Prestack impedance inversions

We perform a prestack seismic impedance inversion on each of the three synthetic surveys. The basis of the method is outlined here, and given in more detail in Lumley and Beydoun (1991), and Lumley (1993). The first step consists of estimating angle-dependent reflectivity via a true-amplitude least-squares Kirchhoff prestack migration. This is obtained by an l₂ estimate of the $\grave{P}\!\acute{P}$ coefficient in constant offset sections at each subsurface point:

 

$$\grave{P}\!\acute{P}({\bf x};{\bf x}_h) \approx \frac{ \in... ...A^2_r \,d{\bf x}_m\,d\omega+ \epsilon^2({\bf x};{\bf x}_h)} \;,$$ (14)

and a separate l₁ estimate of the reflection angles $\bar{\theta}$ directly from the data in constant offset sections at each subsurface point:

 

$$\cos 2\bar{\theta}({\bf x};{\bf x}_h) \approx \frac{ \int_{... ...r}}\, d{\bf x}_m\,d\omega+ \epsilon^2({\bf x};{\bf x}_h) } \;,$$ (15)

where D are the recorded seismic data. To complete the final estimation of $\grave{P}\!\acute{P}(\bar{\theta})$, we make a simple set of mappings from the separate estimates of $\grave{P}\!\acute{P}$ and $\bar{\theta}$.A map of $\bar{\theta}({\bf x};{\bf x}_h)$ can be obtained directly as:

 

$${\cal M}_1 \;\; : \;\; \cos\ 2\bar{\theta}({\bf x};{\bf x}_h) \rightarrow \bar{\theta}({\bf x};{\bf x}_h) \;.$$ (16)

Then, since there is a one-to-one mapping of $\bar{\theta}$ to any point $({\bf x};{\bf x}_h)$, and $\grave{P}\!\acute{P}$ to the same point $({\bf x};{\bf x}_h)$, there is a unique map of $\grave{P}\!\acute{P}({\bf x};{\bf x}_h)$ and $\bar{\theta}({\bf x};{\bf x}_h)$ to $\grave{P}\!\acute{P}(\bar{\theta}({\bf x}))$ such that:

 

$${\cal M}_2 \;\; : \;\; \{ \grave{P}\!\acute{P}({\bf x};{\bf ... ...\} \rightarrow \grave{P}\!\acute{P}(\bar{\theta}({\bf x})) \;,$$ (17)

which is the desired result. This completes the angle-dependent reflectivity estimation process.

Once we have estimated $\grave{P}\!\acute{P}(\bar{\theta}({\bf x}))$,an inverse problem for three isotropic elastic parameters can be posed. Under the assumption that relative contrasts in material properties are small at reflecting boundaries, and the reflection angles are well within the pre-critical region (Aki and Richards, 1980), a linearization of the Zoeppritz plane wave reflection coefficients can be made at every subsurface point ${\bf x}$:

 

$$\grave{P}\!\acute{P}(\bar{\theta}) \approx I_p c_1(\bar{\theta}) + I_s c_2(\bar{\theta}) + D c_3(\bar{\theta}) \;,$$ (18)

where $\{I_p,I_s,D\}$ are the relative contrasts in P impedance, S impedance and density at the reflecting boundary, and $\{c_1, c_2, c_3\}$ are known basis functions which are analytical in $\bar{\theta}$. The three basis functions are plotted in Figure , with c₁ at the top, c₂ at the bottom, and c₃ near the zero axis in the middle, and are given here analytically as:

$$\begin{eqnarray} c_1 & = & 0.5 \sec^2\bar{\theta}\nonumber \\ c_2 & = & -4\gamm... ... \\ c_3 & = & 2\gamma^2 \sin^2\bar{\theta}- 0.5\tan^2\bar{\theta}\end{eqnarray}$$ (19)

where $\gamma$ is the shear to compressional velocity ratio V_s/V_p. We invert (18) at every subsurface location ${\bf x}$by a least-squares method which bootstraps with offset and angle. This yields an output section each of relative P and S impedance contrasts.

 

imprad-ann
Figure 23 Elastic impedance parameterization. Reflectivity unit impulse radiation curves (basis functions) $\{c_1, c_2, c_3\}$ for the impedance parameter set $\{I_p,I_s,D\}$. The uppermost curve is due to a unit perturbation in relative P impedance contrast alone, the lower curve is due to a unit perturbation in relative S impedance contrast alone, and the middle curve is due to a unit perturbation in relative density contrast alone.

Figure shows the P impedance inversion difference section for the two surveys before and after one time step of waterflood. The waterflood zone at the top of the reservoir shows the correct increase in P impedance at the well location due to injection of water which is of higher P impedance than the initial light oil in place at lower pore pressure. Figure shows the impedance inversion difference section for the two surveys before and after two time steps of waterflood production. Again, the waterflood zone at the top of the reservoir shows the correct increase in P impedance at the well, and the correct lateral spatial extent. We note that these P impedance sections resemble the migrated sections because, to first order, a (migrated) stack approximates the normal incidence P-P reflection coefficient in the absence of anomalous AVO. However, the impedance inversion results are more accurate in terms of relative impedance contrast estimates than a simple prestack imaging algorithm in general.

Figure shows the S impedance inversion difference section for the two surveys before and after one time step of waterflood. The waterflood zone at the top of the reservoir shows the correct decrease in S impedance at the well location due to injection of water which is of lower S impedance than the initial light oil in place at lower pore pressure. However, the S impedance results are much noisier than the P impedance results, which is expected since they are most sensitive to the relatively fewer far offset trace data. Figure shows the S impedance inversion difference section for the two surveys before and after two time steps of waterflood production. Again, the waterflood zone at the top of the reservoir shows the correct decrease in S impedance at the well, and the correct lateral spatial extent, although in a somewhat more noisy manner.

Finally, the impedance inversions detected the correct opposite polarity changes in P and S impedance between the pre-waterflood survey and the post-waterflood surveys. These two parameters, instead of one single (potentially ambiguous) migrated or stacked reflection amplitude parameter, may be more diagnostic of changes in reservoir petrophysical properties over time-lapse monitor surveys. Furthermore, the magnitude of the impedance changes was recovered reasonably well in data having a significant noise level, since the change in S impedance between surveys is estimated from the data to be on the order of 1.5 times the magnitude of the change in the P impedance after waterflood production. Estimates of absolute or relative changes in the petrophysical properties themselves could be a very valuable tool in monitoring reservoir production processes.

 

Pimp12
Figure 24 Waterflood 1 - Base P-impedance difference section.

 

Pimp13
Figure 25 Waterflood 2 - Base P-impedance difference section.

 

Simp12
Figure 26 Waterflood 1 - Base S-impedance difference section.

 

Simp13
Figure 27 Waterflood 2 - Base S-impedance difference section.