C ONCLUSIONES

Cross validation is a good way of checking the model paramaters before modeling proceeds, although it can not be limited to this use. Cross validation validates the model by considering a single known value as unknown and estimates the pseudo-unknown value and compares it to the true value. This sample is replaced, and the process is repeated for all of the known values. Plotting the errors on a map illustrates areas of concern by exposing those areas which are plagued by over or under estimation.

The modeling parameters, such as the variogram, should be updated appropriately or the model broken into more appropriate chunks.

The jackknife is simalar to cross validation but in the jackknife the data set is split into non-overlapping sub-data sets. The idea is to estimate the one sub-set of data using another and compare the estimated to the true.

Some useful tools for exposing a problem include:

• A scatterplot of the true values versus the estimated shows where the estimated values deviate from the true without considering location.

• A location map of the errors reveals trouble spots. The errors should be spatially independent of each other.

• The distribution of errors {e(u), i=1,...,N}should be symetric and centered on a zero mean with minimum spread. Deviation of the mean from 0 indicates a bias.

• Summary statistics for the error distribution, i.e. the mean error and the mean squared error.

Bootstrap

The bootstrap is a statistical resampling procedure used to determine the uncertainty in a statistic by using the data itself. Monte Carlo simulation is used to sample the data with replacement as though it were the population. The result is a distribution of uncertainty for the data.

Figure 12.2

Understanding Uncertainty

Ranking and Selecting Realizations

Multiple realizations are boasted as equiprobable and representative of the histogram and variogram, so why would there be a need for ranking and selecting a realization? Running each realization through a flow simulator or considering the viability of every realization is prohibitively expensive in terms of man power and cpu time. Instead the realizations are ranked and only the worst, most likely and the best scenarios are run in a simulator. The ranking can be a result of any criteria the user desires. For example one could rank the realizations by oil-in-place, or by connectivity, or by combinations of any criteria. Why then is it essential to run many simulations only to remove a few candidates for consideration? Only a few realizations are required to assess the average of an attribute, but many are required to assess the extremes of a distribution. It is the average value and the extremes that we are

Risk Qualified Decision Making

Risk has two components: (1) probability of loss, and (2) quantity of loss. Sound decision making entails assessing the risk associated with the gross generalizations made from sample data; the errors made in the model. Geostatistics provides all of the components required for risk qualified decision making. Simulation provides the uncertainty, or the probability of loss. If we knew the truth there would be no loss, the probability of loss would be zero. Greater uncertianty invites greater potential for loss. What remains is an assessment of loss, the loss function. The loss function administers loss commensurate to the magnitude of error. The greater the error, the greater the loss. The loss function is derived from the model. For example over-estimating the oil in place in a reservoir is more punishable than under-estimating because of the investment required to exploit the reservoir. Likewise, it is punishable to under-estimate oil in place as a potential reservoir may be over-looked. The risk for each of these scenarios is different, there is more risk in over-estimating than in under-estimating, therefore, the loss function need not administer the same loss for over-estimation as for under-estimation. Figure 12.3 shows three loss functions. The first shows a loss function that heavily penalizes for under-estimation, the middle loss function administers loss equally, and the last shows a loss function that penalizes over-estimation heavily.

Figure 12.3

The decision making part of risk qualified decision making does not decide whether or not to drill, the user decides that. What it does is decide what the optimal attribute value is based on criteria set forth by the loss function. The criteria for optimality is minimum loss. The selected attribute value represents the "safest" value, or the value that would incur the least loss. This is the attribute value that decisions are made with. The algorithm is best explained by a complete example, the link below is an example using an Excel spread sheet.

Loss Function Example

The algorithm starts by considering one of the possible values within the distribution of uncertainty as the pseudo best estimate and considers the consequences of that decision. If the pseudo best estimate is an over-estimate of the true value then the loss due to over-estimation is tabulated. The loss is considered for all possible truths. This is repeated for all possible values. The value with the lowest loss is the best estimate. The philosophy here is to choose the value that offers the least loss, and hence the greatest profitability. For example a best estimate might suggest X barrels of in situ oil for a particular reservoir. Corporate philosophy may dictate that Y barrels must be present for a reservoir to be considered a viable project. X does not equal Y, therefore the project is turned down.