History Matching in Metric Space

When reviewing metric space methods, we noted an extremely useful aspect is the ability to visualize the set of reservoir models and compare its location with the observed data (red "cross" in figure below, highlighted by crossing blue lines).  One useful application that has been noted is pre-history match screening

For history matching, we can go beyond this screening process and take advantage of the information within the metric space to construct new probabilities of the uncertain parameters in the prior model.   The problem is essentially the same as in geomodeling and geostatistics, when, given measured data at certain points in space (the wells), the geomodelers infer probabilities at unknown points in space (grid blocks around the wells).  In our case though, the points in (metric) space are the reservoir models, the measure data are the parameter values associated with the reservoir models, and the unknown point in space is where the historical data is located in the metric space.  

An illustration of this analysis is shown in the figures below.  220 reservoir models have been created by sampling from probability distributions of 22 unknown parameters (both flow-based and geological parameters).  The figures below show the reservoir models in the first 3 dimensions of the metric space, colored by the value of the Corey maximum relative permeability value for water (left), and the training image (right).  Note how one can visually deduce a spatial distribution of the parameter values.  For krwMax, the darker colors (lower values) are closer to observed values compared to the brighter colors (higher values).  For the TI, the green and yellow colored models are closer to the observed values compared to the orange.  

220 Runs Colored by krwMax 220 Runs Colored by Training Image (TI)

This visual information above can be quantified using a standard algorithm used in spatial statistics called Kriging.  Kriging is an algorithm that can calculate a probability of a value at a (unknown) point given known (measured) values at points a certain distance away.  

Skipping the details, solving the Kriging equations for krwMax and TI gives us the results below.  The black curves are the prior probabilities of the two parameters which are sampled from to create the 220 models above.  The red curves are the result of solving the Kriging equations.  Note that the probabilities have been updated to reflect the spatial information that we could infer visually.  There is a higher probability of low values of krwMax, and a higher probability of TI10 (models colored green).  Note that this information though doesn't come from the 3D plots, but from spatial distribution of the models and history in the high dimension metric space.  

This information can be very useful for model selection (screening), as described here.  We can also iterate using this procedure and create a novel history matching algorithm.  Given a set of new probabilities (which can be created for all 22 parameters in the example above), we resample from them and create new models which provide models closer to the observed data.  The newly created models can then be added to the metric space, and the probabilities for the parameters can be updated again and resampled.