Combining Geostatistical Modelling With Gradient Information for History Matching: The Pilot Point Method
Authors: Bissell R.C. , Dubrule O. , Lamy P. , Swaby P. , Lepine O. 
Published: December 2007

Abstract
The introduction of gradient techniques for history matching has proved to be a useful development. Starting with an initial reservoir description the model is modified until it reproduces the measured production history of the field. It is possible to condition the solution so that it honours prior geostatistical constraints on the reservoir description. This paper evaluates the pilot point method for obtaining these solutions and compares it to an alternative method called the gradzone method where groups of gridcells in the model are modified. The pilot point method described here is adapted from ideas originally proposed by previous workers. A number of improvements are proposed in the technique.

Introduction
The development of methods for history matching reservoir models can perhaps be split into two strands. The earlier work was done by reservoir engineers who, with a background in reservoir simulation, used deterministic methods to obtain history matches. A second methodology was developed later by geostatisticians. Until recently these methods have been used independently. The deterministic methods produced only a few examples of the full set of possible solutions. However, they were efficient at finding these solutions. On the other hand the geostatistical methods were capable of representing the true variability of the solutions but they were inefficient.
It would be useful to combine the methodologies somehow and benefit from the strengths of these two complementary approaches. We adapt a method developed by other workers to solve the problem of generating reservoir descriptions which honour all prior data (measured well data, geostatistical model and production data) in a synthetic case. We have chosen to match porosities rather than permeabilities because we expect the solution to be more degenerate and we wish to demonstrate the very large number of solutions possible to this ill-posed problem.
We briefly discuss the development of deterministic methods and then address geostatistical techniques. We then look at methods for combining them. This is not a review paper and so our literature survey is not comprehensive. We have had to neglect some interesting work for reasons of economy. We leave aside techniques such as simulated annealing, genetic algorithms and neural networks.
The application of optimal control theory to the problem of history matching was presented by Chen et al. and Chavent et al. Chavent et al. minimised an objective function using the steepest-descent algorithm. (A review paper by Dougherty was cited as a guide to the contemporary literature) It was further developed by, amongst others, Watson et al., Lee et al. and Yang et al. who used quasi-Newton and/or variable metric optimisation It has been applied to real field data by Zhang et al. Palatnik et al. and Zhang et al. who compared it to the Gauss-Newton optimisation technique. A second type of optimisation employing derivatives is based on Gauss-Newton optimisation This requires the derivatives of the components of the objective function with respect to the history matching parameters in contrast to the other methods which just require the derivative of the objective function itself. An efficient means of calculating these derivatives was developed by Anterion et al. Tan and Kalogerakis used this technique to solve a model problem. Bissell et al. applied it to real field cases.
Other examples of the use of derivatives for solving inverse problems in petroleum reservoir modelling are given in the reference section.
A complementary method to deterministic methods is that based on stochastic techniques. Here the approach is to model the reservoir description as the outcome or "realisation" of a stochastic process. These realisations must respect all the observations made at the wells and the statistical constraints which quantify the frequency and geometry of facies units. Again there is a very large amount of literature on this subject which we cannot review here.
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