,$$

(A-2b)

where the corresponding parameters $\eta_{mn}$ are determined.

$$\eta_{11} = {{\sin^{4} \theta \cos^{4} \varphi } \mathord{\left/ {\vphantom {{\sin^{4} \theta \cos^{4} \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }} ,$$

(A-3a)

$$\eta_{12} = {{\sin^{4} \theta \sin^{2} \varphi \cos^{2} \varphi } \mathord{\left/ {\vphantom {{\sin^{4} \theta \sin^{2} \varphi \cos^{2} \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3b)

$$\eta_{13} = {{\sin^{2} \theta \cos^{2} \theta \cos^{2} \varphi } \mathord{\left/ {\vphantom {{\sin^{2} \theta \cos^{2} \theta \cos^{2} \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3c)

$$\eta_{14} = {{2\sin^{3} \theta \cos \theta \sin \varphi \cos^{2} \varphi } \mathord{\left/ {\vphantom {{2\sin^{3} \theta \cos \theta \sin \varphi \cos^{2} \varphi } {{\upalpha }_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {{\upalpha }_{{\text{b}}}^{{2}} }},$$

(A-3d)

$$\eta_{15} = {{ - 2\sin^{3} \theta \cos \theta \cos^{3} \varphi } \mathord{\left/ {\vphantom {{ - 2\sin^{3} \theta \cos \theta \cos^{3} \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3e)

$$\eta_{16} = 2{{(\sin^{4} \theta \sin \varphi \cos^{3} \varphi )} \mathord{\left/ {\vphantom {{(\sin^{4} \theta \sin \varphi \cos^{3} \varphi )} {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3f)

$$\eta_{22} = {{\sin^{4} \theta \sin^{4} \varphi } \mathord{\left/ {\vphantom {{\sin^{4} \theta \sin^{4} \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3g)

$$\eta_{23} = {{\sin^{2} \theta \cos^{2} \theta \sin^{2} \varphi } \mathord{\left/ {\vphantom {{\sin^{2} \theta \cos^{2} \theta \sin^{2} \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3h)

$$\eta_{24} = {{ - 2\sin^{3} \theta \cos \theta \sin^{3} \varphi } \mathord{\left/ {\vphantom {{ - 2\sin^{3} \theta \cos \theta \sin^{3} \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3i)

$$\eta_{25} = {{ - 2\sin^{3} \theta \cos \theta \sin^{2} \varphi \cos \varphi } \mathord{\left/ {\vphantom {{ - 2\sin^{3} \theta \cos \theta \sin^{2} \varphi \cos \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3j)

$$\eta_{26} = 2{{(\sin^{4} \theta \sin^{3} \varphi \cos \varphi )} \mathord{\left/ {\vphantom {{(\sin^{4} \theta \sin^{3} \varphi \cos \varphi )} {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3k)

$$\eta_{33} = {{\cos^{4} \theta } \mathord{\left/ {\vphantom {{\cos^{4} \theta } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3l)

$$\eta_{34} = {{2\sin \theta \cos^{3} \theta \sin \varphi } \mathord{\left/ {\vphantom {{2\sin \theta \cos^{3} \theta \sin \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3m)

$$\eta_{35} = {{ - 2\sin \theta \cos^{3} \theta \cos \varphi } \mathord{\left/ {\vphantom {{ - 2\sin \theta \cos^{3} \theta \cos \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3n)

$$\eta_{36} = 2{{(\sin^{2} \theta \cos^{2} \theta \sin \varphi \cos \varphi )} \mathord{\left/ {\vphantom {{(\sin^{2} \theta \cos^{2} \theta \sin \varphi \cos \varphi )} {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3o)

$$\eta_{44} = {{ - 4\sin^{2} \theta \cos^{2} \theta \sin^{2} \varphi } \mathord{\left/ {\vphantom {{ - 4\sin^{2} \theta \cos^{2} \theta \sin^{2} \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3p)

$$\eta_{45} = {{ - 4\sin^{2} \theta \cos^{2} \theta \sin \varphi \cos \varphi } \mathord{\left/ {\vphantom {{ - 4\sin^{2} \theta \cos^{2} \theta \sin \varphi \cos \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3q)

$$\eta_{46} = {{ - 4\sin^{3} \theta \cos \theta \sin^{2} \varphi \cos \varphi } \mathord{\left/ {\vphantom {{ - 4\sin^{3} \theta \cos \theta \sin^{2} \varphi \cos \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3r)

$$\eta_{55} = {{ - 4\sin^{2} \theta \cos^{2} \theta \cos^{2} \varphi } \mathord{\left/ {\vphantom {{ - 4\sin^{2} \theta \cos^{2} \theta \cos^{2} \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3s)

$$\eta_{56} = {{ - 4\sin^{3} \theta \cos \theta \sin \varphi \cos^{2} \varphi } \mathord{\left/ {\vphantom {{ - 4\sin^{3} \theta \cos \theta \sin \varphi \cos^{2} \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3t)

$$\eta_{66} = 4{{\sin^{4} \theta \sin^{2} \varphi \cos^{2} \varphi } \mathord{\left/ {\vphantom {{\sin^{4} \theta \sin^{2} \varphi \cos^{2} \varphi } {a_{{\text{b}}}^{{2}} }}} \right. \kern-0pt} {a_{{\text{b}}}^{{2}} }},$$

(A-3u)

and

$$\eta_{21} = \eta_{12} ,$$

(A-4a)

$$\eta_{31} = \eta_{13} ,$$

(A-4b)

$$\eta_{32} = \eta_{23} ,$$

(A-4c)

$$\eta_{41} = \eta_{14} ,$$

(A-4d)

$$\eta_{42} = \eta_{24} ,$$

(A-4e)

$$\eta_{43} = \eta_{34} ,$$

(A-4f)

$$\eta_{51} = \eta_{15} ,$$

(A-4g)

$$\eta_{52} = \eta_{25} ,$$

(A-4h)

$$\eta_{53} = \eta_{35} ,$$

(A-4i)

$$\eta_{54} = \eta_{45} ,$$

(A-4j)

$$\eta_{61} = \eta_{16} ,$$

(A-4k)

$$\eta_{62} = \eta_{26} ,$$

(A-4l)

$$\eta_{63} = \eta_{36} ,$$

(A-4m)

$$\eta_{64} = \eta_{46} ,$$

(A-4n)

$$\eta_{65} = \eta_{56} ,$$

(A-4o)

where $a_{{\text{b}}}$ represents the P-wave velocity of the host medium. and $\varphi$ denotes the azimuth angle.