A Novel Neural Network for Seismic Anisotropy and Fracture Porosity Measurements in Carbonate Reservoirs

Abstract

Conventional neural networks (NNs) have been extensively used to model the spatial heterogeneity of rock properties from seismic inversion. Nevertheless, these generic NNs have a single network structure, which leads to overfitting and convergence difficulties. Furthermore, for stable predictions, conventional NNs highly depend on the initial weights and bias values. This research focuses on resolving the key problems of the existing NNs. In this paper, we propose and apply a novel neural network based on a multilayer linear calculator (MLLC) to estimate seismic anisotropy and fracture porosity in structurally complex and deeply buried carbonate reservoirs. This method, unlike conventional NNs, develops a nonlinear projection relationship between seismic and well log parameters to predict the spatial variation of seismic anisotropy and fracture porosity. We evaluate inversion effectiveness further by optimizing the MLLC with the particle swarm optimization (PSO) algorithm. We evaluate this new kind of MLLC neural network using computer-based simulations of complex models. After verifying the model′s reliability, we used it to estimate anisotropy and fracture porosity in two case studies from separate regions of China. Focused on anisotropy and fracture porosity estimations, the MLLC neural network outperformed the conventional NNs using backpropagation (BP) neural networks in simulation and field studies. The results indicate that the proposed methodology is considered valid for the anisotropic and porosity prediction of fractured reservoirs in other basins in China with similar geological settings and analogous basins anywhere in the world.

Appendices

Appendix A

1.1 A-1 Estimation of the Anisotropic Parameter from Backus Average

Anisotropy, caused by aligned fractures, is the main attribute for fractured reservoir characterization. Backus [20] has already demonstrated the calculation of stiffness parameters, in which an equivalent elastic medium was described,

$$ \left\{ \begin{gathered} {\varvec{c}}_{11} = \left\langle {{\varvec{\lambda}} + 2{\varvec{\mu}}} \right\rangle \hfill \\ {\varvec{c}}_{13} = \left\langle {\varvec{\lambda}} \right\rangle \hfill \\ {\varvec{c}}_{33} = \left\langle {{\varvec{\lambda}} + 2{\varvec{\mu}}} \right\rangle \hfill \\ {\varvec{c}}_{44} = \left\langle {\varvec{\mu}} \right\rangle \hfill \\ {\varvec{c}}_{66} = \left\langle {\varvec{\mu}} \right\rangle \hfill \\ {\varvec{c}}_{12} = {\varvec{c}}_{11} - 2{\varvec{c}}_{66} = \left\langle {\left( {{\varvec{\lambda}} + 2{\varvec{\mu}}} \right)} \right\rangle - 2\left\langle {\varvec{\mu}} \right\rangle \hfill \\ \end{gathered} \right. $$

(9)

where the angle bracket <> denotes an averaged quantity. The stiffness parameters can be obtained with the knowledge of longitudinal wave velocity, shear wave velocity and density ${\varvec{\rho}}$,

$$ \left\{ \begin{gathered} {\varvec{c}}_{11} = = \left\langle {\varvec{\rho V}_{p}^{2} } \right\rangle \hfill \\ {\varvec{c}}_{13} = \left[ {1 - 2\left\langle {\frac{{{\varvec{V}}_{s}^{2} }}{{{\varvec{V}}_{p}^{2} }}} \right\rangle } \right]\left\langle {\varvec{\rho V}_{p}^{2} } \right\rangle \hfill \\ {\varvec{c}}_{33} = \left\langle {\varvec{\rho V}_{p}^{2} } \right\rangle \hfill \\ {\varvec{c}}_{44} = \left\langle {\varvec{\rho V}_{s}^{2} } \right\rangle \hfill \\ {\varvec{c}}_{66} = \left\langle {\varvec{\rho V}_{s}^{2} } \right\rangle \hfill \\ {\varvec{c}}_{12} = \left\langle {\varvec{\rho V}_{p}^{2} } \right\rangle - 2\left\langle {\varvec{\rho V}_{s}^{2} } \right\rangle \hfill \\ \end{gathered} \right. $$

(10)

According to the relationship between the stiffness parameters and Thomsen parameters [19], Thomsen parameters of the effective medium can be written as follows based on the elastic parameters of ${\varvec{V}}_{{\varvec{p}}}$, ${\varvec{V}}_{{\varvec{s}}}$, ${\varvec{\rho}}$.

