Interpretation of borehole sonic measurements acquired in vertical transversely isotropic formations penetrated by vertical wells

Abstract

Detecting vertical transversely isotropic (VTI) formations and quantifying the magnitude of anisotropy are fundamental for describing organic mudrocks. Methods used to estimate stiffness coefficients of VTI formations often provide discontinuous or spatially averaged results over depth intervals where formation layers are thinner than the receiver aperture of acoustic tools. We have developed an inversion-based method to estimate stiffness coefficients of VTI formations that are continuous over the examined depth interval and that are mitigated for spatial averaging effects. To estimate the coefficients, we use logs of frequency-dependent compressional, Stoneley, and quadrupole/flexural modes measured with wireline or logging-while-drilling (LWD) instruments in vertical wells penetrating horizontal layers. First, we calculate the axial sensitivity functions of borehole sonic modes to stiffness coefficients; next, we use the sensitivity functions to estimate the stiffness coefficients of VTI layers sequentially from frequency-dependent borehole sonic logs. Because sonic logs exhibit spatial averaging effects, we deaverage the logs by calculating layer-by-layer slownesses of formations prior to estimating stiffness coefficients. The method is verified with synthetic models of homogeneous and thinly bedded formations constructed from field examples of organic mudrocks. Results consist of layer-by-layer estimates of [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. We observe three sources of error in the estimated coefficients: (1) bias error originating from deaveraging the sonic logs prior to the sequential inversion, (2) error propagated during the sequential inversion, and (3) error associated with noisy slowness logs. We found that the relative bias and uncertainty of the estimated coefficients are largest for [Formula: see text] and [Formula: see text] because borehole modes exhibit low sensitivity to these two coefficients. The main advantage of our method is that it mitigates spatial averaging effects of sonic logs, while at the same time it detects the presence of anisotropic layers and yields continuous estimations of stiffness coefficients along the depth interval of interest.

Radial profiling of shear slowness from borehole acoustic measurements acquired in thinly laminated formations

“…In a horizontally stratified rock formation penetrated by a vertical borehole, the spatial averaging effects inherent to the borehole acoustic instrument can be derived from wave propagation equations or the ray‐tracing method. The forward model for a layered formation is given by (Huang et al., 2015; Maalouf & Torres‐Verdín, 2018b) $$$$\begin{equation} \bar{s}\left(d,f\right)\approx \sum _{i\hspace*{0.28em}}F\left({z}_{i}\right)s\left({z}_{i},f\right), \end{equation}$$$$where $$\bar s( {d,f} )$$ is the fast modelled frequency‐dependent slowness measured at depth $$d$$ and frequency $$f$$, $$s( {{z_i},f} )$$ is the formation slowness at depth $${z_i}$$ (between the i th and the $$i+1$$th receivers) and frequency $$f$$, while $$F$$ is the axial averaging function introduced by the geometry of the acoustic instrument with m receivers and given by (Maalouf & Torres‐Verdín, 2018b) $$$$\begin{equation} F\hspace*{0.28em}\left({z}_{i}\right)=\frac{6i\left(m-i\right)}{m\left({m}^{2}-1\right)}\hspace*{0.28em}. \end{equation}$$$$Figure 7 shows the axial spatial...…”

“…$$F( {\bf{m}} )$$ is the operator that performs axial averaging; 𝛼, $${{\bf{W}}_{\bf{d}}},$$$${{\bf{W}}_{\bf{e}}}$$ and $${\mathbf{m}}_{\mathrm{ref}}$$ are regularization parameter, regularization weighting matrix, data misfit weighting matrix and reference parameter vector for regularization (when needed), respectively. In the examples considered in this paper, $$\alpha \; = \;0$$ as regularization, that is, regularization is not implemented because the layers are thicker or slightly thinner than the full width at the half‐maximum of function $$F$$ (Maalouf & Torres‐Verdín, 2018b). With much thinner layers identified, first‐ or second‐order Tikhonov regularization for $${{\bf{W}}_{\bf{d}}}$$ can be implemented to bias a preference for a ‘flat’ or ‘smooth’ solution, respectively.…”

“…In a horizontally stratified rock formation penetrated by a vertical borehole, the spatial averaging effects inherent to the borehole acoustic instrument can be derived from wave propagation equations or the ray-tracing method. The forward model for a layered formation is given by (Huang et al, 2015;Maalouf & Torres-Verdín, 2018b)…”

“…where s(𝑑, 𝑓 ) is the fast modelled frequency-dependent slowness measured at depth 𝑑 and frequency 𝑓 , 𝑠(𝑧 𝑖 , 𝑓 ) is the formation slowness at depth 𝑧 𝑖 (between the ith and the 𝑖 + 1th receivers) and frequency 𝑓 , while 𝐹 is the axial averaging function introduced by the geometry of the acoustic instrument with m receivers and given by (Maalouf & Torres-Verdín, 2018b)…”