Using geostatistical constraints in electrical imaging for improved reservoir characterization

Developing predictive models of reservoirs is often complicated by the spatial heterogeneities and the different
scales which control flow and transport processes. In numerous studies over the past two decades, geophysical
imaging techniques have proved very useful for reservoir characterization. However, the loss of resolution and the
non-uniqueness of standard solutions to inverse problems strongly limit the use of such deterministic imaging
approaches. On the other hand, the use of common geostatistical approaches for reservoir characterization, for
instance from logging information, may be a difficult task, since accurate variogram information is difficult to obtain
(dense sampling in the vertical and lateral directions), and also because a high number of conditioned simulations is
needed to remove statistical bias. Combining the high spatial sampling of deterministic geophysical imaging methods
with geostatistical constraints, valid in the whole image plane, appears as a very promising approach to enhance
reservoir characterization. To do so, we use a parameterized model covariance matrix based on standard variogram
functions and a prior model as regularization operator in the inversion of electrical resistance data. This way of
including additional data is not restricted to electrical data but the variogram parameters may be also inferred from for
example available textural or lithological information.
The benefit of the presented approach is twofold: (i) It honors the spatial statistics of the reservoir and (ii) it alters the
posterior model by further reducing model ambiguity inherent to the inversion compared to classical (smooth model)
regularization. The proof of concept is given by synthetic studies carried out on random fields from Gauss simulations
with varying (an)isotropic scale lengths using different model (co)variogram functions. We also demonstrate the
approach on electrical field data combined with borehole electromagnetic data from two artificial sea inlets in the
nature reserve "The Westhoek" near the French-Belgian border. The electromagnetic logs were used to calculate an
experimental vertical variogram characteristic of the study site. The results enabled to determine the extension of the
salt water plume laterally, and significantly enhance its extension in depth, but also in terms of total dissolved solid
content. These observations are in agreement with the hydrogeological situation at the site. A comparison with
borehole data shows that the results are much more plausible than results obtained with a traditional smoothness
constraint used as regularization operator. In conclusion, the incorporation of geostatistical information, vertical
variograms in our case, in the inverse process improves imaging capabilities for reservoir characterization
significantly.