Improved Production Forecasts and History Matching Using Approximate Fluid-Flow Simulators
- Henning Omre (Norwegian U. of Science and Technology) | Ole Petter Lødøen (Norwegian U. of Science and Technology)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 2004
- Document Type
- Journal Paper
- 339 - 351
- 2004. Society of Petroleum Engineers
- 4.1.2 Separation and Treating, 5.5.8 History Matching, 2.2.2 Perforating, 1.2.3 Rock properties, 5.6.5 Tracers, 1.6 Drilling Operations, 5.6.1 Open hole/cased hole log analysis, 5.1.1 Exploration, Development, Structural Geology, 5.5.3 Scaling Methods, 5.1.5 Geologic Modeling, 4.1.5 Processing Equipment, 5.6.9 Production Forecasting
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Forecasts of production with associated uncertainties must be based on a stochastic model of the reservoir variables and a fluid-flow simulator. The latter is usually very computationally demanding to activate. Approximate fluid-flow simulators based on upscaling of the reservoir variables are frequently used to assess the forecasts with uncertainties. The upscaling introduces biases and changes the error structures in the production forecasts, however. A production forecast model that accounts for these biases and changed error structures is defined, and estimators for the model parameters are specified. Forecasts both at the appraisal stage and the production stage including history matching are discussed. The so-called "ranking problem" is formalized and solved as a part of the study. The results are demonstrated and verified on a large case study inspired by the Troll field in the North Sea.
The objective of reservoir evaluation is to forecast production for a given depletion strategy and, eventually, to optimize this strategy with respect to some recovery criterion. Prediction of the reservoir variables will also be important when decisions concerning infill drilling are made. Both general reservoir experience and reservoir-specific observations such as well logs, seismic data, and production history must be used in the evaluation. Quantification of the uncertainty in the forecasts and predictions should be an integral part of the study.
Production forecasts with associated uncertainties are complicated to assess because of huge computational demands by the fluid-flow simulator. The evaluation requires repeated runs of the simulator as a part of a stochastic model. Production history matching is particularly computationally demanding (see Barke et al.1). Approximate fluid-flow simulators based on upscaling of the reservoir variables are frequently used to reduce the computational task. The upscaling is known to introduce biases and to change the error structures in the production forecasts, however. The effect of using an approximate fluid-flow simulator instead of the best simulator should be accounted for in the evaluation. In Glimm et al.,2 a procedure for variance correction is introduced, but bias correction is not discussed. Bias is considered to be the most important factor because it influences the expected volumes to be produced. In statistical literature, several papers on calibration of computer models have appeared recently (see Craig et al.3 and Kennedy and O'Hagan4), but none of these formalizes the use of approximate models.
Focus of the current work is on improving the production forecasts and prediction of reservoir variables, and assessing the associated uncertainties. Evaluations both at the appraisal stage and at the production stage, the latter including production history matching, are discussed. The effect of using an approximate fluid-flow simulator in the evaluation is accounted for. A stochastic model defined in a Bayesian framework is used (see Omre and Tjelmeland5), and the solution is the corresponding posterior probability density function (pdf), which is represented by realizations generated along the lines of the SIR-algorithm (see Omre6). The ranking problem, as described in Saad et al.,7 is also formalized, and a solution procedure is defined. The methodology is demonstrated and verified on the case used in Hegstad and Omre,8 which is inspired by the characteristics of the Troll field in the North Sea.
The stochastic model is defined on a graph (Fig. 1), which appears as a simplification of the graph in Hegstad and Omre.8 The stochastic reservoir variable is termed R, and it contains porosities, permeabilities, saturations, and other variables necessary for evaluation of fluid flow. The reservoir variable R is defined on a grid LD of dimension nD covering the reservoir domain D. Assume that a prior pdf f(r) can be assigned to R (see Hegstad and Omre8 for an example). The stochastic production variable is termed P and contains production rates, production ratios, bottomhole pressures, and other production variables of interest in the wells. The production variable P is defined on a grid LT of dimension nT covering the reservoir production time T :[t0,t1], where t0 is time for production start and t1=a suitably chosen time for production completion. In general, stochastic variables will be represented by capital letters, while the corresponding lower-case letters will represent their associated realizations.
A fluid-flow simulator links the reservoir and production variables
where the symbol ~ links the random variable on the left with the associated pdf on the right (r,p) are the outcome of the stochastic variables (R,P), and q=a given recovery strategy. Note that [P|r ] reads "the production variable P given the outcome of the reservoir variable R=r." The fluid-flow simulator v(r;q ) takes the reservoir variable r, represented on the full grid LD without upscaling, as input. Term v(r;q) is the complete fluid-flow simulator, and one can assume that it reproduces the fluid-flow process without error. Simulations of the fluid flow may take days, or even weeks, to run on a computer, and should be thought of as the best model one can obtain. The expression in Eq. 1 defines the conditional pdf f (p|r) as a Dirac function because no stochastic error term is involved, and it is represented by a double arrow in Fig. 1. In practice, the physical fluid-flow process cannot be exactly represented by a computer implementation of a mathematical model; therefore, there will be a modeling error. This source of error is ignored in the current study, but it could have been modeled by an error term in Eq. 1.
Seismic data and well observations may be available, as in Hegstad and Omre,8 but in order to simplify notation, the conditioning on this information is not indicated in f(r). The important aspect is, however, that R can be efficiently sampled conditional on the available seismic and well observations (see Eide et al.9 and Buland et al.10).
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