Non-Stationary Estimation of Reservoir Properties Using Production Data
Authors: Jansen F.E. , Kelkar M.G. 
Published: December 2007

Abstract
Reservoir property interpolation and estimation has traditionally been a stationary process ignoring the dynamic information available from production data. A proper analysis of these data can provide an estimate of the boundary conditions controlling the flow. A cross correlation between a pair of wells provide a measure of the inter-well relationship or the association between the two wells. This measure of association can be utilized to provide a better estimate of the correct search neighborhood and as a tool to introduce an element of non-stationarity into the reservoir property estimation process. While any interpolation routine can be used when the correct search neighborhood is defined, it is possible to incorporate the cross correlation coefficients directly into the interpolation routine to generate an experimental variogram model. However, while the cross correlation approach produces a matrix describing the spatial association between the sampled locations, this does not guarantee an invertible matrix when applied in a kriging like system. To overcome this problem, a distance weighted interpolation of the reservoir properties is used where the cross correlation coefficients provide the appropriate measure of association without the strict assumption of stationarity. The result is a better structural definition using a simpler interpolation system without the need of a variogram This is a rather unique approach utilizing the information about the flow structure found in the production data and introducing an element of non-stationary element into the reservoir property estimation.

Introduction
Traditional geostatistics apply a stationary approach to data interpolation. The reservoir properties are interpolated between measured data points under the assumption that all of the data belong to the same stationary data set. The samples obtained are often sparse and not randomly sampled, but rather sampled at the locations with the highest expectancy of good reservoir properties. This can lead to a too optimistic and continuos reservoir description.
The assumptions about stationarity in our data set are based on our limited knowledge about the continuity and extent of the reservoir from static measurements. A source of information often excluded in this process is the dynamic information available from wells in the form of pressure and rate measurements. These dynamic data contain information about the extent and the continuity of the reservoir. Production data represent a sampling from the underlying flow process. We would like to be able to utilize these data to describe and characterize the important features of the flow process, there by making a prediction of the controlling reservoir properties. By combining the dynamic and the stationary information it is possible to add an element of non-stationarity to the interpolation. One way of adding these non-stationarity data is through a better definition of the correct search neighborhood for an otherwise stationary interpolation process.
If we look at a regular stationary interpolation process like kriging, it can easily be seen that this represents an averaging which honors the measured data, but which creates too much smoothness and continuity in the interpolated estimates. This is common for most interpolation methods and is caused by a lack of information about the actual variations between the measured data points. This is also a weakness of stochastic methods, thus, common for all existing methods are a lack of information about the extent of the stationarity.
For a good interpolation of the data it therefore becomes critical to select the appropriate sub sets for interpolation. Production data provides a measure of the in-situ flow process and the inter-well relationships.
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