MEDIDAS DE CONTROL ESPECIAL

As discussed before, the top reservoir is characterized by a decrease in P-impedance from the shale to the reservoir sand. The base reservoir is clearly delineated on the seismic by the strong contrast between the sand and shale, as well. The specification of the prior model is the controversial part of the Bayesian inversion. Often the available prior information is not sufficient to define a unique parametric prior distribution.

Figure 3.13 The initial PORO model (left) and the PEM prediction of the P-impedance (right)

on the reservoir model grid, which is used as a prior expectation for the baseline seismic inversion.

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Indeed, when the PEM prediction is introduced, a pragmatic approach is to select a parametric distribution with expectations equal to the sim2seis results. However, the most obvious pitfall with this approach is that the prediction is deterministic and it cannot provide the seis2sim with realistic estimations about the variances. Although the variances can be estimated by looking at the statistics from Figure 3.5, it is rather risky to use information from this single well to represent the entire area. Therefore, fairly big variances are given to the prior estimates, to ensure the inversion covers the full solution space. The estimated standard deviation for the overburden and underburden shales is

2*σshale = 3200 m/s*g/cm3 (), while for the the reservoir sand it is 2*σsand = 2700

m/s*g/cm3. The seis2sim solution for P-impedance is a priori assumed to be Gaussian. This assumption can be graphically evaluated in the well statistics by a Gaussian probability fit, shown in Figure 3.5.

The temporal correlation function of impedance is estimated for certain time lags from the well logs (see Figure 3.14). This function is modelled by an analytic correlation function, 𝛾(𝑡; 𝑎, 𝑏) = 1₂∙ 𝑒−(𝑎𝑡) 2 +1₂∙ (1 −2𝑡_𝑏22) ∙ 𝑒−(𝑡𝑏) 2 , (3.6)

defined by the sum of an exponential correlation function, with a range 𝑎 = 1.8 ms, and a second order term with 𝑏 = 9 ms. The fit to the estimated correlation function is considered to be good for the purpose of stabilising the seis2sim solution.

Figure 3.14 Correlation function estimated from well logs (dots), and an analytical correlation

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The solution of equation (2.3) is obtained by the adaptive Markov chain Monte Carlo (MCMC) simulation proposed in Chapter 2. One trace for the seis2sim inversion at the well location is displayed in Figure 3.15, showing the matches to the observed seismic trace and the posterior realisations of P-impedance traces. The P-impedance is well determined, as the prediction intervals are reduced by up to 95%. The Fangst reservoir is between 2180 to 2244 ms in TWT, inside which the “soft” sand is successfully delineated. Figure 3.16 displays the convergence of the MCMC process. The Markov chain begins to converge after the first 1500 iterations which is usually referred to as the “burn in” stage, after which the “detail balance” of the MCMC is achieved and the samples are representing the posterior probability distribution.

Figure 3.15 The posterior inversion results for 1D baseline seis2sim inversion. The red traces

are the realisations of seismic traces (left) and the corresponding P-impedances (right), where the black trace in the left diagram is the observation. The light blue dashed lines on the right are the prior prediction interval, while the blue and the black is the prior expectation. The range that is covered by the realisations reflects to the posterior uncertainty after inversion.

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Figure 3.16 The MCMC convergence process for the 1D example in Figure 3.15. (a) the overall

misfit evolution with iterations; (b) the evolution of P-impedance of two samples from the overburden (black) and the reservoir (red); (c) the histogram of the posterior realisations of the overburden sample; (d) the histogram of the posterior realisations of the reservoir sample.

In Figure 3.16 (b) there are two different examples from the overburden and the reservoir, with different converging paths. The inversion for the reservoir interval has a shorter “burn-in” period compared to the overburden, which starts to converge from 1500 iterations. After seis2sim inversion, the overburden sample has a mean of 7940 m/s*g/cm3, associated with a standard deviation of 341 m/s*g/cm3 (the prior estimation of the standard deviation is 1600 m/s*g/cm3), while the sample from the reservoir interval has a mean of 4372 m/s*g/cm3, associated with a standard deviation of 230 m/s*g/cm3 (the prior estimation of the standard deviation is 1350 m/s*g/cm3).

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Figure 3.17 The various baseline seis2sim results. (a) observed baseline seismic; (b) synthetic

baseline generated by the posterior mean of the P-impedance; (c) posterior mean of the P- impedance from seis2sim; (d) the posterior standard deviation after inversion.

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The maximum posterior solution for the P-impedance is equal to the posterior expectations which are displayed in Figure 3.17. This maximum a posteriori solution is generally smoother than a single realization in Figure 3.15. A single realization, however, will be a possible solution with the full variability defined mainly from the prior distribution. Figures 3.17 (a) and (b) show the contrast between the observed baseline seismic and the synthetic, from the posterior mean P-impedance volume. The largest mismatch is found along the top reservoir, where the sand to shale contrast lies. Indeed, the details inside the reservoir show a reasonably good match. Figures 3.17 (c) and (d) show the intersections of the posterior mean and the standard deviation of the P- impedance, in which the reservoir sand distribution is better imaged. The biggest

Figure 3.18 Time slices of the baseline seis2sim results. (a) the observed baseline seismic; (b)

the residual error of the synthetic baseline from the inversion; (c) the posterior mean of P- impedance from seis2sim; (d) the posterior standard deviation.

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uncertainties are found along the top reservoir, which could be caused by the default constant models of the overburden. Figure 3.18 shows the time slices of the inverted volumes, where the posterior standard deviation shows a consistent pattern with the residual misfits. This misfit volume will be propagated into the 4D seis2sim, as an additional uncertainty term formulated in the inversion.