Generating Multiple History-Matched Reservoir-Model Realizations Using Wavelets
Authors: Sahni Isha , Horne Roland N. 
Published: June 2006

Summary
This paper focuses on an automated way to generate multiple history-matched reservoir models with the inclusion of both geological uncertainty and varying levels of trust in the production data, using wavelet methods. As opposed to previously developed automated history-matching algorithms, this methodology not only ensures geological consistency in the final models but also includes uncertainty in the production data.
A data distribution, such as a permeability field, can be (reversibly) transformed into wavelet space in which it is fully described by a set of wavelet coefficients. It was found that different subsets of the collection of wavelet coefficients can be constrained separately to (a) the production history and (b) the geological constraints. This means that the history match need be performed only once, after which multiple realizations can be generated by adjusting only the second subset of coefficients.
The ability to include both geological and production-data uncertainty into the reservoir model automatically is of great consequence to reservoir modeling and, hence, to reservoir management, risk analysis, and making key economic decisions. A more complete and realistic reservoir model will lead to better reservoir production and development decisions.

Introduction
Reservoir modeling is an important step in forecasting the performance of a reservoir, forming the basis for reservoir management, risk analysis, and making key economic decisions. A history match, however, is not a sufficient condition for a reservoir to make better predictions for future production. The model should at least conform to all the available data and the geologist's prior conception of the reservoir. Thus, the purpose of reservoir modeling is to use all available sources of information to develop such a reservoir model. This model then can be used to forecast future performance and optimize reservoir-management decisions.
It is essential to integrate all the different sources of data to provide the most complete reservoir model or models (Landa and Horne 1997; Landa 1997; Wang 2001). Our model certainty is always limited by the data available to us. As such, it is never possible to infer or develop a reservoir model with full certainty. However, the optimal use of all consistent data available will yield reservoir models that are less and less uncertain. Herein lies the significance of methodologies that can realistically and efficiently integrate different sources of reservoir information.
Reservoir data are, generally speaking, divided into two categories: production data (e.g., pressure and water-cut histories from wells) and all other sources of data (e.g., core samples, seismic, and well logs). This second category of data depends on reservoir properties like porosity and permeability in a relatively direct way. Core samples can be used to provide porosity and permeability measurements at specific locations (well locations); semivariograms (Deutsch and Journel 1998; Isaaks and Srivastava 1989) obtained from outcrops, for example, act as spatial statistics information, and seismic surveys may provide 3D impedance distributions that can be inverted and used as "soft-conditioning data" at the corresponding locations. These different sources of data can be combined together with different approaches (e.g., Bayesian probability techniques) to give a single set of probabilities.