Abstract
It is common in ensemble-based methods of history matching to evaluate the adequacy of the initial ensemble of models through visual comparison between actual observations and data predictions prior to data assimilation. If the model is appropriate, then the observed data should look plausible when compared to the distribution of realizations of simulated data. The principle of data coverage alone is, however, not an effective method for model criticism, as coverage can often be obtained by increasing the variability in a single model parameter. In this paper, we propose a methodology for determining the suitability of a model before data assimilation, particularly aimed for real cases with large numbers of model parameters, large amounts of data, and correlated observation errors. This model diagnostic is based on an approximation of the Mahalanobis distance between the observations and the ensemble of predictions in high-dimensional spaces. We applied our methodology to two different examples: a Gaussian example which shows that our shrinkage estimate of the covariance matrix is a better discriminator of outliers than the pseudo-inverse and a diagonal approximation of this matrix; and an example using data from the Norne field. In this second test, we used actual production, repeat formation tester, and inverted seismic data to evaluate the suitability of the initial reservoir simulation model and seismic model. Despite the good data coverage, our model diagnostic suggested that model improvement was necessary. After modifying the model, it was validated against the observations and is now ready for history matching to production and seismic data. This shows that the proposed methodology for the evaluation of the adequacy of the model is suitable for large realistic problems.
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Notes
Data for Plith provided by Fridtjof Riis of the Norwegian Petroleum Directorate. Compaction is based on [42].
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Acknowledgements
The authors thank Equinor (operator of the Norne field) and its license partners Eni Norge and Petoro for the release of the Norne data. The authors acknowledge the Center for Integrated Operations at NTNU for cooperation and coordination of the Norne Cases. The view expressed in this paper are the views of the authors and do not necessarily reflect the views of Equinor and the Norne license partners.
We are grateful to Geovariances for providing a license for the use of Isatis for factorial co-kriging, and to Schlumberger for providing Eclipse and Petrel licenses.
Funding
This study is supported by the CIPR/IRIS cooperative research project “4D Seismic History Matching” which is funded by industry partners Eni Norge, Petrobras, and Total, as well as the Research Council of Norway through the Petromaks2 program.
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Appendices
Appendix A: Petro-elastic model
We use the Gassmann model [17, 46] to model the bulk modulus, Ksat, and the shear modulus, μsat, of the saturated rock [31], as follows:
and
where Kdry is the bulk modulus of the dry rock, \(K{{~}_{\min \limits }}\) is the bulk modulus of the mixture of minerals in the rock matrix, Kfluid is the bulk modulus of the mixture of fluids (water, oil and gas) in the porous medium, ϕ is the porosity of the rock, and μdry is the shear modulus of the dry rock.
The P-wave velocity of the saturated rock is obtained from the following relationship:
The acoustic or P-wave impedance of the saturated rock is then computed from the P-wave velocity and the bulk density of the rock as follows:
We model the elastic moduli of the minerals in the rock matrix using the Hashin-Shtrikman bounds [19] for a sand-shale mixture, assuming that the rock matrix is composed of sand and shale, and hence, quartz and clay minerals. Our estimates of \(K{{~}_{\min \limits }}\) (which is required for computation of Ksat (14)) and \(\mu {{~}_{\min \limits }}\) are obtained by arithmetic averaging of upper and lower Hashin-Shtrikman bounds [31]. The mineral density is the volume-weighted average of the densities of quartz and clay minerals. The relative volumes of each mineral are assumed to be related directly to the net-to-gross ratio of the reservoir model.
Elastic moduli of the dry rock are computed using a slightly modified version [4] of the dependence of the dry-rock moduli on porosity [27, 36], coupled with the dependence on changes in effective stress [30]. This combination is used in the context of 4D seismic data, where two or more seismic times, and hence reservoir conditions, are involved. The dynamic part of the elastic dry moduli (Kdry and μdry), which describes the dependence on stress, are computed as follows:
In Eqs. 18 and 19, Peff,init is the effective pressures at initial reservoir conditions, Peff is the effective pressure at a particular reservoir condition (for example, at the time of a certain 4D seismic survey), and Ek, Pk, Eμ, and Pμ are stress sensitivity parameters [30].