$$ \left\{ \begin{gathered} {\varvec{\varepsilon}} = 2\left( {\left\langle {\varvec{\rho V}_{s}^{2} } \right\rangle \left\langle {\frac{1}{{\varvec{\rho V}_{p}^{2} }}} \right\rangle - \left\langle {\frac{{{\varvec{V}}_{s}^{2} }}{{{\varvec{V}}_{p}^{2} }}} \right\rangle } \right) + 2\left( {\left\langle {\varvec{\rho V}_{s}^{2} } \right\rangle \left\langle {\frac{1}{{\varvec{\rho V}_{p}^{2} }}} \right\rangle^{2} - \left\langle {\frac{{{\varvec{V}}_{s}^{2} }}{{{\varvec{V}}_{p}^{2} }}} \right\rangle^{2} \left\langle {\frac{1}{{\varvec{\rho V}_{p}^{2} }}} \right\rangle } \right) \hfill \\ {\varvec{\delta}} = \frac{{2\left( {1 - \left\langle {\frac{{{\varvec{V}}_{s}^{2} }}{{{\varvec{V}}_{p}^{2} }}} \right\rangle } \right)\left( {\left\langle {\frac{1}{{\varvec{\rho V}_{p}^{2} }}} \right\rangle - \left\langle {\frac{{{\varvec{V}}_{s}^{2} }}{{{\varvec{V}}_{p}^{2} }}} \right\rangle \left\langle {\frac{1}{{\varvec{\rho V}_{s}^{2} }}} \right\rangle } \right)}}{{\left( {\left\langle {\frac{1}{{\varvec{\rho V}_{s}^{2} }}} \right\rangle - \left\langle {\frac{1}{{\varvec{\rho V}_{p}^{2} }}} \right\rangle } \right)}} \hfill \\ {\varvec{\gamma}} = \frac{1}{2}\left( {\left\langle {\varvec{\rho V}_{s}^{2} } \right\rangle \left\langle {\frac{1}{{\varvec{\rho V}_{s}^{2} }}} \right\rangle - 1} \right) \hfill \\ \end{gathered} \right. $$

(11)

According to Eq. (11), the target anisotropy parameters are calculated from the well logs (V_p, V_s, ρ), which is particularly useful in detecting seismic anisotropy from seismic data.

1.2 A-2 Fracture Porosity

The volume of shale (V_sh) was estimated from the gamma-ray log using Eq. (12):

$$ V_{{{\text{sh}}}} = \frac{{{\text{GR}}_{\log } - {\text{GR}}_{\min } }}{{{\text{GR}}_{\max } - {\text{GR}}_{\min } }} $$

(12)

where GR_log, GR_min, and GR_max are gamma-ray reading at a depth of interest, 100% clean sand, and 100% shale, respectively (API units).

The total porosity $\varphi_{T}$ was estimated using a density log using Eq. (13):

$$ \varphi_{T} = \frac{{\rho_{{{\text{ma}}}} - \rho_{b} }}{{\rho_{{{\text{ma}}}} - \rho_{f} }} $$

(13)

where ρ_ma is, the matrix density and ρ_f denotes fluid density.

$$ \varphi_{F} = \frac{{{\text{Frac}}\left( {\varphi_{T} - 1} \right)}}{{\left( {v\varphi_{T} - 1} \right)}} $$

(14)

where $\varphi_{F}$ is the fracture porosity (with no vugs) and $v$ denotes porosity partitioning coefficient.

Appendix B

2.1 PSO Algorithm for Optimization

Appendix C

3.1 C-1 Accuracy Analysis of the Simple Model

The inversion of the three anisotropic parameters is carried out using the proposed method, but the parameter γ is lower than the other two parameters. Because the structure of parameter γ is different from the other two parameters (ε, δ), while the seismic response is a comprehensive response with three anisotropic parameters. In this paper, the training samples we choose are CDP50, CDP100, CDP150, CDP200, CDP250, CDP300, CDP350. While CDP50 and CDP100, the correlation between seismic and parameter γ are not very good. So, the precision is low. The precision is affected by the structure of the geological model rather than the stability of the algorithm we mentioned, and we select the same part of the model structure to carry out inversion. Figure 20a gives the complete seismic response, and Fig. 20b gives the partial seismic response, which comes from the red box of complete seismic response, as shown in Fig. 20a. Then, we use this part for the inversion. The inversion results are shown in Fig. 20c–e. The results show that the accuracy of the three anisotropic parameters is the same. The difference in accuracy is due to the design model rather than the stability of the algorithm.

Fig. 20

The model of synthetic and inversion Thomson anisotropic parameters, a synthetic seismic, b part of synthetic seismic c anisotropic parameter of ε, d anisotropic parameter of δ, e anisotropic parameter of γ

3.2 C-2 Case Studies Application Analysis of the BP Neural Network

The comparison results with the backpropagation neural network (BP) are also shown to better reflect the method's credibility in field data examples. The inversion results of anisotropic strength parameters ε, γ, and δ (red curve represents corresponding anisotropic parameter) are shown in Fig. 21. According to the inversion results, both inversion fracture responses (with the red color denoting high value) could satisfy values at the well location. By contrast, the BP-based inversion results have poor lateral continuity of the anisotropy body's seismic event and unclear boundaries. For further comparisons of fracture distribution, the slices of different parameters are shown in Fig. 22. The multilayer linear calculator (MLLC) inversion results indicate that fracture density near Well A and Well B is relatively high, whereas the one surrounding Well D is relatively low, and fractures did not develop near Well C. The results agree with the drilling result in Table 2. However, the BP-based inversion results failed to characterize the lateral fracture distribution, proving the proposed method's stability.

Fig. 21

MLLC inversion of a anisotropic parameter profile of ε; c anisotropic parameter profile of γ; e anisotropic parameter profile of δ; BP inversion of b anisotropic parameter profile of ε; d anisotropic parameter profile of γ; f anisotropic parameter profile of δ

Fig. 22

The slices of anisotropic parameters, MLLC inversion of a anisotropic parameter profile of ε; c anisotropic parameter profile of γ; e anisotropic parameter profile of δ; BP inversion of b anisotropic parameter profile of ε; d anisotropic parameter profile of γ; f anisotropic parameter profile of δ