Channelized reservoir estimation using a low-dimensional parameterization based on high-order singular value decomposition

Abstract

Prior to the estimation process of channelized reservoirs, in the context of any Assisted History Matching method, the parameterization of facies field is a necessary task. The parameterization of the facies field consists of defining a numerical field (parameter field) on the reservoir domain so that, using a projection function, we are able to recover the facies field from the values of parameter field. One of the most important issues encountered is the loss of the multipoint geostatistical properties in the updates (channel continuity). In this study, we start from an initial (global) parameterization of the channelized field and infer from it a low-dimensional parameterization obtained after a high-order singular value decomposition of a tensor built with the parameter fields. We decompose the parameter field as a linear combination of some basis functions with coefficients. The decomposition is followed by a truncation so that we keep the relevant information from the channel continuity perspective, but with a small number of coefficients. The coefficients will represent the low-dimensional parameterization and are further introduced in the estimation process of facies field, using the Ensemble Smoother with Multiple Data Assimilations (ES-MDA). For a fair assessment of the parameterization, we perform a comparison of the results with those obtained by applying the traditional truncated singular value decomposition and the global parameterization. In addition, we compare the parameterization with a low-dimensional parameterization defined with the PCA decomposition. The comparison is done from the perspective of multipoint geostatistical characteristics of the updates and predictions. We show that the new parameterization is able to better keep the multipoint geostatistical structure in the updates than the other parameterizations, while the prediction capabilities are the same.

Appendix: Truncation analysis of the HOSVD of tensor

The reason for truncation of the decomposition from Eq. 5 is to obtain an approximation for the tensor but, with a small loss of information. This approximation generates an approximation for each layer (third mode) of the tensor and consequently, an approximation of the facies fields. Because of the nature of the HOSVD procedure, the relevant information is contained in the superior layers of the core tensor σ, so for a good tensor approximation are necessary only small values for n_x and n_y. However, in our geostatistical context, we have to be very careful when choosing these parameters because we have to take into account the geological meaning of the new facies fields generated with the parameter fields obtained after truncation ($\overline {\theta _{r}}$). Consequently, in order to set values for n_x and n_y, a sensitivity analysis is necessary and we verify the behavior of the transformed facies fields for all possible values of n_x and n_y. In Fig. 27 is shown the mean difference in percents between the original facies fields and the facies fields obtained after the truncation of the parameter $\overline {\theta _{r}}$ ($r\in \overline {1,120})$, for all values of parameters $n_{x}, n_{y}\in \overline {1,35}$. The red curve in the middle of the picture represents the pairs (n_x,n_y) for which the average, from the ensemble of 120 members, is the interval [2.95%, 3.05%]. The values n_x = 30,n_y = 15 are those values that minimize the product n_x ⋅ n_y and yield the minimum coefficients in the truncation. From this picture one can observe that the tensor truncation is only a bit more sensitive in the Ox direction than in the Oy direction (the matrix g associated with the picture has the property g(i, j) ≤ g(j, i) for j ≤ i). In Fig. 28 are shown the differences in percents between the original facies fields and transformed facies fields (for n_x = 30 and n_y = 15) for all ensemble members, and we can see here that the bounds are [2.5%, 3.6%], so we have a small variance. In Fig. 29 is shown the third ensemble member in four situations: the original field (a) (simulated from the training image) and three facies fields obtained after truncation of the parameter field $\overline {\theta _{3}}$ for the cases n_x = 30,n_y = 15 (b), n_x = 15,n_y = 30 (c), and n_x = 15,n_y = 15 (c). The main difference between them consists of the smaller width of the channel for the transformed facies fields compared with the original (the lower width is obtained for small values of n_x and n_y). The conclusion is that with the truncated HOVSD we are able to control better the channel continuity than with the truncated SVD where the channel continuity is broken (see Fig. 6).

Fig. 27

The average of difference (%) between prior facies fields and transformed facies fields

Fig. 28

Difference in percentage between the original fields and the transformed fields

Fig. 29

Facies fields obtained after truncation. a Original. bn_x = 30; n_y = 15. cn_x = 15; n_y = 30. dn_x = 15; n_y = 15