Efficient Uncertainty Quantification of Reservoir Properties for Parameter Estimation and Production Forecasting

Abstract

Three levels of uncertainty and their impact on reservoir estimation and forecasting are examined: (1) definition of reservoir facies geometry derived from uncertain geologic and geophysical information; (2) non-uniqueness of identifying the permeability distribution through inverse parameter estimation using injection and production data; and (3) unknown, fine-scale spatial variation in the heterogeneous rock properties. Inverse parameter estimation with pilot points and kriging is used to create accurate estimates of the permeability field. Efficient sampling of the uncertainty space surrounding these estimates through both probability-field (p-field) simulation and sequential Gaussian simulation (sGs) is demonstrated using a test case reservoir with permeability dominated by a sand-shale, facies distribution. A geologic conceptual model, measured permeabilities at the wells and 2100 days of injection and production data are used to estimate the permeability distribution and create empirical prediction intervals for future production. Diversity of the estimated and sampled fields across all three levels of uncertainty is examined through multi-dimensional scaling. Sampled permeability fields provide precise and accurate parameter estimation from injection and production data. The sGs fields create a wider prediction interval and underestimate the true production rates relative to the p-field samples. For a given estimated variogram model, the p-field samples result in shorter ranges and lower nugget values relative to the sGs fields. Conditional sampling through p-field and sGs simulation provides greater diversity in the solution space than the parameter estimation alone for a fraction of the computational expense.

Appendix: Issues with P-Filed Simulation

Given the theoretical issues of using p-field simulation with conditioning data identified in Pyrcz and Deutsch (2001), the reproduction of the variogram model for all sampled residual fields is checked. Empirical variograms are calculated directly on each of the 8000 sampled fields and examples are shown in Fig. 8. Both sampling approaches generate fields with hole-effect variograms in the Y-direction that are not part of the estimated variogram model, but are created through conditioning on the pilot points. The p-field variograms tend to strongly underestimate the range value, particularly in the X-direction.

Fig. 8

Examples of empirical variograms compared to the estimated variogram models for three different facies models: Models 2 (left), 4 (center) and 6 (right) columns and the X (top) and Y (bottom) row directions. Estimate 4 is shown in each case. In each image, there are 50 empirical variograms created using p-field and 50 created using sGs

To quantify the differences between the empirical and model variograms, a model is fit to every empirical variogram and then parameters of that fitted model are compared to the parameters of the original (estimated) variogram. This approach enables direct comparison of the variogram parameters rather than a single goodness-of-fit measure that does not elucidate the reasons why the empirical variogram deviates from the estimated variogram. In this comparison, a Gaussian variogram model is specified, all sills are normalized to 1.0 and the nugget values are estimated by linear interpolation from the first two points on the empirical variogram back to $h=0$. A one-dimensional search to minimize a weighted root-mean squared error (RMSE) along all possible values of the range is employed

$$\begin{aligned} \mathrm{RMSE}_{\omega } = \sqrt{\frac{\sum _{h=1}^{N_\mathrm{lag}}(\gamma _\mathrm{m} (h) - \gamma _\mathrm{emp} (h))^{2} \cdot \omega (h)}{N_\mathrm{lag}}}, \end{aligned}$$

(5)

where $\gamma _\mathrm{m}$ and $\gamma _\mathrm{emp}$ are the variogram values from the estimated model and the model fit to the empirical variogram, respectively, at lag spacing h. $N_\mathrm{lag}$ is the total number of lag spacings, and the weight for each lag spacing is the fraction of value pairs at that lag distance relative to the total number of pairs across all lag spacings: $\omega (h) = N_\mathrm{pair}(h)/N_\mathrm{pair}(\mathrm{total})$. The weighted RMSE is calculated separately for the X and Y- directions and the range value that minimizes the RMSE is retained.

Results of the variogram fitting are summarized in Fig. 9 with negative errors indicating the parameter used to fit the empirical variogram is less than that of the estimated variogram. For each facies model, variogram parameter differences from all ten estimates are aggregated into a single distribution, i.e., 500 values in each boxplot. In general, both approaches to creating sample fields result in shorter variogram ranges in both directions with p-field simulation consistently creating the shortest ranges. For the nugget value, p-field simulation tends to create locally smoother fields by slightly underestimating the nugget parameter while sGs tends to overestimate the nugget value.

Fig. 9

Differences between model and empirical variogram parameters. Results for the X and Y-direction are in the left and right columns, respectively. The difference between the range (top row) and nugget (bottom row) parameters is calculated on each of the 8000 fields with 500 differences in each distribution. The boxes contain the middle 50% of the values with a horizontal line at the median value. The circles are centered on the mean and the whiskers extend between the 5th and 95th percentiles