# REE4: Stochastic differential equations for oilfield history matching

 Researcher: Lian Duan Team Leader(s): Prof. Ben Hambly, Dr Irene Moroz & Dr Chris Farmer Collaborators: Prof. Ibrahim Hoteit, KAUST Dr Geoff Nicholls, Statistics Ralph Stadie, Paradigm

Project completed June 30, 2012

Background

As oil supplies dwindle, having an accurate representation of an oil reservoir becomes particularly important to maximise production. History matching is an important technique to understand and model the behaviour of oil reservoirs. In history matching, the model of a reservoir is adjusted until it reproduces past behaviour closely – it is an inverse problem, meaning that observations are used to obtain information about the system, thus the accuracy depends on the quality of the model and of the observations.

Techniques and Challenges

Mathematically in history matching, the information about the underlying system can be obtained using filters, which are algorithms that estimate the model parameters based on observations. Reservoir models divide the reservoir into a spatial grid, and the history matching process is used to determine the permeability and/or porosity in each grid cell. Because of the large number of cells in the grid and the relationships between the values at neighbouring grid blocks, history matching problems are generally ill-posed in the sense that many possible combinations of reservoir parameters result in equally good matches to the historical observations.

Oil reservoirs consist of distinct rock types (known as ‘facies’ by geologists) with properties such as permeability or porosity. Most of the information about these properties is indirect. Using mathematical models one infers the properties from measurements. Without making use of a careful summary of our prior geological knowledge, this inference (known as ‘history-matching’) is unstable and non-unique (a feature known as ‘ill-posedness’). Methods for avoiding ill-posedness, that are consistent with accepted statistical principles, is an increasingly important topic in the geosciences.

Researchers at the Oxford Centre for Collaborative Applied Mathematics (OCCAM), with the ultimate aim of improving existing methods for the history matching of facies problems, explored new estimation methods, including developing a new Grid Based Method (GBM), as well as deepening the understanding and application of existing particle filtering methods.

Results

The researchers first focused on gaining insight into their newly developed GBM by testing it on a logistic map – a simple, well-known nonlinear problem. In this work, an algorithm, numerical scheme and predictability analysis of the new GBM method were developed, and it was shown that it reproduced and extended existing results compared to other similar filters. The optimality and order of the accuracy of the new numerical algorithm were assessed. Furthermore, it was compared with the analytical solutions and numerical solutions and a study was made of the breakdown of the filter in the imperfect model scenario due to over-confidence. This problem was previously conjectured to be unsolvable in the literature.

To deepen understanding of existing filters, the researchers explored the underlying relationship between the estimation method of the Regularised Particle Filter (RPF) and of the Ensemble Square Root Filter (EnSRF). They derived and showed that, under certain assumptions, the EnSRF proved to be a special case of the RPF.

The researchers then began work on the history matching problem of facies patterns using existing filters. The boundaries of features were parameterised, and the EnSRF was used to adjust the feature shapes. They found that the facies boundaries were best parameterised using a B-spline, as it captured the underlying physics whilst offering compatibility to the EnSRF.

The numerical results demonstrated good performance and also led to research initiatives for designing a more general filter: the Warping Ensemble Square Root Filter (WEnSRF). Through numerical experiments, the WEnSRF was shown to provide reliable estimates for both simple reservoir models (with a single feature) and large-scale reservoir models (realisations from the Stanford VI model, with channel features).

The Future

The techniques and analysis resulting from this project will no doubt be useful in the area of history matching problems, and also estimation in general.

The GBM developed was shown to be optimal for simple problems, while remaining computationally similar to existing suboptimal filters. Existing filtering methods were explored and a more general filter, the WEnSRF, was developed and shown to be a promising and reliable tool for reservoir modelling problems.

Related publications

[11/34] Luo X., Hoteit I., Duan L., Wang W.: Review of nonlinear Kalman, ensemble and particle filtering with application to the reservoir history matching problem: Nonlinear Estimation and Applications to Industrial Systems Control (Book)

[11/33] Duan L., Farmer C.L., Hoteit I., Luo X., Moroz I.M.: Data Assimilation using Bayesian Filters and B-spline Geological Models, International Inverse Problem Conference, Journal of Physics: Conference Series, Volume 290, 012004, 2011

[10/36] Duan L., Farmer C.L., Moroz I.M.: Sequential Inverse Problems Bayesian Principles and the Logistic Map Example, 8th International Conference of Numerical Analysis and Applied Mathematics, 2010

[10/35] Duan L., Farmer C.L., Moroz I.M.: Regularized Particle Filter with Langevin Resampling Step, 8th International Conference of Numerical Analysis and Applied Mathematics, 2010