Fig 1.
Some types of seismic data and their relation to reservoir petro-physical parameters.
Fig 2.
Workflow of wavelet-based sparse representation.
Fig 3.
The proposed 4D seismic history matching framework.
Fig 4.
Single level decompositions in 2D (left) and 3D (right) decimated discrete wavelet transforms.
Fig 5.
Illustration of sparse representation of a 3D AVA far-offset trace using slices at X = 40, 80, 120 and at Z = 50, 100, 150, 200, respectively.
(a) Reference AVA trace; (b) Noisy AVA trace obtained by adding Gaussian white noise (noise level = 30%) to the reference data; (c) Reconstructed AVA trace obtained by first conducting a 3D DWT on the noisy data, then applying hard thresholding (using the universal threshold value) to wavelet coefficients, and finally reconstructing the data using an inverse 3D DWT based on the modified wavelet coefficients; (d) Wavelet sub-band HHL1 corresponding to the reference AVA data; (e) Wavelet sub-band HHL1 corresponding to the noisy AVA data; (f) Wavelet sub-band HHL1 corresponding to the reconstructed AVA data; (g) Reference noise, defined as noisy AVA data minus reference AVA data; (h) Estimated noise, defined as noisy AVA data minus reconstructed AVA data; (i) Noise difference, defined as estimated noise minus reference noise. All 3D plots are created using the package Sliceomatic (version 1.1) from MATLAB Central (File ID: #764).
Table 1.
Summary of experimental settings in the Brugge benchmark case study.
Fig 6.
Boxplots of production data mismatch as a function of iteration step (scenario S1).
The horizontal dashed line indicates the threshold value (4 × 1400 = 5600) for the stopping criterion (24). For visualization, the vertical axis is in the logarithmic scale. In each box plot, the horizontal line (in red) inside the box denotes the median; the top and bottom of the box represent the 75th and 25th percentiles, respectively; the whiskers indicate the ranges beyond which the data are considered outliers, and the whiskers positions are determined using the default setting of MATLAB R2015b, while the outliers themselves are plotted individually as plus signs (in red).
Fig 7.
Boxplots of RMSEs of (a) log PERMX and (b) PORO as functions of iteration step (scenario S1).
Fig 8.
Profiles of WBHP, WOPR and WWCT of the initial (1st row) and final (2nd row) ensembles at the producer BR-P-5 (scenario S1).
The production data of the reference model are plotted as orange curves, the observed production data at 20 report times as red dots, and the simulated production data of initial and final ensembles as blue curves.
Fig 9.
Log PERMX (top row) and PORO (bottom row) of the reference reservoir model (1st column) and the means of the initial (2nd column) and final (3rd column) ensembles at Layer 2 (scenario S1).
The black dots in the figures represent the locations of injection and production wells (top view).
Fig 10.
Boxplots of seismic data mismatch as functions of iteration step (scenario S2).
Case (a) corresponds to the results with c = 1, for which choice the number of leading wavelet coefficients is 178332, roughly 2.5% of the original data size; Case (b) to the results with c = 5, for which choice the number of leading wavelet coefficients is 3293, more than 2000 times reduction in data size.
Fig 11.
Boxplots of RMSEs of log PERMX (1st column) and PORO (2nd column) as functions of iteration step, with c being 1 (top) and 5 (bottom), respectively (scenario S2).
Fig 12.
Top row: slices of the observed far-offset AVA attributes at X = 80, with respect to the base survey (1st column), the 1st monitor survey (2nd column) and the 2nd monitor survey (3rd column), respectively. Middle row: corresponding reconstructed slices at X = 80 using the leading wavelet coefficients at c = 1 (while all other wavelet coefficients are set to zero). Bottom row: corresponding reconstructed slices at X = 80 using the leading wavelet coefficients at c = 5 (while all other wavelet coefficients are set to zero).
Fig 13.
Top row: slices (at X = 80) of the differences between the reconstructed far-offset AVA attributes using the leading wavelet coefficients (c = 1) of the observed seismic data, and the reconstructed far-offset AVA attributes using the corresponding leading wavelet coefficients (c = 1) of the means of the simulated seismic data of the initial ensemble. From left to right, the three columns correspond to the differences at the base, the 1st monitor, and the 2nd monitor surveys, respectively. Bottom row: as in the top row, except that it is for the differences between the reconstructed far-offset AVA attributes of the observed seismic data, and the reconstructed far-offset AVA attributes of the mean simulated seismic data of the final ensemble.
Fig 14.
As in Fig 9, but for scenario S2 with c = 1.
Fig 15.
Boxplots of production (top) and seismic (bottom) data mismatch as functions of iteration step (scenario S3).
Fig 16.
Boxplots of RMSEs of log PERMX (1st column) and PORO (2nd column) as functions of iteration step, with c being 1 (top) and 5 (bottom), respectively (scenario S3).
Fig 17.
As in Fig 8, but for the production data profiles in scenario S3 with c = 1.
Fig 18.
As in Fig 13, but for the slices (at X = 80) of differences in scenario S3 with c = 1.
Fig 19.
As in Fig 9, but for scenario S3 with c = 1.