The static parts the dry elastic moduli, which describe the dependence of dry-rock elastic properties on porosity are given by the following:
where β is the consolidation factor, which is assumed to be linearly related to the bulk composition [4] as follows:
where Vsand and Vshale are the volumes of sand and shale in the whole rock, respectively, ϕ is the porosity, and a, b, and c are consolidation parameters determining the sensitivity of β to Vsand, Vshale, and ϕ. The elastic moduli of the dry rock (Kdry and μdry) in Eqs. 18 and 19 are required inputs to Gassmann’s relations (14 and 15).
The proportions of sand and shale in the bulk rock are assumed to be determined by porosity and net-to-gross ratio as follows:
Effective pressure (Peff) at each grid cell of the model is computed from Peff = Plith − P where P is fluid pressure and Plith is lithostatic pressure computed from the following formula:Footnote 1
where z is vertical subsea depth in meters.
Elastic moduli of the mixture of fluids at reservoir conditions are modeled using relationships from Batzle and Wang [5]. For the Norne field, the reservoir temperature was assumed to be 98.3 ∘C, water salinity 15000 ppm, oil gravity 32.7 API, and gas gravity 0.645. The bulk modulus of the mixture of fluids (Kfluid) is computed using the Reuss average (Mavko et al., 2009) as follows:
where Sw, So, and Sg, are the saturation of water, oil, and gas, respectively, and Kw, Ko, and Kg are the bulk moduli of water, oil, and gas.
The PEM that we selected (Eqs. 14 to 26) requires that some parameters be estimated and calibrated for the Norne field case. We separated the PEM into two vertical regions comprising the Garn formation (layers 1 to 3 of the Norne simulation model) and the formations underlying the Not shale (Ile to Tofte in layers 5 to 22), as the geological information available on the Norne field suggests that the Garn formation and the formations below the Not shale are reasonably different [44].
Values for the consolidation parameters a, b, and c in the original model were obtained from a weighted average at four different wells in the Norne field [8, 9]. Values for elastic moduli of the mineral parameters in the rock matrix were obtained from well log calibration in the Norne field, and the density of sand and shale in the rock matrix were set to 2.689 g/cc and 2.635 g/cc, respectively [8]. Finally, values of parameters for the pressure sensitivity of dry elastic moduli (Eqs. 18 and 19) are obtained from core measurements from the Schiehallion field [30]. All values used in the PEM are shown in Table 3.
Appendix B: Seismic modeling
We generate realizations of acoustic impedance predictions from the Norne simulation model, using the PEM (Appendix A) and applying a vertical seismic filter to account for differences in resolution between the simulation model and the inverted seismic data.
Due to the gridding in the simulation model, the seismic predictions obtained from application of the PEM to the saturations, pressures, and porosities at the reservoir simulation grid scale may contain high-frequency features that are outside of the observed seismic-frequency spectrum. Although other solutions for dealing with lack of smoothness in synthetic seismic data have been proposed [11, 43], we have applied a vertical seismic filter to the acoustic impedance predictions in the simulation model in order to make them more comparable to actual seismic data [2, 37]. In other words, perturbed impedance predictions and inverted impedance data are compared at the scale of the Norne seismic data.
This filter was created based on the frequency spectrum of the actual inverted acoustic impedance data of the four Norne seismic surveys (Ip2001, Ip2003, Ip2004, and Ip2006), all of which have very similar frequency content. We modeled the observed spectra by a low-pass Ormsby filter [34], which is a trapezoidal filter applied in the frequency domain that removes all the frequencies above some user-defined cut frequencies. To approximately match the frequency spectrum of real impedance data, we set the cut frequencies of the Ormsby filter at 0-0-100-120 Hz. Prior to filtering, we populated the inactive cells of the Norne simulation model with a Not shale acoustic impedance value of 6500 (m/s).(g/cc). Figure 8 shows one realization of the impedance predictions at the time of the 2001 seismic survey (baseline) before and after filtering.
Appendix C: Simulation model
Table 4 summarizes the parameters of the simulation model that were modified in the improved and move-completion models (Section ??). In this table, MULTREGT defines the transmissibility multiplier between flux regions, MULTZ describes the vertical transmissivity multiplier between two layers, and KRW corresponds to the endpoint relative permeability of water.
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Alfonzo, M., Oliver, D.S. Evaluating prior predictions of production and seismic data. Comput Geosci 23, 1331–1347 (2019). https://doi.org/10.1007/s10596-019-09889-6
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DOI: https://doi.org/10.1007/s10596-019-09889-6