Methods for efficient probabilistic history matching and forecasting have been available for complex reservoir studies for nearly 20 years. These require a surprisingly small number of reservoir simulation runs (typically less than 200). Nowadays, the bottleneck for reservoir decision support is building and maintaining a reservoir simulation model. This paper describes an approach which does not require a reservoir simulation model, is data driven, and includes a physics model based on material balance. It can be useful where a full simulation model is not economically justified, or where rapid decisions need to be made.
Previous work has described the use of proxy models and Hamiltonian Markov Chain Monte Carlo to produce valid probabilistic forecasts. To generate a data driven model, we take historical measurements of rates and pressures at each well, and apply multi-variate time series to generate a set of differential-algebraic equations (DAE) which can be integrated over time using a fully implicit solver. We combine the time series models with material balance equations, including a simple PVT and Z factor model. The parameters are adjusted in a fully Bayesian manner to generate an ensemble of models and a probabilistic forecast. The use of a DAE distinguishes the approach from normal time-series analysis, where an ARIMA model or state space model is used, and is normally only reliable for short term forecasting.
We apply these techniques to the Volve reservoir model, and obtain a good history match. Moreover, the effort to build a reservoir model has been removed. We demonstrate the feasibility of simple physics models, and open up the possibility of combinations of physics models and machine learning models, so that the most appropriate approach can be used depending on resources and reservoir complexity. We have bridged the gap between pure machine learning models and full reservoir simulation.
The approach to use multi-variate time series analysis to generate a set of ordinary differential equations is novel. The extension of previously described probabilistic forecasting to a generalised model has many possible applications within and outside the oil and gas industry, and is not restricted to reservoir simulation.